N_{part} \\cdot \\left.","{dn_{pp}}/{d\\eta }\\right|_{\\eta =0}$ , and thus that multiple scattering plays an important role.","Further, the possible scaling with the number of nucleon-nucleon also does not hold $\\left.","{dn_{AA}}/{d\\eta }\\right|_{\\eta =0}\\ll N_{coll} \\cdot \\left.","{dn_{pp}}/{d\\eta }\\right|_{\\eta =0}$ , indicating that coherence effects among the relevant degrees of freedom at the nucleon level are at play during the collision process.","The collision centre-of-mass energy dependence of the charged particle multiplicity in both pp and AA collisions is well reproduced by a power law as suggested by models, e.g.","the Colour Glass Condensate or the String Percolation Model, where coherence effects play an important role.","The logarithmic growth consistent with pre-LHC data is ruled out.","Notably, this dependence is stronger [1] in the AA ($\\left.","{dn_{AA}}/{d\\eta }\\right|_{\\eta =0}\\sim {s}^{0.15}$ ) case than for pp ($\\left.","{dn_{pp}}/{d\\eta }\\right|_{\\eta =0}\\sim {s}^{0.11}$ ).","Several possible explanations for this difference in energy dependence, naively at odds with theoretical expectations, have been put forward recently [2], [3], [4], [5], [6], [7].","The String Percolation Model (SPM) [8], the Dual Parton model [9] including parton saturation effects, describes consistently properties of bulk multiparticle production [10], [11], [12], [13], [14], [15], [16].","This framework is closely related to descriptions based on the Colour Glass Condensate and Glasma flux tubes [17] for which the same issues have been discussed extensively [18], [19], [20].","In this short note, we detail how the SPM can provide for a joint description of the energy dependence of multiplicity in pp and AA collisions once energy-momentum conservation constraints are taken into account.","Further, we show the resulting dependences on the nuclear species $A$ and number of participating nucleon $N_{\\rm part}$ to be fully compatible with available data.","Finally, we give predictions for future higher energy LHC runs and discuss generic expectations at high energy."],["Mid-rapidity multiplicity in the SPM","The Glauber model and its generalizations (see [21] for a review) relate nucleus-nucleus collisions to collisions of their constituent nucleons.","In the single scattering limit the average number of participating nucleons (per nucleon) $N_{A}=N_{\\rm part}/2$ behave incoherently and the mid-rapidity multiplicities in nucleus-nucleus and proton-proton collisions are simply related by: ${\\frac{dn^{AA}}{dy}}\\bigg |_{y=0} \\sim N_{A} \\cdot \\frac{dn^{pp}}{dy}\\bigg |_{y=0} \\, .$ This result, which corresponds to the wounded nucleon model [22], is expected to dominate at sufficiently low centre-of-mass energies.","At higher energies, multiple scattering becomes important, and $\\frac{dn^{AA}}{dy}\\bigg |_{y=0} \\sim \\big (N_{A}^{4/3} -N_A\\big ) \\cdot \\frac{dn^{pp}}{dy}\\bigg |_{y=0} \\, .$ Here, $N_{A}^{4/3}$ is the total number of nucleon-nucleon collisions and single scattering has been subtracted [23], [24].","Energy-momentum conservation constrains the combinatorial factors of the Glauber model at low energy.","In the framework of SPM, these constraints translate into the sharing of energy-momentum of $N_A$ valence strings amongst $N_A^{4/3}$ (mostly sea) strings.","A possible solution to this problem, the reduction of the height of the plateau for sea strings, was pursued in [23], [24].","Here, we proceed differently and account for energy-momentum conservation by reducing the effective number of sea strings via $N_{A}^{4/3}\\rightarrow N_{A}^{1+\\alpha (\\sqrt{s})}\\, ,$ with $\\alpha (\\sqrt{s})=\\frac{1}{3}\\Bigg (1-\\frac{1}{1+\\ln (\\sqrt{s/s_{0}}+1)}\\Bigg )\\, .$ We thus can write $\\frac{dn^{AA}}{dy} \\bigg |_{y=0} \\sim N_A \\big (N_{A}^{\\alpha (\\sqrt{s})} - 1\\big ) \\cdot \\frac{dn^{pp}}{dy}\\bigg |_{y=0} \\, ,$ such that the wounded nucleon model eq.","(REF ) is recovered for $\\sqrt{s}\\ll \\sqrt{s_{0}}$ , and Glauber result eq.","(REF ) follows for $\\sqrt{s} \\gg \\sqrt{s_{0}}$ , $\\alpha (\\sqrt{s}) \\rightarrow \\frac{1}{3}$ .","In the SPM one considers Schwinger strings, which can fuse and percolate [25], [26], [27], [28], as the fundamental degrees of freedom.","Multiparticle production is described in terms of these colour strings which are formed in the collision and stretch between partons in the parting nuclei and are thus longitudinally extended in rapidity.","The multiplicity of produced particles $dn/dy$ is proportional to the average number of such strings (twice the number of elementary collisions) $N^s$ .","Thus, for a generic $N_A N_A$ collision, be it $pp$ or $AA$ , one has $\\frac{dn}{dy} \\bigg |_{y=0} \\sim N^s_{N_A}\\, .$ In the impact parameter plane, the colour content of the strings is confined within a small transverse area $S_1 = \\pi r^2_0$ , with $r_0\\sim 0.2\\div 0.3$ fm.","The strings decay via $q\\bar{q}$ and $qq-\\bar{q}\\bar{q}$ pair production and subsequently hadronize to the observed hadrons.","In the impact parameter plane, the strings appear as disks and with increasing energy-density these disks overlap, fuse and percolate leading to a reduction of the overall charge [29].","A cluster of $n$ strings behaves as a single string with energy-momentum corresponding to the sum of the individual strings and with a higher colour field corresponding to the vectorial sum in colour space of the colour fields of the strings.","In this way, the mean multiplicity $\\langle \\mu _{n}\\rangle $ and the mean transverse momentum squared $\\langle p^{2}_{T}\\rangle $ of the particles produced by a cluster in the limit of random distribution of strings are given by $\\langle \\mu _{n}\\rangle =N^{s}F(\\eta ^t)\\langle \\mu _{1}\\rangle $ and $\\langle p^{2}_{T}\\rangle =\\langle p^{2}_{T,1}\\rangle /F(\\eta ^t)$ where $\\langle \\mu _{1}\\rangle $ and $\\langle p^{2}_{T,1}\\rangle $ are the corresponding quantities for a single string.","For a random distribution of disks, the colour reduction factor $F(\\eta ^t)$ is $F(\\eta ^{t})=\\sqrt{\\frac{1-e^{-\\eta ^{t}}}{\\eta ^{t}}}\\, ,$ where $\\eta ^t$ is the impact parameter transverse density of strings (disks) $\\eta ^{t} \\equiv \\frac{S_1}{S_{N_A}}\\, {N}^{s} = \\frac{\\pi r^2_0}{S_{N_A}}{N}^{s}\\, ,$ with $S_{N_A}$ the area of the impact parameter projected overlap of the interaction.","For a $N_A N_A$ collision, eq.","(REF ) can be recast as $\\frac{dn}{dy} \\bigg |_{y=0} \\sim F(\\eta ^t)\\, N^s_{N_A}\\, , \\qquad N^s_{N_A} = N^s_{p} N_A^{1+\\alpha (\\sqrt{s})}\\, .$ The colour reduction factor $F(\\eta ^t)$ results in a slowdown of the increase of $dn/dy$ with energy and number of participating nucleons.","Note that if nucleons were to act incoherently, as in the limiting case eq.","(REF ), $S_{N_A}$ would be given by the area of a single nucleon $S_p$ , while once coherence is accounted for, eq.","(REF ), $S_{N_A}$ is the overall area of interaction.","In general, $S_{N_A}$ depends on the impact parameter $b$ of the collision ($0\\le b\\le 2 R_A$ ) with $N_A\\rightarrow 0, S_{N_A}\\rightarrow 0$ as $b\\rightarrow R_A$ and $N_A\\rightarrow A, S_{N_A}\\rightarrow S_A\\equiv \\pi R_A^2 \\equiv \\pi R^2_p A^{2/3}$ as $b\\rightarrow 0$ .","These constraints are satisfied by $\\frac{S_{N_{A}}}{S_{A}}=\\bigg (\\frac{N_{A}}{A}\\bigg )^{\\beta }\\, ,$ with $\\beta >0$ .","It follows that $S_{N_{A}}=\\pi R^{2}_{p}\\,A^{2/3}\\bigg (\\frac{N_{A}}{A}\\bigg )^{\\beta }\\, ,$ and $\\eta ^{t}_{N_{A}}=\\eta ^{t}_{p}N_{A}^{\\alpha (\\sqrt{s})}A^{1/3}\\bigg (\\frac{A}{N_{A}}\\bigg )^{\\beta -1}\\, .$ Here, motivated by the scaling limit of the number of vertices in a loop-erased random walk [30], we set $\\beta =5/3$ such that $\\eta ^{t}_{N_{A}}=\\eta ^{t}_{p}N_{A}^{\\alpha (\\sqrt{s})}\\bigg (\\frac{A}{N_{A}^{2/3}}\\bigg )\\, .$ From eqs.","(REF ), (REF ), (REF ), (REF ) we can write $\\frac{1}{N_{A}}\\, \\frac{dn}{dy}\\bigg |_{y=0}= \\frac{dn^{pp}}{dy}\\bigg |_{y=0} \\bigg (1+\\frac{F(\\eta ^{t}_{N_{A}})}{F(\\eta ^{t}_{p})}(N^{\\alpha (\\sqrt{s})}_{A}-1)\\bigg )\\, ,$ The dependence of the multiplicity on the centre-of-mass collision energy $\\sqrt{s}$ is fully specified once the average number of strings in a pp collision $N_p^s$ is known.","At low energy $N_p^s$ , is approximately equal to 2, growing with energy as $(\\sqrt{s}/m_p)^{2\\lambda }$ such that $N_p^s = 2 + 4\\bigg (\\frac{r_0}{R_p}\\bigg )^2 \\bigg (\\frac{\\sqrt{s}}{m_p}\\bigg )^{2\\lambda }\\, ,$ with $R_p$ the proton radius."],["Data description","The energy and number of participants dependences implied by eq.","(REF ) can be tested against the available experimental data.","Before doing so, two remarks on the multiplicity given by eq.","(REF ) are in order: (i) it is for all particles, while experimental data accounts only for charged particles; (ii) it is for mid-rapidity $y\\sim 0$ , while data is collected at mid pseudo-rapidity $\\eta \\sim 0$ and although these coincide, the multiplicity value should be rescaled by Jacobean of the $y\\rightarrow \\eta $ transformation.","Neglecting the dependence of $p_T$ , on which the Jacobean depends, on centre-of-mass energy and the (small) difference between the Jacobean in pp and AA cases, both rescaling factors above can be absorbed into an overall normalization constant $\\kappa $ .","We thus write the charged particle multiplicity at mid-rapidity in the pp case as $\\frac{dn^{pp}_{\\rm ch}}{d\\eta }\\bigg |_{\\eta =0} = \\kappa \\, F(\\eta ^{t}_{p})\\, N_p^s\\, ,$ such that, for the general $N_A N_A$ case, one has $\\frac{1}{N_A} \\frac{dn^{N_AN_A}_{\\rm ch}}{d\\eta }\\bigg |_{\\eta =0} = \\kappa \\, F(\\eta ^{t}_{p})\\, N_p^s \\bigg (1+\\frac{F(\\eta ^{t}_{N_{A}})}{F(\\eta ^{t}_{p})}(N^{\\alpha (\\sqrt{s})}_{A}-1)\\bigg )\\, .$ Eq.","(REF ) depends on three parameters: (i) the normalization $\\kappa $ ; (ii) the threshold scale $\\sqrt{s_0}$ in $\\alpha ({\\sqrt{s}})$ in eq.","(REF ); (iii) the power $\\lambda $ in eq.","(REF ).","We performed a global fit to pp data [31], [32], [33], [34], [35], [36], [37] in the range $ 53\\le \\sqrt{s}\\le 7000$ GeV, and to AA (AuAu, CuCu and PbPb) at different centralities [38], [39] for $19.6\\le \\sqrt{s}\\le 2760$ GeV.","The fitted data sample consists of 116 points (19 for pp and 97 for AA).","The best fit yields the parameter values: $\\kappa = 0.63\\pm 0.01$ , $\\lambda =0.201\\pm 0.003$ , and $\\sqrt{s_0}=245\\pm 29$ GeV.","One notices that while $\\kappa $ and $\\lambda $ are tightly constrained, the threshold scale $\\sqrt{s_0}$ is determined with a sizable associated error of $\\sim $ 10%.","This results from the mild dependence of eq.","(REF ), through eq.","(REF ), on $\\sqrt{s_0}$ compounded with the sizable errors in the existing measurements (see Fig.","REF below).","Fig.","REF shows a comparison of the evolution of the mid-rapidity multiplicity with energy given by eq.","(REF ) with data for pp and AA collisions.","Figure: Multiplicity dependence on s\\sqrt{s}.","pp data from , , , , , , (circles), CuCu (triangles) and AuAu (stars) from , PbPb (star) from .","Curves obtained from eq.","(): (N A =1N_A=1, A=1A=1) for pp (grey line); (N A =50N_A=50, A=63A=63) for CuCu (blue line); and (N A =175N_A=175, A=200A=200) for AuAu/PbPb (red line).","Color online.For reference we quote, in Table REF , the predicted values relevant for future LHC runs.","Table: Predicted multiplicities for pp and PbPb at future LHC energies.The dependence of the multiplicity on the number of participating nucleons implied by eq.","(REF ) is compared with data in Fig.","REF .","Figure: Multiplicity dependence on centrality (the number of participating nucleons N part =2N A N_{\\rm part}=2 N_A where N A N_A is the number of participants per nucleus).","CuCu (triangles) and AuAu (stars) data from , and PbPb (circles) from .","Curves obtained from eq.","(): (s=22.4,62.4,200\\sqrt{s}=22.4, 62.4, 200 GeV) for CuCu (blue); (s=19.6,62.4,130,200\\sqrt{s}=19.6, 62.4, 130, 200 GeV) for AuAu (green); and (s=2.76,3.2,3.9,5.5\\sqrt{s}=2.76, 3.2, 3.9, 5.5 TeV) for PbPb (red).","Color online."],["Conclusions","We have shown that, in the SPM, the power law dependence of the multiplicity on the collision energy is the same in pp and AA collisions.","The slower growth in the AA case is due to finite energy-momentum constraints which tamper string creation, and thus the multiplicity, at low energy.","In the high energy limit, $F(\\eta ^{t})\\rightarrow (\\eta ^t)^{-1/2}$ and $\\alpha (\\sqrt{s})\\rightarrow 1/3$ , eq.","(REF ) can be written $\\frac{1}{N_{A}}\\,\\frac{dn^{N_AN_A}_{\\rm ch}}{d\\eta }\\bigg |_{\\eta =0} = \\frac{dn^{pp}_{\\rm ch}}{d\\eta }\\bigg |_{\\eta =0} \\Bigg (1+\\bigg (\\frac{N_{A}}{A} \\bigg )^{1/2} \\bigg (1 - \\frac{1}{N_A^{1/3}}\\bigg )\\Bigg )\\, .$ With the obtained fit parameters, this asymptotic result becomes a good approximation (within 5% of that given by eq.","(REF )) for $\\sqrt{s}\\gtrsim 500$ GeV.","This indicates that finite energy corrections persist to fairly high energies.","In this high energy limit, the shape of $\\frac{dn^{N_AN_A}}{d\\eta }$ as a function of the number of participants is energy independent, that is to say that the $N_{\\rm part}$ and $\\sqrt{s}$ dependences factorize.","In the SPM, energy-momentum conservation results in violations of this factorization and they are the origin of the observed discrepancy in multiplicity growth with energy in pp and AA.","The arguments put forward in this short note can be readily adapted to the case of asymmetric (proton-nucleus) collisions.","Also in this case, energy-momentum constraints are expected to play an important role."],["Acknowledgments","We thank N. Armesto for useful discussions.","IB is supported by the grant SFRH/BD/51370/2011 from Fundação para a Ciência e a Tecnologia (Portugal).","JDD and JGM acknowledge the support of Fundação para a Ciência e a Tecnologia (Portugal) under project CERN/FP/116379/2010.","IB and CP were partly supported by the project FPA2008-01177 and FPA2011-22776 of MICINN, the Spanish Consolider Ingenio 2010 program CPAN and Conselleria de Educacion Xunta de Galicia."]],"string":"[\n [\n \"Multiplicity in pp and AA collisions: the same power law from\\n energy-momentum constraints in string production\"\n ],\n [\n \"Abstract We show that the dependence of the charged particle multiplicity on the centre-of-mass energy of the collision is, in the String Percolation Model, driven by the same power law behavior in both proton-proton and nucleus- nucleus collisions.\",\n \"The observed different growths are a result of energy- momentum constraints that limit the number of formed strings at low en- ergy.\",\n \"Based on the very good description of the existing data, we provide predictions for future high energy LHC runs.\"\n ],\n [\n \"Introduction\",\n \"Measurements of particle multiplicities constrain the early time properties of colliding systems.\",\n \"In the nucleus-nucleus case, these measurements are an essential ingredient for the estimation of the initial energy and entropy densities and thus the initial conditions from which the system will eventually thermalize and the quark gluon plasma be formed.\",\n \"Data collected at RHIC and the LHC for proton-proton (pp) and nucleus-nucleus (AA) collisions establish unambiguously that nucleus-nucleus collisions are not a simple incoherent superposition of collisions of the participating nucleons, $\\\\left.\",\n \"{dn_{AA}}/{d\\\\eta }\\\\right|_{\\\\eta =0}>N_{part} \\\\cdot \\\\left.\",\n \"{dn_{pp}}/{d\\\\eta }\\\\right|_{\\\\eta =0}$ , and thus that multiple scattering plays an important role.\",\n \"Further, the possible scaling with the number of nucleon-nucleon also does not hold $\\\\left.\",\n \"{dn_{AA}}/{d\\\\eta }\\\\right|_{\\\\eta =0}\\\\ll N_{coll} \\\\cdot \\\\left.\",\n \"{dn_{pp}}/{d\\\\eta }\\\\right|_{\\\\eta =0}$ , indicating that coherence effects among the relevant degrees of freedom at the nucleon level are at play during the collision process.\",\n \"The collision centre-of-mass energy dependence of the charged particle multiplicity in both pp and AA collisions is well reproduced by a power law as suggested by models, e.g.\",\n \"the Colour Glass Condensate or the String Percolation Model, where coherence effects play an important role.\",\n \"The logarithmic growth consistent with pre-LHC data is ruled out.\",\n \"Notably, this dependence is stronger [1] in the AA ($\\\\left.\",\n \"{dn_{AA}}/{d\\\\eta }\\\\right|_{\\\\eta =0}\\\\sim {s}^{0.15}$ ) case than for pp ($\\\\left.\",\n \"{dn_{pp}}/{d\\\\eta }\\\\right|_{\\\\eta =0}\\\\sim {s}^{0.11}$ ).\",\n \"Several possible explanations for this difference in energy dependence, naively at odds with theoretical expectations, have been put forward recently [2], [3], [4], [5], [6], [7].\",\n \"The String Percolation Model (SPM) [8], the Dual Parton model [9] including parton saturation effects, describes consistently properties of bulk multiparticle production [10], [11], [12], [13], [14], [15], [16].\",\n \"This framework is closely related to descriptions based on the Colour Glass Condensate and Glasma flux tubes [17] for which the same issues have been discussed extensively [18], [19], [20].\",\n \"In this short note, we detail how the SPM can provide for a joint description of the energy dependence of multiplicity in pp and AA collisions once energy-momentum conservation constraints are taken into account.\",\n \"Further, we show the resulting dependences on the nuclear species $A$ and number of participating nucleon $N_{\\\\rm part}$ to be fully compatible with available data.\",\n \"Finally, we give predictions for future higher energy LHC runs and discuss generic expectations at high energy.\"\n ],\n [\n \"Mid-rapidity multiplicity in the SPM\",\n \"The Glauber model and its generalizations (see [21] for a review) relate nucleus-nucleus collisions to collisions of their constituent nucleons.\",\n \"In the single scattering limit the average number of participating nucleons (per nucleon) $N_{A}=N_{\\\\rm part}/2$ behave incoherently and the mid-rapidity multiplicities in nucleus-nucleus and proton-proton collisions are simply related by: ${\\\\frac{dn^{AA}}{dy}}\\\\bigg |_{y=0} \\\\sim N_{A} \\\\cdot \\\\frac{dn^{pp}}{dy}\\\\bigg |_{y=0} \\\\, .$ This result, which corresponds to the wounded nucleon model [22], is expected to dominate at sufficiently low centre-of-mass energies.\",\n \"At higher energies, multiple scattering becomes important, and $\\\\frac{dn^{AA}}{dy}\\\\bigg |_{y=0} \\\\sim \\\\big (N_{A}^{4/3} -N_A\\\\big ) \\\\cdot \\\\frac{dn^{pp}}{dy}\\\\bigg |_{y=0} \\\\, .$ Here, $N_{A}^{4/3}$ is the total number of nucleon-nucleon collisions and single scattering has been subtracted [23], [24].\",\n \"Energy-momentum conservation constrains the combinatorial factors of the Glauber model at low energy.\",\n \"In the framework of SPM, these constraints translate into the sharing of energy-momentum of $N_A$ valence strings amongst $N_A^{4/3}$ (mostly sea) strings.\",\n \"A possible solution to this problem, the reduction of the height of the plateau for sea strings, was pursued in [23], [24].\",\n \"Here, we proceed differently and account for energy-momentum conservation by reducing the effective number of sea strings via $N_{A}^{4/3}\\\\rightarrow N_{A}^{1+\\\\alpha (\\\\sqrt{s})}\\\\, ,$ with $\\\\alpha (\\\\sqrt{s})=\\\\frac{1}{3}\\\\Bigg (1-\\\\frac{1}{1+\\\\ln (\\\\sqrt{s/s_{0}}+1)}\\\\Bigg )\\\\, .$ We thus can write $\\\\frac{dn^{AA}}{dy} \\\\bigg |_{y=0} \\\\sim N_A \\\\big (N_{A}^{\\\\alpha (\\\\sqrt{s})} - 1\\\\big ) \\\\cdot \\\\frac{dn^{pp}}{dy}\\\\bigg |_{y=0} \\\\, ,$ such that the wounded nucleon model eq.\",\n \"(REF ) is recovered for $\\\\sqrt{s}\\\\ll \\\\sqrt{s_{0}}$ , and Glauber result eq.\",\n \"(REF ) follows for $\\\\sqrt{s} \\\\gg \\\\sqrt{s_{0}}$ , $\\\\alpha (\\\\sqrt{s}) \\\\rightarrow \\\\frac{1}{3}$ .\",\n \"In the SPM one considers Schwinger strings, which can fuse and percolate [25], [26], [27], [28], as the fundamental degrees of freedom.\",\n \"Multiparticle production is described in terms of these colour strings which are formed in the collision and stretch between partons in the parting nuclei and are thus longitudinally extended in rapidity.\",\n \"The multiplicity of produced particles $dn/dy$ is proportional to the average number of such strings (twice the number of elementary collisions) $N^s$ .\",\n \"Thus, for a generic $N_A N_A$ collision, be it $pp$ or $AA$ , one has $\\\\frac{dn}{dy} \\\\bigg |_{y=0} \\\\sim N^s_{N_A}\\\\, .$ In the impact parameter plane, the colour content of the strings is confined within a small transverse area $S_1 = \\\\pi r^2_0$ , with $r_0\\\\sim 0.2\\\\div 0.3$ fm.\",\n \"The strings decay via $q\\\\bar{q}$ and $qq-\\\\bar{q}\\\\bar{q}$ pair production and subsequently hadronize to the observed hadrons.\",\n \"In the impact parameter plane, the strings appear as disks and with increasing energy-density these disks overlap, fuse and percolate leading to a reduction of the overall charge [29].\",\n \"A cluster of $n$ strings behaves as a single string with energy-momentum corresponding to the sum of the individual strings and with a higher colour field corresponding to the vectorial sum in colour space of the colour fields of the strings.\",\n \"In this way, the mean multiplicity $\\\\langle \\\\mu _{n}\\\\rangle $ and the mean transverse momentum squared $\\\\langle p^{2}_{T}\\\\rangle $ of the particles produced by a cluster in the limit of random distribution of strings are given by $\\\\langle \\\\mu _{n}\\\\rangle =N^{s}F(\\\\eta ^t)\\\\langle \\\\mu _{1}\\\\rangle $ and $\\\\langle p^{2}_{T}\\\\rangle =\\\\langle p^{2}_{T,1}\\\\rangle /F(\\\\eta ^t)$ where $\\\\langle \\\\mu _{1}\\\\rangle $ and $\\\\langle p^{2}_{T,1}\\\\rangle $ are the corresponding quantities for a single string.\",\n \"For a random distribution of disks, the colour reduction factor $F(\\\\eta ^t)$ is $F(\\\\eta ^{t})=\\\\sqrt{\\\\frac{1-e^{-\\\\eta ^{t}}}{\\\\eta ^{t}}}\\\\, ,$ where $\\\\eta ^t$ is the impact parameter transverse density of strings (disks) $\\\\eta ^{t} \\\\equiv \\\\frac{S_1}{S_{N_A}}\\\\, {N}^{s} = \\\\frac{\\\\pi r^2_0}{S_{N_A}}{N}^{s}\\\\, ,$ with $S_{N_A}$ the area of the impact parameter projected overlap of the interaction.\",\n \"For a $N_A N_A$ collision, eq.\",\n \"(REF ) can be recast as $\\\\frac{dn}{dy} \\\\bigg |_{y=0} \\\\sim F(\\\\eta ^t)\\\\, N^s_{N_A}\\\\, , \\\\qquad N^s_{N_A} = N^s_{p} N_A^{1+\\\\alpha (\\\\sqrt{s})}\\\\, .$ The colour reduction factor $F(\\\\eta ^t)$ results in a slowdown of the increase of $dn/dy$ with energy and number of participating nucleons.\",\n \"Note that if nucleons were to act incoherently, as in the limiting case eq.\",\n \"(REF ), $S_{N_A}$ would be given by the area of a single nucleon $S_p$ , while once coherence is accounted for, eq.\",\n \"(REF ), $S_{N_A}$ is the overall area of interaction.\",\n \"In general, $S_{N_A}$ depends on the impact parameter $b$ of the collision ($0\\\\le b\\\\le 2 R_A$ ) with $N_A\\\\rightarrow 0, S_{N_A}\\\\rightarrow 0$ as $b\\\\rightarrow R_A$ and $N_A\\\\rightarrow A, S_{N_A}\\\\rightarrow S_A\\\\equiv \\\\pi R_A^2 \\\\equiv \\\\pi R^2_p A^{2/3}$ as $b\\\\rightarrow 0$ .\",\n \"These constraints are satisfied by $\\\\frac{S_{N_{A}}}{S_{A}}=\\\\bigg (\\\\frac{N_{A}}{A}\\\\bigg )^{\\\\beta }\\\\, ,$ with $\\\\beta >0$ .\",\n \"It follows that $S_{N_{A}}=\\\\pi R^{2}_{p}\\\\,A^{2/3}\\\\bigg (\\\\frac{N_{A}}{A}\\\\bigg )^{\\\\beta }\\\\, ,$ and $\\\\eta ^{t}_{N_{A}}=\\\\eta ^{t}_{p}N_{A}^{\\\\alpha (\\\\sqrt{s})}A^{1/3}\\\\bigg (\\\\frac{A}{N_{A}}\\\\bigg )^{\\\\beta -1}\\\\, .$ Here, motivated by the scaling limit of the number of vertices in a loop-erased random walk [30], we set $\\\\beta =5/3$ such that $\\\\eta ^{t}_{N_{A}}=\\\\eta ^{t}_{p}N_{A}^{\\\\alpha (\\\\sqrt{s})}\\\\bigg (\\\\frac{A}{N_{A}^{2/3}}\\\\bigg )\\\\, .$ From eqs.\",\n \"(REF ), (REF ), (REF ), (REF ) we can write $\\\\frac{1}{N_{A}}\\\\, \\\\frac{dn}{dy}\\\\bigg |_{y=0}= \\\\frac{dn^{pp}}{dy}\\\\bigg |_{y=0} \\\\bigg (1+\\\\frac{F(\\\\eta ^{t}_{N_{A}})}{F(\\\\eta ^{t}_{p})}(N^{\\\\alpha (\\\\sqrt{s})}_{A}-1)\\\\bigg )\\\\, ,$ The dependence of the multiplicity on the centre-of-mass collision energy $\\\\sqrt{s}$ is fully specified once the average number of strings in a pp collision $N_p^s$ is known.\",\n \"At low energy $N_p^s$ , is approximately equal to 2, growing with energy as $(\\\\sqrt{s}/m_p)^{2\\\\lambda }$ such that $N_p^s = 2 + 4\\\\bigg (\\\\frac{r_0}{R_p}\\\\bigg )^2 \\\\bigg (\\\\frac{\\\\sqrt{s}}{m_p}\\\\bigg )^{2\\\\lambda }\\\\, ,$ with $R_p$ the proton radius.\"\n ],\n [\n \"Data description\",\n \"The energy and number of participants dependences implied by eq.\",\n \"(REF ) can be tested against the available experimental data.\",\n \"Before doing so, two remarks on the multiplicity given by eq.\",\n \"(REF ) are in order: (i) it is for all particles, while experimental data accounts only for charged particles; (ii) it is for mid-rapidity $y\\\\sim 0$ , while data is collected at mid pseudo-rapidity $\\\\eta \\\\sim 0$ and although these coincide, the multiplicity value should be rescaled by Jacobean of the $y\\\\rightarrow \\\\eta $ transformation.\",\n \"Neglecting the dependence of $p_T$ , on which the Jacobean depends, on centre-of-mass energy and the (small) difference between the Jacobean in pp and AA cases, both rescaling factors above can be absorbed into an overall normalization constant $\\\\kappa $ .\",\n \"We thus write the charged particle multiplicity at mid-rapidity in the pp case as $\\\\frac{dn^{pp}_{\\\\rm ch}}{d\\\\eta }\\\\bigg |_{\\\\eta =0} = \\\\kappa \\\\, F(\\\\eta ^{t}_{p})\\\\, N_p^s\\\\, ,$ such that, for the general $N_A N_A$ case, one has $\\\\frac{1}{N_A} \\\\frac{dn^{N_AN_A}_{\\\\rm ch}}{d\\\\eta }\\\\bigg |_{\\\\eta =0} = \\\\kappa \\\\, F(\\\\eta ^{t}_{p})\\\\, N_p^s \\\\bigg (1+\\\\frac{F(\\\\eta ^{t}_{N_{A}})}{F(\\\\eta ^{t}_{p})}(N^{\\\\alpha (\\\\sqrt{s})}_{A}-1)\\\\bigg )\\\\, .$ Eq.\",\n \"(REF ) depends on three parameters: (i) the normalization $\\\\kappa $ ; (ii) the threshold scale $\\\\sqrt{s_0}$ in $\\\\alpha ({\\\\sqrt{s}})$ in eq.\",\n \"(REF ); (iii) the power $\\\\lambda $ in eq.\",\n \"(REF ).\",\n \"We performed a global fit to pp data [31], [32], [33], [34], [35], [36], [37] in the range $ 53\\\\le \\\\sqrt{s}\\\\le 7000$ GeV, and to AA (AuAu, CuCu and PbPb) at different centralities [38], [39] for $19.6\\\\le \\\\sqrt{s}\\\\le 2760$ GeV.\",\n \"The fitted data sample consists of 116 points (19 for pp and 97 for AA).\",\n \"The best fit yields the parameter values: $\\\\kappa = 0.63\\\\pm 0.01$ , $\\\\lambda =0.201\\\\pm 0.003$ , and $\\\\sqrt{s_0}=245\\\\pm 29$ GeV.\",\n \"One notices that while $\\\\kappa $ and $\\\\lambda $ are tightly constrained, the threshold scale $\\\\sqrt{s_0}$ is determined with a sizable associated error of $\\\\sim $ 10%.\",\n \"This results from the mild dependence of eq.\",\n \"(REF ), through eq.\",\n \"(REF ), on $\\\\sqrt{s_0}$ compounded with the sizable errors in the existing measurements (see Fig.\",\n \"REF below).\",\n \"Fig.\",\n \"REF shows a comparison of the evolution of the mid-rapidity multiplicity with energy given by eq.\",\n \"(REF ) with data for pp and AA collisions.\",\n \"Figure: Multiplicity dependence on s\\\\sqrt{s}.\",\n \"pp data from , , , , , , (circles), CuCu (triangles) and AuAu (stars) from , PbPb (star) from .\",\n \"Curves obtained from eq.\",\n \"(): (N A =1N_A=1, A=1A=1) for pp (grey line); (N A =50N_A=50, A=63A=63) for CuCu (blue line); and (N A =175N_A=175, A=200A=200) for AuAu/PbPb (red line).\",\n \"Color online.For reference we quote, in Table REF , the predicted values relevant for future LHC runs.\",\n \"Table: Predicted multiplicities for pp and PbPb at future LHC energies.The dependence of the multiplicity on the number of participating nucleons implied by eq.\",\n \"(REF ) is compared with data in Fig.\",\n \"REF .\",\n \"Figure: Multiplicity dependence on centrality (the number of participating nucleons N part =2N A N_{\\\\rm part}=2 N_A where N A N_A is the number of participants per nucleus).\",\n \"CuCu (triangles) and AuAu (stars) data from , and PbPb (circles) from .\",\n \"Curves obtained from eq.\",\n \"(): (s=22.4,62.4,200\\\\sqrt{s}=22.4, 62.4, 200 GeV) for CuCu (blue); (s=19.6,62.4,130,200\\\\sqrt{s}=19.6, 62.4, 130, 200 GeV) for AuAu (green); and (s=2.76,3.2,3.9,5.5\\\\sqrt{s}=2.76, 3.2, 3.9, 5.5 TeV) for PbPb (red).\",\n \"Color online.\"\n ],\n [\n \"Conclusions\",\n \"We have shown that, in the SPM, the power law dependence of the multiplicity on the collision energy is the same in pp and AA collisions.\",\n \"The slower growth in the AA case is due to finite energy-momentum constraints which tamper string creation, and thus the multiplicity, at low energy.\",\n \"In the high energy limit, $F(\\\\eta ^{t})\\\\rightarrow (\\\\eta ^t)^{-1/2}$ and $\\\\alpha (\\\\sqrt{s})\\\\rightarrow 1/3$ , eq.\",\n \"(REF ) can be written $\\\\frac{1}{N_{A}}\\\\,\\\\frac{dn^{N_AN_A}_{\\\\rm ch}}{d\\\\eta }\\\\bigg |_{\\\\eta =0} = \\\\frac{dn^{pp}_{\\\\rm ch}}{d\\\\eta }\\\\bigg |_{\\\\eta =0} \\\\Bigg (1+\\\\bigg (\\\\frac{N_{A}}{A} \\\\bigg )^{1/2} \\\\bigg (1 - \\\\frac{1}{N_A^{1/3}}\\\\bigg )\\\\Bigg )\\\\, .$ With the obtained fit parameters, this asymptotic result becomes a good approximation (within 5% of that given by eq.\",\n \"(REF )) for $\\\\sqrt{s}\\\\gtrsim 500$ GeV.\",\n \"This indicates that finite energy corrections persist to fairly high energies.\",\n \"In this high energy limit, the shape of $\\\\frac{dn^{N_AN_A}}{d\\\\eta }$ as a function of the number of participants is energy independent, that is to say that the $N_{\\\\rm part}$ and $\\\\sqrt{s}$ dependences factorize.\",\n \"In the SPM, energy-momentum conservation results in violations of this factorization and they are the origin of the observed discrepancy in multiplicity growth with energy in pp and AA.\",\n \"The arguments put forward in this short note can be readily adapted to the case of asymmetric (proton-nucleus) collisions.\",\n \"Also in this case, energy-momentum constraints are expected to play an important role.\"\n ],\n [\n \"Acknowledgments\",\n \"We thank N. Armesto for useful discussions.\",\n \"IB is supported by the grant SFRH/BD/51370/2011 from Fundação para a Ciência e a Tecnologia (Portugal).\",\n \"JDD and JGM acknowledge the support of Fundação para a Ciência e a Tecnologia (Portugal) under project CERN/FP/116379/2010.\",\n \"IB and CP were partly supported by the project FPA2008-01177 and FPA2011-22776 of MICINN, the Spanish Consolider Ingenio 2010 program CPAN and Conselleria de Educacion Xunta de Galicia.\"\n ]\n]"},"id":{"kind":"string","value":"1204.1457"}}},{"rowIdx":1866885,"cells":{"text":{"kind":"list like","value":[["A Weiszfeld-like algorithm for a Weber location problem constrained to a\n closed and convex set"],["Abstract The Weber problem consists of finding a point in $\\mathbbm{R}^n$ that minimizes the weighted sum of distances from $m$ points in $\\mathbbm{R}^n$ that are not collinear.","An application that motivated this problem is the optimal location of facilities in the 2-dimensional case.","A classical method to solve the Weber problem, proposed by Weiszfeld in 1937, is based on a fixed point iteration.","In this work a Weber problem constrained to a closed and convex set is considered.","A Weiszfeld-like algorithm, well defined even when an iterate is a vertex, is presented.","The iteration function $Q$ that defines the proposed algorithm, is based mainly on an orthogonal projection over the feasible set, combined with the iteration function of a modified Weiszfeld algorithm presented by Vardi and Zhang in 2001.","It can be seen that the proposed algorithm generates a sequence of feasible iterates that have descent properties.","Under certain hypotheses, the limit of this sequence satisfies the KKT optimality conditions, is a fixed point of the iteration function that defines the algorithm, and is the solution of the constrained minimization problem.","Numerical experiments confirmed the theoretical results."],["Introduction","Let $a^1, \\ldots , a^m$ be $m$ distinct points in the space $\\mathbb {R}^n$ , called vertices, and positive numbers $w_1, \\ldots , w_m$ , called weights.","The function $f : \\mathbb {R}^n \\rightarrow \\mathbb {R}$ defined by $f( x ) = \\sum _{j=1}^m w_j \\left\\Vert x - a^j \\right\\Vert ,$ is called the Weber function, where $\\left\\Vert \\cdot \\right\\Vert $ denotes the Euclidean norm.","It is well-known that this function is not differentiable at the vertices, and strictly convex if the vertices are not collinear (we will assume this hypothesis from now on).","The Weber problem (also known as the Fermat-Weber problem) is to find a point in $\\mathbb {R}^n$ that minimizes the weighted sum of Euclidean distances from the $m$ given points, that is, we have to find the solution of the following unconstrained optimization problem: $\\begin{array}{rl}\\displaystyle \\mathop { \\mathrm {argmin} }_x & \\displaystyle f( x ) \\\\ \\mathrm {subject \\, to} & x \\in \\mathbb {R}^n.\\end{array}$ This problem has a unique solution $x^u$ in $\\mathbb {R}^n$ .","The problem was also stated as a pure mathematical problem by Fermat [44], [27], Cavalieri [37], Steiner [14], Fasbender [20] and many others.","Several solutions, based on geometrical arguments, were proposed by Torricelli and Simpson.","In [30] historical details and geometric aspects were presented by Kupitz and Martini.","In [41] Weber formulated the problem (REF ) from an economical point of view.","The vertices represent customers or demands, the solution to the problem denotes the location of a new facility, and the weights are costs associated with the interactions between the new facility and the customers.","Among several schemes to solve the Weber location problem (see [12], [19], [28], [34]), one of the most popular methods was presented by Weiszfeld in [42], [43].","The Weiszfeld algorithm is an iterative method based on the first-order necessary conditions for a stationary point of the objective function.","If we define $T_0 : \\mathbb {R}^n \\rightarrow \\mathbb {R}^n$ by: $T_0(x) = \\left\\lbrace \\begin{array}{ll} \\frac{ \\displaystyle \\sum _{j=1}^m \\frac{ w_j a^j }{ \\left\\Vert x - a^j \\right\\Vert } }{ \\displaystyle \\sum _{j=1}^m \\frac{ w_j }{ \\left\\Vert x - a^j \\right\\Vert } }, & \\quad \\text{if $x \\ne a^1, \\ldots , a^m$}, \\\\ & \\\\ a^k, & \\quad \\text{if $x = a^k$, $k = 1, \\ldots , m$}, \\end{array} \\right.$ the Weiszfeld algorithm is: $x^{(l)} = T_0 \\left( x^{(l-1)} \\right), \\quad l \\in \\mathbb {N},$ where $x^{(0)} \\in \\mathbb {R}^n$ is a starting point.","The Weiszfeld algorithm (REF ), despite of its simplicity, has a serious problem if some $x^{(l)}$ lands accidentally in a vertex $a^k$ , because the algorithm gets stuck at $a^k$ , even when $a^k$ is not the solution of (REF ).","Many authors studied the set of initial points for which the sequence generated by the Weiszfeld algorithm yields in a vertex (see [29], [11], [6], [9], [7], [3]).","Vardi and Zhang [40] derived a simple but nontrivial modification of the Weiszfeld algorithm in which they solved the problem of landing in a vertex.","Generalizations and new techniques for the Fermat-Weber location problem have been developed in recent years.","In [18] Eckhardt applied the Weiszfeld algorithm to generalized Weber problems in Banach spaces.","An exact algorithm for a Weber problem with attraction and repulsion was presented by Chen et al.","in [13].","Kaplan and Yang [24] proved a duality theorem which includes as special cases a great variety of choices of norms in the terms of the Fermat-Weber sum.","In [10] Carrizosa et al.","studied the so called Regional Weber Problem, which allows the demand not to be concentrated onto a finite set of points, but follows an arbitrary probability measure.","In [17] Drezner and Wesolowsky studied the case where different $l_p$ norms are used for each demand point.","In [23] the so called Complementary Problem (the Weber problem with one negative weight) was studied by Jalal and Krarup, and geometrical solutions were given.","In [15] Drezner presented a Weiszfeld-like iterative procedure and convergence is proved if appropriate conditions hold.","In some practical problems it is necessary to consider barriers (forbidden regions).","Barriers were first introduced to location modeling by Katz and Cooper [25].","There exist several heuristic and iterative algorithms for single-facility location problems for distance computations in the presence of barriers (see [2], [8], [5], [4]).","In [35] Pfeiffer and Klamroth presented a unified formulation for problems with barriers and network location problems.","A complete reference to barriers in location problems can be found in [26].","Barriers can be applied to model real life problems where regions like lakes and mountains are forbidden.","On the other hand, there are location problems whose solution needs to lie within a closed set.","For example, see [39] for a discussion of the case when the solution is constrained to be within a maximum distance of each demand point.","Drezner and Wesolowsky [16] studied the problem of locating an obnoxious facility with rectangular distances ($l_1$ norm), where the facility must lie within some prespecified region (linear constraints).","A primal-dual algorithm to deal with the constrained Fermat-Weber problem using mixed norms was developed in [33] by Idrissi et al..","In [21] Hansen et al.","presented an algorithm for solving the Weber problem when the set of feasible locations is the union of a finite number of convex polygons.","In [36] Pilotta and Torres considered a Weber location problem with box constraints.","Constrained Weber problems arise when we require that the solution is in an area (feasible region) determined by, for example, environmental and/or political reasons.","It could be the case for a facility producing dangerous materials that must be installed in a restricted (constrained) area.","Another example could be the location of a plant in an industrial zone or of a hospital in a non-polluted area.","In this paper a constrained location problem is considered.","An algorithm is proposed to solve the following problem: $\\begin{array}{rl}\\displaystyle \\mathop { \\mathrm {argmin} }_x & \\displaystyle f( x ) \\\\ \\mathrm {subject \\, to} & x \\in \\Omega ,\\end{array}$ where $\\Omega $ is a closed and convex set, generalizing the problem formulated in [36].","Problem (REF ) could be seen as a nonlinear programming problem and solved by standard solvers, but they may fail since the Weber function is not differentiable at the vertices.","It can be proved that problem (REF ) has a unique solution $x^*$ , since the function $f$ is strictly convex and $\\Omega $ is a closed and convex set.","On the other hand, it is well-known that the convex hull of the given vertices $a^1, \\ldots , a^m$ contains the solution $x^u$ of the unconstrained Weber problem (see for instance [29]).","If $\\Omega $ contains the convex hull, both solutions $x^*$ and $x^u$ agree.","In other cases, the solution $x^*$ is not necessarily a projection of $x^u$ over $\\Omega $ (see [36]).","The algorithm is based basically on a slight variation of an orthogonal projection of the Weiszfeld algorithm presented in [40], that is well defined even when an iterate coincides with a vertex.","Properties of the sequence generated by the proposed algorithm related with the minimization problem REF will be proved in the following sections.","The paper is structured as follows: Section describes the results in [40] in which a modified Weiszfeld algorithm is presented and some notation is introduced.","In Section the proposed algorithm is defined.","Section is dedicated to definitions and technical lemmas.","In Section the main results about convergence to optimality are presented.","Numerical experiments are considered in Section .","Finally, conclusions are given in Section .","Some words about notation.","As it was mentioned, we will call $x^u$ the solution of problem (REF ) and $x^*$ the solution of problem (REF ).","The symbols $\\Vert \\cdot \\Vert $ and $\\langle \\cdot , \\cdot \\rangle $ will refer to the standard Euclidean norm and standard inner product in $\\mathbb {R}^n$ , respectively.","For a function $f : \\mathbb {R} \\rightarrow \\mathbb {R}$ we will denote by $f^{\\prime }(a-)$ the left-hand side derivative at $a$ , and by $f^{\\prime }(a+)$ the right-hand side derivative at $a$ ."],["The modified Weiszfeld algorithm","This section reviews the main results presented in [40] in which the authors generalize the Weiszfeld algorithm for the case that an iterate lands on a vertex.","From now on, this algorithm will be referred to as the modified Weiszfeld algorithm.","In order to make notation easier, we define the function $A : \\mathbb {R}^n \\rightarrow \\mathbb {R}$ by: $A(x) = \\left\\lbrace \\begin{array}{ll} \\displaystyle \\sum _{j=1}^m \\frac{ w_j }{ 2 \\left\\Vert x - a^j \\right\\Vert }, & \\quad \\text{if $x \\ne a^1, \\ldots , a^m$}, \\\\ & \\\\ \\displaystyle \\sum _{\\begin{array}{c}j=1 \\\\ j \\ne k\\end{array}}^m \\frac{ w_j }{ 2 \\left\\Vert a^k - a^j \\right\\Vert }, & \\quad \\text{if $x = a^k$, $k = 1, \\ldots , m$}.","\\end{array} \\right.$ Notice that $A(x) > 0$ for all $x \\in \\mathbb {R}^n$ .","In [40], the number $A(a^k)$ was called $A_k$ .","A generalization for the iteration function $T_0$ , defined in (REF ), is given by $\\widetilde{T} : \\mathbb {R}^n \\rightarrow \\mathbb {R}^n$ defined as follows: $\\widetilde{T}(x) = \\left\\lbrace \\begin{array}{ll} \\displaystyle \\frac{ \\displaystyle \\sum _{j=1}^m \\frac{ w_j a^j }{ \\left\\Vert x - a^j \\right\\Vert } }{ 2 A(x) }, & \\quad \\text{if $x \\ne a^1, \\ldots , a^m$}, \\\\ & \\\\ \\displaystyle \\frac{ \\displaystyle \\sum _{\\begin{array}{c}j=1 \\\\ j \\ne k\\end{array}}^m \\frac{ w_j a^j }{ \\left\\Vert a^k - a^j \\right\\Vert } }{ 2 A\\left(a^k\\right) }, & \\quad \\text{if $x = a^k$, $k = 1, \\ldots , m$}.","\\end{array} \\right.$ Notice that $\\widetilde{T}$ coincides with $T_o$ in $\\mathbb {R}^n - \\left\\lbrace a^1, \\ldots , a^m \\right\\rbrace $ .","Let $\\widetilde{R} : \\mathbb {R}^n \\rightarrow \\mathbb {R}^n$ and $r : \\mathbb {R}^n \\rightarrow \\mathbb {R}$ be: $\\widetilde{R}(x) & = & \\left\\lbrace \\begin{array}{ll} \\displaystyle \\sum _{j=1}^m \\frac{ w_j \\left( a^j - x \\right) }{ \\left\\Vert x - a^j \\right\\Vert }, & \\quad \\text{if $x \\ne a^1, \\ldots , a^m$}, \\\\ & \\\\ \\displaystyle \\sum _{\\begin{array}{c}j=1 \\\\ j \\ne k\\end{array}}^m \\frac{ w_j \\left( a^j - a^k \\right) }{ \\left\\Vert a^k - a^j \\right\\Vert }, & \\quad \\text{if $x = a^k$, $k = 1, \\ldots , m$}, \\end{array} \\right.\\\\& \\nonumber \\\\r(x) & = & \\Vert \\widetilde{R}(x) \\Vert , \\qquad \\forall \\, x \\in \\mathbb {R}^n.","\\nonumber $ The function $\\widetilde{R}$ generalizes the negative gradient of the Weber function since, for all $x \\ne a^1, \\ldots , a^m$ , $\\nabla f(x) = - \\widetilde{R}(x).$ The following lemma is very easy to prove (see [40]), and it relates the functionals $\\widetilde{T}$ and $\\widetilde{R}$ .","Lemma 1 For all $x \\in \\mathbb {R}^n$ we have $\\widetilde{R}(x) = 2 A(x) \\left[ \\widetilde{T}(x) - x \\right]$ .","If we define $\\gamma : \\mathbb {R}^n \\rightarrow \\mathbb {R}$ by: $\\gamma (x) = \\left\\lbrace \\begin{array}{ll} 0, & \\quad \\text{if $x \\ne a^1, \\ldots , a^m$}, \\\\ 0, & \\quad \\text{if $x = a^k$ and $r\\left(a^k\\right) = 0$ for some $k = 1, \\ldots , m$}, \\\\ \\displaystyle w_k / r\\left(a^k\\right), & \\quad \\text{if $x = a^k$ and $r\\left(a^k\\right) \\ne 0$ for some $k = 1, \\ldots , m$}, \\end{array} \\right.$ we can see that $\\gamma (x) \\ge 0$ for all $x \\in \\mathbb {R}^n$ .","The modified Weiszfeld algorithm presented in [40] is defined by: $x^{(l)} = T\\left( x^{(l-1)} \\right), \\quad l \\in \\mathbb {N},$ where $x^{(0)} \\in \\mathbb {R}^n$ and $T : \\mathbb {R}^n \\rightarrow \\mathbb {R}^n$ is given by: $T(x) = \\left( 1 - \\beta (x) \\right) \\widetilde{T}(x) + \\beta (x) x,$ where $\\beta : \\mathbb {R}^n \\rightarrow \\mathbb {R}$ is defined by $\\beta (x) = \\min \\left\\lbrace 1, \\gamma (x) \\right\\rbrace $ .","Remark 2 If $x \\ne a^1, \\ldots , a^m$ , then $\\beta (x) = 0$ because $\\gamma (x) = 0$ .","So, we can deduce that $T(x) = \\widetilde{T}(x)$ .","Notice that this fact implies that the functional $T$ is continuous in $\\mathbb {R}^n - \\left\\lbrace a^1, \\ldots , a^m \\right\\rbrace $ .","It can be seen that if $a^k \\ne x^u$ , then $0 < \\beta ( a^k ) < 1$ (see [40]).","From equation (REF ) we obtain that $T(x) - x = \\left( 1 - \\beta (x) \\right) \\left( \\widetilde{T}(x) - x \\right)$ for $x \\in \\mathbb {R}^n$ .","The main result in [40] is: Theorem 3 The following propositions are equivalent: $x = x^u$ .","$T(x) = x$ .","$r(x) \\le \\eta (x)$ .","where $\\eta (x) = \\left\\lbrace \\begin{array}{ll} 0, & \\quad \\text{if $x \\ne a^1, \\ldots , a^m$}, \\\\ w_k, & \\quad \\text{if $x = a^k$, $k = 1, \\ldots , m$}.","\\end{array} \\right.$"],["The proposed algorithm","This section is dedicated to describe the proposed algorithm, introducing some definitions and remarks.","First of all, we can notice that problem (REF ) has a unique solution, due to the fact that $f$ is a non-negative, strictly convex, and continuous function, $\\lim _{\\Vert x \\Vert \\rightarrow \\infty } f(x) = \\infty $ and $\\Omega $ is closed and convex.","In order to define the proposed algorithm at the vertices, we will need to determine which points of the segment that joins $a^k$ and $T(a^k)$ are in the feasible set $\\Omega $ .","If $k = 1, \\ldots , m$ , let the set $\\mathcal {S}_k$ be defined by: $\\mathcal {S}_k = \\left\\lbrace \\lambda \\in [0,1] : ( 1 - \\lambda ) T(a^k) + \\lambda a^k \\in \\Omega \\right\\rbrace .$ Notice that $\\mathcal {S}_k$ could be equal to the empty set in case that $a^k$ and $T(a^k)$ do not belong to $\\Omega $ .","On the other hand, if $a^k \\in \\Omega $ , then $1 \\in \\mathcal {S}_k$ , which means that $\\mathcal {S}_k \\ne \\emptyset $ .","Thus, we can define: $\\lambda ( a^k ) = \\inf \\mathcal {S}_k, \\quad a^k \\in \\Omega .$ In case a vertex $a^k$ is not in $\\Omega $ , there is no need to define the number $\\lambda ( a^k )$ .","In the following lemma, a set of basic properties of $\\lambda ( a^k )$ are listed: Lemma 4 If $k = 1, \\ldots , m$ and $a^k \\in \\Omega $ then: $\\lambda ( a^k ) \\in [0,1]$ .","If $T(a^k) \\in \\Omega $ then $\\lambda ( a^k ) = 0$ .","If $T(a^k) \\notin \\Omega $ then $\\lambda ( a^k ) \\in (0,1]$ .","The proof of (a) follows from the definition of $\\mathcal {S}_k$ .","If $T(a^k) \\in \\Omega $ , then $0 \\in \\mathcal {S}_k$ , so $\\lambda (a^k) = 0$ , and this proves (b).","Finally, for item (c), let us consider that $T(a^k) \\notin \\Omega $ .","Since $\\Omega $ is a closed set, there is an entire ball centered at $T(a^k)$ that does not intersect $\\Omega $ , which implies that there exists $\\epsilon $ such that $( 1 - \\lambda ) T(a^k) + \\lambda a^k \\notin \\Omega $ for all $\\lambda \\in [0,\\epsilon ]$ .","Thus, $\\lambda (a^k) \\in (0,1]$ and this concludes the proof.","$\\Box $ Let us call $P_{\\Omega } : \\mathbb {R}^n \\rightarrow \\Omega $ the orthogonal projection over $\\Omega $ .","Since $\\Omega $ is a nonempty, closed and convex set, the operator $P_{\\Omega }$ is a continuous function [1].","We define the iteration function $Q : \\Omega \\rightarrow \\Omega $ by: $Q(x) = \\left\\lbrace \\begin{array}{ll} P_{\\Omega } \\circ T(x), & \\quad \\text{if $x \\ne a^1, \\ldots , a^m$}, \\\\ \\left( 1 - \\lambda ( a^k ) \\right) T(a^k) + \\lambda ( a^k ) a^k, & \\quad \\text{if $x = a^k \\in \\Omega $, $k = 1, \\ldots , m$}.","\\end{array} \\right.$ There will be no need to define $Q$ outside $\\Omega $ since the proposed algorithm generates a sequence of feasible points.","The iteration function $Q$ at $x \\in \\Omega $ coincides with the orthogonal projection of $T(x)$ over the feasible set when $x$ is different from the vertices.","Only when $x$ is a vertex $a^k$ belonging to $\\Omega $ , $Q(x)$ is defined as the farthest possible feasible point of the segment that joins $x$ with $T(x)$ .","The following remark states some basic properties of the iteration function of the proposed algorithm.","Remark 5 (a) If $a^k \\in \\Omega $ and $T(a^k) \\in \\Omega $ , then $Q(a^k) = T(a^k) = P_{\\Omega } \\circ T(a^k)$ .","(b) If $a^k \\in \\Omega $ , it can be seen that: $Q(a^k) - a^k & = & \\left( 1 - \\lambda ( a^k ) \\right) \\left( T(a^k) - a^k \\right),\\\\Q(a^k) - T(a^k) & = & - \\lambda ( a^k ) \\left( T(a^k) - a^k \\right).$ (c) The functional $Q$ is continuous in $\\mathbb {R}^n - \\left\\lbrace a^1, \\ldots , a^m \\right\\rbrace $ .","The proofs of (a) and (b) are straightforward.","For (c), since $P_{\\Omega }$ is continuous in $\\mathbb {R}^n$ (see [1]) and $T$ is continuous in $\\mathbb {R}^n - \\left\\lbrace a^1, \\ldots , a^m \\right\\rbrace $ (see Remark REF ), we have that $Q$ is continuous in $\\mathbb {R}^n - \\left\\lbrace a^1, \\ldots , a^m \\right\\rbrace $ .","$\\Box $ The proposed algorithm is described below.","Algorithm 6 Let $\\Omega \\subset \\mathbb {R}^n$ be a closed and convex set.","Assume that $x^{(0)} \\in \\Omega $ is an initial approximation such that $f(x^{(0)}) \\le f(a^j)$ for all $j \\in \\left\\lbrace 1, \\ldots , m \\right\\rbrace $ and $a^j \\in \\Omega $ .","Given $\\varepsilon > 0$ a tolerance and $x^{(l-1)} \\in \\Omega $ , do the following steps to compute $x^{(l)}$ : Step 1: Compute: $x^{(l)} = Q \\left( x^{(l-1)} \\right).$ Step 2: Stop the execution if $\\left\\Vert x^{(l)} - x^{(l-1)} \\right\\Vert < \\varepsilon ,$ and declare $x^{(l)}$ as solution to the problem (REF ).","Otherwise return to Step 1.","From the definition of $Q$ it follows that Algorithm REF generates a sequence of feasible iterates.","Also notice that if there are vertices in the feasible set, $x^{(0)}$ can be one of them, for example, a vertex $a_s$ such that $f(a_s) \\le f(a^j)$ for all $a^j \\in \\Omega $ .","On the other hand, if there are no vertices in the feasible set, $x^{(0)}$ can be chosen as the projection over $\\Omega $ of the null vector.","Some definitions and technical results The purpose of this section is to define some entities and prove technical lemmas that will be important in the proof of the main results.","First of all, we will define some useful operators for making notation easier.","If $\\mathcal {A} \\subset \\left\\lbrace 1, \\ldots , n \\right\\rbrace $ , then we define $\\Vert \\cdot \\Vert _{\\mathcal {A}} : \\mathbb {R}^n \\rightarrow \\mathbb {R}$ and $\\langle \\cdot , \\cdot \\rangle _{\\mathcal {A}} : \\mathbb {R}^n \\times \\mathbb {R}^n \\rightarrow \\mathbb {R}$ by: $\\Vert x \\Vert _{\\mathcal {A}} = \\sqrt{ \\sum _{j \\in \\mathcal {A}} x_j^2 }, \\qquad \\langle x , y \\rangle _{\\mathcal {A}} = \\sum _{j \\in \\mathcal {A}} x_j y_j.$ Notice that, when $\\mathcal {A} \\subsetneq \\left\\lbrace 1, \\ldots , n \\right\\rbrace $ , $\\Vert \\cdot \\Vert _{\\mathcal {A}}$ is not necessarily a norm and $\\langle \\cdot , \\cdot \\rangle _{\\mathcal {A}}$ is not necessarily an inner product.","According to this definition, if $\\mathcal {A}$ and $\\mathcal {B}$ are sets such that $\\mathcal {A} \\cap \\mathcal {B} = \\emptyset $ and $\\mathcal {A} \\cup \\mathcal {B} = \\left\\lbrace 1, \\ldots , n \\right\\rbrace $ , it can be seen that: $\\Vert x \\Vert ^2 & = & \\Vert x \\Vert _{\\mathcal {A}}^2 + \\Vert x \\Vert _{\\mathcal {B}}^2,\\\\\\langle x , y \\rangle & = & \\langle x , y \\rangle _{\\mathcal {A}} + \\langle x , y \\rangle _{\\mathcal {B}},\\\\c \\langle x , y \\rangle _{\\mathcal {A}} & = & \\langle cx , y \\rangle _{\\mathcal {A}} = \\langle x , cy \\rangle _{\\mathcal {A}}.$ For $x \\in \\Omega $ , let us define the following sets of indices: $\\mathcal {N}(x) & = & \\left\\lbrace k \\in \\mathbb {N} : \\quad 1 \\le k \\le n, \\quad (T(x))_k \\ne (Q(x))_k \\right\\rbrace ,\\\\\\mathcal {E}(x) & = & \\left\\lbrace k \\in \\mathbb {N} : \\quad 1 \\le k \\le n, \\quad (T(x))_k = (Q(x))_k \\right\\rbrace ,$ Notice that for all $x \\in \\mathbb {R}^n$ we have that $\\mathcal {N}(x) \\cap \\mathcal {E}(x) = \\emptyset $ and $\\mathcal {N}(x) \\cup \\mathcal {E}(x) = \\left\\lbrace 1, \\ldots , n \\right\\rbrace $ .","Let $\\alpha : \\Omega \\rightarrow \\mathbb {R}^n$ be the following function: If $x \\ne a^1, \\ldots , a^m$ : $\\alpha (x) = \\displaystyle \\sum _{j=1}^m \\frac{ w_j }{ \\Vert x - a^j \\Vert } \\left[ Q(x) - a^j \\right].$ If $x = a^k \\in \\Omega $ for some $k = 1, \\ldots , m$ : $\\alpha (x) = \\displaystyle \\sum _{\\begin{array}{c}j=1 \\\\ j \\ne k\\end{array}}^m \\frac{ w_j }{ \\Vert a^k - a^j \\Vert } \\left[ Q(a^k) - (1 - \\beta (a^k)) a^j - \\beta (a^k) a^k \\right].$ It can be seen that the function $\\alpha $ is related to the iteration function $Q$ of the proposed algorithm, and the iteration function $T$ of the modified algorithm.","Lemma 7 If $x \\in \\Omega $ , then $\\alpha (x) = 2 A(x) \\left[ Q(x) - T(x) \\right].$ If $x \\ne a^1, \\ldots , a^m$ , then: $\\alpha (x) & = & \\sum _{j=1}^m \\frac{ w_j }{ \\Vert x - a^j \\Vert } \\left[ Q(x) - a^j \\right] = \\sum _{j=1}^m \\frac{ w_j Q(x) }{ \\Vert x - a^j \\Vert } - \\sum _{j=1}^m \\frac{ w_j a^j }{ \\Vert x - a^j \\Vert }\\\\&&\\\\& = & \\left( \\sum _{j=1}^m \\frac{ w_j }{ \\Vert x - a^j \\Vert } \\right) \\left[ Q(x) - \\frac{ \\sum _{j=1}^m \\frac{ w_j a^j }{ \\Vert x - a^j \\Vert } }{ \\sum _{j=1}^m \\frac{ w_j }{ \\Vert x - a^j \\Vert } } \\right] = 2 A(x) \\left[ Q(x) - \\widetilde{T}(x) \\right]\\\\&&\\\\& = & 2 A(x) \\left[ Q(x) - T(x) \\right].$ where in the last equalities we have used the definition of $\\widetilde{T}$ as in (REF ), and the fact that $\\widetilde{T}(x) = T(x)$ due to Remark REF .","If $x = a^k$ for some $k = 1, \\ldots , m$ , we follow a similar procedure than in the previous case.","$\\Box $ Now, we will define auxiliary functions that take into account the projection $P_{\\Omega }$ in order to prove a descent property of $f$ (see next sections).","If $x \\in \\Omega $ , we define: $E_x : \\mathbb {R}^n \\rightarrow \\mathbb {R}^n$ , where: $\\left( E_x(y) \\right)_k = \\left\\lbrace \\begin{array}{ll} (Q(x))_k, & \\quad \\text{if $k \\in \\mathcal {N}(x)$}, \\\\ y_k, & \\quad \\text{if $k \\in \\mathcal {E}(x)$}.","\\end{array} \\right.$ If $\\mathcal {E}(x) = \\left\\lbrace i_1, \\ldots , i_r \\right\\rbrace \\ne \\emptyset $ define $P_x : \\mathbb {R}^n \\rightarrow \\mathbb {R}^r$ where: $\\left( P_x(y) \\right)_k = y_{i_k}, \\quad k = 1, \\ldots , r.$ A useful property of $E_x$ , that follows from the definition, is pointed out in the following remark.","Remark 8 If $x \\in \\Omega $ then $E_x \\circ Q(x) = Q(x)$ .","The iteration function $Q$ inherits an important property from the orthogonal projection $P_{\\Omega }$ .","Lemma 9 If $x \\in \\Omega $ we have that $\\left\\langle Q(x) - x , Q(x) - T(x) \\right\\rangle \\le 0$ .","If $x \\ne a^1, \\ldots , a^m$ , then $Q(x) = P_{\\Omega } \\circ T(x)$ .","By a property of the orthogonal projection [1] we have that $\\langle Q(x) - x , Q(x) - T(x) \\rangle \\le 0$ .","If $x = a^k$ for some $k = 1, \\ldots , m$ , Remark REF and Lemma REF imply: $\\left\\langle Q(a^k) - a^k , Q(a^k) - T(a^k) \\right\\rangle = - \\lambda ( a^k ) \\left( 1 - \\lambda ( a^k ) \\right) \\left\\Vert T(a^k) - a^k \\right\\Vert ^2 \\le 0,$ and this concludes the proof.","$\\Box $ The next technical lemma will help us to save computations in other lemmas.","Lemma 10 If $x \\in \\Omega $ , $\\mathcal {A}$ is a subset of $\\left\\lbrace 1, \\ldots , n \\right\\rbrace $ and $j \\in \\left\\lbrace 1, \\ldots , m \\right\\rbrace $ , then: $\\left\\Vert Q(x) - a^j \\right\\Vert _{\\mathcal {A}}^2 = \\left\\Vert x - a^j \\right\\Vert _{\\mathcal {A}}^2 - \\left\\Vert Q(x) - x \\right\\Vert _{\\mathcal {A}}^2 + 2 \\left\\langle Q(x) - x, Q(x) - a^j \\right\\rangle _{\\mathcal {A}}.$ If $x \\in \\Omega $ , we have: $\\Vert Q(x) - a^j \\Vert _{\\mathcal {A}}^2 & = & \\langle Q(x) - a^j , Q(x) - a^j \\rangle _{\\mathcal {A}}\\\\& = & \\langle Q(x) - x + x - a^j , Q(x) - x + x - a^j \\rangle _{\\mathcal {A}}\\\\& = & \\Vert Q(x) - x \\Vert _{\\mathcal {A}}^2 + \\Vert x - a^j \\Vert _{\\mathcal {A}}^2 + 2 \\langle Q(x) - x , x - a^j \\rangle _{\\mathcal {A}}\\\\& = & \\Vert Q(x) - x \\Vert _{\\mathcal {A}}^2 + \\Vert x - a^j \\Vert _{\\mathcal {A}}^2 + 2 \\langle Q(x) - x , x - Q(x) \\rangle _{\\mathcal {A}}\\\\& + & 2 \\langle Q(x) - x , Q(x) - a^j \\rangle _{\\mathcal {A}}\\\\& = & \\Vert x - a^j \\Vert _{\\mathcal {A}}^2 - \\Vert Q(x) - x \\Vert _{\\mathcal {A}}^2 + 2 \\langle Q(x) - x , Q(x) - a^j \\rangle _{\\mathcal {A}}.$ $\\Box $ If $x \\in \\Omega $ , let us define $g_x : \\mathbb {R}^n \\rightarrow \\mathbb {R}$ by: $g_x(y) = \\left\\lbrace \\begin{array}{ll} \\displaystyle {\\sum _{j=1}^m} \\frac{ w_j }{ 2 \\Vert x - a^j \\Vert } \\Vert E_x(y) - a^j \\Vert ^2, & \\text{if $x \\ne a^1, \\ldots , a^m$}, \\\\ & \\\\ \\displaystyle { \\sum _{\\begin{array}{c}j=1 \\\\j \\ne k\\end{array}}^m} \\frac{ w_j }{ 2 \\Vert a^k - a^j \\Vert } \\Vert y - a^j \\Vert ^2 + w_k \\Vert y - a^k \\Vert , & \\text{if $x = a^k$}, \\\\ & \\text{$k = 1, \\ldots , m$}.","\\end{array} \\right.$ The values that $g_x$ assumes at $x$ and $Q(x)$ will play an important role in the proof of a property of the objective function $f$ .","Lemma 11 Let $x \\in \\Omega $ be.","If $x \\ne a^1, \\ldots , a^m$ then: $g_x(x) & = & \\frac{1}{2} f(x) + 2 A(x) \\left\\langle Q(x) - x , Q(x) - T(x) \\right\\rangle \\\\& - & A(x) \\left\\Vert Q(x) - x \\right\\Vert _{\\mathcal {N}(x)}^2.$ If $x = a^k$ for some $k = 1, \\ldots , m$ , then $g_{a^k}(a^k) = \\frac{1}{2} f(a^k)$ .","Let us suppose that $x \\ne a^1, \\ldots , a^m$ .","By property (REF ) and (REF ), we have for $j = 1, \\ldots , m$ : $\\left\\Vert E_x(x) - a^j \\right\\Vert ^2 = \\left\\Vert x - a^j \\right\\Vert _{\\mathcal {E}(x)}^2 + \\left\\Vert Q(x) - a^j \\right\\Vert _{\\mathcal {N}(x)}^2.$ Using Lemma REF , we can see that: $g_x(x) & = & \\sum _{j=1}^m \\frac{ w_j }{ 2 \\Vert x - a^j \\Vert } \\left[ \\left\\Vert x - a^j \\right\\Vert _{\\mathcal {E}(x)}^2 + \\left\\Vert x - a^j \\right\\Vert _{\\mathcal {N}(x)}^2 \\right.\\\\& - & \\left.","\\left\\Vert Q(x) - x \\right\\Vert _{\\mathcal {N}(x)}^2 + 2 \\left\\langle Q(x) - x , Q(x) - a^j \\right\\rangle _{\\mathcal {N}(x)} \\right].$ Due to (REF ), the definition of the Weber function $f$ , the definition of $A$ as in (REF ), the property () and the definition of $\\alpha $ as in (REF ), we obtain: $g_x(x) = \\frac{1}{2} f(x) - A(x) \\Vert Q(x) - x \\Vert _{\\mathcal {N}(x)}^2 + \\langle Q(x) - x , \\alpha (x) \\rangle _{\\mathcal {N}(x)}.$ By Lemma REF , the fact that $\\left( Q(x) \\right)_i = \\left( T(x) \\right)_i$ for all $i \\in \\mathcal {E}(x)$ and (), we get: $g_x(x) & = & \\frac{1}{2} f(x) - A(x) \\Vert Q(x) - x \\Vert _{\\mathcal {N}(x)}^2\\\\& + & 2 A(x) \\langle Q(x) - x , Q(x) - T(x) \\rangle ,$ which concludes the proof of (a).","Now, let us assume that $x = a^k$ for some $k = 1, \\ldots , m$ .","Then: $g_{a^k}(a^k) = \\sum _{\\begin{array}{c}j=1 \\\\ j \\ne k\\end{array}}^m \\frac{ w_j }{ 2 \\Vert a^k - a^j \\Vert } \\Vert a^k - a^j \\Vert ^2 = \\frac{1}{2} \\sum _{\\begin{array}{c}j=1 \\\\ j \\ne k\\end{array}}^m w_j \\Vert a^k - a^j \\Vert = \\frac{1}{2} f(a^k).$ This concludes the proof of (b).","$\\Box $ The number $g_x(Q(x))$ can be computed in the next lemma.","Lemma 12 Let $x \\in \\Omega $ be.","If $x \\ne a^1, \\ldots , a^m$ then: $g_x(Q(x)) & = & \\frac{1}{2} f(x) + 2 A(x) \\left\\langle Q(x) - x , Q(x) - T(x) \\right\\rangle \\\\& - & A(x) \\left\\Vert Q(x) - x \\right\\Vert ^2.$ If $x = a^k$ for some $k = 1, \\ldots , m$ , then $g_{a^k}(Q(a^k)) & = & \\frac{1}{2} f(a^k) - A(a^k) \\left\\Vert Q(a^k) - a^k \\right\\Vert ^2\\\\& + & 2 A(a^k) \\left\\langle Q(a^k) - a^k , Q(a^k) - T(a^k) \\right\\rangle \\\\& - & 2 \\beta (a^k) A(a^k) \\left\\langle Q(a^k) - a^k , \\widetilde{T}(a^k) - a^k \\right\\rangle \\\\& + & w_k \\left\\Vert Q(a^k) - a^k \\right\\Vert .$ First, let us consider $x \\ne a^1, \\ldots , a^m$ .","Due to Remark REF we have: $g_x(Q(x)) = \\sum _{j=1}^m \\frac{ w_j }{ 2 \\Vert x - a^j \\Vert } \\Vert Q(x) - a^j \\Vert ^2.$ By Lemma REF we obtain: $g_x(Q(x))& = & \\sum _{j=1}^m \\frac{ w_j }{ 2 \\Vert x - a^j \\Vert } \\left[ \\Vert x - a^j \\Vert ^2 - \\Vert Q(x) - x \\Vert ^2 \\right.\\\\& + & \\left.","2 \\langle Q(x) - x , Q(x) - a^j \\rangle \\right].$ Due to the definition of the Weber function $f$ , the definition of $A$ as in (REF ) and the definition of $\\alpha $ as in (REF ), we deduce that: $g_x(Q(x)) = \\frac{1}{2} f(x) - A(x) \\Vert Q(x) - x \\Vert ^2 + \\langle Q(x) - x , \\alpha (x) \\rangle .$ By Lemma REF we get: $g_x(Q(x)) = \\frac{1}{2} f(x) - A(x) \\Vert Q(x) - x \\Vert ^2 + 2 A(x) \\langle Q(x) - x , Q(x) - T(x) \\rangle ,$ concluding the proof of (a).","Now, consider $x = a^k$ for some $k = 1, \\ldots , m$ .","Due to (REF ) we have: $g_{a^k}(Q(a^k)) = \\sum _{\\begin{array}{c}j=1 \\\\ j \\ne k\\end{array}}^m \\frac{ w_j }{ 2 \\Vert a^k - a^j \\Vert } \\Vert Q(a^k) - a^j \\Vert ^2 + w_k \\Vert Q(a^k) - a^k \\Vert .$ By Lemma REF , the definition of the Weber function $f$ and the definition of $A$ as in (REF ) we obtain: $g_{a^k}\\left( Q(a^k) \\right) & = & \\frac{1}{2} f(a^k) - A(a^k) \\Vert Q(a^k) - a^k \\Vert ^2\\\\& + & \\left\\langle Q(a^k) - a^k , \\sum _{\\begin{array}{c}j=1 \\\\ j \\ne k\\end{array}}^m \\frac{ w_j }{ \\Vert a^k - a^j \\Vert } \\left[ Q(a^k) - a^j \\right] \\right\\rangle \\\\& + & w_k \\Vert Q(a^k) - a^k \\Vert .$ Manipulating algebraically, $Q(a^k) - a^j = Q(a^k) - ( 1 - \\beta (a^k) ) a^j - \\beta (a^k) a^k + \\beta (a^k) ( a^k - a^j ).$ Due to the definition of $\\alpha $ (see (REF )) and the definition of $\\widetilde{R}$ (see (REF )) we get: $g_{a^k} \\left( Q(a^k) \\right)& = & \\frac{1}{2} f(a^k) - A(a^k) \\left\\Vert Q(a^k) - a^k \\right\\Vert ^2\\\\& + & \\left\\langle Q(a^k) - a^k , \\alpha (a^k) \\right\\rangle - \\beta (a^k) \\left\\langle Q(a^k) - a^k , \\widetilde{R}(a^k) \\right\\rangle \\\\& + & w_k \\Vert Q(a^k) - a^k \\Vert .$ By Lemma REF and Lemma REF we have: $g_{a^k} \\left( Q(a^k) \\right) & = & \\frac{1}{2} f(a^k) - A(a^k) \\left\\Vert Q(a^k) - a^k \\right\\Vert ^2\\\\& + & 2 A(a^k) \\left\\langle Q(a^k) - a^k , Q(a^k) - T(a^k) \\right\\rangle \\\\& - & 2 A(a^k) \\beta (a^k) \\left\\langle Q(a^k) - a^k , \\widetilde{T}(a^k) - a^k \\right\\rangle + w_k \\Vert Q(a^k) - a^k \\Vert .$ which concludes the proof.","$\\Box $ The next lemma deals with the last two terms of $g_{a^k}(Q(a^k))$ .","Lemma 13 If $a^k \\in \\Omega $ for some $k = 1, \\ldots , m$ , the number $z = w_k \\left\\Vert Q(a^k) - a^k \\right\\Vert - 2 A(a^k) \\beta (a^k) \\left\\langle Q(a^k) - a^k , \\widetilde{T}(a^k) - a^k \\right\\rangle ,$ is equal to zero.","If $Q(a^k) = a^k$ the result is true.","So, from now on, let us consider that $Q(a^k) \\ne a^k$ .","First, let us check that $a^k \\ne x^u$ .","In case that $a^k = x^u$ , then $T(a^k) = a^k$ by Theorem REF .","Since $a^k \\in \\Omega $ , then $T(a^k) \\in \\Omega $ .","By Remark REF we have that $Q(a^k) = T(a^k) = a^k$ which is a contradiction.","By Remark REF , we have that $\\beta (a^k) \\in (0,1)$ (since $a^k \\ne x^u$ ) and: $z = w_k \\left\\Vert Q(a^k) - a^k \\right\\Vert - \\frac{ 2 A(a^k) \\beta (a^k) }{ 1 - \\beta (a^k) } \\left\\langle Q(a^k) - a^k , T(a^k) - a^k \\right\\rangle .$ Extracting common factors, using Remarks REF and REF , the fact that $T(a^k) \\ne a^k$ (if $T(a^k) = a^k$ then $a^k = x^u$ by Theorem REF ), and the fact that $\\widetilde{T}(a^k) \\ne a^k$ (if $\\widetilde{T}(a^k) = a^k$ then $T(a^k) = a^k$ by definition (REF )) we get that: $z & = & 2 A(a^k) \\left\\Vert Q(a^k) - a^k \\right\\Vert \\left\\Vert \\widetilde{T}(a^k) - a^k \\right\\Vert \\left[ \\frac{ w_k }{ 2 A(a^k) \\left\\Vert \\widetilde{T}(a^k) - a^k \\right\\Vert } \\right.\\\\& - & \\left.","\\beta (a^k) \\left\\langle \\frac{ ( 1 - \\lambda ( a^k ) ) ( T(a^k) - a^k ) }{ \\left\\Vert ( 1 - \\lambda ( a^k ) ) ( T(a^k) - a^k ) \\right\\Vert } , \\frac{ T(a^k) - a^k }{ \\left\\Vert T(a^k) - a^k \\right\\Vert } \\right\\rangle \\right].$ Simplifying and using the definition of $\\beta (a^k)$ we have that: $z = 2 A(a^k) \\left\\Vert Q(a^k) - a^k \\right\\Vert \\left\\Vert \\widetilde{T}(a^k) - a^k \\right\\Vert \\left[ \\beta (a^k) - \\beta (a^k) \\right] = 0,$ which concludes the proof.","$\\Box $ The purpose of the next two lemmas is to determine a strict inequality between the functions $g_x$ and $f$ at suitable points.","First of all, we have to prove the following result.","Lemma 14 Let $x \\in \\Omega $ be such that $x \\ne Q(x)$ .","If $x \\ne a^1, \\ldots , a^m$ , then $g_x(Q(x)) \\le g_x(x)$ .","Besides that, if $\\mathcal {E}(x) \\ne \\emptyset $ and $P_x \\circ Q(x) \\ne P_x(x)$ , then $g_x(Q(x)) < g_x(x)$ .","If $x = a^k$ for some $k = 1, \\ldots , m$ , then $g_{a^k}(Q(a^k)) < g_{a^k}(a^k)$ .","If $x \\ne a^1, \\ldots , a^m$ , then $g_x(Q(x)) - g_x(x) = - A(x) \\Vert Q(x) - x \\Vert _{\\mathcal {E}(x)}^2 \\le 0$ , by Lemma REF and Lemma REF .","Besides that, if $\\mathcal {E}(x) \\ne \\emptyset $ and $P_x \\circ Q(x) \\ne P_x(x)$ we deduce that $\\Vert Q(x) - x \\Vert _{\\mathcal {E}(x)} \\ne 0$ .","Thus, $g_x(Q(x)) < g_x(x)$ .","If $x = a^k$ for some $k = 1, \\ldots , m$ , by Lemmas REF , REF and REF we have: $g_{a^k}(Q(a^k)) - g_{a^k}(a^k) & = & - A(a^k) \\left\\Vert Q(a^k) - a^k \\right\\Vert ^2\\\\& + & 2 A(a^k) \\left\\langle Q(a^k) - a^k , Q(a^k) - T(a^k) \\right\\rangle .$ Due to Lemma REF and the fact that $A > 0$ we obtain: $g_{a^k}(Q(a^k)) - g_{a^k}(a^k) \\le - A(a^k) \\Vert Q(a^k) - a^k \\Vert ^2 < 0,$ and the proof is finished.","$\\Box $ Lemma 15 Let $x \\in \\Omega $ be such that $x \\ne Q(x)$ .","Then $g_x(Q(x)) < \\frac{1}{2} f(x)$ .","Let us consider the case when $x \\ne a^1, \\ldots , a^m$ .","By Lemmas REF , REF and REF we have that: $g_x(Q(x)) & \\le & g_x(x) = \\frac{1}{2} f(x) + 2 A(x) \\left\\langle Q(x) - x , Q(x) - T(x) \\right\\rangle \\\\& - & A(x) \\left\\Vert Q(x) - x \\right\\Vert _{\\mathcal {N}(x)}^2\\\\& \\le & \\frac{1}{2} f(x) - A(x) \\left\\Vert Q(x) - x \\right\\Vert _{\\mathcal {N}(x)}^2.$ If $\\mathcal {E}(x) = \\emptyset $ , then $\\Vert \\cdot \\Vert _{\\mathcal {N}(x)} = \\Vert \\cdot \\Vert $ .","Therefore: $g_x(Q(x)) \\le \\frac{1}{2} f(x) - A(x) \\Vert Q(x) - x \\Vert ^2 < \\frac{1}{2} f(x).$ If $\\mathcal {E}(x) \\ne \\emptyset $ and $P_x \\circ Q(x) = P_x(x)$ , then there exists an index $i \\in \\mathcal {N}(x)$ such that $x_i \\ne (Q(x))_i$ since $x \\ne Q(x)$ .","Thus, $\\Vert Q(x) - x \\Vert _{\\mathcal {N}(x)} \\ne 0$ , which implies: $g_x(Q(x)) \\le \\frac{1}{2} f(x) - A(x) \\left\\Vert Q(x) - x \\right\\Vert _{\\mathcal {N}(x)}^2 < \\frac{1}{2} f(x).$ If $\\mathcal {E}(x) \\ne \\emptyset $ and $P_x \\circ Q(x) \\ne P_x(x)$ , due to Lemmas REF , REF and REF , we have that: $g_x(Q(x)) < g_x(x) \\le \\frac{1}{2} f(x) - A(x) \\left\\Vert Q(x) - x \\right\\Vert _{\\mathcal {N}(x)}^2 \\le \\frac{1}{2} f(x).$ Now, when $x = a^k$ for some $k = 1, \\ldots , m$ , $g_{a^k}(Q(a^k)) < g_{a^k}(a^k) = \\frac{1}{2} f(a^k)$ due to Lemma REF and Lemma REF .","$\\Box $ The next lemma states an equality that relates the Weber function and $g_x$ at appropriate points when $x \\ne a^1, \\ldots , a^m$ .","Besides that, this result will be crucial in the next section.","Lemma 16 Let $x \\ne a^1, \\ldots , a^m$ be such that $x \\in \\Omega $ and $x \\ne Q(x)$ .","Then: $g_x \\circ Q(x) = \\frac{1}{2} f(x) + \\left( f(Q(x)) - f(x) \\right) + \\delta , \\quad \\delta \\ge 0.$ Due to the definition of $g_x$ as in (REF ) and Remark REF we get that: $g_x \\circ Q(x) = \\sum _{j=1}^m \\frac{ w_j }{ 2 \\Vert x - a^j \\Vert } \\Vert Q(x) - a^j \\Vert ^2.$ Adding and subtracting $\\Vert x - a^j \\Vert $ we have: $g_x \\circ Q(x) & = & \\sum _{j=1}^m \\frac{ w_j }{ 2 \\Vert x - a^j \\Vert } \\left[ \\left\\Vert x - a^j \\right\\Vert + \\left( \\left\\Vert Q(x) - a^j \\right\\Vert - \\left\\Vert x - a^j \\right\\Vert \\right) \\right]^2\\\\& = & \\frac{1}{2} \\sum _{j=1}^m w_j \\left\\Vert x - a^j \\right\\Vert + \\sum _{j=1}^m w_j \\left( \\left\\Vert Q(x) - a^j \\right\\Vert - \\left\\Vert x - a^j \\right\\Vert \\right)\\\\& + & \\sum _{j=1}^m \\frac{ w_j }{ 2 \\Vert x - a^j \\Vert } \\left( \\left\\Vert Q(x) - a^j \\right\\Vert - \\left\\Vert x - a^j \\right\\Vert \\right)^2.$ Notice that the first term of the last equality is the Weber function (divided by two), and the last term is a non-negative number, so we will define it as $\\delta $ .","So, using the definition of the Weber function in the middle term we obtain: $g_x \\circ Q(x) = \\frac{1}{2} f(x) + \\left( f(Q(x)) - f(x) \\right) + \\delta .$ $\\Box $ Convergence to optimality results This section states the main results about convergence of the sequence $\\left\\lbrace x^{(l)} \\right\\rbrace $ generated by Algorithm REF .","The next theorem establishes that if a point $x \\in \\Omega $ is not a fixed point of the iteration function, then the function $f$ strictly decreases at the next iterate.","Theorem 17 Let $x \\in \\Omega $ be such that $x \\ne Q(x)$ .","Then $f(Q(x)) < f(x)$ .","Let us consider that $x \\ne a^1, \\ldots , a^m$ .","By Lemma REF , we have that: $g_x \\circ Q(x) < \\frac{1}{2} f(x).$ By Lemma REF we get that: $\\frac{1}{2} f(x) + f(Q(x)) - f(x) + \\delta < \\frac{1}{2} f(x).$ Simplifying the last expression we obtain: $f(Q(x)) - f(x) + \\delta < 0.$ Finally, $f(Q(x)) - f(x) \\le f(Q(x)) - f(x) + \\delta < 0.$ Therefore, $f(Q(x)) < f(x)$ .","Now, consider that $x = a^k$ for some $k = 1, \\ldots , m$ .","Following a reasoning similar than in [40], using Lemma REF we have that: $g_{a^k} \\circ Q(a^k) - g_{a^k}(a^k) < 0.$ By definition of $g_{a^k}$ we know that: $g_{a^k} \\circ Q(a^k) - g_{a^k}(a^k) & = & w_k \\left\\Vert Q(a^k) - a^k \\right\\Vert \\\\& + & \\sum _{\\begin{array}{c}j=1 \\\\ j \\ne k\\end{array}}^m \\frac{ w_j }{ 2 \\left\\Vert a^k - a^j \\right\\Vert } \\left( \\left\\Vert Q(a^k) - a^j \\right\\Vert ^2 - \\left\\Vert a^k - a^j \\right\\Vert ^2 \\right).$ Using the fact that $\\left( a^2 - b^2 \\right)/ (2b) \\ge a - b$ for $a = \\left\\Vert Q(a^k) - a^j \\right\\Vert ^2 \\ge 0$ and $b = \\left\\Vert a^k - a^j \\right\\Vert ^2 > 0$ we obtain that: $g_{a^k} \\circ Q(a^k) - g_{a^k}(a^k) & \\ge & w_k \\left\\Vert Q(a^k) - a^k \\right\\Vert \\\\& - & \\sum _{\\begin{array}{c}j=1 \\\\ j \\ne k\\end{array}}^m w_j \\left\\Vert a^k - a^j \\right\\Vert + \\sum _{\\begin{array}{c}j=1 \\\\ j \\ne k\\end{array}}^m w_j \\left\\Vert Q(a^k) - a^j \\right\\Vert .$ Rearranging terms we deduce that: $0 & > & g_{a^k} \\circ Q(a^k) - g_{a^k}(a^k)\\\\& = & \\sum _{j=1}^m w_j \\left\\Vert Q(a^k) - a^j \\right\\Vert - \\sum _{j=1}^m w_j \\left\\Vert a^k - a^j \\right\\Vert = f(Q(a^k)) - f(a^k),$ and the proof is complete.","$\\Box $ Corollary 18 Let $\\left\\lbrace x^{(l)} \\right\\rbrace $ be the sequence generated by Algorithm REF .","Then the sequence $\\left\\lbrace f \\left( x^{(l)} \\right) \\right\\rbrace $ is not increasing.","Even more, each time $x^{(l)} \\ne Q \\left( x^{(l)} \\right)$ the sequence strictly decreases at the next iterate.","If the sequence $\\left\\lbrace x^{(l)} \\right\\rbrace $ generated by Algorithm REF were not bounded, then we could choose a subsequence $\\left\\lbrace y^{(l)} \\right\\rbrace $ such that $y^{(l)} \\rightarrow \\infty $ .","But this implies that $f(y^{(l)}) \\rightarrow \\infty $ , which is a contradiction since the sequence $\\left\\lbrace f \\left( x^{(l)} \\right) \\right\\rbrace $ is not increasing.","Remark 19 The sequence $\\left\\lbrace x^{(l)} \\right\\rbrace $ generated by Algorithm REF is bounded.","So, there exists a subsequence convergent to a point $x^* \\in \\Omega $ .","Hence, $x^*$ is a feasible point.","Due to the nondifferentiability of $f$ at the vertices $a^1, \\ldots , a^m$ , we can not use the KKT optimality conditions at $a^k$ .","Therefore, if $a^k$ and $z$ are in $\\Omega $ , let us define $G_{a^k}^z : [0,1] \\rightarrow \\mathbb {R}$ by: $G_{a^k}^z(t) = f( a^k + t(z-a^k) ).$ If $a^k \\in \\Omega $ , $z \\in \\Omega $ , $t \\in [0,1]$ and $\\Omega $ convex, we have that $a^k + t(z-a^k) \\in \\Omega $ .","Notice that the right-hand side derivative $G_{a^k}^z(0+)$ (or the directional derivative of $f$ in the direction of $z$ ) exists (see [22]).","Besides that, $G_{a^k}^{z \\ \\prime }(0+) = w_k \\left\\Vert z - a^k \\right\\Vert - \\left\\langle \\widetilde{R}(a^k) , z - a^k \\right\\rangle .$ The next lemma shows that if we are in a vertex $a^k$ , the directional derivative of $f$ at $a^k$ in the direction of $Q(a^k)$ is a descent direction.","Lemma 20 Let $a^k \\in \\Omega $ be such that $T(a^k) \\notin \\Omega $ .","Then: $G_{a^k}^{z \\ \\prime }(0+) \\ge G_{a^k}^{Q(a^k) \\ \\prime }(0+), \\quad \\forall \\, z \\in \\left[ a^k, T(a^k) \\right],$ where: $G_{a^k}^{Q(a^k) \\ \\prime }(0+) = - 2 \\left[ 1 - \\beta (a^k) \\right] A(a^k) \\left\\Vert \\widetilde{T}(a^k) - a^k \\right\\Vert \\left\\Vert Q(a^k) - a^k \\right\\Vert .$ If $T(a^k) = a^k$ then $T(a^k) \\in \\Omega $ , which is a contradiction.","Besides that, if $\\widetilde{T}(a^k) = a^k$ , we would have that $T(a^k) = a^k$ because of (REF ), and again it would be a contradiction.","So, we will consider $\\widetilde{T}(a^k) \\ne a^k$ and $T(a^k) \\ne a^k$ for the rest of the proof.","Since $T(a^k) \\ne a^k$ , then $\\beta (a^k) \\in (0,1)$ (see Remark REF and Theorem REF ).","Let us prove equation (REF ) first.","Now, by (REF ) we can see that: $G_{a^k}^{Q(a^k) \\ \\prime }(0+) = w_k \\left\\Vert Q(a^k) - a^k \\right\\Vert - \\left\\langle \\widetilde{R}(a^k) , Q(a^k) - a^k \\right\\rangle .$ Notice that if $Q(a^k) = a^k$ , equation (REF ) holds.","So, let us consider from now on that $Q(a^k) \\ne a^k$ .","By using Lemma REF we replace $\\widetilde{R}(a^k)$ and get: $G_{a^k}^{Q(a^k) \\ \\prime }(0+) = w_k \\left\\Vert Q(a^k) - a^k \\right\\Vert - 2 A(a^k) \\left\\langle \\widetilde{T}(a^k) - a^k , Q(a^k) - a^k \\right\\rangle .$ Extracting common factors and using the definition of $\\beta $ when it belongs to $(0,1)$ we obtain: $G_{a^k}^{Q(a^k) \\ \\prime }(0+) & = & 2 A(a^k) \\left\\Vert Q(a^k) - a^k \\right\\Vert \\left\\Vert \\widetilde{T}(a^k) - a^k \\right\\Vert \\Bigg [ \\beta (a^k)\\\\& - & \\left.","\\left\\langle \\frac{ \\widetilde{T}(a^k) - a^k }{ \\left\\Vert \\widetilde{T}(a^k) - a^k \\right\\Vert } , \\frac{ Q(a^k) - a^k }{ \\left\\Vert Q(a^k) - a^k \\right\\Vert } \\right\\rangle \\right].$ By Remarks REF and REF the vectors $Q(a^k) - a^k$ and $\\widetilde{T}(a^k) - a^k$ are parallel, so: $G_{a^k}^{Q(a^k) \\ \\prime }(0+) = 2 A(a^k) \\left\\Vert Q(a^k) - a^k \\right\\Vert \\left\\Vert \\widetilde{T}(a^k) - a^k \\right\\Vert \\left[ \\beta (a^k) - 1 \\right].$ which is equivalent to (REF ).","Now, let us prove (REF ).","If $z = a^k$ then $G_{a^k}^z(t) = f(a^k)$ for all $t \\in [0,1]$ , thus $G_{a^k+}^{a^k \\ \\prime }(0) = 0$ , and therefore the inequality (REF ) holds.","So, let us assume that $z \\ne a^k$ for the rest of the proof.","Using (REF ) and due to Lemma REF to replace $\\widetilde{R}(a^k)$ : $G_{a^k}^{z \\ \\prime }(0+) = w_k \\left\\Vert z - a^k \\right\\Vert - 2 A(a^k) \\left\\langle \\widetilde{T}(a^k) - a^k , z - a^k \\right\\rangle .$ Extracting common factors: $G_{a^k}^{z \\ \\prime }(0+) & = & 2 A(a^k) \\left\\Vert z - a^k \\right\\Vert \\left\\Vert \\widetilde{T}(a^k) - a^k \\right\\Vert \\left[ \\frac{ w_k }{ 2 A(a^k) \\left\\Vert \\widetilde{T}(a^k) - a^k \\right\\Vert } \\right.\\\\& - & \\left.","\\left\\langle \\frac{ \\widetilde{T}(a^k) - a^k }{ \\left\\Vert \\widetilde{T}(a^k) - a^k \\right\\Vert } , \\frac{ z - a^k }{ \\left\\Vert z - a^k \\right\\Vert } \\right\\rangle \\right].$ Using the expression for $\\beta (a^k) \\in (0,1)$ we obtain: $G_{a^k}^{z \\ \\prime }(0+) & = & 2 A(a^k) \\left\\Vert z - a^k \\right\\Vert \\left\\Vert \\widetilde{T}(a^k) - a^k \\right\\Vert \\Bigg [ \\beta (a^k)\\\\& - & \\left.","\\left\\langle \\frac{ \\widetilde{T}(a^k) - a^k }{ \\left\\Vert \\widetilde{T}(a^k) - a^k \\right\\Vert } , \\frac{ z - a^k }{ \\left\\Vert z - a^k \\right\\Vert } \\right\\rangle \\right].$ If $z$ belongs to the segment that joins $a^k$ and $T(a^k)$ we have that $z-a^k$ and $\\widetilde{T}(a^k) - a^k$ are parallel vectors, then: $G_{a^k}^{z \\ \\prime }(0+) \\ge - 2 \\left[ 1 - \\beta (a^k) \\right] A(a^k) \\left\\Vert z - a^k \\right\\Vert \\left\\Vert \\widetilde{T}(a^k) - a^k \\right\\Vert .$ We can write $z = (1-\\lambda )T(a^k) + \\lambda a^k$ where $\\lambda \\in [0,1]$ .","Therefore: $G_{a^k}^{z \\ \\prime }(0+) \\ge - 2 \\left[ 1 - \\beta (a^k) \\right] A(a^k) \\left( 1 - \\lambda \\right) \\left\\Vert T(a^k) - a^k \\right\\Vert \\left\\Vert \\widetilde{T}(a^k) - a^k \\right\\Vert .$ for all $\\lambda \\in [0,1]$ .","The minimum value of the right-hand side of the last expression happens when $\\lambda = \\lambda ( a^k )$ , so: $G_{a^k}^{z \\ \\prime }(0+) \\ge - 2 \\left[ 1 - \\beta (a^k) \\right] A(a^k) \\left( 1 - \\lambda ( a^k ) \\right) \\left\\Vert T(a^k) - a^k \\right\\Vert \\left\\Vert \\widetilde{T}(a^k) - a^k \\right\\Vert .$ Using Remark REF we conclude that: $G_{a^k}^{z \\ \\prime }(0+) \\ge - 2 \\left[ 1 - \\beta (a^k) \\right] A(a^k) \\left\\Vert Q(a^k) - a^k \\right\\Vert \\left\\Vert \\widetilde{T}(a^k) - a^k \\right\\Vert .$ $\\Box $ Now we will prove an equivalence that characterizes the solution of (REF ) in terms of the iteration function $Q$ .","Moreover, if $x^*$ is a regular point that is not a vertex, then $x^*$ is a KKT point.","From now on, let us consider that $\\Omega = \\left\\lbrace y \\in \\mathbb {R}^n : g(y) \\le 0, h(y) = 0 \\right\\rbrace ,$ where $g : \\mathbb {R}^n \\rightarrow \\mathbb {R}^s$ is a convex function and $h : \\mathbb {R}^n \\rightarrow \\mathbb {R}^p$ is an affine function.","Theorem 21 Let $\\Omega $ be defined as in (REF ) and $x \\in \\Omega $ .","Consider the following propositions: $x$ is a KKT point.","$x$ is the minimizer of the problem (REF ).","$Q(x) = x$ .","If $x \\ne a^1, \\ldots , a^m$ , $g$ and $h$ are continuously differentiable, and $x$ is a regular point, then (a), (b) and (c) are equivalent.","If $x = a^k$ for some $k = 1, \\ldots , m$ , then (b) implies (c).","Let $x \\ne a^1, \\ldots , a^m$ be.","Since $f$ is strictly convex and $\\Omega $ is convex, the KKT optimality conditions are necessary and sufficient.","Therefore, it holds that (a) is equivalent to (b).","Now we will prove that (b) implies (c).","Let us suppose that $x$ is the minimizer of the problem (REF ).","If $x$ were not a fixed point of the iteration function $Q$ , we would have that $x \\ne Q(x)$ , which means that $f(Q(x)) < f(x)$ by Theorem REF .","This contradicts the hypothesis.","To demonstrate that (c) implies (a), we will assume that $x$ is a fixed point of $Q$ , that is, $x = Q(x)$ .","Since $Q(x) = P_{\\Omega } \\circ T(x)$ , $x$ is the solution of: $\\begin{array}{rl}\\displaystyle \\mathop { \\mathrm {argmin} }_z & \\displaystyle F(z) = \\frac{1}{2} \\left\\Vert z - T(x) \\right\\Vert ^2 \\\\ \\mathrm {subject \\, to} & g(z) \\le 0, \\\\ & h(z) = 0.\\end{array}$ Since $F$ and $g$ are convex, $h$ is affine, and $x$ is a regular point, the KKT optimality conditions hold at $x$ .","That is, there exist multipliers $\\left\\lbrace \\mu _j \\right\\rbrace _{j=1}^s$ and $\\left\\lbrace \\lambda _j \\right\\rbrace _{j=1}^p$ such that (see [31], [38]): $x - T(x) + \\sum _{j=1}^s \\mu _j \\nabla g_j(x) + \\sum _{j=1}^p \\lambda _j \\nabla h_j(x) & = & 0,\\\\\\mu _j g_j( x ) & = & 0, \\quad j = 1, \\ldots , s,\\\\\\mu _j & \\ge & 0, \\quad j = 1, \\ldots , s,\\\\g(x) & \\le & 0,\\\\h(x) & = & 0.$ Multiplying these equations by $2 A(x)$ , using equation (REF ), Lemma REF and Remark REF , we obtain: $\\nabla f(x) + \\sum _{j=1}^s \\left( 2 A(x) \\mu _j \\right) \\nabla g_j(x) + \\sum _{j=1}^p \\left( 2 A(x) \\lambda _j \\right) \\nabla h_j(x) & = & 0,\\\\\\left( 2 A(x) \\mu _j \\right) g_j( x ) & = & 0, \\quad j = 1, \\ldots , s,\\\\\\left( 2 A(x) \\mu _j \\right) & \\ge & 0, \\quad j = 1, \\ldots , s,\\\\g(x) & \\le & 0,\\\\h(x) & = & 0.$ where $\\left\\lbrace 2 A(x) \\mu _j \\right\\rbrace _{j=1}^s$ and $\\left\\lbrace 2 A(x) \\lambda _j \\right\\rbrace _{j=1}^p$ are multipliers.","Therefore, $x$ is a KKT point of the problem (REF ) (see [31], [38]).","Now, let us suppose that $x = a^k$ for some $k = 1, \\ldots , m$ .","As before, if $x$ is a minimizer of the problem (REF ), then $Q(a^k) = a^k$ , otherwise $f(Q(a^k)) < f(a^k)$ , which would be a contradiction.","$\\Box $ Numerical experiments.","The purpose of this section is to discuss the efficiency and robustness of the proposed algorithm versus a solver for nonlinear programming problems.","A prototype code of Algorithm REF was programmed in MATLAB (version R2011a) and executed in a PC running Linux OS, Intel(R) Core(TM) i7 CPU Q720, 1.60GHz.","We have considered a closed and convex set $\\Omega \\subset \\mathbb {R}^2$ defined by the set $\\Omega = \\left\\lbrace y \\in \\mathbb {R}^n : g(y) \\le 0 \\right\\rbrace $ , where $g$ is given by: $g( x ) = \\left[ \\begin{array}{c} \\displaystyle - 4 - \\frac{ 1 }{ 8 } x + \\frac{ 7 }{ 72 } x^2 + \\frac{ 1 }{ 216 } x^2 ( x - 3 ) + y \\\\[3mm] \\displaystyle \\frac{ 4 }{ 5 } x + y - \\frac{ 59 }{ 10 } \\\\[3mm] \\displaystyle x - \\frac{ 11 }{ 2 } \\\\[3mm] \\displaystyle \\frac{ 3 }{ 2 } x - y - \\frac{ 35 }{ 4 } \\\\[3mm] \\displaystyle x - y - \\frac{ 13 }{ 2 } \\\\[3mm] \\displaystyle - 4 + \\frac{ 1 }{ 8 } ( x - 1 ) + \\frac{ 1 }{ 16 } ( x - 1 )^2 + \\frac{ 1 }{ 32 } ( x - 1 )^2 ( x - 3 ) - y \\\\[3mm] \\displaystyle - \\frac{ 1 }{ 3 } x - y - \\frac{ 11 }{ 3 } \\\\[3mm] \\displaystyle - \\frac{ 2 }{ 3 } x - y - \\frac{ 13 }{ 3 } \\\\[3mm] \\displaystyle - 4 x + y - 19 \\end{array} \\right].$ The feasible set is defined by linear and nonlinear constraints, as it can be seen in Figure REF .","Figure: Feasible set Ω\\Omega We have built 1000 different experiments where for each one: The number of vertices was $m = 50$ .","The vertices were normally distributed random vectors, with mean equal to 0 and standard deviation equal to 10.","The weights were uniformly distributed random positive numbers between 0 and 10.","Tolerance was set to $\\varepsilon = 0.00001$ .","On one hand, each experiment was solved using Algorithm REF and, on the other hand, it was considered as a nonlinear programming problem and solved using function $fmincon$ (see [32] and references therein).","Since the Weber function (REF ) is not differentiable at the vertices, nonlinear programming solvers may fail.","Let $x_m(i)$ be the solution of (REF ) obtained by $fmincon$ in experiment $i$ , and $f_{m}(i) = f(x_m(i))$ .","Analogously, let $x_p(i)$ be the solution of (REF ) obtained by Algorithm REF in experiment $i$ , and $f_{p}(i) = f(x_p(i))$ .","Figure REF shows the difference between the arrays $f_{m}$ and $f_{p}$ .","Both methods finished succesfully in all cases, however, Algorithm REF found equal or better results for all experiments.","For example, the difference $f_{m} - f_{p}$ was greater than $0.01$ in 35 experiments (the maximum difference ocurred in experiment 506).","Figure: Difference between minimum values found by Algorithm and fminconfmincon.Feasibility of the solutions $x_p(i)$ can be checked computing $\\max (g(x_m(i)))$ .","Results can be seen in Figure REF Figure: Feasibility of the solution x p (i)x_p(i) obtained by Algorithm .","Conclusions This paper proposes a Weiszfeld-like algorithm for solving the Weber problem constrained to a closed and convex set, and it is well defined even when an iterate is a vertex.","The algorithm consists of two stages: first, iterate using the fixed point modified Weiszfeld iteration (REF ), and second, either project onto the set $\\Omega $ when the iterate is different from the vertices, or, if the iterate is a vertex $a^k$ , take the point belonging to the line that joins $T(a^k)$ with $a^k$ as defined in (REF ).","It is proved that the constrained problem (REF ) has a unique solution.","Besides that, the definition of the iteration function $Q$ allows us to demonstrate that the proposed algorithm produces a sequence $\\left\\lbrace x^{(l)} \\right\\rbrace $ of feasible iterates.","Moreover, the sequence $\\left\\lbrace f \\left( x^{(l)} \\right) \\right\\rbrace $ is not increasing, and when $x^{(l)} \\ne Q\\left( x^{(l)} \\right)$ , the sequence decreases at the next iterate.","It can be seen that if a point $x^*$ is the solution of the problem (REF ) then $x^*$ is a fixed point of the iteration function $Q$ .","Even more, if $x^*$ is different from the vertices, the fact of being $x^*$ a fixed point of $Q$ is equivalent to the fact that $x^*$ satisfies the KKT optimality conditions, and equivalent to the fact that $x^*$ is the solution of the problem (REF ).","These properties allows us to connect the proposed algorithm with the minimization problem.","Numerical experiments showed that the proposed algorithm found equal or better solutions than a well-known standard solver, in a practical example with 1000 random choices of vertices and weights.","That is due to the fact that the proposed algorithm does not use of the existence of derivatives at the vertices, because the Weber function is not differentiable at the vertices."],["Some definitions and technical results","The purpose of this section is to define some entities and prove technical lemmas that will be important in the proof of the main results.","First of all, we will define some useful operators for making notation easier.","If $\\mathcal {A} \\subset \\left\\lbrace 1, \\ldots , n \\right\\rbrace $ , then we define $\\Vert \\cdot \\Vert _{\\mathcal {A}} : \\mathbb {R}^n \\rightarrow \\mathbb {R}$ and $\\langle \\cdot , \\cdot \\rangle _{\\mathcal {A}} : \\mathbb {R}^n \\times \\mathbb {R}^n \\rightarrow \\mathbb {R}$ by: $\\Vert x \\Vert _{\\mathcal {A}} = \\sqrt{ \\sum _{j \\in \\mathcal {A}} x_j^2 }, \\qquad \\langle x , y \\rangle _{\\mathcal {A}} = \\sum _{j \\in \\mathcal {A}} x_j y_j.$ Notice that, when $\\mathcal {A} \\subsetneq \\left\\lbrace 1, \\ldots , n \\right\\rbrace $ , $\\Vert \\cdot \\Vert _{\\mathcal {A}}$ is not necessarily a norm and $\\langle \\cdot , \\cdot \\rangle _{\\mathcal {A}}$ is not necessarily an inner product.","According to this definition, if $\\mathcal {A}$ and $\\mathcal {B}$ are sets such that $\\mathcal {A} \\cap \\mathcal {B} = \\emptyset $ and $\\mathcal {A} \\cup \\mathcal {B} = \\left\\lbrace 1, \\ldots , n \\right\\rbrace $ , it can be seen that: $\\Vert x \\Vert ^2 & = & \\Vert x \\Vert _{\\mathcal {A}}^2 + \\Vert x \\Vert _{\\mathcal {B}}^2,\\\\\\langle x , y \\rangle & = & \\langle x , y \\rangle _{\\mathcal {A}} + \\langle x , y \\rangle _{\\mathcal {B}},\\\\c \\langle x , y \\rangle _{\\mathcal {A}} & = & \\langle cx , y \\rangle _{\\mathcal {A}} = \\langle x , cy \\rangle _{\\mathcal {A}}.$ For $x \\in \\Omega $ , let us define the following sets of indices: $\\mathcal {N}(x) & = & \\left\\lbrace k \\in \\mathbb {N} : \\quad 1 \\le k \\le n, \\quad (T(x))_k \\ne (Q(x))_k \\right\\rbrace ,\\\\\\mathcal {E}(x) & = & \\left\\lbrace k \\in \\mathbb {N} : \\quad 1 \\le k \\le n, \\quad (T(x))_k = (Q(x))_k \\right\\rbrace ,$ Notice that for all $x \\in \\mathbb {R}^n$ we have that $\\mathcal {N}(x) \\cap \\mathcal {E}(x) = \\emptyset $ and $\\mathcal {N}(x) \\cup \\mathcal {E}(x) = \\left\\lbrace 1, \\ldots , n \\right\\rbrace $ .","Let $\\alpha : \\Omega \\rightarrow \\mathbb {R}^n$ be the following function: If $x \\ne a^1, \\ldots , a^m$ : $\\alpha (x) = \\displaystyle \\sum _{j=1}^m \\frac{ w_j }{ \\Vert x - a^j \\Vert } \\left[ Q(x) - a^j \\right].$ If $x = a^k \\in \\Omega $ for some $k = 1, \\ldots , m$ : $\\alpha (x) = \\displaystyle \\sum _{\\begin{array}{c}j=1 \\\\ j \\ne k\\end{array}}^m \\frac{ w_j }{ \\Vert a^k - a^j \\Vert } \\left[ Q(a^k) - (1 - \\beta (a^k)) a^j - \\beta (a^k) a^k \\right].$ It can be seen that the function $\\alpha $ is related to the iteration function $Q$ of the proposed algorithm, and the iteration function $T$ of the modified algorithm.","Lemma 7 If $x \\in \\Omega $ , then $\\alpha (x) = 2 A(x) \\left[ Q(x) - T(x) \\right].$ If $x \\ne a^1, \\ldots , a^m$ , then: $\\alpha (x) & = & \\sum _{j=1}^m \\frac{ w_j }{ \\Vert x - a^j \\Vert } \\left[ Q(x) - a^j \\right] = \\sum _{j=1}^m \\frac{ w_j Q(x) }{ \\Vert x - a^j \\Vert } - \\sum _{j=1}^m \\frac{ w_j a^j }{ \\Vert x - a^j \\Vert }\\\\&&\\\\& = & \\left( \\sum _{j=1}^m \\frac{ w_j }{ \\Vert x - a^j \\Vert } \\right) \\left[ Q(x) - \\frac{ \\sum _{j=1}^m \\frac{ w_j a^j }{ \\Vert x - a^j \\Vert } }{ \\sum _{j=1}^m \\frac{ w_j }{ \\Vert x - a^j \\Vert } } \\right] = 2 A(x) \\left[ Q(x) - \\widetilde{T}(x) \\right]\\\\&&\\\\& = & 2 A(x) \\left[ Q(x) - T(x) \\right].$ where in the last equalities we have used the definition of $\\widetilde{T}$ as in (REF ), and the fact that $\\widetilde{T}(x) = T(x)$ due to Remark REF .","If $x = a^k$ for some $k = 1, \\ldots , m$ , we follow a similar procedure than in the previous case.","$\\Box $ Now, we will define auxiliary functions that take into account the projection $P_{\\Omega }$ in order to prove a descent property of $f$ (see next sections).","If $x \\in \\Omega $ , we define: $E_x : \\mathbb {R}^n \\rightarrow \\mathbb {R}^n$ , where: $\\left( E_x(y) \\right)_k = \\left\\lbrace \\begin{array}{ll} (Q(x))_k, & \\quad \\text{if $k \\in \\mathcal {N}(x)$}, \\\\ y_k, & \\quad \\text{if $k \\in \\mathcal {E}(x)$}.","\\end{array} \\right.$ If $\\mathcal {E}(x) = \\left\\lbrace i_1, \\ldots , i_r \\right\\rbrace \\ne \\emptyset $ define $P_x : \\mathbb {R}^n \\rightarrow \\mathbb {R}^r$ where: $\\left( P_x(y) \\right)_k = y_{i_k}, \\quad k = 1, \\ldots , r.$ A useful property of $E_x$ , that follows from the definition, is pointed out in the following remark.","Remark 8 If $x \\in \\Omega $ then $E_x \\circ Q(x) = Q(x)$ .","The iteration function $Q$ inherits an important property from the orthogonal projection $P_{\\Omega }$ .","Lemma 9 If $x \\in \\Omega $ we have that $\\left\\langle Q(x) - x , Q(x) - T(x) \\right\\rangle \\le 0$ .","If $x \\ne a^1, \\ldots , a^m$ , then $Q(x) = P_{\\Omega } \\circ T(x)$ .","By a property of the orthogonal projection [1] we have that $\\langle Q(x) - x , Q(x) - T(x) \\rangle \\le 0$ .","If $x = a^k$ for some $k = 1, \\ldots , m$ , Remark REF and Lemma REF imply: $\\left\\langle Q(a^k) - a^k , Q(a^k) - T(a^k) \\right\\rangle = - \\lambda ( a^k ) \\left( 1 - \\lambda ( a^k ) \\right) \\left\\Vert T(a^k) - a^k \\right\\Vert ^2 \\le 0,$ and this concludes the proof.","$\\Box $ The next technical lemma will help us to save computations in other lemmas.","Lemma 10 If $x \\in \\Omega $ , $\\mathcal {A}$ is a subset of $\\left\\lbrace 1, \\ldots , n \\right\\rbrace $ and $j \\in \\left\\lbrace 1, \\ldots , m \\right\\rbrace $ , then: $\\left\\Vert Q(x) - a^j \\right\\Vert _{\\mathcal {A}}^2 = \\left\\Vert x - a^j \\right\\Vert _{\\mathcal {A}}^2 - \\left\\Vert Q(x) - x \\right\\Vert _{\\mathcal {A}}^2 + 2 \\left\\langle Q(x) - x, Q(x) - a^j \\right\\rangle _{\\mathcal {A}}.$ If $x \\in \\Omega $ , we have: $\\Vert Q(x) - a^j \\Vert _{\\mathcal {A}}^2 & = & \\langle Q(x) - a^j , Q(x) - a^j \\rangle _{\\mathcal {A}}\\\\& = & \\langle Q(x) - x + x - a^j , Q(x) - x + x - a^j \\rangle _{\\mathcal {A}}\\\\& = & \\Vert Q(x) - x \\Vert _{\\mathcal {A}}^2 + \\Vert x - a^j \\Vert _{\\mathcal {A}}^2 + 2 \\langle Q(x) - x , x - a^j \\rangle _{\\mathcal {A}}\\\\& = & \\Vert Q(x) - x \\Vert _{\\mathcal {A}}^2 + \\Vert x - a^j \\Vert _{\\mathcal {A}}^2 + 2 \\langle Q(x) - x , x - Q(x) \\rangle _{\\mathcal {A}}\\\\& + & 2 \\langle Q(x) - x , Q(x) - a^j \\rangle _{\\mathcal {A}}\\\\& = & \\Vert x - a^j \\Vert _{\\mathcal {A}}^2 - \\Vert Q(x) - x \\Vert _{\\mathcal {A}}^2 + 2 \\langle Q(x) - x , Q(x) - a^j \\rangle _{\\mathcal {A}}.$ $\\Box $ If $x \\in \\Omega $ , let us define $g_x : \\mathbb {R}^n \\rightarrow \\mathbb {R}$ by: $g_x(y) = \\left\\lbrace \\begin{array}{ll} \\displaystyle {\\sum _{j=1}^m} \\frac{ w_j }{ 2 \\Vert x - a^j \\Vert } \\Vert E_x(y) - a^j \\Vert ^2, & \\text{if $x \\ne a^1, \\ldots , a^m$}, \\\\ & \\\\ \\displaystyle { \\sum _{\\begin{array}{c}j=1 \\\\j \\ne k\\end{array}}^m} \\frac{ w_j }{ 2 \\Vert a^k - a^j \\Vert } \\Vert y - a^j \\Vert ^2 + w_k \\Vert y - a^k \\Vert , & \\text{if $x = a^k$}, \\\\ & \\text{$k = 1, \\ldots , m$}.","\\end{array} \\right.$ The values that $g_x$ assumes at $x$ and $Q(x)$ will play an important role in the proof of a property of the objective function $f$ .","Lemma 11 Let $x \\in \\Omega $ be.","If $x \\ne a^1, \\ldots , a^m$ then: $g_x(x) & = & \\frac{1}{2} f(x) + 2 A(x) \\left\\langle Q(x) - x , Q(x) - T(x) \\right\\rangle \\\\& - & A(x) \\left\\Vert Q(x) - x \\right\\Vert _{\\mathcal {N}(x)}^2.$ If $x = a^k$ for some $k = 1, \\ldots , m$ , then $g_{a^k}(a^k) = \\frac{1}{2} f(a^k)$ .","Let us suppose that $x \\ne a^1, \\ldots , a^m$ .","By property (REF ) and (REF ), we have for $j = 1, \\ldots , m$ : $\\left\\Vert E_x(x) - a^j \\right\\Vert ^2 = \\left\\Vert x - a^j \\right\\Vert _{\\mathcal {E}(x)}^2 + \\left\\Vert Q(x) - a^j \\right\\Vert _{\\mathcal {N}(x)}^2.$ Using Lemma REF , we can see that: $g_x(x) & = & \\sum _{j=1}^m \\frac{ w_j }{ 2 \\Vert x - a^j \\Vert } \\left[ \\left\\Vert x - a^j \\right\\Vert _{\\mathcal {E}(x)}^2 + \\left\\Vert x - a^j \\right\\Vert _{\\mathcal {N}(x)}^2 \\right.\\\\& - & \\left.","\\left\\Vert Q(x) - x \\right\\Vert _{\\mathcal {N}(x)}^2 + 2 \\left\\langle Q(x) - x , Q(x) - a^j \\right\\rangle _{\\mathcal {N}(x)} \\right].$ Due to (REF ), the definition of the Weber function $f$ , the definition of $A$ as in (REF ), the property () and the definition of $\\alpha $ as in (REF ), we obtain: $g_x(x) = \\frac{1}{2} f(x) - A(x) \\Vert Q(x) - x \\Vert _{\\mathcal {N}(x)}^2 + \\langle Q(x) - x , \\alpha (x) \\rangle _{\\mathcal {N}(x)}.$ By Lemma REF , the fact that $\\left( Q(x) \\right)_i = \\left( T(x) \\right)_i$ for all $i \\in \\mathcal {E}(x)$ and (), we get: $g_x(x) & = & \\frac{1}{2} f(x) - A(x) \\Vert Q(x) - x \\Vert _{\\mathcal {N}(x)}^2\\\\& + & 2 A(x) \\langle Q(x) - x , Q(x) - T(x) \\rangle ,$ which concludes the proof of (a).","Now, let us assume that $x = a^k$ for some $k = 1, \\ldots , m$ .","Then: $g_{a^k}(a^k) = \\sum _{\\begin{array}{c}j=1 \\\\ j \\ne k\\end{array}}^m \\frac{ w_j }{ 2 \\Vert a^k - a^j \\Vert } \\Vert a^k - a^j \\Vert ^2 = \\frac{1}{2} \\sum _{\\begin{array}{c}j=1 \\\\ j \\ne k\\end{array}}^m w_j \\Vert a^k - a^j \\Vert = \\frac{1}{2} f(a^k).$ This concludes the proof of (b).","$\\Box $ The number $g_x(Q(x))$ can be computed in the next lemma.","Lemma 12 Let $x \\in \\Omega $ be.","If $x \\ne a^1, \\ldots , a^m$ then: $g_x(Q(x)) & = & \\frac{1}{2} f(x) + 2 A(x) \\left\\langle Q(x) - x , Q(x) - T(x) \\right\\rangle \\\\& - & A(x) \\left\\Vert Q(x) - x \\right\\Vert ^2.$ If $x = a^k$ for some $k = 1, \\ldots , m$ , then $g_{a^k}(Q(a^k)) & = & \\frac{1}{2} f(a^k) - A(a^k) \\left\\Vert Q(a^k) - a^k \\right\\Vert ^2\\\\& + & 2 A(a^k) \\left\\langle Q(a^k) - a^k , Q(a^k) - T(a^k) \\right\\rangle \\\\& - & 2 \\beta (a^k) A(a^k) \\left\\langle Q(a^k) - a^k , \\widetilde{T}(a^k) - a^k \\right\\rangle \\\\& + & w_k \\left\\Vert Q(a^k) - a^k \\right\\Vert .$ First, let us consider $x \\ne a^1, \\ldots , a^m$ .","Due to Remark REF we have: $g_x(Q(x)) = \\sum _{j=1}^m \\frac{ w_j }{ 2 \\Vert x - a^j \\Vert } \\Vert Q(x) - a^j \\Vert ^2.$ By Lemma REF we obtain: $g_x(Q(x))& = & \\sum _{j=1}^m \\frac{ w_j }{ 2 \\Vert x - a^j \\Vert } \\left[ \\Vert x - a^j \\Vert ^2 - \\Vert Q(x) - x \\Vert ^2 \\right.\\\\& + & \\left.","2 \\langle Q(x) - x , Q(x) - a^j \\rangle \\right].$ Due to the definition of the Weber function $f$ , the definition of $A$ as in (REF ) and the definition of $\\alpha $ as in (REF ), we deduce that: $g_x(Q(x)) = \\frac{1}{2} f(x) - A(x) \\Vert Q(x) - x \\Vert ^2 + \\langle Q(x) - x , \\alpha (x) \\rangle .$ By Lemma REF we get: $g_x(Q(x)) = \\frac{1}{2} f(x) - A(x) \\Vert Q(x) - x \\Vert ^2 + 2 A(x) \\langle Q(x) - x , Q(x) - T(x) \\rangle ,$ concluding the proof of (a).","Now, consider $x = a^k$ for some $k = 1, \\ldots , m$ .","Due to (REF ) we have: $g_{a^k}(Q(a^k)) = \\sum _{\\begin{array}{c}j=1 \\\\ j \\ne k\\end{array}}^m \\frac{ w_j }{ 2 \\Vert a^k - a^j \\Vert } \\Vert Q(a^k) - a^j \\Vert ^2 + w_k \\Vert Q(a^k) - a^k \\Vert .$ By Lemma REF , the definition of the Weber function $f$ and the definition of $A$ as in (REF ) we obtain: $g_{a^k}\\left( Q(a^k) \\right) & = & \\frac{1}{2} f(a^k) - A(a^k) \\Vert Q(a^k) - a^k \\Vert ^2\\\\& + & \\left\\langle Q(a^k) - a^k , \\sum _{\\begin{array}{c}j=1 \\\\ j \\ne k\\end{array}}^m \\frac{ w_j }{ \\Vert a^k - a^j \\Vert } \\left[ Q(a^k) - a^j \\right] \\right\\rangle \\\\& + & w_k \\Vert Q(a^k) - a^k \\Vert .$ Manipulating algebraically, $Q(a^k) - a^j = Q(a^k) - ( 1 - \\beta (a^k) ) a^j - \\beta (a^k) a^k + \\beta (a^k) ( a^k - a^j ).$ Due to the definition of $\\alpha $ (see (REF )) and the definition of $\\widetilde{R}$ (see (REF )) we get: $g_{a^k} \\left( Q(a^k) \\right)& = & \\frac{1}{2} f(a^k) - A(a^k) \\left\\Vert Q(a^k) - a^k \\right\\Vert ^2\\\\& + & \\left\\langle Q(a^k) - a^k , \\alpha (a^k) \\right\\rangle - \\beta (a^k) \\left\\langle Q(a^k) - a^k , \\widetilde{R}(a^k) \\right\\rangle \\\\& + & w_k \\Vert Q(a^k) - a^k \\Vert .$ By Lemma REF and Lemma REF we have: $g_{a^k} \\left( Q(a^k) \\right) & = & \\frac{1}{2} f(a^k) - A(a^k) \\left\\Vert Q(a^k) - a^k \\right\\Vert ^2\\\\& + & 2 A(a^k) \\left\\langle Q(a^k) - a^k , Q(a^k) - T(a^k) \\right\\rangle \\\\& - & 2 A(a^k) \\beta (a^k) \\left\\langle Q(a^k) - a^k , \\widetilde{T}(a^k) - a^k \\right\\rangle + w_k \\Vert Q(a^k) - a^k \\Vert .$ which concludes the proof.","$\\Box $ The next lemma deals with the last two terms of $g_{a^k}(Q(a^k))$ .","Lemma 13 If $a^k \\in \\Omega $ for some $k = 1, \\ldots , m$ , the number $z = w_k \\left\\Vert Q(a^k) - a^k \\right\\Vert - 2 A(a^k) \\beta (a^k) \\left\\langle Q(a^k) - a^k , \\widetilde{T}(a^k) - a^k \\right\\rangle ,$ is equal to zero.","If $Q(a^k) = a^k$ the result is true.","So, from now on, let us consider that $Q(a^k) \\ne a^k$ .","First, let us check that $a^k \\ne x^u$ .","In case that $a^k = x^u$ , then $T(a^k) = a^k$ by Theorem REF .","Since $a^k \\in \\Omega $ , then $T(a^k) \\in \\Omega $ .","By Remark REF we have that $Q(a^k) = T(a^k) = a^k$ which is a contradiction.","By Remark REF , we have that $\\beta (a^k) \\in (0,1)$ (since $a^k \\ne x^u$ ) and: $z = w_k \\left\\Vert Q(a^k) - a^k \\right\\Vert - \\frac{ 2 A(a^k) \\beta (a^k) }{ 1 - \\beta (a^k) } \\left\\langle Q(a^k) - a^k , T(a^k) - a^k \\right\\rangle .$ Extracting common factors, using Remarks REF and REF , the fact that $T(a^k) \\ne a^k$ (if $T(a^k) = a^k$ then $a^k = x^u$ by Theorem REF ), and the fact that $\\widetilde{T}(a^k) \\ne a^k$ (if $\\widetilde{T}(a^k) = a^k$ then $T(a^k) = a^k$ by definition (REF )) we get that: $z & = & 2 A(a^k) \\left\\Vert Q(a^k) - a^k \\right\\Vert \\left\\Vert \\widetilde{T}(a^k) - a^k \\right\\Vert \\left[ \\frac{ w_k }{ 2 A(a^k) \\left\\Vert \\widetilde{T}(a^k) - a^k \\right\\Vert } \\right.\\\\& - & \\left.","\\beta (a^k) \\left\\langle \\frac{ ( 1 - \\lambda ( a^k ) ) ( T(a^k) - a^k ) }{ \\left\\Vert ( 1 - \\lambda ( a^k ) ) ( T(a^k) - a^k ) \\right\\Vert } , \\frac{ T(a^k) - a^k }{ \\left\\Vert T(a^k) - a^k \\right\\Vert } \\right\\rangle \\right].$ Simplifying and using the definition of $\\beta (a^k)$ we have that: $z = 2 A(a^k) \\left\\Vert Q(a^k) - a^k \\right\\Vert \\left\\Vert \\widetilde{T}(a^k) - a^k \\right\\Vert \\left[ \\beta (a^k) - \\beta (a^k) \\right] = 0,$ which concludes the proof.","$\\Box $ The purpose of the next two lemmas is to determine a strict inequality between the functions $g_x$ and $f$ at suitable points.","First of all, we have to prove the following result.","Lemma 14 Let $x \\in \\Omega $ be such that $x \\ne Q(x)$ .","If $x \\ne a^1, \\ldots , a^m$ , then $g_x(Q(x)) \\le g_x(x)$ .","Besides that, if $\\mathcal {E}(x) \\ne \\emptyset $ and $P_x \\circ Q(x) \\ne P_x(x)$ , then $g_x(Q(x)) < g_x(x)$ .","If $x = a^k$ for some $k = 1, \\ldots , m$ , then $g_{a^k}(Q(a^k)) < g_{a^k}(a^k)$ .","If $x \\ne a^1, \\ldots , a^m$ , then $g_x(Q(x)) - g_x(x) = - A(x) \\Vert Q(x) - x \\Vert _{\\mathcal {E}(x)}^2 \\le 0$ , by Lemma REF and Lemma REF .","Besides that, if $\\mathcal {E}(x) \\ne \\emptyset $ and $P_x \\circ Q(x) \\ne P_x(x)$ we deduce that $\\Vert Q(x) - x \\Vert _{\\mathcal {E}(x)} \\ne 0$ .","Thus, $g_x(Q(x)) < g_x(x)$ .","If $x = a^k$ for some $k = 1, \\ldots , m$ , by Lemmas REF , REF and REF we have: $g_{a^k}(Q(a^k)) - g_{a^k}(a^k) & = & - A(a^k) \\left\\Vert Q(a^k) - a^k \\right\\Vert ^2\\\\& + & 2 A(a^k) \\left\\langle Q(a^k) - a^k , Q(a^k) - T(a^k) \\right\\rangle .$ Due to Lemma REF and the fact that $A > 0$ we obtain: $g_{a^k}(Q(a^k)) - g_{a^k}(a^k) \\le - A(a^k) \\Vert Q(a^k) - a^k \\Vert ^2 < 0,$ and the proof is finished.","$\\Box $ Lemma 15 Let $x \\in \\Omega $ be such that $x \\ne Q(x)$ .","Then $g_x(Q(x)) < \\frac{1}{2} f(x)$ .","Let us consider the case when $x \\ne a^1, \\ldots , a^m$ .","By Lemmas REF , REF and REF we have that: $g_x(Q(x)) & \\le & g_x(x) = \\frac{1}{2} f(x) + 2 A(x) \\left\\langle Q(x) - x , Q(x) - T(x) \\right\\rangle \\\\& - & A(x) \\left\\Vert Q(x) - x \\right\\Vert _{\\mathcal {N}(x)}^2\\\\& \\le & \\frac{1}{2} f(x) - A(x) \\left\\Vert Q(x) - x \\right\\Vert _{\\mathcal {N}(x)}^2.$ If $\\mathcal {E}(x) = \\emptyset $ , then $\\Vert \\cdot \\Vert _{\\mathcal {N}(x)} = \\Vert \\cdot \\Vert $ .","Therefore: $g_x(Q(x)) \\le \\frac{1}{2} f(x) - A(x) \\Vert Q(x) - x \\Vert ^2 < \\frac{1}{2} f(x).$ If $\\mathcal {E}(x) \\ne \\emptyset $ and $P_x \\circ Q(x) = P_x(x)$ , then there exists an index $i \\in \\mathcal {N}(x)$ such that $x_i \\ne (Q(x))_i$ since $x \\ne Q(x)$ .","Thus, $\\Vert Q(x) - x \\Vert _{\\mathcal {N}(x)} \\ne 0$ , which implies: $g_x(Q(x)) \\le \\frac{1}{2} f(x) - A(x) \\left\\Vert Q(x) - x \\right\\Vert _{\\mathcal {N}(x)}^2 < \\frac{1}{2} f(x).$ If $\\mathcal {E}(x) \\ne \\emptyset $ and $P_x \\circ Q(x) \\ne P_x(x)$ , due to Lemmas REF , REF and REF , we have that: $g_x(Q(x)) < g_x(x) \\le \\frac{1}{2} f(x) - A(x) \\left\\Vert Q(x) - x \\right\\Vert _{\\mathcal {N}(x)}^2 \\le \\frac{1}{2} f(x).$ Now, when $x = a^k$ for some $k = 1, \\ldots , m$ , $g_{a^k}(Q(a^k)) < g_{a^k}(a^k) = \\frac{1}{2} f(a^k)$ due to Lemma REF and Lemma REF .","$\\Box $ The next lemma states an equality that relates the Weber function and $g_x$ at appropriate points when $x \\ne a^1, \\ldots , a^m$ .","Besides that, this result will be crucial in the next section.","Lemma 16 Let $x \\ne a^1, \\ldots , a^m$ be such that $x \\in \\Omega $ and $x \\ne Q(x)$ .","Then: $g_x \\circ Q(x) = \\frac{1}{2} f(x) + \\left( f(Q(x)) - f(x) \\right) + \\delta , \\quad \\delta \\ge 0.$ Due to the definition of $g_x$ as in (REF ) and Remark REF we get that: $g_x \\circ Q(x) = \\sum _{j=1}^m \\frac{ w_j }{ 2 \\Vert x - a^j \\Vert } \\Vert Q(x) - a^j \\Vert ^2.$ Adding and subtracting $\\Vert x - a^j \\Vert $ we have: $g_x \\circ Q(x) & = & \\sum _{j=1}^m \\frac{ w_j }{ 2 \\Vert x - a^j \\Vert } \\left[ \\left\\Vert x - a^j \\right\\Vert + \\left( \\left\\Vert Q(x) - a^j \\right\\Vert - \\left\\Vert x - a^j \\right\\Vert \\right) \\right]^2\\\\& = & \\frac{1}{2} \\sum _{j=1}^m w_j \\left\\Vert x - a^j \\right\\Vert + \\sum _{j=1}^m w_j \\left( \\left\\Vert Q(x) - a^j \\right\\Vert - \\left\\Vert x - a^j \\right\\Vert \\right)\\\\& + & \\sum _{j=1}^m \\frac{ w_j }{ 2 \\Vert x - a^j \\Vert } \\left( \\left\\Vert Q(x) - a^j \\right\\Vert - \\left\\Vert x - a^j \\right\\Vert \\right)^2.$ Notice that the first term of the last equality is the Weber function (divided by two), and the last term is a non-negative number, so we will define it as $\\delta $ .","So, using the definition of the Weber function in the middle term we obtain: $g_x \\circ Q(x) = \\frac{1}{2} f(x) + \\left( f(Q(x)) - f(x) \\right) + \\delta .$ $\\Box $"],["Convergence to optimality results","This section states the main results about convergence of the sequence $\\left\\lbrace x^{(l)} \\right\\rbrace $ generated by Algorithm REF .","The next theorem establishes that if a point $x \\in \\Omega $ is not a fixed point of the iteration function, then the function $f$ strictly decreases at the next iterate.","Theorem 17 Let $x \\in \\Omega $ be such that $x \\ne Q(x)$ .","Then $f(Q(x)) < f(x)$ .","Let us consider that $x \\ne a^1, \\ldots , a^m$ .","By Lemma REF , we have that: $g_x \\circ Q(x) < \\frac{1}{2} f(x).$ By Lemma REF we get that: $\\frac{1}{2} f(x) + f(Q(x)) - f(x) + \\delta < \\frac{1}{2} f(x).$ Simplifying the last expression we obtain: $f(Q(x)) - f(x) + \\delta < 0.$ Finally, $f(Q(x)) - f(x) \\le f(Q(x)) - f(x) + \\delta < 0.$ Therefore, $f(Q(x)) < f(x)$ .","Now, consider that $x = a^k$ for some $k = 1, \\ldots , m$ .","Following a reasoning similar than in [40], using Lemma REF we have that: $g_{a^k} \\circ Q(a^k) - g_{a^k}(a^k) < 0.$ By definition of $g_{a^k}$ we know that: $g_{a^k} \\circ Q(a^k) - g_{a^k}(a^k) & = & w_k \\left\\Vert Q(a^k) - a^k \\right\\Vert \\\\& + & \\sum _{\\begin{array}{c}j=1 \\\\ j \\ne k\\end{array}}^m \\frac{ w_j }{ 2 \\left\\Vert a^k - a^j \\right\\Vert } \\left( \\left\\Vert Q(a^k) - a^j \\right\\Vert ^2 - \\left\\Vert a^k - a^j \\right\\Vert ^2 \\right).$ Using the fact that $\\left( a^2 - b^2 \\right)/ (2b) \\ge a - b$ for $a = \\left\\Vert Q(a^k) - a^j \\right\\Vert ^2 \\ge 0$ and $b = \\left\\Vert a^k - a^j \\right\\Vert ^2 > 0$ we obtain that: $g_{a^k} \\circ Q(a^k) - g_{a^k}(a^k) & \\ge & w_k \\left\\Vert Q(a^k) - a^k \\right\\Vert \\\\& - & \\sum _{\\begin{array}{c}j=1 \\\\ j \\ne k\\end{array}}^m w_j \\left\\Vert a^k - a^j \\right\\Vert + \\sum _{\\begin{array}{c}j=1 \\\\ j \\ne k\\end{array}}^m w_j \\left\\Vert Q(a^k) - a^j \\right\\Vert .$ Rearranging terms we deduce that: $0 & > & g_{a^k} \\circ Q(a^k) - g_{a^k}(a^k)\\\\& = & \\sum _{j=1}^m w_j \\left\\Vert Q(a^k) - a^j \\right\\Vert - \\sum _{j=1}^m w_j \\left\\Vert a^k - a^j \\right\\Vert = f(Q(a^k)) - f(a^k),$ and the proof is complete.","$\\Box $ Corollary 18 Let $\\left\\lbrace x^{(l)} \\right\\rbrace $ be the sequence generated by Algorithm REF .","Then the sequence $\\left\\lbrace f \\left( x^{(l)} \\right) \\right\\rbrace $ is not increasing.","Even more, each time $x^{(l)} \\ne Q \\left( x^{(l)} \\right)$ the sequence strictly decreases at the next iterate.","If the sequence $\\left\\lbrace x^{(l)} \\right\\rbrace $ generated by Algorithm REF were not bounded, then we could choose a subsequence $\\left\\lbrace y^{(l)} \\right\\rbrace $ such that $y^{(l)} \\rightarrow \\infty $ .","But this implies that $f(y^{(l)}) \\rightarrow \\infty $ , which is a contradiction since the sequence $\\left\\lbrace f \\left( x^{(l)} \\right) \\right\\rbrace $ is not increasing.","Remark 19 The sequence $\\left\\lbrace x^{(l)} \\right\\rbrace $ generated by Algorithm REF is bounded.","So, there exists a subsequence convergent to a point $x^* \\in \\Omega $ .","Hence, $x^*$ is a feasible point.","Due to the nondifferentiability of $f$ at the vertices $a^1, \\ldots , a^m$ , we can not use the KKT optimality conditions at $a^k$ .","Therefore, if $a^k$ and $z$ are in $\\Omega $ , let us define $G_{a^k}^z : [0,1] \\rightarrow \\mathbb {R}$ by: $G_{a^k}^z(t) = f( a^k + t(z-a^k) ).$ If $a^k \\in \\Omega $ , $z \\in \\Omega $ , $t \\in [0,1]$ and $\\Omega $ convex, we have that $a^k + t(z-a^k) \\in \\Omega $ .","Notice that the right-hand side derivative $G_{a^k}^z(0+)$ (or the directional derivative of $f$ in the direction of $z$ ) exists (see [22]).","Besides that, $G_{a^k}^{z \\ \\prime }(0+) = w_k \\left\\Vert z - a^k \\right\\Vert - \\left\\langle \\widetilde{R}(a^k) , z - a^k \\right\\rangle .$ The next lemma shows that if we are in a vertex $a^k$ , the directional derivative of $f$ at $a^k$ in the direction of $Q(a^k)$ is a descent direction.","Lemma 20 Let $a^k \\in \\Omega $ be such that $T(a^k) \\notin \\Omega $ .","Then: $G_{a^k}^{z \\ \\prime }(0+) \\ge G_{a^k}^{Q(a^k) \\ \\prime }(0+), \\quad \\forall \\, z \\in \\left[ a^k, T(a^k) \\right],$ where: $G_{a^k}^{Q(a^k) \\ \\prime }(0+) = - 2 \\left[ 1 - \\beta (a^k) \\right] A(a^k) \\left\\Vert \\widetilde{T}(a^k) - a^k \\right\\Vert \\left\\Vert Q(a^k) - a^k \\right\\Vert .$ If $T(a^k) = a^k$ then $T(a^k) \\in \\Omega $ , which is a contradiction.","Besides that, if $\\widetilde{T}(a^k) = a^k$ , we would have that $T(a^k) = a^k$ because of (REF ), and again it would be a contradiction.","So, we will consider $\\widetilde{T}(a^k) \\ne a^k$ and $T(a^k) \\ne a^k$ for the rest of the proof.","Since $T(a^k) \\ne a^k$ , then $\\beta (a^k) \\in (0,1)$ (see Remark REF and Theorem REF ).","Let us prove equation (REF ) first.","Now, by (REF ) we can see that: $G_{a^k}^{Q(a^k) \\ \\prime }(0+) = w_k \\left\\Vert Q(a^k) - a^k \\right\\Vert - \\left\\langle \\widetilde{R}(a^k) , Q(a^k) - a^k \\right\\rangle .$ Notice that if $Q(a^k) = a^k$ , equation (REF ) holds.","So, let us consider from now on that $Q(a^k) \\ne a^k$ .","By using Lemma REF we replace $\\widetilde{R}(a^k)$ and get: $G_{a^k}^{Q(a^k) \\ \\prime }(0+) = w_k \\left\\Vert Q(a^k) - a^k \\right\\Vert - 2 A(a^k) \\left\\langle \\widetilde{T}(a^k) - a^k , Q(a^k) - a^k \\right\\rangle .$ Extracting common factors and using the definition of $\\beta $ when it belongs to $(0,1)$ we obtain: $G_{a^k}^{Q(a^k) \\ \\prime }(0+) & = & 2 A(a^k) \\left\\Vert Q(a^k) - a^k \\right\\Vert \\left\\Vert \\widetilde{T}(a^k) - a^k \\right\\Vert \\Bigg [ \\beta (a^k)\\\\& - & \\left.","\\left\\langle \\frac{ \\widetilde{T}(a^k) - a^k }{ \\left\\Vert \\widetilde{T}(a^k) - a^k \\right\\Vert } , \\frac{ Q(a^k) - a^k }{ \\left\\Vert Q(a^k) - a^k \\right\\Vert } \\right\\rangle \\right].$ By Remarks REF and REF the vectors $Q(a^k) - a^k$ and $\\widetilde{T}(a^k) - a^k$ are parallel, so: $G_{a^k}^{Q(a^k) \\ \\prime }(0+) = 2 A(a^k) \\left\\Vert Q(a^k) - a^k \\right\\Vert \\left\\Vert \\widetilde{T}(a^k) - a^k \\right\\Vert \\left[ \\beta (a^k) - 1 \\right].$ which is equivalent to (REF ).","Now, let us prove (REF ).","If $z = a^k$ then $G_{a^k}^z(t) = f(a^k)$ for all $t \\in [0,1]$ , thus $G_{a^k+}^{a^k \\ \\prime }(0) = 0$ , and therefore the inequality (REF ) holds.","So, let us assume that $z \\ne a^k$ for the rest of the proof.","Using (REF ) and due to Lemma REF to replace $\\widetilde{R}(a^k)$ : $G_{a^k}^{z \\ \\prime }(0+) = w_k \\left\\Vert z - a^k \\right\\Vert - 2 A(a^k) \\left\\langle \\widetilde{T}(a^k) - a^k , z - a^k \\right\\rangle .$ Extracting common factors: $G_{a^k}^{z \\ \\prime }(0+) & = & 2 A(a^k) \\left\\Vert z - a^k \\right\\Vert \\left\\Vert \\widetilde{T}(a^k) - a^k \\right\\Vert \\left[ \\frac{ w_k }{ 2 A(a^k) \\left\\Vert \\widetilde{T}(a^k) - a^k \\right\\Vert } \\right.\\\\& - & \\left.","\\left\\langle \\frac{ \\widetilde{T}(a^k) - a^k }{ \\left\\Vert \\widetilde{T}(a^k) - a^k \\right\\Vert } , \\frac{ z - a^k }{ \\left\\Vert z - a^k \\right\\Vert } \\right\\rangle \\right].$ Using the expression for $\\beta (a^k) \\in (0,1)$ we obtain: $G_{a^k}^{z \\ \\prime }(0+) & = & 2 A(a^k) \\left\\Vert z - a^k \\right\\Vert \\left\\Vert \\widetilde{T}(a^k) - a^k \\right\\Vert \\Bigg [ \\beta (a^k)\\\\& - & \\left.","\\left\\langle \\frac{ \\widetilde{T}(a^k) - a^k }{ \\left\\Vert \\widetilde{T}(a^k) - a^k \\right\\Vert } , \\frac{ z - a^k }{ \\left\\Vert z - a^k \\right\\Vert } \\right\\rangle \\right].$ If $z$ belongs to the segment that joins $a^k$ and $T(a^k)$ we have that $z-a^k$ and $\\widetilde{T}(a^k) - a^k$ are parallel vectors, then: $G_{a^k}^{z \\ \\prime }(0+) \\ge - 2 \\left[ 1 - \\beta (a^k) \\right] A(a^k) \\left\\Vert z - a^k \\right\\Vert \\left\\Vert \\widetilde{T}(a^k) - a^k \\right\\Vert .$ We can write $z = (1-\\lambda )T(a^k) + \\lambda a^k$ where $\\lambda \\in [0,1]$ .","Therefore: $G_{a^k}^{z \\ \\prime }(0+) \\ge - 2 \\left[ 1 - \\beta (a^k) \\right] A(a^k) \\left( 1 - \\lambda \\right) \\left\\Vert T(a^k) - a^k \\right\\Vert \\left\\Vert \\widetilde{T}(a^k) - a^k \\right\\Vert .$ for all $\\lambda \\in [0,1]$ .","The minimum value of the right-hand side of the last expression happens when $\\lambda = \\lambda ( a^k )$ , so: $G_{a^k}^{z \\ \\prime }(0+) \\ge - 2 \\left[ 1 - \\beta (a^k) \\right] A(a^k) \\left( 1 - \\lambda ( a^k ) \\right) \\left\\Vert T(a^k) - a^k \\right\\Vert \\left\\Vert \\widetilde{T}(a^k) - a^k \\right\\Vert .$ Using Remark REF we conclude that: $G_{a^k}^{z \\ \\prime }(0+) \\ge - 2 \\left[ 1 - \\beta (a^k) \\right] A(a^k) \\left\\Vert Q(a^k) - a^k \\right\\Vert \\left\\Vert \\widetilde{T}(a^k) - a^k \\right\\Vert .$ $\\Box $ Now we will prove an equivalence that characterizes the solution of (REF ) in terms of the iteration function $Q$ .","Moreover, if $x^*$ is a regular point that is not a vertex, then $x^*$ is a KKT point.","From now on, let us consider that $\\Omega = \\left\\lbrace y \\in \\mathbb {R}^n : g(y) \\le 0, h(y) = 0 \\right\\rbrace ,$ where $g : \\mathbb {R}^n \\rightarrow \\mathbb {R}^s$ is a convex function and $h : \\mathbb {R}^n \\rightarrow \\mathbb {R}^p$ is an affine function.","Theorem 21 Let $\\Omega $ be defined as in (REF ) and $x \\in \\Omega $ .","Consider the following propositions: $x$ is a KKT point.","$x$ is the minimizer of the problem (REF ).","$Q(x) = x$ .","If $x \\ne a^1, \\ldots , a^m$ , $g$ and $h$ are continuously differentiable, and $x$ is a regular point, then (a), (b) and (c) are equivalent.","If $x = a^k$ for some $k = 1, \\ldots , m$ , then (b) implies (c).","Let $x \\ne a^1, \\ldots , a^m$ be.","Since $f$ is strictly convex and $\\Omega $ is convex, the KKT optimality conditions are necessary and sufficient.","Therefore, it holds that (a) is equivalent to (b).","Now we will prove that (b) implies (c).","Let us suppose that $x$ is the minimizer of the problem (REF ).","If $x$ were not a fixed point of the iteration function $Q$ , we would have that $x \\ne Q(x)$ , which means that $f(Q(x)) < f(x)$ by Theorem REF .","This contradicts the hypothesis.","To demonstrate that (c) implies (a), we will assume that $x$ is a fixed point of $Q$ , that is, $x = Q(x)$ .","Since $Q(x) = P_{\\Omega } \\circ T(x)$ , $x$ is the solution of: $\\begin{array}{rl}\\displaystyle \\mathop { \\mathrm {argmin} }_z & \\displaystyle F(z) = \\frac{1}{2} \\left\\Vert z - T(x) \\right\\Vert ^2 \\\\ \\mathrm {subject \\, to} & g(z) \\le 0, \\\\ & h(z) = 0.\\end{array}$ Since $F$ and $g$ are convex, $h$ is affine, and $x$ is a regular point, the KKT optimality conditions hold at $x$ .","That is, there exist multipliers $\\left\\lbrace \\mu _j \\right\\rbrace _{j=1}^s$ and $\\left\\lbrace \\lambda _j \\right\\rbrace _{j=1}^p$ such that (see [31], [38]): $x - T(x) + \\sum _{j=1}^s \\mu _j \\nabla g_j(x) + \\sum _{j=1}^p \\lambda _j \\nabla h_j(x) & = & 0,\\\\\\mu _j g_j( x ) & = & 0, \\quad j = 1, \\ldots , s,\\\\\\mu _j & \\ge & 0, \\quad j = 1, \\ldots , s,\\\\g(x) & \\le & 0,\\\\h(x) & = & 0.$ Multiplying these equations by $2 A(x)$ , using equation (REF ), Lemma REF and Remark REF , we obtain: $\\nabla f(x) + \\sum _{j=1}^s \\left( 2 A(x) \\mu _j \\right) \\nabla g_j(x) + \\sum _{j=1}^p \\left( 2 A(x) \\lambda _j \\right) \\nabla h_j(x) & = & 0,\\\\\\left( 2 A(x) \\mu _j \\right) g_j( x ) & = & 0, \\quad j = 1, \\ldots , s,\\\\\\left( 2 A(x) \\mu _j \\right) & \\ge & 0, \\quad j = 1, \\ldots , s,\\\\g(x) & \\le & 0,\\\\h(x) & = & 0.$ where $\\left\\lbrace 2 A(x) \\mu _j \\right\\rbrace _{j=1}^s$ and $\\left\\lbrace 2 A(x) \\lambda _j \\right\\rbrace _{j=1}^p$ are multipliers.","Therefore, $x$ is a KKT point of the problem (REF ) (see [31], [38]).","Now, let us suppose that $x = a^k$ for some $k = 1, \\ldots , m$ .","As before, if $x$ is a minimizer of the problem (REF ), then $Q(a^k) = a^k$ , otherwise $f(Q(a^k)) < f(a^k)$ , which would be a contradiction.","$\\Box $"],["Numerical experiments.","The purpose of this section is to discuss the efficiency and robustness of the proposed algorithm versus a solver for nonlinear programming problems.","A prototype code of Algorithm REF was programmed in MATLAB (version R2011a) and executed in a PC running Linux OS, Intel(R) Core(TM) i7 CPU Q720, 1.60GHz.","We have considered a closed and convex set $\\Omega \\subset \\mathbb {R}^2$ defined by the set $\\Omega = \\left\\lbrace y \\in \\mathbb {R}^n : g(y) \\le 0 \\right\\rbrace $ , where $g$ is given by: $g( x ) = \\left[ \\begin{array}{c} \\displaystyle - 4 - \\frac{ 1 }{ 8 } x + \\frac{ 7 }{ 72 } x^2 + \\frac{ 1 }{ 216 } x^2 ( x - 3 ) + y \\\\[3mm] \\displaystyle \\frac{ 4 }{ 5 } x + y - \\frac{ 59 }{ 10 } \\\\[3mm] \\displaystyle x - \\frac{ 11 }{ 2 } \\\\[3mm] \\displaystyle \\frac{ 3 }{ 2 } x - y - \\frac{ 35 }{ 4 } \\\\[3mm] \\displaystyle x - y - \\frac{ 13 }{ 2 } \\\\[3mm] \\displaystyle - 4 + \\frac{ 1 }{ 8 } ( x - 1 ) + \\frac{ 1 }{ 16 } ( x - 1 )^2 + \\frac{ 1 }{ 32 } ( x - 1 )^2 ( x - 3 ) - y \\\\[3mm] \\displaystyle - \\frac{ 1 }{ 3 } x - y - \\frac{ 11 }{ 3 } \\\\[3mm] \\displaystyle - \\frac{ 2 }{ 3 } x - y - \\frac{ 13 }{ 3 } \\\\[3mm] \\displaystyle - 4 x + y - 19 \\end{array} \\right].$ The feasible set is defined by linear and nonlinear constraints, as it can be seen in Figure REF .","Figure: Feasible set Ω\\Omega We have built 1000 different experiments where for each one: The number of vertices was $m = 50$ .","The vertices were normally distributed random vectors, with mean equal to 0 and standard deviation equal to 10.","The weights were uniformly distributed random positive numbers between 0 and 10.","Tolerance was set to $\\varepsilon = 0.00001$ .","On one hand, each experiment was solved using Algorithm REF and, on the other hand, it was considered as a nonlinear programming problem and solved using function $fmincon$ (see [32] and references therein).","Since the Weber function (REF ) is not differentiable at the vertices, nonlinear programming solvers may fail.","Let $x_m(i)$ be the solution of (REF ) obtained by $fmincon$ in experiment $i$ , and $f_{m}(i) = f(x_m(i))$ .","Analogously, let $x_p(i)$ be the solution of (REF ) obtained by Algorithm REF in experiment $i$ , and $f_{p}(i) = f(x_p(i))$ .","Figure REF shows the difference between the arrays $f_{m}$ and $f_{p}$ .","Both methods finished succesfully in all cases, however, Algorithm REF found equal or better results for all experiments.","For example, the difference $f_{m} - f_{p}$ was greater than $0.01$ in 35 experiments (the maximum difference ocurred in experiment 506).","Figure: Difference between minimum values found by Algorithm and fminconfmincon.Feasibility of the solutions $x_p(i)$ can be checked computing $\\max (g(x_m(i)))$ .","Results can be seen in Figure REF Figure: Feasibility of the solution x p (i)x_p(i) obtained by Algorithm ."],["Conclusions","This paper proposes a Weiszfeld-like algorithm for solving the Weber problem constrained to a closed and convex set, and it is well defined even when an iterate is a vertex.","The algorithm consists of two stages: first, iterate using the fixed point modified Weiszfeld iteration (REF ), and second, either project onto the set $\\Omega $ when the iterate is different from the vertices, or, if the iterate is a vertex $a^k$ , take the point belonging to the line that joins $T(a^k)$ with $a^k$ as defined in (REF ).","It is proved that the constrained problem (REF ) has a unique solution.","Besides that, the definition of the iteration function $Q$ allows us to demonstrate that the proposed algorithm produces a sequence $\\left\\lbrace x^{(l)} \\right\\rbrace $ of feasible iterates.","Moreover, the sequence $\\left\\lbrace f \\left( x^{(l)} \\right) \\right\\rbrace $ is not increasing, and when $x^{(l)} \\ne Q\\left( x^{(l)} \\right)$ , the sequence decreases at the next iterate.","It can be seen that if a point $x^*$ is the solution of the problem (REF ) then $x^*$ is a fixed point of the iteration function $Q$ .","Even more, if $x^*$ is different from the vertices, the fact of being $x^*$ a fixed point of $Q$ is equivalent to the fact that $x^*$ satisfies the KKT optimality conditions, and equivalent to the fact that $x^*$ is the solution of the problem (REF ).","These properties allows us to connect the proposed algorithm with the minimization problem.","Numerical experiments showed that the proposed algorithm found equal or better solutions than a well-known standard solver, in a practical example with 1000 random choices of vertices and weights.","That is due to the fact that the proposed algorithm does not use of the existence of derivatives at the vertices, because the Weber function is not differentiable at the vertices."]],"string":"[\n [\n \"A Weiszfeld-like algorithm for a Weber location problem constrained to a\\n closed and convex set\"\n ],\n [\n \"Abstract The Weber problem consists of finding a point in $\\\\mathbbm{R}^n$ that minimizes the weighted sum of distances from $m$ points in $\\\\mathbbm{R}^n$ that are not collinear.\",\n \"An application that motivated this problem is the optimal location of facilities in the 2-dimensional case.\",\n \"A classical method to solve the Weber problem, proposed by Weiszfeld in 1937, is based on a fixed point iteration.\",\n \"In this work a Weber problem constrained to a closed and convex set is considered.\",\n \"A Weiszfeld-like algorithm, well defined even when an iterate is a vertex, is presented.\",\n \"The iteration function $Q$ that defines the proposed algorithm, is based mainly on an orthogonal projection over the feasible set, combined with the iteration function of a modified Weiszfeld algorithm presented by Vardi and Zhang in 2001.\",\n \"It can be seen that the proposed algorithm generates a sequence of feasible iterates that have descent properties.\",\n \"Under certain hypotheses, the limit of this sequence satisfies the KKT optimality conditions, is a fixed point of the iteration function that defines the algorithm, and is the solution of the constrained minimization problem.\",\n \"Numerical experiments confirmed the theoretical results.\"\n ],\n [\n \"Introduction\",\n \"Let $a^1, \\\\ldots , a^m$ be $m$ distinct points in the space $\\\\mathbb {R}^n$ , called vertices, and positive numbers $w_1, \\\\ldots , w_m$ , called weights.\",\n \"The function $f : \\\\mathbb {R}^n \\\\rightarrow \\\\mathbb {R}$ defined by $f( x ) = \\\\sum _{j=1}^m w_j \\\\left\\\\Vert x - a^j \\\\right\\\\Vert ,$ is called the Weber function, where $\\\\left\\\\Vert \\\\cdot \\\\right\\\\Vert $ denotes the Euclidean norm.\",\n \"It is well-known that this function is not differentiable at the vertices, and strictly convex if the vertices are not collinear (we will assume this hypothesis from now on).\",\n \"The Weber problem (also known as the Fermat-Weber problem) is to find a point in $\\\\mathbb {R}^n$ that minimizes the weighted sum of Euclidean distances from the $m$ given points, that is, we have to find the solution of the following unconstrained optimization problem: $\\\\begin{array}{rl}\\\\displaystyle \\\\mathop { \\\\mathrm {argmin} }_x & \\\\displaystyle f( x ) \\\\\\\\ \\\\mathrm {subject \\\\, to} & x \\\\in \\\\mathbb {R}^n.\\\\end{array}$ This problem has a unique solution $x^u$ in $\\\\mathbb {R}^n$ .\",\n \"The problem was also stated as a pure mathematical problem by Fermat [44], [27], Cavalieri [37], Steiner [14], Fasbender [20] and many others.\",\n \"Several solutions, based on geometrical arguments, were proposed by Torricelli and Simpson.\",\n \"In [30] historical details and geometric aspects were presented by Kupitz and Martini.\",\n \"In [41] Weber formulated the problem (REF ) from an economical point of view.\",\n \"The vertices represent customers or demands, the solution to the problem denotes the location of a new facility, and the weights are costs associated with the interactions between the new facility and the customers.\",\n \"Among several schemes to solve the Weber location problem (see [12], [19], [28], [34]), one of the most popular methods was presented by Weiszfeld in [42], [43].\",\n \"The Weiszfeld algorithm is an iterative method based on the first-order necessary conditions for a stationary point of the objective function.\",\n \"If we define $T_0 : \\\\mathbb {R}^n \\\\rightarrow \\\\mathbb {R}^n$ by: $T_0(x) = \\\\left\\\\lbrace \\\\begin{array}{ll} \\\\frac{ \\\\displaystyle \\\\sum _{j=1}^m \\\\frac{ w_j a^j }{ \\\\left\\\\Vert x - a^j \\\\right\\\\Vert } }{ \\\\displaystyle \\\\sum _{j=1}^m \\\\frac{ w_j }{ \\\\left\\\\Vert x - a^j \\\\right\\\\Vert } }, & \\\\quad \\\\text{if $x \\\\ne a^1, \\\\ldots , a^m$}, \\\\\\\\ & \\\\\\\\ a^k, & \\\\quad \\\\text{if $x = a^k$, $k = 1, \\\\ldots , m$}, \\\\end{array} \\\\right.$ the Weiszfeld algorithm is: $x^{(l)} = T_0 \\\\left( x^{(l-1)} \\\\right), \\\\quad l \\\\in \\\\mathbb {N},$ where $x^{(0)} \\\\in \\\\mathbb {R}^n$ is a starting point.\",\n \"The Weiszfeld algorithm (REF ), despite of its simplicity, has a serious problem if some $x^{(l)}$ lands accidentally in a vertex $a^k$ , because the algorithm gets stuck at $a^k$ , even when $a^k$ is not the solution of (REF ).\",\n \"Many authors studied the set of initial points for which the sequence generated by the Weiszfeld algorithm yields in a vertex (see [29], [11], [6], [9], [7], [3]).\",\n \"Vardi and Zhang [40] derived a simple but nontrivial modification of the Weiszfeld algorithm in which they solved the problem of landing in a vertex.\",\n \"Generalizations and new techniques for the Fermat-Weber location problem have been developed in recent years.\",\n \"In [18] Eckhardt applied the Weiszfeld algorithm to generalized Weber problems in Banach spaces.\",\n \"An exact algorithm for a Weber problem with attraction and repulsion was presented by Chen et al.\",\n \"in [13].\",\n \"Kaplan and Yang [24] proved a duality theorem which includes as special cases a great variety of choices of norms in the terms of the Fermat-Weber sum.\",\n \"In [10] Carrizosa et al.\",\n \"studied the so called Regional Weber Problem, which allows the demand not to be concentrated onto a finite set of points, but follows an arbitrary probability measure.\",\n \"In [17] Drezner and Wesolowsky studied the case where different $l_p$ norms are used for each demand point.\",\n \"In [23] the so called Complementary Problem (the Weber problem with one negative weight) was studied by Jalal and Krarup, and geometrical solutions were given.\",\n \"In [15] Drezner presented a Weiszfeld-like iterative procedure and convergence is proved if appropriate conditions hold.\",\n \"In some practical problems it is necessary to consider barriers (forbidden regions).\",\n \"Barriers were first introduced to location modeling by Katz and Cooper [25].\",\n \"There exist several heuristic and iterative algorithms for single-facility location problems for distance computations in the presence of barriers (see [2], [8], [5], [4]).\",\n \"In [35] Pfeiffer and Klamroth presented a unified formulation for problems with barriers and network location problems.\",\n \"A complete reference to barriers in location problems can be found in [26].\",\n \"Barriers can be applied to model real life problems where regions like lakes and mountains are forbidden.\",\n \"On the other hand, there are location problems whose solution needs to lie within a closed set.\",\n \"For example, see [39] for a discussion of the case when the solution is constrained to be within a maximum distance of each demand point.\",\n \"Drezner and Wesolowsky [16] studied the problem of locating an obnoxious facility with rectangular distances ($l_1$ norm), where the facility must lie within some prespecified region (linear constraints).\",\n \"A primal-dual algorithm to deal with the constrained Fermat-Weber problem using mixed norms was developed in [33] by Idrissi et al..\",\n \"In [21] Hansen et al.\",\n \"presented an algorithm for solving the Weber problem when the set of feasible locations is the union of a finite number of convex polygons.\",\n \"In [36] Pilotta and Torres considered a Weber location problem with box constraints.\",\n \"Constrained Weber problems arise when we require that the solution is in an area (feasible region) determined by, for example, environmental and/or political reasons.\",\n \"It could be the case for a facility producing dangerous materials that must be installed in a restricted (constrained) area.\",\n \"Another example could be the location of a plant in an industrial zone or of a hospital in a non-polluted area.\",\n \"In this paper a constrained location problem is considered.\",\n \"An algorithm is proposed to solve the following problem: $\\\\begin{array}{rl}\\\\displaystyle \\\\mathop { \\\\mathrm {argmin} }_x & \\\\displaystyle f( x ) \\\\\\\\ \\\\mathrm {subject \\\\, to} & x \\\\in \\\\Omega ,\\\\end{array}$ where $\\\\Omega $ is a closed and convex set, generalizing the problem formulated in [36].\",\n \"Problem (REF ) could be seen as a nonlinear programming problem and solved by standard solvers, but they may fail since the Weber function is not differentiable at the vertices.\",\n \"It can be proved that problem (REF ) has a unique solution $x^*$ , since the function $f$ is strictly convex and $\\\\Omega $ is a closed and convex set.\",\n \"On the other hand, it is well-known that the convex hull of the given vertices $a^1, \\\\ldots , a^m$ contains the solution $x^u$ of the unconstrained Weber problem (see for instance [29]).\",\n \"If $\\\\Omega $ contains the convex hull, both solutions $x^*$ and $x^u$ agree.\",\n \"In other cases, the solution $x^*$ is not necessarily a projection of $x^u$ over $\\\\Omega $ (see [36]).\",\n \"The algorithm is based basically on a slight variation of an orthogonal projection of the Weiszfeld algorithm presented in [40], that is well defined even when an iterate coincides with a vertex.\",\n \"Properties of the sequence generated by the proposed algorithm related with the minimization problem REF will be proved in the following sections.\",\n \"The paper is structured as follows: Section describes the results in [40] in which a modified Weiszfeld algorithm is presented and some notation is introduced.\",\n \"In Section the proposed algorithm is defined.\",\n \"Section is dedicated to definitions and technical lemmas.\",\n \"In Section the main results about convergence to optimality are presented.\",\n \"Numerical experiments are considered in Section .\",\n \"Finally, conclusions are given in Section .\",\n \"Some words about notation.\",\n \"As it was mentioned, we will call $x^u$ the solution of problem (REF ) and $x^*$ the solution of problem (REF ).\",\n \"The symbols $\\\\Vert \\\\cdot \\\\Vert $ and $\\\\langle \\\\cdot , \\\\cdot \\\\rangle $ will refer to the standard Euclidean norm and standard inner product in $\\\\mathbb {R}^n$ , respectively.\",\n \"For a function $f : \\\\mathbb {R} \\\\rightarrow \\\\mathbb {R}$ we will denote by $f^{\\\\prime }(a-)$ the left-hand side derivative at $a$ , and by $f^{\\\\prime }(a+)$ the right-hand side derivative at $a$ .\"\n ],\n [\n \"The modified Weiszfeld algorithm\",\n \"This section reviews the main results presented in [40] in which the authors generalize the Weiszfeld algorithm for the case that an iterate lands on a vertex.\",\n \"From now on, this algorithm will be referred to as the modified Weiszfeld algorithm.\",\n \"In order to make notation easier, we define the function $A : \\\\mathbb {R}^n \\\\rightarrow \\\\mathbb {R}$ by: $A(x) = \\\\left\\\\lbrace \\\\begin{array}{ll} \\\\displaystyle \\\\sum _{j=1}^m \\\\frac{ w_j }{ 2 \\\\left\\\\Vert x - a^j \\\\right\\\\Vert }, & \\\\quad \\\\text{if $x \\\\ne a^1, \\\\ldots , a^m$}, \\\\\\\\ & \\\\\\\\ \\\\displaystyle \\\\sum _{\\\\begin{array}{c}j=1 \\\\\\\\ j \\\\ne k\\\\end{array}}^m \\\\frac{ w_j }{ 2 \\\\left\\\\Vert a^k - a^j \\\\right\\\\Vert }, & \\\\quad \\\\text{if $x = a^k$, $k = 1, \\\\ldots , m$}.\",\n \"\\\\end{array} \\\\right.$ Notice that $A(x) > 0$ for all $x \\\\in \\\\mathbb {R}^n$ .\",\n \"In [40], the number $A(a^k)$ was called $A_k$ .\",\n \"A generalization for the iteration function $T_0$ , defined in (REF ), is given by $\\\\widetilde{T} : \\\\mathbb {R}^n \\\\rightarrow \\\\mathbb {R}^n$ defined as follows: $\\\\widetilde{T}(x) = \\\\left\\\\lbrace \\\\begin{array}{ll} \\\\displaystyle \\\\frac{ \\\\displaystyle \\\\sum _{j=1}^m \\\\frac{ w_j a^j }{ \\\\left\\\\Vert x - a^j \\\\right\\\\Vert } }{ 2 A(x) }, & \\\\quad \\\\text{if $x \\\\ne a^1, \\\\ldots , a^m$}, \\\\\\\\ & \\\\\\\\ \\\\displaystyle \\\\frac{ \\\\displaystyle \\\\sum _{\\\\begin{array}{c}j=1 \\\\\\\\ j \\\\ne k\\\\end{array}}^m \\\\frac{ w_j a^j }{ \\\\left\\\\Vert a^k - a^j \\\\right\\\\Vert } }{ 2 A\\\\left(a^k\\\\right) }, & \\\\quad \\\\text{if $x = a^k$, $k = 1, \\\\ldots , m$}.\",\n \"\\\\end{array} \\\\right.$ Notice that $\\\\widetilde{T}$ coincides with $T_o$ in $\\\\mathbb {R}^n - \\\\left\\\\lbrace a^1, \\\\ldots , a^m \\\\right\\\\rbrace $ .\",\n \"Let $\\\\widetilde{R} : \\\\mathbb {R}^n \\\\rightarrow \\\\mathbb {R}^n$ and $r : \\\\mathbb {R}^n \\\\rightarrow \\\\mathbb {R}$ be: $\\\\widetilde{R}(x) & = & \\\\left\\\\lbrace \\\\begin{array}{ll} \\\\displaystyle \\\\sum _{j=1}^m \\\\frac{ w_j \\\\left( a^j - x \\\\right) }{ \\\\left\\\\Vert x - a^j \\\\right\\\\Vert }, & \\\\quad \\\\text{if $x \\\\ne a^1, \\\\ldots , a^m$}, \\\\\\\\ & \\\\\\\\ \\\\displaystyle \\\\sum _{\\\\begin{array}{c}j=1 \\\\\\\\ j \\\\ne k\\\\end{array}}^m \\\\frac{ w_j \\\\left( a^j - a^k \\\\right) }{ \\\\left\\\\Vert a^k - a^j \\\\right\\\\Vert }, & \\\\quad \\\\text{if $x = a^k$, $k = 1, \\\\ldots , m$}, \\\\end{array} \\\\right.\\\\\\\\& \\\\nonumber \\\\\\\\r(x) & = & \\\\Vert \\\\widetilde{R}(x) \\\\Vert , \\\\qquad \\\\forall \\\\, x \\\\in \\\\mathbb {R}^n.\",\n \"\\\\nonumber $ The function $\\\\widetilde{R}$ generalizes the negative gradient of the Weber function since, for all $x \\\\ne a^1, \\\\ldots , a^m$ , $\\\\nabla f(x) = - \\\\widetilde{R}(x).$ The following lemma is very easy to prove (see [40]), and it relates the functionals $\\\\widetilde{T}$ and $\\\\widetilde{R}$ .\",\n \"Lemma 1 For all $x \\\\in \\\\mathbb {R}^n$ we have $\\\\widetilde{R}(x) = 2 A(x) \\\\left[ \\\\widetilde{T}(x) - x \\\\right]$ .\",\n \"If we define $\\\\gamma : \\\\mathbb {R}^n \\\\rightarrow \\\\mathbb {R}$ by: $\\\\gamma (x) = \\\\left\\\\lbrace \\\\begin{array}{ll} 0, & \\\\quad \\\\text{if $x \\\\ne a^1, \\\\ldots , a^m$}, \\\\\\\\ 0, & \\\\quad \\\\text{if $x = a^k$ and $r\\\\left(a^k\\\\right) = 0$ for some $k = 1, \\\\ldots , m$}, \\\\\\\\ \\\\displaystyle w_k / r\\\\left(a^k\\\\right), & \\\\quad \\\\text{if $x = a^k$ and $r\\\\left(a^k\\\\right) \\\\ne 0$ for some $k = 1, \\\\ldots , m$}, \\\\end{array} \\\\right.$ we can see that $\\\\gamma (x) \\\\ge 0$ for all $x \\\\in \\\\mathbb {R}^n$ .\",\n \"The modified Weiszfeld algorithm presented in [40] is defined by: $x^{(l)} = T\\\\left( x^{(l-1)} \\\\right), \\\\quad l \\\\in \\\\mathbb {N},$ where $x^{(0)} \\\\in \\\\mathbb {R}^n$ and $T : \\\\mathbb {R}^n \\\\rightarrow \\\\mathbb {R}^n$ is given by: $T(x) = \\\\left( 1 - \\\\beta (x) \\\\right) \\\\widetilde{T}(x) + \\\\beta (x) x,$ where $\\\\beta : \\\\mathbb {R}^n \\\\rightarrow \\\\mathbb {R}$ is defined by $\\\\beta (x) = \\\\min \\\\left\\\\lbrace 1, \\\\gamma (x) \\\\right\\\\rbrace $ .\",\n \"Remark 2 If $x \\\\ne a^1, \\\\ldots , a^m$ , then $\\\\beta (x) = 0$ because $\\\\gamma (x) = 0$ .\",\n \"So, we can deduce that $T(x) = \\\\widetilde{T}(x)$ .\",\n \"Notice that this fact implies that the functional $T$ is continuous in $\\\\mathbb {R}^n - \\\\left\\\\lbrace a^1, \\\\ldots , a^m \\\\right\\\\rbrace $ .\",\n \"It can be seen that if $a^k \\\\ne x^u$ , then $0 < \\\\beta ( a^k ) < 1$ (see [40]).\",\n \"From equation (REF ) we obtain that $T(x) - x = \\\\left( 1 - \\\\beta (x) \\\\right) \\\\left( \\\\widetilde{T}(x) - x \\\\right)$ for $x \\\\in \\\\mathbb {R}^n$ .\",\n \"The main result in [40] is: Theorem 3 The following propositions are equivalent: $x = x^u$ .\",\n \"$T(x) = x$ .\",\n \"$r(x) \\\\le \\\\eta (x)$ .\",\n \"where $\\\\eta (x) = \\\\left\\\\lbrace \\\\begin{array}{ll} 0, & \\\\quad \\\\text{if $x \\\\ne a^1, \\\\ldots , a^m$}, \\\\\\\\ w_k, & \\\\quad \\\\text{if $x = a^k$, $k = 1, \\\\ldots , m$}.\",\n \"\\\\end{array} \\\\right.$\"\n ],\n [\n \"The proposed algorithm\",\n \"This section is dedicated to describe the proposed algorithm, introducing some definitions and remarks.\",\n \"First of all, we can notice that problem (REF ) has a unique solution, due to the fact that $f$ is a non-negative, strictly convex, and continuous function, $\\\\lim _{\\\\Vert x \\\\Vert \\\\rightarrow \\\\infty } f(x) = \\\\infty $ and $\\\\Omega $ is closed and convex.\",\n \"In order to define the proposed algorithm at the vertices, we will need to determine which points of the segment that joins $a^k$ and $T(a^k)$ are in the feasible set $\\\\Omega $ .\",\n \"If $k = 1, \\\\ldots , m$ , let the set $\\\\mathcal {S}_k$ be defined by: $\\\\mathcal {S}_k = \\\\left\\\\lbrace \\\\lambda \\\\in [0,1] : ( 1 - \\\\lambda ) T(a^k) + \\\\lambda a^k \\\\in \\\\Omega \\\\right\\\\rbrace .$ Notice that $\\\\mathcal {S}_k$ could be equal to the empty set in case that $a^k$ and $T(a^k)$ do not belong to $\\\\Omega $ .\",\n \"On the other hand, if $a^k \\\\in \\\\Omega $ , then $1 \\\\in \\\\mathcal {S}_k$ , which means that $\\\\mathcal {S}_k \\\\ne \\\\emptyset $ .\",\n \"Thus, we can define: $\\\\lambda ( a^k ) = \\\\inf \\\\mathcal {S}_k, \\\\quad a^k \\\\in \\\\Omega .$ In case a vertex $a^k$ is not in $\\\\Omega $ , there is no need to define the number $\\\\lambda ( a^k )$ .\",\n \"In the following lemma, a set of basic properties of $\\\\lambda ( a^k )$ are listed: Lemma 4 If $k = 1, \\\\ldots , m$ and $a^k \\\\in \\\\Omega $ then: $\\\\lambda ( a^k ) \\\\in [0,1]$ .\",\n \"If $T(a^k) \\\\in \\\\Omega $ then $\\\\lambda ( a^k ) = 0$ .\",\n \"If $T(a^k) \\\\notin \\\\Omega $ then $\\\\lambda ( a^k ) \\\\in (0,1]$ .\",\n \"The proof of (a) follows from the definition of $\\\\mathcal {S}_k$ .\",\n \"If $T(a^k) \\\\in \\\\Omega $ , then $0 \\\\in \\\\mathcal {S}_k$ , so $\\\\lambda (a^k) = 0$ , and this proves (b).\",\n \"Finally, for item (c), let us consider that $T(a^k) \\\\notin \\\\Omega $ .\",\n \"Since $\\\\Omega $ is a closed set, there is an entire ball centered at $T(a^k)$ that does not intersect $\\\\Omega $ , which implies that there exists $\\\\epsilon $ such that $( 1 - \\\\lambda ) T(a^k) + \\\\lambda a^k \\\\notin \\\\Omega $ for all $\\\\lambda \\\\in [0,\\\\epsilon ]$ .\",\n \"Thus, $\\\\lambda (a^k) \\\\in (0,1]$ and this concludes the proof.\",\n \"$\\\\Box $ Let us call $P_{\\\\Omega } : \\\\mathbb {R}^n \\\\rightarrow \\\\Omega $ the orthogonal projection over $\\\\Omega $ .\",\n \"Since $\\\\Omega $ is a nonempty, closed and convex set, the operator $P_{\\\\Omega }$ is a continuous function [1].\",\n \"We define the iteration function $Q : \\\\Omega \\\\rightarrow \\\\Omega $ by: $Q(x) = \\\\left\\\\lbrace \\\\begin{array}{ll} P_{\\\\Omega } \\\\circ T(x), & \\\\quad \\\\text{if $x \\\\ne a^1, \\\\ldots , a^m$}, \\\\\\\\ \\\\left( 1 - \\\\lambda ( a^k ) \\\\right) T(a^k) + \\\\lambda ( a^k ) a^k, & \\\\quad \\\\text{if $x = a^k \\\\in \\\\Omega $, $k = 1, \\\\ldots , m$}.\",\n \"\\\\end{array} \\\\right.$ There will be no need to define $Q$ outside $\\\\Omega $ since the proposed algorithm generates a sequence of feasible points.\",\n \"The iteration function $Q$ at $x \\\\in \\\\Omega $ coincides with the orthogonal projection of $T(x)$ over the feasible set when $x$ is different from the vertices.\",\n \"Only when $x$ is a vertex $a^k$ belonging to $\\\\Omega $ , $Q(x)$ is defined as the farthest possible feasible point of the segment that joins $x$ with $T(x)$ .\",\n \"The following remark states some basic properties of the iteration function of the proposed algorithm.\",\n \"Remark 5 (a) If $a^k \\\\in \\\\Omega $ and $T(a^k) \\\\in \\\\Omega $ , then $Q(a^k) = T(a^k) = P_{\\\\Omega } \\\\circ T(a^k)$ .\",\n \"(b) If $a^k \\\\in \\\\Omega $ , it can be seen that: $Q(a^k) - a^k & = & \\\\left( 1 - \\\\lambda ( a^k ) \\\\right) \\\\left( T(a^k) - a^k \\\\right),\\\\\\\\Q(a^k) - T(a^k) & = & - \\\\lambda ( a^k ) \\\\left( T(a^k) - a^k \\\\right).$ (c) The functional $Q$ is continuous in $\\\\mathbb {R}^n - \\\\left\\\\lbrace a^1, \\\\ldots , a^m \\\\right\\\\rbrace $ .\",\n \"The proofs of (a) and (b) are straightforward.\",\n \"For (c), since $P_{\\\\Omega }$ is continuous in $\\\\mathbb {R}^n$ (see [1]) and $T$ is continuous in $\\\\mathbb {R}^n - \\\\left\\\\lbrace a^1, \\\\ldots , a^m \\\\right\\\\rbrace $ (see Remark REF ), we have that $Q$ is continuous in $\\\\mathbb {R}^n - \\\\left\\\\lbrace a^1, \\\\ldots , a^m \\\\right\\\\rbrace $ .\",\n \"$\\\\Box $ The proposed algorithm is described below.\",\n \"Algorithm 6 Let $\\\\Omega \\\\subset \\\\mathbb {R}^n$ be a closed and convex set.\",\n \"Assume that $x^{(0)} \\\\in \\\\Omega $ is an initial approximation such that $f(x^{(0)}) \\\\le f(a^j)$ for all $j \\\\in \\\\left\\\\lbrace 1, \\\\ldots , m \\\\right\\\\rbrace $ and $a^j \\\\in \\\\Omega $ .\",\n \"Given $\\\\varepsilon > 0$ a tolerance and $x^{(l-1)} \\\\in \\\\Omega $ , do the following steps to compute $x^{(l)}$ : Step 1: Compute: $x^{(l)} = Q \\\\left( x^{(l-1)} \\\\right).$ Step 2: Stop the execution if $\\\\left\\\\Vert x^{(l)} - x^{(l-1)} \\\\right\\\\Vert < \\\\varepsilon ,$ and declare $x^{(l)}$ as solution to the problem (REF ).\",\n \"Otherwise return to Step 1.\",\n \"From the definition of $Q$ it follows that Algorithm REF generates a sequence of feasible iterates.\",\n \"Also notice that if there are vertices in the feasible set, $x^{(0)}$ can be one of them, for example, a vertex $a_s$ such that $f(a_s) \\\\le f(a^j)$ for all $a^j \\\\in \\\\Omega $ .\",\n \"On the other hand, if there are no vertices in the feasible set, $x^{(0)}$ can be chosen as the projection over $\\\\Omega $ of the null vector.\",\n \"Some definitions and technical results The purpose of this section is to define some entities and prove technical lemmas that will be important in the proof of the main results.\",\n \"First of all, we will define some useful operators for making notation easier.\",\n \"If $\\\\mathcal {A} \\\\subset \\\\left\\\\lbrace 1, \\\\ldots , n \\\\right\\\\rbrace $ , then we define $\\\\Vert \\\\cdot \\\\Vert _{\\\\mathcal {A}} : \\\\mathbb {R}^n \\\\rightarrow \\\\mathbb {R}$ and $\\\\langle \\\\cdot , \\\\cdot \\\\rangle _{\\\\mathcal {A}} : \\\\mathbb {R}^n \\\\times \\\\mathbb {R}^n \\\\rightarrow \\\\mathbb {R}$ by: $\\\\Vert x \\\\Vert _{\\\\mathcal {A}} = \\\\sqrt{ \\\\sum _{j \\\\in \\\\mathcal {A}} x_j^2 }, \\\\qquad \\\\langle x , y \\\\rangle _{\\\\mathcal {A}} = \\\\sum _{j \\\\in \\\\mathcal {A}} x_j y_j.$ Notice that, when $\\\\mathcal {A} \\\\subsetneq \\\\left\\\\lbrace 1, \\\\ldots , n \\\\right\\\\rbrace $ , $\\\\Vert \\\\cdot \\\\Vert _{\\\\mathcal {A}}$ is not necessarily a norm and $\\\\langle \\\\cdot , \\\\cdot \\\\rangle _{\\\\mathcal {A}}$ is not necessarily an inner product.\",\n \"According to this definition, if $\\\\mathcal {A}$ and $\\\\mathcal {B}$ are sets such that $\\\\mathcal {A} \\\\cap \\\\mathcal {B} = \\\\emptyset $ and $\\\\mathcal {A} \\\\cup \\\\mathcal {B} = \\\\left\\\\lbrace 1, \\\\ldots , n \\\\right\\\\rbrace $ , it can be seen that: $\\\\Vert x \\\\Vert ^2 & = & \\\\Vert x \\\\Vert _{\\\\mathcal {A}}^2 + \\\\Vert x \\\\Vert _{\\\\mathcal {B}}^2,\\\\\\\\\\\\langle x , y \\\\rangle & = & \\\\langle x , y \\\\rangle _{\\\\mathcal {A}} + \\\\langle x , y \\\\rangle _{\\\\mathcal {B}},\\\\\\\\c \\\\langle x , y \\\\rangle _{\\\\mathcal {A}} & = & \\\\langle cx , y \\\\rangle _{\\\\mathcal {A}} = \\\\langle x , cy \\\\rangle _{\\\\mathcal {A}}.$ For $x \\\\in \\\\Omega $ , let us define the following sets of indices: $\\\\mathcal {N}(x) & = & \\\\left\\\\lbrace k \\\\in \\\\mathbb {N} : \\\\quad 1 \\\\le k \\\\le n, \\\\quad (T(x))_k \\\\ne (Q(x))_k \\\\right\\\\rbrace ,\\\\\\\\\\\\mathcal {E}(x) & = & \\\\left\\\\lbrace k \\\\in \\\\mathbb {N} : \\\\quad 1 \\\\le k \\\\le n, \\\\quad (T(x))_k = (Q(x))_k \\\\right\\\\rbrace ,$ Notice that for all $x \\\\in \\\\mathbb {R}^n$ we have that $\\\\mathcal {N}(x) \\\\cap \\\\mathcal {E}(x) = \\\\emptyset $ and $\\\\mathcal {N}(x) \\\\cup \\\\mathcal {E}(x) = \\\\left\\\\lbrace 1, \\\\ldots , n \\\\right\\\\rbrace $ .\",\n \"Let $\\\\alpha : \\\\Omega \\\\rightarrow \\\\mathbb {R}^n$ be the following function: If $x \\\\ne a^1, \\\\ldots , a^m$ : $\\\\alpha (x) = \\\\displaystyle \\\\sum _{j=1}^m \\\\frac{ w_j }{ \\\\Vert x - a^j \\\\Vert } \\\\left[ Q(x) - a^j \\\\right].$ If $x = a^k \\\\in \\\\Omega $ for some $k = 1, \\\\ldots , m$ : $\\\\alpha (x) = \\\\displaystyle \\\\sum _{\\\\begin{array}{c}j=1 \\\\\\\\ j \\\\ne k\\\\end{array}}^m \\\\frac{ w_j }{ \\\\Vert a^k - a^j \\\\Vert } \\\\left[ Q(a^k) - (1 - \\\\beta (a^k)) a^j - \\\\beta (a^k) a^k \\\\right].$ It can be seen that the function $\\\\alpha $ is related to the iteration function $Q$ of the proposed algorithm, and the iteration function $T$ of the modified algorithm.\",\n \"Lemma 7 If $x \\\\in \\\\Omega $ , then $\\\\alpha (x) = 2 A(x) \\\\left[ Q(x) - T(x) \\\\right].$ If $x \\\\ne a^1, \\\\ldots , a^m$ , then: $\\\\alpha (x) & = & \\\\sum _{j=1}^m \\\\frac{ w_j }{ \\\\Vert x - a^j \\\\Vert } \\\\left[ Q(x) - a^j \\\\right] = \\\\sum _{j=1}^m \\\\frac{ w_j Q(x) }{ \\\\Vert x - a^j \\\\Vert } - \\\\sum _{j=1}^m \\\\frac{ w_j a^j }{ \\\\Vert x - a^j \\\\Vert }\\\\\\\\&&\\\\\\\\& = & \\\\left( \\\\sum _{j=1}^m \\\\frac{ w_j }{ \\\\Vert x - a^j \\\\Vert } \\\\right) \\\\left[ Q(x) - \\\\frac{ \\\\sum _{j=1}^m \\\\frac{ w_j a^j }{ \\\\Vert x - a^j \\\\Vert } }{ \\\\sum _{j=1}^m \\\\frac{ w_j }{ \\\\Vert x - a^j \\\\Vert } } \\\\right] = 2 A(x) \\\\left[ Q(x) - \\\\widetilde{T}(x) \\\\right]\\\\\\\\&&\\\\\\\\& = & 2 A(x) \\\\left[ Q(x) - T(x) \\\\right].$ where in the last equalities we have used the definition of $\\\\widetilde{T}$ as in (REF ), and the fact that $\\\\widetilde{T}(x) = T(x)$ due to Remark REF .\",\n \"If $x = a^k$ for some $k = 1, \\\\ldots , m$ , we follow a similar procedure than in the previous case.\",\n \"$\\\\Box $ Now, we will define auxiliary functions that take into account the projection $P_{\\\\Omega }$ in order to prove a descent property of $f$ (see next sections).\",\n \"If $x \\\\in \\\\Omega $ , we define: $E_x : \\\\mathbb {R}^n \\\\rightarrow \\\\mathbb {R}^n$ , where: $\\\\left( E_x(y) \\\\right)_k = \\\\left\\\\lbrace \\\\begin{array}{ll} (Q(x))_k, & \\\\quad \\\\text{if $k \\\\in \\\\mathcal {N}(x)$}, \\\\\\\\ y_k, & \\\\quad \\\\text{if $k \\\\in \\\\mathcal {E}(x)$}.\",\n \"\\\\end{array} \\\\right.$ If $\\\\mathcal {E}(x) = \\\\left\\\\lbrace i_1, \\\\ldots , i_r \\\\right\\\\rbrace \\\\ne \\\\emptyset $ define $P_x : \\\\mathbb {R}^n \\\\rightarrow \\\\mathbb {R}^r$ where: $\\\\left( P_x(y) \\\\right)_k = y_{i_k}, \\\\quad k = 1, \\\\ldots , r.$ A useful property of $E_x$ , that follows from the definition, is pointed out in the following remark.\",\n \"Remark 8 If $x \\\\in \\\\Omega $ then $E_x \\\\circ Q(x) = Q(x)$ .\",\n \"The iteration function $Q$ inherits an important property from the orthogonal projection $P_{\\\\Omega }$ .\",\n \"Lemma 9 If $x \\\\in \\\\Omega $ we have that $\\\\left\\\\langle Q(x) - x , Q(x) - T(x) \\\\right\\\\rangle \\\\le 0$ .\",\n \"If $x \\\\ne a^1, \\\\ldots , a^m$ , then $Q(x) = P_{\\\\Omega } \\\\circ T(x)$ .\",\n \"By a property of the orthogonal projection [1] we have that $\\\\langle Q(x) - x , Q(x) - T(x) \\\\rangle \\\\le 0$ .\",\n \"If $x = a^k$ for some $k = 1, \\\\ldots , m$ , Remark REF and Lemma REF imply: $\\\\left\\\\langle Q(a^k) - a^k , Q(a^k) - T(a^k) \\\\right\\\\rangle = - \\\\lambda ( a^k ) \\\\left( 1 - \\\\lambda ( a^k ) \\\\right) \\\\left\\\\Vert T(a^k) - a^k \\\\right\\\\Vert ^2 \\\\le 0,$ and this concludes the proof.\",\n \"$\\\\Box $ The next technical lemma will help us to save computations in other lemmas.\",\n \"Lemma 10 If $x \\\\in \\\\Omega $ , $\\\\mathcal {A}$ is a subset of $\\\\left\\\\lbrace 1, \\\\ldots , n \\\\right\\\\rbrace $ and $j \\\\in \\\\left\\\\lbrace 1, \\\\ldots , m \\\\right\\\\rbrace $ , then: $\\\\left\\\\Vert Q(x) - a^j \\\\right\\\\Vert _{\\\\mathcal {A}}^2 = \\\\left\\\\Vert x - a^j \\\\right\\\\Vert _{\\\\mathcal {A}}^2 - \\\\left\\\\Vert Q(x) - x \\\\right\\\\Vert _{\\\\mathcal {A}}^2 + 2 \\\\left\\\\langle Q(x) - x, Q(x) - a^j \\\\right\\\\rangle _{\\\\mathcal {A}}.$ If $x \\\\in \\\\Omega $ , we have: $\\\\Vert Q(x) - a^j \\\\Vert _{\\\\mathcal {A}}^2 & = & \\\\langle Q(x) - a^j , Q(x) - a^j \\\\rangle _{\\\\mathcal {A}}\\\\\\\\& = & \\\\langle Q(x) - x + x - a^j , Q(x) - x + x - a^j \\\\rangle _{\\\\mathcal {A}}\\\\\\\\& = & \\\\Vert Q(x) - x \\\\Vert _{\\\\mathcal {A}}^2 + \\\\Vert x - a^j \\\\Vert _{\\\\mathcal {A}}^2 + 2 \\\\langle Q(x) - x , x - a^j \\\\rangle _{\\\\mathcal {A}}\\\\\\\\& = & \\\\Vert Q(x) - x \\\\Vert _{\\\\mathcal {A}}^2 + \\\\Vert x - a^j \\\\Vert _{\\\\mathcal {A}}^2 + 2 \\\\langle Q(x) - x , x - Q(x) \\\\rangle _{\\\\mathcal {A}}\\\\\\\\& + & 2 \\\\langle Q(x) - x , Q(x) - a^j \\\\rangle _{\\\\mathcal {A}}\\\\\\\\& = & \\\\Vert x - a^j \\\\Vert _{\\\\mathcal {A}}^2 - \\\\Vert Q(x) - x \\\\Vert _{\\\\mathcal {A}}^2 + 2 \\\\langle Q(x) - x , Q(x) - a^j \\\\rangle _{\\\\mathcal {A}}.$ $\\\\Box $ If $x \\\\in \\\\Omega $ , let us define $g_x : \\\\mathbb {R}^n \\\\rightarrow \\\\mathbb {R}$ by: $g_x(y) = \\\\left\\\\lbrace \\\\begin{array}{ll} \\\\displaystyle {\\\\sum _{j=1}^m} \\\\frac{ w_j }{ 2 \\\\Vert x - a^j \\\\Vert } \\\\Vert E_x(y) - a^j \\\\Vert ^2, & \\\\text{if $x \\\\ne a^1, \\\\ldots , a^m$}, \\\\\\\\ & \\\\\\\\ \\\\displaystyle { \\\\sum _{\\\\begin{array}{c}j=1 \\\\\\\\j \\\\ne k\\\\end{array}}^m} \\\\frac{ w_j }{ 2 \\\\Vert a^k - a^j \\\\Vert } \\\\Vert y - a^j \\\\Vert ^2 + w_k \\\\Vert y - a^k \\\\Vert , & \\\\text{if $x = a^k$}, \\\\\\\\ & \\\\text{$k = 1, \\\\ldots , m$}.\",\n \"\\\\end{array} \\\\right.$ The values that $g_x$ assumes at $x$ and $Q(x)$ will play an important role in the proof of a property of the objective function $f$ .\",\n \"Lemma 11 Let $x \\\\in \\\\Omega $ be.\",\n \"If $x \\\\ne a^1, \\\\ldots , a^m$ then: $g_x(x) & = & \\\\frac{1}{2} f(x) + 2 A(x) \\\\left\\\\langle Q(x) - x , Q(x) - T(x) \\\\right\\\\rangle \\\\\\\\& - & A(x) \\\\left\\\\Vert Q(x) - x \\\\right\\\\Vert _{\\\\mathcal {N}(x)}^2.$ If $x = a^k$ for some $k = 1, \\\\ldots , m$ , then $g_{a^k}(a^k) = \\\\frac{1}{2} f(a^k)$ .\",\n \"Let us suppose that $x \\\\ne a^1, \\\\ldots , a^m$ .\",\n \"By property (REF ) and (REF ), we have for $j = 1, \\\\ldots , m$ : $\\\\left\\\\Vert E_x(x) - a^j \\\\right\\\\Vert ^2 = \\\\left\\\\Vert x - a^j \\\\right\\\\Vert _{\\\\mathcal {E}(x)}^2 + \\\\left\\\\Vert Q(x) - a^j \\\\right\\\\Vert _{\\\\mathcal {N}(x)}^2.$ Using Lemma REF , we can see that: $g_x(x) & = & \\\\sum _{j=1}^m \\\\frac{ w_j }{ 2 \\\\Vert x - a^j \\\\Vert } \\\\left[ \\\\left\\\\Vert x - a^j \\\\right\\\\Vert _{\\\\mathcal {E}(x)}^2 + \\\\left\\\\Vert x - a^j \\\\right\\\\Vert _{\\\\mathcal {N}(x)}^2 \\\\right.\\\\\\\\& - & \\\\left.\",\n \"\\\\left\\\\Vert Q(x) - x \\\\right\\\\Vert _{\\\\mathcal {N}(x)}^2 + 2 \\\\left\\\\langle Q(x) - x , Q(x) - a^j \\\\right\\\\rangle _{\\\\mathcal {N}(x)} \\\\right].$ Due to (REF ), the definition of the Weber function $f$ , the definition of $A$ as in (REF ), the property () and the definition of $\\\\alpha $ as in (REF ), we obtain: $g_x(x) = \\\\frac{1}{2} f(x) - A(x) \\\\Vert Q(x) - x \\\\Vert _{\\\\mathcal {N}(x)}^2 + \\\\langle Q(x) - x , \\\\alpha (x) \\\\rangle _{\\\\mathcal {N}(x)}.$ By Lemma REF , the fact that $\\\\left( Q(x) \\\\right)_i = \\\\left( T(x) \\\\right)_i$ for all $i \\\\in \\\\mathcal {E}(x)$ and (), we get: $g_x(x) & = & \\\\frac{1}{2} f(x) - A(x) \\\\Vert Q(x) - x \\\\Vert _{\\\\mathcal {N}(x)}^2\\\\\\\\& + & 2 A(x) \\\\langle Q(x) - x , Q(x) - T(x) \\\\rangle ,$ which concludes the proof of (a).\",\n \"Now, let us assume that $x = a^k$ for some $k = 1, \\\\ldots , m$ .\",\n \"Then: $g_{a^k}(a^k) = \\\\sum _{\\\\begin{array}{c}j=1 \\\\\\\\ j \\\\ne k\\\\end{array}}^m \\\\frac{ w_j }{ 2 \\\\Vert a^k - a^j \\\\Vert } \\\\Vert a^k - a^j \\\\Vert ^2 = \\\\frac{1}{2} \\\\sum _{\\\\begin{array}{c}j=1 \\\\\\\\ j \\\\ne k\\\\end{array}}^m w_j \\\\Vert a^k - a^j \\\\Vert = \\\\frac{1}{2} f(a^k).$ This concludes the proof of (b).\",\n \"$\\\\Box $ The number $g_x(Q(x))$ can be computed in the next lemma.\",\n \"Lemma 12 Let $x \\\\in \\\\Omega $ be.\",\n \"If $x \\\\ne a^1, \\\\ldots , a^m$ then: $g_x(Q(x)) & = & \\\\frac{1}{2} f(x) + 2 A(x) \\\\left\\\\langle Q(x) - x , Q(x) - T(x) \\\\right\\\\rangle \\\\\\\\& - & A(x) \\\\left\\\\Vert Q(x) - x \\\\right\\\\Vert ^2.$ If $x = a^k$ for some $k = 1, \\\\ldots , m$ , then $g_{a^k}(Q(a^k)) & = & \\\\frac{1}{2} f(a^k) - A(a^k) \\\\left\\\\Vert Q(a^k) - a^k \\\\right\\\\Vert ^2\\\\\\\\& + & 2 A(a^k) \\\\left\\\\langle Q(a^k) - a^k , Q(a^k) - T(a^k) \\\\right\\\\rangle \\\\\\\\& - & 2 \\\\beta (a^k) A(a^k) \\\\left\\\\langle Q(a^k) - a^k , \\\\widetilde{T}(a^k) - a^k \\\\right\\\\rangle \\\\\\\\& + & w_k \\\\left\\\\Vert Q(a^k) - a^k \\\\right\\\\Vert .$ First, let us consider $x \\\\ne a^1, \\\\ldots , a^m$ .\",\n \"Due to Remark REF we have: $g_x(Q(x)) = \\\\sum _{j=1}^m \\\\frac{ w_j }{ 2 \\\\Vert x - a^j \\\\Vert } \\\\Vert Q(x) - a^j \\\\Vert ^2.$ By Lemma REF we obtain: $g_x(Q(x))& = & \\\\sum _{j=1}^m \\\\frac{ w_j }{ 2 \\\\Vert x - a^j \\\\Vert } \\\\left[ \\\\Vert x - a^j \\\\Vert ^2 - \\\\Vert Q(x) - x \\\\Vert ^2 \\\\right.\\\\\\\\& + & \\\\left.\",\n \"2 \\\\langle Q(x) - x , Q(x) - a^j \\\\rangle \\\\right].$ Due to the definition of the Weber function $f$ , the definition of $A$ as in (REF ) and the definition of $\\\\alpha $ as in (REF ), we deduce that: $g_x(Q(x)) = \\\\frac{1}{2} f(x) - A(x) \\\\Vert Q(x) - x \\\\Vert ^2 + \\\\langle Q(x) - x , \\\\alpha (x) \\\\rangle .$ By Lemma REF we get: $g_x(Q(x)) = \\\\frac{1}{2} f(x) - A(x) \\\\Vert Q(x) - x \\\\Vert ^2 + 2 A(x) \\\\langle Q(x) - x , Q(x) - T(x) \\\\rangle ,$ concluding the proof of (a).\",\n \"Now, consider $x = a^k$ for some $k = 1, \\\\ldots , m$ .\",\n \"Due to (REF ) we have: $g_{a^k}(Q(a^k)) = \\\\sum _{\\\\begin{array}{c}j=1 \\\\\\\\ j \\\\ne k\\\\end{array}}^m \\\\frac{ w_j }{ 2 \\\\Vert a^k - a^j \\\\Vert } \\\\Vert Q(a^k) - a^j \\\\Vert ^2 + w_k \\\\Vert Q(a^k) - a^k \\\\Vert .$ By Lemma REF , the definition of the Weber function $f$ and the definition of $A$ as in (REF ) we obtain: $g_{a^k}\\\\left( Q(a^k) \\\\right) & = & \\\\frac{1}{2} f(a^k) - A(a^k) \\\\Vert Q(a^k) - a^k \\\\Vert ^2\\\\\\\\& + & \\\\left\\\\langle Q(a^k) - a^k , \\\\sum _{\\\\begin{array}{c}j=1 \\\\\\\\ j \\\\ne k\\\\end{array}}^m \\\\frac{ w_j }{ \\\\Vert a^k - a^j \\\\Vert } \\\\left[ Q(a^k) - a^j \\\\right] \\\\right\\\\rangle \\\\\\\\& + & w_k \\\\Vert Q(a^k) - a^k \\\\Vert .$ Manipulating algebraically, $Q(a^k) - a^j = Q(a^k) - ( 1 - \\\\beta (a^k) ) a^j - \\\\beta (a^k) a^k + \\\\beta (a^k) ( a^k - a^j ).$ Due to the definition of $\\\\alpha $ (see (REF )) and the definition of $\\\\widetilde{R}$ (see (REF )) we get: $g_{a^k} \\\\left( Q(a^k) \\\\right)& = & \\\\frac{1}{2} f(a^k) - A(a^k) \\\\left\\\\Vert Q(a^k) - a^k \\\\right\\\\Vert ^2\\\\\\\\& + & \\\\left\\\\langle Q(a^k) - a^k , \\\\alpha (a^k) \\\\right\\\\rangle - \\\\beta (a^k) \\\\left\\\\langle Q(a^k) - a^k , \\\\widetilde{R}(a^k) \\\\right\\\\rangle \\\\\\\\& + & w_k \\\\Vert Q(a^k) - a^k \\\\Vert .$ By Lemma REF and Lemma REF we have: $g_{a^k} \\\\left( Q(a^k) \\\\right) & = & \\\\frac{1}{2} f(a^k) - A(a^k) \\\\left\\\\Vert Q(a^k) - a^k \\\\right\\\\Vert ^2\\\\\\\\& + & 2 A(a^k) \\\\left\\\\langle Q(a^k) - a^k , Q(a^k) - T(a^k) \\\\right\\\\rangle \\\\\\\\& - & 2 A(a^k) \\\\beta (a^k) \\\\left\\\\langle Q(a^k) - a^k , \\\\widetilde{T}(a^k) - a^k \\\\right\\\\rangle + w_k \\\\Vert Q(a^k) - a^k \\\\Vert .$ which concludes the proof.\",\n \"$\\\\Box $ The next lemma deals with the last two terms of $g_{a^k}(Q(a^k))$ .\",\n \"Lemma 13 If $a^k \\\\in \\\\Omega $ for some $k = 1, \\\\ldots , m$ , the number $z = w_k \\\\left\\\\Vert Q(a^k) - a^k \\\\right\\\\Vert - 2 A(a^k) \\\\beta (a^k) \\\\left\\\\langle Q(a^k) - a^k , \\\\widetilde{T}(a^k) - a^k \\\\right\\\\rangle ,$ is equal to zero.\",\n \"If $Q(a^k) = a^k$ the result is true.\",\n \"So, from now on, let us consider that $Q(a^k) \\\\ne a^k$ .\",\n \"First, let us check that $a^k \\\\ne x^u$ .\",\n \"In case that $a^k = x^u$ , then $T(a^k) = a^k$ by Theorem REF .\",\n \"Since $a^k \\\\in \\\\Omega $ , then $T(a^k) \\\\in \\\\Omega $ .\",\n \"By Remark REF we have that $Q(a^k) = T(a^k) = a^k$ which is a contradiction.\",\n \"By Remark REF , we have that $\\\\beta (a^k) \\\\in (0,1)$ (since $a^k \\\\ne x^u$ ) and: $z = w_k \\\\left\\\\Vert Q(a^k) - a^k \\\\right\\\\Vert - \\\\frac{ 2 A(a^k) \\\\beta (a^k) }{ 1 - \\\\beta (a^k) } \\\\left\\\\langle Q(a^k) - a^k , T(a^k) - a^k \\\\right\\\\rangle .$ Extracting common factors, using Remarks REF and REF , the fact that $T(a^k) \\\\ne a^k$ (if $T(a^k) = a^k$ then $a^k = x^u$ by Theorem REF ), and the fact that $\\\\widetilde{T}(a^k) \\\\ne a^k$ (if $\\\\widetilde{T}(a^k) = a^k$ then $T(a^k) = a^k$ by definition (REF )) we get that: $z & = & 2 A(a^k) \\\\left\\\\Vert Q(a^k) - a^k \\\\right\\\\Vert \\\\left\\\\Vert \\\\widetilde{T}(a^k) - a^k \\\\right\\\\Vert \\\\left[ \\\\frac{ w_k }{ 2 A(a^k) \\\\left\\\\Vert \\\\widetilde{T}(a^k) - a^k \\\\right\\\\Vert } \\\\right.\\\\\\\\& - & \\\\left.\",\n \"\\\\beta (a^k) \\\\left\\\\langle \\\\frac{ ( 1 - \\\\lambda ( a^k ) ) ( T(a^k) - a^k ) }{ \\\\left\\\\Vert ( 1 - \\\\lambda ( a^k ) ) ( T(a^k) - a^k ) \\\\right\\\\Vert } , \\\\frac{ T(a^k) - a^k }{ \\\\left\\\\Vert T(a^k) - a^k \\\\right\\\\Vert } \\\\right\\\\rangle \\\\right].$ Simplifying and using the definition of $\\\\beta (a^k)$ we have that: $z = 2 A(a^k) \\\\left\\\\Vert Q(a^k) - a^k \\\\right\\\\Vert \\\\left\\\\Vert \\\\widetilde{T}(a^k) - a^k \\\\right\\\\Vert \\\\left[ \\\\beta (a^k) - \\\\beta (a^k) \\\\right] = 0,$ which concludes the proof.\",\n \"$\\\\Box $ The purpose of the next two lemmas is to determine a strict inequality between the functions $g_x$ and $f$ at suitable points.\",\n \"First of all, we have to prove the following result.\",\n \"Lemma 14 Let $x \\\\in \\\\Omega $ be such that $x \\\\ne Q(x)$ .\",\n \"If $x \\\\ne a^1, \\\\ldots , a^m$ , then $g_x(Q(x)) \\\\le g_x(x)$ .\",\n \"Besides that, if $\\\\mathcal {E}(x) \\\\ne \\\\emptyset $ and $P_x \\\\circ Q(x) \\\\ne P_x(x)$ , then $g_x(Q(x)) < g_x(x)$ .\",\n \"If $x = a^k$ for some $k = 1, \\\\ldots , m$ , then $g_{a^k}(Q(a^k)) < g_{a^k}(a^k)$ .\",\n \"If $x \\\\ne a^1, \\\\ldots , a^m$ , then $g_x(Q(x)) - g_x(x) = - A(x) \\\\Vert Q(x) - x \\\\Vert _{\\\\mathcal {E}(x)}^2 \\\\le 0$ , by Lemma REF and Lemma REF .\",\n \"Besides that, if $\\\\mathcal {E}(x) \\\\ne \\\\emptyset $ and $P_x \\\\circ Q(x) \\\\ne P_x(x)$ we deduce that $\\\\Vert Q(x) - x \\\\Vert _{\\\\mathcal {E}(x)} \\\\ne 0$ .\",\n \"Thus, $g_x(Q(x)) < g_x(x)$ .\",\n \"If $x = a^k$ for some $k = 1, \\\\ldots , m$ , by Lemmas REF , REF and REF we have: $g_{a^k}(Q(a^k)) - g_{a^k}(a^k) & = & - A(a^k) \\\\left\\\\Vert Q(a^k) - a^k \\\\right\\\\Vert ^2\\\\\\\\& + & 2 A(a^k) \\\\left\\\\langle Q(a^k) - a^k , Q(a^k) - T(a^k) \\\\right\\\\rangle .$ Due to Lemma REF and the fact that $A > 0$ we obtain: $g_{a^k}(Q(a^k)) - g_{a^k}(a^k) \\\\le - A(a^k) \\\\Vert Q(a^k) - a^k \\\\Vert ^2 < 0,$ and the proof is finished.\",\n \"$\\\\Box $ Lemma 15 Let $x \\\\in \\\\Omega $ be such that $x \\\\ne Q(x)$ .\",\n \"Then $g_x(Q(x)) < \\\\frac{1}{2} f(x)$ .\",\n \"Let us consider the case when $x \\\\ne a^1, \\\\ldots , a^m$ .\",\n \"By Lemmas REF , REF and REF we have that: $g_x(Q(x)) & \\\\le & g_x(x) = \\\\frac{1}{2} f(x) + 2 A(x) \\\\left\\\\langle Q(x) - x , Q(x) - T(x) \\\\right\\\\rangle \\\\\\\\& - & A(x) \\\\left\\\\Vert Q(x) - x \\\\right\\\\Vert _{\\\\mathcal {N}(x)}^2\\\\\\\\& \\\\le & \\\\frac{1}{2} f(x) - A(x) \\\\left\\\\Vert Q(x) - x \\\\right\\\\Vert _{\\\\mathcal {N}(x)}^2.$ If $\\\\mathcal {E}(x) = \\\\emptyset $ , then $\\\\Vert \\\\cdot \\\\Vert _{\\\\mathcal {N}(x)} = \\\\Vert \\\\cdot \\\\Vert $ .\",\n \"Therefore: $g_x(Q(x)) \\\\le \\\\frac{1}{2} f(x) - A(x) \\\\Vert Q(x) - x \\\\Vert ^2 < \\\\frac{1}{2} f(x).$ If $\\\\mathcal {E}(x) \\\\ne \\\\emptyset $ and $P_x \\\\circ Q(x) = P_x(x)$ , then there exists an index $i \\\\in \\\\mathcal {N}(x)$ such that $x_i \\\\ne (Q(x))_i$ since $x \\\\ne Q(x)$ .\",\n \"Thus, $\\\\Vert Q(x) - x \\\\Vert _{\\\\mathcal {N}(x)} \\\\ne 0$ , which implies: $g_x(Q(x)) \\\\le \\\\frac{1}{2} f(x) - A(x) \\\\left\\\\Vert Q(x) - x \\\\right\\\\Vert _{\\\\mathcal {N}(x)}^2 < \\\\frac{1}{2} f(x).$ If $\\\\mathcal {E}(x) \\\\ne \\\\emptyset $ and $P_x \\\\circ Q(x) \\\\ne P_x(x)$ , due to Lemmas REF , REF and REF , we have that: $g_x(Q(x)) < g_x(x) \\\\le \\\\frac{1}{2} f(x) - A(x) \\\\left\\\\Vert Q(x) - x \\\\right\\\\Vert _{\\\\mathcal {N}(x)}^2 \\\\le \\\\frac{1}{2} f(x).$ Now, when $x = a^k$ for some $k = 1, \\\\ldots , m$ , $g_{a^k}(Q(a^k)) < g_{a^k}(a^k) = \\\\frac{1}{2} f(a^k)$ due to Lemma REF and Lemma REF .\",\n \"$\\\\Box $ The next lemma states an equality that relates the Weber function and $g_x$ at appropriate points when $x \\\\ne a^1, \\\\ldots , a^m$ .\",\n \"Besides that, this result will be crucial in the next section.\",\n \"Lemma 16 Let $x \\\\ne a^1, \\\\ldots , a^m$ be such that $x \\\\in \\\\Omega $ and $x \\\\ne Q(x)$ .\",\n \"Then: $g_x \\\\circ Q(x) = \\\\frac{1}{2} f(x) + \\\\left( f(Q(x)) - f(x) \\\\right) + \\\\delta , \\\\quad \\\\delta \\\\ge 0.$ Due to the definition of $g_x$ as in (REF ) and Remark REF we get that: $g_x \\\\circ Q(x) = \\\\sum _{j=1}^m \\\\frac{ w_j }{ 2 \\\\Vert x - a^j \\\\Vert } \\\\Vert Q(x) - a^j \\\\Vert ^2.$ Adding and subtracting $\\\\Vert x - a^j \\\\Vert $ we have: $g_x \\\\circ Q(x) & = & \\\\sum _{j=1}^m \\\\frac{ w_j }{ 2 \\\\Vert x - a^j \\\\Vert } \\\\left[ \\\\left\\\\Vert x - a^j \\\\right\\\\Vert + \\\\left( \\\\left\\\\Vert Q(x) - a^j \\\\right\\\\Vert - \\\\left\\\\Vert x - a^j \\\\right\\\\Vert \\\\right) \\\\right]^2\\\\\\\\& = & \\\\frac{1}{2} \\\\sum _{j=1}^m w_j \\\\left\\\\Vert x - a^j \\\\right\\\\Vert + \\\\sum _{j=1}^m w_j \\\\left( \\\\left\\\\Vert Q(x) - a^j \\\\right\\\\Vert - \\\\left\\\\Vert x - a^j \\\\right\\\\Vert \\\\right)\\\\\\\\& + & \\\\sum _{j=1}^m \\\\frac{ w_j }{ 2 \\\\Vert x - a^j \\\\Vert } \\\\left( \\\\left\\\\Vert Q(x) - a^j \\\\right\\\\Vert - \\\\left\\\\Vert x - a^j \\\\right\\\\Vert \\\\right)^2.$ Notice that the first term of the last equality is the Weber function (divided by two), and the last term is a non-negative number, so we will define it as $\\\\delta $ .\",\n \"So, using the definition of the Weber function in the middle term we obtain: $g_x \\\\circ Q(x) = \\\\frac{1}{2} f(x) + \\\\left( f(Q(x)) - f(x) \\\\right) + \\\\delta .$ $\\\\Box $ Convergence to optimality results This section states the main results about convergence of the sequence $\\\\left\\\\lbrace x^{(l)} \\\\right\\\\rbrace $ generated by Algorithm REF .\",\n \"The next theorem establishes that if a point $x \\\\in \\\\Omega $ is not a fixed point of the iteration function, then the function $f$ strictly decreases at the next iterate.\",\n \"Theorem 17 Let $x \\\\in \\\\Omega $ be such that $x \\\\ne Q(x)$ .\",\n \"Then $f(Q(x)) < f(x)$ .\",\n \"Let us consider that $x \\\\ne a^1, \\\\ldots , a^m$ .\",\n \"By Lemma REF , we have that: $g_x \\\\circ Q(x) < \\\\frac{1}{2} f(x).$ By Lemma REF we get that: $\\\\frac{1}{2} f(x) + f(Q(x)) - f(x) + \\\\delta < \\\\frac{1}{2} f(x).$ Simplifying the last expression we obtain: $f(Q(x)) - f(x) + \\\\delta < 0.$ Finally, $f(Q(x)) - f(x) \\\\le f(Q(x)) - f(x) + \\\\delta < 0.$ Therefore, $f(Q(x)) < f(x)$ .\",\n \"Now, consider that $x = a^k$ for some $k = 1, \\\\ldots , m$ .\",\n \"Following a reasoning similar than in [40], using Lemma REF we have that: $g_{a^k} \\\\circ Q(a^k) - g_{a^k}(a^k) < 0.$ By definition of $g_{a^k}$ we know that: $g_{a^k} \\\\circ Q(a^k) - g_{a^k}(a^k) & = & w_k \\\\left\\\\Vert Q(a^k) - a^k \\\\right\\\\Vert \\\\\\\\& + & \\\\sum _{\\\\begin{array}{c}j=1 \\\\\\\\ j \\\\ne k\\\\end{array}}^m \\\\frac{ w_j }{ 2 \\\\left\\\\Vert a^k - a^j \\\\right\\\\Vert } \\\\left( \\\\left\\\\Vert Q(a^k) - a^j \\\\right\\\\Vert ^2 - \\\\left\\\\Vert a^k - a^j \\\\right\\\\Vert ^2 \\\\right).$ Using the fact that $\\\\left( a^2 - b^2 \\\\right)/ (2b) \\\\ge a - b$ for $a = \\\\left\\\\Vert Q(a^k) - a^j \\\\right\\\\Vert ^2 \\\\ge 0$ and $b = \\\\left\\\\Vert a^k - a^j \\\\right\\\\Vert ^2 > 0$ we obtain that: $g_{a^k} \\\\circ Q(a^k) - g_{a^k}(a^k) & \\\\ge & w_k \\\\left\\\\Vert Q(a^k) - a^k \\\\right\\\\Vert \\\\\\\\& - & \\\\sum _{\\\\begin{array}{c}j=1 \\\\\\\\ j \\\\ne k\\\\end{array}}^m w_j \\\\left\\\\Vert a^k - a^j \\\\right\\\\Vert + \\\\sum _{\\\\begin{array}{c}j=1 \\\\\\\\ j \\\\ne k\\\\end{array}}^m w_j \\\\left\\\\Vert Q(a^k) - a^j \\\\right\\\\Vert .$ Rearranging terms we deduce that: $0 & > & g_{a^k} \\\\circ Q(a^k) - g_{a^k}(a^k)\\\\\\\\& = & \\\\sum _{j=1}^m w_j \\\\left\\\\Vert Q(a^k) - a^j \\\\right\\\\Vert - \\\\sum _{j=1}^m w_j \\\\left\\\\Vert a^k - a^j \\\\right\\\\Vert = f(Q(a^k)) - f(a^k),$ and the proof is complete.\",\n \"$\\\\Box $ Corollary 18 Let $\\\\left\\\\lbrace x^{(l)} \\\\right\\\\rbrace $ be the sequence generated by Algorithm REF .\",\n \"Then the sequence $\\\\left\\\\lbrace f \\\\left( x^{(l)} \\\\right) \\\\right\\\\rbrace $ is not increasing.\",\n \"Even more, each time $x^{(l)} \\\\ne Q \\\\left( x^{(l)} \\\\right)$ the sequence strictly decreases at the next iterate.\",\n \"If the sequence $\\\\left\\\\lbrace x^{(l)} \\\\right\\\\rbrace $ generated by Algorithm REF were not bounded, then we could choose a subsequence $\\\\left\\\\lbrace y^{(l)} \\\\right\\\\rbrace $ such that $y^{(l)} \\\\rightarrow \\\\infty $ .\",\n \"But this implies that $f(y^{(l)}) \\\\rightarrow \\\\infty $ , which is a contradiction since the sequence $\\\\left\\\\lbrace f \\\\left( x^{(l)} \\\\right) \\\\right\\\\rbrace $ is not increasing.\",\n \"Remark 19 The sequence $\\\\left\\\\lbrace x^{(l)} \\\\right\\\\rbrace $ generated by Algorithm REF is bounded.\",\n \"So, there exists a subsequence convergent to a point $x^* \\\\in \\\\Omega $ .\",\n \"Hence, $x^*$ is a feasible point.\",\n \"Due to the nondifferentiability of $f$ at the vertices $a^1, \\\\ldots , a^m$ , we can not use the KKT optimality conditions at $a^k$ .\",\n \"Therefore, if $a^k$ and $z$ are in $\\\\Omega $ , let us define $G_{a^k}^z : [0,1] \\\\rightarrow \\\\mathbb {R}$ by: $G_{a^k}^z(t) = f( a^k + t(z-a^k) ).$ If $a^k \\\\in \\\\Omega $ , $z \\\\in \\\\Omega $ , $t \\\\in [0,1]$ and $\\\\Omega $ convex, we have that $a^k + t(z-a^k) \\\\in \\\\Omega $ .\",\n \"Notice that the right-hand side derivative $G_{a^k}^z(0+)$ (or the directional derivative of $f$ in the direction of $z$ ) exists (see [22]).\",\n \"Besides that, $G_{a^k}^{z \\\\ \\\\prime }(0+) = w_k \\\\left\\\\Vert z - a^k \\\\right\\\\Vert - \\\\left\\\\langle \\\\widetilde{R}(a^k) , z - a^k \\\\right\\\\rangle .$ The next lemma shows that if we are in a vertex $a^k$ , the directional derivative of $f$ at $a^k$ in the direction of $Q(a^k)$ is a descent direction.\",\n \"Lemma 20 Let $a^k \\\\in \\\\Omega $ be such that $T(a^k) \\\\notin \\\\Omega $ .\",\n \"Then: $G_{a^k}^{z \\\\ \\\\prime }(0+) \\\\ge G_{a^k}^{Q(a^k) \\\\ \\\\prime }(0+), \\\\quad \\\\forall \\\\, z \\\\in \\\\left[ a^k, T(a^k) \\\\right],$ where: $G_{a^k}^{Q(a^k) \\\\ \\\\prime }(0+) = - 2 \\\\left[ 1 - \\\\beta (a^k) \\\\right] A(a^k) \\\\left\\\\Vert \\\\widetilde{T}(a^k) - a^k \\\\right\\\\Vert \\\\left\\\\Vert Q(a^k) - a^k \\\\right\\\\Vert .$ If $T(a^k) = a^k$ then $T(a^k) \\\\in \\\\Omega $ , which is a contradiction.\",\n \"Besides that, if $\\\\widetilde{T}(a^k) = a^k$ , we would have that $T(a^k) = a^k$ because of (REF ), and again it would be a contradiction.\",\n \"So, we will consider $\\\\widetilde{T}(a^k) \\\\ne a^k$ and $T(a^k) \\\\ne a^k$ for the rest of the proof.\",\n \"Since $T(a^k) \\\\ne a^k$ , then $\\\\beta (a^k) \\\\in (0,1)$ (see Remark REF and Theorem REF ).\",\n \"Let us prove equation (REF ) first.\",\n \"Now, by (REF ) we can see that: $G_{a^k}^{Q(a^k) \\\\ \\\\prime }(0+) = w_k \\\\left\\\\Vert Q(a^k) - a^k \\\\right\\\\Vert - \\\\left\\\\langle \\\\widetilde{R}(a^k) , Q(a^k) - a^k \\\\right\\\\rangle .$ Notice that if $Q(a^k) = a^k$ , equation (REF ) holds.\",\n \"So, let us consider from now on that $Q(a^k) \\\\ne a^k$ .\",\n \"By using Lemma REF we replace $\\\\widetilde{R}(a^k)$ and get: $G_{a^k}^{Q(a^k) \\\\ \\\\prime }(0+) = w_k \\\\left\\\\Vert Q(a^k) - a^k \\\\right\\\\Vert - 2 A(a^k) \\\\left\\\\langle \\\\widetilde{T}(a^k) - a^k , Q(a^k) - a^k \\\\right\\\\rangle .$ Extracting common factors and using the definition of $\\\\beta $ when it belongs to $(0,1)$ we obtain: $G_{a^k}^{Q(a^k) \\\\ \\\\prime }(0+) & = & 2 A(a^k) \\\\left\\\\Vert Q(a^k) - a^k \\\\right\\\\Vert \\\\left\\\\Vert \\\\widetilde{T}(a^k) - a^k \\\\right\\\\Vert \\\\Bigg [ \\\\beta (a^k)\\\\\\\\& - & \\\\left.\",\n \"\\\\left\\\\langle \\\\frac{ \\\\widetilde{T}(a^k) - a^k }{ \\\\left\\\\Vert \\\\widetilde{T}(a^k) - a^k \\\\right\\\\Vert } , \\\\frac{ Q(a^k) - a^k }{ \\\\left\\\\Vert Q(a^k) - a^k \\\\right\\\\Vert } \\\\right\\\\rangle \\\\right].$ By Remarks REF and REF the vectors $Q(a^k) - a^k$ and $\\\\widetilde{T}(a^k) - a^k$ are parallel, so: $G_{a^k}^{Q(a^k) \\\\ \\\\prime }(0+) = 2 A(a^k) \\\\left\\\\Vert Q(a^k) - a^k \\\\right\\\\Vert \\\\left\\\\Vert \\\\widetilde{T}(a^k) - a^k \\\\right\\\\Vert \\\\left[ \\\\beta (a^k) - 1 \\\\right].$ which is equivalent to (REF ).\",\n \"Now, let us prove (REF ).\",\n \"If $z = a^k$ then $G_{a^k}^z(t) = f(a^k)$ for all $t \\\\in [0,1]$ , thus $G_{a^k+}^{a^k \\\\ \\\\prime }(0) = 0$ , and therefore the inequality (REF ) holds.\",\n \"So, let us assume that $z \\\\ne a^k$ for the rest of the proof.\",\n \"Using (REF ) and due to Lemma REF to replace $\\\\widetilde{R}(a^k)$ : $G_{a^k}^{z \\\\ \\\\prime }(0+) = w_k \\\\left\\\\Vert z - a^k \\\\right\\\\Vert - 2 A(a^k) \\\\left\\\\langle \\\\widetilde{T}(a^k) - a^k , z - a^k \\\\right\\\\rangle .$ Extracting common factors: $G_{a^k}^{z \\\\ \\\\prime }(0+) & = & 2 A(a^k) \\\\left\\\\Vert z - a^k \\\\right\\\\Vert \\\\left\\\\Vert \\\\widetilde{T}(a^k) - a^k \\\\right\\\\Vert \\\\left[ \\\\frac{ w_k }{ 2 A(a^k) \\\\left\\\\Vert \\\\widetilde{T}(a^k) - a^k \\\\right\\\\Vert } \\\\right.\\\\\\\\& - & \\\\left.\",\n \"\\\\left\\\\langle \\\\frac{ \\\\widetilde{T}(a^k) - a^k }{ \\\\left\\\\Vert \\\\widetilde{T}(a^k) - a^k \\\\right\\\\Vert } , \\\\frac{ z - a^k }{ \\\\left\\\\Vert z - a^k \\\\right\\\\Vert } \\\\right\\\\rangle \\\\right].$ Using the expression for $\\\\beta (a^k) \\\\in (0,1)$ we obtain: $G_{a^k}^{z \\\\ \\\\prime }(0+) & = & 2 A(a^k) \\\\left\\\\Vert z - a^k \\\\right\\\\Vert \\\\left\\\\Vert \\\\widetilde{T}(a^k) - a^k \\\\right\\\\Vert \\\\Bigg [ \\\\beta (a^k)\\\\\\\\& - & \\\\left.\",\n \"\\\\left\\\\langle \\\\frac{ \\\\widetilde{T}(a^k) - a^k }{ \\\\left\\\\Vert \\\\widetilde{T}(a^k) - a^k \\\\right\\\\Vert } , \\\\frac{ z - a^k }{ \\\\left\\\\Vert z - a^k \\\\right\\\\Vert } \\\\right\\\\rangle \\\\right].$ If $z$ belongs to the segment that joins $a^k$ and $T(a^k)$ we have that $z-a^k$ and $\\\\widetilde{T}(a^k) - a^k$ are parallel vectors, then: $G_{a^k}^{z \\\\ \\\\prime }(0+) \\\\ge - 2 \\\\left[ 1 - \\\\beta (a^k) \\\\right] A(a^k) \\\\left\\\\Vert z - a^k \\\\right\\\\Vert \\\\left\\\\Vert \\\\widetilde{T}(a^k) - a^k \\\\right\\\\Vert .$ We can write $z = (1-\\\\lambda )T(a^k) + \\\\lambda a^k$ where $\\\\lambda \\\\in [0,1]$ .\",\n \"Therefore: $G_{a^k}^{z \\\\ \\\\prime }(0+) \\\\ge - 2 \\\\left[ 1 - \\\\beta (a^k) \\\\right] A(a^k) \\\\left( 1 - \\\\lambda \\\\right) \\\\left\\\\Vert T(a^k) - a^k \\\\right\\\\Vert \\\\left\\\\Vert \\\\widetilde{T}(a^k) - a^k \\\\right\\\\Vert .$ for all $\\\\lambda \\\\in [0,1]$ .\",\n \"The minimum value of the right-hand side of the last expression happens when $\\\\lambda = \\\\lambda ( a^k )$ , so: $G_{a^k}^{z \\\\ \\\\prime }(0+) \\\\ge - 2 \\\\left[ 1 - \\\\beta (a^k) \\\\right] A(a^k) \\\\left( 1 - \\\\lambda ( a^k ) \\\\right) \\\\left\\\\Vert T(a^k) - a^k \\\\right\\\\Vert \\\\left\\\\Vert \\\\widetilde{T}(a^k) - a^k \\\\right\\\\Vert .$ Using Remark REF we conclude that: $G_{a^k}^{z \\\\ \\\\prime }(0+) \\\\ge - 2 \\\\left[ 1 - \\\\beta (a^k) \\\\right] A(a^k) \\\\left\\\\Vert Q(a^k) - a^k \\\\right\\\\Vert \\\\left\\\\Vert \\\\widetilde{T}(a^k) - a^k \\\\right\\\\Vert .$ $\\\\Box $ Now we will prove an equivalence that characterizes the solution of (REF ) in terms of the iteration function $Q$ .\",\n \"Moreover, if $x^*$ is a regular point that is not a vertex, then $x^*$ is a KKT point.\",\n \"From now on, let us consider that $\\\\Omega = \\\\left\\\\lbrace y \\\\in \\\\mathbb {R}^n : g(y) \\\\le 0, h(y) = 0 \\\\right\\\\rbrace ,$ where $g : \\\\mathbb {R}^n \\\\rightarrow \\\\mathbb {R}^s$ is a convex function and $h : \\\\mathbb {R}^n \\\\rightarrow \\\\mathbb {R}^p$ is an affine function.\",\n \"Theorem 21 Let $\\\\Omega $ be defined as in (REF ) and $x \\\\in \\\\Omega $ .\",\n \"Consider the following propositions: $x$ is a KKT point.\",\n \"$x$ is the minimizer of the problem (REF ).\",\n \"$Q(x) = x$ .\",\n \"If $x \\\\ne a^1, \\\\ldots , a^m$ , $g$ and $h$ are continuously differentiable, and $x$ is a regular point, then (a), (b) and (c) are equivalent.\",\n \"If $x = a^k$ for some $k = 1, \\\\ldots , m$ , then (b) implies (c).\",\n \"Let $x \\\\ne a^1, \\\\ldots , a^m$ be.\",\n \"Since $f$ is strictly convex and $\\\\Omega $ is convex, the KKT optimality conditions are necessary and sufficient.\",\n \"Therefore, it holds that (a) is equivalent to (b).\",\n \"Now we will prove that (b) implies (c).\",\n \"Let us suppose that $x$ is the minimizer of the problem (REF ).\",\n \"If $x$ were not a fixed point of the iteration function $Q$ , we would have that $x \\\\ne Q(x)$ , which means that $f(Q(x)) < f(x)$ by Theorem REF .\",\n \"This contradicts the hypothesis.\",\n \"To demonstrate that (c) implies (a), we will assume that $x$ is a fixed point of $Q$ , that is, $x = Q(x)$ .\",\n \"Since $Q(x) = P_{\\\\Omega } \\\\circ T(x)$ , $x$ is the solution of: $\\\\begin{array}{rl}\\\\displaystyle \\\\mathop { \\\\mathrm {argmin} }_z & \\\\displaystyle F(z) = \\\\frac{1}{2} \\\\left\\\\Vert z - T(x) \\\\right\\\\Vert ^2 \\\\\\\\ \\\\mathrm {subject \\\\, to} & g(z) \\\\le 0, \\\\\\\\ & h(z) = 0.\\\\end{array}$ Since $F$ and $g$ are convex, $h$ is affine, and $x$ is a regular point, the KKT optimality conditions hold at $x$ .\",\n \"That is, there exist multipliers $\\\\left\\\\lbrace \\\\mu _j \\\\right\\\\rbrace _{j=1}^s$ and $\\\\left\\\\lbrace \\\\lambda _j \\\\right\\\\rbrace _{j=1}^p$ such that (see [31], [38]): $x - T(x) + \\\\sum _{j=1}^s \\\\mu _j \\\\nabla g_j(x) + \\\\sum _{j=1}^p \\\\lambda _j \\\\nabla h_j(x) & = & 0,\\\\\\\\\\\\mu _j g_j( x ) & = & 0, \\\\quad j = 1, \\\\ldots , s,\\\\\\\\\\\\mu _j & \\\\ge & 0, \\\\quad j = 1, \\\\ldots , s,\\\\\\\\g(x) & \\\\le & 0,\\\\\\\\h(x) & = & 0.$ Multiplying these equations by $2 A(x)$ , using equation (REF ), Lemma REF and Remark REF , we obtain: $\\\\nabla f(x) + \\\\sum _{j=1}^s \\\\left( 2 A(x) \\\\mu _j \\\\right) \\\\nabla g_j(x) + \\\\sum _{j=1}^p \\\\left( 2 A(x) \\\\lambda _j \\\\right) \\\\nabla h_j(x) & = & 0,\\\\\\\\\\\\left( 2 A(x) \\\\mu _j \\\\right) g_j( x ) & = & 0, \\\\quad j = 1, \\\\ldots , s,\\\\\\\\\\\\left( 2 A(x) \\\\mu _j \\\\right) & \\\\ge & 0, \\\\quad j = 1, \\\\ldots , s,\\\\\\\\g(x) & \\\\le & 0,\\\\\\\\h(x) & = & 0.$ where $\\\\left\\\\lbrace 2 A(x) \\\\mu _j \\\\right\\\\rbrace _{j=1}^s$ and $\\\\left\\\\lbrace 2 A(x) \\\\lambda _j \\\\right\\\\rbrace _{j=1}^p$ are multipliers.\",\n \"Therefore, $x$ is a KKT point of the problem (REF ) (see [31], [38]).\",\n \"Now, let us suppose that $x = a^k$ for some $k = 1, \\\\ldots , m$ .\",\n \"As before, if $x$ is a minimizer of the problem (REF ), then $Q(a^k) = a^k$ , otherwise $f(Q(a^k)) < f(a^k)$ , which would be a contradiction.\",\n \"$\\\\Box $ Numerical experiments.\",\n \"The purpose of this section is to discuss the efficiency and robustness of the proposed algorithm versus a solver for nonlinear programming problems.\",\n \"A prototype code of Algorithm REF was programmed in MATLAB (version R2011a) and executed in a PC running Linux OS, Intel(R) Core(TM) i7 CPU Q720, 1.60GHz.\",\n \"We have considered a closed and convex set $\\\\Omega \\\\subset \\\\mathbb {R}^2$ defined by the set $\\\\Omega = \\\\left\\\\lbrace y \\\\in \\\\mathbb {R}^n : g(y) \\\\le 0 \\\\right\\\\rbrace $ , where $g$ is given by: $g( x ) = \\\\left[ \\\\begin{array}{c} \\\\displaystyle - 4 - \\\\frac{ 1 }{ 8 } x + \\\\frac{ 7 }{ 72 } x^2 + \\\\frac{ 1 }{ 216 } x^2 ( x - 3 ) + y \\\\\\\\[3mm] \\\\displaystyle \\\\frac{ 4 }{ 5 } x + y - \\\\frac{ 59 }{ 10 } \\\\\\\\[3mm] \\\\displaystyle x - \\\\frac{ 11 }{ 2 } \\\\\\\\[3mm] \\\\displaystyle \\\\frac{ 3 }{ 2 } x - y - \\\\frac{ 35 }{ 4 } \\\\\\\\[3mm] \\\\displaystyle x - y - \\\\frac{ 13 }{ 2 } \\\\\\\\[3mm] \\\\displaystyle - 4 + \\\\frac{ 1 }{ 8 } ( x - 1 ) + \\\\frac{ 1 }{ 16 } ( x - 1 )^2 + \\\\frac{ 1 }{ 32 } ( x - 1 )^2 ( x - 3 ) - y \\\\\\\\[3mm] \\\\displaystyle - \\\\frac{ 1 }{ 3 } x - y - \\\\frac{ 11 }{ 3 } \\\\\\\\[3mm] \\\\displaystyle - \\\\frac{ 2 }{ 3 } x - y - \\\\frac{ 13 }{ 3 } \\\\\\\\[3mm] \\\\displaystyle - 4 x + y - 19 \\\\end{array} \\\\right].$ The feasible set is defined by linear and nonlinear constraints, as it can be seen in Figure REF .\",\n \"Figure: Feasible set Ω\\\\Omega We have built 1000 different experiments where for each one: The number of vertices was $m = 50$ .\",\n \"The vertices were normally distributed random vectors, with mean equal to 0 and standard deviation equal to 10.\",\n \"The weights were uniformly distributed random positive numbers between 0 and 10.\",\n \"Tolerance was set to $\\\\varepsilon = 0.00001$ .\",\n \"On one hand, each experiment was solved using Algorithm REF and, on the other hand, it was considered as a nonlinear programming problem and solved using function $fmincon$ (see [32] and references therein).\",\n \"Since the Weber function (REF ) is not differentiable at the vertices, nonlinear programming solvers may fail.\",\n \"Let $x_m(i)$ be the solution of (REF ) obtained by $fmincon$ in experiment $i$ , and $f_{m}(i) = f(x_m(i))$ .\",\n \"Analogously, let $x_p(i)$ be the solution of (REF ) obtained by Algorithm REF in experiment $i$ , and $f_{p}(i) = f(x_p(i))$ .\",\n \"Figure REF shows the difference between the arrays $f_{m}$ and $f_{p}$ .\",\n \"Both methods finished succesfully in all cases, however, Algorithm REF found equal or better results for all experiments.\",\n \"For example, the difference $f_{m} - f_{p}$ was greater than $0.01$ in 35 experiments (the maximum difference ocurred in experiment 506).\",\n \"Figure: Difference between minimum values found by Algorithm and fminconfmincon.Feasibility of the solutions $x_p(i)$ can be checked computing $\\\\max (g(x_m(i)))$ .\",\n \"Results can be seen in Figure REF Figure: Feasibility of the solution x p (i)x_p(i) obtained by Algorithm .\",\n \"Conclusions This paper proposes a Weiszfeld-like algorithm for solving the Weber problem constrained to a closed and convex set, and it is well defined even when an iterate is a vertex.\",\n \"The algorithm consists of two stages: first, iterate using the fixed point modified Weiszfeld iteration (REF ), and second, either project onto the set $\\\\Omega $ when the iterate is different from the vertices, or, if the iterate is a vertex $a^k$ , take the point belonging to the line that joins $T(a^k)$ with $a^k$ as defined in (REF ).\",\n \"It is proved that the constrained problem (REF ) has a unique solution.\",\n \"Besides that, the definition of the iteration function $Q$ allows us to demonstrate that the proposed algorithm produces a sequence $\\\\left\\\\lbrace x^{(l)} \\\\right\\\\rbrace $ of feasible iterates.\",\n \"Moreover, the sequence $\\\\left\\\\lbrace f \\\\left( x^{(l)} \\\\right) \\\\right\\\\rbrace $ is not increasing, and when $x^{(l)} \\\\ne Q\\\\left( x^{(l)} \\\\right)$ , the sequence decreases at the next iterate.\",\n \"It can be seen that if a point $x^*$ is the solution of the problem (REF ) then $x^*$ is a fixed point of the iteration function $Q$ .\",\n \"Even more, if $x^*$ is different from the vertices, the fact of being $x^*$ a fixed point of $Q$ is equivalent to the fact that $x^*$ satisfies the KKT optimality conditions, and equivalent to the fact that $x^*$ is the solution of the problem (REF ).\",\n \"These properties allows us to connect the proposed algorithm with the minimization problem.\",\n \"Numerical experiments showed that the proposed algorithm found equal or better solutions than a well-known standard solver, in a practical example with 1000 random choices of vertices and weights.\",\n \"That is due to the fact that the proposed algorithm does not use of the existence of derivatives at the vertices, because the Weber function is not differentiable at the vertices.\"\n ],\n [\n \"Some definitions and technical results\",\n \"The purpose of this section is to define some entities and prove technical lemmas that will be important in the proof of the main results.\",\n \"First of all, we will define some useful operators for making notation easier.\",\n \"If $\\\\mathcal {A} \\\\subset \\\\left\\\\lbrace 1, \\\\ldots , n \\\\right\\\\rbrace $ , then we define $\\\\Vert \\\\cdot \\\\Vert _{\\\\mathcal {A}} : \\\\mathbb {R}^n \\\\rightarrow \\\\mathbb {R}$ and $\\\\langle \\\\cdot , \\\\cdot \\\\rangle _{\\\\mathcal {A}} : \\\\mathbb {R}^n \\\\times \\\\mathbb {R}^n \\\\rightarrow \\\\mathbb {R}$ by: $\\\\Vert x \\\\Vert _{\\\\mathcal {A}} = \\\\sqrt{ \\\\sum _{j \\\\in \\\\mathcal {A}} x_j^2 }, \\\\qquad \\\\langle x , y \\\\rangle _{\\\\mathcal {A}} = \\\\sum _{j \\\\in \\\\mathcal {A}} x_j y_j.$ Notice that, when $\\\\mathcal {A} \\\\subsetneq \\\\left\\\\lbrace 1, \\\\ldots , n \\\\right\\\\rbrace $ , $\\\\Vert \\\\cdot \\\\Vert _{\\\\mathcal {A}}$ is not necessarily a norm and $\\\\langle \\\\cdot , \\\\cdot \\\\rangle _{\\\\mathcal {A}}$ is not necessarily an inner product.\",\n \"According to this definition, if $\\\\mathcal {A}$ and $\\\\mathcal {B}$ are sets such that $\\\\mathcal {A} \\\\cap \\\\mathcal {B} = \\\\emptyset $ and $\\\\mathcal {A} \\\\cup \\\\mathcal {B} = \\\\left\\\\lbrace 1, \\\\ldots , n \\\\right\\\\rbrace $ , it can be seen that: $\\\\Vert x \\\\Vert ^2 & = & \\\\Vert x \\\\Vert _{\\\\mathcal {A}}^2 + \\\\Vert x \\\\Vert _{\\\\mathcal {B}}^2,\\\\\\\\\\\\langle x , y \\\\rangle & = & \\\\langle x , y \\\\rangle _{\\\\mathcal {A}} + \\\\langle x , y \\\\rangle _{\\\\mathcal {B}},\\\\\\\\c \\\\langle x , y \\\\rangle _{\\\\mathcal {A}} & = & \\\\langle cx , y \\\\rangle _{\\\\mathcal {A}} = \\\\langle x , cy \\\\rangle _{\\\\mathcal {A}}.$ For $x \\\\in \\\\Omega $ , let us define the following sets of indices: $\\\\mathcal {N}(x) & = & \\\\left\\\\lbrace k \\\\in \\\\mathbb {N} : \\\\quad 1 \\\\le k \\\\le n, \\\\quad (T(x))_k \\\\ne (Q(x))_k \\\\right\\\\rbrace ,\\\\\\\\\\\\mathcal {E}(x) & = & \\\\left\\\\lbrace k \\\\in \\\\mathbb {N} : \\\\quad 1 \\\\le k \\\\le n, \\\\quad (T(x))_k = (Q(x))_k \\\\right\\\\rbrace ,$ Notice that for all $x \\\\in \\\\mathbb {R}^n$ we have that $\\\\mathcal {N}(x) \\\\cap \\\\mathcal {E}(x) = \\\\emptyset $ and $\\\\mathcal {N}(x) \\\\cup \\\\mathcal {E}(x) = \\\\left\\\\lbrace 1, \\\\ldots , n \\\\right\\\\rbrace $ .\",\n \"Let $\\\\alpha : \\\\Omega \\\\rightarrow \\\\mathbb {R}^n$ be the following function: If $x \\\\ne a^1, \\\\ldots , a^m$ : $\\\\alpha (x) = \\\\displaystyle \\\\sum _{j=1}^m \\\\frac{ w_j }{ \\\\Vert x - a^j \\\\Vert } \\\\left[ Q(x) - a^j \\\\right].$ If $x = a^k \\\\in \\\\Omega $ for some $k = 1, \\\\ldots , m$ : $\\\\alpha (x) = \\\\displaystyle \\\\sum _{\\\\begin{array}{c}j=1 \\\\\\\\ j \\\\ne k\\\\end{array}}^m \\\\frac{ w_j }{ \\\\Vert a^k - a^j \\\\Vert } \\\\left[ Q(a^k) - (1 - \\\\beta (a^k)) a^j - \\\\beta (a^k) a^k \\\\right].$ It can be seen that the function $\\\\alpha $ is related to the iteration function $Q$ of the proposed algorithm, and the iteration function $T$ of the modified algorithm.\",\n \"Lemma 7 If $x \\\\in \\\\Omega $ , then $\\\\alpha (x) = 2 A(x) \\\\left[ Q(x) - T(x) \\\\right].$ If $x \\\\ne a^1, \\\\ldots , a^m$ , then: $\\\\alpha (x) & = & \\\\sum _{j=1}^m \\\\frac{ w_j }{ \\\\Vert x - a^j \\\\Vert } \\\\left[ Q(x) - a^j \\\\right] = \\\\sum _{j=1}^m \\\\frac{ w_j Q(x) }{ \\\\Vert x - a^j \\\\Vert } - \\\\sum _{j=1}^m \\\\frac{ w_j a^j }{ \\\\Vert x - a^j \\\\Vert }\\\\\\\\&&\\\\\\\\& = & \\\\left( \\\\sum _{j=1}^m \\\\frac{ w_j }{ \\\\Vert x - a^j \\\\Vert } \\\\right) \\\\left[ Q(x) - \\\\frac{ \\\\sum _{j=1}^m \\\\frac{ w_j a^j }{ \\\\Vert x - a^j \\\\Vert } }{ \\\\sum _{j=1}^m \\\\frac{ w_j }{ \\\\Vert x - a^j \\\\Vert } } \\\\right] = 2 A(x) \\\\left[ Q(x) - \\\\widetilde{T}(x) \\\\right]\\\\\\\\&&\\\\\\\\& = & 2 A(x) \\\\left[ Q(x) - T(x) \\\\right].$ where in the last equalities we have used the definition of $\\\\widetilde{T}$ as in (REF ), and the fact that $\\\\widetilde{T}(x) = T(x)$ due to Remark REF .\",\n \"If $x = a^k$ for some $k = 1, \\\\ldots , m$ , we follow a similar procedure than in the previous case.\",\n \"$\\\\Box $ Now, we will define auxiliary functions that take into account the projection $P_{\\\\Omega }$ in order to prove a descent property of $f$ (see next sections).\",\n \"If $x \\\\in \\\\Omega $ , we define: $E_x : \\\\mathbb {R}^n \\\\rightarrow \\\\mathbb {R}^n$ , where: $\\\\left( E_x(y) \\\\right)_k = \\\\left\\\\lbrace \\\\begin{array}{ll} (Q(x))_k, & \\\\quad \\\\text{if $k \\\\in \\\\mathcal {N}(x)$}, \\\\\\\\ y_k, & \\\\quad \\\\text{if $k \\\\in \\\\mathcal {E}(x)$}.\",\n \"\\\\end{array} \\\\right.$ If $\\\\mathcal {E}(x) = \\\\left\\\\lbrace i_1, \\\\ldots , i_r \\\\right\\\\rbrace \\\\ne \\\\emptyset $ define $P_x : \\\\mathbb {R}^n \\\\rightarrow \\\\mathbb {R}^r$ where: $\\\\left( P_x(y) \\\\right)_k = y_{i_k}, \\\\quad k = 1, \\\\ldots , r.$ A useful property of $E_x$ , that follows from the definition, is pointed out in the following remark.\",\n \"Remark 8 If $x \\\\in \\\\Omega $ then $E_x \\\\circ Q(x) = Q(x)$ .\",\n \"The iteration function $Q$ inherits an important property from the orthogonal projection $P_{\\\\Omega }$ .\",\n \"Lemma 9 If $x \\\\in \\\\Omega $ we have that $\\\\left\\\\langle Q(x) - x , Q(x) - T(x) \\\\right\\\\rangle \\\\le 0$ .\",\n \"If $x \\\\ne a^1, \\\\ldots , a^m$ , then $Q(x) = P_{\\\\Omega } \\\\circ T(x)$ .\",\n \"By a property of the orthogonal projection [1] we have that $\\\\langle Q(x) - x , Q(x) - T(x) \\\\rangle \\\\le 0$ .\",\n \"If $x = a^k$ for some $k = 1, \\\\ldots , m$ , Remark REF and Lemma REF imply: $\\\\left\\\\langle Q(a^k) - a^k , Q(a^k) - T(a^k) \\\\right\\\\rangle = - \\\\lambda ( a^k ) \\\\left( 1 - \\\\lambda ( a^k ) \\\\right) \\\\left\\\\Vert T(a^k) - a^k \\\\right\\\\Vert ^2 \\\\le 0,$ and this concludes the proof.\",\n \"$\\\\Box $ The next technical lemma will help us to save computations in other lemmas.\",\n \"Lemma 10 If $x \\\\in \\\\Omega $ , $\\\\mathcal {A}$ is a subset of $\\\\left\\\\lbrace 1, \\\\ldots , n \\\\right\\\\rbrace $ and $j \\\\in \\\\left\\\\lbrace 1, \\\\ldots , m \\\\right\\\\rbrace $ , then: $\\\\left\\\\Vert Q(x) - a^j \\\\right\\\\Vert _{\\\\mathcal {A}}^2 = \\\\left\\\\Vert x - a^j \\\\right\\\\Vert _{\\\\mathcal {A}}^2 - \\\\left\\\\Vert Q(x) - x \\\\right\\\\Vert _{\\\\mathcal {A}}^2 + 2 \\\\left\\\\langle Q(x) - x, Q(x) - a^j \\\\right\\\\rangle _{\\\\mathcal {A}}.$ If $x \\\\in \\\\Omega $ , we have: $\\\\Vert Q(x) - a^j \\\\Vert _{\\\\mathcal {A}}^2 & = & \\\\langle Q(x) - a^j , Q(x) - a^j \\\\rangle _{\\\\mathcal {A}}\\\\\\\\& = & \\\\langle Q(x) - x + x - a^j , Q(x) - x + x - a^j \\\\rangle _{\\\\mathcal {A}}\\\\\\\\& = & \\\\Vert Q(x) - x \\\\Vert _{\\\\mathcal {A}}^2 + \\\\Vert x - a^j \\\\Vert _{\\\\mathcal {A}}^2 + 2 \\\\langle Q(x) - x , x - a^j \\\\rangle _{\\\\mathcal {A}}\\\\\\\\& = & \\\\Vert Q(x) - x \\\\Vert _{\\\\mathcal {A}}^2 + \\\\Vert x - a^j \\\\Vert _{\\\\mathcal {A}}^2 + 2 \\\\langle Q(x) - x , x - Q(x) \\\\rangle _{\\\\mathcal {A}}\\\\\\\\& + & 2 \\\\langle Q(x) - x , Q(x) - a^j \\\\rangle _{\\\\mathcal {A}}\\\\\\\\& = & \\\\Vert x - a^j \\\\Vert _{\\\\mathcal {A}}^2 - \\\\Vert Q(x) - x \\\\Vert _{\\\\mathcal {A}}^2 + 2 \\\\langle Q(x) - x , Q(x) - a^j \\\\rangle _{\\\\mathcal {A}}.$ $\\\\Box $ If $x \\\\in \\\\Omega $ , let us define $g_x : \\\\mathbb {R}^n \\\\rightarrow \\\\mathbb {R}$ by: $g_x(y) = \\\\left\\\\lbrace \\\\begin{array}{ll} \\\\displaystyle {\\\\sum _{j=1}^m} \\\\frac{ w_j }{ 2 \\\\Vert x - a^j \\\\Vert } \\\\Vert E_x(y) - a^j \\\\Vert ^2, & \\\\text{if $x \\\\ne a^1, \\\\ldots , a^m$}, \\\\\\\\ & \\\\\\\\ \\\\displaystyle { \\\\sum _{\\\\begin{array}{c}j=1 \\\\\\\\j \\\\ne k\\\\end{array}}^m} \\\\frac{ w_j }{ 2 \\\\Vert a^k - a^j \\\\Vert } \\\\Vert y - a^j \\\\Vert ^2 + w_k \\\\Vert y - a^k \\\\Vert , & \\\\text{if $x = a^k$}, \\\\\\\\ & \\\\text{$k = 1, \\\\ldots , m$}.\",\n \"\\\\end{array} \\\\right.$ The values that $g_x$ assumes at $x$ and $Q(x)$ will play an important role in the proof of a property of the objective function $f$ .\",\n \"Lemma 11 Let $x \\\\in \\\\Omega $ be.\",\n \"If $x \\\\ne a^1, \\\\ldots , a^m$ then: $g_x(x) & = & \\\\frac{1}{2} f(x) + 2 A(x) \\\\left\\\\langle Q(x) - x , Q(x) - T(x) \\\\right\\\\rangle \\\\\\\\& - & A(x) \\\\left\\\\Vert Q(x) - x \\\\right\\\\Vert _{\\\\mathcal {N}(x)}^2.$ If $x = a^k$ for some $k = 1, \\\\ldots , m$ , then $g_{a^k}(a^k) = \\\\frac{1}{2} f(a^k)$ .\",\n \"Let us suppose that $x \\\\ne a^1, \\\\ldots , a^m$ .\",\n \"By property (REF ) and (REF ), we have for $j = 1, \\\\ldots , m$ : $\\\\left\\\\Vert E_x(x) - a^j \\\\right\\\\Vert ^2 = \\\\left\\\\Vert x - a^j \\\\right\\\\Vert _{\\\\mathcal {E}(x)}^2 + \\\\left\\\\Vert Q(x) - a^j \\\\right\\\\Vert _{\\\\mathcal {N}(x)}^2.$ Using Lemma REF , we can see that: $g_x(x) & = & \\\\sum _{j=1}^m \\\\frac{ w_j }{ 2 \\\\Vert x - a^j \\\\Vert } \\\\left[ \\\\left\\\\Vert x - a^j \\\\right\\\\Vert _{\\\\mathcal {E}(x)}^2 + \\\\left\\\\Vert x - a^j \\\\right\\\\Vert _{\\\\mathcal {N}(x)}^2 \\\\right.\\\\\\\\& - & \\\\left.\",\n \"\\\\left\\\\Vert Q(x) - x \\\\right\\\\Vert _{\\\\mathcal {N}(x)}^2 + 2 \\\\left\\\\langle Q(x) - x , Q(x) - a^j \\\\right\\\\rangle _{\\\\mathcal {N}(x)} \\\\right].$ Due to (REF ), the definition of the Weber function $f$ , the definition of $A$ as in (REF ), the property () and the definition of $\\\\alpha $ as in (REF ), we obtain: $g_x(x) = \\\\frac{1}{2} f(x) - A(x) \\\\Vert Q(x) - x \\\\Vert _{\\\\mathcal {N}(x)}^2 + \\\\langle Q(x) - x , \\\\alpha (x) \\\\rangle _{\\\\mathcal {N}(x)}.$ By Lemma REF , the fact that $\\\\left( Q(x) \\\\right)_i = \\\\left( T(x) \\\\right)_i$ for all $i \\\\in \\\\mathcal {E}(x)$ and (), we get: $g_x(x) & = & \\\\frac{1}{2} f(x) - A(x) \\\\Vert Q(x) - x \\\\Vert _{\\\\mathcal {N}(x)}^2\\\\\\\\& + & 2 A(x) \\\\langle Q(x) - x , Q(x) - T(x) \\\\rangle ,$ which concludes the proof of (a).\",\n \"Now, let us assume that $x = a^k$ for some $k = 1, \\\\ldots , m$ .\",\n \"Then: $g_{a^k}(a^k) = \\\\sum _{\\\\begin{array}{c}j=1 \\\\\\\\ j \\\\ne k\\\\end{array}}^m \\\\frac{ w_j }{ 2 \\\\Vert a^k - a^j \\\\Vert } \\\\Vert a^k - a^j \\\\Vert ^2 = \\\\frac{1}{2} \\\\sum _{\\\\begin{array}{c}j=1 \\\\\\\\ j \\\\ne k\\\\end{array}}^m w_j \\\\Vert a^k - a^j \\\\Vert = \\\\frac{1}{2} f(a^k).$ This concludes the proof of (b).\",\n \"$\\\\Box $ The number $g_x(Q(x))$ can be computed in the next lemma.\",\n \"Lemma 12 Let $x \\\\in \\\\Omega $ be.\",\n \"If $x \\\\ne a^1, \\\\ldots , a^m$ then: $g_x(Q(x)) & = & \\\\frac{1}{2} f(x) + 2 A(x) \\\\left\\\\langle Q(x) - x , Q(x) - T(x) \\\\right\\\\rangle \\\\\\\\& - & A(x) \\\\left\\\\Vert Q(x) - x \\\\right\\\\Vert ^2.$ If $x = a^k$ for some $k = 1, \\\\ldots , m$ , then $g_{a^k}(Q(a^k)) & = & \\\\frac{1}{2} f(a^k) - A(a^k) \\\\left\\\\Vert Q(a^k) - a^k \\\\right\\\\Vert ^2\\\\\\\\& + & 2 A(a^k) \\\\left\\\\langle Q(a^k) - a^k , Q(a^k) - T(a^k) \\\\right\\\\rangle \\\\\\\\& - & 2 \\\\beta (a^k) A(a^k) \\\\left\\\\langle Q(a^k) - a^k , \\\\widetilde{T}(a^k) - a^k \\\\right\\\\rangle \\\\\\\\& + & w_k \\\\left\\\\Vert Q(a^k) - a^k \\\\right\\\\Vert .$ First, let us consider $x \\\\ne a^1, \\\\ldots , a^m$ .\",\n \"Due to Remark REF we have: $g_x(Q(x)) = \\\\sum _{j=1}^m \\\\frac{ w_j }{ 2 \\\\Vert x - a^j \\\\Vert } \\\\Vert Q(x) - a^j \\\\Vert ^2.$ By Lemma REF we obtain: $g_x(Q(x))& = & \\\\sum _{j=1}^m \\\\frac{ w_j }{ 2 \\\\Vert x - a^j \\\\Vert } \\\\left[ \\\\Vert x - a^j \\\\Vert ^2 - \\\\Vert Q(x) - x \\\\Vert ^2 \\\\right.\\\\\\\\& + & \\\\left.\",\n \"2 \\\\langle Q(x) - x , Q(x) - a^j \\\\rangle \\\\right].$ Due to the definition of the Weber function $f$ , the definition of $A$ as in (REF ) and the definition of $\\\\alpha $ as in (REF ), we deduce that: $g_x(Q(x)) = \\\\frac{1}{2} f(x) - A(x) \\\\Vert Q(x) - x \\\\Vert ^2 + \\\\langle Q(x) - x , \\\\alpha (x) \\\\rangle .$ By Lemma REF we get: $g_x(Q(x)) = \\\\frac{1}{2} f(x) - A(x) \\\\Vert Q(x) - x \\\\Vert ^2 + 2 A(x) \\\\langle Q(x) - x , Q(x) - T(x) \\\\rangle ,$ concluding the proof of (a).\",\n \"Now, consider $x = a^k$ for some $k = 1, \\\\ldots , m$ .\",\n \"Due to (REF ) we have: $g_{a^k}(Q(a^k)) = \\\\sum _{\\\\begin{array}{c}j=1 \\\\\\\\ j \\\\ne k\\\\end{array}}^m \\\\frac{ w_j }{ 2 \\\\Vert a^k - a^j \\\\Vert } \\\\Vert Q(a^k) - a^j \\\\Vert ^2 + w_k \\\\Vert Q(a^k) - a^k \\\\Vert .$ By Lemma REF , the definition of the Weber function $f$ and the definition of $A$ as in (REF ) we obtain: $g_{a^k}\\\\left( Q(a^k) \\\\right) & = & \\\\frac{1}{2} f(a^k) - A(a^k) \\\\Vert Q(a^k) - a^k \\\\Vert ^2\\\\\\\\& + & \\\\left\\\\langle Q(a^k) - a^k , \\\\sum _{\\\\begin{array}{c}j=1 \\\\\\\\ j \\\\ne k\\\\end{array}}^m \\\\frac{ w_j }{ \\\\Vert a^k - a^j \\\\Vert } \\\\left[ Q(a^k) - a^j \\\\right] \\\\right\\\\rangle \\\\\\\\& + & w_k \\\\Vert Q(a^k) - a^k \\\\Vert .$ Manipulating algebraically, $Q(a^k) - a^j = Q(a^k) - ( 1 - \\\\beta (a^k) ) a^j - \\\\beta (a^k) a^k + \\\\beta (a^k) ( a^k - a^j ).$ Due to the definition of $\\\\alpha $ (see (REF )) and the definition of $\\\\widetilde{R}$ (see (REF )) we get: $g_{a^k} \\\\left( Q(a^k) \\\\right)& = & \\\\frac{1}{2} f(a^k) - A(a^k) \\\\left\\\\Vert Q(a^k) - a^k \\\\right\\\\Vert ^2\\\\\\\\& + & \\\\left\\\\langle Q(a^k) - a^k , \\\\alpha (a^k) \\\\right\\\\rangle - \\\\beta (a^k) \\\\left\\\\langle Q(a^k) - a^k , \\\\widetilde{R}(a^k) \\\\right\\\\rangle \\\\\\\\& + & w_k \\\\Vert Q(a^k) - a^k \\\\Vert .$ By Lemma REF and Lemma REF we have: $g_{a^k} \\\\left( Q(a^k) \\\\right) & = & \\\\frac{1}{2} f(a^k) - A(a^k) \\\\left\\\\Vert Q(a^k) - a^k \\\\right\\\\Vert ^2\\\\\\\\& + & 2 A(a^k) \\\\left\\\\langle Q(a^k) - a^k , Q(a^k) - T(a^k) \\\\right\\\\rangle \\\\\\\\& - & 2 A(a^k) \\\\beta (a^k) \\\\left\\\\langle Q(a^k) - a^k , \\\\widetilde{T}(a^k) - a^k \\\\right\\\\rangle + w_k \\\\Vert Q(a^k) - a^k \\\\Vert .$ which concludes the proof.\",\n \"$\\\\Box $ The next lemma deals with the last two terms of $g_{a^k}(Q(a^k))$ .\",\n \"Lemma 13 If $a^k \\\\in \\\\Omega $ for some $k = 1, \\\\ldots , m$ , the number $z = w_k \\\\left\\\\Vert Q(a^k) - a^k \\\\right\\\\Vert - 2 A(a^k) \\\\beta (a^k) \\\\left\\\\langle Q(a^k) - a^k , \\\\widetilde{T}(a^k) - a^k \\\\right\\\\rangle ,$ is equal to zero.\",\n \"If $Q(a^k) = a^k$ the result is true.\",\n \"So, from now on, let us consider that $Q(a^k) \\\\ne a^k$ .\",\n \"First, let us check that $a^k \\\\ne x^u$ .\",\n \"In case that $a^k = x^u$ , then $T(a^k) = a^k$ by Theorem REF .\",\n \"Since $a^k \\\\in \\\\Omega $ , then $T(a^k) \\\\in \\\\Omega $ .\",\n \"By Remark REF we have that $Q(a^k) = T(a^k) = a^k$ which is a contradiction.\",\n \"By Remark REF , we have that $\\\\beta (a^k) \\\\in (0,1)$ (since $a^k \\\\ne x^u$ ) and: $z = w_k \\\\left\\\\Vert Q(a^k) - a^k \\\\right\\\\Vert - \\\\frac{ 2 A(a^k) \\\\beta (a^k) }{ 1 - \\\\beta (a^k) } \\\\left\\\\langle Q(a^k) - a^k , T(a^k) - a^k \\\\right\\\\rangle .$ Extracting common factors, using Remarks REF and REF , the fact that $T(a^k) \\\\ne a^k$ (if $T(a^k) = a^k$ then $a^k = x^u$ by Theorem REF ), and the fact that $\\\\widetilde{T}(a^k) \\\\ne a^k$ (if $\\\\widetilde{T}(a^k) = a^k$ then $T(a^k) = a^k$ by definition (REF )) we get that: $z & = & 2 A(a^k) \\\\left\\\\Vert Q(a^k) - a^k \\\\right\\\\Vert \\\\left\\\\Vert \\\\widetilde{T}(a^k) - a^k \\\\right\\\\Vert \\\\left[ \\\\frac{ w_k }{ 2 A(a^k) \\\\left\\\\Vert \\\\widetilde{T}(a^k) - a^k \\\\right\\\\Vert } \\\\right.\\\\\\\\& - & \\\\left.\",\n \"\\\\beta (a^k) \\\\left\\\\langle \\\\frac{ ( 1 - \\\\lambda ( a^k ) ) ( T(a^k) - a^k ) }{ \\\\left\\\\Vert ( 1 - \\\\lambda ( a^k ) ) ( T(a^k) - a^k ) \\\\right\\\\Vert } , \\\\frac{ T(a^k) - a^k }{ \\\\left\\\\Vert T(a^k) - a^k \\\\right\\\\Vert } \\\\right\\\\rangle \\\\right].$ Simplifying and using the definition of $\\\\beta (a^k)$ we have that: $z = 2 A(a^k) \\\\left\\\\Vert Q(a^k) - a^k \\\\right\\\\Vert \\\\left\\\\Vert \\\\widetilde{T}(a^k) - a^k \\\\right\\\\Vert \\\\left[ \\\\beta (a^k) - \\\\beta (a^k) \\\\right] = 0,$ which concludes the proof.\",\n \"$\\\\Box $ The purpose of the next two lemmas is to determine a strict inequality between the functions $g_x$ and $f$ at suitable points.\",\n \"First of all, we have to prove the following result.\",\n \"Lemma 14 Let $x \\\\in \\\\Omega $ be such that $x \\\\ne Q(x)$ .\",\n \"If $x \\\\ne a^1, \\\\ldots , a^m$ , then $g_x(Q(x)) \\\\le g_x(x)$ .\",\n \"Besides that, if $\\\\mathcal {E}(x) \\\\ne \\\\emptyset $ and $P_x \\\\circ Q(x) \\\\ne P_x(x)$ , then $g_x(Q(x)) < g_x(x)$ .\",\n \"If $x = a^k$ for some $k = 1, \\\\ldots , m$ , then $g_{a^k}(Q(a^k)) < g_{a^k}(a^k)$ .\",\n \"If $x \\\\ne a^1, \\\\ldots , a^m$ , then $g_x(Q(x)) - g_x(x) = - A(x) \\\\Vert Q(x) - x \\\\Vert _{\\\\mathcal {E}(x)}^2 \\\\le 0$ , by Lemma REF and Lemma REF .\",\n \"Besides that, if $\\\\mathcal {E}(x) \\\\ne \\\\emptyset $ and $P_x \\\\circ Q(x) \\\\ne P_x(x)$ we deduce that $\\\\Vert Q(x) - x \\\\Vert _{\\\\mathcal {E}(x)} \\\\ne 0$ .\",\n \"Thus, $g_x(Q(x)) < g_x(x)$ .\",\n \"If $x = a^k$ for some $k = 1, \\\\ldots , m$ , by Lemmas REF , REF and REF we have: $g_{a^k}(Q(a^k)) - g_{a^k}(a^k) & = & - A(a^k) \\\\left\\\\Vert Q(a^k) - a^k \\\\right\\\\Vert ^2\\\\\\\\& + & 2 A(a^k) \\\\left\\\\langle Q(a^k) - a^k , Q(a^k) - T(a^k) \\\\right\\\\rangle .$ Due to Lemma REF and the fact that $A > 0$ we obtain: $g_{a^k}(Q(a^k)) - g_{a^k}(a^k) \\\\le - A(a^k) \\\\Vert Q(a^k) - a^k \\\\Vert ^2 < 0,$ and the proof is finished.\",\n \"$\\\\Box $ Lemma 15 Let $x \\\\in \\\\Omega $ be such that $x \\\\ne Q(x)$ .\",\n \"Then $g_x(Q(x)) < \\\\frac{1}{2} f(x)$ .\",\n \"Let us consider the case when $x \\\\ne a^1, \\\\ldots , a^m$ .\",\n \"By Lemmas REF , REF and REF we have that: $g_x(Q(x)) & \\\\le & g_x(x) = \\\\frac{1}{2} f(x) + 2 A(x) \\\\left\\\\langle Q(x) - x , Q(x) - T(x) \\\\right\\\\rangle \\\\\\\\& - & A(x) \\\\left\\\\Vert Q(x) - x \\\\right\\\\Vert _{\\\\mathcal {N}(x)}^2\\\\\\\\& \\\\le & \\\\frac{1}{2} f(x) - A(x) \\\\left\\\\Vert Q(x) - x \\\\right\\\\Vert _{\\\\mathcal {N}(x)}^2.$ If $\\\\mathcal {E}(x) = \\\\emptyset $ , then $\\\\Vert \\\\cdot \\\\Vert _{\\\\mathcal {N}(x)} = \\\\Vert \\\\cdot \\\\Vert $ .\",\n \"Therefore: $g_x(Q(x)) \\\\le \\\\frac{1}{2} f(x) - A(x) \\\\Vert Q(x) - x \\\\Vert ^2 < \\\\frac{1}{2} f(x).$ If $\\\\mathcal {E}(x) \\\\ne \\\\emptyset $ and $P_x \\\\circ Q(x) = P_x(x)$ , then there exists an index $i \\\\in \\\\mathcal {N}(x)$ such that $x_i \\\\ne (Q(x))_i$ since $x \\\\ne Q(x)$ .\",\n \"Thus, $\\\\Vert Q(x) - x \\\\Vert _{\\\\mathcal {N}(x)} \\\\ne 0$ , which implies: $g_x(Q(x)) \\\\le \\\\frac{1}{2} f(x) - A(x) \\\\left\\\\Vert Q(x) - x \\\\right\\\\Vert _{\\\\mathcal {N}(x)}^2 < \\\\frac{1}{2} f(x).$ If $\\\\mathcal {E}(x) \\\\ne \\\\emptyset $ and $P_x \\\\circ Q(x) \\\\ne P_x(x)$ , due to Lemmas REF , REF and REF , we have that: $g_x(Q(x)) < g_x(x) \\\\le \\\\frac{1}{2} f(x) - A(x) \\\\left\\\\Vert Q(x) - x \\\\right\\\\Vert _{\\\\mathcal {N}(x)}^2 \\\\le \\\\frac{1}{2} f(x).$ Now, when $x = a^k$ for some $k = 1, \\\\ldots , m$ , $g_{a^k}(Q(a^k)) < g_{a^k}(a^k) = \\\\frac{1}{2} f(a^k)$ due to Lemma REF and Lemma REF .\",\n \"$\\\\Box $ The next lemma states an equality that relates the Weber function and $g_x$ at appropriate points when $x \\\\ne a^1, \\\\ldots , a^m$ .\",\n \"Besides that, this result will be crucial in the next section.\",\n \"Lemma 16 Let $x \\\\ne a^1, \\\\ldots , a^m$ be such that $x \\\\in \\\\Omega $ and $x \\\\ne Q(x)$ .\",\n \"Then: $g_x \\\\circ Q(x) = \\\\frac{1}{2} f(x) + \\\\left( f(Q(x)) - f(x) \\\\right) + \\\\delta , \\\\quad \\\\delta \\\\ge 0.$ Due to the definition of $g_x$ as in (REF ) and Remark REF we get that: $g_x \\\\circ Q(x) = \\\\sum _{j=1}^m \\\\frac{ w_j }{ 2 \\\\Vert x - a^j \\\\Vert } \\\\Vert Q(x) - a^j \\\\Vert ^2.$ Adding and subtracting $\\\\Vert x - a^j \\\\Vert $ we have: $g_x \\\\circ Q(x) & = & \\\\sum _{j=1}^m \\\\frac{ w_j }{ 2 \\\\Vert x - a^j \\\\Vert } \\\\left[ \\\\left\\\\Vert x - a^j \\\\right\\\\Vert + \\\\left( \\\\left\\\\Vert Q(x) - a^j \\\\right\\\\Vert - \\\\left\\\\Vert x - a^j \\\\right\\\\Vert \\\\right) \\\\right]^2\\\\\\\\& = & \\\\frac{1}{2} \\\\sum _{j=1}^m w_j \\\\left\\\\Vert x - a^j \\\\right\\\\Vert + \\\\sum _{j=1}^m w_j \\\\left( \\\\left\\\\Vert Q(x) - a^j \\\\right\\\\Vert - \\\\left\\\\Vert x - a^j \\\\right\\\\Vert \\\\right)\\\\\\\\& + & \\\\sum _{j=1}^m \\\\frac{ w_j }{ 2 \\\\Vert x - a^j \\\\Vert } \\\\left( \\\\left\\\\Vert Q(x) - a^j \\\\right\\\\Vert - \\\\left\\\\Vert x - a^j \\\\right\\\\Vert \\\\right)^2.$ Notice that the first term of the last equality is the Weber function (divided by two), and the last term is a non-negative number, so we will define it as $\\\\delta $ .\",\n \"So, using the definition of the Weber function in the middle term we obtain: $g_x \\\\circ Q(x) = \\\\frac{1}{2} f(x) + \\\\left( f(Q(x)) - f(x) \\\\right) + \\\\delta .$ $\\\\Box $\"\n ],\n [\n \"Convergence to optimality results\",\n \"This section states the main results about convergence of the sequence $\\\\left\\\\lbrace x^{(l)} \\\\right\\\\rbrace $ generated by Algorithm REF .\",\n \"The next theorem establishes that if a point $x \\\\in \\\\Omega $ is not a fixed point of the iteration function, then the function $f$ strictly decreases at the next iterate.\",\n \"Theorem 17 Let $x \\\\in \\\\Omega $ be such that $x \\\\ne Q(x)$ .\",\n \"Then $f(Q(x)) < f(x)$ .\",\n \"Let us consider that $x \\\\ne a^1, \\\\ldots , a^m$ .\",\n \"By Lemma REF , we have that: $g_x \\\\circ Q(x) < \\\\frac{1}{2} f(x).$ By Lemma REF we get that: $\\\\frac{1}{2} f(x) + f(Q(x)) - f(x) + \\\\delta < \\\\frac{1}{2} f(x).$ Simplifying the last expression we obtain: $f(Q(x)) - f(x) + \\\\delta < 0.$ Finally, $f(Q(x)) - f(x) \\\\le f(Q(x)) - f(x) + \\\\delta < 0.$ Therefore, $f(Q(x)) < f(x)$ .\",\n \"Now, consider that $x = a^k$ for some $k = 1, \\\\ldots , m$ .\",\n \"Following a reasoning similar than in [40], using Lemma REF we have that: $g_{a^k} \\\\circ Q(a^k) - g_{a^k}(a^k) < 0.$ By definition of $g_{a^k}$ we know that: $g_{a^k} \\\\circ Q(a^k) - g_{a^k}(a^k) & = & w_k \\\\left\\\\Vert Q(a^k) - a^k \\\\right\\\\Vert \\\\\\\\& + & \\\\sum _{\\\\begin{array}{c}j=1 \\\\\\\\ j \\\\ne k\\\\end{array}}^m \\\\frac{ w_j }{ 2 \\\\left\\\\Vert a^k - a^j \\\\right\\\\Vert } \\\\left( \\\\left\\\\Vert Q(a^k) - a^j \\\\right\\\\Vert ^2 - \\\\left\\\\Vert a^k - a^j \\\\right\\\\Vert ^2 \\\\right).$ Using the fact that $\\\\left( a^2 - b^2 \\\\right)/ (2b) \\\\ge a - b$ for $a = \\\\left\\\\Vert Q(a^k) - a^j \\\\right\\\\Vert ^2 \\\\ge 0$ and $b = \\\\left\\\\Vert a^k - a^j \\\\right\\\\Vert ^2 > 0$ we obtain that: $g_{a^k} \\\\circ Q(a^k) - g_{a^k}(a^k) & \\\\ge & w_k \\\\left\\\\Vert Q(a^k) - a^k \\\\right\\\\Vert \\\\\\\\& - & \\\\sum _{\\\\begin{array}{c}j=1 \\\\\\\\ j \\\\ne k\\\\end{array}}^m w_j \\\\left\\\\Vert a^k - a^j \\\\right\\\\Vert + \\\\sum _{\\\\begin{array}{c}j=1 \\\\\\\\ j \\\\ne k\\\\end{array}}^m w_j \\\\left\\\\Vert Q(a^k) - a^j \\\\right\\\\Vert .$ Rearranging terms we deduce that: $0 & > & g_{a^k} \\\\circ Q(a^k) - g_{a^k}(a^k)\\\\\\\\& = & \\\\sum _{j=1}^m w_j \\\\left\\\\Vert Q(a^k) - a^j \\\\right\\\\Vert - \\\\sum _{j=1}^m w_j \\\\left\\\\Vert a^k - a^j \\\\right\\\\Vert = f(Q(a^k)) - f(a^k),$ and the proof is complete.\",\n \"$\\\\Box $ Corollary 18 Let $\\\\left\\\\lbrace x^{(l)} \\\\right\\\\rbrace $ be the sequence generated by Algorithm REF .\",\n \"Then the sequence $\\\\left\\\\lbrace f \\\\left( x^{(l)} \\\\right) \\\\right\\\\rbrace $ is not increasing.\",\n \"Even more, each time $x^{(l)} \\\\ne Q \\\\left( x^{(l)} \\\\right)$ the sequence strictly decreases at the next iterate.\",\n \"If the sequence $\\\\left\\\\lbrace x^{(l)} \\\\right\\\\rbrace $ generated by Algorithm REF were not bounded, then we could choose a subsequence $\\\\left\\\\lbrace y^{(l)} \\\\right\\\\rbrace $ such that $y^{(l)} \\\\rightarrow \\\\infty $ .\",\n \"But this implies that $f(y^{(l)}) \\\\rightarrow \\\\infty $ , which is a contradiction since the sequence $\\\\left\\\\lbrace f \\\\left( x^{(l)} \\\\right) \\\\right\\\\rbrace $ is not increasing.\",\n \"Remark 19 The sequence $\\\\left\\\\lbrace x^{(l)} \\\\right\\\\rbrace $ generated by Algorithm REF is bounded.\",\n \"So, there exists a subsequence convergent to a point $x^* \\\\in \\\\Omega $ .\",\n \"Hence, $x^*$ is a feasible point.\",\n \"Due to the nondifferentiability of $f$ at the vertices $a^1, \\\\ldots , a^m$ , we can not use the KKT optimality conditions at $a^k$ .\",\n \"Therefore, if $a^k$ and $z$ are in $\\\\Omega $ , let us define $G_{a^k}^z : [0,1] \\\\rightarrow \\\\mathbb {R}$ by: $G_{a^k}^z(t) = f( a^k + t(z-a^k) ).$ If $a^k \\\\in \\\\Omega $ , $z \\\\in \\\\Omega $ , $t \\\\in [0,1]$ and $\\\\Omega $ convex, we have that $a^k + t(z-a^k) \\\\in \\\\Omega $ .\",\n \"Notice that the right-hand side derivative $G_{a^k}^z(0+)$ (or the directional derivative of $f$ in the direction of $z$ ) exists (see [22]).\",\n \"Besides that, $G_{a^k}^{z \\\\ \\\\prime }(0+) = w_k \\\\left\\\\Vert z - a^k \\\\right\\\\Vert - \\\\left\\\\langle \\\\widetilde{R}(a^k) , z - a^k \\\\right\\\\rangle .$ The next lemma shows that if we are in a vertex $a^k$ , the directional derivative of $f$ at $a^k$ in the direction of $Q(a^k)$ is a descent direction.\",\n \"Lemma 20 Let $a^k \\\\in \\\\Omega $ be such that $T(a^k) \\\\notin \\\\Omega $ .\",\n \"Then: $G_{a^k}^{z \\\\ \\\\prime }(0+) \\\\ge G_{a^k}^{Q(a^k) \\\\ \\\\prime }(0+), \\\\quad \\\\forall \\\\, z \\\\in \\\\left[ a^k, T(a^k) \\\\right],$ where: $G_{a^k}^{Q(a^k) \\\\ \\\\prime }(0+) = - 2 \\\\left[ 1 - \\\\beta (a^k) \\\\right] A(a^k) \\\\left\\\\Vert \\\\widetilde{T}(a^k) - a^k \\\\right\\\\Vert \\\\left\\\\Vert Q(a^k) - a^k \\\\right\\\\Vert .$ If $T(a^k) = a^k$ then $T(a^k) \\\\in \\\\Omega $ , which is a contradiction.\",\n \"Besides that, if $\\\\widetilde{T}(a^k) = a^k$ , we would have that $T(a^k) = a^k$ because of (REF ), and again it would be a contradiction.\",\n \"So, we will consider $\\\\widetilde{T}(a^k) \\\\ne a^k$ and $T(a^k) \\\\ne a^k$ for the rest of the proof.\",\n \"Since $T(a^k) \\\\ne a^k$ , then $\\\\beta (a^k) \\\\in (0,1)$ (see Remark REF and Theorem REF ).\",\n \"Let us prove equation (REF ) first.\",\n \"Now, by (REF ) we can see that: $G_{a^k}^{Q(a^k) \\\\ \\\\prime }(0+) = w_k \\\\left\\\\Vert Q(a^k) - a^k \\\\right\\\\Vert - \\\\left\\\\langle \\\\widetilde{R}(a^k) , Q(a^k) - a^k \\\\right\\\\rangle .$ Notice that if $Q(a^k) = a^k$ , equation (REF ) holds.\",\n \"So, let us consider from now on that $Q(a^k) \\\\ne a^k$ .\",\n \"By using Lemma REF we replace $\\\\widetilde{R}(a^k)$ and get: $G_{a^k}^{Q(a^k) \\\\ \\\\prime }(0+) = w_k \\\\left\\\\Vert Q(a^k) - a^k \\\\right\\\\Vert - 2 A(a^k) \\\\left\\\\langle \\\\widetilde{T}(a^k) - a^k , Q(a^k) - a^k \\\\right\\\\rangle .$ Extracting common factors and using the definition of $\\\\beta $ when it belongs to $(0,1)$ we obtain: $G_{a^k}^{Q(a^k) \\\\ \\\\prime }(0+) & = & 2 A(a^k) \\\\left\\\\Vert Q(a^k) - a^k \\\\right\\\\Vert \\\\left\\\\Vert \\\\widetilde{T}(a^k) - a^k \\\\right\\\\Vert \\\\Bigg [ \\\\beta (a^k)\\\\\\\\& - & \\\\left.\",\n \"\\\\left\\\\langle \\\\frac{ \\\\widetilde{T}(a^k) - a^k }{ \\\\left\\\\Vert \\\\widetilde{T}(a^k) - a^k \\\\right\\\\Vert } , \\\\frac{ Q(a^k) - a^k }{ \\\\left\\\\Vert Q(a^k) - a^k \\\\right\\\\Vert } \\\\right\\\\rangle \\\\right].$ By Remarks REF and REF the vectors $Q(a^k) - a^k$ and $\\\\widetilde{T}(a^k) - a^k$ are parallel, so: $G_{a^k}^{Q(a^k) \\\\ \\\\prime }(0+) = 2 A(a^k) \\\\left\\\\Vert Q(a^k) - a^k \\\\right\\\\Vert \\\\left\\\\Vert \\\\widetilde{T}(a^k) - a^k \\\\right\\\\Vert \\\\left[ \\\\beta (a^k) - 1 \\\\right].$ which is equivalent to (REF ).\",\n \"Now, let us prove (REF ).\",\n \"If $z = a^k$ then $G_{a^k}^z(t) = f(a^k)$ for all $t \\\\in [0,1]$ , thus $G_{a^k+}^{a^k \\\\ \\\\prime }(0) = 0$ , and therefore the inequality (REF ) holds.\",\n \"So, let us assume that $z \\\\ne a^k$ for the rest of the proof.\",\n \"Using (REF ) and due to Lemma REF to replace $\\\\widetilde{R}(a^k)$ : $G_{a^k}^{z \\\\ \\\\prime }(0+) = w_k \\\\left\\\\Vert z - a^k \\\\right\\\\Vert - 2 A(a^k) \\\\left\\\\langle \\\\widetilde{T}(a^k) - a^k , z - a^k \\\\right\\\\rangle .$ Extracting common factors: $G_{a^k}^{z \\\\ \\\\prime }(0+) & = & 2 A(a^k) \\\\left\\\\Vert z - a^k \\\\right\\\\Vert \\\\left\\\\Vert \\\\widetilde{T}(a^k) - a^k \\\\right\\\\Vert \\\\left[ \\\\frac{ w_k }{ 2 A(a^k) \\\\left\\\\Vert \\\\widetilde{T}(a^k) - a^k \\\\right\\\\Vert } \\\\right.\\\\\\\\& - & \\\\left.\",\n \"\\\\left\\\\langle \\\\frac{ \\\\widetilde{T}(a^k) - a^k }{ \\\\left\\\\Vert \\\\widetilde{T}(a^k) - a^k \\\\right\\\\Vert } , \\\\frac{ z - a^k }{ \\\\left\\\\Vert z - a^k \\\\right\\\\Vert } \\\\right\\\\rangle \\\\right].$ Using the expression for $\\\\beta (a^k) \\\\in (0,1)$ we obtain: $G_{a^k}^{z \\\\ \\\\prime }(0+) & = & 2 A(a^k) \\\\left\\\\Vert z - a^k \\\\right\\\\Vert \\\\left\\\\Vert \\\\widetilde{T}(a^k) - a^k \\\\right\\\\Vert \\\\Bigg [ \\\\beta (a^k)\\\\\\\\& - & \\\\left.\",\n \"\\\\left\\\\langle \\\\frac{ \\\\widetilde{T}(a^k) - a^k }{ \\\\left\\\\Vert \\\\widetilde{T}(a^k) - a^k \\\\right\\\\Vert } , \\\\frac{ z - a^k }{ \\\\left\\\\Vert z - a^k \\\\right\\\\Vert } \\\\right\\\\rangle \\\\right].$ If $z$ belongs to the segment that joins $a^k$ and $T(a^k)$ we have that $z-a^k$ and $\\\\widetilde{T}(a^k) - a^k$ are parallel vectors, then: $G_{a^k}^{z \\\\ \\\\prime }(0+) \\\\ge - 2 \\\\left[ 1 - \\\\beta (a^k) \\\\right] A(a^k) \\\\left\\\\Vert z - a^k \\\\right\\\\Vert \\\\left\\\\Vert \\\\widetilde{T}(a^k) - a^k \\\\right\\\\Vert .$ We can write $z = (1-\\\\lambda )T(a^k) + \\\\lambda a^k$ where $\\\\lambda \\\\in [0,1]$ .\",\n \"Therefore: $G_{a^k}^{z \\\\ \\\\prime }(0+) \\\\ge - 2 \\\\left[ 1 - \\\\beta (a^k) \\\\right] A(a^k) \\\\left( 1 - \\\\lambda \\\\right) \\\\left\\\\Vert T(a^k) - a^k \\\\right\\\\Vert \\\\left\\\\Vert \\\\widetilde{T}(a^k) - a^k \\\\right\\\\Vert .$ for all $\\\\lambda \\\\in [0,1]$ .\",\n \"The minimum value of the right-hand side of the last expression happens when $\\\\lambda = \\\\lambda ( a^k )$ , so: $G_{a^k}^{z \\\\ \\\\prime }(0+) \\\\ge - 2 \\\\left[ 1 - \\\\beta (a^k) \\\\right] A(a^k) \\\\left( 1 - \\\\lambda ( a^k ) \\\\right) \\\\left\\\\Vert T(a^k) - a^k \\\\right\\\\Vert \\\\left\\\\Vert \\\\widetilde{T}(a^k) - a^k \\\\right\\\\Vert .$ Using Remark REF we conclude that: $G_{a^k}^{z \\\\ \\\\prime }(0+) \\\\ge - 2 \\\\left[ 1 - \\\\beta (a^k) \\\\right] A(a^k) \\\\left\\\\Vert Q(a^k) - a^k \\\\right\\\\Vert \\\\left\\\\Vert \\\\widetilde{T}(a^k) - a^k \\\\right\\\\Vert .$ $\\\\Box $ Now we will prove an equivalence that characterizes the solution of (REF ) in terms of the iteration function $Q$ .\",\n \"Moreover, if $x^*$ is a regular point that is not a vertex, then $x^*$ is a KKT point.\",\n \"From now on, let us consider that $\\\\Omega = \\\\left\\\\lbrace y \\\\in \\\\mathbb {R}^n : g(y) \\\\le 0, h(y) = 0 \\\\right\\\\rbrace ,$ where $g : \\\\mathbb {R}^n \\\\rightarrow \\\\mathbb {R}^s$ is a convex function and $h : \\\\mathbb {R}^n \\\\rightarrow \\\\mathbb {R}^p$ is an affine function.\",\n \"Theorem 21 Let $\\\\Omega $ be defined as in (REF ) and $x \\\\in \\\\Omega $ .\",\n \"Consider the following propositions: $x$ is a KKT point.\",\n \"$x$ is the minimizer of the problem (REF ).\",\n \"$Q(x) = x$ .\",\n \"If $x \\\\ne a^1, \\\\ldots , a^m$ , $g$ and $h$ are continuously differentiable, and $x$ is a regular point, then (a), (b) and (c) are equivalent.\",\n \"If $x = a^k$ for some $k = 1, \\\\ldots , m$ , then (b) implies (c).\",\n \"Let $x \\\\ne a^1, \\\\ldots , a^m$ be.\",\n \"Since $f$ is strictly convex and $\\\\Omega $ is convex, the KKT optimality conditions are necessary and sufficient.\",\n \"Therefore, it holds that (a) is equivalent to (b).\",\n \"Now we will prove that (b) implies (c).\",\n \"Let us suppose that $x$ is the minimizer of the problem (REF ).\",\n \"If $x$ were not a fixed point of the iteration function $Q$ , we would have that $x \\\\ne Q(x)$ , which means that $f(Q(x)) < f(x)$ by Theorem REF .\",\n \"This contradicts the hypothesis.\",\n \"To demonstrate that (c) implies (a), we will assume that $x$ is a fixed point of $Q$ , that is, $x = Q(x)$ .\",\n \"Since $Q(x) = P_{\\\\Omega } \\\\circ T(x)$ , $x$ is the solution of: $\\\\begin{array}{rl}\\\\displaystyle \\\\mathop { \\\\mathrm {argmin} }_z & \\\\displaystyle F(z) = \\\\frac{1}{2} \\\\left\\\\Vert z - T(x) \\\\right\\\\Vert ^2 \\\\\\\\ \\\\mathrm {subject \\\\, to} & g(z) \\\\le 0, \\\\\\\\ & h(z) = 0.\\\\end{array}$ Since $F$ and $g$ are convex, $h$ is affine, and $x$ is a regular point, the KKT optimality conditions hold at $x$ .\",\n \"That is, there exist multipliers $\\\\left\\\\lbrace \\\\mu _j \\\\right\\\\rbrace _{j=1}^s$ and $\\\\left\\\\lbrace \\\\lambda _j \\\\right\\\\rbrace _{j=1}^p$ such that (see [31], [38]): $x - T(x) + \\\\sum _{j=1}^s \\\\mu _j \\\\nabla g_j(x) + \\\\sum _{j=1}^p \\\\lambda _j \\\\nabla h_j(x) & = & 0,\\\\\\\\\\\\mu _j g_j( x ) & = & 0, \\\\quad j = 1, \\\\ldots , s,\\\\\\\\\\\\mu _j & \\\\ge & 0, \\\\quad j = 1, \\\\ldots , s,\\\\\\\\g(x) & \\\\le & 0,\\\\\\\\h(x) & = & 0.$ Multiplying these equations by $2 A(x)$ , using equation (REF ), Lemma REF and Remark REF , we obtain: $\\\\nabla f(x) + \\\\sum _{j=1}^s \\\\left( 2 A(x) \\\\mu _j \\\\right) \\\\nabla g_j(x) + \\\\sum _{j=1}^p \\\\left( 2 A(x) \\\\lambda _j \\\\right) \\\\nabla h_j(x) & = & 0,\\\\\\\\\\\\left( 2 A(x) \\\\mu _j \\\\right) g_j( x ) & = & 0, \\\\quad j = 1, \\\\ldots , s,\\\\\\\\\\\\left( 2 A(x) \\\\mu _j \\\\right) & \\\\ge & 0, \\\\quad j = 1, \\\\ldots , s,\\\\\\\\g(x) & \\\\le & 0,\\\\\\\\h(x) & = & 0.$ where $\\\\left\\\\lbrace 2 A(x) \\\\mu _j \\\\right\\\\rbrace _{j=1}^s$ and $\\\\left\\\\lbrace 2 A(x) \\\\lambda _j \\\\right\\\\rbrace _{j=1}^p$ are multipliers.\",\n \"Therefore, $x$ is a KKT point of the problem (REF ) (see [31], [38]).\",\n \"Now, let us suppose that $x = a^k$ for some $k = 1, \\\\ldots , m$ .\",\n \"As before, if $x$ is a minimizer of the problem (REF ), then $Q(a^k) = a^k$ , otherwise $f(Q(a^k)) < f(a^k)$ , which would be a contradiction.\",\n \"$\\\\Box $\"\n ],\n [\n \"Numerical experiments.\",\n \"The purpose of this section is to discuss the efficiency and robustness of the proposed algorithm versus a solver for nonlinear programming problems.\",\n \"A prototype code of Algorithm REF was programmed in MATLAB (version R2011a) and executed in a PC running Linux OS, Intel(R) Core(TM) i7 CPU Q720, 1.60GHz.\",\n \"We have considered a closed and convex set $\\\\Omega \\\\subset \\\\mathbb {R}^2$ defined by the set $\\\\Omega = \\\\left\\\\lbrace y \\\\in \\\\mathbb {R}^n : g(y) \\\\le 0 \\\\right\\\\rbrace $ , where $g$ is given by: $g( x ) = \\\\left[ \\\\begin{array}{c} \\\\displaystyle - 4 - \\\\frac{ 1 }{ 8 } x + \\\\frac{ 7 }{ 72 } x^2 + \\\\frac{ 1 }{ 216 } x^2 ( x - 3 ) + y \\\\\\\\[3mm] \\\\displaystyle \\\\frac{ 4 }{ 5 } x + y - \\\\frac{ 59 }{ 10 } \\\\\\\\[3mm] \\\\displaystyle x - \\\\frac{ 11 }{ 2 } \\\\\\\\[3mm] \\\\displaystyle \\\\frac{ 3 }{ 2 } x - y - \\\\frac{ 35 }{ 4 } \\\\\\\\[3mm] \\\\displaystyle x - y - \\\\frac{ 13 }{ 2 } \\\\\\\\[3mm] \\\\displaystyle - 4 + \\\\frac{ 1 }{ 8 } ( x - 1 ) + \\\\frac{ 1 }{ 16 } ( x - 1 )^2 + \\\\frac{ 1 }{ 32 } ( x - 1 )^2 ( x - 3 ) - y \\\\\\\\[3mm] \\\\displaystyle - \\\\frac{ 1 }{ 3 } x - y - \\\\frac{ 11 }{ 3 } \\\\\\\\[3mm] \\\\displaystyle - \\\\frac{ 2 }{ 3 } x - y - \\\\frac{ 13 }{ 3 } \\\\\\\\[3mm] \\\\displaystyle - 4 x + y - 19 \\\\end{array} \\\\right].$ The feasible set is defined by linear and nonlinear constraints, as it can be seen in Figure REF .\",\n \"Figure: Feasible set Ω\\\\Omega We have built 1000 different experiments where for each one: The number of vertices was $m = 50$ .\",\n \"The vertices were normally distributed random vectors, with mean equal to 0 and standard deviation equal to 10.\",\n \"The weights were uniformly distributed random positive numbers between 0 and 10.\",\n \"Tolerance was set to $\\\\varepsilon = 0.00001$ .\",\n \"On one hand, each experiment was solved using Algorithm REF and, on the other hand, it was considered as a nonlinear programming problem and solved using function $fmincon$ (see [32] and references therein).\",\n \"Since the Weber function (REF ) is not differentiable at the vertices, nonlinear programming solvers may fail.\",\n \"Let $x_m(i)$ be the solution of (REF ) obtained by $fmincon$ in experiment $i$ , and $f_{m}(i) = f(x_m(i))$ .\",\n \"Analogously, let $x_p(i)$ be the solution of (REF ) obtained by Algorithm REF in experiment $i$ , and $f_{p}(i) = f(x_p(i))$ .\",\n \"Figure REF shows the difference between the arrays $f_{m}$ and $f_{p}$ .\",\n \"Both methods finished succesfully in all cases, however, Algorithm REF found equal or better results for all experiments.\",\n \"For example, the difference $f_{m} - f_{p}$ was greater than $0.01$ in 35 experiments (the maximum difference ocurred in experiment 506).\",\n \"Figure: Difference between minimum values found by Algorithm and fminconfmincon.Feasibility of the solutions $x_p(i)$ can be checked computing $\\\\max (g(x_m(i)))$ .\",\n \"Results can be seen in Figure REF Figure: Feasibility of the solution x p (i)x_p(i) obtained by Algorithm .\"\n ],\n [\n \"Conclusions\",\n \"This paper proposes a Weiszfeld-like algorithm for solving the Weber problem constrained to a closed and convex set, and it is well defined even when an iterate is a vertex.\",\n \"The algorithm consists of two stages: first, iterate using the fixed point modified Weiszfeld iteration (REF ), and second, either project onto the set $\\\\Omega $ when the iterate is different from the vertices, or, if the iterate is a vertex $a^k$ , take the point belonging to the line that joins $T(a^k)$ with $a^k$ as defined in (REF ).\",\n \"It is proved that the constrained problem (REF ) has a unique solution.\",\n \"Besides that, the definition of the iteration function $Q$ allows us to demonstrate that the proposed algorithm produces a sequence $\\\\left\\\\lbrace x^{(l)} \\\\right\\\\rbrace $ of feasible iterates.\",\n \"Moreover, the sequence $\\\\left\\\\lbrace f \\\\left( x^{(l)} \\\\right) \\\\right\\\\rbrace $ is not increasing, and when $x^{(l)} \\\\ne Q\\\\left( x^{(l)} \\\\right)$ , the sequence decreases at the next iterate.\",\n \"It can be seen that if a point $x^*$ is the solution of the problem (REF ) then $x^*$ is a fixed point of the iteration function $Q$ .\",\n \"Even more, if $x^*$ is different from the vertices, the fact of being $x^*$ a fixed point of $Q$ is equivalent to the fact that $x^*$ satisfies the KKT optimality conditions, and equivalent to the fact that $x^*$ is the solution of the problem (REF ).\",\n \"These properties allows us to connect the proposed algorithm with the minimization problem.\",\n \"Numerical experiments showed that the proposed algorithm found equal or better solutions than a well-known standard solver, in a practical example with 1000 random choices of vertices and weights.\",\n \"That is due to the fact that the proposed algorithm does not use of the existence of derivatives at the vertices, because the Weber function is not differentiable at the vertices.\"\n ]\n]"},"id":{"kind":"string","value":"1204.1087"}}},{"rowIdx":1866886,"cells":{"text":{"kind":"list like","value":[["p-Wave holographic superconductors with Weyl corrections"],["Abstract We study the (3+1) dimensional p-wave holographic superconductors with Weyl corrections both numerically and analytically.","We describe numerically the behavior of critical temperature $T_{c}$ with respect to charge density $\\rho$ in a limited range of Weyl coupling parameter $\\gamma$ and we find in general the condensation becomes harder with the increase of parameter $\\gamma$.","In strong coupling limit of Yang-Mills theory, we show that the minimum value of $T_{c}$ obtained from analytical approach is in good agreement with the numerical results, and finally show how we got remarkably a similar result in the critical exponent 1/2 of the chemical potential $\\mu$ and the order parameter$$ with the numerical curves of superconductors."],["Introduction","Anti-de Sitter/ Conformal field theory (AdS/CFT) links a d- dimensional strongly coupled conformal field theory on the boundary to a $(d+1)$ - dimensional weakly coupled dual gravitational description in the bulk[1], [2], [3].","It is necessary to couple a complex scalar field with an Einstein- Maxwell theory to explain the simplest model for holographic superconductors.","In a holographic superconductor, below a critical temperature, the gauge symmetry breaks and a black hole is constructed by the unstable developing scalar hair near the horizon.","According to the AdS/CFT correspondence, the complex scalar field is dual to a charged operator at the boundary, therefore a superconductor phase transition will be occurred by both the $U(1)$ symmetry-breaking in the gravity and a global $U(1)$ symmetry-breaking in the dual boundary theory[4].","Holographic superconductors have been properly considered in two different models , one with an Abelian-Higgs model, which is the gravity dual of an $s$ -wave superconductor with a scalar order parameter.","This model has been studied by many authors see[2], [3], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14].","The other type uses a $SU(2)$ gauge theory [15], [16], [17], [18], [19], [20], [21], [22], [23].","Hartnoll et al.","[3] investigate further this Abelian-Higgs model of superconductivity, and according to AdS/CFT correspondence construct a $s$ -wave holographic superconductor solution with a scalar order parameter displaying the phase transition process at the critical temperature $T_{c}$ below which the charge condensate form.","The Einstein-Yang-Mills (EYM) model of holographic superconductors constructed later by Gubser[24], where spontaneous symmetry-breaking solutions through a condensate of non Abelian gauge fields is presented, and also p-wave and $(p+ip)$ -wave backgrounds have been studied [15].","In this case,the CFT have a global $SU(2)$ symmetry and hence three conserved currents .","The first effort on analytic methods in this topic was the Herzog's work[25], where critical exponent and the expectation values of the dual operators was attained.","Analytical studies of superconductors have been established in two major methods: the small parameter perturbation theory as in[25], and the variational method [26], [27].","In the presented paper, we study the Weyl corrected p-wave holographic superconductor composed of a non-Abelian SU(2) gauge field( the matter sector) and a black hole background( the gravity sector) by using the variational method giving only critical temperature $T_{c}^{Min}$ .","As it has been mentioned in[28], for an Abelian gauge field and large range of the Weyl coupling value $\\frac{-1}{16} < \\gamma < \\frac{1}{24}$ , the universal relation for the critical temperature $T_{c} \\approx \\@root 3 \\of {\\rho }$ has been found.","In this paper we have explored the same validity of the critical temperature relation in a non-Abelian gauge field with Weyl correction numerically and analytically.","The organization of this paper is as follows.","In section II we reconstruct the Weyl corrected superconductor's solution of the EYM theory, which is dual to a p-wave superconductor.","In section III we present numerical results for condensation and critical temperature of the holographic superconductor.","Then we investigate the behavior of critical temperature $T_{c}$ and dual of chemical potential $\\mu $ with respect to Weyl coupling parameter $\\gamma $ , dual of charge density $\\rho $ , and order parameter $\\langle J_{x}^{1} \\rangle $ analytically in section IV.","The conclusion and some discussion are given in section V."],[" Weyl corrected P-wave superconductors","In this section we study the holographic phase transition for the probe $SU(2)$ Yang-Mills (YM) field $A^{a}_{\\mu }$ in a five dimensional space-time.","The bulk action of the Weyl gravity with an $SU(2)$ Yang-Mills in a five dimensional spacetime is: $S=\\int dt d^{4}x\\sqrt{-g}\\lbrace \\frac{1}{16\\pi G_{5}}(R+12)-\\frac{1}{4g^2}F^{a}_{\\mu \\nu }F^{a\\mu \\nu }+\\gamma C^{\\mu \\nu \\rho \\sigma }F^{a}_{\\mu \\nu }F^{a}_{\\rho \\sigma }\\rbrace $ Here $G_{5}$ is the five dimensional gravitational constant, $g$ is the Yang-Mills coupling constant, and the negative cosmological constant is satisfied by the factor $\\frac{12}{l^2},\\ \\ l=1$ in the first parenthesis.","The field strength component is given as below, where $A_{\\mu }^{a}$ 's are the non-Abelian gauge fields.","$F^{a}_{\\mu \\nu }=\\partial _{\\mu }A^{a}_{\\nu }-\\partial _{\\nu }A^{a}_{\\mu }+\\varepsilon ^{abc}A^{b}_{\\mu }A^{c}_{\\nu },\\lbrace a=1,2,3,\\mu =0,1,2,3\\rbrace $ The Weyl's coupling $\\gamma $ is limited such that its value is in the interval $-\\frac{1}{16}<\\gamma <\\frac{1}{24}$ , (for more precise details see [29]).","In the probe limit by neglecting the back reactions , the gravity sector is effectively decoupled from the matter field's sector.","In this probe limit, the background metric is given by an AdS-Schwarzschild black hole: $ds^2=r^2(-fdt^2+dx^idx_i)+\\frac{dr^2}{r^2f}$ where: $f=1-(\\frac{r_{+}}{r})^4$ The black hole horizon is $r=r_{+}$ .","The Smarr-Bekenstein-Hawking temperature of the black hole is determined by the Schwarzschild radius as $T=\\frac{r_{+}}{\\pi }$ .","This is the same as the temperature of the conformal field theory on the boundary of the AdS spacetime.","Applying the Euler-Lagrange equation, we can derive the generalized Yang-Mills equation as[28]: $\\nabla _{\\mu }\\left( F^{a\\mu \\nu } - 4\\gamma C^{\\mu \\nu \\rho \\sigma }F^{a}_{\\rho \\sigma } \\right) =-\\epsilon ^{a}_{bc}A^{b}_{\\mu }F^{c\\mu \\nu }+ 4\\gamma C^{\\mu \\nu \\rho \\sigma }\\epsilon ^{a}_{bc}A^{b}_{\\mu }F^{c}_{\\rho \\sigma }$ where $C_{\\mu \\nu \\rho \\sigma }$ is the Weyl tensor and has the following nonzero components in $AdS^{5}$ : $ C_{0i0j}=f(r)r_{+}^{4} \\delta _{ij},~~C_{0r0r}=-\\frac{3 r_{+}^{4}}{ r^{4}},~~ C_{irjr}=-\\frac{r_{+}^{4}}{r^{4} f(r)} \\delta _{ij},~~ C_{ijkl}=r_{+}^{4} \\delta _{ik}\\delta _{jl}.$ For realization of a holographic p-wave superconductor we take the following anstaz for Yang-Mills gauge field [30]: $A=\\varphi (r)\\sigma ^{3}dt+\\psi (r)\\sigma ^1 dx$ ($\\sigma ^{i}$ Pauli's matrixes).","The condensation of $\\psi (r)$ breaks the $SU(2)$ symmetry and the final state is the superconductor phase transition.","The gauge function $\\psi (r)$ is dual to the $J^1 _x$ operator on the boundary, choosing $x$ axis as a special direction, the condensation phase of $\\psi (r)$ breaks the symmetry and leads to a phase transition, which can be interpreted as a $p$ -wave superconductor phase transition on the boundary.","The resulting Yang-Mills equations for metric (3) are given by: $ \\left( 1 - \\frac{24 \\gamma r_{+}^{4}}{r^{4}}\\right)\\varphi ^{\\prime \\prime } + \\left( \\frac{3}{r} + \\frac{24 \\gamma r_{+}^{4}}{r^{5}} \\right)\\varphi ^{\\prime }- \\left( 1 + \\frac{8 \\gamma r_{+}^{4}}{r^{4}} \\right)\\frac{\\psi ^{2}\\varphi }{r^{4}f}=0\\\\ \\left( 1 - \\frac{8 \\gamma r_{+}^{4}}{r^{4}}\\right)\\psi ^{\\prime \\prime } + \\left[ \\frac{3}{r} + \\frac{f^{\\prime }}{f} - \\frac{8 \\gamma r_{+}^{4}}{r^{4}} \\left( -\\frac{1}{r} + \\frac{f^{\\prime }}{f} \\right) \\right]\\psi ^{\\prime } + \\left( 1 + \\frac{8 \\gamma r_{+}^{4}}{r^{4}} \\right)\\frac{\\varphi ^{2}\\psi }{r^{4}f^{2}}=0$ where the prime denotes derivative with respect to $r$ .","It is more conveint to work in terms of the dimensionless parameter $z=\\frac{r_{+}}{r}$ , in which at the horizon $z=1$ , and the boundary at the infinity locates at $z=0$ .","Then the the equations of motion (REF ) and () can be reexpressed as: $ \\left( 1 - 24 \\gamma z^{4} \\right)\\varphi ^{\\prime \\prime } -\\frac{1}{z} \\left( 1 + 72 \\gamma z^{4} \\right)\\varphi ^{\\prime }- \\left( 1 + 8 \\gamma z^{4} \\right)\\frac{\\psi ^{2}\\varphi }{f}=0\\\\ \\left( 1 - 8 \\gamma z^{4} \\right)\\psi ^{\\prime \\prime } + \\left[-\\frac{1}{z} + \\frac{f^{\\prime }}{f} - 8 \\gamma z^{4} \\left( \\frac{3}{z} +\\frac{f^{\\prime }}{f} \\right) \\right] \\psi ^{\\prime } + \\left( 1 + 8 \\gamma z^{4}\\right) \\frac{\\varphi ^{2}\\psi }{f^{2}}=0$ where the prime now denotes derivative with respect to $z$ .","The boundary conditions at infinity, i.e.","$z\\rightarrow 0 $ , are: $\\varphi \\simeq \\mu -\\rho z^2\\\\\\psi \\simeq \\psi ^{(0)}+\\psi ^{(2)}z^2$ $\\mu $ and $\\rho $ are dual to the chemical potential and charge density of the CFT boundary, $\\psi ^{(0)}$ and $\\psi ^{(2)}$ are dual to the source term and expectation value of the boundary operator $J^1_x$ respectively.","Further to have a normalizable solution, we always set the source $\\psi ^{(0)}$ to zero.","Figure: The condensation as a function oftemperature for the operators .","γ=-0.06,-0.04,-0.02,0,0.02,0.04\\gamma =-0.06, -0.04,-0.02, 0, 0.02, 0.04 from top to bottom and the dotted line is justthe case γ=0\\gamma =0.Table: The critical temperature T c T_{c} for differentvalues of Weyl coupling parameter γ\\gamma (Numerical results)."],["numerical treatment","In this section we will present numerical results for the condensation and critical temperature due to the shooting method.","From EOM's (REF ) and () and the asymptotic behavior of $\\psi $ and $\\varphi $ at infinity (12) and (13), we can obtain the regularity condition as: $\\varphi (1)=\\varphi ^{\\prime }(0)=\\psi ^{\\prime }(0)=\\psi ^{\\prime }(1)=0$ combining boundary condition (12) and (13) with regularity condition (REF ), we can solve EOM's (REF ) and () numerically by using a shooting method, and plot the FIG.1 to demonstrate the condensation as a function of temperature for the operator$< J^1 _x >$ .","The curve in FIG.1 is qualitatively similar to that obtained in the holographic superconductors[28], [32], where the condensation of $< J^1 _x >$ goes to a constant at zero temperature.","As we can see from the FIG.1, it is easy to find that the critical temperatures of Weyl corrected superconductors is increasing as the parameter $\\gamma $ varies in the range of $-0.06$ to $0.04$ , therefore we conclude that when $\\gamma < 0 $ the critical temperature is smaller and the formation of scalar hair is harder and vice versa when $\\gamma > 0$ .","We have also presented the critical temperature $T_{c}$ with different values of the parameter $\\gamma $ in the TABLE I.","According to the results in the TABLE I, we can conclude that by minimizing the coupling parameter $\\gamma $ , the critical temperature decrease smoothly ."],["Analytical treatment","In this section we compute the critical temperature and critical exponent via an analytical method, which has been proposed recently [26].","In this method by defining appropriate equation matching with field's boundary conditions, the field's EOM will be transformed to Sturm-Liouville self adjoint form.","Therefore according to the general variational method to solve the Sturm-Liouville problem ([33] or appendix of [26] ), the eigenvalue $\\lambda ^{2}$ minimizing the Sturm-Liouwille equation can be found.","Using this minimum value of $\\lambda $ , one can obtain the minimum critical temperature $T_{c}^{Min}$ .","In this section we calculate this $T_{c}^{Min}$ and discuss the critical exponent."],["Critical temperature $T_C^{Min}$","Considering the non linear system (10,11).","If there is a second order continuous phase transition at the critical temperature, the solution of the EOMs at the $T_C$ should be: $\\psi (z)=0,\\varphi (z)=\\lambda h_c(1-z^2)$ Here $\\lambda =\\frac{\\rho }{h_c^3}$ , $h_c$ is the radius of the horizon corresponding to $T=T_c$ .","At a temperature slightly below $T_c$ , the EOM for $\\psi $ becomes: $z^2\\frac{d}{dz}((\\frac{(1-z^4)}{2g^2z}+4\\gamma h_c^3z^3)\\frac{d\\psi }{dz})+\\lambda ^2[h_c^3(\\frac{z}{2g^2}+4\\gamma z^5)\\frac{1-z^4}{1+z^4}]\\psi =0$ It is appropriate to define: $\\psi (z)=\\frac{\\langle J^1 _x \\rangle }{h}z^2 F(z)$ Matching the boundary condition at the boundary $z=0$ , we normalize the function as $F(0)=1, F^{\\prime }(0)=0$ .","The equation for F(z) is: $\\frac{d}{dz}(k(z)\\frac{dF(z)}{dz})-p(z)F(z)+\\lambda ^2q(z)F(z)=0$ where: $k(z)=\\frac{z^3(1-z^4)}{2g^2 }+4\\gamma h_c^3 z^7\\\\p(z)=-2z^5(-\\frac{2}{g^2}+16\\gamma h_c^3)\\\\q(z)=h_c^3z^2(\\frac{h_cz}{2g^2}+4\\gamma z^5)\\frac{1-z^4}{1+z^4}$ The eigenvalue $\\lambda $ minimizes the expression (18) is obtained from the following functional: $\\lambda ^2=\\frac{\\int _{0}^{1}(k(z)F^{\\prime }(z)^2+p(z)F(z)^2)dz}{\\int _{0}^{1}q(z)F(z)^2dz}$ To estimate it, we use the trial function $F(z)=1-\\alpha z^2$ .","We then obtain: $\\lambda _{\\alpha }^2=\\frac{2 g^2 \\left(-1.6 h_c^3 \\alpha ^2 \\gamma +8.h_c^3 \\alpha \\gamma -5.33333 h_c^3 \\gamma +\\frac{0.533333 \\alpha ^2}{g ^2}-\\frac{\\alpha }{g ^2}+\\frac{0.666667}{g ^2}\\right)}{h_c^3 \\left(\\alpha ^2\\left(-0.21929 \\gamma g ^2-0.0151862\\right)+\\alpha \\left(0.628086\\gamma g ^2+0.052961\\right)-0.485957 \\gamma g^2-0.0568528\\right)}$ Which attains a minimum at $\\alpha =0.304936$ , and from the $\\lambda =\\frac{\\rho }{h_c^3}$ and $T_c=\\frac{h_c}{\\pi }$ the minimum value of the critical temperature can be read as: $T_c^{Min(\\pm )}=0.256926 \\@root 3 \\of {\\frac{-0.128539\\pm 0.3125 g ^2\\sqrt{\\frac{1.90164\\gamma g ^2 \\rho ^2 \\left(1.00743 \\gamma g^2+0.134769\\right)+0.169187}{g ^4}}}{\\gamma g ^2}}$ We know that in strong-coupling regime of the YM theory, the quantities can be expanded in series of $\\frac{1}{g}$ [34].","Since for some values of $g$ , we may be have $T_{c}^{Min(-)}<0$ , therefore only the $T_{c}^{Min(+)}$ is acceptable and can be read in strong limit of order $\\frac{1}{g^2}$ as: $T_{c}^{Min(+)}\\approx 0.1943040830 \\rho ^{\\frac{1}{3}}+\\frac{0.1497404(-0.128539\\gamma + 0.028931219\\gamma \\rho )}{\\gamma ^{2}\\rho ^\\frac{2}{3}g^{2}}$ In prob limit by neglecting the back reaction, the large values of the YM coupling is accessible.","Comparing equation (25) with the TABLE I we observe that the analytic value of the leading order $T_{c}^{Min}\\approx 0.1943040830 \\rho ^{\\frac{1}{3}}$ obeys the well known role $T_{c}\\propto \\rho ^{\\frac{1}{3}}$ and it is the lower bound for tabulated values of $T_{c}$ , given in TABLE I.","According to the numerical results of TABLE I, the minimum value of $T_{c}$ reads as: $T_{c(numeric)}^{Min(+)}\\approx 0.1701 \\rho ^{\\frac{1}{3}}$ which shows that the analytic values in relation (25) and the numerical estimate in equation (26) are in good agreement with each other.","It seems that there is a deep relation between the strong limit of YM part of the action and the analytical results of the p-wave superconductors ."],["Relation of $ -(\\mu -\\mu _c)$","Now we want to know the behavior of the order parameter at $T_c$ , by solving the equation for the scalar potential close to $T_c$ , therefore by substituting (13) in (10) we have: $\\frac{d}{dz}((-\\frac{1}{2z g^2}+12\\gamma z^3)\\frac{d\\varphi }{dz})+(\\frac{z}{h_c})^2(\\frac{ z}{2(1-z^4)g^2}+\\frac{4\\gamma z^5}{1-z^4 })(\\frac{}{h_c})^2F(z)^2\\varphi =0$ Since near the critical point, the order parameter $$ is small, we can expand $\\varphi $ as a series form of the small parameter as below: $\\varphi =\\mu _c+\\chi (z)+ ...$ The boundary condition imposes that $\\chi (z)$ to be $\\chi (1)= 0$ .The EOM for $\\chi (z)$ can be obtained as below: $\\frac{d}{dz}((-\\frac{1}{2z g^2}+12\\gamma z^3)\\frac{d\\chi (z)}{dz})+(\\frac{z}{h_c})^2(\\frac{ z}{2(1-z^4)g^2}+\\frac{4\\gamma z^5}{1-z^4 })\\frac{J^1 _x}{h_c^2}F(z)^2\\mu _c=0$ By integration from both sides of the (29), the EOM for $\\chi (z)$ can be reduced to: $(-\\frac{1}{2z\\zeta ^2}+12\\gamma z^3)\\frac{d\\chi (z)}{dz}=-\\mu _c\\frac{}{h_c^4}\\int z^2(\\frac{z}{2(1-z^4)g^2 }+\\frac{4\\gamma z^5}{1-z^4 })F(z)^2 dz$ By regularity condition, we have $\\chi ^{\\prime }(0)=0$ , so we take $F(z)=(1-z^2)(1-\\alpha z^2)$ .","Now $\\varphi (z)$ can be expanded as: $\\varphi (z)\\sim \\mu -\\rho z^2\\approx \\mu _c+(\\chi (0)+\\chi ^{\\prime }(0)z+...)$ Now from (31), by comparing the coefficients of $z^0$ term in both sides of the above formula, we obtain: $\\mu -\\mu _c\\approx \\frac{()^2 \\mu _c}{34560 h^4 \\gamma ^2 g^4}(3003.22 \\gamma ^{3/2} g ^3 \\left(8 \\gamma g ^2+1\\right)\\text{Li}_{2}\\left(\\frac{12.","\\sqrt{\\gamma } g }{12.\\sqrt{\\gamma } g -2.44949}\\right)+(9047.79+6359.31 ) \\gamma ^2 g ^4)$ Where $\\alpha =0.304936$ is a parameter minimizing the equation (23), and $\\text{Li}_{2}(z)$ gives the polylogarithm function.","This critical exponent $\\frac{1}{2}$ for the condensation value and $(\\mu -\\mu _c)$ qualitatively match the numerical curves for superconductors with Weyl corrections[28]."],["conclusions","In this letter we have investigated the Weyl corrected p-wave Holographic superconductors at the probe limit using numerical and analytical solutions.","We obtained the behavior of the critical temperature $T_{c}$ as a function of the dual charge density $\\rho $ in different values of Weyl coupling parameter $\\frac{-1}{16}< \\gamma <\\frac{1}{24}$ .","We have found that the critical temperature increases by growing the Weyl coupling parameter, therefore the condensation becomes harder when $\\gamma < 0 $ , and vice versa when $\\gamma > 0$ .","As a final point, obtaining the critical exponent $\\frac{1}{2}$ for the chemical potential $\\mu $ and order parameter$\\langle J^1 _x \\rangle $ show a good agreement with the numerical curves of Weyl corrected s-wave holographic superconductors.","Furthermore, we have shown that in the strong limit of YM theory, the analytical and numerical values of the minimum value of the critical temperature are in good agreement."],["acknowledgement","The authors would like to thank Jian-Pin Wu, from Beijing Normal University (China), for helpful suggestion and recommending useful references, and also we truthfully claim that without his valuable numerical codes and results the substantial improvements of this presentation and outcomes would not have been achieved.","Besides, we thank the referee for good observations and kind guidelines.","NM would like to acknowledge the financial support of University of Tehran for this research under grant No.","02/1/28450."]],"string":"[\n [\n \"p-Wave holographic superconductors with Weyl corrections\"\n ],\n [\n \"Abstract We study the (3+1) dimensional p-wave holographic superconductors with Weyl corrections both numerically and analytically.\",\n \"We describe numerically the behavior of critical temperature $T_{c}$ with respect to charge density $\\\\rho$ in a limited range of Weyl coupling parameter $\\\\gamma$ and we find in general the condensation becomes harder with the increase of parameter $\\\\gamma$.\",\n \"In strong coupling limit of Yang-Mills theory, we show that the minimum value of $T_{c}$ obtained from analytical approach is in good agreement with the numerical results, and finally show how we got remarkably a similar result in the critical exponent 1/2 of the chemical potential $\\\\mu$ and the order parameter$$ with the numerical curves of superconductors.\"\n ],\n [\n \"Introduction\",\n \"Anti-de Sitter/ Conformal field theory (AdS/CFT) links a d- dimensional strongly coupled conformal field theory on the boundary to a $(d+1)$ - dimensional weakly coupled dual gravitational description in the bulk[1], [2], [3].\",\n \"It is necessary to couple a complex scalar field with an Einstein- Maxwell theory to explain the simplest model for holographic superconductors.\",\n \"In a holographic superconductor, below a critical temperature, the gauge symmetry breaks and a black hole is constructed by the unstable developing scalar hair near the horizon.\",\n \"According to the AdS/CFT correspondence, the complex scalar field is dual to a charged operator at the boundary, therefore a superconductor phase transition will be occurred by both the $U(1)$ symmetry-breaking in the gravity and a global $U(1)$ symmetry-breaking in the dual boundary theory[4].\",\n \"Holographic superconductors have been properly considered in two different models , one with an Abelian-Higgs model, which is the gravity dual of an $s$ -wave superconductor with a scalar order parameter.\",\n \"This model has been studied by many authors see[2], [3], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14].\",\n \"The other type uses a $SU(2)$ gauge theory [15], [16], [17], [18], [19], [20], [21], [22], [23].\",\n \"Hartnoll et al.\",\n \"[3] investigate further this Abelian-Higgs model of superconductivity, and according to AdS/CFT correspondence construct a $s$ -wave holographic superconductor solution with a scalar order parameter displaying the phase transition process at the critical temperature $T_{c}$ below which the charge condensate form.\",\n \"The Einstein-Yang-Mills (EYM) model of holographic superconductors constructed later by Gubser[24], where spontaneous symmetry-breaking solutions through a condensate of non Abelian gauge fields is presented, and also p-wave and $(p+ip)$ -wave backgrounds have been studied [15].\",\n \"In this case,the CFT have a global $SU(2)$ symmetry and hence three conserved currents .\",\n \"The first effort on analytic methods in this topic was the Herzog's work[25], where critical exponent and the expectation values of the dual operators was attained.\",\n \"Analytical studies of superconductors have been established in two major methods: the small parameter perturbation theory as in[25], and the variational method [26], [27].\",\n \"In the presented paper, we study the Weyl corrected p-wave holographic superconductor composed of a non-Abelian SU(2) gauge field( the matter sector) and a black hole background( the gravity sector) by using the variational method giving only critical temperature $T_{c}^{Min}$ .\",\n \"As it has been mentioned in[28], for an Abelian gauge field and large range of the Weyl coupling value $\\\\frac{-1}{16} < \\\\gamma < \\\\frac{1}{24}$ , the universal relation for the critical temperature $T_{c} \\\\approx \\\\@root 3 \\\\of {\\\\rho }$ has been found.\",\n \"In this paper we have explored the same validity of the critical temperature relation in a non-Abelian gauge field with Weyl correction numerically and analytically.\",\n \"The organization of this paper is as follows.\",\n \"In section II we reconstruct the Weyl corrected superconductor's solution of the EYM theory, which is dual to a p-wave superconductor.\",\n \"In section III we present numerical results for condensation and critical temperature of the holographic superconductor.\",\n \"Then we investigate the behavior of critical temperature $T_{c}$ and dual of chemical potential $\\\\mu $ with respect to Weyl coupling parameter $\\\\gamma $ , dual of charge density $\\\\rho $ , and order parameter $\\\\langle J_{x}^{1} \\\\rangle $ analytically in section IV.\",\n \"The conclusion and some discussion are given in section V.\"\n ],\n [\n \" Weyl corrected P-wave superconductors\",\n \"In this section we study the holographic phase transition for the probe $SU(2)$ Yang-Mills (YM) field $A^{a}_{\\\\mu }$ in a five dimensional space-time.\",\n \"The bulk action of the Weyl gravity with an $SU(2)$ Yang-Mills in a five dimensional spacetime is: $S=\\\\int dt d^{4}x\\\\sqrt{-g}\\\\lbrace \\\\frac{1}{16\\\\pi G_{5}}(R+12)-\\\\frac{1}{4g^2}F^{a}_{\\\\mu \\\\nu }F^{a\\\\mu \\\\nu }+\\\\gamma C^{\\\\mu \\\\nu \\\\rho \\\\sigma }F^{a}_{\\\\mu \\\\nu }F^{a}_{\\\\rho \\\\sigma }\\\\rbrace $ Here $G_{5}$ is the five dimensional gravitational constant, $g$ is the Yang-Mills coupling constant, and the negative cosmological constant is satisfied by the factor $\\\\frac{12}{l^2},\\\\ \\\\ l=1$ in the first parenthesis.\",\n \"The field strength component is given as below, where $A_{\\\\mu }^{a}$ 's are the non-Abelian gauge fields.\",\n \"$F^{a}_{\\\\mu \\\\nu }=\\\\partial _{\\\\mu }A^{a}_{\\\\nu }-\\\\partial _{\\\\nu }A^{a}_{\\\\mu }+\\\\varepsilon ^{abc}A^{b}_{\\\\mu }A^{c}_{\\\\nu },\\\\lbrace a=1,2,3,\\\\mu =0,1,2,3\\\\rbrace $ The Weyl's coupling $\\\\gamma $ is limited such that its value is in the interval $-\\\\frac{1}{16}<\\\\gamma <\\\\frac{1}{24}$ , (for more precise details see [29]).\",\n \"In the probe limit by neglecting the back reactions , the gravity sector is effectively decoupled from the matter field's sector.\",\n \"In this probe limit, the background metric is given by an AdS-Schwarzschild black hole: $ds^2=r^2(-fdt^2+dx^idx_i)+\\\\frac{dr^2}{r^2f}$ where: $f=1-(\\\\frac{r_{+}}{r})^4$ The black hole horizon is $r=r_{+}$ .\",\n \"The Smarr-Bekenstein-Hawking temperature of the black hole is determined by the Schwarzschild radius as $T=\\\\frac{r_{+}}{\\\\pi }$ .\",\n \"This is the same as the temperature of the conformal field theory on the boundary of the AdS spacetime.\",\n \"Applying the Euler-Lagrange equation, we can derive the generalized Yang-Mills equation as[28]: $\\\\nabla _{\\\\mu }\\\\left( F^{a\\\\mu \\\\nu } - 4\\\\gamma C^{\\\\mu \\\\nu \\\\rho \\\\sigma }F^{a}_{\\\\rho \\\\sigma } \\\\right) =-\\\\epsilon ^{a}_{bc}A^{b}_{\\\\mu }F^{c\\\\mu \\\\nu }+ 4\\\\gamma C^{\\\\mu \\\\nu \\\\rho \\\\sigma }\\\\epsilon ^{a}_{bc}A^{b}_{\\\\mu }F^{c}_{\\\\rho \\\\sigma }$ where $C_{\\\\mu \\\\nu \\\\rho \\\\sigma }$ is the Weyl tensor and has the following nonzero components in $AdS^{5}$ : $ C_{0i0j}=f(r)r_{+}^{4} \\\\delta _{ij},~~C_{0r0r}=-\\\\frac{3 r_{+}^{4}}{ r^{4}},~~ C_{irjr}=-\\\\frac{r_{+}^{4}}{r^{4} f(r)} \\\\delta _{ij},~~ C_{ijkl}=r_{+}^{4} \\\\delta _{ik}\\\\delta _{jl}.$ For realization of a holographic p-wave superconductor we take the following anstaz for Yang-Mills gauge field [30]: $A=\\\\varphi (r)\\\\sigma ^{3}dt+\\\\psi (r)\\\\sigma ^1 dx$ ($\\\\sigma ^{i}$ Pauli's matrixes).\",\n \"The condensation of $\\\\psi (r)$ breaks the $SU(2)$ symmetry and the final state is the superconductor phase transition.\",\n \"The gauge function $\\\\psi (r)$ is dual to the $J^1 _x$ operator on the boundary, choosing $x$ axis as a special direction, the condensation phase of $\\\\psi (r)$ breaks the symmetry and leads to a phase transition, which can be interpreted as a $p$ -wave superconductor phase transition on the boundary.\",\n \"The resulting Yang-Mills equations for metric (3) are given by: $ \\\\left( 1 - \\\\frac{24 \\\\gamma r_{+}^{4}}{r^{4}}\\\\right)\\\\varphi ^{\\\\prime \\\\prime } + \\\\left( \\\\frac{3}{r} + \\\\frac{24 \\\\gamma r_{+}^{4}}{r^{5}} \\\\right)\\\\varphi ^{\\\\prime }- \\\\left( 1 + \\\\frac{8 \\\\gamma r_{+}^{4}}{r^{4}} \\\\right)\\\\frac{\\\\psi ^{2}\\\\varphi }{r^{4}f}=0\\\\\\\\ \\\\left( 1 - \\\\frac{8 \\\\gamma r_{+}^{4}}{r^{4}}\\\\right)\\\\psi ^{\\\\prime \\\\prime } + \\\\left[ \\\\frac{3}{r} + \\\\frac{f^{\\\\prime }}{f} - \\\\frac{8 \\\\gamma r_{+}^{4}}{r^{4}} \\\\left( -\\\\frac{1}{r} + \\\\frac{f^{\\\\prime }}{f} \\\\right) \\\\right]\\\\psi ^{\\\\prime } + \\\\left( 1 + \\\\frac{8 \\\\gamma r_{+}^{4}}{r^{4}} \\\\right)\\\\frac{\\\\varphi ^{2}\\\\psi }{r^{4}f^{2}}=0$ where the prime denotes derivative with respect to $r$ .\",\n \"It is more conveint to work in terms of the dimensionless parameter $z=\\\\frac{r_{+}}{r}$ , in which at the horizon $z=1$ , and the boundary at the infinity locates at $z=0$ .\",\n \"Then the the equations of motion (REF ) and () can be reexpressed as: $ \\\\left( 1 - 24 \\\\gamma z^{4} \\\\right)\\\\varphi ^{\\\\prime \\\\prime } -\\\\frac{1}{z} \\\\left( 1 + 72 \\\\gamma z^{4} \\\\right)\\\\varphi ^{\\\\prime }- \\\\left( 1 + 8 \\\\gamma z^{4} \\\\right)\\\\frac{\\\\psi ^{2}\\\\varphi }{f}=0\\\\\\\\ \\\\left( 1 - 8 \\\\gamma z^{4} \\\\right)\\\\psi ^{\\\\prime \\\\prime } + \\\\left[-\\\\frac{1}{z} + \\\\frac{f^{\\\\prime }}{f} - 8 \\\\gamma z^{4} \\\\left( \\\\frac{3}{z} +\\\\frac{f^{\\\\prime }}{f} \\\\right) \\\\right] \\\\psi ^{\\\\prime } + \\\\left( 1 + 8 \\\\gamma z^{4}\\\\right) \\\\frac{\\\\varphi ^{2}\\\\psi }{f^{2}}=0$ where the prime now denotes derivative with respect to $z$ .\",\n \"The boundary conditions at infinity, i.e.\",\n \"$z\\\\rightarrow 0 $ , are: $\\\\varphi \\\\simeq \\\\mu -\\\\rho z^2\\\\\\\\\\\\psi \\\\simeq \\\\psi ^{(0)}+\\\\psi ^{(2)}z^2$ $\\\\mu $ and $\\\\rho $ are dual to the chemical potential and charge density of the CFT boundary, $\\\\psi ^{(0)}$ and $\\\\psi ^{(2)}$ are dual to the source term and expectation value of the boundary operator $J^1_x$ respectively.\",\n \"Further to have a normalizable solution, we always set the source $\\\\psi ^{(0)}$ to zero.\",\n \"Figure: The condensation as a function oftemperature for the operators .\",\n \"γ=-0.06,-0.04,-0.02,0,0.02,0.04\\\\gamma =-0.06, -0.04,-0.02, 0, 0.02, 0.04 from top to bottom and the dotted line is justthe case γ=0\\\\gamma =0.Table: The critical temperature T c T_{c} for differentvalues of Weyl coupling parameter γ\\\\gamma (Numerical results).\"\n ],\n [\n \"numerical treatment\",\n \"In this section we will present numerical results for the condensation and critical temperature due to the shooting method.\",\n \"From EOM's (REF ) and () and the asymptotic behavior of $\\\\psi $ and $\\\\varphi $ at infinity (12) and (13), we can obtain the regularity condition as: $\\\\varphi (1)=\\\\varphi ^{\\\\prime }(0)=\\\\psi ^{\\\\prime }(0)=\\\\psi ^{\\\\prime }(1)=0$ combining boundary condition (12) and (13) with regularity condition (REF ), we can solve EOM's (REF ) and () numerically by using a shooting method, and plot the FIG.1 to demonstrate the condensation as a function of temperature for the operator$< J^1 _x >$ .\",\n \"The curve in FIG.1 is qualitatively similar to that obtained in the holographic superconductors[28], [32], where the condensation of $< J^1 _x >$ goes to a constant at zero temperature.\",\n \"As we can see from the FIG.1, it is easy to find that the critical temperatures of Weyl corrected superconductors is increasing as the parameter $\\\\gamma $ varies in the range of $-0.06$ to $0.04$ , therefore we conclude that when $\\\\gamma < 0 $ the critical temperature is smaller and the formation of scalar hair is harder and vice versa when $\\\\gamma > 0$ .\",\n \"We have also presented the critical temperature $T_{c}$ with different values of the parameter $\\\\gamma $ in the TABLE I.\",\n \"According to the results in the TABLE I, we can conclude that by minimizing the coupling parameter $\\\\gamma $ , the critical temperature decrease smoothly .\"\n ],\n [\n \"Analytical treatment\",\n \"In this section we compute the critical temperature and critical exponent via an analytical method, which has been proposed recently [26].\",\n \"In this method by defining appropriate equation matching with field's boundary conditions, the field's EOM will be transformed to Sturm-Liouville self adjoint form.\",\n \"Therefore according to the general variational method to solve the Sturm-Liouville problem ([33] or appendix of [26] ), the eigenvalue $\\\\lambda ^{2}$ minimizing the Sturm-Liouwille equation can be found.\",\n \"Using this minimum value of $\\\\lambda $ , one can obtain the minimum critical temperature $T_{c}^{Min}$ .\",\n \"In this section we calculate this $T_{c}^{Min}$ and discuss the critical exponent.\"\n ],\n [\n \"Critical temperature $T_C^{Min}$\",\n \"Considering the non linear system (10,11).\",\n \"If there is a second order continuous phase transition at the critical temperature, the solution of the EOMs at the $T_C$ should be: $\\\\psi (z)=0,\\\\varphi (z)=\\\\lambda h_c(1-z^2)$ Here $\\\\lambda =\\\\frac{\\\\rho }{h_c^3}$ , $h_c$ is the radius of the horizon corresponding to $T=T_c$ .\",\n \"At a temperature slightly below $T_c$ , the EOM for $\\\\psi $ becomes: $z^2\\\\frac{d}{dz}((\\\\frac{(1-z^4)}{2g^2z}+4\\\\gamma h_c^3z^3)\\\\frac{d\\\\psi }{dz})+\\\\lambda ^2[h_c^3(\\\\frac{z}{2g^2}+4\\\\gamma z^5)\\\\frac{1-z^4}{1+z^4}]\\\\psi =0$ It is appropriate to define: $\\\\psi (z)=\\\\frac{\\\\langle J^1 _x \\\\rangle }{h}z^2 F(z)$ Matching the boundary condition at the boundary $z=0$ , we normalize the function as $F(0)=1, F^{\\\\prime }(0)=0$ .\",\n \"The equation for F(z) is: $\\\\frac{d}{dz}(k(z)\\\\frac{dF(z)}{dz})-p(z)F(z)+\\\\lambda ^2q(z)F(z)=0$ where: $k(z)=\\\\frac{z^3(1-z^4)}{2g^2 }+4\\\\gamma h_c^3 z^7\\\\\\\\p(z)=-2z^5(-\\\\frac{2}{g^2}+16\\\\gamma h_c^3)\\\\\\\\q(z)=h_c^3z^2(\\\\frac{h_cz}{2g^2}+4\\\\gamma z^5)\\\\frac{1-z^4}{1+z^4}$ The eigenvalue $\\\\lambda $ minimizes the expression (18) is obtained from the following functional: $\\\\lambda ^2=\\\\frac{\\\\int _{0}^{1}(k(z)F^{\\\\prime }(z)^2+p(z)F(z)^2)dz}{\\\\int _{0}^{1}q(z)F(z)^2dz}$ To estimate it, we use the trial function $F(z)=1-\\\\alpha z^2$ .\",\n \"We then obtain: $\\\\lambda _{\\\\alpha }^2=\\\\frac{2 g^2 \\\\left(-1.6 h_c^3 \\\\alpha ^2 \\\\gamma +8.h_c^3 \\\\alpha \\\\gamma -5.33333 h_c^3 \\\\gamma +\\\\frac{0.533333 \\\\alpha ^2}{g ^2}-\\\\frac{\\\\alpha }{g ^2}+\\\\frac{0.666667}{g ^2}\\\\right)}{h_c^3 \\\\left(\\\\alpha ^2\\\\left(-0.21929 \\\\gamma g ^2-0.0151862\\\\right)+\\\\alpha \\\\left(0.628086\\\\gamma g ^2+0.052961\\\\right)-0.485957 \\\\gamma g^2-0.0568528\\\\right)}$ Which attains a minimum at $\\\\alpha =0.304936$ , and from the $\\\\lambda =\\\\frac{\\\\rho }{h_c^3}$ and $T_c=\\\\frac{h_c}{\\\\pi }$ the minimum value of the critical temperature can be read as: $T_c^{Min(\\\\pm )}=0.256926 \\\\@root 3 \\\\of {\\\\frac{-0.128539\\\\pm 0.3125 g ^2\\\\sqrt{\\\\frac{1.90164\\\\gamma g ^2 \\\\rho ^2 \\\\left(1.00743 \\\\gamma g^2+0.134769\\\\right)+0.169187}{g ^4}}}{\\\\gamma g ^2}}$ We know that in strong-coupling regime of the YM theory, the quantities can be expanded in series of $\\\\frac{1}{g}$ [34].\",\n \"Since for some values of $g$ , we may be have $T_{c}^{Min(-)}<0$ , therefore only the $T_{c}^{Min(+)}$ is acceptable and can be read in strong limit of order $\\\\frac{1}{g^2}$ as: $T_{c}^{Min(+)}\\\\approx 0.1943040830 \\\\rho ^{\\\\frac{1}{3}}+\\\\frac{0.1497404(-0.128539\\\\gamma + 0.028931219\\\\gamma \\\\rho )}{\\\\gamma ^{2}\\\\rho ^\\\\frac{2}{3}g^{2}}$ In prob limit by neglecting the back reaction, the large values of the YM coupling is accessible.\",\n \"Comparing equation (25) with the TABLE I we observe that the analytic value of the leading order $T_{c}^{Min}\\\\approx 0.1943040830 \\\\rho ^{\\\\frac{1}{3}}$ obeys the well known role $T_{c}\\\\propto \\\\rho ^{\\\\frac{1}{3}}$ and it is the lower bound for tabulated values of $T_{c}$ , given in TABLE I.\",\n \"According to the numerical results of TABLE I, the minimum value of $T_{c}$ reads as: $T_{c(numeric)}^{Min(+)}\\\\approx 0.1701 \\\\rho ^{\\\\frac{1}{3}}$ which shows that the analytic values in relation (25) and the numerical estimate in equation (26) are in good agreement with each other.\",\n \"It seems that there is a deep relation between the strong limit of YM part of the action and the analytical results of the p-wave superconductors .\"\n ],\n [\n \"Relation of $ -(\\\\mu -\\\\mu _c)$\",\n \"Now we want to know the behavior of the order parameter at $T_c$ , by solving the equation for the scalar potential close to $T_c$ , therefore by substituting (13) in (10) we have: $\\\\frac{d}{dz}((-\\\\frac{1}{2z g^2}+12\\\\gamma z^3)\\\\frac{d\\\\varphi }{dz})+(\\\\frac{z}{h_c})^2(\\\\frac{ z}{2(1-z^4)g^2}+\\\\frac{4\\\\gamma z^5}{1-z^4 })(\\\\frac{}{h_c})^2F(z)^2\\\\varphi =0$ Since near the critical point, the order parameter $$ is small, we can expand $\\\\varphi $ as a series form of the small parameter as below: $\\\\varphi =\\\\mu _c+\\\\chi (z)+ ...$ The boundary condition imposes that $\\\\chi (z)$ to be $\\\\chi (1)= 0$ .The EOM for $\\\\chi (z)$ can be obtained as below: $\\\\frac{d}{dz}((-\\\\frac{1}{2z g^2}+12\\\\gamma z^3)\\\\frac{d\\\\chi (z)}{dz})+(\\\\frac{z}{h_c})^2(\\\\frac{ z}{2(1-z^4)g^2}+\\\\frac{4\\\\gamma z^5}{1-z^4 })\\\\frac{J^1 _x}{h_c^2}F(z)^2\\\\mu _c=0$ By integration from both sides of the (29), the EOM for $\\\\chi (z)$ can be reduced to: $(-\\\\frac{1}{2z\\\\zeta ^2}+12\\\\gamma z^3)\\\\frac{d\\\\chi (z)}{dz}=-\\\\mu _c\\\\frac{}{h_c^4}\\\\int z^2(\\\\frac{z}{2(1-z^4)g^2 }+\\\\frac{4\\\\gamma z^5}{1-z^4 })F(z)^2 dz$ By regularity condition, we have $\\\\chi ^{\\\\prime }(0)=0$ , so we take $F(z)=(1-z^2)(1-\\\\alpha z^2)$ .\",\n \"Now $\\\\varphi (z)$ can be expanded as: $\\\\varphi (z)\\\\sim \\\\mu -\\\\rho z^2\\\\approx \\\\mu _c+(\\\\chi (0)+\\\\chi ^{\\\\prime }(0)z+...)$ Now from (31), by comparing the coefficients of $z^0$ term in both sides of the above formula, we obtain: $\\\\mu -\\\\mu _c\\\\approx \\\\frac{()^2 \\\\mu _c}{34560 h^4 \\\\gamma ^2 g^4}(3003.22 \\\\gamma ^{3/2} g ^3 \\\\left(8 \\\\gamma g ^2+1\\\\right)\\\\text{Li}_{2}\\\\left(\\\\frac{12.\",\n \"\\\\sqrt{\\\\gamma } g }{12.\\\\sqrt{\\\\gamma } g -2.44949}\\\\right)+(9047.79+6359.31 ) \\\\gamma ^2 g ^4)$ Where $\\\\alpha =0.304936$ is a parameter minimizing the equation (23), and $\\\\text{Li}_{2}(z)$ gives the polylogarithm function.\",\n \"This critical exponent $\\\\frac{1}{2}$ for the condensation value and $(\\\\mu -\\\\mu _c)$ qualitatively match the numerical curves for superconductors with Weyl corrections[28].\"\n ],\n [\n \"conclusions\",\n \"In this letter we have investigated the Weyl corrected p-wave Holographic superconductors at the probe limit using numerical and analytical solutions.\",\n \"We obtained the behavior of the critical temperature $T_{c}$ as a function of the dual charge density $\\\\rho $ in different values of Weyl coupling parameter $\\\\frac{-1}{16}< \\\\gamma <\\\\frac{1}{24}$ .\",\n \"We have found that the critical temperature increases by growing the Weyl coupling parameter, therefore the condensation becomes harder when $\\\\gamma < 0 $ , and vice versa when $\\\\gamma > 0$ .\",\n \"As a final point, obtaining the critical exponent $\\\\frac{1}{2}$ for the chemical potential $\\\\mu $ and order parameter$\\\\langle J^1 _x \\\\rangle $ show a good agreement with the numerical curves of Weyl corrected s-wave holographic superconductors.\",\n \"Furthermore, we have shown that in the strong limit of YM theory, the analytical and numerical values of the minimum value of the critical temperature are in good agreement.\"\n ],\n [\n \"acknowledgement\",\n \"The authors would like to thank Jian-Pin Wu, from Beijing Normal University (China), for helpful suggestion and recommending useful references, and also we truthfully claim that without his valuable numerical codes and results the substantial improvements of this presentation and outcomes would not have been achieved.\",\n \"Besides, we thank the referee for good observations and kind guidelines.\",\n \"NM would like to acknowledge the financial support of University of Tehran for this research under grant No.\",\n \"02/1/28450.\"\n ]\n]"},"id":{"kind":"string","value":"1204.1246"}}},{"rowIdx":1866887,"cells":{"text":{"kind":"list like","value":[["Imaging high-dimensional spatial entanglement with a camera"],["Abstract The light produced by parametric down-conversion shows strong spatial entanglement that leads to violations of EPR criteria for separability.","Historically, such studies have been performed by scanning a single-element, single-photon detector across a detection plane.","Here we show that modern electron-multiplying charge-coupled device cameras can measure correlations in both position and momentum across a multi-pixel field of view.","This capability allows us to observe entanglement of around 2,500 spatial states and demonstrate Einstein-Podolsky-Rosen type correlations by more than two orders of magnitude.","More generally, our work shows that cameras can lead to important new capabilities in quantum optics and quantum information science."],["Results","Imaging system.","The experimental setup used to measure spatial correlations in both position and momentum is shown in Fig.","REF .","A $150\\,\\rm {mW}$ , high-repetition-rate $355\\,\\rm {nm}$ laser is attenuated to $2\\,\\rm {mW}$ and using a simple telescope the Gaussian beam radius is expanded to $\\sigma _{p} = (0.66\\pm 0.05)\\rm {mm}$ .","A $5\\,\\rm {mm}$ long $\\beta $ -Barium Borate (BBO) crystal cut for type-I phase matching was angled to provide near-collinear output for degenerate down-converted photons at $710\\,{\\rm nm}$ .","The subsequent choice of lens configurations was chosen to maintain a constant separation ($700\\,\\rm {mm}$ ) between the BBO crystal and the EMCCD (Andor iXon3), thus providing an efficient system for manual switching between the image and far-field planes.","Figure: Experimental scheme used to measure position and momentum correlations.","(𝐚){\\rm ({\\bf a})} The imaging system used to measure position correlations consists of a telescope with lenses 100 mm 100\\,\\rm {mm} and 250 mm 250\\,\\rm {mm}.","(𝐛){\\rm ({\\bf b})} Momentum correlations are obtained with the implementation of three consecutive Fourier systems with lenses 50 mm 50\\,\\rm {mm}, 100 mm 100\\,\\rm {mm} and 200 mm 200\\,\\rm {mm}.As used by other authors [25], [13], we use a Gaussian model to describe the spatial structure of the two photon field at the crystal: $\\Psi (\\rho _1,\\rho _2) = N\\exp \\left[-\\frac{|\\rho _1+\\rho _2|^2}{4\\sigma _+^2}\\right]\\exp \\left[-\\frac{|\\rho _1-\\rho _2|^2}{4\\sigma _-^2}\\right],$ where $N = 1/(\\pi \\sigma _-\\sigma _+)$ is a normalization constant, $\\rho _i = (x_i,y_i)$ is the transverse position of photon $i$ ($i = 1,2$ ) and $\\sigma _{\\pm }$ are the standard deviations of the two Gaussians, giving the strength of the position and momentum correlations, $\\sigma _-$ and $\\sigma _+^{-1}$ , respectively.","We define $\\sigma _+$ to be equal to the standard deviation of the Gaussian field distribution of the pump laser, $\\sigma _{p}$ , with wavelength $\\lambda _{p}$ and $\\sigma _- = \\sqrt{\\frac{\\alpha L\\lambda _{p}}{2\\pi }},$ where $L$ is the length of the SPDC crystal and $\\alpha = 0.455$ is an adjustment constant [26].","Eq.","(REF ) is referred to as the transverse wave function of the post-selected two-photon field of SPDC for a Gaussian pump beam [27].","The joint detection probability is $\\mathcal {P}(\\rho _1,\\rho _2)\\propto |\\Psi (\\rho _1,\\rho _2)|^2$ .","According to the Gaussian model and the parameters of our system we predict the number of modes for joint detections in both position and momentum to be $\\left(\\sigma _+/\\sigma _-\\right)^2 \\approx 3500$ .","Figure: Position and momentum correlation functions in the x ' x^{\\prime } dimension for an increasing number of images.","(𝐚){\\rm ({\\bf a})} Measured position correlations from imaging plane scheme.","(𝐛){\\rm ({\\bf b})} Measured momentum correlations from far-field scheme.","Blue circles represents the spatial-correlation and red squares represent the reference-correlation.","The asymmetry of the far-field correlation function is the result of a non-uniform near collinear degenerate light cone.When measuring position correlations, we use an imaging system with a magnification of $M = 2.5$ , comprising two lenses of focal lengths $100\\,\\rm {mm}$ and $250\\,\\rm {mm}$ (Fig.","REF a).","The far-field image is obtained by using a composite Fourier system with an effective focal length of $f_{e} = 100\\,{\\rm mm}$ (Fig.","REF b).","Given our imaging and far-field system we predict a transverse correlation length $\\sigma _{{\\rm pos}} = M\\sigma _- \\approx 28\\,\\mu \\rm {m}$ and $\\sigma _{{\\rm mom}} = f_e/(k\\sigma _+) \\approx 17\\,\\mu \\rm {m}$ , where $k = k_{p}/2$ is the wavenumber of the down-converted photons.","Note that the position correlation length is larger than the pixel size of the camera ($16\\,\\mu \\rm {m}$ ) meaning that the position correlated photon pairs have a high probability of being detected in adjacent pixels which removes the need to count multiple photons within the same pixel.","A $10\\,\\rm {nm}$ bandpass filter centred at $710\\,\\rm {nm}$ with a transmission efficiency $\\eta _{{\\rm filter}} = 90\\%$ is located immediately in front of the EMCCD.","The EMCCD camera used in our experiment is a back-illuminated $512\\times 512$ array of $16\\times 16\\,\\mu \\rm {m^2}$ pixels optimized for visible wavelengths.","The quantum efficiency of detection is given as the product of the efficiency of the camera, filter and lenses.","Thus providing that either the signal or idler photon is detected, we predicted a heralding efficiency for detecting its entangled partner to be $\\eta \\approx 80\\%$ .","Image analysis.","A series of $100,000$ images was recorded for image and far-field configurations (see Methods).","The probability distributions $\\mathcal {P}(\\rho _-)$ and $\\mathcal {P}(p_+)$ were calculated respectively by counting the number of coincidences as a function of the difference (imaging plane) or sum (far-field) of the pixel coordinates in both $x^{\\prime }$ and $y^{\\prime }$ directions at the plane of the CCD array.","Here we define $\\rho _- \\equiv \\rho _1-\\rho _2$ and $p_+ \\equiv p_1 + p_2$ , where $p_i = (p_{x_i},p_{y_i})$ is the transverse momentum of photon $i$ .","The transformation from coordinates in the plane of the CCD array to the crystal is performed using the scaling factors $\\gamma _{\\rm {pos}} \\equiv 1/M$ for imaging plane and $\\gamma _{\\rm {mom}} \\equiv (k \\hbar )/f_{e}$ for far-field.","Our result is obtained by averaging the spatial correlation function over all recorded frames.","Within each linear dimension this processing reveals correlations between photon pairs, which are clearly visible on top of a broader correlation arising from uncorrelated events, as shown in Fig.","REF .","We measured the background correlation by performing the same analysis but between consecutive frames to give a reference correlation.","Fig.","REF a shows the measured coincidence counts in the image plane as a function of $x^{\\prime }_{-}$ coordinates, while coincidence counts measured in the far-field as a function of $x^{\\prime }_{+}$ are shown in Fig.","REF b.","A correlation peak in both position and momentum can already be seen in a few images and becomes significantly distinguishable above noise as more images are summed.","Figure: Background subtracted auto-correlation functions in position and momentum for high and low heralding efficiency configurations.","Auto-correlation functions for the image plane (𝐚){\\rm ({\\bf a})} and far-field (𝐛){\\rm ({\\bf b})} of the crystal, summed over 100 000 images and with the measured background correlation removed.","In (𝐜){\\rm ({\\bf c})} and (𝐝){\\rm ({\\bf d})} we show the same measurements performed with a lower heralding efficiency.","Variances along the x ' x^{\\prime }-direction of the high heralding efficiency configuration are indicated.","Due to increased coincidence counts attributed to smearing, the correlation function for y - ' =±1y^{\\prime }_{-} = \\pm 1 were set to 0 in (a){\\rm (a)} and (c){\\rm (c)} for clarity.By subtracting the reference from the spatial correlation function it is possible to remove unwanted pixel defects and obtain a strong correlation in position and anti-correlation in momentum as indicated in Fig.","REF a and Fig.","REF b respectively.","Since we cannot distinguish the detection of more than one photon on the same pixel, we are unable to measure the correlation function for ${\\rho }^{\\prime }_1 = {\\rho }^{\\prime }_2$ , for which a value equal to 0 has been assigned.","To confirm that these correlations arise between photon pairs the neutral density filter used to attenuate the pump beam was placed after the BBO crystal to reduce the heralding efficiency to approximately $2.0\\%$ , whilst ensuring the same photon flux.","As shown in Fig.","REF c and Fig.","REF d, we do not observe a strong correlation in these measurements; however along the $y^{\\prime }_{-}$ axis in the image plane measurements we observe a correlation indicative of charge smearing between pixels in the readout direction.","EPR-type correlations.","From the thresholded images we can calculate the joint probability distributions for both the $x^{\\prime }$ and $y^{\\prime }$ coordinates in the image plane and far-field, as shown in Fig.","REF .","Fig.","REF a and Fig.","REF b show the coincidence counts in the planes $(x_1,x_2)$ and $(y_1,y_2)$ , while Fig.","REF c and Fig.","REF d show the coincidence counts in the planes $(p_{x_1},p_{x_2})$ and $(p_{y_1},p_{y_2})$ .","In the $y^{\\prime }$ direction we observe a strong correlation resulting from charge smearing, thus preventing any meaningful analysis from image plane measurements (Fig.","REF b); however the expected anti-correlation in the far-field (Fig.","REF d) is still evident.","Note that background subtraction has been performed.","Figure: Joint probability distributions for x ' x^{\\prime } and y ' y^{\\prime }-coordinates.","The probability distributions for joint detections in both x ' x^{\\prime } and y ' y^{\\prime }-coordinates in the image plane (𝐚{\\rm ({\\bf a}} and 𝐛){\\rm {\\bf b})} and far-field (𝐜{\\rm ({\\bf c}} and 𝐝){\\rm {\\bf d})} are shown respectively.","Charge smearing gives rise to an artificial correlation in the y ' y^{\\prime } direction (𝐛{\\rm ({\\bf b}} and 𝐝){\\rm {\\bf d})}.It has been shown that EPR-like correlations can be identified by violating the inequality [28], [29] $\\Delta ^2_{\\rm {min}}(x_1|x_2)\\Delta ^2_{\\rm {min}}(p_{x_1}|p_{x_2})>\\frac{\\hbar ^2}{4},$ where $\\Delta ^2_{\\rm {min}}(r_1|r_2)$ is the minimum inferred variance, describing the minimum uncertainty in measuring the variable $r_1$ conditional on the measurement of variable $r_2$ .","The minimum inferred variance for $x$ -coordinates in the far-field measurement is defined as $\\Delta ^2_{\\rm {min}}(p_{x_1}|p_{x_2})=\\int \\mathcal {P}(p_{x_2})\\Delta ^2(p_{x_1}|p_{x_2})dp_{x_2}.$ We define the minimum inferred variance for the $x$ -coordinates in the image plane with a similar approach.","Fitting Gaussians to the joint probabilities in the $x$ -direction (Fig.","REF a and Fig.","REF c) we obtain the variances shown in Table REF .","Substituting these quantities into Eq.","(REF ) we find $\\Delta ^2_{\\rm {min}}(x_1|x_2)\\Delta ^2_{\\rm {min}}(p_{x_1}|p_{x_2})=&(6.6\\pm 1.0)\\times 10^{-4}\\hbar ^2,\\\\\\Delta ^2_{\\rm {min}}(x_2|x_1)\\Delta ^2_{\\rm {min}}(p_{x_2}|p_{x_1})=&(6.2\\pm 0.9)\\times 10^{-4}\\hbar ^2,$ indicating that the transverse DOF of the two photon field exhibits EPR non-locality.","This result is in good agreement with the theoretical prediction (from Eqs.","(REF ) and (REF )) of $\\approx 3\\times 10^{-4}\\hbar ^2$ and, although performed with background subtraction, is an order of magnitude smaller than those in Refs.","[4], [13].","High-dimensional entanglement.","From the joint distributions in Fig.","REF , we can estimate the ratio $(\\sigma _+/\\sigma _-)$ for both $x$ and $y$ transverse DOF, in the image plane (IP) and far-field (FF).","We define the maximum number of joint detections for our position and momentum measurements to be $D^{\\rm max}_{\\rm pos}=[(\\sigma _{x^{\\prime }_+}/\\sigma _{x^{\\prime }_-})(\\sigma _{y^{\\prime }_+}/\\sigma _{y^{\\prime }_-})]_{\\rm {IP}}$ and $D^{\\rm max}_{\\rm mom}=[(\\sigma _{x^{\\prime }_-}/\\sigma _{x^{\\prime }_+})(\\sigma _{y^{\\prime }_-}/\\sigma _{y^{\\prime }_+})]_{\\rm {FF}}$ .","We must also account for the proportion of the beam measured by the detector array which is calculated by integrating the fitted Gaussian functions over the array size.","In all cases we find this gives access to more than $85\\%$ of the available states.","Due to charge smearing in the image plane measurements we must estimate the dimensionality in $y$ based on that of the $x$ dimension, which is supported by the circular symmetry of Fig.","REF a.","Thus, the measured dimensionality in position and momentum is found to be $D_{\\rm pos} = D^{\\rm max}_{\\rm pos} \\times 0.95^2 \\approx 3200$ and $D_{\\rm mom} = D^{\\rm max}_{\\rm mom} \\times 0.95 \\times 0.85 \\approx 2500$ , respectively.","To the best of our knowledge this represents the largest dimensionality for any experiment using entangled spatial states of photons."],["Discussion","We have performed a multimode detection of around 2500 spatially entangled states of photon pairs produced by SPDC.","We have shown that it is possible to utilize an EMCCD camera to measure spatial correlations between photon pairs in both the image plane and far-field of the down-conversion crystal.","After background subtraction, we found that the spatial correlations violate EPR criterion.","We acknowledge that the use of background-subtracted measurements do not meet the strict requirements for tests of EPR non-locality.","If the background was included in the data, it would lead to an overestimate of the standard deviations of the correlations.","Furthermore, in order to satisfy non-locality, the measurement of each photon must be performed outside the light-cone of its entangled partner.","This issue could be solved in a modified setup in which the two-photons are split and sent to two spatially separated cameras.","However, our results are in close agreement with the theoretical predictions, providing evidence that EMCCD technology can already be used in the characterization of spatial correlations of photons.","The ability to access both transverse DOF of photons enables a large state space suggesting the use of spatial states in quantum communication and quantum information processing.","In this case, information should be encoded and processed in an alphabet defined as orthogonal spatial states and subsequently read by projecting the full transverse field of the single photon on to a complete set of orthogonal spatial states.","Thus, low-noise and efficient 2D detector arrays are essential in the use of spatial states in these applications, for which our work is a significant step forward.","Future developments in imaging technologies will likely result in 2D detector arrays with even lower noise and widen their application in quantum optics.","Table: Variances obtained from Gaussian fittings of background subtracted correlation functions shown Fig.","."],["Methods","Thresholding.","There are several noise sources that affect an EMCCD camera.","As well as stray photons, the camera is subject to dark noise; thermally excited electrons, clock induced noise resulting from spurious electrons created during charge transfers between pixels, and readout noise generated by the readout register.","We characterized the level of total noise in our camera when operated at the maximum gain by obtaining $100,000$ images while the laser is blocked but with the shutter on the camera open.","Performing an analysis similar to that of Ref.","[19] we find a Gaussian distribution of readout noise, centred at a value $390\\,e^-$ with a standard deviation $\\sigma _{\\rm noise} = 6\\,e^-$ , in addition to a long tail that decreases exponentially for higher readout values.","Our thresholding is performed by subtracting the mean background value for each pixel and assigning a 1 to pixels with values greater than one standard deviation of noise, and a 0 to all pixels having smaller values.","This binary thresholding was found to reduce the level of noise whilst maximizing the correlation signal strength.","Camera Settings.","The camera was operated at $-85^0\\rm {C}$ , with a horizontal pixel shift readout rate of $1\\,\\rm {MHz}$ , a vertical pixel shift every $0.3\\,\\mu \\rm {s}$ and a vertical clock amplitude voltage of $+4$ above the default factory setting.","The optimum photon flux incident upon the camera was chosen to maximize the strength of the correlation signal relative to noise sources.","In our experiment the image-plane measurements were made with an exposure time of $0.4\\,\\rm {ms}$ , while the far-field measurements are obtained using an exposure time of $1\\,\\rm {ms}$ , in both cases giving an average photon flux of approximately 0.02 photons/pixel, which is equivalent to the level of noise.","A region of interest measuring $201 \\times 201$ pixels was chosen to observe most of the down-converted field, which resulted in a frame rate of approximately $5\\,{\\rm Hz}$ .","Acknowledgements M.J.P.","would like to thank the Royal Society and the Wolfson Foundation.","M.J.P.","and R.W.B.","thank the US DARPA/DSO InPho program.","M.P.E., D.S.T., F.I., R.E.W., G.S.B.","and M.J.P.","acknowledge the financial support from the UK EPSRC.","J.L., M.A., and R.W.B thank Canada's CERC program.","The authors would like to thank N. Radwell for useful discussions.","Author contributions M.J.P., R.W.B., G.S.B.","and J.L.","conceived the experiment.","M.P.E., D.S.T., F.I.","and R.E.W.","designed and performed the experiment.","M.P.E., D.S.T., F.I., J.L., M.A.","and M.J.P.","designed the measurement and analysis programs.","M.P.E., D.S.T., F.I., J.L.","and M.J.P.","analysed the results.","All authors contributed to the writing of the manuscript."]],"string":"[\n [\n \"Imaging high-dimensional spatial entanglement with a camera\"\n ],\n [\n \"Abstract The light produced by parametric down-conversion shows strong spatial entanglement that leads to violations of EPR criteria for separability.\",\n \"Historically, such studies have been performed by scanning a single-element, single-photon detector across a detection plane.\",\n \"Here we show that modern electron-multiplying charge-coupled device cameras can measure correlations in both position and momentum across a multi-pixel field of view.\",\n \"This capability allows us to observe entanglement of around 2,500 spatial states and demonstrate Einstein-Podolsky-Rosen type correlations by more than two orders of magnitude.\",\n \"More generally, our work shows that cameras can lead to important new capabilities in quantum optics and quantum information science.\"\n ],\n [\n \"Results\",\n \"Imaging system.\",\n \"The experimental setup used to measure spatial correlations in both position and momentum is shown in Fig.\",\n \"REF .\",\n \"A $150\\\\,\\\\rm {mW}$ , high-repetition-rate $355\\\\,\\\\rm {nm}$ laser is attenuated to $2\\\\,\\\\rm {mW}$ and using a simple telescope the Gaussian beam radius is expanded to $\\\\sigma _{p} = (0.66\\\\pm 0.05)\\\\rm {mm}$ .\",\n \"A $5\\\\,\\\\rm {mm}$ long $\\\\beta $ -Barium Borate (BBO) crystal cut for type-I phase matching was angled to provide near-collinear output for degenerate down-converted photons at $710\\\\,{\\\\rm nm}$ .\",\n \"The subsequent choice of lens configurations was chosen to maintain a constant separation ($700\\\\,\\\\rm {mm}$ ) between the BBO crystal and the EMCCD (Andor iXon3), thus providing an efficient system for manual switching between the image and far-field planes.\",\n \"Figure: Experimental scheme used to measure position and momentum correlations.\",\n \"(𝐚){\\\\rm ({\\\\bf a})} The imaging system used to measure position correlations consists of a telescope with lenses 100 mm 100\\\\,\\\\rm {mm} and 250 mm 250\\\\,\\\\rm {mm}.\",\n \"(𝐛){\\\\rm ({\\\\bf b})} Momentum correlations are obtained with the implementation of three consecutive Fourier systems with lenses 50 mm 50\\\\,\\\\rm {mm}, 100 mm 100\\\\,\\\\rm {mm} and 200 mm 200\\\\,\\\\rm {mm}.As used by other authors [25], [13], we use a Gaussian model to describe the spatial structure of the two photon field at the crystal: $\\\\Psi (\\\\rho _1,\\\\rho _2) = N\\\\exp \\\\left[-\\\\frac{|\\\\rho _1+\\\\rho _2|^2}{4\\\\sigma _+^2}\\\\right]\\\\exp \\\\left[-\\\\frac{|\\\\rho _1-\\\\rho _2|^2}{4\\\\sigma _-^2}\\\\right],$ where $N = 1/(\\\\pi \\\\sigma _-\\\\sigma _+)$ is a normalization constant, $\\\\rho _i = (x_i,y_i)$ is the transverse position of photon $i$ ($i = 1,2$ ) and $\\\\sigma _{\\\\pm }$ are the standard deviations of the two Gaussians, giving the strength of the position and momentum correlations, $\\\\sigma _-$ and $\\\\sigma _+^{-1}$ , respectively.\",\n \"We define $\\\\sigma _+$ to be equal to the standard deviation of the Gaussian field distribution of the pump laser, $\\\\sigma _{p}$ , with wavelength $\\\\lambda _{p}$ and $\\\\sigma _- = \\\\sqrt{\\\\frac{\\\\alpha L\\\\lambda _{p}}{2\\\\pi }},$ where $L$ is the length of the SPDC crystal and $\\\\alpha = 0.455$ is an adjustment constant [26].\",\n \"Eq.\",\n \"(REF ) is referred to as the transverse wave function of the post-selected two-photon field of SPDC for a Gaussian pump beam [27].\",\n \"The joint detection probability is $\\\\mathcal {P}(\\\\rho _1,\\\\rho _2)\\\\propto |\\\\Psi (\\\\rho _1,\\\\rho _2)|^2$ .\",\n \"According to the Gaussian model and the parameters of our system we predict the number of modes for joint detections in both position and momentum to be $\\\\left(\\\\sigma _+/\\\\sigma _-\\\\right)^2 \\\\approx 3500$ .\",\n \"Figure: Position and momentum correlation functions in the x ' x^{\\\\prime } dimension for an increasing number of images.\",\n \"(𝐚){\\\\rm ({\\\\bf a})} Measured position correlations from imaging plane scheme.\",\n \"(𝐛){\\\\rm ({\\\\bf b})} Measured momentum correlations from far-field scheme.\",\n \"Blue circles represents the spatial-correlation and red squares represent the reference-correlation.\",\n \"The asymmetry of the far-field correlation function is the result of a non-uniform near collinear degenerate light cone.When measuring position correlations, we use an imaging system with a magnification of $M = 2.5$ , comprising two lenses of focal lengths $100\\\\,\\\\rm {mm}$ and $250\\\\,\\\\rm {mm}$ (Fig.\",\n \"REF a).\",\n \"The far-field image is obtained by using a composite Fourier system with an effective focal length of $f_{e} = 100\\\\,{\\\\rm mm}$ (Fig.\",\n \"REF b).\",\n \"Given our imaging and far-field system we predict a transverse correlation length $\\\\sigma _{{\\\\rm pos}} = M\\\\sigma _- \\\\approx 28\\\\,\\\\mu \\\\rm {m}$ and $\\\\sigma _{{\\\\rm mom}} = f_e/(k\\\\sigma _+) \\\\approx 17\\\\,\\\\mu \\\\rm {m}$ , where $k = k_{p}/2$ is the wavenumber of the down-converted photons.\",\n \"Note that the position correlation length is larger than the pixel size of the camera ($16\\\\,\\\\mu \\\\rm {m}$ ) meaning that the position correlated photon pairs have a high probability of being detected in adjacent pixels which removes the need to count multiple photons within the same pixel.\",\n \"A $10\\\\,\\\\rm {nm}$ bandpass filter centred at $710\\\\,\\\\rm {nm}$ with a transmission efficiency $\\\\eta _{{\\\\rm filter}} = 90\\\\%$ is located immediately in front of the EMCCD.\",\n \"The EMCCD camera used in our experiment is a back-illuminated $512\\\\times 512$ array of $16\\\\times 16\\\\,\\\\mu \\\\rm {m^2}$ pixels optimized for visible wavelengths.\",\n \"The quantum efficiency of detection is given as the product of the efficiency of the camera, filter and lenses.\",\n \"Thus providing that either the signal or idler photon is detected, we predicted a heralding efficiency for detecting its entangled partner to be $\\\\eta \\\\approx 80\\\\%$ .\",\n \"Image analysis.\",\n \"A series of $100,000$ images was recorded for image and far-field configurations (see Methods).\",\n \"The probability distributions $\\\\mathcal {P}(\\\\rho _-)$ and $\\\\mathcal {P}(p_+)$ were calculated respectively by counting the number of coincidences as a function of the difference (imaging plane) or sum (far-field) of the pixel coordinates in both $x^{\\\\prime }$ and $y^{\\\\prime }$ directions at the plane of the CCD array.\",\n \"Here we define $\\\\rho _- \\\\equiv \\\\rho _1-\\\\rho _2$ and $p_+ \\\\equiv p_1 + p_2$ , where $p_i = (p_{x_i},p_{y_i})$ is the transverse momentum of photon $i$ .\",\n \"The transformation from coordinates in the plane of the CCD array to the crystal is performed using the scaling factors $\\\\gamma _{\\\\rm {pos}} \\\\equiv 1/M$ for imaging plane and $\\\\gamma _{\\\\rm {mom}} \\\\equiv (k \\\\hbar )/f_{e}$ for far-field.\",\n \"Our result is obtained by averaging the spatial correlation function over all recorded frames.\",\n \"Within each linear dimension this processing reveals correlations between photon pairs, which are clearly visible on top of a broader correlation arising from uncorrelated events, as shown in Fig.\",\n \"REF .\",\n \"We measured the background correlation by performing the same analysis but between consecutive frames to give a reference correlation.\",\n \"Fig.\",\n \"REF a shows the measured coincidence counts in the image plane as a function of $x^{\\\\prime }_{-}$ coordinates, while coincidence counts measured in the far-field as a function of $x^{\\\\prime }_{+}$ are shown in Fig.\",\n \"REF b.\",\n \"A correlation peak in both position and momentum can already be seen in a few images and becomes significantly distinguishable above noise as more images are summed.\",\n \"Figure: Background subtracted auto-correlation functions in position and momentum for high and low heralding efficiency configurations.\",\n \"Auto-correlation functions for the image plane (𝐚){\\\\rm ({\\\\bf a})} and far-field (𝐛){\\\\rm ({\\\\bf b})} of the crystal, summed over 100 000 images and with the measured background correlation removed.\",\n \"In (𝐜){\\\\rm ({\\\\bf c})} and (𝐝){\\\\rm ({\\\\bf d})} we show the same measurements performed with a lower heralding efficiency.\",\n \"Variances along the x ' x^{\\\\prime }-direction of the high heralding efficiency configuration are indicated.\",\n \"Due to increased coincidence counts attributed to smearing, the correlation function for y - ' =±1y^{\\\\prime }_{-} = \\\\pm 1 were set to 0 in (a){\\\\rm (a)} and (c){\\\\rm (c)} for clarity.By subtracting the reference from the spatial correlation function it is possible to remove unwanted pixel defects and obtain a strong correlation in position and anti-correlation in momentum as indicated in Fig.\",\n \"REF a and Fig.\",\n \"REF b respectively.\",\n \"Since we cannot distinguish the detection of more than one photon on the same pixel, we are unable to measure the correlation function for ${\\\\rho }^{\\\\prime }_1 = {\\\\rho }^{\\\\prime }_2$ , for which a value equal to 0 has been assigned.\",\n \"To confirm that these correlations arise between photon pairs the neutral density filter used to attenuate the pump beam was placed after the BBO crystal to reduce the heralding efficiency to approximately $2.0\\\\%$ , whilst ensuring the same photon flux.\",\n \"As shown in Fig.\",\n \"REF c and Fig.\",\n \"REF d, we do not observe a strong correlation in these measurements; however along the $y^{\\\\prime }_{-}$ axis in the image plane measurements we observe a correlation indicative of charge smearing between pixels in the readout direction.\",\n \"EPR-type correlations.\",\n \"From the thresholded images we can calculate the joint probability distributions for both the $x^{\\\\prime }$ and $y^{\\\\prime }$ coordinates in the image plane and far-field, as shown in Fig.\",\n \"REF .\",\n \"Fig.\",\n \"REF a and Fig.\",\n \"REF b show the coincidence counts in the planes $(x_1,x_2)$ and $(y_1,y_2)$ , while Fig.\",\n \"REF c and Fig.\",\n \"REF d show the coincidence counts in the planes $(p_{x_1},p_{x_2})$ and $(p_{y_1},p_{y_2})$ .\",\n \"In the $y^{\\\\prime }$ direction we observe a strong correlation resulting from charge smearing, thus preventing any meaningful analysis from image plane measurements (Fig.\",\n \"REF b); however the expected anti-correlation in the far-field (Fig.\",\n \"REF d) is still evident.\",\n \"Note that background subtraction has been performed.\",\n \"Figure: Joint probability distributions for x ' x^{\\\\prime } and y ' y^{\\\\prime }-coordinates.\",\n \"The probability distributions for joint detections in both x ' x^{\\\\prime } and y ' y^{\\\\prime }-coordinates in the image plane (𝐚{\\\\rm ({\\\\bf a}} and 𝐛){\\\\rm {\\\\bf b})} and far-field (𝐜{\\\\rm ({\\\\bf c}} and 𝐝){\\\\rm {\\\\bf d})} are shown respectively.\",\n \"Charge smearing gives rise to an artificial correlation in the y ' y^{\\\\prime } direction (𝐛{\\\\rm ({\\\\bf b}} and 𝐝){\\\\rm {\\\\bf d})}.It has been shown that EPR-like correlations can be identified by violating the inequality [28], [29] $\\\\Delta ^2_{\\\\rm {min}}(x_1|x_2)\\\\Delta ^2_{\\\\rm {min}}(p_{x_1}|p_{x_2})>\\\\frac{\\\\hbar ^2}{4},$ where $\\\\Delta ^2_{\\\\rm {min}}(r_1|r_2)$ is the minimum inferred variance, describing the minimum uncertainty in measuring the variable $r_1$ conditional on the measurement of variable $r_2$ .\",\n \"The minimum inferred variance for $x$ -coordinates in the far-field measurement is defined as $\\\\Delta ^2_{\\\\rm {min}}(p_{x_1}|p_{x_2})=\\\\int \\\\mathcal {P}(p_{x_2})\\\\Delta ^2(p_{x_1}|p_{x_2})dp_{x_2}.$ We define the minimum inferred variance for the $x$ -coordinates in the image plane with a similar approach.\",\n \"Fitting Gaussians to the joint probabilities in the $x$ -direction (Fig.\",\n \"REF a and Fig.\",\n \"REF c) we obtain the variances shown in Table REF .\",\n \"Substituting these quantities into Eq.\",\n \"(REF ) we find $\\\\Delta ^2_{\\\\rm {min}}(x_1|x_2)\\\\Delta ^2_{\\\\rm {min}}(p_{x_1}|p_{x_2})=&(6.6\\\\pm 1.0)\\\\times 10^{-4}\\\\hbar ^2,\\\\\\\\\\\\Delta ^2_{\\\\rm {min}}(x_2|x_1)\\\\Delta ^2_{\\\\rm {min}}(p_{x_2}|p_{x_1})=&(6.2\\\\pm 0.9)\\\\times 10^{-4}\\\\hbar ^2,$ indicating that the transverse DOF of the two photon field exhibits EPR non-locality.\",\n \"This result is in good agreement with the theoretical prediction (from Eqs.\",\n \"(REF ) and (REF )) of $\\\\approx 3\\\\times 10^{-4}\\\\hbar ^2$ and, although performed with background subtraction, is an order of magnitude smaller than those in Refs.\",\n \"[4], [13].\",\n \"High-dimensional entanglement.\",\n \"From the joint distributions in Fig.\",\n \"REF , we can estimate the ratio $(\\\\sigma _+/\\\\sigma _-)$ for both $x$ and $y$ transverse DOF, in the image plane (IP) and far-field (FF).\",\n \"We define the maximum number of joint detections for our position and momentum measurements to be $D^{\\\\rm max}_{\\\\rm pos}=[(\\\\sigma _{x^{\\\\prime }_+}/\\\\sigma _{x^{\\\\prime }_-})(\\\\sigma _{y^{\\\\prime }_+}/\\\\sigma _{y^{\\\\prime }_-})]_{\\\\rm {IP}}$ and $D^{\\\\rm max}_{\\\\rm mom}=[(\\\\sigma _{x^{\\\\prime }_-}/\\\\sigma _{x^{\\\\prime }_+})(\\\\sigma _{y^{\\\\prime }_-}/\\\\sigma _{y^{\\\\prime }_+})]_{\\\\rm {FF}}$ .\",\n \"We must also account for the proportion of the beam measured by the detector array which is calculated by integrating the fitted Gaussian functions over the array size.\",\n \"In all cases we find this gives access to more than $85\\\\%$ of the available states.\",\n \"Due to charge smearing in the image plane measurements we must estimate the dimensionality in $y$ based on that of the $x$ dimension, which is supported by the circular symmetry of Fig.\",\n \"REF a.\",\n \"Thus, the measured dimensionality in position and momentum is found to be $D_{\\\\rm pos} = D^{\\\\rm max}_{\\\\rm pos} \\\\times 0.95^2 \\\\approx 3200$ and $D_{\\\\rm mom} = D^{\\\\rm max}_{\\\\rm mom} \\\\times 0.95 \\\\times 0.85 \\\\approx 2500$ , respectively.\",\n \"To the best of our knowledge this represents the largest dimensionality for any experiment using entangled spatial states of photons.\"\n ],\n [\n \"Discussion\",\n \"We have performed a multimode detection of around 2500 spatially entangled states of photon pairs produced by SPDC.\",\n \"We have shown that it is possible to utilize an EMCCD camera to measure spatial correlations between photon pairs in both the image plane and far-field of the down-conversion crystal.\",\n \"After background subtraction, we found that the spatial correlations violate EPR criterion.\",\n \"We acknowledge that the use of background-subtracted measurements do not meet the strict requirements for tests of EPR non-locality.\",\n \"If the background was included in the data, it would lead to an overestimate of the standard deviations of the correlations.\",\n \"Furthermore, in order to satisfy non-locality, the measurement of each photon must be performed outside the light-cone of its entangled partner.\",\n \"This issue could be solved in a modified setup in which the two-photons are split and sent to two spatially separated cameras.\",\n \"However, our results are in close agreement with the theoretical predictions, providing evidence that EMCCD technology can already be used in the characterization of spatial correlations of photons.\",\n \"The ability to access both transverse DOF of photons enables a large state space suggesting the use of spatial states in quantum communication and quantum information processing.\",\n \"In this case, information should be encoded and processed in an alphabet defined as orthogonal spatial states and subsequently read by projecting the full transverse field of the single photon on to a complete set of orthogonal spatial states.\",\n \"Thus, low-noise and efficient 2D detector arrays are essential in the use of spatial states in these applications, for which our work is a significant step forward.\",\n \"Future developments in imaging technologies will likely result in 2D detector arrays with even lower noise and widen their application in quantum optics.\",\n \"Table: Variances obtained from Gaussian fittings of background subtracted correlation functions shown Fig.\",\n \".\"\n ],\n [\n \"Methods\",\n \"Thresholding.\",\n \"There are several noise sources that affect an EMCCD camera.\",\n \"As well as stray photons, the camera is subject to dark noise; thermally excited electrons, clock induced noise resulting from spurious electrons created during charge transfers between pixels, and readout noise generated by the readout register.\",\n \"We characterized the level of total noise in our camera when operated at the maximum gain by obtaining $100,000$ images while the laser is blocked but with the shutter on the camera open.\",\n \"Performing an analysis similar to that of Ref.\",\n \"[19] we find a Gaussian distribution of readout noise, centred at a value $390\\\\,e^-$ with a standard deviation $\\\\sigma _{\\\\rm noise} = 6\\\\,e^-$ , in addition to a long tail that decreases exponentially for higher readout values.\",\n \"Our thresholding is performed by subtracting the mean background value for each pixel and assigning a 1 to pixels with values greater than one standard deviation of noise, and a 0 to all pixels having smaller values.\",\n \"This binary thresholding was found to reduce the level of noise whilst maximizing the correlation signal strength.\",\n \"Camera Settings.\",\n \"The camera was operated at $-85^0\\\\rm {C}$ , with a horizontal pixel shift readout rate of $1\\\\,\\\\rm {MHz}$ , a vertical pixel shift every $0.3\\\\,\\\\mu \\\\rm {s}$ and a vertical clock amplitude voltage of $+4$ above the default factory setting.\",\n \"The optimum photon flux incident upon the camera was chosen to maximize the strength of the correlation signal relative to noise sources.\",\n \"In our experiment the image-plane measurements were made with an exposure time of $0.4\\\\,\\\\rm {ms}$ , while the far-field measurements are obtained using an exposure time of $1\\\\,\\\\rm {ms}$ , in both cases giving an average photon flux of approximately 0.02 photons/pixel, which is equivalent to the level of noise.\",\n \"A region of interest measuring $201 \\\\times 201$ pixels was chosen to observe most of the down-converted field, which resulted in a frame rate of approximately $5\\\\,{\\\\rm Hz}$ .\",\n \"Acknowledgements M.J.P.\",\n \"would like to thank the Royal Society and the Wolfson Foundation.\",\n \"M.J.P.\",\n \"and R.W.B.\",\n \"thank the US DARPA/DSO InPho program.\",\n \"M.P.E., D.S.T., F.I., R.E.W., G.S.B.\",\n \"and M.J.P.\",\n \"acknowledge the financial support from the UK EPSRC.\",\n \"J.L., M.A., and R.W.B thank Canada's CERC program.\",\n \"The authors would like to thank N. Radwell for useful discussions.\",\n \"Author contributions M.J.P., R.W.B., G.S.B.\",\n \"and J.L.\",\n \"conceived the experiment.\",\n \"M.P.E., D.S.T., F.I.\",\n \"and R.E.W.\",\n \"designed and performed the experiment.\",\n \"M.P.E., D.S.T., F.I., J.L., M.A.\",\n \"and M.J.P.\",\n \"designed the measurement and analysis programs.\",\n \"M.P.E., D.S.T., F.I., J.L.\",\n \"and M.J.P.\",\n \"analysed the results.\",\n \"All authors contributed to the writing of the manuscript.\"\n ]\n]"},"id":{"kind":"string","value":"1204.1293"}}},{"rowIdx":1866888,"cells":{"text":{"kind":"list like","value":[["Positive stationary solutions for p-Laplacian problems with nonpositive\n perturbation"],["Abstract The paper is devoted to the existence of positive solutions of nonlinear elliptic equations with $p$-Laplacian.","We provide a general topological degree that detects solutions of the problem $$ \\{{array}{l} A(u)=F(u) u\\in M {array}.","$$ where $A:X\\supset D(A)\\to X^*$ is a maximal monotone operator in a Banach space $X$ and $F:M\\to X^*$ is a continuous mapping defined on a closed convex cone $M\\subset X$.","Next, we apply this general framework to a class of partial differential equations with $p$-Laplacian under Dirichlet boundary conditions."],["Introduction","We shall be concerned with solutions to the following nonlinear boundary value problem $\\left\\lbrace \\begin{array}{l}-\\mathrm {div} (|\\nabla u(x)|^{p-2} \\nabla u(x)) = f(x,u (x)), \\ x\\in \\Omega ,\\\\\\ \\ u(x)\\ge 0,\\ x\\in \\Omega ,\\\\\\ \\ u(x) = 0,\\ x\\in \\partial \\Omega \\end{array} \\right.$ where $\\Omega \\subset \\mathbb {R}^N$ ($N\\ge 1$ ) is a bounded domain with the smooth boundary $\\partial \\Omega $ , $p\\ge 2$ and $f:\\Omega \\times [0,+\\infty ) \\rightarrow \\mathbb {R}$ is a Carathéodory function (Recall that we say that $f:\\Omega \\times [0,+\\infty )\\rightarrow \\mathbb {R}$ is a Carathéodory function if $f(\\cdot , s)$ is measurable for all $s\\in [0,+\\infty )$ and $f(x,\\cdot )$ is continuous for almost all $x\\in \\Omega $ .",").","The differential term $\\mathrm {div} (|\\nabla u (x)|^{p-2}\\nabla u(x))$ is referred to as the $p$ -Laplacian of $u$ at a point $x\\in \\Omega $ .","We search for weak solutions in the Sobolev space $W_{0}^{1,p}(\\Omega )$ , i.e.","$u\\in W_{0}^{1,p} (\\Omega )$ such that $\\int _{\\Omega } |\\nabla u(x)|^{p-2} \\nabla u(x) \\cdot \\nabla v(x) \\,\\mathrm {d}x = \\int _{\\Omega } f(x, u(x)) v(x) \\,\\mathrm {d}x \\ \\ \\mbox{ for all } \\ v\\in W_{0}^{1,p} (\\Omega ).$ Such boundary problems with $p$ -Laplace were widely studied by many authors who used various methods.","Let us mention just a few.","Equations with the one dimensional $p$ -Laplacian, i.e.","when $N=1$ , were studied by Manásevich, Njoku i Zanolin [17], Drábek, García-Huidobro and Manásevich [7] and as well as by Kryszewski and the author [5].","In the general case, i.e.","when $N>1$ , positive solutions of $p$ -Laplace problems have been studied by a number of authors, e.g.","Huang [13], Drábek and Pohozaev [6], Cañada, Drábek and Gámez [3], Filippiakis, Gasiński and Papageorgiou [9] or Montreanu D., Montreanu V. V. and Papageorgiou [18], Väth [19].","Generally speaking, in the above mentioned papers, either $N=1$ or $N$ is arbitrary but the right has side of the equation - the function $f$ is assumed to be non-negative or satisfy some monotonicity assumptions.","This makes possible to apply Krasnosel'skii's fixed point theorem (in general, fixed point index in cones) or variational methods.","These assumptions on $f$ seem rather restrictive and sometimes unnatural, especially, when we take into account physical interpretation of the considered boundary value problem.","In this paper, we do not require $f$ to be non-negative or monotone.","A general tool for detection of nonnegative solutions is provided.","It is based on the geometric idea of tangency and using fixed point index in cones.","We construct a topological degree for perturbations of maximal monotone operators with respect to closed convex cones.","Next we prove appropriate index formulae, which together with the homotopy property, allow us to compute the topological degree in specific examples.","It is noteworthy, that this setting does not require variational structure and can be also used for systems of $p$ -Laplace problems.","In this paper, we apply the method to show the following existence criterion Theorem 1.1 Suppose that a Carathéodory function $f:\\Omega \\times [0,+\\infty )\\rightarrow \\mathbb {R}$ and $\\rho _0, \\rho _\\infty \\in L^\\infty (\\Omega )$ satisfy the following conditions $ & & \\mbox{there is } C>0 \\mbox{ such that }|f(x,s)| \\le C(1+s^{p-1}) \\mbox{ for all } s\\ge 0 \\mbox{ and a.a. } x\\in \\Omega ;\\\\ & & \\lim _{s\\rightarrow 0^+} \\frac{f(x,s)}{s^{p-1}} = \\rho _0 (x)\\mbox{ and } \\lim _{s\\rightarrow \\infty } \\frac{f(x,s)}{s^{p-1}} = \\rho _\\infty (x) \\mbox{ uniformly with respect to } x\\in \\Omega .$ If the principal eigenvalue $\\lambda _{1,p}$ of the $p$ -Laplace operator lies between $\\rho _0$ and $\\rho _\\infty $ , i.e.","either $\\rho _0(x) < \\lambda _{1,p} <\\rho _\\infty (x)$ , for a.a. $x\\in \\Omega $ , or $ \\rho _\\infty (x) < \\lambda _{1,p} < \\rho _0 (x)$ , for a.a. $x\\in \\Omega $ , then the problem (REF ) admits a nontrivial weak solution $u\\in W_{0}^{1,p} (\\Omega )$ such that $u(x)\\ge 0$ for a.e.","$x\\in \\Omega $ .","Here the principal eigenvalue $\\lambda _{1,p}$ is the smallest real number $\\lambda $ such that the problem $\\left\\lbrace \\begin{array}{l}-\\mathrm {div} (|\\nabla u(x)|^{p-2} \\nabla u(x)) = \\lambda |u (x)|^{p-2} u(x),\\ x\\in \\Omega \\\\\\ \\ u(x)\\ge 0,\\ x\\in \\Omega \\\\\\ \\ u(x) = 0,\\ x\\in \\partial \\Omega \\end{array} \\right.$ admits a nonzero weak solution (see Remark REF for more details).","Theorem REF corresponds directly to the result of [13], obtained by different methods (the sub-supersolution technique and the existence result for variational inequalities) and under different assumptions corresponding to the inequality $\\rho _\\infty < \\lambda _{1,p} <\\rho _0$ .","Our general method allows us to consider also the case $\\rho _\\infty > \\lambda _{1,p}> \\rho _0$ .","The paper is organized as follows.","In Section 2 we develop a topological degree detecting coincidence points of maximal monotone operators and continuous operators in closed convex cones.","This general tool will be useful if we rewrite the problem (REF ) in the form $\\left\\lbrace \\begin{array}{l} A_p u = N_f (u)\\\\u\\in M_p\\end{array}\\right.$ where $A_p:L^p(\\Omega ) \\supset D(A_p)\\rightarrow L^p (\\Omega )^*$ is the maximal monotone operator determined by the $p$ -Laplacian, $N_f:L^p(\\Omega )\\rightarrow L^p (\\Omega )^*$ is the Nemytzkii type operator associated with $f$ and $M_p$ is the closed convex cone of all non-negative elements in the space $L^p (\\Omega )$ .","Section 3 provides a general setting in which assumptions of Section 2 are verified.","Next, in Section 4 we show that the problem (REF ) falls into the setting and, using our topological degree together with spectral properties of $p$ -Laplacian, we derive topological index formulae.","They turn out to be essential in the proof of Theorem REF , which is provided at the end of Section 4.","Notation If $X$ is a metric space and $B \\subset X$ , then $\\partial B$ and $cl B$ stand for the boundary of $B$ and the closure of $B$ , respectively.","If $x_0\\in X$ and $r>0$ , then $B (x_0,r):=\\lbrace x\\in M\\mid d(x,x_0)0$ , $J_\\alpha (\\tau ):= u, \\ \\mbox{ where } u\\in D(A) \\mbox{ is the unique element such that } \\tau \\in (N+\\alpha A)(u),$ are well defined and continuous; $({\\cal A}_2)$ the mapping ${\\cal J}:X^* \\times (0,+\\infty )\\ni (\\tau , \\alpha )\\mapsto J_{\\alpha } (\\tau )\\in X$ is bounded on bounded sets and such that ${\\cal J}_{\\ |\\, X^* \\times [\\alpha _1, \\alpha _2]}$ is completely continuous if $0<\\alpha _1 \\le \\alpha _2$ ; $({\\cal A}_3)$ $M\\subset X$ is a neighborhood retract of $X$ , $J_\\alpha (N(M)) \\subset M$ for $\\alpha >0$ , and $M^* :=N(M)$ is an ${\\cal L}$ -retract (see [2] and [5]), i.e.","there exist a retraction $r:B(M^*,\\eta )\\rightarrow M^*$ with some $\\eta >0$ and a constant $L>0$ such that $\\Vert r(\\tau )- \\tau \\Vert \\le L d_{M^*} (\\tau ) \\ \\mbox{ for all } \\ \\tau \\in B(M^*, \\eta );$ $({\\cal A}_4)$ $F$ is continuous, bounded on bounded sets and satisfies the tangency condition $F(N^{-1}(\\tau ))\\in T_{M^{*}} (\\tau ), \\ \\mbox{ for } \\tau \\in N(\\overline{U}),$ where $T_{M^*} (\\tau )$ is the Bouligand tangent cone to $M^*$ at the point $\\tau $ , i.e.","$T_{M^*} (\\tau ): = \\left\\lbrace \\theta \\in X^* \\mid \\liminf _{\\alpha \\rightarrow 0^+} \\frac{d_{M^*} (\\tau + \\alpha \\theta ) }{\\alpha } =0 \\right\\rbrace .$ Remark 2.1 Since maximal monotone operators have closed graphs, it can be shown that in order to verify the continuity of the mapping ${\\cal J}_{\\ |\\, X^* \\times [\\alpha _1, \\alpha _2]}$ from condition $({\\cal A}_2)$ it is sufficient to know that it maps bounded sets into relatively compact ones.","Our goal is to transform the problem (REF ) into a fixed point one in $M$ and for which fixed point index theory can be used.","To this end define $\\Phi _{\\alpha }=\\Phi _{\\alpha }^{A,F}:\\overline{U} \\rightarrow M$ by $\\Phi _\\alpha (u):= J_{\\alpha } \\left(r \\left(N(u) + \\alpha F(u)\\right)\\right), \\ \\ u\\in \\overline{U},$ whenever $0<\\alpha < \\eta /\\sup \\lbrace \\Vert F(u)\\Vert \\mid u\\in \\overline{U} \\rbrace $ .","Obviously, it is well defined, since for such $\\alpha $ one has $(N+\\alpha F) (\\overline{U}) \\subset B(M^*,\\eta )$ .","Moreover, observe that due to the assumptions, the mapping $r \\circ (N+\\alpha F)$ is bounded on bounded sets and, by $({\\cal A}_2)$ , $\\Phi _\\alpha $ is compact.","Exploiting the tangency condition (REF ) and the inequality (REF ) together with compactness, we obtain the following localization of fixed points results.","Proposition 2.2 If $K\\subset \\overline{U}$ is a closed set such that $\\lbrace u\\in \\overline{U}\\cap D(A) \\mid 0\\in -Au + F(u)\\rbrace \\cap K = \\emptyset ,$ then, for sufficiently small $\\alpha >0$ , $\\lbrace u\\in \\overline{U} \\mid \\Phi _\\alpha (u)=u\\rbrace \\cap K = \\emptyset .$ Remark 2.3 Actually the tangency condition (REF ) and the continuity of $F\\circ N^{-1}$ imply $F\\left( N^{-1}(\\tau ) \\right) \\in C_{M^*} (\\tau ) := \\left\\lbrace \\theta \\in X^* \\mid \\lim _{\\alpha \\rightarrow 0^+, \\, \\varrho \\rightarrow \\tau , \\, \\varrho \\in M} \\frac{d_M (\\varrho + \\alpha \\theta )}{\\alpha } =0 \\right\\rbrace \\, \\mbox{ for all } \\tau \\in N(U).$ Indeed $F \\left( N^{-1}(\\tau ) \\right)=\\lim _{\\varrho \\rightarrow \\tau } F \\left( N^{-1}(\\varrho ) \\right) \\in \\mathop {\\mathrm {Liminf}\\,}_{\\varrho \\rightarrow \\tau , \\, \\varrho \\in M^*} T_{M^*} (\\varrho ) \\subset C_{M^*} (\\tau ).$ The proof of the latter inclusion can be found in [1].","Lemma 2.4 (i) The graph $\\mathrm {Gr}(A):= \\lbrace (u, \\tau )\\in X\\times X^* \\mid u\\in D(A)\\rbrace $ is closed; (ii) If a sequence of pairs $(u_n ,\\tau _n) \\in \\mathrm {Gr}(A)$ , $n\\ge 1$ , is bounded, then the sequence $(u_n)$ has a convergent subsequence.","Proof: (i) Take any sequence of points $(u_n, \\tau _n)\\in \\mathrm {Gr}(A)$ , $n\\ge 1$ , such that $(u_n, \\tau _n)\\rightarrow (u_0, \\tau _0)$ in $X\\times X^*$ , as $n\\rightarrow +\\infty $ , for some $(u_0, \\tau _0) \\in X\\times X^*$ .","Clearly, $\\tau _n \\in Au_n$ , and this gives $N(u_n) + \\tau _n \\in (N+A)(u_n)$ , which, by $({\\cal A}_1)$ , gives $u_n = J_{1} (N(u_n)+\\tau _n)$ , $n\\ge 1$ .","Hence, using the continuity of $N$ and $J_{1}$ yields $u_n = J_{1}(N(u_n)+\\tau _n)) \\rightarrow J_{1} (N(u_0) + \\tau _0)$ as $n\\rightarrow +\\infty $ , which implies $u_0 = J_{1} (N(u_0) + \\tau _0)$ , i.e.","$\\tau _0\\in Au_0$ .","This shows that $\\mathrm {Gr}(A)$ is closed.","(ii) Note that, for each $n\\ge 1$ , $u_n = J_{1} (N(u_n) + \\tau _n) \\in J_{1} (N(B(0,R))+B(0,R))$ , where $R>0$ is a constant such that $\\Vert u_n\\Vert _X \\le R$ and $\\Vert \\tau _n\\Vert _{X^*} \\le R$ for $n\\ge 1$ .","The boundedness of $N$ and $({\\cal A}_2)$ imply that the set $(u_n)$ is a sequence of elements of the relatively compact set $J_{1} (N(B(0,R))+B(0,R))$ .","$\\square $ Proof of Proposition REF : Suppose to the contrary that there exists a sequence $(\\alpha _n)$ such that $\\alpha _n\\rightarrow 0^+$ such that for each $n\\ge 1$ there is $u_n\\in K$ with $\\Phi _{\\alpha _n} (u_n) = u_n$ , that is $N(u_n)+ \\alpha _n \\tau _n = r(N(u_n) + \\alpha _n F(u_n)) \\mbox{ for some } \\tau _n \\in Au_n.$ In view of (REF ), one has $\\alpha _n \\Vert \\tau _n - F(u_n)\\Vert & = & \\Vert r(N(u_n) + \\alpha _n F(u_n)) - ( N (u_n) + \\alpha _n F (u_n) ) \\Vert \\\\& \\le & L d_{M^*} (N(u_n) + \\alpha _n F(u_n)) \\ \\ \\mbox{ for all } n\\ge 1.","\\nonumber $ This implies $\\Vert \\tau _n \\Vert \\le \\Vert F(u_n)\\Vert + L \\alpha _{n}^{-1} d_{M^*} (N(u_n) + \\alpha _n F(u_n)) \\le (1+L) \\Vert F(u_n)\\Vert , \\ n\\ge 1,$ which means that $(\\tau _n)$ is bounded.","Therefore, by use of Lemma REF (ii), we may assume without loss of generality that $u_n \\rightarrow u_0$ for some $u_0\\in M$ .","Now using (REF ) and putting $p_n:=N(u_n)$ , $n\\ge 0$ , we see that $\\Vert \\tau _n - F(u_n)\\Vert \\le L \\cdot \\frac{d_{M^*} (p_n + \\alpha _n F(N^{-1}(p_0))}{\\alpha _n} + L\\Vert F(u_n)-F(u_0)\\Vert , \\ \\ \\mbox{ for } n\\ge 1.$ By the tangency condition $({\\cal A}_4)$ and Remark REF together with the continuity of $F$ , we get that $\\tau _n \\rightarrow F(u_n)$ as $n\\rightarrow +\\infty $ .","Hence, we have obtained that $(u_n, \\tau _n)\\rightarrow (u_0, F(u_0))$ and, by Lemma REF (i), $(u_0,F(u_0)) \\in \\mathrm {Gr}(A)$ , i.e.","$F(u_0)\\in A u_0$ , a contradiction completing the proof.","$\\square $ Now we put $\\mathrm {Deg}_M ( A, F, U ) := \\lim _{\\alpha \\rightarrow 0^+} \\mathrm {ind}_M ( \\Phi _\\alpha , U ),$ where $\\mathrm {ind}_M$ stands for the fixed point index for compact mappings of absolute neighborhood retracts due to Granas – see [12] or [8] for details.","We call this number as the topological degree of coincidence (or just topological degree) of $A$ and $F$ with respect to $M$.","Theorem 2.5 The coincidence degree defined by (REF ) is well defined and has the following properties: (i) (existence) if $\\mathrm {Deg}_M (A,F, U)\\ne 0$ , then there exists $u\\in U\\cap D(A)$ such that $0\\in -Au+F(u)$ ; (ii) (additivity) if $U_1, U_2$ are open disjoint subsets of a bounded open $U\\subset M$ and $0\\notin (-A+F)(\\overline{U} \\setminus (U_1\\cup U_2))$ , then $\\mathrm {Deg}_M (A,F, U) = \\mathrm {Deg}_M (A,F, U_1) +\\mathrm {Deg}_M (A,F, U_2);$ (iii) (homotopy invariance) if $H:\\overline{U} \\times [0,1]\\rightarrow X^*$ is a continuous and bounded mapping such that $H( N^{-1}(\\tau ), t) \\in T_{M^*} (\\tau ) \\mbox{ for all } \\tau \\in N(\\overline{U}), \\, t>0,$ and $0\\notin -Au+H(u,t)$ for all $u\\in \\partial U\\cap D(A)$ and $t\\in [0,1]$ , then $\\mathrm {Deg}_M (A,H(0,\\cdot ), U) = \\mathrm {Deg}_M (A, H(1,\\cdot ), U);$ (iv) (normalization) if $M$ is bounded and the mapping $\\widetilde{\\cal J}: X^* \\times [0,+\\infty ) \\ni (\\tau , \\alpha ) \\mapsto J^{\\alpha } \\tau \\in X$ with $J^0 = N^{-1}$ is continuous, then $\\mathrm {Deg}_M (A,F, M) = \\chi (M)$ .","Proof: Note that for sufficiently small $\\alpha >0$ it follows from Propostion REF that $\\Phi _\\alpha $ has no fixed point in $\\partial U$ , i.e.","the fixed point index $\\mathrm {ind}_M (\\Phi _\\alpha , U)$ is well defined.","If $\\alpha _1, \\alpha _2>0$ are small enough, then, by $({\\cal A}_2)$ , $\\Phi _{\\alpha _1}$ is homotopic with $\\Phi _{\\alpha _2}$ , which gives $\\mathrm {ind}_M (\\Phi _{\\alpha _1}, U)= \\mathrm {ind}_M (\\Phi _{\\alpha _2}, U)$ , which means that the limit in (REF ) exists.","(i) Suppose to the contrary that there is no $u\\in U \\cap D(A)$ such that $0\\in - Au + F(u)$ .","Then, in view of Proposition REF , for sufficiently small $\\alpha >0$ the mappings $\\Phi _\\alpha $ have no fixed points in $\\overline{U}$ , i.e.","$\\mathrm {Deg}_M (A,F,U) = \\mathrm {ind}_M (\\Phi _\\alpha , U)=0$ , a contradiction.","(ii) Due to Proposition REF , for sufficiently small $\\alpha >0$ , $\\Phi _\\alpha $ has no fixed points in $\\overline{U} \\setminus (U_1\\cup U_2)$ .","Therefore, by the definition of the degree, $\\mathrm {Deg}_M (A, F, U)=\\mathrm {ind}_M (\\Phi _\\alpha , U) \\ \\ \\mbox{ and } \\ \\ \\mathrm {Deg}_M (A, F, U_k)=\\mathrm {ind}_M (\\Phi _\\alpha , U_k) \\mbox{ for } k=1,2.$ By the additivity property of the fixed point index $\\mathrm {ind}_M (\\Phi _\\alpha , U) = \\mathrm {ind}_M (\\Phi _\\alpha , U_1) + \\mathrm {ind}_M (\\Phi _\\alpha , U_2),$ which together with the earlier equalities gives the desired additivity of the degree.","(iii) For sufficiently small $\\alpha >0$ one can define $\\Phi _\\alpha :\\overline{U}\\times [0,1] \\rightarrow M$ by $\\Phi _\\alpha (u,t):= J_{\\alpha }\\left( r (N(u) + \\alpha H(u,t)) \\right), \\ u\\in \\overline{U}, \\ t\\in [0,1].$ Proceeding along the lines of the proof of Proposition REF we can prove that for sufficiently small $\\alpha >0$ $\\Phi _\\alpha (u,t)\\ne u \\mbox{ for all } u\\in \\partial U, \\ t\\in [0,1].$ Hence, by the homotopy invariance of the fixed point index and the formula defining the degree, $\\mathrm {Deg}_M (A,H(\\cdot , 0),U) = \\mathrm {ind}_M (\\Phi _\\alpha (\\cdot ,0), U) = \\mathrm {ind}_M (\\Phi _\\alpha (\\cdot ,1), U)= \\mathrm {Deg}_M (A,H(\\cdot , 1),U).$ (iv) Take small $\\alpha >0$ such that $\\Phi _\\alpha $ is well defined.","Then $\\mathrm {Deg}_M (A, F, M) = \\mathrm {ind}_M (\\Phi _\\alpha , M).$ Note that the normalization property for the fixed point index states that the homomorphism $H_* (\\Phi _\\alpha ):H_* (M)\\rightarrow H_*(M)$ induced on (singular) homology spaces is a Leray endomorphism and $\\mathrm {ind}_M (\\Phi _\\alpha , M) = \\Lambda (\\Phi _\\alpha )$ where $\\Lambda (\\Phi _\\alpha )$ is the generalized Leschetz number of the compact map $\\Phi _\\alpha $ – see [8] or [12].","Further, consider $\\Psi :M\\times [0,1]\\rightarrow M$ given by $\\Psi (u,t) := \\widetilde{\\cal J} (r(N(u)+t\\alpha F(u)),t\\alpha ), \\ \\ u\\in M, \\ \\ t\\in [0,1].$ By the assumption, $\\Psi $ is a continuous homotopy joining $\\Psi (\\cdot , 1) = \\Phi _\\alpha $ with the identity map $\\mathrm {id}_M: M\\rightarrow M$ .","Hence, for the maps induced on homology spaced one has $H_* (\\Phi _\\alpha ) = H_* (\\mathrm {id}_M)=\\mathrm {id}_{H_* (M)}$ and, since $H_* (\\Phi _\\alpha )$ is an endomorphism Leray, we infer that $\\Lambda (\\Phi _\\alpha )=\\sum _{n=0}^{\\infty } (-1)^n \\dim H_n (M) = \\chi (M)$ , which together with (REF ) ends the proof.","$\\square $ We end this section with a general result, which allows us to compute the degree is specific situations (comp.","[5]).","Theorem 2.6 Let $M$ and $M^*$ be closed convex cones and that the mappings $A$ and $N$ are homogeneous with the same degree (i.e.","there exists $\\gamma >0$ such that $A(a u) = a^{\\gamma } A(u)$ , $u\\in D(A)$ , $a>0$ , and $N(au)=a^\\gamma N(u)$ for all $u\\in X$ , $a>0$ .",").","Suppose that there exists $\\lambda _1 \\ge 0$ satisfying the following conditions $({\\cal M}_1)$ $(A-\\lambda N)^{-1}(\\lbrace 0\\rbrace ) \\cap M = \\lbrace 0\\rbrace \\ \\mbox{ for } \\ \\lambda \\ne \\lambda _1;$ $({\\cal M}_2)$ there exists $\\tau _0 \\in M^*$ such that $(A-\\lambda N)^{-1} (\\lbrace \\tau _0 \\rbrace ) \\cap M =\\emptyset $ for $\\lambda >\\lambda _1$ .","Then $\\mathrm {Deg}_M (A, \\lambda N, B_M (0,\\delta )) = \\left\\lbrace \\begin{array}{ll} 1, & \\lambda < \\lambda _1,\\\\0, & \\lambda >\\lambda _1,\\end{array} \\right.$ for any $\\delta >0$ .","Proof: Note that in view of $({\\cal M}_1)$ the topological degree $\\mathrm {Deg}_M (A, \\lambda N, B_M (0, \\delta ))$ is well defined.","Now fix $\\lambda <\\lambda _1$ .","By the very construction, for sufficiently small $\\alpha >0$ , $\\mathrm {Deg}_M (A, \\lambda N, B_M (0,\\delta )) = \\mathrm {ind}_M (\\Phi _\\alpha , B_M (0,\\delta ))$ where $\\Phi _\\alpha :\\overline{B_M (0,\\delta )}\\rightarrow M$ is given by $\\Phi _\\alpha (u):= J_\\alpha (r(N(u) + \\alpha \\lambda N(u))), \\ u \\in \\overline{B_M (0,\\delta )}.$ Define $\\Theta : \\overline{B_M (0,\\delta )}\\times [0,1]\\rightarrow M$ by $\\Theta (u,t):= t \\Phi _\\alpha (u), \\ u \\in \\overline{B_M (0,\\delta )}, \\, t\\in [0,1].$ Suppose there are $u\\ne 0$ and $t\\in [0,1]$ such that $\\Theta (u,t)=u$ .","Then $0 \\in -A(u) + \\mu N(u)$ with $\\mu :=(t^\\gamma -1)/\\alpha + t^\\gamma \\lambda $ , i.e.","$u\\in (A-\\mu N )^{-1} (\\lbrace 0\\rbrace )\\cap M$ , and, since $\\mu =(t^\\gamma -1)/\\alpha + t^\\gamma \\lambda < \\lambda _1$ we get a contradiction with $({\\cal M}_1)$ .","Hence, we can use the homotopy invariance of fixed point index to see that $\\mathrm {ind}_M (\\Phi _\\alpha ,B_M (0,\\delta )) =\\mathrm {ind}_M (0, B_M (0,\\delta )) = 1$ .","This along with (REF ) implies the required equality.","Let us pass to the case when $\\lambda >\\lambda _1$ .","Define $H:M \\times [0,1] \\rightarrow X$ by $H(u,t):= \\lambda N (u) + t\\tau _0$ , $u\\in M$ , $t\\in [0,1]$ .","If $-A(u)+H(u,t)= 0$ , then either $t=0$ and, due to $({\\cal M}_1)$ , $u=0$ or, by the homogeneity $-A(t^{-1/\\gamma }u) + \\lambda N (t^{-1/\\gamma }u) + \\tau _0 = 0$ , where $\\gamma >0$ is the common homogeneity degree for $A$ and $N$ .","The latter equality contradicts $({\\cal M}_2)$ .","Hence, the degrees $\\mathrm {Deg}_M (A, H(\\cdot ,t), B_M (0,\\delta ) )$ , $t\\in [0,1]$ , are well defined and homotopy invariance can be used to obtain $\\mathrm {Deg}_M (A, \\lambda N, B_M (0,\\delta )) = \\mathrm {Deg}_M (A, \\lambda A +\\tau _0, B_M (0,\\delta ).$ Finally the existence property of the degree together with $({\\cal M}_2)$ implies $\\mathrm {Deg}_M (A, \\lambda A +\\tau _0, B_M (0,\\delta ))=0$ , which completes the proof.","$\\square $"],["Abstract setting for $p$ -Laplacian","Now we shall consider an abstract example falling into the setting of Section 2.","It will be used in the sequel for the $p$ -Laplace operator and the cone of positive functions in $L^{p}(\\Omega )$ .","Let $X$ and $Y$ be reflexive normed spaces with a dense and compact linear embedding $i:Y\\rightarrow X$ .","(That is the mapping $i$ is linear and completely continuous with its range $i(Y)$ dense in $X$ .)","Suppose that a closed convex cone $M\\subset X$ and functionals ${a}:Y\\rightarrow \\mathbb {R}$ and ${n}: X\\rightarrow \\mathbb {R}$ satisfy the following conditions: (a1) ${a}$ and ${n}$ are coercive $C^1$ functionals;(By coercivity we mean that counterimages of intervals $(-\\infty ,m)$ , with respect to a given functional, are bounded for all $m\\in \\mathbb {R}$ .)","(a2) there exists a continuous function $\\kappa :[0,+\\infty )\\rightarrow [0, +\\infty )$ such that $\\kappa ^{-1}(\\lbrace 0\\rbrace ) = \\lbrace 0\\rbrace $ , $\\lim \\limits _{s\\rightarrow +\\infty } \\kappa (s) = +\\infty $ and $\\langle D{a} (u_1) - D{a} (u_2), u_1-u_2 \\rangle _{Y} \\ge \\kappa (\\Vert u_1-u_2\\Vert _Y)\\Vert u_1-u_2\\Vert _Y \\ \\ \\mbox{ for all }u_1, u_2\\in Y,\\\\\\langle D{n} (u_1) - D{n} (u_2), u_1-u_2 \\rangle _{X} \\ge \\kappa (\\Vert u_1-u_2\\Vert _X)\\Vert u_1-u_2\\Vert _X \\ \\ \\mbox{ for all }u_1, u_2\\in X;$ (a3) for any $u\\in M$ there exist $u^+, u^-\\in M$ such that $u=u^+ - u^-$ and ${n}(u^+) \\le {n}(u)$ ; if $u\\in i(Y)$ , then $u^+, u^- \\in i(Y)$ and ${a} (i^{-1} u^+) \\le {a} (i^{-1} u)$ ; (a4) $n$ is bounded on bounded sets and monotone with respect to $M$ , i.e.","${n}(u+v) \\ge {n} (u)$ for any $u,v\\in M$ .","Let ${\\cal A}:Y \\rightarrow Y^*$ and $N:X\\rightarrow X^*$ be defined by by $ {\\cal A} : = D{a}$ and $N:=D {n}$ .","Note that that, due to (a2), both ${a}$ and ${n}$ are strictly convex and ${\\cal A}$ and $N$ are monotone operators.","Define $A:D(A)\\rightarrow X^*$ by $D(A):= i \\left( {\\cal A}^{-1} (i^*(X^*))\\right) \\mbox{ and } A u:= (i^*)^{-1}({\\cal A} i^{-1}u), \\mbox{ for } u\\in D(A).$ The above operation of restriction is a generalization of the analogical one that is usually considered in the case of a Gelfand triple $Y\\subset X \\subset Y^*$ where $X$ is a Hilbert space.","Below we show that assumptions $({\\cal A}_1)$ and $({\\cal A}_2)$ of Section 2 are satisfied.","Proposition 3.1 Under the above assumptions (i) $N$ is a homeomorphism which is bounded on bounded sets; (ii) $N(M) = M^*:=\\lbrace \\tau \\in X^* \\mid \\langle \\tau , u \\rangle \\ge 0 \\mbox{ for all } u\\in M \\rbrace $ ; (iii) $A$ is a densely defined maximal monotone operator; (iv) for any $\\alpha >0$ and $\\tau \\in X^*$ there is a unique $u\\in D(A)$ such that $\\tau = (N+\\alpha A) (u)$ ; (v) if $J_\\alpha : X^*\\rightarrow X$ , $\\alpha >0$ , is given by $J_{\\alpha } \\tau := u \\mbox{ where } u\\in D(A) \\mbox{ is such that } N(u)+\\alpha A (u) = \\tau ,$ and ${\\cal J}:X^* \\times [0,+\\infty ) \\rightarrow X$ by ${\\cal J} (u, \\alpha ):= J_{\\alpha } u,$ then ${\\cal J}$ is bounded on bounded sets and ${\\cal J}_{\\ |\\, X^* \\times [\\alpha _1, \\alpha _2]}$ with $0<\\alpha _1 \\le \\alpha _2$ is completely continuous; (vi) $J_{\\alpha }(M^*) \\subset M$ for all $\\alpha >0$ .","Proof: To see (i), first note that $N$ is continuous, since ${n}$ is $C^1$ .","Moreover, as a strictly convex coercive functional on the reflexive Banach space $X$ , for any $\\tau \\in X^{*}$ , ${n}-\\tau $ admits a unique minimum point $u\\in X$ , i.e.","$D{n} (u)-\\tau =0$ , which gives $N(u)=\\tau $ .","Conversely, if $u\\in X$ is such that $N(u)=\\tau $ , then, by the strict convexity, $u$ is the unique minimum point.","Hence, $N$ is bijective.","To see that $N^{-1}$ is continuous, take any $(\\tau _n)$ in $X$ with $\\tau _n \\rightarrow \\tau $ in $X^*$ as $n\\rightarrow +\\infty $ .","Observe that, by (a2), we get $\\langle \\tau _n - \\tau , N^{-1}(\\tau _n)-N^{-1}(\\tau )\\rangle _X \\ge \\kappa (\\Vert N^{-1}(\\tau _n) - N^{-1} (\\tau )\\Vert _X)\\Vert N^{-1}(\\tau _n) - N^{-1} (\\tau )\\Vert _X,$ which yields the inequality $\\Vert \\tau _n-\\tau \\Vert _{X^*}\\ge \\kappa \\left(\\Vert N^{-1}(\\tau _n) - N^{-1} (\\tau )\\Vert _X \\right).$ This in turn means that $N^{-1}(\\tau _n) \\rightarrow N^{-1} (\\tau )$ in $X$ as $n\\rightarrow +\\infty $ , that is $N^{-1}$ is continuous.","To show that $N$ is bounded on bounded sets, we suppose to the contrary that there exists a bounded sequence $(u_n)$ in $X$ such that $\\Vert N(u_n)\\Vert _{X^*}\\rightarrow +\\infty $ as $n\\rightarrow +\\infty $ .","Since $X$ is reflexive, for each $n\\ge 1$ one finds an element $v_n\\in X$ such that $\\Vert v_n - u_n\\Vert _{X} = 1$ and $\\Vert N(u_n)\\Vert _{X^*} = \\langle N(u_n), v_n-u_n\\rangle \\le {n} ( v_n) - {n} (u_n) \\le \\sup _{D_{X}(0, R+1)} {n} - \\inf _{D_X(0,R)} {n}$ where $R>0$ is such that $\\Vert u_n\\Vert \\le R$ for all $n\\ge 1$ .","Thus, a contradiction proving the claim.","To get (ii) take any $u\\in M$ and $v\\in M$ .","In view of (a4) ${n}(u+h v)-{n} (u) \\ge 0 \\ \\mbox{ for any } h>0,$ which, after a division by $h$ and passage to the limit with $h\\rightarrow 0^+$ , yields $\\langle N(u), v\\rangle \\ge 0$ .","Hence $N(M)\\subset M^*$ .","To prove the converse inclusion $M^*\\subset N(M)$ , we take any $\\tau \\in M^*$ .","As we mentioned ${n}-\\tau $ attains the minimum at some $u\\in X$ .","On the other hand, by (a2), ${n}(u^+) -\\tau (u^+) \\le {n}(u) - \\tau (u^+) + \\tau (u^-) = {n}(u) - \\tau (u).$ and, since the minimum point is unique, we infer that $u=u^+ \\in M$ .","To show (iii), take any $u_1, u_2\\in D(A)$ .","Clearly $(A u_k) \\circ i = {\\cal A} (\\tilde{u}_k)$ with $\\tilde{u}_k = i^{-1} (u_k)$ , for $k=1,2$ .","Therefore, by (a2), $\\langle A u_1-Au_2, u_1 - u_2 \\rangle _{X} = [Au_1 - Au_2] i(\\tilde{u}_1-\\tilde{u}_2)= \\langle {\\cal A} (\\tilde{u}_1) - {\\cal A} (\\tilde{u}_2), \\tilde{u}_1-\\tilde{u}_2\\rangle _Y \\ge 0.$ Hence $A$ is monotone and it is left to prove that $A$ is maximal monotone, i.e.","that additionally one has $A (D(A))=X^*$ .","To see it we choose any $\\tau \\in X^*$ and put $\\Phi :={a} - i^*(\\tau )$ .","$\\Phi $ is a convex coercive functional on the reflexive space $Y$ .","Hence it admits a miniumem, i.e.","there is a point $\\bar{u}\\in Y$ such that $D\\Phi (\\bar{u})=0$ , i.e.","${\\cal A}(\\bar{u}) = i^*(\\tau )$ .","This means that $u:=i(\\bar{u})\\in D(A)$ and that $A(u)=\\tau $ .","To show (iv) take any $\\tau \\in X^*$ and $\\alpha >0$ .","We proceed like in (iii), that is we consider a functional $\\Phi := {n}\\circ i + \\alpha {a} - i^*(\\tau )$ on $Y$ .","It is clear that $\\Phi $ – as a strictly convex and coercive functional on a reflexive Banach space – admits a minimum, i.e.","there exists $\\bar{u}\\in Y$ such that $D\\Phi (\\bar{u})=0$ .","This means that $i^*(N(i(\\bar{u}))) + \\alpha {\\cal A} (\\bar{u}) = i^*(\\tau )$ .","Subsequently, we deduce that ${\\cal A}(\\bar{u}) \\in i^*(X^*)$ , i.e.","$u:= i(\\bar{u})\\in D(A)$ and $N(u) + \\alpha A (u) = \\tau $ .","Moreover, observe that for each $u\\in D(A)$ such that $N(u) + \\alpha A (u) = \\tau $ , $i^{-1}(u)$ is a critical point of $\\Phi $ .","Since $\\Phi $ is strictly convex it has to be the unique minimum point.","(v) Suppose that a sequence $(\\tau _n)$ is bounded in $X^*$ and $(\\beta _n)$ is a sequence in $[\\alpha _1, \\alpha _2]$ .","Put $u_n:= J_{\\beta _n} (\\tau _n)$ , $n\\ge 1$ .","Then $i^*N(u_n) + \\beta _n {\\cal A} (\\bar{u}_n) = i^*(\\tau _n)$ , where $\\bar{u}_n:=i^{-1}(u_n)$ , $n\\ge 1$ .","Since $N$ is bounded and $\\beta _n > \\alpha _1>0$ for all $n\\ge 1$ , we infer that $({\\cal A}(\\bar{u}_n))$ is bounded.","Observe that, in view of (a2), $\\langle {\\cal A} (\\bar{u}_n) - {\\cal A}(0), \\bar{u}_n \\rangle _{Y} \\ge \\kappa (\\Vert \\bar{u}_n\\Vert _{Y})\\Vert \\bar{u}_n\\Vert _Y,$ i.e.","$\\Vert {\\cal A}(u_n) - {\\cal A}(0)\\Vert _{Y} \\ge \\kappa (\\Vert \\bar{u}_n\\Vert _{Y})$ .","Hence, by the boundedness of $({\\cal A}(\\bar{u}_n))$ and the assumed property of $\\kappa $ , $(\\bar{u}_n)$ is bounded.","Therefore $(u_n) = (i(\\bar{u}_n))$ is relatively compact, which together with Remark REF proves the assertion.","In order to prove (vi), take any $\\tau \\in M^*$ .","We need to show that $u:=J_{\\alpha } (\\tau ) \\in M$ .","In the proof of (iv) we have showed that $i^{-1} u$ is the unique minimum of the functional $\\Phi ={n}\\circ i + \\alpha {a} - i^*(\\tau )$ on $Y$ .","On the other hand, by use of (a3) and the definition of $M^*$ , one has $\\Phi (i^{-1} u^+ )\\!=\\!","{n} (u^+) \\!+\\!","\\alpha {a}(i^{-1} u^+)\\!","-\\!","\\tau (u^+)\\!\\le \\!","{n}(u) \\!+\\!","\\alpha {a} (i^{-1} u))\\!-\\!\\tau ( u^+) \\!+\\!","\\tau (u^-) = \\Phi (i^{-1}u).$ This means that $i^{-1} u=i^{-1} u^+$ and $u\\in M$ .","$\\square $"],["Elliptic problems with $p$ -Laplacian","Now we shall apply the above abstract setting from the previous section to the $p$ -Laplacian problem.","To this end fix $p>2$ , and put $X_p:= L^p (\\Omega ), \\ Y_p:=W_{0}^{1,p} (\\Omega ) \\mbox{ and } M_p:=\\lbrace u\\in X \\mid u(x)\\ge 0 \\mbox{ for a.e. }","x\\in \\Omega \\rbrace .$ Both, $X_p$ and $Y_p$ are reflexive and, by the Rellich-Kondrachov theorem, the natural embedding $i:Y_p\\rightarrow X_p$ is compact and dense.","It is easy to see that $M_p$ is a closed convex subset of $X_p$ .","Next define functionals ${a}:Y_p\\rightarrow \\mathbb {R}$ and ${n}:X_p\\rightarrow \\mathbb {R}$ by ${a} (u):= \\frac{1}{p} \\int _{\\Omega } |\\nabla u(x)|^p \\,\\mathrm {d}x, \\ u\\in Y_p,\\\\{n}(u):= \\frac{1}{p} \\int _{\\Omega } |u(x)|^p \\,\\mathrm {d}x, \\ u\\in X_p.$ We prove that these objects satisfy the abstract assumptions of the general setting.","Proposition 4.1 The functionals ${a}$ and ${n}$ with the cone $M_p$ satisfy all the assumptions (a1) – (a4) from Section and $\\langle D{a} (u), v\\rangle _{Y} = \\frac{1}{p} \\int _{\\Omega } |\\nabla u(x)|^{p-2} \\nabla u(x)\\cdot \\nabla v(x) \\,\\mathrm {d}x, \\ u, v \\in Y_p, \\\\\\langle D{a} (u) - D{a} (v), u-v \\rangle _Y \\ge 2^{2-p}\\Vert u-v\\Vert _{Y}^{p}, \\, u,v\\in Y_p, \\\\ \\langle D{n} (u), v\\rangle _{X} = \\frac{1}{p} \\int _{\\Omega } |u(x)|^{p-2} u(x) v(x) \\,\\mathrm {d}x, \\ u,v \\in X_p,\\\\\\langle D {n} (u) - D {n} (v), u-v \\rangle _X \\ge 2^{2-p}\\Vert u-v\\Vert _{X}^{p}, \\, u,v\\in X_p.$ Moreover, if $A_p:D(A_p) \\rightarrow X_p$ is defined, in analogy to (REF ), by $D(A_p):= i\\left( (D {a})^{-1} (i^*(X^*))\\right) \\mbox{ and } A_p u:= (i^*)^{-1}( D({a}) i^{-1} u), \\mbox{ for } u\\in D(A),$ then $A_p u= -\\mathrm {div} (|\\nabla u|^{p-2} \\nabla u), \\mbox{ for } u\\in D(A_p),$ where the divergence is meant in the distributional sense and $D(A_p)=\\lbrace u\\in W_{0}^{1,p} (\\Omega ) \\mid \\mathrm {div}(|\\nabla u|^{p-2}\\nabla u) \\mbox{ exists and belongs to } L^{p} (\\Omega )\\rbrace .$ Proof: In order to see (a1), note that the functionals ${a}$ and ${n}$ are clearly Gateaux differentiable with the formulas (REF ) and () satisfied.","Since these Gateaux derivatives are continuous the functionals are Frêchet differentiable.","The coercivity is immediate as ${a}(u) = (1/p)\\Vert u\\Vert _{Y_p}^{p}$ , $u\\in Y$ , and ${n}(u)=(1/p)\\Vert u\\Vert _{X_p}^{p}$ , $u\\in X$ .","One can check the condition (a2), i.e.","() and (), by use of the following inequality $(|x|^{p-2}x-|y|^{p-2}y)\\cdot (x-y)\\ge 2^{2-p} |x-y|^p \\mbox{ for any } x,y\\in \\mathbb {R}^M, \\, M\\ge 1.$ Obviously, for $\\kappa :[0,+\\infty )\\rightarrow [0, +\\infty )$ , given by $\\kappa (s):=2^{2-p} s^p$ , $s\\ge 0$ , one has $\\kappa ^{-1}(\\lbrace 0\\rbrace ) = \\lbrace 0\\rbrace $ , $\\lim \\limits _{s\\rightarrow +\\infty } \\kappa (s) = +\\infty $ .","As for (a3), take any $u\\in X$ .","Then taking $u_+ := \\max \\lbrace u , 0\\rbrace $ and $u_-:=\\max \\lbrace -u,0\\rbrace $ we have $u= u^+ - u^-$ and ${n}(u_+ ) = \\frac{1}{p} \\int _{\\Omega } |u_+ (x)|^p \\,\\mathrm {d}x \\le \\frac{1}{p} \\int _{\\Omega } |u (x)|^p \\,\\mathrm {d}x = {n}(u).$ If $u\\in Y_p = W_{0}^{1,p}(\\Omega )$ , then, due to Lemma 7.6 of [11], $\\nabla u_+(x) = 0$ if $u(x)\\le 0$ and $\\nabla u_+ (x)=\\nabla u(x)$ if $u(x)\\ge 0$ .","Therefore $u_+ \\in Y_p$ and ${a}(u_+) = \\frac{1}{p}\\int _{\\Omega }|\\nabla u_+(x)|^p \\,\\mathrm {d}x \\le \\frac{1}{p}\\int _{\\Omega }|\\nabla u (x)|^p \\,\\mathrm {d}x= {a}(u).$ Finally, (a4) is immediate as, for $u,v\\in M_p$ , $|u|^{p}= u^p \\le (u+v)^{p}=|u+v|^p$ .","$\\square $ In view of Section , the operators $A_p$ , $N_p:=D {n}$ together with $M_p$ and $M_p^*$ satisfy the assumptions made in Section and the topological degree can be applied for perturbations of $A_p$ .","Before we proceed further let us pay attention to the perturbation term.","Proposition 4.2 Let $f:\\Omega \\times [0,+\\infty )\\rightarrow \\mathbb {R}$ satisfy (REF ) and $f(x,0) \\ge 0$ for a.a. $x\\in \\Omega $ .","Then the mapping $F : X_p \\rightarrow X_{p}^{*}$ given by $\\langle F(u), v \\rangle _{X_p}:= \\int _{\\Omega } f(u(x))v(x) \\,\\mathrm {d}x, \\ \\ u\\in M_p, \\, v\\in X_p,$ is well defined, continuous, bounded on bounded sets and $F(N^{-1}(\\tau )) \\in T_{M_{p}^*} (\\tau ) \\ \\ \\mbox{ for any } \\ \\tau \\in M_p^{*}.$ Lemma 4.3 Let $10$ .","This implies that $v_n \\rightarrow v$ in $L^q(\\Omega )$ , i.e.","$v\\in T_{M_q} (u)$ .","In order to show the converse inclusion, observe that $T_u$ is closed and, for any $h>0$ , $h(M-u) \\subset T_u$ .","This clearly implies that $T_{M_q} (u)\\subset T_u$ .","$\\square $ Proof of Proposition REF: Using the Riesz representation isomorphism $\\varrho $ between $L^p (\\Omega )^*$ and $L^q(\\Omega )$ , $1/p +1/q=1$ , the mapping $F\\circ N^{-1}$ can be treated as the mapping $L^q(\\Omega )\\ni u \\mapsto f(\\cdot , \\theta _q (u)) \\in L^q(\\Omega )$ where $\\theta _q:\\mathbb {R}\\rightarrow \\mathbb {R}$ is given by $\\theta _q (s)=|s|^{q-2}s$ , $s\\in \\mathbb {R}$ .","It is well defined as $|(f(x,\\theta _q(s))| \\le C(1+|\\theta _q (s)|^p) = C(1+|s|) \\mbox{ for } s\\ge 0 \\mbox{ and a.e. }","x\\in \\Omega .$ Observe that $f(x,\\theta _q (0))=f(x,0)\\ge 0$ for.","a.e.","$x\\in \\Omega $ , which, by use of Lemma REF , implies that $f(\\cdot , \\theta _q (u(\\cdot )))\\in T_{M_q}(u)$ for all $u\\in M_q$ .","Since $\\varrho (M_p^*)=M_q$ , we infer that (REF ) holds.","$\\square $ Hence we have showed that the problem (REF ) indeed can be formulated as an abstract problem $\\left\\lbrace \\begin{array}{l}A_p (u) = F (u),\\\\u\\in M_p \\cap D(A_p).","\\end{array}\\right.$ In order to take advantage of the topological degree effectively we need some methods of computing it.","Theorem 4.4 If $\\ 2 \\lambda _{1,p}$ for a.e.","$x\\in \\Omega $ , or $\\rho (x)< \\lambda _{1,p}$ for a.e.","$x\\in \\Omega $ , then $\\mathrm {Deg}_{M_p} (A_p, \\rho N_{p}, B_{M_p} (0,R)) = \\left\\lbrace \\begin{array}{cl}1, & \\mbox{ if } \\ \\ \\rho (x) < \\lambda _{1,p} \\ \\ \\mbox{ for a.e. }","\\ \\ x\\in \\Omega ,\\\\0, & \\mbox{ if } \\ \\ \\rho (x) > \\lambda _{1,p} \\ \\ \\mbox{ for a.e. }","\\ \\ x\\in \\Omega .\\end{array}\\right.$ Remark 4.5 Before passing to the proof of Theorem REF , we need to make a comment on the eigenvalue problem relating to the $p$ -Laplace operator.","Solving the nonlinear eigenvalue problem $\\left\\lbrace \\begin{array}{l}A_p (u) = \\lambda N_p(u)\\\\u\\in M_p \\cap D(A_p)\\end{array} \\right.$ reduces to find nonnegative weak solutions $u\\in W^{1,p}(\\Omega )$ of $\\left\\lbrace \\begin{array}{cl}-\\mathrm {div} (|\\nabla u(x)|^{p-2} \\nabla u(x)) = \\lambda |u(x)|^{p-2} u(x), & x\\in \\Omega ,\\\\u(x) = 0, & x\\in \\partial \\Omega .","\\\\\\end{array}\\right.$ It appears that some properties of the eigenvalue problem for the Laplace operator are also valid for the $p$ -Laplace one.","For details we refer to [14], [15] and [16].","In particular, it is known that (REF ) does not admit any nonzero solutions if $\\lambda \\le 0$ , i.e.","the $p$ -Laplace has no nonpositive eigenvalues.","Moreover, there exists the smallest eigenvalue $\\lambda _{1,p}$ given by the Rayleigh formula $\\lambda _{1,p} = \\inf \\limits _{u\\in W_{0}^{1,p} (\\Omega ), u\\ne 0} \\frac{\\int _{\\Omega } |\\nabla u(x)|^p \\,\\mathrm {d}x}{\\int _{\\Omega } |u(x)|^p \\,\\mathrm {d}x}.$ The eigenfunctions corresponding to $\\lambda _{1,p}$ are either strictly positive or negative in $\\Omega $ and belong to $L^\\infty (\\Omega )$ .","Moreover, $\\lambda _{1,p}$ is an isolated eigenvalue and if there are two eigenfunctions $u,v$ for $\\lambda _{1,p}$ , then there exists $\\alpha \\in \\mathbb {R}$ such that $u=\\alpha v$ .","It is also known that if any eigenfunction does not change its sign in $\\Omega $ , then the corresponding eigenvalue must be equal to $\\lambda _{1,p}$ .","$\\square $ In the proof we shall use a few lemmata given below.","Lemma 4.6 There are $ C, s > 0$ such that $\\Vert u\\Vert _{L^p} \\le C |\\tilde{\\Omega }|^{s} \\Vert \\nabla u\\Vert _{L^p}$ for all $u\\in W_{0}^{1,p}(\\Omega )$ and measurable $\\tilde{\\Omega }\\subset \\Omega $ with the property $u(x)=0$ if $x\\notin \\tilde{\\Omega }$ .","Proof: By the Sobolev embedding theorem there exists $q>p$ such that $\\Vert u \\Vert _{L^q} \\le C\\Vert \\nabla u\\Vert _{p} \\mbox{ for all } u\\in W_{0}^{1,p}(\\Omega ).$ On the other hand, by the Hölder inequality, $\\Vert u\\Vert _{L^p} \\le \\Vert u \\Vert _{L^q} |\\tilde{\\Omega }|^{1/p-1/q}.$ Combining the two above inequalities we get the desired one with $s:=1/p-1/q$ .","$\\square $ Lemma 4.7 Let $v$ be a nonnegative weak solution of (REF ) with $\\lambda =\\lambda _{1,p}$ and $\\rho \\in L^{\\infty }(\\Omega )$ .","If $u\\in W_{0}^{1,p}(\\Omega )$ is a weak solution to $-\\mathrm {div} (|\\nabla u|^{p-2} \\nabla u) = \\rho |u|^{p-2} u + |v|^{p-2}v \\, \\mbox{ on } \\Omega ,$ then $u\\in L^{\\infty }(\\Omega )$ .","Proof: Here we adapt the arguments from [15].","Note that without loss of generality we can consider the equation $-\\mathrm {div} (|\\nabla u|^{p-2} \\nabla u) = \\rho |u|^{p-2} u + \\lambda _{1,p} |v|^{p-2} v \\, \\mbox{ on } \\Omega .$ Take any $k>0$ and put $\\eta :=\\max \\lbrace u-v-k,0\\rbrace $ .","Since $\\eta \\in W_{0}^{1,p}(\\Omega )$ , we get $\\int _{\\Omega _k} (|\\nabla u|^{p-2}\\nabla u - |\\nabla v|^{p-2} \\nabla v) \\cdot \\nabla (u-v)\\,\\mathrm {d}x \\le \\Vert \\rho \\Vert _{L^{\\infty }} \\int _{\\Omega _k} u^{p-1}(u-v-k) \\,\\mathrm {d}x$ with $\\Omega _k := \\lbrace x\\in \\Omega \\mid u(x)-v(x)-k > 0 \\rbrace $ .","This, by use of (REF ) and the convexity of the function $s\\mapsto |s |^{p-1}$ , gives $\\int _{\\Omega _k} |\\nabla (u-v)|^p \\,\\mathrm {d}x & \\le & C_1 \\int _{\\Omega _k} u^{p-1} (u-v-k) \\,\\mathrm {d}x \\\\& \\le & C_1 2^{p-2} \\left( \\int _{\\Omega _k} (u-v-k)^p \\,\\mathrm {d}x + \\int _{\\Omega _k} (v+k)^{p-1}(u-v-k) \\,\\mathrm {d}x\\right)$ for some constant $C_1>0$ (here all the constant are to be independent of $k$ ).","By applying Lemma REF , one gets $\\int _{\\Omega _k} (u-v-k)^p \\,\\mathrm {d}x \\le C|\\Omega _k|^{s} \\int _{\\Omega _k} |\\nabla (u-v)|^{p} \\,\\mathrm {d}x,$ which together with the previous inequality yields $(1- C_2|\\Omega _k|^{s}) \\int _{\\Omega _k} (u-v-k)^p \\,\\mathrm {d}x \\le C_2 |\\Omega _k|^{s} \\int _{\\Omega _k} (v+k)^{p-1}(u-v-k) \\,\\mathrm {d}x$ for some $C_2>0$ .","Since $|\\Omega _k|\\rightarrow 0$ as $k\\rightarrow +\\infty $ , there is $k_0$ such that for all $k\\ge k_0$ $1-C_2 |\\Omega _k|^{s} >1/2$ .","Further, for $k\\ge k_0$ , $\\int _{\\Omega _k} (u-v-k)^p \\,\\mathrm {d}x \\le 2 C_2 |\\Omega _k|^{s} (\\Vert v\\Vert _{L^{\\infty }} + k)^{p-1} \\int _{\\Omega _k} (u-v-k) \\,\\mathrm {d}x.$ Next we observe that the Hölder inequality yields $\\int _{\\Omega _k} (u-v-k) \\,\\mathrm {d}x \\le C_4 k |\\Omega _k|^{1+s(p-1)^{-1}} \\mbox{ for all } k\\ge k_0$ and some constant $C_4>0$ .","Now define $j:(0,+\\infty )\\rightarrow [0,+\\infty )$ by $j(k):= \\int _{\\Omega _k} (u-v-k) \\,\\mathrm {d}x, \\, \\, k>0.$ Note that by the Tonelli-Fubini theorem applied to the set $\\lbrace (x,t)\\in \\Omega \\times [0,+\\infty )\\mid u(x)-v(x)>t>k \\rbrace $ one has $j(k) = \\int _{k}^{+\\infty } |\\Omega _t| \\,\\mathrm {d}t, \\,\\, k> 0.$ Obviously, $j$ is nonincreasing and absolutely continuous with $j^{\\prime }(k)=-|\\Omega _k|$ for a.e.","$k\\ge 0$ .","We claim that $j(k)=0$ for some $k>0$ .","If it were not so, then (REF ) could be rewritten as $j(k)^{\\theta } \\le -C_{4}^{\\theta } k^{\\theta } j^{\\prime }(k) \\mbox{ for all } k\\ge k_0$ with $\\theta :=(1+s(p-1)^{-1})^{-1}$ , and consequently $k^{-\\theta } \\le - C_{4}^{\\theta } j(k)^{-\\theta } j^{\\prime }(k) \\mbox{ for all } k\\ge k_0.$ This after integration would give $k^{1-\\theta } + C_{4}^{\\theta } j(k)^{1-\\theta } \\le k_0^{1-\\theta } + C_{4}^{\\theta } j(k_0)^{1-\\theta } \\mbox{ for all } k\\ge k_0,$ which yields a contradiction proving the claim that $j(k)=0$ for some $k>0$ .","Then, for some $k>0$ , $|\\Omega _k|=0$ and $u\\le v+k$ a.e.","on $\\Omega $ .","This shows that $u\\in L^{\\infty }(\\Omega )$ , as $v\\in L^{\\infty }(\\Omega )$ (see Remark REF ).","$\\square $ Lemma 4.8 (see [10]) If $h\\in L^\\infty (\\Omega )$ is nonnegative and nonzero, then the equation $-\\mathrm {div} (|\\nabla u|^{p-2} \\nabla u) = \\lambda _{1,p} |u|^{p-2}u + h, \\, \\mbox{ on } \\Omega ,$ has no nonzero weak solution in $W_{0}^{1,p}(\\Omega )$ .","Lemma 4.9 If $\\rho \\in L^\\infty (\\Omega )$ and either $\\rho (x)> \\lambda _{1,p}$ for a.e.","$x\\in \\Omega $ or $\\rho (x)<\\lambda _{1,p}$ for a.e.","$x\\in \\Omega $ , then the problem $-\\mathrm {div}(|\\nabla u|^{p-2}\\nabla u) = \\rho |u|^{p-2}u \\mbox{ on } \\Omega $ does not admit a nonzero solution $u\\in W_{0}^{1,p}(\\Omega )$ such that $u\\ge 0$ .","Proof: If $\\rho <\\lambda _{1,p}$ a.e.","on $\\Omega $ and $u\\in W_{0}^{1,p}(\\Omega )$ is a nonzero weak solution of (REF ), then $\\int _{\\Omega } |\\nabla u|^{p} \\,\\mathrm {d}x = \\int _{\\Omega } \\rho |u|^{p-2}u \\,\\mathrm {d}x < \\lambda _{1,p} \\int _{\\Omega } |u|^{p-2}u \\,\\mathrm {d}x,$ which gives $\\lambda _{1,p}> \\int _{\\Omega } |\\nabla u|^p \\,\\mathrm {d}x / \\int _{\\Omega } |u|^p \\,\\mathrm {d}x$ , a contradiction with the Rayleigh formula.","In the case $\\rho >\\lambda _{1,p}$ a.e.","on $\\Omega $ , we observe that if $u$ is a weak solution of (REF ), then $u$ is a weak solution of $-\\mathrm {div}(|\\nabla u|^{p-2}\\nabla u) =\\lambda _{1,p} |u|^{p-2}u + h \\ \\mbox{ on } \\ \\Omega $ with $h:=(\\rho - \\lambda _{1,p})|u|^{p-2}u$ .","Clearly, $h\\ge 0$ and $h\\in L^{\\infty }(\\Omega )$ , since $u\\in L^{\\infty }(\\Omega )$ due to Lemma REF .","Hence, Lemma REF leads to a contradiction ending the proof.","$\\square $ Proof of Theorem REF : Assume that $\\rho >\\lambda _{1,p}$ a.e.","on $\\Omega $ and fix $\\tilde{\\lambda }>\\lambda _{1,p}$ .","Define $H:X_p \\times [0,1]\\rightarrow X_p$ by $H (u,t) := (t\\tilde{\\lambda }+ (1-t) \\rho ) N_p(u)$ , $u\\in X_p$ , $t\\in [0,1]$ .","In view of Lemma REF , $-A_p (u)+H(u,t)\\ne 0$ for all $u\\in D(A_p) \\setminus \\lbrace 0 \\rbrace $ and $t\\in [0,1]$ .","Therefore, we can use the homotopy invariance – Theorem REF (iii) to get $\\mathrm {Deg}_{M_p} (A_p, \\rho N_{p}, B_{M_p} (0,R) ) = \\mathrm {Deg}_{M_p} ( A_p, \\tilde{\\lambda }N_p, B_{M_p}(0,R) ).$ In a similar manner one can prove the same formula in the case $\\rho <\\lambda _{1,p}$ a.e.","on $\\Omega $ with $\\tilde{\\lambda }<\\lambda _{1,p}$ .","Now we shall prove that conditions $({\\cal M}_1)$ and $({\\cal M}_2)$ of Theorem REF are satisfied.","Observe that, in view of Lemma REF , for any $\\lambda \\ne \\lambda _{1,p}$ , the eigenvalue problem (REF ) has no nontrivial and nonnegative weak solutions, i.e.","$({\\cal M}_1)$ holds.","To show $({\\cal M}_2)$ let $\\tau _0\\in L^p(\\Omega )$ be the functional determined by $|u_0|^{p-2}u_0$ with $u_0$ being a fixed positive solution of the eigenvalue problem (REF ) with $\\lambda =\\lambda _{1,p}$ .","Suppose that there exists $u\\in (A_p-\\lambda N_p)^{-1}( \\lbrace \\tau _0 \\rbrace ) \\cap M_p$ for some $\\lambda > \\lambda _{1,p}$ .","This means that $u\\in W_{0}^{1,p}(\\Omega )$ is a nonnegative weak solution of $-\\mathrm {div} (|\\nabla u|^{p-2} \\nabla u) = \\lambda _{1,p} |u|^{p-2} u + h \\ \\mbox{ on } \\ \\Omega $ with $h:=(\\lambda -\\lambda _{1,p}) |u|^{p-2} u + |u_0|^{p-2}u_0$ .","It follows from Lemma REF that $h\\in L^{\\infty } (\\Omega )$ .","Since $h\\ge 0$ , Lemma REF implies that such a solution does not exist, a contradiction proving $({\\cal M}_2)$ .","Hence, by Theorem REF and (REF ), the desired formula follows.","$\\square $ The obtained formula results in the following general one.","Theorem 4.10 Let $f$ and $F$ be as in Proposition REF and suppose that (REF ) hold.","(i) If $\\rho _0$ is as in () and either $\\rho _0 (x) < \\lambda _{1,p}$ , for a.e.","$x\\in \\Omega $ , or $\\lambda _{1,p}<\\rho _0 (x)$ , for a.e.","$x\\in \\Omega $ , then there exists $\\delta >0$ such that $A_p (u) \\ne F (u) \\mbox{ for all } u \\in D(A_p) \\cap \\left( B_{M_p} (0,\\delta ) \\setminus \\lbrace 0 \\rbrace \\right)$ and $\\mathrm {Deg}_M (A, F, B_M (0,\\delta )) = \\left\\lbrace \\begin{array}{cl}1, & \\ \\ \\mbox{ if } \\ \\ \\rho _0(x) < \\lambda _{1,p} \\ \\ \\mbox{ for a.e. }","x\\in \\Omega ,\\\\0, & \\ \\ \\mbox{ if } \\ \\ \\rho _0(x) > \\lambda _{1,p} \\ \\ \\mbox{ for a.e. }","x\\in \\Omega .\\end{array}\\right.$ (ii) If $\\rho _\\infty $ is as in () either $\\rho _\\infty (x) < \\lambda _{1,p}$ , for a.e.","$x\\in \\Omega $ , or $\\lambda _{1,p}<\\rho _\\infty (x)$ , for a.e.","$x\\in \\Omega $ , then there exists $R>0$ such that $A_p (u) \\ne F(u) \\mbox{ for all } u \\in D(A_p) \\cap \\left( M_p\\setminus B_{M_p} (0,R)\\right)$ and $\\mathrm {Deg}_{M_p} (A_p, F, B_{M_p} (0,R)) = \\left\\lbrace \\begin{array}{cl}1, & \\mbox{ if } \\ \\ \\rho _\\infty (x)< \\lambda _{1,p} \\ \\ \\mbox{ for a.e. }","x\\in \\Omega ,\\\\0, & \\mbox{ if } \\ \\ \\rho _\\infty (x)> \\lambda _{1,p} \\ \\ \\mbox{ for a.e. }","x\\in \\Omega .\\end{array}\\right.$ Proof: (i) Define $H:M_p\\times [0,1]\\rightarrow X_p$ by $H(u,t):= t F(u) + (1-t) \\rho _0 N_{p} (u),$ $(u,t)\\in M_p \\times [0,1]$ .","By Proposition REF , $H$ is continuous and $F\\circ N_{p}^{-1}$ is tangent to $M^*$ .","Moreover we claim that $\\mbox{ there is $\\delta >0$such that $-A_p (u)+H(u,t) \\ne 0$ for all $u\\in M_p \\cap D(A_p), \\ t\\in [0,1]$.","}$ Suppose to the contrary that there exists $(u_n)$ in $(M_p\\cap D(A_p))\\setminus \\lbrace 0 \\rbrace $ and $(t_n)$ in $[0,1]$ such that $u_n \\rightarrow 0$ in $X_p$ and $-A_p (u_n) + H(u_n, t_n)=0$ , $n\\ge 1$ .","Then clearly, if we put $w_n:= \\Vert u_n\\Vert _{X_p}^{-1}u_n$ and $s_n:=\\Vert u_n\\Vert _{X_p}$ , then $A_p(w_n) = s_{n}^{1-p} H(s_n w_n, t_n)$ , which gives $w_n =J_{1} \\left( N_p (w_n) + s_{n}^{1-p} H(s_n w_n, t_n) \\right), \\ n\\ge 1.$ The growth condition (REF ) and the existence of the first limit in () imply that there exists $C_1>0$ such that $\\Vert N_p (w_n) + s_{n}^{1-p} H(s_n w_n, t_n)\\Vert _{X_{p}^{*}}\\le C_1$ for all $n\\ge 1$ .","Therefore we infer that $(w_n)$ has a subsequence convergent in $X_p$ , since, according to Proposition REF and Proposition REF (v), $J_1$ is completely continuous.","In the sequel, we may assume that $(w_n)$ converges almost everywhere to some $w_0\\in M_p \\setminus \\lbrace 0 \\rbrace $ and that one has $g\\in X_p$ such that $|w_n|\\le g$ a.e.","on $\\Omega $ .","Further, note that if $w_n (x)\\ne 0$ , then $\\frac{f(x,s_n w_n(x))}{s_{n}^{p-1}} = \\frac{f(x, s_n w_n (x))}{(s_n w_n (x))^{p-1}} (w_n(x))^{p-1} \\rightarrow \\rho _0 (x) (w_0(x))^{p-1} \\mbox{ as } n\\rightarrow +\\infty ,$ which, by the dominated convergence theorem, implies that $s_{n}^{1-p} H(s_n w_n, t_n)\\rightarrow \\rho _0 N_p(w_0)$ in $X_p^{*}$ .","Hence, a passage to the limit in (REF ) yields $w_0 = J_1 (N_p (w_0)+ \\rho _0 N_p (w_0) )$ , i.e.","$-A_p w_0 + \\rho _0 N_p (w_0)=0$ .","This is a contradiction due to Lemma REF and (REF ) is proved.","Clearly, (REF ) allows us to use the homotopy invariance – Theorem REF (iii) to see that $\\mathrm {Deg}_{M_p} (A_p, F, B_{M_p} (0,R)) = \\mathrm {Deg}_{M_p} (A_p, \\rho _0 N_p, B_{M_p} (0,R))$ , which together with Theorem REF provides the required formula.","(ii) The proof is analogical to that for part (i) and it is left to the reader.","$\\square $ Proof of Theorem REF: Let $\\delta >0$ and $R>\\delta $ be like in Theorem REF .","Then by use of the additivity property – Theorem REF (ii), we get $\\mathrm {Deg}_{M_p} (A_p, F, B_{M_p} (0,R) \\setminus \\overline{B_{M_p} (0,\\delta )})= \\mathrm {Deg}_{M_p} (A_p, F, B_{M_p} (0,R))\\!","-\\!","\\mathrm {Deg}_{M_p} (A_p, F, B_{M_p} (0,\\delta )) \\\\= \\left\\lbrace \\begin{array}{cl}1, & \\mbox{ if } \\ \\rho _0 (x) > \\lambda _{1,p} > \\rho _\\infty (x) \\ \\mbox{ for a.e. }","x\\in \\Omega ,\\\\-1, & \\mbox{ if } \\ \\rho _0 (x) < \\lambda _{1,p} < \\rho _\\infty (x) \\ \\mbox{ for a.e. }","x\\in \\Omega .\\end{array}\\right.$ Hence the existence property of the topological degree gives the existence of $u\\in B_{M_p} (0,R) \\setminus \\overline{B_{M_p} (0,\\delta )}$ such that $A_p (u) = F(u)$ , which is a required nonzero nonnegative weak solution of (REF ).","$\\square $"]],"string":"[\n [\n \"Positive stationary solutions for p-Laplacian problems with nonpositive\\n perturbation\"\n ],\n [\n \"Abstract The paper is devoted to the existence of positive solutions of nonlinear elliptic equations with $p$-Laplacian.\",\n \"We provide a general topological degree that detects solutions of the problem $$ \\\\{{array}{l} A(u)=F(u) u\\\\in M {array}.\",\n \"$$ where $A:X\\\\supset D(A)\\\\to X^*$ is a maximal monotone operator in a Banach space $X$ and $F:M\\\\to X^*$ is a continuous mapping defined on a closed convex cone $M\\\\subset X$.\",\n \"Next, we apply this general framework to a class of partial differential equations with $p$-Laplacian under Dirichlet boundary conditions.\"\n ],\n [\n \"Introduction\",\n \"We shall be concerned with solutions to the following nonlinear boundary value problem $\\\\left\\\\lbrace \\\\begin{array}{l}-\\\\mathrm {div} (|\\\\nabla u(x)|^{p-2} \\\\nabla u(x)) = f(x,u (x)), \\\\ x\\\\in \\\\Omega ,\\\\\\\\\\\\ \\\\ u(x)\\\\ge 0,\\\\ x\\\\in \\\\Omega ,\\\\\\\\\\\\ \\\\ u(x) = 0,\\\\ x\\\\in \\\\partial \\\\Omega \\\\end{array} \\\\right.$ where $\\\\Omega \\\\subset \\\\mathbb {R}^N$ ($N\\\\ge 1$ ) is a bounded domain with the smooth boundary $\\\\partial \\\\Omega $ , $p\\\\ge 2$ and $f:\\\\Omega \\\\times [0,+\\\\infty ) \\\\rightarrow \\\\mathbb {R}$ is a Carathéodory function (Recall that we say that $f:\\\\Omega \\\\times [0,+\\\\infty )\\\\rightarrow \\\\mathbb {R}$ is a Carathéodory function if $f(\\\\cdot , s)$ is measurable for all $s\\\\in [0,+\\\\infty )$ and $f(x,\\\\cdot )$ is continuous for almost all $x\\\\in \\\\Omega $ .\",\n \").\",\n \"The differential term $\\\\mathrm {div} (|\\\\nabla u (x)|^{p-2}\\\\nabla u(x))$ is referred to as the $p$ -Laplacian of $u$ at a point $x\\\\in \\\\Omega $ .\",\n \"We search for weak solutions in the Sobolev space $W_{0}^{1,p}(\\\\Omega )$ , i.e.\",\n \"$u\\\\in W_{0}^{1,p} (\\\\Omega )$ such that $\\\\int _{\\\\Omega } |\\\\nabla u(x)|^{p-2} \\\\nabla u(x) \\\\cdot \\\\nabla v(x) \\\\,\\\\mathrm {d}x = \\\\int _{\\\\Omega } f(x, u(x)) v(x) \\\\,\\\\mathrm {d}x \\\\ \\\\ \\\\mbox{ for all } \\\\ v\\\\in W_{0}^{1,p} (\\\\Omega ).$ Such boundary problems with $p$ -Laplace were widely studied by many authors who used various methods.\",\n \"Let us mention just a few.\",\n \"Equations with the one dimensional $p$ -Laplacian, i.e.\",\n \"when $N=1$ , were studied by Manásevich, Njoku i Zanolin [17], Drábek, García-Huidobro and Manásevich [7] and as well as by Kryszewski and the author [5].\",\n \"In the general case, i.e.\",\n \"when $N>1$ , positive solutions of $p$ -Laplace problems have been studied by a number of authors, e.g.\",\n \"Huang [13], Drábek and Pohozaev [6], Cañada, Drábek and Gámez [3], Filippiakis, Gasiński and Papageorgiou [9] or Montreanu D., Montreanu V. V. and Papageorgiou [18], Väth [19].\",\n \"Generally speaking, in the above mentioned papers, either $N=1$ or $N$ is arbitrary but the right has side of the equation - the function $f$ is assumed to be non-negative or satisfy some monotonicity assumptions.\",\n \"This makes possible to apply Krasnosel'skii's fixed point theorem (in general, fixed point index in cones) or variational methods.\",\n \"These assumptions on $f$ seem rather restrictive and sometimes unnatural, especially, when we take into account physical interpretation of the considered boundary value problem.\",\n \"In this paper, we do not require $f$ to be non-negative or monotone.\",\n \"A general tool for detection of nonnegative solutions is provided.\",\n \"It is based on the geometric idea of tangency and using fixed point index in cones.\",\n \"We construct a topological degree for perturbations of maximal monotone operators with respect to closed convex cones.\",\n \"Next we prove appropriate index formulae, which together with the homotopy property, allow us to compute the topological degree in specific examples.\",\n \"It is noteworthy, that this setting does not require variational structure and can be also used for systems of $p$ -Laplace problems.\",\n \"In this paper, we apply the method to show the following existence criterion Theorem 1.1 Suppose that a Carathéodory function $f:\\\\Omega \\\\times [0,+\\\\infty )\\\\rightarrow \\\\mathbb {R}$ and $\\\\rho _0, \\\\rho _\\\\infty \\\\in L^\\\\infty (\\\\Omega )$ satisfy the following conditions $ & & \\\\mbox{there is } C>0 \\\\mbox{ such that }|f(x,s)| \\\\le C(1+s^{p-1}) \\\\mbox{ for all } s\\\\ge 0 \\\\mbox{ and a.a. } x\\\\in \\\\Omega ;\\\\\\\\ & & \\\\lim _{s\\\\rightarrow 0^+} \\\\frac{f(x,s)}{s^{p-1}} = \\\\rho _0 (x)\\\\mbox{ and } \\\\lim _{s\\\\rightarrow \\\\infty } \\\\frac{f(x,s)}{s^{p-1}} = \\\\rho _\\\\infty (x) \\\\mbox{ uniformly with respect to } x\\\\in \\\\Omega .$ If the principal eigenvalue $\\\\lambda _{1,p}$ of the $p$ -Laplace operator lies between $\\\\rho _0$ and $\\\\rho _\\\\infty $ , i.e.\",\n \"either $\\\\rho _0(x) < \\\\lambda _{1,p} <\\\\rho _\\\\infty (x)$ , for a.a. $x\\\\in \\\\Omega $ , or $ \\\\rho _\\\\infty (x) < \\\\lambda _{1,p} < \\\\rho _0 (x)$ , for a.a. $x\\\\in \\\\Omega $ , then the problem (REF ) admits a nontrivial weak solution $u\\\\in W_{0}^{1,p} (\\\\Omega )$ such that $u(x)\\\\ge 0$ for a.e.\",\n \"$x\\\\in \\\\Omega $ .\",\n \"Here the principal eigenvalue $\\\\lambda _{1,p}$ is the smallest real number $\\\\lambda $ such that the problem $\\\\left\\\\lbrace \\\\begin{array}{l}-\\\\mathrm {div} (|\\\\nabla u(x)|^{p-2} \\\\nabla u(x)) = \\\\lambda |u (x)|^{p-2} u(x),\\\\ x\\\\in \\\\Omega \\\\\\\\\\\\ \\\\ u(x)\\\\ge 0,\\\\ x\\\\in \\\\Omega \\\\\\\\\\\\ \\\\ u(x) = 0,\\\\ x\\\\in \\\\partial \\\\Omega \\\\end{array} \\\\right.$ admits a nonzero weak solution (see Remark REF for more details).\",\n \"Theorem REF corresponds directly to the result of [13], obtained by different methods (the sub-supersolution technique and the existence result for variational inequalities) and under different assumptions corresponding to the inequality $\\\\rho _\\\\infty < \\\\lambda _{1,p} <\\\\rho _0$ .\",\n \"Our general method allows us to consider also the case $\\\\rho _\\\\infty > \\\\lambda _{1,p}> \\\\rho _0$ .\",\n \"The paper is organized as follows.\",\n \"In Section 2 we develop a topological degree detecting coincidence points of maximal monotone operators and continuous operators in closed convex cones.\",\n \"This general tool will be useful if we rewrite the problem (REF ) in the form $\\\\left\\\\lbrace \\\\begin{array}{l} A_p u = N_f (u)\\\\\\\\u\\\\in M_p\\\\end{array}\\\\right.$ where $A_p:L^p(\\\\Omega ) \\\\supset D(A_p)\\\\rightarrow L^p (\\\\Omega )^*$ is the maximal monotone operator determined by the $p$ -Laplacian, $N_f:L^p(\\\\Omega )\\\\rightarrow L^p (\\\\Omega )^*$ is the Nemytzkii type operator associated with $f$ and $M_p$ is the closed convex cone of all non-negative elements in the space $L^p (\\\\Omega )$ .\",\n \"Section 3 provides a general setting in which assumptions of Section 2 are verified.\",\n \"Next, in Section 4 we show that the problem (REF ) falls into the setting and, using our topological degree together with spectral properties of $p$ -Laplacian, we derive topological index formulae.\",\n \"They turn out to be essential in the proof of Theorem REF , which is provided at the end of Section 4.\",\n \"Notation If $X$ is a metric space and $B \\\\subset X$ , then $\\\\partial B$ and $cl B$ stand for the boundary of $B$ and the closure of $B$ , respectively.\",\n \"If $x_0\\\\in X$ and $r>0$ , then $B (x_0,r):=\\\\lbrace x\\\\in M\\\\mid d(x,x_0)0$ , $J_\\\\alpha (\\\\tau ):= u, \\\\ \\\\mbox{ where } u\\\\in D(A) \\\\mbox{ is the unique element such that } \\\\tau \\\\in (N+\\\\alpha A)(u),$ are well defined and continuous; $({\\\\cal A}_2)$ the mapping ${\\\\cal J}:X^* \\\\times (0,+\\\\infty )\\\\ni (\\\\tau , \\\\alpha )\\\\mapsto J_{\\\\alpha } (\\\\tau )\\\\in X$ is bounded on bounded sets and such that ${\\\\cal J}_{\\\\ |\\\\, X^* \\\\times [\\\\alpha _1, \\\\alpha _2]}$ is completely continuous if $0<\\\\alpha _1 \\\\le \\\\alpha _2$ ; $({\\\\cal A}_3)$ $M\\\\subset X$ is a neighborhood retract of $X$ , $J_\\\\alpha (N(M)) \\\\subset M$ for $\\\\alpha >0$ , and $M^* :=N(M)$ is an ${\\\\cal L}$ -retract (see [2] and [5]), i.e.\",\n \"there exist a retraction $r:B(M^*,\\\\eta )\\\\rightarrow M^*$ with some $\\\\eta >0$ and a constant $L>0$ such that $\\\\Vert r(\\\\tau )- \\\\tau \\\\Vert \\\\le L d_{M^*} (\\\\tau ) \\\\ \\\\mbox{ for all } \\\\ \\\\tau \\\\in B(M^*, \\\\eta );$ $({\\\\cal A}_4)$ $F$ is continuous, bounded on bounded sets and satisfies the tangency condition $F(N^{-1}(\\\\tau ))\\\\in T_{M^{*}} (\\\\tau ), \\\\ \\\\mbox{ for } \\\\tau \\\\in N(\\\\overline{U}),$ where $T_{M^*} (\\\\tau )$ is the Bouligand tangent cone to $M^*$ at the point $\\\\tau $ , i.e.\",\n \"$T_{M^*} (\\\\tau ): = \\\\left\\\\lbrace \\\\theta \\\\in X^* \\\\mid \\\\liminf _{\\\\alpha \\\\rightarrow 0^+} \\\\frac{d_{M^*} (\\\\tau + \\\\alpha \\\\theta ) }{\\\\alpha } =0 \\\\right\\\\rbrace .$ Remark 2.1 Since maximal monotone operators have closed graphs, it can be shown that in order to verify the continuity of the mapping ${\\\\cal J}_{\\\\ |\\\\, X^* \\\\times [\\\\alpha _1, \\\\alpha _2]}$ from condition $({\\\\cal A}_2)$ it is sufficient to know that it maps bounded sets into relatively compact ones.\",\n \"Our goal is to transform the problem (REF ) into a fixed point one in $M$ and for which fixed point index theory can be used.\",\n \"To this end define $\\\\Phi _{\\\\alpha }=\\\\Phi _{\\\\alpha }^{A,F}:\\\\overline{U} \\\\rightarrow M$ by $\\\\Phi _\\\\alpha (u):= J_{\\\\alpha } \\\\left(r \\\\left(N(u) + \\\\alpha F(u)\\\\right)\\\\right), \\\\ \\\\ u\\\\in \\\\overline{U},$ whenever $0<\\\\alpha < \\\\eta /\\\\sup \\\\lbrace \\\\Vert F(u)\\\\Vert \\\\mid u\\\\in \\\\overline{U} \\\\rbrace $ .\",\n \"Obviously, it is well defined, since for such $\\\\alpha $ one has $(N+\\\\alpha F) (\\\\overline{U}) \\\\subset B(M^*,\\\\eta )$ .\",\n \"Moreover, observe that due to the assumptions, the mapping $r \\\\circ (N+\\\\alpha F)$ is bounded on bounded sets and, by $({\\\\cal A}_2)$ , $\\\\Phi _\\\\alpha $ is compact.\",\n \"Exploiting the tangency condition (REF ) and the inequality (REF ) together with compactness, we obtain the following localization of fixed points results.\",\n \"Proposition 2.2 If $K\\\\subset \\\\overline{U}$ is a closed set such that $\\\\lbrace u\\\\in \\\\overline{U}\\\\cap D(A) \\\\mid 0\\\\in -Au + F(u)\\\\rbrace \\\\cap K = \\\\emptyset ,$ then, for sufficiently small $\\\\alpha >0$ , $\\\\lbrace u\\\\in \\\\overline{U} \\\\mid \\\\Phi _\\\\alpha (u)=u\\\\rbrace \\\\cap K = \\\\emptyset .$ Remark 2.3 Actually the tangency condition (REF ) and the continuity of $F\\\\circ N^{-1}$ imply $F\\\\left( N^{-1}(\\\\tau ) \\\\right) \\\\in C_{M^*} (\\\\tau ) := \\\\left\\\\lbrace \\\\theta \\\\in X^* \\\\mid \\\\lim _{\\\\alpha \\\\rightarrow 0^+, \\\\, \\\\varrho \\\\rightarrow \\\\tau , \\\\, \\\\varrho \\\\in M} \\\\frac{d_M (\\\\varrho + \\\\alpha \\\\theta )}{\\\\alpha } =0 \\\\right\\\\rbrace \\\\, \\\\mbox{ for all } \\\\tau \\\\in N(U).$ Indeed $F \\\\left( N^{-1}(\\\\tau ) \\\\right)=\\\\lim _{\\\\varrho \\\\rightarrow \\\\tau } F \\\\left( N^{-1}(\\\\varrho ) \\\\right) \\\\in \\\\mathop {\\\\mathrm {Liminf}\\\\,}_{\\\\varrho \\\\rightarrow \\\\tau , \\\\, \\\\varrho \\\\in M^*} T_{M^*} (\\\\varrho ) \\\\subset C_{M^*} (\\\\tau ).$ The proof of the latter inclusion can be found in [1].\",\n \"Lemma 2.4 (i) The graph $\\\\mathrm {Gr}(A):= \\\\lbrace (u, \\\\tau )\\\\in X\\\\times X^* \\\\mid u\\\\in D(A)\\\\rbrace $ is closed; (ii) If a sequence of pairs $(u_n ,\\\\tau _n) \\\\in \\\\mathrm {Gr}(A)$ , $n\\\\ge 1$ , is bounded, then the sequence $(u_n)$ has a convergent subsequence.\",\n \"Proof: (i) Take any sequence of points $(u_n, \\\\tau _n)\\\\in \\\\mathrm {Gr}(A)$ , $n\\\\ge 1$ , such that $(u_n, \\\\tau _n)\\\\rightarrow (u_0, \\\\tau _0)$ in $X\\\\times X^*$ , as $n\\\\rightarrow +\\\\infty $ , for some $(u_0, \\\\tau _0) \\\\in X\\\\times X^*$ .\",\n \"Clearly, $\\\\tau _n \\\\in Au_n$ , and this gives $N(u_n) + \\\\tau _n \\\\in (N+A)(u_n)$ , which, by $({\\\\cal A}_1)$ , gives $u_n = J_{1} (N(u_n)+\\\\tau _n)$ , $n\\\\ge 1$ .\",\n \"Hence, using the continuity of $N$ and $J_{1}$ yields $u_n = J_{1}(N(u_n)+\\\\tau _n)) \\\\rightarrow J_{1} (N(u_0) + \\\\tau _0)$ as $n\\\\rightarrow +\\\\infty $ , which implies $u_0 = J_{1} (N(u_0) + \\\\tau _0)$ , i.e.\",\n \"$\\\\tau _0\\\\in Au_0$ .\",\n \"This shows that $\\\\mathrm {Gr}(A)$ is closed.\",\n \"(ii) Note that, for each $n\\\\ge 1$ , $u_n = J_{1} (N(u_n) + \\\\tau _n) \\\\in J_{1} (N(B(0,R))+B(0,R))$ , where $R>0$ is a constant such that $\\\\Vert u_n\\\\Vert _X \\\\le R$ and $\\\\Vert \\\\tau _n\\\\Vert _{X^*} \\\\le R$ for $n\\\\ge 1$ .\",\n \"The boundedness of $N$ and $({\\\\cal A}_2)$ imply that the set $(u_n)$ is a sequence of elements of the relatively compact set $J_{1} (N(B(0,R))+B(0,R))$ .\",\n \"$\\\\square $ Proof of Proposition REF : Suppose to the contrary that there exists a sequence $(\\\\alpha _n)$ such that $\\\\alpha _n\\\\rightarrow 0^+$ such that for each $n\\\\ge 1$ there is $u_n\\\\in K$ with $\\\\Phi _{\\\\alpha _n} (u_n) = u_n$ , that is $N(u_n)+ \\\\alpha _n \\\\tau _n = r(N(u_n) + \\\\alpha _n F(u_n)) \\\\mbox{ for some } \\\\tau _n \\\\in Au_n.$ In view of (REF ), one has $\\\\alpha _n \\\\Vert \\\\tau _n - F(u_n)\\\\Vert & = & \\\\Vert r(N(u_n) + \\\\alpha _n F(u_n)) - ( N (u_n) + \\\\alpha _n F (u_n) ) \\\\Vert \\\\\\\\& \\\\le & L d_{M^*} (N(u_n) + \\\\alpha _n F(u_n)) \\\\ \\\\ \\\\mbox{ for all } n\\\\ge 1.\",\n \"\\\\nonumber $ This implies $\\\\Vert \\\\tau _n \\\\Vert \\\\le \\\\Vert F(u_n)\\\\Vert + L \\\\alpha _{n}^{-1} d_{M^*} (N(u_n) + \\\\alpha _n F(u_n)) \\\\le (1+L) \\\\Vert F(u_n)\\\\Vert , \\\\ n\\\\ge 1,$ which means that $(\\\\tau _n)$ is bounded.\",\n \"Therefore, by use of Lemma REF (ii), we may assume without loss of generality that $u_n \\\\rightarrow u_0$ for some $u_0\\\\in M$ .\",\n \"Now using (REF ) and putting $p_n:=N(u_n)$ , $n\\\\ge 0$ , we see that $\\\\Vert \\\\tau _n - F(u_n)\\\\Vert \\\\le L \\\\cdot \\\\frac{d_{M^*} (p_n + \\\\alpha _n F(N^{-1}(p_0))}{\\\\alpha _n} + L\\\\Vert F(u_n)-F(u_0)\\\\Vert , \\\\ \\\\ \\\\mbox{ for } n\\\\ge 1.$ By the tangency condition $({\\\\cal A}_4)$ and Remark REF together with the continuity of $F$ , we get that $\\\\tau _n \\\\rightarrow F(u_n)$ as $n\\\\rightarrow +\\\\infty $ .\",\n \"Hence, we have obtained that $(u_n, \\\\tau _n)\\\\rightarrow (u_0, F(u_0))$ and, by Lemma REF (i), $(u_0,F(u_0)) \\\\in \\\\mathrm {Gr}(A)$ , i.e.\",\n \"$F(u_0)\\\\in A u_0$ , a contradiction completing the proof.\",\n \"$\\\\square $ Now we put $\\\\mathrm {Deg}_M ( A, F, U ) := \\\\lim _{\\\\alpha \\\\rightarrow 0^+} \\\\mathrm {ind}_M ( \\\\Phi _\\\\alpha , U ),$ where $\\\\mathrm {ind}_M$ stands for the fixed point index for compact mappings of absolute neighborhood retracts due to Granas – see [12] or [8] for details.\",\n \"We call this number as the topological degree of coincidence (or just topological degree) of $A$ and $F$ with respect to $M$.\",\n \"Theorem 2.5 The coincidence degree defined by (REF ) is well defined and has the following properties: (i) (existence) if $\\\\mathrm {Deg}_M (A,F, U)\\\\ne 0$ , then there exists $u\\\\in U\\\\cap D(A)$ such that $0\\\\in -Au+F(u)$ ; (ii) (additivity) if $U_1, U_2$ are open disjoint subsets of a bounded open $U\\\\subset M$ and $0\\\\notin (-A+F)(\\\\overline{U} \\\\setminus (U_1\\\\cup U_2))$ , then $\\\\mathrm {Deg}_M (A,F, U) = \\\\mathrm {Deg}_M (A,F, U_1) +\\\\mathrm {Deg}_M (A,F, U_2);$ (iii) (homotopy invariance) if $H:\\\\overline{U} \\\\times [0,1]\\\\rightarrow X^*$ is a continuous and bounded mapping such that $H( N^{-1}(\\\\tau ), t) \\\\in T_{M^*} (\\\\tau ) \\\\mbox{ for all } \\\\tau \\\\in N(\\\\overline{U}), \\\\, t>0,$ and $0\\\\notin -Au+H(u,t)$ for all $u\\\\in \\\\partial U\\\\cap D(A)$ and $t\\\\in [0,1]$ , then $\\\\mathrm {Deg}_M (A,H(0,\\\\cdot ), U) = \\\\mathrm {Deg}_M (A, H(1,\\\\cdot ), U);$ (iv) (normalization) if $M$ is bounded and the mapping $\\\\widetilde{\\\\cal J}: X^* \\\\times [0,+\\\\infty ) \\\\ni (\\\\tau , \\\\alpha ) \\\\mapsto J^{\\\\alpha } \\\\tau \\\\in X$ with $J^0 = N^{-1}$ is continuous, then $\\\\mathrm {Deg}_M (A,F, M) = \\\\chi (M)$ .\",\n \"Proof: Note that for sufficiently small $\\\\alpha >0$ it follows from Propostion REF that $\\\\Phi _\\\\alpha $ has no fixed point in $\\\\partial U$ , i.e.\",\n \"the fixed point index $\\\\mathrm {ind}_M (\\\\Phi _\\\\alpha , U)$ is well defined.\",\n \"If $\\\\alpha _1, \\\\alpha _2>0$ are small enough, then, by $({\\\\cal A}_2)$ , $\\\\Phi _{\\\\alpha _1}$ is homotopic with $\\\\Phi _{\\\\alpha _2}$ , which gives $\\\\mathrm {ind}_M (\\\\Phi _{\\\\alpha _1}, U)= \\\\mathrm {ind}_M (\\\\Phi _{\\\\alpha _2}, U)$ , which means that the limit in (REF ) exists.\",\n \"(i) Suppose to the contrary that there is no $u\\\\in U \\\\cap D(A)$ such that $0\\\\in - Au + F(u)$ .\",\n \"Then, in view of Proposition REF , for sufficiently small $\\\\alpha >0$ the mappings $\\\\Phi _\\\\alpha $ have no fixed points in $\\\\overline{U}$ , i.e.\",\n \"$\\\\mathrm {Deg}_M (A,F,U) = \\\\mathrm {ind}_M (\\\\Phi _\\\\alpha , U)=0$ , a contradiction.\",\n \"(ii) Due to Proposition REF , for sufficiently small $\\\\alpha >0$ , $\\\\Phi _\\\\alpha $ has no fixed points in $\\\\overline{U} \\\\setminus (U_1\\\\cup U_2)$ .\",\n \"Therefore, by the definition of the degree, $\\\\mathrm {Deg}_M (A, F, U)=\\\\mathrm {ind}_M (\\\\Phi _\\\\alpha , U) \\\\ \\\\ \\\\mbox{ and } \\\\ \\\\ \\\\mathrm {Deg}_M (A, F, U_k)=\\\\mathrm {ind}_M (\\\\Phi _\\\\alpha , U_k) \\\\mbox{ for } k=1,2.$ By the additivity property of the fixed point index $\\\\mathrm {ind}_M (\\\\Phi _\\\\alpha , U) = \\\\mathrm {ind}_M (\\\\Phi _\\\\alpha , U_1) + \\\\mathrm {ind}_M (\\\\Phi _\\\\alpha , U_2),$ which together with the earlier equalities gives the desired additivity of the degree.\",\n \"(iii) For sufficiently small $\\\\alpha >0$ one can define $\\\\Phi _\\\\alpha :\\\\overline{U}\\\\times [0,1] \\\\rightarrow M$ by $\\\\Phi _\\\\alpha (u,t):= J_{\\\\alpha }\\\\left( r (N(u) + \\\\alpha H(u,t)) \\\\right), \\\\ u\\\\in \\\\overline{U}, \\\\ t\\\\in [0,1].$ Proceeding along the lines of the proof of Proposition REF we can prove that for sufficiently small $\\\\alpha >0$ $\\\\Phi _\\\\alpha (u,t)\\\\ne u \\\\mbox{ for all } u\\\\in \\\\partial U, \\\\ t\\\\in [0,1].$ Hence, by the homotopy invariance of the fixed point index and the formula defining the degree, $\\\\mathrm {Deg}_M (A,H(\\\\cdot , 0),U) = \\\\mathrm {ind}_M (\\\\Phi _\\\\alpha (\\\\cdot ,0), U) = \\\\mathrm {ind}_M (\\\\Phi _\\\\alpha (\\\\cdot ,1), U)= \\\\mathrm {Deg}_M (A,H(\\\\cdot , 1),U).$ (iv) Take small $\\\\alpha >0$ such that $\\\\Phi _\\\\alpha $ is well defined.\",\n \"Then $\\\\mathrm {Deg}_M (A, F, M) = \\\\mathrm {ind}_M (\\\\Phi _\\\\alpha , M).$ Note that the normalization property for the fixed point index states that the homomorphism $H_* (\\\\Phi _\\\\alpha ):H_* (M)\\\\rightarrow H_*(M)$ induced on (singular) homology spaces is a Leray endomorphism and $\\\\mathrm {ind}_M (\\\\Phi _\\\\alpha , M) = \\\\Lambda (\\\\Phi _\\\\alpha )$ where $\\\\Lambda (\\\\Phi _\\\\alpha )$ is the generalized Leschetz number of the compact map $\\\\Phi _\\\\alpha $ – see [8] or [12].\",\n \"Further, consider $\\\\Psi :M\\\\times [0,1]\\\\rightarrow M$ given by $\\\\Psi (u,t) := \\\\widetilde{\\\\cal J} (r(N(u)+t\\\\alpha F(u)),t\\\\alpha ), \\\\ \\\\ u\\\\in M, \\\\ \\\\ t\\\\in [0,1].$ By the assumption, $\\\\Psi $ is a continuous homotopy joining $\\\\Psi (\\\\cdot , 1) = \\\\Phi _\\\\alpha $ with the identity map $\\\\mathrm {id}_M: M\\\\rightarrow M$ .\",\n \"Hence, for the maps induced on homology spaced one has $H_* (\\\\Phi _\\\\alpha ) = H_* (\\\\mathrm {id}_M)=\\\\mathrm {id}_{H_* (M)}$ and, since $H_* (\\\\Phi _\\\\alpha )$ is an endomorphism Leray, we infer that $\\\\Lambda (\\\\Phi _\\\\alpha )=\\\\sum _{n=0}^{\\\\infty } (-1)^n \\\\dim H_n (M) = \\\\chi (M)$ , which together with (REF ) ends the proof.\",\n \"$\\\\square $ We end this section with a general result, which allows us to compute the degree is specific situations (comp.\",\n \"[5]).\",\n \"Theorem 2.6 Let $M$ and $M^*$ be closed convex cones and that the mappings $A$ and $N$ are homogeneous with the same degree (i.e.\",\n \"there exists $\\\\gamma >0$ such that $A(a u) = a^{\\\\gamma } A(u)$ , $u\\\\in D(A)$ , $a>0$ , and $N(au)=a^\\\\gamma N(u)$ for all $u\\\\in X$ , $a>0$ .\",\n \").\",\n \"Suppose that there exists $\\\\lambda _1 \\\\ge 0$ satisfying the following conditions $({\\\\cal M}_1)$ $(A-\\\\lambda N)^{-1}(\\\\lbrace 0\\\\rbrace ) \\\\cap M = \\\\lbrace 0\\\\rbrace \\\\ \\\\mbox{ for } \\\\ \\\\lambda \\\\ne \\\\lambda _1;$ $({\\\\cal M}_2)$ there exists $\\\\tau _0 \\\\in M^*$ such that $(A-\\\\lambda N)^{-1} (\\\\lbrace \\\\tau _0 \\\\rbrace ) \\\\cap M =\\\\emptyset $ for $\\\\lambda >\\\\lambda _1$ .\",\n \"Then $\\\\mathrm {Deg}_M (A, \\\\lambda N, B_M (0,\\\\delta )) = \\\\left\\\\lbrace \\\\begin{array}{ll} 1, & \\\\lambda < \\\\lambda _1,\\\\\\\\0, & \\\\lambda >\\\\lambda _1,\\\\end{array} \\\\right.$ for any $\\\\delta >0$ .\",\n \"Proof: Note that in view of $({\\\\cal M}_1)$ the topological degree $\\\\mathrm {Deg}_M (A, \\\\lambda N, B_M (0, \\\\delta ))$ is well defined.\",\n \"Now fix $\\\\lambda <\\\\lambda _1$ .\",\n \"By the very construction, for sufficiently small $\\\\alpha >0$ , $\\\\mathrm {Deg}_M (A, \\\\lambda N, B_M (0,\\\\delta )) = \\\\mathrm {ind}_M (\\\\Phi _\\\\alpha , B_M (0,\\\\delta ))$ where $\\\\Phi _\\\\alpha :\\\\overline{B_M (0,\\\\delta )}\\\\rightarrow M$ is given by $\\\\Phi _\\\\alpha (u):= J_\\\\alpha (r(N(u) + \\\\alpha \\\\lambda N(u))), \\\\ u \\\\in \\\\overline{B_M (0,\\\\delta )}.$ Define $\\\\Theta : \\\\overline{B_M (0,\\\\delta )}\\\\times [0,1]\\\\rightarrow M$ by $\\\\Theta (u,t):= t \\\\Phi _\\\\alpha (u), \\\\ u \\\\in \\\\overline{B_M (0,\\\\delta )}, \\\\, t\\\\in [0,1].$ Suppose there are $u\\\\ne 0$ and $t\\\\in [0,1]$ such that $\\\\Theta (u,t)=u$ .\",\n \"Then $0 \\\\in -A(u) + \\\\mu N(u)$ with $\\\\mu :=(t^\\\\gamma -1)/\\\\alpha + t^\\\\gamma \\\\lambda $ , i.e.\",\n \"$u\\\\in (A-\\\\mu N )^{-1} (\\\\lbrace 0\\\\rbrace )\\\\cap M$ , and, since $\\\\mu =(t^\\\\gamma -1)/\\\\alpha + t^\\\\gamma \\\\lambda < \\\\lambda _1$ we get a contradiction with $({\\\\cal M}_1)$ .\",\n \"Hence, we can use the homotopy invariance of fixed point index to see that $\\\\mathrm {ind}_M (\\\\Phi _\\\\alpha ,B_M (0,\\\\delta )) =\\\\mathrm {ind}_M (0, B_M (0,\\\\delta )) = 1$ .\",\n \"This along with (REF ) implies the required equality.\",\n \"Let us pass to the case when $\\\\lambda >\\\\lambda _1$ .\",\n \"Define $H:M \\\\times [0,1] \\\\rightarrow X$ by $H(u,t):= \\\\lambda N (u) + t\\\\tau _0$ , $u\\\\in M$ , $t\\\\in [0,1]$ .\",\n \"If $-A(u)+H(u,t)= 0$ , then either $t=0$ and, due to $({\\\\cal M}_1)$ , $u=0$ or, by the homogeneity $-A(t^{-1/\\\\gamma }u) + \\\\lambda N (t^{-1/\\\\gamma }u) + \\\\tau _0 = 0$ , where $\\\\gamma >0$ is the common homogeneity degree for $A$ and $N$ .\",\n \"The latter equality contradicts $({\\\\cal M}_2)$ .\",\n \"Hence, the degrees $\\\\mathrm {Deg}_M (A, H(\\\\cdot ,t), B_M (0,\\\\delta ) )$ , $t\\\\in [0,1]$ , are well defined and homotopy invariance can be used to obtain $\\\\mathrm {Deg}_M (A, \\\\lambda N, B_M (0,\\\\delta )) = \\\\mathrm {Deg}_M (A, \\\\lambda A +\\\\tau _0, B_M (0,\\\\delta ).$ Finally the existence property of the degree together with $({\\\\cal M}_2)$ implies $\\\\mathrm {Deg}_M (A, \\\\lambda A +\\\\tau _0, B_M (0,\\\\delta ))=0$ , which completes the proof.\",\n \"$\\\\square $\"\n ],\n [\n \"Abstract setting for $p$ -Laplacian\",\n \"Now we shall consider an abstract example falling into the setting of Section 2.\",\n \"It will be used in the sequel for the $p$ -Laplace operator and the cone of positive functions in $L^{p}(\\\\Omega )$ .\",\n \"Let $X$ and $Y$ be reflexive normed spaces with a dense and compact linear embedding $i:Y\\\\rightarrow X$ .\",\n \"(That is the mapping $i$ is linear and completely continuous with its range $i(Y)$ dense in $X$ .)\",\n \"Suppose that a closed convex cone $M\\\\subset X$ and functionals ${a}:Y\\\\rightarrow \\\\mathbb {R}$ and ${n}: X\\\\rightarrow \\\\mathbb {R}$ satisfy the following conditions: (a1) ${a}$ and ${n}$ are coercive $C^1$ functionals;(By coercivity we mean that counterimages of intervals $(-\\\\infty ,m)$ , with respect to a given functional, are bounded for all $m\\\\in \\\\mathbb {R}$ .)\",\n \"(a2) there exists a continuous function $\\\\kappa :[0,+\\\\infty )\\\\rightarrow [0, +\\\\infty )$ such that $\\\\kappa ^{-1}(\\\\lbrace 0\\\\rbrace ) = \\\\lbrace 0\\\\rbrace $ , $\\\\lim \\\\limits _{s\\\\rightarrow +\\\\infty } \\\\kappa (s) = +\\\\infty $ and $\\\\langle D{a} (u_1) - D{a} (u_2), u_1-u_2 \\\\rangle _{Y} \\\\ge \\\\kappa (\\\\Vert u_1-u_2\\\\Vert _Y)\\\\Vert u_1-u_2\\\\Vert _Y \\\\ \\\\ \\\\mbox{ for all }u_1, u_2\\\\in Y,\\\\\\\\\\\\langle D{n} (u_1) - D{n} (u_2), u_1-u_2 \\\\rangle _{X} \\\\ge \\\\kappa (\\\\Vert u_1-u_2\\\\Vert _X)\\\\Vert u_1-u_2\\\\Vert _X \\\\ \\\\ \\\\mbox{ for all }u_1, u_2\\\\in X;$ (a3) for any $u\\\\in M$ there exist $u^+, u^-\\\\in M$ such that $u=u^+ - u^-$ and ${n}(u^+) \\\\le {n}(u)$ ; if $u\\\\in i(Y)$ , then $u^+, u^- \\\\in i(Y)$ and ${a} (i^{-1} u^+) \\\\le {a} (i^{-1} u)$ ; (a4) $n$ is bounded on bounded sets and monotone with respect to $M$ , i.e.\",\n \"${n}(u+v) \\\\ge {n} (u)$ for any $u,v\\\\in M$ .\",\n \"Let ${\\\\cal A}:Y \\\\rightarrow Y^*$ and $N:X\\\\rightarrow X^*$ be defined by by $ {\\\\cal A} : = D{a}$ and $N:=D {n}$ .\",\n \"Note that that, due to (a2), both ${a}$ and ${n}$ are strictly convex and ${\\\\cal A}$ and $N$ are monotone operators.\",\n \"Define $A:D(A)\\\\rightarrow X^*$ by $D(A):= i \\\\left( {\\\\cal A}^{-1} (i^*(X^*))\\\\right) \\\\mbox{ and } A u:= (i^*)^{-1}({\\\\cal A} i^{-1}u), \\\\mbox{ for } u\\\\in D(A).$ The above operation of restriction is a generalization of the analogical one that is usually considered in the case of a Gelfand triple $Y\\\\subset X \\\\subset Y^*$ where $X$ is a Hilbert space.\",\n \"Below we show that assumptions $({\\\\cal A}_1)$ and $({\\\\cal A}_2)$ of Section 2 are satisfied.\",\n \"Proposition 3.1 Under the above assumptions (i) $N$ is a homeomorphism which is bounded on bounded sets; (ii) $N(M) = M^*:=\\\\lbrace \\\\tau \\\\in X^* \\\\mid \\\\langle \\\\tau , u \\\\rangle \\\\ge 0 \\\\mbox{ for all } u\\\\in M \\\\rbrace $ ; (iii) $A$ is a densely defined maximal monotone operator; (iv) for any $\\\\alpha >0$ and $\\\\tau \\\\in X^*$ there is a unique $u\\\\in D(A)$ such that $\\\\tau = (N+\\\\alpha A) (u)$ ; (v) if $J_\\\\alpha : X^*\\\\rightarrow X$ , $\\\\alpha >0$ , is given by $J_{\\\\alpha } \\\\tau := u \\\\mbox{ where } u\\\\in D(A) \\\\mbox{ is such that } N(u)+\\\\alpha A (u) = \\\\tau ,$ and ${\\\\cal J}:X^* \\\\times [0,+\\\\infty ) \\\\rightarrow X$ by ${\\\\cal J} (u, \\\\alpha ):= J_{\\\\alpha } u,$ then ${\\\\cal J}$ is bounded on bounded sets and ${\\\\cal J}_{\\\\ |\\\\, X^* \\\\times [\\\\alpha _1, \\\\alpha _2]}$ with $0<\\\\alpha _1 \\\\le \\\\alpha _2$ is completely continuous; (vi) $J_{\\\\alpha }(M^*) \\\\subset M$ for all $\\\\alpha >0$ .\",\n \"Proof: To see (i), first note that $N$ is continuous, since ${n}$ is $C^1$ .\",\n \"Moreover, as a strictly convex coercive functional on the reflexive Banach space $X$ , for any $\\\\tau \\\\in X^{*}$ , ${n}-\\\\tau $ admits a unique minimum point $u\\\\in X$ , i.e.\",\n \"$D{n} (u)-\\\\tau =0$ , which gives $N(u)=\\\\tau $ .\",\n \"Conversely, if $u\\\\in X$ is such that $N(u)=\\\\tau $ , then, by the strict convexity, $u$ is the unique minimum point.\",\n \"Hence, $N$ is bijective.\",\n \"To see that $N^{-1}$ is continuous, take any $(\\\\tau _n)$ in $X$ with $\\\\tau _n \\\\rightarrow \\\\tau $ in $X^*$ as $n\\\\rightarrow +\\\\infty $ .\",\n \"Observe that, by (a2), we get $\\\\langle \\\\tau _n - \\\\tau , N^{-1}(\\\\tau _n)-N^{-1}(\\\\tau )\\\\rangle _X \\\\ge \\\\kappa (\\\\Vert N^{-1}(\\\\tau _n) - N^{-1} (\\\\tau )\\\\Vert _X)\\\\Vert N^{-1}(\\\\tau _n) - N^{-1} (\\\\tau )\\\\Vert _X,$ which yields the inequality $\\\\Vert \\\\tau _n-\\\\tau \\\\Vert _{X^*}\\\\ge \\\\kappa \\\\left(\\\\Vert N^{-1}(\\\\tau _n) - N^{-1} (\\\\tau )\\\\Vert _X \\\\right).$ This in turn means that $N^{-1}(\\\\tau _n) \\\\rightarrow N^{-1} (\\\\tau )$ in $X$ as $n\\\\rightarrow +\\\\infty $ , that is $N^{-1}$ is continuous.\",\n \"To show that $N$ is bounded on bounded sets, we suppose to the contrary that there exists a bounded sequence $(u_n)$ in $X$ such that $\\\\Vert N(u_n)\\\\Vert _{X^*}\\\\rightarrow +\\\\infty $ as $n\\\\rightarrow +\\\\infty $ .\",\n \"Since $X$ is reflexive, for each $n\\\\ge 1$ one finds an element $v_n\\\\in X$ such that $\\\\Vert v_n - u_n\\\\Vert _{X} = 1$ and $\\\\Vert N(u_n)\\\\Vert _{X^*} = \\\\langle N(u_n), v_n-u_n\\\\rangle \\\\le {n} ( v_n) - {n} (u_n) \\\\le \\\\sup _{D_{X}(0, R+1)} {n} - \\\\inf _{D_X(0,R)} {n}$ where $R>0$ is such that $\\\\Vert u_n\\\\Vert \\\\le R$ for all $n\\\\ge 1$ .\",\n \"Thus, a contradiction proving the claim.\",\n \"To get (ii) take any $u\\\\in M$ and $v\\\\in M$ .\",\n \"In view of (a4) ${n}(u+h v)-{n} (u) \\\\ge 0 \\\\ \\\\mbox{ for any } h>0,$ which, after a division by $h$ and passage to the limit with $h\\\\rightarrow 0^+$ , yields $\\\\langle N(u), v\\\\rangle \\\\ge 0$ .\",\n \"Hence $N(M)\\\\subset M^*$ .\",\n \"To prove the converse inclusion $M^*\\\\subset N(M)$ , we take any $\\\\tau \\\\in M^*$ .\",\n \"As we mentioned ${n}-\\\\tau $ attains the minimum at some $u\\\\in X$ .\",\n \"On the other hand, by (a2), ${n}(u^+) -\\\\tau (u^+) \\\\le {n}(u) - \\\\tau (u^+) + \\\\tau (u^-) = {n}(u) - \\\\tau (u).$ and, since the minimum point is unique, we infer that $u=u^+ \\\\in M$ .\",\n \"To show (iii), take any $u_1, u_2\\\\in D(A)$ .\",\n \"Clearly $(A u_k) \\\\circ i = {\\\\cal A} (\\\\tilde{u}_k)$ with $\\\\tilde{u}_k = i^{-1} (u_k)$ , for $k=1,2$ .\",\n \"Therefore, by (a2), $\\\\langle A u_1-Au_2, u_1 - u_2 \\\\rangle _{X} = [Au_1 - Au_2] i(\\\\tilde{u}_1-\\\\tilde{u}_2)= \\\\langle {\\\\cal A} (\\\\tilde{u}_1) - {\\\\cal A} (\\\\tilde{u}_2), \\\\tilde{u}_1-\\\\tilde{u}_2\\\\rangle _Y \\\\ge 0.$ Hence $A$ is monotone and it is left to prove that $A$ is maximal monotone, i.e.\",\n \"that additionally one has $A (D(A))=X^*$ .\",\n \"To see it we choose any $\\\\tau \\\\in X^*$ and put $\\\\Phi :={a} - i^*(\\\\tau )$ .\",\n \"$\\\\Phi $ is a convex coercive functional on the reflexive space $Y$ .\",\n \"Hence it admits a miniumem, i.e.\",\n \"there is a point $\\\\bar{u}\\\\in Y$ such that $D\\\\Phi (\\\\bar{u})=0$ , i.e.\",\n \"${\\\\cal A}(\\\\bar{u}) = i^*(\\\\tau )$ .\",\n \"This means that $u:=i(\\\\bar{u})\\\\in D(A)$ and that $A(u)=\\\\tau $ .\",\n \"To show (iv) take any $\\\\tau \\\\in X^*$ and $\\\\alpha >0$ .\",\n \"We proceed like in (iii), that is we consider a functional $\\\\Phi := {n}\\\\circ i + \\\\alpha {a} - i^*(\\\\tau )$ on $Y$ .\",\n \"It is clear that $\\\\Phi $ – as a strictly convex and coercive functional on a reflexive Banach space – admits a minimum, i.e.\",\n \"there exists $\\\\bar{u}\\\\in Y$ such that $D\\\\Phi (\\\\bar{u})=0$ .\",\n \"This means that $i^*(N(i(\\\\bar{u}))) + \\\\alpha {\\\\cal A} (\\\\bar{u}) = i^*(\\\\tau )$ .\",\n \"Subsequently, we deduce that ${\\\\cal A}(\\\\bar{u}) \\\\in i^*(X^*)$ , i.e.\",\n \"$u:= i(\\\\bar{u})\\\\in D(A)$ and $N(u) + \\\\alpha A (u) = \\\\tau $ .\",\n \"Moreover, observe that for each $u\\\\in D(A)$ such that $N(u) + \\\\alpha A (u) = \\\\tau $ , $i^{-1}(u)$ is a critical point of $\\\\Phi $ .\",\n \"Since $\\\\Phi $ is strictly convex it has to be the unique minimum point.\",\n \"(v) Suppose that a sequence $(\\\\tau _n)$ is bounded in $X^*$ and $(\\\\beta _n)$ is a sequence in $[\\\\alpha _1, \\\\alpha _2]$ .\",\n \"Put $u_n:= J_{\\\\beta _n} (\\\\tau _n)$ , $n\\\\ge 1$ .\",\n \"Then $i^*N(u_n) + \\\\beta _n {\\\\cal A} (\\\\bar{u}_n) = i^*(\\\\tau _n)$ , where $\\\\bar{u}_n:=i^{-1}(u_n)$ , $n\\\\ge 1$ .\",\n \"Since $N$ is bounded and $\\\\beta _n > \\\\alpha _1>0$ for all $n\\\\ge 1$ , we infer that $({\\\\cal A}(\\\\bar{u}_n))$ is bounded.\",\n \"Observe that, in view of (a2), $\\\\langle {\\\\cal A} (\\\\bar{u}_n) - {\\\\cal A}(0), \\\\bar{u}_n \\\\rangle _{Y} \\\\ge \\\\kappa (\\\\Vert \\\\bar{u}_n\\\\Vert _{Y})\\\\Vert \\\\bar{u}_n\\\\Vert _Y,$ i.e.\",\n \"$\\\\Vert {\\\\cal A}(u_n) - {\\\\cal A}(0)\\\\Vert _{Y} \\\\ge \\\\kappa (\\\\Vert \\\\bar{u}_n\\\\Vert _{Y})$ .\",\n \"Hence, by the boundedness of $({\\\\cal A}(\\\\bar{u}_n))$ and the assumed property of $\\\\kappa $ , $(\\\\bar{u}_n)$ is bounded.\",\n \"Therefore $(u_n) = (i(\\\\bar{u}_n))$ is relatively compact, which together with Remark REF proves the assertion.\",\n \"In order to prove (vi), take any $\\\\tau \\\\in M^*$ .\",\n \"We need to show that $u:=J_{\\\\alpha } (\\\\tau ) \\\\in M$ .\",\n \"In the proof of (iv) we have showed that $i^{-1} u$ is the unique minimum of the functional $\\\\Phi ={n}\\\\circ i + \\\\alpha {a} - i^*(\\\\tau )$ on $Y$ .\",\n \"On the other hand, by use of (a3) and the definition of $M^*$ , one has $\\\\Phi (i^{-1} u^+ )\\\\!=\\\\!\",\n \"{n} (u^+) \\\\!+\\\\!\",\n \"\\\\alpha {a}(i^{-1} u^+)\\\\!\",\n \"-\\\\!\",\n \"\\\\tau (u^+)\\\\!\\\\le \\\\!\",\n \"{n}(u) \\\\!+\\\\!\",\n \"\\\\alpha {a} (i^{-1} u))\\\\!-\\\\!\\\\tau ( u^+) \\\\!+\\\\!\",\n \"\\\\tau (u^-) = \\\\Phi (i^{-1}u).$ This means that $i^{-1} u=i^{-1} u^+$ and $u\\\\in M$ .\",\n \"$\\\\square $\"\n ],\n [\n \"Elliptic problems with $p$ -Laplacian\",\n \"Now we shall apply the above abstract setting from the previous section to the $p$ -Laplacian problem.\",\n \"To this end fix $p>2$ , and put $X_p:= L^p (\\\\Omega ), \\\\ Y_p:=W_{0}^{1,p} (\\\\Omega ) \\\\mbox{ and } M_p:=\\\\lbrace u\\\\in X \\\\mid u(x)\\\\ge 0 \\\\mbox{ for a.e. }\",\n \"x\\\\in \\\\Omega \\\\rbrace .$ Both, $X_p$ and $Y_p$ are reflexive and, by the Rellich-Kondrachov theorem, the natural embedding $i:Y_p\\\\rightarrow X_p$ is compact and dense.\",\n \"It is easy to see that $M_p$ is a closed convex subset of $X_p$ .\",\n \"Next define functionals ${a}:Y_p\\\\rightarrow \\\\mathbb {R}$ and ${n}:X_p\\\\rightarrow \\\\mathbb {R}$ by ${a} (u):= \\\\frac{1}{p} \\\\int _{\\\\Omega } |\\\\nabla u(x)|^p \\\\,\\\\mathrm {d}x, \\\\ u\\\\in Y_p,\\\\\\\\{n}(u):= \\\\frac{1}{p} \\\\int _{\\\\Omega } |u(x)|^p \\\\,\\\\mathrm {d}x, \\\\ u\\\\in X_p.$ We prove that these objects satisfy the abstract assumptions of the general setting.\",\n \"Proposition 4.1 The functionals ${a}$ and ${n}$ with the cone $M_p$ satisfy all the assumptions (a1) – (a4) from Section and $\\\\langle D{a} (u), v\\\\rangle _{Y} = \\\\frac{1}{p} \\\\int _{\\\\Omega } |\\\\nabla u(x)|^{p-2} \\\\nabla u(x)\\\\cdot \\\\nabla v(x) \\\\,\\\\mathrm {d}x, \\\\ u, v \\\\in Y_p, \\\\\\\\\\\\langle D{a} (u) - D{a} (v), u-v \\\\rangle _Y \\\\ge 2^{2-p}\\\\Vert u-v\\\\Vert _{Y}^{p}, \\\\, u,v\\\\in Y_p, \\\\\\\\ \\\\langle D{n} (u), v\\\\rangle _{X} = \\\\frac{1}{p} \\\\int _{\\\\Omega } |u(x)|^{p-2} u(x) v(x) \\\\,\\\\mathrm {d}x, \\\\ u,v \\\\in X_p,\\\\\\\\\\\\langle D {n} (u) - D {n} (v), u-v \\\\rangle _X \\\\ge 2^{2-p}\\\\Vert u-v\\\\Vert _{X}^{p}, \\\\, u,v\\\\in X_p.$ Moreover, if $A_p:D(A_p) \\\\rightarrow X_p$ is defined, in analogy to (REF ), by $D(A_p):= i\\\\left( (D {a})^{-1} (i^*(X^*))\\\\right) \\\\mbox{ and } A_p u:= (i^*)^{-1}( D({a}) i^{-1} u), \\\\mbox{ for } u\\\\in D(A),$ then $A_p u= -\\\\mathrm {div} (|\\\\nabla u|^{p-2} \\\\nabla u), \\\\mbox{ for } u\\\\in D(A_p),$ where the divergence is meant in the distributional sense and $D(A_p)=\\\\lbrace u\\\\in W_{0}^{1,p} (\\\\Omega ) \\\\mid \\\\mathrm {div}(|\\\\nabla u|^{p-2}\\\\nabla u) \\\\mbox{ exists and belongs to } L^{p} (\\\\Omega )\\\\rbrace .$ Proof: In order to see (a1), note that the functionals ${a}$ and ${n}$ are clearly Gateaux differentiable with the formulas (REF ) and () satisfied.\",\n \"Since these Gateaux derivatives are continuous the functionals are Frêchet differentiable.\",\n \"The coercivity is immediate as ${a}(u) = (1/p)\\\\Vert u\\\\Vert _{Y_p}^{p}$ , $u\\\\in Y$ , and ${n}(u)=(1/p)\\\\Vert u\\\\Vert _{X_p}^{p}$ , $u\\\\in X$ .\",\n \"One can check the condition (a2), i.e.\",\n \"() and (), by use of the following inequality $(|x|^{p-2}x-|y|^{p-2}y)\\\\cdot (x-y)\\\\ge 2^{2-p} |x-y|^p \\\\mbox{ for any } x,y\\\\in \\\\mathbb {R}^M, \\\\, M\\\\ge 1.$ Obviously, for $\\\\kappa :[0,+\\\\infty )\\\\rightarrow [0, +\\\\infty )$ , given by $\\\\kappa (s):=2^{2-p} s^p$ , $s\\\\ge 0$ , one has $\\\\kappa ^{-1}(\\\\lbrace 0\\\\rbrace ) = \\\\lbrace 0\\\\rbrace $ , $\\\\lim \\\\limits _{s\\\\rightarrow +\\\\infty } \\\\kappa (s) = +\\\\infty $ .\",\n \"As for (a3), take any $u\\\\in X$ .\",\n \"Then taking $u_+ := \\\\max \\\\lbrace u , 0\\\\rbrace $ and $u_-:=\\\\max \\\\lbrace -u,0\\\\rbrace $ we have $u= u^+ - u^-$ and ${n}(u_+ ) = \\\\frac{1}{p} \\\\int _{\\\\Omega } |u_+ (x)|^p \\\\,\\\\mathrm {d}x \\\\le \\\\frac{1}{p} \\\\int _{\\\\Omega } |u (x)|^p \\\\,\\\\mathrm {d}x = {n}(u).$ If $u\\\\in Y_p = W_{0}^{1,p}(\\\\Omega )$ , then, due to Lemma 7.6 of [11], $\\\\nabla u_+(x) = 0$ if $u(x)\\\\le 0$ and $\\\\nabla u_+ (x)=\\\\nabla u(x)$ if $u(x)\\\\ge 0$ .\",\n \"Therefore $u_+ \\\\in Y_p$ and ${a}(u_+) = \\\\frac{1}{p}\\\\int _{\\\\Omega }|\\\\nabla u_+(x)|^p \\\\,\\\\mathrm {d}x \\\\le \\\\frac{1}{p}\\\\int _{\\\\Omega }|\\\\nabla u (x)|^p \\\\,\\\\mathrm {d}x= {a}(u).$ Finally, (a4) is immediate as, for $u,v\\\\in M_p$ , $|u|^{p}= u^p \\\\le (u+v)^{p}=|u+v|^p$ .\",\n \"$\\\\square $ In view of Section , the operators $A_p$ , $N_p:=D {n}$ together with $M_p$ and $M_p^*$ satisfy the assumptions made in Section and the topological degree can be applied for perturbations of $A_p$ .\",\n \"Before we proceed further let us pay attention to the perturbation term.\",\n \"Proposition 4.2 Let $f:\\\\Omega \\\\times [0,+\\\\infty )\\\\rightarrow \\\\mathbb {R}$ satisfy (REF ) and $f(x,0) \\\\ge 0$ for a.a. $x\\\\in \\\\Omega $ .\",\n \"Then the mapping $F : X_p \\\\rightarrow X_{p}^{*}$ given by $\\\\langle F(u), v \\\\rangle _{X_p}:= \\\\int _{\\\\Omega } f(u(x))v(x) \\\\,\\\\mathrm {d}x, \\\\ \\\\ u\\\\in M_p, \\\\, v\\\\in X_p,$ is well defined, continuous, bounded on bounded sets and $F(N^{-1}(\\\\tau )) \\\\in T_{M_{p}^*} (\\\\tau ) \\\\ \\\\ \\\\mbox{ for any } \\\\ \\\\tau \\\\in M_p^{*}.$ Lemma 4.3 Let $10$ .\",\n \"This implies that $v_n \\\\rightarrow v$ in $L^q(\\\\Omega )$ , i.e.\",\n \"$v\\\\in T_{M_q} (u)$ .\",\n \"In order to show the converse inclusion, observe that $T_u$ is closed and, for any $h>0$ , $h(M-u) \\\\subset T_u$ .\",\n \"This clearly implies that $T_{M_q} (u)\\\\subset T_u$ .\",\n \"$\\\\square $ Proof of Proposition REF: Using the Riesz representation isomorphism $\\\\varrho $ between $L^p (\\\\Omega )^*$ and $L^q(\\\\Omega )$ , $1/p +1/q=1$ , the mapping $F\\\\circ N^{-1}$ can be treated as the mapping $L^q(\\\\Omega )\\\\ni u \\\\mapsto f(\\\\cdot , \\\\theta _q (u)) \\\\in L^q(\\\\Omega )$ where $\\\\theta _q:\\\\mathbb {R}\\\\rightarrow \\\\mathbb {R}$ is given by $\\\\theta _q (s)=|s|^{q-2}s$ , $s\\\\in \\\\mathbb {R}$ .\",\n \"It is well defined as $|(f(x,\\\\theta _q(s))| \\\\le C(1+|\\\\theta _q (s)|^p) = C(1+|s|) \\\\mbox{ for } s\\\\ge 0 \\\\mbox{ and a.e. }\",\n \"x\\\\in \\\\Omega .$ Observe that $f(x,\\\\theta _q (0))=f(x,0)\\\\ge 0$ for.\",\n \"a.e.\",\n \"$x\\\\in \\\\Omega $ , which, by use of Lemma REF , implies that $f(\\\\cdot , \\\\theta _q (u(\\\\cdot )))\\\\in T_{M_q}(u)$ for all $u\\\\in M_q$ .\",\n \"Since $\\\\varrho (M_p^*)=M_q$ , we infer that (REF ) holds.\",\n \"$\\\\square $ Hence we have showed that the problem (REF ) indeed can be formulated as an abstract problem $\\\\left\\\\lbrace \\\\begin{array}{l}A_p (u) = F (u),\\\\\\\\u\\\\in M_p \\\\cap D(A_p).\",\n \"\\\\end{array}\\\\right.$ In order to take advantage of the topological degree effectively we need some methods of computing it.\",\n \"Theorem 4.4 If $\\\\ 2 \\\\lambda _{1,p}$ for a.e.\",\n \"$x\\\\in \\\\Omega $ , or $\\\\rho (x)< \\\\lambda _{1,p}$ for a.e.\",\n \"$x\\\\in \\\\Omega $ , then $\\\\mathrm {Deg}_{M_p} (A_p, \\\\rho N_{p}, B_{M_p} (0,R)) = \\\\left\\\\lbrace \\\\begin{array}{cl}1, & \\\\mbox{ if } \\\\ \\\\ \\\\rho (x) < \\\\lambda _{1,p} \\\\ \\\\ \\\\mbox{ for a.e. }\",\n \"\\\\ \\\\ x\\\\in \\\\Omega ,\\\\\\\\0, & \\\\mbox{ if } \\\\ \\\\ \\\\rho (x) > \\\\lambda _{1,p} \\\\ \\\\ \\\\mbox{ for a.e. }\",\n \"\\\\ \\\\ x\\\\in \\\\Omega .\\\\end{array}\\\\right.$ Remark 4.5 Before passing to the proof of Theorem REF , we need to make a comment on the eigenvalue problem relating to the $p$ -Laplace operator.\",\n \"Solving the nonlinear eigenvalue problem $\\\\left\\\\lbrace \\\\begin{array}{l}A_p (u) = \\\\lambda N_p(u)\\\\\\\\u\\\\in M_p \\\\cap D(A_p)\\\\end{array} \\\\right.$ reduces to find nonnegative weak solutions $u\\\\in W^{1,p}(\\\\Omega )$ of $\\\\left\\\\lbrace \\\\begin{array}{cl}-\\\\mathrm {div} (|\\\\nabla u(x)|^{p-2} \\\\nabla u(x)) = \\\\lambda |u(x)|^{p-2} u(x), & x\\\\in \\\\Omega ,\\\\\\\\u(x) = 0, & x\\\\in \\\\partial \\\\Omega .\",\n \"\\\\\\\\\\\\end{array}\\\\right.$ It appears that some properties of the eigenvalue problem for the Laplace operator are also valid for the $p$ -Laplace one.\",\n \"For details we refer to [14], [15] and [16].\",\n \"In particular, it is known that (REF ) does not admit any nonzero solutions if $\\\\lambda \\\\le 0$ , i.e.\",\n \"the $p$ -Laplace has no nonpositive eigenvalues.\",\n \"Moreover, there exists the smallest eigenvalue $\\\\lambda _{1,p}$ given by the Rayleigh formula $\\\\lambda _{1,p} = \\\\inf \\\\limits _{u\\\\in W_{0}^{1,p} (\\\\Omega ), u\\\\ne 0} \\\\frac{\\\\int _{\\\\Omega } |\\\\nabla u(x)|^p \\\\,\\\\mathrm {d}x}{\\\\int _{\\\\Omega } |u(x)|^p \\\\,\\\\mathrm {d}x}.$ The eigenfunctions corresponding to $\\\\lambda _{1,p}$ are either strictly positive or negative in $\\\\Omega $ and belong to $L^\\\\infty (\\\\Omega )$ .\",\n \"Moreover, $\\\\lambda _{1,p}$ is an isolated eigenvalue and if there are two eigenfunctions $u,v$ for $\\\\lambda _{1,p}$ , then there exists $\\\\alpha \\\\in \\\\mathbb {R}$ such that $u=\\\\alpha v$ .\",\n \"It is also known that if any eigenfunction does not change its sign in $\\\\Omega $ , then the corresponding eigenvalue must be equal to $\\\\lambda _{1,p}$ .\",\n \"$\\\\square $ In the proof we shall use a few lemmata given below.\",\n \"Lemma 4.6 There are $ C, s > 0$ such that $\\\\Vert u\\\\Vert _{L^p} \\\\le C |\\\\tilde{\\\\Omega }|^{s} \\\\Vert \\\\nabla u\\\\Vert _{L^p}$ for all $u\\\\in W_{0}^{1,p}(\\\\Omega )$ and measurable $\\\\tilde{\\\\Omega }\\\\subset \\\\Omega $ with the property $u(x)=0$ if $x\\\\notin \\\\tilde{\\\\Omega }$ .\",\n \"Proof: By the Sobolev embedding theorem there exists $q>p$ such that $\\\\Vert u \\\\Vert _{L^q} \\\\le C\\\\Vert \\\\nabla u\\\\Vert _{p} \\\\mbox{ for all } u\\\\in W_{0}^{1,p}(\\\\Omega ).$ On the other hand, by the Hölder inequality, $\\\\Vert u\\\\Vert _{L^p} \\\\le \\\\Vert u \\\\Vert _{L^q} |\\\\tilde{\\\\Omega }|^{1/p-1/q}.$ Combining the two above inequalities we get the desired one with $s:=1/p-1/q$ .\",\n \"$\\\\square $ Lemma 4.7 Let $v$ be a nonnegative weak solution of (REF ) with $\\\\lambda =\\\\lambda _{1,p}$ and $\\\\rho \\\\in L^{\\\\infty }(\\\\Omega )$ .\",\n \"If $u\\\\in W_{0}^{1,p}(\\\\Omega )$ is a weak solution to $-\\\\mathrm {div} (|\\\\nabla u|^{p-2} \\\\nabla u) = \\\\rho |u|^{p-2} u + |v|^{p-2}v \\\\, \\\\mbox{ on } \\\\Omega ,$ then $u\\\\in L^{\\\\infty }(\\\\Omega )$ .\",\n \"Proof: Here we adapt the arguments from [15].\",\n \"Note that without loss of generality we can consider the equation $-\\\\mathrm {div} (|\\\\nabla u|^{p-2} \\\\nabla u) = \\\\rho |u|^{p-2} u + \\\\lambda _{1,p} |v|^{p-2} v \\\\, \\\\mbox{ on } \\\\Omega .$ Take any $k>0$ and put $\\\\eta :=\\\\max \\\\lbrace u-v-k,0\\\\rbrace $ .\",\n \"Since $\\\\eta \\\\in W_{0}^{1,p}(\\\\Omega )$ , we get $\\\\int _{\\\\Omega _k} (|\\\\nabla u|^{p-2}\\\\nabla u - |\\\\nabla v|^{p-2} \\\\nabla v) \\\\cdot \\\\nabla (u-v)\\\\,\\\\mathrm {d}x \\\\le \\\\Vert \\\\rho \\\\Vert _{L^{\\\\infty }} \\\\int _{\\\\Omega _k} u^{p-1}(u-v-k) \\\\,\\\\mathrm {d}x$ with $\\\\Omega _k := \\\\lbrace x\\\\in \\\\Omega \\\\mid u(x)-v(x)-k > 0 \\\\rbrace $ .\",\n \"This, by use of (REF ) and the convexity of the function $s\\\\mapsto |s |^{p-1}$ , gives $\\\\int _{\\\\Omega _k} |\\\\nabla (u-v)|^p \\\\,\\\\mathrm {d}x & \\\\le & C_1 \\\\int _{\\\\Omega _k} u^{p-1} (u-v-k) \\\\,\\\\mathrm {d}x \\\\\\\\& \\\\le & C_1 2^{p-2} \\\\left( \\\\int _{\\\\Omega _k} (u-v-k)^p \\\\,\\\\mathrm {d}x + \\\\int _{\\\\Omega _k} (v+k)^{p-1}(u-v-k) \\\\,\\\\mathrm {d}x\\\\right)$ for some constant $C_1>0$ (here all the constant are to be independent of $k$ ).\",\n \"By applying Lemma REF , one gets $\\\\int _{\\\\Omega _k} (u-v-k)^p \\\\,\\\\mathrm {d}x \\\\le C|\\\\Omega _k|^{s} \\\\int _{\\\\Omega _k} |\\\\nabla (u-v)|^{p} \\\\,\\\\mathrm {d}x,$ which together with the previous inequality yields $(1- C_2|\\\\Omega _k|^{s}) \\\\int _{\\\\Omega _k} (u-v-k)^p \\\\,\\\\mathrm {d}x \\\\le C_2 |\\\\Omega _k|^{s} \\\\int _{\\\\Omega _k} (v+k)^{p-1}(u-v-k) \\\\,\\\\mathrm {d}x$ for some $C_2>0$ .\",\n \"Since $|\\\\Omega _k|\\\\rightarrow 0$ as $k\\\\rightarrow +\\\\infty $ , there is $k_0$ such that for all $k\\\\ge k_0$ $1-C_2 |\\\\Omega _k|^{s} >1/2$ .\",\n \"Further, for $k\\\\ge k_0$ , $\\\\int _{\\\\Omega _k} (u-v-k)^p \\\\,\\\\mathrm {d}x \\\\le 2 C_2 |\\\\Omega _k|^{s} (\\\\Vert v\\\\Vert _{L^{\\\\infty }} + k)^{p-1} \\\\int _{\\\\Omega _k} (u-v-k) \\\\,\\\\mathrm {d}x.$ Next we observe that the Hölder inequality yields $\\\\int _{\\\\Omega _k} (u-v-k) \\\\,\\\\mathrm {d}x \\\\le C_4 k |\\\\Omega _k|^{1+s(p-1)^{-1}} \\\\mbox{ for all } k\\\\ge k_0$ and some constant $C_4>0$ .\",\n \"Now define $j:(0,+\\\\infty )\\\\rightarrow [0,+\\\\infty )$ by $j(k):= \\\\int _{\\\\Omega _k} (u-v-k) \\\\,\\\\mathrm {d}x, \\\\, \\\\, k>0.$ Note that by the Tonelli-Fubini theorem applied to the set $\\\\lbrace (x,t)\\\\in \\\\Omega \\\\times [0,+\\\\infty )\\\\mid u(x)-v(x)>t>k \\\\rbrace $ one has $j(k) = \\\\int _{k}^{+\\\\infty } |\\\\Omega _t| \\\\,\\\\mathrm {d}t, \\\\,\\\\, k> 0.$ Obviously, $j$ is nonincreasing and absolutely continuous with $j^{\\\\prime }(k)=-|\\\\Omega _k|$ for a.e.\",\n \"$k\\\\ge 0$ .\",\n \"We claim that $j(k)=0$ for some $k>0$ .\",\n \"If it were not so, then (REF ) could be rewritten as $j(k)^{\\\\theta } \\\\le -C_{4}^{\\\\theta } k^{\\\\theta } j^{\\\\prime }(k) \\\\mbox{ for all } k\\\\ge k_0$ with $\\\\theta :=(1+s(p-1)^{-1})^{-1}$ , and consequently $k^{-\\\\theta } \\\\le - C_{4}^{\\\\theta } j(k)^{-\\\\theta } j^{\\\\prime }(k) \\\\mbox{ for all } k\\\\ge k_0.$ This after integration would give $k^{1-\\\\theta } + C_{4}^{\\\\theta } j(k)^{1-\\\\theta } \\\\le k_0^{1-\\\\theta } + C_{4}^{\\\\theta } j(k_0)^{1-\\\\theta } \\\\mbox{ for all } k\\\\ge k_0,$ which yields a contradiction proving the claim that $j(k)=0$ for some $k>0$ .\",\n \"Then, for some $k>0$ , $|\\\\Omega _k|=0$ and $u\\\\le v+k$ a.e.\",\n \"on $\\\\Omega $ .\",\n \"This shows that $u\\\\in L^{\\\\infty }(\\\\Omega )$ , as $v\\\\in L^{\\\\infty }(\\\\Omega )$ (see Remark REF ).\",\n \"$\\\\square $ Lemma 4.8 (see [10]) If $h\\\\in L^\\\\infty (\\\\Omega )$ is nonnegative and nonzero, then the equation $-\\\\mathrm {div} (|\\\\nabla u|^{p-2} \\\\nabla u) = \\\\lambda _{1,p} |u|^{p-2}u + h, \\\\, \\\\mbox{ on } \\\\Omega ,$ has no nonzero weak solution in $W_{0}^{1,p}(\\\\Omega )$ .\",\n \"Lemma 4.9 If $\\\\rho \\\\in L^\\\\infty (\\\\Omega )$ and either $\\\\rho (x)> \\\\lambda _{1,p}$ for a.e.\",\n \"$x\\\\in \\\\Omega $ or $\\\\rho (x)<\\\\lambda _{1,p}$ for a.e.\",\n \"$x\\\\in \\\\Omega $ , then the problem $-\\\\mathrm {div}(|\\\\nabla u|^{p-2}\\\\nabla u) = \\\\rho |u|^{p-2}u \\\\mbox{ on } \\\\Omega $ does not admit a nonzero solution $u\\\\in W_{0}^{1,p}(\\\\Omega )$ such that $u\\\\ge 0$ .\",\n \"Proof: If $\\\\rho <\\\\lambda _{1,p}$ a.e.\",\n \"on $\\\\Omega $ and $u\\\\in W_{0}^{1,p}(\\\\Omega )$ is a nonzero weak solution of (REF ), then $\\\\int _{\\\\Omega } |\\\\nabla u|^{p} \\\\,\\\\mathrm {d}x = \\\\int _{\\\\Omega } \\\\rho |u|^{p-2}u \\\\,\\\\mathrm {d}x < \\\\lambda _{1,p} \\\\int _{\\\\Omega } |u|^{p-2}u \\\\,\\\\mathrm {d}x,$ which gives $\\\\lambda _{1,p}> \\\\int _{\\\\Omega } |\\\\nabla u|^p \\\\,\\\\mathrm {d}x / \\\\int _{\\\\Omega } |u|^p \\\\,\\\\mathrm {d}x$ , a contradiction with the Rayleigh formula.\",\n \"In the case $\\\\rho >\\\\lambda _{1,p}$ a.e.\",\n \"on $\\\\Omega $ , we observe that if $u$ is a weak solution of (REF ), then $u$ is a weak solution of $-\\\\mathrm {div}(|\\\\nabla u|^{p-2}\\\\nabla u) =\\\\lambda _{1,p} |u|^{p-2}u + h \\\\ \\\\mbox{ on } \\\\ \\\\Omega $ with $h:=(\\\\rho - \\\\lambda _{1,p})|u|^{p-2}u$ .\",\n \"Clearly, $h\\\\ge 0$ and $h\\\\in L^{\\\\infty }(\\\\Omega )$ , since $u\\\\in L^{\\\\infty }(\\\\Omega )$ due to Lemma REF .\",\n \"Hence, Lemma REF leads to a contradiction ending the proof.\",\n \"$\\\\square $ Proof of Theorem REF : Assume that $\\\\rho >\\\\lambda _{1,p}$ a.e.\",\n \"on $\\\\Omega $ and fix $\\\\tilde{\\\\lambda }>\\\\lambda _{1,p}$ .\",\n \"Define $H:X_p \\\\times [0,1]\\\\rightarrow X_p$ by $H (u,t) := (t\\\\tilde{\\\\lambda }+ (1-t) \\\\rho ) N_p(u)$ , $u\\\\in X_p$ , $t\\\\in [0,1]$ .\",\n \"In view of Lemma REF , $-A_p (u)+H(u,t)\\\\ne 0$ for all $u\\\\in D(A_p) \\\\setminus \\\\lbrace 0 \\\\rbrace $ and $t\\\\in [0,1]$ .\",\n \"Therefore, we can use the homotopy invariance – Theorem REF (iii) to get $\\\\mathrm {Deg}_{M_p} (A_p, \\\\rho N_{p}, B_{M_p} (0,R) ) = \\\\mathrm {Deg}_{M_p} ( A_p, \\\\tilde{\\\\lambda }N_p, B_{M_p}(0,R) ).$ In a similar manner one can prove the same formula in the case $\\\\rho <\\\\lambda _{1,p}$ a.e.\",\n \"on $\\\\Omega $ with $\\\\tilde{\\\\lambda }<\\\\lambda _{1,p}$ .\",\n \"Now we shall prove that conditions $({\\\\cal M}_1)$ and $({\\\\cal M}_2)$ of Theorem REF are satisfied.\",\n \"Observe that, in view of Lemma REF , for any $\\\\lambda \\\\ne \\\\lambda _{1,p}$ , the eigenvalue problem (REF ) has no nontrivial and nonnegative weak solutions, i.e.\",\n \"$({\\\\cal M}_1)$ holds.\",\n \"To show $({\\\\cal M}_2)$ let $\\\\tau _0\\\\in L^p(\\\\Omega )$ be the functional determined by $|u_0|^{p-2}u_0$ with $u_0$ being a fixed positive solution of the eigenvalue problem (REF ) with $\\\\lambda =\\\\lambda _{1,p}$ .\",\n \"Suppose that there exists $u\\\\in (A_p-\\\\lambda N_p)^{-1}( \\\\lbrace \\\\tau _0 \\\\rbrace ) \\\\cap M_p$ for some $\\\\lambda > \\\\lambda _{1,p}$ .\",\n \"This means that $u\\\\in W_{0}^{1,p}(\\\\Omega )$ is a nonnegative weak solution of $-\\\\mathrm {div} (|\\\\nabla u|^{p-2} \\\\nabla u) = \\\\lambda _{1,p} |u|^{p-2} u + h \\\\ \\\\mbox{ on } \\\\ \\\\Omega $ with $h:=(\\\\lambda -\\\\lambda _{1,p}) |u|^{p-2} u + |u_0|^{p-2}u_0$ .\",\n \"It follows from Lemma REF that $h\\\\in L^{\\\\infty } (\\\\Omega )$ .\",\n \"Since $h\\\\ge 0$ , Lemma REF implies that such a solution does not exist, a contradiction proving $({\\\\cal M}_2)$ .\",\n \"Hence, by Theorem REF and (REF ), the desired formula follows.\",\n \"$\\\\square $ The obtained formula results in the following general one.\",\n \"Theorem 4.10 Let $f$ and $F$ be as in Proposition REF and suppose that (REF ) hold.\",\n \"(i) If $\\\\rho _0$ is as in () and either $\\\\rho _0 (x) < \\\\lambda _{1,p}$ , for a.e.\",\n \"$x\\\\in \\\\Omega $ , or $\\\\lambda _{1,p}<\\\\rho _0 (x)$ , for a.e.\",\n \"$x\\\\in \\\\Omega $ , then there exists $\\\\delta >0$ such that $A_p (u) \\\\ne F (u) \\\\mbox{ for all } u \\\\in D(A_p) \\\\cap \\\\left( B_{M_p} (0,\\\\delta ) \\\\setminus \\\\lbrace 0 \\\\rbrace \\\\right)$ and $\\\\mathrm {Deg}_M (A, F, B_M (0,\\\\delta )) = \\\\left\\\\lbrace \\\\begin{array}{cl}1, & \\\\ \\\\ \\\\mbox{ if } \\\\ \\\\ \\\\rho _0(x) < \\\\lambda _{1,p} \\\\ \\\\ \\\\mbox{ for a.e. }\",\n \"x\\\\in \\\\Omega ,\\\\\\\\0, & \\\\ \\\\ \\\\mbox{ if } \\\\ \\\\ \\\\rho _0(x) > \\\\lambda _{1,p} \\\\ \\\\ \\\\mbox{ for a.e. }\",\n \"x\\\\in \\\\Omega .\\\\end{array}\\\\right.$ (ii) If $\\\\rho _\\\\infty $ is as in () either $\\\\rho _\\\\infty (x) < \\\\lambda _{1,p}$ , for a.e.\",\n \"$x\\\\in \\\\Omega $ , or $\\\\lambda _{1,p}<\\\\rho _\\\\infty (x)$ , for a.e.\",\n \"$x\\\\in \\\\Omega $ , then there exists $R>0$ such that $A_p (u) \\\\ne F(u) \\\\mbox{ for all } u \\\\in D(A_p) \\\\cap \\\\left( M_p\\\\setminus B_{M_p} (0,R)\\\\right)$ and $\\\\mathrm {Deg}_{M_p} (A_p, F, B_{M_p} (0,R)) = \\\\left\\\\lbrace \\\\begin{array}{cl}1, & \\\\mbox{ if } \\\\ \\\\ \\\\rho _\\\\infty (x)< \\\\lambda _{1,p} \\\\ \\\\ \\\\mbox{ for a.e. }\",\n \"x\\\\in \\\\Omega ,\\\\\\\\0, & \\\\mbox{ if } \\\\ \\\\ \\\\rho _\\\\infty (x)> \\\\lambda _{1,p} \\\\ \\\\ \\\\mbox{ for a.e. }\",\n \"x\\\\in \\\\Omega .\\\\end{array}\\\\right.$ Proof: (i) Define $H:M_p\\\\times [0,1]\\\\rightarrow X_p$ by $H(u,t):= t F(u) + (1-t) \\\\rho _0 N_{p} (u),$ $(u,t)\\\\in M_p \\\\times [0,1]$ .\",\n \"By Proposition REF , $H$ is continuous and $F\\\\circ N_{p}^{-1}$ is tangent to $M^*$ .\",\n \"Moreover we claim that $\\\\mbox{ there is $\\\\delta >0$such that $-A_p (u)+H(u,t) \\\\ne 0$ for all $u\\\\in M_p \\\\cap D(A_p), \\\\ t\\\\in [0,1]$.\",\n \"}$ Suppose to the contrary that there exists $(u_n)$ in $(M_p\\\\cap D(A_p))\\\\setminus \\\\lbrace 0 \\\\rbrace $ and $(t_n)$ in $[0,1]$ such that $u_n \\\\rightarrow 0$ in $X_p$ and $-A_p (u_n) + H(u_n, t_n)=0$ , $n\\\\ge 1$ .\",\n \"Then clearly, if we put $w_n:= \\\\Vert u_n\\\\Vert _{X_p}^{-1}u_n$ and $s_n:=\\\\Vert u_n\\\\Vert _{X_p}$ , then $A_p(w_n) = s_{n}^{1-p} H(s_n w_n, t_n)$ , which gives $w_n =J_{1} \\\\left( N_p (w_n) + s_{n}^{1-p} H(s_n w_n, t_n) \\\\right), \\\\ n\\\\ge 1.$ The growth condition (REF ) and the existence of the first limit in () imply that there exists $C_1>0$ such that $\\\\Vert N_p (w_n) + s_{n}^{1-p} H(s_n w_n, t_n)\\\\Vert _{X_{p}^{*}}\\\\le C_1$ for all $n\\\\ge 1$ .\",\n \"Therefore we infer that $(w_n)$ has a subsequence convergent in $X_p$ , since, according to Proposition REF and Proposition REF (v), $J_1$ is completely continuous.\",\n \"In the sequel, we may assume that $(w_n)$ converges almost everywhere to some $w_0\\\\in M_p \\\\setminus \\\\lbrace 0 \\\\rbrace $ and that one has $g\\\\in X_p$ such that $|w_n|\\\\le g$ a.e.\",\n \"on $\\\\Omega $ .\",\n \"Further, note that if $w_n (x)\\\\ne 0$ , then $\\\\frac{f(x,s_n w_n(x))}{s_{n}^{p-1}} = \\\\frac{f(x, s_n w_n (x))}{(s_n w_n (x))^{p-1}} (w_n(x))^{p-1} \\\\rightarrow \\\\rho _0 (x) (w_0(x))^{p-1} \\\\mbox{ as } n\\\\rightarrow +\\\\infty ,$ which, by the dominated convergence theorem, implies that $s_{n}^{1-p} H(s_n w_n, t_n)\\\\rightarrow \\\\rho _0 N_p(w_0)$ in $X_p^{*}$ .\",\n \"Hence, a passage to the limit in (REF ) yields $w_0 = J_1 (N_p (w_0)+ \\\\rho _0 N_p (w_0) )$ , i.e.\",\n \"$-A_p w_0 + \\\\rho _0 N_p (w_0)=0$ .\",\n \"This is a contradiction due to Lemma REF and (REF ) is proved.\",\n \"Clearly, (REF ) allows us to use the homotopy invariance – Theorem REF (iii) to see that $\\\\mathrm {Deg}_{M_p} (A_p, F, B_{M_p} (0,R)) = \\\\mathrm {Deg}_{M_p} (A_p, \\\\rho _0 N_p, B_{M_p} (0,R))$ , which together with Theorem REF provides the required formula.\",\n \"(ii) The proof is analogical to that for part (i) and it is left to the reader.\",\n \"$\\\\square $ Proof of Theorem REF: Let $\\\\delta >0$ and $R>\\\\delta $ be like in Theorem REF .\",\n \"Then by use of the additivity property – Theorem REF (ii), we get $\\\\mathrm {Deg}_{M_p} (A_p, F, B_{M_p} (0,R) \\\\setminus \\\\overline{B_{M_p} (0,\\\\delta )})= \\\\mathrm {Deg}_{M_p} (A_p, F, B_{M_p} (0,R))\\\\!\",\n \"-\\\\!\",\n \"\\\\mathrm {Deg}_{M_p} (A_p, F, B_{M_p} (0,\\\\delta )) \\\\\\\\= \\\\left\\\\lbrace \\\\begin{array}{cl}1, & \\\\mbox{ if } \\\\ \\\\rho _0 (x) > \\\\lambda _{1,p} > \\\\rho _\\\\infty (x) \\\\ \\\\mbox{ for a.e. }\",\n \"x\\\\in \\\\Omega ,\\\\\\\\-1, & \\\\mbox{ if } \\\\ \\\\rho _0 (x) < \\\\lambda _{1,p} < \\\\rho _\\\\infty (x) \\\\ \\\\mbox{ for a.e. }\",\n \"x\\\\in \\\\Omega .\\\\end{array}\\\\right.$ Hence the existence property of the topological degree gives the existence of $u\\\\in B_{M_p} (0,R) \\\\setminus \\\\overline{B_{M_p} (0,\\\\delta )}$ such that $A_p (u) = F(u)$ , which is a required nonzero nonnegative weak solution of (REF ).\",\n \"$\\\\square $\"\n ]\n]"},"id":{"kind":"string","value":"1204.1295"}}},{"rowIdx":1866889,"cells":{"text":{"kind":"list like","value":[["Gauge and moduli hierarchy in a multiply warped braneworld scenario"],["Abstract A generalized Randall sundrum model in six dimensional bulk is studied in presence of non-flat 3-branes at the orbifold fixed points.","The warp factors for this model is determined in terms of multiple moduli and brane cosmological constant.","We show that the requirements of a vanishingly small cosmological constant on the visible brane along with non-hierarchical moduli, each with scale close to Planck length, lead to a scenario where the 3-branes can not have any intermediate scale and have energy scales either close to Tev or close to Planck scale.","Such a scenario can address both the gauge hierarchy as well as fermion mass hierarchy problem in standard model.","Thus simultaneous resolutions to these problems are closely linked with the near flatness condition of our universe without any intermediate hierarchical scale for the moduli."],["Introduction","Large hierarchy of mass scales between the Planck and the electroweak scales results into well-known fine tuning problem in connection with the mass of the Higgs, the only scalar particle in the standard model.","It has been shown that due to large radiative corrections the Higgs mass diverges quadratically and can not be confined within Tev scale unless some unnatural tuning is done order by order in the perturbation theory.","This problem has been addressed in different variants of extra dimensional models.","Among all extra dimensional models the scenario proposed by Randall and Sundrum has drawn lot of attention.","It assumes a warp geometry of the space-time in 5 dimensions [1].","The fifth dimension is compactified on a space $S^1/Z_2$ and the length scale of this extra dimension is of the order of Planck length $r_c$ .","Two 3-branes are located at the two orbifold fixed points.","The exponential warping of the length scale along the fifth dimension naturally suppresses Planck scale quantities of one 3-brane, which we call hidden brane/Planck brane, into electroweak scale on the second 3-brane, which is identified as TeV-brane/visible brane and can be interpreted as our universe without introducing any new hierarchical scale into the theory.","A generalization of RS model have been considered previously by introducing more than one warped extra dimension in the theory [2].","Such models have interesting implications in particle phenomenology in higher dimensional models which can be summarized as : Resolution of the well known fermion mass hierarchy problem among standard model fermions [3].","The localization of massless fermions with a definite chirality on the visible 3-brane [4].","A consistent description of a bulk Higgs and gauge fields with spontaneous symmetry breaking in the bulk, along with proper $W$ and $Z$ boson masses on the visible brane [5].","Provides a stack of branes picture similar to string inspired models [2].","Motivated by the proposal of a non-vanishing cosmological constant as dark energy model for our present universe, in this work we generalize the six dimensional multiple warped model to include non-flat 3-branes.","Such generalization was done earlier for five dimensional scenario in [6].","We find the warp factors solutions for both de-Sitter and anti de-Sitter 3-branes with appropriate brane tensions.","We organize our paper as follows.","In the following section we explain some features of the six dimensional doubly warped model with flat 3-branes.","In sec.II we describe the six dimensional doubly warped model with non vanishing cosmological constant on the 3-branes (i.e with non-flat branes).","In sec.III we explain the correlation between the brane induced cosmological constant, and the ratio of the two extra dimensional moduli in the de-sitter brane.","In sec.IV we present our result and show that the four important aspects, namely 1) the value of cosmological constant in the present universe 2) hierarchical warping along the two extra dimensions, 3) hierarchy between the two extra dimensional moduli and 4) the gauge hierarchy problem, have interesting correlations among themselves.","Here we briefly outline the mechanism for generalizing the 5-dimensional RS model to 6-dimensional doubly warped geometry model with flat 3-branes [2].","The 6-dimensional doubly warped model has six space-time dimensions.","The extra two spatial dimensions are orbifolded successively by $Z_2$ symmetry .","The manifold for such geometry is $[M^{(1,3)}\\times S^1/Z_2]\\times S^1/Z_2$ with four non-compact dimensions denoted by $x^\\mu $ , $\\mu =0,\\cdots ,3$ .","As we are interested in doubly warped model, the metric in this model can be chosen as $ds^2 = b^2(z)[a^2(y)\\eta _{\\mu \\nu }dx^\\mu dx^\\nu +R_y^2dy^2]+r_z^2dz^2$ The angular coordinates $y,z$ represent the extra spatial dimensions with moduli $R_y$ and $r_z$ respectively.","The Minkowski metric in the usual 4-dimensions has the form $\\eta _{\\mu \\nu } = {\\rm diag}(-1,1,1,1)$ .","The functions $a(y),b(z)$ provide the warp factors in the $y$ and $z$ directions respectively.","The total bulk-brane action of this model can be written as: $S &=& S_6+S_5+S_4\\nonumber \\\\S_6 &=& \\int d^4xdydz\\sqrt{-g_6}(R_6-\\Lambda ), \\quad \\nonumber \\\\S_5 &=& \\int d^4xdydz\\sqrt{-g_{5}}[V_1\\delta (y)+V_2\\delta (y-\\pi )]\\nonumber \\\\&&~~+ \\int d^4xdydz\\sqrt{-g_{5}}[V_3\\delta (z)+V_4\\delta (z-\\pi )]\\nonumber \\\\S_4&=& \\int d^4x \\sqrt{-g_{vis}}[{\\cal {L}}-\\hat{V}][\\delta (y)\\delta (z)+\\delta (y)\\delta (z-\\pi )\\nonumber \\\\&&~~+\\delta (y-\\pi )\\delta (z)+\\delta (y-\\pi )\\delta (z-\\pi )]$ Here, $V_{1,2}$ and $V_{3,4}$ are brane tensions of the branes located at $y=0,\\pi $ and $z=0,\\pi $ , respectively.","$\\Lambda $ is the cosmological constant in 6-dimensions.","After solving Einstein's equations, the solutions to the warp factors of the metric as given in eq.","(REF ) [2] are, $a(y) &=& \\exp (-c|y|), \\quad b(z) = \\frac{\\cosh (kz)}{\\cosh (k\\pi )}\\nonumber \\\\{\\rm where}~~ c&\\equiv & \\frac{R_yk}{r_z\\cosh (k\\pi )},\\quad k\\equiv r_z\\sqrt{\\frac{-\\Lambda }{10M_P^4}}$ Here, $M_P$ is the 4-dimensional Planck scale.","The 5-d RS model can be retrieved in the limit $r_{z}\\rightarrow 0$ .","The warp factors $a(y)$ and $b(z)$ provide maximum suppression at $y=\\pi $ and $z=0$ .","For this reason we can interpret the 3-brane formed out of the intersection of 4-branes at $y=\\pi $ and $z=0$ as our standard model brane.","The suppression on the standard model brane can be written as $f=\\frac{\\exp (-c\\pi )}{\\cosh (k\\pi )}$ The desired suppression of the order of $10^{-16}$ on the standard model brane can be obtained for different choices of the parameters $c$ and $k$ .","However from the relation for $c$ in eq.","(REF ) it can be shown that if we want to avoid large hierarchy in the moduli $R_y$ and $r_z$ , the warping in one direction must be large while that in the other direction must be small.","In our analysis we shall explore this feature for non-flat 3-branes over the entire parameter space of $c$ , $k$ and moduli ratio $R_{y}/r_{z}$ for different values of brane cosmological constant.","Previously a generalization of the 5-dimensional RS model has been considered with the non flat 3-branes [6].","In this work we consider doubly compactified six dimensional space-time with $Z_{2}$ orbifolding along each of the compact direction.","Thus the manifold under consideration is [$M^{(1,3)}\\times S^{1}/Z_2$ ]$\\times S^{1}/Z_{2}$ with four non-compact dimensions denoted by $x^\\mu $ , $\\mu =0,\\cdots ,3$ .","We choose a doubly warped general metric as: $ds^2 = b^2(z)[a^2(y)g_{\\mu \\nu }dx^\\mu dx^\\nu +R_y^2dy^2]+r_z^2dz^2$ Since orbifolding, in general, requires a localized concentration of energy we introduce 4-branes [(4+1)-dimensional space-time] at the orbifold fixed points namely at $y=0,\\pi $ and $z=0,\\pi $ .","The total bulk-brane action in this case is given by, $S&=&S_{6}+S_{5}+S_{4}\\nonumber \\\\S_6&=& \\int d^{4}xdydz\\sqrt{-g_{6}}(R_{6}-\\Lambda )\\nonumber \\\\S_{5}&=&\\int d^{4}xdydz\\sqrt{-g_{5}}[V_{1}\\delta (y)+V_{2}\\delta (y-\\pi )]\\nonumber \\\\&&~~+ \\int d^{4}xdydz\\sqrt{-g_{5}}[V_3\\delta (z)+V_{4}\\delta (z-\\pi )]\\nonumber \\\\S_4&=& \\int d^4x \\sqrt{-g_{vis}}[{\\cal {L}}-\\hat{V}][\\delta (y)\\delta (z)+\\delta (y)\\delta (z-\\pi )\\nonumber \\\\&&~~+\\delta (y-\\pi )\\delta (z)+\\delta (y-\\pi )\\delta (z-\\pi )]$ The brane potential terms may be coordinate dependent as, $V_{1,2}=V_{1,2}(z)$ and $V_{3,4}=V_{3,4}(y)$ .","The term $S_{4}$ is the action for 3-branes located at $(y,z)=(0,0),(0,\\pi ),(\\pi ,0),(\\pi ,\\pi )$ .","The full 6-dimensional Einstein's equation can be written as, $-M^{4}\\sqrt{-g_{6}}(R_{MN}-\\frac{R}{2}g_{MN})&=&\\Lambda _{6}\\sqrt{-g_{6}}g_{MN}+\\sqrt{-g_{5}}V_{1}(z)g_{\\alpha \\beta }\\delta ^{\\alpha }_{M}\\delta ^{\\beta }_{N}\\delta (y)\\nonumber \\\\&+&\\sqrt{-g_{5}}V_{2}(z)g_{\\alpha \\beta }\\delta ^{\\alpha }_{M}\\delta ^{\\beta }_{N}\\delta (y-\\pi )\\nonumber \\\\&+&\\sqrt{-\\tilde{g}_{5}}V_{3}(y) \\tilde{g}_{\\tilde{\\alpha }\\tilde{\\beta }}\\delta ^{\\tilde{\\alpha }}_{M}\\delta ^{\\tilde{\\beta }}_{N}\\delta (z)\\nonumber \\\\&+&\\sqrt{-\\tilde{g}_{5}}V_{4}(y) \\tilde{g}_{\\tilde{\\alpha }\\tilde{\\beta }}\\delta ^{\\tilde{\\alpha }}_{M}\\delta ^{\\tilde{\\beta }}_{N}\\delta (z-\\pi ) $ Here M,N are bulk indices, $\\alpha $ , $\\beta $ run over the usual four space-time coordinates ($x^{\\mu }$ ) and the compact coordinate $z$ while $\\tilde{\\alpha }$ , $\\tilde{\\beta }$ run over ($x^{\\mu }$ ) and the compact coordinate $y$ .","$g$ , $\\tilde{g}$ are the respective metrices in these (4+1)-dimensional spaces.","For the metric (REF ), the different components of Einstein's equations reduce to a set of three independent equations, $^{4}G_{\\mu \\nu }+g_{\\mu \\nu }[\\frac{3a^{\\prime }(y)^2}{R_{y}^2}+a(y){\\frac{3a^{\\prime \\prime }(y)}{R_{y}^2}+\\frac{2a(y)}{r_{z}^2}(3b^{\\prime }(z)^2+2b(z)b^{\\prime \\prime }(z))}]\\nonumber \\\\=-\\frac{\\Lambda _{6}}{M^{4}}a(y)^{2}b(z)^{2}g_{\\mu \\nu }$ $6a^{\\prime }(y)^2r_{z}^{2}+a(y)^2[6R_{y}^{2}b^{\\prime }(z)^2+4R_{y}^{2}b(z)b^{\\prime \\prime }(z)]-(1/2)~ ^{4}R~b(z)^{2}a(y)^2R_{y}^{2}r_{z}^{2}\\nonumber \\\\=\\frac{-\\Lambda _{6}}{M^{4}}b(z)^{2}a(y)^2R_{y}^{2}r_{z}^{2}\\\\6a^{\\prime }(y)^2r_{z}^{2}+4r_{z}^{2}a(y)a^{\\prime \\prime }(y)-(1/2)~ ^4R~r_{z}^{2}R_{y}^{2}a(y)^2b(z)^210R_{y}^{2}a(y)^2b^{\\prime }(z)^2\\nonumber \\\\=-\\frac{\\Lambda _{6}}{M^4}b(z)^2a(y)^2R_{y}^{2}r_{z}^{2}$ $^{4}G_{\\mu \\nu }$ and $^4R$ are the four dimensional Einstein tensor and Ricci scalar respectively, defined with respect to $g_{\\mu \\nu }$ .","Dividing both sides of the equation (REF ) by $g_{\\mu \\nu }$ for any $\\mu $ , $\\nu $ and rearranging terms it is seen that one side contains $a(y)$ and $b(z)$ and their derivatives, while the other side depends on the brane coordinates $x^{\\mu }$ only.","Thus we can equate each side to an arbitrary constant $\\Omega $ such that, $^4G_{\\mu \\nu }=-\\Omega g_{\\mu \\nu }\\\\a(y)^2[\\frac{3}{R_{y}^2}\\frac{a^{\\prime \\prime }(y)}{a(y)}+\\frac{3}{R_{y}^2}\\frac{a^{\\prime }(y)^2}{a(y)^2}+\\frac{2}{r_{z}^2}\\nonumber \\\\(3b^{\\prime }(z)^2+2b(z)b^{\\prime \\prime }(z))+\\frac{\\Lambda _{6}}{M^4}b(z)^2]=\\Omega $ From equation(REF ) we identify $\\Omega $ as the effective 4-D cosmological constant on the 3-branes.","We obtain the solutions for the warp factors $a(y)$ and $b(z)$ for de sitter brane from equation () as, $a(y)={\\omega ^{\\prime }{\\rm sinh}}\\left[ {\\rm ln}\\frac{c^{\\prime }_2}{{\\rm \\omega ^{\\prime }}}-cy\\right], \\quad b(z)=\\frac{{\\rm cosh}(kz)}{{\\rm cosh}(k\\pi )}$ Where $c_{2}^{^{\\prime }}$ is an integration constant.","Here $\\omega ^{\\prime }=\\omega {\\rm cosh}(k\\pi )$ , with $\\omega ^2=\\frac{\\Omega }{3k^{\\prime 2}}$ , where $k^{^{\\prime }}= \\sqrt{\\frac{-\\Lambda }{10M^4}}$ , $k=k^{^{\\prime }}r_z$ and $c=\\frac{R_{y}k}{r_{z}{\\rm cosh}(k\\pi )}$ .","Normalizing the warp factor to unity at the orbifold fixed point $y=0$ , we get $c^{\\prime }_2=\\left[ 1+(1+\\omega ^{\\prime 2})^{\\frac{1}{2}}\\right] $ .","Note that the result for RS model generalized to six dimensions with flat branes is recovered in the limit $\\omega \\rightarrow 0$ .","We now focus our attention to the boundary terms to determine the brane tensions.","Using the explicit form of $a(y)$ , $b(z)$ from equation(REF ) and implementing the boundary conditions across the two boundaries at $y=0$ , $y=\\pi $ and $z=0$ , $z=\\pi $ respectively, we obtain: $V_{2}(z)=8M^2\\sqrt{\\frac{-\\Lambda }{10}}\\frac{(\\frac{\\omega ^{^{\\prime }2}}{c^{^{\\prime }2}_{2}}e^{2c\\pi }+1)}{(\\frac{\\omega ^{^{\\prime }2}}{c^{^{\\prime }2}_{2}}e^{2c\\pi }-1)}{\\rm sech}(kz)\\\\V_{1}(z)=8M^2\\sqrt{\\frac{-\\Lambda }{10}}{\\rm sech}(kz)\\frac{(1+\\omega ^{^{\\prime }2}/c^{^{\\prime }2}_{2})}{(1-\\omega ^{^{\\prime }2}/c^{^{\\prime }2}_{})}$ The above two equations imply that the two 4-branes sitting at $y=0$ and $y=\\pi $ have $z$ dependent tensions.","Similarly we find $V_{3}(y)=0$ and $V_{4}(y)=-\\frac{8M^4k}{r_{z}}{\\rm tanh}(k\\pi )$ .","Thus we have determined the tensions for all the 4-branes in this model.","The intersection of two 4-branes give rise to 3-brane.","With this identification, the theory contains four 3-branes located at $(y,z)=(0,0),(0,\\pi )$ , $(\\pi ,0)$ , $(\\pi ,\\pi )$ .","The metric on the 3-brane located at $(y=0,z=\\pi )$ has no warping and can be identified with the Planck brane.","Similarly we identify the standard model brane with the one at $y=\\pi $ , $z=0$ where the warping is maximum.","Finally we obtain the expressions for 3-brane tensions in terms of $\\omega $ , the induced brane cosmological constant as, $V_{vis}=8M^2\\left( \\frac{\\frac{\\omega ^{2}{\\rm cosh}^{2}(k\\pi )e^{2c\\pi }}{(4+2\\omega ^{2}{\\rm cosh}^{2}(k\\pi ))}+1}{\\frac{\\omega ^{2}{\\rm cosh}^{2}(k\\pi )e^{2c\\pi }}{(4+2\\omega ^{2}{\\rm cosh}^{2}(k\\pi ))}-1}\\right)\\sqrt{-\\frac{\\Lambda }{10}}\\\\V_{hid}=8M^2\\left[\\left(\\frac{1+\\frac{\\omega ^{2}{\\rm cosh}^{2}(k\\pi )e^{2c\\pi }}{(4+2\\omega ^{2}{\\rm cosh}^{2}(k\\pi ))}}{1-\\frac{\\omega ^{2}{\\rm cosh}^{2}(k\\pi )e^{2c\\pi }}{(4+2\\omega ^{2}{\\rm cosh}^{2}(k\\pi ))}}\\right){\\rm sech}(k\\pi )-{\\rm tanh}(k\\pi )\\right]\\sqrt{-\\frac{\\Lambda }{10}}$ The two other 3-branes located at $(0,0)$ and $(\\pi ,\\pi )$ have tensions close to hidden brane and visible brane tensions respectively.","Once again limit $\\omega \\rightarrow 0$ reproduces the expression for the 6-D flat 3-brane tensions[2] and with $r_{z}\\rightarrow 0$ (i.e $k=k^{^{\\prime }}r_{z}\\rightarrow 0$ ) we recover 5-D RS brane tensions[1].","Now to solve the gauge hierarchy problem we equate the warp factors at $y=\\pi $ , $z=0$ to the ratio of the mass scale in the two 3-branes given by $10^{-n}$ such that, $a(y)b(z)|_{y=\\pi ,z=0}=10^{-n}\\nonumber \\\\{\\omega ^{\\prime }{\\rm sinh}}\\left[ {\\rm ln}\\frac{c^{\\prime }_2}{{\\rm \\omega ^{\\prime }}}-c \\pi \\right]\\frac{1}{{\\rm cosh}(k\\pi )}=10^{-n}$ At this point, we keep $n$ arbitrary but we will subsequently take it to be $\\simeq 16$ to achieve a Planck to Tev scale warping.","Defining $c\\pi =x$ , the above equation has a positive root for $e^{-x}$ as, $e^{-x}={\\rm cosh}(k\\pi )\\frac{10^{-n}}{c_{2}^{^{\\prime }}}\\left[1+\\left\\lbrace 1+\\omega ^210^{2n}\\right\\rbrace ^{1/2} \\right]$ We re-parametrize the brane cosmological constant $\\omega $ as: $\\omega ^{2}\\equiv 10^{-N}$ .","Equation (REF ) now simplifies to: $-N=\\frac{1}{{\\rm ln}10}{\\rm ln}\\left[ \\frac{4e^{-2x}-4{\\rm cosh}(k\\pi )10^{-n}e^{-x}}{{\\rm cosh}^{2}(k\\pi )[1+10^{-n}{\\rm cosh}(k\\pi )e^{-x}-2e^{-2x}]}\\right] $ This equation relates the three parameters present in this model, $\\omega $ (which has been re-parametrized as $10^{-N}$ ), $k$ , and $x(=c\\pi )$ .","The solution of $x$ , derived from expression (REF ), is obtained as, $x=16{\\rm ln}10-{\\rm ln}{\\rm cosh}(k\\pi )-{\\rm ln}2+{\\rm ln}[2+\\frac{1}{2}10^{-N}{\\rm cosh}^{2}(k\\pi )]-{\\rm ln}[1+\\frac{1}{4}10^{-(N-2n)}] $ Now from expression (REF ) we numerically calculate $x$ i.e $c\\pi $ for different values of $k\\pi $ taking cosmological constant $\\omega ^{2}$ as parameter for $n=16$ .","This value of $n$ ensures the resolution of the gauge hierarchy problem for all the determined values of the parameters in our subsequent analysis.","Also we calculate the corresponding values of the ratio of two moduli $R_{y}/r_{z}$ from the expression $c=\\frac{R_{y}k}{r_{z}{\\rm cosh}(k\\pi )}$ .","The numerical values are given in the following table (REF ): Table: Numerical values of cπc\\pi andR y /r z R_{y}/r_{z} for every kπk\\pi and fixed cosmological constant ω\\omega (in Planck unit)in dS space-timeFor given $k\\pi $ the corresponding values of $c\\pi $ and $R_{y}/r_{z}$ saturate below $\\omega ^{2}<10^{-40}$ .","It may be seen from table(REF ), when the cosmological constant is very large (i.e approximately of the order of 1), for a small $k\\pi \\approx 1.12$ , $c\\pi $ is also small $0.56$ and ratio of the two moduli is $0.84$ .","Hence it is evident that for a very large cosmological constant we can have equal warping along both the extra dimensions and the two extra dimensional moduli are approximately close to $l_{Planck}$ i.e.","without any hierarchical values.","Now if we fix a large $k\\pi =30.12$ , the value of $c\\pi $ turns out to be very very small ($\\approx 10^{-13}$ ) and also the ratio of the two moduli becomes hierarchical.","From table (REF ), it may be noted that when we decrease the value of cosmological constant, for small $k\\pi $ , values of $c\\pi $ become large and the two extra dimensional moduli are again reasonably close to each other with values close to $l_{Planck}$ .","But if we fix a large $k\\pi $ then we can see that for decreasing $\\omega ^{2}$ , $c\\pi $ values become small and the ratio of the two extra dimensional moduli become large.","Thus we conclude that a small but equal warping from both the extra dimensions and the minimum hierarchy between the two extra dimensional moduli can be achieved only when the induced cosmological constant on the brane is very large.","But if we keep on decreasing the brane induced cosmological constant towards its present observed value which is estimated to be of the order of $10^{-120}$ , we cannot have equal warping along both the extra dimensions.","Moreover if we demand that the two extra dimensional moduli are approximately of the same order, then in order to solve the gauge hierarchy problem the most favourable condition consistent with a small value of the brane cosmological constant is small $k\\pi $ and large $c\\pi $ i.e small warping along $z$ direction and large warping along $y$ direction.","This resembles to 5-dimensional RS model perturbed slightly by the additional warping along $z$ direction such that two 3-branes have scales close to Tev while the two other have scales close to Planck scale.","For Anti De-Sitter brane i.e $\\Omega <0$ , we find that there is an upper bound on the brane induced cosmological constant similar to the one found in [6] which is $10^{-32}$ in Planck units.","Repeating the entire analysis in the ADS sector for values of the cosmological constant lower than $10^{-32}$ we determine the correlations among the parameters illustrated in table (REF ) and (REF ).","Table: Numerical values of cπc\\pi andR y /r z R_{y}/r_{z} for different kπk\\pi with fixed cosmological constant ω\\omega in ADS space-time.","The numerical values of ω\\omega are all inPlanckian unitsTable: Numerical values of cπc\\pi andR y /r z R_{y}/r_{z} for different kπk\\pi with fixed cosmological constant ω\\omega in ADS space-time.","The numerical values of ω\\omega are all inPlanckian unitsHere also we come to a similar conclusion that for small cosmological constant, in order to keep minimum hierarchy between the two extra dimensional moduli, the most favourable condition is small $k\\pi \\approx 1.12$ and large $c\\pi \\approx 36.31$ .","In this work the generalization of RS model has been done for a 6-dimensional ADS bulk with non-flat 3- branes.","Requiring the warping from the hidden brane to visible brane $\\sim 10^{-16}$ we find that for de-Sitter 3-brane, the warping along both the directions can be nearly equal with very small hierarchy between the two moduli only when the brane cosmological constant is very large compared to the present value.","To achieve similar warping along both the directions for nearly vanishing cosmological constant we have to introduce large hierarchy between the moduli.","On the contrary for small value of the brane cosmological constant with non-hierarchical small moduli $\\sim l_{Planck}$ , the warping along one direction ( in our case along $y$ ) is very large while the other direction (i.e.","$z$ ) is nearly flat.","The corresponding values of the moduli $c$ and $k$ can be stabilized following the Goldberger-Wise stabilization mechanism [11] by introducing a 6-dimensional bulk scalar field.","This leads to a scenario where the 3-branes can not have any intermediate scale and have energy scales either close to Tev or close to Planck scale.","This remarkable correlations clearly point out that the most favoured condition for small cosmological constant, non-hierarchical moduli and the resolution of the gauge hierarchy problem correspond to a very large warping along $y$ direction with very small warping along z-direction.","This scenario is nothing but a weak perturbation of the original 5-dimensional RS model due to the presence of the sixth dimension which in turn leads to two 3-branes with energy scale close to Tev scale while two other 3-branes having energy scale close to Plank scale.","Such a feature of brane clustering with closely-spaced energy scales enhances when more and more extra warped dimensions are added leading to a stack of Tev scale 3-branes and a stack of Planck scale 3-branes.","It has been shown in [3] that such scenario offers a possible geometric resolution of the fermion mass hierarchy problem among the standard model fermions.","Our work thus explains that in a multiple warped geometry model the requirements of nearly flat 3-brane and non-hierarchical moduli lead naturally to a stack of closely clustered Tev 3-branes which in turn offers a geometric understanding of the fermion mass hierarchy as well as gauge hierarchy problem simultaneously."]],"string":"[\n [\n \"Gauge and moduli hierarchy in a multiply warped braneworld scenario\"\n ],\n [\n \"Abstract A generalized Randall sundrum model in six dimensional bulk is studied in presence of non-flat 3-branes at the orbifold fixed points.\",\n \"The warp factors for this model is determined in terms of multiple moduli and brane cosmological constant.\",\n \"We show that the requirements of a vanishingly small cosmological constant on the visible brane along with non-hierarchical moduli, each with scale close to Planck length, lead to a scenario where the 3-branes can not have any intermediate scale and have energy scales either close to Tev or close to Planck scale.\",\n \"Such a scenario can address both the gauge hierarchy as well as fermion mass hierarchy problem in standard model.\",\n \"Thus simultaneous resolutions to these problems are closely linked with the near flatness condition of our universe without any intermediate hierarchical scale for the moduli.\"\n ],\n [\n \"Introduction\",\n \"Large hierarchy of mass scales between the Planck and the electroweak scales results into well-known fine tuning problem in connection with the mass of the Higgs, the only scalar particle in the standard model.\",\n \"It has been shown that due to large radiative corrections the Higgs mass diverges quadratically and can not be confined within Tev scale unless some unnatural tuning is done order by order in the perturbation theory.\",\n \"This problem has been addressed in different variants of extra dimensional models.\",\n \"Among all extra dimensional models the scenario proposed by Randall and Sundrum has drawn lot of attention.\",\n \"It assumes a warp geometry of the space-time in 5 dimensions [1].\",\n \"The fifth dimension is compactified on a space $S^1/Z_2$ and the length scale of this extra dimension is of the order of Planck length $r_c$ .\",\n \"Two 3-branes are located at the two orbifold fixed points.\",\n \"The exponential warping of the length scale along the fifth dimension naturally suppresses Planck scale quantities of one 3-brane, which we call hidden brane/Planck brane, into electroweak scale on the second 3-brane, which is identified as TeV-brane/visible brane and can be interpreted as our universe without introducing any new hierarchical scale into the theory.\",\n \"A generalization of RS model have been considered previously by introducing more than one warped extra dimension in the theory [2].\",\n \"Such models have interesting implications in particle phenomenology in higher dimensional models which can be summarized as : Resolution of the well known fermion mass hierarchy problem among standard model fermions [3].\",\n \"The localization of massless fermions with a definite chirality on the visible 3-brane [4].\",\n \"A consistent description of a bulk Higgs and gauge fields with spontaneous symmetry breaking in the bulk, along with proper $W$ and $Z$ boson masses on the visible brane [5].\",\n \"Provides a stack of branes picture similar to string inspired models [2].\",\n \"Motivated by the proposal of a non-vanishing cosmological constant as dark energy model for our present universe, in this work we generalize the six dimensional multiple warped model to include non-flat 3-branes.\",\n \"Such generalization was done earlier for five dimensional scenario in [6].\",\n \"We find the warp factors solutions for both de-Sitter and anti de-Sitter 3-branes with appropriate brane tensions.\",\n \"We organize our paper as follows.\",\n \"In the following section we explain some features of the six dimensional doubly warped model with flat 3-branes.\",\n \"In sec.II we describe the six dimensional doubly warped model with non vanishing cosmological constant on the 3-branes (i.e with non-flat branes).\",\n \"In sec.III we explain the correlation between the brane induced cosmological constant, and the ratio of the two extra dimensional moduli in the de-sitter brane.\",\n \"In sec.IV we present our result and show that the four important aspects, namely 1) the value of cosmological constant in the present universe 2) hierarchical warping along the two extra dimensions, 3) hierarchy between the two extra dimensional moduli and 4) the gauge hierarchy problem, have interesting correlations among themselves.\",\n \"Here we briefly outline the mechanism for generalizing the 5-dimensional RS model to 6-dimensional doubly warped geometry model with flat 3-branes [2].\",\n \"The 6-dimensional doubly warped model has six space-time dimensions.\",\n \"The extra two spatial dimensions are orbifolded successively by $Z_2$ symmetry .\",\n \"The manifold for such geometry is $[M^{(1,3)}\\\\times S^1/Z_2]\\\\times S^1/Z_2$ with four non-compact dimensions denoted by $x^\\\\mu $ , $\\\\mu =0,\\\\cdots ,3$ .\",\n \"As we are interested in doubly warped model, the metric in this model can be chosen as $ds^2 = b^2(z)[a^2(y)\\\\eta _{\\\\mu \\\\nu }dx^\\\\mu dx^\\\\nu +R_y^2dy^2]+r_z^2dz^2$ The angular coordinates $y,z$ represent the extra spatial dimensions with moduli $R_y$ and $r_z$ respectively.\",\n \"The Minkowski metric in the usual 4-dimensions has the form $\\\\eta _{\\\\mu \\\\nu } = {\\\\rm diag}(-1,1,1,1)$ .\",\n \"The functions $a(y),b(z)$ provide the warp factors in the $y$ and $z$ directions respectively.\",\n \"The total bulk-brane action of this model can be written as: $S &=& S_6+S_5+S_4\\\\nonumber \\\\\\\\S_6 &=& \\\\int d^4xdydz\\\\sqrt{-g_6}(R_6-\\\\Lambda ), \\\\quad \\\\nonumber \\\\\\\\S_5 &=& \\\\int d^4xdydz\\\\sqrt{-g_{5}}[V_1\\\\delta (y)+V_2\\\\delta (y-\\\\pi )]\\\\nonumber \\\\\\\\&&~~+ \\\\int d^4xdydz\\\\sqrt{-g_{5}}[V_3\\\\delta (z)+V_4\\\\delta (z-\\\\pi )]\\\\nonumber \\\\\\\\S_4&=& \\\\int d^4x \\\\sqrt{-g_{vis}}[{\\\\cal {L}}-\\\\hat{V}][\\\\delta (y)\\\\delta (z)+\\\\delta (y)\\\\delta (z-\\\\pi )\\\\nonumber \\\\\\\\&&~~+\\\\delta (y-\\\\pi )\\\\delta (z)+\\\\delta (y-\\\\pi )\\\\delta (z-\\\\pi )]$ Here, $V_{1,2}$ and $V_{3,4}$ are brane tensions of the branes located at $y=0,\\\\pi $ and $z=0,\\\\pi $ , respectively.\",\n \"$\\\\Lambda $ is the cosmological constant in 6-dimensions.\",\n \"After solving Einstein's equations, the solutions to the warp factors of the metric as given in eq.\",\n \"(REF ) [2] are, $a(y) &=& \\\\exp (-c|y|), \\\\quad b(z) = \\\\frac{\\\\cosh (kz)}{\\\\cosh (k\\\\pi )}\\\\nonumber \\\\\\\\{\\\\rm where}~~ c&\\\\equiv & \\\\frac{R_yk}{r_z\\\\cosh (k\\\\pi )},\\\\quad k\\\\equiv r_z\\\\sqrt{\\\\frac{-\\\\Lambda }{10M_P^4}}$ Here, $M_P$ is the 4-dimensional Planck scale.\",\n \"The 5-d RS model can be retrieved in the limit $r_{z}\\\\rightarrow 0$ .\",\n \"The warp factors $a(y)$ and $b(z)$ provide maximum suppression at $y=\\\\pi $ and $z=0$ .\",\n \"For this reason we can interpret the 3-brane formed out of the intersection of 4-branes at $y=\\\\pi $ and $z=0$ as our standard model brane.\",\n \"The suppression on the standard model brane can be written as $f=\\\\frac{\\\\exp (-c\\\\pi )}{\\\\cosh (k\\\\pi )}$ The desired suppression of the order of $10^{-16}$ on the standard model brane can be obtained for different choices of the parameters $c$ and $k$ .\",\n \"However from the relation for $c$ in eq.\",\n \"(REF ) it can be shown that if we want to avoid large hierarchy in the moduli $R_y$ and $r_z$ , the warping in one direction must be large while that in the other direction must be small.\",\n \"In our analysis we shall explore this feature for non-flat 3-branes over the entire parameter space of $c$ , $k$ and moduli ratio $R_{y}/r_{z}$ for different values of brane cosmological constant.\",\n \"Previously a generalization of the 5-dimensional RS model has been considered with the non flat 3-branes [6].\",\n \"In this work we consider doubly compactified six dimensional space-time with $Z_{2}$ orbifolding along each of the compact direction.\",\n \"Thus the manifold under consideration is [$M^{(1,3)}\\\\times S^{1}/Z_2$ ]$\\\\times S^{1}/Z_{2}$ with four non-compact dimensions denoted by $x^\\\\mu $ , $\\\\mu =0,\\\\cdots ,3$ .\",\n \"We choose a doubly warped general metric as: $ds^2 = b^2(z)[a^2(y)g_{\\\\mu \\\\nu }dx^\\\\mu dx^\\\\nu +R_y^2dy^2]+r_z^2dz^2$ Since orbifolding, in general, requires a localized concentration of energy we introduce 4-branes [(4+1)-dimensional space-time] at the orbifold fixed points namely at $y=0,\\\\pi $ and $z=0,\\\\pi $ .\",\n \"The total bulk-brane action in this case is given by, $S&=&S_{6}+S_{5}+S_{4}\\\\nonumber \\\\\\\\S_6&=& \\\\int d^{4}xdydz\\\\sqrt{-g_{6}}(R_{6}-\\\\Lambda )\\\\nonumber \\\\\\\\S_{5}&=&\\\\int d^{4}xdydz\\\\sqrt{-g_{5}}[V_{1}\\\\delta (y)+V_{2}\\\\delta (y-\\\\pi )]\\\\nonumber \\\\\\\\&&~~+ \\\\int d^{4}xdydz\\\\sqrt{-g_{5}}[V_3\\\\delta (z)+V_{4}\\\\delta (z-\\\\pi )]\\\\nonumber \\\\\\\\S_4&=& \\\\int d^4x \\\\sqrt{-g_{vis}}[{\\\\cal {L}}-\\\\hat{V}][\\\\delta (y)\\\\delta (z)+\\\\delta (y)\\\\delta (z-\\\\pi )\\\\nonumber \\\\\\\\&&~~+\\\\delta (y-\\\\pi )\\\\delta (z)+\\\\delta (y-\\\\pi )\\\\delta (z-\\\\pi )]$ The brane potential terms may be coordinate dependent as, $V_{1,2}=V_{1,2}(z)$ and $V_{3,4}=V_{3,4}(y)$ .\",\n \"The term $S_{4}$ is the action for 3-branes located at $(y,z)=(0,0),(0,\\\\pi ),(\\\\pi ,0),(\\\\pi ,\\\\pi )$ .\",\n \"The full 6-dimensional Einstein's equation can be written as, $-M^{4}\\\\sqrt{-g_{6}}(R_{MN}-\\\\frac{R}{2}g_{MN})&=&\\\\Lambda _{6}\\\\sqrt{-g_{6}}g_{MN}+\\\\sqrt{-g_{5}}V_{1}(z)g_{\\\\alpha \\\\beta }\\\\delta ^{\\\\alpha }_{M}\\\\delta ^{\\\\beta }_{N}\\\\delta (y)\\\\nonumber \\\\\\\\&+&\\\\sqrt{-g_{5}}V_{2}(z)g_{\\\\alpha \\\\beta }\\\\delta ^{\\\\alpha }_{M}\\\\delta ^{\\\\beta }_{N}\\\\delta (y-\\\\pi )\\\\nonumber \\\\\\\\&+&\\\\sqrt{-\\\\tilde{g}_{5}}V_{3}(y) \\\\tilde{g}_{\\\\tilde{\\\\alpha }\\\\tilde{\\\\beta }}\\\\delta ^{\\\\tilde{\\\\alpha }}_{M}\\\\delta ^{\\\\tilde{\\\\beta }}_{N}\\\\delta (z)\\\\nonumber \\\\\\\\&+&\\\\sqrt{-\\\\tilde{g}_{5}}V_{4}(y) \\\\tilde{g}_{\\\\tilde{\\\\alpha }\\\\tilde{\\\\beta }}\\\\delta ^{\\\\tilde{\\\\alpha }}_{M}\\\\delta ^{\\\\tilde{\\\\beta }}_{N}\\\\delta (z-\\\\pi ) $ Here M,N are bulk indices, $\\\\alpha $ , $\\\\beta $ run over the usual four space-time coordinates ($x^{\\\\mu }$ ) and the compact coordinate $z$ while $\\\\tilde{\\\\alpha }$ , $\\\\tilde{\\\\beta }$ run over ($x^{\\\\mu }$ ) and the compact coordinate $y$ .\",\n \"$g$ , $\\\\tilde{g}$ are the respective metrices in these (4+1)-dimensional spaces.\",\n \"For the metric (REF ), the different components of Einstein's equations reduce to a set of three independent equations, $^{4}G_{\\\\mu \\\\nu }+g_{\\\\mu \\\\nu }[\\\\frac{3a^{\\\\prime }(y)^2}{R_{y}^2}+a(y){\\\\frac{3a^{\\\\prime \\\\prime }(y)}{R_{y}^2}+\\\\frac{2a(y)}{r_{z}^2}(3b^{\\\\prime }(z)^2+2b(z)b^{\\\\prime \\\\prime }(z))}]\\\\nonumber \\\\\\\\=-\\\\frac{\\\\Lambda _{6}}{M^{4}}a(y)^{2}b(z)^{2}g_{\\\\mu \\\\nu }$ $6a^{\\\\prime }(y)^2r_{z}^{2}+a(y)^2[6R_{y}^{2}b^{\\\\prime }(z)^2+4R_{y}^{2}b(z)b^{\\\\prime \\\\prime }(z)]-(1/2)~ ^{4}R~b(z)^{2}a(y)^2R_{y}^{2}r_{z}^{2}\\\\nonumber \\\\\\\\=\\\\frac{-\\\\Lambda _{6}}{M^{4}}b(z)^{2}a(y)^2R_{y}^{2}r_{z}^{2}\\\\\\\\6a^{\\\\prime }(y)^2r_{z}^{2}+4r_{z}^{2}a(y)a^{\\\\prime \\\\prime }(y)-(1/2)~ ^4R~r_{z}^{2}R_{y}^{2}a(y)^2b(z)^210R_{y}^{2}a(y)^2b^{\\\\prime }(z)^2\\\\nonumber \\\\\\\\=-\\\\frac{\\\\Lambda _{6}}{M^4}b(z)^2a(y)^2R_{y}^{2}r_{z}^{2}$ $^{4}G_{\\\\mu \\\\nu }$ and $^4R$ are the four dimensional Einstein tensor and Ricci scalar respectively, defined with respect to $g_{\\\\mu \\\\nu }$ .\",\n \"Dividing both sides of the equation (REF ) by $g_{\\\\mu \\\\nu }$ for any $\\\\mu $ , $\\\\nu $ and rearranging terms it is seen that one side contains $a(y)$ and $b(z)$ and their derivatives, while the other side depends on the brane coordinates $x^{\\\\mu }$ only.\",\n \"Thus we can equate each side to an arbitrary constant $\\\\Omega $ such that, $^4G_{\\\\mu \\\\nu }=-\\\\Omega g_{\\\\mu \\\\nu }\\\\\\\\a(y)^2[\\\\frac{3}{R_{y}^2}\\\\frac{a^{\\\\prime \\\\prime }(y)}{a(y)}+\\\\frac{3}{R_{y}^2}\\\\frac{a^{\\\\prime }(y)^2}{a(y)^2}+\\\\frac{2}{r_{z}^2}\\\\nonumber \\\\\\\\(3b^{\\\\prime }(z)^2+2b(z)b^{\\\\prime \\\\prime }(z))+\\\\frac{\\\\Lambda _{6}}{M^4}b(z)^2]=\\\\Omega $ From equation(REF ) we identify $\\\\Omega $ as the effective 4-D cosmological constant on the 3-branes.\",\n \"We obtain the solutions for the warp factors $a(y)$ and $b(z)$ for de sitter brane from equation () as, $a(y)={\\\\omega ^{\\\\prime }{\\\\rm sinh}}\\\\left[ {\\\\rm ln}\\\\frac{c^{\\\\prime }_2}{{\\\\rm \\\\omega ^{\\\\prime }}}-cy\\\\right], \\\\quad b(z)=\\\\frac{{\\\\rm cosh}(kz)}{{\\\\rm cosh}(k\\\\pi )}$ Where $c_{2}^{^{\\\\prime }}$ is an integration constant.\",\n \"Here $\\\\omega ^{\\\\prime }=\\\\omega {\\\\rm cosh}(k\\\\pi )$ , with $\\\\omega ^2=\\\\frac{\\\\Omega }{3k^{\\\\prime 2}}$ , where $k^{^{\\\\prime }}= \\\\sqrt{\\\\frac{-\\\\Lambda }{10M^4}}$ , $k=k^{^{\\\\prime }}r_z$ and $c=\\\\frac{R_{y}k}{r_{z}{\\\\rm cosh}(k\\\\pi )}$ .\",\n \"Normalizing the warp factor to unity at the orbifold fixed point $y=0$ , we get $c^{\\\\prime }_2=\\\\left[ 1+(1+\\\\omega ^{\\\\prime 2})^{\\\\frac{1}{2}}\\\\right] $ .\",\n \"Note that the result for RS model generalized to six dimensions with flat branes is recovered in the limit $\\\\omega \\\\rightarrow 0$ .\",\n \"We now focus our attention to the boundary terms to determine the brane tensions.\",\n \"Using the explicit form of $a(y)$ , $b(z)$ from equation(REF ) and implementing the boundary conditions across the two boundaries at $y=0$ , $y=\\\\pi $ and $z=0$ , $z=\\\\pi $ respectively, we obtain: $V_{2}(z)=8M^2\\\\sqrt{\\\\frac{-\\\\Lambda }{10}}\\\\frac{(\\\\frac{\\\\omega ^{^{\\\\prime }2}}{c^{^{\\\\prime }2}_{2}}e^{2c\\\\pi }+1)}{(\\\\frac{\\\\omega ^{^{\\\\prime }2}}{c^{^{\\\\prime }2}_{2}}e^{2c\\\\pi }-1)}{\\\\rm sech}(kz)\\\\\\\\V_{1}(z)=8M^2\\\\sqrt{\\\\frac{-\\\\Lambda }{10}}{\\\\rm sech}(kz)\\\\frac{(1+\\\\omega ^{^{\\\\prime }2}/c^{^{\\\\prime }2}_{2})}{(1-\\\\omega ^{^{\\\\prime }2}/c^{^{\\\\prime }2}_{})}$ The above two equations imply that the two 4-branes sitting at $y=0$ and $y=\\\\pi $ have $z$ dependent tensions.\",\n \"Similarly we find $V_{3}(y)=0$ and $V_{4}(y)=-\\\\frac{8M^4k}{r_{z}}{\\\\rm tanh}(k\\\\pi )$ .\",\n \"Thus we have determined the tensions for all the 4-branes in this model.\",\n \"The intersection of two 4-branes give rise to 3-brane.\",\n \"With this identification, the theory contains four 3-branes located at $(y,z)=(0,0),(0,\\\\pi )$ , $(\\\\pi ,0)$ , $(\\\\pi ,\\\\pi )$ .\",\n \"The metric on the 3-brane located at $(y=0,z=\\\\pi )$ has no warping and can be identified with the Planck brane.\",\n \"Similarly we identify the standard model brane with the one at $y=\\\\pi $ , $z=0$ where the warping is maximum.\",\n \"Finally we obtain the expressions for 3-brane tensions in terms of $\\\\omega $ , the induced brane cosmological constant as, $V_{vis}=8M^2\\\\left( \\\\frac{\\\\frac{\\\\omega ^{2}{\\\\rm cosh}^{2}(k\\\\pi )e^{2c\\\\pi }}{(4+2\\\\omega ^{2}{\\\\rm cosh}^{2}(k\\\\pi ))}+1}{\\\\frac{\\\\omega ^{2}{\\\\rm cosh}^{2}(k\\\\pi )e^{2c\\\\pi }}{(4+2\\\\omega ^{2}{\\\\rm cosh}^{2}(k\\\\pi ))}-1}\\\\right)\\\\sqrt{-\\\\frac{\\\\Lambda }{10}}\\\\\\\\V_{hid}=8M^2\\\\left[\\\\left(\\\\frac{1+\\\\frac{\\\\omega ^{2}{\\\\rm cosh}^{2}(k\\\\pi )e^{2c\\\\pi }}{(4+2\\\\omega ^{2}{\\\\rm cosh}^{2}(k\\\\pi ))}}{1-\\\\frac{\\\\omega ^{2}{\\\\rm cosh}^{2}(k\\\\pi )e^{2c\\\\pi }}{(4+2\\\\omega ^{2}{\\\\rm cosh}^{2}(k\\\\pi ))}}\\\\right){\\\\rm sech}(k\\\\pi )-{\\\\rm tanh}(k\\\\pi )\\\\right]\\\\sqrt{-\\\\frac{\\\\Lambda }{10}}$ The two other 3-branes located at $(0,0)$ and $(\\\\pi ,\\\\pi )$ have tensions close to hidden brane and visible brane tensions respectively.\",\n \"Once again limit $\\\\omega \\\\rightarrow 0$ reproduces the expression for the 6-D flat 3-brane tensions[2] and with $r_{z}\\\\rightarrow 0$ (i.e $k=k^{^{\\\\prime }}r_{z}\\\\rightarrow 0$ ) we recover 5-D RS brane tensions[1].\",\n \"Now to solve the gauge hierarchy problem we equate the warp factors at $y=\\\\pi $ , $z=0$ to the ratio of the mass scale in the two 3-branes given by $10^{-n}$ such that, $a(y)b(z)|_{y=\\\\pi ,z=0}=10^{-n}\\\\nonumber \\\\\\\\{\\\\omega ^{\\\\prime }{\\\\rm sinh}}\\\\left[ {\\\\rm ln}\\\\frac{c^{\\\\prime }_2}{{\\\\rm \\\\omega ^{\\\\prime }}}-c \\\\pi \\\\right]\\\\frac{1}{{\\\\rm cosh}(k\\\\pi )}=10^{-n}$ At this point, we keep $n$ arbitrary but we will subsequently take it to be $\\\\simeq 16$ to achieve a Planck to Tev scale warping.\",\n \"Defining $c\\\\pi =x$ , the above equation has a positive root for $e^{-x}$ as, $e^{-x}={\\\\rm cosh}(k\\\\pi )\\\\frac{10^{-n}}{c_{2}^{^{\\\\prime }}}\\\\left[1+\\\\left\\\\lbrace 1+\\\\omega ^210^{2n}\\\\right\\\\rbrace ^{1/2} \\\\right]$ We re-parametrize the brane cosmological constant $\\\\omega $ as: $\\\\omega ^{2}\\\\equiv 10^{-N}$ .\",\n \"Equation (REF ) now simplifies to: $-N=\\\\frac{1}{{\\\\rm ln}10}{\\\\rm ln}\\\\left[ \\\\frac{4e^{-2x}-4{\\\\rm cosh}(k\\\\pi )10^{-n}e^{-x}}{{\\\\rm cosh}^{2}(k\\\\pi )[1+10^{-n}{\\\\rm cosh}(k\\\\pi )e^{-x}-2e^{-2x}]}\\\\right] $ This equation relates the three parameters present in this model, $\\\\omega $ (which has been re-parametrized as $10^{-N}$ ), $k$ , and $x(=c\\\\pi )$ .\",\n \"The solution of $x$ , derived from expression (REF ), is obtained as, $x=16{\\\\rm ln}10-{\\\\rm ln}{\\\\rm cosh}(k\\\\pi )-{\\\\rm ln}2+{\\\\rm ln}[2+\\\\frac{1}{2}10^{-N}{\\\\rm cosh}^{2}(k\\\\pi )]-{\\\\rm ln}[1+\\\\frac{1}{4}10^{-(N-2n)}] $ Now from expression (REF ) we numerically calculate $x$ i.e $c\\\\pi $ for different values of $k\\\\pi $ taking cosmological constant $\\\\omega ^{2}$ as parameter for $n=16$ .\",\n \"This value of $n$ ensures the resolution of the gauge hierarchy problem for all the determined values of the parameters in our subsequent analysis.\",\n \"Also we calculate the corresponding values of the ratio of two moduli $R_{y}/r_{z}$ from the expression $c=\\\\frac{R_{y}k}{r_{z}{\\\\rm cosh}(k\\\\pi )}$ .\",\n \"The numerical values are given in the following table (REF ): Table: Numerical values of cπc\\\\pi andR y /r z R_{y}/r_{z} for every kπk\\\\pi and fixed cosmological constant ω\\\\omega (in Planck unit)in dS space-timeFor given $k\\\\pi $ the corresponding values of $c\\\\pi $ and $R_{y}/r_{z}$ saturate below $\\\\omega ^{2}<10^{-40}$ .\",\n \"It may be seen from table(REF ), when the cosmological constant is very large (i.e approximately of the order of 1), for a small $k\\\\pi \\\\approx 1.12$ , $c\\\\pi $ is also small $0.56$ and ratio of the two moduli is $0.84$ .\",\n \"Hence it is evident that for a very large cosmological constant we can have equal warping along both the extra dimensions and the two extra dimensional moduli are approximately close to $l_{Planck}$ i.e.\",\n \"without any hierarchical values.\",\n \"Now if we fix a large $k\\\\pi =30.12$ , the value of $c\\\\pi $ turns out to be very very small ($\\\\approx 10^{-13}$ ) and also the ratio of the two moduli becomes hierarchical.\",\n \"From table (REF ), it may be noted that when we decrease the value of cosmological constant, for small $k\\\\pi $ , values of $c\\\\pi $ become large and the two extra dimensional moduli are again reasonably close to each other with values close to $l_{Planck}$ .\",\n \"But if we fix a large $k\\\\pi $ then we can see that for decreasing $\\\\omega ^{2}$ , $c\\\\pi $ values become small and the ratio of the two extra dimensional moduli become large.\",\n \"Thus we conclude that a small but equal warping from both the extra dimensions and the minimum hierarchy between the two extra dimensional moduli can be achieved only when the induced cosmological constant on the brane is very large.\",\n \"But if we keep on decreasing the brane induced cosmological constant towards its present observed value which is estimated to be of the order of $10^{-120}$ , we cannot have equal warping along both the extra dimensions.\",\n \"Moreover if we demand that the two extra dimensional moduli are approximately of the same order, then in order to solve the gauge hierarchy problem the most favourable condition consistent with a small value of the brane cosmological constant is small $k\\\\pi $ and large $c\\\\pi $ i.e small warping along $z$ direction and large warping along $y$ direction.\",\n \"This resembles to 5-dimensional RS model perturbed slightly by the additional warping along $z$ direction such that two 3-branes have scales close to Tev while the two other have scales close to Planck scale.\",\n \"For Anti De-Sitter brane i.e $\\\\Omega <0$ , we find that there is an upper bound on the brane induced cosmological constant similar to the one found in [6] which is $10^{-32}$ in Planck units.\",\n \"Repeating the entire analysis in the ADS sector for values of the cosmological constant lower than $10^{-32}$ we determine the correlations among the parameters illustrated in table (REF ) and (REF ).\",\n \"Table: Numerical values of cπc\\\\pi andR y /r z R_{y}/r_{z} for different kπk\\\\pi with fixed cosmological constant ω\\\\omega in ADS space-time.\",\n \"The numerical values of ω\\\\omega are all inPlanckian unitsTable: Numerical values of cπc\\\\pi andR y /r z R_{y}/r_{z} for different kπk\\\\pi with fixed cosmological constant ω\\\\omega in ADS space-time.\",\n \"The numerical values of ω\\\\omega are all inPlanckian unitsHere also we come to a similar conclusion that for small cosmological constant, in order to keep minimum hierarchy between the two extra dimensional moduli, the most favourable condition is small $k\\\\pi \\\\approx 1.12$ and large $c\\\\pi \\\\approx 36.31$ .\",\n \"In this work the generalization of RS model has been done for a 6-dimensional ADS bulk with non-flat 3- branes.\",\n \"Requiring the warping from the hidden brane to visible brane $\\\\sim 10^{-16}$ we find that for de-Sitter 3-brane, the warping along both the directions can be nearly equal with very small hierarchy between the two moduli only when the brane cosmological constant is very large compared to the present value.\",\n \"To achieve similar warping along both the directions for nearly vanishing cosmological constant we have to introduce large hierarchy between the moduli.\",\n \"On the contrary for small value of the brane cosmological constant with non-hierarchical small moduli $\\\\sim l_{Planck}$ , the warping along one direction ( in our case along $y$ ) is very large while the other direction (i.e.\",\n \"$z$ ) is nearly flat.\",\n \"The corresponding values of the moduli $c$ and $k$ can be stabilized following the Goldberger-Wise stabilization mechanism [11] by introducing a 6-dimensional bulk scalar field.\",\n \"This leads to a scenario where the 3-branes can not have any intermediate scale and have energy scales either close to Tev or close to Planck scale.\",\n \"This remarkable correlations clearly point out that the most favoured condition for small cosmological constant, non-hierarchical moduli and the resolution of the gauge hierarchy problem correspond to a very large warping along $y$ direction with very small warping along z-direction.\",\n \"This scenario is nothing but a weak perturbation of the original 5-dimensional RS model due to the presence of the sixth dimension which in turn leads to two 3-branes with energy scale close to Tev scale while two other 3-branes having energy scale close to Plank scale.\",\n \"Such a feature of brane clustering with closely-spaced energy scales enhances when more and more extra warped dimensions are added leading to a stack of Tev scale 3-branes and a stack of Planck scale 3-branes.\",\n \"It has been shown in [3] that such scenario offers a possible geometric resolution of the fermion mass hierarchy problem among the standard model fermions.\",\n \"Our work thus explains that in a multiple warped geometry model the requirements of nearly flat 3-brane and non-hierarchical moduli lead naturally to a stack of closely clustered Tev 3-branes which in turn offers a geometric understanding of the fermion mass hierarchy as well as gauge hierarchy problem simultaneously.\"\n ]\n]"},"id":{"kind":"string","value":"1204.1154"}}},{"rowIdx":1866890,"cells":{"text":{"kind":"list like","value":[["Intrinsic instability of electronic interfaces with strong Rashba\n coupling"],["Abstract We consider a model for the two-dimensional electron gas formed at the interface of oxide heterostructures, which includes a Rashba spin-orbit coupling proportional to the electric field perpendicular to the interface.","Based on the standard mechanism of polarity catastrophe, we assume that the electric field is proportional to the electron density.","Under these simple and general assumptions, we show that a phase separation instability occurs for realistic values of the spin-orbit coupling and of the band parameters.","This could provide an intrinsic mechanism for the recently observed inhomogeneous phases at the LaAlO_3/SrTiO_3 or LaTiO_3/SrTiO_3 interfaces."],["The Rashba Spin-Orbit Coupling","A full ab initio derivation of the Rashba coupling is a quite challenging task, which goes beyond the scope of this Letter.","Here, based on standard textbook arguments [1], [2], we simply remind that $\\alpha \\propto \\left< \\psi (z)\\left| \\frac{d}{dz} \\left[\\frac{1}{\\varepsilon ^{\\prime }(z)}-\\frac{1}{\\varepsilon ^{\\prime }(z)+\\Delta } \\right] \\right| \\psi (z)\\right>$ where $\\varepsilon ^{\\prime }(z)=\\varepsilon +V(z)+E_{gap}$ , $\\varepsilon $ is the subband energy relative to the bulk conduction band, $V(z)$ is the band-bending potential, $E_{gap}$ is the band gap, and $\\Delta $ is a measure of the spin-orbit coupling within a Kane ${\\mathbf {k\\cdot p}}$ approach [3].","Thus, since for a uniform electric field $E$ along $z$ the potential is $V(z)=-zE$ , one obtains $\\alpha (E) &\\propto & \\left[\\frac{1}{\\left(\\varepsilon +V(z)+E_{gap}\\right)^2}-\\frac{1}{\\left(\\varepsilon +V(z)+E_{gap}+\\Delta \\right)^2}\\right] \\frac{dV(z)}{dz}\\approx \\frac{ E\\Delta }{\\left(\\varepsilon +E\\overline{z}+E_{gap}\\right)^3}\\nonumber \\\\&=& \\frac{ \\tilde{\\alpha }E}{\\left(1 +\\beta E \\right)^3},$ where $\\overline{z}\\sim 2-6$ nm is the width of the region where the surface reconstruction occurs [4].","This expression has the standard linear behavior at small field, but saturates and then decreases at large fields.","This latter behavior is important to stabilize the system against unphysical unlimited growth of the electric field.","Indeed, in the absence of the denominator, the system would be unstable because it would be energetically (too) convenient to attract large densities of electrons to increase enormously the electric field and the RSO coupling with the consequent formation of deeper and deeper minima in the bands.","This would eventually produce a negative compressibility.","We emphasize, that the negative compressibility found in the Letter is not of this type."],["The anisotropic Rashba band structure","The real STO substrate has three bands, arising from the three $d_{xy}$ , $d_{xz}$ , and $d_{yz}$ orbitals.","Of course the atomic spin-orbit mixes these $t_{2g}$ levels (see, e.g., Ref.","[5] for an insight into possible consequences of this mixing), but here, we will neglect this mixing for the sake of simplicity and because it is unessential for the physical effects at issue in this Letter.","Then the 2DEG may be represented by the prototypical model Hamiltonian $H=\\sum _{k,\\sigma \\varsigma }c_{k\\sigma }^\\dagger H_{\\sigma \\varsigma }(k)c_{k\\varsigma }$ , where $c_{k\\sigma }^{(\\dagger )}$ annihilates (creates) an electron with quasimomentum $k$ and spin projection $\\sigma $ , $H_{\\sigma \\varsigma }(k)=\\left(\\frac{\\hbar ^2k_x^2}{2m_x}+ \\frac{\\hbar ^2k_y^2}{2m_y}+ \\Delta _{xy,xz,yz}\\right)\\delta _{\\sigma \\varsigma }+\\alpha _x k_x\\sigma ^y_{\\sigma \\varsigma }-\\alpha _yk_y\\sigma ^x_{\\sigma \\varsigma },$ and $\\sigma ^{x,y}$ are the Pauli matrices.","The lowest band due to the $d_{xy}$ orbitals has a light isotropic mass, $m\\sim 0.7 m_0$ ($m_0$ is the bare electronic mass) and $\\Delta _{xy}=0$ , $E^{iso}_\\pm (k_x,k_y)=\\frac{\\hbar ^2k_x^2}{2m}+\\frac{\\hbar ^2k_y^2}{2m}\\pm \\sqrt{\\alpha ^2 k_x^2+\\alpha ^2k_y^2}$ while two anisotropic bands due to the $d_{xz}$ and $d_{yz}$ orbitals are present with a shift $\\Delta _{xz,yz}\\equiv \\Delta $ at $k_x=k_y=0$ .","These latter have a heavy mass (as large as $20 m_0$ ) in one direction and a light mass in the other.","These anisotropic bands have dispersions of the form $E^{aniso}_\\pm (k_x,k_y)=\\frac{\\hbar ^2k_x^2}{2m_x}+\\frac{\\hbar ^2k_y^2}{2m_y}\\pm \\sqrt{\\alpha _x^2 k_x^2+\\alpha _y^2k_y^2}+\\Delta ,$ where $\\Delta =50$ meV is the shift with respect to the $d_{xy}$ band and $m_x=20 m_y=20 m$ (for the $d_{yz}$ ) or $m_y=20 m_x=20 m$ (for the $d_{xz}$ ).","The lowest band has four stationary points, which can be minima or saddle points depending on the values of $\\nu \\ge 1$ and $\\frac{1}{\\nu }\\le \\eta \\le 1$ .","In the $\\eta =1/\\nu $ case the bottom of the band occurs at $-\\varepsilon _0+\\Delta $ , where $\\varepsilon _0\\equiv m\\alpha ^2/(2\\hbar ^2)$ , while for $\\eta =1$ it is at $-\\nu \\varepsilon _0+\\Delta $ .","The DOS of the anisotropic $d_{xz,yz}$ bands depends only on the ratio $\\zeta =(\\varepsilon -\\Delta )/\\varepsilon _0$ and its expression can be given analytically in terms of complete elliptic integrals of the first ($K$ ) and third ($\\Pi $ ) kind.","Specifically, when $\\eta ^2\\nu <1$ $N(\\zeta )=\\theta [-\\zeta (\\zeta + 1)]{\\cal {A}}_1(\\zeta )+\\theta [-(\\zeta +\\eta ^2 \\nu )(\\zeta +1)]{\\cal {A}}_2(\\zeta )+\\theta [\\zeta ] N_0 \\sqrt{\\nu }$ where ${\\cal {A}}_1(\\zeta )&=&\\frac{2N_0\\sqrt{\\nu }}{\\pi \\sqrt{\\eta ^2\\nu +\\zeta }}\\left[-\\zeta K\\left(-\\frac{\\zeta (\\eta ^2 \\nu -1)}{\\eta ^2 \\nu +\\zeta }\\right) +(1 + \\zeta ) \\Pi \\left(\\frac{\\eta ^2 \\nu - 1}{\\eta ^2 \\nu + \\zeta },\\frac{-\\zeta (\\eta ^2 \\nu -1)}{\\eta ^2 \\nu +\\zeta }\\right)\\right] \\\\{\\cal {A}}_2(\\zeta )&=& \\frac{2N_0\\sqrt{\\nu }}{\\pi \\sqrt{- \\zeta (\\eta ^2 \\nu -1)}}\\left[-\\zeta K\\left(-\\frac{\\eta ^2 \\nu +\\zeta }{\\zeta (\\eta ^2 \\nu - 1)}\\right) \\right.\\nonumber \\\\&&-\\left.","(1 + \\zeta ) \\Pi \\left(\\frac{\\eta ^2 \\nu + \\zeta }{\\eta ^2 \\nu - 1}, \\frac{\\eta ^2 \\nu + \\zeta }{-\\zeta (\\eta ^2 \\nu -1)}\\right)\\right]$ On the other hand, when $\\eta ^2\\nu >1$ $N(\\zeta )=\\theta [-\\zeta (\\zeta + \\eta ^2\\nu )]{\\cal {A}}_3(\\zeta )+\\theta [-(\\zeta + \\eta ^2 \\nu )(\\zeta +1)]{\\cal {A}}_4(\\zeta )+\\theta [\\zeta ] N_0 \\sqrt{\\nu }$ where ${\\cal {A}}_3(\\zeta )&=&\\frac{2N_0\\sqrt{\\nu }}{\\pi \\eta \\sqrt{1 + \\zeta }}\\Pi \\left(1-\\frac{ 1}{\\eta ^2 \\nu },\\frac{-\\zeta (1-\\eta ^2 \\nu )}{\\eta ^2 \\nu (1+ \\zeta )}\\right) \\\\{\\cal {A}}_4(\\zeta )&=& \\frac{2N_0\\sqrt{\\nu }}{\\pi \\sqrt{-\\zeta (1-\\eta ^2 \\nu )}}\\Pi \\left(\\frac{1+ \\zeta }{\\zeta }, \\frac{\\eta ^2 \\nu (1 + \\zeta )}{\\zeta (\\eta ^2 \\nu -1)}\\right)$ Fig.","REF reports the DOS for various values of $\\nu $ , in the extreme cases of $\\eta =1$ (a) and $\\eta =1/\\nu $ (b).","Figure: (Color online) Density of states for the isotropic case (black, solid line)and for the anisotropic cases (the anisotropic bands do not include theshift Δ\\Delta to allow an easier comparison between the isotropic and anisotropic bandDOS): a) Case with η=1\\eta =1 and ν=3\\nu =3 (red online, dashed),ν=10\\nu =10 (green online, dot-dashed), and ν=20\\nu =20 (blue online, dot-dot-dashed).","b)Case η=1/ν\\eta =1/\\nu and ν=3\\nu =3 (red online, dashed), ν=10\\nu =10 (green online,dot-dashed), and ν=20\\nu =20 (blue online, dot-dot-dashed).While the generic expressions for $\\mu (n)$ are cumbersome, for $\\mu >\\Delta $ the electron density inside each one of the anisotropic bands reads $n=(\\mu -\\Delta ) N_0\\sqrt{\\nu }+2N_0\\varepsilon _0f(\\nu ,\\eta ),$ where $2N_0 f(\\nu ,\\eta )\\equiv \\int ^0_{\\frac{\\varepsilon _{min}-\\Delta }{\\varepsilon _0}}N(\\zeta )d\\zeta $ $\\varepsilon _{min}(\\nu ,\\eta )$ being the bottom of the lowest Rashba split anisotropic band.","We insert the factor $2N_0$ to normalize this function to $ f (1, 1) = 1$ so that in the isotropic case with zero splitting $\\Delta $ the Rashba term is $2N_0\\varepsilon _0$ , as previously found.","$f(\\nu ,\\eta )$ is a (rapidly) increasing function of the mass anisotropy $\\nu $ (see Fig.","REF ).","For the chemical potential one obtains $\\mu =\\Delta +\\frac{1}{\\sqrt{\\nu }}\\left[\\frac{n}{N_0}-2\\varepsilon _0 f(\\nu ,\\eta )\\right].$ The negative term contains the function $f(\\nu ,\\eta )\\gg 1$ , which makes the condition $\\kappa <0$ and the related electronic phase separation much easier to occur.","As shown in Fig.","REF , $f(\\nu ,\\eta )$ depends on the anisotropy of the RSO term $\\eta $ .","To reconstruct the relativistic form of the SO, $({\\bf v}\\times {\\bf \\sigma } )\\cdot \\hat{\\bf E}$ , one should assume $\\alpha _{x,y}\\propto m_{x,y}^{-1}$ , i.e., $\\eta =1/\\nu $ .","In this case, $f(\\nu ,1/\\nu )\\sim \\sqrt{\\nu }$ grows rather slowly with $\\nu $ .","On the other hand, if the RSO term is isotropic ($\\eta =1$ ) despite the mass anisotropy, $f(\\nu ,1)\\sim \\nu ^{\\frac{3}{2}}$ is a rapidly increasing function of $\\nu $ .","The precise RSO coupling in real materials should be determined by first-principles calculations, which are beyond the scope of this Letter, but the behavior of $f(\\nu ,\\eta )$ should be intermediate between these two extreme cases.","Therefore, our analysis shows that the RSO-mediated instability can take a moderate or strong advantage from the mass anisotropy.","Figure: (Color online) Enhancement factor f(μ,ν)f(\\mu ,\\nu ) [cf.","Eq.","()]for different values of the RSO anisotropy parameter η\\eta ."]],"string":"[\n [\n \"Intrinsic instability of electronic interfaces with strong Rashba\\n coupling\"\n ],\n [\n \"Abstract We consider a model for the two-dimensional electron gas formed at the interface of oxide heterostructures, which includes a Rashba spin-orbit coupling proportional to the electric field perpendicular to the interface.\",\n \"Based on the standard mechanism of polarity catastrophe, we assume that the electric field is proportional to the electron density.\",\n \"Under these simple and general assumptions, we show that a phase separation instability occurs for realistic values of the spin-orbit coupling and of the band parameters.\",\n \"This could provide an intrinsic mechanism for the recently observed inhomogeneous phases at the LaAlO_3/SrTiO_3 or LaTiO_3/SrTiO_3 interfaces.\"\n ],\n [\n \"The Rashba Spin-Orbit Coupling\",\n \"A full ab initio derivation of the Rashba coupling is a quite challenging task, which goes beyond the scope of this Letter.\",\n \"Here, based on standard textbook arguments [1], [2], we simply remind that $\\\\alpha \\\\propto \\\\left< \\\\psi (z)\\\\left| \\\\frac{d}{dz} \\\\left[\\\\frac{1}{\\\\varepsilon ^{\\\\prime }(z)}-\\\\frac{1}{\\\\varepsilon ^{\\\\prime }(z)+\\\\Delta } \\\\right] \\\\right| \\\\psi (z)\\\\right>$ where $\\\\varepsilon ^{\\\\prime }(z)=\\\\varepsilon +V(z)+E_{gap}$ , $\\\\varepsilon $ is the subband energy relative to the bulk conduction band, $V(z)$ is the band-bending potential, $E_{gap}$ is the band gap, and $\\\\Delta $ is a measure of the spin-orbit coupling within a Kane ${\\\\mathbf {k\\\\cdot p}}$ approach [3].\",\n \"Thus, since for a uniform electric field $E$ along $z$ the potential is $V(z)=-zE$ , one obtains $\\\\alpha (E) &\\\\propto & \\\\left[\\\\frac{1}{\\\\left(\\\\varepsilon +V(z)+E_{gap}\\\\right)^2}-\\\\frac{1}{\\\\left(\\\\varepsilon +V(z)+E_{gap}+\\\\Delta \\\\right)^2}\\\\right] \\\\frac{dV(z)}{dz}\\\\approx \\\\frac{ E\\\\Delta }{\\\\left(\\\\varepsilon +E\\\\overline{z}+E_{gap}\\\\right)^3}\\\\nonumber \\\\\\\\&=& \\\\frac{ \\\\tilde{\\\\alpha }E}{\\\\left(1 +\\\\beta E \\\\right)^3},$ where $\\\\overline{z}\\\\sim 2-6$ nm is the width of the region where the surface reconstruction occurs [4].\",\n \"This expression has the standard linear behavior at small field, but saturates and then decreases at large fields.\",\n \"This latter behavior is important to stabilize the system against unphysical unlimited growth of the electric field.\",\n \"Indeed, in the absence of the denominator, the system would be unstable because it would be energetically (too) convenient to attract large densities of electrons to increase enormously the electric field and the RSO coupling with the consequent formation of deeper and deeper minima in the bands.\",\n \"This would eventually produce a negative compressibility.\",\n \"We emphasize, that the negative compressibility found in the Letter is not of this type.\"\n ],\n [\n \"The anisotropic Rashba band structure\",\n \"The real STO substrate has three bands, arising from the three $d_{xy}$ , $d_{xz}$ , and $d_{yz}$ orbitals.\",\n \"Of course the atomic spin-orbit mixes these $t_{2g}$ levels (see, e.g., Ref.\",\n \"[5] for an insight into possible consequences of this mixing), but here, we will neglect this mixing for the sake of simplicity and because it is unessential for the physical effects at issue in this Letter.\",\n \"Then the 2DEG may be represented by the prototypical model Hamiltonian $H=\\\\sum _{k,\\\\sigma \\\\varsigma }c_{k\\\\sigma }^\\\\dagger H_{\\\\sigma \\\\varsigma }(k)c_{k\\\\varsigma }$ , where $c_{k\\\\sigma }^{(\\\\dagger )}$ annihilates (creates) an electron with quasimomentum $k$ and spin projection $\\\\sigma $ , $H_{\\\\sigma \\\\varsigma }(k)=\\\\left(\\\\frac{\\\\hbar ^2k_x^2}{2m_x}+ \\\\frac{\\\\hbar ^2k_y^2}{2m_y}+ \\\\Delta _{xy,xz,yz}\\\\right)\\\\delta _{\\\\sigma \\\\varsigma }+\\\\alpha _x k_x\\\\sigma ^y_{\\\\sigma \\\\varsigma }-\\\\alpha _yk_y\\\\sigma ^x_{\\\\sigma \\\\varsigma },$ and $\\\\sigma ^{x,y}$ are the Pauli matrices.\",\n \"The lowest band due to the $d_{xy}$ orbitals has a light isotropic mass, $m\\\\sim 0.7 m_0$ ($m_0$ is the bare electronic mass) and $\\\\Delta _{xy}=0$ , $E^{iso}_\\\\pm (k_x,k_y)=\\\\frac{\\\\hbar ^2k_x^2}{2m}+\\\\frac{\\\\hbar ^2k_y^2}{2m}\\\\pm \\\\sqrt{\\\\alpha ^2 k_x^2+\\\\alpha ^2k_y^2}$ while two anisotropic bands due to the $d_{xz}$ and $d_{yz}$ orbitals are present with a shift $\\\\Delta _{xz,yz}\\\\equiv \\\\Delta $ at $k_x=k_y=0$ .\",\n \"These latter have a heavy mass (as large as $20 m_0$ ) in one direction and a light mass in the other.\",\n \"These anisotropic bands have dispersions of the form $E^{aniso}_\\\\pm (k_x,k_y)=\\\\frac{\\\\hbar ^2k_x^2}{2m_x}+\\\\frac{\\\\hbar ^2k_y^2}{2m_y}\\\\pm \\\\sqrt{\\\\alpha _x^2 k_x^2+\\\\alpha _y^2k_y^2}+\\\\Delta ,$ where $\\\\Delta =50$ meV is the shift with respect to the $d_{xy}$ band and $m_x=20 m_y=20 m$ (for the $d_{yz}$ ) or $m_y=20 m_x=20 m$ (for the $d_{xz}$ ).\",\n \"The lowest band has four stationary points, which can be minima or saddle points depending on the values of $\\\\nu \\\\ge 1$ and $\\\\frac{1}{\\\\nu }\\\\le \\\\eta \\\\le 1$ .\",\n \"In the $\\\\eta =1/\\\\nu $ case the bottom of the band occurs at $-\\\\varepsilon _0+\\\\Delta $ , where $\\\\varepsilon _0\\\\equiv m\\\\alpha ^2/(2\\\\hbar ^2)$ , while for $\\\\eta =1$ it is at $-\\\\nu \\\\varepsilon _0+\\\\Delta $ .\",\n \"The DOS of the anisotropic $d_{xz,yz}$ bands depends only on the ratio $\\\\zeta =(\\\\varepsilon -\\\\Delta )/\\\\varepsilon _0$ and its expression can be given analytically in terms of complete elliptic integrals of the first ($K$ ) and third ($\\\\Pi $ ) kind.\",\n \"Specifically, when $\\\\eta ^2\\\\nu <1$ $N(\\\\zeta )=\\\\theta [-\\\\zeta (\\\\zeta + 1)]{\\\\cal {A}}_1(\\\\zeta )+\\\\theta [-(\\\\zeta +\\\\eta ^2 \\\\nu )(\\\\zeta +1)]{\\\\cal {A}}_2(\\\\zeta )+\\\\theta [\\\\zeta ] N_0 \\\\sqrt{\\\\nu }$ where ${\\\\cal {A}}_1(\\\\zeta )&=&\\\\frac{2N_0\\\\sqrt{\\\\nu }}{\\\\pi \\\\sqrt{\\\\eta ^2\\\\nu +\\\\zeta }}\\\\left[-\\\\zeta K\\\\left(-\\\\frac{\\\\zeta (\\\\eta ^2 \\\\nu -1)}{\\\\eta ^2 \\\\nu +\\\\zeta }\\\\right) +(1 + \\\\zeta ) \\\\Pi \\\\left(\\\\frac{\\\\eta ^2 \\\\nu - 1}{\\\\eta ^2 \\\\nu + \\\\zeta },\\\\frac{-\\\\zeta (\\\\eta ^2 \\\\nu -1)}{\\\\eta ^2 \\\\nu +\\\\zeta }\\\\right)\\\\right] \\\\\\\\{\\\\cal {A}}_2(\\\\zeta )&=& \\\\frac{2N_0\\\\sqrt{\\\\nu }}{\\\\pi \\\\sqrt{- \\\\zeta (\\\\eta ^2 \\\\nu -1)}}\\\\left[-\\\\zeta K\\\\left(-\\\\frac{\\\\eta ^2 \\\\nu +\\\\zeta }{\\\\zeta (\\\\eta ^2 \\\\nu - 1)}\\\\right) \\\\right.\\\\nonumber \\\\\\\\&&-\\\\left.\",\n \"(1 + \\\\zeta ) \\\\Pi \\\\left(\\\\frac{\\\\eta ^2 \\\\nu + \\\\zeta }{\\\\eta ^2 \\\\nu - 1}, \\\\frac{\\\\eta ^2 \\\\nu + \\\\zeta }{-\\\\zeta (\\\\eta ^2 \\\\nu -1)}\\\\right)\\\\right]$ On the other hand, when $\\\\eta ^2\\\\nu >1$ $N(\\\\zeta )=\\\\theta [-\\\\zeta (\\\\zeta + \\\\eta ^2\\\\nu )]{\\\\cal {A}}_3(\\\\zeta )+\\\\theta [-(\\\\zeta + \\\\eta ^2 \\\\nu )(\\\\zeta +1)]{\\\\cal {A}}_4(\\\\zeta )+\\\\theta [\\\\zeta ] N_0 \\\\sqrt{\\\\nu }$ where ${\\\\cal {A}}_3(\\\\zeta )&=&\\\\frac{2N_0\\\\sqrt{\\\\nu }}{\\\\pi \\\\eta \\\\sqrt{1 + \\\\zeta }}\\\\Pi \\\\left(1-\\\\frac{ 1}{\\\\eta ^2 \\\\nu },\\\\frac{-\\\\zeta (1-\\\\eta ^2 \\\\nu )}{\\\\eta ^2 \\\\nu (1+ \\\\zeta )}\\\\right) \\\\\\\\{\\\\cal {A}}_4(\\\\zeta )&=& \\\\frac{2N_0\\\\sqrt{\\\\nu }}{\\\\pi \\\\sqrt{-\\\\zeta (1-\\\\eta ^2 \\\\nu )}}\\\\Pi \\\\left(\\\\frac{1+ \\\\zeta }{\\\\zeta }, \\\\frac{\\\\eta ^2 \\\\nu (1 + \\\\zeta )}{\\\\zeta (\\\\eta ^2 \\\\nu -1)}\\\\right)$ Fig.\",\n \"REF reports the DOS for various values of $\\\\nu $ , in the extreme cases of $\\\\eta =1$ (a) and $\\\\eta =1/\\\\nu $ (b).\",\n \"Figure: (Color online) Density of states for the isotropic case (black, solid line)and for the anisotropic cases (the anisotropic bands do not include theshift Δ\\\\Delta to allow an easier comparison between the isotropic and anisotropic bandDOS): a) Case with η=1\\\\eta =1 and ν=3\\\\nu =3 (red online, dashed),ν=10\\\\nu =10 (green online, dot-dashed), and ν=20\\\\nu =20 (blue online, dot-dot-dashed).\",\n \"b)Case η=1/ν\\\\eta =1/\\\\nu and ν=3\\\\nu =3 (red online, dashed), ν=10\\\\nu =10 (green online,dot-dashed), and ν=20\\\\nu =20 (blue online, dot-dot-dashed).While the generic expressions for $\\\\mu (n)$ are cumbersome, for $\\\\mu >\\\\Delta $ the electron density inside each one of the anisotropic bands reads $n=(\\\\mu -\\\\Delta ) N_0\\\\sqrt{\\\\nu }+2N_0\\\\varepsilon _0f(\\\\nu ,\\\\eta ),$ where $2N_0 f(\\\\nu ,\\\\eta )\\\\equiv \\\\int ^0_{\\\\frac{\\\\varepsilon _{min}-\\\\Delta }{\\\\varepsilon _0}}N(\\\\zeta )d\\\\zeta $ $\\\\varepsilon _{min}(\\\\nu ,\\\\eta )$ being the bottom of the lowest Rashba split anisotropic band.\",\n \"We insert the factor $2N_0$ to normalize this function to $ f (1, 1) = 1$ so that in the isotropic case with zero splitting $\\\\Delta $ the Rashba term is $2N_0\\\\varepsilon _0$ , as previously found.\",\n \"$f(\\\\nu ,\\\\eta )$ is a (rapidly) increasing function of the mass anisotropy $\\\\nu $ (see Fig.\",\n \"REF ).\",\n \"For the chemical potential one obtains $\\\\mu =\\\\Delta +\\\\frac{1}{\\\\sqrt{\\\\nu }}\\\\left[\\\\frac{n}{N_0}-2\\\\varepsilon _0 f(\\\\nu ,\\\\eta )\\\\right].$ The negative term contains the function $f(\\\\nu ,\\\\eta )\\\\gg 1$ , which makes the condition $\\\\kappa <0$ and the related electronic phase separation much easier to occur.\",\n \"As shown in Fig.\",\n \"REF , $f(\\\\nu ,\\\\eta )$ depends on the anisotropy of the RSO term $\\\\eta $ .\",\n \"To reconstruct the relativistic form of the SO, $({\\\\bf v}\\\\times {\\\\bf \\\\sigma } )\\\\cdot \\\\hat{\\\\bf E}$ , one should assume $\\\\alpha _{x,y}\\\\propto m_{x,y}^{-1}$ , i.e., $\\\\eta =1/\\\\nu $ .\",\n \"In this case, $f(\\\\nu ,1/\\\\nu )\\\\sim \\\\sqrt{\\\\nu }$ grows rather slowly with $\\\\nu $ .\",\n \"On the other hand, if the RSO term is isotropic ($\\\\eta =1$ ) despite the mass anisotropy, $f(\\\\nu ,1)\\\\sim \\\\nu ^{\\\\frac{3}{2}}$ is a rapidly increasing function of $\\\\nu $ .\",\n \"The precise RSO coupling in real materials should be determined by first-principles calculations, which are beyond the scope of this Letter, but the behavior of $f(\\\\nu ,\\\\eta )$ should be intermediate between these two extreme cases.\",\n \"Therefore, our analysis shows that the RSO-mediated instability can take a moderate or strong advantage from the mass anisotropy.\",\n \"Figure: (Color online) Enhancement factor f(μ,ν)f(\\\\mu ,\\\\nu ) [cf.\",\n \"Eq.\",\n \"()]for different values of the RSO anisotropy parameter η\\\\eta .\"\n ]\n]"},"id":{"kind":"string","value":"1204.0962"}}},{"rowIdx":1866891,"cells":{"text":{"kind":"list like","value":[["New Sum Rules from Low Energy Compton Scattering on Arbitrary Spin\n Target"],["Abstract We derive two sum rules by studying the low energy Compton scattering on a target of arbitrary (nonzero) spin j.","In the first sum rule, we consider the possibility that the intermediate state in the scattering can have spin |j \\pm 1| and the same mass as the target.","The second sum rule applies if the theory at hand possesses intermediate narrow resonances with masses different from the mass of the scatterer.","These sum rules are generalizations of the Gerasimov-Drell-Hearn-Weinberg sum rule.","Along with the requirement of tree level unitarity, they relate different low energy couplings in the theory.","Using these sum rules, we show that in certain cases the gyromagnetic ratio can differ from the \"natural\" value g=2, even at tree level, without spoiling perturbative unitarity.","These sum rules can be used as constraints applicable to all supergravity and higher-spin theories that contain particles charged under some U(1) gauge field.","In particular, applied to four dimensional N=8 supergravity in a spontaneously broken phase, these sum rules suggest that for the theory to have a good ultraviolet behavior, additional massive states need to be present, such as those coming from the embedding of the N=8 supergravity in type II superstring theory.","We also discuss the possible implications of the sum rules for QCD in the large-N_c limit."],["Introduction, Summary and an Application to $N=8$ Supergravity","Dispersion relations based on analyticity, unitarity and Lorentz invariance of scattering amplitudes can be interpreted as consistency conditions that constrain low-energy data, i.e.","parameters of an effective action.","Conversely, if we know the low energy effective action of a given theory, dispersion relations either constrain any of its possible UV completions, or show that no completion exists.","Beautiful examples of the latter phenomenon are given in Ref. [1].","In this paper we revisit topics addressed years ago by one of us [2] and find their implications on the existence and properties of UV completions of low energy effective theories.","In Ref.","[2] it was argued that when effective field theories of elementary particles admit a perturbative expansion (in some parametrically small dimensionless coupling constant), then the gyromagnetic ratio of the (weakly interacting) particles described by such theory had to be close to a preferred “natural” value: $g=2$ .","This result was obtained in a Lagrangian approach.","Many years earlier, Weinberg [3] also proposed an argument, based on a generalization of the Gerasimov-Drell-Hearn (GDH) [4] sum rule, which similarly selected $g = 2$ as the preferred value in weakly interacting theories.","Given a particle of mass $m$ and electric charge $e$ , the GDH-Weinberg sum rule connects the gyromagnetic ratio $g$ to a dispersion integral.","As usual, $g$ is defined as the ratio of the particle's magnetic moment $\\mu $ to its spin $J$ , so that (in the particle's rest frame): $\\vec{\\mu } = \\frac{e g}{2m}\\vec{J} \\ .$ The main ingredient of the Weinberg sum rule is the low energy forwardAssume that the photon propagates along some $z$ -direction with helicity $\\lambda = \\pm 1$ , and the target has a spin-$z$ projection $J_z$ .","For more details, see Appendix .","Compton scattering amplitude of a photon with energy $\\omega $ and helicity $\\lambda $ off a massive target of spin $J$ .","This amplitude, $f_{\\rm scat} (\\omega , \\lambda )$ , is a real analytic function of the photon's energy $\\omega $ away from the real $\\omega $ -axis, where cuts and poles may exist at $\\omega > 0 $ .","The imaginary part of $f_{\\rm scat}$ is given by the optical theorem: ${\\rm Im} f_{\\rm scat} (\\omega ,\\lambda ) = \\frac{\\omega }{4\\pi }\\sigma _{\\rm tot}(\\omega , \\lambda ) \\ ,$ where $\\sigma _{\\rm tot}$ is the total cross-section for a photon with helicity $\\lambda $ and energy $\\omega $ .","Define now the following function: $&f_-(\\omega ^2) \\equiv \\frac{f_{\\rm scat} (\\omega ,+1)-f_{\\rm scat} (\\omega ,-1)}{2\\omega } \\ .$ When no intermediate (one particle) state exists in the Compton scattering, with either mass or spin different from those of the target, then, it can be checked that (see, e.g., Appendix ): $&f_-(\\omega ^2 \\rightarrow 0) = \\frac{e^2J_z}{16\\pi m^2}(g-2)^2 \\ .$ Using the optical theorem (REF ) and definition (REF ), we have: ${\\rm Im}f_-(\\omega ^2) = \\frac{1}{8\\pi }\\Delta \\sigma (\\omega ) \\ , \\qquad \\Delta \\sigma (\\omega ) \\equiv \\sigma _{\\rm tot}(\\omega ,+1) - \\sigma _{\\rm tot}(\\omega ,-1) \\ .$ Assuming that $f_-(\\omega ^2)$ vanishes when $|\\omega ^2| \\rightarrow \\infty $ , one can write an unsubtracted dispersion relation: $f_-(\\omega ^2) = \\frac{1}{4\\pi ^2}\\int ^{\\infty }_0\\frac{\\Delta \\sigma (\\omega ^{\\prime })}{\\omega ^{\\prime 2} - \\omega ^2 - i \\epsilon } ~\\omega ^{\\prime } d\\omega ^{\\prime } \\ .$ The validity of this assumption is far from obvious and its justification requires some additional assumptions about the UV behavior of the theory.","We will argue below that unsubtracted dispersion relations hold in two important case: superstring theory and any unitary completion of $N=8$ supergravity.","When $\\omega ^2=0$ , Eqs.","(REF ) and (REF ), impliy the GDH-Weinberg sum rule [3], [4]: $\\frac{\\pi e^2 J_z}{4m^2} \\left(g-2\\right)^2 = \\int ^{\\infty }_0\\frac{\\Delta \\sigma (\\omega ^{\\prime })}{\\omega ^{\\prime }} d\\omega ^{\\prime } \\ .$ In weakly interacting systems, the RHS of Eq.","(REF ) is parametrically smaller than the LHS, since in the former the final state contains at least two particles and is thus $O(e^4)$ , while the latter is $O(e^2)$ ; hence $g = 2 + {\\cal O}(e^2)$ .","In truncated $N=2$ supergravity theory (where one drops the cosmological and quartic Fermionic terms from the Lagrangian given in Ref.","[5]) it can be checked that the gravitino has indeed $g=2$ , see also Ref. [6].","On the other hand, in spontaneously broken $N=8$ supergravity theory [7], see also [13], contrary to expectations, one finds that $g=1$ , for both spin-1/2 and spin-3/2 fields.","To show this, we use the dimensionally reduced actionThe $N=8$ supergravity theory with global E(6) and local USp(8) invariance was constructed in five dimensions by Cremmer, Scherk and Schwarz in Ref. [7].","Spontaneous symmetry breaking of $N=8$ supergravity theory is achieved by dimensional reduction to 4D via the Scherk-Schwarz mechanism [8].","given in Ref.","[9], and consider only the relevant terms of the Lagrangians for spin-1/2 and spin-3/2 fields, defined as $\\chi ^{abc}$ and $\\psi _{\\mu }^a$ , respectively.","In our notations (see Appendices), and in units, $\\kappa ^2 = 4\\pi G_N =1$ , where $G_N$ is the 4D Newton's constant, these terms, written in flat space-time, unitary gauge ($\\psi _5^a=0$ ) and near the ground state, take the following form: $&{\\cal L}_{4D}(\\chi ) = \\frac{1}{12}\\bar{\\chi }^{abc}\\biggl [i\\left(\\partial \\hspace{-7.0pt}\\diagup + 2i{\\cal M}_{abc}B\\hspace{-7.0pt}\\diagup \\right)- {\\cal M}_{abc} +\\frac{1}{4}\\sigma ^{\\mu \\nu }B_{\\mu \\nu }\\biggr ]\\chi _{abc} \\ , \\\\ &{\\cal L}_{4D}(\\psi ) = \\frac{1}{2}\\bar{\\psi }^{a}_{\\mu }\\biggl [i\\gamma ^{\\mu \\rho \\nu }\\left(\\partial _{\\nu } + 2i{\\cal M}_{a}B_{\\nu }\\right) - {\\cal M}_{a}\\gamma ^{\\mu \\rho } + \\frac{i}{2}\\left(B^{\\mu \\rho } - i\\gamma _5\\tilde{B}^{\\mu \\rho }\\right)\\biggr ]\\psi _{\\rho a} \\ ,$ where $B_{\\mu }$ is a graviphoton field, ${\\cal M}_{abc}$ and ${\\cal M}_a$ are the mass matrices of $\\chi ^{abc}$ and $\\psi _{\\mu }^a$ correspondingly.","These spinors are $USp(8)$ tensors, and both are charged under the graviphoton, with a charge $e=2\\kappa m$ , where $m$ is the mass of the spinor.","Notice, also that there is no $\\chi $ -$\\psi $ -$B$ mixing.","The Lagrangians above should be compared with the Dirac Lagrangian supplemented with a Pauli term, and the Lagrangian of a charged Rarita-Schwinger field with non-minimal terms [6]: $&{\\cal L}_{DP} = \\bar{\\chi }\\left[i(\\partial \\hspace{-7.0pt}\\diagup + ieB\\hspace{-7.0pt}\\diagup ) - m - \\frac{e(g_{1/2}-2)}{8m}\\sigma ^{\\mu \\nu }B_{\\mu \\nu }\\right]\\chi \\ , \\\\ &{\\cal L}_{RS} = \\bar{\\psi }_{\\mu }\\left[ i\\gamma ^{\\mu \\rho \\nu }(\\partial _{\\nu } + ie B_{\\nu }) - m\\gamma ^{\\mu \\rho } + \\frac{ie}{m}\\left\\lbrace \\frac{3}{4}\\left(g_{3/2}- \\frac{2}{3}\\right)B^{\\mu \\rho } - i\\alpha \\gamma _5\\tilde{B}^{\\mu \\rho }\\right\\rbrace \\right]\\psi _{\\rho } \\ ,$ where $\\alpha $ is a parameter unrelated to $g_{3/2}$ .As was observed in [6], the $\\gamma _5$ matrix in the $\\bar{\\psi }_{\\mu }\\gamma _5 \\tilde{B}^{\\mu \\nu }\\psi _{\\nu }$ term of () mixes the “large” and “small” components of $\\psi _{\\mu }$ , and thus gives contributions of higher order in $\\omega $ .","That is why this term does not contribute to $g_{3/2}$ .","Finally, comparing (REF –) with (REF –), we conclude that in case of the spontaneously broken $N=8$ supergravity: $g_{1/2} = g_{3/2}= 1$ and $\\alpha =1/4$ .Similarly, comparing () with the truncated $N=2$ supergravity [5], one can deduce that: $g_{3/2} = 2$ and $\\alpha = 1$ .","In $N=2$ supergravity with a gauged central charge [11], $g_{3/2}=2$ , however, $g_{1/2} \\ne 2$ or 1.","It appears that $g=1$ for all heavy particles in the Kaluza-Klein theory [10].","This observation was also confirmed within the string theory and on the example of D0-branes in [12].","It is often the case that in supergravity theories, $e \\sim m/M_P$ , where $M_P$ is the Planck mass, suggesting that the charge can be made arbitrarily small.","It is thus natural to ask: why Eq.","(REF ) fails so miserably in case of the spontaneously broken $N=8$ supergravity theory?","One could argue that this happens because $f_-({\\omega ^2})$ does not vanish sufficiently fast, when $|\\omega ^2| \\rightarrow \\infty $ , since supergravity is power counting non-renormalizable.","However, spontaneously broken $N=8$ supergravity can be embedded in type II superstring theory [14]Spontaneously broken $N=4,2,1$ theories can be embedded also in heterotic string theory [15]., where the condition $\\lim _{\\omega ^2\\rightarrow \\infty }f_-(\\omega ^2)= 0$ holds, order-by-order in string perturbation theory.","A more explicit and general argument supporting unsubtracted dispersion relations is that, as argued in Refs.","[16] (see also [17]), the gravitational 2-body elastic scattering amplitude $f(s,t)$ of any theory obeying Hermitian unitarity and crossing symmetry is polynomially bound in the complex-$s$ upper half plane for fixed Mandelstam variable $t$ .","Hermitian unitarity implies $f(s,t)^*=f(s^*,t^*)$ , thus polynomial boundedness in $s$ for the forward elastic scattering amplitude ($t=0$ ).","Furthermore, Eq.","(REF ) suggests that the scattering amplitude $f_-(s,0)$ is not only polynomially bound, but indeed vanishes at large positive $s$ as $\\sim 1/s$ .","Crossing symmetry, Hermitian unitarity, and the same argument used to prove polynomial boundedness, i.e.","the Phragmen Lindelöf theorem, then show that $f_-(s,0)$ vanishes at large $s$ in the whole complex plane.Eq.","(REF ) gives a tree-level $f_-(s,0)$ that does not vanish at $s=\\infty $ ; this fact alone shows that $N=8$ supergravity cannot be a perturbatively complete theory of gravity.","As we will argue, the solution to this puzzle is different and it is one of the main results of this paper.","The point is that the GDH-Weinberg sum rule is modified when the Compton scattering amplitude includes intermediate one-particle states of masses $M_n \\ne m$ .In this case, $\\Delta \\sigma $ becomes a sum of terms proportional to $\\delta (\\omega -\\omega _n)$ plus the contribution from the continuum.","Taking into account this possibility, we find the following generalization of the GDH-Weinberg sum rule: $&{\\qquad }{\\qquad } \\frac{e^2 J_z}{4m^2}(g-2)^2 = [K^{\\dagger },K]_{ii} +\\frac{1}{\\pi }\\int ^{\\infty }_0 \\frac{\\Delta \\sigma }{\\omega } d\\omega \\ , \\\\[5pt] \\nonumber [K^{\\dagger },K]_{ii} \\equiv \\sum _n &\\left[(K_{ni})^{\\dagger }K_{ni} -K_{in}(K_{in})^{\\dagger }\\right] , {\\qquad } K_{ni} \\equiv \\left[2\\omega _n\\sqrt{m(\\omega _n + m)}\\right]^{-1}~\\langle n |\\vec{\\epsilon }_{\\lambda =1}\\vec{{\\cal J}}|i \\rangle \\ ,$ where $m$ is the mass of the target, $M_n$ is the mass of the intermediate state, $\\omega _n = \\frac{1}{2m}(M^2_n - m^2)$ , $\\vec{{\\cal J}}$ is the electromagnetic current, and $\\vec{\\epsilon }_{\\lambda }$ is the polarization vector of the photon with helicity $\\lambda $ .","The matrix $K_{ni}$ describes transition between states of different mass, and possibly spin.","This sum rule should be obeyed not only in string theory, but also in any Lorentz invariant, causal, unitary UV completion of $N=8$ supergravity.","Because $e \\sim m/M_P$ , the LHS of (REF ) is ${\\cal O}(1/M_P^2)$ , independent of the mass $m$ , and thus is the RHS.","Generically this implies that the new intermediate states have masses $M_n$ independent of $m$ ; these masses thus define a cutoff, below which the theory is described by $N=8$ supergravity.","Type II superstring compactified on a 6-torus is a concrete example of this general situation.","In this case the relation between the string mass scale $M_S\\sim 1/\\sqrt{\\alpha ^{\\prime }}$ and the Planck scale is $M_P=M_S^4\\sqrt{V_6}/g_S$ , where $V_6$ is the volume of the 6-torus, and $g_S$ is the string coupling constant.","In the perturbative regime, where $V_6M_S^6 \\gg 1$ and $g_S\\ll 1$ , we have: $M_P\\gg M_S $ .","When all radii of the torus are ${\\cal O}(1/M_S)$ , $M_S$ becomes the only UV cutoff scale and $M_n={\\cal O}(M_S)$ .","The fact that new states, besides those already present in $N=8$ supergravity,In a spontaneously broken phase, where gravitino is massive and charged [7].","are necessary for $N=8$ to admit a UV completion has an immediate consequence.","It implies that $N=8$ supergravity is not perturbatively complete by and in itself.","Such a statement, that is at variance with other remarkable finiteness properties of the theory (see e.g [18]) warrants further discussion.","It follows from two assumptions, besides Lorentz invariance and unitarity, both satisfied by string theory but holding more generally in any perturbative regularization of gravity.","The first is that the cutoff scale $\\Lambda $ is parametrically smaller than $M_P$ : $\\Lambda = \\lambda _c M_P$ , where $\\lambda _c \\ll 1$ is the dimensionless coupling constant of the UV complete theory.","The second assumption is that scattering amplitudes are regular in the limit $m\\rightarrow 0$ ; more precisely, that the mass $m$ appears only with positive powers in scattering amplitudes.","One can then use the fact that forward scattering depends only on the relativistic invariant $s=m^2 + 2m\\omega $ to rewrite the sum rule (REF ) as $\\lambda ^2_c \\frac{\\pi J_z}{4\\Lambda ^2}(g-2)^2 =\\lambda ^2_c \\int _{m^2}^\\infty \\frac{ds}{s-m^2} \\sum _n f_n[m^2,(s-m^2)/\\Lambda ^2]\\delta (s-M_n^2) + O(\\lambda ^4_c).$ Terms ${\\cal O}(\\lambda ^4_c)$ come from the continuum part of the total cross section, in which the final state contains at least two particles.","By assumption, the $m\\rightarrow 0$ limit of Eq.","(REF ) is smooth, so the functions $f_n[0,x]$ are well defined, $m$ -independent and dimensionless.","In the limit $m\\rightarrow 0$ , dividing both sides of Eq.","(REF ) by $\\lambda ^2_c$ , we get: $\\frac{\\pi J_z}{4\\Lambda ^2}(g-2)^2 =\\int _{0}^\\infty \\frac{ds}{s} \\sum _n f_n[0,s/\\Lambda ^2]\\delta (s-M_n^2) + O(\\lambda ^2_c).$ If the LHS of (REF ) is nonzero, then the RHS must contain new states with masses $M_n = {\\cal O}(\\Lambda )$ , because, by assumption, the only massless states in the theory are those of $N=8$ supergravity.Because of $N=8$ supersymmetry, new massless states would necessarily contain an additional massless spin two particle interacting with supergravity particles, in contradiction with the no go theorem in [19].","One may worry that at energies above $M_P$ the cross section $\\Delta \\sigma $ would never be perturbative because contributions from black hole intermediate states would dominate.","In general, these black holes carry mass $M$ , charge $Q$ and angular momentum $J$ .","In such case, the cross section can be crudely approximated by the cross sectional area of a black hole [16]: $\\sigma _J \\sim \\pi r^2_+ \\approx 4\\pi G^2_N M^2-2\\pi G_NQ^2 - 2\\pi \\frac{J^2}{M^2}$ where $r_+ = G_N M + \\sqrt{G^2_NM^2 - J^2/M^2 -G_NQ^2}$ is the outer horizon of the Kerr-Newman black hole, and we assumed $J^2 + G_N M^2 Q^2 \\ll G^2_N M^4 $ or $J \\ll G_NM^2$ , since $Q^2 = e^2 \\sim G_N m^2$ and $m \\ll M$ .","Clearly, the above estimate of the cross section is only applicable in case $r_+ \\sim G_N M \\ge 1/\\Lambda $ , where $\\Lambda $ is a cutoff of the theory (as was mentioned above $\\Lambda = \\lambda _c M_P$ and $\\lambda _c \\ll 1$ ).","Therefore, our estimate is valid, only when $M > \\omega _c$ , where $ \\omega _c \\equiv M_{P}\\left(M_P/\\Lambda \\right) = \\Lambda /\\lambda ^2_c $ .","In this case, the difference between the spin aligned ($J=j+1$ ) and anti-aligned ($J=j-1$ ) cross sections is: $\\Delta \\sigma _{j}(\\omega > \\omega _c) \\equiv \\sigma _{j-1} -\\sigma _{j+1} \\sim \\frac{8\\pi j}{\\omega ^2} \\ ,$ and therefore, the dispersion integral can be divided into two parts as follows: $\\int _{0}^{\\infty }\\frac{\\Delta \\sigma _{j}}{\\omega } d\\omega = \\int _{0}^{\\omega _c}\\frac{\\Delta \\sigma _{j}}{\\omega } d\\omega + \\int _{\\omega _c}^{\\infty }\\frac{\\Delta \\sigma _{j}}{\\omega } d\\omega \\ , {\\qquad } \\ \\int _{\\omega _c}^{\\infty }\\frac{\\Delta \\sigma _{j}}{\\omega } d\\omega \\sim \\frac{4\\pi j}{\\omega ^2_c} \\sim {\\cal O}\\left(\\frac{\\lambda ^2_c}{M^2_P}\\right) \\ ,$ where the second part is due to the exchange of black hole states.","Since, in supergravity theories $e \\sim m/M_P$ , from Eq.","(REF ) it follows that the black hole contribution to the gyromagnetic ratio is: $(g-2)^2_{\\rm B.H.}","\\sim {\\cal O}\\left(\\lambda ^2_c\\right) \\ll 1 \\ .$ If $N=8$ supergravity is a self-complete theory by itself, our argument shows that, unsurprisingly, it must unitarize at the Planck scale, in which case $\\omega _c=M_P$ and the black hole contributions to the dispersive integral becomes ${\\cal O}(1)$ .","This paper also generalizes the Weinberg-GHD sum rule in another way: it allows for off-diagonal couplings in case particles of spin $j$ and $|j\\pm 1|$ are degenerate in mass.","The general sum rule for Compton scattering on a target of arbitrary nonzero spin-$j$ , with additional intermediate mass-degenerate states of spin $|j\\pm 1|$ and possible heavy narrow resonances, can be written as: $&\\frac{e^2J_z}{4m^2} \\left[(g_j -2)^2 - \\left(j+\\frac{3}{2}\\right)h_{j+1/2}^2 + \\left(j-\\frac{1}{2}\\right) h_{j-1/2}^2\\right] = [K^{\\dagger },K]_{ii} + \\frac{1}{\\pi }\\int ^{\\infty }_{0} \\frac{\\Delta \\sigma }{\\omega } d\\omega \\ .$ where $g_j$ is the gyromagnetic ratio for a particle of spin $j$ , and the parameters $h_{j\\pm 1/2}$ are couplings of the theory defined in Eq.","(REF ).Notice, that the LHS of Eq.","(REF ) can be also written as a commutator of some generator like $K$ .","When applied to theories where all states are either proportional to a light mass scale $m$ or to a higher scale $M$ and where charges are proportional to the mass, such as Kaluza-Klein theories or spontaneously broken extended supergravities, this sum rule generalizes Eq.","(REF ) to $\\frac{\\pi J_z}{4M^2}\\left[(g_j -2)^2 - \\left(j+\\frac{3}{2}\\right)h_{j+1/2}^2 + \\left(j-\\frac{1}{2}\\right) h_{j-1/2}^2\\right] =\\int _{0}^\\infty \\frac{ds}{s} \\sum _n f_n[0,s/M^2]\\delta (s-M_n^2) +O(\\lambda ^2).$ Since magnetic dipole couplings are already taken into account by the LHS of Eq.","(REF ), the dispersive integral at low $s$ contains only quadrupole or higher multipole interactions; therefore, $f_n[0,s/M^2]$ is at most $O(s^2/M^4)$ at low $s$ .","This implies that, if the LHS in (REF ) is nonzero, the RHS must contain contributions from the massive states, since the one-particle contribution to the dispersive integral due to states that become massless in the limit $m\\rightarrow 0$ vanishes.","This paper is organized as follows: In section 2, we describe a formalism used by Weinberg to study the low energy Compton scattering.","We also study scattering on a target of spin-1/2, with possible spin-3/2 intermediate state, and deduce some generalization of the GDH-Weinberg sum rule.","In section 3, we consider the generalization of the sum rule to Compton scattering on a target of arbitrary spin.","In section 4, we further generalize the sum rule, assuming that there are other states with masses different from the mass of the scatterer.","In section 5, we discuss possible implications of our sum rule for nucleon to delta electromagnetic transitions, in the large $N_c$ limit."],["Compton Scattering: Weinberg's Approach","Define the non-diagonal vertex for the emission of a single soft photon as follows: $\\langle \\textbf {p}^{\\prime }, s^{\\prime }, \\sigma ^{\\prime } | {\\cal J}^{\\mu }(0)| \\textbf {p}, s, \\sigma \\rangle = \\Gamma _{\\sigma ^{\\prime }\\sigma }^{\\mu }(\\textbf {p}^{\\prime },\\textbf {p}) \\ ,$ where ${\\cal J}^{\\mu }$ is the conserved electromagnetic (EM) current, $| \\textbf {p}, s, \\sigma \\rangle $ is a single-particle state with momentum $\\textbf {p}$ , spin-$s$ and spin $z$ -component $\\sigma $ , as well as energy $E(\\textbf {p}) = \\sqrt{\\textbf {p}^2+m^2}$ .","We adopt the following normalization: $\\langle \\textbf {p}^{\\prime }, s, \\sigma ^{\\prime } | \\textbf {p}, s, \\sigma \\rangle = (2\\pi )^32E_{\\textbf {p}}\\delta _{\\sigma ^{\\prime }\\sigma }\\delta ^{(3)}(\\textbf {p}^{\\prime }-\\textbf {p}) \\ .$ The S-matrix, describing the Compton scattering can be written as: $\\langle \\sigma ^{\\prime }\\textbf {p}^{\\prime };\\lambda ^{\\prime },\\textbf {k}^{\\prime } |S| \\sigma ,\\textbf {p};\\lambda ,\\textbf {k} \\rangle &=i\\epsilon ^{*^{\\prime }}_{\\nu }(\\textbf {k}^{\\prime },\\lambda ^{\\prime })\\epsilon _{\\mu }(\\textbf {k},\\lambda ) ~ (2\\pi )^4\\delta ^{(4)}(p+k-p^{\\prime }-k^{\\prime })M_{\\sigma ^{\\prime }\\sigma }^{\\nu \\mu }(\\textbf {k}; \\textbf {p}^{\\prime }, \\textbf {p}) \\ ,\\\\[5pt]M_{\\sigma ^{\\prime }\\sigma }^{\\nu \\mu }(\\textbf {k}; \\textbf {p}^{\\prime }, \\textbf {p}) &\\equiv i\\int d^4x~e^{ikx} \\left[\\langle \\textbf {p}^{\\prime }, s, \\sigma ^{\\prime } | T\\lbrace {\\cal J}^{\\nu }(0){\\cal J}^{\\mu }(x)\\rbrace |\\textbf {p}, s, \\sigma \\rangle + {\\rm C.T.}","\\right] \\ ,$ where by ${\\rm C.T.","}$ we mean other contact terms (seagulls), such as terms emerging from the interaction of the initial and final photon at a single point.","We will need the pole structure of $M^{\\nu \\mu }$ ($\\sigma $ , $\\sigma ^{\\prime }$ indices are dropped for convenience).","Inserting a complete set of states between the current operators, the time-ordered product can be written as: $&i\\int d^4x ~ e^{ikx}\\langle \\textbf {p}^{\\prime }, s, \\sigma ^{\\prime } |T\\lbrace {\\cal J}^{\\nu }(0){\\cal J}^{\\mu }(x) \\rbrace | \\textbf {p}, s, \\sigma \\rangle \\\\\\nonumber &= \\int \\frac{d^3p_n}{2E_n(\\textbf {p}_n)}\\sum _n \\biggl \\lbrace \\langle \\textbf {p}^{\\prime }, s, \\sigma ^{\\prime } | {\\cal J}^{\\nu }(0)| n \\rangle \\langle n| {\\cal J}^{\\mu }(x)| \\textbf {p}, s, \\sigma \\rangle \\frac{\\delta ^{(3)}(\\textbf {p}_n - \\textbf {p}-\\textbf {k})}{E_n(\\textbf {p}_n) - E(\\textbf {p})- \\omega - i\\epsilon } \\\\ \\nonumber &+\\langle \\textbf {p}^{\\prime }, s,\\sigma ^{\\prime } | {\\cal J}^{\\mu }(0)| n \\rangle \\langle n| {\\cal J}^{\\nu }(x)| \\textbf {p}, s, \\sigma \\rangle \\frac{\\delta ^{(3)}(\\textbf {p}_n - \\textbf {p}^{\\prime }+\\textbf {k})}{E_n(\\textbf {p}_n)-E^{\\prime }(\\textbf {p}^{\\prime }) + \\omega - i\\epsilon } \\biggr \\rbrace \\\\ \\nonumber &=\\sum _n \\biggl \\lbrace \\frac{\\Gamma ^{\\nu }(\\textbf {p}^{\\prime },\\textbf {p}+\\textbf {k})\\Gamma ^{\\mu }(\\textbf {p}+\\textbf {k},\\textbf {p})}{2E_n(\\textbf {p}+\\textbf {k})[E_n(\\textbf {p}+\\textbf {k})-E(\\textbf {p}) - \\omega - i\\epsilon ]}+ \\frac{\\Gamma ^{\\mu }(\\textbf {p}^{\\prime },\\textbf {p}^{\\prime }-\\textbf {k})\\Gamma ^{\\nu }(\\textbf {p}^{\\prime }-\\textbf {k},\\textbf {p})}{2E_n(\\textbf {p}^{\\prime }-\\textbf {k})[E_n(\\textbf {p}^{\\prime }-\\textbf {k})-E^{\\prime }(\\textbf {p}^{\\prime }) + \\omega - i\\epsilon ]} \\biggr \\rbrace \\ .$ If $|n\\rangle $ is a single-particle intermediate state with the same mass $m$ as the target, then both terms in the sum above have a pole at $k^{\\mu }=0$ .","Using Eq.","(REF ), in the forward scattering limit, and near the $\\omega =0$ pole, we get: $M^{\\nu \\mu }(\\textbf {k}; \\textbf {p}^{\\prime }, \\textbf {p}) \\approx &-\\frac{1}{2m\\omega }\\biggl [\\Gamma ^{\\nu }(\\textbf {p}^{\\prime },\\textbf {p}+\\textbf {k})\\Gamma ^{\\mu }(\\textbf {p}+\\textbf {k},\\textbf {p}) - \\Gamma ^{\\mu }(\\textbf {p}^{\\prime },\\textbf {p}^{\\prime }-\\textbf {k})\\Gamma ^{\\nu }(\\textbf {p}^{\\prime }-\\textbf {k},\\textbf {p})\\biggr ] + \\ {\\rm O.T.}","\\ ,$ where by O.T.","we mean other terms that do not contain poles at $\\omega =0$ .","As usual, the scattering amplitude will be defined as: $f_{\\rm scat}(\\textbf {k}^{\\prime },\\lambda ^{\\prime }; \\textbf {k}, \\lambda ) = \\frac{1}{8\\pi m}\\epsilon ^{*^{\\prime }}_j\\epsilon _i M^{ji}(\\textbf {k}^{\\prime },\\textbf {k}, \\omega ) \\ .$ The case when $s=s^{\\prime }$ was considered in detail by Weinberg [3].","Here, we are interested in the case when the intermediate state could be a particle of different spin (but of the same mass) as the target.","To be more specific, we will consider a situation when $|s^{\\prime }-s|= 1$ or 0.","In particular, when $s=1/2$ and $s^{\\prime }=3/2$ , the vertex function is [see also Eq.","(REF )]:This vertex clearly implies the conservation of EM current: $k_{\\nu }\\Gamma ^{\\nu }_{\\sigma \\sigma ^{\\prime }} = 0 $ .","$\\Gamma ^{\\nu }_{\\sigma \\sigma ^{\\prime }}(p^{\\prime },p^{\\prime }+k^{\\prime }) &= \\frac{ie \\kappa _{M}}{2m^2}\\epsilon ^{\\mu \\nu \\alpha \\beta }p^{\\prime }_{\\mu }k^{\\prime }_{\\alpha } \\bar{u}^{\\sigma }(p^{\\prime })\\psi ^{\\sigma ^{\\prime }}_{\\beta }(p^{\\prime }+k^{\\prime }) \\ .$ In case of spin-1/2 target, we take the intermediate states to be either spin-1/2 or spin-3/2 particle state.","Here, we will only need to compute the part of the amplitude with spin-3/2 intermediate state, since the result in case of the spin-1/2 intermediate state is already known.","Using Eqs.","(REF ) and (REF ) [also Eq.","(REF ) to sum over Rarita-Schwinger (RS) states] we compute:From Eq.","(REF ) and current conservation: $k_{\\mu }M^{\\nu \\mu }(\\textbf {k}; \\textbf {p}^{\\prime }, \\textbf {p}) = 0$ , we can deduce that O.T.","in Eq.","(REF ) do not contribute to this part of the amplitude.","$f_{RS}(\\textbf {k}^{\\prime },\\lambda ^{\\prime }; \\textbf {k}, \\lambda ) &= \\frac{i\\omega e^2\\kappa ^2_{M}}{12\\pi m^2} \\left[(\\vec{n}^{\\prime } \\times \\vec{\\epsilon }^{^{\\prime }*})\\times (\\vec{n} \\times \\vec{\\epsilon })\\right]\\vec{J} \\ .$ In the forward-scattering limit, $f_{RS} = f_{RS}(\\omega , \\lambda )$ , and we can define the following amplitude: $&g_{-}(\\omega ^2) \\equiv \\frac{f_{RS}(\\omega , +1) - f_{RS}(\\omega , -1)}{2\\omega } \\ ,$ in which case, it can be checked that: $&g_{-}(\\omega ^2 \\rightarrow 0) = -\\frac{e^2\\kappa _M^2}{12\\pi m^2}~J_z \\ .$ Therefore, assuming $g_-(|\\omega ^2| \\rightarrow \\infty ) \\rightarrow 0$ , the total scattering amplitude will satisfy the following generalized unsubtracted dispersion relation: $4\\pi ^2[f_-(0)+g_-(0)] &= \\frac{\\pi e^2}{m^2} \\left(\\kappa ^2_p - \\frac{1}{3}\\kappa _M^2\\right)J_z= \\int ^{\\infty }_0\\frac{\\sigma _{\\rm tot}(\\omega ^{\\prime },+1) - \\sigma _{\\rm tot}(\\omega ^{\\prime },-1)}{\\omega ^{\\prime }}d\\omega ^{\\prime } \\ ,$ where $\\kappa _p = (g-2)/2$ is the anomalous magnetic moment of the target (see Appendix ).","Eq.","(REF ) is a generalization of the GDH-Weinberg sum rule, when there is a spin-3/2 intermediate state in the Compton scattering process, that has the same mass as the spin-1/2 target.","One of the consequences of this sum rule is that in weakly coupled theories, the gyromagnetic ratio for spin-1/2 particle can be different from its “natural” value $g=2$ .","As can be deduced from Eq.","(REF ), $g = 2\\left(1 \\pm \\frac{1}{\\sqrt{3}}\\kappa _M\\right) + {\\cal O}(e^2) \\ .$ The same result can be obtained using Feynman diagrams in Appendix ."],["Generalization to Spin-$j$ Target: First Sum Rule","The non-diagonal transition vertex for the emission of a single soft photon can be written as in [21]: $&\\Gamma ^{\\mu }(\\textbf {k},\\textbf {0}) \\equiv \\langle \\textbf {k}, j^{\\prime }, \\sigma ^{\\prime } | {\\cal J}^{\\mu }| \\textbf {0}, j, \\sigma \\rangle = -2im~\\epsilon ^{0\\mu \\alpha \\beta }k_{\\alpha } \\langle j^{\\prime }, \\sigma ^{\\prime }|\\mu _{\\beta }|j,\\sigma \\rangle + {\\cal O}(\\omega ^2) \\ ,$ where $\\mu _i$ is the magnetic moment and $j \\ne j^{\\prime }$ .","Applying the Wigner-Eckart theorem, the matrix elements of the magnetic moment can be parametrized as follows: $\\langle j,\\sigma |\\mu _3|j,\\sigma \\rangle = e g_j \\sigma /2m$ and $&\\langle j \\pm 1/2, \\sigma - 1/2|\\mu ^{-}|j\\mp 1/2,\\sigma +1/2\\rangle = \\pm \\frac{e}{2m}h_j\\sqrt{(j\\mp \\sigma +1)(j \\mp \\sigma )} \\ ,$ where $\\mu ^{\\pm } \\equiv \\mu _1 \\pm i\\mu _2$ .","Using Eqs.","(REF ) and (REF ), which are valid for any spin, the non-diagonal contribution to the forward scattering amplitude on a target of arbitrary spin-$j$ ($\\ne 0$ ) is: $&\\tilde{f}_{\\rm scat}= -\\frac{1}{16\\pi m^2\\omega }\\epsilon ^{*^{\\prime }}_{\\nu }\\epsilon _{\\mu }\\biggl [\\Gamma ^{\\nu \\dagger }(\\textbf {k},\\textbf {0})\\Gamma ^{\\mu }(\\textbf {k},\\textbf {0}) - \\Gamma ^{\\mu \\dagger }(-\\textbf {k},0)\\Gamma ^{\\nu }(-\\textbf {k},0)\\biggr ] \\\\ \\nonumber &= -\\frac{\\omega }{4\\pi }\\sum _{j^{\\prime }\\sigma ^{\\prime }}2\\langle j, \\sigma |\\mu _{[i}|j^{\\prime },\\sigma ^{\\prime }\\rangle \\langle j^{\\prime },\\sigma ^{\\prime } |\\mu _{i^{\\prime }]}|j, \\sigma \\rangle (\\vec{n} \\times \\vec{\\epsilon }^{^{\\prime }*})_{i} (\\vec{n} \\times \\vec{\\epsilon })_{i^{\\prime }} \\ .$ Taking into account that: $&\\langle j , \\sigma |\\mu ^{-}|j\\mp 1,\\sigma +1\\rangle = \\pm \\frac{e}{2m}h_{j \\mp 1/2}\\sqrt{(j \\mp \\sigma \\mp 1 + 1)(j \\mp \\sigma \\mp 1)} \\ , \\\\ \\nonumber &\\langle j , \\sigma |\\mu ^{+}|j\\pm 1,\\sigma -1\\rangle = \\pm \\frac{e}{2m}h_{j \\pm 1/2}\\sqrt{(j \\mp \\sigma \\pm 1 + 1)(j \\mp \\sigma \\pm 1)} \\ ,$ and applying the definition in Eq.","(REF ) for $\\tilde{f}_{\\rm scat}(\\omega ,\\lambda )$ , we arrive at the following result in the forward scattering limit: $4\\pi ^2\\tilde{f}_-(0) &= -\\frac{\\pi e^2}{8 m^2}J_z\\left[ (2j+3)h_{j+1/2}^2 - (2j-1)h_{j-1/2}^2 \\right] \\ .$ Therefore, a more general form of the sum rule can be written as: $&\\frac{\\pi e^2J_z}{4m^2} \\left[(g_j -2)^2 - \\left(j+\\frac{3}{2}\\right)h_{j+1/2}^2 + \\left(j-\\frac{1}{2}\\right) h_{j-1/2}^2\\right]= \\int ^{\\infty }_0\\frac{\\Delta \\sigma (\\omega ^{\\prime })}{\\omega ^{\\prime }}d\\omega ^{\\prime } \\ .$ As before, tree-level unitarity demands: $&(g_j -2)^2 + \\frac{(2j-1)}{2}h_{j-1/2}^2 = \\frac{(2j+3)}{2}h_{j+1/2}^2 \\ .$ This is in agreement with the observation made, e.g., in Ref.","[21], proposing that $g=2$ , when $h_j=0$ , for all $j$ .","As a simple illustration, when $j=1/2$ : $(g_{1/2} - 2)^2 = 2h_{1}^2 \\ ,$ suggesting, $h_{1} = \\sqrt{\\frac{2}{3}}\\kappa _M = \\sqrt{2} \\kappa _p$ , where we recalled Eq.","(REF ).","Similarly, when $j=3/2$ , $&(g_{3/2} -2)^2 - 3h_{2}^2 + h_{1}^2 = 0 \\ .$"],["Generalization for Arbitrary Intermediate State: Second Sum Rule","Consider a situation when the intermediate state in the Compton scattering process is a very narrow resonance of mass $M_n$ ($\\ne m$ ).","Then, using Eqs.","(REF ) and (REF ), near each pole the forward scattering amplitude on a target at rest becomes (see also Ref.","[3]): $f^{(n)}(\\omega \\rightarrow \\omega _n, \\lambda ) \\rightarrow \\frac{1}{16\\pi m}\\frac{\\langle \\textbf {0}, s, \\sigma | \\vec{\\epsilon }^{\\ *}_{\\lambda }\\vec{{\\cal J}}|\\textbf {k}, s_n, \\sigma _n \\rangle \\langle \\textbf {k}, s_n, \\sigma _n | \\vec{\\epsilon }_{\\lambda }\\vec{J}|\\textbf {0}, s, \\sigma \\rangle }{\\left(\\sqrt{M^2_n + \\omega ^2} - m - \\omega \\right)\\sqrt{M^2_n + \\omega ^2_n}} \\ ,$ where $\\omega _n = \\frac{1}{2m}(M^2_n - m^2)$ , so that: $\\sqrt{M^2_n + \\omega ^2_n} = m+\\omega _n$ .","Now, defining: $K_{ni} \\equiv \\sqrt{\\frac{m}{2E_n}}~\\frac{\\langle \\textbf {k}, s_n, \\sigma _n | {\\cal J}_x +i {\\cal J}_y| \\textbf {0}, s, \\sigma \\rangle }{M^2_n - m^2} \\ , \\ \\ \\ \\ \\ K_{in} \\equiv \\sqrt{\\frac{m}{2E_n}}~\\frac{\\langle \\textbf {0}, s, \\sigma | {\\cal J}_x + i{\\cal J}_y|\\textbf {k}, s_n, \\sigma _n \\rangle }{M^2_n - m^2} \\ ,$ the amplitude $f^{(n)}_{-}(\\omega ^2)$ , determined using Eq.","(REF ), near each pole, becomes: $&f^{(n)}_{-}(\\omega ^2) = -\\frac{1}{4\\pi } ~\\frac{ \\omega ^2_n}{\\omega ^2 - \\omega ^2_n}\\left[(K_{ni})^{\\dagger }K_{ni} - K_{in}(K_{in})^{\\dagger }\\right] \\ .$ For the total forward scattering amplitude, $f^{\\rm tot}_-(\\omega ^2)$ , with an appropriate contour of integration, we will have: $\\frac{1}{2\\pi i}\\int _C \\frac{f^{\\rm tot}_{-}(\\omega ^{\\prime 2})}{\\omega ^{\\prime 2} - \\omega ^2}d\\omega ^{\\prime 2} &= \\frac{1}{\\pi }\\int ^{\\infty }_0 \\frac{\\mathop {\\rm Im}f^{\\rm tot}_-(\\omega ^{\\prime 2})}{\\omega ^{\\prime 2} - \\omega ^2 - i\\epsilon } d\\omega ^{\\prime 2} \\ .$ In the case $\\omega ^2=0$ and when the integration contour encircles all the particle poles we obtain: $&4\\pi [f_{-}(0)+g_-(0)] - [K^{\\dagger },K]_{ii} = \\frac{1}{\\pi }\\int ^{\\infty }_0 \\frac{\\Delta \\sigma (\\omega ^{\\prime })}{\\omega ^{\\prime }} d\\omega ^{\\prime } \\ , \\\\[5pt] \\nonumber &[K^{\\dagger },K]_{ii} \\equiv \\sum _n \\left[(K_{ni})^{\\dagger }K_{ni} - K_{in}(K_{in})^{\\dagger }\\right]^{\\omega ^2_n}_0 \\ .$ It is also interesting to consider the case when $\\omega ^2 = \\omega ^2_k$ in Eq.","(REF ).","Then one obtains a relation between the derivatives of the matrix elements $K_{ni}$ and the dispersion integral.","Notice that $f_-(0)$ and $g_-(0)$ can be expressed in terms of commutator of some generators like $K_{ni}$ , in which case the left hand side of (REF ) would take a more compact form.","Summarizing, when all other intermediate states in the Compton scattering have masses (and spins) different from the mass (or spin) of the scatterer, we have the second sum rule: $\\frac{e^2 J_z}{4m^2}(g-2)^2 &= \\frac{1}{\\pi }\\int ^{\\infty }_0 \\frac{\\Delta \\sigma }{\\omega } d\\omega + [K^{\\dagger },K]_{ii} \\ge 0 \\ .$ Since the left hand side of Eq.","(REF ) is always non negative, so should be the right hand side.","Ignoring the integral, we can rewrite Eq.","(REF ) at leading order in $e^2$ , as follows: $g-2 &= \\pm \\frac{2m}{e}\\sqrt{\\frac{[K^{\\dagger },K]_{ii}}{J_z}} \\ .$ This means that the gyromagnetic ratio may receive corrections even at tree level, if $[K^{\\dagger },K]_{ii} \\ge 0$ .","In supergravity theories, $e \\sim m/M_P$ , and we expect $g-2$ to be independent on $m$ , therefore, $[K^{\\dagger },K]_{ii} \\sim e^2/m^2$ , suggesting that $g-2 \\sim {\\cal O}(1)$ , as for spontaneously broken $N=8$ supergravity [7].","Finally, allowing for an intermediate state, in a Compton scattering off of a target of spin $j$ , to be any narrow resonance of arbitrary mass and spin, we arrive at the most general sum rule: $&\\frac{e^2J_z}{4m^2} \\left[(g_j -2)^2 - \\left(j+\\frac{3}{2}\\right)h_{j+1/2}^2 + \\left(j-\\frac{1}{2}\\right) h_{j-1/2}^2\\right] = [K^{\\dagger },K]_{ii} + \\frac{1}{\\pi }\\int ^{\\infty }_{0} \\frac{\\Delta \\sigma }{\\omega } d\\omega \\ .$ This sum rule can be extended by taking into account other global charges of the theory at hand.","We will not consider such an extension here, since it depends on the specific form of the theory.","Instead, below we will study a particular case, namely the example of the $N \\rightarrow \\Delta $ magnetic transition."],["Some Applications to QCD","Application of Eq.","(REF ) or (REF ) for $j=1/2$ , directly to Compton scattering on a nucleon ($N(938)$ ) with a possible delta ($\\Delta (1232)$ ) intermediate state would be wrong for two main reasons.","First of all, the mass difference $\\delta $ between $\\Delta $ and $N$ is around $300 \\ {\\rm MeV}$ and the forward scattering amplitude, in the $\\omega \\rightarrow 0 $ limit, would behave as $\\omega ^2/(\\delta - \\omega ) \\sim {\\cal O}(\\omega ^2)$ .","It is only when $\\delta \\rightarrow 0$ that the scattering amplitude becomes of order ${\\cal O}(\\omega )$ and contributes comparably to the forward amplitude.","The second reason is that these baryons are strongly coupled systems and loop corrections may be significant.","However, we can still make a formal use of the first sum rule if we work in the limit when $N_c \\rightarrow \\infty $ , in which case $\\delta \\sim {\\cal O}(1/N_c)$ (see, e.g.","Refs.","[24], [25]).","We also need to take $e \\rightarrow 0$ and $N_c \\rightarrow \\infty $ limits in such a way that if $\\Delta \\sigma \\sim N^k_c$ , then $e^2N^{k+2}_c \\ll 1$ , and the dispersive integral can be safely ignored.","Comparing Eqs.","(REF ) and (REF ) one can deduce that: $\\mu _{p\\rightarrow \\Delta ^+} = g_M(0) = \\sqrt{\\frac{2}{3}}~\\kappa _M = \\pm \\sqrt{2} \\kappa _p \\ ,$ where we work in units of nuclear Bohr magneton, and we took into account the appropriate isospin factor $T^3$ corresponding to the $p \\rightarrow \\Delta ^+$ transition.","Taking the experimental value of the proton's anomalous magnetic moment, $\\kappa _p \\approx 1.79$ , we will obtain $\\mu _{p\\rightarrow \\Delta ^+} \\approx \\pm 2.54$ .","Unfortunately, nothing can be said about the sign, since our sum rule relates the squares of the couplings.","Nevertheless, in absolute value, our result for $\\mu _{p\\rightarrow \\Delta ^+}$ is not very far from similar ones, obtained within the framework of other models, like the Skyrme model [25], and holographic QCD [26], which respectively give the values $2.3$ and $2.58$ (the experimental value is: $\\mu _{p\\rightarrow \\Delta ^+} =3.46 \\pm 0.03$ [27]).","In all models of baryons in the large-$N_c$ limit, baryons are finite size objects, whose sizes ($R$ ) do not scale with $N_c$ , while their masses ($M$ ) do.","As in the case of the GDH-Weinberg sum rule [28], in the zero radius limit, when $M R \\rightarrow 0$ , we expect the magnetic transition moment to approach its canonical value (REF ).","However, in the Skyrme model $R$ is fixed and $M R \\sim {\\cal O}(N_c) $ .","This is not surprising, since the Lagrangian of the Skyrme model [25] is known to behave badly at high energies.","Now, we want to apply the second sum rule to the same system.","In this case, we take the physical masses of $N$ and $\\Delta $ and only assume that these are narrow resonances.","Although in this case the scattering amplitude with $\\Delta $ intermediate state vanishes when $\\omega \\rightarrow 0$ , it contributes to the dispersion integral, when $\\omega \\rightarrow \\omega _n$ .","Employing Ref.","[23], we obtain: $&\\langle \\textbf {k}, 3/2, \\sigma ^{\\prime } | \\vec{\\epsilon }_{\\lambda }\\vec{{\\cal J}}| \\textbf {0}, 1/2, \\sigma \\rangle = \\sqrt{\\frac{3m_N}{2M_{\\Delta }}}~\\frac{ie \\mu _{p\\rightarrow \\Delta ^+} }{(m_N+M_{\\Delta })}\\omega ~\\bar{\\psi }^{\\sigma ^{\\prime }}_r(\\textbf {k})u^{\\sigma }(\\textbf {0})~(\\vec{n} \\times \\vec{\\epsilon }_{\\lambda })_r \\ .$ After straightforward computations, the second sum rule gives: $\\kappa _p = \\pm 2 \\frac{m_N^2}{(m_N+M_{\\Delta })}~\\frac{\\mu _{p\\rightarrow \\Delta ^+}}{\\sqrt{m^2_N + M^2_{\\Delta }}} \\ .$ Numerically, $\\mu _{p\\rightarrow \\Delta ^+} \\approx \\pm 1.91\\kappa _p$ ; therefore, taking $\\kappa _p=1.79$ , we obtain $\\mu _{p\\rightarrow \\Delta ^+} \\approx \\pm 3.42$ .","In absolute value this result is coincidentally close to the experimental value: $\\mu _{p\\rightarrow \\Delta ^+} =3.46 \\pm 0.03$ [27].","Consider another example (also relevant for large-$N_c$ QCD), when the target has spin-1/2 and the intermediate particle is an excited state with the same spin.","Then, using Ref.","[29], the general form of the transition matrix element can be written as: $&\\sqrt{\\frac{m}{2E_n}}~\\frac{\\langle \\textbf {k}, 1/2^*,\\sigma ^{\\prime } |{\\cal J}_x + i \\lambda {\\cal J}_y| \\textbf {0}, 1/2,\\sigma \\rangle }{M^2_n - m^2}= 2e G_{*} \\omega ~\\frac{m}{(M_n-m)}\\sqrt{\\frac{mM_n}{M_n^2+m^2}}~(\\sigma _1 + i\\lambda \\sigma _2)_{\\sigma \\sigma ^{\\prime }}\\ ,$ where $G_*$ is some dimensionful coupling.","Direct computations show that $[K^{\\dagger },K]_{ii} < 0$ , which means that for this theory with excited states to be unitary at tree level, we need $G_*=0$ .","However, this conclusion might change if we include intermediate states with spin-3/2.","This work is supported by NSF grant PHY-0758032 and by ERC Advanced Investigator Grant n.226455 Supersymmetry, Quantum Gravity and Gauge Fields (Superfields).","We would like to thank Sergio Ferrara, Gregory Gabadadze and Sergei Dubovsky for interesting discussions."],["Low Energy Compton Amplitude for a Spin-1/2 Target","Consider the Compton scattering process: $N\\lbrace p,\\sigma \\rbrace + \\gamma \\lbrace k,\\epsilon _{\\mu }(k)\\rbrace \\rightarrow N\\lbrace p^{\\prime },\\sigma ^{\\prime }\\rbrace + \\gamma \\lbrace k^{\\prime },\\epsilon ^{\\prime }_{\\nu }(k^{\\prime })\\rbrace \\ ,$ where $p$ and $p^{\\prime }$ are initial and final 4-momenta of scatterer (which is a spin-1/2 particle), $\\sigma $ and $\\sigma ^{\\prime }$ are projections of initial and final spin along the $z$ -direction.","Analogously, $k$ , $\\epsilon _{\\mu }(k)$ and $k^{\\prime }$ , $\\epsilon ^{\\prime }_{\\nu }(k^{\\prime })$ are the 4-momenta and polarizations of initial and final photons.","We take the external particles to be on-shell, that is: $k_0 = |\\textbf {k}|$ , $k_0^{\\prime } = |\\textbf {k}^{\\prime }|$ , $p_0 = \\sqrt{\\textbf {p}^2 + m^2}$ and $p^{\\prime }_0 = \\sqrt{\\textbf {p}^{\\prime 2} + m^2}$ .","The scattering amplitude corresponding to process in (REF ) is: $\\nonumber {\\cal M} &= -e^2\\bar{u}(p^{\\prime },\\sigma ^{\\prime })\\biggl \\lbrace \\Gamma ^{\\nu }(p^{\\prime },p+k)\\epsilon ^{*^{\\prime }}_{\\nu }(k^{\\prime })\\frac{p\\hspace{-7.0pt}\\diagup + k\\hspace{-7.0pt}\\diagup + m}{(p+k)^2 - m^2}\\Gamma ^{\\mu }(p+k,p)\\epsilon _{\\mu }(k) \\\\&+ \\Gamma ^{\\mu }(p^{\\prime },p-k^{\\prime })\\epsilon _{\\mu }(k)\\frac{p\\hspace{-7.0pt}\\diagup - k\\hspace{-7.0pt}\\diagup ^{\\prime } + m}{(p-k^{\\prime })^2 - m^2}\\Gamma ^{\\nu }(p-k^{\\prime },p)\\epsilon ^{*^{\\prime }}_{\\nu }(k^{\\prime })\\biggr \\rbrace u(p,\\sigma ) \\ , \\\\[8pt]&{\\qquad }{\\qquad } \\Gamma ^{\\mu }(p_2,p_1) \\equiv \\gamma ^{\\mu }F_D(q^2) + \\frac{i\\sigma ^{\\mu \\nu }}{2m}q_{\\nu }F_P(q^2) \\ ,$ where $e$ is the electric charge of scatterer, $m$ is its mass, and $q = p_2 -p_1$ .","Using the notations of Peskin & Schroeder, $a\\hspace{-7.0pt}\\diagup \\equiv a_{\\mu }\\gamma ^{\\mu }$ , $\\sigma ^{\\mu \\nu } =i[\\gamma ^{\\mu },\\gamma ^{\\nu }]/2$ .","Here, $F_D(q^2)$ and $F_P(q^2)$ are Dirac and Pauli form factors, which for $q^2=0$ are: $F_D(0) = 1$ and $F_P(0)=\\kappa _p$ , where $e\\kappa _p/(2m)$ is the anomalous magnetic moment of the scatterer.","The latter arises from the Pauli Lagrangian: ${\\cal L}_{\\rm P} = -\\frac{e\\kappa _p}{4m}~\\bar{u}~\\sigma ^{\\mu \\nu }u ~F_{\\mu \\nu } \\ .$ In what follows, we will adopt a `gauge' in which the initial and final photon are transverely polarized in the laboratory frame.","That is, we choose: $\\epsilon (k) \\cdot k = \\epsilon ^{*^{\\prime }}(k^{\\prime }) \\cdot k^{\\prime } = \\epsilon (k) \\cdot p = \\epsilon ^{*^{\\prime }}(k^{\\prime }) \\cdot p = 0 \\ ,$ where $p=(m,\\textbf {0})$ , implying that $\\epsilon ^0 = \\epsilon ^{\\prime 0} = 0$ and $\\vec{\\epsilon }~\\vec{k} = \\vec{\\epsilon }^{~^{\\prime }}\\vec{k}^{\\prime } = 0$ .","We also adopt the following normalization: $\\epsilon (k) \\cdot \\epsilon ^*(k) = \\epsilon ^{\\prime }(k^{\\prime })\\cdot \\epsilon ^{*^{\\prime }}(k^{\\prime }) = -1$ .","Using the Dirac equation: $(p\\hspace{-7.0pt}\\diagup -m)u(p)=0$ , and after some simplifications, we can rewrite the amplitude as follows: ${\\cal M} &=\\frac{e^2\\mu }{2m}\\bar{u}(p^{\\prime },\\sigma ^{\\prime })\\biggl [ \\left(\\epsilon \\hspace{-7.0pt}\\diagup ^{\\prime } + \\mu ^{\\prime } \\epsilon \\hspace{-7.0pt}\\diagup ^{\\prime }k\\hspace{-7.0pt}\\diagup ^{\\prime }\\right)\\left(\\frac{\\epsilon \\hspace{-7.0pt}\\diagup k\\hspace{-7.0pt}\\diagup }{\\omega } + \\frac{1-\\mu }{\\mu }\\epsilon \\hspace{-7.0pt}\\diagup \\right)+ \\left(\\epsilon \\hspace{-7.0pt}\\diagup - \\mu ^{\\prime } \\epsilon \\hspace{-7.0pt}\\diagup k\\hspace{-7.0pt}\\diagup \\right)\\left(\\frac{\\epsilon \\hspace{-7.0pt}\\diagup ^{\\prime }k\\hspace{-7.0pt}\\diagup ^{\\prime }}{\\omega ^{\\prime }} + \\frac{1-\\mu }{\\mu }\\epsilon \\hspace{-7.0pt}\\diagup ^{\\prime }\\right)\\biggr ] u(p,\\sigma ) \\ ,$ where $\\omega \\equiv pk/m $ , $\\omega ^{\\prime } \\equiv pk^{\\prime }/m$ , $\\epsilon \\hspace{-7.0pt}\\diagup ^{\\prime } \\equiv \\gamma ^{\\mu }\\epsilon ^{*^{\\prime }}_{\\mu }$ , $\\mu ^{\\prime } \\equiv \\kappa _p/(2m)$ and $\\mu = 1 + 2m\\mu ^{\\prime }$ is the magnetic moment.","Since we work in the frame where $p_0=m$ and $\\textbf {p}=\\textbf {0}$ , we have: $\\omega = k_0$ and $\\omega ^{\\prime } = k_0^{\\prime }$ .","The initial state is $u^T(p,\\sigma ) = \\sqrt{m}\\left\\lbrace \\xi ,\\xi \\right\\rbrace $ , where $\\xi $ is a spinor such that $\\xi ^{\\dagger }\\xi = 1$ .","Similarly, for $|\\textbf {p}^{\\prime }| \\ll m $ , $\\bar{u}(p^{\\prime },\\sigma ^{\\prime }) = \\sqrt{m}\\left\\lbrace \\xi ^{\\prime \\dagger }\\left(1 + \\frac{1}{2m}\\vec{\\sigma } \\textbf {p}^{\\prime }\\right), \\xi ^{\\prime \\dagger }\\left( 1 - \\frac{1}{2m}\\vec{\\sigma } \\textbf {p}^{\\prime }\\right)\\right\\rbrace \\ .$ Taking $\\omega = \\omega ^{\\prime }$ and defining $\\vec{n} \\equiv \\vec{k}/\\omega $ and $\\vec{n}^{\\prime } \\equiv \\vec{k}^{\\prime }/\\omega ^{\\prime }$ , we can perform direct matrix and vector multiplications to arrive to the final result.","After tedious but straightforward calculations, the answer can be written in a familiar form [20]: $f_{\\rm scat} &= \\frac{e^2}{4\\pi m}\\biggl \\lbrace -\\vec{\\epsilon }^{^{\\prime }*}\\vec{\\epsilon } +\\frac{i\\kappa _p}{m}\\omega ~\\vec{J}~(\\vec{\\epsilon }^{^{\\prime }*} \\times \\vec{\\epsilon }) - \\frac{i\\mu ^2}{m}\\omega ~\\vec{J}~\\left[(\\vec{n}^{\\prime } \\times \\vec{\\epsilon }^{^{\\prime }*}) \\times (\\vec{n} \\times \\vec{\\epsilon })\\right] \\\\ \\nonumber &- \\frac{i\\mu }{2m}~\\omega ~\\vec{J}~\\left[\\left\\lbrace \\vec{n}(\\vec{n}\\times \\vec{\\epsilon }) + (\\vec{n}\\times \\vec{\\epsilon })\\vec{n}\\right\\rbrace \\vec{\\epsilon }^{^{\\prime }*}-\\lbrace \\vec{n}^{\\prime }(\\vec{n}^{\\prime }\\times \\vec{\\epsilon }^{^{\\prime }*}) + (\\vec{n}^{\\prime }\\times \\vec{\\epsilon }^{^{\\prime }*})\\vec{n}^{\\prime }\\rbrace \\vec{\\epsilon }\\right]\\biggr \\rbrace \\ ,$ where $f_{\\rm scat} \\equiv {\\cal M}/(8\\pi m) $ and $\\vec{J}\\equiv \\delta _{\\sigma ^{\\prime }\\sigma } \\xi ^{\\dagger }_{\\sigma } \\vec{\\sigma } \\xi _{\\sigma }/2$ , with $\\vec{J}$ being the spin of the scatterer (when $\\sigma =\\sigma ^{\\prime }$ )."],["General Properties of Rarita-Schwinger Field","The Lagrangian for a free massive spin-3/2, Rarita-Schwinger (RS) field, $\\psi _{\\mu }$ , can be written as: ${\\cal L}_{\\rm RS} = \\bar{\\psi }_{\\mu }\\left(i \\gamma ^{\\mu \\nu \\rho }\\partial _{\\rho } - m \\gamma ^{\\mu \\nu } \\right)\\psi _{\\nu } \\ ,$ where $\\gamma ^{\\mu \\nu } \\equiv \\gamma ^{[\\mu }\\gamma ^{\\nu ]}$ and $\\gamma ^{\\mu \\nu \\rho } \\equiv \\gamma ^{[\\mu }\\gamma ^{\\nu }\\gamma ^{\\rho ]}$ .","The equations of motion that follow from this Lagrangian can be equivalently written as Dirac equations along with the transversality and tracelessness constraints: $\\left(i\\partial \\hspace{-7.0pt}\\diagup - m\\right)\\psi _{\\mu } = 0 \\ , \\ \\ \\ \\partial ^{\\mu }\\psi _{\\mu } = 0 \\ , \\ \\ \\ \\gamma ^{\\mu }\\psi _{\\mu } = 0 \\ .$ These constraints guarantee that among 16 independent components of RS field only $2s+1 = 4$ physical degrees of freedom will propagate.","The wave function for the RS field can be written as a product of massive spin-1 and spin-1/2 polarizations: $e_{\\mu }(\\vec{p},\\textit {n})$ with $\\textit {n}=-1,0,1$ , and $u(\\vec{p},\\sigma )$ with ($\\sigma =\\pm 1/2$ ), respectively.","More specifically (see, e.g.","Refs.","[30]): $\\psi _{\\mu }(\\vec{p},r) &= \\sum _{\\sigma ,\\textit {n}} \\langle \\left(\\frac{1}{2},\\sigma \\right)\\left(1,\\textit {n}\\right)|\\left(\\frac{3}{2},r\\right)\\rangle ~u(\\vec{p},\\sigma )~e_{\\mu }(\\vec{p},\\textit {n}) \\ ,$ where $u(\\vec{p},\\sigma )$ satisfies equations: $(p\\hspace{-7.0pt}\\diagup -m)u(\\vec{p},\\sigma )=0$ and $\\hat{\\sigma }_z u(\\vec{p},\\sigma ) = \\sigma u(\\vec{p},\\sigma ) $ .","The polarization vectors satisfy the following normalization and transversality conditions: $&e^*_{\\mu }(\\vec{p},\\textit {n})e^{\\mu }(\\vec{p},\\textit {n}^{\\prime }) = -\\delta _{\\textit {n}\\textit {n}^{\\prime }} \\ , {\\qquad }p^{\\mu }e_{\\mu }(\\vec{p},\\textit {n}) = p^{\\mu }e^*_{\\mu }(\\vec{p},\\textit {n}) = 0 \\ .$ Substituting the values of the Clebsch-Gordan coefficients in Eq.","(REF ), that are proportional to $\\delta _{r,\\textit {n}+\\sigma }$ , we will get: $\\psi _{\\alpha }^{\\pm 3/2} &= e_{\\alpha }^{\\pm 1}u^{\\pm 1/2} \\ , {\\qquad }\\psi _{\\alpha }^{\\pm 1/2} = \\frac{1}{\\sqrt{3}}\\left(e_{\\alpha }^{\\pm 1}u^{\\mp 1/2} + \\sqrt{2}e_{\\alpha }^0 u^{\\pm 1/2}\\right) \\ ,$ where $\\psi _{\\alpha }(\\vec{p},r) \\equiv \\psi _{\\alpha }^{r}$ , $u(\\vec{p},\\sigma ) \\equiv u^{\\sigma }$ and $e_{\\mu }(\\vec{p},\\textit {n}) \\equiv e_{\\mu }^\\textit {n}$ .","These solutions satisfy the Dirac equation, as well as transversality and tracelessness constraints.","Moreover, the RS states are normalized as: $&\\bar{\\psi }_{\\alpha }(\\vec{p},r)\\psi ^{\\alpha }(\\vec{p},r^{\\prime }) = -2m \\delta _{rr^{\\prime }} \\ , {\\qquad }\\bar{\\psi }_{\\alpha }(\\vec{p},r)\\gamma _{\\mu }\\psi ^{\\alpha }(\\vec{p},r^{\\prime }) = -2p_{\\mu } \\delta _{rr^{\\prime }} \\ .$ In what follows it would be useful to note that: $P_{\\mu \\nu }(p) &\\equiv \\sum _{r}\\psi _{\\nu }(\\vec{p},r)\\bar{\\psi }_{\\mu }(\\vec{p},r) = -(p\\hspace{-7.0pt}\\diagup + m)\\Pi _{\\mu \\nu }(p) \\ , \\\\ \\nonumber \\Pi _{\\mu \\nu }(p) &\\equiv \\left(g_{\\mu \\nu } -\\frac{p_{\\mu }p_{\\nu }}{m^2}\\right) - \\frac{1}{3}\\left(\\gamma _{\\mu } -\\frac{p_{\\mu }}{m}\\right)\\left(\\gamma _{\\nu } +\\frac{p_{\\nu }}{m}\\right) \\ .$ In the chiral representation of the $\\gamma $ -matrices (as in Peskin & Schroeder), it can be checked that, when $p = (m,0,0,0)$ , the solutions for $u^{\\sigma }$ and $e_{\\mu }^\\textit {n}$ are: $&u^{-1/2} = \\sqrt{m}(0,1,0,1)^T \\ , \\ \\ \\ \\ u^{+1/2} = \\sqrt{m}(1,0,1,0)^T \\ , \\\\ \\nonumber &e^+_{\\mu } = \\frac{1}{\\sqrt{2}}(0,1,i,0) \\ , \\ \\ \\ e^-_{\\mu } = -\\frac{1}{\\sqrt{2}}(0,1,-i,0) \\ , \\ \\ \\ e^0_{\\mu } = (0,0,0,-1) \\ .$ Writing the quantized RS field as, $&\\Psi _{\\mu }(x) = \\int \\frac{d^3p}{(2\\pi )^3}\\frac{1}{2E_{\\textbf {p}}}\\sum _{\\lambda }\\left\\lbrace e^{-ipx}\\psi _{\\mu }(\\vec{p},\\lambda )a_{\\vec{p},\\lambda } + e^{ipx}\\psi ^C_{\\mu }(\\vec{p},\\lambda )a^{\\dagger }_{\\vec{p},\\lambda }\\right\\rbrace \\ , \\\\&\\lbrace a_{\\vec{p},\\lambda }, a^{\\dagger }_{\\vec{p}^{\\prime },\\lambda ^{\\prime }} \\rbrace = (2\\pi )^32E_{\\textbf {p}}\\delta _{\\lambda \\lambda ^{\\prime }}\\delta (\\textbf {p}-\\textbf {p}^{\\prime }) \\ ,{\\qquad }\\lbrace a_{\\vec{p},\\lambda }, a_{\\vec{p}^{\\prime },\\lambda ^{\\prime }} \\rbrace = \\lbrace a^{\\dagger }_{\\vec{p},\\lambda }, a^{\\dagger }_{\\vec{p}^{\\prime },\\lambda ^{\\prime }} \\rbrace = 0 \\ ,$ and using Eq.","(REF ), it can be deduced that: $&\\langle 0| \\Psi _{\\nu }(x) \\bar{\\Psi }_{\\mu }(y) |0 \\rangle = \\int \\frac{d^3p}{(2\\pi )^3}\\frac{1}{2E_{\\textbf {p}}}\\sum _{\\lambda }\\psi _{\\nu }(\\vec{p},\\lambda )\\bar{\\psi }_{\\mu }(\\vec{p},\\lambda ) e^{-ip(x-y)} \\\\ \\nonumber &=-(i\\partial \\hspace{-7.0pt}\\diagup _x+m)\\Pi _{\\mu \\nu }(i\\partial _x) \\int \\frac{d^3p}{(2\\pi )^3}\\frac{1}{2E_{\\textbf {p}}}e^{-ip(x-y)} \\ , \\\\&\\langle 0| \\bar{\\Psi }_{\\mu }(y)\\Psi _{\\nu }(x) |0 \\rangle = \\int \\frac{d^3p}{(2\\pi )^3}\\frac{1}{2E_{\\textbf {p}}}\\sum _{\\lambda }\\psi ^C_{\\nu }(\\vec{p},\\lambda )\\bar{\\psi }^C_{\\mu }(\\vec{p},\\lambda ) e^{-ip(y-x)} \\\\ \\nonumber &=(i\\partial \\hspace{-7.0pt}\\diagup _x+m)\\Pi _{\\mu \\nu }(i\\partial _x) \\int \\frac{d^3p}{(2\\pi )^3}\\frac{1}{2E_{\\textbf {p}}}e^{-ip(y-x)} \\ .$ Since the Feynman propagator is defined as $S^F_{\\mu \\nu }(x-y) \\equiv \\langle 0 |T\\Psi _{\\nu }(x) \\bar{\\Psi }_{\\mu }(y) | 0 \\rangle $ , we have: $&S^F_{\\mu \\nu }(x-y) = -(i\\partial \\hspace{-7.0pt}\\diagup _x+m)\\Pi _{\\mu \\nu }(i\\partial _x) D_F(x-y) \\ , \\\\&D_F(x-y) \\equiv \\int \\frac{d^4p}{(2\\pi )^4}~\\frac{i}{p^2-m^2 +i\\epsilon } e^{-ip(x-y)} \\ ,$ where $D_F(x-y)$ is the Feynman propagator of a free scalar field.","More explicitly, $S^F_{\\alpha \\beta }(p) &= -i\\frac{\\left(p\\hspace{-7.0pt}\\diagup + m\\right)}{p^2-m^2 + i\\epsilon }\\left[g_{\\alpha \\beta } - \\frac{1}{3}\\gamma _{\\alpha }\\gamma _{\\beta } - \\frac{2p_{\\alpha }p_{\\beta }}{3m^2}-\\frac{\\left(\\gamma _{\\alpha }p_{\\beta }- \\gamma _{\\beta }p_{\\alpha }\\right)}{3m}\\right]\\ .$"],["First Sum Rule for Spin-1/2 Target: Alternative Approach","The most general form of the vertex function, describing interactions between photon, spin-1/2 and spin-3/2 particles of the same mass, following Jones and Scadron [22] can be written in terms of magnetic dipole ($g_M$ ), electric quadrupole ($g_E$ ), and Coulomb quadrupole ($g_C$ ) form factors.","This interaction vertex effectively emerges from the following Lagrangian (see, e.g.","[23]): ${\\cal L}_{\\rm int} = \\frac{3ie}{4m^2}~\\bar{u}T^3&\\biggl [g_M \\left(\\partial _{\\mu }\\psi _{\\nu }\\right) \\tilde{F}^{\\mu \\nu } + i g_E \\gamma _5 \\left(\\partial _{\\mu }\\psi _{\\nu }\\right)F^{\\mu \\nu } -\\frac{2g_C}{m}\\gamma _5\\gamma ^{\\alpha }\\partial _{[\\alpha }\\psi _{\\nu ]}\\partial _{\\mu }F^{\\mu \\nu } \\biggr ] + {\\rm h.c.} \\ ,$ where $\\tilde{F}^{\\mu \\nu } = \\epsilon ^{\\mu \\nu \\alpha \\beta }F_{\\alpha \\beta }/2$ and $T^3$ is an operator due to additional internal degrees of freedom (such as global or isospin charge) that the fields could carry.","In the soft momentum transfer (or near forward) limit that we are interested in, only the first term will matter.","We will rewrite the Lagrangian describing the magnetic dipole transition as: ${\\cal L}^{M1}_{\\rm int} &= \\frac{ie\\kappa _{M}}{2m^2}~\\bar{u}\\left(\\partial _{\\mu }\\psi _{\\nu }\\right) \\tilde{F}^{\\mu \\nu } + {\\rm h.c.} \\ .$ The interaction vertex describing the dominant magnetic dipole $3/2 \\rightarrow 1/2$ transition is: $\\Gamma ^{\\nu \\beta }(p^{\\prime },p^{\\prime }+k^{\\prime }) &= \\frac{e \\kappa _{M}}{2m^2}\\epsilon ^{\\mu \\nu \\alpha \\beta }p^{\\prime }_{\\mu }k^{\\prime }_{\\alpha } \\ ,$ where $p^{\\prime }$ is the momentum of spin-1/2 state and $k^{\\prime }$ is the photon momentum.","Similarly, the vertex of $1/2 \\rightarrow 3/2$ transition is: $\\Gamma ^{\\nu \\beta }(p+k,p) &= -\\frac{e\\kappa _{M}}{2m^2}\\epsilon ^{\\mu \\nu \\alpha \\beta }p_{\\mu }k_{\\alpha } \\ ,$ where $p$ is the momentum of spin-1/2 state and $k$ is the photon momentum.","For forward scattering, when $p_1 = p_2 = (m,0,0,0)$ and $\\omega = \\omega ^{\\prime }$ , we have: $\\Gamma ^{\\nu \\beta }(p^{\\prime },p^{\\prime }+k^{\\prime }) = -\\Gamma ^{\\nu \\beta }(p+k,p) &= \\frac{e\\kappa _{M}}{2m}\\omega ~\\epsilon ^{0\\nu \\beta k}n_{k} \\ .$ Consider the scattering in (REF ), where the intermediate state is a spin-3/2 Rarita-Schwinger particle with the same mass as the scatterer.","We want to find the amplitude corresponding to this process.","It can be written as: $\\nonumber i{\\cal M_{RS}} &= \\bar{u}(p^{\\prime },\\sigma ^{\\prime })\\biggl \\lbrace \\Gamma ^{\\nu \\alpha }(p^{\\prime },p+k)\\epsilon ^{*^{\\prime }}_{\\nu }(k^{\\prime })S_{\\alpha \\beta }(p+k)\\Gamma ^{\\mu \\beta }(p+k,p)\\epsilon _{\\mu }(k) \\\\&+ \\Gamma ^{\\mu \\beta }(p^{\\prime },p-k^{\\prime })\\epsilon _{\\mu }(k)S_{\\beta \\alpha }(p-k^{\\prime })\\Gamma ^{\\nu \\alpha }(p-k^{\\prime },p)\\epsilon ^{*^{\\prime }}_{\\nu }(k^{\\prime })\\biggr \\rbrace u(p,\\sigma ) \\ ,$ where vertices are defined in Eqs.","(REF )-(REF ), and propagator of RS field, $S_{\\alpha \\beta }$ , is given by Eq.","(REF ).","Since we are interested in the near forward scattering amplitude up to order ${\\cal O}(\\omega )$ , and using Eqs.","(REF ) and (REF ), direct computations for $f_{RS} = {\\cal M}_{RS}/(8\\pi m)$ give: $\\nonumber f_{RS}(\\omega , \\lambda ) &= \\frac{i}{12\\pi }\\frac{e^2\\kappa _M^2}{m^2}~\\omega \\left[(\\vec{\\epsilon }^{^{\\prime }*} \\times \\vec{n}^{\\prime }) \\times \\left(\\vec{\\epsilon } \\times \\vec{n}\\right)\\right]\\vec{J} = -\\frac{\\lambda \\omega }{12\\pi }\\frac{e^2\\kappa _M^2}{m^2}~J_z \\ .$ Finally, using the definition for $g_-(\\omega ^2)$ , given in Eq.","(REF ), in the forward limit we have: $4\\pi ^2g_-(0) &= -\\frac{\\pi e^2\\kappa ^2_{M}}{3 m^2}J_z \\ ,$ which is in agreement with the previous result (REF ), as should be expected."]],"string":"[\n [\n \"New Sum Rules from Low Energy Compton Scattering on Arbitrary Spin\\n Target\"\n ],\n [\n \"Abstract We derive two sum rules by studying the low energy Compton scattering on a target of arbitrary (nonzero) spin j.\",\n \"In the first sum rule, we consider the possibility that the intermediate state in the scattering can have spin |j \\\\pm 1| and the same mass as the target.\",\n \"The second sum rule applies if the theory at hand possesses intermediate narrow resonances with masses different from the mass of the scatterer.\",\n \"These sum rules are generalizations of the Gerasimov-Drell-Hearn-Weinberg sum rule.\",\n \"Along with the requirement of tree level unitarity, they relate different low energy couplings in the theory.\",\n \"Using these sum rules, we show that in certain cases the gyromagnetic ratio can differ from the \\\"natural\\\" value g=2, even at tree level, without spoiling perturbative unitarity.\",\n \"These sum rules can be used as constraints applicable to all supergravity and higher-spin theories that contain particles charged under some U(1) gauge field.\",\n \"In particular, applied to four dimensional N=8 supergravity in a spontaneously broken phase, these sum rules suggest that for the theory to have a good ultraviolet behavior, additional massive states need to be present, such as those coming from the embedding of the N=8 supergravity in type II superstring theory.\",\n \"We also discuss the possible implications of the sum rules for QCD in the large-N_c limit.\"\n ],\n [\n \"Introduction, Summary and an Application to $N=8$ Supergravity\",\n \"Dispersion relations based on analyticity, unitarity and Lorentz invariance of scattering amplitudes can be interpreted as consistency conditions that constrain low-energy data, i.e.\",\n \"parameters of an effective action.\",\n \"Conversely, if we know the low energy effective action of a given theory, dispersion relations either constrain any of its possible UV completions, or show that no completion exists.\",\n \"Beautiful examples of the latter phenomenon are given in Ref. [1].\",\n \"In this paper we revisit topics addressed years ago by one of us [2] and find their implications on the existence and properties of UV completions of low energy effective theories.\",\n \"In Ref.\",\n \"[2] it was argued that when effective field theories of elementary particles admit a perturbative expansion (in some parametrically small dimensionless coupling constant), then the gyromagnetic ratio of the (weakly interacting) particles described by such theory had to be close to a preferred “natural” value: $g=2$ .\",\n \"This result was obtained in a Lagrangian approach.\",\n \"Many years earlier, Weinberg [3] also proposed an argument, based on a generalization of the Gerasimov-Drell-Hearn (GDH) [4] sum rule, which similarly selected $g = 2$ as the preferred value in weakly interacting theories.\",\n \"Given a particle of mass $m$ and electric charge $e$ , the GDH-Weinberg sum rule connects the gyromagnetic ratio $g$ to a dispersion integral.\",\n \"As usual, $g$ is defined as the ratio of the particle's magnetic moment $\\\\mu $ to its spin $J$ , so that (in the particle's rest frame): $\\\\vec{\\\\mu } = \\\\frac{e g}{2m}\\\\vec{J} \\\\ .$ The main ingredient of the Weinberg sum rule is the low energy forwardAssume that the photon propagates along some $z$ -direction with helicity $\\\\lambda = \\\\pm 1$ , and the target has a spin-$z$ projection $J_z$ .\",\n \"For more details, see Appendix .\",\n \"Compton scattering amplitude of a photon with energy $\\\\omega $ and helicity $\\\\lambda $ off a massive target of spin $J$ .\",\n \"This amplitude, $f_{\\\\rm scat} (\\\\omega , \\\\lambda )$ , is a real analytic function of the photon's energy $\\\\omega $ away from the real $\\\\omega $ -axis, where cuts and poles may exist at $\\\\omega > 0 $ .\",\n \"The imaginary part of $f_{\\\\rm scat}$ is given by the optical theorem: ${\\\\rm Im} f_{\\\\rm scat} (\\\\omega ,\\\\lambda ) = \\\\frac{\\\\omega }{4\\\\pi }\\\\sigma _{\\\\rm tot}(\\\\omega , \\\\lambda ) \\\\ ,$ where $\\\\sigma _{\\\\rm tot}$ is the total cross-section for a photon with helicity $\\\\lambda $ and energy $\\\\omega $ .\",\n \"Define now the following function: $&f_-(\\\\omega ^2) \\\\equiv \\\\frac{f_{\\\\rm scat} (\\\\omega ,+1)-f_{\\\\rm scat} (\\\\omega ,-1)}{2\\\\omega } \\\\ .$ When no intermediate (one particle) state exists in the Compton scattering, with either mass or spin different from those of the target, then, it can be checked that (see, e.g., Appendix ): $&f_-(\\\\omega ^2 \\\\rightarrow 0) = \\\\frac{e^2J_z}{16\\\\pi m^2}(g-2)^2 \\\\ .$ Using the optical theorem (REF ) and definition (REF ), we have: ${\\\\rm Im}f_-(\\\\omega ^2) = \\\\frac{1}{8\\\\pi }\\\\Delta \\\\sigma (\\\\omega ) \\\\ , \\\\qquad \\\\Delta \\\\sigma (\\\\omega ) \\\\equiv \\\\sigma _{\\\\rm tot}(\\\\omega ,+1) - \\\\sigma _{\\\\rm tot}(\\\\omega ,-1) \\\\ .$ Assuming that $f_-(\\\\omega ^2)$ vanishes when $|\\\\omega ^2| \\\\rightarrow \\\\infty $ , one can write an unsubtracted dispersion relation: $f_-(\\\\omega ^2) = \\\\frac{1}{4\\\\pi ^2}\\\\int ^{\\\\infty }_0\\\\frac{\\\\Delta \\\\sigma (\\\\omega ^{\\\\prime })}{\\\\omega ^{\\\\prime 2} - \\\\omega ^2 - i \\\\epsilon } ~\\\\omega ^{\\\\prime } d\\\\omega ^{\\\\prime } \\\\ .$ The validity of this assumption is far from obvious and its justification requires some additional assumptions about the UV behavior of the theory.\",\n \"We will argue below that unsubtracted dispersion relations hold in two important case: superstring theory and any unitary completion of $N=8$ supergravity.\",\n \"When $\\\\omega ^2=0$ , Eqs.\",\n \"(REF ) and (REF ), impliy the GDH-Weinberg sum rule [3], [4]: $\\\\frac{\\\\pi e^2 J_z}{4m^2} \\\\left(g-2\\\\right)^2 = \\\\int ^{\\\\infty }_0\\\\frac{\\\\Delta \\\\sigma (\\\\omega ^{\\\\prime })}{\\\\omega ^{\\\\prime }} d\\\\omega ^{\\\\prime } \\\\ .$ In weakly interacting systems, the RHS of Eq.\",\n \"(REF ) is parametrically smaller than the LHS, since in the former the final state contains at least two particles and is thus $O(e^4)$ , while the latter is $O(e^2)$ ; hence $g = 2 + {\\\\cal O}(e^2)$ .\",\n \"In truncated $N=2$ supergravity theory (where one drops the cosmological and quartic Fermionic terms from the Lagrangian given in Ref.\",\n \"[5]) it can be checked that the gravitino has indeed $g=2$ , see also Ref. [6].\",\n \"On the other hand, in spontaneously broken $N=8$ supergravity theory [7], see also [13], contrary to expectations, one finds that $g=1$ , for both spin-1/2 and spin-3/2 fields.\",\n \"To show this, we use the dimensionally reduced actionThe $N=8$ supergravity theory with global E(6) and local USp(8) invariance was constructed in five dimensions by Cremmer, Scherk and Schwarz in Ref. [7].\",\n \"Spontaneous symmetry breaking of $N=8$ supergravity theory is achieved by dimensional reduction to 4D via the Scherk-Schwarz mechanism [8].\",\n \"given in Ref.\",\n \"[9], and consider only the relevant terms of the Lagrangians for spin-1/2 and spin-3/2 fields, defined as $\\\\chi ^{abc}$ and $\\\\psi _{\\\\mu }^a$ , respectively.\",\n \"In our notations (see Appendices), and in units, $\\\\kappa ^2 = 4\\\\pi G_N =1$ , where $G_N$ is the 4D Newton's constant, these terms, written in flat space-time, unitary gauge ($\\\\psi _5^a=0$ ) and near the ground state, take the following form: $&{\\\\cal L}_{4D}(\\\\chi ) = \\\\frac{1}{12}\\\\bar{\\\\chi }^{abc}\\\\biggl [i\\\\left(\\\\partial \\\\hspace{-7.0pt}\\\\diagup + 2i{\\\\cal M}_{abc}B\\\\hspace{-7.0pt}\\\\diagup \\\\right)- {\\\\cal M}_{abc} +\\\\frac{1}{4}\\\\sigma ^{\\\\mu \\\\nu }B_{\\\\mu \\\\nu }\\\\biggr ]\\\\chi _{abc} \\\\ , \\\\\\\\ &{\\\\cal L}_{4D}(\\\\psi ) = \\\\frac{1}{2}\\\\bar{\\\\psi }^{a}_{\\\\mu }\\\\biggl [i\\\\gamma ^{\\\\mu \\\\rho \\\\nu }\\\\left(\\\\partial _{\\\\nu } + 2i{\\\\cal M}_{a}B_{\\\\nu }\\\\right) - {\\\\cal M}_{a}\\\\gamma ^{\\\\mu \\\\rho } + \\\\frac{i}{2}\\\\left(B^{\\\\mu \\\\rho } - i\\\\gamma _5\\\\tilde{B}^{\\\\mu \\\\rho }\\\\right)\\\\biggr ]\\\\psi _{\\\\rho a} \\\\ ,$ where $B_{\\\\mu }$ is a graviphoton field, ${\\\\cal M}_{abc}$ and ${\\\\cal M}_a$ are the mass matrices of $\\\\chi ^{abc}$ and $\\\\psi _{\\\\mu }^a$ correspondingly.\",\n \"These spinors are $USp(8)$ tensors, and both are charged under the graviphoton, with a charge $e=2\\\\kappa m$ , where $m$ is the mass of the spinor.\",\n \"Notice, also that there is no $\\\\chi $ -$\\\\psi $ -$B$ mixing.\",\n \"The Lagrangians above should be compared with the Dirac Lagrangian supplemented with a Pauli term, and the Lagrangian of a charged Rarita-Schwinger field with non-minimal terms [6]: $&{\\\\cal L}_{DP} = \\\\bar{\\\\chi }\\\\left[i(\\\\partial \\\\hspace{-7.0pt}\\\\diagup + ieB\\\\hspace{-7.0pt}\\\\diagup ) - m - \\\\frac{e(g_{1/2}-2)}{8m}\\\\sigma ^{\\\\mu \\\\nu }B_{\\\\mu \\\\nu }\\\\right]\\\\chi \\\\ , \\\\\\\\ &{\\\\cal L}_{RS} = \\\\bar{\\\\psi }_{\\\\mu }\\\\left[ i\\\\gamma ^{\\\\mu \\\\rho \\\\nu }(\\\\partial _{\\\\nu } + ie B_{\\\\nu }) - m\\\\gamma ^{\\\\mu \\\\rho } + \\\\frac{ie}{m}\\\\left\\\\lbrace \\\\frac{3}{4}\\\\left(g_{3/2}- \\\\frac{2}{3}\\\\right)B^{\\\\mu \\\\rho } - i\\\\alpha \\\\gamma _5\\\\tilde{B}^{\\\\mu \\\\rho }\\\\right\\\\rbrace \\\\right]\\\\psi _{\\\\rho } \\\\ ,$ where $\\\\alpha $ is a parameter unrelated to $g_{3/2}$ .As was observed in [6], the $\\\\gamma _5$ matrix in the $\\\\bar{\\\\psi }_{\\\\mu }\\\\gamma _5 \\\\tilde{B}^{\\\\mu \\\\nu }\\\\psi _{\\\\nu }$ term of () mixes the “large” and “small” components of $\\\\psi _{\\\\mu }$ , and thus gives contributions of higher order in $\\\\omega $ .\",\n \"That is why this term does not contribute to $g_{3/2}$ .\",\n \"Finally, comparing (REF –) with (REF –), we conclude that in case of the spontaneously broken $N=8$ supergravity: $g_{1/2} = g_{3/2}= 1$ and $\\\\alpha =1/4$ .Similarly, comparing () with the truncated $N=2$ supergravity [5], one can deduce that: $g_{3/2} = 2$ and $\\\\alpha = 1$ .\",\n \"In $N=2$ supergravity with a gauged central charge [11], $g_{3/2}=2$ , however, $g_{1/2} \\\\ne 2$ or 1.\",\n \"It appears that $g=1$ for all heavy particles in the Kaluza-Klein theory [10].\",\n \"This observation was also confirmed within the string theory and on the example of D0-branes in [12].\",\n \"It is often the case that in supergravity theories, $e \\\\sim m/M_P$ , where $M_P$ is the Planck mass, suggesting that the charge can be made arbitrarily small.\",\n \"It is thus natural to ask: why Eq.\",\n \"(REF ) fails so miserably in case of the spontaneously broken $N=8$ supergravity theory?\",\n \"One could argue that this happens because $f_-({\\\\omega ^2})$ does not vanish sufficiently fast, when $|\\\\omega ^2| \\\\rightarrow \\\\infty $ , since supergravity is power counting non-renormalizable.\",\n \"However, spontaneously broken $N=8$ supergravity can be embedded in type II superstring theory [14]Spontaneously broken $N=4,2,1$ theories can be embedded also in heterotic string theory [15]., where the condition $\\\\lim _{\\\\omega ^2\\\\rightarrow \\\\infty }f_-(\\\\omega ^2)= 0$ holds, order-by-order in string perturbation theory.\",\n \"A more explicit and general argument supporting unsubtracted dispersion relations is that, as argued in Refs.\",\n \"[16] (see also [17]), the gravitational 2-body elastic scattering amplitude $f(s,t)$ of any theory obeying Hermitian unitarity and crossing symmetry is polynomially bound in the complex-$s$ upper half plane for fixed Mandelstam variable $t$ .\",\n \"Hermitian unitarity implies $f(s,t)^*=f(s^*,t^*)$ , thus polynomial boundedness in $s$ for the forward elastic scattering amplitude ($t=0$ ).\",\n \"Furthermore, Eq.\",\n \"(REF ) suggests that the scattering amplitude $f_-(s,0)$ is not only polynomially bound, but indeed vanishes at large positive $s$ as $\\\\sim 1/s$ .\",\n \"Crossing symmetry, Hermitian unitarity, and the same argument used to prove polynomial boundedness, i.e.\",\n \"the Phragmen Lindelöf theorem, then show that $f_-(s,0)$ vanishes at large $s$ in the whole complex plane.Eq.\",\n \"(REF ) gives a tree-level $f_-(s,0)$ that does not vanish at $s=\\\\infty $ ; this fact alone shows that $N=8$ supergravity cannot be a perturbatively complete theory of gravity.\",\n \"As we will argue, the solution to this puzzle is different and it is one of the main results of this paper.\",\n \"The point is that the GDH-Weinberg sum rule is modified when the Compton scattering amplitude includes intermediate one-particle states of masses $M_n \\\\ne m$ .In this case, $\\\\Delta \\\\sigma $ becomes a sum of terms proportional to $\\\\delta (\\\\omega -\\\\omega _n)$ plus the contribution from the continuum.\",\n \"Taking into account this possibility, we find the following generalization of the GDH-Weinberg sum rule: $&{\\\\qquad }{\\\\qquad } \\\\frac{e^2 J_z}{4m^2}(g-2)^2 = [K^{\\\\dagger },K]_{ii} +\\\\frac{1}{\\\\pi }\\\\int ^{\\\\infty }_0 \\\\frac{\\\\Delta \\\\sigma }{\\\\omega } d\\\\omega \\\\ , \\\\\\\\[5pt] \\\\nonumber [K^{\\\\dagger },K]_{ii} \\\\equiv \\\\sum _n &\\\\left[(K_{ni})^{\\\\dagger }K_{ni} -K_{in}(K_{in})^{\\\\dagger }\\\\right] , {\\\\qquad } K_{ni} \\\\equiv \\\\left[2\\\\omega _n\\\\sqrt{m(\\\\omega _n + m)}\\\\right]^{-1}~\\\\langle n |\\\\vec{\\\\epsilon }_{\\\\lambda =1}\\\\vec{{\\\\cal J}}|i \\\\rangle \\\\ ,$ where $m$ is the mass of the target, $M_n$ is the mass of the intermediate state, $\\\\omega _n = \\\\frac{1}{2m}(M^2_n - m^2)$ , $\\\\vec{{\\\\cal J}}$ is the electromagnetic current, and $\\\\vec{\\\\epsilon }_{\\\\lambda }$ is the polarization vector of the photon with helicity $\\\\lambda $ .\",\n \"The matrix $K_{ni}$ describes transition between states of different mass, and possibly spin.\",\n \"This sum rule should be obeyed not only in string theory, but also in any Lorentz invariant, causal, unitary UV completion of $N=8$ supergravity.\",\n \"Because $e \\\\sim m/M_P$ , the LHS of (REF ) is ${\\\\cal O}(1/M_P^2)$ , independent of the mass $m$ , and thus is the RHS.\",\n \"Generically this implies that the new intermediate states have masses $M_n$ independent of $m$ ; these masses thus define a cutoff, below which the theory is described by $N=8$ supergravity.\",\n \"Type II superstring compactified on a 6-torus is a concrete example of this general situation.\",\n \"In this case the relation between the string mass scale $M_S\\\\sim 1/\\\\sqrt{\\\\alpha ^{\\\\prime }}$ and the Planck scale is $M_P=M_S^4\\\\sqrt{V_6}/g_S$ , where $V_6$ is the volume of the 6-torus, and $g_S$ is the string coupling constant.\",\n \"In the perturbative regime, where $V_6M_S^6 \\\\gg 1$ and $g_S\\\\ll 1$ , we have: $M_P\\\\gg M_S $ .\",\n \"When all radii of the torus are ${\\\\cal O}(1/M_S)$ , $M_S$ becomes the only UV cutoff scale and $M_n={\\\\cal O}(M_S)$ .\",\n \"The fact that new states, besides those already present in $N=8$ supergravity,In a spontaneously broken phase, where gravitino is massive and charged [7].\",\n \"are necessary for $N=8$ to admit a UV completion has an immediate consequence.\",\n \"It implies that $N=8$ supergravity is not perturbatively complete by and in itself.\",\n \"Such a statement, that is at variance with other remarkable finiteness properties of the theory (see e.g [18]) warrants further discussion.\",\n \"It follows from two assumptions, besides Lorentz invariance and unitarity, both satisfied by string theory but holding more generally in any perturbative regularization of gravity.\",\n \"The first is that the cutoff scale $\\\\Lambda $ is parametrically smaller than $M_P$ : $\\\\Lambda = \\\\lambda _c M_P$ , where $\\\\lambda _c \\\\ll 1$ is the dimensionless coupling constant of the UV complete theory.\",\n \"The second assumption is that scattering amplitudes are regular in the limit $m\\\\rightarrow 0$ ; more precisely, that the mass $m$ appears only with positive powers in scattering amplitudes.\",\n \"One can then use the fact that forward scattering depends only on the relativistic invariant $s=m^2 + 2m\\\\omega $ to rewrite the sum rule (REF ) as $\\\\lambda ^2_c \\\\frac{\\\\pi J_z}{4\\\\Lambda ^2}(g-2)^2 =\\\\lambda ^2_c \\\\int _{m^2}^\\\\infty \\\\frac{ds}{s-m^2} \\\\sum _n f_n[m^2,(s-m^2)/\\\\Lambda ^2]\\\\delta (s-M_n^2) + O(\\\\lambda ^4_c).$ Terms ${\\\\cal O}(\\\\lambda ^4_c)$ come from the continuum part of the total cross section, in which the final state contains at least two particles.\",\n \"By assumption, the $m\\\\rightarrow 0$ limit of Eq.\",\n \"(REF ) is smooth, so the functions $f_n[0,x]$ are well defined, $m$ -independent and dimensionless.\",\n \"In the limit $m\\\\rightarrow 0$ , dividing both sides of Eq.\",\n \"(REF ) by $\\\\lambda ^2_c$ , we get: $\\\\frac{\\\\pi J_z}{4\\\\Lambda ^2}(g-2)^2 =\\\\int _{0}^\\\\infty \\\\frac{ds}{s} \\\\sum _n f_n[0,s/\\\\Lambda ^2]\\\\delta (s-M_n^2) + O(\\\\lambda ^2_c).$ If the LHS of (REF ) is nonzero, then the RHS must contain new states with masses $M_n = {\\\\cal O}(\\\\Lambda )$ , because, by assumption, the only massless states in the theory are those of $N=8$ supergravity.Because of $N=8$ supersymmetry, new massless states would necessarily contain an additional massless spin two particle interacting with supergravity particles, in contradiction with the no go theorem in [19].\",\n \"One may worry that at energies above $M_P$ the cross section $\\\\Delta \\\\sigma $ would never be perturbative because contributions from black hole intermediate states would dominate.\",\n \"In general, these black holes carry mass $M$ , charge $Q$ and angular momentum $J$ .\",\n \"In such case, the cross section can be crudely approximated by the cross sectional area of a black hole [16]: $\\\\sigma _J \\\\sim \\\\pi r^2_+ \\\\approx 4\\\\pi G^2_N M^2-2\\\\pi G_NQ^2 - 2\\\\pi \\\\frac{J^2}{M^2}$ where $r_+ = G_N M + \\\\sqrt{G^2_NM^2 - J^2/M^2 -G_NQ^2}$ is the outer horizon of the Kerr-Newman black hole, and we assumed $J^2 + G_N M^2 Q^2 \\\\ll G^2_N M^4 $ or $J \\\\ll G_NM^2$ , since $Q^2 = e^2 \\\\sim G_N m^2$ and $m \\\\ll M$ .\",\n \"Clearly, the above estimate of the cross section is only applicable in case $r_+ \\\\sim G_N M \\\\ge 1/\\\\Lambda $ , where $\\\\Lambda $ is a cutoff of the theory (as was mentioned above $\\\\Lambda = \\\\lambda _c M_P$ and $\\\\lambda _c \\\\ll 1$ ).\",\n \"Therefore, our estimate is valid, only when $M > \\\\omega _c$ , where $ \\\\omega _c \\\\equiv M_{P}\\\\left(M_P/\\\\Lambda \\\\right) = \\\\Lambda /\\\\lambda ^2_c $ .\",\n \"In this case, the difference between the spin aligned ($J=j+1$ ) and anti-aligned ($J=j-1$ ) cross sections is: $\\\\Delta \\\\sigma _{j}(\\\\omega > \\\\omega _c) \\\\equiv \\\\sigma _{j-1} -\\\\sigma _{j+1} \\\\sim \\\\frac{8\\\\pi j}{\\\\omega ^2} \\\\ ,$ and therefore, the dispersion integral can be divided into two parts as follows: $\\\\int _{0}^{\\\\infty }\\\\frac{\\\\Delta \\\\sigma _{j}}{\\\\omega } d\\\\omega = \\\\int _{0}^{\\\\omega _c}\\\\frac{\\\\Delta \\\\sigma _{j}}{\\\\omega } d\\\\omega + \\\\int _{\\\\omega _c}^{\\\\infty }\\\\frac{\\\\Delta \\\\sigma _{j}}{\\\\omega } d\\\\omega \\\\ , {\\\\qquad } \\\\ \\\\int _{\\\\omega _c}^{\\\\infty }\\\\frac{\\\\Delta \\\\sigma _{j}}{\\\\omega } d\\\\omega \\\\sim \\\\frac{4\\\\pi j}{\\\\omega ^2_c} \\\\sim {\\\\cal O}\\\\left(\\\\frac{\\\\lambda ^2_c}{M^2_P}\\\\right) \\\\ ,$ where the second part is due to the exchange of black hole states.\",\n \"Since, in supergravity theories $e \\\\sim m/M_P$ , from Eq.\",\n \"(REF ) it follows that the black hole contribution to the gyromagnetic ratio is: $(g-2)^2_{\\\\rm B.H.}\",\n \"\\\\sim {\\\\cal O}\\\\left(\\\\lambda ^2_c\\\\right) \\\\ll 1 \\\\ .$ If $N=8$ supergravity is a self-complete theory by itself, our argument shows that, unsurprisingly, it must unitarize at the Planck scale, in which case $\\\\omega _c=M_P$ and the black hole contributions to the dispersive integral becomes ${\\\\cal O}(1)$ .\",\n \"This paper also generalizes the Weinberg-GHD sum rule in another way: it allows for off-diagonal couplings in case particles of spin $j$ and $|j\\\\pm 1|$ are degenerate in mass.\",\n \"The general sum rule for Compton scattering on a target of arbitrary nonzero spin-$j$ , with additional intermediate mass-degenerate states of spin $|j\\\\pm 1|$ and possible heavy narrow resonances, can be written as: $&\\\\frac{e^2J_z}{4m^2} \\\\left[(g_j -2)^2 - \\\\left(j+\\\\frac{3}{2}\\\\right)h_{j+1/2}^2 + \\\\left(j-\\\\frac{1}{2}\\\\right) h_{j-1/2}^2\\\\right] = [K^{\\\\dagger },K]_{ii} + \\\\frac{1}{\\\\pi }\\\\int ^{\\\\infty }_{0} \\\\frac{\\\\Delta \\\\sigma }{\\\\omega } d\\\\omega \\\\ .$ where $g_j$ is the gyromagnetic ratio for a particle of spin $j$ , and the parameters $h_{j\\\\pm 1/2}$ are couplings of the theory defined in Eq.\",\n \"(REF ).Notice, that the LHS of Eq.\",\n \"(REF ) can be also written as a commutator of some generator like $K$ .\",\n \"When applied to theories where all states are either proportional to a light mass scale $m$ or to a higher scale $M$ and where charges are proportional to the mass, such as Kaluza-Klein theories or spontaneously broken extended supergravities, this sum rule generalizes Eq.\",\n \"(REF ) to $\\\\frac{\\\\pi J_z}{4M^2}\\\\left[(g_j -2)^2 - \\\\left(j+\\\\frac{3}{2}\\\\right)h_{j+1/2}^2 + \\\\left(j-\\\\frac{1}{2}\\\\right) h_{j-1/2}^2\\\\right] =\\\\int _{0}^\\\\infty \\\\frac{ds}{s} \\\\sum _n f_n[0,s/M^2]\\\\delta (s-M_n^2) +O(\\\\lambda ^2).$ Since magnetic dipole couplings are already taken into account by the LHS of Eq.\",\n \"(REF ), the dispersive integral at low $s$ contains only quadrupole or higher multipole interactions; therefore, $f_n[0,s/M^2]$ is at most $O(s^2/M^4)$ at low $s$ .\",\n \"This implies that, if the LHS in (REF ) is nonzero, the RHS must contain contributions from the massive states, since the one-particle contribution to the dispersive integral due to states that become massless in the limit $m\\\\rightarrow 0$ vanishes.\",\n \"This paper is organized as follows: In section 2, we describe a formalism used by Weinberg to study the low energy Compton scattering.\",\n \"We also study scattering on a target of spin-1/2, with possible spin-3/2 intermediate state, and deduce some generalization of the GDH-Weinberg sum rule.\",\n \"In section 3, we consider the generalization of the sum rule to Compton scattering on a target of arbitrary spin.\",\n \"In section 4, we further generalize the sum rule, assuming that there are other states with masses different from the mass of the scatterer.\",\n \"In section 5, we discuss possible implications of our sum rule for nucleon to delta electromagnetic transitions, in the large $N_c$ limit.\"\n ],\n [\n \"Compton Scattering: Weinberg's Approach\",\n \"Define the non-diagonal vertex for the emission of a single soft photon as follows: $\\\\langle \\\\textbf {p}^{\\\\prime }, s^{\\\\prime }, \\\\sigma ^{\\\\prime } | {\\\\cal J}^{\\\\mu }(0)| \\\\textbf {p}, s, \\\\sigma \\\\rangle = \\\\Gamma _{\\\\sigma ^{\\\\prime }\\\\sigma }^{\\\\mu }(\\\\textbf {p}^{\\\\prime },\\\\textbf {p}) \\\\ ,$ where ${\\\\cal J}^{\\\\mu }$ is the conserved electromagnetic (EM) current, $| \\\\textbf {p}, s, \\\\sigma \\\\rangle $ is a single-particle state with momentum $\\\\textbf {p}$ , spin-$s$ and spin $z$ -component $\\\\sigma $ , as well as energy $E(\\\\textbf {p}) = \\\\sqrt{\\\\textbf {p}^2+m^2}$ .\",\n \"We adopt the following normalization: $\\\\langle \\\\textbf {p}^{\\\\prime }, s, \\\\sigma ^{\\\\prime } | \\\\textbf {p}, s, \\\\sigma \\\\rangle = (2\\\\pi )^32E_{\\\\textbf {p}}\\\\delta _{\\\\sigma ^{\\\\prime }\\\\sigma }\\\\delta ^{(3)}(\\\\textbf {p}^{\\\\prime }-\\\\textbf {p}) \\\\ .$ The S-matrix, describing the Compton scattering can be written as: $\\\\langle \\\\sigma ^{\\\\prime }\\\\textbf {p}^{\\\\prime };\\\\lambda ^{\\\\prime },\\\\textbf {k}^{\\\\prime } |S| \\\\sigma ,\\\\textbf {p};\\\\lambda ,\\\\textbf {k} \\\\rangle &=i\\\\epsilon ^{*^{\\\\prime }}_{\\\\nu }(\\\\textbf {k}^{\\\\prime },\\\\lambda ^{\\\\prime })\\\\epsilon _{\\\\mu }(\\\\textbf {k},\\\\lambda ) ~ (2\\\\pi )^4\\\\delta ^{(4)}(p+k-p^{\\\\prime }-k^{\\\\prime })M_{\\\\sigma ^{\\\\prime }\\\\sigma }^{\\\\nu \\\\mu }(\\\\textbf {k}; \\\\textbf {p}^{\\\\prime }, \\\\textbf {p}) \\\\ ,\\\\\\\\[5pt]M_{\\\\sigma ^{\\\\prime }\\\\sigma }^{\\\\nu \\\\mu }(\\\\textbf {k}; \\\\textbf {p}^{\\\\prime }, \\\\textbf {p}) &\\\\equiv i\\\\int d^4x~e^{ikx} \\\\left[\\\\langle \\\\textbf {p}^{\\\\prime }, s, \\\\sigma ^{\\\\prime } | T\\\\lbrace {\\\\cal J}^{\\\\nu }(0){\\\\cal J}^{\\\\mu }(x)\\\\rbrace |\\\\textbf {p}, s, \\\\sigma \\\\rangle + {\\\\rm C.T.}\",\n \"\\\\right] \\\\ ,$ where by ${\\\\rm C.T.\",\n \"}$ we mean other contact terms (seagulls), such as terms emerging from the interaction of the initial and final photon at a single point.\",\n \"We will need the pole structure of $M^{\\\\nu \\\\mu }$ ($\\\\sigma $ , $\\\\sigma ^{\\\\prime }$ indices are dropped for convenience).\",\n \"Inserting a complete set of states between the current operators, the time-ordered product can be written as: $&i\\\\int d^4x ~ e^{ikx}\\\\langle \\\\textbf {p}^{\\\\prime }, s, \\\\sigma ^{\\\\prime } |T\\\\lbrace {\\\\cal J}^{\\\\nu }(0){\\\\cal J}^{\\\\mu }(x) \\\\rbrace | \\\\textbf {p}, s, \\\\sigma \\\\rangle \\\\\\\\\\\\nonumber &= \\\\int \\\\frac{d^3p_n}{2E_n(\\\\textbf {p}_n)}\\\\sum _n \\\\biggl \\\\lbrace \\\\langle \\\\textbf {p}^{\\\\prime }, s, \\\\sigma ^{\\\\prime } | {\\\\cal J}^{\\\\nu }(0)| n \\\\rangle \\\\langle n| {\\\\cal J}^{\\\\mu }(x)| \\\\textbf {p}, s, \\\\sigma \\\\rangle \\\\frac{\\\\delta ^{(3)}(\\\\textbf {p}_n - \\\\textbf {p}-\\\\textbf {k})}{E_n(\\\\textbf {p}_n) - E(\\\\textbf {p})- \\\\omega - i\\\\epsilon } \\\\\\\\ \\\\nonumber &+\\\\langle \\\\textbf {p}^{\\\\prime }, s,\\\\sigma ^{\\\\prime } | {\\\\cal J}^{\\\\mu }(0)| n \\\\rangle \\\\langle n| {\\\\cal J}^{\\\\nu }(x)| \\\\textbf {p}, s, \\\\sigma \\\\rangle \\\\frac{\\\\delta ^{(3)}(\\\\textbf {p}_n - \\\\textbf {p}^{\\\\prime }+\\\\textbf {k})}{E_n(\\\\textbf {p}_n)-E^{\\\\prime }(\\\\textbf {p}^{\\\\prime }) + \\\\omega - i\\\\epsilon } \\\\biggr \\\\rbrace \\\\\\\\ \\\\nonumber &=\\\\sum _n \\\\biggl \\\\lbrace \\\\frac{\\\\Gamma ^{\\\\nu }(\\\\textbf {p}^{\\\\prime },\\\\textbf {p}+\\\\textbf {k})\\\\Gamma ^{\\\\mu }(\\\\textbf {p}+\\\\textbf {k},\\\\textbf {p})}{2E_n(\\\\textbf {p}+\\\\textbf {k})[E_n(\\\\textbf {p}+\\\\textbf {k})-E(\\\\textbf {p}) - \\\\omega - i\\\\epsilon ]}+ \\\\frac{\\\\Gamma ^{\\\\mu }(\\\\textbf {p}^{\\\\prime },\\\\textbf {p}^{\\\\prime }-\\\\textbf {k})\\\\Gamma ^{\\\\nu }(\\\\textbf {p}^{\\\\prime }-\\\\textbf {k},\\\\textbf {p})}{2E_n(\\\\textbf {p}^{\\\\prime }-\\\\textbf {k})[E_n(\\\\textbf {p}^{\\\\prime }-\\\\textbf {k})-E^{\\\\prime }(\\\\textbf {p}^{\\\\prime }) + \\\\omega - i\\\\epsilon ]} \\\\biggr \\\\rbrace \\\\ .$ If $|n\\\\rangle $ is a single-particle intermediate state with the same mass $m$ as the target, then both terms in the sum above have a pole at $k^{\\\\mu }=0$ .\",\n \"Using Eq.\",\n \"(REF ), in the forward scattering limit, and near the $\\\\omega =0$ pole, we get: $M^{\\\\nu \\\\mu }(\\\\textbf {k}; \\\\textbf {p}^{\\\\prime }, \\\\textbf {p}) \\\\approx &-\\\\frac{1}{2m\\\\omega }\\\\biggl [\\\\Gamma ^{\\\\nu }(\\\\textbf {p}^{\\\\prime },\\\\textbf {p}+\\\\textbf {k})\\\\Gamma ^{\\\\mu }(\\\\textbf {p}+\\\\textbf {k},\\\\textbf {p}) - \\\\Gamma ^{\\\\mu }(\\\\textbf {p}^{\\\\prime },\\\\textbf {p}^{\\\\prime }-\\\\textbf {k})\\\\Gamma ^{\\\\nu }(\\\\textbf {p}^{\\\\prime }-\\\\textbf {k},\\\\textbf {p})\\\\biggr ] + \\\\ {\\\\rm O.T.}\",\n \"\\\\ ,$ where by O.T.\",\n \"we mean other terms that do not contain poles at $\\\\omega =0$ .\",\n \"As usual, the scattering amplitude will be defined as: $f_{\\\\rm scat}(\\\\textbf {k}^{\\\\prime },\\\\lambda ^{\\\\prime }; \\\\textbf {k}, \\\\lambda ) = \\\\frac{1}{8\\\\pi m}\\\\epsilon ^{*^{\\\\prime }}_j\\\\epsilon _i M^{ji}(\\\\textbf {k}^{\\\\prime },\\\\textbf {k}, \\\\omega ) \\\\ .$ The case when $s=s^{\\\\prime }$ was considered in detail by Weinberg [3].\",\n \"Here, we are interested in the case when the intermediate state could be a particle of different spin (but of the same mass) as the target.\",\n \"To be more specific, we will consider a situation when $|s^{\\\\prime }-s|= 1$ or 0.\",\n \"In particular, when $s=1/2$ and $s^{\\\\prime }=3/2$ , the vertex function is [see also Eq.\",\n \"(REF )]:This vertex clearly implies the conservation of EM current: $k_{\\\\nu }\\\\Gamma ^{\\\\nu }_{\\\\sigma \\\\sigma ^{\\\\prime }} = 0 $ .\",\n \"$\\\\Gamma ^{\\\\nu }_{\\\\sigma \\\\sigma ^{\\\\prime }}(p^{\\\\prime },p^{\\\\prime }+k^{\\\\prime }) &= \\\\frac{ie \\\\kappa _{M}}{2m^2}\\\\epsilon ^{\\\\mu \\\\nu \\\\alpha \\\\beta }p^{\\\\prime }_{\\\\mu }k^{\\\\prime }_{\\\\alpha } \\\\bar{u}^{\\\\sigma }(p^{\\\\prime })\\\\psi ^{\\\\sigma ^{\\\\prime }}_{\\\\beta }(p^{\\\\prime }+k^{\\\\prime }) \\\\ .$ In case of spin-1/2 target, we take the intermediate states to be either spin-1/2 or spin-3/2 particle state.\",\n \"Here, we will only need to compute the part of the amplitude with spin-3/2 intermediate state, since the result in case of the spin-1/2 intermediate state is already known.\",\n \"Using Eqs.\",\n \"(REF ) and (REF ) [also Eq.\",\n \"(REF ) to sum over Rarita-Schwinger (RS) states] we compute:From Eq.\",\n \"(REF ) and current conservation: $k_{\\\\mu }M^{\\\\nu \\\\mu }(\\\\textbf {k}; \\\\textbf {p}^{\\\\prime }, \\\\textbf {p}) = 0$ , we can deduce that O.T.\",\n \"in Eq.\",\n \"(REF ) do not contribute to this part of the amplitude.\",\n \"$f_{RS}(\\\\textbf {k}^{\\\\prime },\\\\lambda ^{\\\\prime }; \\\\textbf {k}, \\\\lambda ) &= \\\\frac{i\\\\omega e^2\\\\kappa ^2_{M}}{12\\\\pi m^2} \\\\left[(\\\\vec{n}^{\\\\prime } \\\\times \\\\vec{\\\\epsilon }^{^{\\\\prime }*})\\\\times (\\\\vec{n} \\\\times \\\\vec{\\\\epsilon })\\\\right]\\\\vec{J} \\\\ .$ In the forward-scattering limit, $f_{RS} = f_{RS}(\\\\omega , \\\\lambda )$ , and we can define the following amplitude: $&g_{-}(\\\\omega ^2) \\\\equiv \\\\frac{f_{RS}(\\\\omega , +1) - f_{RS}(\\\\omega , -1)}{2\\\\omega } \\\\ ,$ in which case, it can be checked that: $&g_{-}(\\\\omega ^2 \\\\rightarrow 0) = -\\\\frac{e^2\\\\kappa _M^2}{12\\\\pi m^2}~J_z \\\\ .$ Therefore, assuming $g_-(|\\\\omega ^2| \\\\rightarrow \\\\infty ) \\\\rightarrow 0$ , the total scattering amplitude will satisfy the following generalized unsubtracted dispersion relation: $4\\\\pi ^2[f_-(0)+g_-(0)] &= \\\\frac{\\\\pi e^2}{m^2} \\\\left(\\\\kappa ^2_p - \\\\frac{1}{3}\\\\kappa _M^2\\\\right)J_z= \\\\int ^{\\\\infty }_0\\\\frac{\\\\sigma _{\\\\rm tot}(\\\\omega ^{\\\\prime },+1) - \\\\sigma _{\\\\rm tot}(\\\\omega ^{\\\\prime },-1)}{\\\\omega ^{\\\\prime }}d\\\\omega ^{\\\\prime } \\\\ ,$ where $\\\\kappa _p = (g-2)/2$ is the anomalous magnetic moment of the target (see Appendix ).\",\n \"Eq.\",\n \"(REF ) is a generalization of the GDH-Weinberg sum rule, when there is a spin-3/2 intermediate state in the Compton scattering process, that has the same mass as the spin-1/2 target.\",\n \"One of the consequences of this sum rule is that in weakly coupled theories, the gyromagnetic ratio for spin-1/2 particle can be different from its “natural” value $g=2$ .\",\n \"As can be deduced from Eq.\",\n \"(REF ), $g = 2\\\\left(1 \\\\pm \\\\frac{1}{\\\\sqrt{3}}\\\\kappa _M\\\\right) + {\\\\cal O}(e^2) \\\\ .$ The same result can be obtained using Feynman diagrams in Appendix .\"\n ],\n [\n \"Generalization to Spin-$j$ Target: First Sum Rule\",\n \"The non-diagonal transition vertex for the emission of a single soft photon can be written as in [21]: $&\\\\Gamma ^{\\\\mu }(\\\\textbf {k},\\\\textbf {0}) \\\\equiv \\\\langle \\\\textbf {k}, j^{\\\\prime }, \\\\sigma ^{\\\\prime } | {\\\\cal J}^{\\\\mu }| \\\\textbf {0}, j, \\\\sigma \\\\rangle = -2im~\\\\epsilon ^{0\\\\mu \\\\alpha \\\\beta }k_{\\\\alpha } \\\\langle j^{\\\\prime }, \\\\sigma ^{\\\\prime }|\\\\mu _{\\\\beta }|j,\\\\sigma \\\\rangle + {\\\\cal O}(\\\\omega ^2) \\\\ ,$ where $\\\\mu _i$ is the magnetic moment and $j \\\\ne j^{\\\\prime }$ .\",\n \"Applying the Wigner-Eckart theorem, the matrix elements of the magnetic moment can be parametrized as follows: $\\\\langle j,\\\\sigma |\\\\mu _3|j,\\\\sigma \\\\rangle = e g_j \\\\sigma /2m$ and $&\\\\langle j \\\\pm 1/2, \\\\sigma - 1/2|\\\\mu ^{-}|j\\\\mp 1/2,\\\\sigma +1/2\\\\rangle = \\\\pm \\\\frac{e}{2m}h_j\\\\sqrt{(j\\\\mp \\\\sigma +1)(j \\\\mp \\\\sigma )} \\\\ ,$ where $\\\\mu ^{\\\\pm } \\\\equiv \\\\mu _1 \\\\pm i\\\\mu _2$ .\",\n \"Using Eqs.\",\n \"(REF ) and (REF ), which are valid for any spin, the non-diagonal contribution to the forward scattering amplitude on a target of arbitrary spin-$j$ ($\\\\ne 0$ ) is: $&\\\\tilde{f}_{\\\\rm scat}= -\\\\frac{1}{16\\\\pi m^2\\\\omega }\\\\epsilon ^{*^{\\\\prime }}_{\\\\nu }\\\\epsilon _{\\\\mu }\\\\biggl [\\\\Gamma ^{\\\\nu \\\\dagger }(\\\\textbf {k},\\\\textbf {0})\\\\Gamma ^{\\\\mu }(\\\\textbf {k},\\\\textbf {0}) - \\\\Gamma ^{\\\\mu \\\\dagger }(-\\\\textbf {k},0)\\\\Gamma ^{\\\\nu }(-\\\\textbf {k},0)\\\\biggr ] \\\\\\\\ \\\\nonumber &= -\\\\frac{\\\\omega }{4\\\\pi }\\\\sum _{j^{\\\\prime }\\\\sigma ^{\\\\prime }}2\\\\langle j, \\\\sigma |\\\\mu _{[i}|j^{\\\\prime },\\\\sigma ^{\\\\prime }\\\\rangle \\\\langle j^{\\\\prime },\\\\sigma ^{\\\\prime } |\\\\mu _{i^{\\\\prime }]}|j, \\\\sigma \\\\rangle (\\\\vec{n} \\\\times \\\\vec{\\\\epsilon }^{^{\\\\prime }*})_{i} (\\\\vec{n} \\\\times \\\\vec{\\\\epsilon })_{i^{\\\\prime }} \\\\ .$ Taking into account that: $&\\\\langle j , \\\\sigma |\\\\mu ^{-}|j\\\\mp 1,\\\\sigma +1\\\\rangle = \\\\pm \\\\frac{e}{2m}h_{j \\\\mp 1/2}\\\\sqrt{(j \\\\mp \\\\sigma \\\\mp 1 + 1)(j \\\\mp \\\\sigma \\\\mp 1)} \\\\ , \\\\\\\\ \\\\nonumber &\\\\langle j , \\\\sigma |\\\\mu ^{+}|j\\\\pm 1,\\\\sigma -1\\\\rangle = \\\\pm \\\\frac{e}{2m}h_{j \\\\pm 1/2}\\\\sqrt{(j \\\\mp \\\\sigma \\\\pm 1 + 1)(j \\\\mp \\\\sigma \\\\pm 1)} \\\\ ,$ and applying the definition in Eq.\",\n \"(REF ) for $\\\\tilde{f}_{\\\\rm scat}(\\\\omega ,\\\\lambda )$ , we arrive at the following result in the forward scattering limit: $4\\\\pi ^2\\\\tilde{f}_-(0) &= -\\\\frac{\\\\pi e^2}{8 m^2}J_z\\\\left[ (2j+3)h_{j+1/2}^2 - (2j-1)h_{j-1/2}^2 \\\\right] \\\\ .$ Therefore, a more general form of the sum rule can be written as: $&\\\\frac{\\\\pi e^2J_z}{4m^2} \\\\left[(g_j -2)^2 - \\\\left(j+\\\\frac{3}{2}\\\\right)h_{j+1/2}^2 + \\\\left(j-\\\\frac{1}{2}\\\\right) h_{j-1/2}^2\\\\right]= \\\\int ^{\\\\infty }_0\\\\frac{\\\\Delta \\\\sigma (\\\\omega ^{\\\\prime })}{\\\\omega ^{\\\\prime }}d\\\\omega ^{\\\\prime } \\\\ .$ As before, tree-level unitarity demands: $&(g_j -2)^2 + \\\\frac{(2j-1)}{2}h_{j-1/2}^2 = \\\\frac{(2j+3)}{2}h_{j+1/2}^2 \\\\ .$ This is in agreement with the observation made, e.g., in Ref.\",\n \"[21], proposing that $g=2$ , when $h_j=0$ , for all $j$ .\",\n \"As a simple illustration, when $j=1/2$ : $(g_{1/2} - 2)^2 = 2h_{1}^2 \\\\ ,$ suggesting, $h_{1} = \\\\sqrt{\\\\frac{2}{3}}\\\\kappa _M = \\\\sqrt{2} \\\\kappa _p$ , where we recalled Eq.\",\n \"(REF ).\",\n \"Similarly, when $j=3/2$ , $&(g_{3/2} -2)^2 - 3h_{2}^2 + h_{1}^2 = 0 \\\\ .$\"\n ],\n [\n \"Generalization for Arbitrary Intermediate State: Second Sum Rule\",\n \"Consider a situation when the intermediate state in the Compton scattering process is a very narrow resonance of mass $M_n$ ($\\\\ne m$ ).\",\n \"Then, using Eqs.\",\n \"(REF ) and (REF ), near each pole the forward scattering amplitude on a target at rest becomes (see also Ref.\",\n \"[3]): $f^{(n)}(\\\\omega \\\\rightarrow \\\\omega _n, \\\\lambda ) \\\\rightarrow \\\\frac{1}{16\\\\pi m}\\\\frac{\\\\langle \\\\textbf {0}, s, \\\\sigma | \\\\vec{\\\\epsilon }^{\\\\ *}_{\\\\lambda }\\\\vec{{\\\\cal J}}|\\\\textbf {k}, s_n, \\\\sigma _n \\\\rangle \\\\langle \\\\textbf {k}, s_n, \\\\sigma _n | \\\\vec{\\\\epsilon }_{\\\\lambda }\\\\vec{J}|\\\\textbf {0}, s, \\\\sigma \\\\rangle }{\\\\left(\\\\sqrt{M^2_n + \\\\omega ^2} - m - \\\\omega \\\\right)\\\\sqrt{M^2_n + \\\\omega ^2_n}} \\\\ ,$ where $\\\\omega _n = \\\\frac{1}{2m}(M^2_n - m^2)$ , so that: $\\\\sqrt{M^2_n + \\\\omega ^2_n} = m+\\\\omega _n$ .\",\n \"Now, defining: $K_{ni} \\\\equiv \\\\sqrt{\\\\frac{m}{2E_n}}~\\\\frac{\\\\langle \\\\textbf {k}, s_n, \\\\sigma _n | {\\\\cal J}_x +i {\\\\cal J}_y| \\\\textbf {0}, s, \\\\sigma \\\\rangle }{M^2_n - m^2} \\\\ , \\\\ \\\\ \\\\ \\\\ \\\\ K_{in} \\\\equiv \\\\sqrt{\\\\frac{m}{2E_n}}~\\\\frac{\\\\langle \\\\textbf {0}, s, \\\\sigma | {\\\\cal J}_x + i{\\\\cal J}_y|\\\\textbf {k}, s_n, \\\\sigma _n \\\\rangle }{M^2_n - m^2} \\\\ ,$ the amplitude $f^{(n)}_{-}(\\\\omega ^2)$ , determined using Eq.\",\n \"(REF ), near each pole, becomes: $&f^{(n)}_{-}(\\\\omega ^2) = -\\\\frac{1}{4\\\\pi } ~\\\\frac{ \\\\omega ^2_n}{\\\\omega ^2 - \\\\omega ^2_n}\\\\left[(K_{ni})^{\\\\dagger }K_{ni} - K_{in}(K_{in})^{\\\\dagger }\\\\right] \\\\ .$ For the total forward scattering amplitude, $f^{\\\\rm tot}_-(\\\\omega ^2)$ , with an appropriate contour of integration, we will have: $\\\\frac{1}{2\\\\pi i}\\\\int _C \\\\frac{f^{\\\\rm tot}_{-}(\\\\omega ^{\\\\prime 2})}{\\\\omega ^{\\\\prime 2} - \\\\omega ^2}d\\\\omega ^{\\\\prime 2} &= \\\\frac{1}{\\\\pi }\\\\int ^{\\\\infty }_0 \\\\frac{\\\\mathop {\\\\rm Im}f^{\\\\rm tot}_-(\\\\omega ^{\\\\prime 2})}{\\\\omega ^{\\\\prime 2} - \\\\omega ^2 - i\\\\epsilon } d\\\\omega ^{\\\\prime 2} \\\\ .$ In the case $\\\\omega ^2=0$ and when the integration contour encircles all the particle poles we obtain: $&4\\\\pi [f_{-}(0)+g_-(0)] - [K^{\\\\dagger },K]_{ii} = \\\\frac{1}{\\\\pi }\\\\int ^{\\\\infty }_0 \\\\frac{\\\\Delta \\\\sigma (\\\\omega ^{\\\\prime })}{\\\\omega ^{\\\\prime }} d\\\\omega ^{\\\\prime } \\\\ , \\\\\\\\[5pt] \\\\nonumber &[K^{\\\\dagger },K]_{ii} \\\\equiv \\\\sum _n \\\\left[(K_{ni})^{\\\\dagger }K_{ni} - K_{in}(K_{in})^{\\\\dagger }\\\\right]^{\\\\omega ^2_n}_0 \\\\ .$ It is also interesting to consider the case when $\\\\omega ^2 = \\\\omega ^2_k$ in Eq.\",\n \"(REF ).\",\n \"Then one obtains a relation between the derivatives of the matrix elements $K_{ni}$ and the dispersion integral.\",\n \"Notice that $f_-(0)$ and $g_-(0)$ can be expressed in terms of commutator of some generators like $K_{ni}$ , in which case the left hand side of (REF ) would take a more compact form.\",\n \"Summarizing, when all other intermediate states in the Compton scattering have masses (and spins) different from the mass (or spin) of the scatterer, we have the second sum rule: $\\\\frac{e^2 J_z}{4m^2}(g-2)^2 &= \\\\frac{1}{\\\\pi }\\\\int ^{\\\\infty }_0 \\\\frac{\\\\Delta \\\\sigma }{\\\\omega } d\\\\omega + [K^{\\\\dagger },K]_{ii} \\\\ge 0 \\\\ .$ Since the left hand side of Eq.\",\n \"(REF ) is always non negative, so should be the right hand side.\",\n \"Ignoring the integral, we can rewrite Eq.\",\n \"(REF ) at leading order in $e^2$ , as follows: $g-2 &= \\\\pm \\\\frac{2m}{e}\\\\sqrt{\\\\frac{[K^{\\\\dagger },K]_{ii}}{J_z}} \\\\ .$ This means that the gyromagnetic ratio may receive corrections even at tree level, if $[K^{\\\\dagger },K]_{ii} \\\\ge 0$ .\",\n \"In supergravity theories, $e \\\\sim m/M_P$ , and we expect $g-2$ to be independent on $m$ , therefore, $[K^{\\\\dagger },K]_{ii} \\\\sim e^2/m^2$ , suggesting that $g-2 \\\\sim {\\\\cal O}(1)$ , as for spontaneously broken $N=8$ supergravity [7].\",\n \"Finally, allowing for an intermediate state, in a Compton scattering off of a target of spin $j$ , to be any narrow resonance of arbitrary mass and spin, we arrive at the most general sum rule: $&\\\\frac{e^2J_z}{4m^2} \\\\left[(g_j -2)^2 - \\\\left(j+\\\\frac{3}{2}\\\\right)h_{j+1/2}^2 + \\\\left(j-\\\\frac{1}{2}\\\\right) h_{j-1/2}^2\\\\right] = [K^{\\\\dagger },K]_{ii} + \\\\frac{1}{\\\\pi }\\\\int ^{\\\\infty }_{0} \\\\frac{\\\\Delta \\\\sigma }{\\\\omega } d\\\\omega \\\\ .$ This sum rule can be extended by taking into account other global charges of the theory at hand.\",\n \"We will not consider such an extension here, since it depends on the specific form of the theory.\",\n \"Instead, below we will study a particular case, namely the example of the $N \\\\rightarrow \\\\Delta $ magnetic transition.\"\n ],\n [\n \"Some Applications to QCD\",\n \"Application of Eq.\",\n \"(REF ) or (REF ) for $j=1/2$ , directly to Compton scattering on a nucleon ($N(938)$ ) with a possible delta ($\\\\Delta (1232)$ ) intermediate state would be wrong for two main reasons.\",\n \"First of all, the mass difference $\\\\delta $ between $\\\\Delta $ and $N$ is around $300 \\\\ {\\\\rm MeV}$ and the forward scattering amplitude, in the $\\\\omega \\\\rightarrow 0 $ limit, would behave as $\\\\omega ^2/(\\\\delta - \\\\omega ) \\\\sim {\\\\cal O}(\\\\omega ^2)$ .\",\n \"It is only when $\\\\delta \\\\rightarrow 0$ that the scattering amplitude becomes of order ${\\\\cal O}(\\\\omega )$ and contributes comparably to the forward amplitude.\",\n \"The second reason is that these baryons are strongly coupled systems and loop corrections may be significant.\",\n \"However, we can still make a formal use of the first sum rule if we work in the limit when $N_c \\\\rightarrow \\\\infty $ , in which case $\\\\delta \\\\sim {\\\\cal O}(1/N_c)$ (see, e.g.\",\n \"Refs.\",\n \"[24], [25]).\",\n \"We also need to take $e \\\\rightarrow 0$ and $N_c \\\\rightarrow \\\\infty $ limits in such a way that if $\\\\Delta \\\\sigma \\\\sim N^k_c$ , then $e^2N^{k+2}_c \\\\ll 1$ , and the dispersive integral can be safely ignored.\",\n \"Comparing Eqs.\",\n \"(REF ) and (REF ) one can deduce that: $\\\\mu _{p\\\\rightarrow \\\\Delta ^+} = g_M(0) = \\\\sqrt{\\\\frac{2}{3}}~\\\\kappa _M = \\\\pm \\\\sqrt{2} \\\\kappa _p \\\\ ,$ where we work in units of nuclear Bohr magneton, and we took into account the appropriate isospin factor $T^3$ corresponding to the $p \\\\rightarrow \\\\Delta ^+$ transition.\",\n \"Taking the experimental value of the proton's anomalous magnetic moment, $\\\\kappa _p \\\\approx 1.79$ , we will obtain $\\\\mu _{p\\\\rightarrow \\\\Delta ^+} \\\\approx \\\\pm 2.54$ .\",\n \"Unfortunately, nothing can be said about the sign, since our sum rule relates the squares of the couplings.\",\n \"Nevertheless, in absolute value, our result for $\\\\mu _{p\\\\rightarrow \\\\Delta ^+}$ is not very far from similar ones, obtained within the framework of other models, like the Skyrme model [25], and holographic QCD [26], which respectively give the values $2.3$ and $2.58$ (the experimental value is: $\\\\mu _{p\\\\rightarrow \\\\Delta ^+} =3.46 \\\\pm 0.03$ [27]).\",\n \"In all models of baryons in the large-$N_c$ limit, baryons are finite size objects, whose sizes ($R$ ) do not scale with $N_c$ , while their masses ($M$ ) do.\",\n \"As in the case of the GDH-Weinberg sum rule [28], in the zero radius limit, when $M R \\\\rightarrow 0$ , we expect the magnetic transition moment to approach its canonical value (REF ).\",\n \"However, in the Skyrme model $R$ is fixed and $M R \\\\sim {\\\\cal O}(N_c) $ .\",\n \"This is not surprising, since the Lagrangian of the Skyrme model [25] is known to behave badly at high energies.\",\n \"Now, we want to apply the second sum rule to the same system.\",\n \"In this case, we take the physical masses of $N$ and $\\\\Delta $ and only assume that these are narrow resonances.\",\n \"Although in this case the scattering amplitude with $\\\\Delta $ intermediate state vanishes when $\\\\omega \\\\rightarrow 0$ , it contributes to the dispersion integral, when $\\\\omega \\\\rightarrow \\\\omega _n$ .\",\n \"Employing Ref.\",\n \"[23], we obtain: $&\\\\langle \\\\textbf {k}, 3/2, \\\\sigma ^{\\\\prime } | \\\\vec{\\\\epsilon }_{\\\\lambda }\\\\vec{{\\\\cal J}}| \\\\textbf {0}, 1/2, \\\\sigma \\\\rangle = \\\\sqrt{\\\\frac{3m_N}{2M_{\\\\Delta }}}~\\\\frac{ie \\\\mu _{p\\\\rightarrow \\\\Delta ^+} }{(m_N+M_{\\\\Delta })}\\\\omega ~\\\\bar{\\\\psi }^{\\\\sigma ^{\\\\prime }}_r(\\\\textbf {k})u^{\\\\sigma }(\\\\textbf {0})~(\\\\vec{n} \\\\times \\\\vec{\\\\epsilon }_{\\\\lambda })_r \\\\ .$ After straightforward computations, the second sum rule gives: $\\\\kappa _p = \\\\pm 2 \\\\frac{m_N^2}{(m_N+M_{\\\\Delta })}~\\\\frac{\\\\mu _{p\\\\rightarrow \\\\Delta ^+}}{\\\\sqrt{m^2_N + M^2_{\\\\Delta }}} \\\\ .$ Numerically, $\\\\mu _{p\\\\rightarrow \\\\Delta ^+} \\\\approx \\\\pm 1.91\\\\kappa _p$ ; therefore, taking $\\\\kappa _p=1.79$ , we obtain $\\\\mu _{p\\\\rightarrow \\\\Delta ^+} \\\\approx \\\\pm 3.42$ .\",\n \"In absolute value this result is coincidentally close to the experimental value: $\\\\mu _{p\\\\rightarrow \\\\Delta ^+} =3.46 \\\\pm 0.03$ [27].\",\n \"Consider another example (also relevant for large-$N_c$ QCD), when the target has spin-1/2 and the intermediate particle is an excited state with the same spin.\",\n \"Then, using Ref.\",\n \"[29], the general form of the transition matrix element can be written as: $&\\\\sqrt{\\\\frac{m}{2E_n}}~\\\\frac{\\\\langle \\\\textbf {k}, 1/2^*,\\\\sigma ^{\\\\prime } |{\\\\cal J}_x + i \\\\lambda {\\\\cal J}_y| \\\\textbf {0}, 1/2,\\\\sigma \\\\rangle }{M^2_n - m^2}= 2e G_{*} \\\\omega ~\\\\frac{m}{(M_n-m)}\\\\sqrt{\\\\frac{mM_n}{M_n^2+m^2}}~(\\\\sigma _1 + i\\\\lambda \\\\sigma _2)_{\\\\sigma \\\\sigma ^{\\\\prime }}\\\\ ,$ where $G_*$ is some dimensionful coupling.\",\n \"Direct computations show that $[K^{\\\\dagger },K]_{ii} < 0$ , which means that for this theory with excited states to be unitary at tree level, we need $G_*=0$ .\",\n \"However, this conclusion might change if we include intermediate states with spin-3/2.\",\n \"This work is supported by NSF grant PHY-0758032 and by ERC Advanced Investigator Grant n.226455 Supersymmetry, Quantum Gravity and Gauge Fields (Superfields).\",\n \"We would like to thank Sergio Ferrara, Gregory Gabadadze and Sergei Dubovsky for interesting discussions.\"\n ],\n [\n \"Low Energy Compton Amplitude for a Spin-1/2 Target\",\n \"Consider the Compton scattering process: $N\\\\lbrace p,\\\\sigma \\\\rbrace + \\\\gamma \\\\lbrace k,\\\\epsilon _{\\\\mu }(k)\\\\rbrace \\\\rightarrow N\\\\lbrace p^{\\\\prime },\\\\sigma ^{\\\\prime }\\\\rbrace + \\\\gamma \\\\lbrace k^{\\\\prime },\\\\epsilon ^{\\\\prime }_{\\\\nu }(k^{\\\\prime })\\\\rbrace \\\\ ,$ where $p$ and $p^{\\\\prime }$ are initial and final 4-momenta of scatterer (which is a spin-1/2 particle), $\\\\sigma $ and $\\\\sigma ^{\\\\prime }$ are projections of initial and final spin along the $z$ -direction.\",\n \"Analogously, $k$ , $\\\\epsilon _{\\\\mu }(k)$ and $k^{\\\\prime }$ , $\\\\epsilon ^{\\\\prime }_{\\\\nu }(k^{\\\\prime })$ are the 4-momenta and polarizations of initial and final photons.\",\n \"We take the external particles to be on-shell, that is: $k_0 = |\\\\textbf {k}|$ , $k_0^{\\\\prime } = |\\\\textbf {k}^{\\\\prime }|$ , $p_0 = \\\\sqrt{\\\\textbf {p}^2 + m^2}$ and $p^{\\\\prime }_0 = \\\\sqrt{\\\\textbf {p}^{\\\\prime 2} + m^2}$ .\",\n \"The scattering amplitude corresponding to process in (REF ) is: $\\\\nonumber {\\\\cal M} &= -e^2\\\\bar{u}(p^{\\\\prime },\\\\sigma ^{\\\\prime })\\\\biggl \\\\lbrace \\\\Gamma ^{\\\\nu }(p^{\\\\prime },p+k)\\\\epsilon ^{*^{\\\\prime }}_{\\\\nu }(k^{\\\\prime })\\\\frac{p\\\\hspace{-7.0pt}\\\\diagup + k\\\\hspace{-7.0pt}\\\\diagup + m}{(p+k)^2 - m^2}\\\\Gamma ^{\\\\mu }(p+k,p)\\\\epsilon _{\\\\mu }(k) \\\\\\\\&+ \\\\Gamma ^{\\\\mu }(p^{\\\\prime },p-k^{\\\\prime })\\\\epsilon _{\\\\mu }(k)\\\\frac{p\\\\hspace{-7.0pt}\\\\diagup - k\\\\hspace{-7.0pt}\\\\diagup ^{\\\\prime } + m}{(p-k^{\\\\prime })^2 - m^2}\\\\Gamma ^{\\\\nu }(p-k^{\\\\prime },p)\\\\epsilon ^{*^{\\\\prime }}_{\\\\nu }(k^{\\\\prime })\\\\biggr \\\\rbrace u(p,\\\\sigma ) \\\\ , \\\\\\\\[8pt]&{\\\\qquad }{\\\\qquad } \\\\Gamma ^{\\\\mu }(p_2,p_1) \\\\equiv \\\\gamma ^{\\\\mu }F_D(q^2) + \\\\frac{i\\\\sigma ^{\\\\mu \\\\nu }}{2m}q_{\\\\nu }F_P(q^2) \\\\ ,$ where $e$ is the electric charge of scatterer, $m$ is its mass, and $q = p_2 -p_1$ .\",\n \"Using the notations of Peskin & Schroeder, $a\\\\hspace{-7.0pt}\\\\diagup \\\\equiv a_{\\\\mu }\\\\gamma ^{\\\\mu }$ , $\\\\sigma ^{\\\\mu \\\\nu } =i[\\\\gamma ^{\\\\mu },\\\\gamma ^{\\\\nu }]/2$ .\",\n \"Here, $F_D(q^2)$ and $F_P(q^2)$ are Dirac and Pauli form factors, which for $q^2=0$ are: $F_D(0) = 1$ and $F_P(0)=\\\\kappa _p$ , where $e\\\\kappa _p/(2m)$ is the anomalous magnetic moment of the scatterer.\",\n \"The latter arises from the Pauli Lagrangian: ${\\\\cal L}_{\\\\rm P} = -\\\\frac{e\\\\kappa _p}{4m}~\\\\bar{u}~\\\\sigma ^{\\\\mu \\\\nu }u ~F_{\\\\mu \\\\nu } \\\\ .$ In what follows, we will adopt a `gauge' in which the initial and final photon are transverely polarized in the laboratory frame.\",\n \"That is, we choose: $\\\\epsilon (k) \\\\cdot k = \\\\epsilon ^{*^{\\\\prime }}(k^{\\\\prime }) \\\\cdot k^{\\\\prime } = \\\\epsilon (k) \\\\cdot p = \\\\epsilon ^{*^{\\\\prime }}(k^{\\\\prime }) \\\\cdot p = 0 \\\\ ,$ where $p=(m,\\\\textbf {0})$ , implying that $\\\\epsilon ^0 = \\\\epsilon ^{\\\\prime 0} = 0$ and $\\\\vec{\\\\epsilon }~\\\\vec{k} = \\\\vec{\\\\epsilon }^{~^{\\\\prime }}\\\\vec{k}^{\\\\prime } = 0$ .\",\n \"We also adopt the following normalization: $\\\\epsilon (k) \\\\cdot \\\\epsilon ^*(k) = \\\\epsilon ^{\\\\prime }(k^{\\\\prime })\\\\cdot \\\\epsilon ^{*^{\\\\prime }}(k^{\\\\prime }) = -1$ .\",\n \"Using the Dirac equation: $(p\\\\hspace{-7.0pt}\\\\diagup -m)u(p)=0$ , and after some simplifications, we can rewrite the amplitude as follows: ${\\\\cal M} &=\\\\frac{e^2\\\\mu }{2m}\\\\bar{u}(p^{\\\\prime },\\\\sigma ^{\\\\prime })\\\\biggl [ \\\\left(\\\\epsilon \\\\hspace{-7.0pt}\\\\diagup ^{\\\\prime } + \\\\mu ^{\\\\prime } \\\\epsilon \\\\hspace{-7.0pt}\\\\diagup ^{\\\\prime }k\\\\hspace{-7.0pt}\\\\diagup ^{\\\\prime }\\\\right)\\\\left(\\\\frac{\\\\epsilon \\\\hspace{-7.0pt}\\\\diagup k\\\\hspace{-7.0pt}\\\\diagup }{\\\\omega } + \\\\frac{1-\\\\mu }{\\\\mu }\\\\epsilon \\\\hspace{-7.0pt}\\\\diagup \\\\right)+ \\\\left(\\\\epsilon \\\\hspace{-7.0pt}\\\\diagup - \\\\mu ^{\\\\prime } \\\\epsilon \\\\hspace{-7.0pt}\\\\diagup k\\\\hspace{-7.0pt}\\\\diagup \\\\right)\\\\left(\\\\frac{\\\\epsilon \\\\hspace{-7.0pt}\\\\diagup ^{\\\\prime }k\\\\hspace{-7.0pt}\\\\diagup ^{\\\\prime }}{\\\\omega ^{\\\\prime }} + \\\\frac{1-\\\\mu }{\\\\mu }\\\\epsilon \\\\hspace{-7.0pt}\\\\diagup ^{\\\\prime }\\\\right)\\\\biggr ] u(p,\\\\sigma ) \\\\ ,$ where $\\\\omega \\\\equiv pk/m $ , $\\\\omega ^{\\\\prime } \\\\equiv pk^{\\\\prime }/m$ , $\\\\epsilon \\\\hspace{-7.0pt}\\\\diagup ^{\\\\prime } \\\\equiv \\\\gamma ^{\\\\mu }\\\\epsilon ^{*^{\\\\prime }}_{\\\\mu }$ , $\\\\mu ^{\\\\prime } \\\\equiv \\\\kappa _p/(2m)$ and $\\\\mu = 1 + 2m\\\\mu ^{\\\\prime }$ is the magnetic moment.\",\n \"Since we work in the frame where $p_0=m$ and $\\\\textbf {p}=\\\\textbf {0}$ , we have: $\\\\omega = k_0$ and $\\\\omega ^{\\\\prime } = k_0^{\\\\prime }$ .\",\n \"The initial state is $u^T(p,\\\\sigma ) = \\\\sqrt{m}\\\\left\\\\lbrace \\\\xi ,\\\\xi \\\\right\\\\rbrace $ , where $\\\\xi $ is a spinor such that $\\\\xi ^{\\\\dagger }\\\\xi = 1$ .\",\n \"Similarly, for $|\\\\textbf {p}^{\\\\prime }| \\\\ll m $ , $\\\\bar{u}(p^{\\\\prime },\\\\sigma ^{\\\\prime }) = \\\\sqrt{m}\\\\left\\\\lbrace \\\\xi ^{\\\\prime \\\\dagger }\\\\left(1 + \\\\frac{1}{2m}\\\\vec{\\\\sigma } \\\\textbf {p}^{\\\\prime }\\\\right), \\\\xi ^{\\\\prime \\\\dagger }\\\\left( 1 - \\\\frac{1}{2m}\\\\vec{\\\\sigma } \\\\textbf {p}^{\\\\prime }\\\\right)\\\\right\\\\rbrace \\\\ .$ Taking $\\\\omega = \\\\omega ^{\\\\prime }$ and defining $\\\\vec{n} \\\\equiv \\\\vec{k}/\\\\omega $ and $\\\\vec{n}^{\\\\prime } \\\\equiv \\\\vec{k}^{\\\\prime }/\\\\omega ^{\\\\prime }$ , we can perform direct matrix and vector multiplications to arrive to the final result.\",\n \"After tedious but straightforward calculations, the answer can be written in a familiar form [20]: $f_{\\\\rm scat} &= \\\\frac{e^2}{4\\\\pi m}\\\\biggl \\\\lbrace -\\\\vec{\\\\epsilon }^{^{\\\\prime }*}\\\\vec{\\\\epsilon } +\\\\frac{i\\\\kappa _p}{m}\\\\omega ~\\\\vec{J}~(\\\\vec{\\\\epsilon }^{^{\\\\prime }*} \\\\times \\\\vec{\\\\epsilon }) - \\\\frac{i\\\\mu ^2}{m}\\\\omega ~\\\\vec{J}~\\\\left[(\\\\vec{n}^{\\\\prime } \\\\times \\\\vec{\\\\epsilon }^{^{\\\\prime }*}) \\\\times (\\\\vec{n} \\\\times \\\\vec{\\\\epsilon })\\\\right] \\\\\\\\ \\\\nonumber &- \\\\frac{i\\\\mu }{2m}~\\\\omega ~\\\\vec{J}~\\\\left[\\\\left\\\\lbrace \\\\vec{n}(\\\\vec{n}\\\\times \\\\vec{\\\\epsilon }) + (\\\\vec{n}\\\\times \\\\vec{\\\\epsilon })\\\\vec{n}\\\\right\\\\rbrace \\\\vec{\\\\epsilon }^{^{\\\\prime }*}-\\\\lbrace \\\\vec{n}^{\\\\prime }(\\\\vec{n}^{\\\\prime }\\\\times \\\\vec{\\\\epsilon }^{^{\\\\prime }*}) + (\\\\vec{n}^{\\\\prime }\\\\times \\\\vec{\\\\epsilon }^{^{\\\\prime }*})\\\\vec{n}^{\\\\prime }\\\\rbrace \\\\vec{\\\\epsilon }\\\\right]\\\\biggr \\\\rbrace \\\\ ,$ where $f_{\\\\rm scat} \\\\equiv {\\\\cal M}/(8\\\\pi m) $ and $\\\\vec{J}\\\\equiv \\\\delta _{\\\\sigma ^{\\\\prime }\\\\sigma } \\\\xi ^{\\\\dagger }_{\\\\sigma } \\\\vec{\\\\sigma } \\\\xi _{\\\\sigma }/2$ , with $\\\\vec{J}$ being the spin of the scatterer (when $\\\\sigma =\\\\sigma ^{\\\\prime }$ ).\"\n ],\n [\n \"General Properties of Rarita-Schwinger Field\",\n \"The Lagrangian for a free massive spin-3/2, Rarita-Schwinger (RS) field, $\\\\psi _{\\\\mu }$ , can be written as: ${\\\\cal L}_{\\\\rm RS} = \\\\bar{\\\\psi }_{\\\\mu }\\\\left(i \\\\gamma ^{\\\\mu \\\\nu \\\\rho }\\\\partial _{\\\\rho } - m \\\\gamma ^{\\\\mu \\\\nu } \\\\right)\\\\psi _{\\\\nu } \\\\ ,$ where $\\\\gamma ^{\\\\mu \\\\nu } \\\\equiv \\\\gamma ^{[\\\\mu }\\\\gamma ^{\\\\nu ]}$ and $\\\\gamma ^{\\\\mu \\\\nu \\\\rho } \\\\equiv \\\\gamma ^{[\\\\mu }\\\\gamma ^{\\\\nu }\\\\gamma ^{\\\\rho ]}$ .\",\n \"The equations of motion that follow from this Lagrangian can be equivalently written as Dirac equations along with the transversality and tracelessness constraints: $\\\\left(i\\\\partial \\\\hspace{-7.0pt}\\\\diagup - m\\\\right)\\\\psi _{\\\\mu } = 0 \\\\ , \\\\ \\\\ \\\\ \\\\partial ^{\\\\mu }\\\\psi _{\\\\mu } = 0 \\\\ , \\\\ \\\\ \\\\ \\\\gamma ^{\\\\mu }\\\\psi _{\\\\mu } = 0 \\\\ .$ These constraints guarantee that among 16 independent components of RS field only $2s+1 = 4$ physical degrees of freedom will propagate.\",\n \"The wave function for the RS field can be written as a product of massive spin-1 and spin-1/2 polarizations: $e_{\\\\mu }(\\\\vec{p},\\\\textit {n})$ with $\\\\textit {n}=-1,0,1$ , and $u(\\\\vec{p},\\\\sigma )$ with ($\\\\sigma =\\\\pm 1/2$ ), respectively.\",\n \"More specifically (see, e.g.\",\n \"Refs.\",\n \"[30]): $\\\\psi _{\\\\mu }(\\\\vec{p},r) &= \\\\sum _{\\\\sigma ,\\\\textit {n}} \\\\langle \\\\left(\\\\frac{1}{2},\\\\sigma \\\\right)\\\\left(1,\\\\textit {n}\\\\right)|\\\\left(\\\\frac{3}{2},r\\\\right)\\\\rangle ~u(\\\\vec{p},\\\\sigma )~e_{\\\\mu }(\\\\vec{p},\\\\textit {n}) \\\\ ,$ where $u(\\\\vec{p},\\\\sigma )$ satisfies equations: $(p\\\\hspace{-7.0pt}\\\\diagup -m)u(\\\\vec{p},\\\\sigma )=0$ and $\\\\hat{\\\\sigma }_z u(\\\\vec{p},\\\\sigma ) = \\\\sigma u(\\\\vec{p},\\\\sigma ) $ .\",\n \"The polarization vectors satisfy the following normalization and transversality conditions: $&e^*_{\\\\mu }(\\\\vec{p},\\\\textit {n})e^{\\\\mu }(\\\\vec{p},\\\\textit {n}^{\\\\prime }) = -\\\\delta _{\\\\textit {n}\\\\textit {n}^{\\\\prime }} \\\\ , {\\\\qquad }p^{\\\\mu }e_{\\\\mu }(\\\\vec{p},\\\\textit {n}) = p^{\\\\mu }e^*_{\\\\mu }(\\\\vec{p},\\\\textit {n}) = 0 \\\\ .$ Substituting the values of the Clebsch-Gordan coefficients in Eq.\",\n \"(REF ), that are proportional to $\\\\delta _{r,\\\\textit {n}+\\\\sigma }$ , we will get: $\\\\psi _{\\\\alpha }^{\\\\pm 3/2} &= e_{\\\\alpha }^{\\\\pm 1}u^{\\\\pm 1/2} \\\\ , {\\\\qquad }\\\\psi _{\\\\alpha }^{\\\\pm 1/2} = \\\\frac{1}{\\\\sqrt{3}}\\\\left(e_{\\\\alpha }^{\\\\pm 1}u^{\\\\mp 1/2} + \\\\sqrt{2}e_{\\\\alpha }^0 u^{\\\\pm 1/2}\\\\right) \\\\ ,$ where $\\\\psi _{\\\\alpha }(\\\\vec{p},r) \\\\equiv \\\\psi _{\\\\alpha }^{r}$ , $u(\\\\vec{p},\\\\sigma ) \\\\equiv u^{\\\\sigma }$ and $e_{\\\\mu }(\\\\vec{p},\\\\textit {n}) \\\\equiv e_{\\\\mu }^\\\\textit {n}$ .\",\n \"These solutions satisfy the Dirac equation, as well as transversality and tracelessness constraints.\",\n \"Moreover, the RS states are normalized as: $&\\\\bar{\\\\psi }_{\\\\alpha }(\\\\vec{p},r)\\\\psi ^{\\\\alpha }(\\\\vec{p},r^{\\\\prime }) = -2m \\\\delta _{rr^{\\\\prime }} \\\\ , {\\\\qquad }\\\\bar{\\\\psi }_{\\\\alpha }(\\\\vec{p},r)\\\\gamma _{\\\\mu }\\\\psi ^{\\\\alpha }(\\\\vec{p},r^{\\\\prime }) = -2p_{\\\\mu } \\\\delta _{rr^{\\\\prime }} \\\\ .$ In what follows it would be useful to note that: $P_{\\\\mu \\\\nu }(p) &\\\\equiv \\\\sum _{r}\\\\psi _{\\\\nu }(\\\\vec{p},r)\\\\bar{\\\\psi }_{\\\\mu }(\\\\vec{p},r) = -(p\\\\hspace{-7.0pt}\\\\diagup + m)\\\\Pi _{\\\\mu \\\\nu }(p) \\\\ , \\\\\\\\ \\\\nonumber \\\\Pi _{\\\\mu \\\\nu }(p) &\\\\equiv \\\\left(g_{\\\\mu \\\\nu } -\\\\frac{p_{\\\\mu }p_{\\\\nu }}{m^2}\\\\right) - \\\\frac{1}{3}\\\\left(\\\\gamma _{\\\\mu } -\\\\frac{p_{\\\\mu }}{m}\\\\right)\\\\left(\\\\gamma _{\\\\nu } +\\\\frac{p_{\\\\nu }}{m}\\\\right) \\\\ .$ In the chiral representation of the $\\\\gamma $ -matrices (as in Peskin & Schroeder), it can be checked that, when $p = (m,0,0,0)$ , the solutions for $u^{\\\\sigma }$ and $e_{\\\\mu }^\\\\textit {n}$ are: $&u^{-1/2} = \\\\sqrt{m}(0,1,0,1)^T \\\\ , \\\\ \\\\ \\\\ \\\\ u^{+1/2} = \\\\sqrt{m}(1,0,1,0)^T \\\\ , \\\\\\\\ \\\\nonumber &e^+_{\\\\mu } = \\\\frac{1}{\\\\sqrt{2}}(0,1,i,0) \\\\ , \\\\ \\\\ \\\\ e^-_{\\\\mu } = -\\\\frac{1}{\\\\sqrt{2}}(0,1,-i,0) \\\\ , \\\\ \\\\ \\\\ e^0_{\\\\mu } = (0,0,0,-1) \\\\ .$ Writing the quantized RS field as, $&\\\\Psi _{\\\\mu }(x) = \\\\int \\\\frac{d^3p}{(2\\\\pi )^3}\\\\frac{1}{2E_{\\\\textbf {p}}}\\\\sum _{\\\\lambda }\\\\left\\\\lbrace e^{-ipx}\\\\psi _{\\\\mu }(\\\\vec{p},\\\\lambda )a_{\\\\vec{p},\\\\lambda } + e^{ipx}\\\\psi ^C_{\\\\mu }(\\\\vec{p},\\\\lambda )a^{\\\\dagger }_{\\\\vec{p},\\\\lambda }\\\\right\\\\rbrace \\\\ , \\\\\\\\&\\\\lbrace a_{\\\\vec{p},\\\\lambda }, a^{\\\\dagger }_{\\\\vec{p}^{\\\\prime },\\\\lambda ^{\\\\prime }} \\\\rbrace = (2\\\\pi )^32E_{\\\\textbf {p}}\\\\delta _{\\\\lambda \\\\lambda ^{\\\\prime }}\\\\delta (\\\\textbf {p}-\\\\textbf {p}^{\\\\prime }) \\\\ ,{\\\\qquad }\\\\lbrace a_{\\\\vec{p},\\\\lambda }, a_{\\\\vec{p}^{\\\\prime },\\\\lambda ^{\\\\prime }} \\\\rbrace = \\\\lbrace a^{\\\\dagger }_{\\\\vec{p},\\\\lambda }, a^{\\\\dagger }_{\\\\vec{p}^{\\\\prime },\\\\lambda ^{\\\\prime }} \\\\rbrace = 0 \\\\ ,$ and using Eq.\",\n \"(REF ), it can be deduced that: $&\\\\langle 0| \\\\Psi _{\\\\nu }(x) \\\\bar{\\\\Psi }_{\\\\mu }(y) |0 \\\\rangle = \\\\int \\\\frac{d^3p}{(2\\\\pi )^3}\\\\frac{1}{2E_{\\\\textbf {p}}}\\\\sum _{\\\\lambda }\\\\psi _{\\\\nu }(\\\\vec{p},\\\\lambda )\\\\bar{\\\\psi }_{\\\\mu }(\\\\vec{p},\\\\lambda ) e^{-ip(x-y)} \\\\\\\\ \\\\nonumber &=-(i\\\\partial \\\\hspace{-7.0pt}\\\\diagup _x+m)\\\\Pi _{\\\\mu \\\\nu }(i\\\\partial _x) \\\\int \\\\frac{d^3p}{(2\\\\pi )^3}\\\\frac{1}{2E_{\\\\textbf {p}}}e^{-ip(x-y)} \\\\ , \\\\\\\\&\\\\langle 0| \\\\bar{\\\\Psi }_{\\\\mu }(y)\\\\Psi _{\\\\nu }(x) |0 \\\\rangle = \\\\int \\\\frac{d^3p}{(2\\\\pi )^3}\\\\frac{1}{2E_{\\\\textbf {p}}}\\\\sum _{\\\\lambda }\\\\psi ^C_{\\\\nu }(\\\\vec{p},\\\\lambda )\\\\bar{\\\\psi }^C_{\\\\mu }(\\\\vec{p},\\\\lambda ) e^{-ip(y-x)} \\\\\\\\ \\\\nonumber &=(i\\\\partial \\\\hspace{-7.0pt}\\\\diagup _x+m)\\\\Pi _{\\\\mu \\\\nu }(i\\\\partial _x) \\\\int \\\\frac{d^3p}{(2\\\\pi )^3}\\\\frac{1}{2E_{\\\\textbf {p}}}e^{-ip(y-x)} \\\\ .$ Since the Feynman propagator is defined as $S^F_{\\\\mu \\\\nu }(x-y) \\\\equiv \\\\langle 0 |T\\\\Psi _{\\\\nu }(x) \\\\bar{\\\\Psi }_{\\\\mu }(y) | 0 \\\\rangle $ , we have: $&S^F_{\\\\mu \\\\nu }(x-y) = -(i\\\\partial \\\\hspace{-7.0pt}\\\\diagup _x+m)\\\\Pi _{\\\\mu \\\\nu }(i\\\\partial _x) D_F(x-y) \\\\ , \\\\\\\\&D_F(x-y) \\\\equiv \\\\int \\\\frac{d^4p}{(2\\\\pi )^4}~\\\\frac{i}{p^2-m^2 +i\\\\epsilon } e^{-ip(x-y)} \\\\ ,$ where $D_F(x-y)$ is the Feynman propagator of a free scalar field.\",\n \"More explicitly, $S^F_{\\\\alpha \\\\beta }(p) &= -i\\\\frac{\\\\left(p\\\\hspace{-7.0pt}\\\\diagup + m\\\\right)}{p^2-m^2 + i\\\\epsilon }\\\\left[g_{\\\\alpha \\\\beta } - \\\\frac{1}{3}\\\\gamma _{\\\\alpha }\\\\gamma _{\\\\beta } - \\\\frac{2p_{\\\\alpha }p_{\\\\beta }}{3m^2}-\\\\frac{\\\\left(\\\\gamma _{\\\\alpha }p_{\\\\beta }- \\\\gamma _{\\\\beta }p_{\\\\alpha }\\\\right)}{3m}\\\\right]\\\\ .$\"\n ],\n [\n \"First Sum Rule for Spin-1/2 Target: Alternative Approach\",\n \"The most general form of the vertex function, describing interactions between photon, spin-1/2 and spin-3/2 particles of the same mass, following Jones and Scadron [22] can be written in terms of magnetic dipole ($g_M$ ), electric quadrupole ($g_E$ ), and Coulomb quadrupole ($g_C$ ) form factors.\",\n \"This interaction vertex effectively emerges from the following Lagrangian (see, e.g.\",\n \"[23]): ${\\\\cal L}_{\\\\rm int} = \\\\frac{3ie}{4m^2}~\\\\bar{u}T^3&\\\\biggl [g_M \\\\left(\\\\partial _{\\\\mu }\\\\psi _{\\\\nu }\\\\right) \\\\tilde{F}^{\\\\mu \\\\nu } + i g_E \\\\gamma _5 \\\\left(\\\\partial _{\\\\mu }\\\\psi _{\\\\nu }\\\\right)F^{\\\\mu \\\\nu } -\\\\frac{2g_C}{m}\\\\gamma _5\\\\gamma ^{\\\\alpha }\\\\partial _{[\\\\alpha }\\\\psi _{\\\\nu ]}\\\\partial _{\\\\mu }F^{\\\\mu \\\\nu } \\\\biggr ] + {\\\\rm h.c.} \\\\ ,$ where $\\\\tilde{F}^{\\\\mu \\\\nu } = \\\\epsilon ^{\\\\mu \\\\nu \\\\alpha \\\\beta }F_{\\\\alpha \\\\beta }/2$ and $T^3$ is an operator due to additional internal degrees of freedom (such as global or isospin charge) that the fields could carry.\",\n \"In the soft momentum transfer (or near forward) limit that we are interested in, only the first term will matter.\",\n \"We will rewrite the Lagrangian describing the magnetic dipole transition as: ${\\\\cal L}^{M1}_{\\\\rm int} &= \\\\frac{ie\\\\kappa _{M}}{2m^2}~\\\\bar{u}\\\\left(\\\\partial _{\\\\mu }\\\\psi _{\\\\nu }\\\\right) \\\\tilde{F}^{\\\\mu \\\\nu } + {\\\\rm h.c.} \\\\ .$ The interaction vertex describing the dominant magnetic dipole $3/2 \\\\rightarrow 1/2$ transition is: $\\\\Gamma ^{\\\\nu \\\\beta }(p^{\\\\prime },p^{\\\\prime }+k^{\\\\prime }) &= \\\\frac{e \\\\kappa _{M}}{2m^2}\\\\epsilon ^{\\\\mu \\\\nu \\\\alpha \\\\beta }p^{\\\\prime }_{\\\\mu }k^{\\\\prime }_{\\\\alpha } \\\\ ,$ where $p^{\\\\prime }$ is the momentum of spin-1/2 state and $k^{\\\\prime }$ is the photon momentum.\",\n \"Similarly, the vertex of $1/2 \\\\rightarrow 3/2$ transition is: $\\\\Gamma ^{\\\\nu \\\\beta }(p+k,p) &= -\\\\frac{e\\\\kappa _{M}}{2m^2}\\\\epsilon ^{\\\\mu \\\\nu \\\\alpha \\\\beta }p_{\\\\mu }k_{\\\\alpha } \\\\ ,$ where $p$ is the momentum of spin-1/2 state and $k$ is the photon momentum.\",\n \"For forward scattering, when $p_1 = p_2 = (m,0,0,0)$ and $\\\\omega = \\\\omega ^{\\\\prime }$ , we have: $\\\\Gamma ^{\\\\nu \\\\beta }(p^{\\\\prime },p^{\\\\prime }+k^{\\\\prime }) = -\\\\Gamma ^{\\\\nu \\\\beta }(p+k,p) &= \\\\frac{e\\\\kappa _{M}}{2m}\\\\omega ~\\\\epsilon ^{0\\\\nu \\\\beta k}n_{k} \\\\ .$ Consider the scattering in (REF ), where the intermediate state is a spin-3/2 Rarita-Schwinger particle with the same mass as the scatterer.\",\n \"We want to find the amplitude corresponding to this process.\",\n \"It can be written as: $\\\\nonumber i{\\\\cal M_{RS}} &= \\\\bar{u}(p^{\\\\prime },\\\\sigma ^{\\\\prime })\\\\biggl \\\\lbrace \\\\Gamma ^{\\\\nu \\\\alpha }(p^{\\\\prime },p+k)\\\\epsilon ^{*^{\\\\prime }}_{\\\\nu }(k^{\\\\prime })S_{\\\\alpha \\\\beta }(p+k)\\\\Gamma ^{\\\\mu \\\\beta }(p+k,p)\\\\epsilon _{\\\\mu }(k) \\\\\\\\&+ \\\\Gamma ^{\\\\mu \\\\beta }(p^{\\\\prime },p-k^{\\\\prime })\\\\epsilon _{\\\\mu }(k)S_{\\\\beta \\\\alpha }(p-k^{\\\\prime })\\\\Gamma ^{\\\\nu \\\\alpha }(p-k^{\\\\prime },p)\\\\epsilon ^{*^{\\\\prime }}_{\\\\nu }(k^{\\\\prime })\\\\biggr \\\\rbrace u(p,\\\\sigma ) \\\\ ,$ where vertices are defined in Eqs.\",\n \"(REF )-(REF ), and propagator of RS field, $S_{\\\\alpha \\\\beta }$ , is given by Eq.\",\n \"(REF ).\",\n \"Since we are interested in the near forward scattering amplitude up to order ${\\\\cal O}(\\\\omega )$ , and using Eqs.\",\n \"(REF ) and (REF ), direct computations for $f_{RS} = {\\\\cal M}_{RS}/(8\\\\pi m)$ give: $\\\\nonumber f_{RS}(\\\\omega , \\\\lambda ) &= \\\\frac{i}{12\\\\pi }\\\\frac{e^2\\\\kappa _M^2}{m^2}~\\\\omega \\\\left[(\\\\vec{\\\\epsilon }^{^{\\\\prime }*} \\\\times \\\\vec{n}^{\\\\prime }) \\\\times \\\\left(\\\\vec{\\\\epsilon } \\\\times \\\\vec{n}\\\\right)\\\\right]\\\\vec{J} = -\\\\frac{\\\\lambda \\\\omega }{12\\\\pi }\\\\frac{e^2\\\\kappa _M^2}{m^2}~J_z \\\\ .$ Finally, using the definition for $g_-(\\\\omega ^2)$ , given in Eq.\",\n \"(REF ), in the forward limit we have: $4\\\\pi ^2g_-(0) &= -\\\\frac{\\\\pi e^2\\\\kappa ^2_{M}}{3 m^2}J_z \\\\ ,$ which is in agreement with the previous result (REF ), as should be expected.\"\n ]\n]"},"id":{"kind":"string","value":"1204.1064"}}},{"rowIdx":1866892,"cells":{"text":{"kind":"list like","value":[["Spectrum of Higher Derivative 6D Chiral Supergravity"],["Abstract Gauged off-shell Maxwell-Einstein supergravity in six dimensions with N=(1,0) supersymmetry has a higher derivative extension afforded by a supersymmetrized Riemann squared term.","This theory admits a supersymmetric Minkowski x S^2 compactification with a U(1) monopole of unit charge on S^2.","We determine the full spectrum of the theory on this background.","We also determine the spectrum on a non-supersymmetric version of this compactification in which the monopole charge is different from unity, and we find the peculiar feature that there are massless gravitini in a representation of the S^2 isometry group determined by the monopole charge."],["Introduction","Higher-derivative supergravities are of considerable interest, especially when they arise as low-energy effective actions of string theories with higher-derivative corrections proportional to powers of the slope parameter $\\alpha ^{\\prime }$ .","However, their construction is notoriously difficult, in part due to the fact that supergravities exist only on-shell in ten dimensions.","In view of this difficulty, the compactifications of these theories are rarely studied.","In order to gain insights into the compactification of higher-derivative theories, it is instructive to investigate the issue in the simpler situation of lower-dimensional supergravities with higher-derivative terms, postponing for the present the question of how they may arise from ten dimensions.","An important technical advantage is that in some lower-dimensional cases, off-shell formulations of the supergravity theories exist.","This leads us to consider in particular ${\\cal N}=(1,0)$ supergravity in six dimensions, which is the highest dimension, and the highest degree of supersymmetry, for which a supergravity with an off-shell formulation is known.","The off-shell formulation of this supergravity was constructed in [1], [2], and a higher-derivative extension with an off-shell supersymmetrized Riemann-squared term was obtained in [3], [4].","The gauging of the $U(1)$ R-symmetry in the presence of this higher-derivative extension has also recently been obtained [5].","The model has two parameters, namely an overall coefficient $M^{-2}$ in front of the higher-derivative superinvariant in the action, and the gauge-coupling constant $g$ .","In the present paper, we shall study the six-dimensional gauged ${\\cal N}=(1,0)$ theory with the Riemann-squared term constructed in [5].","In the absence of the curvature-squared terms the model is an (off-shell) version of the Salam-Sezgin theory constructed long ago [6].","It was shown in [6] that the model had the unusual feature of admitting a supersymmetric Minkowski$_4\\times S^2$ vacuum, in which there is a $U(1)$ monopole flux with charge $q=\\pm 1$ on the $S^2$ internal space.","A remarkable feature of the theory with the Riemann-squared extension is that the Minkowski$_4 \\times S^2$ background continues to be a supersymmetric solution [5].","It also admits non-supersymmetric Minkowski$_4\\times S^2$ backgrounds in which the quantised monopole charge $q$ is larger than 1.","Our focus in this paper is to study the spectrum of the Kaluza-Klein states in the fluctuations around the Minkowski$_4\\times S^2$ background.","As far as we are aware, such a Kaluza-Klein spectral analysis of a higher-derivative supergravity around a background with non-abelian symmetries has not previously been carried out.","Even in the much simpler $S^2$ reduction of the Salam-Sezgin model discussed in [6], the situation is of considerable interest because of the very unusual feature of obtaining non-abelian symmetries from a sphere reduction, whilst obtaining a Poincaré rather than AdS supergravity in the lower dimension.","As expected, the states assemble into ${\\cal N}=1$ four-dimensional supermultiplets.","In the model constructed in [5] with the higher-order Riemann-squared extension, we find a number of novel features associated with the occurrence of higher-order wave operators, and the fact that certain fields that were purely auxiliary prior to the inclusion of the higher-order terms now become dynamical.","In particular, we find that certain four-dimensional vector supermultiplets have wave operators that give rise to masses $m$ that are determined by a non-trivial polynomial of fourth order in $m^2$ .","This leads to mass-squared values that are not simply linear in the eigenvalues of the Laplace operators on the internal space, but, rather, involve non-trivial roots of the associated quartic equation.","One consequence of this is that the values of $m^2$ can be negative or even complex, thus implying that there will be instabilities.","The occurrence of such states might at first sight seem surprising in a supersymmetric vacuum.","A standard argument for positive semi-definiteness of the energy, first given in [7], uses the fact that if a state $|\\psi \\rangle $ is annihilated by the supercharge $Q$ , then the superalgebra $\\lbrace Q,Q\\rbrace \\sim P$ implies that $P_0\\ge 0$ .","However, a crucial ingredient in this argument is that the norm on the states $|\\psi \\rangle $ is positive definite [8] In our case, the higher-derivative terms in the six-dimensional theory lead to ghost modes in the spectrum, and thus the assumptions required for the positivity result in [7] are violated.","The detailed structure of the quartic polynomial in $m^2$ for the vector multiplets implies that two of the four roots are always real and positive, while the remaining two can be complex.","The conditions under which this occurs are governed by the ratio $M^2/g^2$ and by the Kaluza-Klein level number $\\ell $ of the harmonics on $S^2$ .","As $M^2$ becomes larger, the non-positivity and complexity of the two roots sets in at larger and larger values of the level number $\\ell $ .","$M^2$ must at least satisfy $M^2\\ge 8(5+2 \\sqrt{6})g^2$ in order for the roots to be real and positive even at the lowest level $\\ell =0$ .","We also study the spectrum of the modes in the non-supersymmetric Minkowski$_4\\times S^2$ vacua that arise for $S^2$ monopole charges $q$ greater than 1.","An interesting feature in these cases is that the spectrum includes an $SU(2)$ multiplet of massless spin-${\\textstyle {\\frac{\\scriptstyle 3}{\\scriptstyle 2} } }$ fields at level $\\ell ={\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } }(|q|-3)$ .","The organisation of the paper is as follows.","In section 2 we review the six-dimensional gauged ${\\cal N}=(1,0)$ off-shell $R+ |\\hbox{Riem}|^2$ supergravity that was recently constructed in [5].","In section 3 we study the complete linearised spectrum of Kaluza-Klein modes in the supersymmetric Minkowski$_4\\times S^2$ vacuum, which has a monopole charge $q=1$ on the $S^2$ internal space, and exists for any value of the coupling $M^{-2}$ of the Riemann-squared invariant.","In section 4 we repeat the analysis for the non-supersymmetric Minkowski$_4\\times S^2$ vacua, which have arbitrary integer monopole charge $|q|\\ge 2$ , and which exist only for a special value of the ratio $g^2/M^2$ .","For this analysis we need many results on the properties of spin-weighted spherical harmonics on $S^2$ , since these are needed for the expansions in the monopole background of the fermion fields and certain vector fields that carry charges.","We present a detailed discussion of these harmonics in appendix B.","In appendix A we give our spinor conventions, and in appendix C we summarise some results for spin projection operators in four dimensions."],["The Theory","The off-shell $6D$ $(1,0)$ supergravity multiplet consists of the fields [1] $\\left(e_\\mu {}^a,\\ V_\\mu ^{\\prime ij},\\ V_\\mu , \\ B_{\\mu \\nu },\\ L,\\ C_{\\mu \\nu \\rho \\sigma },\\ \\psi _\\mu ^i,\\ \\chi ^i \\right)$ where $V_\\mu ^{\\prime ij}$ is symmetric and traceless in its $Sp(1)$ doublet indices, $B$ and $C$ are antisymmetric tensor fields, $L$ is a real scalar, and the spinors are symplectic Majorana-Weyl.","The above fields have $(15,12,5,10,1,5, 40,8)$ degrees of freedom.","In addition, we shall consider the off-shell Maxwell multiplet consisting of the fields $\\left( A_\\mu , \\ Y^{ij},\\ \\lambda ^i\\right)\\ ,$ where $Y^{ij}$ is symmetric in its indices and the fermion is symplectic Majorana Weyl.","These fields have $(5,3,8)$ degrees of freedom.","The total Lagrangian we shall study is given by ${\\cal L}= {\\cal L}_R -\\frac{1}{8M^2} {\\cal L}_{R^2}\\ ,$ where the $U(1)_R$ gauged off-shell supergravity Lagrangian, up to quartic fermion terms, is [1], [5]We have let $g\\rightarrow 4g$ and $A_\\mu \\rightarrow A_\\mu /{\\sqrt{2}}$ in the results of [5].","$e^{-1}{\\cal L}_R &=&\\frac{1}{2} L R +\\frac{1}{2} L^{-1}\\partial _\\mu L\\partial ^\\mu L+2{\\sqrt{2}}g L \\delta ^{ij}Y_{ij} -\\frac{1}{24} L H_{\\mu \\nu \\rho }H^{\\mu \\nu \\rho }\\nonumber \\\\&& +L V^{\\prime }_\\mu {}^{ij}V^{^{\\prime }\\mu }{}_{ij} -\\frac{1}{4} L^{-1}E^\\mu E_\\mu +\\frac{1}{\\sqrt{2}} E^\\mu \\left( V_\\mu +2g A_\\mu \\right)\\nonumber \\\\&& +Y^{ij} Y_{ij}-\\frac{1}{8}F_{\\mu \\nu }F^{\\mu \\nu }-\\frac{1}{16}\\varepsilon ^{\\mu \\nu \\rho \\sigma \\lambda \\tau }B_{\\mu \\nu }F_{\\rho \\sigma }F_{\\lambda \\tau }\\nonumber \\\\[0.2cm]&& -\\frac{1}{2}L \\bar{\\psi }_{\\rho }\\gamma ^{\\mu \\nu \\rho }D_{\\mu }\\psi _{\\nu }-{\\sqrt{2}} {\\bar{\\chi }}_i\\gamma ^{\\mu \\nu }D_\\mu \\psi _{\\nu j} \\delta ^{ij}+ L^{-1}{\\bar{\\chi }} {D}\\chi \\nonumber \\\\&& -\\frac{1}{2} {\\bar{\\psi }}^\\mu \\gamma ^\\nu \\psi _\\nu \\partial _\\mu L-\\frac{1}{\\sqrt{2}} \\delta _{ij}{\\bar{\\psi }}_\\nu ^i\\gamma ^\\mu \\gamma ^\\nu \\chi ^j L^{-1}\\partial _\\mu L-2{\\sqrt{2}}g L {\\bar{\\lambda }}_i\\gamma ^\\mu \\psi _{\\mu j} \\delta ^{ij}\\nonumber \\\\[0.2cm]&&+ 2 g {\\bar{\\lambda }}\\chi +\\frac{1}{2} V_\\mu ^{^{\\prime }ij}\\left( {2\\sqrt{2}}{\\bar{\\chi }}^k\\psi ^\\mu _i\\delta _{jk}-3L^{-1}{\\bar{\\chi }}_i\\gamma ^\\mu \\chi _j\\right)\\nonumber \\\\&& -\\frac{1}{48}L H_{\\mu \\nu \\rho }\\left(\\bar{\\psi }^\\lambda \\gamma _{[\\lambda } \\gamma ^{\\mu \\nu \\rho }\\gamma _{\\tau ]}\\psi ^\\tau +2{\\sqrt{2}} L^{-1}{\\bar{\\psi }}_{\\lambda i}\\gamma ^{\\lambda \\mu \\nu \\rho }\\chi _j \\delta ^{ij}-2L^{-2}{\\bar{\\chi }}\\gamma ^{\\mu \\nu \\rho }\\chi \\right)\\nonumber \\\\&& -\\frac{1}{4\\sqrt{2}} E_\\rho \\left(\\psi _\\mu ^i \\gamma ^{\\rho \\mu \\nu }\\psi _\\nu ^j \\delta _{ij}-2{\\sqrt{2}} L^{-1} {\\bar{\\psi }}_\\sigma \\gamma ^{\\rho }\\gamma ^{\\sigma }\\chi +2 L^{-2}{\\bar{\\chi }}_i\\gamma ^\\rho \\chi _j \\delta ^{ij} \\right)\\nonumber \\\\&&-2{\\bar{\\lambda }}{D}\\lambda +\\frac{1}{12}H_{\\mu \\nu \\rho }{\\bar{\\lambda }}\\gamma ^{\\mu \\nu \\rho }\\lambda +\\frac{1}{2\\sqrt{2} } F_{\\mu \\nu } {\\bar{\\lambda }}\\gamma ^\\rho \\gamma ^{\\mu \\nu }\\psi _\\rho \\ ,$ where $H_{\\mu \\nu \\rho }=3\\partial _{[\\mu } B_{\\nu \\rho ]}$ and $E^\\mu &=& \\frac{1}{24}\\varepsilon ^{\\mu \\nu _1\\cdots \\nu _5} \\partial _{[\\nu _1}C_{\\nu _2\\cdots \\nu _5]}\\ .\\\\[0.2cm]D_\\mu \\psi _\\nu ^i &=& (\\partial _\\mu + \\frac{1}{4}\\omega _{\\mu }{}^{ab}\\gamma _{ab} )\\psi _\\nu ^i-\\frac{1}{2} V_\\mu \\delta ^{ij}\\psi _{\\nu j}\\ ,\\\\[0.2cm]D_\\mu \\chi ^i &=& (\\partial _\\mu + \\frac{1}{4}\\omega _{\\mu }{}^{ab}\\gamma _{ab})\\chi ^i-\\frac{1}{2} V_\\mu \\delta ^{ij}\\chi _j + V_\\mu {}^{^{\\prime }i}{}_j \\chi ^j\\ .$ Note the presence of arbitrary coupling constant in ${\\cal L}_R$ .","In fact, the sum of all the terms in this Lagrangian that depend on $g$ separately have the off-shell supersymmetry.","Thus, the total Lagrangian is a sum of three separately off-shell supersymmetric pieces.","The Lagrangian for the supersymmetrized Riemann squared term, up to quartic fermion terms, is given by [3], [4] $e^{-1} {\\mathcal {L}}_{\\rm R^2} &=& R_{\\mu \\nu }{}^{ab}(\\omega _-)R^{\\mu \\nu }{}_{ab}(\\omega _-)-2G^{ab}G_{ab} -4G_{\\mu \\nu }^{^{\\prime }ij}G^{^{\\prime }\\mu \\nu }_{ij}\\ ,\\nonumber \\\\&& +\\frac{1}{4} \\varepsilon ^{\\mu \\nu \\rho \\sigma \\lambda \\tau }B_{\\mu \\nu }R_{\\rho \\sigma ab}(\\omega _-)R_{\\lambda \\tau }{}_{ab}(\\omega _-)\\nonumber \\\\[0.2cm]&& +2{\\bar{\\psi }}^{ab}(\\omega _+)\\gamma ^\\mu D_\\mu (\\omega ,\\omega _-)\\psi _{ab}(\\omega _+)- R_{\\nu \\rho }{}^{ab}(\\omega _-){\\bar{\\psi }}_{ab}(\\omega _+)\\gamma ^\\mu \\gamma ^{\\nu \\rho }\\psi _\\mu \\nonumber \\\\[0.2cm]&& -8G_{\\mu \\nu }^{ij}\\left({\\bar{\\psi }}^\\mu _i \\gamma _\\lambda \\psi ^{\\lambda \\nu }_j(\\omega _+)+\\frac{1}{6} {\\bar{\\psi }}^\\mu _i \\gamma \\cdot H\\psi ^\\nu _j\\right)-\\frac{1}{12}{\\bar{\\psi }}^{ab}(\\omega _+)\\gamma \\cdot H \\psi _{ab}(\\omega _+)\\nonumber \\\\&&-\\frac{1}{2} \\Bigl [ D_\\mu (\\omega _-,\\Gamma _+) R^{\\mu \\nu ab} (\\omega _-)-2 H_{\\mu \\nu }{}^\\rho R^{\\mu \\nu ab} (\\omega _-)\\Bigr ] {\\bar{\\psi }}^a\\gamma _\\rho \\psi _b\\ ,$ where $G_{\\mu \\nu }^{^{\\prime }ij}$ and $G_{\\mu \\nu }$ are the field strengths associated with $V_\\mu ^{^{\\prime }ij}$ and $V_\\mu $ , which can be combined as $V_\\mu ^{ij} = V_\\mu ^{^{\\prime }ij} + {\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } } \\delta ^{ij} V_\\mu $ .","Furthermore $\\psi _{\\mu \\nu }(\\omega _+) =2D_{[\\mu }(\\omega _+) \\psi _{\\nu ]}$ and $D_\\mu (\\omega ,\\omega _-)\\psi ^{ab i} &=&(\\partial _\\mu +\\frac{1}{4}\\omega _\\mu {}^{cd}\\gamma _{cd} )\\psi ^{ab i}+2 \\omega _{\\mu -}{}^{c[a} \\psi ^{b]i}{}_c +V_\\mu ^i{}_j \\psi ^{ab i}\\ ,\\nonumber \\\\[0.2cm]\\omega _{\\mu \\pm }{}^{ab} &=& \\omega _\\mu {}^{ab} \\pm {\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } } H_\\mu {}^{ab}\\ ,\\qquad \\Gamma _{\\mu \\nu \\pm }^\\rho =\\Gamma _{\\mu \\nu }^\\rho \\pm {\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } } H_\\mu {}^{\\nu \\rho }\\ .$ The off-shell resulting supersymmetry transformations of the Poincaré multiplet, up to cubic fermion terms, are [1], [3], [5] $\\delta e_{\\mu }{}^a&=&\\frac{1}{2}{\\bar{\\epsilon }}\\gamma ^a\\psi _{\\mu }\\ ,\\nonumber \\\\\\delta \\psi _{\\mu }{}^i&=& (\\partial _{\\mu } +\\frac{1}{4}\\omega _{\\mu ab}\\gamma ^{ab})\\epsilon ^i+V_\\mu {}^i{}_j\\epsilon ^j +\\frac{1}{8} H_{\\mu \\nu \\rho }\\gamma ^{\\nu \\rho }\\epsilon ^i\\ ,\\nonumber \\\\\\delta B_{\\mu \\nu }&=&-{\\bar{\\epsilon }}\\gamma _{[\\mu }\\psi _{\\nu ]}\\ ,\\nonumber \\\\\\delta \\chi ^i &=& \\frac{1}{2\\sqrt{2}} \\gamma ^\\mu \\delta ^{ij}\\partial _\\mu L \\epsilon _j-\\frac{1}{4} \\gamma ^\\mu E_\\mu \\epsilon ^i+\\frac{1}{\\sqrt{2}}\\gamma ^\\mu V^{\\prime }{}_\\mu ^{(i}{}_k \\delta ^{j)k} L\\epsilon _j - \\frac{1}{12\\sqrt{2}}L\\delta ^{ij}\\gamma \\cdot H \\epsilon _j \\ ,\\nonumber \\\\[0.2cm]\\delta L &=& \\frac{1}{\\sqrt{2}} {\\bar{\\epsilon }}^i \\chi ^j\\delta _{ij} \\ ,\\nonumber \\\\[0.2cm]\\delta C_{\\mu \\nu \\rho \\sigma } &=& L{\\bar{\\epsilon }}^i\\gamma _{[\\mu \\nu \\rho }\\psi _{\\sigma ]}^j\\delta _{ij} -\\frac{1}{2\\sqrt{2}} {\\bar{\\epsilon }}\\gamma _{\\mu \\nu \\rho \\sigma }\\chi \\ ,\\nonumber \\\\[0.2cm]\\delta V_{\\mu }{}^{ij}&=& \\frac{1}{2}\\bar{\\epsilon }^{(i}\\gamma ^{\\rho }\\psi _{\\mu \\rho }^{j)}+\\frac{1}{12}\\bar{\\epsilon }^{(i}\\gamma \\cdot H\\psi _{\\mu }^{j)} +\\frac{1}{8}\\sigma ^{-1}\\bar{\\epsilon }^{(i}\\gamma ^{\\rho }\\Bigl (H_{[\\mu }{}^{ab}\\gamma _{ab}\\psi _{\\rho ]}^{j)}\\Bigr )$ and the off-shell supersymmetry transformations of the vector multiplet are $\\delta A_{\\mu }&=&-\\bar{\\epsilon }\\gamma _{\\mu }\\lambda \\ ,\\nonumber \\\\\\delta \\lambda ^{i}&=&\\frac{1}{8\\sqrt{2}}\\gamma ^{\\mu \\nu }F_{\\mu \\nu }\\epsilon ^i-\\frac{1}{2}Y^{ij}\\epsilon _j\\ ,\\nonumber \\\\\\delta Y^{ij} &=& -\\bar{\\epsilon }^{(i}\\gamma ^{\\mu }D_{\\mu }\\lambda ^{j)}+\\frac{1}{8} {\\bar{\\epsilon }}^{(i} \\gamma ^\\mu \\gamma \\cdot H \\psi _\\mu ^{j)}-\\frac{1}{24} {\\bar{\\lambda }}^i \\gamma \\cdot H \\lambda ^j-\\frac{1}{2} Y^{k(i} {\\bar{\\epsilon }}^{j)} \\gamma ^\\mu \\psi _{\\mu k}\\ .$ Of the auxiliary fields of the Poincaré supergravity, $V_\\mu ^{^{\\prime }ij}$ and $V_\\mu $ can no longer be eliminated algebraically due to the presence of the Riemann squared invariant but $Y^{ij}$ and $C_{\\mu \\nu \\rho \\sigma }$ can still be eliminated by means of their field equations as $Y^{ij} = -{\\sqrt{2}} g L\\delta ^{ij}\\ ,\\qquad E_\\mu = {\\sqrt{2}} L\\left(V_\\mu +2g A_\\mu \\right)\\ .$ The total Lagrangian we shall study here is given by ${\\cal L}= L_{\\rm R} -\\frac{1}{8M^2} {\\cal L}_{\\rm R^2}\\ ,$ where $M$ is an arbitrary mass parameter."],["Supersymmetric $\\rm {Minkowski}_4\\times S^2$ background","We shall study the compactification on the one half supersymmetric vacuum solution with the geometry of $\\rm {Minkowski}_4\\times S^2$ .","From here on, the 6D coordinates will be denoted by $x^M$ and they will be split as $(x^\\mu ,y^m)$ to denote the coordinates of 4D spacetime and the internal two-dimensional space.","The supersymmetric $\\rm {Minkowski}_4\\times S^2$ vacuum solution given by [5] $\\bar{R}_{\\mu \\nu \\lambda \\rho } &=0\\ , &\\qquad \\bar{R}_{mn} &= \\alpha ^2\\bar{g}_{mn}\\ , \\qquad \\bar{L} &=1\\ , \\nonumber \\\\\\bar{F}_{\\mu \\nu } &=0\\ , & \\qquad \\bar{F}_{mn} &= 4g\\epsilon _{mn}\\ , \\qquad & \\nonumber \\\\\\bar{G}_{\\mu \\nu } &=0\\ , & \\qquad \\bar{G}_{mn} &= -\\alpha ^2\\epsilon _{mn}\\ , \\qquad &$ where $\\alpha ^2\\equiv 8g^2$ , $\\bar{g}_{mn}$ is the metric on $S^2$ with radius 1/$\\alpha $ , and $\\epsilon _{mn}$ is the Levi-Civita tensor on the same $S^2$ .","We define the complex vectors ${\\hat{Z}}_M = {\\hat{V}}^{^{\\prime }11}_M+i {\\hat{V}}^{^{\\prime }12}_M\\ ,$ and parametrize the linearized fluctuations around above background as follows $\\hat{g}_{MN} &=\\bar{g}_{MN}+\\hat{h}_{MN}\\ ,&\\qquad \\hat{L} &=1+\\hat{\\phi }\\ ,&\\qquad \\hat{A}_{M} &=\\bar{A}_M+\\hat{a}_M\\ ,\\nonumber \\\\\\hat{V}_{M} &=\\bar{V}_M+\\hat{v}_{M}\\ , &\\qquad ~~{\\hat{Z}}_{M} &={\\hat{z}}_M\\ , &\\qquad \\hat{B}_{MN} &=\\hat{b}_{MN}\\ ,$ where we use “$\\rm {hat}$ ” to stand for six dimensional quantities and “$\\rm {bar}$ ” to denote quantities evaluated in the vacuum background.","In the background specified above, the linearized six dimensional bosonic and fermionic gauge symmetries are expressed asFor later convenience, starting from the USp(2) symplectic-Majorana-Weyl spinors we have defined Weyl spinors by complexifying as $\\psi =\\psi _1+i\\psi _2 $ and rescaled $\\hat{\\chi }$ and $\\hat{\\lambda }$ used in [5] by $\\hat{\\chi }\\rightarrow \\sqrt{2}\\hat{\\chi }$ , $\\hat{\\lambda }\\rightarrow \\sqrt{2}\\hat{\\lambda }$ .","$\\delta \\hat{h}_{MN} &= \\bar{\\nabla }_{M}\\hat{\\xi }_N+ \\bar{\\nabla }_{N}\\hat{\\xi }_M\\ , & \\qquad \\delta \\hat{ a}_{M} &=\\hat{\\xi }^{N}\\bar{F}_{NM}+\\partial _{M}\\hat{\\Lambda }, & \\nonumber \\\\\\delta \\hat{v}_{M}&=\\hat{\\xi }^{N}\\bar{G}_{NM}-2g\\partial _{M}\\hat{\\Lambda }\\ , &\\qquad \\delta \\hat{b}_{MN} &=\\partial _{M}\\hat{\\Lambda }_{N}-\\partial _{N}\\hat{\\Lambda }_{M}\\ , & \\nonumber \\\\\\delta \\hat{\\psi }_M &=\\bar{D}_M\\hat{\\epsilon }, &\\qquad \\delta \\hat{\\lambda } &={\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 16} } }{\\bar{\\Gamma }}^{MN}\\bar{F}_{MN}\\hat{\\epsilon }+{\\textstyle {\\frac{\\scriptstyle i}{\\scriptstyle 2} } }g\\hat{\\epsilon }\\ , & \\qquad \\delta \\hat{\\chi }& =0\\ .$ This background preserves half supersymmetry because it admits a Killing spinor $\\hat{\\eta }$ which has the following properties $\\delta \\hat{\\psi }_M=\\bar{D}_M\\hat{\\eta }=0,\\qquad \\delta \\hat{\\chi }=0,\\qquad \\delta \\hat{\\lambda }=({\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 16} } }{\\bar{\\Gamma }}^{MN}\\bar{F}_{MN}+{\\textstyle {\\frac{\\scriptstyle i}{\\scriptstyle 2} } }g)\\hat{\\eta }=0\\ ,$ and by choosing the six dimensional gamma matrices as in Appendix, it can be shown that $\\hat{\\eta }=\\epsilon \\otimes \\eta ,\\qquad \\eta =\\left(\\begin{array}{c}0 \\\\1 \\\\\\end{array}\\right),$ where $\\epsilon $ is a constant four dimensional Weyl spinor with appropriate chirality inherited from six dimensions."],["Bosonic Sector","In this section, we shall drop the “bar” on the covariant derivatives for simplicity in notation.","The linearized bosonic field equations are given as follows $(\\hat{R}^{(L)}_{MN}+\\hat{\\phi }\\bar{R}_{MN}) &=&{\\hat{\\nabla }}_M{\\hat{\\nabla }}_N\\hat{\\phi }+\\alpha ^2\\bar{g}_{MN}\\hat{\\phi }+{\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } }(\\hat{F}^{(L)}_{MP}\\bar{F}_{N}^{~P}+ \\hat{F}^{(L)}_{NP}\\bar{F}_{M}^{~P} -\\bar{F}_{MP}\\bar{F}_{NQ}\\hat{h}^{PQ})\\\\[0.2cm]&&+{\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } }\\hat{h}_{MN}(\\alpha ^2-{\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 4} } }\\bar{F}^{PQ}\\bar{F}_{PQ})-{\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 4} } }\\bar{g}_{MN}(\\hat{F}^{(L)}_{PQ}\\bar{F}^{PQ}-\\bar{F}_{P}^{~Q}\\bar{F}^{PT}\\hat{h}_{QT}),\\nonumber \\\\&&-{\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 8M^2} } }S^{(L)}_{MN}\\ ,\\nonumber \\\\[0.2cm]\\hat{R}^{(L)} &=& 2\\alpha ^2\\hat{\\phi }+2\\hat{\\Box }\\hat{\\phi }\\ ,\\\\[0.2cm]\\hat{\\nabla }^{P}\\hat{H}^{(L)}_{PMN} &=& {\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } }\\varepsilon _{MN}^{~~~~PQST}({\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } }\\hat{F}^{(L)}_{PQ}\\bar{F}_{ST}-\\frac{1}{2M^2}\\tilde{R}^{(L)J}_{~~~~KPQ}\\bar{R}^{K}_{~JST})+\\frac{1}{M^2}\\hat{\\nabla }^P\\hat{\\Box } \\hat{H}^{(L)}_{PMN}, \\nonumber \\\\&&+3{M^2}\\hat{\\nabla }^P(\\hat{H}^{(L)ST}_{+~~~~[P}\\bar{R}_{MN]ST})\\ ,$ $0&=& \\hat{\\nabla }^{P}\\hat{F}^{(L)}_{PM}-\\hat{\\nabla }^P\\hat{h}_{PQ}\\bar{F}^{Q}_{~M}-\\hat{\\nabla }_P\\hat{h}_{QM}\\bar{F}^{PQ}+{\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } }\\hat{\\nabla }_P\\hat{h}\\bar{F}^{P}_{~M}+4g(\\hat{v}_M+2g\\hat{a}_M)-{\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } }*\\hat{H}^{(L)}_{MPQ}\\bar{F}^{PQ}\\ ,\\nonumber \\\\&&\\\\0&=& \\hat{\\nabla }^P\\hat{G}^{(L)}_{PM}-\\hat{\\nabla }^P\\hat{h}_{PQ}\\bar{G}^{Q}_{~M}-\\hat{\\nabla }_P\\hat{h}_{QM}\\bar{G}^{PQ}+{\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } }\\hat{\\nabla }_P\\hat{h}\\bar{G}^{P}_{~M}-M^2({\\hat{v}}_M+2g {\\hat{a}}_M) \\ ,\\\\[0.2cm]0&=& \\left( {\\hat{\\nabla }}^P - i{\\bar{V}}^P\\right) \\hat{G}_{PM}^{^{\\prime }(L)}- i{\\bar{G}}_{MN} {\\hat{z}}^N-M^2 {\\hat{z}}_M \\ ,$ where ${\\hat{G}}_{MN}^{^{\\prime }(L)} &=& 2{\\hat{D}}_{[M}{\\hat{z}}_{N]}\\ ,\\qquad {\\hat{D}}_M {\\hat{z}}_N \\equiv ( {\\hat{\\nabla }}_M - i{\\bar{V}}_M ){\\hat{z}}_N\\ ,\\nonumber \\\\[0.2cm]R^{(L)P}_{~~~~MNQ} &=&\\hat{R}^{(L)P}_{~~~~MNQ}-\\hat{\\nabla }_{[N}H^{(L)P}_{Q]~~M}\\ ,\\nonumber \\\\[0.2cm]S^{(L)}_{MN} &=& 8\\left(\\hat{G}^{(L)}_{MP}\\bar{G}_{N}^{~P}+\\hat{G}^{(L)}_{NP}\\bar{G}_{M}^{~P}-\\bar{G}_{M}^{~P}\\bar{G}_{N}^{~Q}\\hat{h}_{PQ}\\right)-4\\bar{g}_{MN}(\\hat{G}^{(L)}_{PQ}\\bar{G}^{PQ}-\\bar{G}_{P}^{~Q}\\bar{G}^{PT}\\hat{h}_{QT})\\nonumber \\\\&& +4(\\tilde{R}^{(L)S}_{~~~~QMP}\\bar{R}^{Q~~~P}_{~SN}+\\bar{R}^{S~~~P}_{~QM}\\tilde{R}^{(L)Q}_{~~~~SNP}-\\hat{h}^{PQ}\\bar{R}^{S}_{~TMP}\\bar{R}^{T}_{~SNQ})-2\\hat{h}_{MN}\\bar{G}^{PQ}\\bar{G}_{PQ} \\nonumber \\\\&&+\\hat{h}_{MN}\\bar{R}^{PQST}\\bar{R}_{PQST}-2\\bar{g}_{MN}(\\tilde{R}^{(L)S}_{~~~~TPQ}\\bar{R}^{T~PQ}_{~S} +\\bar{R}_{JKSP}\\bar{R}^{JKS}_{~~~~Q}\\hat{h}^{PQ})\\nonumber \\\\&&+8(\\hat{\\nabla }^{P}\\tilde{\\nabla }^{Q}\\tilde{R}_{P(MN)Q})^{(L)}+8\\hat{\\nabla }^{S}(\\bar{R}_{S(M}^{~~~~PQ}\\hat{H}^{+{(L)}}_{N)PQ}),$ and the penultimate term takes the form $(\\hat{\\nabla }^{P}\\tilde{\\nabla }^{Q}\\tilde{R}_{P(MN)Q})^{(L)} &=&\\bar{R}^{P~~~Q}_{~(MN)}(\\bar{R}_{P}^{~S}\\hat{h}_{SQ}-\\bar{R}^{S~T}_{~P~Q}\\hat{h}_{ST}-{\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } }\\hat{\\Box } \\hat{h}_{PQ})+\\hat{\\nabla }_{P}\\hat{\\nabla }_{Q}\\hat{h}_{S(M}\\bar{R}^{P~~QS}_{~N)}\\nonumber \\\\&&-{\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } }(\\hat{\\nabla }^{P}\\hat{\\nabla }_{(M}\\hat{h}^{QS}\\bar{R}_{N)SPQ}+\\hat{\\nabla }^{P}\\hat{\\nabla }^{S}\\hat{h}^{Q}_{~(M}\\bar{R}_{N)SPQ}\\nonumber \\\\&&-{\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } }\\bar{R}^{PQS}_{~~~~~T}\\hat{h}^{T}_{~(M}\\bar{R}_{N)SPQ}+{\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } }\\bar{R}^{PQ}_{~~~T(M}\\bar{R}_{N)SPQ}\\hat{h}^{ST} )\\nonumber \\\\&& +{\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } }\\bar{R}_{PMNQ}\\hat{\\nabla }^{P}\\hat{\\nabla }^{Q}\\hat{h}+\\hat{\\nabla }_{P}\\hat{\\nabla }^{Q}\\tilde{R}^{(L)P}_{~~~~(MN)Q}\\nonumber \\\\&& -{\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } }(\\hat{\\nabla }_P \\hat{H}^{(L)}_{QS(M}\\bar{R}^{PQ~S}_{~~~N)}+\\hat{\\nabla }_P\\hat{H}^{(L)}_{QS(M}\\bar{R}^{P~QS}_{~N)}).$ The covariant derivative ${\\tilde{\\nabla }}_M$ is defined with respect to the connection ${\\widetilde{\\Gamma }}_{\\mu \\nu }^\\rho $ containing bosonic torsion as ${\\widetilde{\\Gamma }}_{\\mu \\nu }^\\rho = \\left\\lbrace \\begin{array}{c}\\rho \\\\\\mu \\nu \\\\\\end{array}\\right\\rbrace +\\frac{1}{2} H_{\\mu \\nu }{}^\\rho \\ .$ Note that we are using $\\hat{G}^{^{\\prime }(L)}_{MN}$ to denote the covariant field strength of the complex vector field $\\hat{z}_M$ , and $G^{(L)}_{MN}$ to denote the field strength of the real vector $v_M$ .","There are no transverse traceless spin-2 harmonics on $S^2$ , and the transverse spin-1 harmonics are related to spin-0 harmonics by $Y^{(\\ell )}_m=\\epsilon _m{}^n\\nabla _nY^{(\\ell )}\\ .$ We can expand the six-dimensional bosonic fields in terms of $S^2$ harmonics as follows $\\hat{h}_{\\mu \\nu }& =&\\sum _{\\ell \\ge 0 }h_{\\mu \\nu }^{(\\ell )}Y^{(\\ell )} \\ ,\\nonumber \\\\\\hat{h}_{mn}&=&\\sum _{\\ell \\ge 2 }\\left(L^{(\\ell )}\\nabla _{\\lbrace m}Y^{(\\ell )}_{n\\rbrace }+\\tilde{L}^{(\\ell )}\\nabla _{\\lbrace m}\\nabla _{n\\rbrace }Y^{(\\ell )}\\right)+ \\bar{g}_{mn}\\sum _{\\ell \\ge 0}N^{(\\ell )}Y^{(\\ell )}\\ ,\\nonumber \\\\\\hat{h}_{\\mu m}&=&\\sum _{\\ell \\ge 1}(k_{\\mu }^{(\\ell )}Y_{m}^{(\\ell )}+\\tilde{k}_{\\mu }^{(\\ell )}\\nabla _mY^{(\\ell )})\\ ,\\nonumber \\\\\\hat{\\phi } &=&\\sum _{\\ell \\ge 0}\\phi ^{(\\ell )}Y^{(\\ell )}\\ ,\\nonumber \\\\\\hat{a}_{\\mu }&=&\\sum _{\\ell \\ge 0 }a_{\\mu }^{(\\ell )}Y^{(\\ell )},\\qquad \\hat{a}_{m}=\\sum _{\\ell \\ge 1}\\left(a^{(\\ell )}Y_m^{(\\ell )}+\\tilde{a}^{(\\ell )}\\nabla _mY^{(\\ell )}\\right)\\ ,\\nonumber \\\\\\hat{v}_{\\mu }&=&\\sum _{\\ell \\ge 0}v^{(\\ell )}_{\\mu }Y^{(\\ell )},\\qquad \\hat{v}_{m}=\\sum _{\\ell \\ge 1}\\left(v^{(\\ell )}Y^{(\\ell )}_{m}+{\\tilde{v}}^{(\\ell )}\\nabla _m Y^{(\\ell )}\\right)\\ ,\\nonumber \\\\{\\hat{z}}_\\mu &=& \\sum _{\\ell \\ge 1} z^{(\\ell )}_\\mu {}_{-1}{Y}^{(\\ell )} \\ ,\\nonumber \\\\{\\hat{z}}_m =&& \\sum _{\\ell =0,1} z^{(\\ell )} {}_{-1}V_{m}^{(\\ell )}+\\sum _{\\ell \\ge 2} \\left(z^{(\\ell )} D_m {}_{-1}{ Y}^{(\\ell )} +i{\\tilde{z}}^{(\\ell )} \\epsilon _m{}^n D_n {}_{-1}{Y}^{(\\ell )}\\right)\\ ,\\nonumber \\\\\\hat{b}_{\\mu \\nu }&=&\\sum _{\\ell \\ge 0}b_{\\mu \\nu }^{(\\ell )}Y^{(\\ell )},\\qquad \\hat{b}_{mn}=\\epsilon _{mn}\\sum _{\\ell \\ge 0}b^{(\\ell )}Y^{(\\ell )}\\ ,\\nonumber \\\\\\hat{b}_{\\mu m}&=&\\sum _{\\ell \\ge 1}\\left(b_{\\mu }^{(\\ell )}Y^{(\\ell )}_{m}+\\tilde{b}_{\\mu }^{(\\ell )}\\nabla _mY^{(\\ell )}\\right)\\ ,$ where the notation $\\lbrace mn\\rbrace $ means “symmetric and traceless,” and in the $\\hat{z}_m$ expansion ${}_{-1}V_{m}^{(0)}$ and ${}_{-1}V_{m}^{(1)}$ are level $\\ell =0$ and $\\ell =1$ complex anti-self dual vector harmonics with charge $-1$ on the 2-sphere, whose explicit forms are given in Appendix B.2.","$D_m$ is the $U(1)$ covariant derivative on the 2-sphere, and ${}_{-1}{Y}^{(\\ell )}$ are the charged harmonics which are described in some detail in Appendix B.1.","Furthermore, the scalar harmonics $Y^{(\\ell )}$ employed above satisfy $\\Box _2Y^{(\\ell )}=- \\alpha ^2c_\\ell Y^{(\\ell )}\\ ,$ where $\\Box _2$ is the d'Alembertian on $S^2$ with radius $1/\\alpha $ and $\\boxed{{c_\\ell \\equiv \\ell (\\ell +1)}} \\ ,\\qquad \\boxed{\\alpha ^2\\equiv 8g^2}\\ .$ We have also used the spin-1 harmonics $Y_m^{(\\ell )}$ which satisfy the relations $\\Box _2Y^{(\\ell )}_n=-(c_\\ell -1)\\alpha ^2 Y^{(\\ell )}_n\\ ,\\qquad \\epsilon ^{mn}\\nabla _m Y^{(\\ell )}_n=\\alpha ^2c_\\ell \\, Y^{(\\ell )}\\ .$ Utilizing the six dimensional gauge symmetries (REF ), we impose the following gauge condition on the linearized fields [9] $\\hat{\\nabla }^{m}\\hat{h}_{\\lbrace mn\\rbrace }&=&0,\\qquad \\hat{\\nabla }^{m}\\hat{h}_{m\\mu }=0,\\nonumber \\\\\\hat{ \\nabla }^m\\hat{a}_m&=&0,\\qquad \\hat{ \\nabla }^m\\hat{b}_{mM}=0.$ Upon the use of these gauge conditions, the harmonic expansions (REF ) simplify to $\\hat{h}_{\\mu \\nu }&=&\\sum _{\\ell \\ge 0}h_{\\mu \\nu }^{(\\ell )}Y^{(\\ell )}\\ ,\\qquad \\ \\ \\hat{h}_{\\mu m}=\\sum _{\\ell \\ge 1}k_{\\mu }^{(\\ell )}Y_{m}^{(\\ell )}\\ ,\\nonumber \\\\\\hat{h}_{mn}&=&\\bar{g}_{mn}\\sum _{\\ell \\ge 0}N^{(\\ell )}Y^{(\\ell )},\\qquad \\hat{\\phi }=\\sum _{\\ell \\ge 0}\\phi ^{(\\ell )}Y^{(\\ell )}\\ ,\\nonumber \\\\\\hat{a}_{\\mu }&=&\\sum _{\\ell \\ge 0}a_{\\mu }^{(\\ell )}Y^{(\\ell )}\\ ,\\qquad \\quad \\hat{a}_{m}=\\sum _{\\ell \\ge 1}a^{(\\ell )}Y^{(\\ell )}_m\\ ,\\nonumber \\\\\\hat{v}_{\\mu } & =&\\sum _{\\ell \\ge 0}v^{(\\ell )}_{\\mu }Y^{(\\ell )}\\ ,\\qquad \\quad \\hat{v}_{m}=\\sum _{\\ell \\ge 1}\\left(v^{(\\ell )}Y^{(\\ell )}_{m}+{\\tilde{v}}^{(\\ell )}\\nabla _m Y^{(\\ell )}\\right)\\ ,\\nonumber \\\\{\\hat{z}}_\\mu &=& \\sum _{\\ell \\ge 1} z^{(\\ell )}_\\mu \\ _{-1} Y^{(\\ell )} \\ ,\\nonumber \\\\{\\hat{z}}_m =&& \\sum _{\\ell =0,1} z^{(\\ell )} {}_{-1}V_{m}^{(\\ell )}+\\sum _{\\ell \\ge 2} \\left(z^{(\\ell )} D_m {}_{-1}{Y}^{(\\ell )} +i{\\tilde{z}}^{(\\ell )} \\epsilon _m{}^n D_n{}_{-1}{Y}^{(\\ell )}\\right)\\ ,\\nonumber \\\\\\hat{b}_{\\mu \\nu }&=&\\sum _{\\ell \\ge 0}b_{\\mu \\nu }^{(\\ell )}Y^{(\\ell )}\\ ,\\qquad \\ \\ \\hat{ b}_{\\mu m}=\\sum _{\\ell \\ge 1}b_{\\mu }^{(\\ell )}Y^{(\\ell )}_{m}\\ ,\\qquad \\hat{b}_{mn}=\\epsilon _{mn}b^{(0)}Y^{(0)}\\ .$ The de Donder-Lorentz gauge (REF ) does not fix all the gauge symmetries, and consequently there are some residual ones generated by harmonic zero modes, $S^2$ Killing vector $Y^{(1)}_m$ and conformal Killing vectors $\\nabla _mY^{(1)}$ .","Specifically, these residual gauge symmetries are: The four dimensional coordinate transformation generated by $\\hat{\\xi }_{\\mu }=\\xi ^{(0)}_{\\mu }Y^{(0)}$ $\\delta h_{\\mu \\nu }^{(0)}=\\partial _{\\mu }\\xi ^{(0)}_{\\nu }+\\partial _{\\nu }\\xi ^{(0)}_{\\mu }\\ .$ The Stueckelberg shift symmetries generated by $\\hat{\\xi }_{m}=\\xi ^{(1)}\\nabla _{m}Y^{(1)}$ $\\delta h^{(1)}_{\\mu \\nu }&=&-\\partial _{\\mu }\\partial _{\\nu }\\xi ^{(1)},\\qquad \\delta N^{(1)}=-2\\xi ^{(1)} \\nonumber \\\\\\delta a^{(1)}&=&4g\\xi ^{(1)},\\qquad \\delta v^{(1)}=-\\alpha ^2\\xi ^{(1)}\\ .$ Linearized $SU(2)$ symmetry generated by $\\hat{\\xi }_{m}=\\xi ^{\\prime (1)}Y^{(1)}_m$ and $\\hat{\\Lambda }=-4g\\xi ^{\\prime (1)}Y^{(1)}$ $\\delta k_{\\mu }^{(1)}=\\partial _{\\mu }\\xi ^{\\prime (1)}\\ .$ Four dimensional $U(1)_{\\rm {R}}$ symmetry generated by $\\hat{\\Lambda }=\\Lambda ^{(0)} Y^{(0)}$ $\\delta a^{(0)}_{\\mu }=\\partial _{\\mu }\\Lambda ^{(0)}\\ .$ Abelian 2-form symmetry generated by $\\hat{\\Lambda }_{\\mu }=\\Lambda ^{(0)}_{\\mu } Y^{(0)}$ $\\delta b_{\\mu \\nu }^{(0)}=\\partial _{\\mu }\\Lambda _{\\nu }^{(0)}-\\partial _{\\nu }\\Lambda _{\\mu }^{(0)}\\ .$ We shall take into account these symmetries in the analysis of the spectrum below, where we treat the spin-2, spin-1 and spin-0 sectors separately.","In doing so we shall encounter the following wave operators ${\\cal O}_1 &\\equiv &\\hat{\\Box }_0 + \\alpha ^2 - M^2\\ ,\\nonumber \\\\[0.2cm]{\\cal O}_2 &\\equiv & \\hat{\\Box }_0^2 -M^2 \\hat{\\Box }_0 -\\alpha ^4\\,c_\\ell \\ ,\\nonumber \\\\[0.2cm]{\\cal O}_4 &\\equiv & \\hat{\\Box }_0^4 + (\\alpha ^2-M^2)\\, \\hat{\\Box }_0^3-2\\alpha ^2(\\alpha ^2c_\\ell - M^2)\\, \\hat{\\Box }_0^2 -4c_\\ell \\,\\alpha ^4 (\\alpha ^2 - M^2)\\, \\hat{\\Box }_0 -2\\alpha ^8\\, c_\\ell ^2\\ ,$ where $\\boxed{\\hat{\\Box }_0 \\equiv \\Box - \\alpha ^2c_\\ell }\\ .$ In particular, the operator ${\\cal O}_4$ has the property that for $\\ell =1$ it factorizes as ${\\cal O}_4|_{\\ell =1} &=& \\Box \\,{\\cal O}_3\\ ,\\nonumber \\\\[0.2cm]{\\cal O}_3 & \\equiv & \\Box ^3 -(M^2+7\\alpha ^2)\\Box ^2 +2\\alpha ^2 (4M^2+7\\alpha ^2)\\Box -12\\alpha ^4(M^2+\\alpha ^2)\\ .$ In the $\\ell =0$ sector, we will encounter the wave operator $\\widetilde{\\cal O}_2 = \\Box (\\Box +\\alpha ^2) -M^2(\\Box -2\\alpha ^2)\\ .$"],["The spin-2 sector contains only the transverse and traceless gravitons, which upon the use of the spin projector operators provided in the Appendix C, and for $\\ell \\ge 1$ , satisfy the following equation $\\ell \\ge 1:\\qquad {\\cal O}_2 \\left({{\\cal P}}^{2}h\\right)^{(\\ell )}_{\\mu \\nu }=0\\ ,$ where $P^2$ is the spin-2 projector defined in Appendix C. This equation describes two massive gravitons with mass squared $\\ell \\ge 1: \\qquad m^2_{\\pm } (\\ell )= \\frac{1}{2} \\left(M^2 +2\\alpha ^2c_\\ell \\pm \\sqrt{M^4+4\\alpha ^4 c_\\ell }\\right)\\ .$ The $\\ell =0$ needs to be treated separately, and in this case the gravitons satisfy $(\\Box -M^2)R^{L(0)}_{\\mu \\nu }=-M^2\\left( \\partial _{\\mu }\\partial _{\\nu }S^{(0)}+\\alpha ^2\\eta _{\\mu \\nu } S^{(0)}\\right)+\\partial _{\\mu }\\partial _{\\nu }(\\Box +\\alpha ^2) S^{(0)},$ where $S^{(0)}=\\phi ^{(0)}+N^{(0)}$ .","The solutions of this equation can be expressed as $h^{(0)}_{\\mu \\nu }=h^{\\prime (0)}_{\\mu \\nu }+h^{\\prime \\prime (0)}_{\\mu \\nu }$ , where $h^{\\prime \\prime (0)}_{\\mu \\nu }$ is completely determined by $S^{(0)}$ while $h^{\\prime (0)}_{\\mu \\nu }$ is the solution to the following equations modulo the gauge symmetry (REF ): $\\ell =0:\\qquad \\Box (\\Box -M^2)h^{\\prime (0)}_{\\mu \\nu }=0,\\qquad \\,R^{\\prime (0)}=0\\ ,$ which describe a massless graviton and massive graviton with $m^2=M^2$ .","Let $\\ell \\ge 2$ .","Then, the spin-1 sector consists of eight vectors $(P^{1}h)_{\\mu \\nu },\\partial ^{\\nu }b_{\\mu \\nu }, z^{T}_{\\mu }, k^{T}_{\\mu },a^{T}_{\\mu },v^{T}_{\\mu },b_{\\mu \\nu }^{T},b^{T}_{\\mu })$ , where “T” indicates the transverse part and $(P^{1}h)_{\\mu \\nu }=P^1_{\\mu \\nu }{}^{\\rho \\sigma }h_{\\rho \\sigma }$ (see Appendix B).","Of these eight vectors, $(k^{T}_{\\mu },a^{T}_{\\mu },v^{T}_{\\mu },b_{\\mu \\nu }^{T},b^{T}_{\\mu })$ have mixing with each other through the following equations $0&=&{\\cal O}_2 b^{T(\\ell )}_{\\mu \\nu }+4gM^2\\star F^{(\\ell )}_{\\mu \\nu }(a)-\\alpha ^4c_\\ell \\left(\\star F^{(\\ell )}_{\\mu \\nu }(k)-\\star F^{(\\ell )}_{\\mu \\nu }(b)\\right)\\ ,\\\\0&=& {\\cal O}_1 \\hat{\\Box }_0 b^{T(\\ell )}_{\\mu }+{\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } }\\alpha ^2\\epsilon _{\\mu }^{~\\nu \\lambda \\rho }\\partial _{\\nu }b_{\\lambda \\rho }^{T(\\ell )}\\ ,\\\\0&=&(\\hat{\\Box }_0+\\alpha ^2)a^{T(\\ell )}_{\\mu }-4g\\alpha ^2c_\\ell k^{T(\\ell )}_{\\mu }+4gv_{\\mu }^{T(\\ell )}-2g\\epsilon _{\\mu }^{~\\nu \\lambda \\rho }\\partial _{\\nu }b_{\\lambda \\rho }^{T(\\ell )}\\ ,\\\\0&=&(\\hat{\\Box }_0-M^2)v^{T(\\ell )}_{\\mu }+\\alpha ^2c_\\ell k^{T(\\ell )}_{\\mu }+{\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 4} } }gM^2a^{(\\ell )}_{\\mu }\\ ,\\\\0&=&\\biggl ( {\\cal O}_2+\\alpha ^2(\\hat{\\Box }_0-\\alpha ^2c_\\ell )\\biggr )k^{T(\\ell )}_{\\mu }+4gM^2a^{T(\\ell )}_{\\mu }+{\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } } \\alpha ^2\\left(4 v^{T(\\ell )}_{\\mu }-\\epsilon _{\\mu }^{~\\nu \\lambda \\rho }\\partial _{\\nu }b_{\\lambda \\rho }^{T(\\ell )}\\right)\\ .$ Diagonalising the associated $5\\times 5$ operator-valued matrix, we find that the modes are annihilated by ${\\cal O}_1^2\\,{\\cal O}_2\\,{\\cal O}_4$ .","In particular, the linear combinations with coefficients $(-1,-4g\\alpha ^2c_\\ell /M^2,0,0,1)$ and $(2,4g\\alpha ^2c_\\ell /M^2,1,0,0)$ are annihilated by ${\\cal O}_1$ .","The remaining vectors, namely, $ \\left( (P^{1} h)_{\\mu \\nu },\\partial ^{\\nu }b_{\\mu \\nu }, z_\\mu \\right)$ are separately annihilated by ${\\cal O}_1$ as well.","In summary, for $\\ell \\ge 2$ the total wave operator can be denoted by $\\ell \\ge 2:\\qquad {\\cal O}^{(1)} = {\\cal O}_1^6\\,{\\cal O}_2\\,{\\cal O}_4\\ ,$ implying six massive vectors with mass squared $\\ell \\ge 2:\\qquad m^2(\\ell ) = M^2+ \\alpha ^2 (c_\\ell -1)\\ ,$ two massive vectors with mass squared defined in Eq.","(REF ) and four massive vectors whose squared masses are given by the roots of the polynomial $&& x^4 + ax^3+bx^2+cx+d=0\\ ,\\nonumber \\\\[0.2cm]&& a= -M^2-(4\\ell ^2+4\\ell -1)\\alpha ^2\\ ,\\nonumber \\\\[0.2cm]&& b= \\alpha ^2\\left[ 2M^2+\\ell (\\ell +1)\\left(\\,(6\\ell ^2+6\\ell -5)\\alpha ^2+3M^2\\right)\\,\\right] \\ ,\\nonumber \\\\[0.2cm]&& c=-\\ell (\\ell +1)\\alpha ^2 \\left\\lbrace \\alpha ^2\\left[4+\\ell (\\ell +1)(4\\ell ^2+4\\ell -7)\\right]\\alpha ^2+3\\ell (\\ell +1)M^2\\right\\rbrace \\ ,\\nonumber \\\\[0.2cm]&& d=\\ell ^2(\\ell +1)^2(\\ell -1)(\\ell +2)\\alpha ^6 \\left[(\\ell ^2+\\ell -1)\\alpha ^2+M^2\\right]\\ .$ Next, consider the case $\\ell =1$ .","Recalling the factorization result given in (REF ), the total wave operator becomes $\\ell =1:\\qquad {\\cal O}^{(1)} = {\\cal O}_1^6 {\\cal O}_2|_{\\ell =1} \\Box {\\cal O}_3\\ .$ In particular the massless vector is a linear combination of $(k^{T(1)}_{\\mu },a^{T(1)}_{\\mu },v^{T(1)}_{\\mu },b_{\\mu \\nu }^{T(1)},b^{T(1)}_{\\mu })$ with mixing coefficients $(1,-4g,\\alpha ^2,1,0)$ .","The squared masses associated with ${\\cal O}_1^6 {\\cal O}_2|_{\\ell =1}$ can be read of from (REF ) and (REF ) by setting $\\ell =1$ , and those associated with ${\\cal O}_3$ are the roots of the following polynomial $x^3 -(M^2+7\\alpha ^2)x^2 +2\\alpha ^2 (4M^2+7\\alpha ^2)x -12\\alpha ^4(M^2+\\alpha ^2)=0\\ .$ There remains the case of $\\ell =0$ .","The only vector fluctuations at this level are $(b_{\\mu \\nu }^{T(0)},a^{T(0)}_{\\mu },v^{T(0)}_{\\mu })$ .","Upon diagonalising the associated $3\\times 3$ operator-valued matrix, we find that the modes are annihilated by the following partially-factorising operator polynomial $\\ell =0:\\qquad {\\cal O}^{(1)} = \\Box (\\Box -M^2) \\widetilde{\\cal O}_2 \\ ,$ where the would-be massless vector annihilated by $\\Box $ is eaten by the two form and the operator ${\\tilde{O}}_2$ is defined in (REF ).","Thus, for $\\ell =0$ there are no massless vector modes, a massive vector with mass $M$ and two massive vectors with squared masses given by $\\widetilde{m}^2_{\\pm } = \\frac{1}{2}\\left(M^2-\\alpha ^2 \\pm \\sqrt{M^4-10M^2\\alpha ^2 +\\alpha ^4}\\, \\right)\\ .$ We start with the case $\\ell \\ge 2$ .","Defining $\\tilde{\\varphi }=\\omega _{\\mu \\nu }h^{\\mu \\nu }$ and $\\varphi ={\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 3} } }\\theta _{\\mu \\nu }h^{\\mu \\nu }$ (see Appendix C), this sector consists of thirteen scalars $\\left(\\phi , N,\\varphi , \\tilde{\\varphi },a, v, \\partial ^{\\mu }k_{\\mu },\\partial ^\\mu b_{\\mu },\\partial ^\\mu a_{\\mu },\\partial ^\\mu v_{\\mu }, \\partial ^\\mu z_{\\mu }, z, \\tilde{z}\\right)$ .","The first six scalars $(\\phi ,N, \\varphi , \\tilde{\\varphi },a, v)$ mix as follows $\\ell \\ge 2:\\ \\ 0&=&2(\\hat{\\Box }_0+\\alpha ^2)\\phi ^{(\\ell )}+(2\\hat{\\Box }_0+2\\alpha ^2+ \\alpha ^2c_\\ell )N^{(\\ell )}+3\\hat{\\Box }_0\\varphi ^{(\\ell )}-\\alpha ^2c_\\ell \\tilde{\\varphi }^{(\\ell )}\\ ,\\\\0&=&{\\cal O}_2\\varphi ^{(\\ell )}+2\\alpha ^2M^2 \\left[\\phi ^{(\\ell )}-2gc_\\ell a^{(\\ell )}\\right]-2\\alpha ^2(M^2-\\alpha ^2c_\\ell )N^{(\\ell )}+2\\alpha ^3c_\\ell v^{(\\ell )}\\ ,\\\\0&=&\\biggl (M^2-(\\alpha ^2+\\Box )\\biggr )\\tilde{\\varphi }^{(\\ell )} +3(M^2-\\alpha ^2)\\varphi ^{(\\ell )}+2M^2\\phi ^{(\\ell )}+(2\\hat{\\Box }_0 +2\\alpha ^2+\\alpha ^2c_\\ell )N^{(\\ell )}\\ ,\\nonumber \\\\&& \\\\0&=& (\\hat{\\Box _0}+\\alpha ^2)a^{(\\ell )}+4gN^{(\\ell )}-2g(\\tilde{\\varphi }^{(\\ell )}+3\\varphi ^{(\\ell )})+4gv^{(\\ell )}\\ ,\\\\0&=&(\\hat{\\Box }_0-M^2)v^{(\\ell )}-\\alpha ^2 N^{(\\ell )}+\\frac{1}{2}\\alpha ^2(\\tilde{\\varphi }^{(\\ell )} +3\\varphi ^{(\\ell )})-2 gM^2a^{(\\ell )}\\ ,\\\\0&=&\\biggl (M^2-(\\alpha ^2-\\Box )\\biggr )N^{(\\ell )}+3M^2\\varphi ^{(\\ell )}+2M^2\\phi ^{(\\ell )}-4gM^2a^{(\\ell )}-2\\alpha ^2 v^{(\\ell )}-\\alpha ^2(c_\\ell -1)\\tilde{\\varphi }^{(\\ell )}\\ .\\nonumber \\\\&&$ Diagonalising the associated $6\\times 6$ operator-valued matrix, we find that the modes are annihilated by ${\\cal O}_1^3\\,{\\cal O}_4$ .","Of the remaining scalars, $(\\partial ^{\\mu }k_{\\mu },\\partial ^\\mu b_{\\mu },\\partial ^\\mu a_{\\mu },\\partial ^\\mu v_{\\mu },\\tilde{v})$ mix but only three of them are dynamical.","We choose these to be $(\\partial ^{\\mu }k_{\\mu },\\partial ^\\mu b_{\\mu },\\tilde{v})$ which are separately annihilated by ${\\cal O}_1$ , while $(\\partial ^\\mu a_{\\mu },\\partial ^\\mu v_{\\mu })$ are determined by $\\partial ^\\mu a_\\mu ^{(\\ell )} &=& \\frac{\\alpha ^2}{2g}\\left( {\\tilde{v}}^{(\\ell )} - \\partial ^\\mu k_\\mu ^{(\\ell )}\\right)\\ ,\\nonumber \\\\[0.2cm]\\partial ^\\mu v_\\mu ^{(\\ell )} &=& \\alpha ^2 (c_\\ell -1){\\tilde{v}}^{(\\ell )} +\\alpha ^2 \\partial ^\\mu k_\\mu ^{(\\ell )}\\ .$ Finally, the remaining scalars $(z, \\tilde{z})$ are annihilated by ${\\cal O}_1$ , and the longitudinal modes $\\partial ^{\\mu }z_{\\mu }$ are given in terms of $z$ and $\\tilde{z}$ , by virtue of the equation $\\hat{D}^{M}\\hat{z}_M=0$ .","Thus, for $\\ell \\ge 2$ the total wave operator is given by $\\ell \\ge 2:\\qquad {\\cal O}^{(0)} = {\\cal O}_1^{10}\\,{\\cal O}_4\\ .$ Of these, the three linear combinations of $(\\phi , N,\\varphi , \\tilde{\\varphi },a, v)$ with coefficients $(2+\\alpha ^2{c_\\ell }/{M^2},-2,0,0,0,2)$ , $(-2-\\alpha ^2c_\\ell /{M^2},2,0,0,-{8g}/{M^2},0)$ and $(-2+\\alpha ^2{c_\\ell }/{M^2},2,0,4,0,0)$ are annihilated by ${\\cal O}_1$ .","In the case of $\\ell =1$ , utilizing the residual symmetry (REF ) and (REF ), one can eliminate $N^{(1)}$ and $\\partial ^\\mu k_{\\mu }^{(1)}$ .","Taking into account the fact that the harmonic expansion of ${\\hat{z}}_m$ contributes only one complex scalar for $\\ell =1$ , namely $z^{(1)}$ , we find that for $\\ell =1$ the total wave operator for the scalar fields is given by $\\ell =1:\\qquad {\\cal O}^{(0)} = {\\cal O}_1^6|_{\\ell =1}\\,{\\cal O}_3\\ .$ There remains the case of $\\ell =0$ .","Of the remaining scalars, $(\\phi ^{(0)},N^{(0)},b^{(0)})$ satisfy the equations $ \\widetilde{\\cal O}_2\\,S^{(0)}=0$ where $S^{(0)}=\\phi ^{(0)}+N^{(0)}$ , $\\Box \\,{{\\cal O}_1}|_{\\ell =0}\\, b^{(0)}=0$ and $\\Box \\,{{\\cal O}_1}|_{\\ell =0}\\, N^{(0)}= 0$ .","Finally, there is a complex scalar $z^{(0)}$ annihilated by ${\\cal O}_1|_{\\ell =0}$ .","Thus, for $\\ell =0$ the total wave operator or the scalar fields is given by $\\ell =0: \\qquad {\\cal O}^{(0)} = \\Box ^2\\,{\\cal O}_1^4|_{\\ell =0}\\, \\widetilde{\\cal O}_2\\ ,$ implying two massless and six massive scalars in this sector."],["Fermionic sector","In terms of the complex spinor, the linearized equations of fermions around the background (REF ) are given by $0&=& {\\bar{\\Gamma }}^{MPQ}\\bar{D}_{P}\\psi _{Q}+2 i{\\bar{\\Gamma }}^{NM}\\bar{D}_N\\chi -4g i{\\bar{\\Gamma }}^M\\lambda -{\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } }\\bar{F}_{PQ}{\\bar{\\Gamma }}^{PQ}{\\bar{\\Gamma }}^{M}\\lambda -\\frac{1}{8M^2}\\Theta ^{M}\\ ,\\\\0&=&{\\bar{\\Gamma }}^{MN}\\bar{D}_M\\psi _{N}-2i{\\bar{\\Gamma }}^{M}\\bar{D}_{M}\\chi -8gi\\lambda ,\\\\0&=& 2gi{\\bar{\\Gamma }}^M\\psi _{M}+8g\\chi -4{\\bar{\\Gamma }}^M {\\bar{D}}_M\\lambda +{\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 4} } }\\bar{F}_{PQ}{\\bar{\\Gamma }}^{M}{\\bar{\\Gamma }}^{PQ}\\psi _{M}\\ ,$ where $\\bar{D}_M\\psi &=&(\\partial _M+{\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 4} } }\\bar{\\omega }_M^{~AB}{\\bar{\\Gamma }}_{AB})\\psi -{\\textstyle {\\frac{\\scriptstyle i}{\\scriptstyle 2} } }\\bar{V}_M\\psi \\ ,\\nonumber \\\\[0.2cm]\\Theta ^{M}&=& 8{\\bar{\\Gamma }}^P\\bar{D}_Q\\bar{ D}_P\\psi ^{QM} -2\\bar{R}^{PM}_{~~~ST}{\\bar{\\Gamma }}^Q{\\bar{\\Gamma }}^{ST}\\bar{D}_{P}\\psi _{Q}+2\\bar{R}^{PQ}_{~~~ST}{\\bar{\\Gamma }}^{ST}{\\bar{\\Gamma }}^{M}\\bar{D}_{P}\\psi _{Q}\\nonumber \\\\&& +8i\\bar{G}^{P[M}{\\bar{\\Gamma }}^{Q]}\\bar{D}_{Q}\\psi _{P}-8i\\bar{G}^{M[P}{\\bar{\\Gamma }}^{Q]}\\bar{D}_{Q}\\psi _{P}\\ .$ In the remainder of this section, we shall drop the “bar” on the covariant derivatives as well as the $\\Gamma $ -matrices for simplicity in notation.","Since the (REF ) contains gauge field, we adopt the spin-weighted harmonics $ _{s-{\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } }}\\eta ^{(\\ell )} $ , which are described in detail in appendix B, as the expansion basis.","In this section we will need the harmonics for $s=0$ , namely $ _{-{\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } }} \\eta ^{(\\ell )} $ which we will denote as $\\eta ^{(\\ell )}$ for brevity in notation.","These harmonics satisfy the relations, $\\eta _-^{(0)}=\\eta \\ ,\\qquad \\eta ^{(\\ell )}_+ = \\frac{1}{i\\alpha \\sqrt{c_\\ell }}\\nabla _n Y^{(\\ell )}\\sigma ^n\\eta \\ ,\\qquad \\eta ^{(\\ell )}_- = Y^{(\\ell )}\\eta ,\\qquad \\ell \\ge 1\\ ,$ and have the following properties $\\sigma _3 {\\eta _{\\pm }^{(\\ell )}}&=\\pm {\\eta _{\\pm }^{(\\ell )}}\\ ,&\\qquad \\quad \\sigma ^n {D}_n{\\eta _{\\pm }^{(\\ell )}}& =i\\alpha \\sqrt{c_\\ell }\\,{\\eta _{\\mp }^{(\\ell )}}\\ ,\\nonumber \\\\[{D}_m, {D}_n]{\\eta _-^{(\\ell )}}&=0\\ , &\\qquad \\quad [{D}_m,{D}_n]{\\eta _+^{(\\ell )}} &=i\\alpha ^2\\epsilon _{mn}{\\eta _+^{(\\ell )}} ,\\nonumber \\\\{D}^n{D}_n{\\eta _-^{(\\ell )}}&=-\\alpha ^2c_\\ell {\\eta _-^{(\\ell )}}, &\\qquad \\quad {D}^n{D}_n{\\eta _+^{(\\ell )}}&=-\\alpha ^2(c_\\ell -1){\\eta _+^{(\\ell )}}\\ .$ The Killing spinor $\\eta $ also has the property $\\sigma _3\\eta =-\\eta $ .","Furthermore, given that $\\Gamma _7=\\gamma _5\\times \\sigma _3$ (see Appendix A), the chirality property of a spinor in $6D$ correlates the $4D$ and $\\sigma _3$ chiralities.","Since there is no gamma traceless and transverse spin-3/2 harmonics on the $S^2$ , generically, the harmonic expansion are carried out as $\\hat{\\psi }_{\\mu } &=&\\psi _{\\mu -}^{(0)}\\otimes {\\eta ^{(0)}}+\\sum _{\\ell \\ge 1}\\left(\\psi _{\\mu +}^{(\\ell )}\\otimes {\\eta _+^{(\\ell )}}+\\psi _{\\mu -}^{(\\ell )}\\otimes {\\eta _-^{(\\ell )}}\\right)\\ ,\\nonumber \\\\\\hat{\\psi }_m &=& \\Gamma _m\\psi ^{(0)}_+\\otimes {\\eta ^{(0)}}+\\Gamma _m\\sum _{\\ell \\ge 1}\\left(\\psi _-^{(\\ell )}\\otimes {\\eta _+^{(\\ell )}}+\\psi _+^{(\\ell )}\\otimes {\\eta _-^{(\\ell )}}\\right)\\nonumber \\\\&&+\\sum _{\\ell \\ge 1}\\left(\\tilde{\\psi }_+^{(\\ell )}\\otimes D_{\\lbrace m\\rbrace }{\\eta _+^{(\\ell )}}+\\tilde{\\psi }_-^{(\\ell )}\\otimes D_{\\lbrace m\\rbrace }{\\eta _-^{(\\ell )}}\\right)\\ ,\\nonumber \\\\\\hat{\\chi }&=&\\chi ^{(0)}_+ \\otimes {\\eta ^{(0)}}+ \\sum _{\\ell \\ge 1} \\left(\\chi _-^{(\\ell )}\\otimes {\\eta _+^{(\\ell )}}+\\chi _+^{(\\ell )}\\otimes {\\eta _-^{(\\ell )}}\\right)\\ ,\\nonumber \\\\\\hat{\\lambda } &=& \\lambda ^{(0)}_-\\otimes \\eta ^{(0)}+\\sum _{\\ell \\ge 1}\\left( \\lambda _+^{(\\ell )} \\eta _+^{(\\ell )}+\\lambda _-^{(\\ell )} \\eta _-^{(\\ell )}\\right)\\ ,$ where $D_{\\lbrace m\\rbrace }$ is the gamma traceless covariant derivative and the $\\pm $ subscripts denote chirality property under $\\gamma _5$ .","Using the 6D linearized fermionic gauge symmetry (REF ), one can impose the following gauge condition $\\hat{\\psi }_{\\lbrace m\\rbrace }=0\\ ,$ where $\\lbrace m\\rbrace $ means $\\Gamma $ -traceless.","As a consequence, the expansion takes the following simpler forms $\\hat{\\psi }_{\\mu } &=&\\psi _{\\mu -}^{(0)}\\otimes {\\eta ^{(0)}}+\\sum _{\\ell \\ge 1}\\left(\\psi _{\\mu +}^{(\\ell )}\\otimes {\\eta _+^{(\\ell )}}+\\psi _{\\mu -}^{(\\ell )}\\otimes {\\eta _-^{(\\ell )}}\\right)\\ ,\\\\\\hat{\\psi }_m &=& \\Gamma _m\\psi ^{(0)}_+\\otimes {\\eta ^{(0)}}+\\Gamma _m\\sum _{\\ell \\ge 1}\\left(\\psi _-^{(\\ell )}\\otimes {\\eta _+^{(\\ell )}}+\\psi _+^{(\\ell )}\\otimes {\\eta _-^{(\\ell )}}\\right)\\ ,\\\\\\hat{\\chi } &=& \\chi ^{(0)}_+\\otimes {\\eta ^{(0)}}+ \\sum _{\\ell \\ge 1}\\left(\\chi _-^{(\\ell )}\\otimes {\\eta _+^{(\\ell )}} +\\chi _+^{(\\ell )}\\otimes {\\eta _-^{(\\ell )}}\\right)\\ ,\\\\\\hat{\\lambda } &=& \\lambda ^{(0)}_-\\otimes {\\eta ^{(0)}}+\\sum _{\\ell \\ge 1} \\left(\\lambda _+^{(\\ell )}\\otimes {\\eta _+^{(\\ell )}}+\\lambda _-^{(\\ell )}\\otimes {\\eta _-^{(\\ell )}}\\right)\\ .$ The gauge choice (REF ) does not fix all the gauge symmetries, we find the following residual symmetry transformations Generated by $\\hat{\\epsilon }^{(0)}=\\epsilon ^{(0)}{\\eta ^{(0)}}$ : $\\delta \\psi ^{(0)}_\\mu =\\partial _{\\mu }\\epsilon ^{(0)},$ Generated by $\\hat{\\epsilon }=\\epsilon _+^{(1)}{\\eta _+^{(1)}}$ : $\\delta \\psi _{\\mu +}^{(1)}&=&\\partial _{\\mu }\\epsilon _+^{(1)}+\\frac{i}{\\sqrt{2}}\\,\\alpha \\,\\epsilon _+^{(1)}\\ ,\\\\\\delta \\lambda _+^{(1)}&=&ig\\epsilon _+^{(1)}\\ .$ We shall take into account these symmetries in the analysis of the spectrum below, where we treat the spin-3/2, spin-1/2 sectors separately."],["Let us begin with the restriction $\\ell \\ge 1$ .","This sector contains only the gravitino fields which satisfy the following equations $\\ell \\ge 1:\\quad &&{\\partial }\\Big ((\\hat{\\Box }_0+\\alpha ^2)-M^2\\Big ) P^{3/2}\\psi _{\\mu +}^{(\\ell )}+i\\alpha \\sqrt{c_\\ell }\\Big ((\\hat{\\Box }_0+\\alpha ^2)-M^2\\Big )P^{3/2}\\psi _{\\mu -}^{(\\ell )}=0\\ ,\\nonumber \\\\&&i\\alpha \\sqrt{c_\\ell }\\Big ((\\hat{\\Box }_0+\\alpha ^2)-M^2\\Big )P^{3/2}\\psi _{\\mu +}^{(\\ell )}-{\\partial }\\Big (\\hat{\\Box }_0-M^2\\Big )P^{3/2}\\psi _{\\mu -}^{(\\ell )}=0\\ ,$ where $P^{3/2}$ is the spin-3/2 projector operator defined in Appendix C. Diagonalising the associated $2\\times 2$ operator-valued matrix, we find that the modes are annihilated by the partially-factorising operator polynomial, of sixth order in ${\\partial }$ , given by $\\ell \\ge 1:\\qquad {\\cal O}^{(3/2)} = {\\cal O}_1 {\\cal O}_2\\ .$ Next, consider the case of $\\ell =0$ .","By choosing the gauge $\\gamma ^{\\mu }\\psi ^{(0)}_{\\mu }=0$ , the gravitino equation can be written as ${\\partial }\\left(\\Box -M^2\\right)\\psi ^{(0)\\mu }=-\\left({\\partial }\\partial ^{\\mu } - M^2\\gamma ^{\\mu }\\right)\\Psi ^{(0)}-2M^2\\left( \\gamma ^{\\mu \\nu }\\partial _{\\nu }\\psi ^{(0)}-i\\gamma ^{\\mu \\nu }\\partial _{\\nu }\\chi ^{(0)}\\right)\\ .$ The solutions of above equation can be expressed as $\\psi ^{(0)}_{\\mu }=\\psi ^{\\prime (0)}_{\\mu }+\\psi ^{\\prime \\prime (0)}_{\\mu }$ where $\\psi ^{\\prime \\prime (0)}_{\\mu }$ is completely determined by $\\psi ^{(0)}$ and $\\chi ^{(0)}$ while $\\psi ^{\\prime (0)}_{\\mu }$ is the solution to the following equations modular gauge symmetry (REF ) $\\ell =0:\\qquad {\\partial }(\\Box -M^2)\\psi ^{\\prime (0)}_{\\mu }=0\\ ,\\qquad \\gamma ^{\\mu }\\psi ^{\\prime (0)}_{\\mu }=0\\ ,\\qquad \\partial ^{\\mu }\\psi ^{\\prime (0)}_{\\mu }=0\\ .$ It describes a massless and two massive gravitini.","The $\\ell \\ge 2$ sector consists of ten spin 1/2 fields $(\\Lambda _+,\\Psi _+,\\Lambda _-,\\Psi _-,$ $\\psi _+,\\psi _-,\\chi _+,\\chi _-,\\lambda _+,\\lambda _-)$ , where $\\Psi ^{(\\ell )}\\equiv \\partial ^{\\mu }\\psi _{\\mu }^{(\\ell )}$ and $\\gamma ^{\\mu }\\psi _{\\mu }^{(\\ell )}\\equiv \\Lambda ^{(\\ell )}$ .","The linearized equations describing their mixing are $0&=&{\\partial }\\Lambda ^{(\\ell )}_{-}-\\Psi _+^{(\\ell )} +i\\alpha \\sqrt{c_\\ell }\\psi _+^{(\\ell )}-i\\alpha \\sqrt{c_\\ell }\\Lambda _+^{(\\ell )} +2{\\partial }\\psi _-^{(\\ell )}-2i{\\partial }\\chi _-^{(\\ell )} +2\\alpha \\sqrt{c_\\ell }\\chi _+^{(\\ell )}-8gi\\lambda _+^{(\\ell )}\\ ,\\\\[0.2cm]0&=&{\\partial }\\Lambda _+^{(\\ell )} -\\Psi _-^{(\\ell )}+i\\alpha \\sqrt{c_\\ell }\\psi _-^{(\\ell )}+i\\alpha \\sqrt{c_\\ell }\\Lambda _-^{(\\ell )} -2{\\partial }\\psi _+^{(\\ell )}+2i{\\partial }\\chi _+^{(\\ell )} +2\\alpha \\sqrt{c_\\ell }\\chi _-^{(\\ell )}-8gi\\lambda _-^{(\\ell )}\\ ,\\\\[0.2cm]0&=&i g\\Lambda _-^{(\\ell )}+2g\\chi _-^{(\\ell )} -{\\partial }\\lambda _+^{(\\ell )}-i\\alpha \\sqrt{c_\\ell }\\lambda _-^{(\\ell )}\\ ,\\\\[0.2cm]0&=&2i g\\psi _+^{(\\ell )}+2g\\chi _+^{(\\ell )}+{\\partial }\\lambda _-^{(\\ell )}-i\\alpha \\sqrt{c_\\ell }\\lambda _+^{(\\ell )}\\ ,\\\\[0.2cm]0&=&i\\alpha \\sqrt{c_\\ell }{\\partial }(\\Box +M^2)\\psi _+^{(\\ell )}-iM^2\\alpha \\sqrt{c_\\ell }{\\partial }\\Lambda _+^{(\\ell )}+i\\alpha \\sqrt{c_\\ell }\\biggl (M^2-\\alpha ^2+\\alpha ^2c_\\ell \\biggr )\\Psi _-^{(\\ell )}+2M^2\\alpha \\sqrt{c_\\ell }{\\partial }\\chi _+^{(\\ell )}\\nonumber \\\\&&-8igM^2{\\partial }\\lambda _+^{(\\ell )}+\\alpha ^2(c_\\ell -1){\\partial }\\Psi _+^{(\\ell )}+\\alpha ^2(2-c_\\ell )\\Box \\psi _-^{(\\ell )}\\ ,\\\\[0.2cm]0&=&-i{\\partial }\\biggl (\\Box +M^2-\\alpha ^2)\\biggr )\\psi _-^{(\\ell )}-iM^2{\\partial }\\Lambda _-^{(\\ell )}+i\\biggl (\\alpha ^2+\\alpha ^2c_\\ell - M^2\\biggr )\\Psi _+^{(\\ell )}-2M^2{\\partial }\\chi _-^{(\\ell )}\\nonumber \\\\&&-\\alpha \\sqrt{c_\\ell }\\left( {\\partial }\\Psi _-^{(\\ell )} +\\Box \\psi _+^{(\\ell )}\\right)\\ ,\\\\[0.2cm]0&=&{\\partial }\\biggl (\\Box +\\alpha ^2-\\alpha ^2c_\\ell +2M^2\\biggr )\\Lambda _-^{(\\ell )}-\\biggl (\\Box +2\\alpha ^2-2\\alpha ^2c_\\ell +2M^2\\biggr )\\Psi _+^{(\\ell )}+i\\alpha \\sqrt{c_\\ell }(\\Box +4M^2)\\psi _+^{(\\ell )}\\nonumber \\\\&&+{\\partial }\\biggl (6M^2+2\\alpha ^2-\\alpha ^2c_\\ell )\\biggr )\\psi _-^{(\\ell )}-i\\alpha \\sqrt{c_\\ell }\\biggl (\\Box +\\alpha ^2-\\alpha ^2c_\\ell +3M^2\\biggr )\\Lambda _+^{(\\ell )}-6iM^2{\\partial }\\chi _-^{(\\ell )}\\nonumber \\\\&&+8\\alpha M^2\\sqrt{c_\\ell }\\chi _+^{(\\ell )}-32igM^2\\lambda _+^{(\\ell )}+i\\alpha \\sqrt{c_\\ell }{\\partial }\\Psi _-^{(\\ell )}\\ ,\\\\[0.2cm]0&=&-{\\partial }\\biggl (\\Box -\\alpha ^2c_\\ell +2M^2)\\biggr )\\Lambda _+^{(\\ell )}+\\biggl (\\Box -2\\alpha ^2c_\\ell +2M^2)\\biggr )\\Psi _-^{(\\ell )} -i\\alpha \\sqrt{c_\\ell }\\biggl (\\Box -4\\alpha ^2+4M^2)\\biggr )\\psi _-^{(\\ell )}\\nonumber \\\\&&+{\\partial }(6M^2-\\alpha ^2c_\\ell )\\psi _+^{(\\ell )}-i\\alpha \\sqrt{c_\\ell }\\biggl (\\Box +\\alpha ^2-\\alpha ^2c_\\ell +3M^2\\biggr )\\Lambda _-^{(\\ell )}-6iM^2{\\partial }\\chi _+^{(\\ell )}-8\\alpha M^2\\sqrt{c_\\ell }\\chi _-^{(\\ell )}\\nonumber \\\\&&+i\\alpha \\sqrt{c_\\ell }{\\partial }\\Psi _+^{(\\ell )}\\ ,\\\\[0.2cm]0&=& M^2\\left(-\\Lambda _-^{(\\ell )}+2i\\chi _-^{(\\ell )}\\right)-\\left(2\\Box -\\alpha ^2c_\\ell \\right)\\psi _-^{(\\ell )}+{\\partial }\\Psi _+^{(\\ell )} +i\\alpha \\sqrt{c_\\ell }\\Psi _-^{(\\ell )}+i\\alpha \\sqrt{c_\\ell }{\\partial }\\psi _+^{(\\ell )}\\ ,\\\\[0.2cm]0&=&(\\alpha ^2-M^2)\\Lambda _+^{(\\ell )}-2iM^2\\chi _+^{(\\ell )} +\\left(2\\Box +2\\alpha ^2- \\alpha ^2c_\\ell \\right)\\psi _+^{(\\ell )}-i\\alpha \\sqrt{c_\\ell }\\Psi _+^{(\\ell )}+{\\partial }\\Psi _-^{(\\ell )}+i\\alpha \\sqrt{c_\\ell }{\\partial }\\psi _-^{(\\ell )}\\ .\\nonumber \\\\&&$ Diagonalising the associated $10\\times 10$ operator-valued matrix, we find that the modes are annihilated by the partially-factorising operator polynomial, of eighteenth order in ${\\partial }$ , given by $\\ell \\ge 2:\\qquad {\\cal O}^{1/2} = {\\cal O}_1^5\\, {\\cal O}_4\\ .$ Next we consider the case of $\\ell =1$ .","In this case, one can use the fermionic shift symmetry (REF ) to eliminate $\\psi _+^{(1)}$ .","Consequently we get 9 by 9 mixing and we find that the modes are annihilated by the partially-factorising operator polynomial, of fifteenth order in ${\\partial }$ , given by $\\ell =1:\\qquad {\\cal O}^{1/2} = {\\cal O}_1^4|_{\\ell =1}\\, {\\partial }\\, {\\cal O}_3\\ ,$ where ${\\cal O}_3$ is defined in (REF ).","The factor ${\\partial }$ demonstrates that there is massless spin-1/2 mode.","This massless mode corresponds to linear combinations of $(\\Lambda ^{(1)}_+,\\Psi ^{(1)}_+,\\Lambda ^{(1)}_-,\\Psi ^{(1)}_-,\\psi ^{(1)}_-,\\chi ^{(1)}_+,\\chi ^{(1)}_-,\\lambda ^{(1)}_+,\\lambda ^{(1)}_-)$ with mixing coefficients $(8,0,0,0,-2,-2i,0,0,1)$ .","There remains the case of $\\ell =0$ .","In this case, we have $\\ell =0:\\qquad 0&=&-\\Psi ^{(0)}-2{\\partial }\\psi ^{(0)}+2i{\\partial }\\chi ^{(0)}-8gi\\lambda ^{(0)}\\ ,\\\\0&=& 2gi\\psi ^{(0)}+2g\\chi ^{(0)}+{\\partial }\\lambda ^{(0)}\\ ,\\\\0&=& (1+\\frac{1}{2M^2}\\Box )\\Psi ^{(0)}+3{\\partial }\\psi ^{(0)}-3i{\\partial }\\chi ^{(0)}\\ ,\\\\0&=& -\\Psi ^{(0)}-{\\partial }\\left(1+\\frac{1}{M^2} (\\Box +\\alpha ^2)\\right)\\psi ^{(0)}+2i{\\partial }\\chi ^{(0)}-8gi\\lambda ^{(0)}\\ .$ Diagonalising the associated $4\\times 4$ operator-valued matrix, we find that the modes are annihilated by the partially-factorising operator polynomial, of seventh order in ${\\partial }$ , given by $\\ell =0:\\qquad {\\cal O}^{(1/2)} = {\\cal O}_1|_{\\ell =0}\\,{\\partial }\\,\\widetilde{\\cal O}_2\\ .$ Thus, at the $\\ell =0$ level, there is only one massless spin-1/2 modes given by $\\Psi ^{(0)}=0,\\lambda ^{(0)}=0,i\\psi ^{(0)}+\\chi ^{(0)}=0$ ."],["The supermultiplet structure and stability","In arranging the full spectrum described above into a collection of supermultiplet structure, it is useful to recall that following massive supermultiplets: $&{\\rm massive\\ supergravity\\ multiplet:} & \\qquad (h_{\\mu \\nu }, A_\\mu , \\Psi _\\mu )\\ ,\\nonumber \\\\&{\\rm massive\\ gravitino \\ multiplet:} & \\qquad (\\psi _\\mu , Z_\\mu , \\chi )\\ ,\\nonumber \\\\&{\\rm massive\\ vector multiplet\\ multiplet:} & \\qquad (A_\\mu , \\phi , \\lambda )\\ ,\\nonumber \\\\&{\\rm massive\\ scalar\\ multiplet:} & \\qquad (Z, \\psi )\\ ,$ where $A_\\mu $ is a real and $Z_\\mu $ is a complex vector, $\\phi $ is a real and $Z$ is a complex scalar, and $\\Psi _\\mu $ is Dirac and $\\psi _\\mu ,\\chi , \\psi $ are Majorana.","The Dirac gravitino can be written as $\\Psi =\\psi _{\\mu +}^1 + \\psi _{\\mu -}^2$ , where the two terms represent Weyl spinors that are independent of each other, and consequently $\\Psi _\\mu $ on-shell describes 8 real degrees of freedom.","The Majorana gravitino, on the other hand can be written as $\\psi _\\mu =\\psi _{\\mu +} + \\psi _{\\mu -}$ where $\\psi _{\\mu -}= (\\psi _{\\mu +})^*$ .","Thus, on shell $\\psi _\\mu $ describes 4 real degrees of freedom.","With this information at hand, we can now tabulate the supermultiplet structure of the full spectrum.","It is convenient to do so by specifying the wave operators for different spin fields and consider the cases of $\\ell =0$ , $\\ell =1$ and $\\ell \\ge 2$ separately.","The results are given in Table 1, Table 2 and Table 3.","Table: The spectrum of wave operators for ℓ=0\\ell =0.","The operator 𝒪 1 {\\cal O}_1 is to be evaluated for ℓ=0\\ell =0.","There is one massless spin-2 and one massless spin-0 multiplet, a massive spin-2 multiplet with mass MM, two spin-0 multiplets with squared mass m 2 =M 2 -α 2 m^2=M^2-\\alpha ^2 and two massive spin-1 multiplets with mass 2 ^2 given in ().Table: The spectrum of wave operators for ℓ=1\\ell =1.","The operators 𝒪 1 {\\cal O}_1 and 𝒪 2 {\\cal O}_2 are to be evaluated for ℓ=1\\ell =1.","There are two spin-2 multiplets with squared masses given in () for ℓ=1\\ell =1, two spin-3/2, two spin-1 multiplets and two spin-0 multiplets with squared mass m 2 =M 2 +α 2 m^2=M^2+\\alpha ^2, three spin-1 multiplets with squared masses given by the roots of the polynomial given in () and a massless vector multiplet.Table: The spectrum of wave operators for ℓ≥2\\ell \\ge 2.","For each integer ℓ\\ell , there are two spin-2 multiplets with squared mass m ± 2 (ℓ)m^2_\\pm (\\ell ) given in (), two spin-3/2, two spin-1 multiplets and four spin-0 multiplets with squared mass m 2 (ℓ)m^2(\\ell ) given in (), and four spin-1 multiplets with squared masses given by the roots of the polynomial given in ().Having established the full spectrum of states in the four-dimensional theory, we may now examine the question of stability, which is governed by the mass values for the massive fields.","We begin with the $\\ell =0$ level, given in Table 1.","In addition to the massless graviton and scalar multiplets, there are massive graviton, vector and scalar multiplets at this level.","The massive graviton multiplet has $m^2=M^2$ , the scalar multiplet has $m^2=M^2-\\alpha ^2$ , and the vector multiplet has masses given by (REF ).","These imply, respectively, that stability requires $M^2>0$ , $M^2>\\alpha ^2$ and $M^2\\ge (5+2\\sqrt{6})\\alpha ^2 \\approx 9.89898 \\alpha ^2$ .","At the level $\\ell =1$ , in addition to the massless vector multiplet, there are massive graviton, gravitino, vector and scalar multiplets.","The gravitino multiplet, two of the massive vector multiplets and the scalar multiplet have masses given by the operator ${\\cal O}_1$ , implying $m^2=M^2+\\alpha ^2$ , which therefore impose no new conditions.","The massive graviton multiplet with mass operator ${\\cal O}_2$ has $m^2$ given by equation (REF ) with $\\ell $ set equal to 1.","This implies $m^2= (M^2+4\\alpha ^2\\pm \\sqrt{M^4+8\\alpha ^4})/2$ , and hence gives no further restriction.","There remains the massive vector multiplet with mass operator ${\\cal O}_3$ given in (REF ).","This gives a cubic polynomial in $m^2$ , and we find that this has three real (and positive) roots for $m^2$ provided that $\\mu \\equiv M^2/\\alpha ^2$ satisfies the condition $4\\mu ^4 - 64\\mu ^3 + 153 \\mu ^2 - 26\\mu - 139\\ge 0\\,.$ This implies we must have $\\mu \\ge \\mu _{\\rm min}$ , where $\\mu _{\\rm min}\\approx 13.1425$ .","In other words, at level $\\ell =1$ stability requires that $M^2$ should exceed approximately $13.1425 \\alpha ^2$ .","For levels $\\ell \\ge 2$ , the multiplets are given in Table 3.","The gravitino, vector and scalar multiplets with mass operator ${\\cal O}_1$ have $m^2$ given in equation (REF ), and these are always positive for all values of $\\ell \\ge 2$ .","Likewise, for the graviton multiplet with mass operator ${\\cal O}_2$ , $m^2$ , given in (REF ), is positive for all $\\ell \\ge 2$ .","There remains the vector multiplet with mass operator ${\\cal O}_4$ .","This leads to a quartic polynomial in $m^2$ , which can be read off from (REF ) and (REF ).","One can show that this polynomial necessarily has at least two real roots, which are positive, and that if the four roots for $m^2$ are real then they are also positive.","The condition for having four real roots is that a rather complicated discriminant of sixth order in $\\mu =M^2/\\alpha ^2$ should be positive.","This discriminant also depends on the level $\\ell $ .","For a few representative values of $\\ell $ , we find the requirement $\\mu \\ge \\mu _{\\rm min}(\\ell )$ : Table: Minimum values of μ=M 2 /α 2 \\mu =M^2/\\alpha ^2 necessary to achieve realpositive mass-squared values for the 𝒪 4 {\\cal O}_4 vector multiplet at levelℓ\\ell .In the limit of large $\\ell $ , we find that to leading order, $\\mu _{\\rm min}(\\ell )$ grows linearly with $\\ell $ , with $\\mu _{\\rm min}(\\ell ) \\sim 4.05874 \\, \\ell +\\cdots $ This implies that for any given ratio $\\mu =M^2/\\alpha ^2$ , there is a a critical level $\\ell _{\\rm max}$ beyond which the Kaluza-Klein tower must be truncated in order not to have modes with complex masses, which would be associated with instabilities."],["Non-supersymmetric $\\rm {Minkowski}_4\\times S^2$ background","In addition to supersymmetric vacuum solution discussed in previous section, the theory [5] also possesses non-supersymmetric $\\rm {Minkowski}_4\\times S^2$ vacua when $M^2=\\alpha ^2$ , with the curvature and flux given by $\\bar{R}_{\\mu \\nu \\lambda \\rho } &=0 \\ ,&\\qquad \\bar{R}_{mn} &= \\alpha ^2\\bar{g}_{mn}\\ , \\qquad \\bar{L} &=1\\ , \\nonumber \\\\\\bar{F}_{\\mu \\nu } &=0\\ , & \\qquad \\bar{F}_{mn} &= 4q g\\epsilon _{mn}\\ , \\qquad & \\nonumber \\\\\\bar{G}_{\\mu \\nu } &=0\\ , & \\qquad \\bar{G}_{mn} &= -q\\alpha ^2\\epsilon _{mn}\\ , \\qquad &$ where $q$ plays the role of monopole charge and is quantized to be $q=0,\\pm 1,\\pm 2\\ldots $ .","The supersymmetric vacua correspond to $q=\\pm 1$ ."],["Bosonic sector","We will perform a similar spectrum analysis around the non-supersymmetric background.","The harmonic expansion (REF ) for the uncharged fields after the gauge fixing is still valid, and the residual gauge symmetries are almost the same except that some terms related to background flux should be multiplied by the monopole charge.","We present the results for the spectrum below.","While we shall use the same notation for operators such as $\\hat{\\Box }_0$ and others, it is understood that they are to be evaluated for $M^2=\\alpha ^2$ ."],["The equations of motion satisfied by graviton for $\\ell \\ge 1$ is $\\ell \\ge 1:\\qquad ( \\hat{\\Box }_0^2-\\alpha ^2\\hat{\\Box }_0-\\alpha ^4\\, c_\\ell )({{\\cal P}}^{2} h)^{(\\ell )}_{\\mu \\nu }=0\\ ,$ describing massive gravitons with square masses $m^2_\\pm (\\ell ) = \\frac{1}{2} \\alpha ^2\\left(1+c_\\ell \\pm \\sqrt{1+4c_\\ell }\\right)\\ .$ For $\\ell =0$ , the linearized field equation is $\\ell =0:\\qquad (\\Box -\\alpha ^2)R^{L(0)}_{\\mu \\nu }=-\\alpha ^2\\partial _{\\mu }\\partial _{\\nu }S^{(0)}-\\alpha ^4\\eta _{\\mu \\nu }S^{(0)}+\\partial _{\\mu }\\partial _{\\nu }(\\Box +\\alpha ^2)S^{(0)}\\ ,$ where $S^{(0)}=\\phi ^{(0)}+N^{(0)}$ .","It describes a massless graviton and massive graviton with squared mass $m^2=\\alpha ^2$ .","For $\\ell \\ge 2$ the mixing among the five vector fields $(k^{T}_{\\mu },a^{T}_{\\mu },v^{T}_{\\mu }$ $,b_{\\mu \\nu }^{T},b^{T}_{\\mu })$ now have the following form $\\ell \\ge 2:\\quad 0&=&(2c_\\ell \\alpha ^4-\\hat{\\Box }_0^2)k^{T(\\ell )}_{\\mu }-4g\\alpha ^2q a^{T(\\ell )}_{\\mu }-\\frac{\\alpha ^2}{2}(4 q v^{T(\\ell )}_{\\mu }-\\epsilon _{\\mu }^{~\\nu \\lambda \\rho }\\partial _{\\nu }b_{\\lambda \\rho }^{T(\\ell )})\\ ,\\\\[0.2cm]0&=&(\\alpha ^2\\,\\Box -\\hat{\\Box }^2_0)b^{T(\\ell )}_{\\mu \\nu }-4q g\\alpha ^2\\star F^{(\\ell )}_{\\mu \\nu }(a)+\\alpha ^4c_\\ell (\\star F^{(\\ell )}_{\\mu \\nu }(k)-\\star F^{(\\ell )}_{\\mu \\nu }(b))\\ ,\\\\[0.2cm]0&=& \\hat{\\Box }_0^2b^{T(\\ell )}_{\\mu }+{\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } }\\alpha ^2\\epsilon _{\\mu }^{~\\nu \\lambda \\rho }\\partial _{\\nu }b_{\\lambda \\rho }^{T(\\ell )}\\ ,\\\\[0.2cm]0&=& (\\hat{\\Box }_0+\\alpha ^2)a^{T(\\ell )}_{\\mu } -4q g\\alpha ^2c_\\ell k^{T(\\ell )}_{\\mu }+4gv_{\\mu }^{T(\\ell )}-2q g\\epsilon _{\\mu }^{~\\nu \\lambda \\rho }\\partial _{\\nu }b_{\\lambda \\rho }^{T(\\ell )}\\ ,\\\\[0.2cm]0&=& (\\hat{\\Box }_0-\\alpha ^2)v^{T(\\ell )}_{\\mu }+\\alpha ^4qc_\\ell k^{T(\\ell )}_{\\mu }-2g\\alpha ^2 a^{(\\ell )}_{\\mu }\\ ,$ Diagonalising the associated $5\\times 5$ operator-valued matrix, we find that the modes are annihilated by the partially-factorising operator polynomial, of eighth order in $\\hat{\\Box }_0$ , given by $\\hat{\\Box }_0^2\\,{\\cal O}_6$ .","The explicit form of ${\\cal O}_6$ can be obtained straightforwardly from (REF ), and one finds that it is symmetric under $q\\rightarrow -q$ meaning that vector spectrum is symmetric under the sign change of monopole charge.","Of the remaining vectors $(({{\\cal P}}^{1}h)_{\\mu \\nu }, \\partial ^\\mu b_{\\mu \\nu })$ are annihilated by $\\hat{\\Box }_0$ .","Thus, apart from the charged vectors which will be treated separately below, the total wave operator for $\\ell \\ge 2$ is given by $\\ell \\ge 2:\\qquad {\\cal O}^{(1)} = \\hat{\\Box }_0^4|_{M^2=\\alpha ^2}\\,{\\cal O}_6\\ ,$ implying four massive vectors with squared masses $m^2=\\alpha ^2 c_\\ell $ , and six massive vectors whose squared masses $m^2$ correspond to the roots ${\\cal O}_6$ in which $\\Box $ is to be replaced by $m^2$ .","In the case of $\\ell =1$ , again excluding the charged vector, we find that a massless vector appears since for $c_\\ell =2$ , the operator ${\\cal O}_6 $ factorizes as ${\\cal O}_6= \\Box \\, {\\cal O}_5$ , and the total wave operator becomes $\\ell =1:\\qquad {\\cal O}^{(1)} = \\hat{\\Box }_0^4|_{\\ell =1}\\,\\Box \\, {\\cal O}_5\\ .$ The massless vector is composed from a linear combination of $(k^{T(1)}_{\\mu },a^{T(1)}_{\\mu },v^{T(1)}_{\\mu },b_{\\mu \\nu }^{T(1)},b^{T(1)}_{\\mu })$ with mixing coefficients $\\left({\\textstyle {\\frac{\\scriptstyle 2}{\\scriptstyle 1+q^2} } },-{\\textstyle {\\frac{\\scriptstyle 8q g}{\\scriptstyle 1+q^2} } },{\\textstyle {\\frac{\\scriptstyle 2q\\alpha ^2}{\\scriptstyle 1+q^2} } },1,0\\right)$ .","In the uncharged vector sector, there remains the case of $\\ell =0$ , for which the relevant vector fields are $(b_{\\mu \\nu }^{T(0)},a^{T(0)}_{\\mu },v^{T(0)}_{\\mu })$ .","Upon diagonalising the associated $3\\times 3$ operator-valued matrix, we find that the modes are annihilated by the following partially-factorising operator polynomial $\\ell =0: \\qquad {\\cal O}^{(1)}= \\Box (\\Box -\\alpha ^2)(\\Box ^2+2\\alpha ^4q^2)\\ .$ As before, we find that the would-be massless modes annihilated by $\\Box $ is eaten by the two form.","Thus there are no massless vector modes at $\\ell =0$ .","Finally, we turn to the treatment of the complex vector $\\hat{z}_{\\mu }$ .","This field $\\hat{z}_{\\mu }$ is expanded in terms of charge “-$q$ ” scalar harmonics starting from $\\ell =|q|$ as follows: ${\\hat{z}}_\\mu = \\sum _{\\ell \\ge q} z^{(\\ell )}_\\mu \\, {}_{-q}{Y}^{(\\ell )}\\ ,$ The resulting linearized field equation is $\\ell \\ge |q|:\\qquad \\left(\\Box -\\alpha ^2c_\\ell +\\alpha ^2q^2-M^2\\right)\\, z^{T(\\ell )}_{\\mu }=0\\ .$ For $\\ell \\ge 2$ , the equations describing the mixing between $(\\phi , N, \\varphi , \\tilde{\\varphi },a, v)$ take the following form $\\ell \\ge 2:\\quad 0&=&2(\\hat{\\Box }_0+\\alpha ^2)\\phi ^{(\\ell )}+(2\\hat{\\Box }_0+2\\alpha ^2+\\alpha ^2c_\\ell )N^{(\\ell )}+3\\hat{\\Box }_0\\varphi ^{(\\ell )}-\\alpha ^2c_\\ell \\tilde{\\varphi }^{(\\ell )}\\ ,\\\\[0.2cm]0&=&\\alpha ^2(3\\varphi ^{(\\ell )}+2\\phi ^{(\\ell )}-4gq a^{(\\ell )}-2q v^{(\\ell )})+\\alpha ^2(1-c_\\ell )\\tilde{\\varphi }^{(\\ell )}+\\Box \\,N^{(\\ell )}\\ ,\\\\[0.2cm]0&=&2\\alpha ^2\\phi ^{(\\ell )}+(2\\hat{\\Box }_0+2\\alpha ^2+\\alpha ^2c_\\ell )N^{(\\ell )}-\\Box \\tilde{\\varphi }^{(\\ell )}\\ ,\\\\[0.2cm]0&=& (\\hat{\\Box _0}+\\alpha ^2)a^{(\\ell )}+4gq N^{(\\ell )}-2gq (3\\varphi ^{(\\ell )}+\\tilde{\\varphi }^{(\\ell )})+4gv^{(\\ell )}\\ ,\\\\[0.2cm]0&=&(\\hat{ \\Box }_0-\\alpha ^2)v^{(\\ell )}-\\alpha ^2 q N^{(\\ell )}+\\frac{\\alpha ^2}{2}q(3\\varphi ^{(\\ell )}+\\tilde{\\varphi }^{\\ell })-2g\\alpha ^2 a^{(\\ell )}\\ ,\\\\[0.2cm]0&=&(\\alpha ^2\\Box -\\hat{\\Box }^2_0)\\varphi ^{(\\ell )}+2\\alpha ^4\\phi ^{(\\ell )}+\\alpha ^4(2-c_\\ell )N^{(\\ell )}-4q gc_\\ell \\alpha ^4 a^{(\\ell )}-2c_\\ell \\alpha ^4q v^{(\\ell )} \\ .$ Diagonalising the associated $6\\times 6$ operator-valued matrix, we find that the modes are annihilated by the partially-factorising operator polynomial, of seventh order in $\\hat{\\Box }_0$ , given by ${\\cal O}_4\\hat{\\Box }_0^3|_{M^2=\\alpha ^2}$ Of the remaining scalars, $(\\partial ^{\\mu }k_{\\mu },\\partial ^\\mu b_{\\mu },\\partial ^\\mu a_{\\mu },\\partial ^\\mu v_{\\mu },\\tilde{v})$ , three of them namely $(\\partial ^{\\mu }k_{\\mu },\\partial ^\\mu b_{\\mu },\\tilde{v}) $ are annihilated by ${\\hat{\\Box }}_0$ , and the remaining two are determined in terms of them.","Thus, in total, apart from the complex scalars which will be treated separately below, the wave operator for the scalar fields is given by ${\\cal O}^{(0)} = {\\hat{\\Box }}_0^6\\,{\\cal O}_4\\ .$ Next, consider the case $\\ell =1$ .","Utilizing the residual symmetry (REF ) and (REF ), one can eliminate $N^{(1)}$ and $\\partial ^\\mu k_{\\mu }^{(1)}$ .","The mass operator coming from the mixing among $(\\phi ,\\varphi , \\tilde{\\varphi },a, v)$ takes the form $\\hat{\\Box }_0^2|_{\\ell =1}{\\cal O}_3$ .","Taking into account $\\partial ^\\mu b_\\mu ^{(1)}$ and $\\widetilde{v}^{(1)}$ , the total wave operator, again, excluding the complex scalar sector, is given by $\\ell =1:\\qquad {\\cal O}^{(0)} = \\hat{\\Box }_0^4\\,{\\cal O}_3\\ .$ There remaining the case of $\\ell =0$ .","In this case, the relevant scalar fields are $(\\phi ^{(0)},N^{(0)},b^{(0)})$ , and they satisfy the following equations respectively $\\ell =0:\\quad &&(\\Box ^2+2\\alpha ^4)S^{(0)}=0\\ ,\\qquad S^{(0)}=\\phi ^{(0)}+N^{(0)}\\ ,\\nonumber \\\\&& \\Box ^2b^{(0)}=0\\ ,\\nonumber \\\\&&\\Box ^2N^{(0)}= 0\\ .$ Thus besides two massless modes, we also have modes with linear time coordinate dependence.","Finally, we discuss the complex scalars originating from $\\hat{z}_m$ .","For positive monopole charge, the harmonic expansion of $\\hat{z}_m$ is given by $\\hat{z}_m=z^{(q-1)} {}_{-q}V_m^{(q-1)}+z^{(q)} {}_{-q}V_m^{(q)}+\\sum _{\\ell >q}(z^{(\\ell )}D_m{}_{-q}Y^{(\\ell )}+\\tilde{z}^{(\\ell )}\\epsilon _m^{~n}D_n{}_{-q}Y^{(\\ell )})\\ .$ Thus we have $&&\\ell >q &:\\qquad z^{(\\ell )}\\ ,\\ \\tilde{z}^{(\\ell )} &\\qquad \\mbox{with} &\\qquad m^2=\\alpha ^2c_\\ell -\\alpha ^2q+M^2\\ ,\\nonumber \\\\&&\\ell =q &:\\qquad z^{(q)}\\ , &\\qquad \\mbox{with} &\\qquad m^2=\\alpha ^2q+M^2\\ ,\\nonumber \\\\&&\\ell =q-1 &:\\qquad z^{(q-1)}\\ , &\\qquad \\mbox{with} &\\qquad m^2=M^2-\\alpha ^2q\\ .$"],["Fermionic sector","The analysis of the fermionic spectrum in a non-supersymmetric background (REF ) is more subtle than that in supersymmetric background.","Since the non-supersymmetric background do not posses Killing spinor, we will use spin-weighted harmonics $ _{s-{\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } }}\\eta ^{(\\ell )}$ , described in detail in appendix B, as basis of expansion.","For brevity, we shall use the notation $_{s-{\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } }}\\eta ^{(\\ell )} \\equiv \\tilde{\\eta }^{(\\ell )}\\ ,$ where $s={\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } }(1-q)$ .","These harmonics satisfy the relations $&&\\tilde{\\eta }^{(\\ell )}_+=\\eta _+ \\left(_{s-1}Y^{(\\ell )} \\right)\\ , \\qquad \\tilde{\\eta }^{(\\ell )}_- = \\eta _- \\left( _{s}Y^{(\\ell )} \\right)\\ ,\\\\&& (\\frac{d}{d\\theta }+m\\csc \\theta +s\\cot \\theta ){}\\left(_{s}Y^{(\\ell )}\\right)=\\sqrt{(\\ell +s)(\\ell +1-s)}{}\\left(_{s-1}Y^{(\\ell )}\\right)\\ ,\\\\&&(\\frac{d}{d\\theta }-m\\csc \\theta -(s-1)\\cot \\theta ){}\\left(_{s-1}Y^{(\\ell )}\\right)=-\\sqrt{(\\ell +s)(\\ell +1-s)}{} \\left(_{s}Y^{(\\ell )}\\right)\\ .$ The lowest level would have definite chirality when $\\ell =-s$ for $q>0$ and $\\ell =s-1$ for $q<0$ .","The spin weighted harmonics satisfy the following properties $\\sigma _3\\tilde{\\eta }_{\\pm }^{(\\ell )} &=\\pm \\tilde{\\eta }_{\\pm }^{(\\ell )}\\ , &\\qquad \\sigma ^n D_n\\tilde{\\eta }_{\\pm }^{(\\ell )} &=i\\alpha \\sqrt{\\tilde{c_\\ell }}\\,\\tilde{\\eta }_{\\mp }^{(\\ell )}\\ ,\\nonumber \\\\[D_m, D_n]\\tilde{\\eta }_-^{(\\ell )} &=-is\\alpha ^2\\epsilon _{mn}\\tilde{\\eta }_-^{(\\ell )}\\ ,&\\qquad [D_m,D_n]{\\tilde{\\eta }}_+^{(\\ell )} &=i(1-s)\\alpha ^2\\epsilon _{mn}{\\eta _+^{(\\ell )}}\\ ,\\nonumber \\\\D^nD_n\\tilde{\\eta }_-^{(\\ell )} &=-\\alpha ^2(\\tilde{c_\\ell }-s)\\tilde{\\eta }_-^{(\\ell )}\\ ,&\\qquad D^nD_n\\tilde{\\eta }_+^{(\\ell )} &= \\alpha ^2 (1-s-\\tilde{c_\\ell })\\tilde{\\eta }_+^{(\\ell )}\\ ,$ where $\\tilde{c_\\ell }=\\biggl (c_\\ell -s(s-1)\\biggr )\\ .$ The harmonic expansion for 6D spin-$1/2$ fields follows the same procedure as in supersymmetric case by using the spin weighted harmonics, while it is more subtle when expanding the 6D gravitini.","It can be checked that the linearized equations have the following discreet symmetry $q\\rightarrow -q\\ , &\\qquad \\psi _{\\mu +}\\rightarrow -\\psi _{\\mu -},& \\qquad \\psi _{\\mu -}\\rightarrow \\psi _{\\mu +}\\ ,\\nonumber \\\\\\psi _-\\rightarrow \\psi _+\\ , &\\qquad \\psi _+\\rightarrow -\\psi _-\\ , &\\qquad \\chi _-\\rightarrow \\chi _+\\ ,\\nonumber \\\\\\chi _+\\rightarrow -\\chi _-\\ , &\\qquad \\lambda _+\\rightarrow -\\lambda _-\\ ,& \\qquad \\lambda _-\\rightarrow \\lambda _+\\ ,$ which implies that the spectrum keeps the same under the sign change of monopole charge.","In the following, we will focus on the case with positive monopole charge and use $|s|$ to denote ${\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } }(q-1)$ ."],["The gravitini satisfy $\\ell \\ge |s|+1:\\qquad &&{\\partial }\\Big (\\Box +\\alpha ^2|s|-\\alpha ^2\\tilde{c_\\ell }\\Big )\\psi _{\\mu +}^{(\\ell )}+i\\alpha \\sqrt{\\tilde{c_\\ell }}\\Big (\\Box -\\alpha ^2\\tilde{c_\\ell }\\Big )\\psi _{\\mu -}^{(\\ell )}=0\\ ,\\nonumber \\\\&&i\\alpha \\sqrt{\\tilde{c_\\ell }}\\Big (\\Box -\\alpha ^2\\tilde{c_\\ell }\\Big )\\psi _{\\mu +}^{(\\ell )}-{\\partial }\\Big (\\Box +(|s|+1-{\\tilde{c}}_\\ell )\\alpha ^2\\Big )\\psi _{\\mu -}^{(\\ell )}=0\\ .$ Diagonalising the associated $2\\times 2$ operator-valued matrix, we find that the modes are annihilated by the partially-factorising operator polynomial, of third order in $\\Box $ , given by ${\\cal O}^{(3/2)} = \\tilde{{\\cal O}}_3\\ ,$ where the explicit form of $\\tilde{{\\cal O}}_3$ can be deduced from (REF ).","Next, we consider the case of $\\ell =|s|$ .","In this case, we find that the quadratic action for the lowest level fermionic fields is proportional to ${\\cal L}^{(2)}&\\propto & -i\\bar{\\chi }\\gamma ^{\\mu \\nu }\\partial _{\\mu }\\psi _{\\nu }-i\\bar{\\psi }_{\\mu }\\gamma ^{\\mu \\nu }\\partial _{\\nu }\\chi +2i\\bar{\\chi }{\\partial }\\psi -2i\\bar{\\psi }{\\partial }\\chi +2\\bar{\\chi }{\\partial }\\chi \\nonumber \\\\&&-8g\\bar{\\chi }\\lambda -8g\\bar{\\lambda }\\chi -4\\bar{\\lambda }{\\partial }\\lambda -4i g|s|\\bar{\\lambda }\\gamma ^{\\mu }\\psi _{\\mu }-4i g|s|\\bar{\\psi }_{\\mu }\\gamma ^{\\mu }\\lambda \\nonumber \\\\&& -8g(1+|s|)i\\bar{\\lambda }\\psi +8g(1+|s|)i\\bar{\\psi }\\lambda -{\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } }\\bar{\\psi }_{\\mu }\\gamma ^{\\mu \\nu \\lambda }\\partial _{\\nu }\\psi _{\\lambda }+\\bar{\\psi }{\\partial }\\psi \\nonumber \\\\&&+\\bar{\\psi }_{\\mu }\\gamma ^{\\mu \\nu }\\partial _{\\nu }\\psi -\\psi \\gamma ^{\\mu \\nu }\\partial _{\\mu }\\psi _{\\nu }-\\frac{1}{\\alpha ^2}\\biggl ({\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 4} } }\\bar{\\psi }_{\\mu \\nu }{\\partial }\\psi ^{\\mu \\nu }-\\bar{\\psi }{\\partial }\\Box \\psi \\nonumber \\\\&& +{\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } }|s|\\alpha ^2\\bar{\\psi }_{\\mu }{\\partial }\\psi ^{\\mu }-|s|\\alpha ^2\\bar{\\psi }\\partial _{\\mu }\\psi ^{\\mu }+|s|\\alpha ^2\\bar{\\psi }_{\\mu }\\partial ^{\\mu }\\psi -(1+2|s|)\\alpha ^2\\bar{\\psi }{\\partial }\\psi \\biggr )\\ .$ Unlike the supersymmetric case, we see the appearance of the terms $\\bar{\\lambda }\\gamma ^{\\mu }\\psi _{\\mu }$ , $\\bar{\\psi }_{\\mu }{\\partial }\\psi ^{\\mu }$ and $\\bar{\\psi }_{\\mu }\\partial ^{\\mu }\\psi $ which break the fermionic gauge symmetry.","The homogeneous solutions for gravitini satisfy ${\\partial }(\\Box -\\alpha ^2-|s|\\alpha ^2)\\psi _{\\mu }^{(|s|)}=0,\\qquad \\gamma ^{\\mu }\\psi ^{(|s|)}_{\\mu }=0,\\qquad \\partial ^{\\mu }\\psi _{\\mu }^{(|s|)}=0.$ Thus, due to the lack of fermionic gauge symmetry, the longitudinal mode $\\psi _{\\mu }\\propto \\,p_{\\mu }e^{ipx}$ with $p^2=0$ becomes a dynamical degree of freedom.","The $\\ell \\ge |s|+2$ sector consists of ten spin-1/2 fields $(\\Lambda _+,\\Psi _+,\\Lambda _-,\\Psi _-,$ $\\psi _+,\\psi _-,\\chi _+,\\chi _-,\\lambda _+,\\lambda _-)$ .","The linearized equations describing their mixing are $0&=&{\\partial }\\Lambda ^{(\\ell )}_{-}-\\Psi _+^{(\\ell )}+i\\alpha \\sqrt{\\tilde{c_\\ell }}\\psi _+^{(\\ell )}-i\\alpha \\sqrt{\\tilde{c_\\ell }}\\Lambda _+^{(\\ell )}+2{\\partial }\\psi _-^{(\\ell )}-2i{\\partial }\\chi _-^{(\\ell )}+2\\alpha \\sqrt{\\tilde{c_\\ell }}\\chi _+^{(\\ell )}-8gi\\lambda _+^{(\\ell )}\\ ,\\\\0&=&{\\partial }\\Lambda _+^{(\\ell )}-\\Psi _-^{(\\ell )}+i\\alpha \\sqrt{\\tilde{c_\\ell }}\\psi _-^{(\\ell )}+i\\alpha \\sqrt{\\tilde{c_\\ell }}\\Lambda _-^{(\\ell )}-2{\\partial }\\psi _+^{(\\ell )}+2i{\\partial }\\chi _+^{(\\ell )} +2\\alpha \\sqrt{\\tilde{c_\\ell }}\\chi _-^{(\\ell )}-8gi\\lambda _-^{(\\ell )}\\ ,\\\\0&=& g(1+|s|)i\\Lambda _-^{(\\ell )}+2g\\chi _-^{(\\ell )}-{\\partial }\\lambda _+^{(\\ell )}-i\\alpha \\sqrt{\\tilde{c_\\ell }}\\lambda _-^{(\\ell )}\\ ,\\\\0&=&2i g|s|\\psi _+^{(\\ell )}+2g\\chi _+^{(\\ell )}+{\\partial }\\lambda _-^{(\\ell )}-i\\alpha \\sqrt{\\tilde{c_\\ell }}\\lambda _+^{(\\ell )}\\ ,\\\\0&=&i\\sqrt{\\tilde{c_\\ell }}{\\partial }(\\Box +\\alpha ^2+|s|\\alpha ^2)\\psi _+^{(\\ell )}-i\\alpha ^2\\sqrt{\\tilde{c_\\ell }}{\\partial }\\Lambda _+^{(\\ell )}+i\\alpha ^2\\tilde{c_\\ell }^{3/2}\\Psi _-^{(\\ell )}+2\\alpha ^2\\sqrt{\\tilde{c_\\ell }}{\\partial }\\chi _+^{(\\ell )}\\nonumber \\\\&& -8g(1+|s|)\\alpha \\,i{\\partial }\\lambda _+^{(\\ell )}+\\alpha (\\tilde{c_\\ell }-1-|s|){\\partial }\\Psi _+^{(\\ell )}+\\alpha (2+2|s|-\\tilde{c_\\ell })\\Box \\psi _-^{(\\ell )}\\ ,\\\\0&=& i\\sqrt{\\tilde{c_\\ell }}{\\partial }(\\Box -|s|\\alpha ^2)\\psi _-^{(\\ell )}+i\\alpha ^2\\sqrt{\\tilde{c_\\ell }}{\\partial }\\Lambda _-^{(\\ell )}-i\\alpha ^2\\tilde{c_\\ell }^{3/2}\\Psi _+^{(\\ell )}+2\\alpha ^2\\sqrt{\\tilde{c_\\ell }}{\\partial }\\chi _-^{(\\ell )}+8ig|s|\\alpha {\\partial }\\lambda _{-}^{(\\ell )}\\nonumber \\\\&&+\\alpha (\\tilde{c_\\ell }+|s|){\\partial }\\Psi _-^{(\\ell )}+\\alpha (\\tilde{c_\\ell }+2|s|)\\Box \\psi _+^{(\\ell )}\\ ,\\\\0&=&{\\partial }(\\Box +3\\alpha ^2+|s|\\alpha ^2-\\alpha ^2\\tilde{c_\\ell })\\Lambda _-^{(\\ell )}-(\\Box +4\\alpha ^2+2|s|\\alpha ^2-2\\alpha ^2\\tilde{c_\\ell })\\Psi _+^{(\\ell )}+i\\alpha \\sqrt{\\tilde{c_\\ell }}(\\Box +4\\alpha ^2+4|s|\\alpha ^2)\\psi _+^{(\\ell )}\\nonumber \\\\&&+\\alpha ^2{\\partial }(8+2|s|-\\tilde{c_\\ell })\\psi _-^{(\\ell )}-i\\alpha \\sqrt{\\tilde{c_\\ell }}(\\Box +4\\alpha ^2-\\alpha ^2\\tilde{c_\\ell })\\Lambda _+^{(\\ell )}-6i\\alpha ^2{\\partial }\\chi _-^{(\\ell )}+8\\alpha ^3\\sqrt{\\tilde{c_\\ell }}\\chi _+^{(\\ell )}\\nonumber \\\\&&-32ig\\alpha ^2\\,(1+|s|)\\lambda _+^{(\\ell )} +i\\alpha \\sqrt{\\tilde{c_\\ell }}{\\partial }\\Psi _-^{(\\ell )}\\ ,\\\\0&=&-{\\partial }(\\Box +2\\alpha ^2-|s|\\alpha ^2-\\alpha ^2\\tilde{c_\\ell })\\Lambda _+^{(\\ell )}+(\\Box +2\\alpha ^2-2|s|-2\\alpha ^2\\tilde{c_\\ell })\\Psi _-^{(\\ell )}- i\\alpha \\sqrt{\\tilde{c_\\ell }}(\\Box -4|s|\\alpha ^2)\\psi _-^{(\\ell )}\\nonumber \\\\&&+\\alpha ^2{\\partial }(6-2|s|-\\tilde{c_\\ell })\\psi _+^{(\\ell )}-i\\alpha \\sqrt{\\tilde{c_\\ell }}(\\Box +4\\alpha ^2-\\alpha ^2\\tilde{c_\\ell })\\Lambda _-^{(\\ell )}-6i\\alpha ^2 {\\partial }\\chi _+^{(\\ell )}-8\\alpha ^3\\sqrt{\\tilde{c_\\ell }}\\chi _-^{(\\ell )}\\nonumber \\\\&&-32ig|s|\\alpha ^2 \\lambda _-^{(\\ell )}+i\\alpha \\sqrt{\\tilde{c_\\ell }} {\\partial }\\Psi _+^{(\\ell )}\\ ,\\\\0&=&(1+|s|)\\alpha ^2\\Lambda _-^{(\\ell )}-2i\\alpha ^2 \\chi _-^{(\\ell )}+(2\\Box -2 |s|\\alpha ^2-\\alpha ^2\\tilde{c_\\ell })\\psi _-^{(\\ell )}-{\\partial }\\Psi _+^{(\\ell )}-i\\alpha \\sqrt{\\tilde{c_\\ell }}\\Psi _-^{(\\ell )}-i\\alpha \\sqrt{\\tilde{c_\\ell }}{\\partial }\\psi _+^{(\\ell )}\\ ,\\nonumber \\\\&& \\\\0&=&|s|\\alpha ^2\\Lambda _+^{(\\ell )}-2i\\alpha ^2 \\chi _+^{(\\ell )}+(2\\Box +2\\alpha ^2+2|s|\\alpha ^2-\\alpha ^2\\tilde{c_\\ell })\\psi _+^{(\\ell )} -i\\alpha \\sqrt{\\tilde{c_\\ell }}\\Psi _+^{(\\ell )}+{\\partial }\\Psi _-^{(\\ell )}+i\\alpha \\sqrt{\\tilde{c_\\ell }}{\\partial }\\psi _-^{(\\ell )}\\ .\\nonumber \\\\$ Diagonalising the associated $10\\times 10$ operator-valued matrix, we find that the modes are annihilated by the partially-factorising operator polynomial, of ninth order in $\\Box $ , given by $\\ell \\ge |s|+2:\\qquad {\\cal O}^{(0)} = \\hat{\\Box }_0^3\\,\\tilde{{\\cal O}}_6\\ ,$ where the explicit form of the operator ${\\cal O}_6$ can be determined from the linearized spin 1/2 field equations listed above.","$\\ell =|s|+1$ At this level, since $D_m{\\eta _+^{(|s|+1)}}=\\frac{i}{2}\\alpha \\sqrt{\\tilde{c_\\ell }}\\sigma _m{\\eta _-^{(|s|+1)}}$ , there emerges a fermionic gauge symmetry generated by $\\hat{\\epsilon }=\\epsilon _+^{(|s|+1)}{\\eta _+^{(|s|+1)}}$ $\\delta \\psi _{\\mu +}^{(|s|+1)}&=&\\partial _{\\mu }\\epsilon _+^{(|s|+1)}+i\\alpha \\sqrt{\\frac{(1+|s|)}{2}}\\,\\epsilon _+^{(|s|+1)}\\ ,\\nonumber \\\\\\delta \\lambda _+^{(|s|+1,m)}&=&ig(1+|s|)\\epsilon _+^{(|s|+1)}\\ .$ Using this gauge symmetry, one can eliminate $\\psi _+^{(|s|+1)}$ , such that the 10 by 10 mixing becomes 9 by 9 mixing.","Diagonalising this system, we find that the modes are determined by the partially-factorising operator polynomial, of fifteenth order in ${\\partial }$ , given by $\\ell =|s|+1:\\qquad {\\partial }\\Big (\\Box - (|s|+2)\\alpha ^2\\Big )^2 \\tilde{{\\cal O}}_5\\ ,$ where the explicit form of $\\tilde{\\cal O}_5$ can be deduced from the mixing equations.","It is clear that is a massless spin-1/2 mode.","Explicitly, it is a linear combination of $(\\Lambda ^{(|s|+1)}_+,\\Psi ^{(|s|+1)}_+,\\Lambda ^{(|s|+1)}_-,\\Psi ^{(|s|+1)}_-,\\psi ^{(|s|+1)}_-,$ $\\chi ^{(|s|+1)}_-, \\chi ^{(|s|+1)}_+,\\lambda ^{(|s|+1)}_+,\\lambda ^{(|s|+1)}_-)$ with mixing coefficients $(8(1+2|s|),0,0,0,-2,-2i(1+2|s|),0,0,\\sqrt{1+|s|}(1+4|s|).$ $\\ell =|s|$ The coupled system of linearized field equations for the spin-1/2 fields $(\\Lambda _+^{(|s|)},\\Psi _-^{(|s|)}, \\psi _+^{(|s|)},\\chi _+^{(|s|)},\\lambda _-^{(|s|)})$ are $0&=&{\\partial }\\Lambda _+^{(|s|)}-\\Psi _-^{(|s|)}-2{\\partial }\\psi _+^{(|s|)}+2i{\\partial }\\chi _+^{(|s|)}-8gi\\lambda _-^{(|s|)}\\ ,\\\\0&=&i g |s| \\Lambda _+^{(|s|)}+2g(1+|s|)i\\psi _+^{(|s|)}+2g\\chi _+^{(|s|)}+{\\partial }\\lambda _-^{(|s|)}=0,\\\\0&=&8gi{\\partial }\\lambda _-^{(|s|)}+{\\partial }\\Psi _-^{(|s|)}+2\\Box \\psi _+^{(|s|)}=0,\\\\0&=&(\\Box +2\\alpha ^2-|s|\\alpha ^2){\\partial }\\Lambda _+^{(|s| )}-(\\Box +2\\alpha ^2-2|s|\\alpha ^2)\\Psi _-^{(|s| )}+(2|s|-6)\\alpha ^2{\\partial }\\psi _+^{(|s| )}+6\\alpha ^2\\,i{\\partial }\\chi _+^{(|s| )}\\nonumber \\\\&&+32g|s|\\alpha ^2\\,i\\lambda _-^{(|s|) }\\ ,\\\\0&=&\\alpha ^2{\\partial }\\Lambda _+^{(|s| )}-(1+|s|)\\alpha ^2\\Psi _-^{(|s| )}-{\\partial }(\\Box +2\\alpha ^2+2|s|\\alpha ^2)\\psi _+^{|s|}-8g(1+|s|)\\alpha ^2\\,i\\lambda _-^{(|s|)}+2\\alpha ^2\\,i{\\partial }\\chi _+^{(|s|)}\\nonumber .\\\\$ Diagonalising this system, we find two massless modes which satisfy the relations $\\Psi _-^{(|s|)}=0\\ ,\\qquad \\lambda _-^{(|s|)}=0\\ ,\\qquad |s|\\Lambda _+^{(|s|)}-2i\\chi _+^{(|s|)}+2(1+|s|)\\psi _+^{(|s|)}=0\\ .$ $\\ell =|s|-1$ When monopole charge $q\\ge 3$ , there exist charge “$-q/2$ ” vector-spinor harmonics $\\eta _m$ on $S^2$ possessing following properties $D^m\\left( _{s-{\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } }}\\eta ^{(|s|-1)}\\right)_m=0\\ , \\qquad \\sigma ^m\\left( _{s-{\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } }}\\eta ^{(|s|-1)}\\right)_m =0\\ ,$ where $s=(1-q)/2$ .","It follows that $\\sigma ^m D_m \\left( _{s-{\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } }}\\eta ^{(|s|-1)}\\right)_n=0\\ .$ Associated to this harmonics, there is a new spin-1/2 field $\\tilde{\\psi }^{(|s|-1)}$ satisfying ${\\partial }\\Box \\tilde{\\psi }^{(|s|-1)}=0\\ .$"],["Remarks on the non-supersymmetric spectrum","We saw in the supersymmetric vacuum that even at the $\\ell =0$ level in the Kaluza-Klein harmonic expansions, avoiding tachyons in the four-dimensional spectrum required imposing the condition $M^2\\ge (5+2\\sqrt{6})\\alpha ^2$ on the parameter $M$ in the six-dimensional Lagrangian.","In the non-supersymmetric vacua we necessarily have $M^2=\\alpha ^2$ , and in fact having larger values for the background monopole charge $q$ then the $q=\\pm 1$ supersymmetric case only makes the tachyon problem worse, as can be seen from the vector mass operator (REF ).","For this reason, we shall not explore further the precise details of the occurrence of tachyonic states in the non-supersymmetric backgrounds.","One feature of interest that we shall, however, comment on is the occurrence of massless fermions in the non-supersymmetric backgrounds.","As can be seen from the spin-${\\textstyle {\\frac{\\scriptstyle 3}{\\scriptstyle 2} } }$ operator in (REF ), there will be massless spin-${\\textstyle {\\frac{\\scriptstyle 3}{\\scriptstyle 2} } }$ fields at level $\\ell =|s|={\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } }(q-1)$ ; thus these will occur in an $SU(2)$ multiplet of dimension $2\\ell +1 = q$ .","At $\\ell =|s|+1$ , $\\ell =|s|$ and $\\ell =|s|-1$ there will also be massless spin-${\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } }$ modes, as was discussed in the spin-${\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } }$ section above."],["Conclusions","In this paper we have studied the complete linearised spectrum in the $S^2$ Kaluza-Klein reduction of off-shell six-dimensional ${\\cal N}=(1,0)$ gauged supergravity extended by a Riemann-squared superinvariant.","The higher-derivative terms in the six-dimensional theory can be expected to imply the occurrence of ghosts.","As discussed in the introduction, the usual argument for the positivity of the energies of states in a supersymmetric background breaks down, and indeed we found that states in the Kaluza-Klein spectrum could now have complex energies, thus implying instabilities.","One way to understand the occurrence of complex masses is that there are mixings between four-dimensional ghostlike and non-ghostlike modes.","This may be illustrated by the following simple example.","Consider a set of fields $\\phi _i$ with the Lagrangian ${\\cal L}= -{\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } } K_{ij}\\, \\partial \\phi _i\\, \\partial \\phi _j -{\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } } V_{ij}\\phi _i\\phi _j\\,,$ where $K_{ij}$ and $V_{ij}$ are constant symmetric matrices.","If the eigenvalues of $K_{ij}$ are all positive, then $K_{ij}$ and $V_{ij}$ can be simultaneously diagonalised, by means of orthogonal transformations combined with rescalings of the fields.","However, if $K_{ij}$ has negative as well as positive eigenvalues, then the rescalings will introduce factors of $\\sqrt{-1}$ and the diagonalised mass matrix will be complex.","One way to avoid the ghost problems of the higher-derivative theory is to treat it not as an exact model in its own right, but rather as an effective theory valid at energy scales $\\sqrt{\\Lambda }$ much smaller than $M$ .","In this case the propagators are governed by the leading-order theory without the higher-derivative terms, and these terms are treated as interactions.","The Kaluza-Klein spectrum in the reduced four-dimensional theory would then be simply that of the original Salam-Sezgin model corresponding to the $M^2\\rightarrow \\infty $ limit.","This spectrum has been given in [6] and our results agree in the $M^2\\rightarrow \\infty $ limit.","However, the Kaluza-Klein level number $\\ell $ would have to be restricted to lie below some maximum value, in order to satisfy the $\\Lambda <0$ , $M^2>\\\\alpha ^2$ and $M^2\\\\ge (5+2\\\\sqrt{6})\\\\alpha ^2 \\\\approx 9.89898 \\\\alpha ^2$ .\",\n \"At the level $\\\\ell =1$ , in addition to the massless vector multiplet, there are massive graviton, gravitino, vector and scalar multiplets.\",\n \"The gravitino multiplet, two of the massive vector multiplets and the scalar multiplet have masses given by the operator ${\\\\cal O}_1$ , implying $m^2=M^2+\\\\alpha ^2$ , which therefore impose no new conditions.\",\n \"The massive graviton multiplet with mass operator ${\\\\cal O}_2$ has $m^2$ given by equation (REF ) with $\\\\ell $ set equal to 1.\",\n \"This implies $m^2= (M^2+4\\\\alpha ^2\\\\pm \\\\sqrt{M^4+8\\\\alpha ^4})/2$ , and hence gives no further restriction.\",\n \"There remains the massive vector multiplet with mass operator ${\\\\cal O}_3$ given in (REF ).\",\n \"This gives a cubic polynomial in $m^2$ , and we find that this has three real (and positive) roots for $m^2$ provided that $\\\\mu \\\\equiv M^2/\\\\alpha ^2$ satisfies the condition $4\\\\mu ^4 - 64\\\\mu ^3 + 153 \\\\mu ^2 - 26\\\\mu - 139\\\\ge 0\\\\,.$ This implies we must have $\\\\mu \\\\ge \\\\mu _{\\\\rm min}$ , where $\\\\mu _{\\\\rm min}\\\\approx 13.1425$ .\",\n \"In other words, at level $\\\\ell =1$ stability requires that $M^2$ should exceed approximately $13.1425 \\\\alpha ^2$ .\",\n \"For levels $\\\\ell \\\\ge 2$ , the multiplets are given in Table 3.\",\n \"The gravitino, vector and scalar multiplets with mass operator ${\\\\cal O}_1$ have $m^2$ given in equation (REF ), and these are always positive for all values of $\\\\ell \\\\ge 2$ .\",\n \"Likewise, for the graviton multiplet with mass operator ${\\\\cal O}_2$ , $m^2$ , given in (REF ), is positive for all $\\\\ell \\\\ge 2$ .\",\n \"There remains the vector multiplet with mass operator ${\\\\cal O}_4$ .\",\n \"This leads to a quartic polynomial in $m^2$ , which can be read off from (REF ) and (REF ).\",\n \"One can show that this polynomial necessarily has at least two real roots, which are positive, and that if the four roots for $m^2$ are real then they are also positive.\",\n \"The condition for having four real roots is that a rather complicated discriminant of sixth order in $\\\\mu =M^2/\\\\alpha ^2$ should be positive.\",\n \"This discriminant also depends on the level $\\\\ell $ .\",\n \"For a few representative values of $\\\\ell $ , we find the requirement $\\\\mu \\\\ge \\\\mu _{\\\\rm min}(\\\\ell )$ : Table: Minimum values of μ=M 2 /α 2 \\\\mu =M^2/\\\\alpha ^2 necessary to achieve realpositive mass-squared values for the 𝒪 4 {\\\\cal O}_4 vector multiplet at levelℓ\\\\ell .In the limit of large $\\\\ell $ , we find that to leading order, $\\\\mu _{\\\\rm min}(\\\\ell )$ grows linearly with $\\\\ell $ , with $\\\\mu _{\\\\rm min}(\\\\ell ) \\\\sim 4.05874 \\\\, \\\\ell +\\\\cdots $ This implies that for any given ratio $\\\\mu =M^2/\\\\alpha ^2$ , there is a a critical level $\\\\ell _{\\\\rm max}$ beyond which the Kaluza-Klein tower must be truncated in order not to have modes with complex masses, which would be associated with instabilities.\"\n ],\n [\n \"Non-supersymmetric $\\\\rm {Minkowski}_4\\\\times S^2$ background\",\n \"In addition to supersymmetric vacuum solution discussed in previous section, the theory [5] also possesses non-supersymmetric $\\\\rm {Minkowski}_4\\\\times S^2$ vacua when $M^2=\\\\alpha ^2$ , with the curvature and flux given by $\\\\bar{R}_{\\\\mu \\\\nu \\\\lambda \\\\rho } &=0 \\\\ ,&\\\\qquad \\\\bar{R}_{mn} &= \\\\alpha ^2\\\\bar{g}_{mn}\\\\ , \\\\qquad \\\\bar{L} &=1\\\\ , \\\\nonumber \\\\\\\\\\\\bar{F}_{\\\\mu \\\\nu } &=0\\\\ , & \\\\qquad \\\\bar{F}_{mn} &= 4q g\\\\epsilon _{mn}\\\\ , \\\\qquad & \\\\nonumber \\\\\\\\\\\\bar{G}_{\\\\mu \\\\nu } &=0\\\\ , & \\\\qquad \\\\bar{G}_{mn} &= -q\\\\alpha ^2\\\\epsilon _{mn}\\\\ , \\\\qquad &$ where $q$ plays the role of monopole charge and is quantized to be $q=0,\\\\pm 1,\\\\pm 2\\\\ldots $ .\",\n \"The supersymmetric vacua correspond to $q=\\\\pm 1$ .\"\n ],\n [\n \"Bosonic sector\",\n \"We will perform a similar spectrum analysis around the non-supersymmetric background.\",\n \"The harmonic expansion (REF ) for the uncharged fields after the gauge fixing is still valid, and the residual gauge symmetries are almost the same except that some terms related to background flux should be multiplied by the monopole charge.\",\n \"We present the results for the spectrum below.\",\n \"While we shall use the same notation for operators such as $\\\\hat{\\\\Box }_0$ and others, it is understood that they are to be evaluated for $M^2=\\\\alpha ^2$ .\"\n ],\n [\n \"The equations of motion satisfied by graviton for $\\\\ell \\\\ge 1$ is $\\\\ell \\\\ge 1:\\\\qquad ( \\\\hat{\\\\Box }_0^2-\\\\alpha ^2\\\\hat{\\\\Box }_0-\\\\alpha ^4\\\\, c_\\\\ell )({{\\\\cal P}}^{2} h)^{(\\\\ell )}_{\\\\mu \\\\nu }=0\\\\ ,$ describing massive gravitons with square masses $m^2_\\\\pm (\\\\ell ) = \\\\frac{1}{2} \\\\alpha ^2\\\\left(1+c_\\\\ell \\\\pm \\\\sqrt{1+4c_\\\\ell }\\\\right)\\\\ .$ For $\\\\ell =0$ , the linearized field equation is $\\\\ell =0:\\\\qquad (\\\\Box -\\\\alpha ^2)R^{L(0)}_{\\\\mu \\\\nu }=-\\\\alpha ^2\\\\partial _{\\\\mu }\\\\partial _{\\\\nu }S^{(0)}-\\\\alpha ^4\\\\eta _{\\\\mu \\\\nu }S^{(0)}+\\\\partial _{\\\\mu }\\\\partial _{\\\\nu }(\\\\Box +\\\\alpha ^2)S^{(0)}\\\\ ,$ where $S^{(0)}=\\\\phi ^{(0)}+N^{(0)}$ .\",\n \"It describes a massless graviton and massive graviton with squared mass $m^2=\\\\alpha ^2$ .\",\n \"For $\\\\ell \\\\ge 2$ the mixing among the five vector fields $(k^{T}_{\\\\mu },a^{T}_{\\\\mu },v^{T}_{\\\\mu }$ $,b_{\\\\mu \\\\nu }^{T},b^{T}_{\\\\mu })$ now have the following form $\\\\ell \\\\ge 2:\\\\quad 0&=&(2c_\\\\ell \\\\alpha ^4-\\\\hat{\\\\Box }_0^2)k^{T(\\\\ell )}_{\\\\mu }-4g\\\\alpha ^2q a^{T(\\\\ell )}_{\\\\mu }-\\\\frac{\\\\alpha ^2}{2}(4 q v^{T(\\\\ell )}_{\\\\mu }-\\\\epsilon _{\\\\mu }^{~\\\\nu \\\\lambda \\\\rho }\\\\partial _{\\\\nu }b_{\\\\lambda \\\\rho }^{T(\\\\ell )})\\\\ ,\\\\\\\\[0.2cm]0&=&(\\\\alpha ^2\\\\,\\\\Box -\\\\hat{\\\\Box }^2_0)b^{T(\\\\ell )}_{\\\\mu \\\\nu }-4q g\\\\alpha ^2\\\\star F^{(\\\\ell )}_{\\\\mu \\\\nu }(a)+\\\\alpha ^4c_\\\\ell (\\\\star F^{(\\\\ell )}_{\\\\mu \\\\nu }(k)-\\\\star F^{(\\\\ell )}_{\\\\mu \\\\nu }(b))\\\\ ,\\\\\\\\[0.2cm]0&=& \\\\hat{\\\\Box }_0^2b^{T(\\\\ell )}_{\\\\mu }+{\\\\textstyle {\\\\frac{\\\\scriptstyle 1}{\\\\scriptstyle 2} } }\\\\alpha ^2\\\\epsilon _{\\\\mu }^{~\\\\nu \\\\lambda \\\\rho }\\\\partial _{\\\\nu }b_{\\\\lambda \\\\rho }^{T(\\\\ell )}\\\\ ,\\\\\\\\[0.2cm]0&=& (\\\\hat{\\\\Box }_0+\\\\alpha ^2)a^{T(\\\\ell )}_{\\\\mu } -4q g\\\\alpha ^2c_\\\\ell k^{T(\\\\ell )}_{\\\\mu }+4gv_{\\\\mu }^{T(\\\\ell )}-2q g\\\\epsilon _{\\\\mu }^{~\\\\nu \\\\lambda \\\\rho }\\\\partial _{\\\\nu }b_{\\\\lambda \\\\rho }^{T(\\\\ell )}\\\\ ,\\\\\\\\[0.2cm]0&=& (\\\\hat{\\\\Box }_0-\\\\alpha ^2)v^{T(\\\\ell )}_{\\\\mu }+\\\\alpha ^4qc_\\\\ell k^{T(\\\\ell )}_{\\\\mu }-2g\\\\alpha ^2 a^{(\\\\ell )}_{\\\\mu }\\\\ ,$ Diagonalising the associated $5\\\\times 5$ operator-valued matrix, we find that the modes are annihilated by the partially-factorising operator polynomial, of eighth order in $\\\\hat{\\\\Box }_0$ , given by $\\\\hat{\\\\Box }_0^2\\\\,{\\\\cal O}_6$ .\",\n \"The explicit form of ${\\\\cal O}_6$ can be obtained straightforwardly from (REF ), and one finds that it is symmetric under $q\\\\rightarrow -q$ meaning that vector spectrum is symmetric under the sign change of monopole charge.\",\n \"Of the remaining vectors $(({{\\\\cal P}}^{1}h)_{\\\\mu \\\\nu }, \\\\partial ^\\\\mu b_{\\\\mu \\\\nu })$ are annihilated by $\\\\hat{\\\\Box }_0$ .\",\n \"Thus, apart from the charged vectors which will be treated separately below, the total wave operator for $\\\\ell \\\\ge 2$ is given by $\\\\ell \\\\ge 2:\\\\qquad {\\\\cal O}^{(1)} = \\\\hat{\\\\Box }_0^4|_{M^2=\\\\alpha ^2}\\\\,{\\\\cal O}_6\\\\ ,$ implying four massive vectors with squared masses $m^2=\\\\alpha ^2 c_\\\\ell $ , and six massive vectors whose squared masses $m^2$ correspond to the roots ${\\\\cal O}_6$ in which $\\\\Box $ is to be replaced by $m^2$ .\",\n \"In the case of $\\\\ell =1$ , again excluding the charged vector, we find that a massless vector appears since for $c_\\\\ell =2$ , the operator ${\\\\cal O}_6 $ factorizes as ${\\\\cal O}_6= \\\\Box \\\\, {\\\\cal O}_5$ , and the total wave operator becomes $\\\\ell =1:\\\\qquad {\\\\cal O}^{(1)} = \\\\hat{\\\\Box }_0^4|_{\\\\ell =1}\\\\,\\\\Box \\\\, {\\\\cal O}_5\\\\ .$ The massless vector is composed from a linear combination of $(k^{T(1)}_{\\\\mu },a^{T(1)}_{\\\\mu },v^{T(1)}_{\\\\mu },b_{\\\\mu \\\\nu }^{T(1)},b^{T(1)}_{\\\\mu })$ with mixing coefficients $\\\\left({\\\\textstyle {\\\\frac{\\\\scriptstyle 2}{\\\\scriptstyle 1+q^2} } },-{\\\\textstyle {\\\\frac{\\\\scriptstyle 8q g}{\\\\scriptstyle 1+q^2} } },{\\\\textstyle {\\\\frac{\\\\scriptstyle 2q\\\\alpha ^2}{\\\\scriptstyle 1+q^2} } },1,0\\\\right)$ .\",\n \"In the uncharged vector sector, there remains the case of $\\\\ell =0$ , for which the relevant vector fields are $(b_{\\\\mu \\\\nu }^{T(0)},a^{T(0)}_{\\\\mu },v^{T(0)}_{\\\\mu })$ .\",\n \"Upon diagonalising the associated $3\\\\times 3$ operator-valued matrix, we find that the modes are annihilated by the following partially-factorising operator polynomial $\\\\ell =0: \\\\qquad {\\\\cal O}^{(1)}= \\\\Box (\\\\Box -\\\\alpha ^2)(\\\\Box ^2+2\\\\alpha ^4q^2)\\\\ .$ As before, we find that the would-be massless modes annihilated by $\\\\Box $ is eaten by the two form.\",\n \"Thus there are no massless vector modes at $\\\\ell =0$ .\",\n \"Finally, we turn to the treatment of the complex vector $\\\\hat{z}_{\\\\mu }$ .\",\n \"This field $\\\\hat{z}_{\\\\mu }$ is expanded in terms of charge “-$q$ ” scalar harmonics starting from $\\\\ell =|q|$ as follows: ${\\\\hat{z}}_\\\\mu = \\\\sum _{\\\\ell \\\\ge q} z^{(\\\\ell )}_\\\\mu \\\\, {}_{-q}{Y}^{(\\\\ell )}\\\\ ,$ The resulting linearized field equation is $\\\\ell \\\\ge |q|:\\\\qquad \\\\left(\\\\Box -\\\\alpha ^2c_\\\\ell +\\\\alpha ^2q^2-M^2\\\\right)\\\\, z^{T(\\\\ell )}_{\\\\mu }=0\\\\ .$ For $\\\\ell \\\\ge 2$ , the equations describing the mixing between $(\\\\phi , N, \\\\varphi , \\\\tilde{\\\\varphi },a, v)$ take the following form $\\\\ell \\\\ge 2:\\\\quad 0&=&2(\\\\hat{\\\\Box }_0+\\\\alpha ^2)\\\\phi ^{(\\\\ell )}+(2\\\\hat{\\\\Box }_0+2\\\\alpha ^2+\\\\alpha ^2c_\\\\ell )N^{(\\\\ell )}+3\\\\hat{\\\\Box }_0\\\\varphi ^{(\\\\ell )}-\\\\alpha ^2c_\\\\ell \\\\tilde{\\\\varphi }^{(\\\\ell )}\\\\ ,\\\\\\\\[0.2cm]0&=&\\\\alpha ^2(3\\\\varphi ^{(\\\\ell )}+2\\\\phi ^{(\\\\ell )}-4gq a^{(\\\\ell )}-2q v^{(\\\\ell )})+\\\\alpha ^2(1-c_\\\\ell )\\\\tilde{\\\\varphi }^{(\\\\ell )}+\\\\Box \\\\,N^{(\\\\ell )}\\\\ ,\\\\\\\\[0.2cm]0&=&2\\\\alpha ^2\\\\phi ^{(\\\\ell )}+(2\\\\hat{\\\\Box }_0+2\\\\alpha ^2+\\\\alpha ^2c_\\\\ell )N^{(\\\\ell )}-\\\\Box \\\\tilde{\\\\varphi }^{(\\\\ell )}\\\\ ,\\\\\\\\[0.2cm]0&=& (\\\\hat{\\\\Box _0}+\\\\alpha ^2)a^{(\\\\ell )}+4gq N^{(\\\\ell )}-2gq (3\\\\varphi ^{(\\\\ell )}+\\\\tilde{\\\\varphi }^{(\\\\ell )})+4gv^{(\\\\ell )}\\\\ ,\\\\\\\\[0.2cm]0&=&(\\\\hat{ \\\\Box }_0-\\\\alpha ^2)v^{(\\\\ell )}-\\\\alpha ^2 q N^{(\\\\ell )}+\\\\frac{\\\\alpha ^2}{2}q(3\\\\varphi ^{(\\\\ell )}+\\\\tilde{\\\\varphi }^{\\\\ell })-2g\\\\alpha ^2 a^{(\\\\ell )}\\\\ ,\\\\\\\\[0.2cm]0&=&(\\\\alpha ^2\\\\Box -\\\\hat{\\\\Box }^2_0)\\\\varphi ^{(\\\\ell )}+2\\\\alpha ^4\\\\phi ^{(\\\\ell )}+\\\\alpha ^4(2-c_\\\\ell )N^{(\\\\ell )}-4q gc_\\\\ell \\\\alpha ^4 a^{(\\\\ell )}-2c_\\\\ell \\\\alpha ^4q v^{(\\\\ell )} \\\\ .$ Diagonalising the associated $6\\\\times 6$ operator-valued matrix, we find that the modes are annihilated by the partially-factorising operator polynomial, of seventh order in $\\\\hat{\\\\Box }_0$ , given by ${\\\\cal O}_4\\\\hat{\\\\Box }_0^3|_{M^2=\\\\alpha ^2}$ Of the remaining scalars, $(\\\\partial ^{\\\\mu }k_{\\\\mu },\\\\partial ^\\\\mu b_{\\\\mu },\\\\partial ^\\\\mu a_{\\\\mu },\\\\partial ^\\\\mu v_{\\\\mu },\\\\tilde{v})$ , three of them namely $(\\\\partial ^{\\\\mu }k_{\\\\mu },\\\\partial ^\\\\mu b_{\\\\mu },\\\\tilde{v}) $ are annihilated by ${\\\\hat{\\\\Box }}_0$ , and the remaining two are determined in terms of them.\",\n \"Thus, in total, apart from the complex scalars which will be treated separately below, the wave operator for the scalar fields is given by ${\\\\cal O}^{(0)} = {\\\\hat{\\\\Box }}_0^6\\\\,{\\\\cal O}_4\\\\ .$ Next, consider the case $\\\\ell =1$ .\",\n \"Utilizing the residual symmetry (REF ) and (REF ), one can eliminate $N^{(1)}$ and $\\\\partial ^\\\\mu k_{\\\\mu }^{(1)}$ .\",\n \"The mass operator coming from the mixing among $(\\\\phi ,\\\\varphi , \\\\tilde{\\\\varphi },a, v)$ takes the form $\\\\hat{\\\\Box }_0^2|_{\\\\ell =1}{\\\\cal O}_3$ .\",\n \"Taking into account $\\\\partial ^\\\\mu b_\\\\mu ^{(1)}$ and $\\\\widetilde{v}^{(1)}$ , the total wave operator, again, excluding the complex scalar sector, is given by $\\\\ell =1:\\\\qquad {\\\\cal O}^{(0)} = \\\\hat{\\\\Box }_0^4\\\\,{\\\\cal O}_3\\\\ .$ There remaining the case of $\\\\ell =0$ .\",\n \"In this case, the relevant scalar fields are $(\\\\phi ^{(0)},N^{(0)},b^{(0)})$ , and they satisfy the following equations respectively $\\\\ell =0:\\\\quad &&(\\\\Box ^2+2\\\\alpha ^4)S^{(0)}=0\\\\ ,\\\\qquad S^{(0)}=\\\\phi ^{(0)}+N^{(0)}\\\\ ,\\\\nonumber \\\\\\\\&& \\\\Box ^2b^{(0)}=0\\\\ ,\\\\nonumber \\\\\\\\&&\\\\Box ^2N^{(0)}= 0\\\\ .$ Thus besides two massless modes, we also have modes with linear time coordinate dependence.\",\n \"Finally, we discuss the complex scalars originating from $\\\\hat{z}_m$ .\",\n \"For positive monopole charge, the harmonic expansion of $\\\\hat{z}_m$ is given by $\\\\hat{z}_m=z^{(q-1)} {}_{-q}V_m^{(q-1)}+z^{(q)} {}_{-q}V_m^{(q)}+\\\\sum _{\\\\ell >q}(z^{(\\\\ell )}D_m{}_{-q}Y^{(\\\\ell )}+\\\\tilde{z}^{(\\\\ell )}\\\\epsilon _m^{~n}D_n{}_{-q}Y^{(\\\\ell )})\\\\ .$ Thus we have $&&\\\\ell >q &:\\\\qquad z^{(\\\\ell )}\\\\ ,\\\\ \\\\tilde{z}^{(\\\\ell )} &\\\\qquad \\\\mbox{with} &\\\\qquad m^2=\\\\alpha ^2c_\\\\ell -\\\\alpha ^2q+M^2\\\\ ,\\\\nonumber \\\\\\\\&&\\\\ell =q &:\\\\qquad z^{(q)}\\\\ , &\\\\qquad \\\\mbox{with} &\\\\qquad m^2=\\\\alpha ^2q+M^2\\\\ ,\\\\nonumber \\\\\\\\&&\\\\ell =q-1 &:\\\\qquad z^{(q-1)}\\\\ , &\\\\qquad \\\\mbox{with} &\\\\qquad m^2=M^2-\\\\alpha ^2q\\\\ .$\"\n ],\n [\n \"Fermionic sector\",\n \"The analysis of the fermionic spectrum in a non-supersymmetric background (REF ) is more subtle than that in supersymmetric background.\",\n \"Since the non-supersymmetric background do not posses Killing spinor, we will use spin-weighted harmonics $ _{s-{\\\\textstyle {\\\\frac{\\\\scriptstyle 1}{\\\\scriptstyle 2} } }}\\\\eta ^{(\\\\ell )}$ , described in detail in appendix B, as basis of expansion.\",\n \"For brevity, we shall use the notation $_{s-{\\\\textstyle {\\\\frac{\\\\scriptstyle 1}{\\\\scriptstyle 2} } }}\\\\eta ^{(\\\\ell )} \\\\equiv \\\\tilde{\\\\eta }^{(\\\\ell )}\\\\ ,$ where $s={\\\\textstyle {\\\\frac{\\\\scriptstyle 1}{\\\\scriptstyle 2} } }(1-q)$ .\",\n \"These harmonics satisfy the relations $&&\\\\tilde{\\\\eta }^{(\\\\ell )}_+=\\\\eta _+ \\\\left(_{s-1}Y^{(\\\\ell )} \\\\right)\\\\ , \\\\qquad \\\\tilde{\\\\eta }^{(\\\\ell )}_- = \\\\eta _- \\\\left( _{s}Y^{(\\\\ell )} \\\\right)\\\\ ,\\\\\\\\&& (\\\\frac{d}{d\\\\theta }+m\\\\csc \\\\theta +s\\\\cot \\\\theta ){}\\\\left(_{s}Y^{(\\\\ell )}\\\\right)=\\\\sqrt{(\\\\ell +s)(\\\\ell +1-s)}{}\\\\left(_{s-1}Y^{(\\\\ell )}\\\\right)\\\\ ,\\\\\\\\&&(\\\\frac{d}{d\\\\theta }-m\\\\csc \\\\theta -(s-1)\\\\cot \\\\theta ){}\\\\left(_{s-1}Y^{(\\\\ell )}\\\\right)=-\\\\sqrt{(\\\\ell +s)(\\\\ell +1-s)}{} \\\\left(_{s}Y^{(\\\\ell )}\\\\right)\\\\ .$ The lowest level would have definite chirality when $\\\\ell =-s$ for $q>0$ and $\\\\ell =s-1$ for $q<0$ .\",\n \"The spin weighted harmonics satisfy the following properties $\\\\sigma _3\\\\tilde{\\\\eta }_{\\\\pm }^{(\\\\ell )} &=\\\\pm \\\\tilde{\\\\eta }_{\\\\pm }^{(\\\\ell )}\\\\ , &\\\\qquad \\\\sigma ^n D_n\\\\tilde{\\\\eta }_{\\\\pm }^{(\\\\ell )} &=i\\\\alpha \\\\sqrt{\\\\tilde{c_\\\\ell }}\\\\,\\\\tilde{\\\\eta }_{\\\\mp }^{(\\\\ell )}\\\\ ,\\\\nonumber \\\\\\\\[D_m, D_n]\\\\tilde{\\\\eta }_-^{(\\\\ell )} &=-is\\\\alpha ^2\\\\epsilon _{mn}\\\\tilde{\\\\eta }_-^{(\\\\ell )}\\\\ ,&\\\\qquad [D_m,D_n]{\\\\tilde{\\\\eta }}_+^{(\\\\ell )} &=i(1-s)\\\\alpha ^2\\\\epsilon _{mn}{\\\\eta _+^{(\\\\ell )}}\\\\ ,\\\\nonumber \\\\\\\\D^nD_n\\\\tilde{\\\\eta }_-^{(\\\\ell )} &=-\\\\alpha ^2(\\\\tilde{c_\\\\ell }-s)\\\\tilde{\\\\eta }_-^{(\\\\ell )}\\\\ ,&\\\\qquad D^nD_n\\\\tilde{\\\\eta }_+^{(\\\\ell )} &= \\\\alpha ^2 (1-s-\\\\tilde{c_\\\\ell })\\\\tilde{\\\\eta }_+^{(\\\\ell )}\\\\ ,$ where $\\\\tilde{c_\\\\ell }=\\\\biggl (c_\\\\ell -s(s-1)\\\\biggr )\\\\ .$ The harmonic expansion for 6D spin-$1/2$ fields follows the same procedure as in supersymmetric case by using the spin weighted harmonics, while it is more subtle when expanding the 6D gravitini.\",\n \"It can be checked that the linearized equations have the following discreet symmetry $q\\\\rightarrow -q\\\\ , &\\\\qquad \\\\psi _{\\\\mu +}\\\\rightarrow -\\\\psi _{\\\\mu -},& \\\\qquad \\\\psi _{\\\\mu -}\\\\rightarrow \\\\psi _{\\\\mu +}\\\\ ,\\\\nonumber \\\\\\\\\\\\psi _-\\\\rightarrow \\\\psi _+\\\\ , &\\\\qquad \\\\psi _+\\\\rightarrow -\\\\psi _-\\\\ , &\\\\qquad \\\\chi _-\\\\rightarrow \\\\chi _+\\\\ ,\\\\nonumber \\\\\\\\\\\\chi _+\\\\rightarrow -\\\\chi _-\\\\ , &\\\\qquad \\\\lambda _+\\\\rightarrow -\\\\lambda _-\\\\ ,& \\\\qquad \\\\lambda _-\\\\rightarrow \\\\lambda _+\\\\ ,$ which implies that the spectrum keeps the same under the sign change of monopole charge.\",\n \"In the following, we will focus on the case with positive monopole charge and use $|s|$ to denote ${\\\\textstyle {\\\\frac{\\\\scriptstyle 1}{\\\\scriptstyle 2} } }(q-1)$ .\"\n ],\n [\n \"The gravitini satisfy $\\\\ell \\\\ge |s|+1:\\\\qquad &&{\\\\partial }\\\\Big (\\\\Box +\\\\alpha ^2|s|-\\\\alpha ^2\\\\tilde{c_\\\\ell }\\\\Big )\\\\psi _{\\\\mu +}^{(\\\\ell )}+i\\\\alpha \\\\sqrt{\\\\tilde{c_\\\\ell }}\\\\Big (\\\\Box -\\\\alpha ^2\\\\tilde{c_\\\\ell }\\\\Big )\\\\psi _{\\\\mu -}^{(\\\\ell )}=0\\\\ ,\\\\nonumber \\\\\\\\&&i\\\\alpha \\\\sqrt{\\\\tilde{c_\\\\ell }}\\\\Big (\\\\Box -\\\\alpha ^2\\\\tilde{c_\\\\ell }\\\\Big )\\\\psi _{\\\\mu +}^{(\\\\ell )}-{\\\\partial }\\\\Big (\\\\Box +(|s|+1-{\\\\tilde{c}}_\\\\ell )\\\\alpha ^2\\\\Big )\\\\psi _{\\\\mu -}^{(\\\\ell )}=0\\\\ .$ Diagonalising the associated $2\\\\times 2$ operator-valued matrix, we find that the modes are annihilated by the partially-factorising operator polynomial, of third order in $\\\\Box $ , given by ${\\\\cal O}^{(3/2)} = \\\\tilde{{\\\\cal O}}_3\\\\ ,$ where the explicit form of $\\\\tilde{{\\\\cal O}}_3$ can be deduced from (REF ).\",\n \"Next, we consider the case of $\\\\ell =|s|$ .\",\n \"In this case, we find that the quadratic action for the lowest level fermionic fields is proportional to ${\\\\cal L}^{(2)}&\\\\propto & -i\\\\bar{\\\\chi }\\\\gamma ^{\\\\mu \\\\nu }\\\\partial _{\\\\mu }\\\\psi _{\\\\nu }-i\\\\bar{\\\\psi }_{\\\\mu }\\\\gamma ^{\\\\mu \\\\nu }\\\\partial _{\\\\nu }\\\\chi +2i\\\\bar{\\\\chi }{\\\\partial }\\\\psi -2i\\\\bar{\\\\psi }{\\\\partial }\\\\chi +2\\\\bar{\\\\chi }{\\\\partial }\\\\chi \\\\nonumber \\\\\\\\&&-8g\\\\bar{\\\\chi }\\\\lambda -8g\\\\bar{\\\\lambda }\\\\chi -4\\\\bar{\\\\lambda }{\\\\partial }\\\\lambda -4i g|s|\\\\bar{\\\\lambda }\\\\gamma ^{\\\\mu }\\\\psi _{\\\\mu }-4i g|s|\\\\bar{\\\\psi }_{\\\\mu }\\\\gamma ^{\\\\mu }\\\\lambda \\\\nonumber \\\\\\\\&& -8g(1+|s|)i\\\\bar{\\\\lambda }\\\\psi +8g(1+|s|)i\\\\bar{\\\\psi }\\\\lambda -{\\\\textstyle {\\\\frac{\\\\scriptstyle 1}{\\\\scriptstyle 2} } }\\\\bar{\\\\psi }_{\\\\mu }\\\\gamma ^{\\\\mu \\\\nu \\\\lambda }\\\\partial _{\\\\nu }\\\\psi _{\\\\lambda }+\\\\bar{\\\\psi }{\\\\partial }\\\\psi \\\\nonumber \\\\\\\\&&+\\\\bar{\\\\psi }_{\\\\mu }\\\\gamma ^{\\\\mu \\\\nu }\\\\partial _{\\\\nu }\\\\psi -\\\\psi \\\\gamma ^{\\\\mu \\\\nu }\\\\partial _{\\\\mu }\\\\psi _{\\\\nu }-\\\\frac{1}{\\\\alpha ^2}\\\\biggl ({\\\\textstyle {\\\\frac{\\\\scriptstyle 1}{\\\\scriptstyle 4} } }\\\\bar{\\\\psi }_{\\\\mu \\\\nu }{\\\\partial }\\\\psi ^{\\\\mu \\\\nu }-\\\\bar{\\\\psi }{\\\\partial }\\\\Box \\\\psi \\\\nonumber \\\\\\\\&& +{\\\\textstyle {\\\\frac{\\\\scriptstyle 1}{\\\\scriptstyle 2} } }|s|\\\\alpha ^2\\\\bar{\\\\psi }_{\\\\mu }{\\\\partial }\\\\psi ^{\\\\mu }-|s|\\\\alpha ^2\\\\bar{\\\\psi }\\\\partial _{\\\\mu }\\\\psi ^{\\\\mu }+|s|\\\\alpha ^2\\\\bar{\\\\psi }_{\\\\mu }\\\\partial ^{\\\\mu }\\\\psi -(1+2|s|)\\\\alpha ^2\\\\bar{\\\\psi }{\\\\partial }\\\\psi \\\\biggr )\\\\ .$ Unlike the supersymmetric case, we see the appearance of the terms $\\\\bar{\\\\lambda }\\\\gamma ^{\\\\mu }\\\\psi _{\\\\mu }$ , $\\\\bar{\\\\psi }_{\\\\mu }{\\\\partial }\\\\psi ^{\\\\mu }$ and $\\\\bar{\\\\psi }_{\\\\mu }\\\\partial ^{\\\\mu }\\\\psi $ which break the fermionic gauge symmetry.\",\n \"The homogeneous solutions for gravitini satisfy ${\\\\partial }(\\\\Box -\\\\alpha ^2-|s|\\\\alpha ^2)\\\\psi _{\\\\mu }^{(|s|)}=0,\\\\qquad \\\\gamma ^{\\\\mu }\\\\psi ^{(|s|)}_{\\\\mu }=0,\\\\qquad \\\\partial ^{\\\\mu }\\\\psi _{\\\\mu }^{(|s|)}=0.$ Thus, due to the lack of fermionic gauge symmetry, the longitudinal mode $\\\\psi _{\\\\mu }\\\\propto \\\\,p_{\\\\mu }e^{ipx}$ with $p^2=0$ becomes a dynamical degree of freedom.\",\n \"The $\\\\ell \\\\ge |s|+2$ sector consists of ten spin-1/2 fields $(\\\\Lambda _+,\\\\Psi _+,\\\\Lambda _-,\\\\Psi _-,$ $\\\\psi _+,\\\\psi _-,\\\\chi _+,\\\\chi _-,\\\\lambda _+,\\\\lambda _-)$ .\",\n \"The linearized equations describing their mixing are $0&=&{\\\\partial }\\\\Lambda ^{(\\\\ell )}_{-}-\\\\Psi _+^{(\\\\ell )}+i\\\\alpha \\\\sqrt{\\\\tilde{c_\\\\ell }}\\\\psi _+^{(\\\\ell )}-i\\\\alpha \\\\sqrt{\\\\tilde{c_\\\\ell }}\\\\Lambda _+^{(\\\\ell )}+2{\\\\partial }\\\\psi _-^{(\\\\ell )}-2i{\\\\partial }\\\\chi _-^{(\\\\ell )}+2\\\\alpha \\\\sqrt{\\\\tilde{c_\\\\ell }}\\\\chi _+^{(\\\\ell )}-8gi\\\\lambda _+^{(\\\\ell )}\\\\ ,\\\\\\\\0&=&{\\\\partial }\\\\Lambda _+^{(\\\\ell )}-\\\\Psi _-^{(\\\\ell )}+i\\\\alpha \\\\sqrt{\\\\tilde{c_\\\\ell }}\\\\psi _-^{(\\\\ell )}+i\\\\alpha \\\\sqrt{\\\\tilde{c_\\\\ell }}\\\\Lambda _-^{(\\\\ell )}-2{\\\\partial }\\\\psi _+^{(\\\\ell )}+2i{\\\\partial }\\\\chi _+^{(\\\\ell )} +2\\\\alpha \\\\sqrt{\\\\tilde{c_\\\\ell }}\\\\chi _-^{(\\\\ell )}-8gi\\\\lambda _-^{(\\\\ell )}\\\\ ,\\\\\\\\0&=& g(1+|s|)i\\\\Lambda _-^{(\\\\ell )}+2g\\\\chi _-^{(\\\\ell )}-{\\\\partial }\\\\lambda _+^{(\\\\ell )}-i\\\\alpha \\\\sqrt{\\\\tilde{c_\\\\ell }}\\\\lambda _-^{(\\\\ell )}\\\\ ,\\\\\\\\0&=&2i g|s|\\\\psi _+^{(\\\\ell )}+2g\\\\chi _+^{(\\\\ell )}+{\\\\partial }\\\\lambda _-^{(\\\\ell )}-i\\\\alpha \\\\sqrt{\\\\tilde{c_\\\\ell }}\\\\lambda _+^{(\\\\ell )}\\\\ ,\\\\\\\\0&=&i\\\\sqrt{\\\\tilde{c_\\\\ell }}{\\\\partial }(\\\\Box +\\\\alpha ^2+|s|\\\\alpha ^2)\\\\psi _+^{(\\\\ell )}-i\\\\alpha ^2\\\\sqrt{\\\\tilde{c_\\\\ell }}{\\\\partial }\\\\Lambda _+^{(\\\\ell )}+i\\\\alpha ^2\\\\tilde{c_\\\\ell }^{3/2}\\\\Psi _-^{(\\\\ell )}+2\\\\alpha ^2\\\\sqrt{\\\\tilde{c_\\\\ell }}{\\\\partial }\\\\chi _+^{(\\\\ell )}\\\\nonumber \\\\\\\\&& -8g(1+|s|)\\\\alpha \\\\,i{\\\\partial }\\\\lambda _+^{(\\\\ell )}+\\\\alpha (\\\\tilde{c_\\\\ell }-1-|s|){\\\\partial }\\\\Psi _+^{(\\\\ell )}+\\\\alpha (2+2|s|-\\\\tilde{c_\\\\ell })\\\\Box \\\\psi _-^{(\\\\ell )}\\\\ ,\\\\\\\\0&=& i\\\\sqrt{\\\\tilde{c_\\\\ell }}{\\\\partial }(\\\\Box -|s|\\\\alpha ^2)\\\\psi _-^{(\\\\ell )}+i\\\\alpha ^2\\\\sqrt{\\\\tilde{c_\\\\ell }}{\\\\partial }\\\\Lambda _-^{(\\\\ell )}-i\\\\alpha ^2\\\\tilde{c_\\\\ell }^{3/2}\\\\Psi _+^{(\\\\ell )}+2\\\\alpha ^2\\\\sqrt{\\\\tilde{c_\\\\ell }}{\\\\partial }\\\\chi _-^{(\\\\ell )}+8ig|s|\\\\alpha {\\\\partial }\\\\lambda _{-}^{(\\\\ell )}\\\\nonumber \\\\\\\\&&+\\\\alpha (\\\\tilde{c_\\\\ell }+|s|){\\\\partial }\\\\Psi _-^{(\\\\ell )}+\\\\alpha (\\\\tilde{c_\\\\ell }+2|s|)\\\\Box \\\\psi _+^{(\\\\ell )}\\\\ ,\\\\\\\\0&=&{\\\\partial }(\\\\Box +3\\\\alpha ^2+|s|\\\\alpha ^2-\\\\alpha ^2\\\\tilde{c_\\\\ell })\\\\Lambda _-^{(\\\\ell )}-(\\\\Box +4\\\\alpha ^2+2|s|\\\\alpha ^2-2\\\\alpha ^2\\\\tilde{c_\\\\ell })\\\\Psi _+^{(\\\\ell )}+i\\\\alpha \\\\sqrt{\\\\tilde{c_\\\\ell }}(\\\\Box +4\\\\alpha ^2+4|s|\\\\alpha ^2)\\\\psi _+^{(\\\\ell )}\\\\nonumber \\\\\\\\&&+\\\\alpha ^2{\\\\partial }(8+2|s|-\\\\tilde{c_\\\\ell })\\\\psi _-^{(\\\\ell )}-i\\\\alpha \\\\sqrt{\\\\tilde{c_\\\\ell }}(\\\\Box +4\\\\alpha ^2-\\\\alpha ^2\\\\tilde{c_\\\\ell })\\\\Lambda _+^{(\\\\ell )}-6i\\\\alpha ^2{\\\\partial }\\\\chi _-^{(\\\\ell )}+8\\\\alpha ^3\\\\sqrt{\\\\tilde{c_\\\\ell }}\\\\chi _+^{(\\\\ell )}\\\\nonumber \\\\\\\\&&-32ig\\\\alpha ^2\\\\,(1+|s|)\\\\lambda _+^{(\\\\ell )} +i\\\\alpha \\\\sqrt{\\\\tilde{c_\\\\ell }}{\\\\partial }\\\\Psi _-^{(\\\\ell )}\\\\ ,\\\\\\\\0&=&-{\\\\partial }(\\\\Box +2\\\\alpha ^2-|s|\\\\alpha ^2-\\\\alpha ^2\\\\tilde{c_\\\\ell })\\\\Lambda _+^{(\\\\ell )}+(\\\\Box +2\\\\alpha ^2-2|s|-2\\\\alpha ^2\\\\tilde{c_\\\\ell })\\\\Psi _-^{(\\\\ell )}- i\\\\alpha \\\\sqrt{\\\\tilde{c_\\\\ell }}(\\\\Box -4|s|\\\\alpha ^2)\\\\psi _-^{(\\\\ell )}\\\\nonumber \\\\\\\\&&+\\\\alpha ^2{\\\\partial }(6-2|s|-\\\\tilde{c_\\\\ell })\\\\psi _+^{(\\\\ell )}-i\\\\alpha \\\\sqrt{\\\\tilde{c_\\\\ell }}(\\\\Box +4\\\\alpha ^2-\\\\alpha ^2\\\\tilde{c_\\\\ell })\\\\Lambda _-^{(\\\\ell )}-6i\\\\alpha ^2 {\\\\partial }\\\\chi _+^{(\\\\ell )}-8\\\\alpha ^3\\\\sqrt{\\\\tilde{c_\\\\ell }}\\\\chi _-^{(\\\\ell )}\\\\nonumber \\\\\\\\&&-32ig|s|\\\\alpha ^2 \\\\lambda _-^{(\\\\ell )}+i\\\\alpha \\\\sqrt{\\\\tilde{c_\\\\ell }} {\\\\partial }\\\\Psi _+^{(\\\\ell )}\\\\ ,\\\\\\\\0&=&(1+|s|)\\\\alpha ^2\\\\Lambda _-^{(\\\\ell )}-2i\\\\alpha ^2 \\\\chi _-^{(\\\\ell )}+(2\\\\Box -2 |s|\\\\alpha ^2-\\\\alpha ^2\\\\tilde{c_\\\\ell })\\\\psi _-^{(\\\\ell )}-{\\\\partial }\\\\Psi _+^{(\\\\ell )}-i\\\\alpha \\\\sqrt{\\\\tilde{c_\\\\ell }}\\\\Psi _-^{(\\\\ell )}-i\\\\alpha \\\\sqrt{\\\\tilde{c_\\\\ell }}{\\\\partial }\\\\psi _+^{(\\\\ell )}\\\\ ,\\\\nonumber \\\\\\\\&& \\\\\\\\0&=&|s|\\\\alpha ^2\\\\Lambda _+^{(\\\\ell )}-2i\\\\alpha ^2 \\\\chi _+^{(\\\\ell )}+(2\\\\Box +2\\\\alpha ^2+2|s|\\\\alpha ^2-\\\\alpha ^2\\\\tilde{c_\\\\ell })\\\\psi _+^{(\\\\ell )} -i\\\\alpha \\\\sqrt{\\\\tilde{c_\\\\ell }}\\\\Psi _+^{(\\\\ell )}+{\\\\partial }\\\\Psi _-^{(\\\\ell )}+i\\\\alpha \\\\sqrt{\\\\tilde{c_\\\\ell }}{\\\\partial }\\\\psi _-^{(\\\\ell )}\\\\ .\\\\nonumber \\\\\\\\$ Diagonalising the associated $10\\\\times 10$ operator-valued matrix, we find that the modes are annihilated by the partially-factorising operator polynomial, of ninth order in $\\\\Box $ , given by $\\\\ell \\\\ge |s|+2:\\\\qquad {\\\\cal O}^{(0)} = \\\\hat{\\\\Box }_0^3\\\\,\\\\tilde{{\\\\cal O}}_6\\\\ ,$ where the explicit form of the operator ${\\\\cal O}_6$ can be determined from the linearized spin 1/2 field equations listed above.\",\n \"$\\\\ell =|s|+1$ At this level, since $D_m{\\\\eta _+^{(|s|+1)}}=\\\\frac{i}{2}\\\\alpha \\\\sqrt{\\\\tilde{c_\\\\ell }}\\\\sigma _m{\\\\eta _-^{(|s|+1)}}$ , there emerges a fermionic gauge symmetry generated by $\\\\hat{\\\\epsilon }=\\\\epsilon _+^{(|s|+1)}{\\\\eta _+^{(|s|+1)}}$ $\\\\delta \\\\psi _{\\\\mu +}^{(|s|+1)}&=&\\\\partial _{\\\\mu }\\\\epsilon _+^{(|s|+1)}+i\\\\alpha \\\\sqrt{\\\\frac{(1+|s|)}{2}}\\\\,\\\\epsilon _+^{(|s|+1)}\\\\ ,\\\\nonumber \\\\\\\\\\\\delta \\\\lambda _+^{(|s|+1,m)}&=&ig(1+|s|)\\\\epsilon _+^{(|s|+1)}\\\\ .$ Using this gauge symmetry, one can eliminate $\\\\psi _+^{(|s|+1)}$ , such that the 10 by 10 mixing becomes 9 by 9 mixing.\",\n \"Diagonalising this system, we find that the modes are determined by the partially-factorising operator polynomial, of fifteenth order in ${\\\\partial }$ , given by $\\\\ell =|s|+1:\\\\qquad {\\\\partial }\\\\Big (\\\\Box - (|s|+2)\\\\alpha ^2\\\\Big )^2 \\\\tilde{{\\\\cal O}}_5\\\\ ,$ where the explicit form of $\\\\tilde{\\\\cal O}_5$ can be deduced from the mixing equations.\",\n \"It is clear that is a massless spin-1/2 mode.\",\n \"Explicitly, it is a linear combination of $(\\\\Lambda ^{(|s|+1)}_+,\\\\Psi ^{(|s|+1)}_+,\\\\Lambda ^{(|s|+1)}_-,\\\\Psi ^{(|s|+1)}_-,\\\\psi ^{(|s|+1)}_-,$ $\\\\chi ^{(|s|+1)}_-, \\\\chi ^{(|s|+1)}_+,\\\\lambda ^{(|s|+1)}_+,\\\\lambda ^{(|s|+1)}_-)$ with mixing coefficients $(8(1+2|s|),0,0,0,-2,-2i(1+2|s|),0,0,\\\\sqrt{1+|s|}(1+4|s|).$ $\\\\ell =|s|$ The coupled system of linearized field equations for the spin-1/2 fields $(\\\\Lambda _+^{(|s|)},\\\\Psi _-^{(|s|)}, \\\\psi _+^{(|s|)},\\\\chi _+^{(|s|)},\\\\lambda _-^{(|s|)})$ are $0&=&{\\\\partial }\\\\Lambda _+^{(|s|)}-\\\\Psi _-^{(|s|)}-2{\\\\partial }\\\\psi _+^{(|s|)}+2i{\\\\partial }\\\\chi _+^{(|s|)}-8gi\\\\lambda _-^{(|s|)}\\\\ ,\\\\\\\\0&=&i g |s| \\\\Lambda _+^{(|s|)}+2g(1+|s|)i\\\\psi _+^{(|s|)}+2g\\\\chi _+^{(|s|)}+{\\\\partial }\\\\lambda _-^{(|s|)}=0,\\\\\\\\0&=&8gi{\\\\partial }\\\\lambda _-^{(|s|)}+{\\\\partial }\\\\Psi _-^{(|s|)}+2\\\\Box \\\\psi _+^{(|s|)}=0,\\\\\\\\0&=&(\\\\Box +2\\\\alpha ^2-|s|\\\\alpha ^2){\\\\partial }\\\\Lambda _+^{(|s| )}-(\\\\Box +2\\\\alpha ^2-2|s|\\\\alpha ^2)\\\\Psi _-^{(|s| )}+(2|s|-6)\\\\alpha ^2{\\\\partial }\\\\psi _+^{(|s| )}+6\\\\alpha ^2\\\\,i{\\\\partial }\\\\chi _+^{(|s| )}\\\\nonumber \\\\\\\\&&+32g|s|\\\\alpha ^2\\\\,i\\\\lambda _-^{(|s|) }\\\\ ,\\\\\\\\0&=&\\\\alpha ^2{\\\\partial }\\\\Lambda _+^{(|s| )}-(1+|s|)\\\\alpha ^2\\\\Psi _-^{(|s| )}-{\\\\partial }(\\\\Box +2\\\\alpha ^2+2|s|\\\\alpha ^2)\\\\psi _+^{|s|}-8g(1+|s|)\\\\alpha ^2\\\\,i\\\\lambda _-^{(|s|)}+2\\\\alpha ^2\\\\,i{\\\\partial }\\\\chi _+^{(|s|)}\\\\nonumber .\\\\\\\\$ Diagonalising this system, we find two massless modes which satisfy the relations $\\\\Psi _-^{(|s|)}=0\\\\ ,\\\\qquad \\\\lambda _-^{(|s|)}=0\\\\ ,\\\\qquad |s|\\\\Lambda _+^{(|s|)}-2i\\\\chi _+^{(|s|)}+2(1+|s|)\\\\psi _+^{(|s|)}=0\\\\ .$ $\\\\ell =|s|-1$ When monopole charge $q\\\\ge 3$ , there exist charge “$-q/2$ ” vector-spinor harmonics $\\\\eta _m$ on $S^2$ possessing following properties $D^m\\\\left( _{s-{\\\\textstyle {\\\\frac{\\\\scriptstyle 1}{\\\\scriptstyle 2} } }}\\\\eta ^{(|s|-1)}\\\\right)_m=0\\\\ , \\\\qquad \\\\sigma ^m\\\\left( _{s-{\\\\textstyle {\\\\frac{\\\\scriptstyle 1}{\\\\scriptstyle 2} } }}\\\\eta ^{(|s|-1)}\\\\right)_m =0\\\\ ,$ where $s=(1-q)/2$ .\",\n \"It follows that $\\\\sigma ^m D_m \\\\left( _{s-{\\\\textstyle {\\\\frac{\\\\scriptstyle 1}{\\\\scriptstyle 2} } }}\\\\eta ^{(|s|-1)}\\\\right)_n=0\\\\ .$ Associated to this harmonics, there is a new spin-1/2 field $\\\\tilde{\\\\psi }^{(|s|-1)}$ satisfying ${\\\\partial }\\\\Box \\\\tilde{\\\\psi }^{(|s|-1)}=0\\\\ .$\"\n ],\n [\n \"Remarks on the non-supersymmetric spectrum\",\n \"We saw in the supersymmetric vacuum that even at the $\\\\ell =0$ level in the Kaluza-Klein harmonic expansions, avoiding tachyons in the four-dimensional spectrum required imposing the condition $M^2\\\\ge (5+2\\\\sqrt{6})\\\\alpha ^2$ on the parameter $M$ in the six-dimensional Lagrangian.\",\n \"In the non-supersymmetric vacua we necessarily have $M^2=\\\\alpha ^2$ , and in fact having larger values for the background monopole charge $q$ then the $q=\\\\pm 1$ supersymmetric case only makes the tachyon problem worse, as can be seen from the vector mass operator (REF ).\",\n \"For this reason, we shall not explore further the precise details of the occurrence of tachyonic states in the non-supersymmetric backgrounds.\",\n \"One feature of interest that we shall, however, comment on is the occurrence of massless fermions in the non-supersymmetric backgrounds.\",\n \"As can be seen from the spin-${\\\\textstyle {\\\\frac{\\\\scriptstyle 3}{\\\\scriptstyle 2} } }$ operator in (REF ), there will be massless spin-${\\\\textstyle {\\\\frac{\\\\scriptstyle 3}{\\\\scriptstyle 2} } }$ fields at level $\\\\ell =|s|={\\\\textstyle {\\\\frac{\\\\scriptstyle 1}{\\\\scriptstyle 2} } }(q-1)$ ; thus these will occur in an $SU(2)$ multiplet of dimension $2\\\\ell +1 = q$ .\",\n \"At $\\\\ell =|s|+1$ , $\\\\ell =|s|$ and $\\\\ell =|s|-1$ there will also be massless spin-${\\\\textstyle {\\\\frac{\\\\scriptstyle 1}{\\\\scriptstyle 2} } }$ modes, as was discussed in the spin-${\\\\textstyle {\\\\frac{\\\\scriptstyle 1}{\\\\scriptstyle 2} } }$ section above.\"\n ],\n [\n \"Conclusions\",\n \"In this paper we have studied the complete linearised spectrum in the $S^2$ Kaluza-Klein reduction of off-shell six-dimensional ${\\\\cal N}=(1,0)$ gauged supergravity extended by a Riemann-squared superinvariant.\",\n \"The higher-derivative terms in the six-dimensional theory can be expected to imply the occurrence of ghosts.\",\n \"As discussed in the introduction, the usual argument for the positivity of the energies of states in a supersymmetric background breaks down, and indeed we found that states in the Kaluza-Klein spectrum could now have complex energies, thus implying instabilities.\",\n \"One way to understand the occurrence of complex masses is that there are mixings between four-dimensional ghostlike and non-ghostlike modes.\",\n \"This may be illustrated by the following simple example.\",\n \"Consider a set of fields $\\\\phi _i$ with the Lagrangian ${\\\\cal L}= -{\\\\textstyle {\\\\frac{\\\\scriptstyle 1}{\\\\scriptstyle 2} } } K_{ij}\\\\, \\\\partial \\\\phi _i\\\\, \\\\partial \\\\phi _j -{\\\\textstyle {\\\\frac{\\\\scriptstyle 1}{\\\\scriptstyle 2} } } V_{ij}\\\\phi _i\\\\phi _j\\\\,,$ where $K_{ij}$ and $V_{ij}$ are constant symmetric matrices.\",\n \"If the eigenvalues of $K_{ij}$ are all positive, then $K_{ij}$ and $V_{ij}$ can be simultaneously diagonalised, by means of orthogonal transformations combined with rescalings of the fields.\",\n \"However, if $K_{ij}$ has negative as well as positive eigenvalues, then the rescalings will introduce factors of $\\\\sqrt{-1}$ and the diagonalised mass matrix will be complex.\",\n \"One way to avoid the ghost problems of the higher-derivative theory is to treat it not as an exact model in its own right, but rather as an effective theory valid at energy scales $\\\\sqrt{\\\\Lambda }$ much smaller than $M$ .\",\n \"In this case the propagators are governed by the leading-order theory without the higher-derivative terms, and these terms are treated as interactions.\",\n \"The Kaluza-Klein spectrum in the reduced four-dimensional theory would then be simply that of the original Salam-Sezgin model corresponding to the $M^2\\\\rightarrow \\\\infty $ limit.\",\n \"This spectrum has been given in [6] and our results agree in the $M^2\\\\rightarrow \\\\infty $ limit.\",\n \"However, the Kaluza-Klein level number $\\\\ell $ would have to be restricted to lie below some maximum value, in order to satisfy the $\\\\Lambda <0$ .","We recall that classifying such compact manifolds is a challenging open problem, only solved for $n=3$ using Perelman's techniques.","We also mention [5] where M. Dahl and C. Bär deduce many topological properties of compact manifolds from a careful study of the eigenvalues $\\lambda _i$ of the Yamabe operator $L_g$ .","The sign of $\\lambda _1$ can also be read in terms of existence or non-existence of positive solutions of the Yamabe equation: $ L_g u = \\epsilon |u|^{N-2} u,$ where $N := \\frac{2n}{n-2}$ and $\\epsilon \\in \\lbrace -1;0;1\\rbrace $ .","Inspired by this observation, B. Ammann and E. Humbert [1] enlighted the role of $\\lambda _2$ in the existence of nodal solutions (i.e.","having a changing sign) of the Yamabe equation (REF ).","See again Section for more explanations.","In this paper, we establish various properties of the eigenvalues of the Yamabe operator.","First of all, we extend their definition to what we call generalized metrics when possible (see Paragraph ) and prove that their sign is a conformal invariant (see Paragraph REF ).","This paper initiates the study of the relationships between these conformal invariants and the topology of the manifold by showing that their negativity is not topologically obstructed (see Paragraph REF ).","These investigations will be treated much more deeply in [7].","The main point of this article is to complete the results of B. Ammann and E. Humbert [1] and to study how the sign of the second eigenvalue of the Yamabe operator can be related to the existence of nodal solutions of the Yamabe equation (REF ), in particular when the Yamabe constant of $(M,g)$ is negative.","Our main result is to prove that under this condition, such a solution always exists with $\\epsilon = sign (\\lambda _2(g))$ .","This is the object of Theorem REF .","The author would like to thank Emmanuel Humbert for his support and encouragements."],["Eigenvalues in conformal metrics","In the whole paper, we will deal with the behavior of the eigenvalues of the Yamabe operator in a fixed conformal class.","It will be usefull to express their definition relatively to a fixed metric.","This is the goal of this section."],["Smooth metrics","Let $(M,g)$ be a compact Riemannian manifold of dimension $n\\ge 3$ , we keep the notations of the introduction and for any metric $\\widetilde{g}$ , we will denote by ${\\lambda }_1(\\widetilde{g}) < {\\lambda }_2(\\widetilde{g}) \\le {\\lambda }_3(\\widetilde{g}) \\le \\cdots \\le {\\lambda }_k(\\widetilde{g}) \\cdots \\rightarrow +\\infty ,$ the eigenvalues of the Yamabe operator.","We will deal with the case where $\\widetilde{g}$ is conformal to $g$ , i.e.","when $\\widetilde{g} =u^{N-2} g,$ where $u$ is a positive function of class $C^\\infty $ .","By referring to [1], one sees that the $i-$ th eigenvalue ${\\lambda }_i(\\widetilde{g})$ is given by $\\lambda _i (\\widetilde{g}) = \\inf _{V\\in Gr_{i}({H_1^2(M)})} \\sup _{v \\in V\\setminus \\lbrace 0\\rbrace }\\frac{\\int _M c_n |v|^2 + S_g v^2 \\, dv_g }{\\int _M v^2 \\ u^{N-2} \\ dv_g},$ where $\\widetilde{g} = u^{N-2}g;$ $u\\in C^\\infty (M),$ $u>0$ and $Gr_{i}({H_1^2(M)})$ stands for the set of all $i$ -dimensional subspaces of $H_1^2(M).$"],["Generalized metrics","Reducing to smooth metrics will too restrictive for our investigations.","We will need to work with generalized metrics, i.e.","metrics of the form $\\widetilde{g} = u^{N-2}g$ with $u\\in L^N(M),$ $u\\ge 0$ and $u\\lnot \\equiv 0$ .","The Yamabe operator $L_{\\widetilde{g}}$ has no meaning any more but the definition of $\\lambda _i(\\widetilde{g})$ can anyway be extended to this case by using (REF ) as it was done in [1] when the Yamabe constant was non-negative i.e.","when $\\lambda _1(g) \\ge 0$ .","When $\\lambda _1(g) < 0$ , the situation is a little bit different: $\\lambda _i(\\widetilde{g})$ defined by (REF ) can be $-\\infty $ as proved by the following proposition.","Proposition 2.1 Assume that $\\lambda _1(g)<0$ , then there exists $u\\in L^N(M)$ , $u \\lnot \\equiv 0$ , $u\\ge 0$ such that $\\lambda _1(\\widetilde{g}) = -\\infty $ , where $\\widetilde{g} = u^{N-2}g$ .","This proposition will be proved in Paragraph REF .","To make sure that $\\lambda _1(\\widetilde{g})$ is finite, one has to assume in addition that $u$ is positive.","Proposition 2.2 Let $u$ be a positive function in $L^N(M)$ .","Suppose that ${\\lambda }_1(g)<0$ .","Then, we have ${\\lambda }_1(\\widetilde{g})> -\\infty .$ The proposition is proved in Paragraph REF .","Notation 2.3 The $i^{th}$ eigenvalue of $L_g,$ ${\\lambda }_i(\\widetilde{g}) = {\\lambda }_i(u^{N-2}g)$ will be denoted by ${\\lambda }_i(u)$ when there is no ambiguity about $g$ ."],["Proof of Proposition ","We have ${\\lambda }_1(g)<0$ , this implies that there exists a function $v\\in C^{\\infty }(M)$ such that $\\int _M (L_g v)v\\,dv_g < 0.$ Let $P$ be a point of $M$ .","For ${\\varepsilon }>0$ , we define $\\eta _{{\\varepsilon }}$ as follows $\\left\\lbrace \\begin{array}{c}0 \\le \\eta _{{\\varepsilon }} \\le 1,\\\\\\\\\\eta _{{\\varepsilon }} = 0 \\text{ on }B_{{\\varepsilon }}(P),\\\\\\\\\\eta _{{\\varepsilon }} = 1\\text{ on } M\\backslash B_{2{\\varepsilon }}(P),\\\\\\\\\\vert \\nabla \\eta _{{\\varepsilon }} \\vert \\le \\frac{2}{{\\varepsilon }}.\\end{array}\\right.$ where $B_{\\delta }(P)$ stands for the ball of center $P$ and radius $\\delta $ in the metric $g$ .","Then one easily cheks $\\lim _{{\\varepsilon }\\rightarrow 0}\\int _M (L_g(\\eta _{{\\varepsilon }}v))(\\eta _{{\\varepsilon }}v)\\,dv_g = \\int _M (L_g v)v\\,dv_g.$ We define $w:= \\eta _{{\\varepsilon }}v$ .","Therefore, for a fixed small ${\\varepsilon }> 0$ , we have $\\int _M (L_g w)w\\, dv_g < 0.$ Let $u\\ge 0$ , $u\\lnot \\equiv 0$ of class $C^{\\infty }$ with support in $B_{{\\varepsilon }}(P).$ For $\\alpha >0$ , since $(w+\\alpha )u \\lnot \\equiv 0$ , we can write ${\\lambda }_1(\\widetilde{g}) &=& \\inf _{v^{\\prime }} \\frac{\\int _M (L_g v^{\\prime })v^{\\prime }\\,dv_g}{\\int _M u^{N-2} v^{\\prime 2} \\,dv_g}\\\\&\\le & \\lim _{\\alpha \\rightarrow 0^+} \\frac{\\int _M (L_g (w+\\alpha ))(w+\\alpha )\\,dv_g}{\\int _M u^{N-2} (w+\\alpha )^2 \\,dv_g}.$ Moreover, we have $\\lim _{\\alpha \\rightarrow 0^+}\\int _M (L_g (w+\\alpha ))(w+\\alpha )\\,dv_g=\\int _M (L_g (w))w\\,dv_g <0,$ and $ \\lim _{\\alpha \\rightarrow 0^+} \\int _M u^{N-2} (w+\\alpha )^2 \\,dv_g = 0$ which gives that $\\lim _{\\alpha \\rightarrow 0}\\frac{\\int _M (L_g (w+\\alpha ))(w+\\alpha )\\,dv_g}{\\int _M u^{N-2} (w+\\alpha )^2 \\,dv_g} = -\\infty .$ This ends the proof of Proposition REF ."],["Proof of Proposition ","Let $(v_m)_m$ be a minimizing sequence for ${\\lambda }_1(u),$ i.e.","$v_m\\in H_1^2(M)$ such that $\\lim _{m\\longrightarrow \\infty } \\frac{\\int _M c_n\\vert \\nabla v_m\\vert ^2 + S_gv_m^2 \\ dv_g}{\\int _M \\vert u\\vert ^{N-2} v_m^2 \\ dv_g} = \\lim _{m\\longrightarrow \\infty } {\\lambda }_m = {\\lambda }_1(u) < 0.$ Since $(\\vert v_m\\vert )_m$ is also a minimizing sequence for ${\\lambda }_1(u),$ we can assume that $v_m\\ge 0.$ We normalize $v_m$ by $\\int _M \\vert u\\vert ^{N-2} v_m^2 \\ dv_g = 1.$ Here we show that $(v_m)_m$ is bounded in $H_1^2(M).$ Indeed, suppose that $(v_m)_m$ is not bounded in $H_1^2(M)$ and let $v_m^\\prime = \\frac{v_m}{\\parallel v_m\\parallel _{H_1^2(M)}}.$ $(v_m^\\prime )_m$ is bounded in $H_1^2(M),$ and his norm is equal to 1, then there exists $v^\\prime \\in H_1^2(M),$ (after restriction to a subsequence) such that $v_m^\\prime \\rightharpoonup v^\\prime \\text{ in } H_1^2(M),$ $v_m^\\prime \\longrightarrow v^\\prime \\text{ in }L^2(M).$ We have $c_n \\int _M \\left|\\nabla v_m^{\\prime }\\right|^2\\, dv_g + \\int _M S_g v_m^{\\prime 2} \\, dv_g = {\\lambda }_m \\int _M \\left|u\\right|^{N-2}v_m^{\\prime 2}\\, dv_g.$ Moreover, $\\int _M \\left|u\\right|^{N-2}v^{\\prime 2}\\, dv_g \\le \\int _M \\left|u\\right|^{N-2}v_m^{\\prime 2}\\, dv_g \\rightarrow _{m\\longrightarrow \\infty } 0$ since $\\left\\Vert v_m\\right\\Vert _{H_1^2(M)}\\longrightarrow \\infty .$ It follows that $\\int _M \\left|u\\right|^{N-2}v^{\\prime 2}\\, dv_g = 0$ and since $u$ is positive, $v^{\\prime } = 0.$ Now, we write $1 = \\int _M \\left|\\nabla v_m^{\\prime }\\right|^2\\, dv_g + \\underbrace{\\int _M \\left|v_m^{\\prime }\\right|^2 \\, dv_g}_{\\longrightarrow 0}.$ We deduce that $\\lim _{m\\rightarrow \\infty } \\int _M \\left|\\nabla v_m^{\\prime }\\right|^2\\, dv_g=1,$ giving the desired contradiction: $c_n \\underbrace{\\int _M \\left|\\nabla v_m^{\\prime }\\right|^2\\, dv_g}_{\\longrightarrow 1} + \\underbrace{\\int _M S_g v_m^{\\prime 2} \\, dv_g}_{\\longrightarrow 0} = {\\lambda }_m \\int _M \\left|u\\right|^{N-2}v_m^{\\prime 2}\\, dv_g \\le 0.$ This proves that $(v_m)_m$ is bounded in $H_1^2(M),$ and implies that ${\\lambda }_m \\ge C.$ We finally get ${\\lambda }_1(\\widetilde{g}) > -\\infty .$"],["PDE associated to ${\\lambda }_i$","Proposition 2.4 For any non-negative function $u\\in L^N(M),$ such that ${\\lambda }_1(u)>-\\infty $ , there exists functions $v_1>0, v_2, \\ldots , v_k \\in H_1^2(M)$ having a changing sign, such that in the sense of distributions, we have $L_g v_1 = {\\lambda }_1(u) \\vert u\\vert ^{N-2} v_1,$ and $L_g v_k = {\\lambda }_k(u) \\vert u\\vert ^{N-2} v_k.$ Moreover, we can normalize the $v_k$ by $ \\int _M \\vert u\\vert ^{N-2} v_k^2 \\ dv_g = 1 \\text{ and }\\int _M\\vert u\\vert ^{N-2} v_i v_j \\ dv_g =0 \\hspace{5.69046pt}\\forall i \\ne j.$ Proof: Let $(v_m)_m$ be a minimizing sequence for ${\\lambda }_1(u),$ i.e.","$v_m\\in H_1^2(M)$ such that $\\lim _{m\\longrightarrow \\infty } \\frac{\\int _M c_n\\vert \\nabla v_m\\vert ^2 + S_gv_m^2 \\ dv_g}{\\int _M \\vert u\\vert ^{N-2} v_m^2 \\ dv_g} = {\\lambda }_1(u).$ According to the Paragraph REF , we get that $(v_m)_m$ is bounded in $H_1^2(M)$ and there exists $v\\ge 0$ in $H_1^2(M)$ such that $v_m$ converges to $v$ weakly in $H_1^2(M)$ and strongly in $L^2(M)$ (after restriction to a subsequence).","We now want to prove $\\int _M \\vert u\\vert ^{N-2} v^2 \\ dv_g = \\lim _{m\\longrightarrow \\infty } \\int _M\\vert u\\vert ^{N-2} v_m^2 \\ dv_g = 1.$ If $u$ is smooth, this relation is clear.","So let us assume that $u\\in Ł^N(M),$ let $A$ be a large real number and set $u_A = \\inf \\left\\lbrace u,A\\right\\rbrace .$ By Hölder inequality, we write $\\left| \\int _M u^{N-2} \\left(v_m^2 - v^2\\right) \\,dv_g \\right| &=& \\left| \\int _M\\left(u^{N-2} -u_A^{N-2} +u_A^{N-2}\\right)\\left(v_m^2 - v^2\\right) \\,dv_g \\right|\\\\& \\le & \\left( \\int _M u_A^{N-2} |v_m^2 - v^2| \\,dv_g + \\int _M (u^{N-2}- u_A^{N-2}) (|v_m|+|v|)^2 \\,dv_g\\right) \\\\& \\le & A^{N-2} \\int _M |v_m^2 - v^2| \\,dv_g\\\\&&+ {\\left( \\int _M (u^{N-2}-u_A^{N-2})^\\frac{N}{N-2} \\,dv_g\\right)}^{\\frac{N-2}{N}} {\\left( \\int _M (|v_m|+|v|)^N \\,dv_g\\right)}^{\\frac{2}{N}}.$ $(v_m)_m$ is bounded in $H_1^2(M),$ it is bounded in $L^N(M).$ Hence there exists a constant $C$ such that $\\int _M \\left(|v_m|+|v|\\right)^N \\,dv_g \\le C.$ The convergence in $L^2(M)$ gives $\\lim _{m\\longrightarrow \\infty }\\int _M |v_m^2 - v^2| \\ dv_g = 0.$ By dominated convergence theorem, we have $\\lim _{A\\longrightarrow \\infty }\\int _M \\left(u^{N-2}-u_A^{N-2}\\right)^\\frac{N}{N-2}\\ dv_g= 0.$ Hence, we get (REF ).","Since $\\lim _m \\int _M \\langle \\nabla v_m, \\nabla {\\varphi }\\rangle \\ dv_g = \\int _M\\langle \\nabla v, \\nabla {\\varphi }\\rangle \\ dv_g,$ $\\lim _m \\int _M S_g v_m {\\varphi }\\ dv_g = \\int _M S_g v {\\varphi }\\ dv_g$ and $\\lim _m \\int _M \\vert u\\vert ^{N-2} v_m {\\varphi }\\ dv_g = \\int _M \\vert u\\vert ^{N-2} v{\\varphi }\\ dv_g,$ (by strong convergence in $L^2(M)$ ), we obtain that in the sense of distributions $v$ verifies $L_gv = {\\lambda }_1(u) \\vert u\\vert ^{N-2} v.$ Now we define ${\\lambda }_k^\\prime (u) = \\inf _{{\\scriptstyle v_k; \\vert u\\vert ^{\\frac{N-2}{2}} v_k\\lnot \\equiv 0\\atop \\scriptstyle \\int _M \\vert u\\vert ^{N-2} v_i v_k \\ dv_g = 0 \\forall i 0,$ we know that ${\\lambda }_i (u) = \\inf _{u_1,\\ldots , u_i}\\sup _{{\\lambda }_1,\\ldots ,{\\lambda }_i}\\frac{\\int _ML_g({\\lambda }_1 u_1+ \\cdots +{\\lambda }_i u_i)({\\lambda }_1 u_1+ \\cdots +{\\lambda }_i u_i) \\ dv_g}{\\int _M ({\\lambda }_1u_1+ \\cdots +{\\lambda }_i u_i)^2 u^{N-2} \\ dv_g},$ and ${\\lambda }_i (1) = \\inf _{u_1,\\ldots , u_i}\\sup _{{\\lambda }_1,\\ldots ,{\\lambda }_i}\\frac{\\int _ML_g({\\lambda }_1 u_1+ \\cdots +{\\lambda }_i u_i)({\\lambda }_1 u_1+ \\cdots +{\\lambda }_i u_i) \\ dv_g}{\\int _M ({\\lambda }_1u_1+ \\cdots +{\\lambda }_i u_i)^2 \\ dv_g}.$ Suppose that ${\\lambda }_i(u)$ is attained by $v_1,\\ldots ,v_i$ .","Since denominators of this expressions are positive, then $\\sup _{{\\lambda }_1,\\cdots ,{\\lambda }_i}\\int _M L_g({\\lambda }_1 v_1+ \\cdots +{\\lambda }_i v_i)({\\lambda }_1 v_1+\\cdots +{\\lambda }_i v_i) \\ dv_g = 0.$ So ${\\lambda }_i(1) \\le \\sup _{{\\lambda }_1,\\ldots ,{\\lambda }_i}\\frac{\\int _M L_g({\\lambda }_1 v_1+ \\cdots +{\\lambda }_iv_i)({\\lambda }_1 v_1+ \\cdots +{\\lambda }_i v_i) \\ dv_g}{\\int _M ({\\lambda }_1 v_1+ \\cdots +{\\lambda }_i v_i)^2u^{N-2} \\ dv_g} = 0,$ which gives a contradiction.","The remaining cases are treated similarly."],["The negativity of ${\\lambda }_k$ is not topologically obstructed","In this paragraph, we will see that on each manifold, there exists a metric which has a negative $k^{th}$ -eigenvalue.","Proposition 3.2 On any compact Riemannian manifold $M,$ and for all $k\\ge 1$ there exists a metric $g$ such that ${\\lambda }_k(g)<0.$ Proof: Let $M$ be a compact Riemannian manifold of dimension $n,$ and we take $k$ spheres of dimension $n=dim (M).$ We equip each sphere $\\mathbb {S}^n$ by the same metric $g,$ such that $\\mu (g)<0.$ We can do this by referring to [3] (Theorem [1] page 38).","Let $P\\in M,$ since $\\mu (g)<0,$ for all ${\\varepsilon }, \\delta >0$ we can find a function $u$ supported in $\\mathbb {S}^n\\backslash {B_{{\\varepsilon }}(P)}$ such that $\\frac{\\int _M c_n\\vert \\nabla u\\vert ^2 + S_g u^2 \\ dv_g}{\\int _M u^2 \\ dv_g}<-\\delta .$ Indeed, let $\\eta _{\\varepsilon }$ be a smooth cut-off function such that $0\\le \\eta _{\\varepsilon }\\le 1,$ $\\eta _{\\varepsilon }(B_{{\\varepsilon }}(P))=0,$ $\\eta _{\\varepsilon }(\\mathbb {S}^n\\backslash {B_{2{\\varepsilon }}(P)})=1,$ $\\vert \\nabla \\eta _{\\varepsilon }\\vert \\le \\frac{2}{{\\varepsilon }}$ and a function $v$ satisfying $I_g(v) = \\frac{\\int _Mc_n\\vert \\nabla v\\vert ^2 + S_g v^2 \\ dv_g}{\\int _M v^2 \\ dv_g}< -2\\delta .$ Note that the existence of $v$ is given by the fact that $\\mu (g)<0.$ The desired function $u$ will be given by $\\eta _{\\varepsilon }v,$ where ${\\varepsilon }>0$ is sufficiently small.","Indeed, it suffices to notice that, as easily checked, $\\lim _{{\\varepsilon }\\longrightarrow 0}I_g(\\eta _{\\varepsilon }v) = I_g(v).$ Let $P_1,\\cdots ,P_k$ be points of $M.$ We consider the following connected sum $M^\\prime = M \\# (\\mathbb {S}^n)_1 \\# \\ldots \\# (\\mathbb {S}^n)_k,$ where the $(\\mathbb {S}^n)_i$ are attached at $P$ on the spheres $\\mathbb {S}^n$ and at $P_i$ on $M$ so that the handles are attached in $B_{{\\varepsilon }}(P)$ and $B_{{\\varepsilon }}(P_i)$ .","Note that $M^\\prime $ is diffeomorphic to $M.$ Moreover, the above construction allows to see $(\\mathbb {S}^n)_i\\backslash B_{{\\varepsilon }}(P_i)$ as a part of $M^\\prime .$ We take on $M^\\prime $ any metric $h$ satisfying $h\\vert _{(\\mathbb {S}^n)_i\\backslash B_{{\\varepsilon }}(P_i)}= g.$ On $M^\\prime ,$ we define the following function $u_i=\\left|\\;\\begin{matrix}u\\hfill & \\hbox{on } (\\mathbb {S}^n)_i\\backslash B_{{\\varepsilon }}(P_i)\\\\\\\\0\\hfill & \\hbox{otherwise} .\\end{matrix}\\right.$ Since the $u_i$ have disjoint supports, we get ${\\lambda }_k(1)&\\le & \\sup _{{\\lambda }_1,\\ldots ,{\\lambda }_k}\\frac{\\int _{M^\\prime } L_g({\\lambda }_1 u_1+\\cdots +{\\lambda }_ku_k)({\\lambda }_1 u_1+ \\cdots +{\\lambda }_k u_k) \\ dv_h}{\\int _{M^\\prime } ({\\lambda }_1 u_1+ \\cdots +{\\lambda }_ku_k)^2 \\ dv_h}\\\\\\\\&\\le & \\sup _{{\\lambda }_1,\\ldots ,{\\lambda }_k}\\frac{({\\lambda }_1^2+ \\ldots + {\\lambda }_k^2)\\int _M (L_gu)u \\ dv_g}{({\\lambda }_1^2+ \\ldots + {\\lambda }_k^2)\\int _M u^2 \\ dv_g}\\\\\\\\&\\le & \\frac{\\int _M (L_gu)u \\ dv_g}{\\int _M u^2 \\ dv_g}< -\\delta .$"],["Nodal solutions of the Yamabe equations","A famous problem in Riemannian geometry is the Yamabe problem, solved between 1960 and 1984 by Yamabe, Trüdinger, Aubin and Schoen, [15], [13], [2], [12].","The reader can also refer to [11], [8], [3].","The Yamabe problem consists in finding a metric $\\widetilde{g}$ conformal to $g$ such that the scalar curvature $S_{\\widetilde{g}}$ of $\\widetilde{g}$ is constant.","Solving this problem is equivalent to finding a positive smooth function and a number $C_0 \\in \\mathbb {R}$ such that $L_g(u) = C_0 \\vert u\\vert ^{N-2} u,$ where $N = \\frac{2n}{n-2}.$ In order to obtain solutions of the Yamabe equation we define the Yamabe invariant by $\\mu (M,g):= \\inf _ {u\\ne 0, u\\in C^\\infty (M)}Y(u),$ where $Y(u) = \\frac{\\int _M c_n\\vert \\nabla u\\vert ^2+S_g u^2 \\ dv_g}{\\left(\\int _M \\vert u\\vert ^N \\ dv_g\\right)^\\frac{2}{N}}.$ The works of Yamabe, Trüdinger, Aubin and Schoen provides a positive smooth minimizer $u$ of $Y$ , satisfying, if normalized by $\\Vert u\\Vert _{L^N(M)} = 1,$ $L_gu = \\mu (M,g) \\vert u\\vert ^{N-2}u.$ The metric $\\widetilde{g} = u^{N-2} g$ is the desired metric: its scalar curvature is constant equal to $\\mu (M,g)$ .","If we set $u^{\\prime } = \\mu (M,g)^{\\frac{n-2}{4}} u$ , we obtain a positive solution of $L_gu^{\\prime } = {\\varepsilon }\\vert u^{\\prime }\\vert ^{N-2}u^{\\prime }$ where ${\\varepsilon }= \\hbox{ sign } (\\mu (M,g))= \\hbox{ sign}(\\lambda _1(g))$ .","Now, if $\\mu (M,g)\\ge 0,$ it is easy to chek that $\\mu (M,g) = \\inf _{\\widetilde{g}\\in \\left[ g\\right]} {\\lambda }_1(\\widetilde{g}) vol(M,\\widetilde{g})^\\frac{2}{n},$ where $\\left[ g\\right]$ is the conformal class of $g$ and ${\\lambda }_1$ is the first eigenvalue of the Yamabe operator $L_g$ .","Inspired by this approach, in their paper [1], B. Ammann et E. Humbert introduced the second Yamabe invariant defined by $\\mu _2(M,g)&=& \\inf _{\\widetilde{g}} {\\lambda }_2(\\widetilde{g}) vol(M,\\widetilde{g})^\\frac{2}{n}\\\\&=& \\inf _u {\\lambda }_2(u^{N-2}g) \\left(\\int _M u^N \\ dv_g\\right)^\\frac{2}{n}.$ They studied this invariant in the case where $\\mu (M,g)\\ge 0$ , and they proved that $\\mu _2$ is attained by a generalized metric, (i.e.","a metric of the form $u^{N-2}g$ where $u\\in L^N(M)$ , $u\\ge 0$ which may vanish), in the following two cases $\\bullet $ $\\mu (M,g)>0$ , $(M,g)$ is not locally conformally flat and $n\\ge 11$ .","$\\bullet $ $\\mu (M,g)=0$ , $(M,g)$ is not locally conformally flat and $n\\ge 9$ .","In this context, they proved that $u$ is the absolute value of a changing sign function $w$ of class $C^{3,\\alpha }(M)$ , which verifies the following equation $L_g w = \\mu _2(M,g)\\vert w\\vert ^{N-2} w.$ Many works are devoted to the study of this kind of solutions, for example [1], [6], [9], [10], [14].","See also [4] for an analogue study for the Paneitz-Branson operator.","Setting again $w^{\\prime } = \\mu _2(M,g)^{\\frac{n-2}{4}} w$ , we obtain a solution of $L_g w^{\\prime }= {\\varepsilon }\\vert w^{\\prime }\\vert ^{N-2} w^{\\prime }$ with $\\epsilon =1= \\hbox{ sign }(\\mu _2(M,g))= \\hbox{ sign }(\\lambda _2(g)).$ The goal of this section is to study if this result extends to metrics where the sign of ${\\lambda }_2(g)$ is arbitrary.","The answer is yes when ${\\lambda }_2 < 0$ without any other condition, we obtain this result by a method different than the one of [1].","Notice that this situation occurs for a large number of metrics (see Proposition REF ).","When ${\\lambda }_2\\ge 0$ , we show that the methods in [1] can be extended to the case where $\\mu (M,g)<0$ .","Namely, the main result of this paper is: Theorem 4.1 Let $(M,g)$ be a compact Riemannian manifold of dimension $n\\ge 3$ whose Yamabe invariant $\\mu (M,g)$ is strictly negative, we denote by ${\\lambda }_2$ the second eigenvalue of $L_g.$ Then, if ${\\lambda }_2\\le 0$ or if ${\\lambda }_2>0$ , $(M,g)$ not locally conformally flat and $n\\ge 6$ : There exists a function $w$ changing sign, solution of the equation $L_g w = {\\varepsilon }\\vert w \\vert ^{N-2}w,$ where ${\\varepsilon }=+1$ if ${\\lambda }_2>0,$ ${\\varepsilon }= -1$ if ${\\lambda }_2< 0$ and ${\\varepsilon }=0$ if ${\\lambda }_2 = 0.$ Moreover, $w \\in C^{3,\\alpha }(M)$ , for all $\\alpha < N-2.$"],["The case ${\\lambda }_2 = 0$","This case is obvious: indeed, Proposition REF provides the existence of a nodal solution $v$ of $L_g v = 0= {\\varepsilon }\\left|v\\right|^{N-2}v$ where ${\\varepsilon }= 0 = \\hbox{ sign }(\\lambda _2(g))$ ."],["The case ${\\lambda }_2 > 0$","As in [1], we introduce the second Yamabe invariant given by $\\mu _2(M,g)&=& \\inf _{\\widetilde{g}}{\\lambda }_2(\\widetilde{g}) vol(M,\\widetilde{g})^\\frac{2}{n}\\\\&=& \\inf _{u>0} {\\lambda }_2(u)\\left(\\int _M u^N \\ dv_g\\right)^\\frac{2}{n}.$ By Proposition REF below, the problem reduces to finding a minimizer of $\\mu _2(M,g)$ .","The case where $\\mu (M,g) \\ge 0$ have been treated in [1].","We will then focus on the case where $\\mu (M,g) < 0$ (i.e.","${\\lambda }_1(g) <0$ ).","We will see that the method of Ammann and Humbert remains valid in this case and the following three propositions answer our questions.","Proposition 4.2 Let $(M,g)$ be a compact Riemannian manifold of dimension $n\\ge 3$ , such that ${\\lambda }_2>0$ .","If $\\mu _2(M,g)<\\mu (\\mathbb {S}^n),$ with $\\mu (\\mathbb {S}^n)=n(n-1)\\omega _n^{\\frac{2}{n}}$ , where $\\omega _n$ stands for the volume of the standard sphere $\\mathbb {S}^n$ , then the second Yamabe invariant is attained by a non-negative function $u\\in L^N(M)$ that we normalize by $\\int _M u^N \\ dv_g = 1$ .","There exists a function $w$ having a changing sign which verifies in the sense of distributions the following equation $L_gw = \\mu _2(M,g) \\vert u\\vert ^{N-2} w.$ The functions $u$ and $w$ will be normalized by $\\int _M u^N \\ dv_g = 1, \\hspace{5.69046pt}\\int _M u^{N-2} \\ w^2 \\ dv_g = 1.$ Proposition 4.3 The two functions $u$ and $w$ given by Proposition REF satisfy $u = \\vert w\\vert .$ Finally, we give a condition under which assumption (REF ) is satisfied: Proposition 4.4 Let $(M,g)$ be a compact Riemannian manifold of dimension $n\\ge 6,$ suppose that $M$ is not locally conformally flat and his Yamabe invariant $\\mu (M,g)<0,$ then $\\mu _2(M,g)< \\mu (\\mathbb {S}^n).$"],["Proof of Proposition ","The case where $\\mu (M,g) \\ge 0$ is done in [1], hence we consider here the case where $\\mu (M,g) < 0$ .","By the solution of the Yamabe problem, we can assume without loss of generality, that $S_g = -1$ .","Let $(u_m)_m$ be a minimizing sequence for $\\mu _2(M,g),$ i.e., $u_m$ is positive, smooth and $\\lim _{m\\longrightarrow \\infty }\\lambda _2(u_m)\\left(\\int _Mu_m^Ndv_g\\right)^\\frac{2}{n} =\\mu _2(M,g).$ The sequence $(u_m)_m$ will be choosen such that $\\int _M u_m^N \\ dv_g=1$ , hence $\\mu _2(M,g)=\\lim _{m\\longrightarrow \\infty }\\lambda _2(u_m).$ For each $u_m,$ Proposition REF provides the existence of a function $w_m \\in H_1^2(M)$ such that $L_g w_m = {\\lambda }_2(u_m)\\vert u_m\\vert ^{N-2}w_m.$ Moreover, the sequence $(w_m)_m$ can be normalized by $\\int _M\\vert u_m\\vert ^{N-2}w_m^2 \\ dv_g=1.$ Since $\\int _Mu_m^Ndv_g=1,$ $(u_m)_m$ is bounded in $L^N(M)$ which is a reflexive space, there exists $u\\in L^N(M)$ such that $u_m$ converges weakly to $u$ in $L^N(M)$ , we have $u_m\\rightharpoonup u \\text{ in }L^N(M).$ $\\bullet $ The sequence $(w_m)_m$ is bounded in $H_1^2(M).$ We proceed by contradiction and assume that $\\Vert w_m\\Vert _{H_1^2(M)}\\longrightarrow \\infty .$ Let $w^\\prime _m=\\frac{w_m}{\\Vert w_m\\Vert _{H_1^2(M)}}.$ $\\Vert w^\\prime _m\\Vert _{H_1^2(M)}=1,$ hence $(w^\\prime _m)_m$ is bounded in $H_1^2(M).$ Since $H_1^2(M)$ is a reflexive space, this implies using Kondrakov and Banach-Alaoglu theorems, that there exists a subsequence $(w_m^\\prime )_m$ and $w^\\prime \\in H_1^2(M)$ such that $w^\\prime _m\\rightharpoonup w^\\prime \\text{ in } H_1^2(M),$ and $w^\\prime _m\\longrightarrow w^\\prime \\text{ in } L^2(M).$ Equation (REF ) is linear, so $w^\\prime _m$ satisfies $L_gw_m^{\\prime } = \\lambda _2(u_m)\\left|u_m \\right|^{N-2}w^\\prime _m.$ Hence for all $\\varphi \\in C^\\infty (M),$ we have: $c_n \\int _M \\left\\langle \\nabla w^\\prime _m,\\nabla \\varphi \\right\\rangle dv_g+\\int _MS_gw^\\prime _m\\varphi dv_g=\\int _M\\lambda _2(u_m)\\left|u_m\\right|^{N-2}w^\\prime _m \\varphi dv_g.$ Since $w^\\prime _m \\rightharpoonup w^\\prime $ in $H_1^2(M)$ and $w \\longmapsto \\left\\langle \\nabla w ,\\nabla \\varphi \\right\\rangle $ is a linear form on $H_1^2(M),$ then $c_n\\int _M \\left\\langle \\nabla w_m^\\prime ,\\nabla \\varphi \\right\\rangle dv_g\\longrightarrow c_n \\int _M\\left\\langle \\nabla w^{\\prime },\\nabla \\varphi \\right\\rangle dv_g.$ The sequence $w_m^\\prime $ converges strongly to $w^\\prime $ in $L^2(M)$ .","This gives that $\\int _M S_gw_m^\\prime \\varphi dv_g\\longrightarrow \\int _M S_g w^\\prime \\varphi dv_g.$ Using Hölder inequality, we obtain that $\\int _M\\left|u_m\\right|^{N-2}w^\\prime _m\\varphi \\ dv_g\\longrightarrow 0.$ Indeed, $\\left|\\int _M\\left|u_m\\right|^{N-2}w^\\prime _m\\varphi dv_g\\right|&\\le &\\left\\Vert \\varphi \\right\\Vert _{\\infty }\\int _M\\left|u_m\\right|^\\frac{N-2}{2}\\left|w_m^\\prime \\right|\\left|u_m\\right|^\\frac{N-2}{2}\\ dv_g\\\\&\\le &\\left\\Vert \\varphi \\right\\Vert _{\\infty }\\left(\\int _M\\left|u_m\\right|^{N-2}{w^\\prime _m}^2 \\ dv_g\\right)^\\frac{1}{2}\\left(\\int _M\\left|u_m\\right|^{N-2} \\ dv_g\\right)^\\frac{1}{2}\\\\&\\le &\\left\\Vert \\varphi \\right\\Vert _{\\infty }\\frac{\\left(\\int _M\\left|u_m\\right|^{N-2}{w_m}^2 \\ dv_g\\right)^\\frac{1}{2}}{\\left\\Vert w_m\\right\\Vert _{H_1^2(M)}}\\left(\\int _M\\left|u_m\\right|^{N}\\ dv_g\\right)^\\frac{N-2}{2N}\\left(\\rm vol(M,g)^{1-\\frac{N-2}{N}}\\right)^\\frac{1}{2}\\\\&\\le &\\left\\Vert \\varphi \\right\\Vert _{\\infty } \\frac{1}{\\left\\Vert w_m\\right\\Vert _{H_1^2(M)}}\\left(\\rm vol(M,g)\\right)^\\frac{1}{N} \\longrightarrow _{m\\longrightarrow +\\infty }0.$ Then $c_n \\int _M\\left\\langle \\nabla w^{\\prime },\\nabla \\varphi \\right\\rangle \\ dv_g+\\int _MS_g w^\\prime \\varphi \\ dv_g=0,$ which means that in the sense of distributions, we have $L_g w^\\prime =0.$ Since ${\\lambda }_1(1)<0$ and ${\\lambda }_2(1)$ is positive, $0\\notin Sp(L_g)$ .","It follows that $w^\\prime =0.$ Now, we also have $\\int _M c_n\\left|\\nabla w^\\prime _m\\right|^2 \\ dv_g+\\int _M S_g{w^\\prime _m}^2 \\ dv_g=\\lambda _2(u_m)\\int _M\\left|u_m\\right|^{N-2}{w_m^\\prime }^2 \\ dv_g,$ with $\\lambda _2(u_m)\\int _M\\left|u_m\\right|^{N-2}{w_m^\\prime }^2 \\ dv_g=\\frac{\\lambda _2(u_m)}{\\left\\Vert w_m\\right\\Vert ^2_{H_1^2(M)}}\\longrightarrow 0$ and $\\int _M S_g{w^\\prime _m}^2 \\ dv_g\\longrightarrow \\int _M S_g{w^\\prime }^2 \\ dv_g = 0.$ Hence $\\int _M\\left|\\nabla w^\\prime _m\\right|^2 \\ dv_g\\longrightarrow 0.$ Finally, we get that $\\left\\Vert w^\\prime _m\\right\\Vert ^2_{H_1^2(M)}=1=\\int _M\\left|\\nabla w_m^\\prime \\right|^2 \\ dv_g+\\int _M{w^\\prime _m}^2 \\ dv_g\\longrightarrow 0,$ which gives the desired contradiction.","We obtain that $(w_m)_m$ is a bounded sequence in $H_1^2(M).$ Then there exists $w\\in H_1^2(M)$ such that: $w_m\\rightharpoonup w \\text{ in } H_1^2(M),$ $w_m\\longrightarrow w \\text{ in }L^2(M).$ It follows that in the sense of distributions, we have $L_gw=\\mu _2(M,g)\\left|u\\right|^{N-2}w.$ It remains to show that $w$ changes sign and is different from zero.","$\\bullet $ Suppose that $w$ does not change sign.","Without loss of generality, we can assume that $w\\ge 0.$ In the sense of distributions, we have $c_n\\Delta _gw+S_gw=\\mu _2(M,g)\\left|u\\right|^{N-2}w.$ It was already mentioned at the beginning of this section that we can assume that $S_g<0,$ because $\\mu (M,g)<0.$ Integrating (REF ) over $M$ , we get: $\\underbrace{\\int _Mc_n\\Delta _gw \\ dv_g}_{=0}+\\underbrace{\\int _MS_gw \\ dv_g}_{<0}=\\underbrace{\\mu _2(M,g)\\int _M\\left|u\\right|^{N-2}w \\ dv_g}_{\\ge 0}.$ This gives a contradiction unless $w \\equiv 0$ which is prohibited by what follows.","$\\bullet $ Assume that $w=0.$ By referring to [8] and [2] we have the following theorem: If $(M,g)$ is a Riemannian manifold of dimension $n\\ge 3,$ for all $\\epsilon >0,$ there exists $B_{\\epsilon }$ such that for any $u\\in H_1^2(M),$ we have $\\left(\\int _M\\left|u\\right|^N \\ dv_g \\right)^\\frac{2}{N}\\le (\\mu (\\mathbb {S}^n)^{-1}+\\epsilon )\\left(\\int _M c_n\\left|\\nabla u\\right|^2 \\ dv_g+B_\\epsilon \\int _Mu^2 \\ dv_g\\right).$ We obtain $c_n\\int _M\\left|\\nabla w_m\\right|^2 \\ dv_g+S_g\\int _Mw_m^2 \\ dv_g&=&\\mu _2(M,g)\\int _M\\left|u_m\\right|^{N-2}w_m^2 \\ dv_g\\\\&\\le &\\mu _2(M,g)\\underbrace{\\left(\\int _M\\left|u_m\\right|^{N} \\ dv_g\\right)^\\frac{N-2}{N}}_{=1}\\left(\\int _M\\left|w_m\\right|^{N} \\ dv_g\\right)^\\frac{2}{N}\\\\&\\le &\\underbrace{\\mu _2(M,g)(\\mu (\\mathbb {S}^n)^{-1}+\\epsilon )}_{<1(\\text{if}\\hspace{2.84544pt}{\\varepsilon }\\hspace{2.84544pt}\\text{is small enough})}\\left(\\int _Mc_n\\left|\\nabla w_m\\right|^2 \\ dv_g+B_\\epsilon \\int _Mw_m^2 \\ dv_g\\right).$ Hence $c_n\\underbrace{\\left[1-\\mu _2(M,g)(\\mu (\\mathbb {S}^n)^{-1}+\\epsilon )\\right]}_{>0}\\int _M\\left|\\nabla w_m\\right|^2 \\ dv_g\\le c\\underbrace{\\int _Mw_m^2 \\ dv_g}_{\\longrightarrow 0},$ then $\\int _M\\left|\\nabla w_m\\right|^2 \\ dv_g\\longrightarrow 0,$ so $\\left\\Vert w_m\\right\\Vert _{H_1^2(M)}\\longrightarrow 0.$ This shows that $w_m\\longrightarrow 0$ in $H_1^2(M).$ We finally get that $1=\\int _M\\left|u_m\\right|^{N-2}w_m^2 \\ dv_g\\le \\left(\\int _M\\left|u_m\\right|^N \\ dv_g\\right)^\\frac{N-2}{2}\\underbrace{\\int _Mw_m^N \\ dv_g}_{\\longrightarrow 0}.$ This gives a contradiction, then $w\\ne 0.$"],["Proof of Proposition ","Since ${\\lambda }_2(g)>0$ , then $\\mu _2(M,g) = \\inf _{u>0} {\\lambda }_2(u)\\left(\\int _M u^N \\ dv_g\\right)^\\frac{2}{n} = \\inf _{u\\ge 0} {\\lambda }_2(u)\\left(\\int _M u^N \\ dv_g\\right)^\\frac{2}{n}.$ We mimic the proof of Lemma 3.3 in [1] by taking $w_1 = w_+ = \\sup \\left\\lbrace 0,w\\right\\rbrace $ and $w_2 = w_- = \\sup \\left\\lbrace 0, -w\\right\\rbrace $ .","This gives that $u = aw_+ + b w_-,$ where $a,b>0$ .","By Lemma 3.1, $w\\in C^{2,\\alpha }$ , $u\\in C^{0,\\alpha }$ and Step 4 of the proof of Theorem 3.4 in [1] then shows that $u = \\vert w\\vert .$ Since $w$ is in $H_1^2(M),$ Lemma 3.1 of [1] says that $w\\in L^{N + {\\varepsilon }}(M),$ because $w$ satisfies the equation $L_gw =\\mu _2\\vert w\\vert ^{N-2} w,$ and standard bootstrap arguments gives that $w\\in C^{3,\\alpha }(M)$ for all $\\alpha 0$ we define the following function $v_{\\epsilon } =c_{\\epsilon } \\eta (\\epsilon +r^2)^\\frac{2-n}{2},$ where $c_\\epsilon $ is choosen such that $\\int _M v_{\\epsilon }^N \\ dv_g=1.$ By referring to [2] $\\lim _{\\epsilon \\longrightarrow 0}Y(v_{\\epsilon })=\\mu _1(\\mathbb {S}^n),$ where $Y(u)$ is the Yamabe functional defined by $Y(u)=\\frac{\\int _M c_n\\vert \\nabla u\\vert ^2+S_g u^2 \\ dv_g}{\\left(\\int _M u^N \\ dv_g\\right)^\\frac{2}{N}}.$ If $(M,g)$ is not locally conformally flat, by a calculation made in [2], there exists a constant $C(M)>0$ such that $Y(v_{{\\varepsilon }})=\\left|\\;\\begin{matrix}\\mu _1(\\doba {S}^n) - C(M) {\\varepsilon }^2 + o({\\varepsilon }^2)\\hfill & \\hbox{ if } n > 6 \\\\\\\\\\mu _1(\\doba {S}^n) - C(M) {\\varepsilon }^2 |\\ln ({\\varepsilon })| + o({\\varepsilon }^2 |\\ln ({\\varepsilon })|)\\hfill & \\hbox{ if } n = 6.\\end{matrix}\\right.$ Again from [2] there exists constants $a,$ $b,$ $C_1,$ $C_2>0$ , such that $a {\\varepsilon }^{\\frac{n-2}{4} } \\le c_{{\\varepsilon }} \\le b {\\varepsilon }^{\\frac{n-2}{4} },$ and $ C_1 {\\alpha }_{p,{\\varepsilon }} \\le \\int _M v_{{\\varepsilon }}^p \\,dv_g \\le C_2 {\\alpha }_{p,{\\varepsilon }}$ where $ {\\alpha }_{p,{\\varepsilon }} = \\left| \\begin{array}{lll}{\\varepsilon }^{\\frac{2n - (n-2) p}{4}} & \\hbox{if} & p> \\frac{n}{n-2};\\\\\\\\|\\ln ({\\varepsilon })| {\\varepsilon }^{\\frac{n}{4}} & \\hbox{if} & p= \\frac{n}{n-2};\\\\\\\\{\\varepsilon }^{ \\frac{(n-2) p}{4}} & \\hbox{if}& p< \\frac{n}{n-2}\\end{array} \\right.","$ We have $\\mu _2(M,g)&=&\\inf _u\\lambda _2(u)\\left(\\int _Mu^N \\ dv_g\\right)^{\\frac{2}{n}}\\\\\\\\&=& \\inf _{\\scriptstyle u \\atop \\scriptstyle w,w^\\prime } \\sup _{\\lambda ,\\mu } \\frac{\\int _M L_g(\\lambda w+\\mu w^\\prime )(\\lambda w+\\mu w^\\prime ) \\ dv_g}{\\int _M u^{N-2}(\\lambda w+\\mu w^\\prime )^2 \\ dv_g}\\left(\\int _Mu^N \\ dv_g\\right)^{\\frac{2}{n}}\\\\\\\\&=& \\inf _{\\scriptstyle u \\atop \\scriptstyle w,w^\\prime } \\sup _{\\lambda ,\\mu }F(u,\\lambda w+\\mu w^\\prime ).$ Let $\\lambda _\\epsilon ,$ $\\mu _\\epsilon $ such that $\\lambda _\\epsilon ^2+\\mu _\\epsilon ^2=1$ and $F(v_\\epsilon , \\lambda _\\epsilon v+\\mu _\\epsilon v_\\epsilon )=\\sup _{(\\lambda ,\\mu )\\in \\mathbb {R}^2\\backslash \\left\\lbrace (0,0)\\right\\rbrace } F(v_\\epsilon , \\lambda v +\\mu v_\\epsilon ),$ where $v$ is the function defined in the equation (REF ).","Calculating $F(v_\\epsilon , \\lambda _\\epsilon v+\\mu _\\epsilon v_\\epsilon ),$ we get $F(v_\\epsilon , \\lambda _\\epsilon v+\\mu _\\epsilon v_\\epsilon )&=& \\frac{\\int _ML_g(\\lambda _\\epsilon v+\\mu _\\epsilon v_\\epsilon )(\\lambda _\\epsilon v+\\mu _\\epsilon v_\\epsilon ) \\ dv_g}{\\int _M v_\\epsilon ^{N-2}(\\lambda _\\epsilon v+\\mu _\\epsilon v_\\epsilon )^2 \\ dv_g}\\left(\\int _Mv_\\epsilon ^N \\ dv_g\\right)^{\\frac{2}{n}}\\\\\\\\&=& \\frac{\\lambda _\\epsilon ^2 \\mu (M,g)+\\mu _\\epsilon ^2Y(v_\\epsilon )+2\\lambda _\\epsilon \\mu _\\epsilon \\mu (M,g)\\int _M v^{N-1}v_\\epsilon \\ dv_g}{\\lambda _\\epsilon ^2\\int _Mv_\\epsilon ^{N-2}v^2 \\ dv_g+\\mu _\\epsilon ^2+2\\lambda _\\epsilon \\mu _\\epsilon \\int _M v_\\epsilon ^{N-1}v \\ dv_g}\\\\\\\\&=&\\frac{A_\\epsilon }{B_\\epsilon }.$ If ${\\lambda }_{\\varepsilon }\\longrightarrow {\\lambda }\\ne 0, \\hspace{5.69046pt}\\mu _{\\varepsilon }\\longrightarrow \\mu \\ne 0,$ then $F(v_{\\varepsilon },{\\lambda }_{\\varepsilon }v + \\mu _{\\varepsilon }v_{\\varepsilon })\\longrightarrow \\frac{{\\lambda }^2 \\mu (M,g) + \\mu ^2\\mu (\\mathbb {S}^n)}{\\mu ^2}<\\mu (\\mathbb {S}^n).$ Similarly, if $\\mu = 0,$ ${\\lambda }^2 = 1,$ then the numerator $A_{\\varepsilon }\\sim \\mu (M,g)< 0,$ while the denominator $B_{\\varepsilon }$ remains positive, which gives again that $F(v_{\\varepsilon },{\\lambda }_{\\varepsilon }v + \\mu _{\\varepsilon }v_{\\varepsilon })\\le 0<\\mu (\\mathbb {S}^n),$ which gives the desired inequality.","Then, in the sequel, we assume that ${\\lambda }_{\\varepsilon }\\longrightarrow 0$ and $\\mu _{\\varepsilon }\\longrightarrow \\pm 1.$ The case $n>6$ Using (REF ) we have $ \\int _M v^{N-1} v_{{\\varepsilon }} \\,dv_g \\sim _{{\\varepsilon }\\rightarrow 0} C {\\varepsilon }^{\\frac{n-2}{4}},$ $\\int _M v_{{\\varepsilon }}^{N-2} v^2 \\,dv_g \\sim _{{\\varepsilon }\\rightarrow 0} C {\\varepsilon },$ and $\\int _M v_{{\\varepsilon }}^{N-1} v \\,dv_g \\sim _{{\\varepsilon }\\rightarrow 0} C {\\varepsilon }^{\\frac{n-2}{4}},$ where $C$ denotes a constant that might change its value from line to line.","We distinguish two cases $\\bullet $ there exists a constant $C>0$ such that $\\vert {\\lambda }_{\\varepsilon }\\vert \\le C {\\varepsilon }^\\frac{n-2}{4},$ or $\\bullet $ there exists $\\alpha _{\\varepsilon }$ such that $\\vert \\lambda _{\\varepsilon }\\vert = \\alpha _{\\varepsilon }{\\varepsilon }^{\\frac{n-2}{4}},$ and $\\alpha _{\\varepsilon }\\longrightarrow +\\infty .$ (possibly extracting a subsequence).","Suppose first that (REF ) is verified.","Then we have $\\vert {\\lambda }_{\\varepsilon }\\vert \\le C {\\varepsilon }^\\frac{n-2}{4}.$ Hence ${\\lambda }_{\\varepsilon }^2=O({\\varepsilon }^\\frac{n-2}{2}),$ so $\\mu ^2_{\\varepsilon }=1-{\\lambda }^2_{\\varepsilon }=1+O({\\varepsilon }^\\frac{n-2}{2}).$ Therefore $\\mu _{\\varepsilon }=1+O({\\varepsilon }^\\frac{n-2}{2}).$ This gives $A_\\epsilon &=& O({\\varepsilon }^{\\frac{n-2}{2}}) + ( 1 +O({\\varepsilon }^{\\frac{n-2}{2}})) \\Big ( \\mu (\\mathbb {S}^n) -C(M){\\varepsilon }^2 +o({\\varepsilon }^2)\\Big ) + O({\\varepsilon }^{\\frac{n-2}{2}})\\\\&=& \\mu (\\mathbb {S}^n)-C(M){\\varepsilon }^2 + O({\\varepsilon }^{\\frac{n-2}{2}}) +o({\\varepsilon }^2).$ Since $\\frac{n-2}{2} > 2$ , $A_{\\varepsilon }=\\mu (\\mathbb {S}^n) -C(M){\\varepsilon }^2 +o({\\varepsilon }^2),$ and $B_{\\varepsilon }= O({\\varepsilon }^{\\frac{n-2}{2}+1}) + 1 + O({\\varepsilon }^{\\frac{n-2}{2}})+O({\\varepsilon }^{\\frac{n-2}{2}}) = 1+ o({\\varepsilon }^2).$ Then, $\\frac{A_{\\varepsilon }}{B_{\\varepsilon }} = \\mu (\\mathbb {S}^n)- C(M) {\\varepsilon }^2 + o({\\varepsilon }^2) < \\mu (\\mathbb {S}^n).$ Assume now that (REF ) is fulfilled.","In this case $\\frac{A_{\\varepsilon }}{B_{\\varepsilon }} &=& \\frac{\\lambda ^2_{\\varepsilon }\\mu (M, g) + (1 - \\lambda ^2_{\\varepsilon })Y(v_{\\varepsilon }) + \\lambda _{\\varepsilon }O({\\varepsilon }^{\\frac{n-2}{4}})}{ \\lambda ^2_{\\varepsilon }O({\\varepsilon }) + (1- \\lambda ^2_{\\varepsilon }) + 2\\lambda _{\\varepsilon }\\mu _{\\varepsilon }O({\\varepsilon }^{\\frac{n-2}{4}})}\\\\&=& \\frac{\\lambda ^2_{\\varepsilon }\\mu (M, g) + (1 - \\lambda ^2_{\\varepsilon }) Y(v_{\\varepsilon }) +o({\\lambda }^2_{\\varepsilon })}{o({\\lambda }^2_{\\varepsilon }) + (1-{\\lambda }^2_{\\varepsilon }) + o({\\lambda }^2_{\\varepsilon })}\\\\&=& \\frac{{\\lambda }^2_{\\varepsilon }\\mu (M, g)}{1-{\\lambda }_{\\varepsilon }^2 + o({\\lambda }^2_{\\varepsilon })} + \\frac{Y(v_{\\varepsilon })}{ 1 +\\frac{o({\\lambda }_{\\varepsilon }^2)}{\\mu ^2_{\\varepsilon }}} + o({\\lambda }_{\\varepsilon }^2)\\\\\\\\&\\le & \\mu (M,g) {\\lambda }_{\\varepsilon }^2 + \\mu (\\mathbb {S}^n)(1 + o({\\lambda }_{\\varepsilon }^2)) + o({\\lambda }_{\\varepsilon }^2)\\\\&\\le & \\mu (\\mathbb {S}^n) + \\mu (M,g) {\\lambda }_{\\varepsilon }^2 + o({\\lambda }_{\\varepsilon }^2)\\\\&<& \\mu (\\mathbb {S}^n),$ because $\\mu (M,g)<0$ and $Y(v_{\\varepsilon }) \\le \\mu (\\mathbb {S}^n).$ The case $n =6$ Since $\\int _M v_{\\varepsilon }^{N-2} v^2 dv_g &\\sim _{{\\varepsilon }\\rightarrow 0}& C {\\varepsilon },\\\\\\int _M v^{N-1}v_{\\varepsilon }dv_g &\\sim _{{\\varepsilon }\\rightarrow 0}& C {\\varepsilon },\\\\\\int _M v_{\\varepsilon }^{N-1} v dv_g &\\sim _{{\\varepsilon }\\rightarrow 0}& C {\\varepsilon },$ then $A_{\\varepsilon }= {\\lambda }_{\\varepsilon }^2 \\mu (M,g) + \\mu ^2_{\\varepsilon }Y(v_{\\varepsilon }) + 2{\\lambda }_{\\varepsilon }\\mu _{\\varepsilon }O({\\varepsilon }),$ $B_{\\varepsilon }= {\\lambda }^2_{\\varepsilon }O({\\varepsilon }) + \\mu ^2_{\\varepsilon }+ 2{\\lambda }_{\\varepsilon }\\mu _{\\varepsilon }O({\\varepsilon }).$ Again, we have two cases to study If $\\vert {\\lambda }_{\\varepsilon }\\vert \\le C{\\varepsilon },$ then ${\\lambda }_{\\varepsilon }^2\\le C{\\varepsilon }^2.$ This implies $A_{\\varepsilon }= \\mu (\\mathbb {S}^n) - C{\\varepsilon }^2\\vert \\ln ({\\varepsilon })\\vert + o({\\varepsilon }^2\\vert \\ln ({\\varepsilon })\\vert )$ and $B_{\\varepsilon }= 1+O({\\varepsilon }^2) = 1 + o({\\varepsilon }^2 \\vert \\ln ({\\varepsilon })\\vert ).$ Hence $\\frac{A_{\\varepsilon }}{B_{\\varepsilon }}< \\mu (\\mathbb {S}^n).$ If $\\vert {\\lambda }_{\\varepsilon }\\vert = \\alpha _{\\varepsilon }{\\varepsilon },$ with $\\alpha _{\\varepsilon }\\longrightarrow +\\infty .$ Since $Y(v_{\\varepsilon }) \\le \\mu (\\mathbb {S}^n),$ therefore $A_{\\varepsilon }= \\alpha _{\\varepsilon }^2 {\\varepsilon }^2 \\mu (M,g) + \\mu ^2_{\\varepsilon }\\mu (\\mathbb {S}^n) + o(\\alpha _{\\varepsilon }^2{\\varepsilon }^2),$ and $B_{\\varepsilon }= \\mu ^2_{\\varepsilon }+ o(\\alpha _{\\varepsilon }^2 {\\varepsilon }^2).$ Therefore $\\frac{A_{\\varepsilon }}{B_{\\varepsilon }} &=& \\mu (M,g) \\frac{\\alpha _{\\varepsilon }^2 {\\varepsilon }^2}{1+o(1)}+ \\frac{\\mu (\\mathbb {S}^n)}{1 + o(\\alpha _{\\varepsilon }^2 {\\varepsilon }^2)} + \\frac{o(\\alpha _{\\varepsilon }^2 {\\varepsilon }^2)}{1+ o(1)}\\\\&=& \\mu (M,g)\\alpha _{\\varepsilon }^2 {\\varepsilon }^2 + \\mu (\\mathbb {S}^n) + o(\\alpha _{\\varepsilon }^2 {\\varepsilon }^2)\\\\&<& \\mu (\\mathbb {S}^n).$ This ends the proof of Propositon REF .","$\\square $ So we get a solution $w$ having a changing sign of the equation $L_g w = \\mu _2 \\vert w\\vert ^{N-2}w.$ Finally, to obtain the resultat announced in Theorem REF , it suffices to set $w^\\prime = \\mu _2^\\frac{n-2}{4} w,$ then $w^\\prime $ verifies $L_gw^\\prime = {\\varepsilon }\\vert w^\\prime \\vert ^{N-2} w^\\prime ,$ with ${\\varepsilon }= 1 = \\hbox{ sign}({\\lambda }_2(g))$ ."],["The case ${\\lambda }_2<0$","In this section, we will show that in all cases, there exists a nodal solution of the equation $L_g w = C_0 \\vert w\\vert ^{N-2}w,$ where $C_0$ is a negative constant.","First, since $\\mu <0,$ we assume in the whole section that the metric $g$ is such that $S_g=-1.$ In this context, the approach will be different.","Indeed, the second Yamabe invariant is not well defined as shown in the following proposition: Proposition 5.1 Let $M$ be a compact Riemannian manifold of dimension $n\\ge 3$ .","Suppose that ${\\lambda }_2< 0$ , then $\\inf _u {\\lambda }_2(u)\\left(\\int _M u^N \\ dv_g\\right)^\\frac{2}{n} = -\\infty .$ The proof will be detailed in Subsection REF .","We will use a new functional $I_g(u)= \\frac{\\left(\\int _M\\vert L_gu\\vert ^\\frac{2n}{n+2} \\ dv_g\\right)^\\frac{n+2}{n}}{\\vert \\int _M uL_gu \\ dv_g\\vert }.$ We study $\\alpha := \\inf I_g(u)$ where the infimum is taken over the functions $u\\in H_2^\\frac{2n}{n+2}(M)$ such that $\\int _M uL_gu \\ dv_g <0,$ and with the following constraint $\\int _M \\vert u \\vert ^{N-2} u \\ v \\ dv_g=0,$ for any function $v\\in \\ker L_g.$ We will show that $\\alpha $ is a conformal invariant.","We obtain also that the infimum of this functional is attained by a function $u.$ We set $v = \\vert L_gu\\vert ^\\frac{-4}{n+2} L_gu,$ and we will observe that $v$ has the following properties: $\\bullet $ $v$ is a solution of the equation $L_g v = \\alpha ^\\prime \\vert v\\vert ^{N-2}v,$ where $\\alpha ^\\prime <0$ (i.e.","has same sign than ${\\lambda }_2$ ).","$\\bullet $ $v$ has a changing sign.","$\\bullet $ $v$ is of class $C^{3,\\alpha }(M)$ ($\\alpha < N-2$ )."],["Conformal invariance of $\\alpha $","Let $\\widetilde{g} = {\\varphi }^\\frac{4}{n-2} g$ be a conformal metric, ${\\varphi }$ a smooth positive function.","Then $dv_{\\widetilde{g}} = {\\varphi }^\\frac{2n}{n-2} dv_g,$ and $L_{\\widetilde{g}}u = {\\varphi }^{-\\frac{n+2}{n-2}}L_g(u{\\varphi }),$ for all functions $u.$ Remark that $I_{\\widetilde{g}}(u) = I_g(u{\\varphi }).$ $I_{\\widetilde{g}} (u)&=& \\frac{\\left(\\int _M \\vert L_{\\widetilde{g}}u\\vert ^\\frac{2n}{n+2} \\ dv_{\\widetilde{g}}\\right)^\\frac{n+2}{n}}{\\vert \\int _M uL_{\\widetilde{g}}u \\ dv_{\\widetilde{g}}\\vert }\\\\\\\\&=& \\frac{\\left(\\int _M \\vert {\\varphi }\\vert ^\\frac{-2n}{n-2} \\vert L_g(u{\\varphi })\\vert ^\\frac{2n}{n+2} {\\varphi }^\\frac{2n}{n-2} \\ dv_g\\right)^\\frac{n+2}{n}}{\\vert \\int _M u{\\varphi }^\\frac{-(n+2)}{n-2} L_g (u{\\varphi }){\\varphi }^\\frac{2n}{n-2} \\ dv_g\\vert }\\\\\\\\&=& \\frac{\\left(\\int _M \\vert L_g(u{\\varphi })\\vert ^\\frac{2n}{n-2} \\ dv_g\\right)^\\frac{n+2}{n}}{\\vert \\int _M u{\\varphi }L_g(u{\\varphi }) \\ dv_g\\vert }\\\\\\\\&=& I_g(u{\\varphi }),$ where we have used $\\int _M uL_{\\widetilde{g}}u \\ dv_{\\widetilde{g}} &=& \\int _M u{\\varphi }^{-\\frac{n+2}{n-2}}L_g(u{\\varphi }){\\varphi }^\\frac{2n}{n-2} \\ dv_g\\\\&=& \\int _M (u{\\varphi })L_g(u{\\varphi }) \\ dv_g.$ Assume that for any $v\\in \\ker L_{\\widetilde{g}},$ we have $\\int _M \\vert u\\vert ^{N-2} uv \\ dv_{\\widetilde{g}} = 0.$ Then, for any $v^\\prime \\in \\ker L_g,$ we obtain $\\int _M \\vert u{\\varphi }\\vert ^{N-2} (u{\\varphi })v^\\prime \\ dv_g = \\int _M \\vert u\\vert ^{N-2} u (v^\\prime {\\varphi }^{-1}) \\ dv_{\\widetilde{g}} = 0,$ since $L_{\\widetilde{g}}(v^\\prime {\\varphi }^{-1}) = {\\varphi }^{-\\frac{n+2}{n-2}}L_g(v^\\prime ) = 0,$ i.e.","$v^\\prime {\\varphi }^{-1}\\in Ker L_{\\widetilde{g}}.$ $\\square $"],["Proof of Proposition ","Assume that ${\\lambda }_2(g)< 0,$ and choose $u>0.$ By Lemma REF , there exists two functions $v_1$ and $v_2$ solutions of the following equations $L_g v_1 = {\\lambda }_1(u) \\vert u\\vert ^{N-2} v_1,$ and $L_g v_2 = {\\lambda }_2(u) \\vert u\\vert ^{N-2} v_2,$ such that $\\int _M \\vert u\\vert ^{N-2} v_1v_2 \\ dv_g=0.$ Let $v_{\\varepsilon }$ the function defined in Section REF , and let $V=\\left\\lbrace v_1,v_2\\right\\rbrace .$ For all $v\\in V,$ we get $\\lim _{{\\varepsilon }\\longrightarrow 0}\\int _Mv_{\\varepsilon }^{N-2} v^2 \\ dv_g = 0.$ Since ${\\lambda }_1(u)<0$ and ${\\lambda }_2(u)<0,$ then for ${\\varepsilon }$ sufficiently small, we have $\\lim _{{\\varepsilon }\\rightarrow 0}\\left(\\sup _{v\\in V}\\frac{\\int _M (L_gv)(v) \\ dv_g}{\\int _M v_{\\varepsilon }^{N-2}v^2 \\ dv_g}\\right) =-\\infty ,$ hence $\\lim _{{\\varepsilon }\\rightarrow 0}\\left(\\inf _u {\\lambda }_2(u)\\left(\\int _M u^N \\ dv_g\\right)^\\frac{2}{n}\\right) =-\\infty .$ $\\square $"],["The infimum of the functional $I_g$ is attained","Let $(u_m)_m$ be a minimizing sequence, i.e., $\\lim _{m\\longrightarrow \\infty }I_g(u_m) = \\alpha ,$ with $\\int _M \\vert u_m \\vert ^{N-2} \\ u_m \\ v \\ dv_g=0, \\hspace{5.69046pt}\\forall \\ v\\in \\ker L_g.$ We can assume that $\\int _M u_mL_gu_m \\ dv_g = -1.$ Then $\\alpha = \\lim _{m\\longrightarrow \\infty } \\left(\\int _M\\vert L_gu_m\\vert ^\\frac{2n}{n+2} \\ dv_g\\right)^\\frac{n+2}{n}.$ Now we show that $(u_m)_m$ is a bounded sequence in $H_2^\\frac{2n}{n+2}(M).$ We proceed by contradiction and we assume that, up to a subsequence, $\\lim \\Vert u_m\\Vert _{H_2^\\frac{2n}{n+2}(M)}= +\\infty $ .","Let $v_m = \\frac{u_m}{\\Vert u_m\\Vert _{H_2^\\frac{2n}{n+2}(M)}}.$ Since $\\Vert v_m\\Vert _{H_2^\\frac{2n}{n+2}(M)} = 1,$ $(v_m)_m$ is a bounded sequence in $H_2^\\frac{2n}{n+2}(M),$ and therefore there exists $v \\in H_2^\\frac{2n}{n+2}(M)$ such that after restriction to a subsequence $v_m\\rightharpoonup v\\text{ in }H_2^\\frac{2n}{n+2}(M),$ $v_m\\longrightarrow v\\text{ in }L^2(M).$ By standard arguments, we get $\\left(\\int _M\\vert L_gv\\vert ^\\frac{2n}{n+2} \\ dv_g\\right)^\\frac{n+2}{n}\\le \\underbrace{\\liminf _m \\left(\\int _M\\vert L_gv_m\\vert ^\\frac{2n}{n+2} \\ dv_g\\right)^\\frac{n+2}{n}}_{=0}.$ This gives $L_gv = 0,$ hence $v\\in \\ker L_g.$ We have for all function $v^\\prime \\in \\ker L_g,$ $\\int _M \\vert v_m\\vert ^{N-2}v_m v^\\prime \\ dv_g = \\frac{\\int _M \\vert u_m\\vert ^{N-2}u_m v^\\prime \\ dv_g}{\\Vert u_m\\Vert ^{N-1}_{H_2^\\frac{2n}{n+2}}} = 0.$ In particular for $v^\\prime = v$ , $\\int _M \\vert v_m\\vert ^{N-2}v_m v \\ dv_g = 0\\longrightarrow _{m\\rightarrow \\infty }\\int _Mv^N \\ dv_g,$ so $v=0.$ According to the regularity Theorem 3.75 in [3], we have $1 = \\Vert v_m\\Vert _{H_2^\\frac{2n}{n+2}}\\le C\\left[\\underbrace{\\Vert L_gv_m\\Vert _{L^\\frac{2n}{n+2}}}_{\\longrightarrow 0} +\\Vert v_m\\Vert _{L^\\frac{2n}{n+2}}\\right].$ Passing to the limit, we obtain $\\int _M v^\\frac{2n}{n+2} \\ dv_g\\ge \\frac{1}{C},$ which gives a contradiction.","We deduce that $(u_m)_m$ is a bounded sequence in $H_2^\\frac{2n}{n+2}(M).$ Then, after restriction to a subsequence, there exists $u$ in $H_2^\\frac{2n}{n+2}(M)$ such that $u_m\\rightharpoonup u \\text{ in } H_2^\\frac{2n}{n+2}(M),$ $u_m\\rightharpoonup u \\text{ in }H_1^2(M),$ $u_m\\longrightarrow u \\text{ in } L^2(M).$ Further, we have $\\left(\\int _M \\vert L_gu\\vert ^\\frac{2n}{n+2} \\ dv_g\\right)^\\frac{n+2}{n}\\le \\liminf _m \\left(\\int _M \\vert L_gu_m\\vert ^\\frac{2n}{n+2} \\ dv_g\\right)^\\frac{n+2}{n}.$ Moreover, $\\int _M uL_gu \\ dv_g =& \\int _M \\vert \\nabla u\\vert ^2 \\ dv_g - \\int _M u^2 \\ dv_g \\nonumber \\\\\\le & \\liminf _m \\int _M \\vert \\nabla u_m\\vert ^2 \\ dv_g - \\int _M u_m^2 \\ dv_g\\nonumber \\\\=& \\liminf _m \\int _M u_mL_gu_m \\ dv_g = -1.$ Therefore $ \\int _M uL_gu \\ dv_g < 0,$ and $u\\ne 0.$ Finally, with (REF ) $I_g(u) &=& \\frac{\\left(\\int _M \\vert L_gu\\vert ^\\frac{2n}{n+2} \\ dv_g\\right)^\\frac{n+2}{n}}{\\vert \\int _M uL_gu \\ dv_g\\vert }\\\\&\\le & \\liminf _m \\frac{\\left(\\int _M \\vert L_gu_m\\vert ^\\frac{2n}{n+2} \\ dv_g\\right)^\\frac{n+2}{n}}{\\vert \\int _M u_mL_gu_m \\ dv_g\\vert }= \\liminf _m I_g(u_m) =\\alpha .$ Hence the result is proved i.e.","$I_g(u) = \\alpha $ .","Euler equation Notice that $\\int _M \\vert u\\vert ^{N-2} \\ u \\ v^\\prime \\ dv_g = 0, \\text{ for anyfunction }v^\\prime \\in \\ker L_g.$ In particular, ${\\alpha }\\ne 0$ .","Remark also that $\\int _M uL_gu \\ dv_g = -1.$ Indeed, the relation $\\int _M uL_gu \\ dv_g < -1$ would imply that $I_g(u)< \\lim I_g(u_m) = {\\alpha }$ .","We now write Euler equation of $u$ .","Let $\\left\\lbrace u_1, \\ldots ,u_k\\right\\rbrace $ be a base of $\\ker L_g.$ By the Lagrange multipliers theorem, there exists real numbers $ {\\lambda }_1,\\ldots ,{\\lambda }_k$ for which, for all function ${\\varphi }\\in C^\\infty (M),$ we get $\\frac{d}{dt}|_{t=0}I_g(u+t{\\varphi }) = \\Sigma _i {\\lambda }_i\\frac{d}{dt}|_{t=0} g_i(u+t{\\varphi }),$ where $g_i(u) = \\int _M \\vert u \\vert ^{N-2} \\ u \\ u_i \\ dv_g.$ Setting $a = \\left(\\int _M \\vert L_gu\\vert ^\\frac{2n}{n+2} \\ dv_g\\right)^\\frac{n+2}{n}$ , one checks $2 a^\\frac{2}{n+2}\\int _M\\vert L_gu\\vert ^\\frac{-4}{n+2}L_gu L_g{\\varphi }\\ dv_g + 2 a\\int _M {\\varphi }L_gu \\ dv_g = (N-1) \\Sigma _i{\\lambda }_i\\int _M \\vert u \\vert ^{N-2} \\ {\\varphi }\\ u_i \\ dv_g.$ If ${\\varphi }\\in \\ker L_g,$ this last equation implies that $\\Sigma _i {\\lambda }_i\\int _M \\vert u \\vert ^{N-2} \\ {\\varphi }\\ u_i \\ dv_g = 0.$ Then, for ${\\varphi }= \\Sigma _i{\\lambda }_i \\ u_i \\in \\ker L_g,$ we have $\\int _M \\vert u\\vert ^{N-2} \\ {\\varphi }^2 \\ dv_g = 0.$ Therefore $\\vert u\\vert ^{N-2} \\ {\\varphi }^2 = 0&\\Rightarrow & \\vert u\\vert ^{N-2} \\ {\\varphi }= 0\\\\&\\Rightarrow &\\Sigma _i \\ {\\lambda }_i \\ \\vert u\\vert ^{N-2} \\ u_i = 0.$ This gives, for any function ${\\varphi }$ (in $\\ker L_g$ or not), that $\\Sigma _i {\\lambda }_i\\int _M \\vert u\\vert ^{N-2} {\\varphi }\\ u_i \\ dv_g = 0.$ Then $u$ verifies in the sense of distributions the following equation $ L_g\\left(\\vert L_gu\\vert ^\\frac{-4}{n+2} L_gu\\right) = \\alpha ^\\prime L_gu,$ where $\\alpha ^\\prime = - \\alpha ^\\frac{n}{n+2}= -\\int _M \\vert L_gu\\vert ^\\frac{2n}{n+2} \\ dv_g.$ We set $v = \\vert L_gu\\vert ^\\frac{-4}{n+2} L_gu \\in L^N(M),$ then $\\vert v\\vert = \\vert L_gu\\vert ^{1-\\frac{4}{n+2}} = \\vert L_gu\\vert ^\\frac{n-2}{n+2}.$ Hence, $ L_gu = \\vert v\\vert ^{N-2} \\ v.$ Replacing each term by its value in Equation (REF ), we obtain $L_gv = \\alpha ^\\prime \\vert v\\vert ^{N-2} \\ v.$ Regularity of $v$ We have $u\\in H_2^\\frac{2n}{n+2}(M),$ then $L_gu \\in L^\\frac{2n}{n+2}(M).$ Therefore $v\\in L^N(M),$ since $\\vert v\\vert ^N = \\vert L_g u\\vert ^\\frac{2n}{n+2}.$ Moreover, in the sense of distributions $L_gv = \\alpha ^\\prime \\vert v\\vert ^{N-2} \\ v,$ this implies that $\\vert L_g v \\vert = \\vert \\alpha ^\\prime \\vert \\vert v\\vert ^{N-1},$ hence $L_gv \\in L^\\frac{N}{N-1} (M)= L^\\frac{2n}{n+2}(M),$ therefore $v \\in H_2^\\frac{2n}{n+2}(M)\\subset H_1^2(M).$ Using Lemma 3.1 of [1], we get $v\\in L^{N+{\\varepsilon }}(M),$ By a standard bootstrap argument, we show that $v\\in C^{3,\\alpha }(M) (\\alpha 0$ .","In addition, Equation (REF ) says that there exists an $i$ such that $\\alpha ^\\prime = {\\lambda }_i(v)$ .","The only positive eigenfunctions are the ones associated to $\\lambda _1$ and hence ${\\alpha }^{\\prime }=\\lambda _1(v)$ .","By Proposition REF , there exists a function $w$ solution of the following equation $L_gw = {\\lambda }_2(v) \\vert v\\vert ^{N-2} w.$ $I_g(w) &=& \\frac{\\left(\\int _M \\vert L_gw\\vert ^\\frac{2n}{n+2} \\ dv_g\\right)^\\frac{n+2}{n}}{\\vert \\int _M wL_gw \\ dv_g\\vert }\\\\\\\\&=& \\frac{\\vert {\\lambda }_2(v) \\vert ^2 \\left(\\int _M \\vert v\\vert ^{(N-2)\\times \\frac{2n}{n+2}} \\vert w\\vert ^\\frac{2n}{n+2} \\ dv_g\\right)^\\frac{n+2}{n}}{\\vert {\\lambda }_2(v)\\vert \\int _M \\vert v\\vert ^{N-2} \\ w^2 \\ dv_g}.$ By applying the Hölder inequality with $p = \\frac{n+2}{n}$ and $q = \\frac{n+2}{2},$ we get $\\int _M \\vert v\\vert ^{(N-2)\\times \\frac{2n}{n+2}} \\ \\vert w\\vert ^\\frac{2n}{n+2}\\ dv_g &=& \\int _M \\vert v\\vert ^{(N-2)\\times \\frac{n}{n+2}} \\ \\vert w\\vert ^\\frac{2n}{n+2} \\ \\vert v\\vert ^{(N-2)\\times \\frac{n}{n+2}} \\ dv_g\\\\\\\\&\\le & \\left(\\int _M \\vert v\\vert ^{N-2} \\ w^2 \\ dv_g\\right)^\\frac{n}{n+2}\\left(\\int _M \\vert v\\vert ^{\\frac{4}{n-2}\\times \\frac{n}{2}} \\ dv_g\\right)^\\frac{2}{n+2}.$ Therefore $I_g(w) &\\le & \\vert {\\lambda }_2(v) \\vert \\left(\\int _M \\vert v \\vert ^\\frac{2n}{n-2} \\ dv_g\\right)^\\frac{2}{n}\\\\&=& \\vert {\\lambda }_1(v) \\vert \\left(\\int _M \\vert L_gu \\vert ^\\frac{2n}{n+2} \\ dv_g\\right)^\\frac{2}{n}\\\\&=& \\alpha ^\\frac{n}{n+2}\\alpha ^\\frac{2}{n+2} = \\alpha ,$ since by assumption ${\\lambda }_1(v) = \\alpha ^\\prime = \\alpha ^\\frac{n}{n+2},$ which gives a contradiction.","Then $v$ is a nodal solution of the equation $L_g v = \\alpha ^{\\prime } \\vert v\\vert ^{N-2} v,$ where $\\alpha ^{\\prime } <0$ .","Setting $v^\\prime : = \\vert \\alpha \\vert ^\\frac{n-2}{4},$ we obtain that $v^\\prime $ is a solution of the equation $L_g v^\\prime = {\\varepsilon }\\vert v^\\prime \\vert ^{N-2} v^\\prime $ with ${\\varepsilon }= -1 \\hbox{ sign }(\\lambda _2(g))$ .","This ends the proof of Theorem REF ."]],"string":"[\n [\n \"Second Eigenvalue of the Yamabe Operator and Applications\"\n ],\n [\n \"Abstract Let $(M, g)$ be a compact Riemannian manifold of dimension $n \\\\geq 3$.\",\n \"In this paper, we give various properties of the eigenvalues of the Yamabe operator $L_g$.\",\n \"In particular, we show how the second eigenvalue of $L_g$ is related to the existence of nodal solutions of the equation $L_g u = \\\\epsilon | u|^{N-2} u$, where $\\\\epsilon = +1, 0,$ or -1.\"\n ],\n [\n \"Introduction\",\n \"This paper is part of a Phd thesis whose purpose is to study the relationships between the eigenvalues of the Yamabe operator, in particular their sign, and analytic, geometrical or topological properties of compact manifolds of dimension $n \\\\ge 3$ : Let $(M,g)$ be a $n$ -dimensional compact Riemannian manifold ($n \\\\ge 3$ ).\",\n \"The Yamabe operator or conformal Laplacian operator $L_g$ is defined by $L_g(u):=c_n\\\\Delta _gu+S_g u,$ where $\\\\Delta _g$ is the Laplace-Beltrami operator, $c_n=\\\\frac{4(n-1)}{n-2}$ and $S_g$ the scalar curvature of $g$ .\",\n \"The Yamabe operator $L_g$ has discrete spectrum $\\\\rm spec(L_g)=\\\\left\\\\lbrace \\\\lambda _1(g), \\\\lambda _2(g),\\\\cdots \\\\right\\\\rbrace ,$ where the eigenvalues are such that ${\\\\lambda }_1(g) < {\\\\lambda }_2(g) \\\\le {\\\\lambda }_3(g) \\\\le \\\\cdots \\\\le {\\\\lambda }_k(g) \\\\cdots \\\\rightarrow +\\\\infty .$ The $i-$ th eigenvalue ${\\\\lambda }_i(g)$ is characterized by $ \\\\lambda _i (g) = \\\\inf _{V\\\\in Gr_{i}({H_1^2(M)})} \\\\sup _{v \\\\in V\\\\setminus \\\\lbrace 0\\\\rbrace }\\\\frac{\\\\int _M v L_{g} v \\\\ dv_g }{\\\\int _M v^2 \\\\ dv_g},$ where $Gr_{i}({H_1^2(M)})$ stands for the set of all $i$ -dimensional subspaces of $H_1^2(M).$ Our project is to understand what we can deduce from the sign of $\\\\lambda _i$ .\",\n \"Now, we summarize what is known about this question and explain our motivations.\",\n \"At first, it is straightforward to see that the sign of $\\\\lambda _1(g)$ is the same as the sign of the Yamabe constant $\\\\mu (M,g)$ of $(M,g)$ (and as a consequence is conformally invariant).\",\n \"See Section for more informations.\",\n \"Hence the positivity of $\\\\lambda _1(g)$ has many consequences usually stated in terms of positivity of the Yamabe constant.\",\n \"For instance, we obtain Proposition 1.1 A compact manifold $M$ of dimension $n \\\\ge 3$ carries a metric with positive scalar curvature if and only if it carries a metric $g$ such that $\\\\lambda _1(g) >0$ .\",\n \"We recall that classifying such compact manifolds is a challenging open problem, only solved for $n=3$ using Perelman's techniques.\",\n \"We also mention [5] where M. Dahl and C. Bär deduce many topological properties of compact manifolds from a careful study of the eigenvalues $\\\\lambda _i$ of the Yamabe operator $L_g$ .\",\n \"The sign of $\\\\lambda _1$ can also be read in terms of existence or non-existence of positive solutions of the Yamabe equation: $ L_g u = \\\\epsilon |u|^{N-2} u,$ where $N := \\\\frac{2n}{n-2}$ and $\\\\epsilon \\\\in \\\\lbrace -1;0;1\\\\rbrace $ .\",\n \"Inspired by this observation, B. Ammann and E. Humbert [1] enlighted the role of $\\\\lambda _2$ in the existence of nodal solutions (i.e.\",\n \"having a changing sign) of the Yamabe equation (REF ).\",\n \"See again Section for more explanations.\",\n \"In this paper, we establish various properties of the eigenvalues of the Yamabe operator.\",\n \"First of all, we extend their definition to what we call generalized metrics when possible (see Paragraph ) and prove that their sign is a conformal invariant (see Paragraph REF ).\",\n \"This paper initiates the study of the relationships between these conformal invariants and the topology of the manifold by showing that their negativity is not topologically obstructed (see Paragraph REF ).\",\n \"These investigations will be treated much more deeply in [7].\",\n \"The main point of this article is to complete the results of B. Ammann and E. Humbert [1] and to study how the sign of the second eigenvalue of the Yamabe operator can be related to the existence of nodal solutions of the Yamabe equation (REF ), in particular when the Yamabe constant of $(M,g)$ is negative.\",\n \"Our main result is to prove that under this condition, such a solution always exists with $\\\\epsilon = sign (\\\\lambda _2(g))$ .\",\n \"This is the object of Theorem REF .\",\n \"The author would like to thank Emmanuel Humbert for his support and encouragements.\"\n ],\n [\n \"Eigenvalues in conformal metrics\",\n \"In the whole paper, we will deal with the behavior of the eigenvalues of the Yamabe operator in a fixed conformal class.\",\n \"It will be usefull to express their definition relatively to a fixed metric.\",\n \"This is the goal of this section.\"\n ],\n [\n \"Smooth metrics\",\n \"Let $(M,g)$ be a compact Riemannian manifold of dimension $n\\\\ge 3$ , we keep the notations of the introduction and for any metric $\\\\widetilde{g}$ , we will denote by ${\\\\lambda }_1(\\\\widetilde{g}) < {\\\\lambda }_2(\\\\widetilde{g}) \\\\le {\\\\lambda }_3(\\\\widetilde{g}) \\\\le \\\\cdots \\\\le {\\\\lambda }_k(\\\\widetilde{g}) \\\\cdots \\\\rightarrow +\\\\infty ,$ the eigenvalues of the Yamabe operator.\",\n \"We will deal with the case where $\\\\widetilde{g}$ is conformal to $g$ , i.e.\",\n \"when $\\\\widetilde{g} =u^{N-2} g,$ where $u$ is a positive function of class $C^\\\\infty $ .\",\n \"By referring to [1], one sees that the $i-$ th eigenvalue ${\\\\lambda }_i(\\\\widetilde{g})$ is given by $\\\\lambda _i (\\\\widetilde{g}) = \\\\inf _{V\\\\in Gr_{i}({H_1^2(M)})} \\\\sup _{v \\\\in V\\\\setminus \\\\lbrace 0\\\\rbrace }\\\\frac{\\\\int _M c_n |v|^2 + S_g v^2 \\\\, dv_g }{\\\\int _M v^2 \\\\ u^{N-2} \\\\ dv_g},$ where $\\\\widetilde{g} = u^{N-2}g;$ $u\\\\in C^\\\\infty (M),$ $u>0$ and $Gr_{i}({H_1^2(M)})$ stands for the set of all $i$ -dimensional subspaces of $H_1^2(M).$\"\n ],\n [\n \"Generalized metrics\",\n \"Reducing to smooth metrics will too restrictive for our investigations.\",\n \"We will need to work with generalized metrics, i.e.\",\n \"metrics of the form $\\\\widetilde{g} = u^{N-2}g$ with $u\\\\in L^N(M),$ $u\\\\ge 0$ and $u\\\\lnot \\\\equiv 0$ .\",\n \"The Yamabe operator $L_{\\\\widetilde{g}}$ has no meaning any more but the definition of $\\\\lambda _i(\\\\widetilde{g})$ can anyway be extended to this case by using (REF ) as it was done in [1] when the Yamabe constant was non-negative i.e.\",\n \"when $\\\\lambda _1(g) \\\\ge 0$ .\",\n \"When $\\\\lambda _1(g) < 0$ , the situation is a little bit different: $\\\\lambda _i(\\\\widetilde{g})$ defined by (REF ) can be $-\\\\infty $ as proved by the following proposition.\",\n \"Proposition 2.1 Assume that $\\\\lambda _1(g)<0$ , then there exists $u\\\\in L^N(M)$ , $u \\\\lnot \\\\equiv 0$ , $u\\\\ge 0$ such that $\\\\lambda _1(\\\\widetilde{g}) = -\\\\infty $ , where $\\\\widetilde{g} = u^{N-2}g$ .\",\n \"This proposition will be proved in Paragraph REF .\",\n \"To make sure that $\\\\lambda _1(\\\\widetilde{g})$ is finite, one has to assume in addition that $u$ is positive.\",\n \"Proposition 2.2 Let $u$ be a positive function in $L^N(M)$ .\",\n \"Suppose that ${\\\\lambda }_1(g)<0$ .\",\n \"Then, we have ${\\\\lambda }_1(\\\\widetilde{g})> -\\\\infty .$ The proposition is proved in Paragraph REF .\",\n \"Notation 2.3 The $i^{th}$ eigenvalue of $L_g,$ ${\\\\lambda }_i(\\\\widetilde{g}) = {\\\\lambda }_i(u^{N-2}g)$ will be denoted by ${\\\\lambda }_i(u)$ when there is no ambiguity about $g$ .\"\n ],\n [\n \"Proof of Proposition \",\n \"We have ${\\\\lambda }_1(g)<0$ , this implies that there exists a function $v\\\\in C^{\\\\infty }(M)$ such that $\\\\int _M (L_g v)v\\\\,dv_g < 0.$ Let $P$ be a point of $M$ .\",\n \"For ${\\\\varepsilon }>0$ , we define $\\\\eta _{{\\\\varepsilon }}$ as follows $\\\\left\\\\lbrace \\\\begin{array}{c}0 \\\\le \\\\eta _{{\\\\varepsilon }} \\\\le 1,\\\\\\\\\\\\\\\\\\\\eta _{{\\\\varepsilon }} = 0 \\\\text{ on }B_{{\\\\varepsilon }}(P),\\\\\\\\\\\\\\\\\\\\eta _{{\\\\varepsilon }} = 1\\\\text{ on } M\\\\backslash B_{2{\\\\varepsilon }}(P),\\\\\\\\\\\\\\\\\\\\vert \\\\nabla \\\\eta _{{\\\\varepsilon }} \\\\vert \\\\le \\\\frac{2}{{\\\\varepsilon }}.\\\\end{array}\\\\right.$ where $B_{\\\\delta }(P)$ stands for the ball of center $P$ and radius $\\\\delta $ in the metric $g$ .\",\n \"Then one easily cheks $\\\\lim _{{\\\\varepsilon }\\\\rightarrow 0}\\\\int _M (L_g(\\\\eta _{{\\\\varepsilon }}v))(\\\\eta _{{\\\\varepsilon }}v)\\\\,dv_g = \\\\int _M (L_g v)v\\\\,dv_g.$ We define $w:= \\\\eta _{{\\\\varepsilon }}v$ .\",\n \"Therefore, for a fixed small ${\\\\varepsilon }> 0$ , we have $\\\\int _M (L_g w)w\\\\, dv_g < 0.$ Let $u\\\\ge 0$ , $u\\\\lnot \\\\equiv 0$ of class $C^{\\\\infty }$ with support in $B_{{\\\\varepsilon }}(P).$ For $\\\\alpha >0$ , since $(w+\\\\alpha )u \\\\lnot \\\\equiv 0$ , we can write ${\\\\lambda }_1(\\\\widetilde{g}) &=& \\\\inf _{v^{\\\\prime }} \\\\frac{\\\\int _M (L_g v^{\\\\prime })v^{\\\\prime }\\\\,dv_g}{\\\\int _M u^{N-2} v^{\\\\prime 2} \\\\,dv_g}\\\\\\\\&\\\\le & \\\\lim _{\\\\alpha \\\\rightarrow 0^+} \\\\frac{\\\\int _M (L_g (w+\\\\alpha ))(w+\\\\alpha )\\\\,dv_g}{\\\\int _M u^{N-2} (w+\\\\alpha )^2 \\\\,dv_g}.$ Moreover, we have $\\\\lim _{\\\\alpha \\\\rightarrow 0^+}\\\\int _M (L_g (w+\\\\alpha ))(w+\\\\alpha )\\\\,dv_g=\\\\int _M (L_g (w))w\\\\,dv_g <0,$ and $ \\\\lim _{\\\\alpha \\\\rightarrow 0^+} \\\\int _M u^{N-2} (w+\\\\alpha )^2 \\\\,dv_g = 0$ which gives that $\\\\lim _{\\\\alpha \\\\rightarrow 0}\\\\frac{\\\\int _M (L_g (w+\\\\alpha ))(w+\\\\alpha )\\\\,dv_g}{\\\\int _M u^{N-2} (w+\\\\alpha )^2 \\\\,dv_g} = -\\\\infty .$ This ends the proof of Proposition REF .\"\n ],\n [\n \"Proof of Proposition \",\n \"Let $(v_m)_m$ be a minimizing sequence for ${\\\\lambda }_1(u),$ i.e.\",\n \"$v_m\\\\in H_1^2(M)$ such that $\\\\lim _{m\\\\longrightarrow \\\\infty } \\\\frac{\\\\int _M c_n\\\\vert \\\\nabla v_m\\\\vert ^2 + S_gv_m^2 \\\\ dv_g}{\\\\int _M \\\\vert u\\\\vert ^{N-2} v_m^2 \\\\ dv_g} = \\\\lim _{m\\\\longrightarrow \\\\infty } {\\\\lambda }_m = {\\\\lambda }_1(u) < 0.$ Since $(\\\\vert v_m\\\\vert )_m$ is also a minimizing sequence for ${\\\\lambda }_1(u),$ we can assume that $v_m\\\\ge 0.$ We normalize $v_m$ by $\\\\int _M \\\\vert u\\\\vert ^{N-2} v_m^2 \\\\ dv_g = 1.$ Here we show that $(v_m)_m$ is bounded in $H_1^2(M).$ Indeed, suppose that $(v_m)_m$ is not bounded in $H_1^2(M)$ and let $v_m^\\\\prime = \\\\frac{v_m}{\\\\parallel v_m\\\\parallel _{H_1^2(M)}}.$ $(v_m^\\\\prime )_m$ is bounded in $H_1^2(M),$ and his norm is equal to 1, then there exists $v^\\\\prime \\\\in H_1^2(M),$ (after restriction to a subsequence) such that $v_m^\\\\prime \\\\rightharpoonup v^\\\\prime \\\\text{ in } H_1^2(M),$ $v_m^\\\\prime \\\\longrightarrow v^\\\\prime \\\\text{ in }L^2(M).$ We have $c_n \\\\int _M \\\\left|\\\\nabla v_m^{\\\\prime }\\\\right|^2\\\\, dv_g + \\\\int _M S_g v_m^{\\\\prime 2} \\\\, dv_g = {\\\\lambda }_m \\\\int _M \\\\left|u\\\\right|^{N-2}v_m^{\\\\prime 2}\\\\, dv_g.$ Moreover, $\\\\int _M \\\\left|u\\\\right|^{N-2}v^{\\\\prime 2}\\\\, dv_g \\\\le \\\\int _M \\\\left|u\\\\right|^{N-2}v_m^{\\\\prime 2}\\\\, dv_g \\\\rightarrow _{m\\\\longrightarrow \\\\infty } 0$ since $\\\\left\\\\Vert v_m\\\\right\\\\Vert _{H_1^2(M)}\\\\longrightarrow \\\\infty .$ It follows that $\\\\int _M \\\\left|u\\\\right|^{N-2}v^{\\\\prime 2}\\\\, dv_g = 0$ and since $u$ is positive, $v^{\\\\prime } = 0.$ Now, we write $1 = \\\\int _M \\\\left|\\\\nabla v_m^{\\\\prime }\\\\right|^2\\\\, dv_g + \\\\underbrace{\\\\int _M \\\\left|v_m^{\\\\prime }\\\\right|^2 \\\\, dv_g}_{\\\\longrightarrow 0}.$ We deduce that $\\\\lim _{m\\\\rightarrow \\\\infty } \\\\int _M \\\\left|\\\\nabla v_m^{\\\\prime }\\\\right|^2\\\\, dv_g=1,$ giving the desired contradiction: $c_n \\\\underbrace{\\\\int _M \\\\left|\\\\nabla v_m^{\\\\prime }\\\\right|^2\\\\, dv_g}_{\\\\longrightarrow 1} + \\\\underbrace{\\\\int _M S_g v_m^{\\\\prime 2} \\\\, dv_g}_{\\\\longrightarrow 0} = {\\\\lambda }_m \\\\int _M \\\\left|u\\\\right|^{N-2}v_m^{\\\\prime 2}\\\\, dv_g \\\\le 0.$ This proves that $(v_m)_m$ is bounded in $H_1^2(M),$ and implies that ${\\\\lambda }_m \\\\ge C.$ We finally get ${\\\\lambda }_1(\\\\widetilde{g}) > -\\\\infty .$\"\n ],\n [\n \"PDE associated to ${\\\\lambda }_i$\",\n \"Proposition 2.4 For any non-negative function $u\\\\in L^N(M),$ such that ${\\\\lambda }_1(u)>-\\\\infty $ , there exists functions $v_1>0, v_2, \\\\ldots , v_k \\\\in H_1^2(M)$ having a changing sign, such that in the sense of distributions, we have $L_g v_1 = {\\\\lambda }_1(u) \\\\vert u\\\\vert ^{N-2} v_1,$ and $L_g v_k = {\\\\lambda }_k(u) \\\\vert u\\\\vert ^{N-2} v_k.$ Moreover, we can normalize the $v_k$ by $ \\\\int _M \\\\vert u\\\\vert ^{N-2} v_k^2 \\\\ dv_g = 1 \\\\text{ and }\\\\int _M\\\\vert u\\\\vert ^{N-2} v_i v_j \\\\ dv_g =0 \\\\hspace{5.69046pt}\\\\forall i \\\\ne j.$ Proof: Let $(v_m)_m$ be a minimizing sequence for ${\\\\lambda }_1(u),$ i.e.\",\n \"$v_m\\\\in H_1^2(M)$ such that $\\\\lim _{m\\\\longrightarrow \\\\infty } \\\\frac{\\\\int _M c_n\\\\vert \\\\nabla v_m\\\\vert ^2 + S_gv_m^2 \\\\ dv_g}{\\\\int _M \\\\vert u\\\\vert ^{N-2} v_m^2 \\\\ dv_g} = {\\\\lambda }_1(u).$ According to the Paragraph REF , we get that $(v_m)_m$ is bounded in $H_1^2(M)$ and there exists $v\\\\ge 0$ in $H_1^2(M)$ such that $v_m$ converges to $v$ weakly in $H_1^2(M)$ and strongly in $L^2(M)$ (after restriction to a subsequence).\",\n \"We now want to prove $\\\\int _M \\\\vert u\\\\vert ^{N-2} v^2 \\\\ dv_g = \\\\lim _{m\\\\longrightarrow \\\\infty } \\\\int _M\\\\vert u\\\\vert ^{N-2} v_m^2 \\\\ dv_g = 1.$ If $u$ is smooth, this relation is clear.\",\n \"So let us assume that $u\\\\in Ł^N(M),$ let $A$ be a large real number and set $u_A = \\\\inf \\\\left\\\\lbrace u,A\\\\right\\\\rbrace .$ By Hölder inequality, we write $\\\\left| \\\\int _M u^{N-2} \\\\left(v_m^2 - v^2\\\\right) \\\\,dv_g \\\\right| &=& \\\\left| \\\\int _M\\\\left(u^{N-2} -u_A^{N-2} +u_A^{N-2}\\\\right)\\\\left(v_m^2 - v^2\\\\right) \\\\,dv_g \\\\right|\\\\\\\\& \\\\le & \\\\left( \\\\int _M u_A^{N-2} |v_m^2 - v^2| \\\\,dv_g + \\\\int _M (u^{N-2}- u_A^{N-2}) (|v_m|+|v|)^2 \\\\,dv_g\\\\right) \\\\\\\\& \\\\le & A^{N-2} \\\\int _M |v_m^2 - v^2| \\\\,dv_g\\\\\\\\&&+ {\\\\left( \\\\int _M (u^{N-2}-u_A^{N-2})^\\\\frac{N}{N-2} \\\\,dv_g\\\\right)}^{\\\\frac{N-2}{N}} {\\\\left( \\\\int _M (|v_m|+|v|)^N \\\\,dv_g\\\\right)}^{\\\\frac{2}{N}}.$ $(v_m)_m$ is bounded in $H_1^2(M),$ it is bounded in $L^N(M).$ Hence there exists a constant $C$ such that $\\\\int _M \\\\left(|v_m|+|v|\\\\right)^N \\\\,dv_g \\\\le C.$ The convergence in $L^2(M)$ gives $\\\\lim _{m\\\\longrightarrow \\\\infty }\\\\int _M |v_m^2 - v^2| \\\\ dv_g = 0.$ By dominated convergence theorem, we have $\\\\lim _{A\\\\longrightarrow \\\\infty }\\\\int _M \\\\left(u^{N-2}-u_A^{N-2}\\\\right)^\\\\frac{N}{N-2}\\\\ dv_g= 0.$ Hence, we get (REF ).\",\n \"Since $\\\\lim _m \\\\int _M \\\\langle \\\\nabla v_m, \\\\nabla {\\\\varphi }\\\\rangle \\\\ dv_g = \\\\int _M\\\\langle \\\\nabla v, \\\\nabla {\\\\varphi }\\\\rangle \\\\ dv_g,$ $\\\\lim _m \\\\int _M S_g v_m {\\\\varphi }\\\\ dv_g = \\\\int _M S_g v {\\\\varphi }\\\\ dv_g$ and $\\\\lim _m \\\\int _M \\\\vert u\\\\vert ^{N-2} v_m {\\\\varphi }\\\\ dv_g = \\\\int _M \\\\vert u\\\\vert ^{N-2} v{\\\\varphi }\\\\ dv_g,$ (by strong convergence in $L^2(M)$ ), we obtain that in the sense of distributions $v$ verifies $L_gv = {\\\\lambda }_1(u) \\\\vert u\\\\vert ^{N-2} v.$ Now we define ${\\\\lambda }_k^\\\\prime (u) = \\\\inf _{{\\\\scriptstyle v_k; \\\\vert u\\\\vert ^{\\\\frac{N-2}{2}} v_k\\\\lnot \\\\equiv 0\\\\atop \\\\scriptstyle \\\\int _M \\\\vert u\\\\vert ^{N-2} v_i v_k \\\\ dv_g = 0 \\\\forall i 0,$ we know that ${\\\\lambda }_i (u) = \\\\inf _{u_1,\\\\ldots , u_i}\\\\sup _{{\\\\lambda }_1,\\\\ldots ,{\\\\lambda }_i}\\\\frac{\\\\int _ML_g({\\\\lambda }_1 u_1+ \\\\cdots +{\\\\lambda }_i u_i)({\\\\lambda }_1 u_1+ \\\\cdots +{\\\\lambda }_i u_i) \\\\ dv_g}{\\\\int _M ({\\\\lambda }_1u_1+ \\\\cdots +{\\\\lambda }_i u_i)^2 u^{N-2} \\\\ dv_g},$ and ${\\\\lambda }_i (1) = \\\\inf _{u_1,\\\\ldots , u_i}\\\\sup _{{\\\\lambda }_1,\\\\ldots ,{\\\\lambda }_i}\\\\frac{\\\\int _ML_g({\\\\lambda }_1 u_1+ \\\\cdots +{\\\\lambda }_i u_i)({\\\\lambda }_1 u_1+ \\\\cdots +{\\\\lambda }_i u_i) \\\\ dv_g}{\\\\int _M ({\\\\lambda }_1u_1+ \\\\cdots +{\\\\lambda }_i u_i)^2 \\\\ dv_g}.$ Suppose that ${\\\\lambda }_i(u)$ is attained by $v_1,\\\\ldots ,v_i$ .\",\n \"Since denominators of this expressions are positive, then $\\\\sup _{{\\\\lambda }_1,\\\\cdots ,{\\\\lambda }_i}\\\\int _M L_g({\\\\lambda }_1 v_1+ \\\\cdots +{\\\\lambda }_i v_i)({\\\\lambda }_1 v_1+\\\\cdots +{\\\\lambda }_i v_i) \\\\ dv_g = 0.$ So ${\\\\lambda }_i(1) \\\\le \\\\sup _{{\\\\lambda }_1,\\\\ldots ,{\\\\lambda }_i}\\\\frac{\\\\int _M L_g({\\\\lambda }_1 v_1+ \\\\cdots +{\\\\lambda }_iv_i)({\\\\lambda }_1 v_1+ \\\\cdots +{\\\\lambda }_i v_i) \\\\ dv_g}{\\\\int _M ({\\\\lambda }_1 v_1+ \\\\cdots +{\\\\lambda }_i v_i)^2u^{N-2} \\\\ dv_g} = 0,$ which gives a contradiction.\",\n \"The remaining cases are treated similarly.\"\n ],\n [\n \"The negativity of ${\\\\lambda }_k$ is not topologically obstructed\",\n \"In this paragraph, we will see that on each manifold, there exists a metric which has a negative $k^{th}$ -eigenvalue.\",\n \"Proposition 3.2 On any compact Riemannian manifold $M,$ and for all $k\\\\ge 1$ there exists a metric $g$ such that ${\\\\lambda }_k(g)<0.$ Proof: Let $M$ be a compact Riemannian manifold of dimension $n,$ and we take $k$ spheres of dimension $n=dim (M).$ We equip each sphere $\\\\mathbb {S}^n$ by the same metric $g,$ such that $\\\\mu (g)<0.$ We can do this by referring to [3] (Theorem [1] page 38).\",\n \"Let $P\\\\in M,$ since $\\\\mu (g)<0,$ for all ${\\\\varepsilon }, \\\\delta >0$ we can find a function $u$ supported in $\\\\mathbb {S}^n\\\\backslash {B_{{\\\\varepsilon }}(P)}$ such that $\\\\frac{\\\\int _M c_n\\\\vert \\\\nabla u\\\\vert ^2 + S_g u^2 \\\\ dv_g}{\\\\int _M u^2 \\\\ dv_g}<-\\\\delta .$ Indeed, let $\\\\eta _{\\\\varepsilon }$ be a smooth cut-off function such that $0\\\\le \\\\eta _{\\\\varepsilon }\\\\le 1,$ $\\\\eta _{\\\\varepsilon }(B_{{\\\\varepsilon }}(P))=0,$ $\\\\eta _{\\\\varepsilon }(\\\\mathbb {S}^n\\\\backslash {B_{2{\\\\varepsilon }}(P)})=1,$ $\\\\vert \\\\nabla \\\\eta _{\\\\varepsilon }\\\\vert \\\\le \\\\frac{2}{{\\\\varepsilon }}$ and a function $v$ satisfying $I_g(v) = \\\\frac{\\\\int _Mc_n\\\\vert \\\\nabla v\\\\vert ^2 + S_g v^2 \\\\ dv_g}{\\\\int _M v^2 \\\\ dv_g}< -2\\\\delta .$ Note that the existence of $v$ is given by the fact that $\\\\mu (g)<0.$ The desired function $u$ will be given by $\\\\eta _{\\\\varepsilon }v,$ where ${\\\\varepsilon }>0$ is sufficiently small.\",\n \"Indeed, it suffices to notice that, as easily checked, $\\\\lim _{{\\\\varepsilon }\\\\longrightarrow 0}I_g(\\\\eta _{\\\\varepsilon }v) = I_g(v).$ Let $P_1,\\\\cdots ,P_k$ be points of $M.$ We consider the following connected sum $M^\\\\prime = M \\\\# (\\\\mathbb {S}^n)_1 \\\\# \\\\ldots \\\\# (\\\\mathbb {S}^n)_k,$ where the $(\\\\mathbb {S}^n)_i$ are attached at $P$ on the spheres $\\\\mathbb {S}^n$ and at $P_i$ on $M$ so that the handles are attached in $B_{{\\\\varepsilon }}(P)$ and $B_{{\\\\varepsilon }}(P_i)$ .\",\n \"Note that $M^\\\\prime $ is diffeomorphic to $M.$ Moreover, the above construction allows to see $(\\\\mathbb {S}^n)_i\\\\backslash B_{{\\\\varepsilon }}(P_i)$ as a part of $M^\\\\prime .$ We take on $M^\\\\prime $ any metric $h$ satisfying $h\\\\vert _{(\\\\mathbb {S}^n)_i\\\\backslash B_{{\\\\varepsilon }}(P_i)}= g.$ On $M^\\\\prime ,$ we define the following function $u_i=\\\\left|\\\\;\\\\begin{matrix}u\\\\hfill & \\\\hbox{on } (\\\\mathbb {S}^n)_i\\\\backslash B_{{\\\\varepsilon }}(P_i)\\\\\\\\\\\\\\\\0\\\\hfill & \\\\hbox{otherwise} .\\\\end{matrix}\\\\right.$ Since the $u_i$ have disjoint supports, we get ${\\\\lambda }_k(1)&\\\\le & \\\\sup _{{\\\\lambda }_1,\\\\ldots ,{\\\\lambda }_k}\\\\frac{\\\\int _{M^\\\\prime } L_g({\\\\lambda }_1 u_1+\\\\cdots +{\\\\lambda }_ku_k)({\\\\lambda }_1 u_1+ \\\\cdots +{\\\\lambda }_k u_k) \\\\ dv_h}{\\\\int _{M^\\\\prime } ({\\\\lambda }_1 u_1+ \\\\cdots +{\\\\lambda }_ku_k)^2 \\\\ dv_h}\\\\\\\\\\\\\\\\&\\\\le & \\\\sup _{{\\\\lambda }_1,\\\\ldots ,{\\\\lambda }_k}\\\\frac{({\\\\lambda }_1^2+ \\\\ldots + {\\\\lambda }_k^2)\\\\int _M (L_gu)u \\\\ dv_g}{({\\\\lambda }_1^2+ \\\\ldots + {\\\\lambda }_k^2)\\\\int _M u^2 \\\\ dv_g}\\\\\\\\\\\\\\\\&\\\\le & \\\\frac{\\\\int _M (L_gu)u \\\\ dv_g}{\\\\int _M u^2 \\\\ dv_g}< -\\\\delta .$\"\n ],\n [\n \"Nodal solutions of the Yamabe equations\",\n \"A famous problem in Riemannian geometry is the Yamabe problem, solved between 1960 and 1984 by Yamabe, Trüdinger, Aubin and Schoen, [15], [13], [2], [12].\",\n \"The reader can also refer to [11], [8], [3].\",\n \"The Yamabe problem consists in finding a metric $\\\\widetilde{g}$ conformal to $g$ such that the scalar curvature $S_{\\\\widetilde{g}}$ of $\\\\widetilde{g}$ is constant.\",\n \"Solving this problem is equivalent to finding a positive smooth function and a number $C_0 \\\\in \\\\mathbb {R}$ such that $L_g(u) = C_0 \\\\vert u\\\\vert ^{N-2} u,$ where $N = \\\\frac{2n}{n-2}.$ In order to obtain solutions of the Yamabe equation we define the Yamabe invariant by $\\\\mu (M,g):= \\\\inf _ {u\\\\ne 0, u\\\\in C^\\\\infty (M)}Y(u),$ where $Y(u) = \\\\frac{\\\\int _M c_n\\\\vert \\\\nabla u\\\\vert ^2+S_g u^2 \\\\ dv_g}{\\\\left(\\\\int _M \\\\vert u\\\\vert ^N \\\\ dv_g\\\\right)^\\\\frac{2}{N}}.$ The works of Yamabe, Trüdinger, Aubin and Schoen provides a positive smooth minimizer $u$ of $Y$ , satisfying, if normalized by $\\\\Vert u\\\\Vert _{L^N(M)} = 1,$ $L_gu = \\\\mu (M,g) \\\\vert u\\\\vert ^{N-2}u.$ The metric $\\\\widetilde{g} = u^{N-2} g$ is the desired metric: its scalar curvature is constant equal to $\\\\mu (M,g)$ .\",\n \"If we set $u^{\\\\prime } = \\\\mu (M,g)^{\\\\frac{n-2}{4}} u$ , we obtain a positive solution of $L_gu^{\\\\prime } = {\\\\varepsilon }\\\\vert u^{\\\\prime }\\\\vert ^{N-2}u^{\\\\prime }$ where ${\\\\varepsilon }= \\\\hbox{ sign } (\\\\mu (M,g))= \\\\hbox{ sign}(\\\\lambda _1(g))$ .\",\n \"Now, if $\\\\mu (M,g)\\\\ge 0,$ it is easy to chek that $\\\\mu (M,g) = \\\\inf _{\\\\widetilde{g}\\\\in \\\\left[ g\\\\right]} {\\\\lambda }_1(\\\\widetilde{g}) vol(M,\\\\widetilde{g})^\\\\frac{2}{n},$ where $\\\\left[ g\\\\right]$ is the conformal class of $g$ and ${\\\\lambda }_1$ is the first eigenvalue of the Yamabe operator $L_g$ .\",\n \"Inspired by this approach, in their paper [1], B. Ammann et E. Humbert introduced the second Yamabe invariant defined by $\\\\mu _2(M,g)&=& \\\\inf _{\\\\widetilde{g}} {\\\\lambda }_2(\\\\widetilde{g}) vol(M,\\\\widetilde{g})^\\\\frac{2}{n}\\\\\\\\&=& \\\\inf _u {\\\\lambda }_2(u^{N-2}g) \\\\left(\\\\int _M u^N \\\\ dv_g\\\\right)^\\\\frac{2}{n}.$ They studied this invariant in the case where $\\\\mu (M,g)\\\\ge 0$ , and they proved that $\\\\mu _2$ is attained by a generalized metric, (i.e.\",\n \"a metric of the form $u^{N-2}g$ where $u\\\\in L^N(M)$ , $u\\\\ge 0$ which may vanish), in the following two cases $\\\\bullet $ $\\\\mu (M,g)>0$ , $(M,g)$ is not locally conformally flat and $n\\\\ge 11$ .\",\n \"$\\\\bullet $ $\\\\mu (M,g)=0$ , $(M,g)$ is not locally conformally flat and $n\\\\ge 9$ .\",\n \"In this context, they proved that $u$ is the absolute value of a changing sign function $w$ of class $C^{3,\\\\alpha }(M)$ , which verifies the following equation $L_g w = \\\\mu _2(M,g)\\\\vert w\\\\vert ^{N-2} w.$ Many works are devoted to the study of this kind of solutions, for example [1], [6], [9], [10], [14].\",\n \"See also [4] for an analogue study for the Paneitz-Branson operator.\",\n \"Setting again $w^{\\\\prime } = \\\\mu _2(M,g)^{\\\\frac{n-2}{4}} w$ , we obtain a solution of $L_g w^{\\\\prime }= {\\\\varepsilon }\\\\vert w^{\\\\prime }\\\\vert ^{N-2} w^{\\\\prime }$ with $\\\\epsilon =1= \\\\hbox{ sign }(\\\\mu _2(M,g))= \\\\hbox{ sign }(\\\\lambda _2(g)).$ The goal of this section is to study if this result extends to metrics where the sign of ${\\\\lambda }_2(g)$ is arbitrary.\",\n \"The answer is yes when ${\\\\lambda }_2 < 0$ without any other condition, we obtain this result by a method different than the one of [1].\",\n \"Notice that this situation occurs for a large number of metrics (see Proposition REF ).\",\n \"When ${\\\\lambda }_2\\\\ge 0$ , we show that the methods in [1] can be extended to the case where $\\\\mu (M,g)<0$ .\",\n \"Namely, the main result of this paper is: Theorem 4.1 Let $(M,g)$ be a compact Riemannian manifold of dimension $n\\\\ge 3$ whose Yamabe invariant $\\\\mu (M,g)$ is strictly negative, we denote by ${\\\\lambda }_2$ the second eigenvalue of $L_g.$ Then, if ${\\\\lambda }_2\\\\le 0$ or if ${\\\\lambda }_2>0$ , $(M,g)$ not locally conformally flat and $n\\\\ge 6$ : There exists a function $w$ changing sign, solution of the equation $L_g w = {\\\\varepsilon }\\\\vert w \\\\vert ^{N-2}w,$ where ${\\\\varepsilon }=+1$ if ${\\\\lambda }_2>0,$ ${\\\\varepsilon }= -1$ if ${\\\\lambda }_2< 0$ and ${\\\\varepsilon }=0$ if ${\\\\lambda }_2 = 0.$ Moreover, $w \\\\in C^{3,\\\\alpha }(M)$ , for all $\\\\alpha < N-2.$\"\n ],\n [\n \"The case ${\\\\lambda }_2 = 0$\",\n \"This case is obvious: indeed, Proposition REF provides the existence of a nodal solution $v$ of $L_g v = 0= {\\\\varepsilon }\\\\left|v\\\\right|^{N-2}v$ where ${\\\\varepsilon }= 0 = \\\\hbox{ sign }(\\\\lambda _2(g))$ .\"\n ],\n [\n \"The case ${\\\\lambda }_2 > 0$\",\n \"As in [1], we introduce the second Yamabe invariant given by $\\\\mu _2(M,g)&=& \\\\inf _{\\\\widetilde{g}}{\\\\lambda }_2(\\\\widetilde{g}) vol(M,\\\\widetilde{g})^\\\\frac{2}{n}\\\\\\\\&=& \\\\inf _{u>0} {\\\\lambda }_2(u)\\\\left(\\\\int _M u^N \\\\ dv_g\\\\right)^\\\\frac{2}{n}.$ By Proposition REF below, the problem reduces to finding a minimizer of $\\\\mu _2(M,g)$ .\",\n \"The case where $\\\\mu (M,g) \\\\ge 0$ have been treated in [1].\",\n \"We will then focus on the case where $\\\\mu (M,g) < 0$ (i.e.\",\n \"${\\\\lambda }_1(g) <0$ ).\",\n \"We will see that the method of Ammann and Humbert remains valid in this case and the following three propositions answer our questions.\",\n \"Proposition 4.2 Let $(M,g)$ be a compact Riemannian manifold of dimension $n\\\\ge 3$ , such that ${\\\\lambda }_2>0$ .\",\n \"If $\\\\mu _2(M,g)<\\\\mu (\\\\mathbb {S}^n),$ with $\\\\mu (\\\\mathbb {S}^n)=n(n-1)\\\\omega _n^{\\\\frac{2}{n}}$ , where $\\\\omega _n$ stands for the volume of the standard sphere $\\\\mathbb {S}^n$ , then the second Yamabe invariant is attained by a non-negative function $u\\\\in L^N(M)$ that we normalize by $\\\\int _M u^N \\\\ dv_g = 1$ .\",\n \"There exists a function $w$ having a changing sign which verifies in the sense of distributions the following equation $L_gw = \\\\mu _2(M,g) \\\\vert u\\\\vert ^{N-2} w.$ The functions $u$ and $w$ will be normalized by $\\\\int _M u^N \\\\ dv_g = 1, \\\\hspace{5.69046pt}\\\\int _M u^{N-2} \\\\ w^2 \\\\ dv_g = 1.$ Proposition 4.3 The two functions $u$ and $w$ given by Proposition REF satisfy $u = \\\\vert w\\\\vert .$ Finally, we give a condition under which assumption (REF ) is satisfied: Proposition 4.4 Let $(M,g)$ be a compact Riemannian manifold of dimension $n\\\\ge 6,$ suppose that $M$ is not locally conformally flat and his Yamabe invariant $\\\\mu (M,g)<0,$ then $\\\\mu _2(M,g)< \\\\mu (\\\\mathbb {S}^n).$\"\n ],\n [\n \"Proof of Proposition \",\n \"The case where $\\\\mu (M,g) \\\\ge 0$ is done in [1], hence we consider here the case where $\\\\mu (M,g) < 0$ .\",\n \"By the solution of the Yamabe problem, we can assume without loss of generality, that $S_g = -1$ .\",\n \"Let $(u_m)_m$ be a minimizing sequence for $\\\\mu _2(M,g),$ i.e., $u_m$ is positive, smooth and $\\\\lim _{m\\\\longrightarrow \\\\infty }\\\\lambda _2(u_m)\\\\left(\\\\int _Mu_m^Ndv_g\\\\right)^\\\\frac{2}{n} =\\\\mu _2(M,g).$ The sequence $(u_m)_m$ will be choosen such that $\\\\int _M u_m^N \\\\ dv_g=1$ , hence $\\\\mu _2(M,g)=\\\\lim _{m\\\\longrightarrow \\\\infty }\\\\lambda _2(u_m).$ For each $u_m,$ Proposition REF provides the existence of a function $w_m \\\\in H_1^2(M)$ such that $L_g w_m = {\\\\lambda }_2(u_m)\\\\vert u_m\\\\vert ^{N-2}w_m.$ Moreover, the sequence $(w_m)_m$ can be normalized by $\\\\int _M\\\\vert u_m\\\\vert ^{N-2}w_m^2 \\\\ dv_g=1.$ Since $\\\\int _Mu_m^Ndv_g=1,$ $(u_m)_m$ is bounded in $L^N(M)$ which is a reflexive space, there exists $u\\\\in L^N(M)$ such that $u_m$ converges weakly to $u$ in $L^N(M)$ , we have $u_m\\\\rightharpoonup u \\\\text{ in }L^N(M).$ $\\\\bullet $ The sequence $(w_m)_m$ is bounded in $H_1^2(M).$ We proceed by contradiction and assume that $\\\\Vert w_m\\\\Vert _{H_1^2(M)}\\\\longrightarrow \\\\infty .$ Let $w^\\\\prime _m=\\\\frac{w_m}{\\\\Vert w_m\\\\Vert _{H_1^2(M)}}.$ $\\\\Vert w^\\\\prime _m\\\\Vert _{H_1^2(M)}=1,$ hence $(w^\\\\prime _m)_m$ is bounded in $H_1^2(M).$ Since $H_1^2(M)$ is a reflexive space, this implies using Kondrakov and Banach-Alaoglu theorems, that there exists a subsequence $(w_m^\\\\prime )_m$ and $w^\\\\prime \\\\in H_1^2(M)$ such that $w^\\\\prime _m\\\\rightharpoonup w^\\\\prime \\\\text{ in } H_1^2(M),$ and $w^\\\\prime _m\\\\longrightarrow w^\\\\prime \\\\text{ in } L^2(M).$ Equation (REF ) is linear, so $w^\\\\prime _m$ satisfies $L_gw_m^{\\\\prime } = \\\\lambda _2(u_m)\\\\left|u_m \\\\right|^{N-2}w^\\\\prime _m.$ Hence for all $\\\\varphi \\\\in C^\\\\infty (M),$ we have: $c_n \\\\int _M \\\\left\\\\langle \\\\nabla w^\\\\prime _m,\\\\nabla \\\\varphi \\\\right\\\\rangle dv_g+\\\\int _MS_gw^\\\\prime _m\\\\varphi dv_g=\\\\int _M\\\\lambda _2(u_m)\\\\left|u_m\\\\right|^{N-2}w^\\\\prime _m \\\\varphi dv_g.$ Since $w^\\\\prime _m \\\\rightharpoonup w^\\\\prime $ in $H_1^2(M)$ and $w \\\\longmapsto \\\\left\\\\langle \\\\nabla w ,\\\\nabla \\\\varphi \\\\right\\\\rangle $ is a linear form on $H_1^2(M),$ then $c_n\\\\int _M \\\\left\\\\langle \\\\nabla w_m^\\\\prime ,\\\\nabla \\\\varphi \\\\right\\\\rangle dv_g\\\\longrightarrow c_n \\\\int _M\\\\left\\\\langle \\\\nabla w^{\\\\prime },\\\\nabla \\\\varphi \\\\right\\\\rangle dv_g.$ The sequence $w_m^\\\\prime $ converges strongly to $w^\\\\prime $ in $L^2(M)$ .\",\n \"This gives that $\\\\int _M S_gw_m^\\\\prime \\\\varphi dv_g\\\\longrightarrow \\\\int _M S_g w^\\\\prime \\\\varphi dv_g.$ Using Hölder inequality, we obtain that $\\\\int _M\\\\left|u_m\\\\right|^{N-2}w^\\\\prime _m\\\\varphi \\\\ dv_g\\\\longrightarrow 0.$ Indeed, $\\\\left|\\\\int _M\\\\left|u_m\\\\right|^{N-2}w^\\\\prime _m\\\\varphi dv_g\\\\right|&\\\\le &\\\\left\\\\Vert \\\\varphi \\\\right\\\\Vert _{\\\\infty }\\\\int _M\\\\left|u_m\\\\right|^\\\\frac{N-2}{2}\\\\left|w_m^\\\\prime \\\\right|\\\\left|u_m\\\\right|^\\\\frac{N-2}{2}\\\\ dv_g\\\\\\\\&\\\\le &\\\\left\\\\Vert \\\\varphi \\\\right\\\\Vert _{\\\\infty }\\\\left(\\\\int _M\\\\left|u_m\\\\right|^{N-2}{w^\\\\prime _m}^2 \\\\ dv_g\\\\right)^\\\\frac{1}{2}\\\\left(\\\\int _M\\\\left|u_m\\\\right|^{N-2} \\\\ dv_g\\\\right)^\\\\frac{1}{2}\\\\\\\\&\\\\le &\\\\left\\\\Vert \\\\varphi \\\\right\\\\Vert _{\\\\infty }\\\\frac{\\\\left(\\\\int _M\\\\left|u_m\\\\right|^{N-2}{w_m}^2 \\\\ dv_g\\\\right)^\\\\frac{1}{2}}{\\\\left\\\\Vert w_m\\\\right\\\\Vert _{H_1^2(M)}}\\\\left(\\\\int _M\\\\left|u_m\\\\right|^{N}\\\\ dv_g\\\\right)^\\\\frac{N-2}{2N}\\\\left(\\\\rm vol(M,g)^{1-\\\\frac{N-2}{N}}\\\\right)^\\\\frac{1}{2}\\\\\\\\&\\\\le &\\\\left\\\\Vert \\\\varphi \\\\right\\\\Vert _{\\\\infty } \\\\frac{1}{\\\\left\\\\Vert w_m\\\\right\\\\Vert _{H_1^2(M)}}\\\\left(\\\\rm vol(M,g)\\\\right)^\\\\frac{1}{N} \\\\longrightarrow _{m\\\\longrightarrow +\\\\infty }0.$ Then $c_n \\\\int _M\\\\left\\\\langle \\\\nabla w^{\\\\prime },\\\\nabla \\\\varphi \\\\right\\\\rangle \\\\ dv_g+\\\\int _MS_g w^\\\\prime \\\\varphi \\\\ dv_g=0,$ which means that in the sense of distributions, we have $L_g w^\\\\prime =0.$ Since ${\\\\lambda }_1(1)<0$ and ${\\\\lambda }_2(1)$ is positive, $0\\\\notin Sp(L_g)$ .\",\n \"It follows that $w^\\\\prime =0.$ Now, we also have $\\\\int _M c_n\\\\left|\\\\nabla w^\\\\prime _m\\\\right|^2 \\\\ dv_g+\\\\int _M S_g{w^\\\\prime _m}^2 \\\\ dv_g=\\\\lambda _2(u_m)\\\\int _M\\\\left|u_m\\\\right|^{N-2}{w_m^\\\\prime }^2 \\\\ dv_g,$ with $\\\\lambda _2(u_m)\\\\int _M\\\\left|u_m\\\\right|^{N-2}{w_m^\\\\prime }^2 \\\\ dv_g=\\\\frac{\\\\lambda _2(u_m)}{\\\\left\\\\Vert w_m\\\\right\\\\Vert ^2_{H_1^2(M)}}\\\\longrightarrow 0$ and $\\\\int _M S_g{w^\\\\prime _m}^2 \\\\ dv_g\\\\longrightarrow \\\\int _M S_g{w^\\\\prime }^2 \\\\ dv_g = 0.$ Hence $\\\\int _M\\\\left|\\\\nabla w^\\\\prime _m\\\\right|^2 \\\\ dv_g\\\\longrightarrow 0.$ Finally, we get that $\\\\left\\\\Vert w^\\\\prime _m\\\\right\\\\Vert ^2_{H_1^2(M)}=1=\\\\int _M\\\\left|\\\\nabla w_m^\\\\prime \\\\right|^2 \\\\ dv_g+\\\\int _M{w^\\\\prime _m}^2 \\\\ dv_g\\\\longrightarrow 0,$ which gives the desired contradiction.\",\n \"We obtain that $(w_m)_m$ is a bounded sequence in $H_1^2(M).$ Then there exists $w\\\\in H_1^2(M)$ such that: $w_m\\\\rightharpoonup w \\\\text{ in } H_1^2(M),$ $w_m\\\\longrightarrow w \\\\text{ in }L^2(M).$ It follows that in the sense of distributions, we have $L_gw=\\\\mu _2(M,g)\\\\left|u\\\\right|^{N-2}w.$ It remains to show that $w$ changes sign and is different from zero.\",\n \"$\\\\bullet $ Suppose that $w$ does not change sign.\",\n \"Without loss of generality, we can assume that $w\\\\ge 0.$ In the sense of distributions, we have $c_n\\\\Delta _gw+S_gw=\\\\mu _2(M,g)\\\\left|u\\\\right|^{N-2}w.$ It was already mentioned at the beginning of this section that we can assume that $S_g<0,$ because $\\\\mu (M,g)<0.$ Integrating (REF ) over $M$ , we get: $\\\\underbrace{\\\\int _Mc_n\\\\Delta _gw \\\\ dv_g}_{=0}+\\\\underbrace{\\\\int _MS_gw \\\\ dv_g}_{<0}=\\\\underbrace{\\\\mu _2(M,g)\\\\int _M\\\\left|u\\\\right|^{N-2}w \\\\ dv_g}_{\\\\ge 0}.$ This gives a contradiction unless $w \\\\equiv 0$ which is prohibited by what follows.\",\n \"$\\\\bullet $ Assume that $w=0.$ By referring to [8] and [2] we have the following theorem: If $(M,g)$ is a Riemannian manifold of dimension $n\\\\ge 3,$ for all $\\\\epsilon >0,$ there exists $B_{\\\\epsilon }$ such that for any $u\\\\in H_1^2(M),$ we have $\\\\left(\\\\int _M\\\\left|u\\\\right|^N \\\\ dv_g \\\\right)^\\\\frac{2}{N}\\\\le (\\\\mu (\\\\mathbb {S}^n)^{-1}+\\\\epsilon )\\\\left(\\\\int _M c_n\\\\left|\\\\nabla u\\\\right|^2 \\\\ dv_g+B_\\\\epsilon \\\\int _Mu^2 \\\\ dv_g\\\\right).$ We obtain $c_n\\\\int _M\\\\left|\\\\nabla w_m\\\\right|^2 \\\\ dv_g+S_g\\\\int _Mw_m^2 \\\\ dv_g&=&\\\\mu _2(M,g)\\\\int _M\\\\left|u_m\\\\right|^{N-2}w_m^2 \\\\ dv_g\\\\\\\\&\\\\le &\\\\mu _2(M,g)\\\\underbrace{\\\\left(\\\\int _M\\\\left|u_m\\\\right|^{N} \\\\ dv_g\\\\right)^\\\\frac{N-2}{N}}_{=1}\\\\left(\\\\int _M\\\\left|w_m\\\\right|^{N} \\\\ dv_g\\\\right)^\\\\frac{2}{N}\\\\\\\\&\\\\le &\\\\underbrace{\\\\mu _2(M,g)(\\\\mu (\\\\mathbb {S}^n)^{-1}+\\\\epsilon )}_{<1(\\\\text{if}\\\\hspace{2.84544pt}{\\\\varepsilon }\\\\hspace{2.84544pt}\\\\text{is small enough})}\\\\left(\\\\int _Mc_n\\\\left|\\\\nabla w_m\\\\right|^2 \\\\ dv_g+B_\\\\epsilon \\\\int _Mw_m^2 \\\\ dv_g\\\\right).$ Hence $c_n\\\\underbrace{\\\\left[1-\\\\mu _2(M,g)(\\\\mu (\\\\mathbb {S}^n)^{-1}+\\\\epsilon )\\\\right]}_{>0}\\\\int _M\\\\left|\\\\nabla w_m\\\\right|^2 \\\\ dv_g\\\\le c\\\\underbrace{\\\\int _Mw_m^2 \\\\ dv_g}_{\\\\longrightarrow 0},$ then $\\\\int _M\\\\left|\\\\nabla w_m\\\\right|^2 \\\\ dv_g\\\\longrightarrow 0,$ so $\\\\left\\\\Vert w_m\\\\right\\\\Vert _{H_1^2(M)}\\\\longrightarrow 0.$ This shows that $w_m\\\\longrightarrow 0$ in $H_1^2(M).$ We finally get that $1=\\\\int _M\\\\left|u_m\\\\right|^{N-2}w_m^2 \\\\ dv_g\\\\le \\\\left(\\\\int _M\\\\left|u_m\\\\right|^N \\\\ dv_g\\\\right)^\\\\frac{N-2}{2}\\\\underbrace{\\\\int _Mw_m^N \\\\ dv_g}_{\\\\longrightarrow 0}.$ This gives a contradiction, then $w\\\\ne 0.$\"\n ],\n [\n \"Proof of Proposition \",\n \"Since ${\\\\lambda }_2(g)>0$ , then $\\\\mu _2(M,g) = \\\\inf _{u>0} {\\\\lambda }_2(u)\\\\left(\\\\int _M u^N \\\\ dv_g\\\\right)^\\\\frac{2}{n} = \\\\inf _{u\\\\ge 0} {\\\\lambda }_2(u)\\\\left(\\\\int _M u^N \\\\ dv_g\\\\right)^\\\\frac{2}{n}.$ We mimic the proof of Lemma 3.3 in [1] by taking $w_1 = w_+ = \\\\sup \\\\left\\\\lbrace 0,w\\\\right\\\\rbrace $ and $w_2 = w_- = \\\\sup \\\\left\\\\lbrace 0, -w\\\\right\\\\rbrace $ .\",\n \"This gives that $u = aw_+ + b w_-,$ where $a,b>0$ .\",\n \"By Lemma 3.1, $w\\\\in C^{2,\\\\alpha }$ , $u\\\\in C^{0,\\\\alpha }$ and Step 4 of the proof of Theorem 3.4 in [1] then shows that $u = \\\\vert w\\\\vert .$ Since $w$ is in $H_1^2(M),$ Lemma 3.1 of [1] says that $w\\\\in L^{N + {\\\\varepsilon }}(M),$ because $w$ satisfies the equation $L_gw =\\\\mu _2\\\\vert w\\\\vert ^{N-2} w,$ and standard bootstrap arguments gives that $w\\\\in C^{3,\\\\alpha }(M)$ for all $\\\\alpha 0$ we define the following function $v_{\\\\epsilon } =c_{\\\\epsilon } \\\\eta (\\\\epsilon +r^2)^\\\\frac{2-n}{2},$ where $c_\\\\epsilon $ is choosen such that $\\\\int _M v_{\\\\epsilon }^N \\\\ dv_g=1.$ By referring to [2] $\\\\lim _{\\\\epsilon \\\\longrightarrow 0}Y(v_{\\\\epsilon })=\\\\mu _1(\\\\mathbb {S}^n),$ where $Y(u)$ is the Yamabe functional defined by $Y(u)=\\\\frac{\\\\int _M c_n\\\\vert \\\\nabla u\\\\vert ^2+S_g u^2 \\\\ dv_g}{\\\\left(\\\\int _M u^N \\\\ dv_g\\\\right)^\\\\frac{2}{N}}.$ If $(M,g)$ is not locally conformally flat, by a calculation made in [2], there exists a constant $C(M)>0$ such that $Y(v_{{\\\\varepsilon }})=\\\\left|\\\\;\\\\begin{matrix}\\\\mu _1(\\\\doba {S}^n) - C(M) {\\\\varepsilon }^2 + o({\\\\varepsilon }^2)\\\\hfill & \\\\hbox{ if } n > 6 \\\\\\\\\\\\\\\\\\\\mu _1(\\\\doba {S}^n) - C(M) {\\\\varepsilon }^2 |\\\\ln ({\\\\varepsilon })| + o({\\\\varepsilon }^2 |\\\\ln ({\\\\varepsilon })|)\\\\hfill & \\\\hbox{ if } n = 6.\\\\end{matrix}\\\\right.$ Again from [2] there exists constants $a,$ $b,$ $C_1,$ $C_2>0$ , such that $a {\\\\varepsilon }^{\\\\frac{n-2}{4} } \\\\le c_{{\\\\varepsilon }} \\\\le b {\\\\varepsilon }^{\\\\frac{n-2}{4} },$ and $ C_1 {\\\\alpha }_{p,{\\\\varepsilon }} \\\\le \\\\int _M v_{{\\\\varepsilon }}^p \\\\,dv_g \\\\le C_2 {\\\\alpha }_{p,{\\\\varepsilon }}$ where $ {\\\\alpha }_{p,{\\\\varepsilon }} = \\\\left| \\\\begin{array}{lll}{\\\\varepsilon }^{\\\\frac{2n - (n-2) p}{4}} & \\\\hbox{if} & p> \\\\frac{n}{n-2};\\\\\\\\\\\\\\\\|\\\\ln ({\\\\varepsilon })| {\\\\varepsilon }^{\\\\frac{n}{4}} & \\\\hbox{if} & p= \\\\frac{n}{n-2};\\\\\\\\\\\\\\\\{\\\\varepsilon }^{ \\\\frac{(n-2) p}{4}} & \\\\hbox{if}& p< \\\\frac{n}{n-2}\\\\end{array} \\\\right.\",\n \"$ We have $\\\\mu _2(M,g)&=&\\\\inf _u\\\\lambda _2(u)\\\\left(\\\\int _Mu^N \\\\ dv_g\\\\right)^{\\\\frac{2}{n}}\\\\\\\\\\\\\\\\&=& \\\\inf _{\\\\scriptstyle u \\\\atop \\\\scriptstyle w,w^\\\\prime } \\\\sup _{\\\\lambda ,\\\\mu } \\\\frac{\\\\int _M L_g(\\\\lambda w+\\\\mu w^\\\\prime )(\\\\lambda w+\\\\mu w^\\\\prime ) \\\\ dv_g}{\\\\int _M u^{N-2}(\\\\lambda w+\\\\mu w^\\\\prime )^2 \\\\ dv_g}\\\\left(\\\\int _Mu^N \\\\ dv_g\\\\right)^{\\\\frac{2}{n}}\\\\\\\\\\\\\\\\&=& \\\\inf _{\\\\scriptstyle u \\\\atop \\\\scriptstyle w,w^\\\\prime } \\\\sup _{\\\\lambda ,\\\\mu }F(u,\\\\lambda w+\\\\mu w^\\\\prime ).$ Let $\\\\lambda _\\\\epsilon ,$ $\\\\mu _\\\\epsilon $ such that $\\\\lambda _\\\\epsilon ^2+\\\\mu _\\\\epsilon ^2=1$ and $F(v_\\\\epsilon , \\\\lambda _\\\\epsilon v+\\\\mu _\\\\epsilon v_\\\\epsilon )=\\\\sup _{(\\\\lambda ,\\\\mu )\\\\in \\\\mathbb {R}^2\\\\backslash \\\\left\\\\lbrace (0,0)\\\\right\\\\rbrace } F(v_\\\\epsilon , \\\\lambda v +\\\\mu v_\\\\epsilon ),$ where $v$ is the function defined in the equation (REF ).\",\n \"Calculating $F(v_\\\\epsilon , \\\\lambda _\\\\epsilon v+\\\\mu _\\\\epsilon v_\\\\epsilon ),$ we get $F(v_\\\\epsilon , \\\\lambda _\\\\epsilon v+\\\\mu _\\\\epsilon v_\\\\epsilon )&=& \\\\frac{\\\\int _ML_g(\\\\lambda _\\\\epsilon v+\\\\mu _\\\\epsilon v_\\\\epsilon )(\\\\lambda _\\\\epsilon v+\\\\mu _\\\\epsilon v_\\\\epsilon ) \\\\ dv_g}{\\\\int _M v_\\\\epsilon ^{N-2}(\\\\lambda _\\\\epsilon v+\\\\mu _\\\\epsilon v_\\\\epsilon )^2 \\\\ dv_g}\\\\left(\\\\int _Mv_\\\\epsilon ^N \\\\ dv_g\\\\right)^{\\\\frac{2}{n}}\\\\\\\\\\\\\\\\&=& \\\\frac{\\\\lambda _\\\\epsilon ^2 \\\\mu (M,g)+\\\\mu _\\\\epsilon ^2Y(v_\\\\epsilon )+2\\\\lambda _\\\\epsilon \\\\mu _\\\\epsilon \\\\mu (M,g)\\\\int _M v^{N-1}v_\\\\epsilon \\\\ dv_g}{\\\\lambda _\\\\epsilon ^2\\\\int _Mv_\\\\epsilon ^{N-2}v^2 \\\\ dv_g+\\\\mu _\\\\epsilon ^2+2\\\\lambda _\\\\epsilon \\\\mu _\\\\epsilon \\\\int _M v_\\\\epsilon ^{N-1}v \\\\ dv_g}\\\\\\\\\\\\\\\\&=&\\\\frac{A_\\\\epsilon }{B_\\\\epsilon }.$ If ${\\\\lambda }_{\\\\varepsilon }\\\\longrightarrow {\\\\lambda }\\\\ne 0, \\\\hspace{5.69046pt}\\\\mu _{\\\\varepsilon }\\\\longrightarrow \\\\mu \\\\ne 0,$ then $F(v_{\\\\varepsilon },{\\\\lambda }_{\\\\varepsilon }v + \\\\mu _{\\\\varepsilon }v_{\\\\varepsilon })\\\\longrightarrow \\\\frac{{\\\\lambda }^2 \\\\mu (M,g) + \\\\mu ^2\\\\mu (\\\\mathbb {S}^n)}{\\\\mu ^2}<\\\\mu (\\\\mathbb {S}^n).$ Similarly, if $\\\\mu = 0,$ ${\\\\lambda }^2 = 1,$ then the numerator $A_{\\\\varepsilon }\\\\sim \\\\mu (M,g)< 0,$ while the denominator $B_{\\\\varepsilon }$ remains positive, which gives again that $F(v_{\\\\varepsilon },{\\\\lambda }_{\\\\varepsilon }v + \\\\mu _{\\\\varepsilon }v_{\\\\varepsilon })\\\\le 0<\\\\mu (\\\\mathbb {S}^n),$ which gives the desired inequality.\",\n \"Then, in the sequel, we assume that ${\\\\lambda }_{\\\\varepsilon }\\\\longrightarrow 0$ and $\\\\mu _{\\\\varepsilon }\\\\longrightarrow \\\\pm 1.$ The case $n>6$ Using (REF ) we have $ \\\\int _M v^{N-1} v_{{\\\\varepsilon }} \\\\,dv_g \\\\sim _{{\\\\varepsilon }\\\\rightarrow 0} C {\\\\varepsilon }^{\\\\frac{n-2}{4}},$ $\\\\int _M v_{{\\\\varepsilon }}^{N-2} v^2 \\\\,dv_g \\\\sim _{{\\\\varepsilon }\\\\rightarrow 0} C {\\\\varepsilon },$ and $\\\\int _M v_{{\\\\varepsilon }}^{N-1} v \\\\,dv_g \\\\sim _{{\\\\varepsilon }\\\\rightarrow 0} C {\\\\varepsilon }^{\\\\frac{n-2}{4}},$ where $C$ denotes a constant that might change its value from line to line.\",\n \"We distinguish two cases $\\\\bullet $ there exists a constant $C>0$ such that $\\\\vert {\\\\lambda }_{\\\\varepsilon }\\\\vert \\\\le C {\\\\varepsilon }^\\\\frac{n-2}{4},$ or $\\\\bullet $ there exists $\\\\alpha _{\\\\varepsilon }$ such that $\\\\vert \\\\lambda _{\\\\varepsilon }\\\\vert = \\\\alpha _{\\\\varepsilon }{\\\\varepsilon }^{\\\\frac{n-2}{4}},$ and $\\\\alpha _{\\\\varepsilon }\\\\longrightarrow +\\\\infty .$ (possibly extracting a subsequence).\",\n \"Suppose first that (REF ) is verified.\",\n \"Then we have $\\\\vert {\\\\lambda }_{\\\\varepsilon }\\\\vert \\\\le C {\\\\varepsilon }^\\\\frac{n-2}{4}.$ Hence ${\\\\lambda }_{\\\\varepsilon }^2=O({\\\\varepsilon }^\\\\frac{n-2}{2}),$ so $\\\\mu ^2_{\\\\varepsilon }=1-{\\\\lambda }^2_{\\\\varepsilon }=1+O({\\\\varepsilon }^\\\\frac{n-2}{2}).$ Therefore $\\\\mu _{\\\\varepsilon }=1+O({\\\\varepsilon }^\\\\frac{n-2}{2}).$ This gives $A_\\\\epsilon &=& O({\\\\varepsilon }^{\\\\frac{n-2}{2}}) + ( 1 +O({\\\\varepsilon }^{\\\\frac{n-2}{2}})) \\\\Big ( \\\\mu (\\\\mathbb {S}^n) -C(M){\\\\varepsilon }^2 +o({\\\\varepsilon }^2)\\\\Big ) + O({\\\\varepsilon }^{\\\\frac{n-2}{2}})\\\\\\\\&=& \\\\mu (\\\\mathbb {S}^n)-C(M){\\\\varepsilon }^2 + O({\\\\varepsilon }^{\\\\frac{n-2}{2}}) +o({\\\\varepsilon }^2).$ Since $\\\\frac{n-2}{2} > 2$ , $A_{\\\\varepsilon }=\\\\mu (\\\\mathbb {S}^n) -C(M){\\\\varepsilon }^2 +o({\\\\varepsilon }^2),$ and $B_{\\\\varepsilon }= O({\\\\varepsilon }^{\\\\frac{n-2}{2}+1}) + 1 + O({\\\\varepsilon }^{\\\\frac{n-2}{2}})+O({\\\\varepsilon }^{\\\\frac{n-2}{2}}) = 1+ o({\\\\varepsilon }^2).$ Then, $\\\\frac{A_{\\\\varepsilon }}{B_{\\\\varepsilon }} = \\\\mu (\\\\mathbb {S}^n)- C(M) {\\\\varepsilon }^2 + o({\\\\varepsilon }^2) < \\\\mu (\\\\mathbb {S}^n).$ Assume now that (REF ) is fulfilled.\",\n \"In this case $\\\\frac{A_{\\\\varepsilon }}{B_{\\\\varepsilon }} &=& \\\\frac{\\\\lambda ^2_{\\\\varepsilon }\\\\mu (M, g) + (1 - \\\\lambda ^2_{\\\\varepsilon })Y(v_{\\\\varepsilon }) + \\\\lambda _{\\\\varepsilon }O({\\\\varepsilon }^{\\\\frac{n-2}{4}})}{ \\\\lambda ^2_{\\\\varepsilon }O({\\\\varepsilon }) + (1- \\\\lambda ^2_{\\\\varepsilon }) + 2\\\\lambda _{\\\\varepsilon }\\\\mu _{\\\\varepsilon }O({\\\\varepsilon }^{\\\\frac{n-2}{4}})}\\\\\\\\&=& \\\\frac{\\\\lambda ^2_{\\\\varepsilon }\\\\mu (M, g) + (1 - \\\\lambda ^2_{\\\\varepsilon }) Y(v_{\\\\varepsilon }) +o({\\\\lambda }^2_{\\\\varepsilon })}{o({\\\\lambda }^2_{\\\\varepsilon }) + (1-{\\\\lambda }^2_{\\\\varepsilon }) + o({\\\\lambda }^2_{\\\\varepsilon })}\\\\\\\\&=& \\\\frac{{\\\\lambda }^2_{\\\\varepsilon }\\\\mu (M, g)}{1-{\\\\lambda }_{\\\\varepsilon }^2 + o({\\\\lambda }^2_{\\\\varepsilon })} + \\\\frac{Y(v_{\\\\varepsilon })}{ 1 +\\\\frac{o({\\\\lambda }_{\\\\varepsilon }^2)}{\\\\mu ^2_{\\\\varepsilon }}} + o({\\\\lambda }_{\\\\varepsilon }^2)\\\\\\\\\\\\\\\\&\\\\le & \\\\mu (M,g) {\\\\lambda }_{\\\\varepsilon }^2 + \\\\mu (\\\\mathbb {S}^n)(1 + o({\\\\lambda }_{\\\\varepsilon }^2)) + o({\\\\lambda }_{\\\\varepsilon }^2)\\\\\\\\&\\\\le & \\\\mu (\\\\mathbb {S}^n) + \\\\mu (M,g) {\\\\lambda }_{\\\\varepsilon }^2 + o({\\\\lambda }_{\\\\varepsilon }^2)\\\\\\\\&<& \\\\mu (\\\\mathbb {S}^n),$ because $\\\\mu (M,g)<0$ and $Y(v_{\\\\varepsilon }) \\\\le \\\\mu (\\\\mathbb {S}^n).$ The case $n =6$ Since $\\\\int _M v_{\\\\varepsilon }^{N-2} v^2 dv_g &\\\\sim _{{\\\\varepsilon }\\\\rightarrow 0}& C {\\\\varepsilon },\\\\\\\\\\\\int _M v^{N-1}v_{\\\\varepsilon }dv_g &\\\\sim _{{\\\\varepsilon }\\\\rightarrow 0}& C {\\\\varepsilon },\\\\\\\\\\\\int _M v_{\\\\varepsilon }^{N-1} v dv_g &\\\\sim _{{\\\\varepsilon }\\\\rightarrow 0}& C {\\\\varepsilon },$ then $A_{\\\\varepsilon }= {\\\\lambda }_{\\\\varepsilon }^2 \\\\mu (M,g) + \\\\mu ^2_{\\\\varepsilon }Y(v_{\\\\varepsilon }) + 2{\\\\lambda }_{\\\\varepsilon }\\\\mu _{\\\\varepsilon }O({\\\\varepsilon }),$ $B_{\\\\varepsilon }= {\\\\lambda }^2_{\\\\varepsilon }O({\\\\varepsilon }) + \\\\mu ^2_{\\\\varepsilon }+ 2{\\\\lambda }_{\\\\varepsilon }\\\\mu _{\\\\varepsilon }O({\\\\varepsilon }).$ Again, we have two cases to study If $\\\\vert {\\\\lambda }_{\\\\varepsilon }\\\\vert \\\\le C{\\\\varepsilon },$ then ${\\\\lambda }_{\\\\varepsilon }^2\\\\le C{\\\\varepsilon }^2.$ This implies $A_{\\\\varepsilon }= \\\\mu (\\\\mathbb {S}^n) - C{\\\\varepsilon }^2\\\\vert \\\\ln ({\\\\varepsilon })\\\\vert + o({\\\\varepsilon }^2\\\\vert \\\\ln ({\\\\varepsilon })\\\\vert )$ and $B_{\\\\varepsilon }= 1+O({\\\\varepsilon }^2) = 1 + o({\\\\varepsilon }^2 \\\\vert \\\\ln ({\\\\varepsilon })\\\\vert ).$ Hence $\\\\frac{A_{\\\\varepsilon }}{B_{\\\\varepsilon }}< \\\\mu (\\\\mathbb {S}^n).$ If $\\\\vert {\\\\lambda }_{\\\\varepsilon }\\\\vert = \\\\alpha _{\\\\varepsilon }{\\\\varepsilon },$ with $\\\\alpha _{\\\\varepsilon }\\\\longrightarrow +\\\\infty .$ Since $Y(v_{\\\\varepsilon }) \\\\le \\\\mu (\\\\mathbb {S}^n),$ therefore $A_{\\\\varepsilon }= \\\\alpha _{\\\\varepsilon }^2 {\\\\varepsilon }^2 \\\\mu (M,g) + \\\\mu ^2_{\\\\varepsilon }\\\\mu (\\\\mathbb {S}^n) + o(\\\\alpha _{\\\\varepsilon }^2{\\\\varepsilon }^2),$ and $B_{\\\\varepsilon }= \\\\mu ^2_{\\\\varepsilon }+ o(\\\\alpha _{\\\\varepsilon }^2 {\\\\varepsilon }^2).$ Therefore $\\\\frac{A_{\\\\varepsilon }}{B_{\\\\varepsilon }} &=& \\\\mu (M,g) \\\\frac{\\\\alpha _{\\\\varepsilon }^2 {\\\\varepsilon }^2}{1+o(1)}+ \\\\frac{\\\\mu (\\\\mathbb {S}^n)}{1 + o(\\\\alpha _{\\\\varepsilon }^2 {\\\\varepsilon }^2)} + \\\\frac{o(\\\\alpha _{\\\\varepsilon }^2 {\\\\varepsilon }^2)}{1+ o(1)}\\\\\\\\&=& \\\\mu (M,g)\\\\alpha _{\\\\varepsilon }^2 {\\\\varepsilon }^2 + \\\\mu (\\\\mathbb {S}^n) + o(\\\\alpha _{\\\\varepsilon }^2 {\\\\varepsilon }^2)\\\\\\\\&<& \\\\mu (\\\\mathbb {S}^n).$ This ends the proof of Propositon REF .\",\n \"$\\\\square $ So we get a solution $w$ having a changing sign of the equation $L_g w = \\\\mu _2 \\\\vert w\\\\vert ^{N-2}w.$ Finally, to obtain the resultat announced in Theorem REF , it suffices to set $w^\\\\prime = \\\\mu _2^\\\\frac{n-2}{4} w,$ then $w^\\\\prime $ verifies $L_gw^\\\\prime = {\\\\varepsilon }\\\\vert w^\\\\prime \\\\vert ^{N-2} w^\\\\prime ,$ with ${\\\\varepsilon }= 1 = \\\\hbox{ sign}({\\\\lambda }_2(g))$ .\"\n ],\n [\n \"The case ${\\\\lambda }_2<0$\",\n \"In this section, we will show that in all cases, there exists a nodal solution of the equation $L_g w = C_0 \\\\vert w\\\\vert ^{N-2}w,$ where $C_0$ is a negative constant.\",\n \"First, since $\\\\mu <0,$ we assume in the whole section that the metric $g$ is such that $S_g=-1.$ In this context, the approach will be different.\",\n \"Indeed, the second Yamabe invariant is not well defined as shown in the following proposition: Proposition 5.1 Let $M$ be a compact Riemannian manifold of dimension $n\\\\ge 3$ .\",\n \"Suppose that ${\\\\lambda }_2< 0$ , then $\\\\inf _u {\\\\lambda }_2(u)\\\\left(\\\\int _M u^N \\\\ dv_g\\\\right)^\\\\frac{2}{n} = -\\\\infty .$ The proof will be detailed in Subsection REF .\",\n \"We will use a new functional $I_g(u)= \\\\frac{\\\\left(\\\\int _M\\\\vert L_gu\\\\vert ^\\\\frac{2n}{n+2} \\\\ dv_g\\\\right)^\\\\frac{n+2}{n}}{\\\\vert \\\\int _M uL_gu \\\\ dv_g\\\\vert }.$ We study $\\\\alpha := \\\\inf I_g(u)$ where the infimum is taken over the functions $u\\\\in H_2^\\\\frac{2n}{n+2}(M)$ such that $\\\\int _M uL_gu \\\\ dv_g <0,$ and with the following constraint $\\\\int _M \\\\vert u \\\\vert ^{N-2} u \\\\ v \\\\ dv_g=0,$ for any function $v\\\\in \\\\ker L_g.$ We will show that $\\\\alpha $ is a conformal invariant.\",\n \"We obtain also that the infimum of this functional is attained by a function $u.$ We set $v = \\\\vert L_gu\\\\vert ^\\\\frac{-4}{n+2} L_gu,$ and we will observe that $v$ has the following properties: $\\\\bullet $ $v$ is a solution of the equation $L_g v = \\\\alpha ^\\\\prime \\\\vert v\\\\vert ^{N-2}v,$ where $\\\\alpha ^\\\\prime <0$ (i.e.\",\n \"has same sign than ${\\\\lambda }_2$ ).\",\n \"$\\\\bullet $ $v$ has a changing sign.\",\n \"$\\\\bullet $ $v$ is of class $C^{3,\\\\alpha }(M)$ ($\\\\alpha < N-2$ ).\"\n ],\n [\n \"Conformal invariance of $\\\\alpha $\",\n \"Let $\\\\widetilde{g} = {\\\\varphi }^\\\\frac{4}{n-2} g$ be a conformal metric, ${\\\\varphi }$ a smooth positive function.\",\n \"Then $dv_{\\\\widetilde{g}} = {\\\\varphi }^\\\\frac{2n}{n-2} dv_g,$ and $L_{\\\\widetilde{g}}u = {\\\\varphi }^{-\\\\frac{n+2}{n-2}}L_g(u{\\\\varphi }),$ for all functions $u.$ Remark that $I_{\\\\widetilde{g}}(u) = I_g(u{\\\\varphi }).$ $I_{\\\\widetilde{g}} (u)&=& \\\\frac{\\\\left(\\\\int _M \\\\vert L_{\\\\widetilde{g}}u\\\\vert ^\\\\frac{2n}{n+2} \\\\ dv_{\\\\widetilde{g}}\\\\right)^\\\\frac{n+2}{n}}{\\\\vert \\\\int _M uL_{\\\\widetilde{g}}u \\\\ dv_{\\\\widetilde{g}}\\\\vert }\\\\\\\\\\\\\\\\&=& \\\\frac{\\\\left(\\\\int _M \\\\vert {\\\\varphi }\\\\vert ^\\\\frac{-2n}{n-2} \\\\vert L_g(u{\\\\varphi })\\\\vert ^\\\\frac{2n}{n+2} {\\\\varphi }^\\\\frac{2n}{n-2} \\\\ dv_g\\\\right)^\\\\frac{n+2}{n}}{\\\\vert \\\\int _M u{\\\\varphi }^\\\\frac{-(n+2)}{n-2} L_g (u{\\\\varphi }){\\\\varphi }^\\\\frac{2n}{n-2} \\\\ dv_g\\\\vert }\\\\\\\\\\\\\\\\&=& \\\\frac{\\\\left(\\\\int _M \\\\vert L_g(u{\\\\varphi })\\\\vert ^\\\\frac{2n}{n-2} \\\\ dv_g\\\\right)^\\\\frac{n+2}{n}}{\\\\vert \\\\int _M u{\\\\varphi }L_g(u{\\\\varphi }) \\\\ dv_g\\\\vert }\\\\\\\\\\\\\\\\&=& I_g(u{\\\\varphi }),$ where we have used $\\\\int _M uL_{\\\\widetilde{g}}u \\\\ dv_{\\\\widetilde{g}} &=& \\\\int _M u{\\\\varphi }^{-\\\\frac{n+2}{n-2}}L_g(u{\\\\varphi }){\\\\varphi }^\\\\frac{2n}{n-2} \\\\ dv_g\\\\\\\\&=& \\\\int _M (u{\\\\varphi })L_g(u{\\\\varphi }) \\\\ dv_g.$ Assume that for any $v\\\\in \\\\ker L_{\\\\widetilde{g}},$ we have $\\\\int _M \\\\vert u\\\\vert ^{N-2} uv \\\\ dv_{\\\\widetilde{g}} = 0.$ Then, for any $v^\\\\prime \\\\in \\\\ker L_g,$ we obtain $\\\\int _M \\\\vert u{\\\\varphi }\\\\vert ^{N-2} (u{\\\\varphi })v^\\\\prime \\\\ dv_g = \\\\int _M \\\\vert u\\\\vert ^{N-2} u (v^\\\\prime {\\\\varphi }^{-1}) \\\\ dv_{\\\\widetilde{g}} = 0,$ since $L_{\\\\widetilde{g}}(v^\\\\prime {\\\\varphi }^{-1}) = {\\\\varphi }^{-\\\\frac{n+2}{n-2}}L_g(v^\\\\prime ) = 0,$ i.e.\",\n \"$v^\\\\prime {\\\\varphi }^{-1}\\\\in Ker L_{\\\\widetilde{g}}.$ $\\\\square $\"\n ],\n [\n \"Proof of Proposition \",\n \"Assume that ${\\\\lambda }_2(g)< 0,$ and choose $u>0.$ By Lemma REF , there exists two functions $v_1$ and $v_2$ solutions of the following equations $L_g v_1 = {\\\\lambda }_1(u) \\\\vert u\\\\vert ^{N-2} v_1,$ and $L_g v_2 = {\\\\lambda }_2(u) \\\\vert u\\\\vert ^{N-2} v_2,$ such that $\\\\int _M \\\\vert u\\\\vert ^{N-2} v_1v_2 \\\\ dv_g=0.$ Let $v_{\\\\varepsilon }$ the function defined in Section REF , and let $V=\\\\left\\\\lbrace v_1,v_2\\\\right\\\\rbrace .$ For all $v\\\\in V,$ we get $\\\\lim _{{\\\\varepsilon }\\\\longrightarrow 0}\\\\int _Mv_{\\\\varepsilon }^{N-2} v^2 \\\\ dv_g = 0.$ Since ${\\\\lambda }_1(u)<0$ and ${\\\\lambda }_2(u)<0,$ then for ${\\\\varepsilon }$ sufficiently small, we have $\\\\lim _{{\\\\varepsilon }\\\\rightarrow 0}\\\\left(\\\\sup _{v\\\\in V}\\\\frac{\\\\int _M (L_gv)(v) \\\\ dv_g}{\\\\int _M v_{\\\\varepsilon }^{N-2}v^2 \\\\ dv_g}\\\\right) =-\\\\infty ,$ hence $\\\\lim _{{\\\\varepsilon }\\\\rightarrow 0}\\\\left(\\\\inf _u {\\\\lambda }_2(u)\\\\left(\\\\int _M u^N \\\\ dv_g\\\\right)^\\\\frac{2}{n}\\\\right) =-\\\\infty .$ $\\\\square $\"\n ],\n [\n \"The infimum of the functional $I_g$ is attained\",\n \"Let $(u_m)_m$ be a minimizing sequence, i.e., $\\\\lim _{m\\\\longrightarrow \\\\infty }I_g(u_m) = \\\\alpha ,$ with $\\\\int _M \\\\vert u_m \\\\vert ^{N-2} \\\\ u_m \\\\ v \\\\ dv_g=0, \\\\hspace{5.69046pt}\\\\forall \\\\ v\\\\in \\\\ker L_g.$ We can assume that $\\\\int _M u_mL_gu_m \\\\ dv_g = -1.$ Then $\\\\alpha = \\\\lim _{m\\\\longrightarrow \\\\infty } \\\\left(\\\\int _M\\\\vert L_gu_m\\\\vert ^\\\\frac{2n}{n+2} \\\\ dv_g\\\\right)^\\\\frac{n+2}{n}.$ Now we show that $(u_m)_m$ is a bounded sequence in $H_2^\\\\frac{2n}{n+2}(M).$ We proceed by contradiction and we assume that, up to a subsequence, $\\\\lim \\\\Vert u_m\\\\Vert _{H_2^\\\\frac{2n}{n+2}(M)}= +\\\\infty $ .\",\n \"Let $v_m = \\\\frac{u_m}{\\\\Vert u_m\\\\Vert _{H_2^\\\\frac{2n}{n+2}(M)}}.$ Since $\\\\Vert v_m\\\\Vert _{H_2^\\\\frac{2n}{n+2}(M)} = 1,$ $(v_m)_m$ is a bounded sequence in $H_2^\\\\frac{2n}{n+2}(M),$ and therefore there exists $v \\\\in H_2^\\\\frac{2n}{n+2}(M)$ such that after restriction to a subsequence $v_m\\\\rightharpoonup v\\\\text{ in }H_2^\\\\frac{2n}{n+2}(M),$ $v_m\\\\longrightarrow v\\\\text{ in }L^2(M).$ By standard arguments, we get $\\\\left(\\\\int _M\\\\vert L_gv\\\\vert ^\\\\frac{2n}{n+2} \\\\ dv_g\\\\right)^\\\\frac{n+2}{n}\\\\le \\\\underbrace{\\\\liminf _m \\\\left(\\\\int _M\\\\vert L_gv_m\\\\vert ^\\\\frac{2n}{n+2} \\\\ dv_g\\\\right)^\\\\frac{n+2}{n}}_{=0}.$ This gives $L_gv = 0,$ hence $v\\\\in \\\\ker L_g.$ We have for all function $v^\\\\prime \\\\in \\\\ker L_g,$ $\\\\int _M \\\\vert v_m\\\\vert ^{N-2}v_m v^\\\\prime \\\\ dv_g = \\\\frac{\\\\int _M \\\\vert u_m\\\\vert ^{N-2}u_m v^\\\\prime \\\\ dv_g}{\\\\Vert u_m\\\\Vert ^{N-1}_{H_2^\\\\frac{2n}{n+2}}} = 0.$ In particular for $v^\\\\prime = v$ , $\\\\int _M \\\\vert v_m\\\\vert ^{N-2}v_m v \\\\ dv_g = 0\\\\longrightarrow _{m\\\\rightarrow \\\\infty }\\\\int _Mv^N \\\\ dv_g,$ so $v=0.$ According to the regularity Theorem 3.75 in [3], we have $1 = \\\\Vert v_m\\\\Vert _{H_2^\\\\frac{2n}{n+2}}\\\\le C\\\\left[\\\\underbrace{\\\\Vert L_gv_m\\\\Vert _{L^\\\\frac{2n}{n+2}}}_{\\\\longrightarrow 0} +\\\\Vert v_m\\\\Vert _{L^\\\\frac{2n}{n+2}}\\\\right].$ Passing to the limit, we obtain $\\\\int _M v^\\\\frac{2n}{n+2} \\\\ dv_g\\\\ge \\\\frac{1}{C},$ which gives a contradiction.\",\n \"We deduce that $(u_m)_m$ is a bounded sequence in $H_2^\\\\frac{2n}{n+2}(M).$ Then, after restriction to a subsequence, there exists $u$ in $H_2^\\\\frac{2n}{n+2}(M)$ such that $u_m\\\\rightharpoonup u \\\\text{ in } H_2^\\\\frac{2n}{n+2}(M),$ $u_m\\\\rightharpoonup u \\\\text{ in }H_1^2(M),$ $u_m\\\\longrightarrow u \\\\text{ in } L^2(M).$ Further, we have $\\\\left(\\\\int _M \\\\vert L_gu\\\\vert ^\\\\frac{2n}{n+2} \\\\ dv_g\\\\right)^\\\\frac{n+2}{n}\\\\le \\\\liminf _m \\\\left(\\\\int _M \\\\vert L_gu_m\\\\vert ^\\\\frac{2n}{n+2} \\\\ dv_g\\\\right)^\\\\frac{n+2}{n}.$ Moreover, $\\\\int _M uL_gu \\\\ dv_g =& \\\\int _M \\\\vert \\\\nabla u\\\\vert ^2 \\\\ dv_g - \\\\int _M u^2 \\\\ dv_g \\\\nonumber \\\\\\\\\\\\le & \\\\liminf _m \\\\int _M \\\\vert \\\\nabla u_m\\\\vert ^2 \\\\ dv_g - \\\\int _M u_m^2 \\\\ dv_g\\\\nonumber \\\\\\\\=& \\\\liminf _m \\\\int _M u_mL_gu_m \\\\ dv_g = -1.$ Therefore $ \\\\int _M uL_gu \\\\ dv_g < 0,$ and $u\\\\ne 0.$ Finally, with (REF ) $I_g(u) &=& \\\\frac{\\\\left(\\\\int _M \\\\vert L_gu\\\\vert ^\\\\frac{2n}{n+2} \\\\ dv_g\\\\right)^\\\\frac{n+2}{n}}{\\\\vert \\\\int _M uL_gu \\\\ dv_g\\\\vert }\\\\\\\\&\\\\le & \\\\liminf _m \\\\frac{\\\\left(\\\\int _M \\\\vert L_gu_m\\\\vert ^\\\\frac{2n}{n+2} \\\\ dv_g\\\\right)^\\\\frac{n+2}{n}}{\\\\vert \\\\int _M u_mL_gu_m \\\\ dv_g\\\\vert }= \\\\liminf _m I_g(u_m) =\\\\alpha .$ Hence the result is proved i.e.\",\n \"$I_g(u) = \\\\alpha $ .\",\n \"Euler equation Notice that $\\\\int _M \\\\vert u\\\\vert ^{N-2} \\\\ u \\\\ v^\\\\prime \\\\ dv_g = 0, \\\\text{ for anyfunction }v^\\\\prime \\\\in \\\\ker L_g.$ In particular, ${\\\\alpha }\\\\ne 0$ .\",\n \"Remark also that $\\\\int _M uL_gu \\\\ dv_g = -1.$ Indeed, the relation $\\\\int _M uL_gu \\\\ dv_g < -1$ would imply that $I_g(u)< \\\\lim I_g(u_m) = {\\\\alpha }$ .\",\n \"We now write Euler equation of $u$ .\",\n \"Let $\\\\left\\\\lbrace u_1, \\\\ldots ,u_k\\\\right\\\\rbrace $ be a base of $\\\\ker L_g.$ By the Lagrange multipliers theorem, there exists real numbers $ {\\\\lambda }_1,\\\\ldots ,{\\\\lambda }_k$ for which, for all function ${\\\\varphi }\\\\in C^\\\\infty (M),$ we get $\\\\frac{d}{dt}|_{t=0}I_g(u+t{\\\\varphi }) = \\\\Sigma _i {\\\\lambda }_i\\\\frac{d}{dt}|_{t=0} g_i(u+t{\\\\varphi }),$ where $g_i(u) = \\\\int _M \\\\vert u \\\\vert ^{N-2} \\\\ u \\\\ u_i \\\\ dv_g.$ Setting $a = \\\\left(\\\\int _M \\\\vert L_gu\\\\vert ^\\\\frac{2n}{n+2} \\\\ dv_g\\\\right)^\\\\frac{n+2}{n}$ , one checks $2 a^\\\\frac{2}{n+2}\\\\int _M\\\\vert L_gu\\\\vert ^\\\\frac{-4}{n+2}L_gu L_g{\\\\varphi }\\\\ dv_g + 2 a\\\\int _M {\\\\varphi }L_gu \\\\ dv_g = (N-1) \\\\Sigma _i{\\\\lambda }_i\\\\int _M \\\\vert u \\\\vert ^{N-2} \\\\ {\\\\varphi }\\\\ u_i \\\\ dv_g.$ If ${\\\\varphi }\\\\in \\\\ker L_g,$ this last equation implies that $\\\\Sigma _i {\\\\lambda }_i\\\\int _M \\\\vert u \\\\vert ^{N-2} \\\\ {\\\\varphi }\\\\ u_i \\\\ dv_g = 0.$ Then, for ${\\\\varphi }= \\\\Sigma _i{\\\\lambda }_i \\\\ u_i \\\\in \\\\ker L_g,$ we have $\\\\int _M \\\\vert u\\\\vert ^{N-2} \\\\ {\\\\varphi }^2 \\\\ dv_g = 0.$ Therefore $\\\\vert u\\\\vert ^{N-2} \\\\ {\\\\varphi }^2 = 0&\\\\Rightarrow & \\\\vert u\\\\vert ^{N-2} \\\\ {\\\\varphi }= 0\\\\\\\\&\\\\Rightarrow &\\\\Sigma _i \\\\ {\\\\lambda }_i \\\\ \\\\vert u\\\\vert ^{N-2} \\\\ u_i = 0.$ This gives, for any function ${\\\\varphi }$ (in $\\\\ker L_g$ or not), that $\\\\Sigma _i {\\\\lambda }_i\\\\int _M \\\\vert u\\\\vert ^{N-2} {\\\\varphi }\\\\ u_i \\\\ dv_g = 0.$ Then $u$ verifies in the sense of distributions the following equation $ L_g\\\\left(\\\\vert L_gu\\\\vert ^\\\\frac{-4}{n+2} L_gu\\\\right) = \\\\alpha ^\\\\prime L_gu,$ where $\\\\alpha ^\\\\prime = - \\\\alpha ^\\\\frac{n}{n+2}= -\\\\int _M \\\\vert L_gu\\\\vert ^\\\\frac{2n}{n+2} \\\\ dv_g.$ We set $v = \\\\vert L_gu\\\\vert ^\\\\frac{-4}{n+2} L_gu \\\\in L^N(M),$ then $\\\\vert v\\\\vert = \\\\vert L_gu\\\\vert ^{1-\\\\frac{4}{n+2}} = \\\\vert L_gu\\\\vert ^\\\\frac{n-2}{n+2}.$ Hence, $ L_gu = \\\\vert v\\\\vert ^{N-2} \\\\ v.$ Replacing each term by its value in Equation (REF ), we obtain $L_gv = \\\\alpha ^\\\\prime \\\\vert v\\\\vert ^{N-2} \\\\ v.$ Regularity of $v$ We have $u\\\\in H_2^\\\\frac{2n}{n+2}(M),$ then $L_gu \\\\in L^\\\\frac{2n}{n+2}(M).$ Therefore $v\\\\in L^N(M),$ since $\\\\vert v\\\\vert ^N = \\\\vert L_g u\\\\vert ^\\\\frac{2n}{n+2}.$ Moreover, in the sense of distributions $L_gv = \\\\alpha ^\\\\prime \\\\vert v\\\\vert ^{N-2} \\\\ v,$ this implies that $\\\\vert L_g v \\\\vert = \\\\vert \\\\alpha ^\\\\prime \\\\vert \\\\vert v\\\\vert ^{N-1},$ hence $L_gv \\\\in L^\\\\frac{N}{N-1} (M)= L^\\\\frac{2n}{n+2}(M),$ therefore $v \\\\in H_2^\\\\frac{2n}{n+2}(M)\\\\subset H_1^2(M).$ Using Lemma 3.1 of [1], we get $v\\\\in L^{N+{\\\\varepsilon }}(M),$ By a standard bootstrap argument, we show that $v\\\\in C^{3,\\\\alpha }(M) (\\\\alpha 0$ .\",\n \"In addition, Equation (REF ) says that there exists an $i$ such that $\\\\alpha ^\\\\prime = {\\\\lambda }_i(v)$ .\",\n \"The only positive eigenfunctions are the ones associated to $\\\\lambda _1$ and hence ${\\\\alpha }^{\\\\prime }=\\\\lambda _1(v)$ .\",\n \"By Proposition REF , there exists a function $w$ solution of the following equation $L_gw = {\\\\lambda }_2(v) \\\\vert v\\\\vert ^{N-2} w.$ $I_g(w) &=& \\\\frac{\\\\left(\\\\int _M \\\\vert L_gw\\\\vert ^\\\\frac{2n}{n+2} \\\\ dv_g\\\\right)^\\\\frac{n+2}{n}}{\\\\vert \\\\int _M wL_gw \\\\ dv_g\\\\vert }\\\\\\\\\\\\\\\\&=& \\\\frac{\\\\vert {\\\\lambda }_2(v) \\\\vert ^2 \\\\left(\\\\int _M \\\\vert v\\\\vert ^{(N-2)\\\\times \\\\frac{2n}{n+2}} \\\\vert w\\\\vert ^\\\\frac{2n}{n+2} \\\\ dv_g\\\\right)^\\\\frac{n+2}{n}}{\\\\vert {\\\\lambda }_2(v)\\\\vert \\\\int _M \\\\vert v\\\\vert ^{N-2} \\\\ w^2 \\\\ dv_g}.$ By applying the Hölder inequality with $p = \\\\frac{n+2}{n}$ and $q = \\\\frac{n+2}{2},$ we get $\\\\int _M \\\\vert v\\\\vert ^{(N-2)\\\\times \\\\frac{2n}{n+2}} \\\\ \\\\vert w\\\\vert ^\\\\frac{2n}{n+2}\\\\ dv_g &=& \\\\int _M \\\\vert v\\\\vert ^{(N-2)\\\\times \\\\frac{n}{n+2}} \\\\ \\\\vert w\\\\vert ^\\\\frac{2n}{n+2} \\\\ \\\\vert v\\\\vert ^{(N-2)\\\\times \\\\frac{n}{n+2}} \\\\ dv_g\\\\\\\\\\\\\\\\&\\\\le & \\\\left(\\\\int _M \\\\vert v\\\\vert ^{N-2} \\\\ w^2 \\\\ dv_g\\\\right)^\\\\frac{n}{n+2}\\\\left(\\\\int _M \\\\vert v\\\\vert ^{\\\\frac{4}{n-2}\\\\times \\\\frac{n}{2}} \\\\ dv_g\\\\right)^\\\\frac{2}{n+2}.$ Therefore $I_g(w) &\\\\le & \\\\vert {\\\\lambda }_2(v) \\\\vert \\\\left(\\\\int _M \\\\vert v \\\\vert ^\\\\frac{2n}{n-2} \\\\ dv_g\\\\right)^\\\\frac{2}{n}\\\\\\\\&=& \\\\vert {\\\\lambda }_1(v) \\\\vert \\\\left(\\\\int _M \\\\vert L_gu \\\\vert ^\\\\frac{2n}{n+2} \\\\ dv_g\\\\right)^\\\\frac{2}{n}\\\\\\\\&=& \\\\alpha ^\\\\frac{n}{n+2}\\\\alpha ^\\\\frac{2}{n+2} = \\\\alpha ,$ since by assumption ${\\\\lambda }_1(v) = \\\\alpha ^\\\\prime = \\\\alpha ^\\\\frac{n}{n+2},$ which gives a contradiction.\",\n \"Then $v$ is a nodal solution of the equation $L_g v = \\\\alpha ^{\\\\prime } \\\\vert v\\\\vert ^{N-2} v,$ where $\\\\alpha ^{\\\\prime } <0$ .\",\n \"Setting $v^\\\\prime : = \\\\vert \\\\alpha \\\\vert ^\\\\frac{n-2}{4},$ we obtain that $v^\\\\prime $ is a solution of the equation $L_g v^\\\\prime = {\\\\varepsilon }\\\\vert v^\\\\prime \\\\vert ^{N-2} v^\\\\prime $ with ${\\\\varepsilon }= -1 \\\\hbox{ sign }(\\\\lambda _2(g))$ .\",\n \"This ends the proof of Theorem REF .\"\n ]\n]"},"id":{"kind":"string","value":"1204.1268"}}},{"rowIdx":1866894,"cells":{"text":{"kind":"list like","value":[["Experimental implementation of a fully controllable depolarizing quantum\n operation"],["Abstract The depolarizing quantum operation plays an important role in studying the quantum noise effect and implementing general quantum operations.","In this work, we report a scheme which implements a fully controllable input-state independent depolarizing quantum operation for a photonic polarization qubit."],["Experimental implementation of a fully controllable depolarizing quantum operation Youn-Chang Jeongw31400@gmail.com Jong-Chan Lee Yoon-Ho Kimyoonho72@gmail.com Department of Physics, Pohang University of Science and Technology (POSTECH), Pohang, 790-784, Korea The depolarizing quantum operation plays an important role in studying the quantum noise effect and implementing general quantum operations.","In this work, we report a scheme which implements a fully controllable input-state independent depolarizing quantum operation for a photonic polarization qubit.","03.67.Dd, 03.67.Hk, 42.79.Sz Depolarizing quantum operation, Quantum channel, Quantum noise One of the main challenges in experimental quantum information is to deal with decoherence.","To preserve a qubit state, it is often necessary to isolate the qubit from the environment but to be able to do information processing tasks, interactions with an external (classical or quantum) system is necessary.","More often than not, unwanted interactions with the environment leave the qubit in a different quantum state or even cause the qubit to lose coherence.","Such unwanted quantum state transformation can be described by the quantum process due to a noisy quantum channel, in other words, a noisy quantum process, which can be understood with a few basic single-qubit noise operations including the bit-flip, phase-flip, depolarization, and amplitude damping [1].","Understanding and being able to implement each of the basic quantum noise processes are important in theoretical and experimental quantum information science.","First, it would allow us to simulate and quantify quantum noise processes.","Second, the bit-flip and the phase-flip operations are essential for quantum error correction protocols.","Third, the depolarization and the amplitude damping operations can describe decoherence.","The depolarization operation is of particular interest because, in addition to being relevant to a number of practical quantum communication and computation scenarios [2], [3], [4], [5], [6], it is also an essential operation for optimally approximating non-physical quantum operations [7], [8] and for generating exotic quantum states including Werner states [9] and bound entangled states [10].","Clearly, from the experimental point of view, it is important to develop a method to achieve a fully controllable input-independent depolarization quantum operation.","As such, there have been many reports on experimental implementation of a depolarization quantum channel.","In Ref.","[11], optical scattering media were used to achieve the depolarization quantum channel but the scheme naturally induces the spread in the photon momenta and it is difficult to control the degree of depolarization in this scheme.","A controllable depolarization channel was demonstrated in Ref.","[12], [13] but the output was dependent on the input polarization and the scheme also introduced the momentum spread for the photon.","Input-independent depolarizing quantum operation was demonstrated in Ref.","[3], [14], but here the depolarizing operation was achieved by time averaging of many `fast' operations, i.e., incoherent sum of many different pure quantum states.","In this paper, we report an experimental implementation of a fully-controllable depolarization quantum operation for a photonic polarization qubit.","The scheme is completely input-state independent so that it is possible to introduce any desired degree of depolarization regardless of the state of the input qubit.","Furthermore, our scheme does not rely on time- or spatial-averaging so that neither the measurement duration nor the measurement area affect the output quantum state.","In other words, our scheme achieves a truly observer-independent depolarizing quantum operation.","Figure: Experimental schematic for a fully-controllable input-independent depolarizing quantum operation for a photonic polarization qubit ρ\\rho .The depolarizing quantum operation is described as $\\mathcal {E}(\\rho ) = \\frac{p I}{2}+(1-p)\\rho ,$ where $p$ is the degree of decoherence ($0\\le p\\le 1$ ), $\\rho $ is the input state of the photonic polarization qubit, and $I$ is the two-dimensional identity matrix.","We implement the depolarizing operation $\\mathcal {E}(\\rho )$ by using a modified displaced Sagnac interferometer setup as shown in Fig.","REF .","The displaced Sagnac interferometer consists of a polarizing beam splitter (PBS) and two half-wave plates that can be oriented at an angle $\\theta $ (@$\\theta $ in Fig.","REF ).","Another half-wave plate fixed at 45$^\\circ $ at the output mode $B$ makes the polarization state of the two output modes $A$ and $B$ identical.","It is not difficult to see that the displaced Sagnac interferometer in Fig.","REF acts as a continuously variable non-polarizing beam splitter in which the output ratio $A:B=1-p:p$ can be linearly varied by setting the angle $\\theta $ of the two half-wave plates.","Note that the splitting parameter $p = \\sin ^{2}(2 \\theta )$ .","This is confirmed in the experiment as shown in Fig.","REF .","In the experiment, we prepared the single-photon polarization qubit $\\rho $ by using the heralded single-photon state generated from spontaneous parametric down-conversion (SPDC) in a 3 mm type-II BBO crystal.","The wavelengths of the pump and that of the SPDC photon are 405 nm and 810 nm.","The input polarization qubit was prepared in a pure state $\\rho = |\\psi \\rangle \\langle \\psi |$ by using a half-wave and a quarter-wave plate.","We then measured the count rates at the two single-photon detectors placed at the output ports $A$ and $B$ and the data are shown in Fig.","REF .","As expected, the data show the linear splitting ratio between two outputs $A$ and $B$ with the splitting parameter $p$ .","In Fig.","REF , we plot the averaged normalized outputs $A$ and $B$ for six input polarization states, $|H\\rangle $ (horizontal), $|V\\rangle $ (vertical), $|D\\rangle \\equiv (|H\\rangle + |V\\rangle )/\\sqrt{2}$ , $|A\\rangle \\equiv (|H\\rangle - |V\\rangle )/\\sqrt{2}$ , $|R\\rangle \\equiv (|H\\rangle - i|V\\rangle )/\\sqrt{2}$ , and $|L\\rangle \\equiv (|H\\rangle +i|V\\rangle )/\\sqrt{2}$ .","The normalized outputs $A$ and $B$ for each polarization state look almost identical to the averaged result shown in Fig.","REF .","We also point out that the linearity and the splitting ratio are very stable over time due to the Sagnac geometry.","Figure: Data showing the linear splitting ratio A:B=1-p:pA:B = 1-p:p where p=sin 2 (2θ)p = \\sin ^2(2\\theta ) for the displaced Sagnac.","Each data point represents the averaged normalized output of six input polarization states, |H〉|H\\rangle , |V〉|V\\rangle , |D〉|D\\rangle , |A〉|A\\rangle , |R〉|R\\rangle , and |L〉|L\\rangle .","The normalized output of each polarization state appears almost identical to the averaged result shown here.Previously reported continuously variable non-polarizing beam splitters by using a prism pair [17], a sapphire disk [18], and a phase grating [19] were all input-polarization dependent.","However, our scheme based on the displaced Sagnac interferometer is completely input-polarization independent as demonstrated in Fig.","REF .","To be sure that the displaced Sagnac splits the input probability amplitude into two spatial modes $A$ and $B$ without affecting the quantum state $\\rho $ , we have also performed the quantum state tomography of the input qubit as well as the output qubit in modes $A$ and $B$ [15].","We observed that the fidelity between the input and the output quantum states is better than $0.982 \\pm 0.003$ for all polarization qubit states we have tested.","Thus, the displaced Sagnac acts as a nearly ideal identity operation for the polarization qubit except that it diverts the amplitude into two different spatial modes $A$ and $B$ .","Figure: Experimental data.","As pp is increased, the qubit states become more mixed, hence moving toward the center of the Bloch sphere.","The outer sphere represents p=0p=0 (pure states) and the inner sphere representsp=0.5p=0.5.","The arrows point the data points (i.e., qubit states) on the inner sphere.Equation (1) clearly states that, to implement the depolarizing quantum operation $\\mathcal {E}(\\rho )$ , it is necessary to achieve a mixture of the input quantum state $\\rho $ and the unpolarized state $I/2$ with the weighting factors $1-p$ and $p$ , respectively.","Note now that the qubit states found at the outputs $A$ and $B$ are identical to the input $\\rho $ but with the probability amplitudes $1-p$ and $p$ , respectively.","Thus, we first couple the output mode $B$ into a 2-m long multi-mode fiber to transform a polarized input to a completely unpolarized state.","We then combine the output of the multi-mode fiber (after collimation) and that of $A$ at a beam splitter (BS).","The 2-m long multi-mode fiber ensures that the beam combination at BS is a completely incoherent process since the path length difference is orders of magnitude larger than the single-photon coherence time, which is on the order of hundreds of femtoseconds.","Therefore, the quantum state found at the two outputs of the BS is described precisely as $pI/2 + (1-p)\\rho $ , indicating that the the input state $\\rho $ has gone through the depolarizing quantum operation $\\mathcal {E}(\\rho )$ in eq. (1).","To demonstrate that the outputs of the BS indeed correspond to the quantum state after the depolarizing quantum operation, we performed quantum state tomography on the output states for six different input qubit states.","The experimental data are shown in Fig.","REF .","It is clear that, by increasing $p$ , the qubit states become more mixed, moving toward the center of the Bloch sphere.","It is important to note that the depolarizing quantum operation is an isotropic operation so that the output state purity $Tr[\\rho ^2]$ does not depend on the input state: it only depends on the choice $p$ .","As such, the depolarizing quantum operation should only shrink the size of the Bloch sphere, rather than making it asymmetric.","This feature is well demonstrated in Fig.","REF : all the data points corresponding to the depolarizing quantum operation of the same $p$ , regardless of the input state, should reside on the same sphere.","In Fig.","REF , the arrows represent the qubit states after the depolarizing quantum operation $\\mathcal {E}(\\rho )$ for $p=0.5$ and they all lie on the inner sphere representing all qubit states undergone $\\mathcal {E}(\\rho )$ with $p=0.5$ .","We also note that changing $p$ is quite easy in our setup as it requires only the rotation of the waveplates due to the relation $p=\\sin ^2(2\\theta )$ .","Figure: Real part of χ\\chi matrices of the depolarizing quantumchannel.","II refers to the two dimensional identity matrix.","XX, YY, and ZZ correspond to Pauli operators.The process fidelities (in comparison to the ideal operation) are (a) 0.966,(b) 0.994, (c) 0.998, and (d) 0.997.","The imaginary part of the χ\\chi matrices are almost zero, hence not shown.It is known that a quantum channel that implements a particular quantum operation can be fully characterized by performing quantum process tomography [16].","We have carried out quantum process tomography for the depolarizing quantum operation with various $p$ and the resulting $\\chi $ matrices are shown in Fig.","REF .","For a single-qubit operation as this one, it is often best to use the Pauli-basis for decomposing the quantum process as done in the figure.","Clearly, when $p=0$ , the quantum process corresponds to an identity operation as it should be, see Fig.","REF (a).","As $p$ is increased, contributions from the Pauli operations rise, see Fig.","REF (b) and REF (c), and when $p=1$ , it is clear that the output state will be a fully mixed state regardless of the input state, see Fig.","REF (d).","The high process fidelities for various $p$ values indicate the robustness of our setup to faithfully implement the fully controllable depolarizing quantum operation.","In summary, we have reported an experimental realization of a fully controllable depolarizing quantum operation for a single-photon polarization qubit.","Our scheme not only allows continuous adjustment of the degree of depolarization but also is independent of the input quantum state, as demonstrated with quantum state tomography and quantum process tomography.","A versatile depolarizing quantum channel like the one reported in this paper should find applications in many areas of photonic quantum information research, including generating exotic quantum states, studying the quantum noise processes, approximating non-physical quantum operations, etc.","This work was supported in part by the National Research Foundation of Korea (2009-0070668 and 2011-0021452)."]],"string":"[\n [\n \"Experimental implementation of a fully controllable depolarizing quantum\\n operation\"\n ],\n [\n \"Abstract The depolarizing quantum operation plays an important role in studying the quantum noise effect and implementing general quantum operations.\",\n \"In this work, we report a scheme which implements a fully controllable input-state independent depolarizing quantum operation for a photonic polarization qubit.\"\n ],\n [\n \"Experimental implementation of a fully controllable depolarizing quantum operation Youn-Chang Jeongw31400@gmail.com Jong-Chan Lee Yoon-Ho Kimyoonho72@gmail.com Department of Physics, Pohang University of Science and Technology (POSTECH), Pohang, 790-784, Korea The depolarizing quantum operation plays an important role in studying the quantum noise effect and implementing general quantum operations.\",\n \"In this work, we report a scheme which implements a fully controllable input-state independent depolarizing quantum operation for a photonic polarization qubit.\",\n \"03.67.Dd, 03.67.Hk, 42.79.Sz Depolarizing quantum operation, Quantum channel, Quantum noise One of the main challenges in experimental quantum information is to deal with decoherence.\",\n \"To preserve a qubit state, it is often necessary to isolate the qubit from the environment but to be able to do information processing tasks, interactions with an external (classical or quantum) system is necessary.\",\n \"More often than not, unwanted interactions with the environment leave the qubit in a different quantum state or even cause the qubit to lose coherence.\",\n \"Such unwanted quantum state transformation can be described by the quantum process due to a noisy quantum channel, in other words, a noisy quantum process, which can be understood with a few basic single-qubit noise operations including the bit-flip, phase-flip, depolarization, and amplitude damping [1].\",\n \"Understanding and being able to implement each of the basic quantum noise processes are important in theoretical and experimental quantum information science.\",\n \"First, it would allow us to simulate and quantify quantum noise processes.\",\n \"Second, the bit-flip and the phase-flip operations are essential for quantum error correction protocols.\",\n \"Third, the depolarization and the amplitude damping operations can describe decoherence.\",\n \"The depolarization operation is of particular interest because, in addition to being relevant to a number of practical quantum communication and computation scenarios [2], [3], [4], [5], [6], it is also an essential operation for optimally approximating non-physical quantum operations [7], [8] and for generating exotic quantum states including Werner states [9] and bound entangled states [10].\",\n \"Clearly, from the experimental point of view, it is important to develop a method to achieve a fully controllable input-independent depolarization quantum operation.\",\n \"As such, there have been many reports on experimental implementation of a depolarization quantum channel.\",\n \"In Ref.\",\n \"[11], optical scattering media were used to achieve the depolarization quantum channel but the scheme naturally induces the spread in the photon momenta and it is difficult to control the degree of depolarization in this scheme.\",\n \"A controllable depolarization channel was demonstrated in Ref.\",\n \"[12], [13] but the output was dependent on the input polarization and the scheme also introduced the momentum spread for the photon.\",\n \"Input-independent depolarizing quantum operation was demonstrated in Ref.\",\n \"[3], [14], but here the depolarizing operation was achieved by time averaging of many `fast' operations, i.e., incoherent sum of many different pure quantum states.\",\n \"In this paper, we report an experimental implementation of a fully-controllable depolarization quantum operation for a photonic polarization qubit.\",\n \"The scheme is completely input-state independent so that it is possible to introduce any desired degree of depolarization regardless of the state of the input qubit.\",\n \"Furthermore, our scheme does not rely on time- or spatial-averaging so that neither the measurement duration nor the measurement area affect the output quantum state.\",\n \"In other words, our scheme achieves a truly observer-independent depolarizing quantum operation.\",\n \"Figure: Experimental schematic for a fully-controllable input-independent depolarizing quantum operation for a photonic polarization qubit ρ\\\\rho .The depolarizing quantum operation is described as $\\\\mathcal {E}(\\\\rho ) = \\\\frac{p I}{2}+(1-p)\\\\rho ,$ where $p$ is the degree of decoherence ($0\\\\le p\\\\le 1$ ), $\\\\rho $ is the input state of the photonic polarization qubit, and $I$ is the two-dimensional identity matrix.\",\n \"We implement the depolarizing operation $\\\\mathcal {E}(\\\\rho )$ by using a modified displaced Sagnac interferometer setup as shown in Fig.\",\n \"REF .\",\n \"The displaced Sagnac interferometer consists of a polarizing beam splitter (PBS) and two half-wave plates that can be oriented at an angle $\\\\theta $ (@$\\\\theta $ in Fig.\",\n \"REF ).\",\n \"Another half-wave plate fixed at 45$^\\\\circ $ at the output mode $B$ makes the polarization state of the two output modes $A$ and $B$ identical.\",\n \"It is not difficult to see that the displaced Sagnac interferometer in Fig.\",\n \"REF acts as a continuously variable non-polarizing beam splitter in which the output ratio $A:B=1-p:p$ can be linearly varied by setting the angle $\\\\theta $ of the two half-wave plates.\",\n \"Note that the splitting parameter $p = \\\\sin ^{2}(2 \\\\theta )$ .\",\n \"This is confirmed in the experiment as shown in Fig.\",\n \"REF .\",\n \"In the experiment, we prepared the single-photon polarization qubit $\\\\rho $ by using the heralded single-photon state generated from spontaneous parametric down-conversion (SPDC) in a 3 mm type-II BBO crystal.\",\n \"The wavelengths of the pump and that of the SPDC photon are 405 nm and 810 nm.\",\n \"The input polarization qubit was prepared in a pure state $\\\\rho = |\\\\psi \\\\rangle \\\\langle \\\\psi |$ by using a half-wave and a quarter-wave plate.\",\n \"We then measured the count rates at the two single-photon detectors placed at the output ports $A$ and $B$ and the data are shown in Fig.\",\n \"REF .\",\n \"As expected, the data show the linear splitting ratio between two outputs $A$ and $B$ with the splitting parameter $p$ .\",\n \"In Fig.\",\n \"REF , we plot the averaged normalized outputs $A$ and $B$ for six input polarization states, $|H\\\\rangle $ (horizontal), $|V\\\\rangle $ (vertical), $|D\\\\rangle \\\\equiv (|H\\\\rangle + |V\\\\rangle )/\\\\sqrt{2}$ , $|A\\\\rangle \\\\equiv (|H\\\\rangle - |V\\\\rangle )/\\\\sqrt{2}$ , $|R\\\\rangle \\\\equiv (|H\\\\rangle - i|V\\\\rangle )/\\\\sqrt{2}$ , and $|L\\\\rangle \\\\equiv (|H\\\\rangle +i|V\\\\rangle )/\\\\sqrt{2}$ .\",\n \"The normalized outputs $A$ and $B$ for each polarization state look almost identical to the averaged result shown in Fig.\",\n \"REF .\",\n \"We also point out that the linearity and the splitting ratio are very stable over time due to the Sagnac geometry.\",\n \"Figure: Data showing the linear splitting ratio A:B=1-p:pA:B = 1-p:p where p=sin 2 (2θ)p = \\\\sin ^2(2\\\\theta ) for the displaced Sagnac.\",\n \"Each data point represents the averaged normalized output of six input polarization states, |H〉|H\\\\rangle , |V〉|V\\\\rangle , |D〉|D\\\\rangle , |A〉|A\\\\rangle , |R〉|R\\\\rangle , and |L〉|L\\\\rangle .\",\n \"The normalized output of each polarization state appears almost identical to the averaged result shown here.Previously reported continuously variable non-polarizing beam splitters by using a prism pair [17], a sapphire disk [18], and a phase grating [19] were all input-polarization dependent.\",\n \"However, our scheme based on the displaced Sagnac interferometer is completely input-polarization independent as demonstrated in Fig.\",\n \"REF .\",\n \"To be sure that the displaced Sagnac splits the input probability amplitude into two spatial modes $A$ and $B$ without affecting the quantum state $\\\\rho $ , we have also performed the quantum state tomography of the input qubit as well as the output qubit in modes $A$ and $B$ [15].\",\n \"We observed that the fidelity between the input and the output quantum states is better than $0.982 \\\\pm 0.003$ for all polarization qubit states we have tested.\",\n \"Thus, the displaced Sagnac acts as a nearly ideal identity operation for the polarization qubit except that it diverts the amplitude into two different spatial modes $A$ and $B$ .\",\n \"Figure: Experimental data.\",\n \"As pp is increased, the qubit states become more mixed, hence moving toward the center of the Bloch sphere.\",\n \"The outer sphere represents p=0p=0 (pure states) and the inner sphere representsp=0.5p=0.5.\",\n \"The arrows point the data points (i.e., qubit states) on the inner sphere.Equation (1) clearly states that, to implement the depolarizing quantum operation $\\\\mathcal {E}(\\\\rho )$ , it is necessary to achieve a mixture of the input quantum state $\\\\rho $ and the unpolarized state $I/2$ with the weighting factors $1-p$ and $p$ , respectively.\",\n \"Note now that the qubit states found at the outputs $A$ and $B$ are identical to the input $\\\\rho $ but with the probability amplitudes $1-p$ and $p$ , respectively.\",\n \"Thus, we first couple the output mode $B$ into a 2-m long multi-mode fiber to transform a polarized input to a completely unpolarized state.\",\n \"We then combine the output of the multi-mode fiber (after collimation) and that of $A$ at a beam splitter (BS).\",\n \"The 2-m long multi-mode fiber ensures that the beam combination at BS is a completely incoherent process since the path length difference is orders of magnitude larger than the single-photon coherence time, which is on the order of hundreds of femtoseconds.\",\n \"Therefore, the quantum state found at the two outputs of the BS is described precisely as $pI/2 + (1-p)\\\\rho $ , indicating that the the input state $\\\\rho $ has gone through the depolarizing quantum operation $\\\\mathcal {E}(\\\\rho )$ in eq. (1).\",\n \"To demonstrate that the outputs of the BS indeed correspond to the quantum state after the depolarizing quantum operation, we performed quantum state tomography on the output states for six different input qubit states.\",\n \"The experimental data are shown in Fig.\",\n \"REF .\",\n \"It is clear that, by increasing $p$ , the qubit states become more mixed, moving toward the center of the Bloch sphere.\",\n \"It is important to note that the depolarizing quantum operation is an isotropic operation so that the output state purity $Tr[\\\\rho ^2]$ does not depend on the input state: it only depends on the choice $p$ .\",\n \"As such, the depolarizing quantum operation should only shrink the size of the Bloch sphere, rather than making it asymmetric.\",\n \"This feature is well demonstrated in Fig.\",\n \"REF : all the data points corresponding to the depolarizing quantum operation of the same $p$ , regardless of the input state, should reside on the same sphere.\",\n \"In Fig.\",\n \"REF , the arrows represent the qubit states after the depolarizing quantum operation $\\\\mathcal {E}(\\\\rho )$ for $p=0.5$ and they all lie on the inner sphere representing all qubit states undergone $\\\\mathcal {E}(\\\\rho )$ with $p=0.5$ .\",\n \"We also note that changing $p$ is quite easy in our setup as it requires only the rotation of the waveplates due to the relation $p=\\\\sin ^2(2\\\\theta )$ .\",\n \"Figure: Real part of χ\\\\chi matrices of the depolarizing quantumchannel.\",\n \"II refers to the two dimensional identity matrix.\",\n \"XX, YY, and ZZ correspond to Pauli operators.The process fidelities (in comparison to the ideal operation) are (a) 0.966,(b) 0.994, (c) 0.998, and (d) 0.997.\",\n \"The imaginary part of the χ\\\\chi matrices are almost zero, hence not shown.It is known that a quantum channel that implements a particular quantum operation can be fully characterized by performing quantum process tomography [16].\",\n \"We have carried out quantum process tomography for the depolarizing quantum operation with various $p$ and the resulting $\\\\chi $ matrices are shown in Fig.\",\n \"REF .\",\n \"For a single-qubit operation as this one, it is often best to use the Pauli-basis for decomposing the quantum process as done in the figure.\",\n \"Clearly, when $p=0$ , the quantum process corresponds to an identity operation as it should be, see Fig.\",\n \"REF (a).\",\n \"As $p$ is increased, contributions from the Pauli operations rise, see Fig.\",\n \"REF (b) and REF (c), and when $p=1$ , it is clear that the output state will be a fully mixed state regardless of the input state, see Fig.\",\n \"REF (d).\",\n \"The high process fidelities for various $p$ values indicate the robustness of our setup to faithfully implement the fully controllable depolarizing quantum operation.\",\n \"In summary, we have reported an experimental realization of a fully controllable depolarizing quantum operation for a single-photon polarization qubit.\",\n \"Our scheme not only allows continuous adjustment of the degree of depolarization but also is independent of the input quantum state, as demonstrated with quantum state tomography and quantum process tomography.\",\n \"A versatile depolarizing quantum channel like the one reported in this paper should find applications in many areas of photonic quantum information research, including generating exotic quantum states, studying the quantum noise processes, approximating non-physical quantum operations, etc.\",\n \"This work was supported in part by the National Research Foundation of Korea (2009-0070668 and 2011-0021452).\"\n ]\n]"},"id":{"kind":"string","value":"1204.0850"}}},{"rowIdx":1866895,"cells":{"text":{"kind":"list like","value":[["Measurements of the branching fractions of the decays $B^{0}_{s} \\to\n D^{\\mp}_{s} K^{\\pm} $ and $B^{0}_{s} \\to D^{-}_{s} \\pi^{+}$"],["Abstract The decay mode $\\B^{0}_{s} \\to D^{\\mp}_{s} K^{\\pm} $ allows for one of the theoretically cleanest measurements of the CKM angle $\\gamma$ through the study of time-dependent $\\ensuremath{CP}\\xspace$ violation.","This paper reports a measurement of its branching fraction relative to the Cabibbo-favoured mode $\\B^{0}_{s} \\to D^{-}_{s} \\pi^{+}$ based on a data sample of 0.37 fb$^{-1}$ proton-proton collisions at $\\sqrt{s} = 7$ TeV collected in 2011 with the LHCb detector.","In addition, the ratio of $\\ensuremath{\\mathrm{B}}\\xspace$ meson production fractions $\\ensuremath{f_s/f_d}$, determined from semileptonic decays, together with the known branching fraction of the control channel $B^{0} \\to D^{-} \\pi^{+}$, is used to perform an absolute measurement of the branching fractions: $B (\\B^0_s \\to D^-_s \\pi^+) \\;= (2.95 \\pm 0.05 \\pm 0.17^{\\,+\\,0.18}_{\\,-\\,0.22}) \\times 10^{-3}$, $B (\\B^0_s \\to D^\\mp_s K^\\pm) = (1.90 \\pm 0.12 \\pm 0.13^{\\,+\\,0.12}_{\\,-\\,0.14}) \\times 10^{-4}\\,$, where the first uncertainty is statistical, the second the experimental systematic uncertainty, and the third the uncertainty due to $f_s/f_d$."],["Introduction","Unlike the flavour-specific decay $B ^0_s \\!\\rightarrow D ^-_s \\pi ^+ $ , the Cabibbo-suppressed decay $B ^0_s \\!\\rightarrow D_s^\\mp K^\\pm $ proceeds through two different tree-level amplitudes of similar strength: a $\\bar{b} \\rightarrow \\bar{c} u \\bar{s}$ transition leading to $B^0_s \\rightarrow D^-_s K^+$ and a $\\bar{b} \\rightarrow \\bar{u} c \\bar{s}$ transition leading to $B^0_s \\rightarrow D^+_sK^-$ .","These two decay amplitudes can have a large $C\\!P$ -violating interference via $B^0_s-\\bar{B}^0_s$ mixing, allowing the determination of the CKM angle $\\gamma $ with negligible theoretical uncertainties through the measurement of tagged and untagged time-dependent decay rates to both the $D^-_s K^+$ and $D^+_s K^-$ final states [1].","Although the $B ^0_s \\!\\rightarrow D_s^\\mp K^\\pm $ decay mode has been observed by the CDF [2] and BELLE [3] collaborations, only the LHCb experiment has both the necessary decay time resolution and access to large enough signal yields to perform the time-dependent $C\\!P$ measurement.","In this analysis, the $B ^0_s \\!\\rightarrow D_s^\\mp K^\\pm $ branching fraction is determined relative to $B ^0_s \\!\\rightarrow D ^-_s \\pi ^+ $ , and the absolute $B ^0_s \\!\\rightarrow D ^-_s \\pi ^+ $ branching fraction is determined using the known branching fraction of $B ^0 \\!\\rightarrow D ^- \\pi ^+ $ and the production fraction ratio $f_s/f_d$ [4].","The two measurements are then combined to obtain the absolute branching fraction of the decay $B ^0_s \\!\\rightarrow D_s^\\mp K^\\pm $ .","Charge conjugate modes are implied throughout.","Our notation $B ^0 \\!\\rightarrow D ^- \\pi ^+ $ , which matches that of Ref.","[5], encompasses both the Cabibbo-favoured $B^0 \\rightarrow D^-\\pi ^+$ mode and the doubly-Cabibbo-suppressed $B^0 \\rightarrow D^+\\pi ^-$ mode.","The LHCb detector [6] is a single-arm forward spectrometer covering the pseudo-rapidity range $2<\\eta <5$ , designed for studing particles containing $b $ or $c $ quarks.","The detector includes a high-precision tracking system consisting of a silicon-strip vertex detector surrounding the $pp$ interaction region, a large-area silicon-strip detector located upstream of a dipole magnet with a bending power of about $4{\\rm \\,Tm}$ , and three stations of silicon-strip detectors and straw drift tubes placed downstream.","The combined tracking system has a momentum resolution $\\Delta p/p$ that varies from 0.4% at 5${\\mathrm {\\,Ge\\hspace{-1.00006pt}V\\!/}c}$ to 0.6% at 100${\\mathrm {\\,Ge\\hspace{-1.00006pt}V\\!/}c}$ , an impact parameter resolution of 20$\\,\\rm m$ for tracks with high transverse momentum, and a decay time resolution of 50 fs.","Charged hadrons are identified using two ring-imaging Cherenkov detectors.","Photon, electron and hadron candidates are identified by a calorimeter system consisting of scintillating-pad and pre-shower detectors, an electromagnetic calorimeter, and a hadronic calorimeter.","Muons are identified by a muon system composed of alternating layers of iron and multiwire proportional chambers.","The LHCb trigger consists of a hardware stage, based on information from the calorimeter and muon systems, followed by a software stage which applies a full event reconstruction.","Two categories of events are recognised based on the hardware trigger decision.","The first category are events triggered by tracks from signal decays which have an associated cluster in the calorimeters, and the second category are events triggered independently of the signal decay particles.","Events which do not fall into either of these two categories are not used in the subsequent analysis.","The second, software, trigger stage requires a two-, three- or four-track secondary vertex with a large value of the scalar sum of the transverse momenta ($p_{\\rm T}$ ) of the tracks, and a significant displacement from the primary interaction.","At least one of the tracks used to form this vertex is required to have $p_{\\rm T}> 1.7$ GeV$/c$ , an impact parameter $\\chi ^2$ $>16$ , and a track fit $\\chi ^2 $ per degree of freedom $\\chi ^2/\\textrm {ndf}$ $< 2$ .","A multivariate algorithm is used for the identification of the secondary vertices [7].","Each input variable is binned to minimise the effect of systematic differences between the trigger behaviour on data and simulated events.","The samples of simulated events used in this analysis are based on the Pythia 6.4 generator [8], with a choice of parameters specifically configured for LHCb [9].","The EvtGen package [10] describes the decay of the $B $ mesons, and the Geant4 package [11] simulates the detector response.","QED radiative corrections are generated with the Photos package [12].","The analysis is based on a sample of $pp$ collisions corresponding to an integrated luminosity of 0.37 fb$^{-1}$ , collected at the LHC in 2011 at a centre-of-mass energy $\\sqrt{s} = 7$ TeV.","The decay modes $B ^0_s \\!\\rightarrow D ^-_s \\pi ^+ $ and $B ^0_s \\!\\rightarrow D_s^\\mp K^\\pm $ are topologically identical and are selected using identical geometric and kinematic criteria, thereby minimising efficiency corrections in the ratio of branching fractions.","The decay mode $B ^0 \\!\\rightarrow D ^- \\pi ^+ $ has a similar topology to the other two, differing only in the Dalitz plot structure of the $D $ decay and the lifetime of the $D $ meson.","These differences are verified, using simulated events, to alter the selection efficiency at the level of a few percent, and are taken into account.","$B ^0_s$ ($B ^0$ ) candidates are reconstructed from a $D ^-_s$ ($D ^-$ ) candidate and an additional pion or kaon (the “bachelor” particle), with the $D ^-_s$ ($D ^-$ ) meson decaying in the $K^+ K^- \\pi ^-$ ($K^+ \\pi ^-\\pi ^-$ ) mode.","All selection criteria will now be specified for the $B ^0_s$ decays, and are implied to be identical for the $B ^0$ decay unless explicitly stated otherwise.","All final-state particles are required to satisfy a track fit $\\chi ^2 / \\textrm {ndf} < 4$ and to have a high transverse momentum and a large impact parameter $\\chi ^2$ with respect to all primary vertices in the event.","In order to remove backgrounds which contain the same final-state particles as the signal decay, and therefore have the same mass lineshape, but do not proceed through the decay of a charmed meson, the flight distance $\\chi ^2$ of the $D ^-_s$ from the $B ^0_s$ is required to be larger than 2.","Only $D ^-_s$ and bachelor candidates forming a vertex with a $\\chi ^2 / \\textrm {ndf} < 9$ are considered as $B ^0_s$ candidates.","The same vertex quality criterion is applied to the $D ^-_s$ candidates.","The $B ^0_s$ candidate is further required to point to the primary vertex imposing $\\theta _{\\textrm {flight}} < 0.8$ degrees, where $\\theta _{\\textrm {flight}}$ is the angle between the candidate momentum vector and the line between the primary vertex and the $B ^0_s$ vertex.","The $B ^0_s$ candidates are also required to have a $\\chi ^2$ of their impact parameter with respect to the primary vertex less than 16.","Further suppression of combinatorial backgrounds is achieved using a gradient boosted decision tree technique [13] identical to the decision tree used in the previously published determination of $f_s/f_d$ with the hadronic decays [14].","The optimal working point is evaluated directly from a sub-sample of $B ^0_s \\!\\rightarrow D ^-_s \\pi ^+ $ events, corresponding to 10$\\%$ of the full dataset used, distributed evenly over the data taking period and selected using particle identification and trigger requirements.","The chosen figure of merit is the significance of the $B ^0_s \\!\\rightarrow D_s^\\mp K^\\pm $ signal, scaled according to the Cabibbo suppression relative to the $B ^0_s \\!\\rightarrow D ^-_s \\pi ^+ $ signal, with respect to the combinatorial background.","The significance exhibits a wide plateau around its maximum, and the optimal working point is chosen at the point in the plateau which maximizes the signal yield.","Multiple candidates occur in about $2\\%$ of the events and in such cases a single candidate is selected at random."],["Particle identification","Particle identification (PID) criteria serve two purposes in the selection of the three signal decays $B ^0 \\!\\rightarrow D ^- \\pi ^+ $ , $B ^0_s \\!\\rightarrow D ^-_s \\pi ^+ $ and $B ^0_s \\!\\rightarrow D_s^\\mp K^\\pm $ .","When applied to the decay products of the $D ^-_s$ or $D ^-$ , they suppress misidentified backgrounds which have the same bachelor particle as the signal mode under consideration, henceforth the “cross-feed” backgrounds.","When applied to the bachelor particle (pion or kaon) they separate the Cabibbo-favoured from the Cabibbo-suppressed decay modes.","All PID criteria are based on the differences in log-likelihood (DLL) between the kaon, proton, or pion hypotheses.","Their efficiencies are obtained from calibration samples of $D ^{*+} \\rightarrow (D ^0 \\rightarrow K^- \\pi ^+) \\pi ^+ $ and $焃 \\rightarrow p\\pi ^- $ signals, which are themselves selected without any PID requirements.","These samples are split according to the magnet polarity, binned in momentum and $p_{\\rm T}$ , and then reweighted to have the same momentum and $p_{\\rm T}$ distributions as the signal decays under study.","The selection of a pure $B ^0 \\!\\rightarrow D ^- \\pi ^+ $ sample can be accomplished with minimal PID requirements since all cross-feed backgrounds are less abundant than the signal.","The $\\overline{焃} ^0_b \\!\\rightarrow \\overline{焃} _c^- \\pi ^+ $ background is suppressed by requiring that both pions produced in the $D ^-$ decay satisfy $\\textrm {DLL}_{\\pi -p}>-10$ , and the $B ^0 \\!\\rightarrow D ^- K ^+ $ background is suppressed by requiring that the bachelor pion satisfies $\\textrm {DLL}_{K-\\pi }<0$ .","The selection of a pure $B ^0_s \\!\\rightarrow D ^-_s \\pi ^+ $ or $B ^0_s \\!\\rightarrow D_s^\\mp K^\\pm $ sample requires the suppression of the $B ^0 \\!\\rightarrow D ^- \\pi ^+ $ and $\\overline{焃} ^0_b \\!\\rightarrow \\overline{焃} _c^- \\pi ^+ $ backgrounds, whereas the combinatorial background contributes to a lesser extent.","The $D^{-}$ contamination in the $D^{-}_{s}$ data sample is reduced by requiring that the kaon which has the same charge as the pion in $D ^-_s \\rightarrow K^+ K^- \\pi ^- $ satisfies $\\textrm {DLL}_{K-\\pi }>5$ .","In addition, the other kaon is required to satisfy $\\textrm {DLL}_{K-\\pi }>0$ .","This helps to suppress combinatorial as well as doubly misidentified backgrounds.","For the same reason the pion is required to have $\\textrm {DLL}_{K-\\pi }<5$ .","The contamination of $\\overline{焃} ^0_b \\!\\rightarrow \\overline{焃} _c^- \\pi ^+ $ , $\\overline{焃} _c^- \\rightarrow \\overline{p } K^+ \\pi ^-$ is reduced by applying a requirement of $\\textrm {DLL}_{K-p}>0$ to the candidates that, when reconstructed under the $\\overline{焃} _c^- \\rightarrow \\overline{p } K^+ \\pi ^-$ mass hypothesis, lie within $\\pm 21$ MeV$/c^2$ of the $\\overline{焃} _c^- $ mass.","Because of its larger branching fraction, $B ^0_s \\!\\rightarrow D ^-_s \\pi ^+ $ is a significant background to $B ^0_s \\!\\rightarrow D_s^\\mp K^\\pm $ .","It is suppressed by demanding that the bachelor satisfies the criterion $\\textrm {DLL}_{K-\\pi }>5$ .","Conversely, a sample of $B ^0_s \\!\\rightarrow D ^-_s \\pi ^+ $ , free of $B ^0_s \\!\\rightarrow D_s^\\mp K^\\pm $ contamination, is obtained by requiring that the bachelor satisfies $\\textrm {DLL}_{K-\\pi }<0$ .","The efficiency and misidentification probabilities for the PID criterion used to select the bachelor, $D ^-$ , and $D ^-_s$ candidates are summarised in Table REF .","Table: PID efficiency and misidentification probabilities, separated according to the up (U) and down (D) magnetpolarities.","The first two lines refer to the bachelor track selection,the third line is the D - D ^- efficiency and the fourth the D s - D ^-_s efficiency.","Probabilitiesare obtained from the efficiencies in the D *+ D ^{*+} calibration sample,binned in momentum and p T p_{\\rm T}.","Only bachelor tracks with momentum below100 GeV/c/c are considered.","The uncertainties shown are the statistical uncertainties due tothe finite number of signal events in the PID calibration samples."],["Mass fits","The fits to the invariant mass distributions of the $B ^0_s \\!\\rightarrow D ^-_s \\pi ^+ $ and $B ^0_s \\!\\rightarrow D_s^\\mp K^\\pm $ candidates require knowledge of the signal and background shapes.","The signal lineshape is taken from a fit to simulated signal events which had the full trigger, reconstruction, and selection chain applied to them.","Various lineshape parameterisations have been examined.","The best fit to the simulated event distributions is obtained with the sum of two Crystal Ball functions [15] with a common peak position and width, and opposite side power-law tails.","Mass shifts in the signal peaks relative to world average values [5], arising from an imperfect detector alignment [16], are observed in the data and are accounted for.","A constraint on the $D ^-_s$ meson mass is used to improve the $B ^0_s$ mass resolution.","Three kinds of backgrounds need to be considered: fully reconstructed (misidentified) backgrounds, partially reconstructed backgrounds with or without misidentification (e.g.","$B ^0_s \\!\\rightarrow D ^{*-}_s K ^+ $ or $B ^0_s \\!\\rightarrow D ^-_s \\rho ^{+}$ ), and combinatorial backgrounds.","The three most important fully reconstructed backgrounds are $B ^0 \\!\\rightarrow D ^-_s K ^+ $ and $B ^0_s \\!\\rightarrow D ^-_s \\pi ^+ $ for $B ^0_s \\!\\rightarrow D_s^\\mp K^\\pm $ , and $B ^0 \\!\\rightarrow D ^- \\pi ^+ $ for $B ^0_s \\!\\rightarrow D ^-_s \\pi ^+ $ .","The mass distribution of the $B ^0 \\!\\rightarrow D ^- \\pi ^+ $ events does not suffer from fully reconstructed backgrounds.","In the case of the $B ^0 \\!\\rightarrow D ^-_s K ^+ $ decay, which is fully reconstructed under its own mass hypothesis, the signal shape is fixed to be the same as for $B ^0_s \\!\\rightarrow D_s^\\mp K^\\pm $ and the peak position is varied.","The shapes of the misidentified backgrounds $B ^0 \\!\\rightarrow D ^- \\pi ^+ $ and $B ^0_s \\!\\rightarrow D ^-_s \\pi ^+ $ are taken from data using a reweighting procedure.","First, a clean signal sample of $B ^0 \\!\\rightarrow D ^- \\pi ^+ $ and $B ^0_s \\!\\rightarrow D ^-_s \\pi ^+ $ decays is obtained by applying the PID selection for the bachelor track given in Sect. .","The invariant mass of these decays under the wrong mass hypothesis ($B ^0_s \\!\\rightarrow D ^-_s \\pi ^+ $ or $B ^0_s \\!\\rightarrow D_s^\\mp K^\\pm $ ) depends on the momentum of the misidentified particle.","This momentum distribution must therefore be reweighted by taking into account the momentum dependence of the misidentifaction rate.","This dependence is obtained using a dedicated calibration sample of prompt $D ^{*+}$ decays.","The mass distributions under the wrong mass hypothesis are then reweighted using this momentum distribution to obtain the $B ^0 \\!\\rightarrow D ^- \\pi ^+ $ and $B ^0_s \\!\\rightarrow D ^-_s \\pi ^+ $ mass shapes under the $B ^0_s \\!\\rightarrow D ^-_s \\pi ^+ $ and $B ^0_s \\!\\rightarrow D_s^\\mp K^\\pm $ mass hypotheses, respectively.","For partially reconstructed backgrounds, the probability density functions (PDFs) of the invariant mass distributions are taken from samples of simulated events generated in specific exclusive modes and are corrected for mass shifts, momentum spectra, and PID efficiencies in data.","The use of simulated events is justified by the observed good agreement between data and simulation.","The combinatorial background in the $B ^0_s \\!\\rightarrow D ^-_s \\pi ^+ $ and $B ^0 \\!\\rightarrow D ^- \\pi ^+ $ fits is modelled by an exponential function where the exponent is allowed to vary in the fit.","The resulting shape and normalisation of the combinatorial backgrounds are in agreement within one standard deviation with the distribution of a wrong-sign control sample (where the $D^{-}_{s}$ and the bachelor track have the same charges).","The shape of the combinatorial background in the $B ^0_s \\!\\rightarrow D_s^\\mp K^\\pm $ fit cannot be left free because of the partially reconstructed backgrounds which dominate in the mass region below the signal peak.","In this case, therefore, the combinatorial slope is fixed to be flat, as measured from the wrong sign events.","In the $B ^0_s \\!\\rightarrow D_s^\\mp K^\\pm $ fit, an additional complication arises due to backgrounds from $焃 ^0_b \\!\\rightarrow D ^-_s p$ and $焃 ^0_b \\!\\rightarrow D ^{*-}_s p$ , which fall in the signal region when misreconstructed.","To avoid a loss of $B ^0_s \\!\\rightarrow D_s^\\mp K^\\pm $ signal, no requirement is made on the $\\textrm {DLL}_{K-p}$ of the bachelor particle.","Instead, the $焃 ^0_b \\!\\rightarrow D ^-_s p$ mass shape is obtained from simulated $焃 ^0_b \\!\\rightarrow D ^-_s p$ decays, which are reweighted in momentum using the efficiency of the $\\textrm {DLL}_{K-\\pi }> 5$ requirement on protons.","The $焃 ^0_b \\!\\rightarrow D ^{*-}_s p$ mass shape is obtained by shifting the $焃 ^0_b \\!\\rightarrow D ^-_s p$ mass shape downwards by 200 MeV$/c^2$ .","The branching fractions of $焃 ^0_b \\!\\rightarrow D ^-_s p$ and $焃 ^0_b \\!\\rightarrow D ^{*-}_s p$ are assumed to be equal, motivated by the fact that the decays $B^0\\rightarrow D^-D_s^+$ and $B^0\\rightarrow D^-D_s^{*+}$ (dominated by similar tree topologies) have almost equal branching fractions.","Therefore the overall mass shape is formed by summing the $焃 ^0_b \\!\\rightarrow D ^-_s p$ and $焃 ^0_b \\!\\rightarrow D ^{*-}_s p$ shapes with equal weight.","The signal yields are obtained from unbinned extended maximum likelihood fits to the data.","In order to achieve the highest sensitivity, the sample is separated according to the two magnet polarities, allowing for possible differences in PID performance and in running conditions.","A simultaneous fit to the two magnet polarities is performed for each decay, with the peak position and width of each signal, as well as the combinatorial background shape, shared between the two.","The fit under the $B ^0_s \\!\\rightarrow D ^-_s \\pi ^+ $ hypothesis requires a description of the $B ^0 \\!\\rightarrow D ^- \\pi ^+ $ background.","A fit to the $B ^0 \\!\\rightarrow D ^- \\pi ^+ $ spectrum is first performed to determine the yield of signal $B ^0 \\!\\rightarrow D ^- \\pi ^+ $ events, shown in Fig.","REF .","The expected $B ^0 \\!\\rightarrow D ^- \\pi ^+ $ contribution under the $B ^0_s \\!\\rightarrow D ^-_s \\pi ^+ $ hypothesis is subsequently constrained with a $10\\%$ uncertainty to account for uncertainties on the PID efficiencies.","The fits to the $B ^0_s \\!\\rightarrow D ^-_s \\pi ^+ $ candidates are shown in Fig.","REF and the fit results for both decay modes are summarised in Table REF .","The peak position of the signal shape is varied, as are the yields of the different partially reconstructed backgrounds (except $B ^0 \\!\\rightarrow D ^- \\pi ^+ $ ) and the shape of the combinatorial background.","The width of the signal is fixed to the values found in the $B ^0 \\!\\rightarrow D ^- \\pi ^+ $ fit ($17.2$ MeV$/c^2$ ), scaled by the ratio of widths observed in simulated events between $B ^0 \\!\\rightarrow D ^- \\pi ^+ $ and $B ^0_s \\!\\rightarrow D ^-_s \\pi ^+ $ decays ($0.987$ ).","The accuracy of these fixed parameters is evaluated using ensembles of simulated experiments described in Sect. .","The yield of $B ^0 \\!\\rightarrow D ^-_s \\pi ^+ $ is fixed to be 2.9% of the $B ^0_s \\!\\rightarrow D ^-_s \\pi ^+ $ signal yield, based on the world average branching fraction of $B ^0 \\!\\rightarrow D ^-_s \\pi ^+ $ of $(2.16 \\pm 0.26) \\times 10^{-5}$ , the value of $f_s/f_d$ given in [4], and the value of the branching fraction computed in this paper.","The shape used to fit this component is the sum of two Crystal Ball functions obtained from the $B ^0_s \\!\\rightarrow D ^-_s \\pi ^+ $ sample with the peak position fixed to the value obtained with the fit of the $B ^0 \\!\\rightarrow D ^- \\pi ^+ $ data sample and the width fixed to the width of the $B ^0_s \\!\\rightarrow D ^-_s \\pi ^+ $ peak.","Figure: Mass distribution of the B 0 →D - π + B ^0 \\!\\rightarrow D ^- \\pi ^+ candidates (top) and B s 0 →D s - π + B ^0_s \\!\\rightarrow D ^-_s \\pi ^+ candidates (bottom).","The stacked background shapes follow the same top-to-bottom order in the legend and the plot.","For illustration purposes the plot includes events from both magnet polarities, but they are fitted separately as described in the text.The $\\overline{焃} ^0_b \\!\\rightarrow \\overline{焃} _c^- \\pi ^+ $ background is negligible in this fit owing to the effectiveness of the veto procedure described earlier.","Nevertheless, a $\\overline{焃} ^0_b \\!\\rightarrow \\overline{焃} _c^- \\pi ^+ $ component, whose yield is allowed to vary, is included in the fit (with the mass shape obtained using the reweighting procedure on simulated events described previously) and results in a negligible contribution, as expected.","Table: Results of the mass fits to the B 0 →D - π + B ^0 \\!\\rightarrow D ^- \\pi ^+ , B s 0 →D s - π + B ^0_s \\!\\rightarrow D ^-_s \\pi ^+ , and B s 0 →D s ∓ K ± B ^0_s \\!\\rightarrow D_s^\\mp K^\\pm candidates separated according to the up (U) and down (D) magnetpolarities.","In the B s 0 →D s ∓ K ± B ^0_s \\!\\rightarrow D_s^\\mp K^\\pm case,the number quoted for B s 0 →D s - π + B ^0_s \\!\\rightarrow D ^-_s \\pi ^+ also includes a smallnumber of B 0 →D - π + B ^0 \\!\\rightarrow D ^- \\pi ^+ events which have the same mass shape (20 events from the expected misidentification).See Table 3 for the constrained values used in the B s 0 →D s ∓ K ± B ^0_s \\!\\rightarrow D_s^\\mp K^\\pm decay fit for the partially reconstructed backgrounds and the B 0 →D - K + B ^0 \\!\\rightarrow D ^- K ^+ decay channel.The fits for the $B ^0_s \\!\\rightarrow D_s^\\mp K^\\pm $ candidates are shown in Fig.","REF and the fit results are collected in Table REF .","There are numerous reflections which contribute to the mass distribution.","The most important reflection is $B ^0_s \\!\\rightarrow D ^-_s \\pi ^+ $ , whose shape is taken from the earlier $B ^0_s \\!\\rightarrow D ^-_s \\pi ^+ $ signal fit, reweighted according to the efficiencies of the applied PID requirements.","Furthermore, the yield of the $B ^0 \\!\\rightarrow D ^- K ^+ $ reflection is constrained to the values in Table REF .","In addition, there is potential cross-feed from partially reconstructed modes with a misidentified pion such as $B ^0_s \\!\\rightarrow D ^-_s \\rho ^{+}$ , as well as several small contributions from partially reconstructed backgrounds with similar mass shapes.","The yields of these modes, whose branching fractions are known or can be estimated (e.g.","$B ^0_s \\!\\rightarrow D ^-_s \\rho ^{+}$ , $B ^0_s \\!\\rightarrow D ^-_s K^{*+}$ ), are constrained to the values in Table REF , based on criteria such as relative branching fractions and reconstruction efficiencies and PID probabilities.","An important cross-check is performed by comparing the fitted value of the yield of misidentified $B ^0_s \\!\\rightarrow D ^-_s \\pi ^+ $ events ($318\\pm 30$ ) to the yield expected from PID efficiencies ($370\\pm 11$ ) and an agreement is found.","Figure: Mass distribution of the B s 0 →D s ∓ K ± B ^0_s \\!\\rightarrow D_s^\\mp K^\\pm candidates.","The stacked background shapes follow the same top-to-bottom order in the legend and the plot.","For illustration purposes the plot includesevents from both magnet polarities, but they are fitted separately as described in the text.Table: Gaussian constraints on the yields of partially reconstructed and misidentified backgrounds applied in the B s 0 →D s ∓ K ± B ^0_s \\!\\rightarrow D_s^\\mp K^\\pm fit,separated according to the up (U) and down (D) magnet polarities."],["Systematic uncertainties","The major systematic uncertainities on the measurement of the relative branching fraction of $B ^0_s \\!\\rightarrow D_s^\\mp K^\\pm $ and $B ^0_s \\!\\rightarrow D ^-_s \\pi ^+ $ are related to the fit, PID calibration, and trigger and offline selection efficiency corrections.","Systematic uncertainties related to the fit are evaluated by generating large sets of simulated experiments using the nominal fit, and then fitting them with a model where certain parameters are varied.","To give two examples, the signal width is deliberately fixed to a value different from the width used in the generation, or the combinatorial background slope in the $B ^0_s \\!\\rightarrow D_s^\\mp K^\\pm $ fit is fixed to the combinatorial background slope found in the $B ^0_s \\!\\rightarrow D ^-_s \\pi ^+ $ fit.","The deviations of the peak position of the pull distributions from zero are then included in the systematic uncertainty.","Table: Relative systematic uncertainities on the branching fraction ratios.In the case of the $B ^0_s \\!\\rightarrow D_s^\\mp K^\\pm $ fit the presence of constraints for the partially reconstructed backgrounds must be considered.","The generic extended likelihood function can be written as $\\mathcal {L} = e^{-N} N^{N_{\\rm obs}}\\times \\prod _j G(N^j; N^j_c, \\sigma _{N^j_0})\\times \\prod _{i=1}^{N_{\\rm obs}} P(m_i; \\vec{\\lambda })\\,,$ where the first factor is the extended Poissonian likelihood in which $N$ is the total number of fitted events, given by the sum of the fitted component yields $N = \\sum _k N_k$ .","The fitted data sample contains $N_{\\rm obs}$ events.","The second factor is the product of the $j$ external constraints on the yields, $j 5$ .","The observed signal yields are corrected by the difference observed in the (non-PID) selection efficiencies of different modes as measured from simulated events: $\\epsilon (B ^0_s \\!\\rightarrow D ^-_s \\pi ^+ )/\\epsilon (B ^0 \\!\\rightarrow D ^- \\pi ^+ ) &=& 1.015\\;,\\\\\\epsilon (B ^0_s \\!\\rightarrow D ^-_s \\pi ^+ )/\\epsilon (B ^0_s \\!\\rightarrow D_s^\\mp K^\\pm ) &=& 1.061\\;.$ A systematic uncertainty is assigned on the ratio to account for percent level differences between the data and the simulation.","These are dominated by the simulation of the hardware trigger.","All sources of systematic uncertainty are summarized in Table REF ."],["Determination of the branching fractions","The $B ^0_s \\!\\rightarrow D_s^\\mp K^\\pm $ branching fraction relative to $B ^0_s \\!\\rightarrow D ^-_s \\pi ^+ $ is obtained by correcting the raw signal yields for PID and selection efficiency differences $\\frac{{\\cal B}\\left(B ^0_s \\!\\rightarrow D_s^\\mp K^\\pm \\right)}{{\\cal B}\\left(B ^0_s \\!\\rightarrow D ^-_s \\pi ^+ \\right)} =\\frac{N_{B ^0_s \\!\\rightarrow D_s^\\mp K^\\pm }}{N_{B ^0_s \\!\\rightarrow D ^-_s \\pi ^+ }}\\frac{\\epsilon ^{\\textrm {PID}}_{B ^0_s \\!\\rightarrow D ^-_s \\pi ^+ }}{\\epsilon ^{\\textrm {PID}}_{B ^0_s \\!\\rightarrow D_s^\\mp K^\\pm }}\\frac{\\epsilon ^{\\textrm {Sel}}_{B ^0_s \\!\\rightarrow D ^-_s \\pi ^+ }}{\\epsilon ^{\\textrm {Sel}}_{B ^0_s \\!\\rightarrow D_s^\\mp K^\\pm }}\\;,$ where $\\epsilon _{X}$ is the efficiency to reconstruct decay mode $X$ and $N_{X}$ is the number of observed events in this decay mode.","The PID efficiencies are given in Table REF , and the ratio of the two selection efficiencies is $0.943\\pm 0.013$ .","The ratio of the branching fractions of $B ^0_s \\!\\rightarrow D_s^\\mp K^\\pm $ relative to $B ^0_s \\!\\rightarrow D ^-_s \\pi ^+ $ is determined separately for the down ($0.0601\\pm 0.0056$ ) and up ($0.0694\\pm 0.0066$ ) magnet polarities and the two results are in good agreement.","The quoted errors are purely statistical.","The combined result is $\\frac{{\\cal B}\\left(B ^0_s \\!\\rightarrow D_s^\\mp K^\\pm \\right)}{{\\cal B}\\left(B ^0_s \\!\\rightarrow D ^-_s \\pi ^+ \\right)} = 0.0646\\pm 0.0043 \\pm 0.0025 \\;,$ where the first uncertainty is statistical and the second is the total systematic uncertainty from Table REF .","The relative yields of $B ^0_s \\!\\rightarrow D ^-_s \\pi ^+ $ and $B ^0 \\!\\rightarrow D ^- \\pi ^+ $ are used to extract the branching fraction of $B ^0_s \\!\\rightarrow D ^-_s \\pi ^+ $ from the following relation ${\\cal B}(B ^0_s \\!\\rightarrow D ^-_s \\pi ^+ )= {\\cal B}\\left(B ^0 \\!\\rightarrow D ^- \\pi ^+ \\right)\\frac{\\epsilon _{B ^0 \\!\\rightarrow D ^- \\pi ^+ }}{\\epsilon _{B ^0_s \\!\\rightarrow D ^-_s \\pi ^+ }}\\frac{N_{B ^0_s \\!\\rightarrow D ^-_s \\pi ^+ }{\\cal B}\\left(D^- \\rightarrow K^+ \\pi ^-\\pi ^-\\right)}{\\frac{f_s}{f_d}N_{B ^0 \\!\\rightarrow D ^- \\pi ^+ }{\\cal B}\\left(D^-_s \\rightarrow K^- K^+ \\pi ^-\\right)}\\;,$ using the recent $f_s/f_d$ measurement from semileptonic decays [4] $\\frac{f_s}{f_d}= 0.268 \\pm 0.008^{+0.022}_{-0.020}\\;,$ where the first uncertainty is statistical and the second systematic.","Only the semileptonic result is used since the hadronic determination of $f_s/f_d$ relies on theoretical assumptions about the ratio of the branching fractions of the $B ^0_s \\!\\rightarrow D ^-_s \\pi ^+ $ and $B ^0 \\!\\rightarrow D ^- \\pi ^+ $ decays.","In addition, the following world average values [5] for the $B$ and $D$ branching fractions are used ${\\cal B}(B ^0 \\!\\rightarrow D ^- \\pi ^+ ) &=& \\left(2.68\\pm 0.13\\right)\\times 10^{-3}\\;,\\\\{\\cal B}(D ^- \\rightarrow K^+ \\pi ^- \\pi ^-) &=& \\left(9.13\\pm 0.19\\right)\\times 10^{-2}\\;,\\\\{\\cal B}(D ^-_s \\rightarrow K^+ K^- \\pi ^-) &=& \\left(5.49\\pm 0.27\\right)\\times 10^{-2}\\;,$ leading to ${\\cal B}(B ^0_s \\!\\rightarrow D ^-_s \\pi ^+ ) &=& (2.95 \\pm 0.05 \\pm 0.17^{+0.18}_{-0.22})\\times 10^{-3}\\;,\\\\{\\cal B}(B ^0_s \\!\\rightarrow D_s^\\mp K^\\pm ) &=& (1.90 \\pm 0.12 \\pm 0.13^{+0.12}_{-0.14})\\times 10^{-4}\\;,$ where the first uncertainty is statistical, the second is the experimental systematics (as listed in Table REF ) plus the uncertainty arising from the $B ^0 \\!\\rightarrow D ^- \\pi ^+ $ branching fraction, and the third is the uncertainty (statistical and systematic) from the semileptonic $f_s/f_d$ measurement.","Both measurements are significantly more precise than the existing world averages [5]."],["Acknowledgments","We express our gratitude to our colleagues in the CERN accelerator departments for the excellent performance of the LHC.","We thank the technical and administrative staff at CERN and at the LHCb institutes, and acknowledge support from the National Agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); CERN; NSFC (China); CNRS/IN2P3 (France); BMBF, DFG, HGF and MPG (Germany); SFI (Ireland); INFN (Italy); FOM and NWO (The Netherlands); SCSR (Poland); ANCS (Romania); MinES of Russia and Rosatom (Russia); MICINN, XuntaGal and GENCAT (Spain); SNSF and SER (Switzerland); NAS Ukraine (Ukraine); STFC (United Kingdom); NSF (USA).","We also acknowledge the support received from the ERC under FP7 and the Region Auvergne.","tocsectionReferences 10 n subitem) R. Fleischer, New strategies to obtain insights into $CP$ violation through $B_s\\rightarrow D_s^{\\pm }K^\\mp ,D_s^{\\ast \\pm }K^\\mp , ...$ and $B_d\\rightarrow D^{\\pm }\\pi ^\\mp , D^{\\ast \\pm }\\pi ^\\mp , ...$ decays, Nucl.","Phys.","B671 (2003) 459, arXiv:hep-ph/0304027 CDF collaboration, T. Aaltonen et al., First observation of $\\bar{B}^0_s\\rightarrow D_s^\\pm K^\\mp $ and measurement of the ratio of branching fractions B$(\\bar{B}^0_s\\rightarrow D_s^\\pm K^\\mp )$ /B$(\\bar{B}^0_s\\rightarrow D_s^+ \\pi ^-)$, Phys.","Rev.","Lett.","103 (2009) 191802, arXiv:0809.0080 Belle collaboration, R. Louvot et al., Measurement of the decay $B_s^0\\rightarrow D_s^- \\pi ^{+}$ and evidence for $B_s^0 \\rightarrow D_s^\\mp K^\\pm $ in $e^+ e^-$ annihilation at $\\sqrt{s}\\sim 10.87$ GeV, Phys.","Rev.","Lett.","102 (2009) 021801, arXiv:0809.2526 LHCb collaboration, R. Aaij et al., Measurement of b hadron production fractions in 7 TeV pp collisions, Phys.","Rev.","D 85 (2012) 032008, arXiv:1111.2357 Particle Data Group, K. Nakamura et al., Review of particle physics, J. Phys.","G37 (2010) 075021 LHCb collaboration, A. A.","Alves Jr. et al., The LHCb detector at the LHC, JINST 3 (2008) S08005 V. V. Gligorov, C. Thomas, and M. Williams, The HLT inclusive $B$ triggers, LHCb-PUB-2011-016 T. Sjöstrand, S. Mrenna, and P. Skands, PYTHIA 6.4 Physics and manual, JHEP 05 (2006) 026, arXiv:hep-ph/0603175 M. Clemencic et al., The LHCb Simulation Application, Gauss: Design, Evolution and Experience, Journal of Physics: Conference Series 331 (2011), no.","3 032023 D. J. Lange, The EvtGen particle decay simulation package, Nucl.","Instrum.","Meth.","A462 (2001) 152 GEANT4 collaboration, S. Agostinelli et al., GEANT4: A simulation toolkit, Nucl.","Instrum.","Meth.","A506 (2003) 250 P. Golonka and Z.","Was, PHOTOS Monte Carlo: A Precision tool for QED corrections in Z and W decays, Eur.","Phys.","J. C45 (2006) 97, arXiv:hep-ph/0506026 A. Hoecker et al., TMVA: Toolkit for multivariate data analysis, PoS ACAT (2007) 040, arXiv:physics/0703039 LHCb collaboration, R. Aaij et al., Determination of ${f}_{s}/{f}_{d}$ for 7 TeV $pp$ collisions and measurement of the ${B}^{0}\\rightarrow {D}^{-}{K}^{+}$ branching fraction, Phys.","Rev.","Lett.","107 (2011) 211801, arXiv:1106.4435 T. Skwarnicki, A study of the radiative cascade transitions between the Upsilon-prime and Upsilon resonances.","PhD thesis, Institute of Nuclear Physics, Krakow, 1986, DESY-F31-86-02 LHCb collaboration, R. Aaij et al., Measurement of $b $ -hadron masses, Phys.","Lett.","B708 (2012) 241, arXiv:1112.4896"]],"string":"[\n [\n \"Measurements of the branching fractions of the decays $B^{0}_{s} \\\\to\\n D^{\\\\mp}_{s} K^{\\\\pm} $ and $B^{0}_{s} \\\\to D^{-}_{s} \\\\pi^{+}$\"\n ],\n [\n \"Abstract The decay mode $\\\\B^{0}_{s} \\\\to D^{\\\\mp}_{s} K^{\\\\pm} $ allows for one of the theoretically cleanest measurements of the CKM angle $\\\\gamma$ through the study of time-dependent $\\\\ensuremath{CP}\\\\xspace$ violation.\",\n \"This paper reports a measurement of its branching fraction relative to the Cabibbo-favoured mode $\\\\B^{0}_{s} \\\\to D^{-}_{s} \\\\pi^{+}$ based on a data sample of 0.37 fb$^{-1}$ proton-proton collisions at $\\\\sqrt{s} = 7$ TeV collected in 2011 with the LHCb detector.\",\n \"In addition, the ratio of $\\\\ensuremath{\\\\mathrm{B}}\\\\xspace$ meson production fractions $\\\\ensuremath{f_s/f_d}$, determined from semileptonic decays, together with the known branching fraction of the control channel $B^{0} \\\\to D^{-} \\\\pi^{+}$, is used to perform an absolute measurement of the branching fractions: $B (\\\\B^0_s \\\\to D^-_s \\\\pi^+) \\\\;= (2.95 \\\\pm 0.05 \\\\pm 0.17^{\\\\,+\\\\,0.18}_{\\\\,-\\\\,0.22}) \\\\times 10^{-3}$, $B (\\\\B^0_s \\\\to D^\\\\mp_s K^\\\\pm) = (1.90 \\\\pm 0.12 \\\\pm 0.13^{\\\\,+\\\\,0.12}_{\\\\,-\\\\,0.14}) \\\\times 10^{-4}\\\\,$, where the first uncertainty is statistical, the second the experimental systematic uncertainty, and the third the uncertainty due to $f_s/f_d$.\"\n ],\n [\n \"Introduction\",\n \"Unlike the flavour-specific decay $B ^0_s \\\\!\\\\rightarrow D ^-_s \\\\pi ^+ $ , the Cabibbo-suppressed decay $B ^0_s \\\\!\\\\rightarrow D_s^\\\\mp K^\\\\pm $ proceeds through two different tree-level amplitudes of similar strength: a $\\\\bar{b} \\\\rightarrow \\\\bar{c} u \\\\bar{s}$ transition leading to $B^0_s \\\\rightarrow D^-_s K^+$ and a $\\\\bar{b} \\\\rightarrow \\\\bar{u} c \\\\bar{s}$ transition leading to $B^0_s \\\\rightarrow D^+_sK^-$ .\",\n \"These two decay amplitudes can have a large $C\\\\!P$ -violating interference via $B^0_s-\\\\bar{B}^0_s$ mixing, allowing the determination of the CKM angle $\\\\gamma $ with negligible theoretical uncertainties through the measurement of tagged and untagged time-dependent decay rates to both the $D^-_s K^+$ and $D^+_s K^-$ final states [1].\",\n \"Although the $B ^0_s \\\\!\\\\rightarrow D_s^\\\\mp K^\\\\pm $ decay mode has been observed by the CDF [2] and BELLE [3] collaborations, only the LHCb experiment has both the necessary decay time resolution and access to large enough signal yields to perform the time-dependent $C\\\\!P$ measurement.\",\n \"In this analysis, the $B ^0_s \\\\!\\\\rightarrow D_s^\\\\mp K^\\\\pm $ branching fraction is determined relative to $B ^0_s \\\\!\\\\rightarrow D ^-_s \\\\pi ^+ $ , and the absolute $B ^0_s \\\\!\\\\rightarrow D ^-_s \\\\pi ^+ $ branching fraction is determined using the known branching fraction of $B ^0 \\\\!\\\\rightarrow D ^- \\\\pi ^+ $ and the production fraction ratio $f_s/f_d$ [4].\",\n \"The two measurements are then combined to obtain the absolute branching fraction of the decay $B ^0_s \\\\!\\\\rightarrow D_s^\\\\mp K^\\\\pm $ .\",\n \"Charge conjugate modes are implied throughout.\",\n \"Our notation $B ^0 \\\\!\\\\rightarrow D ^- \\\\pi ^+ $ , which matches that of Ref.\",\n \"[5], encompasses both the Cabibbo-favoured $B^0 \\\\rightarrow D^-\\\\pi ^+$ mode and the doubly-Cabibbo-suppressed $B^0 \\\\rightarrow D^+\\\\pi ^-$ mode.\",\n \"The LHCb detector [6] is a single-arm forward spectrometer covering the pseudo-rapidity range $2<\\\\eta <5$ , designed for studing particles containing $b $ or $c $ quarks.\",\n \"The detector includes a high-precision tracking system consisting of a silicon-strip vertex detector surrounding the $pp$ interaction region, a large-area silicon-strip detector located upstream of a dipole magnet with a bending power of about $4{\\\\rm \\\\,Tm}$ , and three stations of silicon-strip detectors and straw drift tubes placed downstream.\",\n \"The combined tracking system has a momentum resolution $\\\\Delta p/p$ that varies from 0.4% at 5${\\\\mathrm {\\\\,Ge\\\\hspace{-1.00006pt}V\\\\!/}c}$ to 0.6% at 100${\\\\mathrm {\\\\,Ge\\\\hspace{-1.00006pt}V\\\\!/}c}$ , an impact parameter resolution of 20$\\\\,\\\\rm m$ for tracks with high transverse momentum, and a decay time resolution of 50 fs.\",\n \"Charged hadrons are identified using two ring-imaging Cherenkov detectors.\",\n \"Photon, electron and hadron candidates are identified by a calorimeter system consisting of scintillating-pad and pre-shower detectors, an electromagnetic calorimeter, and a hadronic calorimeter.\",\n \"Muons are identified by a muon system composed of alternating layers of iron and multiwire proportional chambers.\",\n \"The LHCb trigger consists of a hardware stage, based on information from the calorimeter and muon systems, followed by a software stage which applies a full event reconstruction.\",\n \"Two categories of events are recognised based on the hardware trigger decision.\",\n \"The first category are events triggered by tracks from signal decays which have an associated cluster in the calorimeters, and the second category are events triggered independently of the signal decay particles.\",\n \"Events which do not fall into either of these two categories are not used in the subsequent analysis.\",\n \"The second, software, trigger stage requires a two-, three- or four-track secondary vertex with a large value of the scalar sum of the transverse momenta ($p_{\\\\rm T}$ ) of the tracks, and a significant displacement from the primary interaction.\",\n \"At least one of the tracks used to form this vertex is required to have $p_{\\\\rm T}> 1.7$ GeV$/c$ , an impact parameter $\\\\chi ^2$ $>16$ , and a track fit $\\\\chi ^2 $ per degree of freedom $\\\\chi ^2/\\\\textrm {ndf}$ $< 2$ .\",\n \"A multivariate algorithm is used for the identification of the secondary vertices [7].\",\n \"Each input variable is binned to minimise the effect of systematic differences between the trigger behaviour on data and simulated events.\",\n \"The samples of simulated events used in this analysis are based on the Pythia 6.4 generator [8], with a choice of parameters specifically configured for LHCb [9].\",\n \"The EvtGen package [10] describes the decay of the $B $ mesons, and the Geant4 package [11] simulates the detector response.\",\n \"QED radiative corrections are generated with the Photos package [12].\",\n \"The analysis is based on a sample of $pp$ collisions corresponding to an integrated luminosity of 0.37 fb$^{-1}$ , collected at the LHC in 2011 at a centre-of-mass energy $\\\\sqrt{s} = 7$ TeV.\",\n \"The decay modes $B ^0_s \\\\!\\\\rightarrow D ^-_s \\\\pi ^+ $ and $B ^0_s \\\\!\\\\rightarrow D_s^\\\\mp K^\\\\pm $ are topologically identical and are selected using identical geometric and kinematic criteria, thereby minimising efficiency corrections in the ratio of branching fractions.\",\n \"The decay mode $B ^0 \\\\!\\\\rightarrow D ^- \\\\pi ^+ $ has a similar topology to the other two, differing only in the Dalitz plot structure of the $D $ decay and the lifetime of the $D $ meson.\",\n \"These differences are verified, using simulated events, to alter the selection efficiency at the level of a few percent, and are taken into account.\",\n \"$B ^0_s$ ($B ^0$ ) candidates are reconstructed from a $D ^-_s$ ($D ^-$ ) candidate and an additional pion or kaon (the “bachelor” particle), with the $D ^-_s$ ($D ^-$ ) meson decaying in the $K^+ K^- \\\\pi ^-$ ($K^+ \\\\pi ^-\\\\pi ^-$ ) mode.\",\n \"All selection criteria will now be specified for the $B ^0_s$ decays, and are implied to be identical for the $B ^0$ decay unless explicitly stated otherwise.\",\n \"All final-state particles are required to satisfy a track fit $\\\\chi ^2 / \\\\textrm {ndf} < 4$ and to have a high transverse momentum and a large impact parameter $\\\\chi ^2$ with respect to all primary vertices in the event.\",\n \"In order to remove backgrounds which contain the same final-state particles as the signal decay, and therefore have the same mass lineshape, but do not proceed through the decay of a charmed meson, the flight distance $\\\\chi ^2$ of the $D ^-_s$ from the $B ^0_s$ is required to be larger than 2.\",\n \"Only $D ^-_s$ and bachelor candidates forming a vertex with a $\\\\chi ^2 / \\\\textrm {ndf} < 9$ are considered as $B ^0_s$ candidates.\",\n \"The same vertex quality criterion is applied to the $D ^-_s$ candidates.\",\n \"The $B ^0_s$ candidate is further required to point to the primary vertex imposing $\\\\theta _{\\\\textrm {flight}} < 0.8$ degrees, where $\\\\theta _{\\\\textrm {flight}}$ is the angle between the candidate momentum vector and the line between the primary vertex and the $B ^0_s$ vertex.\",\n \"The $B ^0_s$ candidates are also required to have a $\\\\chi ^2$ of their impact parameter with respect to the primary vertex less than 16.\",\n \"Further suppression of combinatorial backgrounds is achieved using a gradient boosted decision tree technique [13] identical to the decision tree used in the previously published determination of $f_s/f_d$ with the hadronic decays [14].\",\n \"The optimal working point is evaluated directly from a sub-sample of $B ^0_s \\\\!\\\\rightarrow D ^-_s \\\\pi ^+ $ events, corresponding to 10$\\\\%$ of the full dataset used, distributed evenly over the data taking period and selected using particle identification and trigger requirements.\",\n \"The chosen figure of merit is the significance of the $B ^0_s \\\\!\\\\rightarrow D_s^\\\\mp K^\\\\pm $ signal, scaled according to the Cabibbo suppression relative to the $B ^0_s \\\\!\\\\rightarrow D ^-_s \\\\pi ^+ $ signal, with respect to the combinatorial background.\",\n \"The significance exhibits a wide plateau around its maximum, and the optimal working point is chosen at the point in the plateau which maximizes the signal yield.\",\n \"Multiple candidates occur in about $2\\\\%$ of the events and in such cases a single candidate is selected at random.\"\n ],\n [\n \"Particle identification\",\n \"Particle identification (PID) criteria serve two purposes in the selection of the three signal decays $B ^0 \\\\!\\\\rightarrow D ^- \\\\pi ^+ $ , $B ^0_s \\\\!\\\\rightarrow D ^-_s \\\\pi ^+ $ and $B ^0_s \\\\!\\\\rightarrow D_s^\\\\mp K^\\\\pm $ .\",\n \"When applied to the decay products of the $D ^-_s$ or $D ^-$ , they suppress misidentified backgrounds which have the same bachelor particle as the signal mode under consideration, henceforth the “cross-feed” backgrounds.\",\n \"When applied to the bachelor particle (pion or kaon) they separate the Cabibbo-favoured from the Cabibbo-suppressed decay modes.\",\n \"All PID criteria are based on the differences in log-likelihood (DLL) between the kaon, proton, or pion hypotheses.\",\n \"Their efficiencies are obtained from calibration samples of $D ^{*+} \\\\rightarrow (D ^0 \\\\rightarrow K^- \\\\pi ^+) \\\\pi ^+ $ and $焃 \\\\rightarrow p\\\\pi ^- $ signals, which are themselves selected without any PID requirements.\",\n \"These samples are split according to the magnet polarity, binned in momentum and $p_{\\\\rm T}$ , and then reweighted to have the same momentum and $p_{\\\\rm T}$ distributions as the signal decays under study.\",\n \"The selection of a pure $B ^0 \\\\!\\\\rightarrow D ^- \\\\pi ^+ $ sample can be accomplished with minimal PID requirements since all cross-feed backgrounds are less abundant than the signal.\",\n \"The $\\\\overline{焃} ^0_b \\\\!\\\\rightarrow \\\\overline{焃} _c^- \\\\pi ^+ $ background is suppressed by requiring that both pions produced in the $D ^-$ decay satisfy $\\\\textrm {DLL}_{\\\\pi -p}>-10$ , and the $B ^0 \\\\!\\\\rightarrow D ^- K ^+ $ background is suppressed by requiring that the bachelor pion satisfies $\\\\textrm {DLL}_{K-\\\\pi }<0$ .\",\n \"The selection of a pure $B ^0_s \\\\!\\\\rightarrow D ^-_s \\\\pi ^+ $ or $B ^0_s \\\\!\\\\rightarrow D_s^\\\\mp K^\\\\pm $ sample requires the suppression of the $B ^0 \\\\!\\\\rightarrow D ^- \\\\pi ^+ $ and $\\\\overline{焃} ^0_b \\\\!\\\\rightarrow \\\\overline{焃} _c^- \\\\pi ^+ $ backgrounds, whereas the combinatorial background contributes to a lesser extent.\",\n \"The $D^{-}$ contamination in the $D^{-}_{s}$ data sample is reduced by requiring that the kaon which has the same charge as the pion in $D ^-_s \\\\rightarrow K^+ K^- \\\\pi ^- $ satisfies $\\\\textrm {DLL}_{K-\\\\pi }>5$ .\",\n \"In addition, the other kaon is required to satisfy $\\\\textrm {DLL}_{K-\\\\pi }>0$ .\",\n \"This helps to suppress combinatorial as well as doubly misidentified backgrounds.\",\n \"For the same reason the pion is required to have $\\\\textrm {DLL}_{K-\\\\pi }<5$ .\",\n \"The contamination of $\\\\overline{焃} ^0_b \\\\!\\\\rightarrow \\\\overline{焃} _c^- \\\\pi ^+ $ , $\\\\overline{焃} _c^- \\\\rightarrow \\\\overline{p } K^+ \\\\pi ^-$ is reduced by applying a requirement of $\\\\textrm {DLL}_{K-p}>0$ to the candidates that, when reconstructed under the $\\\\overline{焃} _c^- \\\\rightarrow \\\\overline{p } K^+ \\\\pi ^-$ mass hypothesis, lie within $\\\\pm 21$ MeV$/c^2$ of the $\\\\overline{焃} _c^- $ mass.\",\n \"Because of its larger branching fraction, $B ^0_s \\\\!\\\\rightarrow D ^-_s \\\\pi ^+ $ is a significant background to $B ^0_s \\\\!\\\\rightarrow D_s^\\\\mp K^\\\\pm $ .\",\n \"It is suppressed by demanding that the bachelor satisfies the criterion $\\\\textrm {DLL}_{K-\\\\pi }>5$ .\",\n \"Conversely, a sample of $B ^0_s \\\\!\\\\rightarrow D ^-_s \\\\pi ^+ $ , free of $B ^0_s \\\\!\\\\rightarrow D_s^\\\\mp K^\\\\pm $ contamination, is obtained by requiring that the bachelor satisfies $\\\\textrm {DLL}_{K-\\\\pi }<0$ .\",\n \"The efficiency and misidentification probabilities for the PID criterion used to select the bachelor, $D ^-$ , and $D ^-_s$ candidates are summarised in Table REF .\",\n \"Table: PID efficiency and misidentification probabilities, separated according to the up (U) and down (D) magnetpolarities.\",\n \"The first two lines refer to the bachelor track selection,the third line is the D - D ^- efficiency and the fourth the D s - D ^-_s efficiency.\",\n \"Probabilitiesare obtained from the efficiencies in the D *+ D ^{*+} calibration sample,binned in momentum and p T p_{\\\\rm T}.\",\n \"Only bachelor tracks with momentum below100 GeV/c/c are considered.\",\n \"The uncertainties shown are the statistical uncertainties due tothe finite number of signal events in the PID calibration samples.\"\n ],\n [\n \"Mass fits\",\n \"The fits to the invariant mass distributions of the $B ^0_s \\\\!\\\\rightarrow D ^-_s \\\\pi ^+ $ and $B ^0_s \\\\!\\\\rightarrow D_s^\\\\mp K^\\\\pm $ candidates require knowledge of the signal and background shapes.\",\n \"The signal lineshape is taken from a fit to simulated signal events which had the full trigger, reconstruction, and selection chain applied to them.\",\n \"Various lineshape parameterisations have been examined.\",\n \"The best fit to the simulated event distributions is obtained with the sum of two Crystal Ball functions [15] with a common peak position and width, and opposite side power-law tails.\",\n \"Mass shifts in the signal peaks relative to world average values [5], arising from an imperfect detector alignment [16], are observed in the data and are accounted for.\",\n \"A constraint on the $D ^-_s$ meson mass is used to improve the $B ^0_s$ mass resolution.\",\n \"Three kinds of backgrounds need to be considered: fully reconstructed (misidentified) backgrounds, partially reconstructed backgrounds with or without misidentification (e.g.\",\n \"$B ^0_s \\\\!\\\\rightarrow D ^{*-}_s K ^+ $ or $B ^0_s \\\\!\\\\rightarrow D ^-_s \\\\rho ^{+}$ ), and combinatorial backgrounds.\",\n \"The three most important fully reconstructed backgrounds are $B ^0 \\\\!\\\\rightarrow D ^-_s K ^+ $ and $B ^0_s \\\\!\\\\rightarrow D ^-_s \\\\pi ^+ $ for $B ^0_s \\\\!\\\\rightarrow D_s^\\\\mp K^\\\\pm $ , and $B ^0 \\\\!\\\\rightarrow D ^- \\\\pi ^+ $ for $B ^0_s \\\\!\\\\rightarrow D ^-_s \\\\pi ^+ $ .\",\n \"The mass distribution of the $B ^0 \\\\!\\\\rightarrow D ^- \\\\pi ^+ $ events does not suffer from fully reconstructed backgrounds.\",\n \"In the case of the $B ^0 \\\\!\\\\rightarrow D ^-_s K ^+ $ decay, which is fully reconstructed under its own mass hypothesis, the signal shape is fixed to be the same as for $B ^0_s \\\\!\\\\rightarrow D_s^\\\\mp K^\\\\pm $ and the peak position is varied.\",\n \"The shapes of the misidentified backgrounds $B ^0 \\\\!\\\\rightarrow D ^- \\\\pi ^+ $ and $B ^0_s \\\\!\\\\rightarrow D ^-_s \\\\pi ^+ $ are taken from data using a reweighting procedure.\",\n \"First, a clean signal sample of $B ^0 \\\\!\\\\rightarrow D ^- \\\\pi ^+ $ and $B ^0_s \\\\!\\\\rightarrow D ^-_s \\\\pi ^+ $ decays is obtained by applying the PID selection for the bachelor track given in Sect. .\",\n \"The invariant mass of these decays under the wrong mass hypothesis ($B ^0_s \\\\!\\\\rightarrow D ^-_s \\\\pi ^+ $ or $B ^0_s \\\\!\\\\rightarrow D_s^\\\\mp K^\\\\pm $ ) depends on the momentum of the misidentified particle.\",\n \"This momentum distribution must therefore be reweighted by taking into account the momentum dependence of the misidentifaction rate.\",\n \"This dependence is obtained using a dedicated calibration sample of prompt $D ^{*+}$ decays.\",\n \"The mass distributions under the wrong mass hypothesis are then reweighted using this momentum distribution to obtain the $B ^0 \\\\!\\\\rightarrow D ^- \\\\pi ^+ $ and $B ^0_s \\\\!\\\\rightarrow D ^-_s \\\\pi ^+ $ mass shapes under the $B ^0_s \\\\!\\\\rightarrow D ^-_s \\\\pi ^+ $ and $B ^0_s \\\\!\\\\rightarrow D_s^\\\\mp K^\\\\pm $ mass hypotheses, respectively.\",\n \"For partially reconstructed backgrounds, the probability density functions (PDFs) of the invariant mass distributions are taken from samples of simulated events generated in specific exclusive modes and are corrected for mass shifts, momentum spectra, and PID efficiencies in data.\",\n \"The use of simulated events is justified by the observed good agreement between data and simulation.\",\n \"The combinatorial background in the $B ^0_s \\\\!\\\\rightarrow D ^-_s \\\\pi ^+ $ and $B ^0 \\\\!\\\\rightarrow D ^- \\\\pi ^+ $ fits is modelled by an exponential function where the exponent is allowed to vary in the fit.\",\n \"The resulting shape and normalisation of the combinatorial backgrounds are in agreement within one standard deviation with the distribution of a wrong-sign control sample (where the $D^{-}_{s}$ and the bachelor track have the same charges).\",\n \"The shape of the combinatorial background in the $B ^0_s \\\\!\\\\rightarrow D_s^\\\\mp K^\\\\pm $ fit cannot be left free because of the partially reconstructed backgrounds which dominate in the mass region below the signal peak.\",\n \"In this case, therefore, the combinatorial slope is fixed to be flat, as measured from the wrong sign events.\",\n \"In the $B ^0_s \\\\!\\\\rightarrow D_s^\\\\mp K^\\\\pm $ fit, an additional complication arises due to backgrounds from $焃 ^0_b \\\\!\\\\rightarrow D ^-_s p$ and $焃 ^0_b \\\\!\\\\rightarrow D ^{*-}_s p$ , which fall in the signal region when misreconstructed.\",\n \"To avoid a loss of $B ^0_s \\\\!\\\\rightarrow D_s^\\\\mp K^\\\\pm $ signal, no requirement is made on the $\\\\textrm {DLL}_{K-p}$ of the bachelor particle.\",\n \"Instead, the $焃 ^0_b \\\\!\\\\rightarrow D ^-_s p$ mass shape is obtained from simulated $焃 ^0_b \\\\!\\\\rightarrow D ^-_s p$ decays, which are reweighted in momentum using the efficiency of the $\\\\textrm {DLL}_{K-\\\\pi }> 5$ requirement on protons.\",\n \"The $焃 ^0_b \\\\!\\\\rightarrow D ^{*-}_s p$ mass shape is obtained by shifting the $焃 ^0_b \\\\!\\\\rightarrow D ^-_s p$ mass shape downwards by 200 MeV$/c^2$ .\",\n \"The branching fractions of $焃 ^0_b \\\\!\\\\rightarrow D ^-_s p$ and $焃 ^0_b \\\\!\\\\rightarrow D ^{*-}_s p$ are assumed to be equal, motivated by the fact that the decays $B^0\\\\rightarrow D^-D_s^+$ and $B^0\\\\rightarrow D^-D_s^{*+}$ (dominated by similar tree topologies) have almost equal branching fractions.\",\n \"Therefore the overall mass shape is formed by summing the $焃 ^0_b \\\\!\\\\rightarrow D ^-_s p$ and $焃 ^0_b \\\\!\\\\rightarrow D ^{*-}_s p$ shapes with equal weight.\",\n \"The signal yields are obtained from unbinned extended maximum likelihood fits to the data.\",\n \"In order to achieve the highest sensitivity, the sample is separated according to the two magnet polarities, allowing for possible differences in PID performance and in running conditions.\",\n \"A simultaneous fit to the two magnet polarities is performed for each decay, with the peak position and width of each signal, as well as the combinatorial background shape, shared between the two.\",\n \"The fit under the $B ^0_s \\\\!\\\\rightarrow D ^-_s \\\\pi ^+ $ hypothesis requires a description of the $B ^0 \\\\!\\\\rightarrow D ^- \\\\pi ^+ $ background.\",\n \"A fit to the $B ^0 \\\\!\\\\rightarrow D ^- \\\\pi ^+ $ spectrum is first performed to determine the yield of signal $B ^0 \\\\!\\\\rightarrow D ^- \\\\pi ^+ $ events, shown in Fig.\",\n \"REF .\",\n \"The expected $B ^0 \\\\!\\\\rightarrow D ^- \\\\pi ^+ $ contribution under the $B ^0_s \\\\!\\\\rightarrow D ^-_s \\\\pi ^+ $ hypothesis is subsequently constrained with a $10\\\\%$ uncertainty to account for uncertainties on the PID efficiencies.\",\n \"The fits to the $B ^0_s \\\\!\\\\rightarrow D ^-_s \\\\pi ^+ $ candidates are shown in Fig.\",\n \"REF and the fit results for both decay modes are summarised in Table REF .\",\n \"The peak position of the signal shape is varied, as are the yields of the different partially reconstructed backgrounds (except $B ^0 \\\\!\\\\rightarrow D ^- \\\\pi ^+ $ ) and the shape of the combinatorial background.\",\n \"The width of the signal is fixed to the values found in the $B ^0 \\\\!\\\\rightarrow D ^- \\\\pi ^+ $ fit ($17.2$ MeV$/c^2$ ), scaled by the ratio of widths observed in simulated events between $B ^0 \\\\!\\\\rightarrow D ^- \\\\pi ^+ $ and $B ^0_s \\\\!\\\\rightarrow D ^-_s \\\\pi ^+ $ decays ($0.987$ ).\",\n \"The accuracy of these fixed parameters is evaluated using ensembles of simulated experiments described in Sect. .\",\n \"The yield of $B ^0 \\\\!\\\\rightarrow D ^-_s \\\\pi ^+ $ is fixed to be 2.9% of the $B ^0_s \\\\!\\\\rightarrow D ^-_s \\\\pi ^+ $ signal yield, based on the world average branching fraction of $B ^0 \\\\!\\\\rightarrow D ^-_s \\\\pi ^+ $ of $(2.16 \\\\pm 0.26) \\\\times 10^{-5}$ , the value of $f_s/f_d$ given in [4], and the value of the branching fraction computed in this paper.\",\n \"The shape used to fit this component is the sum of two Crystal Ball functions obtained from the $B ^0_s \\\\!\\\\rightarrow D ^-_s \\\\pi ^+ $ sample with the peak position fixed to the value obtained with the fit of the $B ^0 \\\\!\\\\rightarrow D ^- \\\\pi ^+ $ data sample and the width fixed to the width of the $B ^0_s \\\\!\\\\rightarrow D ^-_s \\\\pi ^+ $ peak.\",\n \"Figure: Mass distribution of the B 0 →D - π + B ^0 \\\\!\\\\rightarrow D ^- \\\\pi ^+ candidates (top) and B s 0 →D s - π + B ^0_s \\\\!\\\\rightarrow D ^-_s \\\\pi ^+ candidates (bottom).\",\n \"The stacked background shapes follow the same top-to-bottom order in the legend and the plot.\",\n \"For illustration purposes the plot includes events from both magnet polarities, but they are fitted separately as described in the text.The $\\\\overline{焃} ^0_b \\\\!\\\\rightarrow \\\\overline{焃} _c^- \\\\pi ^+ $ background is negligible in this fit owing to the effectiveness of the veto procedure described earlier.\",\n \"Nevertheless, a $\\\\overline{焃} ^0_b \\\\!\\\\rightarrow \\\\overline{焃} _c^- \\\\pi ^+ $ component, whose yield is allowed to vary, is included in the fit (with the mass shape obtained using the reweighting procedure on simulated events described previously) and results in a negligible contribution, as expected.\",\n \"Table: Results of the mass fits to the B 0 →D - π + B ^0 \\\\!\\\\rightarrow D ^- \\\\pi ^+ , B s 0 →D s - π + B ^0_s \\\\!\\\\rightarrow D ^-_s \\\\pi ^+ , and B s 0 →D s ∓ K ± B ^0_s \\\\!\\\\rightarrow D_s^\\\\mp K^\\\\pm candidates separated according to the up (U) and down (D) magnetpolarities.\",\n \"In the B s 0 →D s ∓ K ± B ^0_s \\\\!\\\\rightarrow D_s^\\\\mp K^\\\\pm case,the number quoted for B s 0 →D s - π + B ^0_s \\\\!\\\\rightarrow D ^-_s \\\\pi ^+ also includes a smallnumber of B 0 →D - π + B ^0 \\\\!\\\\rightarrow D ^- \\\\pi ^+ events which have the same mass shape (20 events from the expected misidentification).See Table 3 for the constrained values used in the B s 0 →D s ∓ K ± B ^0_s \\\\!\\\\rightarrow D_s^\\\\mp K^\\\\pm decay fit for the partially reconstructed backgrounds and the B 0 →D - K + B ^0 \\\\!\\\\rightarrow D ^- K ^+ decay channel.The fits for the $B ^0_s \\\\!\\\\rightarrow D_s^\\\\mp K^\\\\pm $ candidates are shown in Fig.\",\n \"REF and the fit results are collected in Table REF .\",\n \"There are numerous reflections which contribute to the mass distribution.\",\n \"The most important reflection is $B ^0_s \\\\!\\\\rightarrow D ^-_s \\\\pi ^+ $ , whose shape is taken from the earlier $B ^0_s \\\\!\\\\rightarrow D ^-_s \\\\pi ^+ $ signal fit, reweighted according to the efficiencies of the applied PID requirements.\",\n \"Furthermore, the yield of the $B ^0 \\\\!\\\\rightarrow D ^- K ^+ $ reflection is constrained to the values in Table REF .\",\n \"In addition, there is potential cross-feed from partially reconstructed modes with a misidentified pion such as $B ^0_s \\\\!\\\\rightarrow D ^-_s \\\\rho ^{+}$ , as well as several small contributions from partially reconstructed backgrounds with similar mass shapes.\",\n \"The yields of these modes, whose branching fractions are known or can be estimated (e.g.\",\n \"$B ^0_s \\\\!\\\\rightarrow D ^-_s \\\\rho ^{+}$ , $B ^0_s \\\\!\\\\rightarrow D ^-_s K^{*+}$ ), are constrained to the values in Table REF , based on criteria such as relative branching fractions and reconstruction efficiencies and PID probabilities.\",\n \"An important cross-check is performed by comparing the fitted value of the yield of misidentified $B ^0_s \\\\!\\\\rightarrow D ^-_s \\\\pi ^+ $ events ($318\\\\pm 30$ ) to the yield expected from PID efficiencies ($370\\\\pm 11$ ) and an agreement is found.\",\n \"Figure: Mass distribution of the B s 0 →D s ∓ K ± B ^0_s \\\\!\\\\rightarrow D_s^\\\\mp K^\\\\pm candidates.\",\n \"The stacked background shapes follow the same top-to-bottom order in the legend and the plot.\",\n \"For illustration purposes the plot includesevents from both magnet polarities, but they are fitted separately as described in the text.Table: Gaussian constraints on the yields of partially reconstructed and misidentified backgrounds applied in the B s 0 →D s ∓ K ± B ^0_s \\\\!\\\\rightarrow D_s^\\\\mp K^\\\\pm fit,separated according to the up (U) and down (D) magnet polarities.\"\n ],\n [\n \"Systematic uncertainties\",\n \"The major systematic uncertainities on the measurement of the relative branching fraction of $B ^0_s \\\\!\\\\rightarrow D_s^\\\\mp K^\\\\pm $ and $B ^0_s \\\\!\\\\rightarrow D ^-_s \\\\pi ^+ $ are related to the fit, PID calibration, and trigger and offline selection efficiency corrections.\",\n \"Systematic uncertainties related to the fit are evaluated by generating large sets of simulated experiments using the nominal fit, and then fitting them with a model where certain parameters are varied.\",\n \"To give two examples, the signal width is deliberately fixed to a value different from the width used in the generation, or the combinatorial background slope in the $B ^0_s \\\\!\\\\rightarrow D_s^\\\\mp K^\\\\pm $ fit is fixed to the combinatorial background slope found in the $B ^0_s \\\\!\\\\rightarrow D ^-_s \\\\pi ^+ $ fit.\",\n \"The deviations of the peak position of the pull distributions from zero are then included in the systematic uncertainty.\",\n \"Table: Relative systematic uncertainities on the branching fraction ratios.In the case of the $B ^0_s \\\\!\\\\rightarrow D_s^\\\\mp K^\\\\pm $ fit the presence of constraints for the partially reconstructed backgrounds must be considered.\",\n \"The generic extended likelihood function can be written as $\\\\mathcal {L} = e^{-N} N^{N_{\\\\rm obs}}\\\\times \\\\prod _j G(N^j; N^j_c, \\\\sigma _{N^j_0})\\\\times \\\\prod _{i=1}^{N_{\\\\rm obs}} P(m_i; \\\\vec{\\\\lambda })\\\\,,$ where the first factor is the extended Poissonian likelihood in which $N$ is the total number of fitted events, given by the sum of the fitted component yields $N = \\\\sum _k N_k$ .\",\n \"The fitted data sample contains $N_{\\\\rm obs}$ events.\",\n \"The second factor is the product of the $j$ external constraints on the yields, $j 5$ .\",\n \"The observed signal yields are corrected by the difference observed in the (non-PID) selection efficiencies of different modes as measured from simulated events: $\\\\epsilon (B ^0_s \\\\!\\\\rightarrow D ^-_s \\\\pi ^+ )/\\\\epsilon (B ^0 \\\\!\\\\rightarrow D ^- \\\\pi ^+ ) &=& 1.015\\\\;,\\\\\\\\\\\\epsilon (B ^0_s \\\\!\\\\rightarrow D ^-_s \\\\pi ^+ )/\\\\epsilon (B ^0_s \\\\!\\\\rightarrow D_s^\\\\mp K^\\\\pm ) &=& 1.061\\\\;.$ A systematic uncertainty is assigned on the ratio to account for percent level differences between the data and the simulation.\",\n \"These are dominated by the simulation of the hardware trigger.\",\n \"All sources of systematic uncertainty are summarized in Table REF .\"\n ],\n [\n \"Determination of the branching fractions\",\n \"The $B ^0_s \\\\!\\\\rightarrow D_s^\\\\mp K^\\\\pm $ branching fraction relative to $B ^0_s \\\\!\\\\rightarrow D ^-_s \\\\pi ^+ $ is obtained by correcting the raw signal yields for PID and selection efficiency differences $\\\\frac{{\\\\cal B}\\\\left(B ^0_s \\\\!\\\\rightarrow D_s^\\\\mp K^\\\\pm \\\\right)}{{\\\\cal B}\\\\left(B ^0_s \\\\!\\\\rightarrow D ^-_s \\\\pi ^+ \\\\right)} =\\\\frac{N_{B ^0_s \\\\!\\\\rightarrow D_s^\\\\mp K^\\\\pm }}{N_{B ^0_s \\\\!\\\\rightarrow D ^-_s \\\\pi ^+ }}\\\\frac{\\\\epsilon ^{\\\\textrm {PID}}_{B ^0_s \\\\!\\\\rightarrow D ^-_s \\\\pi ^+ }}{\\\\epsilon ^{\\\\textrm {PID}}_{B ^0_s \\\\!\\\\rightarrow D_s^\\\\mp K^\\\\pm }}\\\\frac{\\\\epsilon ^{\\\\textrm {Sel}}_{B ^0_s \\\\!\\\\rightarrow D ^-_s \\\\pi ^+ }}{\\\\epsilon ^{\\\\textrm {Sel}}_{B ^0_s \\\\!\\\\rightarrow D_s^\\\\mp K^\\\\pm }}\\\\;,$ where $\\\\epsilon _{X}$ is the efficiency to reconstruct decay mode $X$ and $N_{X}$ is the number of observed events in this decay mode.\",\n \"The PID efficiencies are given in Table REF , and the ratio of the two selection efficiencies is $0.943\\\\pm 0.013$ .\",\n \"The ratio of the branching fractions of $B ^0_s \\\\!\\\\rightarrow D_s^\\\\mp K^\\\\pm $ relative to $B ^0_s \\\\!\\\\rightarrow D ^-_s \\\\pi ^+ $ is determined separately for the down ($0.0601\\\\pm 0.0056$ ) and up ($0.0694\\\\pm 0.0066$ ) magnet polarities and the two results are in good agreement.\",\n \"The quoted errors are purely statistical.\",\n \"The combined result is $\\\\frac{{\\\\cal B}\\\\left(B ^0_s \\\\!\\\\rightarrow D_s^\\\\mp K^\\\\pm \\\\right)}{{\\\\cal B}\\\\left(B ^0_s \\\\!\\\\rightarrow D ^-_s \\\\pi ^+ \\\\right)} = 0.0646\\\\pm 0.0043 \\\\pm 0.0025 \\\\;,$ where the first uncertainty is statistical and the second is the total systematic uncertainty from Table REF .\",\n \"The relative yields of $B ^0_s \\\\!\\\\rightarrow D ^-_s \\\\pi ^+ $ and $B ^0 \\\\!\\\\rightarrow D ^- \\\\pi ^+ $ are used to extract the branching fraction of $B ^0_s \\\\!\\\\rightarrow D ^-_s \\\\pi ^+ $ from the following relation ${\\\\cal B}(B ^0_s \\\\!\\\\rightarrow D ^-_s \\\\pi ^+ )= {\\\\cal B}\\\\left(B ^0 \\\\!\\\\rightarrow D ^- \\\\pi ^+ \\\\right)\\\\frac{\\\\epsilon _{B ^0 \\\\!\\\\rightarrow D ^- \\\\pi ^+ }}{\\\\epsilon _{B ^0_s \\\\!\\\\rightarrow D ^-_s \\\\pi ^+ }}\\\\frac{N_{B ^0_s \\\\!\\\\rightarrow D ^-_s \\\\pi ^+ }{\\\\cal B}\\\\left(D^- \\\\rightarrow K^+ \\\\pi ^-\\\\pi ^-\\\\right)}{\\\\frac{f_s}{f_d}N_{B ^0 \\\\!\\\\rightarrow D ^- \\\\pi ^+ }{\\\\cal B}\\\\left(D^-_s \\\\rightarrow K^- K^+ \\\\pi ^-\\\\right)}\\\\;,$ using the recent $f_s/f_d$ measurement from semileptonic decays [4] $\\\\frac{f_s}{f_d}= 0.268 \\\\pm 0.008^{+0.022}_{-0.020}\\\\;,$ where the first uncertainty is statistical and the second systematic.\",\n \"Only the semileptonic result is used since the hadronic determination of $f_s/f_d$ relies on theoretical assumptions about the ratio of the branching fractions of the $B ^0_s \\\\!\\\\rightarrow D ^-_s \\\\pi ^+ $ and $B ^0 \\\\!\\\\rightarrow D ^- \\\\pi ^+ $ decays.\",\n \"In addition, the following world average values [5] for the $B$ and $D$ branching fractions are used ${\\\\cal B}(B ^0 \\\\!\\\\rightarrow D ^- \\\\pi ^+ ) &=& \\\\left(2.68\\\\pm 0.13\\\\right)\\\\times 10^{-3}\\\\;,\\\\\\\\{\\\\cal B}(D ^- \\\\rightarrow K^+ \\\\pi ^- \\\\pi ^-) &=& \\\\left(9.13\\\\pm 0.19\\\\right)\\\\times 10^{-2}\\\\;,\\\\\\\\{\\\\cal B}(D ^-_s \\\\rightarrow K^+ K^- \\\\pi ^-) &=& \\\\left(5.49\\\\pm 0.27\\\\right)\\\\times 10^{-2}\\\\;,$ leading to ${\\\\cal B}(B ^0_s \\\\!\\\\rightarrow D ^-_s \\\\pi ^+ ) &=& (2.95 \\\\pm 0.05 \\\\pm 0.17^{+0.18}_{-0.22})\\\\times 10^{-3}\\\\;,\\\\\\\\{\\\\cal B}(B ^0_s \\\\!\\\\rightarrow D_s^\\\\mp K^\\\\pm ) &=& (1.90 \\\\pm 0.12 \\\\pm 0.13^{+0.12}_{-0.14})\\\\times 10^{-4}\\\\;,$ where the first uncertainty is statistical, the second is the experimental systematics (as listed in Table REF ) plus the uncertainty arising from the $B ^0 \\\\!\\\\rightarrow D ^- \\\\pi ^+ $ branching fraction, and the third is the uncertainty (statistical and systematic) from the semileptonic $f_s/f_d$ measurement.\",\n \"Both measurements are significantly more precise than the existing world averages [5].\"\n ],\n [\n \"Acknowledgments\",\n \"We express our gratitude to our colleagues in the CERN accelerator departments for the excellent performance of the LHC.\",\n \"We thank the technical and administrative staff at CERN and at the LHCb institutes, and acknowledge support from the National Agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); CERN; NSFC (China); CNRS/IN2P3 (France); BMBF, DFG, HGF and MPG (Germany); SFI (Ireland); INFN (Italy); FOM and NWO (The Netherlands); SCSR (Poland); ANCS (Romania); MinES of Russia and Rosatom (Russia); MICINN, XuntaGal and GENCAT (Spain); SNSF and SER (Switzerland); NAS Ukraine (Ukraine); STFC (United Kingdom); NSF (USA).\",\n \"We also acknowledge the support received from the ERC under FP7 and the Region Auvergne.\",\n \"tocsectionReferences 10 n subitem) R. Fleischer, New strategies to obtain insights into $CP$ violation through $B_s\\\\rightarrow D_s^{\\\\pm }K^\\\\mp ,D_s^{\\\\ast \\\\pm }K^\\\\mp , ...$ and $B_d\\\\rightarrow D^{\\\\pm }\\\\pi ^\\\\mp , D^{\\\\ast \\\\pm }\\\\pi ^\\\\mp , ...$ decays, Nucl.\",\n \"Phys.\",\n \"B671 (2003) 459, arXiv:hep-ph/0304027 CDF collaboration, T. Aaltonen et al., First observation of $\\\\bar{B}^0_s\\\\rightarrow D_s^\\\\pm K^\\\\mp $ and measurement of the ratio of branching fractions B$(\\\\bar{B}^0_s\\\\rightarrow D_s^\\\\pm K^\\\\mp )$ /B$(\\\\bar{B}^0_s\\\\rightarrow D_s^+ \\\\pi ^-)$, Phys.\",\n \"Rev.\",\n \"Lett.\",\n \"103 (2009) 191802, arXiv:0809.0080 Belle collaboration, R. Louvot et al., Measurement of the decay $B_s^0\\\\rightarrow D_s^- \\\\pi ^{+}$ and evidence for $B_s^0 \\\\rightarrow D_s^\\\\mp K^\\\\pm $ in $e^+ e^-$ annihilation at $\\\\sqrt{s}\\\\sim 10.87$ GeV, Phys.\",\n \"Rev.\",\n \"Lett.\",\n \"102 (2009) 021801, arXiv:0809.2526 LHCb collaboration, R. 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Golonka and Z.\",\n \"Was, PHOTOS Monte Carlo: A Precision tool for QED corrections in Z and W decays, Eur.\",\n \"Phys.\",\n \"J. C45 (2006) 97, arXiv:hep-ph/0506026 A. Hoecker et al., TMVA: Toolkit for multivariate data analysis, PoS ACAT (2007) 040, arXiv:physics/0703039 LHCb collaboration, R. Aaij et al., Determination of ${f}_{s}/{f}_{d}$ for 7 TeV $pp$ collisions and measurement of the ${B}^{0}\\\\rightarrow {D}^{-}{K}^{+}$ branching fraction, Phys.\",\n \"Rev.\",\n \"Lett.\",\n \"107 (2011) 211801, arXiv:1106.4435 T. Skwarnicki, A study of the radiative cascade transitions between the Upsilon-prime and Upsilon resonances.\",\n \"PhD thesis, Institute of Nuclear Physics, Krakow, 1986, DESY-F31-86-02 LHCb collaboration, R. Aaij et al., Measurement of $b $ -hadron masses, Phys.\",\n \"Lett.\",\n \"B708 (2012) 241, arXiv:1112.4896\"\n ]\n]"},"id":{"kind":"string","value":"1204.1237"}}},{"rowIdx":1866896,"cells":{"text":{"kind":"list like","value":[["Diffusion along transition chains of invariant tori and Aubry-Mather\n sets"],["Abstract We describe a topological mechanism for the existence of diffusing orbits in a dynamical system satisfying the following assumptions: (i) the phase space contains a normally hyperbolic invariant manifold diffeomorphic to a two-dimensional annulus, (ii) the restriction of the dynamics to the annulus is an area preserving monotone twist map, (iii) the annulus contains sequences of invariant one-dimensional tori that form transition chains, i.e., the unstable manifold of each torus has a topologically transverse intersection with the stable manifold of the next torus in the sequence, (iv) the transition chains of tori are interspersed with gaps created by resonances, (v) within each gap there is prescribed a finite collection of Aubry-Mather sets.","Under these assumptions, there exist trajectories that follow the transition chains, cross over the gaps, and follow the Aubry-Mather sets within each gap, in any specified order.","This mechanism is related to the Arnold diffusion problem in Hamiltonian systems.","In particular, we prove the existence of diffusing trajectories in the large gap problem of Hamiltonian systems.","The argument is topological and constructive."],["Introduction","In this paper we study a topological mechanism of diffusion and chaotic orbits related to the Arnold diffusion problem.","We consider a normally hyperbolic invariant manifold diffeomorphic to a two-dimensional annulus.","We assume that the dynamics restricted to the annulus is given by an area preserving monotone twist map.","We assume that inside the annulus there exist invariant primary one-dimensional tori (homotopically nontrivial invariant closed curves) with the property that the unstable manifold of each torus has topologically transverse intersections with the stable manifolds of all sufficiently close tori.","Sequences of such tori and their heteroclinic connections form transition chains of tori.","The successive transition chains of tori are interspersed with gaps.","We assume that each gap is a region in the annulus bounded by two invariant primary tori of one of the following types: (i) a Birkhoff Zone of Instability, i.e., it contains no invariant primary torus in its interior (ii) it contains only finitely many invariant primary tori in its interior, such that each torus is either isolated, or it consists of a hyperbolic periodic orbit together with its stable and unstable manifolds that are assumed to coincide.","Within each gap we prescribe a finite collection of Aubry-Mather sets.","We prove the existence of orbits that shadow the primary tori in each transition chain, cross over the gaps that separate the successive transition chains, and also shadow the specified Aubry-Mather sets within each gap.","The motivation for this result is the Arnold diffusion problem of Hamiltonian systems.","This problem asserts that all sufficiently small perturbations of generic, integrable Hamiltonian systems exhibit orbits along which the action variable changes substantially; also, there exist chaotic orbits that can be coded through symbolic dynamics.","A classification of nearly integrable systems proposed in [13] distinguishes between a priori stable systems, in which the unperturbed system can be expressed in terms of action-angle variables only, and a priori unstable systems, in which the unperturbed system contains both action-angle and hyperbolic variables.","In the a priori stable case analytical results have been announced in [47].","In the a priori unstable case there have been several analytical results in the last several years (see [13], [54], [51], [52], [11], [18], [3], [12]).","Some of the approaches involve variational methods, geometric methods, or topological methods.","See [20] for an overview on the Arnold diffusion problem, applications, and additional references.","It is relevant in applications to detect, combine, and compare different mechanisms of diffusion displayed by concrete systems.","In many models, as well as in numerical experiments, diffusion can only be observed for perturbations of sizes much larger than those considered by the analytical approaches [14], [40], [39], [36], [28].","For these types of problems, the geometric and topological approaches are particularly advantageous as they yield constructive methods to detect diffusion, quantitative estimates on diffusing orbits, and explicit conditions that can be verified in concrete examples.","In this paper we describe a general method to establish the existence of diffusing orbits for a large class of dynamical systems.","The dynamical systems under consideration are not assumed to be small perturbations of integrable Hamiltonians.","Moreover, some systems that are not Hamiltonian can be considered.","Our method requires the existence of certain geometric objects that organize the dynamics, and employs topological tools to establish the existence of diffusing orbits.","The existence of the geometric objects can be verified in concrete systems through analytical methods, or through numerical methods, or through a combination of thereof.","We illustrate our method in the case of a Hamiltonian system consisting of a pendulum and a rotator with a small periodic coupling.","We give an analytic argument for the existence of diffusing orbits.","In [15] the topological method is applied to show, with the aid of a computer, the existence of diffusing orbits in the spatial restricted three-body problem, where the two primaries are the Sun and the Earth; this model is not nearly integrable.","Similar ideas appear in [10], [21], [22].","The diffusing orbits detected by this approach follow transition chains of invariant tori up to the gaps between these transition chains, and then cross the gaps following the inner dynamics restricted to the annulus.","The orbits that cross the gaps follow Birkhoff connecting orbits that go from one boundary of the gap to the other, or Mather connecting orbits, that shadow a prescribed sequence of Aubry-Mather set inside the gap, or homoclinic orbits.","Related approaches, that use Birkhoff's ideas to understand generic drift properties of random iterations of exact-symplectic twist maps, appear in [48], [38].","These yield the concept of a polysystem, which is a locally constant skew-product over a Bernoulli shift [43].","Polysystems are a key ingredient in the approach of the instability problem in a priori unstable systems in [6].","Arnold diffusion-type orbits for random iterations of flows of families of Tonelli Hamiltonians are studied in [42], based on an extension of the pseudograph method from [3].","The diffusing orbits that we obtain in this paper are similar to those found through variational methods, as in [11], [12]; they are however not necessarily action minimizing.","Also, we do not need the unperturbed Hamiltonian to have positive-definite normal torsion.","This paper completes the investigations undertaken in [25], [26] where some non-generic conditions, or some conditions difficult to verify in concrete systems, were assumed.","The present paper assumes very general conditions and provides a new topological mechanism of diffusion based on Aubry-Mather sets."],["Main result","In this section we state the assumptions and the main result of this paper.","After the statement, the assumptions and the main result are explained and exemplified.","(A1) $M$ is a $n$ -dimensional $C^r$ -differentiable Riemannian manifold, and $f:M\\rightarrow M$ is a $C^r$ -smooth map, for some $r\\ge 2$ .","(A2) There exists a 2-dimensional submanifold $\\Lambda $ in $M$ , diffeormorphic to an annulus $\\Lambda \\simeq \\mathbb {T}^1\\times [0,1]$ .","We assume that $f$ is 2-normally hyperbolic to $\\Lambda $ in $M$ (see Subsection REF for the definition).","Since $f$ is 2-normally hyperbolic to $\\Lambda $ , $W^s(\\Lambda )$ , $W^u(\\Lambda )$ , and $\\Lambda $ are all $C^2$ -differentiable.","Denote the dimensions of the stable and unstable manifolds of a point $x \\in \\Lambda $ by $\\text{dim}(W^s(x)) = n_s$ and $\\text{dim}(W^u(x)) = n_u$ .","Then, $n = 2 + n_s + n_u$ .","(A3) On $\\Lambda $ there is a system of angle-action coordinates $(\\phi ,I)$ , with $\\phi \\in {\\mathbb {T}}^1$ and $I \\in [0,1]$ .","The restriction $f|_{\\Lambda }$ of $f$ to $\\Lambda $ is a boundary component preserving, area preserving, monotone twist map, with respect to the angle-action coordinates $(\\phi ,I)$ .","(A4) The stable and unstable manifolds of $\\Lambda $ , $W^s(\\Lambda )$ and $W^u(\\Lambda )$ , have a differentiably transverse intersection along a 2-dimensional homoclinic channel $\\Gamma $ .","Since the manifolds $W^s(\\Lambda )$ and $W^u(\\Lambda )$ are $C^2$ and transverse, $\\Gamma $ is $C^2$ .","We assume that the scattering map $S$ associated to $\\Gamma $ is well defined, and hence is $C^1$ .","See Subsection REF .","(A5) There exists a bi-infinite sequence of Lipschitz primary invariant tori $\\lbrace T_i\\rbrace _{i\\in \\mathbb {Z}}$ in $\\Lambda $ , and a bi-infinite, increasing sequence of integers $\\lbrace i_k\\rbrace _{k\\in \\mathbb {Z}}$ with the following properties: (i) Each torus $T_i$ intersects the domain $U^-$ and the range $U^+$ of the scattering map $S$ associated to $\\Gamma $ .","(ii) For each $i\\in \\lbrace {i_{k}+1}, \\ldots , {i_{k+1}-1}\\rbrace $ , the image of $T_i\\cap U^-$ under the scattering map $S$ is topologically transverse to $T_{i+1}$ .","(iii) For each torus $T_i$ with $i\\in \\lbrace {i_{k}+2}, \\ldots , {i_{k+1}-1}\\rbrace $ , the restriction of $f$ to $T_i$ is topologically transitive.","(iv) Each torus $T_i$ with $i\\in \\lbrace {i_{k}+2}, \\ldots , {i_{k+1}-1}\\rbrace $ , can be $C^0$ -approximated from both sides by other primary invariant tori from $\\Lambda $ .","We will refer to a finite sequence $\\lbrace T_i\\rbrace _{i=i_k+1,\\ldots , i_{k+1}}$ as above as a transition chain of tori.","(A6) The region in $\\Lambda $ between $T_{i_k}$ and $T_{i_{k}+1}$ contains no invariant primary torus in its interior.","(A7) Inside each region between $T_{i_k}$ and $T_{i_{k}+1}$ there is prescribed a finite collection of Aubry-Mather sets $\\lbrace \\Sigma _{\\omega ^k_1}, \\Sigma _{\\omega ^k_2},\\ldots , \\Sigma _{\\omega ^k_{s_k}}\\rbrace $ , where $s_k\\ge 1$ , and $\\omega ^k_s$ denotes the rotation number of $\\Sigma _{\\omega ^k_s}$ .","Each Aubry-Mather set $\\Sigma _{\\omega ^k_s}$ is assumed to lie on some essential circle $C_{\\omega ^k_s}$ , with the circles $C_{\\omega ^k_s}$ mutually disjoint for all $s\\in \\lbrace 1,\\ldots ,s_k\\rbrace $ , and $C_{\\omega ^k_s}$ below $C_{\\omega ^k_{s^{\\prime }}}$ for all $\\omega ^k_s<\\omega ^k_{s^{\\prime }}$ .","The vertical ordering of these circles is relative to the $I$ -coordinate on the annulus.","Instead of (A6) we can consider the following condition: (A6$^{\\prime }$ ) The region $\\Lambda _k$ in $\\Lambda $ between $T_{i_k}$ and $T_{i_{k}+1}$ contains finitely many invariant primary tori $\\lbrace \\Upsilon _{h^k_1}, \\ldots , \\Upsilon _{h^k_{l_k}}\\rbrace $ , where $l_k\\ge 1$ , satisfying the following properties: (i) Each $\\Upsilon _{h^k_j}$ falls in one of the following two cases: (a) $\\Upsilon _{h^k_j}$ is an isolated invariant primary torus, i.e., an invariant torus that has a neighborhood in the annulus that does not contain any other invariant primary torus inside.","(b) There exists a hyperbolic periodic orbit in $\\Lambda $ such that its stable and unstable manifolds coincide.","Each invariant manifold has two branches, an upper branch and a lower branch (where the vertical ordering is relative to the $I$ -coordinate of the annulus).","Then $\\Upsilon _{h^k_j}$ is an invariant primary torus consisting of the hyperbolic periodic orbit together with the upper branches of the invariant manifolds, or consisting of the hyperbolic periodic orbit together with the lower branches of the invariant manifolds.","(ii) The invariant primary tori $\\lbrace \\Upsilon _{h^k_1}, \\ldots , \\Upsilon _{h^k_{l_k}}\\rbrace $ are vertically ordered, in the sense that $\\Upsilon _{h^k_j}$ is below $\\Upsilon _{h^k_{j+1}}$ , for all $j=1,\\ldots , l_k-1$ .","The vertical ordering of these tori is relative to the $I$ -coordinate on the annulus.","(iii) For each $\\Upsilon _{h^k_j}$ , $j=1,\\ldots , l_k$ , the inverse image $S^{-1}(\\Upsilon _{h^k_j}\\cap U^+)$ forms with $\\Upsilon _{h^k_j}$ a topological disk $D_{h^k_j} \\subseteq U^-$ below $\\Upsilon _{h^k_j}$ , such that $S(D_{h^k_j})\\subseteq U^+$ is a topological disk above $\\Upsilon _{h^k_j}$ , which is bounded by $\\Upsilon _{h^k_j}$ and $S(\\Upsilon _{h^k_j}\\cap U^-)$ .","See Fig.","REF .","Now we state the main result of the paper.","Theorem 2.1 Let $f:M\\rightarrow M$ be a $C^r$ -differentiable map, and let $(T_i)_{i\\in \\mathbb {Z}}$ be a sequence of invariant primary tori in $\\Lambda $ , satisfying the properties (A1) – (A6), or (A1)-(A5) and (A6 $^{\\prime }$ ), from above.","Then for each sequence $(\\epsilon _i)_{i\\in \\mathbb {Z}}$ of positive real numbers, there exist a point $z\\in M$ and a bi-infinite increasing sequence of integers $(N_i)_{i\\in \\mathbb {Z}}$ such that $ d(f^{N_i}(z), T_{i})<\\epsilon _i, \\textrm { for all }i\\in \\mathbb {Z}.$ In addition, if condition (A7) is assumed, and some positive integers $\\lbrace n^k_s\\rbrace _{s=1,\\ldots , s_k}$ , $k\\in \\mathbb {Z}$ are given, then there exist $z\\in M$ and $(N_i)_{i\\in \\mathbb {Z}}$ as in (REF ), and positive integers $\\lbrace m^k_s\\rbrace _{s=1,\\ldots , s_k}$ , $k\\in \\mathbb {Z}$ , such that, for each $k$ and each $s\\in \\lbrace 1,\\ldots , s_k\\rbrace $ , we have $\\pi _\\phi (f^j(w^k_s))<\\pi _\\phi (f^j(z))<\\pi _\\phi (f^j(\\bar{w}^k_s)),$ for some $w^k_s,\\bar{w}^k_s\\in \\Sigma _{\\omega ^k_s}$ and for all $j$ with $N_{i_k}+\\sum _{t=0}^{s-1} n^k_t+\\sum _{t=0}^{s-1}m^k_t\\le j\\le N_{i_k}+\\sum _{t=0}^{s} n^k_t+\\sum _{t=0}^{s-1}m^k_t.$ Figure: An illustration of the condition (A6 ' ^{\\prime }).","In the figure, inside the region between T i k T_{i_k} and T i k +1 T_{i_{k}+1}, three invariant primary tori are shown, Υ h j k \\Upsilon _{h^k_j}, Υ h j+1 k \\Upsilon _{h^k_{j+1}}, Υ h j+2 k \\Upsilon _{h^k_{j+2}}, where Υ h j k \\Upsilon _{h^k_j} is as in (A6 ' ^{\\prime }-i-a), and Υ h j+1 k \\Upsilon _{h^k_{j+1}}, Υ h j+2 k \\Upsilon _{h^k_{j+2}} are as in (A6 ' ^{\\prime }-i-b).","The vertical ordering of the tori stated condition (A6 ' ^{\\prime }-ii) is depicted.","For each of the tori Υ h j k \\Upsilon _{h^k_j}, Υ h j+1 k \\Upsilon _{h^k_{j+1}}, Υ h j+2 k \\Upsilon _{h^k_{j+2}}, the condition (A6 ' ^{\\prime }-iii) is also illustrated.Theorem REF asserts that if the conditions (A1)-(A6), or (A1)-(A5) and (A6$^{\\prime }$ ) are satisfied, then there exists an orbit that shadows all tori in the transition chains in the prescribed order, and also crosses over the large gaps that separate the successive transition chains.","In particular, there exists an orbit that travels arbitrarily far with respect to the action variable, and there also exists an orbit that executes chaotic excursions.","Additionally, if some Aubry-Mather sets are prescribed inside each gap that separates the successive transition chains, as in condition (A7), then there exists an orbit that, besides shadowing the transition chains, it also shadows the Aubry-Mather sets in the prescribed order.","Note that the tori in the transition chains are shadowed in the sense that the diffusing orbit gets arbitrarily close to these tori.","However, the Aubry-Mather sets are shadowed in the sense of the cyclical ordering: for each prescribed Aubry-Mather set, the diffusing orbit stays between the orbits of two points in the Aubry-Mather set, relative to the $\\phi $ -coordinate, for any prescribed time interval.","Now we explain each assumption.","Assumption (A1) describes a $C^r$ -differentiable, discrete dynamical system.","In applications, the map $f$ represents the first return map to a Poincaré section associated to a flow.","In many examples of interest the flow is a Hamiltonian flow.","Assumption (A2) prescribes the existence of a normally hyperbolic invariant manifold $\\Lambda $ for $f$ , which is diffeomorphic to an annulus.","The differentiability class $r$ and the contraction and expansion rates along the stable and unstable bundles on $\\Lambda $ are chosen so that the manifolds $\\Lambda $ , $W^u(\\Lambda )$ , $W^s(\\Lambda )$ , $\\Gamma $ are at least $C^2$ -differentiable, and the scattering map $S$ associated to $\\Gamma $ is at least $C^1$ -differentiable.","The relations between the rates and the differentiability of these objects is given explicitly in Subsection REF .","In many examples of nearly integrable Hamiltonian systems, e.g.","[2], one can identify a normally hyperbolic invariant manifold $\\Lambda _0$ in the unperturbed system, and use the standard theory of normal hyperbolicity to establish the persistence of a normally hyperbolic invariant manifold $\\Lambda _\\varepsilon $ diffeomorphic to $\\Lambda _0$ for the perturbed system, for all sufficiently small perturbation parameters $\\varepsilon \\ne 0$ .","There also exist examples, e.g.","from celestial mechanics, were the existence of a normally hyperbolic manifold can be established through a computer assisted proof (see [9]).","We note that the assumption (A2) does not require that the stable and unstable manifolds of $\\Lambda $ have equal dimensions, thus our setting includes dynamical systems that are not Hamiltonian.","Assumption (A3) is satisfied automatically in examples like the weakly-coupled pendulum-rotator system considered in [18], or the periodically perturbed geodesic flow on a torus considered in [17], [33].","Some properties of area preserving, monotone twist maps of the annulus are reviewed in Section .","Assumption (A4) asserts that the stable and unstable manifolds of $\\Lambda $ have a transverse intersection $\\Gamma $ along a homoclinic manifold $\\Gamma $ , such that the scattering map associated to $\\Gamma $ is well defined, and hence is $C^1$ .","See Subsection REF .","In perturbed systems one often uses a Melnikov method to establish the existence, and the persistence for all sufficiently small values of the perturbation, of a transverse intersection of the invariant manifolds.","Assumption (A5) prescribes the existence of a bi-infinite collection $\\lbrace T_{i}\\rbrace _{i\\in \\mathbb {Z}}$ of invariant primary Lipschitz tori that can be grouped into transition chains of the type $\\lbrace T_i\\rbrace _{i=i_k+1,\\ldots ,i_{k+1}}$ .","Assumption (A5)-(i) requires that each torus $T_i$ intersects the domain and the range of the scattering map.","Assumption (A5)-(ii) requires that each torus in a transition chain$\\lbrace T_i\\rbrace _{i=i_k+1,\\ldots ,i_{k+1}}$ is mapped by the scattering map topologically transversally across the next torus in the chain.","Since the tori are only Lipschitz, topological transversality (topological crossing) is used in place of differentiable transversality.","The definition of topological crossing can be found in [8].","Roughly speaking, two manifolds are topologically crossing if they can be made differentiably transverse with non-zero oriented intersection number by the means of a sufficiently small homotopy.","In our case the two manifolds are two 1-dimensional arcs of $S(T_i)$ and of $T_{i+1}$ in $\\Lambda $ .","From (A5)-(ii), it follows as in [19] that $W^u(T_i)$ has a topologically transverse intersection point with $W^s(T_{i+1})$ .","Condition (A5)-(iii) requires that each torus in $\\lbrace T_{i}\\rbrace _{i=i_{k}+1, \\ldots , i_{k+1}}$ , except for the end tori, are topologically transitive.","Assumption (A5)-(iv) says that all tori in the transition chain except for the end ones can be $C^0$ -approximated from both ways by some other invariant primary tori in $\\Lambda $ , not necessarily from the transition chain.","This means that for each $i\\in \\lbrace i_{k}+2, \\ldots , i_{k+1}-1\\rbrace $ there exist two sequences of invariant primary tori $(T_{j^-_l(i)})_{l\\ge 1},(T_{j^+_l(i)})_{l\\ge 1}$ in $\\Lambda $ that approach $T_i$ in the $C^0$ -topology, such that the annulus bounded by $T_{j^-_l(i)}$ and $T_{j^+_l(i)}$ contains $T_i$ in its interior for all $l$ .","Assumption (A6) says that every pair of successive transition chains$\\lbrace T_{i}\\rbrace _{i=i_{k-1}+1, \\ldots , i_{k}}$ and $\\lbrace T_{i}\\rbrace _{i=i_{k}+1, \\ldots , i_{k+1}}$ is separated by the region between $T_{i_{k}}$ and $T_{i_{k}+1}$ which contains no invariant primary torus in its interior.","A region in an annulus that is bounded by two invariant primary tori and contains no invariant primary torus in its interior is referred as a Birkhoff Zone of Instability (BZI).","The boundary tori have in general only Lipschitz regularity.","Assumptions (A5) and (A6) describe a geometric structure that is typical for the large gap problem for a priori unstable Hamiltonian systems.","In such systems, Melnikov theory implies that $W^u(\\Lambda )$ intersects transversally $W^s(\\Lambda )$ at an angle of order $\\varepsilon $ , where $\\varepsilon $ is the size of the perturbation.","The KAM theorem yields a Cantor family of smooth, invariant primary tori that survives the perturbation.","The family of tori is interrupted by `large gaps' of order $\\varepsilon ^{1/2}$ located at the resonant regions.","Using the transverse intersection between $W^u(\\Lambda )$ and $W^s(\\Lambda )$ , one can find heteroclinic connections between KAM tori that are sufficiently close, within order $\\varepsilon $ , from one another, and thus form transition chains of tori.","Since the large gaps are of order $\\varepsilon ^{1/2}$ and the splitting size of $W^u(\\Lambda )$ , $W^s(\\Lambda )$ is only order $\\varepsilon $ , the transition chain mechanism cannot be extended across the large gaps.","In our model, the large gaps are modeled by BZI's, as in assumption (A6).","This can be achieved by extending the transition chains to maximal transition chains, that go from the boundary of one large gap to the boundary of the next large gap.","The intermediate tori in the chain can be chosen as KAM tori: therefore the assumption that these tori are topologically transitive and are $C^0$ -approximable from both sides by other tori is satisfied in such cases.","This may not be the case for the tori at the ends of the transition chains.","Assumption (A7) says that inside each BZI between $T_{i_k}$ and $T_{i_{k}+1}$ there is a prescribed collection of Aubry-Mather sets $\\lbrace \\Sigma _{\\omega ^k_1}, \\Sigma _{\\omega ^k_2},\\ldots , \\Sigma _{\\omega ^k_{s_k}}\\rbrace $ that is vertically ordered.","The vertical ordering means that the Aubry-Mather sets lie on essential (non-invariant) circles $C_{\\omega ^k_s}$ that are graphs over the $\\phi $ -coordinate of the annulus, and with $C_{\\omega ^k_s}$ below $C_{\\omega ^k_{s^{\\prime }}}$ provided $\\omega ^k_s<\\omega ^k_{s^{\\prime }}$ ; we write $C_{\\omega ^k_s}\\prec C_{\\omega ^k_{s^{\\prime }}}$ .","The vertical ordering of the Aubry-Mather sets is shown for example in [27].","Assumption (A6$^{\\prime }$ ) is a relaxation of (A6).","Instead of requiring that the region in $\\Lambda $ between $T_{i_{k}}$ and $T_{i_{k}+1}$ is a BZI, it allows the existence of finitely many invariant primary tori $\\lbrace \\Upsilon _{h^k_j}\\rbrace _{j=1, \\ldots , l_k}$ that separate the region into disjoint components.","These invariant primary tori are either isolated or else they consist of hyperbolic periodic points together with branches of their stable and unstable manifolds which are assumed to coincide.","We require that the image of each $\\Upsilon _{h^k_j}$ under $S$ satisfies a certain transversality condition with $\\Upsilon _{h^k_{j+1}}$ that allows one to use the scattering map in order to move points from one side of the set to the other side of the set.","We note that isolated invariant tori and hyperbolic periodic points whose stable and unstable manifolds coincide do not occur in generic systems."],["Application","We apply Theorem REF to show the existence of diffusing orbits in an example of a nearly integrable Hamiltonian system.","Let $H_\\varepsilon (p, q, I,\\phi ,t)&=&h_0(I)\\pm \\left(\\frac{1}{2}p^2+V(q)\\right)+\\varepsilon h(p, q,I,\\phi , t; \\varepsilon ),$ where $(p, q, I,\\phi , t)\\in \\mathbb {R}\\times \\mathbb {T}^1\\times \\mathbb {R}\\times \\mathbb {T}^1\\times \\mathbb {T}^1$ and $h$ is a trigonometric polynomial in $(\\phi ,t)$ .","Here $h_0(I)$ represents a rotator, $P_{\\pm }(p,q)=\\pm (\\frac{1}{2}p^2+V(q))$ represents a pendulum, and $\\varepsilon h$ a small, periodic coupling.","We assume that $V$ , $h_0$ and $h$ are uniformly $C^r$ for some $r$ sufficiently large.","We assume that $V$ is periodic in $q$ of period 1 and has a unique non-degenerate global maximum; this implies that the pendulum has a homoclinic orbit $(p^0 (\\sigma ),q^0 (\\sigma ))$ to $(0,0)$ , with $\\sigma \\in \\mathbb {R}$ .","We also assume that $h_0$ satisfies a uniform twist condition $\\partial ^2h_0/\\partial I^2>\\theta $ , for some $\\theta >0$ , and for all $I$ in some interval $(I^-, I^+)$ , with $I^-0$ .","Condition (ii) implies that there are points in the domain of the scattering map $S$ whose $I_\\varepsilon $ -coordinate is increased by $O(\\varepsilon )$ under $S$ .","We can use these estimates to construct transition chains of invariant primary tori alternating with gaps, as in (A5) and (A6), or as in (A5) and (A6$^{\\prime }$ ).","For $\\varepsilon $ sufficiently small and fixed, we let $M:=M_\\varepsilon $ , $f:=f_\\varepsilon $ , and $\\Lambda $ be the annulus in $\\Lambda _\\varepsilon $ bounded by a pair of tori $T_{I_a}$ , $T_{I_b}$ with $I^-0$ at all points in the annulus.","We note that the above properties imply that $f$ is exact symplectic, i.e.","$ f$ has zero flux, meaning that for any rotational curve $\\gamma $ the area of the regions above $\\gamma $ and below $f(\\gamma )$ equals the area below $\\gamma $ and above $f(\\gamma )$ .","In the sequel we will assume that $f$ is a positive twist, meaning that$\\partial (\\pi _{x}\\circ f)/\\partial y>0$ at all points.","The map $ f$ restricted to the boundary components $\\mathbb {T}^1 \\times \\lbrace 0\\rbrace $ , $\\mathbb {T}^1 \\times \\lbrace 1\\rbrace $ of the annulus has well defined rotation numbers $\\omega _-,\\omega _+$ , respectively, with $\\omega ^-<\\omega ^+$ .","We will assume that $\\omega ^-,\\omega ^+>0$ .","By an invariant primary torus (essential invariant circle) we mean a 1-dimensional torus $T$ invariant under $f$ that cannot be homotopically deformed into a point inside the annulus.","Since $f$ is a monotone twist map, each invariant primary torus $T$ is the graph of some Lipschitz function (see [4], [5]).","A region in $A$ between two invariant primary tori $T_1$ and $T_2$ is called a Birkhoff Zone of Instability (BZI) provided that there is no invariant primary torus in the interior of the region.","It is known that, for an area preserving monotone twist map $f$ of $A$ , given a BZI, there exist Birkhoff connecting orbits that go from any neighborhood of one boundary torus to any neighborhood of the other boundary torus (see [4], [5], [37]).","We have the following results: Theorem 4.1 (Birkhoff Connecting Theorem) Suppose that $T_1$ and $T_2$ bound a BZI.","For every pair of neighborhoods $U$ of $T_1$ and $V$ of $T_2$ there exist a point $z\\in U$ and an integer $N>0$ such that $f^{N}(z)\\in V$ .","Corollary 4.2 Suppose that $T_1$ and $T_2$ bound a BZI, and that the restrictions of $f$ to $T_1$ and $T_2$ are topologically transitive.","For every $\\zeta _1\\in T_1,\\zeta _2\\in T_2$ and every pair of neighborhoods $U$ of $\\zeta _1$ and $V$ of $\\zeta _2$ , there exist a point $z\\in U$ and an integer $N>0$ such that $f^{N}(z)\\in V$ .","A subset $M\\subseteq A$ is said to be monotone (cyclically ordered) if $\\pi _x(z_1)<\\pi _x(z_2)$ implies $\\pi _x(f(z_1))<\\pi _x(f(z_2))$ for all $z_1,z_2\\in M$ .","For $z\\in A$ the extended orbit of $z$ is the set $EO(z)=\\lbrace f^n(z)+(j,0)\\,:\\,n,j\\in \\mathbb {Z}\\rbrace $ .","The orbit of $z$ is said to be monotone (cyclically ordered) if the set $EO(z)$ is monotone.","If the orbit of $z\\in A$ is monotone, then the rotation number $\\rho (z)=\\lim _{n\\rightarrow \\infty }(\\pi _x(f^n(z))/n)$ exists.","We denote $\\textrm {Rot}(\\omega )=\\lbrace z\\in A\\,:\\,\\rho (z)=\\omega \\rbrace $ .","All points in the same monotone set have the same rotation number.","Definition 4.3 An Aubry-Mather set for $\\omega \\in \\mathbb {T}^1$ is a minimal, monotone, $f$ -invariant subset of $\\textrm {Rot}(\\omega )$ .","Here by a minimal set we mean a closed invariant set that does not contain any proper closed invariant subsets.","(Equivalently, the orbit of every point in the set is dense in the set.)","This should not be confused with action-minimizing or $h$ -minimal sets, where $h$ is a generating function for $f$ .","Theorem 4.4 (Aubry-Mather Theorem) For every $\\omega \\in [\\omega ^-,\\omega ^+]$ , there exists a non-empty Aubry-Mather set $\\Sigma _\\omega $ in $\\textrm {Rot}(\\omega )$ .","Aubry-Mather sets defined as above can be obtained as limits of monotone Birkhoff periodic orbits [34].","There may be many Aubry-Mather sets with the same rotation number [45].","On the other hand, if one requires Aubry-Mather sets to be action minizing, there exists a unique recurrent Aubry-Mather set for any given irrational rotation number.","In the sequel we will use the following result on the vertical ordering of Aubry-Mather sets from [27].","Theorem 4.5 There exists a family of essential circles $C_\\omega $ in $A$ for $\\omega \\in [\\omega ^-,\\omega ^+]$ such that: (i) Each $C_\\omega $ is a graph over $y=0$ ; (ii) The circles $C_\\omega $ are mutually disjoint, and if $\\omega ^{\\prime }>\\omega $ then $C_{\\omega ^{\\prime }}$ is above $C_{\\omega }$ ; (iii) Each $C_\\omega $ contains an Aubry-Mather set $\\Sigma _\\omega $ .","The above circles have Lipschitz regularity, and are projections of so called `ghost circles' that are objects in $\\mathbb {R}^\\mathbb {Z}$ .","See [27] for details.","A similar result to Theorem REF appears in [35] who find Aubry-Mather sets lying on pseudo-graphs that are (not strictly) vertically ordered.","There are some analogues of the Birkhoff Connecting Theorem for Aubry-Mather sets.","The following lemma is used in [33] to provide a topological proof for Mather Connecting Theorem stated below.","Lemma 4.6 Suppose that $T_1$ and $T_2$ bound a BZI.","Let $\\Sigma _\\omega $ be an Aubry-Mather set of rotation number $\\omega $ inside the BZI.","Let $p$ be a recurrent point in $\\Sigma _\\omega $ and $W(p)$ be a neighborhood of $p$ inside the BZI.","The following hold true: (i) For some positive number $n^+$ (resp.","$n^-$ ) depending on $W(p)$ the set$\\bigcup _{j=0}^{n^+}f^j(W(p))$ (resp.","$\\bigcup _{j=0}^{n^-}f^{-j}(W(p))$ ) separates the cylinder.","(ii) The set $W^{+\\infty }:=\\bigcup _{j=0}^{\\infty }f^j(W (p))$ (resp.","the set $W^{-\\infty }:=\\bigcup _{j=0}^{\\infty }f^{-j}(W(p) )$ ), is connected and open.","(iii) The closure of $W^{+\\infty }$ (resp.","$W ^{-\\infty }$ ) contains both boundary tori $T_1$ and $T_2$ .","(iv) The set $W ^\\infty :=\\bigcup _{j=-\\infty }^{\\infty }f^j(W(p))$ is invariant, and both $W^{+\\infty } $ and $W^{-\\infty } $ are open and dense in $W ^\\infty $ .","The following result says that there exist orbits that visit any prescribed bi-infinite sequence of Aubry-Mather sets inside a BZI (see [46], [53], [29], [35], [33]).","Theorem 4.7 (Mather Connecting Theorem) Suppose that $T_1$ and $T_2$ bound a BZI, and $\\lbrace \\Sigma _{\\omega _i}\\rbrace _{i\\in \\mathbb {Z}}$ is a bi-infinite sequence of Aubry-Mather sets inside the BZI.","Let $\\varepsilon _i>0$ for $i\\in \\mathbb {Z}$ .","Then there exist a point $z$ inside the BZI and an increasing bi-infinite sequence of integers $\\lbrace j_i\\rbrace _{i\\in \\mathbb {Z}}$ such that $f^{j_i}(z)$ is within $\\varepsilon _i$ from $\\Sigma _{\\omega _i}$ for all $i\\in \\mathbb {Z}$ .","The Aubry-Mather sets in Theorem REF are action minimizing.","The following topological version of Mather Connecting Theorem, due to Hall [29], provides shadowing orbits of Aubry-Mather sets that are not necessarily action minimizing.","This approach can be implemented in rigorous computer experiments [32].","Theorem 4.8 Suppose that $T_1$ and $T_2$ bound a BZI, and $\\lbrace z_s\\rbrace _{s\\in \\mathbb {Z}}$ is a bi-infinite sequence of monotone $(p_s/q_s)$ -periodic points, with the rotation numbers $p_s/q_s$ mutually distinct, inside the BZI.","Given a bi-infinite sequence $\\lbrace n_s\\rbrace _{s\\in \\mathbb {Z}}$ of positive integers, then there exist a point $z$ and a bi-infinite sequence $\\lbrace m_s\\rbrace _{s\\in \\mathbb {Z}}$ of positive integers such that, for each $s\\ge 0$ , there exist some points $w_s,\\bar{w}_s$ in the extended orbit of $z_s$ such that $\\begin{split}\\pi _x(f^j(w_s))<\\pi _x(f^j(z))<\\pi _x(f^j(\\bar{w}_s))\\textrm { for }\\\\\\sum _{t=0}^{s-1} n_t+\\sum _{t=0}^{s-1}m_t\\le j\\le \\sum _{t=0}^{s} n_t+\\sum _{t=0}^{s-1}m_t.\\end{split}$ A similar statement holds for each $s<0$ .","In the above, $n_s$ represents the number of iterates for which the orbit of $z$ shadows – in the sense of the cyclical ordering – the extended orbit of $z_s$ , and $m_s$ represents the number of iterates it takes the orbit of $z$ to pass from the extended orbit of $z_s$ to the extended orbit of $z_{s+1}$ .","They main tool used in Hall's arguments is that of a positive (negative) diagonal.","Denote by $\\mathcal {Z}$ the BZI bounded by the tori $T_1$ and $T_2$ .","Let $I_z&=&\\lbrace w\\in \\mathcal {Z}\\,|\\, \\pi _x(w)=\\pi _x(z)\\rbrace ,\\\\I^+_z&=&\\lbrace w\\in I_z\\,|\\, \\pi _y(w)\\ge \\pi _y(z)\\rbrace ,\\\\I^-_z&=&\\lbrace w\\in I_z\\,|\\, \\pi _y(w)\\le \\pi _y(z)\\rbrace ,\\\\B_{z_0,z_1}&=&\\lbrace w\\in \\mathcal {Z}\\,|\\, \\pi _x(z_0)< \\pi _x(w)< \\pi _x(z_1)\\rbrace ,$ where $z,z_0,z_1$ are points in the annulus.","A positive diagonal $D$ in $B_{z_0,z_1}$ is a set $D\\subseteq \\textrm {cl}(B_{z_0,z_1})$ such that (i) $D$ is simply connected and the closure of its interior; (ii) $\\partial D \\cap \\textrm {cl}(B_{z_0,z_1})\\subseteq I^-_{z_0}\\cup I^+_{z_1} \\cup T_1\\cup T_2$ ; (iii) $\\partial D\\cap I^-_{z_0}\\ne \\emptyset $ and $\\partial D\\cap I^+_{z_1}\\ne \\emptyset $ .","The set $\\partial D\\cap B_{z_0,z_1}$ has exactly two components connecting $I^-_{z_0}\\cup T_1$ to $I^+_{z_1}\\cup T_2$ , which are called the upper and lower edges of $D$ , respectively.","We informally say that these components `stretch across' $B_{z_0,z_1}$ .","See Fig.","REF .","A negative diagonal and its upper and lower edges are defined similarly.","Figure: Two positive diagonal sets.","The positive diagonal set on the left has its upper and lower edges marked.","The positive diagonal set on the right is obtained by intersecting f(B w 0 ,w 1 )f(B_{w_0,w_1}) with B f(w 0 ),f(w 1 ) B_{f(w_0),f(w_1)}.","The upper edge of this diagonal set is contained in f(I w 0 + )f(I^+_{w_0}) and the lower edge is contained in f(I w 1 - )f(I^-_{w_1}).An important feature of positive diagonals is the following hereditary property.","Given $z_0,z_0$ such that $\\pi _x(z_0)<\\pi _x(z_1)$ and $\\pi _x(f(z_0))<\\pi _x(f(z_1))$ , if $D$ is a positive diagonal in $B_{z_0,z_1}$ , then $f(D)\\cap B_{f(z_0),f(z_1)}$ has a component $D^{\\prime }$ that is a positive diagonal in $B_{f(z_0),f(z_1)}$ .","One way to generate a positive diagonal set is by taking a component of the intersection between $f^k(B_{w_0,w_1})$ and $B_{f^k(w_0),f^k(w_1)}$ .","In this case, there exists a positive diagonal in $B_{f^k(w_0),f^k(w_1)}$ whose upper edge is contained on $f^k(I_{w_0}^+)$ and lower edge is contained in $f^k(I_{w_1}^-)$ .","More general, one has the following important property.","If $D$ has the upper edge contained in $f^k(I^+_{w_0})$ and the lower edge contained in $f^k(I^-_{w_1})$ , and $\\partial D\\cap B_{z_0,z_1}\\subseteq f^k(I^+_{w_0}\\cup I^-_{w_1})$ , for some $w_0,w_1$ with $\\pi _x(w_0)<\\pi _x(w_1)$ and some $k>0$ , then then $f(D)\\cap B_{f(z_0),f(z_1)}$ has a component $D^{\\prime }$ which can be chosen so that its upper edge is contained in $f^{k+1}(I^+_{w_0})$ and its lower edge is contained in $f^{k+1}(I^-_{w_1})$ .","See Fig.","REF .","A similar property holds for negative diagonals.","The proof of Theorem REF in [29] is an inductive argument which, for a given pair of adjacent points $w_0,\\bar{w}_0$ in the extended orbit of $z_0$ , and for each $\\sigma \\ge 0$ , produces a nested sequence $D_0\\supseteq D_1\\supseteq \\ldots \\supseteq D_{\\sigma }$ of negative diagonals of $B_{w_0,\\bar{w}_0}$ such that, for each $s\\in \\lbrace 0,\\ldots ,\\sigma \\rbrace $ , the following hold: (a) the orbit of each point $z \\in D_s$ satisfies the ordering relation (REF ), and (b) there is a sufficiently large $j_s>0$ such that $f^{j_s+j}(D_s)$ contains a component that is a positive diagonal in $B_{f^j(w_s),f^j(\\bar{w}_s)}$ , for some adjacent points $w_s,\\bar{w}_s\\in EO(z_s)$ , and for all $j=1,\\ldots , n_s$ .","In the above, $j_s=\\sum _{t=0}^{s-1} n_t+\\sum _{t=0}^{s-1}m_t$ .","Moreover, in this inductive argument one can choose the diagonal sets $D_s$ so that $f^{j_s+j}(D_s)$ has the upper edge contained in $f^{j_s+j}(I^+_{w_0})$ , lower edge contained in $f^{j_s+j}(I^-_{\\bar{w}_0})$ , and $\\partial f^{j_s+j}(D_s)\\cap B_{f^{j_s+j}(w_0),f^{j_s+j}(\\bar{w}_0)} \\subseteq f^{j_s+j} (I^+_{w_0}\\cup I^-_{\\bar{w}_0})$ .","For the basis step, starting with $w_0,\\bar{w}_0$ and applying the hereditary property from above $n_0$ times, one obtains a negative diagonal set $D_0$ of $B_{w_0,\\bar{w}_0}$ with the properties that each point $z \\in D_0$ satisfies the ordering relation (REF ) for $s=0$ , and $f^{n_0}(D_0)$ has a component that is a positive diagonal of $B_{f^{n_0}(w_0),f^{n_0}(\\bar{w}_0)}$ .","For the inductive step, one assumes a negative diagonal $D_\\sigma $ of $B_{w_0,\\bar{w}_0}$ as above, and wants to produce a negative diagonal $D_{\\sigma +1}\\subseteq D_\\sigma $ of $B_{w_0,\\bar{w}_0}$ which fulfils the corresponding properties.","The key idea is to use the existence of points near $y=0$ that get near $y=1$ , and of points near $y=1$ that get near $y=0$ , as provided by Theorem REF , in order to show that for some $j_\\sigma $ sufficiently large $f^{j_\\sigma }(D_\\sigma )$ contains a component that stretches all the way across a fundamental interval of the annulus.","Hence $f^{j_\\sigma }(D_\\sigma )$ contains a subset that is a positive diagonal of $B_{w_{\\sigma +1}, \\bar{w}_{\\sigma +1}}$ for two adjacent points $w_{\\sigma +1}, \\bar{w}_{\\sigma +1}\\in EO(z_{\\sigma +1})$ .","From this it follows that $f^{j_\\sigma +j}(D_\\sigma )$ contains a component that is a positive diagonal in $B_{f^j(w_{\\sigma +1}),f^j(\\bar{w}_{\\sigma +1})}$ for all $j=1,\\ldots , n_{\\sigma +1}$ .","This completes the inductive step.","Applying a similar argument for the negative iterates of $f$ produces a nested sequence of positive diagonals of $B_{w_0,\\bar{w}_0}$ .","A positive diagonal of $B_{w_0,\\bar{w}_0}$ always has a non-empty intersection with a negative diagonal of $B_{w_0,\\bar{w}_0}$ .","This implies the existence of points $z$ whose forward orbits satisfy the ordering conditions in (REF ) and whose backwards orbits satisfy similar ordering conditions.","Using limit arguments as in [34], one can obtain shadowing of Aubry-Mather sets of irrational rotation numbers as well.","These topological ideas will be used in the proof of Theorem REF below.","Remark 4.9 An immediate consequence of Lemma REF is that, given a neighborhood $W$ of a point $p\\in \\Sigma _\\omega $ , where $\\Sigma _\\omega $ is an Aubry-Mather set inside a BZI bounded by $T_1$ and T$_2$ , and given a neighborhood $U$ of $T_1$ or $T_2$ , there exists an arbitrarily large $j>0$ such that $f^j(W(p))\\cap U\\ne \\emptyset $ .","Also, there exists an arbitrarily large $j^{\\prime }>0$ such that $f^{-j^{\\prime }}(W(p))\\cap U\\ne \\emptyset $ .","Remark 4.10 The results in this section hold if we replace conditions (i) and (ii) from the definition of an area preserving, monotone twist map with the following weaker conditions: (i') $f$ satisfies the following `condition B': for every pair of neighborhoods $U_1$ of $T_1$ and $U_2$ of $T_2$ , there exist $z_1,z_2\\in A$ and $n_1,n_2>0$ such that $z_1\\in U_1$ and $f^{n_1}(z_1)\\in U_2$ , and $z_2\\in U_2$ and $f^{n_2}(z_2)\\in U_1$ .","Note that this condition only makes sense if we restrict the dynamics to a BZI.","(ii') $f$ satisfies the following positive tilt condition: if we denote by $\\theta _{z}$ the angle deviation from the vertical, measured from the vertical vector $(0,1)$ to $Df_{z}(0,1)$ , with the clockwise direction taken as the positive direction, and defined in such a way that $\\theta _{(x,0)}\\in [-\\pi /2,\\pi /2]$ and $\\theta $ is continuous, then $\\theta _{z}> 0$ at all points.","See Fig.","REF .","Compositions of positive twist maps, are for example, positive tilt maps.","We shall note that the Aubry-Mather theory applies to positive tilt maps as well (see [31]).","Figure: Positively tilted map.Remark 4.11 Aubry-Mather theory and the above shadowing result also hold for generalized twist maps of the higher dimensional annulus $S^1\\times \\mathbb {R}^n$ .","See [1].","Remark 4.12 The topological approach in this section does not yield trajectories that get close to each set in the prescribed collection of Aubry-Mather sets, as in Theorem REF .","It seems possible, however, that these topological methods can be combined with the variational methods in [46] to obtain trajectories that go very close to each Aubry-Mather set in the given collection."],["Scattering map","The scattering map acts on the normally hyperbolic invariant manifold $\\Lambda $ and relates the past asymptotic trajectory of each orbit in the homoclinic manifold to its future asymptotic behavior.","We review its properties following [19].","For the general case, we consider a $C^r$ -differentiable manifold $M$ , a $C^r$ -differentiable map $f:M\\rightarrow M$ , and an $l$ -dimensional normally hyperbolic invariant manifold $\\Lambda $ for $f$ .","By the definition of normal hyperbolicity, there exists a splitting of the tangent bundle of $TM$ into sub-bundles $TM=E^u\\oplus E^s\\oplus T\\Lambda ,$ that are invariant under $df$ , and there exist a constant $C>0$ and rates $0<\\lambda <\\mu ^{-1}<1$ , such that for all $x\\in \\Lambda $ we have $\\begin{split}v\\in E^s_x \\Leftrightarrow \\Vert Df^k_x(v)\\Vert \\le C\\lambda ^k\\Vert v\\Vert \\textrm { for all } k\\ge 0,\\\\v\\in E^u_x \\Leftrightarrow \\Vert Df^k_x(v)\\Vert \\le C\\lambda ^{-k}\\Vert v\\Vert \\textrm { for all } k\\le 0,\\\\v\\in T_x\\Lambda \\Leftrightarrow \\Vert Df^k_x(v)\\Vert \\le C\\mu ^{|k|}\\Vert v\\Vert \\textrm { for all } k\\in \\mathbb {Z}.\\end{split}$ The smoothness of the invariant objects defined by the normally hyperbolic structure depends on the rates $\\lambda $ and $\\mu $ .","The map $f$ is said to be $\\ell $ -normally hyperbolic along $\\Lambda $ provided that $1 \\le \\ell \\le r$ is an integer satisfying $\\lambda \\,\\mu ^\\ell <1$ , i.e., $\\ell < (\\log \\lambda ^{-1})(\\log \\mu )^{-1}$ .","Then the stable and unstable manifolds $W^s(\\Lambda )$ and $W^u(\\Lambda )$ and the normally hyperbolic manifold $\\Lambda $ are all $C^\\ell $ -differentiable.","The splitting $E^s_z=T_z(W^s(x))$ depends $C^{\\ell -1}$ smoothly on $z$ in $W^s(\\Lambda )$ so $\\lbrace \\,W^s(x)\\, |\\, x \\in \\Lambda \\,\\rbrace $ is a $C^{\\ell -1}$ foliation of $W^s(\\Lambda )$ .","(This is stated explicitly in [50] and follows from the $C^r$ Section Theorem of [30].)","Similarly, $\\lbrace \\,W^u(x)\\, |\\, x \\in \\Lambda \\,\\rbrace $ is a $C^{\\ell -1}$ foliation of $W^u(\\Lambda )$ .","Since the stable (resp.","unstable) manifolds of $\\Lambda $ are foliated by stable (resp.","unstable) manifolds of points, we have that for each $x\\in W^s(\\Lambda )$ (resp.","$x\\in W^u(\\Lambda )$ ), there exists a unique $x^+ \\in \\Lambda $ (resp.","$x^- \\in \\Lambda $ ) such that $x \\in W^s(x^+)$ (resp.","$x \\in W^u(x^-)$ ).","We define the maps $\\Omega ^+ : W^s(\\Lambda ) \\rightarrow \\Lambda $ by $\\Omega ^+(x) = x^+$ and $\\Omega ^- : W^u(\\Lambda ) \\rightarrow \\Lambda $ by $\\Omega ^-(x) = x^-$ .","The maps $\\Omega ^+$ and $\\Omega ^-$ are $C^{\\ell -1}$ -smooth since the foliations are smooth.","We now describe the scattering map.","Assume that $W^u(\\Lambda )$ and $W^s(\\Lambda )$ have a differentiably transverse intersection along a homoclinic $l$ -dimensional $C^{\\ell -1}$ -differentiable manifold $\\Gamma $ .","This means that $\\Gamma \\subseteq W^u(\\Lambda ) \\cap W^s(\\Lambda )$ and, for each $x\\in \\Gamma $ , we have $ \\begin{split}T_xM=T_xW^u(\\Lambda )+T_xW^s(\\Lambda ),\\\\T_x\\Gamma =T_xW^u(\\Lambda )\\cap T_xW^s(\\Lambda ).\\end{split} $ We assume the additional condition that for each $x\\in \\Gamma $ we have $\\begin{split}T_xW^s(\\Lambda )=T_xW^s(x^+)\\oplus T_x(\\Gamma ),\\\\T_xW^u(\\Lambda )=T_xW^u(x^-)\\oplus T_x(\\Gamma ),\\end{split} $ where $x^-,x^+$ are the uniquely defined points in $\\Lambda $ corresponding to $x$ .","The restrictions $\\Omega ^+_\\Gamma ,\\Omega ^-_\\Gamma $ of $\\Omega ^+,\\Omega ^-$ to $\\Gamma $ are local $C^{\\ell -1}$ -diffeomorphisms.","By replacing $\\Gamma $ to a submanifold of it (which, with an abuse of notation, we still denote $\\Gamma $ ) we can ensure that $\\Omega ^+_\\Gamma :\\Gamma \\rightarrow U^+,\\Omega ^-_\\Gamma :\\Gamma \\rightarrow U^-$ are $C^{\\ell -1}$ -diffeomorphisms from $\\Gamma $ to the open sets $U^+,U^-$ in $\\Lambda $ , respectively.","Definition 5.1 A homoclinic manifold $\\Gamma $ satisfying (REF ) and (REF ), and for which the corresponding restrictions of the wave maps are $C^{\\ell -1}$ -diffeomorphisms, is referred as a homoclinic channel.","Definition 5.2 Given a homoclinic channel $\\Gamma $ , the scattering map associated to $\\Gamma $ is the $C^{\\ell -1}$ -diffeomorphism $S_\\Gamma =\\Omega _\\Gamma ^+\\circ (\\Omega _\\Gamma ^-)^{-1}$ from the open subset $U^-:=\\Omega _\\Gamma ^-(\\Gamma )$ in $\\Lambda $ to the open subset $U^+:=\\Omega _\\Gamma ^+(\\Gamma )$ in $\\Lambda $ .","In the sequel we will regard $S_\\Gamma $ as a partially defined map, so the image of a set $A$ by $S_\\Gamma $ means the set $S_\\Gamma (A\\cap U^-)$ .","In this paper, we need the following property of the scattering map.","Proposition 5.3 Assume that $T_1$ and $T_2$ are two invariant submanifolds of complementary dimensions in $\\Lambda $ .","Then $W^u(T_1)$ has a topologically transverse intersection with $W^s(T_2)$ inside $\\Gamma $ if and only if $S_\\Gamma (T_1\\cap U^-)$ has a topologically transverse intersection with $T_2\\cap U^+$ in $\\Lambda $ .","Let $\\bar{T}_1=(\\Omega ^-_\\Gamma )^{-1}(T_1\\cap U^-)\\subseteq \\Gamma $ and $\\bar{T}_2=(\\Omega ^+_\\Gamma )^{-1}(T_2\\cap U^+)\\subseteq \\Gamma $ .","A topologically transverse intersection of $W^u(T_1)$ with $W^s(T_2)$ in $\\Gamma $ occurs if and only if $\\bar{T}_1$ intersects $\\bar{T}_2$ topologically transversally in $\\Gamma $ , which is equivalent to $S_\\Gamma (T_1\\cap U^-)$ intersects topologically transversally $T_2\\cap U^+$ in $\\Lambda $ .","For the definition of topological transversality (topological crossing) see [8].","For the main result of this paper the normally hyperbolic invariant manifold $\\Lambda $ is assumed to be 2-dimensional, i.e., $l=2$ , and the invariant submanifolds $T_1,T_2$ in Proposition REF are 1-dimensional invariant tori."],["Topological method of correctly aligned windows","We describe briefly the topological method of correctly aligned windows.","We follow [55].","See also [24], [23], [41].","Definition 5.4 An $(n_1,n_2)$ -window in an $n$ -dimensional manifold $M$ , where $n_1+n_2=n$ , is a compact subset $W$ of $M$ together with a homeomorphism $\\chi $ from some open neighborhood of $[0,1]^{n_1}\\times [0,1]^{n_2}$ in $\\mathbb {R}^{n_1}\\times \\mathbb {R}^{n_2}$ to an open subset of $M$ , such that $W=\\chi ([0,1]^{n_1}\\times [0,1]^{n_2}),$ and with a choice of an `exit set' $W^{\\rm exit} =\\chi \\left(\\partial [0,1]^{n_1}\\times [0,1]^{n_2} \\right)$ and of an `entry set' $W^{\\rm entry}=\\chi \\left([0,1]^{n_1}\\times \\partial [0,1]^{n_2}\\right).$ Denote by $\\pi _{1}: \\mathbb {R}^{n_1}\\times \\mathbb {R}^{n_2}\\rightarrow \\mathbb {R}^{n_1}$ the projection onto the first component, and by $\\pi _{2}: \\mathbb {R}^{n_1}\\times \\mathbb {R}^{n_2}\\rightarrow \\mathbb {R}^{n_2}$ the projection onto the second component.","Definition 5.5 Let $W_1$ and $W_2$ be $(n_1,n_2)$ -windows, and let $\\chi _1$ and $\\chi _2$ be the corresponding local parametrizations.","Let $f$ be a continuous map on $M$ with $f(\\textrm {im}(\\chi _1))\\subseteq \\textrm {im}(\\chi _2)$ .","We say that $W_1$ is correctly aligned with $W_2$ under $f$ if the following conditions are satisfied: (i) $f(W^{\\rm exit}_1) \\cap (W_2) = \\emptyset $ and $f(W_1) \\cap W_2^{\\rm entry} = \\emptyset $ ; (ii) There exists $y_0 \\in [0,1]^{n_2}$ such that the curve $x\\in [0,1]^{n_1} \\mapsto \\hat{f} (x,y_0)$ , where $\\hat{f}:=\\chi _2^{-1}\\circ f\\circ \\chi _1$ , has the following properties: $\\hat{f}_{y_0}\\left( [0,1]^{n_1}\\right)\\subseteq \\mathbb {R}^{n_1}\\times (0,1)^{n_2},\\\\\\hat{f}_{y_0}\\left( \\partial [0,1]^{n_1}\\right)\\subseteq (\\mathbb {R}^{n_1}\\setminus [0,1]^{n_1})\\times (0,1)^{n_2}=\\emptyset ,\\\\\\deg (\\pi _{1}\\circ \\hat{f}_{y_0},0)=w\\ne 0.$ We call the integer $w\\ne 0$ in the above definition the degree of the alignment.","The following result is a topological version of the Shadowing Lemma.","Theorem 5.6 Let $\\lbrace W_i\\rbrace _{i\\in \\mathbb {Z}}$ , be a collection of $(n_1,n_2)$ -windows in $M$ , and let $f_i$ be a collection of continuous maps on $M$ .","If for each $i\\in \\mathbb {Z}$ , $W_i$ is correctly aligned with $W_{i+1}$ under $f_i$ , then there exists a point $p\\in W_0$ such that $(f_{i}\\circ \\cdots \\circ f_{0})(p)\\in W_{i+1}, \\textrm { for all } i\\in \\mathbb {Z}.","$ Moreover, assuming that there exists $k>0$ such that $W_{i}=W_{(i\\,{\\rm mod}\\, k)}$ and $f_{i}=f_{(i\\,{\\rm mod}\\, k)}$ for all $i\\in \\mathbb {Z}$ , then there exists a point $p$ as above that is periodic in the sense $(f_{k-1}\\circ \\cdots \\circ f_{0})(p)=p.$ The correct alignment of windows is robust, in the sense that if two windows are correctly aligned under a map, then they remain correctly aligned under a sufficiently small $C^0$ -perturbation of the map.","Robustness makes the method of correctly aligned windows appropriate for perturbative arguments, as well as for rigorous numerical experiments.","Also, the correct alignment satisfies a natural product property.","Given two windows and a map, if each window can be written as a product of window components, and if the components of the first window are correctly aligned with the corresponding components of the second window under the appropriate components of the map, then the first window is correctly aligned with the second window under the given map.","For example, if we consider a pair of windows in a neighborhood of a normally hyperbolic invariant manifold, if the center components of the windows are correctly aligned and the hyperbolic components of the windows are also correctly aligned, then the windows are correctly aligned.","Although the product property is quite intuitive, its rigorous statement is rather technical, so we will omit it here.","The details can be found in [23].","In Sections and , we will consider various windows lying on one of the manifolds $\\Lambda $ , $W^u(\\Lambda )$ , $W^s(\\Lambda )$ , or $M$ .","Without explicit mention, every such a window will be represented by the image of a rectangle through a local parametrization of the appropriate manifold.","We will also consider correct alignment relations of windows lying on the same manifold.","All the correct alignment relations in the arguments presented later in this paper have degree $w=1$ ."],["Existence of Birkhoff connecting orbits","In this section we state and prove an extension of the Corollary REF of the Birkhoff connecting orbit theorem, and an extension of Mather's theorem on shadowing of Aubry-Mather sets, specifically of Theorem REF .","The methodology is based on the topological approach of Hall and on the Jordan Curve Theorem.","The statements below will be used in the proof of the main theorem.","Throughout the section we assume that $f$ is an orientation preserving, boundary preserving, area preserving, monotone twist map of the annulus $A$ , as in Section , and we adopt the notation conventions from that section.","In Corollary REF , it was assumed that the restrictions of the map to the boundary tori of the BZI are topologically transitive, and it was inferred the existence of connecting orbits from an arbitrarily small neighborhood of some prescribed point on one boundary torus to an arbitrarily small neighborhood of some prescribed point on the other boundary torus.","In the statements below we prove the same result without the topological transitivity assumption.","Theorem 6.1 Suppose that $T_1$ and $T_2$ bound a BZI $\\mathcal {Z}$ .","Assume that $\\zeta _1\\in T_1$ and $\\zeta _2\\in T_2$ .","Fix a pair of neighborhoods $U$ of $\\zeta _1$ and $V$ of $\\zeta _2$ .","Then there exists a point $z\\in U$ and an integer $N>0$ such that $f^{N}(z)\\in V$ for some $N>0$ that can be chosen arbitrarily large.","Moreover, if we assume that $U,V$ are chosen so that $U\\cap \\mathcal {Z},V\\cap \\mathcal {Z}$ are topological disks, then there exists a point $z^{\\prime }\\in \\partial U$ such that $f^N(z^{\\prime })\\in \\partial V$ .","Consider the neighborhood $U$ of $\\zeta _1\\in T_1$ .","Choose two points $\\zeta ^{\\prime }_1,\\zeta ^{\\prime \\prime }_1\\in T_1\\cap U$ such that $\\zeta _1$ is between $\\zeta ^{\\prime }_1$ and $\\zeta ^{\\prime \\prime }_1$ and the portion of $T_1$ between $\\zeta ^{\\prime }_1$ and $\\zeta ^{\\prime \\prime }_1$ is contained in $\\textrm {int}(U)$ .","Choose a simple curve $\\gamma _0$ inside $U$ , with endpoints at $\\zeta ^{\\prime }_1$ and $\\zeta ^{\\prime \\prime }_1$ .","The curve $\\gamma _0$ together with the portion of $T_1$ between $\\zeta ^{\\prime }_1$ and $\\zeta ^{\\prime \\prime }_1$ determines a closed topological disk $U_0\\subseteq U$ , which is a one-sided compact neighborhood of $\\zeta _1$ in $\\mathcal {Z}$ .","Similarly, we can choose a one-sided compact neighborhood $V_0\\subseteq V$ of $\\zeta _2\\in T_2$ , whose boundary consists of a simple curve $\\eta _0$ connecting two points $\\zeta ^{\\prime }_2,\\zeta ^{\\prime \\prime }_2\\in T_2$ , with $\\zeta _2$ between $\\zeta ^{\\prime }_2$ and $\\zeta ^{\\prime \\prime }_2$ , and the portion of $T_2$ between $\\zeta ^{\\prime }_2$ and $\\zeta ^{\\prime \\prime }_2$ .","In this way, for proving the theorem we can consider the one-sided neighborhoods $U_0,V_0$ instead of $U,V$ , respectively.","Assume first that the interior of $U_0$ meets some Aubry-Mather set $\\Sigma _{\\rho _1}\\subseteq \\mathcal {Z}$ , and that the interior of $V_0$ meets some Aubry-Mather set $\\Sigma _{\\rho _2}\\subseteq \\mathcal {Z}$ .","Since $U_0$ and $V_0$ are neighborhoods of points in the Aubry-Mather sets $\\Sigma _{\\rho _1}$ and $\\Sigma _{\\rho _2}$ respectively, Theorem REF yields the existence of a forward orbit that goes from $U_0$ to $V_0$ .","Hence there exists $N>0$ such that $f^N(U_0)\\cap V_0\\ne \\emptyset $ .","Since $f^N(U_0)$ and $V_0$ are open topological disks that have common points as well as non-common points (e.g., the points in $T_1$ and $T_2$ , respectively), the Jordan Curve Theorem implies that $f^N(\\partial U_0)\\cap \\partial V_0\\ne \\emptyset $ .","Assume now that the interiors of $U_0,V_0$ do not meet any Aubry-Mather set.","We choose three Aubry-Mather sets $\\Sigma _{\\rho _1}$ , $\\Sigma _{\\rho _1^{\\prime }}$ , $\\Sigma _{\\rho _1^{\\prime \\prime }}$ in $\\mathcal {Z}$ , lying on three essential circles $C_{\\rho _1},C_{\\rho ^{\\prime }_1},C_{\\rho ^{\\prime \\prime }_1}$ , respectively, with ${\\rho _1}<{\\rho ^{\\prime }_1}<{\\rho ^{\\prime \\prime }_1}$ irrational rotation numbers, and $C_{\\rho _1}\\prec C_{\\rho ^{\\prime }_1}\\prec C_{\\rho ^{\\prime \\prime }_1}$ .","The existence of such vertically ordered Aubry-Mather sets follows from Theorem REF .","The proof of the theorem uses the following intermediate step.","Claim.","There exist $j^{\\prime }_*>j_*>0$ such that $\\mathcal {Z}\\setminus [f^{j_*}(U_0)\\cup f^{j^{\\prime }_*}(U_0)]$ contains a component $\\mathcal {U}$ which is an open topological disk that is a neighborhood of some point in $\\Sigma _{\\rho _1}$ .","Moreover, $j_*,j^{\\prime }_*$ can be chosen arbitrarily large.","A similar statement holds for $T_2$ .","Proof of the claim.","Let $p_1$ be a point in $\\Sigma _{\\rho ^{\\prime \\prime }_1}$ .","Let $W(p_1)$ be a small neighborhood of $p_1$ inside the BZI, which does not intersect $\\Sigma _{\\rho _1}$ and $\\Sigma _{\\rho _1^{\\prime }}$ .","By assumption, $U_0$ does not meet any of the sets $\\Sigma _{\\rho _1}$ , $\\Sigma _{\\rho ^{\\prime }_1}$ , $\\Sigma _{\\rho ^{\\prime \\prime }_1}$ .","By Lemma REF (iii) the closure of $\\bigcup _{j=0}^{\\infty }f^{-j}(W(p_1))$ contains $T_1$ , and in particular $\\zeta _1$ .","Since $U_0$ is a one-sided neighborhood of $\\zeta _1$ , there exists $j_1>0$ such that $f^{j_1}(U_0)\\cap W(p_1)\\ne \\emptyset $ .","In the covering space of the annulus, $f^{j_1}(U_0)$ intersects some copy $W^{h_1}:=W(p_1)+(h_1,0)$ of $W(p_1)$ , where $h_1$ is some positive integer.","Since $U_0$ does not intersect $\\Sigma _{\\rho _1}$ and $\\Sigma _{\\rho ^{\\prime }_1}$ , it follows that $f^{j_1}(U_0)$ does not intersect $\\Sigma _{\\rho _1}$ and $\\Sigma _{\\rho ^{\\prime }_1}$ .","On the other hand, $f^{j_1}(U_0)$ intersects the essential circles $C_{\\rho _1},C_{\\rho ^{\\prime }_1}$ , containing the Aubry-Mather sets $\\Sigma _{\\rho _1},\\Sigma _{\\rho ^{\\prime }_1}$ , respectively.","Let $\\gamma _{1}:[0,1]\\rightarrow U_0$ be a vertical curve, i.e., $\\gamma _1(0)\\in T_1$ and $\\pi _x(\\gamma _1(t))=\\pi _x(\\gamma _1(0))$ for all $t$ , such that $f^{j_1}(\\gamma _1(1))$ is an intersection point of $f^{j_1}(U_0)$ with $W^{h_1}$ .","The curve $f^{j_1}(\\gamma _1)$ is a positively tilted curve which crosses both essential circles $C_{\\rho _1}$ and $C_{\\rho ^{\\prime }_1}$ .","(See Remark REF .)","Since $f^{j_1}(U_0)$ is disjoint from $\\Sigma _{\\rho _1}$ , the intersections between $f^{j_1}(\\gamma _1)$ and $C_{\\rho _1}$ occur within the `gaps' of $\\Sigma _{\\rho _1}$ , i.e., within the open interval components of $C_{\\rho _1}\\setminus \\Sigma _{\\rho _1}$ .","We can assign an oriented intersection number between $f^{j_1}(\\gamma _1 )$ and each gap of $\\Sigma _{\\rho _1}$ .","Consider a homotopy $h_s:A\\rightarrow A$ , $s\\in [0,1]$ , such that $f^{j_1}(h_s(\\gamma _1))$ keeps the endpoints of $f^{j_1}(\\gamma _1)$ fixed for all $s$ , $f^{j_1}(h_s(\\gamma _1))$ does not intersect $\\Sigma _{\\rho _1}$ for any $s\\in [0,1]$ , and $f^{j_1}(h_1(\\gamma _1))$ is transverse to $C_{\\rho _1}$ (see [8]).","We set the oriented intersection number of $f^{j_1}(h_1(\\gamma _1))$ with $C_{\\rho _1}$ to be $+1$ at a point where the curve moves from below $C_{\\rho _1}$ to above $C_{\\rho _1}$ as $t$ increases, and to be $-1$ at a point where the curve moves from above $C_{\\rho _1}$ to below $C_{\\rho _1}$ as $t$ increases.","Then we assign an oriented intersection number between $f^{j_1}(h_1(\\gamma _1))$ and a gap of $\\Sigma _{\\rho _1}$ , by adding the oriented intersection numbers for all of the intersection points within that gap.","Since the oriented intersection number is preserved by homotopy, the oriented intersection number between $f^{j_1}( \\gamma _1)$ and a gap is, by definition, the oriented intersection number between $f^{j_1}(h_1(\\gamma _1))$ and that gap.","Then the oriented intersection number between $f^{j_1}( \\gamma _1 )$ and $C_{\\rho _1}$ is the sum of the oriented intersection numbers over all gaps.","Since the curve $f^{j_1}(\\gamma _1)$ starts from below $C_{\\rho _1}$ and ends above $C_{\\rho _1}$ , there exists a gap for which the oriented intersection number with $f^{j_1}(\\gamma _1)$ is positive.","We follow the curve $t\\mapsto f^{j_1}(\\gamma _1(t))$ starting with $t=0$ and we mark the first gap of $\\Sigma _{\\rho _1}$ that is crossed by $f^{j_1}(\\gamma _1)$ with a positive oriented intersection number; we denote by $a^1_{\\rho _1},b^1_{\\rho _1}$ the endpoints of this gap.","This means that if $f^{j_1}(\\gamma _1)$ crosses other gaps of $\\Sigma _{\\rho _1}$ that are to the left of this gap, it does so with 0 oriented intersection number.","(If the first gap crossed would be crossed with negative oriented intersection number, it would violate the positive tilt condition of $f^{j_1}(\\gamma _1)$ .)","Thus, when the curve $f^{j_1}(\\gamma _1)$ crosses the gap between $a^1_{\\rho _1},b^1_{\\rho _1}$ , it comes from below the circle $C_{\\rho _1}$ .","Following the curve segment of $f^{j_1}(\\gamma _1(t))$ after crossing the gap of endpoints $a^1_{\\rho _1},b^1_{\\rho _1}$ , we mark the first gap of $\\Sigma _{\\rho ^{\\prime }_1}$ that is crossed by $f^{j_1}(\\gamma _1)$ with positive oriented intersection number, and we denote by $a^1_{\\rho ^{\\prime }_1},b^1_{\\rho ^{\\prime }_1}$ its endpoints.","Similarly, when the curve $f^{j_1}(\\gamma _1)$ crosses the gap between $a^1_{\\rho ^{\\prime }_1},b^1_{\\rho ^{\\prime }_1}$ , it comes from below the circle $C_{\\rho ^{\\prime }_1}$ .","We claim that the left endpoint of the lower gap is to the right of the right endpoint of the upper gap, i.e., $\\pi _x(a^1_{\\rho _1})<\\pi _x(b^1_{\\rho ^{\\prime }_1})$ .","Otherwise, if $\\pi _x(a^1_{\\rho _1})\\ge \\pi _x(b^1_{\\rho ^{\\prime }_1})$ , then there exists an arc $f^{j_1}(\\gamma _1(s))$ , $s\\in [s_1,s_2]$ , such that $\\pi _x(f^{j_1}(\\gamma _1(s_1))=\\pi _x(f^{j_1}(\\gamma _1(s_2))=\\pi _x(a^1_{\\rho _1})$ , $\\pi _y(f^{j_1}(\\gamma _1(s_1))<\\pi _y(f^{j_1}(\\gamma _1(s_2))$ , and $\\pi _x(f^{j_1}(\\gamma _1(s))\\ge \\pi _x(a^1_{\\rho _1})$ for all $s\\in (s_1,s_2)$ .","This implies that either the angle deviation from the vertical $\\theta (s)$ along the curve $f^{j_1}(\\gamma _1)$ becomes non-positive for some $s\\in (s_1,s_2)$ , or that there exists another arc $f^{j_1}(\\gamma _1(\\tau ))$ , $\\tau \\in [\\tau _1,\\tau _2]$ , with $\\tau _1<\\tau _2\\pi _y(f^{j_1}(\\gamma _1(\\tau _2))$ , and $\\pi _x(f^{j_1}(\\gamma _1(s))\\le \\pi _x(f^{j_1}(\\gamma _1(\\tau ))$ for all $s\\in (s_1,s_2)$ and $\\tau \\in (\\tau _1,\\tau _2)$ .","In the first case we obtain a contradiction with the positive tilt condition on the curve $f^{j_1}(\\gamma _1)$ .","See Fig.","REF .","In the second case we obtain a contradiction with the fact that $f^{j_1}(\\gamma _1)$ comes from below $C_{\\rho _1}$ before crossing the gap of endpoints $a^1_{\\rho _1},b^1_{\\rho _1}$ .","The conclusion of this step is that the curve $f^{j_1}(\\gamma _1(t))$ passes through the gap between $a^1_{\\rho _1}$ and $b^1_{\\rho _1}$ of $\\Sigma _{\\rho _1}$ , from below $C_{\\rho _1}$ to above $C_{\\rho _1}$ , then it passes through the gap between $a^1_{\\rho ^{\\prime }_1}$ and $b^1_{\\rho ^{\\prime }_1}$ of $\\Sigma _{\\rho ^{\\prime }_1}$ , from below $C_{\\rho ^{\\prime }_1}$ to above $C_{\\rho ^{\\prime }_1}$ , and $\\pi _x(a^1_{\\rho _1})<\\pi _x(b^1_{\\rho ^{\\prime }_1})$ .","Figure: Violation of the positive tilt condition.Now we consider a one-sided rectangular neighborhood $U_1\\subseteq U_0$ of some point in $T_1$ , bounded below by $T_1$ , to the left by $\\gamma _1$ , and to the right by some other vertical curve segment $\\gamma _1^{\\prime }$ .","If $\\gamma ^{\\prime }_1$ is sufficiently close to $\\gamma _1$ , then, by continuity, the image of each vertical curve in $U_1$ under $f^{j_1}$ crosses the gap between $a^1_{\\rho _1}$ and $b^1_{\\rho _1}$ of $\\Sigma _{\\rho _1}$ with positive oriented intersection number, and crosses the gap between $a^1_{\\rho ^{\\prime }_1}$ and $b^1_{\\rho ^{\\prime }_1}$ of $\\Sigma _{\\rho ^{\\prime }_1}$ with positive oriented intersection number.","We choose and fix a set $U_1\\subseteq U_0$ with these properties.","By Lemma REF (iii) (see also Remark REF ) the closure of $\\bigcup _{j=0}^{\\infty }f^{-j}(W(p_1))$ contains $T_1$ , so there exists $j_2>j_1$ such that, in the annulus, $f^{j_2}(U_1)\\cap W(p_1)\\ne \\emptyset $ , and, in the covering space of the annulus, $f^{j_2}(U_1)$ intersects some copy $W^{h_2}:=W(p_1)+(h_2,0)$ of $W(p_1)$ for some positive integer $h_2>h_1$ .","(Due to the positive twist condition on $f$ and the assumption that the rotation numbers on the boundary components of the annulus are positive, the vertical line $\\lbrace x=\\pi _x(p_1)+h_1\\rbrace $ is mapped by $f^{j_2-j_1}$ to a positive tilted map to the right of $\\lbrace x=\\pi _x(p_1)+h_1\\rbrace $ , hence, if $j_2$ is large enough, we can choose $h_2>h_1$ .)","Then there exists a vertical curve $\\gamma _2:[0,1]\\rightarrow U_1$ , such that $f^{j_2}(\\gamma _2(1))$ is an intersection point of $f^{j_2}(U_1)$ with $W^{h_2}$ .","The curve $f^{j_2}(\\gamma _2(t))$ crosses $C_{\\rho _1}$ and $C_{\\rho ^{\\prime }_1}$ .","Let $a^2_{\\rho _1},b^2_{\\rho _1}$ be the endpoints of the leftmost gap of $\\Sigma _{\\rho _1}$ that is crossed by $f^{j_2}(\\gamma _2(t))$ with positive oriented intersection number equal, and let $a^2_{\\rho ^{\\prime }_1},b^2_{\\rho ^{\\prime }_1}$ be the endpoints of the leftmost gap of $C_{\\rho ^{\\prime }_1}$ that is crossed by $f^{j_2}(\\gamma _2(t))$ with positive oriented intersection number.","The image curve $f^{j_2}(\\gamma _2)$ is a positively tilted curve located on the `right side' of the positively tilted curve $f^{j_2}(\\gamma _1)$ , in the sense that any graph over $x$ that intersects both $f^{j_2}(\\gamma _1)$ and $f^{j_2}(\\gamma _2)$ has the leftmost intersection point with the $f^{j_1}(\\gamma _1)$ .","Therefore, the gap endpoints $a^2_{\\rho _1},b^2_{\\rho _1}$ are either the image under $f^{j_2-j_1}$ of the gap endpoints $a^1_{\\rho _1},b^1_{\\rho _1}$ found at the previous step, or are the image under $f^{j_2-j_1}$ of some other gap endpoints of $\\Sigma _{\\rho _1}$ located to the right of the gap between $a^1_{\\rho _1}$ and $b^1_{\\rho _1}$ .","Similarly, the gap endpoints $a^2_{\\rho ^{\\prime }_1},b^2_{\\rho ^{\\prime }_1}$ are either the image under $f^{j_2-j_1}$ of the gap endpoints $a^1_{\\rho ^{\\prime }_1},b^1_{\\rho ^{\\prime }_1}$ from the previous step, or are the image under $f^{j_2-j_1}$ of some other gap of $\\Sigma _{\\rho ^{\\prime }_1}$ located to the right of the gap between $a^1_{\\rho ^{\\prime }_1}$ and $b^1_{\\rho ^{\\prime }_1}$ .","Then, there exists a one-sided rectangular neighborhood $U_2\\subseteq U_1$ of some point in $T_1$ , bounded below by $T_1$ , to the left by $\\gamma _2$ , and to the right by some other vertical curve segment $\\gamma ^{\\prime }_2$ , such that the image of each vertical curve in $U_2$ under $f^{j_2}$ crosses the gap between $a^2_{\\rho _1}$ and $b^2_{\\rho _1}$ of $\\Sigma _{\\rho _2}$ with positive oriented intersection number, and crosses the gap between $a^2_{\\rho ^{\\prime }_1}$ and $b^2_{\\rho ^{\\prime }_1}$ of $\\Sigma _{\\rho ^{\\prime }_1}$ with positive oriented intersection number.","Recursively, we obtain a nested sequence of one-sided neighborhoods of points in $T_1$ , denoted $U_1\\supseteq U_2\\supseteq \\ldots U_m\\supseteq \\ldots $ , all contained in $U_0$ , and two sequences of positive integers $j_1\\rho _1$ .","Then, for some sufficiently large iterate $j_m$ the order of the gaps gets reversed in the annulus.","That is, we get the following ordering in terms of the angle coordinate in the covering space of the annulus: (i) $\\pi _x(a^m_{\\rho _1})<\\pi _x(b^m_{\\rho _1})< \\pi _x(a^1_{\\rho _1})+h_m<\\pi _x(b^1_{\\rho _1})+h_m$ , (ii) $\\pi _x(a^1_{\\rho ^{\\prime }_1})+h_m<\\pi _x(b^1_{\\rho ^{\\prime }_1})+h_m<\\pi _x(a^m_{\\rho ^{\\prime }_1})<\\pi _x(b^m_{\\rho ^{\\prime }_1})$ .","Since $f^{j_1}(U_1)$ and $f^{j_m}(U_m)$ are connected, the above ordering of the crossings with $C_{\\rho _1}$ and $C_{\\rho ^{\\prime }_1}$ implies that $f^{j_1}(U_1)$ and $f^{j_m}(U_m)$ have an intersection point above $C_{\\rho _1}$ .","As $U_1, U_m\\subseteq U_0$ , it follows that $f^{j_1}(U_0)$ and $f^{j_m}(U_0)$ have an intersection point above $C_{\\rho _1}$ .","Since $f^{j_1}(U_0)$ and $f^{j_m}(U_0)$ are connected and disjoint from the Aubry-Mather set $\\Sigma _{\\rho _1}\\subseteq C_{\\rho _1}$ , there exists a component $\\mathcal {U}$ of the complement $\\mathcal {Z}\\setminus [f^{j_1}(U_0)\\cup f^{j_m}(U_0)]$ which is an open topological disk containing some point $\\xi _1\\in \\Sigma _{\\rho _1}$ .","The boundary of $\\mathcal {U}$ consists of a finite union of sub-arcs of the boundaries of $f^{j_1}(U_0)$ and $f^{j_m}(U_0)$ , and possibly of curve segments of $T_1$ .","See Figure REF .","Letting $j_*=j_1$ and $j^{\\prime }_*=j_m$ ends the proof of the claim.","Figure: An open topological disk forming a neighborhood of a point in the Aubry-Mather set Σ ρ 1 \\Sigma _{\\rho _1}.We now apply the statement of the claim to $T_2$ , starting with the one-sided neighborhood $V_0$ of $\\zeta _2\\in T_2$ .","We choose three Aubry-Mather sets $\\Sigma _{\\rho _2}$ , $\\Sigma _{\\rho _2^{\\prime }}$ , $\\Sigma _{\\rho _2^{\\prime \\prime }}$ in $\\mathcal {Z}$ , lying on three essential circles $C_{\\rho _2},C_{\\rho ^{\\prime }_2},C_{\\rho ^{\\prime \\prime }_2}$ , respectively, with ${\\rho _2}>{\\rho ^{\\prime }_2}>{\\rho ^{\\prime \\prime }_2}$ irrational rotation numbers, and $C_{\\rho _2} \\succ C_{\\rho ^{\\prime }_2} \\succ C_{\\rho ^{\\prime \\prime }_2}$ .","The statement in the claim implies that there exists a neighborhood $\\mathcal {V}$ of some point $\\xi _2\\in \\Sigma _{\\rho _2}$ , homoeomorphic to an open disk, whose boundary consists of a finite union of sub-arcs in the boundaries of $f^{-l_*}(V_0)$ and $f^{-l^{\\prime }_*}(V_0)$ , for some $l_*0$ such that $f^N(\\textrm {int}(U _0))\\cap \\textrm {int}(V_0) \\ne \\emptyset $ .","Since the sets $f^{N}(U_0)$ and $V_0$ are topological disks have interior points in common, but also points that are not in common (namely the points lying on $T_1$ and $T_2$ , respectively), the Jordan Curve Theorem implies that $f^{N}(\\partial U _0)\\cap \\partial V_0\\ne \\emptyset $ .","The remaining case of the proof, when the interior of $U_0$ does intersect some Aubry-Mather set and the interior of $V_0$ does not, or when the interior of $U_0$ does intersect some Aubry-Mather set and the interior of $V_0$ does not, follows easily from the above arguments.","The next statement says that given two points on the boundary tori of a BZI, and a finite sequence of Aubry-Mather sets inside the zone, there exists an orbit that starts in a prescribed neighborhood of the point on the lower boundary torus, then moves on and shadows, in the sense of the ordering of the orbit, each Aubry-Mather set in the sequence, and ends in a prescribed neighborhood of the point on the upper boundary torus.","This result extends Theorem REF , and relies on the topological argument of Hall.","As in the previous theorem, we do not need any extra conditions on the dynamics on the boundary tori.","The resulting shadowing orbits are not necessarily minimal.","Theorem 6.2 Suppose that $T_1$ and $T_2$ bound a BZI $\\mathcal {Z}$ .","Let $\\zeta _1\\in T_1,\\zeta _2\\in T_2$ , $U$ be a neighborhood of $\\zeta _1$ , and $V$ a neighborhood of $\\zeta _2$ .","Let $\\lbrace \\Sigma _{\\omega _s}\\rbrace _{s\\in \\lbrace 1,\\ldots ,\\sigma \\rbrace }$ be a finite collection of Aubry-Mather sets inside $\\mathcal {Z}$ such that each $\\Sigma _{\\omega _s}$ lies on some essential circle $C_{\\omega _s}$ that is a graph over the $x$ -coordinate, with $C_{\\omega _s}\\prec C_{\\omega _{s^{\\prime }}}$ provided $\\omega _s<\\omega _{s^{\\prime }}$ .","Let $\\lbrace n_s\\rbrace _{s=1,\\ldots , \\sigma }$ be sequence of positive integers.","Then there exist a point $z\\in U$ , and a sequence of positive integers $\\lbrace m_s\\rbrace _{s=0,\\ldots ,\\sigma }$ , such that, for each $s\\in \\lbrace 1,\\ldots ,\\sigma \\rbrace $ , $\\begin{split}\\pi _x(f^j(w_s))<\\pi _x(f^j(z))<\\pi _x(f^j(\\bar{w}_s))\\textrm { for } \\\\ \\sum _{t=1}^{s-1} n_t+\\sum _{t=0}^{s-1}m_t\\le j\\le \\sum _{t=1}^{s} n_t+\\sum _{t=0}^{s-1}m_t,\\end{split}$ where $w_s$ and $\\bar{w}_s$ are some points in the Aubry-Mather set $\\Sigma _{\\omega _s}$ , and $f^{N}(z)\\in V$ for $N=\\sum _{t=1}^{\\sigma } n_t+\\sum _{t=0}^{\\sigma }m_t$ .","The number $N$ can be chosen arbitrarily large.","Moreover, if $U,V$ are chosen so that $U\\cap \\mathcal {Z},V\\cap \\mathcal {Z}$ are topological disks, then there exists a point $z^{\\prime }\\in \\partial U$ satisfying the ordering condition (REF ) such that $f^N(z^{\\prime })\\in \\partial V$ .","We use the construction of diagonal sets described in the sketch of the proof of Theorem REF ; for details see [29].","Part 1.","As in the proof of Theorem REF we can choose one sided compact neighborhoods $U_0$ of $\\zeta _1\\in T_1$ and $V_0$ of $\\zeta _2\\in T_2$ , such that $U_0\\cap \\mathcal {Z}$ and $V_0\\cap \\mathcal {Z}$ are topological disks.","We first prove the existence of a point $z\\in U_0$ satisfying (REF ) and such that $f^N(z)\\in V_0$ .","Let $C_{\\omega _1}$ be the essential circle containing $\\Sigma _{\\omega _1}$ .","We choose an Aubry-Mather set $\\Sigma _{\\rho _1}$ lying on some essential circle $C_{\\rho _1}$ , such that $C_{\\omega _1}\\prec C_{\\rho _1}$ .","We choose a point $p_1\\in \\Sigma _{\\rho _1}$ and a small neighborhood $W(p_1)$ of $p_1$ which does not intersect $\\Sigma _{\\omega _1}$ .","We assume that the interior of $U_0$ does not meet $\\Sigma _{\\omega _1}$ and $\\Sigma _{\\rho _1}$ , otherwise the proof follows as in [29].","By Lemma REF (iii) the closure of $\\bigcup _{j=0}^{\\infty }f^{-j}(W(p_1))$ contains $T_1$ , and in particular $\\zeta _1$ .","Proceeding as in the proof of Theorem REF , we obtain a nested sequence $U_1\\supseteq U_2\\supseteq \\ldots \\supseteq U_i$ of one-sided neighborhoods of points in $T_1$ , all contained in $U_0$ , and two sequences of positive integers $j_10$ and a point $q\\in U_0$ depending on $l$ such that $f^l(q)$ is in some prescribed neighborhood of a point $r\\in T_2$ , where $l$ can be chosen arbitrarily large.","Since the points of $T_1$ and $T_2$ have different rotation numbers hence move apart under iteration, there exists $l_0$ sufficiently large such that $f^{l_0}(T_1\\cap U_0)$ and $f^{l_0}(q)$ are separated by a fundamental interval of the annulus, i.e., $\\pi _x(f^{l_0}(r))-\\pi _x(f^{l_0}(q))>1$ for all $r\\in T_1\\cap U_0$ .","Let $x_0,\\bar{x}_0\\in \\mathbb {R}$ be such that $\\pi _x(f^{l_0}(r))\\bar{y}^{\\prime \\prime }_0\\rbrace $ is a neighborhood of an arc in $T_1$ , with the property that each point $(\\bar{x}^{\\prime \\prime }_0,\\bar{y}^{\\prime \\prime }_0)\\in \\bar{W}_0$ has the $y$ -coordinate smaller than that of any point in $f^{l_0}(\\textrm {cl}(U_0))\\cap \\lbrace x=\\bar{x}^{\\prime \\prime }_0\\rbrace $ .","Let $\\Sigma _{\\rho _1}$ be an Aubry-Mather set lying on an essential circle $C_{\\rho _1}$ that is below the essential circle $C_{\\omega _1}$ containing $\\Sigma _{\\omega _1}$ , let $p_1\\in C_{\\rho _1}$ , and let $W(p_1)$ be a small neighborhood of $p_1$ that does not intersect $\\Sigma _{\\omega _1}$ .","Then there exists $z_0\\in W_0$ and ${j_0}$ sufficiently large such that $f^{j_0}(z_0)\\in W(p_1)$ .","Since the curve $f^{j_0}(I^+_{z_0})$ is a positively tilted curve emerging from $T_2$ , the arguments used in Part 1 show that there is a gap of the Aubry-Mather set $\\Sigma _{\\omega _1}$ , between a pair of points $w_1,w^{\\prime }_1\\in \\Sigma _{\\omega _1}$ , such that $f^{j_0}(I^+_{z_0})$ crosses this gap with negative intersection number (where the parametrization of $f^{j_0}(I^+_{z_0})$ is chosen so that to $t=0$ it corresponds a point on $T_2$ ) and has its first intersection with $I_{w_1}$ below the point $w_1$ , provided ${j_0}$ is chosen large enough.","This implies that the image of the diagonal component of $f^{l_0}(\\textrm {cl}(U_0))\\cap B_{z_0,\\bar{w}_0}$ under $f^{j_0}$ has a component $\\Delta _0$ in $B_{w_1,w^{\\prime }_1}$ that satisfies the positive diagonal set conditions relative to its left side.","See Fig.","REF .","Figure: Construction of a positive diagonal set.Now, for all $j^{\\prime }>0$ , the image of $\\Delta _0$ under $f^{j^{\\prime }}$ also has a component that satisfies the positive diagonal set conditions relative to the left side of $B_{f^{j^{\\prime }}(w_1),f^{j^{\\prime }}(w^{\\prime }_1)}$ .","(This follows from the hereditary property of diagonal sets, see Section .)","In a similar fashion, there exist a point $\\bar{z}_0\\in \\bar{W}_0$ and $j^{\\prime }_0$ sufficiently large such that the positively tilted curve $f^{j^{\\prime }_0+j_0}(I^-_{\\bar{z}_0})$ , emerging from $T_1$ , crosses a gap of the Aubry-Mather set $\\Sigma _{\\omega _1}$ between a pair of points $\\bar{w}_1,\\bar{w}^{\\prime }_1\\in \\Sigma _{\\omega _1}$ , the oriented intersection number between $f^{j^{\\prime }_0+j_0}(I^-_{\\bar{z}_0})$ and this gap is positive, and the first intersection with $I_{\\bar{w}^{\\prime }_1}$ occurs above the point $\\bar{w}^{\\prime }_1$ .","Thus, the image of $\\Delta _0$ under $f^{j^{\\prime }_0}$ has a component that satisfies the positive diagonal set conditions relative to its right side.","In summary, the image of the diagonal set component of $f^{l_0}(\\textrm {cl}(U_0))\\cap B_{z_0, \\bar{z}_0}$ under $f^{j_0+j^{\\prime }_0}$ contains a component $\\Delta ^{\\prime }_0$ that is a positive diagonal in $B_{w_1,\\bar{w}^{\\prime }_1}$ , where $w_1,\\bar{w}^{\\prime }_1$ are two points in the Aubry-Mather set $\\Sigma _{\\omega _1}$ .","Moreover, the upper edge and the lower edge of $\\Delta ^{\\prime }_0$ are contained in $f^{j^{\\prime }_0+j_0+l_0}(\\partial U_0)$ .","We apply an analogous argument at the other boundary torus $T_2$ .","Given $V_0$ a neighborhood of a point $\\zeta _2\\in T_2$ , there exist $L>0$ and a pair of points $w_\\sigma ,\\bar{w}_\\sigma \\in \\Sigma _{\\omega _\\sigma }$ such that $f^{-L}(\\textrm {cl}(V_0))\\cap B_{w_\\sigma ,\\bar{w}_\\sigma }$ has a component $D^{\\prime \\prime }_\\sigma $ that is a negative diagonal in $B_{w_\\sigma ,\\bar{w}_\\sigma }$ .","The upper edge and the lower edge of this positive diagonal set are contained in $f^{-L}(\\partial V_0)$ .","Now we apply the argument from Part 1.","There exists a negative diagonal set $D_\\sigma $ in $B_{w_1,\\bar{w}_1}$ such that all points $z\\in D_\\sigma $ satisfy (REF ).","By the argument for Theorem REF , the upper and lower edge of $D_{\\sigma }$ lie on $f^{-j_\\sigma }(I^-_{w_\\sigma }),f^{-j_\\sigma }(I^+_{\\bar{w}_\\sigma })$ where $j_\\sigma =\\sum _{s=0}^{\\sigma }n_s+\\sum _{s=0}^{\\sigma -1}m_s$ .","Since negative and positive diagonal sets in the same vertical strip always intersect, the negative diagonal set $D_\\sigma $ intersects $\\Delta ^{\\prime }_0$ , and in particular it intersects its upper and lower edges that are contained in $f^{j^{\\prime }_0+j_0+l_0}(\\partial U_0)$ .","Iterating $\\Delta ^{\\prime }_0$ forward for $j_\\sigma $ times yields a positive diagonal set $D^{\\prime }_\\sigma $ in $B_{w_\\sigma ,\\bar{w}_\\sigma }$ .","The upper and lower edges of $D^{\\prime }_\\sigma $ are contained in $f^{j_\\sigma +j^{\\prime }_0+j_0+l_0}(\\partial U_0)$ .","The positive diagonal set $D^{\\prime }_\\sigma $ intersects the negative diagonal component $D^{\\prime \\prime }_\\sigma $ of $f^{-L}(\\textrm {cl}(V_0))\\cap B_{w_\\sigma ,\\bar{w}_\\sigma }$ .","In particular the upper and lower edges of $D^{\\prime }_{\\sigma }$ , that are contained in $f^{j_\\sigma +j^{\\prime }_0+j_0+l_0}(\\partial U_0)$ , intersect the upper and lower edges of $D^{\\prime \\prime }_\\sigma $ that are contained in $f^{-L}(\\partial V_0)$ .","Thus, there exists a point $z^{\\prime }\\in \\partial U_0$ that is taken by $f^{j_\\sigma +j^{\\prime }_0+j_0+l_0}$ to $\\partial V_0$ and satisfies the ordering relations (REF )."],["A shadowing lemma in normally hyperbolic invariant manifolds","In this section we present a shadowing lemma-type of result saying that, given a sequence of windows within a normally hyperbolic invariant manifold, consisting of pairs of windows correctly aligned under the scattering map, alternating with pairs of windows correctly aligned under some iterate of the inner map, then there exists a true orbit in the full space dynamics that follows these windows.","This result reduces the construction of windows within the full dimensional phase space to the construction of lower dimensional windows within the normally hyperbolic invariant manifold.","For this section, we assume a diffeomorphism $f:M\\rightarrow M$ on a manifold $M$ , and an $l$ -dimensional normally hyperbolic invariant manifold $\\Lambda \\subseteq M$ as in Subsection REF .","In the subsequent sections, we will apply Lemma REF only in the case when $\\Lambda $ is a 2-dimensional normally hyperbolic invariant manifold in $M$ , i.e., $l=2$ .","A more general version of this lemma which does not involve the scattering map, and some additional details and applications, appear in [16].","Lemma 7.1 Let $\\lbrace R_i,R^{\\prime }_i\\rbrace _{i\\in \\mathbb {Z}}$ be a bi-infinite sequence of $l$ -dimensional windows contained in $\\Lambda $ .","Assume that the following properties hold for all $i\\in \\mathbb {Z}$ : (i) $R_{i}\\subseteq U^-$ and $R^{\\prime }_{i}\\subseteq U^+$ .","(ii) $R_{i}$ is correctly aligned with $R^{\\prime }_{i+1}$ under the scattering map $S$ .","(iii) for each pair $R^{\\prime }_{i+1},R_{i+1}$ and for each $L>0$ there exists $L^{\\prime }>L$ such that $R^{\\prime }_{i+1}$ is correctly aligned with $R_{i+1}$ under the iterate $f_{\\mid \\Lambda }^{L^{\\prime }}$ of the restriction $f_{\\mid \\Lambda }$ of $f$ to $\\Lambda $ .","Fix any bi-infinite sequence of positive real numbers $\\lbrace \\varepsilon _i\\rbrace _{i\\in \\mathbb {Z}}$ .","Then there exist an orbit $(f^{n}(z))_{n\\in \\mathbb {Z}}$ of some point $z\\in M$ , an increasing sequence of integers $(n_i)_{i\\in \\mathbb {Z}}$ , and some sequences of positive integers $\\lbrace N_i\\rbrace _{i\\in \\mathbb {Z}}, \\lbrace K_i\\rbrace _{i\\in \\mathbb {Z}},\\lbrace M_i\\rbrace _{i\\in \\mathbb {Z}}$ , such that, for all $i\\in \\mathbb {Z}$ : $ \\begin{split}d(f^{n_i}(z),\\Gamma )<\\varepsilon _i,\\\\d(f^{n_i+N_{i+1}}(z), f_{\\mid \\Lambda }^{N_{i+1}}(R^{\\prime }_{i+1}))<\\varepsilon _{i+1},\\\\d(f^{n_{i}-M_{i}}(z), f_{\\mid \\Lambda }^{-M_{i}}(R_{i}))<\\varepsilon _{i},\\\\n_{i+1}=n_i+N_{i+1}+K_{i+1}+M_{i+1}.\\end{split}$ The idea of this proof is to `thicken' some appropriate iterates of the windows $R_{i},R^{\\prime }_{i}$ in $\\Lambda $ to full dimensional windows $W_i,W^{\\prime }_i$ in $M$ , so that $\\lbrace W_i,W^{\\prime }_i\\rbrace _{i\\in \\mathbb {Z}}$ form a sequence of windows that are correctly aligned under some appropriate maps.","We start with a brief sketch of the construction before we proceed to the formal proof.","We constructs some copies $\\bar{R}_{i},\\bar{R}^{\\prime }_{i+1}$ in $\\Gamma $ of $R_{i},R^{\\prime }_{i+1}$ , respectively, through the inverses of the wave maps (see Section REF ).","We then expand the rectangles $\\bar{R}_{i},\\bar{R}^{\\prime }_{i+1}$ into the hyperbolic directions to produce a pair of windows $\\bar{W}_{i},\\bar{W}^{\\prime }_{i+1}$ , respectively, which are correctly aligned under the identity map.","Then we take a backwards iterate of $\\bar{W}_{i}$ such that $f^{-M_{i}} (\\bar{W}_{i})$ is sufficiently close to $\\Lambda $ , and we construct a new window $W_i$ about $f^{-M_i}(R_i)$ such that $W_i$ is correctly aligned with $\\bar{W}_i$ under $f^{M_i}$ .","Similarly, we construct a window $W^{\\prime }_{i+1}$ about $f^{N_i}(R^{\\prime }_{i+1})$ such that $\\bar{W}^{\\prime }_{i+1}$ is correctly aligned with $W^{\\prime }_{i+1}$ under $f^{N_{i+1}}$ .","We are given that we can align $R^{\\prime }_{i+1}$ with $R_{i+1}$ under some high enough iterate.","Hence we can align $W^{\\prime }_{i+1}$ with $W_{i+1}$ under some iterate $f^{K_{i+1}}$ .","This construction can be continued inductively.","Notationwise, the $N_i$ 's are associated to forward iterations along the stable manifold, the $M_i$ 's to backwards iterations along the unstable manifold, and the $K_i$ 's to iterations following the inner dynamics of $f$ restricted to $\\Lambda $ .","See Fig.","REF .","Figure: Schematic illustration of the construction of windows for the shadowing lemma.","The windows R i ,R i+1 ' R_i,R^{\\prime }_{i+1} are depicted as large dots.Step 1.","Let $\\lbrace R,R^{\\prime }\\rbrace $ be a pair of $l$ -dimensional windows of the type $\\lbrace R_{i},R^{\\prime }_{i+1}\\rbrace $ , and let $\\lbrace \\varepsilon ,\\varepsilon ^{\\prime }\\rbrace $ stand for the corresponding $\\lbrace \\varepsilon _i,\\varepsilon _{i+1}\\rbrace $ .","Let $\\bar{R} =(\\Omega ^-_\\Gamma )^{-1}(R )$ and $\\bar{R}^{\\prime } =(\\Omega ^+_\\Gamma )^{-1}(R^{\\prime } )$ be the copies of $R $ and $R^{\\prime } $ , respectively, in the homoclinic channel $\\Gamma $ .","By making some arbitrarily small changes in the sizes of their exit and entry directions, we can alter the windows $\\bar{R} $ and $\\bar{R}^{\\prime } $ such that $R $ is correctly aligned with $\\bar{R} $ under $(\\Omega ^-_\\Gamma )^{-1}$ , $\\bar{R}$ is correctly aligned with $\\bar{R}^{\\prime } $ under the identity mapping, and $\\bar{R}^{\\prime } $ is correctly aligned with $R $ under $\\Omega ^+_\\Gamma $ .","We `thicken' the $l$ -dimensional windows $\\bar{R} $ and $\\bar{R}^{\\prime } $ in $\\Gamma $ , which are correctly aligned under the identity mapping, to $(l+n_u+n_s)$ -dimensional windows $\\bar{W} $ and $\\bar{W}^{\\prime } $ , respectively, that are correctly aligned in $M$ under the identity mapping as well.","We now explain the `thickening' procedure.","First, we describe how to thicken $\\bar{R} $ to a full dimensional window $\\bar{W} $ .","We choose some $0<\\bar{\\delta }<\\varepsilon $ and $0<\\bar{\\eta }<\\varepsilon $ .","At each point $x \\in \\bar{R} $ we choose an $n_u$ -dimensional closed ball $\\bar{B}_{\\bar{\\delta }}(x )$ of radius $\\bar{\\delta }$ centered at $x $ and contained in $W^u({x^- })$ , where $x^- =\\Omega ^-_\\Gamma (x )$ .","Let $\\bar{\\Delta }:=\\bigcup _{x \\in \\bar{R} }\\bar{B}^{u}_{\\bar{\\delta }}(x )$ .","Note that $\\bar{\\Delta }$ is contained in $W^u(\\Lambda )$ and is homeomorphic to a $(l+n_u)$ -dimensional rectangle.","We define the exit set and the entry set of this rectangle as follows: $\\begin{split}(\\bar{\\Delta })^{\\rm exit}:=\\bigcup _{x \\in (\\bar{R} )^{\\rm exit}}\\bar{B}^{u}_{\\bar{\\delta }}(x ) \\cup \\bigcup _{x \\in \\bar{R} }\\partial \\bar{B}^{u}_{\\bar{\\delta }}(x ),\\\\(\\bar{\\Delta })^{\\rm entry}:=\\bigcup _{x \\in (\\bar{R} )^{\\rm entry}}\\bar{B}^{u}_{\\bar{\\delta }}(x ).\\end{split}$ We consider the normal bundle $N$ to $W^u(\\Lambda )$ .","At each point $y \\in \\bar{\\Delta }$ , we choose an $n_s$ -dimensional closed ball $\\bar{B}^s_{\\bar{\\eta }}(y)$ centered at $y$ and contained in the image of $N_y$ under the exponential map $\\exp _{y }:T_{y }M\\rightarrow M$ .","We let $\\bar{W} :=\\bigcup _{y \\in \\bar{\\Delta }}\\bar{B}^{s}_{\\bar{\\eta }}(y )$ .","By the Tubular Neighborhood Theorem (see, e.g., [7]), we have that for $\\bar{\\eta }$ sufficiently small, $\\bar{W} $ is a homeomorphic copy of a $(l+n_u+n_s)$ -dimensional rectangle.","We now define the exit set and the entry set of $\\bar{W} $ as follows: $\\begin{split} (\\bar{W} )^{\\rm exit}:=\\bigcup _{y \\in (\\bar{\\Delta })^{\\rm exit}}\\bar{B}^{s}_{\\bar{\\eta }}(y ),\\\\(\\bar{W} )^{\\rm entry}:=\\bigcup _{y \\in (\\bar{\\Delta })^{\\rm entry}}\\bar{B}^{s}_{\\bar{\\eta }}(y ) \\cup \\bigcup _{y \\in (\\bar{\\Delta })}\\partial \\bar{B}^{s}_{\\bar{\\eta }}(y ).\\end{split}$ Second, we describe in a similar fashion how to thicken $\\bar{R}^{\\prime } $ to a full dimensional window $\\bar{W}^{\\prime } $ .","We choose $0<\\bar{\\delta }^{\\prime } <\\varepsilon ^{\\prime } $ and $0<\\bar{\\eta }^{\\prime } <\\varepsilon ^{\\prime }$ .","We consider the $(l+n_s)$ -dimensional rectangle $\\bar{\\Delta }^{\\prime } :=\\bigcup _{x^{\\prime } \\in \\bar{R}^{\\prime } }\\bar{B}^{s}_{\\bar{\\eta }^{\\prime } }(x^{\\prime })\\subseteq W^s(\\Lambda )$ , where $\\bar{B}^s_{\\bar{\\eta }^{\\prime } }(x^{\\prime } )$ is the $n$ -dimensional closed ball of radius $\\bar{\\eta }^{\\prime } $ centered at $x^{\\prime } $ and contained in $W^s(x^+ )$ , with $x^+ =\\Omega ^+_\\Gamma (x^{\\prime } )$ .","Its exit set and entry sets are defined as follows: $\\begin{split}(\\bar{\\Delta }^{\\prime } )^{\\rm exit}:=\\bigcup _{x \\in (\\bar{R}^{\\prime } )^{\\rm exit}}\\bar{B}^{s}_{\\bar{\\eta }^{\\prime } }(x^{\\prime } ),\\\\(\\bar{\\Delta }^{\\prime } )^{\\rm entry}:=\\bigcup _{x^{\\prime } \\in (\\bar{R}^{\\prime } )^{\\rm entry}}\\bar{B}^{s}_{\\bar{\\eta }^{\\prime } }(x^{\\prime } )\\cup \\bigcup _{x\\in (\\bar{R}^{\\prime } )}\\partial \\bar{B}^{s}_{\\bar{\\eta }^{\\prime } }(x^{\\prime } ).\\end{split}$ We define $\\bar{W}^{\\prime } :=\\bigcup _{y^{\\prime } \\in \\bar{\\Delta }^{\\prime } }\\bar{B}^{u}_{\\bar{\\delta }^{\\prime } }(y^{\\prime } )$ , where $\\bar{B}^u_{\\bar{\\delta }^{\\prime } }(y^{\\prime } )$ is the $n$ -dimensional closed ball centered at $y^{\\prime } $ and contained in the image of $N^{\\prime }_{y^{\\prime } }$ under the exponential map $\\exp _{y^{\\prime } }:T_{y^{\\prime } }M\\rightarrow M$ , with $N^{\\prime }$ being the normal bundle to $W^s(\\Lambda )$ .","For $\\bar{\\delta }^{\\prime } >0$ sufficiently small $\\bar{W}^{\\prime }$ is a homeomorphic copy of a $(l+n_u+n_s)$ -dimensional rectangle.","The exit set and the entry set of $\\bar{W}^{\\prime } $ are defined by: $\\begin{split} (\\bar{W}^{\\prime } )^{\\rm exit}:=\\bigcup _{y^{\\prime } \\in (\\bar{\\Delta }^{\\prime } )^{\\rm exit}}\\bar{B}^{u}_{\\bar{\\delta }^{\\prime } }(y^{\\prime } )\\cup \\bigcup _{y^{\\prime } \\in (\\bar{\\Delta }^{\\prime } )}\\partial \\bar{B}^{u}_{\\bar{\\delta }^{\\prime } }(y^{\\prime } ),\\\\(W^{\\prime } )^{\\rm entry}:=\\bigcup _{y^{\\prime } \\in (\\bar{\\Delta }^{\\prime } )^{\\rm entry}}\\bar{B}^{u}_{\\bar{\\delta }^{\\prime }_{i+1}}(y^{\\prime } ).\\end{split}$ This completes the description of the thickening of the $l$ -dimensional window $\\bar{R} $ into a $(l+n_u+n_s)$ -dimensional window $\\bar{W} $ , and of the thickening of the $l$ -dimensional window $\\bar{R}^{\\prime } $ into a $(l+n_u+n_s)$ -dimensional window $\\bar{W}^{\\prime } $ .","Note that by construction $\\bar{W} $ is contained in an $\\varepsilon $ -neighborhood of $\\Lambda $ and $\\bar{W}^{\\prime } $ is contained in an $\\varepsilon ^{\\prime }$ -neighborhood of $\\Gamma $ .","In order to make $\\bar{W} $ correctly aligned with $\\bar{W}^{\\prime } $ under the identity map, we choose $\\bar{\\delta }^{\\prime } $ sufficiently small relative to $\\bar{\\delta }$ , and $\\bar{\\eta }$ sufficiently small relative to $\\bar{\\eta }^{\\prime } $ .","Step 2.","We take a negative iterate $f^{-M}(\\bar{R})$ of $\\bar{R}$ , where $M>0$ .","We have that $f^{-M}(\\Gamma )$ is $\\varepsilon $ -close to $\\Lambda $ in the $C^1$ -topology, for all $M$ sufficiently large.","The vectors tangent to the fibers $W^u(x^-)$ in $\\bar{R}$ are contracted, and the vectors transverse to $W^u(\\Lambda )$ along $\\bar{R}$ are expanded by the derivative of $f^{-M}$ .","We choose $M$ sufficiently large so that $f^{-M}(\\bar{R})$ is $\\varepsilon $ -close to $f^{-M}(R)$ .","We now construct a window $W$ about $f^{-M}(R)$ that is correctly aligned with $f^{-M}(\\bar{W})$ under the identity.","Note that each closed ball $\\bar{B}^{u}_{\\delta }(x)$ , which is a part of $\\bar{\\Delta }$ , gets exponentially contracted as it is mapped onto $W^u(f^{-{M }}(x^-))$ by $f^{-{M }}$ .","By the Lambda Lemma (see the version in [44]), each closed ball $\\bar{B}^s_{\\eta }(y)$ , $y\\in \\bar{\\Delta }$ , $C^1$ -approaches some subset of $W^s(f^{-{M }}(y^-))$ under $f^{-{M }}$ , as $M \\rightarrow \\infty $ .","For $M $ sufficiently large, we can assume that $f^{-{M }}(\\bar{B}^s_{\\bar{\\eta }}(y))$ is $\\varepsilon $ -close to some ball in $W^s(f^{-{M }}(y^-))$ in the $C^1$ -topology, for all $y\\in \\bar{\\Delta }$ .","We fix $M$ with the above properties.","As $R $ is correctly aligned with $\\bar{R}$ under $(\\Omega ^-_\\Gamma )^{-1}$ , we have that $f^{-M }(R )$ is correctly aligned with $f^{-M }(\\bar{R} )$ under $({\\Omega ^-}_{f^{-M }(\\Gamma )})^{-1}$ .","In other words, $f^{-M }(R )$ is correctly aligned under the identity mapping with the projection of $f^{-M }(\\bar{R} )$ onto $\\Lambda $ along the unstable fibers.","To define the window $W$ , we use a local linearization of the normally hyperbolic invariant manifold.","By Theorem 1 in [49], there exists a homeomorphism $h$ of an open neighborhood of $(E^u\\oplus E^s)_{ \\mid \\Lambda }$ to an open neighborhood of $\\Lambda $ in $M$ such that $h\\circ Df = f\\circ h$ .","We select a point $x \\in f^{-M}( R )$ .","Since $R$ is contractible the bundles are trivial on $f^{-M}(R)$ and we can identify $(E^u\\oplus E^s)_{\\mid f^{-M }(R)}$ with $f^{-M }(R) \\times E^u_{x } \\times E^s_{x }$ .","Let us consider $0<\\delta <\\varepsilon $ and $0<\\eta <\\varepsilon $ .","We define a window $W $ as $W =h(f^{-M }( R ) \\times \\bar{B}^u_{\\delta }(0) \\times \\bar{B}^s_{\\eta }(0)),$ where $\\bar{B}^u_{\\delta }(0)$ is the closed ball centered at 0 of radius $\\delta $ in $E^u_{x }$ and $\\bar{B}^s_{\\eta }(0)$ is the closed ball centered at 0 of radius $\\eta $ in $E^s_{x }$ .","We define the exit set of $W $ as $(W )^{\\text{exit}}= h(f^{-M }(\\bar{R} ) \\times \\partial \\bar{B}^u_{\\delta }(0) \\times \\bar{B}^s_{\\eta }(0))\\cup h(f^{-M}(\\bar{R}^{\\text{exit}}) \\times B^u_{\\delta }(0) \\times \\bar{B}^s_{\\eta }(0)).$ Similarly, the entry set of $W$ is defined as $(W)^{\\text{entry}}= h(f^{-M}(\\bar{R}) \\times \\bar{B}^u_{\\delta }(0) \\times \\partial \\bar{B}^s_{\\eta }(0))\\cup h(f^{-M}(\\bar{R}^{\\text{entry}}) \\times B^u_{\\delta }(0) \\times \\bar{B}^s_{\\eta }(0)).$ In order to ensure the correct alignment of $W$ with $f^{-M}(\\bar{W})$ under the identity map, we choose $\\delta ,\\eta $ such that $h(f^{-M}( R) \\times \\bar{B}^u_{\\delta }(0) \\times \\lbrace 0\\rbrace )$ is correctly aligned with $f^{-M }(\\bar{\\Delta })$ under the identity map (the exit sets of both windows being in the unstable directions), and that each closed ball $f^{-M }(\\bar{B}^s_{\\eta })$ intersects $W$ in a closed ball that is contained in the interior of $f^{-M }(\\bar{B}^s_{\\eta })$ .","The existence of suitable $\\delta ,\\eta $ follows from the exponential contraction of $\\bar{\\Delta }$ under negative iteration, and from the Lambda Lemma applied to $\\bar{B}^s_{\\eta }(y)$ under negative iteration.","In a similar fashion, we construct a window $W^{\\prime }$ contained in an $\\varepsilon ^{\\prime }$ -neighborhood of $\\Lambda $ such that $\\bar{W}^{\\prime }$ is correctly aligned with $W^{\\prime }$ under $f^{N}$ .","The window $W^{\\prime }$ , and its entry and exit sets, are defined by: $\\begin{split}W^{\\prime } =&h(f^{N}( R^{\\prime }) \\times \\bar{B}^u_{\\delta ^{\\prime }}(0) \\times \\bar{B}^s_{\\eta ^{\\prime }}(0)),\\\\(W^{\\prime } )^{\\text{exit}}=& h(f^{N}( R^{\\prime }) \\times \\partial \\bar{B}^u_{\\delta ^{\\prime }}(0) \\times \\bar{B}^s_{\\eta ^{\\prime }}(0))\\\\&\\cup h(f^{N}( (R^{\\prime })^\\text{exit}) \\times \\bar{B}^u_{\\delta ^{\\prime }}(0) \\times \\bar{B}^s_{\\eta ^{\\prime }}(0)),\\\\(W^{\\prime } )^{\\text{entry}}=&h(f^{N}( R^{\\prime }) \\times \\bar{B}^u_{\\delta ^{\\prime }}(0) \\times \\partial \\bar{B}^s_{\\eta ^{\\prime }}(0))\\\\&\\cup h(f^{N}(( R^{\\prime })^\\text{entry}) \\times \\bar{B}^u_{\\delta ^{\\prime }}(0) \\times \\bar{B}^s_{\\eta ^{\\prime }}(0)),\\end{split}$ for some appropriate choices of radii $0<\\delta ^{\\prime } ,\\eta ^{\\prime } <\\varepsilon ^{\\prime }$ .","Step 3.","Suppose that we have constructed, as in Step 2, a window $W^{\\prime } $ about the $l$ -dimensional rectangle $f^{N }(R^{\\prime } )\\subseteq \\Lambda $ , and a window $W $ about the $l$ -dimensional rectangle $f^{-M }(R )\\subseteq \\Lambda $ .","Under positive iterations, the rectangle $\\bar{B}^u_{\\delta ^{\\prime } }(0)\\times \\bar{B}^s_{\\eta ^{\\prime }}(0)\\subseteq E^u\\oplus E^s$ gets exponentially expanded in the unstable direction and exponentially contracted in the stable direction by $Df$ .","Thus $\\bar{B}^u_{\\delta ^{\\prime } }(0)\\times \\bar{B}^s_{\\eta ^{\\prime } }(0)$ gets correctly aligned with $\\bar{B}^u_{\\delta }(0)\\times \\bar{B}^s_{\\eta }(0)$ under some power $Df^{L}$ of $Df$ , provided $L $ is sufficiently large.","This implies that $f^{L }(h(\\lbrace x \\rbrace \\times \\bar{B}^u_{\\delta ^{\\prime } }(0)\\times \\bar{B}^s_{\\eta ^{\\prime } }(0)))$ is correctly aligned with $h({f^{L }(x )}\\times \\bar{B}^u_{\\delta }(0)\\times \\bar{B}^s_{\\eta }(0))$ under the identity map (both rectangles are contained in $h({f^{L }(x )}\\times E^u\\times E^s)$ .","By assumption (iii), there exists $L^{\\prime } >\\max \\lbrace L ,N +M \\rbrace $ such that $R^{\\prime } $ is correctly aligned with $R $ under $f^{L^{\\prime } }$ .","This means that $f^{N }(R^{\\prime } )$ is correctly aligned with $f^{-M }(R )$ under $f^{K }$ with $K :=L^{\\prime } -N -M >0$ .","The product property of correctly aligned windows implies that $W^{\\prime } $ is correctly aligned with $W $ under $f^{K }$ , provided that $K $ is chosen as above.","Step 4.","We will now describe the process of constructing, based on Steps 1, 2, and 3, two bi-infinite sequences of windows $\\lbrace W_{i},W^{\\prime }_{i}\\rbrace _{i\\in \\mathbb {Z}}$ and $\\lbrace \\bar{W}_{i},\\bar{W}^{\\prime }_{i}\\rbrace _{i\\in \\mathbb {Z}}$ such that, for all $i\\in \\mathbb {Z}$ , $W_i$ is correctly aligned with $\\bar{W}_i$ under $f^{N_i}$ , $\\bar{W}_i$ is correctly aligned with $\\bar{W}^{\\prime }_{i+1}$ under the identity mapping, $\\bar{W}^{\\prime }_{i+1}$ is correctly aligned with $W^{\\prime }_{i+1}$ under $f^{M_i}$ , and $W^{\\prime }_{i+1}$ is correctly aligned with $W_{i+1}$ under $f^{K_{i+1}}$ .","The point of this step is that we can repeatedly choose the rectangles and the parameters at Steps 1, 2, and 3, in a consistent way, in order to produce infinite sequences of correctly aligned windows.","Staring with $i=0$ and continuing for all $i\\ge 0$ , we do the following.","For a given pair of $l$ -dimensional windows $\\lbrace R_i,R^{\\prime }_{i+1}\\rbrace $ , we consider the corresponding copies $\\lbrace \\bar{R}_i,\\bar{R}^{\\prime }_{i+1}\\rbrace $ in $\\Gamma $ such that $\\bar{R}_i$ is correctly aligned with $\\bar{R}^{\\prime }_{i+1}$ under the identity.","As in Step 1, we construct a pair of $(l+n_u+n_s)$ -dimensional windows $\\bar{W}_i$ about $\\bar{R}_i$ , and $\\bar{W}^{\\prime }_{i+1}$ about $\\bar{R}^{\\prime }_{i+1}$ .","The size of $\\bar{W}_i$ in the hyperbolic directions is given by some disks radii $\\bar{\\delta }_i,\\bar{\\eta }_i<\\varepsilon _i$ , and that of $\\bar{W}^{\\prime }_{i+1}$ by some disk radii $\\bar{\\delta }^{\\prime }_{i+1}, \\bar{\\eta }^{\\prime }_{i+1}<\\varepsilon _{i+1}$ .","We choose the quantities $\\bar{\\delta }_i,\\bar{\\eta }_i, \\bar{\\delta }^{\\prime }_{i+1}, \\bar{\\eta }^{\\prime }_{i+1}$ such that $\\bar{W}_i$ is correctly aligned with $\\bar{W}^{\\prime }_{i+1}$ under the identity.","Then, we consider the pair of $l$ -dimensional windows $\\lbrace R_{i+1},R^{\\prime }_{i+2}\\rbrace $ in $\\Lambda $ , and their corresponding copies $\\lbrace \\bar{R}_{i+1},\\bar{R}^{\\prime }_{i+2}\\rbrace $ in $\\Gamma $ .","As in Step 1, we construct a pair of $(l+n_u+n_s)$ -dimensional windows $\\bar{W}_{i+1}$ about $\\bar{R}_{i+1}$ , and $\\bar{W}^{\\prime }_{i+2}$ about $\\bar{R}^{\\prime }_{i+2}$ .","By choosing the quantities $\\bar{\\delta }_{i+1},\\bar{\\eta }_{i+1}, \\bar{\\delta }^{\\prime }_{i+2}, \\bar{\\eta }^{\\prime }_{i+2}$ as in Step 1 we can ensure that $\\bar{W}_{i+1}$ is correctly aligned with $\\bar{W}^{\\prime }_{i+2}$ .","We choose $N_{i+1},M_{i+1}$ large enough so that $f^{N_{i+1}}(\\bar{W}^{\\prime }_{i+1})$ is contained in an $\\varepsilon _{i+1}$ -neighborhood of $f^{N_{i+1}}(R^{\\prime }_{i+1})$ , and $f^{-M_{i+1}}(\\bar{W}_{i+1})$ is contained in an $\\varepsilon _{i+1}$ -neighborhood of $f^{-M_{i+1}}(R_{i+1})$ .","As in Step 2, we construct a window $W^{\\prime }_{i+1}$ about $f^{N_1}(R^{\\prime }_{i+1})$ such that $\\bar{W}^{\\prime }_{i+1}$ correctly aligned with $W^{\\prime }_{i+1}$ under $f^{N_{i+1}}$ , and a window $W_{i+1}$ about $f^{-M_1}(R_{i+1})$ such that $W_{i+1}$ correctly aligned with $\\bar{W}_{i+1}$ under $f^{M_{i+1}}$ .","This amounts to choosing the quantities $\\delta ^{\\prime }_{i+1}, \\eta ^{\\prime }_{i+1},\\delta _{i+1},\\eta _{i+1}$ as in Step 2 in order to ensure the correct alignment of the windows.","Then, we choose $K_{i+1}$ sufficiently large, and at least as large as $N_{i+1}+M_{i+1}$ , such that $W^{\\prime }_{i+1}$ is correctly aligned with $W_{i+1}$ under $f^{K_{i+1}}$ .","At this point, we have that $\\bar{W}^{\\prime }_{i+1}$ is correctly aligned with $W^{\\prime }_{i+1}$ under $f^{N_{i+1}}$ , $ W^{\\prime }_{i+1}$ is correctly aligned with $W_{i+1}$ under $f^{K_{i+1}}$ , and $W_{i+1}$ is correctly aligned with $\\bar{W}_{i+1}$ under $f^{M_{i+1}}$ .","Inductively, we obtain two sequences of windows $\\lbrace W_{i},W^{\\prime }_{i}\\rbrace _{i\\ge 0}$ and $\\lbrace \\bar{W}_{i},\\bar{W}^{\\prime }_{i}\\rbrace _{i\\ge 0}$ that satisfy the desired correct alignment conditions for all $i\\ge 0$ .","A similar inductive construction of windows can be done backwards starting with $W_0, W^{\\prime }_1$ .","In the end, we obtain two bi-infinite sequences of windows $\\lbrace W_{i},W^{\\prime }_{i}\\rbrace _{i\\in \\mathbb {Z}}$ and $\\lbrace \\bar{W}_{i},\\bar{W}^{\\prime }_{i}\\rbrace _{i\\ge 0}$ that satisfy the desired correct alignment conditions for all $i\\in \\mathbb {Z}$ .","By Theorem REF , there exists an orbit $\\lbrace f^n(z)\\rbrace _{n\\in \\mathbb {Z}}$ with $f^{n_i}(z)\\in \\bar{W}_i\\cap \\bar{W}^{\\prime }_{i+1}$ , $f^{n_i+N_{i+1}}(z)\\in W^{\\prime }_{i+1}$ , $f^{n_i+N_{i+1}+K_{i+1}}(z)\\in W_{i+1}$ , $f^{n_i+N_{i+1}+K_{i+1}+M_{i+1}}(z)\\in \\bar{W}_{i+1}\\cap \\bar{W}^{\\prime }_{i+2}$ , for all $i\\in \\mathbb {Z}$ .","Thus $n_{i+1}=n_i+N_{i+1}+K_{i+1}+M_{i+1}$ .","Such an orbit $\\lbrace f^n(z)\\rbrace _{n\\in \\mathbb {Z}}$ satisfies the properties required by Lemma REF ."],["Construction of correctly aligned windows","In this section we prove Theorem REF .","The methodology consists of constructing 2-dimensional windows in $\\Lambda $ about the prescribed invariant primary tori, BZI's, and Aubry-Mather sets inside the BZI's.","The successive pairs of windows are correctly aligned under the scattering map alternatively with powers of the inner map.","Lemma REF implies that there exist trajectories that follow these windows."],["Construction of correctly aligned windows across a BZI","In this section we will construct correctly aligned windows across a BZI between two successive transition chains of tori.","On each side of the BZI we will choose a one-sided neighborhood of a point on the boundary torus, and we will use Theorem REF or Theorem REF to cross over the BZI.","These one-sided neighborhoods are of a special type: their boundaries are images of some transition tori under the inner or outer dynamics.","We will construct some windows about the boundaries of these one-sided neighborhoods.","Then we will construct some other windows about the corresponding transition tori; these windows will be chosen so that they have a pair of sides lying on some nearby tori.","We will use this feature later to connect sequence of windows across the BZI's with sequences of windows along the transition chains.","Consider an annular region $\\Lambda _k$ in $\\Lambda $ that is a BZI, and is between two transition chains of invariant tori, as in (A5).","To simplify notation, we denote the tori at the boundary of $\\Lambda _k$ by $T_{a}$ and $T_{b}$ .","We choose a pair of transition tori $T_{i}, T_{j}$ in $\\Lambda $ as in (A5), ordered as follows: $T_{j}\\prec T_{i}\\prec T_{a}$ .","These tori are outside of the BZI $\\Lambda _k$ and on the same side of it as $T_{a}$ .","By (A5-iv) there exist $T_{i^{\\prime }}\\prec T_i$ and $T_{j^{\\prime }}\\prec T_j$ such that $T_{i^{\\prime }}$ is $\\varepsilon _i$ -close to $T_i$ and $T_{j^{\\prime }}$ is $\\varepsilon _j$ -close to $T_j$ , in the $C^0$ topology.","By (A5-ii) $S(T_j)$ intersects $T_i$ in a topologically transverse manner, so both $S(T_j)$ and $S(T_{j^{\\prime }})$ intersect both $T_i$ and $T_{i^{\\prime }}$ in a topologically transverse manner, provided $T_{i^{\\prime }}, T_{j^{\\prime }}$ are sufficiently close to $T_i,T_j$ , respectively.","Since $S$ is a diffeomorphism, $T_{i}, T_{i^{\\prime }}$ and the images of $T_{j}, T_{j^{\\prime }}$ under $S$ form a topological rectangle $D_{iji^{\\prime }j^{\\prime }}$ in $\\Lambda $ .","This rectangle may not be contained in the domain $U^-$ of the scattering map $S$ .","Provided that we choose the tori $T_{i}, T_{i^{\\prime }}$ sufficiently $C^0$ -close to one another, and also $T_{j}, T_{j^{\\prime }}$ sufficiently $C^0$ -close to one another, the rectangle $D_{iji^{\\prime }j^{\\prime }}$ will be sufficiently small so that some iterate $f_{\\mid \\Lambda }^{K_a}$ of $f_{\\mid \\Lambda }$ takes the rectangle $D_{iji^{\\prime }j^{\\prime }}$ into a rectangle $f_{\\mid \\Lambda }^{K_a}(D_{iji^{\\prime }j^{\\prime }})$ inside $U^-$ .","This is possible since each torus intersects $U^-$ by (A5-i), and the motion on the tori is topologically transitive by (A5-iii).","The rectangle $f_{\\mid \\Lambda }^{K_a}(D_{iji^{\\prime }j^{\\prime }})$ has a pair of sides lying on the tori $T_{i}, T_{i^{\\prime }}$ , and the other pair of sides on $(f_{\\mid \\Lambda }^{K_a}\\circ S)(T_{j}),(f_{\\mid \\Lambda }^{K_a}\\circ S)(T_{j^{\\prime }})$ .","The curves $(f_{\\mid \\Lambda }^{K_a}\\circ S)(T_{j}),(f_{\\mid \\Lambda }^{K_a}\\circ S)(T_{j^{\\prime }})$ are topologically transverse to both $T_{i},T_{i^{\\prime }}$ .","By assumption (A5-ii), $S(T_i)$ topologically crosses $T_{a}$ , so $S^{-1}(T_{a})$ topologically crosses $T_{i}$ .","We can ensure that the interior of $f_{\\mid \\Lambda }^{K_a}(D_{iji^{\\prime }j^{\\prime }})$ intersects $S^{-1}(T_{a})$ by choosing $K_a$ sufficiently large, and the tori in each pair $T_{i}, T_{i^{\\prime }}$ and $T_{j}, T_{j^{\\prime }}$ sufficiently $C^0$ -close to one another.","This implies that the image of $f_{\\mid \\Lambda }^{K_a}(D_{iji^{\\prime }j^{\\prime }})$ under $S$ is a topological rectangle in $\\Lambda $ which intersects $T_{a}$ , and the intersection of $S(f_{\\mid \\Lambda }^{K_a}(D_{iji^{\\prime }j^{\\prime }}))$ with $\\Lambda _k$ forms a one-sided neighborhood in $\\Lambda _k$ of some part of $T_{a}$ .","The boundary of $S(f_{\\mid \\Lambda }^{K_a}(D_{iji^{\\prime }j^{\\prime }}))\\cap \\Lambda _k$ consists of arcs of the curves $T_{a}, S(T_i), S(T_{i^{\\prime }}),S\\circ f_{\\mid \\Lambda }^{K_a}\\circ S(T_j),S\\circ f_{\\mid \\Lambda }^{K_a}\\circ S(T_{j^{\\prime }})$ .","We make a similar construction on the other side of the BZI $\\Lambda _k$ .","We choose a pair of transition tori $T_{k}, T_{l}$ with $ T_{b}\\prec T_{k}\\prec T_{l}$ , outside of the BZI $\\Lambda _k$ and on the same side as $T_{b}$ .","We also choose $T_{k}\\prec T_{k^{\\prime }}$ and $T_{l}\\prec T_{l^{\\prime }}$ such that $T_{k^{\\prime }}$ is $\\varepsilon _k$ -close to $T_k$ and $T_{l^{\\prime }}$ is $\\varepsilon _l$ -close to $T_l$ , and $S^{-1}(T_l), S^{-1}(T_{l^{\\prime }})$ are topologically transverse to both $T_k,T_{k^{\\prime }}$ , and so they form a a topological rectangle $D_{klk^{\\prime }l^{\\prime }}$ .","There exists $K_b$ sufficiently large such that $f_{\\mid \\Lambda }^{-K_b}(D_{klk^{\\prime }l^{\\prime }})\\subseteq U^-$ , its interior intersects $S(T_{b})$ , and $S^{-1}(f_{\\mid \\Lambda }^{-K_b}(D_{klk^{\\prime }l^{\\prime }}))$ forms a one-sided neighborhood in $\\Lambda _k$ of some part of $T_{b}$ .","The boundary of $S^{-1}(f_{\\mid \\Lambda }^{K_b}(D_{klk^{\\prime }l^{\\prime }}))\\cap \\Lambda _k$ consists of arcs of the curves $T_{b}, S(T_k), S(T_{k^{\\prime }}),S^{-1}\\circ f_{\\mid \\Lambda }^{K_b}\\circ S^{-1}(T_l),S^{-1}\\circ f_{\\mid \\Lambda }^{K_b}\\circ S^{-1}(T_{l^{\\prime }})$ .","Figure: Orbits across a BZIAt this stage we have obtained in $\\Lambda _k$ a one-sided neighborhood $(S\\circ f_{\\mid \\Lambda }^{K_a})(D_{iji^{\\prime }j^{\\prime }})$ of an arc in $T_{a}$ , and a one-sided neighborhood $(S^{-1}\\circ f_{\\mid \\Lambda }^{-K_b})(D_{klk^{\\prime }l^{\\prime }})$ of an arc in $T_{b}$ .","If we are under the assumptions (A1)-(A6), Theorem REF yields a point $x_a\\in \\partial (S\\circ f_{\\mid \\Lambda }^{K_a})(D_{iji^{\\prime }j^{\\prime }})$ whose image $x_b=f_{\\mid \\Lambda }^{K_{ab}}(x_a)$ under some power $f_{\\mid \\Lambda }^{K_{ab}}$ lies on $\\partial (S^{-1}\\circ f_{\\mid \\Lambda }^{-K_b})(D_{klk^{\\prime }l^{\\prime }})$ .","If we also assume (A7), Theorem REF yields a point $x_a\\in \\partial (S\\circ f_{\\mid \\Lambda }^{K_a})(D_{iji^{\\prime }j^{\\prime }})$ with $x_b=f_{\\mid \\Lambda }^{K_{ab}}(x_a)\\in \\partial (S^{-1}\\circ f_{\\mid \\Lambda }^{-K_b})(D_{klk^{\\prime }l^{\\prime }})$ as above, satisfying the additional conditions $\\pi _\\phi (f^j(w^k_s))<\\pi _\\phi (f^j(x_a))<\\pi _\\phi (f^j(\\bar{w}^k_s)), $ for each $s\\in \\lbrace 1,\\ldots , s_k\\rbrace $ , where $w^k_s,\\bar{w}^k_s\\in \\Sigma _{\\omega ^k_s}$ , and for all $j$ within a certain interval of integers.","The trajectories of all points sufficiently close to $x_a$ will satisfy these conditions as well.","In either case, there exist an arc $\\bar{e}^{\\prime }_a\\subseteq \\partial (S\\circ f_{\\mid \\Lambda }^{K_a})(D_{iji^{\\prime }j^{\\prime }})$ containing $x_a$ , and another arc $\\bar{e}_b\\subseteq \\partial (S^{-1}\\circ f_{\\mid \\Lambda }^{-K_b})(D_{klk^{\\prime }l^{\\prime }})$ containing $x_b$ , such that $f_{\\mid \\Lambda }^{K_{ab}}(\\bar{e}^{\\prime }_a)$ is topologically transverse to $\\bar{e}_b$ at $x_b$ .","The arc $\\bar{e}^{\\prime }_a$ lies on one of the sets $S({T_{i}}), S({T_{i^{\\prime }}}), (S\\circ f_{\\mid \\Lambda }^{K_{a}}\\circ S)(T_{j}), (S\\circ f_{\\mid \\Lambda }^{K_{a}}\\circ S)(T_{j^{\\prime }})$ .","Similarly, the arc $\\bar{e}_b$ lies on one of the sets $S^{-1}(T_{k}), S^{-1}({T_{k^{\\prime }}}), (S^{-1}\\circ f_{\\mid \\Lambda }^{-K_{b}}\\circ S^{-1})(T_{l}), (S^{-1}\\circ f_{\\mid \\Lambda }^{-K_{b}}\\circ S^{-1})(T_{l^{\\prime }})$ .","We define a 2-dimensional window $R^{\\prime }_a$ about $\\bar{e}^{\\prime }_a$ , and a 2-dimensional window $R_b$ about $\\bar{e}_b$ such that $R^{\\prime }_a$ is correctly aligned with $R_b$ under $f_{\\mid \\Lambda }^{K_{ab}}$ .","Informally, the exit direction of $R^{\\prime }_a$ is along $\\bar{e}^{\\prime }_a$ , and the exit direction of $R_b$ is across $\\bar{e}_b$ .","The formal construction now follows.","Since the arc $\\bar{e}^{\\prime }_a$ is an embedded 1-dimensional $C^0$ -submanifold of $\\Lambda $ , there exists a $C^0$ -local parametrization $\\chi ^{\\prime }_a:\\mathbb {R}^2\\rightarrow \\Lambda $ such that $\\chi ^{\\prime }_a([0,1]\\times \\lbrace 0\\rbrace )=\\bar{e}^{\\prime }_a$ , provided $\\bar{e}^{\\prime }_a$ is sufficiently small.","Then $R^{\\prime }_a=\\chi _a([0,1]\\times [-\\eta ^{\\prime }_a,\\eta ^{\\prime }_a]),$ is a topological rectangle.","We define the exit set of $R^{\\prime }_a$ as ${R^{\\prime }}_a^\\textrm {exit}=\\chi _a(\\partial [0,1]\\times [-\\eta ^{\\prime }_a,\\eta ^{\\prime }_a]).$ The definition of the entry set of of $R^{\\prime }_a$ follows by default.","Similarly, there exists a $C^0$ -local parametrization $\\chi _b:\\mathbb {R}^2\\rightarrow \\Lambda $ such that $\\chi _b(\\lbrace 0\\rbrace \\times [0,1])=\\bar{e}_b$ , and $R_b=\\chi _b([-\\delta _b,\\delta _b]\\times [0,1]),$ is a topological rectangle.","We define the exit set of $R_b$ as ${R}_b^\\textrm {exit}=\\chi _b(\\partial [-\\delta _b,\\delta _b]\\times [0,1]).$ By choosing $\\eta ^{\\prime }_a, \\delta _b$ sufficiently small, we ensure that $R^{\\prime }_a$ is correctly aligned with $R_b$ under $f_{\\mid \\Lambda }^{K_{ab}}$ (see Definition REF ).","See Figure REF .","We now construct other windows outside the BZI $\\Lambda _k$ .","We consider two cases: first case, when the arc $\\bar{e}^{\\prime }_{a}$ is a part of $S(T_{i})$ or $S(T_{i^{\\prime }})$ ; second case, when $\\bar{e}^{\\prime }_a$ is a part of $(S\\circ f_{\\mid \\Lambda }^{K_a}\\circ S)(T_{j})$ or $(S\\circ f_{\\mid \\Lambda }^{K_a}\\circ S)(T_{j^{\\prime }})$ .","Case 1.","In the first case, when $\\bar{e}^{\\prime }_a$ is a part of $S(T_{i})$ or $S(T_{i^{\\prime }})$ we proceed with the construction as follows.","Let us say that $\\bar{e}^{\\prime }_a$ is a part of $S(T_{i})$ .","We take the inverse image of $R^{\\prime }_a$ by $S$ .","This is a topological rectangle $S^{-1}(R^{\\prime }_a)$ about the torus $T_{i}$ .","We construct a window $R_{i}$ about an arc $\\bar{e}_{i}$ in $T_{i}$ , such that $R_{i}$ is correctly aligned with $S^{-1}(R^{\\prime }_{a})$ under the identity map, and with the exit direction of $R_{i}$ in the direction of $T_i$ .","Let $\\bar{e}_{i}$ be an arc in $T_{i}$ , and $\\chi _{i}:\\mathbb {R}^2\\rightarrow \\Lambda $ be a local parametrization with $\\chi _{i}([0,1]\\times \\lbrace 0\\rbrace )= \\bar{e}_{i}$ .","We define $R_{i}=\\chi _{i}([0,1]\\times [-\\eta _i,\\eta _i])\\\\{R}^\\textrm {exit}_{i}=\\chi _{i}(\\partial [0,1]\\times [-\\eta _i,\\eta _i]),$ where $\\eta _{i}>0$ is sufficiently small.","By choosing the arc $\\bar{e}_{i}$ sufficiently large so that $\\bar{e}_{i}\\supseteq S^{-1}(\\bar{e}^{\\prime }_a)$ , and by choosing $\\eta _{i}>0$ sufficiently small, we can ensure that $R_{i}$ is correctly aligned with $S^{-1}(R^{\\prime }_{a})$ under the identity map, or, equivalently, $R_{i}$ is correctly aligned with $R^{\\prime }_{a}$ under $S$ .","See Figure REF .","Figure: Construction of windows near the boundaries of a BZI – case 1We construct another window $R^{\\prime }_{i}$ about $T_{i}$ such that $R^{\\prime }_{i}$ is correctly aligned with $R_{i}$ under some power $f_{\\mid \\Lambda }^{K_{i}}$ of $f_{\\mid \\Lambda }$ , with $K_{i}>0$ .","The window $R^{\\prime }_{i}$ will have its exit direction across the torus $T_{i}$ .","We choose a pair of invariant primary tori $T^{\\rm lower}_{i}\\prec T_{i}\\prec T^{\\rm upper}_{i}$ , with $T^{\\rm lower}_{i}, T^{\\rm upper}_{i}$ located $\\varepsilon _{i}$ -close to $T_{i}$ in the $C^0$ -topology.","Such tori exist due to the assumption that $T_{i}$ is not an end torus in the transition chain, as in (A5-iv).","Moreover, we choose the tori $T^{\\rm lower}_{i}, T^{\\rm upper}_{i}$ sufficiently close to $T_{i}$ so that the components of $R_{i}^{\\rm entry}$ are outside the annulus between $T^{\\rm lower}_{i}$ and $ T^{\\rm upper}_{i}$ , with one component on one side and the other component on the other side of this annulus.","We choose an arc $\\bar{e}^{\\prime }_i\\subseteq T_i$ that lies on an edge of the topological rectangle $D_{iji^{\\prime }j^{\\prime }}$ .","Let $\\chi ^{\\prime }_{i}:\\mathbb {R}^2\\rightarrow \\Lambda $ be a $C^0$ -local parametrization with $\\chi ^{\\prime }_{i}(\\lbrace 0\\rbrace \\times [0,1])= \\bar{e}^{\\prime }_{i}$ , such that for some $\\delta ^{\\prime }_i$ sufficiently small, $\\chi ^{\\prime }_i (\\lbrace -\\delta ^{\\prime }_i\\rbrace \\times [0,1])\\subseteq T_i^{\\rm lower}$ and $\\chi ^{\\prime }_i (\\lbrace \\delta ^{\\prime }_i\\rbrace \\times [0,1])\\subseteq T_i^{\\rm upper}$ .","We define $R^{\\prime }_{i}$ by: $R^{\\prime }_{i}=\\chi ^{\\prime }_i ([-\\delta ^{\\prime }_i,\\delta ^{\\prime }_i]\\times [0,1]),\\\\{R^{\\prime }}_{i}^\\textrm {exit}=\\chi ^{\\prime }_i (\\partial [-\\delta ^{\\prime }_i,\\delta ^{\\prime }_i]\\times [0,1]).$ The exit set components of $R^{\\prime }_i$ lie on the tori $T_i^{\\rm upper}, T_i^{\\rm lower}$ neighboring $T_i$ .","Since the motion on the tori is topologically transitive, by (A5-iii), there exists $K_{i}>0$ such that $R^{\\prime }_{i}$ is correctly aligned with $R_{i}$ under $f_{\\mid \\Lambda }^{K_{i}}$ .","Indeed, to achieve correct alignment of these windows in the covering space of the annulus we only have to choose $K_{i}$ sufficiently large so that the two components of ${R^{\\prime }}_{i}^\\textrm {exit}$ , which are two arcs in $T^{\\rm lower}_{i}$ and $T^{\\rm upper}_{i}$ , are mapped by $f_{\\mid \\Lambda }^{K_{i}}$ on the opposite sides of the part of $R_{i}$ between $T^{\\rm lower}_{i}$ and $T^{\\rm upper}_{i}$ .","We note that the number of iterates $K_{i}$ of $f_{\\mid \\Lambda }$ needed to make $R^{\\prime }_{i}$ correctly aligned with $R_{i}$ may be different than the number of iterates $K_a$ which takes the topological rectangle $D_{iji^{\\prime }j^{\\prime }}$ onto $f_{\\mid \\Lambda }^{K_a}(D_{ihi^{\\prime }j^{\\prime }})$ .","See Figure REF .","The conclusion of this step is that we obtain the window $R^{\\prime }_{i}$ around $T_{i}$ , with its exit direction across $T_{i}$ , such that $R^{\\prime }_{i}$ is correctly aligned with $R_{i}$ under $f_{\\mid \\Lambda }^{K_{i}}$ .","Both windows $R^{\\prime }_i,R_i$ are contained in an $\\varepsilon _i$ -neighborhood of $T_i$ .","In the case when the edge $\\bar{e}^{\\prime }_a$ of $D_{iji^{\\prime }j^{\\prime }}$ is a part of $S(T_{i^{\\prime }})$ instead of $S(T_{i})$ , the construction goes similarly to the one above.","Case 2.","We now consider the second case, when the arc $\\bar{e}^{\\prime }_a$ of $\\partial (S\\circ f_{\\mid \\Lambda }^{K_a})(D_{iji^{\\prime }j^{\\prime }})$ is a part of $(S\\circ f_{\\mid \\Lambda }^{K_a}\\circ S)(T_{j})$ or $(S\\circ f_{\\mid \\Lambda }^{K_a}\\circ S)(T_{j^{\\prime }})$ .","Let us say that $\\bar{e}^{\\prime }_a$ is a part of $(S\\circ f_{\\mid \\Lambda }^{K_a}\\circ S)(T_{j})$ .","We construct a window $R^{\\prime }_a$ in $\\Lambda $ about $\\bar{e}^{\\prime }_a$ as before; the exit set of $R^{\\prime }_a$ is in a direction along $\\bar{e}^{\\prime }_a$ , and the size of $R^{\\prime }_a$ in the direction across $R_a$ is given by some parameter $\\eta ^{\\prime }_a$ .","See Figure REF .","We consider the inverse image $S^{-1}(R^{\\prime }_a)$ of $R^{\\prime }_a$ by $S$ , which is a topological rectangle about $(f_{\\mid \\Lambda }^{K_a}\\circ S)(T_{j})$ .","Let $\\bar{e}^{\\prime \\prime }_j$ be an arc in $(f_{\\mid \\Lambda }^{K_a}\\circ S)(T_{j})$ such that $S(\\bar{e}^{\\prime \\prime }_j)\\supset \\bar{e}^{\\prime }_a$ , and let $\\chi ^{\\prime \\prime }_j :\\mathbb {R}^2\\rightarrow \\Lambda $ be a local parametrization with $\\chi ^{\\prime \\prime }_{j}([0,1]\\times \\lbrace 0\\rbrace )= \\bar{e}^{\\prime \\prime }_{j}$ .","We define $R^{\\prime \\prime }_{j}$ by: $R^{\\prime \\prime }_{j}=\\chi ^{\\prime \\prime }_j ([0,1]\\times [-\\eta ^{\\prime \\prime }_j,\\eta ^{\\prime \\prime }_j]) ,\\\\{R^{\\prime \\prime }}_{j}^\\textrm {exit}=\\chi ^{\\prime \\prime }_j (\\partial [0,1]\\times [-\\eta ^{\\prime \\prime }_j,\\eta ^{\\prime \\prime }_j]).$ By choosing $\\eta ^{\\prime \\prime }_{j}>0$ sufficiently small, we can ensure that $R^{\\prime \\prime }_{j}$ is correctly aligned with aligned with $R^{\\prime }_{a}$ under $S$ .","The window $R^{\\prime \\prime }_{j}$ is in a neighborhood of an edge of the topological rectangle $f_{\\mid \\Lambda }^{K_a}(D_{iji^{\\prime }j^{\\prime }})$ .","The exit set of $R^{\\prime \\prime }_{j}$ is in the direction of $(f_{\\mid \\Lambda }^{K_a}\\circ S)(T_{j})$ .","See Figure REF .","Figure: Construction of windows near the boundaries of a BZI – case 2We construct a window $R^{\\prime \\prime \\prime }_{j}$ about $S(T_j)$ such that $R^{\\prime \\prime \\prime }_{j}$ is correctly aligned with $R^{\\prime \\prime }_{j}$ under $f_{\\mid \\Lambda }^{K_{a}}$ .","We define $R^{\\prime \\prime \\prime }_{j} =\\chi ^{\\prime \\prime \\prime }_j ([0,1]\\times [-\\eta ^{\\prime \\prime \\prime }_j,\\eta ^{\\prime \\prime \\prime }_j]) ,\\\\{R^{\\prime \\prime \\prime }}_{j}^\\textrm {exit}=\\chi ^{\\prime \\prime \\prime }_j (\\partial [0,1]\\times [-\\eta ^{\\prime \\prime \\prime }_j,\\eta ^{\\prime \\prime \\prime }_j]),$ where $\\bar{e}^{\\prime \\prime \\prime }_{j}$ is an arc of $S(T_{j})$ with $f_{\\mid \\Lambda }^{K_a}(\\bar{e}^{\\prime \\prime \\prime }_{j})\\supseteq \\bar{e}^{\\prime \\prime }_j$ , $\\chi ^{\\prime \\prime \\prime }_j :\\mathbb {R}^2\\rightarrow \\Lambda $ is a local parametrization with $\\chi ^{\\prime \\prime \\prime }_{j}([0,1]\\times \\lbrace 0\\rbrace )= \\bar{e}^{\\prime \\prime \\prime }_{j}$ , and $\\eta ^{\\prime \\prime \\prime }_j>0$ is sufficiently small.","The arc $\\bar{e}^{\\prime \\prime \\prime }_{j}$ is contained in one of the edges of the topological rectangle $D_{iji^{\\prime }j^{\\prime }}$ .","For suitable $\\eta ^{\\prime \\prime \\prime }_{j}>0$ , we can ensure that $R^{\\prime \\prime \\prime }_{j}$ is correctly aligned with aligned with $R^{\\prime \\prime }_{j}$ under $f_{\\mid \\Lambda }^{K_a}$ .","Moreover, we choose $R^{\\prime \\prime \\prime }_{j},R^{\\prime \\prime }_{j}$ so that these windows are both contained in an $\\varepsilon _j$ neighborhood of $T_j$ .","The exit set of $R^{\\prime \\prime \\prime }_{j}$ is in a direction along $S(T_{j})$ .","We take the inverse image $S^{-1}(R^{\\prime \\prime \\prime }_{j})$ of $R^{\\prime \\prime \\prime }_{j}$ under $S$ , which is a topological rectangle about an arc in $T_{j}$ .","In a fashion similar to Case 1, we construct two windows $R^{\\prime }_j,R_j$ about $T_j$ such that $R^{\\prime }_j$ is correctly aligned with $R_j$ under some power $f_{\\mid \\Lambda }^{K_j}$ of $f_{\\mid \\Lambda }$ , and $R_j$ is correctly aligned with $R^{\\prime \\prime \\prime }_j$ under $S$ .","The exit of $R^{\\prime }_j$ is chosen in a direction across $T_j$ , and the exit set components ${R^{\\prime }}^{\\rm exit}_j$ of $R^{\\prime }_j$ lie on two invariant primary tori $T_j^{\\rm upper}$ , $T_j^{\\rm lower}$ neighboring $T_j$ , that are $\\varepsilon _j$ -close to $T_j$ and contain $T_j$ between them.","The exit of $R_j$ is in a direction along $T_j$ .","The windows $R^{\\prime }_j,R_j$ are chosen to lie in an $\\varepsilon _j$ neighborhood of $T_j$ relative to the $C^0$ -topology.","The case when the arc $\\bar{e}^{\\prime }_a$ is a part of $(S\\circ f_{\\mid \\Lambda }^{K_a}\\circ S)(T_{j^{\\prime }})$ is treated similarly.","This concludes the construction of correctly aligned windows in $\\Lambda $ , starting with the window $R^{\\prime }_{a}$ about $T_{a}$ , and moving backwards along transition tori that are on the same side as $T_{a}$ of the BZI $\\Lambda _k$ .","This construction yields a sequence of windows of the type $R^{\\prime }_{i}, R_{i}, R^{\\prime }_{a},$ in the first case, or $R^{\\prime }_{j}, R_{j}, R^{\\prime \\prime \\prime }_{j}, R^{\\prime \\prime }_{j}, R^{\\prime }_{a},$ in the second case.","In the first case, $R^{\\prime }_{i}$ is correctly aligned with $R_{i}$ under some power $f_{\\mid \\Lambda }^{K_{i}}$ of the inner map, and $R_{i}$ is correctly aligned with $R^{\\prime }_{a}$ under the outer map $S$ .","The exit direction of $R^{\\prime }_{i}$ is across the torus $T_{i}$ , and its exit set components lie on some invariant primary tori that are $\\varepsilon _i$ -close to $T_{i}$ .","The windows $R^{\\prime }_i,R_i$ are contained in an $\\varepsilon _i$ neighborhood of $T_i$ .","In the second case, $R^{\\prime }_{j}$ is correctly aligned with $R_{j}$ under some power $f_{\\mid \\Lambda }^{K_{j}}$ of the inner map, $R_{j}$ is correctly aligned with $R^{\\prime \\prime \\prime }_{j}$ under the outer map $S$ , $R^{\\prime \\prime \\prime }_{j}$ is correctly aligned with $R^{\\prime \\prime }_{j}$ under some power $f_{\\mid \\Lambda }^{K_{a}}$ of the inner map, and $R^{\\prime \\prime }_{j}$ is correctly aligned with $R^{\\prime }_{a}$ under the outer map $S$ .","The exit direction of $R_{j}$ is across the torus $T_{j}$ , and its exit set components lie on some invariant primary tori that are $\\varepsilon _j$ -close to $T_{j}$ .","The windows $R^{\\prime }_j,R_j$ are contained in an $\\varepsilon _j$ neighborhood of $T_j$ , and the windows $R^{\\prime \\prime \\prime }_j,R^{\\prime \\prime }_j$ are contained in an $\\varepsilon _i$ neighborhood of $T_i$ .","We proceed with a similar construction on the other side of the BZI between $\\Lambda _k$ , that is, on the same side of the BZI as $T_b$ .","We have already defined the window $R_b$ about $T_b$ that $R^{\\prime }_a$ is correctly aligned with $R_b$ under $f_{\\mid \\Lambda }^{K_{ab}}$ .","Starting with the window $R_{b}$ and moving forward along the transition chain $T_b,T_k,T_l$ , we construct a sequence of windows of the type $R_{b}, R^{\\prime }_{k} $ or of the type $R_{b}, R^{\\prime \\prime \\prime }_{k}, R^{\\prime \\prime }_{k}, R^{\\prime }_{l},$ satisfying the correct alignment conditions below.","In the first case, $R_{b}$ is correctly aligned with $R^{\\prime }_{k}$ under the outer map $S$ .","The exit direction of $R^{\\prime }_k$ is across $T_k$ , and the exit set components lie on two invariant primary tori $\\varepsilon _k$ -close to $T_k$ .","Moreover, $R^{\\prime }_{k}$ is contained in an $\\varepsilon _k$ -neighborhood of $T_k$ .","In the second case, $R^{\\prime }_{b}$ is correctly aligned with $R^{\\prime \\prime \\prime }_{k}$ under the outer map $S$ , $R^{\\prime \\prime \\prime }_{k}$ is correctly aligned with $R^{\\prime \\prime }_{k}$ under some power $f_{\\mid \\Lambda }^{K_b}$ of the inner map, and $R^{\\prime \\prime }_{k}$ is correctly aligned with $R^{\\prime }_{l}$ under the outer map $S$ .","The exit direction of $R^{\\prime }_{l}$ is across $T_{l}$ , and its exit set components lie on some invariant primary tori that are $\\varepsilon _l$ -close to $T_{l}$ .","The windows $R^{\\prime \\prime \\prime }_{k}, R^{\\prime \\prime }_{k}$ are contained in an $\\varepsilon _k$ -neighborhood of $T_k$ , and $R^{\\prime }_l$ is contained in an $\\varepsilon _l$ -neighborhood of $T_l$ .","Similar statements apply when instead of windows around $T_{i},T_{j},T_{k},T_{l}$ we construct windows about $T_{i^{\\prime }},T_{j^{\\prime }},T_{k^{\\prime }},T_{l^{\\prime }}$ , respectively.","The conclusion of this section is that, by combining a sequence of correctly aligned window of the type (REF ) or (REF ) with a sequence of correctly aligned window of the type (REF ) or (REF ) we obtain a finite sequence of correctly aligned windows that crosses the BZI $\\Lambda _k$ .","The shadowing lemma-type of result Lemma REF yields an orbit that visits some prescribed neighborhood in the phase space of each window in $\\Lambda $ .","In particular, the shadowing orbit has points that go close to the transition tori."],["Construction of correctly aligned windows across annular regions separated by invariant tori","We consider an annular region in $\\Lambda $ between two transition chains of invariant tori.","Inside this annular region, we assume the existence of a finite collection of invariant tori that separate the region, as in (A6$^{\\prime }$ -i), and are vertically ordered as in (A6$^{\\prime }$ -ii).","Thus, the annular region between the transition chains is not a BZI.","We also assume that the scattering map satisfies the non-degeneracy condition (A6$^{\\prime }$ -iii) on these invariant tori.","The situation presented in this section is non-generic.","Assume that the annular region in $\\Lambda $ is bounded by a pair of invariant Lipschitz tori $T_{a}$ and $T_{b}$ .","Let $T_a$ be the end torus of the transition chain of one side, and $T_b$ be the end torus of the transition chain on the other side.","We assume that inside the region in the annulus bounded by $T_a$ and $T_b$ there exist a finite collection of invariant tori $\\lbrace \\Upsilon _{h}\\rbrace _{h=1,\\ldots ,k-1}$ .","Each $\\Upsilon _{h}$ is either an isolated invariant primary torus, or consists of a hyperbolic periodic orbit together with the upper branches, or with the lower branches, of its stable and unstable manifolds; the stable and unstable manifolds are assumed to coincide.","In the second case, there is another invariant torus, say $\\Upsilon _{h+1}$ , formed by the remaining branches of the same hyperbolic periodic orbit as for $\\Upsilon _{h}$ .","$\\Upsilon _{h+1}$ is assumed to be above $\\Upsilon _{h}$ , relative to the $I$ -coordinate, and is assumed to share with $\\Upsilon _{h}$ only the points of the periodic orbit.","The region in the annulus bounded by $\\Upsilon _{h}$ and $\\Upsilon _{h+1}$ is referred to as a resonant region.","Thus, the region in the annulus between $T_a$ and $T_b$ is divided into a finite number of BZI's and resonant regions.","The constructions in Subsection REF provide a one-sided neighborhood of the type $(S\\circ f_{\\mid \\Lambda }^{K_a})(D_{iji^{\\prime }j^{\\prime }})$ of some point in $T_a$ , and a one-sided neighborhood $(S^{-1}\\circ f_{\\mid \\Lambda }^{-K_b})(D_{klk^{\\prime }l^{\\prime }})$ of some point in $T_b$ .","Let us denote $D_0=(S\\circ f_{\\mid \\Lambda }^{K_a})(D_{iji^{\\prime }j^{\\prime }})$ and $D_{k+1}=(S^{-1}\\circ f_{\\mid \\Lambda }^{-K_b})(D_{klk^{\\prime }l^{\\prime }})$ .","Assume that $\\Upsilon _1$ is an isolated invariant primary torus.","We have that $S^{-1}(\\Upsilon _{1})$ forms with $\\Upsilon _{1}$ a topological disk $D_1$ between $T_a$ and $\\Upsilon _1$ , which is mapped by $S$ onto a topological disk $S(D_{1})$ between $\\Upsilon _1$ and $\\Upsilon _{2}$ .","Theorem REF provides us a trajectory that starts from $\\partial D_0$ and ends at $\\partial D_1$ .","In particular, there exist $K_1>0$ and a component of $f^{K_1}(D_{0})\\cap D_1$ that is a topological disk $D^{\\prime }_1$ whose boundary contains an arc of $S^{-1}(T_1)$ .","The image of $D^{\\prime }_1$ under $S$ is a one-sided neighborhood $D^{\\prime \\prime }_1\\subseteq S(D_1)$ of some point in $\\Upsilon _1$ , such that $D^{\\prime \\prime }_1$ is contained in the region between $\\Upsilon _1$ and $\\Upsilon _2$ .","The boundary of $S(D^{\\prime }_1)$ consists of an arc in $\\Upsilon _1$ and an arc in $(S\\circ f^{K_1}(\\partial D_{0}))$ .","See Fig.","REF .","Figure: Crossing over an isolated invariant primary torus.Now assume that $\\Upsilon _1$ consists of a hyperbolic periodic orbit, together with the lower branches of its stable and unstable manifolds, and that $\\Upsilon _2$ consists of the same hyperbolic periodic orbit, together with the upper branches of the stable and unstable manifolds.","The stable and unstable manifolds are assumed to coincide.","Excepting for the common points, $\\Upsilon _{1}$ is below $\\Upsilon _{2}$ .","Thus, $\\Upsilon _1$ and $\\Upsilon _2$ enclose a resonant region within the annulus.","We have that $S^{-1}(\\Upsilon _{1})$ forms with $\\Upsilon _{1}$ a topological disk $D_1$ between $T_a$ and $\\Upsilon _1$ , which is mapped by $S$ onto a topological disk $S(D_{1})$ between $\\Upsilon _1$ and $\\Upsilon _{2}$ .","We also have that $S^{-1}(\\Upsilon _{2})$ forms with $\\Upsilon _{2}$ a topological disk $D_2$ between $\\Upsilon _1$ and $\\Upsilon _2$ , which is mapped by $S$ onto a topological disk $S(D_{2})$ between $\\Upsilon _2$ and $\\Upsilon _3$ .","By Theorem REF there exist $K_1>0$ and a component of $f^{K_1}(D_{0})\\cap D_1$ that is a topological disk $D^{\\prime }_1 $ whose boundary contains an arc of $S^{-1}(\\Upsilon _1)$ .","The image of $D^{\\prime }_1$ under $S$ is a one-sided neighborhood $D^{\\prime \\prime }_1\\subseteq S(D_1)$ of some point in $\\Upsilon _1$ , contained in the region between $\\Upsilon _1 $ and $\\Upsilon _2$ .","The boundary of ${D^{\\prime \\prime }}_1$ consists of an arc in $\\Upsilon _1$ and an arc in $S(f^{K_1}(\\partial D_{0}))$ .","By the argument in the Homoclinic Orbit Theorem (see e.g.","[8]), there exists $N_1$ such that $f^{N_1}({D^{\\prime \\prime }}_1)$ intersects $D_2$ .","Let ${D^{\\prime }}_2$ be a component of this intersection that is a topological disk, and whose boundary contains an arc of $S^{-1}(\\Upsilon _2)$ .","Then the image of ${D^{\\prime }}_2$ under $S$ is a one sided neighborhood $D^{\\prime \\prime }_2$ of some point in $\\Upsilon _2$ that is contained in the region between $\\Upsilon _2$ and $\\Upsilon _3$ .","The boundary of $D^{\\prime \\prime }_2$ consists of an arc in $\\Upsilon _2$ and an arc in $(S\\circ f^{N_1}\\circ S\\circ f^{K_1})(\\partial D_{0})$ .","See Figure REF .","Figure: Construction of windows across a resonance.The main point of this construction is that, using the inner and outer dynamics, one can cross the successive BZI's and resonance regions determined by $\\lbrace \\Upsilon _h\\rbrace _{h=1,\\ldots ,k}$ one at a time, and obtain at each step a one-sided neighborhood of some point in some $\\Upsilon _h$ , which is between $\\Upsilon _h$ and $\\Upsilon _{h+1}$ , such that the boundary of that neighborhood contains the image of $\\partial D_0$ under a suitable composition of $S$ and powers of $f$ .","This argument can be repeated for each invariant torus $\\Upsilon _h$ with $2\\le h\\le k$ , yielding a point $x_a\\in \\partial D_{0}$ that is mapped by some appropriate composition of $S$ and powers of $f$ onto a point $x_b\\in \\partial D_{k+1}$ .","Then the constructions from Subsection REF can be applied to obtain a window $R^{\\prime }_a$ about $\\partial D_{0}$ , and a window $R^{\\prime }_b$ about $\\partial D_{k+1}$ , such that $R^{\\prime }_a$ is correctly aligned with $R_b$ under the appropriate composition of $S$ and powers of $f$ .","Then there exists a shadowing orbit that goes from a neighborhood of $R_a$ in $M$ to a neighborhood of $R^{\\prime }_b$ in $M$ .","If the region between $T_a$ and $T_b$ contains some prescribed collection of Aubry-Mather sets, then Theorem REF yields a shadowing orbit that crosses the region between $T_a$ and $T_b$ and follows the prescribed Aubry-Mather sets as in (REF )."],["Construction of correctly aligned windows along transition chains of tori","We consider a finite transition chain of invariant primary tori $\\lbrace T_1,T_2,\\ldots , T_n\\rbrace $ in the annulus $\\Lambda $ satisfying (A5).","All tori $T_k$ , $k=1, \\ldots , n$ , intersect the subset $U^-_+$ of the domain of the scattering map $S$ where $S$ moves points upwards in the annulus $\\Lambda $ .","The motion on each torus $T_k$ is topologically transitive.","Each torus $T_k$ with $2\\le k\\le n-1$ is an `interior' torus, i.e.","it can be $C^0$ -approximated from both sides in $\\Lambda $ by other invariant primary tori.","For each $k\\in \\lbrace 1,\\ldots ,n-1\\rbrace $ , the image $S(T_k)$ of the torus $T_k$ under the scattering map $S$ has a topologically transverse intersection with the torus $T_{k+1}$ .","We would like to show that there exists an orbit that visits some $\\varepsilon _k$ -neighborhood of each torus $T_k$ , $k=1,\\ldots ,n$ , in the prescribed order.","To this end we will construct a sequence of 2-dimensional windows in $\\Lambda $ that are correctly aligned one with another under the scattering map alternatively with some iterates of the inner map, as in Lemma REF .","Then the lemma will imply the existence of a shadowing orbit to the transition chain.","For each $k\\in \\lbrace 1,\\ldots ,n-1\\rbrace $ , we choose and fix a pair of points $x^-_{k,k+1}\\in T_k$ and $x^+_{k,k+1}\\in T_{k+1}$ such that $S(x^-_{k,k+1})=x^+_{k,k+1}$ and $S(T_{k})$ intersects $T_{k+1}$ transversally at $x^+_{k,k+1}$ .","We construct inductively a sequence of correctly aligned windows in $\\Lambda $ along the tori $\\lbrace T_1,T_2,\\ldots , T_n\\rbrace $ , such that each window is correctly aligned with the next window in the sequence either by the outer map or by some sufficiently large power of the inner map.","Moreover, each window will be contained in some $\\varepsilon $ -neighborhood of some transition torus.","We start the inductive construction at $T_1$ .","Consider the point $x^-_{1,2}\\in T_1$ with $S(x^-_{1,2})=x^+_{1,2}\\in T_{2}$ as above.","We construct a window $R^{\\prime }_1$ about $T_1$ as follows.","Let $\\bar{e}^{\\prime }_1$ be an arc contained in $T_1$ , and $\\chi ^{\\prime }_1:\\mathbb {R}^2\\rightarrow \\Lambda $ a $C^0$ -local parametrization such that $\\chi ^{\\prime }_1([0,1]\\times \\lbrace 0\\rbrace )=\\bar{e}^{\\prime }_1$ .","Then we define $R^{\\prime }_{1}=\\chi ^{\\prime }_1([0,1]\\times [-\\delta ^{\\prime }_1,\\delta ^{\\prime }_1]),\\\\ {R^{\\prime }}_{1}^\\textrm {exit}=\\chi ^{\\prime }_1(\\partial [0,1]\\times [-\\delta ^{\\prime }_1,\\delta ^{\\prime }_1]),$ where $0<\\delta ^{\\prime }_1<\\varepsilon _1$ .","We choose $\\bar{e}^{\\prime }_1$ and $\\delta ^{\\prime }_1$ sufficiently small so that and $S(\\bar{e}^{\\prime }_1)$ intersects $T_2$ only at $x^+_{1,2}$ , and also so that $R^{\\prime }_1$ defined as above is contained in ${U}^-_+$ .","The exit direction of $R^{\\prime }_1$ is in the direction of the torus $T_1$ .","The image $S(R^{\\prime }_1)$ of $R^{\\prime }_1$ under the scattering map is a topological rectangle.","Since $S(\\bar{e}^{\\prime }_1)$ is transverse to $T_2$ at $x^+_{1,2}$ , then by choosing $\\delta ^{\\prime }_1$ sufficiently small we ensure that the components of $S({R^{\\prime }}_{1}^\\textrm {exit})$ lie on opposite sides of $T_2$ .","Thus, the exit direction of $S(R^{\\prime }_1)$ is across the torus $T_2$ .","Next we consider the point $x^-_{2,3}\\in T_2$ with $S(x^-_{2,3})=x^+_{2,3}\\in T_{3}$ as above.","We construct a window $R^{\\prime }_2$ about $T_2$ in a manner similar to $R^{\\prime }_1$ : $R^{\\prime }_{2}=\\chi ^{\\prime }_2([0,1]\\times [-\\delta ^{\\prime }_2,\\delta ^{\\prime }_2]),\\\\ {R^{\\prime }}_{2}^\\textrm {exit}=\\chi ^{\\prime }_2(\\partial [0,1]\\times [-\\delta ^{\\prime }_2,\\delta ^{\\prime }_2]),$ where $\\bar{e}^{\\prime }_2$ is an arc contained in $T_2$ , $\\chi ^{\\prime }_2:\\mathbb {R}^2\\rightarrow \\Lambda $ is a $C^0$ -local parametrization such that $\\chi ^{\\prime }_2([0,1]\\times \\lbrace 0\\rbrace )=\\bar{e}^{\\prime }_2$ , and $0<\\delta ^{\\prime }_2<\\varepsilon _2$ is chosen sufficiently small.","Now we construct a window $R_2$ about the point $x^+_{1,2}$ such that $R^{\\prime }_1$ is correctly aligned with $R_2$ under $S$ and $R_2$ is correctly aligned with $R^{\\prime }_2$ under some power of $f_{\\mid \\Lambda }$ .","We choose an arc $\\bar{e}_2$ in $T_2$ containing $x^+_{1,2}$ such that $\\bar{e}_2 \\supseteq S(R^{\\prime }_1)\\cap T_2$ .","We choose a pair of invariant tori $T_2^{\\rm lower}$ and $T_2^{\\rm upper}$ such that $T_2^{\\rm lower} \\prec T_2 \\prec T_2^{\\rm upper}$ and the exit set components of $S(R^{\\prime }_1)$ , as well as the entry set components of $R^{\\prime }_2$ , are outside of the annulus bounded by $T_2^{\\rm lower}$ and $T_2^{\\rm upper}$ , on the both sides of the annulus.","Furthermore, we require that $T_2^{\\rm lower}$ and $T_2^{\\rm upper}$ should be $\\varepsilon _2$ -close to $T_2$ in the $C^0$ -topology.","The existence of such neighboring tori $T_2^{\\rm lower}$ and $T_2^{\\rm upper}$ to $T_2$ is guaranteed by (A5-iv).","Let $\\chi _2:\\mathbb {R}^2\\rightarrow \\Lambda $ be a $C^0$ -local parametrization such that $\\chi _2( \\lbrace 0\\rbrace \\times [0,1])=\\bar{e}_2$ , $\\chi _2( \\lbrace -\\delta _2\\rbrace \\times [0,1])\\subseteq T_2^{\\rm lower}$ and $\\chi _2( \\lbrace \\delta _2\\rbrace \\times [0,1])\\subseteq T_2^{\\rm upper}$ , for some $0<\\delta _2<\\varepsilon _2$ sufficiently small.","Define $R_{2}=\\chi _2([-\\delta _2,\\delta _2]\\times [0,1]),\\\\ {R}_{2}^\\textrm {exit}=\\chi _2(\\partial [-\\delta _2,\\delta _2]\\times [0,1]).$ The exit set components of $R_2$ lie on the invariant tori $T_2^{\\rm lower},T_2^{\\rm upper}$ neighboring $T_2$ .","Since the motion on the torus $T_2$ is topological transitive, there exists a power $f_{\\mid \\Lambda }^{K_2}$ of $f_{\\mid \\Lambda }$ such that $R_2$ is correctly aligned with $R^{\\prime }_2$ under $f_{\\mid \\Lambda }^{K_2}$ .","See Figure REF .","We have obtained the windows $R^{\\prime }_1$ about $T_1$ and $R^{\\prime }_2,R_2$ about $T_2$ such that $R^{\\prime }_1$ is correctly aligned with $R_2$ under $S$ and $R_2$ is correctly aligned with $R^{\\prime }_2$ under some power of $f_{\\mid \\Lambda }$ .","This ends the initial step of the inductive construction.","The inductive step goes on similarly, yielding the windows $R^{\\prime }_k$ about $T_k$ and $R^{\\prime }_{k+1},R_{k+1}$ about $T_{k+1}$ such that $R^{\\prime }_k$ is correctly aligned with $R_{k+1}$ under $S$ and $R_{k+1}$ is correctly aligned with $R^{\\prime }_{k+1}$ under some power of $f_{\\mid \\Lambda }$ .","See Figure REF .","Figure: Construction of windows along a transition chain.The construction proceeds inductively until a sequence of windows in $\\Lambda $ of the following type is obtained $ R^{\\prime }_1,R_2,R^{\\prime }_2, \\ldots , R_k,R^{\\prime }_k, R_{k+1}, R^{\\prime }_{k+1}, \\ldots , R_n, $ where for each $k\\in \\lbrace 1,\\ldots ,n-1\\rbrace $ we have that $R^{\\prime }_k$ is correctly aligned with $R_{k+1}$ under $S$ , and $R_{k+1}$ is correctly aligned with $R^{\\prime }_{k+1}$ under $f_{\\mid \\Lambda }^{K_{k+1}}$ for some $K_{k+1}>0$ .","Each window $R_k,R^{\\prime }_k$ is contained in an $\\varepsilon _k$ -neighborhood of the torus $T_k$ .","Applying the shadowing lemma-type of result Lemma REF provides an orbit that visits an $\\varepsilon _k$ -neighborhood in the phase space of each window in the sequence, and in particular of each torus in the transition chain."],["Gluing correctly aligned windows across BZI's with correctly aligned windows along transition chains of tori","We consider three transition chains of tori $\\lbrace T_i\\rbrace _{i=i_{k-1}+1,\\ldots , i_{{k}}}$ , $\\lbrace T_i\\rbrace _{i=i_k+1,\\ldots , i_{k+1}}$ , $\\lbrace T_i\\rbrace _{i=i_{k+1}+1,\\ldots , i_{{k+2}}}$ , with the property that each one of the regions between $T_{i_{k}}$ and $T_{i_{{k}}+1}$ , and between $T_{i_{k+1}}$ and $T_{i_{k+1}+1}$ , is either a BZI as in (A6), or it contains a finite number of invariant tori that separate the region as in (A6$^{\\prime }$ ).","Inside each region there is a prescribed collection of Aubry-Mather sets as in (A7) The constructions in Subsection REF and in Subsection REF yield correctly aligned windows in $\\Lambda $ that cross the region between $T_{i_{k}}$ and $T_{i_{{k}}+1}$ , and correctly aligned windows in $\\Lambda $ that cross the region between $T_{i_{k+1}}$ and $T_{i_{k+1}+1}$ .","The construction in Subsection REF yield sequences of correctly aligned windows along the adjacent transition chains.","The choices of the windows constructed along the transition chains depend on the choices of the windows that cross the region between $T_{i_{k}}$ and $T_{i_{{k}}+1}$ , and of the windows that cross the region between $T_{i_{k+1}}$ and $T_{i_{k+1}+1}$ .","Propagating the construction of windows starting from $T_{i_k+1}$ and moving forward along the transition chain $\\lbrace T_i\\rbrace _{i=i_k+1,\\ldots , i_{k+1}}$ , and the construction of windows starting from $T_{i_{k+1}}$ and moving backward along the same transition chain $\\lbrace T_i\\rbrace _{i=i_k+1,\\ldots , i_{k+1}}$ , may result in a pair of windows about some intermediate torus that are not correctly aligned.","We would like to glue this sequences of windows in a manner that is correctly aligned, without having to revise the windows constructed to that point.","Assume that $T_j$ is one of the tori $\\lbrace T_i\\rbrace _{i=i_k+1,\\ldots , i_{k+1}}$ , with $j\\in \\lbrace i_k+2,\\ldots , i_{k+1}-1\\rbrace $ .","Assume that one has already constructed a window $R_j$ about $T_j$ by propagating the construction from $T_{i_k+1}$ and moving forward along the transition chain, and another window $R^{\\prime }_j$ about $T_j$ by propagating the construction from $T_{i_{k+1}}$ and moving backwards along the transition chain.","The window $R_j$ is of the form $R _{j}=\\chi _j ([-\\delta _j,\\delta _j]\\times [0,1]),\\\\{R }_{j}^\\textrm {exit}=\\chi _j (\\partial [-\\delta _j,\\delta _j]\\times [0,1]),$ where $\\chi _{j}:\\mathbb {R}^2\\rightarrow \\Lambda $ is a $C^0$ -local parametrization with $\\chi _{j}(\\lbrace 0\\rbrace \\times [0,1])\\subseteq T_j$ , and $\\chi _j (\\lbrace -\\delta _j\\rbrace \\times [0,1])\\subseteq T_j^{\\rm lower}$ and $\\chi _j (\\lbrace \\delta _j\\rbrace \\times [0,1])\\subseteq T_j^{\\rm upper}$ , where $T_j^{\\rm lower}$ and $T_j^{\\rm upper}$ are two primary invariant tori on the opposite sides of $T_j$ .","The window $R^{\\prime }_j$ is of the form $R^{\\prime }_{j}=\\chi ^{\\prime }_j ([0,1]\\times [-\\delta ^{\\prime }_j,\\delta ^{\\prime }_j]),\\\\{R^{\\prime }}_{j}^\\textrm {exit}=\\chi ^{\\prime }_j (\\partial [0,1]\\times [-\\delta ^{\\prime }_j,\\delta ^{\\prime }_j]),$ where $\\chi ^{\\prime }_{j}:\\mathbb {R}^2\\rightarrow \\Lambda $ is a $C^0$ -local parametrization with $\\chi ^{\\prime }_{j}([0,1]\\times \\lbrace 0\\rbrace )\\subseteq T_j$ , and $\\chi ^{\\prime }_j ([0,1]\\times \\lbrace -\\delta ^{\\prime }_j\\rbrace )\\subseteq T_j^{\\rm lower}$ and $\\chi ^{\\prime }_j ([0,1]\\times \\lbrace \\delta ^{\\prime }_j\\rbrace )\\subseteq {T^{\\prime }}_j^{\\rm upper}$ , where ${T^{\\prime }}_j^{\\rm lower}$ and ${T^{\\prime }}_j^{\\rm upper}$ are two primary invariant tori on the opposite sides of $T_j$ .","Let us assume that the annular region between ${T^{\\prime }}_j^{\\rm lower}$ and ${T^{\\prime }}_j^{\\rm upper}$ is inside the region between $T_j^{\\rm lower}$ and $T_j^{\\rm upper}$ .","We construct a new window $R^{\\prime \\prime }_j$ about $T_j$ , such that $R_j$ is correctly aligned with $R^{\\prime \\prime }_j$ under the identity map, and $R^{\\prime \\prime }_j$ is correctly aligned with $R^{\\prime }_j$ under some power of $f$ .","We let $R^{\\prime \\prime }_j$ is of the form $R^{\\prime \\prime }_{j}=\\chi ^{\\prime \\prime }_j ([-\\delta ^{\\prime \\prime }_j,\\delta ^{\\prime \\prime }_j]\\times [0,1]),\\\\{R^{\\prime \\prime }}_{j}^\\textrm {exit}=\\chi ^{\\prime \\prime }_j (\\partial [-\\delta ^{\\prime \\prime }_j,\\delta ^{\\prime \\prime }_j]\\times [0,1]),$ where $\\chi ^{\\prime \\prime }_{j}:\\mathbb {R}^2\\rightarrow \\Lambda $ is a $C^0$ -local parametrization with $\\chi ^{\\prime \\prime }_{j}(\\lbrace 0\\rbrace \\times [0,1])\\supseteq \\chi ^{\\prime }_{j}(\\lbrace 0\\rbrace \\times [0,1])$ , and $\\chi ^{\\prime \\prime }_j (\\lbrace -\\delta ^{\\prime \\prime }_j\\rbrace \\times [0,1])\\subseteq {T^{\\prime }}_j^{\\rm lower}$ and $\\chi ^{\\prime \\prime }_j (\\lbrace \\delta ^{\\prime \\prime }_j\\rbrace \\times [0,1])\\subseteq {T^{\\prime }}_j^{\\rm upper}$ , for some $\\delta ^{\\prime \\prime }_j>0$ .","Since the motion on the torus $T_j$ is topological transitive, there exists a power $f_{\\mid \\Lambda }^{K_j}$ of $f_{\\mid \\Lambda }$ such that $R^{\\prime \\prime }_j$ is correctly aligned with $R^{\\prime }_j$ under $f_{\\mid \\Lambda }^{K_j}$ .","See Figure REF .","We have obtained that $R_j$ is correctly aligned with $R^{\\prime \\prime }_j$ under the identity map, and $R^{\\prime \\prime }_j$ is correctly aligned with $R^{\\prime }_j$ under some power $f_{\\mid \\Lambda }^{K_j}$ of $f_{\\mid \\Lambda }$ .","Figure: Gluing sequences of correctly aligned windows along a transition chain.The case when the annular region between $T_j^{\\rm lower}$ and $T_j^{\\rm upper}$ is inside the region between ${T^{\\prime }}_j^{\\rm lower}$ and ${T^{\\prime }}_j^{\\rm upper}$ results in the window $R_j$ correctly aligned with $R^{\\prime }_j$ under some power of $f$ , without the need of creating an intermediate window $R^{\\prime \\prime }_j$ .","The case when neither annular region is contained in the other annular region can be dealt similarly by constructing an intermediate window $R^{\\prime \\prime }_j$ .","Through this process, a sequence of correctly windows constructed forward along a transition chain can be glued, in a correctly aligned manner, with a sequence of correctly windows constructed backwards along the same transition chain."],["Proof of Theorem ","To summarize, in Subsection REF and in Subsection REF , we described the construction of correctly aligned windows that cross an annular regions between two consecutive transition chains of invariant tori.","These annular regions are BZI's or else they are separated by some finite collection of invariant tori.","If each annular region has some prescribed collection od Aubry-Mather sets, the construction yields windows that follow these Aubry-Mather sets.","In Subsection REF yield sequences of correctly aligned windows each transition chain.","In Subsection REF the construction of a sequence of correctly windows constructed forward along a transition chain can be glued, in a correctly aligned manner, with a the construction of a sequence of correctly windows constructed backwards along the same transition chain.","Thus, starting from some initial annular region, one can construct sequences of correctly aligned windows, forward and backwards, along infinitely many topological transition chains interspersed with annular regions.","The Shadowing Lemma-type of result Theorem REF implies the existence of an orbit that gets $\\varepsilon _i$ -close to each transition torus $T_i$ , and also follows each Aubry-Mather set $\\Sigma _i$ for a prescribed time $n_i$ ."]],"string":"[\n [\n \"Diffusion along transition chains of invariant tori and Aubry-Mather\\n sets\"\n ],\n [\n \"Abstract We describe a topological mechanism for the existence of diffusing orbits in a dynamical system satisfying the following assumptions: (i) the phase space contains a normally hyperbolic invariant manifold diffeomorphic to a two-dimensional annulus, (ii) the restriction of the dynamics to the annulus is an area preserving monotone twist map, (iii) the annulus contains sequences of invariant one-dimensional tori that form transition chains, i.e., the unstable manifold of each torus has a topologically transverse intersection with the stable manifold of the next torus in the sequence, (iv) the transition chains of tori are interspersed with gaps created by resonances, (v) within each gap there is prescribed a finite collection of Aubry-Mather sets.\",\n \"Under these assumptions, there exist trajectories that follow the transition chains, cross over the gaps, and follow the Aubry-Mather sets within each gap, in any specified order.\",\n \"This mechanism is related to the Arnold diffusion problem in Hamiltonian systems.\",\n \"In particular, we prove the existence of diffusing trajectories in the large gap problem of Hamiltonian systems.\",\n \"The argument is topological and constructive.\"\n ],\n [\n \"Introduction\",\n \"In this paper we study a topological mechanism of diffusion and chaotic orbits related to the Arnold diffusion problem.\",\n \"We consider a normally hyperbolic invariant manifold diffeomorphic to a two-dimensional annulus.\",\n \"We assume that the dynamics restricted to the annulus is given by an area preserving monotone twist map.\",\n \"We assume that inside the annulus there exist invariant primary one-dimensional tori (homotopically nontrivial invariant closed curves) with the property that the unstable manifold of each torus has topologically transverse intersections with the stable manifolds of all sufficiently close tori.\",\n \"Sequences of such tori and their heteroclinic connections form transition chains of tori.\",\n \"The successive transition chains of tori are interspersed with gaps.\",\n \"We assume that each gap is a region in the annulus bounded by two invariant primary tori of one of the following types: (i) a Birkhoff Zone of Instability, i.e., it contains no invariant primary torus in its interior (ii) it contains only finitely many invariant primary tori in its interior, such that each torus is either isolated, or it consists of a hyperbolic periodic orbit together with its stable and unstable manifolds that are assumed to coincide.\",\n \"Within each gap we prescribe a finite collection of Aubry-Mather sets.\",\n \"We prove the existence of orbits that shadow the primary tori in each transition chain, cross over the gaps that separate the successive transition chains, and also shadow the specified Aubry-Mather sets within each gap.\",\n \"The motivation for this result is the Arnold diffusion problem of Hamiltonian systems.\",\n \"This problem asserts that all sufficiently small perturbations of generic, integrable Hamiltonian systems exhibit orbits along which the action variable changes substantially; also, there exist chaotic orbits that can be coded through symbolic dynamics.\",\n \"A classification of nearly integrable systems proposed in [13] distinguishes between a priori stable systems, in which the unperturbed system can be expressed in terms of action-angle variables only, and a priori unstable systems, in which the unperturbed system contains both action-angle and hyperbolic variables.\",\n \"In the a priori stable case analytical results have been announced in [47].\",\n \"In the a priori unstable case there have been several analytical results in the last several years (see [13], [54], [51], [52], [11], [18], [3], [12]).\",\n \"Some of the approaches involve variational methods, geometric methods, or topological methods.\",\n \"See [20] for an overview on the Arnold diffusion problem, applications, and additional references.\",\n \"It is relevant in applications to detect, combine, and compare different mechanisms of diffusion displayed by concrete systems.\",\n \"In many models, as well as in numerical experiments, diffusion can only be observed for perturbations of sizes much larger than those considered by the analytical approaches [14], [40], [39], [36], [28].\",\n \"For these types of problems, the geometric and topological approaches are particularly advantageous as they yield constructive methods to detect diffusion, quantitative estimates on diffusing orbits, and explicit conditions that can be verified in concrete examples.\",\n \"In this paper we describe a general method to establish the existence of diffusing orbits for a large class of dynamical systems.\",\n \"The dynamical systems under consideration are not assumed to be small perturbations of integrable Hamiltonians.\",\n \"Moreover, some systems that are not Hamiltonian can be considered.\",\n \"Our method requires the existence of certain geometric objects that organize the dynamics, and employs topological tools to establish the existence of diffusing orbits.\",\n \"The existence of the geometric objects can be verified in concrete systems through analytical methods, or through numerical methods, or through a combination of thereof.\",\n \"We illustrate our method in the case of a Hamiltonian system consisting of a pendulum and a rotator with a small periodic coupling.\",\n \"We give an analytic argument for the existence of diffusing orbits.\",\n \"In [15] the topological method is applied to show, with the aid of a computer, the existence of diffusing orbits in the spatial restricted three-body problem, where the two primaries are the Sun and the Earth; this model is not nearly integrable.\",\n \"Similar ideas appear in [10], [21], [22].\",\n \"The diffusing orbits detected by this approach follow transition chains of invariant tori up to the gaps between these transition chains, and then cross the gaps following the inner dynamics restricted to the annulus.\",\n \"The orbits that cross the gaps follow Birkhoff connecting orbits that go from one boundary of the gap to the other, or Mather connecting orbits, that shadow a prescribed sequence of Aubry-Mather set inside the gap, or homoclinic orbits.\",\n \"Related approaches, that use Birkhoff's ideas to understand generic drift properties of random iterations of exact-symplectic twist maps, appear in [48], [38].\",\n \"These yield the concept of a polysystem, which is a locally constant skew-product over a Bernoulli shift [43].\",\n \"Polysystems are a key ingredient in the approach of the instability problem in a priori unstable systems in [6].\",\n \"Arnold diffusion-type orbits for random iterations of flows of families of Tonelli Hamiltonians are studied in [42], based on an extension of the pseudograph method from [3].\",\n \"The diffusing orbits that we obtain in this paper are similar to those found through variational methods, as in [11], [12]; they are however not necessarily action minimizing.\",\n \"Also, we do not need the unperturbed Hamiltonian to have positive-definite normal torsion.\",\n \"This paper completes the investigations undertaken in [25], [26] where some non-generic conditions, or some conditions difficult to verify in concrete systems, were assumed.\",\n \"The present paper assumes very general conditions and provides a new topological mechanism of diffusion based on Aubry-Mather sets.\"\n ],\n [\n \"Main result\",\n \"In this section we state the assumptions and the main result of this paper.\",\n \"After the statement, the assumptions and the main result are explained and exemplified.\",\n \"(A1) $M$ is a $n$ -dimensional $C^r$ -differentiable Riemannian manifold, and $f:M\\\\rightarrow M$ is a $C^r$ -smooth map, for some $r\\\\ge 2$ .\",\n \"(A2) There exists a 2-dimensional submanifold $\\\\Lambda $ in $M$ , diffeormorphic to an annulus $\\\\Lambda \\\\simeq \\\\mathbb {T}^1\\\\times [0,1]$ .\",\n \"We assume that $f$ is 2-normally hyperbolic to $\\\\Lambda $ in $M$ (see Subsection REF for the definition).\",\n \"Since $f$ is 2-normally hyperbolic to $\\\\Lambda $ , $W^s(\\\\Lambda )$ , $W^u(\\\\Lambda )$ , and $\\\\Lambda $ are all $C^2$ -differentiable.\",\n \"Denote the dimensions of the stable and unstable manifolds of a point $x \\\\in \\\\Lambda $ by $\\\\text{dim}(W^s(x)) = n_s$ and $\\\\text{dim}(W^u(x)) = n_u$ .\",\n \"Then, $n = 2 + n_s + n_u$ .\",\n \"(A3) On $\\\\Lambda $ there is a system of angle-action coordinates $(\\\\phi ,I)$ , with $\\\\phi \\\\in {\\\\mathbb {T}}^1$ and $I \\\\in [0,1]$ .\",\n \"The restriction $f|_{\\\\Lambda }$ of $f$ to $\\\\Lambda $ is a boundary component preserving, area preserving, monotone twist map, with respect to the angle-action coordinates $(\\\\phi ,I)$ .\",\n \"(A4) The stable and unstable manifolds of $\\\\Lambda $ , $W^s(\\\\Lambda )$ and $W^u(\\\\Lambda )$ , have a differentiably transverse intersection along a 2-dimensional homoclinic channel $\\\\Gamma $ .\",\n \"Since the manifolds $W^s(\\\\Lambda )$ and $W^u(\\\\Lambda )$ are $C^2$ and transverse, $\\\\Gamma $ is $C^2$ .\",\n \"We assume that the scattering map $S$ associated to $\\\\Gamma $ is well defined, and hence is $C^1$ .\",\n \"See Subsection REF .\",\n \"(A5) There exists a bi-infinite sequence of Lipschitz primary invariant tori $\\\\lbrace T_i\\\\rbrace _{i\\\\in \\\\mathbb {Z}}$ in $\\\\Lambda $ , and a bi-infinite, increasing sequence of integers $\\\\lbrace i_k\\\\rbrace _{k\\\\in \\\\mathbb {Z}}$ with the following properties: (i) Each torus $T_i$ intersects the domain $U^-$ and the range $U^+$ of the scattering map $S$ associated to $\\\\Gamma $ .\",\n \"(ii) For each $i\\\\in \\\\lbrace {i_{k}+1}, \\\\ldots , {i_{k+1}-1}\\\\rbrace $ , the image of $T_i\\\\cap U^-$ under the scattering map $S$ is topologically transverse to $T_{i+1}$ .\",\n \"(iii) For each torus $T_i$ with $i\\\\in \\\\lbrace {i_{k}+2}, \\\\ldots , {i_{k+1}-1}\\\\rbrace $ , the restriction of $f$ to $T_i$ is topologically transitive.\",\n \"(iv) Each torus $T_i$ with $i\\\\in \\\\lbrace {i_{k}+2}, \\\\ldots , {i_{k+1}-1}\\\\rbrace $ , can be $C^0$ -approximated from both sides by other primary invariant tori from $\\\\Lambda $ .\",\n \"We will refer to a finite sequence $\\\\lbrace T_i\\\\rbrace _{i=i_k+1,\\\\ldots , i_{k+1}}$ as above as a transition chain of tori.\",\n \"(A6) The region in $\\\\Lambda $ between $T_{i_k}$ and $T_{i_{k}+1}$ contains no invariant primary torus in its interior.\",\n \"(A7) Inside each region between $T_{i_k}$ and $T_{i_{k}+1}$ there is prescribed a finite collection of Aubry-Mather sets $\\\\lbrace \\\\Sigma _{\\\\omega ^k_1}, \\\\Sigma _{\\\\omega ^k_2},\\\\ldots , \\\\Sigma _{\\\\omega ^k_{s_k}}\\\\rbrace $ , where $s_k\\\\ge 1$ , and $\\\\omega ^k_s$ denotes the rotation number of $\\\\Sigma _{\\\\omega ^k_s}$ .\",\n \"Each Aubry-Mather set $\\\\Sigma _{\\\\omega ^k_s}$ is assumed to lie on some essential circle $C_{\\\\omega ^k_s}$ , with the circles $C_{\\\\omega ^k_s}$ mutually disjoint for all $s\\\\in \\\\lbrace 1,\\\\ldots ,s_k\\\\rbrace $ , and $C_{\\\\omega ^k_s}$ below $C_{\\\\omega ^k_{s^{\\\\prime }}}$ for all $\\\\omega ^k_s<\\\\omega ^k_{s^{\\\\prime }}$ .\",\n \"The vertical ordering of these circles is relative to the $I$ -coordinate on the annulus.\",\n \"Instead of (A6) we can consider the following condition: (A6$^{\\\\prime }$ ) The region $\\\\Lambda _k$ in $\\\\Lambda $ between $T_{i_k}$ and $T_{i_{k}+1}$ contains finitely many invariant primary tori $\\\\lbrace \\\\Upsilon _{h^k_1}, \\\\ldots , \\\\Upsilon _{h^k_{l_k}}\\\\rbrace $ , where $l_k\\\\ge 1$ , satisfying the following properties: (i) Each $\\\\Upsilon _{h^k_j}$ falls in one of the following two cases: (a) $\\\\Upsilon _{h^k_j}$ is an isolated invariant primary torus, i.e., an invariant torus that has a neighborhood in the annulus that does not contain any other invariant primary torus inside.\",\n \"(b) There exists a hyperbolic periodic orbit in $\\\\Lambda $ such that its stable and unstable manifolds coincide.\",\n \"Each invariant manifold has two branches, an upper branch and a lower branch (where the vertical ordering is relative to the $I$ -coordinate of the annulus).\",\n \"Then $\\\\Upsilon _{h^k_j}$ is an invariant primary torus consisting of the hyperbolic periodic orbit together with the upper branches of the invariant manifolds, or consisting of the hyperbolic periodic orbit together with the lower branches of the invariant manifolds.\",\n \"(ii) The invariant primary tori $\\\\lbrace \\\\Upsilon _{h^k_1}, \\\\ldots , \\\\Upsilon _{h^k_{l_k}}\\\\rbrace $ are vertically ordered, in the sense that $\\\\Upsilon _{h^k_j}$ is below $\\\\Upsilon _{h^k_{j+1}}$ , for all $j=1,\\\\ldots , l_k-1$ .\",\n \"The vertical ordering of these tori is relative to the $I$ -coordinate on the annulus.\",\n \"(iii) For each $\\\\Upsilon _{h^k_j}$ , $j=1,\\\\ldots , l_k$ , the inverse image $S^{-1}(\\\\Upsilon _{h^k_j}\\\\cap U^+)$ forms with $\\\\Upsilon _{h^k_j}$ a topological disk $D_{h^k_j} \\\\subseteq U^-$ below $\\\\Upsilon _{h^k_j}$ , such that $S(D_{h^k_j})\\\\subseteq U^+$ is a topological disk above $\\\\Upsilon _{h^k_j}$ , which is bounded by $\\\\Upsilon _{h^k_j}$ and $S(\\\\Upsilon _{h^k_j}\\\\cap U^-)$ .\",\n \"See Fig.\",\n \"REF .\",\n \"Now we state the main result of the paper.\",\n \"Theorem 2.1 Let $f:M\\\\rightarrow M$ be a $C^r$ -differentiable map, and let $(T_i)_{i\\\\in \\\\mathbb {Z}}$ be a sequence of invariant primary tori in $\\\\Lambda $ , satisfying the properties (A1) – (A6), or (A1)-(A5) and (A6 $^{\\\\prime }$ ), from above.\",\n \"Then for each sequence $(\\\\epsilon _i)_{i\\\\in \\\\mathbb {Z}}$ of positive real numbers, there exist a point $z\\\\in M$ and a bi-infinite increasing sequence of integers $(N_i)_{i\\\\in \\\\mathbb {Z}}$ such that $ d(f^{N_i}(z), T_{i})<\\\\epsilon _i, \\\\textrm { for all }i\\\\in \\\\mathbb {Z}.$ In addition, if condition (A7) is assumed, and some positive integers $\\\\lbrace n^k_s\\\\rbrace _{s=1,\\\\ldots , s_k}$ , $k\\\\in \\\\mathbb {Z}$ are given, then there exist $z\\\\in M$ and $(N_i)_{i\\\\in \\\\mathbb {Z}}$ as in (REF ), and positive integers $\\\\lbrace m^k_s\\\\rbrace _{s=1,\\\\ldots , s_k}$ , $k\\\\in \\\\mathbb {Z}$ , such that, for each $k$ and each $s\\\\in \\\\lbrace 1,\\\\ldots , s_k\\\\rbrace $ , we have $\\\\pi _\\\\phi (f^j(w^k_s))<\\\\pi _\\\\phi (f^j(z))<\\\\pi _\\\\phi (f^j(\\\\bar{w}^k_s)),$ for some $w^k_s,\\\\bar{w}^k_s\\\\in \\\\Sigma _{\\\\omega ^k_s}$ and for all $j$ with $N_{i_k}+\\\\sum _{t=0}^{s-1} n^k_t+\\\\sum _{t=0}^{s-1}m^k_t\\\\le j\\\\le N_{i_k}+\\\\sum _{t=0}^{s} n^k_t+\\\\sum _{t=0}^{s-1}m^k_t.$ Figure: An illustration of the condition (A6 ' ^{\\\\prime }).\",\n \"In the figure, inside the region between T i k T_{i_k} and T i k +1 T_{i_{k}+1}, three invariant primary tori are shown, Υ h j k \\\\Upsilon _{h^k_j}, Υ h j+1 k \\\\Upsilon _{h^k_{j+1}}, Υ h j+2 k \\\\Upsilon _{h^k_{j+2}}, where Υ h j k \\\\Upsilon _{h^k_j} is as in (A6 ' ^{\\\\prime }-i-a), and Υ h j+1 k \\\\Upsilon _{h^k_{j+1}}, Υ h j+2 k \\\\Upsilon _{h^k_{j+2}} are as in (A6 ' ^{\\\\prime }-i-b).\",\n \"The vertical ordering of the tori stated condition (A6 ' ^{\\\\prime }-ii) is depicted.\",\n \"For each of the tori Υ h j k \\\\Upsilon _{h^k_j}, Υ h j+1 k \\\\Upsilon _{h^k_{j+1}}, Υ h j+2 k \\\\Upsilon _{h^k_{j+2}}, the condition (A6 ' ^{\\\\prime }-iii) is also illustrated.Theorem REF asserts that if the conditions (A1)-(A6), or (A1)-(A5) and (A6$^{\\\\prime }$ ) are satisfied, then there exists an orbit that shadows all tori in the transition chains in the prescribed order, and also crosses over the large gaps that separate the successive transition chains.\",\n \"In particular, there exists an orbit that travels arbitrarily far with respect to the action variable, and there also exists an orbit that executes chaotic excursions.\",\n \"Additionally, if some Aubry-Mather sets are prescribed inside each gap that separates the successive transition chains, as in condition (A7), then there exists an orbit that, besides shadowing the transition chains, it also shadows the Aubry-Mather sets in the prescribed order.\",\n \"Note that the tori in the transition chains are shadowed in the sense that the diffusing orbit gets arbitrarily close to these tori.\",\n \"However, the Aubry-Mather sets are shadowed in the sense of the cyclical ordering: for each prescribed Aubry-Mather set, the diffusing orbit stays between the orbits of two points in the Aubry-Mather set, relative to the $\\\\phi $ -coordinate, for any prescribed time interval.\",\n \"Now we explain each assumption.\",\n \"Assumption (A1) describes a $C^r$ -differentiable, discrete dynamical system.\",\n \"In applications, the map $f$ represents the first return map to a Poincaré section associated to a flow.\",\n \"In many examples of interest the flow is a Hamiltonian flow.\",\n \"Assumption (A2) prescribes the existence of a normally hyperbolic invariant manifold $\\\\Lambda $ for $f$ , which is diffeomorphic to an annulus.\",\n \"The differentiability class $r$ and the contraction and expansion rates along the stable and unstable bundles on $\\\\Lambda $ are chosen so that the manifolds $\\\\Lambda $ , $W^u(\\\\Lambda )$ , $W^s(\\\\Lambda )$ , $\\\\Gamma $ are at least $C^2$ -differentiable, and the scattering map $S$ associated to $\\\\Gamma $ is at least $C^1$ -differentiable.\",\n \"The relations between the rates and the differentiability of these objects is given explicitly in Subsection REF .\",\n \"In many examples of nearly integrable Hamiltonian systems, e.g.\",\n \"[2], one can identify a normally hyperbolic invariant manifold $\\\\Lambda _0$ in the unperturbed system, and use the standard theory of normal hyperbolicity to establish the persistence of a normally hyperbolic invariant manifold $\\\\Lambda _\\\\varepsilon $ diffeomorphic to $\\\\Lambda _0$ for the perturbed system, for all sufficiently small perturbation parameters $\\\\varepsilon \\\\ne 0$ .\",\n \"There also exist examples, e.g.\",\n \"from celestial mechanics, were the existence of a normally hyperbolic manifold can be established through a computer assisted proof (see [9]).\",\n \"We note that the assumption (A2) does not require that the stable and unstable manifolds of $\\\\Lambda $ have equal dimensions, thus our setting includes dynamical systems that are not Hamiltonian.\",\n \"Assumption (A3) is satisfied automatically in examples like the weakly-coupled pendulum-rotator system considered in [18], or the periodically perturbed geodesic flow on a torus considered in [17], [33].\",\n \"Some properties of area preserving, monotone twist maps of the annulus are reviewed in Section .\",\n \"Assumption (A4) asserts that the stable and unstable manifolds of $\\\\Lambda $ have a transverse intersection $\\\\Gamma $ along a homoclinic manifold $\\\\Gamma $ , such that the scattering map associated to $\\\\Gamma $ is well defined, and hence is $C^1$ .\",\n \"See Subsection REF .\",\n \"In perturbed systems one often uses a Melnikov method to establish the existence, and the persistence for all sufficiently small values of the perturbation, of a transverse intersection of the invariant manifolds.\",\n \"Assumption (A5) prescribes the existence of a bi-infinite collection $\\\\lbrace T_{i}\\\\rbrace _{i\\\\in \\\\mathbb {Z}}$ of invariant primary Lipschitz tori that can be grouped into transition chains of the type $\\\\lbrace T_i\\\\rbrace _{i=i_k+1,\\\\ldots ,i_{k+1}}$ .\",\n \"Assumption (A5)-(i) requires that each torus $T_i$ intersects the domain and the range of the scattering map.\",\n \"Assumption (A5)-(ii) requires that each torus in a transition chain$\\\\lbrace T_i\\\\rbrace _{i=i_k+1,\\\\ldots ,i_{k+1}}$ is mapped by the scattering map topologically transversally across the next torus in the chain.\",\n \"Since the tori are only Lipschitz, topological transversality (topological crossing) is used in place of differentiable transversality.\",\n \"The definition of topological crossing can be found in [8].\",\n \"Roughly speaking, two manifolds are topologically crossing if they can be made differentiably transverse with non-zero oriented intersection number by the means of a sufficiently small homotopy.\",\n \"In our case the two manifolds are two 1-dimensional arcs of $S(T_i)$ and of $T_{i+1}$ in $\\\\Lambda $ .\",\n \"From (A5)-(ii), it follows as in [19] that $W^u(T_i)$ has a topologically transverse intersection point with $W^s(T_{i+1})$ .\",\n \"Condition (A5)-(iii) requires that each torus in $\\\\lbrace T_{i}\\\\rbrace _{i=i_{k}+1, \\\\ldots , i_{k+1}}$ , except for the end tori, are topologically transitive.\",\n \"Assumption (A5)-(iv) says that all tori in the transition chain except for the end ones can be $C^0$ -approximated from both ways by some other invariant primary tori in $\\\\Lambda $ , not necessarily from the transition chain.\",\n \"This means that for each $i\\\\in \\\\lbrace i_{k}+2, \\\\ldots , i_{k+1}-1\\\\rbrace $ there exist two sequences of invariant primary tori $(T_{j^-_l(i)})_{l\\\\ge 1},(T_{j^+_l(i)})_{l\\\\ge 1}$ in $\\\\Lambda $ that approach $T_i$ in the $C^0$ -topology, such that the annulus bounded by $T_{j^-_l(i)}$ and $T_{j^+_l(i)}$ contains $T_i$ in its interior for all $l$ .\",\n \"Assumption (A6) says that every pair of successive transition chains$\\\\lbrace T_{i}\\\\rbrace _{i=i_{k-1}+1, \\\\ldots , i_{k}}$ and $\\\\lbrace T_{i}\\\\rbrace _{i=i_{k}+1, \\\\ldots , i_{k+1}}$ is separated by the region between $T_{i_{k}}$ and $T_{i_{k}+1}$ which contains no invariant primary torus in its interior.\",\n \"A region in an annulus that is bounded by two invariant primary tori and contains no invariant primary torus in its interior is referred as a Birkhoff Zone of Instability (BZI).\",\n \"The boundary tori have in general only Lipschitz regularity.\",\n \"Assumptions (A5) and (A6) describe a geometric structure that is typical for the large gap problem for a priori unstable Hamiltonian systems.\",\n \"In such systems, Melnikov theory implies that $W^u(\\\\Lambda )$ intersects transversally $W^s(\\\\Lambda )$ at an angle of order $\\\\varepsilon $ , where $\\\\varepsilon $ is the size of the perturbation.\",\n \"The KAM theorem yields a Cantor family of smooth, invariant primary tori that survives the perturbation.\",\n \"The family of tori is interrupted by `large gaps' of order $\\\\varepsilon ^{1/2}$ located at the resonant regions.\",\n \"Using the transverse intersection between $W^u(\\\\Lambda )$ and $W^s(\\\\Lambda )$ , one can find heteroclinic connections between KAM tori that are sufficiently close, within order $\\\\varepsilon $ , from one another, and thus form transition chains of tori.\",\n \"Since the large gaps are of order $\\\\varepsilon ^{1/2}$ and the splitting size of $W^u(\\\\Lambda )$ , $W^s(\\\\Lambda )$ is only order $\\\\varepsilon $ , the transition chain mechanism cannot be extended across the large gaps.\",\n \"In our model, the large gaps are modeled by BZI's, as in assumption (A6).\",\n \"This can be achieved by extending the transition chains to maximal transition chains, that go from the boundary of one large gap to the boundary of the next large gap.\",\n \"The intermediate tori in the chain can be chosen as KAM tori: therefore the assumption that these tori are topologically transitive and are $C^0$ -approximable from both sides by other tori is satisfied in such cases.\",\n \"This may not be the case for the tori at the ends of the transition chains.\",\n \"Assumption (A7) says that inside each BZI between $T_{i_k}$ and $T_{i_{k}+1}$ there is a prescribed collection of Aubry-Mather sets $\\\\lbrace \\\\Sigma _{\\\\omega ^k_1}, \\\\Sigma _{\\\\omega ^k_2},\\\\ldots , \\\\Sigma _{\\\\omega ^k_{s_k}}\\\\rbrace $ that is vertically ordered.\",\n \"The vertical ordering means that the Aubry-Mather sets lie on essential (non-invariant) circles $C_{\\\\omega ^k_s}$ that are graphs over the $\\\\phi $ -coordinate of the annulus, and with $C_{\\\\omega ^k_s}$ below $C_{\\\\omega ^k_{s^{\\\\prime }}}$ provided $\\\\omega ^k_s<\\\\omega ^k_{s^{\\\\prime }}$ ; we write $C_{\\\\omega ^k_s}\\\\prec C_{\\\\omega ^k_{s^{\\\\prime }}}$ .\",\n \"The vertical ordering of the Aubry-Mather sets is shown for example in [27].\",\n \"Assumption (A6$^{\\\\prime }$ ) is a relaxation of (A6).\",\n \"Instead of requiring that the region in $\\\\Lambda $ between $T_{i_{k}}$ and $T_{i_{k}+1}$ is a BZI, it allows the existence of finitely many invariant primary tori $\\\\lbrace \\\\Upsilon _{h^k_j}\\\\rbrace _{j=1, \\\\ldots , l_k}$ that separate the region into disjoint components.\",\n \"These invariant primary tori are either isolated or else they consist of hyperbolic periodic points together with branches of their stable and unstable manifolds which are assumed to coincide.\",\n \"We require that the image of each $\\\\Upsilon _{h^k_j}$ under $S$ satisfies a certain transversality condition with $\\\\Upsilon _{h^k_{j+1}}$ that allows one to use the scattering map in order to move points from one side of the set to the other side of the set.\",\n \"We note that isolated invariant tori and hyperbolic periodic points whose stable and unstable manifolds coincide do not occur in generic systems.\"\n ],\n [\n \"Application\",\n \"We apply Theorem REF to show the existence of diffusing orbits in an example of a nearly integrable Hamiltonian system.\",\n \"Let $H_\\\\varepsilon (p, q, I,\\\\phi ,t)&=&h_0(I)\\\\pm \\\\left(\\\\frac{1}{2}p^2+V(q)\\\\right)+\\\\varepsilon h(p, q,I,\\\\phi , t; \\\\varepsilon ),$ where $(p, q, I,\\\\phi , t)\\\\in \\\\mathbb {R}\\\\times \\\\mathbb {T}^1\\\\times \\\\mathbb {R}\\\\times \\\\mathbb {T}^1\\\\times \\\\mathbb {T}^1$ and $h$ is a trigonometric polynomial in $(\\\\phi ,t)$ .\",\n \"Here $h_0(I)$ represents a rotator, $P_{\\\\pm }(p,q)=\\\\pm (\\\\frac{1}{2}p^2+V(q))$ represents a pendulum, and $\\\\varepsilon h$ a small, periodic coupling.\",\n \"We assume that $V$ , $h_0$ and $h$ are uniformly $C^r$ for some $r$ sufficiently large.\",\n \"We assume that $V$ is periodic in $q$ of period 1 and has a unique non-degenerate global maximum; this implies that the pendulum has a homoclinic orbit $(p^0 (\\\\sigma ),q^0 (\\\\sigma ))$ to $(0,0)$ , with $\\\\sigma \\\\in \\\\mathbb {R}$ .\",\n \"We also assume that $h_0$ satisfies a uniform twist condition $\\\\partial ^2h_0/\\\\partial I^2>\\\\theta $ , for some $\\\\theta >0$ , and for all $I$ in some interval $(I^-, I^+)$ , with $I^-0$ .\",\n \"Condition (ii) implies that there are points in the domain of the scattering map $S$ whose $I_\\\\varepsilon $ -coordinate is increased by $O(\\\\varepsilon )$ under $S$ .\",\n \"We can use these estimates to construct transition chains of invariant primary tori alternating with gaps, as in (A5) and (A6), or as in (A5) and (A6$^{\\\\prime }$ ).\",\n \"For $\\\\varepsilon $ sufficiently small and fixed, we let $M:=M_\\\\varepsilon $ , $f:=f_\\\\varepsilon $ , and $\\\\Lambda $ be the annulus in $\\\\Lambda _\\\\varepsilon $ bounded by a pair of tori $T_{I_a}$ , $T_{I_b}$ with $I^-0$ at all points in the annulus.\",\n \"We note that the above properties imply that $f$ is exact symplectic, i.e.\",\n \"$ f$ has zero flux, meaning that for any rotational curve $\\\\gamma $ the area of the regions above $\\\\gamma $ and below $f(\\\\gamma )$ equals the area below $\\\\gamma $ and above $f(\\\\gamma )$ .\",\n \"In the sequel we will assume that $f$ is a positive twist, meaning that$\\\\partial (\\\\pi _{x}\\\\circ f)/\\\\partial y>0$ at all points.\",\n \"The map $ f$ restricted to the boundary components $\\\\mathbb {T}^1 \\\\times \\\\lbrace 0\\\\rbrace $ , $\\\\mathbb {T}^1 \\\\times \\\\lbrace 1\\\\rbrace $ of the annulus has well defined rotation numbers $\\\\omega _-,\\\\omega _+$ , respectively, with $\\\\omega ^-<\\\\omega ^+$ .\",\n \"We will assume that $\\\\omega ^-,\\\\omega ^+>0$ .\",\n \"By an invariant primary torus (essential invariant circle) we mean a 1-dimensional torus $T$ invariant under $f$ that cannot be homotopically deformed into a point inside the annulus.\",\n \"Since $f$ is a monotone twist map, each invariant primary torus $T$ is the graph of some Lipschitz function (see [4], [5]).\",\n \"A region in $A$ between two invariant primary tori $T_1$ and $T_2$ is called a Birkhoff Zone of Instability (BZI) provided that there is no invariant primary torus in the interior of the region.\",\n \"It is known that, for an area preserving monotone twist map $f$ of $A$ , given a BZI, there exist Birkhoff connecting orbits that go from any neighborhood of one boundary torus to any neighborhood of the other boundary torus (see [4], [5], [37]).\",\n \"We have the following results: Theorem 4.1 (Birkhoff Connecting Theorem) Suppose that $T_1$ and $T_2$ bound a BZI.\",\n \"For every pair of neighborhoods $U$ of $T_1$ and $V$ of $T_2$ there exist a point $z\\\\in U$ and an integer $N>0$ such that $f^{N}(z)\\\\in V$ .\",\n \"Corollary 4.2 Suppose that $T_1$ and $T_2$ bound a BZI, and that the restrictions of $f$ to $T_1$ and $T_2$ are topologically transitive.\",\n \"For every $\\\\zeta _1\\\\in T_1,\\\\zeta _2\\\\in T_2$ and every pair of neighborhoods $U$ of $\\\\zeta _1$ and $V$ of $\\\\zeta _2$ , there exist a point $z\\\\in U$ and an integer $N>0$ such that $f^{N}(z)\\\\in V$ .\",\n \"A subset $M\\\\subseteq A$ is said to be monotone (cyclically ordered) if $\\\\pi _x(z_1)<\\\\pi _x(z_2)$ implies $\\\\pi _x(f(z_1))<\\\\pi _x(f(z_2))$ for all $z_1,z_2\\\\in M$ .\",\n \"For $z\\\\in A$ the extended orbit of $z$ is the set $EO(z)=\\\\lbrace f^n(z)+(j,0)\\\\,:\\\\,n,j\\\\in \\\\mathbb {Z}\\\\rbrace $ .\",\n \"The orbit of $z$ is said to be monotone (cyclically ordered) if the set $EO(z)$ is monotone.\",\n \"If the orbit of $z\\\\in A$ is monotone, then the rotation number $\\\\rho (z)=\\\\lim _{n\\\\rightarrow \\\\infty }(\\\\pi _x(f^n(z))/n)$ exists.\",\n \"We denote $\\\\textrm {Rot}(\\\\omega )=\\\\lbrace z\\\\in A\\\\,:\\\\,\\\\rho (z)=\\\\omega \\\\rbrace $ .\",\n \"All points in the same monotone set have the same rotation number.\",\n \"Definition 4.3 An Aubry-Mather set for $\\\\omega \\\\in \\\\mathbb {T}^1$ is a minimal, monotone, $f$ -invariant subset of $\\\\textrm {Rot}(\\\\omega )$ .\",\n \"Here by a minimal set we mean a closed invariant set that does not contain any proper closed invariant subsets.\",\n \"(Equivalently, the orbit of every point in the set is dense in the set.)\",\n \"This should not be confused with action-minimizing or $h$ -minimal sets, where $h$ is a generating function for $f$ .\",\n \"Theorem 4.4 (Aubry-Mather Theorem) For every $\\\\omega \\\\in [\\\\omega ^-,\\\\omega ^+]$ , there exists a non-empty Aubry-Mather set $\\\\Sigma _\\\\omega $ in $\\\\textrm {Rot}(\\\\omega )$ .\",\n \"Aubry-Mather sets defined as above can be obtained as limits of monotone Birkhoff periodic orbits [34].\",\n \"There may be many Aubry-Mather sets with the same rotation number [45].\",\n \"On the other hand, if one requires Aubry-Mather sets to be action minizing, there exists a unique recurrent Aubry-Mather set for any given irrational rotation number.\",\n \"In the sequel we will use the following result on the vertical ordering of Aubry-Mather sets from [27].\",\n \"Theorem 4.5 There exists a family of essential circles $C_\\\\omega $ in $A$ for $\\\\omega \\\\in [\\\\omega ^-,\\\\omega ^+]$ such that: (i) Each $C_\\\\omega $ is a graph over $y=0$ ; (ii) The circles $C_\\\\omega $ are mutually disjoint, and if $\\\\omega ^{\\\\prime }>\\\\omega $ then $C_{\\\\omega ^{\\\\prime }}$ is above $C_{\\\\omega }$ ; (iii) Each $C_\\\\omega $ contains an Aubry-Mather set $\\\\Sigma _\\\\omega $ .\",\n \"The above circles have Lipschitz regularity, and are projections of so called `ghost circles' that are objects in $\\\\mathbb {R}^\\\\mathbb {Z}$ .\",\n \"See [27] for details.\",\n \"A similar result to Theorem REF appears in [35] who find Aubry-Mather sets lying on pseudo-graphs that are (not strictly) vertically ordered.\",\n \"There are some analogues of the Birkhoff Connecting Theorem for Aubry-Mather sets.\",\n \"The following lemma is used in [33] to provide a topological proof for Mather Connecting Theorem stated below.\",\n \"Lemma 4.6 Suppose that $T_1$ and $T_2$ bound a BZI.\",\n \"Let $\\\\Sigma _\\\\omega $ be an Aubry-Mather set of rotation number $\\\\omega $ inside the BZI.\",\n \"Let $p$ be a recurrent point in $\\\\Sigma _\\\\omega $ and $W(p)$ be a neighborhood of $p$ inside the BZI.\",\n \"The following hold true: (i) For some positive number $n^+$ (resp.\",\n \"$n^-$ ) depending on $W(p)$ the set$\\\\bigcup _{j=0}^{n^+}f^j(W(p))$ (resp.\",\n \"$\\\\bigcup _{j=0}^{n^-}f^{-j}(W(p))$ ) separates the cylinder.\",\n \"(ii) The set $W^{+\\\\infty }:=\\\\bigcup _{j=0}^{\\\\infty }f^j(W (p))$ (resp.\",\n \"the set $W^{-\\\\infty }:=\\\\bigcup _{j=0}^{\\\\infty }f^{-j}(W(p) )$ ), is connected and open.\",\n \"(iii) The closure of $W^{+\\\\infty }$ (resp.\",\n \"$W ^{-\\\\infty }$ ) contains both boundary tori $T_1$ and $T_2$ .\",\n \"(iv) The set $W ^\\\\infty :=\\\\bigcup _{j=-\\\\infty }^{\\\\infty }f^j(W(p))$ is invariant, and both $W^{+\\\\infty } $ and $W^{-\\\\infty } $ are open and dense in $W ^\\\\infty $ .\",\n \"The following result says that there exist orbits that visit any prescribed bi-infinite sequence of Aubry-Mather sets inside a BZI (see [46], [53], [29], [35], [33]).\",\n \"Theorem 4.7 (Mather Connecting Theorem) Suppose that $T_1$ and $T_2$ bound a BZI, and $\\\\lbrace \\\\Sigma _{\\\\omega _i}\\\\rbrace _{i\\\\in \\\\mathbb {Z}}$ is a bi-infinite sequence of Aubry-Mather sets inside the BZI.\",\n \"Let $\\\\varepsilon _i>0$ for $i\\\\in \\\\mathbb {Z}$ .\",\n \"Then there exist a point $z$ inside the BZI and an increasing bi-infinite sequence of integers $\\\\lbrace j_i\\\\rbrace _{i\\\\in \\\\mathbb {Z}}$ such that $f^{j_i}(z)$ is within $\\\\varepsilon _i$ from $\\\\Sigma _{\\\\omega _i}$ for all $i\\\\in \\\\mathbb {Z}$ .\",\n \"The Aubry-Mather sets in Theorem REF are action minimizing.\",\n \"The following topological version of Mather Connecting Theorem, due to Hall [29], provides shadowing orbits of Aubry-Mather sets that are not necessarily action minimizing.\",\n \"This approach can be implemented in rigorous computer experiments [32].\",\n \"Theorem 4.8 Suppose that $T_1$ and $T_2$ bound a BZI, and $\\\\lbrace z_s\\\\rbrace _{s\\\\in \\\\mathbb {Z}}$ is a bi-infinite sequence of monotone $(p_s/q_s)$ -periodic points, with the rotation numbers $p_s/q_s$ mutually distinct, inside the BZI.\",\n \"Given a bi-infinite sequence $\\\\lbrace n_s\\\\rbrace _{s\\\\in \\\\mathbb {Z}}$ of positive integers, then there exist a point $z$ and a bi-infinite sequence $\\\\lbrace m_s\\\\rbrace _{s\\\\in \\\\mathbb {Z}}$ of positive integers such that, for each $s\\\\ge 0$ , there exist some points $w_s,\\\\bar{w}_s$ in the extended orbit of $z_s$ such that $\\\\begin{split}\\\\pi _x(f^j(w_s))<\\\\pi _x(f^j(z))<\\\\pi _x(f^j(\\\\bar{w}_s))\\\\textrm { for }\\\\\\\\\\\\sum _{t=0}^{s-1} n_t+\\\\sum _{t=0}^{s-1}m_t\\\\le j\\\\le \\\\sum _{t=0}^{s} n_t+\\\\sum _{t=0}^{s-1}m_t.\\\\end{split}$ A similar statement holds for each $s<0$ .\",\n \"In the above, $n_s$ represents the number of iterates for which the orbit of $z$ shadows – in the sense of the cyclical ordering – the extended orbit of $z_s$ , and $m_s$ represents the number of iterates it takes the orbit of $z$ to pass from the extended orbit of $z_s$ to the extended orbit of $z_{s+1}$ .\",\n \"They main tool used in Hall's arguments is that of a positive (negative) diagonal.\",\n \"Denote by $\\\\mathcal {Z}$ the BZI bounded by the tori $T_1$ and $T_2$ .\",\n \"Let $I_z&=&\\\\lbrace w\\\\in \\\\mathcal {Z}\\\\,|\\\\, \\\\pi _x(w)=\\\\pi _x(z)\\\\rbrace ,\\\\\\\\I^+_z&=&\\\\lbrace w\\\\in I_z\\\\,|\\\\, \\\\pi _y(w)\\\\ge \\\\pi _y(z)\\\\rbrace ,\\\\\\\\I^-_z&=&\\\\lbrace w\\\\in I_z\\\\,|\\\\, \\\\pi _y(w)\\\\le \\\\pi _y(z)\\\\rbrace ,\\\\\\\\B_{z_0,z_1}&=&\\\\lbrace w\\\\in \\\\mathcal {Z}\\\\,|\\\\, \\\\pi _x(z_0)< \\\\pi _x(w)< \\\\pi _x(z_1)\\\\rbrace ,$ where $z,z_0,z_1$ are points in the annulus.\",\n \"A positive diagonal $D$ in $B_{z_0,z_1}$ is a set $D\\\\subseteq \\\\textrm {cl}(B_{z_0,z_1})$ such that (i) $D$ is simply connected and the closure of its interior; (ii) $\\\\partial D \\\\cap \\\\textrm {cl}(B_{z_0,z_1})\\\\subseteq I^-_{z_0}\\\\cup I^+_{z_1} \\\\cup T_1\\\\cup T_2$ ; (iii) $\\\\partial D\\\\cap I^-_{z_0}\\\\ne \\\\emptyset $ and $\\\\partial D\\\\cap I^+_{z_1}\\\\ne \\\\emptyset $ .\",\n \"The set $\\\\partial D\\\\cap B_{z_0,z_1}$ has exactly two components connecting $I^-_{z_0}\\\\cup T_1$ to $I^+_{z_1}\\\\cup T_2$ , which are called the upper and lower edges of $D$ , respectively.\",\n \"We informally say that these components `stretch across' $B_{z_0,z_1}$ .\",\n \"See Fig.\",\n \"REF .\",\n \"A negative diagonal and its upper and lower edges are defined similarly.\",\n \"Figure: Two positive diagonal sets.\",\n \"The positive diagonal set on the left has its upper and lower edges marked.\",\n \"The positive diagonal set on the right is obtained by intersecting f(B w 0 ,w 1 )f(B_{w_0,w_1}) with B f(w 0 ),f(w 1 ) B_{f(w_0),f(w_1)}.\",\n \"The upper edge of this diagonal set is contained in f(I w 0 + )f(I^+_{w_0}) and the lower edge is contained in f(I w 1 - )f(I^-_{w_1}).An important feature of positive diagonals is the following hereditary property.\",\n \"Given $z_0,z_0$ such that $\\\\pi _x(z_0)<\\\\pi _x(z_1)$ and $\\\\pi _x(f(z_0))<\\\\pi _x(f(z_1))$ , if $D$ is a positive diagonal in $B_{z_0,z_1}$ , then $f(D)\\\\cap B_{f(z_0),f(z_1)}$ has a component $D^{\\\\prime }$ that is a positive diagonal in $B_{f(z_0),f(z_1)}$ .\",\n \"One way to generate a positive diagonal set is by taking a component of the intersection between $f^k(B_{w_0,w_1})$ and $B_{f^k(w_0),f^k(w_1)}$ .\",\n \"In this case, there exists a positive diagonal in $B_{f^k(w_0),f^k(w_1)}$ whose upper edge is contained on $f^k(I_{w_0}^+)$ and lower edge is contained in $f^k(I_{w_1}^-)$ .\",\n \"More general, one has the following important property.\",\n \"If $D$ has the upper edge contained in $f^k(I^+_{w_0})$ and the lower edge contained in $f^k(I^-_{w_1})$ , and $\\\\partial D\\\\cap B_{z_0,z_1}\\\\subseteq f^k(I^+_{w_0}\\\\cup I^-_{w_1})$ , for some $w_0,w_1$ with $\\\\pi _x(w_0)<\\\\pi _x(w_1)$ and some $k>0$ , then then $f(D)\\\\cap B_{f(z_0),f(z_1)}$ has a component $D^{\\\\prime }$ which can be chosen so that its upper edge is contained in $f^{k+1}(I^+_{w_0})$ and its lower edge is contained in $f^{k+1}(I^-_{w_1})$ .\",\n \"See Fig.\",\n \"REF .\",\n \"A similar property holds for negative diagonals.\",\n \"The proof of Theorem REF in [29] is an inductive argument which, for a given pair of adjacent points $w_0,\\\\bar{w}_0$ in the extended orbit of $z_0$ , and for each $\\\\sigma \\\\ge 0$ , produces a nested sequence $D_0\\\\supseteq D_1\\\\supseteq \\\\ldots \\\\supseteq D_{\\\\sigma }$ of negative diagonals of $B_{w_0,\\\\bar{w}_0}$ such that, for each $s\\\\in \\\\lbrace 0,\\\\ldots ,\\\\sigma \\\\rbrace $ , the following hold: (a) the orbit of each point $z \\\\in D_s$ satisfies the ordering relation (REF ), and (b) there is a sufficiently large $j_s>0$ such that $f^{j_s+j}(D_s)$ contains a component that is a positive diagonal in $B_{f^j(w_s),f^j(\\\\bar{w}_s)}$ , for some adjacent points $w_s,\\\\bar{w}_s\\\\in EO(z_s)$ , and for all $j=1,\\\\ldots , n_s$ .\",\n \"In the above, $j_s=\\\\sum _{t=0}^{s-1} n_t+\\\\sum _{t=0}^{s-1}m_t$ .\",\n \"Moreover, in this inductive argument one can choose the diagonal sets $D_s$ so that $f^{j_s+j}(D_s)$ has the upper edge contained in $f^{j_s+j}(I^+_{w_0})$ , lower edge contained in $f^{j_s+j}(I^-_{\\\\bar{w}_0})$ , and $\\\\partial f^{j_s+j}(D_s)\\\\cap B_{f^{j_s+j}(w_0),f^{j_s+j}(\\\\bar{w}_0)} \\\\subseteq f^{j_s+j} (I^+_{w_0}\\\\cup I^-_{\\\\bar{w}_0})$ .\",\n \"For the basis step, starting with $w_0,\\\\bar{w}_0$ and applying the hereditary property from above $n_0$ times, one obtains a negative diagonal set $D_0$ of $B_{w_0,\\\\bar{w}_0}$ with the properties that each point $z \\\\in D_0$ satisfies the ordering relation (REF ) for $s=0$ , and $f^{n_0}(D_0)$ has a component that is a positive diagonal of $B_{f^{n_0}(w_0),f^{n_0}(\\\\bar{w}_0)}$ .\",\n \"For the inductive step, one assumes a negative diagonal $D_\\\\sigma $ of $B_{w_0,\\\\bar{w}_0}$ as above, and wants to produce a negative diagonal $D_{\\\\sigma +1}\\\\subseteq D_\\\\sigma $ of $B_{w_0,\\\\bar{w}_0}$ which fulfils the corresponding properties.\",\n \"The key idea is to use the existence of points near $y=0$ that get near $y=1$ , and of points near $y=1$ that get near $y=0$ , as provided by Theorem REF , in order to show that for some $j_\\\\sigma $ sufficiently large $f^{j_\\\\sigma }(D_\\\\sigma )$ contains a component that stretches all the way across a fundamental interval of the annulus.\",\n \"Hence $f^{j_\\\\sigma }(D_\\\\sigma )$ contains a subset that is a positive diagonal of $B_{w_{\\\\sigma +1}, \\\\bar{w}_{\\\\sigma +1}}$ for two adjacent points $w_{\\\\sigma +1}, \\\\bar{w}_{\\\\sigma +1}\\\\in EO(z_{\\\\sigma +1})$ .\",\n \"From this it follows that $f^{j_\\\\sigma +j}(D_\\\\sigma )$ contains a component that is a positive diagonal in $B_{f^j(w_{\\\\sigma +1}),f^j(\\\\bar{w}_{\\\\sigma +1})}$ for all $j=1,\\\\ldots , n_{\\\\sigma +1}$ .\",\n \"This completes the inductive step.\",\n \"Applying a similar argument for the negative iterates of $f$ produces a nested sequence of positive diagonals of $B_{w_0,\\\\bar{w}_0}$ .\",\n \"A positive diagonal of $B_{w_0,\\\\bar{w}_0}$ always has a non-empty intersection with a negative diagonal of $B_{w_0,\\\\bar{w}_0}$ .\",\n \"This implies the existence of points $z$ whose forward orbits satisfy the ordering conditions in (REF ) and whose backwards orbits satisfy similar ordering conditions.\",\n \"Using limit arguments as in [34], one can obtain shadowing of Aubry-Mather sets of irrational rotation numbers as well.\",\n \"These topological ideas will be used in the proof of Theorem REF below.\",\n \"Remark 4.9 An immediate consequence of Lemma REF is that, given a neighborhood $W$ of a point $p\\\\in \\\\Sigma _\\\\omega $ , where $\\\\Sigma _\\\\omega $ is an Aubry-Mather set inside a BZI bounded by $T_1$ and T$_2$ , and given a neighborhood $U$ of $T_1$ or $T_2$ , there exists an arbitrarily large $j>0$ such that $f^j(W(p))\\\\cap U\\\\ne \\\\emptyset $ .\",\n \"Also, there exists an arbitrarily large $j^{\\\\prime }>0$ such that $f^{-j^{\\\\prime }}(W(p))\\\\cap U\\\\ne \\\\emptyset $ .\",\n \"Remark 4.10 The results in this section hold if we replace conditions (i) and (ii) from the definition of an area preserving, monotone twist map with the following weaker conditions: (i') $f$ satisfies the following `condition B': for every pair of neighborhoods $U_1$ of $T_1$ and $U_2$ of $T_2$ , there exist $z_1,z_2\\\\in A$ and $n_1,n_2>0$ such that $z_1\\\\in U_1$ and $f^{n_1}(z_1)\\\\in U_2$ , and $z_2\\\\in U_2$ and $f^{n_2}(z_2)\\\\in U_1$ .\",\n \"Note that this condition only makes sense if we restrict the dynamics to a BZI.\",\n \"(ii') $f$ satisfies the following positive tilt condition: if we denote by $\\\\theta _{z}$ the angle deviation from the vertical, measured from the vertical vector $(0,1)$ to $Df_{z}(0,1)$ , with the clockwise direction taken as the positive direction, and defined in such a way that $\\\\theta _{(x,0)}\\\\in [-\\\\pi /2,\\\\pi /2]$ and $\\\\theta $ is continuous, then $\\\\theta _{z}> 0$ at all points.\",\n \"See Fig.\",\n \"REF .\",\n \"Compositions of positive twist maps, are for example, positive tilt maps.\",\n \"We shall note that the Aubry-Mather theory applies to positive tilt maps as well (see [31]).\",\n \"Figure: Positively tilted map.Remark 4.11 Aubry-Mather theory and the above shadowing result also hold for generalized twist maps of the higher dimensional annulus $S^1\\\\times \\\\mathbb {R}^n$ .\",\n \"See [1].\",\n \"Remark 4.12 The topological approach in this section does not yield trajectories that get close to each set in the prescribed collection of Aubry-Mather sets, as in Theorem REF .\",\n \"It seems possible, however, that these topological methods can be combined with the variational methods in [46] to obtain trajectories that go very close to each Aubry-Mather set in the given collection.\"\n ],\n [\n \"Scattering map\",\n \"The scattering map acts on the normally hyperbolic invariant manifold $\\\\Lambda $ and relates the past asymptotic trajectory of each orbit in the homoclinic manifold to its future asymptotic behavior.\",\n \"We review its properties following [19].\",\n \"For the general case, we consider a $C^r$ -differentiable manifold $M$ , a $C^r$ -differentiable map $f:M\\\\rightarrow M$ , and an $l$ -dimensional normally hyperbolic invariant manifold $\\\\Lambda $ for $f$ .\",\n \"By the definition of normal hyperbolicity, there exists a splitting of the tangent bundle of $TM$ into sub-bundles $TM=E^u\\\\oplus E^s\\\\oplus T\\\\Lambda ,$ that are invariant under $df$ , and there exist a constant $C>0$ and rates $0<\\\\lambda <\\\\mu ^{-1}<1$ , such that for all $x\\\\in \\\\Lambda $ we have $\\\\begin{split}v\\\\in E^s_x \\\\Leftrightarrow \\\\Vert Df^k_x(v)\\\\Vert \\\\le C\\\\lambda ^k\\\\Vert v\\\\Vert \\\\textrm { for all } k\\\\ge 0,\\\\\\\\v\\\\in E^u_x \\\\Leftrightarrow \\\\Vert Df^k_x(v)\\\\Vert \\\\le C\\\\lambda ^{-k}\\\\Vert v\\\\Vert \\\\textrm { for all } k\\\\le 0,\\\\\\\\v\\\\in T_x\\\\Lambda \\\\Leftrightarrow \\\\Vert Df^k_x(v)\\\\Vert \\\\le C\\\\mu ^{|k|}\\\\Vert v\\\\Vert \\\\textrm { for all } k\\\\in \\\\mathbb {Z}.\\\\end{split}$ The smoothness of the invariant objects defined by the normally hyperbolic structure depends on the rates $\\\\lambda $ and $\\\\mu $ .\",\n \"The map $f$ is said to be $\\\\ell $ -normally hyperbolic along $\\\\Lambda $ provided that $1 \\\\le \\\\ell \\\\le r$ is an integer satisfying $\\\\lambda \\\\,\\\\mu ^\\\\ell <1$ , i.e., $\\\\ell < (\\\\log \\\\lambda ^{-1})(\\\\log \\\\mu )^{-1}$ .\",\n \"Then the stable and unstable manifolds $W^s(\\\\Lambda )$ and $W^u(\\\\Lambda )$ and the normally hyperbolic manifold $\\\\Lambda $ are all $C^\\\\ell $ -differentiable.\",\n \"The splitting $E^s_z=T_z(W^s(x))$ depends $C^{\\\\ell -1}$ smoothly on $z$ in $W^s(\\\\Lambda )$ so $\\\\lbrace \\\\,W^s(x)\\\\, |\\\\, x \\\\in \\\\Lambda \\\\,\\\\rbrace $ is a $C^{\\\\ell -1}$ foliation of $W^s(\\\\Lambda )$ .\",\n \"(This is stated explicitly in [50] and follows from the $C^r$ Section Theorem of [30].)\",\n \"Similarly, $\\\\lbrace \\\\,W^u(x)\\\\, |\\\\, x \\\\in \\\\Lambda \\\\,\\\\rbrace $ is a $C^{\\\\ell -1}$ foliation of $W^u(\\\\Lambda )$ .\",\n \"Since the stable (resp.\",\n \"unstable) manifolds of $\\\\Lambda $ are foliated by stable (resp.\",\n \"unstable) manifolds of points, we have that for each $x\\\\in W^s(\\\\Lambda )$ (resp.\",\n \"$x\\\\in W^u(\\\\Lambda )$ ), there exists a unique $x^+ \\\\in \\\\Lambda $ (resp.\",\n \"$x^- \\\\in \\\\Lambda $ ) such that $x \\\\in W^s(x^+)$ (resp.\",\n \"$x \\\\in W^u(x^-)$ ).\",\n \"We define the maps $\\\\Omega ^+ : W^s(\\\\Lambda ) \\\\rightarrow \\\\Lambda $ by $\\\\Omega ^+(x) = x^+$ and $\\\\Omega ^- : W^u(\\\\Lambda ) \\\\rightarrow \\\\Lambda $ by $\\\\Omega ^-(x) = x^-$ .\",\n \"The maps $\\\\Omega ^+$ and $\\\\Omega ^-$ are $C^{\\\\ell -1}$ -smooth since the foliations are smooth.\",\n \"We now describe the scattering map.\",\n \"Assume that $W^u(\\\\Lambda )$ and $W^s(\\\\Lambda )$ have a differentiably transverse intersection along a homoclinic $l$ -dimensional $C^{\\\\ell -1}$ -differentiable manifold $\\\\Gamma $ .\",\n \"This means that $\\\\Gamma \\\\subseteq W^u(\\\\Lambda ) \\\\cap W^s(\\\\Lambda )$ and, for each $x\\\\in \\\\Gamma $ , we have $ \\\\begin{split}T_xM=T_xW^u(\\\\Lambda )+T_xW^s(\\\\Lambda ),\\\\\\\\T_x\\\\Gamma =T_xW^u(\\\\Lambda )\\\\cap T_xW^s(\\\\Lambda ).\\\\end{split} $ We assume the additional condition that for each $x\\\\in \\\\Gamma $ we have $\\\\begin{split}T_xW^s(\\\\Lambda )=T_xW^s(x^+)\\\\oplus T_x(\\\\Gamma ),\\\\\\\\T_xW^u(\\\\Lambda )=T_xW^u(x^-)\\\\oplus T_x(\\\\Gamma ),\\\\end{split} $ where $x^-,x^+$ are the uniquely defined points in $\\\\Lambda $ corresponding to $x$ .\",\n \"The restrictions $\\\\Omega ^+_\\\\Gamma ,\\\\Omega ^-_\\\\Gamma $ of $\\\\Omega ^+,\\\\Omega ^-$ to $\\\\Gamma $ are local $C^{\\\\ell -1}$ -diffeomorphisms.\",\n \"By replacing $\\\\Gamma $ to a submanifold of it (which, with an abuse of notation, we still denote $\\\\Gamma $ ) we can ensure that $\\\\Omega ^+_\\\\Gamma :\\\\Gamma \\\\rightarrow U^+,\\\\Omega ^-_\\\\Gamma :\\\\Gamma \\\\rightarrow U^-$ are $C^{\\\\ell -1}$ -diffeomorphisms from $\\\\Gamma $ to the open sets $U^+,U^-$ in $\\\\Lambda $ , respectively.\",\n \"Definition 5.1 A homoclinic manifold $\\\\Gamma $ satisfying (REF ) and (REF ), and for which the corresponding restrictions of the wave maps are $C^{\\\\ell -1}$ -diffeomorphisms, is referred as a homoclinic channel.\",\n \"Definition 5.2 Given a homoclinic channel $\\\\Gamma $ , the scattering map associated to $\\\\Gamma $ is the $C^{\\\\ell -1}$ -diffeomorphism $S_\\\\Gamma =\\\\Omega _\\\\Gamma ^+\\\\circ (\\\\Omega _\\\\Gamma ^-)^{-1}$ from the open subset $U^-:=\\\\Omega _\\\\Gamma ^-(\\\\Gamma )$ in $\\\\Lambda $ to the open subset $U^+:=\\\\Omega _\\\\Gamma ^+(\\\\Gamma )$ in $\\\\Lambda $ .\",\n \"In the sequel we will regard $S_\\\\Gamma $ as a partially defined map, so the image of a set $A$ by $S_\\\\Gamma $ means the set $S_\\\\Gamma (A\\\\cap U^-)$ .\",\n \"In this paper, we need the following property of the scattering map.\",\n \"Proposition 5.3 Assume that $T_1$ and $T_2$ are two invariant submanifolds of complementary dimensions in $\\\\Lambda $ .\",\n \"Then $W^u(T_1)$ has a topologically transverse intersection with $W^s(T_2)$ inside $\\\\Gamma $ if and only if $S_\\\\Gamma (T_1\\\\cap U^-)$ has a topologically transverse intersection with $T_2\\\\cap U^+$ in $\\\\Lambda $ .\",\n \"Let $\\\\bar{T}_1=(\\\\Omega ^-_\\\\Gamma )^{-1}(T_1\\\\cap U^-)\\\\subseteq \\\\Gamma $ and $\\\\bar{T}_2=(\\\\Omega ^+_\\\\Gamma )^{-1}(T_2\\\\cap U^+)\\\\subseteq \\\\Gamma $ .\",\n \"A topologically transverse intersection of $W^u(T_1)$ with $W^s(T_2)$ in $\\\\Gamma $ occurs if and only if $\\\\bar{T}_1$ intersects $\\\\bar{T}_2$ topologically transversally in $\\\\Gamma $ , which is equivalent to $S_\\\\Gamma (T_1\\\\cap U^-)$ intersects topologically transversally $T_2\\\\cap U^+$ in $\\\\Lambda $ .\",\n \"For the definition of topological transversality (topological crossing) see [8].\",\n \"For the main result of this paper the normally hyperbolic invariant manifold $\\\\Lambda $ is assumed to be 2-dimensional, i.e., $l=2$ , and the invariant submanifolds $T_1,T_2$ in Proposition REF are 1-dimensional invariant tori.\"\n ],\n [\n \"Topological method of correctly aligned windows\",\n \"We describe briefly the topological method of correctly aligned windows.\",\n \"We follow [55].\",\n \"See also [24], [23], [41].\",\n \"Definition 5.4 An $(n_1,n_2)$ -window in an $n$ -dimensional manifold $M$ , where $n_1+n_2=n$ , is a compact subset $W$ of $M$ together with a homeomorphism $\\\\chi $ from some open neighborhood of $[0,1]^{n_1}\\\\times [0,1]^{n_2}$ in $\\\\mathbb {R}^{n_1}\\\\times \\\\mathbb {R}^{n_2}$ to an open subset of $M$ , such that $W=\\\\chi ([0,1]^{n_1}\\\\times [0,1]^{n_2}),$ and with a choice of an `exit set' $W^{\\\\rm exit} =\\\\chi \\\\left(\\\\partial [0,1]^{n_1}\\\\times [0,1]^{n_2} \\\\right)$ and of an `entry set' $W^{\\\\rm entry}=\\\\chi \\\\left([0,1]^{n_1}\\\\times \\\\partial [0,1]^{n_2}\\\\right).$ Denote by $\\\\pi _{1}: \\\\mathbb {R}^{n_1}\\\\times \\\\mathbb {R}^{n_2}\\\\rightarrow \\\\mathbb {R}^{n_1}$ the projection onto the first component, and by $\\\\pi _{2}: \\\\mathbb {R}^{n_1}\\\\times \\\\mathbb {R}^{n_2}\\\\rightarrow \\\\mathbb {R}^{n_2}$ the projection onto the second component.\",\n \"Definition 5.5 Let $W_1$ and $W_2$ be $(n_1,n_2)$ -windows, and let $\\\\chi _1$ and $\\\\chi _2$ be the corresponding local parametrizations.\",\n \"Let $f$ be a continuous map on $M$ with $f(\\\\textrm {im}(\\\\chi _1))\\\\subseteq \\\\textrm {im}(\\\\chi _2)$ .\",\n \"We say that $W_1$ is correctly aligned with $W_2$ under $f$ if the following conditions are satisfied: (i) $f(W^{\\\\rm exit}_1) \\\\cap (W_2) = \\\\emptyset $ and $f(W_1) \\\\cap W_2^{\\\\rm entry} = \\\\emptyset $ ; (ii) There exists $y_0 \\\\in [0,1]^{n_2}$ such that the curve $x\\\\in [0,1]^{n_1} \\\\mapsto \\\\hat{f} (x,y_0)$ , where $\\\\hat{f}:=\\\\chi _2^{-1}\\\\circ f\\\\circ \\\\chi _1$ , has the following properties: $\\\\hat{f}_{y_0}\\\\left( [0,1]^{n_1}\\\\right)\\\\subseteq \\\\mathbb {R}^{n_1}\\\\times (0,1)^{n_2},\\\\\\\\\\\\hat{f}_{y_0}\\\\left( \\\\partial [0,1]^{n_1}\\\\right)\\\\subseteq (\\\\mathbb {R}^{n_1}\\\\setminus [0,1]^{n_1})\\\\times (0,1)^{n_2}=\\\\emptyset ,\\\\\\\\\\\\deg (\\\\pi _{1}\\\\circ \\\\hat{f}_{y_0},0)=w\\\\ne 0.$ We call the integer $w\\\\ne 0$ in the above definition the degree of the alignment.\",\n \"The following result is a topological version of the Shadowing Lemma.\",\n \"Theorem 5.6 Let $\\\\lbrace W_i\\\\rbrace _{i\\\\in \\\\mathbb {Z}}$ , be a collection of $(n_1,n_2)$ -windows in $M$ , and let $f_i$ be a collection of continuous maps on $M$ .\",\n \"If for each $i\\\\in \\\\mathbb {Z}$ , $W_i$ is correctly aligned with $W_{i+1}$ under $f_i$ , then there exists a point $p\\\\in W_0$ such that $(f_{i}\\\\circ \\\\cdots \\\\circ f_{0})(p)\\\\in W_{i+1}, \\\\textrm { for all } i\\\\in \\\\mathbb {Z}.\",\n \"$ Moreover, assuming that there exists $k>0$ such that $W_{i}=W_{(i\\\\,{\\\\rm mod}\\\\, k)}$ and $f_{i}=f_{(i\\\\,{\\\\rm mod}\\\\, k)}$ for all $i\\\\in \\\\mathbb {Z}$ , then there exists a point $p$ as above that is periodic in the sense $(f_{k-1}\\\\circ \\\\cdots \\\\circ f_{0})(p)=p.$ The correct alignment of windows is robust, in the sense that if two windows are correctly aligned under a map, then they remain correctly aligned under a sufficiently small $C^0$ -perturbation of the map.\",\n \"Robustness makes the method of correctly aligned windows appropriate for perturbative arguments, as well as for rigorous numerical experiments.\",\n \"Also, the correct alignment satisfies a natural product property.\",\n \"Given two windows and a map, if each window can be written as a product of window components, and if the components of the first window are correctly aligned with the corresponding components of the second window under the appropriate components of the map, then the first window is correctly aligned with the second window under the given map.\",\n \"For example, if we consider a pair of windows in a neighborhood of a normally hyperbolic invariant manifold, if the center components of the windows are correctly aligned and the hyperbolic components of the windows are also correctly aligned, then the windows are correctly aligned.\",\n \"Although the product property is quite intuitive, its rigorous statement is rather technical, so we will omit it here.\",\n \"The details can be found in [23].\",\n \"In Sections and , we will consider various windows lying on one of the manifolds $\\\\Lambda $ , $W^u(\\\\Lambda )$ , $W^s(\\\\Lambda )$ , or $M$ .\",\n \"Without explicit mention, every such a window will be represented by the image of a rectangle through a local parametrization of the appropriate manifold.\",\n \"We will also consider correct alignment relations of windows lying on the same manifold.\",\n \"All the correct alignment relations in the arguments presented later in this paper have degree $w=1$ .\"\n ],\n [\n \"Existence of Birkhoff connecting orbits\",\n \"In this section we state and prove an extension of the Corollary REF of the Birkhoff connecting orbit theorem, and an extension of Mather's theorem on shadowing of Aubry-Mather sets, specifically of Theorem REF .\",\n \"The methodology is based on the topological approach of Hall and on the Jordan Curve Theorem.\",\n \"The statements below will be used in the proof of the main theorem.\",\n \"Throughout the section we assume that $f$ is an orientation preserving, boundary preserving, area preserving, monotone twist map of the annulus $A$ , as in Section , and we adopt the notation conventions from that section.\",\n \"In Corollary REF , it was assumed that the restrictions of the map to the boundary tori of the BZI are topologically transitive, and it was inferred the existence of connecting orbits from an arbitrarily small neighborhood of some prescribed point on one boundary torus to an arbitrarily small neighborhood of some prescribed point on the other boundary torus.\",\n \"In the statements below we prove the same result without the topological transitivity assumption.\",\n \"Theorem 6.1 Suppose that $T_1$ and $T_2$ bound a BZI $\\\\mathcal {Z}$ .\",\n \"Assume that $\\\\zeta _1\\\\in T_1$ and $\\\\zeta _2\\\\in T_2$ .\",\n \"Fix a pair of neighborhoods $U$ of $\\\\zeta _1$ and $V$ of $\\\\zeta _2$ .\",\n \"Then there exists a point $z\\\\in U$ and an integer $N>0$ such that $f^{N}(z)\\\\in V$ for some $N>0$ that can be chosen arbitrarily large.\",\n \"Moreover, if we assume that $U,V$ are chosen so that $U\\\\cap \\\\mathcal {Z},V\\\\cap \\\\mathcal {Z}$ are topological disks, then there exists a point $z^{\\\\prime }\\\\in \\\\partial U$ such that $f^N(z^{\\\\prime })\\\\in \\\\partial V$ .\",\n \"Consider the neighborhood $U$ of $\\\\zeta _1\\\\in T_1$ .\",\n \"Choose two points $\\\\zeta ^{\\\\prime }_1,\\\\zeta ^{\\\\prime \\\\prime }_1\\\\in T_1\\\\cap U$ such that $\\\\zeta _1$ is between $\\\\zeta ^{\\\\prime }_1$ and $\\\\zeta ^{\\\\prime \\\\prime }_1$ and the portion of $T_1$ between $\\\\zeta ^{\\\\prime }_1$ and $\\\\zeta ^{\\\\prime \\\\prime }_1$ is contained in $\\\\textrm {int}(U)$ .\",\n \"Choose a simple curve $\\\\gamma _0$ inside $U$ , with endpoints at $\\\\zeta ^{\\\\prime }_1$ and $\\\\zeta ^{\\\\prime \\\\prime }_1$ .\",\n \"The curve $\\\\gamma _0$ together with the portion of $T_1$ between $\\\\zeta ^{\\\\prime }_1$ and $\\\\zeta ^{\\\\prime \\\\prime }_1$ determines a closed topological disk $U_0\\\\subseteq U$ , which is a one-sided compact neighborhood of $\\\\zeta _1$ in $\\\\mathcal {Z}$ .\",\n \"Similarly, we can choose a one-sided compact neighborhood $V_0\\\\subseteq V$ of $\\\\zeta _2\\\\in T_2$ , whose boundary consists of a simple curve $\\\\eta _0$ connecting two points $\\\\zeta ^{\\\\prime }_2,\\\\zeta ^{\\\\prime \\\\prime }_2\\\\in T_2$ , with $\\\\zeta _2$ between $\\\\zeta ^{\\\\prime }_2$ and $\\\\zeta ^{\\\\prime \\\\prime }_2$ , and the portion of $T_2$ between $\\\\zeta ^{\\\\prime }_2$ and $\\\\zeta ^{\\\\prime \\\\prime }_2$ .\",\n \"In this way, for proving the theorem we can consider the one-sided neighborhoods $U_0,V_0$ instead of $U,V$ , respectively.\",\n \"Assume first that the interior of $U_0$ meets some Aubry-Mather set $\\\\Sigma _{\\\\rho _1}\\\\subseteq \\\\mathcal {Z}$ , and that the interior of $V_0$ meets some Aubry-Mather set $\\\\Sigma _{\\\\rho _2}\\\\subseteq \\\\mathcal {Z}$ .\",\n \"Since $U_0$ and $V_0$ are neighborhoods of points in the Aubry-Mather sets $\\\\Sigma _{\\\\rho _1}$ and $\\\\Sigma _{\\\\rho _2}$ respectively, Theorem REF yields the existence of a forward orbit that goes from $U_0$ to $V_0$ .\",\n \"Hence there exists $N>0$ such that $f^N(U_0)\\\\cap V_0\\\\ne \\\\emptyset $ .\",\n \"Since $f^N(U_0)$ and $V_0$ are open topological disks that have common points as well as non-common points (e.g., the points in $T_1$ and $T_2$ , respectively), the Jordan Curve Theorem implies that $f^N(\\\\partial U_0)\\\\cap \\\\partial V_0\\\\ne \\\\emptyset $ .\",\n \"Assume now that the interiors of $U_0,V_0$ do not meet any Aubry-Mather set.\",\n \"We choose three Aubry-Mather sets $\\\\Sigma _{\\\\rho _1}$ , $\\\\Sigma _{\\\\rho _1^{\\\\prime }}$ , $\\\\Sigma _{\\\\rho _1^{\\\\prime \\\\prime }}$ in $\\\\mathcal {Z}$ , lying on three essential circles $C_{\\\\rho _1},C_{\\\\rho ^{\\\\prime }_1},C_{\\\\rho ^{\\\\prime \\\\prime }_1}$ , respectively, with ${\\\\rho _1}<{\\\\rho ^{\\\\prime }_1}<{\\\\rho ^{\\\\prime \\\\prime }_1}$ irrational rotation numbers, and $C_{\\\\rho _1}\\\\prec C_{\\\\rho ^{\\\\prime }_1}\\\\prec C_{\\\\rho ^{\\\\prime \\\\prime }_1}$ .\",\n \"The existence of such vertically ordered Aubry-Mather sets follows from Theorem REF .\",\n \"The proof of the theorem uses the following intermediate step.\",\n \"Claim.\",\n \"There exist $j^{\\\\prime }_*>j_*>0$ such that $\\\\mathcal {Z}\\\\setminus [f^{j_*}(U_0)\\\\cup f^{j^{\\\\prime }_*}(U_0)]$ contains a component $\\\\mathcal {U}$ which is an open topological disk that is a neighborhood of some point in $\\\\Sigma _{\\\\rho _1}$ .\",\n \"Moreover, $j_*,j^{\\\\prime }_*$ can be chosen arbitrarily large.\",\n \"A similar statement holds for $T_2$ .\",\n \"Proof of the claim.\",\n \"Let $p_1$ be a point in $\\\\Sigma _{\\\\rho ^{\\\\prime \\\\prime }_1}$ .\",\n \"Let $W(p_1)$ be a small neighborhood of $p_1$ inside the BZI, which does not intersect $\\\\Sigma _{\\\\rho _1}$ and $\\\\Sigma _{\\\\rho _1^{\\\\prime }}$ .\",\n \"By assumption, $U_0$ does not meet any of the sets $\\\\Sigma _{\\\\rho _1}$ , $\\\\Sigma _{\\\\rho ^{\\\\prime }_1}$ , $\\\\Sigma _{\\\\rho ^{\\\\prime \\\\prime }_1}$ .\",\n \"By Lemma REF (iii) the closure of $\\\\bigcup _{j=0}^{\\\\infty }f^{-j}(W(p_1))$ contains $T_1$ , and in particular $\\\\zeta _1$ .\",\n \"Since $U_0$ is a one-sided neighborhood of $\\\\zeta _1$ , there exists $j_1>0$ such that $f^{j_1}(U_0)\\\\cap W(p_1)\\\\ne \\\\emptyset $ .\",\n \"In the covering space of the annulus, $f^{j_1}(U_0)$ intersects some copy $W^{h_1}:=W(p_1)+(h_1,0)$ of $W(p_1)$ , where $h_1$ is some positive integer.\",\n \"Since $U_0$ does not intersect $\\\\Sigma _{\\\\rho _1}$ and $\\\\Sigma _{\\\\rho ^{\\\\prime }_1}$ , it follows that $f^{j_1}(U_0)$ does not intersect $\\\\Sigma _{\\\\rho _1}$ and $\\\\Sigma _{\\\\rho ^{\\\\prime }_1}$ .\",\n \"On the other hand, $f^{j_1}(U_0)$ intersects the essential circles $C_{\\\\rho _1},C_{\\\\rho ^{\\\\prime }_1}$ , containing the Aubry-Mather sets $\\\\Sigma _{\\\\rho _1},\\\\Sigma _{\\\\rho ^{\\\\prime }_1}$ , respectively.\",\n \"Let $\\\\gamma _{1}:[0,1]\\\\rightarrow U_0$ be a vertical curve, i.e., $\\\\gamma _1(0)\\\\in T_1$ and $\\\\pi _x(\\\\gamma _1(t))=\\\\pi _x(\\\\gamma _1(0))$ for all $t$ , such that $f^{j_1}(\\\\gamma _1(1))$ is an intersection point of $f^{j_1}(U_0)$ with $W^{h_1}$ .\",\n \"The curve $f^{j_1}(\\\\gamma _1)$ is a positively tilted curve which crosses both essential circles $C_{\\\\rho _1}$ and $C_{\\\\rho ^{\\\\prime }_1}$ .\",\n \"(See Remark REF .)\",\n \"Since $f^{j_1}(U_0)$ is disjoint from $\\\\Sigma _{\\\\rho _1}$ , the intersections between $f^{j_1}(\\\\gamma _1)$ and $C_{\\\\rho _1}$ occur within the `gaps' of $\\\\Sigma _{\\\\rho _1}$ , i.e., within the open interval components of $C_{\\\\rho _1}\\\\setminus \\\\Sigma _{\\\\rho _1}$ .\",\n \"We can assign an oriented intersection number between $f^{j_1}(\\\\gamma _1 )$ and each gap of $\\\\Sigma _{\\\\rho _1}$ .\",\n \"Consider a homotopy $h_s:A\\\\rightarrow A$ , $s\\\\in [0,1]$ , such that $f^{j_1}(h_s(\\\\gamma _1))$ keeps the endpoints of $f^{j_1}(\\\\gamma _1)$ fixed for all $s$ , $f^{j_1}(h_s(\\\\gamma _1))$ does not intersect $\\\\Sigma _{\\\\rho _1}$ for any $s\\\\in [0,1]$ , and $f^{j_1}(h_1(\\\\gamma _1))$ is transverse to $C_{\\\\rho _1}$ (see [8]).\",\n \"We set the oriented intersection number of $f^{j_1}(h_1(\\\\gamma _1))$ with $C_{\\\\rho _1}$ to be $+1$ at a point where the curve moves from below $C_{\\\\rho _1}$ to above $C_{\\\\rho _1}$ as $t$ increases, and to be $-1$ at a point where the curve moves from above $C_{\\\\rho _1}$ to below $C_{\\\\rho _1}$ as $t$ increases.\",\n \"Then we assign an oriented intersection number between $f^{j_1}(h_1(\\\\gamma _1))$ and a gap of $\\\\Sigma _{\\\\rho _1}$ , by adding the oriented intersection numbers for all of the intersection points within that gap.\",\n \"Since the oriented intersection number is preserved by homotopy, the oriented intersection number between $f^{j_1}( \\\\gamma _1)$ and a gap is, by definition, the oriented intersection number between $f^{j_1}(h_1(\\\\gamma _1))$ and that gap.\",\n \"Then the oriented intersection number between $f^{j_1}( \\\\gamma _1 )$ and $C_{\\\\rho _1}$ is the sum of the oriented intersection numbers over all gaps.\",\n \"Since the curve $f^{j_1}(\\\\gamma _1)$ starts from below $C_{\\\\rho _1}$ and ends above $C_{\\\\rho _1}$ , there exists a gap for which the oriented intersection number with $f^{j_1}(\\\\gamma _1)$ is positive.\",\n \"We follow the curve $t\\\\mapsto f^{j_1}(\\\\gamma _1(t))$ starting with $t=0$ and we mark the first gap of $\\\\Sigma _{\\\\rho _1}$ that is crossed by $f^{j_1}(\\\\gamma _1)$ with a positive oriented intersection number; we denote by $a^1_{\\\\rho _1},b^1_{\\\\rho _1}$ the endpoints of this gap.\",\n \"This means that if $f^{j_1}(\\\\gamma _1)$ crosses other gaps of $\\\\Sigma _{\\\\rho _1}$ that are to the left of this gap, it does so with 0 oriented intersection number.\",\n \"(If the first gap crossed would be crossed with negative oriented intersection number, it would violate the positive tilt condition of $f^{j_1}(\\\\gamma _1)$ .)\",\n \"Thus, when the curve $f^{j_1}(\\\\gamma _1)$ crosses the gap between $a^1_{\\\\rho _1},b^1_{\\\\rho _1}$ , it comes from below the circle $C_{\\\\rho _1}$ .\",\n \"Following the curve segment of $f^{j_1}(\\\\gamma _1(t))$ after crossing the gap of endpoints $a^1_{\\\\rho _1},b^1_{\\\\rho _1}$ , we mark the first gap of $\\\\Sigma _{\\\\rho ^{\\\\prime }_1}$ that is crossed by $f^{j_1}(\\\\gamma _1)$ with positive oriented intersection number, and we denote by $a^1_{\\\\rho ^{\\\\prime }_1},b^1_{\\\\rho ^{\\\\prime }_1}$ its endpoints.\",\n \"Similarly, when the curve $f^{j_1}(\\\\gamma _1)$ crosses the gap between $a^1_{\\\\rho ^{\\\\prime }_1},b^1_{\\\\rho ^{\\\\prime }_1}$ , it comes from below the circle $C_{\\\\rho ^{\\\\prime }_1}$ .\",\n \"We claim that the left endpoint of the lower gap is to the right of the right endpoint of the upper gap, i.e., $\\\\pi _x(a^1_{\\\\rho _1})<\\\\pi _x(b^1_{\\\\rho ^{\\\\prime }_1})$ .\",\n \"Otherwise, if $\\\\pi _x(a^1_{\\\\rho _1})\\\\ge \\\\pi _x(b^1_{\\\\rho ^{\\\\prime }_1})$ , then there exists an arc $f^{j_1}(\\\\gamma _1(s))$ , $s\\\\in [s_1,s_2]$ , such that $\\\\pi _x(f^{j_1}(\\\\gamma _1(s_1))=\\\\pi _x(f^{j_1}(\\\\gamma _1(s_2))=\\\\pi _x(a^1_{\\\\rho _1})$ , $\\\\pi _y(f^{j_1}(\\\\gamma _1(s_1))<\\\\pi _y(f^{j_1}(\\\\gamma _1(s_2))$ , and $\\\\pi _x(f^{j_1}(\\\\gamma _1(s))\\\\ge \\\\pi _x(a^1_{\\\\rho _1})$ for all $s\\\\in (s_1,s_2)$ .\",\n \"This implies that either the angle deviation from the vertical $\\\\theta (s)$ along the curve $f^{j_1}(\\\\gamma _1)$ becomes non-positive for some $s\\\\in (s_1,s_2)$ , or that there exists another arc $f^{j_1}(\\\\gamma _1(\\\\tau ))$ , $\\\\tau \\\\in [\\\\tau _1,\\\\tau _2]$ , with $\\\\tau _1<\\\\tau _2\\\\pi _y(f^{j_1}(\\\\gamma _1(\\\\tau _2))$ , and $\\\\pi _x(f^{j_1}(\\\\gamma _1(s))\\\\le \\\\pi _x(f^{j_1}(\\\\gamma _1(\\\\tau ))$ for all $s\\\\in (s_1,s_2)$ and $\\\\tau \\\\in (\\\\tau _1,\\\\tau _2)$ .\",\n \"In the first case we obtain a contradiction with the positive tilt condition on the curve $f^{j_1}(\\\\gamma _1)$ .\",\n \"See Fig.\",\n \"REF .\",\n \"In the second case we obtain a contradiction with the fact that $f^{j_1}(\\\\gamma _1)$ comes from below $C_{\\\\rho _1}$ before crossing the gap of endpoints $a^1_{\\\\rho _1},b^1_{\\\\rho _1}$ .\",\n \"The conclusion of this step is that the curve $f^{j_1}(\\\\gamma _1(t))$ passes through the gap between $a^1_{\\\\rho _1}$ and $b^1_{\\\\rho _1}$ of $\\\\Sigma _{\\\\rho _1}$ , from below $C_{\\\\rho _1}$ to above $C_{\\\\rho _1}$ , then it passes through the gap between $a^1_{\\\\rho ^{\\\\prime }_1}$ and $b^1_{\\\\rho ^{\\\\prime }_1}$ of $\\\\Sigma _{\\\\rho ^{\\\\prime }_1}$ , from below $C_{\\\\rho ^{\\\\prime }_1}$ to above $C_{\\\\rho ^{\\\\prime }_1}$ , and $\\\\pi _x(a^1_{\\\\rho _1})<\\\\pi _x(b^1_{\\\\rho ^{\\\\prime }_1})$ .\",\n \"Figure: Violation of the positive tilt condition.Now we consider a one-sided rectangular neighborhood $U_1\\\\subseteq U_0$ of some point in $T_1$ , bounded below by $T_1$ , to the left by $\\\\gamma _1$ , and to the right by some other vertical curve segment $\\\\gamma _1^{\\\\prime }$ .\",\n \"If $\\\\gamma ^{\\\\prime }_1$ is sufficiently close to $\\\\gamma _1$ , then, by continuity, the image of each vertical curve in $U_1$ under $f^{j_1}$ crosses the gap between $a^1_{\\\\rho _1}$ and $b^1_{\\\\rho _1}$ of $\\\\Sigma _{\\\\rho _1}$ with positive oriented intersection number, and crosses the gap between $a^1_{\\\\rho ^{\\\\prime }_1}$ and $b^1_{\\\\rho ^{\\\\prime }_1}$ of $\\\\Sigma _{\\\\rho ^{\\\\prime }_1}$ with positive oriented intersection number.\",\n \"We choose and fix a set $U_1\\\\subseteq U_0$ with these properties.\",\n \"By Lemma REF (iii) (see also Remark REF ) the closure of $\\\\bigcup _{j=0}^{\\\\infty }f^{-j}(W(p_1))$ contains $T_1$ , so there exists $j_2>j_1$ such that, in the annulus, $f^{j_2}(U_1)\\\\cap W(p_1)\\\\ne \\\\emptyset $ , and, in the covering space of the annulus, $f^{j_2}(U_1)$ intersects some copy $W^{h_2}:=W(p_1)+(h_2,0)$ of $W(p_1)$ for some positive integer $h_2>h_1$ .\",\n \"(Due to the positive twist condition on $f$ and the assumption that the rotation numbers on the boundary components of the annulus are positive, the vertical line $\\\\lbrace x=\\\\pi _x(p_1)+h_1\\\\rbrace $ is mapped by $f^{j_2-j_1}$ to a positive tilted map to the right of $\\\\lbrace x=\\\\pi _x(p_1)+h_1\\\\rbrace $ , hence, if $j_2$ is large enough, we can choose $h_2>h_1$ .)\",\n \"Then there exists a vertical curve $\\\\gamma _2:[0,1]\\\\rightarrow U_1$ , such that $f^{j_2}(\\\\gamma _2(1))$ is an intersection point of $f^{j_2}(U_1)$ with $W^{h_2}$ .\",\n \"The curve $f^{j_2}(\\\\gamma _2(t))$ crosses $C_{\\\\rho _1}$ and $C_{\\\\rho ^{\\\\prime }_1}$ .\",\n \"Let $a^2_{\\\\rho _1},b^2_{\\\\rho _1}$ be the endpoints of the leftmost gap of $\\\\Sigma _{\\\\rho _1}$ that is crossed by $f^{j_2}(\\\\gamma _2(t))$ with positive oriented intersection number equal, and let $a^2_{\\\\rho ^{\\\\prime }_1},b^2_{\\\\rho ^{\\\\prime }_1}$ be the endpoints of the leftmost gap of $C_{\\\\rho ^{\\\\prime }_1}$ that is crossed by $f^{j_2}(\\\\gamma _2(t))$ with positive oriented intersection number.\",\n \"The image curve $f^{j_2}(\\\\gamma _2)$ is a positively tilted curve located on the `right side' of the positively tilted curve $f^{j_2}(\\\\gamma _1)$ , in the sense that any graph over $x$ that intersects both $f^{j_2}(\\\\gamma _1)$ and $f^{j_2}(\\\\gamma _2)$ has the leftmost intersection point with the $f^{j_1}(\\\\gamma _1)$ .\",\n \"Therefore, the gap endpoints $a^2_{\\\\rho _1},b^2_{\\\\rho _1}$ are either the image under $f^{j_2-j_1}$ of the gap endpoints $a^1_{\\\\rho _1},b^1_{\\\\rho _1}$ found at the previous step, or are the image under $f^{j_2-j_1}$ of some other gap endpoints of $\\\\Sigma _{\\\\rho _1}$ located to the right of the gap between $a^1_{\\\\rho _1}$ and $b^1_{\\\\rho _1}$ .\",\n \"Similarly, the gap endpoints $a^2_{\\\\rho ^{\\\\prime }_1},b^2_{\\\\rho ^{\\\\prime }_1}$ are either the image under $f^{j_2-j_1}$ of the gap endpoints $a^1_{\\\\rho ^{\\\\prime }_1},b^1_{\\\\rho ^{\\\\prime }_1}$ from the previous step, or are the image under $f^{j_2-j_1}$ of some other gap of $\\\\Sigma _{\\\\rho ^{\\\\prime }_1}$ located to the right of the gap between $a^1_{\\\\rho ^{\\\\prime }_1}$ and $b^1_{\\\\rho ^{\\\\prime }_1}$ .\",\n \"Then, there exists a one-sided rectangular neighborhood $U_2\\\\subseteq U_1$ of some point in $T_1$ , bounded below by $T_1$ , to the left by $\\\\gamma _2$ , and to the right by some other vertical curve segment $\\\\gamma ^{\\\\prime }_2$ , such that the image of each vertical curve in $U_2$ under $f^{j_2}$ crosses the gap between $a^2_{\\\\rho _1}$ and $b^2_{\\\\rho _1}$ of $\\\\Sigma _{\\\\rho _2}$ with positive oriented intersection number, and crosses the gap between $a^2_{\\\\rho ^{\\\\prime }_1}$ and $b^2_{\\\\rho ^{\\\\prime }_1}$ of $\\\\Sigma _{\\\\rho ^{\\\\prime }_1}$ with positive oriented intersection number.\",\n \"Recursively, we obtain a nested sequence of one-sided neighborhoods of points in $T_1$ , denoted $U_1\\\\supseteq U_2\\\\supseteq \\\\ldots U_m\\\\supseteq \\\\ldots $ , all contained in $U_0$ , and two sequences of positive integers $j_1\\\\rho _1$ .\",\n \"Then, for some sufficiently large iterate $j_m$ the order of the gaps gets reversed in the annulus.\",\n \"That is, we get the following ordering in terms of the angle coordinate in the covering space of the annulus: (i) $\\\\pi _x(a^m_{\\\\rho _1})<\\\\pi _x(b^m_{\\\\rho _1})< \\\\pi _x(a^1_{\\\\rho _1})+h_m<\\\\pi _x(b^1_{\\\\rho _1})+h_m$ , (ii) $\\\\pi _x(a^1_{\\\\rho ^{\\\\prime }_1})+h_m<\\\\pi _x(b^1_{\\\\rho ^{\\\\prime }_1})+h_m<\\\\pi _x(a^m_{\\\\rho ^{\\\\prime }_1})<\\\\pi _x(b^m_{\\\\rho ^{\\\\prime }_1})$ .\",\n \"Since $f^{j_1}(U_1)$ and $f^{j_m}(U_m)$ are connected, the above ordering of the crossings with $C_{\\\\rho _1}$ and $C_{\\\\rho ^{\\\\prime }_1}$ implies that $f^{j_1}(U_1)$ and $f^{j_m}(U_m)$ have an intersection point above $C_{\\\\rho _1}$ .\",\n \"As $U_1, U_m\\\\subseteq U_0$ , it follows that $f^{j_1}(U_0)$ and $f^{j_m}(U_0)$ have an intersection point above $C_{\\\\rho _1}$ .\",\n \"Since $f^{j_1}(U_0)$ and $f^{j_m}(U_0)$ are connected and disjoint from the Aubry-Mather set $\\\\Sigma _{\\\\rho _1}\\\\subseteq C_{\\\\rho _1}$ , there exists a component $\\\\mathcal {U}$ of the complement $\\\\mathcal {Z}\\\\setminus [f^{j_1}(U_0)\\\\cup f^{j_m}(U_0)]$ which is an open topological disk containing some point $\\\\xi _1\\\\in \\\\Sigma _{\\\\rho _1}$ .\",\n \"The boundary of $\\\\mathcal {U}$ consists of a finite union of sub-arcs of the boundaries of $f^{j_1}(U_0)$ and $f^{j_m}(U_0)$ , and possibly of curve segments of $T_1$ .\",\n \"See Figure REF .\",\n \"Letting $j_*=j_1$ and $j^{\\\\prime }_*=j_m$ ends the proof of the claim.\",\n \"Figure: An open topological disk forming a neighborhood of a point in the Aubry-Mather set Σ ρ 1 \\\\Sigma _{\\\\rho _1}.We now apply the statement of the claim to $T_2$ , starting with the one-sided neighborhood $V_0$ of $\\\\zeta _2\\\\in T_2$ .\",\n \"We choose three Aubry-Mather sets $\\\\Sigma _{\\\\rho _2}$ , $\\\\Sigma _{\\\\rho _2^{\\\\prime }}$ , $\\\\Sigma _{\\\\rho _2^{\\\\prime \\\\prime }}$ in $\\\\mathcal {Z}$ , lying on three essential circles $C_{\\\\rho _2},C_{\\\\rho ^{\\\\prime }_2},C_{\\\\rho ^{\\\\prime \\\\prime }_2}$ , respectively, with ${\\\\rho _2}>{\\\\rho ^{\\\\prime }_2}>{\\\\rho ^{\\\\prime \\\\prime }_2}$ irrational rotation numbers, and $C_{\\\\rho _2} \\\\succ C_{\\\\rho ^{\\\\prime }_2} \\\\succ C_{\\\\rho ^{\\\\prime \\\\prime }_2}$ .\",\n \"The statement in the claim implies that there exists a neighborhood $\\\\mathcal {V}$ of some point $\\\\xi _2\\\\in \\\\Sigma _{\\\\rho _2}$ , homoeomorphic to an open disk, whose boundary consists of a finite union of sub-arcs in the boundaries of $f^{-l_*}(V_0)$ and $f^{-l^{\\\\prime }_*}(V_0)$ , for some $l_*0$ such that $f^N(\\\\textrm {int}(U _0))\\\\cap \\\\textrm {int}(V_0) \\\\ne \\\\emptyset $ .\",\n \"Since the sets $f^{N}(U_0)$ and $V_0$ are topological disks have interior points in common, but also points that are not in common (namely the points lying on $T_1$ and $T_2$ , respectively), the Jordan Curve Theorem implies that $f^{N}(\\\\partial U _0)\\\\cap \\\\partial V_0\\\\ne \\\\emptyset $ .\",\n \"The remaining case of the proof, when the interior of $U_0$ does intersect some Aubry-Mather set and the interior of $V_0$ does not, or when the interior of $U_0$ does intersect some Aubry-Mather set and the interior of $V_0$ does not, follows easily from the above arguments.\",\n \"The next statement says that given two points on the boundary tori of a BZI, and a finite sequence of Aubry-Mather sets inside the zone, there exists an orbit that starts in a prescribed neighborhood of the point on the lower boundary torus, then moves on and shadows, in the sense of the ordering of the orbit, each Aubry-Mather set in the sequence, and ends in a prescribed neighborhood of the point on the upper boundary torus.\",\n \"This result extends Theorem REF , and relies on the topological argument of Hall.\",\n \"As in the previous theorem, we do not need any extra conditions on the dynamics on the boundary tori.\",\n \"The resulting shadowing orbits are not necessarily minimal.\",\n \"Theorem 6.2 Suppose that $T_1$ and $T_2$ bound a BZI $\\\\mathcal {Z}$ .\",\n \"Let $\\\\zeta _1\\\\in T_1,\\\\zeta _2\\\\in T_2$ , $U$ be a neighborhood of $\\\\zeta _1$ , and $V$ a neighborhood of $\\\\zeta _2$ .\",\n \"Let $\\\\lbrace \\\\Sigma _{\\\\omega _s}\\\\rbrace _{s\\\\in \\\\lbrace 1,\\\\ldots ,\\\\sigma \\\\rbrace }$ be a finite collection of Aubry-Mather sets inside $\\\\mathcal {Z}$ such that each $\\\\Sigma _{\\\\omega _s}$ lies on some essential circle $C_{\\\\omega _s}$ that is a graph over the $x$ -coordinate, with $C_{\\\\omega _s}\\\\prec C_{\\\\omega _{s^{\\\\prime }}}$ provided $\\\\omega _s<\\\\omega _{s^{\\\\prime }}$ .\",\n \"Let $\\\\lbrace n_s\\\\rbrace _{s=1,\\\\ldots , \\\\sigma }$ be sequence of positive integers.\",\n \"Then there exist a point $z\\\\in U$ , and a sequence of positive integers $\\\\lbrace m_s\\\\rbrace _{s=0,\\\\ldots ,\\\\sigma }$ , such that, for each $s\\\\in \\\\lbrace 1,\\\\ldots ,\\\\sigma \\\\rbrace $ , $\\\\begin{split}\\\\pi _x(f^j(w_s))<\\\\pi _x(f^j(z))<\\\\pi _x(f^j(\\\\bar{w}_s))\\\\textrm { for } \\\\\\\\ \\\\sum _{t=1}^{s-1} n_t+\\\\sum _{t=0}^{s-1}m_t\\\\le j\\\\le \\\\sum _{t=1}^{s} n_t+\\\\sum _{t=0}^{s-1}m_t,\\\\end{split}$ where $w_s$ and $\\\\bar{w}_s$ are some points in the Aubry-Mather set $\\\\Sigma _{\\\\omega _s}$ , and $f^{N}(z)\\\\in V$ for $N=\\\\sum _{t=1}^{\\\\sigma } n_t+\\\\sum _{t=0}^{\\\\sigma }m_t$ .\",\n \"The number $N$ can be chosen arbitrarily large.\",\n \"Moreover, if $U,V$ are chosen so that $U\\\\cap \\\\mathcal {Z},V\\\\cap \\\\mathcal {Z}$ are topological disks, then there exists a point $z^{\\\\prime }\\\\in \\\\partial U$ satisfying the ordering condition (REF ) such that $f^N(z^{\\\\prime })\\\\in \\\\partial V$ .\",\n \"We use the construction of diagonal sets described in the sketch of the proof of Theorem REF ; for details see [29].\",\n \"Part 1.\",\n \"As in the proof of Theorem REF we can choose one sided compact neighborhoods $U_0$ of $\\\\zeta _1\\\\in T_1$ and $V_0$ of $\\\\zeta _2\\\\in T_2$ , such that $U_0\\\\cap \\\\mathcal {Z}$ and $V_0\\\\cap \\\\mathcal {Z}$ are topological disks.\",\n \"We first prove the existence of a point $z\\\\in U_0$ satisfying (REF ) and such that $f^N(z)\\\\in V_0$ .\",\n \"Let $C_{\\\\omega _1}$ be the essential circle containing $\\\\Sigma _{\\\\omega _1}$ .\",\n \"We choose an Aubry-Mather set $\\\\Sigma _{\\\\rho _1}$ lying on some essential circle $C_{\\\\rho _1}$ , such that $C_{\\\\omega _1}\\\\prec C_{\\\\rho _1}$ .\",\n \"We choose a point $p_1\\\\in \\\\Sigma _{\\\\rho _1}$ and a small neighborhood $W(p_1)$ of $p_1$ which does not intersect $\\\\Sigma _{\\\\omega _1}$ .\",\n \"We assume that the interior of $U_0$ does not meet $\\\\Sigma _{\\\\omega _1}$ and $\\\\Sigma _{\\\\rho _1}$ , otherwise the proof follows as in [29].\",\n \"By Lemma REF (iii) the closure of $\\\\bigcup _{j=0}^{\\\\infty }f^{-j}(W(p_1))$ contains $T_1$ , and in particular $\\\\zeta _1$ .\",\n \"Proceeding as in the proof of Theorem REF , we obtain a nested sequence $U_1\\\\supseteq U_2\\\\supseteq \\\\ldots \\\\supseteq U_i$ of one-sided neighborhoods of points in $T_1$ , all contained in $U_0$ , and two sequences of positive integers $j_10$ and a point $q\\\\in U_0$ depending on $l$ such that $f^l(q)$ is in some prescribed neighborhood of a point $r\\\\in T_2$ , where $l$ can be chosen arbitrarily large.\",\n \"Since the points of $T_1$ and $T_2$ have different rotation numbers hence move apart under iteration, there exists $l_0$ sufficiently large such that $f^{l_0}(T_1\\\\cap U_0)$ and $f^{l_0}(q)$ are separated by a fundamental interval of the annulus, i.e., $\\\\pi _x(f^{l_0}(r))-\\\\pi _x(f^{l_0}(q))>1$ for all $r\\\\in T_1\\\\cap U_0$ .\",\n \"Let $x_0,\\\\bar{x}_0\\\\in \\\\mathbb {R}$ be such that $\\\\pi _x(f^{l_0}(r))\\\\bar{y}^{\\\\prime \\\\prime }_0\\\\rbrace $ is a neighborhood of an arc in $T_1$ , with the property that each point $(\\\\bar{x}^{\\\\prime \\\\prime }_0,\\\\bar{y}^{\\\\prime \\\\prime }_0)\\\\in \\\\bar{W}_0$ has the $y$ -coordinate smaller than that of any point in $f^{l_0}(\\\\textrm {cl}(U_0))\\\\cap \\\\lbrace x=\\\\bar{x}^{\\\\prime \\\\prime }_0\\\\rbrace $ .\",\n \"Let $\\\\Sigma _{\\\\rho _1}$ be an Aubry-Mather set lying on an essential circle $C_{\\\\rho _1}$ that is below the essential circle $C_{\\\\omega _1}$ containing $\\\\Sigma _{\\\\omega _1}$ , let $p_1\\\\in C_{\\\\rho _1}$ , and let $W(p_1)$ be a small neighborhood of $p_1$ that does not intersect $\\\\Sigma _{\\\\omega _1}$ .\",\n \"Then there exists $z_0\\\\in W_0$ and ${j_0}$ sufficiently large such that $f^{j_0}(z_0)\\\\in W(p_1)$ .\",\n \"Since the curve $f^{j_0}(I^+_{z_0})$ is a positively tilted curve emerging from $T_2$ , the arguments used in Part 1 show that there is a gap of the Aubry-Mather set $\\\\Sigma _{\\\\omega _1}$ , between a pair of points $w_1,w^{\\\\prime }_1\\\\in \\\\Sigma _{\\\\omega _1}$ , such that $f^{j_0}(I^+_{z_0})$ crosses this gap with negative intersection number (where the parametrization of $f^{j_0}(I^+_{z_0})$ is chosen so that to $t=0$ it corresponds a point on $T_2$ ) and has its first intersection with $I_{w_1}$ below the point $w_1$ , provided ${j_0}$ is chosen large enough.\",\n \"This implies that the image of the diagonal component of $f^{l_0}(\\\\textrm {cl}(U_0))\\\\cap B_{z_0,\\\\bar{w}_0}$ under $f^{j_0}$ has a component $\\\\Delta _0$ in $B_{w_1,w^{\\\\prime }_1}$ that satisfies the positive diagonal set conditions relative to its left side.\",\n \"See Fig.\",\n \"REF .\",\n \"Figure: Construction of a positive diagonal set.Now, for all $j^{\\\\prime }>0$ , the image of $\\\\Delta _0$ under $f^{j^{\\\\prime }}$ also has a component that satisfies the positive diagonal set conditions relative to the left side of $B_{f^{j^{\\\\prime }}(w_1),f^{j^{\\\\prime }}(w^{\\\\prime }_1)}$ .\",\n \"(This follows from the hereditary property of diagonal sets, see Section .)\",\n \"In a similar fashion, there exist a point $\\\\bar{z}_0\\\\in \\\\bar{W}_0$ and $j^{\\\\prime }_0$ sufficiently large such that the positively tilted curve $f^{j^{\\\\prime }_0+j_0}(I^-_{\\\\bar{z}_0})$ , emerging from $T_1$ , crosses a gap of the Aubry-Mather set $\\\\Sigma _{\\\\omega _1}$ between a pair of points $\\\\bar{w}_1,\\\\bar{w}^{\\\\prime }_1\\\\in \\\\Sigma _{\\\\omega _1}$ , the oriented intersection number between $f^{j^{\\\\prime }_0+j_0}(I^-_{\\\\bar{z}_0})$ and this gap is positive, and the first intersection with $I_{\\\\bar{w}^{\\\\prime }_1}$ occurs above the point $\\\\bar{w}^{\\\\prime }_1$ .\",\n \"Thus, the image of $\\\\Delta _0$ under $f^{j^{\\\\prime }_0}$ has a component that satisfies the positive diagonal set conditions relative to its right side.\",\n \"In summary, the image of the diagonal set component of $f^{l_0}(\\\\textrm {cl}(U_0))\\\\cap B_{z_0, \\\\bar{z}_0}$ under $f^{j_0+j^{\\\\prime }_0}$ contains a component $\\\\Delta ^{\\\\prime }_0$ that is a positive diagonal in $B_{w_1,\\\\bar{w}^{\\\\prime }_1}$ , where $w_1,\\\\bar{w}^{\\\\prime }_1$ are two points in the Aubry-Mather set $\\\\Sigma _{\\\\omega _1}$ .\",\n \"Moreover, the upper edge and the lower edge of $\\\\Delta ^{\\\\prime }_0$ are contained in $f^{j^{\\\\prime }_0+j_0+l_0}(\\\\partial U_0)$ .\",\n \"We apply an analogous argument at the other boundary torus $T_2$ .\",\n \"Given $V_0$ a neighborhood of a point $\\\\zeta _2\\\\in T_2$ , there exist $L>0$ and a pair of points $w_\\\\sigma ,\\\\bar{w}_\\\\sigma \\\\in \\\\Sigma _{\\\\omega _\\\\sigma }$ such that $f^{-L}(\\\\textrm {cl}(V_0))\\\\cap B_{w_\\\\sigma ,\\\\bar{w}_\\\\sigma }$ has a component $D^{\\\\prime \\\\prime }_\\\\sigma $ that is a negative diagonal in $B_{w_\\\\sigma ,\\\\bar{w}_\\\\sigma }$ .\",\n \"The upper edge and the lower edge of this positive diagonal set are contained in $f^{-L}(\\\\partial V_0)$ .\",\n \"Now we apply the argument from Part 1.\",\n \"There exists a negative diagonal set $D_\\\\sigma $ in $B_{w_1,\\\\bar{w}_1}$ such that all points $z\\\\in D_\\\\sigma $ satisfy (REF ).\",\n \"By the argument for Theorem REF , the upper and lower edge of $D_{\\\\sigma }$ lie on $f^{-j_\\\\sigma }(I^-_{w_\\\\sigma }),f^{-j_\\\\sigma }(I^+_{\\\\bar{w}_\\\\sigma })$ where $j_\\\\sigma =\\\\sum _{s=0}^{\\\\sigma }n_s+\\\\sum _{s=0}^{\\\\sigma -1}m_s$ .\",\n \"Since negative and positive diagonal sets in the same vertical strip always intersect, the negative diagonal set $D_\\\\sigma $ intersects $\\\\Delta ^{\\\\prime }_0$ , and in particular it intersects its upper and lower edges that are contained in $f^{j^{\\\\prime }_0+j_0+l_0}(\\\\partial U_0)$ .\",\n \"Iterating $\\\\Delta ^{\\\\prime }_0$ forward for $j_\\\\sigma $ times yields a positive diagonal set $D^{\\\\prime }_\\\\sigma $ in $B_{w_\\\\sigma ,\\\\bar{w}_\\\\sigma }$ .\",\n \"The upper and lower edges of $D^{\\\\prime }_\\\\sigma $ are contained in $f^{j_\\\\sigma +j^{\\\\prime }_0+j_0+l_0}(\\\\partial U_0)$ .\",\n \"The positive diagonal set $D^{\\\\prime }_\\\\sigma $ intersects the negative diagonal component $D^{\\\\prime \\\\prime }_\\\\sigma $ of $f^{-L}(\\\\textrm {cl}(V_0))\\\\cap B_{w_\\\\sigma ,\\\\bar{w}_\\\\sigma }$ .\",\n \"In particular the upper and lower edges of $D^{\\\\prime }_{\\\\sigma }$ , that are contained in $f^{j_\\\\sigma +j^{\\\\prime }_0+j_0+l_0}(\\\\partial U_0)$ , intersect the upper and lower edges of $D^{\\\\prime \\\\prime }_\\\\sigma $ that are contained in $f^{-L}(\\\\partial V_0)$ .\",\n \"Thus, there exists a point $z^{\\\\prime }\\\\in \\\\partial U_0$ that is taken by $f^{j_\\\\sigma +j^{\\\\prime }_0+j_0+l_0}$ to $\\\\partial V_0$ and satisfies the ordering relations (REF ).\"\n ],\n [\n \"A shadowing lemma in normally hyperbolic invariant manifolds\",\n \"In this section we present a shadowing lemma-type of result saying that, given a sequence of windows within a normally hyperbolic invariant manifold, consisting of pairs of windows correctly aligned under the scattering map, alternating with pairs of windows correctly aligned under some iterate of the inner map, then there exists a true orbit in the full space dynamics that follows these windows.\",\n \"This result reduces the construction of windows within the full dimensional phase space to the construction of lower dimensional windows within the normally hyperbolic invariant manifold.\",\n \"For this section, we assume a diffeomorphism $f:M\\\\rightarrow M$ on a manifold $M$ , and an $l$ -dimensional normally hyperbolic invariant manifold $\\\\Lambda \\\\subseteq M$ as in Subsection REF .\",\n \"In the subsequent sections, we will apply Lemma REF only in the case when $\\\\Lambda $ is a 2-dimensional normally hyperbolic invariant manifold in $M$ , i.e., $l=2$ .\",\n \"A more general version of this lemma which does not involve the scattering map, and some additional details and applications, appear in [16].\",\n \"Lemma 7.1 Let $\\\\lbrace R_i,R^{\\\\prime }_i\\\\rbrace _{i\\\\in \\\\mathbb {Z}}$ be a bi-infinite sequence of $l$ -dimensional windows contained in $\\\\Lambda $ .\",\n \"Assume that the following properties hold for all $i\\\\in \\\\mathbb {Z}$ : (i) $R_{i}\\\\subseteq U^-$ and $R^{\\\\prime }_{i}\\\\subseteq U^+$ .\",\n \"(ii) $R_{i}$ is correctly aligned with $R^{\\\\prime }_{i+1}$ under the scattering map $S$ .\",\n \"(iii) for each pair $R^{\\\\prime }_{i+1},R_{i+1}$ and for each $L>0$ there exists $L^{\\\\prime }>L$ such that $R^{\\\\prime }_{i+1}$ is correctly aligned with $R_{i+1}$ under the iterate $f_{\\\\mid \\\\Lambda }^{L^{\\\\prime }}$ of the restriction $f_{\\\\mid \\\\Lambda }$ of $f$ to $\\\\Lambda $ .\",\n \"Fix any bi-infinite sequence of positive real numbers $\\\\lbrace \\\\varepsilon _i\\\\rbrace _{i\\\\in \\\\mathbb {Z}}$ .\",\n \"Then there exist an orbit $(f^{n}(z))_{n\\\\in \\\\mathbb {Z}}$ of some point $z\\\\in M$ , an increasing sequence of integers $(n_i)_{i\\\\in \\\\mathbb {Z}}$ , and some sequences of positive integers $\\\\lbrace N_i\\\\rbrace _{i\\\\in \\\\mathbb {Z}}, \\\\lbrace K_i\\\\rbrace _{i\\\\in \\\\mathbb {Z}},\\\\lbrace M_i\\\\rbrace _{i\\\\in \\\\mathbb {Z}}$ , such that, for all $i\\\\in \\\\mathbb {Z}$ : $ \\\\begin{split}d(f^{n_i}(z),\\\\Gamma )<\\\\varepsilon _i,\\\\\\\\d(f^{n_i+N_{i+1}}(z), f_{\\\\mid \\\\Lambda }^{N_{i+1}}(R^{\\\\prime }_{i+1}))<\\\\varepsilon _{i+1},\\\\\\\\d(f^{n_{i}-M_{i}}(z), f_{\\\\mid \\\\Lambda }^{-M_{i}}(R_{i}))<\\\\varepsilon _{i},\\\\\\\\n_{i+1}=n_i+N_{i+1}+K_{i+1}+M_{i+1}.\\\\end{split}$ The idea of this proof is to `thicken' some appropriate iterates of the windows $R_{i},R^{\\\\prime }_{i}$ in $\\\\Lambda $ to full dimensional windows $W_i,W^{\\\\prime }_i$ in $M$ , so that $\\\\lbrace W_i,W^{\\\\prime }_i\\\\rbrace _{i\\\\in \\\\mathbb {Z}}$ form a sequence of windows that are correctly aligned under some appropriate maps.\",\n \"We start with a brief sketch of the construction before we proceed to the formal proof.\",\n \"We constructs some copies $\\\\bar{R}_{i},\\\\bar{R}^{\\\\prime }_{i+1}$ in $\\\\Gamma $ of $R_{i},R^{\\\\prime }_{i+1}$ , respectively, through the inverses of the wave maps (see Section REF ).\",\n \"We then expand the rectangles $\\\\bar{R}_{i},\\\\bar{R}^{\\\\prime }_{i+1}$ into the hyperbolic directions to produce a pair of windows $\\\\bar{W}_{i},\\\\bar{W}^{\\\\prime }_{i+1}$ , respectively, which are correctly aligned under the identity map.\",\n \"Then we take a backwards iterate of $\\\\bar{W}_{i}$ such that $f^{-M_{i}} (\\\\bar{W}_{i})$ is sufficiently close to $\\\\Lambda $ , and we construct a new window $W_i$ about $f^{-M_i}(R_i)$ such that $W_i$ is correctly aligned with $\\\\bar{W}_i$ under $f^{M_i}$ .\",\n \"Similarly, we construct a window $W^{\\\\prime }_{i+1}$ about $f^{N_i}(R^{\\\\prime }_{i+1})$ such that $\\\\bar{W}^{\\\\prime }_{i+1}$ is correctly aligned with $W^{\\\\prime }_{i+1}$ under $f^{N_{i+1}}$ .\",\n \"We are given that we can align $R^{\\\\prime }_{i+1}$ with $R_{i+1}$ under some high enough iterate.\",\n \"Hence we can align $W^{\\\\prime }_{i+1}$ with $W_{i+1}$ under some iterate $f^{K_{i+1}}$ .\",\n \"This construction can be continued inductively.\",\n \"Notationwise, the $N_i$ 's are associated to forward iterations along the stable manifold, the $M_i$ 's to backwards iterations along the unstable manifold, and the $K_i$ 's to iterations following the inner dynamics of $f$ restricted to $\\\\Lambda $ .\",\n \"See Fig.\",\n \"REF .\",\n \"Figure: Schematic illustration of the construction of windows for the shadowing lemma.\",\n \"The windows R i ,R i+1 ' R_i,R^{\\\\prime }_{i+1} are depicted as large dots.Step 1.\",\n \"Let $\\\\lbrace R,R^{\\\\prime }\\\\rbrace $ be a pair of $l$ -dimensional windows of the type $\\\\lbrace R_{i},R^{\\\\prime }_{i+1}\\\\rbrace $ , and let $\\\\lbrace \\\\varepsilon ,\\\\varepsilon ^{\\\\prime }\\\\rbrace $ stand for the corresponding $\\\\lbrace \\\\varepsilon _i,\\\\varepsilon _{i+1}\\\\rbrace $ .\",\n \"Let $\\\\bar{R} =(\\\\Omega ^-_\\\\Gamma )^{-1}(R )$ and $\\\\bar{R}^{\\\\prime } =(\\\\Omega ^+_\\\\Gamma )^{-1}(R^{\\\\prime } )$ be the copies of $R $ and $R^{\\\\prime } $ , respectively, in the homoclinic channel $\\\\Gamma $ .\",\n \"By making some arbitrarily small changes in the sizes of their exit and entry directions, we can alter the windows $\\\\bar{R} $ and $\\\\bar{R}^{\\\\prime } $ such that $R $ is correctly aligned with $\\\\bar{R} $ under $(\\\\Omega ^-_\\\\Gamma )^{-1}$ , $\\\\bar{R}$ is correctly aligned with $\\\\bar{R}^{\\\\prime } $ under the identity mapping, and $\\\\bar{R}^{\\\\prime } $ is correctly aligned with $R $ under $\\\\Omega ^+_\\\\Gamma $ .\",\n \"We `thicken' the $l$ -dimensional windows $\\\\bar{R} $ and $\\\\bar{R}^{\\\\prime } $ in $\\\\Gamma $ , which are correctly aligned under the identity mapping, to $(l+n_u+n_s)$ -dimensional windows $\\\\bar{W} $ and $\\\\bar{W}^{\\\\prime } $ , respectively, that are correctly aligned in $M$ under the identity mapping as well.\",\n \"We now explain the `thickening' procedure.\",\n \"First, we describe how to thicken $\\\\bar{R} $ to a full dimensional window $\\\\bar{W} $ .\",\n \"We choose some $0<\\\\bar{\\\\delta }<\\\\varepsilon $ and $0<\\\\bar{\\\\eta }<\\\\varepsilon $ .\",\n \"At each point $x \\\\in \\\\bar{R} $ we choose an $n_u$ -dimensional closed ball $\\\\bar{B}_{\\\\bar{\\\\delta }}(x )$ of radius $\\\\bar{\\\\delta }$ centered at $x $ and contained in $W^u({x^- })$ , where $x^- =\\\\Omega ^-_\\\\Gamma (x )$ .\",\n \"Let $\\\\bar{\\\\Delta }:=\\\\bigcup _{x \\\\in \\\\bar{R} }\\\\bar{B}^{u}_{\\\\bar{\\\\delta }}(x )$ .\",\n \"Note that $\\\\bar{\\\\Delta }$ is contained in $W^u(\\\\Lambda )$ and is homeomorphic to a $(l+n_u)$ -dimensional rectangle.\",\n \"We define the exit set and the entry set of this rectangle as follows: $\\\\begin{split}(\\\\bar{\\\\Delta })^{\\\\rm exit}:=\\\\bigcup _{x \\\\in (\\\\bar{R} )^{\\\\rm exit}}\\\\bar{B}^{u}_{\\\\bar{\\\\delta }}(x ) \\\\cup \\\\bigcup _{x \\\\in \\\\bar{R} }\\\\partial \\\\bar{B}^{u}_{\\\\bar{\\\\delta }}(x ),\\\\\\\\(\\\\bar{\\\\Delta })^{\\\\rm entry}:=\\\\bigcup _{x \\\\in (\\\\bar{R} )^{\\\\rm entry}}\\\\bar{B}^{u}_{\\\\bar{\\\\delta }}(x ).\\\\end{split}$ We consider the normal bundle $N$ to $W^u(\\\\Lambda )$ .\",\n \"At each point $y \\\\in \\\\bar{\\\\Delta }$ , we choose an $n_s$ -dimensional closed ball $\\\\bar{B}^s_{\\\\bar{\\\\eta }}(y)$ centered at $y$ and contained in the image of $N_y$ under the exponential map $\\\\exp _{y }:T_{y }M\\\\rightarrow M$ .\",\n \"We let $\\\\bar{W} :=\\\\bigcup _{y \\\\in \\\\bar{\\\\Delta }}\\\\bar{B}^{s}_{\\\\bar{\\\\eta }}(y )$ .\",\n \"By the Tubular Neighborhood Theorem (see, e.g., [7]), we have that for $\\\\bar{\\\\eta }$ sufficiently small, $\\\\bar{W} $ is a homeomorphic copy of a $(l+n_u+n_s)$ -dimensional rectangle.\",\n \"We now define the exit set and the entry set of $\\\\bar{W} $ as follows: $\\\\begin{split} (\\\\bar{W} )^{\\\\rm exit}:=\\\\bigcup _{y \\\\in (\\\\bar{\\\\Delta })^{\\\\rm exit}}\\\\bar{B}^{s}_{\\\\bar{\\\\eta }}(y ),\\\\\\\\(\\\\bar{W} )^{\\\\rm entry}:=\\\\bigcup _{y \\\\in (\\\\bar{\\\\Delta })^{\\\\rm entry}}\\\\bar{B}^{s}_{\\\\bar{\\\\eta }}(y ) \\\\cup \\\\bigcup _{y \\\\in (\\\\bar{\\\\Delta })}\\\\partial \\\\bar{B}^{s}_{\\\\bar{\\\\eta }}(y ).\\\\end{split}$ Second, we describe in a similar fashion how to thicken $\\\\bar{R}^{\\\\prime } $ to a full dimensional window $\\\\bar{W}^{\\\\prime } $ .\",\n \"We choose $0<\\\\bar{\\\\delta }^{\\\\prime } <\\\\varepsilon ^{\\\\prime } $ and $0<\\\\bar{\\\\eta }^{\\\\prime } <\\\\varepsilon ^{\\\\prime }$ .\",\n \"We consider the $(l+n_s)$ -dimensional rectangle $\\\\bar{\\\\Delta }^{\\\\prime } :=\\\\bigcup _{x^{\\\\prime } \\\\in \\\\bar{R}^{\\\\prime } }\\\\bar{B}^{s}_{\\\\bar{\\\\eta }^{\\\\prime } }(x^{\\\\prime })\\\\subseteq W^s(\\\\Lambda )$ , where $\\\\bar{B}^s_{\\\\bar{\\\\eta }^{\\\\prime } }(x^{\\\\prime } )$ is the $n$ -dimensional closed ball of radius $\\\\bar{\\\\eta }^{\\\\prime } $ centered at $x^{\\\\prime } $ and contained in $W^s(x^+ )$ , with $x^+ =\\\\Omega ^+_\\\\Gamma (x^{\\\\prime } )$ .\",\n \"Its exit set and entry sets are defined as follows: $\\\\begin{split}(\\\\bar{\\\\Delta }^{\\\\prime } )^{\\\\rm exit}:=\\\\bigcup _{x \\\\in (\\\\bar{R}^{\\\\prime } )^{\\\\rm exit}}\\\\bar{B}^{s}_{\\\\bar{\\\\eta }^{\\\\prime } }(x^{\\\\prime } ),\\\\\\\\(\\\\bar{\\\\Delta }^{\\\\prime } )^{\\\\rm entry}:=\\\\bigcup _{x^{\\\\prime } \\\\in (\\\\bar{R}^{\\\\prime } )^{\\\\rm entry}}\\\\bar{B}^{s}_{\\\\bar{\\\\eta }^{\\\\prime } }(x^{\\\\prime } )\\\\cup \\\\bigcup _{x\\\\in (\\\\bar{R}^{\\\\prime } )}\\\\partial \\\\bar{B}^{s}_{\\\\bar{\\\\eta }^{\\\\prime } }(x^{\\\\prime } ).\\\\end{split}$ We define $\\\\bar{W}^{\\\\prime } :=\\\\bigcup _{y^{\\\\prime } \\\\in \\\\bar{\\\\Delta }^{\\\\prime } }\\\\bar{B}^{u}_{\\\\bar{\\\\delta }^{\\\\prime } }(y^{\\\\prime } )$ , where $\\\\bar{B}^u_{\\\\bar{\\\\delta }^{\\\\prime } }(y^{\\\\prime } )$ is the $n$ -dimensional closed ball centered at $y^{\\\\prime } $ and contained in the image of $N^{\\\\prime }_{y^{\\\\prime } }$ under the exponential map $\\\\exp _{y^{\\\\prime } }:T_{y^{\\\\prime } }M\\\\rightarrow M$ , with $N^{\\\\prime }$ being the normal bundle to $W^s(\\\\Lambda )$ .\",\n \"For $\\\\bar{\\\\delta }^{\\\\prime } >0$ sufficiently small $\\\\bar{W}^{\\\\prime }$ is a homeomorphic copy of a $(l+n_u+n_s)$ -dimensional rectangle.\",\n \"The exit set and the entry set of $\\\\bar{W}^{\\\\prime } $ are defined by: $\\\\begin{split} (\\\\bar{W}^{\\\\prime } )^{\\\\rm exit}:=\\\\bigcup _{y^{\\\\prime } \\\\in (\\\\bar{\\\\Delta }^{\\\\prime } )^{\\\\rm exit}}\\\\bar{B}^{u}_{\\\\bar{\\\\delta }^{\\\\prime } }(y^{\\\\prime } )\\\\cup \\\\bigcup _{y^{\\\\prime } \\\\in (\\\\bar{\\\\Delta }^{\\\\prime } )}\\\\partial \\\\bar{B}^{u}_{\\\\bar{\\\\delta }^{\\\\prime } }(y^{\\\\prime } ),\\\\\\\\(W^{\\\\prime } )^{\\\\rm entry}:=\\\\bigcup _{y^{\\\\prime } \\\\in (\\\\bar{\\\\Delta }^{\\\\prime } )^{\\\\rm entry}}\\\\bar{B}^{u}_{\\\\bar{\\\\delta }^{\\\\prime }_{i+1}}(y^{\\\\prime } ).\\\\end{split}$ This completes the description of the thickening of the $l$ -dimensional window $\\\\bar{R} $ into a $(l+n_u+n_s)$ -dimensional window $\\\\bar{W} $ , and of the thickening of the $l$ -dimensional window $\\\\bar{R}^{\\\\prime } $ into a $(l+n_u+n_s)$ -dimensional window $\\\\bar{W}^{\\\\prime } $ .\",\n \"Note that by construction $\\\\bar{W} $ is contained in an $\\\\varepsilon $ -neighborhood of $\\\\Lambda $ and $\\\\bar{W}^{\\\\prime } $ is contained in an $\\\\varepsilon ^{\\\\prime }$ -neighborhood of $\\\\Gamma $ .\",\n \"In order to make $\\\\bar{W} $ correctly aligned with $\\\\bar{W}^{\\\\prime } $ under the identity map, we choose $\\\\bar{\\\\delta }^{\\\\prime } $ sufficiently small relative to $\\\\bar{\\\\delta }$ , and $\\\\bar{\\\\eta }$ sufficiently small relative to $\\\\bar{\\\\eta }^{\\\\prime } $ .\",\n \"Step 2.\",\n \"We take a negative iterate $f^{-M}(\\\\bar{R})$ of $\\\\bar{R}$ , where $M>0$ .\",\n \"We have that $f^{-M}(\\\\Gamma )$ is $\\\\varepsilon $ -close to $\\\\Lambda $ in the $C^1$ -topology, for all $M$ sufficiently large.\",\n \"The vectors tangent to the fibers $W^u(x^-)$ in $\\\\bar{R}$ are contracted, and the vectors transverse to $W^u(\\\\Lambda )$ along $\\\\bar{R}$ are expanded by the derivative of $f^{-M}$ .\",\n \"We choose $M$ sufficiently large so that $f^{-M}(\\\\bar{R})$ is $\\\\varepsilon $ -close to $f^{-M}(R)$ .\",\n \"We now construct a window $W$ about $f^{-M}(R)$ that is correctly aligned with $f^{-M}(\\\\bar{W})$ under the identity.\",\n \"Note that each closed ball $\\\\bar{B}^{u}_{\\\\delta }(x)$ , which is a part of $\\\\bar{\\\\Delta }$ , gets exponentially contracted as it is mapped onto $W^u(f^{-{M }}(x^-))$ by $f^{-{M }}$ .\",\n \"By the Lambda Lemma (see the version in [44]), each closed ball $\\\\bar{B}^s_{\\\\eta }(y)$ , $y\\\\in \\\\bar{\\\\Delta }$ , $C^1$ -approaches some subset of $W^s(f^{-{M }}(y^-))$ under $f^{-{M }}$ , as $M \\\\rightarrow \\\\infty $ .\",\n \"For $M $ sufficiently large, we can assume that $f^{-{M }}(\\\\bar{B}^s_{\\\\bar{\\\\eta }}(y))$ is $\\\\varepsilon $ -close to some ball in $W^s(f^{-{M }}(y^-))$ in the $C^1$ -topology, for all $y\\\\in \\\\bar{\\\\Delta }$ .\",\n \"We fix $M$ with the above properties.\",\n \"As $R $ is correctly aligned with $\\\\bar{R}$ under $(\\\\Omega ^-_\\\\Gamma )^{-1}$ , we have that $f^{-M }(R )$ is correctly aligned with $f^{-M }(\\\\bar{R} )$ under $({\\\\Omega ^-}_{f^{-M }(\\\\Gamma )})^{-1}$ .\",\n \"In other words, $f^{-M }(R )$ is correctly aligned under the identity mapping with the projection of $f^{-M }(\\\\bar{R} )$ onto $\\\\Lambda $ along the unstable fibers.\",\n \"To define the window $W$ , we use a local linearization of the normally hyperbolic invariant manifold.\",\n \"By Theorem 1 in [49], there exists a homeomorphism $h$ of an open neighborhood of $(E^u\\\\oplus E^s)_{ \\\\mid \\\\Lambda }$ to an open neighborhood of $\\\\Lambda $ in $M$ such that $h\\\\circ Df = f\\\\circ h$ .\",\n \"We select a point $x \\\\in f^{-M}( R )$ .\",\n \"Since $R$ is contractible the bundles are trivial on $f^{-M}(R)$ and we can identify $(E^u\\\\oplus E^s)_{\\\\mid f^{-M }(R)}$ with $f^{-M }(R) \\\\times E^u_{x } \\\\times E^s_{x }$ .\",\n \"Let us consider $0<\\\\delta <\\\\varepsilon $ and $0<\\\\eta <\\\\varepsilon $ .\",\n \"We define a window $W $ as $W =h(f^{-M }( R ) \\\\times \\\\bar{B}^u_{\\\\delta }(0) \\\\times \\\\bar{B}^s_{\\\\eta }(0)),$ where $\\\\bar{B}^u_{\\\\delta }(0)$ is the closed ball centered at 0 of radius $\\\\delta $ in $E^u_{x }$ and $\\\\bar{B}^s_{\\\\eta }(0)$ is the closed ball centered at 0 of radius $\\\\eta $ in $E^s_{x }$ .\",\n \"We define the exit set of $W $ as $(W )^{\\\\text{exit}}= h(f^{-M }(\\\\bar{R} ) \\\\times \\\\partial \\\\bar{B}^u_{\\\\delta }(0) \\\\times \\\\bar{B}^s_{\\\\eta }(0))\\\\cup h(f^{-M}(\\\\bar{R}^{\\\\text{exit}}) \\\\times B^u_{\\\\delta }(0) \\\\times \\\\bar{B}^s_{\\\\eta }(0)).$ Similarly, the entry set of $W$ is defined as $(W)^{\\\\text{entry}}= h(f^{-M}(\\\\bar{R}) \\\\times \\\\bar{B}^u_{\\\\delta }(0) \\\\times \\\\partial \\\\bar{B}^s_{\\\\eta }(0))\\\\cup h(f^{-M}(\\\\bar{R}^{\\\\text{entry}}) \\\\times B^u_{\\\\delta }(0) \\\\times \\\\bar{B}^s_{\\\\eta }(0)).$ In order to ensure the correct alignment of $W$ with $f^{-M}(\\\\bar{W})$ under the identity map, we choose $\\\\delta ,\\\\eta $ such that $h(f^{-M}( R) \\\\times \\\\bar{B}^u_{\\\\delta }(0) \\\\times \\\\lbrace 0\\\\rbrace )$ is correctly aligned with $f^{-M }(\\\\bar{\\\\Delta })$ under the identity map (the exit sets of both windows being in the unstable directions), and that each closed ball $f^{-M }(\\\\bar{B}^s_{\\\\eta })$ intersects $W$ in a closed ball that is contained in the interior of $f^{-M }(\\\\bar{B}^s_{\\\\eta })$ .\",\n \"The existence of suitable $\\\\delta ,\\\\eta $ follows from the exponential contraction of $\\\\bar{\\\\Delta }$ under negative iteration, and from the Lambda Lemma applied to $\\\\bar{B}^s_{\\\\eta }(y)$ under negative iteration.\",\n \"In a similar fashion, we construct a window $W^{\\\\prime }$ contained in an $\\\\varepsilon ^{\\\\prime }$ -neighborhood of $\\\\Lambda $ such that $\\\\bar{W}^{\\\\prime }$ is correctly aligned with $W^{\\\\prime }$ under $f^{N}$ .\",\n \"The window $W^{\\\\prime }$ , and its entry and exit sets, are defined by: $\\\\begin{split}W^{\\\\prime } =&h(f^{N}( R^{\\\\prime }) \\\\times \\\\bar{B}^u_{\\\\delta ^{\\\\prime }}(0) \\\\times \\\\bar{B}^s_{\\\\eta ^{\\\\prime }}(0)),\\\\\\\\(W^{\\\\prime } )^{\\\\text{exit}}=& h(f^{N}( R^{\\\\prime }) \\\\times \\\\partial \\\\bar{B}^u_{\\\\delta ^{\\\\prime }}(0) \\\\times \\\\bar{B}^s_{\\\\eta ^{\\\\prime }}(0))\\\\\\\\&\\\\cup h(f^{N}( (R^{\\\\prime })^\\\\text{exit}) \\\\times \\\\bar{B}^u_{\\\\delta ^{\\\\prime }}(0) \\\\times \\\\bar{B}^s_{\\\\eta ^{\\\\prime }}(0)),\\\\\\\\(W^{\\\\prime } )^{\\\\text{entry}}=&h(f^{N}( R^{\\\\prime }) \\\\times \\\\bar{B}^u_{\\\\delta ^{\\\\prime }}(0) \\\\times \\\\partial \\\\bar{B}^s_{\\\\eta ^{\\\\prime }}(0))\\\\\\\\&\\\\cup h(f^{N}(( R^{\\\\prime })^\\\\text{entry}) \\\\times \\\\bar{B}^u_{\\\\delta ^{\\\\prime }}(0) \\\\times \\\\bar{B}^s_{\\\\eta ^{\\\\prime }}(0)),\\\\end{split}$ for some appropriate choices of radii $0<\\\\delta ^{\\\\prime } ,\\\\eta ^{\\\\prime } <\\\\varepsilon ^{\\\\prime }$ .\",\n \"Step 3.\",\n \"Suppose that we have constructed, as in Step 2, a window $W^{\\\\prime } $ about the $l$ -dimensional rectangle $f^{N }(R^{\\\\prime } )\\\\subseteq \\\\Lambda $ , and a window $W $ about the $l$ -dimensional rectangle $f^{-M }(R )\\\\subseteq \\\\Lambda $ .\",\n \"Under positive iterations, the rectangle $\\\\bar{B}^u_{\\\\delta ^{\\\\prime } }(0)\\\\times \\\\bar{B}^s_{\\\\eta ^{\\\\prime }}(0)\\\\subseteq E^u\\\\oplus E^s$ gets exponentially expanded in the unstable direction and exponentially contracted in the stable direction by $Df$ .\",\n \"Thus $\\\\bar{B}^u_{\\\\delta ^{\\\\prime } }(0)\\\\times \\\\bar{B}^s_{\\\\eta ^{\\\\prime } }(0)$ gets correctly aligned with $\\\\bar{B}^u_{\\\\delta }(0)\\\\times \\\\bar{B}^s_{\\\\eta }(0)$ under some power $Df^{L}$ of $Df$ , provided $L $ is sufficiently large.\",\n \"This implies that $f^{L }(h(\\\\lbrace x \\\\rbrace \\\\times \\\\bar{B}^u_{\\\\delta ^{\\\\prime } }(0)\\\\times \\\\bar{B}^s_{\\\\eta ^{\\\\prime } }(0)))$ is correctly aligned with $h({f^{L }(x )}\\\\times \\\\bar{B}^u_{\\\\delta }(0)\\\\times \\\\bar{B}^s_{\\\\eta }(0))$ under the identity map (both rectangles are contained in $h({f^{L }(x )}\\\\times E^u\\\\times E^s)$ .\",\n \"By assumption (iii), there exists $L^{\\\\prime } >\\\\max \\\\lbrace L ,N +M \\\\rbrace $ such that $R^{\\\\prime } $ is correctly aligned with $R $ under $f^{L^{\\\\prime } }$ .\",\n \"This means that $f^{N }(R^{\\\\prime } )$ is correctly aligned with $f^{-M }(R )$ under $f^{K }$ with $K :=L^{\\\\prime } -N -M >0$ .\",\n \"The product property of correctly aligned windows implies that $W^{\\\\prime } $ is correctly aligned with $W $ under $f^{K }$ , provided that $K $ is chosen as above.\",\n \"Step 4.\",\n \"We will now describe the process of constructing, based on Steps 1, 2, and 3, two bi-infinite sequences of windows $\\\\lbrace W_{i},W^{\\\\prime }_{i}\\\\rbrace _{i\\\\in \\\\mathbb {Z}}$ and $\\\\lbrace \\\\bar{W}_{i},\\\\bar{W}^{\\\\prime }_{i}\\\\rbrace _{i\\\\in \\\\mathbb {Z}}$ such that, for all $i\\\\in \\\\mathbb {Z}$ , $W_i$ is correctly aligned with $\\\\bar{W}_i$ under $f^{N_i}$ , $\\\\bar{W}_i$ is correctly aligned with $\\\\bar{W}^{\\\\prime }_{i+1}$ under the identity mapping, $\\\\bar{W}^{\\\\prime }_{i+1}$ is correctly aligned with $W^{\\\\prime }_{i+1}$ under $f^{M_i}$ , and $W^{\\\\prime }_{i+1}$ is correctly aligned with $W_{i+1}$ under $f^{K_{i+1}}$ .\",\n \"The point of this step is that we can repeatedly choose the rectangles and the parameters at Steps 1, 2, and 3, in a consistent way, in order to produce infinite sequences of correctly aligned windows.\",\n \"Staring with $i=0$ and continuing for all $i\\\\ge 0$ , we do the following.\",\n \"For a given pair of $l$ -dimensional windows $\\\\lbrace R_i,R^{\\\\prime }_{i+1}\\\\rbrace $ , we consider the corresponding copies $\\\\lbrace \\\\bar{R}_i,\\\\bar{R}^{\\\\prime }_{i+1}\\\\rbrace $ in $\\\\Gamma $ such that $\\\\bar{R}_i$ is correctly aligned with $\\\\bar{R}^{\\\\prime }_{i+1}$ under the identity.\",\n \"As in Step 1, we construct a pair of $(l+n_u+n_s)$ -dimensional windows $\\\\bar{W}_i$ about $\\\\bar{R}_i$ , and $\\\\bar{W}^{\\\\prime }_{i+1}$ about $\\\\bar{R}^{\\\\prime }_{i+1}$ .\",\n \"The size of $\\\\bar{W}_i$ in the hyperbolic directions is given by some disks radii $\\\\bar{\\\\delta }_i,\\\\bar{\\\\eta }_i<\\\\varepsilon _i$ , and that of $\\\\bar{W}^{\\\\prime }_{i+1}$ by some disk radii $\\\\bar{\\\\delta }^{\\\\prime }_{i+1}, \\\\bar{\\\\eta }^{\\\\prime }_{i+1}<\\\\varepsilon _{i+1}$ .\",\n \"We choose the quantities $\\\\bar{\\\\delta }_i,\\\\bar{\\\\eta }_i, \\\\bar{\\\\delta }^{\\\\prime }_{i+1}, \\\\bar{\\\\eta }^{\\\\prime }_{i+1}$ such that $\\\\bar{W}_i$ is correctly aligned with $\\\\bar{W}^{\\\\prime }_{i+1}$ under the identity.\",\n \"Then, we consider the pair of $l$ -dimensional windows $\\\\lbrace R_{i+1},R^{\\\\prime }_{i+2}\\\\rbrace $ in $\\\\Lambda $ , and their corresponding copies $\\\\lbrace \\\\bar{R}_{i+1},\\\\bar{R}^{\\\\prime }_{i+2}\\\\rbrace $ in $\\\\Gamma $ .\",\n \"As in Step 1, we construct a pair of $(l+n_u+n_s)$ -dimensional windows $\\\\bar{W}_{i+1}$ about $\\\\bar{R}_{i+1}$ , and $\\\\bar{W}^{\\\\prime }_{i+2}$ about $\\\\bar{R}^{\\\\prime }_{i+2}$ .\",\n \"By choosing the quantities $\\\\bar{\\\\delta }_{i+1},\\\\bar{\\\\eta }_{i+1}, \\\\bar{\\\\delta }^{\\\\prime }_{i+2}, \\\\bar{\\\\eta }^{\\\\prime }_{i+2}$ as in Step 1 we can ensure that $\\\\bar{W}_{i+1}$ is correctly aligned with $\\\\bar{W}^{\\\\prime }_{i+2}$ .\",\n \"We choose $N_{i+1},M_{i+1}$ large enough so that $f^{N_{i+1}}(\\\\bar{W}^{\\\\prime }_{i+1})$ is contained in an $\\\\varepsilon _{i+1}$ -neighborhood of $f^{N_{i+1}}(R^{\\\\prime }_{i+1})$ , and $f^{-M_{i+1}}(\\\\bar{W}_{i+1})$ is contained in an $\\\\varepsilon _{i+1}$ -neighborhood of $f^{-M_{i+1}}(R_{i+1})$ .\",\n \"As in Step 2, we construct a window $W^{\\\\prime }_{i+1}$ about $f^{N_1}(R^{\\\\prime }_{i+1})$ such that $\\\\bar{W}^{\\\\prime }_{i+1}$ correctly aligned with $W^{\\\\prime }_{i+1}$ under $f^{N_{i+1}}$ , and a window $W_{i+1}$ about $f^{-M_1}(R_{i+1})$ such that $W_{i+1}$ correctly aligned with $\\\\bar{W}_{i+1}$ under $f^{M_{i+1}}$ .\",\n \"This amounts to choosing the quantities $\\\\delta ^{\\\\prime }_{i+1}, \\\\eta ^{\\\\prime }_{i+1},\\\\delta _{i+1},\\\\eta _{i+1}$ as in Step 2 in order to ensure the correct alignment of the windows.\",\n \"Then, we choose $K_{i+1}$ sufficiently large, and at least as large as $N_{i+1}+M_{i+1}$ , such that $W^{\\\\prime }_{i+1}$ is correctly aligned with $W_{i+1}$ under $f^{K_{i+1}}$ .\",\n \"At this point, we have that $\\\\bar{W}^{\\\\prime }_{i+1}$ is correctly aligned with $W^{\\\\prime }_{i+1}$ under $f^{N_{i+1}}$ , $ W^{\\\\prime }_{i+1}$ is correctly aligned with $W_{i+1}$ under $f^{K_{i+1}}$ , and $W_{i+1}$ is correctly aligned with $\\\\bar{W}_{i+1}$ under $f^{M_{i+1}}$ .\",\n \"Inductively, we obtain two sequences of windows $\\\\lbrace W_{i},W^{\\\\prime }_{i}\\\\rbrace _{i\\\\ge 0}$ and $\\\\lbrace \\\\bar{W}_{i},\\\\bar{W}^{\\\\prime }_{i}\\\\rbrace _{i\\\\ge 0}$ that satisfy the desired correct alignment conditions for all $i\\\\ge 0$ .\",\n \"A similar inductive construction of windows can be done backwards starting with $W_0, W^{\\\\prime }_1$ .\",\n \"In the end, we obtain two bi-infinite sequences of windows $\\\\lbrace W_{i},W^{\\\\prime }_{i}\\\\rbrace _{i\\\\in \\\\mathbb {Z}}$ and $\\\\lbrace \\\\bar{W}_{i},\\\\bar{W}^{\\\\prime }_{i}\\\\rbrace _{i\\\\ge 0}$ that satisfy the desired correct alignment conditions for all $i\\\\in \\\\mathbb {Z}$ .\",\n \"By Theorem REF , there exists an orbit $\\\\lbrace f^n(z)\\\\rbrace _{n\\\\in \\\\mathbb {Z}}$ with $f^{n_i}(z)\\\\in \\\\bar{W}_i\\\\cap \\\\bar{W}^{\\\\prime }_{i+1}$ , $f^{n_i+N_{i+1}}(z)\\\\in W^{\\\\prime }_{i+1}$ , $f^{n_i+N_{i+1}+K_{i+1}}(z)\\\\in W_{i+1}$ , $f^{n_i+N_{i+1}+K_{i+1}+M_{i+1}}(z)\\\\in \\\\bar{W}_{i+1}\\\\cap \\\\bar{W}^{\\\\prime }_{i+2}$ , for all $i\\\\in \\\\mathbb {Z}$ .\",\n \"Thus $n_{i+1}=n_i+N_{i+1}+K_{i+1}+M_{i+1}$ .\",\n \"Such an orbit $\\\\lbrace f^n(z)\\\\rbrace _{n\\\\in \\\\mathbb {Z}}$ satisfies the properties required by Lemma REF .\"\n ],\n [\n \"Construction of correctly aligned windows\",\n \"In this section we prove Theorem REF .\",\n \"The methodology consists of constructing 2-dimensional windows in $\\\\Lambda $ about the prescribed invariant primary tori, BZI's, and Aubry-Mather sets inside the BZI's.\",\n \"The successive pairs of windows are correctly aligned under the scattering map alternatively with powers of the inner map.\",\n \"Lemma REF implies that there exist trajectories that follow these windows.\"\n ],\n [\n \"Construction of correctly aligned windows across a BZI\",\n \"In this section we will construct correctly aligned windows across a BZI between two successive transition chains of tori.\",\n \"On each side of the BZI we will choose a one-sided neighborhood of a point on the boundary torus, and we will use Theorem REF or Theorem REF to cross over the BZI.\",\n \"These one-sided neighborhoods are of a special type: their boundaries are images of some transition tori under the inner or outer dynamics.\",\n \"We will construct some windows about the boundaries of these one-sided neighborhoods.\",\n \"Then we will construct some other windows about the corresponding transition tori; these windows will be chosen so that they have a pair of sides lying on some nearby tori.\",\n \"We will use this feature later to connect sequence of windows across the BZI's with sequences of windows along the transition chains.\",\n \"Consider an annular region $\\\\Lambda _k$ in $\\\\Lambda $ that is a BZI, and is between two transition chains of invariant tori, as in (A5).\",\n \"To simplify notation, we denote the tori at the boundary of $\\\\Lambda _k$ by $T_{a}$ and $T_{b}$ .\",\n \"We choose a pair of transition tori $T_{i}, T_{j}$ in $\\\\Lambda $ as in (A5), ordered as follows: $T_{j}\\\\prec T_{i}\\\\prec T_{a}$ .\",\n \"These tori are outside of the BZI $\\\\Lambda _k$ and on the same side of it as $T_{a}$ .\",\n \"By (A5-iv) there exist $T_{i^{\\\\prime }}\\\\prec T_i$ and $T_{j^{\\\\prime }}\\\\prec T_j$ such that $T_{i^{\\\\prime }}$ is $\\\\varepsilon _i$ -close to $T_i$ and $T_{j^{\\\\prime }}$ is $\\\\varepsilon _j$ -close to $T_j$ , in the $C^0$ topology.\",\n \"By (A5-ii) $S(T_j)$ intersects $T_i$ in a topologically transverse manner, so both $S(T_j)$ and $S(T_{j^{\\\\prime }})$ intersect both $T_i$ and $T_{i^{\\\\prime }}$ in a topologically transverse manner, provided $T_{i^{\\\\prime }}, T_{j^{\\\\prime }}$ are sufficiently close to $T_i,T_j$ , respectively.\",\n \"Since $S$ is a diffeomorphism, $T_{i}, T_{i^{\\\\prime }}$ and the images of $T_{j}, T_{j^{\\\\prime }}$ under $S$ form a topological rectangle $D_{iji^{\\\\prime }j^{\\\\prime }}$ in $\\\\Lambda $ .\",\n \"This rectangle may not be contained in the domain $U^-$ of the scattering map $S$ .\",\n \"Provided that we choose the tori $T_{i}, T_{i^{\\\\prime }}$ sufficiently $C^0$ -close to one another, and also $T_{j}, T_{j^{\\\\prime }}$ sufficiently $C^0$ -close to one another, the rectangle $D_{iji^{\\\\prime }j^{\\\\prime }}$ will be sufficiently small so that some iterate $f_{\\\\mid \\\\Lambda }^{K_a}$ of $f_{\\\\mid \\\\Lambda }$ takes the rectangle $D_{iji^{\\\\prime }j^{\\\\prime }}$ into a rectangle $f_{\\\\mid \\\\Lambda }^{K_a}(D_{iji^{\\\\prime }j^{\\\\prime }})$ inside $U^-$ .\",\n \"This is possible since each torus intersects $U^-$ by (A5-i), and the motion on the tori is topologically transitive by (A5-iii).\",\n \"The rectangle $f_{\\\\mid \\\\Lambda }^{K_a}(D_{iji^{\\\\prime }j^{\\\\prime }})$ has a pair of sides lying on the tori $T_{i}, T_{i^{\\\\prime }}$ , and the other pair of sides on $(f_{\\\\mid \\\\Lambda }^{K_a}\\\\circ S)(T_{j}),(f_{\\\\mid \\\\Lambda }^{K_a}\\\\circ S)(T_{j^{\\\\prime }})$ .\",\n \"The curves $(f_{\\\\mid \\\\Lambda }^{K_a}\\\\circ S)(T_{j}),(f_{\\\\mid \\\\Lambda }^{K_a}\\\\circ S)(T_{j^{\\\\prime }})$ are topologically transverse to both $T_{i},T_{i^{\\\\prime }}$ .\",\n \"By assumption (A5-ii), $S(T_i)$ topologically crosses $T_{a}$ , so $S^{-1}(T_{a})$ topologically crosses $T_{i}$ .\",\n \"We can ensure that the interior of $f_{\\\\mid \\\\Lambda }^{K_a}(D_{iji^{\\\\prime }j^{\\\\prime }})$ intersects $S^{-1}(T_{a})$ by choosing $K_a$ sufficiently large, and the tori in each pair $T_{i}, T_{i^{\\\\prime }}$ and $T_{j}, T_{j^{\\\\prime }}$ sufficiently $C^0$ -close to one another.\",\n \"This implies that the image of $f_{\\\\mid \\\\Lambda }^{K_a}(D_{iji^{\\\\prime }j^{\\\\prime }})$ under $S$ is a topological rectangle in $\\\\Lambda $ which intersects $T_{a}$ , and the intersection of $S(f_{\\\\mid \\\\Lambda }^{K_a}(D_{iji^{\\\\prime }j^{\\\\prime }}))$ with $\\\\Lambda _k$ forms a one-sided neighborhood in $\\\\Lambda _k$ of some part of $T_{a}$ .\",\n \"The boundary of $S(f_{\\\\mid \\\\Lambda }^{K_a}(D_{iji^{\\\\prime }j^{\\\\prime }}))\\\\cap \\\\Lambda _k$ consists of arcs of the curves $T_{a}, S(T_i), S(T_{i^{\\\\prime }}),S\\\\circ f_{\\\\mid \\\\Lambda }^{K_a}\\\\circ S(T_j),S\\\\circ f_{\\\\mid \\\\Lambda }^{K_a}\\\\circ S(T_{j^{\\\\prime }})$ .\",\n \"We make a similar construction on the other side of the BZI $\\\\Lambda _k$ .\",\n \"We choose a pair of transition tori $T_{k}, T_{l}$ with $ T_{b}\\\\prec T_{k}\\\\prec T_{l}$ , outside of the BZI $\\\\Lambda _k$ and on the same side as $T_{b}$ .\",\n \"We also choose $T_{k}\\\\prec T_{k^{\\\\prime }}$ and $T_{l}\\\\prec T_{l^{\\\\prime }}$ such that $T_{k^{\\\\prime }}$ is $\\\\varepsilon _k$ -close to $T_k$ and $T_{l^{\\\\prime }}$ is $\\\\varepsilon _l$ -close to $T_l$ , and $S^{-1}(T_l), S^{-1}(T_{l^{\\\\prime }})$ are topologically transverse to both $T_k,T_{k^{\\\\prime }}$ , and so they form a a topological rectangle $D_{klk^{\\\\prime }l^{\\\\prime }}$ .\",\n \"There exists $K_b$ sufficiently large such that $f_{\\\\mid \\\\Lambda }^{-K_b}(D_{klk^{\\\\prime }l^{\\\\prime }})\\\\subseteq U^-$ , its interior intersects $S(T_{b})$ , and $S^{-1}(f_{\\\\mid \\\\Lambda }^{-K_b}(D_{klk^{\\\\prime }l^{\\\\prime }}))$ forms a one-sided neighborhood in $\\\\Lambda _k$ of some part of $T_{b}$ .\",\n \"The boundary of $S^{-1}(f_{\\\\mid \\\\Lambda }^{K_b}(D_{klk^{\\\\prime }l^{\\\\prime }}))\\\\cap \\\\Lambda _k$ consists of arcs of the curves $T_{b}, S(T_k), S(T_{k^{\\\\prime }}),S^{-1}\\\\circ f_{\\\\mid \\\\Lambda }^{K_b}\\\\circ S^{-1}(T_l),S^{-1}\\\\circ f_{\\\\mid \\\\Lambda }^{K_b}\\\\circ S^{-1}(T_{l^{\\\\prime }})$ .\",\n \"Figure: Orbits across a BZIAt this stage we have obtained in $\\\\Lambda _k$ a one-sided neighborhood $(S\\\\circ f_{\\\\mid \\\\Lambda }^{K_a})(D_{iji^{\\\\prime }j^{\\\\prime }})$ of an arc in $T_{a}$ , and a one-sided neighborhood $(S^{-1}\\\\circ f_{\\\\mid \\\\Lambda }^{-K_b})(D_{klk^{\\\\prime }l^{\\\\prime }})$ of an arc in $T_{b}$ .\",\n \"If we are under the assumptions (A1)-(A6), Theorem REF yields a point $x_a\\\\in \\\\partial (S\\\\circ f_{\\\\mid \\\\Lambda }^{K_a})(D_{iji^{\\\\prime }j^{\\\\prime }})$ whose image $x_b=f_{\\\\mid \\\\Lambda }^{K_{ab}}(x_a)$ under some power $f_{\\\\mid \\\\Lambda }^{K_{ab}}$ lies on $\\\\partial (S^{-1}\\\\circ f_{\\\\mid \\\\Lambda }^{-K_b})(D_{klk^{\\\\prime }l^{\\\\prime }})$ .\",\n \"If we also assume (A7), Theorem REF yields a point $x_a\\\\in \\\\partial (S\\\\circ f_{\\\\mid \\\\Lambda }^{K_a})(D_{iji^{\\\\prime }j^{\\\\prime }})$ with $x_b=f_{\\\\mid \\\\Lambda }^{K_{ab}}(x_a)\\\\in \\\\partial (S^{-1}\\\\circ f_{\\\\mid \\\\Lambda }^{-K_b})(D_{klk^{\\\\prime }l^{\\\\prime }})$ as above, satisfying the additional conditions $\\\\pi _\\\\phi (f^j(w^k_s))<\\\\pi _\\\\phi (f^j(x_a))<\\\\pi _\\\\phi (f^j(\\\\bar{w}^k_s)), $ for each $s\\\\in \\\\lbrace 1,\\\\ldots , s_k\\\\rbrace $ , where $w^k_s,\\\\bar{w}^k_s\\\\in \\\\Sigma _{\\\\omega ^k_s}$ , and for all $j$ within a certain interval of integers.\",\n \"The trajectories of all points sufficiently close to $x_a$ will satisfy these conditions as well.\",\n \"In either case, there exist an arc $\\\\bar{e}^{\\\\prime }_a\\\\subseteq \\\\partial (S\\\\circ f_{\\\\mid \\\\Lambda }^{K_a})(D_{iji^{\\\\prime }j^{\\\\prime }})$ containing $x_a$ , and another arc $\\\\bar{e}_b\\\\subseteq \\\\partial (S^{-1}\\\\circ f_{\\\\mid \\\\Lambda }^{-K_b})(D_{klk^{\\\\prime }l^{\\\\prime }})$ containing $x_b$ , such that $f_{\\\\mid \\\\Lambda }^{K_{ab}}(\\\\bar{e}^{\\\\prime }_a)$ is topologically transverse to $\\\\bar{e}_b$ at $x_b$ .\",\n \"The arc $\\\\bar{e}^{\\\\prime }_a$ lies on one of the sets $S({T_{i}}), S({T_{i^{\\\\prime }}}), (S\\\\circ f_{\\\\mid \\\\Lambda }^{K_{a}}\\\\circ S)(T_{j}), (S\\\\circ f_{\\\\mid \\\\Lambda }^{K_{a}}\\\\circ S)(T_{j^{\\\\prime }})$ .\",\n \"Similarly, the arc $\\\\bar{e}_b$ lies on one of the sets $S^{-1}(T_{k}), S^{-1}({T_{k^{\\\\prime }}}), (S^{-1}\\\\circ f_{\\\\mid \\\\Lambda }^{-K_{b}}\\\\circ S^{-1})(T_{l}), (S^{-1}\\\\circ f_{\\\\mid \\\\Lambda }^{-K_{b}}\\\\circ S^{-1})(T_{l^{\\\\prime }})$ .\",\n \"We define a 2-dimensional window $R^{\\\\prime }_a$ about $\\\\bar{e}^{\\\\prime }_a$ , and a 2-dimensional window $R_b$ about $\\\\bar{e}_b$ such that $R^{\\\\prime }_a$ is correctly aligned with $R_b$ under $f_{\\\\mid \\\\Lambda }^{K_{ab}}$ .\",\n \"Informally, the exit direction of $R^{\\\\prime }_a$ is along $\\\\bar{e}^{\\\\prime }_a$ , and the exit direction of $R_b$ is across $\\\\bar{e}_b$ .\",\n \"The formal construction now follows.\",\n \"Since the arc $\\\\bar{e}^{\\\\prime }_a$ is an embedded 1-dimensional $C^0$ -submanifold of $\\\\Lambda $ , there exists a $C^0$ -local parametrization $\\\\chi ^{\\\\prime }_a:\\\\mathbb {R}^2\\\\rightarrow \\\\Lambda $ such that $\\\\chi ^{\\\\prime }_a([0,1]\\\\times \\\\lbrace 0\\\\rbrace )=\\\\bar{e}^{\\\\prime }_a$ , provided $\\\\bar{e}^{\\\\prime }_a$ is sufficiently small.\",\n \"Then $R^{\\\\prime }_a=\\\\chi _a([0,1]\\\\times [-\\\\eta ^{\\\\prime }_a,\\\\eta ^{\\\\prime }_a]),$ is a topological rectangle.\",\n \"We define the exit set of $R^{\\\\prime }_a$ as ${R^{\\\\prime }}_a^\\\\textrm {exit}=\\\\chi _a(\\\\partial [0,1]\\\\times [-\\\\eta ^{\\\\prime }_a,\\\\eta ^{\\\\prime }_a]).$ The definition of the entry set of of $R^{\\\\prime }_a$ follows by default.\",\n \"Similarly, there exists a $C^0$ -local parametrization $\\\\chi _b:\\\\mathbb {R}^2\\\\rightarrow \\\\Lambda $ such that $\\\\chi _b(\\\\lbrace 0\\\\rbrace \\\\times [0,1])=\\\\bar{e}_b$ , and $R_b=\\\\chi _b([-\\\\delta _b,\\\\delta _b]\\\\times [0,1]),$ is a topological rectangle.\",\n \"We define the exit set of $R_b$ as ${R}_b^\\\\textrm {exit}=\\\\chi _b(\\\\partial [-\\\\delta _b,\\\\delta _b]\\\\times [0,1]).$ By choosing $\\\\eta ^{\\\\prime }_a, \\\\delta _b$ sufficiently small, we ensure that $R^{\\\\prime }_a$ is correctly aligned with $R_b$ under $f_{\\\\mid \\\\Lambda }^{K_{ab}}$ (see Definition REF ).\",\n \"See Figure REF .\",\n \"We now construct other windows outside the BZI $\\\\Lambda _k$ .\",\n \"We consider two cases: first case, when the arc $\\\\bar{e}^{\\\\prime }_{a}$ is a part of $S(T_{i})$ or $S(T_{i^{\\\\prime }})$ ; second case, when $\\\\bar{e}^{\\\\prime }_a$ is a part of $(S\\\\circ f_{\\\\mid \\\\Lambda }^{K_a}\\\\circ S)(T_{j})$ or $(S\\\\circ f_{\\\\mid \\\\Lambda }^{K_a}\\\\circ S)(T_{j^{\\\\prime }})$ .\",\n \"Case 1.\",\n \"In the first case, when $\\\\bar{e}^{\\\\prime }_a$ is a part of $S(T_{i})$ or $S(T_{i^{\\\\prime }})$ we proceed with the construction as follows.\",\n \"Let us say that $\\\\bar{e}^{\\\\prime }_a$ is a part of $S(T_{i})$ .\",\n \"We take the inverse image of $R^{\\\\prime }_a$ by $S$ .\",\n \"This is a topological rectangle $S^{-1}(R^{\\\\prime }_a)$ about the torus $T_{i}$ .\",\n \"We construct a window $R_{i}$ about an arc $\\\\bar{e}_{i}$ in $T_{i}$ , such that $R_{i}$ is correctly aligned with $S^{-1}(R^{\\\\prime }_{a})$ under the identity map, and with the exit direction of $R_{i}$ in the direction of $T_i$ .\",\n \"Let $\\\\bar{e}_{i}$ be an arc in $T_{i}$ , and $\\\\chi _{i}:\\\\mathbb {R}^2\\\\rightarrow \\\\Lambda $ be a local parametrization with $\\\\chi _{i}([0,1]\\\\times \\\\lbrace 0\\\\rbrace )= \\\\bar{e}_{i}$ .\",\n \"We define $R_{i}=\\\\chi _{i}([0,1]\\\\times [-\\\\eta _i,\\\\eta _i])\\\\\\\\{R}^\\\\textrm {exit}_{i}=\\\\chi _{i}(\\\\partial [0,1]\\\\times [-\\\\eta _i,\\\\eta _i]),$ where $\\\\eta _{i}>0$ is sufficiently small.\",\n \"By choosing the arc $\\\\bar{e}_{i}$ sufficiently large so that $\\\\bar{e}_{i}\\\\supseteq S^{-1}(\\\\bar{e}^{\\\\prime }_a)$ , and by choosing $\\\\eta _{i}>0$ sufficiently small, we can ensure that $R_{i}$ is correctly aligned with $S^{-1}(R^{\\\\prime }_{a})$ under the identity map, or, equivalently, $R_{i}$ is correctly aligned with $R^{\\\\prime }_{a}$ under $S$ .\",\n \"See Figure REF .\",\n \"Figure: Construction of windows near the boundaries of a BZI – case 1We construct another window $R^{\\\\prime }_{i}$ about $T_{i}$ such that $R^{\\\\prime }_{i}$ is correctly aligned with $R_{i}$ under some power $f_{\\\\mid \\\\Lambda }^{K_{i}}$ of $f_{\\\\mid \\\\Lambda }$ , with $K_{i}>0$ .\",\n \"The window $R^{\\\\prime }_{i}$ will have its exit direction across the torus $T_{i}$ .\",\n \"We choose a pair of invariant primary tori $T^{\\\\rm lower}_{i}\\\\prec T_{i}\\\\prec T^{\\\\rm upper}_{i}$ , with $T^{\\\\rm lower}_{i}, T^{\\\\rm upper}_{i}$ located $\\\\varepsilon _{i}$ -close to $T_{i}$ in the $C^0$ -topology.\",\n \"Such tori exist due to the assumption that $T_{i}$ is not an end torus in the transition chain, as in (A5-iv).\",\n \"Moreover, we choose the tori $T^{\\\\rm lower}_{i}, T^{\\\\rm upper}_{i}$ sufficiently close to $T_{i}$ so that the components of $R_{i}^{\\\\rm entry}$ are outside the annulus between $T^{\\\\rm lower}_{i}$ and $ T^{\\\\rm upper}_{i}$ , with one component on one side and the other component on the other side of this annulus.\",\n \"We choose an arc $\\\\bar{e}^{\\\\prime }_i\\\\subseteq T_i$ that lies on an edge of the topological rectangle $D_{iji^{\\\\prime }j^{\\\\prime }}$ .\",\n \"Let $\\\\chi ^{\\\\prime }_{i}:\\\\mathbb {R}^2\\\\rightarrow \\\\Lambda $ be a $C^0$ -local parametrization with $\\\\chi ^{\\\\prime }_{i}(\\\\lbrace 0\\\\rbrace \\\\times [0,1])= \\\\bar{e}^{\\\\prime }_{i}$ , such that for some $\\\\delta ^{\\\\prime }_i$ sufficiently small, $\\\\chi ^{\\\\prime }_i (\\\\lbrace -\\\\delta ^{\\\\prime }_i\\\\rbrace \\\\times [0,1])\\\\subseteq T_i^{\\\\rm lower}$ and $\\\\chi ^{\\\\prime }_i (\\\\lbrace \\\\delta ^{\\\\prime }_i\\\\rbrace \\\\times [0,1])\\\\subseteq T_i^{\\\\rm upper}$ .\",\n \"We define $R^{\\\\prime }_{i}$ by: $R^{\\\\prime }_{i}=\\\\chi ^{\\\\prime }_i ([-\\\\delta ^{\\\\prime }_i,\\\\delta ^{\\\\prime }_i]\\\\times [0,1]),\\\\\\\\{R^{\\\\prime }}_{i}^\\\\textrm {exit}=\\\\chi ^{\\\\prime }_i (\\\\partial [-\\\\delta ^{\\\\prime }_i,\\\\delta ^{\\\\prime }_i]\\\\times [0,1]).$ The exit set components of $R^{\\\\prime }_i$ lie on the tori $T_i^{\\\\rm upper}, T_i^{\\\\rm lower}$ neighboring $T_i$ .\",\n \"Since the motion on the tori is topologically transitive, by (A5-iii), there exists $K_{i}>0$ such that $R^{\\\\prime }_{i}$ is correctly aligned with $R_{i}$ under $f_{\\\\mid \\\\Lambda }^{K_{i}}$ .\",\n \"Indeed, to achieve correct alignment of these windows in the covering space of the annulus we only have to choose $K_{i}$ sufficiently large so that the two components of ${R^{\\\\prime }}_{i}^\\\\textrm {exit}$ , which are two arcs in $T^{\\\\rm lower}_{i}$ and $T^{\\\\rm upper}_{i}$ , are mapped by $f_{\\\\mid \\\\Lambda }^{K_{i}}$ on the opposite sides of the part of $R_{i}$ between $T^{\\\\rm lower}_{i}$ and $T^{\\\\rm upper}_{i}$ .\",\n \"We note that the number of iterates $K_{i}$ of $f_{\\\\mid \\\\Lambda }$ needed to make $R^{\\\\prime }_{i}$ correctly aligned with $R_{i}$ may be different than the number of iterates $K_a$ which takes the topological rectangle $D_{iji^{\\\\prime }j^{\\\\prime }}$ onto $f_{\\\\mid \\\\Lambda }^{K_a}(D_{ihi^{\\\\prime }j^{\\\\prime }})$ .\",\n \"See Figure REF .\",\n \"The conclusion of this step is that we obtain the window $R^{\\\\prime }_{i}$ around $T_{i}$ , with its exit direction across $T_{i}$ , such that $R^{\\\\prime }_{i}$ is correctly aligned with $R_{i}$ under $f_{\\\\mid \\\\Lambda }^{K_{i}}$ .\",\n \"Both windows $R^{\\\\prime }_i,R_i$ are contained in an $\\\\varepsilon _i$ -neighborhood of $T_i$ .\",\n \"In the case when the edge $\\\\bar{e}^{\\\\prime }_a$ of $D_{iji^{\\\\prime }j^{\\\\prime }}$ is a part of $S(T_{i^{\\\\prime }})$ instead of $S(T_{i})$ , the construction goes similarly to the one above.\",\n \"Case 2.\",\n \"We now consider the second case, when the arc $\\\\bar{e}^{\\\\prime }_a$ of $\\\\partial (S\\\\circ f_{\\\\mid \\\\Lambda }^{K_a})(D_{iji^{\\\\prime }j^{\\\\prime }})$ is a part of $(S\\\\circ f_{\\\\mid \\\\Lambda }^{K_a}\\\\circ S)(T_{j})$ or $(S\\\\circ f_{\\\\mid \\\\Lambda }^{K_a}\\\\circ S)(T_{j^{\\\\prime }})$ .\",\n \"Let us say that $\\\\bar{e}^{\\\\prime }_a$ is a part of $(S\\\\circ f_{\\\\mid \\\\Lambda }^{K_a}\\\\circ S)(T_{j})$ .\",\n \"We construct a window $R^{\\\\prime }_a$ in $\\\\Lambda $ about $\\\\bar{e}^{\\\\prime }_a$ as before; the exit set of $R^{\\\\prime }_a$ is in a direction along $\\\\bar{e}^{\\\\prime }_a$ , and the size of $R^{\\\\prime }_a$ in the direction across $R_a$ is given by some parameter $\\\\eta ^{\\\\prime }_a$ .\",\n \"See Figure REF .\",\n \"We consider the inverse image $S^{-1}(R^{\\\\prime }_a)$ of $R^{\\\\prime }_a$ by $S$ , which is a topological rectangle about $(f_{\\\\mid \\\\Lambda }^{K_a}\\\\circ S)(T_{j})$ .\",\n \"Let $\\\\bar{e}^{\\\\prime \\\\prime }_j$ be an arc in $(f_{\\\\mid \\\\Lambda }^{K_a}\\\\circ S)(T_{j})$ such that $S(\\\\bar{e}^{\\\\prime \\\\prime }_j)\\\\supset \\\\bar{e}^{\\\\prime }_a$ , and let $\\\\chi ^{\\\\prime \\\\prime }_j :\\\\mathbb {R}^2\\\\rightarrow \\\\Lambda $ be a local parametrization with $\\\\chi ^{\\\\prime \\\\prime }_{j}([0,1]\\\\times \\\\lbrace 0\\\\rbrace )= \\\\bar{e}^{\\\\prime \\\\prime }_{j}$ .\",\n \"We define $R^{\\\\prime \\\\prime }_{j}$ by: $R^{\\\\prime \\\\prime }_{j}=\\\\chi ^{\\\\prime \\\\prime }_j ([0,1]\\\\times [-\\\\eta ^{\\\\prime \\\\prime }_j,\\\\eta ^{\\\\prime \\\\prime }_j]) ,\\\\\\\\{R^{\\\\prime \\\\prime }}_{j}^\\\\textrm {exit}=\\\\chi ^{\\\\prime \\\\prime }_j (\\\\partial [0,1]\\\\times [-\\\\eta ^{\\\\prime \\\\prime }_j,\\\\eta ^{\\\\prime \\\\prime }_j]).$ By choosing $\\\\eta ^{\\\\prime \\\\prime }_{j}>0$ sufficiently small, we can ensure that $R^{\\\\prime \\\\prime }_{j}$ is correctly aligned with aligned with $R^{\\\\prime }_{a}$ under $S$ .\",\n \"The window $R^{\\\\prime \\\\prime }_{j}$ is in a neighborhood of an edge of the topological rectangle $f_{\\\\mid \\\\Lambda }^{K_a}(D_{iji^{\\\\prime }j^{\\\\prime }})$ .\",\n \"The exit set of $R^{\\\\prime \\\\prime }_{j}$ is in the direction of $(f_{\\\\mid \\\\Lambda }^{K_a}\\\\circ S)(T_{j})$ .\",\n \"See Figure REF .\",\n \"Figure: Construction of windows near the boundaries of a BZI – case 2We construct a window $R^{\\\\prime \\\\prime \\\\prime }_{j}$ about $S(T_j)$ such that $R^{\\\\prime \\\\prime \\\\prime }_{j}$ is correctly aligned with $R^{\\\\prime \\\\prime }_{j}$ under $f_{\\\\mid \\\\Lambda }^{K_{a}}$ .\",\n \"We define $R^{\\\\prime \\\\prime \\\\prime }_{j} =\\\\chi ^{\\\\prime \\\\prime \\\\prime }_j ([0,1]\\\\times [-\\\\eta ^{\\\\prime \\\\prime \\\\prime }_j,\\\\eta ^{\\\\prime \\\\prime \\\\prime }_j]) ,\\\\\\\\{R^{\\\\prime \\\\prime \\\\prime }}_{j}^\\\\textrm {exit}=\\\\chi ^{\\\\prime \\\\prime \\\\prime }_j (\\\\partial [0,1]\\\\times [-\\\\eta ^{\\\\prime \\\\prime \\\\prime }_j,\\\\eta ^{\\\\prime \\\\prime \\\\prime }_j]),$ where $\\\\bar{e}^{\\\\prime \\\\prime \\\\prime }_{j}$ is an arc of $S(T_{j})$ with $f_{\\\\mid \\\\Lambda }^{K_a}(\\\\bar{e}^{\\\\prime \\\\prime \\\\prime }_{j})\\\\supseteq \\\\bar{e}^{\\\\prime \\\\prime }_j$ , $\\\\chi ^{\\\\prime \\\\prime \\\\prime }_j :\\\\mathbb {R}^2\\\\rightarrow \\\\Lambda $ is a local parametrization with $\\\\chi ^{\\\\prime \\\\prime \\\\prime }_{j}([0,1]\\\\times \\\\lbrace 0\\\\rbrace )= \\\\bar{e}^{\\\\prime \\\\prime \\\\prime }_{j}$ , and $\\\\eta ^{\\\\prime \\\\prime \\\\prime }_j>0$ is sufficiently small.\",\n \"The arc $\\\\bar{e}^{\\\\prime \\\\prime \\\\prime }_{j}$ is contained in one of the edges of the topological rectangle $D_{iji^{\\\\prime }j^{\\\\prime }}$ .\",\n \"For suitable $\\\\eta ^{\\\\prime \\\\prime \\\\prime }_{j}>0$ , we can ensure that $R^{\\\\prime \\\\prime \\\\prime }_{j}$ is correctly aligned with aligned with $R^{\\\\prime \\\\prime }_{j}$ under $f_{\\\\mid \\\\Lambda }^{K_a}$ .\",\n \"Moreover, we choose $R^{\\\\prime \\\\prime \\\\prime }_{j},R^{\\\\prime \\\\prime }_{j}$ so that these windows are both contained in an $\\\\varepsilon _j$ neighborhood of $T_j$ .\",\n \"The exit set of $R^{\\\\prime \\\\prime \\\\prime }_{j}$ is in a direction along $S(T_{j})$ .\",\n \"We take the inverse image $S^{-1}(R^{\\\\prime \\\\prime \\\\prime }_{j})$ of $R^{\\\\prime \\\\prime \\\\prime }_{j}$ under $S$ , which is a topological rectangle about an arc in $T_{j}$ .\",\n \"In a fashion similar to Case 1, we construct two windows $R^{\\\\prime }_j,R_j$ about $T_j$ such that $R^{\\\\prime }_j$ is correctly aligned with $R_j$ under some power $f_{\\\\mid \\\\Lambda }^{K_j}$ of $f_{\\\\mid \\\\Lambda }$ , and $R_j$ is correctly aligned with $R^{\\\\prime \\\\prime \\\\prime }_j$ under $S$ .\",\n \"The exit of $R^{\\\\prime }_j$ is chosen in a direction across $T_j$ , and the exit set components ${R^{\\\\prime }}^{\\\\rm exit}_j$ of $R^{\\\\prime }_j$ lie on two invariant primary tori $T_j^{\\\\rm upper}$ , $T_j^{\\\\rm lower}$ neighboring $T_j$ , that are $\\\\varepsilon _j$ -close to $T_j$ and contain $T_j$ between them.\",\n \"The exit of $R_j$ is in a direction along $T_j$ .\",\n \"The windows $R^{\\\\prime }_j,R_j$ are chosen to lie in an $\\\\varepsilon _j$ neighborhood of $T_j$ relative to the $C^0$ -topology.\",\n \"The case when the arc $\\\\bar{e}^{\\\\prime }_a$ is a part of $(S\\\\circ f_{\\\\mid \\\\Lambda }^{K_a}\\\\circ S)(T_{j^{\\\\prime }})$ is treated similarly.\",\n \"This concludes the construction of correctly aligned windows in $\\\\Lambda $ , starting with the window $R^{\\\\prime }_{a}$ about $T_{a}$ , and moving backwards along transition tori that are on the same side as $T_{a}$ of the BZI $\\\\Lambda _k$ .\",\n \"This construction yields a sequence of windows of the type $R^{\\\\prime }_{i}, R_{i}, R^{\\\\prime }_{a},$ in the first case, or $R^{\\\\prime }_{j}, R_{j}, R^{\\\\prime \\\\prime \\\\prime }_{j}, R^{\\\\prime \\\\prime }_{j}, R^{\\\\prime }_{a},$ in the second case.\",\n \"In the first case, $R^{\\\\prime }_{i}$ is correctly aligned with $R_{i}$ under some power $f_{\\\\mid \\\\Lambda }^{K_{i}}$ of the inner map, and $R_{i}$ is correctly aligned with $R^{\\\\prime }_{a}$ under the outer map $S$ .\",\n \"The exit direction of $R^{\\\\prime }_{i}$ is across the torus $T_{i}$ , and its exit set components lie on some invariant primary tori that are $\\\\varepsilon _i$ -close to $T_{i}$ .\",\n \"The windows $R^{\\\\prime }_i,R_i$ are contained in an $\\\\varepsilon _i$ neighborhood of $T_i$ .\",\n \"In the second case, $R^{\\\\prime }_{j}$ is correctly aligned with $R_{j}$ under some power $f_{\\\\mid \\\\Lambda }^{K_{j}}$ of the inner map, $R_{j}$ is correctly aligned with $R^{\\\\prime \\\\prime \\\\prime }_{j}$ under the outer map $S$ , $R^{\\\\prime \\\\prime \\\\prime }_{j}$ is correctly aligned with $R^{\\\\prime \\\\prime }_{j}$ under some power $f_{\\\\mid \\\\Lambda }^{K_{a}}$ of the inner map, and $R^{\\\\prime \\\\prime }_{j}$ is correctly aligned with $R^{\\\\prime }_{a}$ under the outer map $S$ .\",\n \"The exit direction of $R_{j}$ is across the torus $T_{j}$ , and its exit set components lie on some invariant primary tori that are $\\\\varepsilon _j$ -close to $T_{j}$ .\",\n \"The windows $R^{\\\\prime }_j,R_j$ are contained in an $\\\\varepsilon _j$ neighborhood of $T_j$ , and the windows $R^{\\\\prime \\\\prime \\\\prime }_j,R^{\\\\prime \\\\prime }_j$ are contained in an $\\\\varepsilon _i$ neighborhood of $T_i$ .\",\n \"We proceed with a similar construction on the other side of the BZI between $\\\\Lambda _k$ , that is, on the same side of the BZI as $T_b$ .\",\n \"We have already defined the window $R_b$ about $T_b$ that $R^{\\\\prime }_a$ is correctly aligned with $R_b$ under $f_{\\\\mid \\\\Lambda }^{K_{ab}}$ .\",\n \"Starting with the window $R_{b}$ and moving forward along the transition chain $T_b,T_k,T_l$ , we construct a sequence of windows of the type $R_{b}, R^{\\\\prime }_{k} $ or of the type $R_{b}, R^{\\\\prime \\\\prime \\\\prime }_{k}, R^{\\\\prime \\\\prime }_{k}, R^{\\\\prime }_{l},$ satisfying the correct alignment conditions below.\",\n \"In the first case, $R_{b}$ is correctly aligned with $R^{\\\\prime }_{k}$ under the outer map $S$ .\",\n \"The exit direction of $R^{\\\\prime }_k$ is across $T_k$ , and the exit set components lie on two invariant primary tori $\\\\varepsilon _k$ -close to $T_k$ .\",\n \"Moreover, $R^{\\\\prime }_{k}$ is contained in an $\\\\varepsilon _k$ -neighborhood of $T_k$ .\",\n \"In the second case, $R^{\\\\prime }_{b}$ is correctly aligned with $R^{\\\\prime \\\\prime \\\\prime }_{k}$ under the outer map $S$ , $R^{\\\\prime \\\\prime \\\\prime }_{k}$ is correctly aligned with $R^{\\\\prime \\\\prime }_{k}$ under some power $f_{\\\\mid \\\\Lambda }^{K_b}$ of the inner map, and $R^{\\\\prime \\\\prime }_{k}$ is correctly aligned with $R^{\\\\prime }_{l}$ under the outer map $S$ .\",\n \"The exit direction of $R^{\\\\prime }_{l}$ is across $T_{l}$ , and its exit set components lie on some invariant primary tori that are $\\\\varepsilon _l$ -close to $T_{l}$ .\",\n \"The windows $R^{\\\\prime \\\\prime \\\\prime }_{k}, R^{\\\\prime \\\\prime }_{k}$ are contained in an $\\\\varepsilon _k$ -neighborhood of $T_k$ , and $R^{\\\\prime }_l$ is contained in an $\\\\varepsilon _l$ -neighborhood of $T_l$ .\",\n \"Similar statements apply when instead of windows around $T_{i},T_{j},T_{k},T_{l}$ we construct windows about $T_{i^{\\\\prime }},T_{j^{\\\\prime }},T_{k^{\\\\prime }},T_{l^{\\\\prime }}$ , respectively.\",\n \"The conclusion of this section is that, by combining a sequence of correctly aligned window of the type (REF ) or (REF ) with a sequence of correctly aligned window of the type (REF ) or (REF ) we obtain a finite sequence of correctly aligned windows that crosses the BZI $\\\\Lambda _k$ .\",\n \"The shadowing lemma-type of result Lemma REF yields an orbit that visits some prescribed neighborhood in the phase space of each window in $\\\\Lambda $ .\",\n \"In particular, the shadowing orbit has points that go close to the transition tori.\"\n ],\n [\n \"Construction of correctly aligned windows across annular regions separated by invariant tori\",\n \"We consider an annular region in $\\\\Lambda $ between two transition chains of invariant tori.\",\n \"Inside this annular region, we assume the existence of a finite collection of invariant tori that separate the region, as in (A6$^{\\\\prime }$ -i), and are vertically ordered as in (A6$^{\\\\prime }$ -ii).\",\n \"Thus, the annular region between the transition chains is not a BZI.\",\n \"We also assume that the scattering map satisfies the non-degeneracy condition (A6$^{\\\\prime }$ -iii) on these invariant tori.\",\n \"The situation presented in this section is non-generic.\",\n \"Assume that the annular region in $\\\\Lambda $ is bounded by a pair of invariant Lipschitz tori $T_{a}$ and $T_{b}$ .\",\n \"Let $T_a$ be the end torus of the transition chain of one side, and $T_b$ be the end torus of the transition chain on the other side.\",\n \"We assume that inside the region in the annulus bounded by $T_a$ and $T_b$ there exist a finite collection of invariant tori $\\\\lbrace \\\\Upsilon _{h}\\\\rbrace _{h=1,\\\\ldots ,k-1}$ .\",\n \"Each $\\\\Upsilon _{h}$ is either an isolated invariant primary torus, or consists of a hyperbolic periodic orbit together with the upper branches, or with the lower branches, of its stable and unstable manifolds; the stable and unstable manifolds are assumed to coincide.\",\n \"In the second case, there is another invariant torus, say $\\\\Upsilon _{h+1}$ , formed by the remaining branches of the same hyperbolic periodic orbit as for $\\\\Upsilon _{h}$ .\",\n \"$\\\\Upsilon _{h+1}$ is assumed to be above $\\\\Upsilon _{h}$ , relative to the $I$ -coordinate, and is assumed to share with $\\\\Upsilon _{h}$ only the points of the periodic orbit.\",\n \"The region in the annulus bounded by $\\\\Upsilon _{h}$ and $\\\\Upsilon _{h+1}$ is referred to as a resonant region.\",\n \"Thus, the region in the annulus between $T_a$ and $T_b$ is divided into a finite number of BZI's and resonant regions.\",\n \"The constructions in Subsection REF provide a one-sided neighborhood of the type $(S\\\\circ f_{\\\\mid \\\\Lambda }^{K_a})(D_{iji^{\\\\prime }j^{\\\\prime }})$ of some point in $T_a$ , and a one-sided neighborhood $(S^{-1}\\\\circ f_{\\\\mid \\\\Lambda }^{-K_b})(D_{klk^{\\\\prime }l^{\\\\prime }})$ of some point in $T_b$ .\",\n \"Let us denote $D_0=(S\\\\circ f_{\\\\mid \\\\Lambda }^{K_a})(D_{iji^{\\\\prime }j^{\\\\prime }})$ and $D_{k+1}=(S^{-1}\\\\circ f_{\\\\mid \\\\Lambda }^{-K_b})(D_{klk^{\\\\prime }l^{\\\\prime }})$ .\",\n \"Assume that $\\\\Upsilon _1$ is an isolated invariant primary torus.\",\n \"We have that $S^{-1}(\\\\Upsilon _{1})$ forms with $\\\\Upsilon _{1}$ a topological disk $D_1$ between $T_a$ and $\\\\Upsilon _1$ , which is mapped by $S$ onto a topological disk $S(D_{1})$ between $\\\\Upsilon _1$ and $\\\\Upsilon _{2}$ .\",\n \"Theorem REF provides us a trajectory that starts from $\\\\partial D_0$ and ends at $\\\\partial D_1$ .\",\n \"In particular, there exist $K_1>0$ and a component of $f^{K_1}(D_{0})\\\\cap D_1$ that is a topological disk $D^{\\\\prime }_1$ whose boundary contains an arc of $S^{-1}(T_1)$ .\",\n \"The image of $D^{\\\\prime }_1$ under $S$ is a one-sided neighborhood $D^{\\\\prime \\\\prime }_1\\\\subseteq S(D_1)$ of some point in $\\\\Upsilon _1$ , such that $D^{\\\\prime \\\\prime }_1$ is contained in the region between $\\\\Upsilon _1$ and $\\\\Upsilon _2$ .\",\n \"The boundary of $S(D^{\\\\prime }_1)$ consists of an arc in $\\\\Upsilon _1$ and an arc in $(S\\\\circ f^{K_1}(\\\\partial D_{0}))$ .\",\n \"See Fig.\",\n \"REF .\",\n \"Figure: Crossing over an isolated invariant primary torus.Now assume that $\\\\Upsilon _1$ consists of a hyperbolic periodic orbit, together with the lower branches of its stable and unstable manifolds, and that $\\\\Upsilon _2$ consists of the same hyperbolic periodic orbit, together with the upper branches of the stable and unstable manifolds.\",\n \"The stable and unstable manifolds are assumed to coincide.\",\n \"Excepting for the common points, $\\\\Upsilon _{1}$ is below $\\\\Upsilon _{2}$ .\",\n \"Thus, $\\\\Upsilon _1$ and $\\\\Upsilon _2$ enclose a resonant region within the annulus.\",\n \"We have that $S^{-1}(\\\\Upsilon _{1})$ forms with $\\\\Upsilon _{1}$ a topological disk $D_1$ between $T_a$ and $\\\\Upsilon _1$ , which is mapped by $S$ onto a topological disk $S(D_{1})$ between $\\\\Upsilon _1$ and $\\\\Upsilon _{2}$ .\",\n \"We also have that $S^{-1}(\\\\Upsilon _{2})$ forms with $\\\\Upsilon _{2}$ a topological disk $D_2$ between $\\\\Upsilon _1$ and $\\\\Upsilon _2$ , which is mapped by $S$ onto a topological disk $S(D_{2})$ between $\\\\Upsilon _2$ and $\\\\Upsilon _3$ .\",\n \"By Theorem REF there exist $K_1>0$ and a component of $f^{K_1}(D_{0})\\\\cap D_1$ that is a topological disk $D^{\\\\prime }_1 $ whose boundary contains an arc of $S^{-1}(\\\\Upsilon _1)$ .\",\n \"The image of $D^{\\\\prime }_1$ under $S$ is a one-sided neighborhood $D^{\\\\prime \\\\prime }_1\\\\subseteq S(D_1)$ of some point in $\\\\Upsilon _1$ , contained in the region between $\\\\Upsilon _1 $ and $\\\\Upsilon _2$ .\",\n \"The boundary of ${D^{\\\\prime \\\\prime }}_1$ consists of an arc in $\\\\Upsilon _1$ and an arc in $S(f^{K_1}(\\\\partial D_{0}))$ .\",\n \"By the argument in the Homoclinic Orbit Theorem (see e.g.\",\n \"[8]), there exists $N_1$ such that $f^{N_1}({D^{\\\\prime \\\\prime }}_1)$ intersects $D_2$ .\",\n \"Let ${D^{\\\\prime }}_2$ be a component of this intersection that is a topological disk, and whose boundary contains an arc of $S^{-1}(\\\\Upsilon _2)$ .\",\n \"Then the image of ${D^{\\\\prime }}_2$ under $S$ is a one sided neighborhood $D^{\\\\prime \\\\prime }_2$ of some point in $\\\\Upsilon _2$ that is contained in the region between $\\\\Upsilon _2$ and $\\\\Upsilon _3$ .\",\n \"The boundary of $D^{\\\\prime \\\\prime }_2$ consists of an arc in $\\\\Upsilon _2$ and an arc in $(S\\\\circ f^{N_1}\\\\circ S\\\\circ f^{K_1})(\\\\partial D_{0})$ .\",\n \"See Figure REF .\",\n \"Figure: Construction of windows across a resonance.The main point of this construction is that, using the inner and outer dynamics, one can cross the successive BZI's and resonance regions determined by $\\\\lbrace \\\\Upsilon _h\\\\rbrace _{h=1,\\\\ldots ,k}$ one at a time, and obtain at each step a one-sided neighborhood of some point in some $\\\\Upsilon _h$ , which is between $\\\\Upsilon _h$ and $\\\\Upsilon _{h+1}$ , such that the boundary of that neighborhood contains the image of $\\\\partial D_0$ under a suitable composition of $S$ and powers of $f$ .\",\n \"This argument can be repeated for each invariant torus $\\\\Upsilon _h$ with $2\\\\le h\\\\le k$ , yielding a point $x_a\\\\in \\\\partial D_{0}$ that is mapped by some appropriate composition of $S$ and powers of $f$ onto a point $x_b\\\\in \\\\partial D_{k+1}$ .\",\n \"Then the constructions from Subsection REF can be applied to obtain a window $R^{\\\\prime }_a$ about $\\\\partial D_{0}$ , and a window $R^{\\\\prime }_b$ about $\\\\partial D_{k+1}$ , such that $R^{\\\\prime }_a$ is correctly aligned with $R_b$ under the appropriate composition of $S$ and powers of $f$ .\",\n \"Then there exists a shadowing orbit that goes from a neighborhood of $R_a$ in $M$ to a neighborhood of $R^{\\\\prime }_b$ in $M$ .\",\n \"If the region between $T_a$ and $T_b$ contains some prescribed collection of Aubry-Mather sets, then Theorem REF yields a shadowing orbit that crosses the region between $T_a$ and $T_b$ and follows the prescribed Aubry-Mather sets as in (REF ).\"\n ],\n [\n \"Construction of correctly aligned windows along transition chains of tori\",\n \"We consider a finite transition chain of invariant primary tori $\\\\lbrace T_1,T_2,\\\\ldots , T_n\\\\rbrace $ in the annulus $\\\\Lambda $ satisfying (A5).\",\n \"All tori $T_k$ , $k=1, \\\\ldots , n$ , intersect the subset $U^-_+$ of the domain of the scattering map $S$ where $S$ moves points upwards in the annulus $\\\\Lambda $ .\",\n \"The motion on each torus $T_k$ is topologically transitive.\",\n \"Each torus $T_k$ with $2\\\\le k\\\\le n-1$ is an `interior' torus, i.e.\",\n \"it can be $C^0$ -approximated from both sides in $\\\\Lambda $ by other invariant primary tori.\",\n \"For each $k\\\\in \\\\lbrace 1,\\\\ldots ,n-1\\\\rbrace $ , the image $S(T_k)$ of the torus $T_k$ under the scattering map $S$ has a topologically transverse intersection with the torus $T_{k+1}$ .\",\n \"We would like to show that there exists an orbit that visits some $\\\\varepsilon _k$ -neighborhood of each torus $T_k$ , $k=1,\\\\ldots ,n$ , in the prescribed order.\",\n \"To this end we will construct a sequence of 2-dimensional windows in $\\\\Lambda $ that are correctly aligned one with another under the scattering map alternatively with some iterates of the inner map, as in Lemma REF .\",\n \"Then the lemma will imply the existence of a shadowing orbit to the transition chain.\",\n \"For each $k\\\\in \\\\lbrace 1,\\\\ldots ,n-1\\\\rbrace $ , we choose and fix a pair of points $x^-_{k,k+1}\\\\in T_k$ and $x^+_{k,k+1}\\\\in T_{k+1}$ such that $S(x^-_{k,k+1})=x^+_{k,k+1}$ and $S(T_{k})$ intersects $T_{k+1}$ transversally at $x^+_{k,k+1}$ .\",\n \"We construct inductively a sequence of correctly aligned windows in $\\\\Lambda $ along the tori $\\\\lbrace T_1,T_2,\\\\ldots , T_n\\\\rbrace $ , such that each window is correctly aligned with the next window in the sequence either by the outer map or by some sufficiently large power of the inner map.\",\n \"Moreover, each window will be contained in some $\\\\varepsilon $ -neighborhood of some transition torus.\",\n \"We start the inductive construction at $T_1$ .\",\n \"Consider the point $x^-_{1,2}\\\\in T_1$ with $S(x^-_{1,2})=x^+_{1,2}\\\\in T_{2}$ as above.\",\n \"We construct a window $R^{\\\\prime }_1$ about $T_1$ as follows.\",\n \"Let $\\\\bar{e}^{\\\\prime }_1$ be an arc contained in $T_1$ , and $\\\\chi ^{\\\\prime }_1:\\\\mathbb {R}^2\\\\rightarrow \\\\Lambda $ a $C^0$ -local parametrization such that $\\\\chi ^{\\\\prime }_1([0,1]\\\\times \\\\lbrace 0\\\\rbrace )=\\\\bar{e}^{\\\\prime }_1$ .\",\n \"Then we define $R^{\\\\prime }_{1}=\\\\chi ^{\\\\prime }_1([0,1]\\\\times [-\\\\delta ^{\\\\prime }_1,\\\\delta ^{\\\\prime }_1]),\\\\\\\\ {R^{\\\\prime }}_{1}^\\\\textrm {exit}=\\\\chi ^{\\\\prime }_1(\\\\partial [0,1]\\\\times [-\\\\delta ^{\\\\prime }_1,\\\\delta ^{\\\\prime }_1]),$ where $0<\\\\delta ^{\\\\prime }_1<\\\\varepsilon _1$ .\",\n \"We choose $\\\\bar{e}^{\\\\prime }_1$ and $\\\\delta ^{\\\\prime }_1$ sufficiently small so that and $S(\\\\bar{e}^{\\\\prime }_1)$ intersects $T_2$ only at $x^+_{1,2}$ , and also so that $R^{\\\\prime }_1$ defined as above is contained in ${U}^-_+$ .\",\n \"The exit direction of $R^{\\\\prime }_1$ is in the direction of the torus $T_1$ .\",\n \"The image $S(R^{\\\\prime }_1)$ of $R^{\\\\prime }_1$ under the scattering map is a topological rectangle.\",\n \"Since $S(\\\\bar{e}^{\\\\prime }_1)$ is transverse to $T_2$ at $x^+_{1,2}$ , then by choosing $\\\\delta ^{\\\\prime }_1$ sufficiently small we ensure that the components of $S({R^{\\\\prime }}_{1}^\\\\textrm {exit})$ lie on opposite sides of $T_2$ .\",\n \"Thus, the exit direction of $S(R^{\\\\prime }_1)$ is across the torus $T_2$ .\",\n \"Next we consider the point $x^-_{2,3}\\\\in T_2$ with $S(x^-_{2,3})=x^+_{2,3}\\\\in T_{3}$ as above.\",\n \"We construct a window $R^{\\\\prime }_2$ about $T_2$ in a manner similar to $R^{\\\\prime }_1$ : $R^{\\\\prime }_{2}=\\\\chi ^{\\\\prime }_2([0,1]\\\\times [-\\\\delta ^{\\\\prime }_2,\\\\delta ^{\\\\prime }_2]),\\\\\\\\ {R^{\\\\prime }}_{2}^\\\\textrm {exit}=\\\\chi ^{\\\\prime }_2(\\\\partial [0,1]\\\\times [-\\\\delta ^{\\\\prime }_2,\\\\delta ^{\\\\prime }_2]),$ where $\\\\bar{e}^{\\\\prime }_2$ is an arc contained in $T_2$ , $\\\\chi ^{\\\\prime }_2:\\\\mathbb {R}^2\\\\rightarrow \\\\Lambda $ is a $C^0$ -local parametrization such that $\\\\chi ^{\\\\prime }_2([0,1]\\\\times \\\\lbrace 0\\\\rbrace )=\\\\bar{e}^{\\\\prime }_2$ , and $0<\\\\delta ^{\\\\prime }_2<\\\\varepsilon _2$ is chosen sufficiently small.\",\n \"Now we construct a window $R_2$ about the point $x^+_{1,2}$ such that $R^{\\\\prime }_1$ is correctly aligned with $R_2$ under $S$ and $R_2$ is correctly aligned with $R^{\\\\prime }_2$ under some power of $f_{\\\\mid \\\\Lambda }$ .\",\n \"We choose an arc $\\\\bar{e}_2$ in $T_2$ containing $x^+_{1,2}$ such that $\\\\bar{e}_2 \\\\supseteq S(R^{\\\\prime }_1)\\\\cap T_2$ .\",\n \"We choose a pair of invariant tori $T_2^{\\\\rm lower}$ and $T_2^{\\\\rm upper}$ such that $T_2^{\\\\rm lower} \\\\prec T_2 \\\\prec T_2^{\\\\rm upper}$ and the exit set components of $S(R^{\\\\prime }_1)$ , as well as the entry set components of $R^{\\\\prime }_2$ , are outside of the annulus bounded by $T_2^{\\\\rm lower}$ and $T_2^{\\\\rm upper}$ , on the both sides of the annulus.\",\n \"Furthermore, we require that $T_2^{\\\\rm lower}$ and $T_2^{\\\\rm upper}$ should be $\\\\varepsilon _2$ -close to $T_2$ in the $C^0$ -topology.\",\n \"The existence of such neighboring tori $T_2^{\\\\rm lower}$ and $T_2^{\\\\rm upper}$ to $T_2$ is guaranteed by (A5-iv).\",\n \"Let $\\\\chi _2:\\\\mathbb {R}^2\\\\rightarrow \\\\Lambda $ be a $C^0$ -local parametrization such that $\\\\chi _2( \\\\lbrace 0\\\\rbrace \\\\times [0,1])=\\\\bar{e}_2$ , $\\\\chi _2( \\\\lbrace -\\\\delta _2\\\\rbrace \\\\times [0,1])\\\\subseteq T_2^{\\\\rm lower}$ and $\\\\chi _2( \\\\lbrace \\\\delta _2\\\\rbrace \\\\times [0,1])\\\\subseteq T_2^{\\\\rm upper}$ , for some $0<\\\\delta _2<\\\\varepsilon _2$ sufficiently small.\",\n \"Define $R_{2}=\\\\chi _2([-\\\\delta _2,\\\\delta _2]\\\\times [0,1]),\\\\\\\\ {R}_{2}^\\\\textrm {exit}=\\\\chi _2(\\\\partial [-\\\\delta _2,\\\\delta _2]\\\\times [0,1]).$ The exit set components of $R_2$ lie on the invariant tori $T_2^{\\\\rm lower},T_2^{\\\\rm upper}$ neighboring $T_2$ .\",\n \"Since the motion on the torus $T_2$ is topological transitive, there exists a power $f_{\\\\mid \\\\Lambda }^{K_2}$ of $f_{\\\\mid \\\\Lambda }$ such that $R_2$ is correctly aligned with $R^{\\\\prime }_2$ under $f_{\\\\mid \\\\Lambda }^{K_2}$ .\",\n \"See Figure REF .\",\n \"We have obtained the windows $R^{\\\\prime }_1$ about $T_1$ and $R^{\\\\prime }_2,R_2$ about $T_2$ such that $R^{\\\\prime }_1$ is correctly aligned with $R_2$ under $S$ and $R_2$ is correctly aligned with $R^{\\\\prime }_2$ under some power of $f_{\\\\mid \\\\Lambda }$ .\",\n \"This ends the initial step of the inductive construction.\",\n \"The inductive step goes on similarly, yielding the windows $R^{\\\\prime }_k$ about $T_k$ and $R^{\\\\prime }_{k+1},R_{k+1}$ about $T_{k+1}$ such that $R^{\\\\prime }_k$ is correctly aligned with $R_{k+1}$ under $S$ and $R_{k+1}$ is correctly aligned with $R^{\\\\prime }_{k+1}$ under some power of $f_{\\\\mid \\\\Lambda }$ .\",\n \"See Figure REF .\",\n \"Figure: Construction of windows along a transition chain.The construction proceeds inductively until a sequence of windows in $\\\\Lambda $ of the following type is obtained $ R^{\\\\prime }_1,R_2,R^{\\\\prime }_2, \\\\ldots , R_k,R^{\\\\prime }_k, R_{k+1}, R^{\\\\prime }_{k+1}, \\\\ldots , R_n, $ where for each $k\\\\in \\\\lbrace 1,\\\\ldots ,n-1\\\\rbrace $ we have that $R^{\\\\prime }_k$ is correctly aligned with $R_{k+1}$ under $S$ , and $R_{k+1}$ is correctly aligned with $R^{\\\\prime }_{k+1}$ under $f_{\\\\mid \\\\Lambda }^{K_{k+1}}$ for some $K_{k+1}>0$ .\",\n \"Each window $R_k,R^{\\\\prime }_k$ is contained in an $\\\\varepsilon _k$ -neighborhood of the torus $T_k$ .\",\n \"Applying the shadowing lemma-type of result Lemma REF provides an orbit that visits an $\\\\varepsilon _k$ -neighborhood in the phase space of each window in the sequence, and in particular of each torus in the transition chain.\"\n ],\n [\n \"Gluing correctly aligned windows across BZI's with correctly aligned windows along transition chains of tori\",\n \"We consider three transition chains of tori $\\\\lbrace T_i\\\\rbrace _{i=i_{k-1}+1,\\\\ldots , i_{{k}}}$ , $\\\\lbrace T_i\\\\rbrace _{i=i_k+1,\\\\ldots , i_{k+1}}$ , $\\\\lbrace T_i\\\\rbrace _{i=i_{k+1}+1,\\\\ldots , i_{{k+2}}}$ , with the property that each one of the regions between $T_{i_{k}}$ and $T_{i_{{k}}+1}$ , and between $T_{i_{k+1}}$ and $T_{i_{k+1}+1}$ , is either a BZI as in (A6), or it contains a finite number of invariant tori that separate the region as in (A6$^{\\\\prime }$ ).\",\n \"Inside each region there is a prescribed collection of Aubry-Mather sets as in (A7) The constructions in Subsection REF and in Subsection REF yield correctly aligned windows in $\\\\Lambda $ that cross the region between $T_{i_{k}}$ and $T_{i_{{k}}+1}$ , and correctly aligned windows in $\\\\Lambda $ that cross the region between $T_{i_{k+1}}$ and $T_{i_{k+1}+1}$ .\",\n \"The construction in Subsection REF yield sequences of correctly aligned windows along the adjacent transition chains.\",\n \"The choices of the windows constructed along the transition chains depend on the choices of the windows that cross the region between $T_{i_{k}}$ and $T_{i_{{k}}+1}$ , and of the windows that cross the region between $T_{i_{k+1}}$ and $T_{i_{k+1}+1}$ .\",\n \"Propagating the construction of windows starting from $T_{i_k+1}$ and moving forward along the transition chain $\\\\lbrace T_i\\\\rbrace _{i=i_k+1,\\\\ldots , i_{k+1}}$ , and the construction of windows starting from $T_{i_{k+1}}$ and moving backward along the same transition chain $\\\\lbrace T_i\\\\rbrace _{i=i_k+1,\\\\ldots , i_{k+1}}$ , may result in a pair of windows about some intermediate torus that are not correctly aligned.\",\n \"We would like to glue this sequences of windows in a manner that is correctly aligned, without having to revise the windows constructed to that point.\",\n \"Assume that $T_j$ is one of the tori $\\\\lbrace T_i\\\\rbrace _{i=i_k+1,\\\\ldots , i_{k+1}}$ , with $j\\\\in \\\\lbrace i_k+2,\\\\ldots , i_{k+1}-1\\\\rbrace $ .\",\n \"Assume that one has already constructed a window $R_j$ about $T_j$ by propagating the construction from $T_{i_k+1}$ and moving forward along the transition chain, and another window $R^{\\\\prime }_j$ about $T_j$ by propagating the construction from $T_{i_{k+1}}$ and moving backwards along the transition chain.\",\n \"The window $R_j$ is of the form $R _{j}=\\\\chi _j ([-\\\\delta _j,\\\\delta _j]\\\\times [0,1]),\\\\\\\\{R }_{j}^\\\\textrm {exit}=\\\\chi _j (\\\\partial [-\\\\delta _j,\\\\delta _j]\\\\times [0,1]),$ where $\\\\chi _{j}:\\\\mathbb {R}^2\\\\rightarrow \\\\Lambda $ is a $C^0$ -local parametrization with $\\\\chi _{j}(\\\\lbrace 0\\\\rbrace \\\\times [0,1])\\\\subseteq T_j$ , and $\\\\chi _j (\\\\lbrace -\\\\delta _j\\\\rbrace \\\\times [0,1])\\\\subseteq T_j^{\\\\rm lower}$ and $\\\\chi _j (\\\\lbrace \\\\delta _j\\\\rbrace \\\\times [0,1])\\\\subseteq T_j^{\\\\rm upper}$ , where $T_j^{\\\\rm lower}$ and $T_j^{\\\\rm upper}$ are two primary invariant tori on the opposite sides of $T_j$ .\",\n \"The window $R^{\\\\prime }_j$ is of the form $R^{\\\\prime }_{j}=\\\\chi ^{\\\\prime }_j ([0,1]\\\\times [-\\\\delta ^{\\\\prime }_j,\\\\delta ^{\\\\prime }_j]),\\\\\\\\{R^{\\\\prime }}_{j}^\\\\textrm {exit}=\\\\chi ^{\\\\prime }_j (\\\\partial [0,1]\\\\times [-\\\\delta ^{\\\\prime }_j,\\\\delta ^{\\\\prime }_j]),$ where $\\\\chi ^{\\\\prime }_{j}:\\\\mathbb {R}^2\\\\rightarrow \\\\Lambda $ is a $C^0$ -local parametrization with $\\\\chi ^{\\\\prime }_{j}([0,1]\\\\times \\\\lbrace 0\\\\rbrace )\\\\subseteq T_j$ , and $\\\\chi ^{\\\\prime }_j ([0,1]\\\\times \\\\lbrace -\\\\delta ^{\\\\prime }_j\\\\rbrace )\\\\subseteq T_j^{\\\\rm lower}$ and $\\\\chi ^{\\\\prime }_j ([0,1]\\\\times \\\\lbrace \\\\delta ^{\\\\prime }_j\\\\rbrace )\\\\subseteq {T^{\\\\prime }}_j^{\\\\rm upper}$ , where ${T^{\\\\prime }}_j^{\\\\rm lower}$ and ${T^{\\\\prime }}_j^{\\\\rm upper}$ are two primary invariant tori on the opposite sides of $T_j$ .\",\n \"Let us assume that the annular region between ${T^{\\\\prime }}_j^{\\\\rm lower}$ and ${T^{\\\\prime }}_j^{\\\\rm upper}$ is inside the region between $T_j^{\\\\rm lower}$ and $T_j^{\\\\rm upper}$ .\",\n \"We construct a new window $R^{\\\\prime \\\\prime }_j$ about $T_j$ , such that $R_j$ is correctly aligned with $R^{\\\\prime \\\\prime }_j$ under the identity map, and $R^{\\\\prime \\\\prime }_j$ is correctly aligned with $R^{\\\\prime }_j$ under some power of $f$ .\",\n \"We let $R^{\\\\prime \\\\prime }_j$ is of the form $R^{\\\\prime \\\\prime }_{j}=\\\\chi ^{\\\\prime \\\\prime }_j ([-\\\\delta ^{\\\\prime \\\\prime }_j,\\\\delta ^{\\\\prime \\\\prime }_j]\\\\times [0,1]),\\\\\\\\{R^{\\\\prime \\\\prime }}_{j}^\\\\textrm {exit}=\\\\chi ^{\\\\prime \\\\prime }_j (\\\\partial [-\\\\delta ^{\\\\prime \\\\prime }_j,\\\\delta ^{\\\\prime \\\\prime }_j]\\\\times [0,1]),$ where $\\\\chi ^{\\\\prime \\\\prime }_{j}:\\\\mathbb {R}^2\\\\rightarrow \\\\Lambda $ is a $C^0$ -local parametrization with $\\\\chi ^{\\\\prime \\\\prime }_{j}(\\\\lbrace 0\\\\rbrace \\\\times [0,1])\\\\supseteq \\\\chi ^{\\\\prime }_{j}(\\\\lbrace 0\\\\rbrace \\\\times [0,1])$ , and $\\\\chi ^{\\\\prime \\\\prime }_j (\\\\lbrace -\\\\delta ^{\\\\prime \\\\prime }_j\\\\rbrace \\\\times [0,1])\\\\subseteq {T^{\\\\prime }}_j^{\\\\rm lower}$ and $\\\\chi ^{\\\\prime \\\\prime }_j (\\\\lbrace \\\\delta ^{\\\\prime \\\\prime }_j\\\\rbrace \\\\times [0,1])\\\\subseteq {T^{\\\\prime }}_j^{\\\\rm upper}$ , for some $\\\\delta ^{\\\\prime \\\\prime }_j>0$ .\",\n \"Since the motion on the torus $T_j$ is topological transitive, there exists a power $f_{\\\\mid \\\\Lambda }^{K_j}$ of $f_{\\\\mid \\\\Lambda }$ such that $R^{\\\\prime \\\\prime }_j$ is correctly aligned with $R^{\\\\prime }_j$ under $f_{\\\\mid \\\\Lambda }^{K_j}$ .\",\n \"See Figure REF .\",\n \"We have obtained that $R_j$ is correctly aligned with $R^{\\\\prime \\\\prime }_j$ under the identity map, and $R^{\\\\prime \\\\prime }_j$ is correctly aligned with $R^{\\\\prime }_j$ under some power $f_{\\\\mid \\\\Lambda }^{K_j}$ of $f_{\\\\mid \\\\Lambda }$ .\",\n \"Figure: Gluing sequences of correctly aligned windows along a transition chain.The case when the annular region between $T_j^{\\\\rm lower}$ and $T_j^{\\\\rm upper}$ is inside the region between ${T^{\\\\prime }}_j^{\\\\rm lower}$ and ${T^{\\\\prime }}_j^{\\\\rm upper}$ results in the window $R_j$ correctly aligned with $R^{\\\\prime }_j$ under some power of $f$ , without the need of creating an intermediate window $R^{\\\\prime \\\\prime }_j$ .\",\n \"The case when neither annular region is contained in the other annular region can be dealt similarly by constructing an intermediate window $R^{\\\\prime \\\\prime }_j$ .\",\n \"Through this process, a sequence of correctly windows constructed forward along a transition chain can be glued, in a correctly aligned manner, with a sequence of correctly windows constructed backwards along the same transition chain.\"\n ],\n [\n \"Proof of Theorem \",\n \"To summarize, in Subsection REF and in Subsection REF , we described the construction of correctly aligned windows that cross an annular regions between two consecutive transition chains of invariant tori.\",\n \"These annular regions are BZI's or else they are separated by some finite collection of invariant tori.\",\n \"If each annular region has some prescribed collection od Aubry-Mather sets, the construction yields windows that follow these Aubry-Mather sets.\",\n \"In Subsection REF yield sequences of correctly aligned windows each transition chain.\",\n \"In Subsection REF the construction of a sequence of correctly windows constructed forward along a transition chain can be glued, in a correctly aligned manner, with a the construction of a sequence of correctly windows constructed backwards along the same transition chain.\",\n \"Thus, starting from some initial annular region, one can construct sequences of correctly aligned windows, forward and backwards, along infinitely many topological transition chains interspersed with annular regions.\",\n \"The Shadowing Lemma-type of result Theorem REF implies the existence of an orbit that gets $\\\\varepsilon _i$ -close to each transition torus $T_i$ , and also follows each Aubry-Mather set $\\\\Sigma _i$ for a prescribed time $n_i$ .\"\n ]\n]"},"id":{"kind":"string","value":"1204.1503"}}},{"rowIdx":1866897,"cells":{"text":{"kind":"list like","value":[["Stochastic Turing Patterns on a Network"],["Abstract The process of stochastic Turing instability on a network is discussed for a specific case study, the stochastic Brusselator model.","The system is shown to spontaneously differentiate into activator-rich and activator-poor nodes, outside the region of parameters classically deputed to the deterministic Turing instability.","This phenomenon, as revealed by direct stochastic simulations, is explained analytically, and eventually traced back to the finite size corrections stemming from the inherent graininess of the scrutinized medium."],["At the next-to-leading order approximation in the van Kampen expansion, one obtains a Fokker Planck equation for the probability distribution of fluctuations, which is equivalent to the Langevin equation (REF ).","For each fixed pair of nodes, $i$ , $j$ , the $2 \\times 2$ matrix $\\mathbf {\\mathcal {M}_{sr,ij}}$ may be decomposed as the sum of two contributions, one relative to the activator-inhibitor reactions (non spatial components (NS)), and the other associated to the diffusion process (spatial components (SP)): $\\mathbf {\\mathcal {M}_{sr,ij}}=\\mathbf {\\mathcal {M}_{sr}}^{(NS)}+\\mathbf {\\mathcal {M}_{sr}}^{(SP)}\\Delta _{ij}$ .","The above matrices are evaluated at the mean-field fixed points $\\phi ^{*}$ , $\\psi ^{*}$ .","After a lengthy calculation [16] one obtains the following entries for matrix $\\mathbf {\\mathcal {M}}^{(NS)}$ : $\\mathcal {M}^{(NS)}_{11}&=&-a-b-d+2c\\phi ^*\\psi ^*, \\nonumber \\\\\\mathcal {M}^{(NS)}_{12}&=&-a+c\\phi ^{*^2}, \\nonumber \\\\\\mathcal {M}^{(NS)}_{21}&=&b-2c\\phi ^*\\psi ^*, \\nonumber \\\\\\mathcal {M}^{(NS)}_{22}&=&-c\\phi ^{*^2}.","\\nonumber $ The elements of matrix $\\mathbf {\\mathcal {M}}^{(SP)}$ read instead: $\\mathcal {M}^{(SP)}_{11}&=&2\\mu (1-\\psi ^*), \\\\\\mathcal {M}^{(SP)}_{12}&=&2\\mu \\phi ^*, \\\\\\mathcal {M}^{(SP)}_{21}&=&2\\delta \\psi ^*, \\\\\\mathcal {M}^{(SP)}_{22}&=&2\\delta (1-\\phi ^*).$ As concerns the matrix $\\mathbf {\\mathcal {B}}$ , one finds: $\\mathcal {B}_{11,ii} &=& D_1 + \\tilde{k}_i H_1, \\\\\\mathcal {B}_{12,ii}&=&\\mathcal {B}_{21, ii}=C \\\\\\mathcal {B}_{22,ii}&=& D_2 + \\tilde{k}_i H_2 \\\\\\mathcal {B}_{11, ij}&=&-\\left(\\frac{1}{k_i}+\\frac{1}{k_j}\\right) H_1 \\\\\\mathcal {B}_{12, ij}&=&\\mathcal {B}_{21, ij}=0.","\\\\\\mathcal {B}_{22, ij}&=&-\\left(\\frac{1}{k_i}+\\frac{1}{k_j}\\right) H_2$ where: $D_1 &=& a(1-\\phi ^*-\\psi ^*)+\\phi ^*\\left(b+c\\phi ^*\\psi ^*+d\\right) \\\\H_1 &=& 4\\mu \\phi ^*(1-\\phi ^*-\\psi ^*) \\nonumber \\\\C &=& -\\phi ^*\\left(b+c\\phi ^*\\psi ^*\\right) \\nonumber \\\\D_2 &=& \\phi ^*\\left(b+c\\phi ^*\\psi ^*\\right) \\nonumber \\\\H_2 &=& 4\\delta \\psi ^*(1-\\phi ^*-\\psi ^*) \\nonumber \\\\$ Matrix $\\mathcal {B}$ can be hence cast in the equivalent form: $\\mathcal {B}_{ss,ij} &=& (D_s+\\tilde{k}_i H_s) \\delta _{ij} - \\left(\\frac{1}{k_i}+\\frac{1}{k_j}\\right) W_{ij} H_s \\nonumber \\\\\\mathcal {B}_{rs,ij} &=& \\mathcal {B}_{sr,ij} = C \\delta _{ij}$ for $r,s=1,2$ .","Performing the transformation (REF ) on both sides of the Langevin equation (REF ), we get $\\mathtt {j} \\omega \\tilde{\\xi }_{s}^\\alpha = \\sum _{r=1}^2\\left(\\mathcal {M}_{sr}^{(NS)}+\\mathcal {M}_{sr}^{(SP)}\\Lambda _\\alpha \\right)\\tilde{\\xi }_r^\\alpha +\\tilde{\\eta }_s^\\alpha $ where the term $\\tilde{\\eta }_s^\\alpha $ denotes the transform of the noise.","Introducing the matrix $\\Phi _{sr}=\\mathtt {j} \\omega \\delta _{sr}-\\left(\\mathcal {M}_{sr}^{(NS)}+\\mathcal {M}_{sr}^{(SP)}\\Lambda _\\alpha \\right)$ we get $\\tilde{\\xi }_s^\\alpha =\\sum _{r=1}^2 \\Phi _{sr}^{-1}\\tilde{\\eta }_r^\\alpha $ , and thus the power spectrum of the $s$ -th species is given by $P_s(\\omega ,\\Lambda _\\alpha )=\\langle |\\tilde{\\xi }^\\alpha _s|^2\\rangle =\\sum _{r,k=1}^2 \\Phi _{sr}^{-1}\\langle \\tilde{\\eta }^\\alpha _r \\tilde{\\eta }_k^{\\alpha }\\rangle (\\Phi _{ks}^{\\dag })^{-1}$ It can be shown that $\\langle \\tilde{\\eta }^\\alpha _r \\tilde{\\eta }_k^{\\alpha } \\rangle = \\sum _{i,j}\\mathcal {B}_{rk,ij}v_i^{(\\alpha )}v_{j}^{(\\alpha )} $ .","An explicit form for the $2\\times 2$ matrix $\\langle \\tilde{\\eta }^\\alpha _r \\tilde{\\eta }_k^{\\alpha } \\rangle $ can be derived by making use of Eqs (REF ).","Let us focus on the non trivial contribution $r=k$ : $&&\\sum _{i,j}\\mathcal {B}_{rr,ij}v_i^{(\\alpha )}v_{j}^{(\\alpha )} =\\\\ &&\\sum _{i,j}\\left[ (D_r+\\tilde{k}_i H_r) \\delta _{ij} - \\left(\\frac{1}{k_i}+\\frac{1}{k_j}\\right) W_{ij} H_r \\right]v_i^{(\\alpha )}v_{j}^{(\\alpha )} =\\\\&& D_r \\sum _{i} v_i^{(\\alpha )} v_{i}^{(\\alpha )} - H_r \\sum _{i,j} \\Delta _{i,j} v_i^{(\\alpha )} v_{i}^{(\\alpha )} =D_r - H_r \\Lambda _{\\alpha }$ where in the last step we made use of $\\sum _{i} v_i^{(\\alpha )} v_{i}^{(\\alpha )}=1$ and $\\sum _j \\Delta _{ij} v_j^{(\\alpha )}=\\Lambda _\\alpha v_i^{\\alpha }$ .","The component $r \\ne k$ is trivially equal to $C$ .","The power spectrum is fully specified as function of $\\Lambda _\\alpha $ and $\\omega $ .","Figure REF is obtained by setting $\\omega =0$ in the above formulae.","We emphasize that the same result can be recovered from the continuum medium power spectrum [16] provided $-k^2$ is replaced by the discrete eigenvalue $\\Lambda _{\\alpha }$ .","We end this Appendix by providing a list of explanatory captions to the movies annexed as supplementary material.","mf_inside_X.mov and mf_inside_Y.mov show respectively the time evolution of the mean field concentrations $\\phi _i$ and $\\psi _i$ .","The data are obtained by integrating the governing mean field equations (REF ).","Parameters are chosen so to yield a Turing instability (magenta circle in Fig.","REF ).","At time $t=0$ a small perturbation is applied to perturb the homogeneous fixed point.","Then, the system evolves toward a stable, non-homogeneous stationary configuration.","st_inside_X.mov and st_inside_Y.mov show the result of the stochastic simulations (fluctuating blue circles), for respectively $n_i/N$ and $m_i/N$ .","The red symbols refer to a late time snapshot of the deterministic dynamics.","Parameters are set as specified above: the system is hence inside the region of Turing instability.","Notice that the noise that takes the system away from the homogeneous fixed point is now endogenous to the system and not externally imposed.","st_outside_X.mov and st_outside_Y.mov report on the stochastic simulations (blue symbols), for respectively $n_i/N$ and $m_i/N$ .","The parameters are now assigned so to fall outside the region of Turing order (blue diamond in Fig.","REF ).","The mean field solutions are not destabilized and converge to the stable fixed point.","At variance, the stochastic simulations evolve toward a non homogeneous state.","Financial support from Ente Cassa di Risparmio di Firenze and the Program Prin2009 funded by Italian MIUR is acknowledged.","D.F.","thanks Tommaso Biancalani for pointing out reference [18] and for stimulating discussions.","F.D.P.","thanks Alessio Cardillo for providing the code to generate the network."]],"string":"[\n [\n \"Stochastic Turing Patterns on a Network\"\n ],\n [\n \"Abstract The process of stochastic Turing instability on a network is discussed for a specific case study, the stochastic Brusselator model.\",\n \"The system is shown to spontaneously differentiate into activator-rich and activator-poor nodes, outside the region of parameters classically deputed to the deterministic Turing instability.\",\n \"This phenomenon, as revealed by direct stochastic simulations, is explained analytically, and eventually traced back to the finite size corrections stemming from the inherent graininess of the scrutinized medium.\"\n ],\n [\n \"At the next-to-leading order approximation in the van Kampen expansion, one obtains a Fokker Planck equation for the probability distribution of fluctuations, which is equivalent to the Langevin equation (REF ).\",\n \"For each fixed pair of nodes, $i$ , $j$ , the $2 \\\\times 2$ matrix $\\\\mathbf {\\\\mathcal {M}_{sr,ij}}$ may be decomposed as the sum of two contributions, one relative to the activator-inhibitor reactions (non spatial components (NS)), and the other associated to the diffusion process (spatial components (SP)): $\\\\mathbf {\\\\mathcal {M}_{sr,ij}}=\\\\mathbf {\\\\mathcal {M}_{sr}}^{(NS)}+\\\\mathbf {\\\\mathcal {M}_{sr}}^{(SP)}\\\\Delta _{ij}$ .\",\n \"The above matrices are evaluated at the mean-field fixed points $\\\\phi ^{*}$ , $\\\\psi ^{*}$ .\",\n \"After a lengthy calculation [16] one obtains the following entries for matrix $\\\\mathbf {\\\\mathcal {M}}^{(NS)}$ : $\\\\mathcal {M}^{(NS)}_{11}&=&-a-b-d+2c\\\\phi ^*\\\\psi ^*, \\\\nonumber \\\\\\\\\\\\mathcal {M}^{(NS)}_{12}&=&-a+c\\\\phi ^{*^2}, \\\\nonumber \\\\\\\\\\\\mathcal {M}^{(NS)}_{21}&=&b-2c\\\\phi ^*\\\\psi ^*, \\\\nonumber \\\\\\\\\\\\mathcal {M}^{(NS)}_{22}&=&-c\\\\phi ^{*^2}.\",\n \"\\\\nonumber $ The elements of matrix $\\\\mathbf {\\\\mathcal {M}}^{(SP)}$ read instead: $\\\\mathcal {M}^{(SP)}_{11}&=&2\\\\mu (1-\\\\psi ^*), \\\\\\\\\\\\mathcal {M}^{(SP)}_{12}&=&2\\\\mu \\\\phi ^*, \\\\\\\\\\\\mathcal {M}^{(SP)}_{21}&=&2\\\\delta \\\\psi ^*, \\\\\\\\\\\\mathcal {M}^{(SP)}_{22}&=&2\\\\delta (1-\\\\phi ^*).$ As concerns the matrix $\\\\mathbf {\\\\mathcal {B}}$ , one finds: $\\\\mathcal {B}_{11,ii} &=& D_1 + \\\\tilde{k}_i H_1, \\\\\\\\\\\\mathcal {B}_{12,ii}&=&\\\\mathcal {B}_{21, ii}=C \\\\\\\\\\\\mathcal {B}_{22,ii}&=& D_2 + \\\\tilde{k}_i H_2 \\\\\\\\\\\\mathcal {B}_{11, ij}&=&-\\\\left(\\\\frac{1}{k_i}+\\\\frac{1}{k_j}\\\\right) H_1 \\\\\\\\\\\\mathcal {B}_{12, ij}&=&\\\\mathcal {B}_{21, ij}=0.\",\n \"\\\\\\\\\\\\mathcal {B}_{22, ij}&=&-\\\\left(\\\\frac{1}{k_i}+\\\\frac{1}{k_j}\\\\right) H_2$ where: $D_1 &=& a(1-\\\\phi ^*-\\\\psi ^*)+\\\\phi ^*\\\\left(b+c\\\\phi ^*\\\\psi ^*+d\\\\right) \\\\\\\\H_1 &=& 4\\\\mu \\\\phi ^*(1-\\\\phi ^*-\\\\psi ^*) \\\\nonumber \\\\\\\\C &=& -\\\\phi ^*\\\\left(b+c\\\\phi ^*\\\\psi ^*\\\\right) \\\\nonumber \\\\\\\\D_2 &=& \\\\phi ^*\\\\left(b+c\\\\phi ^*\\\\psi ^*\\\\right) \\\\nonumber \\\\\\\\H_2 &=& 4\\\\delta \\\\psi ^*(1-\\\\phi ^*-\\\\psi ^*) \\\\nonumber \\\\\\\\$ Matrix $\\\\mathcal {B}$ can be hence cast in the equivalent form: $\\\\mathcal {B}_{ss,ij} &=& (D_s+\\\\tilde{k}_i H_s) \\\\delta _{ij} - \\\\left(\\\\frac{1}{k_i}+\\\\frac{1}{k_j}\\\\right) W_{ij} H_s \\\\nonumber \\\\\\\\\\\\mathcal {B}_{rs,ij} &=& \\\\mathcal {B}_{sr,ij} = C \\\\delta _{ij}$ for $r,s=1,2$ .\",\n \"Performing the transformation (REF ) on both sides of the Langevin equation (REF ), we get $\\\\mathtt {j} \\\\omega \\\\tilde{\\\\xi }_{s}^\\\\alpha = \\\\sum _{r=1}^2\\\\left(\\\\mathcal {M}_{sr}^{(NS)}+\\\\mathcal {M}_{sr}^{(SP)}\\\\Lambda _\\\\alpha \\\\right)\\\\tilde{\\\\xi }_r^\\\\alpha +\\\\tilde{\\\\eta }_s^\\\\alpha $ where the term $\\\\tilde{\\\\eta }_s^\\\\alpha $ denotes the transform of the noise.\",\n \"Introducing the matrix $\\\\Phi _{sr}=\\\\mathtt {j} \\\\omega \\\\delta _{sr}-\\\\left(\\\\mathcal {M}_{sr}^{(NS)}+\\\\mathcal {M}_{sr}^{(SP)}\\\\Lambda _\\\\alpha \\\\right)$ we get $\\\\tilde{\\\\xi }_s^\\\\alpha =\\\\sum _{r=1}^2 \\\\Phi _{sr}^{-1}\\\\tilde{\\\\eta }_r^\\\\alpha $ , and thus the power spectrum of the $s$ -th species is given by $P_s(\\\\omega ,\\\\Lambda _\\\\alpha )=\\\\langle |\\\\tilde{\\\\xi }^\\\\alpha _s|^2\\\\rangle =\\\\sum _{r,k=1}^2 \\\\Phi _{sr}^{-1}\\\\langle \\\\tilde{\\\\eta }^\\\\alpha _r \\\\tilde{\\\\eta }_k^{\\\\alpha }\\\\rangle (\\\\Phi _{ks}^{\\\\dag })^{-1}$ It can be shown that $\\\\langle \\\\tilde{\\\\eta }^\\\\alpha _r \\\\tilde{\\\\eta }_k^{\\\\alpha } \\\\rangle = \\\\sum _{i,j}\\\\mathcal {B}_{rk,ij}v_i^{(\\\\alpha )}v_{j}^{(\\\\alpha )} $ .\",\n \"An explicit form for the $2\\\\times 2$ matrix $\\\\langle \\\\tilde{\\\\eta }^\\\\alpha _r \\\\tilde{\\\\eta }_k^{\\\\alpha } \\\\rangle $ can be derived by making use of Eqs (REF ).\",\n \"Let us focus on the non trivial contribution $r=k$ : $&&\\\\sum _{i,j}\\\\mathcal {B}_{rr,ij}v_i^{(\\\\alpha )}v_{j}^{(\\\\alpha )} =\\\\\\\\ &&\\\\sum _{i,j}\\\\left[ (D_r+\\\\tilde{k}_i H_r) \\\\delta _{ij} - \\\\left(\\\\frac{1}{k_i}+\\\\frac{1}{k_j}\\\\right) W_{ij} H_r \\\\right]v_i^{(\\\\alpha )}v_{j}^{(\\\\alpha )} =\\\\\\\\&& D_r \\\\sum _{i} v_i^{(\\\\alpha )} v_{i}^{(\\\\alpha )} - H_r \\\\sum _{i,j} \\\\Delta _{i,j} v_i^{(\\\\alpha )} v_{i}^{(\\\\alpha )} =D_r - H_r \\\\Lambda _{\\\\alpha }$ where in the last step we made use of $\\\\sum _{i} v_i^{(\\\\alpha )} v_{i}^{(\\\\alpha )}=1$ and $\\\\sum _j \\\\Delta _{ij} v_j^{(\\\\alpha )}=\\\\Lambda _\\\\alpha v_i^{\\\\alpha }$ .\",\n \"The component $r \\\\ne k$ is trivially equal to $C$ .\",\n \"The power spectrum is fully specified as function of $\\\\Lambda _\\\\alpha $ and $\\\\omega $ .\",\n \"Figure REF is obtained by setting $\\\\omega =0$ in the above formulae.\",\n \"We emphasize that the same result can be recovered from the continuum medium power spectrum [16] provided $-k^2$ is replaced by the discrete eigenvalue $\\\\Lambda _{\\\\alpha }$ .\",\n \"We end this Appendix by providing a list of explanatory captions to the movies annexed as supplementary material.\",\n \"mf_inside_X.mov and mf_inside_Y.mov show respectively the time evolution of the mean field concentrations $\\\\phi _i$ and $\\\\psi _i$ .\",\n \"The data are obtained by integrating the governing mean field equations (REF ).\",\n \"Parameters are chosen so to yield a Turing instability (magenta circle in Fig.\",\n \"REF ).\",\n \"At time $t=0$ a small perturbation is applied to perturb the homogeneous fixed point.\",\n \"Then, the system evolves toward a stable, non-homogeneous stationary configuration.\",\n \"st_inside_X.mov and st_inside_Y.mov show the result of the stochastic simulations (fluctuating blue circles), for respectively $n_i/N$ and $m_i/N$ .\",\n \"The red symbols refer to a late time snapshot of the deterministic dynamics.\",\n \"Parameters are set as specified above: the system is hence inside the region of Turing instability.\",\n \"Notice that the noise that takes the system away from the homogeneous fixed point is now endogenous to the system and not externally imposed.\",\n \"st_outside_X.mov and st_outside_Y.mov report on the stochastic simulations (blue symbols), for respectively $n_i/N$ and $m_i/N$ .\",\n \"The parameters are now assigned so to fall outside the region of Turing order (blue diamond in Fig.\",\n \"REF ).\",\n \"The mean field solutions are not destabilized and converge to the stable fixed point.\",\n \"At variance, the stochastic simulations evolve toward a non homogeneous state.\",\n \"Financial support from Ente Cassa di Risparmio di Firenze and the Program Prin2009 funded by Italian MIUR is acknowledged.\",\n \"D.F.\",\n \"thanks Tommaso Biancalani for pointing out reference [18] and for stimulating discussions.\",\n \"F.D.P.\",\n \"thanks Alessio Cardillo for providing the code to generate the network.\"\n ]\n]"},"id":{"kind":"string","value":"1204.1475"}}},{"rowIdx":1866898,"cells":{"text":{"kind":"list like","value":[["Metabasins - a State Space Aggregation for highly disordered Energy\n Landscapes"],["Abstract Glass-forming systems, which are characterized by a highly disordered energy landscape, have been studied in physics by a simulation-based state space aggregation.","The purpose of this article is to develop a path-independent approach within the framework of aperiodic, reversible Markov chains with exponentially small transition probabilities which depend on some energy function.","This will lead to the definition of certain metastates, also called metabasins in physics.","More precisely, our aggregation procedure will provide a sequence of state space partitions such that on an appropriate aggregation level certain properties (see Properties 1--4 of the Introduction) are fulfilled.","Roughly speaking, this will be the case for the finest aggregation such that transitions back to an already visited (meta-)state are very unlikely within a moderate time frame."],["Introduction","tocsectionIntroduction Supercooled liquids of glass forming systems are typical examples of high-dimensional systems with highly disordered energy landscapes and our main concern behind this work.","Simulations have shown that many important characteristics of such a system are better described by a process on the set of so-called metabasins (MB) than by the more common process on the set of visited minima of the energy landscape (see [12]).","Those MB are formed in the following way by aggregation of suitable states of the describing process $(X_{n})_{n\\ge 0}$ along a simulated trajectory: Fixing a reasonable observation time $T$ , define $\\chi _0\\equiv 0$ and then recursively for $n\\ge 1$ $\\chi _n:=\\inf \\big \\lbrace k>\\chi _{n-1}\\,|\\,\\lbrace X_{k}{,}...,X_{T}\\rbrace \\cap \\lbrace X_{0}{,}...,X_{k-1}\\rbrace =\\emptyset \\big \\rbrace .$ Then the MB up to $\\upsilon :=\\sup \\lbrace n\\ge 0\\,|\\,\\chi _n\\le T\\rbrace $ are chosen as $\\mathcal {V}_{n}:=\\lbrace X_{\\chi _n}{,}...,X_{\\chi _{n+1}-1}\\rbrace ,\\quad 0\\le n\\le \\upsilon .$ Simulation studies have shown that local sampling within a MB does not affect typical parameters of the process like the diffusion coefficient or the time to reach equilibrium.","Dynamical aspects are therefore fully characterized by the MB-valued process.","Furthermore, this model reduction by aggregation, as proposed in [12] and [18], offers several advantages (referred to as Properties 1–5 hereafter): The probability of a transition from one MB to any other one does not depend on the state at which this MB is entered.","There are basically no reciprocating jumps between two MB.","This is in strong contrast to the unaggregated process where such jumps occur very often: The system falls back to a minimum many times before eventually cresting a high energy barrier and then falling into a new valley, where it will again take many unsuccessful trials to escape.","These reciprocating jumps are not only irrelevant for the actual motion on the state space but also complicating the estimation of parameters like the diffusion coefficient or the relaxation time.","The expected time spent in a MB is proportional to its depth.","Thus there is a strong and explicit relation between dynamics and thermodynamics, not in terms of the absolute but the relative energy.","The energy barriers between any two MB are approximately of the same height, that is, there is an energy threshold $E_0$ such that, for a small $\\varepsilon $ , it requires a crossing of at least $E_0-\\varepsilon $ and at most $E_0+\\varepsilon $ to make a transition from one MB to another.","Such systems with $\\varepsilon =0$ are called trap models (see [3]).","The sojourn times and the jump distances between successively visited MB (measured in Euclidean distance) form sequences of weakly or even uncorrelated random variables, and are also mutually independent, at least approximately.","Therefore, the aggregated process can be well approximated by a continuous time random walk, which in turn simplifies its analysis and thus the analysis of the whole process.","Despite these advantages, the suggested definition of MB has the obvious blemish that it depends on the realization of the considered process and may thus vary from simulation to simulation.","To provide a mathematically stringent definition of a path-independent aggregation of the state space, which maintains the above properties and is based on the well-established notion of metastable states, is therefore our principal concern here with the main results being Theorem REF and Theorem REF .","In this endeavor, we will draw on some of the ideas developed by Bovier in [4] and by Scoppola in [21], most notably her definition of metastable states.","Metastability, a phenomenon of ongoing interest for complex physical systems described by finite Markov processes on very large state spaces, can be defined and dealt with in several ways.","It has been derived from a renormalization procedure in [20], by a pathwise approach in [6], and via energy landscapes in [4], the latter being also our approach hereafter.","To characterize a supercooled liquid, i.e.","a glass forming system at low temperature, via its energy landscape was first done by Goldstein in 1969 [11] and has by now become a common method.","The general task when studying metastability, as well originally raised in physics ([13], [19] or [18]), is to provide mathematical tools for an analysis of the property of thermodynamical systems to evolve in state space along a trajectory of unstable or metastable states with very long sojourn times.","Inspired by simulations of glass forming systems at very low temperatures with the Metropolis algorithm, we will study (as in [21]) finite Markov chains with exponentially small transition probabilities which are determined by an energy function and a parameter $\\beta >0$ .","This parameter can be understood as the inverse temperature and we are thus interested in the behavior of the process as $\\beta \\rightarrow \\infty $ .","We envisage an energy function of highly complex order and without the hierarchical ordering that is typical in spin glass models.","A good picture is provided by randomly chosen energies with correlations between neighbors or by an energy landscape that looks like a real mountain landscape.","We will show that, towards an aggregation outlined above, the metastable states as defined in [21] are quite appropriate because they have an ordering from a kind of “weak” to a kind of “strong” metastability.","Around those states we will define and then study connected valleys [Definition REF ] characterized by minimal energy barriers.","In the limit of low temperatures, any such barrier will determine the speed, respectively probability of a transition between the two valleys it separates.","More precisely, in the limit $\\beta \\rightarrow \\infty $ , the process, when starting in a state $x$ , will almost surely reach a state with lower barrier earlier than a state with higher barrier [Theorem REF ].","In the limit of low temperatures, the bottom (minimum) of an entered valley will therefore almost surely be reached before that valley is left again [Proposition REF ].","As a consequence, the probability for a transition from one valley to another is asymptotically independent of the state where the valley is entered.","This is Property 1 above.","Furthermore, since valleys as well as metastable states have a hierarchical ordering, we can build valleys of higher order by a successive merger of valleys of lower order [Proposition REF ].","Given an appropriate energy landscape, this procedure can annihilate (on the macroscopic scale) the accumulation of reciprocating jumps by merging valleys exhibiting such jumps into a single valley [Subsection REF ].","Hence, valleys of sufficiently high order will have Property 2.","Beside the macroscopic process [Section ], which describes the transitions between valleys, one can also analyze the microscopic process [Section ], that is, the system behavior when moving within a fixed valley.","Here we will give a formula for the exit time and connect it with its parameters [Theorem REF ].","This will confirm Property 3.","Having thus established Properties 1–4 [Theorem REF ], we will finally proceed to a comparison of our path-independent definition of MB with the path-dependent one given above.","It will be shown [Theorem REF ] that both coincide with high probability under some reasonable conditions on the connectivity of valleys which, in essence, ensure the existence of reasonable path-dependent MB.","We will also briefly touch on the phenomenon of quasi-stationarity [Proposition REF ] which is a large area [17] but to our best knowledge less studied in connection with the aggregation of states of large physical systems driven by energy landscapes.","Let us mention two further publications which, despite having a different thrust, provide definitions of valleys, called basins of attraction or metastates there, to deal with related questions.","Olivieri & Scoppola [16] fully describe the tube of exit from a domain in terms of which basins of attraction of increasing order are visited during a stay in that domain and for how long these basins are visited.","In a very recent publication, Beltrán & Landim [2], by working with transition rates instead of energies, aim at finding a universal depth (and time scale) for all metastates.","However, we rather aim at the finest aggregation such that transitions back to an already visited metastate are very unlikely within a time frame used in simulations.","This finest aggregation will lead to valleys of very variable depth just as simulations do not exhibit a universal depth or timescale."],["Valleys","Let $X$ be a Markov chain on a finite set $\\mathcal {S}$ with transition matrix $\\mathbf {P}=(p(r,s))_{r,s\\in \\mathcal {S}}$ and stationary distribution $\\pi $ , and let $E:\\mathcal {S}\\rightarrow \\mathbb {R}$ be an energy function such that the following conditions hold: Irreducibility: $\\mathbf {P}$ is irreducible with $p(s,s)>0$ and $p(r,s)>0$ iff $p(s,r)>0$ for all $r,s\\in \\mathcal {S}$ .","Transition Probabilities: There exist parameters $\\beta >0$ and $\\gamma _{\\beta }>0$ with $\\gamma _{\\beta }\\rightarrow 0, \\beta \\gamma _{\\beta }\\rightarrow \\infty $ as $\\beta \\rightarrow \\infty $ such that $e^{-\\beta ((E(s)-E(r))^++\\gamma _{\\beta }/|\\mathcal {S}|)}\\le p(r,s)\\le e^{-\\beta ((E(s)-E(r))^+-\\gamma _{\\beta }/|\\mathcal {S}|)}$ for all distinct $r,s\\in \\mathcal {S}$ with $p(r,s)>0$ .","Furthermore, $p^{*}(r,s):=\\lim _{\\beta \\rightarrow \\infty }p(r,s)$ exists for all $r,s\\in \\mathcal {S}$ , is positive if $E(r)\\ge E(s)$ and $=0$ otherwise.","Reversibility: The pair $(\\pi ,\\mathbf {P})$ satisfies the detailed balance equations, i.e.","$\\pi (r)p(r,s)=\\pi (s)p(s,r)$ for all $r,s\\in \\mathcal {S}$ .","Non-Degeneracy: $E(r)\\ne E(s)$ for all $r,s \\in \\mathcal {S}, r\\ne s$ .","We are thus dealing with a reversible Markov chain with exponentially small transition probabilities driven by an energy landscape.","As an example, which is also the main motivation behind this work, one can think of a Metropolis chain with transition probabilities of the form $p(r,s)=\\frac{1}{C(r)}e^{-\\beta (E(s)-E(r))^+}.$ Here $\\beta $ is the inverse temperature and $C(r), r\\in \\mathcal {S}$ , is a parameter, independent from $\\beta $ , giving the number of neighbors of $r$ .","For $\\gamma _{\\beta }/|\\mathcal {S}|:=\\max _{r\\in \\mathcal {S}}\\ln (C(r))(\\beta +1)^{-1/2}$ the above conditions are fulfilled.","Let us start with the following basic result for the ratios of the stationary distribution.","Lemma 1.1 For any two states $r,s\\in \\mathcal {S}$ with $E(r)>E(s)$ , we have $e^{-\\beta (E(r)-E(s)+2\\gamma _{\\beta })}\\le \\frac{\\pi (r)}{\\pi (s)}\\le e^{-\\beta (E(r)-E(s)-2\\gamma _{\\beta })}.$ [Proof:] To start with, assume $r\\sim s$ .","Reversibility and the assumptions on the transition probabilities imply $\\frac{\\pi (r)}{\\pi (s)}\\ &=\\ \\frac{p(s,r)}{p(r,s)}\\ \\le \\ \\frac{e^{-\\beta ((E(r)-E(s))^+-\\gamma _{\\beta }/|\\mathcal {S}|)}}{e^{-\\beta ((E(s)-E(r))^++\\gamma _{\\beta }/|\\mathcal {S}|)}}\\ =\\ e^{-\\beta (E(r)-E(s)-2\\gamma _{\\beta }/|\\mathcal {S}|)}.$ and $\\frac{\\pi (r)}{\\pi (s)}\\ &=\\ \\frac{p(s,r)}{p(r,s)}\\ \\ge \\ \\frac{e^{-\\beta ((E(r)-E(s))^++\\gamma _{\\beta }/|\\mathcal {S}|)}}{e^{-\\beta ((E(s)-E(r))^+-\\gamma _{\\beta }/|\\mathcal {S}|)}}\\ =\\ e^{-\\beta (E(r)-E(s)+2\\gamma _{\\beta }/|\\mathcal {S}|)}.$ Now let $r$ and $s$ be arbitrary.","By the irreducibility, there is a path $r=r_0,r_1{,}...,r_n=s$ from $r$ to $s$ of neighboring states with $\\pi (r_i)/\\pi (r_{i+1})\\in [e^{-\\beta (E(r_i)-E(r_{i+1})+2\\gamma _{\\beta }/|\\mathcal {S}|)},e^{-\\beta (E(r_i)-E(r_{i+1})-2\\gamma _{\\beta }/|\\mathcal {S}|)}],\\, 0\\le i\\le n-1$ .","This finishes the proof.","Under the stated assumptions, Scoppola [21] has shown the existence of a successive filtration (aggregation) $\\mathcal {S}=M^{(0)}\\supset M^{(1)}\\supset ... \\supset M^{(\\mathfrak {n})}=\\lbrace s_0\\rbrace $ , $\\mathfrak {n}\\in \\mathbb {N}$ , of the state space such that the elements of each set $M^{(i)}, 1\\le i\\le \\mathfrak {n}$ , can be called metastable in the following sense: They arise from the local minima of the energy function or certain modifications of it.","There is a lower bound on the expected time needed for a transition from $m_1$ to $m_2$ for any $m_1,m_2\\in M^{(i)}$ which increases very fast with $i$ .","There exists a constant $C$ such that $\\mathbb {P}_m(X_t\\notin M^{(i+1)})\\le e^{-C\\beta }$ for all $m\\in M^{(i)}, 0\\le i\\le \\mathfrak {n}-1,$ and sufficiently large $t$ .","This filtration starts with $M^{(1)}:=\\lbrace s\\in \\mathcal {S}| E(s)0 \\textrm { for } 0\\le i\\le k-1,\\,i\\ne j\\rbrace $ be the set of all finite self-avoiding paths from $r$ to $s$ having positive probability.","For any such path $\\gamma =(\\gamma _0{,}...,\\gamma _k)\\in \\Gamma (r,s)$ , let $|\\gamma |:=k$ be its length.","We further write $t\\in \\gamma $ if $t\\in \\lbrace \\gamma _{1}{,}...,\\gamma _{k}\\rbrace $ .","(b) A self-avoiding path $\\gamma =(\\gamma _{1}{,}...,\\gamma _{k})$ from $r$ to $s$ is called minimal if its maximal energy $\\max _{1\\le i\\le k} E(\\gamma _i)$ is minimal among all $\\gamma ^{\\prime }\\in \\Gamma (r,s)$ .","The set of these paths is denoted $\\Gamma ^*(r,s)$ .","(c) The essential saddle $z^*(r,s)$ between $r$ and $s$ is then defined as $z^*(r,s):=\\mathop {\\mathrm {argmax}}_{t\\in \\gamma }E(t)\\in \\mathcal {S}$ for any $\\gamma \\in \\Gamma ^{*}(r,s)$ .","As for (c), it is to be noted that, due to the assumed non-degeneracy of the energy function, the essential saddle is unique, which means that it does not depend on (as it must) which minimal path we choose in the definition of $z^*(r,s)$ .","There may indeed be several minimal paths, every single one thus crossing the saddle at some time.","With the help of these notions the valleys can now be defined in a quite concrete way.","Let us label the local minima as $m^{(1)}{,}...,m^{(\\mathfrak {n})}$ , so that $M^{(i)}=\\lbrace m^{(i)}{,}...,m^{(\\mathfrak {n})}\\rbrace $ for each $i=1{,}...,\\mathfrak {n}$ .","Definition 1.4 For each $m \\in M^{(i)}$ , $1\\le i\\le \\mathfrak {n}$ , let $V^{(i)}_<(m):=\\left\\lbrace s\\in \\mathcal {S}\\Big |E(z^*(s,m))< E(z^*(s,m^{\\prime }))\\text{ for all }m^{\\prime }\\in M^{(i)}\\backslash \\lbrace m\\rbrace \\right\\rbrace .$ We say that state $s$ is attracted by $m$ at level $i$, expressed as $s\\leadsto m$ at level $i$ , if $ E(z^*(s,m))=\\min _{n\\in M^{(i)}}E(z^*(s,n))$ and every minimal path from $s$ to a state $m^{\\prime }\\in M^{(i)}\\backslash \\lbrace m\\rbrace $ with $E(z^*(s,m^{\\prime }))=E(z^*(s,m))$ hits $V^{(i)}_{<}(m)$ at some time.","Finally, let $l(i):=\\inf \\big \\lbrace i2$ ), it remains untouched until reaching level $l(1)$ where its bottom state $m^{(1)}$ becomes attracted by some $m^{\\prime }\\in M^{(l(1))}$ causing its valley to be merged into $V^{(l(1))}(m^{\\prime })$ .","This procedure continues in the now obvious recursive manner until at level $\\mathfrak {n}$ all states have been merged into one valley.","Obviously, valleys of the same order are pairwise disjoint.","Also, valleys once formed at some level can only be merged as a whole and will thus never be ripped apart during the recursive construction.","For the energy function depicted in Figure REF , the successively derived valleys of order $i=1{,}...,7$ are shown in Figure REF .","Before proceeding to results on the general shape of valleys, we collect some basic, mostly technical properties of essential saddles which will be useful thereafter.","Proposition 1.6 For any $r,s,u\\in \\mathcal {S}$ , $0\\le i\\le \\mathfrak {n}$ , $m_1,m_2\\in M^{(i)}, m_1\\ne m_2,$ and $x_1,x_2\\in \\mathcal {S}$ with $x_1\\in V^{(i)}_<(m_1)$ and $x_2\\in V^{(i)}(m_2)$ , we have (a) $z^*(r,s)=z^*(s,r)$ .","(b) $E(z^*(r,s))\\le E(z^*(r,u))\\vee E(z^*(u,s))$ .","(c) $E(z^*(x_2,m_2))\\le E(z^*(x_2,m^{\\prime }))$ for all $m^{\\prime }\\in M^{(i)}$ .","(d) $E(z^*(x_1,m_2))=E(z^*(m_1,m_2))$ .","(e) $E(z^*(x_1,x_2))\\ge E(z^*(m_1,m_2))$ .","(f) $z^*(x_1,x_2)\\ne x_1$ .","[Proof:] Parts (a) and (b) are obvious.","For (c) we use an induction over $i$ and note that there is nothing to show when $i=1$ .","For general $i$ , we must only verify that $E(z^{*}(x_{2},m_{2}))\\le E(z^{*}(x_{2},m^{\\prime }))$ for all $m^{\\prime }\\in M^{(i)}$ if $x_{2}\\in V^{(j)}(m^{(j)})$ for some $jE(z^*(x_1,m_1))$ , which in combination with (a) and (b) implies $E(z^*(m_1,m_2))\\le E(z^*(m_1,x_1))\\vee E(z^*(x_1,m_2))=E(z^*(x_1,m_2))$ and then further $E(z^*(x_1,m_2))\\le \\underbrace{E(z^*(x_1,m_1))}_{E(z^*(x_1,m_1))\\ge E(x_1)$ and thus $z^*(x_1,x_2)\\ne x_1$ as claimed in (f).","Remark 1.7 It is useful to point out the following consequence of the previous proposition.","If, for an arbitrary state $s$ and any two distinct metastable states $m,n\\in M^{(i)}$ , there exists a minimal path $\\gamma $ from $s$ to $n$ that hits a state $r$ with $E(z^*(r,m))E(z^*(x,m))$ .","[Proof:] Suppose there is a minimal path $\\gamma $ from $x$ to $y$ avoiding $V^{(i)}_<(m)$ .","Since $y\\notin V^{(i)}(m)$ , it is not attracted by $m$ at level $i$ implying the existence of some $m^{\\prime }\\in M^{(i)}$ with $E(z^*(y,m^{\\prime }))\\le E(z^*(y,m))$ and of some $\\tau \\in \\Gamma ^*(y,m^{\\prime })$ avoiding $V^{(i)}_<(m)$ .","Hence, the concatenation $\\gamma \\tau $ avoids $V^{(i)}_<(m)$ and must therefore have maximal energy larger than $E(z^*(x,m))$ .","Consequently, $E(z^*(x,m))&E(z^*(x,m))$ .","In order to state the second lemma, let us define the outer part $\\partial ^{+}V$ of a valley $V$ to be the set of those states outside $V$ which are adjacent to a state in $V$ .","With the help of the previous result, we can easily show that $\\partial ^{+}V$ contains only non-assigned states at any level where $V$ has not yet been merged into a larger valley.","Lemma 1.15 For any $1\\le i, j\\le \\mathfrak {n}$ and $m=m^{(i)}$ with $l(i)> j$ , the outer part $\\partial ^{+}V$ of the valley $V:=V^{(j\\wedge i)}(m)$ is a subset of $N^{(j)}$ and $E(z^*(s,m))=E(s)$ for every $s\\in \\partial ^+V$ .","[Proof:] First, let $s\\in \\partial ^{+}V$ and suppose that $s\\notin N^{(j)}$ .","Then $s\\leadsto m^{\\prime }$ at level $k$ , in particular $s\\in V^{(k)}(m^{\\prime })$ for some $m^{\\prime }\\in M^{(k)}$ and $k\\le j$ .","Pick any $r\\in V$ with $r\\sim s$ and note that $r\\in \\partial ^{+}V^{(k)}(m^{\\prime })$ .","W.l.o.g.","we may assume that $r\\leadsto m$ at level $j\\wedge i$ .","Then Lemma REF (with $x=r$ and $y=s$ ) ensures that either $E(z^{*}(r,s))>E(z^{*}(r,m)\\ge E(r)$ , thus $z^{*}(r,s)=s$ and $E(r)E(s)$ which is clearly impossible.","Consequently, $s$ must be non-assigned at level $j$ as claimed.","For the second assertion take again $s\\in \\partial ^+V$ and a minimal path $\\gamma =(s,r{,}...,m)$ from $s$ to $m$ with $r\\in V$ .","Again, by use of Lemma REF , we find either $E(z^*(r,m))0\\rbrace .$ Two states $x,y$ with $p(x,y)>0$ are called neighbors ($x\\sim y$ ) and $\\mathcal {N}(x)$ the neighborhood of $x$ .","Hence, $\\mathcal {S}$ may (and will) be viewed as a graph hereafter with edge set $\\lbrace (x,y)|x\\sim y\\rbrace $ .","Given any subgraph $\\Delta $ , we will write $\\widetilde{\\mathbf {P}}$ for the transition matrix of the chain restricted to $\\Delta $ ($\\tilde{p}(r,s)=p(r,s)$ for all distinct $r,s\\in \\Delta $ , $\\tilde{p}(r,r)=1-\\sum _{s\\in \\Delta , s\\ne r}\\tilde{p}(r,s)$ ) and $\\widetilde{\\mathbb {P}}_{x}$ for probabilities when regarding this restricted chain starting at $x\\in \\Delta $ .","Theorem 2.1 Let $x, y, z\\in \\mathcal {S}$ be any pairwise distinct states satisfying $E(z^*(x,z))>E(z^*(x,y))$ and $z^*(x,z)\\ne x$ .","Then there exist nonnegative constants $K(\\beta )$ satisfying $\\sup _{\\beta >0}K(\\beta )<\\infty $ (and given more explicitly in Proposition REF below) such that $\\mathbb {P}_{x}(\\tau _{z}<\\tau _{y})\\le K(\\beta )\\,e^{-\\beta (E(z^*(x,z))-E(z^*(x,y))-7\\gamma _{\\beta })}=:\\varepsilon (x,y,z,\\beta ) \\overset{\\beta \\rightarrow \\infty }{\\longrightarrow } 0.$ Thus in the limit of low temperatures $(\\beta \\rightarrow \\infty )$ , only the smallest of all possible energy barriers affects the speed of a transition.","In particular, we have the following result which is preliminary to the subsequent one.","Theorem 2.2 Given distinct $x,y\\in \\mathcal {S}$ and $m\\in M^{(i)}$ such that $x\\leadsto m$ at level $i$ and $y\\notin V^{(i)}(m)$ , let $B:=\\lbrace z\\in \\mathcal {S}|E(z^*(x,z))>E(z^*(x,m))\\rbrace $ .","Then it holds true that $\\mathbb {P}_{x}&(\\tau _{y}<\\tau _{m})\\\\&\\le \\ \\varepsilon (x,m,y,\\beta )\\,{1}_{B}(y)+\\left(\\sum _{z:E(z)>E(z^*(x,y))}\\varepsilon (x,y,z,\\beta )+\\sum _{z\\in V^{(i)}_<(m)}\\varepsilon (z,m,y,\\beta )\\right)\\,{1}_{B^{c}}(y)\\\\&=:\\ \\tilde{\\varepsilon }(x,m,y,\\beta )\\ \\overset{\\beta \\rightarrow \\infty }{\\longrightarrow }\\ 0.$ Theorem 2.3 Given $m\\in M^{(i)}$ , $x\\in V^{(i)}(m)$ and $y\\notin V^{(i)}(m)$ , let $k\\le i$ be such that $&m_{0}:=x\\leadsto m_{1}\\text{ at level }l_{1},\\quad m_{1}\\leadsto m_{2}\\text{ at level }l_{2},\\quad ...\\quad m_{k-1}\\leadsto m_{k}=m\\text{ at level }l_{k}$ for suitable $1\\le l_{1}<...E(b)}(E(a)-E(b))-2\\gamma _{\\beta })}+1\\right)$ .","The proof requires several lemmata, the first of which may already be found in [14] and is stated here in the notation of [5].","Lemma 2.5 (see Theorem 2.1 in [5]) Defining $\\mathcal {H}^x_z:=\\lbrace h:\\mathcal {S}\\rightarrow [0,1]\\,|\\, h(x)=0,h(z)=1\\rbrace $ and the Dirichlet form $\\mathcal {E}(h):=\\sum _{r\\sim s\\in \\mathcal {S}}\\pi (r)p(r,s) (h(r)-h(s))^2,$ we have $\\mathbb {P}_x(\\tau _z<\\tau _x)=\\frac{1}{2\\pi (x)}\\inf _{h\\in \\mathcal {H}^x_z}\\mathcal {E}(h).$ Lemma 2.6 (see Lemma 2.2 in [5]) For any subgraph $\\Delta \\subset \\mathcal {S}$ containing $x,y$ and corresponding transition matrix $\\widetilde{\\mathbf {P}}$ , we have $\\mathbb {P}_x(\\tau _y<\\tau _x)\\ge \\widetilde{\\mathbb {P}}_x(\\tau _y<\\tau _x).$ Lemma 2.7 (see Lemma 2.5 in [5]) If $\\Delta =(\\omega _0{,}...,\\omega _k)$ is any one-dimensional subgraph of $\\mathcal {S}$ , then $\\widetilde{\\mathbb {P}}_{\\omega _0}(\\tau _{\\omega _k}<\\tau _{\\omega _0}) =\\left(\\sum _{i=1}^k\\frac{\\pi (\\omega _0)}{\\pi (\\omega _i)}\\frac{1}{p(\\omega _i,\\omega _{i-1})}\\right)^{-1}.$ [Proof:](of Proposition REF ) In view of Lemma REF , we must find an appropriate function $h$ for the upper bound.","Let us define $\\mathcal {R}:=\\lbrace s\\in \\mathcal {S}\\,|\\,\\exists ~ \\gamma \\in \\Gamma (x,z):\\mathop {\\mathrm {argmax}}_{i=1{,}... , |\\gamma |}E(\\gamma _i)=s\\rbrace ,$ the set of all peak states (with respect to the energy function) along self-avoiding paths from $x$ to $z$ .","Obviously $z^*(x,z)\\in \\mathcal {R}$ and, by non-degeneracy, $E(z^*(x,z))E(b)} (E(a)- E(b))-2\\gamma _{\\beta })}+|\\mathcal {N}(z^*(x,z))|\\right)\\\\&\\le \\ e^{-\\beta (E(z^*(x,z))-E(x)-2\\gamma _{\\beta })}\\\\&\\hspace*{28.45274pt}\\times \\,\\max _{r\\in \\mathcal {S}}|\\mathcal {N}(r)|\\,\\left(|\\mathcal {S}|e^{-\\beta (\\min _{a\\ne b:E(a)>E(b)}(E(a)- E(b))-2\\gamma _{\\beta })}+1\\right).$ Lemmata REF and REF will enter in the proof of the lower bound.","Consider the chain restricted to the one-dimensional subgraph given by a minimal path $\\rho =(s_{1}{,}...,s_{|\\rho |})$ from $x$ to $y$ .","Then $\\mathbb {P}_x(\\tau _y<\\tau _x)\\ &\\ge \\ \\left(\\sum _{i=1}^{|\\rho |}\\frac{\\pi (x)}{\\pi (s_i)}\\frac{1}{p(s_i,s_{i-1})}\\right)^{-1}\\\\&\\ge \\ \\frac{\\pi (z^*(x,y))}{\\pi (x)}\\left(\\sum _{i=1}^{|\\rho |}\\frac{\\pi (z^*(x,y))}{\\pi (s_i)}e^{\\beta ((E(s_{i-1})-E(s_i))^{+} +\\gamma _{\\beta })}\\right)^{-1}\\\\&\\ge \\ e^{-\\beta (E(z^*(x,y))-E(x) +2\\gamma _{\\beta })}\\left(\\sum _{i=1}^{|\\rho |}e^{-\\beta (E(z^*(x,y))-E(s_i)-(E(s_{i-1})-E(s_i))^{+}-3\\gamma _{\\beta })}\\right)^{-1}\\\\&\\ge \\ e^{-\\beta (E(z^*(x,y))-E(x)+5\\gamma _{\\beta })}\\,\\frac{1}{|\\mathcal {S}|}.$ This completes the proof of Proposition REF .","We proceed to the second proposition needed to prove Theorem REF .","Proposition 2.8 (see Corollary 1.6 in [5]) Given $I\\subset \\mathcal {S}$ and distinct $x,z\\in \\mathcal {S}\\backslash I$ , $\\mathbb {P}_x(\\tau _z<\\tau _I)=\\frac{\\mathbb {P}_x(\\tau _z<\\tau _{I\\cup \\lbrace x\\rbrace })}{\\mathbb {P}_x(\\tau _{I\\cup \\lbrace z\\rbrace }<\\tau _x)}$ holds true.","With the help of Propositions REF and REF , the proof of Theorem REF can now be given quite easily.","[Proof:](of Theorem REF ) By first using the previous result and then Proposition REF (with $K(\\beta )$ as defined there), we find $\\mathbb {P}_{x}(\\tau _{z}<\\tau _{y})\\ &=\\ \\frac{\\mathbb {P}_{x}(\\tau _{z}<\\tau _{\\lbrace x,y\\rbrace })}{\\mathbb {P}_{x}(\\tau _{\\lbrace z,y\\rbrace }<\\tau _{x})}\\\\&\\le \\ \\frac{\\mathbb {P}_{x}(\\tau _{z}<\\tau _{x})}{\\mathbb {P}_{x}(\\tau _{y}<\\tau _{x})}\\\\&\\le \\ K(\\beta )\\,e^{-\\beta (E(z^*(x,z))-E(z^*(x,y))-7\\gamma _{\\beta })}.$ The argument is completed by noting that $E(z^*(x,z))>E(z^*(x,y))$ and $K(\\beta )\\ge 0$ converges to $|\\mathcal {S}|\\max _{r\\in \\mathcal {S}}|\\mathcal {N}(r)|$ as $\\beta \\rightarrow \\infty $ .","[Proof:](of Theorem REF ) If $B$ occurs, the asserted bound follows directly from Theorem REF .","Proceeding to the case when $B^{c}$ occurs, i.e.","$E(z^*(x,y))\\le E(z^*(x,m))$ , we first point out that $\\mathbb {P}_{x}(\\tau _{y}<\\tau _{m})\\ &=\\ \\mathbb {P}_{x}(\\tau _{y}<\\tau _{m}, E(X_n)>E(z^*(x,y)) \\text{ for some }n\\le \\tau _{y})\\\\&\\qquad +\\ \\mathbb {P}_{x}(\\tau _{y}<\\tau _{m},E(X_n)\\le E(z^*(x,y))\\text{ for all }n\\le \\tau _{y})\\\\&=:\\ P_{1}+P_{2}.$ For all $z\\in \\mathcal {S}$ with $E(z)>E(z^*(x,y))$ , we have $z^*(x,z)\\ne x$ and $E(z^*(x,z))>E(z^*(x,y))$ , for $E(z^*(x,z))\\ge E(z)>E(z^*(x,y))\\ge E(x).$ Therefore, by an appeal to Theorem REF , $P_{1}\\ &\\le \\ \\mathbb {P}_{x}(\\tau _{z}<\\tau _{y}\\text{ for some $z$ with $E(z)>E(z^*(x,y))$})\\\\&\\le \\ \\sum _{z:E(z)>E(z^*(x,y))}\\mathbb {P}_{x}(\\tau _z<\\tau _{y})\\\\&\\le \\ \\sum _{z:E(z)>E(z^*(x,y))}\\varepsilon (x,y,z,\\beta ).$ To get an estimate for $P_{2}$ , note that every minimal path from $x$ to $y$ must pass through $V^{(i)}_<(m)$ (Lemma REF ).","With this observation and by another appeal to Theorem REF , we infer $\\mathbb {P}_{x}(\\tau _{y}<\\tau _{m},E(X_n)\\le E(z^*(x,y))\\text{ for all }n\\le \\tau _{y})\\ &\\le \\sum _{z\\in V^{(i)}_<(m)}\\mathbb {P}_z(\\tau _{y}<\\tau _ {m})\\\\&\\le \\ \\sum _{z\\in V^{(i)}_<(m)}\\varepsilon (z,m,y,\\beta ),$ having further utilized that (by Proposition REF (f) and (e)) $z^*(z,y)\\ne z$ and $E(z^*(z,m))&n)=\\nu (j)\\quad \\text{for all }j\\in V,$ but also satisfies $\\lim _{n\\rightarrow \\infty }\\,\\mathbb {P}_{\\mu }(X_{n}=j|\\tau _{\\mathcal {S}\\backslash V}>n)=\\nu (j)\\quad \\text{for all }j\\in V$ if $\\mu $ is an arbitrary distribution with $\\mu (V)=1$ .","The latter property renders uniqueness of $\\nu $ .","Since $\\mathcal {S}$ is finite, existence of $\\nu $ follows by an old result due to Darroch & Seneta [7].","It is obtained as the normalized eigenvector of the Perron-Frobenius eigenvalue $\\lambda =\\lambda (V)$ of a modification of $\\mathbf {P}$ .","This eigenvalue $\\lambda $ is also the probability for the chain to stay in $V$ at least one step when started with $\\nu $ , thus $\\mathbb {P}_{\\nu }(\\tau _{V^{c}}>1)=\\lambda $ .","As an immediate consequence, one finds that the exit time $\\tau _{V^{c}}$ has a geometric distribution with parameter $1-\\lambda $ under $\\mathbb {P}_{\\nu }$ .","In the present context, this naturally raises the question how the parameter $\\lambda $ relates to the transition probabilities or the energies of the valley $V$ .","A simple probabilistic argument shows the following basic and intuitively obvious result concerning the eigenvalues associated with the nesting $V^{(1)}(m)\\subset ...\\subset V^{(i)}(m)$ (Proposition REF ) for any $1\\le i\\le \\mathfrak {n}$ and $m\\in M^{(i)}$ .","Proposition 2.9 Fixing any $1\\le i\\le \\mathfrak {n}$ and $m\\in M^{(i)}$ , let $\\lambda ^{(j)}:=\\lambda (V^{(j)}(m))$ for $j=1{,}...,i$ .","Then $\\lambda ^{(1)}\\le ...\\le \\lambda ^{(i)}$ .","[Proof:] Write $\\nu _{j}$ as shorthand for the quasi-stationary distribution on $V^{(j)}(m)$ and $T_{j}$ for $\\tau _{\\mathcal {S}\\backslash V^{(j)}(m)}$ .","Plainly, $T_{j}\\le T_{j+1}$ $(\\lambda ^{(j)})^{n}\\ =\\ \\mathbb {P}_{\\nu _{j}}(T_{j}>n)\\ &\\le \\ \\mathbb {P}_{\\nu _{j}}(T_{j+1}>n)\\nonumber \\\\&=\\ \\int _{\\lbrace T_{j+1}>k\\rbrace }\\mathbb {P}_{X_{k}}(T_{j+1}>n-k)\\ d\\mathbb {P}_{\\nu _{j}}\\nonumber \\\\&=\\ \\mathbb {P}_{\\nu _{j}}(T_{j+1}>k)\\,\\mathbb {P}_{\\mu _{k}}(T_{j+1}>n-k),$ where $\\mu _{k}(x):=\\mathbb {P}_{\\nu _{j}}(X_{k}=x|T_{j+1}>k)$ for $x\\in V^{(j+1)}$ .","Since $\\mathcal {S}$ is finite and by virtue of (REF ), we have that $\\mu _{k}\\le 2\\nu _{j+1}$ when choosing $k$ sufficiently large.","For any such $k$ , we find that (REF ) has upper bound $2\\,\\mathbb {P}_{\\nu _{j}}(T_{j+1}>k)\\,\\mathbb {P}_{\\nu _{j+1}}(T_{j+1}>n-k)\\ =\\ 2\\,\\mathbb {P}_{\\nu _{j}}(T_{j+1}>k)\\,(\\lambda ^{(j+1)})^{n-k}.$ Hence, we finally conclude $\\lambda ^{(j)}\\ \\le \\ \\Big (2\\,\\mathbb {P}_{\\nu _{j}}(T_{j+1}>k)\\,(\\lambda ^{(j+1)})^{-k}\\Big )^{1/n}\\,\\lambda ^{(j+1)}$ and thereby the assertion upon letting $n\\rightarrow \\infty $ .","An alternative matrix-analytic proof draws on an old result by Frobenius [9], here cited from [10].","Lemma 2.10 If $A=(a_{ij})$ and $C=(c_{ij})$ denote two real $k\\times k$ -matrices such that $A$ is nonnegative and irreducible with maximal eigenvalue $\\lambda _A^*$ and $|c_{ij}|\\le a_{ij}$ for all $1\\le i,j\\le k$ , then $|\\lambda |\\le \\lambda _A^*$ for all eigenvalues $\\lambda $ of $C$ .","Second proof of Proposition REF : For any fixed valley $V$ , collaps all states $s\\notin V$ into an absorbing state (grave) $\\Delta $ which leaves transition probabilities between states in $V$ unchanged.","A proper rearrangement of states allows us to assume that the new transition matrix has the form $\\mathbf {P}=\\begin{pmatrix} 1 & {\\bf 0} \\\\ \\mathbf {p} & \\mathbf {Q}\\end{pmatrix}$ for a $|V|\\times 1$ -column vector $\\mathbf {p}\\ne 0$ and a nonnegative, substochastic and irreducible $|V|\\times |V|$ -matrix $\\mathbf {Q}$ .","Now, for any $2\\le j\\le i$ , let $A$ be this matrix $\\mathbf {Q}$ when $V=V^{(j)}(m)$ , and $D$ be this matrix when $V=V^{(j-1)}(m)$ .","Then, obviously, $A:=\\begin{pmatrix} A_1 & A_2 \\\\ A_3 & D \\end{pmatrix}$ and $A$ is irreducible and nonnegative with maximal eigenvalue $\\lambda ^{(j)}$ .","Defining further $C:=\\begin{pmatrix} 0 & 0 \\\\ 0 & D \\end{pmatrix}.$ the largest eigenvalue of $C$ equals the largest eigenvalue of $D$ , thus $\\lambda ^{(j-1)}$ .","Finally, the desired conclusion follows from the previous lemma, since $|c_{ij}|=c_{ij}\\le a_{ij}$ for all $i\\le i,j\\le k$ .$\\square $ Another question is how long a given valley is visited and thus about its exit time.","There is an extensive literature on exit problems for different kinds of stochastic processes.","We mention [22] and [8] as two related to our work.","The latter one studies perturbed systems on a continuous space.","We can discretize their argument to get, with use of the main theorem in [21], a nice result on the time needed to leave a valley $V^{(i)}(m)$ for any fixed $1\\le i\\le \\mathfrak {n}$ and $m\\in M^{(i)}$ .","This result is more explicit than the one in [16].","Definition 2.11 For $1\\le i\\le \\mathfrak {n},\\, N:=N^{(i)}$ , we define the following stopping (entrance/exit) times: $\\xi _0^{(i)}&:=\\tau _{N^c}\\\\\\zeta _{n}^{(i)}&:=\\inf \\left\\lbrace k\\ge \\xi _n^{(i)}|X_k\\in N\\right\\rbrace \\\\\\xi _{n+1}^{(i)}&:=\\inf \\left\\lbrace k\\ge \\zeta _{n}^{(i)}|X_k\\in N^c\\right\\rbrace , ~n\\ge 0.$ The entrance times $\\xi ^{(i)}_n$ mark the epochs when a new valley is visited, while the exit times $\\zeta ^{(i)}_n$ are the epochs at which a valley is left.","The reader should notice that we do not restrict ourselves to valleys of order $i$ but include those valleys which up to order $i$ have not yet been absorbed by some larger valley.","Exit and entrance times never coincide since there is no way to go from one valley to another without hitting a non-assigned state - crests are always non-assigned (see Lemma REF ).","In this section, we will focus on $\\zeta _0^{(i)}$ for any fixed $i$ , thus writing $\\zeta _0:=\\zeta _0^{(i)}$ hereafter, but later for the macroscopic process the other times will be needed as well.","For each valley $V^{(i)}(m),\\,m\\in M^{(i)}$ , let us define $s_m\\ =\\ s_m^{(i)}\\ :=\\ \\mathop {\\mathrm {argmin}}_{s\\in \\partial ^+V^{(i)}(m)}E(s)\\ =\\ \\mathop {\\mathrm {argmin}}_{s\\in \\partial ^+V^{(i)}(m)}E(z^*(m,s)),$ where the second equality follows from Lemma REF .","Theorem 2.12 Let $m\\in M^{(i)}$ .","Then $\\lim _{\\beta \\rightarrow \\infty }\\frac{1}{\\beta }\\ln \\mathbb {E}_{r}\\zeta _0\\ =\\ E(s_m)-E(m)$ for any $r\\in V^{(i)}(m)$ .","For the upper bound, we need a result from [21], which in our notation is: Proposition 2.13 (see Main Theorem (iv) in [21]) For any $1\\le i\\le \\mathfrak {n},\\, \\beta $ sufficiently large and $t> 2^{i-1}\\exp (\\beta (E(s_{m^{(i-1)}})-E(m^{(i-1)})+2i|\\mathcal {S}|\\gamma _{\\beta }))$ , $\\sup _{x\\in M^{(i-1)}}\\mathbb {P}_x(\\tau _{M^{(i)}}>t)\\le \\exp (-\\Delta \\beta )$ holds true with a positive constant $\\Delta $ , where $M^{(0)}=\\mathcal {S}$ should be recalled.","This will now be used to show the following result.","Lemma 2.14 Fix $1\\le i\\le \\mathfrak {n},\\, m\\in M^{(i)}$ and $r\\in V^{(i)}(m)$ .","Then, for any $\\beta $ sufficiently large and $t>2^{i}\\exp (\\beta (E(s_{m})-E(m)+2(i+1)|\\mathcal {S}|\\gamma _{\\beta }))$ , it holds true that $\\mathbb {P}_r(\\zeta _0<(i+1)t)\\ \\ge \\ \\frac{1}{4}.$ [Proof:] Let us first note that we can always arrange for $m$ being equal to $m^{(i)}$ by sufficiently decreasing the energy function at any $m^{\\prime }\\in M^{(i)}\\backslash \\lbrace m\\rbrace $ so as to make $E(s_{m})-E(m)$ minimal among all states in $M^{(i)}$ .","This affects neither the valley $V^{(i)}(m)$ and its outer boundary nor the distribution of $\\zeta _0$ when starting in $m$ , for this distribution does not depend on the energy landscape outside of $V^{(i)}(m)\\cup \\partial ^+V^{(i)}(m)$ .","When applying the previous proposition, the constant $\\Delta $ may have changed but is still positive which suffices for our purposes.","So let $m=m^{(i)}$ hereafter.","Fix $t>2^i\\exp (\\beta (E(s_{m})-E(m)+2(i+1)|\\mathcal {S}|\\gamma _{\\beta }))$ and $T:=it$ .","Since $E(s_{m})-E(m)\\ge E(s_{m^{(j)}})-E(m^{(j)})$ for every $1\\le j\\le i$ , we infer $\\mathbb {P}_r(\\tau _{M^{(i)}}\\le T)\\ &\\ge \\ \\mathbb {P}_r(\\tau _{M^{(i)}}\\le T, \\tau _{M^{(1)}}\\le t)\\\\&\\ge \\ \\mathbb {P}_r(\\tau _{M^{(1)}}\\le t)\\,\\inf _{x\\in M^{(1)}}\\mathbb {P}_{x}(\\tau _{M^{(i)}}\\le (i-1)t)\\\\&\\ge \\ \\mathbb {P}_r(\\tau _{M^{(1)}}\\le t)\\,\\inf _{x\\in M^{(1)}}\\mathbb {P}_{x}(\\tau _{M^{(2)}}\\le t)\\,\\inf _{x\\in M^{(2)}}\\mathbb {P}_{x}(\\tau _{M^{(i)}}\\le (i-2)t)\\\\&\\hspace*{28.45274pt}\\vdots \\\\&\\ge \\ \\prod _{j=1}^{i}\\inf _{x\\in M^{(j-1)}}\\mathbb {P}_{x}(\\tau _{M^{(j)}}\\le t)\\\\&\\ge \\ \\big (1-\\exp (-\\Delta \\beta )\\big )^i\\\\&\\ge \\ \\frac{3}{4}$ for $\\beta $ sufficiently large.","Furthermore, for $\\beta $ so large that $\\mathbb {P}_r(\\tau _{M^{(i)}}<\\tau _m)\\le 1/4$ , we find that $\\mathbb {P}_r(\\tau _{M^{(i)}}\\le T)\\ &=\\ \\mathbb {P}_r(\\tau _{M^{(i)}}=\\tau _m\\le T)+\\mathbb {P}_r(\\tau _{M^{(i)}}\\le T, \\tau _{M^{(i)}}<\\tau _m)\\\\&\\le \\ \\mathbb {P}_r(\\tau _m\\le T)+\\mathbb {P}_r(\\tau _{M^{(i)}}<\\tau _m)\\\\&\\le \\ \\mathbb {P}_r(\\tau _m\\le T)+\\frac{1}{4}.$ By combining both estimates, we obtain $\\mathbb {P}_r(\\tau _m\\le T)\\ \\ge \\ \\mathbb {P}_r(\\tau _{M^{(i)}}\\le T)-\\frac{1}{4}\\ \\ge \\ \\frac{1}{2}.$ Hence, state $m$ is hit in time $T$ with at least probability $1/2$ when starting in $r$ .","Since $m=m^{(i)}$ , we further have $\\mathbb {P}_m(\\zeta _0\\le t)\\ \\ge \\ \\mathbb {P}_m(\\tau _{M^{(i+1)}}\\le t)\\ \\ge \\ 1-\\exp (-\\Delta \\beta )\\ \\ge \\ \\frac{1}{2}$ for $\\beta $ sufficiently large.","Hence, state $s_m$ is hit in time $t$ with at least probability $1/2$ when starting in $m$ .","By combining the estimates, we finally obtain $\\mathbb {P}_{r}(\\zeta _0\\le (i+1)t)\\ &\\ge \\ \\mathbb {P}_{r}(\\zeta _0\\le T+t|\\tau _{m}\\le T)\\,\\mathbb {P}_{r}(\\tau _{m}\\le T)\\\\&\\ge \\ \\mathbb {P}_{r}(\\tau _{m}\\le T)\\,\\mathbb {P}_{m}(\\zeta _0\\le t)\\\\&\\ge \\ \\frac{1}{4},$ which proves our claim.","[Proof:](of Theorem REF ) Using the lemma just shown, we infer $\\mathbb {E}_{r}(\\zeta _0)\\ &\\le \\ (i+1)t\\sum _{n\\ge 0}(n+1)\\,\\mathbb {P}_{r}\\left(n(i+1)t\\le \\zeta _0<(n+1)(i+1)t\\right)\\\\&=\\ (i+1)t\\sum _{n\\ge 0}(n+1)\\Big (\\mathbb {P}_{r}\\left(\\zeta _0\\ge n(i+1)t\\right)-\\mathbb {P}_{r}\\left(\\zeta _0\\ge (n+1)(i+1)t\\right)\\Big )\\\\&=\\ (i+1)t\\sum _{n\\ge 0}\\mathbb {P}_{r}\\left(\\zeta _0\\ge n(i+1)t\\right)\\\\&\\le \\ (i+1)t\\sum _{n\\ge 0}\\left(\\max _{x\\in V}\\mathbb {P}_x\\left(\\zeta _0\\ge (i+1)t\\right)\\right)^n\\\\&\\le \\ (i+1)t\\sum _{n\\ge 0}\\left(\\frac{3}{4}\\right)^n\\\\&=\\ 4(i+1)t,$ where $t:=2^i\\exp (\\beta (E(s_{m})-E(m)+2(i+1)|\\mathcal {S}|\\gamma _{\\beta }))+1$ .","Since $\\gamma _{\\beta }\\rightarrow 0$ , we get in the limit $\\lim _{\\beta \\rightarrow \\infty }\\frac{1}{\\beta }\\ln \\mathbb {E}_{r}\\zeta _0\\ \\le \\ E(s_m)-E(m)$ for all $r\\in V^{(i)}(m)$ .","Turning to the lower bound, define a sequence of stopping times, viz.","$\\rho _0:=0$ and $\\rho _n:=\\inf \\lbrace k>\\rho _{n-1}|X_k=m\\text{ or }X_{k}\\in \\partial ^+ V\\rbrace $ for $n\\ge 1$ .","Then $Z_n:=X_{\\rho _n}$ , $n\\ge 0$ , forms a Markov chain the transition probabilities of which when starting in $m$ can be estimated with the help of Proposition REF , namely $\\mathbb {P}(Z_1\\in \\partial ^+ V^{(i)}(m)|Z_0=m)\\ &=\\ \\mathbb {P}_{m}(\\rho _1=\\zeta _0)\\\\&=\\ \\mathbb {P}_{m}(\\zeta _0<\\tau _m)\\\\&\\le \\ \\sum _{s\\in \\partial ^+V}\\mathbb {P}_m(\\tau _s<\\tau _m)\\\\&\\le \\ K(\\beta )\\,e^{-\\beta (\\min _{s\\in \\partial ^+V}E(z^*(m,s))-E(m)-2\\gamma _{\\beta })}\\\\&=\\ K(\\beta )\\,e^{-\\beta (E(s_m)-E(m)-2\\gamma _{\\beta })}$ where $K(\\beta )\\rightarrow K\\in (0,\\infty )$ as $\\beta \\rightarrow \\infty $ .","Further defining $\\nu :=\\inf \\lbrace k\\ge 0| Z_k\\in \\partial ^+ V\\rbrace $ , this implies in combination with a geometric trials argument that $\\mathbb {P}_{m}(\\nu >n)\\ \\ge \\ \\left(1-K(\\beta )\\,e^{-\\beta (E(s_m)-E(m)-2\\gamma _{\\beta })}\\right)^{n-1}.$ As a consequence, $\\mathbb {E}_{m}\\zeta _0\\ &=\\ \\sum _{n\\ge 1} \\mathbb {E}_{m}(\\underbrace{\\rho _n-\\rho _{n-1}}_{\\ge 1}){1}_{\\lbrace \\nu \\ge n\\rbrace }\\ \\ge \\ \\sum _{n\\ge 1} \\mathbb {P}_{m}(\\nu \\ge n)\\ \\ge \\ K(\\beta )^{-1}\\,e^{\\beta (E(s_m)-E(m)-2\\gamma _{\\beta })}.$ For arbitrary $r\\in V$ , we now infer $\\mathbb {E}_{r}\\zeta _0\\ &=\\ \\mathbb {E}_{r}\\zeta _0{1}_{\\lbrace \\zeta _0\\le \\rho _1\\rbrace }+\\mathbb {E}_{r}\\zeta _0{1}_{\\lbrace \\zeta _0>\\rho _1\\rbrace }\\\\&\\ge \\ \\mathbb {E}_{r}\\big (\\mathbb {E}_r(\\zeta _0{1}_{\\lbrace \\zeta _0>\\rho _1\\rbrace }|X_{\\rho _{1}}=m)\\big )\\\\&\\ge \\ \\mathbb {E}_{r}{1}_{\\lbrace \\zeta _0>\\rho _1\\rbrace }\\mathbb {E}_{m}\\zeta _0\\\\&\\ge \\ \\mathbb {P}_{r}(\\zeta _0>\\rho _1)K(\\beta )^{-1}\\,e^{\\beta (E(s_m)-E(m)-2\\gamma _{\\beta })}\\\\&\\ge \\ \\frac{1}{2}\\,K(\\beta )^{-1}\\,e^{\\beta (E(s_m)-E(m)-2\\gamma _{\\beta })}$ for all sufficiently large $\\beta $ , because $\\lim _{\\beta \\rightarrow \\infty }\\mathbb {P}_{r}(\\zeta _0>\\rho _1)=1$ (Theorem REF ).","Finally, by taking logarithms and letting $\\beta $ tend to $\\infty $ , we arrive at the inequality $\\lim _{\\beta \\rightarrow \\infty }\\frac{1}{\\beta }\\ln \\mathbb {E}_{r}\\zeta _0\\ \\ge \\ E(s_m)-E(m)$ which completes the proof.","In [12], $E(s_m)-E(m),\\,m\\in M^{(i)},$ is referred to as the depth of the valley $V(m)$ .","Therefore, Property 3 from the Introduction holds true and we can relate thermodynamics of the system (energies) to dynamics of the chain (holding times) in a very precise way.","Especially, there is no universal scale for the times spent in different valleys because in general they differ exponentially."],["Macroscopic View: Transitions between valleys","With the help of the nested state space decompositions into valleys of different orders and around bottom states of different stability, we will now be able to provide an appropriate definition of the metabasins (MB) that has been announced and to some extent discussed in the Introduction.","We will further define and study macroscopic versions of the original process $X=(X_{n})_{n\\ge 0}$ .","These are obtained by choosing different levels of aggregation in the sense that they keep track only of the valleys of a chosen level that are visited by $X$ .","The motivation behind this approach is, on the one hand, to exhibit strong relations between properties of the energy landscape and the behavior of $X$ (as in Theorem REF ) and, on the other hand, to describe essential features of this process by looking at suitable macroscopic scales.","In the subsequent definition of aggregated versions of $X$ , we will distinguish between two variants: A time-scale preserving aggregation that, for a fixed level and each $n$ , keeps track of the valley the original chain visits at time $n$ and thus only blinds its exact location within a valley.","An accelerated version that, while also keeping track of the visited valleys, further blinds the sojourn times within a valley by counting a visit just once.","Actually, the definition of these aggregations at a chosen level $i$ is a little more complicated because their state space, denoted $\\mathcal {S}^{(i)}$ below and the elements of which we call level $i$ metastates, also comprises the non-assigned states at level $i$ as well as the minima of those valleys that were formed at an earlier level and whose merger is pending at level $i$ because their minima are not attracted at this level.","Definition 3.1 Fix $1\\le i\\le \\mathfrak {n}$ , let $\\mathcal {S}^{(i)}:=\\lbrace m^{(j)}\\in M^{(1)}|\\,l(j)>i\\rbrace \\cup N^{(i)}$ and $V^{(i)}(s):={\\left\\lbrace \\begin{array}{ll}V^{(i)}(m^{(j)}),&\\text{if }s=m^{(j)} \\textrm { for some j\\ge i}\\\\V^{(j)}(m^{(j)}),&\\text{if }s=m^{(j)} \\textrm { for some j\\sigma _{n-1}\\Big |\\overline{Y}_k^{(i)}\\ne \\overline{Y}_{k-1}^{(i)}\\right\\rbrace $ for $n\\ge 1$ .","We call $\\overline{Y}^{(i)}=(\\overline{Y}_n^{(i)})_{n\\ge 0}$ and $Y^{(i)}=(Y_{n}^{(i)})_{n\\ge 0}$ the aggregated chain (AC) and the accelerated aggregated chain (AAC) (at level $i$ ) associated with $X=(X_n)_{n\\ge 0}$ .","So, starting in an arbitrary valley, the original chain stays there for a time $\\zeta _0=\\zeta _{0}^{(i)}$ (as defined in Definition REF ) before it jumps via some non-assigned states $k_1{,}..., k_l$ (staying a geometric time in each of these states) to another valley at time $\\xi _1=\\xi _{1}^{(i)}$ .","There it stays for $\\zeta _1-\\xi _1$ time units before it moves on in a similar manner.","By going from $X$ to its aggregation $\\overline{Y}^{(i)}$ at level $i$ , we regard the whole valley $V^{(i)}(s)$ for $s\\in \\mathcal {S}^{(i)}$ as one single metastate and therefore give up information about the exact location of $X$ within a valley.","$\\overline{Y}^{(i)}$ is a jump process on $\\mathcal {S}^{(i)}$ with successive sojourn times $\\sigma _{n+1}-\\sigma _n,\\,n\\ge 0$ , which do not only depend on the valley but also on the states of entrance and exit.","The AAC then is the embedded chain, viz.","$\\overline{Y}^{(i)}_n=\\sum _{j\\ge 0}Y^{(i)}_j{1}_{\\lbrace \\sigma _j\\le n<\\sigma _{j+1}\\rbrace },$ giving the states only at jumps epochs: starting from the minimum of a first valley it moves to states $k_1{,}...,k_l\\in N^{(i)}$ and then proceeds to the minimum of a second valley, and so on.","Of course, at small temperatures the time spent in a non-assigned state or in a valley around a low order metastable state is very small compared to the time spent in a valley around a metastable state of higher order.","Thus, such states can be seen as instantaneous and of little importance for the evolution of the process.","We account for them nonetheless for two reasons.","First, in the path-dependent definition mentioned in the Introduction and used in Physics, they build small MB of great transitional activity of the process and are thus relevant in view of our goal to provide a definition of MB that conforms as much as possible to a path-dependent one.","Second, a complete partitioning of the state space that is an assignment of every $s\\in \\mathcal {S}$ to a metastate via a global algorithm fails when merely focusing on $\\left\\lbrace V^{(i)}(m), m\\in M^{(i)}\\right\\rbrace $ because there is neither an obvious nor natural way how to assign non-assigned states to them.","The incoherent scattering function and its associated relaxation time, for $X$ defined by $S(q,n):=\\mathbb {E}_{\\pi }\\cos \\big (q|X_n-X_0|\\big )$ (with $|\\cdot |$ being Euclidean distance in phase space) and $\\tau _q(\\varepsilon ):=\\inf \\lbrace n|S(q,n)\\le \\varepsilon \\rbrace ,\\quad \\varepsilon >0,$ respectively, may serve as an example which shows the strong relation between the behavior of the original process and its macroscopic versions.","For more detailed information on the meaning and relevance of $S(q,n)$ as a measure of incoherent scattering between the initial state of a glass-forming system and its state $n$ time steps onward, we refer to the survey by Heuer [12].","Proposition 3.2 For each $1\\le i\\le \\mathfrak {n}$ there is a constant $\\Delta (i)$ such that $\\sup _{n\\ge 0}\\,\\mathbb {P}_{\\pi }\\left(X_n\\ne \\overline{Y}^{(i)}_n\\right)\\le e^{-\\Delta (i)\\beta }.$ As a consequence, for any given $\\varepsilon >0$ , the incoherent scattering functions of $X$ and $\\overline{Y}^{(i)}$ differ by at most $4\\varepsilon $ for $\\beta $ sufficiently large.","[Proof:] Use Lemma REF to infer $\\mathbb {P}_{\\pi }(X_n\\ne \\overline{Y}^{(i)}_n)&=\\sum _{s\\in \\mathcal {S}^{(i)}\\backslash N^{(i)}}\\mathbb {P}_{\\pi }(\\overline{Y}^{(i)}_n=s)\\sum _{x\\in V^{(i)}(s)\\backslash \\lbrace s\\rbrace }\\frac{\\pi (x)}{\\pi (V^{(i)}(s))}\\\\&\\le \\sum _{s\\in \\mathcal {S}^{(i)}\\backslash N^{(i)}}\\mathbb {P}_{\\pi }(\\overline{Y}^{(i)}_n=s)\\sum _{x\\in V^{(i)}(s)\\backslash \\lbrace s\\rbrace }\\frac{\\pi (x)}{\\pi (s)}\\\\&\\le \\sum _{s\\in \\mathcal {S}^{(i)}\\backslash N^{(i)}}\\mathbb {P}_{\\pi }(\\overline{Y}^{(i)}_n=s)\\sum _{x\\in V^{(i)}(s)\\backslash \\lbrace s\\rbrace }e^{-\\beta (E(x)-E(s)-2\\gamma _{\\beta })}\\\\&\\le \\max _{s\\in \\mathcal {S}^{(i)}\\backslash N^{(i)}}|V^{(i)}(s)|\\max _{x\\in V^{(i)}(s)\\backslash \\lbrace s\\rbrace }e^{-\\beta (E(x)-E(s)-2\\gamma _{\\beta })}.$ This proves equation (REF ) because $E(x)>E(s)$ for each $x\\in V^{(i)}(s)\\backslash \\lbrace s\\rbrace , s\\in \\mathcal {S}^{(i)}\\backslash N^{(i)}$ .","Now let $\\beta $ be so large that $e^{-\\Delta (i)\\beta }\\le \\varepsilon $ for a given $\\varepsilon >0$ and observe that $\\mathbb {E}_{\\pi }&\\cos \\big (q|X_n-X_0|\\big )\\\\&=\\ \\int _{\\lbrace X_n=\\overline{Y}_{n}^{(i)},X_0=\\overline{Y}^{(i)}_0\\rbrace }\\cos \\big (q|X_n-X_0|\\big )\\ d\\mathbb {P}_{\\pi }\\ +\\ \\int _{\\lbrace X_n\\ne \\overline{Y}_n^{(i)}\\rbrace \\cup \\lbrace X_0\\ne \\overline{Y}^{(i)}_0\\rbrace }\\cos \\big (q|X_n-X_0|\\big )\\ d\\mathbb {P}_{\\pi }\\\\&\\le \\ \\int _{\\lbrace X_n=\\overline{Y}_n^{(i)},X_0=\\overline{Y}^{(i)}_0\\rbrace }\\cos \\big (q|X_n-X_0|\\big )\\ d\\mathbb {P}_{\\pi }\\ +\\ \\mathbb {P}_{\\pi }(X_n\\ne \\overline{Y}_n^{(i)})\\ +\\ \\mathbb {P}_{\\pi }(X_0\\ne \\overline{Y}^{(i)}_0)\\\\&=\\ \\mathbb {E}_{\\pi }\\cos \\left(q|\\overline{Y}_n^{(i)}-\\overline{Y}_0^{(i)}|\\right)\\ -\\ \\int _{\\lbrace X_n\\ne \\overline{Y}_n^{(i)}\\rbrace \\cup \\lbrace X_0\\ne \\overline{Y}^{(i)}_0\\rbrace }\\cos \\left(q|\\overline{Y}_n^{(i)}-\\overline{Y}_0^{(i)}|\\right)\\ d\\mathbb {P}_{\\pi } +\\ 2\\varepsilon \\\\&\\le \\ \\mathbb {E}_{\\pi }\\cos \\left(q|\\overline{Y}_n^{(i)}-\\overline{Y}_0^{(i)}|\\right)\\ +\\ 4\\varepsilon $ and, by a similar argument, $\\mathbb {E}_{\\pi }\\cos \\left(q|\\overline{Y}_n^{(i)}-\\overline{Y}_0^{(i)}|\\right)\\ \\le \\ \\mathbb {E}_{\\pi }&\\cos \\big (q|X_n-X_0|\\big )\\ +\\ 4\\varepsilon .$ This completes the proof."],["(Semi-)Markov Property","In general, both aggregated chains are no longer Markovian.","Transition probabilities of the AAC not only depend on the current state, i.e.","the current valley, but also on the entrance state into that valley, whereas transition probabilities of the AC depend on the current sojourn times which in turn depend on the previous, the present and the next state.","On the other hand, since valleys are defined in such a way that asymptotically almost surely (a.a.s.","), i.e., with probability tending to one as $\\beta \\rightarrow \\infty $ , the minimum will be reached from anywhere inside the valley before the valley is left, and since, furthermore, the exit state on the outer boundary a.a.s.","equals the one with the smallest energy, the AAC will be shown below to converge to a certain Markov chain on $\\mathcal {S}^{(i)}$ .","Also, the sojourn times depend on the past only via the last and the current state.","This means that the AC converges to a semi-Markov chain (for semi-Markov chains see for example [1]): Definition 3.3 Given any nonempty countable set $\\mathcal {S}$ , let $(M_{n},T_{n})_{n\\ge 0}$ be a bivariate temporally homogeneous Markov chain on $\\mathcal {S}\\times \\mathbb {N}$ , with transition kernel $Q(s,\\cdot )$ only depending on the first component, viz., for all $n\\ge 0, s\\in S$ and $t\\ge 0$ , $\\mathbb {P}(M_{n+1}=s, T_{n+1}\\le t|M_{n},T_{n})=Q(M_{n},\\lbrace s\\rbrace \\times [0,t])$ holds.","Put $S_n:=\\sum _{i=0}^{n}T_i$ for $n\\ge 0$ and $\\nu (t):=\\max \\lbrace n\\ge 0|S_n\\le t\\rbrace \\ (\\max \\emptyset :=0)$ for $t\\ge 0$ .","Then $Z_n:=M_{\\nu (n)}, n\\ge 0,$ is called semi-Markov chain with embedded Markov chain $(M_{n})_{n\\ge 0}$ and sojourn or holding times $T_{0},T_{1}{,}...$ .","Note that equation (REF ) holds iff $M=(M_{n})_{n\\ge 0}$ forms a temporally homogeneous Markov chain and the $(T_{n})_{n\\ge 0}$ are conditionally independent given $M$ such that the distribution of $T_{n}$ only depends on $M_{n-1},M_{n}$ for $n\\ge 1$ (in a temporally homogeneous manner), and on $M_{0}$ for $n=0$ .","Note further that we have specialized to the case where holding times take values in $\\mathbb {N}$ only (instead of $(0,\\infty )$ ).","Recall from (REF ) the definition of $s_m$ for $m\\in \\mathcal {S}^{(i)}\\backslash N^{(i)}$ and notice that the second equality there entails $E(z^{*}(m,s_{m}))\\sigma _1)\\\\&=\\ \\mathbb {P}_{r}(Y_1=s)\\,\\mathbb {P}_x(\\tau _{r}<\\sigma _1)+\\mathbb {P}_x(Y_1=s,\\tau _{r}>\\sigma _1).$ The last two summands can further be bounded by $\\mathbb {P}_{r}(Y_1=s)\\,\\mathbb {P}_x(\\tau _{r}<\\sigma _1)\\ \\le \\ \\mathbb {P}_{r}(Y_1=s)\\quad \\text{and}\\quad \\mathbb {P}_x(Y_1=s,\\tau _{r}>\\sigma _1)\\ \\le \\ \\mathbb {P}_x(\\sigma _1<\\tau _{r}).$ For the last probability, Theorem REF ensures $\\mathbb {P}_x(\\sigma _1<\\tau _{r})\\ &\\le \\ \\sum _{z\\in \\partial ^+ V^{(i)}(r)}\\mathbb {P}_x(\\tau _z<\\tau _{r})\\ \\le \\ \\sum _{z\\in \\partial ^+ V^{(i)}(r)} \\tilde{\\varepsilon }(x,r,z,\\beta )\\ \\stackrel{\\beta \\rightarrow \\infty }{\\longrightarrow }\\ 0.$ Consequently, as $\\beta \\rightarrow \\infty $ , $\\mathbb {P}_{r}(Y_1=s)\\ &=\\ \\left(1-\\sum _{z\\in \\partial ^+ V^{(i)}(r)} \\tilde{\\varepsilon }(x,r,z,\\beta )\\right)\\mathbb {P}_{r}(Y_1=s)+o(1)\\\\&\\le \\ \\big (1-\\mathbb {P}_x(\\sigma _1<\\tau _{r})\\big )\\,\\mathbb {P}_{r}(Y_1=s)+o(1)\\\\&\\le \\ \\mathbb {P}(Y_{n+1}=s|X_{\\sigma _n}=x,Y_n=r)+o(1)\\\\&\\le \\ \\mathbb {P}_{r}(Y_1=s)+\\sum _{z\\in \\partial ^+ V^{(i)}(r)} \\tilde{\\varepsilon }(x,r,z,\\beta )+o(1)\\\\&=\\ \\mathbb {P}_{r}(Y_1=s)+o(1),$ and therefore $\\mathbb {P}&(Y_{n+1}=s|Y_{n}=r,Y_{n-1}=m_{n-1}{,}...,Y_0=m_0)\\\\&=\\ \\sum _{x\\in V^{(i)}(r)}\\mathbb {P}(Y_{n+1}=s|X_{\\sigma _n}=x, Y_n=r)\\,\\mathbb {P}(X_{\\sigma _n}=x|Y_{n}=r,Y_{n-1}=m_{n-1}{,}...,Y_0=m_0)\\\\&=\\ \\mathbb {P}_{r}(Y_1=s)+o(1).$ It remains to verify that $\\mathbb {P}_{r}(Y_1=s)={\\widehat{p}}(r,s)+o(1)$ for any $r,s\\in \\mathcal {S}^{(i)}$ .","If $r\\in \\mathcal {S}^{(i)}\\backslash N^{(i)}$ , then $\\sigma _{1}=\\tau _{N^{(i)}}$ and $Y_{1}=X_{\\tau _{N^{(i)}}}$ .","Since $E(z^*(r,s_r))0$ , the sojourn times $\\sigma _{n+1}-\\sigma _{n}$ , $n\\ge 0$ , of the AC $\\overline{Y}^{(i)}$ are conditionally independent given $Y^{(i)}$ .","The conditional law of $\\sigma _{n+1}-\\sigma _{n}$ depends only on $(Y_{n-1}^{(i)},Y_{n}^{(i)},Y_{n+1}^{(i)})$ and satisfies $&\\mathbb {P}\\big (\\sigma _{n+1}-\\sigma _{n}\\in \\cdot ~|Y^{(i)}_{n-1}=x,Y^{(i)}_{n}=y,Y^{(i)}_{n+1}=z\\big )\\nonumber \\\\&\\hspace{14.22636pt}=\\ Q((x,y,z),\\cdot \\, )\\ :=\\ {\\left\\lbrace \\begin{array}{ll}\\hfill \\textit {Geom}(1-p(y,y)),&\\text{if }y\\in N^{(i)}\\\\\\sum _{s\\in V^{(i)}(y),s\\sim x}\\mathbb {P}_s(\\sigma _1\\in \\cdot ~|Y^{(i)}_1=z)\\,\\mathbb {P}_{x}(X_{\\sigma _1}=s),&\\text{if }y\\notin N^{(i)}\\end{array}\\right.","}$ for all $x,y,z\\in \\mathcal {S}^{(i)}$ with $\\mathbb {P}(Y^{(i)}_{n-1}=x,Y^{(i)}_{n}=y,Y^{(i)}_{n+1}=z)>0$ and $n\\ge 1$ .","[Proof:] The assertions follow easily when observing that, on the one hand, at least one state $y\\in N^{(i)}$ must be visited between two states $x,z\\in \\mathcal {S}^{(i)}\\backslash N^{(i)}$ (Lemma REF ) and that, on the other hand, the original chain $X$ and its aggregation $\\overline{Y}^{(i)}$ coincide at any epoch where a non-assigned state is hit, which renders the Markov property of $\\overline{Y}^{(i)}$ at these epochs.","Further details are omitted.","In order to formulate the next result, let $0={\\widehat{\\sigma }}_{0}<{\\widehat{\\sigma }}_{1}<...$ be an increasing sequence of random variables such that its increments ${\\widehat{\\sigma }}_{n+1}-{\\widehat{\\sigma }}_{n}, n\\ge 0$ , are conditionally independent given the asymptotic jump chain ${\\widehat{Y}}^{(i)}$ .","Moreover, let the conditional law of ${\\widehat{\\sigma }}_{n+1}-{\\widehat{\\sigma }}_{n}$ depend only on $\\big ({\\widehat{Y}}^{(i)}_{n-1},{\\widehat{Y}}^{(i)}_{n},{\\widehat{Y}}^{(i)}_{n+1}\\big )$ and be equal to $Q\\big (\\big ({\\widehat{Y}}^{(i)}_{n-1},{\\widehat{Y}}^{(i)}_{n},{\\widehat{Y}}^{(i)}_{n+1}\\big ),\\cdot \\,\\big )$ , with $Q$ as defined in (REF ).","Then $(({\\widehat{Y}}^{(i)}_{n},{\\widehat{Y}}^{(i)}_{n+1}),{\\widehat{\\sigma }}_{n+1})_{n\\ge 0}$ forms a Markov renewal process and $\\big ({\\widehat{Y}}^{(i)}_{{\\widehat{\\nu }}(n)},{\\widehat{Y}}^{(i)}_{{\\widehat{\\nu }}(n+1)}\\big )_{n\\ge 0}$ a semi-Markov chain, where ${\\widehat{\\nu }}(n):=\\sup \\lbrace k\\ge 0|{\\widehat{\\sigma }}_{k}\\le n\\rbrace $ .","Proposition 3.6 For each $1\\le i\\le \\mathfrak {n}$ , $((Y^{(i)}_{n},Y^{(i)}_{n+1}),\\sigma _{n+1})_{n\\ge 0}$ converges to the Markov renewal process $(({\\widehat{Y}}^{(i)}_{n},{\\widehat{Y}}^{(i)}_{n+1}),{\\widehat{\\sigma }}_{n+1})_{n\\ge 0}$ in the sense that $\\lim _{\\beta \\rightarrow \\infty }\\frac{\\mathbb {P}_{y_0}\\big (\\big (Y^{(i)}_{k},Y^{(i)}_{k+1}\\big )=(y_{k},y_{k+1}),\\,\\sigma _{k+1}=i_{k+1},\\,0\\le k\\le n\\big )}{\\mathbb {P}_{y_0}\\big (\\big ({\\widehat{Y}}^{(i)}_{k},{\\widehat{Y}}^{(i)}_{k+1}\\big )=(y_{k},y_{k+1}),\\,{\\widehat{\\sigma }}_{k+1}=i_{k+1},\\,0\\le k\\le n\\big )}\\ =\\ 1$ for all $y_{0}{,}...,y_{n+1}\\in \\mathcal {S}^{(i)}$ , $00$ if there exists a nonempty subset $A\\mathcal {S}^{(i)}\\backslash N^{(i)}$ with the following property: For each $m_{1}\\in A$ , there exists $m_{2}\\in A$ such that $\\lim _{\\beta \\rightarrow \\infty }\\frac{1}{\\beta }\\left(\\ln \\left(\\mathbb {P}_{m_1}\\left(X_{\\xi _1}\\in V^{(i)}(m_2)\\right)\\right)-\\ln \\left(\\mathbb {P}_{m_1}\\left(X_{\\xi _1}\\in V^{(i)}(m)\\right)\\right)\\right)\\ge \\varepsilon $ for all $m\\in \\mathcal {S}^{(i)}\\backslash (N^{(i)}\\cup A)$ .","In other words, it is exponentially more likely to stay in $A$ than to leave it (ignoring intermediate visits to non-assigned states).","In view of our principal goal to give a path-independent definition of MBs, we must point out that, by irreducibility, reciprocating jumps always occur with positive probability at any nontrivial level of aggregation and can therefore never be ruled out completely.","This is in contrast to the path-dependent version by Heuer [12] in which the non-occurrence of reciprocating jumps appears to be the crucial requirement.","As a consequence, Definition REF provides an alternative, probabilistic and verifiable criterion for reciprocating jumps to be sufficiently unlikely in a chosen aggregation.","The following proposition contains further information on which valleys are visited successively by providing the probabilities of making a transition from $V^{(i)}(m)$ to $V^{(i)}(m^{\\prime })$ for any $m,m^{\\prime }\\in \\mathcal {S}^{(i)}\\backslash N^{(i)}$ .","It is a direct consequence of the asymptotic results in the previous subsection, notably Proposition REF .","Proposition 3.8 Let $m\\in \\mathcal {S}^{(i)}\\backslash N^{(i)}, s_m\\in \\partial ^+V^{(i)}(m)$ be as defined in (REF ) (i.e., the state on the outer boundary of $V^{(i)}(m)$ with minimal energy).","Then $\\lim _{\\beta \\rightarrow \\infty }\\mathbb {P}_m(X_{\\xi _1}\\in V^{(i)}(m))\\ =\\ {\\widehat{p}}(s_m,m)\\ =\\ \\frac{\\sum _{r\\in \\mathcal {N}(s_m)\\cap V^{(i)}(m)}p^{*}(s_m,r)}{1-p^{*}(s_m,s_m)},$ while $\\lim _{\\beta \\rightarrow \\infty }\\mathbb {P}_m(X_{\\xi _1}\\in V^{(i)}(m^{\\prime }))\\ =\\ {\\widehat{p}}(s_{m},m^{\\prime })+\\sum _{n\\ge 1}\\sum _{r_{1}{,}...,r_{n}\\in N^{(i)}}{\\widehat{p}}(s,r_{1})\\cdot ...\\cdot {\\widehat{p}}(r_{n-1},r_{n})\\,{\\widehat{p}}(r_{n},m^{\\prime })$ for any other $m^{\\prime }\\in \\mathcal {S}^{(i)}\\backslash N^{(i)}$ .","The reader should notice that, as ${\\widehat{p}}(s,r)=0$ whenever $E(s)E(r_{1})>...>E(r_{n})>E(m^{\\prime })$ .","[Proof:] Let us first point out that $\\mathbb {P}_{r}(X_{\\xi _{0}}\\in V^{(i)}(m))=o(1)$ as $\\beta \\rightarrow \\infty $ for any $r\\in N^{(i)}$ such that $E(r)E(r)$ , whence the probability for such a path goes to zero as $\\beta \\rightarrow \\infty $ .","Noting further that $\\mathbb {P}_m(Y_1^{(i)}\\ne s_m)=o(1)$ as $\\beta \\rightarrow \\infty $ by Proposition REF , we now infer (with $\\xi _{n}=\\xi _{n}^{(i)}$ ) $\\mathbb {P}_m(X_{\\xi _1}\\in V^{(i)}(m))\\ &=\\ \\mathbb {P}_{s_m}(X_{\\xi _0}\\in V^{(i)}(m))+o(1)\\\\&=\\ \\mathbb {P}_{s_m}(Y^{(i)}_1=m)+\\sum _{r\\in \\mathcal {N}(s_{m})\\cap N^{(i)}}\\frac{p(s_{m},r)}{1-p(s_m,s_m)}\\,\\mathbb {P}_{r}(X_{\\xi _0}\\in V^{(i)}(m))+o(1)\\\\&=\\ {\\widehat{p}}(s_{m},m)+o(1).$ The expression for ${\\widehat{p}}(s_{m},m)$ in terms of the $p^{*}(s_{m},r)$ may be read off directly from the formula given in Proposition REF .","For $m^{\\prime }\\ne m$ , $m^{\\prime }\\in \\mathcal {S}^{(i)}\\backslash N^{(i)}$ , we obtain in a similar manner $\\mathbb {P}_m&(X_{\\xi _1}\\in V^{(i)}(m^{\\prime }))\\\\&=\\ \\mathbb {P}_{s_{m}}(Y_{1}^{(i)}=m^{\\prime })\\ +\\ \\sum _{n\\ge 1}\\sum _{r_{1}{,}...,r_{n}\\in N^{(i)}} \\mathbb {P}_{s_{m}}(Y_{1}^{(i)}=r_{1}{,}...,Y_{n}^{(i)}=r_{n},Y_{n+1}^{(i)}=m^{\\prime })+o(1)\\\\&=\\ {\\widehat{p}}(s_{m},m^{\\prime })+\\sum _{n\\ge 1}\\sum _{r_{1}{,}...,r_{n}\\in N^{(i)}}{\\widehat{p}}(s_m,r_{1})\\cdot ...\\cdot {\\widehat{p}}(r_{n-1},r_{n})\\,{\\widehat{p}}(r_{n},m^{\\prime })+o(1),$ the last line by another appeal to the afore-mentioned proposition.","In essence, the previous result tells us that a valley $V^{(i)}(m^{\\prime })$ is neighbored to $V^{(i)}(m)$ , that is, reachable with positive probability by the asymptotic jump chain ${\\widehat{Y}}^{(i)}$ (and thus by $Y^{(i)}$ at any temperature level $\\beta $ ) without intermediately hitting any other valley, iff there exists at least one (in terms of energies) decreasing path in $N^{(i)}$ from $s_{m}$ to $m^{\\prime }$ .","For any other such pair of valleys, connected by a path through states in $N^{(i)}$ , the transition probability decreases to zero exponentially in $\\beta $ .","If this path can be chosen to be unimodal, here called uphill-downhill-path, this can be stated in a very precise way as the next result shows.","Lemma 3.9 Let $m_0,m_1\\in \\mathcal {S}^{(i)}\\backslash N^{(i)}$ be two distinct local minima for some $0\\le i\\le \\mathfrak {n}$ .","Suppose there exists a minimal path $\\gamma =(\\gamma _0{,}...,\\gamma _k)$ from $s_{m_0}$ to $m_1$ not hitting any other valley but $V^{(i)}(m_{1})$ and such that $I(\\gamma _0{,}...,\\gamma _{k})=E(z^*(s,m_1))-E(s_{m_0})$ .","Then $\\lim _{\\beta \\rightarrow \\infty }\\frac{1}{\\beta }\\ln \\mathbb {P}_{m_0}(X_{\\xi _1}\\in V^{(i)}(m_1))\\ =\\ -\\left(E(z^*(m_0,m_1))-E(s_{m_0})\\right).$ Without assuming the existence of $\\gamma $ as stated, the result remains valid when replacing $=$ with $\\le $ .","Note that $I(\\gamma _0{,}... ,\\gamma _{k})=E(z^*(s_{m_0},m_1))-E(s_{m_0})$ does indeed imply the already mentioned property that $E(\\gamma _i)&>E(\\gamma _{i-1})\\quad \\textrm {for 1\\le i\\le j}\\\\\\text{and}\\quad E(\\gamma _i)&E(b)}(E(a)-E(b))+2\\gamma _{\\beta }\\beta \\right)+1\\right)\\max _{r\\in \\mathcal {S}}|\\mathcal {N}(r)|.$ For any $r\\in V^{(i)}(m_{1})$ , we have $E(z^{*}(m_{0},r))\\ge E(z^{*}(m_{1},r))$ and therefore $E(z^{*}(m_{0},m_{1}))\\ \\le \\ E(z^{*}(m_{0},r))\\vee E(z^{*}(r,m_{1}))\\ =\\ E(z^{*}(m_{0},r)).$ Using this in (REF ), we obtain $\\frac{\\mathbb {P}_{m_0}(\\xi ^{(i)}_1=\\tau _{V^{(i)}(m_1)}<\\tau _{m_0})}{\\mathbb {P}_{m_0}(\\zeta _0<\\tau _{m_0})}\\ \\le \\ K(\\beta )\\, |\\mathcal {S}|\\, e^{-\\beta (E(z^*(m_0,m_1))-E(z^*(m_0,x))-7\\gamma _{\\beta })}$ and then, upon choosing $x=s_{m_0}$ and noting that $E(z^{*}(m_{0},s_{m_0}))=E(s_{m_0})$ , $\\mathbb {P}_{m_0}(X_{\\xi _1}\\in V^{(i)}(m_1))\\ \\le \\ K(\\beta )\\,|\\mathcal {S}|\\,e^{-\\beta \\left(E\\left(z^*\\left(m_0,m_1\\right)\\right)-E(s_{m_0})-7\\gamma _{\\beta }\\right)}.$ By combining all previous results, we finally conclude $\\lim _{\\beta \\rightarrow \\infty }\\frac{1}{\\beta }\\ln \\mathbb {P}_{m_0}(X_{\\xi _1}\\in V^{(i)}(m_1))\\ =\\ -\\left(E(z^*(m_0,m_1))-E(s_{m_0})\\right)$ as asserted.","To summarize, which valleys are visited consecutively depends on (a) their spatial arrangement and (b) the energy barriers between them: A transition from one valley $V^{(i)}(m_0)$ to another valley $V^{(i)}(m_1)$ is only possible, if there exists a path from $s_{m_0}$ to $V^{(i)}(m_1)$ , not hitting any other valley.","This transition is made at small temperatures (i.e.","large $\\beta $ ) if the additional energy barrier $E(z^*(s_{m_0},m_1))-E(s_{m_0})$ is sufficiently small or in other words the energy barrier $E(z^*(s_{m_0},m_1))$ is approximately of the same height as all other energy barriers, including the barrier $E(z^*(s_{m_0},m_0))=E(s_{m_0})$ .","A result similar to the previous lemma holds true for transitions from $m\\in \\mathcal {S}^{(i)}\\backslash N^{(i)}$ to any $s\\in \\partial ^+V(m)$ .","Lemma 3.10 Let $m\\in \\mathcal {S}^{(i)}\\backslash N^{(i)}$ and $s\\in \\partial ^+V^{(i)}(m)$ .","Then $\\lim _{\\beta \\rightarrow \\infty }\\frac{1}{\\beta }\\ln \\mathbb {P}_m(Y_1=s)=-(E(s)-E(s_m)).$ [Proof:] For the proof, decompose again the event $\\lbrace Y_1=s\\rbrace $ with respect to the number of visits to $m$ before $V(m)=V^{(i)}(m)$ is left (or use Proposition REF ), giving $\\mathbb {P}_m(Y_1=s)\\ =\\ \\frac{\\mathbb {P}_m(\\sigma _1=\\tau _s<\\tau _m)}{\\mathbb {P}_m(\\sigma _1<\\tau _m)}.$ The proof of the lower bound is much more technical.","Let $\\gamma =(\\gamma _1{,}...,\\gamma _n)\\in \\Gamma ^*(m,s)$ be a minimal path which leaves $V(m)$ only in the last step and such that for any $\\gamma _i,\\gamma _j\\in \\gamma $ both, the subpath from $\\gamma _i$ to $\\gamma _j$ , and the inversed path from $\\gamma _j$ to $\\gamma _i$ , are minimal.","Define $r_0:=m \\quad \\textrm {and}\\quad r_1:=\\gamma _{i_0}\\quad \\textrm {with}\\quad i_0:=\\inf \\lbrace 0\\le i\\le n-1|E(\\gamma _{i+1})\\ge E(s_m)\\rbrace .$ In particular, $E(r_1)E(\\gamma _{i})\\rbrace \\Big \\rbrace ,$ and then successively for $k\\ge 1$ with $s_k=\\gamma _{i_k}\\ne s$ the records $s_{k+1}:=\\gamma _{i_{k+1}}$ with $i_{k+1}\\ :=\\ \\inf \\Big \\lbrace n\\ge i\\ge \\inf &\\lbrace j\\ge i_k|E(\\gamma _j)E(\\gamma _{i})\\rbrace \\Big \\rbrace .$ Note that the energy of these records is increasing.","Let $s_{k-1}$ be the last record defined in this way and $s_k:=s$ .","Since $E(z^*(m,s))=E(s)$ , $s_k$ is as well a record.","Given the records $s_1{,}...,s_k$ , for $1\\le i\\le k-1$ let $r_{2i}$ be the first minimum along $\\gamma $ after $s_i$ and $r_{2i+1}$ the last minimum along $\\gamma $ before $s_{i+1}$ .","Here a minimum along $\\gamma $ is some $\\gamma _i\\in \\gamma $ such that it is a minimum of $E$ restricted to $\\gamma $ .","Finally, let $r_{2k}:=s_k=s$ .","In the following we will proof that (a) $\\mathbb {P}_{r_{2j}}(\\tau _{r_{2j+1}}<\\zeta _0)\\rightarrow 1$ as $\\beta \\rightarrow \\infty $ for any $0\\le j\\le k-1$ , (b) $\\mathbb {P}_{r_1}(\\tau _{r_2}<\\zeta _0)\\ge e^{-\\beta (E(z^*(r_1,r_2))-E(s_m)+o(1))}$ , (c) $\\mathbb {P}_{r_{2j+1}}(\\tau _{r_{2j+2}}<\\zeta _0)\\ge e^{-\\beta (E(z^*(r_{2j+1},r_{2j+2}))-E(z^*(r_{2j-1},r_{2j}))+o(1))}$ for any $1\\le j\\le k-2$ , (d) $\\mathbb {P}_{r_{2k-1}}(\\tau _{r_{2k}}=\\zeta _0)\\ge e^{-\\beta (E(z^*(r_{2k-1},r_{2k}))-E(z^*(r_{2k-3},r_{2k-2}))+o(1))}$ .","This gives for $\\beta $ large enough $\\mathbb {P}_m(Y_1=s)\\ &\\ge \\ \\left(\\prod _{j=0}^{2k-2}\\mathbb {P}_{r_j}(\\tau _{r_{j+1}}<\\zeta _0)\\right)\\mathbb {P}_{r_{2k-1}}(\\tau _{r_{2k}}=\\zeta _0)\\\\&\\ge \\ \\frac{1}{2}\\left(\\prod _{j=0}^{k-2}\\mathbb {P}_{r_{2j+1}}(\\tau _{r_{2j+2}}<\\zeta _0)\\right)\\mathbb {P}_{r_{2k-1}}(\\tau _{r_{2k}}=\\zeta _0)\\\\&\\ge \\ \\frac{1}{2} e^{-\\beta (E(z^*(r_1,r_2))-E(s_m)+o(1))}\\cdot \\prod _{j=1}^{k-1}e^{-\\beta (E(z^*(r_{2j+1},r_{2j+2}))-E(z^*(r_{2j-1},r_{2j}))+o(1))}\\\\&=\\ \\frac{1}{2}e^{-\\beta (E(z^*(r_{2k-1},r_{2k}))-E(s_m)+o(1))}\\\\&=\\ e^{-\\beta (E(s)-E(s_m)+o(1))},$ and thus the assertion.","(a) For $j=0$ this is obvious since $E(z^*(r_0,r_1))E(r_1)\\quad \\textrm { and }\\quad E(z^*(r_1,r_2))\\ \\ge \\ E(s_m)>E(r_1).$ Therefore, $\\mathbb {P}_{r_1}((\\tau _{r_2}\\wedge \\zeta _0)<\\tau _{r_1})\\ &\\le \\ \\sum _{r^{\\prime }\\in \\partial ^+V(m)\\cup \\lbrace r_2\\rbrace }\\mathbb {P}_{r_1}(\\tau _{r^{\\prime }}<\\tau _{r_1})\\\\&\\le \\ \\sum _{r^{\\prime }\\in \\partial ^+V(m)\\cup \\lbrace r_2\\rbrace }e^{-\\beta (E(z^*(r_1,r^{\\prime }))-E(r_1)+o(1))}\\\\&\\le \\ e^{-\\beta (E(s_m)-E(r_1)+o(1))}.$ Combining the two estimates, we get $\\mathbb {P}_{r_1}(\\tau _{r_2}<\\zeta _0)\\ &=\\ \\frac{\\mathbb {P}_{r_1}(\\tau _{r_2}<(\\zeta _0\\wedge \\tau _{r_1}))}{\\mathbb {P}_{r_1}((\\tau _{r_2}\\wedge \\zeta _0)<\\tau _{r_1})}\\ \\ge \\ e^{-\\beta (E(z^*(r_1,r_2))-E(s_m)+o(1))}.$ (c) Let $1\\le j\\le k-2$ .","We use the same strategy as in the proof of (b).","So, again, by the definition of $r_{2j+1}$ and $r_{2j+2}$ , there is a unimodal path between them with cumulative activation energy $E(z^*(r_{2j+1},r_{2j+2}))-E(r_{2j+1})$ along this path, and $\\mathbb {P}_{r_{2j+1}}(\\tau _{r_{2j+2}}<(\\zeta _0\\wedge \\tau _{r_{2j+1}}))\\ \\ge \\ e^{-\\beta (E(z^*(r_{2j+1},r_{2j+2}))-E(r_{2j+1})+o(1))}.$ Furthermore, $E(z^*(r_{2j-1},r_{2j}))\\ \\le \\ E(z^*(r_{2j+1},r_{2j+2}))\\quad \\textrm { and }\\quad E(r_{2j+1})\\ <\\ E(z^*(r_{2j+1},r_{2j+2})).$ Finally, for any $r^{\\prime }\\in \\partial ^+V(m)$ , by use of Equation (REF ), $E(r_{2j+1})\\ &\\le E(z^*(r_{2j+1},r_{2j}))\\\\&<\\ E(z^*(r_{2j},r^{\\prime }))\\\\&\\le \\ E(z^*(r_{2j},r_{2j+1}))\\vee E(z^*(r_{2j+1},r^{\\prime }))\\\\&=\\ E(z^*(r_{2j+1},r^{\\prime })).$ Thus, $\\mathbb {P}_{r_{2j+1}}((\\tau _{r_{2j+2}}\\wedge \\zeta _0)<\\tau _{r_{2j+1}})\\ &\\le \\ \\sum _{r^{\\prime }\\in \\partial ^+V(m)\\cup \\lbrace r_{2j+2}\\rbrace }\\mathbb {P}_{r_{2j+1}}(\\tau _{r^{\\prime }}<\\tau _{r_{2j+1}})\\\\&\\le \\ \\sum _{r^{\\prime }\\in \\partial ^+V(m)\\cup \\lbrace r_{2j+2}\\rbrace }e^{-\\beta (E(z^*(r_{2j+1},r^{\\prime }))-E(r_{2j+1})+o(1))}\\\\&\\le \\ e^{-\\beta (E(r_{2j-1},r_{2j})-E(r_{2j+1})+o(1))}.$ Combining the two estimates, we get $\\mathbb {P}_{r_{2j+1}}(\\tau _{r_{2j+2}}<\\zeta _0)\\ &=\\ \\frac{\\mathbb {P}_{r_{2j+1}}(\\tau _{r_{2j+2}}<(\\zeta _0\\wedge \\tau _{r_{2j+1}}))}{\\mathbb {P}_{r_{2j+1}}((\\tau _{r_{2j+2}}\\wedge \\zeta _0)<\\tau _{r_{2j+1}})}\\ \\ge \\ e^{-\\beta (E(z^*(r_{2j+1},r_{2j+2}))-E(r_{2j-1},r_{2j})+o(1))}.$ (d) All bounds for energies in (c) can be proved in the very same way (here $r^{\\prime }\\in \\partial ^+V(m)\\backslash \\lbrace s\\rbrace $ ), so that $\\mathbb {P}_{r_{2k-1}}(\\tau _{r_{2k}}=\\zeta _0)\\ &=\\ \\frac{\\mathbb {P}_{r_{2k-1}}(\\tau _{r_{2k}}=(\\zeta _0\\wedge \\tau _{r_{2k-1}}))}{\\mathbb {P}_{r_{2k-1}}((\\tau _{r_{2k}}\\wedge \\zeta _0)<\\tau _{r_{2k-1}})}\\ \\ge \\ e^{-\\beta (E(z^*(r_{2k-1},r_{2k}))-E(r_{2k-3},r_{2k-2})+o(1))}.$ Now we see for the reciprocating jumps in the accelerated chain: Proposition 3.11 Fix $1\\le i\\le \\mathfrak {n}$ and $\\varepsilon >0$ .","Then the AAC at level $i$ exhibits no reciprocating jumps of order $\\varepsilon $ if the following three conditions hold true: (1) $E(z^*(m_0,m_1))-E(s_{m_0})\\le \\varepsilon $ for all distinct $m_0,m_1\\in \\mathcal {S}^{(i)}\\backslash N^{(i)}$ .","(2) For each $m\\in \\mathcal {S}^{(i)}\\backslash N^{(i)}$ , there exist at least two distinct $m_1,m_2\\in \\mathcal {S}^{(i)}\\backslash N^{(i)}, m\\ne m_1,m_2,$ such that $\\mathbb {P}_m(X_{\\xi _1}\\in V^{(i)}(m_j))>0$ for $j=1,2$ .","(3) For each pair $m_0,m_1\\in \\mathcal {S}^{(i)}\\backslash N^{(i)}$ with $\\mathbb {P}_{m_0}(X_{\\xi _1}\\in V^{(i)}(m_1))>0$ , there exists a minimal uphill-downhill-path from $s_{m_0}$ to $m_1$ not hitting any valley but $V^{(i)}(m_1)$ .","The origin of our endeavor to define aggregations with no reciprocating jumps of an order larger than a small $\\varepsilon $ is to obtain an associated process with (almost) decorrelated increments (in Euclidean state space), for this and a proper centering causes the variance up to the $n$ -th jump to grow with $n$ instead of $n^2$ .","This is known as diffusive behavior in physics.","Without aggregation increments are highly correlated due to the following argument: at any given time, the process is with high probability in a minimum and when leaving it, say by making a positive jump, the next increment is most likely negative because there is a drift back to the minimum.","Likewise, the increments of the asymptotic jump chain are neither uncorrelated nor having mean zero since trajectories of ${\\widehat{Y}}$ on an irreducibility class are almost surely of the form $m_1\\rightarrow s\\rightarrow m_2\\rightarrow s\\rightarrow m_3\\rightarrow ...$ , where $s\\in N^{(i)}$ and $m_1,m_2{,}...\\in \\mathcal {S}^{(i)}\\backslash N^{(i)}$ .","Thus, given the previous increments, it is in general easy to predict the next increment and they do not have mean zero.","On the other hand, if we can choose $\\beta $ and $i$ such that, for any $m_0,m_1$ , we have $\\mathbb {P}_{m_0}(Y_1=m_1)\\in \\lbrace p_{m_0}\\pm \\varepsilon \\rbrace \\cup [0,\\varepsilon ]$ for $\\varepsilon \\ll p_{m_0}$ , we obtain an AAC which behaves roughly like a RW on a graph.","Such a RW is diffusive if we assume periodic boundary conditions (or sufficiently large state space compared to the observation time $n$ ) and an energy landscape $E$ which is homogeneous enough to ensure zero-mean increments.","In particular $\\lbrace m|\\mathbb {P}_{m_0}(Y_1=m)=p_{m_0}\\pm \\varepsilon \\rbrace $ has to comprise at least two states."],["Metabasins","A path-independent definition of metabasins can now be given on the basis of the previous considerations.","Definition 3.12 A finite Markov chain $X$ driven by an energy function $E$ satisfying the assumptions stated at the beginning of Section has metabasins of order $\\varepsilon >0$ if there exists an aggregation level $i<\\mathfrak {n}-1$ such that the following conditions are fulfilled for each $m\\in \\mathcal {S}^{(i)}\\backslash N^{(i)}$ : (MB4) (MB1) $\\sup _{m^{\\prime }\\in \\mathcal {S}^{(i)}\\backslash (N^{(i)}\\cup \\lbrace m\\rbrace )}E(z^*(m,m^{\\prime }))-E(s_m)\\le \\varepsilon $ .","(MB2) There are at least two distinct $m_1,m_2\\in \\mathcal {S}^{(i)}\\backslash (N^{(i)}\\cup \\lbrace m\\rbrace )$ with a minimal uphill-downhill-path from $s_m$ to $m_k$ not hitting any other valley but $V^{(i)}(m_{k})$ for $k=1,2$ .","In this case, the valleys $(V^{(i)}(m))_{m\\in \\mathcal {S}^{(i)}}$ are called metabasins (MB) of order $\\varepsilon $.","The reader should notice that each singleton set $\\lbrace s\\rbrace $ consisting of a non-assigned state $s\\in \\mathcal {N}^{(i)}$ forms a MB.","The conditions (MB1) and (MB2) ensure the good nature of (a) the energy barriers and (b) the spatial arrangement of minima.","As already pointed out, this determines which valleys are visited consecutively.","Properties of MB which can be concluded from the results of the previous sections are summarized in the next theorem.","The reader is reminded of Properties 1–4 stated in the Introduction.","Theorem 3.13 For MB as defined in Definition REF we have (4) (1) The transition probabilities for jumps between MB do not depend on the point of entrance as $\\beta \\rightarrow \\infty $ (Property 1).","(2) There are no reciprocating jumps of order $\\varepsilon $ (Property 2).","(3) The expected residence time in a MB depends on $E$ only via the depth of the MB as $\\beta \\rightarrow \\infty $ (Property 3).","(4) Regarding only MB pertaining to local minima, the system is a trap model (Property 4).","[Proof:] (1) follows from Proposition REF , (2) from Proposition REF , (3) from Theorem REF , and (4) directly from the definition.","It should not be surprising that the path-dependent definition of MB by Heuer [12] and stated in the Introduction differs from our path-independent one.","Figure: Example of an energy landscape with a tree-like structure.For example, the energy landscape in Figure REF has no reasonable path-dependent MB because every transition between two branches of the shown tree must pass through the state $x$ .","For a typical trajectory, there will be at most three MB: the states visited before the first occurrence of $x$ , the states visited between the first and the last occurrence of $x$ , and the states visited after the last occurrence of $x$ .","The reason for this poor performance is the tree-like structure of the energy landscape or, more generally, the fact that the connectivity between the branches is too small to allow a selfavoiding walk through more than two branches.","This results in a small recurrence time for $x$ (compared to the number of states visited in between).","However, every branch constitutes a MB when using the path-independent definition for sufficiently small $\\varepsilon $ , in which case the AAC forms a Markov chain and, given the Metropolis algorithm, even a RW on the graph.","Having thus exemplified that the two definitions of MB do not necessarily coincide, where the path-independent approach applies to a wider class of energy landscapes, we turn to the question about conditions for them to coincide with high probability.","As already pointed out, we have to assume a sufficient connectivity between the metastates to ensure the existence of reasonable path-dependent MB.","In terms of this connectivity (for a precise definition see Definition REF ) and the parameter $\\beta $ and $\\varepsilon $ , our last result, Theorem REF below, provides lower bounds for the probability that both definitions yield the same partition of the state space.","The first step towards this end is to identify and count, for each $m\\in \\mathcal {S}^{(i)}$ and a given $\\beta $ , the states $s\\in \\mathcal {S}^{(i)}$ for which a transition of $Y$ from $m$ to $s$ is likely.","This leads to the announced connectivity parameter.","Definition 3.14 Let $\\varepsilon >0$ and suppose that $X$ has MB of order $\\varepsilon >0$ at level $i$ .","Define the connectivity parameters $\\eta _1\\ &=\\ \\eta _{1,\\varepsilon }\\ :=\\ \\min _{n\\in N^{(i)},r\\in \\mathcal {S}^{(i)}:V^{(i)}(r)\\cap \\mathcal {N}(n)\\ne \\emptyset }\\left|\\left\\lbrace s\\in \\mathcal {N}(n)\\backslash V^{(i)}(r)\\big |\\,E(s)\\le E(n)+\\varepsilon \\right\\rbrace \\right|,\\nonumber \\\\\\eta _2\\ &=\\ \\eta _{2,\\varepsilon }\\ :=\\ \\min _{m\\in \\mathcal {S}^{(i)}\\backslash N^{(i)}}\\big |\\lbrace s\\in \\partial ^+V^{(i)}(m)|E(s)\\le E(s_m)+\\varepsilon \\rbrace \\big |,\\\\\\eta _3\\ &=\\ \\eta _{3,\\varepsilon }\\ :=\\ \\min _{n\\in N^{(i)}}\\big |\\lbrace s\\in \\mathcal {S}^{(i)}| E(x)\\le E(n)+\\varepsilon \\textrm { for some x\\in V^{(i)}(s)\\cap \\mathcal {N}(n)}\\rbrace \\big |.\\nonumber $ $\\eta _1$ is the minimal number of neighboring sites of a non-assigned state $n$ which do not belong to a particular neighboring valley and whose energy is at most $\\varepsilon $ plus the energy of $n$ .","$\\eta _2$ is the minimal number of neighboring sites/valleys of a valley $V^{(i)}(m)$ whose energy is at most $\\varepsilon $ plus the energy of $s_m$ and which can be reached via an uphill-path from $m$ .","Finally, $\\eta _3$ is the minimal number of neighboring valleys of a non-assigned state $n$ which comprise a state with energy of at most $\\varepsilon $ plus the energy of $n$ .","$\\eta _1$ and $\\eta _3$ are always at least 2 and in fact quite large in the very complex energy landscapes of structural glasses.","For very small $\\varepsilon $ , $\\eta _2$ may be 1, but if $X$ has MB of order $\\varepsilon $ in a high dimensional energy landscape, then $\\eta _2$ can be assumed to be quite large as well.","That transitions to states counted above have reasonable large probabilities is content of the following lemma.","Thus, the defined parameters do in fact measure the connectivity of the MB.","Lemma 3.15 Let $\\varepsilon >0$ and suppose that $X$ has MB of order $\\varepsilon >0$ at level $i$ with connectivity parameters defined in (REF ).","Writing $Y_{k}$ for $Y_{k}^{(i)}$ and $V(m)$ for $V^{(i)}(m)$ , $m\\in \\mathcal {S}^{(i)}$ , the following assertions hold true for all sufficiently large $\\beta $ : (a) If $m\\in \\mathcal {S}^{(i)}\\backslash N^{(i)}$ and $s\\in \\partial ^{+}V(m)\\cap \\lbrace x|E(x)-E(s_{m})\\le \\varepsilon \\rbrace $ , or $m\\in N^{(i)}$ and $s\\in \\mathcal {S}^{(i)}$ satisfies $V(s)\\cap \\lbrace x\\in \\mathcal {N}(m)|E(x)-E(m)\\le \\varepsilon \\rbrace \\ne \\emptyset $ , then $\\mathbb {P}_{m}(Y_{1}=s)\\ \\ge \\ e^{-2\\beta \\varepsilon }.$ (b) For any distinct $m\\in N^{(i)}$ and $s\\in \\mathcal {S}^{(i)}$ , $\\mathbb {P}_{m}(Y_1\\ne s)\\ \\ge \\ \\eta _1\\,e^{-2\\beta \\varepsilon },$ (c) For any distinct $m\\in \\mathcal {S}^{(i)}\\backslash N^{(i)}$ and $s\\in \\mathcal {S}^{(i)}$ , $\\mathbb {P}_{m}(Y_1\\ne s)\\ \\ge \\ (\\eta _2-1)\\,e^{-2\\beta \\varepsilon },$ We see that, for $\\varepsilon $ small enough compared to $\\beta $ , transitions with an energy barrier of at most $\\varepsilon $ are still quite likely and thus a jump to a particular valley quite unlikely in the case of high connectivity.","[Proof:] (a) Choose $\\beta _{0}>0$ so large that, for $\\beta \\ge \\beta _{0}$ , $\\gamma _{\\beta }\\le \\varepsilon $ and $\\mathbb {P}_m(Y_1=s)\\ge e^{-2\\beta (E(s)-E(s_m))}$ for any $m\\in \\mathcal {S}^{(i)}\\backslash N^{(i)}$ and $s\\in \\partial ^{+}V(m)$ , the latter being possible by Lemma REF .","Then for any such $m$ and $s$ , we infer $\\mathbb {P}_{m}(Y_1=s)\\ge e^{-2\\varepsilon \\beta }$ provided that additionally $E(s)\\le E(s_m)+\\varepsilon $ holds true.","If $m\\in N^{(i)}$ , then $\\mathbb {P}_{m}(Y_1=s)\\ge e^{-2\\varepsilon \\beta }$ for any $s\\in \\mathcal {S}^{(i)}$ such that $E(x)\\le E(s)+\\varepsilon $ for some $x\\in V(s)\\cap \\mathcal {N}(m)$ , for $\\mathbb {P}_{m}(Y_1=s)\\ \\ge \\ \\mathbb {P}_{m}(X_{\\sigma _{1}}=x)\\ =\\ p(m,x)\\ \\ge \\ e^{-\\beta ((E(x)-E(m))^{+}+\\gamma _{\\beta })}.$ (b) Pick again $\\beta _{0}$ so large that $\\gamma _{\\beta }\\le \\varepsilon $ for all $\\beta \\ge \\beta _{0}$ .","Then, $\\mathbb {P}_{m}(Y_1\\ne s)\\ &\\ge \\ \\sum _{x\\in \\mathcal {N}(m),x\\notin V(s)}p(m,x)\\\\&\\ge \\ \\sum _{x\\in \\mathcal {N}(m),x\\notin V(s), E(x)\\le E(m)+\\varepsilon }\\exp \\big (-\\beta ((E(x)-E(m))^{+}+\\gamma _{\\beta })\\big )\\\\&\\ge \\ \\eta _1\\exp (-2\\beta \\varepsilon ),$ by definition of $\\eta _1$ .","(c) Fix $\\beta _{0}$ so large that $\\mathbb {P}_m(Y_1=x)\\ge e^{-2\\beta (E(x)-E(s_m))}$ for any $x\\in \\partial ^{+}V(m)$ and $\\beta \\ge \\beta _{0}$ .","In the very same way as in part (b), we then get for all $\\beta \\ge \\beta _0$ $\\mathbb {P}_{m}(Y_1\\ne s)\\ &\\ge \\ \\sum _{x\\in \\partial ^+V(m),x\\ne s}\\mathbb {P}_m(Y_1=x)\\\\&\\ge \\ \\sum _{x\\in \\partial ^+V(m),x\\ne s, E(x)\\le E(s_m)+\\varepsilon }\\exp \\big (-2\\beta (E(x)-E(s_m))\\big )\\\\&\\ge \\ (\\eta _2-1)\\exp (-2\\beta \\varepsilon ),$ by definition of $\\eta _2$ .","The above result motivates that in the case of high connectivity the probability to revisit a particular valley within a fixed time $T$ is quite small, or in other words, the probability for the AAC to jump along a selfavoiding path is quite high.","This is the main step towards the announced theorem and stated below.","The observation time $T$ of course has to be small compared to the cover time of the process.","Lemma 3.16 Let $\\varepsilon >0$ and suppose that $X$ has MB of order $\\varepsilon >0$ at level $i$ with connectivity parameters defined in (REF ).","Writing $Y_{k}$ for $Y_{k}^{(i)}$ and $V(m)$ for $V^{(i)}(m)$ , $m\\in \\mathcal {S}^{(i)}$ , define $\\tau _{V(m)}^{(i)}:=\\inf \\lbrace k\\ge 1|Y_{k}=m\\rbrace .$ Then the following assertions hold true for all sufficiently large $\\beta $ : (a) For any $0<\\delta <1-\\mathbb {P}_{m}(Y_{2}=m)$ and $1\\le T\\le T(m,\\beta )+1$ , $\\mathbb {P}_{m}\\left(\\tau _{V(m)}^{(i)}>T\\right)\\ \\ge \\ \\delta ,$ where $T(m,\\beta )\\ :=\\ \\frac{\\ln \\delta }{\\ln \\!\\big (\\min _{m^{\\prime }\\ne m}\\mathbb {P}_{m^{\\prime }}(Y_1\\ne m)(1-{1}_{\\lbrace m^{\\prime }\\notin N^{(i)}\\rbrace }\\,\\delta (m^{\\prime },\\beta ))\\big )}$ and $\\delta (m^{\\prime },\\beta )\\ :=\\ \\max _{x\\in V(m^{\\prime })}\\sum _{z\\in \\partial ^+ V(m^{\\prime })}\\tilde{\\varepsilon }(x,m^{\\prime },z,\\beta ).$ In particular, if $\\delta \\le \\left((\\eta _1\\wedge (\\eta _2-2))e^{-2\\beta \\varepsilon }\\right)^{T}$ for some $T>0$ , then $T(m,\\beta )\\ge T$ .","(b) For each $k\\ge 1$ and $m_{0}\\in \\mathcal {S}^{(i)}$ , $\\sum _{m_{1}{,}...,m_{k}}\\prod _{j=0}^{k-1}\\mathbb {P}_{m_j}(Y_1=m_{j+1})\\ \\ge \\ [\\eta _2\\wedge \\eta _3]_{k}\\,e^{-2k\\varepsilon \\beta }$ where summation ranges over all pairwise distinct $m_{1}{,}...,m_{k}\\in \\mathcal {S}^{(i)}\\backslash \\lbrace m_0\\rbrace $ and for $N\\in \\mathbb {N}$ we write $[N]_{k}:=N(N-1)\\cdot ...\\cdot (N-k+1)$ .","It should be noticed that $\\mathbb {P}_{m}(\\tau _{V(m)}^{(i)}>1)=1$ (the AAC never stays put) and $\\mathbb {P}_{m}(\\tau _{V(m)}^{(i)}>T)\\ \\le \\ \\mathbb {P}_{m}(\\tau _{V(m)}^{(i)}>2)\\ =\\ 1-\\mathbb {P}_{m}(Y^{(i)}_2=m)$ for every $T\\ge 2$ with equality holding only if $T=2$ .","We thus see that $\\mathbb {P}_{m}(\\tau _{V(m)}^{(i)}>T)\\ge \\delta $ entails $\\delta <1-\\mathbb {P}_{m}(Y_2=m)$ , the latter being typically large.","Furthermore, the bound on the number of self-avoiding path of length $k$ is very crude and can be improved when knowing more about the spatial arrangement of the metastable states.","[Proof:] (a) Recall from the first part of the proof of Proposition REF that $\\mathbb {P}_{m}(Y_{n+1}\\ne z|Y_n=y,X_{\\sigma _n}=x)\\ &\\ge \\ \\mathbb {P}_{y}(Y_1\\ne z)\\left(1-{1}_{\\lbrace y\\notin N^{(i)}\\rbrace }\\sum _{r\\in \\partial ^{+}V(y)}\\mathbb {P}_{x}(\\tau _{r}<\\tau _{y})\\right)\\\\&\\ge \\ \\mathbb {P}_{y}(Y_1\\ne z)\\left(1-{1}_{\\lbrace y\\notin N^{(i)}\\rbrace } \\sum _{r\\in \\partial ^+V(y)}\\tilde{\\varepsilon }(x,y,r,\\beta )\\right)\\\\&\\ge \\ \\mathbb {P}_{y}(Y_1\\ne z)\\left(1-{1}_{\\lbrace y\\notin N^{(i)}\\rbrace }\\,\\delta (y,\\beta )\\right)\\\\$ holds true for all $y,z\\in \\mathcal {S}^{(i)},\\,x\\in V(y)$ and $\\beta >0$ .","This will now be used repeatedly to show that $\\mathbb {P}_{m}\\Big (\\tau _{V(m)}^{(i)}>T\\Big )\\ \\ge \\ \\left(\\min _{m^{\\prime }\\ne m}\\mathbb {P}_{m^{\\prime }}(Y_1\\ne m)(1-{1}_{\\lbrace m^{\\prime }\\notin N^{(i)}\\rbrace }\\,\\delta (m^{\\prime },\\beta ))\\right)^{T-1}$ for each $T>2$ .","Putting $\\mathfrak {m}(x):=m^{\\prime }$ if $x\\in V(m^{\\prime })$ for $m^{\\prime }\\in \\mathcal {S}^{(i)}$ , we obtain $\\mathbb {P}_{m}&\\Big (\\tau _{V(m)}^{(i)}>T\\Big )\\\\&=\\ \\mathbb {P}_{m}(Y_{1}\\ne m{,}...,Y_{T}\\ne m)\\\\&=\\ \\sum _{x_1{,}...,x_{T-1}\\notin V^{(i)}(m)}\\mathbb {P}_{m}(X_{\\sigma _1}=x_1)\\prod _{k=1}^{T-2}\\mathbb {P}_{m}\\big (X_{\\sigma _{k+1}}=x_{k+1}|X_{\\sigma _{k}}=x_{k}, Y_{k}=\\mathfrak {m}(x_k)\\big )\\\\&\\hspace*{76.82234pt}\\times \\mathbb {P}_{m}(Y_{T}\\ne m|X_{\\sigma _{T-1}}=x_{T-1}, Y_{T-1}=\\mathfrak {m}(x_{T-1}))\\\\&\\ge \\ \\sum _{x_1{,}...,x_{T-1}\\notin V^{(i)}(m)}\\mathbb {P}_{m}(X_{\\sigma _1}=x_1)\\prod _{k=1}^{T-2}\\mathbb {P}_{m}\\big (X_{\\sigma _{k+1}}=x_{k+1}|X_{\\sigma _{k}}=x_{k}, Y_{k}=\\mathfrak {m}(x_k)\\big )\\\\&\\hspace{48.36958pt}\\times \\left(\\min _{m^{\\prime }\\ne m}\\left(\\mathbb {P}_{m^{\\prime }}(Y_1\\ne m)(1-{1}_{\\lbrace m^{\\prime }\\notin N^{(i)}\\rbrace }\\,\\delta (m^{\\prime },\\beta ))\\right)\\right)\\\\&\\hspace{5.69046pt}\\vdots \\\\&\\ge \\ \\min _{m^{\\prime }\\ne m}\\Big (\\mathbb {P}_{m^{\\prime }}(Y_1\\ne m)(1-{1}_{\\lbrace m^{\\prime }\\notin N^{(i)}\\rbrace }\\,\\delta (m^{\\prime },\\beta ))\\Big )^{T-1}.\\\\$ But this establishes the asserted inequality when finally observing that the last expression is $\\ge \\delta $ iff $T\\le T(m,\\beta )+1$ .","Having just said that $T(m,\\beta )\\ge T$ holds iff $\\min _{m^{\\prime }\\ne m}\\Big (\\mathbb {P}_{m^{\\prime }}(Y_1\\ne m)(1-{1}_{\\lbrace m^{\\prime }\\notin N^{(i)}\\rbrace }\\,\\delta (m^{\\prime },\\beta ))\\Big )^{T}\\ \\ge \\ \\delta ,$ it suffices to note that, as $\\beta \\rightarrow \\infty $ , $\\delta (m^{\\prime },\\beta )\\rightarrow 0$ holds true if $m^{\\prime }\\in \\mathcal {S}^{(i)}\\backslash N^{(i)}$ , giving $1-\\delta (m^{\\prime },\\beta )\\ \\ge \\ \\frac{\\eta _2-2}{\\eta _2-1}$ for sufficiently large $\\beta $ .","Together with Lemma REF (b), this further yields $\\min _{m^{\\prime }\\ne m}\\Big (\\mathbb {P}_{m^{\\prime }}(Y_1\\ne m)(1-{1}_{\\lbrace m^{\\prime }\\notin N^{(i)}\\rbrace }\\,\\delta (m^{\\prime },\\beta ))\\Big )^{T}&\\ge \\ \\left((\\eta _1\\wedge (\\eta _2-2))e^{-2\\beta \\varepsilon }\\right)^{T}$ and then the assertion.","(c) Here it suffices to notice that, by (a), $[\\eta _2\\wedge \\eta _3]_{k}$ forms a lower bound for the number of self-avoiding paths $(m_{0}{,}...,m_{k})$ such that $\\mathbb {P}_{m_{j}}(Y_{1}=m_{j+1})\\ge e^{-2\\beta \\varepsilon }$ for each $j=0{,}...,k-1$ .","We proceed with the announced result about the relation between path-dependent and path-independent MB.","To this end, we fix $T=\\sigma _{K}$ for some $K\\in \\mathbb {N}$ .","Let $\\mathcal {V}_{k}$ for $k=1{,}...,\\upsilon $ denote the random number of MB obtained from $X_{0}{,}...,X_{T}$ as defined in the Introduction.","For $x\\in \\mathcal {S}$ , we further let $\\mathcal {V}(x)$ denote the MB $\\mathcal {V}_{k}$ containing $x$ and put $\\mathcal {V}(x):=\\emptyset $ if no such MB exists which is the case iff $x\\notin \\lbrace X_{0}{,}...,X_{T}\\rbrace $ .","Theorem 3.17 Let $\\varepsilon >0$ and suppose that $X$ has MB of order $\\varepsilon >0$ at level $i$ with connectivity parameters defined in (REF ).","Fix $K\\in \\mathbb {N}$ , $T=\\sigma _{K}$ and $0<\\delta \\le \\left((\\eta _1\\wedge (\\eta _2-1)-1)e^{-2\\beta \\varepsilon }\\right)^{K}$ .","Then, for each $0\\le k0$ such that for all $\\beta \\ge \\beta _0$ (a) $\\mathbb {P}_{m_0}\\!\\left(V^{(i)}_<(Y_{k})\\subseteq \\mathcal {V}(Y_{k})\\right)\\ge 1-(\\max _{m\\in \\mathcal {S}^{(i)}\\backslash N^{(i)}}|V_<(m)|+2)\\max _{m\\in \\mathcal {S}^{(i)}\\backslash N^{(i)}}\\delta (m,\\beta )$ , where $V_{<}^{(i)}(s):=\\lbrace s\\rbrace $ if $s\\in N^{(i)}$ .","(b) $\\mathbb {P}_{m_0}(\\mathcal {V}(Y_{j})\\subseteq V^{(i)}(Y_{j}),\\,0\\le jK-1$ , then $\\mathbb {P}_{m_0}(\\mathcal {V}(Y_j)\\subseteq V^{(i)}(Y_j),\\,0\\le j\\le K-1)\\ \\ge \\ [\\eta _2\\wedge \\eta _3]_{K}\\left(1-\\max _{m\\in \\mathcal {S}^{(i)}\\backslash N^{(i)}}\\delta (m,\\beta )\\right)^{K-1}e^{-2K\\varepsilon \\beta }.$ For the occurring bounds to be significant, two requirements must be met.","First, $K$ must be small compared to the cover time of the AAC and $\\varepsilon $ must be small compared to $\\beta _0$ to ensure $\\exp (-2\\beta \\varepsilon )\\gg 0$ .","Second, the connectivity must be high to ensure $1-\\delta \\ll 1$ and $[\\eta _2\\wedge \\eta _3]_{K}e^{-2K\\varepsilon \\beta }\\gg 0$ .","Typically, the inclusions in parts (b) and (c) are strict because of high energy states within a valley that will probably be missed during one simulation run and therefore not belong to any path-dependent MB.","On the other hand, since our approach strives to cover the state space as completely as possible by valleys the latter comprise such high energy states whenever they are assignable in the sense described in Section .","[Proof:] With $i$ being fixed, let us write as earlier $V(m)$ for $V^{(i)}(m)$ , and also $V_{<}(m)$ for $V_{<}^{(i)}(m)$ .","(a) For a given $0\\le k \\sigma _1 \\textrm { for some \\right.x\\in V_<(m)}-\\mathbb {P}_m\\big (\\textrm {X_l\\ne m for each \\max _{x\\in V_<(m)}\\tau _x\\le l<\\sigma _1}\\big ).$ The second probability in the preceding line can further be bounded with the help of (REF ), viz.","$\\mathbb {P}_m&(\\textrm {X_l\\ne m for each \\max _{x\\in V_<(m)}\\tau _x\\le l<\\sigma _1})\\\\&=\\ \\sum _{y\\in V_<(m)}\\mathbb {P}_m(\\max _{x\\in V_<(m)}\\tau _x=\\tau _y,\\textrm { X_l\\ne m for every \\max _{x\\in V_<(m)}\\tau _x\\le l<\\sigma _1})\\\\&\\le \\ \\sum _{y\\in V_<(m)}\\mathbb {P}_m(\\max _{x\\in V_<(m)}\\tau _x=\\tau _y)\\mathbb {P}_y(\\tau _m>\\sigma _1)\\\\&\\le \\ \\delta _{\\max },$ while for the first probability, we obtain with the help of Theorem REF $\\mathbb {P}_{m}\\left(\\tau _x> \\sigma _1 \\textrm { for some \\right.x\\in V_<(m)}\\ &\\le \\sum _{x\\in V_<(m)}\\mathbb {P}_{m}(\\sigma _1<\\tau _x)\\nonumber \\\\&\\le \\sum _{x\\in V_<(m)}\\sum _{y\\in \\partial ^+V(m)}\\mathbb {P}_m(\\tau _y<\\tau _x)\\nonumber \\\\&\\le \\sum _{x\\in V_<(m)}\\sum _{y\\in \\partial ^+V(m)}\\varepsilon (m,x,y,\\beta ),$ because $E(z^*(m,y))>E(z^*(m,x))$ for $x\\in V_<(m)$ and $y\\in \\partial ^+V(m)$ .","The latter can be seen as follows: It has been shown in the proof of Theorem REF that $E(z^*(x,m))0$ such that $\\max _{m\\in \\mathcal {S}^{(i)}}\\mathbb {P}_{m}\\left(\\tau _{V(m)}^{(i)}\\le K\\right)\\ \\le \\ 1-\\delta $ for each $\\beta \\ge \\beta _{0}$ .","By using (REF ) and (REF ), we now infer $\\mathbb {P}_{m_0}\\!&\\left(Y_l= Y_k\\textrm { for some }k+1\\le l\\le K\\right)\\\\&=\\ \\sum _{s\\in \\mathcal {S}}\\mathbb {P}_{m_0}\\!\\left(Y_l= Y_k\\textrm { for some }k+1\\le l\\le K,X_{\\sigma _{k}}=s\\right)\\\\&\\le \\ \\sum _{s\\in \\mathcal {S}}\\mathbb {P}_{m_0}(X_{\\sigma _{k}}=s)\\Big (\\mathbb {P}_{s}(\\tau _{V(Y_0)}^{(i)}\\le K-k,\\,X_{j}=\\mathfrak {m}(s) \\text{ for some }0\\le j< \\sigma _1)\\\\&\\hspace*{48.36958pt}+{1}_{\\lbrace s\\notin N^{(i)}\\rbrace }\\,\\mathbb {P}_{s}(\\tau _{V(Y_0)}^{(i)}\\le K-k, X_{j}\\ne \\mathfrak {m}(s) \\text{ for all }0\\le j< \\sigma _1)\\Big )\\\\&\\le \\ \\sum _{s\\in \\mathcal {S}}\\mathbb {P}_{m_0}(X_{\\sigma _{k}}=s)\\,\\mathbb {P}_{\\mathfrak {m}(s)}(\\tau _{V(Y_0)}^{(i)}\\le K-k)+\\sum _{s\\notin N^{(i)}}\\mathbb {P}_{m_0}(X_{\\sigma _{k}}=s)\\,\\mathbb {P}_{s}(\\sigma _1<\\tau _{\\mathfrak {m}(s)})\\\\&\\le \\ 1-\\delta +\\delta _{\\max },$ and finally $\\mathbb {P}_{m_0}(\\mathcal {V}(Y_j)\\subset V^{(i)}(Y_j),\\,0\\le j< k)\\ &\\ge \\ \\mathbb {P}_{m_0}\\left(\\bigcap _{j=0}^{k-1}\\left\\lbrace Y_l\\ne Y_j,j+1\\le l\\le K\\right\\rbrace \\right)\\\\&\\ge \\ 1-\\sum _{j=0}^{k-1}\\mathbb {P}_{m_0}(Y_l= Y_j\\textrm { for some }j+1\\le l\\le K)\\\\&\\ge \\ 1-k(\\delta _{\\max }+(1-\\delta )).$ (c) In the following calculation, let $r_{0}=m_{0}$ , $\\sum _{m_{j}}$ range over all $K$ -vectors $(m_{1},...,m_{K})$ with pairwise distinct components in $\\mathcal {S}^{(i)}\\backslash \\lbrace m_0\\rbrace $ and, for each $k\\\\chi _{n-1}\\\\,|\\\\,\\\\lbrace X_{k}{,}...,X_{T}\\\\rbrace \\\\cap \\\\lbrace X_{0}{,}...,X_{k-1}\\\\rbrace =\\\\emptyset \\\\big \\\\rbrace .$ Then the MB up to $\\\\upsilon :=\\\\sup \\\\lbrace n\\\\ge 0\\\\,|\\\\,\\\\chi _n\\\\le T\\\\rbrace $ are chosen as $\\\\mathcal {V}_{n}:=\\\\lbrace X_{\\\\chi _n}{,}...,X_{\\\\chi _{n+1}-1}\\\\rbrace ,\\\\quad 0\\\\le n\\\\le \\\\upsilon .$ Simulation studies have shown that local sampling within a MB does not affect typical parameters of the process like the diffusion coefficient or the time to reach equilibrium.\",\n \"Dynamical aspects are therefore fully characterized by the MB-valued process.\",\n \"Furthermore, this model reduction by aggregation, as proposed in [12] and [18], offers several advantages (referred to as Properties 1–5 hereafter): The probability of a transition from one MB to any other one does not depend on the state at which this MB is entered.\",\n \"There are basically no reciprocating jumps between two MB.\",\n \"This is in strong contrast to the unaggregated process where such jumps occur very often: The system falls back to a minimum many times before eventually cresting a high energy barrier and then falling into a new valley, where it will again take many unsuccessful trials to escape.\",\n \"These reciprocating jumps are not only irrelevant for the actual motion on the state space but also complicating the estimation of parameters like the diffusion coefficient or the relaxation time.\",\n \"The expected time spent in a MB is proportional to its depth.\",\n \"Thus there is a strong and explicit relation between dynamics and thermodynamics, not in terms of the absolute but the relative energy.\",\n \"The energy barriers between any two MB are approximately of the same height, that is, there is an energy threshold $E_0$ such that, for a small $\\\\varepsilon $ , it requires a crossing of at least $E_0-\\\\varepsilon $ and at most $E_0+\\\\varepsilon $ to make a transition from one MB to another.\",\n \"Such systems with $\\\\varepsilon =0$ are called trap models (see [3]).\",\n \"The sojourn times and the jump distances between successively visited MB (measured in Euclidean distance) form sequences of weakly or even uncorrelated random variables, and are also mutually independent, at least approximately.\",\n \"Therefore, the aggregated process can be well approximated by a continuous time random walk, which in turn simplifies its analysis and thus the analysis of the whole process.\",\n \"Despite these advantages, the suggested definition of MB has the obvious blemish that it depends on the realization of the considered process and may thus vary from simulation to simulation.\",\n \"To provide a mathematically stringent definition of a path-independent aggregation of the state space, which maintains the above properties and is based on the well-established notion of metastable states, is therefore our principal concern here with the main results being Theorem REF and Theorem REF .\",\n \"In this endeavor, we will draw on some of the ideas developed by Bovier in [4] and by Scoppola in [21], most notably her definition of metastable states.\",\n \"Metastability, a phenomenon of ongoing interest for complex physical systems described by finite Markov processes on very large state spaces, can be defined and dealt with in several ways.\",\n \"It has been derived from a renormalization procedure in [20], by a pathwise approach in [6], and via energy landscapes in [4], the latter being also our approach hereafter.\",\n \"To characterize a supercooled liquid, i.e.\",\n \"a glass forming system at low temperature, via its energy landscape was first done by Goldstein in 1969 [11] and has by now become a common method.\",\n \"The general task when studying metastability, as well originally raised in physics ([13], [19] or [18]), is to provide mathematical tools for an analysis of the property of thermodynamical systems to evolve in state space along a trajectory of unstable or metastable states with very long sojourn times.\",\n \"Inspired by simulations of glass forming systems at very low temperatures with the Metropolis algorithm, we will study (as in [21]) finite Markov chains with exponentially small transition probabilities which are determined by an energy function and a parameter $\\\\beta >0$ .\",\n \"This parameter can be understood as the inverse temperature and we are thus interested in the behavior of the process as $\\\\beta \\\\rightarrow \\\\infty $ .\",\n \"We envisage an energy function of highly complex order and without the hierarchical ordering that is typical in spin glass models.\",\n \"A good picture is provided by randomly chosen energies with correlations between neighbors or by an energy landscape that looks like a real mountain landscape.\",\n \"We will show that, towards an aggregation outlined above, the metastable states as defined in [21] are quite appropriate because they have an ordering from a kind of “weak” to a kind of “strong” metastability.\",\n \"Around those states we will define and then study connected valleys [Definition REF ] characterized by minimal energy barriers.\",\n \"In the limit of low temperatures, any such barrier will determine the speed, respectively probability of a transition between the two valleys it separates.\",\n \"More precisely, in the limit $\\\\beta \\\\rightarrow \\\\infty $ , the process, when starting in a state $x$ , will almost surely reach a state with lower barrier earlier than a state with higher barrier [Theorem REF ].\",\n \"In the limit of low temperatures, the bottom (minimum) of an entered valley will therefore almost surely be reached before that valley is left again [Proposition REF ].\",\n \"As a consequence, the probability for a transition from one valley to another is asymptotically independent of the state where the valley is entered.\",\n \"This is Property 1 above.\",\n \"Furthermore, since valleys as well as metastable states have a hierarchical ordering, we can build valleys of higher order by a successive merger of valleys of lower order [Proposition REF ].\",\n \"Given an appropriate energy landscape, this procedure can annihilate (on the macroscopic scale) the accumulation of reciprocating jumps by merging valleys exhibiting such jumps into a single valley [Subsection REF ].\",\n \"Hence, valleys of sufficiently high order will have Property 2.\",\n \"Beside the macroscopic process [Section ], which describes the transitions between valleys, one can also analyze the microscopic process [Section ], that is, the system behavior when moving within a fixed valley.\",\n \"Here we will give a formula for the exit time and connect it with its parameters [Theorem REF ].\",\n \"This will confirm Property 3.\",\n \"Having thus established Properties 1–4 [Theorem REF ], we will finally proceed to a comparison of our path-independent definition of MB with the path-dependent one given above.\",\n \"It will be shown [Theorem REF ] that both coincide with high probability under some reasonable conditions on the connectivity of valleys which, in essence, ensure the existence of reasonable path-dependent MB.\",\n \"We will also briefly touch on the phenomenon of quasi-stationarity [Proposition REF ] which is a large area [17] but to our best knowledge less studied in connection with the aggregation of states of large physical systems driven by energy landscapes.\",\n \"Let us mention two further publications which, despite having a different thrust, provide definitions of valleys, called basins of attraction or metastates there, to deal with related questions.\",\n \"Olivieri & Scoppola [16] fully describe the tube of exit from a domain in terms of which basins of attraction of increasing order are visited during a stay in that domain and for how long these basins are visited.\",\n \"In a very recent publication, Beltrán & Landim [2], by working with transition rates instead of energies, aim at finding a universal depth (and time scale) for all metastates.\",\n \"However, we rather aim at the finest aggregation such that transitions back to an already visited metastate are very unlikely within a time frame used in simulations.\",\n \"This finest aggregation will lead to valleys of very variable depth just as simulations do not exhibit a universal depth or timescale.\"\n ],\n [\n \"Valleys\",\n \"Let $X$ be a Markov chain on a finite set $\\\\mathcal {S}$ with transition matrix $\\\\mathbf {P}=(p(r,s))_{r,s\\\\in \\\\mathcal {S}}$ and stationary distribution $\\\\pi $ , and let $E:\\\\mathcal {S}\\\\rightarrow \\\\mathbb {R}$ be an energy function such that the following conditions hold: Irreducibility: $\\\\mathbf {P}$ is irreducible with $p(s,s)>0$ and $p(r,s)>0$ iff $p(s,r)>0$ for all $r,s\\\\in \\\\mathcal {S}$ .\",\n \"Transition Probabilities: There exist parameters $\\\\beta >0$ and $\\\\gamma _{\\\\beta }>0$ with $\\\\gamma _{\\\\beta }\\\\rightarrow 0, \\\\beta \\\\gamma _{\\\\beta }\\\\rightarrow \\\\infty $ as $\\\\beta \\\\rightarrow \\\\infty $ such that $e^{-\\\\beta ((E(s)-E(r))^++\\\\gamma _{\\\\beta }/|\\\\mathcal {S}|)}\\\\le p(r,s)\\\\le e^{-\\\\beta ((E(s)-E(r))^+-\\\\gamma _{\\\\beta }/|\\\\mathcal {S}|)}$ for all distinct $r,s\\\\in \\\\mathcal {S}$ with $p(r,s)>0$ .\",\n \"Furthermore, $p^{*}(r,s):=\\\\lim _{\\\\beta \\\\rightarrow \\\\infty }p(r,s)$ exists for all $r,s\\\\in \\\\mathcal {S}$ , is positive if $E(r)\\\\ge E(s)$ and $=0$ otherwise.\",\n \"Reversibility: The pair $(\\\\pi ,\\\\mathbf {P})$ satisfies the detailed balance equations, i.e.\",\n \"$\\\\pi (r)p(r,s)=\\\\pi (s)p(s,r)$ for all $r,s\\\\in \\\\mathcal {S}$ .\",\n \"Non-Degeneracy: $E(r)\\\\ne E(s)$ for all $r,s \\\\in \\\\mathcal {S}, r\\\\ne s$ .\",\n \"We are thus dealing with a reversible Markov chain with exponentially small transition probabilities driven by an energy landscape.\",\n \"As an example, which is also the main motivation behind this work, one can think of a Metropolis chain with transition probabilities of the form $p(r,s)=\\\\frac{1}{C(r)}e^{-\\\\beta (E(s)-E(r))^+}.$ Here $\\\\beta $ is the inverse temperature and $C(r), r\\\\in \\\\mathcal {S}$ , is a parameter, independent from $\\\\beta $ , giving the number of neighbors of $r$ .\",\n \"For $\\\\gamma _{\\\\beta }/|\\\\mathcal {S}|:=\\\\max _{r\\\\in \\\\mathcal {S}}\\\\ln (C(r))(\\\\beta +1)^{-1/2}$ the above conditions are fulfilled.\",\n \"Let us start with the following basic result for the ratios of the stationary distribution.\",\n \"Lemma 1.1 For any two states $r,s\\\\in \\\\mathcal {S}$ with $E(r)>E(s)$ , we have $e^{-\\\\beta (E(r)-E(s)+2\\\\gamma _{\\\\beta })}\\\\le \\\\frac{\\\\pi (r)}{\\\\pi (s)}\\\\le e^{-\\\\beta (E(r)-E(s)-2\\\\gamma _{\\\\beta })}.$ [Proof:] To start with, assume $r\\\\sim s$ .\",\n \"Reversibility and the assumptions on the transition probabilities imply $\\\\frac{\\\\pi (r)}{\\\\pi (s)}\\\\ &=\\\\ \\\\frac{p(s,r)}{p(r,s)}\\\\ \\\\le \\\\ \\\\frac{e^{-\\\\beta ((E(r)-E(s))^+-\\\\gamma _{\\\\beta }/|\\\\mathcal {S}|)}}{e^{-\\\\beta ((E(s)-E(r))^++\\\\gamma _{\\\\beta }/|\\\\mathcal {S}|)}}\\\\ =\\\\ e^{-\\\\beta (E(r)-E(s)-2\\\\gamma _{\\\\beta }/|\\\\mathcal {S}|)}.$ and $\\\\frac{\\\\pi (r)}{\\\\pi (s)}\\\\ &=\\\\ \\\\frac{p(s,r)}{p(r,s)}\\\\ \\\\ge \\\\ \\\\frac{e^{-\\\\beta ((E(r)-E(s))^++\\\\gamma _{\\\\beta }/|\\\\mathcal {S}|)}}{e^{-\\\\beta ((E(s)-E(r))^+-\\\\gamma _{\\\\beta }/|\\\\mathcal {S}|)}}\\\\ =\\\\ e^{-\\\\beta (E(r)-E(s)+2\\\\gamma _{\\\\beta }/|\\\\mathcal {S}|)}.$ Now let $r$ and $s$ be arbitrary.\",\n \"By the irreducibility, there is a path $r=r_0,r_1{,}...,r_n=s$ from $r$ to $s$ of neighboring states with $\\\\pi (r_i)/\\\\pi (r_{i+1})\\\\in [e^{-\\\\beta (E(r_i)-E(r_{i+1})+2\\\\gamma _{\\\\beta }/|\\\\mathcal {S}|)},e^{-\\\\beta (E(r_i)-E(r_{i+1})-2\\\\gamma _{\\\\beta }/|\\\\mathcal {S}|)}],\\\\, 0\\\\le i\\\\le n-1$ .\",\n \"This finishes the proof.\",\n \"Under the stated assumptions, Scoppola [21] has shown the existence of a successive filtration (aggregation) $\\\\mathcal {S}=M^{(0)}\\\\supset M^{(1)}\\\\supset ... \\\\supset M^{(\\\\mathfrak {n})}=\\\\lbrace s_0\\\\rbrace $ , $\\\\mathfrak {n}\\\\in \\\\mathbb {N}$ , of the state space such that the elements of each set $M^{(i)}, 1\\\\le i\\\\le \\\\mathfrak {n}$ , can be called metastable in the following sense: They arise from the local minima of the energy function or certain modifications of it.\",\n \"There is a lower bound on the expected time needed for a transition from $m_1$ to $m_2$ for any $m_1,m_2\\\\in M^{(i)}$ which increases very fast with $i$ .\",\n \"There exists a constant $C$ such that $\\\\mathbb {P}_m(X_t\\\\notin M^{(i+1)})\\\\le e^{-C\\\\beta }$ for all $m\\\\in M^{(i)}, 0\\\\le i\\\\le \\\\mathfrak {n}-1,$ and sufficiently large $t$ .\",\n \"This filtration starts with $M^{(1)}:=\\\\lbrace s\\\\in \\\\mathcal {S}| E(s)0 \\\\textrm { for } 0\\\\le i\\\\le k-1,\\\\,i\\\\ne j\\\\rbrace $ be the set of all finite self-avoiding paths from $r$ to $s$ having positive probability.\",\n \"For any such path $\\\\gamma =(\\\\gamma _0{,}...,\\\\gamma _k)\\\\in \\\\Gamma (r,s)$ , let $|\\\\gamma |:=k$ be its length.\",\n \"We further write $t\\\\in \\\\gamma $ if $t\\\\in \\\\lbrace \\\\gamma _{1}{,}...,\\\\gamma _{k}\\\\rbrace $ .\",\n \"(b) A self-avoiding path $\\\\gamma =(\\\\gamma _{1}{,}...,\\\\gamma _{k})$ from $r$ to $s$ is called minimal if its maximal energy $\\\\max _{1\\\\le i\\\\le k} E(\\\\gamma _i)$ is minimal among all $\\\\gamma ^{\\\\prime }\\\\in \\\\Gamma (r,s)$ .\",\n \"The set of these paths is denoted $\\\\Gamma ^*(r,s)$ .\",\n \"(c) The essential saddle $z^*(r,s)$ between $r$ and $s$ is then defined as $z^*(r,s):=\\\\mathop {\\\\mathrm {argmax}}_{t\\\\in \\\\gamma }E(t)\\\\in \\\\mathcal {S}$ for any $\\\\gamma \\\\in \\\\Gamma ^{*}(r,s)$ .\",\n \"As for (c), it is to be noted that, due to the assumed non-degeneracy of the energy function, the essential saddle is unique, which means that it does not depend on (as it must) which minimal path we choose in the definition of $z^*(r,s)$ .\",\n \"There may indeed be several minimal paths, every single one thus crossing the saddle at some time.\",\n \"With the help of these notions the valleys can now be defined in a quite concrete way.\",\n \"Let us label the local minima as $m^{(1)}{,}...,m^{(\\\\mathfrak {n})}$ , so that $M^{(i)}=\\\\lbrace m^{(i)}{,}...,m^{(\\\\mathfrak {n})}\\\\rbrace $ for each $i=1{,}...,\\\\mathfrak {n}$ .\",\n \"Definition 1.4 For each $m \\\\in M^{(i)}$ , $1\\\\le i\\\\le \\\\mathfrak {n}$ , let $V^{(i)}_<(m):=\\\\left\\\\lbrace s\\\\in \\\\mathcal {S}\\\\Big |E(z^*(s,m))< E(z^*(s,m^{\\\\prime }))\\\\text{ for all }m^{\\\\prime }\\\\in M^{(i)}\\\\backslash \\\\lbrace m\\\\rbrace \\\\right\\\\rbrace .$ We say that state $s$ is attracted by $m$ at level $i$, expressed as $s\\\\leadsto m$ at level $i$ , if $ E(z^*(s,m))=\\\\min _{n\\\\in M^{(i)}}E(z^*(s,n))$ and every minimal path from $s$ to a state $m^{\\\\prime }\\\\in M^{(i)}\\\\backslash \\\\lbrace m\\\\rbrace $ with $E(z^*(s,m^{\\\\prime }))=E(z^*(s,m))$ hits $V^{(i)}_{<}(m)$ at some time.\",\n \"Finally, let $l(i):=\\\\inf \\\\big \\\\lbrace i2$ ), it remains untouched until reaching level $l(1)$ where its bottom state $m^{(1)}$ becomes attracted by some $m^{\\\\prime }\\\\in M^{(l(1))}$ causing its valley to be merged into $V^{(l(1))}(m^{\\\\prime })$ .\",\n \"This procedure continues in the now obvious recursive manner until at level $\\\\mathfrak {n}$ all states have been merged into one valley.\",\n \"Obviously, valleys of the same order are pairwise disjoint.\",\n \"Also, valleys once formed at some level can only be merged as a whole and will thus never be ripped apart during the recursive construction.\",\n \"For the energy function depicted in Figure REF , the successively derived valleys of order $i=1{,}...,7$ are shown in Figure REF .\",\n \"Before proceeding to results on the general shape of valleys, we collect some basic, mostly technical properties of essential saddles which will be useful thereafter.\",\n \"Proposition 1.6 For any $r,s,u\\\\in \\\\mathcal {S}$ , $0\\\\le i\\\\le \\\\mathfrak {n}$ , $m_1,m_2\\\\in M^{(i)}, m_1\\\\ne m_2,$ and $x_1,x_2\\\\in \\\\mathcal {S}$ with $x_1\\\\in V^{(i)}_<(m_1)$ and $x_2\\\\in V^{(i)}(m_2)$ , we have (a) $z^*(r,s)=z^*(s,r)$ .\",\n \"(b) $E(z^*(r,s))\\\\le E(z^*(r,u))\\\\vee E(z^*(u,s))$ .\",\n \"(c) $E(z^*(x_2,m_2))\\\\le E(z^*(x_2,m^{\\\\prime }))$ for all $m^{\\\\prime }\\\\in M^{(i)}$ .\",\n \"(d) $E(z^*(x_1,m_2))=E(z^*(m_1,m_2))$ .\",\n \"(e) $E(z^*(x_1,x_2))\\\\ge E(z^*(m_1,m_2))$ .\",\n \"(f) $z^*(x_1,x_2)\\\\ne x_1$ .\",\n \"[Proof:] Parts (a) and (b) are obvious.\",\n \"For (c) we use an induction over $i$ and note that there is nothing to show when $i=1$ .\",\n \"For general $i$ , we must only verify that $E(z^{*}(x_{2},m_{2}))\\\\le E(z^{*}(x_{2},m^{\\\\prime }))$ for all $m^{\\\\prime }\\\\in M^{(i)}$ if $x_{2}\\\\in V^{(j)}(m^{(j)})$ for some $jE(z^*(x_1,m_1))$ , which in combination with (a) and (b) implies $E(z^*(m_1,m_2))\\\\le E(z^*(m_1,x_1))\\\\vee E(z^*(x_1,m_2))=E(z^*(x_1,m_2))$ and then further $E(z^*(x_1,m_2))\\\\le \\\\underbrace{E(z^*(x_1,m_1))}_{E(z^*(x_1,m_1))\\\\ge E(x_1)$ and thus $z^*(x_1,x_2)\\\\ne x_1$ as claimed in (f).\",\n \"Remark 1.7 It is useful to point out the following consequence of the previous proposition.\",\n \"If, for an arbitrary state $s$ and any two distinct metastable states $m,n\\\\in M^{(i)}$ , there exists a minimal path $\\\\gamma $ from $s$ to $n$ that hits a state $r$ with $E(z^*(r,m))E(z^*(x,m))$ .\",\n \"[Proof:] Suppose there is a minimal path $\\\\gamma $ from $x$ to $y$ avoiding $V^{(i)}_<(m)$ .\",\n \"Since $y\\\\notin V^{(i)}(m)$ , it is not attracted by $m$ at level $i$ implying the existence of some $m^{\\\\prime }\\\\in M^{(i)}$ with $E(z^*(y,m^{\\\\prime }))\\\\le E(z^*(y,m))$ and of some $\\\\tau \\\\in \\\\Gamma ^*(y,m^{\\\\prime })$ avoiding $V^{(i)}_<(m)$ .\",\n \"Hence, the concatenation $\\\\gamma \\\\tau $ avoids $V^{(i)}_<(m)$ and must therefore have maximal energy larger than $E(z^*(x,m))$ .\",\n \"Consequently, $E(z^*(x,m))&E(z^*(x,m))$ .\",\n \"In order to state the second lemma, let us define the outer part $\\\\partial ^{+}V$ of a valley $V$ to be the set of those states outside $V$ which are adjacent to a state in $V$ .\",\n \"With the help of the previous result, we can easily show that $\\\\partial ^{+}V$ contains only non-assigned states at any level where $V$ has not yet been merged into a larger valley.\",\n \"Lemma 1.15 For any $1\\\\le i, j\\\\le \\\\mathfrak {n}$ and $m=m^{(i)}$ with $l(i)> j$ , the outer part $\\\\partial ^{+}V$ of the valley $V:=V^{(j\\\\wedge i)}(m)$ is a subset of $N^{(j)}$ and $E(z^*(s,m))=E(s)$ for every $s\\\\in \\\\partial ^+V$ .\",\n \"[Proof:] First, let $s\\\\in \\\\partial ^{+}V$ and suppose that $s\\\\notin N^{(j)}$ .\",\n \"Then $s\\\\leadsto m^{\\\\prime }$ at level $k$ , in particular $s\\\\in V^{(k)}(m^{\\\\prime })$ for some $m^{\\\\prime }\\\\in M^{(k)}$ and $k\\\\le j$ .\",\n \"Pick any $r\\\\in V$ with $r\\\\sim s$ and note that $r\\\\in \\\\partial ^{+}V^{(k)}(m^{\\\\prime })$ .\",\n \"W.l.o.g.\",\n \"we may assume that $r\\\\leadsto m$ at level $j\\\\wedge i$ .\",\n \"Then Lemma REF (with $x=r$ and $y=s$ ) ensures that either $E(z^{*}(r,s))>E(z^{*}(r,m)\\\\ge E(r)$ , thus $z^{*}(r,s)=s$ and $E(r)E(s)$ which is clearly impossible.\",\n \"Consequently, $s$ must be non-assigned at level $j$ as claimed.\",\n \"For the second assertion take again $s\\\\in \\\\partial ^+V$ and a minimal path $\\\\gamma =(s,r{,}...,m)$ from $s$ to $m$ with $r\\\\in V$ .\",\n \"Again, by use of Lemma REF , we find either $E(z^*(r,m))0\\\\rbrace .$ Two states $x,y$ with $p(x,y)>0$ are called neighbors ($x\\\\sim y$ ) and $\\\\mathcal {N}(x)$ the neighborhood of $x$ .\",\n \"Hence, $\\\\mathcal {S}$ may (and will) be viewed as a graph hereafter with edge set $\\\\lbrace (x,y)|x\\\\sim y\\\\rbrace $ .\",\n \"Given any subgraph $\\\\Delta $ , we will write $\\\\widetilde{\\\\mathbf {P}}$ for the transition matrix of the chain restricted to $\\\\Delta $ ($\\\\tilde{p}(r,s)=p(r,s)$ for all distinct $r,s\\\\in \\\\Delta $ , $\\\\tilde{p}(r,r)=1-\\\\sum _{s\\\\in \\\\Delta , s\\\\ne r}\\\\tilde{p}(r,s)$ ) and $\\\\widetilde{\\\\mathbb {P}}_{x}$ for probabilities when regarding this restricted chain starting at $x\\\\in \\\\Delta $ .\",\n \"Theorem 2.1 Let $x, y, z\\\\in \\\\mathcal {S}$ be any pairwise distinct states satisfying $E(z^*(x,z))>E(z^*(x,y))$ and $z^*(x,z)\\\\ne x$ .\",\n \"Then there exist nonnegative constants $K(\\\\beta )$ satisfying $\\\\sup _{\\\\beta >0}K(\\\\beta )<\\\\infty $ (and given more explicitly in Proposition REF below) such that $\\\\mathbb {P}_{x}(\\\\tau _{z}<\\\\tau _{y})\\\\le K(\\\\beta )\\\\,e^{-\\\\beta (E(z^*(x,z))-E(z^*(x,y))-7\\\\gamma _{\\\\beta })}=:\\\\varepsilon (x,y,z,\\\\beta ) \\\\overset{\\\\beta \\\\rightarrow \\\\infty }{\\\\longrightarrow } 0.$ Thus in the limit of low temperatures $(\\\\beta \\\\rightarrow \\\\infty )$ , only the smallest of all possible energy barriers affects the speed of a transition.\",\n \"In particular, we have the following result which is preliminary to the subsequent one.\",\n \"Theorem 2.2 Given distinct $x,y\\\\in \\\\mathcal {S}$ and $m\\\\in M^{(i)}$ such that $x\\\\leadsto m$ at level $i$ and $y\\\\notin V^{(i)}(m)$ , let $B:=\\\\lbrace z\\\\in \\\\mathcal {S}|E(z^*(x,z))>E(z^*(x,m))\\\\rbrace $ .\",\n \"Then it holds true that $\\\\mathbb {P}_{x}&(\\\\tau _{y}<\\\\tau _{m})\\\\\\\\&\\\\le \\\\ \\\\varepsilon (x,m,y,\\\\beta )\\\\,{1}_{B}(y)+\\\\left(\\\\sum _{z:E(z)>E(z^*(x,y))}\\\\varepsilon (x,y,z,\\\\beta )+\\\\sum _{z\\\\in V^{(i)}_<(m)}\\\\varepsilon (z,m,y,\\\\beta )\\\\right)\\\\,{1}_{B^{c}}(y)\\\\\\\\&=:\\\\ \\\\tilde{\\\\varepsilon }(x,m,y,\\\\beta )\\\\ \\\\overset{\\\\beta \\\\rightarrow \\\\infty }{\\\\longrightarrow }\\\\ 0.$ Theorem 2.3 Given $m\\\\in M^{(i)}$ , $x\\\\in V^{(i)}(m)$ and $y\\\\notin V^{(i)}(m)$ , let $k\\\\le i$ be such that $&m_{0}:=x\\\\leadsto m_{1}\\\\text{ at level }l_{1},\\\\quad m_{1}\\\\leadsto m_{2}\\\\text{ at level }l_{2},\\\\quad ...\\\\quad m_{k-1}\\\\leadsto m_{k}=m\\\\text{ at level }l_{k}$ for suitable $1\\\\le l_{1}<...E(b)}(E(a)-E(b))-2\\\\gamma _{\\\\beta })}+1\\\\right)$ .\",\n \"The proof requires several lemmata, the first of which may already be found in [14] and is stated here in the notation of [5].\",\n \"Lemma 2.5 (see Theorem 2.1 in [5]) Defining $\\\\mathcal {H}^x_z:=\\\\lbrace h:\\\\mathcal {S}\\\\rightarrow [0,1]\\\\,|\\\\, h(x)=0,h(z)=1\\\\rbrace $ and the Dirichlet form $\\\\mathcal {E}(h):=\\\\sum _{r\\\\sim s\\\\in \\\\mathcal {S}}\\\\pi (r)p(r,s) (h(r)-h(s))^2,$ we have $\\\\mathbb {P}_x(\\\\tau _z<\\\\tau _x)=\\\\frac{1}{2\\\\pi (x)}\\\\inf _{h\\\\in \\\\mathcal {H}^x_z}\\\\mathcal {E}(h).$ Lemma 2.6 (see Lemma 2.2 in [5]) For any subgraph $\\\\Delta \\\\subset \\\\mathcal {S}$ containing $x,y$ and corresponding transition matrix $\\\\widetilde{\\\\mathbf {P}}$ , we have $\\\\mathbb {P}_x(\\\\tau _y<\\\\tau _x)\\\\ge \\\\widetilde{\\\\mathbb {P}}_x(\\\\tau _y<\\\\tau _x).$ Lemma 2.7 (see Lemma 2.5 in [5]) If $\\\\Delta =(\\\\omega _0{,}...,\\\\omega _k)$ is any one-dimensional subgraph of $\\\\mathcal {S}$ , then $\\\\widetilde{\\\\mathbb {P}}_{\\\\omega _0}(\\\\tau _{\\\\omega _k}<\\\\tau _{\\\\omega _0}) =\\\\left(\\\\sum _{i=1}^k\\\\frac{\\\\pi (\\\\omega _0)}{\\\\pi (\\\\omega _i)}\\\\frac{1}{p(\\\\omega _i,\\\\omega _{i-1})}\\\\right)^{-1}.$ [Proof:](of Proposition REF ) In view of Lemma REF , we must find an appropriate function $h$ for the upper bound.\",\n \"Let us define $\\\\mathcal {R}:=\\\\lbrace s\\\\in \\\\mathcal {S}\\\\,|\\\\,\\\\exists ~ \\\\gamma \\\\in \\\\Gamma (x,z):\\\\mathop {\\\\mathrm {argmax}}_{i=1{,}... , |\\\\gamma |}E(\\\\gamma _i)=s\\\\rbrace ,$ the set of all peak states (with respect to the energy function) along self-avoiding paths from $x$ to $z$ .\",\n \"Obviously $z^*(x,z)\\\\in \\\\mathcal {R}$ and, by non-degeneracy, $E(z^*(x,z))E(b)} (E(a)- E(b))-2\\\\gamma _{\\\\beta })}+|\\\\mathcal {N}(z^*(x,z))|\\\\right)\\\\\\\\&\\\\le \\\\ e^{-\\\\beta (E(z^*(x,z))-E(x)-2\\\\gamma _{\\\\beta })}\\\\\\\\&\\\\hspace*{28.45274pt}\\\\times \\\\,\\\\max _{r\\\\in \\\\mathcal {S}}|\\\\mathcal {N}(r)|\\\\,\\\\left(|\\\\mathcal {S}|e^{-\\\\beta (\\\\min _{a\\\\ne b:E(a)>E(b)}(E(a)- E(b))-2\\\\gamma _{\\\\beta })}+1\\\\right).$ Lemmata REF and REF will enter in the proof of the lower bound.\",\n \"Consider the chain restricted to the one-dimensional subgraph given by a minimal path $\\\\rho =(s_{1}{,}...,s_{|\\\\rho |})$ from $x$ to $y$ .\",\n \"Then $\\\\mathbb {P}_x(\\\\tau _y<\\\\tau _x)\\\\ &\\\\ge \\\\ \\\\left(\\\\sum _{i=1}^{|\\\\rho |}\\\\frac{\\\\pi (x)}{\\\\pi (s_i)}\\\\frac{1}{p(s_i,s_{i-1})}\\\\right)^{-1}\\\\\\\\&\\\\ge \\\\ \\\\frac{\\\\pi (z^*(x,y))}{\\\\pi (x)}\\\\left(\\\\sum _{i=1}^{|\\\\rho |}\\\\frac{\\\\pi (z^*(x,y))}{\\\\pi (s_i)}e^{\\\\beta ((E(s_{i-1})-E(s_i))^{+} +\\\\gamma _{\\\\beta })}\\\\right)^{-1}\\\\\\\\&\\\\ge \\\\ e^{-\\\\beta (E(z^*(x,y))-E(x) +2\\\\gamma _{\\\\beta })}\\\\left(\\\\sum _{i=1}^{|\\\\rho |}e^{-\\\\beta (E(z^*(x,y))-E(s_i)-(E(s_{i-1})-E(s_i))^{+}-3\\\\gamma _{\\\\beta })}\\\\right)^{-1}\\\\\\\\&\\\\ge \\\\ e^{-\\\\beta (E(z^*(x,y))-E(x)+5\\\\gamma _{\\\\beta })}\\\\,\\\\frac{1}{|\\\\mathcal {S}|}.$ This completes the proof of Proposition REF .\",\n \"We proceed to the second proposition needed to prove Theorem REF .\",\n \"Proposition 2.8 (see Corollary 1.6 in [5]) Given $I\\\\subset \\\\mathcal {S}$ and distinct $x,z\\\\in \\\\mathcal {S}\\\\backslash I$ , $\\\\mathbb {P}_x(\\\\tau _z<\\\\tau _I)=\\\\frac{\\\\mathbb {P}_x(\\\\tau _z<\\\\tau _{I\\\\cup \\\\lbrace x\\\\rbrace })}{\\\\mathbb {P}_x(\\\\tau _{I\\\\cup \\\\lbrace z\\\\rbrace }<\\\\tau _x)}$ holds true.\",\n \"With the help of Propositions REF and REF , the proof of Theorem REF can now be given quite easily.\",\n \"[Proof:](of Theorem REF ) By first using the previous result and then Proposition REF (with $K(\\\\beta )$ as defined there), we find $\\\\mathbb {P}_{x}(\\\\tau _{z}<\\\\tau _{y})\\\\ &=\\\\ \\\\frac{\\\\mathbb {P}_{x}(\\\\tau _{z}<\\\\tau _{\\\\lbrace x,y\\\\rbrace })}{\\\\mathbb {P}_{x}(\\\\tau _{\\\\lbrace z,y\\\\rbrace }<\\\\tau _{x})}\\\\\\\\&\\\\le \\\\ \\\\frac{\\\\mathbb {P}_{x}(\\\\tau _{z}<\\\\tau _{x})}{\\\\mathbb {P}_{x}(\\\\tau _{y}<\\\\tau _{x})}\\\\\\\\&\\\\le \\\\ K(\\\\beta )\\\\,e^{-\\\\beta (E(z^*(x,z))-E(z^*(x,y))-7\\\\gamma _{\\\\beta })}.$ The argument is completed by noting that $E(z^*(x,z))>E(z^*(x,y))$ and $K(\\\\beta )\\\\ge 0$ converges to $|\\\\mathcal {S}|\\\\max _{r\\\\in \\\\mathcal {S}}|\\\\mathcal {N}(r)|$ as $\\\\beta \\\\rightarrow \\\\infty $ .\",\n \"[Proof:](of Theorem REF ) If $B$ occurs, the asserted bound follows directly from Theorem REF .\",\n \"Proceeding to the case when $B^{c}$ occurs, i.e.\",\n \"$E(z^*(x,y))\\\\le E(z^*(x,m))$ , we first point out that $\\\\mathbb {P}_{x}(\\\\tau _{y}<\\\\tau _{m})\\\\ &=\\\\ \\\\mathbb {P}_{x}(\\\\tau _{y}<\\\\tau _{m}, E(X_n)>E(z^*(x,y)) \\\\text{ for some }n\\\\le \\\\tau _{y})\\\\\\\\&\\\\qquad +\\\\ \\\\mathbb {P}_{x}(\\\\tau _{y}<\\\\tau _{m},E(X_n)\\\\le E(z^*(x,y))\\\\text{ for all }n\\\\le \\\\tau _{y})\\\\\\\\&=:\\\\ P_{1}+P_{2}.$ For all $z\\\\in \\\\mathcal {S}$ with $E(z)>E(z^*(x,y))$ , we have $z^*(x,z)\\\\ne x$ and $E(z^*(x,z))>E(z^*(x,y))$ , for $E(z^*(x,z))\\\\ge E(z)>E(z^*(x,y))\\\\ge E(x).$ Therefore, by an appeal to Theorem REF , $P_{1}\\\\ &\\\\le \\\\ \\\\mathbb {P}_{x}(\\\\tau _{z}<\\\\tau _{y}\\\\text{ for some $z$ with $E(z)>E(z^*(x,y))$})\\\\\\\\&\\\\le \\\\ \\\\sum _{z:E(z)>E(z^*(x,y))}\\\\mathbb {P}_{x}(\\\\tau _z<\\\\tau _{y})\\\\\\\\&\\\\le \\\\ \\\\sum _{z:E(z)>E(z^*(x,y))}\\\\varepsilon (x,y,z,\\\\beta ).$ To get an estimate for $P_{2}$ , note that every minimal path from $x$ to $y$ must pass through $V^{(i)}_<(m)$ (Lemma REF ).\",\n \"With this observation and by another appeal to Theorem REF , we infer $\\\\mathbb {P}_{x}(\\\\tau _{y}<\\\\tau _{m},E(X_n)\\\\le E(z^*(x,y))\\\\text{ for all }n\\\\le \\\\tau _{y})\\\\ &\\\\le \\\\sum _{z\\\\in V^{(i)}_<(m)}\\\\mathbb {P}_z(\\\\tau _{y}<\\\\tau _ {m})\\\\\\\\&\\\\le \\\\ \\\\sum _{z\\\\in V^{(i)}_<(m)}\\\\varepsilon (z,m,y,\\\\beta ),$ having further utilized that (by Proposition REF (f) and (e)) $z^*(z,y)\\\\ne z$ and $E(z^*(z,m))&n)=\\\\nu (j)\\\\quad \\\\text{for all }j\\\\in V,$ but also satisfies $\\\\lim _{n\\\\rightarrow \\\\infty }\\\\,\\\\mathbb {P}_{\\\\mu }(X_{n}=j|\\\\tau _{\\\\mathcal {S}\\\\backslash V}>n)=\\\\nu (j)\\\\quad \\\\text{for all }j\\\\in V$ if $\\\\mu $ is an arbitrary distribution with $\\\\mu (V)=1$ .\",\n \"The latter property renders uniqueness of $\\\\nu $ .\",\n \"Since $\\\\mathcal {S}$ is finite, existence of $\\\\nu $ follows by an old result due to Darroch & Seneta [7].\",\n \"It is obtained as the normalized eigenvector of the Perron-Frobenius eigenvalue $\\\\lambda =\\\\lambda (V)$ of a modification of $\\\\mathbf {P}$ .\",\n \"This eigenvalue $\\\\lambda $ is also the probability for the chain to stay in $V$ at least one step when started with $\\\\nu $ , thus $\\\\mathbb {P}_{\\\\nu }(\\\\tau _{V^{c}}>1)=\\\\lambda $ .\",\n \"As an immediate consequence, one finds that the exit time $\\\\tau _{V^{c}}$ has a geometric distribution with parameter $1-\\\\lambda $ under $\\\\mathbb {P}_{\\\\nu }$ .\",\n \"In the present context, this naturally raises the question how the parameter $\\\\lambda $ relates to the transition probabilities or the energies of the valley $V$ .\",\n \"A simple probabilistic argument shows the following basic and intuitively obvious result concerning the eigenvalues associated with the nesting $V^{(1)}(m)\\\\subset ...\\\\subset V^{(i)}(m)$ (Proposition REF ) for any $1\\\\le i\\\\le \\\\mathfrak {n}$ and $m\\\\in M^{(i)}$ .\",\n \"Proposition 2.9 Fixing any $1\\\\le i\\\\le \\\\mathfrak {n}$ and $m\\\\in M^{(i)}$ , let $\\\\lambda ^{(j)}:=\\\\lambda (V^{(j)}(m))$ for $j=1{,}...,i$ .\",\n \"Then $\\\\lambda ^{(1)}\\\\le ...\\\\le \\\\lambda ^{(i)}$ .\",\n \"[Proof:] Write $\\\\nu _{j}$ as shorthand for the quasi-stationary distribution on $V^{(j)}(m)$ and $T_{j}$ for $\\\\tau _{\\\\mathcal {S}\\\\backslash V^{(j)}(m)}$ .\",\n \"Plainly, $T_{j}\\\\le T_{j+1}$ $(\\\\lambda ^{(j)})^{n}\\\\ =\\\\ \\\\mathbb {P}_{\\\\nu _{j}}(T_{j}>n)\\\\ &\\\\le \\\\ \\\\mathbb {P}_{\\\\nu _{j}}(T_{j+1}>n)\\\\nonumber \\\\\\\\&=\\\\ \\\\int _{\\\\lbrace T_{j+1}>k\\\\rbrace }\\\\mathbb {P}_{X_{k}}(T_{j+1}>n-k)\\\\ d\\\\mathbb {P}_{\\\\nu _{j}}\\\\nonumber \\\\\\\\&=\\\\ \\\\mathbb {P}_{\\\\nu _{j}}(T_{j+1}>k)\\\\,\\\\mathbb {P}_{\\\\mu _{k}}(T_{j+1}>n-k),$ where $\\\\mu _{k}(x):=\\\\mathbb {P}_{\\\\nu _{j}}(X_{k}=x|T_{j+1}>k)$ for $x\\\\in V^{(j+1)}$ .\",\n \"Since $\\\\mathcal {S}$ is finite and by virtue of (REF ), we have that $\\\\mu _{k}\\\\le 2\\\\nu _{j+1}$ when choosing $k$ sufficiently large.\",\n \"For any such $k$ , we find that (REF ) has upper bound $2\\\\,\\\\mathbb {P}_{\\\\nu _{j}}(T_{j+1}>k)\\\\,\\\\mathbb {P}_{\\\\nu _{j+1}}(T_{j+1}>n-k)\\\\ =\\\\ 2\\\\,\\\\mathbb {P}_{\\\\nu _{j}}(T_{j+1}>k)\\\\,(\\\\lambda ^{(j+1)})^{n-k}.$ Hence, we finally conclude $\\\\lambda ^{(j)}\\\\ \\\\le \\\\ \\\\Big (2\\\\,\\\\mathbb {P}_{\\\\nu _{j}}(T_{j+1}>k)\\\\,(\\\\lambda ^{(j+1)})^{-k}\\\\Big )^{1/n}\\\\,\\\\lambda ^{(j+1)}$ and thereby the assertion upon letting $n\\\\rightarrow \\\\infty $ .\",\n \"An alternative matrix-analytic proof draws on an old result by Frobenius [9], here cited from [10].\",\n \"Lemma 2.10 If $A=(a_{ij})$ and $C=(c_{ij})$ denote two real $k\\\\times k$ -matrices such that $A$ is nonnegative and irreducible with maximal eigenvalue $\\\\lambda _A^*$ and $|c_{ij}|\\\\le a_{ij}$ for all $1\\\\le i,j\\\\le k$ , then $|\\\\lambda |\\\\le \\\\lambda _A^*$ for all eigenvalues $\\\\lambda $ of $C$ .\",\n \"Second proof of Proposition REF : For any fixed valley $V$ , collaps all states $s\\\\notin V$ into an absorbing state (grave) $\\\\Delta $ which leaves transition probabilities between states in $V$ unchanged.\",\n \"A proper rearrangement of states allows us to assume that the new transition matrix has the form $\\\\mathbf {P}=\\\\begin{pmatrix} 1 & {\\\\bf 0} \\\\\\\\ \\\\mathbf {p} & \\\\mathbf {Q}\\\\end{pmatrix}$ for a $|V|\\\\times 1$ -column vector $\\\\mathbf {p}\\\\ne 0$ and a nonnegative, substochastic and irreducible $|V|\\\\times |V|$ -matrix $\\\\mathbf {Q}$ .\",\n \"Now, for any $2\\\\le j\\\\le i$ , let $A$ be this matrix $\\\\mathbf {Q}$ when $V=V^{(j)}(m)$ , and $D$ be this matrix when $V=V^{(j-1)}(m)$ .\",\n \"Then, obviously, $A:=\\\\begin{pmatrix} A_1 & A_2 \\\\\\\\ A_3 & D \\\\end{pmatrix}$ and $A$ is irreducible and nonnegative with maximal eigenvalue $\\\\lambda ^{(j)}$ .\",\n \"Defining further $C:=\\\\begin{pmatrix} 0 & 0 \\\\\\\\ 0 & D \\\\end{pmatrix}.$ the largest eigenvalue of $C$ equals the largest eigenvalue of $D$ , thus $\\\\lambda ^{(j-1)}$ .\",\n \"Finally, the desired conclusion follows from the previous lemma, since $|c_{ij}|=c_{ij}\\\\le a_{ij}$ for all $i\\\\le i,j\\\\le k$ .$\\\\square $ Another question is how long a given valley is visited and thus about its exit time.\",\n \"There is an extensive literature on exit problems for different kinds of stochastic processes.\",\n \"We mention [22] and [8] as two related to our work.\",\n \"The latter one studies perturbed systems on a continuous space.\",\n \"We can discretize their argument to get, with use of the main theorem in [21], a nice result on the time needed to leave a valley $V^{(i)}(m)$ for any fixed $1\\\\le i\\\\le \\\\mathfrak {n}$ and $m\\\\in M^{(i)}$ .\",\n \"This result is more explicit than the one in [16].\",\n \"Definition 2.11 For $1\\\\le i\\\\le \\\\mathfrak {n},\\\\, N:=N^{(i)}$ , we define the following stopping (entrance/exit) times: $\\\\xi _0^{(i)}&:=\\\\tau _{N^c}\\\\\\\\\\\\zeta _{n}^{(i)}&:=\\\\inf \\\\left\\\\lbrace k\\\\ge \\\\xi _n^{(i)}|X_k\\\\in N\\\\right\\\\rbrace \\\\\\\\\\\\xi _{n+1}^{(i)}&:=\\\\inf \\\\left\\\\lbrace k\\\\ge \\\\zeta _{n}^{(i)}|X_k\\\\in N^c\\\\right\\\\rbrace , ~n\\\\ge 0.$ The entrance times $\\\\xi ^{(i)}_n$ mark the epochs when a new valley is visited, while the exit times $\\\\zeta ^{(i)}_n$ are the epochs at which a valley is left.\",\n \"The reader should notice that we do not restrict ourselves to valleys of order $i$ but include those valleys which up to order $i$ have not yet been absorbed by some larger valley.\",\n \"Exit and entrance times never coincide since there is no way to go from one valley to another without hitting a non-assigned state - crests are always non-assigned (see Lemma REF ).\",\n \"In this section, we will focus on $\\\\zeta _0^{(i)}$ for any fixed $i$ , thus writing $\\\\zeta _0:=\\\\zeta _0^{(i)}$ hereafter, but later for the macroscopic process the other times will be needed as well.\",\n \"For each valley $V^{(i)}(m),\\\\,m\\\\in M^{(i)}$ , let us define $s_m\\\\ =\\\\ s_m^{(i)}\\\\ :=\\\\ \\\\mathop {\\\\mathrm {argmin}}_{s\\\\in \\\\partial ^+V^{(i)}(m)}E(s)\\\\ =\\\\ \\\\mathop {\\\\mathrm {argmin}}_{s\\\\in \\\\partial ^+V^{(i)}(m)}E(z^*(m,s)),$ where the second equality follows from Lemma REF .\",\n \"Theorem 2.12 Let $m\\\\in M^{(i)}$ .\",\n \"Then $\\\\lim _{\\\\beta \\\\rightarrow \\\\infty }\\\\frac{1}{\\\\beta }\\\\ln \\\\mathbb {E}_{r}\\\\zeta _0\\\\ =\\\\ E(s_m)-E(m)$ for any $r\\\\in V^{(i)}(m)$ .\",\n \"For the upper bound, we need a result from [21], which in our notation is: Proposition 2.13 (see Main Theorem (iv) in [21]) For any $1\\\\le i\\\\le \\\\mathfrak {n},\\\\, \\\\beta $ sufficiently large and $t> 2^{i-1}\\\\exp (\\\\beta (E(s_{m^{(i-1)}})-E(m^{(i-1)})+2i|\\\\mathcal {S}|\\\\gamma _{\\\\beta }))$ , $\\\\sup _{x\\\\in M^{(i-1)}}\\\\mathbb {P}_x(\\\\tau _{M^{(i)}}>t)\\\\le \\\\exp (-\\\\Delta \\\\beta )$ holds true with a positive constant $\\\\Delta $ , where $M^{(0)}=\\\\mathcal {S}$ should be recalled.\",\n \"This will now be used to show the following result.\",\n \"Lemma 2.14 Fix $1\\\\le i\\\\le \\\\mathfrak {n},\\\\, m\\\\in M^{(i)}$ and $r\\\\in V^{(i)}(m)$ .\",\n \"Then, for any $\\\\beta $ sufficiently large and $t>2^{i}\\\\exp (\\\\beta (E(s_{m})-E(m)+2(i+1)|\\\\mathcal {S}|\\\\gamma _{\\\\beta }))$ , it holds true that $\\\\mathbb {P}_r(\\\\zeta _0<(i+1)t)\\\\ \\\\ge \\\\ \\\\frac{1}{4}.$ [Proof:] Let us first note that we can always arrange for $m$ being equal to $m^{(i)}$ by sufficiently decreasing the energy function at any $m^{\\\\prime }\\\\in M^{(i)}\\\\backslash \\\\lbrace m\\\\rbrace $ so as to make $E(s_{m})-E(m)$ minimal among all states in $M^{(i)}$ .\",\n \"This affects neither the valley $V^{(i)}(m)$ and its outer boundary nor the distribution of $\\\\zeta _0$ when starting in $m$ , for this distribution does not depend on the energy landscape outside of $V^{(i)}(m)\\\\cup \\\\partial ^+V^{(i)}(m)$ .\",\n \"When applying the previous proposition, the constant $\\\\Delta $ may have changed but is still positive which suffices for our purposes.\",\n \"So let $m=m^{(i)}$ hereafter.\",\n \"Fix $t>2^i\\\\exp (\\\\beta (E(s_{m})-E(m)+2(i+1)|\\\\mathcal {S}|\\\\gamma _{\\\\beta }))$ and $T:=it$ .\",\n \"Since $E(s_{m})-E(m)\\\\ge E(s_{m^{(j)}})-E(m^{(j)})$ for every $1\\\\le j\\\\le i$ , we infer $\\\\mathbb {P}_r(\\\\tau _{M^{(i)}}\\\\le T)\\\\ &\\\\ge \\\\ \\\\mathbb {P}_r(\\\\tau _{M^{(i)}}\\\\le T, \\\\tau _{M^{(1)}}\\\\le t)\\\\\\\\&\\\\ge \\\\ \\\\mathbb {P}_r(\\\\tau _{M^{(1)}}\\\\le t)\\\\,\\\\inf _{x\\\\in M^{(1)}}\\\\mathbb {P}_{x}(\\\\tau _{M^{(i)}}\\\\le (i-1)t)\\\\\\\\&\\\\ge \\\\ \\\\mathbb {P}_r(\\\\tau _{M^{(1)}}\\\\le t)\\\\,\\\\inf _{x\\\\in M^{(1)}}\\\\mathbb {P}_{x}(\\\\tau _{M^{(2)}}\\\\le t)\\\\,\\\\inf _{x\\\\in M^{(2)}}\\\\mathbb {P}_{x}(\\\\tau _{M^{(i)}}\\\\le (i-2)t)\\\\\\\\&\\\\hspace*{28.45274pt}\\\\vdots \\\\\\\\&\\\\ge \\\\ \\\\prod _{j=1}^{i}\\\\inf _{x\\\\in M^{(j-1)}}\\\\mathbb {P}_{x}(\\\\tau _{M^{(j)}}\\\\le t)\\\\\\\\&\\\\ge \\\\ \\\\big (1-\\\\exp (-\\\\Delta \\\\beta )\\\\big )^i\\\\\\\\&\\\\ge \\\\ \\\\frac{3}{4}$ for $\\\\beta $ sufficiently large.\",\n \"Furthermore, for $\\\\beta $ so large that $\\\\mathbb {P}_r(\\\\tau _{M^{(i)}}<\\\\tau _m)\\\\le 1/4$ , we find that $\\\\mathbb {P}_r(\\\\tau _{M^{(i)}}\\\\le T)\\\\ &=\\\\ \\\\mathbb {P}_r(\\\\tau _{M^{(i)}}=\\\\tau _m\\\\le T)+\\\\mathbb {P}_r(\\\\tau _{M^{(i)}}\\\\le T, \\\\tau _{M^{(i)}}<\\\\tau _m)\\\\\\\\&\\\\le \\\\ \\\\mathbb {P}_r(\\\\tau _m\\\\le T)+\\\\mathbb {P}_r(\\\\tau _{M^{(i)}}<\\\\tau _m)\\\\\\\\&\\\\le \\\\ \\\\mathbb {P}_r(\\\\tau _m\\\\le T)+\\\\frac{1}{4}.$ By combining both estimates, we obtain $\\\\mathbb {P}_r(\\\\tau _m\\\\le T)\\\\ \\\\ge \\\\ \\\\mathbb {P}_r(\\\\tau _{M^{(i)}}\\\\le T)-\\\\frac{1}{4}\\\\ \\\\ge \\\\ \\\\frac{1}{2}.$ Hence, state $m$ is hit in time $T$ with at least probability $1/2$ when starting in $r$ .\",\n \"Since $m=m^{(i)}$ , we further have $\\\\mathbb {P}_m(\\\\zeta _0\\\\le t)\\\\ \\\\ge \\\\ \\\\mathbb {P}_m(\\\\tau _{M^{(i+1)}}\\\\le t)\\\\ \\\\ge \\\\ 1-\\\\exp (-\\\\Delta \\\\beta )\\\\ \\\\ge \\\\ \\\\frac{1}{2}$ for $\\\\beta $ sufficiently large.\",\n \"Hence, state $s_m$ is hit in time $t$ with at least probability $1/2$ when starting in $m$ .\",\n \"By combining the estimates, we finally obtain $\\\\mathbb {P}_{r}(\\\\zeta _0\\\\le (i+1)t)\\\\ &\\\\ge \\\\ \\\\mathbb {P}_{r}(\\\\zeta _0\\\\le T+t|\\\\tau _{m}\\\\le T)\\\\,\\\\mathbb {P}_{r}(\\\\tau _{m}\\\\le T)\\\\\\\\&\\\\ge \\\\ \\\\mathbb {P}_{r}(\\\\tau _{m}\\\\le T)\\\\,\\\\mathbb {P}_{m}(\\\\zeta _0\\\\le t)\\\\\\\\&\\\\ge \\\\ \\\\frac{1}{4},$ which proves our claim.\",\n \"[Proof:](of Theorem REF ) Using the lemma just shown, we infer $\\\\mathbb {E}_{r}(\\\\zeta _0)\\\\ &\\\\le \\\\ (i+1)t\\\\sum _{n\\\\ge 0}(n+1)\\\\,\\\\mathbb {P}_{r}\\\\left(n(i+1)t\\\\le \\\\zeta _0<(n+1)(i+1)t\\\\right)\\\\\\\\&=\\\\ (i+1)t\\\\sum _{n\\\\ge 0}(n+1)\\\\Big (\\\\mathbb {P}_{r}\\\\left(\\\\zeta _0\\\\ge n(i+1)t\\\\right)-\\\\mathbb {P}_{r}\\\\left(\\\\zeta _0\\\\ge (n+1)(i+1)t\\\\right)\\\\Big )\\\\\\\\&=\\\\ (i+1)t\\\\sum _{n\\\\ge 0}\\\\mathbb {P}_{r}\\\\left(\\\\zeta _0\\\\ge n(i+1)t\\\\right)\\\\\\\\&\\\\le \\\\ (i+1)t\\\\sum _{n\\\\ge 0}\\\\left(\\\\max _{x\\\\in V}\\\\mathbb {P}_x\\\\left(\\\\zeta _0\\\\ge (i+1)t\\\\right)\\\\right)^n\\\\\\\\&\\\\le \\\\ (i+1)t\\\\sum _{n\\\\ge 0}\\\\left(\\\\frac{3}{4}\\\\right)^n\\\\\\\\&=\\\\ 4(i+1)t,$ where $t:=2^i\\\\exp (\\\\beta (E(s_{m})-E(m)+2(i+1)|\\\\mathcal {S}|\\\\gamma _{\\\\beta }))+1$ .\",\n \"Since $\\\\gamma _{\\\\beta }\\\\rightarrow 0$ , we get in the limit $\\\\lim _{\\\\beta \\\\rightarrow \\\\infty }\\\\frac{1}{\\\\beta }\\\\ln \\\\mathbb {E}_{r}\\\\zeta _0\\\\ \\\\le \\\\ E(s_m)-E(m)$ for all $r\\\\in V^{(i)}(m)$ .\",\n \"Turning to the lower bound, define a sequence of stopping times, viz.\",\n \"$\\\\rho _0:=0$ and $\\\\rho _n:=\\\\inf \\\\lbrace k>\\\\rho _{n-1}|X_k=m\\\\text{ or }X_{k}\\\\in \\\\partial ^+ V\\\\rbrace $ for $n\\\\ge 1$ .\",\n \"Then $Z_n:=X_{\\\\rho _n}$ , $n\\\\ge 0$ , forms a Markov chain the transition probabilities of which when starting in $m$ can be estimated with the help of Proposition REF , namely $\\\\mathbb {P}(Z_1\\\\in \\\\partial ^+ V^{(i)}(m)|Z_0=m)\\\\ &=\\\\ \\\\mathbb {P}_{m}(\\\\rho _1=\\\\zeta _0)\\\\\\\\&=\\\\ \\\\mathbb {P}_{m}(\\\\zeta _0<\\\\tau _m)\\\\\\\\&\\\\le \\\\ \\\\sum _{s\\\\in \\\\partial ^+V}\\\\mathbb {P}_m(\\\\tau _s<\\\\tau _m)\\\\\\\\&\\\\le \\\\ K(\\\\beta )\\\\,e^{-\\\\beta (\\\\min _{s\\\\in \\\\partial ^+V}E(z^*(m,s))-E(m)-2\\\\gamma _{\\\\beta })}\\\\\\\\&=\\\\ K(\\\\beta )\\\\,e^{-\\\\beta (E(s_m)-E(m)-2\\\\gamma _{\\\\beta })}$ where $K(\\\\beta )\\\\rightarrow K\\\\in (0,\\\\infty )$ as $\\\\beta \\\\rightarrow \\\\infty $ .\",\n \"Further defining $\\\\nu :=\\\\inf \\\\lbrace k\\\\ge 0| Z_k\\\\in \\\\partial ^+ V\\\\rbrace $ , this implies in combination with a geometric trials argument that $\\\\mathbb {P}_{m}(\\\\nu >n)\\\\ \\\\ge \\\\ \\\\left(1-K(\\\\beta )\\\\,e^{-\\\\beta (E(s_m)-E(m)-2\\\\gamma _{\\\\beta })}\\\\right)^{n-1}.$ As a consequence, $\\\\mathbb {E}_{m}\\\\zeta _0\\\\ &=\\\\ \\\\sum _{n\\\\ge 1} \\\\mathbb {E}_{m}(\\\\underbrace{\\\\rho _n-\\\\rho _{n-1}}_{\\\\ge 1}){1}_{\\\\lbrace \\\\nu \\\\ge n\\\\rbrace }\\\\ \\\\ge \\\\ \\\\sum _{n\\\\ge 1} \\\\mathbb {P}_{m}(\\\\nu \\\\ge n)\\\\ \\\\ge \\\\ K(\\\\beta )^{-1}\\\\,e^{\\\\beta (E(s_m)-E(m)-2\\\\gamma _{\\\\beta })}.$ For arbitrary $r\\\\in V$ , we now infer $\\\\mathbb {E}_{r}\\\\zeta _0\\\\ &=\\\\ \\\\mathbb {E}_{r}\\\\zeta _0{1}_{\\\\lbrace \\\\zeta _0\\\\le \\\\rho _1\\\\rbrace }+\\\\mathbb {E}_{r}\\\\zeta _0{1}_{\\\\lbrace \\\\zeta _0>\\\\rho _1\\\\rbrace }\\\\\\\\&\\\\ge \\\\ \\\\mathbb {E}_{r}\\\\big (\\\\mathbb {E}_r(\\\\zeta _0{1}_{\\\\lbrace \\\\zeta _0>\\\\rho _1\\\\rbrace }|X_{\\\\rho _{1}}=m)\\\\big )\\\\\\\\&\\\\ge \\\\ \\\\mathbb {E}_{r}{1}_{\\\\lbrace \\\\zeta _0>\\\\rho _1\\\\rbrace }\\\\mathbb {E}_{m}\\\\zeta _0\\\\\\\\&\\\\ge \\\\ \\\\mathbb {P}_{r}(\\\\zeta _0>\\\\rho _1)K(\\\\beta )^{-1}\\\\,e^{\\\\beta (E(s_m)-E(m)-2\\\\gamma _{\\\\beta })}\\\\\\\\&\\\\ge \\\\ \\\\frac{1}{2}\\\\,K(\\\\beta )^{-1}\\\\,e^{\\\\beta (E(s_m)-E(m)-2\\\\gamma _{\\\\beta })}$ for all sufficiently large $\\\\beta $ , because $\\\\lim _{\\\\beta \\\\rightarrow \\\\infty }\\\\mathbb {P}_{r}(\\\\zeta _0>\\\\rho _1)=1$ (Theorem REF ).\",\n \"Finally, by taking logarithms and letting $\\\\beta $ tend to $\\\\infty $ , we arrive at the inequality $\\\\lim _{\\\\beta \\\\rightarrow \\\\infty }\\\\frac{1}{\\\\beta }\\\\ln \\\\mathbb {E}_{r}\\\\zeta _0\\\\ \\\\ge \\\\ E(s_m)-E(m)$ which completes the proof.\",\n \"In [12], $E(s_m)-E(m),\\\\,m\\\\in M^{(i)},$ is referred to as the depth of the valley $V(m)$ .\",\n \"Therefore, Property 3 from the Introduction holds true and we can relate thermodynamics of the system (energies) to dynamics of the chain (holding times) in a very precise way.\",\n \"Especially, there is no universal scale for the times spent in different valleys because in general they differ exponentially.\"\n ],\n [\n \"Macroscopic View: Transitions between valleys\",\n \"With the help of the nested state space decompositions into valleys of different orders and around bottom states of different stability, we will now be able to provide an appropriate definition of the metabasins (MB) that has been announced and to some extent discussed in the Introduction.\",\n \"We will further define and study macroscopic versions of the original process $X=(X_{n})_{n\\\\ge 0}$ .\",\n \"These are obtained by choosing different levels of aggregation in the sense that they keep track only of the valleys of a chosen level that are visited by $X$ .\",\n \"The motivation behind this approach is, on the one hand, to exhibit strong relations between properties of the energy landscape and the behavior of $X$ (as in Theorem REF ) and, on the other hand, to describe essential features of this process by looking at suitable macroscopic scales.\",\n \"In the subsequent definition of aggregated versions of $X$ , we will distinguish between two variants: A time-scale preserving aggregation that, for a fixed level and each $n$ , keeps track of the valley the original chain visits at time $n$ and thus only blinds its exact location within a valley.\",\n \"An accelerated version that, while also keeping track of the visited valleys, further blinds the sojourn times within a valley by counting a visit just once.\",\n \"Actually, the definition of these aggregations at a chosen level $i$ is a little more complicated because their state space, denoted $\\\\mathcal {S}^{(i)}$ below and the elements of which we call level $i$ metastates, also comprises the non-assigned states at level $i$ as well as the minima of those valleys that were formed at an earlier level and whose merger is pending at level $i$ because their minima are not attracted at this level.\",\n \"Definition 3.1 Fix $1\\\\le i\\\\le \\\\mathfrak {n}$ , let $\\\\mathcal {S}^{(i)}:=\\\\lbrace m^{(j)}\\\\in M^{(1)}|\\\\,l(j)>i\\\\rbrace \\\\cup N^{(i)}$ and $V^{(i)}(s):={\\\\left\\\\lbrace \\\\begin{array}{ll}V^{(i)}(m^{(j)}),&\\\\text{if }s=m^{(j)} \\\\textrm { for some j\\\\ge i}\\\\\\\\V^{(j)}(m^{(j)}),&\\\\text{if }s=m^{(j)} \\\\textrm { for some j\\\\sigma _{n-1}\\\\Big |\\\\overline{Y}_k^{(i)}\\\\ne \\\\overline{Y}_{k-1}^{(i)}\\\\right\\\\rbrace $ for $n\\\\ge 1$ .\",\n \"We call $\\\\overline{Y}^{(i)}=(\\\\overline{Y}_n^{(i)})_{n\\\\ge 0}$ and $Y^{(i)}=(Y_{n}^{(i)})_{n\\\\ge 0}$ the aggregated chain (AC) and the accelerated aggregated chain (AAC) (at level $i$ ) associated with $X=(X_n)_{n\\\\ge 0}$ .\",\n \"So, starting in an arbitrary valley, the original chain stays there for a time $\\\\zeta _0=\\\\zeta _{0}^{(i)}$ (as defined in Definition REF ) before it jumps via some non-assigned states $k_1{,}..., k_l$ (staying a geometric time in each of these states) to another valley at time $\\\\xi _1=\\\\xi _{1}^{(i)}$ .\",\n \"There it stays for $\\\\zeta _1-\\\\xi _1$ time units before it moves on in a similar manner.\",\n \"By going from $X$ to its aggregation $\\\\overline{Y}^{(i)}$ at level $i$ , we regard the whole valley $V^{(i)}(s)$ for $s\\\\in \\\\mathcal {S}^{(i)}$ as one single metastate and therefore give up information about the exact location of $X$ within a valley.\",\n \"$\\\\overline{Y}^{(i)}$ is a jump process on $\\\\mathcal {S}^{(i)}$ with successive sojourn times $\\\\sigma _{n+1}-\\\\sigma _n,\\\\,n\\\\ge 0$ , which do not only depend on the valley but also on the states of entrance and exit.\",\n \"The AAC then is the embedded chain, viz.\",\n \"$\\\\overline{Y}^{(i)}_n=\\\\sum _{j\\\\ge 0}Y^{(i)}_j{1}_{\\\\lbrace \\\\sigma _j\\\\le n<\\\\sigma _{j+1}\\\\rbrace },$ giving the states only at jumps epochs: starting from the minimum of a first valley it moves to states $k_1{,}...,k_l\\\\in N^{(i)}$ and then proceeds to the minimum of a second valley, and so on.\",\n \"Of course, at small temperatures the time spent in a non-assigned state or in a valley around a low order metastable state is very small compared to the time spent in a valley around a metastable state of higher order.\",\n \"Thus, such states can be seen as instantaneous and of little importance for the evolution of the process.\",\n \"We account for them nonetheless for two reasons.\",\n \"First, in the path-dependent definition mentioned in the Introduction and used in Physics, they build small MB of great transitional activity of the process and are thus relevant in view of our goal to provide a definition of MB that conforms as much as possible to a path-dependent one.\",\n \"Second, a complete partitioning of the state space that is an assignment of every $s\\\\in \\\\mathcal {S}$ to a metastate via a global algorithm fails when merely focusing on $\\\\left\\\\lbrace V^{(i)}(m), m\\\\in M^{(i)}\\\\right\\\\rbrace $ because there is neither an obvious nor natural way how to assign non-assigned states to them.\",\n \"The incoherent scattering function and its associated relaxation time, for $X$ defined by $S(q,n):=\\\\mathbb {E}_{\\\\pi }\\\\cos \\\\big (q|X_n-X_0|\\\\big )$ (with $|\\\\cdot |$ being Euclidean distance in phase space) and $\\\\tau _q(\\\\varepsilon ):=\\\\inf \\\\lbrace n|S(q,n)\\\\le \\\\varepsilon \\\\rbrace ,\\\\quad \\\\varepsilon >0,$ respectively, may serve as an example which shows the strong relation between the behavior of the original process and its macroscopic versions.\",\n \"For more detailed information on the meaning and relevance of $S(q,n)$ as a measure of incoherent scattering between the initial state of a glass-forming system and its state $n$ time steps onward, we refer to the survey by Heuer [12].\",\n \"Proposition 3.2 For each $1\\\\le i\\\\le \\\\mathfrak {n}$ there is a constant $\\\\Delta (i)$ such that $\\\\sup _{n\\\\ge 0}\\\\,\\\\mathbb {P}_{\\\\pi }\\\\left(X_n\\\\ne \\\\overline{Y}^{(i)}_n\\\\right)\\\\le e^{-\\\\Delta (i)\\\\beta }.$ As a consequence, for any given $\\\\varepsilon >0$ , the incoherent scattering functions of $X$ and $\\\\overline{Y}^{(i)}$ differ by at most $4\\\\varepsilon $ for $\\\\beta $ sufficiently large.\",\n \"[Proof:] Use Lemma REF to infer $\\\\mathbb {P}_{\\\\pi }(X_n\\\\ne \\\\overline{Y}^{(i)}_n)&=\\\\sum _{s\\\\in \\\\mathcal {S}^{(i)}\\\\backslash N^{(i)}}\\\\mathbb {P}_{\\\\pi }(\\\\overline{Y}^{(i)}_n=s)\\\\sum _{x\\\\in V^{(i)}(s)\\\\backslash \\\\lbrace s\\\\rbrace }\\\\frac{\\\\pi (x)}{\\\\pi (V^{(i)}(s))}\\\\\\\\&\\\\le \\\\sum _{s\\\\in \\\\mathcal {S}^{(i)}\\\\backslash N^{(i)}}\\\\mathbb {P}_{\\\\pi }(\\\\overline{Y}^{(i)}_n=s)\\\\sum _{x\\\\in V^{(i)}(s)\\\\backslash \\\\lbrace s\\\\rbrace }\\\\frac{\\\\pi (x)}{\\\\pi (s)}\\\\\\\\&\\\\le \\\\sum _{s\\\\in \\\\mathcal {S}^{(i)}\\\\backslash N^{(i)}}\\\\mathbb {P}_{\\\\pi }(\\\\overline{Y}^{(i)}_n=s)\\\\sum _{x\\\\in V^{(i)}(s)\\\\backslash \\\\lbrace s\\\\rbrace }e^{-\\\\beta (E(x)-E(s)-2\\\\gamma _{\\\\beta })}\\\\\\\\&\\\\le \\\\max _{s\\\\in \\\\mathcal {S}^{(i)}\\\\backslash N^{(i)}}|V^{(i)}(s)|\\\\max _{x\\\\in V^{(i)}(s)\\\\backslash \\\\lbrace s\\\\rbrace }e^{-\\\\beta (E(x)-E(s)-2\\\\gamma _{\\\\beta })}.$ This proves equation (REF ) because $E(x)>E(s)$ for each $x\\\\in V^{(i)}(s)\\\\backslash \\\\lbrace s\\\\rbrace , s\\\\in \\\\mathcal {S}^{(i)}\\\\backslash N^{(i)}$ .\",\n \"Now let $\\\\beta $ be so large that $e^{-\\\\Delta (i)\\\\beta }\\\\le \\\\varepsilon $ for a given $\\\\varepsilon >0$ and observe that $\\\\mathbb {E}_{\\\\pi }&\\\\cos \\\\big (q|X_n-X_0|\\\\big )\\\\\\\\&=\\\\ \\\\int _{\\\\lbrace X_n=\\\\overline{Y}_{n}^{(i)},X_0=\\\\overline{Y}^{(i)}_0\\\\rbrace }\\\\cos \\\\big (q|X_n-X_0|\\\\big )\\\\ d\\\\mathbb {P}_{\\\\pi }\\\\ +\\\\ \\\\int _{\\\\lbrace X_n\\\\ne \\\\overline{Y}_n^{(i)}\\\\rbrace \\\\cup \\\\lbrace X_0\\\\ne \\\\overline{Y}^{(i)}_0\\\\rbrace }\\\\cos \\\\big (q|X_n-X_0|\\\\big )\\\\ d\\\\mathbb {P}_{\\\\pi }\\\\\\\\&\\\\le \\\\ \\\\int _{\\\\lbrace X_n=\\\\overline{Y}_n^{(i)},X_0=\\\\overline{Y}^{(i)}_0\\\\rbrace }\\\\cos \\\\big (q|X_n-X_0|\\\\big )\\\\ d\\\\mathbb {P}_{\\\\pi }\\\\ +\\\\ \\\\mathbb {P}_{\\\\pi }(X_n\\\\ne \\\\overline{Y}_n^{(i)})\\\\ +\\\\ \\\\mathbb {P}_{\\\\pi }(X_0\\\\ne \\\\overline{Y}^{(i)}_0)\\\\\\\\&=\\\\ \\\\mathbb {E}_{\\\\pi }\\\\cos \\\\left(q|\\\\overline{Y}_n^{(i)}-\\\\overline{Y}_0^{(i)}|\\\\right)\\\\ -\\\\ \\\\int _{\\\\lbrace X_n\\\\ne \\\\overline{Y}_n^{(i)}\\\\rbrace \\\\cup \\\\lbrace X_0\\\\ne \\\\overline{Y}^{(i)}_0\\\\rbrace }\\\\cos \\\\left(q|\\\\overline{Y}_n^{(i)}-\\\\overline{Y}_0^{(i)}|\\\\right)\\\\ d\\\\mathbb {P}_{\\\\pi } +\\\\ 2\\\\varepsilon \\\\\\\\&\\\\le \\\\ \\\\mathbb {E}_{\\\\pi }\\\\cos \\\\left(q|\\\\overline{Y}_n^{(i)}-\\\\overline{Y}_0^{(i)}|\\\\right)\\\\ +\\\\ 4\\\\varepsilon $ and, by a similar argument, $\\\\mathbb {E}_{\\\\pi }\\\\cos \\\\left(q|\\\\overline{Y}_n^{(i)}-\\\\overline{Y}_0^{(i)}|\\\\right)\\\\ \\\\le \\\\ \\\\mathbb {E}_{\\\\pi }&\\\\cos \\\\big (q|X_n-X_0|\\\\big )\\\\ +\\\\ 4\\\\varepsilon .$ This completes the proof.\"\n ],\n [\n \"(Semi-)Markov Property\",\n \"In general, both aggregated chains are no longer Markovian.\",\n \"Transition probabilities of the AAC not only depend on the current state, i.e.\",\n \"the current valley, but also on the entrance state into that valley, whereas transition probabilities of the AC depend on the current sojourn times which in turn depend on the previous, the present and the next state.\",\n \"On the other hand, since valleys are defined in such a way that asymptotically almost surely (a.a.s.\",\n \"), i.e., with probability tending to one as $\\\\beta \\\\rightarrow \\\\infty $ , the minimum will be reached from anywhere inside the valley before the valley is left, and since, furthermore, the exit state on the outer boundary a.a.s.\",\n \"equals the one with the smallest energy, the AAC will be shown below to converge to a certain Markov chain on $\\\\mathcal {S}^{(i)}$ .\",\n \"Also, the sojourn times depend on the past only via the last and the current state.\",\n \"This means that the AC converges to a semi-Markov chain (for semi-Markov chains see for example [1]): Definition 3.3 Given any nonempty countable set $\\\\mathcal {S}$ , let $(M_{n},T_{n})_{n\\\\ge 0}$ be a bivariate temporally homogeneous Markov chain on $\\\\mathcal {S}\\\\times \\\\mathbb {N}$ , with transition kernel $Q(s,\\\\cdot )$ only depending on the first component, viz., for all $n\\\\ge 0, s\\\\in S$ and $t\\\\ge 0$ , $\\\\mathbb {P}(M_{n+1}=s, T_{n+1}\\\\le t|M_{n},T_{n})=Q(M_{n},\\\\lbrace s\\\\rbrace \\\\times [0,t])$ holds.\",\n \"Put $S_n:=\\\\sum _{i=0}^{n}T_i$ for $n\\\\ge 0$ and $\\\\nu (t):=\\\\max \\\\lbrace n\\\\ge 0|S_n\\\\le t\\\\rbrace \\\\ (\\\\max \\\\emptyset :=0)$ for $t\\\\ge 0$ .\",\n \"Then $Z_n:=M_{\\\\nu (n)}, n\\\\ge 0,$ is called semi-Markov chain with embedded Markov chain $(M_{n})_{n\\\\ge 0}$ and sojourn or holding times $T_{0},T_{1}{,}...$ .\",\n \"Note that equation (REF ) holds iff $M=(M_{n})_{n\\\\ge 0}$ forms a temporally homogeneous Markov chain and the $(T_{n})_{n\\\\ge 0}$ are conditionally independent given $M$ such that the distribution of $T_{n}$ only depends on $M_{n-1},M_{n}$ for $n\\\\ge 1$ (in a temporally homogeneous manner), and on $M_{0}$ for $n=0$ .\",\n \"Note further that we have specialized to the case where holding times take values in $\\\\mathbb {N}$ only (instead of $(0,\\\\infty )$ ).\",\n \"Recall from (REF ) the definition of $s_m$ for $m\\\\in \\\\mathcal {S}^{(i)}\\\\backslash N^{(i)}$ and notice that the second equality there entails $E(z^{*}(m,s_{m}))\\\\sigma _1)\\\\\\\\&=\\\\ \\\\mathbb {P}_{r}(Y_1=s)\\\\,\\\\mathbb {P}_x(\\\\tau _{r}<\\\\sigma _1)+\\\\mathbb {P}_x(Y_1=s,\\\\tau _{r}>\\\\sigma _1).$ The last two summands can further be bounded by $\\\\mathbb {P}_{r}(Y_1=s)\\\\,\\\\mathbb {P}_x(\\\\tau _{r}<\\\\sigma _1)\\\\ \\\\le \\\\ \\\\mathbb {P}_{r}(Y_1=s)\\\\quad \\\\text{and}\\\\quad \\\\mathbb {P}_x(Y_1=s,\\\\tau _{r}>\\\\sigma _1)\\\\ \\\\le \\\\ \\\\mathbb {P}_x(\\\\sigma _1<\\\\tau _{r}).$ For the last probability, Theorem REF ensures $\\\\mathbb {P}_x(\\\\sigma _1<\\\\tau _{r})\\\\ &\\\\le \\\\ \\\\sum _{z\\\\in \\\\partial ^+ V^{(i)}(r)}\\\\mathbb {P}_x(\\\\tau _z<\\\\tau _{r})\\\\ \\\\le \\\\ \\\\sum _{z\\\\in \\\\partial ^+ V^{(i)}(r)} \\\\tilde{\\\\varepsilon }(x,r,z,\\\\beta )\\\\ \\\\stackrel{\\\\beta \\\\rightarrow \\\\infty }{\\\\longrightarrow }\\\\ 0.$ Consequently, as $\\\\beta \\\\rightarrow \\\\infty $ , $\\\\mathbb {P}_{r}(Y_1=s)\\\\ &=\\\\ \\\\left(1-\\\\sum _{z\\\\in \\\\partial ^+ V^{(i)}(r)} \\\\tilde{\\\\varepsilon }(x,r,z,\\\\beta )\\\\right)\\\\mathbb {P}_{r}(Y_1=s)+o(1)\\\\\\\\&\\\\le \\\\ \\\\big (1-\\\\mathbb {P}_x(\\\\sigma _1<\\\\tau _{r})\\\\big )\\\\,\\\\mathbb {P}_{r}(Y_1=s)+o(1)\\\\\\\\&\\\\le \\\\ \\\\mathbb {P}(Y_{n+1}=s|X_{\\\\sigma _n}=x,Y_n=r)+o(1)\\\\\\\\&\\\\le \\\\ \\\\mathbb {P}_{r}(Y_1=s)+\\\\sum _{z\\\\in \\\\partial ^+ V^{(i)}(r)} \\\\tilde{\\\\varepsilon }(x,r,z,\\\\beta )+o(1)\\\\\\\\&=\\\\ \\\\mathbb {P}_{r}(Y_1=s)+o(1),$ and therefore $\\\\mathbb {P}&(Y_{n+1}=s|Y_{n}=r,Y_{n-1}=m_{n-1}{,}...,Y_0=m_0)\\\\\\\\&=\\\\ \\\\sum _{x\\\\in V^{(i)}(r)}\\\\mathbb {P}(Y_{n+1}=s|X_{\\\\sigma _n}=x, Y_n=r)\\\\,\\\\mathbb {P}(X_{\\\\sigma _n}=x|Y_{n}=r,Y_{n-1}=m_{n-1}{,}...,Y_0=m_0)\\\\\\\\&=\\\\ \\\\mathbb {P}_{r}(Y_1=s)+o(1).$ It remains to verify that $\\\\mathbb {P}_{r}(Y_1=s)={\\\\widehat{p}}(r,s)+o(1)$ for any $r,s\\\\in \\\\mathcal {S}^{(i)}$ .\",\n \"If $r\\\\in \\\\mathcal {S}^{(i)}\\\\backslash N^{(i)}$ , then $\\\\sigma _{1}=\\\\tau _{N^{(i)}}$ and $Y_{1}=X_{\\\\tau _{N^{(i)}}}$ .\",\n \"Since $E(z^*(r,s_r))0$ , the sojourn times $\\\\sigma _{n+1}-\\\\sigma _{n}$ , $n\\\\ge 0$ , of the AC $\\\\overline{Y}^{(i)}$ are conditionally independent given $Y^{(i)}$ .\",\n \"The conditional law of $\\\\sigma _{n+1}-\\\\sigma _{n}$ depends only on $(Y_{n-1}^{(i)},Y_{n}^{(i)},Y_{n+1}^{(i)})$ and satisfies $&\\\\mathbb {P}\\\\big (\\\\sigma _{n+1}-\\\\sigma _{n}\\\\in \\\\cdot ~|Y^{(i)}_{n-1}=x,Y^{(i)}_{n}=y,Y^{(i)}_{n+1}=z\\\\big )\\\\nonumber \\\\\\\\&\\\\hspace{14.22636pt}=\\\\ Q((x,y,z),\\\\cdot \\\\, )\\\\ :=\\\\ {\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\hfill \\\\textit {Geom}(1-p(y,y)),&\\\\text{if }y\\\\in N^{(i)}\\\\\\\\\\\\sum _{s\\\\in V^{(i)}(y),s\\\\sim x}\\\\mathbb {P}_s(\\\\sigma _1\\\\in \\\\cdot ~|Y^{(i)}_1=z)\\\\,\\\\mathbb {P}_{x}(X_{\\\\sigma _1}=s),&\\\\text{if }y\\\\notin N^{(i)}\\\\end{array}\\\\right.\",\n \"}$ for all $x,y,z\\\\in \\\\mathcal {S}^{(i)}$ with $\\\\mathbb {P}(Y^{(i)}_{n-1}=x,Y^{(i)}_{n}=y,Y^{(i)}_{n+1}=z)>0$ and $n\\\\ge 1$ .\",\n \"[Proof:] The assertions follow easily when observing that, on the one hand, at least one state $y\\\\in N^{(i)}$ must be visited between two states $x,z\\\\in \\\\mathcal {S}^{(i)}\\\\backslash N^{(i)}$ (Lemma REF ) and that, on the other hand, the original chain $X$ and its aggregation $\\\\overline{Y}^{(i)}$ coincide at any epoch where a non-assigned state is hit, which renders the Markov property of $\\\\overline{Y}^{(i)}$ at these epochs.\",\n \"Further details are omitted.\",\n \"In order to formulate the next result, let $0={\\\\widehat{\\\\sigma }}_{0}<{\\\\widehat{\\\\sigma }}_{1}<...$ be an increasing sequence of random variables such that its increments ${\\\\widehat{\\\\sigma }}_{n+1}-{\\\\widehat{\\\\sigma }}_{n}, n\\\\ge 0$ , are conditionally independent given the asymptotic jump chain ${\\\\widehat{Y}}^{(i)}$ .\",\n \"Moreover, let the conditional law of ${\\\\widehat{\\\\sigma }}_{n+1}-{\\\\widehat{\\\\sigma }}_{n}$ depend only on $\\\\big ({\\\\widehat{Y}}^{(i)}_{n-1},{\\\\widehat{Y}}^{(i)}_{n},{\\\\widehat{Y}}^{(i)}_{n+1}\\\\big )$ and be equal to $Q\\\\big (\\\\big ({\\\\widehat{Y}}^{(i)}_{n-1},{\\\\widehat{Y}}^{(i)}_{n},{\\\\widehat{Y}}^{(i)}_{n+1}\\\\big ),\\\\cdot \\\\,\\\\big )$ , with $Q$ as defined in (REF ).\",\n \"Then $(({\\\\widehat{Y}}^{(i)}_{n},{\\\\widehat{Y}}^{(i)}_{n+1}),{\\\\widehat{\\\\sigma }}_{n+1})_{n\\\\ge 0}$ forms a Markov renewal process and $\\\\big ({\\\\widehat{Y}}^{(i)}_{{\\\\widehat{\\\\nu }}(n)},{\\\\widehat{Y}}^{(i)}_{{\\\\widehat{\\\\nu }}(n+1)}\\\\big )_{n\\\\ge 0}$ a semi-Markov chain, where ${\\\\widehat{\\\\nu }}(n):=\\\\sup \\\\lbrace k\\\\ge 0|{\\\\widehat{\\\\sigma }}_{k}\\\\le n\\\\rbrace $ .\",\n \"Proposition 3.6 For each $1\\\\le i\\\\le \\\\mathfrak {n}$ , $((Y^{(i)}_{n},Y^{(i)}_{n+1}),\\\\sigma _{n+1})_{n\\\\ge 0}$ converges to the Markov renewal process $(({\\\\widehat{Y}}^{(i)}_{n},{\\\\widehat{Y}}^{(i)}_{n+1}),{\\\\widehat{\\\\sigma }}_{n+1})_{n\\\\ge 0}$ in the sense that $\\\\lim _{\\\\beta \\\\rightarrow \\\\infty }\\\\frac{\\\\mathbb {P}_{y_0}\\\\big (\\\\big (Y^{(i)}_{k},Y^{(i)}_{k+1}\\\\big )=(y_{k},y_{k+1}),\\\\,\\\\sigma _{k+1}=i_{k+1},\\\\,0\\\\le k\\\\le n\\\\big )}{\\\\mathbb {P}_{y_0}\\\\big (\\\\big ({\\\\widehat{Y}}^{(i)}_{k},{\\\\widehat{Y}}^{(i)}_{k+1}\\\\big )=(y_{k},y_{k+1}),\\\\,{\\\\widehat{\\\\sigma }}_{k+1}=i_{k+1},\\\\,0\\\\le k\\\\le n\\\\big )}\\\\ =\\\\ 1$ for all $y_{0}{,}...,y_{n+1}\\\\in \\\\mathcal {S}^{(i)}$ , $00$ if there exists a nonempty subset $A\\\\mathcal {S}^{(i)}\\\\backslash N^{(i)}$ with the following property: For each $m_{1}\\\\in A$ , there exists $m_{2}\\\\in A$ such that $\\\\lim _{\\\\beta \\\\rightarrow \\\\infty }\\\\frac{1}{\\\\beta }\\\\left(\\\\ln \\\\left(\\\\mathbb {P}_{m_1}\\\\left(X_{\\\\xi _1}\\\\in V^{(i)}(m_2)\\\\right)\\\\right)-\\\\ln \\\\left(\\\\mathbb {P}_{m_1}\\\\left(X_{\\\\xi _1}\\\\in V^{(i)}(m)\\\\right)\\\\right)\\\\right)\\\\ge \\\\varepsilon $ for all $m\\\\in \\\\mathcal {S}^{(i)}\\\\backslash (N^{(i)}\\\\cup A)$ .\",\n \"In other words, it is exponentially more likely to stay in $A$ than to leave it (ignoring intermediate visits to non-assigned states).\",\n \"In view of our principal goal to give a path-independent definition of MBs, we must point out that, by irreducibility, reciprocating jumps always occur with positive probability at any nontrivial level of aggregation and can therefore never be ruled out completely.\",\n \"This is in contrast to the path-dependent version by Heuer [12] in which the non-occurrence of reciprocating jumps appears to be the crucial requirement.\",\n \"As a consequence, Definition REF provides an alternative, probabilistic and verifiable criterion for reciprocating jumps to be sufficiently unlikely in a chosen aggregation.\",\n \"The following proposition contains further information on which valleys are visited successively by providing the probabilities of making a transition from $V^{(i)}(m)$ to $V^{(i)}(m^{\\\\prime })$ for any $m,m^{\\\\prime }\\\\in \\\\mathcal {S}^{(i)}\\\\backslash N^{(i)}$ .\",\n \"It is a direct consequence of the asymptotic results in the previous subsection, notably Proposition REF .\",\n \"Proposition 3.8 Let $m\\\\in \\\\mathcal {S}^{(i)}\\\\backslash N^{(i)}, s_m\\\\in \\\\partial ^+V^{(i)}(m)$ be as defined in (REF ) (i.e., the state on the outer boundary of $V^{(i)}(m)$ with minimal energy).\",\n \"Then $\\\\lim _{\\\\beta \\\\rightarrow \\\\infty }\\\\mathbb {P}_m(X_{\\\\xi _1}\\\\in V^{(i)}(m))\\\\ =\\\\ {\\\\widehat{p}}(s_m,m)\\\\ =\\\\ \\\\frac{\\\\sum _{r\\\\in \\\\mathcal {N}(s_m)\\\\cap V^{(i)}(m)}p^{*}(s_m,r)}{1-p^{*}(s_m,s_m)},$ while $\\\\lim _{\\\\beta \\\\rightarrow \\\\infty }\\\\mathbb {P}_m(X_{\\\\xi _1}\\\\in V^{(i)}(m^{\\\\prime }))\\\\ =\\\\ {\\\\widehat{p}}(s_{m},m^{\\\\prime })+\\\\sum _{n\\\\ge 1}\\\\sum _{r_{1}{,}...,r_{n}\\\\in N^{(i)}}{\\\\widehat{p}}(s,r_{1})\\\\cdot ...\\\\cdot {\\\\widehat{p}}(r_{n-1},r_{n})\\\\,{\\\\widehat{p}}(r_{n},m^{\\\\prime })$ for any other $m^{\\\\prime }\\\\in \\\\mathcal {S}^{(i)}\\\\backslash N^{(i)}$ .\",\n \"The reader should notice that, as ${\\\\widehat{p}}(s,r)=0$ whenever $E(s)E(r_{1})>...>E(r_{n})>E(m^{\\\\prime })$ .\",\n \"[Proof:] Let us first point out that $\\\\mathbb {P}_{r}(X_{\\\\xi _{0}}\\\\in V^{(i)}(m))=o(1)$ as $\\\\beta \\\\rightarrow \\\\infty $ for any $r\\\\in N^{(i)}$ such that $E(r)E(r)$ , whence the probability for such a path goes to zero as $\\\\beta \\\\rightarrow \\\\infty $ .\",\n \"Noting further that $\\\\mathbb {P}_m(Y_1^{(i)}\\\\ne s_m)=o(1)$ as $\\\\beta \\\\rightarrow \\\\infty $ by Proposition REF , we now infer (with $\\\\xi _{n}=\\\\xi _{n}^{(i)}$ ) $\\\\mathbb {P}_m(X_{\\\\xi _1}\\\\in V^{(i)}(m))\\\\ &=\\\\ \\\\mathbb {P}_{s_m}(X_{\\\\xi _0}\\\\in V^{(i)}(m))+o(1)\\\\\\\\&=\\\\ \\\\mathbb {P}_{s_m}(Y^{(i)}_1=m)+\\\\sum _{r\\\\in \\\\mathcal {N}(s_{m})\\\\cap N^{(i)}}\\\\frac{p(s_{m},r)}{1-p(s_m,s_m)}\\\\,\\\\mathbb {P}_{r}(X_{\\\\xi _0}\\\\in V^{(i)}(m))+o(1)\\\\\\\\&=\\\\ {\\\\widehat{p}}(s_{m},m)+o(1).$ The expression for ${\\\\widehat{p}}(s_{m},m)$ in terms of the $p^{*}(s_{m},r)$ may be read off directly from the formula given in Proposition REF .\",\n \"For $m^{\\\\prime }\\\\ne m$ , $m^{\\\\prime }\\\\in \\\\mathcal {S}^{(i)}\\\\backslash N^{(i)}$ , we obtain in a similar manner $\\\\mathbb {P}_m&(X_{\\\\xi _1}\\\\in V^{(i)}(m^{\\\\prime }))\\\\\\\\&=\\\\ \\\\mathbb {P}_{s_{m}}(Y_{1}^{(i)}=m^{\\\\prime })\\\\ +\\\\ \\\\sum _{n\\\\ge 1}\\\\sum _{r_{1}{,}...,r_{n}\\\\in N^{(i)}} \\\\mathbb {P}_{s_{m}}(Y_{1}^{(i)}=r_{1}{,}...,Y_{n}^{(i)}=r_{n},Y_{n+1}^{(i)}=m^{\\\\prime })+o(1)\\\\\\\\&=\\\\ {\\\\widehat{p}}(s_{m},m^{\\\\prime })+\\\\sum _{n\\\\ge 1}\\\\sum _{r_{1}{,}...,r_{n}\\\\in N^{(i)}}{\\\\widehat{p}}(s_m,r_{1})\\\\cdot ...\\\\cdot {\\\\widehat{p}}(r_{n-1},r_{n})\\\\,{\\\\widehat{p}}(r_{n},m^{\\\\prime })+o(1),$ the last line by another appeal to the afore-mentioned proposition.\",\n \"In essence, the previous result tells us that a valley $V^{(i)}(m^{\\\\prime })$ is neighbored to $V^{(i)}(m)$ , that is, reachable with positive probability by the asymptotic jump chain ${\\\\widehat{Y}}^{(i)}$ (and thus by $Y^{(i)}$ at any temperature level $\\\\beta $ ) without intermediately hitting any other valley, iff there exists at least one (in terms of energies) decreasing path in $N^{(i)}$ from $s_{m}$ to $m^{\\\\prime }$ .\",\n \"For any other such pair of valleys, connected by a path through states in $N^{(i)}$ , the transition probability decreases to zero exponentially in $\\\\beta $ .\",\n \"If this path can be chosen to be unimodal, here called uphill-downhill-path, this can be stated in a very precise way as the next result shows.\",\n \"Lemma 3.9 Let $m_0,m_1\\\\in \\\\mathcal {S}^{(i)}\\\\backslash N^{(i)}$ be two distinct local minima for some $0\\\\le i\\\\le \\\\mathfrak {n}$ .\",\n \"Suppose there exists a minimal path $\\\\gamma =(\\\\gamma _0{,}...,\\\\gamma _k)$ from $s_{m_0}$ to $m_1$ not hitting any other valley but $V^{(i)}(m_{1})$ and such that $I(\\\\gamma _0{,}...,\\\\gamma _{k})=E(z^*(s,m_1))-E(s_{m_0})$ .\",\n \"Then $\\\\lim _{\\\\beta \\\\rightarrow \\\\infty }\\\\frac{1}{\\\\beta }\\\\ln \\\\mathbb {P}_{m_0}(X_{\\\\xi _1}\\\\in V^{(i)}(m_1))\\\\ =\\\\ -\\\\left(E(z^*(m_0,m_1))-E(s_{m_0})\\\\right).$ Without assuming the existence of $\\\\gamma $ as stated, the result remains valid when replacing $=$ with $\\\\le $ .\",\n \"Note that $I(\\\\gamma _0{,}... ,\\\\gamma _{k})=E(z^*(s_{m_0},m_1))-E(s_{m_0})$ does indeed imply the already mentioned property that $E(\\\\gamma _i)&>E(\\\\gamma _{i-1})\\\\quad \\\\textrm {for 1\\\\le i\\\\le j}\\\\\\\\\\\\text{and}\\\\quad E(\\\\gamma _i)&E(b)}(E(a)-E(b))+2\\\\gamma _{\\\\beta }\\\\beta \\\\right)+1\\\\right)\\\\max _{r\\\\in \\\\mathcal {S}}|\\\\mathcal {N}(r)|.$ For any $r\\\\in V^{(i)}(m_{1})$ , we have $E(z^{*}(m_{0},r))\\\\ge E(z^{*}(m_{1},r))$ and therefore $E(z^{*}(m_{0},m_{1}))\\\\ \\\\le \\\\ E(z^{*}(m_{0},r))\\\\vee E(z^{*}(r,m_{1}))\\\\ =\\\\ E(z^{*}(m_{0},r)).$ Using this in (REF ), we obtain $\\\\frac{\\\\mathbb {P}_{m_0}(\\\\xi ^{(i)}_1=\\\\tau _{V^{(i)}(m_1)}<\\\\tau _{m_0})}{\\\\mathbb {P}_{m_0}(\\\\zeta _0<\\\\tau _{m_0})}\\\\ \\\\le \\\\ K(\\\\beta )\\\\, |\\\\mathcal {S}|\\\\, e^{-\\\\beta (E(z^*(m_0,m_1))-E(z^*(m_0,x))-7\\\\gamma _{\\\\beta })}$ and then, upon choosing $x=s_{m_0}$ and noting that $E(z^{*}(m_{0},s_{m_0}))=E(s_{m_0})$ , $\\\\mathbb {P}_{m_0}(X_{\\\\xi _1}\\\\in V^{(i)}(m_1))\\\\ \\\\le \\\\ K(\\\\beta )\\\\,|\\\\mathcal {S}|\\\\,e^{-\\\\beta \\\\left(E\\\\left(z^*\\\\left(m_0,m_1\\\\right)\\\\right)-E(s_{m_0})-7\\\\gamma _{\\\\beta }\\\\right)}.$ By combining all previous results, we finally conclude $\\\\lim _{\\\\beta \\\\rightarrow \\\\infty }\\\\frac{1}{\\\\beta }\\\\ln \\\\mathbb {P}_{m_0}(X_{\\\\xi _1}\\\\in V^{(i)}(m_1))\\\\ =\\\\ -\\\\left(E(z^*(m_0,m_1))-E(s_{m_0})\\\\right)$ as asserted.\",\n \"To summarize, which valleys are visited consecutively depends on (a) their spatial arrangement and (b) the energy barriers between them: A transition from one valley $V^{(i)}(m_0)$ to another valley $V^{(i)}(m_1)$ is only possible, if there exists a path from $s_{m_0}$ to $V^{(i)}(m_1)$ , not hitting any other valley.\",\n \"This transition is made at small temperatures (i.e.\",\n \"large $\\\\beta $ ) if the additional energy barrier $E(z^*(s_{m_0},m_1))-E(s_{m_0})$ is sufficiently small or in other words the energy barrier $E(z^*(s_{m_0},m_1))$ is approximately of the same height as all other energy barriers, including the barrier $E(z^*(s_{m_0},m_0))=E(s_{m_0})$ .\",\n \"A result similar to the previous lemma holds true for transitions from $m\\\\in \\\\mathcal {S}^{(i)}\\\\backslash N^{(i)}$ to any $s\\\\in \\\\partial ^+V(m)$ .\",\n \"Lemma 3.10 Let $m\\\\in \\\\mathcal {S}^{(i)}\\\\backslash N^{(i)}$ and $s\\\\in \\\\partial ^+V^{(i)}(m)$ .\",\n \"Then $\\\\lim _{\\\\beta \\\\rightarrow \\\\infty }\\\\frac{1}{\\\\beta }\\\\ln \\\\mathbb {P}_m(Y_1=s)=-(E(s)-E(s_m)).$ [Proof:] For the proof, decompose again the event $\\\\lbrace Y_1=s\\\\rbrace $ with respect to the number of visits to $m$ before $V(m)=V^{(i)}(m)$ is left (or use Proposition REF ), giving $\\\\mathbb {P}_m(Y_1=s)\\\\ =\\\\ \\\\frac{\\\\mathbb {P}_m(\\\\sigma _1=\\\\tau _s<\\\\tau _m)}{\\\\mathbb {P}_m(\\\\sigma _1<\\\\tau _m)}.$ The proof of the lower bound is much more technical.\",\n \"Let $\\\\gamma =(\\\\gamma _1{,}...,\\\\gamma _n)\\\\in \\\\Gamma ^*(m,s)$ be a minimal path which leaves $V(m)$ only in the last step and such that for any $\\\\gamma _i,\\\\gamma _j\\\\in \\\\gamma $ both, the subpath from $\\\\gamma _i$ to $\\\\gamma _j$ , and the inversed path from $\\\\gamma _j$ to $\\\\gamma _i$ , are minimal.\",\n \"Define $r_0:=m \\\\quad \\\\textrm {and}\\\\quad r_1:=\\\\gamma _{i_0}\\\\quad \\\\textrm {with}\\\\quad i_0:=\\\\inf \\\\lbrace 0\\\\le i\\\\le n-1|E(\\\\gamma _{i+1})\\\\ge E(s_m)\\\\rbrace .$ In particular, $E(r_1)E(\\\\gamma _{i})\\\\rbrace \\\\Big \\\\rbrace ,$ and then successively for $k\\\\ge 1$ with $s_k=\\\\gamma _{i_k}\\\\ne s$ the records $s_{k+1}:=\\\\gamma _{i_{k+1}}$ with $i_{k+1}\\\\ :=\\\\ \\\\inf \\\\Big \\\\lbrace n\\\\ge i\\\\ge \\\\inf &\\\\lbrace j\\\\ge i_k|E(\\\\gamma _j)E(\\\\gamma _{i})\\\\rbrace \\\\Big \\\\rbrace .$ Note that the energy of these records is increasing.\",\n \"Let $s_{k-1}$ be the last record defined in this way and $s_k:=s$ .\",\n \"Since $E(z^*(m,s))=E(s)$ , $s_k$ is as well a record.\",\n \"Given the records $s_1{,}...,s_k$ , for $1\\\\le i\\\\le k-1$ let $r_{2i}$ be the first minimum along $\\\\gamma $ after $s_i$ and $r_{2i+1}$ the last minimum along $\\\\gamma $ before $s_{i+1}$ .\",\n \"Here a minimum along $\\\\gamma $ is some $\\\\gamma _i\\\\in \\\\gamma $ such that it is a minimum of $E$ restricted to $\\\\gamma $ .\",\n \"Finally, let $r_{2k}:=s_k=s$ .\",\n \"In the following we will proof that (a) $\\\\mathbb {P}_{r_{2j}}(\\\\tau _{r_{2j+1}}<\\\\zeta _0)\\\\rightarrow 1$ as $\\\\beta \\\\rightarrow \\\\infty $ for any $0\\\\le j\\\\le k-1$ , (b) $\\\\mathbb {P}_{r_1}(\\\\tau _{r_2}<\\\\zeta _0)\\\\ge e^{-\\\\beta (E(z^*(r_1,r_2))-E(s_m)+o(1))}$ , (c) $\\\\mathbb {P}_{r_{2j+1}}(\\\\tau _{r_{2j+2}}<\\\\zeta _0)\\\\ge e^{-\\\\beta (E(z^*(r_{2j+1},r_{2j+2}))-E(z^*(r_{2j-1},r_{2j}))+o(1))}$ for any $1\\\\le j\\\\le k-2$ , (d) $\\\\mathbb {P}_{r_{2k-1}}(\\\\tau _{r_{2k}}=\\\\zeta _0)\\\\ge e^{-\\\\beta (E(z^*(r_{2k-1},r_{2k}))-E(z^*(r_{2k-3},r_{2k-2}))+o(1))}$ .\",\n \"This gives for $\\\\beta $ large enough $\\\\mathbb {P}_m(Y_1=s)\\\\ &\\\\ge \\\\ \\\\left(\\\\prod _{j=0}^{2k-2}\\\\mathbb {P}_{r_j}(\\\\tau _{r_{j+1}}<\\\\zeta _0)\\\\right)\\\\mathbb {P}_{r_{2k-1}}(\\\\tau _{r_{2k}}=\\\\zeta _0)\\\\\\\\&\\\\ge \\\\ \\\\frac{1}{2}\\\\left(\\\\prod _{j=0}^{k-2}\\\\mathbb {P}_{r_{2j+1}}(\\\\tau _{r_{2j+2}}<\\\\zeta _0)\\\\right)\\\\mathbb {P}_{r_{2k-1}}(\\\\tau _{r_{2k}}=\\\\zeta _0)\\\\\\\\&\\\\ge \\\\ \\\\frac{1}{2} e^{-\\\\beta (E(z^*(r_1,r_2))-E(s_m)+o(1))}\\\\cdot \\\\prod _{j=1}^{k-1}e^{-\\\\beta (E(z^*(r_{2j+1},r_{2j+2}))-E(z^*(r_{2j-1},r_{2j}))+o(1))}\\\\\\\\&=\\\\ \\\\frac{1}{2}e^{-\\\\beta (E(z^*(r_{2k-1},r_{2k}))-E(s_m)+o(1))}\\\\\\\\&=\\\\ e^{-\\\\beta (E(s)-E(s_m)+o(1))},$ and thus the assertion.\",\n \"(a) For $j=0$ this is obvious since $E(z^*(r_0,r_1))E(r_1)\\\\quad \\\\textrm { and }\\\\quad E(z^*(r_1,r_2))\\\\ \\\\ge \\\\ E(s_m)>E(r_1).$ Therefore, $\\\\mathbb {P}_{r_1}((\\\\tau _{r_2}\\\\wedge \\\\zeta _0)<\\\\tau _{r_1})\\\\ &\\\\le \\\\ \\\\sum _{r^{\\\\prime }\\\\in \\\\partial ^+V(m)\\\\cup \\\\lbrace r_2\\\\rbrace }\\\\mathbb {P}_{r_1}(\\\\tau _{r^{\\\\prime }}<\\\\tau _{r_1})\\\\\\\\&\\\\le \\\\ \\\\sum _{r^{\\\\prime }\\\\in \\\\partial ^+V(m)\\\\cup \\\\lbrace r_2\\\\rbrace }e^{-\\\\beta (E(z^*(r_1,r^{\\\\prime }))-E(r_1)+o(1))}\\\\\\\\&\\\\le \\\\ e^{-\\\\beta (E(s_m)-E(r_1)+o(1))}.$ Combining the two estimates, we get $\\\\mathbb {P}_{r_1}(\\\\tau _{r_2}<\\\\zeta _0)\\\\ &=\\\\ \\\\frac{\\\\mathbb {P}_{r_1}(\\\\tau _{r_2}<(\\\\zeta _0\\\\wedge \\\\tau _{r_1}))}{\\\\mathbb {P}_{r_1}((\\\\tau _{r_2}\\\\wedge \\\\zeta _0)<\\\\tau _{r_1})}\\\\ \\\\ge \\\\ e^{-\\\\beta (E(z^*(r_1,r_2))-E(s_m)+o(1))}.$ (c) Let $1\\\\le j\\\\le k-2$ .\",\n \"We use the same strategy as in the proof of (b).\",\n \"So, again, by the definition of $r_{2j+1}$ and $r_{2j+2}$ , there is a unimodal path between them with cumulative activation energy $E(z^*(r_{2j+1},r_{2j+2}))-E(r_{2j+1})$ along this path, and $\\\\mathbb {P}_{r_{2j+1}}(\\\\tau _{r_{2j+2}}<(\\\\zeta _0\\\\wedge \\\\tau _{r_{2j+1}}))\\\\ \\\\ge \\\\ e^{-\\\\beta (E(z^*(r_{2j+1},r_{2j+2}))-E(r_{2j+1})+o(1))}.$ Furthermore, $E(z^*(r_{2j-1},r_{2j}))\\\\ \\\\le \\\\ E(z^*(r_{2j+1},r_{2j+2}))\\\\quad \\\\textrm { and }\\\\quad E(r_{2j+1})\\\\ <\\\\ E(z^*(r_{2j+1},r_{2j+2})).$ Finally, for any $r^{\\\\prime }\\\\in \\\\partial ^+V(m)$ , by use of Equation (REF ), $E(r_{2j+1})\\\\ &\\\\le E(z^*(r_{2j+1},r_{2j}))\\\\\\\\&<\\\\ E(z^*(r_{2j},r^{\\\\prime }))\\\\\\\\&\\\\le \\\\ E(z^*(r_{2j},r_{2j+1}))\\\\vee E(z^*(r_{2j+1},r^{\\\\prime }))\\\\\\\\&=\\\\ E(z^*(r_{2j+1},r^{\\\\prime })).$ Thus, $\\\\mathbb {P}_{r_{2j+1}}((\\\\tau _{r_{2j+2}}\\\\wedge \\\\zeta _0)<\\\\tau _{r_{2j+1}})\\\\ &\\\\le \\\\ \\\\sum _{r^{\\\\prime }\\\\in \\\\partial ^+V(m)\\\\cup \\\\lbrace r_{2j+2}\\\\rbrace }\\\\mathbb {P}_{r_{2j+1}}(\\\\tau _{r^{\\\\prime }}<\\\\tau _{r_{2j+1}})\\\\\\\\&\\\\le \\\\ \\\\sum _{r^{\\\\prime }\\\\in \\\\partial ^+V(m)\\\\cup \\\\lbrace r_{2j+2}\\\\rbrace }e^{-\\\\beta (E(z^*(r_{2j+1},r^{\\\\prime }))-E(r_{2j+1})+o(1))}\\\\\\\\&\\\\le \\\\ e^{-\\\\beta (E(r_{2j-1},r_{2j})-E(r_{2j+1})+o(1))}.$ Combining the two estimates, we get $\\\\mathbb {P}_{r_{2j+1}}(\\\\tau _{r_{2j+2}}<\\\\zeta _0)\\\\ &=\\\\ \\\\frac{\\\\mathbb {P}_{r_{2j+1}}(\\\\tau _{r_{2j+2}}<(\\\\zeta _0\\\\wedge \\\\tau _{r_{2j+1}}))}{\\\\mathbb {P}_{r_{2j+1}}((\\\\tau _{r_{2j+2}}\\\\wedge \\\\zeta _0)<\\\\tau _{r_{2j+1}})}\\\\ \\\\ge \\\\ e^{-\\\\beta (E(z^*(r_{2j+1},r_{2j+2}))-E(r_{2j-1},r_{2j})+o(1))}.$ (d) All bounds for energies in (c) can be proved in the very same way (here $r^{\\\\prime }\\\\in \\\\partial ^+V(m)\\\\backslash \\\\lbrace s\\\\rbrace $ ), so that $\\\\mathbb {P}_{r_{2k-1}}(\\\\tau _{r_{2k}}=\\\\zeta _0)\\\\ &=\\\\ \\\\frac{\\\\mathbb {P}_{r_{2k-1}}(\\\\tau _{r_{2k}}=(\\\\zeta _0\\\\wedge \\\\tau _{r_{2k-1}}))}{\\\\mathbb {P}_{r_{2k-1}}((\\\\tau _{r_{2k}}\\\\wedge \\\\zeta _0)<\\\\tau _{r_{2k-1}})}\\\\ \\\\ge \\\\ e^{-\\\\beta (E(z^*(r_{2k-1},r_{2k}))-E(r_{2k-3},r_{2k-2})+o(1))}.$ Now we see for the reciprocating jumps in the accelerated chain: Proposition 3.11 Fix $1\\\\le i\\\\le \\\\mathfrak {n}$ and $\\\\varepsilon >0$ .\",\n \"Then the AAC at level $i$ exhibits no reciprocating jumps of order $\\\\varepsilon $ if the following three conditions hold true: (1) $E(z^*(m_0,m_1))-E(s_{m_0})\\\\le \\\\varepsilon $ for all distinct $m_0,m_1\\\\in \\\\mathcal {S}^{(i)}\\\\backslash N^{(i)}$ .\",\n \"(2) For each $m\\\\in \\\\mathcal {S}^{(i)}\\\\backslash N^{(i)}$ , there exist at least two distinct $m_1,m_2\\\\in \\\\mathcal {S}^{(i)}\\\\backslash N^{(i)}, m\\\\ne m_1,m_2,$ such that $\\\\mathbb {P}_m(X_{\\\\xi _1}\\\\in V^{(i)}(m_j))>0$ for $j=1,2$ .\",\n \"(3) For each pair $m_0,m_1\\\\in \\\\mathcal {S}^{(i)}\\\\backslash N^{(i)}$ with $\\\\mathbb {P}_{m_0}(X_{\\\\xi _1}\\\\in V^{(i)}(m_1))>0$ , there exists a minimal uphill-downhill-path from $s_{m_0}$ to $m_1$ not hitting any valley but $V^{(i)}(m_1)$ .\",\n \"The origin of our endeavor to define aggregations with no reciprocating jumps of an order larger than a small $\\\\varepsilon $ is to obtain an associated process with (almost) decorrelated increments (in Euclidean state space), for this and a proper centering causes the variance up to the $n$ -th jump to grow with $n$ instead of $n^2$ .\",\n \"This is known as diffusive behavior in physics.\",\n \"Without aggregation increments are highly correlated due to the following argument: at any given time, the process is with high probability in a minimum and when leaving it, say by making a positive jump, the next increment is most likely negative because there is a drift back to the minimum.\",\n \"Likewise, the increments of the asymptotic jump chain are neither uncorrelated nor having mean zero since trajectories of ${\\\\widehat{Y}}$ on an irreducibility class are almost surely of the form $m_1\\\\rightarrow s\\\\rightarrow m_2\\\\rightarrow s\\\\rightarrow m_3\\\\rightarrow ...$ , where $s\\\\in N^{(i)}$ and $m_1,m_2{,}...\\\\in \\\\mathcal {S}^{(i)}\\\\backslash N^{(i)}$ .\",\n \"Thus, given the previous increments, it is in general easy to predict the next increment and they do not have mean zero.\",\n \"On the other hand, if we can choose $\\\\beta $ and $i$ such that, for any $m_0,m_1$ , we have $\\\\mathbb {P}_{m_0}(Y_1=m_1)\\\\in \\\\lbrace p_{m_0}\\\\pm \\\\varepsilon \\\\rbrace \\\\cup [0,\\\\varepsilon ]$ for $\\\\varepsilon \\\\ll p_{m_0}$ , we obtain an AAC which behaves roughly like a RW on a graph.\",\n \"Such a RW is diffusive if we assume periodic boundary conditions (or sufficiently large state space compared to the observation time $n$ ) and an energy landscape $E$ which is homogeneous enough to ensure zero-mean increments.\",\n \"In particular $\\\\lbrace m|\\\\mathbb {P}_{m_0}(Y_1=m)=p_{m_0}\\\\pm \\\\varepsilon \\\\rbrace $ has to comprise at least two states.\"\n ],\n [\n \"Metabasins\",\n \"A path-independent definition of metabasins can now be given on the basis of the previous considerations.\",\n \"Definition 3.12 A finite Markov chain $X$ driven by an energy function $E$ satisfying the assumptions stated at the beginning of Section has metabasins of order $\\\\varepsilon >0$ if there exists an aggregation level $i<\\\\mathfrak {n}-1$ such that the following conditions are fulfilled for each $m\\\\in \\\\mathcal {S}^{(i)}\\\\backslash N^{(i)}$ : (MB4) (MB1) $\\\\sup _{m^{\\\\prime }\\\\in \\\\mathcal {S}^{(i)}\\\\backslash (N^{(i)}\\\\cup \\\\lbrace m\\\\rbrace )}E(z^*(m,m^{\\\\prime }))-E(s_m)\\\\le \\\\varepsilon $ .\",\n \"(MB2) There are at least two distinct $m_1,m_2\\\\in \\\\mathcal {S}^{(i)}\\\\backslash (N^{(i)}\\\\cup \\\\lbrace m\\\\rbrace )$ with a minimal uphill-downhill-path from $s_m$ to $m_k$ not hitting any other valley but $V^{(i)}(m_{k})$ for $k=1,2$ .\",\n \"In this case, the valleys $(V^{(i)}(m))_{m\\\\in \\\\mathcal {S}^{(i)}}$ are called metabasins (MB) of order $\\\\varepsilon $.\",\n \"The reader should notice that each singleton set $\\\\lbrace s\\\\rbrace $ consisting of a non-assigned state $s\\\\in \\\\mathcal {N}^{(i)}$ forms a MB.\",\n \"The conditions (MB1) and (MB2) ensure the good nature of (a) the energy barriers and (b) the spatial arrangement of minima.\",\n \"As already pointed out, this determines which valleys are visited consecutively.\",\n \"Properties of MB which can be concluded from the results of the previous sections are summarized in the next theorem.\",\n \"The reader is reminded of Properties 1–4 stated in the Introduction.\",\n \"Theorem 3.13 For MB as defined in Definition REF we have (4) (1) The transition probabilities for jumps between MB do not depend on the point of entrance as $\\\\beta \\\\rightarrow \\\\infty $ (Property 1).\",\n \"(2) There are no reciprocating jumps of order $\\\\varepsilon $ (Property 2).\",\n \"(3) The expected residence time in a MB depends on $E$ only via the depth of the MB as $\\\\beta \\\\rightarrow \\\\infty $ (Property 3).\",\n \"(4) Regarding only MB pertaining to local minima, the system is a trap model (Property 4).\",\n \"[Proof:] (1) follows from Proposition REF , (2) from Proposition REF , (3) from Theorem REF , and (4) directly from the definition.\",\n \"It should not be surprising that the path-dependent definition of MB by Heuer [12] and stated in the Introduction differs from our path-independent one.\",\n \"Figure: Example of an energy landscape with a tree-like structure.For example, the energy landscape in Figure REF has no reasonable path-dependent MB because every transition between two branches of the shown tree must pass through the state $x$ .\",\n \"For a typical trajectory, there will be at most three MB: the states visited before the first occurrence of $x$ , the states visited between the first and the last occurrence of $x$ , and the states visited after the last occurrence of $x$ .\",\n \"The reason for this poor performance is the tree-like structure of the energy landscape or, more generally, the fact that the connectivity between the branches is too small to allow a selfavoiding walk through more than two branches.\",\n \"This results in a small recurrence time for $x$ (compared to the number of states visited in between).\",\n \"However, every branch constitutes a MB when using the path-independent definition for sufficiently small $\\\\varepsilon $ , in which case the AAC forms a Markov chain and, given the Metropolis algorithm, even a RW on the graph.\",\n \"Having thus exemplified that the two definitions of MB do not necessarily coincide, where the path-independent approach applies to a wider class of energy landscapes, we turn to the question about conditions for them to coincide with high probability.\",\n \"As already pointed out, we have to assume a sufficient connectivity between the metastates to ensure the existence of reasonable path-dependent MB.\",\n \"In terms of this connectivity (for a precise definition see Definition REF ) and the parameter $\\\\beta $ and $\\\\varepsilon $ , our last result, Theorem REF below, provides lower bounds for the probability that both definitions yield the same partition of the state space.\",\n \"The first step towards this end is to identify and count, for each $m\\\\in \\\\mathcal {S}^{(i)}$ and a given $\\\\beta $ , the states $s\\\\in \\\\mathcal {S}^{(i)}$ for which a transition of $Y$ from $m$ to $s$ is likely.\",\n \"This leads to the announced connectivity parameter.\",\n \"Definition 3.14 Let $\\\\varepsilon >0$ and suppose that $X$ has MB of order $\\\\varepsilon >0$ at level $i$ .\",\n \"Define the connectivity parameters $\\\\eta _1\\\\ &=\\\\ \\\\eta _{1,\\\\varepsilon }\\\\ :=\\\\ \\\\min _{n\\\\in N^{(i)},r\\\\in \\\\mathcal {S}^{(i)}:V^{(i)}(r)\\\\cap \\\\mathcal {N}(n)\\\\ne \\\\emptyset }\\\\left|\\\\left\\\\lbrace s\\\\in \\\\mathcal {N}(n)\\\\backslash V^{(i)}(r)\\\\big |\\\\,E(s)\\\\le E(n)+\\\\varepsilon \\\\right\\\\rbrace \\\\right|,\\\\nonumber \\\\\\\\\\\\eta _2\\\\ &=\\\\ \\\\eta _{2,\\\\varepsilon }\\\\ :=\\\\ \\\\min _{m\\\\in \\\\mathcal {S}^{(i)}\\\\backslash N^{(i)}}\\\\big |\\\\lbrace s\\\\in \\\\partial ^+V^{(i)}(m)|E(s)\\\\le E(s_m)+\\\\varepsilon \\\\rbrace \\\\big |,\\\\\\\\\\\\eta _3\\\\ &=\\\\ \\\\eta _{3,\\\\varepsilon }\\\\ :=\\\\ \\\\min _{n\\\\in N^{(i)}}\\\\big |\\\\lbrace s\\\\in \\\\mathcal {S}^{(i)}| E(x)\\\\le E(n)+\\\\varepsilon \\\\textrm { for some x\\\\in V^{(i)}(s)\\\\cap \\\\mathcal {N}(n)}\\\\rbrace \\\\big |.\\\\nonumber $ $\\\\eta _1$ is the minimal number of neighboring sites of a non-assigned state $n$ which do not belong to a particular neighboring valley and whose energy is at most $\\\\varepsilon $ plus the energy of $n$ .\",\n \"$\\\\eta _2$ is the minimal number of neighboring sites/valleys of a valley $V^{(i)}(m)$ whose energy is at most $\\\\varepsilon $ plus the energy of $s_m$ and which can be reached via an uphill-path from $m$ .\",\n \"Finally, $\\\\eta _3$ is the minimal number of neighboring valleys of a non-assigned state $n$ which comprise a state with energy of at most $\\\\varepsilon $ plus the energy of $n$ .\",\n \"$\\\\eta _1$ and $\\\\eta _3$ are always at least 2 and in fact quite large in the very complex energy landscapes of structural glasses.\",\n \"For very small $\\\\varepsilon $ , $\\\\eta _2$ may be 1, but if $X$ has MB of order $\\\\varepsilon $ in a high dimensional energy landscape, then $\\\\eta _2$ can be assumed to be quite large as well.\",\n \"That transitions to states counted above have reasonable large probabilities is content of the following lemma.\",\n \"Thus, the defined parameters do in fact measure the connectivity of the MB.\",\n \"Lemma 3.15 Let $\\\\varepsilon >0$ and suppose that $X$ has MB of order $\\\\varepsilon >0$ at level $i$ with connectivity parameters defined in (REF ).\",\n \"Writing $Y_{k}$ for $Y_{k}^{(i)}$ and $V(m)$ for $V^{(i)}(m)$ , $m\\\\in \\\\mathcal {S}^{(i)}$ , the following assertions hold true for all sufficiently large $\\\\beta $ : (a) If $m\\\\in \\\\mathcal {S}^{(i)}\\\\backslash N^{(i)}$ and $s\\\\in \\\\partial ^{+}V(m)\\\\cap \\\\lbrace x|E(x)-E(s_{m})\\\\le \\\\varepsilon \\\\rbrace $ , or $m\\\\in N^{(i)}$ and $s\\\\in \\\\mathcal {S}^{(i)}$ satisfies $V(s)\\\\cap \\\\lbrace x\\\\in \\\\mathcal {N}(m)|E(x)-E(m)\\\\le \\\\varepsilon \\\\rbrace \\\\ne \\\\emptyset $ , then $\\\\mathbb {P}_{m}(Y_{1}=s)\\\\ \\\\ge \\\\ e^{-2\\\\beta \\\\varepsilon }.$ (b) For any distinct $m\\\\in N^{(i)}$ and $s\\\\in \\\\mathcal {S}^{(i)}$ , $\\\\mathbb {P}_{m}(Y_1\\\\ne s)\\\\ \\\\ge \\\\ \\\\eta _1\\\\,e^{-2\\\\beta \\\\varepsilon },$ (c) For any distinct $m\\\\in \\\\mathcal {S}^{(i)}\\\\backslash N^{(i)}$ and $s\\\\in \\\\mathcal {S}^{(i)}$ , $\\\\mathbb {P}_{m}(Y_1\\\\ne s)\\\\ \\\\ge \\\\ (\\\\eta _2-1)\\\\,e^{-2\\\\beta \\\\varepsilon },$ We see that, for $\\\\varepsilon $ small enough compared to $\\\\beta $ , transitions with an energy barrier of at most $\\\\varepsilon $ are still quite likely and thus a jump to a particular valley quite unlikely in the case of high connectivity.\",\n \"[Proof:] (a) Choose $\\\\beta _{0}>0$ so large that, for $\\\\beta \\\\ge \\\\beta _{0}$ , $\\\\gamma _{\\\\beta }\\\\le \\\\varepsilon $ and $\\\\mathbb {P}_m(Y_1=s)\\\\ge e^{-2\\\\beta (E(s)-E(s_m))}$ for any $m\\\\in \\\\mathcal {S}^{(i)}\\\\backslash N^{(i)}$ and $s\\\\in \\\\partial ^{+}V(m)$ , the latter being possible by Lemma REF .\",\n \"Then for any such $m$ and $s$ , we infer $\\\\mathbb {P}_{m}(Y_1=s)\\\\ge e^{-2\\\\varepsilon \\\\beta }$ provided that additionally $E(s)\\\\le E(s_m)+\\\\varepsilon $ holds true.\",\n \"If $m\\\\in N^{(i)}$ , then $\\\\mathbb {P}_{m}(Y_1=s)\\\\ge e^{-2\\\\varepsilon \\\\beta }$ for any $s\\\\in \\\\mathcal {S}^{(i)}$ such that $E(x)\\\\le E(s)+\\\\varepsilon $ for some $x\\\\in V(s)\\\\cap \\\\mathcal {N}(m)$ , for $\\\\mathbb {P}_{m}(Y_1=s)\\\\ \\\\ge \\\\ \\\\mathbb {P}_{m}(X_{\\\\sigma _{1}}=x)\\\\ =\\\\ p(m,x)\\\\ \\\\ge \\\\ e^{-\\\\beta ((E(x)-E(m))^{+}+\\\\gamma _{\\\\beta })}.$ (b) Pick again $\\\\beta _{0}$ so large that $\\\\gamma _{\\\\beta }\\\\le \\\\varepsilon $ for all $\\\\beta \\\\ge \\\\beta _{0}$ .\",\n \"Then, $\\\\mathbb {P}_{m}(Y_1\\\\ne s)\\\\ &\\\\ge \\\\ \\\\sum _{x\\\\in \\\\mathcal {N}(m),x\\\\notin V(s)}p(m,x)\\\\\\\\&\\\\ge \\\\ \\\\sum _{x\\\\in \\\\mathcal {N}(m),x\\\\notin V(s), E(x)\\\\le E(m)+\\\\varepsilon }\\\\exp \\\\big (-\\\\beta ((E(x)-E(m))^{+}+\\\\gamma _{\\\\beta })\\\\big )\\\\\\\\&\\\\ge \\\\ \\\\eta _1\\\\exp (-2\\\\beta \\\\varepsilon ),$ by definition of $\\\\eta _1$ .\",\n \"(c) Fix $\\\\beta _{0}$ so large that $\\\\mathbb {P}_m(Y_1=x)\\\\ge e^{-2\\\\beta (E(x)-E(s_m))}$ for any $x\\\\in \\\\partial ^{+}V(m)$ and $\\\\beta \\\\ge \\\\beta _{0}$ .\",\n \"In the very same way as in part (b), we then get for all $\\\\beta \\\\ge \\\\beta _0$ $\\\\mathbb {P}_{m}(Y_1\\\\ne s)\\\\ &\\\\ge \\\\ \\\\sum _{x\\\\in \\\\partial ^+V(m),x\\\\ne s}\\\\mathbb {P}_m(Y_1=x)\\\\\\\\&\\\\ge \\\\ \\\\sum _{x\\\\in \\\\partial ^+V(m),x\\\\ne s, E(x)\\\\le E(s_m)+\\\\varepsilon }\\\\exp \\\\big (-2\\\\beta (E(x)-E(s_m))\\\\big )\\\\\\\\&\\\\ge \\\\ (\\\\eta _2-1)\\\\exp (-2\\\\beta \\\\varepsilon ),$ by definition of $\\\\eta _2$ .\",\n \"The above result motivates that in the case of high connectivity the probability to revisit a particular valley within a fixed time $T$ is quite small, or in other words, the probability for the AAC to jump along a selfavoiding path is quite high.\",\n \"This is the main step towards the announced theorem and stated below.\",\n \"The observation time $T$ of course has to be small compared to the cover time of the process.\",\n \"Lemma 3.16 Let $\\\\varepsilon >0$ and suppose that $X$ has MB of order $\\\\varepsilon >0$ at level $i$ with connectivity parameters defined in (REF ).\",\n \"Writing $Y_{k}$ for $Y_{k}^{(i)}$ and $V(m)$ for $V^{(i)}(m)$ , $m\\\\in \\\\mathcal {S}^{(i)}$ , define $\\\\tau _{V(m)}^{(i)}:=\\\\inf \\\\lbrace k\\\\ge 1|Y_{k}=m\\\\rbrace .$ Then the following assertions hold true for all sufficiently large $\\\\beta $ : (a) For any $0<\\\\delta <1-\\\\mathbb {P}_{m}(Y_{2}=m)$ and $1\\\\le T\\\\le T(m,\\\\beta )+1$ , $\\\\mathbb {P}_{m}\\\\left(\\\\tau _{V(m)}^{(i)}>T\\\\right)\\\\ \\\\ge \\\\ \\\\delta ,$ where $T(m,\\\\beta )\\\\ :=\\\\ \\\\frac{\\\\ln \\\\delta }{\\\\ln \\\\!\\\\big (\\\\min _{m^{\\\\prime }\\\\ne m}\\\\mathbb {P}_{m^{\\\\prime }}(Y_1\\\\ne m)(1-{1}_{\\\\lbrace m^{\\\\prime }\\\\notin N^{(i)}\\\\rbrace }\\\\,\\\\delta (m^{\\\\prime },\\\\beta ))\\\\big )}$ and $\\\\delta (m^{\\\\prime },\\\\beta )\\\\ :=\\\\ \\\\max _{x\\\\in V(m^{\\\\prime })}\\\\sum _{z\\\\in \\\\partial ^+ V(m^{\\\\prime })}\\\\tilde{\\\\varepsilon }(x,m^{\\\\prime },z,\\\\beta ).$ In particular, if $\\\\delta \\\\le \\\\left((\\\\eta _1\\\\wedge (\\\\eta _2-2))e^{-2\\\\beta \\\\varepsilon }\\\\right)^{T}$ for some $T>0$ , then $T(m,\\\\beta )\\\\ge T$ .\",\n \"(b) For each $k\\\\ge 1$ and $m_{0}\\\\in \\\\mathcal {S}^{(i)}$ , $\\\\sum _{m_{1}{,}...,m_{k}}\\\\prod _{j=0}^{k-1}\\\\mathbb {P}_{m_j}(Y_1=m_{j+1})\\\\ \\\\ge \\\\ [\\\\eta _2\\\\wedge \\\\eta _3]_{k}\\\\,e^{-2k\\\\varepsilon \\\\beta }$ where summation ranges over all pairwise distinct $m_{1}{,}...,m_{k}\\\\in \\\\mathcal {S}^{(i)}\\\\backslash \\\\lbrace m_0\\\\rbrace $ and for $N\\\\in \\\\mathbb {N}$ we write $[N]_{k}:=N(N-1)\\\\cdot ...\\\\cdot (N-k+1)$ .\",\n \"It should be noticed that $\\\\mathbb {P}_{m}(\\\\tau _{V(m)}^{(i)}>1)=1$ (the AAC never stays put) and $\\\\mathbb {P}_{m}(\\\\tau _{V(m)}^{(i)}>T)\\\\ \\\\le \\\\ \\\\mathbb {P}_{m}(\\\\tau _{V(m)}^{(i)}>2)\\\\ =\\\\ 1-\\\\mathbb {P}_{m}(Y^{(i)}_2=m)$ for every $T\\\\ge 2$ with equality holding only if $T=2$ .\",\n \"We thus see that $\\\\mathbb {P}_{m}(\\\\tau _{V(m)}^{(i)}>T)\\\\ge \\\\delta $ entails $\\\\delta <1-\\\\mathbb {P}_{m}(Y_2=m)$ , the latter being typically large.\",\n \"Furthermore, the bound on the number of self-avoiding path of length $k$ is very crude and can be improved when knowing more about the spatial arrangement of the metastable states.\",\n \"[Proof:] (a) Recall from the first part of the proof of Proposition REF that $\\\\mathbb {P}_{m}(Y_{n+1}\\\\ne z|Y_n=y,X_{\\\\sigma _n}=x)\\\\ &\\\\ge \\\\ \\\\mathbb {P}_{y}(Y_1\\\\ne z)\\\\left(1-{1}_{\\\\lbrace y\\\\notin N^{(i)}\\\\rbrace }\\\\sum _{r\\\\in \\\\partial ^{+}V(y)}\\\\mathbb {P}_{x}(\\\\tau _{r}<\\\\tau _{y})\\\\right)\\\\\\\\&\\\\ge \\\\ \\\\mathbb {P}_{y}(Y_1\\\\ne z)\\\\left(1-{1}_{\\\\lbrace y\\\\notin N^{(i)}\\\\rbrace } \\\\sum _{r\\\\in \\\\partial ^+V(y)}\\\\tilde{\\\\varepsilon }(x,y,r,\\\\beta )\\\\right)\\\\\\\\&\\\\ge \\\\ \\\\mathbb {P}_{y}(Y_1\\\\ne z)\\\\left(1-{1}_{\\\\lbrace y\\\\notin N^{(i)}\\\\rbrace }\\\\,\\\\delta (y,\\\\beta )\\\\right)\\\\\\\\$ holds true for all $y,z\\\\in \\\\mathcal {S}^{(i)},\\\\,x\\\\in V(y)$ and $\\\\beta >0$ .\",\n \"This will now be used repeatedly to show that $\\\\mathbb {P}_{m}\\\\Big (\\\\tau _{V(m)}^{(i)}>T\\\\Big )\\\\ \\\\ge \\\\ \\\\left(\\\\min _{m^{\\\\prime }\\\\ne m}\\\\mathbb {P}_{m^{\\\\prime }}(Y_1\\\\ne m)(1-{1}_{\\\\lbrace m^{\\\\prime }\\\\notin N^{(i)}\\\\rbrace }\\\\,\\\\delta (m^{\\\\prime },\\\\beta ))\\\\right)^{T-1}$ for each $T>2$ .\",\n \"Putting $\\\\mathfrak {m}(x):=m^{\\\\prime }$ if $x\\\\in V(m^{\\\\prime })$ for $m^{\\\\prime }\\\\in \\\\mathcal {S}^{(i)}$ , we obtain $\\\\mathbb {P}_{m}&\\\\Big (\\\\tau _{V(m)}^{(i)}>T\\\\Big )\\\\\\\\&=\\\\ \\\\mathbb {P}_{m}(Y_{1}\\\\ne m{,}...,Y_{T}\\\\ne m)\\\\\\\\&=\\\\ \\\\sum _{x_1{,}...,x_{T-1}\\\\notin V^{(i)}(m)}\\\\mathbb {P}_{m}(X_{\\\\sigma _1}=x_1)\\\\prod _{k=1}^{T-2}\\\\mathbb {P}_{m}\\\\big (X_{\\\\sigma _{k+1}}=x_{k+1}|X_{\\\\sigma _{k}}=x_{k}, Y_{k}=\\\\mathfrak {m}(x_k)\\\\big )\\\\\\\\&\\\\hspace*{76.82234pt}\\\\times \\\\mathbb {P}_{m}(Y_{T}\\\\ne m|X_{\\\\sigma _{T-1}}=x_{T-1}, Y_{T-1}=\\\\mathfrak {m}(x_{T-1}))\\\\\\\\&\\\\ge \\\\ \\\\sum _{x_1{,}...,x_{T-1}\\\\notin V^{(i)}(m)}\\\\mathbb {P}_{m}(X_{\\\\sigma _1}=x_1)\\\\prod _{k=1}^{T-2}\\\\mathbb {P}_{m}\\\\big (X_{\\\\sigma _{k+1}}=x_{k+1}|X_{\\\\sigma _{k}}=x_{k}, Y_{k}=\\\\mathfrak {m}(x_k)\\\\big )\\\\\\\\&\\\\hspace{48.36958pt}\\\\times \\\\left(\\\\min _{m^{\\\\prime }\\\\ne m}\\\\left(\\\\mathbb {P}_{m^{\\\\prime }}(Y_1\\\\ne m)(1-{1}_{\\\\lbrace m^{\\\\prime }\\\\notin N^{(i)}\\\\rbrace }\\\\,\\\\delta (m^{\\\\prime },\\\\beta ))\\\\right)\\\\right)\\\\\\\\&\\\\hspace{5.69046pt}\\\\vdots \\\\\\\\&\\\\ge \\\\ \\\\min _{m^{\\\\prime }\\\\ne m}\\\\Big (\\\\mathbb {P}_{m^{\\\\prime }}(Y_1\\\\ne m)(1-{1}_{\\\\lbrace m^{\\\\prime }\\\\notin N^{(i)}\\\\rbrace }\\\\,\\\\delta (m^{\\\\prime },\\\\beta ))\\\\Big )^{T-1}.\\\\\\\\$ But this establishes the asserted inequality when finally observing that the last expression is $\\\\ge \\\\delta $ iff $T\\\\le T(m,\\\\beta )+1$ .\",\n \"Having just said that $T(m,\\\\beta )\\\\ge T$ holds iff $\\\\min _{m^{\\\\prime }\\\\ne m}\\\\Big (\\\\mathbb {P}_{m^{\\\\prime }}(Y_1\\\\ne m)(1-{1}_{\\\\lbrace m^{\\\\prime }\\\\notin N^{(i)}\\\\rbrace }\\\\,\\\\delta (m^{\\\\prime },\\\\beta ))\\\\Big )^{T}\\\\ \\\\ge \\\\ \\\\delta ,$ it suffices to note that, as $\\\\beta \\\\rightarrow \\\\infty $ , $\\\\delta (m^{\\\\prime },\\\\beta )\\\\rightarrow 0$ holds true if $m^{\\\\prime }\\\\in \\\\mathcal {S}^{(i)}\\\\backslash N^{(i)}$ , giving $1-\\\\delta (m^{\\\\prime },\\\\beta )\\\\ \\\\ge \\\\ \\\\frac{\\\\eta _2-2}{\\\\eta _2-1}$ for sufficiently large $\\\\beta $ .\",\n \"Together with Lemma REF (b), this further yields $\\\\min _{m^{\\\\prime }\\\\ne m}\\\\Big (\\\\mathbb {P}_{m^{\\\\prime }}(Y_1\\\\ne m)(1-{1}_{\\\\lbrace m^{\\\\prime }\\\\notin N^{(i)}\\\\rbrace }\\\\,\\\\delta (m^{\\\\prime },\\\\beta ))\\\\Big )^{T}&\\\\ge \\\\ \\\\left((\\\\eta _1\\\\wedge (\\\\eta _2-2))e^{-2\\\\beta \\\\varepsilon }\\\\right)^{T}$ and then the assertion.\",\n \"(c) Here it suffices to notice that, by (a), $[\\\\eta _2\\\\wedge \\\\eta _3]_{k}$ forms a lower bound for the number of self-avoiding paths $(m_{0}{,}...,m_{k})$ such that $\\\\mathbb {P}_{m_{j}}(Y_{1}=m_{j+1})\\\\ge e^{-2\\\\beta \\\\varepsilon }$ for each $j=0{,}...,k-1$ .\",\n \"We proceed with the announced result about the relation between path-dependent and path-independent MB.\",\n \"To this end, we fix $T=\\\\sigma _{K}$ for some $K\\\\in \\\\mathbb {N}$ .\",\n \"Let $\\\\mathcal {V}_{k}$ for $k=1{,}...,\\\\upsilon $ denote the random number of MB obtained from $X_{0}{,}...,X_{T}$ as defined in the Introduction.\",\n \"For $x\\\\in \\\\mathcal {S}$ , we further let $\\\\mathcal {V}(x)$ denote the MB $\\\\mathcal {V}_{k}$ containing $x$ and put $\\\\mathcal {V}(x):=\\\\emptyset $ if no such MB exists which is the case iff $x\\\\notin \\\\lbrace X_{0}{,}...,X_{T}\\\\rbrace $ .\",\n \"Theorem 3.17 Let $\\\\varepsilon >0$ and suppose that $X$ has MB of order $\\\\varepsilon >0$ at level $i$ with connectivity parameters defined in (REF ).\",\n \"Fix $K\\\\in \\\\mathbb {N}$ , $T=\\\\sigma _{K}$ and $0<\\\\delta \\\\le \\\\left((\\\\eta _1\\\\wedge (\\\\eta _2-1)-1)e^{-2\\\\beta \\\\varepsilon }\\\\right)^{K}$ .\",\n \"Then, for each $0\\\\le k0$ such that for all $\\\\beta \\\\ge \\\\beta _0$ (a) $\\\\mathbb {P}_{m_0}\\\\!\\\\left(V^{(i)}_<(Y_{k})\\\\subseteq \\\\mathcal {V}(Y_{k})\\\\right)\\\\ge 1-(\\\\max _{m\\\\in \\\\mathcal {S}^{(i)}\\\\backslash N^{(i)}}|V_<(m)|+2)\\\\max _{m\\\\in \\\\mathcal {S}^{(i)}\\\\backslash N^{(i)}}\\\\delta (m,\\\\beta )$ , where $V_{<}^{(i)}(s):=\\\\lbrace s\\\\rbrace $ if $s\\\\in N^{(i)}$ .\",\n \"(b) $\\\\mathbb {P}_{m_0}(\\\\mathcal {V}(Y_{j})\\\\subseteq V^{(i)}(Y_{j}),\\\\,0\\\\le jK-1$ , then $\\\\mathbb {P}_{m_0}(\\\\mathcal {V}(Y_j)\\\\subseteq V^{(i)}(Y_j),\\\\,0\\\\le j\\\\le K-1)\\\\ \\\\ge \\\\ [\\\\eta _2\\\\wedge \\\\eta _3]_{K}\\\\left(1-\\\\max _{m\\\\in \\\\mathcal {S}^{(i)}\\\\backslash N^{(i)}}\\\\delta (m,\\\\beta )\\\\right)^{K-1}e^{-2K\\\\varepsilon \\\\beta }.$ For the occurring bounds to be significant, two requirements must be met.\",\n \"First, $K$ must be small compared to the cover time of the AAC and $\\\\varepsilon $ must be small compared to $\\\\beta _0$ to ensure $\\\\exp (-2\\\\beta \\\\varepsilon )\\\\gg 0$ .\",\n \"Second, the connectivity must be high to ensure $1-\\\\delta \\\\ll 1$ and $[\\\\eta _2\\\\wedge \\\\eta _3]_{K}e^{-2K\\\\varepsilon \\\\beta }\\\\gg 0$ .\",\n \"Typically, the inclusions in parts (b) and (c) are strict because of high energy states within a valley that will probably be missed during one simulation run and therefore not belong to any path-dependent MB.\",\n \"On the other hand, since our approach strives to cover the state space as completely as possible by valleys the latter comprise such high energy states whenever they are assignable in the sense described in Section .\",\n \"[Proof:] With $i$ being fixed, let us write as earlier $V(m)$ for $V^{(i)}(m)$ , and also $V_{<}(m)$ for $V_{<}^{(i)}(m)$ .\",\n \"(a) For a given $0\\\\le k \\\\sigma _1 \\\\textrm { for some \\\\right.x\\\\in V_<(m)}-\\\\mathbb {P}_m\\\\big (\\\\textrm {X_l\\\\ne m for each \\\\max _{x\\\\in V_<(m)}\\\\tau _x\\\\le l<\\\\sigma _1}\\\\big ).$ The second probability in the preceding line can further be bounded with the help of (REF ), viz.\",\n \"$\\\\mathbb {P}_m&(\\\\textrm {X_l\\\\ne m for each \\\\max _{x\\\\in V_<(m)}\\\\tau _x\\\\le l<\\\\sigma _1})\\\\\\\\&=\\\\ \\\\sum _{y\\\\in V_<(m)}\\\\mathbb {P}_m(\\\\max _{x\\\\in V_<(m)}\\\\tau _x=\\\\tau _y,\\\\textrm { X_l\\\\ne m for every \\\\max _{x\\\\in V_<(m)}\\\\tau _x\\\\le l<\\\\sigma _1})\\\\\\\\&\\\\le \\\\ \\\\sum _{y\\\\in V_<(m)}\\\\mathbb {P}_m(\\\\max _{x\\\\in V_<(m)}\\\\tau _x=\\\\tau _y)\\\\mathbb {P}_y(\\\\tau _m>\\\\sigma _1)\\\\\\\\&\\\\le \\\\ \\\\delta _{\\\\max },$ while for the first probability, we obtain with the help of Theorem REF $\\\\mathbb {P}_{m}\\\\left(\\\\tau _x> \\\\sigma _1 \\\\textrm { for some \\\\right.x\\\\in V_<(m)}\\\\ &\\\\le \\\\sum _{x\\\\in V_<(m)}\\\\mathbb {P}_{m}(\\\\sigma _1<\\\\tau _x)\\\\nonumber \\\\\\\\&\\\\le \\\\sum _{x\\\\in V_<(m)}\\\\sum _{y\\\\in \\\\partial ^+V(m)}\\\\mathbb {P}_m(\\\\tau _y<\\\\tau _x)\\\\nonumber \\\\\\\\&\\\\le \\\\sum _{x\\\\in V_<(m)}\\\\sum _{y\\\\in \\\\partial ^+V(m)}\\\\varepsilon (m,x,y,\\\\beta ),$ because $E(z^*(m,y))>E(z^*(m,x))$ for $x\\\\in V_<(m)$ and $y\\\\in \\\\partial ^+V(m)$ .\",\n \"The latter can be seen as follows: It has been shown in the proof of Theorem REF that $E(z^*(x,m))0$ such that $\\\\max _{m\\\\in \\\\mathcal {S}^{(i)}}\\\\mathbb {P}_{m}\\\\left(\\\\tau _{V(m)}^{(i)}\\\\le K\\\\right)\\\\ \\\\le \\\\ 1-\\\\delta $ for each $\\\\beta \\\\ge \\\\beta _{0}$ .\",\n \"By using (REF ) and (REF ), we now infer $\\\\mathbb {P}_{m_0}\\\\!&\\\\left(Y_l= Y_k\\\\textrm { for some }k+1\\\\le l\\\\le K\\\\right)\\\\\\\\&=\\\\ \\\\sum _{s\\\\in \\\\mathcal {S}}\\\\mathbb {P}_{m_0}\\\\!\\\\left(Y_l= Y_k\\\\textrm { for some }k+1\\\\le l\\\\le K,X_{\\\\sigma _{k}}=s\\\\right)\\\\\\\\&\\\\le \\\\ \\\\sum _{s\\\\in \\\\mathcal {S}}\\\\mathbb {P}_{m_0}(X_{\\\\sigma _{k}}=s)\\\\Big (\\\\mathbb {P}_{s}(\\\\tau _{V(Y_0)}^{(i)}\\\\le K-k,\\\\,X_{j}=\\\\mathfrak {m}(s) \\\\text{ for some }0\\\\le j< \\\\sigma _1)\\\\\\\\&\\\\hspace*{48.36958pt}+{1}_{\\\\lbrace s\\\\notin N^{(i)}\\\\rbrace }\\\\,\\\\mathbb {P}_{s}(\\\\tau _{V(Y_0)}^{(i)}\\\\le K-k, X_{j}\\\\ne \\\\mathfrak {m}(s) \\\\text{ for all }0\\\\le j< \\\\sigma _1)\\\\Big )\\\\\\\\&\\\\le \\\\ \\\\sum _{s\\\\in \\\\mathcal {S}}\\\\mathbb {P}_{m_0}(X_{\\\\sigma _{k}}=s)\\\\,\\\\mathbb {P}_{\\\\mathfrak {m}(s)}(\\\\tau _{V(Y_0)}^{(i)}\\\\le K-k)+\\\\sum _{s\\\\notin N^{(i)}}\\\\mathbb {P}_{m_0}(X_{\\\\sigma _{k}}=s)\\\\,\\\\mathbb {P}_{s}(\\\\sigma _1<\\\\tau _{\\\\mathfrak {m}(s)})\\\\\\\\&\\\\le \\\\ 1-\\\\delta +\\\\delta _{\\\\max },$ and finally $\\\\mathbb {P}_{m_0}(\\\\mathcal {V}(Y_j)\\\\subset V^{(i)}(Y_j),\\\\,0\\\\le j< k)\\\\ &\\\\ge \\\\ \\\\mathbb {P}_{m_0}\\\\left(\\\\bigcap _{j=0}^{k-1}\\\\left\\\\lbrace Y_l\\\\ne Y_j,j+1\\\\le l\\\\le K\\\\right\\\\rbrace \\\\right)\\\\\\\\&\\\\ge \\\\ 1-\\\\sum _{j=0}^{k-1}\\\\mathbb {P}_{m_0}(Y_l= Y_j\\\\textrm { for some }j+1\\\\le l\\\\le K)\\\\\\\\&\\\\ge \\\\ 1-k(\\\\delta _{\\\\max }+(1-\\\\delta )).$ (c) In the following calculation, let $r_{0}=m_{0}$ , $\\\\sum _{m_{j}}$ range over all $K$ -vectors $(m_{1},...,m_{K})$ with pairwise distinct components in $\\\\mathcal {S}^{(i)}\\\\backslash \\\\lbrace m_0\\\\rbrace $ and, for each $k1.2\\phi \\left(t_{\\rm bounce}\\right) > 1.2."],["Acknowledgments","This work was supported in part by the NSF grants PHY0854743 and PHY1068743."]],"string":"[\n [\n \"Perturbations in loop quantum cosmology\"\n ],\n [\n \"Abstract The era of precision cosmology has allowed us to accurately determine many important cosmological parameters, in particular via the CMB.\",\n \"Confronting Loop Quantum Cosmology with these observations provides us with a powerful test of the theory.\",\n \"For this to be possible we need a detailed understanding of the generation and evolution of inhomogeneous perturbations during the early, Quantum Gravity, phase of the universe.\",\n \"Here we describe how Loop Quantum Cosmology provides a completion of the inflationary paradigm, that is consistent with the observed power spectra of the CMB.\"\n ],\n [\n \"Open issues with inflation\",\n \"The standard picture we have of early universe cosmology is that of inflation.\",\n \"This beautiful idea contains very few basic assumptions, yet it produces, with spectacular agreement, the power spectra of density fluctuations observed in the Cosmic Microwave Background (CMB).\",\n \"Nevertheless, inflation remains a paradigm in search of a model and there are several key questions that remain unanswered.\",\n \"Broadly speaking, these can be split into two categories: difficulties facing the particles physics interpretation of inflation, and those we expect to be related to quantum gravity.\",\n \"Examples of each type are given in Table (REF ).\",\n \"Inflation is a period of quasi-de Sitter expansion and typically the assumption is that slow-roll inflation began with the quantum state describing perturbations in its `natural' vacuum state: the Bunch-Davies vacuum.\",\n \"A priori this is an odd assumption, since it says that the quantum state is tuned to the subsequent (quasi-de Sitter) evolution of the geometry.\",\n \"Essentially this implies that the pre-inflationary dynamics of the universe and the `true' initial state conspired in such a way as to ensure that we arrived at the onset of slow-roll inflation with no particles present (relative to the Bunch-Davies vacuum).\",\n \"The intuition behind this assumption was that even if there were particles present at the onset of inflation, the exponential expansion would rapidly dilute them and hence their consequences can safely be ignored.\",\n \"This intuition misses the important fact that quantum fields in a dynamical space-time experience both spontaneous and simulated creation of particles [2].\",\n \"The latter effect actually compensates for the exponential growth of the volume in such a way that the particle number density remains approximately constant.\",\n \"In classical general relativity inflation is inevitably preceded by the Big-Bang and the only natural place to give initial conditions is at this singularity.\",\n \"An important open question then is: can one find a quantum gravity completion to the inflationary paradigm?\",\n \"To be viably, such a completion should be non-singular and agree with current observations.\",\n \"It would also open up the exciting possibility that we can directly observe the pre-inflationary universe.\",\n \"If it were possible to see observational consequences of the quantum state at the onset of slow-roll, we would be able to probe the dynamics of the quantum gravity era of the universe.\",\n \"Table: Some examples of the unresolved issues facing inflation.\"\n ],\n [\n \"Loop Quantum Cosmology\",\n \"Loop Quantum Gravity is a particularly well developed approach to quantising gravity [3].\",\n \"It is a Hamiltonian quantisation that maintains the fundamental relationship between geometry and gravity and is fully non-perturbative.\",\n \"However a rigorous understanding of the dynamics of the theory are still lacking (see Madhavan Varadarajan's talk in this session).\",\n \"One useful way to make progress is to consider simplified (truncated) systems of the full theory, which, on the one hand, can be completely understood, and on the other are physically interesting.\",\n \"This approach has been applied with great success to the study of black-holes (see Fernando Barbero's talk in this session) and graviton propagators (see Carlo Rovelli's talk in this session) within Loop Quantum Gravity.\",\n \"One can also consider the truncation of full general relativity to homogeneous systems and then use Loop Quantum Gravity techniques to quantise these.\",\n \"This leads to Loop Quantum Cosmology (LQC)[4], which has turned out to be very successful at answering many of the difficulties facing classical cosmology.\",\n \"In particular, it has been shown in detail how the classical singularity is replaced by a `Big-Bounce' and how the late time, low energy, limit reproduces the standard expectations of general relativity.\",\n \"It has also been shown that the probability of having a sufficiently long phase of (single scalar field driven) slow-roll inflation, within LQC, is very close to one [5].\",\n \"However in order to make predictions about LQC effects on the CMB, one first has to extend the underlying approach to include (peturbative) inhomogeneities.\",\n \"There have been several promising attempts to include such inhomogeneities (see Jakub Mielczarek's talk in this session) based on consistent alterations of the homogeneous formulation of LQC.\",\n \"Here we describe a systematic approach to extending the underlying formulation.\"\n ],\n [\n \"LQC and perturbations\",\n \"The first step is to find a suitable truncation of the phase-space of classical general relativity that allows for cosmologies with perturbative inhomogeneities.\",\n \"Since we work in the Hamiltonian theory, we restrict our attention to cosmologies whose spatial slices are (flat) tori.\",\n \"One can then show that the full phase-space decomposes into homogeneous and purely inhomogeneous parts, $\\\\Gamma _{\\\\rm Full} = \\\\Gamma _{\\\\rm H} \\\\times \\\\Gamma _{\\\\rm IH}~.$ The canonical variables of the homogeneous phase-space ($\\\\Gamma _{\\\\rm H}$ ) are $\\\\left( a,\\\\pi _a,\\\\phi ,\\\\pi _\\\\phi \\\\right)$ , i.e.\",\n \"the scale factor $a$ and the scalar field $\\\\phi $ , and their conjugate momenta.\",\n \"The canonical variables of the inhomogeneous phase-space ($\\\\Gamma _{\\\\rm IH}$ ) are the corresponding perturbations, $\\\\left( h_{ab}\\\\left(x\\\\right),\\\\pi ^{ab}\\\\left(x\\\\right),\\\\varphi \\\\left(x\\\\right),\\\\pi _\\\\varphi \\\\left(x\\\\right)\\\\right)$ , all of which are purely inhomogeneous.\",\n \"All the canonical structures of the phase-space decompose in this way, in particular the symplectic structure and the Possion brackets factor.\",\n \"One can now consider the inhomogeneous fields as perturbations, expand the constraints and find gauge invariant degrees of freedom [6].\",\n \"In particular the gauge invariant scalar modes of the perturbations satisfy the constraint, ${\\\\cal C}= \\\\frac{\\\\pi _\\\\phi ^2}{2l^3} + \\\\frac{m^2V^2}{2l^3} - \\\\frac{3}{8\\\\pi G l^3} b^2 V^2+\\\\int {\\\\rm d}^3 k\\\\left( \\\\frac{1}{2} P_k^2 + f\\\\left( V,b,\\\\phi ,\\\\pi _\\\\phi ; k\\\\right) Q^2_k\\\\right) \\\\approx 0~,$ where $V\\\\sim a^3$ and $b\\\\sim \\\\pi _a/a^2$ are canonically conjugate background variables, $l$ is the coordinate size of the 3-torus and $\\\\left( P_k, Q_k \\\\right)$ are the gauge invariant, scalar degree of freedom (related to the Mukhanov variable).\",\n \"The important point is that gauge invariant scalar (and tensor) perturbations behave exactly as test scalar fields with a time dependent mass $f\\\\left( V,b,\\\\phi ,\\\\pi _\\\\phi ; k\\\\right)$ .\",\n \"In [7] it was shown how to quantise such a system, and we schematically sketch this procedure here.\",\n \"Writing the background and perturbation parts of Eq.\",\n \"(REF ) as, ${\\\\cal C} = \\\\frac{\\\\pi _\\\\phi ^2}{2 l^3}+H^2_0 + \\\\int {\\\\rm d}^3 k H_{\\\\tau ,k}~,$ where $H_0$ is the Hamiltonian for the background geometry (and the potential term) and $H_{\\\\tau ,k}$ is the (time dependent) Hamiltonian for the perturbations.\",\n \"These are then promoted to operators and we deparameterise with respect to the scalar field $\\\\phi $ , to find, $-i \\\\hbar \\\\partial _\\\\phi \\\\Psi = \\\\left( \\\\widehat{H_0^2} + \\\\int {\\\\rm d}^3 k\\\\widehat{H}_{\\\\tau ,k} \\\\right)^{1/2} \\\\Psi \\\\approx \\\\left(\\\\widehat{H}_0 + \\\\widehat{H}_0^{-1/2}\\\\left( \\\\int {\\\\rm d}^3 k\\\\widehat{H}_{\\\\tau ,k}\\\\right) \\\\widehat{H}_0^{-1/2} \\\\right) \\\\Psi ~,$ where $\\\\widehat{H_0^2} = \\\\hbar ^2 \\\\widehat{\\\\Theta }_0$ is the operator that governs the LQC evolution of the background [4], and the right hand side has been approximated via a series expansion.\",\n \"This is the only approximation that is made in the procedure and it can be viewed as a test-field approximation.\",\n \"Essentially, one restrictions ones attention to those states in which the $\\\\widehat{H}_0$ is dominant and $\\\\widehat{H}_{\\\\tau ,k}$ does not alter the background dynamicsThere are many important issues to do with ensuring that the integral over $k$ results in a well defined operator that are not being described here.\",\n \"See [1] for details..\",\n \"The second term on the right hand side of Eq.\",\n \"(REF ) is precisely the Hamiltonian for the perturbations associated to the relational time defined by $\\\\phi $ .\",\n \"Finally, one decomposes the wave-function as a tensor product of back-ground and perturbation pieces, $\\\\Psi \\\\left( V, Q_k,\\\\phi \\\\right) = \\\\Psi _0\\\\left( V,\\\\phi \\\\right) \\\\otimes \\\\psi \\\\left( Q_k, \\\\phi \\\\right)~,$ and considers states for which $\\\\Psi _0$ is sharply peaked with respect to some particular classical background geometry at late times i.e.\",\n \"a semi-classical background geometry.\",\n \"Taking expectation values of Eq.\",\n \"(REF ) with respect to $\\\\Psi _0$ , one arrives the standard Hamiltonian for test quantum fields in a curved space-time, with the background scale factor $a\\\\left(\\\\phi \\\\right)$ replaced by $\\\\langle \\\\Psi _0\\\\big |\\\\hat{a}\\\\big |\\\\Psi _0\\\\rangle \\\\big |_{\\\\phi }$ .\"\n ],\n [\n \"Results and conclusions\",\n \"In the previous section we briefly sketched how one can extend the formulation of LQC to include (perturbative) inhomogeneities.\",\n \"With this in hand, one can now look for consequences of the pre-inflationary era and in particular, we can look for deviations from the standard initial conditions for inflation.\",\n \"The standard approach in inflation is to assume the existence of a scalar field with a suitable potential, give oneself some initial conditions for the quantum state of the perturbations and hence calculate the late time power spectra.\",\n \"Here we will take exactly the same approach: we specify initial conditions at the Big-Bounce and calculate the resulting power-spectra (here we concentrate only on tensor perturbations).\",\n \"Since, at the bounce, the wavelength of all the observable modes is much smaller than the curvature scale, we take the initial state (for these modes) to be the Minkowski vacuum (see [1] for a discussion on this point).\",\n \"Figure (REF ) shows the resulting late time (dimensionless) power spectra, $\\\\Delta ^2_R$ , defined as, $\\\\delta \\\\left({\\\\bf k} + {\\\\bf k}^{\\\\prime }\\\\right) \\\\frac{\\\\Delta ^2_R\\\\left(k\\\\right)}{4\\\\pi |{\\\\bf k}|^3} =\\\\langle 0| \\\\hat{R}_{\\\\bf k} \\\\hat{R}_{\\\\bf k^{\\\\prime }}|0\\\\rangle ~,$ where the variable $R_{\\\\bf k}$ is related to the the Mukhanov variable $Q_{\\\\bf k}$ by $R_{\\\\bf k} = \\\\frac{\\\\dot{\\\\phi }}{H} Q_{\\\\bf k}$ , and represents the gravitational potential on constant $\\\\phi $ hyper-surfaces.\",\n \"The important result here is that provided the background scalar field, $\\\\phi $ , at the bounce is large enough, i.e.\",\n \"$\\\\phi \\\\left(t_{\\\\rm bounce}\\\\right) \\\\ge 1.2$ , the power spectra is entirely consistent with that of standard inflation.\",\n \"Hence we have a completion of the inflationary paradigm, including the quantum gravity era, which agree with the observations.\",\n \"Note that there remains a (small but important) window ($0.95 < \\\\phi \\\\left(t_{\\\\rm bounce}\\\\right)< 1.2$ ) in which we may hope to see corrections to the largest scales (smallest $k$ ) of the observed power spectra, and hence directly observe pre-inflationary (i.e.\",\n \"quantum gravity) physics.\",\n \"Figure: The power-spectra for tensor perturbations at the end of slow-roll inflation, given an initialvacuum at the bounce.\",\n \"We have set at bounce =1a\\\\left(t_{\\\\rm bounce}\\\\right) =1 and hence the physicalscale at (for example) decoupling depend on the amount of inflation that has occurred.\",\n \"Note in particular the agreementwith the predictions of standard inflation provided φt bounce >1.2\\\\phi \\\\left(t_{\\\\rm bounce}\\\\right) > 1.2.\"\n ],\n [\n \"Acknowledgments\",\n \"This work was supported in part by the NSF grants PHY0854743 and PHY1068743.\"\n ]\n]"},"id":{"kind":"string","value":"1204.1288"}}}],"truncated":false,"partial":false},"paginationData":{"pageIndex":18668,"numItemsPerPage":100,"numTotalItems":1867602,"offset":1866800,"length":100}},"jwt":"eyJhbGciOiJFZERTQSJ9.eyJyZWFkIjp0cnVlLCJwZXJtaXNzaW9ucyI6eyJyZXBvLmNvbnRlbnQucmVhZCI6dHJ1ZX0sImlhdCI6MTc0MTg1MjE5Niwic3ViIjoiL2RhdGFzZXRzL2hvd2V5L3VuYXJYaXZlIiwiZXhwIjoxNzQxODU1Nzk2LCJpc3MiOiJodHRwczovL2h1Z2dpbmdmYWNlLmNvIn0.zSuuRXHesG7mrgiLwFGdDaDf85yc26QYMA5V2aYbNLU2uZnYycNLb0exFoCZMRB3GH60j_ht3iQpR7AJJRpvBw","displayUrls":true},"discussionsStats":{"closed":0,"open":1,"total":1},"fullWidth":true,"hasGatedAccess":true,"hasFullAccess":true,"isEmbedded":false}">
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"Filamentary Diffusion of Cosmic Rays on Small Scales"
],
[
"Abstract We investigate the diffusion of cosmic rays (CR) close to their sources.",
"Propagating individual CRs in purely isotropic turbulent magnetic fields with maximal scale of spatial variations Lmax, we find that CRs diffuse anisotropically at distances r <~ Lmax from their sources.",
"As a result, the CR densities around the sources are strongly irregular and show filamentary structures.",
"We determine the transition time t* to standard diffusion as t* ~ 10^4 yr (Lmax/150 pc)^b (E/PeV)^(-g) (Brms/4 muG)^g, with b ~ 2 and g = 0.25-0.5 for a turbulent field with Kolmogorov power spectrum.",
"We calculate the photon emission due to CR interactions with gas and the resulting irregular source images."
],
[
"Filamentary Diffusion of Cosmic Rays on Small Scales G. Giacinti$^{1}$ M. Kachelrieß$^{1}$ D. V. Semikoz$^{2,3}$ $^1$ Institutt for fysikk, NTNU, Trondheim, Norway $^2$ AstroParticle and Cosmology (APC), Paris, France $^{3}$ Institute for Nuclear Research of the Russian Academy of Sciences, Moscow, Russia We investigate the diffusion of cosmic rays (CR) close to their sources.",
"Propagating individual CRs in purely isotropic turbulent magnetic fields with maximal scale of spatial variations $l_{\\max }$ , we find that CRs diffuse anisotropically at distances $r$ <$\\hspace{-7.5pt}$$$ l$ from their sources.As a result, the CR densities around the sources are strongly irregular and showfilamentary structures.",
"We determine the transition time $ t$to standard diffusion as$ t104 yr (l/150 pc) (E/PeV)- (Brms/4 G)$, with$ 2$ and $ =0.25$--$ 0.5$ for a turbulent field with Kolmogorov power spectrum.",
"We calculate the photon emissiondue to CR interactions with gas and the resulting irregular source images.$ Introduction.—The suggestion that Galactic cosmic rays (CR) are accelerated using the energy released in supernova (SN) explosions dates back to the 1930's [1].",
"This idea was supported initially mainly by the argument that SNe inject sufficient energy into the Galaxy to maintain the observed CR energy density, while later the radio emission observed from SN remnants (SNR) was interpreted as indication for the acceleration of high-energy electrons.",
"Until present, a clear proof for both the acceleration of hadrons and the identity of their sources is still missing [2].",
"Alternatively, the required power could be provided by acceleration processes which operate at distances and time scales larger than for individual SNRs, e.g.",
"in superbubbles [3].",
"Main obstacle for the identification of CR sources is the diffusion of CRs in the Galactic magnetic field (GMF), erasing directional information on the position of their sources.",
"The GMF has a turbulent component which varies on scales between $l_{\\min }\\lesssim 1$ AU and $l_{\\max }\\sim {\\rm few~to~} 200$ pc.",
"Since CRs scatter on inhomogeneities with variation scales comparable to their Larmor radius, the propagation of Galactic CRs in the GMF resembles a random walk and is well described by the diffusion approximation [4], [5].",
"However, CRs around young sources do not have time to diffuse far away.",
"They should produce an extended gamma-ray halo, which can be detected using Cherenkov telescopes and gamma-ray satellites.",
"If SNe power the Galactic CR population, about ten sources with degree extension should be detected at energies $E_{\\gamma }>100$ GeV in the Galactic plane assuming the sensitivity of Fermi-LAT, while Ref.",
"[6] found 18.",
"Most of these sources were observed also as extended sources up to energies $E_{\\gamma }$ >$\\hspace{-7.5pt}$$$ 10$\\,TeV by the HESSexperiment~\\cite {HESS} and have a non-spherical shape.Similarly, the Veritas observationsof Tycho show a clear asymmetric extension of TeV photons towardsthe north of the SNR~\\cite {VERITAS}.$ The diffusion approximation cannot predict local phenomena which arise below the CR mean free path $\\lambda \\ll l_{\\max }$ , when the local configuration of the turbulent field must lead to observable imprints [9].",
"Before the present study it has remained unclear to which extent the diffusion approximation is satisfied on intermediate scales $\\lambda \\ll l \\lesssim l_{\\max }$ , where large-scale fluctuations of the field lead to local anisotropies, which in turn can explain the irregular images of extended sources found in Refs.",
"[6], [7], [8].",
"We study therefore in this letter the diffusion of CRs on scales comparable to the coherence length of the turbulent GMF, ${\\cal O}(100\\,{\\rm pc})$ .",
"In contrast to earlier studies, we calculate the diffusion tensor propagating individual CRs in specific realizations of the turbulent magnetic field.",
"We find that diffusion is anisotropic even for an isotropic random field and can lead to a filamentary structure of the CR density around young sources.",
"Responsible for these anisotropies are turbulent field modes with variation scales much larger than the Larmor radius of CRs which mimic a regular field.",
"This effect can be confirmed via the observation of irregular gamma-ray emissions around CR sources.",
"Cosmic rays in magnetic turbulence.—Since we want to test effects beyond the diffusion approximation, we propagate individual CRs in turbulent magnetic fields ${B}({k})\\propto \\exp (-ikx)$ using the numerical code described in [10], [11].",
"The validity of this code was checked reproducing earlier results from [12], [13].",
"We assume that the spectrum $\\mathcal {P}(k)$ of magnetic field fluctuations is static and follows a power-law, $\\mathcal {P}(k)\\propto k^{-\\alpha }$ .",
"The former assumption is justified, because the changes introduced by a finite Alfvén velocity are small for the considered time scales.",
"We fix the mean magnetic field strength $B_{\\rm rms}^2\\equiv \\langle {B}^2({x})\\rangle $ as $B_{\\rm rms}=4\\,\\mu $ G, normalising $B_{\\rm rms}$ for fluctuations bounded by $l_{\\min }=1$ AU and $l_{\\max }=150$ pc.",
"In the numerical simulations, we choose $l_{\\min }$ sufficiently small compared to the CR Larmor radius $R_L$ .",
"We also adopt an isotropic spectrum of fluctuations, as we want to demonstrate that even in this case CRs diffuse initially anisotropically.",
"The spectral index of the turbulent GMF is only weakly constrained, and both Kolmogorov ($\\alpha =5/3$ ) and Kraichnan ($\\alpha = 3/2$ ) spectra are consistent with observations [5], [14].",
"As our results do not vary much between these two cases, we present them for a Kolmogorov spectrum.",
"The turbulence is expected to have a Bohm spectrum ($\\alpha =1$ ) only close to shocks, where efficient CR acceleration requires a diffusion coefficient $D(E)\\sim R_L$ .",
"As we are interested in time-scales when CRs have already escaped from the acceleration zone and diffuse on scales ${\\cal O}(100\\,{\\rm pc})$ , we do not address here the question of CR diffusion in the shock region.",
"This allows us also to neglect the backreaction of CRs on the turbulent field, discussed e.g.",
"in [15], which modifies the magnetic field only in a thin layer in front of the shock.",
"We also only consider CRs with $E\\gtrsim 1$ TeV, which are those relevant for gamma ray observations above 100 GeV.",
"CR diffusion and diffusion tensor—Investigating the propagation of CRs close to their source requires to inject them localised in space, say at $x=0$ , and to propagate them in a given concrete realization of the turbulent field.",
"We find that diffusion can be strongly anisotropic in a specific field realization, as long as the distance between CRs and the source is $$ <$\\hspace{-7.5pt}$$$ l$.",
"This anisotropy is washed out averaging over manyrealizations of the turbulent field.",
"Thus the correct procedureto calculate the diffusion tensor in this case is to compute$ Dij(b)= 12Nta=1N xi(a)xj(a) $for $ N$ particles (labeled by the subscript $ a$) and injected at$ x=0$ in one single realization $ b$.For each of the $ M$ realizations one diagonalizes $ Dij(b)$ and finds itseigenvalues $ di(b)$.",
"Then one averages the ordered eigenvalues,$ d1(b)<d2(b)<d3(b)$, over the $ M$ realizations,$ di= 1Mb=1M di(b)$.$ Figure: Eigenvalues d i d_i (solid lines) of the diffusion tensor D ij =〈x i x j 〉/(2t)D_{ij}=\\langle x_ix_j\\rangle /(2t)together with the average diffusion coefficient DD (dashed line)as function of time tt.",
"For B rms =4μB_{\\rm rms}=4\\,\\mu G, l max =150l_{\\max }=150 pc, α=5/3\\alpha =5/3and CR energy E=10 15 E=10^{15} eV.In Fig.",
"REF , we show the three eigenvalues $d_i$ of $D_{ij}$ as function of time for the case of CRs with energy $E=10^{15}$ eV.",
"We used $M=10$ realizations, propagating for each $N=10^{4}$ particles.",
"At early times $t \\lesssim t_\\ast $ , the diffusion tensor is strongly anisotropic, and the ratio $d_3/d_1$ between its largest and smallest eigenvalue can reach a factor of a few hundreds in some turbulent field realizations, while it is on average around a factor of a few tens.",
"For $t\\gtrsim t_\\ast $ , CRs propagate more and more isotropically, approaching the predictions of the diffusion approximation for a purely isotropic turbulent field.",
"Modes with large scale variations contain most of the power in Kolmogorov turbulence, compared to the smaller scale variation modes which are responsible for the diffusion of TeV–PeV CRs.",
"Particles see modes with large variation scales as local uniform fields and diffuse therefore anisotropically.",
"Once a sufficient fraction of particles moved beyond $|{x}|\\sim l_{\\max }$ , the anisotropies of different “cells” of size $l_{\\max }^3$ are averaged out and the CR densities around sources tend towards the limit predicted by the diffusion approximation.",
"Adding a large scale regular field $B_0$ on top of the turbulence would increase the anisotropies.",
"In this case, CRs are known to diffuse faster in the direction of $B_0$ than in the perpendicular direction.",
"The dashed line in Fig.",
"REF represents the average diffusion coefficient $D= 1/M \\sum _{b=1}^M D^{(b)}$ , with $D^{(b)}$ defined as $D^{(b)}= 1/(6Nt) \\sum _{a=1}^N {x}^{(a)}\\cdot {x}^{(a)}$ for the magnetic field realization $b$ .",
"For the largest times numerically reachable, $t=10^5$ yr, the eigenvalues in Fig.",
"REF and the average $D$ approach a common value, $D \\approx 5.5 \\times 10^{28}$ cm$^2/$ s. Interestingly, $2\\times 10^4$ yr corresponds to $\\langle {r}^2\\rangle ^{1/2}=\\sqrt{6Dt}\\approx l_{\\max }$ valid in the isotropic limit.",
"We verified that the limiting value for the average $D$ agrees with the value of the diffusion coefficient computed using CRs with random starting positions.",
"It is consistent, among others, with the computations of [12] for pure random fields.",
"The value of $D(E)$ in our Galaxy is currently only known within a factor $\\approx 50$ at $E_0\\sim 10$ GeV, and $5.5 \\times 10^{28}$ cm$^2/$ s is in the acceptable range extrapolating $D(E)$ to $E=10^{15}$ eV for a Kolmogorov spectrum [16], [17].",
"It is a factor $\\approx 4$ smaller than the value used in Ref. [5].",
"Earlier works [4], [12], [13], [18] did not report anisotropic diffusion for several reasons: Diffusion coefficients were computed averaging over several configurations, with random initial positions for particles, or the considered space or time scales were too large.",
"For instance, [4] and [12] find isotropic diffusion in the limit of a vanishing uniform field, $B_0\\ll B_{\\rm rms}$ .",
"In both works the diffusion coefficient are calculated averaging over many realisations of the turbulent field.",
"As a result, any anisotropy is averaged out, if the random field is isotropic.",
"We also stress that $t_\\ast $ is much larger than the transition time $\\tau _{\\rm diff}\\sim 4D/c^2$ from the ballistic to the diffusive regime [18].",
"Figure: Relative cosmic ray densities around their source projected in the panel planes, for energies E=100E=100 TeV (upper row), 1 PeV (middle row), 10 PeV (lower row) and times t=500t=500 yr (left column), 2 kyr (middle column), 7 kyr (right column).",
"Same field realization in each panel.",
"Each panel corresponds to a 600 pc ×400 pc 600\\,{\\rm pc}\\times 400\\,{\\rm pc} field-of-view, with the source located in the center.Cosmic ray intensity and extrapolation to low E.— In the middle row of Fig.",
"REF , we show the projection of the number density of 1 PeV CRs on an arbitrarily chosen plane of size $600\\,{\\rm pc}\\times 400\\,{\\rm pc}$ containing the injection point ${x}={0}$ in the center.",
"We consider here one given realization of the turbulent field, out of the ten used for Fig.",
"REF .",
"The diffusion is confirmed to be strongly anisotropic at early times, 500 yr (left) and 2000 yr (middle panel), and even strongly filamentary.",
"At 7000 yr (right panel), the distribution of CRs slowly tends towards the spherical limit expected for true isotropic diffusion.",
"Most of the nine other configurations display similar filamentary structures at early times.",
"For a few of them, no thin filaments are visible, but the CR distribution around the source is still asymmetric, showing wind-like structures, also visible for some of the observed sources [7].",
"The upper and lower rows of Fig.",
"REF present results for $E=100$ TeV and $E=10$ PeV, respectively.",
"A comparison of the three rows shows that the period of anisotropic diffusion lasts longer for CRs with lower energy.",
"For instance, the panel with $t=2$ kyr and $E=1$ PeV is very similar to the one with $t=7$ kyr and $E=100$ TeV, which suggests that the expected scaling $t\\propto 1/D(E) \\propto E^{-1/3}$ also holds in the case of anisotropic diffusion.",
"To determine this scaling law more quantitatively, we plot the values of $d_1^{(b)}$ , $d_2^{(b)}$ , $d_3^{(b)}$ and $D^{(b)}$ as function of time for given realizations $b$ in Fig.",
"REF and look for times with similar $d_3^{(b)}/d_1^{(b)}$ : Numerically we find $t_\\ast \\propto E^{-\\gamma }$ with $\\gamma =0.25$ –$0.5$ , i.e.",
"a value of $\\gamma $ consistent with the theoretical expectation.",
"Figure: Eigenvalues d i (b) d_i^{(b)} of the diffusion tensor D ij D_{ij} as function of time tt for energies E=100E=100 TeV (left), 1 PeV (middle) and 10 PeV (right).",
"Same field realization as in Fig. .",
"Dashed lines for the average diffusion coefficient D (b) ≃DD^{(b)} \\simeq D.Transition time.—For the parameters of Fig.",
"REF one estimates $t_{\\ast }\\sim 10^4$ yr.",
"This value is mostly determined by $l_{\\max }$ , and we find that the naive expectation $t_{\\ast } \\propto l_{\\max }^2$ holds in a first approximation.",
"Reducing $l_{\\max }$ by a factor 6, to 25 pc, the transition happens at 200–$300\\simeq 10^4/6^2$ yr in all ten tested configurations.",
"Therefore, our numerical results suggest that the diffusion approximation predictions become valid at $t_{\\ast } \\sim 10^4 \\,{\\rm yr}\\; \\left( l_{\\max }/150\\,{\\rm pc}\\right)^{\\beta }\\left( E/{\\rm PeV} \\right)^{-\\gamma }\\left( B_{\\rm rms} /4\\,{\\rm \\mu G} \\right)^{\\gamma }$ with $\\beta \\simeq 2$ and $\\gamma =0.25$ –$0.5$ for Kolmogorov turbulence.",
"$B_{\\rm rms} \\simeq 4\\,\\mu G$ corresponds to the estimated local value around the Earth from unpolarized synchrotron radiation data [19].",
"However, the large uncertainties of the turbulent field parameters, in particular of $l_{\\max }$ , imply that $t_{\\ast }$ may differ significantly from $10^4$ yr for PeV CRs.",
"The spectral index $\\alpha $ does not have a strong impact on $t_{\\ast }$ .",
"In contrast, it influences the ratio $d_3/d_1$ at early times.",
"For a Bohm spectrum, we find that the anisotropy at $t \\lesssim t_{\\ast }$ is reduced.",
"Indeed, more power is concentrated in small scale modes than in a Kolmogorov spectrum, resulting in a better isotropization of PeV CRs.",
"After averaging over five realizations of the field, we find a factor $d_3/d_1 \\approx 3$ –4 at early times.",
"Out of these five configurations, only one is Kolmogorov-like with $d_3^{(b)}/d_1^{(b)} \\sim 10$ and visible filaments.",
"The others are just slightly anisotropic.",
"Gamma-ray emission.—High energy protons can scatter on protons of the interstellar gas, producing secondaries which in turn decay into photons.",
"We simulate cross sections and the final state of proton-proton interactions using QGSJET-II [20], while we use SIBYLL 2.1 [21] for the subsequent decays of unstable particles.",
"The emission of secondaries is strongly forward beamed and CRs in filaments have an anisotropic distribution of momenta.",
"We may expect that the emitted $\\gamma $ -rays are a good tracker of the underlying CR anisotropies.",
"For simplicity, we assume a uniform gas density around the source.",
"Then we place an observer at 500 pc distance from the source and integrate the photon flux emitted along the line-of-sight towards the observer.",
"We model the observer as a sphere of 5 pc, which is the smallest size providing reasonable statistics and introducing only a small amount of artificial 'fuzziness'.",
"The resulting source image is shown in the right panel of Fig.",
"REF .",
"The comparison to the corresponding CR intensity (left panel) shows that the latter can be used to predict the shape of the gamma ray halo.",
"Note that gamma rays emitted via Compton scattering by electrons would also display such anisotropic patterns.",
"Figure: Left panel: Relative cosmic ray densities along the lines-of-sight as seen from a specific observer located 500 pc away from the source.",
"E=1E=1 PeV and t=1t=1 kyr, for a given magnetic field realization.",
"Box size is 20 ∘ ×20 ∘ 20^{\\circ } \\times 20^{\\circ }, with the source at (0,0).",
"Right panel: Corresponding relative surface brightness in γ\\gamma -rays with energies E γ ≥100E_{\\gamma } \\ge 100 GeV.Conclusions.— We studied the diffusion of TeV–PeV CRs on scales $l$ smaller or comparable to the largest scales of magnetic field fluctuations, $l$ <$\\hspace{-7.5pt}$$$ l$.The propagation of such CRs close to their sources has to be studied insingle realizations of the turbulent field.",
"We showed that CRsdiffuse anisotropically at early times $ tt$, with $ t$ from Eq.~(\\ref {tast}), leading to a filamentary structure of the CR density around their sources.",
"Turbulent field modes with variation scales much larger than the Larmor radius of CRs are responsible for this anisotropic diffusion regime.This effect can explain the observations of irregular gamma-ray halos~\\cite {NS12,HESS,VERITAS} around CR proton and electron sources.",
"If CRs propagate distances $ l$\\;>$$\\sim \\;$ l$, these anisotropies are averaged out.",
"CR densities around sources become isotropic and tend towards those expected from the diffusion approximation.$ GG acknowledges support from the Research Council of Norway through an Yggdrasil grant."
]
] | 1204.1271 |
[
[
"Search for pair production of a new quark that decays to a Z boson and a\n bottom quark with the ATLAS detector"
],
[
"Abstract A search is reported for the pair production of a new quark, b', with at least one b' decaying to a Z boson and a bottom quark.",
"The data, corresponding to 2.0 fb^-1 of integrated luminosity, were collected from pp collisions at sqrt(s) = 7 TeV with the ATLAS detector at the CERN Large Hadron Collider.",
"Using events with a b-tagged jet and a Z boson reconstructed from opposite-charge electrons, the mass distribution of large transverse momentum b' candidates is tested for an enhancement.",
"No evidence for a b' signal is detected in the observed mass distribution, resulting in the exclusion at 95% confidence level of b' quarks with masses m_{b'} < 400 GeV that decay entirely via b' to Z+b.",
"In the case of a vector-like singlet b' mixing solely with the third Standard Model generation, masses m_{b'} < 358 GeV are excluded."
],
[
"Search for pair production of a new quark that decays to a $Z$ boson and a bottom quark with the ATLAS detector CERN-PH-EP-2012-073 A search is reported for the pair production of a new quark, $b^{\\prime }$ , with at least one $b^{\\prime }$ decaying to a $Z$ boson and a bottom quark.",
"The data, corresponding to 2.0 fb$^{-1}$ of integrated luminosity, were collected from $pp$ collisions at $\\sqrt{s}=7$ TeV with the ATLAS detector at the CERN Large Hadron Collider.",
"Using events with a $b$ -tagged jet and a $Z$ boson reconstructed from opposite-charge electrons, the mass distribution of large transverse momentum $b^{\\prime }$ candidates is tested for an enhancement.",
"No evidence for a $b^{\\prime }$ signal is detected in the observed mass distribution, resulting in the exclusion at 95% confidence level of $b^{\\prime }$ quarks with masses $m_{b^{\\prime }}<400$ GeV that decay entirely via $b^{\\prime } \\rightarrow Z+b$.",
"In the case of a vector-like singlet $b^{\\prime }$ mixing solely with the third Standard Model generation, masses $m_{b^{\\prime }}<358$ GeV are excluded.",
"Physical Review Letters Search for pair production of a new quark that decays to a $Z$ boson and a bottom quark with the ATLAS detector The ATLAS Collaboration A search is reported for the pair production of a new quark, $b^{\\prime }$ , with at least one $b^{\\prime }$ decaying to a $Z$ boson and a bottom quark.",
"The data, corresponding to 2.0 fb$^{-1}$ of integrated luminosity, were collected from $pp$ collisions at $\\sqrt{s}=7$ TeV with the ATLAS detector at the CERN Large Hadron Collider.",
"Using events with a $b$ -tagged jet and a $Z$ boson reconstructed from opposite-charge electrons, the mass distribution of large transverse momentum $b^{\\prime }$ candidates is tested for an enhancement.",
"No evidence for a $b^{\\prime }$ signal is detected in the observed mass distribution, resulting in the exclusion at 95% confidence level of $b^{\\prime }$ quarks with masses $m_{b^{\\prime }}<400$ GeV that decay entirely via $b^{\\prime } \\rightarrow Z+b$.",
"In the case of a vector-like singlet $b^{\\prime }$ mixing solely with the third Standard Model generation, masses $m_{b^{\\prime }}<358$ GeV are excluded.",
"14.65.Fy, 14.65.Jk, 12.60.-i The matter sector of the Standard Model (SM) consists of three generations of chiral fermions, with each generation containing a quark doublet and a lepton doublet.",
"A natural question is whether quarks and leptons exist beyond the third generation [1].",
"In this Letter we present a search for the pair production of a new quark with electric charge $-1/3$ , denoted $b^{\\prime }$ , using data collected by the ATLAS experiment at the Large Hadron Collider.",
"New quarks appear in a variety of models that address shortcomings of the SM [1], [2], [3], [4], [5].",
"In addition to signaling a richer matter content at high energy, their existence would impact lower-scale physics, such as altering Higgs boson ($H$ ) phenomenology [6], and providing new sources of CP violation potentially sufficient to generate the baryon asymmetry in the universe [7].",
"Several collaborations have previously searched for a chiral $b^{\\prime }$ .",
"A search by D0 [8] for the decay $b^{\\prime } \\rightarrow \\gamma + b$ excludes $b^{\\prime }$ quarks with masses below $m_{Z}+m_{b}=96$ GeV.",
"CDF [9] searches for the decay $b^{\\prime } \\rightarrow Z+b$ exclude masses below $m_{W} + m_{t}=256$ GeV.",
"These limits apply to prompt $b^{\\prime }$ decays.",
"CDF and D0 have also searched for non-prompt $b^{\\prime } \\rightarrow Z+b$ decays [10], excluding, for example, $b^{\\prime }$ masses below 180 GeV for $c\\tau = 20$ cm [11].",
"More recently, CDF [12], CMS [13], and ATLAS [14] have searched for the prompt charged-current decay $b^{\\prime } \\rightarrow W+t$ .",
"This decay mode is dominant for a chiral $b^{\\prime }$ with mass in excess of $m_{W} + m_{t}$ , as the neutral-current modes only occur through loop diagrams [1].",
"The ATLAS result excludes chiral $b^{\\prime }$ quarks with masses below 480 GeV.",
"Extensions to the SM often propose new quarks transforming as vector-like representations of the electroweak gauge groups [2], [3], [4], [5].",
"The decay of a vector-like $b^{\\prime }$ to a $Z$ boson and a bottom quark is a tree-level process with a branching ratio comparable to that of the decay $b^{\\prime } \\rightarrow W+t$ .",
"In particular, the branching ratios $Wt:Zb:Hb$ approach the proportion $2:1:1$ in the limit of large $b^{\\prime }$ mass as a consequence of the Goldstone boson equivalence theorem [2], [5].",
"Furthermore, if a signal were observed in the $WtWt$ final state, a search for a resonant $Z+b$ signal would aid in establishing the charge of the new quark.",
"In light of these observations, this search explores the $Z+b$ -jet final state for the presence of a $b^{\\prime }$ quark.",
"The ATLAS detector [15] consists of particle-tracking detectors, electromagnetic and hadronic calorimeters, and a muon spectrometer.",
"At small radii transverse to the beamline, the inner tracking system utilizes fine-granularity pixel and microstrip detectors designed to provide precision track impact parameter and secondary vertex measurements.",
"These silicon-based detectors cover the pseudorapidity [16] range $|\\eta |<2.5$ .",
"A gas-filled straw tube tracker complements the silicon tracker at larger radii.",
"The tracking detectors are immersed in a 2 T magnetic field produced by a thin superconducting solenoid located in the same cryostat as the barrel electromagnetic (EM) calorimeter.",
"The EM calorimeters employ lead absorbers and utilize liquid argon as the active medium.",
"The barrel EM calorimeter covers $|\\eta |<1.5$ , and the end-cap EM calorimeters $1.4<|\\eta |<3.2$ .",
"Hadronic calorimetry in the region $|\\eta |<1.7$ is achieved using steel absorbers and scintillating tiles as the active medium.",
"Liquid argon calorimetry with copper absorbers is employed in the hadronic end-cap calorimeters, which cover the region $1.5<|\\eta |<3.2$ .",
"The search for the decay $b^{\\prime }\\rightarrow Z+b$ is performed in the final state with the $Z$ boson decaying to an electron-positron pair ($e^{+}e^{-}$ ) using a dataset collected in 2011 corresponding to an integrated luminosity of $1.98\\pm 0.07~\\mathrm {fb}^{-1}$ [17].",
"The selected events were recorded with a single-electron trigger that is over 95% efficient for reconstructed electrons [18] with momentum transverse to the beam direction, $p_{\\mathrm {T}}$ , exceeding 25 GeV.",
"At least two opposite-charge electron candidates are required, each satisfying $p_{\\mathrm {T}}>25$ GeV and reconstructed in the pseudorapidity region $|\\eta |<2.47$ , excluding the barrel to end-cap calorimeter transition region, $1.37< |\\eta | < 1.52$.",
"In addition, the electron candidates satisfy medium quality requirements [18] on the reconstructed track and properties of the electromagnetic shower.",
"The two opposite-charge electron candidates yielding an invariant mass, $m_{ee}$ , that satisfies $|m_{ee}-m_{Z}|<15$ GeV and is closest to the $Z$ boson mass define the $Z$ candidate.",
"Approximately 475,000 events pass the $Z \\rightarrow e^{+}e^{-}$ selection criteria.",
"Jets are reconstructed using the anti-$k_{t}$ clustering algorithm [19] with a distance parameter of 0.4.",
"The inputs to the algorithm are three-dimensional clusters formed from calorimeter energy deposits.",
"Jets are calibrated using $p_{\\mathrm {T}}$ - and $\\eta $ -dependent factors determined from simulation and validated with data [20].",
"Jets are rejected if they do not satisfy quality criteria to suppress noise and non-collision backgrounds, as are jets whose axis is within $\\Delta R=\\sqrt{(\\Delta \\eta )^{2}+(\\Delta \\phi )^{2}}=0.5$ of a reconstructed electron associated with the $Z$ candidate.",
"A requirement is made to ensure at least $75\\%$ of the total $p_{\\mathrm {T}}$ of all tracks associated with the jet be attributed to tracks also associated with the selected $pp$ collision vertex [21].",
"Lastly, jets in this analysis are restricted to the region covered by the tracking detectors, $|\\eta |<2.5$ , and satisfy $p_{\\mathrm {T}}>25$ GeV.",
"Approximately 81,000 events pass the $Z \\rightarrow e^{+}e^{-}$ candidate selection and contain at least one selected jet.",
"The SM production of $Z$ bosons in association with jets accounts for most events passing the $Z+\\ge 1$ jet selection.",
"Two leading-order Monte Carlo (MC) generators, alpgen [22] and sherpa [23], are used to assess the background arising from this process, with alpgen providing the baseline prediction.",
"A description of the generation of these samples, in particular in regard to differences between alpgen and sherpa in the modeling of $Z$ boson production in association with $b$ -jets, is detailed in Ref. [24].",
"The predictions of both are normalized such that the inclusive $Z$ boson cross section is equal to a next-to-next-to-leading-order (NNLO) calculation [25].",
"All MC samples fully simulate the ATLAS detector [26] and are reconstructed with the same algorithms as those applied to data.",
"The $Z$ +bottom background category comprises simulated $Z+\\mathrm {jet(s)}$ events in which a generated $p_{\\mathrm {T}}>5$ GeV bottom quark is matched to a selected reconstructed jet.",
"Similarly, events with a jet matched to a charm quark, but not a bottom quark, constitute the $Z$ +charm category.",
"In the $Z$ +light category, none of the selected jets are matched to a bottom or charm quark.",
"Additional SM backgrounds modeled with MC events include top quark pair production ($t\\bar{t}$ ), single top production, heavy vector boson pair (diboson) production, $Z( \\rightarrow \\tau \\tau )+$ jet(s) events, and $W(\\rightarrow e\\nu )+$ jet(s) events.",
"Processes with a top quark are simulated with mc@nlo [27], [28].",
"The $t\\bar{t}$ cross section used is the hathor [29] approximate NNLO value, while mc@nlo [28] values are used for the single top processes.",
"herwig [30] models the contribution of diboson events, with the cross sections set by the mcfm [31] NLO predictions.",
"The remaining $W/Z+$ jet(s) backgrounds are simulated with alpgen, and normalized using single vector boson production NNLO cross sections [25].",
"The multi-jet background is estimated using a data sample with both electron candidates passing loose criteria [18] but failing the slightly tighter medium criteria.",
"This sample is normalized to the difference in the inclusive $Z$ sample between the data and all other backgrounds in the region $50<m_{ee}<65$ GeV.",
"The small single top, diboson, $Z\\rightarrow \\tau \\tau $ , $W\\rightarrow e\\nu $ , and multi-jet contributions are combined and denoted Other SM.",
"Figure: e + e - e^{+}e^{-} invariant mass distribution for events passing the Z+≥1Z+\\ge 1 jet selection, before imposing the |m ee -m Z |<15|m_{ee}-m_{Z}|<15 GeV requirement.",
"The predicted contributions of the SM background sources are shown stacked.",
"The lower panel shows the ratio of the data to the SM prediction, and the solid yellow band denotes the systematic uncertainty on the SM prediction.Figure: e + e - e^{+}e^{-} invariant mass distribution for events passing the Z+≥1Z+\\ge 1 bb-jet selection, before imposing the |m ee -m Z |<15|m_{ee}-m_{Z}|<15 GeV requirement.Table: Number of predicted and observed events at three stages in the event selection.",
"The contributions from SM backgrounds are shown individually, as well as combined into the total SM prediction.",
"The uncertainties on the predicted number of events combine all sources of uncertainty.",
"The number of expected signal events is also listed for two representative b ' b^{\\prime } masses in the case where BR(b ' →Zb)=1BR(b^{\\prime } \\rightarrow Zb)=1.Figure REF presents the $e^{+}e^{-}$ invariant mass distribution for events passing the $Z+\\ge 1$ jet selection, before imposing the $|m_{ee}-m_{Z}|<15$ GeV requirement, together with the SM prediction.",
"The observed and predicted number of events are listed in Table REF for this and two other stages of the event selection.",
"Most events passing the $Z+\\ge 1$ jet selection arise from the $Z$ +light category.",
"The appreciable lifetime of the $b$ -hadron originating from the bottom quark in the decay $b^{\\prime } \\rightarrow Z+b$ provides a means to reduce this background source.",
"A $b$ -jet tagging algorithm referred to as IP3D+SV1 [32] is utilized to select events with at least one $b$ -jet from the $Z+\\ge 1$ jet sample.",
"The discriminant combines two likelihood variables based on the tracks associated with a jet.",
"The first employs the longitudinal and transverse track impact parameters, while the second utilizes properties of a reconstructed secondary vertex.",
"In a simulated $t\\bar{t}$ sample, the requirement on the discriminant defining a $b$ -jet is $60\\%$ efficient for jets with a $b$ -hadron, and yields a light flavor jet rejection rate of 300 [32].",
"A total of 3,466 events satisfy the $Z+\\ge 1$ $b$ -jet selection.",
"Figure REF presents the $e^{+}e^{-}$ invariant mass distribution in this sample and the SM prediction, before imposing the $|m_{ee}-m_{Z}|<15$ GeV requirement.",
"The accurate modeling of the mass distribution for values beyond the $Z$ boson mass supports the prediction of $t\\bar{t}$ and Other SM background events.",
"Within the window around the $Z$ boson mass, alpgen and sherpa agree to within $1\\%$ and $7\\%$ in the prediction of the number of $Z$ +light and $Z$ +charm events, respectively.",
"However, alpgen and sherpa disagree in the prediction of the $Z$ +bottom contribution, a fact previously reported in an ATLAS cross section measurement of $Z$ bosons produced in association with $b$ -jets using a smaller dataset [24].",
"The alpgen and sherpa $Z$ +bottom predictions are scaled to account for the difference between data and all other predicted backgrounds in a subsample of the $Z+\\ge 1$ $b$ -jet sample that contains events failing the requirement discussed below on the transverse momentum of the $b^{\\prime }$ candidate.",
"The scale factors are consistent with those measured in Ref.",
"[24], and the invariant mass distribution of secondary vertex tracks is used to confirm the validity of the resulting prediction for the flavor composition in the $Z+\\ge 1$ $b$ -jet sample [24].",
"Simulated $b^{\\prime }\\bar{b}^{\\prime }$ events are generated for a range of $b^{\\prime } $ masses using madgraph [33] with the G4LHC extension [6].",
"pythia [34] performs fragmentation and hadronization of the parton-level events.",
"The signal cross sections are obtained with hathor [29], and vary from 80 pb to 30 fb over the range $m_{b^{\\prime }}=200-700$ GeV.",
"In each sample, one $b^{\\prime }$ decays in the mode $b^{\\prime } \\rightarrow Z+b$ , with the $Z$ boson decaying via $Z\\rightarrow e^{+}e^{-}$ .",
"Two separate samples are produced for each mass value, with the other $b^{\\prime }$ decaying either via $b^{\\prime } \\rightarrow Z+b$ or $b^{\\prime } \\rightarrow W+t$ , and with all decay modes of the $Z$ and $W$ bosons allowed.",
"The factor $\\beta = 2\\times BR(b^{\\prime }\\rightarrow Zb)-BR(b^{\\prime }\\rightarrow Zb)^{2}$ characterizes the fraction of signal events with at least one $b^{\\prime }\\rightarrow Z+b$ decay as a function of the branching ratio.",
"The case $\\beta =1$ is equivalent to previous measurements [9] which assumed $BR(b^{\\prime }\\rightarrow Zb)=1$ .",
"The case of a vector-like singlet (VLS) mixing solely with the third SM generation is also considered by computing $\\beta $ as a function of $b^{\\prime }$ mass [5].",
"Over the range $m_{b^{\\prime }}=200-700$ GeV, $\\beta $ varies from $0.9$ to $0.5$ .",
"A SM Higgs of mass 125 GeV is assumed.",
"Figure: Transverse momentum distribution of the b ' b^{\\prime } candidate in events passing the Z+≥1Z+\\ge 1 bb-jet selection.",
"The predicted contributions of the SM background sources are stacked, while the distributions for the two signal scenarios described in the text are overlaid.The $b^{\\prime }$ candidate is formed from the $e^{+}e^{-}$ pair and the highest $p_{\\mathrm {T}}$ $b$ -jet.",
"The mass of the $b^{\\prime }$ candidate, $m(Zb)$ , is the discriminant distinguishing the background-only and signal-plus-background hypotheses.",
"In $b^{\\prime }$ pair production, the new quarks are typically produced with large transverse momentum, $p_{\\mathrm {T}}(Zb)$ .",
"Therefore, a $p_{\\mathrm {T}}(Zb)>150$ GeV requirement is applied to increase the signal sensitivity.",
"Figure REF presents the $p_{\\mathrm {T}}(Zb)$ distribution for data and the predicted SM backgrounds.",
"Additionally, the signal distribution is overlaid for a $b^{\\prime }$ mass of 350 GeV, assuming the VLS scenario value $\\beta =0.63$ , and for a mass of 450 GeV, assuming $\\beta =1$ .",
"Figure: Mass distribution of the b ' b^{\\prime } candidate in events passing the Z+≥1Z+\\ge 1 bb-jet selection and satisfying p T (Zb)>150p_{\\mathrm {T}}(Zb)>150 GeV.",
"The highest mass bin also includes the data and prediction for m(Zb)>1m(Zb)>1 TeV.The fraction of signal events passing all requirements varies from $7\\%$ to $43\\%$ between $m_{b^{\\prime }}=200-700$ GeV, assuming $\\beta =1$, with the efficiency to pass the minimum $p_{\\mathrm {T}}(Zb)$ requirement contributing most to the degree of variation.",
"The requirement $p_{\\mathrm {T}}(Zb)>150$ GeV was determined by assessing the signal sensitivity for different minimum $p_{\\mathrm {T}}(Zb)$ values, as quantified by the expected cross section exclusion limit.",
"The limit is computed using a binned Poisson likelihood ratio test [35] of the $m(Zb)$ distribution for different $m_{b^{\\prime }}$ hypotheses.",
"Pseudo-experiments are generated according to the background-only and signal-plus-background hypotheses, and incorporate the impact of systematic uncertainties.",
"The cross section limit is evaluated using the $\\mathrm {CL}_{s}$ modified frequentist approach [35].",
"The impact of each systematic uncertainty on the normalization and shape of the $m(Zb)$ distribution is assessed for each SM background source and the expected $b^{\\prime } $ signal.",
"The fractional uncertainty on the total number of background events passing the $p_{\\mathrm {T}}(Zb)>150$ GeV requirement is 27%.",
"Significant contributions arise from uncertainties in the $p_{\\mathrm {T}}(Zb)$ distribution shape in $Z+\\mathrm {jet(s)}$ events.",
"Such sources of uncertainty include the renormalization and factorization scale choice (14%, evaluated using mcfm [36]), shape differences observed between alpgen and sherpa (12%), and variations in the degree of initial and final state QCD radiation (9%).",
"The uncertainty in the efficiency of the $b$ -tagging requirement contributes an additional 12%.",
"Other sources of uncertainty contributing at the level of 6% or less include the jet energy scale [20] , parton distribution functions (PDF), MC sample sizes, electron identification efficiency, $Z$ boson cross section, luminosity, $b$ -jet mis-tag rate, $t\\bar{t}$ cross section, jet energy resolution, trigger efficiency, and the Other SM event yield.",
"Most of the above uncertainties, with the notable exception of the $p_{\\mathrm {T}}(Zb)$ modeling uncertainties in $Z+\\mathrm {jet(s)}$ events, contribute to the total uncertainty on the signal normalization, which varies between 11% and 14% depending on the $b^{\\prime }$ mass.",
"Figure: The expected and observed 95%95\\% C.L.",
"cross section limits as a function of b ' b^{\\prime } mass.",
"The signal cross section is shown with uncertainties arising from PDFs and renormalization and factorization scale choice.",
"The prediction is also multiplied by the β\\beta factors described in the text.Figure REF presents the $b^{\\prime }$ candidate mass distribution after requiring $p_{\\mathrm {T}}(Zb)>150$ GeV and the predicted SM background.",
"The distributions for the signal scenarios depicted in Fig.",
"REF are shown overlaid.",
"The data are in agreement with the SM prediction over the full range of $m(Zb)$ values.",
"In the absence of evidence of an enhancement, 95% confidence level (C.L.)",
"cross section exclusion limits are derived.",
"Figure REF presents the expected and observed cross section limits as a function of $m_{b^{\\prime }}$ , computed under the assumption $\\beta =1$ .",
"The expected cross section limit was checked to be stable to within $15\\%$ over the full mass range considered using the signal samples in which one $b^{\\prime }$ quark decays via $b^{\\prime } \\rightarrow Z+b$ and the other decays via $b^{\\prime } \\rightarrow W+t$ .",
"The approximate NNLO $b^{\\prime }\\bar{b}^{\\prime }$ cross section prediction is shown multiplied by $\\beta =1$ , as well as by the VLS $\\beta $ value, with the shaded region representing the total uncertainty arising from PDF uncertainties and the factorization and renormalization scale choice.",
"From the intersection of the observed cross section limit and the theoretical prediction, $b^{\\prime }$ quarks with masses $m_{b^{\\prime }}<400$ GeV decaying entirely via $b^{\\prime } \\rightarrow Z+b$ are excluded at 95% C.L., representing a significant improvement with respect to the previous best limit of 268 GeV [9].",
"In the case of a vector-like singlet $b^{\\prime }$ mixing solely with the third SM generation, masses $m_{b^{\\prime }}<358$ GeV are excluded.",
"In conclusion, a search with 2.0 fb$^{-1}$ of ATLAS data is presented for $b^{\\prime }$ quark pair production, with at least one $b^{\\prime }$ decaying to a $Z$ boson and a bottom quark.",
"This decay mode is particularly relevant in the context of vector-like quarks and is an essential complement to searches in the mode with both $b^{\\prime }$ decaying to a $W$ boson and a top quark.",
"No evidence for a $b^{\\prime }$ is observed in the $Z+b$ -jet final state, and new limits are derived on the mass of a $b^{\\prime }$ quark decaying via $b^{\\prime } \\rightarrow Z+b$ .",
"We thank CERN for the very successful operation of the LHC, as well as the support staff from our institutions without whom ATLAS could not be operated efficiently.",
"We acknowledge the support of ANPCyT, Argentina; YerPhI, Armenia; ARC, Australia; BMWF, Austria; ANAS, Azerbaijan; SSTC, Belarus; CNPq and FAPESP, Brazil; NSERC, NRC and CFI, Canada; CERN; CONICYT, Chile; CAS, MOST and NSFC, China; COLCIENCIAS, Colombia; MSMT CR, MPO CR and VSC CR, Czech Republic; DNRF, DNSRC and Lundbeck Foundation, Denmark; EPLANET and ERC, European Union; IN2P3-CNRS, CEA-DSM/IRFU, France; GNAS, Georgia; BMBF, DFG, HGF, MPG and AvH Foundation, Germany; GSRT, Greece; ISF, MINERVA, GIF, DIP and Benoziyo Center, Israel; INFN, Italy; MEXT and JSPS, Japan; CNRST, Morocco; FOM and NWO, Netherlands; RCN, Norway; MNiSW, Poland; GRICES and FCT, Portugal; MERYS (MECTS), Romania; MES of Russia and ROSATOM, Russian Federation; JINR; MSTD, Serbia; MSSR, Slovakia; ARRS and MVZT, Slovenia; DST/NRF, South Africa; MICINN, Spain; SRC and Wallenberg Foundation, Sweden; SER, SNSF and Cantons of Bern and Geneva, Switzerland; NSC, Taiwan; TAEK, Turkey; STFC, the Royal Society and Leverhulme Trust, United Kingdom; DOE and NSF, United States of America.",
"The crucial computing support from all WLCG partners is acknowledged gratefully, in particular from CERN and the ATLAS Tier-1 facilities at TRIUMF (Canada), NDGF (Denmark, Norway, Sweden), CC-IN2P3 (France), KIT/GridKA (Germany), INFN-CNAF (Italy), NL-T1 (Netherlands), PIC (Spain), ASGC (Taiwan), RAL (UK) and BNL (USA) and in the Tier-2 facilities worldwide.",
"The ATLAS Collaboration G. Aad$^{\\rm 48}$ , B. Abbott$^{\\rm 112}$ , J. Abdallah$^{\\rm 11}$ , S. Abdel Khalek$^{\\rm 116}$ , A.A. Abdelalim$^{\\rm 49}$ , A. Abdesselam$^{\\rm 119}$ , O. Abdinov$^{\\rm 10}$ , B. Abi$^{\\rm 113}$ , M. Abolins$^{\\rm 89}$ , O.S.",
"AbouZeid$^{\\rm 159}$ , H. Abramowicz$^{\\rm 154}$ , H. Abreu$^{\\rm 137}$ , E. Acerbi$^{\\rm 90a,90b}$ , B.S.",
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"Castro$^{\\rm 125a}$ , G. Cataldi$^{\\rm 73a}$ , P. Catastini$^{\\rm 57}$ , A. Catinaccio$^{\\rm 29}$ , J.R. Catmore$^{\\rm 29}$ , A. Cattai$^{\\rm 29}$ , G. Cattani$^{\\rm 134a,134b}$ , S. Caughron$^{\\rm 89}$ , D. Cauz$^{\\rm 165a,165c}$ , P. Cavalleri$^{\\rm 79}$ , D. Cavalli$^{\\rm 90a}$ , M. Cavalli-Sforza$^{\\rm 11}$ , V. Cavasinni$^{\\rm 123a,123b}$ , F. Ceradini$^{\\rm 135a,135b}$ , A.S. Cerqueira$^{\\rm 23b}$ , A. Cerri$^{\\rm 29}$ , L. Cerrito$^{\\rm 76}$ , F. Cerutti$^{\\rm 47}$ , S.A. Cetin$^{\\rm 18b}$ , F. Cevenini$^{\\rm 103a,103b}$ , A. Chafaq$^{\\rm 136a}$ , D. Chakraborty$^{\\rm 107}$ , I. Chalupkova$^{\\rm 127}$ , K. Chan$^{\\rm 2}$ , B. Chapleau$^{\\rm 86}$ , J.D.",
"Chapman$^{\\rm 27}$ , J.W.",
"Chapman$^{\\rm 88}$ , E. Chareyre$^{\\rm 79}$ , D.G.",
"Charlton$^{\\rm 17}$ , V. Chavda$^{\\rm 83}$ , C.A.",
"Chavez Barajas$^{\\rm 29}$ , S. Cheatham$^{\\rm 86}$ , S. Chekanov$^{\\rm 5}$ , S.V.",
"Chekulaev$^{\\rm 160a}$ , G.A.",
"Chelkov$^{\\rm 65}$ , M.A.",
"Chelstowska$^{\\rm 105}$ , C. Chen$^{\\rm 64}$ , H. Chen$^{\\rm 24}$ , S. Chen$^{\\rm 32c}$ , T. Chen$^{\\rm 32c}$ , X. Chen$^{\\rm 174}$ , S. Cheng$^{\\rm 32a}$ , A. Cheplakov$^{\\rm 65}$ , V.F.",
"Chepurnov$^{\\rm 65}$ , R. Cherkaoui El Moursli$^{\\rm 136e}$ , V. Chernyatin$^{\\rm 24}$ , E. Cheu$^{\\rm 6}$ , S.L.",
"Cheung$^{\\rm 159}$ , L. Chevalier$^{\\rm 137}$ , G. Chiefari$^{\\rm 103a,103b}$ , L. Chikovani$^{\\rm 51a}$ , J.T.",
"Childers$^{\\rm 29}$ , A. Chilingarov$^{\\rm 72}$ , G. Chiodini$^{\\rm 73a}$ , A.S. Chisholm$^{\\rm 17}$ , R.T. Chislett$^{\\rm 78}$ , M.V.",
"Chizhov$^{\\rm 65}$ , G. Choudalakis$^{\\rm 30}$ , S. Chouridou$^{\\rm 138}$ , I.A.",
"Christidi$^{\\rm 78}$ , A. Christov$^{\\rm 48}$ , D. Chromek-Burckhart$^{\\rm 29}$ , M.L.",
"Chu$^{\\rm 152}$ , J. Chudoba$^{\\rm 126}$ , G. Ciapetti$^{\\rm 133a,133b}$ , A.K.",
"Ciftci$^{\\rm 3a}$ , R. Ciftci$^{\\rm 3a}$ , D. Cinca$^{\\rm 33}$ , V. Cindro$^{\\rm 75}$ , C. Ciocca$^{\\rm 19a}$ , A. Ciocio$^{\\rm 14}$ , M. Cirilli$^{\\rm 88}$ , M. Citterio$^{\\rm 90a}$ , M. Ciubancan$^{\\rm 25a}$ , A. Clark$^{\\rm 49}$ , P.J.",
"Clark$^{\\rm 45}$ , W. Cleland$^{\\rm 124}$ , J.C. Clemens$^{\\rm 84}$ , B. Clement$^{\\rm 55}$ , C. Clement$^{\\rm 147a,147b}$ , Y. Coadou$^{\\rm 84}$ , M. Cobal$^{\\rm 165a,165c}$ , A. Coccaro$^{\\rm 139}$ , J. Cochran$^{\\rm 64}$ , P. Coe$^{\\rm 119}$ , J.G.",
"Cogan$^{\\rm 144}$ , J. Coggeshall$^{\\rm 166}$ , E. Cogneras$^{\\rm 179}$ , J. Colas$^{\\rm 4}$ , A.P.",
"Colijn$^{\\rm 106}$ , N.J. Collins$^{\\rm 17}$ , C. Collins-Tooth$^{\\rm 53}$ , J. Collot$^{\\rm 55}$ , G. Colon$^{\\rm 85}$ , P. Conde Muiño$^{\\rm 125a}$ , E. Coniavitis$^{\\rm 119}$ , M.C.",
"Conidi$^{\\rm 11}$ , M. Consonni$^{\\rm 105}$ , S.M.",
"Consonni$^{\\rm 90a,90b}$ , V. Consorti$^{\\rm 48}$ , S. Constantinescu$^{\\rm 25a}$ , C. Conta$^{\\rm 120a,120b}$ , G. Conti$^{\\rm 57}$ , F. Conventi$^{\\rm 103a}$$^{,i}$ , J. Cook$^{\\rm 29}$ , M. Cooke$^{\\rm 14}$ , B.D.",
"Cooper$^{\\rm 78}$ , A.M. Cooper-Sarkar$^{\\rm 119}$ , K. Copic$^{\\rm 14}$ , T. Cornelissen$^{\\rm 176}$ , M. Corradi$^{\\rm 19a}$ , F. Corriveau$^{\\rm 86}$$^{,j}$ , A. Cortes-Gonzalez$^{\\rm 166}$ , G. Cortiana$^{\\rm 100}$ , G. Costa$^{\\rm 90a}$ , M.J. Costa$^{\\rm 168}$ , D. Costanzo$^{\\rm 140}$ , T. Costin$^{\\rm 30}$ , D. Côté$^{\\rm 29}$ , L. Courneyea$^{\\rm 170}$ , G. Cowan$^{\\rm 77}$ , C. Cowden$^{\\rm 27}$ , B.E.",
"Cox$^{\\rm 83}$ , K. Cranmer$^{\\rm 109}$ , F. Crescioli$^{\\rm 123a,123b}$ , M. Cristinziani$^{\\rm 20}$ , G. Crosetti$^{\\rm 36a,36b}$ , R. Crupi$^{\\rm 73a,73b}$ , S. Crépé-Renaudin$^{\\rm 55}$ , C.-M. Cuciuc$^{\\rm 25a}$ , C. Cuenca Almenar$^{\\rm 177}$ , T. Cuhadar Donszelmann$^{\\rm 140}$ , M. Curatolo$^{\\rm 47}$ , C.J.",
"Curtis$^{\\rm 17}$ , C. Cuthbert$^{\\rm 151}$ , P. Cwetanski$^{\\rm 61}$ , H. Czirr$^{\\rm 142}$ , P. Czodrowski$^{\\rm 43}$ , Z. Czyczula$^{\\rm 177}$ , S. D'Auria$^{\\rm 53}$ , M. D'Onofrio$^{\\rm 74}$ , A.",
"D'Orazio$^{\\rm 133a,133b}$ , P.V.M.",
"Da Silva$^{\\rm 23a}$ , C. Da Via$^{\\rm 83}$ , W. Dabrowski$^{\\rm 37}$ , A. Dafinca$^{\\rm 119}$ , T. Dai$^{\\rm 88}$ , C. Dallapiccola$^{\\rm 85}$ , M. Dam$^{\\rm 35}$ , M. Dameri$^{\\rm 50a,50b}$ , D.S.",
"Damiani$^{\\rm 138}$ , H.O.",
"Danielsson$^{\\rm 29}$ , D. Dannheim$^{\\rm 100}$ , V. Dao$^{\\rm 49}$ , G. Darbo$^{\\rm 50a}$ , G.L.",
"Darlea$^{\\rm 25b}$ , W. Davey$^{\\rm 20}$ , T. Davidek$^{\\rm 127}$ , N. Davidson$^{\\rm 87}$ , R. Davidson$^{\\rm 72}$ , E. Davies$^{\\rm 119}$$^{,c}$ , M. Davies$^{\\rm 94}$ , A.R.",
"Davison$^{\\rm 78}$ , Y. Davygora$^{\\rm 58a}$ , E. Dawe$^{\\rm 143}$ , I. Dawson$^{\\rm 140}$ , J.W.",
"Dawson$^{\\rm 5}$$^{,*}$ , R.K. Daya-Ishmukhametova$^{\\rm 22}$ , K. De$^{\\rm 7}$ , R. de Asmundis$^{\\rm 103a}$ , S. De Castro$^{\\rm 19a,19b}$ , P.E.",
"De Castro Faria Salgado$^{\\rm 24}$ , S. De Cecco$^{\\rm 79}$ , J. de Graat$^{\\rm 99}$ , N. De Groot$^{\\rm 105}$ , P. de Jong$^{\\rm 106}$ , C. De La Taille$^{\\rm 116}$ , H. De la Torre$^{\\rm 81}$ , F. De Lorenzi$^{\\rm 64}$ , B.",
"De Lotto$^{\\rm 165a,165c}$ , L. de Mora$^{\\rm 72}$ , L. De Nooij$^{\\rm 106}$ , D. De Pedis$^{\\rm 133a}$ , A.",
"De Salvo$^{\\rm 133a}$ , U.",
"De Sanctis$^{\\rm 165a,165c}$ , A.",
"De Santo$^{\\rm 150}$ , J.B. De Vivie De Regie$^{\\rm 116}$ , G. De Zorzi$^{\\rm 133a,133b}$ , S. Dean$^{\\rm 78}$ , W.J.",
"Dearnaley$^{\\rm 72}$ , R. Debbe$^{\\rm 24}$ , C. Debenedetti$^{\\rm 45}$ , B. Dechenaux$^{\\rm 55}$ , D.V.",
"Dedovich$^{\\rm 65}$ , J. Degenhardt$^{\\rm 121}$ , C. Del Papa$^{\\rm 165a,165c}$ , J. Del Peso$^{\\rm 81}$ , T. Del Prete$^{\\rm 123a,123b}$ , T. Delemontex$^{\\rm 55}$ , M. Deliyergiyev$^{\\rm 75}$ , A. Dell'Acqua$^{\\rm 29}$ , L. Dell'Asta$^{\\rm 21}$ , M. Della Pietra$^{\\rm 103a}$$^{,i}$ , D. della Volpe$^{\\rm 103a,103b}$ , M. Delmastro$^{\\rm 4}$ , N. Delruelle$^{\\rm 29}$ , P.A.",
"Delsart$^{\\rm 55}$ , C. Deluca$^{\\rm 149}$ , S. Demers$^{\\rm 177}$ , M. Demichev$^{\\rm 65}$ , B. Demirkoz$^{\\rm 11}$$^{,k}$ , J. Deng$^{\\rm 164}$ , S.P.",
"Denisov$^{\\rm 129}$ , D. Derendarz$^{\\rm 38}$ , J.E.",
"Derkaoui$^{\\rm 136d}$ , F. Derue$^{\\rm 79}$ , P. Dervan$^{\\rm 74}$ , K. Desch$^{\\rm 20}$ , E. Devetak$^{\\rm 149}$ , P.O.",
"Deviveiros$^{\\rm 106}$ , A. Dewhurst$^{\\rm 130}$ , B. DeWilde$^{\\rm 149}$ , S. Dhaliwal$^{\\rm 159}$ , R. Dhullipudi$^{\\rm 24}$$^{,l}$ , A.",
"Di Ciaccio$^{\\rm 134a,134b}$ , L. Di Ciaccio$^{\\rm 4}$ , A.",
"Di Girolamo$^{\\rm 29}$ , B.",
"Di Girolamo$^{\\rm 29}$ , S. Di Luise$^{\\rm 135a,135b}$ , A.",
"Di Mattia$^{\\rm 174}$ , B.",
"Di Micco$^{\\rm 29}$ , R. Di Nardo$^{\\rm 47}$ , A.",
"Di Simone$^{\\rm 134a,134b}$ , R. Di Sipio$^{\\rm 19a,19b}$ , M.A.",
"Diaz$^{\\rm 31a}$ , F. Diblen$^{\\rm 18c}$ , E.B.",
"Diehl$^{\\rm 88}$ , J. Dietrich$^{\\rm 41}$ , T.A.",
"Dietzsch$^{\\rm 58a}$ , S. Diglio$^{\\rm 87}$ , K. Dindar Yagci$^{\\rm 39}$ , J. Dingfelder$^{\\rm 20}$ , C. Dionisi$^{\\rm 133a,133b}$ , P. Dita$^{\\rm 25a}$ , S. Dita$^{\\rm 25a}$ , F. Dittus$^{\\rm 29}$ , F. Djama$^{\\rm 84}$ , T. Djobava$^{\\rm 51b}$ , M.A.B.",
"do Vale$^{\\rm 23c}$ , A.",
"Do Valle Wemans$^{\\rm 125a}$ , T.K.O.",
"Doan$^{\\rm 4}$ , M. Dobbs$^{\\rm 86}$ , R. Dobinson $^{\\rm 29}$$^{,*}$ , D. Dobos$^{\\rm 29}$ , E. Dobson$^{\\rm 29}$$^{,m}$ , J. Dodd$^{\\rm 34}$ , C. Doglioni$^{\\rm 49}$ , T. Doherty$^{\\rm 53}$ , Y. Doi$^{\\rm 66}$$^{,*}$ , J. Dolejsi$^{\\rm 127}$ , I. Dolenc$^{\\rm 75}$ , Z. Dolezal$^{\\rm 127}$ , B.A.",
"Dolgoshein$^{\\rm 97}$$^{,*}$ , T. Dohmae$^{\\rm 156}$ , M. Donadelli$^{\\rm 23d}$ , M. Donega$^{\\rm 121}$ , J. Donini$^{\\rm 33}$ , J. Dopke$^{\\rm 29}$ , A. Doria$^{\\rm 103a}$ , A. Dos Anjos$^{\\rm 174}$ , M. Dosil$^{\\rm 11}$ , A. Dotti$^{\\rm 123a,123b}$ , M.T.",
"Dova$^{\\rm 71}$ , A.D. Doxiadis$^{\\rm 106}$ , A.T. Doyle$^{\\rm 53}$ , Z. Drasal$^{\\rm 127}$ , N. Dressnandt$^{\\rm 121}$ , C. Driouichi$^{\\rm 35}$ , M. Dris$^{\\rm 9}$ , J. Dubbert$^{\\rm 100}$ , S. Dube$^{\\rm 14}$ , E. Duchovni$^{\\rm 173}$ , G. Duckeck$^{\\rm 99}$ , A. Dudarev$^{\\rm 29}$ , F. Dudziak$^{\\rm 64}$ , M. Dührssen $^{\\rm 29}$ , I.P.",
"Duerdoth$^{\\rm 83}$ , L. Duflot$^{\\rm 116}$ , M-A.",
"Dufour$^{\\rm 86}$ , M. Dunford$^{\\rm 29}$ , H. Duran Yildiz$^{\\rm 3a}$ , R. Duxfield$^{\\rm 140}$ , M. Dwuznik$^{\\rm 37}$ , F. Dydak $^{\\rm 29}$ , M. Düren$^{\\rm 52}$ , W.L.",
"Ebenstein$^{\\rm 44}$ , J. Ebke$^{\\rm 99}$ , S. Eckweiler$^{\\rm 82}$ , K. Edmonds$^{\\rm 82}$ , C.A.",
"Edwards$^{\\rm 77}$ , N.C. Edwards$^{\\rm 53}$ , W. Ehrenfeld$^{\\rm 41}$ , T. Ehrich$^{\\rm 100}$ , T. Eifert$^{\\rm 144}$ , G. Eigen$^{\\rm 13}$ , K. Einsweiler$^{\\rm 14}$ , E. Eisenhandler$^{\\rm 76}$ , T. Ekelof$^{\\rm 167}$ , M. El Kacimi$^{\\rm 136c}$ , M. Ellert$^{\\rm 167}$ , S. Elles$^{\\rm 4}$ , F. Ellinghaus$^{\\rm 82}$ , K. Ellis$^{\\rm 76}$ , N. Ellis$^{\\rm 29}$ , J. Elmsheuser$^{\\rm 99}$ , M. Elsing$^{\\rm 29}$ , D. Emeliyanov$^{\\rm 130}$ , R. Engelmann$^{\\rm 149}$ , A. Engl$^{\\rm 99}$ , B. Epp$^{\\rm 62}$ , A. Eppig$^{\\rm 88}$ , J. Erdmann$^{\\rm 54}$ , A. Ereditato$^{\\rm 16}$ , D. Eriksson$^{\\rm 147a}$ , J. Ernst$^{\\rm 1}$ , M. Ernst$^{\\rm 24}$ , J. Ernwein$^{\\rm 137}$ , D. Errede$^{\\rm 166}$ , S. Errede$^{\\rm 166}$ , E. Ertel$^{\\rm 82}$ , M. Escalier$^{\\rm 116}$ , C. Escobar$^{\\rm 124}$ , X. Espinal Curull$^{\\rm 11}$ , B. Esposito$^{\\rm 47}$ , F. Etienne$^{\\rm 84}$ , A.I.",
"Etienvre$^{\\rm 137}$ , E. Etzion$^{\\rm 154}$ , D. Evangelakou$^{\\rm 54}$ , H. Evans$^{\\rm 61}$ , L. Fabbri$^{\\rm 19a,19b}$ , C. Fabre$^{\\rm 29}$ , R.M.",
"Fakhrutdinov$^{\\rm 129}$ , S. Falciano$^{\\rm 133a}$ , Y. Fang$^{\\rm 174}$ , M. Fanti$^{\\rm 90a,90b}$ , A. Farbin$^{\\rm 7}$ , A. Farilla$^{\\rm 135a}$ , J. Farley$^{\\rm 149}$ , T. Farooque$^{\\rm 159}$ , S. Farrell$^{\\rm 164}$ , S.M.",
"Farrington$^{\\rm 119}$ , P. Farthouat$^{\\rm 29}$ , P. Fassnacht$^{\\rm 29}$ , D. Fassouliotis$^{\\rm 8}$ , B. Fatholahzadeh$^{\\rm 159}$ , A. Favareto$^{\\rm 90a,90b}$ , L. Fayard$^{\\rm 116}$ , S. Fazio$^{\\rm 36a,36b}$ , R. Febbraro$^{\\rm 33}$ , P. Federic$^{\\rm 145a}$ , O.L.",
"Fedin$^{\\rm 122}$ , W. Fedorko$^{\\rm 89}$ , M. Fehling-Kaschek$^{\\rm 48}$ , L. Feligioni$^{\\rm 84}$ , D. Fellmann$^{\\rm 5}$ , C. Feng$^{\\rm 32d}$ , E.J.",
"Feng$^{\\rm 30}$ , A.B.",
"Fenyuk$^{\\rm 129}$ , J. Ferencei$^{\\rm 145b}$ , J. Ferland$^{\\rm 94}$ , W. Fernando$^{\\rm 5}$ , S. Ferrag$^{\\rm 53}$ , J. Ferrando$^{\\rm 53}$ , V. Ferrara$^{\\rm 41}$ , A. Ferrari$^{\\rm 167}$ , P. Ferrari$^{\\rm 106}$ , R. Ferrari$^{\\rm 120a}$ , D.E.",
"Ferreira de Lima$^{\\rm 53}$ , A. Ferrer$^{\\rm 168}$ , M.L.",
"Ferrer$^{\\rm 47}$ , D. Ferrere$^{\\rm 49}$ , C. Ferretti$^{\\rm 88}$ , A. Ferretto Parodi$^{\\rm 50a,50b}$ , M. Fiascaris$^{\\rm 30}$ , F. Fiedler$^{\\rm 82}$ , A. Filipčič$^{\\rm 75}$ , A. Filippas$^{\\rm 9}$ , F. Filthaut$^{\\rm 105}$ , M. Fincke-Keeler$^{\\rm 170}$ , M.C.N.",
"Fiolhais$^{\\rm 125a}$$^{,h}$ , L. Fiorini$^{\\rm 168}$ , A. Firan$^{\\rm 39}$ , G. Fischer$^{\\rm 41}$ , M.J. Fisher$^{\\rm 110}$ , M. Flechl$^{\\rm 48}$ , I. Fleck$^{\\rm 142}$ , J. Fleckner$^{\\rm 82}$ , P. Fleischmann$^{\\rm 175}$ , S. Fleischmann$^{\\rm 176}$ , T. Flick$^{\\rm 176}$ , A. Floderus$^{\\rm 80}$ , L.R.",
"Flores Castillo$^{\\rm 174}$ , M.J. Flowerdew$^{\\rm 100}$ , M. Fokitis$^{\\rm 9}$ , T. Fonseca Martin$^{\\rm 16}$ , D.A.",
"Forbush$^{\\rm 139}$ , A. Formica$^{\\rm 137}$ , A. Forti$^{\\rm 83}$ , D. Fortin$^{\\rm 160a}$ , J.M.",
"Foster$^{\\rm 83}$ , D. Fournier$^{\\rm 116}$ , A. Foussat$^{\\rm 29}$ , A.J.",
"Fowler$^{\\rm 44}$ , K. Fowler$^{\\rm 138}$ , H. Fox$^{\\rm 72}$ , P. Francavilla$^{\\rm 11}$ , S. Franchino$^{\\rm 120a,120b}$ , D. Francis$^{\\rm 29}$ , T. Frank$^{\\rm 173}$ , M. Franklin$^{\\rm 57}$ , S. Franz$^{\\rm 29}$ , M. Fraternali$^{\\rm 120a,120b}$ , S. Fratina$^{\\rm 121}$ , S.T.",
"French$^{\\rm 27}$ , C. Friedrich$^{\\rm 41}$ , F. Friedrich $^{\\rm 43}$ , R. Froeschl$^{\\rm 29}$ , D. Froidevaux$^{\\rm 29}$ , J.A.",
"Frost$^{\\rm 27}$ , C. Fukunaga$^{\\rm 157}$ , E. Fullana Torregrosa$^{\\rm 29}$ , B.G.",
"Fulsom$^{\\rm 144}$ , J. Fuster$^{\\rm 168}$ , C. Gabaldon$^{\\rm 29}$ , O. Gabizon$^{\\rm 173}$ , T. Gadfort$^{\\rm 24}$ , S. Gadomski$^{\\rm 49}$ , G. Gagliardi$^{\\rm 50a,50b}$ , P. Gagnon$^{\\rm 61}$ , C. Galea$^{\\rm 99}$ , E.J.",
"Gallas$^{\\rm 119}$ , V. Gallo$^{\\rm 16}$ , B.J.",
"Gallop$^{\\rm 130}$ , P. Gallus$^{\\rm 126}$ , K.K.",
"Gan$^{\\rm 110}$ , Y.S.",
"Gao$^{\\rm 144}$$^{,e}$ , V.A.",
"Gapienko$^{\\rm 129}$ , A. Gaponenko$^{\\rm 14}$ , F. Garberson$^{\\rm 177}$ , M. Garcia-Sciveres$^{\\rm 14}$ , C. García$^{\\rm 168}$ , J.E.",
"García Navarro$^{\\rm 168}$ , R.W.",
"Gardner$^{\\rm 30}$ , N. Garelli$^{\\rm 29}$ , H. Garitaonandia$^{\\rm 106}$ , V. Garonne$^{\\rm 29}$ , J. Garvey$^{\\rm 17}$ , C. Gatti$^{\\rm 47}$ , G. Gaudio$^{\\rm 120a}$ , B. Gaur$^{\\rm 142}$ , L. Gauthier$^{\\rm 137}$ , P. Gauzzi$^{\\rm 133a,133b}$ , I.L.",
"Gavrilenko$^{\\rm 95}$ , C. Gay$^{\\rm 169}$ , G. Gaycken$^{\\rm 20}$ , J-C. Gayde$^{\\rm 29}$ , E.N.",
"Gazis$^{\\rm 9}$ , P. Ge$^{\\rm 32d}$ , Z. Gecse$^{\\rm 169}$ , C.N.P.",
"Gee$^{\\rm 130}$ , D.A.A.",
"Geerts$^{\\rm 106}$ , Ch.",
"Geich-Gimbel$^{\\rm 20}$ , K. Gellerstedt$^{\\rm 147a,147b}$ , C. Gemme$^{\\rm 50a}$ , A. Gemmell$^{\\rm 53}$ , M.H.",
"Genest$^{\\rm 55}$ , S. Gentile$^{\\rm 133a,133b}$ , M. George$^{\\rm 54}$ , S. George$^{\\rm 77}$ , P. Gerlach$^{\\rm 176}$ , A. Gershon$^{\\rm 154}$ , C. Geweniger$^{\\rm 58a}$ , H. Ghazlane$^{\\rm 136b}$ , N. Ghodbane$^{\\rm 33}$ , B. Giacobbe$^{\\rm 19a}$ , S. Giagu$^{\\rm 133a,133b}$ , V. Giakoumopoulou$^{\\rm 8}$ , V. Giangiobbe$^{\\rm 11}$ , F. Gianotti$^{\\rm 29}$ , B. Gibbard$^{\\rm 24}$ , A. Gibson$^{\\rm 159}$ , S.M.",
"Gibson$^{\\rm 29}$ , L.M.",
"Gilbert$^{\\rm 119}$ , V. Gilewsky$^{\\rm 92}$ , D. Gillberg$^{\\rm 28}$ , A.R.",
"Gillman$^{\\rm 130}$ , D.M.",
"Gingrich$^{\\rm 2}$$^{,d}$ , J. Ginzburg$^{\\rm 154}$ , N. Giokaris$^{\\rm 8}$ , M.P.",
"Giordani$^{\\rm 165c}$ , R. Giordano$^{\\rm 103a,103b}$ , F.M.",
"Giorgi$^{\\rm 15}$ , P. Giovannini$^{\\rm 100}$ , P.F.",
"Giraud$^{\\rm 137}$ , D. Giugni$^{\\rm 90a}$ , M. Giunta$^{\\rm 94}$ , P. Giusti$^{\\rm 19a}$ , B.K.",
"Gjelsten$^{\\rm 118}$ , L.K.",
"Gladilin$^{\\rm 98}$ , C. Glasman$^{\\rm 81}$ , J. Glatzer$^{\\rm 48}$ , A. Glazov$^{\\rm 41}$ , K.W.",
"Glitza$^{\\rm 176}$ , G.L.",
"Glonti$^{\\rm 65}$ , J.R. Goddard$^{\\rm 76}$ , J. Godfrey$^{\\rm 143}$ , J. Godlewski$^{\\rm 29}$ , M. Goebel$^{\\rm 41}$ , T. Göpfert$^{\\rm 43}$ , C. Goeringer$^{\\rm 82}$ , C. Gössling$^{\\rm 42}$ , T. Göttfert$^{\\rm 100}$ , S. Goldfarb$^{\\rm 88}$ , T. Golling$^{\\rm 177}$ , A. Gomes$^{\\rm 125a}$$^{,b}$ , L.S.",
"Gomez Fajardo$^{\\rm 41}$ , R. Gonçalo$^{\\rm 77}$ , J. Goncalves Pinto Firmino Da Costa$^{\\rm 41}$ , L. Gonella$^{\\rm 20}$ , A. Gonidec$^{\\rm 29}$ , S. Gonzalez$^{\\rm 174}$ , S. González de la Hoz$^{\\rm 168}$ , G. Gonzalez Parra$^{\\rm 11}$ , M.L.",
"Gonzalez Silva$^{\\rm 26}$ , S. Gonzalez-Sevilla$^{\\rm 49}$ , J.J. Goodson$^{\\rm 149}$ , L. Goossens$^{\\rm 29}$ , P.A.",
"Gorbounov$^{\\rm 96}$ , H.A.",
"Gordon$^{\\rm 24}$ , I. Gorelov$^{\\rm 104}$ , G. Gorfine$^{\\rm 176}$ , B. Gorini$^{\\rm 29}$ , E. Gorini$^{\\rm 73a,73b}$ , A. Gorišek$^{\\rm 75}$ , E. Gornicki$^{\\rm 38}$ , V.N.",
"Goryachev$^{\\rm 129}$ , B. Gosdzik$^{\\rm 41}$ , A.T. Goshaw$^{\\rm 5}$ , M. Gosselink$^{\\rm 106}$ , M.I.",
"Gostkin$^{\\rm 65}$ , I. Gough Eschrich$^{\\rm 164}$ , M. Gouighri$^{\\rm 136a}$ , D. Goujdami$^{\\rm 136c}$ , M.P.",
"Goulette$^{\\rm 49}$ , A.G. Goussiou$^{\\rm 139}$ , C. Goy$^{\\rm 4}$ , S. Gozpinar$^{\\rm 22}$ , I. Grabowska-Bold$^{\\rm 37}$ , P. Grafström$^{\\rm 29}$ , K-J.",
"Grahn$^{\\rm 41}$ , F. Grancagnolo$^{\\rm 73a}$ , S. Grancagnolo$^{\\rm 15}$ , V. Grassi$^{\\rm 149}$ , V. Gratchev$^{\\rm 122}$ , N. Grau$^{\\rm 34}$ , H.M. Gray$^{\\rm 29}$ , J.A.",
"Gray$^{\\rm 149}$ , E. Graziani$^{\\rm 135a}$ , O.G.",
"Grebenyuk$^{\\rm 122}$ , T. Greenshaw$^{\\rm 74}$ , Z.D.",
"Greenwood$^{\\rm 24}$$^{,l}$ , K. Gregersen$^{\\rm 35}$ , I.M.",
"Gregor$^{\\rm 41}$ , P. Grenier$^{\\rm 144}$ , J. Griffiths$^{\\rm 139}$ , N. Grigalashvili$^{\\rm 65}$ , A.A. Grillo$^{\\rm 138}$ , S. Grinstein$^{\\rm 11}$ , Y.V.",
"Grishkevich$^{\\rm 98}$ , J.-F. Grivaz$^{\\rm 116}$ , E. Gross$^{\\rm 173}$ , J. Grosse-Knetter$^{\\rm 54}$ , J. Groth-Jensen$^{\\rm 173}$ , K. Grybel$^{\\rm 142}$ , V.J.",
"Guarino$^{\\rm 5}$ , D. Guest$^{\\rm 177}$ , C. Guicheney$^{\\rm 33}$ , A. Guida$^{\\rm 73a,73b}$ , S. Guindon$^{\\rm 54}$ , H. Guler$^{\\rm 86}$$^{,n}$ , J. Gunther$^{\\rm 126}$ , B. Guo$^{\\rm 159}$ , J. Guo$^{\\rm 34}$ , A. Gupta$^{\\rm 30}$ , Y. Gusakov$^{\\rm 65}$ , V.N.",
"Gushchin$^{\\rm 129}$ , P. Gutierrez$^{\\rm 112}$ , N. Guttman$^{\\rm 154}$ , O. Gutzwiller$^{\\rm 174}$ , C. Guyot$^{\\rm 137}$ , C. Gwenlan$^{\\rm 119}$ , C.B.",
"Gwilliam$^{\\rm 74}$ , A. Haas$^{\\rm 144}$ , S. Haas$^{\\rm 29}$ , C. Haber$^{\\rm 14}$ , H.K.",
"Hadavand$^{\\rm 39}$ , D.R.",
"Hadley$^{\\rm 17}$ , P. Haefner$^{\\rm 100}$ , F. Hahn$^{\\rm 29}$ , S. Haider$^{\\rm 29}$ , Z. Hajduk$^{\\rm 38}$ , H. Hakobyan$^{\\rm 178}$ , D. Hall$^{\\rm 119}$ , J. Haller$^{\\rm 54}$ , K. Hamacher$^{\\rm 176}$ , P. Hamal$^{\\rm 114}$ , M. Hamer$^{\\rm 54}$ , A. Hamilton$^{\\rm 146b}$$^{,o}$ , S. Hamilton$^{\\rm 162}$ , H. Han$^{\\rm 32a}$ , L. Han$^{\\rm 32b}$ , K. Hanagaki$^{\\rm 117}$ , K. Hanawa$^{\\rm 161}$ , M. Hance$^{\\rm 14}$ , C. Handel$^{\\rm 82}$ , P. Hanke$^{\\rm 58a}$ , J.R. Hansen$^{\\rm 35}$ , J.B. Hansen$^{\\rm 35}$ , J.D.",
"Hansen$^{\\rm 35}$ , P.H.",
"Hansen$^{\\rm 35}$ , P. Hansson$^{\\rm 144}$ , K. Hara$^{\\rm 161}$ , G.A.",
"Hare$^{\\rm 138}$ , T. Harenberg$^{\\rm 176}$ , S. Harkusha$^{\\rm 91}$ , D. Harper$^{\\rm 88}$ , R.D.",
"Harrington$^{\\rm 45}$ , O.M.",
"Harris$^{\\rm 139}$ , K. Harrison$^{\\rm 17}$ , J. Hartert$^{\\rm 48}$ , F. Hartjes$^{\\rm 106}$ , T. Haruyama$^{\\rm 66}$ , A. Harvey$^{\\rm 56}$ , S. Hasegawa$^{\\rm 102}$ , Y. Hasegawa$^{\\rm 141}$ , S. Hassani$^{\\rm 137}$ , M. Hatch$^{\\rm 29}$ , D. Hauff$^{\\rm 100}$ , S. Haug$^{\\rm 16}$ , M. Hauschild$^{\\rm 29}$ , R. Hauser$^{\\rm 89}$ , M. Havranek$^{\\rm 20}$ , C.M.",
"Hawkes$^{\\rm 17}$ , R.J. Hawkings$^{\\rm 29}$ , A.D. Hawkins$^{\\rm 80}$ , D. Hawkins$^{\\rm 164}$ , T. Hayakawa$^{\\rm 67}$ , T. Hayashi$^{\\rm 161}$ , D. Hayden$^{\\rm 77}$ , H.S.",
"Hayward$^{\\rm 74}$ , S.J.",
"Haywood$^{\\rm 130}$ , E. Hazen$^{\\rm 21}$ , M. He$^{\\rm 32d}$ , S.J.",
"Head$^{\\rm 17}$ , V. Hedberg$^{\\rm 80}$ , L. Heelan$^{\\rm 7}$ , S. Heim$^{\\rm 89}$ , B. Heinemann$^{\\rm 14}$ , S. Heisterkamp$^{\\rm 35}$ , L. Helary$^{\\rm 4}$ , C. Heller$^{\\rm 99}$ , M. Heller$^{\\rm 29}$ , S. Hellman$^{\\rm 147a,147b}$ , D. Hellmich$^{\\rm 20}$ , C. Helsens$^{\\rm 11}$ , R.C.W.",
"Henderson$^{\\rm 72}$ , M. Henke$^{\\rm 58a}$ , A. Henrichs$^{\\rm 54}$ , A.M. Henriques Correia$^{\\rm 29}$ , S. Henrot-Versille$^{\\rm 116}$ , F. Henry-Couannier$^{\\rm 84}$ , C. Hensel$^{\\rm 54}$ , T. Henß$^{\\rm 176}$ , C.M.",
"Hernandez$^{\\rm 7}$ , Y. Hernández Jiménez$^{\\rm 168}$ , R. Herrberg$^{\\rm 15}$ , G. Herten$^{\\rm 48}$ , R. Hertenberger$^{\\rm 99}$ , L. Hervas$^{\\rm 29}$ , G.G.",
"Hesketh$^{\\rm 78}$ , N.P.",
"Hessey$^{\\rm 106}$ , E. Higón-Rodriguez$^{\\rm 168}$ , D. Hill$^{\\rm 5}$$^{,*}$ , J.C. Hill$^{\\rm 27}$ , N. Hill$^{\\rm 5}$ , K.H.",
"Hiller$^{\\rm 41}$ , S. Hillert$^{\\rm 20}$ , S.J.",
"Hillier$^{\\rm 17}$ , I. Hinchliffe$^{\\rm 14}$ , E. Hines$^{\\rm 121}$ , M. Hirose$^{\\rm 117}$ , F. Hirsch$^{\\rm 42}$ , D. Hirschbuehl$^{\\rm 176}$ , J. Hobbs$^{\\rm 149}$ , N. Hod$^{\\rm 154}$ , M.C.",
"Hodgkinson$^{\\rm 140}$ , P. Hodgson$^{\\rm 140}$ , A. Hoecker$^{\\rm 29}$ , M.R.",
"Hoeferkamp$^{\\rm 104}$ , J. Hoffman$^{\\rm 39}$ , D. Hoffmann$^{\\rm 84}$ , M. Hohlfeld$^{\\rm 82}$ , M. Holder$^{\\rm 142}$ , S.O.",
"Holmgren$^{\\rm 147a}$ , T. Holy$^{\\rm 128}$ , J.L.",
"Holzbauer$^{\\rm 89}$ , Y. Homma$^{\\rm 67}$ , T.M.",
"Hong$^{\\rm 121}$ , L. Hooft van Huysduynen$^{\\rm 109}$ , T. Horazdovsky$^{\\rm 128}$ , C. Horn$^{\\rm 144}$ , S. Horner$^{\\rm 48}$ , J-Y.",
"Hostachy$^{\\rm 55}$ , S. Hou$^{\\rm 152}$ , M.A.",
"Houlden$^{\\rm 74}$ , A. Hoummada$^{\\rm 136a}$ , J. Howarth$^{\\rm 83}$ , D.F.",
"Howell$^{\\rm 119}$ , I. Hristova $^{\\rm 15}$ , J. Hrivnac$^{\\rm 116}$ , I. Hruska$^{\\rm 126}$ , T. Hryn'ova$^{\\rm 4}$ , P.J.",
"Hsu$^{\\rm 82}$ , S.-C. Hsu$^{\\rm 14}$ , G.S.",
"Huang$^{\\rm 112}$ , Z. Hubacek$^{\\rm 128}$ , F. Hubaut$^{\\rm 84}$ , F. Huegging$^{\\rm 20}$ , A. Huettmann$^{\\rm 41}$ , T.B.",
"Huffman$^{\\rm 119}$ , E.W.",
"Hughes$^{\\rm 34}$ , G. Hughes$^{\\rm 72}$ , R.E.",
"Hughes-Jones$^{\\rm 83}$ , M. Huhtinen$^{\\rm 29}$ , P. Hurst$^{\\rm 57}$ , M. Hurwitz$^{\\rm 14}$ , U. Husemann$^{\\rm 41}$ , N. Huseynov$^{\\rm 65}$$^{,p}$ , J. Huston$^{\\rm 89}$ , J. Huth$^{\\rm 57}$ , G. Iacobucci$^{\\rm 49}$ , G. Iakovidis$^{\\rm 9}$ , M. Ibbotson$^{\\rm 83}$ , I. Ibragimov$^{\\rm 142}$ , L. Iconomidou-Fayard$^{\\rm 116}$ , J. Idarraga$^{\\rm 116}$ , P. Iengo$^{\\rm 103a}$ , O. Igonkina$^{\\rm 106}$ , Y. Ikegami$^{\\rm 66}$ , M. Ikeno$^{\\rm 66}$ , D. Iliadis$^{\\rm 155}$ , N. Ilic$^{\\rm 159}$ , M. Imori$^{\\rm 156}$ , T. Ince$^{\\rm 20}$ , J. Inigo-Golfin$^{\\rm 29}$ , P. Ioannou$^{\\rm 8}$ , M. Iodice$^{\\rm 135a}$ , K. Iordanidou$^{\\rm 8}$ , V. Ippolito$^{\\rm 133a,133b}$ , A. Irles Quiles$^{\\rm 168}$ , C. Isaksson$^{\\rm 167}$ , A. Ishikawa$^{\\rm 67}$ , M. Ishino$^{\\rm 68}$ , R. Ishmukhametov$^{\\rm 39}$ , C. Issever$^{\\rm 119}$ , S. Istin$^{\\rm 18a}$ , A.V.",
"Ivashin$^{\\rm 129}$ , W. Iwanski$^{\\rm 38}$ , H. Iwasaki$^{\\rm 66}$ , J.M.",
"Izen$^{\\rm 40}$ , V. Izzo$^{\\rm 103a}$ , B. Jackson$^{\\rm 121}$ , J.N.",
"Jackson$^{\\rm 74}$ , P. Jackson$^{\\rm 144}$ , M.R.",
"Jaekel$^{\\rm 29}$ , V. Jain$^{\\rm 61}$ , K. Jakobs$^{\\rm 48}$ , S. Jakobsen$^{\\rm 35}$ , J. Jakubek$^{\\rm 128}$ , D.K.",
"Jana$^{\\rm 112}$ , E. Jansen$^{\\rm 78}$ , H. Jansen$^{\\rm 29}$ , A. Jantsch$^{\\rm 100}$ , M. Janus$^{\\rm 48}$ , G. Jarlskog$^{\\rm 80}$ , L. Jeanty$^{\\rm 57}$ , K. Jelen$^{\\rm 37}$ , I. Jen-La Plante$^{\\rm 30}$ , P. Jenni$^{\\rm 29}$ , A. Jeremie$^{\\rm 4}$ , P. Jež$^{\\rm 35}$ , S. Jézéquel$^{\\rm 4}$ , M.K.",
"Jha$^{\\rm 19a}$ , H. Ji$^{\\rm 174}$ , W. Ji$^{\\rm 82}$ , J. Jia$^{\\rm 149}$ , Y. Jiang$^{\\rm 32b}$ , M. Jimenez Belenguer$^{\\rm 41}$ , G. Jin$^{\\rm 32b}$ , S. Jin$^{\\rm 32a}$ , O. Jinnouchi$^{\\rm 158}$ , M.D.",
"Joergensen$^{\\rm 35}$ , D. Joffe$^{\\rm 39}$ , L.G.",
"Johansen$^{\\rm 13}$ , M. Johansen$^{\\rm 147a,147b}$ , K.E.",
"Johansson$^{\\rm 147a}$ , P. Johansson$^{\\rm 140}$ , S. Johnert$^{\\rm 41}$ , K.A.",
"Johns$^{\\rm 6}$ , K. Jon-And$^{\\rm 147a,147b}$ , G. Jones$^{\\rm 119}$ , R.W.L.",
"Jones$^{\\rm 72}$ , T.W.",
"Jones$^{\\rm 78}$ , T.J. Jones$^{\\rm 74}$ , O. Jonsson$^{\\rm 29}$ , C. Joram$^{\\rm 29}$ , P.M. Jorge$^{\\rm 125a}$ , J. Joseph$^{\\rm 14}$ , K.D.",
"Joshi$^{\\rm 83}$ , J. Jovicevic$^{\\rm 148}$ , T. Jovin$^{\\rm 12b}$ , X. Ju$^{\\rm 174}$ , C.A.",
"Jung$^{\\rm 42}$ , R.M.",
"Jungst$^{\\rm 29}$ , V. Juranek$^{\\rm 126}$ , P. Jussel$^{\\rm 62}$ , A. Juste Rozas$^{\\rm 11}$ , V.V.",
"Kabachenko$^{\\rm 129}$ , S. Kabana$^{\\rm 16}$ , M. Kaci$^{\\rm 168}$ , A. Kaczmarska$^{\\rm 38}$ , P. Kadlecik$^{\\rm 35}$ , M. Kado$^{\\rm 116}$ , H. Kagan$^{\\rm 110}$ , M. Kagan$^{\\rm 57}$ , S. Kaiser$^{\\rm 100}$ , E. Kajomovitz$^{\\rm 153}$ , S. Kalinin$^{\\rm 176}$ , L.V.",
"Kalinovskaya$^{\\rm 65}$ , S. Kama$^{\\rm 39}$ , N. Kanaya$^{\\rm 156}$ , M. Kaneda$^{\\rm 29}$ , S. Kaneti$^{\\rm 27}$ , T. Kanno$^{\\rm 158}$ , V.A.",
"Kantserov$^{\\rm 97}$ , J. Kanzaki$^{\\rm 66}$ , B. Kaplan$^{\\rm 177}$ , A. Kapliy$^{\\rm 30}$ , J. Kaplon$^{\\rm 29}$ , D. Kar$^{\\rm 53}$ , M. Karagounis$^{\\rm 20}$ , M. Karagoz$^{\\rm 119}$ , M. Karnevskiy$^{\\rm 41}$ , V. Kartvelishvili$^{\\rm 72}$ , A.N.",
"Karyukhin$^{\\rm 129}$ , L. Kashif$^{\\rm 174}$ , G. Kasieczka$^{\\rm 58b}$ , R.D.",
"Kass$^{\\rm 110}$ , A. Kastanas$^{\\rm 13}$ , M. Kataoka$^{\\rm 4}$ , Y. Kataoka$^{\\rm 156}$ , E. Katsoufis$^{\\rm 9}$ , J. Katzy$^{\\rm 41}$ , V. Kaushik$^{\\rm 6}$ , K. Kawagoe$^{\\rm 70}$ , T. Kawamoto$^{\\rm 156}$ , G. Kawamura$^{\\rm 82}$ , M.S.",
"Kayl$^{\\rm 106}$ , V.A.",
"Kazanin$^{\\rm 108}$ , M.Y.",
"Kazarinov$^{\\rm 65}$ , R. Keeler$^{\\rm 170}$ , R. Kehoe$^{\\rm 39}$ , M. Keil$^{\\rm 54}$ , G.D. Kekelidze$^{\\rm 65}$ , J.S.",
"Keller$^{\\rm 139}$ , J. Kennedy$^{\\rm 99}$ , M. Kenyon$^{\\rm 53}$ , O. Kepka$^{\\rm 126}$ , N. Kerschen$^{\\rm 29}$ , B.P.",
"Kerševan$^{\\rm 75}$ , S. Kersten$^{\\rm 176}$ , K. Kessoku$^{\\rm 156}$ , J. Keung$^{\\rm 159}$ , F. Khalil-zada$^{\\rm 10}$ , H. Khandanyan$^{\\rm 166}$ , A. Khanov$^{\\rm 113}$ , D. Kharchenko$^{\\rm 65}$ , A. Khodinov$^{\\rm 97}$ , A.G. Kholodenko$^{\\rm 129}$ , A. Khomich$^{\\rm 58a}$ , T.J. Khoo$^{\\rm 27}$ , G. Khoriauli$^{\\rm 20}$ , A. Khoroshilov$^{\\rm 176}$ , N. Khovanskiy$^{\\rm 65}$ , V. Khovanskiy$^{\\rm 96}$ , E. Khramov$^{\\rm 65}$ , J. Khubua$^{\\rm 51b}$ , H. Kim$^{\\rm 147a,147b}$ , M.S.",
"Kim$^{\\rm 2}$ , S.H.",
"Kim$^{\\rm 161}$ , N. Kimura$^{\\rm 172}$ , O. Kind$^{\\rm 15}$ , B.T.",
"King$^{\\rm 74}$ , M. King$^{\\rm 67}$ , R.S.B.",
"King$^{\\rm 119}$ , J. Kirk$^{\\rm 130}$ , L.E.",
"Kirsch$^{\\rm 22}$ , A.E.",
"Kiryunin$^{\\rm 100}$ , T. Kishimoto$^{\\rm 67}$ , D. Kisielewska$^{\\rm 37}$ , T. Kittelmann$^{\\rm 124}$ , A.M. Kiver$^{\\rm 129}$ , E. Kladiva$^{\\rm 145b}$ , M. Klein$^{\\rm 74}$ , U. Klein$^{\\rm 74}$ , K. Kleinknecht$^{\\rm 82}$ , M. Klemetti$^{\\rm 86}$ , A. Klier$^{\\rm 173}$ , P. Klimek$^{\\rm 147a,147b}$ , A. Klimentov$^{\\rm 24}$ , R. Klingenberg$^{\\rm 42}$ , J.A.",
"Klinger$^{\\rm 83}$ , E.B.",
"Klinkby$^{\\rm 35}$ , T. Klioutchnikova$^{\\rm 29}$ , P.F.",
"Klok$^{\\rm 105}$ , S. Klous$^{\\rm 106}$ , E.-E. Kluge$^{\\rm 58a}$ , T. Kluge$^{\\rm 74}$ , P. Kluit$^{\\rm 106}$ , S. Kluth$^{\\rm 100}$ , N.S.",
"Knecht$^{\\rm 159}$ , E. Kneringer$^{\\rm 62}$ , J. Knobloch$^{\\rm 29}$ , E.B.F.G.",
"Knoops$^{\\rm 84}$ , A. Knue$^{\\rm 54}$ , B.R.",
"Ko$^{\\rm 44}$ , T. Kobayashi$^{\\rm 156}$ , M. Kobel$^{\\rm 43}$ , M. Kocian$^{\\rm 144}$ , P. Kodys$^{\\rm 127}$ , K. Köneke$^{\\rm 29}$ , A.C. König$^{\\rm 105}$ , S. Koenig$^{\\rm 82}$ , L. Köpke$^{\\rm 82}$ , F. Koetsveld$^{\\rm 105}$ , P. Koevesarki$^{\\rm 20}$ , T. Koffas$^{\\rm 28}$ , E. Koffeman$^{\\rm 106}$ , L.A. Kogan$^{\\rm 119}$ , S. Kohlmann$^{\\rm 176}$ , F. Kohn$^{\\rm 54}$ , Z. Kohout$^{\\rm 128}$ , T. Kohriki$^{\\rm 66}$ , T. Koi$^{\\rm 144}$ , T. Kokott$^{\\rm 20}$ , G.M.",
"Kolachev$^{\\rm 108}$ , H. Kolanoski$^{\\rm 15}$ , V. Kolesnikov$^{\\rm 65}$ , I. Koletsou$^{\\rm 90a}$ , J. Koll$^{\\rm 89}$ , M. Kollefrath$^{\\rm 48}$ , S.D.",
"Kolya$^{\\rm 83}$ , A.A. Komar$^{\\rm 95}$ , Y. Komori$^{\\rm 156}$ , T. Kondo$^{\\rm 66}$ , T. Kono$^{\\rm 41}$$^{,q}$ , A.I.",
"Kononov$^{\\rm 48}$ , R. Konoplich$^{\\rm 109}$$^{,r}$ , N. Konstantinidis$^{\\rm 78}$ , A. Kootz$^{\\rm 176}$ , S. Koperny$^{\\rm 37}$ , K. Korcyl$^{\\rm 38}$ , K. Kordas$^{\\rm 155}$ , V. Koreshev$^{\\rm 129}$ , A. Korn$^{\\rm 119}$ , A. Korol$^{\\rm 108}$ , I. Korolkov$^{\\rm 11}$ , E.V.",
"Korolkova$^{\\rm 140}$ , V.A.",
"Korotkov$^{\\rm 129}$ , O. Kortner$^{\\rm 100}$ , S. Kortner$^{\\rm 100}$ , V.V.",
"Kostyukhin$^{\\rm 20}$ , M.J. Kotamäki$^{\\rm 29}$ , S. Kotov$^{\\rm 100}$ , V.M.",
"Kotov$^{\\rm 65}$ , A. Kotwal$^{\\rm 44}$ , C. Kourkoumelis$^{\\rm 8}$ , V. Kouskoura$^{\\rm 155}$ , A. Koutsman$^{\\rm 160a}$ , R. Kowalewski$^{\\rm 170}$ , T.Z.",
"Kowalski$^{\\rm 37}$ , W. Kozanecki$^{\\rm 137}$ , A.S. Kozhin$^{\\rm 129}$ , V. Kral$^{\\rm 128}$ , V.A.",
"Kramarenko$^{\\rm 98}$ , G. Kramberger$^{\\rm 75}$ , M.W.",
"Krasny$^{\\rm 79}$ , A. Krasznahorkay$^{\\rm 109}$ , J. Kraus$^{\\rm 89}$ , J.K. Kraus$^{\\rm 20}$ , F. Krejci$^{\\rm 128}$ , J. Kretzschmar$^{\\rm 74}$ , N. Krieger$^{\\rm 54}$ , P. Krieger$^{\\rm 159}$ , K. Kroeninger$^{\\rm 54}$ , H. Kroha$^{\\rm 100}$ , J. Kroll$^{\\rm 121}$ , J. Kroseberg$^{\\rm 20}$ , J. Krstic$^{\\rm 12a}$ , U. Kruchonak$^{\\rm 65}$ , H. Krüger$^{\\rm 20}$ , T. Kruker$^{\\rm 16}$ , N. Krumnack$^{\\rm 64}$ , Z.V.",
"Krumshteyn$^{\\rm 65}$ , A. Kruth$^{\\rm 20}$ , T. Kubota$^{\\rm 87}$ , S. Kuday$^{\\rm 3a}$ , S. Kuehn$^{\\rm 48}$ , A. Kugel$^{\\rm 58c}$ , T. Kuhl$^{\\rm 41}$ , D. Kuhn$^{\\rm 62}$ , V. Kukhtin$^{\\rm 65}$ , Y. Kulchitsky$^{\\rm 91}$ , S. Kuleshov$^{\\rm 31b}$ , C. Kummer$^{\\rm 99}$ , M. Kuna$^{\\rm 79}$ , J. Kunkle$^{\\rm 121}$ , A. Kupco$^{\\rm 126}$ , H. Kurashige$^{\\rm 67}$ , M. Kurata$^{\\rm 161}$ , Y.A.",
"Kurochkin$^{\\rm 91}$ , V. Kus$^{\\rm 126}$ , E.S.",
"Kuwertz$^{\\rm 148}$ , M. Kuze$^{\\rm 158}$ , J. Kvita$^{\\rm 143}$ , R. Kwee$^{\\rm 15}$ , A.",
"La Rosa$^{\\rm 49}$ , L. La Rotonda$^{\\rm 36a,36b}$ , L. Labarga$^{\\rm 81}$ , J. Labbe$^{\\rm 4}$ , S. Lablak$^{\\rm 136a}$ , C. Lacasta$^{\\rm 168}$ , F. Lacava$^{\\rm 133a,133b}$ , H. Lacker$^{\\rm 15}$ , D. Lacour$^{\\rm 79}$ , V.R.",
"Lacuesta$^{\\rm 168}$ , E. Ladygin$^{\\rm 65}$ , R. Lafaye$^{\\rm 4}$ , B. Laforge$^{\\rm 79}$ , T. Lagouri$^{\\rm 81}$ , S. Lai$^{\\rm 48}$ , E. Laisne$^{\\rm 55}$ , M. Lamanna$^{\\rm 29}$ , L. Lambourne$^{\\rm 78}$ , C.L.",
"Lampen$^{\\rm 6}$ , W. Lampl$^{\\rm 6}$ , E. Lancon$^{\\rm 137}$ , U. Landgraf$^{\\rm 48}$ , M.P.J.",
"Landon$^{\\rm 76}$ , J.L.",
"Lane$^{\\rm 83}$ , C. Lange$^{\\rm 41}$ , A.J.",
"Lankford$^{\\rm 164}$ , F. Lanni$^{\\rm 24}$ , K. Lantzsch$^{\\rm 176}$ , S. Laplace$^{\\rm 79}$ , C. Lapoire$^{\\rm 20}$ , J.F.",
"Laporte$^{\\rm 137}$ , T. Lari$^{\\rm 90a}$ , A.V.",
"Larionov $^{\\rm 129}$ , A. Larner$^{\\rm 119}$ , C. Lasseur$^{\\rm 29}$ , M. Lassnig$^{\\rm 29}$ , P. Laurelli$^{\\rm 47}$ , V. Lavorini$^{\\rm 36a,36b}$ , W. Lavrijsen$^{\\rm 14}$ , P. Laycock$^{\\rm 74}$ , A.B.",
"Lazarev$^{\\rm 65}$ , O.",
"Le Dortz$^{\\rm 79}$ , E. Le Guirriec$^{\\rm 84}$ , C. Le Maner$^{\\rm 159}$ , E. Le Menedeu$^{\\rm 11}$ , C. Lebel$^{\\rm 94}$ , T. LeCompte$^{\\rm 5}$ , F. Ledroit-Guillon$^{\\rm 55}$ , H. Lee$^{\\rm 106}$ , J.S.H.",
"Lee$^{\\rm 117}$ , S.C. Lee$^{\\rm 152}$ , L. Lee$^{\\rm 177}$ , M. Lefebvre$^{\\rm 170}$ , M. Legendre$^{\\rm 137}$ , A. Leger$^{\\rm 49}$ , B.C.",
"LeGeyt$^{\\rm 121}$ , F. Legger$^{\\rm 99}$ , C. Leggett$^{\\rm 14}$ , M. Lehmacher$^{\\rm 20}$ , G. Lehmann Miotto$^{\\rm 29}$ , X. Lei$^{\\rm 6}$ , M.A.L.",
"Leite$^{\\rm 23d}$ , R. Leitner$^{\\rm 127}$ , D. Lellouch$^{\\rm 173}$ , M. Leltchouk$^{\\rm 34}$ , B. Lemmer$^{\\rm 54}$ , V. Lendermann$^{\\rm 58a}$ , K.J.C.",
"Leney$^{\\rm 146b}$ , T. Lenz$^{\\rm 106}$ , G. Lenzen$^{\\rm 176}$ , B. Lenzi$^{\\rm 29}$ , K. Leonhardt$^{\\rm 43}$ , S. Leontsinis$^{\\rm 9}$ , F. Lepold$^{\\rm 58a}$ , C. Leroy$^{\\rm 94}$ , J-R. Lessard$^{\\rm 170}$ , C.G.",
"Lester$^{\\rm 27}$ , C.M.",
"Lester$^{\\rm 121}$ , J. Levêque$^{\\rm 4}$ , D. Levin$^{\\rm 88}$ , L.J.",
"Levinson$^{\\rm 173}$ , M.S.",
"Levitski$^{\\rm 129}$ , A. Lewis$^{\\rm 119}$ , G.H.",
"Lewis$^{\\rm 109}$ , A.M. Leyko$^{\\rm 20}$ , M. Leyton$^{\\rm 15}$ , B. Li$^{\\rm 84}$ , H. Li$^{\\rm 174}$$^{,s}$ , S. Li$^{\\rm 32b}$$^{,t}$ , X. Li$^{\\rm 88}$ , Z. Liang$^{\\rm 119}$$^{,u}$ , H. Liao$^{\\rm 33}$ , B. Liberti$^{\\rm 134a}$ , P. Lichard$^{\\rm 29}$ , M. Lichtnecker$^{\\rm 99}$ , K. Lie$^{\\rm 166}$ , W. Liebig$^{\\rm 13}$ , C. Limbach$^{\\rm 20}$ , A. Limosani$^{\\rm 87}$ , M. Limper$^{\\rm 63}$ , S.C. Lin$^{\\rm 152}$$^{,v}$ , F. Linde$^{\\rm 106}$ , J.T.",
"Linnemann$^{\\rm 89}$ , E. Lipeles$^{\\rm 121}$ , L. Lipinsky$^{\\rm 126}$ , A. Lipniacka$^{\\rm 13}$ , T.M.",
"Liss$^{\\rm 166}$ , D. Lissauer$^{\\rm 24}$ , A. Lister$^{\\rm 49}$ , A.M. Litke$^{\\rm 138}$ , C. Liu$^{\\rm 28}$ , D. Liu$^{\\rm 152}$ , H. Liu$^{\\rm 88}$ , J.B. Liu$^{\\rm 88}$ , M. Liu$^{\\rm 32b}$ , Y. Liu$^{\\rm 32b}$ , M. Livan$^{\\rm 120a,120b}$ , S.S.A.",
"Livermore$^{\\rm 119}$ , A. Lleres$^{\\rm 55}$ , J. Llorente Merino$^{\\rm 81}$ , S.L.",
"Lloyd$^{\\rm 76}$ , E. Lobodzinska$^{\\rm 41}$ , P. Loch$^{\\rm 6}$ , W.S.",
"Lockman$^{\\rm 138}$ , T. Loddenkoetter$^{\\rm 20}$ , F.K.",
"Loebinger$^{\\rm 83}$ , A. Loginov$^{\\rm 177}$ , C.W.",
"Loh$^{\\rm 169}$ , T. Lohse$^{\\rm 15}$ , K. Lohwasser$^{\\rm 48}$ , M. Lokajicek$^{\\rm 126}$ , J. Loken $^{\\rm 119}$ , V.P.",
"Lombardo$^{\\rm 4}$ , R.E.",
"Long$^{\\rm 72}$ , L. Lopes$^{\\rm 125a}$ , D. Lopez Mateos$^{\\rm 57}$ , J. Lorenz$^{\\rm 99}$ , N. Lorenzo Martinez$^{\\rm 116}$ , M. Losada$^{\\rm 163}$ , P. Loscutoff$^{\\rm 14}$ , F. Lo Sterzo$^{\\rm 133a,133b}$ , M.J. Losty$^{\\rm 160a}$ , X. Lou$^{\\rm 40}$ , A. Lounis$^{\\rm 116}$ , K.F.",
"Loureiro$^{\\rm 163}$ , J. Love$^{\\rm 21}$ , P.A.",
"Love$^{\\rm 72}$ , A.J.",
"Lowe$^{\\rm 144}$$^{,e}$ , F. Lu$^{\\rm 32a}$ , H.J.",
"Lubatti$^{\\rm 139}$ , C. Luci$^{\\rm 133a,133b}$ , A. Lucotte$^{\\rm 55}$ , A. Ludwig$^{\\rm 43}$ , D. Ludwig$^{\\rm 41}$ , I. Ludwig$^{\\rm 48}$ , J. Ludwig$^{\\rm 48}$ , F. Luehring$^{\\rm 61}$ , G. Luijckx$^{\\rm 106}$ , W. Lukas$^{\\rm 62}$ , D. Lumb$^{\\rm 48}$ , L. Luminari$^{\\rm 133a}$ , E. Lund$^{\\rm 118}$ , B. Lund-Jensen$^{\\rm 148}$ , B. Lundberg$^{\\rm 80}$ , J. Lundberg$^{\\rm 147a,147b}$ , J. Lundquist$^{\\rm 35}$ , M. Lungwitz$^{\\rm 82}$ , G. Lutz$^{\\rm 100}$ , D. Lynn$^{\\rm 24}$ , J. Lys$^{\\rm 14}$ , E. Lytken$^{\\rm 80}$ , H. Ma$^{\\rm 24}$ , L.L.",
"Ma$^{\\rm 174}$ , J.A.",
"Macana Goia$^{\\rm 94}$ , G. Maccarrone$^{\\rm 47}$ , A. Macchiolo$^{\\rm 100}$ , B. Maček$^{\\rm 75}$ , J. Machado Miguens$^{\\rm 125a}$ , R. Mackeprang$^{\\rm 35}$ , R.J. Madaras$^{\\rm 14}$ , W.F.",
"Mader$^{\\rm 43}$ , R. Maenner$^{\\rm 58c}$ , T. Maeno$^{\\rm 24}$ , P. Mättig$^{\\rm 176}$ , S. Mättig$^{\\rm 41}$ , L. Magnoni$^{\\rm 29}$ , E. Magradze$^{\\rm 54}$ , Y. Mahalalel$^{\\rm 154}$ , K. Mahboubi$^{\\rm 48}$ , S. Mahmoud$^{\\rm 74}$ , G. Mahout$^{\\rm 17}$ , C. Maiani$^{\\rm 133a,133b}$ , C. Maidantchik$^{\\rm 23a}$ , A. Maio$^{\\rm 125a}$$^{,b}$ , S. Majewski$^{\\rm 24}$ , Y. Makida$^{\\rm 66}$ , N. Makovec$^{\\rm 116}$ , P. Mal$^{\\rm 137}$ , B. Malaescu$^{\\rm 29}$ , Pa. Malecki$^{\\rm 38}$ , P. Malecki$^{\\rm 38}$ , V.P.",
"Maleev$^{\\rm 122}$ , F. Malek$^{\\rm 55}$ , U. Mallik$^{\\rm 63}$ , D. Malon$^{\\rm 5}$ , C. Malone$^{\\rm 144}$ , S. Maltezos$^{\\rm 9}$ , V. Malyshev$^{\\rm 108}$ , S. Malyukov$^{\\rm 29}$ , R. Mameghani$^{\\rm 99}$ , J. Mamuzic$^{\\rm 12b}$ , A. Manabe$^{\\rm 66}$ , L. Mandelli$^{\\rm 90a}$ , I. Mandić$^{\\rm 75}$ , R. Mandrysch$^{\\rm 15}$ , J. Maneira$^{\\rm 125a}$ , P.S.",
"Mangeard$^{\\rm 89}$ , L. Manhaes de Andrade Filho$^{\\rm 23a}$ , I.D.",
"Manjavidze$^{\\rm 65}$ , A. Mann$^{\\rm 54}$ , P.M. Manning$^{\\rm 138}$ , A. Manousakis-Katsikakis$^{\\rm 8}$ , B. Mansoulie$^{\\rm 137}$ , A. Manz$^{\\rm 100}$ , A. Mapelli$^{\\rm 29}$ , L. Mapelli$^{\\rm 29}$ , L. March $^{\\rm 81}$ , J.F.",
"Marchand$^{\\rm 28}$ , F. Marchese$^{\\rm 134a,134b}$ , G. Marchiori$^{\\rm 79}$ , M. Marcisovsky$^{\\rm 126}$ , C.P.",
"Marino$^{\\rm 170}$ , F. Marroquim$^{\\rm 23a}$ , R. Marshall$^{\\rm 83}$ , Z. Marshall$^{\\rm 29}$ , F.K.",
"Martens$^{\\rm 159}$ , S. Marti-Garcia$^{\\rm 168}$ , A.J.",
"Martin$^{\\rm 177}$ , B. Martin$^{\\rm 29}$ , B. Martin$^{\\rm 89}$ , F.F.",
"Martin$^{\\rm 121}$ , J.P. Martin$^{\\rm 94}$ , Ph.",
"Martin$^{\\rm 55}$ , T.A.",
"Martin$^{\\rm 17}$ , V.J.",
"Martin$^{\\rm 45}$ , B. Martin dit Latour$^{\\rm 49}$ , S. Martin-Haugh$^{\\rm 150}$ , M. Martinez$^{\\rm 11}$ , V. Martinez Outschoorn$^{\\rm 57}$ , A.C. Martyniuk$^{\\rm 170}$ , M. Marx$^{\\rm 83}$ , F. Marzano$^{\\rm 133a}$ , A. Marzin$^{\\rm 112}$ , L. Masetti$^{\\rm 82}$ , T. Mashimo$^{\\rm 156}$ , R. Mashinistov$^{\\rm 95}$ , J. Masik$^{\\rm 83}$ , A.L.",
"Maslennikov$^{\\rm 108}$ , I. Massa$^{\\rm 19a,19b}$ , G. Massaro$^{\\rm 106}$ , N. Massol$^{\\rm 4}$ , P. Mastrandrea$^{\\rm 133a,133b}$ , A. Mastroberardino$^{\\rm 36a,36b}$ , T. Masubuchi$^{\\rm 156}$ , P. Matricon$^{\\rm 116}$ , H. Matsunaga$^{\\rm 156}$ , T. Matsushita$^{\\rm 67}$ , C. Mattravers$^{\\rm 119}$$^{,c}$ , J.M.",
"Maugain$^{\\rm 29}$ , J. Maurer$^{\\rm 84}$ , S.J.",
"Maxfield$^{\\rm 74}$ , E.N.",
"May$^{\\rm 5}$ , A. Mayne$^{\\rm 140}$ , R. Mazini$^{\\rm 152}$ , M. Mazur$^{\\rm 20}$ , L. Mazzaferro$^{\\rm 134a,134b}$ , M. Mazzanti$^{\\rm 90a}$ , S.P.",
"Mc Kee$^{\\rm 88}$ , A. McCarn$^{\\rm 166}$ , R.L.",
"McCarthy$^{\\rm 149}$ , T.G.",
"McCarthy$^{\\rm 28}$ , N.A.",
"McCubbin$^{\\rm 130}$ , K.W.",
"McFarlane$^{\\rm 56}$ , J.A.",
"Mcfayden$^{\\rm 140}$ , H. McGlone$^{\\rm 53}$ , G. Mchedlidze$^{\\rm 51b}$ , R.A. McLaren$^{\\rm 29}$ , T. Mclaughlan$^{\\rm 17}$ , S.J.",
"McMahon$^{\\rm 130}$ , R.A. McPherson$^{\\rm 170}$$^{,j}$ , A. Meade$^{\\rm 85}$ , J. Mechnich$^{\\rm 106}$ , M. Mechtel$^{\\rm 176}$ , M. Medinnis$^{\\rm 41}$ , R. Meera-Lebbai$^{\\rm 112}$ , T. Meguro$^{\\rm 117}$ , R. Mehdiyev$^{\\rm 94}$ , S. Mehlhase$^{\\rm 35}$ , A. Mehta$^{\\rm 74}$ , K. Meier$^{\\rm 58a}$ , B. Meirose$^{\\rm 80}$ , C. Melachrinos$^{\\rm 30}$ , B.R.",
"Mellado Garcia$^{\\rm 174}$ , F. Meloni$^{\\rm 90a,90b}$ , L. Mendoza Navas$^{\\rm 163}$ , Z. Meng$^{\\rm 152}$$^{,s}$ , A. Mengarelli$^{\\rm 19a,19b}$ , S. Menke$^{\\rm 100}$ , C. Menot$^{\\rm 29}$ , E. Meoni$^{\\rm 11}$ , K.M.",
"Mercurio$^{\\rm 57}$ , P. Mermod$^{\\rm 49}$ , L. Merola$^{\\rm 103a,103b}$ , C. Meroni$^{\\rm 90a}$ , F.S.",
"Merritt$^{\\rm 30}$ , H. Merritt$^{\\rm 110}$ , A. Messina$^{\\rm 29}$ , J. Metcalfe$^{\\rm 104}$ , A.S. Mete$^{\\rm 64}$ , C. Meyer$^{\\rm 82}$ , C. Meyer$^{\\rm 30}$ , J-P. Meyer$^{\\rm 137}$ , J. Meyer$^{\\rm 175}$ , J. Meyer$^{\\rm 54}$ , T.C.",
"Meyer$^{\\rm 29}$ , W.T.",
"Meyer$^{\\rm 64}$ , J. Miao$^{\\rm 32d}$ , S. Michal$^{\\rm 29}$ , L. Micu$^{\\rm 25a}$ , R.P.",
"Middleton$^{\\rm 130}$ , S. Migas$^{\\rm 74}$ , L. Mijović$^{\\rm 41}$ , G. Mikenberg$^{\\rm 173}$ , M. Mikestikova$^{\\rm 126}$ , M. Mikuž$^{\\rm 75}$ , D.W. Miller$^{\\rm 30}$ , R.J. Miller$^{\\rm 89}$ , W.J.",
"Mills$^{\\rm 169}$ , C. Mills$^{\\rm 57}$ , A. Milov$^{\\rm 173}$ , D.A.",
"Milstead$^{\\rm 147a,147b}$ , D. Milstein$^{\\rm 173}$ , A.A. Minaenko$^{\\rm 129}$ , M. Miñano Moya$^{\\rm 168}$ , I.A.",
"Minashvili$^{\\rm 65}$ , A.I.",
"Mincer$^{\\rm 109}$ , B. Mindur$^{\\rm 37}$ , M. Mineev$^{\\rm 65}$ , Y. Ming$^{\\rm 174}$ , L.M.",
"Mir$^{\\rm 11}$ , G. Mirabelli$^{\\rm 133a}$ , L. Miralles Verge$^{\\rm 11}$ , A. Misiejuk$^{\\rm 77}$ , J. Mitrevski$^{\\rm 138}$ , G.Y.",
"Mitrofanov$^{\\rm 129}$ , V.A.",
"Mitsou$^{\\rm 168}$ , S. Mitsui$^{\\rm 66}$ , P.S.",
"Miyagawa$^{\\rm 140}$ , K. Miyazaki$^{\\rm 67}$ , J.U.",
"Mjörnmark$^{\\rm 80}$ , T. Moa$^{\\rm 147a,147b}$ , P. Mockett$^{\\rm 139}$ , S. Moed$^{\\rm 57}$ , V. Moeller$^{\\rm 27}$ , K. Mönig$^{\\rm 41}$ , N. Möser$^{\\rm 20}$ , S. Mohapatra$^{\\rm 149}$ , W. Mohr$^{\\rm 48}$ , S. Mohrdieck-Möck$^{\\rm 100}$ , R. Moles-Valls$^{\\rm 168}$ , J. Molina-Perez$^{\\rm 29}$ , J. Monk$^{\\rm 78}$ , E. Monnier$^{\\rm 84}$ , S. Montesano$^{\\rm 90a,90b}$ , F. Monticelli$^{\\rm 71}$ , S. Monzani$^{\\rm 19a,19b}$ , R.W.",
"Moore$^{\\rm 2}$ , G.F. Moorhead$^{\\rm 87}$ , C. Mora Herrera$^{\\rm 49}$ , A. Moraes$^{\\rm 53}$ , N. Morange$^{\\rm 137}$ , J. Morel$^{\\rm 54}$ , G. Morello$^{\\rm 36a,36b}$ , D. Moreno$^{\\rm 82}$ , M. Moreno Llácer$^{\\rm 168}$ , P. Morettini$^{\\rm 50a}$ , M. Morgenstern$^{\\rm 43}$ , M. Morii$^{\\rm 57}$ , J. Morin$^{\\rm 76}$ , A.K.",
"Morley$^{\\rm 29}$ , G. Mornacchi$^{\\rm 29}$ , S.V.",
"Morozov$^{\\rm 97}$ , J.D.",
"Morris$^{\\rm 76}$ , L. Morvaj$^{\\rm 102}$ , H.G.",
"Moser$^{\\rm 100}$ , M. Mosidze$^{\\rm 51b}$ , J. Moss$^{\\rm 110}$ , R. Mount$^{\\rm 144}$ , E. Mountricha$^{\\rm 9}$$^{,w}$ , S.V.",
"Mouraviev$^{\\rm 95}$ , E.J.W.",
"Moyse$^{\\rm 85}$ , M. Mudrinic$^{\\rm 12b}$ , F. Mueller$^{\\rm 58a}$ , J. Mueller$^{\\rm 124}$ , K. Mueller$^{\\rm 20}$ , T.A.",
"Müller$^{\\rm 99}$ , T. Mueller$^{\\rm 82}$ , D. Muenstermann$^{\\rm 29}$ , Y. Munwes$^{\\rm 154}$ , W.J.",
"Murray$^{\\rm 130}$ , I. Mussche$^{\\rm 106}$ , E. Musto$^{\\rm 103a,103b}$ , A.G. Myagkov$^{\\rm 129}$ , M. Myska$^{\\rm 126}$ , J. Nadal$^{\\rm 11}$ , K. Nagai$^{\\rm 161}$ , K. Nagano$^{\\rm 66}$ , A. Nagarkar$^{\\rm 110}$ , Y. Nagasaka$^{\\rm 60}$ , M. Nagel$^{\\rm 100}$ , A.M. Nairz$^{\\rm 29}$ , Y. Nakahama$^{\\rm 29}$ , K. Nakamura$^{\\rm 156}$ , T. Nakamura$^{\\rm 156}$ , I. Nakano$^{\\rm 111}$ , G. Nanava$^{\\rm 20}$ , A. Napier$^{\\rm 162}$ , R. Narayan$^{\\rm 58b}$ , M. Nash$^{\\rm 78}$$^{,c}$ , N.R.",
"Nation$^{\\rm 21}$ , T. Nattermann$^{\\rm 20}$ , T. Naumann$^{\\rm 41}$ , G. Navarro$^{\\rm 163}$ , H.A.",
"Neal$^{\\rm 88}$ , E. Nebot$^{\\rm 81}$ , P.Yu.",
"Nechaeva$^{\\rm 95}$ , T.J. Neep$^{\\rm 83}$ , A. Negri$^{\\rm 120a,120b}$ , G. Negri$^{\\rm 29}$ , S. Nektarijevic$^{\\rm 49}$ , A. Nelson$^{\\rm 164}$ , T.K.",
"Nelson$^{\\rm 144}$ , S. Nemecek$^{\\rm 126}$ , P. Nemethy$^{\\rm 109}$ , A.A. Nepomuceno$^{\\rm 23a}$ , M. Nessi$^{\\rm 29}$$^{,x}$ , M.S.",
"Neubauer$^{\\rm 166}$ , A. Neusiedl$^{\\rm 82}$ , R.M.",
"Neves$^{\\rm 109}$ , P. Nevski$^{\\rm 24}$ , P.R.",
"Newman$^{\\rm 17}$ , V. Nguyen Thi Hong$^{\\rm 137}$ , R.B.",
"Nickerson$^{\\rm 119}$ , R. Nicolaidou$^{\\rm 137}$ , L. Nicolas$^{\\rm 140}$ , B. Nicquevert$^{\\rm 29}$ , F. Niedercorn$^{\\rm 116}$ , J. Nielsen$^{\\rm 138}$ , T. Niinikoski$^{\\rm 29}$ , N. Nikiforou$^{\\rm 34}$ , A. Nikiforov$^{\\rm 15}$ , V. Nikolaenko$^{\\rm 129}$ , K. Nikolaev$^{\\rm 65}$ , I. Nikolic-Audit$^{\\rm 79}$ , K. Nikolics$^{\\rm 49}$ , K. Nikolopoulos$^{\\rm 24}$ , H. Nilsen$^{\\rm 48}$ , P. Nilsson$^{\\rm 7}$ , Y. Ninomiya $^{\\rm 156}$ , A. Nisati$^{\\rm 133a}$ , T. Nishiyama$^{\\rm 67}$ , R. Nisius$^{\\rm 100}$ , L. Nodulman$^{\\rm 5}$ , M. Nomachi$^{\\rm 117}$ , I. Nomidis$^{\\rm 155}$ , M. Nordberg$^{\\rm 29}$ , P.R.",
"Norton$^{\\rm 130}$ , J. Novakova$^{\\rm 127}$ , M. Nozaki$^{\\rm 66}$ , L. Nozka$^{\\rm 114}$ , I.M.",
"Nugent$^{\\rm 160a}$ , A.-E. Nuncio-Quiroz$^{\\rm 20}$ , G. Nunes Hanninger$^{\\rm 87}$ , T. Nunnemann$^{\\rm 99}$ , E. Nurse$^{\\rm 78}$ , B.J.",
"O'Brien$^{\\rm 45}$ , S.W.",
"O'Neale$^{\\rm 17}$$^{,*}$ , D.C. O'Neil$^{\\rm 143}$ , V. O'Shea$^{\\rm 53}$ , L.B.",
"Oakes$^{\\rm 99}$ , F.G. Oakham$^{\\rm 28}$$^{,d}$ , H. Oberlack$^{\\rm 100}$ , J. Ocariz$^{\\rm 79}$ , A. Ochi$^{\\rm 67}$ , S. Oda$^{\\rm 156}$ , S. Odaka$^{\\rm 66}$ , J. Odier$^{\\rm 84}$ , H. Ogren$^{\\rm 61}$ , A. Oh$^{\\rm 83}$ , S.H.",
"Oh$^{\\rm 44}$ , C.C.",
"Ohm$^{\\rm 147a,147b}$ , T. Ohshima$^{\\rm 102}$ , H. Ohshita$^{\\rm 141}$ , S. Okada$^{\\rm 67}$ , H. Okawa$^{\\rm 164}$ , Y. Okumura$^{\\rm 102}$ , T. Okuyama$^{\\rm 156}$ , A. Olariu$^{\\rm 25a}$ , M. Olcese$^{\\rm 50a}$ , A.G. Olchevski$^{\\rm 65}$ , S.A. Olivares Pino$^{\\rm 31a}$ , M. Oliveira$^{\\rm 125a}$$^{,h}$ , D. Oliveira Damazio$^{\\rm 24}$ , E. Oliver Garcia$^{\\rm 168}$ , D. Olivito$^{\\rm 121}$ , A. Olszewski$^{\\rm 38}$ , J. Olszowska$^{\\rm 38}$ , C. Omachi$^{\\rm 67}$ , A. Onofre$^{\\rm 125a}$$^{,y}$ , P.U.E.",
"Onyisi$^{\\rm 30}$ , C.J.",
"Oram$^{\\rm 160a}$ , M.J. Oreglia$^{\\rm 30}$ , Y. Oren$^{\\rm 154}$ , D. Orestano$^{\\rm 135a,135b}$ , N. Orlando$^{\\rm 73a,73b}$ , I. Orlov$^{\\rm 108}$ , C. Oropeza Barrera$^{\\rm 53}$ , R.S.",
"Orr$^{\\rm 159}$ , B. Osculati$^{\\rm 50a,50b}$ , R. Ospanov$^{\\rm 121}$ , C. Osuna$^{\\rm 11}$ , G. Otero y Garzon$^{\\rm 26}$ , J.P. Ottersbach$^{\\rm 106}$ , M. Ouchrif$^{\\rm 136d}$ , E.A.",
"Ouellette$^{\\rm 170}$ , F. Ould-Saada$^{\\rm 118}$ , A. Ouraou$^{\\rm 137}$ , Q. Ouyang$^{\\rm 32a}$ , A. Ovcharova$^{\\rm 14}$ , M. Owen$^{\\rm 83}$ , S. Owen$^{\\rm 140}$ , V.E.",
"Ozcan$^{\\rm 18a}$ , N. Ozturk$^{\\rm 7}$ , A. Pacheco Pages$^{\\rm 11}$ , C. Padilla Aranda$^{\\rm 11}$ , S. Pagan Griso$^{\\rm 14}$ , E. Paganis$^{\\rm 140}$ , F. Paige$^{\\rm 24}$ , P. Pais$^{\\rm 85}$ , K. Pajchel$^{\\rm 118}$ , G. Palacino$^{\\rm 160b}$ , C.P.",
"Paleari$^{\\rm 6}$ , S. Palestini$^{\\rm 29}$ , D. Pallin$^{\\rm 33}$ , A. Palma$^{\\rm 125a}$ , J.D.",
"Palmer$^{\\rm 17}$ , Y.B.",
"Pan$^{\\rm 174}$ , E. Panagiotopoulou$^{\\rm 9}$ , N. Panikashvili$^{\\rm 88}$ , S. Panitkin$^{\\rm 24}$ , D. Pantea$^{\\rm 25a}$ , M. Panuskova$^{\\rm 126}$ , V. Paolone$^{\\rm 124}$ , A. Papadelis$^{\\rm 147a}$ , Th.D.",
"Papadopoulou$^{\\rm 9}$ , A. Paramonov$^{\\rm 5}$ , D. Paredes Hernandez$^{\\rm 33}$ , W. Park$^{\\rm 24}$$^{,z}$ , M.A.",
"Parker$^{\\rm 27}$ , F. Parodi$^{\\rm 50a,50b}$ , J.A.",
"Parsons$^{\\rm 34}$ , U. Parzefall$^{\\rm 48}$ , S. Pashapour$^{\\rm 54}$ , E. Pasqualucci$^{\\rm 133a}$ , S. Passaggio$^{\\rm 50a}$ , A. Passeri$^{\\rm 135a}$ , F. Pastore$^{\\rm 135a,135b}$ , Fr.",
"Pastore$^{\\rm 77}$ , G. Pásztor $^{\\rm 49}$$^{,aa}$ , S. Pataraia$^{\\rm 176}$ , N. Patel$^{\\rm 151}$ , J.R. Pater$^{\\rm 83}$ , S. Patricelli$^{\\rm 103a,103b}$ , T. Pauly$^{\\rm 29}$ , M. Pecsy$^{\\rm 145a}$ , M.I.",
"Pedraza Morales$^{\\rm 174}$ , S.V.",
"Peleganchuk$^{\\rm 108}$ , D. Pelikan$^{\\rm 167}$ , H. Peng$^{\\rm 32b}$ , B. Penning$^{\\rm 30}$ , A. Penson$^{\\rm 34}$ , J. Penwell$^{\\rm 61}$ , M. Perantoni$^{\\rm 23a}$ , K. Perez$^{\\rm 34}$$^{,ab}$ , T. Perez Cavalcanti$^{\\rm 41}$ , E. Perez Codina$^{\\rm 160a}$ , M.T.",
"Pérez García-Estañ$^{\\rm 168}$ , V. Perez Reale$^{\\rm 34}$ , L. Perini$^{\\rm 90a,90b}$ , H. Pernegger$^{\\rm 29}$ , R. Perrino$^{\\rm 73a}$ , P. Perrodo$^{\\rm 4}$ , S. Persembe$^{\\rm 3a}$ , V.D.",
"Peshekhonov$^{\\rm 65}$ , K. Peters$^{\\rm 29}$ , B.A.",
"Petersen$^{\\rm 29}$ , J. Petersen$^{\\rm 29}$ , T.C.",
"Petersen$^{\\rm 35}$ , E. Petit$^{\\rm 4}$ , A. Petridis$^{\\rm 155}$ , C. Petridou$^{\\rm 155}$ , E. Petrolo$^{\\rm 133a}$ , F. Petrucci$^{\\rm 135a,135b}$ , D. Petschull$^{\\rm 41}$ , M. Petteni$^{\\rm 143}$ , R. Pezoa$^{\\rm 31b}$ , A. Phan$^{\\rm 87}$ , P.W.",
"Phillips$^{\\rm 130}$ , G. Piacquadio$^{\\rm 29}$ , A. Picazio$^{\\rm 49}$ , E. Piccaro$^{\\rm 76}$ , M. Piccinini$^{\\rm 19a,19b}$ , S.M.",
"Piec$^{\\rm 41}$ , R. Piegaia$^{\\rm 26}$ , D.T.",
"Pignotti$^{\\rm 110}$ , J.E.",
"Pilcher$^{\\rm 30}$ , A.D. Pilkington$^{\\rm 83}$ , J. Pina$^{\\rm 125a}$$^{,b}$ , M. Pinamonti$^{\\rm 165a,165c}$ , A. Pinder$^{\\rm 119}$ , J.L.",
"Pinfold$^{\\rm 2}$ , J. Ping$^{\\rm 32c}$ , B. Pinto$^{\\rm 125a}$ , C. Pizio$^{\\rm 90a,90b}$ , R. Placakyte$^{\\rm 41}$ , M. Plamondon$^{\\rm 170}$ , M.-A.",
"Pleier$^{\\rm 24}$ , A.V.",
"Pleskach$^{\\rm 129}$ , E. Plotnikova$^{\\rm 65}$ , A. Poblaguev$^{\\rm 24}$ , S. Poddar$^{\\rm 58a}$ , F. Podlyski$^{\\rm 33}$ , L. Poggioli$^{\\rm 116}$ , T. Poghosyan$^{\\rm 20}$ , M. Pohl$^{\\rm 49}$ , F. Polci$^{\\rm 55}$ , G. Polesello$^{\\rm 120a}$ , A. Policicchio$^{\\rm 36a,36b}$ , A. Polini$^{\\rm 19a}$ , J. Poll$^{\\rm 76}$ , V. Polychronakos$^{\\rm 24}$ , D.M.",
"Pomarede$^{\\rm 137}$ , D. Pomeroy$^{\\rm 22}$ , K. Pommès$^{\\rm 29}$ , L. Pontecorvo$^{\\rm 133a}$ , B.G.",
"Pope$^{\\rm 89}$ , G.A.",
"Popeneciu$^{\\rm 25a}$ , D.S.",
"Popovic$^{\\rm 12a}$ , A. Poppleton$^{\\rm 29}$ , X. Portell Bueso$^{\\rm 29}$ , C. Posch$^{\\rm 21}$ , G.E.",
"Pospelov$^{\\rm 100}$ , S. Pospisil$^{\\rm 128}$ , I.N.",
"Potrap$^{\\rm 100}$ , C.J.",
"Potter$^{\\rm 150}$ , C.T.",
"Potter$^{\\rm 115}$ , G. Poulard$^{\\rm 29}$ , J. Poveda$^{\\rm 174}$ , V. Pozdnyakov$^{\\rm 65}$ , R. Prabhu$^{\\rm 78}$ , P. Pralavorio$^{\\rm 84}$ , A. Pranko$^{\\rm 14}$ , S. Prasad$^{\\rm 29}$ , R. Pravahan$^{\\rm 24}$ , S. Prell$^{\\rm 64}$ , K. Pretzl$^{\\rm 16}$ , L. Pribyl$^{\\rm 29}$ , D. Price$^{\\rm 61}$ , J. Price$^{\\rm 74}$ , L.E.",
"Price$^{\\rm 5}$ , M.J. Price$^{\\rm 29}$ , D. Prieur$^{\\rm 124}$ , M. Primavera$^{\\rm 73a}$ , K. Prokofiev$^{\\rm 109}$ , F. Prokoshin$^{\\rm 31b}$ , S. Protopopescu$^{\\rm 24}$ , J. Proudfoot$^{\\rm 5}$ , X. Prudent$^{\\rm 43}$ , M. Przybycien$^{\\rm 37}$ , H. Przysiezniak$^{\\rm 4}$ , S. Psoroulas$^{\\rm 20}$ , E. Ptacek$^{\\rm 115}$ , E. Pueschel$^{\\rm 85}$ , J. Purdham$^{\\rm 88}$ , M. Purohit$^{\\rm 24}$$^{,z}$ , P. Puzo$^{\\rm 116}$ , Y. Pylypchenko$^{\\rm 63}$ , J. Qian$^{\\rm 88}$ , Z. Qian$^{\\rm 84}$ , Z. Qin$^{\\rm 41}$ , A. Quadt$^{\\rm 54}$ , D.R.",
"Quarrie$^{\\rm 14}$ , W.B.",
"Quayle$^{\\rm 174}$ , F. Quinonez$^{\\rm 31a}$ , M. Raas$^{\\rm 105}$ , V. Radescu$^{\\rm 41}$ , B. Radics$^{\\rm 20}$ , P. Radloff$^{\\rm 115}$ , T. Rador$^{\\rm 18a}$ , F. Ragusa$^{\\rm 90a,90b}$ , G. Rahal$^{\\rm 179}$ , A.M. Rahimi$^{\\rm 110}$ , D. Rahm$^{\\rm 24}$ , S. Rajagopalan$^{\\rm 24}$ , M. Rammensee$^{\\rm 48}$ , M. Rammes$^{\\rm 142}$ , A.S. Randle-Conde$^{\\rm 39}$ , K. Randrianarivony$^{\\rm 28}$ , P.N.",
"Ratoff$^{\\rm 72}$ , F. Rauscher$^{\\rm 99}$ , T.C.",
"Rave$^{\\rm 48}$ , M. Raymond$^{\\rm 29}$ , A.L.",
"Read$^{\\rm 118}$ , D.M.",
"Rebuzzi$^{\\rm 120a,120b}$ , A. Redelbach$^{\\rm 175}$ , G. Redlinger$^{\\rm 24}$ , R. Reece$^{\\rm 121}$ , K. Reeves$^{\\rm 40}$ , A. Reichold$^{\\rm 106}$ , E. Reinherz-Aronis$^{\\rm 154}$ , A. Reinsch$^{\\rm 115}$ , I. Reisinger$^{\\rm 42}$ , C. Rembser$^{\\rm 29}$ , Z.L.",
"Ren$^{\\rm 152}$ , A. Renaud$^{\\rm 116}$ , M. Rescigno$^{\\rm 133a}$ , S. Resconi$^{\\rm 90a}$ , B. Resende$^{\\rm 137}$ , P. Reznicek$^{\\rm 99}$ , R. Rezvani$^{\\rm 159}$ , A. Richards$^{\\rm 78}$ , R. Richter$^{\\rm 100}$ , E. Richter-Was$^{\\rm 4}$$^{,ac}$ , M. Ridel$^{\\rm 79}$ , M. Rijpstra$^{\\rm 106}$ , M. Rijssenbeek$^{\\rm 149}$ , A. Rimoldi$^{\\rm 120a,120b}$ , L. Rinaldi$^{\\rm 19a}$ , R.R.",
"Rios$^{\\rm 39}$ , I. Riu$^{\\rm 11}$ , G. Rivoltella$^{\\rm 90a,90b}$ , F. Rizatdinova$^{\\rm 113}$ , E. Rizvi$^{\\rm 76}$ , S.H.",
"Robertson$^{\\rm 86}$$^{,j}$ , A. Robichaud-Veronneau$^{\\rm 119}$ , D. Robinson$^{\\rm 27}$ , J.E.M.",
"Robinson$^{\\rm 78}$ , A. Robson$^{\\rm 53}$ , J.G.",
"Rocha de Lima$^{\\rm 107}$ , C. Roda$^{\\rm 123a,123b}$ , D. Roda Dos Santos$^{\\rm 29}$ , D. Rodriguez$^{\\rm 163}$ , A. Roe$^{\\rm 54}$ , S. Roe$^{\\rm 29}$ , O. Røhne$^{\\rm 118}$ , V. Rojo$^{\\rm 1}$ , S. Rolli$^{\\rm 162}$ , A. Romaniouk$^{\\rm 97}$ , M. Romano$^{\\rm 19a,19b}$ , V.M.",
"Romanov$^{\\rm 65}$ , G. Romeo$^{\\rm 26}$ , E. Romero Adam$^{\\rm 168}$ , L. Roos$^{\\rm 79}$ , E. Ros$^{\\rm 168}$ , S. Rosati$^{\\rm 133a}$ , K. Rosbach$^{\\rm 49}$ , A. Rose$^{\\rm 150}$ , M. Rose$^{\\rm 77}$ , G.A.",
"Rosenbaum$^{\\rm 159}$ , E.I.",
"Rosenberg$^{\\rm 64}$ , P.L.",
"Rosendahl$^{\\rm 13}$ , O. Rosenthal$^{\\rm 142}$ , L. Rosselet$^{\\rm 49}$ , V. Rossetti$^{\\rm 11}$ , E. Rossi$^{\\rm 133a,133b}$ , L.P. Rossi$^{\\rm 50a}$ , M. Rotaru$^{\\rm 25a}$ , I. Roth$^{\\rm 173}$ , J. Rothberg$^{\\rm 139}$ , D. Rousseau$^{\\rm 116}$ , C.R.",
"Royon$^{\\rm 137}$ , A. Rozanov$^{\\rm 84}$ , Y. Rozen$^{\\rm 153}$ , X. Ruan$^{\\rm 32a}$$^{,ad}$ , F. Rubbo$^{\\rm 11}$ , I. Rubinskiy$^{\\rm 41}$ , B. Ruckert$^{\\rm 99}$ , N. Ruckstuhl$^{\\rm 106}$ , V.I.",
"Rud$^{\\rm 98}$ , C. Rudolph$^{\\rm 43}$ , G. Rudolph$^{\\rm 62}$ , F. Rühr$^{\\rm 6}$ , F. Ruggieri$^{\\rm 135a,135b}$ , A. Ruiz-Martinez$^{\\rm 64}$ , V. Rumiantsev$^{\\rm 92}$$^{,*}$ , L. Rumyantsev$^{\\rm 65}$ , K. Runge$^{\\rm 48}$ , Z. Rurikova$^{\\rm 48}$ , N.A.",
"Rusakovich$^{\\rm 65}$ , J.P. Rutherfoord$^{\\rm 6}$ , C. Ruwiedel$^{\\rm 14}$ , P. Ruzicka$^{\\rm 126}$ , Y.F.",
"Ryabov$^{\\rm 122}$ , V. Ryadovikov$^{\\rm 129}$ , P. Ryan$^{\\rm 89}$ , M. Rybar$^{\\rm 127}$ , G. Rybkin$^{\\rm 116}$ , N.C. Ryder$^{\\rm 119}$ , S. Rzaeva$^{\\rm 10}$ , A.F.",
"Saavedra$^{\\rm 151}$ , I. Sadeh$^{\\rm 154}$ , H.F-W. Sadrozinski$^{\\rm 138}$ , R. Sadykov$^{\\rm 65}$ , F. Safai Tehrani$^{\\rm 133a}$ , H. Sakamoto$^{\\rm 156}$ , G. Salamanna$^{\\rm 76}$ , A. Salamon$^{\\rm 134a}$ , M. Saleem$^{\\rm 112}$ , D. Salek$^{\\rm 29}$ , D. Salihagic$^{\\rm 100}$ , A. Salnikov$^{\\rm 144}$ , J. Salt$^{\\rm 168}$ , B.M.",
"Salvachua Ferrando$^{\\rm 5}$ , D. Salvatore$^{\\rm 36a,36b}$ , F. Salvatore$^{\\rm 150}$ , A. Salvucci$^{\\rm 105}$ , A. Salzburger$^{\\rm 29}$ , D. Sampsonidis$^{\\rm 155}$ , B.H.",
"Samset$^{\\rm 118}$ , A. Sanchez$^{\\rm 103a,103b}$ , V. Sanchez Martinez$^{\\rm 168}$ , H. Sandaker$^{\\rm 13}$ , H.G.",
"Sander$^{\\rm 82}$ , M.P.",
"Sanders$^{\\rm 99}$ , M. Sandhoff$^{\\rm 176}$ , T. Sandoval$^{\\rm 27}$ , C. Sandoval $^{\\rm 163}$ , R. Sandstroem$^{\\rm 100}$ , S. Sandvoss$^{\\rm 176}$ , D.P.C.",
"Sankey$^{\\rm 130}$ , A. Sansoni$^{\\rm 47}$ , C. Santamarina Rios$^{\\rm 86}$ , C. Santoni$^{\\rm 33}$ , R. Santonico$^{\\rm 134a,134b}$ , H. Santos$^{\\rm 125a}$ , J.G.",
"Saraiva$^{\\rm 125a}$ , T. Sarangi$^{\\rm 174}$ , E. Sarkisyan-Grinbaum$^{\\rm 7}$ , F. Sarri$^{\\rm 123a,123b}$ , G. Sartisohn$^{\\rm 176}$ , O. Sasaki$^{\\rm 66}$ , N. Sasao$^{\\rm 68}$ , I. Satsounkevitch$^{\\rm 91}$ , G. Sauvage$^{\\rm 4}$ , E. Sauvan$^{\\rm 4}$ , J.B. Sauvan$^{\\rm 116}$ , P. Savard$^{\\rm 159}$$^{,d}$ , V. Savinov$^{\\rm 124}$ , D.O.",
"Savu$^{\\rm 29}$ , L. Sawyer$^{\\rm 24}$$^{,l}$ , D.H. Saxon$^{\\rm 53}$ , J. Saxon$^{\\rm 121}$ , L.P. Says$^{\\rm 33}$ , C. Sbarra$^{\\rm 19a}$ , A. Sbrizzi$^{\\rm 19a,19b}$ , O. Scallon$^{\\rm 94}$ , D.A.",
"Scannicchio$^{\\rm 164}$ , M. Scarcella$^{\\rm 151}$ , J. Schaarschmidt$^{\\rm 116}$ , P. Schacht$^{\\rm 100}$ , D. Schaefer$^{\\rm 121}$ , U. Schäfer$^{\\rm 82}$ , S. Schaepe$^{\\rm 20}$ , S. Schaetzel$^{\\rm 58b}$ , A.C. Schaffer$^{\\rm 116}$ , D. Schaile$^{\\rm 99}$ , R.D.",
"Schamberger$^{\\rm 149}$ , A.G. Schamov$^{\\rm 108}$ , V. Scharf$^{\\rm 58a}$ , V.A.",
"Schegelsky$^{\\rm 122}$ , D. Scheirich$^{\\rm 88}$ , M. Schernau$^{\\rm 164}$ , M.I.",
"Scherzer$^{\\rm 34}$ , C. Schiavi$^{\\rm 50a,50b}$ , J. Schieck$^{\\rm 99}$ , M. Schioppa$^{\\rm 36a,36b}$ , S. Schlenker$^{\\rm 29}$ , J.L.",
"Schlereth$^{\\rm 5}$ , E. Schmidt$^{\\rm 48}$ , K. Schmieden$^{\\rm 20}$ , C. Schmitt$^{\\rm 82}$ , S. Schmitt$^{\\rm 58b}$ , M. Schmitz$^{\\rm 20}$ , A. Schöning$^{\\rm 58b}$ , M. Schott$^{\\rm 29}$ , D. Schouten$^{\\rm 160a}$ , J. Schovancova$^{\\rm 126}$ , M. Schram$^{\\rm 86}$ , C. Schroeder$^{\\rm 82}$ , N. Schroer$^{\\rm 58c}$ , G. Schuler$^{\\rm 29}$ , M.J. Schultens$^{\\rm 20}$ , J. Schultes$^{\\rm 176}$ , H.-C. Schultz-Coulon$^{\\rm 58a}$ , H. Schulz$^{\\rm 15}$ , J.W.",
"Schumacher$^{\\rm 20}$ , M. Schumacher$^{\\rm 48}$ , B.A.",
"Schumm$^{\\rm 138}$ , Ph.",
"Schune$^{\\rm 137}$ , C. Schwanenberger$^{\\rm 83}$ , A. Schwartzman$^{\\rm 144}$ , Ph.",
"Schwemling$^{\\rm 79}$ , R. Schwienhorst$^{\\rm 89}$ , R. Schwierz$^{\\rm 43}$ , J. Schwindling$^{\\rm 137}$ , T. Schwindt$^{\\rm 20}$ , M. Schwoerer$^{\\rm 4}$ , G. Sciolla$^{\\rm 22}$ , W.G.",
"Scott$^{\\rm 130}$ , J. Searcy$^{\\rm 115}$ , G. Sedov$^{\\rm 41}$ , E. Sedykh$^{\\rm 122}$ , E. Segura$^{\\rm 11}$ , S.C. Seidel$^{\\rm 104}$ , A. Seiden$^{\\rm 138}$ , F. Seifert$^{\\rm 43}$ , J.M.",
"Seixas$^{\\rm 23a}$ , G. Sekhniaidze$^{\\rm 103a}$ , S.J.",
"Sekula$^{\\rm 39}$ , K.E.",
"Selbach$^{\\rm 45}$ , D.M.",
"Seliverstov$^{\\rm 122}$ , B. Sellden$^{\\rm 147a}$ , G. Sellers$^{\\rm 74}$ , M. Seman$^{\\rm 145b}$ , N. Semprini-Cesari$^{\\rm 19a,19b}$ , C. Serfon$^{\\rm 99}$ , L. Serin$^{\\rm 116}$ , L. Serkin$^{\\rm 54}$ , R. Seuster$^{\\rm 100}$ , H. Severini$^{\\rm 112}$ , M.E.",
"Sevior$^{\\rm 87}$ , A. Sfyrla$^{\\rm 29}$ , E. Shabalina$^{\\rm 54}$ , M. Shamim$^{\\rm 115}$ , L.Y.",
"Shan$^{\\rm 32a}$ , J.T.",
"Shank$^{\\rm 21}$ , Q.T.",
"Shao$^{\\rm 87}$ , M. Shapiro$^{\\rm 14}$ , P.B.",
"Shatalov$^{\\rm 96}$ , L. Shaver$^{\\rm 6}$ , K. Shaw$^{\\rm 165a,165c}$ , D. Sherman$^{\\rm 177}$ , P. Sherwood$^{\\rm 78}$ , A. Shibata$^{\\rm 109}$ , H. Shichi$^{\\rm 102}$ , S. Shimizu$^{\\rm 29}$ , M. Shimojima$^{\\rm 101}$ , T. Shin$^{\\rm 56}$ , M. Shiyakova$^{\\rm 65}$ , A. Shmeleva$^{\\rm 95}$ , M.J. Shochet$^{\\rm 30}$ , D. Short$^{\\rm 119}$ , S. Shrestha$^{\\rm 64}$ , E. Shulga$^{\\rm 97}$ , M.A.",
"Shupe$^{\\rm 6}$ , P. Sicho$^{\\rm 126}$ , A. Sidoti$^{\\rm 133a}$ , F. Siegert$^{\\rm 48}$ , Dj.",
"Sijacki$^{\\rm 12a}$ , O. Silbert$^{\\rm 173}$ , J. Silva$^{\\rm 125a}$ , Y. Silver$^{\\rm 154}$ , D. Silverstein$^{\\rm 144}$ , S.B.",
"Silverstein$^{\\rm 147a}$ , V. Simak$^{\\rm 128}$ , O. Simard$^{\\rm 137}$ , Lj.",
"Simic$^{\\rm 12a}$ , S. Simion$^{\\rm 116}$ , B. Simmons$^{\\rm 78}$ , R. Simoniello$^{\\rm 90a,90b}$ , M. Simonyan$^{\\rm 35}$ , P. Sinervo$^{\\rm 159}$ , N.B.",
"Sinev$^{\\rm 115}$ , V. Sipica$^{\\rm 142}$ , G. Siragusa$^{\\rm 175}$ , A. Sircar$^{\\rm 24}$ , A.N.",
"Sisakyan$^{\\rm 65}$ , S.Yu.",
"Sivoklokov$^{\\rm 98}$ , J. Sjölin$^{\\rm 147a,147b}$ , T.B.",
"Sjursen$^{\\rm 13}$ , L.A. Skinnari$^{\\rm 14}$ , H.P.",
"Skottowe$^{\\rm 57}$ , K. Skovpen$^{\\rm 108}$ , P. Skubic$^{\\rm 112}$ , N. Skvorodnev$^{\\rm 22}$ , M. Slater$^{\\rm 17}$ , T. Slavicek$^{\\rm 128}$ , K. Sliwa$^{\\rm 162}$ , J. Sloper$^{\\rm 29}$ , V. Smakhtin$^{\\rm 173}$ , B.H.",
"Smart$^{\\rm 45}$ , S.Yu.",
"Smirnov$^{\\rm 97}$ , Y. Smirnov$^{\\rm 97}$ , L.N.",
"Smirnova$^{\\rm 98}$ , O. Smirnova$^{\\rm 80}$ , B.C.",
"Smith$^{\\rm 57}$ , D. Smith$^{\\rm 144}$ , K.M.",
"Smith$^{\\rm 53}$ , M. Smizanska$^{\\rm 72}$ , K. Smolek$^{\\rm 128}$ , A.A. Snesarev$^{\\rm 95}$ , S.W.",
"Snow$^{\\rm 83}$ , J. Snow$^{\\rm 112}$ , S. Snyder$^{\\rm 24}$ , R. Sobie$^{\\rm 170}$$^{,j}$ , J. Sodomka$^{\\rm 128}$ , A. Soffer$^{\\rm 154}$ , C.A.",
"Solans$^{\\rm 168}$ , M. Solar$^{\\rm 128}$ , J. Solc$^{\\rm 128}$ , E. Soldatov$^{\\rm 97}$ , U. Soldevila$^{\\rm 168}$ , E. Solfaroli Camillocci$^{\\rm 133a,133b}$ , A.A. Solodkov$^{\\rm 129}$ , O.V.",
"Solovyanov$^{\\rm 129}$ , N. Soni$^{\\rm 2}$ , V. Sopko$^{\\rm 128}$ , B. Sopko$^{\\rm 128}$ , M. Sosebee$^{\\rm 7}$ , R. Soualah$^{\\rm 165a,165c}$ , A. Soukharev$^{\\rm 108}$ , S. Spagnolo$^{\\rm 73a,73b}$ , F. Spanò$^{\\rm 77}$ , R. Spighi$^{\\rm 19a}$ , G. Spigo$^{\\rm 29}$ , F. Spila$^{\\rm 133a,133b}$ , R. Spiwoks$^{\\rm 29}$ , M. Spousta$^{\\rm 127}$ , T. Spreitzer$^{\\rm 159}$ , B. Spurlock$^{\\rm 7}$ , R.D. St.",
"Denis$^{\\rm 53}$ , J. Stahlman$^{\\rm 121}$ , R. Stamen$^{\\rm 58a}$ , E. Stanecka$^{\\rm 38}$ , R.W.",
"Stanek$^{\\rm 5}$ , C. Stanescu$^{\\rm 135a}$ , M. Stanescu-Bellu$^{\\rm 41}$ , S. Stapnes$^{\\rm 118}$ , E.A.",
"Starchenko$^{\\rm 129}$ , J. Stark$^{\\rm 55}$ , P. Staroba$^{\\rm 126}$ , P. Starovoitov$^{\\rm 41}$ , A. Staude$^{\\rm 99}$ , P. Stavina$^{\\rm 145a}$ , G. Steele$^{\\rm 53}$ , P. Steinbach$^{\\rm 43}$ , P. Steinberg$^{\\rm 24}$ , I. Stekl$^{\\rm 128}$ , B. Stelzer$^{\\rm 143}$ , H.J.",
"Stelzer$^{\\rm 89}$ , O. Stelzer-Chilton$^{\\rm 160a}$ , H. Stenzel$^{\\rm 52}$ , S. Stern$^{\\rm 100}$ , K. Stevenson$^{\\rm 76}$ , G.A.",
"Stewart$^{\\rm 29}$ , J.A.",
"Stillings$^{\\rm 20}$ , M.C.",
"Stockton$^{\\rm 86}$ , K. Stoerig$^{\\rm 48}$ , G. Stoicea$^{\\rm 25a}$ , S. Stonjek$^{\\rm 100}$ , P. Strachota$^{\\rm 127}$ , A.R.",
"Stradling$^{\\rm 7}$ , A. Straessner$^{\\rm 43}$ , J. Strandberg$^{\\rm 148}$ , S. Strandberg$^{\\rm 147a,147b}$ , A. Strandlie$^{\\rm 118}$ , M. Strang$^{\\rm 110}$ , E. Strauss$^{\\rm 144}$ , M. Strauss$^{\\rm 112}$ , P. Strizenec$^{\\rm 145b}$ , R. Ströhmer$^{\\rm 175}$ , D.M.",
"Strom$^{\\rm 115}$ , J.A.",
"Strong$^{\\rm 77}$$^{,*}$ , R. Stroynowski$^{\\rm 39}$ , J. Strube$^{\\rm 130}$ , B. Stugu$^{\\rm 13}$ , I. Stumer$^{\\rm 24}$$^{,*}$ , J. Stupak$^{\\rm 149}$ , P. Sturm$^{\\rm 176}$ , N.A.",
"Styles$^{\\rm 41}$ , D.A.",
"Soh$^{\\rm 152}$$^{,u}$ , D. Su$^{\\rm 144}$ , HS.",
"Subramania$^{\\rm 2}$ , A. Succurro$^{\\rm 11}$ , Y. Sugaya$^{\\rm 117}$ , T. Sugimoto$^{\\rm 102}$ , C. Suhr$^{\\rm 107}$ , K. Suita$^{\\rm 67}$ , M. Suk$^{\\rm 127}$ , V.V.",
"Sulin$^{\\rm 95}$ , S. Sultansoy$^{\\rm 3d}$ , T. Sumida$^{\\rm 68}$ , X. Sun$^{\\rm 55}$ , J.E.",
"Sundermann$^{\\rm 48}$ , K. Suruliz$^{\\rm 140}$ , S. Sushkov$^{\\rm 11}$ , G. Susinno$^{\\rm 36a,36b}$ , M.R.",
"Sutton$^{\\rm 150}$ , Y. Suzuki$^{\\rm 66}$ , Y. Suzuki$^{\\rm 67}$ , M. Svatos$^{\\rm 126}$ , Yu.M.",
"Sviridov$^{\\rm 129}$ , S. Swedish$^{\\rm 169}$ , I. Sykora$^{\\rm 145a}$ , T. Sykora$^{\\rm 127}$ , B. Szeless$^{\\rm 29}$ , J. Sánchez$^{\\rm 168}$ , D. Ta$^{\\rm 106}$ , K. Tackmann$^{\\rm 41}$ , A. Taffard$^{\\rm 164}$ , R. Tafirout$^{\\rm 160a}$ , N. Taiblum$^{\\rm 154}$ , Y. Takahashi$^{\\rm 102}$ , H. Takai$^{\\rm 24}$ , R. Takashima$^{\\rm 69}$ , H. Takeda$^{\\rm 67}$ , T. Takeshita$^{\\rm 141}$ , Y. Takubo$^{\\rm 66}$ , M. Talby$^{\\rm 84}$ , A. Talyshev$^{\\rm 108}$$^{,f}$ , M.C.",
"Tamsett$^{\\rm 24}$ , J. Tanaka$^{\\rm 156}$ , R. Tanaka$^{\\rm 116}$ , S. Tanaka$^{\\rm 132}$ , S. Tanaka$^{\\rm 66}$ , Y. Tanaka$^{\\rm 101}$ , A.J.",
"Tanasijczuk$^{\\rm 143}$ , K. Tani$^{\\rm 67}$ , N. Tannoury$^{\\rm 84}$ , G.P.",
"Tappern$^{\\rm 29}$ , S. Tapprogge$^{\\rm 82}$ , D. Tardif$^{\\rm 159}$ , S. Tarem$^{\\rm 153}$ , F. Tarrade$^{\\rm 28}$ , G.F. Tartarelli$^{\\rm 90a}$ , P. Tas$^{\\rm 127}$ , M. Tasevsky$^{\\rm 126}$ , E. Tassi$^{\\rm 36a,36b}$ , M. Tatarkhanov$^{\\rm 14}$ , Y. Tayalati$^{\\rm 136d}$ , C. Taylor$^{\\rm 78}$ , F.E.",
"Taylor$^{\\rm 93}$ , G.N.",
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"Thompson$^{\\rm 34}$ , P.D.",
"Thompson$^{\\rm 17}$ , P.D.",
"Thompson$^{\\rm 159}$ , A.S. Thompson$^{\\rm 53}$ , L.A. Thomsen$^{\\rm 35}$ , E. Thomson$^{\\rm 121}$ , M. Thomson$^{\\rm 27}$ , R.P.",
"Thun$^{\\rm 88}$ , F. Tian$^{\\rm 34}$ , M.J. Tibbetts$^{\\rm 14}$ , T. Tic$^{\\rm 126}$ , V.O.",
"Tikhomirov$^{\\rm 95}$ , Y.A.",
"Tikhonov$^{\\rm 108}$$^{,f}$ , S Timoshenko$^{\\rm 97}$ , P. Tipton$^{\\rm 177}$ , F.J. Tique Aires Viegas$^{\\rm 29}$ , S. Tisserant$^{\\rm 84}$ , B. Toczek$^{\\rm 37}$ , T. Todorov$^{\\rm 4}$ , S. Todorova-Nova$^{\\rm 162}$ , B. Toggerson$^{\\rm 164}$ , J. Tojo$^{\\rm 70}$ , S. Tokár$^{\\rm 145a}$ , K. Tokunaga$^{\\rm 67}$ , K. Tokushuku$^{\\rm 66}$ , K. Tollefson$^{\\rm 89}$ , M. Tomoto$^{\\rm 102}$ , L. Tompkins$^{\\rm 30}$ , K. Toms$^{\\rm 104}$ , G. Tong$^{\\rm 32a}$ , A. Tonoyan$^{\\rm 13}$ , C. Topfel$^{\\rm 16}$ , N.D. Topilin$^{\\rm 65}$ , I. Torchiani$^{\\rm 29}$ , E. Torrence$^{\\rm 115}$ , H. Torres$^{\\rm 79}$ , E. Torró Pastor$^{\\rm 168}$ , J. Toth$^{\\rm 84}$$^{,aa}$ , F. Touchard$^{\\rm 84}$ , D.R.",
"Tovey$^{\\rm 140}$ , T. Trefzger$^{\\rm 175}$ , L. Tremblet$^{\\rm 29}$ , A. Tricoli$^{\\rm 29}$ , I.M.",
"Trigger$^{\\rm 160a}$ , S. Trincaz-Duvoid$^{\\rm 79}$ , M.F.",
"Tripiana$^{\\rm 71}$ , W. Trischuk$^{\\rm 159}$ , A. Trivedi$^{\\rm 24}$$^{,z}$ , B. Trocmé$^{\\rm 55}$ , C. Troncon$^{\\rm 90a}$ , M. Trottier-McDonald$^{\\rm 143}$ , M. Trzebinski$^{\\rm 38}$ , A. Trzupek$^{\\rm 38}$ , C. Tsarouchas$^{\\rm 29}$ , J.C-L. Tseng$^{\\rm 119}$ , M. Tsiakiris$^{\\rm 106}$ , P.V.",
"Tsiareshka$^{\\rm 91}$ , D. Tsionou$^{\\rm 4}$$^{,ae}$ , G. Tsipolitis$^{\\rm 9}$ , V. Tsiskaridze$^{\\rm 48}$ , E.G.",
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"Tsukerman$^{\\rm 96}$ , V. Tsulaia$^{\\rm 14}$ , J.-W. Tsung$^{\\rm 20}$ , S. Tsuno$^{\\rm 66}$ , D. Tsybychev$^{\\rm 149}$ , A. Tua$^{\\rm 140}$ , A. Tudorache$^{\\rm 25a}$ , V. Tudorache$^{\\rm 25a}$ , J.M.",
"Tuggle$^{\\rm 30}$ , M. Turala$^{\\rm 38}$ , D. Turecek$^{\\rm 128}$ , I. Turk Cakir$^{\\rm 3e}$ , E. Turlay$^{\\rm 106}$ , R. Turra$^{\\rm 90a,90b}$ , P.M. Tuts$^{\\rm 34}$ , A. Tykhonov$^{\\rm 75}$ , M. Tylmad$^{\\rm 147a,147b}$ , M. Tyndel$^{\\rm 130}$ , G. Tzanakos$^{\\rm 8}$ , K. Uchida$^{\\rm 20}$ , I. Ueda$^{\\rm 156}$ , R. Ueno$^{\\rm 28}$ , M. Ugland$^{\\rm 13}$ , M. Uhlenbrock$^{\\rm 20}$ , M. Uhrmacher$^{\\rm 54}$ , F. Ukegawa$^{\\rm 161}$ , G. Unal$^{\\rm 29}$ , D.G.",
"Underwood$^{\\rm 5}$ , A. Undrus$^{\\rm 24}$ , G. Unel$^{\\rm 164}$ , Y. Unno$^{\\rm 66}$ , D. Urbaniec$^{\\rm 34}$ , G. Usai$^{\\rm 7}$ , M. Uslenghi$^{\\rm 120a,120b}$ , L. Vacavant$^{\\rm 84}$ , V. Vacek$^{\\rm 128}$ , B. Vachon$^{\\rm 86}$ , S. Vahsen$^{\\rm 14}$ , J. Valenta$^{\\rm 126}$ , P. Valente$^{\\rm 133a}$ , S. Valentinetti$^{\\rm 19a,19b}$ , S. Valkar$^{\\rm 127}$ , E. Valladolid Gallego$^{\\rm 168}$ , S. Vallecorsa$^{\\rm 153}$ , J.A.",
"Valls Ferrer$^{\\rm 168}$ , H. van der Graaf$^{\\rm 106}$ , E. van der Kraaij$^{\\rm 106}$ , R. Van Der Leeuw$^{\\rm 106}$ , E. van der Poel$^{\\rm 106}$ , D. van der Ster$^{\\rm 29}$ , N. van Eldik$^{\\rm 85}$ , P. van Gemmeren$^{\\rm 5}$ , Z. van Kesteren$^{\\rm 106}$ , I. van Vulpen$^{\\rm 106}$ , M. Vanadia$^{\\rm 100}$ , W. Vandelli$^{\\rm 29}$ , G. Vandoni$^{\\rm 29}$ , A. Vaniachine$^{\\rm 5}$ , P. Vankov$^{\\rm 41}$ , F. Vannucci$^{\\rm 79}$ , F. Varela Rodriguez$^{\\rm 29}$ , R. Vari$^{\\rm 133a}$ , T. Varol$^{\\rm 85}$ , D. Varouchas$^{\\rm 14}$ , A. Vartapetian$^{\\rm 7}$ , K.E.",
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"Vassilakopoulos$^{\\rm 56}$ , F. Vazeille$^{\\rm 33}$ , T. Vazquez Schroeder$^{\\rm 54}$ , G. Vegni$^{\\rm 90a,90b}$ , J.J. Veillet$^{\\rm 116}$ , C. Vellidis$^{\\rm 8}$ , F. Veloso$^{\\rm 125a}$ , R. Veness$^{\\rm 29}$ , S. Veneziano$^{\\rm 133a}$ , A. Ventura$^{\\rm 73a,73b}$ , D. Ventura$^{\\rm 139}$ , M. Venturi$^{\\rm 48}$ , N. Venturi$^{\\rm 159}$ , V. Vercesi$^{\\rm 120a}$ , M. Verducci$^{\\rm 139}$ , W. Verkerke$^{\\rm 106}$ , J.C. Vermeulen$^{\\rm 106}$ , A. Vest$^{\\rm 43}$ , M.C.",
"Vetterli$^{\\rm 143}$$^{,d}$ , I. Vichou$^{\\rm 166}$ , T. Vickey$^{\\rm 146b}$$^{,af}$ , O.E.",
"Vickey Boeriu$^{\\rm 146b}$ , G.H.A.",
"Viehhauser$^{\\rm 119}$ , S. Viel$^{\\rm 169}$ , M. Villa$^{\\rm 19a,19b}$ , M. Villaplana Perez$^{\\rm 168}$ , E. Vilucchi$^{\\rm 47}$ , M.G.",
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"Vinogradov$^{\\rm 65}$ , M. Virchaux$^{\\rm 137}$$^{,*}$ , J. Virzi$^{\\rm 14}$ , O. Vitells$^{\\rm 173}$ , M. Viti$^{\\rm 41}$ , I. Vivarelli$^{\\rm 48}$ , F. Vives Vaque$^{\\rm 2}$ , S. Vlachos$^{\\rm 9}$ , D. Vladoiu$^{\\rm 99}$ , M. Vlasak$^{\\rm 128}$ , N. Vlasov$^{\\rm 20}$ , A. Vogel$^{\\rm 20}$ , P. Vokac$^{\\rm 128}$ , G. Volpi$^{\\rm 47}$ , M. Volpi$^{\\rm 87}$ , G. Volpini$^{\\rm 90a}$ , H. von der Schmitt$^{\\rm 100}$ , J. von Loeben$^{\\rm 100}$ , H. von Radziewski$^{\\rm 48}$ , E. von Toerne$^{\\rm 20}$ , V. Vorobel$^{\\rm 127}$ , A.P.",
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"Voss$^{\\rm 176}$ , J.H.",
"Vossebeld$^{\\rm 74}$ , N. Vranjes$^{\\rm 137}$ , M. Vranjes Milosavljevic$^{\\rm 106}$ , V. Vrba$^{\\rm 126}$ , M. Vreeswijk$^{\\rm 106}$ , T. Vu Anh$^{\\rm 48}$ , R. Vuillermet$^{\\rm 29}$ , I. Vukotic$^{\\rm 116}$ , W. Wagner$^{\\rm 176}$ , P. Wagner$^{\\rm 121}$ , H. Wahlen$^{\\rm 176}$ , J. Wakabayashi$^{\\rm 102}$ , S. Walch$^{\\rm 88}$ , J. Walder$^{\\rm 72}$ , R. Walker$^{\\rm 99}$ , W. Walkowiak$^{\\rm 142}$ , R. Wall$^{\\rm 177}$ , P. Waller$^{\\rm 74}$ , C. Wang$^{\\rm 44}$ , H. Wang$^{\\rm 174}$ , H. Wang$^{\\rm 32b}$$^{,ag}$ , J. Wang$^{\\rm 152}$ , J. Wang$^{\\rm 55}$ , J.C. Wang$^{\\rm 139}$ , R. Wang$^{\\rm 104}$ , S.M.",
"Wang$^{\\rm 152}$ , T. Wang$^{\\rm 20}$ , A. Warburton$^{\\rm 86}$ , C.P.",
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"Watson$^{\\rm 151}$ , M.F.",
"Watson$^{\\rm 17}$ , G. Watts$^{\\rm 139}$ , S. Watts$^{\\rm 83}$ , A.T. Waugh$^{\\rm 151}$ , B.M.",
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"Weidberg$^{\\rm 119}$ , P. Weigell$^{\\rm 100}$ , J. Weingarten$^{\\rm 54}$ , C. Weiser$^{\\rm 48}$ , H. Wellenstein$^{\\rm 22}$ , P.S.",
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"Wheeler-Ellis$^{\\rm 164}$ , S.P.",
"Whitaker$^{\\rm 21}$ , A. White$^{\\rm 7}$ , M.J. White$^{\\rm 87}$ , S. White$^{\\rm 123a,123b}$ , S.R.",
"Whitehead$^{\\rm 119}$ , D. Whiteson$^{\\rm 164}$ , D. Whittington$^{\\rm 61}$ , F. Wicek$^{\\rm 116}$ , D. Wicke$^{\\rm 176}$ , F.J. Wickens$^{\\rm 130}$ , W. Wiedenmann$^{\\rm 174}$ , M. Wielers$^{\\rm 130}$ , P. Wienemann$^{\\rm 20}$ , C. Wiglesworth$^{\\rm 76}$ , L.A.M.",
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"Wijeratne$^{\\rm 78}$ , A. Wildauer$^{\\rm 168}$ , M.A.",
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"Wilson$^{\\rm 17}$ , M.G.",
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"Wolter$^{\\rm 38}$ , H. Wolters$^{\\rm 125a}$$^{,h}$ , W.C. Wong$^{\\rm 40}$ , G. Wooden$^{\\rm 88}$ , B.K.",
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"Ybeles Smit$^{\\rm 131}$ , J. Ye$^{\\rm 39}$ , S. Ye$^{\\rm 24}$ , M. Yilmaz$^{\\rm 3c}$ , R. Yoosoofmiya$^{\\rm 124}$ , K. Yorita$^{\\rm 172}$ , R. Yoshida$^{\\rm 5}$ , C. Young$^{\\rm 144}$ , C.J.",
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"Zhu$^{\\rm 32d}$ , H. Zhu$^{\\rm 41}$ , J. Zhu$^{\\rm 88}$ , Y. Zhu$^{\\rm 32b}$ , X. Zhuang$^{\\rm 99}$ , V. Zhuravlov$^{\\rm 100}$ , D. Zieminska$^{\\rm 61}$ , R. Zimmermann$^{\\rm 20}$ , S. Zimmermann$^{\\rm 20}$ , S. Zimmermann$^{\\rm 48}$ , M. Ziolkowski$^{\\rm 142}$ , R. Zitoun$^{\\rm 4}$ , L. Živković$^{\\rm 34}$ , V.V.",
"Zmouchko$^{\\rm 129}$$^{,*}$ , G. Zobernig$^{\\rm 174}$ , A. Zoccoli$^{\\rm 19a,19b}$ , M. zur Nedden$^{\\rm 15}$ , V. Zutshi$^{\\rm 107}$ , L. Zwalinski$^{\\rm 29}$ .",
"$^{1}$ University at Albany, Albany NY, United States of America $^{2}$ Department of Physics, University of Alberta, Edmonton AB, Canada $^{3}$ $^{(a)}$ Department of Physics, Ankara University, Ankara; $^{(b)}$ Department of Physics, Dumlupinar University, Kutahya; $^{(c)}$ Department of Physics, Gazi University, Ankara; $^{(d)}$ Division of Physics, TOBB University of Economics and Technology, Ankara; $^{(e)}$ Turkish Atomic Energy Authority, Ankara, Turkey $^{4}$ LAPP, CNRS/IN2P3 and Université de Savoie, Annecy-le-Vieux, France $^{5}$ High Energy Physics Division, Argonne National Laboratory, Argonne IL, United States of America $^{6}$ Department of Physics, University of Arizona, Tucson AZ, United States of America $^{7}$ Department of Physics, The University of Texas at Arlington, Arlington TX, United States of America $^{8}$ Physics Department, University of Athens, Athens, Greece $^{9}$ Physics Department, National Technical University of Athens, Zografou, Greece $^{10}$ Institute of Physics, Azerbaijan Academy of Sciences, Baku, Azerbaijan $^{11}$ Institut de Física d'Altes Energies and Departament de Física de la Universitat Autònoma de Barcelona and ICREA, Barcelona, Spain $^{12}$ $^{(a)}$ Institute of Physics, University of Belgrade, Belgrade; $^{(b)}$ Vinca Institute of Nuclear Sciences, University of Belgrade, Belgrade, Serbia $^{13}$ Department for Physics and Technology, University of Bergen, Bergen, Norway $^{14}$ Physics Division, Lawrence Berkeley National Laboratory and University of California, Berkeley CA, United States of America $^{15}$ Department of Physics, Humboldt University, Berlin, Germany $^{16}$ Albert Einstein Center for Fundamental Physics and Laboratory for High Energy Physics, University of Bern, Bern, Switzerland $^{17}$ School of Physics and Astronomy, University of Birmingham, Birmingham, United Kingdom $^{18}$ $^{(a)}$ Department of Physics, Bogazici University, Istanbul; $^{(b)}$ Division of Physics, Dogus University, Istanbul; $^{(c)}$ Department of Physics Engineering, Gaziantep University, Gaziantep; $^{(d)}$ Department of Physics, Istanbul Technical University, Istanbul, Turkey $^{19}$ $^{(a)}$ INFN Sezione di Bologna; $^{(b)}$ Dipartimento di Fisica, Università di Bologna, Bologna, Italy $^{20}$ Physikalisches Institut, University of Bonn, Bonn, Germany $^{21}$ Department of Physics, Boston University, Boston MA, United States of America $^{22}$ Department of Physics, Brandeis University, Waltham MA, United States of America $^{23}$ $^{(a)}$ Universidade Federal do Rio De Janeiro COPPE/EE/IF, Rio de Janeiro; $^{(b)}$ Federal University of Juiz de Fora (UFJF), Juiz de Fora; $^{(c)}$ Federal University of Sao Joao del Rei (UFSJ), Sao Joao del Rei; $^{(d)}$ Instituto de Fisica, Universidade de Sao Paulo, Sao Paulo, Brazil $^{24}$ Physics Department, Brookhaven National Laboratory, Upton NY, United States of America $^{25}$ $^{(a)}$ National Institute of Physics and Nuclear Engineering, Bucharest; $^{(b)}$ University Politehnica Bucharest, Bucharest; $^{(c)}$ West University in Timisoara, Timisoara, Romania $^{26}$ Departamento de Física, Universidad de Buenos Aires, Buenos Aires, Argentina $^{27}$ Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom $^{28}$ Department of Physics, Carleton University, Ottawa ON, Canada $^{29}$ CERN, Geneva, Switzerland $^{30}$ Enrico Fermi Institute, University of Chicago, Chicago IL, United States of America $^{31}$ $^{(a)}$ Departamento de Fisica, Pontificia Universidad Católica de Chile, Santiago; $^{(b)}$ Departamento de Física, Universidad Técnica Federico Santa María, Valparaíso, Chile $^{32}$ $^{(a)}$ Institute of High Energy Physics, Chinese Academy of Sciences, Beijing; $^{(b)}$ Department of Modern Physics, University of Science and Technology of China, Anhui; $^{(c)}$ Department of Physics, Nanjing University, Jiangsu; $^{(d)}$ School of Physics, Shandong University, Shandong, China $^{33}$ Laboratoire de Physique Corpusculaire, Clermont Université and Université Blaise Pascal and CNRS/IN2P3, Aubiere Cedex, France $^{34}$ Nevis Laboratory, Columbia University, Irvington NY, United States of America $^{35}$ Niels Bohr Institute, University of Copenhagen, Kobenhavn, Denmark $^{36}$ $^{(a)}$ INFN Gruppo Collegato di Cosenza; $^{(b)}$ Dipartimento di Fisica, Università della Calabria, Arcavata di Rende, Italy $^{37}$ AGH University of Science and Technology, Faculty of Physics and Applied Computer Science, Krakow, Poland $^{38}$ The Henryk Niewodniczanski Institute of Nuclear Physics, Polish Academy of Sciences, Krakow, Poland $^{39}$ Physics Department, Southern Methodist University, Dallas TX, United States of America $^{40}$ Physics Department, University of Texas at Dallas, Richardson TX, United States of America $^{41}$ DESY, Hamburg and Zeuthen, Germany $^{42}$ Institut für Experimentelle Physik IV, Technische Universität Dortmund, Dortmund, Germany $^{43}$ Institut für Kern- und Teilchenphysik, Technical University Dresden, Dresden, Germany $^{44}$ Department of Physics, Duke University, Durham NC, United States of America $^{45}$ SUPA - School of Physics and Astronomy, University of Edinburgh, Edinburgh, United Kingdom $^{46}$ Fachhochschule Wiener Neustadt, Johannes Gutenbergstrasse 3 2700 Wiener Neustadt, Austria $^{47}$ INFN Laboratori Nazionali di Frascati, Frascati, Italy $^{48}$ Fakultät für Mathematik und Physik, Albert-Ludwigs-Universität, Freiburg i.Br., Germany $^{49}$ Section de Physique, Université de Genève, Geneva, Switzerland $^{50}$ $^{(a)}$ INFN Sezione di Genova; $^{(b)}$ Dipartimento di Fisica, Università di Genova, Genova, Italy $^{51}$ $^{(a)}$ E.Andronikashvili Institute of Physics, Tbilisi State University, Tbilisi; $^{(b)}$ High Energy Physics Institute, Tbilisi State University, Tbilisi, Georgia $^{52}$ II Physikalisches Institut, Justus-Liebig-Universität Giessen, Giessen, Germany $^{53}$ SUPA - School of Physics and Astronomy, University of Glasgow, Glasgow, United Kingdom $^{54}$ II Physikalisches Institut, Georg-August-Universität, Göttingen, Germany $^{55}$ Laboratoire de Physique Subatomique et de Cosmologie, Université Joseph Fourier and CNRS/IN2P3 and Institut National Polytechnique de Grenoble, Grenoble, France $^{56}$ Department of Physics, Hampton University, Hampton VA, United States of America $^{57}$ Laboratory for Particle Physics and Cosmology, Harvard University, Cambridge MA, United States of America $^{58}$ $^{(a)}$ Kirchhoff-Institut für Physik, Ruprecht-Karls-Universität Heidelberg, Heidelberg; $^{(b)}$ Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg; $^{(c)}$ ZITI Institut für technische Informatik, Ruprecht-Karls-Universität Heidelberg, Mannheim, Germany $^{59}$ .",
"$^{60}$ Faculty of Applied Information Science, Hiroshima Institute of Technology, Hiroshima, Japan $^{61}$ Department of Physics, Indiana University, Bloomington IN, United States of America $^{62}$ Institut für Astro- und Teilchenphysik, Leopold-Franzens-Universität, Innsbruck, Austria $^{63}$ University of Iowa, Iowa City IA, United States of America $^{64}$ Department of Physics and Astronomy, Iowa State University, Ames IA, United States of America $^{65}$ Joint Institute for Nuclear Research, JINR Dubna, Dubna, Russia $^{66}$ KEK, High Energy Accelerator Research Organization, Tsukuba, Japan $^{67}$ Graduate School of Science, Kobe University, Kobe, Japan $^{68}$ Faculty of Science, Kyoto University, Kyoto, Japan $^{69}$ Kyoto University of Education, Kyoto, Japan $^{70}$ Department of Physics, Kyushu University, Fukuoka, Japan $^{71}$ Instituto de Física La Plata, Universidad Nacional de La Plata and CONICET, La Plata, Argentina $^{72}$ Physics Department, Lancaster University, Lancaster, United Kingdom $^{73}$ $^{(a)}$ INFN Sezione di Lecce; $^{(b)}$ Dipartimento di Fisica, Università del Salento, Lecce, Italy $^{74}$ Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom $^{75}$ Department of Physics, Jožef Stefan Institute and University of Ljubljana, Ljubljana, Slovenia $^{76}$ School of Physics and Astronomy, Queen Mary University of London, London, United Kingdom $^{77}$ Department of Physics, Royal Holloway University of London, Surrey, United Kingdom $^{78}$ Department of Physics and Astronomy, University College London, London, United Kingdom $^{79}$ Laboratoire de Physique Nucléaire et de Hautes Energies, UPMC and Université Paris-Diderot and CNRS/IN2P3, Paris, France $^{80}$ Fysiska institutionen, Lunds universitet, Lund, Sweden $^{81}$ Departamento de Fisica Teorica C-15, Universidad Autonoma de Madrid, Madrid, Spain $^{82}$ Institut für Physik, Universität Mainz, Mainz, Germany $^{83}$ School of Physics and Astronomy, University of Manchester, Manchester, United Kingdom $^{84}$ CPPM, Aix-Marseille Université and CNRS/IN2P3, Marseille, France $^{85}$ Department of Physics, University of Massachusetts, Amherst MA, United States of America $^{86}$ Department of Physics, McGill University, Montreal QC, Canada $^{87}$ School of Physics, University of Melbourne, Victoria, Australia $^{88}$ Department of Physics, The University of Michigan, Ann Arbor MI, United States of America $^{89}$ Department of Physics and Astronomy, Michigan State University, East Lansing MI, United States of America $^{90}$ $^{(a)}$ INFN Sezione di Milano; $^{(b)}$ Dipartimento di Fisica, Università di Milano, Milano, Italy $^{91}$ B.I.",
"Stepanov Institute of Physics, National Academy of Sciences of Belarus, Minsk, Republic of Belarus $^{92}$ National Scientific and Educational Centre for Particle and High Energy Physics, Minsk, Republic of Belarus $^{93}$ Department of Physics, Massachusetts Institute of Technology, Cambridge MA, United States of America $^{94}$ Group of Particle Physics, University of Montreal, Montreal QC, Canada $^{95}$ P.N.",
"Lebedev Institute of Physics, Academy of Sciences, Moscow, Russia $^{96}$ Institute for Theoretical and Experimental Physics (ITEP), Moscow, Russia $^{97}$ Moscow Engineering and Physics Institute (MEPhI), Moscow, Russia $^{98}$ Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University, Moscow, Russia $^{99}$ Fakultät für Physik, Ludwig-Maximilians-Universität München, München, Germany $^{100}$ Max-Planck-Institut für Physik (Werner-Heisenberg-Institut), München, Germany $^{101}$ Nagasaki Institute of Applied Science, Nagasaki, Japan $^{102}$ Graduate School of Science, Nagoya University, Nagoya, Japan $^{103}$ $^{(a)}$ INFN Sezione di Napoli; $^{(b)}$ Dipartimento di Scienze Fisiche, Università di Napoli, Napoli, Italy $^{104}$ Department of Physics and Astronomy, University of New Mexico, Albuquerque NM, United States of America $^{105}$ Institute for Mathematics, Astrophysics and Particle Physics, Radboud University Nijmegen/Nikhef, Nijmegen, Netherlands $^{106}$ Nikhef National Institute for Subatomic Physics and University of Amsterdam, Amsterdam, Netherlands $^{107}$ Department of Physics, Northern Illinois University, DeKalb IL, United States of America $^{108}$ Budker Institute of Nuclear Physics, SB RAS, Novosibirsk, Russia $^{109}$ Department of Physics, New York University, New York NY, United States of America $^{110}$ Ohio State University, Columbus OH, United States of America $^{111}$ Faculty of Science, Okayama University, Okayama, Japan $^{112}$ Homer L. Dodge Department of Physics and Astronomy, University of Oklahoma, Norman OK, United States of America $^{113}$ Department of Physics, Oklahoma State University, Stillwater OK, United States of America $^{114}$ Palacký University, RCPTM, Olomouc, Czech Republic $^{115}$ Center for High Energy Physics, University of Oregon, Eugene OR, United States of America $^{116}$ LAL, Univ.",
"Paris-Sud and CNRS/IN2P3, Orsay, France $^{117}$ Graduate School of Science, Osaka University, Osaka, Japan $^{118}$ Department of Physics, University of Oslo, Oslo, Norway $^{119}$ Department of Physics, Oxford University, Oxford, United Kingdom $^{120}$ $^{(a)}$ INFN Sezione di Pavia; $^{(b)}$ Dipartimento di Fisica, Università di Pavia, Pavia, Italy $^{121}$ Department of Physics, University of Pennsylvania, Philadelphia PA, United States of America $^{122}$ Petersburg Nuclear Physics Institute, Gatchina, Russia $^{123}$ $^{(a)}$ INFN Sezione di Pisa; $^{(b)}$ Dipartimento di Fisica E. Fermi, Università di Pisa, Pisa, Italy $^{124}$ Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh PA, United States of America $^{125}$ $^{(a)}$ Laboratorio de Instrumentacao e Fisica Experimental de Particulas - LIP, Lisboa, Portugal; $^{(b)}$ Departamento de Fisica Teorica y del Cosmos and CAFPE, Universidad de Granada, Granada, Spain $^{126}$ Institute of Physics, Academy of Sciences of the Czech Republic, Praha, Czech Republic $^{127}$ Faculty of Mathematics and Physics, Charles University in Prague, Praha, Czech Republic $^{128}$ Czech Technical University in Prague, Praha, Czech Republic $^{129}$ State Research Center Institute for High Energy Physics, Protvino, Russia $^{130}$ Particle Physics Department, Rutherford Appleton Laboratory, Didcot, United Kingdom $^{131}$ Physics Department, University of Regina, Regina SK, Canada $^{132}$ Ritsumeikan University, Kusatsu, Shiga, Japan $^{133}$ $^{(a)}$ INFN Sezione di Roma I; $^{(b)}$ Dipartimento di Fisica, Università La Sapienza, Roma, Italy $^{134}$ $^{(a)}$ INFN Sezione di Roma Tor Vergata; $^{(b)}$ Dipartimento di Fisica, Università di Roma Tor Vergata, Roma, Italy $^{135}$ $^{(a)}$ INFN Sezione di Roma Tre; $^{(b)}$ Dipartimento di Fisica, Università Roma Tre, Roma, Italy $^{136}$ $^{(a)}$ Faculté des Sciences Ain Chock, Réseau Universitaire de Physique des Hautes Energies - Université Hassan II, Casablanca; $^{(b)}$ Centre National de l'Energie des Sciences Techniques Nucleaires, Rabat; $^{(c)}$ Faculté des Sciences Semlalia, Université Cadi Ayyad, LPHEA-Marrakech; $^{(d)}$ Faculté des Sciences, Université Mohamed Premier and LPTPM, Oujda; $^{(e)}$ Faculté des Sciences, Université Mohammed V- Agdal, Rabat, Morocco $^{137}$ DSM/IRFU (Institut de Recherches sur les Lois Fondamentales de l'Univers), CEA Saclay (Commissariat a l'Energie Atomique), Gif-sur-Yvette, France $^{138}$ Santa Cruz Institute for Particle Physics, University of California Santa Cruz, Santa Cruz CA, United States of America $^{139}$ Department of Physics, University of Washington, Seattle WA, United States of America $^{140}$ Department of Physics and Astronomy, University of Sheffield, Sheffield, United Kingdom $^{141}$ Department of Physics, Shinshu University, Nagano, Japan $^{142}$ Fachbereich Physik, Universität Siegen, Siegen, Germany $^{143}$ Department of Physics, Simon Fraser University, Burnaby BC, Canada $^{144}$ SLAC National Accelerator Laboratory, Stanford CA, United States of America $^{145}$ $^{(a)}$ Faculty of Mathematics, Physics & Informatics, Comenius University, Bratislava; $^{(b)}$ Department of Subnuclear Physics, Institute of Experimental Physics of the Slovak Academy of Sciences, Kosice, Slovak Republic $^{146}$ $^{(a)}$ Department of Physics, University of Johannesburg, Johannesburg; $^{(b)}$ School of Physics, University of the Witwatersrand, Johannesburg, South Africa $^{147}$ $^{(a)}$ Department of Physics, Stockholm University; $^{(b)}$ The Oskar Klein Centre, Stockholm, Sweden $^{148}$ Physics Department, Royal Institute of Technology, Stockholm, Sweden $^{149}$ Departments of Physics & Astronomy and Chemistry, Stony Brook University, Stony Brook NY, United States of America $^{150}$ Department of Physics and Astronomy, University of Sussex, Brighton, United Kingdom $^{151}$ School of Physics, University of Sydney, Sydney, Australia $^{152}$ Institute of Physics, Academia Sinica, Taipei, Taiwan $^{153}$ Department of Physics, Technion: Israel Inst.",
"of Technology, Haifa, Israel $^{154}$ Raymond and Beverly Sackler School of Physics and Astronomy, Tel Aviv University, Tel Aviv, Israel $^{155}$ Department of Physics, Aristotle University of Thessaloniki, Thessaloniki, Greece $^{156}$ International Center for Elementary Particle Physics and Department of Physics, The University of Tokyo, Tokyo, Japan $^{157}$ Graduate School of Science and Technology, Tokyo Metropolitan University, Tokyo, Japan $^{158}$ Department of Physics, Tokyo Institute of Technology, Tokyo, Japan $^{159}$ Department of Physics, University of Toronto, Toronto ON, Canada $^{160}$ $^{(a)}$ TRIUMF, Vancouver BC; $^{(b)}$ Department of Physics and Astronomy, York University, Toronto ON, Canada $^{161}$ Institute of Pure and Applied Sciences, University of Tsukuba,1-1-1 Tennodai,Tsukuba, Ibaraki 305-8571, Japan $^{162}$ Science and Technology Center, Tufts University, Medford MA, United States of America $^{163}$ Centro de Investigaciones, Universidad Antonio Narino, Bogota, Colombia $^{164}$ Department of Physics and Astronomy, University of California Irvine, Irvine CA, United States of America $^{165}$ $^{(a)}$ INFN Gruppo Collegato di Udine; $^{(b)}$ ICTP, Trieste; $^{(c)}$ Dipartimento di Chimica, Fisica e Ambiente, Università di Udine, Udine, Italy $^{166}$ Department of Physics, University of Illinois, Urbana IL, United States of America $^{167}$ Department of Physics and Astronomy, University of Uppsala, Uppsala, Sweden $^{168}$ Instituto de Física Corpuscular (IFIC) and Departamento de Física Atómica, Molecular y Nuclear and Departamento de Ingeniería Electrónica and Instituto de Microelectrónica de Barcelona (IMB-CNM), University of Valencia and CSIC, Valencia, Spain $^{169}$ Department of Physics, University of British Columbia, Vancouver BC, Canada $^{170}$ Department of Physics and Astronomy, University of Victoria, Victoria BC, Canada $^{171}$ Department of Physics, University of Warwick, Coventry, United Kingdom $^{172}$ Waseda University, Tokyo, Japan $^{173}$ Department of Particle Physics, The Weizmann Institute of Science, Rehovot, Israel $^{174}$ Department of Physics, University of Wisconsin, Madison WI, United States of America $^{175}$ Fakultät für Physik und Astronomie, Julius-Maximilians-Universität, Würzburg, Germany $^{176}$ Fachbereich C Physik, Bergische Universität Wuppertal, Wuppertal, Germany $^{177}$ Department of Physics, Yale University, New Haven CT, United States of America $^{178}$ Yerevan Physics Institute, Yerevan, Armenia $^{179}$ Domaine scientifique de la Doua, Centre de Calcul CNRS/IN2P3, Villeurbanne Cedex, France $^{a}$ Also at Laboratorio de Instrumentacao e Fisica Experimental de Particulas - LIP, Lisboa, Portugal $^{b}$ Also at Faculdade de Ciencias and CFNUL, Universidade de Lisboa, Lisboa, Portugal $^{c}$ Also at Particle Physics Department, Rutherford Appleton Laboratory, Didcot, United Kingdom $^{d}$ Also at TRIUMF, Vancouver BC, Canada e Also at Department of Physics, California State University, Fresno CA, United States of America $^{f}$ Also at Novosibirsk State University, Novosibirsk, Russia $^{g}$ Also at Fermilab, Batavia IL, United States of America $^{h}$ Also at Department of Physics, University of Coimbra, Coimbra, Portugal $^{i}$ Also at Università di Napoli Parthenope, Napoli, Italy $^{j}$ Also at Institute of Particle Physics (IPP), Canada $^{k}$ Also at Department of Physics, Middle East Technical University, Ankara, Turkey $^{l}$ Also at Louisiana Tech University, Ruston LA, United States of America $^{m}$ Also at Department of Physics and Astronomy, University College London, London, United Kingdom $^{n}$ Also at Group of Particle Physics, University of Montreal, Montreal QC, Canada $^{o}$ Also at Department of Physics, University of Cape Town, Cape Town, South Africa $^{p}$ Also at Institute of Physics, Azerbaijan Academy of Sciences, Baku, Azerbaijan $^{q}$ Also at Institut für Experimentalphysik, Universität Hamburg, Hamburg, Germany $^{r}$ Also at Manhattan College, New York NY, United States of America $^{s}$ Also at School of Physics, Shandong University, Shandong, China $^{t}$ Also at CPPM, Aix-Marseille Université and CNRS/IN2P3, Marseille, France $^{u}$ Also at School of Physics and Engineering, Sun Yat-sen University, Guanzhou, China $^{v}$ Also at Academia Sinica Grid Computing, Institute of Physics, Academia Sinica, Taipei, Taiwan $^{w}$ Also at DSM/IRFU (Institut de Recherches sur les Lois Fondamentales de l'Univers), CEA Saclay (Commissariat a l'Energie Atomique), Gif-sur-Yvette, France $^{x}$ Also at Section de Physique, Université de Genève, Geneva, Switzerland $^{y}$ Also at Departamento de Fisica, Universidade de Minho, Braga, Portugal $^{z}$ Also at Department of Physics and Astronomy, University of South Carolina, Columbia SC, United States of America $^{aa}$ Also at Institute for Particle and Nuclear Physics, Wigner Research Centre for Physics, Budapest, Hungary $^{ab}$ Also at California Institute of Technology, Pasadena CA, United States of America $^{ac}$ Also at Institute of Physics, Jagiellonian University, Krakow, Poland $^{ad}$ Also at LAL, Univ.",
"Paris-Sud and CNRS/IN2P3, Orsay, France $^{ae}$ Also at Department of Physics and Astronomy, University of Sheffield, Sheffield, United Kingdom $^{af}$ Also at Department of Physics, Oxford University, Oxford, United Kingdom $^{ag}$ Also at Institute of Physics, Academia Sinica, Taipei, Taiwan $^{ah}$ Also at Department of Physics, The University of Michigan, Ann Arbor MI, United States of America $^{*}$ Deceased"
]
] | 1204.1265 |
[
[
"Prediction of Catastrophes: an experimental model"
],
[
"Abstract Catastrophes of all kinds can be roughly defined as short duration-large amplitude events following and followed by long periods of \"ripening\".",
"Major earthquakes surely belong to the class of 'catastrophic' events.",
"Because of the space-time scales involved, an experimental approach is often difficult, not to say impossible, however desirable it could be.",
"Described in this article is a \"laboratory\" setup that yields data of a type that is amenable to theoretical methods of prediction.",
"Observations are made of a critical slowing down in the noisy signal of a solder wire creeping under constant stress.",
"This effect is shown to be a fair signal of the forthcoming catastrophe in both of two dynamical models.",
"The first is an \"abstract\" model in which a time dependent quantity drifts slowly but makes quick jumps from time to time.",
"The second is a realistic physical model for the collective motion of dislocations (the Ananthakrishna set of equations for creep).",
"Hope thus exists that similar changes in the response to noise could forewarn catastrophes in other situations, where such precursor effects should manifest early enough."
],
[
"Introduction",
"Catastrophes as defined in the abstract are related to a class of phenomena sometimes called “relaxation” oscillations.",
"The observation of two widely separated time scales makes relaxation oscillations a good a priori subject of theoretical investigation, because one may suspect that their formation results from the existence of a (more or less hidden) small parameter.",
"This gives some hope of a “general” theory based upon the small size of the parameter.",
"To take an example, some major earthquakes lasting a few tens of seconds occur in the same general area about every hundred years.",
"They thus involve a ratio of `typical' times in the neighborhood of $10^{-9}$ , which is a very small number.",
"In the present study a laboratory experiment was devised which displays similar relaxation oscillations, but acting time-wise on a scale that is convenient for investigation.",
"Experimental observations are explained in light of a two-part dynamical bifurcation model of catastrophe.",
"Comprising the local form of a physical model shown to be valid when close to the catastrophe, the striking result is as follows.",
"In response to an external source of noise the two models (the local one and the physical one) predict fluctuations with a correlation time that increases before the catastrophe, just as is observed experimentally.",
"The important point is that this well known critical slowing down phenomenon occurs significantly before the transition and could be used to forewarn it.",
"We study the creeping of a soft metal under constant stress.",
"Accurate time records show that this creeping actually displays the following time dependent noisy component.",
"The wire typically lengthens slowly with background (non-thermal) noise, with sometimes a “large” sliding event, followed again by a noisy slow lengthening regime, etc.",
"Plastic deformation of solids is a complex phenomenon, not yet fully understood [1].",
"It has long been observed to take place in a non-smooth manner.",
"Most studies have focused on the Portevin-Le Chatelier effect observed under constant strain rate conditions, whereas the present experiment is concerned with creep at constant stress.",
"In both cases large steps (cyclic slips) are embedded in a noisy background.",
"There is a general agreement that this is due to the complex dynamics of networks of dislocations whose motion is a means for the stressed solid to flow.",
"As reported in section below, careful observations of the creep in strained Sn-Pb quasi-eutectic material (a solder wire) show the following : i) On average a sample under constant stress lengthens at constant rate.",
"ii) Continuous monitoring shows time dependent fluctuations of this length superposed on its secular increase.",
"iii) From time to time the length jumps by steps.",
"Afterwards a noisy and steady (on average) length increase is recovered until the next jump, etc.",
"Explaining all this remains a challenge for the common models of creep.",
"Nevertheless it is of great interest because it can be seen as a laboratory model of other far less accessible phenomena like earthquakes, where on average there is also a continuous slow sliding with random microseismic noise, interrupted by large fast sliding steps characterizing major earthquakes.",
"We recently introduced the idea [2] that in such systems the slow to fast transition can be described by a saddle-node bifurcation in a dynamical system evolving slowly with time.",
"We pointed out the interest in such a modelization, which allows one to theoretically predict the response of this dynamical system to a noise source.",
"It was shown that the induced fluctuations drift toward low frequencies (i.e., toward large correlation times) before the transition, and so could be used as a forewarning.",
"Here we show that the dynamical saddle-node bifurcation model is the reduced form of a set of equations previously used to describe the Portevin-Le Chatelier effect in metals, or metallic alloys, i.e., the Ananthakrishna (AK) model [1].",
"The saddle-node model describes fast transitions resulting from the intrinsic dynamics of the original system which can be described locally (close to the step) as a slowly rocking potential system.",
"Using the AK model as well as the saddle-node reduced model we show that, with an added external source of noise, the correlation time of the fluctuations increases before the transition, following the classical scenario of slowing down at bifurcation points.",
"As was shown long ago by Dorodnytsin [3], the same local dynamics describe the transition from slow manifold to fast transients in relaxation oscillations of dynamical models, like the van der Pol equation in the strongly nonlinear limit.",
"By looking at the fluctuations of the length of our samples, we found this behavior near the step-like transitions, with the characteristic drift to low frequencies before the transition.",
"Unlike recent publications that have presented the idea [4] that precursors of earthquakes could be found in the mechanical response to external perturbations (such as the increase of the fluctuations and their slowing down near transitions) our idea goes further, by giving an order of magnitude of the precursor time.",
"The authors of [4] do not introduce the effect of a given time dependence of the parameters and consider only systems with steady parameter values on both sides of the bifurcation.",
"To quote reference [5] “The suggested approach to analytical study of any kind of catastrophes is based essentially on the solution of a stationary problem of the possibility and conditions of the unstable equilibrium state in the system in question”.",
"Without taking into account explicitly the time dependence of the parameters sweeping the bifurcation set, it is impossible to get the time scale for predictions.",
"As we show, this scale depends crucially on the rate of change of the parameters near the bifurcation, which may be estimated from the knowledge of the ratio of the two time scales (fast and slow ones) for saddle-node models.",
"An important challenge lies in the difficulty of stating a suitable model for a given catastrophic event.",
"Actually there is more than one class of possible slow-to-fast transitions in dynamical systems.",
"The dynamical saddle-node can be seen as belonging to the class of systems with an equilibrium point losing stability as a parameter changes.",
"It is not a loss of stability, but rather a loss of existence of the equilibrium point, occurring at the folding point of the slow manifold.",
"However slow-to-fast transition may happen without any folding of the slow manifold.",
"As shown in [6] exploring a very often used mathematical representation of stick-slip behavior, the Dieterich-Ruina equations, the slow-fast transition can originate from the finite time singularity of the slow dynamics itself.",
"In that case, critical speed-up, or drift of fluctuations toward large frequencies, is found to replace the critical slowing-down effect.",
"This nonlinear phenomenon is obviously outside the class of phenomena explainable by a stability analysis of equilibria of dynamical systems.",
"We refer the interested reader to the paper on this subject [6].",
"In the present paper we do not consider this case, because it is clearly not the one observed in the creeping experiments.",
"Note that at this time it is unknown if real earthquakes (as well as other observed catastrophes) belong to the saddle-node case with a slowing-down near the transition or to the finite time singularity case with speed-up expected.",
"The creeping experiment is described in section together with the striking spectral observations.",
"In the two sections following section , we present our theoretical approach.",
"We introduce in section an abstract dynamical model showing fast jumps.",
"In this model, one assumes that, as a dynamical system, the jump follows a “saddle-node” bifurcation.",
"There a pair of fixed points, one locally stable, the other locally unstable, merge and disappear as a control parameter changes.",
"Afterwards, the system moves quickly to a new equilibrium state that is at finite distance (in phase space) from previous equilibria, whence the jump.",
"After reviewing the saddle-node bifurcation in this light, we assume that the parameter changes with time, namely that the parameter defining the bifurcation is itself a slow function of time.",
"When, by this change, the parameter crosses the bifurcation value, the dynamical system makes an abrupt transition and jumps “generically” from one equilibrium state to another.",
"In section we show that this “abstract” model is pertinent for describing the slow-fast transition in the relaxation regime of a set of equations derived by Ananthakrishna for creeping in the relaxation regime.",
"This set of equations describes the dynamics of populations of dislocations in the creeping solid.",
"The model shows relaxation oscillations in which slow drift is interrupted by fast variations.",
"Near the slow-fast transition, we show that this model reduces to the generic equation quoted above, for a certain range of parameters.",
"Therefore it displays a typical critical slowing-down in its response to external noise, in agreement with the real data reported in section .",
"Of particular interest is the fact that, from the experimental data, one can predict in advance a “large slip” event.",
"The event is preceded by a shift toward low frequencies in the random fluctuations of specimen length."
],
[
" Creeping experiment",
"Choose a uniform“wire” that creeps under small stress at room temperature, fabricated from a much studied material, Sn-Pb solder, alloyed to be nearly eutectic.",
"To establish creep at virtually constant stress, let one end of the wire be fixed and at the other end establish a constant force of tension.",
"By this means we found the length of the wire to increase on average at constant rate.",
"By highly accurate monitoring of this length vs time we observed a small time dependent part with the following pattern: on a background of fluctuations, from time to time a large slip event was observed, after which the continuous lengthening with a small background noise was recovered.",
"The observed noise is non-thermal, since thermal noise has a far too small amplitude to be relevant.",
"It is assumed to result from rearrangement of defect structures in the poly-crystalline structure of the quasi-eutectic, involving dislocation dynamics.",
"We assume that this noise originates from a source that is to first-order independent of the overall lengthening of the wire.",
"This is equivalent to saying that it is due to ongoing micro-scale events triggered by the imposed stress, independent of the global creeping.",
"Therefore we analyzed the response to this noise source according to the AK equations subject to an external noise (see below)."
],
[
"apparatus",
"The instrument used in these experiments, which is pictured in Fig.REF , is an extensometer [8].",
"Young's modulus can be accurately measured with a wire specimen, by placing different size masses on the weight pan.",
"The trace of the vertical wire holding up the boom/weight pan in Fig.REF has been enforced (colored in black) to be visible in the image.",
"Though not presently used, the black clamp was for purpose of holding a power resistor that was employed to measure the specimen's thermal coefficient of expansion.",
"To measure temperature changes, a solid state thermometer was placed down into the sample space, through the hole seen near the top knurled clamp.",
"Figure: Extensometer used in the experiment.To calibrate the instrument a tungsten wire of diameter 0.1 mm was mounted in the extensometer, and signal output level changes were recorded as various gram-mass standards were placed on the weight pan.",
"By using the known Young's modulus for tungsten, the resulting measurements yielded a constant of 1.0 nm per analog to digital count, for the 24 bit adc employed, which is sold by Symmetric Research [11].",
"This constant is applicable to the data presently reported.",
"A different measurement technique yielded essentially the same calibration constant.",
"A He-Ne laser was used with a mirror, operating as an optical lever, to measure boom position change as different masses were placed on the pan.",
"On the basis of two factors, ordinary solder was chosen for the present study.",
"First of all, an unusual property of tin is well known, when an ingot of the metal is strained by large amounts.",
"The sound which it then emits is well described by the German word “zinngeschrei\", which translates “tin cries”.",
"The initiative for this work was also influenced by the observation of unusual spectral features in the output from a novel seismograph [12].",
"The unusual low-frequency motions of the Earth's crust that were then observed to precede an earthquake [13] are readily seen by the VolksMeter.",
"This is due to the instrument's use of a “displacement” sensor, rather than the “velocity” sensor used by conventional seismometers.",
"It was therefore natural to consider an alloy of tin, with the expectation that its defect properties should be more like those of the earth than is possible for a pure metal.",
"The tin alloy for our study was ordinary soft solder (60% Sn/ 40% Pb), used universally in the electronics industry.",
"Although the pure `Eutectic' alloy is actually 63% Sn/37% Pb, we will nevertheless use this word to describe our specimen in the discussions that follow.",
"The stress level due to the load placed on the Pb-Sn wire used in the present experiments was considerably smaller than the typically $10 MPa$ used in usual creep studies.",
"The present load was due solely to the weight of aluminum comprising the boom plus empty weight pan of the extensometer.",
"This stress value was estimated to be $0.5 MPa$ , based on considerations that include the distance of the wire's attachment point from the position of the (fine) roller bearing in the unpright housing, that supports the boom on its end opposite the pan.",
"Figure: (a) Elongation of the wire versus time, raw signal x(n)x(n).",
"(b) Filtered signal y(n)y(n) at the exit of the single pole high-pass filter ().During a typical avalanche, the elongation of a $13.5 cm$ long, $1.6 mm$ diameter specimen, would be about $30 \\mu m$ .",
"Before the avalanche, typical lengthening velocity is $5\\mu m/s$ and typical rms fluctuations observed in the wire length (with secular term removed by high-pass filtering) would be about $50 nm$ .",
"An example of creep record (length growth versus time) is shown in Fig.REF (a).",
"The fluctuations of the raw signal are not visible because they are much smaller than the average length variations."
],
[
"Data analysis",
"To eliminate the average growth, we use a standard technique of filtering.",
"At the exit of a low-pass single pole filter, the filtered signal $y(n)$ is given by the recursive formula $y_n = \\frac{1+p}{2}(x_n-x_{n-1})+py_{n-1}\\mathrm {,}$ where $x(n)$ stands for the raw creep signal.",
"In other words the filtered signal is the convolution product of the raw signal derivative by an exponential response function $R(t)=\\exp {-2\\pi f_c t}$ , where $f_c$ is the corner frequency of the filter.",
"In equation (REF ) the parameter $p$ is given by $p=\\exp {-2\\pi f_c \\delta t}$ where $\\delta t $ is the sampling time of the record, equal to $\\frac{1}{130} sec$ in the experiment.",
"Using a corner frequency $f_c=50 mHz$ , the filtered signal takes the form shown in Fig.REF (b).",
"Figure: (a) and (b) Normalized spectral density of y(n)y(n) before the second step (the step starts at n=26370n=26370), for the twotime intervals (1-2) indicated in Figure ()-a, at the abscissa 18370<n<2037018370<n<20370 and 24370<n=2637024370<n=26370respectively, each for a time interval of N=2000N=2000 counts.",
"(c-d) Cumulative spectra corresponding to figures (a-b) respectively(e)Width w(t)w(t) of the cumulative spectrum versus time (at height equal to 75 per cent ofthe maximum), the vertical arrows indicate the very beginning of the fast steps.In order to analyze the spectral properties of the fluctuations, and see how they evolve with time, we must consider separate sequences of a given time duration.",
"One significant difficulty in this analysis is the proper choice for the magnitude of this time duration.",
"The window cannot be too large, lest the spectrum becomes invariant with time.",
"Conversely, if the window is too narrow, there will be excessive noise in the spectral (or correlation) signal.",
"We believe the time windows have been judiciously chosen; so that there is a significant increase of the correlation time before a “burst”.",
"Such increase of the correlation time (actually the decrease of the frequency width of the signal) was found to happen before every sliding event, giving some hope that this “critical slowing down\" is pertinent for predicting the “catastrophe\" before it occurs.",
"For the present work a time duration of order $1/10~th$ of the interval between adjacent bursts was chosen.",
"The spectral density of a given sequence ($n_i,n_i+N$ ) of the filtered signal $S(n) = |\\frac{1}{\\sqrt{N}}\\sum _{n_i}^{n_i+N} y(k)\\exp {2i\\pi nk}|^2\\mathrm {,}$ changes significantly during the creep process.",
"The striking effect is the shift of the spectral density toward low frequencies in the last stage of the slow regime, i.e., just before the burst.",
"Examples of this phenomenon are presented in the figures (REF )(a-b), which show the spectra corresponding to the time intervals ($1,2)$ indicated in Figure (REF )-a.",
"The two spectra clearly differ.",
"The first spectral density (a) displays a large number of components; whereas the second spectral density, corresponding to the time interval just before the burst (figure b), is concentrated close to zero frequency.",
"To quantify this effect, we calculate the cumulative spectrum $C_S(n) = \\sum _1^n S(j)\\mathrm {,}$ which is a smooth curve whose asymptotic value (for n=N) gives the experimental variance of the fluctuations during the sequence considered $\\sigma ^2 = C_S(N)\\mathrm {.",
"}$ Figure: (a) Creep data versus time, (b) Width of the cumulative spectra before the burst with N=2000N=2000.",
"The horizontal segments ending at each point indicates the time interval (before this point) over which the width is calculated.The shift of the density spectrum toward low frequencies is visually more clear in its associated cumulative spectrum, as seen in figures (c-d).",
"We define the “low frequency extension\" $w$ of a sequence by the width of the cumulative spectrum corresponding to 75 per cent of its maximum value, $C_S(w)= \\frac{3}{4} C_S(N)\\mathrm {.",
"}$ Shown in figure 3(e) is the evolution of the spectral width for the whole experimental record.",
"A drop in $w$ is clearly seen to occur before each burst.",
"The important point is that the decrease of the width before the burst is experimentally foreseeable because it occurs during a “precursor\" time which is noticeably larger than the duration of a sequence $N \\delta t$ .",
"This is illustrated in Fig.REF (b) which displays a zoom of the spectral width evolution before the burst.",
"The two following sections are devoted to theoretical models, the first one shows the main features of what is measured in the experiment on creeping, namely the critical slowing down of the fluctuation spectrum occurring before the sliding event, the second one is a model of creeping found in the literature which consists in a set of nonlinear coupled ordinary differential equations, derived by Ananthakrishna and collaborators [15]."
],
[
"Dynamical saddle-node bifurcation",
"This section explains why a saddle-node bifurcation with a slow sweeping of the bifurcation parameter exhibits a slowing down in its response to a source of noise in a window of time extending well before the bifurcation itself.",
"This “abstract' model has no direct connection with the physical phenomenon of creeping.",
"In the section afterwards, however, we explain that a model of creeping shows the same slowing down in a range of parameters, linked to a local saddle-node bifurcation.",
"Consider first the saddle-node bifurcation of a “gradient flow\", that is a damped dynamical system such that a coordinate $x(t)$ is a solution of the equation of motion of the form $\\frac{{\\mathrm {d}}x}{{\\mathrm {d}}t} = -\\frac{\\partial V}{\\partial x}\\mathrm {.",
"}$ In this equation $V(x)$ is a potential, and the dynamics tends to everywhere lower the value of $V(x)$ .",
"The catastrophe theory of Thom and Arnol'd [16] studies how steady equilibria of equations like (REF ) change under smooth deformations of the potential $V(x)$ .",
"Below we consider a different kind of question, namely what happens to the solution of equation (REF ) when the potential $V(x)$ becomes itself a slowly varying function of time, and particularly when a pair of equilibrium points disappears by a saddle-node bifurcation.",
"Indeed this question of the sweeping across bifurcations has been already widely studied [14] with various applications in mind (by \"sweeping\" we mean crossing of a transition point with a time dependent parameter in the equation(s) of motion).",
"However, to the best of our knowledge the occurrence of an intermediate time scale in the case of slow sweeping has been overlooked, although we believe it to be crucial for a strategy of foretelling catastrophes in the real world."
],
[
"local cubic potential",
"The equation (REF ) is too general to be very helpful.",
"However, it may describe a saddle-node bifurcation where a stable equilibrium disappears, assuming that $V$ depends slowly on time in a prescribed way, to become a function $V(x,t)$ .",
"Near the transition, one may use a mathematical picture which is correct for a short time around the transition if the potential $V(x,t)$ is a smooth function (see below for what happens afterwards).",
"Assume first that $V(x)$ does not depend explicitly on time and takes the form $V(x) =- (\\frac{1}{3} x^3 + b x)\\mathrm {,}$ with $b$ real constant (for the moment).",
"For $b$ negative $V(x)$ has two real extrema (i.e.",
"the roots of $\\frac{\\partial V}{\\partial x} =0$ ), one $-\\sqrt{-b}$ is a stable equilibrium, the other, $\\sqrt{-b}$ , is an unstable equilibrium.",
"For $b =0$ the two equilibria merge and disappear for $b$ positive, see Fig.REF (a).",
"This is the saddle-node bifurcation.",
"The shape of $V(x)$ near $x =0$ and for $b$ small is universal: for a given smooth $V(x)$ showing this saddle-node bifurcation, one can always rescale $x$ and the external parameter to obtain the \"local' problem in this form.",
"The extension to a time dependent control parameter $b$ goes as follows.",
"If $b$ is a smooth function of time, one can assume that $b(t)$ crosses the critical value, i.e.",
"zero in the present case, at time zero in such a way that $ b(t) = a t +...$ with $a$ non zero constant, dots being for higher terms in the Taylor expansion of $b(t)$ .",
"For $t$ and $x$ close to zero, after rescaling, one can represent the dynamical system (REF ), close to the saddle-node bifurcation, by the \"universal\" parameterless equation $\\frac{{\\mathrm {d}}x}{{\\mathrm {d}}t} = x^2 + t\\mathrm {.",
"}$ Figure: (a) Cubic potential for b=-1,0,1b =-1,0,1.",
"(b) Quartic potential, b=-1,0,1 b=-1,0,1Outside of the neighborhood of $x=0$ , the solution of (REF ) depends on other parameters defining $V(.",
")$ for finite values of $x$ , as explained below.",
"Although this is not obvious from its formulation, this model is also valid for the transition from slow to fast motion in the van der Pol oscillator in the limit of large non linearities, as shown by Dorodnitsyn [3].",
"Let us sketch the proof of this (interesting) point.",
"Actually we shall look at the formally more general situation of relaxation oscillations, namely at solutions of a set of coupled ODE's (Ordinary differential equations) with a large parameter in the form: $\\dot{x} = \\eta F(x,y)\\mathrm {,}$ and $\\dot{y} = G(x,y)\\mathrm {.",
"}$ In this set of equations dots are for time derivatives and the functions $F()$ and $G()$ are smooth with values of order 1 when their argument is also of order one.",
"Moreover, $\\eta $ is a large parameter.",
"The slow manifold is defined by the condition that, in the limit $\\eta $ large, the function $F(x,y)$ must be close to zero over at least part of the trajectory.",
"The Cartesian equation $F(x,y) = 0$ defines a curve in the plane $(x,y)$ which allows to find $y$ as a function of $x$ , at least locally.",
"This defines the equation of motion (along the slow manifold) $ \\dot{y} = G(x (y),y) \\mathrm {,}$ where $x(y)$ is such that $F(x(y), y) =0 \\mathrm {.",
"}$ The slow trajectory so defined stops at \"folds\" where the function $x(y)$ ceases to be well defined, namely for values of $(x,y)$ such that $\\frac{\\partial F}{\\partial y} = F_{,y} = 0$ , $F(x,y) = 0$ , $F_{,yy}$ being not zero.",
"This defines a discrete set of points.",
"Near those points, the equation of motion can be solved by taking $\\delta x = x - x_0$ and $\\delta y = y - y_0$ where $(x_0 , y_0)$ are the Cartesian coordinates of the point such that $F = F_{,x} = 0\\mathrm {.",
"}$ Let us look at the solution of the coupled equations (REF ), (REF ) near $(x_0 , y_0)$ .",
"The equation (REF ) is not singular at this point and so can be solved for small variation of $y$ and for $t$ small, like $\\delta y = t G_0$ , $t = 0$ being the (arbitrary) time where the trajectory is in $O$ and $G(x_0,y_0) = G_0$ .",
"Consider now the first equation (REF ), and expand its right-hand side for $\\delta x$ and $\\delta y$ small.",
"Because $F_{,x} = 0$ at $(x_0 , y_0)$ , the first nontrivial term in the Taylor expansion of $F(.",
")$ near $(x_0, y_0)$ with respect to $x$ is $\\frac{1}{2} (F_{,xx})_0 \\delta x^2$ .",
"On the other hand the first term coming from the expansion with respect to $\\delta y$ is $F_{,y} \\delta y = (F_{,y})_0 G_0t$ .",
"Therefore, for $t$ and $\\delta x$ small, the equation for $\\dot{\\delta }x $ derived from (REF ) reads: $\\delta \\dot{x} = \\eta \\left( \\frac{1}{2} (F_{,xx})_0 \\delta x^2 + (F_{,y})_0 G_0 t \\right)\\mathrm {,}$ One can check that all terms not written explicitly there are effectively negligible compared to the ones kept.",
"Standard rescalings allows to transform the equation (REF ) into the \"universal\" equation (REF ), provided various constraints of sign are satisfied by the quantities $F_{,xx}$ , $F_{,y}$ and $G(.",
")$ all computed at $(x_0 , y_0)$ .",
"This is a short version of the classical calculation by Dorodnycsin, showing that the \"universal\" equation (REF ) is also relevant for the slow to fast transitions in relaxation oscillations.",
"It is worth mentioning also that this derivation does not really assume that the overall motion is periodic, as it may be extended quite easily to a system of many more coupled ODE's with only one fast variable.",
"This shows that such large jumps are also possible in non periodic dynamics, because the dynamics on the slow manifold may be chaotic in between the jumps if this slow manifold has a number of dimensions sufficiently large.",
"Now we shall focus on the slowing down near the dynamical saddle-node bifurcation described by equation (REF ).",
"We shall first give its explicit solution.",
"We look for a solution of equation (REF ) transiting from the \"stable\" fixed point at \"large\" negative times to the rolling down towards positive value of $x$ at positive times.",
"This solution behaves like $x(t) \\approx - \\sqrt{-t}$ at large negative times.",
"The equation (REF ) is of the Riccati type and can be integrated by introducing the function $y(t)$ such that $x(t) = -\\frac{\\dot{y}}{y}$ where $\\dot{y} = \\frac{{\\mathrm {d}}y}{{\\mathrm {d}}t}$ and $y(t)$ is a solution of Airy's equation $\\ddot{y} + t y = 0\\mathrm {.",
"}$ The solution of equation (REF ) relevant with the given condition at $t\\rightarrow -\\infty $ is drawn on Figure(REF -a).",
"In terms of the variable $y(t)$ it is the Airy function $Ai(-t)$ which writes $Y(t) = Ai(-t)=\\int _0^{+\\infty } \\cos (\\frac{u^3}{3} - ut) {\\mathrm {d}}u\\mathrm {.",
"}$ Yet we have only solved the transient problem near the saddle-node bifurcation.",
"The transition ends-up when $t$ becomes equal to the first zero of the Airy function $Ai(-t)$ , i.e.",
"the smallest root of the equation $Y(t) = 0$ , a pure number, about $t_c \\approx 2.338$ .",
"It corresponds to a divergence of $x(t) = -\\frac{y^{\\prime }}{y}$ , which behaves as $x(t) \\approx \\frac{1}{t_c - t} -\\frac{t_c}{3}(t_c-t)+...\\mathrm {.",
"}$ just before this transition, as derived by expanding $Y(t)$ close to $t_c$ .",
"Therefore the \"generic\" equation (REF ) for the dynamical saddle-node bifurcation displays a finite time singularity.",
"Let us precise the following mathematical subtlety.",
"This property of the local flow, which results from the folding of the slow manifold, differs qualitatively from the finite time singularity found in the Dieterich-Ruina equations [6], where it was a property of the flow reduced to the slow manifold which is everywhere convex, as discussed in the introduction.",
"In the case of the dynamical saddle-node bifurcation the singularity requires one to consider both the dynamics on and off the slow manifold, and this happens because the geometry forbids the continuation of an exact trajectory on this folded slow manifold.",
"Actually the solution (REF ) loses its physical meaning sometime before the singularity since the \"universal\" dynamical equation (REF ) was derived under the assumption that $x$ remains close to zero.",
"This local theory cannot deal with finite variations away from the critical values, therefore we shall need to add finite amplitude effects to limit the growth of the instability after the transition, see the subsection REF .",
"We shall study now two questions, first the response of this dynamical system to an external noise, then the dynamics of a system showing a saddle-node bifurcation of the type just studied and reaching a new stable fixed point after this bifurcation.",
"We explore first the response of our system to a small external noise, and look for qualitative changes in this response which could be a signal that occurs before the transition.",
"Let us consider the equation (REF ) with a small noise added, so that equation (REF ) is replaced by $\\dot{x} = x^2 + t + \\epsilon \\xi (t)\\mathrm {,}$ where $\\xi (t)$ is a random function of time, and $\\epsilon $ a small coefficient.",
"In the limit $\\epsilon $ small, one can solve equation (REF ) by expansion in powers of $\\epsilon $ , $ x(t) = x_0 (t) + \\epsilon x_1(t) + ...\\mathrm {.",
"}$ where $ x_0(t) = -\\frac{Y^{\\prime }(t)}{Y(t)}\\mathrm {,}$ The linear response to the noise is $x_1 (t) = \\frac{1}{Y^2(t)} \\int _{t_0}^t {\\mathrm {d}}\\tilde{t} \\,\\xi (\\tilde{t}) \\ Y^2(\\tilde{t})\\mathrm {.",
"}$ Because $Y^2(\\tilde{t})$ tends rapidly to zero as $(\\tilde{t})$ tends to minus infinity, one can take $t_0 = -\\infty $ to get rid of the effect of the initial conditions.",
"To make the developments above more concrete, let us take a delta-correlated (or white) noise, such that $< \\xi (t_a) \\xi (t_b)> = \\delta (t_a - t_b)\\mathrm {.",
"}$ The pair correlation of $x_1(t)$ is given by $ <x_1 (t) x_1(t^{\\prime })> = \\frac{1}{Y^2(t) Y^2(t^{\\prime })} \\int _{-\\infty }^{\\inf (t,t^{\\prime })} {\\mathrm {d}}\\tilde{t} Y^4(\\tilde{t}) \\mathrm {,}$ where ${\\inf (t,t^{\\prime })}$ is the smallest of the two real numbers $t$ and $t^{\\prime }$ .",
"The behavior of this pair correlation for large negative values of both $t$ and $t^{\\prime }$ , is derived from the asymptotic expression of Airy's function, $Ai(-t) \\approx \\frac{e^{ - \\frac{2}{3} (-t)^{3/2}}}{2\\sqrt{\\pi }(-t)^{1/4}}$ .",
"Setting $w=\\frac{\\tilde{t}}{t}$ , and $F(w)=1-w^{3/2}$ , the variance of the fluctuations writes $<x_1 (t)^2> \\approx (-t) \\int _{1}^{\\infty } \\frac{{\\mathrm {d}}w }{w}e^{\\frac{8}{3} (-t)^{3/2} F(w)}\\mathrm {.",
"}$ In the limit $(-t) \\rightarrow \\infty $ the integral is concentrated near $w = 1$ so that $<x_1 (t)^2> \\approx \\frac{1}{4}(-t)^{-\\frac{1}{2}}\\mathrm {,}$ which shows that the amplitude of the fluctuations increases some time before the transition itself.",
"As the transition approaches, the standard deviation of the fluctuations, $ \\sigma (t)=\\sqrt{<(x(t)-x_0(t))^2>}\\mathrm {,}$ increases close to the critical time $t_c$ , because $Y(t_c)=0$ ."
],
[
"Quartic potential",
"As the zeroth order solution diverges at $t=t_c$ , it does not make sense to describe the dynamical behavior of the fluctuations due to the external noise very close to $t_c$ , as shown above.",
"As said before, this unbounded growth of the fluctuations is a consequence of the local cubic form of $V(x)$ when expanded near $x =0$ , in obvious contradiction with the fact that $x(t)$ tends to infinity.",
"To suppress the divergence of $x(t)$ after the saddle-node bifurcation we add a stabilizing (positive) term to the potential $V(x)$ which becomes quartic, $V_q(x) = - \\frac{x^3}{3} - b x + \\frac{x^4}{4}\\mathrm {,}$ as drawn in Fig.REF (b).",
"Because of the growth of $V_q(x)$ at infinity, like $x^4$ , the solution of the differential equation $\\dot{x} = b + x^2 - x^3\\mathrm {.",
"}$ does not diverge at finite time.",
"Notice that, formally this does not apply to cases where the local equation (REF ) describes a slow-to-fast transition in a limit cycle, instead of a gradient flow dynamics.",
"However, unless one insists to look at what happens well after the saddle-node bifurcation, the details of the dynamics after the fast slide are not significant, because one can consider that the point of landing on the slow manifold after the fast drift is like a new equilibrium for a gradient flow, this neglecting the slow motion on this manifold.",
"The equation (REF ) can be written in the given scaled form for any quartic potential provided the coefficient of $x^4$ is positive.",
"For such a potential one parameter only remains.",
"In equation (REF ), we choosed to keep as explicit coefficient the coefficient $b$ of the linear term in equation (REF ).",
"For $b = 0$ the dynamical system (REF ) is exactly at the saddle-node bifurcation, because at $b = x =0$ both the first and second derivative of $V_q(x)$ vanish, but not the third derivative.",
"Contrary to the case of the pure cubic potential, this system has always, that is for any value of $b$ , a stable fixed point beyond the pair of fixed points collapsing at the saddle-node bifurcation.",
"This makes it a fair candidate for describing the dynamical saddle-node bifurcation without blow-up.",
"As in the previous case, we shall take now a time dependent $b$ , that will be taken as $ b = a t$ with $a$ positive constant.",
"Because of the rescaling of the cubic and quartic term, the parameter $a$ cannot be eliminated (another possibility would be to put a parameter in front of the cubic term).",
"For the potential $V_q(x) = -\\frac{x^3}{3} - a t x + \\frac{x^4}{4}$ we shall analyze the solution of the dynamical equation $\\dot{x} = a t + x^2 - x^3\\mathrm {,}$ tending at large negative and positive times to the equilibrium point $ x = (a t)^{1/3}$ , $t$ being considered as a parameter, see Fig.REF (a).",
"Moreover we consider the limit $a$ small, $a$ being related to the ratio of small to large time scales, it can be estimated from experimental data.",
"We shall prove that in this limit $a$ small, there are three characteristic time intervals, depending how $x$ is close to zero.",
"The long time scale is the time lapse between successive major slips, typically of order $150-300 s$ in our experiments on creeping.",
"In our model it is the time needed for the potential $V_q(x,t)$ to change significantly, to move from a pair of fixed points to a saddle-node bifurcation.",
"Because time enters in $V_q(x,t)$ through the combination $(at)$ , the a-dimensional time needed for a change of shape of $V_q$ is of order $t_{creep} \\sim a^{-1}\\mathrm {.",
"}$ The short time $t_{slip}$ is of order unity in units of our model equation (REF ) as shown in the next paragraph.",
"It is the duration of the abrupt change of slope of the function of time $x(t)$ , the rising of $x(t)$ at $t_c$ .",
"In the creep experiment reported above, it is about $1 s$ .",
"Therefore the ratio of these two time scales $t_{slip}/t_{creep}$ is as small as $10^{-2}-10^{-3}$ in our experiments.",
"There is another time scale, $t_0$ , the time interval standing before the transition, and close to it, during which the potential is very flat, while the solution has not yet jumped.",
"During this time, $x$ and $at$ are much smaller than unity, then the cubic term on the right-hand side of equation (REF ) is negligible.",
"In this range one recovers the universal equation of the dynamical saddle-node bifurcation (REF ) by taking $X = x a^{-1/3}$ and $T = t a^{1/3}$ , with the boundary condition $X(t) \\approx - \\sqrt{-T}$ at $T$ tending to minus infinity.",
"This property concerns the rectangular domain drawn on Fig.",
"REF -(a), where $x$ is small, $x \\sim a^{1/3}$ , and $t$ extends from $-a^{-1/3}$ to $t \\sim a^{-1/3}$ , located before the abrupt increase.",
"Therefore the time extension of this domain introduces the intermediate time scale, $t_0 \\sim a^{-1/3}\\mathrm {,}$ long compared to the short time $t_{slip}$ (which is of order unity, see below) and small compared to $t_{creep}=a^{-1}$ the average time between slips.",
"We now show that the short time is of order unity, by matching the solution $X(T)$ of the universal equation to the solution of equation (REF ) below, in the vicinity of the critical point $t_c(a)=a^{-1/3} t_c$ .",
"Because $X(T)$ behaves like$\\frac{1}{t_c-T}$ before it diverges, it follows that the solution $x(t)$ behaves as $\\approx \\frac{1}{a^{-1/3}t_c- t}$ for \"large\" values of $\\delta t = a^{-1/3}t_c-t$ before the critical time.",
"This becomes of order one when $\\delta t$ becomes also of order one.",
"When this happens, the term $at$ in equation (REF ) is negligible, therefore the solution of this equation, which can be matched with the solution near the bifurcation, is the solution of the integrable equation $\\dot{x} = x^2 - x^3\\mathrm {,}$ with the asymptotic behavior for very large negative times $x(\\delta t)\\sim - \\frac{1}{\\delta t}$ .",
"This equation shows that, in our model, the fast time scale is of order one, because it has no explicit dependence with respect to the small parameter $a$ .",
"This result is confirmed by the numerics, see Fig.REF (a).",
"More precisely defining the rising time of $x(t)$ by $1/5$ of the width of the time derivative $\\dot{x}$ , we get $t_{slip} \\sim 1\\mathrm {,}$ independently of the value of $a$ , for $a$ small.",
"In the experiment, the intermediate time scale is $t_0^{phys} \\sim a^{-1/3}t_{slip}^{phys}$ , which gives about $5s$ for $a=10^{-3}$ , $t_{slip}^{phys}=1s$ .",
"In summary , by matching the two solutions in the range $1 \\ll (-\\delta t) \\ll a^{-1/3}$ , we show that the catastrophe takes place during this time $t_{slip}$ which is of order one, because the displacement is then of order one, compared to the small displacement of order $a^{1/3}$ taking place during time $t_0=a^{-1/3}$ typical of the \"universal\" transition process.",
"From this understanding of the various scales in the deterministic part of the dynamical equations, we can now look at the response to noise of this system, particularly at the range of time where something like a \"critical slowing down\" could be observed, and which actually happens in our experiments.",
"With a noise source added, the dynamical equation (REF ) becomes, $\\dot{x} = x^2 - x^3 + a t +\\epsilon \\xi (t)\\mathrm {.",
"}$ Actually the effective noise amplitude is not equal to $\\epsilon $ close to the saddle-node, as it depends on the value of the parameter $a$ .",
"Indeed for $|t|\\le t_{0}$ , the cubic term in equation (REF ) is negligible, and the equation reduces to $\\dot{x} = x^2 + a t +\\epsilon \\xi (t)\\mathrm {.",
"}$ which may be written as $\\frac{{\\mathrm {d}}X}{{\\mathrm {d}}T} = X^2 + T +\\tilde{\\epsilon }(a) \\xi (t)$ , by setting $X=x a^{-1/3}$ , $T=ta^{1/3}$ , and $\\tilde{\\epsilon }(a)=\\epsilon a^{-2/3}$ .",
"Therefore the effective noise is larger than $\\epsilon $ , by a factor $a^{-2/3}$ in the rectangular domain of fig.REF .",
"Figure: (a) solutions of equation (), with and without noise,ϵ=0\\epsilon =0 (smooth curve) and large amplitude noise ϵ=1\\epsilon =1.",
"(b) Solution of equation () for a=10 -3 a=10^{-3}.",
"The rectangle around the origin defines the region -t 0 <t<t 0 -t_0 < t <t_0 and -1/t 0 <x<1/t 0 -1/t_0 < x <1/t_0, with t 0 ∼a -1/3 t_0\\sim a^{-1/3}.",
"The critical time is t c ∼2.34t_c\\sim 2.34 t 0 t_0.",
"The two vertical lines inserted between the two arrows delimitate the large slope time duration, of order unity.",
"(c) Solution of equation () with large amplitude noise ϵ=1\\epsilon =1Let us consider now the fluctuations of the solution $x(t)$ of equation (REF ).",
"For a small noise source, the solution may be expanded in power of $\\epsilon $ as above.",
"At first order it gives $\\dot{x}_1 = [2x_0(t) -3 x_0^2(t)] x_1(t) + \\xi (t)\\mathrm {,}$ whose solution is formally $x_1 (t) = \\int _{t_0}^t {\\mathrm {d}}\\tilde{t} \\ \\xi (\\tilde{t}) \\exp [g(t)-g(\\tilde{t})]\\mathrm {.",
"}$ In general $g(t)$ is the time integral of the second derivative of the potential $-\\frac{d^2V_q(x)}{dx^2}$ , which yields with our choice of $V(.",
")$ : $g(t)=\\int _{t_0}^t [2x_0(u) -3 x_0^2(u)] \\mathrm {.",
"}$ The standard deviation $\\sigma _{x_1}(t)$ has to be calculated numerically.",
"We expect it to display the same behavior as for the cubic case in the whole domain where $x(t) <<1$ , i.e.",
"a little before the transition and close to it, because the potential is cubic in this range.",
"After the transition, we expect that the fluctuation decreases, because the solution without noise becomes quasi-steady.",
"This is confirmed by the numerics: the amplitude of the fluctuations strongly increases close to the critical time $t_c$ , its maximum occurring at time $t_c$ , then it decreases.",
"More precisely the standard deviation behaves exactly as $\\dot{x}_0(t)$ , the red curve in Fig.",
"(REF (a), for small noise.",
"Therefore the strong increase of the variance of the signal fluctuations cannot be used as a precursor for predicting the transition because it occurs simultaneously with the signal itself close to the transition.",
"Note that this observation seems to contradict the currently found statement that fluctuation enhancement precedes the transition and can be used as a precursor.",
"In the case of the saddle-node bifurcation model, we have indeed observed a \"precursor\" growth of the fluctuations, but only in the case of \"large\" amplitude noise, see Fig.REF (b).",
"Let us focus on the case of small noise.",
"In this case we show below that the correlation time of the fluctuations changes much earlier than the onset of amplitude growth.",
"This slowing-down can be used to foretell the event itself, in considerable advance of its occurrence.",
"It is seen to consistently happen in the creeping experiment described below.",
"Figure: (a-b) width (in arbitrary units) of the correlation functions of the fluctuation x(t)-x 0 (t)x(t)-x_0(t) for the saddle-node model () with a=10 -3 a=10^{-3}.",
"The correlations functions are defined by equation () and () for curves (a) and (b) respectively; (c) Spectral width (a.",
"u.)",
"calculated with w g =t 0 w_g=t_0, for a=10 -3 a=10^{-3}.",
"(d) g(t/t c )g(t/t_c) for a=10 -3 a=10^{-3} (blue),for a=10 -6 a=10^{-6} (red).Consider the case of small effective noise, where the correlation function and the spectrum of the fluctuations $(x(t)-x_0(t))$ are well described by the correlation function and spectrum of $x_1(t)$ , respectively.",
"As noted in the previous section, the calculation of these functions requires some care; because the system is not in a statistically steady state.",
"Therefore the spectral density of the fluctuations depends on time and the correlation function $\\Gamma _{x_1}(t,\\tau )= <x_1 (t-\\tau ) x_1(t)>$ depends on both $t$ and on the time difference $\\tau $ .",
"More precisely, the correct definition of the correlation function is actually given by $\\Gamma _{x_1}(t,\\tau )= \\frac{<x_1 (t-\\tau ) x_1(t)>-<x_1 (t-\\tau )>< x_1(t)>}{\\sigma _{x_1(t-\\tau )}\\sigma _{x_1(t)}}\\mathrm {,}$ The latter definition of correlation is not readily accessible in experimental situations, because it requires knowledge of the time dependent variance, which is difficult to estimate from a single sample.",
"A more accessible tool is often used; it is given by the expression $\\Gamma ^{\\prime } _{x_1}(t,\\tau )^{\\prime }= \\frac{<x_1 (t-\\tau ) x_1(t)>-<x_1 (t-\\tau )>< x_1(t)>}{\\sigma _{x_1(t)^2}}\\mathrm {,}$ which coincides with the correct expression if the variance is the same at time $t$ and $t-\\tau $ only.",
"If the variance changes noticeably during the time interval of duration $\\tau $ , the latter expression is biased, since $\\Gamma =\\Gamma ^{\\prime } \\frac{\\sigma (t)}{\\sigma (t-\\tau )}$ .",
"We expect such a discrepancy to manifest itself close to the catastrophe, since the variance increases there by a large amount.",
"To illustrate this point we show in Fig.REF (a-b) the correlation functions of the fluctuation $x(t)-x_0(t)$ for the solution of the saddle-node model, as defined by equations (REF ) and (REF ) respectively.",
"The strong increase of the correlation time before the catastrophe is noticeably truncated when using the biased expression (REF ).",
"Using the correct definition of the correlation function (REF ), the correlation time increases by a factor of ten over a time interval of order $t_0$ before the catastrophe, while the enhancement is only about 3 when using the biased expression (REF ).",
"Moreover the enhancement is followed by a drop in the latter case.",
"Nevertheless both curves display well the critical slowing down effect, which manifests as a growth of the correlation time close to the transition.",
"The increase of the correlation length before the catastrophe is understood by looking at the formal expression (REF ).",
"The second derivative of the potential vanishes at $t=t_0$ , and that leads to the flatness of $g(t)$ in the time domain $0 <t <t_c(a)$ , as shown in Fig.REF (d).",
"This time domain could therefore be identified as a \"precursor time\", of order few $t_0$ .",
"Now let us consider the spectrum of the fluctuations in order to compare with the experimental results.",
"A time dependent spectrum can be defined formally by the (real) Wigner transform $\\mathrm {S} _{x_1}(t,f)= < \\int _{-\\infty }^{-\\infty } {\\mathrm {d}}\\tau \\ <x_1 (t-\\tau ) x_1(t)> e^{-2i\\pi f \\tau }\\mathrm {,}$ that has to be modified for numerical applications, either by using a filtering procedure like the one used in the previous section or by introducing a slipping window.",
"In this section we use the latter process.",
"Choosing a Gaussian window function of width $w_g$ , the numerical spectrum is given by $\\mathrm {S} _{x_1}(t,\\nu )= \\langle | \\int _{t_a}^{t_b}{\\mathrm {d}}\\tau e^{-(\\frac{t- \\tau }{w_g})^2} x_1 (\\tau ) e^{-2i\\pi \\nu \\tau }|^2 \\rangle \\mathrm {,}$ where $t_{a,b}$ are the numerical integration time boundaries.",
"This expression does not take into account the variability of the variance; therefore we expect the observed change in the spectral properties to be biased, as was the case for the correlation function $\\Gamma ^{\\prime }$ .",
"The evolution of the spectral width $\\Delta f$ (half-height width) is reported in Fig.REF (c) in a range of a few times $t_0$ around the transition.",
"The solution ${\\dot{x}}_0(t)$ is hightlighted by the solid red line.",
"The (c) part of the figure illustrates the same effect as the correlation function $\\Gamma ^{\\prime }$ , but in the Fourier space.",
"The spectral width decreases noticeably from negative time of order few $t_0$ , until the time $t\\sim 1.5*t_0$ , where it grows, because of the bias due to the variance increase close to the burst.",
"The decrease of $\\Delta f$ corresponds to a shift of the spectrum towards low frequencies.",
"The important result is that this shift occurs well before the transition.",
"It occurs over a time interval of order few $t_0$ .",
"This result agrees with our experiment where $t_0$ was estimated to be about $10 s$ and the decrease of the spectral width keeps on for about $30s$ ( see Fig.REF where $10s$ corresponds to 1300 counts in the abscissa scale).",
"The growth of the fluctuations and their shift to lower frequencies can be understood as follows.",
"As the transition approaches, the potential $V(x,t)$ becomes flatter and flatter, making weaker and weaker the restoring force toward equilibrium.",
"Therefore, at constant noise source, the amplitude of the fluctuations driven by this noise source will grow because the damping is ever less efficient.",
"Moreover, the typical time scale for this damping will get increasingly larger because of the decreasing stiffness of the potential, thus favoring noise at lower and lower frequencies."
],
[
"Ananthakrishna model",
"This section is devoted to an analysis of solutions of a set of equations devised for describing creeping in solids.",
"More precisely our purpose is to show that, in a range of parameters this equation exhibits a dynamical saddle node bifurcation.",
"As in the “abstract\" model of the previous section, this bifurcation is also preceded by a slowing-down of the fluctuations triggered by an external source of noise.",
"Creeping phenomena in real materials are complex and difficult to predict quantitatively, despite decades of efforts on theoretical models.",
"We have chosen to consider a model developed by Ananthakrishna for creeping in strained solids.",
"To make things simpler, we have only used its version without space dependence in the quantities involved.",
"Note that the introduction of space variables would lead to an aperiodic creep signal more realistic than the periodic signals of the present model, however we conjecture that it should not affect the main result of our study (the emphasis of a precursor signal over a given time interval).",
"The AK model considered here is a set of three coupled non linear ordinary equations with three dimensionless parameters ($a$ , $b$ and $c$ , where the letters $a$ and $b$ have no connection with the same symbols used previously).",
"The unknown time dependent quantities are three scaled variables corresponding to three density types of dislocations, $x(t)$ , $y(t)$ and $z(t)$ , representing respectively mobile, immobile, and those with clouds of solute atoms that mimic Cottrell's idea.",
"The model equations write, $b\\dot{x}(t) = G(x,y)= (1-a)x(t)-b x(t)^2-x(t)y(t)+y(t)\\mathrm {.",
"}$ $\\dot{y} (t) = F(x,y,z)= b x(t)^2-x(t)y(t)-y(t)+az(t)\\mathrm {.",
"}$ $b \\dot{z} (t) = H(z,x)= c(x(t)-z(t))\\mathrm {.",
"}$ where the variable $x(t)$ stands for the elongation rate.",
"As one can check, if the three variables are positive at time zero, they remain so at later times if $a$ , $b$ and $c$ are positive, as assumed.",
"The relative elongation (or strain, or creep) $L(t)$ of the solder wire is the time integral of $x(t)$ , $L(t)= \\int _0 ^t x(t^{\\prime })dt^{\\prime }$ .",
"Solutions of equations (REF )-(REF ) have been extensively studied [1], their shape and duration versus the parameter values are given in [15].",
"Recall that relaxation oscillations are depicted for small values of the parameter $b$ only.",
"For $c$ larger than a certain critical value $c_{cr}$ , depending on $a,b$ and not written here, the solution is stable.",
"The fixed point coordinates are given by the expressions $\\left\\lbrace \\begin{array}{l}x_s=\\frac{1-2a}{2b} \\, \\, \\, \\, \\, \\, \\, \\, \\, \\, \\, \\, \\, \\, \\, \\, \\, \\, \\, \\, \\, \\, \\, \\, \\, \\, \\, \\, \\, \\, \\, \\, \\, \\, \\mathrm {for} \\, \\, \\, \\, a< \\frac{1}{2}\\\\x_s=(1-a)(1+\\sqrt{2})\\, \\, \\, \\, \\, \\mathrm {for} \\, \\, \\, \\, a> \\frac{1}{2}\\\\y_s=\\frac{1}{2}\\mathrm {,}\\end{array}\\right.",
"$ at lowest order for the small parameter $b$ .",
"This fixed point becomes unstable by a Poincaré-Andronov bifurcation for small values of $b$ , only under the condition that $c$ becomes smaller than $c_{cr}$ .",
"In a large range of parameter values the limit cycle associated with the variable $x(t)$ displays relaxation oscillations characterized by slow steps and fast bursts.",
"In this range of $b$ and $c$ small, we focus on the case $c<<b$ , for two reasons.",
"First because it allows the set of equations to be reduced to the generic saddle-node equation introduced in the previous section.",
"Secondly because it leads to $x(t)$ solutions looking approximately like our experimental data (the strain rate $x(t)$ has to be compared with the data files labeled $y(n)$ in section ).",
"An example of the numerical solution of equations (REF )-(REF ) is given in Fig.REF .",
"In this case the burst amplitude drawn in figure (b)) is noticeably larger than the experimental one shown in Fig.REF -(a), nevertheless we have chosen the parameter values for pedagogical reasons, in order to give a clear representation of the full cycle of the relaxation oscillations.",
"Consider the evolution of the elongation rate $x(t)$ .",
"The limit cycle of duration $T=180$ displays four stages.",
"In the first step, $(0<t<130$ , the flow evolves slowly and $x(t)$ takes values of order unity.",
"This stage is followed by a fast jump of $x(t)$ at $t\\sim 130$ , then by a short slow stage $(130<t<144)$ with high values of $x(t)$ , of order $1/b$ , finally followed by a fast decrease of $x(t)$ at $t\\sim 144$ .",
"Figure: (a) Solution of the 3D flow()-() for a=0.65a=0.65 , b=410 -3 b=4 10^{-3}, and c=b/100c=b/100 .",
"The three curves are x(t)/20x(t)/20 (blue), y(t)y(t) (purple) and z(t)/8z(t)/8 (yellow), (b) Creep signal L(t)L(t)."
],
[
"Stability of the 3D flow",
"We first consider the linear stability of the 3D flow, especially along the first stage described just above, which precedes the fast jump.",
"In the next subsection we derive the local form of the model close to the burst.",
"We shall prove that both approaches provide the same information, namely an estimation of the precursor time value.",
"In the slow regime, assuming a 3D flow of the form $ x(t)=x_0(t) + \\delta x \\exp { \\lambda t}$ (and similar expressions for $y(t), z(t)$ ) close to the trajectory $x_0(t),y_0(t),z_0(t)$ , the exponents $\\lambda $ are the eigenvalues of the Jacobian matrix $\\left(\\begin{array}{ccc}( 1-a-2bx_0-y_0)/b & (1-x_0)/b & 0 \\\\2bx_0-y_0 & -(x_0+1) & a \\\\c/b & 0 & -c/b \\\\\\end{array}\\right)\\mathrm {,}$ or solutions of the equation $\\lambda ^3+ a_2\\lambda ^2+a_1\\lambda +a_0=0\\mathrm {,}$ with $a_2=(c-(1-a-y_0))/b+3x_0+1$ , $a_1=(2x_0+(y_0+a-1)/b)(x_0+1+c/b)+(x_0+1)c/b-(x_0-1)(y_0-2bx_0)/b$ , and $a_0=-c(1+x_0)(2bx_0+y_0+a-1)-(y_0-2bx_0)(x_0-1)+a(x-0-1)/b^2$ .",
"Figure: (a) Real and (b) imaginary part of one of the two eigenvalues crossing the real axis before the burst for the case of the 3D flow illustrated in figure (1).One of the eigenvalues is real and negative all along the trajectory, while the other two become complex conjugates in the slow regime, their real part crossing zero before the burst at time $t_{lyap} \\sim t_c-3$ , see Fig.REF .",
"The time interval $t_c-t_{lyap}$ depends on the values of the parameters $a,b,c$ .",
"It is generally a small fraction of the limit cycle period.",
"Below it is shown that the numerical value of $t_c-t_{lyap}$ is nearly equal to the intermediate time scale $t_0$ for the model."
],
[
"Normal form close to the burst",
"We now consider the behavior of the 3D flow in the vicinity of the burst, and derive the normal form of the AK model close to $B$ ."
],
[
"slow manifold",
"We focus on the first part of the limit cycle, preceding the burst of $x(t)$ , and try to understand how the trajectory leaves the slow manifold (SM) that we define now.",
"Because $c<<b<<1$ , the slow stages are described by canceling the right-hand side of equations (REF )-(REF ), that reduces the 3D flow to the 1D flow $\\left\\lbrace \\begin{array}{l}G(x,y)=0\\\\F(x,y,z)=0\\\\\\dot{z} (t) = \\frac{c}{b}(x(t)-z(t))\\mathrm {.",
"}\\end{array}\\right.$ From the first equation the variables $x(t)$ is an explicit function of $y(t)$ , $x(y) = \\frac{-(y-1+a)+\\sqrt{(y-1+a)^2+4by}}{2b}\\mathrm {.",
"}$ Inserting this expression into the second equation (REF ), allows to also express the variable $z(t)$ in terms of $y(t)$ , $z(y) = \\frac{-bx(y)^2 +y(x(y)+1)}{a}\\mathrm {,}$ In the phase space $(z,y)$ expression (REF ) defines the slow manifold, which is illustrated in Fig.",
"(REF ), red curve.",
"On the positive slope parts of the slow manifold the 1D flow obeys the differential equation $ \\dot{y} (t) = \\frac{c}{b} \\frac{1}{z_{,y}} (x(y(t))-z(y(t)))$ , which becomes singular at critical points defined (on the SM) by the relation $z_{,y}=0\\mathrm {.",
"}$ The 3D flow (closed blue curve) follows the path $A \\rightarrow B \\rightarrow C \\rightarrow D$ .",
"At the critical point $B$ , the trajectory leaves the slow manifold, jumps to point $C$ (fast stage $B \\rightarrow C$ ), then it follows the portion ($C \\rightarrow D$ ), and finally returns to the SM in $A$ .",
"We consider below the exit of the SM close to the critical point $B$ .",
"Figure: SM z(y)z(y) (red curve) for a=0.65a=0.65 , b=4.10 -3 b=4 .10^{-3}, and parametric plot of the 3D flow (blue curve, numerical solution z(x(y(t)),y(t))z(x(y(t)),y(t)) of equations ()-() for c=b/100c=b/100."
],
[
"Critical point B",
"The coordinates of the critical point $B$ in the phase space $x,y,z$ are solutions of equations (REF ) and (REF ).",
"From equation (REF ) we derive an expression for $ x_{,y}=\\frac{\\partial x}{\\partial y}$ $x_{,y}= -\\frac{(x-1)^2}{1-a+bx^2-2bx}\\mathrm {.",
"}$ which should be identical to the expression of $x_{,y}$ taken from the relation (REF ), $(x_{,y})_B= \\frac{x^2-1}{3bx^2-(2b+1-a)x}\\mathrm {.",
"}$ These two expressions are identical if $x_B$ is a root of the cubic polynomial equation $x(x-1)(1+2\\tilde{b}-3\\tilde{b}x)= (x+1)(1-2\\tilde{b}x+\\tilde{b}x^2)\\mathrm {.",
"}$ where $\\tilde{b}=b/(1-a)$ .",
"The positive solution $x_B$ and the corresponding value of $y_B=((1-a)x_B-bx_B^2)/(x_B-1)$ and $z_B$ satisfying equations (REF )-(REF ) respectively are given at leading order with respect to the small parameter $\\tilde{b}$ by $\\left\\lbrace \\begin{array}{l}x_B=\\sqrt{2}+1 + \\frac{8+5\\sqrt{2}}{2}\\frac{b}{1-a}\\\\y_B=(1-a)(1+\\sqrt{2})-\\frac{8+11\\sqrt{2}}{2}\\frac{b}{1-a}\\\\z_B=\\frac{(3+2\\sqrt{2})(1-a)}{a}-b \\frac{10+7\\sqrt{2}}{a}\\mathrm {,}\\end{array}\\right.$ valid in the parameter range $b<<(1-a)$ .",
"We next derive the normal form describing the dynamical behavior of the solution close to the critical point.",
"Assuming that the variable $x$ follows adiabatically the variable $y$ according to equation (REF ), the original system reduces to $\\left\\lbrace \\begin{array}{l}\\dot{y}=F(y,z)\\\\\\dot{z}=\\frac{c}{b}(x(y)-z)\\mathrm {,}\\end{array}\\right.$ defining the dynamics for the portion of the trajectory near $B$ , including the burst $B \\rightarrow C$ .",
"This is confirmed by the numerics: starting from the point B , with numerical initial conditions (REF ), the behavior of the solutions of equation (REF ) agree well with the 3D flow, see Fig.",
"REF .",
"Figure: Solution from point B to point C, (a) Solution of the 1D flow equation () with relations ()-(), (b) solution of the 3D flow equations ()-().Since we are interested in the description of the solution before and at the burst, in the intermediate regime where $x$ remains close to $x_B$ , we shall pursue our analysis by canceling the terms $bx^2$ in (REF ) and in $F$ , because $bx_B<<1$ .",
"This leads to the system $\\left\\lbrace \\begin{array}{l}x=\\frac{y}{y-1+a}\\\\\\dot{y}=\\tilde{F}(y,z)\\\\\\dot{z}=\\frac{c}{b}(x(y)-z)\\mathrm {,}\\end{array}\\right.$ where $\\tilde{F}=-xy-y+az$ .",
"Close to $B$ and during the burst, the variable $z$ is essentially constant, due to the small value of the ratio $c/b$ , whereas the variable $y$ jumps toward smaller values, as shown in Fig.",
"REF .",
"Therefore at leading order with respect to the small parameter $c/b$ , the solution of the last equation (REF ) is given by $z ( t) =z_B+ \\frac{c}{b}(x_B-z_B) t\\mathrm {.",
"}$ in the vicinity of $B$ .",
"Inserting this local solution (REF ) for $z(t)$ into the system (REF ), the exit from the SM is then described by a single equation for the local variation $\\delta y= y-y_B$ of $y$ of the variable $y$ , which is of the form ${\\delta \\dot{y}} = \\sum _{n>2} \\frac{1}{n!",
"}(\\tilde{F}_{,y^n})_B \\delta y^n +\\gamma \\delta t\\mathrm {,}$ where $\\gamma =(ac/b)(x_B-z_B)\\mathrm {,}$ and $\\tilde{F}^{(n)}_B$ sets for the $n^{th}$ derivative of $\\tilde{F}$ with respect to the variable $y$ , taken at point $B$ ( note that $(F_{,y})_B=0$ at the critical point $B$ ).",
"Using the reduced system (REF ), for the first two derivatives of $\\tilde{F}$ with respect to $y$ we write $\\left\\lbrace \\begin{array}{l}\\tilde{F}_{,y^2}=-yx_{,y^2} -2x_{,y} \\\\\\tilde{F}_{,y^3}=-y x_{,y^3}- 3 x_{,y^2}\\mathrm {.}\\end{array}\\right.",
"$ Expanding all expressions close to $B$ at leading order with respect to the small parameter $b$ , one obtains $\\left\\lbrace \\begin{array}{l}(x_{,y})_B=-\\frac{2}{1-a}\\\\(x_{,y^2})_B=-\\frac{4\\sqrt{2}}{(1-a)^2}\\\\(\\tilde{F}_{,y^2})_B=-\\frac{4\\sqrt{2}}{1-a}\\\\(\\tilde{F}_{,y^3})_B=\\frac{24}{(1-a)^2}\\mathrm {.}\\end{array}\\right.",
"$ At this stage we can limit the series expansion in equation (REF ).",
"A rough approximation for the series to converge is given by $ (\\tilde{F}_{,y^3})_B \\delta y < 3 (\\tilde{F}_{,y^2})_B $ .",
"Using expressions (REF ) and the first equation (REF ) gives the range of variation $\\left\\lbrace \\begin{array}{l}|\\delta y |<\\frac{3}{\\sqrt{2}}(1-a)\\\\|\\delta x |<\\sqrt{2}\\mathrm {.}\\end{array}\\right.",
"$ In this range the local form of the AK equations close to $B$ becomes $\\delta \\dot{y} = \\frac{1}{2}(\\tilde{F}_{,y^2})_B \\delta y^2 +\\gamma \\delta t\\mathrm {.",
"}$ which is identical to equation (REF ).",
"Finally , setting $T=(\\frac{-\\gamma (\\tilde{F}_{,y^2})_B }{2})^{1/3}\\delta t $ , and $Y=[\\frac{((\\tilde{F}_{,y^2})_B )^2}{4\\gamma }]^{1/3} \\delta y$ , the relation (REF ) takes the generic form (REF ) without any parameter recently proposed as a possible description of a signal before a catastrophe [2].",
"$\\dot{Y} =-Y^2- T\\mathrm {.",
"}$ As shown in section the solution of equation (REF ) displays an intermediate time scale of order $t_0 =(\\frac{2}{\\gamma (\\tilde{F}_{,y^2})_B })^{1/3}\\mathrm {.",
"}$ which could be used to predict the burst, because the critical slowing down effect occurs during this time interval.",
"In the AK equations the intermediate time scale, or precursor time, is given by the relation $t_0 \\sim (\\frac{1-a}{2\\sqrt{2}\\gamma })^{1/3}\\mathrm {,}$ where the parameter $\\gamma $ is given by (REF ).",
"Recall that $t_0=t_c - t_{Lyap}$ was shown to be the time delay between the catastrophe and the time $t_{Lyap}$ at which the instability builds up along the slow manifold.",
"For the parameter values of the above Figures, we have $t_0 \\sim 3.3$ , which agrees well with the Lyapunov analysis, see Fig.",
"REF , and also with the precursor time deduced from the spectral analysis presented in the next subsection.",
"In summary we have proven that the AK model, although formally different from the van der Pol model, has a normal form close to the critical point that is consistent with the dynamical saddle-mode model equation studied in ([2]), with an intermediate time scale given by expression (REF ).",
"Consequently, as for the saddle-node model, the AK model should display a response to noise with a strong increase of the correlation time occurring with a few times $t_0$ before the burst."
],
[
"Response to noise before B",
"The response to noise of the system (REF )-(REF ) is studied by setting ($x=x_0(t)+u_b(t)$ , $ y(t)=y_0+v_b(t)$ , $z(t)=z_0(t)+ \\theta _b(t)$ ), where $x_0(t),y_0(t),z_0(t)$ is the solution of the noiseless AK equations, and the vector $V(t)= u_b(t),v_b(t),\\theta _b(t)$ characterizes the fluctuations of the response to a noise source.",
"These fluctuations result from the introduction of noise terms (either multiplicative or additive) in the original system.",
"In the case of multiplicative noise sources, the response of the AK equations is a solution of the system $\\left\\lbrace \\begin{array}{l}\\dot{x}(t) =(1/b) [(1-a)x(t)-b x(t)^2-x(t)y(t)+y(t)](1+\\epsilon _x f_x(t))\\\\\\dot{y} (t) = [b x(t)^2-x(t)y(t)-y(t)+az(t)](1 +\\epsilon _y f_y(t))\\\\\\dot{z} (t) = (c/b)[x(t)-z(t)](1 +\\epsilon _z f_z(t))\\mathrm {.}\\end{array}\\right.",
"$ We report below the result of the numerical study of the correlation function given by expressions (REF ) and (REF ) for small amplitude noise.",
"In this case the variance increases slowly along the SM, following approximately the evolution of the variable $x_0(t)$ , as reported in Fig.",
"REF (a).",
"Figures (b-c) displays the evolution of the correlation function before the burst, along the path $B\\rightarrow C$ .",
"The width of the correlation function $\\Gamma $ clearly increases as the burst is approached, showing an evident critical slowing down effect.",
"The maximum growth of the correlation time occurs close to the time $t \\sim t_{lyap}$ where the SM becomes linearly unstable, persisting until the burst.",
"This result is in agreement with the experimental result given in section showing a shift of the spectrum towards low frequencies before the burst.",
"Figure: Multiplicative noise: (a) variance of the response (red curve) as a function of time, compared to x 0 x_0 (blue curve), in Log scale.",
"(b) Correlation functions Γ(t,τ)\\Gamma (t,\\tau ) at timest=50t=50 -green , t=100t=100-orange and t=129t=129-black curve versus τ\\tau .",
"(c) Half-height-half-with of the correlation function Γ(t,τ)\\Gamma (t,\\tau ) versus time tt.",
"The input data are those of Figure (), noise amplitudes are ϵ i =10 -5 \\epsilon _i=10^{-5} for the three independent noise sources f i (t)f_i(t).For noise of additive type the width of the correlation function $\\Gamma (t,\\tau )$ behaves similarly, as shown in Fig.",
"REF .",
"As pointed out in a previous section, using the expression (REF ) to calculate the correlation time of the solution gives a biased result close to the burst where the variance is time dependent.",
"In this case the increase of the width before the burst is reduced as illustrated in Fig.",
"REF (b,d).",
"Figure: With additive noise.",
"(a-b) Correlation functions at times t=50,100,129t=50,100,129 (dashed, Blue, Red curves respectively), (a) Γ(t,τ)\\Gamma (t,\\tau ), (b) Γ ' (t,τ)\\Gamma ^{\\prime }(t,\\tau ) versus τ\\tau .",
"(c-d) Evolution of the half-width of the correlation functions along the trajectory before the burst, (c) half-width of Γ(t,τ)\\Gamma (t,\\tau ) (d) half-width of Γ ' (t,τ)\\Gamma ^{\\prime }(t,\\tau ) both curves drawn versus tt.",
"Same parameters and input data as in Figures ().",
"Noise amplitudes are ϵ i =10 -4 ,10 -6 ,10 -7 \\epsilon _i=10^{-4},10^{-6},10^{-7} for the three independent noise sources f i (t)f_i(t).",
"(e) Evolution of the variance along the path (B→CB \\rightarrow C)In order to present the AK model results by using the same analytical tools as used for the experiment, we have calculated the cumulative spectrum of the response $u_b(t)$ , $\\textrm {C}_S(t,\\nu )= \\int _{0}^{\\nu }{d\\nu ^{\\prime } S(t,\\nu ^{\\prime })}$ where $S(t,\\nu )= <|\\int _{t-\\Delta t}^{t}{dt^{\\prime } u_b(t^{\\prime })\\exp ^{2i\\pi \\nu t^{\\prime }}}|^2>$ , is the spectrum of the fluctuations for the response signal sampled during the time interval $(t-\\Delta t, t)$ , in analogy with equations (REF )-(REF ).",
"The characteristic spectral width $\\omega $ defined by the relation (REF ) evolves in time as illustrated in Fig.",
"REF for the AK model with additive noise.",
"The horizontal segments corresponding to abscissa ($t-\\Delta t, t$ ), have ordinates $\\omega $ , the numerical value of the spectral width calculated during this time interval.",
"The decrease of $\\omega $ close to the burst shows well the expected shift of the spectrum towards low-frequencies, observed also in the experiments.",
"Figure: Cumulative spectral width ω\\omega of the response to additive noise u b (t)u_b(t), versus time before the burst, same data as in other figures .In conclusion the AK model displays a range of parameter values where one can see the critical slowing-down observed also in the experiments.",
"We have studied the Ananthakrishna model for the case of small $b$ and $c$ parameter values, with $c\\ll b$ .",
"In this case the burst occurs close to a critical point, as noticed in [15].",
"Close to this critical point, the 3D flow (namely a set of three coupled ODE's) can be reduced to the 2D system (REF ) with a sort of Langevin-like source term (noise term with small amplitude noise)of the form (REF )-(REF ) discussed in section .",
"The present system differs from the van der Pol equation; however, close to the critical point both models take the form of the dynamical saddle-mode model equation studied in [2].",
"The Ananthakrishna model displays a response to noise with a strong increase of the correlation time occurring close to the burst, or a shift of the spectrum towards low-frequency components.",
"For this model the “precursor time\" is a time interval of order several times $t_c-t_{Lyap}$ , also equal to several times the intermediate time scale which depends on the values of the parameters $a,b,c$ as given by equation (REF ).",
"A detailed quantitative comparison between the AK model and the creep experiment will require a more complete study, which is beyond the scope of the present work.",
"In particular it should be noted that including spatial variables in the AK model leads to chaotic solutions, which is in better agreement with our experiment where the limit cycle period may indeed vary by a factor of 3 from one recorded data set to another."
],
[
"Conclusion",
"We have shown that the dynamical model of saddle-node transitions recently proposed to foretell catastrophes is applicable to describe the physics of collective dislocations.",
"The experimental signal of the plastic deformation of the eutectic mixture of Sn-Pb subjected to a constant stress presented above clearly displays the well-known critical slowing down effect, with a precursor time of order $1/10$ of the relaxation oscillation period.",
"This observation is shown to agree with the response to noise of a physical model proposed by Ananthakrishna for describing creeping in ductile materials.",
"This physical model has 3 parameters, therefore it would be quasi-impossible to make a quantitative comparison between the experimental data and the theoretical model.",
"Here we show the AK model has a range of parameter values for which the slow-fast transition is described by the dynamical saddle-node model mentioned above.",
"When adding a small noise to the AK model, we show that it indeed displays the critical slowing down scenario, and we derive the expression for the precursor time in terms of the 3 parameter values.",
"This gives a relation between the parameter values of the AK model in which the precursor time (i.e.",
"the time during which the most intense spectral components clearly shift toward low frequencies) is of order one tenth of the relaxation oscillation period, as observed in the experiment.",
"The important result is the following: for such systems which display relaxation oscillations with saddle-node transition, we show that there exists an intermediate time scale between the slow and fast regimes, which can be used to foretell catastrophes.",
"Unlike the majority of precursor tools that have focused on signals in the time domain, we have looked at fluctuations in the frequency domain.",
"These spectral precursors are prone to fewer false alarms."
]
] | 1204.1551 |
[
[
"Free energy of a copolymer in a micro-emulsion"
],
[
"Abstract In this paper we consider a two-dimensional model of a copolymer consisting of a random concatenation of hydrophilic and hydrophobic monomers, immersed in a micro-emulsion of random droplets of oil and water.",
"The copolymer interacts with the micro-emulsion through an interaction Hamiltonian that favors matches and disfavors mismatches between the monomers and the solvents, in such a way that the interaction with the oil is stronger than with the water.",
"The configurations of the copolymers are directed self-avoiding paths in which only steps up, down and right are allowed.",
"The configurations of the micro-emulsion are square blocks with oil and water arranged in percolation-type fashion.",
"The only restriction imposed on the path is that in every column of blocks its vertical displacement on the block scale is bounded.",
"The way in which the copolymer enters and exits successive columns of blocks is a directed self-avoiding path as well, but on the block scale.",
"We refer to this path as the coarse-grained self-avoiding path.",
"We are interested in the limit as the copolymer and the blocks become large, in such a way that the copolymer spends a long time in each block yet visits many blocks.",
"This is a coarse-graining limit in which the space-time scales of the copolymer and of the micro-emulsion become separated.",
"We derive a variational formula for the quenched free energy per monomer, where quenched means that the disorder in the copolymer and the disorder in the micro-emulsion are both frozen.",
"In a sequel paper we will analyze this variational formula and identify the phase diagram.",
"It turns out that there are two regimes, supercritical and subcritical, depending on whether the oil blocks percolate or not along the coarse-grained self-avoiding path.",
"The phase diagrams in the two regimes turn out to be completely different."
],
[
"Introduction and main result",
"In Section REF we define the model.",
"In Section REF we state our main result, a variational formula for the quenched free energy per monomer of a random copolymer in a random emulsion (Theorem REF below).",
"In Section REF we discuss the significance of this variational formula and place it in a broader context.",
"Section gives a precise definition of the various ingredients in the variational formula, and states some key properties of these ingredients formulated in terms of a number of propositions.",
"The proof of these propositions is deferred to Section .",
"The proof of the variational formula is given in Section .",
"Appendices – contain a number of technical facts that are needed in Sections –.",
"For a general overview on polymers with disorder, we refer the reader to the monographs by Giacomin [1] and den Hollander [2]."
],
[
"Model and free energy",
"To build our model, we distinguish between three scales: (1) the microscopic scale associated with the size of the monomers in the copolymer ($=1$ , by convention); (2) the mesoscopic scale associated with the size of the droplets in the micro-emulsion ($L_n\\gg 1$ ); (3) the macroscopic scale associated with the size of the copolymer ($n\\gg L_n$ ).",
"Copolymer configurations.",
"Pick $n\\in {\\mathbb {N}}\\cup \\lbrace \\infty \\rbrace $ and let ${\\mathcal {W}}_n$ be the set of $n$ -step directed self-avoiding paths starting at the origin and being allowed to move upwards, downwards and to the right, i.e., $\\nonumber {\\mathcal {W}}_n &= \\big \\lbrace \\pi =(\\pi _i)_{i=0}^n \\in ({\\mathbb {N}}_0\\times {\\mathbb {Z}})^{n+1}\\colon \\,\\pi _0=(0,1),\\\\&\\qquad \\pi _{i+1}-\\pi _i\\in \\lbrace (1,0),(0,1),(0,-1)\\rbrace \\,\\,\\forall \\,0\\le i< n,\\,\\pi _i\\ne \\pi _j\\,\\,\\forall \\,0\\le i<j \\le n\\big \\rbrace .$ The copolymer is associated with the path $\\pi $ .",
"The $i$ -th monomer is associated with the bond $(\\pi _{i-1},\\pi _i)$ .",
"The starting point $\\pi _0$ is located at $(0,1)$ for technical convenience only.",
"Figure: Microscopic disorder ω\\omega in the copolymer.",
"Dashed edges representmonomers of type AA (hydrophobic), drawn edges represent monomers of type BB(hydrophilic).Microscopic disorder in the copolymer.",
"Each monomer is randomly labelled $A$ (hydrophobic) or $B$ (hydrophilic), with probability $\\frac{1}{2}$ each, independently for different monomers.",
"The resulting labelling is denoted by $\\omega = \\lbrace \\omega _i \\colon \\, i \\in {\\mathbb {N}}\\rbrace \\in \\lbrace A,B\\rbrace ^{\\mathbb {N}}$ and represents the randomness of the copolymer, i.e., $\\omega _i=A$ (respectively, $\\omega _i=B$ ) means that the $i$ -th monomer is of type $A$ (respectively, $B$ ); see Fig.",
"REF .",
"Figure: Mesoscopic disorder Ω\\Omega in the micro-emulsion.",
"Light shaded blocksrepresent droplets of type AA (oil), dark shaded blocks represent droplets oftype BB (water).",
"Drawn is also the copolymer, but without an indication of themicroscopic disorder ω\\omega attached to it.Mesoscopic disorder in the micro-emulsion.",
"Fix $p \\in (0,1)$ and $L_n \\in {\\mathbb {N}}$ .",
"Partition $(0,\\infty )\\times {\\mathbb {R}}$ into square blocks of size $L_n$ : $(0,\\infty )\\times {\\mathbb {R}}= \\bigcup _{x \\in {\\mathbb {N}}_0 \\times {\\mathbb {Z}}} \\Lambda _{L_n}(x), \\qquad \\Lambda _{L_n}(x) = xL_n + (0,L_n]^2.$ Each block is randomly labelled $A$ (oil) or $B$ (water), with probability $p$ , respectively, $1-p$ , independently for different blocks.",
"The resulting labelling is denoted by $\\Omega = \\lbrace \\Omega (x)\\colon \\,x \\in {\\mathbb {N}}_0\\times {\\mathbb {Z}}\\rbrace \\in \\lbrace A,B\\rbrace ^{{\\mathbb {N}}_0\\times {\\mathbb {Z}}}$ and represents the randomness of the micro-emulsion, i.e., $\\Omega (x)=A$ (respectively, $\\Omega (x)=B$ ) means that the $x$ -th block is of type $A$ (repectively, $B$ ); see Fig.",
"REF .",
"The size of the blocks $L_n$ is assumed to be non-decreasing and to satisfy $\\lim _{n\\rightarrow \\infty } L_n = \\infty \\quad \\text{and} \\quad \\lim _{n\\rightarrow \\infty } \\tfrac{L_n}{n} = 0,$ i.e., the blocks are large compared to the monomer size but (sufficiently) small compared to the copolymer size.",
"For convenience we assume that if an $A$ -block and a $B$ -block are on top of each other, then the interface belongs to the $A$ -block.",
"Path restriction.",
"We bound the vertical displacement on the block scale in each column of blocks by $M\\in {\\mathbb {N}}$ .",
"The value of $M$ will be arbitrary but fixed.",
"In other words, instead of considering the full set of trajectories ${\\mathcal {W}}_n$ , we consider only trajectories that exit a column through a block at most $M$ above or $M$ below the block where the column was entered (see Fig.",
"REF ).",
"Formally, we partition $(0,\\infty )\\times {\\mathbb {R}}$ into columns of blocks of width $L_n$ , i.e., $(0,\\infty )\\times {\\mathbb {R}}= \\cup _{j\\in {\\mathbb {N}}_0} {\\mathcal {C}}_{j,L_n}, \\qquad {\\mathcal {C}}_{j,L_n}=\\cup _{k\\in {\\mathbb {Z}}} \\Lambda _{L_n}(j,k),$ where $C_{j,L_n}$ is the $j$ -th column.",
"For each $\\pi \\in {\\mathcal {W}}_n$ , we let $\\tau _j$ be the time at which $\\pi $ leaves the $(j-1)$ -th column and enters the $j$ -th column, i.e., $\\tau _j = \\sup \\lbrace i\\in {\\mathbb {N}}_{0} \\colon \\,\\pi _i\\in {\\mathcal {C}}_{j-1,n}\\rbrace = \\inf \\lbrace i\\in {\\mathbb {N}}_0 \\colon \\, \\pi _i\\in {\\mathcal {C}}_{j,n} \\rbrace -1,\\qquad j = 1,\\dots ,N_\\pi -1,$ where $N_\\pi $ indicates how many columns have been visited by $\\pi $ .",
"Finally, we let $v_{-1}(\\pi )=0$ and, for $j \\in \\lbrace 0,\\dots ,N_\\pi -1\\rbrace $ , we let $v_{j}(\\pi ) \\in {\\mathbb {Z}}$ be such that the block containing the last step of the copolymer in ${\\mathcal {C}}_{j,n}$ is labelled by $(j,v_j(\\pi ))$ , i.e., $(\\pi _{\\tau _{j+1}-1},\\pi _{\\tau _{j+1}}) \\in \\Lambda _{L_N}(j,v_j(\\pi ))$ .",
"Thus, we restrict ${\\mathcal {W}}_{n}$ to the subset ${\\mathcal {W}}_{n,M}$ defined as ${\\mathcal {W}}_{n,M} = \\big \\lbrace \\pi \\in {\\mathcal {W}}_n \\colon \\, |v_j(\\pi )-v_{j-1}(\\pi )|\\le M\\,\\,\\forall \\,j\\in \\lbrace 0,\\dots ,N_\\pi -1\\rbrace \\big \\rbrace .$ Figure: Example of a trajectory π∈𝒲 n,M \\pi \\in {\\mathcal {W}}_{n,M} with M=2M=2 crossing the column𝒞 0,L n {\\mathcal {C}}_{0,L_n} with v 0 (π)=2v_0(\\pi )=2.Hamiltonian and free energy.",
"Given $\\omega ,\\Omega ,M$ and $n$ , with each path $\\pi \\in {\\mathcal {W}}_{n,M}$ we associate an energy given by the Hamiltonian $H_{n,L_n}^{\\omega ,\\Omega }(\\pi )= \\sum _{i=1}^n \\Big (\\alpha \\, 1\\Big \\lbrace \\omega _i=\\Omega ^{L_n}_{(\\pi _{i-1},\\pi _i)}=A\\Big \\rbrace + \\beta \\, 1\\left\\lbrace \\omega _i=\\Omega ^{L_n}_{(\\pi _{i-1},\\pi _i)}=B\\right\\rbrace \\Big ),$ where $\\Omega ^{L_n}_{(\\pi _{i-1},\\pi _i)}$ denotes the label of the block the step $(\\pi _{i-1},\\pi _i)$ lies in.",
"What this Hamiltonian does is count the number of $AA$ -matches and $BB$ -matches and assign them energy $\\alpha $ and $\\beta $ , respectively, where $\\alpha ,\\beta \\in {\\mathbb {R}}$ .",
"(Note that the interaction is assigned to bonds rather than to sites, and that we do not follow the convention of putting a minus sign in front of the Hamiltonian.)",
"Similarly to what was done in our earlier papers [3], [4], [5], [6], without loss of generality we may restrict the interaction parameters to the cone ${\\hbox{\\footnotesize \\rm CONE}}= \\lbrace (\\alpha ,\\beta )\\in {\\mathbb {R}}^2\\colon \\,\\alpha \\ge |\\beta |\\rbrace .$ For $n\\in {\\mathbb {N}}$ , the free energy per monomer is defined as $f_{n}^{\\omega ,\\Omega }(M;\\alpha ,\\beta )=\\tfrac{1}{n}\\log Z_{n,L_n}^{\\omega ,\\Omega }(M;\\alpha ,\\beta )\\quad \\text{with}\\quad Z_{n,L_n}^{\\omega ,\\Omega }(M)=\\sum _{\\pi \\in {\\mathcal {W}}_{n,M}} e^{H_{n,L_n}^{\\omega ,\\Omega }(\\pi )},$ and in the limit as $n\\rightarrow \\infty $ the free energy per monomer is given by $f(M;\\alpha ,\\beta ) = \\lim _{n\\rightarrow \\infty }f_{n,L_n}^{\\omega ,\\Omega }(M;\\alpha ,\\beta ),$ provided this limit exists.",
"Henceforth, we subtract from the Hamiltonian the quantity $\\alpha \\sum _{i=1}^n1\\left\\lbrace \\omega _i=A\\right\\rbrace $ , which by the law of large numbers is $\\tfrac{\\alpha }{2}n (1+o(1))$ as $n\\rightarrow \\infty $ and corresponds to a shift of $-\\tfrac{\\alpha }{2}$ in the free energy.",
"The latter transformation allows us to lighten the notation, starting with the Hamiltonian, which becomes $H_{n,L_n}^{\\omega ,\\Omega }(\\pi )= \\sum _{i=1}^n \\Big (\\beta \\, 1\\left\\lbrace \\omega _i=B\\right\\rbrace -\\alpha \\,1\\left\\lbrace \\omega _i=A\\right\\rbrace \\Big )\\,1\\left\\lbrace \\Omega ^{L_n}_{(\\pi _{i-1},\\pi _i)}=B\\right\\rbrace .$"
],
[
"Variational formula for the quenched free energy",
"Theorem REF below is the main result of our paper.",
"It expresses the quenched free energy per monomer in the form of a variational formula.",
"To state this variational formula, we need to define some quantities that capture the way in which the copolymer moves inside single columns of blocks and samples different columns.",
"A precise definition of these quantities will be given in Section .",
"Given $M\\in {\\mathbb {N}}$ , the type of a column is denoted by $\\Theta $ and takes values in a type space $\\overline{{\\mathcal {V}}}_M$ , defined in Section REF .",
"The type indicates both the vertical displacement of the copolymer in the column and the mesoscopic disorder seen relative to the block where the copolymer enters the column.",
"In Section REF we further associate with each $\\Theta \\in \\overline{{\\mathcal {V}}}_M$ a quantity $u_\\Theta \\in [t_\\Theta ,\\infty )$ that indicates how many steps on scale $L_n$ the copolymer makes in columns of type $\\Theta $ , where $t_\\Theta $ is the minimal number of steps required to cross a column of type $\\Theta $ .",
"These numbers are gathered into the set ${\\mathcal {B}}_{\\overline{{\\mathcal {V}}}_M}=\\big \\lbrace (u_\\Theta )_{\\Theta \\in \\overline{{\\mathcal {V}}}_M}\\in {\\mathbb {R}}^{\\overline{{\\mathcal {V}}}_M}\\colon \\,u_\\Theta \\ge t_\\Theta \\,\\,\\forall \\,\\Theta \\in \\overline{{\\mathcal {V}}}_M,\\,\\Theta \\mapsto u_\\Theta \\mbox{ continuous}\\big \\rbrace .$ In Section REF we introduce the free energy per step $\\psi (\\Theta ,u_\\Theta ;\\alpha ,\\beta )$ associated with the copolymer when crossing a column of type $\\Theta $ in $u_\\Theta $ steps, which depends on the parameters $\\alpha ,\\beta $ .",
"After that it remains to define the family of frequencies with which successive pairs of different types of columns can be visited by the copolymer.",
"This is done in Section REF and is given by a family of probability laws $\\rho $ in ${\\mathcal {M}}_1(\\overline{{\\mathcal {V}}}_M)$ , the set of probability measures on $\\overline{{\\mathcal {V}}}_M$ , forming a set ${\\mathcal {R}}_{p,M} \\subset {\\mathcal {M}}_1(\\overline{{\\mathcal {V}}}_M),$ which depends on $M$ and on the parameter $p$ .",
"Theorem 1.1 For every $(\\alpha ,\\beta )\\in {\\hbox{\\footnotesize \\rm CONE}}$ , $M\\in {\\mathbb {N}}$ and $p \\in (0,1)$ the free energy in (REF ) exists for ${\\mathbb {P}}$ -a.e.",
"$(\\omega ,\\Omega )$ and in $L^1({\\mathbb {P}})$ , and is given by $f(M;\\,\\alpha ,\\beta ) =\\sup _{\\rho \\in {\\mathcal {R}}_{p,M}}\\,\\sup _{(u_\\Theta )_{\\Theta \\in \\overline{{\\mathcal {V}}}_M}\\,\\in \\,{\\mathcal {B}}_{\\,\\overline{{\\mathcal {V}}}_M}} V(\\rho ,u)$ with $V(\\rho ,u) = \\frac{\\int _{\\overline{{\\mathcal {V}}}_M}\\,u_\\Theta \\,\\psi (\\Theta ,u_{\\Theta };\\alpha ,\\beta )\\,\\rho (d\\Theta )}{\\int _{\\overline{{\\mathcal {V}}}_M}\\,u_\\Theta \\, \\rho (d\\Theta )}\\quad \\text{if}\\quad \\int _{\\overline{{\\mathcal {V}}}_M}\\,u_\\Theta \\, \\rho (d\\Theta )=\\infty ,$ and $V(\\rho ,u)=-\\infty $ otherwise."
],
[
"Discussion",
"Structure of the variational formula.",
"The variational formula in (REF ) has a simple structure: each column type $\\Theta $ has its own number of monomers $u_\\Theta $ and its own free energy per monomer $\\psi (\\Theta ,u_\\Theta ;\\alpha ,\\beta )$ (both on the mesoscopic scale), and the total free energy per monomer is obtained by weighting each column type with the frequency $\\rho _1(d\\Theta )$ at which it is visited by the copolymer.",
"The numerator is the total free energy, the denominator is the total number of monomers (both on the mesoscopic scale).",
"The variational formula optimizes over $(u_\\Theta )_{\\Theta \\in \\overline{{\\mathcal {V}}}_M} \\in {\\mathcal {B}}_{\\,\\overline{{\\mathcal {V}}}_M}$ and $\\rho \\in {\\mathcal {R}}_{p,M}$ .",
"The reason why these two suprema appear in (REF ) is that, as a consequence of assumption (REF ), the mesoscopic scale carries no entropy: all the entropy comes from the microscopic scale, through the free energy per monomer in single columns.",
"In Section we will see that $\\psi (\\Theta ,u_\\Theta ;\\alpha ,\\beta )$ in turn is given by a variational formula that involves the entropy of the copolymer inside a single column (for which an explicit expression is available) and the quenched free energy per monomer of a copolymer near a single linear interface (for which there is an abundant literature).",
"Consequently, the free energy of our model with a random geometry is directly linked to the free energy of a model with a non-random geometry.",
"This will be crucial for our analysis of the free energy in the sequel paper.",
"Removal of the corner restriction.",
"In our earlier papers [3], [4], [5], [6], we allowed the configurations of the copolymer to be given by the subset of ${\\mathcal {W}}_n$ consisting of those paths that enter pairs of blocks through a common corner, exit them at one of the two corners diagonally opposite and in between stay confined to the two blocks that are seen upon entering.",
"The latter is an unphysical restriction that was adopted to simplify the model.",
"In these papers we derived a variational formula for the free energy per step that had a simpler structure.",
"We analyzed this variational formula as a function of $\\alpha ,\\beta ,p$ and found that there are two regimes, supercritical and subcritical, depending on whether the oil blocks percolate or not along the coarse-grained self-avoiding path.",
"In the supercritical regime the phase diagram turned out to have two phases, in the subcritical regime it turned out to have four phases, meeting at two tricritical points.",
"In a sequel paper we will show that the phase diagrams found in the restricted model are largely robust against the removal of the corner restriction, despite the fact that the variational formula is more complicated.",
"In particular, there are again two types of phases: localized phases (where the copolymer spends a positive fraction of its time near the $AB$ -interfaces) and delocalized phases (where it spends a zero fraction near the $AB$ -interfaces).",
"Which of these phases occurs depends on the parameters $\\alpha ,\\beta ,p$ .",
"It is energetically favorable for the copolymer to stay close to the $AB$ -interfaces, where it has the possibility of placing more than half of its monomers in their preferred solvent (by switching sides when necessary), but this comes with a loss of entropy.",
"The competition between energy and entropy is controlled by the energy parameters $\\alpha ,\\beta $ (determining the reward of switching sides) and by the density parameter $p$ (determining the density of the $AB$ -interfaces).",
"Figure: Picture of a directed polymer with bulk disorder.",
"The different shadesof black, grey and white represent different values of the disorder.Comparison with the directed polymer with bulk disorder.",
"A model of a polymer with disorder that has been studied intensively in the literature is the directed polymer with bulk disorder.",
"Here, the set of paths is ${\\mathcal {W}}_n = \\big \\lbrace \\pi =(i,\\pi _i)_{i=0}^n \\in ({\\mathbb {N}}_0\\times {\\mathbb {Z}}^d)^{n+1}\\colon \\,\\pi _0=0,\\,\\Vert \\pi _{i+1}-\\pi _i\\Vert =1\\,\\,\\forall \\,0 \\le i<n\\big \\rbrace ,$ where $\\Vert \\cdot \\Vert $ is the Euclidean norm on ${\\mathbb {Z}}^d$ , and the Hamiltonian is $H^\\omega _n(\\pi ) = \\lambda \\sum _{i=1}^n \\omega (i,\\pi _i),$ where $\\lambda >0$ is a parameter and $\\omega = \\lbrace \\omega (i,x)\\colon \\,i\\in {\\mathbb {N}},\\,x\\in {\\mathbb {Z}}^d\\rbrace $ is a field of i.i.d.",
"${\\mathbb {R}}$ -valued random variables with zero mean, unit variance and finite moment generating function, where ${\\mathbb {N}}$ is time and ${\\mathbb {Z}}^d$ is space (see Fig.",
"REF ).",
"This model can be viewed as a version of a copolymer in a micro-emulsion where the droplets are of the same size as the monomers.",
"For this model no variational formula is known for the free energy, and the analysis relies on the application of martingale techniques (for details, see e.g.",
"den Hollander [2], Chapter 12).",
"In our model (which is restricted to $d=1$ and has self-avoiding paths that may move north, south and east instead of north-east and south-east), the droplets are much larger than the monomers.",
"This causes a self-averaging of the microscopic disorder, both when the copolymer moves inside one of the solvents and when it moves near an interface.",
"Moreover, since the copolymer is much larger than the droplets, also self-averaging of the mesoscopic disorder occurs.",
"This is why the free energy can be expressed in terms of a variational formula, as in Theorem REF .",
"In the sequel paper we will see that this variational formula acts as a jumpboard for a detailed analysis of the phase diagram.",
"Such a detailed analysis is lacking for the directed polymer with bulk disorder.",
"The directed polymer in random environment has two phases: a weak disorder phase (where the quenched and the annealed free energy are asymptotically comparable) and a strong disorder phase (where the quenched free energy is asymptotically smaller than the annealed free energy).",
"The strong disorder phase occurs in dimension $d=1,2$ for all $\\lambda >0$ and in dimension $d \\ge 3$ for $\\lambda >\\lambda _c$ , with $\\lambda _c \\in [0,\\infty ]$ a critical value that depends on $d$ and on the law of the disorder.",
"It is predicted that in the strong disorder phase the copolymer moves within a narrow corridor that carries sites with high energy (recall our convention of not putting a minus sign in front of the Hamiltonian), resulting in superdiffusive behavior in the spatial direction.",
"We expect a similar behavior to occur in the localized phases of our model, where the polymer targets the $AB$ -interfaces.",
"It would be interesting to find out how far the coarsed-grained path in our model travels vertically as a function of $n$ ."
],
[
"Key ingredients of the variational formula",
"In this section we give a precise definition of the various ingredients in Theorem REF .",
"In Section REF we define the entropy of the copolymer inside a single column (Proposition REF ) and the quenched free energy per monomer for a random copolymer near a single linear interface (Proposition REF ), which serve as the key microscopic ingredients.",
"In Section REF these quantities are used to derive variational formulas for the quenched free energy per monomer in a single column (Proposition REF ).",
"These variational formulas come in two varieties (Propositions REF and REF ).",
"In Section REF we define certain percolation frequencies describing how the copolymer samples the droplets in the emulsion (Proposition REF ), which serve as the key mesoscopic ingredients.",
"Propositions REF –REF will be proved in Section .",
"The results in Sections REF –REF will be used in Section to prove our variational formula in Theorem REF for the copolymer in the emulsion, which is our main macroscopic object of interest."
],
[
"Path entropies and free energy along a single linear interface",
"Path entropies.",
"We begin by defining the entropy of a path crossing a single column.",
"Let $\\nonumber {\\mathcal {H}}&= \\lbrace (u,l)\\in [0,\\infty ) \\times {\\mathbb {R}}\\colon \\, u\\ge 1+|l|\\rbrace ,\\\\{\\mathcal {H}}_L &= \\big \\lbrace (u,l)\\in {\\mathcal {H}}\\colon \\,l \\in \\tfrac{{\\mathbb {Z}}}{L},\\,u\\in 1+|l|+\\tfrac{2{\\mathbb {N}}}{L}\\big \\rbrace , \\qquad L \\in {\\mathbb {N}},$ and note that ${\\mathcal {H}}\\cap \\mathbb {Q}^2=\\cup _{L\\in {\\mathbb {N}}} {\\mathcal {H}}_L$ .",
"For $(u,l) \\in {\\mathcal {H}}_L$ , denote by ${\\mathcal {W}}_L(u,l)$ (see Fig.",
"REF ) the set containing those paths $\\pi =(0,-1)+\\widetilde{\\pi }$ with $\\widetilde{\\pi }\\in {\\mathcal {W}}_{uL}$ (recall (REF )) for which $\\pi _{uL}= (L,lL)$ .",
"The entropy per step associated with the paths in ${\\mathcal {W}}_L(u,l)$ is given by $\\tilde{\\kappa }_L(u,l)=\\tfrac{1}{uL} \\log |{\\mathcal {W}}_L(u,l)|.$ Figure: A trajectory in 𝒲 L (u,l){\\mathcal {W}}_L(u,l).The following proposition will be proved in Appendix .",
"Proposition 2.1 For all $(u,l)\\in {\\mathcal {H}}\\cap \\mathbb {Q}^2$ there exists a $\\tilde{\\kappa }(u,l) \\in [0,\\log 3]$ such that $\\lim _{ {L\\rightarrow \\infty } \\atop {(u,l)\\in {\\mathcal {H}}_L} } \\tilde{\\kappa }_L(u,l)= \\sup _{ {L\\in {\\mathbb {N}}} \\atop {(u,l)\\in {\\mathcal {H}}_L}} \\tilde{\\kappa }_L(u,l)= \\tilde{\\kappa }(u,l).$ An explicit formula is available for $\\tilde{\\kappa }(u,l)$ , namely, $\\tilde{\\kappa }(u,l) = \\left\\lbrace \\begin{array}{ll}\\kappa (u/|l|,1/|l|), &ł\\ne 0,\\\\\\hat{\\kappa }(u), &l = 0,\\end{array}\\right.$ where $\\kappa (a,b)$ , $a\\ge 1+b$ , $b\\ge 0$ , and $\\hat{\\kappa }(\\mu )$ , $\\mu \\ge 1$ , are given in [3], Section 2.1, in terms of elementary variational formulas involving entropies (see [3], proof of Lemmas 2.1.1–2.1.2).",
"Free energy along a single linear interface.",
"To analyze the free energy per monomer in a single column we need to first analyse the free energy per monomer when the path moves in the vicinity of an $AB$ -interface.",
"To that end we consider a single linear interface ${\\mathcal {I}}$ separating a liquid $B$ in the lower halfplane from a liquid $A$ in the upper halfplane (including the interface itself).",
"For $L\\in {\\mathbb {N}}$ and $\\mu \\in 1+\\frac{2{\\mathbb {N}}}{L}$ , let ${\\mathcal {W}}^{\\mathcal {I}}_L(\\mu )={\\mathcal {W}}_L(\\mu ,0)$ denote the set of $\\mu L$ -step directed self-avoiding paths starting at $(0,0)$ and ending at $(L,0)$ .",
"Define $\\phi ^{\\omega ,{\\mathcal {I}}}_L(\\mu ) = \\frac{1}{\\mu L} \\log Z^{\\omega ,{\\mathcal {I}}}_{L,\\mu }\\quad \\text{ and } \\quad \\phi ^{\\mathcal {I}}_L(\\mu )={\\mathbb {E}}[\\phi ^{\\omega ,{\\mathcal {I}}}_L(\\mu )] ,$ with $\\begin{aligned}Z^{\\omega ,{\\mathcal {I}}}_{L,\\mu }&= \\sum _{\\pi \\in {\\mathcal {W}}_L^{\\mathcal {I}}(\\mu )} \\exp \\left[H^{\\omega ,{\\mathcal {I}}}_{L}(\\pi )\\right],\\\\H^{\\omega ,{\\mathcal {I}}}_{L}(\\pi )&= \\sum _{i=1}^{\\mu L}\\big (\\beta \\, 1\\lbrace \\omega _i=B\\rbrace -\\alpha \\,1\\lbrace \\omega _i=A\\rbrace \\big )\\ 1\\lbrace (\\pi _{i-1},\\pi _i) < 0\\rbrace ,\\end{aligned}$ where $(\\pi _{i-1},\\pi _i) < 0$ means that the $i$ -th step lies in the lower halfplane, strictly below the interface (see Fig.",
"REF ).",
"The following proposition was derived in [3], Section 2.2.2.",
"Proposition 2.2 For all $(\\alpha ,\\beta )\\in {\\hbox{\\footnotesize \\rm CONE}}$ and $\\mu \\in \\mathbb {Q}\\cap [1,\\infty )$ there exists a $\\phi ^{\\mathcal {I}}(\\mu )=\\phi ^{\\mathcal {I}}(\\mu ;\\alpha ,\\beta )$ $\\in {\\mathbb {R}}$ such that $\\lim _{ {L\\rightarrow \\infty } \\atop {\\mu \\in 1+\\frac{2{\\mathbb {N}}}{L}} }\\phi ^{\\omega ,{\\mathcal {I}}}_L(\\mu ) = \\phi ^{\\mathcal {I}}(\\mu ) = \\phi ^{\\mathcal {I}}(\\mu ;\\alpha ,\\beta )\\quad \\text{for ${\\mathbb {P}}$-a.e.",
"$\\omega $ and in $L^1({\\mathbb {P}})$.",
"}$ It is easy to check (via concatenation of trajectories) that $\\mu \\mapsto \\mu \\phi ^{{\\mathcal {I}}}(\\mu ;\\alpha ,\\beta )$ is concave.",
"For technical reasons we need to assume that it is strictly concave, a property which we believe to be true but are unable to verify: Assumption 2.3 For all $(\\alpha ,\\beta )\\in {\\hbox{\\footnotesize \\rm CONE}}$ the function $\\mu \\mapsto \\mu \\phi ^{{\\mathcal {I}}}(\\mu ;\\alpha ,\\beta )$ is strictly concave on $[1,\\infty )$ .",
"Figure: Copolymer near a single linear interface."
],
[
"Free energy in a single column and variational formulas",
"In this section we use Propositions REF –REF to derive a variational formula for the free energy per step in a single column (Proposition REF ).",
"The variational formula comes in three varieties (Propositions REF and REF ), depending on whether there is or is not an $AB$ -interface between the heights where the copolymer enters and exits the column, and in the latter case whether an $AB$ -interface is reached or not.",
"In what follows we need to consider the randomness in a single column.",
"To that aim, we recall (REF ), we pick $L\\in {\\mathbb {N}}$ and once $\\Omega $ is chosen, we can record the randomness of $C_{j,L}$ as $\\Omega _{(j,~\\cdot ~)}=\\lbrace \\Omega _{(j,l)}\\colon \\, l\\in {\\mathbb {Z}}\\rbrace .$ We will also need to consider the randomness of the $j$ -th column seen by a trajectory that enters ${\\mathcal {C}}_{j,L}$ through the block $\\Lambda _{j,k}$ with $k\\ne 0$ instead of $k=0$ .",
"In this case, the randomness of ${\\mathcal {C}}_{j,L}$ is recorded as $\\Omega _{(j,k+~\\cdot ~)}=\\lbrace \\Omega _{(j,k+l)}\\colon \\, l\\in {\\mathbb {Z}}\\rbrace .$ Pick $L\\in {\\mathbb {N}}$ , $\\chi \\in \\lbrace A,B\\rbrace ^{\\mathbb {Z}}$ and consider ${\\mathcal {C}}_{0,L}$ endowed with the disorder $\\chi $ , i.e., $\\Omega (0,\\cdot )=\\chi $ .",
"Let $(n_i)_{i\\in {\\mathbb {Z}}}\\in {\\mathbb {Z}}^{\\mathbb {Z}}$ be the successive heights of the $AB$ -interfaces in ${\\mathcal {C}}_{0,L}$ divided by $L$ , i.e., $\\dots <n_{-1}<n_0\\le 0< n_1<n_2<\\dots .$ and the $j$ -th interface of ${\\mathcal {C}}_{0,L}$ is ${\\mathcal {I}}_j=\\lbrace 0,\\dots ,L\\rbrace \\times \\lbrace n_j L\\rbrace $ (see Fig.",
"REF ).",
"Next, for $r\\in {\\mathbb {N}}_0$ we set $k_{r,\\chi }=&\\,0 \\text{ if } n_1>r \\text{ and }k_{r,\\chi }=\\max \\lbrace i\\ge 1\\colon n_i\\le r\\rbrace \\text{ otherwise},$ while for $r\\in -{\\mathbb {N}}$ we set $\\quad \\quad \\quad \\quad k_{r,\\chi }=&\\, 0 \\text{ if }n_0\\le r\\text{ and } k_{r,\\chi }=\\min \\lbrace i\\le 0\\colon n_{i}\\ge r+1\\rbrace -1 \\text{ otherwise}.$ Thus, $|k_{r,\\chi }|$ is the number of $AB$ -interfaces between heigths 1 and $r L$ in ${\\mathcal {C}}_{0,L}$ .",
"Figure: Example of a column with disorder χ=(⋯,χ(-3),χ(-2),χ(-1),χ(0),χ(1),χ(2),\\chi =(\\dots ,\\chi (-3),\\chi (-2),\\chi (-1),\\chi (0),\\chi (1),\\chi (2), ⋯)=(⋯,B,A,B,B,B,A,,⋯)\\dots )=(\\dots ,B,A,B,B,B,A,,\\dots ).",
"In this example, forinstance, k -2,χ =-1k_{-2,\\chi }=-1 and k 1,χ =0k_{1,\\chi }=0."
],
[
"Free energy in a single column",
"Column crossing characteristics.",
"Pick $L,M\\in {\\mathbb {N}}$ , and consider the first column ${\\mathcal {C}}_{0,L}$ .",
"The type of ${\\mathcal {C}}_{0,L}$ is determined by $\\Theta =(\\chi ,\\Xi ,x)$ , where $\\chi =(\\chi _j)_{j\\in {\\mathbb {Z}}}$ encodes the type of each block in ${\\mathcal {C}}_{0,L}$ , i.e., $\\chi _j=\\Omega _{(0,j)}$ for $j\\in {\\mathbb {Z}}$ , and $(\\Xi ,x)$ indicates which trajectories $\\pi $ are taken into account.",
"In the latter, $\\Xi $ is given by $(\\Delta \\Pi ,b_0,b_1)$ such that the vertical increment in ${\\mathcal {C}}_{0,L}$ on the block scale is $\\Delta \\Pi $ and satisfies $|\\Delta \\Pi |\\le M$ , i.e., $\\pi $ enters ${\\mathcal {C}}_{0,L}$ at $(0,b_0 L)$ and exits ${\\mathcal {C}}_{0,L}$ at $(L,(\\Delta \\Pi +b_1)L)$ .",
"As in (REF ) and (REF ), we set $k_\\Theta =k_{\\Delta \\Pi ,\\chi }$ and we let ${\\mathcal {V}}_\\mathrm {int}$ be the set containing those $\\Theta $ satisfying $k_\\Theta \\ne 0$ .",
"Thus, $\\Theta \\in {\\mathcal {V}}_\\mathrm {int}$ means that the trajectories crossing ${\\mathcal {C}}_{0,L}$ from $(0,b_0 L)$ to $(L,(\\Delta \\Pi +b_1)L)$ necessarily hit an $AB$ -interface, and in this case we set $x=1$ .",
"If, on the other hand, $\\Theta \\in {\\mathcal {V}}_{\\mathrm {nint}}={\\mathcal {V}}\\setminus {\\mathcal {V}}_{\\mathrm {int}}$ , then we have $k_\\Theta = 0$ and we set $x=1$ when the set of trajectories crossing ${\\mathcal {C}}_{0,L}$ from $(0,b_0 L)$ to $(L,(\\Delta \\Pi +b_1)L)$ is restricted to those that do not reach an $AB$ -interface before exiting ${\\mathcal {C}}_{0,L}$ , while we set $x=2$ when it is restricted to those trajectories that reach at least one $AB$ -interface before exiting ${\\mathcal {C}}_{0,L}$ .",
"To fix the possible values taken by $\\Theta =(\\chi ,\\Xi ,x)$ in a column of width $L$ , we put ${\\mathcal {V}}_{L,M}={\\mathcal {V}}_{\\mathrm {int},L,M}\\cup {\\mathcal {V}}_{\\mathrm {nint},L,M}$ with $\\nonumber {\\mathcal {V}}_{\\mathrm {int},L,M}&=\\big \\lbrace (\\chi ,\\Delta \\Pi ,b_0,b_1,x)\\in \\lbrace A,B\\rbrace ^{\\mathbb {Z}}\\times {\\mathbb {Z}}\\times \\big \\lbrace \\tfrac{1}{L},\\tfrac{2}{L},\\dots ,1\\big \\rbrace ^2\\times \\lbrace 1\\rbrace \\colon \\\\\\nonumber &\\hspace{170.71652pt}\\qquad \\qquad \\qquad \\qquad |\\Delta \\Pi |\\le M,\\,k_{\\Delta \\Pi ,\\chi }\\ne 0\\big \\rbrace ,\\\\\\nonumber {\\mathcal {V}}_{\\mathrm {nint},L,M}&=\\big \\lbrace (\\chi ,\\Delta \\Pi ,b_0,b_1,x)\\in \\lbrace A,B\\rbrace ^{\\mathbb {Z}}\\times {\\mathbb {Z}}\\times \\big \\lbrace \\tfrac{1}{L},\\tfrac{2}{L},\\dots ,1\\big \\rbrace ^2\\times \\lbrace 1,2\\rbrace \\colon \\\\&\\hspace{170.71652pt}\\qquad \\qquad \\qquad \\qquad |\\Delta \\Pi |\\le M,\\, k_{\\Delta \\Pi ,\\chi }= 0\\big \\rbrace .$ Thus, the set of all possible values of $\\Theta $ is ${\\mathcal {V}}_M=\\cup _{L\\ge 1} {\\mathcal {V}}_{L,M}$ , which we partition into ${\\mathcal {V}}_M={\\mathcal {V}}_{\\mathrm {int},M}\\cup {\\mathcal {V}}_{\\mathrm {nint},M}$ (see Fig.",
"REF ) with $\\nonumber {\\mathcal {V}}_{\\mathrm {int},M}&=\\cup _{L\\in {\\mathbb {N}}}\\ {\\mathcal {V}}_{\\mathrm {int},L,M}\\\\\\nonumber &= \\big \\lbrace (\\chi ,\\Delta \\Pi ,b_0,b_1,x)\\in \\lbrace A,B\\rbrace ^{\\mathbb {Z}}\\times {\\mathbb {Z}}\\times (\\mathbb {Q}_{(0,1]})^2\\times \\lbrace 1\\rbrace \\colon \\,\\,|\\Delta \\Pi |\\le M,\\, k_{\\Delta \\Pi ,\\chi }\\ne 0\\big \\rbrace ,\\\\\\nonumber {\\mathcal {V}}_{\\mathrm {nint},M}&=\\cup _{L\\in {\\mathbb {N}}}\\ {\\mathcal {V}}_{\\mathrm {nint},L,M}\\\\&= \\big \\lbrace (\\chi ,\\Delta \\Pi ,b_0,b_1,x)\\in \\lbrace A,B\\rbrace ^{\\mathbb {Z}}\\times {\\mathbb {Z}}\\times (\\mathbb {Q}_{(0,1]})^2\\times \\lbrace 1,2\\rbrace \\colon \\,|\\Delta \\Pi |\\le M,\\, k_{\\Delta \\Pi ,\\chi }= 0\\big \\rbrace ,$ where, for all $I\\subset {\\mathbb {R}}$ , we set $\\mathbb {Q}_{I}=I\\cap \\mathbb {Q}$ .",
"We define the closure of ${\\mathcal {V}}_M$ as $\\overline{{\\mathcal {V}}}_M=\\overline{{\\mathcal {V}}}_{\\mathrm {int},M}\\cup \\overline{{\\mathcal {V}}}_{\\mathrm {nint},M}$ with $\\nonumber &\\overline{{\\mathcal {V}}}_{\\mathrm {int},M}= \\big \\lbrace (\\chi ,\\Delta \\Pi ,b_0,b_1,x)\\in \\lbrace A,B\\rbrace ^{\\mathbb {Z}}\\times {\\mathbb {Z}}\\times [0,1]^2\\times \\lbrace 1\\rbrace \\colon \\,|\\Delta \\Pi |\\le M,\\, k_{\\Delta \\Pi ,\\chi }\\ne 0\\big \\rbrace ,\\\\&\\overline{{\\mathcal {V}}}_{\\mathrm {nint},M}=\\big \\lbrace (\\chi ,\\Delta \\Pi ,b_0,b_1,x)\\in \\lbrace A,B\\rbrace ^{\\mathbb {Z}}\\times {\\mathbb {Z}}\\times [0,1]^2\\times \\lbrace 1,2\\rbrace \\colon \\,|\\Delta \\Pi |\\le M,\\, k_{\\Delta \\Pi ,\\chi }= 0\\big \\rbrace .$ Figure: Labelling of coarse-grained paths and columns.",
"On the left the type of thecolumn is in 𝒱 int ,M {\\mathcal {V}}_{\\mathrm {int},M}, on the right it is in 𝒱 nint ,M {\\mathcal {V}}_{\\mathrm {nint},M} (with M≥6M\\ge 6).Time spent in columns.",
"We pick $L,M\\in {\\mathbb {N}}$ , $\\Theta =(\\chi ,\\Delta \\Pi ,b_0,b_1,x)\\in {\\mathcal {V}}_{L,M}$ and we specify the total number of steps that a trajectory crossing the column ${\\mathcal {C}}_{0,L}$ of type $\\Theta $ is allowed to make.",
"For $\\Theta =(\\chi ,\\Delta \\Pi ,b_0,b_1,1)$ , set $t_{\\Theta }= 1+\\mathrm {sign} (\\Delta \\Pi )\\,(\\Delta \\Pi + b_1-b_0) \\,{1}_{\\lbrace \\Delta \\Pi \\ne 0\\rbrace } + |b_1-b_0|\\,{1}_{\\lbrace \\Delta \\Pi =0\\rbrace },$ so that a trajectory $\\pi $ crossing a column of width $L$ from $(0,b_0 L)$ to $(L,(\\Delta \\Pi +b_1)L)$ makes a total of $uL$ steps with $u\\in t_{\\Theta }+\\tfrac{2{\\mathbb {N}}}{L}$ .",
"For $\\Theta =(\\chi ,\\Delta \\Pi ,b_0,b_1,2)$ in turn, recall (REF ) and let $t_{\\Theta }= 1+\\mathrm {min}\\lbrace 2 n_1-b_0-b_1-\\Delta \\Pi , 2 |n_0| +b_0+b_1+\\Delta \\Pi \\rbrace ,$ so that a trajectory $\\pi $ crossing a column of width $L$ and type $\\Theta \\in {\\mathcal {V}}_{\\mathrm {nint},L,M}$ from $(0,b_0 L)$ to $(L,(\\Delta \\Pi +b_1)L)$ and reaching an $AB$ -interface makes a total of $uL$ steps with $u\\in t_{\\Theta }+\\tfrac{2{\\mathbb {N}}}{L}$ .",
"Figure: Example of a uLuL-step path inside a column of type (χ,ΔΠ,b 0 ,b 1 ,1)∈𝒱 int ,L (\\chi ,\\Delta \\Pi ,b_0,b_1,1)\\in {\\mathcal {V}}_{\\mathrm {int},L} with disorder χ=(⋯,χ(0),χ(1),χ(2),⋯)=(⋯,A,B,A,⋯)\\chi =(\\dots ,\\chi (0),\\chi (1),\\chi (2),\\dots )=(\\dots ,A,B,A,\\dots ), vertical displacement ΔΠ=2\\Delta \\Pi =2, entrance height b 0 b_0 and exit heightb 1 b_1.Figure: Two examples of a uLuL-step path inside a column of type (χ,ΔΠ,b 0 ,b 1 ,1)∈𝒱 nint ,L (\\chi ,\\Delta \\Pi ,b_0,b_1,1)\\in {\\mathcal {V}}_{\\mathrm {nint},L} (left picture) and (χ,ΔΠ,b 0 ,b 1 ,2)∈𝒱 nint ,L (\\chi ,\\Delta \\Pi ,b_0,b_1,2)\\in {\\mathcal {V}}_{\\mathrm {nint},L}(right picture) with disorder χ=(⋯,χ(0),χ(1),χ(2),χ(3),χ(4),⋯)=(⋯,B,B,B,B,A,⋯)\\chi =(\\dots ,\\chi (0),\\chi (1),\\chi (2),\\chi (3),\\chi (4),\\dots )=(\\dots ,B,B,B,B,A,\\dots ), vertical displacement ΔΠ=2\\Delta \\Pi =2, entranceheight b 0 b_0 and exit height b 1 b_1.At this stage, we can fully determine the set ${\\mathcal {W}}_{\\Theta ,u,L}$ consisting of the $uL$ -step trajectories $\\pi $ that are considered in a column of width $L$ and type $\\Theta $ .",
"To that end, for $\\Theta \\in {\\mathcal {V}}_{\\mathrm {int},L,M}$ we map the trajectories $\\pi \\in {\\mathcal {W}}_{L}(u,\\Delta \\Pi +b_1-b_0)$ onto ${\\mathcal {C}}_{0,L}$ such that $\\pi $ enters ${\\mathcal {C}}_{0,L}$ at $(0,b_0 L)$ and exits ${\\mathcal {C}}_{0,L}$ at $(L,(\\Delta \\Pi +b_1)L)$ (see Fig.",
"REF ), and for $\\Theta \\in {\\mathcal {V}}_{\\mathrm {nint},L,M}$ we remove, dependencing on $x\\in \\lbrace 1,2\\rbrace $ , those trajectories that reach or do not reach an $AB$ -interface in the column (see Fig.",
"REF ).",
"Thus, for $\\Theta \\in {\\mathcal {V}}_{\\mathrm {int},L,M}$ and $u\\in t_{\\Theta }+\\tfrac{2{\\mathbb {N}}}{L}$ , we let ${\\mathcal {W}}_{\\Theta ,u,L}=\\big \\lbrace \\pi =(0,b_0 L)+\\widetilde{\\pi }\\colon \\,\\widetilde{\\pi }\\in {\\mathcal {W}}_{L}(u,\\Delta \\Pi +b_1-b_0) \\big \\rbrace ,$ and, for $\\Theta \\in {\\mathcal {V}}_{\\mathrm {nint},L,M}$ and $u\\in t_{\\Theta }+\\tfrac{2{\\mathbb {N}}}{L}$ , $\\nonumber {\\mathcal {W}}_{\\Theta ,u,L}&=\\big \\lbrace \\pi \\in (0,b_0 L)+ {\\mathcal {W}}_{L}(u,\\Delta \\Pi +b_1-b_0)\\colon \\,\\text{$\\pi $ reaches no $AB$-interface} \\big \\rbrace \\ \\text{if}\\ x_\\Theta =1,\\\\{\\mathcal {W}}_{\\Theta ,u,L}&=\\big \\lbrace \\pi \\in (0,b_0 L)+{\\mathcal {W}}_{L}(u,\\Delta \\Pi +b_1-b_0)\\colon \\,\\text{$\\pi $ reaches an $AB$-interface} \\big \\rbrace \\ \\text{if}\\ x_\\Theta =2,$ with $x_\\Theta $ the last coordinate of $\\Theta \\in {\\mathcal {V}}_M$ .",
"Next, we set $\\nonumber {\\mathcal {V}}_{L,M}^* &= \\Big \\lbrace (\\Theta ,u)\\in {\\mathcal {V}}_{L,M} \\times [0,\\infty )\\colon \\,u\\in t_{\\Theta }+\\tfrac{2{\\mathbb {N}}}{L}\\Big \\rbrace ,\\\\\\nonumber {\\mathcal {V}}_M^*&=\\big \\lbrace (\\Theta ,u)\\in {\\mathcal {V}}_M \\times \\mathbb {Q}_{[1,\\infty )}\\colon \\,u\\ge t_{\\Theta }\\big \\rbrace ,\\\\\\overline{{\\mathcal {V}}}_M^{*}&=\\big \\lbrace (\\Theta ,u)\\in \\overline{{\\mathcal {V}}}_M\\times [1,\\infty ) \\colon \\,u\\ge t_{\\Theta }\\big \\rbrace ,$ which we partition into ${\\mathcal {V}}^{*}_{\\mathrm {int},L,M}\\cup {\\mathcal {V}}^{*}_{\\mathrm {nint},L,M}$ , ${\\mathcal {V}}^*_{\\mathrm {int},M}\\cup {\\mathcal {V}}^*_{\\mathrm {nint},M}$ and $\\overline{{\\mathcal {V}}}^{*}_{\\mathrm {int},M}\\cup \\overline{{\\mathcal {V}}}^{*}_{\\mathrm {nint},M}$ .",
"Note that for every $(\\Theta ,u)\\in {\\mathcal {V}}^*_M$ there are infinitely many $L\\in {\\mathbb {N}}$ such that $(\\Theta ,u)\\in {\\mathcal {V}}_{L,M}^*$ , because $(\\Theta ,u)\\in {\\mathcal {V}}_{qL,M}^*$ for all $q\\in {\\mathbb {N}}$ as soon as $(\\Theta ,u)\\in {\\mathcal {V}}_{L,M}^*$ .",
"Restriction on the number of steps per column.",
"In what follows, we set ${\\hbox{\\footnotesize \\rm EIGH}}= \\lbrace (M,m)\\in {\\mathbb {N}}\\times {\\mathbb {N}}\\colon \\,m\\ge M+2\\rbrace ,$ and, for $(M,m)\\in {\\hbox{\\footnotesize \\rm EIGH}}$ , we consider the situation where the number of steps $uL$ made by a trajectory $\\pi $ in a column of width $L\\in {\\mathbb {N}}$ is bounded by $m L$ .",
"Thus, we restrict the set ${\\mathcal {V}}_{L,M}$ to the subset ${\\mathcal {V}}_{L,M}^{\\,m}$ containing only those types of columns $\\Theta $ that can be crossed in less than $m L$ steps, i.e., ${\\mathcal {V}}_{L,M}^{\\,m}=\\lbrace \\Theta \\in {\\mathcal {V}}_{L,M}\\colon \\, t_\\Theta \\le m\\rbrace .$ Note that the latter restriction only conconcerns those $\\Theta $ satisfying $x_\\Theta =2$ .",
"When $x_\\Theta =1$ a quick look at (REF ) suffices to state that $t_\\Theta \\le M+2\\le m$ .",
"Thus, we set ${\\mathcal {V}}_{L,M}^{\\,m}={\\mathcal {V}}_{\\mathrm {int},L,M}^{\\,m}\\cup {\\mathcal {V}}_{\\mathrm {nint},L,M}^{\\,m}$ with ${\\mathcal {V}}_{\\mathrm {int},L,M}^{\\,m}={\\mathcal {V}}_{\\mathrm {int},L,M}$ and with $\\nonumber {\\mathcal {V}}_{\\mathrm {nint},L,M}^{\\,m}= \\Big \\lbrace \\Theta \\in \\lbrace A,B\\rbrace ^{{\\mathbb {Z}}}\\times {\\mathbb {Z}}\\times \\big \\lbrace \\tfrac{1}{L},\\tfrac{2}{L},\\dots ,1\\big \\rbrace ^2&\\times \\lbrace 1,2\\rbrace \\colon \\\\& |\\Delta \\Pi | \\le M,\\ k_{\\Theta }= 0\\ \\ \\text{and}\\ t_\\Theta \\le m\\Big \\rbrace .$ The sets ${\\mathcal {V}}_{M}^{\\,m}={\\mathcal {V}}_{\\mathrm {int},M}^{\\,m}\\cup {\\mathcal {V}}_{\\mathrm {nint},M}^{\\,m}$ and $\\overline{{\\mathcal {V}}}_{M}^{\\,m}=\\overline{{\\mathcal {V}}}_{\\mathrm {int},M}^{\\,m}\\cup \\overline{{\\mathcal {V}}}_{\\mathrm {nint},M}^{\\,m}$ are obtained by mimicking (REF –REF ).",
"In the same spirit, we restrict ${\\mathcal {V}}_{L,M}^{*}$ to ${\\mathcal {V}}_{L,M}^{*,\\,m}=\\lbrace (\\Theta ,u)\\in {\\mathcal {V}}_{L,M}^{*}\\colon \\, \\Theta \\in {\\mathcal {V}}_{L,M}^{\\,m}, u\\le m\\rbrace $ and ${\\mathcal {V}}_{L,M}^*={\\mathcal {V}}_{\\mathrm {int},L,M}^*\\cup {\\mathcal {V}}_{\\mathrm {nint},L,M}^*$ with $\\begin{aligned}{\\mathcal {V}}_{\\mathrm {int},L,M}^{*,\\,m} &= \\Big \\lbrace (\\Theta ,u)&\\in {\\mathcal {V}}_{\\mathrm {int},L,M}^{\\,m} \\times [1,m]\\colon \\,u\\in t_{\\Theta }+\\tfrac{2{\\mathbb {N}}}{L}\\Big \\rbrace ,\\\\{\\mathcal {V}}_{\\mathrm {nint},L,M}^{*\\,m} &= \\Big \\lbrace (\\Theta ,u)&\\in {\\mathcal {V}}_{\\mathrm {nint},L,M}^{\\,m}\\times [1,m]\\colon \\,u\\in t_{\\Theta }+\\tfrac{2{\\mathbb {N}}}{L} \\Big \\rbrace .\\end{aligned}$ We set also ${\\mathcal {V}}_M^{*,\\,m}={\\mathcal {V}}^{*,\\,m}_{\\mathrm {int},M}\\cup {\\mathcal {V}}^{*,\\,m}_{\\mathrm {nint},M}$ with ${\\mathcal {V}}_{\\mathrm {int},M}^{*,\\,m}=\\cup _{L\\in {\\mathbb {N}}}{\\mathcal {V}}^{*,\\,m}_{\\mathrm {int}, L,M}$ and ${\\mathcal {V}}^{*,\\,m}_{\\mathrm {nint},M}=\\cup _{L\\in {\\mathbb {N}}} {\\mathcal {V}}^{*,\\,m}_{\\mathrm {nint},L,M}$ , and rewrite these as $\\nonumber {\\mathcal {V}}^{*,\\,m}_{\\mathrm {int},M}=\\big \\lbrace (\\Theta ,u)&\\in {\\mathcal {V}}_{\\mathrm {int},M}^{\\,m} \\times \\mathbb {Q}_{[1,m]}\\colon \\,u\\ge t_{\\Theta }\\big \\rbrace ,\\\\{\\mathcal {V}}_{\\mathrm {nint},M}^{*,\\,m}= \\big \\lbrace (\\Theta ,u)&\\in {\\mathcal {V}}_{\\mathrm {nint},M}^{\\,m} \\times \\mathbb {Q}_{[1,m]}\\colon \\,u\\ge t_{\\Theta }\\big \\rbrace .$ We further set $\\overline{{\\mathcal {V}}}^{\\,*}_M = \\overline{{\\mathcal {V}}}^{\\,*,\\,m}_{\\mathrm {int},M}\\cup \\overline{{\\mathcal {V}}}^{\\,*,\\,m}_{\\mathrm {nint},M}$ with $\\begin{aligned}\\overline{{\\mathcal {V}}}^{\\,*,\\,m}_{\\mathrm {int},M}&=\\big \\lbrace (\\Theta ,u)\\in \\overline{{\\mathcal {V}}}_{\\mathrm {int},M}^{\\,m} \\times [1,m]\\colon \\,u\\ge t_{\\Theta }\\big \\rbrace ,\\\\\\overline{{\\mathcal {V}}}_{\\mathrm {nint},M}^{\\,*,\\,m}&= \\Big \\lbrace (\\Theta ,u) \\in \\overline{{\\mathcal {V}}}_{\\mathrm {nint},M}^{\\,m} \\times [1,m]\\colon \\,u\\ge t_{\\Theta }\\Big \\rbrace .\\end{aligned}$ Existence and uniform convergence of free energy per column.",
"Recall (REF ), (REF ) and, for $L\\in {\\mathbb {N}}$ , $\\omega \\in \\lbrace A,B\\rbrace ^{\\mathbb {N}}$ and $(\\Theta ,u)\\in {\\mathcal {V}}^{\\,*}_{L,M}$ , we associate with each $\\pi \\in {\\mathcal {W}}_{\\Theta ,u,L}$ the energy $H_{uL,L}^{\\omega ,\\chi }(\\pi )= \\sum _{i=1}^{uL} \\big (\\beta \\, 1\\left\\lbrace \\omega _i=B\\right\\rbrace -\\alpha \\, 1\\left\\lbrace \\omega _i=A\\right\\rbrace \\big )\\,1\\Big \\lbrace \\chi ^{L}_{(\\pi _{i-1},\\pi _i)}=B\\Big \\rbrace ,$ where $\\chi ^{L}_{(\\pi _{i-1},\\pi _i)}$ indicates the label of the block containing $(\\pi _{i-1},\\pi _i)$ in a column with disorder $\\chi $ of width $L$ .",
"(Recall that the disorder in the block is part of the type of the block.)",
"The latter allows us to define the quenched free energy per monomer in a column of type $\\Theta $ and size $L$ as $\\psi ^{\\omega }_L(\\Theta ,u)= \\frac{1}{u L} \\log Z^{\\omega }_L(\\Theta ,u)\\quad \\text{with} \\quad Z^{\\omega }_L(\\Theta ,u)=\\sum _{\\pi \\in {\\mathcal {W}}_{\\Theta ,u,L}} e^{\\,H_{uL,L}^{\\omega ,\\chi }(\\pi )}.$ Abbreviate $\\psi _L(\\Theta ,u)={\\mathbb {E}}[\\psi ^\\omega _L(\\Theta ,u)]$ , and note that for $M\\in {\\mathbb {N}}$ , $m\\ge M+2$ and $(\\Theta ,u)\\in {\\mathcal {V}}_{L,M}^{\\,*,\\,m}$ all $\\pi \\in {\\mathcal {W}}_{\\Theta ,u,L}$ necessarily remain in the blocks $\\Lambda _L(0,i)$ with $i\\in \\lbrace -m+1,\\dots ,m-1\\rbrace $ .",
"Consequently, the dependence on $\\chi $ of $\\psi ^{\\omega }_L(\\Theta ,u)$ is restricted to those coordinates of $\\chi $ indexed by $\\lbrace -m+1,\\dots ,m-1\\rbrace $ .",
"The following proposition will be proven in Section .",
"Proposition 2.4 For every $M\\in {\\mathbb {N}}$ and $(\\Theta ,u)\\in {\\mathcal {V}}_M^*$ there exists a $\\psi (\\Theta ,u)\\in {\\mathbb {R}}$ such that $\\lim _{ {L\\rightarrow \\infty } \\atop {(\\Theta ,u)\\in {\\mathcal {V}}_{L,M}^*} } \\psi ^{\\omega }_L(\\Theta ,u)= \\psi (\\Theta ,u) = \\psi (\\Theta ,u;\\alpha ,\\beta ) \\quad \\omega -a.s.$ Moreover, for every $(M,m)\\in {\\hbox{\\footnotesize \\rm EIGH}}$ the convergence is uniform in $(\\Theta ,u)\\in {\\mathcal {V}}^{*,\\,m}_M$ .",
"Uniform bound on the free energies.",
"Pick $(\\alpha ,\\beta )\\in {\\hbox{\\footnotesize \\rm CONE}}$ , $n\\in {\\mathbb {N}}$ , $\\omega \\in \\lbrace A,B\\rbrace ^{\\mathbb {N}}$ , $\\Omega \\in \\lbrace A,B\\rbrace ^{{\\mathbb {N}}_0\\times {\\mathbb {Z}}}$ , and let $\\bar{{\\mathcal {W}}}_n$ be any non-empty subset of ${\\mathcal {W}}_n$ (recall (REF )).",
"Note that the quenched free energies per monomer introduced until now are all of the form $\\psi _n=\\tfrac{1}{n}\\log \\sum _{\\pi \\in \\bar{{\\mathcal {W}}}_n} e^{\\,H_n(\\pi )},$ where $H_n(\\pi )$ may depend on $\\omega $ and $\\Omega $ and satisfies $-\\alpha n\\le H_n(\\pi )\\le \\alpha n$ for all $\\pi \\in \\bar{{\\mathcal {W}}}_n$ (recall that $|\\beta |\\le \\alpha $ in ${\\hbox{\\footnotesize \\rm CONE}}$ ).",
"Since $1\\le |\\bar{{\\mathcal {W}}}_n|\\le |{\\mathcal {W}}_n|\\le 3^n$ , we have $|\\psi _n|\\le \\log 3+\\alpha =^\\mathrm {def} C_{\\text{uf}}(\\alpha ).$ The uniformity of this bound in $n$ , $\\omega $ and $\\Omega $ allows us to average over $\\omega $ and/or $\\Omega $ or to let $n\\rightarrow \\infty $ ."
],
[
"Variational formulas for the free energy in a single column",
"We next show how the free energies per column can be expressed in terms of two variational formulas involving the path entropy and the single interface free energy defined in Section REF .",
"Note that $M\\in {\\mathbb {N}}$ is given until the end of the section.",
"Free energy in columns of class $\\mathrm {int}$ .",
"Pick $\\Theta \\in {\\mathcal {V}}_{\\mathrm {int},M}$ and put $\\nonumber l_1&=1_{\\lbrace \\Delta \\Pi >0\\rbrace }(n_1-b_0)+1_{\\lbrace \\Delta \\Pi <0\\rbrace } (b_0-n_0),\\\\\\nonumber l_j&=1_{\\lbrace \\Delta \\Pi >0\\rbrace }(n_j-n_{j-1})+1_{\\lbrace \\Delta \\Pi <0\\rbrace } (n_{-j+2}-n_{-j+1}) \\quad \\text{for} \\quad j\\in \\lbrace 2,\\dots ,|k_\\Theta |\\rbrace ,\\\\l_{|k_\\Theta |+1}&=1_{\\lbrace \\Delta \\Pi >0\\rbrace }(\\Delta \\Pi +b_1-n_{k_\\Theta })+1_{\\lbrace \\Delta \\Pi <0\\rbrace } (n_{k_\\Theta +1}-\\Delta \\Pi -b_1),$ i.e., $l_1$ is the vertical distance between the entrance point and the first interface, $l_{i}$ is the vertical distance between the $i$ -th interface and the $(i+1)$ -th interface, and $l_{|k_\\Theta |+1}$ is the vertical distance between the last interface and the exit point.",
"Denote by $(h)$ and $(a)$ the triples $(h_A,h_B,h^{\\mathcal {I}})$ and $(a_A,a_B,a^{\\mathcal {I}})$ .",
"For $(l_A,l_B)\\in (0,\\infty )^2$ and $u\\ge l_A+l_B+1$ , put $\\nonumber {\\mathcal {L}}(l_A,l_B; u)=\\big \\lbrace (h),(a)\\in [0,1]^3\\times [0,\\infty )^3\\colon \\,&h_A+h_B+h^{\\mathcal {I}}=1,\\, a_A+a_B+a^{\\mathcal {I}}=u \\\\& a_A\\ge h_A+l_A,\\, a_B\\ge h_B+l_B,\\, a^{\\mathcal {I}}\\ge h^{\\mathcal {I}}\\big \\rbrace .$ With the help of (REF ) and (REF ) we can now provide a variational characterization of the free energy in columns of type $\\Theta $ of class $\\mathrm {int}$ .",
"Let $l_A(\\chi ,\\Delta \\Pi ,b_0,b_1)$ and $l_B(\\chi ,\\Delta \\Pi ,b_0,b_1)$ correspond to the minimal vertical distance the copolymer must cross in blocks of type $A$ and $B$ , respectively, in a column with disorder $\\chi $ when going from $(0,b_0)$ to $(1,\\Delta \\Pi +b_1)$ , i.e., $\\nonumber l_A(\\chi ,\\Delta \\Pi ,b_0,b_1)&=1_{\\lbrace \\Delta \\Pi >0\\rbrace } \\sum _{j=1}^{|k_\\Theta |+1} l_j 1_{\\lbrace \\chi (n_{j-1})=A\\rbrace }+1_{\\lbrace \\Delta \\Pi <0\\rbrace } \\sum _{j=1}^{|k_\\Theta |+1} l_j 1_{\\lbrace \\chi (n_{-j+1})=A\\rbrace }, \\\\l_B(\\chi ,\\Delta \\Pi ,b_0,b_1)&=1_{\\lbrace \\Delta \\Pi >0\\rbrace } \\sum _{j=1}^{|k_\\Theta |+1} l_j 1_{\\lbrace \\chi (n_{j-1})=B\\rbrace }+1_{\\lbrace \\Delta \\Pi <0\\rbrace } \\sum _{j=1}^{|k_\\Theta |+1} l_j 1_{\\lbrace \\chi (n_{-j+1})=B\\rbrace }.$ The following proposition will be proven in Section .",
"Proposition 2.5 For $(\\Theta ,u)\\in {\\mathcal {V}}^*_{\\mathrm {int},M}$ , $\\nonumber \\psi (\\Theta ,u)&=\\psi _{\\mathrm {int}}(u,l_A,l_B)\\\\&=\\sup _{(h),(a) \\in {\\mathcal {L}}(l_A,\\, l_B;\\,u)}\\frac{a_A\\, \\tilde{\\kappa }\\big (\\tfrac{a_A}{h_A},\\tfrac{l_A}{h_A}\\big )+a_B\\,\\big [\\tilde{\\kappa }\\big (\\tfrac{a_B}{h_B},\\tfrac{l_B}{h_B}\\big )+\\tfrac{\\beta -\\alpha }{2}\\big ]+a^{\\mathcal {I}}\\,\\phi ^{\\mathcal {I}}(\\tfrac{a^{\\mathcal {I}}}{h^{\\mathcal {I}}})}{u}.$ Free energy in columns of class $\\mathrm {nint}$ .",
"Pick $\\Theta \\in {\\mathcal {V}}_{\\mathrm {nint},M}$ .",
"In this case, there is no $AB$ -interface between $b_0$ and $\\Delta \\Pi +b_1$ , which means that $\\Delta \\Pi < n_1$ if $\\Delta \\Pi \\ge 0$ and $\\Delta \\Pi \\ge n_0$ if $\\Delta \\Pi <0$ ($n_0$ and $n_1$ being defined in (REF )).",
"Let $l_{\\mathrm {nint}}(\\Delta \\Pi ,b_0,b_1)$ be the vertical distance between the entrance point $(0,b_0)$ and the exit point $(1,\\Delta \\Pi +b_1)$ , i.e., $l_{\\mathrm {nint}}(\\Delta \\Pi ,b_0,b_1) &= 1_{\\lbrace \\Delta \\Pi \\ge 0\\rbrace } (\\Delta \\Pi -b_0+b_1)+ 1_{\\lbrace \\Delta \\Pi <0\\rbrace } (|\\Delta \\Pi |+b_0-b_1)+ 1_{\\lbrace \\Delta \\Pi =0\\rbrace } |b_1-b_0|,$ and let $l_\\mathrm {int}(\\chi ,\\Delta \\Pi ,b_0,b_1)$ be the minimal vertical distance a trajectory has to cross in a column with disorder $\\chi $ , starting from $(0,b_0)$ , to reach the closest $AB$ -interface before exiting at $(1,\\Delta \\Pi +b_1)$ , i.e., $l_\\mathrm {int}(\\chi ,\\Delta \\Pi ,b_0,b_1)&=\\mathrm {min}\\lbrace 2 n_1-b_0-b_1-\\Delta \\Pi , 2 |n_0| +b_0+b_1+\\Delta \\Pi \\rbrace .$ The following proposition will be proved in Section .",
"Proposition 2.6 For $(\\Theta ,u)\\in {\\mathcal {V}}^*_{\\mathrm {nint},M}$ such that $x_\\Theta =1$ , $\\psi (\\Theta ,u)=\\tilde{\\kappa }(u,l_{\\mathrm {nint}})+\\tfrac{\\beta -\\alpha }{2}\\,1_{\\lbrace \\chi (0)=B\\rbrace }.$ For $(\\Theta ,u)\\in {\\mathcal {V}}^*_{\\mathrm {nint},M}$ such that $x_\\Theta =2$ , $\\nonumber \\psi (\\Theta ,u)&=\\psi _{\\mathrm {nint}}(u,l_\\mathrm {int};\\,\\chi (0))\\\\&=\\sup _{\\stackrel{h^{\\mathcal {I}}\\in [0,1],}{u^{\\mathcal {I}}\\in [h^{\\mathcal {I}}, u+h^{\\mathcal {I}}-1-l_\\mathrm {int}]}}\\frac{(u-u^{\\mathcal {I}}) \\big [\\tilde{\\kappa }\\big (\\tfrac{u-u^{\\mathcal {I}}}{1-h^{\\mathcal {I}}},\\tfrac{l_\\mathrm {int}}{1-h^{\\mathcal {I}}}\\big )+\\tfrac{\\beta -\\alpha }{2}\\,1_{\\lbrace \\chi (0)=B\\rbrace }\\big ]+u^{\\mathcal {I}}\\phi ^{\\mathcal {I}}(\\tfrac{u^{\\mathcal {I}}}{h^{\\mathcal {I}}})}{u}.$ The importance of Propositions REF –REF is that they express the free energy in a single column in terms of the path entropy in a single column $\\tilde{\\kappa }$ and the free energy along a single linear interface $\\phi ^{\\mathcal {I}}$, which were defined in Section REF and are well understood."
],
[
"Mesoscopic percolation frequencies",
"In this section, we define a set of probability laws providing the frequencies with which each type of column can be crossed by the copolymer.",
"Coarse-grained paths.",
"For $x\\in {\\mathbb {N}}_0\\times {\\mathbb {Z}}$ and $n\\in {\\mathbb {N}}$ , let $c_{x,n}$ denote the center of the block $\\Lambda _{L_n}(x)$ defined in (REF ), i.e., $c_{x,n}=x L_n+(\\tfrac{1}{2},\\tfrac{1}{2}) L_n,$ and abbreviate $({\\mathbb {N}}_0\\times {\\mathbb {Z}})_n=\\lbrace c_{x,n}\\colon \\, x\\in {\\mathbb {N}}_0\\times {\\mathbb {Z}}\\rbrace .$ Let $\\widehat{{\\mathcal {W}}}$ be the set of coarse-grained paths on $({\\mathbb {N}}_0\\times {\\mathbb {Z}})_n$ that start at $c_{0,n}$ , are self-avoiding and are allowed to jump up, down and to the right between neighboring sites of $({\\mathbb {N}}_0\\times {\\mathbb {Z}})_n$ , i.e., the increments of $\\widehat{\\Pi } = (\\widehat{\\Pi }_j)_{j\\in {\\mathbb {N}}_0} \\in \\widehat{{\\mathcal {W}}}$ are $(0,L_n),(0,-L_n)$ and $(L_n,0)$ .",
"(These paths are the coarse-grained counterparts of the paths $\\pi $ introduced in (REF ).)",
"For $l\\in {\\mathbb {N}}\\cup \\lbrace \\infty \\rbrace $ , let $\\widehat{{\\mathcal {W}}}_l$ be the set of $l$ -step coarse-grained paths.",
"Recall, for $\\pi \\in {\\mathcal {W}}_{n}$ , the definitions of $N_\\pi $ and $(v_j(\\pi ))_{j\\le N_\\pi -1}$ given below (REF ).",
"With $\\pi $ we associate a coarse-grained path $\\widehat{\\Pi }\\in \\widehat{{\\mathcal {W}}}_{N_\\pi }$ that describes how $\\pi $ moves with respect to the blocks.",
"The construction of $\\widehat{\\Pi }$ is done as follows: $\\widehat{\\Pi }_0=c_{(0,0)}$ , $\\widehat{\\Pi }$ moves vertically until it reaches $c_{(0,v_0)}$ , moves one step to the right to $c_{(1,v_0)}$ , moves vertically until it reaches $c_{(1,v_1)}$ , moves one step to the right to $c_{(2,v_1)}$ , and so on.",
"The vertical increment of $\\widehat{\\Pi }$ in the $j$ -th column is $\\Delta \\widehat{\\Pi }_j=(v_j-v_{j-1}) L_n$ (see Figs.",
"REF –REF ).",
"Figure: Example of a coarse-grained path.To characterize a path $\\pi $ , we will often use the sequence of vertical increments of its associated coarse-grained path $\\widehat{\\Pi }$ , modified in such a way that it does not depend on $L_n$ anymore.",
"To that end, with every $\\pi \\in {\\mathcal {W}}_n$ we associate $\\Pi =(\\Pi _k)_{k=0}^{N_\\pi -1}$ such that $\\Pi _0=0$ and, $\\Pi _k=\\sum _{j=0}^{k-1} \\Delta \\Pi _j\\quad \\text{with} \\quad \\Delta \\Pi _j=\\frac{1}{L_n} \\Delta \\widehat{\\Pi }_j, \\qquad j = 0,\\dots ,N_\\pi -1.$ Pick $M\\in {\\mathbb {N}}$ and note that $\\pi \\in {\\mathcal {W}}_{n,M}$ if and only if $|\\Delta \\Pi _j|\\le M$ for all $j\\in \\lbrace 0,\\dots , N_\\pi -1\\rbrace $ .",
"Percolation frequencies along coarse-grained paths.",
"Given $M\\in {\\mathbb {N}}$ , we denote by ${\\mathcal {M}}_1(\\overline{{\\mathcal {V}}}_M)$ the set of probability measures on $\\overline{{\\mathcal {V}}}_M$ .",
"Pick $\\Omega \\in \\lbrace A,B\\rbrace ^{{\\mathbb {N}}_0\\times {\\mathbb {Z}}}$ , $\\Pi \\in {\\mathbb {Z}}^{{\\mathbb {N}}_0}$ such that $\\Pi _0=0$ and $|\\Delta \\Pi _i|\\le M$ for all $i\\ge 0$ and $b=(b_j)_{j\\in {\\mathbb {N}}_0}\\in (\\mathbb {Q}_{(0,1]})^{{\\mathbb {N}}_0}$ .",
"Set $\\Theta _{\\text{traj}}=(\\Xi _j)_{j\\in {\\mathbb {N}}_0}$ with $\\Xi _j=\\big (\\Delta \\Pi _j,b_j,b_{j+1}\\big ), \\qquad j\\in {\\mathbb {N}}_0,$ let ${\\mathcal {X}}_{\\,\\Pi ,\\Omega }=\\big \\lbrace x\\in \\lbrace 1,2\\rbrace ^{{\\mathbb {N}}_0}\\colon \\,(\\Omega (i,\\Pi _i+\\cdot ),\\Xi _i,x_i)\\in {\\mathcal {V}}_M \\,\\,\\, \\forall \\,i\\in {\\mathbb {N}}_0\\big \\rbrace ,$ and for $x\\in {\\mathcal {X}}_{\\,\\Pi ,\\Omega }$ set $\\Theta _j=\\big (\\Omega (j,\\Pi _j+\\cdot ),\\Delta \\Pi _j,b_j,b_{j+1},x_j\\big ), \\qquad j\\in {\\mathbb {N}}_0.$ With the help of (REF ), we can define the empirical distribution $\\rho _N(\\Omega ,\\Pi ,b,x)(\\Theta ) = \\frac{1}{N} \\sum _{j=0}^{N-1}1_{\\lbrace \\Theta _j=\\Theta \\rbrace },\\quad N\\in {\\mathbb {N}},\\,\\Theta \\in \\overline{{\\mathcal {V}}}_M,$ Definition 2.7 For $\\Omega \\in \\lbrace A,B\\rbrace ^{{\\mathbb {N}}_0\\times {\\mathbb {Z}}}$ and $M\\in {\\mathbb {N}}$ , let $\\begin{aligned}{\\mathcal {R}}^\\Omega _{M,N} &= \\big \\lbrace \\rho _N(\\Omega ,\\Pi ,b,x)\\ \\text{with}\\ b=(b_j)_{j\\in {\\mathbb {N}}_0} \\in (\\mathbb {Q}_{(0,1]})^{{\\mathbb {N}}_0},\\\\&\\qquad \\Pi =(\\Pi _j)_{j\\in {\\mathbb {N}}_0} \\in \\lbrace 0\\rbrace \\times {\\mathbb {Z}}^{{\\mathbb {N}}} \\colon \\,|\\Delta \\Pi _j|\\le M\\,\\ \\ \\forall \\,j\\in {\\mathbb {N}}_0,\\\\&\\qquad x=(x_j)_{j\\in {\\mathbb {N}}_0} \\in \\lbrace 1,2\\rbrace ^{N_0}\\colon \\, \\big (\\Omega (j,\\Pi _j+\\cdot ),\\Delta \\Pi _j,b_j,b_{j+1},x_j\\big )\\in {\\mathcal {V}}_M\\big \\rbrace \\end{aligned}$ and ${\\mathcal {R}}^\\Omega _M = \\mathrm {closure}\\Big (\\cap _{N^{\\prime }\\in {\\mathbb {N}}} \\cup _{N \\ge N^{\\prime }}\\,{\\mathcal {R}}^\\Omega _{M,N}\\Big ),$ both of which are subsets of ${\\mathcal {M}}_1(\\overline{{\\mathcal {V}}}_M)$ .",
"Proposition 2.8 For every $p \\in (0,1)$ and $M\\in {\\mathbb {N}}$ there exists a closed set ${\\mathcal {R}}_{p,M}\\subsetneq {\\mathcal {M}}_1(\\overline{{\\mathcal {V}}}_M)$ such that ${\\mathcal {R}}_{M}^\\Omega ={\\mathcal {R}}_{p,M} \\text{ for } {\\mathbb {P}}\\text{-a.e.",
"}\\,\\Omega .$ Proof.",
"Note that, for every $\\Omega \\in \\lbrace A,B\\rbrace ^{{\\mathbb {N}}_0\\times {\\mathbb {Z}}}$ , the set ${\\mathcal {R}}_{M}^\\Omega $ does not change when finitely many variables in $\\Omega $ are changed.",
"Therefore ${\\mathcal {R}}_{M}^\\Omega $ is measurable with respect to the tail $\\sigma $ -algebra of $\\Omega $ .",
"Since $\\Omega $ is an i.i.d.",
"random field, the claim follows from Kolmogorov's zero-one law.",
"Because of the constraint on the vertical displacement, ${\\mathcal {R}}_{p,M}$ does not coincide with ${\\mathcal {M}}_1(\\overline{{\\mathcal {V}}}_M)$ .",
"$\\square $"
],
[
"Proof of Propositions ",
"In this section we prove Propositions REF and REF –REF , which were stated in Sections REF –REF and contain the precise definition of the key ingredients of the variational formula in Theorem REF .",
"In Section we will use these propositions to prove Theorem REF .",
"In Section REF we associate with each trajectory $\\pi $ in a column a sequence recording the indices of the $AB$ -interfaces successively visited by $\\pi $ .",
"The latter allows us to state a key proposition, Proposition REF below, from which Propositions REF and REF –REF are straightforward consequences.",
"In Section REF we give an outline of the proof of Proposition REF , in Sections REF –REF we provide the details."
],
[
"The order of the visits to the interfaces",
"Pick $(M,m)\\in {\\hbox{\\footnotesize \\rm EIGH}}$ .",
"To prove Proposition REF , instead of considering $(\\Theta ,u)\\in {\\mathcal {V}}^{*,\\,m}_M$ , we will restrict to $(\\Theta ,u)\\in {\\mathcal {V}}^{*,\\,m}_{\\text{int},M}$ .",
"Our proof can be easily extended to $(\\Theta ,u)\\in {\\mathcal {V}}^{*,\\,m}_{\\text{nint},M}$ .",
"Pick $(\\Theta ,u)\\in {\\mathcal {V}}^{*,\\,m}_{\\mathrm {int},M}$ , recall (REF ) and set ${\\mathcal {J}}_{\\Theta ,u}=\\lbrace {\\mathcal {N}}_{\\Theta ,u}^{\\downarrow },\\dots ,{\\mathcal {N}}_{\\Theta ,u}^{\\uparrow }\\rbrace ,$ with ${\\mathcal {N}}^{\\uparrow }_{\\Theta ,u}= &\\max \\lbrace i\\ge 1\\colon n_i\\le u\\rbrace \\quad \\text{and}\\quad {\\mathcal {N}}^{\\uparrow }_{\\Theta ,u}=0\\quad \\text{if}\\quad n_1>u.",
"\\\\\\nonumber {\\mathcal {N}}^{\\downarrow }_{\\Theta ,u}=&\\min \\lbrace i\\le 0\\colon |n_i|\\le u\\rbrace \\quad \\text{and}\\quad {\\mathcal {N}}^{\\downarrow }_{\\Theta ,u}=1\\quad \\text{if}\\quad |n_0|>u.$ Next pick $L\\in {\\mathbb {N}}$ so that $(\\Theta ,u)\\in {\\mathcal {V}}^{*}_{\\mathrm {int},L,M}$ and recall that for $j\\in {\\mathcal {J}}_{\\Theta ,u}$ the $j$ -th interface of the $\\Theta $ -column is ${\\mathcal {I}}_j=\\lbrace 0,\\dots ,L\\rbrace \\times \\lbrace n_j L\\rbrace $ .",
"Note also that $\\pi \\in {\\mathcal {W}}_{\\Theta ,u,L}$ makes $uL$ steps inside the column and therefore can not reach the $AB$ -interfaces labelled outside $\\lbrace {\\mathcal {N}}^{\\downarrow }_{\\Theta ,u},\\dots ,{\\mathcal {N}}^{\\uparrow }_{\\Theta ,u}\\rbrace $ .",
"First, we associate with each trajectory $\\pi \\in {\\mathcal {W}}_{\\Theta ,u,L}$ the sequence $J(\\pi )$ that records the indices of the interfaces that are successively visited by $\\pi $ .",
"Next, we pick $\\pi \\in {\\mathcal {W}}_{\\Theta ,u,L}$ , and define $\\tau _1, J_1$ as $\\tau _1=\\inf \\lbrace i\\in {\\mathbb {N}}\\colon \\, \\exists j\\in {\\mathcal {J}}_{\\Theta ,u}\\colon \\,\\pi _i\\in {\\mathcal {I}}_j \\rbrace , \\qquad \\pi _{\\tau _1}\\in {\\mathcal {I}}_{J_1},$ so that $J_1=0$ (respectively, $J_1=1$ ) if the first interface reached by $\\pi $ is ${\\mathcal {I}}_0$ (respectively, ${\\mathcal {I}}_1$ ).",
"For $i\\in {\\mathbb {N}}\\setminus \\lbrace 1\\rbrace $ , we define $\\tau _i,J_i$ as $\\tau _i=\\inf \\big \\lbrace t>\\tau _{i-1}\\colon \\,\\exists j\\in {\\mathcal {J}}_{\\Theta ,u}\\setminus \\lbrace J_{i-1}\\rbrace , \\pi _i\\in {\\mathcal {I}}_j\\big \\rbrace ,\\qquad \\pi _{\\tau _i}\\in {\\mathcal {I}}_{J_i},$ so that the increments of $J(\\pi )$ are restricted to $-1$ or 1.",
"The length of $J(\\pi )$ is denoted by $m(\\pi )$ and corresponds to the number of jumps made by $\\pi $ between neighboring interfaces before time $uL$ , i.e., $J(\\pi )=(J_i)_{i=1}^{m(\\pi )}$ with $m(\\pi )=\\max \\lbrace i\\in {\\mathbb {N}}\\colon \\,\\tau _i\\le uL\\rbrace .$ Note that $(\\Theta ,u)\\in {\\mathcal {V}}_{\\mathrm {int},M}^{*,\\,m}$ necessarily implies $k_{\\Theta }\\le m(\\pi )\\le u\\le m$ .",
"Set ${\\mathcal {S}}_r=\\lbrace j=(j_i)_{i=1}^r\\in {\\mathbb {Z}}^{\\mathbb {N}}\\colon \\, j_1\\in \\lbrace 0,1\\rbrace ,\\,j_{i+1}-j_i\\in \\lbrace -1,1\\rbrace \\,\\,\\forall \\,1\\le i\\le r-1\\rbrace , \\qquad r\\in {\\mathbb {N}},$ and, for $\\Theta \\in {\\mathcal {V}}$ , $r\\in \\lbrace 1,\\dots , m\\rbrace $ and $j\\in {\\mathcal {S}}_r$ , define $\\nonumber l_1&=1_{\\lbrace j_1=1\\rbrace } (n_{1}-b_0)+1_{\\lbrace j_1= 0\\rbrace } (b_0-n_{0}),\\\\\\nonumber l_i&=|n_{j_i}-n_{j_{i-1}}| \\text{ for } i\\in \\lbrace 2,\\dots ,r\\rbrace ,\\\\l_{r+1}&=1_{\\lbrace j_r=k_\\Theta +1\\rbrace } (n_{k_\\Theta +1}-\\Delta \\Pi -b_1)+1_{\\lbrace j_r= k_\\Theta \\rbrace } (\\Delta \\Pi +b_1-n_{k_\\Theta }),$ so that $(l_i)_{i\\in \\lbrace 1,\\dots ,r+1\\rbrace }$ depends on $\\Theta $ and $j$ .",
"Set ${\\mathcal {A}}_{\\Theta ,j}&=\\lbrace i\\in \\lbrace 1,\\dots ,r+1\\rbrace \\colon \\,\\text{$A$ between}\\,{\\mathcal {I}}_{j_{i-1}}\\,\\text{and}\\,{\\mathcal {I}}_{j_{i}}\\rbrace ,\\\\\\nonumber {\\mathcal {B}}_{\\Theta ,j}&=\\lbrace i\\in \\lbrace 1,\\dots ,r+1\\rbrace \\colon \\,\\text{$B$ between}\\,{\\mathcal {I}}_{j_{i-1}}\\,\\text{and} \\ {\\mathcal {I}}_{j_{i}}\\rbrace ,$ and set $l_{\\Theta ,j}=(l_{A,\\Theta ,j},l_{B,\\Theta ,j})$ with $l_{A,\\Theta ,j}&={\\textstyle \\sum _{i\\in {\\mathcal {A}}_{\\Theta ,j}}} l_i,\\,\\,l_{B,\\Theta ,j}={\\textstyle \\sum _{i\\in {\\mathcal {B}}_{\\Theta ,j}}} l_i.$ For $L\\in {\\mathbb {N}}$ and $(\\Theta ,u)\\in {\\mathcal {V}}^{*,\\,m}_{\\mathrm {int},L,M}$ , we denote by ${\\mathcal {S}}_{\\Theta ,u,L}$ the set $\\lbrace J(\\pi ), \\pi \\in {\\mathcal {W}}_{\\Theta ,u,L}\\rbrace $ .",
"It is not difficult to see that a sequence $j\\in {\\mathcal {S}}_r$ belongs to ${\\mathcal {S}}_{\\Theta ,u,L}$ if and only if it satisfies the two following conditions.",
"First, $j_r \\in \\lbrace k_\\Theta ,k_\\Theta +1\\rbrace $ , since $j_r$ is the index of the interface last visited before the $\\Theta $ -column is exited.",
"Second, $u\\ge 1+l_{A,\\Theta ,j}+l_{B,\\Theta ,j}$ because the number of steps taken by a trajectory $\\pi \\in {\\mathcal {W}}_{\\Theta ,u,L}$ satisfying $J(\\pi )=j$ must be large enough to ensures that all interfaces ${\\mathcal {I}}_{j_s}$ , $s\\in \\lbrace 1,\\dots ,r\\rbrace $ , can be visited by $\\pi $ before time $uL$ .",
"Consequently, ${\\mathcal {S}}_{\\Theta ,u,L}$ does not depend on $L$ and can be written as ${\\mathcal {S}}_{\\Theta ,u}= \\cup _{r=1}^{m}{\\mathcal {S}}_{\\Theta ,u,r}$ , where ${\\mathcal {S}}_{\\Theta ,u,r}=\\lbrace j\\in {\\mathcal {S}}_r\\colon j_r\\in \\lbrace k_\\Theta ,k_\\Theta +1\\rbrace ,u \\ge 1+l_{A,\\Theta ,j}+l_{B,\\Theta ,j}\\rbrace .$ Thus, we partition ${\\mathcal {W}}_{\\Theta ,u,L}$ according to the value taken by $J(\\pi )$ , i.e., ${\\mathcal {W}}_{\\Theta ,u,L}=\\bigcup _{r=1}^{m} \\ \\bigcup _{j\\in {\\mathcal {S}}_{\\Theta ,u,r}} \\ {\\mathcal {W}}_{\\Theta ,u,L,j},$ where ${\\mathcal {W}}_{\\Theta ,u,L,j}$ contains those trajectories $\\pi \\in {\\mathcal {W}}_{\\Theta ,u,L}$ for which $J(\\pi )=j$ .",
"Next, for $j \\in {\\mathcal {S}}_{\\Theta ,u}$ , we define (recall (REF )) $\\psi ^{\\omega }_L(\\Theta ,u,j)= \\frac{1}{u L} \\log Z^{\\omega }_L(\\Theta ,u,j), \\qquad \\psi _L(\\Theta ,u,j)={\\mathbb {E}}\\big [\\psi ^{\\omega }_L(\\Theta ,u,j)\\big ],$ with $Z^{\\omega }_L(\\Theta ,u,j)=\\sum _{\\pi \\in {\\mathcal {W}}_{\\Theta ,u,L,j}} e^{H_{uL,L}^{\\omega ,\\chi }(\\pi )}.$ For each $L\\in {\\mathbb {N}}$ satisfying $(\\Theta ,u)\\in {\\mathcal {V}}^{*,\\,m}_{\\mathrm {int},L,M}$ and each $j\\in {\\mathcal {S}}_{\\Theta ,u}$ , the quantity $l_{A,\\Theta ,j} L$ (respectively, $l_{B,\\Theta ,j} L$ ) corresponds to the minimal vertical distance a trajectory $\\pi \\in {\\mathcal {W}}_{\\Theta ,u,L,j}$ has to cross in solvent $A$ (respectively, $B$ )."
],
[
"Key proposition",
"Recalling (REF ) and (REF ), we define the free energy associated with $\\Theta ,u,j$ as $\\psi (\\Theta ,u,j)&=\\psi _{\\mathrm {int}}(u,l_{\\Theta ,j})\\\\ \\nonumber &=\\sup _{(h),(u) \\in {\\mathcal {L}}(l_{\\Theta ,j};\\, u)}\\frac{u_A\\, \\tilde{\\kappa }\\big (\\tfrac{u_A}{h_A},\\tfrac{l_{A,\\Theta ,j}}{h_A}\\big )+u_B\\,\\big [\\tilde{\\kappa }\\big (\\tfrac{u_B}{h_B},\\tfrac{l_{B,\\Theta ,j}}{h_B}\\big )+\\tfrac{\\beta -\\alpha }{2}\\big ]+u_I\\, \\phi (\\tfrac{u^I}{h^I})}{u}.$ Proposition REF below states that $\\lim _{L\\rightarrow \\infty } \\psi _L(\\Theta ,u,j)=\\psi (\\Theta ,u,j)$ uniformly in $(\\Theta ,u) \\in {\\mathcal {V}}_{\\mathrm {int},M}^{*,\\,m}$ and $j\\in {\\mathcal {S}}_{\\Theta ,u}$ .",
"Proposition 3.1 For every $M,m\\in {\\mathbb {N}}$ such that $m\\ge M+2$ and every ${\\varepsilon }>0$ there exists an $L_{\\varepsilon }\\in {\\mathbb {N}}$ such that $\\big |\\psi _L(\\Theta ,u,j)-\\psi (\\Theta ,u,j)\\big |\\le {\\varepsilon }\\quad \\forall \\,(\\Theta ,u)\\in {\\mathcal {V}}^{*,\\,m}_{\\mathrm {int},L,M},\\ \\,j\\in {\\mathcal {S}}_{\\Theta ,u},\\ \\, L\\ge L_{\\varepsilon }.$ Proof of Propositions REF and REF –REF subject to Proposition REF.",
"Pick ${\\varepsilon }>0$ , $L\\in {\\mathbb {N}}$ and $(\\Theta ,u)\\in {\\mathcal {V}}^{*,\\,m}_{\\mathrm {int},L,M}$ .",
"Recall (REF ) and note that $l_A(\\Theta )L$ and $l_B(\\Theta )L$ are the minimal vertical distances the trajectories of ${\\mathcal {W}}_{\\Theta ,u,L}$ have to cross in blocks of type $A$ , respectively, $B$ .",
"For simplicity, in what follows the $\\Theta $ -dependence of $l_A$ and $l_B$ will be suppressed.",
"In other words, $l_A$ and $l_B$ are the two coordinates of $l_{\\Theta ,f}$ (recall (REF )) with $f=(1,2,\\dots ,|k_{\\Theta }|)$ when $\\Delta \\Pi \\ge 0$ and $f=(0,-1,\\dots ,-|k_{\\Theta }|+1)$ when $\\Delta \\Pi < 0$ , so (REF ) and (REF ) imply $\\psi _{\\mathrm {int}}(u,l_A,l_B)= \\psi (\\Theta ,u,f).$ Hence Propositions REF and REF will be proven once we show that $\\lim _{L\\rightarrow \\infty } \\psi _L(\\Theta ,u)=\\psi (\\Theta ,u,f)$ uniformly in $(\\Theta ,u)\\in {\\mathcal {V}}^{*,\\,m}_{\\mathrm {int},L,M}$ .",
"Moreover, a look at (REF ), (REF ) and (REF ) allows us to assert that for every $j\\in {\\mathcal {S}}_{\\Theta ,u}$ we have $\\psi (\\Theta ,u,j)\\le \\psi (\\Theta ,u,f)$ .",
"The latter is a consequence of the fact that $l\\mapsto \\tilde{\\kappa }(u,l)$ decreases on $[0,u-1]$ (see Lemma REF (ii) in Appendix ) and that $\\nonumber l_{A}&=l_{A,\\Theta ,f}=\\min \\lbrace l_{A,\\Theta ,j} \\colon \\, j\\in {\\mathcal {S}}_{\\Theta ,u}\\rbrace ,\\\\l_{B}&=l_{B,\\Theta ,f}=\\min \\lbrace l_{B,\\Theta ,j} \\colon \\, j\\in {\\mathcal {S}}_{\\Theta ,u}\\rbrace .$ By applying Proposition REF we have, for $L\\ge L_{\\varepsilon }$ , $\\nonumber \\psi _L(\\Theta ,u,j)&\\le \\psi (\\Theta ,u,f)+{\\varepsilon }\\qquad \\forall \\,(\\Theta ,u)\\in {\\mathcal {V}}^{*,\\,m}_{\\mathrm {int},L,M},\\ \\forall \\,j\\in {\\mathcal {S}}_{\\Theta ,u},\\\\\\psi _L(\\Theta ,u,f)&\\ge \\psi (\\Theta ,u,f)-{\\varepsilon }\\qquad \\forall \\,(\\Theta ,u)\\in {\\mathcal {V}}^{*,\\,m}_{\\mathrm {int},L,M}.$ The second inequality in (REF ) allows us to write, for $L\\ge L_{\\varepsilon }$ , $\\psi (\\Theta ,u,f)-{\\varepsilon }\\le \\psi _L(\\Theta ,u,f)\\le \\psi _L(\\Theta ,u) \\qquad \\forall \\,(\\Theta ,u)\\in {\\mathcal {V}}^{*,\\,m}_{\\mathrm {int},L,M}.$ To obtain the upper bound we introduce ${\\mathcal {A}}_{L,{\\varepsilon }}=\\Big \\lbrace \\omega \\colon \\, |\\psi ^\\omega _L(\\Theta ,u,j)-\\psi _L(\\Theta ,u,j)|\\le {\\varepsilon }\\ \\quad \\forall \\,(\\Theta ,u)\\in {\\mathcal {V}}^{*,\\,m}_{\\mathrm {int},L,M},\\, \\forall \\,j\\in {\\mathcal {S}}_{\\Theta ,u}\\Big \\rbrace ,$ so that $\\psi _L(\\Theta ,u)&\\le {\\mathbb {E}}\\big [ 1_{{\\mathcal {A}}^c_{L,{\\varepsilon }}}\\, \\psi ^\\omega _L(\\Theta ,u)\\big ]+ {\\mathbb {E}}\\big [ 1_{{\\mathcal {A}}_{L,{\\varepsilon }}}\\, \\psi ^\\omega _L(\\Theta ,u)\\big ]\\\\\\nonumber &\\le C_{\\text{uf}}(\\alpha )\\,{\\mathbb {P}}({\\mathcal {A}}_{L,{\\varepsilon }}^c)+\\tfrac{1}{uL}{\\mathbb {E}}\\Big [ 1_{{\\mathcal {A}}_{L,{\\varepsilon }}}\\,\\log {\\textstyle \\sum _{j\\in {\\mathcal {S}}_{\\Theta ,u}}}\\,e^{uL (\\psi _L(\\Theta ,u,j)+{\\varepsilon })}\\Big ],$ where we use (REF ) to bound the first term in the right-hand side, and the definition of ${\\mathcal {A}}_{L,{\\varepsilon }}$ to bound the second term.",
"Next, with the help of the first inequality in (REF ) we can rewrite (REF ) for $L\\ge L_{\\varepsilon }$ and $(\\Theta ,u)\\in {\\mathcal {V}}^{*,\\,m}_{\\mathrm {int},L,M}$ in the form $\\psi _L(\\Theta ,u)&\\le C_{\\text{uf}}(\\alpha )\\, {\\mathbb {P}}({\\mathcal {A}}_{L,{\\varepsilon }}^c)+\\tfrac{1}{uL}\\log |\\cup _{r=1}^{m} {\\mathcal {S}}_{r}|+ \\psi (\\Theta ,u,f)+2{\\varepsilon }.$ At this stage we want to prove that $\\lim _{L\\rightarrow \\infty } {\\mathbb {P}}({\\mathcal {A}}^c_{L,{\\varepsilon }})=0$ .",
"To that end, we use the concentration of measure property in (REF ) in Appendix with $l=uL$ , $\\Gamma ={\\mathcal {W}}_{\\Theta ,u,L,j}$ , $\\eta ={\\varepsilon }uL$ , $\\xi _i=-\\alpha 1\\lbrace \\omega _i=A\\rbrace +\\beta 1\\lbrace \\omega _i=B\\rbrace $ for all $i\\in {\\mathbb {N}}$ and $T(x,y)=1\\lbrace \\chi ^{L_n}_{(x,y)}=B\\rbrace $ .",
"We then obtain that there exist $C_{1},C_{2}>0$ such that, for all $L\\in {\\mathbb {N}}$ , $(\\Theta ,u)\\in {\\mathcal {V}}^{*,\\,m}_{\\mathrm {int},L,M}$ and $j\\in {\\mathcal {S}}_{\\Theta ,u}$ , ${\\mathbb {P}}\\big ( |\\psi ^\\omega _L(\\Theta ,u,j)-\\psi _L(\\Theta ,u,j)|> {\\varepsilon }\\big )\\le C_{1} \\, e^{-C_{2}\\, {\\varepsilon }^2\\, uL}.$ The latter inequality, combined with the fact that $|{\\mathcal {V}}^{*,\\,m}_{\\mathrm {int},L,M}|$ grows polynomialy in $L$ , allows us to assert that $\\lim _{L\\rightarrow \\infty } {\\mathbb {P}}({\\mathcal {A}}^c_{L,{\\varepsilon }})=0$ .",
"Next, we note that $|\\cup _{r=1}^{m} {\\mathcal {S}}_{r}|<\\infty $ , so that for $L_{\\varepsilon }$ large enough we obtain from (REF ) that, for $L\\ge L_{\\varepsilon }$ , $\\psi _L(\\Theta ,u) \\le \\psi (\\Theta ,u,f)+3{\\varepsilon }\\qquad \\forall \\,(\\Theta ,u)\\in {\\mathcal {V}}^{*,\\,m}_{\\mathrm {int},L,M}.$ Now (REF ) and (REF ) are sufficient to complete the proof of Propositions REF –REF .",
"The proof of Proposition REF follows in a similar manner after minor modifications.",
"$\\square $"
],
[
"Structure of the proof of Proposition ",
"Intermediate column free energies.",
"Let $G_M^{\\,m}=\\big \\lbrace (L,\\Theta ,u,j)\\colon \\,(\\Theta ,u)\\in {\\mathcal {V}}^{*,\\,m}_{\\mathrm {int},L,M},\\, j\\in {\\mathcal {S}}_{\\Theta ,u}\\big \\rbrace ,$ and define the following order relation.",
"Definition 3.2 For $g,\\widetilde{g}\\colon \\,G_M^{\\,m}\\mapsto {\\mathbb {R}}$ , write $g\\prec \\widetilde{g}$ when for every ${\\varepsilon }>0$ there exists an $L_{\\varepsilon }\\in {\\mathbb {N}}$ such that $g(L,\\Theta ,u,j)\\le \\widetilde{g}(L,\\Theta ,u,j)+{\\varepsilon }\\qquad \\forall \\,(L,\\Theta ,u,j)\\in G_M^{\\,m}\\colon \\, L\\ge L_{\\varepsilon }.$ Recall (REF ) and (REF ), set $\\psi _1(L,\\Theta ,u,j) = \\psi _L(\\Theta ,u,j), \\qquad \\psi _4(L,\\Theta ,u,j)=\\psi (\\Theta ,u,j),$ and note that the proof of Proposition REF will be complete once we show that $\\psi _1\\prec \\psi _4$ and $\\psi _4\\prec \\psi _1$ .",
"In what follows, we will focus on $\\psi _1\\prec \\psi _4$ .",
"Each step of the proof can be adapted to obtain $\\psi _4\\prec \\psi _1$ without additional difficulty.",
"In the proof we need to define two intermediate free energies $\\psi _2$ and $\\psi _3$ , in addition to $\\psi _1$ and $\\psi _4$ above.",
"Our proof is divided into 3 steps, organized in Sections REF –REF , and consists of showing that $\\psi _1\\prec \\psi _2\\prec \\psi _3\\prec \\psi _4$ .",
"Additional notation.",
"Before stating Step 1, we need some further notation.",
"First, we partition ${\\mathcal {W}}_{\\Theta ,u,L,j}$ according to the total number of steps and the number of horizontal steps made by a trajectory along and in between $AB$ -interfaces.",
"To that end, we assume that $j\\in {\\mathcal {S}}_{\\Theta ,u,r}$ with $r\\in \\lbrace 1,\\dots ,m\\rbrace $ , we recall (REF ) and we let $\\nonumber {\\mathcal {D}}_{\\Theta ,L,j}&=\\big \\lbrace (d_i,t_i)_{i=1}^{r+1}\\colon \\, d_i\\in {\\mathbb {N}}\\,\\,\\text{and}\\,\\,t_i\\in d_i+l_i L+2{\\mathbb {N}}_0\\,\\,\\forall \\,1\\le i\\le r+1\\big \\rbrace ,\\\\{\\mathcal {D}}_{r}^{\\mathcal {I}}&=\\big \\lbrace (d_i^{\\mathcal {I}},t_i^{\\mathcal {I}})_{i=1}^r\\colon \\,d_i^{\\mathcal {I}}\\in {\\mathbb {N}}\\,\\,\\text{and}\\,\\,t_i^{\\mathcal {I}}\\in d_i^{\\mathcal {I}}+2{\\mathbb {N}}_0\\,\\,\\forall \\, 1\\le i\\le r\\big \\rbrace ,$ where $d_i,t_i$ denote the number of horizontal steps and the total number of steps made by the trajectory between the $(i-1)$ -th and $i$ -th interfaces, and $d_i^{\\mathcal {I}},t_i^{\\mathcal {I}}$ denote the number of horizontal steps and the total number of steps made by the trajectory along the $i$ -th interface.",
"For $(d,t)\\in {\\mathcal {D}}_{\\Theta ,L,j}$ , $(d^{\\mathcal {I}},t^{\\mathcal {I}})\\in {\\mathcal {D}}_{r}^{\\mathcal {I}}$ and $1\\le i\\le r$ , we set $T_0=0$ and $\\nonumber V_{i}&=\\sum _{j=1}^i t_j+\\sum _{j=1}^{i-1} t_j^{{\\mathcal {I}}}, \\qquad i = 1,\\dots ,r,\\\\T_{i}&=\\sum _{j=1}^{i} t_j+\\sum _{j=1}^{i} t_j^{{\\mathcal {I}}}, \\qquad i = 1,\\dots ,r,$ so that $V_i$ , respectively, $T_i$ indicates the number of steps made by the trajectory when reaching, respectively, leaving the $i$ -th interface.",
"Next, we let $\\theta \\colon \\,{\\mathbb {R}}^{\\mathbb {N}}\\mapsto {\\mathbb {R}}^{\\mathbb {N}}$ be the left-shift acting on infinite sequences of real numbers and, for $u\\in {\\mathbb {N}}$ and $\\omega \\in \\lbrace A,B\\rbrace ^{\\mathbb {N}}$ , we put $H_u^\\omega (B)=\\sum _{i=1}^u \\big [\\beta \\,1_{\\lbrace \\omega _i=B\\rbrace }-\\alpha \\,1_{\\lbrace \\omega _i=A\\rbrace }\\big ].$ Finally, we recall that $\\psi _1(L,\\Theta ,u,j)=\\tfrac{1}{ uL}\\,\\mathbb {E}[\\log Z_1^{\\omega }(L,\\Theta ,u,j)],$ where the partition function defined in (REF ) has been renamed $Z_1$ and can be written in the form $Z^{\\omega }_1(L,\\Theta ,u,j)=\\sum _{(d,t)\\in {\\mathcal {D}}_{\\Theta ,L,j}}\\,\\sum _{(d^{\\mathcal {I}},t^{\\mathcal {I}})\\in {\\mathcal {D}}_{r}^{\\mathcal {I}}} A_1\\,B_1\\,C_1,$ where (recall (REF ) and (REF )) $A_1 &=\\prod _{i\\in {\\mathcal {A}}_{\\Theta ,j}}\\,e^{t_{i}\\, \\tilde{\\kappa }_{d_i}\\big (\\tfrac{t_{i}}{d_i},\\,\\tfrac{l_{i} L}{d_i}\\big )}\\,\\prod _{i\\in {\\mathcal {B}}_{\\Theta ,j}}\\,e^{t_{i}\\,\\tilde{\\kappa }_{d_i}\\big (\\tfrac{t_{i}}{d_i},\\,\\tfrac{ l_{i} L}{d_i}\\big )}\\,e^{H^{\\theta ^{T_{i-1}}(w)}_{t_{i}}(B)},\\\\\\nonumber B _1&=\\prod _{i=1}^{r}\\,e^{t_{i}^{\\mathcal {I}}\\,\\phi ^{\\theta ^{V_{i}}(w)}_{d^{{\\mathcal {I}}}_{i}}\\big (\\tfrac{t^{{\\mathcal {I}}}_{i}}{d_{i}^{\\mathcal {I}}}\\big )},\\\\\\nonumber C _1&= {1}_{\\big \\lbrace \\sum _{i=1}^{r+1} d_i+\\sum _{i=1}^{r} d_i^{\\mathcal {I}}=L\\big \\rbrace }\\,{1}_{\\big \\lbrace \\sum _{i=1}^{r+1} t_i+\\sum _{i=1}^{r} t_i^{\\mathcal {I}}=u L\\big \\rbrace }.$ It is important to note that a simplification has been made in the term $A_1$ in (REF ).",
"Indeed, this term is not $\\tilde{\\kappa }_{d_i}(\\cdot ,\\cdot )$ defined in (REF ), since the latter does not take into account the vertical restrictions on the path when it moves from one interface to the next.",
"However, the fact that two neighboring $AB$ -interfaces are necessarily separated by a distance at least $L$ allows us to apply Lemma REF in Appendix REF , which ensures that these vertical restrictions can be removed at the cost of a negligible error.",
"To show that $\\psi _1\\prec \\psi _2\\prec \\psi _3\\prec \\psi _4$ , we fix $(M,m)\\in {\\hbox{\\footnotesize \\rm EIGH}}$ and ${\\varepsilon }>0$ , and we show that there exists an $L_{\\varepsilon }\\in {\\mathbb {N}}$ s such that $\\psi _k(L,\\Theta ,u,j)\\le \\psi _{k+1}(L,\\Theta ,u,j)+{\\varepsilon }$ for all $(L,\\Theta ,u,j)\\in G_M^{\\,m}$ and $L\\ge L_{\\varepsilon }$ .",
"The latter will complete the proof of Proposition REF ."
],
[
"Step 1",
"In this step, we remove the $\\omega $ -dependence from $Z_1^{\\,\\omega }(L,\\Theta ,u,j)$ .",
"To that aim, we put $\\psi _2(L,\\Theta ,u,j)=\\frac{1}{ uL} \\log Z_2(L,\\Theta ,u,j)$ with $Z_2(L,\\Theta ,u,j) = \\sum _{(d,t)\\in {\\mathcal {D}}_{\\Theta ,L,j}}\\,\\sum _{(d^{\\mathcal {I}},t^{\\mathcal {I}})\\in {\\mathcal {D}}_{r}^{\\mathcal {I}}} A_2 \\ B_2 \\ C_2,$ where $A_2 &= \\prod _{i\\in {\\mathcal {A}}_{\\Theta ,j}}\\,e^{t_{i}\\, \\tilde{\\kappa }_{d_i}\\big (\\tfrac{t_{i}}{d_i},\\,\\tfrac{ l_{i} L}{d_i}\\big )}\\prod _{i\\in {\\mathcal {B}}_{\\Theta ,j}}\\,e^{t_{i}\\,\\tilde{\\kappa }_{d_i}\\big (\\tfrac{t_{i}}{d_i},\\,\\tfrac{ l_{i} L}{d_i}\\big )}\\,e^{\\tfrac{\\beta -\\alpha }{2}\\, t_{i}},\\\\\\nonumber B_2 &= \\prod _{i=1}^{r}\\,e^{t_{i}^{\\mathcal {I}}\\,\\phi _{d^{\\mathcal {I}}_{i}}\\Big (\\tfrac{t^{\\mathcal {I}}_{i}}{d^{\\mathcal {I}}_{i}}\\Big )},\\\\\\nonumber C_2 &= C_1.$ Next, for $n\\in {\\mathbb {N}}$ we define $\\nonumber {\\mathcal {A}}_{{\\varepsilon },n} &= \\Big \\lbrace \\exists \\,0\\le t,s\\le n\\colon \\, t\\ge {\\varepsilon }n,\\,\\big |H_t^{\\theta ^s(\\omega )}(B)-\\tfrac{\\beta -\\alpha }{2} t\\big |>{\\varepsilon }t\\Big \\rbrace ,\\\\{\\mathcal {B}}_{{\\varepsilon },n} &= \\Big \\lbrace \\exists \\,0\\le t,d,s\\le n\\colon \\,t\\in d+2{\\mathbb {N}}_0,\\,t\\ge {\\varepsilon }n,\\,\\big |\\phi ^{\\theta ^s(w)}_d(\\tfrac{t}{d})-\\phi _d(\\tfrac{t}{d})\\big |>{\\varepsilon }\\Big \\rbrace .$ By applying Cramér's theorem for i.i.d.",
"random variables (see e.g.",
"den Hollander [2], Chapter 1), we obtain that there exist $C_{1}({\\varepsilon }),C_{2}({\\varepsilon })>0$ such that ${\\mathbb {P}}\\big (\\big |H_t^{\\theta ^s(w)}(B)-\\tfrac{\\beta -\\alpha }{2} t \\big |>{\\varepsilon }t\\big )\\le C_{1}({\\varepsilon })\\, e^{-C_2({\\varepsilon }) t}, \\qquad t,s\\in {\\mathbb {N}}.$ By using the concentration of measure property in (REF ) in Appendix with $l=t$ , $\\Gamma ={\\mathcal {W}}^{\\mathcal {I}}_{d}(\\tfrac{t}{d})$ , $T(x,y)=1\\lbrace (x,y)< 0\\rbrace $ , $\\eta ={\\varepsilon }t$ and $\\xi _i=-\\alpha 1\\lbrace \\omega _i=A\\rbrace +\\beta 1\\lbrace \\omega _i=B\\rbrace $ for all $i\\in {\\mathbb {N}}$ , we find that there exist $C_{1},C_{2}>0$ such that ${\\mathbb {P}}\\big (\\big |\\phi ^{\\theta ^s(w)}_d(\\tfrac{t}{d})-\\phi _d(\\tfrac{t}{d})\\big |>{\\varepsilon }\\big |\\big )\\le C_{1} \\, e^{-C_{2}\\, {\\varepsilon }^2 t}, \\qquad t,d,s \\in {\\mathbb {N}},\\,t\\in d+2{\\mathbb {N}}_0.$ With the help of (REF ) and (REF ) we may write, for $(L,\\Theta ,u,j)\\in G_M^{\\,m}$ , $\\psi _1(L,\\Theta ,u,j)\\le C_{\\text{uf}}(\\alpha )\\,{\\mathbb {P}}\\big ({\\mathcal {A}}_{{\\varepsilon },m L}\\cup {\\mathcal {B}}_{{\\varepsilon },m L}\\big )+\\tfrac{1}{uL}\\,{\\mathbb {E}}\\big [1_{\\lbrace {\\mathcal {A}}^c_{{\\varepsilon },m L}\\cap {\\mathcal {B}}^c_{{\\varepsilon },m L}\\rbrace }\\,\\log Z_1^\\omega (L,\\Theta ,u,j)\\big ].$ With the help of (REF ) and (REF ), we get that ${\\mathbb {P}}({\\mathcal {A}}_{{\\varepsilon },m L})\\rightarrow 0$ and ${\\mathbb {P}}({\\mathcal {B}}_{{\\varepsilon },m L})\\rightarrow 0$ as $L\\rightarrow \\infty $ .",
"Moreover, from ((REF )-(REF )) it follows that, for $(L,\\Theta ,u,j)\\in G_M^{\\,m}$ and $\\omega \\in {\\mathcal {A}}^c_{{\\varepsilon },m L}\\cap {\\mathcal {B}}^c_{{\\varepsilon },ML}$ , $Z^\\omega _1(L,\\Theta ,u,j)\\le Z_2(L,\\Theta ,u,j)\\,e^{{\\varepsilon }u L}.$ The latter completes the proof of $\\psi _1\\prec \\psi _2$ ."
],
[
"Step 2",
"In this step, we concatenate the pieces of trajectories that travel in $A$ -blocks, respectively, $B$ -blocks, respectively, along the $AB$ -interfaces and replace the finite-size entropies and free energies by their infinite-size counterparts.",
"Recall the definition of $l_{A,\\Theta ,j}$ and $l_{B,\\Theta ,j}$ in (REF ) and define, for $(L,\\Theta ,u,j)\\in G_M^{\\,m}$ , the sets ${\\mathcal {J}}_{\\Theta ,L,j}&=\\Big \\lbrace \\big (a_A,h_A,a_B,h_B\\big )\\in {\\mathbb {N}}^4\\colon \\, a_A\\in l_{A,\\Theta ,j} L+h_A+2{\\mathbb {N}}_0,\\,a_B\\in l_{B,\\Theta ,j} L+h_B+2 {\\mathbb {N}}_0\\Big \\rbrace ,\\\\\\nonumber {\\mathcal {J}}^{\\mathcal {I}}&=\\Big \\lbrace \\big (a^{\\mathcal {I}},h^{\\mathcal {I}}\\big )\\in {\\mathbb {N}}^2\\colon \\, a^{\\mathcal {I}}\\in h^{\\mathcal {I}}+ 2{\\mathbb {N}}_0\\Big \\rbrace ,$ and put $\\psi _3(L,\\Theta ,u,j)=\\frac{1}{ uL} \\log Z_3(L,\\Theta ,u,j)$ with $Z_3(L,\\Theta ,u,j)=\\sum _{(a,h)\\in {\\mathcal {J}}_{\\Theta ,L,j}}&\\sum _{(a^{\\mathcal {I}},h^{\\mathcal {I}})\\in {\\mathcal {J}}^{\\mathcal {I}}} A_3\\,B_3\\,C_3,$ where $\\nonumber A_3 &= e^{a_A\\,\\tilde{\\kappa }\\Big (\\tfrac{a_A}{h_A},\\,\\tfrac{l_{A,\\Theta ,j} L}{h_A}\\Big )}\\,e^{a_B\\, \\tilde{\\kappa }\\Big (\\tfrac{a_B}{h_B},\\,\\tfrac{l_{B,\\Theta ,j} L}{h_B}\\Big )}\\,e^{\\tfrac{\\beta -\\alpha }{2}\\, a_B},\\\\\\nonumber B_3 &= e^{a^{\\mathcal {I}}\\, \\phi \\big (\\tfrac{a^{\\mathcal {I}}}{h^{\\mathcal {I}}}\\big )},\\\\C_3 &= 1_{\\lbrace a_A+a_B+a^{\\mathcal {I}}=uL\\rbrace }\\,1_{\\lbrace h_A+h_B+h^{\\mathcal {I}}=L\\rbrace }.$ In order to establish a link between $\\psi _2$ and $\\psi _3$ we define, for $(a,h)\\in {\\mathcal {J}}_{\\Theta ,L,j}$ and $(a^{\\mathcal {I}},h^{\\mathcal {I}})\\in {\\mathcal {J}}^{\\mathcal {I}}$ , $\\nonumber {\\mathcal {P}}_{(a,h)} &= \\big \\lbrace (t,d)\\in {\\mathcal {D}}_{\\Theta ,L,j}\\colon \\,\\textstyle \\sum _{i\\in {\\mathcal {A}}_{\\Theta ,j}} (t_{i},d_{i})=(a_A,h_A),\\,\\sum _{i\\in {\\mathcal {B}}_{\\Theta ,j}} (t_{i},d_{i})=(a_B,h_B)\\big \\rbrace ,\\\\{\\mathcal {Q}}_{(a^{\\mathcal {I}},h^{\\mathcal {I}})}&=\\big \\lbrace (t^{\\mathcal {I}},d^{\\mathcal {I}})\\in {\\mathcal {D}}_{r}^{\\mathcal {I}}\\colon \\,\\textstyle \\sum _{i=1}^{r}(t^{\\mathcal {I}}_{i},d^{\\mathcal {I}}_{i})=(a^{\\mathcal {I}},h^{\\mathcal {I}})\\big \\rbrace .$ Then we can rewrite $Z_2$ as $Z_2(L,\\Theta ,u,j)=\\sum _{(a,h)\\in {\\mathcal {J}}_{\\Theta ,L,j}}&\\sum _{(a^{\\mathcal {I}},h^{\\mathcal {I}})\\in {\\mathcal {J}}^{\\mathcal {I}}} C_3\\sum _{(t,d)\\in {\\mathcal {P}}_{(a,h)}}\\sum _{(t^{\\mathcal {I}},d^{\\mathcal {I}})\\in {\\mathcal {Q}}_{(a^{\\mathcal {I}},h^{\\mathcal {I}})}} A_2\\,B_2.$ To prove that $\\psi _2\\prec \\psi _3$ , we need the following lemma.",
"Lemma 3.3 For every $\\eta >0$ there exists an $L_\\eta \\in {\\mathbb {N}}$ such that, for every $(L,\\Theta ,u,j)\\in G_M^{\\,m}$ with $L\\ge L_\\eta $ and every $(d,t)\\in {\\mathcal {D}}_{\\Theta ,L,j}$ and $(d^{\\mathcal {I}},t^{\\mathcal {I}})\\in {\\mathcal {D}}_{r}^{\\mathcal {I}}$ satisfying $\\sum _{i=1}^{r+1} d_i+\\sum _{i=1}^{r}d_i^{\\mathcal {I}}=L$ and $\\sum _{i=1}^{r+1} t_i+\\sum _{i=1}^{r} t_i^{\\mathcal {I}}=u L$ , $t_i\\,\\tilde{\\kappa }\\big (\\tfrac{t_i}{d_i},\\,\\tfrac{l_i L}{d_i}\\big )-\\eta uL&\\le t_i\\,\\tilde{\\kappa }_{d_i}\\big (\\tfrac{t_i}{d_i},\\,\\tfrac{l_i L}{d_i}\\big )\\le t_i\\,\\tilde{\\kappa }\\big (\\tfrac{t_i}{d_i},\\,\\tfrac{l_i L}{d_i}\\big )+\\eta uL \\quad i = 1,\\dots ,r+1,\\\\\\nonumber t_i^{\\mathcal {I}}\\phi (\\tfrac{t_i^{\\mathcal {I}}}{d_i^{\\mathcal {I}}})-\\eta uL&\\le t_i^{\\mathcal {I}}\\phi _{d^{\\mathcal {I}}_i}(\\tfrac{t_i^{\\mathcal {I}}}{d_i^{\\mathcal {I}}})\\le t_i^{\\mathcal {I}}\\phi (\\tfrac{t_i^{\\mathcal {I}}}{d_i^{\\mathcal {I}}})+\\eta uL \\quad i = 1,\\dots ,r.$ Proof.",
"By using Lemmas REF and REF in Appendix , we have that there exists a $\\tilde{L}_\\eta \\in {\\mathbb {N}}$ such that, for $L\\ge \\tilde{L}_\\eta $ , $(u,l)\\in {\\mathcal {H}}_L$ and $\\mu \\in 1+\\frac{2{\\mathbb {N}}}{L}$ , $|\\tilde{\\kappa }_L(u,l)-\\tilde{\\kappa }(u,l)|\\le \\eta , \\qquad |\\phi ^{\\mathcal {I}}_L(\\mu )- \\phi ^{\\mathcal {I}}(\\mu )|\\le \\eta .$ Moreover, Lemmas REF , REF (ii–iii), REF (ii) and REF ensure that there exists a $v_\\eta >1$ such that, for $L\\ge 1$ , $(u,l)\\in {\\mathcal {H}}_L$ with $u\\ge v_\\eta $ and $\\mu \\in 1+\\frac{2{\\mathbb {N}}}{L}$ with $\\mu \\ge v_\\eta $ , $0\\le \\tilde{\\kappa }_L(u,l)\\le \\eta , \\qquad 0\\le \\phi _L(\\mu )\\le \\eta .$ Note that the two inequalities in (REF ) remain valid when $L=\\infty $ .",
"Next, we set $r_\\eta =\\eta /(2 v_\\eta C_{\\text{uf}})$ and $L_\\eta =\\tilde{L}_\\eta /r_\\eta $ , and we consider $L\\ge L_\\eta $ .",
"Because of the left-hand side of (REF ), the two inequalities in the first line of (REF ) hold when $d_i \\ge r_\\eta L\\ge \\tilde{L}_\\eta $ .",
"We deal with the case $d_i \\le r_\\eta L$ by considering first the case $t_i\\le \\eta u L/2 C_{\\text{uf}}$ , which is easy because $\\tilde{\\kappa }_{d_i}$ and $\\tilde{\\kappa }$ are uniformly bounded by $C_{\\text{uf}}$ (see (REF )).",
"The case $t_i\\ge \\eta u L/2 C_{\\text{uf}}$ gives $t_i/d_i\\ge u v_\\eta \\ge v_\\eta $ , which by the left-hand side of (REF ) completes the proof of the first line in (REF ).",
"The same observations applied to $t_i^{\\mathcal {I}}, d_i^{\\mathcal {I}}$ combined with the right-hand side of (REF ) and (REF ) provide the two inequalities in the second line in (REF ).",
"$\\square $ To prove that $\\psi _2\\prec \\psi _3$ , we apply Lemma REF with $\\eta ={\\varepsilon }/(2m+1)$ and we use (REF ) to obtain, for $L\\ge L_{{\\varepsilon }/(2m+1)}$ , $(d,t)\\in {\\mathcal {D}}_{\\Theta ,L,j}$ and $(d^{\\mathcal {I}},t^{\\mathcal {I}})\\in {\\mathcal {D}}_{r}^{\\mathcal {I}}$ , $A_2 &\\le \\prod _{i\\in {\\mathcal {A}}_{\\Theta ,j}} e^{ t_{i}\\, \\tilde{\\kappa }\\big (\\tfrac{t_{i}}{d_i}, \\,\\tfrac{ l_{i} L}{d_i}\\big )+ \\tfrac{{\\varepsilon }uL}{2m+1}} \\prod _{i\\in {\\mathcal {B}}_{\\Theta ,j}}\\,e^{t_{i}\\,\\tilde{\\kappa }\\big (\\tfrac{t_{i}}{d_i},\\,\\tfrac{ l_{i} L}{d_i}\\big ) + t_i\\,\\tfrac{\\beta -\\alpha }{2}+ \\tfrac{{\\varepsilon }uL}{2m+1}},\\\\\\nonumber B_2 &\\le \\prod _{i=1}^{r}\\, e^{t_{i}^{\\mathcal {I}}\\,\\phi \\Big (\\tfrac{t^{\\mathcal {I}}_{i}}{d^{\\mathcal {I}}_{i}}\\Big )+ \\tfrac{{\\varepsilon }uL}{2m+1}}.$ Next, we pick $(a,h)\\in {\\mathcal {J}}_{\\Theta ,L,j}$ , $(a^{\\mathcal {I}},h^{\\mathcal {I}})\\in {\\mathcal {J}}^{\\mathcal {I}}$ , $(t,d)\\in {\\mathcal {P}}_{(a,h)}$ and $(t^{\\mathcal {I}},d^{\\mathcal {I}})\\in {\\mathcal {Q}}_{(a^{\\mathcal {I}},h^{\\mathcal {I}})}$ , and we use the concavity of $(a,b)\\mapsto a \\tilde{\\kappa }(a,b)$ and $\\mu \\mapsto \\phi ^{\\mathcal {I}}(\\mu )$ (see Lemma REF in Appendix and Lemma REF in Appendix ) to rewrite (REF ) as $A_2 &\\le e^{a_A\\,\\tilde{\\kappa }\\big (\\tfrac{a_A}{h_A},\\,\\tfrac{l_{A,\\Theta ,j} L}{h_A}\\big ) + a_B \\, \\tilde{\\kappa }\\big (\\tfrac{a_B}{h_B},\\,\\tfrac{l_{B,\\Theta ,j} L}{h_B}\\big ) + \\tfrac{\\beta -\\alpha }{2} a_B+ \\tfrac{{\\varepsilon }(r+1)uL}{2m+1}}= A_3\\,e^{\\tfrac{{\\varepsilon }(r+1) u L}{2m+1}},\\\\\\nonumber B_2 &\\le e^{a^{\\mathcal {I}}\\, \\phi ^{\\mathcal {I}}\\big (\\tfrac{a^{\\mathcal {I}}}{h^{\\mathcal {I}}}\\big )+ \\tfrac{{\\varepsilon }r uL}{2m+1}} = B_3\\, e^{\\tfrac{{\\varepsilon }r uL}{2m+1}}.$ Moreover, $r$ , which is the number of $AB$ interfaces crossed by the trajectories in ${\\mathcal {W}}_{\\Theta ,u,j,L}$ , is at most $m$ (see (REF )), so that (REF ) allows us to rewrite (REF ) as $Z_2(L,\\Theta ,u,j) \\le e^{{\\varepsilon }u L} \\sum _{(a,h)\\in {\\mathcal {J}}_{\\Theta ,L,j}}&\\sum _{(a^{\\mathcal {I}},h^{\\mathcal {I}})\\in {\\mathcal {J}}^{\\mathcal {I}}} C_3\\,|{\\mathcal {P}}_{(a,h)}|\\, |{\\mathcal {Q}}_{(a^{\\mathcal {I}},h^{\\mathcal {I}})}|\\, A_3\\,B_3.$ Finally, it turns out that $|{\\mathcal {P}}_{(a,h)}|\\le (uL)^{8r}$ and $|{\\mathcal {Q}}_{(a^{\\mathcal {I}},h^{\\mathcal {I}})}|\\le (uL)^{8r}$ .",
"Therefore, since $r\\le m$ , (REF ) and (REF ) allow us to write, for $(L,\\Theta ,u,j)\\in G_M^{\\,m}$ and $L\\ge L_{{\\varepsilon }/2m+1}$ , $Z_2(L,\\Theta ,u,j)\\le (mL)^{16 m} Z_3(L,\\Theta ,u,j).$ The latter is sufficient to conclude that $\\psi _2\\prec \\psi _3$ ."
],
[
"Step 3",
"For every $(L,\\Theta ,u,j)\\in G_M^{\\,m}$ we have, by the definition of ${\\mathcal {L}}(l_{A,\\Theta ,j},l_{B,\\Theta ,j};u)$ in (REF ), that $(a,h)\\in {\\mathcal {J}}_{\\Theta ,L,j}$ and $(a^{\\mathcal {I}},h^{\\mathcal {I}})\\in {\\mathcal {J}}^{\\mathcal {I}}$ satisfying $a_A+a_B+a^{\\mathcal {I}}=uL$ and $h_A+h_B+h^{\\mathcal {I}}=L$ also satisfy $\\Big (\\big (\\tfrac{a_A}{L},\\tfrac{a_B}{L},\\tfrac{a^{\\mathcal {I}}}{L}\\big ),\\big (\\tfrac{h_A}{L},\\tfrac{h_B}{L},\\tfrac{h^{\\mathcal {I}}}{L}\\big )\\Big )\\in {\\mathcal {L}}(l_{A,\\Theta ,j},l_{B,\\Theta ,j};u).$ Hence, (REF ) and the definition of $\\psi _{\\mathcal {I}}$ in (REF ) ensure that, for this choice of $(a,h)$ and $(a^{\\mathcal {I}},h^{\\mathcal {I}})$ , $A_3 B_3\\le e^{uL \\psi _{\\mathcal {I}}(u,\\,l_{A,\\Theta ,j},\\,l_{B,\\Theta ,j})}.$ Because of $C_3$ , the summation in (REF ) is restricted to those $(a,h)\\in {\\mathcal {J}}_{\\Theta ,L,j}$ and $(a^{\\mathcal {I}},h^{\\mathcal {I}})\\in {\\mathcal {J}}^{\\mathcal {I}}$ for which $a_A,a_B,a^{\\mathcal {I}}\\le uL$ and $h_A,h_B,h^{\\mathcal {I}}\\le L$ .",
"Hence, the summation is restricted to a set of cardinality at most $(uL)^3 L^3$ .",
"Consequently, for all $(L,\\Theta ,u,j)\\in G_M^{\\,m}$ we have $Z_3(L,\\Theta ,u,j)&=\\sum _{(a,h)\\in {\\mathcal {J}}_{\\Theta ,L,j}}\\sum _{(a^{\\mathcal {I}},h^{\\mathcal {I}})\\in {\\mathcal {J}}^{\\mathcal {I}}} A_4\\, B_4\\, C_4\\le (m L)^3 L^3\\,e^{u\\,L\\, \\psi _{\\mathcal {I}}(u,\\,l_{A,\\Theta ,j},\\,l_{B,\\Theta ,j})}.$ The latter implies that $\\psi _3\\prec \\psi _4$ since $\\psi _4=\\psi _{\\mathcal {I}}(u,\\,l_{A,\\Theta ,j},\\,l_{B,\\Theta ,j})$ by definition (recall (REF ) and (REF ))."
],
[
"Proof of Theorem ",
"This section is technically involved because it goes through a sequence of approximation steps in which the self-averaging of the free energy with respect to $\\omega $ and $\\Omega $ in the limit as $n\\rightarrow \\infty $ is proven, and the various ingredients of the variational formula in Theorem REF that were constructed in Section are put together.",
"In Section REF we introduce additional notation and state Propositions REF , REF and REF from which Theorem REF is a straightforward consequence.",
"Proposition REF , which deals with $(M,m)\\in {\\hbox{\\footnotesize \\rm EIGH}}$ , is proven in Section REF and the details of the proof are worked out in Sections REF –REF , organized into 5 Steps that link intermediate free energies.",
"We pass to the limit $m\\rightarrow \\infty $ with Propositions REF and REF which are proven in Section REF and REF , respectively."
],
[
"Additional notation",
"Pick $(M,m)\\in {\\hbox{\\footnotesize \\rm EIGH}}$ and recall that $\\Omega $ and $\\omega $ are independent, i.e., ${\\mathbb {P}}={\\mathbb {P}}_\\omega \\times {\\mathbb {P}}_\\Omega $ .",
"For $\\Omega \\in \\lbrace A,B\\rbrace ^{{\\mathbb {N}}_0\\times {\\mathbb {Z}}}$ , $\\omega \\in \\lbrace A,B\\rbrace ^{\\mathbb {N}}$ , $n\\in {\\mathbb {N}}$ and $(\\alpha ,\\beta )\\in {\\hbox{\\footnotesize \\rm CONE}}$ , define $f^{\\omega ,\\Omega }_{1,n}(M,m;\\alpha ,\\beta )= \\tfrac{1}{n} \\log Z_{1,n,L_n}^{\\,\\omega ,\\Omega }(M,m)\\quad \\text{with}\\quad Z_{1,n,L_n}^{\\,\\omega ,\\Omega }(M,m)=\\sum _{\\pi \\in {\\mathcal {W}}_{n,M,}^{\\,m}} e^{\\,H_{n,L_n}^{\\omega ,\\Omega }(\\pi )},$ where ${\\mathcal {W}}_{n,M}^{\\,m}$ contains those paths in ${\\mathcal {W}}_{n,M}$ that, in each column, make at most $m L_n$ steps.",
"We also restrict the set ${\\mathcal {R}}_{p,M}$ in (REF ) to those limiting empirical measures whose support is included in $\\overline{{\\mathcal {V}}}_{M}^{\\,m}$ , i.e., those measures charging the types of column that can be crossed in less than $m L_n$ steps only.",
"To that aim we recall (REF ) and define, for $\\Omega \\in \\lbrace A,B\\rbrace ^{{\\mathbb {N}}_0\\times {\\mathbb {Z}}}$ and $N\\in {\\mathbb {N}}$ , $\\begin{aligned}{\\mathcal {R}}^{\\Omega ,m}_{M,N} &= \\big \\lbrace \\rho _N(\\Omega ,\\Pi ,b,x)\\ \\text{with}\\ b=(b_j)_{j\\in {\\mathbb {N}}_0} \\in (\\mathbb {Q}_{(0,1]})^{{\\mathbb {N}}_0},\\\\&\\qquad \\Pi =(\\Pi _j)_{j\\in {\\mathbb {N}}_0} \\in \\lbrace 0\\rbrace \\times {\\mathbb {Z}}^{{\\mathbb {N}}} \\colon \\,|\\Delta \\Pi _j|\\le M\\,\\ \\ \\forall \\,j\\in {\\mathbb {N}}_0,\\\\&\\qquad x=(x_j)_{j\\in {\\mathbb {N}}_0} \\in \\lbrace 1,2\\rbrace ^{N_0}\\colon \\, \\big (\\Omega (j,\\Pi _j+\\cdot ),\\Delta \\Pi _j,b_j,b_{j+1},x_j\\big )\\in {\\mathcal {V}}_M^{m}\\big \\rbrace \\end{aligned}$ which is a subset of ${\\mathcal {R}}_{M,N}^{\\Omega }$ and allows us to define ${\\mathcal {R}}^{\\Omega ,m}_M = \\mathrm {closure}\\Big (\\cap _{N^{\\prime }\\in {\\mathbb {N}}} \\cup _{N \\ge N^{\\prime }}\\,{\\mathcal {R}}^{\\Omega ,m}_{M,N}\\Big ),$ which, for ${\\mathbb {P}}$ -a.e.",
"$\\Omega $ is equal to ${\\mathcal {R}}_{p,M}^{m}\\subsetneq {\\mathcal {R}}_{p,M}$ .",
"At this stage, we further define, $f(M,m;\\alpha ,\\beta )= \\sup _{\\rho \\in {\\mathcal {R}}_{p,M}^{\\,m}}\\,\\sup _{(u_\\Theta )_{\\Theta \\in \\overline{{\\mathcal {V}}}_M^{\\,m}}\\in {\\mathcal {B}}_{\\,\\overline{{\\mathcal {V}}}_M^{\\,m}}}V(\\rho ,u),$ where $V(\\rho ,u)=\\frac{\\int _{\\overline{{\\mathcal {V}}}_M^{\\,m}}\\,u_\\Theta \\,\\psi (\\Theta ,u_{\\Theta };\\alpha ,\\beta )\\,\\rho (d\\Theta )}{\\int _{\\overline{{\\mathcal {V}}}_M^{\\,m}}\\,u_\\Theta \\, \\rho (d\\Theta )},$ where (recall (REF )) ${\\mathcal {B}}_{\\overline{{\\mathcal {V}}}_M^{\\,m}}= \\Big \\lbrace (u_\\Theta )_{\\Theta \\in \\overline{{\\mathcal {V}}}_M^{\\,m}}\\in {\\mathbb {R}}^{\\overline{{\\mathcal {V}}}_M^{\\,m}}\\colon \\Theta \\mapsto u_\\Theta \\in {\\mathcal {C}}^0\\big (\\overline{{\\mathcal {V}}}_M^{\\,m},{\\mathbb {R}}\\big ),\\,t_\\Theta \\le u_\\Theta \\le m\\,\\,\\forall \\,\\Theta \\in \\overline{{\\mathcal {V}}}_M^{\\,m} \\Big \\rbrace ,$ and where $\\overline{{\\mathcal {V}}}_M^{\\,m}$ is endowed with the distance $d_M$ defined in (REF ) in Appendix REF .",
"Let ${\\mathcal {W}}^{*,m}_{n,M}\\subset {\\mathcal {W}}_{n,M}^{\\,m}$ be the subset consisting of those paths whose endpoint lies at the boundary between two columns of blocks, i.e., satisfies $\\pi _{n,1}\\in {\\mathbb {N}}L_n$ .",
"Recall (REF ), and define $Z^{*,\\omega ,\\Omega }_{n,L_n}(M)$ and $f^{*,\\omega ,\\Omega }_{1,n}(M,m;\\alpha ,\\beta )$ as the counterparts of $Z^{\\omega ,\\Omega }_{n,L_n}(M,m)$ and $f^{\\omega ,\\Omega }_{1,n}(M,m;\\alpha ,\\beta )$ when ${\\mathcal {W}}_{n,M}^{\\,m}$ is replaced by ${\\mathcal {W}}_{n,M}^{*,m}$ .",
"Then there exists a constant $c>0$ , depending on $\\alpha $ and $\\beta $ only, such that $\\begin{aligned}&Z^{\\omega ,\\Omega }_{1,n,L_n}(M,m) e^{-c L_n}\\le Z^{*,\\omega ,\\Omega }_{1,n,L_n}(M,m)\\le Z^{\\omega ,\\Omega }_{1,n,L_n}(M,m),\\\\&n\\in {\\mathbb {N}}, \\,\\omega \\in \\lbrace A,B\\rbrace ^{\\mathbb {N}},\\, \\Omega \\in \\lbrace A,B\\rbrace ^{{\\mathbb {N}}_0\\times {\\mathbb {Z}}}.\\end{aligned}$ The left-hand side of the latter inequality is obtained by changing the last $L_n$ steps of each trajectory in ${\\mathcal {W}}_{n,M}^{\\,m}$ to make sure that the endpoint falls in $L_n{\\mathbb {N}}$ .",
"The energetic and entropic cost of this change are obviously $O(L_n)$ .",
"By assumption, $\\lim _{n\\rightarrow \\infty } L_n/n=0$ , which together with (REF ) implies that the limits of $f^{\\omega ,\\Omega }_{1,n}(M,m;\\alpha ,\\beta )$ and $f^{*,\\omega ,\\Omega }_{1,n}(M,m;\\alpha ,\\beta )$ as $n\\rightarrow \\infty $ are the same.",
"In the sequel we will therefore restrict the summation in the partition function to ${\\mathcal {W}}_{n,M}^{*,m}$ and drop the $*$ from the notations.",
"Finally, let $\\begin{aligned}f_{1,n}^{\\Omega }(M,m;\\alpha ,\\beta )&= {\\mathbb {E}}_\\omega \\big [f^{\\omega ,\\Omega }_{1,n}(M,m;\\alpha ,\\beta )\\big ],\\\\f_{1,n}(M,m;\\alpha ,\\beta )&= {\\mathbb {E}}_{\\omega ,\\Omega }\\big [f^{\\omega ,\\Omega }_{1,n}(M,m;\\alpha ,\\beta )\\big ],\\end{aligned}$ and recall (REF ) to set $f_n^\\Omega (M;\\alpha ,\\beta )={\\mathbb {E}}_\\omega [f_n^{\\omega ,\\Omega }(M;\\alpha ,\\beta )]$ ."
],
[
"Key Propositions",
"Theorem REF is a consequence of Propositions REF , REF and REF stated below and proven in Sections REF –REF , Sections REF –REF and Section REF , respectively.",
"Proposition 4.1 For all $(M,m)\\in {\\hbox{\\footnotesize \\rm EIGH}}$ , $\\lim _{n\\rightarrow \\infty } f^{\\Omega }_{1,n}(M,m;\\alpha ,\\beta )=f(M,m;\\alpha ,\\beta )\\quad \\text{ for } {\\mathbb {P}}-a.e.\\,\\Omega .$ Proposition 4.2 For all $M\\in {\\mathbb {N}}$ , $\\lim _{n\\rightarrow \\infty } f^{\\Omega }_{n}(M;\\alpha ,\\beta )=\\sup _{m\\ge M+2} f(M,m;\\alpha ,\\beta )\\quad \\text{ for } {\\mathbb {P}}-a.e.\\,\\Omega .$ Proposition 4.3 For all $M\\in {\\mathbb {N}}$ , $\\sup _{m\\ge M+2} f(M,m;\\alpha ,\\beta )=\\sup _{\\rho \\in {\\mathcal {R}}_{p,M}}\\,\\sup _{(u_\\Theta )_{\\Theta \\in \\overline{{\\mathcal {V}}}_M}\\,\\in \\,{\\mathcal {B}}_{\\,\\overline{{\\mathcal {V}}}_M}} V(\\rho ,u),$ where, in the righthand side of (REF ), we recognize the variational formula of Theorem REF and with ${\\mathcal {B}}_{\\overline{{\\mathcal {V}}}_M}$ defined in (REF ).",
"Proof of Theorem REF subject to Propositions REF , REF and REF .",
"The proof of Theorem REF will be complete once we show that for all $(M,m)\\in {\\hbox{\\footnotesize \\rm EIGH}}$ $\\lim _{n\\rightarrow \\infty } |f_{n}^{\\omega ,\\Omega }(M,m;\\alpha ,\\beta )-f_{n}^{\\Omega }(M,m;\\alpha ,\\beta )|=0\\quad \\text{ for } {\\mathbb {P}}-a.e.\\,\\,(\\omega ,\\Omega ).$ To that aim, we note that for all $n\\in {\\mathbb {N}}$ the $\\Omega $ -dependence of $f_{n}^{\\omega ,\\Omega }(M,m;\\alpha ,\\beta )$ is restricted to $\\big \\lbrace \\Omega _x\\colon \\,x\\in G_n\\big \\rbrace $ with $G_n=\\lbrace 0,\\dots ,\\frac{n}{L_n}\\rbrace \\times \\lbrace -\\frac{n}{L_n},\\dots ,\\frac{n}{L_n}\\rbrace $ .",
"Thus, for $n\\in {\\mathbb {N}}$ and ${\\varepsilon }>0$ we set $A_{{\\varepsilon },n}=\\lbrace | f_{n}^{\\omega ,\\Omega }(M;\\alpha ,\\beta )- f_{n}^{\\Omega }(M;\\alpha ,\\beta )|>{\\varepsilon })\\rbrace ,$ and by independence of $\\omega $ and $\\Omega $ we can write ${\\textstyle \\nonumber {\\mathbb {P}}_{\\omega ,\\Omega }(A_{{\\varepsilon },n})}&{\\textstyle =\\sum _{\\Upsilon \\in \\lbrace A,B\\rbrace ^{G_n}}{\\mathbb {P}}_{\\omega ,\\Omega }(A_{{\\varepsilon },n}\\cap \\lbrace \\Omega _{G_n}=\\Upsilon \\rbrace )}\\\\&{\\textstyle =\\sum _{\\Upsilon \\in \\lbrace A,B\\rbrace ^{G_n}}{\\mathbb {P}}_{\\omega }(| f_{n}^{\\omega ,\\Upsilon }(M;\\alpha ,\\beta )- f_{n}^{\\Upsilon }(M;\\alpha ,\\beta )|>{\\varepsilon })\\ {\\mathbb {P}}_{\\Omega }(\\lbrace \\Omega _{G_n}=\\Upsilon \\rbrace )}.$ At this stage, for each $n\\in {\\mathbb {N}}$ we can apply the concentration inequality $(\\ref {concmesut})$ in Appendix with $\\Gamma ={\\mathcal {W}}_{n,M}^{\\,m}$ , $l=n$ , $\\eta ={\\varepsilon }n$ , $\\xi _i = -\\alpha \\, 1\\lbrace \\omega _i=A\\rbrace +\\beta \\,1\\lbrace \\omega _i=B\\rbrace , \\qquad i\\in {\\mathbb {N}},$ and with $T(x,y)$ indicating in which block step $(x,y)$ lies in.",
"Therefore, there exist $C_1, C_2>0$ such that for all $n\\in {\\mathbb {N}}$ and all $\\Upsilon \\in \\lbrace A,B\\rbrace ^{G_n}$ we have ${\\mathbb {P}}_{\\omega }(| f_{n}^{\\omega ,\\Upsilon }(M;\\alpha ,\\beta )- f_{n}^{\\Upsilon }(M;\\alpha ,\\beta )|>{\\varepsilon }) \\le C_1 e^{-C_2 {\\varepsilon }^2 n},$ which, together with (REF ) yields ${\\mathbb {P}}_{\\omega ,\\Omega }(A_{{\\varepsilon },n})\\le C_1 e^{-C_2 {\\varepsilon }^2 n}$ for all $n\\in {\\mathbb {N}}$ .",
"By using the Borel-Cantelli Lemma, we obtain (REF ).",
"$\\square $"
],
[
"Proof of Proposition ",
"Pick $(M,m)\\in {\\hbox{\\footnotesize \\rm EIGH}}$ and $(\\alpha ,\\beta )\\in {\\hbox{\\footnotesize \\rm CONE}}$ .",
"In Steps 1–2 in Sections REF –REF we introduce an intermediate free energy $f_{3,n}^{\\Omega }(M,m;\\alpha ,\\beta )$ and show that $\\lim _{n\\rightarrow \\infty } |f_{1,n}^{\\Omega }(M,m;\\alpha ,\\beta )- f_{3,n}^{\\Omega }(M,m;\\alpha ,\\beta )|=0 \\qquad \\forall \\,\\Omega \\in \\lbrace A,B\\rbrace ^{N_0\\times {\\mathbb {Z}}}.$ Next, in Steps 3–4 in Sections REF –REF we show that $\\limsup _{n\\rightarrow \\infty } f_{3,n}^{\\Omega }(M,m;\\alpha ,\\beta )= f(M,m;\\alpha ,\\beta )\\qquad \\text{for}\\,\\,{\\mathbb {P}}-a.e.\\,\\, \\Omega ,$ while in Step 5 in Section REF we prove that $\\liminf _{n\\rightarrow \\infty } f_{3,n}^{\\Omega }(M,m;\\alpha ,\\beta )= \\limsup _{n\\rightarrow \\infty } f_{3,n}^{\\Omega }(M,m;\\alpha ,\\beta )\\qquad \\text{for}\\,\\,{\\mathbb {P}}-a.e.\\,\\,\\Omega .$ Combing (REF –REF ) we get $\\liminf _{n\\rightarrow \\infty } f_{1,n}^{\\Omega }(M,m;\\alpha ,\\beta )= \\limsup _{n\\rightarrow \\infty } f_{1,n}^{\\Omega }(M,m;\\alpha ,\\beta )=f(M,m;\\alpha ,\\beta )\\qquad \\text{for}\\,\\,{\\mathbb {P}}-a.e.\\,\\, \\Omega ,$ which completes the proof of Proposition REF .",
"In the proof we need the following order relation.",
"Definition 4.4 For $g,\\widetilde{g}\\colon \\,{\\mathbb {N}}^3\\times {\\hbox{\\footnotesize \\rm CONE}}\\mapsto {\\mathbb {R}}$ , write $g\\prec \\widetilde{g}$ if for all $(M,m)\\in {\\hbox{\\footnotesize \\rm EIGH}}$ , $(\\alpha ,\\beta )\\in {\\hbox{\\footnotesize \\rm CONE}}$ and ${\\varepsilon }>0$ there exists an $n_{{\\varepsilon }}\\in {\\mathbb {N}}$ such that $g(n,M,m; \\alpha ,\\beta ) \\le \\widetilde{g}(n,M,m; \\alpha ,\\beta )+{\\varepsilon }\\qquad \\forall \\,n\\ge n_{{\\varepsilon }}.$ The proof of (REF ) will be complete once we show that $f_1^\\Omega \\prec f_3^\\Omega $ and $f_3^\\Omega \\prec f_1^\\Omega $ for all $\\Omega \\in \\lbrace A,B\\rbrace ^{{\\mathbb {N}}_0\\times {\\mathbb {Z}}}$ .",
"We will focus on $f_1^\\Omega \\prec f_3^\\Omega $ , since the proof of the latter can be easily adapted to obtain $f_3^\\Omega \\prec f_1^\\Omega $ .",
"To prove $f_1^\\Omega \\prec f_3^\\Omega $ we introduce another intermediate free energy $f_2^\\Omega $ , and we show that $f_1^\\Omega \\prec f_2^\\Omega $ and $f_2^\\Omega \\prec f_3^\\Omega $ .",
"For $L\\in {\\mathbb {N}}$ , let ${\\mathcal {D}}_L^M=\\left\\lbrace \\Xi =(\\Delta \\Pi ,b_0,b_1)\\in \\lbrace -M,\\dots ,M\\rbrace \\times \\lbrace \\tfrac{1}{L},\\tfrac{2}{L},\\dots ,1\\rbrace ^2 \\right\\rbrace .",
"$ For $L,N\\in {\\mathbb {N}}$ , let $\\widetilde{{\\mathcal {D}}}_{L,N}^M&= \\Big \\lbrace \\Theta _{\\text{traj}}=(\\Xi _i)_{i\\in \\lbrace 0,\\dots , N-1\\rbrace }\\in ({\\mathcal {D}}_L^M)^N\\colon \\, b_{0,0}=\\tfrac{1}{L},\\,b_{0,i}=b_{1,i-1}\\,\\,\\forall \\,1\\le i\\le N-1\\Big \\rbrace ,$ and with each $\\Theta _{\\text{traj}}\\in \\widetilde{{\\mathcal {D}}}_{L,N}^M$ associate the sequence $(\\Pi _i)_{i=0}^N$ defined by $\\Pi _0=0$ and $\\Pi _i=\\sum _{j=0}^{i-1} \\Delta \\Pi _j$ for $1\\le i\\le N$ .",
"Next, for $\\Omega \\in \\lbrace A,B\\rbrace ^{{\\mathbb {N}}_0\\times {\\mathbb {Z}}}$ and $\\Theta _{\\text{traj}}\\in \\widetilde{{\\mathcal {D}}}_{L,N}^M$ , set ${\\mathcal {X}}_{\\Theta _{\\text{traj}},\\Omega }^{M,m}=\\big \\lbrace x\\in \\lbrace 1,2\\rbrace ^{\\lbrace 0,\\dots ,N-1\\rbrace }\\colon (\\Omega (i,\\Pi _i+\\cdot ),\\Xi _i,x_i)\\in {\\mathcal {V}}_{M}^{m} \\,\\,\\forall \\,0\\le i\\le N-1\\big \\rbrace ,$ and, for $x\\in {\\mathcal {X}}_{\\Theta _{\\text{traj}},\\Omega }^{M,m}$ , set $\\Theta _i=(\\Omega (i,\\Pi _i+\\cdot ),\\Xi _i,x_i) \\quad \\text{for}\\quad i\\in \\lbrace 0,\\dots ,N-1\\rbrace $ and $\\begin{aligned}{\\mathcal {U}}_{\\,\\Theta _{\\text{traj}},x,n}^{\\,M,m,L} &= \\Big \\lbrace u=(u_i)_{ i\\in \\lbrace 0,\\dots , N-1\\rbrace }\\in [1,m]^{N}\\colon u_i\\in t_{\\Theta _i}+\\tfrac{2{\\mathbb {N}}}{L}\\ \\,\\,\\forall \\,0\\le i\\le N-1,\\,\\sum _{i=0}^{N-1} u_i=\\tfrac{n}{L}\\Big \\rbrace .\\end{aligned}$ Note that ${\\mathcal {U}}_{\\,\\Theta _{\\text{traj}},x,n}^{\\,M,m,L}$ is empty when $N\\notin \\big [\\frac{n}{m L},\\frac{n}{L}\\big ]$ .",
"For $\\pi \\in {\\mathcal {W}}_{n,M}^{\\,m}$ , we let $N_\\pi $ be the number of columns crossed by $\\pi $ after $n$ steps.",
"We denote by $(u_0(\\pi ),\\dots ,u_{N_\\pi -1}(\\pi ))$ the time spent by $\\pi $ in each column divided by $L_n$ , and we set $\\widetilde{u}_0(\\pi )=0$ and $\\widetilde{u}_{j}(\\pi )=\\sum _{k=0}^{j-1} u_k(\\pi )$ for $1\\le j\\le N_\\pi $ .",
"With these notations, the partition function in (REF ) can be rewritten as $Z_{1,n,L_n}^{\\,\\omega ,\\Omega }(M,m)=\\sum _{N=n/m L_n}^{n/L_n}\\,\\,\\sum _{\\Theta _{\\text{traj}}\\in \\widetilde{{\\mathcal {D}}}_{L_n,N}^M }\\,\\sum _{x\\in {\\mathcal {X}}_{\\Theta _{\\text{traj}},\\Omega }^{M,m} }\\,\\sum _{u\\in \\,{\\mathcal {U}}_{\\,\\Theta _{\\text{traj}},x,n}^{\\,M,m,L_n}}\\,A_1,$ with (recall (REF )) $A_1=\\prod _{i=0}^{N-1}\\, Z^{\\theta ^{\\widetilde{u}_{i} L_n}(\\omega )}_{L_{n}}(\\Omega (i,\\Pi _{i}+\\cdot ),\\Xi _i,x_i, u_i).$"
],
[
"Step 1",
"In this step we average over the disorder $\\omega $ in each column.",
"To that end, we set $f_{2,n}^\\Omega (M,m;\\alpha ,\\beta ) = \\tfrac{1}{n} \\log Z_{2,n,L_n}^{\\Omega }(M,m)$ with $Z_{2,n,L_n}^{\\Omega }(M,m)=\\sum _{N=n/m L_n}^{n/L_n} \\ \\,\\sum _{\\Theta _{\\text{traj}}\\in \\widetilde{{\\mathcal {D}}}_{L_n,N}^M }\\,\\sum _{x\\in {\\mathcal {X}}_{\\Theta _{\\text{traj}},\\Omega }^{M,m}}\\,\\sum _{u\\in \\,{\\mathcal {U}}_{\\,\\Theta _{\\text{traj}},x,n}^{\\,M,m,L_n}}\\,A_2,$ where $A_2 &=\\prod _{i=0}^{N-1} e^{{\\mathbb {E}}_\\omega \\big [\\log Z^{\\theta ^{\\widetilde{u}_{i}}(\\omega )}_{L_{n}}(\\Omega (i,\\Pi _{i}+\\cdot ),\\Xi _i, x_i, u_i)\\big ]}= \\prod _{i=0}^{N-1}\\,e^{u_i L_n \\psi _{L_n}(\\Omega (i,\\Pi _{i}+\\cdot ),\\Xi _i,x_i, u_i)}.$ Note that the $\\omega $ -dependence has been removed from $Z^\\Omega _{2,n,L_n}(M,m)$ .",
"To prove that $f_1^\\Omega \\prec f_2^\\Omega $ , we need to show that for all ${\\varepsilon }>0$ there exists an $n_{{\\varepsilon }}\\in {\\mathbb {N}}$ such that, for $n\\ge n_{{\\varepsilon }}$ and all $\\Omega $ , ${\\mathbb {E}}_\\omega \\big [\\log Z_{1,n,L_n}^{\\,\\omega ,\\Omega }(M,m)\\big ]\\le \\log Z_{2,n,L_n}^{\\Omega }(M,m)+{\\varepsilon }n.$ To this end, we rewrite $Z_{1,n,L_n}^{\\,\\omega ,\\Omega }(M,m)$ as $Z_{1,n,L_n}^{\\,\\omega ,\\Omega }(M,m)=\\sum _{N=n/m L_n}^{n/L_n}\\sum _{\\Theta _{\\text{traj}}\\in \\widetilde{{\\mathcal {D}}}_{L_n,N}^M }\\,\\sum _{x\\in {\\mathcal {X}}_{\\Theta _{\\text{traj}},\\Omega }^{M,m} }\\,\\sum _{u\\in \\,{\\mathcal {U}}_{\\Theta _{\\text{traj}},x,n}^{\\,M,m,L_n}}A_2\\,\\frac{A_1}{A_2},$ where we note that $\\frac{A_1}{A_2}=\\prod _{i=0}^{N-1}\\,e^{u_i L_n \\big [\\psi ^{\\theta ^{\\widetilde{u}_{i}L_n}(\\omega )}_{L_n}(\\Omega (i,\\Pi _{i}+\\cdot ),\\Xi _i,x_i,u_i)-\\psi _{L_n}(\\Omega (i,\\Pi _{i}+\\cdot ),\\Xi _i,x_i,u_i)\\big ]}.$ In order to average over $\\omega $ , we apply a concentration of measure inequality.",
"Set $\\mathcal {K}_n = \\bigcup _{N=n/m L_n}^{n/L_n}\\bigcup _{\\Theta _{\\text{traj}}\\in \\widetilde{{\\mathcal {D}}}_{L_n,N}^M}\\,\\bigcup _{x\\in {\\mathcal {X}}_{\\Theta _{\\text{traj}},\\Omega }^{M,m} }\\,\\bigcup _{u\\in \\,{\\mathcal {U}}_{\\Theta _{\\text{traj}},x,n}^{\\,M,m,L_n}}\\Big \\lbrace |\\log A_1 -\\log A_2| \\ge {\\varepsilon }n\\Big \\rbrace ,$ and note that $\\omega \\in \\mathcal {K}_n^c$ implies that $Z_{1,n,L_n}^{\\,\\omega ,\\Omega }(M,m)\\le e^{{\\varepsilon }n} Z_{2,n,L_n}^{\\Omega }(M,m)$ .",
"Consequently, we can write $\\nonumber {\\mathbb {E}}_\\omega \\big [\\log Z_{1,n,L_n}^{\\,\\omega ,\\Omega }(M,m)\\big ]&= {\\mathbb {E}}_\\omega \\big [\\log Z_{1,n,L_n}^{\\,\\omega ,\\Omega }(M,m)\\,1_{\\lbrace \\mathcal {K}_n\\rbrace } \\big ]+{\\mathbb {E}}_\\omega \\big [\\log Z_{1,n,L_n}^{\\,\\omega ,\\Omega }(M,m)\\,1_{\\lbrace \\mathcal {K}_n^c\\rbrace }\\big ]\\\\&\\le {\\mathbb {E}}_\\omega \\big [\\log Z_{1,n,L_n}^{\\,\\omega ,\\Omega }(M,m)\\,1_{\\lbrace \\mathcal {K}_n\\rbrace } \\big ]+ \\log Z_{2,n,L_n}^{\\,\\Omega }(M,m)\\,+ {\\varepsilon }n.$ We can now use the uniform bound in (REF ) to control the first term in the right-hand side of (REF ), to obtain ${\\mathbb {E}}_\\omega \\big [\\log Z_{1,n,L_n}^{\\,\\omega ,\\Omega }(M,m)\\big ]\\le & \\log Z_{2,n,L_n}^{\\,\\Omega }(M,m) +{\\varepsilon }n+ C_{\\text{uf}}(\\alpha )\\,n\\,{\\mathbb {P}}_\\omega (\\mathcal {K}_n).$ Therefore the proof of this step will be complete once we show that ${\\mathbb {P}}_\\omega (\\mathcal {K}_n)$ vanishes as $n\\rightarrow \\infty $ .",
"Lemma 4.5 There exist $C_1,C_2>0$ such that, for all ${\\varepsilon }>0$ , $n\\in {\\mathbb {N}}$ , $N\\in \\big \\lbrace \\tfrac{n}{m L_n},\\dots ,\\frac{n}{L_n}\\big \\rbrace $ , $\\Omega \\in \\lbrace A,B\\rbrace ^{{\\mathbb {N}}_0\\times {\\mathbb {Z}}}$ , $\\Theta _{\\mathrm {traj}}\\in \\widetilde{{\\mathcal {D}}}_{L_n,N}^M$ , $x\\in {\\mathcal {X}}_{\\Theta _{\\mathrm {traj}},\\Omega }^{M,m}$ and $u\\in \\,{\\mathcal {U}}_{\\Theta _{\\mathrm {traj}},x,n}^{\\,M,m,L_n}$ , ${\\mathbb {P}}_{\\omega }(|\\log A_1-\\log A_2|\\ge {\\varepsilon }n)\\le C_1 e^{-C_2{\\varepsilon }^2 n}.$ Proof.",
"Pick $\\Theta _{\\text{traj}}\\in \\widetilde{{\\mathcal {D}}}_{L_n,N}^M$ , $x\\in {\\mathcal {X}}_{\\Theta _{\\text{traj},\\Omega }}^{M,m}$ and $u\\in \\,{\\mathcal {U}}_{\\Theta _{\\text{traj}},x,n}^{\\,M,m,L_n}$ , and consider the subset $\\Gamma $ of ${\\mathcal {W}}_{n,M}^{\\,m}$ consisting of those paths of length $n$ that first cross the $(\\Omega (0,\\cdot ),\\Xi _{0},x_0)$ column such that $\\pi _0=(0,1)$ and $\\pi _{\\widetilde{u}_{1}L_n}=(1,\\Pi _{1}+b_{1,0}) L_n$ , then cross the $(\\Omega (1,\\cdot ),\\Xi _{1},x_1)$ column such that $\\pi _{\\widetilde{u}_{1}L_n+1}=(1+1/L_n,\\Pi _{1}+b_{1,0})L_n$ and $\\pi _{\\widetilde{u}_{2}L_n}=(2,\\Pi _{2}+b_{1,1}) L_n$ , and so on.",
"We can apply the concentration of measure inequality stated in (REF ) to the set $\\Gamma $ defined above, with $l=n$ , $\\eta ={\\varepsilon }n$ , $\\xi _i = -\\alpha \\, 1\\lbrace \\omega _i=A\\rbrace +\\beta \\,1\\lbrace \\omega _i=B\\rbrace , \\qquad i\\in {\\mathbb {N}},$ and with $T(x,y)$ indicating in which block step $(x,y)$ lies in.",
"After noting that ${\\mathbb {E}}_\\omega (\\log A_1)= \\log A_2$ , we obtain that there exist $C_{1},C_{2}>0$ such that, for all $n\\in {\\mathbb {N}}$ , $N\\in \\big \\lbrace \\tfrac{n}{m L_n},\\dots ,\\frac{n}{L_n}\\big \\rbrace $ , $\\Omega \\in \\lbrace A,B\\rbrace ^{{\\mathbb {N}}_0\\times {\\mathbb {Z}}}$ , $\\Theta _{\\mathrm {traj}}\\in \\widetilde{{\\mathcal {D}}}_{L_n,N}^M$ , $x\\in {\\mathcal {X}}_{\\Theta _{\\text{traj},\\Omega }}^{M,m}$ and $u\\in \\,{\\mathcal {U}}_{\\Theta _{\\mathrm {traj}},x,n}^{\\,M,m,L_n}$ , ${\\mathbb {P}}\\big (|\\log A_1 -\\log A_2| \\ge {\\varepsilon }\\,n \\big ) \\le C_{1}\\,e^{-C_{2}\\,{\\varepsilon }^3\\, n}.$ $\\square $ It now suffices to remark that $\\big |\\lbrace (N,\\Theta _{\\text{traj}},x,u)\\colon \\,N\\in \\lbrace \\tfrac{n}{m L_n},\\dots ,\\tfrac{n}{L_n}\\rbrace , \\Theta _{\\text{traj}}\\in \\widetilde{{\\mathcal {D}}}_{L_n,N}^M,\\, x\\in {\\mathcal {X}}_{\\Theta _{\\text{traj},\\Omega }}^{M,m},u\\in \\,{\\mathcal {U}}_{\\Theta _{\\mathrm {traj}},x,n}^{\\,M,m,L_n}\\rbrace \\big |$ grows subexponentially in $n$ to obtain that $f_1^\\Omega \\prec f_2^\\Omega $ for all $\\Omega $ ."
],
[
"Step 2",
"In this step we replace the finite-size free energy $\\psi _{L_n}$ by its limit $\\psi $ .",
"To do so we introduce a third intermediate free energy, $f_{3,n}^\\Omega (M,m;\\alpha ,\\beta ) ={\\mathbb {E}}\\big [\\tfrac{1}{n} \\log Z_{3,n,L_n}^{\\Omega }(M,m)\\big ],$ where $Z_{3,n,L_n}^{\\Omega }(M,m)=\\sum _{N=n/m L_n}^{n/L_n}\\ \\sum _{\\Theta _{\\text{traj}}\\in \\widetilde{{\\mathcal {D}}}_{L_n,N}^M } \\,\\sum _{x\\in {\\mathcal {X}}_{\\Theta _{\\text{traj}},\\Omega }^{M,m} }\\,\\sum _{u\\in \\,{\\mathcal {U}}_{\\Theta _{\\text{traj}},x,n}^{\\,M,m,L_n}}A_3$ with $A_{3} &=\\prod _{i=0}^{N-1}\\,e^{u_i L_n \\psi (\\Omega (i,\\Pi _{i}+\\cdot ),\\Xi _i,x_i,u_i)}.$ For all $\\Omega $ , $\\frac{A_2}{A_3} = \\prod _{i=0}^{N-1}\\,e^{u_i L_n \\big [\\psi _{L_n}(\\Omega (i,\\Pi _{i}+\\cdot ),\\Xi _i,x_i,u_i)-\\psi (\\Omega (i,\\Pi _{i}+\\cdot ),\\Xi _i,x_i,u_i)\\big ]},$ and, for all $i\\in \\lbrace 0,\\dots ,N-1\\rbrace $ , we have $(\\Omega (i,\\Pi _{i}+\\cdot ),\\Xi _i,x_i,u_i)\\in {\\mathcal {V}}^{*,\\,m}_M$ , so that Proposition REF can be applied."
],
[
"Step 3",
"In this step we want the variational formula (REF ) to appear.",
"Recall (REF ) and define, for $n\\in {\\mathbb {N}}$ , $(M,m)\\in {\\hbox{\\footnotesize \\rm EIGH}}$ , $N\\in \\lbrace \\frac{n}{m L_n},\\dots ,\\frac{n}{L_n}\\rbrace $ , $\\Theta _{\\text{traj}}\\in \\widetilde{{\\mathcal {D}}}_{L_n,N}^M$ and $x\\in {\\mathcal {X}}_{\\Theta _{\\text{traj}},\\Omega }^{M,m}$ , $\\Theta _j=(\\Omega (j,\\Pi _{j}+\\cdot ),\\Xi _j,x_j),\\qquad j = 0,\\dots ,N-1,$ and $\\rho ^\\Omega _{\\Theta _{\\text{traj}},x}\\big (\\Theta ,\\Theta ^{^{\\prime }}\\big )=\\frac{1}{N} \\sum _{j=1}^{N}1_{\\big \\lbrace (\\Theta _{j-1},\\Theta _{j})=(\\Theta ,\\Theta ^{^{\\prime }})\\big \\rbrace },$ and, for $u\\in \\,{\\mathcal {U}}_{\\Theta _{\\text{traj}},x,n}^{\\,M,m,L_n}$ , $H^\\Omega (\\Theta _{\\text{traj}},x,u)=\\sum _{j=0}^{N-1} u_j\\,\\psi (\\Theta _j, u_{j}).$ In terms of these quantities we can rewrite $Z_{3,n,L_n}^\\Omega (M,m)$ in (REF ) as $Z_{3,n,L_n}^{\\Omega }(M,m)&=\\sum _{N=n/m L_n}^{n/L_n} \\sum _{\\Theta _{\\text{traj}}\\in \\widetilde{{\\mathcal {D}}}_{L_n,N}^M }\\,\\sum _{x\\in {\\mathcal {X}}_{\\Theta _{\\text{traj}},\\Omega }^{M,m} }\\,\\sum _{u\\in \\,{\\mathcal {U}}_{\\Theta _{\\text{traj}},x,n}^{\\,M,m,L_n}}e^{L_n\\, H^\\Omega (\\Theta _{\\text{traj}},x,u) }.$ For $n\\in {\\mathbb {N}}$ , denote by $N_n^{\\Omega }, \\qquad \\Theta _{\\text{traj},n}^{\\Omega }\\in \\widetilde{{\\mathcal {D}}}_{L_n,N^{\\Omega }_n}^M,\\qquad x_n^\\Omega \\in {\\mathcal {X}}_{\\Theta _{\\text{traj},n}^\\Omega ,\\Omega }^{M,m},\\qquad u^\\Omega _n \\in {\\mathcal {U}}_{\\Theta _{\\text{traj},n}^\\Omega ,x_n^\\Omega ,n}^{\\,M,m,L_n},$ the indices in the summation set of (REF ) that maximize $H^\\Omega (\\Theta _{\\text{traj}},x,u)$ .",
"For ease of notation we put $\\Theta _{\\text{traj},n}^{\\Omega }=(\\Xi _{j}^n)_{j=0}^{N_n^{\\Omega }-1},\\quad x^\\Omega _n=(x_j^n)_{j=0}^{N_n^{\\Omega }-1}, \\quad u^\\Omega _n=(u_j^n)_{j=0}^{N_n^{\\Omega }-1},$ and $c_n =\\big |\\lbrace (N,\\Theta _{\\text{traj}},x,u)\\colon \\,\\tfrac{n}{m L_n} \\le N\\le \\tfrac{n}{L_n},\\, \\Theta _{\\text{traj}}\\in \\widetilde{{\\mathcal {D}}}^M_{L_n,N},\\, x\\in {\\mathcal {X}}_{\\Theta _{\\text{traj}},\\Omega }^{M,m},\\,u\\in \\,{\\mathcal {U}}_{\\Theta _{\\text{traj}},x,n}^{\\,M,m,L_n}\\rbrace \\big |.$ Then we can estimate $\\frac{1}{n} \\log Z_{3,n,L_n}^{\\Omega }(M,m)\\le \\frac{1}{n} \\log c_n+ \\tfrac{L_n}{n}\\sum _{j=0}^{N_n^{\\Omega }-1} u_j^n\\,\\psi (\\Theta _j^n, u_{j}^n).$ We next note that $u\\mapsto u \\psi (\\Theta ,u)$ is concave for all $\\Theta \\in \\overline{{\\mathcal {V}}}_M$ (see Lemma REF ).",
"Hence, after setting $v_\\Theta ^n=\\sum _{j=0}^{N_n^{\\Omega }-1} 1_{\\lbrace \\Theta _j^n=\\Theta \\rbrace }\\,u_j^n, \\qquad d_\\Theta ^n=\\sum _{j=0}^{N_n^{\\Omega }-1}1_{\\lbrace \\Theta _j^n=\\Theta \\rbrace }, \\qquad \\Theta \\in \\overline{{\\mathcal {V}}}_M^{\\,m},$ we can estimate $\\sum _{j=0}^{N_n^{\\Omega }-1} 1_{\\lbrace \\Theta _j^n=\\Theta \\rbrace }\\,u_{j}^n\\, \\psi (\\Theta _j^n, u_{j}^n)\\le v_\\Theta ^n\\,\\psi \\big (\\Theta , \\tfrac{ v_\\Theta ^n}{d_\\Theta ^n}\\big )\\quad \\text{for} \\quad \\Theta \\in \\overline{{\\mathcal {V}}}_M^{\\,m}\\colon d_\\Theta ^n\\ge 1.$ Next, we recall (REF ) and we set $\\rho _{n}=\\rho _{\\Theta _{\\text{traj},n}^{\\Omega },x_n^\\Omega }^\\Omega $ , so that $\\rho _{n,1}(\\Theta )=d_\\Theta ^n/N_n^\\Omega $ for all $\\Theta \\in \\overline{{\\mathcal {V}}}_M^{\\,m}$ .",
"Since $\\lbrace \\Theta \\in \\overline{{\\mathcal {V}}}_M^{\\,m}\\colon \\,d_\\Theta ^n\\ge 1\\rbrace $ is a finite subset of $\\overline{{\\mathcal {V}}}_M^{\\,m}$ , we can easily extend $\\Theta \\mapsto v_\\Theta ^n/d_\\Theta ^n$ from $\\lbrace \\Theta \\in \\overline{{\\mathcal {V}}}_M\\colon \\,d_\\Theta ^n\\ge 1\\rbrace $ to $\\overline{{\\mathcal {V}}}_M^{\\,m}$ as a continuous function.",
"Moreover, $\\sum _{j=0}^{N_n^\\Omega -1} u_j^n=n/L_n$ implies that $N_n^\\Omega \\int _{\\overline{{\\mathcal {V}}}_M^{\\,m}}v_\\Theta ^n/d_\\Theta ^n\\,\\rho _{n,1} (d\\Theta )=n/L_n,$ which, together with (REF ) and (REF ) gives $\\tfrac{1}{n} \\log Z_{3,n,L_n}^{\\Omega }(M,m)&\\le \\sup _{u \\in {\\mathcal {B}}_{\\,\\overline{{\\mathcal {V}}}_M^{\\,m}}}\\frac{\\int _{\\overline{{\\mathcal {V}}}_M^{\\,m}}u_{\\Theta }\\, \\psi (\\Theta ,u_\\Theta )\\,\\rho _{n}(d\\Theta )}{\\int _{\\overline{{\\mathcal {V}}}_M^{\\,m}} u_{\\Theta }\\,\\rho _{n}(d\\Theta )}+ o(1), \\qquad n\\rightarrow \\infty ,$ where we use that $\\lim _{n\\rightarrow \\infty } \\frac{1}{n} \\log c_n=0$ .",
"In what follows, we abbreviate the first term in the right-hand side of the last display by $l_n$ .",
"We want to show that $\\limsup _{n\\rightarrow \\infty } \\tfrac{1}{n} \\log Z_{3,n,L_n}^{\\Omega }(M,m)$ $\\le f(M,m;\\alpha ,\\beta )$ .",
"To that end, we assume that $\\tfrac{1}{n} \\log Z_{3,n,L_n}^{\\Omega }(M,m)$ converges to some $t\\in {\\mathbb {R}}$ and we prove that $t\\le f(M,m;\\alpha ,\\beta )$ .",
"Since $(l_n)_{n\\in {\\mathbb {N}}}$ is bounded and $\\overline{{\\mathcal {V}}}_M^{\\,m}$ is compact, it follows from the definition of $l_n$ that along an appropriate subsequence both $l_n \\rightarrow l_\\infty \\ge t$ and $\\rho _n\\rightarrow \\rho _\\infty \\in {\\mathcal {R}}_{p,M}^{\\,m}$ as $n\\rightarrow \\infty $ .",
"Hence, the proof will be complete once we show that $l_\\infty \\le \\sup _{u\\in {\\mathcal {B}}_{\\,\\overline{{\\mathcal {V}}}_M^{\\,m}}} V(\\rho _{\\infty },u),$ because the right-hand side in (REF ) is bounded from above by $f(M,m;\\alpha ,\\beta )$ .",
"Recall (REF ) and, for $\\Theta \\in \\overline{{\\mathcal {V}}}_M^{\\,m}$ and $y\\in {\\mathbb {R}}$ , define $u_\\Theta ^{M,m}(y)= \\left\\lbrace \\begin{array}{ll}\\vspace{2.84544pt}t_\\Theta & \\mbox{if } \\partial ^+_u (u\\,\\psi (\\Theta ,u))(t_\\Theta ) \\le y, \\\\\\vspace{2.84544pt}m& \\mbox{if } \\partial ^-_u (u\\,\\psi (\\Theta ,u))(m) \\ge y,\\\\z&\\mbox{otherwise, with } z \\mbox{ such that } \\partial ^-_u (u\\,\\psi (\\Theta ,u))(z)\\ge y \\ge \\partial ^+_u (u\\,\\psi (\\Theta ,u))(z),\\end{array}\\right.$ where $z$ is unique by strict concavity of $u\\rightarrow u\\psi (\\Theta ,u)$ (see Lemma REF ).",
"Lemma 4.6 (i) For all $y\\in {\\mathbb {R}}$ and $(M,m)\\in {\\hbox{\\footnotesize \\rm EIGH}}$ , $\\Theta \\mapsto u_\\Theta ^{M,m}(y)$ is continuous on $(\\overline{{\\mathcal {V}}}_M^{\\,m},d_M)$ , where $d_M$ is defined in (REF ) in Appendix .",
"(ii) For all $(M,m)\\in {\\hbox{\\footnotesize \\rm EIGH}}$ and $\\Theta \\in \\overline{{\\mathcal {V}}}_M^{\\,m}$ , $y \\mapsto u_\\Theta ^{M,m}(y)$ is continuous on ${\\mathbb {R}}$ .",
"Proof.",
"The proof uses the strict concavity of $u\\rightarrow u\\psi (\\Theta ,u)$ (see Lemma REF ).",
"(i) The proof is by contradiction.",
"Pick $y\\in {\\mathbb {R}}$ , and pick a sequence $(\\Theta _n)_{n\\in {\\mathbb {N}}}$ in $\\overline{{\\mathcal {V}}}_M^{\\,m}$ such that $\\lim _{n\\rightarrow \\infty } \\Theta _n=\\Theta _\\infty \\in \\overline{{\\mathcal {V}}}_M^{\\,m}$ .",
"Suppose that $u_{\\Theta _n}^{M,m}(y)$ does not tend to $u_{\\Theta _\\infty }^{M,m}(y)$ as $n\\rightarrow \\infty $ .",
"Then, by choosing an appropriate subsequence, we may assume that $\\lim _{n\\rightarrow \\infty } u_{\\Theta _n}^{M,m}(y) = u_1\\in [t_{\\Theta _\\infty },m]$ with $u_1<u_{\\Theta _\\infty }^{M,m}(y)$ .",
"The case $u_1>u_{\\Theta _\\infty }^{M,m}(y)$ can be handled similarly.",
"Pick $u_2\\in (u_1,u_{\\Theta _\\infty }^{M,m}(y))$ .",
"For $n$ large enough, we have $u_{\\Theta _n}^{M,m}(y)< u_2<u_{\\Theta _\\infty }^{M,m}(y)$ .",
"By the definition of $u_{\\Theta _n}^{M,m}(y)$ in (REF ) and the strict concavity of $u\\mapsto u \\psi (\\Theta _n,u)$ we have, for $n$ large enough, $\\partial ^+_u (u\\,\\psi (\\Theta _n,u))(u_{\\Theta _n}^{M,m}(y))> \\frac{u_{\\Theta _\\infty }^{M,m}(y)\\psi (\\Theta _n,u_{\\Theta _\\infty }^{M,m}(y))-u_2\\psi (\\Theta _n,u_2)}{u_{\\Theta _\\infty }^{M,m}(y)-u_2}.$ Let $n\\rightarrow \\infty $ in (REF ) and use the strict concavity once again, to get $\\liminf _{n\\rightarrow \\infty } \\partial ^+_u (u\\,\\psi (\\Theta _n,u))(u_{\\Theta _n}^{M,m}(y))&>\\partial ^-_u (u\\,\\psi (\\Theta _\\infty ,u))(u_{\\Theta _\\infty }^{M,m}(y)).$ If $u_{\\Theta _\\infty }^{M,m}(y)\\in (t_{\\Theta _\\infty },m]$ , then (REF ) implies that the right-hand side of (REF ) is not smaller than $y$ .",
"Hence (REF ) yields that $\\partial ^+_u (u\\,\\psi (\\Theta _n,u))(u_{\\Theta _n}^{M,m}(y))>y$ for $n$ large enough, which implies that $u_{\\Theta _n}^{M,m}(y)=m$ by (REF ).",
"However, the latter inequality contradicts the fact that $u_{\\Theta _n}^{M,m}(y)<u_2<u_{\\Theta _\\infty }^{M,m}(y)$ for $n$ large enough.",
"If $u_{\\Theta _\\infty }^{M,m}(y)= t_{\\Theta _\\infty }$ , then we note that $\\lim _{n\\rightarrow \\infty } t_{\\Theta _n}=t_{\\Theta _\\infty }$ , which again contradicts that $t_{\\Theta _n}\\le u_{\\Theta _n}^{M,m}(y)< u_2<u_{\\Theta _\\infty }^{M,m}(y)$ for $n$ large enough.",
"(ii) The proof is again by contradiction.",
"Pick $\\Theta \\in \\overline{{\\mathcal {V}}}_M^{\\,m}$ , and pick an infinite sequence $(y_n)_{n\\in {\\mathbb {N}}}$ such that $\\lim _{n\\rightarrow \\infty } y_n=y_\\infty \\in {\\mathbb {R}}$ and such that $u_{\\Theta }^{M,m}(y_n)$ does not converge to $u_{\\Theta }^{M,m}(y_\\infty )$ .",
"Then, by choosing an appropriate subsequence, we may assume that there exists a $u_1 <u_{\\Theta }^{M,m}(y_\\infty )$ such that $\\lim _{n\\rightarrow \\infty }u_{\\Theta }^{M,m}(y_n)=u_1$ .",
"The case $u_1>u_{\\Theta }^{M,m}(y_\\infty )$ can be treated similarly.",
"Pick $u_2,u_3\\in (u_1, u_{\\Theta }^{M,m}(y_\\infty ))$ such that $u_2<u_3$ .",
"Then, for $n$ large enough, we have $t_\\Theta \\le u_{\\Theta }^{M,m}(y_n)<u_2<u_3<u_{\\Theta }^{M,m}(y_\\infty )\\le m.$ Combining (REF ) and (REF ) with the strict concavity of $u\\mapsto u\\psi (\\Theta ,u)$ we get, for $n$ large enough, $y_n>\\partial ^+_u (u\\,\\psi (\\Theta ,u))(u_2) >\\partial ^-_u (u\\,\\psi (\\Theta ,u))(u_3)> y_\\infty ,$ which contradicts $\\lim _{n\\rightarrow \\infty } y_n=y_\\infty $ .",
"$\\square $ We resume the line of proof.",
"Recall that $\\rho _{n,1}$ , $n\\in {\\mathbb {N}}$ , charges finitely many $\\Theta \\in \\overline{{\\mathcal {V}}}_M^{\\,m}$ .",
"Therefore the continuity and the strict concavity of $u\\mapsto u\\psi (\\Theta ,u)$ on $[t_\\Theta ,m]$ for all $\\Theta \\in \\overline{{\\mathcal {V}}}_M^{\\,m}$ (see Lemma REF ) imply that the supremum in (REF ) is attained at some $u_{n}^{M,m}\\in {\\mathcal {B}}_{\\,\\overline{{\\mathcal {V}}}_M^{\\,m}}$ that satisfies $u_{n}^{M,m}(\\Theta )= u_\\Theta ^{M,m}(l_n)$ for $\\Theta \\in \\overline{{\\mathcal {V}}}_M^{\\,m}$ .",
"Set $u^{M,m}_{\\infty }(\\Theta ) = u^{M,m}_{\\Theta }(l_\\infty )$ for $\\Theta \\in \\overline{{\\mathcal {V}}}_M^{\\,m}$ and note that $(l_n)_{n\\in {\\mathbb {N}}}$ may be assumed to be monotone, say, non-decreasing.",
"Then the concavity of $u\\mapsto u\\psi (\\Theta ,u)$ for $\\Theta \\in \\overline{{\\mathcal {V}}}_M^{\\,m}$ implies that $(u_n^{M,m})_{n\\in {\\mathbb {N}}}$ is a non-increasing sequence of functions on $\\overline{{\\mathcal {V}}}_M^{\\,m}$ .",
"Moreover, $\\overline{{\\mathcal {V}}}_M^{\\,m}$ is a compact set and, by Lemma REF (ii), $\\lim _{n\\rightarrow \\infty } u_n^{M,m}(\\Theta )=u_\\infty ^{M,m}(\\Theta )$ for $\\Theta \\in \\overline{{\\mathcal {V}}}_M^{\\,m}$ .",
"Therefore Dini's theorem implies that $\\lim _{n\\rightarrow \\infty } u_n^{M,m}=u_\\infty ^{M,m}$ uniformly on $\\overline{{\\mathcal {V}}}_M^{\\,m}$ .",
"We estimate $\\nonumber &\\left|l_n-\\int _{\\overline{{\\mathcal {V}}}_M^{\\,m}}u^{M,m}_{\\infty }(\\Theta )\\,\\psi (\\Theta ,u^{M,m}_{\\infty }(\\Theta ))\\rho _{\\infty }(d\\Theta )\\right|\\\\&\\qquad \\le \\int _{\\overline{{\\mathcal {V}}}_M^{\\,m}} \\Big |u^{M,m}_{n}(\\Theta )\\,\\psi (\\Theta ,u^{M,m}_{n}(\\Theta ))-u^{M,m}_{\\infty }(\\Theta )\\,\\psi (\\Theta ,u^{M,m}_{\\infty }(\\Theta ))\\Big |\\,\\rho _{n}(d\\Theta )\\\\\\nonumber &\\qquad +\\Big |\\int _{\\overline{{\\mathcal {V}}}_M^{\\,m}} u^{M,m}_{\\infty }(\\Theta )\\,\\psi (\\Theta ,u^{M,m}_{\\infty }(\\Theta ))\\,\\rho _{n}(d\\Theta )-\\int _{\\overline{{\\mathcal {V}}}_M^{m}} u^{M,m}_{\\infty }(\\Theta )\\,\\psi (\\Theta ,u^{M,m}_{\\infty }(\\Theta ))\\, \\rho _{\\infty }(d\\Theta )\\Big |.$ The second term in the right-hand side of (REF ) tends to zero as $n\\rightarrow \\infty $ because, by Lemma REF (i), $\\Theta \\mapsto u_{\\infty }^{M,m}(\\Theta )$ is continuous on $\\overline{{\\mathcal {V}}}_M^{\\,m}$ and because $\\rho _n$ converges in law to $\\rho _\\infty $ as $n\\rightarrow \\infty $ .",
"The first term in the right-hand side of (REF ) tends to zero as well, because $(\\Theta ,u)\\mapsto u\\psi (\\Theta ,u)$ is uniformly continuous on $\\overline{{\\mathcal {V}}}^{\\,*,m}_M$ (see Lemma REF ) and because we have proved above that $u^{M,m}_{n}$ converges to $u^{M,m}_{\\infty }$ uniformly on $\\overline{{\\mathcal {V}}}_M^{\\,m}$ .",
"This proves (REF ), and so Step 3 is complete."
],
[
"Step 4",
"In this step we prove that $\\limsup _{n\\rightarrow \\infty } f_{3,n}^{\\Omega }(M,m;\\alpha ,\\beta )\\ge f(M,m;\\alpha ,\\beta )\\,\\text{ for } {\\mathbb {P}}-a.e.\\ \\Omega .$ Note that the proof will be complete once we show that $\\limsup _{n\\rightarrow \\infty } f_{3,n}^\\Omega (M,m,\\alpha ,\\beta )\\ge V(\\rho ,u) \\ \\text{for} \\ \\rho \\in {\\mathcal {R}}_{p,M}^m,u\\in {\\mathcal {B}}_{\\,\\overline{{\\mathcal {V}}}_M^{\\,m}}.$ Pick $\\Omega \\in \\lbrace A,B\\rbrace ^{{\\mathbb {N}}_0\\times {\\mathbb {Z}}}$ , $\\rho \\in {\\mathcal {R}}^{\\Omega ,m}_{p,M}$ and $u\\in {\\mathcal {B}}_{\\,\\overline{{\\mathcal {V}}}_M^{\\,m}}$ .",
"By the definition of ${\\mathcal {R}}_{p,M}^{\\Omega ,m}$ , there exists a strictly increasing subsequence $(n_k)_{k\\in {\\mathbb {N}}}\\in {\\mathbb {N}}^{\\mathbb {N}}$ such that, for all $k\\in {\\mathbb {N}}$ , there exists an $N_k\\in \\left\\lbrace \\frac{n_k}{m L_{n_k}},\\dots ,\\frac{n_k}{L_{n_k}}\\right\\rbrace ,$ a $\\Theta _{\\text{traj}}^k\\in \\widetilde{{\\mathcal {D}}}_{L_{n_k},N_k}^M$ and a $x^k\\in {\\mathcal {X}}_{\\Theta _{\\text{traj}}^k,\\Omega }^{M,m}$ such that $\\rho _k=^\\mathrm {def}\\rho _{\\Theta _{\\text{traj}}^k,x^k}^\\Omega $ (see (REF )) converges in law to $\\rho $ as $k\\rightarrow \\infty $ .",
"Recall (REF ), and note that $\\Xi _j^k=\\big (\\Delta \\Pi ^k_j,\\,b^k_j,\\,b_{j+1}^k\\big ),\\quad \\text{$j=0,\\dots ,N_k-1$},$ with $\\Delta \\Pi ^k_j\\in \\lbrace -M,\\dots ,M\\rbrace $ and $b_j^k\\in (0,1]\\cap \\frac{{\\mathbb {N}}}{L_{n_k}}$ for $j=0,\\dots ,N_k$ .",
"For ease of notation we define $\\Theta _{j}^k=\\big (\\Omega (j,\\Pi ^k_j+\\cdot ),\\Xi _j^k,x_j^k\\big )\\quad \\text{with} \\quad \\Pi _j^k=\\sum _{i=0}^{j-1} \\Delta \\Pi _i^k, \\qquad j=0,\\dots ,N_k-1,$ and $v_k=N_k \\int _{\\Theta \\in {\\mathcal {V}}_M^{\\,m}} u_\\Theta \\,\\rho _{k,1}(d\\Theta )= \\sum _{j=0}^{N_k-1} u_{\\Theta _j^k},$ where we recall that $u=(u_\\Theta )_{\\Theta \\in \\overline{{\\mathcal {V}}}_M^{\\,m}}$ was fixed at the beginning of the section.",
"Next, we recall that $\\lim _{n\\rightarrow \\infty } n/L_n=\\infty $ and that $L_n$ is non-decreasing.",
"Together with the fact that $\\lim _{n\\rightarrow \\infty } L_n/n=0$ , this implies that $L_n$ is constant on intervals.",
"On those intervals, $n/L_n$ takes constant increments.",
"The latter implies that there exists an $\\widetilde{n}_k\\in {\\mathbb {N}}$ satisfying $0\\le v_k-\\tfrac{\\widetilde{n}_k}{L_{\\widetilde{n}_k}}\\le \\tfrac{1}{L_{\\widetilde{n}_k}}\\quad \\text{and therefore}\\quad 0\\le v_k L_{\\widetilde{n}_k}-\\widetilde{n}_k \\le 1.$ Next, for $j=0,\\dots ,N_k-1$ we pick $\\overline{b_j^k}\\in (0,1] \\cap \\frac{N}{L_{\\widetilde{n}_k}}$ such that $|\\overline{b_j^k}-b_j^k|\\le \\tfrac{1}{L_{\\widetilde{n}_k}}$ , define $\\overline{\\Xi _j^k} = \\big (\\Delta \\Pi ^k_j,\\overline{b_j^k},\\overline{b_{j+1}^k}\\,\\big ),\\quad \\overline{\\Theta _j^k}= \\big (\\Omega (j,\\Pi ^k_j+\\cdot ),\\overline{\\Xi _j^k},x_j^k\\,\\big ),$ and pick $s_j^k\\in t_{\\,\\overline{\\Theta _j^k}}+\\frac{2{\\mathbb {N}}}{L_{\\widetilde{n}_k}}\\quad \\text{such that} \\quad |s_j^k-u_{\\Theta _j^k}|\\le 2/L_{\\widetilde{n}_k}.$ We use (REF ) to write $L_{\\widetilde{n}_k} \\sum _{j=0}^{N_k-1} s^k_j=L_{\\widetilde{n}_k}\\bigg (v_k +\\sum _{j=0}^{N_k-1} (s_j^k-u_{\\Theta _j^k})\\bigg )= L_{\\widetilde{n}_k} (I+II).$ Next, we note that (REF ) and (REF ) imply that $|L_{\\widetilde{n}_k}I-\\widetilde{n}_k|\\le 1$ and $|L_{\\widetilde{n}_k} II|\\le 2 N_k$ .",
"The latter in turn implies that, by adding or subtracting at most 3 steps per colum, the quantities $s_j^k$ for $j=0,\\dots ,N_k-1$ can be chosen in such a way that $\\sum _{j=0}^{N_k-1} s_j^k=\\widetilde{n}_k/L_{\\widetilde{n}_k}$ .",
"Next, set $\\overline{\\Theta _{\\text{traj}}^k}=(\\overline{\\Xi _j^k})_{j=0}^{N_k-1}\\in \\widetilde{{\\mathcal {D}}}_{L_{\\widetilde{n}_k},N_k }^M,\\quad s^k=(s^k_j)_{j=0}^{N_k-1}\\in {\\mathcal {U}}_{\\overline{\\Theta _{\\text{traj}}^k},\\,x^k,\\widetilde{n}_k}^{M,m,L_{\\widetilde{n}_k}},$ and recall (REF ) to get $f_3^\\Omega (\\widetilde{n}_k,M)\\ge R_k$ with $R_k=\\frac{L_{\\widetilde{n}_k} \\,H^\\Omega \\big (\\,\\overline{\\Theta _{\\text{traj}}^k},x^k,s^k\\,\\big )}{\\widetilde{n}_k}= \\frac{\\sum _{j=0}^{N_k-1}\\, s_j^k\\,\\psi \\Big (\\overline{\\Theta _j^k},\\,s_j^k\\Big )}{\\sum _{j=0}^{N_k-1} s_j^k}=\\frac{R_{\\text{nu}}^k}{R_{\\text{de}}^k}.$ Further set $R^{^{\\prime }}_k=\\frac{R^{^{\\prime } k}_{\\text{nu}}}{R^{^{\\prime } k}_{\\text{de}}}=\\frac{\\int _{\\overline{{\\mathcal {V}}}_M^{\\,m}} u_\\Theta \\, \\psi (\\Theta ,u_\\Theta )\\rho _{k}(d\\Theta )}{\\int _{\\overline{{\\mathcal {V}}}_M^{\\,m}} u_\\Theta \\,\\rho _{k}(d\\Theta )},$ and note that $\\lim _{k\\rightarrow \\infty } R^{^{\\prime }}_k=V(\\rho ,u)$ , since $\\lim _{k\\rightarrow \\infty }\\rho _k=\\rho $ by assumption and $\\Theta \\mapsto u_{\\Theta }$ is continuous on ${\\mathcal {V}}_M^{\\,m}$ .",
"We note that $R^{^{\\prime }}_k$ can be rewritten in the form $R^{^{\\prime }}_k=\\frac{R^{^{\\prime } k}_{\\text{nu}}}{R^{^{\\prime } k}_{\\text{de}}}= \\frac{\\sum _{j=0}^{N_k-1}\\, u_{\\Theta _j^k}\\,\\psi \\big (\\Theta _j^k,\\,u_{\\Theta _j^k}\\big )}{\\sum _{j=0}^{N_k-1}\\, u_{\\Theta _j^k}}.$ Now recall that $\\lim _{k\\rightarrow \\infty } n_k=\\infty $ .",
"Since $N_k\\ge n_k/M L_{n_k}$ , it follows that $\\lim _{k\\rightarrow \\infty } N_k=\\infty $ as well.",
"Moreover, $N_k\\le \\widetilde{n}_k/L_{\\widetilde{n}_k}$ with $\\lim _{k\\rightarrow \\infty } \\widetilde{n}_k=\\infty $ .",
"Therefore (REF –REF ) allow us to conclude that $R_{\\text{de}}^k=\\widetilde{n}_k/L_{\\widetilde{n}_k}=R^{^{\\prime }k}_{\\text{de}}[1+o(1)]$ .",
"Next, note that ${\\mathcal {H}}_M$ is compact, and that $(\\Theta ,u)\\mapsto u \\psi (\\Theta ,u)$ is continuous on ${\\mathcal {H}}_M$ and therefore is uniformly continuous.",
"Consequently, for all ${\\varepsilon }>0$ there exists an $\\eta >0$ such that, for all $(\\Theta ,u),(\\Theta ^{^{\\prime }},u^{^{\\prime }})\\in {\\mathcal {H}}_M$ satisfying $|\\Theta -\\Theta ^{^{\\prime }}|\\le \\eta $ and $|u-u^{^{\\prime }}|\\le \\eta $ , $|u\\psi (\\Theta ,u)-u^{^{\\prime }}\\psi (\\Theta ^{^{\\prime }},u^{^{\\prime }})|\\le {\\varepsilon }.$ We recall (REF ), which implies that $d_M(\\overline{\\Theta _j^k},\\Theta _j)\\le 2/L_{\\widetilde{n}_k}$ for all $j\\in \\lbrace 0,\\dots ,N_k-1\\rbrace $ , we choose $k$ large enough to ensure that $2/L_{\\widetilde{n}_k}\\le \\eta $ , and we use (REF ), to obtain $R^{k}_{\\text{nu}}=\\sum _{j=0}^{N_k-1}\\,s_j^k\\, \\psi \\Big (\\overline{\\Theta _j^k},\\,s_j^k\\Big )= \\sum _{j=0}^{N_k-1}\\, u_{\\Theta _j^k}\\,\\psi \\big (\\Theta _j^k,\\,u_{\\Theta _j^k}\\big )+T=R^{^{\\prime } k}_{\\text{nu}}+T,$ with $|T| \\le {\\varepsilon }N_k$ .",
"Since $\\lim _{k\\rightarrow \\infty } R^{^{\\prime }}_k=V(\\rho ,u)$ and $\\sum _{j=0}^{N_k-1} u_{\\Theta _j^k}=v_k\\ge \\widetilde{n}_k/L_{\\widetilde{n}_k}$ (see (REF )), if $ V(\\rho ,u)\\ne 0$ , then $\\big |R^{^{\\prime } k}_{\\text{nu}}\\big |\\ge \\mathrm {Cst.",
"}\\,\\,\\widetilde{n}_k/L_{\\widetilde{n}_k}$ , whereas $|T|\\le {\\varepsilon }N_k\\le {\\varepsilon }\\widetilde{n}_k/L_{\\widetilde{n}_k} $ for $k$ large enough.",
"Hence $T=o(R^{^{\\prime } k}_{\\text{nu}})$ and $\\frac{R^{k}_{\\text{nu}}}{R^{k}_{\\text{de}}}= \\frac{R^{^{\\prime } k}_{\\text{nu}}\\, [1+o(1)]}{R^{^{\\prime }k}_{\\text{de}}\\,[1+o(1)]}\\rightarrow V(\\rho ,u), \\qquad k\\rightarrow \\infty .$ Finally, if $ V(\\rho ,u)= 0$ , then $R^{^{\\prime } k}_{\\text{nu}}=o(R^{^{\\prime }k}_{\\text{de}})$ and $T=o(R^{^{\\prime }k}_{\\text{de}})$ , so that $R_k$ tends to 0.",
"This completes the proof of Step 4."
],
[
"Step 5",
"In this step we prove (REF ), suppressing the $(\\alpha ,\\beta )$ -dependence from the notation.",
"For $\\Omega \\in \\lbrace A,B\\rbrace ^{{\\mathbb {N}}_0\\times {\\mathbb {Z}}^2}$ , $n\\in {\\mathbb {N}}$ , $N\\in \\lbrace n/m L_n,\\dots ,n/L_n\\rbrace $ and $r\\in \\lbrace -N M,\\dots ,N M\\rbrace $ , we recall (REF ) and define $\\widetilde{{\\mathcal {D}}}_{L,N}^{M,m,r}= \\Big \\lbrace \\Theta _{\\text{traj}}\\in \\widetilde{{\\mathcal {D}}}_{L,N}^{M,m}\\colon \\Pi _N=r\\Big \\rbrace ,$ where we recall that $\\Pi _N=\\sum _{j=0}^{N-1} \\Delta \\Pi _j$ .",
"We set $f_{3,n}^\\Omega (M,m,N,r) = \\tfrac{1}{n} \\log Z_{3,n,L_n}^{\\Omega }(N,M,m,r)$ with $Z_{3,n,L_n}^{\\Omega }(N,M,m,r)= \\sum _{\\Theta _{\\text{traj}}\\in \\widetilde{{\\mathcal {D}}}_{L_n,N}^{M,m,r} }\\,\\sum _{x\\in {\\mathcal {X}}_{\\Theta _{\\text{traj}},\\Omega }^{M,m} }\\,\\sum _{u\\in \\,{\\mathcal {U}}_{\\Theta _{\\text{traj}},n}^{\\,M,m,L_n}} A_3,$ where $A_3$ is defined in (REF ).",
"We further set $f_3(\\cdot )={\\mathbb {E}}_\\Omega \\big (f_3^\\Omega (\\cdot )\\big )$ ."
],
[
"Concentration of measure",
"In the first part of this step we prove that for all $(M,m,\\alpha ,\\beta )\\in {\\hbox{\\footnotesize \\rm EIGH}}\\times {\\hbox{\\footnotesize \\rm CONE}}$ there exist $c_1,c_2>0$ (depending on $(M,m,\\alpha ,\\beta )$ only) such that, for all $n\\in {\\mathbb {N}}$ , $N\\in \\lbrace n/(m L_n),\\dots n/L_n\\rbrace $ and $r\\in \\lbrace -N M,\\dots ,N M\\rbrace $ , $&{\\mathbb {P}}_{\\Omega }\\big (\\big |f_{3,n}^{\\Omega }(M,m)-f_{3,n}(M,m)\\big |>{\\varepsilon }\\big )\\le c_1\\ e^{-\\frac{c_2 {\\varepsilon }^2 n}{L_n}}, \\qquad \\\\\\nonumber &{\\mathbb {P}}_{\\Omega }\\big (\\big |f_{3,n}^{\\Omega }(M,m,N,r)-f_{3,n}(M,m,N,r)\\big |>{\\varepsilon }\\big )\\le c_1\\ e^{-\\frac{c_2 {\\varepsilon }^2 n}{L_n}}.$ We only give the proof of the first inequality.",
"The second inequality is proved in a similar manner.",
"The proof uses Theorem REF .",
"Before we start we note that, for all $n\\in {\\mathbb {N}}$ , $(M,m)\\in {\\hbox{\\footnotesize \\rm EIGH}}$ and $\\Omega \\in \\lbrace A,B\\rbrace ^{{\\mathbb {N}}_0\\times {\\mathbb {Z}}}$ , $f_{3,n}^\\Omega (M,m)$ only depends on ${\\mathcal {C}}_{0,L_n}^\\Omega ,\\dots ,{\\mathcal {C}}_{n/L_n,L_n}^\\Omega \\quad \\text{with}\\quad {\\mathcal {C}}_{j,L_n}^\\Omega =(\\Omega (j,i))_{i=-n/L_n}^{n/L_n}.$ We apply Theorem REF with ${\\mathcal {S}}=\\lbrace 0,\\dots ,n/L_n\\rbrace $ , with $X_i=\\lbrace A,B\\rbrace ^{\\lbrace -\\frac{n}{L_n},\\dots ,\\frac{n}{L_n}\\rbrace }$ and with $\\mu _i$ the uniform measure on $X_i$ for all $i\\in {\\mathcal {S}}$ .",
"Note that $|f_{3,n}^{\\Omega _1}(M,m)-f_{3,n}^{\\Omega _2}(M,m)|\\le 2 C_{\\text{uf}}(\\alpha ) m \\tfrac{L_n}{n}$ for all $i\\in {\\mathcal {S}}$ and all $\\Omega _1,\\Omega _2$ satisfying ${\\mathcal {C}}_{j,n}^{\\Omega _1}={\\mathcal {C}}_{j,n}^{\\Omega _2}$ for all $j\\ne i$ .",
"After we set $c=2 C_{\\text{uf}}(\\alpha ) m$ we can apply Theorem REF with $D=c^2 L_n/n$ to get (REF ).",
"Next, we note that the first inequality in (REF ), the Borel-Cantelli lemma and the fact that $\\lim _{n\\rightarrow \\infty } n/L_n \\log n = \\infty $ imply that, for all $(M,m)\\in {\\hbox{\\footnotesize \\rm EIGH}}$ , $\\lim _{n\\rightarrow \\infty } \\Big [f_{3,n}^\\Omega (M,m)- f_{3,n}(M,m)\\Big ]=0\\quad \\text{for } {\\mathbb {P}}-a.e.~\\Omega .$ Therefore (REF ) will be proved once we show that $\\liminf _{n\\rightarrow \\infty } f_{3,n}(M,m)= \\limsup _{n\\rightarrow \\infty } f_{3,n}(M,m).$ To that end, we first prove that, for all $n\\in {\\mathbb {N}}$ and all $(M,m)\\in {\\hbox{\\footnotesize \\rm EIGH}}$ , there exist an $N_n\\in \\lbrace n/m L_n,\\dots ,n/L_n\\rbrace $ and an $r_n\\in \\lbrace -M N_n,\\dots ,M N_n\\rbrace $ such that $\\lim _{n\\rightarrow \\infty } \\Big [f_{3,n}(M,m)-f_{3,n}(M,m,N_n,r_n)\\Big ] = 0.$ The proof of (REF ) is done as follows.",
"Pick ${\\varepsilon }>0$ , and for $\\Omega \\in \\lbrace A,B\\rbrace ^{{\\mathbb {N}}_0\\times {\\mathbb {Z}}}$ , $n\\in {\\mathbb {N}}$ and $(M,m)\\in {\\hbox{\\footnotesize \\rm EIGH}}$ , denote by $N_n^\\Omega $ and $r_n^{\\Omega }$ the maximizers of $f^\\Omega _{3,n}(M,m,N,r)$ .",
"Then $f^\\Omega _{3,n} \\big (M,m,N_n^{\\Omega },r_n^{\\Omega }\\big )\\le f^\\Omega _{3,n}(M,m)\\le \\tfrac{1}{n} \\log (\\tfrac{n^2}{L_n^2})+ f^\\Omega _{3,n}\\big (M,m,N_n^{\\Omega },r_n^{\\Omega }\\big ),$ so that, for $n$ large enough and every $\\Omega $ , $0\\le f^\\Omega _{3,n}(M,m)-f^\\Omega _{3,n}\\big (M,m,N_n^{\\Omega },r_n^{\\Omega }\\big )\\le {\\varepsilon }.$ For $n\\in {\\mathbb {N}}$ , $N\\in \\lbrace n/m L_n,\\dots , n/L_n\\rbrace $ and $r\\in \\lbrace -N M,\\dots , N M\\rbrace ,$ we set $A_{n,N,r}=\\lbrace \\Omega \\colon (N_n^\\Omega ,r_n^\\Omega )=(N,r)\\rbrace .$ Next, denote by $N_n,r_n$ the maximizers of ${\\mathbb {P}}(A_{n,N,r})$ .",
"Note that (REF ) will be proved once we show that, for all ${\\varepsilon }>0$ , $|f_{3,n}(M,m)-f_{3,n}(M,m,N_n,r_n)|\\le {\\varepsilon }$ for $n$ large enough.",
"Further note that ${\\mathbb {P}}(A_{n,N_n,r_n})\\ge L_n^2/n^2$ for all $n\\in {\\mathbb {N}}$ .",
"For every $\\Omega $ we can therefore estimate $|f_{3,n}(M,m)-f_{3,n}(M,m,N_n,r_n)|\\le I+II+III$ with $&I=|f_{3,n}(M,m)-f_{3,n}^\\Omega (M,m)|,\\\\\\nonumber & II=|f_{3,n}^\\Omega (M,m)-f_{3,n}^\\Omega (M,m,N_n,r_n)|, \\\\\\nonumber &III=|f_{3,n}^\\Omega (M,m,N_n,r_n)-f_{3,n}(M,m,N_n,r_n)|.$ Hence, the proof of (REF ) will be complete once we show that, for $n$ large enough, there exists an $\\Omega _{{\\varepsilon },n}$ for which $I,II$ and $III$ in (REF ) are bounded from above by ${\\varepsilon }/3$ .",
"To that end, note that, because of (REF ), the probabilities ${\\mathbb {P}}(\\lbrace I> {\\varepsilon }/3\\rbrace )$ and ${\\mathbb {P}}(\\lbrace III> {\\varepsilon }/3\\rbrace )$ are bounded from above by $c_1 e^{-c_2 {\\varepsilon }^2 n/9 L_n}$ , while ${\\mathbb {P}}(\\lbrace II>{\\varepsilon }\\rbrace )\\le {\\mathbb {P}}(A_{n,N_n,r_n}^c)\\le 1-(L_n^2/n^2), \\qquad n\\in {\\mathbb {N}}.$ Since $\\lim _{n\\rightarrow \\infty } n/L_n \\log n=\\infty $ , we have ${\\mathbb {P}}(\\lbrace I,II,III \\le {\\varepsilon }/3\\rbrace )>0$ for $n$ large enough.",
"Consequently, the set $\\lbrace I,II,III\\le {\\varepsilon }/3\\rbrace $ is non-empty and (REF ) is proven."
],
[
"Convergence",
"It remains to prove (REF ).",
"Assume that there exist two strictly increasing subsequences $(n_k)_{k\\in {\\mathbb {N}}}$ and $(t_k)_{k\\in {\\mathbb {N}}}$ and two limits $l_2>l_1$ such that $\\lim _{k\\rightarrow \\infty }f_{3,n_k}(M,m)=l_2$ and $\\lim _{k\\rightarrow \\infty }$ $f_{3,t_k}(M,m) =l_1$ .",
"By using (REF ), we have that for every $k\\in {\\mathbb {N}}$ there exist $N_k\\in \\lbrace n_k/m L_{n_k},$ $\\dots ,n_k/L_{n_k}\\rbrace $ and $r_k\\in \\lbrace -M N_k,\\dots ,M N_k\\rbrace $ such that $\\lim _{k\\rightarrow \\infty }$ $f_{3,n_k}(M,m,N_k,r_k)=l_2$ .",
"Denote by $(\\Theta _{\\text{traj,max}}^{k,\\Omega },x_{\\text{max}}^{k,\\Omega },u_{\\text{max}}^{k,\\Omega })\\in \\widetilde{{\\mathcal {D}}}_{L_{n_k},N_k}^{M,r_k}\\times {\\mathcal {X}}_{\\Theta _{\\text{traj,max}}^{k,\\Omega },\\Omega }^{M,m}\\times {\\mathcal {U}}_{\\Theta _{\\text{traj,max}}^{k,\\Omega },x_{\\text{max}}^{k,\\Omega },n_k}^{\\,M,m,L_n}$ the maximizer of $H^\\Omega (\\Theta _{\\text{traj}},x,u)$ .",
"We recall that $\\Theta _{\\text{traj}},x$ and $u$ take their values in sets that grow subexponentially fast in $n_k$ , and therefore $\\lim _{k\\rightarrow \\infty } \\tfrac{L_{n_k}}{n_k}\\,{\\mathbb {E}}_\\Omega \\big [H^\\Omega (\\Theta _{\\text{traj,max}}^{k,\\Omega },x_{\\text{max}}^{k,\\Omega },u_{\\text{max}}^{k,\\Omega })\\big ]=l_2.$ Since $l_2>l_1$ , we can use (REF ) and the fact that $\\lim _{k\\rightarrow \\infty } n_k/L_{n_k}=\\infty $ to obtain, for $k$ large enough, ${\\mathbb {E}}_\\Omega \\big [H^\\Omega (\\Theta _{\\text{traj,max}}^{k,\\Omega },x_{\\text{max}}^{k,\\Omega },u_{\\text{max}}^{k,\\Omega })\\big ]+(\\beta -\\alpha )\\ge \\tfrac{n_k}{L_{n_k}} \\big (l_1+\\tfrac{l_2-l_1}{2}\\big ).$ (The term $\\beta -\\alpha $ in the left-hand side of (REF ) is introduced for later convenience only.)",
"Next, pick $k_0\\in {\\mathbb {N}}$ satisfying (REF ), whose value will be specified later.",
"Similarly to what we did in (REF ) and (REF ), for $\\Omega \\in \\lbrace A,B\\rbrace ^{{\\mathbb {N}}_0\\times {\\mathbb {Z}}}$ and $k\\in {\\mathbb {N}}$ we associate with $\\Theta _{\\text{traj,max}}^{k_0,\\Omega }=\\big (\\Delta \\Pi _{\\,j}^{k_0,\\Omega }, b^{k_0,\\Omega }_{\\,0,j},b_{\\,1,j}^{k_0,\\Omega }\\big )_{j=0}^{N_{k_0}-1}\\in \\widetilde{{\\mathcal {D}}}_{L_{n_{k_0}},N_{k_0}}^{M,r_{k_0}}$ and $x_{\\text{max}}^{k_0,\\Omega }=\\big (x_{\\,j}^{k_0,\\Omega }\\big )_{j=0}^{N_{k_0}-1}\\in {\\mathcal {X}}_{\\Theta _{\\text{traj,max}}^{k_0,\\Omega },\\Omega }^{M,m}$ and $u_{\\text{max}}^{k_0,\\Omega }=\\big (u_{\\,j}^{k_0,\\Omega }\\big )_{j=0}^{N_{k_0}-1}\\in {\\mathcal {U}}_{\\Theta _{\\text{traj,max}}^{k_0,\\Omega },x_{\\text{max}}^{k_0,\\Omega },n_{k_0}}^{M,m,L_{n_{k_0}}}$ the quantities $\\overline{\\Theta }^{\\,k,\\Omega }_{\\text{traj}}= \\big (\\Delta \\Pi _{\\,j}^{k_0,\\Omega }, \\overline{b}_{\\,0,j}^{\\,k,\\,\\Omega },\\overline{b}_{\\,1,j}^{\\,k,\\,\\Omega }\\big )_{j=0}^{N_{k_0}-1}\\in \\widetilde{{\\mathcal {D}}}_{L_{t_{k}},N_{k_0}}^{M,r_{k_0}}$ and $\\overline{u}^{\\,k,\\Omega }=\\big (\\overline{u}_{\\,j}^{\\,k,\\Omega }\\big )_{j=0}^{N_{k_0}-1}\\in {\\mathcal {U}}_{\\overline{\\Theta }_{\\text{traj}}^{\\,k,\\Omega },x_{\\text{max}}^{k_0,\\Omega },*}^{M,m,L_{t_{k}}}$ (where $*$ will be specified later), so that $\\big |\\overline{b}_{\\,0,j}^{\\,k,\\,\\Omega }-b_{\\,0,j}^{k_0,\\Omega }\\big |\\le \\tfrac{1}{L_{t_k}},\\ \\ \\big |\\overline{b}_{\\,1,j}^{\\,k,\\,\\Omega }-b_{\\,1,j}^{k_0,\\Omega }\\big |\\le \\tfrac{1}{L_{t_k}},\\ \\ \\ \\ \\big |\\overline{u}_{\\,j}^{\\,k,\\,\\Omega }-u_{\\,j}^{k_0,\\Omega }\\big |\\le \\tfrac{2}{L_{t_k}},\\ \\ \\ j=0,\\dots ,N_{k_0}-1.$ Next, put $\\overline{s}_{k}^{\\,\\Omega }=L_{t_k} \\sum _{j=0}^{N_{k_0}-1}\\overline{u}_j^{\\,k,\\,\\Omega }$ , which we substitute for $*$ above.",
"The uniform continuity in Lemma REF allows us to claim that, for $k$ large enough and for all $\\Omega $ , $\\Big |\\overline{u}_j^{\\,k,\\,\\Omega } \\ \\psi \\Big (\\overline{\\Theta }_j^{\\,k,\\,\\Omega },\\overline{u}_j^{\\,k,\\,\\Omega }\\Big )-u_{\\,j}^{k_0,\\Omega }\\ \\psi \\Big (\\Theta _{\\,j}^{k_0,\\Omega },u_{\\,j}^{k_0,\\Omega }\\Big )\\Big |\\le \\tfrac{l_2-l_1}{4},$ where we recall that, as in (REF ), for all $j=0,\\dots ,N_{k_0}-1$ , $\\overline{\\Theta }_j^{\\,k,\\,\\Omega }&=\\Big (\\Omega \\big (j,\\Pi ^{k_0,\\Omega }_{\\,j}+\\cdot \\big ),\\,\\Delta \\Pi _{\\,j}^{k_0,\\Omega },\\,\\overline{b}_{\\,0,j}^{\\,k,\\,\\Omega },\\,\\overline{b}_{\\,1,j}^{\\,k,\\,\\Omega },\\,x_j^{k_0,\\Omega }\\Big ),\\\\\\nonumber \\Theta _{\\,j}^{k_0,\\Omega }&=\\Big (\\Omega \\big (j,\\Pi ^{k_0,\\Omega }_{\\,j}+\\cdot \\big ),\\,\\Delta \\Pi _j^{k_0,\\Omega },\\,b_{\\,0,j}^{k_0,\\Omega },\\, b_{\\,1,j}^{k_0,\\Omega },\\,x_j^{k_0,\\Omega }\\Big ).$ Recall (REF ).",
"An immediate consequence of (REF ) is that $\\big |H^\\Omega (\\overline{\\Theta }_{\\text{traj}}^{\\,k,\\Omega },x_{\\text{max}}^{k_0,\\Omega },\\overline{u}^{\\,k,\\Omega })-H^\\Omega (\\Theta _{\\text{traj,max}}^{k_0,\\Omega },x_{\\text{max}}^{k_0,\\Omega },u_{\\text{max}}^{k_0,\\Omega })\\big |\\le N_{k_0} \\tfrac{l_2-l_1}{4}.$ Hence we can use (REF ), (REF ) and the fact that $N_{k_0}\\le n_{k_0}/L_{n_{k_0}}$ , to conclude that, for $k$ large enough, ${\\mathbb {E}}_\\Omega \\big [H^\\Omega (\\overline{\\Theta }_{\\text{traj}}^{\\,k,\\Omega },x_{\\text{max}}^{k_0,\\Omega },\\overline{u}^{\\,k,\\Omega })\\big ]+(\\beta -\\alpha )\\ge \\tfrac{n_{k_0}}{L_{n_{k_0}}}\\big (l_1+\\tfrac{l_2-l_1}{4}\\big ).$ At this stage we add a column at the end of the group of $N_{k_0}$ columns in such a way that the conditions $\\widehat{b}^{k,\\Omega }_{1, N_{k_0}-1}=\\widehat{b}^{k,\\Omega }_{0, N_{k_0}}$ and $\\widehat{b}^{k,\\Omega }_{1, N_{k_0}}=1/L_{t_k}$ are satisfied.",
"We put $\\widehat{\\Xi }_{N_{k_0}}^{k,\\,\\Omega }=\\big (\\Delta \\Pi _{N_{{k_0}}}^{k_0,\\Omega },\\widehat{b}_{0,N_{k_0}}^{k,\\,\\Omega },\\widehat{b}_{1,N_{k_0}}^{k,\\,\\Omega }\\big )=\\big (0,\\widehat{b}_{1,N_{k_0}-1}^{k,\\,\\Omega },\\tfrac{1}{L_{t_k}}\\big ),$ and we let $\\widehat{\\Theta }_{\\text{traj}}^{k,\\,\\Omega }\\in \\widetilde{{\\mathcal {D}}}_{L_{t_k},\\,N_{k_0}+1}^{M,\\,r_{k_0}}$ be the concatenation of $\\overline{\\Theta }_{\\text{traj}}^{k,\\Omega }$ (see (REF )) and $\\widehat{\\Xi }_{N_{k_0}}^{k,\\Omega }$ .",
"We let $\\widehat{x}^{k_0,\\Omega }\\in {\\mathcal {X}}_{\\widehat{\\Theta }_{\\text{traj}}^{k,\\Omega },\\Omega }^{M,m}$ be the concatenation of $x_{\\text{max}}^{k_0,\\Omega }$ and 0.",
"We further let $\\widehat{s}_k^{\\,\\Omega }=\\overline{s}_k^{\\,\\Omega }+\\Big [1+b_{1,N_{k_0}-1}^{k,\\Omega }-\\tfrac{1}{L_{t_k}}\\Big ] L_{t_k},$ and we let $\\widehat{u}^{k,\\Omega }\\in {\\mathcal {U}}_{\\,\\widehat{\\Theta }_{\\text{traj}}^{k,\\ \\Omega },\\, \\widehat{x}^{k_0,\\Omega },\\,\\widehat{s}_k^{\\,\\Omega }}^{M,m,\\,L_{t_k}}$ be the concatenation of $\\overline{u}^{\\,k,\\Omega }$ (see (REF )) and $\\widehat{u}_{N_{k_0}}^{k,\\Omega }=1+(b_{1,N_{k_0}-1}^{k,\\Omega }-\\tfrac{1}{L_{t_k}}).$ Next, we note that the right-most inequality in (REF ), together with the fact that $\\sum _{j=0}^{N_{k_{0}}-1} u_{\\,j}^{k_0,\\Omega } = n_{k_0}/L_{n_{k_0}},$ allow us to asset that $|\\overline{s}_k^{\\,\\Omega }-L_{t_k} n_{k_0}/L_{n_{k_0}}|\\le 2 N_{k_0}$ .",
"Therefore the definition of $\\widehat{s}_k^{\\,\\Omega }$ in (REF ) implies that $\\widehat{s}_k^{\\,\\Omega }=L_{t_k} \\frac{n_{k_0}}{L_{n_{k_0}}}+\\widehat{m}_k^\\Omega \\quad \\text{with} \\quad |\\widehat{m}_k^{\\Omega }|\\le 2 N_{k_0}+2 L_{t_k}.$ Moreover, $H^\\Omega \\big (\\widehat{\\Theta }_{\\text{traj}}^{k,\\Omega },\\widehat{x}^{k_0,\\Omega },\\widehat{u}^{k,\\Omega }\\big )\\ge H^\\Omega \\big (\\overline{\\Theta }_{\\text{traj}}^{\\,k,\\Omega },x_{\\text{max}}^{k_0,\\Omega }, \\overline{u}^{\\,k,\\Omega }\\big )+(\\beta -\\alpha ),$ because $\\widehat{u}_{N_{k_0}}^{k,\\Omega }\\le 2$ by definition (see (REF )) and the free energies per columns are all bounded from below by $(\\beta -\\alpha )/2$ .",
"Hence, (REF ) and (REF ) give that for all $\\Omega $ there exist a $\\widehat{\\Theta }^{k,\\Omega }_{\\text{traj}}\\in \\widetilde{{\\mathcal {D}}}_{L_{t_{k}},\\,N_{k_0}+1}^{\\,M,\\,r_{k_0}}\\colon \\,b_{1,N_{k_0}}=\\tfrac{1}{L_{t_k}},$ an $\\widehat{x}^{k_0,\\Omega }\\in {\\mathcal {X}}_{\\widehat{\\Theta }_{\\text{traj}}^{k,\\Omega },\\Omega }^{M,m}$ and a $\\widehat{u}^{k,\\Omega }\\in {\\mathcal {U}}_{\\,\\widehat{\\Theta }_{\\text{traj}}^{k,\\,\\Omega },\\,\\widehat{x}^{k_0,\\Omega },\\,\\widehat{s}_k^{\\,\\Omega }}^{M,m,\\,L_{t_k}}$ such that, for $k$ large enough, ${\\mathbb {E}}_{\\,\\Omega }\\big [H(\\widehat{\\Theta }_{\\text{traj}}^{k,\\Omega },\\widehat{x}^{k_0,\\Omega },\\widehat{u}^{k,\\Omega })\\big ]\\ge \\tfrac{n_{k_0}}{L_{n_{k_0}}}\\big (l_1+\\tfrac{l_2-l_1}{4}).$ Next, we subdivide the disorder $\\Omega $ into groups of $N_{k_0}+1$ consecutive columns that are successively translated by $r_{k_0}$ in the vertical direction, i.e., $\\Omega =(\\Omega _1,\\Omega _2,\\dots )$ with (recall (REF )) $\\Omega _j=\\big (\\Omega (i,\\,(j-1)\\, r_{k_0}+\\cdot )\\big )_{i=(j-1)(N_{k_0}+1)}^{\\,j (N_{k_0}+1)-1},$ and we let $q_k^\\Omega $ be the unique integer satisfying $\\widehat{s}_k^{\\, \\Omega _1}+\\widehat{s}_k^{\\, \\Omega _2}+ \\dots +\\widehat{s}_k^{\\,\\Omega _{q_k}}\\le t_k<\\widehat{s}_k^{\\,\\Omega _1}+\\dots +\\widehat{s}_k^{\\,\\Omega _{q_k+1}},$ where we suppress the $\\Omega $ -dependence of $q_k$ .",
"We recall that $f_{3,t_k}^\\Omega (M,m) ={\\mathbb {E}}\\Bigg [\\frac{1}{t_k} \\log \\sum _{N=t_k/m L_{t_k}}^{t_k/L_{t_k}}\\ \\sum _{\\Theta _{\\text{traj}}\\in \\widetilde{{\\mathcal {D}}}_{L_{t_k},N}^{M}}\\ \\,\\sum _{x\\in {\\mathcal {X}}_{\\Theta _{\\text{traj}},\\Omega }^{M,m} }\\,\\sum _{u\\in \\,{\\mathcal {U}}_{\\,\\Theta _{\\text{traj}},\\,x,\\,t_k}^{M,m,\\,L_{t_k}}} e^{L_{t_k}\\,H^\\Omega (\\Theta _{\\text{traj}},x,u)}\\Bigg ],$ set $\\widetilde{t}_k^{\\,\\Omega }=\\widehat{s}_k^{\\,\\Omega _1}+\\widehat{s}_k^{\\,\\Omega _2}+\\dots +\\widehat{s}_k^{\\,\\Omega _{q_k}}$ , and concatenate $\\widehat{\\Theta }^{k,\\Omega }_{\\text{traj,tot}}=\\Big (\\widehat{\\Theta }^{k,\\Omega _1}_{\\text{traj}},\\widehat{\\Theta }^{k,\\Omega _2}_{\\text{traj}},\\dots ,\\widehat{\\Theta }^{k,\\Omega _{q_k}}_{\\text{traj}}\\Big )\\in \\widetilde{{\\mathcal {D}}}_{L_{t_k},\\,q_k (N_{k_0}+1),\\,}^{M,}$ and $\\widehat{x}^{k,\\Omega }_{\\text{tot}}=\\big (\\widehat{x}^{k_0,\\Omega _1},\\widehat{x}^{k_0,\\Omega _2},\\dots ,\\widehat{x}^{k_0,\\Omega _{q_k}}\\big )\\in {\\mathcal {X}}_{\\widehat{\\Theta }^{k,\\Omega }_{\\text{traj,tot}}\\Omega }^{M,m}.$ and $\\widehat{u}^{k,\\Omega }_{\\text{tot}}=\\big (\\widehat{u}^{k,\\Omega _1},\\widehat{u}^{k,\\Omega _2},\\dots ,\\widehat{u}^{k,\\Omega _{q_k}}\\big )\\in {\\mathcal {U}}_{\\widehat{\\Theta }^{k,\\Omega }_{\\text{traj,tot}},\\widehat{x}^{k,\\Omega }_{\\text{tot}},\\widetilde{t}_k^{\\,\\Omega }}^{M,m,L_{t_k}}.$ It still remains to complete $\\widehat{\\Theta }^{k,\\Omega }_{\\text{traj,tot}}$ , $\\widehat{x}^{k,\\Omega }_{\\text{tot}}$ and $\\widehat{u}^{k,\\Omega }_{\\text{tot}}$ such that the latter becomes an element of ${\\mathcal {U}}_{\\widehat{\\Theta }^{k,\\Omega }_{\\text{traj,tot}},\\widehat{x}^{k,\\Omega }_{\\text{tot}},t_k}^{M,m,L_{t_k}}$ .",
"To that end, we recall (REF ), which gives $t_k-\\widetilde{t}_k^{\\,\\Omega }\\le \\widehat{s}_k^{\\Omega _{q_k+1}}$ .",
"Then, using (REF ), we have that there exists a $c>0$ such that $t_k-\\widetilde{t}_k^{\\,\\Omega }\\le c L_{t_k} \\tfrac{n_{k_0}}{L_{n_{k_0}}}.$ Therefore we can complete $\\widehat{\\Theta }^{k,\\Omega }_{\\text{traj,tot}}$ , $\\widehat{x}^{k,\\Omega }_{\\text{tot}}$ and $\\widehat{u}^{k,\\Omega }_{\\text{tot}}$ with $\\Theta _{\\text{rest}}\\in {\\mathcal {D}}_{L_{t_k},\\,g_k^\\Omega }^M,\\qquad x_{\\text{rest}}\\in {\\mathcal {X}}_{\\Theta _{\\text{rest}},\\Omega }^{M,m},\\qquad u_{\\text{rest}}\\in {\\mathcal {U}}_{\\Theta _{\\text{rest}}, x_{\\text{rest}}, t_k-\\widetilde{t}_k^{\\,\\Omega }}^{M,m,L_{t_k}},$ such that, by (REF ), the number of columns $g_{k}^\\Omega $ involved in $\\Theta _{\\text{rest}}$ satisfies $g_k^\\Omega \\le c n_{k_0}/L_{n_{k_0}}$ .",
"Henceforth $\\widehat{\\Theta }^{k,\\Omega }_{\\text{traj,tot}}$ , $\\widehat{x}^{k,\\Omega }_{\\text{tot}}$ and $\\widehat{u}^{k,\\Omega }_{\\text{tot}}$ stand for the quantities defined in (REF ) and (REF ), and concatenated with $\\Theta _{\\text{rest}},x_{\\text{rest}}$ and $u_{\\text{rest}}$ so that they become elements of ${\\mathcal {D}}_{L_{t_k},\\,q_k (N_{k_0}+1)+g_k^\\Omega }^M,\\qquad {\\mathcal {X}}_{\\widehat{\\Theta }^{k,\\Omega }_{\\text{traj,tot}},\\Omega }^{M,m},\\qquad {\\mathcal {U}}_{\\widehat{\\Theta }^{k,\\Omega }_{\\text{traj,tot}},\\widehat{x}^{k,\\Omega }_{\\text{tot}},t_k}^{M,m,L_{t_k}},$ respectively.",
"By restricting the summation in (REF ) to $\\widehat{\\Theta }^{k,\\Omega }_{\\text{traj,tot}}$ , $\\widehat{x}^{k,\\Omega }_{\\text{tot}}$ and $\\widehat{u}^{k,\\Omega }_{\\text{tot}}$ , we get $f_{3,t_k}(M,m)\\ge \\frac{L_{t_k}}{t_k} {\\mathbb {E}}_{\\Omega }\\bigg [\\sum _{j=1}^{q_k} H^{\\Omega _j}(\\widehat{\\Theta }^{k,\\Omega _j}_{\\text{traj}},\\widehat{x}^{k_0,\\Omega _j},\\widehat{u}^{k,\\Omega _j})+ H(\\Theta _{\\text{rest}}, x_{\\text{rest}}, u_{\\text{rest}})\\bigg ],$ where the term $H(\\Theta _{\\text{rest}},x_{\\text{rest}}, u_{\\text{rest}})$ is negligible because, by (REF ), $(t_k-\\widetilde{t}_k^{\\,\\Omega })/t_k$ vanishes as $k\\rightarrow \\infty $ , while all free energies per column are bounded from below by $(\\beta -\\alpha )/2$ .",
"Pick ${\\varepsilon }>0$ and recall (REF ).",
"Choose $k_0$ such that $2 L_{n_{k_0}}/n_{k_0} \\le {\\varepsilon }/2$ and note that, for $k$ large enough, $\\widehat{s}_k^{\\,\\Omega }\\in \\Big [L_{t_k}\\tfrac{n_{k_0}}{L_{n_{k_0}}}(1-{\\varepsilon }),L_{t_k}\\tfrac{n_{k_0}}{L_{n_{k_0}}}(1+{\\varepsilon })\\Big ].$ By (REF ), we therefore have $q_k\\in \\Big [\\tfrac{t_k L_{n_{k_0}}}{L_{t_k} n_{k_0}} \\tfrac{1}{1+{\\varepsilon }},\\tfrac{t_k L_{n_{k_0}}}{L_{t_k} n_{k_0}} \\tfrac{1}{1-{\\varepsilon }}\\Big ]=[a,b].$ Recalling (REF ), we obtain $f_{3,t_k}(M,m)\\ge \\frac{L_{t_k}}{t_k} {\\mathbb {E}}_{\\Omega }\\bigg [\\sum _{j=1}^{a}H^{\\Omega _j}(\\widehat{\\Theta }^{k,\\Omega _j}_{\\text{traj}}, \\widehat{x}^{k_0,\\Omega _j},\\widehat{u}^{k,\\Omega _j})-\\sum _{j=a}^{b} \\Big | H^{\\Omega _j}(\\widehat{\\Theta }^{k,\\Omega _j}_{\\text{traj}},\\widehat{x}^{k_0,\\Omega _j},\\widehat{u}^{k,\\Omega _j})\\Big |\\bigg ],$ and, consequently, $f_{3,t_k}(M,m)\\ge \\tfrac{L_{n_{k_0}}}{n_{k_0} (1+{\\varepsilon })}\\,{\\mathbb {E}}_{\\,\\Omega }\\Big [H^{\\Omega }(\\widehat{\\Theta }^{k,\\Omega }_{\\text{traj}},\\widehat{x}^{k_0,\\Omega },\\widehat{u}^{k,\\Omega })\\Big ]-\\frac{L_{t_k}}{t_k}(b-a) (N_{k_0}+1) m \\tfrac{\\beta -\\alpha }{2},$ and, by (REF ), $f_{3,t_k}(M,m)\\ge \\tfrac{l_1+\\tfrac{l_2-l_1}{4}}{1+{\\varepsilon }}-(\\tfrac{1}{1-{\\varepsilon }}-\\tfrac{1}{1+{\\varepsilon }}) (b-a) m \\tfrac{\\beta -\\alpha }{2}.$ After taking ${\\varepsilon }$ small enough, we may conclude that $\\liminf _{k\\rightarrow \\infty }f_{3,t_k}(M,m) >l_1$ , which completes the proof."
],
[
"Proof of Proposition ",
"Pick $(M,m)\\in {\\hbox{\\footnotesize \\rm EIGH}}$ and note that, for every $n\\in {\\mathbb {N}}$ , the set ${\\mathcal {W}}_{n,M}^{\\,m}$ is contained in ${\\mathcal {W}}_{n,M}$ .",
"Thus, by using Proposition REF we obtain $\\nonumber \\liminf _{n\\rightarrow \\infty } f^{\\Omega }_{1,n}(M;\\alpha ,\\beta )&\\ge \\sup _{m\\ge M+2} \\liminf _{n\\rightarrow \\infty } f^{\\Omega }_{1,n}(M,m;\\alpha ,\\beta )\\\\&= \\sup _{m\\ge M+2} f(M,m;\\alpha ,\\beta )\\quad \\text{ for } {\\mathbb {P}}-a.e.\\,\\Omega .$ Therefore, the proof of Proposition REF will be complete once we show that $\\limsup _{n\\rightarrow \\infty } f^{\\Omega }_{1,n}(M;\\alpha ,\\beta )\\le \\sup _{m\\ge M+2} \\limsup _{n\\rightarrow \\infty }f^{\\Omega }_{1,n}(M,m;\\alpha ,\\beta )\\quad \\text{ for } {\\mathbb {P}}-a.e.\\,\\Omega .$ We will not prove (REF ) in full detail, but only give the main steps in the proof.",
"The proof consists in showing that, for $m$ large enough, the pieces of the trajectory in a column that exeed $m L_n$ steps do not contribute substantially to the free energy.",
"Recall (REF –REF ) and use (REF ) with $m=\\infty $ , i.e., $Z_{n,L_n}^{\\omega ,\\Omega }(M)&=\\sum _{N=1}^{n/L_n} \\sum _{\\Theta _{\\text{traj}}\\in \\widetilde{{\\mathcal {D}}}_{L_n,N}^{M} }\\,\\sum _{x\\in {\\mathcal {X}}_{\\Theta _{\\text{traj}},\\Omega }^{M,\\infty } }\\,\\sum _{u\\in \\,{\\mathcal {U}}_{\\Theta _{\\text{traj}},x,n}^{\\,M,\\infty ,L_n}} A_1.$ With each $(N,\\Theta _{\\text{traj}},x,u)$ in (REF ), we associate the trajectories obtained by concatenating $N$ shorter trajectories $(\\pi _i)_{i\\in \\lbrace 0,\\dots ,N-1\\rbrace }$ chosen in $({\\mathcal {W}}_{\\Theta _i,u_i,L_n})_{i\\in \\lbrace 0,\\dots ,N-1\\rbrace }$ , respectively.",
"Thus, the quantity $A_1$ in (REF ) corresponds to the restriction of the partition function to the trajectories associated with $(N,\\Theta _{\\text{traj}},x,u)$ .",
"In order to discriminate between the columns in which more than $m L_n$ steps are taken and those in which less are taken, we rewrite $A_1$ as $A_2 \\widetilde{A}_2$ with $A_2&=\\prod _{i\\in V_{u,m} }\\,Z_{L_n}^{\\omega _{I_i}}(\\Theta _i,u_i),\\qquad \\widetilde{A}_2=\\prod _{i\\in \\widetilde{V}_{u,m} }\\,Z_{L_n}^{\\omega _{I_i}}(\\Theta _i,u_i),$ with $\\widetilde{u}_{i}=\\sum _{k=0}^{i-1} u_k$ , $\\Theta _i=(\\Omega (i,\\Pi _{i}+\\cdot ),\\Xi _i,x_i)$ and $I_i=\\lbrace \\widetilde{u}_{i} L_n,\\dots ,\\widetilde{u}_{i+1} L_n-1\\rbrace $ for $i\\in \\lbrace 0,\\dots ,N-1\\rbrace $ , with $\\omega _{I}=(\\omega _i)_{i\\in I}$ for $I\\subset {\\mathbb {N}}$ , where $\\lbrace 0,\\dots , N-1\\rbrace $ is partitioned into $\\widetilde{V}_{u,m}\\cup V_{u,m}\\quad \\text{with}\\quad \\widetilde{V}_{u,m}=\\lbrace i\\in \\lbrace 0,\\dots ,N-1\\rbrace \\colon \\; u_i>m\\rbrace .$ For all $(N,\\Theta _{\\text{traj}},x,u)$ , we rewrite $\\widetilde{V}_{u,m}$ in the form of an increasing sequence $\\lbrace i_1,\\dots ,i_{\\widetilde{k}}\\rbrace $ and we drop the $(u,m)$ -dependence of $\\widetilde{k}$ for simplicity.",
"We also set $\\widetilde{u}=u_{i_1}+\\dots +u_{i_{\\widetilde{k}}}$ , which is the total number of steps taken by a trajectory associated with $(N,\\Theta _{\\text{traj}},x,u)$ in those columns where more than $m L_n$ steps are taken.",
"Finally, for $s\\in \\lbrace 1,\\dots ,\\widetilde{k}\\rbrace $ we partition $I_{i_s}$ into $J_{i_s}\\cup \\widetilde{J}_{i_s} \\quad \\text{with}\\quad J_{i_s}&=\\lbrace \\widetilde{u}_{i_s} L_n,\\dots ,(\\widetilde{u}_{i_s}+M+2) L_n\\rbrace , \\\\\\quad \\widetilde{J}_{i_s}&=\\lbrace (\\widetilde{u}_{i_s}+M+2) L_n+1,\\dots ,\\widetilde{u}_{i_{s}+1} L_n-1\\rbrace ,$ and we partition $\\lbrace 1,\\dots ,n\\rbrace $ into $& J\\cup \\widetilde{J} \\quad \\text{with} \\quad \\widetilde{J}=\\cup _{s=1}^{\\widetilde{k}} \\widetilde{J}_{i_s},\\qquad J=\\lbrace 1,\\dots ,n\\rbrace \\setminus \\widetilde{J},$ so that $\\widetilde{J}$ contains the label of the steps constituting the pieces of trajectory exeeding $(M+2)L_n$ steps in those columns where more than $m L_n$ steps are taken."
],
[
"Step 1",
"In this step we replace the pieces of trajectories in the columns indexed in $\\widetilde{V}_{u,m}$ by shorter trajectories of length $(M+2) L_n$ .",
"To that aim, for every $(N,\\Theta _{\\text{traj}},x,u)$ we set $\\widehat{A}_2=\\prod _{i\\in \\widetilde{V}_{u,m} }\\,Z_{L_n}^{\\,\\omega _{J_i}}(\\Theta ^{^{\\prime }}_i,M+2)$ with $\\Theta ^{^{\\prime }}_i=(\\Omega (i,\\Pi _{i}+\\cdot ),\\Xi _i,1)$ .",
"We will show that for all ${\\varepsilon }>0$ and for $m$ large enough, the event $B_n=\\lbrace \\omega \\colon \\, \\widetilde{A}_2\\le \\widehat{A}_2\\, e^{3{\\varepsilon }n} \\ \\text{for all}\\ (N,\\Theta _{\\text{traj}},x,u)\\rbrace $ satisfies ${\\mathbb {P}}_{\\omega }(B_n)\\rightarrow 1$ as $n\\rightarrow \\infty $ .",
"Pick, for each $s\\in \\lbrace 1,\\dots ,\\widetilde{k}\\rbrace $ , a trajectory $\\pi _s$ in the set ${\\mathcal {W}}_{\\Theta _{i_s},u_{i_s},L_n}$ .",
"By concatenating them we obtain a trajectory in ${\\mathcal {W}}_{\\widetilde{u} L_n}$ satisfying $\\pi _{\\widetilde{u} L_n,1}=\\widetilde{k}L_n$ .",
"Thus, the total entropy carried by those pieces of trajectories crossing the columns indexed in $\\lbrace i_1,\\dots ,i_ {\\widetilde{k}}\\rbrace $ is bounded above by ${\\textstyle \\prod _{s=1}^{\\widetilde{k}}\\,|{\\mathcal {W}}_{\\Theta _{i_s},u_{i_s},L_n}|\\le \\big |\\lbrace \\pi \\in {\\mathcal {W}}_{\\widetilde{u}L_n}\\colon \\,\\pi _{\\widetilde{u}L_n,1}=\\widetilde{k}L_n\\rbrace \\big |.",
"}$ Since $\\widetilde{u}/\\widetilde{k}\\ge m$ , we can use Lemma REF in Appendix to assert that, for $m$ large enough, the right-hand side of (REF ) is bounded above by $e^{{\\varepsilon }n}$ .",
"Moreover, we note that an $\\widetilde{u} L_n$ -step trajectory satisfying $\\pi _{\\widetilde{u}L_n,1}=\\widetilde{k}L_n$ makes at most $\\widetilde{k} L_n+ \\widetilde{u}$ excursions in the $B$ solvent because such an excursion requires at least one horizontal step or at least $L_n$ vertical steps.",
"Therefore, by using the inequalities $\\widetilde{k} L_n\\le n/m$ and $\\widetilde{u}\\le n/L_n$ we obtain that, for $n$ large enough, the sum of the Hamiltonians associated with $(\\pi _1,\\dots ,\\pi _{\\widetilde{k}})$ is bounded from above, uniformly in $(N,\\Theta _{\\text{traj}},x,u)$ and $(\\pi _1,\\dots ,\\pi _{\\widetilde{k}})$ , by ${\\textstyle \\sum _{s=1}^{\\widetilde{k}}H_{u_{i_s} L_n,L_n}^{\\omega _{I_{i_s}},\\Omega (i_s,\\Pi _{i_s}+\\cdot )}(\\pi _s)\\le \\max \\lbrace \\sum _{i\\in I} \\xi _i\\colon \\, I\\in \\cup _{r=1}^{2n/m}{\\mathcal {E}}_{n,r}\\rbrace },$ with ${\\mathcal {E}}_{n,r}$ defined in (REF ) in Appendix and $\\xi _i=\\beta 1_{\\lbrace \\omega _i=A\\rbrace }-\\alpha 1_{\\lbrace \\omega _i=B\\rbrace }$ for $i\\in {\\mathbb {N}}$ .",
"At this stage we use the definition in (REF ) and note that, for all $\\omega \\in {\\mathcal {Q}}_{n,m}^{{\\varepsilon }/\\beta ,(\\alpha -\\beta )/2+{\\varepsilon }}$ , the right-hand side in (REF ) is smaller than ${\\varepsilon }n$ .",
"Consequently, for $m$ and $n$ large enough we have that, for all $\\omega \\in {\\mathcal {Q}}_{n,m}^{{\\varepsilon }/\\beta ,(\\alpha -\\beta )/2+{\\varepsilon }}$ , $\\widetilde{A}_2\\le e^{2{\\varepsilon }n}\\quad \\text{for all}\\quad (N,\\Theta _{\\text{traj}},x,u).$ Recalling (REF ) and noting that $\\widetilde{k} L_n\\le n/m$ , we can write $\\widehat{A}_2\\ge e^{-\\widetilde{k} (M+2) L_n C_{\\text{uf}}(\\alpha )}\\ge e^{-n \\tfrac{M+2}{m} C_{\\text{uf}}(\\alpha )},$ and therefore, for $m$ large enough, for all $n$ and all $(N,\\Theta _{\\text{traj}},x,u)$ we have $ \\widehat{A}_2\\ge e^{-{\\varepsilon }n}$ .",
"Finally, use (REF ) and (REF ) to conclude that, for $m$ and $n$ large enough, ${\\mathcal {Q}}_{n,m}^{{\\varepsilon }/\\beta ,(\\alpha -\\beta )/2+{\\varepsilon }}$ is a subset of $B_n$ .",
"Thus, Lemma REF ensures that, for $m$ large enough, $\\lim _{n\\rightarrow \\infty } P_\\omega (B_n)=1$ ."
],
[
"Step 2",
"Let $(\\widetilde{w}_i)_{i\\in {\\mathbb {N}}}$ be an i.i.d.",
"sequence of Bernouilli trials, independent of $\\omega ,\\Omega $ .",
"For $(N,\\Theta _{\\text{traj}},x,u)$ we set $\\widehat{u}=\\widetilde{u}-\\widetilde{k} (M+2)$ .",
"In Step 1 we have removed $\\widehat{u}L_n$ steps from the trajectories associated with $(N,\\Theta _{\\text{traj}},x,u)$ so that they have become trajectories associated with $(N,\\Theta _{\\text{traj}},x^{^{\\prime }},u)$ .",
"In this step, we will concatenate the trajectories associated with $(N,\\Theta _{\\text{traj}},x^{^{\\prime }},u)$ with an $\\widehat{u} L_n$ -step trajectory to recover a trajectory that belongs to ${\\mathcal {W}}_{n,M}^{\\,m}$ .",
"For $\\Omega \\in \\lbrace A,B\\rbrace ^{{\\mathbb {N}}_0\\times {\\mathbb {Z}}}$ , $t,N\\in {\\mathbb {N}}$ and $k\\in {\\mathbb {Z}}$ , let $P_{A}^\\Omega (N,k)(t)=\\frac{1}{t} \\sum _{j=0}^{t-1} 1_{\\lbrace \\Omega (N+j,k)=A\\rbrace }$ be the proportion of $A$ -blocks on the $k^{\\text{th}}$ line and between the $N^{\\text{th}}$ and the $(N+t-1)^{\\text{th}}$ column of $\\Omega $ .",
"Pick $\\eta >0$ and $j\\in {\\mathbb {N}}$ , and set $S_{\\eta ,j}=\\bigcup _{N=0}^{j}\\bigcup _{k=-m_1 N}^{m_1 N}\\bigcup _{t\\ge \\eta j}\\Big \\lbrace P_{A}^\\Omega (N,k)(t)\\le \\frac{p}{2}\\Big \\rbrace .$ By a straightforward application of Cramer's Theorem for i.i.d.",
"random variables, we have that $\\sum _{j\\in {\\mathbb {N}}} P_\\Omega (S_{\\eta ,j})<\\infty $ .",
"Therefore, using the Borel-Cantelli Lemma, it follows that for ${\\mathbb {P}}_\\Omega $ -a.e.",
"$\\Omega $ , there exists a $j_\\eta (\\Omega )\\in {\\mathbb {N}}$ such that $\\Omega \\notin S_{\\eta ,j}$ as soon as $j\\ge j_\\eta (\\Omega )$ .",
"In what follows, we consider $\\eta ={\\varepsilon }/\\alpha m$ and we take $n$ large enough so that $n/L_n\\ge j_{{\\varepsilon }/\\alpha m}(\\Omega )$ , and therefore $\\Omega \\notin S_{\\frac{n}{L_n},\\frac{{\\varepsilon }}{\\alpha m}}$ .",
"Pick $(N,\\Theta ,x,u)$ and consider one trajectory $\\widehat{\\pi }$ , of length $\\widehat{u} L_n$ , starting from $(N,\\Pi _N+b_N)L_n$ , staying in the coarsed-grained line at height $\\Pi _N$ , crossing the $B$ -blocks in a straight line and the $A$ -blocks in $m L_n$ steps.",
"The number of columns crossed by $\\widehat{\\pi }$ is denoted by $\\widehat{N}$ and satisfies $\\widehat{N}\\ge \\widehat{u}/m$ .",
"If $\\widehat{u} L_n \\le {\\varepsilon }n/\\alpha $ , then the Hamiltonian associated with $\\widehat{\\pi }$ is clearly larger than $-{\\varepsilon }n$ .",
"If $\\widehat{u} L_n \\ge {\\varepsilon }n/\\alpha $ in turn, then $H_{\\widehat{u} L_n,L_n}^{\\,\\widetilde{w},\\Omega (N+\\cdot ,\\Pi _N)}(\\widehat{\\pi })\\ge -\\alpha L_n \\widehat{N} \\big [1-P_{A}^\\Omega (N,\\Pi _N)(\\widehat{N})\\big ].$ Since $N\\le n/L_n$ , $|\\Pi _N|\\le m_1 N$ and $\\widehat{N}\\ge {\\varepsilon }n/(\\alpha m L_n)$ , we can use the fact that $\\Omega \\notin S_{\\frac{n}{L_n},\\frac{{\\varepsilon }}{\\alpha m}}$ to obtain $P_{A}^\\Omega (N,\\Pi _N)(\\widehat{N})\\ge \\frac{p}{2}.$ At this point it remains to bound $\\widehat{N}$ from above, which is done by noting that $\\widehat{N} \\big [m P_{A}^\\Omega (N,\\Pi _N)(\\widehat{N})+1-P_{A}^\\Omega (N,\\Pi _N)(\\widehat{N})\\big ]=\\widehat{u}\\le \\tfrac{n}{L_n}.$ Hence, using (REF ) and (REF ), we obtain $\\widehat{N}\\le 2 n/p m L_n$ and therefore the right-hand side of (REF ) is bounded from below by $-\\alpha (2-p)n/p m$ , which for $m$ large enough is larger than $-{\\varepsilon }n$ .",
"Thus, for $n$ and $m$ large enough and for all $(N,\\Theta ,x,u)$ , we have a trajectory $\\widehat{\\pi }$ at which the Hamiltonian is bounded from below by $-{\\varepsilon }n$ that can be concatenated with all trajectories associated with $(N,\\Theta ,x{^{\\prime }},u)$ to obtain a trajectory in ${\\mathcal {W}}_{n,M}^{\\,m}$ .",
"Consequently, recalling (REF ), for $n$ and $m$ large enough we have $A_2\\widehat{A_2}\\le e^{{\\varepsilon }n}Z_{\\,n,L_n}^{(\\omega _{J}\\,,\\,\\widetilde{\\omega }), \\Omega } (M,m)\\qquad \\forall \\, (N,\\Theta ,x,u).$"
],
[
" Step 3",
"In this step, we average over the microscopic disorders $\\omega ,\\widetilde{\\omega }$ .",
"Use (REF ) to note that, for $n$ and $m$ large enough and all $\\omega \\in B_n$ , we have $Z_{n,L_n}^{\\omega ,\\Omega }(M)&\\le e^{4{\\varepsilon }n} \\sum _{N=1}^{n/L_n}\\sum _{\\Theta _{\\text{traj}}\\in \\widetilde{{\\mathcal {D}}}_{L_n,N}^{M} }\\,\\sum _{x\\in {\\mathcal {X}}_{\\Theta _{\\text{traj}},\\Omega }^{M,\\infty } }\\,\\sum _{u\\in \\,{\\mathcal {U}}_{\\Theta _{\\text{traj}},x,n}^{\\,M,\\infty ,L_n}}Z_{\\,n,L_n}^{(\\omega _{J}\\,,\\,\\widetilde{\\omega }), \\Omega } (M,m).$ We use (REF ) to claim that there exists $C_1,C_2>0$ so that for all $n\\in {\\mathbb {N}}$ , all $m\\in {\\mathbb {N}}$ and all $J$ , ${\\mathbb {P}}_{\\omega ,\\widetilde{\\omega }}\\Big (\\Big |\\tfrac{1}{n}\\log Z_{n,L_n}^{(\\omega _{J}\\,,\\,\\widetilde{\\omega }),\\Omega }(M,m)-f_{1,n}^{\\Omega }(M,m)\\Big |\\ge {\\varepsilon }\\Big )\\le C_1 e^{-C_2 {\\varepsilon }^2 n}.$ We set also $D_{n}=\\bigcap _{(N,\\Theta _{\\text{traj}},x,u)}\\Big \\lbrace \\Big |\\tfrac{1}{n}\\log Z_{n,L_n}^{(\\omega _{J}\\,,\\,\\widetilde{\\omega }),\\Omega }(M,m)-f_{1,n}^{\\Omega }(M,m)\\Big |\\le {\\varepsilon }\\Big \\rbrace ,$ recall the definition of $c_n$ in (REF ) (used with $(M,\\infty )$ ), and use (REF ) and the fact that $c_n$ grows subexponentially, to obtain $\\lim _{n\\rightarrow \\infty }{\\mathbb {P}}_{\\omega ,\\widetilde{\\omega }}(D_n^c)= 0$ .",
"For all $(\\omega ,\\widetilde{\\omega })$ satisfying $\\omega \\in B_n$ and $(\\omega ,\\widetilde{\\omega })\\in D_n$ , we can rewrite (REF ) as $Z_{n,L_n}^{\\omega ,\\Omega }(M)&\\le c_n\\ e^{n f_{1,n}^{\\Omega }(M,m) +5{\\varepsilon }n}.$ As a consequence, recalling (REF ), for $m$ large enough we have $f^{\\Omega }_{n}(M;\\alpha ,\\beta )\\le {\\mathbb {P}}(B_n^c\\cup D_n^c)\\, C_{\\text{uf}}(\\alpha )+ \\frac{\\log c_n}{n}+\\frac{1}{n}{\\mathbb {E}}\\Big (1_{\\lbrace B_n\\cup D_n\\rbrace } \\big (n f_{1,n}^{\\Omega }(M,m) +5{\\varepsilon }n\\big )\\Big ).$ Since ${\\mathbb {P}}(B_n^c\\cup D_n^c)$ and $(\\log c_n)/n$ vanish when $n\\rightarrow \\infty $ , it suffices to apply Proposition REF and to let ${\\varepsilon }\\rightarrow 0$ to obtain (REF ).",
"This completes the proof of Proposition REF ."
],
[
"Proof of Proposition ",
"Note that, for all $m\\ge M+2$ , we have ${\\mathcal {R}}_{p,M}^{m}\\subset {\\mathcal {R}}_{p,M}$ .",
"Moreover, any $(u_\\Theta )_{\\Theta \\in \\overline{{\\mathcal {V}}}_M^{\\,m}}\\in {\\mathcal {B}}_{\\overline{{\\mathcal {V}}}_M^{\\,m}}$ can be extended to $\\overline{{\\mathcal {V}}}_M$ so that it belongs to ${\\mathcal {B}}_{\\overline{{\\mathcal {V}}}_M}$ .",
"Thus, $\\sup _{m\\ge M+2} f(M,m;\\alpha ,\\beta )\\le \\sup _{\\rho \\in {\\mathcal {R}}_{p,M}}\\sup _{(u)\\in {\\mathcal {B}}_{\\overline{{\\mathcal {V}}}_M} } V(\\rho ,u).$ As a consequence, it suffices to show that for all $\\rho \\in {\\mathcal {R}}_{p,M}$ and $(u_\\Theta )_{\\Theta \\in \\overline{{\\mathcal {V}}}_M}\\in {\\mathcal {B}}_{\\overline{{\\mathcal {V}}}_M}$ , $V(\\rho ,u)\\le \\sup _{m\\ge M+2} \\sup _{\\rho \\in {\\mathcal {R}}_{p,M}^{m}}\\sup _{(u)\\in {\\mathcal {B}}_{\\overline{{\\mathcal {V}}}_M^{\\,m}} } V(\\rho ,u).$ If $\\int _{\\overline{{\\mathcal {V}}}_M} u_\\Theta \\,\\rho (d\\Theta )=\\infty $ , then (REF ) is trivially satisfied since $V(\\rho ,u)=-\\infty $ .",
"Thus, we can assume that $\\rho (\\overline{{\\mathcal {V}}}_M\\setminus D_M)=1$ , where $D_M=\\lbrace \\Theta \\in \\overline{{\\mathcal {V}}}_M\\colon \\,\\chi _\\Theta \\in \\lbrace A^{{\\mathbb {Z}}},B^{{\\mathbb {Z}}}\\rbrace , x_\\Theta =2\\rbrace $ .",
"Since $\\int _{\\overline{{\\mathcal {V}}}_M} u_\\Theta \\,\\rho (d\\Theta )<\\infty $ and since (recall (REF )) $\\psi (\\Theta ,u)$ is uniformly bounded by $C_{\\text{uf}}(\\alpha )$ on $(\\Theta ,u)\\in \\overline{{\\mathcal {V}}}_M^{\\,*}$ , we have by dominated convergence that for all ${\\varepsilon }>0$ there exists an $m_0\\ge M+2$ such that, for all $m\\ge m_0$ , $V(\\rho ,u)\\le \\frac{\\int _{\\overline{{\\mathcal {V}}}_M^{\\,m}}u_\\Theta \\psi (\\Theta ,u_\\Theta ) \\rho (d\\Theta )}{\\int _{\\overline{{\\mathcal {V}}}_M^{\\,m}}u_\\Theta \\rho (d\\Theta )}+\\tfrac{{\\varepsilon }}{2}.$ Since $\\rho (\\overline{{\\mathcal {V}}}_M\\setminus D_M)=1$ and since $\\cup _{m\\ge M+2} \\overline{{\\mathcal {V}}}_M^{\\,m}= \\overline{{\\mathcal {V}}}_M\\setminus D_M$ , we have $\\lim _{m\\rightarrow \\infty }\\rho (\\overline{{\\mathcal {V}}}_M^{\\,m})=1$ .",
"Moreover, for all $m\\ge m_0$ there exists a $\\widehat{\\rho }_{m}\\in {\\mathcal {R}}_{p,M}^{m}$ such that $\\widehat{\\rho }_{m}=\\rho _{m}+\\overline{\\rho }_{m}$ , with $\\rho _{m}$ the restriction of $\\rho $ to $\\overline{{\\mathcal {V}}}_M^{\\,m}$ and $\\overline{\\rho }_m$ charging only those $\\Theta $ satisfying $x_\\Theta =1$ .",
"Since all $\\Theta \\in \\overline{{\\mathcal {V}}}_M$ with $x_\\Theta =1$ also belong to $\\overline{{\\mathcal {V}}}_M^{\\,M+2}$ , we can state that $\\overline{\\rho }_m$ only charges $\\overline{{\\mathcal {V}}}_M^{\\,M+2}$ .",
"Therefore $V(\\widehat{\\rho }_m,u)=\\frac{\\int _{\\overline{{\\mathcal {V}}}_M^{\\,m}} u_\\Theta \\psi (\\Theta ,u_\\Theta ) \\rho (d\\Theta )+\\int _{\\overline{{\\mathcal {V}}}_M^{\\,M+2}} u_\\Theta \\psi (\\Theta ,u_\\Theta )\\overline{\\rho }_m(d\\Theta )}{\\int _{\\overline{{\\mathcal {V}}}_M^{\\,m}}u_\\Theta \\rho (d\\Theta )+\\int _{\\overline{{\\mathcal {V}}}_M^{\\,M+2}} u_\\Theta \\overline{\\rho }_m(d\\Theta )}.$ Since $\\Theta \\mapsto u_\\Theta $ is continuous on $\\overline{{\\mathcal {V}}}_M$ , there exists an $R>0$ such that $u_\\Theta \\le R$ for all $\\Theta \\in \\overline{{\\mathcal {V}}}_M^{M+2}$ .",
"Therefore we can use (REF ) and (REF ) to obtain, for $m\\ge m_0$ , $V(\\widehat{\\rho }_m,u)\\ge (V(\\rho ,u)-\\tfrac{{\\varepsilon }}{2})\\frac{\\int _{\\overline{{\\mathcal {V}}}_M^{\\,m}} u_\\Theta \\rho (d\\Theta )}{\\int _{\\overline{{\\mathcal {V}}}_M^{\\,m}}u_\\Theta \\rho (d\\Theta )+\\int _{\\overline{{\\mathcal {V}}}_M^{\\,M+2}}u_\\Theta \\overline{\\rho }_m(d\\Theta )}- R\\, C_{\\text{uf}}(\\alpha )\\, (1-\\rho (\\overline{{\\mathcal {V}}}_M^{\\,m})).$ The fact that $\\overline{\\rho }_m({\\mathcal {V}}_M^{M+2})=\\rho (\\overline{{\\mathcal {V}}}_M\\setminus \\overline{{\\mathcal {V}}}_M^{\\,m})$ for all $m\\ge m_0$ impliess that $\\lim _{m\\rightarrow \\infty }\\overline{\\rho }_m({\\mathcal {V}}_M^{M+2})=0$ .",
"Consequently, the right-hand side in (REF ) tends to $V(\\rho ,u)-{\\varepsilon }/2$ as $m\\rightarrow \\infty $ .",
"Thus, there exists a $m_1\\ge m_0$ such that $V(\\widehat{\\rho }_{m_1},u)\\ge V(\\rho ,u)-{\\varepsilon }$ .",
"Finally, we note that there exists a $m_2\\ge m_1+1$ such that $u_\\Theta \\le m_2$ for all $\\Theta \\in \\overline{{\\mathcal {V}}}_M^{\\,m_1}$ , which allows us to extend $(u_\\Theta )_{\\Theta \\in \\overline{{\\mathcal {V}}}_M^{\\,m_1}}$ to $\\overline{{\\mathcal {V}}}_M^{\\,m_2}$ such that $(u_\\Theta )_{\\Theta \\in \\overline{{\\mathcal {V}}}_M^{\\,m_2}}\\in {\\mathcal {B}}_{\\overline{{\\mathcal {V}}}_M^{\\,m_2}}$ .",
"It suffices to note that $\\widehat{\\rho }_{m_1}\\in {\\mathcal {R}}_{p,M}^{m_1}\\subset {\\mathcal {R}}_{p,M}^{m_2}$ to conclude that $V(\\rho ,u)\\le f(M,m_2;\\,\\alpha ,\\beta )+{\\varepsilon }.$"
],
[
"Properties of path entropies",
"In Appendix REF we state a basic lemma (Lemma REF ) about uniform convergence of path entropies in a single column.",
"This lemma is proved with the help of three additional lemmas (Lemmas REF –REF ), which are proved in Appendix REF .",
"The latter ends with an elementary lemma (Lemma REF ) that allows us to extend path entropies from rational to irrational parameter values.",
"In Appendix REF , we extend Lemma REF to entropies associated with sets of paths fullfilling certain restrictions on their vertical displacement."
],
[
"Basic lemma",
"We recall the definition of $\\widetilde{\\kappa }_L$ , $L\\in {\\mathbb {N}}$ , in (REF ) and $\\widetilde{\\kappa }$ in (REF ).",
"Lemma A.1 For every ${\\varepsilon }>0$ there exists an $L_{\\varepsilon }\\in {\\mathbb {N}}$ such that $|\\tilde{\\kappa }_L(u,l)-\\tilde{\\kappa }(u,l)|&\\le {\\varepsilon }\\text{ for } L\\ge L_{\\varepsilon }\\text{ and } (u,l)\\in {\\mathcal {H}}_L.$ Proof.",
"With the help of Lemma REF below we get rid of those $(u,l) \\in {\\mathcal {H}}\\cap \\mathbb {Q}^2$ with $u$ large, i.e., we prove that $\\lim _{u\\rightarrow \\infty }\\kappa _L(u,l)=0$ uniformly in $L\\in {\\mathbb {N}}$ and $(u,l)\\in {\\mathcal {H}}_L$ .",
"Lemma REF in turn deals with the moderate values of $u$ , i.e., $u$ bounded away from infinity and $1+|l|$ .",
"Finally, with Lemma REF we take into account the small values of $u$ , i.e., $u$ close to $1+|l|$ .",
"To ease the notation we set, for $\\eta \\ge 0$ and $M>1$ , ${\\mathcal {H}}_{L,\\eta ,M}=\\lbrace (u,l)\\in {\\mathcal {H}}_L\\colon 1+|l|+\\eta \\le u\\le M\\rbrace ,\\qquad {\\mathcal {H}}_{\\eta ,M}=\\lbrace (u,l)\\in {\\mathcal {H}}\\colon 1+|l|+\\eta \\le u\\le M\\rbrace .$ Lemma A.2 For every ${\\varepsilon }>0$ there exists an $M_{\\varepsilon }>1$ such that $\\tfrac{1}{uL}\\log \\big |\\lbrace \\pi \\in {\\mathcal {W}}_{uL}\\colon \\pi _{uL,1}=L\\rbrace \\big |\\le {\\varepsilon }\\qquad \\forall \\,L\\in {\\mathbb {N}},\\,u\\in 1+\\tfrac{{\\mathbb {N}}}{L}\\colon u\\ge M_{\\varepsilon }.$ Lemma A.3 For every ${\\varepsilon }>0$ , $\\eta >0$ and $M>1$ there exists an $L_{{\\varepsilon },\\eta ,M}\\in {\\mathbb {N}}$ such that $|\\tilde{\\kappa }_L(u,l)-\\tilde{\\kappa }(u,l)|&\\le {\\varepsilon }\\qquad \\forall \\,L\\ge L_{{\\varepsilon },\\eta ,M},\\,(u,l)\\in {\\mathcal {H}}_{L,\\eta ,M}.$ Lemma A.4 For every ${\\varepsilon }>0$ there exist $\\eta _{\\varepsilon }\\in (0,\\tfrac{1}{2})$ and $L_{\\varepsilon }\\in {\\mathbb {N}}$ such that $|\\tilde{\\kappa }_L(u,l)-\\tilde{\\kappa }_{L}(u+\\eta ,l)|\\le {\\varepsilon }\\qquad \\forall \\,L\\ge L_{\\varepsilon },\\,(u,l)\\in {\\mathcal {H}}_{L},\\,\\eta \\in (0,\\eta _{\\varepsilon })\\cap \\tfrac{2{\\mathbb {N}}}{L}.$ Note that, after letting $L\\rightarrow \\infty $ in Lemma REF , we get $|\\tilde{\\kappa }(u,l)-\\tilde{\\kappa }(u+\\eta ,l)|\\le {\\varepsilon }\\qquad \\forall \\,(u,l)\\in {\\mathcal {H}}\\cap \\mathbb {Q}^2,\\,\\eta \\in (0,\\eta _{\\varepsilon })\\cap \\mathbb {Q}.$ Pick ${\\varepsilon }>0$ and $\\eta _{\\varepsilon }\\in (0,\\tfrac{1}{2})$ as in Lemma REF .",
"Note that Lemmas REF –REF yield that, for $L$ large enough, (REF ) holds on $\\lbrace (u,l)\\in {\\mathcal {H}}_L\\colon u\\ge 1+|l|+\\frac{\\eta _{\\varepsilon }}{2}\\rbrace $ .",
"Next, pick $L\\in {\\mathbb {N}}$ , $(u,l)\\in {\\mathcal {H}}_{L}\\colon u\\le 1+|l|+\\frac{\\eta _{\\varepsilon }}{2}$ and $\\eta _L\\in (\\frac{\\eta _{\\varepsilon }}{2},\\eta _{\\varepsilon })\\cap \\frac{2{\\mathbb {N}}}{L}$ , and write $|\\tilde{\\kappa }_L(u,l)-\\tilde{\\kappa }(u,l)|\\le A+B+C,$ where $A=|\\tilde{\\kappa }_L(u,l)-\\tilde{\\kappa }_L(u+\\eta _L,l)|,\\quad B=|\\tilde{\\kappa }_L(u+\\eta _L,l)-\\tilde{\\kappa }(u+\\eta _L,l)|,\\quad C=|\\tilde{\\kappa }(u+\\eta _L,l)-\\tilde{\\kappa }(u,l)|.$ By (REF ), it follows that $C\\le {\\varepsilon }$ .",
"As mentioned above, the fact that $(u+\\eta _L,l)\\in {\\mathcal {H}}_L$ and $u+\\eta _L\\ge |l|+\\frac{\\eta _{\\varepsilon }}{2}$ implies that, for $L$ large enough, $B\\le {\\varepsilon }$ uniformly in $(u,l)\\in {\\mathcal {H}}_{L}\\colon \\,u\\le 1+|l|+\\frac{\\eta _{\\varepsilon }}{2}$ .",
"Finally, from Lemma REF we obtain that $A\\le {\\varepsilon }$ for $L$ large enough, uniformly in $(u,l)\\in {\\mathcal {H}}_{L}\\colon \\,u\\le 1+|l|+\\frac{\\eta _{\\varepsilon }}{2}$ .",
"This completes the proof of Lemma REF .",
"$\\square $"
],
[
"Proof of Lemma ",
"The proof relies on the following expression: $v_{u,L} = \\big |\\lbrace \\pi \\in {\\mathcal {W}}_{uL}\\colon \\pi _{uL,1}=L\\rbrace \\big |=\\sum _{r=1}^{L+1} \\binom{L+1}{r} \\binom{(u-1) L}{r} 2^r,$ where $r$ stands for the number of vertical stretches made by the trajectory (a vertical stretch being a maximal sequence of consecutive vertical steps).",
"Stirling's formula allows us to assert that there exists a $g\\colon \\,[1,\\infty )\\rightarrow (0,\\infty )$ satisfying $\\lim _{u\\rightarrow \\infty }g(u)=0$ such that $\\binom{uL}{L}\\le e^{g(u) uL}, \\qquad u\\ge 1,\\,L\\in {\\mathbb {N}}.$ Equations (REF –REF ) complete the proof."
],
[
"Proof of Lemma ",
"We first note that, since $u$ is bounded from above, it is equivalent to prove (REF ) with $\\tilde{\\kappa }_L$ and $\\tilde{\\kappa }$ , or with $G_L$ and $G$ given by $G(u,l) = u\\tilde{\\kappa }(u,l),\\qquad G_L(u,l) = u\\tilde{\\kappa }_L(u,l),\\quad (u,l)\\in {\\mathcal {H}}_L.$ Via concatenation of trajectories, it is straightforward to prove that $G$ is $\\mathbb {Q}$ -concave on ${\\mathcal {H}}\\cap \\mathbb {Q}^2$ , i.e., $G(\\lambda (u_1,l_1)+(1-\\lambda )(u_2,l_2))\\ge \\lambda G(u_1,l_1)+(1-\\lambda ) G(u_2,l_2),\\ \\ \\lambda \\in \\mathbb {Q}_{[0,1]},\\,(u_1,l_1),(u_2,l_2)\\in {\\mathcal {H}}\\cap \\mathbb {Q}^2.$ Therefore $G$ is Lipschitz on every $K\\cap {\\mathcal {H}}\\cap \\mathbb {Q}^2$ with $K \\subset {\\mathcal {H}}^0$ (the interior of ${\\mathcal {H}}$ ) compact.",
"Thus, $G$ can be extended on ${\\mathcal {H}}^0$ to a function that is Lipschitz on every compact subset in ${\\mathcal {H}}^0$ .",
"Pick $\\eta >0$ , $M>1$ , ${\\varepsilon }>0$ , and choose $L_{\\varepsilon }\\in {\\mathbb {N}}$ such that $1/L_{\\varepsilon }\\le {\\varepsilon }$ .",
"Since ${\\mathcal {H}}_{\\eta ,M}\\subset {\\mathcal {H}}^0$ is compact, there exists a $c>0$ (depending on $\\eta ,M$ ) such that $G$ is $c$ -Lipschitz on ${\\mathcal {H}}_{\\eta ,M}$ .",
"Moreover, any point in ${\\mathcal {H}}_{\\eta ,M}$ is at distance at most ${\\varepsilon }$ from the finite lattice ${\\mathcal {H}}_{L_{\\varepsilon },\\eta ,M}$ .",
"Lemma REF therefore implies that there exists a $q_{\\varepsilon }\\in {\\mathbb {N}}$ satisfying $|G_{qL_{\\varepsilon }}(u,l)-G(u,l)|\\le {\\varepsilon }\\qquad \\forall \\,(u,l)\\in {\\mathcal {H}}_{L_{\\varepsilon },\\eta ,M},\\, q\\ge q_{\\varepsilon }.$ Let $L^{\\prime }=q_{\\varepsilon }L_{\\varepsilon }$ , and pick $q\\in {\\mathbb {N}}$ to be specified later.",
"Then, for $L\\ge q L^{\\prime }$ and $(u,l)\\in {\\mathcal {H}}_{L,\\eta ,M}$ , there exists an $(u^{\\prime },l^{\\prime })\\in {\\mathcal {H}}_{L_{{\\varepsilon }},\\eta ,M}$ such that $|(u,l)-(u^{\\prime },l^{\\prime })|_\\infty \\le {\\varepsilon }$ , $u>u^{\\prime }$ , $|l|\\ge |l^{\\prime }|$ and $u-u^{\\prime }\\ge |l|-|l^{\\prime }|$ .",
"We recall (REF ) and write $0\\le G(u,l)-G_L(u,l)\\le A+B+C,$ with $A= |G(u,l)-G(u^{\\prime },l^{\\prime })|,\\quad B= |G(u^{\\prime },l^{\\prime })-G_{L^{\\prime }}(u^{\\prime },l^{\\prime })|,\\quad C= G_{L^{\\prime }}(u^{\\prime },l^{\\prime })-G_L(u,l).$ Since $G$ is $c$ -Lipschitz on ${\\mathcal {H}}_{\\eta ,M}$ , and since $|(u,l)-(u^{\\prime },l^{\\prime })|_\\infty \\le {\\varepsilon }$ , we have $A\\le c{\\varepsilon }$ .",
"By (REF ) we have that $B\\le {\\varepsilon }$ .",
"Therefore only $C$ remains to be considered.",
"By Euclidean division, we get that $L=sL^{\\prime }+r$ , where $s\\ge q$ and $r\\in \\lbrace 0,\\dots ,L^{\\prime }-1\\rbrace $ .",
"Pick $\\pi _1,\\pi _2,\\dots ,\\pi _s\\in {\\mathcal {W}}_{L^{\\prime }}(u^{\\prime },|l^{\\prime }|)$ , and concatenate them to obtain a trajectory in ${\\mathcal {W}}_{sL^{\\prime }}(u^{\\prime },|l^{\\prime }|)$ .",
"Moreover, note that $uL-u^{\\prime }sL^{\\prime }&=(u-u^{\\prime })sL^{\\prime }+ur\\\\\\nonumber &\\ge (|l|-|l^{\\prime }|)sL^{\\prime }+(1+|l|)r= (L-sL^{\\prime })+ (|l|L-s|l^{\\prime }|L^{\\prime }),$ where we use that $L-sL^{\\prime }=r$ , $u-u^{\\prime }\\ge |l|-|l^{\\prime }|$ and $u\\ge 1+|l|$ .",
"Thus, (REF ) implies that any trajectory in ${\\mathcal {W}}_{L^{\\prime }}(u^{\\prime },|l^{\\prime }|)$ can be concatenated with an ($uL-u^{\\prime }sL^{\\prime }$ )-step trajectory, starting at $(sL^{\\prime },s|l^{\\prime }|L^{\\prime })$ and ending at $(L,|l|L)$ , to obtain a trajectory in ${\\mathcal {W}}_{L}(u,|l|)$ .",
"Consequently, $G_{L}(u,l)\\ge \\tfrac{s}{L}\\log \\kappa _{L^{\\prime }}(u^{\\prime },l^{\\prime })\\ge \\tfrac{s}{s+1} G_{L^{\\prime }}(u^{\\prime },l^{\\prime }).$ But $s\\ge q$ and therefore $G_{L^{\\prime }}(u^{\\prime },l^{\\prime })-G_{L}(u,l)\\le \\tfrac{1}{q} G_{L^{\\prime }}(u^{\\prime },l^{\\prime })\\le \\tfrac{1}{q} M \\log 3$ (recall that $\\log 3$ is an upper bound for all entropies per step).",
"Thus, by taking $q$ large enough, we complete the proof."
],
[
"Proof of Lemma ",
"Pick $L\\in {\\mathbb {N}}$ , $(u,l)\\in {\\mathcal {H}}_L$ , $\\eta \\in \\frac{2{\\mathbb {N}}}{L}$ , and define the map $T\\colon {\\mathcal {W}}_{L}(u,l)\\mapsto {\\mathcal {W}}_L(u+\\eta ,l)$ as follows.",
"Pick $\\pi \\in {\\mathcal {W}}_{L}(u,l)$ , find its first vertical stretch, and extend this stretch by $\\tfrac{\\eta L}{2}$ steps.",
"Then, find the first vertical stretch in the opposite direction of the stretch just extended, and extend this stretch by $\\tfrac{\\eta L}{2}$ steps.",
"The result of this map is $T(\\pi )\\in {\\mathcal {W}}_L(u+\\eta ,l)$ , and it is easy to verify that $T$ is an injection, so that $|{\\mathcal {W}}_{L}(u,l)|\\le | {\\mathcal {W}}_{L}(u+\\eta ,l)|$ .",
"Next, define a map $\\widetilde{T}\\colon \\,{\\mathcal {W}}_{L}(u+\\eta ,l)\\mapsto {\\mathcal {W}}_L(u,l)$ as follows.",
"Pick $\\pi \\in {\\mathcal {W}}_{L}(u+\\eta ,l)$ and remove its first $\\tfrac{\\eta L}{2}$ steps north and its first $\\tfrac{\\eta L}{2}$ steps south.",
"The result is $\\widetilde{T}(\\pi )\\in {\\mathcal {W}}_L(u,l)$ , but $\\widetilde{T}$ is not injective.",
"However, we can easily prove that for every ${\\varepsilon }>0$ there exist $\\eta _{\\varepsilon }>0$ and $L_{\\varepsilon }\\in {\\mathbb {N}}$ such that, for all $\\eta <\\eta _{\\varepsilon }$ and all $L\\ge l_{\\varepsilon }$ , the number of trajectories in ${\\mathcal {W}}_{L}(u+\\eta ,l)$ that are mapped by $\\widetilde{T}$ to a particular trajectory in $\\pi \\in {\\mathcal {W}}_{L}(u,l)$ is bounded from above by $e^{{\\varepsilon }L}$ , uniformly in $(u,l)\\in {\\mathcal {H}}_L$ and $\\pi \\in {\\mathcal {W}}_{L},(u,l)$ .",
"This completes the proof of Lemmas REF –REF ."
],
[
"Observation",
"We close this appendix with the following observation.",
"Recall Lemma REF , where $(u,l)\\mapsto \\tilde{\\kappa }(u,l)$ is defined on ${\\mathcal {H}}\\cap \\mathbb {Q}^2$ .",
"Lemma A.5 (i) $(u,l)\\mapsto u\\tilde{\\kappa }(u,l)$ extends to a continuous and strictly concave function on ${\\mathcal {H}}$ .",
"(ii) $l\\mapsto \\tilde{\\kappa }(u,l)$ is increasing on $[-u+1,0]$ and decreasing on $[0,u-1]$ , (iii) $\\lim _{u\\rightarrow \\infty } \\tilde{\\kappa }(u,0)=0$ .",
"(iv) $u\\mapsto u\\tilde{\\kappa }(u,l)$ is strictly increasing on $[1+|l|,\\infty )$ and $\\lim _{u\\rightarrow \\infty } u \\tilde{\\kappa }(u,l)=\\infty $ .",
"Proof.",
"(i) In the proof of Lemma REF we have shown that $\\tilde{\\kappa }$ can be extended to ${\\mathcal {H}}^0$ in such a way that $(u,l)\\mapsto u\\tilde{\\kappa }(u,l)$ is continuous and concave on ${\\mathcal {H}}^0$ .",
"Lemma REF allows us to extend $\\tilde{\\kappa }$ to the boundary of ${\\mathcal {H}}$ , in such a way that continuity and concavity of $(u,l)\\mapsto u\\tilde{\\kappa }(u,l)$ hold on all of ${\\mathcal {H}}$ .",
"To obtain the strict concavity, we recall the formula in (REF ), i.e., $u\\tilde{\\kappa }(u,l) = \\left\\lbrace \\begin{array}{ll}u\\kappa (u/|l|,1/|l|), &ł\\ne 0,\\\\u\\hat{\\kappa }(u), &l = 0,\\end{array}\\right.$ where $(a,b)\\mapsto a\\kappa (a,b)$ , $a\\ge 1+b$ , $b\\ge 0$ , and $\\mu \\mapsto \\mu \\hat{\\kappa }(\\mu )$ , $\\mu \\ge 1$ , are given in [3], Section 2.1, and are strictly concave.",
"In the case $l\\ne 0$ , (REF ) provides strict concavity of $(u,l)\\mapsto u\\tilde{\\kappa }(u,l)$ on ${\\mathcal {H}}^+=\\lbrace (u,l)\\in {\\mathcal {H}}\\colon l>0\\rbrace $ and on ${\\mathcal {H}}^-=\\lbrace (u,l)\\in {\\mathcal {H}}\\colon l<0\\rbrace $ , while in the case $l=0$ it provides strict concavity on $\\overline{{\\mathcal {H}}}=\\lbrace (u,0),u\\ge 1\\rbrace $ .",
"We already know that $(u,l)\\mapsto u\\tilde{\\kappa }(u,l)$ is concave on ${\\mathcal {H}}$ , which, by the strict concavity on ${\\mathcal {H}}^+$ , ${\\mathcal {H}}^-$ and $\\overline{{\\mathcal {H}}}$ , implies strict concavity of $(u,l)\\mapsto u\\tilde{\\kappa }(u,l)$ on ${\\mathcal {H}}$ .",
"(ii) This follows from concavity of $l\\mapsto \\tilde{\\kappa }(u,l)$ and the fact that $\\tilde{\\kappa }(u,l) = \\tilde{\\kappa }(u,-l)$ .",
"(iii) This is a direct consequence of Lemma REF .",
"(iv) By (i) we have that $u\\mapsto u\\tilde{\\kappa }(u,l)$ is strictly concave on $[1+|l|,\\infty )$ .",
"Therefore, proving that $\\lim _{u\\rightarrow \\infty } u \\tilde{\\kappa }(u,l)=\\infty $ is sufficient to obtain that $u\\mapsto u\\tilde{\\kappa }(u,l)$ is strictly increasing.",
"It is proven in [3], Lemma 2.1.2 (iii), that $\\lim _{\\mu \\rightarrow \\infty } u\\hat{\\kappa }(u)=\\infty $ , so that (REF ) completes the proof for $l=0$ .",
"If $l\\ne 0$ , then we use (REF ) again and the variational formula in the proof of [3], Lemma 2.1.1, to check that $\\lim _{a\\rightarrow \\infty } a \\kappa (a,b)=\\infty $ for all $b>0$ .",
"$\\square $"
],
[
"A generalization of Lemma ",
"In Section we sometimes needed to deal with subsets of trajectories of the following form.",
"Recall (REF ), pick $L\\in {\\mathbb {N}}$ , $(u,l)\\in {\\mathcal {H}}_L$ and $B_0, B_1\\in \\tfrac{Z}{L}$ such that $B_1\\,\\ge 0\\vee l\\,\\ge \\, 0\\wedge l\\,\\ge B_0 \\quad \\text{and}\\quad B_1-B_0\\ge 1.$ Denote by $\\widetilde{{\\mathcal {W}}}_L(u,l,B_0,B_1)$ the subset of ${\\mathcal {W}}_L(u,l)$ containing those trajectories that are constrained to remain above $B_0 L$ and below $B_1L$ (see Fig.",
"REF ), i.e., $\\widetilde{{\\mathcal {W}}}_L(u,l,B_0,B_1) &= \\big \\lbrace \\pi \\in {\\mathcal {W}}_L(u,l) \\colon \\,B_0 L< \\pi _{i,2}<B_1 L \\text{ for } i\\in \\lbrace 1,\\dots ,uL-1\\rbrace \\big \\rbrace ,$ and let $\\widetilde{\\kappa }_L(u,l,B_0,B_1) = \\frac{1}{uL} \\log |\\widetilde{{\\mathcal {W}}}_L(u,l,B_0,B_1) |$ be the entropy per step carried by the trajectories in $\\widetilde{{\\mathcal {W}}}_L(u,l,B_0,B_1)$ .",
"With Lemma REF below we prove that the effect on the entropy of the restriction induced by $B_0$ and $B_1$ in the set $\\widetilde{{\\mathcal {W}}}_L(u,l)$ vanishes uniformly as $L\\rightarrow \\infty $ .",
"Figure: A trajectory in 𝒲 ˜ L (u,l,B 0 ,B 1 )\\widetilde{{\\mathcal {W}}}_L(u,l,B_0,B_1).Lemma A.6 For every ${\\varepsilon }>0$ there exists an $L_{\\varepsilon }\\in {\\mathbb {N}}$ such that, for $L\\ge L_{\\varepsilon }$ , $(u,l)\\in {\\mathcal {H}}_L$ and $B_0,B_1\\in {\\mathbb {Z}}/L$ satisfying $B_1-B_0\\ge 1$ , $B_1\\,\\ge \\max \\lbrace 0,l\\rbrace $ and $B_0\\le \\min \\lbrace 0,l\\rbrace $ , $& |\\tilde{\\kappa }_L(u,l,B_0,B_1)-\\tilde{\\kappa }_L(u,l)|\\le {\\varepsilon }.$ Proof.",
"The key fact is that $B_1-B_0 \\ge 1$ .",
"The vertical restrictions $B_1\\,\\ge \\max \\lbrace 0,l\\rbrace $ and $B_0\\le \\min \\lbrace 0,l\\rbrace $ gives polynomial corrections in the computation of the entropy, but these corrections are harmless because $(B_1-B_0)L$ is large.",
"$\\square $"
],
[
"Free energy along a single linear interface",
"Also the free energy $\\mu \\mapsto \\phi ^{\\mathcal {I}}(\\mu ;\\alpha ,\\beta )$ defined in Proposition REF can be extended from $\\mathbb {Q}\\cap [1,\\infty )$ to $[1,\\infty )$ , in such a way that $\\mu \\mapsto \\mu \\phi ^{\\mathcal {I}}(\\mu ;\\alpha ,\\beta )$ is concave and continous on $[1,\\infty )$ .",
"By concatenating trajectories, we can indeed check that $\\mu \\mapsto \\mu \\phi ^{\\mathcal {I}}(\\mu ;\\alpha ,\\beta )$ is concave on $\\mathbb {Q}\\cap [1,\\infty )$ .",
"Therefore it is Lipschitz on every compact subset of $(1,\\infty )$ and can be extended to a concave and continuous function on $(1,\\infty )$ .",
"The continuity at $\\mu =1$ comes from the fact that $\\phi ^{\\mathcal {I}}(1;\\alpha ,\\beta )=0$ and $\\lim _{\\mu \\downarrow 1}\\phi ^{\\mathcal {I}}(\\mu )=0$ , which is obtained by using Lemma REF below.",
"Lemma B.1 For all $(\\alpha ,\\beta )\\in {\\hbox{\\footnotesize \\rm CONE}}$ : (i) $\\mu \\mapsto \\mu \\phi ^{\\mathcal {I}}(\\mu ;\\alpha ,\\beta )$ is strictly increasing on $[1,\\infty )$ and $\\lim _{\\mu \\rightarrow \\infty } \\mu \\phi ^{\\mathcal {I}}(\\mu ;\\alpha ,\\beta )=\\infty $ .",
"(ii) $\\lim _{\\mu \\rightarrow \\infty }\\phi ^{\\mathcal {I}}(\\mu ;\\alpha ,\\beta )=0$ .",
"Proof.",
"(i) Clearly, $\\phi ^{\\mathcal {I}}(\\mu ;\\alpha ,\\beta )\\ge \\widetilde{\\kappa }(\\mu ,0)$ for $\\mu \\ge 1$ .",
"Therefore Lemma REF (iv) implies that $\\lim _{\\mu \\rightarrow \\infty } \\mu \\phi ^{\\mathcal {I}}(\\mu ;\\alpha ,\\beta )=\\infty $ .",
"Thus, the concavity of $\\mu \\mapsto \\mu \\phi ^{\\mathcal {I}}(\\mu ;\\alpha ,\\beta )$ is sufficient to obtain that it is strictly increasing on $[1,\\infty )$ .",
"(ii) See [4], Lemma 2.4.1(i).",
"$\\square $ Recall Assumption REF , in which we assumed that $\\mu \\mapsto \\mu \\phi ^{\\mathcal {I}}(\\mu ;\\alpha ,\\beta )$ is strictly concave on $[1,\\infty )$ .",
"The next lemma states that the convergence of the average quenched free energy $\\phi ^{\\mathcal {I}}_L$ to $\\phi ^{\\mathcal {I}}$ as $L\\rightarrow \\infty $ is uniform on $\\mathbb {Q} \\cap [1,\\infty )$ .",
"Lemma B.2 For every $(\\alpha ,\\beta )\\in {\\hbox{\\footnotesize \\rm CONE}}$ and ${\\varepsilon }>0$ there exists an $L_{\\varepsilon }\\in {\\mathbb {N}}$ such that $|\\phi _L(\\mu )-\\phi (\\mu )|\\le {\\varepsilon }\\qquad \\forall \\,\\mu \\in 1+\\tfrac{2{\\mathbb {N}}}{L},\\,L\\ge L_{\\varepsilon }.$ Proof.",
"Similarly to what we did for Lemma REF , the proof can be done by treating separately the cases $\\mu $ large, moderate and small.",
"We leave the details to the reader.",
"$\\square $"
],
[
"Free energy in a single column",
"We can extend $(\\Theta ,u)\\mapsto \\psi (\\Theta ,u)$ from ${\\mathcal {V}}_M^{*}$ to $\\overline{{\\mathcal {V}}}_M^{*}$ by using the variational formulas in (REF ) and (REF ) and by recalling that $\\widetilde{\\kappa }$ and $\\phi ^{{\\mathcal {I}}}$ have been extended to ${\\mathcal {H}}$ and $[1,\\infty )$ in Appendices REF and REF .",
"Pick $M\\in {\\mathbb {N}}$ and recall (REF ).",
"Define a distance $d_M$ on $\\overline{{\\mathcal {V}}}_M$ as follows.",
"Pick $\\Theta _1,\\Theta _2 \\in \\overline{{\\mathcal {V}}}_M$ , abbreviate $\\Theta _1=(\\chi _1,\\Delta \\Pi _1, b_{0,1},b_{1,1},x_1),\\qquad \\Theta _2=(\\chi _2,\\Delta \\Pi _2, b_{0,2},b_{1,2},x_2),$ and define $d_M(\\Theta _1,\\Theta _2)= \\sum _{j\\in {\\mathbb {Z}}} \\frac{1_{\\lbrace \\chi _1(j)\\ne \\chi _2(j)\\rbrace }}{2^{|j|}} +|\\Delta \\Pi _1-\\Delta \\Pi _2|+|b_{0,1}-b_{0,2}|+|b_{1,1}-b_{1,2}|$ so that $\\widetilde{d}_M((\\Theta _1,u_1),(\\Theta _2,u_2))=\\max \\lbrace |u_1-u_2|,d_M(\\Theta _1,\\Theta _2)\\rbrace $ is a distance on $\\overline{{\\mathcal {V}}}^{\\,*,m}_M$ for which $\\overline{{\\mathcal {V}}}^{\\,*,m}_M$ is compact.",
"Lemma B.3 For every $(M,m)\\in {\\hbox{\\footnotesize \\rm EIGH}}$ and $(\\alpha ,\\beta )\\in {\\hbox{\\footnotesize \\rm CONE}}$ , $(u,\\Theta ) \\mapsto u\\,\\psi (\\Theta ,u;\\alpha ,\\beta )$ is uniformly continuous on $\\overline{{\\mathcal {V}}}^{\\,*,m}_M$ endowed with $\\widetilde{d}_M$ .",
"Proof.",
"Pick $(M,m)\\in {\\hbox{\\footnotesize \\rm EIGH}}$ .",
"By the compactness of $\\overline{{\\mathcal {V}}}^{\\,*,m}_M$ , it suffices to show that $(u,\\Theta ) \\mapsto u\\,\\psi (\\Theta ,u)$ is continuous on $\\overline{{\\mathcal {V}}}^{\\,*,m}_M$ .",
"Let $(\\Theta _n,u_n)=(\\chi _n,\\Delta \\Pi _n,b_{0,n},b_{1,n},u_n)$ be the general term of an infinite sequence that tends to $(\\Theta ,u)=(\\chi ,\\Delta \\Pi ,b_{0},b_{1},u)$ in $(\\overline{{\\mathcal {V}}}^{\\,*,m}_M,\\widetilde{d}_M)$ .",
"We want to show that $\\lim _{n\\rightarrow \\infty }u_n\\psi (\\Theta _n,u_n)=u \\psi (\\Theta ,u)$ .",
"By the definition of $\\widetilde{d}_M$ , we have $\\chi _n=\\chi $ and $\\Delta \\Pi _n=\\Delta \\Pi $ for $n$ large enough.",
"We assume that $\\Theta \\in {\\mathcal {V}}_{\\mathrm {int}}$ , so that $\\Theta _n\\in {\\mathcal {V}}_{\\mathrm {int}}$ for $n$ large enough as well.",
"The case $\\Theta \\in {\\mathcal {V}}_{\\text{nint}}$ can be treated similarly.",
"Set ${\\mathcal {R}}_m=\\lbrace (a,h,l)\\in [0,m]\\times [0,1]\\times {\\mathbb {R}}\\colon h+|l|\\le a\\rbrace $ and note that ${\\mathcal {R}}_m$ is a compact set.",
"Let $g\\colon \\,{\\mathcal {R}}_m\\mapsto [0,\\infty )$ be defined as $g(a,h,l)=a\\,\\widetilde{\\kappa }(\\tfrac{a}{h},\\tfrac{l}{h})$ if $h>0$ and $g(a,h,l)=0$ if $h=0$ .",
"The continuity of $\\widetilde{\\kappa }$ , stated in Lemma REF (i), ensures that $g$ is continuous on $\\lbrace (a,h,l)\\in {\\mathcal {R}}_m\\colon h>0\\rbrace $ .",
"The continuity at all $(a,0,l)\\in {\\mathcal {R}}_m$ is obtained by recalling that $\\lim _{u\\rightarrow \\infty }\\tilde{\\kappa }(u,l)=0$ uniformly in $l\\in [-u+1,u-1]$ (see Lemma REF (ii-iii)) and that $\\widetilde{\\kappa }$ is bounded on ${\\mathcal {H}}$ .",
"In the same spirit, we may set ${\\mathcal {R}}^{\\prime }_m=\\lbrace (u,h)\\in [0,m]\\times [0,1]\\colon \\,h\\le u\\rbrace $ and define $g^{\\prime }\\colon \\,{\\mathcal {R}}^{\\prime }_m\\mapsto [0,\\infty )$ as $g^{\\prime }(u,h)= u\\,\\phi ^{{\\mathcal {I}}}(\\tfrac{u}{h})$ for $h>0$ and $g^{\\prime }(u,h)=0$ for $h=0$ .",
"With the help of Lemma REF we obtain the continuity of $g^{\\prime }$ on ${\\mathcal {R}}^{\\prime }_m$ by mimicking the proof of the continuity of $g$ on ${\\mathcal {R}}_m$ .",
"Note that the variational formula in (REF ) can be rewriten as $u \\,\\psi (\\Theta ,u)&=\\sup _{(h),(a) \\in {\\mathcal {L}}(l_A,\\, l_B;\\,u)} Q((h),(a),l_A,l_B),$ with $Q((h),(a),l_A,l_B)=g(a_A,h_A,l_A)+g(a_B,h_B,l_B)+a_B \\,\\tfrac{\\beta -\\alpha }{2}+g^{^{\\prime }}(a^{\\mathcal {I}},h^{\\mathcal {I}}),$ and with $l_A$ and $l_B$ defined in (REF ).",
"Note that ${\\mathcal {L}}(l_A,\\,l_B;\\,u)$ is compact, and that $(h),(a)\\mapsto Q((h),(a),l_A,l_B)$ is continuous on ${\\mathcal {L}}(l_A,\\, l_B;\\,u)$ because $g$ and $g^{\\prime }$ are continuous on ${\\mathcal {R}}_m$ and ${\\mathcal {R}}^{^{\\prime }}_m$ , respectively.",
"Hence, the supremum in (REF ) is attained.",
"Pick ${\\varepsilon }>0$ , and note that $g$ and $g^{\\prime }$ are uniformly continuous on ${\\mathcal {R}}_m$ and ${\\mathcal {R}}^{\\prime }_m$ , which are compact sets.",
"Hence there exists an $\\eta _{\\varepsilon }>0$ such that $|g(a,h,l)-g(a^{\\prime },h^{\\prime },l^{\\prime })|\\le {\\varepsilon }$ and $|g^{\\prime }(u,b)-g^{\\prime }(u^{\\prime },b^{\\prime })| \\le {\\varepsilon }$ when $(a,h,l),(a^{\\prime },h^{\\prime },l^{\\prime })\\in {\\mathcal {R}}_m$ and $(u,b),(u^{\\prime },b^{\\prime })\\in {\\mathcal {R}}^{\\prime }_m$ are such that $|a-a^{\\prime }|,|h-h^{\\prime }|,|l-l^{\\prime }|,|u-u^{\\prime }|$ and $|b-b^{\\prime }|$ are bounded from above by $\\eta _{\\varepsilon }$ .",
"Since $\\lim _{n\\rightarrow \\infty }(\\Theta _n,u_n)=(\\Theta ,u)$ we also have that $\\lim _{n\\rightarrow \\infty } b_{0,n}=b_0$ , $\\lim _{n\\rightarrow \\infty } b_{1,n}=b_1$ and $\\lim _{n\\rightarrow \\infty } u_{n}=u$ .",
"Thus, $\\lim _{n\\rightarrow \\infty } l_{A,n}= l_A$ and $\\lim _{n\\rightarrow \\infty } l_{B,n}= l_B$ , and therefore $|l_{A,n}- l_A|\\le \\eta _{\\varepsilon }$ , $|l_{B,n}- l_B|\\le \\eta _{\\varepsilon }$ and $|u_n-u|\\le \\eta _{\\varepsilon }$ for $n\\ge n_{\\varepsilon }$ large enough.",
"For $n\\in {\\mathbb {N}}$ , let $(h_n),(a_n)\\in {\\mathcal {L}}(l_{A,n},\\, l_{B,n};\\,u_n)$ be a maximizer of (REF ) at $(\\Theta _n,u_n)$ , and note that, for $n\\ge n_{\\varepsilon }$ , we can choose $(\\widetilde{h}_n),(\\widetilde{a}_n)\\in {\\mathcal {L}}(l_A,\\,l_B;\\,u)$ such that $|\\widetilde{a}_{A,n}-a_{A,n}|$ , $|\\widetilde{a}_{B,n}-a_{B,n}|$ , $|\\widetilde{a}_{n}^{\\mathcal {I}}-a_{n}^{\\mathcal {I}}|$ , $|\\widetilde{h}_{A,n}-h_{A,n}|$ , $|\\widetilde{h}_{B,n}-h_{B,n}|$ and $|\\widetilde{h}_{n}^{\\mathcal {I}}-h_{n}^{\\mathcal {I}}|$ are bounded above by $\\eta _{\\varepsilon }$ .",
"Consequently, $u_n \\psi (\\Theta _n,u_n)- u\\psi (\\Theta ,u)\\le Q((h_n),(a_n), l_{A,n},l_{B,n})-Q((\\widetilde{h}_n),(\\widetilde{a}_n), l_{A},l_{B})\\le 3{\\varepsilon }.$ We bound $u\\psi (\\Theta ,u)-u_n \\psi (\\Theta _n,u_n)$ from above in a similar manner, and this suffices to obtain the claim.",
"$\\square $ Lemma B.4 For every $\\Theta \\in \\overline{{\\mathcal {V}}}_M$ , the function $u \\mapsto u\\psi (\\Theta ,u)$ is continuous and strictly concave on $[t_\\Theta ,\\infty )$ .",
"Proof.",
"The continuity is a straightforward consequence of Lemma REF : simply fix $\\Theta $ and let $m\\rightarrow \\infty $ .",
"To prove the strict concavity, we note that the cases $\\Theta \\in {\\mathcal {V}}_{\\mathrm {int}}$ and $\\Theta \\in {\\mathcal {V}}_{\\text{nint}}$ can be treated similarly.",
"We will therefore focus on $\\Theta \\in {\\mathcal {V}}_{\\mathrm {int}}$ .",
"For $l\\in {\\mathbb {R}}$ , let ${\\mathcal {N}}_l=\\lbrace (a,h)\\in [0,\\infty ) \\times [0,1]\\colon a\\ge h+|l|\\rbrace ,\\quad {\\mathcal {N}}_l^+=\\lbrace (a,h)\\in {\\mathcal {N}}_l\\colon h>0\\rbrace ,$ and let $g_l\\colon \\,{\\mathcal {N}}_l\\mapsto [0,\\infty )$ be defined as $g_l(a,h)=a\\,\\widetilde{\\kappa }(\\tfrac{a}{h},\\tfrac{l}{h})$ for $h>0$ and $g_l(a,h)=0$ for $h=0$ .",
"The strict concavity of $(u,l)\\mapsto u\\widetilde{\\kappa }(u,l)$ on ${\\mathcal {H}}$ , stated in Lemma REF (i), immediately yields that $g_l$ is strictly concave on ${\\mathcal {N}}_l^+$ and concave on ${\\mathcal {N}}_l$ .",
"Consequently, for all $(a_1,h_1)\\in {\\mathcal {N}}_l^+$ and $(a_2,h_2){\\mathcal {N}}_l\\setminus {\\mathcal {N}}_l^+ $ , $g_l$ is strictly concave on the segment $[(u_1,h_1),(u_2,h_2)]$ .",
"Let $\\widetilde{{\\mathcal {N}}}=\\lbrace (u,h)\\in [0,\\infty )\\times [0,1]\\colon \\, h\\le u\\rbrace $ and define $\\widetilde{g}\\colon \\,\\widetilde{{\\mathcal {N}}}\\mapsto [0,\\infty )$ as $\\widetilde{g}(u,h)= u\\,\\phi ^{{\\mathcal {I}}}(\\tfrac{u}{h})$ for $h>0$ and $\\widetilde{g}(u,h)=0$ for $h=0$ .",
"The strict concavity of $u\\mapsto u\\phi ^{{\\mathcal {I}}}(u)$ on $[1,\\infty )$ , stated in Assumption REF , immediately yields that $\\widetilde{g}$ is strictly concave on $\\widetilde{{\\mathcal {N}}}^+=\\lbrace (u,h)\\in \\widetilde{{\\mathcal {N}}}\\colon h>0\\rbrace $ and concave on $\\widetilde{{\\mathcal {N}}}$ .",
"Consequently, for all $(u_1,h_1)\\in \\widetilde{{\\mathcal {N}}}^+$ and $(u_2,h_2)\\in \\widetilde{{\\mathcal {N}}}\\setminus \\widetilde{{\\mathcal {N}}}^+$ , $\\widetilde{g}$ is strictly concave on the segment $[(u_1,h_1),(u_2,h_2)]$ .",
"Similarly to what we did in (REF ), we can rewrite the variational formula in (REF ) as $u \\,\\psi (\\Theta ,u)&=\\sup _{(h),(a) \\in {\\mathcal {L}}(l_A,\\, l_B;\\,u)} \\widetilde{Q}((h),(a))$ with $\\widetilde{Q}((h),(a))=g_{l_A}(a_A,h_A)+g_{l_B}(a_B,h_B)+a_B \\,\\tfrac{\\beta -\\alpha }{2}+\\widetilde{g}(u-a_A-a_B,1-h_A-h_B),$ and the supremum in (REF ) is attained.",
"Next we show that if $(h),(a)\\in {\\mathcal {L}}(l_A,\\,l_B;\\,u)$ realizes the maximum in (REF ), then $(h),(a)\\notin \\widetilde{{\\mathcal {L}}}(l_A,\\,l_B;\\,u)$ with $\\widetilde{{\\mathcal {L}}}(l_A,\\, l_B;\\,u)= \\widetilde{{\\mathcal {L}}}_A(l_A,\\, l_B;\\,u)\\cup \\widetilde{{\\mathcal {L}}}_B(l_A,\\, l_B;\\,u)\\cup \\widetilde{{\\mathcal {L}}}^{\\, {\\mathcal {I}}}(l_A,\\, l_B;\\,u)$ and $\\nonumber \\widetilde{{\\mathcal {L}}}_A(l_A,\\, l_B;\\,u)&=\\lbrace (h),(a)\\in {\\mathcal {L}}(l_A,\\, l_B;\\,u)\\colon \\,h_A=0 \\ \\ \\text{and}\\ \\ a_A>l_A\\rbrace ,\\\\\\nonumber \\widetilde{{\\mathcal {L}}}_B(l_A,\\, l_B;\\,u)&=\\lbrace (h),(a)\\in {\\mathcal {L}}(l_A,\\, l_B;\\,u)\\colon \\, h_B=0 \\ \\ \\text{and}\\ \\ a_B>l_B\\rbrace ,\\\\\\widetilde{{\\mathcal {L}}}^{\\,{\\mathcal {I}}}(l_A,\\, l_B;\\,u)&=\\lbrace (h),(a)\\in {\\mathcal {L}}(l_A,\\, l_B;\\,u)\\colon \\, h_I=0 \\ \\ \\text{and}\\ \\ a_I>0\\rbrace .$ Assume that $(h),(a)\\in \\widetilde{{\\mathcal {L}}}(l_A,\\, l_B;\\,u)$ , and that $h_A>0$ or $h^{\\mathcal {I}}>0$ .",
"For instance, $(h),(a)\\in \\widetilde{{\\mathcal {L}}}^{\\mathcal {I}}(l_A,\\, l_B;\\,u)$ and $h_A>0$ .",
"Then, by Lemma REF (iv), $\\widetilde{Q}$ strictly increases when $a_A$ is replaced by $a_A+a^{\\mathcal {I}}$ and $a^{\\mathcal {I}}$ by 0.",
"This contradicts the fact that $(h),(a)$ is a maximizer.",
"Next, if $(h),(a)\\in \\widetilde{{\\mathcal {L}}}(l_A,\\,l_B;\\,u)$ and $h_A=h^{\\mathcal {I}}=0$ , then $h_B=1$ , and the first case is $(h),(a)\\in \\widetilde{{\\mathcal {L}}}_A(l_A,\\,l_B;\\,u)$ , while the second case is $(h),(a)\\in \\widetilde{{\\mathcal {L}}}^{\\mathcal {I}}(l_A,\\, l_B;\\,u)$ .",
"In the second case, as before, we replace $a_A$ by $a_A+a^{\\mathcal {I}}$ and $a^{\\mathcal {I}}$ by 0, which does not change $\\widetilde{Q}$ but yields that $a_A>l_A$ and therefore brings us back to the first case.",
"In this first case, we are left with an expression of the form $Q((h),(a))=g_{l_B}(a_B,1) + a_B\\, \\tfrac{\\beta -\\alpha }{2}$ with $h_A=h^{\\mathcal {I}}=0$ and $a_A>l_A$ .",
"Thus, if we can show that there exists an $x\\in (0,1)$ such that $g_{l_A}(a_A,x)+g_{l_B}(a_B,1-x)>g_{l_B}(a_B,1),$ then we can claim that $(h),(a)$ is not a maximizer of (REF ) and the proof for $(h),(a)\\notin \\widetilde{{\\mathcal {L}}}(l_A,\\,l_B;\\,u)$ will be complete.",
"To that end, we recall (REF ), which allows us to rewrite the left-hand side in (REF ) as $g_{l_A}(a_A,x)+g_{l_B}(a_B,1-x)=a_A \\,\\kappa \\big (\\tfrac{a_A}{l_A},\\tfrac{x}{l_A}\\big )+a_B \\,\\kappa \\big (\\tfrac{a_B}{l_B},\\tfrac{1-x}{l_B}\\big )+ a_B\\, \\tfrac{\\beta -\\alpha }{2}.$ We recall [3], Lemma 2.1.1, which claims that $\\kappa $ is defined on ${\\hbox{\\footnotesize \\rm DOM}}=\\lbrace (a,b)\\colon a\\ge 1+b, b\\ge 0\\rbrace $ , is analytic on the interior of ${\\hbox{\\footnotesize \\rm DOM}}$ and is continuous on ${\\hbox{\\footnotesize \\rm DOM}}$ .",
"Moreover, in the proof of this lemma, an expression for $\\partial _b\\, \\kappa (a,b)$ is provided, which is valid on the interior of ${\\hbox{\\footnotesize \\rm DOM}}$ .",
"From this expression we can easily check that if $a>1$ , then $\\lim _{b\\rightarrow 0} \\partial _b\\,\\kappa (a,b)=\\infty $ .",
"Therefore, by the continuity of $\\kappa $ on $(a_A/l_A,0)$ with $a_A/l_A>1$ we can assert that the derivative with respect to $x$ of the left-hand side in (REF ) at $x=0$ is infinite, and therefore there exists an $x>0$ such that (REF ) is satisfied.",
"Pick $u_1>u_2\\ge t_\\Theta $ , and let $(h_1),(a_1) \\in {\\mathcal {L}}(l_A,\\, l_B;\\,u_1)$ and $(h_2),(a_2) \\in {\\mathcal {L}}(l_A,\\, l_B;\\,u_2)$ be maximizers of (REF ) for $u_1$ and $u_2$ , respectively.",
"We can write $\\nonumber (a_1),(h_1)&=\\big (a_{A,1},a_{B,1},a^{{\\mathcal {I}}}_1),(h_{A,1},h_{B,1},h^{{\\mathcal {I}}}_1\\big ),\\\\(a_2),(h_2)&=\\big (a_{A,2},a_{B,2},a^{{\\mathcal {I}}}_2),(h_{A,2},h_{B,2},h^{{\\mathcal {I}}}_2\\big ).$ Thus, $(\\tfrac{a_1+a_2}{2}),(\\tfrac{h_1+h_2}{2})\\in {\\mathcal {L}}(l_A,\\,l_B;\\,\\tfrac{u_1+u_2}{2})$ and, with the help of the concavity of $g_{l_A}, g_{l_B},\\widetilde{g}$ proven above, we can write $\\tfrac{u_1+u_2}{2}\\,\\psi (\\Theta ,\\tfrac{u_1+u_2}{2})\\ge \\widetilde{Q}((\\tfrac{a_1+a_2}{2}), (\\tfrac{h_1+h_2}{2}))\\ge \\tfrac{1}{2} \\big (u_1\\,\\psi (\\Theta ,u_1)+u_2\\,\\psi (\\Theta ,u_2)\\big ).$ We have proven above that $(a_1),(h_1) \\notin \\widetilde{{\\mathcal {L}}}(l_A,\\, l_B;\\,u_1)$ and $(a_2),(h_2) \\notin \\widetilde{{\\mathcal {L}}}(l_A,\\, l_B;\\,u_2)$ .",
"Thus, we can use (REF ) and the strict concavity of $g_{l_A}, g_{l_B},\\widetilde{g}$ on ${\\mathcal {N}}_{l_A}^+,{\\mathcal {N}}_{l_B}^+\\widetilde{{\\mathcal {N}}}^+$ , to conclude that the right-most inequality in (REF ) is an equality only if $\\begin{aligned}&(a_{A,1},h_{A,1}) = (a_{A,2},h_{A,2}), \\quad (a_{B,1},h_{B,1}) = (a_{B,2},h_{B,2}),\\\\&(u_1-a_{A,1}-a_{B,1},1-h_{A,1}-h_{B,1}) = (u_2-a_{A,2}-a_{B,2},1-h_{A,2}-h_{B,2}),\\end{aligned}$ which clearly is not possible because $u_1>u_2$ .",
"$\\square $"
],
[
"Concentration of measure",
"Let ${\\mathcal {S}}$ be a finite set and let $(X_i,{\\mathcal {A}}_i,\\mu _i)_{i\\in {\\mathcal {S}}}$ be a family of probability spaces.",
"Consider the product space $X=\\prod _{i\\in {\\mathcal {S}}} X_i$ endowed with the product $\\sigma $ -field ${\\mathcal {A}}=\\otimes _{i\\in {\\mathcal {S}}}{\\mathcal {A}}_i$ and with the product probability measure $\\mu =\\otimes _{i\\in {\\mathcal {S}}} \\mu _i$ .",
"Theorem C.1 (Talagrand [7]) Let $f\\colon \\,X\\mapsto {\\mathbb {R}}$ be integrable with respect to $({\\mathcal {A}},\\mu )$ and, for $i\\in {\\mathcal {S}}$ , let $d_i>0$ be such that $|f(x)-f(y)|\\le d_i$ when $x,y\\in X$ differ in the $i$ -th coordinate only.",
"Let $D=\\sum _{i\\in {\\mathcal {S}}} d_i^2$ .",
"Then, for all ${\\varepsilon }>0$ , $\\mu \\left\\lbrace x\\in X\\colon \\left|f(x)-\\int fd\\mu \\right|>{\\varepsilon }\\right\\rbrace \\le 2 e^{-\\frac{{\\varepsilon }^2}{2D}}.$ The following corollary of Theorem REF was used several times in the paper.",
"Let $(\\alpha ,\\beta )\\in {\\hbox{\\footnotesize \\rm CONE}}$ and let $(\\xi _i)_{i\\in {\\mathbb {N}}}$ be an i.i.d.",
"sequence of Bernouilli trials taking the values $-\\alpha $ and $\\beta $ with probability $\\tfrac{1}{2}$ each.",
"Let $l\\in {\\mathbb {N}}$ , $T\\colon \\,\\,\\lbrace (x,y)\\in {\\mathbb {Z}}^2\\times {\\mathbb {Z}}^2\\colon |x-y|=1\\rbrace \\rightarrow \\lbrace 0,1\\rbrace $ and $\\Gamma \\subset {\\mathcal {W}}_l$ (recall (REF )).",
"Let $F_l\\colon \\,[-\\alpha ,\\alpha ]^l\\rightarrow {\\mathbb {R}}$ be such that $F_l(x_1,\\dots ,x_l) = \\log \\sum _{\\pi \\in \\Gamma } e^{\\sum _{i=1}^{l}x_i\\, T( (\\pi _{i-1},\\pi _i))}.$ For all $x,y\\in [-\\alpha ,\\alpha ]^l$ that differ in one coordinate only we have $|F_l(x)-F_l(y)|\\le 2\\alpha $ .",
"Therefore we can use Theorem REF with ${\\mathcal {S}}=\\lbrace 1,\\dots ,l\\rbrace $ , $X_i=[-\\alpha ,\\alpha ]$ and $\\mu _i=\\tfrac{1}{2} (\\delta _{-\\alpha }+ \\delta _{\\beta })$ for all $i\\in {\\mathcal {S}}$ , and $D=4\\alpha ^2 l$ , to obtain that there exist $C_1,C_2>0$ such that, for every $l\\in {\\mathbb {N}}$ , $\\Gamma \\subset {\\mathcal {W}}_n$ and $T\\colon \\,\\lbrace (x,y)\\in {\\mathbb {Z}}^2\\times {\\mathbb {Z}}^2\\colon |x-y|=1\\rbrace \\rightarrow \\lbrace 0,1\\rbrace $ , ${\\mathbb {P}}\\big (|F_l(\\xi _1,\\dots ,\\xi _m)-{\\mathbb {E}}(F_l(\\xi _1,\\dots ,\\xi _m))|>\\eta \\big )\\le C_1e^{-\\tfrac{C_2\\eta ^2}{l}}.$"
],
[
"Large deviation estimate",
"Let $(\\xi _i)_{i\\in {\\mathbb {N}}}$ be an i.i.d.",
"sequence of Bernouilli trials taking values $\\beta $ and $-\\alpha $ with probability $\\frac{1}{2}$ each.",
"For $N\\le n\\in {\\mathbb {N}}$ , denote by ${\\mathcal {E}}_{n,N}$ the set of all ordered sequences of $N$ disjoint and non-empty intervals included in $\\lbrace 1,\\dots ,n\\rbrace $ , i.e., $\\nonumber {\\mathcal {E}}_{n,N} &= \\big \\lbrace (I_j)_{1\\le j\\le N}\\subset \\lbrace 1,\\dots ,n\\rbrace \\colon I_j=\\lbrace \\min I_j,\\dots ,\\max I_j\\rbrace \\,\\,\\forall \\,1\\le j\\le N,\\\\& \\max I_j<\\min I_{j+1}\\,\\,\\forall \\,1\\le j\\le N-1\\ \\text{and}\\ I_j \\ne \\emptyset \\,\\,\\forall \\,1\\le j\\le N\\big \\rbrace .$ For $(I)\\in {\\mathcal {E}}_{n,N}$ , let $T(I)=\\sum _{j=1}^N |I_j|$ be the cumulative length of the intervals making up $(I)$ .",
"Pick $\\gamma >0$ and $M\\in {\\mathbb {N}}$ , and denote by $\\widehat{{\\mathcal {E}}}_{n,M}^{\\,\\gamma }$ the set of those $(I)$ in $\\cup _{1\\le N\\le (n/M)}\\,{\\mathcal {E}}_{n,N}$ that have a cumulative length larger than $\\gamma n$ , i.e., $\\widehat{{\\mathcal {E}}}_{n,M}^{\\,\\gamma } = \\cup _{N=1}^{n/M}\\big \\lbrace (I)\\in {\\mathcal {E}}_{n,N}\\colon \\,T(I)\\ge \\gamma n\\big \\rbrace .$ Next, for $\\eta >0$ set ${\\mathcal {Q}}_{n,M}^{\\gamma ,\\eta } = \\bigcap _{\\,(I)\\in \\widehat{{\\mathcal {E}}}_{n,M}^{\\,\\gamma }}\\left\\lbrace \\sum _{j=1}^N \\sum _{i\\in I_j} \\xi _i\\le (\\tfrac{\\beta -\\alpha }{2}+\\eta )\\,T(I)\\right\\rbrace .$ Lemma D.1 For all $(\\alpha ,\\beta )\\in {\\hbox{\\footnotesize \\rm CONE}}$ , $\\gamma >0$ and $\\eta >0$ there exists an $\\widehat{M}\\in {\\mathbb {N}}$ such that, for all $M\\ge \\widehat{M}$ , $\\lim _{n\\rightarrow \\infty } P(({\\mathcal {Q}}_{n,M}^{\\gamma ,\\eta })^c)=0.$ Proof.",
"An application of Cramér's theorem for i.i.d.",
"random variables gives that there exists a $c_\\eta >0$ such that, for every $(I)\\in \\widehat{{\\mathcal {E}}}_{n,M}^{\\,\\gamma }$ , ${\\mathbb {P}}_\\xi \\bigg (\\sum _{j=1}^N \\sum _{i\\in I_j}\\xi _i\\ge (\\tfrac{\\beta -\\alpha }{2}+\\eta )\\, T(I)\\bigg )\\le \\,e^{-c_\\eta T(I)}\\le \\, e^{-c_\\eta \\gamma n},$ where we use that $T(I)\\ge \\gamma n$ for every $(I)\\in \\widehat{{\\mathcal {E}}}_{n,M}^{\\,\\gamma }$ .",
"Therefore ${\\mathbb {P}}_\\xi (({\\mathcal {Q}}_{n,M}^{\\gamma ,\\eta })^c)&\\le |\\widehat{{\\mathcal {E}}}_{n,M}^{\\,\\gamma }|e^{-c(\\eta )\\gamma n },$ and it remains to bound $|\\widehat{{\\mathcal {E}}}_{n,M}^{\\,\\gamma }|$ as $\\widehat{{\\mathcal {E}}}_{n,M}^{\\,\\gamma } = \\sum _{N=1}^{n/M}\\big |\\big \\lbrace (I)\\in {\\mathcal {E}}_{n,N}\\colon T(I)\\ge \\gamma n\\big \\rbrace \\big |\\le \\sum _{N=1}^{n/M} \\binom{n}{2N},$ where we use that choosing $(I)\\in {\\mathcal {E}}_{n,N}$ amounts to choosing in $\\lbrace 1,\\dots ,n\\rbrace $ the end points of the $N$ disjoint intervals.",
"Thus, the right-hand side of (REF ) is at most $(n/M) \\binom{n}{2n/M}$ , which for $M$ large enough is $o( e^{c(\\eta )\\gamma n})$ as $n\\rightarrow \\infty $ .",
"$\\square $"
]
] | 1204.1234 |
[
[
"Crossover between two different Kondo couplings in side-coupled double\n quantum dots"
],
[
"Abstract We study the Kondo effect in side-coupled double quantum dots with particular focus on the crossover between two distinct singlet ground states, using the numerical renormalization group.",
"The crossover occurs as the quantized energy level of the embedded dot, which is connected directly to the leads, is varied.",
"In the parameter region where the embedded dot becomes almost empty or doubly occupied, the local moment emerging in the other dot at the side of the path for the current is screened via a superexchange process by the conduction electrons tunneling through the embedded dot.",
"In contrast, in the other region where the embedded dot is occupied by a single electron, the local moment emerges also in the embedded dot, and forms a singlet bond with the moment in the side dot.",
"Furthermore, we derive two different Kondo Hamiltonians for these limits carrying out the Schrieffer-Wolff transformation, and show that they describe the essential feature of the screening for each case."
],
[
"Introduction",
"The Kondo effect is a prototypical many-body phenomenon that is caused by the interaction between a localized spin and conduction electrons.",
"Since the Kondo effect was observed in a quantum dot (QD) system,[1], [2] effects of electron correlation on quantum transport have attracted much attention.",
"Moreover, the recent experimental advancement enables one to examine the Kondo physics in a variety of systems, such as an Aharonov-Bohm ring with a QD and double quantum dots (DQD).",
"In these systems multiple paths for electron propagation also affect the tunneling currents, and give rise to a Fano-type asymmetry in the line shapes of conductance.",
"A side-coupled DQD system with a T-shape configuration, as shown in Fig.",
"REF , is a typical system in which the interplay between the Kondo effect and the interference effect occurs.",
"For this type of DQD, a number of theoretical studies have been carried out so far.",
"[3], [4], [6], [7], [8], [9], [10], [11], [12], [15], [14], [5], [13] It also becomes experimentally possible to fabricate this kind of geometry.",
"[16] In the side-coupled DQD system shown in Fig.",
"REF , one of the dots (QD2), which is referred to as a side dot, has no direct coupling to the leads, but is coupled to the embedded dot (QD1).",
"Because of this unique geometry with the multiple paths, intriguing phenomena can occur in this system.",
"For example, a two-stage Kondo effect can occur for the small interdot coupling $t$ , where each of the dots is occupied by a single electron.",
"[6], [9], [14], [11], [10], [12], [15] In this case, the local moment in the QD1 is screened first by the conduction electrons at higher temperature, and then the moment in the QD2 is screened at lower temperature to form a singlet ground state.",
"Most of the preceding studies of a side-coupled DQD system focused on the two-stage Kondo effect taking place in this situation.",
"In contrast, for large $t$ , the two adjacent local moments screen each other to form a molecular-type singlet.",
"The gate voltage that is applied to the dots can further change the charge and spin distributions in the DQD, and evolve the system toward the mixed-valence regime.",
"Specifically, as the energy level $\\varepsilon _1$ of the QD1 is varied, another typical singlet state appears in the parameter region where the QD1 becomes almost empty or doubly occupied.",
"It is a singlet bond between the local moment at QD2 and the conduction electrons, and is formed by a superexchange mechanism.",
"Some numerical indications that this type of singlet state is formed were seen in data of previous works.",
"[5], [9] Maruyama et al.",
"obtained an asymmetric conductance peak of the Fano shape for finite $\\varepsilon _1$ in the Kondo regime.",
"[5] However, their study is focused mainly on the transport properties.",
"Žitko et al.",
"[9] examined a similar situation, and showed that the Kondo temperature becomes small in this case.",
"The precise features of the singlet state, however, were not fully examined.",
"Some groups have studied the gate-voltage dependence of the conductance where the energy levels of the two dots are moved simultaneously.",
"[6], [9], [10] However, it has still not been clarified in detail how the singlet ground state evolves across the crossover region between the singlet state due to the two-stage Kondo effect and the one formed by the superexchange mechanism between the QD2 and the conduction electrons.",
"In this paper, we re-examine a side-coupled DQD system shown in Fig.",
"REF , and study how the Kondo singlet bond is deformed as the energy level $\\varepsilon _1$ in the QD1 is changed.",
"We find that as $\\varepsilon _1$ moves away from the Fermi energy, the electrons at the QD1 cannot contribute to the screening of the local moment at the QD2, and the conduction electrons tunneling into the QD2 virtually via $\\varepsilon _1$ screen the local moment.",
"This electron tunneling process is similar to the superexchange mechanism seen in transition metal oxides such as MnO and CuO.",
"[17], [18] In the present case, the Kondo singlet bond is formed between the QD2 and the leads, mediated by the quantized level $\\varepsilon _1$ at the QD1.",
"This mechanism is quite different from the Kondo screening in the case of $\\varepsilon _1 \\simeq 0$ where a singlet bond between the QD1 and QD2 plays a dominant role.",
"[3], [4], [6], [7], [8], [9], [10], [11], [12], [15], [14], [5], [13] Therefore, the gate voltage applied to the QD1 deforms the Kondo cloud, and it can be probed through the variation in the phase shift of the DQD.",
"In order to clarify these features, we calculate the phase shift using the numerical renormalization group.",
"Furthermore, we calculate the spin susceptibility to obtain the Kondo temperature, which shows good agreement with the one obtained from an effective Kondo Hamiltonian we have derived in this work.",
"In the presence of the Coulomb interaction $U_1$ at the QD1, we also find that the two-stage Kondo screening changes to a single-stage process as $\\varepsilon _1$ moves away from the electron-hole symmetric point $\\varepsilon _1 \\simeq -U_1/2$ .",
"This paper is organized as follows.",
"In Sec.",
", we give the Hamiltonian of our system.",
"In Sec.",
", we derive the effective Hamiltonian using the perturbation theory in the tunneling matrix elements, which is identical to that derived from the Schrieffer-Wolff transformation.",
"[19] In Sec.",
", we show the numerical results, and discuss how the Kondo singlet state evolves as $\\varepsilon _1$ varies.",
"A summary and discussions are given in Sec.",
"."
],
[
"Model",
"The Hamiltonian of a side-coupled DQD system shown in Fig.",
"REF reads $\\,H = H_{QD1} + H_{QD2} + H_{\\rm int} +\\!\\!\\!\\sum _{\\nu \\in \\lbrace L,R \\rbrace }\\!\\!\\!",
"(H_{\\nu } + H_{T,\\nu }),$ where $&H_{QDi}=\\varepsilon _{i} \\sum _{\\sigma } n_{i,\\sigma }+ U_i n_{i,\\uparrow }n_{i,\\downarrow } ,\\nonumber \\\\&H_{\\rm int}=t\\,\\sum _{\\sigma }\\left(d_{1\\sigma }^{\\dag }d_{2\\sigma }^{}+\\textrm {H.c.}\\right),\\,\\,\\,H_{\\nu }=\\sum _{k,\\sigma }\\varepsilon _{k}c_{\\nu ,k\\sigma }^\\dag c_{\\nu ,k\\sigma }^{} ,\\nonumber \\\\&H_{T,\\nu } = \\sum _{k,\\sigma } \\frac{V_{\\nu }}{\\sqrt{\\mathcal {N}}}\\left(c_{\\nu ,k\\sigma }^\\dag d_{1\\sigma }^{} + \\textrm {H.c.} \\right),\\quad \\ \\ \\nu =L, R.$ $H_{QDi}$ describes the QD1 for $i=1$ , and the QD2 for $i=2$ , $\\varepsilon _{i}$ the energy level, $U_{i}$ the Coulomb interaction, and $n_{i,\\sigma }=d^{\\dag }_{i\\sigma }d^{}_{i\\sigma }$ .",
"$H_{\\rm int}$ denotes the interdot coupling with the hopping matrix element $t$ .",
"$H_{L/R}$ describes the normal lead of the left/right side.",
"$V_{L/R}$ is the tunneling matrix element between the QD1 and the left/right lead.",
"We assume that $\\Gamma _{L/R}(\\varepsilon ) \\equiv \\pi V_{L/R}^{2} \\sum _k \\delta (\\varepsilon -\\varepsilon _{k})/\\mathcal {N}$ is a constant independent of the energy $\\varepsilon $ , where $\\mathcal {N}$ is the number of the states in each lead.",
"For convenience of the following discussions, we apply a unitary transformation to the leads, using the inversion symmetry, $&& \\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!s_{k\\sigma }=\\frac{V_L c_{L,k\\sigma }+V_R c_{R,k\\sigma }}{V_s}, \\,a_{k\\sigma }=\\frac{V_L c_{L,k\\sigma }-V_R c_{R,k\\sigma }}{V_s},\\nonumber \\\\&& \\qquad \\qquad V_s= \\sqrt{{V_L}^2+{V_R}^2}.$ Then, the Hamiltonian for the leads, $\\sum _{\\nu \\in \\lbrace L,R \\rbrace } (H_{\\nu } + H_{T,\\nu })$ , can be rewritten as $\\sum _{\\nu \\in \\lbrace L,R \\rbrace }\\left( H_{\\nu } + H_{T,\\nu } \\right)&=&H_s + H_a + H_{T,s} ,$ with $&&H_s = \\sum _{k,\\sigma }\\varepsilon _{k}s_{k\\sigma }^\\dag s_{k\\sigma }^{},\\quad H_a = \\sum _{k,\\sigma }\\varepsilon _{k}a_{k\\sigma }^\\dag a_{k\\sigma }^{},\\\\&&H_{T,s} =\\sum _{k,\\sigma } \\frac{V_{s}}{\\sqrt{\\mathcal {N}}}\\left(s_{k\\sigma }^\\dag d_{1\\sigma }^{} + \\textrm {H.c.} \\right).$ Note that the operator $s_{k\\sigma }$ couples to the QD1, while the operator $a_{k\\sigma }$ is decoupled.",
"Therefore, we can map the original model given by Eq.",
"(REF ) to the two-impurity Anderson model (TIAM), $\\mathcal {H}_{\\rm TIAM} = H_{QD1} + H_{QD2} + H_{\\rm int} +H_{s} + H_{T,s} .$ Using this model, we will discuss how the singlet state due to the Kondo effect changes as the energy level $\\varepsilon _1$ at the QD1 varies."
],
[
"Kondo Hamiltonians",
"In this section, we derive the effective Hamiltonians in order to study the Kondo behavior in the two opposite cases, $\\varepsilon _1 \\simeq 0$ and large $\\varepsilon _1$ .",
"For this purpose, we use the perturbation theory in the tunneling matrix elements $t$ and $V_S$ , which is equivalent to the Schrieffer-Wolff transformation.",
"[19] We assume that the Coulomb interaction at the QD1 is zero, $U_1=0$ , in this section in order to focus on the effects of $\\varepsilon _1$ .",
"We also consider the effects of $U_1$ in Sec.",
"B and the Appendix.",
"Figure: (Color online)Schematic energy diagram of a side-coupled DQD.",
"(a) ε 1 ≃0\\varepsilon _{1} \\simeq 0, and(b) ε 1 >0\\varepsilon _{1} >0 such that〈n 1,σ 〉≃0\\langle n_{1,\\sigma } \\rangle \\simeq 0 is satisfied.E F E_F denotes the Fermi energy of the leads.Figure REF shows a schematic energy diagram of a side-coupled DQD for $\\varepsilon _{2} + U_{2}/2 \\simeq 0$ .",
"Let us first consider the case, shown in Fig.",
"REF (a), where the energy level at the QD1 is located around the Fermi energy $\\varepsilon _{1} \\simeq 0$ .",
"When the Coulomb interaction $U_2$ is much larger than the interdot coupling $t$ , the local spin moment arises at the QD2.",
"Carrying out the perturbation expansion in the tunneling matrix element $t$ , we obtain the Kondo Hamiltonian of the form $\\mathcal {H}_{\\rm K}^{\\rm (a)}=J_{12}\\sum _{\\sigma , \\sigma ^{\\prime }}d^{\\dag }_{1\\sigma } \\frac{\\vec{\\tau }_{\\sigma \\sigma ^{\\prime }}}{2}d^{}_{1\\sigma ^{\\prime }} \\cdot \\vec{S}_2 + H_{QD1} + H_{s} + H_{T,s},$ where $J_{12} &=2 t^2 \\left\\lbrace -\\frac{1}{\\varepsilon _2-\\varepsilon _1}+\\frac{1}{(\\varepsilon _2+U_2)-\\varepsilon _1}\\right\\rbrace \\nonumber \\\\&\\xrightarrow{}\\frac{8t^2}{U_2}.$ $\\vec{\\tau }_{\\sigma \\sigma ^{\\prime }} = (\\tau _{\\sigma \\sigma ^{\\prime }}^x, \\tau _{\\sigma \\sigma ^{\\prime }}^y, \\tau _{\\sigma \\sigma ^{\\prime }}^z)$ is the vector representation of the Pauli matrix, and $\\vec{S}_2$ is the spin operator of the local spin at the QD2.",
"In the electron-hole symmetric case, $\\varepsilon _1=0$ and $\\varepsilon _2 = -U_2/2$ , the exchange coupling is given by $J_{12} =8t^2/U_2$ .",
"This type of exchange coupling appears in the usual Kondo model, and is referred to as the “interdot exchange (IE)\" coupling in the following, in order to distinguish it from the superexchange coupling discussed later.",
"In the case where $J_{12}$ is larger than $\\Gamma $ ($\\equiv \\Gamma _L+\\Gamma _R$ ), the local moments at the QD1 and QD2 form a molecular-type singlet bond.",
"In the opposite case $J_{12} < \\Gamma $ where the QD1 is coupled more strongly to the leads, the spectral weight of the electron at the QD1 is broadened, and the local moment $\\vec{S}_2$ at the QD2 is screened by the electrons with this broadened density of states.",
"In both of these cases, the singlet state is formed mainly between the moments in the QD1 and QD2.",
"Similar considerations also make sense for finite $U_1$ ,[6], [9] just by replacing $\\Gamma $ with the Kondo temperature $T_K^{QD1}$ for the QD1.",
"Namely, the two-stage Kondo effect occurs for $J_{12}<T_K^{QD1}$ , whereas a singlet bond becomes the molecular-type one for $J_{12}>T_K^{QD1}$ .",
"Next, we consider the situation shown in Fig.",
"REF (b), where the energy level at the QD1 is away from the Fermi energy.",
"The Kondo effect in this situation is a main focus of this paper.",
"In this case, the QD1 is almost empty (doubly occupied) for $\\varepsilon _1>0$ ($\\varepsilon _1<0$ ), and the spin degree of freedom disappears at the QD1.",
"Thus, the electrons at the QD1 can not contribute to the screening of the local moment at the QD2, and $\\varepsilon _1$ works as a potential barrier that disturbs charge transfer between the QD2 and the leads.",
"Maruyama et al.",
"[5] and Žitko et al.",
"[9] also considered the $|\\varepsilon _1| \\rightarrow \\infty $ limit.",
"However, the precise features of the screening process due to the superexchange mechanism were not examined in detail.",
"A similar situation to Fig.",
"REF (b) also arises in transition metal oxides such as MnO and CuO, where the superexchange interaction describes the coupling between the local spins in magnetic ions mediated by nonmagnetic oxygen anions.",
"[17], [18] In order to clarify the Kondo screening in the situation shown in Fig.",
"REF (b), we derive the effective Hamiltonian from the perturbation expansion with respect to the tunneling elements $t$ and $V_{s}$ to the fourth order (see the Appendix).",
"The result can be expressed in the form $\\mathcal {H}_{\\rm K}^{\\rm (b)} =J_{\\rm SE} \\sum _{k, k^{\\prime }} \\sum _{\\sigma , \\sigma ^{\\prime }}s^{\\dag }_{k\\sigma } \\frac{\\vec{\\tau }_{\\sigma \\sigma ^{\\prime }}}{2}s^{}_{k^{\\prime }\\sigma ^{\\prime }} \\cdot \\vec{S}_2+ H_{s},$ where $J_{\\rm SE} =2 \\left( \\frac{V_s t}{\\varepsilon _1} \\right)^2\\!\\!\\left(-\\frac{1}{\\varepsilon _2}+\\frac{1}{\\varepsilon _2+U_2}\\right)\\xrightarrow{}\\frac{8}{U_2}\\left( \\frac{V_s t}{\\varepsilon _1} \\right)^2_.$ The coupling constant $J_{\\rm SE}$ depends on the energy level $\\varepsilon _{1}$ , which is caused by a virtual process with a single electron passing through the QD1.",
"This term appears as the $V_s^2 t^2 $ -type contribution in the fourth order perturbation expansion with respect to tunneling elements.",
"The screening of the local moment $\\vec{S}_2$ is achieved for large $\\varepsilon _1$ by the conduction electrons tunneling virtually through the QD1.",
"This screening mechanism is essentially the same as the one due to the superexchange interaction mentioned above, and the singlet bond becomes long compared to that in the case of $\\varepsilon _1 \\simeq 0$ .",
"Therefore, $J_{\\rm SE}$ is referred to as the “superexchange (SE)\" Kondo coupling in the following.",
"Note that a similar screening occurs also for negative $\\varepsilon _{1} (<0)$ , although Fig.",
"REF (b) describes only the situation for positive $\\varepsilon _{1} (>0)$ .",
"For negative large $\\varepsilon _1$ , the QD1 is almost doubly occupied and the effective Hamiltonian takes the same form $\\mathcal {H}_{\\rm K}^{\\rm (b)}$ in Eq.",
"(REF )."
],
[
"Crossover between the IE and SE Kondo screenings",
"To confirm the above discussions more precisely, we calculate the phase shift, the average number of electrons in each of the dots, and the spin susceptibility for the two-impurity Anderson model $\\mathcal {H}_{\\rm TIAM}$ using the numerical renormalization group (NRG).",
"[20] We first show the numerical results of the phase shift due to the DQD, which is helpful to clarify the formation of the singlet state because the phase shift reflects an electron scattering at the DQD.",
"From the phase shift $\\varphi $ , we can also deduce the total number of electrons $N_{DQD} \\equiv \\sum _{\\sigma } \\langle n_{1,\\sigma } + n_{2,\\sigma }\\rangle $ in the DQD, using the Friedel sum rule [21], [22] $N_{DQD} \\,=\\, \\frac{2}{\\pi }\\varphi \\;.$ Figure: (Color online)Total number of electrons in the DQD, N DQD N_{DQD},and each number of the electrons at the dots.We set U 1 =0U_1=0, U 2 /t=6U_2/t=6, ε 2 =-U 2 /2\\varepsilon _2=-U_2/2,and Γ L /t=Γ R /t=0.1\\Gamma _L/t=\\Gamma _R/t=0.1.The inserted figures illustrate a formation ofa dominant singlet bond which is described by the (green) dashed line.Figure REF shows $N_{DQD}$ as a function of $\\varepsilon _1/t$ , and the number of electrons in each of the dots, $\\langle n_{1(2)} \\rangle = \\sum _{\\sigma } \\langle n_{1(2),\\sigma } \\rangle $ .",
"Let us first look at a region around $\\varepsilon _1=0$ .",
"In this region, namely $N_{DQD} \\simeq 2$ , both of the dots are nearly half-filled ($\\langle n_{1(2)} \\rangle \\simeq 1$ ), and the phase shift takes the value of $\\varphi \\simeq \\pi $ from Eq.",
"(REF ).",
"The singlet state is formed dominantly inside the DQD, and thereby the conduction electrons at the leads are not scattered by the local spin at the dots.",
"As $\\varepsilon _1$ increases, $N_{DQD}$ shows a sharp drop around $\\varepsilon _1/t=0.8$ and approaches $N_{DQD} \\simeq 1$ .",
"From Eq.",
"(REF ), we see that the phase shift $\\varphi $ also changes from $\\pi $ to $\\pi /2$ .",
"In the region around $\\varepsilon _1/t=2.0$ , $n_{1}$ goes to zero while $n_{2}$ almost remains unchanged.",
"It indicates that the local spin appears only at the QD2 for large $\\varepsilon _1$ , and the SE Kondo coupling can be described by the Kondo Hamiltonian given in Eq.",
"(REF ).",
"Therefore, the kink behavior of the phase shift with the height $\\pi /2$ , seen in Fig.",
"REF , signifies the crossover between the IE Kondo screening described by the Hamiltonian $\\mathcal {H}_{\\rm K}^{\\rm (a)}$ with $J_{12}$ and the SE one described by $\\mathcal {H}_{\\rm K}^{\\rm (b)}$ with $J_{\\rm SE}$ .",
"In order to discuss the change of the low-energy states for the energy level $\\varepsilon _1$ in more detail, it is helpful to use the fixed-point Hamiltonian in terms of the renormalized parameters [23] $& \\widetilde{H}_{qp}^{(0)} =\\widetilde{\\varepsilon }_{2} n_2+\\varepsilon _{1} n_1+\\widetilde{t}\\, \\sum _{\\sigma }\\left(d_{1\\sigma }^\\dag d_{2\\sigma }^{}+\\textrm {H.c.}\\right)\\nonumber \\\\& \\qquad \\quad +\\sum _{\\nu } (H_{\\nu } + H_{T,\\nu }) ,$ where $\\widetilde{\\varepsilon }_{2} \\equiv Z \\,(\\varepsilon _{2} + \\Sigma _{2}(0)) , \\quad \\widetilde{t} \\equiv \\sqrt{Z} \\, t ,\\nonumber \\\\Z \\equiv \\left(1- \\left.\\!\\frac{\\partial \\Sigma _{2}(\\varepsilon )}{\\partial \\varepsilon }\\right|_{\\varepsilon =0}\\right)^{-1}.$ $\\Sigma _{2}(\\varepsilon )$ is the self energy due to the Coulomb interaction $U_2$ .",
"Using this fixed-point Hamiltonian, a unified analysis for the $\\varepsilon _{1}$ dependence becomes possible.",
"The crossover between the two opposite limits, at $\\varepsilon _1 = 0$ and $\\varepsilon _1 \\rightarrow \\infty $ , can be described as a continuous change of the parameter values of the fixed-point Hamiltonian.",
"We can calculate the renormalized parameters $\\widetilde{\\varepsilon }_{2}$ and $\\widetilde{t}$ using the NRG.",
"[24] The results are shown in Fig.",
"REF (a).",
"Figure: (Color online)(a)Renormalized parameters ε ˜ 2 \\widetilde{\\varepsilon }_{2}, t ˜\\widetilde{t}and(b) conductance at zero temperature as a function of ε 1 /t\\varepsilon _{1}/t.The parameters of the system are the same as in Fig.",
".Inset of (a):Phase boundary between singlet and doublet ground statesfor the isolated DQD system (Γ L/R =0\\Gamma _{L/R}=0),where we set U 1 =0U_1=0 and ε 2 =-U 2 /2\\varepsilon _2=-U_2/2.It is noteworthy that both $\\widetilde{\\varepsilon }_{2}$ and $\\widetilde{t}$ show a sharp decrease around $\\varepsilon _{1}/t \\simeq 0.8$ , which indicates the crossover between the two different singlet bonds, namely the one due to the IE coupling $J_{12}$ and the other due to the SE Kondo coupling $J_{\\rm SE}$ .",
"Furthermore, from these parameters we can deduce the phase shift $\\varphi $ of the DQD and the conductance $G$ at zero temperature: [6] $\\varphi &=&\\frac{\\pi }{2}+{\\rm tan}^{-1}\\left(\\frac{\\widetilde{t}^2-\\widetilde{\\varepsilon }_{2} \\varepsilon _{1}}{\\widetilde{\\varepsilon }_{2}\\Gamma }\\right)\\, ,\\\\G&=&\\frac{2e^2}{h} \\sin ^2 \\varphi =\\frac{2e^2}{h}\\left\\lbrace 1+\\left( \\frac{\\widetilde{t}^2-\\widetilde{\\varepsilon }_{2} \\varepsilon _{1}}{\\widetilde{\\varepsilon }_{2}\\Gamma }\\right)^2\\right\\rbrace ^{-1}_,$ where $\\Gamma \\equiv \\Gamma _L+\\Gamma _R$ .",
"Note that this expression for $G$ is exact at zero temperature for the symmetric coupling $\\Gamma _L=\\Gamma _R$ , and can be obtained, for instance, by using the Meir-Wingreen formula[25] for the Hamiltonian in Eq.",
"(REF ).",
"The phase shift $\\varphi $ and the conductance $G$ can be deduced from the exact NRG results for the renormalized parameters.",
"Figure REF (b) shows the result of the conductance as a function of $\\varepsilon _{1}$ for $\\varepsilon _2=-U_2/2$ .",
"We see that the conductance shows an upturn around $\\varepsilon _{1}/t \\simeq 0.8$ , and at this value the crossover between the ground state due to the IE coupling and that due to the SE Kondo coupling occurs.",
"The behavior of the conductance in Fig.",
"REF (b) can also be explained in terms of the Fano-Kondo effect.",
"This is because the energy level $\\varepsilon _{1}$ at the QD1 varies the asymmetric parameter $q$ for a Fano line shape, as discussed by Maruyama et al.",
"[5] and Žitko.",
"[14] Indeed, the conductance decreases at $\\varepsilon _{1}/t \\simeq 0$ , where $q\\simeq 0$ , due to the destructive interference effect while the conductance approaches $2e^2/h$ in the limit of $\\varepsilon _{1}/t \\rightarrow \\infty $ where $q\\rightarrow \\infty $ .",
"The nature of the crossover can also be related to a level crossing taking place in a molecule limit $\\Gamma _{L/R}=0$ , where the QD1 is decoupled from the lead.",
"In this limit the isolated DQD is described by a Hamiltonian $H_{QD1} + H_{QD2} + H_{\\rm int}$ , and the ground state of the molecule becomes a singlet or doublet, depending on the value of $|\\varepsilon _1|/t$ and $U_2/t$ .",
"As shown in the inset of Fig.",
"REF (a) for $U_{1}=0$ , the ground state is a spin singlet if either $|\\varepsilon _1|/t$ or $U_2/t$ is small.",
"In the opposite case, a spin doublet becomes the ground state.",
"Note that in the doublet region, nearly one electron occupies the QD2, whereas the QD1 is almost empty or doubly occupied.",
"Thus, the local moment emerges mainly at the QD2.",
"We see in the phase diagram in Fig.",
"REF (a) that the transition takes place in this molecule limit at $|\\varepsilon _1|/t \\simeq 0.8$ for $U_2/t=6$ , and it agrees well with the position where $\\widetilde{\\varepsilon }_{2}$ and $\\widetilde{t}$ show a sharp decrease.",
"For finite $\\Gamma _{L/R}$ , the conduction electrons can tunnel from the lead to the QD2 via the QD1.",
"However, the electrons at the QD1 cannot contribute to the screening of the moment at the QD2, because the QD1 is almost empty or doubly occupied, and has no local spin moment.",
"Then, the local spin at the QD2 is screened by the conduction electrons from the leads over the QD1, which is the SE Kondo screening discussed above.",
"Therefore, using the phase diagram in Fig.",
"REF (a), we can estimate the value $|\\varepsilon _1|/t$ and $U_2/t$ , at which a crossover between two distinct singlet states occurs.",
"In order to estimate the Kondo temperature due to the SE process, we calculate the contribution of the QD2 to the impurity susceptibility, defined by $\\chi _{2}=\\frac{(g\\mu _B)^2}{k_B T}\\left( \\langle S_z^2 \\rangle - \\langle S_z^2 \\rangle _0 \\right),$ where $\\langle S_z^2 \\rangle $ ($\\langle S_z^2 \\rangle _0$ ) is the $z$ component of the total spin of the system with (without) the QD2.",
"Figure: (Color online)Plots of k B Tχ 2 /(gμ B ) 2 k_B T \\chi _{2}/(g\\mu _B)^2 vs k B T/Dk_B T/D for several values ofε 1 /t\\varepsilon _{1}/t.We set U 2 /t=6,ε 2 =-U 2 /2U_2/t=6, \\varepsilon _{2}=-U_2/2, Γ L /t=Γ R /t=0.1\\Gamma _L/t=\\Gamma _R/t=0.1,and t/D=10 -3 t/D=10^{-3}.The arrows with dashed lines indicate theKondo temperature T K (J SE )T_K(J_{\\rm SE}) obtained from Eq.",
"()for ε 1 /t=1.2\\varepsilon _1/t=1.2, 1.61.6, and 2.02.0 (see Table ).Figure REF shows the results of $\\chi _{2}$ for several values of $\\varepsilon _1$ .",
"We can estimate the Kondo temperature from the slope of these plots.",
"[20] In particular, for a large value of $\\varepsilon _1$ , we can compare the Kondo temperature estimated from $T\\chi _{2}$ with that obtained from the Kondo Hamiltonian $\\mathcal {H}_{\\rm K}^{\\rm (b)}$ with a formula[20] $T_K(J_{\\rm SE})= D \\sqrt{\\rho J_{\\rm SE}} \\,\\exp (-1/\\rho J_{\\rm SE}),$ where $D$ is the half-bandwidth of the leads and $\\rho =1/2D$ .",
"Table: The Kondo temperatures T K (J SE )T_K(J_{\\rm SE}) from Eq.",
"().The Kondo temperature $T_K(J_{\\rm SE})$ , which is obtained from Eq.",
"(REF ), is listed in Table REF for several values of $\\varepsilon _1/t$ .",
"Furthermore, these values of $T_K(J_{\\rm SE})$ are indicated by the arrows with dashed lines in Fig.",
"REF .",
"We see that $T\\chi _{2}$ obtained from the NRG for $\\varepsilon _1/t=1.2$ , $1.6$ , and $2.0$ decreases rapidly showing a clear crossover to the Kondo regime around the temperature indicated by the arrows.",
"This agreement demonstrates that the Kondo screening for a large value of $\\varepsilon _1$ is mainly owing to the SE one described by the Kondo Hamiltonian $\\mathcal {H}_{\\rm K}^{\\rm (b)}$ ."
],
[
"Coulomb interaction $U_1$ at QD1",
"So far, we have assumed that the QD1 is noninteracting, $U_1=0$ .",
"In this subsection, we discuss the effects of the Coulomb interaction $U_1$ at the QD1 on the energy scale of the Kondo screening."
],
[
"Finite $U_1$ for {{formula:f5a042a3-cad2-487c-b8e9-f1c47ad621b0}}",
"We first introduce the Coulomb interaction $U_1$ for positive $\\varepsilon _1>0$ .",
"Figure: (Color online)Plots of k B Tχ 2 /(gμ B ) 2 k_B T \\chi _{2}/(g\\mu _B)^2 vs k B T/Dk_B T/D for several values ofε 1 /t\\varepsilon _{1}/t and U 1 /tU_1/t, where we setU 2 /t=6,ε 2 =-U 2 /2U_2/t=6, \\varepsilon _{2}=-U_2/2, Γ L /t=Γ R /t=0.1\\Gamma _L/t=\\Gamma _R/t=0.1,and t/D=10 -3 t/D=10^{-3}.Figure REF shows the impurity susceptibility at the QD2, $\\chi _{2}$ , for several values of $\\varepsilon _1$ and $U_1$ .",
"The other parameters are the same as those of Fig.",
"REF .",
"Since we set $\\varepsilon _1$ in a way such that there is almost no local spin moment at the QD1, the SE Kondo screening occurs in these examples.",
"In Fig.",
"REF , we see that the crossover temperature, corresponding to the Kondo energy scale, decreases as $U_1$ increases.",
"This is because the QD1 becomes almost empty for large positive $\\varepsilon _1$ and large Coulomb repulsion $U_1$ , and the virtual electron tunneling from the leads to QD2 is suppressed significantly.",
"Specifically, in the case where $\\varepsilon _1$ is away from the Fermi energy ($\\varepsilon _1>t, \\Gamma $ ), the SE Kondo coupling $J_{\\rm SE}$ between the QD2 and the leads given in Eq.",
"(REF ) can be expressed in a more general form, by taking into account the effect of the Coulomb repulsion $U_1$ , as $J_{\\rm SE}=2 \\!\\left( \\frac{V_s t}{\\varepsilon _1} \\right)^{\\!",
"2}\\!\\!\\left\\lbrace \\frac{\\varepsilon _1^2 U_1- (\\varepsilon _2-2\\varepsilon _1)(\\varepsilon _1-\\varepsilon _2)^2 }{\\varepsilon _2 (\\varepsilon _2-2\\varepsilon _1-U_1)(\\varepsilon _1-\\varepsilon _2)^2}+\\frac{1}{\\varepsilon _2+U_2}\\right\\rbrace _.\\nonumber \\\\$ This $J_{\\rm SE}$ monotonically decreases with increasing $U_1$ , so that the Kondo temperature $T_K(J_{\\rm SE})$ defined by Eq.",
"(REF ) also becomes small as $U_1$ increases.",
"We can see the corresponding shift of the crossover temperature in Fig.",
"REF for the two-impurity Anderson model $\\mathcal {H}_{\\rm TIAM}$ ."
],
[
"\nCrossover between the two-stage and single-stage Kondo screenings\n",
"Next we consider another case: the crossover between the two-stage Kondo screening and a single-stage Kondo screening.",
"It has been discussed previously that the two-stage Kondo effect can occur in the case where each of the two dots has a local moment and the Kondo temperature $T_{K}^{QD1}$ for the QD1 is larger than the exchange coupling $J_{12}$ between the dots.",
"[6], [9] It takes place typically near the electron-hole symmetric point, where $\\varepsilon _{1(2)} + U_{1(2)}/2 \\simeq 0$ .",
"Therefore, as the energy level of the QD1 moves away from the electron-hole symmetric point $\\varepsilon _1 \\simeq -U_1/2$ , the SE Kondo screening can arise because the local spin moment at the QD1 disappears.",
"Furthermore, the two-stage Kondo screening near the symmetric point changes to the single-stage Kondo screening.",
"This was also discussed partly by Žitko et al.",
"They showed that the Kondo temperature rapidly drops as $\\varepsilon _1$ moves away from the electron-hole symmetric point.",
"[9] However, how the singlet ground state evolves in the crossover region has not been clarified in detail.",
"In order to confirm the precise features of the Kondo screening, we calculate the susceptibility for the two dots as well as that for the QD2, $\\chi _{\\rm DQD}=\\frac{(g\\mu _B)^2}{k_B T}\\left( \\langle S_z^2 \\rangle - \\langle S_z^2 \\rangle _{\\rm lead} \\right).$ Here, $\\langle S_z^2 \\rangle $ is the $z$ component of the total spin of the whole system including the DQD, and $\\langle S_z^2 \\rangle _{\\rm lead}$ is the same quantity without the DQD.",
"Furthermore, we also calculate the entropy of the DQD defined by $S_{\\rm DQD}=\\frac{1}{k_B T}\\Bigl \\lbrace (E-F) - (E_{\\rm lead} -F_{\\rm lead}) \\Bigr \\rbrace .$ Here, $E={\\rm Tr}[H e^{-H/(k_BT)}]/{\\rm Tr}[e^{-H/(k_BT)}]$ and $F=-k_BT\\,{\\rm ln}{\\rm Tr}[e^{-H/(k_BT)}]$ are the internal energy and free energy of the whole system consisting of the DQD and the leads, while $E_{\\rm lead}$ and $F_{\\rm lead}$ are those for the unconnected leads.",
"Figure: (Color online)Spin susceptibility (a) k B Tχ DQD /(gμ B ) 2 k_B T \\chi _{\\rm DQD}/(g\\mu _B)^2,(b) k B Tχ 2 /(gμ B ) 2 k_B T \\chi _{\\rm 2}/(g\\mu _B)^2, and(c) entropy S DQD /k B S_{\\rm DQD}/k_B as a function of k B T/Dk_B T/Dfor several values of ε 1 /Γ\\varepsilon _1/\\Gamma .We set t/Γ=0.3t/\\Gamma =0.3, U 1 /Γ=U 2 /Γ=6U_1/\\Gamma =U_2/\\Gamma =6,ε 2 /U 2 =-0.5\\varepsilon _{2}/U_2=-0.5, and Γ/D=10 -3 \\Gamma /D=10^{-3}.Figure: (Color online)The results presented in Fig.",
"are replotted over a wide temperature range.These results confirm that the moment is eventuallyscreened in the cases of ε 1 /Γ=1\\varepsilon _1/\\Gamma =1 and 2although the screening temperature becomes exponentially small.Therefore, to observe the screening process due to the SE Kondo couplingJ SE J_{\\rm SE} defined by Eq.",
"()in a realistic temperature, ε 1 \\varepsilon _1 should be not so large.Figure REF (a) shows the spin susceptibility $\\chi _{\\rm DQD}$ for several values of $\\varepsilon _1/\\Gamma $ .",
"For $\\varepsilon _1/\\Gamma =-3$ , which corresponds to the electron-hole symmetric point $\\varepsilon _1+U_1/2=0$ , $T \\chi _{\\rm DQD}$ shows a peak around $T/D \\sim 10^{-3}$ .",
"This peak indicates that each of the dots is occupied by a single electron, and the local moment is well developed.",
"In this case, a two-stage screening occurs as temperature decreases: the first stage can be seen at $T/D \\sim 10^{-4}$ , and the second one at $T/D \\sim 10^{-7}$ .",
"These two energy scales correspond to the Kondo temperature for the first stage $T_K^{\\rm 1st}$ and that for the second stage $T_K^{\\rm 2nd}$ , respectively.",
"If the Coulomb interaction at the dots is much larger than the tunneling constants $\\Gamma $ and $t$ , then the peak of $T \\chi _{\\rm DQD}$ approaches $0.5$ in units of $(g\\mu _\\mathrm {B})^2$ and the structure that emerges at $T_K^{\\rm 1st}$ becomes clear.",
"For $\\varepsilon _1/\\Gamma =-2$ and $-1$ , which are still negative but closer to the Fermi energy, the two-stage Kondo effect can be seen more clearly.",
"The Kondo temperature $T_K^{\\rm 2nd}$ for the second stage becomes lower: $T_K^{\\rm 2nd}/D \\sim 10^{-8}$ for $\\varepsilon _1/\\Gamma =-2$ and $T_K^{\\rm 2nd}/D \\sim 10^{-11}$ for $\\varepsilon _1/\\Gamma =-1$ .",
"In Fig.",
"REF (c), we see that the two-stage behavior can be observed more sharply in the temperature dependence of the entropy than that of the susceptibility.",
"As $\\varepsilon _1$ crosses the Fermi energy and takes a positive value, the peak of $T \\chi _{\\rm DQD}$ seen in Fig.",
"REF (a) at $T/D \\sim 10^{-3}$ is suppressed.",
"This indicates that the local moment at the QD1 disappears as the QD1 becomes almost empty.",
"Then, the local moment at the QD2 is screened by the SE process via a single stage as the temperature approaches zero.",
"We see in Fig.",
"REF (b) that $T \\chi _{2}$ is almost constant (0.25) in a wide temperature range below $T/D \\sim 10^{-3}$ .",
"This indicates that the local spin moment at the QD2 remains almost free in this temperature region.",
"Furthermore, the entropy for the DQD shown in Fig.",
"REF (c) is locked at the value of ${\\rm ln}2$ below $T/D \\sim 10^{-4}$ , which is caused by the unscreened spin moment at the QD2.",
"This free moment must be screened eventually at low temperatures by the SE Kondo coupling $J_{\\rm SE}$ although the Coulomb interaction $U_1$ makes the Kondo temperature very small.",
"In order to see the low temperature region, $\\chi _{\\rm DQD}$ , $\\chi _{2}$ , and $S_{\\rm DQD}$ are shown in a wide temperature range in Fig.",
"REF .",
"We see that $T\\chi _{\\rm DQD}$ for $\\varepsilon _1/\\Gamma =1$ and $\\varepsilon _1/\\Gamma =2$ shows the decrease at $T/D \\sim 10^{-45}$ and $T/D \\sim 10^{-90}$ , respectively.",
"These results confirm clearly that the local moment at the QD2 is really screened although the energy scale for the SE Kondo screening becomes small for large $U_1$ and $\\varepsilon _1$ ."
],
[
"Summary and discussion",
"We have studied the Kondo effect in a side-coupled DQD system with focus on how the Kondo singlet state changes by varying the energy level at the embedded dot (QD1).",
"We have found that when the side dot (QD2) is in the Kondo regime, two distinct singlet states appear; one is due to the IE coupling between the QD1 and the QD2, and the other is caused by the SE Kondo coupling between the QD2 and the leads via the QD1.",
"In this sense, the latter is a different type of the singlet state from the former which has been studied in the side-coupled DQD systems so far.",
"In order to clarify the screening process, we have obtained the effective Kondo Hamiltonians using the perturbation expansion with respect to the tunneling matrix elements.",
"From these Kondo Hamiltonians, we have shown that in the case where the QD1 is almost empty and doubly occupied the screening is caused by a superexchange mechanism, and the singlet bond becomes long.",
"Moreover, we have calculated the phase shift and the conductance using the NRG method, and have obtained the relation between the phase shift and the conductance.",
"We have found that the conductance is enhanced at the crossover region between the singlet ground state due to the IE coupling and that due to the SE Kondo coupling.",
"We have also calculated the local spin susceptibility and have estimated the Kondo temperature, which shows good agreement with that obtained from the effective Kondo Hamiltonian.",
"Furthermore, we have demonstrated precisely how the two-stage Kondo screening changes to a single-stage process as the energy level at the QD1 moves away from the electron-hole symmetric point.",
"In closing, we would like to make some comments on the SE Kondo screening.",
"The scenario of the SE Kondo screening is not limited to our side-coupled DQD system, but is more generic for the Anderson model where the impurity spin and the conduction electrons are connected via the discrete energy level.",
"In usual cases of the Kondo problem, a magnetic impurity is coupled directly to the conduction electrons, for instance, a bulk system with a magnetic impurity.",
"On the other hand, in the two-impurity Anderson model we have considered, a local spin moment is coupled indirectly to the conduction electrons via a discrete energy level.",
"The resulting screening process shows a unique feature, which has clearly been demonstrated in this paper with the help of the effective Kondo Hamiltonians and the NRG method.",
"The SE Kondo screening discussed in this paper also appears in a side-coupled DQD system coupled to normal and superconducting leads, which we have previously studied.",
"[26] In this case, the superconducting proximity to the embedded dot quenches the local spin moment because this proximity tends to make a singlet consisting of a linear combination of the empty and doubly occupied states.",
"Thus, the superconducting proximity to the embedded dot plays the role of a potential barrier between the side dot and the normal lead, which can cause the SE Kondo screening.",
"Recently, the side-coupled DQD system has been fabricated in experiments.",
"[16] We thus expect that in the near future, it may become possible to observe the SE Kondo screening discussed in this paper, providing further interesting examples of correlation effects in the context of electron transport in nanoscale systems.",
"Y.T.",
"was supported by the Special Postdoctoral Researchers Program of RIKEN.",
"N.K.",
"is supported by JSPS FIRST-Program, the Grant-in-Aid for Scientific Research [Grant Nos.",
"21540359 and 20102008], and the Global COE Program “The Next Generation of Physics, Spun from Universality and Emergence\" from MEXT of Japan.",
"A.O.",
"is supported by JSPS Grant-in-Aid for Scientific Research (C) (Grant No.",
"23540375)."
],
[
"Appendix: Derivation of the effective Kondo Hamiltonian",
"We outline the derivation of the effective Hamiltonian for the SE Kondo coupling given in Eq.",
"(REF ) by using the perturbation theory in the tunneling matrix elements.",
"[27] The unperturbed ground state is chosen to be the one for $t=V_{s}=0$ and $-\\varepsilon _2,\\, \\varepsilon _2+U_2,\\,\\varepsilon _1 \\gg E_F\\,(\\equiv 0)$ .",
"In this situation, the ground state of the two-impurity Anderson model defined in Eq.",
"(REF ) is described by the singly occupied state at the QD2 and the empty one at the QD1.",
"We thus choose the unperturbed Hamiltonian to be $H_0 = H_{QD1} + H_{QD2} + H_{s}.$ The unperturbed ground state is described by $d_{2\\sigma }^\\dag |F\\rangle $ , where $|F\\rangle $ is the Fermi sea of the conduction band.",
"The tunneling terms $H_{T,s}$ and $H_{\\rm int}$ are taken to be as the perturbation Hamiltonian, as illustrated in Fig.",
"REF .",
"Figure: (Color online)Schematic energy diagram of a side-coupled DQDfor ε 1 >0\\varepsilon _{1} >0.",
"This situationis the same as that for Fig.",
"(b).In this figure, we describe the virtual tunneling eventsby the (blue) dashed arrows, which are labeled bya,b,ca, b, c and dd.The bold arrow at the QD2 indicates the local spin.The superexchange mechanism via the energy level $\\varepsilon _1$ at the QD1 is described by the fourth order perturbation with respect to the tunneling matrix elements $t$ and $V_{s}$ .",
"In the fourth order perturbation, there are six processes.",
"Using the label for virtual electron tunnelings shown in Fig.",
"REF , these six processes are given by $({\\rm i}) & \\quad a \\rightarrow b \\rightarrow c \\rightarrow d &\\qquad ({\\rm iv}) & \\quad c \\rightarrow d \\rightarrow a \\rightarrow b \\nonumber \\\\({\\rm ii}) & \\quad a \\rightarrow c \\rightarrow b \\rightarrow d&\\qquad ({\\rm v}) & \\quad c \\rightarrow a \\rightarrow b \\rightarrow d \\nonumber \\\\({\\rm iii}) & \\quad a \\rightarrow c \\rightarrow d \\rightarrow b &\\qquad ({\\rm vi}) & \\quad c \\rightarrow a \\rightarrow d \\rightarrow b .",
"\\nonumber $ The virtual electron tunnelings described by the labels $a$ and $d$ are caused by $H_{\\rm int}$ , whereas those described by labels $b$ and $c$ are caused by $H_{T,s}$ .",
"Therefore, the fourth order perturbation takes the form $H^{\\prime }_{1} \\frac{1}{E-H_0} H^{\\prime }_{2} \\frac{1}{E-H_0}H^{\\prime }_{3} \\frac{1}{E-H_0} H^{\\prime }_{4},$ where $H^{\\prime }_{j}$ for $j \\in \\lbrace 1,2,3,4 \\rbrace $ depends on the virtual processes, namely $H^{\\prime }_j=H_{\\rm int}$ for the virtual electron tunnelings labeled $a$ and $d$ , whereas $H^{\\prime }_j=H_{T,s}$ for those labeled $b$ and $c$ .",
"As a representation of Eq.",
"(REF ) in the subspace of $d_{2\\sigma }^\\dag |F\\rangle $ , we obtain the effective Hamiltonian in the form $H_{\\rm eff}^{\\lambda }=W^{\\lambda } \\sum _{k,k^{\\prime }}\\sum _{\\sigma } s_{k\\sigma }^\\dag s_{k^{\\prime }\\sigma }^{}+J^{\\lambda } \\sum _{k,k^{\\prime }}\\sum _{\\sigma ,\\sigma ^{\\prime }}s^{\\dag }_{k\\sigma } \\frac{\\vec{\\tau }_{\\sigma \\sigma ^{\\prime }}}{2} s^{}_{k^{\\prime }\\sigma ^{\\prime }} \\cdot \\vec{S}_2,\\nonumber \\\\$ where $\\lambda $ runs over $({\\rm i})$ to $({\\rm vi})$ .",
"The couplings $W^{\\lambda }$ and $J^{\\lambda }$ are given by $&&W^{({\\rm i})}\\!=\\!",
"-\\frac{(V_s t)^2}{2\\varepsilon _2 (\\varepsilon _2-\\varepsilon _1)^2}, \\quad J^{({\\rm i})}\\!=\\!4W^{({\\rm i})},\\\\&&W^{({\\rm ii})}\\!=\\!",
"\\frac{(V_s t)^2}{2(\\varepsilon _2-2\\varepsilon _1)(\\varepsilon _2-\\varepsilon _1)^2},\\,\\, J^{({\\rm ii})}\\!=\\!-4W^{({\\rm ii})},\\\\&&W^{({\\rm iii})} \\!=\\!",
"-\\frac{(V_s t)^2}{2\\varepsilon _1 (\\varepsilon _2-2\\varepsilon _1)(\\varepsilon _2-\\varepsilon _1)},\\,\\, J^{({\\rm iii})}\\!=\\!-4W^{({\\rm iii})},\\nonumber \\\\&& \\\\&&W^{({\\rm iv})}\\!=\\!",
"-\\frac{(V_s t)^2}{2(\\varepsilon _1)^2 (\\varepsilon _2+U_2)}, \\quad J^{({\\rm iv})}\\!=\\!-4W^{({\\rm iv})},\\\\&&W^{({\\rm v})}\\!=\\!",
"W^{({\\rm iii})},\\quad J^{({\\rm v})}\\!=\\!J^{({\\rm iii})},\\\\&&W^{({\\rm vi})}\\!=\\!",
"\\frac{(V_s t)^2}{2(\\varepsilon _2-2\\varepsilon _1)(\\varepsilon _1)^2},\\quad J^{({\\rm vi})}\\!=\\!-4W^{({\\rm vi})}.$ Summing up these six processes, we obtain the effective Hamiltonian for the SE Kondo coupling, corresponding to $\\mathcal {H}_{\\rm K}^{\\rm (b)}$ given in Eq.",
"(REF ).",
"For the finite Coulomb interaction $U_1$ at the QD1, the processes of (ii), (iii), (v), and (vi) depend on $U_1$ since the QD1 is doubly occupied in the intermediate state.",
"Thus for finite $U_1$ , $W^{\\lambda }$ takes the form $W^{({\\rm ii})}\\!\\!&=&\\!\\!",
"\\frac{(V_s t)^2}{2(\\varepsilon _2-2\\varepsilon _1-U_1)(\\varepsilon _2-\\varepsilon _1)^2},\\\\W^{({\\rm iii})}\\!\\!&=&\\!\\!",
"-\\frac{(V_s t)^2}{2\\varepsilon _1(\\varepsilon _2-2\\varepsilon _1-U_1)(\\varepsilon _2-\\varepsilon _1)},\\\\W^{({\\rm v})}\\!\\!&=&\\!\\!",
"W^{({\\rm iii})},\\\\W^{({\\rm vi})}\\!\\!&=&\\!\\!",
"\\frac{(V_s t)^2}{2(\\varepsilon _2-2\\varepsilon _1-U_1)(\\varepsilon _1)^2}.$ Correspondingly, $J^{\\lambda }$ for $\\lambda =({\\rm ii})$ , (iii), (v), and (vi) is given by $J^{\\lambda }=-4W^{\\lambda }$ , and thus we can obtain the exchange coupling $J_{\\rm SE}$ given in Eq.",
"(REF )."
]
] | 1204.0970 |
[
[
"Supersymmetric \\Delta A_{CP}"
],
[
"Abstract There is experimental evidence for a direct CP asymmetry in singly Cabibbo suppressed D decays, \\Delta A_{CP} \\sim 0.006.",
"Naive expectations are that the Standard Model contribution to \\Delta A_{CP} is an order of magnitude smaller.",
"We explore the possibility that a major part of the symmetry comes from supersymmetric contributions.",
"The leading candidates are models where the flavor structure of the trilinear scalar couplings is related to the structure of the Yukawa couplings via approximate flavor symmetries, particularly U(1), [U(1)]^2 and U(2).",
"The recent hints for a lightest neutral Higgs boson with mass around 125 GeV support the requisite order one trilinear terms.",
"The typical value of the supersymmetric contribution to the asymmetry is \\Delta A_{CP}^{SUSY}\\sim 0.001, but it could be accidentally enhanced by order one coefficients."
],
[
"Introduction",
"The world average for the direct CP asymmetry in singly Cabibbo suppressed $D$ decays, based on measurements by E687 [1], CLEO [2], [3], E791 [4], FOCUS [5], BaBar [6], Belle [7], CDF [8], [9] and LHCb [10], is now $4.3\\sigma $ away from zero [11]: $\\Delta A_{CP}&\\equiv &A_{CP}(K^+K^-)-A_{CP}(\\pi ^+\\pi ^-)\\numero \\\\&=&-0.00656\\pm 0.00154.$ Here, $A_{CP}(f)=\\frac{\\Gamma (D^0\\rightarrow f)-\\Gamma (\\overline{D}^0\\rightarrow f)}{\\Gamma (D^0\\rightarrow f)+\\Gamma (\\overline{D}^0\\rightarrow f)}.$ In $\\Delta A_{CP}$ , that is the difference between asymmetries, effects of indirect CP violation largely cancel out [12].",
"Thus, $\\Delta A_{CP}$ is a manifestation of CP violation in decay.",
"The Standard Model (SM) contribution to the individual asymmetries is suppressed by a CKM factor of order $2{\\cal I}m\\left(\\frac{V_{ub}V_{cb}^*}{V_{us}V_{cs}^*}\\right)\\approx 1.2\\times 10^{-3}$ , and by a loop factor of order $\\alpha _s(m_c)/\\pi \\sim 0.1$ .",
"While one cannot exclude an enhancement factor of order 30 from hadronic physics [13], [14], [15], [16], [17], [19], [20], [18], [21], in which case (REF ) will be accounted for by SM physics, it is interesting to explore the possibility that new physics contributes a major part of $\\Delta A_{CP}$ .",
"The size of new physics contributions to $\\Delta A_{CP}$ is often constrained by other flavor-related observables, such as $D^0-\\overline{D}^0$ mixing or $\\epsilon ^\\prime /\\epsilon $ [22].",
"Supersymmetric models, via their contribution to the chromomagnetic operator, can generate large enough asymmetry in $D$ decays without conflicting with these observables [12], [23], [24].",
"In this work, we investigate whether this scenario is likely to be realized in supersymmetric models with viable and natural flavor structure."
],
[
"The supersymmetric parameters",
"The $6\\times 6$ mass-squared matrix for the up- and down-type squarks can be decomposed into $3\\times 3$ blocks, $q=u,d$ , $\\tilde{M}^{2q}=\\left(\\begin{array} {cc}\\tilde{M}^{2q}_{LL} & \\tilde{M}^{2q}_{LR} \\\\\\tilde{M}^{2q}_{RL} & \\tilde{M}^{2q}_{RR} \\end{array}\\right) +D,F\\mbox{-terms}, $ where $L$ and $R$ denote SU(2) doublets and singlets, respectively.",
"We denote the average squark mass by $\\tilde{m}$ .",
"Then, it is convenient to parameterize the supersymmetric contributions to flavor changing processes in terms of dimensionless parameters, $(\\delta ^q_{MN})_{ij}=\\frac{(\\tilde{M}^{2q}_{MN})_{ij}}{\\tilde{m}^2},$ where $M,N=L,R$ .",
"When, to a good approximation, only two squark generations are involved, one can express these parameters in terms of the supersymmetric mixing angles, $(K^q_M)_{ij}$ , and the mass-squared splittings between squarks, $\\Delta \\tilde{m}^2_{ij}$ : $(\\delta ^q_{MN})_{ij}=\\frac{\\Delta \\tilde{m}^2_{q_{Mi}q_{Nj}}}{\\tilde{m}^2}(K^q_M)_{ij}(K^q_N)_{jj}.",
"$ The parameters that are most relevant to $\\Delta A_{CP}$ are $\\delta _{LL}\\equiv (\\delta ^u_{LL})_{12}$ and $\\delta _{LR}\\equiv (\\delta ^u_{LR})_{12}$ , which generate the chromomagnetic operator with Wilson coefficient given by $C_{8g}=F(x)\\delta _{LL}+G(x)\\frac{m_{\\tilde{g}}}{m_c}\\delta _{LR},$ where $x=(m_{\\tilde{g}}^2/\\tilde{m}^2)$ , and the functions $F$ and $G$ can be found, for example, in Ref.",
"[12].",
"Given that $G(x)$ is larger than $F(x)$ by a factor of a few, and the enhancement factor of $m_{\\tilde{g}}/m_c$ , the dominant contribution in the models that we consider comes from $\\delta _{LR}$ .",
"It can be estimated as follows [23]: $\\Delta A_{CP}^{\\rm SUSY} \\sim 0.006\\ \\frac{{\\cal I}m(\\delta _{LR})}{0.001}\\ \\frac{1\\ {\\rm TeV}}{\\tilde{m}}.$ In the following sections, we investigate whether ${\\cal I}m(\\delta _{LR})\\sim 0.001$ can plausibly arise in supersymmetric flavor models."
],
[
"Supersymmetric flavor models",
"If the soft supersymmetry breaking terms had a generic flavor structure (“anarchy\"), the supersymmetric contributions to flavor changing neutral current processes would exceed experimental constraints by orders of magnitude.",
"Thus, these terms must have a special structure.",
"The most extreme solution to this “supersymmetric flavor puzzle\" is a constrained version of minimal flavor violation (MFV): at the supersymmetry breaking mediation scale, squark masses are universal and the trilinear scalar couplings (the $A$ terms) are proportional to the corresponding Yukawa matrices.",
"Such a situation arises naturally in various mediation schemes, most notably gauge- and anomaly-mediated supersymmetry breaking.",
"The renormalization group evolution does generate flavor changing effects in the soft breaking terms, but MFV implies a very strong flavor suppression of these effects, $\\delta _{LR} \\propto \\frac{m_c}{\\tilde{m}}(V_{us} V_{cs}^* y_s^2 +V_{ub} V_{cb}^* y_b^2 ) \\lesssim {\\cal {O}}(10^{-7}).$ (For the exact expression in anomaly mediation, see [25].)",
"It is possible, however, that the supersymmetric flavor structure is related to that of the Standard Model, but is not MFV.",
"This is the case in models where an approximate flavor symmetry dictates the structure of all flavor changing couplings.",
"In what follows, we examine several such symmetries – $U(1)$ , $[U(1)]^2$ , $U(2)$ and $[U(2)]^3$ – with regard to their implications for ${\\cal I}m(\\delta _{LR})$ ."
],
[
"Abelian Symmetries",
"The Froggatt-Nielsen (FN) framework [26] postulates an approximate $U(1)$ symmetry, broken by a spurion with value $\\ll 1$ .",
"Assigning different charges to the different quark generations results in parameterically suppressed quark mass ratios and mixing angles.",
"In supersymmetric FN models [27], [28], the squark spectrum is anarchical, up to some level of degeneracy between the first two generations from renormalization group evolution (RGE) effects, but the mixing angles are small.",
"With a single $U(1)$ symmetry, the parametric suppression of squark flavor parameters is related to quark flavor parameters, independent of details of the model such as the size of the spurion and the charge assignments.",
"In particular, the following relations hold for the entries that are relevant to $c\\rightarrow u$ transitions [29]: $(\\delta ^u_{LL})_{12}&\\sim &\\frac{|V_{us}|}{r_3},\\\\(\\delta ^u_{RR})_{12}&\\sim &\\frac{m_u}{r_3 m_c|V_{us}|},\\\\(\\delta ^u_{LR})_{12}&\\sim &\\frac{\\tilde{a}}{\\tilde{m}}\\frac{m_c|V_{us}|}{\\tilde{m}},$ where $\\tilde{a}$ is the typical scale of the $A$ -terms.",
"The $1/r_3$ factor, defined in Ref.",
"[30], represents the gluino-related RGE effect which generates some level of degeneracy between the first two squark generations, $\\Delta \\tilde{m}^2_{12}/\\tilde{m}^2\\sim 1/r_3$ .",
"Throughout this paper we assume unsuppressed CP phases and $x \\approx 1$ .",
"The supersymmetric contribution to $\\epsilon _K$ that is proportional to $(\\delta ^d_{LL})_{12}(\\delta ^d_{RR})_{12}\\sim m_d/(r_3^2 m_s)$ is too large, unless $r_3\\mathrel {\\unknown.",
"{\\hspace{1.0pt}\\sim }}$ >$$ 440(TeV/m)$ (or, equivalently, in the language used by CMSSMpractitioners, $ m1/2/m0$\\sim $ $>$ 7$).",
"Assuming that this is indeed thecase, the model is viable, and provides\\begin{equation}{\\cal I}m(\\delta _{LR})\\sim 1.5\\times 10^{-4}\\ \\frac{\\tilde{a}}{\\tilde{m}}\\ \\frac{1\\ {\\rm TeV}}{\\tilde{m}}.\\end{equation}Comparing Eq.",
"(\\ref {eq:avail}) to Eq.",
"(\\ref {eq:desired}), we learnthat for supersymmetry to account for $ ACP$, the ratio$ (a/m)$ should be large.",
"Taking $ a0,m0$ and $ m1/2$to stand for the Planck scale values of the $ A$-terms, squark massesand gluino mass, respectively, and using the approximations of Ref.\\cite {Hiller:2008sv}, we obtain\\begin{equation}\\frac{\\tilde{a}}{\\tilde{m}}\\sim \\frac{3a_0}{m_0\\sqrt{1+8(m_{1/2}/m_0)^2}}\\rightarrow \\left\\lbrace \\begin{array}{cc}a_0/m_{1/2} & (m_{1/2}\\gg m_0),\\\\3a_0/m_{0}& (m_{1/2}\\ll m_0).\\end{array}\\right.\\end{equation}Given that the $ U(1)$ models are only viable if $ m1/2$\\sim $ $>$ 7m0$,the optimal enhancement occurs for $ a0>m1/2 m0$ which might,however, lead to negative squark masses-squared.$ The single $U(1)$ models lead to the following simple parametric relation between the up and down sectors: $\\frac{(\\delta ^u_{LR})_{12}}{(\\delta ^d_{LR})_{12}}\\sim \\frac{m_c}{m_s}.$ The $(\\delta ^d_{LR})_{12}$ parameter is constrained, however, by $\\epsilon ^\\prime /\\epsilon $ : $(\\delta ^d_{LR})_{12}\\mathrel {\\unknown.",
"{\\hspace{1.0pt}\\sim }}$ <$$ 410-5 ( m/TeV)$ (see \\cite {Hiller:2008sv,Isidori:2010kg} and referencestherein).",
"We thus obtain an upper bound that is independent of theflavor-diagonal scales,\\begin{equation}(\\delta ^u_{LR})_{12}\\mathrel {\\unknown.",
"{\\hspace{1.0pt}\\sim }}<\\end{equation}$ 510-4 mTeV.",
"Moreover, the approximate symmetry relates $(\\delta ^u_{LR})_{12}$ to flavor diagonal parameters, $\\frac{{\\cal I}m(\\delta ^u_{LR})_{12}}{{\\cal I}m(\\delta ^q_{LR})_{11}}\\sim \\frac{m_c|V_{us}|}{m_q},\\ \\ \\ (q=u,d).$ Assuming phases of order one (which we must do to explain $\\Delta A_{CP}$ ), these flavor diagonal parameters are bounded by electric dipole moment (EDM) constraints (see [30], [31] and references therein), $(\\delta ^u_{LR})_{11}\\mathrel {\\unknown.",
"{\\hspace{1.0pt}\\sim }}$ <$$ 310-6(m/TeV)$ and$ (dLR)11$\\sim $ $<$ 210-6(m/TeV)$.",
"Theresulting bounds are\\begin{eqnarray}(\\delta ^u_{LR})_{12}\\mathrel {\\unknown.",
"{\\hspace{1.0pt}\\sim }}<\\end{eqnarray}$ 310-4 mTeV (from (uLR)11), (uLR)12$\\sim $ $<$ 810-5 mTeV (from (dLR)11).",
"We conclude that FN models with a single $U(1)$ are unlikely to account for $\\Delta A_{CP}\\gg 0.001$ .",
"Of course, since the FN mechanism only dictates the parametric suppression, it is impossible to exclude an accidental enhancement of $(\\delta ^u_{LR})_{12}$ by the order-one coefficient.",
"Models with an FN symmetry $[U(1)]^2$ allow one to take advantage of the holomorphicity of the superpotential to obtain vanishing entries in the Yukawa and $A$ matrices and to strongly suppress entries in the squark mass-squared matrices (compared to the single $U(1)$ case).",
"This feature was first employed in Refs.",
"[27], [28] to align in a very precise way the squark and quark mass matrices in the down sector.",
"The flavor structure of the Yukawa and $A$ terms in these models can be written as follows: $Y^d&\\sim & \\frac{\\tilde{M}^{2d}_{LR}}{\\tilde{a} v_d}\\sim \\left(\\begin{array}{ccc}y_d & 0 & y_b|V_{ub}| \\\\ 0 & y_s & y_b|V_{cb}| \\\\ 0 & 0 & y_b\\end{array}\\right),\\\\Y^u&\\sim & \\frac{\\tilde{M}^{2u}_{LR}}{\\tilde{a} v_u}\\sim \\left(\\begin{array}{ccc}y_u & y_c|V_{us}| & y_t|V_{ub}| \\\\ Y^u_{21} & y_c & y_t|V_{cb}| \\\\ Y^u_{31} & y_c/|V_{cb}| & y_t\\end{array}\\right).$ The four holomorphic zeros in the down sector are essential to obtain an effective alignment [29].",
"The $(21)$ and $(31)$ entries in the up sector either (i) get their naive parametric suppression (of order $y_u/|V_{us}|$ and $y_u/|V_{ub}|$ , respectively) or (ii) vanish.",
"In both cases, the contribution to $\\epsilon _K$ from $(\\delta ^d_{LL})_{12}(\\delta ^d_{RR})_{12}$ for unsuppressed phases is too large, unless RGE generates degeneracy, $r_3\\mathrel {\\unknown.",
"{\\hspace{1.0pt}\\sim }}$ >$$ 18(TeV/m)$.",
"(For moderate suppression of phases, $ D0-D0$ mixing provides the strongest constraint, requiring a milder degeneracy \\cite {Gedalia:2012pi}:In case (i),the estimates (\\ref {eq:LL}) and (\\ref {eq:RR}) hold, and the contribution from $ (uLL)12(uRR)12$ is too large, unless RGE generates degeneracy, $ r3$\\sim $ $>$ 7(TeV/m)$ \\cite {Hiller:2008sv}.",
"In case (ii),$ (uRR)12$ is suppressed compared to Eq.",
"(\\ref {eq:RR}),and the contribution from $ [(uLL)12]2$ only requires avery mild degeneracy.",
")$ In either case, the parametric suppression of $(\\delta ^u_{LR})_{12}$ is as in Eq.",
"(), and the numerical estimate is as in Eq.",
"().",
"The parametric relation between the up and down sectors of Eq.",
"(REF ) does not hold in the $[U(1)]^2$ models since $(\\delta ^d_{LR})_{12}$ is further suppressed, and so the constraint of Eq.",
"() does not hold.",
"The constraints of Eqs.",
"() and (REF ) hold, leading to the same conclusion as in the single $U(1)$ case: The parametric suppression is such that the contribution to $\\Delta A_{CP}$ falls an order of magnitude short compared to the benchmark value of Eq.",
"(REF ).",
"However, one cannot exclude the possibility that: (i) the large hadronic uncertainties in the EDM calculation are such that the bounds are weaker by an order of magnitude; or (ii) the order one uncertainty from the unknown coefficients provides further enhancement; or a combination of the above."
],
[
"Non-Abelian Symmetries",
"In $U(2)$ models, the first two generations are in a doublet and the third generation in a singlet of the $U(2)$ symmetry [33], [34], [35], [36].",
"With a two-stage symmetry breaking, the structure of the Yukawa, $A$ and $\\tilde{M}^2$ matrices is as follows (see, for example, [36]): $Y^q&\\sim &\\frac{\\tilde{M}^{2q}_{LR}}{\\tilde{a} v_q}\\sim \\left(\\begin{array}{ccc}0 & \\epsilon _1 & 0 \\\\ -\\epsilon _1 & \\epsilon _2 & \\epsilon _2 \\\\0 & \\epsilon _2 & 1\\end{array}\\right),\\numero \\\\\\tilde{M}^{2q}_{NN}&\\sim &\\left(\\begin{array}{ccc}m_1^2 & 0 & 0 \\\\ 0 & m_1^2(1+\\epsilon _2^2) & \\epsilon _2 m_4^{2*} \\\\0 & \\epsilon _2 m_4^2 & m_3^2\\end{array}\\right).$ As in the Abelian case, all non-vanishing entries have unknown coefficients of order one, but $Y^q_{12}=-Y^q_{21}$ and there are relations for the $\\tilde{M}^2_{NN}$ matrices that follow from hermiticity.",
"As concerns the $\\delta _{LR}^q$ parameters, their parametric suppression is similar to the $U(1)$ model.",
"Hence, Eqs.",
"(), (), () and (REF ) all hold.",
"The main phenomenological difference of the $U(2)$ model with respect to the $U(1)$ model is that the first two squark generations are quasi-degenerate already at the mediation scale, with a mass splitting $\\Delta \\tilde{m}^2_{12}/\\tilde{m}^2\\sim \\epsilon _2^2\\sim 10^{-3}$ .",
"Hence, the model is viable even without invoking flavor-universal RGE effects.",
"A flavor $[U(2)]^3$ symmetry [37] is motivated by the tension between the measured value of the CP asymmetry $S_{\\psi K}$ , and its theoretical value in the Standard Model derived from a global CKM fit.",
"To alleviate this tension, a new physics contribution to $B^0-\\overline{B}^0$ mixing of order 10 percent of the total amplitude is required.",
"In a $U(2)$ model, such a contribution entails a contribution to $\\epsilon _K$ of order 100 percent, which is unacceptable.",
"A $U(2)_Q\\times U(2)_U\\times U(2)_D$ model, with minimal spurion content – $V(2,1,1)$ , $\\Delta Y_u(2,\\bar{2},1)$ and $\\Delta Y_d(2,1,\\bar{2})$ – allows one to suppress the contribution to $\\epsilon _K$ .",
"The structure of the Yukawa and $A$ matrices is as follows [37]: $Y^q&\\sim &\\frac{\\tilde{M}^{2q}_{LR}}{\\tilde{a} v_q}\\sim y_{q_3} \\left(\\begin{array}{cc}\\Delta Y_q & x_q V \\\\ 0 & 1\\end{array}\\right),$ where $y_{q_3}=y_t(y_b)$ for $q=u(d)$ , $x_q$ is a complex free parameter of order one, $\\Delta Y_q$ is a $2\\times 2$ matrix, and $V$ is a $2\\times 1$ vector.",
"This structure is quite unique in that one and the same spurion, $\\Delta Y_q$ , determines the structure of the $2\\times 2$ upper left block of both $Y^q$ and $\\tilde{M}^{2q}_{LR}$ .",
"Consequently, to leading order in the breaking parameters, $(\\delta ^q_{LR})_{12}=0$ , and the supersymmetric contribution to $\\Delta A_{CP}$ vanishes.",
"Corrections to $\\delta _{LR}$ arise at the order $y_c y_t V_{cb}^* V_{ub}$ and are negligible."
],
[
"Hybrid Mediation",
"In models of hybrid mediation, the dominant source of supersymmetry breaking is MFV, but there are non-negligible contributions from Planck scale physics that do not obey the MFV principle.",
"Examples include high-scale gauge mediation [38] and a class of models with anomaly mediation [39].",
"At the messenger scale, the relative size between the soft masses-squared arising from gravity and MFV physics is given by $r\\sim \\tilde{m}_{\\rm grav}^2/\\tilde{m}_{\\rm MFV}^2$ .",
"The $c \\rightarrow u$ couplings are given by $(\\delta ^u_{LL})_{12}&\\sim &\\frac{r |V_{us}|}{r_3},\\\\(\\delta ^u_{RR})_{12}&\\sim &\\frac{r\\, m_u}{r_3 m_c|V_{us}|},$ and the expressions for $\\delta _{LR}$ remain as in the pure gravity case, Eqs.",
"() and ().",
"The EDM constraints of Eqs.",
"() and (REF ) hold.",
"One can now ask what further constraints arise when linking the trilinear terms and the soft masses as is characteristic in hybrid models.",
"If all gravity soft terms are dictated by a single scale, then $a_0\\sim \\sqrt{r}\\; \\tilde{m}_{\\rm MFV}$ , where $\\tilde{m}_{\\rm MFV}$ is the typical messenger scale MFV soft mass.",
"The relevant combination entering $\\delta _{LR}$ is then $\\tilde{a}/\\tilde{m} \\lesssim 3\\sqrt{r/r_3}$ , where the numerical factor stems from RGE and is largest for high scale mediation.",
"In the following we analyze this single gravity scale scenario in the FN context.",
"In models with a single $U(1)$ and order one phases kaon mixing requires $r/r_3 \\lesssim 0.002\\;(\\tilde{m}/{\\rm TeV})$ [30].",
"Therefore, ${\\cal I}m(\\delta _{LR})\\lesssim 0.2 \\times 10^{-4}\\ \\sqrt{\\frac{{\\rm TeV}}{\\tilde{m}}},$ a stronger constraint than the EDM bound of Eq.",
"(REF ).",
"In $[U(1)]^2$ models the kaon system constrains $r/r_3 \\lesssim 0.06\\;(\\tilde{m}/{\\rm TeV})$ , and so ${\\cal I}m(\\delta _{LR})\\lesssim 1 \\times 10^{-4}\\ \\sqrt{\\frac{{\\rm TeV}}{\\tilde{m}}},$ close to the upper bound from EDMs, Eq.",
"(REF ).",
"We note that it is possible in specific $U(1)^2$ models to further suppress the contribution to the kaon system, such that the strongest bound comes from $D^0-\\overline{D}^0$ mixing and gives $r/r_3\\mathrel {\\unknown.",
"{\\hspace{1.0pt}\\sim }}$ <$$ 0.8(m/TeV)$, relaxing the constraint~(\\ref {eq:availhu12}) by a factor of 4.$ Both $U(2)$ and $[U(2)]^3$ models do not require further flavor suppression, and are viable for $r \\lesssim 1$ , with predictions as in the non-hybrid models.",
"We conclude that hybrid models with a $[U(1)]^2$ , $U(2)$ or $U(2)^3$ symmetry generate $\\Delta A_{CP}$ of the same size as non-hybrid models.",
"The size of $\\delta _{LR}$ allowed by hybrid models with a single $U(1)$ is somewhat smaller."
],
[
"$A$ terms and the lightest Higgs mass",
"In supersymmetry $\\Delta A_{CP}$ can be interpreted via the left-right mixing $\\delta _{LR}$ which requires unsuppressed trilinear couplings with respect to the squark masses, $\\tilde{a}/\\tilde{m}\\sim {\\cal O}(1)$ , see Eq. ().",
"At the same time the recent hints from ATLAS and CMS of a neutral Higgs boson with mass near 125 GeV [40] implies that the stops – if not decoupled – are largely mixed as well [41], [42]: $|A_t/y_t -\\mu /\\tan \\beta | \\gtrsim M_S,$ where $M_S$ denotes the geometric mean of the stop masses.",
"In the FN models, where the flavor structure of the $A$ terms is parametrically similar to that of the Yukawas, $A \\sim Y$ , the stop $A$ terms at the weak scale can be written as [43] $A_t/y_t \\simeq \\tilde{a} -\\Delta a, ~~ \\Delta a=y_t^2 b a_0+m_{1/2}c,$ with positive RGE-induced coefficients $b,c$ of order one.",
"For positive $\\mu $ or sufficient $\\tan \\beta $ suppression one obtains from Eq.",
"(REF ) a lower bound that supports a sizeable supersymmetric $\\Delta A_{CP}$ , $\\tilde{a}/\\tilde{m} \\gtrsim M_S/\\tilde{m}$ for $A_t >0$ and unsplit spectrum where the stops are not too far away from the other squarks.",
"Negative $A_t<0$ can arise in scenarios with tiny or vanishing $a_0$ such as gauge mediation, which lead to acceptable phenomenology only for sufficiently large gluino masses.",
"While the Higgs signal needs to be consolidated, it is interesting that if confirmed, the current mass of $\\sim 125$ GeV points to a similar region in supersymmetric parameter space as the interpretation of $\\Delta A_{CP}$ ."
],
[
"Conclusions",
"Supersymmetric models can contribute to direct CP violation in singly Cabibbo suppressed $D$ decays at the level observed by experiments, $\\Delta A_{CP}^{\\rm SUSY}\\sim 0.006$ , without conflicting with phenomenological constraints from $D^0-\\overline{D}^0$ mixing or $\\epsilon ^\\prime /\\epsilon $ .",
"This is naturally the case if the flavor changing parameter $(\\delta ^u_{LR})_{12}$ , generated by trilinear scalar couplings, is of order $10^{-3}$ .",
"In minimally flavor violating supersymmetric models, such as those of gauge mediation and anomaly mediation, $(\\delta ^u_{LR})_{12}$ is orders of magnitude too small.",
"Thus, to account for $\\Delta A_{CP}$ , one has to go beyond minimal flavor violation.",
"We examined models where the flavor structure of the soft breaking terms is dictated by an approximate flavor symmetry.",
"We found that quite generically in such models, $(\\delta ^u_{LR})_{12}$ is flavor-suppressed by $(m_c|V_{us}|/\\tilde{m})$ , which is of order a few times $10^{-4}$ .",
"There is however additional dependence on the ratio between flavor-diagonal parameters, $\\tilde{a}/\\tilde{m}$ , and on unknown coefficients of order one, that can provide enhancement by a factor of a few.",
"In most such models, however, the selection rules that set the flavor structure of the soft breaking terms, relate $(\\delta ^u_{LR})_{12}$ to $(\\delta ^d_{LR})_{12}$ and to $(\\delta ^{u,d}_{LR})_{11}$ , which are bounded from above by, respectively, $\\epsilon ^{\\prime }/\\epsilon $ and EDM constraints.",
"Since both $\\epsilon ^{\\prime }/\\epsilon $ and EDMs suffer from hadronic uncertainties, small enhancement due to the flavor-diagonal supersymmetric parameters cannot be ruled out.",
"Additionally, it is still possible that $(\\delta ^u_{LR})_{12}$ is accidentally enhanced by the order one coefficient.",
"Chirality-flipping couplings between the first and second generation up squarks can effectively arise also via chirality-flipping in the third generation [23].",
"In all flavor models considered here, the effective $\\delta _{LR}$ generated is at most parametrically of the same order as the direct contribution, accompanied by an additional $1/r_3^2$ .",
"For $U(1)$ and $[U(1)]^2$ models this provides extra suppression, while for $U(2)$ models the effective contribution to $\\delta _{LR}$ is flavor-suppressed with respect to the direct one.",
"In any event, the constraints remain the same and the analysis stands.",
"We conclude that it is possible to accommodate $\\Delta A_{CP}\\sim 0.006$ in supersymmetric models that are non-minimally flavor violating, but – barring hadronic enhancements in charm decays – it takes a fortuitous accident to lift the supersymmetric contribution above the permil level."
],
[
"Acknowledgments",
"We thank Gino Isidori for useful discussions.",
"This project is supported by the German-Israeli foundation for scientific research and development (GIF).",
"YN is the Amos de-Shalit chair of theoretical physics and supported by the Israel Science Foundation."
]
] | 1204.1046 |
[
[
"Foliation Theory and it's applications"
],
[
"Abstract Subject of present paper is the review of results of authors on foliation theory and applications of foliation theory in control systems.",
"The paper consists of two parts.",
"In the first part the results of authors on foliation theory are presented, in the second part the results on applications of foliation theory in the qualitative theory of control systems are given.",
"In paper everywhere smoothness of a class $C^\\infty$ is considered."
],
[
"FOLIATION THEORY AND IT'S APPLICATIONS A.Ya.Narmanov,G.Kaypnazarova National University of Uzbekistan [email protected],[email protected] Subject of present paper is the review of results of authors on foliation theory and applications of foliation theory in control systems.",
"The paper consists of two parts.",
"In the first part the results of authors on foliation theory are presented, in the second part the results on applications of foliation theory in the qualitative theory of control systems are given.",
"In paper everywhere smoothness of a class $C^\\infty $ is considered.",
"2000 Mathematics Subject classification:Primary 53C12; Secondary 57R30; 93C15 Keywords.",
"a riemannian manifold, a foliation, a leaf, holonomy group, local stability theorem, level surfaces, metric function, total geodesic submanifold, connection, foliation with singularities, a orbit, control system,controllability set.",
"Research supported by grant OT-F1-096 of the Ministry of higher and secondary specialized education of Republic of Uzbekistan.",
"1.",
"Topology of foliations The foliation theory is a branch of the geometry which has arisen in the second half of the XX-th century on a joint of ordinary differential equations and the differential topology.",
"Basic works on the foliation theory belong to the French mathematicians A. Haefliger [8],[9], G. Ehresman [5],[6], G.Reeb [45],[46], H. Rosenberg [48],[49],G. Lamoureux [21], [22],R. Langevin [23],[24].",
"Important contribution to foliation theory was made by known mathematicians - as well as I.Tamura [53], R. Herman [10] [11],[12] ,[13],T.Inaba [14],[15], W.Turston [54],[55], P. Molino [26], P.Novikov [42], Ph.",
"Tondeur [56],[57],B. Reinhart [47].",
"Now the foliation theory is intensively developed, has wide applications in various areas of mathematics - such, as the optimal control theory, the theory of dynamic polysystems.",
"There are numerous researches on the foliation theory.",
"The review of the last scientific works on the foliation theory and very big bibliography is presented in work of Ph.",
"Tondeur [57].",
"Definition-1.1 Let $(M,A)$ be a smooth manifold of dimension $n$ , where $A $ is a $C^r$ atlas, $r\\ge 1$ , $0<k<n$ .",
"A family $F=\\lbrace L_\\alpha :\\alpha \\in B\\rbrace $ of path-wise connected subsets of $M$ is called $ k$ - dimensional $C^r-$ foliation of if it satisfies to the following three conditions: $F_I :\\ $ $ \\bigcup \\limits _{\\alpha \\in B} L_\\alpha =M $ ; $F_{II}:$ for every $ \\alpha ,\\beta \\in B $ if $ \\alpha \\ne \\beta $ then $ L_\\alpha \\bigcap L_\\beta = \\emptyset $ ; $F_{III}:$ For any point $p \\in M $ there exists a local chart ( local coordinate system ) $ (U, \\varphi ) \\in A, \\ p \\in U $ so that if $ U \\bigcap L_\\alpha \\ne \\emptyset $ for some $\\alpha \\in B $ the components of $ \\varphi (U \\bigcap L_\\alpha )$ are following subsets of parallel affine planes $ \\lbrace ( x_1, x_2, ..., x_n ) \\in \\varphi (U) : x_{k+1}= c_{k+1},x_{k+2}= c_{k+2},...,x_{n}= c_{n} \\rbrace $ where numbers $ c_{k+1}, c_{k+2},...,c_n $ are constant on components (Figure-1,[53], p. 121).",
"The most simple examples of a foliation are given by integral curves of a vector field and by level surfaces of differentiable functions.",
"If the $ X $ vector field without singular points is given on manifold $ M $ under the theorem of straightening of a vector field (under the theorem of existence of the solution of the differential equation) integral curves generate one-dimensional foliation on $M $ .",
"Figure: NO_CAPTION Figure-1 Let $M$ be a smooth manifold of dimension $n$ , $f:M\\rightarrow R^1$ be a differentiable function.",
"Let $p_0\\in M$ ,$ f(p_0)=c_0 $ and the level set $L=\\lbrace p\\in M:f(p)=c_0\\rbrace $ does not contain critical points.",
"Then the level set is a smooth submanifold of dimension $n-1$ .",
"If we will assume that differentiable function has no critical points, partition of $M$ into level surfaces of function is a $n-1$ - dimensional foliation (codimension one foliation).Codimension one foliations generated by level surfaces were studied in papers [2],[17],[18], [19],[30], [31], [32], [47], [57].",
"The following theorem gives to us a simple example of foliation.",
"Theorem-1.1.",
"Let $f:M\\rightarrow N$ be a differentiable mapping of the maximum rank, where $M$ is a smooth manifold of dimension $n$ ,$N$ is a smooth manifold of dimension $m$ , where $n>m$ .",
"Then for each point $q\\in N$ a level set $L_q= \\lbrace p\\in M: f(p)=f(q)\\rbrace $ is a manifold of dimension $n-m$ and partition of $M$ into connection components of the manifolds $L_q$ is a $n-m$ - dimensional foliation.",
"Using the condition 3 of definition 1.1 it is easy to establish that there is a differential structure on each leaf such that a leaf is immersed $k$ -dimensional submanifold of $M$ , i.e the canonical injection is a immersing map(a map of the maximum rank).",
"Thus on each leaf there are two topology: the topology $\\tau _M$ induced from $M$ and it's own topology $\\tau _F$ as a submanifold .",
"These two topologies are generally different.",
"The topology $\\tau _F$ is stronger than topology $\\tau _M$ , i.e.",
"each open subset of $L_\\alpha $ in topology $\\tau _M$ is open in $\\tau _F$ .",
"A leaf $L_\\alpha $ is called compact if $(L_\\alpha , \\tau _F)$ is compact topological space.",
"It is obvious that the compact leaf is a compact subset of manifold $M$ .",
"The leaf is $L_\\alpha $ called as proper if the topology $\\tau _F$ coincides with the topology $\\tau _M$ induced from $M$ .If these two these topology on $L_\\alpha $ do not coincide, the leaf is called a non- proper leaf.",
"It is easy to prove that the compact leaf is proper leaf.",
"In work [1] the following assertion is proved which takes place for foliations with singularities too which we will discuss in the second part of this paper.",
"Proposition.",
"If a leaf is a closed subset of $M$ then it is a proper leaf.",
"Let $L$ be a leaf of $F$ .",
"Point $y\\in M$ is called a limit point of the leaf $L$ if there is a sequence of points $y_m$ from $L$ which converges to $y$ in topology of manifold $M$ and does not converge to this point in the topology of the leaf $L$ [36].",
"The set of all limit points of the leaf $L$ we will denote by $\\Omega (L)$ .",
"It is easy to show that the limit set consists of the whole leaves, i. e. if $y\\in \\Omega (L)$ then $L(y)\\subset \\Omega (L)$ , where $L(y)$ is a leaf containing $y$ .",
"Generally the set $\\Omega (L)$ can be empty or can coincide with all manifold.",
"It can already take place for trajectories of dynamic systems.",
"For example, if $L$ is closed the set $\\Omega (L)$ is empty, and in case of an irrational winding of torus each trajectory everywhere is dense and consequently its limit set coincides with all torus.",
"Studying of limit sets of leaves of foliation includes studying of limit sets of trajectories of the differential equations and it is the important problem of the foliation theory.",
"On these subjects there are numerous researches[1],[14],[25],[33]- [39].",
"In work [1] following properties of a leaf are proved which also takes place for foliation with singularities too.",
"Theorem-1.2.",
"(1).",
"A leaf $L_0$ is proper leaf if and only if $L_0\\bigcap \\Omega (L_0)=0$ ; (2).",
"A leaf is $L_0$ is not proper leaf if and only if $\\Omega (L_0)=\\overline{L}_0 $ where $\\overline{L}_0$ is the closure of $L_0$ in manifold $M$ .",
"For two leaves $L_1$ and $L_2$ we will write in $ L_1\\le L_2$ only in a case when $L_1\\subset \\Omega (L_2)$ .",
"The inequality $L_1<L_2$ means $L_1\\le L_2$ and $L_1\\ne L_2$ .",
"The relation $\\le $ on the set of leaves has been entered by the Japanese mathematician T.Nisimori in the paper [41].",
"We will denote by $(M/F,\\le )$ set of leaves with the entered relation on it.",
"It is obvious that the $\\le $ on $M/F$ reflective and is transitive, but in many cases this relation is not asymmetric, therefore generally the set $(M/F,\\le )$ is not partially ordered.T.Nisimori was interested in the case where $(M/F,\\le )$ is a partially ordered set.",
"Except that T.Nisimori has entered concepts of depth of a leaf $L$ and depth of foliation $F$ as follows: $ d(L)=\\sup \\lbrace k:$ there exist leaves $L_1,L_2,...,L_k $ such that $ L_1<L_2<...<L_k=L\\rbrace $ , $d(F)=\\sup \\lbrace d(L): L\\in M/F\\rbrace $ .",
"A leaf $L$ , being the closed subset, has the depth equal to one.",
"It is easy to construct one dimensional foliation of Euclid plane with leaves of depth equal to two.",
"In work of [41] Nishimori has proved the following theorem which shows that for each positive integer $k$ there exists two-dimensional foliation with leaves of the depth equal to $k$ .",
"Theorem-1.3.",
"Let $S_2$ be a closed surface of a genus 2.For all positive integer $k$ there is a codimension one foliation $F$ on $M=S_2\\times [0,1]$ satisfying the following conditions (1),(2)and(3).",
"(1) All leaves of $F$ are proper and transverse to ${x}\\times [0,1]$ for all $ x\\in S_2 $ .",
"$S_2\\times {0}$ and $S_2\\times {1}$ are compact leaves.",
"(2) $d(F)=k$ .",
"(3) All holonomy groups of $F$ are abelian.",
"The following theorem is proved in paper [1] shows that there exists one dimensional analytical foliation generated by integral curves of analytical vector field which have leaves of depth equal to 1,2 and 3.",
"Theorem-1.4.",
"Let $S^k$ be a $k$ dimensional sphere.On the manifold $ S^2\\times S^1 $ there exists an analytical vector field without singular points and with three pairwise different integral curves $\\alpha ,\\beta ,\\gamma $ such that $\\alpha \\subset \\Omega (\\beta )$ ,$\\beta \\subset \\Omega (\\gamma )$ ,where $\\alpha $ is a closed trajectory, $\\Omega (\\beta ) $ consists of only closed trajectories,$\\Omega (\\gamma )$ consists of only the trajectories of depth equal to two.",
"This vector field generates one-dimensional foliation of the depth equal to 3.",
"Remark.The example of not analytical dynamic system of a class $C^{\\infty }$ was constructed in the paper [25] for which there is an infinite chain of not closed trajectories $ L_i$ such that each trajectory $ L_{i+1} $ is in the limit set of $L_i $ .",
"In the paper [41] for codimension one foliation the following theorem is proved: Theorem-1.5.",
"(Nishimori).If $ d(F)<\\infty $ or all leaves of foliation $F$ are proper,then the set $(M/F,\\le )$ is partially ordered.",
"Nishimori, studying property codimension one foliation in the case when the set $(M/F,\\le )$ is a partially ordered, has delivered following questions which are of interest for foliation with singularities too [41]: 1.",
"Are all leaves of foliation $F$ proper under the assumption that the set$(M/F,\\le )$ is partially ordered?",
"2.",
"Is a leaf $L$ proper under the assumption that $dL<\\infty $ ?",
"A.Narmanov studied the relation $\\le $ for foliation with singularities in the paper [33].",
"In particular, he proved the following theorems which solves problems 1, 2 delivered by Nishimori.",
"Theorem-1.5.",
"Let $M/F$ be the set of leaves of foliation $F$ with singularities .Then the set $(M/F,\\le )$ is a partially ordered if and only if all leaves are proper.",
"Theorem-1.6.If the depth of a leaf is finite, then it is proper leaf.",
"It is known that the limit leaf of compact leaves codimension one foliation on compact manifold is a compact leaf and the limit set of each leaf contains finite number of compact leaves.",
"The following theorems are generalizations of these facts for leaves with finite depth [34].",
"Theorem-1.7.Let $F$ be a transversely oriented codimension one foliation on compact manifold $M$ , $L_i$ - a leaf of foliation $F$ , and $x_i\\longrightarrow x$ , where $x_i\\in L_i$ .",
"If $dL_i\\le k $ for each $i$ then $ dL(x)\\le k $ .",
"Theorem-1.8.",
"Let $F$ is a transversely oriented codimension one foliation on compact manifold $M$ ,$L^{0}$ - some leaf of foliation $F$ .",
"Then for each $k\\ge 1$ the set $C_k=\\lbrace L:L<L^{0},dL=k\\rbrace $ either is empty, or consists of finite number of leaves.",
"Let's remind that transversally orientability of $F$ means that there exists smooth non-degenerated vector field $X$ on $M$ , which is transversal to leaves of foliation $ F$ .",
"Let $ x \\in M $ , $L(x) $ is a leaf foliation,containing the point $x$ , $T_x$ is a manifold dimension of $n-k$ transversal to $L(x)$ such that $ T_x \\bigcap L(x)=x $ .To each the closed continuous curve in $ L(x) $ beginning and the ending at the point $ x\\in M$ corresponds a local diffeomorphism $ g $ of the manifold $ T_x $ , given in some neighborhood of the point $ x $ in $ T _x $ such that $ g(x)=x $ .",
"The set of such diffeomorfisms forms the pseudogroup $ \\Gamma _x(L) $ of the leaf $ L $ at the point $x$ , and germs of these diffeomorphisms form holonomy group $ H $ of the leaf $ L(x)$ .For different points from $L$ corresponding holonomy groups are isomorphic[53].",
"The important results in foliation theory are received by G. Reeb.",
"One of his theorems is called as the theorem of local stability which can be formulated as follows.",
"Theorem-1.9.",
"[53] Let $ L_0 $ a compact leaf foliation $ F $ with finite holonomy group.",
"Then there is an open saturated set $ V $ which contains $ L_0 $ and consists of compact leaves.",
"Let's notice that a saturated set $ S\\subset M $ on a foliated manifold is a subset which is the union of leaves.",
"In 1976 in Rio de Janeiro at the international conference the attention to the question on possibility of the proof of theorems on local stability for noncompact leaves [50] has been brought.",
"In 1977 the Japanese mathematician T.Inaba has constructed a counterexample which shows that if codimension of foliation is not equal to one G.Reeb's theorem cannot be generalized for noncompact leaves [14].",
"Let's bring the theorem on a neighborhood of a leaf with finite depth which is generalization of the theorem of J.Reeb on local stability for transversely oriented codimension one foliation.",
"Let $F$ transversely oriented, $X$ a smooth vector field on $M$ , transversely to leaves of $F$ .",
"Let $x\\in M,t\\rightarrow X^{t}(x)$ , - the integral curve of a vector field $X$ passing through the point $x$ at $t=0$ .",
"Let's put $T_x=\\lbrace X^{t}(x):-a<t<a \\rbrace $ .",
"In further will write $T_x\\approx (-a,a)$ and as usually, to replace subsets of $ T_x $ by their images from $(-a,a)$ .",
"The point is $y\\in T_x$ called as a motionless point of pseudo-group $\\Gamma =\\Gamma _x(L)$ , if $g(y)=y$ for each $g\\in \\Gamma $ , advanced in a point $y\\in T_x$ .",
"If there exists $\\varepsilon > 0$ such that each point from $( -\\varepsilon ,\\varepsilon )$ is a motionless point of pseudogroup $\\Gamma $ we will say that the pseudo-group is $ \\Gamma $ trivial.",
"Let $F$ be a codimension one foliation, $L$ be a some leaf of $F$ with finite depth ,$\\rho $ - distance function defined by some fixed riemannian metric on $M$ .",
"Let's enter set $ U_r=\\lbrace y\\in M:\\rho (y,L)<r\\rbrace ,r>0 $ , where $\\rho (y,L) $ - distance from the point $ y $ to the leaf $ L $ .",
"Theorem-1.10([35]).",
"Let $ F $ be a transversely oriented codimension one foliation on compact manifold $M$ .",
"If the holonomy pseudogroup $ \\Gamma $ the leaf $L$ is trivial,then for each $r>0$ there is a invariant open set $V$ containing $L$ and consisting of leaves diffeomorphic to $ L $ which satisfies to following conditions: 1)$ V\\subset U_r $ ; 2) $ dL_\\alpha =dL $ for each leaf $ L_\\alpha \\subset V $ .",
"One more generalization of G. Reeb theorem for a noncompact leaf is resulted below.For this purpose we will bring some definitions.",
"Let $ M $ be smooth connected complete riemannian manifold of dimension$ n $ with riemannian metric $ g $ , $ F $ -smooth foliation of dimensions $ k $ on $ M $ .",
"Let's denote through $ L(p)$ a leaf of $ F $ passing through a point $ p $ , $ F(p) $ - tangent space of leaf at the point $p$ , $H(p)$ - orthogonal complementary of $ F(p)$ in $ T_p M $ , $p\\in M $ .",
"There are two subbundle (smooth distributions),$TF=\\lbrace F(p):p\\in M\\rbrace $ ,$ H=\\lbrace H(p):p\\in M\\rbrace $ of tangent bundle $ TM $ such ,that $ TM=TF\\oplus H $ where is $ F$ orthogonal addition $ TF $ .",
"Piecewise smooth curve $ \\gamma :[0,1]\\rightarrow M $ we name horizontal, if $\\dot{\\gamma }(t)\\in H(\\gamma (t)) $ for each $t\\in [0,1] $ .Piecewise smooth curve which lies in a leaf foliation $F$ is called as vertical.",
"Let $I =[0,1] $ , $ \\nu :I\\rightarrow M $ a vertical curve,$ h:I\\rightarrow M $ a horizontal curve and $ h(0)=\\nu (0)$ .Piecewise smooth mapping $ P:I\\times I\\rightarrow M $ is called as vertical-horizontal homotopy for pair $v,h $ if $ t\\rightarrow P(t,s) $ is a vertical curve for each $ s\\in I $ , $ s\\rightarrow P(t,s) $ is a horizontal curve for each $ t\\in I $ ,and $P(t,0)=\\nu (t) $ for $ t\\in I $ , $ P(0,s)=h(s) $ for $ s\\in I $ .",
"If for each pair of vertical and horizontal curves $ \\nu , h:I\\rightarrow M $ with $ h(0)=\\nu (0) $ there exists corresponding vertical-horizontal homotopy $ P $ ,we say that distribution $ H $ is Ehresman connection for $ F $ [3].",
"Let $ L_0 $ a leaf of codimension one foliation $ F $ , $ U_r=\\lbrace x\\in M: \\rho (x;L_0)< r\\rbrace $ , where $ \\rho (x,L_0) $ - distance from the point$ x $ to a leaf $ L_0 $ .",
"We will assume that there is such number $ r_0 >0 $ that for each horizontal curve $ h:[0,1]\\rightarrow U_{r_0}$ and for each vertical curve $ \\nu :[0,1]\\rightarrow L_0 $ such that $ h(0)=v(0) $ there exists vertically-horizontal homotopy for pair $(\\nu ,h ) $ .",
"At this assumption we formulate generalization of the theorem of J.Reeb [36].",
"Theorem-1.11.",
"Let $ F $ a transversely oriented codimension one foliation , $ L_0 $ be a relatively compact proper leaf leaf with finitely generated fundamental group.",
"Then if holonomy group of the leaf $ L_0 $ is trivial then for each $ r>0 $ there is an saturated set $ V $ such that $ L_0 \\subset V\\subset U_r $ and restriction of $ F $ on $ V $ is a fibrarion over $ R^{1}$ with the leaf $ L_0 $ .",
"From the geometrical point of view, the important classes of foliation are total geodesic and riemannian foliations.",
"Foliation $F$ on riemannian manifold $ M $ is called total geodesic if each leaf of foliation $F$ is a total geodesic submanifold,i.e every geodesic tangent to a leaf foliation $ F $ at one point, remains on this leaf.",
"The geometry of total geodesic foliations is studied in works [13], [16], [38],[4].",
"Foliation $F $ on a riemannian manifold $ M $ is called riemannian if each geodesic, orthogonal at some point to a leaf of foliation $F $ , remains orthogonal at all points to leaves of $F$ [47].",
"Riemannian foliation without singularities for the first time have been entered and studied by Reinhart in work [47].",
"This class foliation naturally arising at studying of bundles and level surfaces.",
"Riemannian foliation are studied by many mathematicians, in particular,in works of R.Herman [10],[11],[12],P.Molino [26],A Morgan [27], Ph.Tondeur [57] .",
"The most simple examples of Riemannian foliation are partition of $ R^{n} $ into parallel planes or into concentric hyperspheres.",
"Riemannian foliation with singularities have been entered and studied in works of P.Molino [26], they also were studied by A.Narmanov in the papers [38],[40].",
"Let $ M $ smooth connected complete riemannian reducible manifold.",
"Then on $ M $ there are two parallel foliations, mutually additional on orthogonality [20].",
"If $ M $ simple connected manifold then the de Rham theorem takes place which asserts that $ M $ is isometric to direct product of any two leaves from different foliations [20].",
"In this case both foliations are riemannian and total geodesic simultaneously.",
"Below it is presented results of authors on geometry of riemannian and totally geodesic foliations.",
"Assume that $ F $ is a riemannian foliation with respect to riemannian metric $ g $ on $ M $ .",
"Let $ \\pi _1 : TM \\rightarrow TF $ , $ \\pi _2 :TM \\rightarrow H $ be orthogonal projections, $V(M) $ ,$ V(F)$ , $ V( H) $ be the sets of smooth sections of bundles $ TM $ ,$ TF $ and $H$ accordingly.",
"If $X\\in V(F) $ ($X\\in V(H) $ ) $ X$ is a called a vertical (horizontal) field.",
"Now we will assume that each leaf of $ F $ is a total geodesic submanifolds of $M$ .",
"It is equivalent to that, $\\nabla _X Y \\in V(F)$ for all $ X,Y \\in V(F) $ [47], where $\\nabla $ - Levi-Civita connection.",
"In this case $ F $ is a riemannian foliation with total geodesic leaves.",
"Then on bundles $ TF $ and $H $ are given metric connections $\\nabla ^{1}$ and $ \\nabla ^{2}$ as follows.If $X\\in V(F)$ ,$ Y\\in V(H) $ , $ Z\\in V(M),$ we will put $\\nabla ^1_Z=\\pi _1 (\\nabla _Z(X)),\\ \\nabla ^2_Z(Y)=\\pi _2[Z_1,\\widetilde{Y}]+\\pi _2[\\nabla _{Z_{2}},Y],$ where $ Z=Z_{1}\\oplus Z_{2}$ ,$Z_{1}\\in V(F),Z_{2}\\in V(H),\\widetilde{Y}\\in V(M) ,\\pi _{2}\\widetilde{Y}=Y$ .",
"Here $[Z_{1},\\widetilde{Y}]$ - Lie bracket of vector fields $Z_{1}$ and $ \\widetilde{Y}$ .",
"$ \\nabla _{2}$ is a metric connection only in case when $ F $ is riemanian.",
"Owing to that $F$ is a totally geodesic foliation, connection $ \\nabla ^{1}$ also is metric [38], [47].",
"Let $ p\\in M $ , $ S(p) $ be the set of points $M$ ,which can be connected by horizontal curves with $p$ .",
"Owing to that foliation $F$ is total geodesic, for each the $ p\\in M $ set $ S(p) $ has topology and differentiable structure, in relation to which $ S(P)$ is a immersed submanifold of $M$ [4].It is easy to prove the following assertion.",
"Lemma-1.1.",
"$ dimS(p)\\ge k $ for every point $p\\in M$ .",
"Owing to that the manifold $M$ is complete, and considered foliation is a riemannian, the distribution $H$ is a Ehresmann connection for $F $ [3].",
"Therefore for each piece-wise smooth curve $\\gamma :I\\rightarrow M$ there exists unique vertical-horizontal homotopy $P_\\gamma :I\\times I\\rightarrow M$ such that $\\gamma (t)=P_\\gamma (t,t)$ for $t\\in I$ .",
"Let $P:I\\times I\\rightarrow M$ be a vertical-horizontal homotopy.",
"We will denote by $D_t(P(t,s))$ the tangent vector of the curve $t\\rightarrow P(t,s)$ at the point $P(t,s)$ , by $D_s (P(t,s))$ the tangent vector of the curve $s\\rightarrow P(t,s)$ at the point $P(t,s)$ .",
"Lemma-1.2.",
"Let $X(t,s)=D_s (P(t,s))$ , $Y(t,s)=D_t(P(t,s))$ for $(t,s)\\in I\\times I$ .",
"Then $ \\nabla ^1_X Y=0 $ and $\\nabla ^2_Y X=0 $ .",
"In the proof of the de Rham theorem projections of a curve $\\gamma $ in $ L(p_0) $ and in $ S(p_0) $ are defined with connection $ \\nabla $ and it is shown that these projections coincide with curves $ P_\\gamma (\\cdot ,0 ) :I \\rightarrow M $ and $ P_\\gamma (0 ,\\cdot ) :I \\rightarrow M $ accordingly.",
"In this case it is used that distribution $H $ is complete integrable [20].",
"In the paper [38] the similar fact is proved for projections of a curve $\\gamma $ without the assumption that $H $ is complete integrable by means of metric connection $\\widetilde{\\nabla }$ which is entered below.",
"Metric connection $\\widetilde{\\nabla }$ is defined as follows: $\\widetilde{\\nabla } _Z X=\\nabla ^1_Z X_1 +\\nabla ^2_Z X_2,$ where $X,Z\\in V(M)$ , $X_i=\\pi _i(X)$ , $i=1,2$ .It is not difficult to check up that distributions $TF$ and $H$ are parallel with respect to $\\widetilde{\\nabla }$ .",
"Let $ \\gamma :I\\rightarrow M $ be a smooth curve,$ \\gamma (0)=p_0 $ and $\\gamma (1)=p$ , $ C:I \\rightarrow T_{p_0} M $ is a development of the curve $ \\gamma $ in $ T_{p_0} M $ defined by connection $\\widetilde{\\nabla }$ .",
"(See definition of development in [[20],p.129].",
"(Here for convenience tangent vector space $ T_{p_0}M $ is identified with affine tangent space at the point $p_0.",
")$ Let $ C(t)=(A(t),B(t)) $ where $ A(t)\\in F(p_0)$ ,$ B(t)\\in H(p_0)$ for $ t\\in I $ .",
"As $M$ is a complete riemannian manifold, $\\widetilde{\\nabla }$ - metric connection, there are smooth curves $\\gamma _1,\\gamma _2 :I\\rightarrow M $ , which are developed on curves $ t\\rightarrow A(t)$ and $ t\\rightarrow B(t) $ accordingly ([20], p.167, the theorem 4.1).",
"According to the proposition 4.1 in ([20],p.129) $\\gamma _i$ is a such curve that result of parallel transport $\\dot{\\gamma }_i$ to the point $p_0$ along $\\gamma _i^{-1}$ defined by connection $\\widetilde{\\nabla }$ coincides with the result of parallel transport of $\\pi _i(\\dot{\\gamma }(t))$ to the point $p_0$ along $\\gamma ^{-1}$ defined by connection $\\widetilde{\\nabla }$ too.That is why $\\gamma _1$ is a vertical curve, $\\gamma _2$ is a horizontal curve.",
"Curves $\\gamma _1$ , $\\gamma _2$ are called projections of the curve $\\gamma $ in $L(p_0)$ and in $S(p_0)$ accordingly.",
"The following theorems are proved in the work [38].",
"Theorem-1.12.",
"The projection of curve $\\gamma :I\\rightarrow V$ in $L(q_0)$ (in $ S(q_0)$ ) is a curve $ P(\\cdot ,0):I\\rightarrow L(q_0)$ (accordingly $P(0,\\cdot ):I\\rightarrow S(q_0)$ ).",
"The following theorem shows that if distribution $ H $ is complete integrable if and only if connection $\\widetilde{\\nabla }$ coincides with Levi-Civita connection $ \\nabla $ .",
"Theorem-1.13.",
"Following assertions are equivalent.",
"1.Distribution $H$ is complete integrable (i.e $dimS(p)=n-k$ for each $p\\in M$ ).",
"2.",
"$\\widetilde{\\nabla }$ is connection without torsion (i.e $\\widetilde{\\nabla }=\\nabla $ ).",
"Remark .",
"As shows known Hopf fibration of on three-dimensional sphere,the distribution $ H $ it is not always complete integrable.",
"In a case when $ H $ is complete integrable de Rham theorem takes place: if $ M $ is simple connected,it is isometric to product $ L(p)\\times S(p) $ for each $ p\\in M $ .",
"In this case $S(p)$ is a leaf of foliation $F ^\\bot $ generated by distribution $H$ .",
"Projections of any point $ p\\in M $ in $L(p_0)$ and in $S(p_0)$ are defined as follows.",
"Let $\\gamma :I\\rightarrow M$ is a smooth curve, $\\gamma (0)=p_0, \\ \\gamma (1)=p ,\\ \\nu $ , $h$ are projections of $\\gamma $ in $L(p_0)$ and in $S(p_0)$ .",
"The points $\\nu (1)$ , $h(1)$ are called as projections of $p$ in $L(p_0)$ and in $S(p_0)$ accordingly.",
"Owing to that the distribution $H$ is completely integrable, the projection of $p$ depends only on the homotopy class of the curve $\\gamma $ .That is why when $M$ is simple connected,the mapping $f:p \\rightarrow (p_1,p(_2)$ is correctly defined.",
"Under theorems 1.12 and 1.13 mapping $f$ is a isometric immersing.",
"Since $dimM=dim\\lbrace L(p_0)\\times S(p_0)\\rbrace $ the mapping $f$ is covering mapping, hence, it is an isometry ([20],p.134).",
"In the known monograph [57] Ph.",
"Tondeur studied foliation, generated by level surfaces of functions of a certain class.",
"He considered function $f:M \\rightarrow R ^1$ without critical points on Riemannian manifold $M$ for which length of a gradient is constant on each level surface.",
"For such functions he has proven that foliation generated by level surfaces of such function, is a Riemannian foliation .",
"Authors of the present article studied geometry of foliation generated by level surfaces of the functions considered in the monograph of professor Ph.",
"Tondeur without the assumption of absence of critical points.",
"Definition -1.2.Let $M$ be a smooth manifold of dimension $n$ .Function $f:M \\rightarrow R ^1$ of the class $C^2(M,R^1)$ for which length of a gradient is constant on connection components of level sets is called a metric function.",
"Let $f:R^n \\rightarrow R ^1$ be a metric function.",
"We will consider system of the differential equations $\\dot{x}=gradf(x)$ (1.1) As, the right part of system (1.1) is differentiable, for each point $x_0\\in M$ there is a unique solution of system (1.1) with the initial condition $x(0)=x_0$ .",
"The trajectory of system (1.1) is called gradient line of the function $f$ .",
"Theorem-1.14[17].Curvature of each gradient line of metric function is equal to zero.",
"In the papers [17],[19] topology of level surfaces are studied under the assumption that each of a connection components of the set of critical points of metric function is either a point, or is a regular surface and every component is isolated from others.",
"The following theorem gives complete classification of foliations generated by level surfaces of metric function [19].",
"Theorem-1.15.",
"Let $f:M \\rightarrow R ^1$ be a metric function given in $R^n$ .",
"Then level surfaces of function $f$ form foliation which has one of following $n$ types : 1) Foliation $F$ consists of parallel hyperplanes; 2) Foliation $F$ consists of concentric hyperspheres and the point (the center of hyperspheres); 3) Foliation $F$ consists of concentric cylinders of the kind $S^{n-k-1}\\times R^k$ and the singular leaf $R^k$ (which arises at degeneration of spheres to a point), where $k$ - the minimum of dimensions of critical level surfaces,$1\\le k \\le {n-2}$ .",
"At the proof of the theorem-1.1 4 the following theorem is used which also is proved in work [19].",
"Theorem-1.16.",
"Let $L$ be a regular surface of dimension $r$ which is the closed subset $R^n$ , where $1\\le r\\le {n-1}$ .",
"If the normal planes passing through various points $L$ are not crossed,then $L$ is $r$ - a dimensional plane.",
"This theorem represents independent interest for the course of differential geometry.",
"The following theorem shows internal link between geometry of riemannian manifold and property of metric function which is given on it.",
"Theorem-1.17 [32] Let $(M,g)$ be a smooth riemannian manifold of dimension $n$ , $f:M \\rightarrow R ^1$ metric function.",
"Then each gradient line of the function $f$ is a geodesic line of the riemannian manifold $M$ .",
"For the metric functions given on a riemannian manifold , it is difficult to get classification theorems, as in a case $M=R^n$ .",
"Here one much depends on topology of riemannian manifold on which function is given.",
"For example, in Euclid case if metric function has no critical points, then as shown [19] all level surfaces are hyperplanes.",
"It is easy to construct metric function without critical points on the two-dimensional cylinder ( with the metric induced from Euclid structure of $R^3$ ) level lines of which are circles (compact sets).",
"The following theorem, is the classification theorem for level lines of the metric function given on two-dimensional riemannian manifold.",
"Theorem-1.18[32] Let $M$ be two-dimensional riemannian manifold, $f:M \\rightarrow R ^1$ be a metric function without critical points.",
"Then all level lines are homeoomorphic to circle, or all level lines are homeoomorphic to a straight line.",
"The following theorem shows that if the metric function is given on complete simple connected riemannian manifold and does not have critical points, then it has no compact level surfaces.",
"Theorem-1.19 [32] Let $M$ be a smooth complete simple connected riemannian manifold, $f:M \\rightarrow R ^1$ be a metric function without critical points.",
"Then level surfaces are mutually diffeomorphic noncompact submanifolds of $M$ .",
"2.",
"Applications of foliation theory in control systems The last years methods and results of foliation theory began to be used widely in the qualitative theory of optimal control.It was promoted by works of the American mathematician of G.Sussmann [52] and the English mathematician P.Stefan [51] which have shown that a orbit of family of smooth vector fields is a smooth immersed submanifold.",
"Besides, they have shown that if dimensions of orbits are the same, partition of phase space into orbits is a foliation.",
"Papers [33], [35], [36], [37], [7], [39], [40], [43], [44] are devoted to applications of foliation theory in control theory.",
"Let $D$ be a family of the smooth vector fields on $M$ , and $X\\in D$ .",
"Then for a point $x\\in M$ by $X^t(x)$ we will denote the integral curve of a vector field $X$ passing through the point $x$ at $t=0$ .",
"Mapping $t\\rightarrow X^t(x )$ is defined in some domain $I(x)$ which generally depends not only on a field $X$ , but also from the point $x$ .",
"Further everywhere in formulas kind of $X^t(x)$ we will consider that $t\\in I(x)$ .",
"The orbit $L(x)$ passing through a point $x$ of the family of the vector fields $D$ is defined as a set of points $y$ from $M$ for which there are real numbers $t_1,t_2,\\ldots ,t_k$ and vector fields $X_{i_1},X_{i_2},\\ldots X_{i_k}$ from (where $k$ is a natural number) such that $y=X^{t_k}_{i_k}(X^{t_{k-1}}_{i_{k-1}}(\\ldots (X^{t_1}_{i_1})\\ldots ))$ .",
"Now we will bring a definition of foliation with singularities [51].",
"A subset $L$ of manifold $M$ is called as a leaf if 1) there is a differential structure $ \\sigma $ on $ L $ such that $(L,\\sigma )$ is a connected $k$ - dimensional immersed submanifold of $M$ .",
"2) for locally connected topological space $ N $ and for continuous mapping $ f:N\\rightarrow M $ such that $ f(N)\\subset L$ the mapping $f:N\\rightarrow (L,\\sigma )$ is continuous.",
"Partition $F$ of manifold $M$ into leaves is called smooth (of the class $C^r$ ) foliation with singularities if following conditions are satisfied: 1) For each point $x\\in M$ there exists a local $C^r$ - chart $(\\psi ,U)$ containing the point $x$ such that $\\psi (U) =V_1\\times V_2$ where $V_1$ is a neighborhood of origin in $R^k$ , $V_2$ - a neighborhood of origin in $R^{n-k}$ ,$k$ - dimension of the leaf containing the point $x$ ; 2) $\\psi (x)=(0,0)$ ; 3)For each leaf $L$ such that $L\\cap U\\ne \\emptyset $ it takes place equality $L\\cap U=\\psi ^{-1}(V_1\\times l)$ where $l=\\lbrace \\nu \\in V_2:\\psi ^{-1}(0,\\nu )\\in L\\rbrace $ .",
"By definition 1.1 each regular foliation is a foliation in sense of the above-stated definition.",
"In this case every connection component of the set $l $ is a point.",
"If dimensions of leaves of a foliation with singularities are the same as noted above, it is a foliation in sense of the definition 1.1.",
"Thus, the conception of foliation with singularities is a generalization of classical notion of a foliation (now which is called as regular foliation).",
"In the literature instead of \"foliation with singularities\" the term \"singular foliation\" is used also [26].To the studying of a foliation with singularities are devoted papers [1],[26], [36], [37], [51].",
"Now we will consider some applications of the foliation theory in problems of the qualitative theory of control systems.",
"Let's consider a control system $ \\dot{x}=f(x,u)$ ,$ x\\in M $ , $ u\\in U\\subset R^{m}$ $ (2.1)$ where $ M $ is a smooth (class$ C^{\\infty }$ ) connected manifold of dimension $ n $ with some riemannian metric $ g $ , $ U $ is a compact set, for each the $ u\\in U $ vector field $ f(x,u )$ is a field of class $ C^{\\infty }$ , and mapping $ f:M\\times U\\longrightarrow TM $ ,where $ TM $ is the tangent bundle of $ M$ , is continuously differentiable.",
"It means that there is such open set $ V $ such that $ U\\subset V \\subset R ^{m} $ , and continuously differentiable mapping $ \\bar{f} : M\\times V\\longrightarrow TM $ , restriction of which on $ M\\times U $ coincides with $ f(x,u )$ .",
"Admissible controls are defined as piecewise-constant functions $ u:[0,T]\\longrightarrow U $ , where $ 0<T<\\infty $ .",
"Thus, the trajectories of system (2.1) corresponding to admissible controls, represent piecewise smooth mapping $ x:[0,T]\\rightarrow M $ .",
"The purpose of control is a bring of the system to some fixed (target) point $ \\eta \\in M $ .",
"We will say that the point $x_{0}\\in M$ is controllable from a point $\\eta $ in time $T>0$ , if there is such trajectory of $x:[0,T]\\rightarrow M$ of system (2.1) that $x(0)=x_{0}, x(T)=\\eta $ .",
"Let's denote by $G_{\\eta }(<T)$ a set of points of $ M $ which are controllable from a point $ \\eta $ for time, smaller than $ T $ .We assume that $ \\eta \\in G{\\eta }(<T) $ for each $ T> 0 $ .",
"The set of all points $ M $ ,which are controllable from a point $ \\eta $ , is called as set of controllability with a target point $ \\eta $ and is denoted by $G_{\\eta }$ .",
"We will denote by $ T=T_{\\eta }(x) $ function of Bellman given on set $G_{\\eta }$ for the optimal time problem.",
"It is known that a set of smooth vector fields on a smooth manifold can be transformed into Lie algebra in which as product of vector fields $ X $ and $ Y $ serves their Lie bracket $[ X,Y] $ .",
"Let's denote by $ D $ set of vector fields $\\lbrace f(\\cdot ,u):u\\in U\\rbrace $ , by $ A(D) $ minimal Lie subalgebra, containing $D$ , by $A_{x}(D)$ the subspace tangent spaces at a point $ x\\in M $ , consisting of all vectors $\\lbrace X(x): X\\in A(D)\\rbrace $ .",
"If we will denote by $ L(\\eta ) $ a orbit of family $ D $ containing the point $ \\eta $ , then it follows from definition of the orbit that $G_{\\eta } \\subset L(\\eta ) $ for all $ \\eta \\in M $ .",
"The following assertion [43] takes place.",
"Theorem-2.1.",
"If $ dimA_{\\eta }(D)=dimL(\\eta ) $ then $ intG_{\\eta } \\ne \\varnothing $ in topology of $ L(\\eta ) $ .",
"Now let us give following definitions.",
"Definition-2.1.",
"We will say that the system (2.1) is completely controllable on $ L(\\eta _{0}) $ , if for all $ \\eta \\in L(\\eta _{0})$ it takes place equality $ G_{\\eta }=L(\\eta _{0})$ .",
"Definition- 2.2.",
"The system (2.1) is called normally -locally controllable (or, more shortly, $N$ - locally controllable) near a point $ \\eta $ if for any $T>0 $ there is a neighborhood $ V $ of the point $ \\eta $ in $ L(\\eta ) $ such that each point from $ V $ is controllable from $ \\eta $ in time, smaller $ T $ .",
"If the system is $ N $ - locally controllable near each point of $L(\\eta )$ we will say that it is $ N $ - locally controllable on $ L(\\eta )$ (see [52]).",
"Definition - 2.3.",
"We will say that the system (2.1) is completely ($ N $ -locally) controllable on invariant set $ S $ if it is completely controllable ( $ N $ - locally controllable) on each a leaf of $ S $ .",
"Assume that $ dimA_{x}(D)=k $ for every $ x\\in M $ , where $0<k<n$ , $ A_{x} (D)=\\lbrace X(x):X\\in A(D)\\rbrace $ .",
"In this case splitting of $ F $ manifold $ M $ into orbits family $ D$ is a $k$ -dimensional foliation, i.e.orbits are $ k $ dimensional submanifolds of $M$ .",
"Let's consider the following question: if the system (2.1) has property of complete controllability on one fixed leaf of the foliation $ F $ , under what conditions the system (2.1) has this property on leaves close to a given leaf?",
"This question closely related with problems of the qualitative theory of foliations on local stability of a leaf in sense of J.Reeb(see [53]).",
"In the paper [7] the answer is given to this question in the case when the leaf $ L_{0} $ of $F $ in neighborhood of which the system (2.1) is studied,is compact set.",
"In this case conditions of the theorem of J.Reeb on local stability is required.",
"Namely, the following theorem is proved.",
"Theorem-2.2.",
"Let $ L_{0}$ be a compact leaf of $ F $ with finite holonomy group.If the system (2.1) is complete controllable on $ L_{0} $ then it is is complete controllable on leaves,close to $ L_{0} $ .",
"Thus, existence of such saturated(invariant) neighborhood $ V $ of a leaf $L_0$ gives the sufficient condition for stability of the complete controllable system (2.1) on $ L_{0} $ , when $ L_{0} $ is a compact leaf.",
"As Example 3 in [7] shows, complete controllability on close leaves does not follow from the fact that the system (2.1) is complete controlled on a noncompact proper leaf $L_0$ having a neighborhood $V$ described in theorem 2.1.",
"Therefore, in a case when $L_0$ is a noncompact leaf, we need additional conditions that guarantee stability of the complete controlled system (2.1) on $L_0$ .",
"The theorem 1.11 gives the possibility to get sufficient conditions for stability of the complete controllable system (2.1) on $ L_{0} $ , when $ L_{0} $ is a noncompact leaf.",
"Theorem-2.3 [32] Let $ F $ be a transversely oriented codimension one foliation, $ L_{0} $ be a relatively compact proper leaf with finitely generated fundamental group and with trivial holonomy group .",
"Then if the system (2.1) is $ N$ -locally controllable on $\\overline{L}_0 $ (the closure in $ M$ ) then there is an open saturated neighborhood $ V $ of the leaf $ L_{0} $ such that on each leaf from $ V $ the system (2.1) is complete controllable.",
"Remark 1.",
"The closure of each leaf is an invariant set (see ([53],Theorem 4.9).",
"Remark 2.",
"If the system (2.1) is $N$ -locally controlled on a leaf $L$ then it is complete controlled on $L$ (see [39]).",
"Let now $ dimA_{x}(D)=k $ for every $x\\in M$ where $0<k<n$ , $F$ is a riemannian foliation with respect to riemannian metric $g$ .",
"We will remind that foliation $ F $ is called riemannian, if each geodesic orthogonal at some point to a leaf foliation $ F $ , remains orthogonal to leaves at all points.",
"Theorem-2.4 [39] Let $(M,g)$ be a complete riemannian manifold, $ L_{0} $ be a relatively compact proper leaf of $ F $ .Then if the system (2.1) $ N $ -is locally controllable on $\\overline{L}_0 $ (on the closure of $ L_{0} $ in $M$ )then there is an invariant neighborhood $ V $ of the leaf $ L_{0} $ such that system (1) is complete controllable on each leaf from $V $ .",
"In the paper [26] it is given the necessary and sufficient condition for singular foliation $ F $ to be riemannian foliation.",
"This condition deals with vector fields from $ A(D) $ and riemannian metric $ g $ .",
"Let now $F$ be $k $ -dimensional riemannian foliation with respect to riemannian metric $ g $ , where $0<k<n$ .",
"It is possible to present each vector field $ X\\in V(M) $ as $ X=X_{P}+X_{H} $ where $ X_{P},X_{H}$ orthogonal projections of $X $ on $ P $ and$H $ accordingly.",
"If$ X_{H}=0 $ , then $ X\\in V(F) $ and the vector field $ X$ is called tangent vector field , if $ X_{P}=0 $ then $ X\\in V(H)$ and the vector field $X$ is called horizontal field.",
"For vector fields $X,Y$ we will consider the bilinear symmetric form $g_T(X,Y)=g(X_H,Y_H)$ on $V(M)$ , kernel of which coincides with $V(F)$ .",
"We will study properties of this form.",
"By the definition of foliation for each point $p\\in M$ there is a neighborhood $U$ of the point $p$ and local system of coordinates $x^1,x^2,\\ldots x^k,y^1,y^2,\\ldots y^{n-k}$ on $U$ such that $\\frac{\\partial }{\\partial x^1},\\frac{\\partial }{\\partial x^2},\\ldots ,\\frac{\\partial }{\\partial x^k}$ form basis of sections of $ TF| _U $ .The basis $\\nu _{k+1},\\nu _{k+2},\\ldots ,\\nu _{n}$ for sections $H| _U$ can be chosen in such a manner that brackets $[X,\\nu _j]$ will be tangent vector fields to foliation $F$ for each section $X$ of the bundle $H|_U$ .Now assume that foliation $F$ is a riemannian.",
"Then for each tangent vector $ X\\in V(F)|_U$ it takes place $X_g(\\nu _i,\\nu _j)=0, \\ i,j={k+1},\\ldots ,n$ [40].",
"By using this fact, it is easy to show that for each vector field $ X\\in V(F) $ takes place $Xg_{T}(Y,Z)=g_{T}([X,Y],Z)+g_{T}([Y,[X,Z]])$ , where $Y,Z\\in V(M)$ .",
"In this case is $g_T$ called transversal metrics for foliation $F$ , defined by the riemannian metric $g$ ([26], see p.77).",
"Notice that a transversal metrics $g_T$ determines the local distance between the leaves,since it defines the length of the perpendicular geodesics.As follows from ( [26] the assertion 3.2), it is true also the converse fact i.e.",
"if it is given $k$ -dimensional foliation $F$ on riemannian manifold and riemannian metric $g$ defines transversal metric for $F$ then $F$ is a riemannian foliation with respect to riemannian metric $g$ .",
"Authors proved the similar fact for foliations with singularities.",
"Let $F$ be a foliation with singularities, $L$ is a leaf of the foliation $F$ , $Q $ is a normal bundle of $L$ .",
"Then riemannian metric $g$ defines the metric $g^L_T$ on $Q$ as follows: if $\\nu _1,\\nu _2:L\\rightarrow Q$ are smooth (of the class $C^\\infty $ ) sections of normal bundle $Q$ we will put $g^L_T(\\nu _1,\\nu _2)=g(X,Y)$ , where $X,Y\\in V(M)$ ,the restrictions of $X,Y$ on $L$ coincides with $\\nu _1,\\nu _2$ accordingly.",
"The metric $g^L_T$ is called transversal metric for $F$ on $L$ , if for each $X\\in V(F)$ at points of the leaf $L$ takes place $X{g^L_T}(Y,Z)=g^L_T([X,Y],Z)+g^L_T([Y,[X,Z]])$ where $Y,Z\\in V(M)$ , $g^L_T(Y,Z)=g(\\pi Y,\\pi Z)$ , $\\pi :TM\\rightarrow Q$ is the orthogonal projection considered over $L$ .",
"We will notice that riemannian foliation with singularities has no $n$ -dimensional leaves.There is an assumption in ([26], p. 201) that if complete riemannian metric defines on each leaf foliation $F$ transversal metric,then foliation $F$ will be a riemannian.",
"This problem is solved positively by following theorem.",
"Theorem-2.5.",
"[40] Let $M$ be a complete riemannian manifold with riemannian metric $g$ , $F$ is a singular foliation on $M$ which has no $n$ - dimensional leaves.",
"Then the foliation $F$ is a riemannian if and only if riemannian metric $g$ defines on each leaf of the foliation $F$ transversal metric.",
"Now we will assume that $x_0\\in M$ , $L_0=L(x_0)$ is a proper leaf with trivial holonomy group and the system (2.1) is completely controllable on $L_0$ .",
"Theorem-2.6 [1] Let the mapping $x\\rightarrow L(x)$ be continuous at a point $x_0$ .",
"Then the system (2.1)is completely controllable on the orbits,sufficiently close to $L_0$ .",
"Remark.",
"Multiple-valued mapping $x\\rightarrow L(x)$ is called to be lower semicontinuous at a point $x_0$ if for each open set $V$ such that $V\\bigcap L(x_0)\\ne \\emptyset $ there exists neighborhood $B_{x_0}$ of the point $x_0$ such that $L(x)\\bigcap V\\ne \\emptyset $ for $x\\in B_{x_0}$ .",
"Multiple-valued mapping $x\\rightarrow L(x)$ is called to be upper semicontinuous at a point $x_0$ if for each open set $V$ such that $L(x_0)\\subset V $ , there is a neighborhood $B_{x_0}$ of the point $x_0$ such that $L(x)\\subset V$ for all $x\\in B_{x_0}$ .",
"Multiple-valued mapping is continuous at a point $x_0$ if it simultaneously lower and upper semicontinuous at a point $x_0$ .In our case is easy to prove that mapping $x\\rightarrow L(x)$ is lower semicontinuous at each point of $M$ [36].",
"Sufficient conditions at which mapping $x\\rightarrow L(x)$ is continuous, are studied in papers of A.Narmanov [36], [37].",
"We will bring some of them.",
"The following sufficient condition on continuity of multiple-valued mapping $x\\rightarrow L(x)$ follows directly from the theorem 1.9 of the part one.",
"Theorem-2.7.",
"Assume that $dimA_x(D)=k$ for every $x\\in M$ , where $0<k<n$ .If the set $L(x_0)$ is a compact leaf with finite holonomy group then mapping $x\\rightarrow L(x)$ is continuous at the point $x_0$ .",
"The following theorem shows that if foliation $F$ is a singular riemannian foliation then the mapping $x\\rightarrow L(x)$ is continuous at each point $x_0$ .",
"Theorem-2.8 ([36]) Let $F$ is a riemannian foliation with singularities.",
"Then multiple-valued mapping $x\\rightarrow L(x)$ is continuous at each point of the manifold $M$ .",
"Now we will consider the problem on a continuity Bellman function for a optimal time problem.",
"We will remind that Bellman function $T_\\eta (x):G_\\eta \\rightarrow R^1$ is defined as follows: $T_\\eta (\\eta )=0$ , $T_\\eta (x)=inf{(\\tau : there \\ exists \\ trajectory \\ \\alpha :[0,\\tau ]\\rightarrow M }$ ${of \\ the \\ system (2.1) \\ such\\ that \\,\\alpha (0)=x, \\ \\alpha (\\tau )=\\eta )}$ .",
"The structure of set of controllability generally can be rather difficult.",
"Now we will determine a class of control systems for which a set of controllability $G_\\eta $ of the system (2.1) for all $\\eta \\in M$ coincides with a orbit $L(\\eta )$ of family of the vector fields $D=\\lbrace f(\\cdot ,u):u\\in U\\rbrace $ .",
"Definition-2.4 [43] We will say that the system (2.1) is continuously-balanced at a point $x\\in M$ if for each vector field $X\\in D$ there are vector fields $X_1,X_2,\\ldots ,X_k$ from $D$ , a neighborhood $V(x)$ of the point $x$ and the positive continuous functions $\\lambda _1(y),\\lambda _2(y),\\ldots ,\\lambda _k(y)$ which are given in this neighborhood such that for all $y\\in V(x)$ takes place equality: $X(y)+\\sum \\ {\\lambda _i(y)X_i(y)}=0.$ If we assume that the system (2.1) is continuously-balanced at each point $ x \\in M $ by means of results of work [51] it is possible to show that for each $\\eta \\in M $ the set of controllability $ G_\\eta $ of system (2.1) coincides with the orbit $L (\\eta ) $ of the family $ D=\\lbrace f(\\cdot ,u):u\\in U\\rbrace $ .",
"Definition-2.5.",
"We will say that function $ T=T_\\eta (x)$ is continuous at the point $ x_0\\in G_\\eta $ if for every $\\varepsilon >0 $ there is such neighborhood $V $ of the point $ x_0$ in topology of $ M $ that for any point $ x\\in G_\\eta \\bigcap V $ takes place inequality $\\mid T_\\eta (x)-T_\\eta (x_0)\\mid <\\varepsilon $ .",
"For the system (2.1) given on compact manifold the following result is obtained by professor N.N.Petrov [44].",
"Theorem-2.9.Let $M$ be compact manifold.",
"Then following assertions are equivalent: 1) System (2.1) is $N$ - locally controllable near the point $\\eta $ .",
"2) For each $T>0$ the set $G_\\eta (<T)$ is a domain in the manifold $L(\\eta )$ .",
"3) For each $T>0$ the level ${x\\in M:T_\\eta (x)=T}$ is the border of the set$G_\\eta (<T)$ .",
"4) Bellman function $ T=T_\\eta (x)$ is continuous at every point of $G_\\eta $ .",
"Thus,for compact manifold the problem on a continuity of Bellman function is reduced to a question about $N$ - local controllability of system (2.1).",
"For noncompact manifolds the following theorem gives the necessary and sufficient conditions of a continuity of Bellman function which is presented in [44].",
"Theorem-2.10.",
"Let the system (2.1)is continuous-balanced at each point of $M$ , for each $T>0$ the set $G_\\eta (\\le T)$ has compact closure and $dimA_x(D)=const$ for every $x\\in G_\\eta $ .Then Bellman function is continuous on $G_\\eta $ if and only if $G_\\eta $ is a proper leaf of the foliation generated by orbits of $D$ .",
"In a case when manifold $M$ is a analytic and the set $D$ consists of analytical vector fields owing to theorem Nagano [28] the condition $dimA_x(D)=const$ for all $x\\in G_\\eta $ is always satisfied.",
"Generally $dimA_x(D)$ can vary from a point to a point on $G_\\eta $ and always $ dimA_x(D)\\le dimL_\\eta $ for $x\\in G_\\eta $ .",
"For continuously-balanced control systems the following theorem takes place [37].",
"Theorem-2.11.",
"The set $G_\\eta $ is a proper leaf of the foliation generated by orbits of the family $D$ if and only if the set $G_\\eta $ is a set of type $F_\\sigma $ and $G_\\delta $ simultaneously in topology of manifold $M$ ."
]
] | 1204.0861 |
[
[
"Local linear estimator for stochastic differential equations driven by\n $\\alpha$-stable L\\'{e}vy motions"
],
[
"Abstract We study the local linear estimator for the drift coefficient of stochastic differential equations driven by $\\alpha$-stable L\\'{e}vy motions observed at discrete instants letting $T \\rightarrow \\infty$.",
"Under regular conditions, we derive the weak consistency and central limit theorem of the estimator.",
"Compare with Nadaraya-Watson estimator, the local linear estimator has a bias reduction whether kernel function is symmetric or not under different schemes."
],
[
"Continuous-time models play an important role in the study of financial time series.",
"Especially, many models in economics and finance, like those for an interest rate or an asset price involve continuous-time diffusion processes.",
"Particularly, their theoretical and empirical applications to finance are quite extensive (see Jacod and Shiryaev [18]).",
"However, growing evidence shows that stochastic processes with jumps are becoming more and more important (see Andersen et al.",
"[2]; Baskshi et al.",
"[7]; Duffie et al.",
"[11]).",
"Recently, stochastic processes with jumps as an intension of continuous-path ones have been studied by more and more statisticians since the financial phenomena can be better characterized (see A$\\ddot{1}$ t-Sahalia and Jacod [1]; Bandi and Nguyen [3]).",
"A diffusion model with continuous paths is represented by the following stochastic differential equation: $dX_{t}=\\mu (X_{t})dt+\\sigma (X_{t})dW_{t},\\qquad \\mathrm {(1.1)}$ where $W_{t}$ is a standard Brownian motion, $\\mu : \\mathbb {R}\\rightarrow \\mathbb {R}$ is an unknown measurable function and $\\sigma : \\mathbb {R} \\rightarrow \\mathbb {R}_{+}$ is an unknown positive function.",
"Many authors have investigated nonparametric estimations for the drift function $\\mu (x)$ and the diffusion function $\\sigma (x)$ , which to some extend prevent the misspecification of the model (1.1) compare with the parametric estimations.",
"Prakasa Rao [27] constructed a non-parametric estimator similar as the Nadaraya-Watson estimator for $\\mu (x)$ .",
"Bandi and Phillips [4] discuss the Nadaraya-Watson estimator for these functions of non-stationary recurrent diffusion processes.",
"Fan and Zhang [15] proposed local linear estimators for them and obtained bias reduction properties.",
"In a finite sample, Xu [34]extended re-weighted idea proposed by Hall and Presnell [16] to estimate $\\sigma (x)$ under recurrence.",
"Xu [33] discussed the empirical likelihood-based inference for nonparametric recurrent diffusions to construct confidence intervals.",
"Furthermore, Bandi and Phillips [5] proposed a simple and robust approach to specify a parameter class of diffusions and estimate the parameters of interest by minimizing criteria based on the integrated squared difference between kernel estimates of the drift and diffusion functions and their parametric counterparts.",
"Recently, stochastic processes with jumps have been paid more attention in various applications, for instance, financial time series to reflect discontinuity of asset return (see Baskshi et al.",
"[7]; Duffie et al.",
"[11]; Johannes [20]; Bandi and Nguyen [5]).",
"In this paper, we consider the stochastic process with jumps through the stochastic differential equation driven by an $\\alpha $ -stable Lévy motion (1 $< \\alpha < 2$ ): $d X_{t} = \\mu (X_{t-}) dt + \\sigma (X_{t-}) d Z_{t},~~~~~X_{0} = \\eta , \\qquad \\mathrm {(1.2)}$ where $\\lbrace Z_{t}, t \\ge 0\\rbrace $ is a standard $\\alpha $ -stable Lévy motion defined on a probability space $(\\Omega , {F}, P)$ equipped with a right continuous and increasing family of $\\sigma $ -algebras $\\lbrace {F}_{t}, t \\ge 0\\rbrace $ and $\\eta $ is a random variable independent of $\\lbrace Z_{t}\\rbrace $ .",
"$Z_{1}$ has a $\\alpha $ - stable distribution $S_{\\alpha }(1, \\beta , 0)$ with the characteristic function: $ E \\textsf {exp}\\lbrace i u Z_{1}\\rbrace = \\textsf {exp}\\left\\lbrace -|u|^{\\alpha }\\left(1 - i \\beta sgn(u) \\tan \\frac{\\alpha \\pi }{2}\\right)\\right\\rbrace , u \\in \\mathbb {R},\\qquad \\mathrm {(1.3)}$ where $\\beta \\in [-1, 1]$ is the skewness parameter.",
"One can refer to Sato [30], Barndorff-Nielsen et al.",
"[6] for more detailed properties on stable distributions.",
"Usually, we get observations $\\lbrace X_{t_{i}}, t_{i} = i \\Delta _{n}, i = 0, 1, \\ldots ,n\\rbrace $ for model (1.2), where $\\Delta _{n}$ is the time frequency for observation and $n$ is the sample size.",
"This paper is devoted to the nonparametric estimation of the unknown drift function.",
"Our estimation procedure for model (1.2) should be based on $\\lbrace X_{t_{i}}, t_{i} = i \\Delta _{n}, i = 0, 1, \\ldots , n\\rbrace .$ The stochastic differential equation driven by Lévy motion has received growing interest from both theoreticians and practitioners recently, such as applications to finance, climate dynamics et al.. Masuda ([23], [24]) proved some probabilistic properties of a multidimensional diffusion processes with jumps and provided mild regularity conditions for a multidimensional Ornstein-Uhlenbeck process driven by a general Lévy process for any initial distribution to be exponential $\\beta $ - mixing.",
"When model (1.2) is specially a mean-reverted Ornstein-Uhlenbeck process driven by a Lévy process, i.e.",
"$\\mu (x)$ is known to be linear with the form $\\mu (x) = \\gamma - \\lambda x$ and $\\sigma = 1,$ where $(\\gamma ,\\lambda )$ is unknown parameters to be estimated.",
"Based on $\\lbrace X_{t_{i}}, t_{i} = i \\Delta _{n}, i = 0, 1, \\ldots , n\\rbrace ,$ Hu and Long [17] studied the least-squares estimator for $\\lambda > 0$ when Z is symmetric $\\alpha $ -stable and $\\gamma = 0$ .",
"Masuda [25]considered an approximate self-weighted least absolute deviation type estimator for $(\\gamma , \\lambda ).$ Zhou and Yu [37]proved the asymptotic distributions of the least squares estimator of the mean reversion parameter $\\lambda $ allowing for nonlinearity in the diffusion function under three sampling schemes.",
"However, in model (1.2), the drift function $\\mu (x)$ is seldom known and the diffusion function may be nonlinear in reality.",
"With no prior specified form of the drift function, Long and Qian [22] discussed the Nadaraya-Watson estimator for it and obtained the weak consistency and central limit theorem.",
"The Nadaraya-Watson estimator given for $\\mu (x)$ is locally approximating $\\mu (x)$ by a constant (a zero-degree polynomial).",
"However, in the context of nonparametric estimator with finite-dimensional auxiliary variables, local polynomial smoothing has become the ¡°golden standard¡± (see Fan [12], Wand and Jones [36]).",
"The local polynomial estimator is known to share the simplicity and consistency of the kernel estimators as Nadaraya-Watson or Gasser-Müller estimators but avoids boundary effects, at least when convergence rates are concerned.",
"Local polynomial smoothing at a point x fits a polynomial to the pairs $(X_{i}; Y_{i})$ for those $X_{i}$ falling in a neighborhood of $x$ determined by a smoothing parameter $h$ .",
"The local polynomial estimator has received increasing attention and it has gained acceptance as an attractive method of nonparametric estimation function and its derivatives.",
"This smoothing method has become a powerful and useful diagnostic tool for data analysis.",
"In particular, the local linear estimator locally fits a polynomial of degree one.",
"In this paper, we propose the local linear estimators for drift function in model (1.2).",
"As a nonparametric methodology, local polynomial estimator makes use of the observation information to estimate corresponding functions not assuming the function form.",
"The estimator is obtained by locally fitting a polynomial of degree one to the data via weighted least squares and it shows advantages compared with Nadaraya-Watson approach (see Fan and Gijbels [13]).",
"For further motivation and study of the local linear estimator, see Fan and Gijbels [14], Ruppert and Wand [29], Stone [32], Cleveland [10].",
"The remainder of this paper is organized as follows.",
"In Section 2, local linear estimator and appropriate assumptions for model (1.2) are introduced.",
"In Section 3, we present some technical lemmas and asymptotic results.",
"The proofs will be collected in Section 4."
],
[
"Local Linear Estimator and Assumptions",
"We lay out some notations.",
"For simplify, $X_{i}$ denotes $X_{t_{i}}$ and we shall omit the subscript $n$ in the notation if no confusion will be caused.",
"We will use notation “$\\stackrel{p}{\\rightarrow }$ ” to denote “convergence in probability”, notation “$\\stackrel{a.s.}{\\rightarrow }$ ” to denote “convergence almost surely” and notation “$\\Rightarrow $ ” to denote “convergence in distribution”.",
"Local polynomial estimator firstly introduced in Fan [12] has been widely used in regression analysis and time series analysis.",
"It has gained acceptance as an attractive method of nonparametric estimation of regression function and its derivatives.",
"The estimator is obtained by locally fitting $p$ -th polynomial to the data via weighted least squares and it shows advantages compared with other kernel nonparametric regression estimators.",
"The idea of weighted local polynomial regression is the following: under some smoothness conditions of the curve $m(x)$ , we can expand $m(x)$ in a neighborhood of the point $x_{0}$ as follows: $m(x) & \\approx & m(x_{0}) + m^{^{\\prime }}(x_{0})(x - x_{0}) +\\frac{m^{^{\\prime \\prime }}(x_{0})}{2!",
"}(x - x_{0})^{2} + \\cdots +\\frac{m^{(p)}(x_{0})}{p~!",
"}(x - x_{0})^{p}\\\\& \\equiv & \\sum _{j=0}^{p}\\beta _{j}(x - x_{0})^{j},$ where $\\beta _{j} = \\frac{m^{(j)}(x_{0})}{j~!",
"}.$ Thus, the problem of estimating infinite dimensional $m(x)$ is equivalent to estimating the $p$ -dimensional parameter $\\beta _{0},\\beta _{0}, \\cdots , \\beta _{p}.$ Consider a weighted local polynomial regression: $\\arg \\min _{\\beta _{0},\\beta _{1},\\cdots ,\\beta _{p}} \\sum _{i=0}^{n-1}\\Big \\lbrace Y_{i} - \\sum _{j=0}^{p}\\beta _{j}(X_{i} -x)^{j}\\Big \\rbrace ^{2}K_{h_{n}}(X_{i} - x),$ where $Y_{i} = \\frac{X_{i+1}- X_{i}}{\\Delta }$ and $K_{h_{n}}(\\cdot )=\\frac{1}{h_{n}}K(\\frac{\\cdot }{h_{n}}).$ is kernel function with $h_{n}$ the bandwidth.",
"What we are interested in is to estimate $\\hat{\\mu }(x) =\\hat{\\beta }_{0}$ , hence as Fan and Gijbels [14] remarked, it is reasonable for us to discuss $p = 1$ : the local linear estimator for the drift function $\\mu (\\cdot )$ in this paper.",
"The local linear estimator for $\\mu (x)$ is the solution $\\beta _{0}$ of the optimal problem: $\\arg \\min _{\\beta _{0},\\beta _{1}} \\sum _{i=0}^{n-1}\\Big \\lbrace Y_{i} - \\sum _{j=0}^{1}\\beta _{j}(X_{i} -x)^{j}\\Big \\rbrace ^{2}K_{h}(X_{i} - x).$ The solution of $\\beta _{0}$ is $\\hat{\\mu }(x) = \\frac{\\sum \\limits _{i=0}^{n-1}K_{h}(X_{i} - x)\\left\\lbrace \\frac{S_{n,2}}{h^{2}} - (\\frac{X_{i} -x}{h})\\frac{S_{n1}}{h}\\right\\rbrace (X_{i+1} -X_{i})}{\\Delta \\sum \\limits _{i=0}^{n-1}K_{h}(X_{i} -x)\\left\\lbrace \\frac{S_{n,2}}{h^{2}} - (\\frac{X_{i} -x}{h})\\frac{S_{n1}}{h}\\right\\rbrace },$ where $S_{n,k} =\\sum \\limits _{i=0}^{n-1}{K}_{h}(X_{i} - x)(X_{i} - x)^{k} , k=1,2.$ We can also write $\\hat{\\mu }(x) =\\frac{\\hat{g}_{n}(x)}{\\hat{h}_{n}(x)},$ $\\hat{h}_{n}(x) =\\frac{1}{n}\\sum \\limits _{i=0}^{n-1}K_{h}(X_{i} - x)\\left\\lbrace \\frac{S_{n,2}}{nh^{2}} - \\left(\\frac{X_{i} -x}{h}\\right)\\frac{S_{n1}}{nh}\\right\\rbrace ,$ $\\hat{g}_{n}(x) =\\frac{1}{n\\Delta }\\sum \\limits _{i=0}^{n-1}K_{h}(X_{i} - x)\\left\\lbrace \\frac{S_{n,2}}{nh^{2}} - \\left(\\frac{X_{i} -x}{h}\\right)\\frac{S_{n1}}{nh}\\right\\rbrace (X_{i+1} - X_{i}).$ We now present some assumptions used in this paper.",
"$\\bf {(A.1).",
"}$ The drift function $\\mu (\\cdot )$ is twice continuously differentiable with bounded first and second order derivatives; the diffusion function $\\sigma (\\cdot )$ satisfy a global Lipschitz condition, there exists a positive constant C $>$ 0 such that $|\\sigma (y) - \\sigma (x)| \\le C |y - x|,~ y,~x~\\in \\mathbb {R}.$ $\\bf {(A.2).",
"}$ There exist positive constants $\\sigma _{0}$ and $\\sigma _{1}$ such that $0 < \\sigma _{0} \\le \\sigma (x) \\le \\sigma _{1}$ for each $x~\\in \\mathbb {R}.$ $\\bf {(A.3).",
"}$ The solution $X_{t}$ admits a unique invariant distribution $F(x)$ and is geometrically strong mixing, i.e.",
"there exists $c_{0} > 0$ and $\\rho \\in (0 , 1)$ such that $\\alpha _{X}(t)\\le c_{0}\\rho ^{t},~t \\ge 0.$ $\\bf {(A.4).",
"}$ The density function $f(x)$ of the stationary distribution $F(x)$ is continuously differentiable and $f(x)$ $>$ 0.",
"$\\bf {(A.5).",
"}$ The kernel function $K(\\cdot )$ is nonnegative probability density function with compact support satisfying: $K_{2}:= \\int _{-\\infty }^{+\\infty }u^{2}K(u)du <\\infty ,~~~\\int _{-\\infty }^{+\\infty }K^{2}(u)du < \\infty .$ $\\bf {(A.6).",
"}$ As $n \\rightarrow \\infty ,~ h \\rightarrow 0,~ \\Delta \\rightarrow 0,~ \\and ~ nh\\Delta \\rightarrow \\infty .$ $\\bf {Remark 2.1.",
"}$ The condition (A.1) ensures that (1.2) admits a unique non-plosive càdlàg adapted solution, see Jacod and Shiryaev [18].",
"(A.3) implies $X_{t}$ is ergodic and stationary.",
"The mixing property of a stochastic process describes the temporal dependence in data.",
"One can refer to Bradley [9] for different kinds of mixing properties.",
"For some sufficient conditions which guarantee (A.3), one can refer to Masuda [24].",
"The kernel function is not necessarily to be symmetric.",
"Sometimes, unilateral kernel function may make predictor easier (see Fan and Zhang [15])."
],
[
"Some Technical Lemmas and Asymptotic Results",
"We say that a continuous function $G~:[0, \\infty ) \\rightarrow [0,\\infty )$ grows more slowly than $u^{\\alpha }$ ($\\alpha > 0$ ) if there exist positive constants $c, \\lambda _{0} \\and \\alpha _{0} < \\alpha $ such that $G(\\lambda u) \\le c\\lambda ^{\\alpha _{0}}G(u)$ for all $u> 0$ and all $\\lambda \\ge \\lambda _{0}$ .",
"$\\bf {Lemma~ 3.1.",
"}$ Let $\\phi (t)$ be a predictable process satisfying $\\int _{0}^{T}|\\phi (t)|^{\\alpha }dt < \\infty $ almost surely for $T < \\infty .$ We assume that either $\\phi $ is nonnegative or $Z$ is symmetric.",
"If $G(u)$ grows more slowly than $u^{\\alpha }$ , then there exist positive constants $c_{1}$ and $c_{2}$ depending only on $\\alpha , \\alpha _{0}, c ~and \\lambda _{0}$ such that for each $T> 0$ $c_{1}E[G((\\int _{0}^{T}|\\phi (t)|^{\\alpha }dt)^{1/{\\alpha }})] \\le E[G(\\sup _{t \\le T}|\\int _{0}^{t}\\phi (s)dZ_{s}|)]\\le c_{2}E[G((\\int _{0}^{T}|\\phi (t)|^{\\alpha }dt)^{1/{\\alpha }})].$ $\\bf {Remark~ 3.1.",
"}$ This lemma can be viewed as a generalization of Theorem 3.2 in Rosonski and Woyczynski [28], where they only dealt with the case that $Z$ is symmetric.",
"$\\bf {Lemma~ 3.2.",
"}$ Suppose that there is a deterministic and nonnegative function $\\Phi $ such that $\\Phi ^{\\alpha }(T)\\int _{0}^{T}|\\phi (t)|^{\\alpha }dt \\stackrel{p}{\\rightarrow }1 ~~~as~~~ T \\rightarrow \\infty .$ Then, we have $\\Phi (T)\\int _{0}^{T}|\\phi (t)|dZ_{t} \\Longrightarrow S_{\\alpha }(1, \\beta , 0).$ $\\bf {Remark~3.2.",
"}$ This lemma can be regarded as an extending to $\\alpha $ -stable case of Theorem 1.19 in Kutoyants [21].",
"$\\bf {Lemma~ 3.3.",
"}$ Assumptions (A.1) - (A.6) lead to the following result: $\\frac{1}{n}\\sum _{i=0}^{n-1}K_{h}(X_{i} - x)\\left(\\frac{X_{i} - x}{h}\\right)^{k}\\stackrel{a.s.}{\\longrightarrow } f(x)\\int _{- \\infty }^{+\\infty }u^{k}K(u)du$ $\\bf {Remark~3.3.",
"}$ In Long and Qian [22], they proved a weaker case: $\\frac{1}{n}\\sum \\limits _{i=0}^{n-1}K_{h}(X_{i} - x)\\stackrel{p}{\\longrightarrow } f(x).$ One can easily obtain $\\hat{h}_{n}(x) \\stackrel{a.s.}{\\longrightarrow } K_{2}f^{2}(x) -(K_{1}f(x))^{2}$ based on this lemma.",
"$\\bf {Theorem~ 3.1.",
"}$ Assume that (A.1)-(A.6) hold and $\\alpha \\in (1, 2)$ , then $\\hat{\\mu }(x)\\stackrel{p}{\\longrightarrow } \\mu (x)$ as $n \\rightarrow \\infty .$ $\\bf {Theorem~ 3.2.",
"}$ Let $\\alpha \\in [1, 2)$ and assume that (A.1) - (A.6) are satisfied.",
"$(i)$ If $(n \\Delta h)^{1 - \\frac{1}{\\alpha }} h = O(1)$ and $(n\\Delta h)^{1 - \\frac{1}{\\alpha }} \\Delta ^{\\frac{1}{\\kappa }} = O(1)$ for some $\\kappa > \\alpha $ , then $(n \\Delta h)^{1 - \\frac{1}{\\alpha }} \\Lambda (x) (\\hat{\\mu }(x) - \\mu (x)) \\Rightarrow S_{\\alpha }(1, \\beta , 0)$ $(ii)$ If $(n \\Delta h)^{1 - \\frac{1}{\\alpha }} h^{2} = O(1)$ and $(n\\Delta h)^{1 - \\frac{1}{\\alpha }} \\Delta ^{\\frac{1}{\\kappa }} = O(1)$ for some $\\kappa > \\alpha $ , then $(n \\Delta h)^{1 - \\frac{1}{\\alpha }} \\Lambda (x) (\\hat{\\mu }(x) - \\mu (x) - h^{2}\\Gamma _{\\mu }(x)) \\Rightarrow S_{\\alpha }(1, \\beta , 0)$ where $\\Lambda (x) = \\frac{[K_{2} - (K_{1})^{2}]f(x)^{1 -\\frac{1}{\\alpha }}}{\\sigma (x)\\big (\\int _{- \\infty }^{+\\infty }K^{\\alpha }(u)\\lbrace K_{2} - uK_{1}\\rbrace ^{\\alpha }du\\big )^{\\frac{1}{\\alpha }}},$ and $\\Gamma _{\\mu }(x) =\\frac{\\mu ^{^{\\prime \\prime }}(x)[(K_{2})^{2} - K_{1}K_{3}]}{2 \\big (K_{2} -(K_{1})^{2}\\big )}.$ $\\bf {Remark~3.4.",
"}$ In Long and Qian [22], they showed the following results under the Assumptions in this paper: $(i)$ If $(n\\Delta h)^{1 - \\frac{1}{\\alpha }} h = O(1)$ and $(n \\Delta h)^{1 -\\frac{1}{\\alpha }} \\Delta ^{\\frac{1}{\\kappa }} = O(1)$ for some $\\kappa > \\alpha $ , then $(n \\Delta h)^{1 - \\frac{1}{\\alpha }} \\Lambda (x) (\\hat{\\mu }(x) - \\mu (x)- hK_{1}) \\Rightarrow S_{\\alpha }(1, \\beta , 0)$ $(ii)$ If $(n \\Delta h)^{1 - \\frac{1}{\\alpha }} h^{2} =O(1)$ and $(n \\Delta h)^{1 - \\frac{1}{\\alpha }}\\Delta ^{\\frac{1}{\\kappa }} = O(1)$ for some $\\kappa > \\alpha $ , $K(\\cdot )$ is symmetric,, then $(n \\Delta h)^{1 - \\frac{1}{\\alpha }} \\Lambda (x) (\\hat{\\mu }(x) - \\mu (x) - h^{2}\\Gamma _{\\mu }(x)) \\Rightarrow S_{\\alpha }(1, \\beta , 0)$ where $\\Lambda (x) = \\frac{f(x)^{1 -\\frac{1}{\\alpha }}}{\\sigma (x)\\big (\\int _{- \\infty }^{+\\infty }K^{\\alpha }(u)\\big )^{\\frac{1}{\\alpha }}},$ and $\\Gamma _{\\mu }(x)= \\big [\\mu ^{^{\\prime }}(x)\\frac{f^{^{\\prime }}(x)}{f(x)} +\\frac{1}{2}\\mu ^{^{\\prime \\prime }}(x)\\big ]K_{2}.$ We can easily observe that the bias in the local linear case is smaller than the one in the Nadaraya-Watson case in comparison to the results between this paper and Long & Qian [22] whether $K(\\cdot )$ is symmetric or not.",
"Furthermore, when $\\alpha = 1$ , $\\hat{\\mu }(x)$ is inconsistent with $\\mu (x)$ easily obtained from Theorem 3.2."
],
[
"Proofs",
"$\\bf {Proof ~of ~Lemma ~3.1.",
"}$ See Long and Qian (Lemma 2.7).",
"$\\bf {Proof ~of ~Lemma ~3.2.",
"}$ See Long and Qian (Lemma 2.6).",
"$\\bf {Proof ~of ~Lemma ~3.3.",
"}$ We first note that $& ~ &\\frac{1}{n}\\sum _{i=0}^{n-1}K_{h}(X_{i} - x)\\left(\\frac{X_{i} -x}{h}\\right)^{k} - f(x)\\int _{- \\infty }^{+ \\infty }u^{k}K(u)du\\\\& = & \\frac{1}{n}\\sum _{i=0}^{n-1}K_{h}(X_{i} - x)\\left(\\frac{X_{i} -x}{h}\\right)^{k} - \\frac{1}{n}\\sum _{i=0}^{n-1}E\\left[K_{h}(X_{i} -x)\\left(\\frac{X_{i} -x}{h}\\right)^{k}\\right]\\\\& ~ & + \\frac{1}{n}\\sum _{i=0}^{n-1}E\\left[K_{h}(X_{i} -x)\\left(\\frac{X_{i} - x}{h}\\right)^{k}\\right] - f(x)\\int _{-\\infty }^{+ \\infty }u^{k}K(u)du.",
"\\hspace{170.71652pt}(4.1)$ From the stationarity of $X_{t}$ , we have: $\\frac{1}{n}\\sum _{i=0}^{n-1}E\\left[K_{h}(X_{i} - x)\\left(\\frac{X_{i}- x}{h}\\right)^{k}\\right] & = & E\\left[K_{h}(X_{1} - x)(\\frac{X_{1}- x}{h})^{k}\\right]\\\\& = & \\int _{- \\infty }^{+ \\infty }K_{h}(y - x)\\left(\\frac{y -x}{h}\\right)^{k}f(y)dy\\\\& = & \\int _{- \\infty }^{+ \\infty }K(u)u^{k}f(x + uh)du\\\\& \\rightarrow & f(x)\\int _{- \\infty }^{+ \\infty }u^{k}K(u)du.\\hspace{190.63345pt}(4.2)$ Thus, from (4.1) and (4.2) it suffices to prove that $& ~ &\\frac{1}{n}\\sum _{i=0}^{n-1}K_{h}(X_{i} - x)\\left(\\frac{X_{i} -x}{h}\\right)^{k} - \\frac{1}{n}\\sum _{i=0}^{n-1}E\\left[K_{h}(X_{i} -x)\\left(\\frac{X_{i} - x}{h}\\right)^{k}\\right]\\\\& = & \\frac{1}{n}\\sum _{i=0}^{n-1}\\left\\lbrace K_{h}(X_{i} -x)\\left(\\frac{X_{i} - x}{h}\\right)^{k} - E\\left[K_{h}(X_{i} -x)\\left(\\frac{X_{i} - x}{h}\\right)^{k}\\right]\\right\\rbrace \\\\& =: & \\frac{1}{n}\\sum _{i=0}^{n-1} \\delta _{n,i}(x) \\stackrel{a.s.}{\\longrightarrow } 0.",
"\\hspace{367.04054pt}(4.3)$ Note that $\\sup \\limits _{0 \\le i \\le n-1} |\\delta _{n,i}(x)| \\le C_{0}h^{-1}$ a.s. for some positive constant $C_{0} < \\infty $ by the compact support of $K(\\cdot ).$ Applying Theorem 1.3 (2) in Bosq [8], we have for each integer $q \\in [1 , \\frac{n}{2}]$ and each $\\varepsilon > 0$ $P\\left(\\frac{1}{n}\\left|\\sum _{i=0}^{n-1}\\delta _{n,i}(x)\\right|> \\varepsilon \\right) \\le 4 \\textsf {exp}\\left( -\\frac{\\varepsilon ^{2}q}{8\\nu ^{2}(q)}\\right) + 22\\left( 1 + \\frac{4C_{0}h^{-1}}{\\varepsilon }\\right)^{1/2} q\\alpha _{X}([p]\\Delta ),\\qquad \\mathrm {(4.4)}$ where $\\nu ^{2}(q) = \\frac{2}{p^{2}} s(q) +\\frac{C_{0}h^{-1}\\varepsilon }{2}$ with $p = \\frac{n}{2q}$ and $s(q) & = & \\max _{0 \\le j \\le 2q -1} E[([jp] + 1-jp)\\delta _{n,[jp]+1}(x) + \\delta _{n,[jp]+2}(x) + \\cdots \\\\ & ~ & +~\\delta _{n,[(j+1)p]}(x) + ((j + 1)p - [(j + 1)p])\\delta _{n,[(j +1)p]+1}(x)]^{2}.$ Using the Hölder inequality and stationarity of $X_{i}$ , one can easily obtain that $s(q) = O(p^{2} h^{-1})$ .",
"By choosing $q = [\\sqrt{n\\Delta } / \\sqrt{h}]$ and $p = \\frac{n}{2q}= O(\\sqrt{nh} / \\sqrt{\\Delta })$ , we get $\\frac{\\varepsilon ^{2}q}{8\\nu ^{2}(q)} = \\varepsilon ^{2} \\cdot O(qh) = O(\\varepsilon ^{2}\\sqrt{n \\Delta h}).\\qquad \\mathrm {(4.5)}$ Moreover, we can obtain $22\\left( 1 + \\frac{4C_{0}h^{-1}}{\\varepsilon }\\right)^{1/2} q \\alpha _{X}([p]\\Delta ) \\le C(\\varepsilon ) \\textsf {exp}(- O(\\varepsilon ^{2}\\sqrt{n\\Delta h}))\\qquad \\mathrm {(4.6)}$ under the mixing properties of $X_{t}$ in (A.3) and (A.6).",
"(4.4), (4.5) and (4.6) imply $P\\left(\\frac{1}{n}\\left|\\sum _{i=0}^{n-1}\\delta _{n,i}(x)\\right|> \\varepsilon \\right) \\le C(\\varepsilon )\\textsf {exp}(- O(\\varepsilon ^{2}\\sqrt{n\\Delta h})).$ Therefore, $\\frac{1}{n}\\sum \\limits _{i=0}^{n-1}K_{h}(X_{i}- x)\\left(\\frac{X_{i} - x}{h}\\right)^{k} -\\frac{1}{n}\\sum \\limits _{i=0}^{n-1}E\\left[K_{h}(X_{i} -x)\\left(\\frac{X_{i} - x}{h}\\right)^{k}\\right] \\stackrel{a.s.}{\\longrightarrow } 0$ based on Borel-Cantelli lemma and (A.6).",
"$\\bf {Proof ~of ~Theorem ~3.1.",
"}$ It suffices to prove that $\\hat{g}_{n}(x) \\stackrel{p}{\\longrightarrow } [K_{2}f^{2}(x) - (K_{1}f(x))^{2}]\\mu (x).$ By (1.2), we first note that $\\hat{g}_{n}(x)\\!\\!\\!",
"& = &\\!\\!\\!\\frac{1}{n\\Delta }\\sum \\limits _{i=0}^{n-1}K_{h}(X_{i} - x)\\left\\lbrace \\frac{S_{n,2}}{nh^{2}} - \\left(\\frac{X_{i} -x}{h}\\right)\\frac{S_{n1}}{nh}\\right\\rbrace (X_{i+1} - X_{i})\\\\& = & \\!\\!\\!\\frac{1}{n\\Delta }\\sum \\limits _{i=0}^{n-1}K_{h}(X_{i} - x)\\left\\lbrace \\frac{S_{n,2}}{nh^{2}} - \\left(\\frac{X_{i} -x}{h}\\right)\\frac{S_{n1}}{nh}\\right\\rbrace \\left(\\int _{t_{i}}^{t_{i+1}}\\mu (X_{s-}) ds~ + ~\\int _{t_{i}}^{t_{i+1}}\\sigma (X_{s-}) dZ_{s})\\right)\\\\& = & \\!\\!\\!\\frac{1}{n\\Delta }\\sum \\limits _{i=0}^{n-1}K_{h}(X_{i} - x)\\left\\lbrace \\frac{S_{n,2}}{nh^{2}} - \\left(\\frac{X_{i} -x}{h}\\right)\\frac{S_{n1}}{nh}\\right\\rbrace \\left(\\mu (X_{i})\\Delta +\\int _{t_{i}}^{t_{i+1}}(\\mu (X_{s-}) - \\mu (X_{i})) ds + \\int _{t_{i}}^{t_{i+1}}\\sigma (X_{s-}) dZ_{s}\\right)\\\\& =: & g_{n,1}(x) + g_{n,2}(x) + g_{n,3}(x).$ To show the convergence of $\\hat{g}_{n}(x)$ , we should prove the following three results: (i) $g_{n,1}(x) \\stackrel{p}{\\longrightarrow } [K_{2}f^{2}(x) -(K_{1}f(x))^{2}]\\mu (x),~ as ~ n \\rightarrow \\infty ~;$ (ii) $g_{n,2}(x) \\stackrel{p}{\\longrightarrow } 0,~ as ~ n\\rightarrow \\infty ~;$ (iii) $g_{n,3}(x) \\stackrel{p}{\\longrightarrow } 0,~ as ~ n\\rightarrow \\infty ~;$ $\\bf {Proof~of~(i):}$ $g_{n,1}(x) & = & \\mu (x)\\frac{1}{n}\\sum \\limits _{i=0}^{n-1}K_{h}(X_{i}- x) \\left\\lbrace \\frac{S_{n,2}}{nh^{2}} - \\left(\\frac{X_{i} -x}{h}\\right)\\frac{S_{n1}}{nh}\\right\\rbrace \\\\& ~ & + \\frac{1}{n}\\sum \\limits _{i=0}^{n-1}K_{h}(X_{i} - x)\\left\\lbrace \\frac{S_{n,2}}{nh^{2}} - \\left(\\frac{X_{i} -x}{h}\\right)\\frac{S_{n1}}{nh}\\right\\rbrace (\\mu (X_{i}) - \\mu (x))\\\\& =: & A_{n,1}(x) + A_{n,2}(x).",
"\\hspace{335.74251pt}(4.7)$ Using Lemma 3.3, it is obviously that $A_{n,1}(x)\\stackrel{a.s.}{\\longrightarrow } \\mu (x)[K_{2}f^{2}(x) -(K_{1}f(x))^{2}].\\qquad \\mathrm {(4.8)}$ By the Lipschitz property of $\\mu (x)$ and the stationarity of $X_{t}$ , we have $|A_{n,2}(x)| \\le \\frac{L}{n}\\sum _{i=0}^{n-1}|X_{i} - x|K_{h}(X_{i}- x)\\left|\\frac{S_{n,2}}{nh^{2}}\\right| +\\frac{L}{n}\\sum _{i=0}^{n-1}|X_{i} - x|K_{h}(X_{i} -x)\\left|\\frac{X_{i} - x}{h}\\right|\\left|\\frac{S_{n,1}}{nh}\\right|,\\qquad \\mathrm {(4.9)}$ where $L$ denotes the bound of the first derivative of $\\mu (x)$ .",
"The two components of the right part are dealt with in the same way, so we only deal with the first one for convenience.",
"$\\frac{1}{n}\\sum _{i=0}^{n-1}|X_{i} - x|K_{h}(X_{i} -x)\\left|\\frac{S_{n,2}}{nh^{2}}\\right|\\!\\!\\!\\!\\!\\!",
"&=&\\!\\!\\!\\!\\!\\!\\frac{1}{n}\\sum _{i=0}^{n-1}\\left(|X_{i} - x|K_{h}(X_{i} -x)\\left|\\frac{S_{n,2}}{nh^{2}}\\right| - E\\left[|X_{i} -x|K_{h}(X_{i} -x)\\left|\\frac{S_{n,2}}{nh^{2}}\\right|\\right]\\right)\\\\& &\\!\\!\\!\\!\\!\\!+ \\frac{1}{n}\\sum _{i=0}^{n-1}E\\left[|X_{i} -x|K_{h}(X_{i} -x)\\left|\\frac{S_{n,2}}{nh^{2}}\\right|\\right].\\hspace{133.72795pt}(4.10)$ We find that $|X_{i} - x|K_{h}(X_{i} -x)\\left|\\frac{S_{n,2}}{nh^{2}}\\right| - E\\left[|X_{i} -x|K_{h}(X_{i} - x)\\left|\\frac{S_{n,2}}{nh^{2}}\\right|\\right]$ is a.s. uniformly bounded for each i by Lemma 3.3 and the compact support of $K(\\cdot )$ .",
"Similar as the proof of (4.3), we can show that $\\frac{1}{n}\\sum _{i=0}^{n-1}\\left(|X_{i} - x|K_{h}(X_{i} -x)\\left|\\frac{S_{n,2}}{nh^{2}}\\right| - E\\left[|X_{i} -x|K_{h}(X_{i} - x)\\left|\\frac{S_{n,2}}{nh^{2}}\\right|\\right]\\right)\\stackrel{p}{\\longrightarrow } 0.\\qquad \\mathrm {(4.11)}$ As for the second part, $\\lim _{h \\rightarrow 0}\\frac{1}{n}\\sum _{i=0}^{n-1}E\\left[|X_{i} -x|K_{h}(X_{i} - x)\\left|\\frac{S_{n,2}}{nh^{2}}\\right|\\right] \\!\\!\\!&= &\\!\\!\\!",
"\\lim _{h \\rightarrow 0}E\\left[|X_{1} - x|K_{h}(X_{1} -x)\\left|\\frac{S_{n,2}}{nh^{2}}\\right|\\right]\\\\& = & \\lim _{h \\rightarrow 0}K_{2}f(x)E\\left[|X_{1} -x|K_{h}(X_{1} - x)\\right]\\\\& = & \\lim _{h \\rightarrow 0}hK_{2}f(x)\\int _{- \\infty }^{+\\infty }|u|K(u)f(x + uh)du\\\\& = & \\lim _{h \\rightarrow 0}hK_{2}f^{2}(x)\\int _{- \\infty }^{+\\infty }|u|K(u)du \\rightarrow 0.",
"\\hspace{93.89409pt}(4.12)$ It follows that $A_{n,2} \\stackrel{p}{\\longrightarrow } 0 $ as $n \\rightarrow \\infty $ .",
"Hence $g_{n,1}(x)\\stackrel{p}{\\longrightarrow } [K_{2}f^{2}(x) -(K_{1}f(x))^{2}]\\mu (x)$ by (4.7)-(4.12).",
"$\\bf {Proof~of~(ii):}$ We first introduce a basic inequality for (1.2): $\\sup _{t_{i} \\le t \\le t_{i+1}}|X_{t} - X_{t_{i}}| \\le e^{L\\Delta }\\left(|\\mu (X_{i})|\\Delta + \\sup _{t_{i} \\le t \\le t_{i+1}}\\left|\\int _{t_{i}}^{t}\\sigma (X_{s-})dZ_{s}\\right|\\right),\\qquad \\mathrm {(4.13)}$ which one can refer to Long & Qian [22], Shimizu & Yoshida [31] and Jacod & Protter [19].",
"$|g_{n,2}(x)| & \\le &\\frac{1}{n\\Delta }\\sum \\limits _{i=0}^{n-1}K_{h}(X_{i} - x)\\left\\lbrace \\left|\\frac{S_{n,2}}{nh^{2}}\\right| + \\left|\\frac{X_{i} -x}{h}\\right|\\left|\\frac{S_{n1}}{nh}\\right|\\right\\rbrace \\int _{t_{i}}^{t_{i+1}} |\\mu (X_{s-}) - \\mu (X_{i})| ds\\\\& \\le & \\frac{L}{n\\Delta }\\sum \\limits _{i=0}^{n-1}K_{h}(X_{i} - x)\\left\\lbrace \\left|\\frac{S_{n,2}}{nh^{2}}\\right| + \\left|\\frac{X_{i} -x}{h}\\right|\\left|\\frac{S_{n1}}{nh}\\right|\\right\\rbrace \\int _{t_{i}}^{t_{i+1}} |X_{s-} - X_{i}| ds\\\\& \\le & \\frac{L}{n}\\sum \\limits _{i=0}^{n-1}K_{h}(X_{i} - x)\\left\\lbrace \\left|\\frac{S_{n,2}}{nh^{2}}\\right| + \\left|\\frac{X_{i} -x}{h}\\right|\\left|\\frac{S_{n1}}{nh}\\right|\\right\\rbrace \\sup _{t_{i} \\le t\\le t_{i+1}}|X_{t} - X_{t_{i}}|\\\\& \\le & \\frac{Le^{L\\Delta }}{n}\\sum \\limits _{i=0}^{n-1}K_{h}(X_{i} -x) \\left\\lbrace \\left|\\frac{S_{n,2}}{nh^{2}}\\right| + \\left|\\frac{X_{i} -x}{h}\\right|\\left|\\frac{S_{n1}}{nh}\\right|\\right\\rbrace \\left(|\\mu (X_{i})|\\Delta + \\sup _{t_{i} \\le t \\le t_{i+1}}\\left|\\int _{t_{i}}^{t}\\sigma (X_{s-})dZ_{s}\\right|\\right)\\\\& =: & A_{n,3}(x) + A_{n,4}(x) + A_{n,5}(x).",
"\\hspace{278.837pt}(4.14)$ Similarly as the proof of (4.1), we know that $\\frac{L\\Delta e^{L\\Delta }}{n}\\sum \\limits _{i=0}^{n-1}K_{h}(X_{i}- x) \\left\\lbrace \\left|\\frac{S_{n,2}}{nh^{2}}\\right| + \\left|\\frac{X_{i}- x}{h}\\right|\\left|\\frac{S_{n1}}{nh}\\right|\\right\\rbrace |\\mu (X_{i})| -Q(x) \\stackrel{p}{\\rightarrow } 0,$ where $Q(x) = L\\Delta e^{L\\Delta }|\\mu (x)|\\left[K_{2}f^{2}(x) + \\left(\\int _{- \\infty }^{+\\infty }|u|K(u)du\\right)^{2}f^{2}(x)\\right] \\rightarrow 0,$ that is $\\frac{L\\Delta e^{L\\Delta }}{n}\\sum \\limits _{i=0}^{n-1}K_{h}(X_{i} - x)\\left\\lbrace \\left|\\frac{S_{n,2}}{nh^{2}}\\right| + \\left|\\frac{X_{i} -x}{h}\\right|\\left|\\frac{S_{n1}}{nh}\\right|\\right\\rbrace |\\mu (X_{i})|\\stackrel{p}{\\rightarrow } 0.\\qquad \\mathrm {(4.15)}$ $A_{n,4}(x), A_{n,5}(x)$ are dealt with the same approach, hence here we only verify $A_{n,4}(x) \\stackrel{p}{\\rightarrow } 0.$ As for $A_{n,4}(x)$ , according to Lemma 3.3, we only need to verify $\\frac{1}{n}\\sum \\limits _{i=0}^{n-1}K_{h}(X_{i} -x)\\sup _{t_{i} \\le t \\le t_{i+1}}\\left|\\int _{t_{i}}^{t}\\sigma (X_{s-})dZ_{s}\\right|\\stackrel{p}{\\rightarrow } 0.\\qquad \\mathrm {(4.16)}$ By Markov inequality and Lemma 3.1, we have $P\\left(\\frac{1}{n}\\sum \\limits _{i=0}^{n-1}K_{h}(X_{i} - x)\\sup _{t_{i}\\le t \\le t_{i+1}}\\left|\\int _{t_{i}}^{t}\\sigma (X_{s-})dZ_{s}\\right| >\\varepsilon \\right) & \\le &\\frac{1}{n\\varepsilon }\\sum _{i=0}^{n-1}E\\left[\\sup _{t_{i} \\le t \\le t_{i+1}}\\left|\\int _{t_{i}}^{t}K_{h}(X_{i} -x)\\sigma (X_{s-})dZ_{s}\\right|\\right]\\\\& \\le &\\frac{c_{1}}{n\\varepsilon }\\sum _{i=0}^{n-1}E\\left[\\left(\\int _{t_{i}}^{t_{i+1}}K_{h}^{\\alpha }(X_{i}- x)\\sigma ^{\\alpha }(X_{s-})ds\\right)^{1/\\alpha }\\right]\\\\& \\le &\\frac{c_{1}}{n\\varepsilon }\\sum _{i=0}^{n-1}E\\left[K_{h}(X_{i} - x)\\sigma _{1}\\Delta ^{1/\\alpha }\\right]\\\\& \\le & O(\\Delta ^{1/\\alpha }) \\rightarrow 0.\\hspace{139.4185pt}(4.17)\\\\$ Now, $g_{n,2}(x) \\stackrel{p}{\\longrightarrow } 0$ by (4.14), (4.15) and (4.16).",
"$\\bf {Proof~of~(iii):}$ $|g_{n,3}(x)| & = &\\left|\\frac{1}{n\\Delta }\\sum \\limits _{i=0}^{n-1}K_{h}(X_{i} - x)\\left\\lbrace \\frac{S_{n,2}}{nh^{2}} - \\left(\\frac{X_{i} -x}{h}\\right)\\frac{S_{n1}}{nh}\\right\\rbrace \\left(\\int _{t_{i}}^{t_{i+1}}\\sigma (X_{s-}) dZ_{s}\\right)\\right|\\\\& \\le & \\left|\\frac{S_{n,2}}{nh^{2}}\\right|\\cdot \\frac{1}{n\\Delta }\\left|\\sum \\limits _{i=0}^{n-1}K_{h}(X_{i} -x)\\int _{t_{i}}^{t_{i+1}}\\sigma (X_{s-}) dZ_{s}\\right|\\\\ & ~ & +\\left|\\frac{S_{n,1}}{nh}\\right|\\cdot \\frac{1}{n\\Delta }\\left|\\sum \\limits _{i=0}^{n-1}K_{h}(X_{i} -x)\\left(\\frac{X_{i} -x}{h}\\right)\\int _{t_{i}}^{t_{i+1}}\\sigma (X_{s-})dZ_{s}\\right|.\\hspace{128.0374pt}(4.18)$ By Lemma 3.3, we only need to prove $\\frac{1}{n\\Delta }\\left|\\sum \\limits _{i=0}^{n-1}K_{h}(X_{i} -x)\\int _{t_{i}}^{t_{i+1}}\\sigma (X_{s-}) dZ_{s}\\right|\\stackrel{p}{\\rightarrow } 0 \\qquad \\mathrm {(4.19)}$ and $\\frac{1}{n\\Delta }\\left|\\sum \\limits _{i=0}^{n-1}K_{h}(X_{i} -x)\\left(\\frac{X_{i} -x}{h}\\right)\\int _{t_{i}}^{t_{i+1}}\\sigma (X_{s-}) dZ_{s}\\right|\\stackrel{p}{\\rightarrow } 0.\\qquad \\mathrm {(4.20)}$ We only prove (4.19) for simplicity.",
"Denote $\\phi _{n,1}(t, x) = \\sum _{i=0}^{n-1}\\frac{1}{h^{1/\\alpha }}K\\left(\\frac{X_{i} -x}{h}\\right)\\sigma (X_{t-}) {1}_{(t_{i}, t_{i+1}]}(t),$ we have $\\frac{1}{n\\Delta }\\left|\\sum \\limits _{i=0}^{n-1}K_{h}(X_{i} -x)\\int _{t_{i}}^{t_{i+1}}\\sigma (X_{s-}) dZ_{s}\\right| =\\frac{1}{n\\Delta h^{\\frac{\\alpha -1}{\\alpha }}}\\left|\\int _{0}^{t_{n}}\\phi _{n,1}(t, x) dZ_{t}\\right|.$ With the same argument as the proof of $A_{n, 4}(x)$ , we have that: $P\\left(\\frac{1}{n\\Delta }\\left|\\sum \\limits _{i=0}^{n-1}K_{h}(X_{i} -x)\\int _{t_{i}}^{t_{i+1}}\\sigma (X_{s-}) dZ_{s}\\right| >\\varepsilon \\right) = O \\left((n\\Delta h)^{\\frac{1-\\alpha }{\\alpha }}\\right).\\qquad \\mathrm {(4.21)}$ We get $g_{n,3}(x) \\stackrel{p}{\\longrightarrow } 0$ by (4.18)-(4.21) and Assumption (A.6).",
"$\\bf {Proof ~of ~Theorem ~3.2.",
"}$ Note that $(n\\Delta h)^{1 - \\frac{1}{\\alpha }}\\Lambda (x) (\\hat{\\mu }(x) - \\mu (x)) =\\frac{(n\\Delta h)^{1 -\\frac{1}{\\alpha }}\\Lambda (x)\\left[\\hat{g}_{n}(x) -\\mu (x)\\hat{h}_{n}(x)\\right]}{\\hat{h}_{n}(x)} =:\\frac{B_{n}(x)}{\\hat{h}_{n}(x)}.$ We have obtained $\\hat{h}_{n}(x) \\rightarrow [K_{2} -K_{1}^{2}]f^{2}(x)$ applying Lemma 3.3, so it is enough to study the asymptotic behavior of $B_{n}(x)$ .",
"$B_{n}(x) & = & (n\\Delta h)^{1 -\\frac{1}{\\alpha }}\\Lambda (x)\\left[g_{n,1}(x) -\\mu (x)\\hat{h}_{n}(x)\\right] + (n\\Delta h)^{1 -\\frac{1}{\\alpha }}\\Lambda (x)g_{n,2}(x) + (n\\Delta h)^{1 -\\frac{1}{\\alpha }}\\Lambda (x)g_{n,3}(x)\\\\& =: & B_{n,1}(x) + B_{n,2}(x) + B_{n,3}(x).\\hspace{281.68228pt}(4.22)$ $\\bf {Proof~of~ B_{n,1}(x):}$ We can express $ B_{n,1}(x)$ as $ B_{n,1}(x) = (n\\Delta h)^{1 -\\frac{1}{\\alpha }}\\Lambda (x)\\frac{1}{n}\\sum \\limits _{i=0}^{n-1}K_{h}(X_{i} - x)\\left\\lbrace \\frac{S_{n,2}}{nh^{2}} - \\left(\\frac{X_{i} -x}{h}\\right)\\frac{S_{n1}}{nh}\\right\\rbrace (\\mu (X_{i}) - \\mu (x)).$ Using Taylor's expansion, we get $\\mu (X_{i}) - \\mu (x) = \\mu ^{^{\\prime }}(x)(X_{i} - x) + \\frac{1}{2}\\mu ^{^{\\prime \\prime }}(x + \\theta _{i}(X_{i} - x))(X_{i} - x)^{2},$ where $\\theta _{i}$ is some random variable satisfying $\\theta _{i}\\in [0, 1].$ Under a simple calculus, we have $\\sum \\limits _{i=0}^{n-1}K_{h}(X_{i} - x)\\left\\lbrace \\frac{S_{n,2}}{nh^{2}} - \\left(\\frac{X_{i} -x}{h}\\right)\\frac{S_{n1}}{nh}\\right\\rbrace \\cdot (X_{i}- x) \\equiv 0,$ so $B_{n,1}(x) & = & (n\\Delta h)^{1 - \\frac{1}{\\alpha }}\\Lambda (x)\\frac{1}{n}\\sum \\limits _{i=0}^{n-1}K_{h}(X_{i} - x)\\left\\lbrace \\frac{S_{n,2}}{nh^{2}} - \\left(\\frac{X_{i} -x}{h}\\right)\\frac{S_{n1}}{nh}\\right\\rbrace \\frac{1}{2}\\mu ^{^{\\prime \\prime }}(x)(X_{i} -x)^{2}\\\\& ~ & + (n\\Delta h)^{1 - \\frac{1}{\\alpha }}\\Lambda (x)\\frac{1}{n}\\sum \\limits _{i=0}^{n-1}K_{h}(X_{i} - x)\\left\\lbrace \\frac{S_{n,2}}{nh^{2}} - \\left(\\frac{X_{i} -x}{h}\\right)\\frac{S_{n1}}{nh}\\right\\rbrace (X_{i} -x)^{2}\\frac{1}{2}[\\mu ^{^{\\prime \\prime }}(x + \\theta _{i}(X_{i} - x))-\\mu ^{^{\\prime \\prime }}(x)]\\\\& =: & B_{n,1}^{(1)}(x) + B_{n,1}^{(2)}(x).",
"\\hspace{330.05196pt}(4.23)$ By the stationary of $X_{t}$ , we get $B_{n,1}^{(1)}(x) & = & \\frac{1}{2}\\mu ^{^{\\prime \\prime }}(x)(n\\Delta h)^{1 -\\frac{1}{\\alpha }}\\Lambda (x)E\\left[K_{h}(X_{1} - x)\\left\\lbrace \\frac{S_{n,2}}{nh^{2}} - \\left(\\frac{X_{1} -x}{h}\\right)\\frac{S_{n1}}{nh}\\right\\rbrace (X_{1} - x)^{2}\\right]\\\\& ~ & + \\frac{1}{2}\\mu ^{^{\\prime \\prime }}(x)(n\\Delta h)^{1 -\\frac{1}{\\alpha }}\\Lambda (x)\\frac{1}{n}\\sum _{i=0}^{n-1}\\Big \\lbrace K_{h}(X_{i}- x) \\left\\lbrace \\frac{S_{n,2}}{nh^{2}} - \\left(\\frac{X_{i} -x}{h}\\right)\\frac{S_{n1}}{nh}\\right\\rbrace (X_{i} - x)^{2}\\\\ & ~ & -E\\left[K_{h}(X_{i} - x) \\left\\lbrace \\frac{S_{n,2}}{nh^{2}} -\\left(\\frac{X_{i} - x}{h}\\right)\\frac{S_{n1}}{nh}\\right\\rbrace (X_{i} -x)^{2}\\right]\\Big \\rbrace \\\\& = & D_{n,1}(x) + D_{n,2}(x).",
"\\hspace{330.05196pt}(4.24)$ One can easily obtain $D_{n,1}(x) = \\frac{1}{2}\\mu ^{^{\\prime \\prime }}(x)\\Lambda (x)[(K_{2})^{2} - K_{1}K_{3}](f(x))^{2}(n\\Delta h)^{1 -\\frac{1}{\\alpha }}h^{2}(1 + o(1)).\\qquad \\mathrm {(4.25)}$ Denote $D_{n,2}(x) := \\frac{1}{2}\\mu ^{^{\\prime \\prime }}(x)\\Lambda (x)\\frac{1}{n}\\sum _{i=0}^{n-1}\\xi _{n,i}.\\qquad \\mathrm {(4.26)}$ Note that $\\sup \\limits _{0 \\le i \\le n-1}|\\xi _{n,i}| \\le M_{1}(n\\Delta h)^{1 - \\frac{1}{\\alpha }}h$ a.s. for some positive constant $M_{1} < \\infty .$ It follows from the proof of (4.3) that $\\frac{1}{n}\\sum \\limits _{i=0}^{n-1}\\xi _{n,i}\\stackrel{p}{\\rightarrow } 0$ in exponential rate.",
"However, in addition, we should calculate $\\tilde{s}(q)$ , which may be a little different from $s(q).$ $\\tilde{s}(q) & = & \\max _{0 \\le j \\le 2q -1} E[([jp] + 1-jp)\\xi _{n,[jp]+1}(x) + \\xi _{n,[jp]+2}(x) + \\cdots \\\\ & ~ & +~\\xi _{n,[(j+1)p]}(x) + ((j + 1)p - [(j + 1)p])\\xi _{n,[(j +1)p]+1}(x)]^{2}.$ Using Billingsley's inequality in Bosq [8], $\\tilde{s}(q)= O(p(n\\Delta h)^{2\\left(1 -\\frac{1}{\\alpha }\\right)}\\Delta ^{-1}h^{2}).$ $|B_{n,1}^{(2)}(x)| & \\le & \\frac{1}{2}\\Lambda (x)\\sup \\limits _{|x -y| \\le M h} |\\mu ^{^{\\prime \\prime }}(x) - \\mu ^{^{\\prime \\prime }}(y)|(n\\Delta h)^{1 -\\frac{1}{\\alpha }}\\Big (\\frac{1}{n}\\sum \\limits _{i=0}^{n-1}K_{h}(X_{i}- x) (X_{i} - x)^{2}\\left|\\frac{S_{n,2}}{nh^{2}}\\right|\\\\& & + \\frac{1}{nh}\\sum \\limits _{i=0}^{n-1}K_{h}(X_{i} - x)|X_{i} -x|^{3}\\left|\\frac{S_{n1}}{nh}\\right|\\Big )\\\\& = & o(1)(n\\Delta h)^{1 - \\frac{1}{\\alpha }} h^{2} \\cdot \\frac{1}{n}\\sum \\limits _{i=0}^{n-1}K_{h}(X_{i} - x)\\\\& = & o_{p}(1) (n\\Delta h)^{1 - \\frac{1}{\\alpha }} h^{2}.\\hspace{327.20668pt}(4.27)$ In conclusion, it follows from (4.23)-(4.27) that $B_{n,1}(x) = o_{p}(1) + o_{p}(1) (n\\Delta h)^{1 -\\frac{1}{\\alpha }} h^{2} +\\frac{1}{2}\\mu ^{^{\\prime \\prime }}(x)\\Lambda (x)[(K_{2})^{2} -K_{1}K_{3}](f(x))^{2}(n\\Delta h)^{1 - \\frac{1}{\\alpha }}h^{2}(1 +o(1)).\\qquad \\mathrm {(4.28)}$ $\\bf {Proof~of~ B_{n,2}(x):}$ By (4.13), we have $B_{n,2}(x) & = & (n\\Delta h)^{1 -\\frac{1}{\\alpha }}\\Lambda (x)g_{n,2}(x)\\\\& \\le & (n\\Delta h)^{1 - \\frac{1}{\\alpha }}\\Lambda (x)\\Big [L\\Delta e^{L\\Delta }\\frac{1}{n}\\sum \\limits _{i=0}^{n-1}K_{h}(X_{i} - x)\\left\\lbrace \\left|\\frac{S_{n,2}}{nh^{2}}\\right| + \\left|\\frac{X_{i} -x}{h}\\right|\\left|\\frac{S_{n1}}{nh}\\right|\\right\\rbrace |\\mu (X_{i})|\\\\ & ~&+L\\Delta ^{\\frac{1}{\\kappa }}\\frac{1}{n\\Delta ^{\\frac{1}{\\kappa }}}\\sum \\limits _{i=0}^{n-1}K_{h}(X_{i}- x) \\left\\lbrace \\left|\\frac{S_{n,2}}{nh^{2}}\\right| + \\left|\\frac{X_{i}- x}{h}\\right|\\left|\\frac{S_{n1}}{nh}\\right|\\right\\rbrace \\sup _{t_{i} \\le t \\le t_{i+1}}\\left|\\int _{t_{i}}^{t}\\sigma (X_{s-})dZ_{s}\\right|\\Big ].\\hspace{28.45274pt}(4.29)$ Similar to the proof of (4.1), under given conditions and Lemma 3.3, we can obtain $\\frac{1}{n}\\sum \\limits _{i=0}^{n-1}K_{h}(X_{i} - x)\\left\\lbrace \\left|\\frac{S_{n,2}}{nh^{2}}\\right| + \\left|\\frac{X_{i} -x}{h}\\right|\\left|\\frac{S_{n1}}{nh}\\right|\\right\\rbrace |\\mu (X_{i})|\\stackrel{p}{\\rightarrow } \\left[K_{2}f^{2}(x) +|K_{1}|f^{2}(x)\\int _{- \\infty }^{+ \\infty }|u|K(u)du\\right]|\\mu (x)|.\\qquad \\mathrm {(4.30)}$ As in the proof of (4.17) and Lemma 3.3, we can show that $P\\left(\\frac{1}{n\\Delta ^{\\frac{1}{\\kappa }}}\\sum \\limits _{i=0}^{n-1}K_{h}(X_{i}- x) \\left\\lbrace \\left|\\frac{S_{n,2}}{nh^{2}}\\right| + \\left|\\frac{X_{i}- x}{h}\\right|\\left|\\frac{S_{n1}}{nh}\\right|\\right\\rbrace \\sup _{t_{i} \\le t \\le t_{i+1}}\\left|\\int _{t_{i}}^{t}\\sigma (X_{s-})dZ_{s}\\right| >\\varepsilon \\right) \\le O(\\Delta ^{\\frac{1}{\\alpha } -\\frac{1}{\\kappa }}).\\qquad \\mathrm {(4.31)}$ It follows from (4.29)-(4.31) and $\\kappa > \\alpha $ that $B_{n,2}(x) = O_{p}(1)\\cdot (n\\Delta h)^{1 -\\frac{1}{\\alpha }}\\Delta + o_{p}(1)\\cdot (n\\Delta h)^{1 -\\frac{1}{\\alpha }}\\Delta ^{\\frac{1}{\\kappa }}.\\qquad \\mathrm {(4.32)}$ $\\bf {Proof~of~ B_{n,3}(x):}$ According to (4.18), $B_{n,3}(x) & = & (n\\Delta h)^{1 -\\frac{1}{\\alpha }}\\Lambda (x)\\frac{1}{n\\Delta }\\sum \\limits _{i=0}^{n-1}K_{h}(X_{i}- x) \\left\\lbrace \\frac{S_{n,2}}{nh^{2}} - \\left(\\frac{X_{i} -x}{h}\\right)\\frac{S_{n1}}{nh}\\right\\rbrace \\left(\\int _{t_{i}}^{t_{i+1}}\\sigma (X_{s-}) dZ_{s}\\right)\\\\& =: & (n\\Delta h)^{1 - \\frac{1}{\\alpha }}\\Lambda (x) B_{n,3}^{^{\\prime }}(x).\\hspace{304.44447pt}(4.33)$ Denote $\\phi _{n,2}(t, x) = \\sum _{i=0}^{n-1}\\frac{1}{h^{1/\\alpha }}K\\left(\\frac{X_{i} -x}{h}\\right)\\left\\lbrace \\frac{S_{n,2}}{nh^{2}} - \\left(\\frac{X_{i} -x}{h}\\right)\\frac{S_{n1}}{nh}\\right\\rbrace \\sigma (X_{t-}){1}_{(t_{i},t_{i+1}]}(t),$ then $B_{n,3}^{^{\\prime }}(x) = \\int _{0}^{t_{n}}\\phi _{n,2}(t, x) dZ_{t}.$ Let $\\Phi _{t_{n}} = \\left[t_{n} \\sigma ^{\\alpha }(x)f(x)\\int _{- \\infty }^{+ \\infty }K^{\\alpha }(u)\\left\\lbrace K_{2}f(x) -uK_{1}f(x)\\right\\rbrace ^{\\alpha }du\\right]^{-\\frac{1}{\\alpha }}.$ Then, we have $\\Phi ^{\\alpha }_{t_{n}} \\cdot \\int _{0}^{t_{n}} \\phi _{n,2}^{\\alpha }(t,x) dt & = & \\Phi ^{\\alpha }_{t_{n}} \\cdot \\sum _{i=0}^{n-1}\\frac{1}{h}K^{\\alpha }\\left(\\frac{X_{i} -x}{h}\\right)\\left\\lbrace \\frac{S_{n,2}}{nh^{2}} - \\left(\\frac{X_{i} -x}{h}\\right)\\frac{S_{n1}}{nh}\\right\\rbrace ^{\\alpha }\\int _{t_{i}}^{t_{i +1}}\\sigma ^{\\alpha }(X_{s-})ds\\\\& = & \\Phi ^{\\alpha }_{t_{n}} \\cdot \\sum _{i=0}^{n-1}\\frac{1}{h}K^{\\alpha }\\left(\\frac{X_{i} -x}{h}\\right)\\left\\lbrace \\frac{S_{n,2}}{nh^{2}} - \\left(\\frac{X_{i} -x}{h}\\right)\\frac{S_{n1}}{nh}\\right\\rbrace ^{\\alpha }\\sigma ^{\\alpha }(X_{i})\\Delta \\\\& ~ & + \\Phi ^{\\alpha }_{t_{n}} \\cdot \\sum _{i=0}^{n-1}\\frac{1}{h}K^{\\alpha }\\left(\\frac{X_{i} -x}{h}\\right)\\left\\lbrace \\frac{S_{n,2}}{nh^{2}} - \\left(\\frac{X_{i} -x}{h}\\right)\\frac{S_{n1}}{nh}\\right\\rbrace ^{\\alpha }\\int _{t_{i}}^{t_{i +1}}(\\sigma ^{\\alpha }(X_{s-}) -\\sigma ^{\\alpha }(X_{i}))ds\\\\& =: & I + J.\\hspace{344.27834pt}(4.34)$ It follows from the proof of (4.1) that $\\frac{1}{n}\\sum _{i=0}^{n-1}\\frac{1}{h}K^{\\alpha }\\left(\\frac{X_{i} -x}{h}\\right)\\left\\lbrace \\frac{S_{n,2}}{nh^{2}} - \\left(\\frac{X_{i} -x}{h}\\right)\\frac{S_{n1}}{nh}\\right\\rbrace ^{\\alpha }\\sigma ^{\\alpha }(X_{i})\\stackrel{p}{\\rightarrow } \\sigma ^{\\alpha }(x)f(x)\\int _{- \\infty }^{+\\infty } K^{\\alpha }(u)\\left\\lbrace K_{2}f(x) -uK_{1}f(x)\\right\\rbrace ^{\\alpha }du.$ Therefore, we have $I & = & \\frac{1}{\\sigma ^{\\alpha }(x)f(x)\\int _{- \\infty }^{+ \\infty }K^{\\alpha }(u)\\left\\lbrace K_{2}f(x) - uK_{1}f(x)\\right\\rbrace ^{\\alpha }du} \\cdot \\frac{1}{n}\\sum _{i=0}^{n-1}\\frac{1}{h}K^{\\alpha }\\left(\\frac{X_{i}-x}{h}\\right) \\left\\lbrace \\frac{S_{n,2}}{nh^{2}} - \\left(\\frac{X_{i} -x}{h}\\right)\\frac{S_{n1}}{nh}\\right\\rbrace ^{\\alpha }\\sigma ^{\\alpha }(X_{i})\\\\& \\stackrel{p}{\\rightarrow } & 1.\\hspace{446.70827pt}(4.35)$ Next we deal with the second term J.",
"By the mean-value theorem, the Lipschitz condition (A.1) and (4.13), we have $|J| & = &\\Phi _{t_{n}}^{\\alpha }\\left|\\sum _{i=0}^{n-1}\\frac{1}{h}K^{\\alpha }\\left(\\frac{X_{i}-x}{h}\\right) \\left\\lbrace \\frac{S_{n,2}}{nh^{2}} - \\left(\\frac{X_{i} -x}{h}\\right)\\frac{S_{n1}}{nh}\\right\\rbrace ^{\\alpha }\\int _{t_{i}}^{t_{i +1}}(\\sigma ^{\\alpha }(X_{s-}) - \\sigma ^{\\alpha }(X_{i}))ds\\right|\\\\& \\le &\\Phi _{t_{n}}^{\\alpha }\\sum _{i=0}^{n-1}\\frac{1}{h}K^{\\alpha }\\left(\\frac{X_{i}-x}{h}\\right) \\left\\lbrace \\frac{S_{n,2}}{nh^{2}} + \\left|\\frac{X_{i} -x}{h}\\right|\\left|\\frac{S_{n1}}{nh}\\right|\\right\\rbrace ^{\\alpha }\\int _{t_{i}}^{t_{i+ 1}}|\\sigma ^{\\alpha }(X_{s-}) - \\sigma ^{\\alpha }(X_{i})|ds\\\\& \\le &\\frac{C}{n\\Delta }\\sum _{i=0}^{n-1}\\frac{1}{h}K^{\\alpha }\\left(\\frac{X_{i}-x}{h}\\right) \\left\\lbrace \\frac{S_{n,2}}{nh^{2}} + \\left|\\frac{X_{i} -x}{h}\\right|\\left|\\frac{S_{n1}}{nh}\\right|\\right\\rbrace ^{\\alpha }\\int _{t_{i}}^{t_{i+ 1}}\\alpha |\\zeta ^{\\alpha - 1}||\\sigma (X_{s-}) - \\sigma (X_{i})|ds\\\\& \\le &\\frac{C}{n\\Delta }\\sum _{i=0}^{n-1}\\frac{1}{h}K^{\\alpha }\\left(\\frac{X_{i}-x}{h}\\right) \\left\\lbrace \\frac{S_{n,2}}{nh^{2}} + \\left|\\frac{X_{i} -x}{h}\\right|\\left|\\frac{S_{n1}}{nh}\\right|\\right\\rbrace ^{\\alpha } \\cdot \\Delta \\cdot \\sup \\limits _{t_{i}\\le t \\le t_{i+1}}|X_{t} -X{i}|\\\\& \\le & \\frac{Ce^{L\\Delta }}{n}\\sum _{i=0}^{n-1}\\frac{1}{h}K^{\\alpha }\\left(\\frac{X_{i}-x}{h}\\right) \\left\\lbrace \\frac{S_{n,2}}{nh^{2}} + \\left|\\frac{X_{i} -x}{h}\\right|\\left|\\frac{S_{n1}}{nh}\\right|\\right\\rbrace ^{\\alpha }|\\mu (X_{i})|\\Delta \\\\& ~ & + \\frac{Ce^{L\\Delta }}{n}\\sum _{i=0}^{n-1}\\frac{1}{h}K^{\\alpha }\\left(\\frac{X_{i}-x}{h}\\right) \\left\\lbrace \\frac{S_{n,2}}{nh^{2}} + \\left|\\frac{X_{i} -x}{h}\\right|\\left|\\frac{S_{n1}}{nh}\\right|\\right\\rbrace ^{\\alpha }\\sup \\limits _{t_{i}\\le t \\le t_{i+1}}|\\int _{t_{i}}^{t}\\sigma (X_{s-})dZ_{s}|\\\\& =: & J_{1} + J_{2}, \\hspace{401.18385pt}(4.36)$ where $\\zeta $ is some random variable satisfying $\\zeta \\in [\\sigma (X_{s-}) , \\sigma (X_{i})]$ or $\\zeta \\in [\\sigma (X_{i}) ,\\sigma (X_{s-})].$ Applying Lemma 3.3, and the similar proof of Lemma 3.3, we can obtain $\\frac{1}{n}\\sum _{i=0}^{n-1}\\frac{1}{h}K^{\\alpha }\\left(\\frac{X_{i}-x}{h}\\right) \\left\\lbrace \\frac{S_{n,2}}{nh^{2}} + \\left|\\frac{X_{i} -x}{h}\\right|\\left|\\frac{S_{n1}}{nh}\\right|\\right\\rbrace ^{\\alpha }|\\mu (X_{i})|\\stackrel{p}{\\rightarrow } |\\mu (x)|f(x) \\int _{- \\infty }^{+ \\infty }K^{\\alpha }(u)\\left\\lbrace K_{2}f(x) + |u||K_{1}|f(x)\\right\\rbrace ^{\\alpha }du,$ hence $J_{1} \\stackrel{p}{\\rightarrow } 0.$ As for $J_{2}\\stackrel{p}{\\rightarrow } 0$ , one can refer to (4.31).",
"From (4.33)-(4.36) and Lemma 3.2, we have $\\Phi _{t_{n}} B^{^{\\prime }}_{n,3}(x) \\Rightarrow S_{\\alpha }(1, \\beta , 0).\\qquad \\mathrm {(4.37)}$ Therefore, it follows from (4.33), (4.37) and Lemma 3.3 that $B_{n,3}(x) & = & [K_{2}f^{2}(x) - (K_{1}f(x))^{2}] \\cdot \\Phi _{t_{n}} \\cdot B^{^{\\prime }}_{n,3}(x)\\\\& \\Rightarrow & [K_{2}f^{2}(x) - (K_{1}f(x))^{2}]S_{\\alpha }(1,\\beta , 0).$ By Slutsky's theorem and remark 3.3, we find $& ~ &(n \\Delta h)^{1 - \\frac{1}{\\alpha }} \\Lambda (x) (\\hat{\\mu }(x) -\\mu (x) - h^{2}\\Gamma _{\\mu }(x))\\\\& = & \\frac{B_{n}(x)}{\\hat{h}_{n}(x)} - (n \\Delta h)^{1 -\\frac{1}{\\alpha }} h^{2} \\Lambda (x)\\Gamma _{\\mu }(x)\\\\& = & \\frac{B_{n}(x) - (n \\Delta h)^{1 - \\frac{1}{\\alpha }} h^{2}\\Lambda (x)\\Gamma _{\\mu }(x)h(x)}{\\hat{h}_{n}(x)} + (n \\Delta h)^{1 -\\frac{1}{\\alpha }} h^{2}\\Lambda (x)\\Gamma _{\\mu }(x)\\left(\\frac{h(x)}{\\hat{h}_{n}(x)} -1\\right)\\\\& \\Rightarrow & S_{\\alpha }(1, \\beta , 0).$ This complete the proof of (ii) in Theorem 3.2.",
"The proof of (i) is similar."
],
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] | 1204.1454 |
[
[
"Phase Transition and Anisotropic Deformations of Neutron Star Matter"
],
[
"Abstract The Skyrme model is a low energy, effective field theory for QCD which when coupled to a gravitational field provides an ideal semi-classical model to describe neutron stars.",
"We use the Skyrme crystal solution composed of a lattice of $\\alpha$-like particles as a building block to construct minimum energy neutron star configurations, allowing the crystal to be strained anisotropically.",
"We find that below 1.49 solar masses the stars' crystal deforms isotropically and that above this critical mass, it undergoes anisotropic strain.",
"We then find that the maximum mass allowed for a neutron star is 1.90 solar masses, in close agreement with a recent observation of the most massive neutron star yet found.",
"The radii of the computed solutions also match the experimentally estimated values of approximately 10km."
],
[
"Introduction",
"Neutron stars are stars that have collapsed under intense self gravitational pressure to the point where all electrons are squeezed into nuclei, hence forming a large cluster of neutrons with a typical radius of about 10km.",
"A neutron star can thus be seen as a gigantic nuclei that is electrically neutral but is strongly affected by the gravitational field that it generates.",
"For lack of a unified theory of strong interactions and gravity one has to resort to finding an approximate theory that allows us to describe such a system.",
"One such theory is the Skyrme model.",
"Originally proposed by Skyrme in 1961 [1], [2] as a nonlinear theory of pions to describe strong interactions, it was later shown by Witten [3] to be an approximate, low energy, effective field theory for QCD which becomes more exact as the number of quark colours becomes large.",
"Each solution of the Skyrme model is characterised by an integer valued topological charge which can be identified with the baryon number $B$ .",
"The simplest solution, $B=1$ , is made out of a so-called Skyrmion and corresponds to a proton or neutron.",
"At the semi-classical level, the Skyrme model does not distinguish between a neutron and a proton.",
"Moreover, as the model does not include the electroweak interaction, all Skyrmions are electrically neutral.",
"The $B=1$ solution of the Skyrme model is the only exact stable solution that can be computed easily [1].",
"Solutions with larger values of $B$ can only be computed numerically [4],[5] and these solutions have been shown to successfully describe various nuclei and their properties [6].",
"Moreover, one can also compute crystal-like solutions made out of an infinite number of Skyrmions.",
"In particular, it has been shown that the Skyrme solution with the lowest energy per Skyrmion corresponds to a cubic lattice where each lattice unit has a topological charge $B=4$ [7].",
"These solutions can thus be seen as a crystal of $\\alpha $ particles.",
"This lattice of Skyrmions looks thus as the best building block to describe a neutron star as its has the lowest possible energy per baryon.",
"Yet, one must first estimate if a star could instead be made out of a liquid or gas of Skyrmions.",
"The temperature of a neutron star, a few years after its creation, cools down to an approximate temperature of about $100\\mbox{eV}\\approx 10^6K$ [8].",
"While this looks like a very high temperature compared to the binding energy of an electron around a nucleus, this energy is quite small from a nuclear point of view.",
"Indeed, the lowest excited state of an $\\alpha $ particle, for example, is $23.3$ MeV [9] and the lowest vibration mode of a $B=4$ Skyrmion is of the order of $100\\mbox{MeV}$ [10],[11].",
"Even under intense gravitational energy, Walhout [12] showed that the excitation energy of a lattice of $B=1$ Skyrmions is also of the order of $100\\mbox{MeV}$ .",
"This points out that the neutron star will be in a solid phase rather than a liquid or a gas and that the thermal energy will only excite acoustic phonon modes.",
"It is thus natural to model a neutron star as a lattice of $B=4$ Skyrmions.",
"Before we proceed we must also question the possibility of having an atmosphere around the star, and to estimate its height if it turns out not to be small.",
"At the surface of a neutron star twice the mass of the sun, the gravitational acceleration is $g\\approx 2.6\\times 10^{12}\\mbox{ms}^{-2}$ .",
"It is then easy to compute that the average height that an $\\alpha $ particle with a thermal energy of $100\\mbox{eV}$ will be able to jump is of the order of $1mm$ , i.e.",
"much smaller than the radius of the star.",
"We can thus consider that such an atmosphere is extremely thin and assume in our model that the neutron star is fully made out of a solid.",
"Having established that we can use a Skyrme crystal as the building block to describe a neutron star we will proceed as follows.",
"First of all, we will use the equations of state computed by Castillejo et al.",
"[7] for the $B=4$ crystal when the lattice is deformed asymmetrically.",
"Following Walhout [12] we will then use a Tolman-Oppenheimer-Volkoff (TOV) equation [13], [14], generalising it to allow for matter to be anisotropic [15].",
"The TOV equation describes the static equilibrium between matter forces within a solid or fluid and the gravitational forces self-generated by the matter for a spherically symmetric body.",
"Combining the TOV equation with the equations of state of the Skyrme crystal, we will be able to find configurations that are spherically symmetric distributions of anisotropically deformed matter in static equilibrium and so are suitable to model neutron stars.",
"Solving these equations numerically for large stars we will show that below a critical mass of $1.49$ solar masses ($M_\\odot =1.98892\\times 10^{30}$ kg) all neutron/Skyrmion stars are made out of an isotropically strained crystal.",
"We will then show that at this critical mass, there is a phase transition and that heavier stars are made out of an anisotropically deformed crystal that is less strained radially than tangentially.",
"We will also show that these stars can have a mass of up to $1.90M_\\odot $ .",
"Finally, we will investigate the impact of adding a mass term to the Skyrme model and describe what happens to a star when its mass is increased above its maximum value.",
"Using Skyrmions to model neutron stars is not new and has been performed previously in several ways.",
"First of all, Walhout used a lattice of $B=1$ Skyrmions [12] to describe a neutron star.",
"He then improved his results by considering a lattice of $B=4$ Skyrmions [16].",
"In both cases he assumed an isotropic compression of the lattice, assuming a gas-like phase, and he used numerical solutions of the model to estimate the stress tensor.",
"The maximum mass he obtained for the neutron star was $2.57M_\\odot $ .",
"Later, Jaikumar and Ouyed [17] considered the equation of state for a neutron star based on a Skyrme fluid and obtained a maximum mass of $3.6M_\\odot $ .",
"The main difference between these 2 approaches and ours is that they assumed an isotropic fluid of Skyrmions whereas we consider a solid crystal allowed to deform anisotropically, i.e.",
"be compressed differently in the radial and tangential directions of the star.",
"In our previous papers [18], [19], we computed minimal energy Skyrmion stars made out of layers of 2 dimensional Skyrme lattices.",
"This allowed us to use the rational map ansatz [20] to minimise the energy directly but resulted in relatively small stars with a maximum mass of $0.574M_\\odot $ .",
"This was mainly due to the fact that the field transition between the different layers in our ansatz over estimated the energy of the configuration and that the energy per baryon in each layer of the ansatz was also larger than that of the crystal of $B=4$ Skyrmions."
],
[
"Skyrme Crystals",
"The Skyrme model [1], [2] is described by the Lagrangian $\\mathcal {L}_{Sk} = \\frac{F_{\\pi } ^2}{16}{\\rm {Tr}} (\\nabla _{\\mu } U \\nabla ^{\\mu } U^{-1})+ \\frac{1}{32 e^2} {\\rm {Tr}} [(\\nabla _{\\mu } U) U^{-1},(\\nabla _{\\nu } U) U^{-1}]^2 + \\frac{m_{\\pi }^2F_{\\pi }^2}{8}{\\rm {Tr}}(U-1),$ where here $U$ , the Skyrme field, is an $SU(2)$ matrix and $F_{\\pi }$ , $e$ and $m_{\\pi }$ are the pion decay constant, the Skyrme coupling and the pion mass term respectively.",
"In the Lagrangian (REF ) the $\\nabla $ are ordinary partial derivatives in the absence of a gravitational field and become covarient derivatives when the Skyrme field is coupled to gravity.",
"The results summarised in this section all relate to the pure Skyrme model without gravity.",
"The Skyrme field is a map from $\\mathbb {R}^{3}$ to $S^3$ , the group manifold of $SU(2)$ , but finite energy considerations imply that the field at spatial infinity should map to the same point, meaning the Skyrme field is a map between two three-spheres.",
"Such maps fall into homotopy classes indexed by an integer, known as the topological charge, which is interpreted as the baryon number, $B$ .",
"The topological soliton solutions, known as Skyrmions, are identified as baryons with an $\\alpha $ particle described by a $B=4$ Skyrmion solution.",
"Here we will be considering the zero pion mass case where $m_{\\pi }=0$ with section $4.3$ describing the effects of its inclusion.",
"The two other Skyrme parameters, $F_\\pi $ and $e$ can be obtained in different ways.",
"Skyrme first evaluated them by taking the experimental value of the pion decay constant $F_\\pi =186\\mbox{MeV}$ and then fitting the mass of a Skyrmion to that of a proton and obtained $e=4.84$ .",
"Later Adkins, Nappi and Witten [21] quantised the $B=1$ Skyrmion to fit the parameter values to the mass of the nucleon and the delta excitation and obtained $F_\\pi =129\\mbox{MeV}$ and $e=5.45$ .",
"These later values were the ones used by Castillejo et al.",
"[7] to compute the energy of the deformed $B=4$ crystal and we will thus use them too.",
"The solution of the Skyrme model with the lowest energy per baryon has been shown to be a face-centred cubic (fcc) lattice of Skyrmions [22], [7].",
"Each unit cell is a cube of side length $a$ with a baryon number of $B=4$ and can therefore be considered as an $\\alpha $ particle.",
"In the context of a neutron star, we will be able to interpret each $B=4$ crystal component as being 4 neutrons as the Skyrme model does not distinguish between neutrons and protons.",
"Castillejo et al.",
"[7] also investigated the energy of dense Skyrmion crystals where the configuration was not a face-centred cubic lattice but rather a lattice where the aspect ratio of the unit cell, $B=4$ $\\alpha $ particle, of side $a$ was altered so that it becomes rectangular with aspect ratio $r^3$ .",
"This means that in the $x$ and $y$ directions the lattice size becomes $ra$ and in the $z$ direction, $a/r^2$ .",
"As in Castillejo et al.",
"we use the measure $p=r-1/r$ to describe the deviation away from the face-centred cubic lattice symmetries which have $p=0$ .",
"The numerical solutions found in [7] provide an equation for the dependence of the energy of a single Skyrmion, $E(L,p)$ , on its size, $L=n ^{-1/3}$ , where $n$ is the Skyrmion number density, and its aspect ratio measure, $p$ .",
"$E(L,p)=E_{p=0}(L)+E_0[\\alpha (L)p^2+\\beta (L)p^3+\\gamma (L)p^4+\\delta (L)p^5+...],$ where the coefficients are given by $E_{p=0}(L)&=&E_0\\left[0.474\\left(\\frac{L}{L_0}+\\frac{L_0}{L}\\right)+0.0515\\right],\\\\\\alpha (L)&=&0.649-0.487\\frac{L}{L_0}+0.089\\frac{L_0}{L},\\\\\\beta (L)&=&0.300+0.006\\frac{L}{L_0}-0.119\\frac{L_0}{L},\\\\\\gamma (L)&=&-1.64+0.78\\frac{L}{L_0}+0.71\\frac{L_0}{L},\\\\\\delta (L)&=&0.53-0.55\\frac{L}{L_0}.$ Here $E_0=727.4$ MeV and $L_0=1.666\\times 10^{-15}$ m. The equation can be extended to include lower densities [7] but they are not of interest here where we are only considering densities higher than the minimal energy crystal.",
"Notice that for any value of $L$ the minimum energy occurs at the face-centred cubic lattice configuration, $p=0$ , and the global minimum is reached for $L=L_0$ ."
],
[
"TOV Equation for Skyrmion Stars",
"Using equation (REF ) relating the energy of a Skyrmion to its size and aspect ratio we will now investigate how one can describe a neutron star using a Skyrme crystal and how this crystal is deformed under the high gravitational field it experiences.",
"In our numerical work we denote $\\lambda _r$ as the Skyrmion length in the radial direction of the star and $\\lambda _t$ as the Skyrmion length in the tangential direction.",
"These parameters and the parameters $L$ and $p$ used in (REF ) are related as follows $L=(\\lambda _r\\lambda _t\\lambda _t)^{\\frac{1}{3}},\\ {\\rm {and}}\\ p=\\left(\\frac{\\lambda _t}{\\lambda _r}\\right)^\\frac{1}{3}-\\left(\\frac{\\lambda _r}{\\lambda _t}\\right)^\\frac{1}{3}.$ To construct a neutron star we consider a spherically symmetric distribution of matter in static equilibrium with a stress tensor that is in general locally anisotropic.",
"Spherical symmetry demands that the stress tensor, $T_\\nu ^\\mu $ , is diagonal and that all the components are a function of the radial coordinate only.",
"We denote this stress tensor as $T_\\nu ^\\mu ={\\rm diag}(\\rho (r),p_r(r),p_\\theta (r),p_\\phi (r)),$ and consider that, again due to spherical symmetry, $p_\\theta (r) =p_\\phi (r)$ which we will denote by $p_t(r)=p_\\theta (r) =p_\\phi (r)$ .",
"The quantities $p_r(r)$ and $p_t(r)$ describe the stresses in the radial and tangential directions respectively while the quantity $\\rho (r)$ is the mass density.",
"A generalised TOV equation [13], [14] to describe a spherically symmetric star composed of anisotropically deformed matter in static equilibrium has been studied previously [15] and we summarise it now.",
"The metric for the static spherically symmetric distribution of matter can be written in Schwarzschild coordinates as $ds^2=e^{\\nu (r)} dt^2-e^{\\lambda (r)} dr^2-r^2d\\theta ^2-r^2\\sin ^2\\theta d\\phi ^2,$ where $e^{\\nu (r)}$ and $e^{\\lambda (r)}$ are functions of the radial coordinate that need to be determined.",
"The combination of this metric and the matter distribution, described by the stress tensor (REF ), must be a solution of Einstein's equations $G_{ab}=R_{ab}-\\frac{1}{2}Rg_{ab}=8\\pi T_{ab},$ where we have set $G=c=1$ .",
"After calculating the Ricci tensor and Ricci scalar from the metric we find $e^{\\lambda } \\left( \\frac{{\\lambda }^{\\prime }}{r} - \\frac{1}{r^2} \\right)+ \\frac{1}{r^2} &=& 8{\\pi }{\\rho }\\\\e^{-\\lambda } \\left( \\frac{{\\nu }^{\\prime }}{r}+ \\frac{1}{r^2} \\right)- \\frac{1}{r^2} &=& 8{\\pi }p_{r}\\\\e^{-\\lambda } \\left( \\frac{1}{2} {\\nu }^{\\prime \\prime }- \\frac{1}{4}{\\lambda }^{\\prime }{\\nu }^{\\prime }+ \\frac{1}{4} \\left({\\nu }^{\\prime } \\right)^{2}+ \\frac{\\left({\\nu }^{\\prime } -{\\lambda }^{\\prime } \\right)}{2r} \\right)&=& 8{\\pi }p_{t}~.$ Equation (REF ) can be rewritten as $(re^{-\\lambda })^{\\prime }=1-8\\pi \\rho r^2$ and integrated to give $e^{-\\lambda }=1-\\frac{2m}{r}$ where $m = m(r)$ is defined as the gravitational mass contained within the radius $r$ and can be calculated by $m=\\int ^r_04\\pi r^2\\rho dr.$ We can now substitute equation (REF ) for $e^{-\\lambda }$ into equation () to find $\\frac{1}{2}\\nu ^{\\prime }=\\frac{m+4\\pi r^3 p_r}{r(r-2m)}.$ The generalised TOV equation that we will use to find suitable neutron star configurations can now be obtained by differentiating equation () with respect to $r$ and adding it to equation () to find $\\frac{dp_{r}}{dr} = -(\\rho + p_{r})\\frac{{\\nu }^{\\prime }}{2}+ \\frac{2}{r}(p_{t} - p_{r})~.$ Now, substituting (REF ) into (REF ), we get $\\frac{dp_{r}}{dr} = -(\\rho + p_{r})\\frac{m+4\\pi r^3 p_r}{r(r-2m)}+ \\frac{2}{r}(p_{t} - p_{r})~.$ For this generalised TOV equation to be solvable two equations of state need to be specified, ${p_{r} =p_{r}(\\rho )}$ and ${p_{t} = p_{t}(\\rho )}$ , where, as argued above, we are able to use a zero temperature assumption.",
"We also need to specify appropriate boundary conditions.",
"First, we must require that the solution is regular at the origin and impose that $m(r)\\rightarrow 0$ as $r \\rightarrow 0$ .",
"Then $p_{r}$ must be finite at the centre of the star implying that ${\\nu }^{\\prime } \\rightarrow 0$ as $r \\rightarrow 0$ .",
"Moreover, the gradient $dp_{r}/dr$ must be finite at the origin too and so $(p_{t} - p_{r})$ must vanish at least as rapidly as $r$ when $r\\rightarrow 0$ .",
"This implies that we need to impose the boundary condition $p_{t}=p_{r}$ at the centre of the star.",
"The radius of the star, $R$ , is determined by the condition $p_{r}(R)= 0$ as the radial stress for the Skyrmions on the surface of the star will be negligibly small.",
"The equations, however, do not impose that $p_{t}(R)$ vanishes at the surface.",
"One should also point out that physically relevant solutions will all have $p_{r} ,p_{t} \\ge 0 $ for $r \\le R$ .",
"We note that an exterior vacuum Schwarzschild metric can always be matched to our metric for the interior of the star across the boundary $r=R$ as long as $p_{r}(R) = 0$ , even though $p_{t}(R)$ and $\\rho _{r}(R)$ may be discontinuous, implying that the star can have a sharp edge, as expected from a solid rather than gaseous star.",
"As we are considering Skyrmion matter at zero temperature the equations of state that will be used in finding suitable neutron star configurations can be calculated from equation (REF ) which depends on the lattice scale $L$ , and aspect ratio, $p$ , which are both functions of the radial distance form the centre of the star, $r$ .",
"From the theory of elasticity we then find that the radial and the tangential stresses are related to the energy per Skryrmion, Eq (REF ), as follows $p_r=-\\frac{1}{\\lambda _t^2}\\frac{\\partial E}{\\partial \\lambda _r},\\ {\\rm {and}}\\ p_t=-\\frac{1}{\\lambda _r}\\frac{\\partial E}{\\partial \\lambda _t^2}.$ Using the generalised TOV equation (REF ) and the two equations of state (REF ), a minimum energy configuration for various values of the total baryon number can be calculated numerically.",
"The minimum energy configuration is defined as the minimum value of the gravitational mass, $M_G$ , $M_G=m(R)=m(\\infty )=\\int _0^R4\\pi r^2\\rho dr,$ where $R$ is the total radius of the star and $\\rho = \\frac{E}{\\lambda _r \\lambda _t^2 c^2}.$ We now need to minimise $M_G$ as a function of $\\lambda _r$ and $\\lambda _t$ which both depend on $r$ .",
"To achieve this, we will first assume a profile for $\\lambda _t(r)$ and compute $M_G$ for this profile as described below.",
"We will then determine the configuration of the neutron star, with a specific baryon charge, by minimising $M_G$ over the field $\\lambda _t$ .",
"This can be easily done using the simulated annealing algorithm.",
"To compute $M_G$ we notice that at the origin, one can use (REF ) to determine $p_r(0)$ and $p_t(0)$ from the initial values of $\\lambda _r(0)$ and $\\lambda _t(0)$ .",
"Then the integration steps can be performed as follows.",
"Knowing $\\lambda _r(r)$ and $\\lambda _t(r)$ one computes $\\rho (r)$ using (REF ) and $m(r)$ using (REF ).",
"Then, knowing $p_r(r)$ , $p_t(r)$ , $\\rho (r)$ and $m(r)$ one can integrate (REF ) by one step to determine $p_r(r+dr)$ .",
"One can then use (REF ) to determine $\\lambda _r(r+dr)$ and as the profile for $\\lambda _t(r)$ is fixed, one can proceed with the next integration step.",
"One then integrates (REF ) up to the radius $R$ for which $p_r(R)=0$ ; this sets the radius of the star.",
"In our integration, we used a radial step of 50m.",
"One must then evaluate the total baryon charge of the star using $B = \\int _0^R \\frac{4\\pi r^2 n(r)}{(1-\\frac{2 G m}{c^2 r})^{1/2}}dr$ where $n(r) = \\frac{1}{\\lambda _r(r)\\lambda _t(r)^2}$ and rescale $\\lambda _t$ to restore the baryon number to the desired value.",
"One then repeats the integration procedures until the baryons charge reaches the correct value without needing any rescaling."
],
[
"Stars Made of Isotropically Deformed Skyrme Crystal",
"We found that up to a baryon number of $2.61\\times 10^{57}$ , equivalent to $1.49M_\\odot $ , the minimum energy configurations are all composed of Skyrme crystals that are isotropically deformed, with $\\lambda _t(r)=\\lambda _r(r)$ across the whole radius of the star.",
"It can be shown that this indeed has to be the case as we can prove that if it is possible to find an isotropic Skyrme crystal solution then that solution will be the minimum energy configuration.",
"Such isotropic solutions can only be found up to a baryon number of $2.61\\times 10^{57}$ .",
"Corchero [23] used a similar proof for a quantum model of neutron stars and we adapt this here for our Skyrme crystal model: If there is a locally isotropic, stable solution to the generalised TOV equation (REF ) with mass $M$ and total baryon number $N$ , then all locally anisotropic solutions that have the same total baryon number $N$ , in the neighbourhood of that stable solution, will have a mass not smaller than $M$ .",
"To prove this, we first note that stable solutions for a given baryon number at zero temperature, by definition, have a mass that is not greater than that which could be achieved by any variation of the density that preserves the baryon number [24].",
"We then consider two changes to a stable solution with mass $M_1$ , baryon number $N_1$ , density $\\rho _1(r)$ and number density $n_1(r)$ .",
"The first involves changing the density from $\\rho _1(r)$ to $\\rho _2(r)$ while keeping the total baryon number constant and preserving locally isotropy.",
"This will result in a configuration that has a mass $M_2$ that is greater than or equal to the mass $M_1$ of our initial configuration as that was defined as the minimum mass solution.",
"This new configuration will have a different number density, $n_2(r)$ , but the same total baryon number, $N_1$ , by assumption.",
"The second change involves introducing local anisotropy while the mass, $M_2$ , remains the same, as does the density, $\\rho _2(r)$ .",
"In order to keep $\\rho _2(r)$ constant when we alter the configuration so that it is now made of anisotropic Skyrme crystal the number density must also be altered.",
"A change from an isotropic to an anisotropic Skyrme crystal involves increasing its energy, and therefore mass, so to keep its mass density constant we need to reduce the number of Skyrmions to the new number density $n_3(r)\\le n_2(r)$ , meaning the total baryon number is now $N_3\\le N_1$ .",
"The two changes described have the effect of firstly increasing $M$ without changing $N$ and then, secondly, decreasing $N$ without altering $M$ .",
"We know that $M$ is a monotonically increasing function of $N$ for isotropic Skyrme crystal stable star configurations, so we have proved that moving from isotropic to anisotropic Skyrme crystal configurations increases the energy for a given baryon number so does not produce a minimum energy solution.",
"Figure: Total baryon number as a function of the size of theSkyrmions at the centre of the star, L(r=0)L(r=0).The above proof however does not rule out the existence of anisotropic Skyrme crystal solutions for those baryon numbers for which there does not exist an isotropic Skyrme crystal solution and such configurations will be discussed in the next section.",
"To confirm the results obtained for isotropically deformed crystals, we will now determine the properties of these symmetric stars by imposing that symmetry, i.e.",
"$p_t=p_r$ .",
"In this case the problem simplifies greatly and the TOV equation (REF ) reduces to $\\frac{dp_{r}}{dr} = -(\\rho + p_{r})\\frac{m+4\\pi r^3 p_r}{r(r-2m)}$ Using this standard TOV equation, a central Skyrmion length $\\lambda _{t}(r=0)=\\lambda _{r}(r=0)=L(r=0)$ can be specified at the centre of the star.",
"The equation can then be numerically integrated over the radius of the star using the Skyrmion energy equation (REF ) with $p_r=-\\frac{\\partial E}{\\partial \\lambda _r^3},$ where, as we are only considering isotropic Skyrme crystal deformations, $\\lambda _{t}=\\lambda _{r}$ and $p=0$ in the energy equation.",
"This was done using a fourth order Runge Kutta method over points every 20m.",
"Notice that this did not require the explicit minimisation of $M_G$ .",
"Figure REF shows a plot of the total baryon number of the star against its Skyrmion length at the center, $L(r=0)$ , calculated using this method.",
"We found that isotropic Skyrme crystal solutions can be found up to a baryon number of $2.61\\times 10^{57}$ , which is equivalent to a mass of $1.49M_\\odot $ .",
"This agrees with the results that we found from our minimisation procedure using the generalised TOV equation that allows for anisotropic Skyrme crystal deformations.",
"Table: Properties of the isotropic minimum energy neutron starconfigurations for various baryon numbers.Table REF shows some of the properties of the minimum energy solutions for various baryon numbers obtained from the energy minimisation of the generalised TOV equation.",
"The results are in perfect agreement with the results obtained by solving the isotropic TOV equation (REF ).",
"The quantity $S_{min}$ is the minimum value, over the radius of the star, of $S(r)=e^{-\\lambda (r)}=1-\\frac{2m(r)}{r},$ which appears in the static, spherically symmetric metric (REF ) that we are considering.",
"The zeros of $S(r)$ correspond to singularities in the metric, or in other words, to horizons.",
"Had $S_{min}$ been negative, we would have concluded that the neutron star would have collapsed into a black hole, but this never occurred.",
"We note that the solutions are energetically favourable as the energy per baryon decreases when the total baryon number increases, indicating that the solutions are stable.",
"They correspond to the solutions to the right of the maximum in figure REF with solutions to the left being unstable with a higher energy per baryon for a given baryon number, and therefore not found by the energy minimisation procedure.",
"The neutron star solutions which have masses larger than the mass of the Sun have radii of about 10km, which very much matches the experimental estimates of the radii of observed neutrons stars.",
"Notice also that the largest neutron star, in our model, has a mass of approximately $1.28M_\\odot $ , and above that value, the radius of the stars decreases with their mass (see table REF and figure REF ).",
"Figure: Radius of the neutron star solutions as a function oftheir mass (solid line), and that of the maximum mass solution (cross).Figure: Variation of the size of the isotropic Skyrmions, L(r)L(r),(solid line) and of the metric function S(r)S(r) (dotted line) over theradius of a star of mass 1.40M ⊙ 1.40M_\\odot .We now consider the structures of these isotropic Skyrme crystal stars, in particular we consider the case of a star with a mass of $1.40M_\\odot $ , a typical mass for a realistic neutron star, equivalent to a baryon number of $2.44\\times 10^{57}$ , although all the isotropic Skyrme crystal minimum energy solutions show the same qualitative behaviour.",
"Figure REF shows the size of the Skyrmions, $L(r)$ , over the radius of the star.",
"As expected the Skyrmions are deformed more towards the centre of the star than at the edge, increasing the Skyrmion mass density by a factor of $4.44$ .",
"Due to this decrease in the size of the Skyrmions as we reach the centre of the star the stress is higher at the centre and decreases towards zero at the edge of the star as imposed by the boundary conditions.",
"The isotropic Skyrme crystal solutions have a $S_{min}$ that is always greater than zero so the configurations do not collapse into black holes.",
"Figure REF also shows how the value of $S(r)$ varies over the radius of the star."
],
[
"Stars Made of Anisotropically Deformed Skyrme Crystal",
"Having shown in the previous section that no isotropic Skyrme crystal solutions exist for baryon numbers larger than $2.61\\times 10^{57}$ , we will now show that anisotropic solutions do exist.",
"Table REF shows some of the properties of the anisotropic minimum energy Skyrme crystal solutions for various baryon numbers obtained using the generalised TOV equation.",
"We found solutions in this way up to a baryon number of $3.25\\times 10^{57}$ , corresponding to $1.81M_\\odot $ , after which the numerical energy minimisation procedure became difficult to implement.",
"However by using a similar simulated annealing process to maximise the baryon number, rather than minimise the energy for a particular baryon number, we found anisotropic Skyrme crystal solutions up to a baryon number of $3.41\\times 10^{57}$ , equivalent to $1.90M_\\odot $ .",
"At this maximum baryon number solution there is only one possible configuration of the Skyrmions, as any modification to it results in a decrease in the baryon number, hence it is the minimum energy solution.",
"Above this baryon number, solutions do not exist.",
"Table: Properties of the anisotropic minimum energy neutron starconfigurations for various baryon numbers.As in the case of isotropic Skyrme crystal deformations we find that the solutions are energetically favourable as the energy per baryon decreases as the total baryon number increases, indicating stable solutions.",
"As the baryon number is increased towards its maximum value of $3.41\\times 10^{57}$ the energy per baryon begins to level off and we find that the maximum baryon number has the lowest energy per baryon, as in the isotropic case.",
"We can see that the configurations we have constructed do not collapse into a black hole by noticing that the values of $S_{min}$ are always positive, as shown in figure REF .",
"Figure: S min S_{min} of the neutron star solutions as a function oftheir mass.",
"The maximum mass solution is shown as a cross.Figure REF shows a plot of the mass radius curve for both the isotropic and anisotropic Skyrme crystal cases, with the mass in units of $M_\\odot $ .",
"As stated above, large isotropic crystal neutron stars have a radius that decreases as the mass increases.",
"We can clearly see in figure REF , that at the critical mass of $1.49M_\\odot $ , the radius keeps decreasing as the mass of the star increases.",
"Moreover, we also observe a sharp drop of radius just over $1.5M_\\odot $ followed by a plateau at about $9.5$ km.",
"Figure: Skyrmion lengths λ r (r)\\lambda _r(r) (solid line), λ t (r)\\lambda _t(r) (dashed line)and L(r)L(r) (dotted line) fora) Largest neutron star (R=10.8R=10.8km): M=1.28M ⊙ M=1.28M_\\odot b) Heaviest isotropic neutron star: M=1.49M ⊙ M=1.49M_\\odot (all lengths coincide as they are made of isotropically deformed crystal);c) Densest neutron star: M=1.54M ⊙ M=1.54M_\\odot ;d) Heaviest neutron star: M=1.90M ⊙ M=1.90M_\\odot .By considering anisotropic as well as isotropic Skyrme crystal solutions we have extended the mass range over which solutions can be found, finding masses up to $28\\%$ above the maximum mass of the isotropic case.",
"This is an interesting finding because isotropy of matter is often taken as an assumption when studying neutron star models, including the Skyrme crystal case considered in [12],[16],[17] and a maximum mass is then derived.",
"We have shown that by not assuming isotropy and instead allowing anisotropic matter configurations the maximum mass can be increased by a significant amount.",
"In this simple Skyrme crystal model the maximum mass found is equivalent to $1.90M_\\odot $ and the recent discovery of a $1.97 \\pm 0.04\\,M_\\odot $ neutron star [25], the highest neutron star mass ever determined, makes this an encouraging finding, especially when we consider that including the effects of rotation into our model will increase the maximum mass found, by up to $2\\%$ for a star with a typical $3.15$ ms spin period [26].",
"Figure REF shows a selection of plots of the Skyrmion lengths $\\lambda _r$ and $\\lambda _t$ and the Skyrmion size $L$ , equation (REF ), over the radius of the star for four special stars: the largest star, with radius $R=10.8$ km and mass $M=1.28M_\\odot $ (figure REF ); the heaviest isotropically deformed star $M=1.49M_\\odot $ (figure REF ); the densest neutron star, $M=1.54M_\\odot $ (figure REF ) and the heaviest neutron star, $M=1.90M_\\odot $ (figure REF ).",
"The first two are made out of an isotropically deformed crystal, while the last two are anisotropically deformed and one notices that the amount of anisotropy increases as the mass increases (the divergence between $\\lambda _r$ and $\\lambda _t$ increases).",
"Throughout this paper, we will use these four special stars as examples to illustrate various properties of the neutron stars.",
"As the maximum mass is approached the gradient of the profile of tangential Skyrmion lengths over the radius of the star becomes smaller and we note that physically meaningful stars composed of anisotropically deformed crystal should have $d\\lambda _t/dr\\ge 0$ [27].",
"This confirms that the minimum energy solution for the maximum mass found, $1.90M_\\odot $ , for anisotropic Skyrme crystal solutions is the configuration with a constant tangential Skyrmion length as illustrated in figure REF .",
"The generalised TOV equation imposes that the sizes of the Skyrmions are equal in all directions at the centre of the star, but away from the centre, for all the anisotropic Skyrme crystal solutions, we find that the amount of Skyrmion anisotropy increases as we move towards the edge of the star, reaching the maximum at the edge.",
"The Skyrmions are deformed to a greater extent in the tangential direction in agreement with the value of the aspect ratio, $p$ , being negative over the values where $\\lambda _r \\ne \\lambda _t$ .",
"Figure: Mass density ρ(r)\\rho (r) for:a) Largest neutron star (R=10.8R=10.8km): M=1.28M ⊙ M=1.28M_\\odot (solid line)b) Heaviest isotropic neutron star: M=1.49M ⊙ M=1.49M_\\odot (dashed line);c) Densest neutron star: M=1.54M ⊙ M=1.54M_\\odot (dotted line);d) Heaviest neutron star: M=1.90M ⊙ M=1.90M_\\odot (dash dotted line).As expected, the profiles for $\\lambda _r$ and $\\lambda _t$ show that the mass density at the centre of the star is higher than at the edge, decreasing monotonically as the radial distance increases.",
"This is shown by figure REF for the largest, heaviest isotropic, densest and maximum mass solutions.",
"In figure REF one can see how the lengths of the Skyrme crystal $\\lambda _r$ and $\\lambda _t$ vary with the mass of the star both at the center ($r=0$ ) and the edge of the star ($r=R$ ).",
"For isotropically deformed stars, $\\lambda _r(R)=\\lambda _t(R)$ is constant and corresponds to the minimum energy Skyrme crystal in the absence of gravity.",
"Not surprisingly, $\\lambda _r(0)=\\lambda _t(0)$ decreases steadily as the mass of the star increases, showing that the density at the center of the star increases.",
"Once the phase transition has taken place and the star is too heavy to remain isotropically deformed, we observe that $\\lambda _r(0)=\\lambda _t(0)$ drops sharply to a local minimum, reached for $M\\approx 1.54M_\\odot $ .",
"Meanwhile, $\\lambda _r(R)$ and $\\lambda _t(R)$ remain nearly identical.",
"Beyond the minimum of $\\lambda _{r,t}(0)$ , $\\lambda _r(R)$ and $\\lambda _t(R)$ start to diverge sharply; $\\lambda _r(R)$ decreases slightly in value while $\\lambda _t(R)$ decreases rapidly.",
"These stars are thus much more compressed in the tangential direction than in the radial one.",
"As seen on figure REF , $\\lambda _t(R)=\\lambda _t(0)$ for the maximum mass neutron star.",
"Figure: Skyrmion lengths at the edge of the star,λ r (R)\\lambda _r(R) (solid line)and λ t (R)\\lambda _t(R) (dashed line), and at the center of the star,λ r (0)=λ t (0)\\lambda _r(0)=\\lambda _t(0) (dotted line), as a function of the star mass.Figure: Radial speed of sound, v r (r)v_r(r) fora) Largest neutron star (R=10.8R=10.8km): M=1.28M ⊙ M=1.28M_\\odot (solid line)b) Heaviest isotropic neutron star: M=1.49M ⊙ M=1.49M_\\odot (dashed line);c) Densest neutron star: M=1.54M ⊙ M=1.54M_\\odot (dotted line);d) Heaviest neutron star: M=1.90M ⊙ M=1.90M_\\odot (dash dotted line).Another property of a neutron star worth considering is the speed of sound.",
"To compute it one needs to know how the energy of the crystal varies when it is deformed in the direction of wave propagation.",
"Using (REF ) we can thus compute the speed of sound in the $z$ direction.",
"To compute the speed of sound in the $x$ and $y$ directions when the crystal is deformed we need to know how the energy of the crystal varies when the crystal is deformed in all three directions independently, an expression we do not have.",
"We are thus only able to compute the radial speed of sound inside a neutron star and it is given by $v_r = \\left(\\frac{d p_r}{d \\lambda _r}\\left(\\frac{d \\rho }{d\\lambda _r} \\right)^{-1}\\right)^{1/2}$ where both $p_r$ and $\\rho $ are functions of $\\lambda _r$ and $\\lambda _t$ given respectively by (REF ) and (REF ).",
"Obviously, when the crystal inside the star is isotropically deformed, the speed of sound is the same in all 3 directions.",
"First of all it is interesting to notice that the speed of sound in the minimum energy Skyrme crystal, in the absence of a gravitational field, is amazingly large: $v=0.57\\,c$ .",
"This is the speed of sound at the surface of a neutron star when it is deformed isotropically.",
"From figure REF one sees that $v_r$ increases as one moves towards the center of the star.",
"As $v_r$ is directly related to the density of the star, it is not surprising to find that the maximum radial speed, $v_r= 0.78c$ , is reached at the center of the densest neutron star, i.e.",
"the one with $M=1.54M_\\odot $ .",
"As expected, $v_r < c$ everywhere.",
"Figure: The function S(r)S(r) for:a) Largest neutron star (R=10.8R=10.8km): M=1.28M ⊙ M=1.28M_\\odot (solid line)b) Heaviest isotropic neutron star: M=1.49M ⊙ M=1.49M_\\odot (dashed line);c) Densest neutron star: M=1.54M ⊙ M=1.54M_\\odot (dotted line);d) Heaviest neutron star: M=1.90M ⊙ M=1.90M_\\odot (dash dotted line).Figure REF shows how the value of $S(r)$ varies over the radius of the star for, again, the largest, heaviest isotropic, densest and maximum mass solutions, showing how the metric is altered as $r$ varies.",
"The minimum value of $S(r)$ is always located at the edge of the star, i.e.",
"$S_{min}=S(R)$ , and it is presented in figure REF as a function of the star masses.",
"One sees that $S_{min}$ decreases monotonically as the mass increases, and exhibits a sharp decrease just over $1.5M_\\odot $ , i.e.",
"just above the critical mass.",
"However $S_{min}$ always remains positive, indicating that no black hole is formed.",
"Figure: Mass of the neutron star solutions as a function oftheir baryon number.",
"The maximum mass solution is shown as a cross.Figure REF shows how the total baryon number and the mass of all the solutions found are related.",
"As the baryon number increases the effects of gravitational attraction increase, resulting in a slightly lower gravitational mass per baryon than expected from a linear relation.",
"We note that the minimum value of the aspect ratio, $p$ , for the minimum energy configurations found is $-0.283$ and the minimum value of $L$ is $8.11\\times 10^{-16}$ , both of which are within the valid range of values for equation (REF ) [7]."
],
[
"Inclusion of the Pion Mass",
"Throughout the work described we have assumed a zero pion mass.",
"The inclusion of a non-zero pion mass can be considered by including the pion mass term, $\\int \\frac{m_{\\pi }^2F_{\\pi }^2}{8}{\\rm {Tr}}(U-1)d^3x,$ in the static Skyrme Lagrangian (REF ), where $U$ is the Skyrme field, $F_\\pi $ is the pion decay constant and $m_\\pi $ is the pion mass.",
"Using the cubic lattice of $\\alpha $ -like Skyrmions that has been considered above one finds that ${\\rm {Tr}}(U-1)=-2$ , meaning that the energy $E_\\pi $ arising from the pion mass term reduces to $E_\\pi =\\frac{1}{4} m_{\\pi }^2F_{\\pi }^2L^3,$ an energy term proportional to the volume of the Skyrmions.",
"Figure: Mass of the star as a function of the size of the Skyrmionsat the centre, L 0 L_0, for zero pion mass (solid line) and m=138m=138MeV(dashed line).It can be seen in figure REF that including a pion mass of $m=138$ MeV decreases the maximum mass of the star by a very small amount from $1.49$ to $1.47M_\\odot $ while also slightly decreasing the central density at which this occurs.",
"Including a pion mass of $m=138$ MeV in the simulated annealing process used to find the maximum baryon number for the anisotropic Skyrme crystal solutions results in a maximum baryon number of $3.34\\times 10^{57}$ , equivalent to $1.88M_\\odot $ , a decrease of $0.02M_\\odot $ from the maximum mass found in the case without a pion mass.",
"This gives an indication as to how the pion mass affects the structures of the neutron star configurations that can be constructed, and a similar reduction in the maximum mass is expected for all the anisotropic crystal solutions, however when the pion mass is included it also has the effect of driving the Skyrme crystal lattice away from the half-Skyrmion symmetry [7].",
"This will be a small effect for the dense Skyrme crystals that we are considering because while the pion mass term is the dominant term in the Lagrangian far away from the centres of the Skyrmions when they are well separated, in the dense Skyrme crystal there is no space away from the centres of the Skyrmions so it becomes less important in affecting the field distributions.",
"Its effect will be to reduce the pion mass term, Eq.",
"(REF ), by a small amount."
],
[
"Stars above the Maximum Mass",
"As in other studies of neutron starts based on the Skyrme model, we found a critical mass above which solutions do not exist.",
"In other words, when the star is too massive, the crystal of which it is made is not capable of counterbalancing the gravitation pull and the star then collapses into a black hole.",
"This is indeed what we observed when trying to construct solutions above the critical mass: the energy of the configuration kept decreasing as the radius of the star decreased and the $S_{min}$ function became negative, indicating the formation of an horizon, and hence a black hole.",
"Throughout this work we have assumed a spherically symmetric metric and stress tensor, however, these assumptions could be removed and it may be that higher mass solutions could be found.",
"We could instead consider an axially symmetric metric, the most general form [28] being $ds^2=\\alpha ^2(d\\rho ^2+dz^2)+\\beta ^2d\\phi ^2-\\gamma ^2dt^2,$ when written in cylindrical coordinates.",
"The stress tensor, $T_\\nu ^\\mu ={\\rm diag}(\\rho ,p_1,p_2,p_3),$ could then be completely anisotropic with $p_1\\ne p_2\\ne p_3$ .",
"Minimum energy solutions to Einstein's equations for such a metric and stress tensor could be found by direct minimisation of the action of the Skyrme model coupled to gravity or by using an, as yet undetermined, axisymmetric form of the TOV equation.",
"Another approach to investigate such solutions would be to perturb the spherically symmetric solutions that we have found.",
"Following the procedure for doing so described in [28] the exterior metric for an axially symmetric solution can be written in Schwarzschild coordinates and, after comparing the exterior spherically symmetric Schwarzschild solution to our solutions for the interior metric of the star and finding the substitutions necessary to move from one to the other, we can make to same substitutions to the axially symmetric exterior metric.",
"This allows us to then describe approximately both the metric and the stress energy tensor of the axially symmetric solution.",
"To carry out such investigations into axially symmetric static configurations an equation analogous to (REF ) which would relate the energy of the Skyrme crystal to its size and deformation in all three directions independently would need to be considered.",
"We have also assumed that the stress tensor, $T_\\nu ^\\mu ={\\rm diag}(\\rho ,p_r,p_\\theta ,p_\\phi )$ , is diagonal, however, if shear strains are included in our model off diagonal components would have to be introduced.",
"This would also remove the assumption of spherical symmetry altering the configurations found.",
"Spherical symmetry also needs to be removed to consider rotating stars.",
"This will result in configurations above the maximum mass found in this work, by up to $2\\%$ for a star with a typical $3.15$ ms spin period [26], and as neutron stars are known to be rotating, this is an important effect to consider."
],
[
"Conclusions",
"Neutron stars are large bodies of matter where the electrons, instead of circling atoms, are forced to merge with the nuclei, resulting in extremely dense stars made entirely of neutrons.",
"Their temperature, from a nuclear point of view, is very low and this means nuclear matter must be considered as a solid rather than a fluid.",
"Moreover, the gravitational pull of the star is so strong that the “atmospheric” fluid one might expect at the surface is of negligible height.",
"In this context, the Skyrme model, known to be a low energy effective field theory for QCD [3], is an ideal candidate to describe neutron stars once the model is coupled to gravity.",
"The minimum energy configuration of large numbers of Skyrmions is a cubic crystal made of $B=4$ Skyrmions which correspond to a crystal of $\\alpha $ -like particles.",
"We have thus used these solutions as a building block to describe the neutron star by combining the deformation energy computed in [7] and a generalised version of the TOV equation [13], [14], [15] which describes the static equilibrium between matter forces, within a solid or fluid, and the gravitational forces self-generated by the matter for a spherically symmetric body.",
"The key feature of our approach to the problem was to consider the star as a solid that could potentially deform itself anisotropically.",
"We then found that below $1.49M_\\odot $ , all stars were made of a crystal deformed isotropically, i.e.",
"the radial strain was identical to the tangential one.",
"Above that critical value, the neutron star undergoes a critical phase transition and the lattice of Skyrmions compresses anisotropically: the Skyrmions are more compressed tangentially than radially.",
"Stars were shown to exist up to a critical mass of $1.90M_\\odot $ , a result that closely matches the recent discovery of Demorest et al.",
"[25] who measured the mass of the heaviest neutron star found to date, PSR J1614-2230, to be $1.97M_\\odot $ .",
"We also observed that the maximum radius for a Skyrmion star was approximately 11km, a figure that matches well the experimental estimations.",
"In our model we did not consider the rotational energy of the star which is approximated at about 2% of its total energy.",
"If we included that extra energy, our upper bound would thus just fit above the mass of PSR J1614-2230.",
"Finally we have also shown that if the mass of a neutron star was to be raised to cross the critical mass threshold, it would collapse into a black hole."
],
[
"Acknowledgements",
"BP was supported by the STFC Consolidated Grant ST/J000426/1 and SN by an EPSRC studentship."
]
] | 1204.0910 |
[
[
"Symmetric competition as a general model for single-species adaptive\n dynamics"
],
[
"Abstract Adaptive dynamics is a widely used framework for modeling long-term evolution of continuous phenotypes.",
"It is based on invasion fitness functions, which determine selection gradients and the canonical equation of adaptive dynamics.",
"Even though the derivation of the adaptive dynamics from a given invasion fitness function is general and model-independent, the derivation of the invasion fitness function itself requires specification of an underlying ecological model.",
"Therefore, evolutionary insights gained from adaptive dynamics models are generally model-dependent.",
"Logistic models for symmetric, frequency-dependent competition are widely used in this context.",
"Such models have the property that the selection gradients derived from them are gradients of scalar functions, which reflects a certain gradient property of the corresponding invasion fitness function.",
"We show that any adaptive dynamics model that is based on an invasion fitness functions with this gradient property can be transformed into a generalized symmetric competition model.",
"This provides a precise delineation of the generality of results derived from competition models.",
"Roughly speaking, to understand the adaptive dynamics of the class of models satisfying a certain gradient condition, one only needs a complete understanding of the adaptive dynamics of symmetric, frequency-dependent competition.",
"We show how this result can be applied to number of basic issues in evolutionary theory."
],
[
"Introduction",
"Adaptive dynamics ([19], [10], [4]) has emerged as a widely used framework for modeling long-term evolution of continuous phenotypes.",
"The basic ingredient of an adaptive dynamics model is the invasion fitness function ([20]), which describes the ecological growth rate of rare mutant phenotypes in a given resident community, which is assumed to persist on a community-dynamical attractor.",
"The invasion fitness function determines the selection gradients, which are in turn the core ingredient for deriving the canonical equation ([4]) for the adaptive dynamics of the phenotypes under consideration.",
"Following this basic recipe, adaptive dynamics models have been constructed for a plethora of different ecological settings, and have been used to analyze a number of interesting and fundamental evolutionary scenarios, such as evolutionary cycling in predator-prey arms races ([5], [17]), evolutionary diversification ([2], [3], [6]) and evolutionary suicide [11], [22].",
"Even though the derivation of the adaptive dynamics from a given invasion fitness function is general and model-independent, the derivation of the invasion fitness function itself requires specification of an underlying ecological model.",
"Therefore, evolutionary insights gained from adaptive dynamics models are generally tied to a specific ecological setting, and hence model-dependent.",
"One particular ecological model that has been often used to derive invasion fitness functions and adaptive dynamics is the symmetric logistic competition model, which in fact is the most popular ecological model in the theory of ecology and evolution.",
"The basic form of this model is $\\frac{dN}{dt}=rN\\left(1-\\frac{N}{K}\\right),$ where $N$ is population density, and $r$ and $K$ are parameters describing the intrinsic per capita growth rate and the equilibrium population size, respectively.",
"$K$ is often called the carrying capacity of the population, but it useful to note that $K$ can also be interpreted as a property of individuals, i.e., as a measure of how well individuals cope with competition, as expressed in the per capita death rate $rN/K$ .",
"The ecological model (1) can be used to construct a well-known adaptive dynamics model by assuming that the carrying capacity $K(x)$ is a positive function that depends on a continuous, 1-dimensional phenotype $x$ (e.g., body size), and that the competitive impact between individuals of phenotypes $x$ and $y$ is given by a competition kernel $\\alpha (x,y)$ .",
"For simplicity, it is often assumed that the intrinsic growth rate $r$ is independent of the phenotype $x$ , and hence is set to $r=1$ .",
"To derive the corresponding invasion fitness function, it is assumed that a resident type $x$ is at its ecological equilibrium density $K(x)$ .",
"The dynamics of the density $N(y)$ of a mutant type $y$ is then given by $\\frac{dN(y)}{dt}=N(y)\\left(1-\\frac{\\alpha (x,y)K(x)+\\alpha (y,y) N(y)}{K(y)}\\right),$ where $\\alpha (x,y)K(x)$ is the competitive impact that the resident population exerts on mutant individuals.",
"Assuming that the mutant is rare, $N(y)\\approx 0$ , the per capita growth rate of the mutant type $y$ in the resident $x$ is $f(x,y)=1-\\frac{\\alpha (x,y)K(x)}{K(y)}.$ This is the invasion fitness function for the given ecological scenario ([6]).",
"The corresponding selection gradient is $s(x)=\\left.\\frac{\\partial f(x,y)}{\\partial y}\\right|_{y=x}.$ The selection gradient in turn is the main determinant of the adaptive dynamics.",
"More precisely, for 1-dimensional traits $x$ the adaptive dynamics is $\\frac{dx}{dt}=m(x)\\cdot s(x),$ where $m(x)>0$ is a scalar quantity that reflects the rate at which mutations occur ([19], [10], [4]).",
"In particular, singular points, i.e., equilibrium points of the adaptive dynamics, are solutions $x^*$ of $s(x^*)=0$ .",
"Symmetric competition models are characterized by the assumption that $\\partial \\alpha (x,y)/\\partial y\\vert _{y=x}=0$ for all $x$ , it is also usually assumed that $\\alpha (x,x)=1$ .",
"In this case, $s(x)=K^{\\prime }(x)/K(x)$ , and singular points $x^*$ are given as solutions of $K^{\\prime }(x^*)=0$ , and hence are given as local maxima and minima of the carrying capacity.",
"Moreover, singular points that are local maxima are attractors of the adaptive dynamics, i.e., $ds/dx(x^*)<0$ .",
"For logistic models it is often assumed that $K(x)$ is unimodal, attaining a unique maximum at some trait value $x_0$ .",
"$K(x)$ then represents a (global) stabilizing component of selection for $x_0$ , and the adaptive dynamics (5) converges to $x_0$ .",
"In contrast, the competition kernel $\\alpha (x,y)$ generally represents the frequency-dependent component of selection.",
"For symmetric competition it is usually assumed that the effect of competition decreases with increasing phenotypic distance $\\vert x-y\\vert $ , which implies negative frequency dependence, because it confers a competitive advantage to rare phenotypes.",
"However, the opposite is also possible, so that $\\alpha (x,y)$ has a local minimum as a function of $y$ at $y=x$ , which can occur for example in models with explicit resource dynamics ([1]).",
"While the carrying capacity is the sole determinant of the singular points and their convergence stability for the adaptive dynamics (5), the competition kernel comes into play when determining the evolutionary stability of the singular point $x^*$ .",
"Evolutionary stability of $x^*$ is determined by the second derivative of the invasion fitness function at the singular point: $\\left.\\frac{\\partial ^2 f(x^*,y)}{\\partial y^2}\\right|_{y=x^*}=\\frac{K^{\\prime \\prime }(x^*)}{K(x^*)}-\\left.\\frac{\\partial ^2 \\alpha (x^*,y)}{\\partial y^2}\\right|_{y=x^*}.$ The singular point $x^*$ is evolutionarily stable if and only if expression (6) is negative, and it is clear that if $\\partial ^2 \\alpha (x^*,y)/\\partial y^2\\vert _{y=x^*}$ is negative enough, then this condition will not be satisfied, and instead $x^*$ will be evolutionarily unstable.",
"In particular, the distinction between convergence stability and evolutionary stability makes it clear that it is possible for the singular point $x^*$ to be both convergent stable and evolutionarily unstable.",
"In this case, $x^*$ is called an evolutionary branching point, for such points are potential starting points for evolutionary diversification.",
"The phenomenon of evolutionary branching is an iconic feature of adaptive dynamics and has been studied extensively ([10], [6]).",
"It is straightforward to extend the symmetric logistic competition model to multi-dimensional phenotype spaces ([4], [16], [8], [6]).",
"In this case, $x\\in \\mathbb {R}^{m}$ is a $m$ -dimensional vector, where $m$ is the dimension of phenotype space, and $K(x):\\mathbb {R}^{m}\\rightarrow \\mathbb {R}$ is a scalar function, as is the competition kernel $\\alpha (x,y):\\mathbb {R}^{m}\\times \\mathbb {R}^{m}\\rightarrow \\mathbb {R}$ .",
"For symmetric competition, it is assumed that the partial derivatives $\\partial \\alpha (x,y)/\\partial y_i\\vert _{y=x}=0$ for all $x$ and all $i=1,...,m$ (as well as, without loss of generality, $\\alpha (x,x)=1$ for all $x$ ).",
"The corresponding invasion fitness function $f(x,y):\\mathbb {R}^{m}\\times \\mathbb {R}^{m}\\rightarrow \\mathbb {R}$ again has the form (3), and the selection gradient is given as a vector-valued function $s(x)=(s_{1}(x),\\ldots ,s_{m}(x)),:\\mathbb {R}^{m}\\rightarrow \\mathbb {R}$ , where $s_i(x)=\\left.\\frac{\\partial f(x,y)}{\\partial y_i}\\right|_{y=x},\\quad i=1,...,m$ The adaptive dynamics is then given by a system of $m$ coupled differential equations $\\frac{dx}{dt}=M(x)\\cdot s(x),$ where $M(x)$ is the $m\\times m$ mutational variance-covariance matrix, specifying the rate and magnitude at which mutations occur in the various trait components $x_i$ , as well how mutations in different trait components are correlated ([16], [6]).",
"The matrix $M(x)$ is typically assumed to be symmetric and positive definite.",
"For symmetric competition models, it is easy to see that the $m$ components of the selection gradient are $s_i(x)=\\left.\\frac{\\partial f(x,y)}{\\partial y_i}\\right|_{y=x}=\\frac{1}{K(x)}\\cdot \\frac{\\partial K(x)}{\\partial x_i},$ for $i=1,...,m$ .",
"Thus, the selection gradient is the gradient of a scalar function, $s(x)=\\nabla S(x),$ with $S(x)=\\ln (K(x))$ .",
"Thus, just as in the 1-dimensional symmetric competition model, the selection gradients in multi-dimensional generalizations of symmetric competition are essentially gradients of the carrying capacity function $K(x)$ (i.e., gradients of the stabilizing component of selection).",
"The selection gradients therefore induce an evolutionary hill-climbing process towards local maxima of the carrying capacity.",
"The adaptive dynamics (8), resulting from applying the mutational variance-covariance matrix to the selection gradient, is a “warped” version of the hill-climbing process generated by the selection gradients.",
"If the mutational matrix $M(x)$ is positive definite, this warped hill-climbing process is essentially equivalent to the unwarped version defined by the selection gradients alone.",
"In particular, in this case the adaptive dynamics (8) also converges to local maxima of $K(x)$ .",
"It is worth noting that the structure of the canonical equation (8) is similar to other general equations for evolutionary dynamics, such as those introduced by [14] and those introduced by [21].",
"But the assumptions underlying the canonical equation (8), and in particular the notion of invasion fitness, are unique features of adaptive dynamics.",
"In general, invasion fitness functions $f(x,y)$ can be derived for multidimensional phenotypes in each of a number of interacting and coexisting species for many different ecological scenarios ([4], [16], [6]).",
"In each of the interacting species, invasion fitness is the long-term per capita growth rate of rare mutant types $y$ in a population in equilibrium that is monomorphic for the resident type $x$ .",
"Here we consider the following question: under what conditions does an arbitrary invasion fitness function for multi-dimensional phenotypes in a single species has the form of an invasion fitness function derived from a logistic symmetric competition model?",
"This is a relevant question because symmetric competition models have been used for a long time as a basic metaphor to generate ecological and evolutionary insights.",
"It is therefore of interest to understand how universal such models are.",
"For example, in [8] it has recently been shown that evolutionary branching, and hence adaptive diversification, becomes more likely in symmetric competition models if the dimension of phenotype space is increased, and it would be useful to know whether this applies to other models.",
"In the present paper we give a general and precise condition for any given invasion fitness to be equivalent to the invasion fitness function derived from a symmetric competition model.",
"We also show how this result can be applied to shed light on some general issues in adaptive dynamics theory, such as the notion of frequency-dependent selection, the relationship between symmetric and asymmetric competition, the existence of complicated evolutionary dynamics, and the problem of evolutionary stability in single-species models."
],
[
"A condition for universality",
"We consider an invasion fitness function $f(x,y)$ for a single species, in which the multidimensional trait $x=(x_1,\\ldots ,x_m)$ denotes the resident trait, and the vector $y=(y_1,\\ldots ,y_m)$ denotes the mutant trait.",
"Recall that the selection gradient $s(x)$ is a vector $s(x)=(s_1(x),\\ldots ,s_m(x)),$ where $s_i(x)=\\left.\\frac{\\partial f(x,y)}{\\partial y_i}\\right|_{y=x}$ [4], [10].",
"Thus, for a given resident trait vector $x$ , the selection gradient $s(x)$ is the gradient of the fitness landscape given by $f(x,y)$ , but it is important to note that $s(x)$ , which is a vector-valued function on an $m$ -dimensional space, is defined as the gradient with respect to $y$ of a scalar function defined on a $2m$ -dimensional $(x,y)$ space.",
"In particular, the vector field $s(x)$ is in general not the gradient field of a scalar function defined on $m$ variables.",
"The latter case is captured in the following Definition: We call the selection gradient $s(x)$ a gradient field if it can be obtained as the derivative of a scalar function $S(x)$ , i.e., if there is a function $S(x)$ such that $s(x)=\\nabla S(x),$ i.e., such that $s_i(x)=\\frac{\\partial S(x)}{\\partial x_i}\\quad \\text{for}\\quad i=1,...,m.$ The following proposition describes the conditions under which a given invasion fitness function is equivalent to an invasion fitness function derived from a symmetric competition model.",
"Proposition 2.1 The selection gradient $s(x)$ of an invasion fitness function $f(x,y)$ is a gradient field if and only if the invasion fitness function is of the form $f(x,y)=1-\\frac{\\tilde{\\alpha }(x,y)\\tilde{K}(x)}{\\tilde{K}(y)}$ for some scalar functions $\\tilde{K}(x)$ and $\\tilde{\\alpha }(x,y)$ , such that $\\tilde{K}(x)>0$ , $\\tilde{\\alpha }(x,x)=1$ for all $x$ , and $\\partial \\tilde{\\alpha }(x,y)/\\partial y\\vert _{y=x}=0$ for all $x$ .",
"Proof If the invasion fitness function $f(x,y)$ has the form (15), let $S(x)=\\ln \\tilde{K}(x)$ .",
"Then it is easy to calculate that $s(x)=\\frac{\\nabla \\tilde{K}(x)}{\\tilde{K}(x)}=\\nabla S(x),$ where $\\nabla $ is short for $(\\partial /\\partial x_1,\\ldots ,\\partial /\\partial x_m)$ .",
"Hence the selection gradient is a gradient field.",
"Conversely, if the selection gradient is a gradient field, $s(x)=\\nabla S(x)$ , then let $\\tilde{K}(x)=\\exp \\left[S(x)\\right],$ and $\\tilde{\\alpha }(x,y)=\\left[1-f(x,y)\\right]\\frac{\\tilde{K}(y)}{\\tilde{K}(x)}.$ Note that $\\tilde{K}(x)>0$ and $\\tilde{\\alpha }(x,x)=1$ for all $x$ , because $f(x,x)=0$ for any invasion fitness function (i.e., the long-term per capita growth rate of individuals with the resident phenotype must be 0, because the resident, assumed to exist on an equilibrium, neither goes extinct nor increases without bounds).",
"Moreover, using the fact that $s(x)=\\nabla S(x)=\\nabla \\tilde{K}(x)\\cdot \\tilde{K}(x)^{-1}$ by construction, it is easy to check that $\\partial \\tilde{\\alpha }(x,y)/\\partial y\\vert _{y=x}=0$ for all $x$ , and that expression (15) holds for the invasion fitness function $f(x,y)$ .",
"$\\Box $ Thus, any invasion fitness function for which the selection gradient is a gradient field can be viewed as the invasion fitness function of a generalized Lotka-Volterra model for symmetric, frequency-dependent competition.",
"As mentioned in the previous section, in this case the adaptive dynamics is a hill-climbing process on a fixed fitness landscape determined by the (generalized) carrying capacity $\\tilde{K}(x)$ defined by (17), and singular points of the adaptive dynamics are given as local extrema of $\\tilde{K}(x)$ .",
"In particular, a singular point of the adaptive dynamics is convergent stable if and only if it is a local maximum of $\\tilde{K}(x)$ .",
"Next, we derive a general condition for a selection gradients to be a gradient field.",
"For this we recall that a geometric object is called simply connected if any closed path can be shrunk continuously to a single point within that object.",
"Thus, roughly speaking an object is simply connected if it doesn't have any holes.",
"A typical example is Euclidean space $\\mathbb {R}^n$ for any $n$ .",
"Proposition 2.2 Assume that the phenotype space (i.e., the set of all attainable, m-dimensional phenotype vectors $x$ ) is simply connected.",
"Then the selection gradient $s(x)$ of an invasion fitness function $f(x,y)$ is a gradient field if and only if the invasion fitness function satisfies the following gradient condition: $\\left.\\frac{\\partial ^2 f(x,y)}{\\partial x_i\\partial y_j}\\right|_{y=x}= \\left.\\frac{\\partial ^2 f(x,y)}{\\partial x_j\\partial y_i}\\right|_{y=x}$ for all $i,j$ .",
"Proof First of all, we note that in any case $\\frac{\\partial s_i(x)}{\\partial x_j}=\\left.\\frac{\\partial ^2 f}{\\partial x_j\\partial y_i}\\right|_{y=x}+\\left.\\frac{\\partial ^2 f}{\\partial y_j\\partial y_i}\\right|_{y=x}.$ Therefore, $D_{ij}(x):=\\frac{\\partial s_i(x)}{\\partial x_j}-\\frac{\\partial s_j(x)}{\\partial x_i}=\\left.\\frac{\\partial ^2 f}{\\partial x_j\\partial y_i}\\right|_{y=x}-\\left.\\frac{\\partial ^2 f}{\\partial x_i\\partial y_j}\\right|_{y=x}.$ If the selection gradient is a gradient field, $s(x)=\\nabla S(x)$ , then $D_{ij}(x)=\\frac{\\partial ^2S(x)}{\\partial x_j\\partial x_i}-\\frac{\\partial ^2S(x)}{\\partial x_i\\partial x_j}=0$ for all $x$ , as claimed.",
"Conversely, according to basic theory of differential forms ([23]), the condition $D_{ij}=0$ is equivalent to saying that the differential of the selection gradient is 0, i.e., that the selection gradient is a closed form.",
"A closed form is exact if it is the differential of a function, and exactness of closed forms is captured by the first De Rham cohomology group ([23]).",
"For simply connected spaces this group is trivial, which implies that every closed form is exact, i.e., every closed form is the differential of a function, and hence a gradient field.",
"$\\Box $ Together, the two propositions above yield the following Corollary 2.3 Let $f(x,y)$ be an invasion fitness function defined for a single species on a simply connected phenotype space of arbitrary dimension.",
"Then the function $f(x,y)$ is equivalent to the invasion fitness derived from a generalized Lotka-Volterra model for symmetric, frequency-dependent competition if and only if $\\left.\\frac{\\partial ^2 f(x,y)}{\\partial x_i\\partial y_j}\\right|_{y=x}= \\left.\\frac{\\partial ^2 f(x,y)}{\\partial x_j\\partial y_i}\\right|_{y=x}$ for all $i,j$ .",
"If this condition is satisfied, the generalized carrying capacity and competition kernel are given by expressions (17) and (18).",
"Applications We illustrate the potential usefulness of the theory presented in the previous section with some examples.",
"Definition of frequency-dependent selection The first application concerns the conceptual issue of frequency-dependent selection, a fundamental and much debated topic in evolutionary theory.",
"For adaptive dynamics models, it has been argued by [12] that selection should be considered frequency-independent (or trivially frequency-dependent) if the resident phenotype $x$ only enters the invasion fitness function through a scalar function describing the resident's effect on the environment (e.g., the resident population size).",
"Otherwise, selection is frequency-dependent.",
"If an adaptive dynamics model is given by an invasion fitness function whose selection gradient is a gradient field, this notion of frequency dependence can be made mathematically precise in the context of the generalized competition model (15): selection is frequency-independent if and only if the generalized competition kernel $\\tilde{\\alpha }(x,y)$ given by (18) is a constant (equal to 1), i.e., if and only if $f(x,y)=1-\\frac{\\tilde{K}(x)}{\\tilde{K}(y)},$ where $\\tilde{K}(x)$ is the generalized carrying capacity (17).",
"Because the generalized carrying capacity is entirely defined in terms of the invasion fitness, this leads to a model-independent definition of frequency independence (and hence of frequency dependence).",
"Universality of symmetric competition in 1-dimensional phenotype space The second application concerns the adaptive dynamics in 1-dimensional phenotype spaces.",
"In this case, any differentiable invasion fitness function trivially satisfies the gradient condition (19), i.e., the selection gradient $s(x)$ is always an integrable function, and hence any 1-dimensional adaptive dynamics model that is based on a differentiable invasion fitness function is equivalent to the adaptive dynamics of symmetric logistic competition.",
"We illustrate this by considering the invasion fitness function for a 1-dimensional trait under asymmetric competition, $f(x,y)=1-\\frac{\\alpha (x,y)K(x)}{K(y)},$ where the competition kernel $\\alpha (x,y)$ still has the property that $\\alpha (x,x)=1$ for all $x$ , but $\\partial \\alpha (x,y)/\\partial y\\vert _{y=x}\\ne 0$ in general.",
"Such models have been considered in the literature (e.g.",
"[24], [13], [7], [6]) and reflect the assumption of an intrinsic advantage of one phenotypic direction, such as an intrinsic advantage to being higher or larger when competition between plant individuals is affected by access to sunlight.",
"As in symmetric competition models, the function $K(x)$ is the carrying capacity function.",
"In this case, it follows that $s(x)=\\left.\\frac{\\partial f(x,y)}{\\partial y}\\right|_{y=x}=-\\left.\\frac{\\partial \\alpha (x,y)}{\\partial y}\\right|_{y=x}+\\frac{K^{\\prime }(x)}{K(x)}.$ Let $A(x)=\\left.\\frac{\\partial \\alpha (x,y)}{\\partial y}\\right|_{y=x}$ and consider the generalized carrying capacity $\\tilde{K}(x)=\\exp \\left[-\\int ^x A(x^{\\prime })dx^{\\prime }+\\ln \\left(K(x)\\right)\\right].$ and the generalized competition kernel $\\tilde{\\alpha }(x,y)=\\left(1-f(x,y)\\right)\\frac{\\tilde{K}(y)}{\\tilde{K}(x)}.$ Then $\\tilde{K}(x)>0$ for all $x$ , $\\tilde{\\alpha }(x,x)=1$ for all $x$ , and $\\partial \\tilde{\\alpha }(x,y)/\\partial y\\vert _{y=x}=0$ for all $x$ .",
"Also $f(x,y)=1-\\frac{\\tilde{\\alpha }(x,y)\\tilde{K}(x)}{\\tilde{K}(y)},$ and hence the invasion fitness function for asymmetric competition is equivalent to the invasion fitness function of a symmetric competition model, in which the original asymmetry in the competition kernel is shifted onto the generalized carrying capacity function $\\tilde{K}(x)$ .",
"Cyclic adaptive dynamics in asymmetric single-species models Third, to further illustrate the implications of the gradient condition (19), we present an example of the adaptive dynamics that may occur when the gradient condition (19) is not satisfied.",
"For this we consider an asymmetric competition model with a 2-dimensional phenotype space defined by a carrying capacity function $K(x)$ and a competition kernel $\\alpha (x,y)$ , where $x=(x_1,x_2)$ and $y=(y_1,y_2)$ are 2-dimensional traits, as follows: $K(x)=\\exp \\left(-\\frac{x_1^4+x_2^4}{2\\sigma _K^4}\\right)$ $\\alpha (x,y)=&\\exp \\left(-\\frac{(x_1-y_1)^2+(x_2-y_2)^2}{2\\sigma _\\alpha ^2}\\right)\\\\\\nonumber &\\times \\exp \\left(c_1(x_1y_2-x_2y_1)+c_2(x_1(y_1-x_1)+x_2(y_2-x_2))\\right).$ We will not try to biologically justify this choice of functions, except to note that the competition kernel (32) has been chosen so that for $i=1,2$ , $\\partial \\alpha (x,y)/\\partial {y_i}\\vert _{y=x}\\ne 0$ .",
"Biologically, this means that there are epistatic interactions between the trait components $x_1$ and $x_2$ , and as we will see, such interactions can be the source of complicated evolutionary dynamics even in a single evolving species.",
"Note that the carrying capacity (31) has a maximum at $(0,0)$ , that the competition kernel $\\alpha $ given by (32) retains the property that $\\alpha (x,x)=1$ for all $x$ , and that for $c_1=c_2=0$ , the competition kernel becomes symmetric.",
"For $c_1\\ne 0$ or $c_2\\ne 0$ , and in contrast to the 1-dimensional case for asymmetric competition kernels discussed above, the resulting invasion fitness function $f(x_1,x_2,y_1,y_2)$ does not satisfy the gradient condition (19), and hence is not equivalent to the invasion fitness of a symmetric competition model.",
"This can be seen by directly checking condition (19), or by considering the corresponding adaptive dynamics given by the selection gradients $s_1(x)=&\\left.\\frac{\\partial f(x,y)}{\\partial y_1}\\right|_{y=x}=\\left.-\\frac{\\partial \\alpha (x,y)}{\\partial y_1}\\right|_{y=x}+\\frac{\\partial K(x)}{\\partial x_1}\\cdot \\frac{1}{K(x)},\\\\s_2(x)=&\\left.\\frac{\\partial f(x,y)}{\\partial y_2}\\right|_{y=x}=\\left.-\\frac{\\partial \\alpha (x,y)}{\\partial y_2}\\right|_{y=x}+\\frac{\\partial K(x)}{\\partial x_2}\\cdot \\frac{1}{K(x)}.$ It is easy to see that this adaptive dynamical system has a singular point at $x^*=(0,0)$ (note that $\\partial \\alpha (x,y)/\\partial {y_i}\\vert _{y=x=x^*}=0$ for $i=1,2$ ), and that the Jacobian at the singular point $x^*$ has complex eigenvalues $\\pm ic_1 - c_2$ .",
"In particular, if $c_1\\ne 0$ , the adaptive dynamics has a cyclic component in the vicinity of the singular point $x^*$ , and it follows that if $c_1\\ne 0$ , the selection gradient $s=(s_1,s_2)$ cannot be a gradient field because the gradient condition is not satisfied.",
"In fact, the adaptive dynamics given by the selection gradients (33) and (34) can exhibit a stable limit cycle, as is illustrated in Figure 1.",
"Cyclic evolutionary dynamics are known to occur in the adaptive dynamics of multiple interacting species ([17], [15], [7]), but the figure shows that such dynamics can also occur in the adaptive dynamics of a single species with multidimensional phenotype.",
"The occurrence of cyclic evolutionary dynamics in a single species implies that the corresponding adaptive dynamics model is not equivalent to a symmetric competition model (even if it is an asymmetric competition model, as in the example above).",
"Local universality of symmetric competition models for determining evolutionary stability So far we have considered the possible equivalence of general single-species adaptive dynamics models in a given dimension to the adaptive dynamics of symmetric competition models.",
"If the adaptive dynamics convergence to a singular point, the question of evolutionary stability of the singular point arises, and we may then ask whether the evolutionary stability of singular points of generic single-species adaptive dynamics models can be understood in terms of the evolutionary stability of singular points in symmetric competition models.",
"In this section, we show that in fact, the evolutionary stability of singular points of any generic single-species adaptive dynamics models can always be understood by means of symmetric competition models.",
"Consider an arbitrary invasion fitness function $f(x,y)$ in the neighbourhood of a singular point $x^*$ , which we assume to be $x^*=0$ without loss of generality.",
"Because $f(x^*,x^*)=0$ as always, and $\\partial f(x^*,y)/\\partial y\\vert _{y=x^*}=(\\partial f(x^*,y)/\\partial y_1\\vert _{y=x^*},...,\\partial f(x^*,y)/\\partial y_m\\vert _{y=x^*})=0$ by assumption of singularity of $x^*$ , the Taylor expansion of the function $f$ in $x$ and $y$ in a neighbourhood of the point $(x^*,x^*)$ has the following form: $f(x,y)=g(x)+xAy^T+\\frac{1}{2}yHy^T+h.o.t.$ where $g(x)$ is some function of $x$ , $A$ is a square matrix, and $H$ is the symmetric Hessian matrix $H=\\begin{pmatrix}\\left.\\frac{\\partial ^2 f(x,y)}{\\partial y_1^2}\\right|_{y=x=x^*} & \\ldots & \\left.\\frac{\\partial ^2 f(x,y)}{\\partial y_1\\partial y_m}\\right|_{y=x=x^*} \\\\& \\ldots & \\\\\\left.\\frac{\\partial ^2 f(x,y)}{\\partial y_1\\partial y_m}\\right|_{y=x=x^*} & \\ldots & \\left.\\frac{\\partial ^2 f(x,y)}{\\partial y_m^2}\\right|_{y=x=x^*}\\end{pmatrix}$ Here $h.o.t.$ denotes terms of order $>2$ in $y$ , and $x^T$ and $y^T$ denote the transpose of the vectors $x=(x_1,...,x_m)$ and $y=(y_1,...,y_m)$ , where $m$ is the dimension of phenotype space.",
"In generic models the evolutionary stability of the singular point is determined by the Hessian matrix $H$ ([16], [6]).",
"Here we call a model generic if the Hessian $H$ is non-degenerate (i.e., has a trivial kernel).",
"Now consider the modified invasion fitness function $\\tilde{f}(x,y)=g(x)+\\frac{1}{2}yHy^T,$ also defined in a neighbourhood of $x^*$ .",
"Then the Jacobian $\\tilde{J}$ of the corresponding selection gradient $\\tilde{s}$ is simply the matrix $H$ , which by construction is symmetric and hence has real eigenvalues.",
"In particular, the linear dynamics defined by $\\tilde{s}$ has no cyclic component, and hence $\\tilde{s}$ is a gradient field ([23]).",
"It therefore follows from Proposition 2.1 that the invasion fitness function $\\tilde{f}$ is equivalent to the invasion fitness of a symmetric competition model.",
"On the other hand, the evolutionary stability of the singular point $x^*$ for the invasion fitness function $\\tilde{f}$ is also given by the matrix $H$ , and hence is exactly the same as the evolutionary stability of the singular point $x^*$ for the original invasion fitness function $f$ .",
"This proves the following Proposition 4.1 Given any single-species invasion fitness function $f$ with a non-degenerate Hessian matrix $H$ , and a singular point $x^*$ of the corresponding adaptive dynamics model, then in a neighbourhood of $x^*$ , there is an invasion fitness function $\\tilde{f}$ derived from a symmetric competition model for which $x^*$ is a singular point, and such that the evolutionary stability of $x^*$ is the same for $f$ and $\\tilde{f}$ .$\\Box $ It is important to note that while the evolutionary stability is the same for $f$ and $\\tilde{f}$ , convergence stability of $x^*$ in the adaptive dynamics models defined by $f$ and $\\tilde{f}$ (i.e., by $s$ and $\\tilde{s}$ in the construction above) is generally not the same.",
"We note that even if it were, convergence stability generally depends on the mutational variance-covariance matrix ([16]).",
"Nevertheless, the proposition shows that the evolutionary stability of singular points of any single species adaptive dynamics model can be fully understood in terms of the evolutionary stability of singular points in symmetric competition models.",
"We again note that evolutionary stability is not affected by the mutational variance-covariance matrix even in high-dimensional phenotype spaces ([16]), hence this result is important for generalizing results already known for evolutionary stability in competition models, such as the finding in [8] that increasing the dimension of phenotype space generally increases the likelihood of evolutionary branching in symmetric competition models.",
"Conclusions We have shown that any model for the adaptive dynamics of a single species that is defined on a simply connected phenotype space and satisfies the gradient condition (19) is equivalent to an adaptive dynamics model for symmetric frequency-dependent competition.",
"Specifically, expression (15) can be considered a “normal form” for any given invasion fitness function satisfying the gradient condition (19).",
"The obvious advantage of having such a normal form is that results obtained for the normal form are general and hold for any adaptive dynamics model that can be transformed into this normal form.",
"To illustrate this, we have shown that any single-species adaptive dynamics model in a 1-dimensional phenotype space is equivalent to a symmetric competition model.",
"In addition, we have shown that the evolutionary stability of single-species adaptive dynamics models with arbitrary phenotypic dimension can be understood in terms of symmetric competition models.",
"Attempts at finding normal forms of invasion fitness functions have been made previously.",
"For example, [9] showed that for every single-resident fitness function there exists a Lotka-Volterra competition model that has the same single-resident invasion fitness function.",
"However, the interaction function was not partitioned into a frequency-dependent competition kernel and a frequency-independent carrying capacity.",
"The approach presented here appears to be at the same time simpler and more specific, which is probably due to the fact that [9] dealt with the more complicated issue of normal forms describing the transition from monomorphic to polymorphic populations.",
"Our normal form (15) is simpler because it only considers selection gradients in monomorphic populations, and hence only requires the definition of the generalized carrying capacity function (17) and the generalized competition kernel (18).",
"Even though our normal form is only valid for invasion fitness functions satisfying (19) and for the monomorphic resident population, for those conditions it is general because it holds globally, i.e., everywhere in phenotype space, rather than just in the neighbourhood of singular points.",
"And it is more specific because it disentangles the frequency-dependent and the frequency-independent components of selection.",
"Generalized competition function without separation of frequency-dependent and frequency-independent components have also been considered by [18] for polymorphic populations.",
"Essentially, the normal form (15) holds for any frequency-dependent adaptive dynamics model whose selection gradient is a gradient field, and hence whose dynamics can be described as a hill-climbing process on a fixed landscape.",
"Because such hill-climbing processes cannot capture oscillatory behaviour, this makes it clear that the normal form cannot hold for any adaptive dynamics model exhibiting cyclic dynamics, e.g.",
"evolutionary arms races in predator-prey systems ([7], [6]).",
"To illustrate this, we have given an example of cyclic dynamics in a single-species adaptive dynamics model for asymmetric competition in 2-dimensional phenotype space, which therefore does not have the normal form (15).",
"In general, the normal form can be applied to models of single species with high-dimensional phenotype spaces, and it is important to note that the generalized carrying capacity (17) and competition kernel (18) may be complicated functions in general.",
"For example, the generalized carrying capacity may have multiple local maxima and minima (each representing a singular point of the adaptive dynamics), and the generalized competition kernel may have positive curvature at $x=y$ (representing positive frequency dependence).",
"Accordingly, the adaptive dynamics resulting from a normal form may exhibit repellers and dependence on initial conditions.",
"The normal form (15) does not generally apply to the adaptive dynamics of multiple species.",
"For example, if a single species adaptive dynamics model has a normal form (15), and if that normal form predicts evolutionary branching, then evolutionary branching does indeed occur in the given adaptive dynamics model, but the normal form cannot be used to derive the adaptive dynamics after evolutionary diversification has occurred.",
"This is because to derive the adaptive dynamics ensuing after evolutionary branching, one has to know the ecological attractor of coexisting phenotypes, i.e., one has to have explicit information about the ecological dynamics underlying the adaptive dynamics model, which the normal form does not contain.",
"We leave it as a challenge for future research to derive normal forms for the adaptive dynamics of multiple interacting species.",
"Acknowledgements: We thank P. Krapivsky and M. Plyushchay for discussions.",
"M.D.",
"acknowledges the support of NSERC (Canada) and of the Human Frontier Science Program.",
"I. I. acknowledges the support of FONDECYT (Chile).",
"Author contributions: M.D and I. I. contributed equally to this work.",
"Figure: NO_CAPTIONFigure legend Figure 1: Example of cyclic adaptive dynamics in a single species with 2-dimensional phenotype space.",
"The figure shows a numerical solution of the dynamical system $dx_1/dt=s_1(x_1,x_2)$ and $dx_2/dt=s_2(x_1,x_2)$ , where $s_1$ and $s_2$ are given by (33) and (34).",
"This system reflects the simplifying assumption that the mutational variance-covariance matrix ([16], [6]) is the identity matrix.",
"Panel (a) shows the two traits $x_1$ and $x_2$ as a function of time, and panel (b) shows the corresponding phase diagram, illustrating convergence to a limit cycle from two different initial conditions both inside and outside the limit cycle.",
"Parameter values were $\\sigma _K=\\sigma _\\alpha =1$ , $c_1=-1$ and $c_2=-0.1$ ."
],
[
"Applications",
"We illustrate the potential usefulness of the theory presented in the previous section with some examples."
],
[
"Definition of frequency-dependent selection",
"The first application concerns the conceptual issue of frequency-dependent selection, a fundamental and much debated topic in evolutionary theory.",
"For adaptive dynamics models, it has been argued by [12] that selection should be considered frequency-independent (or trivially frequency-dependent) if the resident phenotype $x$ only enters the invasion fitness function through a scalar function describing the resident's effect on the environment (e.g., the resident population size).",
"Otherwise, selection is frequency-dependent.",
"If an adaptive dynamics model is given by an invasion fitness function whose selection gradient is a gradient field, this notion of frequency dependence can be made mathematically precise in the context of the generalized competition model (15): selection is frequency-independent if and only if the generalized competition kernel $\\tilde{\\alpha }(x,y)$ given by (18) is a constant (equal to 1), i.e., if and only if $f(x,y)=1-\\frac{\\tilde{K}(x)}{\\tilde{K}(y)},$ where $\\tilde{K}(x)$ is the generalized carrying capacity (17).",
"Because the generalized carrying capacity is entirely defined in terms of the invasion fitness, this leads to a model-independent definition of frequency independence (and hence of frequency dependence)."
],
[
"Universality of symmetric competition in 1-dimensional phenotype space",
"The second application concerns the adaptive dynamics in 1-dimensional phenotype spaces.",
"In this case, any differentiable invasion fitness function trivially satisfies the gradient condition (19), i.e., the selection gradient $s(x)$ is always an integrable function, and hence any 1-dimensional adaptive dynamics model that is based on a differentiable invasion fitness function is equivalent to the adaptive dynamics of symmetric logistic competition.",
"We illustrate this by considering the invasion fitness function for a 1-dimensional trait under asymmetric competition, $f(x,y)=1-\\frac{\\alpha (x,y)K(x)}{K(y)},$ where the competition kernel $\\alpha (x,y)$ still has the property that $\\alpha (x,x)=1$ for all $x$ , but $\\partial \\alpha (x,y)/\\partial y\\vert _{y=x}\\ne 0$ in general.",
"Such models have been considered in the literature (e.g.",
"[24], [13], [7], [6]) and reflect the assumption of an intrinsic advantage of one phenotypic direction, such as an intrinsic advantage to being higher or larger when competition between plant individuals is affected by access to sunlight.",
"As in symmetric competition models, the function $K(x)$ is the carrying capacity function.",
"In this case, it follows that $s(x)=\\left.\\frac{\\partial f(x,y)}{\\partial y}\\right|_{y=x}=-\\left.\\frac{\\partial \\alpha (x,y)}{\\partial y}\\right|_{y=x}+\\frac{K^{\\prime }(x)}{K(x)}.$ Let $A(x)=\\left.\\frac{\\partial \\alpha (x,y)}{\\partial y}\\right|_{y=x}$ and consider the generalized carrying capacity $\\tilde{K}(x)=\\exp \\left[-\\int ^x A(x^{\\prime })dx^{\\prime }+\\ln \\left(K(x)\\right)\\right].$ and the generalized competition kernel $\\tilde{\\alpha }(x,y)=\\left(1-f(x,y)\\right)\\frac{\\tilde{K}(y)}{\\tilde{K}(x)}.$ Then $\\tilde{K}(x)>0$ for all $x$ , $\\tilde{\\alpha }(x,x)=1$ for all $x$ , and $\\partial \\tilde{\\alpha }(x,y)/\\partial y\\vert _{y=x}=0$ for all $x$ .",
"Also $f(x,y)=1-\\frac{\\tilde{\\alpha }(x,y)\\tilde{K}(x)}{\\tilde{K}(y)},$ and hence the invasion fitness function for asymmetric competition is equivalent to the invasion fitness function of a symmetric competition model, in which the original asymmetry in the competition kernel is shifted onto the generalized carrying capacity function $\\tilde{K}(x)$ ."
],
[
"Cyclic adaptive dynamics in asymmetric single-species models",
"Third, to further illustrate the implications of the gradient condition (19), we present an example of the adaptive dynamics that may occur when the gradient condition (19) is not satisfied.",
"For this we consider an asymmetric competition model with a 2-dimensional phenotype space defined by a carrying capacity function $K(x)$ and a competition kernel $\\alpha (x,y)$ , where $x=(x_1,x_2)$ and $y=(y_1,y_2)$ are 2-dimensional traits, as follows: $K(x)=\\exp \\left(-\\frac{x_1^4+x_2^4}{2\\sigma _K^4}\\right)$ $\\alpha (x,y)=&\\exp \\left(-\\frac{(x_1-y_1)^2+(x_2-y_2)^2}{2\\sigma _\\alpha ^2}\\right)\\\\\\nonumber &\\times \\exp \\left(c_1(x_1y_2-x_2y_1)+c_2(x_1(y_1-x_1)+x_2(y_2-x_2))\\right).$ We will not try to biologically justify this choice of functions, except to note that the competition kernel (32) has been chosen so that for $i=1,2$ , $\\partial \\alpha (x,y)/\\partial {y_i}\\vert _{y=x}\\ne 0$ .",
"Biologically, this means that there are epistatic interactions between the trait components $x_1$ and $x_2$ , and as we will see, such interactions can be the source of complicated evolutionary dynamics even in a single evolving species.",
"Note that the carrying capacity (31) has a maximum at $(0,0)$ , that the competition kernel $\\alpha $ given by (32) retains the property that $\\alpha (x,x)=1$ for all $x$ , and that for $c_1=c_2=0$ , the competition kernel becomes symmetric.",
"For $c_1\\ne 0$ or $c_2\\ne 0$ , and in contrast to the 1-dimensional case for asymmetric competition kernels discussed above, the resulting invasion fitness function $f(x_1,x_2,y_1,y_2)$ does not satisfy the gradient condition (19), and hence is not equivalent to the invasion fitness of a symmetric competition model.",
"This can be seen by directly checking condition (19), or by considering the corresponding adaptive dynamics given by the selection gradients $s_1(x)=&\\left.\\frac{\\partial f(x,y)}{\\partial y_1}\\right|_{y=x}=\\left.-\\frac{\\partial \\alpha (x,y)}{\\partial y_1}\\right|_{y=x}+\\frac{\\partial K(x)}{\\partial x_1}\\cdot \\frac{1}{K(x)},\\\\s_2(x)=&\\left.\\frac{\\partial f(x,y)}{\\partial y_2}\\right|_{y=x}=\\left.-\\frac{\\partial \\alpha (x,y)}{\\partial y_2}\\right|_{y=x}+\\frac{\\partial K(x)}{\\partial x_2}\\cdot \\frac{1}{K(x)}.$ It is easy to see that this adaptive dynamical system has a singular point at $x^*=(0,0)$ (note that $\\partial \\alpha (x,y)/\\partial {y_i}\\vert _{y=x=x^*}=0$ for $i=1,2$ ), and that the Jacobian at the singular point $x^*$ has complex eigenvalues $\\pm ic_1 - c_2$ .",
"In particular, if $c_1\\ne 0$ , the adaptive dynamics has a cyclic component in the vicinity of the singular point $x^*$ , and it follows that if $c_1\\ne 0$ , the selection gradient $s=(s_1,s_2)$ cannot be a gradient field because the gradient condition is not satisfied.",
"In fact, the adaptive dynamics given by the selection gradients (33) and (34) can exhibit a stable limit cycle, as is illustrated in Figure 1.",
"Cyclic evolutionary dynamics are known to occur in the adaptive dynamics of multiple interacting species ([17], [15], [7]), but the figure shows that such dynamics can also occur in the adaptive dynamics of a single species with multidimensional phenotype.",
"The occurrence of cyclic evolutionary dynamics in a single species implies that the corresponding adaptive dynamics model is not equivalent to a symmetric competition model (even if it is an asymmetric competition model, as in the example above)."
],
[
"Local universality of symmetric competition models for determining evolutionary stability",
"So far we have considered the possible equivalence of general single-species adaptive dynamics models in a given dimension to the adaptive dynamics of symmetric competition models.",
"If the adaptive dynamics convergence to a singular point, the question of evolutionary stability of the singular point arises, and we may then ask whether the evolutionary stability of singular points of generic single-species adaptive dynamics models can be understood in terms of the evolutionary stability of singular points in symmetric competition models.",
"In this section, we show that in fact, the evolutionary stability of singular points of any generic single-species adaptive dynamics models can always be understood by means of symmetric competition models.",
"Consider an arbitrary invasion fitness function $f(x,y)$ in the neighbourhood of a singular point $x^*$ , which we assume to be $x^*=0$ without loss of generality.",
"Because $f(x^*,x^*)=0$ as always, and $\\partial f(x^*,y)/\\partial y\\vert _{y=x^*}=(\\partial f(x^*,y)/\\partial y_1\\vert _{y=x^*},...,\\partial f(x^*,y)/\\partial y_m\\vert _{y=x^*})=0$ by assumption of singularity of $x^*$ , the Taylor expansion of the function $f$ in $x$ and $y$ in a neighbourhood of the point $(x^*,x^*)$ has the following form: $f(x,y)=g(x)+xAy^T+\\frac{1}{2}yHy^T+h.o.t.$ where $g(x)$ is some function of $x$ , $A$ is a square matrix, and $H$ is the symmetric Hessian matrix $H=\\begin{pmatrix}\\left.\\frac{\\partial ^2 f(x,y)}{\\partial y_1^2}\\right|_{y=x=x^*} & \\ldots & \\left.\\frac{\\partial ^2 f(x,y)}{\\partial y_1\\partial y_m}\\right|_{y=x=x^*} \\\\& \\ldots & \\\\\\left.\\frac{\\partial ^2 f(x,y)}{\\partial y_1\\partial y_m}\\right|_{y=x=x^*} & \\ldots & \\left.\\frac{\\partial ^2 f(x,y)}{\\partial y_m^2}\\right|_{y=x=x^*}\\end{pmatrix}$ Here $h.o.t.$ denotes terms of order $>2$ in $y$ , and $x^T$ and $y^T$ denote the transpose of the vectors $x=(x_1,...,x_m)$ and $y=(y_1,...,y_m)$ , where $m$ is the dimension of phenotype space.",
"In generic models the evolutionary stability of the singular point is determined by the Hessian matrix $H$ ([16], [6]).",
"Here we call a model generic if the Hessian $H$ is non-degenerate (i.e., has a trivial kernel).",
"Now consider the modified invasion fitness function $\\tilde{f}(x,y)=g(x)+\\frac{1}{2}yHy^T,$ also defined in a neighbourhood of $x^*$ .",
"Then the Jacobian $\\tilde{J}$ of the corresponding selection gradient $\\tilde{s}$ is simply the matrix $H$ , which by construction is symmetric and hence has real eigenvalues.",
"In particular, the linear dynamics defined by $\\tilde{s}$ has no cyclic component, and hence $\\tilde{s}$ is a gradient field ([23]).",
"It therefore follows from Proposition 2.1 that the invasion fitness function $\\tilde{f}$ is equivalent to the invasion fitness of a symmetric competition model.",
"On the other hand, the evolutionary stability of the singular point $x^*$ for the invasion fitness function $\\tilde{f}$ is also given by the matrix $H$ , and hence is exactly the same as the evolutionary stability of the singular point $x^*$ for the original invasion fitness function $f$ .",
"This proves the following Proposition 4.1 Given any single-species invasion fitness function $f$ with a non-degenerate Hessian matrix $H$ , and a singular point $x^*$ of the corresponding adaptive dynamics model, then in a neighbourhood of $x^*$ , there is an invasion fitness function $\\tilde{f}$ derived from a symmetric competition model for which $x^*$ is a singular point, and such that the evolutionary stability of $x^*$ is the same for $f$ and $\\tilde{f}$ .$\\Box $ It is important to note that while the evolutionary stability is the same for $f$ and $\\tilde{f}$ , convergence stability of $x^*$ in the adaptive dynamics models defined by $f$ and $\\tilde{f}$ (i.e., by $s$ and $\\tilde{s}$ in the construction above) is generally not the same.",
"We note that even if it were, convergence stability generally depends on the mutational variance-covariance matrix ([16]).",
"Nevertheless, the proposition shows that the evolutionary stability of singular points of any single species adaptive dynamics model can be fully understood in terms of the evolutionary stability of singular points in symmetric competition models.",
"We again note that evolutionary stability is not affected by the mutational variance-covariance matrix even in high-dimensional phenotype spaces ([16]), hence this result is important for generalizing results already known for evolutionary stability in competition models, such as the finding in [8] that increasing the dimension of phenotype space generally increases the likelihood of evolutionary branching in symmetric competition models.",
"Conclusions We have shown that any model for the adaptive dynamics of a single species that is defined on a simply connected phenotype space and satisfies the gradient condition (19) is equivalent to an adaptive dynamics model for symmetric frequency-dependent competition.",
"Specifically, expression (15) can be considered a “normal form” for any given invasion fitness function satisfying the gradient condition (19).",
"The obvious advantage of having such a normal form is that results obtained for the normal form are general and hold for any adaptive dynamics model that can be transformed into this normal form.",
"To illustrate this, we have shown that any single-species adaptive dynamics model in a 1-dimensional phenotype space is equivalent to a symmetric competition model.",
"In addition, we have shown that the evolutionary stability of single-species adaptive dynamics models with arbitrary phenotypic dimension can be understood in terms of symmetric competition models.",
"Attempts at finding normal forms of invasion fitness functions have been made previously.",
"For example, [9] showed that for every single-resident fitness function there exists a Lotka-Volterra competition model that has the same single-resident invasion fitness function.",
"However, the interaction function was not partitioned into a frequency-dependent competition kernel and a frequency-independent carrying capacity.",
"The approach presented here appears to be at the same time simpler and more specific, which is probably due to the fact that [9] dealt with the more complicated issue of normal forms describing the transition from monomorphic to polymorphic populations.",
"Our normal form (15) is simpler because it only considers selection gradients in monomorphic populations, and hence only requires the definition of the generalized carrying capacity function (17) and the generalized competition kernel (18).",
"Even though our normal form is only valid for invasion fitness functions satisfying (19) and for the monomorphic resident population, for those conditions it is general because it holds globally, i.e., everywhere in phenotype space, rather than just in the neighbourhood of singular points.",
"And it is more specific because it disentangles the frequency-dependent and the frequency-independent components of selection.",
"Generalized competition function without separation of frequency-dependent and frequency-independent components have also been considered by [18] for polymorphic populations.",
"Essentially, the normal form (15) holds for any frequency-dependent adaptive dynamics model whose selection gradient is a gradient field, and hence whose dynamics can be described as a hill-climbing process on a fixed landscape.",
"Because such hill-climbing processes cannot capture oscillatory behaviour, this makes it clear that the normal form cannot hold for any adaptive dynamics model exhibiting cyclic dynamics, e.g.",
"evolutionary arms races in predator-prey systems ([7], [6]).",
"To illustrate this, we have given an example of cyclic dynamics in a single-species adaptive dynamics model for asymmetric competition in 2-dimensional phenotype space, which therefore does not have the normal form (15).",
"In general, the normal form can be applied to models of single species with high-dimensional phenotype spaces, and it is important to note that the generalized carrying capacity (17) and competition kernel (18) may be complicated functions in general.",
"For example, the generalized carrying capacity may have multiple local maxima and minima (each representing a singular point of the adaptive dynamics), and the generalized competition kernel may have positive curvature at $x=y$ (representing positive frequency dependence).",
"Accordingly, the adaptive dynamics resulting from a normal form may exhibit repellers and dependence on initial conditions.",
"The normal form (15) does not generally apply to the adaptive dynamics of multiple species.",
"For example, if a single species adaptive dynamics model has a normal form (15), and if that normal form predicts evolutionary branching, then evolutionary branching does indeed occur in the given adaptive dynamics model, but the normal form cannot be used to derive the adaptive dynamics after evolutionary diversification has occurred.",
"This is because to derive the adaptive dynamics ensuing after evolutionary branching, one has to know the ecological attractor of coexisting phenotypes, i.e., one has to have explicit information about the ecological dynamics underlying the adaptive dynamics model, which the normal form does not contain.",
"We leave it as a challenge for future research to derive normal forms for the adaptive dynamics of multiple interacting species.",
"Acknowledgements: We thank P. Krapivsky and M. Plyushchay for discussions.",
"M.D.",
"acknowledges the support of NSERC (Canada) and of the Human Frontier Science Program.",
"I. I. acknowledges the support of FONDECYT (Chile).",
"Author contributions: M.D and I. I. contributed equally to this work.",
"Figure legend Figure 1: Example of cyclic adaptive dynamics in a single species with 2-dimensional phenotype space.",
"The figure shows a numerical solution of the dynamical system $dx_1/dt=s_1(x_1,x_2)$ and $dx_2/dt=s_2(x_1,x_2)$ , where $s_1$ and $s_2$ are given by (33) and (34).",
"This system reflects the simplifying assumption that the mutational variance-covariance matrix ([16], [6]) is the identity matrix.",
"Panel (a) shows the two traits $x_1$ and $x_2$ as a function of time, and panel (b) shows the corresponding phase diagram, illustrating convergence to a limit cycle from two different initial conditions both inside and outside the limit cycle.",
"Parameter values were $\\sigma _K=\\sigma _\\alpha =1$ , $c_1=-1$ and $c_2=-0.1$ ."
]
] | 1204.0831 |
[
[
"Characterising Vainshtein Solutions in Massive Gravity"
],
[
"Abstract We study static, spherically symmetric solutions in a recently proposed ghost-free model of non-linear massive gravity.",
"We focus on a branch of solutions where the helicity-0 mode can be strongly coupled within certain radial regions, giving rise to the Vainshtein effect.",
"We truncate the analysis to scales below the gravitational Compton wavelength, and consider the weak field limit for the gravitational potentials, while keeping all non-linearities of the helicity-0 mode.",
"We determine analytically the number and properties of local solutions which exist asymptotically on large scales, and of local (inner) solutions which exist on small scales.",
"We find two kinds of asymptotic solutions, one of which is asymptotically flat, while the other one is not, and also two types of inner solutions, one of which displays the Vainshtein mechanism, while the other exhibits a self-shielding behaviour of the gravitational field.",
"We analyse in detail in which cases the solutions match in an intermediate region.",
"The asymptotically flat solutions connect only to inner configurations displaying the Vainshtein mechanism, while the non asymptotically flat solutions can connect with both kinds of inner solutions.",
"We show furthermore that there are some regions in the parameter space where global solutions do not exist, and characterise precisely in which regions of the phase space the Vainshtein mechanism takes place."
],
[
"Introduction",
"Promoting Einstein's gravity to a classical theory of a massive graviton is a theoretical challenge.",
"The appearance of a ghost instability and the fact that in the massless limit solutions may not reduce to those of General Relativity (GR) have been the two main obstacles for a successful Lagrangian construction [1].",
"Both of these problems are related to the scalar sector of the theory.",
"In recent years, new developments have opened a window into a ghost-free model for massive gravity [2], [4], [5], based on the request that the equation of motion for the scalar mode does not contain derivatives of order higher than two [6].",
"However, with respect to the second issue the story is not entirely settled, as we will discuss here.",
"In this massive gravity theory, spherically symmetric solutions do not obey a uniqueness principle, like Birkhoff's theorem in General Relativity.",
"As a result, the theory exhibits two classes of static spherically symmetric solutions, which differ in how the scalar sector couples to graviton.",
"The first class of solutions naturally contains (Anti) de Sitter-Schwarzschild solutions, where the cosmological constant is proportional to the graviton's mass [7], [8], [9].",
"Some of these solutions are very similar to those found many years ago in the simple Fierz-Pauli model [10], and what characterises them is that the scalar mode is strongly coupled everywhere, which avoids a discontinuous connection to GR in the massless limit.",
"Therefore, at the level of the background, these solutions are indistinguishable from their counterparts in General Relativity, but do present differences at first orders in perturbations.",
"Interestingly, in some of these solutions, scalar and vector perturbations are not dynamical at first order in perturbations, and only the tensor modes differ significantly from General Relativity [11], [12], [13].",
"If the scalar and vector modes remain non-dynamical at higher orders in perturbation theory, then only the detection of gravitational waves can rule out these solutions in favour of $\\Lambda $ CDM.",
"In the second class of solutions the story is not that clear.",
"The scalar sector is not strongly coupled everywhere, and the solutions behave very differently as a function of the radial coordinate.",
"The purpose of this paper is to exhibit the rich structure of such class of solutions, and show whether one can recover GR solutions in the massless limit, at least within some radial region.",
"This phenomenon of recovering GR predictions within some macroscopic radius from a mass source is known as the Vainshtein mechanism [14].",
"In the range where the solution reduces to GR the scalar field becomes strongly coupled.",
"Therefore, it is imperative to include non-linearities in the scalar sector in order to study the Vainshtein mechanism.",
"Solutions presenting these behaviour in massive gravity where found in [15], and more recently in the ghost-free model considered here in [7], [8], [17], [18].",
"Unfortunately, these solutions do not cover the whole parameter space of the theory as we will describe here.",
"We organise the paper as follows: in Section 2 we give an overview of the main equations describing the theory and spherically symmetric solutions, with particular attention to this second branch.",
"In Section 3 we present an exhaustive analysis of the vacuum equations, with particular attention to the behaviour of their solutions near the origin and towards infinity.",
"In Section 4 we present our main results, including the parameter space in which GR solutions are recovered via the Vainshtein mechanism and numerical solutions of the vacuum equations for some representative cases.",
"Finally, we present some conclusions in Section 5."
],
[
"Ghost-free massive gravity and spherically symmetric ansätze",
"We consider the following Lagrangian for massive gravity, a non-linear extension of Fierz-Pauli theory proposed in [4] ${\\cal L} = \\frac{M_{Pl}^2}{2}\\,\\sqrt{-g}\\left( R - {\\cal U}\\right).$ The potential ${\\cal U}$ depends on a dimension-full parameter $m$ , which sets the graviton mass scale, and on two dimensionless parameters $\\alpha _3$ and $\\alpha _4$ .",
"It has the following functional form ${\\cal U}= -m^2\\left[{\\cal U}_2+\\alpha _3\\, {\\cal U}_3+\\alpha _4\\, {\\cal U}_4\\right],$ with ${\\cal U}_2&=&({\\rm tr}\\,{\\mathcal {K}})^2-{\\rm tr}\\,({\\mathcal {K}}^2),\\nonumber \\\\{\\cal U}_3&=&({\\rm tr}\\,{\\mathcal {K}})^3 - 3 ({\\rm tr}\\,{\\mathcal {K}})({\\rm tr}\\,{\\mathcal {K}}^2) + 2 {\\rm tr}\\,{\\mathcal {K}}^3,\\nonumber \\\\{\\cal U}_4 &=& ({\\rm tr}\\,{\\mathcal {K}})^4 - 6 ({\\rm tr}\\,{\\mathcal {K}})^2 ({\\rm tr}\\,{\\mathcal {K}}^2)+ 8 ({\\rm tr}\\,{\\mathcal {K}})({\\rm tr}\\,{\\mathcal {K}}^3) + 3 ({\\rm tr}\\,{\\mathcal {K}}^2)^2 - 6 {\\rm tr}\\,{\\mathcal {K}}^4 .\\nonumber $ The tensor ${\\cal K}_{\\mu }^{\\ \\nu }$ is defined as [4] ${\\mathcal {K}}_{\\mu }^{\\ \\nu } &\\equiv &\\delta _{\\mu }^{\\ \\nu }-\\left(\\sqrt{g^{-1} \\Sigma }\\right)_{\\mu }^{\\ \\nu }\\,\\,,$ where the square root of a tensor is formally understood as $\\sqrt{{\\cal M}}_{\\mu }^{\\ \\alpha }\\sqrt{\\cal M}_{\\alpha }^{\\ \\nu }={\\cal M}_{\\mu }^{\\ \\nu }$ , for any tensor ${\\cal M}_\\mu ^{\\,\\,\\nu }$ .",
"The tensor $\\Sigma _{\\mu \\nu }$ is a fiducial metric, which also makes the theory reparametrisation invariant by means of four scalars $\\phi ^\\mu $ , and it is given by $\\Sigma _{\\mu \\nu }=\\partial _\\mu \\phi ^a\\partial _\\nu \\phi ^b\\eta _{ab}.$ The theory defined by (REF ) has Minkowski spacetime as solution, hence one can rewrite the metric $g_{\\mu \\nu }$ and the scalars $\\phi ^\\mu $ as deviations from flat space, namely $g_{\\mu \\nu }=\\eta _{\\mu \\nu }+h_{\\mu \\nu },\\qquad \\qquad \\phi ^\\mu =x^\\mu +\\pi ^\\mu ,$ where $x^\\mu $ are the usual cartesian coordinates spanning $\\eta _{\\mu \\nu }$ .",
"Therefore, a change of coordinates $x^{\\mu } \\rightarrow x^{\\mu } + \\xi ^{\\mu }$ should be accompanied by the following transformation of the Stückelberg field $\\pi ^\\mu $ , $\\pi ^{\\mu } \\rightarrow \\pi ^{\\mu } + \\xi ^{\\mu },$ in order to recover full diffeomorphism invariance.",
"Moreover, we choose the unitary gauge, where $\\pi ^\\mu =0$ , and the potential (REF ) considerably simplifies.",
"Since we are intested in spherically symmetric solutions, we use the most general static ansätze, given by $d s^2\\,=\\,-C(r) \\,d t^2+A(r)\\, d r^2 +2 D(r)\\, dt dr+B(r) d \\Omega ^2,$ where $d \\Omega ^2 = d \\theta ^2 + \\sin ^2 \\theta d \\phi ^2$ .",
"We choose to write the non-dynamical flat metric as $ds^2 = -dt^2 + dr^2 + r^2 d \\Omega ^2$ .",
"It should be noticed that this is not a coordinate choice, but a way to simplify the expressions.",
"We plug the previous metric into the Einstein equations $G_{\\mu \\nu }=T^{{\\cal U}}_{\\mu \\nu }$ , where the energy momentum tensor is defined as $T^{{\\cal U}}_{\\mu \\nu }\\,=\\frac{m^2}{\\sqrt{-g}}\\,\\frac{ \\delta \\sqrt{-g}\\ {\\cal U}}{\\delta g^{\\mu \\nu }}$ .",
"The Einstein tensor $G_{\\mu \\nu }$ satisfies the identity $D(r)\\, G_{tt}+C(r)\\,G_{tr}\\,=\\,0$ , which implies the algebraic constraint $0= D(r)\\, T^{{\\cal U}}_{tt}+C(r)\\,T^{{\\cal U}}_{tr}$ .",
"This last equation reduces to $D(r)\\left(b_0 r-\\sqrt{B(r)}\\right)$ , where $b_0$ is a function of $\\alpha _3$ and $\\alpha _4$ only [12].",
"This constraint is solved in two possible ways, defining two class of solutions: either the metric is diagonal $D=0$ , or $B=b_0^2 r^2$ .",
"Notice that this classification only holds in the unitary gauge, since one can always map the metric from one class to the other by a coordinate transformation, but to the price of exciting components of $\\pi ^\\mu $ .",
"The class of solutions with a non-diagonal metric leads to Schwarzschild or Schwarzschild-de Sitter solutions [7], [8], [9], as explained in the introduction.",
"In this sector of the theory GR is recovered by means of a strongly coupled $\\pi ^\\mu $ everywhere [11], [12].",
"However, in the other class of solutions, where the metric is diagonal in the unitary gauge, the situation is different.",
"As we discuss in the next section, $\\pi ^\\mu $ may or may not be strongly coupled: it could be strongly coupled within certain radial region, leading to a Vainshtein effect.",
"This branch is the one that concerns us here, so from now on we will only consider static diagonal spherically symmetric solutions in the unitary gauge (see [16] for a similar discussion on bimetric solutions in the unitary gauge).",
"The problem of finding exact vacuum solutions in this branch is an open question, but one can make progresses by considering perturbations (not necessarily small) from flat space, and the following ansatz results adequate for this purpose, $ds^2 = - \\Big (1+N(r)\\Big )^2 dt^2 + \\Big (1+F(r)\\Big )^{-1} dr^2 + r^2 \\Big (1+H(r)\\Big )^{-2} d \\Omega ^2.$ In order to analyse the system, it is convenient to introduce a new radial coordinate $\\rho = \\frac{r}{1+H(r)}\\,,$ so that the linearised metric is expressed as $ds^2 = - (1 + n) dt^2 + (1 - f) d\\rho ^2+ \\rho ^2 d \\Omega ^2,$ where $f(\\rho ) = F\\big (r(\\rho )\\big ) - 2 h(\\rho ) - 2 \\rho h^{\\prime }(\\rho )$ , $n(\\rho )=2N\\big (r(\\rho )\\big )$ , $h(\\rho )=H\\big (r(\\rho )\\big )$ and a prime denotes a derivative with respect to $\\rho $ .",
"As discussed above, one should be careful with this change of coordinates, since, after fixing a gauge, a change of frame in the metric modifies the Stückelberg field $\\pi ^\\mu $ as well.",
"It turns out that this coordinate transformation excites the radial component of $\\pi ^\\mu $ , which explicitly is $\\pi ^\\rho =\\rho h$ .",
"Therefore, from now on one can think of $h$ as simply being the only non-zero component of the Stückelberg field $\\pi ^\\mu $ .",
"At linear order, the equations for the functions $n(\\rho )$ , $f(\\rho )$ and $h(\\rho )$ in the new variable $\\rho $ are $0&=& \\left(m^2\\rho ^2+2\\right)f+2 \\rho \\left(f^{\\prime }+m^2 \\rho ^2h^{\\prime }+3 \\, m^2 \\rho h \\right), \\\\0&=& \\frac{1}{2}m^2 \\rho ^2 (n-4 h) -\\rho \\, n^{\\prime }-f, \\\\0&=& f +\\frac{1}{2}\\rho \\, n^{\\prime }.$ In this linear expansion, the solutions for $n$ and $f$ are $n &=& - \\frac{8 G M}{3 \\rho } e^{- m \\rho } \\\\f &=& -\\frac{4 G M}{3 \\rho } (1 + m \\rho ) e^{- m \\rho } $ where we fix the integration constant so that $M$ is the mass of a point particle at the origin, and $8 \\pi G = M_{pl}^{-2}$ .",
"These solutions exhibit the vDVZ discontinuity, since the post-Newtonian parameter $\\gamma =f/n$ is $\\gamma =\\frac{1}{2}(1+m\\rho )$ , which in the massless limit reduces to $\\gamma =1/2$ , in disagreement with GR and Solar system observations.",
"However, in order to understand what really happens in this limit, we must also analyse the behaviour of $h$ , or equivalently $\\pi ^\\rho $ , as $m\\rightarrow 0$ .",
"To do this, we consider scales below the Compton wavelength $m \\rho \\ll 1$ , and at the same time ignore higher order terms in $G M$ .",
"Under these approximations, the equations of motion can still be truncated to linear order in $f$ and $n$ , but since $h$ is not necessarily small, we have to keep all non-linear terms in $h$ .",
"In other words, we take the usual weak field limit for the metric fields, but keep all non-linearities in the Stückelberg field, since we expect regions where this field is strongly coupled.",
"As shown in [8], the field equations reduce to the following system of coupled equations for the fields $f$ , $n$ , $h$ : $f = - 2 \\frac{G M}{\\rho } - (m \\rho )^2 \\left[h - (1+ 3\\alpha _3)h^2+(\\alpha _3+4\\alpha _4)h^3\\right] \\\\[2mm]\\rho \\, n^{\\prime } = \\frac{2 G M}{\\rho } - (m \\rho )^2 \\left[h - (\\alpha _3 +4\\alpha _4) h^3\\right] \\\\[2mm]\\begin{split}\\frac{G M}{\\rho } \\left[1 - 3(\\alpha _3+4\\alpha _4)h^2\\right] &= - (m \\rho )^2\\Big \\lbrace \\frac{3}{2} h - 3 (1 + 3\\alpha _3)h^2 +\\left[(1 + 3\\alpha _3)^2 + 2(\\alpha _3 +4\\alpha _4)\\right]h^3 \\Big .",
"\\\\& \\hspace{59.75095pt} \\Big .",
"- \\frac{3}{2}(\\alpha _3 +4\\alpha _4)^2 h^5 \\Big \\rbrace \\end{split}$ These equations can also be obtained directly from the decoupling theory [2], [3], as it was shown in [8].",
"The previous expressions are the starting point of our analysis and classifications of solutions, which we will discuss in the next section."
],
[
"Classifying Solutions",
"Our aim is to scan the $(\\alpha _3, \\alpha _4)$ parameter phase space of theories, to understand how many solutions the system (REF )-() admits, and characterise their (asymptotic) geometrical properties.",
"Since the last equation () does not contain $f$ and $n$ , the field $h$ obeys a decoupled equation, thus once a solution of this equation is found, the gravitational potentials $f$ , $n$ are uniquely determined (up to an integration constant) by the other two equations (REF ) and ().",
"Therefore, we first focus on classifying the number and properties of solutions of () in every point of the phase space, and then discuss the behaviour of the gravitational potentials correspondent to these solutions."
],
[
"The quintic equation",
"For notational convenience, we define $\\alpha \\equiv 1 + 3 \\, \\alpha _3$ and $\\beta \\equiv \\alpha _3 + 4 \\, \\alpha _4 \\,$ .",
"Therefore, the system that determines the gravitational potentials in terms of $h$ takes the form $f &= - 2 \\, \\frac{G M}{\\rho } - (m \\rho )^2 \\Big (h - \\alpha h^2 + \\beta h^3 \\Big ) \\\\[2mm]n^{\\prime } &= 2 \\, \\frac{G M}{\\rho ^2} - m^2 \\rho \\, \\Big ( h - \\beta h^3 \\Big )$ while the equation which determines $h$ reduces to $\\frac{3}{2} \\, \\beta ^{2}\\, h^{5}(\\rho ) - \\Big ( \\alpha ^{2}+ 2 \\beta \\Big ) \\, h^{3}(\\rho ) + 3 \\, \\Big ( \\alpha + \\beta A(\\rho ) \\Big ) \\, h^{2}(\\rho )- \\frac{3}{2} \\, h(\\rho ) - A(\\rho ) = 0$ where $A(\\rho ) = \\big ( \\rho _{v} / \\rho \\big )^{3}$ and $\\rho _{v}$ is the Vainshtein radius defined as $\\rho _{v} \\equiv \\big ( G M / m^{2} \\big )^{\\!",
"1/3}$ .",
"The last equation is an algebraic equation in $h$ , $A$ , $\\alpha $ and $\\beta $ ; at fixed $\\rho $ , $\\alpha $ and $\\beta $ it is, in fact, a polynomial equation of fifth degree in $h$ , except for the special case $\\beta = 0$ .",
"In this particular case of $\\beta =0$ , the equation for $h$ becomes a cubic equation and it is possible to obtain solutions for $h$ and the metric perturbations exactly.",
"These solutions were studied in [7], [8] and it was shown that the solutions exhibit the Vainshtein mechanism.",
"Therefore, in what follows, we assume $\\beta \\ne 0$ .",
"It is difficult to find analytical solutions to this quintic equation, and it is not easy even to understand how many solutions it admits.",
"In fact, even if a local solution is found in a radial interval, it is not, in general, possible to extend it to the whole radial domain, as we will explicitly see in the next section.",
"Nevertheless, it is possible to determine exactly how many local solutions exist in a neighbourhood of $\\rho = +\\infty $ , which we refer as asymptotic solutions, and also how many local solutions exist in a neighbourhood of $\\rho = 0^{+}$ , which we call inner solutions.",
"Furthermore, we can find analytically their leading behaviour as a function of $\\rho $ .",
"Any global solution of (REF ) should necessarily interpolate between one of the asymptotic solutions and one of the inner solutions.",
"Therefore, our aim is to understand, for each point in the $(\\alpha , \\beta )$ phase space, whether and how the above solutions match.",
"A systematic approach to Vainshtein effects in covariant Galileon theory was performed in [19], and in general scalar-tensor theories [20].",
"However, in massive gravity a systematic treatment was not fully performed.",
"The main difference between massive gravity and standard Galileon theories is that in the latter case the fifth power of $h$ in equation () is absent.",
"The reason of this becomes particularly transparent when focussing on the decoupling limit.",
"Then one can see that, for $\\beta \\ne 0$ , there is a mixing between $h_{\\mu \\nu }$ and a combination of derivatives of the scalar graviton polarization, that cannot be removed by field redefinitions.",
"Such derivative mixing of the scalar with gravity is not included in standard Galileon models, and the corresponding equations of motion consequently miss its effect.",
"Implementing a suitable coordinate transformation [8], one can recognize that precisely the aforementioned mixing gives rise to the quintic term in $h$ in the equation ().",
"As we will discuss in what follows, the analysis associated with the quintic equation turning on a $\\beta \\ne 0$ enriches considerably the properties of the phase space with respect to the case $\\beta =0$ ."
],
[
"Phase space analysis",
"To be able to describe how the matching works in all the phase space, in principle we should study separately every point ($\\alpha $ , $\\beta $ ).",
"However, this is not necessary since equation (REF ) obeys a remarkable symmetry: defining the quintic function as $q \\, \\big ( h; A, \\alpha , \\beta \\big ) \\equiv \\frac{3}{2} \\, \\beta ^{2}\\, h^{5} - \\big ( \\alpha ^{2}+ 2 \\beta \\big ) \\, h^{3} + 3 \\, \\big ( \\alpha + \\beta A \\big ) \\, h^{2}- \\frac{3}{2} \\, h - A$ it is simple to see that $q \\, \\Big ( \\frac{h}{k} ; \\frac{A}{k} , k \\, \\alpha , k^2 \\beta \\Big ) = \\frac{1}{k} \\, q \\, \\big ( h; A, \\alpha , \\beta \\big )$ Therefore if a local solution of (REF ) exists for a given $(\\alpha ,\\beta )$ within a certain radial interval, it would also be present for $(k \\alpha ,k^2 \\beta )$ , for $k>0$ , with $h$ being replaced by $h/k$ and the radial interval rescaled by $1/\\@root 3 \\of {k}$ .",
"As a result, each point belonging to the $\\alpha >0$ part of the parabola $\\beta = c \\, \\alpha ^{2}$ of the phase space (with $c$ any non-vanishing constant) shares the same physics, hence having the same number of global solutions and matching properties.",
"The same is true for the points belonging the $\\alpha <0$ part of the parabola.",
"So, to understand the global structure of the phase space, it is sufficient to analyse one point for each of the half parabolas present in the phase space.",
"It is worthwhile to point out that our starting equations (REF )-(REF ) were constructed assuming $GM < \\rho < 1/m$ , but in the following analysis we use the whole radial domain $0 < \\rho < + \\infty $ .",
"On one hand, this allows us to characterise exactly the number and properties of solutions on large and small scales.",
"On the other hand, the picture we have in mind is that the Compton wavelength of the gravitational field $\\rho _c = 1/m$ is of the same order of the Hubble radius today, and that there is a huge hierarchy between $\\rho _c$ and the gravitational radiusWe are using units where the speed of light speed has unitary value.",
"$\\rho _g = GM$ , i.e.",
"$\\rho _c / \\rho _g \\ggg 1$ .",
"Therefore, we expect that extending the analysis to the whole radial domain captures the correct physical results."
],
[
"Asymptotic and inner solutions",
"We sum up here the results obtained in the appendix on the existence and properties of asymptotic and inner solutions of eq.",
"(REF ).",
"We refer the reader to the appendix for the details."
],
[
"Asymptotic solutions",
"In a neighbourhood of $\\rho \\rightarrow +\\infty $ there are, depending on the value of $(\\alpha , \\beta )$ , three or five solutions to eq.",
"(REF ).",
"In particular: - There is always a decaying solution, which we indicate with $\\textbf {L}$ .",
"Its asymptotic behaviour is $h(\\rho ) = - \\frac{2}{3} \\left( \\frac{\\rho _{v}}{\\rho } \\right)^{\\!",
"3} + \\, R(\\rho )$ where $\\lim _{\\rho \\rightarrow +\\infty } \\, \\rho ^3 R(\\rho ) = 0$ .",
"This solution corresponds to a spacetime which is asymptotically flat, as one can see from eqs.",
"(REF )-().",
"- Additionally, there are two or four solutions to eq.",
"(REF ) which tend to a finite, nonzero value as $\\rho \\rightarrow +\\infty $ .",
"We name these solutions with $\\textbf {C}_{+}$ , $\\textbf {C}_{-}$ , $\\textbf {P}_{1}$ and $\\textbf {P}_{2}$ (details about this denomination are given in the appendix).",
"Their asymptotic behaviour is $h(\\rho ) = C + \\, R(\\rho )$ where $\\lim _{\\rho \\rightarrow +\\infty } \\, R(\\rho ) = 0$ and $C$ is a root of the reduced asymptotic equation (REF ).",
"From eqs.",
"(REF )-(), one can get convinced that these solutions correspond to spacetimes which are asymptotically non-flat.",
"Interestingly, the leading term in the gravitational potentials scales as $\\rho ^2$ for large radii, the same scaling which we find in a de Sitter spacetime.",
"It is worthwhile to point out that, since we are working on scales below the Compton wavelength of the gravitational field, “asymptotically non-flat” really means that (from the non-truncated theory point of view) the spacetime correspondent to this solution tends to a non-flat spacetime when the Compton wavelength is approached.",
"To understand the “true” asymptotic behaviour of this solution, one should use the non-truncated equations.",
"Note that, even if $C$ (and so $h$ ) is much smaller than one, the gravitational potentials $n$ and $f$ can be very large (as they behave like $\\propto \\rho ^2$ far from the origin in this case): therefore, the linear approximation of the non-truncated theory we used to obtain eqs.",
"(REF )-() is not valid.",
"Instead, the asymptotic fate of the solution is dictated by the nonlinear behaviour of the non-truncated equations.",
"This seems not too easy to predict without a separate analysis, and we don't attempt to address this interesting problem in the present paper."
],
[
"Inner solutions",
"In a neighbourhood of $\\rho \\rightarrow 0^+$ there are either one or three solutions to eq.",
"(REF ).",
"For $\\beta > 0$ there are exactly three inner solutions, while for $\\beta < 0$ there is only one inner solution.",
"In particular: - There is always a diverging solution, which we denote by $\\textbf {D}$ .",
"Its leading behaviour is $h(\\rho ) = - \\, \\@root 3 \\of {\\frac{2}{\\beta }} \\, \\frac{\\rho _v}{\\rho } + R(\\rho )$ where $\\lim _{\\rho \\rightarrow 0^+} \\, (R(\\rho )/\\rho )$ is finite.",
"This solution exists for both $\\beta > 0$ and $\\beta < 0$ , with opposite signs for each case.",
"Using this solution in eqs.",
"(REF )-(), one realises that the $h^3$ term cancels the $GM/\\rho $ term, so the gravitational field is self-shielded and does not diverge as $\\rho \\rightarrow 0^+$ .",
"This solution is in strong disagreement with gravitational observations.",
"- For $\\beta > 0$ , there are two additional solutions to eq.",
"(REF ), which tend to a finite, non-zero value as $\\rho \\rightarrow 0^+$ .",
"We indicate these solutions by $\\textbf {F}_{+}$ and $\\textbf {F}_{-} \\,$ .",
"Their leading behaviour is $h(\\rho ) = \\pm \\sqrt{\\frac{1}{3 \\, \\beta }} + \\, R(\\rho )$ where $\\lim _{\\rho \\rightarrow 0^+} \\, R = 0$ .",
"Notice that for $\\beta < 0$ there are no solutions to eq.",
"(REF ) which tend to a finite value as $\\rho \\rightarrow 0^+$ .",
"The expressions (REF )-() for the gravitational potentials imply that the metric associated to these solutions ($\\textbf {F}_{+}$ and $\\textbf {F}_{-}$ ) approximate the linearised Schwarzschild metric as $\\rho \\rightarrow 0^+$ .",
"From the behaviour of the inner solutions, one concludes that only in the $\\beta >0$ part of the phase space solutions may exhibit the Vainshtein mechanism, but not necessarily for all values of $\\alpha $ .",
"In the next subsection we see more in detail how this mechanism works."
],
[
"Vainshtein mechanism",
"In order to study where in the phase space the Vainshtein mechanism works, it is useful to compare the gravitational potentials $f$ and $n$ with their counterparts in the GR case.",
"In the weak field limit, the Schwarzschild solution of GR reads $ds^2 = - \\bigg ( 1 - \\frac{2 G M}{\\rho } \\bigg ) \\, dt^2 + \\bigg ( 1 + \\frac{2 G M}{\\rho } \\bigg ) \\, d\\rho ^2+ \\rho ^2 \\, d \\Omega ^2$ so by calling $f_{GR} = n_{GR} = - 2 G M / \\rho $ we obtain $\\frac{f}{f_{GR}} &= 1 + \\frac{1}{2} \\, \\bigg ( \\frac{\\rho }{\\rho _{v}} \\bigg )^3 \\, \\Big ( h - \\alpha h^2 + \\beta h^3 \\Big ) \\\\[2mm]\\frac{n^{\\, \\prime }}{n_{GR}^{\\, \\prime }} &= 1 - \\frac{1}{2} \\, \\bigg ( \\frac{\\rho }{\\rho _{v}} \\bigg )^3 \\, \\Big ( h - \\beta h^3 \\Big )$ Let us now first discuss the asymptotic solutions.",
"For the decaying solution $\\textbf {L}$ , we have that the linear contribution in $h$ rescales the coefficients of the Schwarzschild-like terms, so we obtain $f / f_{GR} \\rightarrow 2/3$ and $n^{\\, \\prime } / n_{GR}^{\\, \\prime } \\rightarrow 4/3$ for $\\rho \\rightarrow +\\infty $ .",
"For the non-decaying solutions $\\textbf {C}_{\\pm }$ and $\\textbf {P}_{1,2}$ , the leading behaviour for $f / f_{GR}$ and $n^{\\, \\prime } / n_{GR}^{\\, \\prime }$ is proportional to $( \\rho / \\rho _v )^3$ in both cases, however the proportionality coefficients generally differ since they have a different functional dependence on $\\alpha $ and $\\beta $ .",
"There are some special cases for $(\\alpha ,\\beta )$ where these asymptotic solutions lead to $f/n\\rightarrow 1$ as $\\rho \\rightarrow +\\infty $ , and therefore have the same behaviour as in a de Sitter spacetime.",
"Consider instead the inner solutions.",
"For the finite solutions $\\textbf {F}_{\\pm }$ we obtain $(f / f_{GR}) \\rightarrow 1$ and $(n^{\\, \\prime } / n_{GR}^{\\, \\prime }) \\rightarrow 1$ as $\\rho \\rightarrow 0^+$ , where the corrections scale like $\\rho ^3$ .",
"On the contrary, for the diverging solution $\\textbf {D}$ , the cubic terms in $h$ cancel out the contribution coming from the Schwarzschild-like terms, as explained above, and so $(f / f_{GR}) \\rightarrow 0$ and $(n^{\\, \\prime } / n_{GR}^{\\, \\prime }) \\rightarrow 0$ when $\\rho \\rightarrow 0^+$ .",
"In this case, corrections are linear in $\\rho $ .",
"Therefore, any global solution of equation (REF ) which interpolates between $\\textbf {L}$ and $\\textbf {F}_{\\pm }$ provides a realisation of the Vainshtein mechanism in an asymptotically flat spacetime, whereas an interpolation between $\\textbf {C}_{\\pm }$ or $\\textbf {P}_{1,2}$ with $\\textbf {F}_{\\pm }$ exhibits the Vainshtein mechanism in an asymptotically non-flat spacetime.",
"Furthermore, notice that any asymptotic solution which interpolates with the inner solution $\\textbf {D}$ does no lead to the Vainshtein mechanism.",
"These matchings will be explicitly exposed in the next section."
],
[
"Solutions matching",
"The phase space diagram which displays our results about solution matching is given in figure REF .",
"We discuss separately the $\\beta > 0$ and $\\beta < 0$ part of the phase space, and refer to the figure for the numbering of the regions.",
"The notation $\\textbf {I}\\leftrightarrow \\textbf {A}$ means that there is matching between the inner solution $\\textbf {I}$ and the asymptotic solution $\\textbf {A}$ .",
"Figure: Phase space diagram in (α,β)(\\alpha ,\\beta ) for the solutions to the quintic equation() in hh, where the different regions show different matching of innersolutions to asymptotic ones.",
"The lines splitting the regions are half parabolas(β∝α 2 \\beta \\propto \\alpha ^{2}, with α>0\\alpha >0 or α<0\\alpha <0) due to rescaling symmetry of eq.",
"().$\\beta < 0$ In this part of the phase space, there is only one inner solution, $\\textbf {D}$ , so there can be at most one global solution to (REF ).",
"There are three distinct regions which differ in the way the matching works: - region 1: $\\textbf {D} \\leftrightarrow \\textbf {C}_{+}$ .",
"In this region, there are three or five asymptotic solutions, and only one of them, $\\textbf {C}_{+}$ , is positive.",
"This solution is the one which connects with the inner solution $\\textbf {D}$ , which is also positive, leading to the only global solution of eq.",
"(REF ).",
"The boundaries of this region are the line $\\beta = 0$ for $\\alpha <0$ and the parabola $\\beta = c_{12} \\, \\alpha ^2$ for $\\alpha >0$ , where $c_{12}$ is the negativeThe equation $-4 - 8 \\, y + 88 \\, y^2 - 1076 \\, y^3 + 2883 \\, y^4 = 0$ has only two real roots, one positive and one negative.",
"root of the equation $-4 - 8 \\, y + 88 \\, y^2 - 1076 \\, y^3 + 2883 \\, y^4 = 0$ (approximatively, $c_{12} \\simeq -0.1124$ ).",
"On the boundary $\\beta = c_{12} \\, \\alpha ^2$ the matching $\\textbf {D} \\leftrightarrow \\textbf {C}_{+}$ still holds, however the solution $h(\\rho )$ displays an inflection point with vertical tangent.",
"- region 2: $\\textbf {No matching}$ .",
"In this region there are three asymptotic solutions.",
"However, none of them can be extended all the way to $\\rho \\rightarrow 0^+$ , and so, despite the fact that local solutions exist both at infinity and near the origin, equation (REF ) does not admit any global solution.",
"The boundaries of this region are the parabola $\\beta = c_{12} \\, \\alpha ^2$ and the (negative) five-roots-at-infinity parabola $\\beta = c_{-} \\, \\alpha ^2$ , where $c_{-}$ is the only real root of the equation $8 + 48 \\, y - 435 \\, y^2 + 676 \\, y^3 = 0$ (approximatively, $c_{-} \\simeq -0.0876$ ).",
"- region 3: $\\textbf {D} \\leftrightarrow \\textbf {P}_{2}$ .",
"This region coincides with the $\\alpha > 0$ , $\\beta < 0$ part of the five roots at infinity region of the phase space (see fig.",
"REF ).",
"The largest positive asymptotic solution, $\\textbf {P}_{2}$ , is the one which connects to $\\textbf {D}$ , leading to the only global solution of eq.",
"(REF ).",
"On the boundary $\\beta = c_{-} \\, \\alpha ^2$ the matching $\\textbf {D} \\leftrightarrow \\textbf {P}_{2}$ still holds, but the solution $h$ seen as a function of $A$ has infinite derivative in $A=0$ .",
"$\\beta > 0$ In this part of the phase space, there are three inner solutions, $\\textbf {D}$ , $\\textbf {F}_{+}$ and $\\textbf {F}_{-}$ , so there can be at most three global solutions to eq.",
"(REF ).",
"There are six distinct regions with different matching properties: - region 4: $\\textbf {F}_{-} \\leftrightarrow \\textbf {L}$ , $\\textbf {D} \\leftrightarrow \\textbf {C}_{-}$ .",
"This region lies inside the $\\alpha > 0$ , $\\beta > 0$ part of the five roots at infinity region of the phase space (see fig.",
"REF ), so there are five asymptotic solutions.",
"Of the five asymptotic solution, $\\textbf {C}_{-}$ and $\\textbf {L}$ can always be extended to $\\rho \\rightarrow 0^+$ , while $\\textbf {C}_{+}$ , $\\textbf {P}_{1}$ and $\\textbf {P}_{2}$ cannot.",
"So there are just two global solutions to eq.",
"(REF ).",
"The boundaries of this region are the parabola $\\beta = c_{45} \\, \\alpha ^2$ , where $c_{45} = 1/12 \\simeq 0.0833$ , and the line $\\beta = 0$ .",
"On the boundary $\\beta = c_{45} \\, \\alpha ^2$ there is the additional matching $\\textbf {F}_{+} \\leftrightarrow \\textbf {C}_{+}$ , and the correspondent solution is $h(\\rho ) = const = + \\sqrt{1/\\,3\\,\\beta }\\,$ .",
"- region 5: $\\textbf {F}_{+} \\leftrightarrow \\textbf {C}_{+}$ , $\\textbf {F}_{-} \\leftrightarrow \\textbf {L}$ , $\\textbf {D} \\leftrightarrow \\textbf {C}_{-}$ .",
"In this region there are three or five asymptotic solutions; $\\textbf {C}_{-}$ , $\\textbf {C}_{+}$ and $\\textbf {L}$ can always be extended to $\\rho \\rightarrow 0^+$ , while $\\textbf {P}_{1}$ and $\\textbf {P}_{2}$ , where present, cannot.",
"So there are three global solutions to (REF ).",
"The boundaries of this region are the parabola $\\beta = c_{45} \\, \\alpha ^2$ for $\\alpha > 0$ and the parabola $\\beta = c_{56} \\, \\alpha ^2$ for $\\alpha < 0$ , where $c_{56} = (5 + \\sqrt{13})/24 \\simeq 0.3586$ .",
"On the $\\alpha < 0$ boundary $\\beta = c_{56} \\, \\alpha ^2$ the matching works as in the rest of the region, but the solution $\\textbf {F}_{-} \\leftrightarrow \\textbf {L}$ has an inflection point with vertical tangent.",
"- region 6: $\\textbf {D} \\leftrightarrow \\textbf {C}_{-}$ , $\\textbf {F}_{+} \\leftrightarrow \\textbf {C}_{+}$ .",
"In this region there are three asymptotic solutions, however only two of them can be extended to $\\rho \\rightarrow 0^+$ , while $ \\textbf {L}$ cannot.",
"Therefore, there are just two global solutions to eq.",
"(REF ).",
"The boundaries of this region are the parabolas $\\beta = c_{56} \\, \\alpha ^2$ and $\\beta = c_{67} \\, \\alpha ^2$ , where $c_{67}$ is the positive root of the equation $-4 - 8 \\, y + 88 \\, y^2 - 1076 \\, y^3 + 2883 \\, y^4 = 0$ (approximatively, $c_{67} \\simeq 0.3423$ ).",
"On the boundary $\\beta = c_{67} \\, \\alpha ^2$ the matching works as in the rest of the region, but the solution $\\textbf {D} \\leftrightarrow \\textbf {C}_{-}$ has an inflection point with vertical tangent.",
"- region 7: $\\textbf {F}_{+} \\leftrightarrow \\textbf {C}_{+}$ .",
"In this region there are three asymptotic solutions, however only one of them can be extended to $\\rho \\rightarrow 0^+$ , while $\\textbf {L}$ and $\\textbf {C}_{-}$ cannot.",
"The boundaries of this region are the parabola $\\beta = c_{67} \\, \\alpha ^2$ and the (positive) five-roots-at-infinity parabola $\\beta = c_{+} \\, \\alpha ^2$ , where $c_{+} = 1/4$ .",
"Note that on the ($\\alpha < 0$ ) part of the parabola $\\beta = 1/3 \\, \\alpha ^{2}$ there is the additional matching $\\textbf {F}_{-} \\leftrightarrow \\textbf {C}_{-}$ , so for these points there are two global solutions to eq.",
"(REF ).",
"On the boundary $\\beta = c_{+} \\, \\alpha ^2$ there are the additional matchings $\\textbf {F}_{-} \\leftrightarrow \\textbf {P}_{1}$ , $\\textbf {D} \\leftrightarrow \\textbf {P}_{2}$ , and the solutions corresponding to both of these additional matchings, seen as functions of $A$ , display an infinite derivative in $A=0$ .",
"- region 8: $\\textbf {F}_{+} \\leftrightarrow \\textbf {C}_{+}$ , $\\textbf {F}_{-} \\leftrightarrow \\textbf {P}_{1}$ , $\\textbf {D} \\leftrightarrow \\textbf {P}_{2}$ .",
"This region lies inside the $\\alpha < 0$ , $\\beta > 0$ part of the five roots at infinity region of the phase space (see fig.",
"REF ), so there are five asymptotic solutions.",
"Only three of them can be extended to $\\rho \\rightarrow 0^+$ , while $ \\textbf {C}_{-}$ and $\\textbf {L}$ cannot.",
"The boundaries of this region are the parabolas $\\beta = c_{+} \\, \\alpha ^2$ and $\\beta = c_{89} \\, \\alpha ^2$ , where $c_{89} = (5 - \\sqrt{13})/24 \\simeq 0.0581$ .",
"On the boundary $\\beta = c_{89} \\, \\alpha ^2$ the matchings are the same as in the rest of the region, but the solution $h(\\rho )$ correspondent to the matching $\\textbf {F}_{+} \\leftrightarrow \\textbf {C}_{+}$ has an inflection point with vertical tangent.",
"- region 9: $\\textbf {F}_{-} \\leftrightarrow \\textbf {P}_{1}$ , $\\textbf {D} \\leftrightarrow \\textbf {P}_{2}$ .",
"This region lies inside the $\\alpha < 0$ , $\\beta > 0$ part of the five roots at infinity region of the phase space (see fig.",
"REF ), so there are again five asymptotic solutions.",
"The matching is similar to that of region 8, apart from the fact that $\\textbf {C}_{+}$ cannot be extended to $\\rho \\rightarrow 0^+$ anymore; hence there are just two global solutions to eq.",
"(REF ).",
"The boundaries of this region are the parabola $\\beta = c_{89} \\, \\alpha ^2$ and line $\\beta = 0$ .",
"We note that the decaying solution $\\textbf {L}$ never connects to the diverging one $\\textbf {D}$ , so we cannot have a spacetime which is asymptotically flat and exhibit the self-shielding of the gravitational field at the origin.",
"On the other hand, finite non-zero asymptotic solutions ($\\textbf {C}_{\\pm }$ or $\\textbf {P}_{1,2}$ ) can connect to both finite and diverging inner solutions.",
"Therefore, one can have an asymptotically non-flat spacetime which presents self-shielding at the origin, or an asymptotically non-flat spacetime which tends to Schwarzschild spacetime for small radii.",
"More precisely, for $\\beta < 0$ there are only solutions displaying the self-shielding of the gravitational field, apart from region 2 where there are no global solutions.",
"Therefore the Vainshtein mechanism never works for $\\beta < 0$ .",
"In contrast, for $\\beta > 0$ all three kinds of global solutions are present.",
"Solutions with asymptotic flatness and the Vainshtein mechanism are present in regions 4 and 5, while solutions which are asymptotically non-flat and exhibit the Vainshtein mechanism do exist in all ($\\beta > 0$ ) regions but region 4.",
"Finally, solutions which display the self-shielding of the gravitational field are present in all ($\\beta > 0$ ) regions but region 7."
],
[
"Numerical solutions",
"We present here the numerical solutions for the $h$ field and the gravitational potentials in some representative cases.",
"We choose a specific realisation for each of the three physically distinct cases, namely asymptotic flatness with Vainshtein mechanism, asymptotically non-flat spacetime with Vainshtein mechanism, and asymptotically non-flat spacetime with self-shielded gravitational field at the origin.",
"In addition, we consider the case in which there are no global solutions to eq.",
"(REF ).",
"This provides an illustration of what happens, in general, to local solutions of eq.",
"(REF ) which cannot be extended to the whole radial domain, and give an insight on the phenomenology of equation (REF ).",
"Asymptotic flatness with Vainshtein mechanism Let's consider the case in which the solution of eq.",
"(REF ) connects to the decaying solution at infinity $\\textbf {L}$ and to a finite inner solution (in this case $\\textbf {F}_{-}$ ).",
"In figure REF , the numerical solutions for $h$ (dashed line), $f / f_{GR}$ (bottom continuous line) and $n^{\\, \\prime } / n_{GR}^{\\, \\prime }$ (top continuous line) are plotted as functions of the dimensionless radial coordinate $x \\equiv \\rho / \\rho _v$ .",
"These solutions correspond to the point $(\\alpha , \\beta ) = (0 \\, , 0.1)$ of the phase space.",
"Figure: Numerical solutions for the case 𝐅 - ↔𝐋\\textbf {F}_{-} \\leftrightarrow \\textbf {L}.This plot displays very clearly the presence of the vDVZ discontinuity and its resolution via the Vainshtein mechanism.",
"For large scales, $h$ is small and the gravitational potentials behave like the Schwarzschild one, however their ratio is different from one, unlike the massless case.",
"Note that the ratio of the two potentials for $\\rho \\gg \\rho _v$ is independent of $m$ , so does not approach one as $m \\rightarrow 0$ (vDVZ discontinuity).",
"However, on small scales $h$ is strongly coupled, and well inside the Vainshtein radius the two potentials scale again as the Schwarzschild one, but their ratio is now one even if $m \\ne 0$ .",
"So, the strong coupling of the $h$ field on small scales restores the agreement with GR (Vainshtein mechanism).",
"Asymptotically non-flat spacetime with Vainshtein mechanism Let's consider now the case in which the solution of eq.",
"(REF ) connects to a finite solution at infinity and to a finite inner solution.",
"We consider for definiteness the phase space point $(\\alpha , \\beta ) = (0 \\, , 0.1)$ .",
"In figure REF , we plot the numerical results for the gravitational potentials (normalised to their GR values) and the global solution of eq.",
"(REF ) which interpolates between the inner solution $\\textbf {F}_{+}$ and the asymptotic solution $\\textbf {C}_{+}$ .",
"Figure: Numerical solutions for the case 𝐅 + ↔𝐂 + \\textbf {F}_{+} \\leftrightarrow \\textbf {C}_{+}.We can see that, on large scales, the gravitational potentials are not only different one from the other but also behave very differently compared to the GR case.",
"However, on small scales there is a macroscopic region where the two potentials agree, and their ratio with the Schwarzschild potential stays nearly constant and equal to one.",
"Therefore, also in this case the small scale behaviour of $h$ guarantees that GR results are recovered, even if the spacetime is not asymptotically flat.",
"This behaviour provide then, in a more general sense, a realisation of the Vainshtein mechanism.",
"Asymptotically non-flat spacetime with self-shielding We turn now to the case where the solution of eq.",
"(REF ) connects to a finite solution at infinity and to the diverging inner solution.",
"In figure REF , we plot the global solution $h$ and the associated gravitational potentials, normalised to their GR values, correspondent to the phase space point $(\\alpha , \\beta ) = (-1 \\, , -0.5)$ .",
"It is apparent that there are no regions where the solutions behave like in the GR case.",
"Figure: Numerical solutions for the case 𝐃↔𝐂 + \\textbf {D} \\leftrightarrow \\textbf {C}_{+}.To see that the gravitational potentials are indeed finite at the origin, we plot in figure REF the potentials $f$ and $n^{\\prime }$ themselves, as functions of $\\rho /\\rho _v$ .",
"We choose for definiteness the following ratio between the Compton wavelength and the gravitational radius $\\rho _{c} / \\rho _{g} = 10^{6}$ , and plot the potentials for $0.01 < \\rho /\\rho _v < 2$ .",
"Note that, since in this case $\\rho _{c} / \\rho _{v}= \\@root 3 \\of {\\rho _{c} / \\rho _{g}} = 10^{2}$ , the range where the functions are plotted is well inside the range of validity of our approximations.",
"We can see that the potentials approach a finite value as $\\rho \\rightarrow 0^+$ , and so indeed the gravitational field does not diverge at the origin.",
"Figure: Numerical solutions for the gravitational potentials, for the case𝐃↔𝐂 + \\textbf {D} \\leftrightarrow \\textbf {C}_{+}.No matching Finally, we consider the case in which equations (REF ) $-$ (REF ) do not admit global solutions.",
"We consider for definiteness the phase space point $(\\alpha , \\beta ) = (1 \\, , -0.092)$ .",
"In figure REF we plot all the local solutions of the quintic equation (REF ) as functions of the dimensionless radial coordinate $x \\equiv \\rho / \\rho _v$ .",
"Figure: Numerical results for all local solutions of eq.",
"() in the case where there is nomatching.For $0 < x < 0.4$ , there is only one local solution (the top continuous curve), which connects to the diverging inner solution $\\textbf {D}$ .",
"At $x \\simeq 0.4$ a pair of solutions is created (dashed and continuous negative valued curves), and at $x \\simeq 0.9$ , another pair of solutions is created (positive valued dashed curve and positive valued bottom continuous curve).",
"However, at $x \\simeq 1.3$ one of the newly created functions (the positive valued dashed curve) annihilates with the solution with connects to the inner solution, so for $x > 1.3$ there are three local solutions, which finally connect with the asymptotic solutions $\\textbf {C}_{-}$ , $\\textbf {L}$ and $\\textbf {C}_{+}$ .",
"Therefore, the number of existing local solutions is one for $0 < x < 0.4$ , three for $0.4 < x < 0.9$ , five for $0.9 < x < 1.3$ and three for $x > 1.3$ .",
"We can see that, despite the fact that for every $\\rho $ there is at least one local solution, there does not exist a solution which extends over the whole radial domain.",
"Note that the solutions are created and annihilated in pairs; furthermore, the pairs of solutions have infinite slope when they are created and when they annihilate, which results in the gravitational potentials having diverging derivatives when the creation/annihilation points are approached.",
"These are found to be general features of the phenomenology of equation (REF ).",
"In fact, in most part of the phase space some of the asymptotic/inner solutions cannot be extended to the whole radial domain $0 < \\rho < +\\infty $ .",
"However, it never happens that the value of the local solutions diverges at a finite radius, while they always disappear or are created in pairs, with their values remaining bounded but with derivatives which diverge."
],
[
"Conclusions",
"Recently, a ghost-free model of non-linear massive gravity has been proposed.",
"We studied static, spherically symmetric solutions of this theory inside the Compton radius of the gravitational field, and considered the weak field limit for the gravitational potentials, while keeping all non-linearities of the helicity-0 mode.",
"For every point of the two free parameter phase space, we characterised completely the number and properties of asymptotic solutions on large scales and also of inner solutions on small scales.",
"In particular, there are two kinds of asymptotic solutions, where one of them is asymptotically flat and the other one is not.",
"There are also two kinds of inner solutions, one which displays the Vainshtein mechanism and the other which exhibits the self-shielding of the gravitational field near the origin.",
"We described under which circumstances the theory admits global solutions interpolating between the asymptotic and inner solutions, and found that the asymptotically flat solution connects only to inner solutions displaying the Vainshtein mechanism, while solutions which diverge asymptotically can connect to both kinds of inner solutions.",
"Furthermore, we showed that there are some regions in the parameter space where global solutions do not exist, and characterised precisely in which regions of the phase space the Vainshtein mechanism is working.",
"Our study embraces all of the phase space spanned by the two parameters of the theory.",
"Notably, we found that, within our approximations, the asymptotic and inner solutions cannot in general be extended to the whole radial domain.",
"In particular, we exhibited extreme cases in which global solutions do not exist at all.",
"This happens because at a finite radius the derivatives of the metric components diverge, while the metric components themselves remain bounded.",
"When the derivatives of the metric cease to be small, the approximations we used to derive the equations (REF ) $-$ (REF ) break down.",
"It would be interesting to study what happens at this radius in the full theory."
],
[
"Acknowledgments",
"GN and KK are supported by the European Research Council.",
"KK is also supported by the STFC (grant no.",
"ST/H002774/1) and the Leverhulme trust.",
"GT is supported by an STFC Advanced Fellowship ST/H005498/1."
],
[
"Asymptotic behaviour of solutions of the quintic equation",
"In this appendix we study the local solutions of the quintic equation (REF ) in a neighbourhood of $\\rho = 0^+$ and also in a neighbourhood of $\\rho = +\\infty $ .",
"As previously mentioned we consider only the $\\beta \\ne 0$ case, since for $\\beta = 0$ the equation becomes a cubic and it is possible to find all local solutions analytically (see [8])."
],
[
"Asymptotic solutions",
"Suppose that a solution $h(\\rho )$ of the quintic equation (REF ), which for convenience we rewrite here, $\\frac{3}{2} \\, \\beta ^{2}\\, h^{5} - \\Big ( \\alpha ^{2}+ 2 \\beta \\Big ) \\, h^{3} + 3 \\, \\Big ( \\alpha + \\beta A \\Big ) \\, h^{2}- \\frac{3}{2} \\, h - A = 0$ exists in a neighbourhood of $\\rho = + \\infty \\,$ , and that it has a well defined limit as $\\rho \\rightarrow + \\infty $ .",
"Then such a solution cannot be divergent.",
"To see this, suppose that indeed the solution is divergent $|\\lim _{\\rho \\rightarrow + \\infty } h(\\rho )| = + \\infty \\,$ : it is then possible to divide the quintic equation by $h^{5}$ (in a neighbourhood of $\\rho = + \\infty \\,$ ), obtaining the following equation in $v = 1/h$ $\\frac{3}{2} \\, \\beta ^{2}- \\big ( \\alpha ^{2}+ 2 \\beta \\big ) \\, v^{2} + 3 \\, \\big ( \\alpha + \\beta A \\big ) \\, v^{3} - \\frac{3}{2}\\, v^{4} - A \\, v^{5} = 0$ Taking the $\\rho \\rightarrow + \\infty $ limit of both sides of this expression one obtains $\\beta = 0$ , which is precisely against our initial assumption.",
"Suppose now that $\\lim _{\\rho \\rightarrow + \\infty } h(\\rho )$ is finite, and let's call it $C$ .",
"Then both of the sides of the quintic equation itself have a finite limit when $\\rho \\rightarrow + \\infty \\,$ , and taking this limit one gets $\\frac{3}{2} \\, \\beta ^{2}\\, C^{5} - \\big ( \\alpha ^{2}+ 2 \\beta \\big ) \\, C^{3} + 3 \\, \\alpha \\, C^{2} - \\frac{3}{2} \\, C = 0$ It follows then that the allowed asymptotic values at infinity for $h(\\rho )$ are the roots of the following equation, which we call the asymptotic equation ${A}(y) \\equiv \\frac{3}{2} \\, \\beta ^{2}\\, y^{5} - \\big ( \\alpha ^{2}+ 2 \\beta \\big ) \\, y^{3} + 3 \\, \\alpha \\, y^{2}- \\frac{3}{2} \\, y = 0$ Note that $y = 0$ is always a root of this equation, and in fact a simple root (i.e.",
"a root of multiplicity one) since $\\frac{d}{dy} {A}(0) \\ne 0 \\,$ .",
"Dividing by $y$ , one obtains that the other asymptotic values for $h(\\rho )$ are the roots of the reduced asymptotic equation ${A}_{r}(y) \\equiv \\frac{3}{2} \\, \\beta ^{2}\\, y^{4} - \\big ( \\alpha ^{2}+ 2 \\beta \\big ) \\, y^{2}+ 3 \\, \\alpha \\, y - \\frac{3}{2} = 0$ This last equation is a quartic, so it can have up to 4 (real) roots, depending on the specific values of $\\alpha $ and $\\beta $ .",
"Since $\\lim _{h \\rightarrow \\pm \\infty } {A}_{r}(h) =\\pm \\infty $ and ${A}_{r}(0) = - 3/2 < 0 \\,$ , it has always at least two roots, one positive and one negative.",
"For the same reason, it cannot have two positive and two negative roots, since at each simple root the quartic function changes sign.",
"Since the function ${A}(h)$ changes smoothly with $\\alpha $ and $\\beta $ , the boundaries between regions where there are five roots and regions where there are three roots are found enforcing that ${A}(h)$ has a multiple root.",
"This condition is satisfied only by the points belonging to the parabolas $\\beta = c_{+} \\, \\alpha ^2$ and $\\beta = c_{-} \\, \\alpha ^2$ , where $c_{+} = 1/4$ and $c_{-}$ is the only real root of the equation $8 + 48 \\, y - 435 \\, y^2 + 676 \\, y^3 = 0$ .",
"The regions above the positive parabola and below the negative one have only three roots, which are simple roots, while the regions between the two parabolas (except $\\beta = 0$ ) have five roots, which are again simple roots.",
"On the boundaries $\\beta = c_{\\pm } \\, \\alpha ^2$ between the three-roots regions and the five-roots regions there are four roots, one of which is a root of multiplicity two.",
"This is summarised in figure REF .",
"Figure: phase space diagram for the number of asymptotic solutionsWe name the roots in the following way: the $y=0$ root is denoted as $\\textbf {L}$ .",
"For the phase space points where there are just three roots, the positive root is denoted as $\\textbf {C}_{+}$ and the negative one as $\\textbf {C}_{-}$ .",
"For points in the five-roots regions, we adopt the following convention.",
"Be $(\\alpha _5, \\beta _5)$ a point where there are five roots.",
"In the same quadrant of the phase space, take another point $(\\alpha _3, \\beta _3)$ where there are three roots, and a path ${C}$ which connects the two points.",
"Following the path ${C}$ , two of the four non-zero roots of $(\\alpha _5, \\beta _5)$ smoothly flow to the non-zero roots of $(\\alpha _3, \\beta _3)$ , and are denoted as $\\textbf {C}_{+}$ and $\\textbf {C}_{-}$ themselves.",
"The other two non-zero roots of $(\\alpha _5, \\beta _5)$ , instead, disappear when (following ${C}$ ) the boundary of the five-roots region is crossed, and are denoted as $\\textbf {P}_{1}$ and $\\textbf {P}_{2}$ .",
"We adopt the convention that $\\vert \\textbf {P}_{1}\\vert \\le \\vert \\textbf {P}_{2}\\vert $ .",
"The definition is independent of the particular choice of the point $(\\alpha _3, \\beta _3)$ and of the path ${C}$ used.",
"A careful study of the asymptotic equation and of its derivatives permits to show that we have $\\textbf {C}_{-} < \\textbf {C}_{+} < \\textbf {P}_{1} < \\textbf {P}_{2}$ for $\\alpha > 0$ and $\\textbf {P}_{2} <\\textbf {P}_{1} < \\textbf {C}_{-} < \\textbf {C}_{+}$ for $\\alpha < 0$ .",
"On the boundaries $\\beta = c_{\\pm } \\, \\alpha ^2$ we have $\\textbf {P}_{1} = \\textbf {P}_{2} \\equiv \\textbf {P}$ .",
"Suppose now that a solution of the quintic equation exists in a neighborhood of $\\rho = 0^+$ (possibly not defined in $\\rho = 0$ ), and that it has a well defined limit when $\\rho \\rightarrow 0^+$ .",
"Then such a solution cannot tend to zero as $\\rho \\rightarrow 0^+$ .",
"In fact, for $\\rho \\ne 0$ we can divide (REF ) by $A$ , and indicating $x \\equiv \\rho / \\rho _v$ we obtain $x^3 \\, \\bigg ( \\frac{3}{2} \\, \\beta ^{2}\\, h^{5} - \\big ( \\alpha ^{2}+ 2 \\beta \\big ) \\, h^{3} + 3 \\, \\alpha \\, h^{2}- \\frac{3}{2} \\, h \\bigg ) + 3 \\, \\beta \\, h^{2} - 1 = 0$ If we had $h \\rightarrow 0$ as $\\rho \\rightarrow 0^+$ , then we would get $-1=0$ when taking the $\\rho \\rightarrow 0^+$ limit in the previous equation.",
"Therefore, $\\lim _{\\rho \\rightarrow 0^+} h (\\rho ) \\ne 0$ .",
"In a neighbourhood of $\\rho = 0^+ \\,$ , we can further divide the equation above by $h^{5}$ , obtaining the following equation in $v = 1 / h$ $v^{5} + \\frac{3}{2} \\, x^3 \\, v^{4} - 3 \\, \\big ( \\beta + \\alpha \\, x^3 \\big ) \\, v^{3} + \\big ( \\alpha ^{2}+ 2 \\beta \\big )\\, x^3 \\, v^{2} - \\frac{3}{2} \\, \\beta ^{2}\\, x^3 = 0$ The permitted limiting values for $v$ are then the roots of the equation obtained from the previous one setting $x = 0$ , namely $v^{5} - 3 \\, \\beta \\, v^{3} = 0$ For $\\beta > 0$ there are three roots, namely $v_{0} = 0$ , $v_{+} = + \\sqrt{3 \\, \\beta }$ and $v_{-} = - \\sqrt{3 \\, \\beta }$ ; for $\\beta < 0$ , instead, there is only the root $v = 0$ .",
"Therefore, the permitted limiting behaviours for $h$ when $\\rho \\rightarrow 0^+$ are $\\vert h(\\rho )\\vert \\rightarrow + \\infty $ for $\\beta \\ne 0$ , and $h \\rightarrow \\pm \\sqrt{\\frac{1}{3 \\, \\beta }}$ only for $\\beta > 0$ ."
],
[
"Leading behaviours",
"Note that so far we have not proved that inner and asymptotic solutions exist, but just found the values that have to be the limit of these solutions if they exist.",
"The existence and uniqueness of solutions can be proved applying the implicit function theorem (known also as Dini's theorem, see for example [21]) to the equation (REF ) around $A = 0$ and to the equation (REF ) around $x = 0$ .",
"One can then prove that, to each of the roots $v_0$ , $v_{\\pm }$ of (REF ), it is possible to associate a local solution of (REF ) defined in a neighbourhood of $\\rho = 0^+$ , which are respectively the diverging inner solution $\\textbf {D}$ and the finite inner solutions $\\textbf {F}_{\\pm }$ .",
"Likewise, it can be shown that it is possible to associate a local solution of (REF ) defined in a neighbourhood of $\\rho = + \\infty $ , to each of the simple roots of (REF ).",
"These solutions are the asymptotic solutions $\\textbf {L}$ , $\\textbf {C}_{+}$ , $\\textbf {C}_{-}$ , $\\textbf {P}_{1}$ and $\\textbf {P}_{2}$ .",
"In the case $\\textbf {P}_{1} = \\textbf {P}_{2} \\equiv \\textbf {P}$ , a separate analysis is needed.",
"It can be shown that for $\\alpha > 0$ and $\\beta = c_{+}\\, \\alpha ^{2}$ there are no local solutions of (REF ) which tend to $\\textbf {P}$ when $\\rho \\rightarrow + \\infty $ , and the same holds for $\\alpha < 0$ and $\\beta =c_{-}\\, \\alpha ^{2}$ .",
"On the other hand, for $\\alpha > 0$ and $\\beta = c_{-}\\, \\alpha ^{2}$ there are two different local solutions of (REF ) which tend to $\\textbf {P}$ when $\\rho \\rightarrow + \\infty $ , and the same holds for $\\alpha < 0$ and $\\beta = c_{+}\\, \\alpha ^{2}$ .",
"Despite having the same limit for $\\rho \\rightarrow + \\infty $ , these two local solutions are different when $A \\ne 0$ : we then call $\\textbf {P}_{1}$ the solution which in absolute value is smaller, and $\\textbf {P}_{2}$ the solution which in absolute value is bigger.",
"Therefore, on the boundaries between the three-roots-at-infinity regions and the five-roots-at-infinity regions, for $\\alpha \\gtrless 0$ , $\\beta = c_{\\pm }\\, \\alpha ^{2}$ there are three asymptotic solutions of (REF ), while for $\\alpha \\gtrless 0$ , $\\beta =c_{\\mp }\\, \\alpha ^{2}$ there are five asymptotic solutions of (REF ).",
"We now turn to the discussion of the leading behaviours of the inner and asymptotic solutions.",
"For the finite inner solutions $\\textbf {F}_{\\pm }$ and finite non-zero asymptotic solutions $\\textbf {C}_{\\pm }$ and $\\textbf {P}_{1,2}$ , the behaviour is $h(\\rho ) = C + R(\\rho )$ where $C \\ne 0$ is their limiting value, and $R$ is respectively such that $\\lim _{\\rho \\rightarrow 0^+}R = 0$ (inner solutions) and $\\lim _{\\rho \\rightarrow + \\infty } R = 0$ (asymptotic solutions).",
"For the asymptotic decaying solution $\\mathbf {L}$ and the inner diverging solution $\\mathbf {D}$ , a more detailed study is worthwhile.",
"Asymptotic decaying solution $\\mathbf {L}$ Let's consider the solution $\\mathbf {L}$ , which satisfies $\\lim _{\\rho \\rightarrow +\\infty } h(\\rho ) = 0$ .",
"Dividing the quintic equation (REF ) by $h$ , we get $\\frac{3}{2} \\, \\beta ^{2}\\, h^{4} - \\big ( \\alpha ^{2}+ 2 \\beta \\big ) \\, h^{2}+ 3 \\, \\alpha \\, h - \\frac{3}{2} = \\bigg ( \\frac{\\rho _v}{\\rho } \\bigg )^{\\!",
"3} \\, \\bigg ( \\frac{1}{h} - 3 \\, \\beta \\, h \\bigg )$ The left hand side has a finite limit when $\\rho \\rightarrow +\\infty $ , so the same has to hold for the right hand side: taking this limit in the equation above gives $\\lim _{\\rho \\rightarrow +\\infty } \\bigg ( \\frac{\\rho _v}{\\rho } \\bigg )^{\\!",
"3} \\, \\frac{1}{h} = - \\frac{3}{2}$ which implies that $h(\\rho ) = - \\frac{2}{3} \\, \\bigg ( \\frac{\\rho _v}{\\rho } \\bigg )^{\\!",
"3} + R(\\rho )$ with $\\lim _{\\rho \\rightarrow +\\infty } \\rho ^3 R(\\rho ) = 0$ .",
"Inner diverging solution $\\mathbf {D}$ Let's consider now the solution $\\mathbf {D}$ , which satisfies $\\lim _{\\rho \\rightarrow 0^+} \\vert h(\\rho )\\vert = +\\infty $ .",
"Dividing the equation (REF ) by $v^3$ , one finds that $v^{2} - 3 \\, \\beta = \\bigg ( \\frac{\\rho }{\\rho _v} \\bigg )^{\\!",
"3} \\, \\frac{1}{v^{3}} \\, \\Big ( - \\frac{3}{2} \\, v^{4}+ 3 \\, \\alpha \\, v^{3} - \\big ( \\alpha ^{2}+ 2 \\beta \\big ) \\, v^{2} + \\frac{3}{2} \\, \\beta ^{2}\\Big )$ One more time, the left hand side has a finite limit when $\\rho \\rightarrow 0^+$ , so the same should hold for the right hand side.",
"Therefore, the $\\rho \\rightarrow 0^+$ limit in the equation above gives $\\lim _{\\rho \\rightarrow 0^+} \\bigg ( \\frac{\\rho }{\\rho _v} \\bigg )^{\\!",
"3} \\, \\frac{1}{v^{3}} = - \\frac{2}{\\beta }$ and so $v(\\rho ) = - \\@root 3 \\of {\\frac{\\beta }{2}} \\, \\frac{\\rho }{\\rho _v} + \\mathrm {R}(\\rho )$ with $\\lim _{\\rho \\rightarrow 0^+} \\mathrm {R}(\\rho )/\\rho = 0$ .",
"To understand the behaviour of the gravitational potentials (REF )-() in this case, it is useful to calculate the next to leading order behaviour.",
"In fact, it turns out that, after going back to $h = 1/ v$ , the leading behaviour precisely cancels the Schwarzschild-like contribution, so to understand if the gravitational potentials are finite at the origin it is essential to know how $\\mathrm {R}$ behaves for very small radii.",
"Inserting (REF ) into (REF ) and dividing by $x^5$ , one obtains taking the limit $\\rho \\rightarrow 0^+$ that $\\lim _{\\rho \\rightarrow 0^+} \\frac{\\mathrm {R}}{x^3} = \\frac{1}{9 \\, \\beta } \\bigg ( \\alpha ^{2}+ \\frac{3}{2} \\, \\beta \\bigg )$ where $x = \\rho / \\rho _v$ .",
"We have then $v(\\rho ) = - \\@root 3 \\of {\\frac{\\beta }{2}} \\, \\frac{\\rho }{\\rho _v} + \\mathcal {N} \\, \\Big ( \\frac{\\rho }{\\rho _v} \\Big )^{\\!",
"3} + \\mathcal {R}(\\rho )$ where $\\mathcal {N} = \\frac{1}{9 \\, \\beta } \\, \\Big ( \\alpha ^{2}+ \\frac{3}{2} \\, \\beta \\Big )$ and $\\lim _{\\rho \\rightarrow 0^+} ( \\mathcal {R}(\\rho ) / \\rho ^3 ) = 0$ .",
"Finally, going back to the function $h$ we get $h(\\rho ) = - \\@root 3 \\of {\\frac{2}{\\beta }} \\, \\frac{\\rho _v}{\\rho } - \\mathcal {M} \\, \\frac{\\rho }{\\rho _v} + {R}(\\rho )$ where $\\mathcal {M} = \\frac{1}{9} \\, \\@root 3 \\of {\\frac{4}{\\beta ^5}} \\, \\bigg ( \\alpha ^{2}+ \\frac{3}{2} \\, \\beta \\bigg )$ and $\\lim _{\\rho \\rightarrow 0^+} ({R}(\\rho )/\\rho ) = 0$ .",
"It can be shown that in the special case $\\alpha ^{2}+ 3 \\, \\beta / 2 = 0$ , the next to leading order term scales as $\\rho ^2$ instead of $\\rho $ , and that $\\lim _{\\rho \\rightarrow 0^+} ({R}(\\rho )/\\rho ^2) = 0$ .",
"Therefore, we can conclude that in general the diverging inner solution $\\textbf {D}$ is such that $h(\\rho ) = - \\@root 3 \\of {\\frac{2}{\\beta }} \\, \\frac{\\rho _v}{\\rho } + R(\\rho )$ where $\\lim _{\\rho \\rightarrow 0^+} (R(\\rho )/\\rho )$ is finite."
]
] | 1204.1193 |
[
[
"McGenus: A Monte Carlo algorithm to predict RNA secondary structures\n with pseudoknots"
],
[
"Abstract We present McGenus, an algorithm to predict RNA secondary structures with pseudoknots.",
"The method is based on a classification of RNA structures according to their topological genus.",
"McGenus can treat sequences of up to 1000 bases and performs an advanced stochastic search of their minimum free energy structure allowing for non trivial pseudoknot topologies.",
"Specifically, McGenus employs a multiple Markov chain scheme for minimizing a general scoring function which includes not only free energy contributions for pair stacking, loop penalties, etc.",
"but also a phenomenological penalty for the genus of the pairing graph.",
"The good performance of the stochastic search strategy was successfully validated against TT2NE which uses the same free energy parametrization and performs exhaustive or partially exhaustive structure search, albeit for much shorter sequences (up to 200 bases).",
"Next, the method was applied to other RNA sets, including an extensive tmRNA database, yielding results that are competitive with existing algorithms.",
"Finally, it is shown that McGenus highlights possible limitations in the free energy scoring function.",
"The algorithm is available as a web-server at http://ipht.cea.fr/rna/mcgenus.php ."
],
[
"Introduction",
"In the past twenty years, there has been a tremendous increase of interest in RNA by the biological community.",
"This biopolymer, which was at first merely considered as a simple information carrier, was gradually proven to be a major actor in the biology of the cell [1].",
"Since the RNA functionality is mostly determined by its three-dimensional conformation, the accurate prediction of RNA folding from the nucleotide sequence is a central issue [2].",
"It is strongly believed that the biological activity of RNA (be it enzymatic or regulatory), is implemented through the binding of some unpaired bases of the RNA with their ligand.",
"It is thus crucial to have a precise and reliable map of all the pairings taking place in RNA and to correctly identify loops.",
"The complete list of all Watson-Crick and Wobble base pairs in RNA is called the secondary structure of RNA.",
"In this paper, we stick to the standard assumption that there is an effective free energy which governs the formation of secondary structures, so that the optimal folding of an RNA sequence is found as the minimum free energy structure (MFE for short).",
"The problem of finding the MFE structure given a certain sequence has been conceptually solved provided the MFE is planar, i. e. the MFE structure contains no pair ($i$ ,$j$ ), ($k$ ,$l$ ) such that $i<k<j<l$ or $k<i<l<j$ .",
"In that case, polynomial algorithms which can treat long RNAs assuming a mostly linear free energy model have been proposed [3], [4], [5].",
"Otherwise, the MFE structure is said to contain pseudoknots and finding it has been shown to be an NP-complete problem with respect to the sequence length [6].",
"In a previous paper [7], we proposed an algorithm, TT2NE, which consists in searching for the exact MFE structure for a certain form of the energy function, where pseudoknots are penalized according to a topological index, namely their genus.",
"TT2NE relies on the “maximum weighted independent set\" (WIS) formalism.",
"In this approach, an RNA structure is viewed as a collection of stem-like structures (helices possibly comprising bulges of size 1 or internal loops of size $1 \\times 1$ ), called “helipoints\" [7], defined in the next section.",
"Given a certain sequence, the set of all possible helipoints is enumerated and used to build a weighted graph.",
"The graph vertices are the helipoints and their weight is given by -1 times the helipoint free energy.",
"Two vertices are linked by an arc if and only if the corresponding helipoints are not compatible in the same secondary structure.",
"Incompatibilities arise, for example, when two helipoints share one or more bases as this could imply the formation of base triplets, which is forbidden.",
"Finding the MFE structure thus amounts to finding the maximum weighted independent set of the graph, i. e. the set of pairwise compatible helipoints for which the overall free energy is minimum.",
"Both McGenus and TT2NE utilize the same energy function, defined in terms of helipoints and genus penalty as well as the same initial graph.",
"The difference between the two lies in the search algorithm for the MFE.",
"While in TT2NE the secondary structure is built by adding or removing helipoints in a deterministic order, in McGenus, they are added or removed one at a time according to a stochastic Monte Carlo Metropolis scheme.",
"As in TT2NE, there is no restriction on the pseudoknot topology that McGenus can generate.",
"A server implementation of McGenus can be found at http://ipht.cea.fr/rna/mcgenus.php.",
"In the following and in the numerical implementation of McGenus, we will restrict ourselves to the energy function and genus penalty described in detail in [7].",
"While in TT2NE, the energy form was dictated by the requirement to allow for a branch and bound procedure, here in McGenus we insist that there is no such restriction on the form of the energy function.",
"It can for instance include loop and pseudoknot entropies.",
"Furthermore, the penalty for pseudoknots needs not be proportional to the genus as in TT2NE, but may depend also on the topology of each individual pseudoknot (see below).",
"Therefore, by modifying the energy function, it is possible to improve on the results that we will present below.",
"As stated in the introduction, the initial graph is generated in the same way as in [7]."
],
[
"Materials and methods",
"In the present framework, the folded structure of a given RNA sequence is given by the set of helipoints which minimizes the free energy.",
"We recall that a helipoint is an ensemble of helices (defined as a stack of base pairs possibly comprising bulges of size 1 or internal loops of size $1 \\times 1$ ) that are demarcated by the same extremal (initial and terminal) base pairs.",
"Given two extremal pairs $(i,j)$ and $(k,l)$ , the set $\\omega ^{ij}_{kl}$ of all helices that end with these two pairs can be generated and their individual energies calculated according to a given energy model.",
"The free energy $\\Delta F^{ij}_{kl}$ of the helipoint is then computed as $&\\exp &{(-\\beta \\Delta F^{ij}_{kl})} = \\sum _{\\begin{array}{c}h \\in \\omega ^{ij}_{kl}\\end{array}} \\exp {(-\\beta e(h))} \\\\&&\\mbox{with} \\ \\beta = (k_B T)^{-1} \\nonumber $ where $e(h)$ is the free energy of formation of helix $h$ .",
"In our implementation, to speed up the computation of this sum, helices of non-negative (i. e. unfavorable) energies are neglected, since their Boltzmann weight would strongly suppress their contribution.",
"Helipoints are stem-like structural building blocks which account for all possible internal pairing possibilities that occur between their extremal pairs.",
"We shall denote by $\\lbrace h_1, ... , h_N\\rbrace $ the set of all helipoints that can possibly arise from the pairings of nucleotides in the given sequence (their total number $N$ , is clearly sequence dependent).",
"We stress that the set of enumerated helipoints comprises all possible helipoints, and hence is not restricted to maximal ones.",
"Clearly, a given RNA structure $S$ is fully specified by a collection of compatible helipoints.",
"It is therefore convenient to identify $S$ with a binary vector, $\\vec{\\sigma }^S$ , of length $N$ and whose $i$ -th component, $\\sigma ^S_i$ takes on the value 0 or 1 according to whether helipoint $h_i$ belongs to $S$ .",
"The free energy of $S$ can accordingly be written as: $F_S = \\sum _{i=1}^{N} \\sigma ^S_i\\, \\Delta F(h_{i}) + \\mu \\, g (S) \\ .$ The first term is the additive contribution of the free energy $\\Delta F$ of individual helipoints, and is parametrized as in [7].",
"The second term weights the topological complexity of the structure, measured by its genus $g$ [8], [9].",
"Unlike the first term which is local, the genus, which is a non-negative integer, depends globally on all the helipoints.",
"The parameter $\\mu \\ge 0$ is used to penalize structures with excessively large values of the genus, in agreement with the phenomenological observation that the genus of most naturally-occurring RNA structures of size up to 600 bases, is smaller than 4.",
"Based on previous studies [7], the default value of the genus penalty $\\mu $ is set equal to 1.5 kcal/mol.",
"It is implicitly assumed that the free energy of incompatible sets of helipoints is infinite."
],
[
"Advanced Monte Carlo search of MFE structures",
"The minimization of the free energy (REF ) is carried out by a Monte Carlo (MC) exploration of structure space, that is over the set of possible $\\vec{\\sigma }$ vectors.",
"Starting from a structure $S$ where only one helipoint is present, at each Monte Carlo step, one of the helipoints $h_i$ is added ($\\sigma _i=0 \\rightarrow \\sigma _i=1$ ) or removed ($\\sigma _i=1 \\rightarrow \\sigma _i=0$ ).",
"The helipoint to be modified is picked with a biased probability favoring the addition (resp.",
"removal) of helipoints with low (resp.",
"high) free energy $e$ .",
"The biasing is inspired by the heat-bath MC algorithm.",
"Specifically, the a priori probability to pick helipoint $h_i$ to be changed in structure $S$ is given by: $w^S_i = {\\sigma _i^S + (1 - \\sigma ^S_i) \\, e^{-\\beta \\Delta F(h_i)} \\over \\sum ^\\prime _{j=1..N} \\, \\sigma _j^S + (1 - \\sigma ^S_j) \\, e^{-\\beta \\Delta F(h_j)}}$ where the prime superscript indicates that helipoints incompatible with $S$ are not considered.",
"Changing the state of $h_i$ defines a trial structure, $S^\\prime $ , which is accepted with probability $\\min \\left[1,{ w^{S^\\prime }_i \\over w^S_i} e^{-\\beta (F_{S^\\prime } - F_S)}\\right]\\ .$ The above acceptance criterion is a generalization of the standard Metropolis rule and ensures that, in the long run, the generated structures are sampled with probability given by the canonical weight $\\exp [-\\beta F_S]$ .",
"The stochastic generation of structures is carried out within a Monte Carlo algorithm with replica exchange where several simulations are run in parallel at different inverse temperatures $\\beta $ .",
"The values of $\\beta $ are chosen so as to cover a range of thermal energies ${1 / \\beta }$ , going from about one tenth of the smallest helipoint energy up to the largest helipoint energy.",
"At regular time intervals, swaps are proposed between structures at neighboring temperatures and are accepted with the generalized Metropolis criterion described in ref. [10].",
"The Markov replicas at the lowest temperature progressively populate structures of low free-energy, and a record is kept of the lowest energy structures which are finally provided as output.",
"Finally, we point out that the Monte Carlo optimization can be performed not only within the whole space of secondary structures (unconstrained search) but is straightforwardly restricted to topologically-constrained subspaces.",
"In particular, by introducing ad hoc “infinite\" energy penalties in eq.",
"REF , the search can be restricted to structures whose genus, topology or extent of pseudoknots satisfy some preassigned constraints.",
"The web-server interface allows the user to set such thresholds, e.g.",
"to account for knowledge based constraints."
],
[
"Generalized Topological Penalties",
"As we have previously reported [11], [12], any RNA complex pseudoknot structure may be built from of a set of building blocks, called primitive pseudoknots.",
"A pseudoknot is termed primitive if it is (i) irreducible, i.e.",
"its standard diagrammatic representation cannot be disconnected by cutting one backbone line and (ii) contains no nested pseudoknot, that is it cannot be disconnected by cutting two backbone lines, see Fig.",
"REF .",
"An arbitrary pseudoknotted structure can be decomposed in a collection of primitive pseudoknots and its total genus is the sum of the genii of its primitive constituents [11].",
"Figure: The only four primitive pseudoknots of genus 1 .",
"Therefore, it makes sense to assign different penalties to pseudoknots having same genus but with different primitive components.",
"For example, all tmRNAs have total genus 3 or 4 and contain no primitive pseudoknots of genus larger than 1.",
"In the present implementation, we propose only two options: i) we forbid primitive pseudoknots of genus larger than 1 (by assigning them an infinite penalty) but the overall structure can have any total genus or ii) we assign a global penalty proportional to the total genus and do not take into account the decomposition of the structure into primitive blocks."
],
[
"RESULTS AND DISCUSSION",
"We have carried out an extensive comparison of McGenus predictions against those of other methods.",
"For this purpose we used hundreds of RNA sequences from various sets, including: the dataset previously used for TT2NE [7], an extensive set of tmRNAs [13] and the more limited set of pseudoknotted RNA molecules for which the structural data is available in the protein databank (PDB).",
"Over such diverse datasets, the predictive performance is aptly conveyed by the sensitivity of the method, that is the fraction of pairs in the reference (native) structure that are correctly predicted by the method.",
"Depending on the context we shall also report on the positive predicted value (PPV).",
"The PPV corresponds to the fraction of predicted pairs that are found in the native structure, and hence measures the incidence of false positives in the predicted contacts.",
"We have considered this measure for the PDB set, but not for the tmRNA set whose entries, often corresponding to putative native structures derived from homology, are known to potentially lack several native contacts, as in the paradigmatic case of Aste.yell._TRW-322098_1-426 [13].",
"A visual representation of this structure can be found in the RNA STRAND database [14] under the reference TMR_00037.",
"From an overall point of view, the tests are aimed at elucidating two issues that are central to any MFE-based method.",
"The first issue, regards the algorithmic effectiveness of the energy minimization, while the second regards the viability of the energy parametrization within the considered space of secondary structures.",
"The former is most clearly ascertained by comparing algorithms employing the same energy parametrization.",
"This step is crucial for the second aspect too.",
"In fact, the appropriateness or the limitations of a given energy parametrization and/or of the considered secondary structure space, can be exposed in a non-ambiguous way only if the minimization algorithm is well-performing.",
"Following the above-mentioned logical order, we started by comparing the predictions of McGenus against TT2NE on a database of 47 short sequences ($<$ 209 bases) used in [7].",
"Because McGenus and TT2NE rely on the same energy parametrization[16], the comparison provides a stringent test of the effectiveness of the energy-minimization procedure.",
"In fact, we recall that TT2NE is based on an exhaustive, or nearly exhaustive search in sequence space.",
"Despite the stochastic, non-exhaustive and much faster McGenus searches, its performance turned out to be optimal.",
"Over the full data set, McGenus returned exactly the same MFE structures as TT2NE, as well as all the suboptimal structures.",
"To extend the assessment of McGenus minimization performance for longer chains, that cannot be addressed by TT2NE, we considered UNAFold [4], a MFE-based algorithm restricted to secondary structures without pseudoknots.",
"We used a customized version of UNAFold which employs the same energy parametrization as McGenus.",
"However, it cannot yet be compared to McGenus since it outputs secondary structures in terms of base pairs rather than helipoints.",
"To circumvent this difficulty, we generated all the lowest lying secondary structures (within 1kCal/mol from the lowest energy structure) using the algorithm presented in ref. [15].",
"To match the description of the structure in terms of helipoints, we made clusters of secondary structures sharing the same extremities of their helical fragments.",
"We then resummed them (in terms of their Boltzmann weights) and as a result the energy discrepancy between the two approaches is negligible.",
"In the sequel, we will refer to this process as cUNAFold.",
"The comparison was carried out over the complete set of 590 sequences of genus 3, 4 or 5 from the tmRNA database [13] with lengths in the 200-500 range.",
"To assess the efficiency of the minimization algorithm of McGenus, we ran it over our sample of 590 sequences, with the constraint $g_{max}=0$ and compared it with the output of cUNAFold.",
"The average MFE from McGenus with $g_{max}=0$ is -105.1 kCal/mol while that of cUNAFold is -106.7 kCal/mol.",
"Interestingly enough, out of the 590 sequences, 191 sequences are predicted to have identical secondary structures by both algorithms.",
"This comparison shows the good efficiency of McGenus minimization algorithm.",
"In the non-zero genus case, for each of the 590 sequences, McGenus returned structures with lower free energy than cUNAFold.",
"On the average, the free energy of the McGenus predicted structures was -125 kcal/mol.",
"These two tests prove the effectiveness of the energy-minimization scheme adopted by McGenus and we accordingly turned our attention to the overall predictive performance of the method (sensitivity).",
"For this purpose we used again the 590 sequences of genus 3, 4 or 5 from the tmRNA database [13] and compared McGenus predictions against McQfold [17], HotKnots [18], ProbKnot [19] and UNAFold [22] on this set.",
"We did not compare McGenus against PKnots [20] and gfold [21], as the original articles claim that they cannot handle sequences longer than 200 bases.",
"We recall that UNAFold predictions are restricted to secondary structures free of pseudoknots, while ProbKnot and McQfold can output any topology of pseudoknot.",
"The genus of each of McGenus prediction was enforced not to exceed the genus of the native structures of the dataset.",
"As discussed in [7], the setting of the corresponding parameter $g_{max}$ can be decided by the user.",
"In this report, for each test sequence, we chose to set $g_{max}$ to the appropriate, native, value to illustrate the performance of McGenus when it is driven in the appropriate secondary structure search space.",
"The total number of base pairs to be predicted in the set is 56740.",
"The UNAFold, McQfold, ProbKnot, HotKnots and McGenus arithmetic averages of the sensitivity over all sequences are respectively 37%, 42%, 43%, 39% and 43%, with a respective standard deviation of 14%, 15%, 14%, 14% and 16%.",
"A closer look at the secondary structures output by ProbKnot and HotKnots showed that none of them contained any pseudoknot.",
"Therefore the performance of McGenus is not inferior to that of the few methods that can handle sequences of comparable length.",
"Even without resorting to advanced comparative tests [23], [24], the consistent sensitivity of these 5 algorithms allows to conclude that their performance is very similar.",
"The fact that the average sensitivity of the five methods is below 50% poses the question of whether it can be improved by tweaking the energy parameters or by suitably further constraining the space of secondary structures over which the minimization is performed.",
"We focus on the latter aspect as the first has been already discussed in [7].",
"The space of secondary structures considered by prediction schemes based on abstract, graph-theoretical representations, include structures that are unphysical, i.e.",
"that cannot be realized in a three-dimensional space because of chain connectivity constraints.",
"The impact of this major difficulty can be lessened by excluding from further considerations those structures that present physically-unviable or atypical levels of entanglement.",
"To illustrate this point, we note that, in the mentioned dataset of 590 molecules, only H-pseudoknots which span less than 70 bases are present.",
"By enforcing such knowledge-based constraint on the search space, the sensitivity of McGenus is boosted from 43% to 53% with a standard deviation of 18%.",
"To assess the statistical significance of this improvement, we performed the Welch t-test.",
"We find a t-value of $t=10$ , which with a total of 1168 degrees of freedom implies a $p-$ value smaller than $10^{-7}$ , i.e.",
"the improvement is definitely significant.",
"Introducing the constraint in structure space clearly results in higher energies for the predicted structures.",
"In fact the average free energy was -125 kcal/mol without the constraint while it is -114 kcal/mol with the restriction of the pseudoknot length.",
"Notwithstanding the reduction of the search space due to the pseudoknot-length constraint, the structures returned by McGenus have an energy that is significantly lower than the reference, (putative) native structures, which is about - 73kcal/mol.",
"The free energy difference appears too large to be accounted for by the neglected contribution of loop entropy, missing chain-connectivity constraints or imperfect parametrization of the potentials, which are well established.",
"A more plausible source of discrepancy could the missing contacts in the homology-derived native structure of the tmRNA database.",
"To check this last point, we have studied the unconstrained version McGenus on a set of 4 sequences from the protein databank (PDB) with $g_{max}$ being fixed to the native genus.",
"Their PDB ids are: 1Y0Q (length=229, g=1), 3EOH (length=412, g=1), 2A64 (length=417, g=1) and 2H0W (length=151, g=2).",
"The structures of these entries are unambiguously known from X-ray scattering data and contain very few long and non-hybridized RNA sequences (i.e.",
"not bound to proteins, DNA or other molecules).",
"Accordingly, the McGenus performance on this set was higher than for the tmRNA set.",
"The sensitivity for 1Y0Q, 3EOH, 2A64 and 2H0W was equal to 87%, 39%, 50% and 72%, respectively while the PPV was equal to 90%, 38%, 35% and 84%, respectively.",
"Again, the structures predicted by McGenus have a lower free energy than the native ones.",
"This indicates that, besides accounting for topological effects, further improvements of secondary structure predictions would probably require a better parametrization of the free energy.",
"The generality and flexibility of the McGenus search algorithm ought to allow for incorporating any such modifications in a transparent way.",
"Finally let us discuss the choice of a maximum genus.",
"Ideally, one should perform the computation with a completely unconstrained genus.",
"However, there are two difficulties to this approach.",
"First, since steric constraints are only limitedly accounted for by available pseudoknot prediction algorithms (including McGenus), the predicted structures can be sterically impossible and hence associated to an excessively high genus.",
"Secondly, the computational time required to explore the unrestricted genus space could be impractical.",
"To overcome these difficulties and restrict the search space one can profitably introduce knowledge-based constraints.",
"In particular, the statistical PDB analysis of ref.",
"[11] provides a quantitative indication for the dependence of the genus on the length of naturally-occurring RNA sequences.",
"The data can be clearly used to provide a phenomenological upper bound to $g_{max}$ .",
"Alternatively, a user could explore a few different increasing values of $g_{max}$ and carry out a supervised evaluation of the results by taking into account (i) the phenomenological constraints and (ii) the possibility that structures with excessively large genus value are returned because of the imperfect treatment of steric constraints.",
"To illustrate this last point, we ran McGenus on a set of 792 5S rRNA sequences of length around 150, with no pseudoknot.",
"We set $g_{max}=3$ which according to the study of ref.",
"[11] (see Fig.",
"10 therein) is very large.",
"The number of sequences predicted with genus 0 (i.e.",
"without pseudoknots) is 258, with genus 1 is 500, with genus 2 is 34 and with genus 3 is 0.",
"Consistently with the remarks made in the context of H-pseudoknots, the results indicate that performance of pseudoknot prediction algorithms could certainly benefit by improving the current handling of chain connectivity and excluded volume constraints."
],
[
"CPU time",
"The CPU time required by McGenus to fold an RNA sequence depends on the total number of Monte Carlo steps.",
"For a tm-RNA of length 400, the typical number of helipoints is 3500.",
"For each sequence, we use 10 replicas, and overall 3000 $\\times $ number of helipoints steps to achieve these results.",
"The result is typically returned in 15 minutes on a parallel quadcore computer (Intel Xeon CPU @2.66GHz).",
"The current implementation of McGenus on the server is not parallelized."
],
[
"Conclusion",
"In this article, we presented McGenus, an efficient algorithm for RNA pseudoknot prediction, which proves that classifying pseudoknots according to their genus is a relevant and successful concept.",
"We showed that on a set of RNA structures from the tm-RNA database [13], McGenus allows treatment of sequences of sizes up to 1000 with a typical CPU time of 15 minutes for a 500 long sequence on a quadcore CPU, with a performance that is comparable or better than the few methods that can treat sequences with comparable length.",
"In order to further improve the performance of McGenus, we see 3 main directions: I) improvement on the computing techniques, in particular on the parallelization of the algorithm.",
"II) improvement of the functional form and parametrization of the energy model (likely to have an impact also on the parametrization of pseudoknot-free methods such as UNAFold).",
"III) inclusion of steric constraints."
],
[
"Acknowledgements",
"We acknowledge financial support from the Italian Ministry of research, grant FIRB - Futuro in Ricerca N. RBFR102PY5.",
"The authors wish to thank A. Capdepon for setting up the McGenus server at http://ipht.cea.fr/rna/mcgenus.php."
]
] | 1204.0993 |
[
[
"Anelastic tidal dissipation in multi-layer planets"
],
[
"Abstract Earth-like planets have viscoelastic mantles, whereas giant planets may have viscoelastic cores.",
"The tidal dissipation of such solid regions, gravitationally perturbed by a companion body, highly depends on their rheology and on the tidal frequency.",
"Therefore, modelling tidal interactions presents a high interest to provide constraints on planets' properties and to understand their history and their evolution, in our Solar System or in exoplanetary systems.",
"We examine the equilibrium tide in the anelastic parts of a planet whatever the rheology, taking into account the presence of a fluid envelope of constant density.",
"We show how to obtain the different Love numbers that describe its tidal deformation.",
"Thus, we discuss how the tidal dissipation in solid parts depends on the planet's internal structure and rheology.",
"Finally, we show how the results may be implemented to describe the dynamical evolution of planetary systems.",
"The first manifestation of the tide is to distort the shape of the planet adiabatically along the line of centers.",
"Then, the response potential of the body to the tidal potential defines the complex Love numbers whose real part corresponds to the purely adiabatic elastic deformation, while its imaginary part accounts for dissipation.",
"This dissipation is responsible for the imaginary part of the disturbing function, which is implemented in the dynamical evolution equations, from which we derive the characteristic evolution times.",
"The rate at which the system evolves depends on the physical properties of tidal dissipation, and specifically on how the shear modulus varies with tidal frequency, on the radius and also the rheological properties of the solid core.",
"The quantification of the tidal dissipation in solid cores of giant planets reveals a possible high dissipation which may compete with dissipation in fluid layers."
],
[
"Introduction and general context",
"Since 1995 a large number of extrasolar planets have been discovered, which display a wide diversity of physical parameters (Santos & et al., 2007).",
"Quite naturally the question arose of their habitability, i.e.",
"whether they could allow the development of life.",
"Determining factors are the presence of liquid water and of a protective magnetic field, properties which are closely linked to the values of the rotational and orbital elements of the planetary systems.",
"And these elements strongly depend on the action of tides.",
"Once a planetary system is formed in a turbulent accretion disk, its fate is determined by the initial conditions and the mass ratio between planet and hosting star.",
"Through tidal interaction between components, the system evolves either to a stable state of minimum energy (where all spins are aligned, the orbits are circular and the rotation of each body is synchronized with the orbital motion) or the companion tends to spiral into the parent body.",
"Indeed, by converting kinetic energy into heat through internal friction, tidal interactions modify the orbital and rotational properties of the components of the considered system, and thus their structure through internal heating.",
"This mechanism depends sensitively on the internal structure and dynamics of the perturbed body.",
"Recent studies have been carried out on tidal effects in fluid bodies like stars and envelopes of giant planets (Ogilvie & Lin 2004-2007; Ogilvie 2009; Remus, Mathis & Zahn 2012).",
"However the planetary solid regions, such as mantles of Earth-like planets or rocky cores of giant planets, if present (e.g.",
"Guillot 1999, Gaulme et al.",
"2011), may contribute likewise to tidal dissipation.",
"The first study of a tidally deformed elastic body was done by Lord Kelvin (1863) who applied it to an incompressible homogeneous Earth.",
"Further developments were made by Love (1911), who introduced a set of dimensionless numbers, the so-called Love numbers, to quantify the tidal perturbation.",
"More recently Greff-Lefftz (2005) generalized these results in the case of a spheroidal rotating Earth.",
"In the meanwhile Dermott (1979) considered a two-layer model and he studied the impact of a tidally deformed static fluid shell on the deformation of an elastic solid core.",
"If the body is not perfectly elastic, i.e.",
"if its internal structure is anelastic, the tidal deformation presents a lag with respect to the tension exerted by the tidal force, and causes the dissipation of kinetic energy.",
"Several recent studies addressed this problem of tidal dissipation using linear viscoelastic models.",
"Peale & Cassen (1978) evaluated the tidal dissipation in the Moon considering various models of internal structure.",
"Tobie et al.",
"(2005) applied the Maxwell rheological model to evaluate the dissipation in Titan and Europa; Ross & Schubert (1986) compared three different linear models of viscoelasticity (Kelvin-Voigt, Maxwell, Standard Anelastic Solid), to which Henning et al.",
"(2009) added the Burgers body.",
"All these studies suscitate our interest in the tidal dissipation resulting from the anelastic deformation of the solid parts of a planet when perturbed by a companion.",
"We shall study here the tidal dissipation in a planet which possesses an anelastic core, made of a mix of ice and rock, surrounded by a fluid envelope, such as an ocean.",
"The planet is part of a binary system where what we call the companion (or perturber) may be either the hosting star or a satellite of the planet.",
"Due to the tide exerted by the companion, the core of the two-layer planet is deformed elastically, but because of the anelasticity of the material composing the core, this deformation is accompanied by a viscous dissipation that we evaluate whatever the rheology.",
"As an illustration, the results will be given for a Maxwell body.",
"We then compare the value of the tidal dissipation in presence of a fluid envelope with that achieved by the fully solid planet, and we examine the dependence of the results on the relative sizes of the core and the planet, the relative densities, and the viscoelastic parameters.",
"In the last section, we establish the equations governing the dynamical evolution of the system, from which we deduce the caracteristic times of circularization, synchronization and spin alignments.",
"We consider a planet $A$ of mass $M_A$ , consisting of a rocky (or icy) core and a fluid envelope, rotating at the angular velocity $\\Omega $ and tidally perturbed by a second body of mass $M_B$ , assumed to be ponctual, moving around $A$ on a Keplerian orbit, of semi-major axis $a$ and eccentricity $e$ , at the mean motion $n$ .",
"We locate any point $M$ in space by its usual spherical coordinates ${\\left( r, \\theta , \\varphi \\right)}$ in a spin equatorial reference frame ${\\mathcal {R}_E: \\lbrace A, \\bf X_E, \\bf Y_E, \\bf Z_E \\rbrace }$ centered on body $A$ and whose axis ${\\left(A,\\bf Z_E\\right)}$ has the direction of the rotation axis of $A$ .",
"The corresponding configuration is illustrated on Figure REF .",
"Figure: Spherical coordinates system attached to the equatorial reference frame ℛ E :{A,𝐗 𝐄 ,𝐘 𝐄 ,𝐙 𝐄 }{\\mathcal {R}_E: \\lbrace A, \\bf X_E, \\bf Y_E, \\bf Z_E \\rbrace } associated to body AA.",
"A point MM is located by 𝐫≡r,θ,ϕ{\\mathbf {r} \\equiv \\left( r,\\theta ,\\varphi \\right)}; the point mass body BB by 𝐫 B ≡r B ,θ B ,ϕ B {\\mathbf {r}_B \\equiv \\left( r_B,\\theta _B,\\varphi _B \\right)}.In this section, we first assume that planet $A$ has no fluid layer and its internal structure is supposed to be perfectly elastic.",
"We will then denote by $\\rho $ its density and $R$ its mean radius."
],
[
"The tidal potential",
"The planet is submitted to a tidal force, exerted by the perturber $B$ , which derives from a perturbing time dependent potential $U\\left(\\mathbf {r},t\\right)$ .",
"Following Zahn (1966a-b) and generalizing it by using Kaula (1962), Lambeck (1980), Yoder (1995-1997) and Mathis & Le Poncin (2009) (hereafter MLP09) in the present case of a close binary system where spins are not aligned, the components are not synchronized with the orbital motion and where the orbit is not circular, we expand the tidal potential $U$ in spherical harmonics ${Y_l^m(\\theta ,\\varphi )}$ in $\\mathcal {R}_E$ .",
"Before we proceed, we need to define the Euler angles that link the spin equatorial frame ${\\mathcal {R}_E: \\lbrace A, \\bf X_E, \\bf Y_E, \\bf Z_E \\rbrace }$ of the central body $A$ , on one hand, and the orbital frame ${\\mathcal {R}_O: \\lbrace A, \\bf X_O, \\bf Y_O, \\bf Z_O \\rbrace }$ , on the other hand, to the quasi-inertial frame ${\\mathcal {R}_R: \\lbrace A, \\bf X_R, \\bf Y_R, \\bf Z_R \\rbrace }$ whose axis $\\bf Z_R$ has the direction of the total angular momentum of the whole system.",
"We need the three following Euler angles to locate the orbital reference frame $\\mathcal {R}_O$ with respect to $\\mathcal {R}_R$ : $I$ , the inclination of the orbital plane of $B$ ; $\\omega ^*$ , the argument of the orbit pericenter; $\\Omega ^*$ , the longitude of the orbit ascending node.",
"The equatorial reference frame $\\mathcal {R}_E$ is defined by three other Euler angles with respect to $\\mathcal {R}_R$ : $\\varepsilon $ , the obliquity of the rotation axis of $A$ ; $\\Theta ^*$ , the mean sideral angle defined by ${\\Omega = {\\mathrm {d} \\Theta ^*}/{\\mathrm {d}t}}$ ; $\\phi ^*$ , the general precession angle.",
"Refer to Figure REF for an illustration of the relative position of these three reference frames and the associated angles.",
"For convenience, all the following developpements will be done in the spin equatorial frame $\\mathcal {R}_E$ of $A$ (as it has been done in MLP09).",
"Figure: Inertial reference (ℛ R \\mathcal {R}_R), orbital (ℛ O \\mathcal {R}_O), and equatorial (ℛ E \\mathcal {R}_E) rotating frames, and associated Euler’s angles of orientation.All following results are derived from the Kaula's transform (Kaula 1962), used to explicitly express the whole generic multipole expansion in spherical harmonics of the perturbing potential $U$ in terms of the Keplerian orbital elements of $B$ in the equatorial $A$ -frame: $\\frac{1}{r_B^{l+1}} \\, P_{l}^{m}(\\cos \\theta _B) \\, e^{im\\varphi _B}= \\frac{1}{a^{l+1}} \\sum _{j=-l}^{l} \\sum _{p=0}^{l} \\sum _{q\\in \\mathbb {Z}}\\left\\lbrace \\sqrt{ \\frac{2l+1}{4\\pi }\\,\\frac{(l-|j|)!}{(l+|j|)!}",
"} \\right.",
"\\\\\\left.",
"\\times d_{j,m}^{l} (\\varepsilon ) \\, F_{l,j,p}(I) \\, G_{l,p,q}(e) \\, e^{i\\Psi _{l,m,j,p,q}} \\vphantom{\\sqrt{ \\frac{2l+1}{4\\pi }\\,\\frac{(l-|j|)!}{(l+|j|)!}",
"}}\\right\\rbrace $ where $\\theta _B$ and $\\varphi _B$ are respectively the colatitude and the longitude of the point mass perturber $B$ , and where the phase argument is given by: $\\Psi _{l,m,j,p,q}(t) = \\sigma _{l,m,p,q} (n,\\Omega ) \\, t + \\tau _{l,m,j,p,q}\\left( \\omega ^*, \\Omega ^*, \\phi ^* \\right) \\:.$ We have defined here the tidal frequency: $\\sigma _{l,m,p,q} (n,\\Omega ) = (l-2p+q) \\, n - m \\Omega \\:,$ and the phase $\\tau _{l,m,j,p,q}$ : $\\tau _{l,m,j,p,q} = (l-2p)\\omega ^* + j(\\Omega ^*-\\phi ^*) + (l-m)\\frac{\\pi }{2}.$ We study here binary systems close enough for the tidal interaction to play a role, but we also consider that the companion is far (or small) enough to be treated as a point mass.",
"We then are allowed to assume the quadrupolar approximation, where we only keep the first mode of the potential, $l=2$ : $U(r, \\theta , \\varphi , t) = \\mathcal {R} \\!e \\ \\left\\lbrace \\sum _{m=-2}^{2} \\sum _{j=-2}^{2} \\sum _{p=0}^{2} \\sum _{q \\in \\mathbb {Z}} \\right.",
"\\\\\\left.",
"U_{m,j,p,q}(r) \\, P_2^m(\\cos \\theta ) \\, e^{i \\, \\Phi _{2,m,j,p,q}(\\varphi ,t)} \\vphantom{\\sum _{m=-2}^{2}} \\right\\rbrace $ where $\\Phi _{2,m,j,p,q}(\\varphi ,t) = m\\varphi + \\Psi _{2,m,j,p,q}(t)$ The functions ${U_{m,j,p,q}\\left(r,\\theta \\right)}$ may be expressed in terms of the Keplerian elements (the semi-major axis $a$ of the orbit, its eccentricity $e$ and its inclination $I$ ) and the obliquity $\\varepsilon $ of the rotation axis of $A$ , as: $U_{m,j,p,q}(r)= (-1)^m \\, \\sqrt{ \\frac{(2-m)!",
"\\, (2-|j|)!}{(2+m)!",
"\\, (2+|j|)!}",
"} \\\\\\times \\frac{ \\mathcal {G} \\, M_B}{a^3} \\,\\left[ d_{j,m}^{2}(\\varepsilon ) \\, F_{2,j,p}(I) \\, G_{2,p,q}(e) \\right] \\, r^2 \\:,$ where $\\mathcal {G}$ is the gravitational constant.",
"The obliquity function ${d_{j,m}^{2}(\\gamma )}$ is defined, for $j\\geqslant m$ , by: $d_{j,m}^{2}(\\gamma ) = (-1)^{j-m} \\left[\\frac{ (2+j)!",
"(2-j)!",
"}{ (2+m)!",
"(2-m)!",
"}\\right]^{1\\over 2} \\\\\\times \\left[ \\cos \\left(\\frac{\\gamma }{2}\\right) \\right]^{j+m} \\, \\left[ \\sin \\left(\\frac{\\gamma }{2}\\right) \\right]^{j-m} \\, \\mathrm {P}_{2-j}^{(j-m , j+m)} (\\cos \\gamma ) \\,,$ where ${\\mathrm {P}_l^{\\alpha ,\\beta } (x)}$ are the Jacobi polynomials (cf.",
"MLP09).",
"The values of these functions, for indices $j < m$ , are deduced from ${d_{j,m}^{2}(\\pi +\\gamma ) = (-1)^{2-j} d_{-j,m}^{2}(\\gamma )}$ or from their symmetry properties: ${d_{j,m}^{2}(\\gamma ) = (-1)^{j-m} d_{-j,-m}^{2}(\\gamma ) = d_{m,j}^{2}(-\\gamma )}$ ; moreover, we have: ${d_{j,m}^{2}(0) = \\delta _{j,m}}$ .",
"Values are given in Table REF .",
"Table: Values of the obliquity function d j,m 2 ε{d_{j,m}^{2}\\left(\\varepsilon \\right)} in the case where j⩾mj\\geqslant m obtained from Eq.",
"(), (from MLP09).We also define, the inclination function $F_{2,j,p}(I)$ : $F_{2,j,p}(I) = (-1)^j \\, \\frac{(2+j)!",
"}{ 4\\, p!",
"\\, (2-p)! }",
"\\\\\\times \\left[ \\cos \\left(\\frac{I}{2}\\right) \\right]^{j-2p+2} \\, \\left[ \\sin \\left(\\frac{I}{2}\\right) \\right]^{j+2p-2} \\\\\\times \\mathrm {P}_{2-j}^{(j+2p-2 , j-2p+2)} (\\cos I) \\,,$ with the symmetry property: $F_{2,-j,p}(I) = \\left[ (-1)^{2-j} \\frac{\\left(2-j\\right)!}{\\left(2+j\\right)!",
"}\\right]F_{2,j,p}\\left(I\\right).$ Values are given in Table REF .",
"Table: Values of the inclination function F 2,j,p (I)F_{2,j,p}(I).",
"Values for indices j<0j < 0 can be deduced from Eq.",
"(), (from MLP09).The eccentricity functions $G_{2,p,q}(e)$ are polynomial functions having $e^q$ for argument (see Kaula, 1962).",
"Their values for the usual sets $\\lbrace 2,p,q\\rbrace $ are given in Table REF , knowing that in the case of weakly eccentric orbits, the summation over a small number of values of $q$ is sufficient ($ q \\in \\textrm {\\xbox {char}{\\XMLaddatt {name}{lbrackdbl}}}-2,2\\textrm {\\xbox {char}{\\XMLaddatt {name}{rbrackdbl}}} $ ).",
"In the following, let us denote by ${\\mathbb {I} = \\textrm {\\xbox {char}{\\XMLaddatt {name}{lbrackdbl}}}-2,2\\textrm {\\xbox {char}{\\XMLaddatt {name}{rbrackdbl}}} \\times \\textrm {\\xbox {char}{\\XMLaddatt {name}{lbrackdbl}}}-2,2\\textrm {\\xbox {char}{\\XMLaddatt {name}{rbrackdbl}}} \\times \\textrm {\\xbox {char}{\\XMLaddatt {name}{lbrackdbl}}}0,2\\textrm {\\xbox {char}{\\XMLaddatt {name}{rbrackdbl}}} \\times \\mathbb {Z}}$ the set in which the quadruple $\\lbrace m,j,p,q\\rbrace $ takes its values.",
"Table: Values of the eccentricity function G 2,p,q (e)G_{2,p,q}(e), (from MLP09).If we simplify the expansion of the potential in the case where spins are aligned and perpendicular to the orbital plan, where obliquity $\\varepsilon $ and orbital inclination $I$ are zero, Eq.",
"(REF ) reduces to the expression of the potential given by Zahn (1977).",
"The tidal force induces a displacement of each particule constituting the planet, thus causing some deformations.",
"In particular, the core's surface is deformed as described by the Love theory (Love 1911)."
],
[
"Dynamical equations for a solid body and their boundary conditions",
"To describe the internal evolution of the main component $A$ submitted to the perturbations induced by the tidal potential presented above, we use the Eulerian formalism (Dahlen et al.",
"1999).",
"The system of equations, needed to follow the motion of a particule, is composed by the Eulerian momentum (REF ) and mass () conservation laws, and the Poisson equation () satisfied by the potential $\\Phi $ of self-gravitation: $&\\rho \\, \\frac{\\partial ^2 \\bf s}{\\partial t^2} = {\\nabla }\\cdot {\\bar{ \\bar{\\sigma } }} + \\rho \\, {\\nabla }\\left( \\Phi + U \\right) \\:, \\\\&\\frac{\\partial \\rho }{\\partial t} + {\\nabla }\\cdot \\left( \\rho \\, \\frac{\\partial \\mathbf {s}}{\\partial t} \\right) = 0 \\:, \\\\&{\\nabla }^2 \\Phi = - 4 \\pi \\mathcal {G} \\rho \\:, $ where $\\bf s$ designates the displacement vector and $ {\\bar{ \\bar{\\sigma } }} $ is the stress tensor.",
"We complete this system with the constitutive equation to link the stress exerted on the body to the resulting deformation.",
"Assuming that tidal effect corresponds to a traction applied on the body, with no rotational contribution, the deformation tensor reduces to the strain tensor $ {\\bar{ \\bar{\\epsilon } }} $ : $ {\\bar{ \\bar{\\epsilon } }} = \\frac{1}{2} \\left[ {\\nabla }\\mathbf {s} + \\left( {\\nabla }\\mathbf {s} \\right)^{\\mathrm {T}} \\right] \\:,$ where $ {\\bar{ \\bar{h} }} ^{\\mathrm {T}}$ designates the transposed tensor of $ {\\bar{ \\bar{h} }} $ .",
"We then get a relation linking the stress tensor $ {\\bar{ \\bar{\\sigma } }} $ to the strain tensor $ {\\bar{ \\bar{\\epsilon } }} $ that accounts for the rheology of the body, and that we represent by a function ${\\mathcal {F}_\\mathrm {rh}}$ : $ {\\bar{ \\bar{\\sigma } }} = {\\mathcal {F}_\\mathrm {rh}} ( {\\bar{ \\bar{\\epsilon } }} ) \\:.$ To solve this system (REF ), we need to apply boundary conditions to the five previous equations, assuming that there is no displacement (REF ) neither attraction () at the center of mass $r=0$ , the gravitational potential has to be continuous () and the Lagrangian traction has to vanish () at the surface $r=R$ : $&\\left.",
"\\mathbf {s} \\right|_{r=0}= \\mathbf {0} \\:, \\\\&\\left.",
"\\left( \\Phi + U \\right) \\right|_{r=0} = \\mathbf {0} \\:, \\\\&\\left[ \\Phi \\right]_{R^{-}}^{R^{+}} = 0 \\quad ,\\quad \\text{i.e.",
": }\\quad \\left[ \\frac{\\partial \\Phi }{\\partial r} + 4\\pi \\, \\mathcal {G} \\, \\rho \\, s_r \\right]_{R^{-}}^{R^{+}} = 0 \\:, \\\\&\\left.",
"\\left( {\\mathbf {e}_r} \\cdot {\\bar{ \\bar{\\sigma } }} \\right) \\right|_ {r=R} = 0 \\:.$"
],
[
"Linearization of the system",
"Assuming that tidal effects, and thus the resulting elastic deformation, are small amplitude perturbations to the hydrostatic equilbrium, we are allowed to linearize the system (REF ) and its boundary conditions (REF ).",
"To do so, we expand a scalar quantity $X$ as: $X \\left(r,\\theta ,\\varphi ,t\\right) = X_0(r) + X^{\\prime } \\left(r,\\theta ,\\varphi ,t\\right) \\, ;$ $X_0$ designates the spherically symmetrical profile of $X$ , and $X^{\\prime }$ represents the perturbation due to the tidal potential.",
"The displacement $\\mathbf {s}$ and the tidal potential $U$ are also considered as perturbations.",
"Thus, correct to first order in $|| \\mathbf {s} ||$ , we obtain the following form of system (REF ): $\\rho _0 \\, \\frac{\\partial ^2 \\bf s}{\\partial t^2} &= {\\nabla }\\cdot {\\bar{ \\bar{\\sigma } }} + \\rho _0 \\, {\\nabla }\\left( \\Phi ^{\\prime } + U \\right) + \\rho ^{\\prime } \\, {\\nabla }\\Phi _0 \\:, \\\\\\rho ^{\\prime } + {\\nabla }\\cdot \\left( \\rho _0 \\, \\bf s \\right) &= 0 \\:, \\\\{\\nabla }^2 \\Phi ^{\\prime } &= - 4 \\pi \\mathcal {G} \\rho ^{\\prime } \\:, \\\\ {\\bar{ \\bar{\\sigma } }} &= \\left( K - \\frac{2}{3} \\mu \\right) \\mathrm {tr} \\left( {\\bar{ \\bar{\\epsilon } }} \\right) \\, {\\bar{ \\bar{I} }} + 2 \\mu {\\bar{ \\bar{\\epsilon } }} \\:, $ where we made use of the Hooke's law (), which is a linear constitutive law that governs elastic materials as long as the load does not exceed the material's elastic limit, in the case of an isotropic material (i.e.",
": whose properties are independent of direction in space).",
"It means that strain is directly proportional to stress, through the bulk modulus $K$ and the shear modulus $\\mu $ .",
"The reference state is drawn from an up-to-date planetary structure model.",
"It is governed by the following Poisson equation and the static momentum equation: ${\\nabla }^2 \\Phi _0 &= - 4 \\pi \\mathcal {G} \\rho _0 \\:, \\\\{\\nabla }P_0 &= \\rho _0 \\, {\\nabla }\\Phi _0 \\:,$ where we made use of the following convention for the gravity: ${\\mathbf {g}_0 = {\\nabla }\\Phi _0}$ ."
],
[
"Analytical solutions for an homogeneous incompressible body",
"To solve the linear system (REF ), we expand all scalar quantities in spherical harmonics $Y_l^m(\\theta , \\varphi )$ .",
"Moreover, as all vectorial quantities that intervene in Eqs.",
"(REF -) are poloidal, we may expand them in the basis of vectorial spherical harmonics ${\\left[ \\mathbf {R}_l^m(\\theta , \\varphi ) , \\mathbf {S}_l^m(\\theta , \\varphi ) \\right]}$ , where $\\mathbf {R}$ refers to the radial part and $\\mathbf {S}$ to the spheroidal part of a given vector (Rieutord 1987, Mathis & Zahn 2005): [left= (l,m) N-l,l, ] flalign Rlm(, ) = Ylm(, ) er , Slm(, ) = S [ Ylm(, ) ] , where ${\\nabla }_{\\mathrm {S}}$ designates the horizontal gradient: ${\\nabla }_{\\mathrm {S}} = \\frac{\\partial \\, \\cdot }{\\partial \\theta } \\, \\mathbf {e}_{\\theta } + \\frac{1}{\\sin \\theta } \\frac{\\partial \\, \\cdot }{\\partial \\varphi } \\, \\mathbf {e}_{\\varphi } \\:.$ We introduce six radial functions $y_{\\lbrace 1...6\\rbrace }^m(r)$ (Takeuchi & Saito 1972) to expand all quantities in spherical harmonics, at the quadrupolar approximation ($l=2$ ): the displacement: $\\mathbf {s} (r,\\theta ,\\varphi ,t) = \\sum _{(m,j,p,q) \\in \\mathbb {I}} \\left[y_1^m(r) \\, \\mathbf {R}_2^m(\\theta , \\varphi ) \\right.",
"\\\\\\left.",
"+ y_3^m(r) \\, \\mathbf {S}_2^m(\\theta , \\varphi ) \\right] \\, e^{i\\Psi _{2,m,j,p,q}(t)} \\:,$ the total potential: $ (U+\\Phi ^{\\prime }) (r,\\theta ,\\varphi ,t) = \\sum _{(m,j,p,q) \\in \\mathbb {I}} y_5^m(r) \\, Y_2^m(\\theta , \\varphi ) \\, e^{i\\Psi _{2,m,j,p,q}(t)} \\:,$ the Lagrangian traction: $\\mathbf {e}_r \\cdot \\, {\\bar{ \\bar{\\sigma } }} (r,\\theta ,\\varphi ,t) = \\sum _{(m,j,p,q) \\in \\mathbb {I}} \\left[y_2^m(r) \\, \\mathbf {R}_2^m(\\theta , \\varphi ) \\right.",
"\\\\\\left.",
"+ y_4^m(r) \\, \\mathbf {S}_2^m(\\theta , \\varphi ) \\right] \\, e^{i\\Psi _{2,m,j,p,q}(t)} \\:,$ the Lagrangian attraction (introduced to express the continuity of the gradient of the potential): $\\forall m \\in \\textrm {\\xbox {char}{\\XMLaddatt {name}{lbrackdbl}}}-2,2\\textrm {\\xbox {char}{\\XMLaddatt {name}{rbrackdbl}}} , \\:y_6^m(r) = \\frac{\\mathrm {d}}{\\mathrm {d} r} \\left[ y_5^m(r) \\right] - 4\\pi \\mathcal {G} \\rho _0 y_1^m(r) \\:.$ The linear system governing the radial functions $y_{\\lbrace 1...6\\rbrace }^m(r)$ is given in Appendix.",
"In the case of an incompressible (${K \\rightarrow +\\infty }$ ) and homogeneous body (${\\mu , \\, \\rho _0 = {cst}}$ ), the system (REF ) constrained by boundary conditions (REF ) admits the following solutions, considering the expansion (REF ): $\\displaystyle \\forall m \\in \\textrm {\\xbox {char}{\\XMLaddatt {name}{lbrackdbl}}}-2,2\\textrm {\\xbox {char}{\\XMLaddatt {name}{rbrackdbl}}} $ , $y_1^m(r) &= \\sum _{j,p,q} \\, \\frac{k_2}{r\\,R\\,g_s} \\, \\left( \\frac{8}{3} R^2 - r^2 \\right) \\, U_{m,j,p,q}(r) \\:, \\\\y_2^m(r) &= \\sum _{j,p,q} \\, \\left[ 2\\mu \\dfrac{k_2}{r^2\\,R\\,g_s} \\left( \\dfrac{8}{3} R^2 + \\dfrac{1}{2} r^2 \\right) \\right.",
"\\nonumber \\\\&\\qquad \\quad + \\dfrac{4}{3} \\pi \\mathcal {G} \\rho ^2 \\dfrac{k_2}{R\\,g_s} \\left( \\dfrac{8}{3} R^2 - \\dfrac{1}{2} r^2 \\right)\\nonumber \\\\&\\qquad \\qquad \\qquad - \\rho (1+k_2) \\Bigg ] U_{m,j,p,q}(r) \\:, \\\\ \\nonumber \\\\y_3^m(r) &= \\sum _{j,p,q} \\, \\frac{k_2}{r\\,R\\,g_s} \\, \\left( \\frac{4}{3} R^2 - \\frac{5}{6} r^2 \\right) \\,U_{m,j,p,q}(r)\\:, \\\\y_4^m(r) &= \\sum _{j,p,q} \\frac{8\\mu }{3} \\, \\frac{k_2}{r\\,R\\,g_s} \\, (R^2 - r^2) \\, U_{m,j,p,q}(r) \\:, \\\\y_5^m(r) &= \\sum _{j,p,q} \\, (1 + k_2) \\, U_{m,j,p,q}(r) \\:, \\\\y_6^m(r) &= \\sum _{j,p,q} \\, \\left[ \\frac{2 (1+k_2)}{r} \\right.",
"\\nonumber \\\\& \\left.",
"+ 4\\pi \\mathcal {G} \\rho _0 \\, \\frac{k_2}{r\\,R\\,g_s} \\, \\left( r^2 - \\frac{8}{3} R^2 \\right) \\right] U_{m,j,p,q}(r) \\:,$ where we have introduced the acceleration of gravity at the surface $g_s$ and the second-order Love number $k_2$ .",
"The latter compares the perturbed part $\\Phi ^{\\prime }(R)$ of the self-gravitational potential at the surface of a fully-solid planet of mean radius $R$ , deformed by tidal force, with the tidal perturbing potential $U(R)$ : $k_2 \\stackrel{\\text{def}}{=} \\frac{\\Phi ^{\\prime }(R)}{U(R)} \\: .$ The expression of $k_2$ is established in Sect.",
", for an ocean-free planet (Eq.",
"REF ) or a two-layer planet (Eq.",
"REF ).",
"Figure: Left: tidal displacement 𝐬\\mathbf {s}.",
"Middle: equatorial slice of 𝐬\\mathbf {s}.",
"Right: meridional slice of 𝐬\\mathbf {s}.",
"The orange arrow indicates the direction of the perturber BB, the red one corresponds to the rotation axis of AA.",
"The two slices are planes of symmetry."
],
[
"Modified elastic tidal theory in presence of a fluid envelope",
"We now assume that planet $A$ is not entirely solid, but has a fluid envelope.",
"We follow the method proposed by Dermott (1979) to evaluate how the anelastic dissipation is modified by the presence of a fluid layer surrounding the solid region.",
"The first step consists in determining the behaviour of the elastic response in this configuration.",
"We denote by $R_c$ (resp.",
"$R_p$ ) the mean radius of the solid core (resp.",
"of the whole planet, including the height of the fluid layer); $\\rho _c$ and $\\rho _o$ designate the density of the core and the ocean, both assumed to be uniform, as a first step.",
"More generally, all quantities will be written with a “$c$ \" subscript when evaluated at the core boundary, and with a “$p$ \" subscript if taken at the surface of the planet.",
"The evolution of the system is described in the orbital frame ${\\mathcal {R}_O: \\lbrace A, X_O, Y_O, Z_O\\rbrace }$ centered on $A$ and comoving with the perturber $B$ .",
"We will use polar coordinates $(r,\\Theta )$ to locate a point $P$ , where $r$ is the distance to the center of $A$ , and $\\Theta $ is the angle formed by the radial vector and the line of centers."
],
[
"Vertical deformation at the boundary of the core",
"In $\\mathcal {R}_O$ , the tidal potential takes the form (Dermott 1979) $U(\\mathbf {r}) = - \\zeta (r) \\, g(r) \\, P_2(\\cos \\Theta ) = - \\zeta _c \\, g_c \\, \\frac{r^2}{R_c^2} \\, P_2(\\cos \\Theta ) \\:,$ where we have introduced the tidal height $\\zeta (r) = \\frac{M_B}{M(r)} \\, \\left( \\frac{r}{a} \\right)^3 \\, r \\:,$ and the gravity $g(r) = \\frac{\\mathcal {G} M(r)}{r^2} \\:,$ $M(r)$ being the fraction of mass of the planet inside the radius $r$ .",
"The expression of the tidal potential in the rotating frame of $B$ (Eq.",
"REF ) is linked to its expression in the equatorial inertial frame (Eq.",
"REF ), through the Kaula's transform (Eq.",
"REF ).",
"Indeed, the Legendre polynomia summation formula $P_2(\\cos \\Theta ) =\\mathcal {R} \\!e \\ \\left[ \\sum _{m=0}^2 \\frac{(2-m)!}{(2+m)!}",
"(2-\\delta _{0,m}) \\, P_2^m(\\cos \\theta ) \\, {\\mathbf {e}}^{i m\\varphi } \\right.",
"\\\\\\times \\left.",
"P_2^m(\\cos \\theta _B) \\, {\\mathbf {e}}^{- i m \\varphi _B} \\vphantom{\\sum _n} \\right]$ involves the term ${P_2^m(\\cos \\theta _B) \\, {\\mathbf {e}}^{- i m \\varphi _B}}$ in Eq.",
"(REF ) that has to be transformed following Eq.",
"(REF -REF ) to obtain Eq.",
"(REF ).",
"In this section we are interested in the mofication of Love numbers due to the presence of a fluid envelope on top of the solid core.",
"Thus, we will focus on the deformations of the core's surface, and particularly on the vertical displacements.",
"Love (1911) proved that tidal deformations could be described by the same harmonic function than the tidal potential which causes it.",
"Therefore the equations of the core and planet boundaries are respectively of the form $r_c \\equiv R_c + s_r(R_c) &= R_c \\left[ 1 + S_2 \\, P_2(\\cos \\Theta ) \\right] \\:, \\\\r_p &= R_p \\left[ 1 + T_2 \\, P_2(\\cos \\Theta ) \\right] \\:.",
"$ Thus, ${s_r(R_c) = R_c \\, S_2 \\, P_2(\\cos \\Theta )}$ represents the radial displacement at the core's boundary corresponding to the vertical tidal deformation of amplitude ${S_2 \\, P_2(\\cos \\Theta )}$ .",
"In 1909, Love defined the number $h_2$ , as the ratio between the amplitude of the vertical displacement at the surface of the planet and the equilibrium tidal height (disturbing potential divided by undisturbed surface gravity, both taken at the surface of the core) in the case of a fully-solid planet.",
"Solving the whole system of equations, he determined its expression as $h_2 \\stackrel{\\text{def}}{=} \\frac{s_r(R_c)}{U(R_c) / g_c} \\equiv \\frac{R_c \\, S_2}{\\zeta _c} = \\frac{5}{2} \\, \\frac{1}{1+\\bar{\\mu }} \\: ,$ where $\\bar{\\mu }$ is called the effective rigidity, in the sense that it evaluates the relative importance of elastic and gravitational forces: $\\bar{\\mu } = \\frac{19 \\mu }{2 \\rho _c g R_c} \\: .$ In presence of the fluid envelope, the ratio between the amplitude of the tidal surface vertical displacement and the tidal height will be modulated by a multiplicative factor $F$ , due to the additional loading exerted by the tidally-deformed fluid layer.",
"We may then introduce a new notation $h^F_2$ for the modified Love number in presence of a fluid envelope: $h^F_2 \\stackrel{\\text{def}}{=} \\frac{s_r(R_c)}{U(R_c)/g_c} \\equiv \\frac{R_c \\, S_2}{\\zeta _c} = F \\, h_2 = F \\times \\frac{5/2}{1+\\bar{\\mu }} \\: .$ We have now to express this factor in function of the parameters of the system.",
"To do so, we have to list all the forces acting on the surface of the core.",
"Before carrying out the study of these forces, let us introduce a specific notation.",
"All physical quantities $X(\\mathbf {r})$ will be separated in two terms: the first corresponds to the constant part that does not depend on where the quantity is calculated; the second one (called the \"effective deforming\" contribution and denoted $X^{\\prime }(\\mathbf {r})$ ) is a term proportional to the spherical surface harmonic $P_2$ (see REF )."
],
[
"Gravitational forces acting on the surface of the core",
"The planet is not only subjected to the direct action of the tidal potential $U$ , but also to the self-gravitational potential $\\Phi $ perturbed by the first.",
"In calculating the latter, we have to consider both contributions of the solid core and the fluid envelope, $\\Phi _c$ and $\\Phi _o$ respectively.",
"At any point $\\mathbf {r}$ of the core, $\\Phi _c (\\mathbf {r})$ corresponds to the internal potential created by the core : $\\Phi _c (\\mathbf {r}) = - \\frac{g_c}{R_c} \\left( \\frac{3 R_c^2 - r^2}{2} + \\frac{3}{5} \\, r^2 \\, S_2 \\, P_2 \\right) \\: .$ At the same point $\\mathbf {r}$ , $\\Phi _o (\\mathbf {r})$ is the internal potential created by the fluid shell of density $\\rho _o$ and of lower and upper surface boundaries $r_c$ and $r_p$ respectively: $\\Phi _o (\\mathbf {r}) = - \\frac{\\rho _o}{\\rho _c} \\, \\frac{g_c}{R_c} \\, \\left[\\left( \\frac{3 R_p^2 - r^2}{2} + \\frac{3}{5} \\, r^2 \\, T_2 \\, P_2 \\right) \\right.\\\\\\left.",
"- \\left( \\frac{3 R_c^2 - r^2}{2} + \\frac{3}{5} \\, r^2 \\, S_2 \\, P_2 \\right)\\right] \\: .$ Therefore ${V(\\mathbf {r}) = U(\\mathbf {r}) + \\Phi _c(\\mathbf {r}) + \\Phi _o(\\mathbf {r})}$ has the following expression, at any point $\\mathbf {r}$ inside the core: $V(\\mathbf {r}) = - \\frac{1}{2} \\, \\frac{g_c}{R_c} \\, \\left[ -r^2 + 3 R_c^2 \\left( 1 - \\frac{\\rho _o}{\\rho _c} + \\frac{\\rho _o}{\\rho _c} \\, \\frac{R_p^2}{R_c^2} \\right) \\right] - V^{\\prime }(\\mathbf {r}) \\: ,$ where the effective deforming potential is expressed by $V^{\\prime }(\\mathbf {r}) = - Z \\, r^2 \\, P_2 \\: ,$ $Z$ being a constant that depends on the characteristics of the planet: $Z = \\frac{g_c}{R_c} \\, \\left[ \\frac{\\zeta _c}{R_c} + \\frac{3}{5} \\, \\frac{\\rho _o}{\\rho _c} (T_2-S_2) + \\frac{3}{5} S_2 \\right] \\: .$ We then obtain its expression, correct to first order in $S_2 P_2$ or $T_2 P_2$ , at any point ${\\mathbf {r_c} = r_c \\, \\mathbf {e_r}}$ of the surface of the core: $V(\\mathbf {r_c}) = - g_c \\, R_c \\, \\left[ \\left( 1 - \\frac{3}{2} \\frac{\\rho _o}{\\rho _c} \\right) + \\frac{3}{2} \\frac{\\rho _o}{\\rho _c} \\, \\frac{R_p^2}{R_c^2} \\right] + V^{\\prime }(\\mathbf {r_c}) \\: ,$ where $V^{\\prime }(\\mathbf {r_c}) = - Z_c \\, R_c^2 \\, P_2 \\: ,$ $Z_c$ being a constant that depends on the characteristics of the planet: $Z_c = \\frac{g_c}{R_c} \\, \\left[ \\frac{\\zeta _c}{R_c} + \\frac{3}{5} \\, \\frac{\\rho _o}{\\rho _c} (T_2-S_2) - \\frac{2}{5} S_2 \\right] \\: .$ Chree (1896) showed that the deformation of the core's surface under the gravitational forces (that derive from the effective deforming potential $V^{\\prime }$ ) is the same as that which would result from the outward normal traction $f^{T_N}_1$ applied at its mean surface $r=R_c$ : $f^{T_N}_1(R_c) = \\rho _c \\, Z \\, R_c^2 \\, P_2 \\: .$"
],
[
"Total effective normal traction acting on the surface of the core",
"The mean surface of the core is subjected to two additional forces induced by both the loading of the core and the loading of the ocean, tidally deformed.",
"First, the pressure due to the differential overloading of the deformed elastoviscous matter on the mean surface of radius $R_c$ is given by the product of the local gravity $g_c$ , the density of the core $\\rho _c$ and the solid tidal height $R_c S_2$ : $f^{T_N}_2(R_c) = - \\rho _c \\, g_c \\, R_c \\, S_2 \\, P_2 \\: .$ We also have to take into account the oceanic hydrostatic pressure.",
"Following Zahn (1966) and Remus et al.",
"(2012), we express all scalar quantities ${X(r,\\Theta )}$ as the sum of their spherically symmetrical profile $X^0(r)$ and their perturbation ${X^{\\prime }(r,\\Theta )}$ due to the tidal potential ${U(r,\\Theta ) \\propto P_2(\\cos \\Theta )}$ : $X(r,\\Theta ) = X^0(r) + X^{\\prime }(r,\\Theta ) \\equiv X^0(r) + \\hat{X}(r) \\, P_2(\\cos \\Theta )\\:.$ The perturbations of pressure ${P^{\\prime }(r,\\theta )}$ obey the relation of the hydrostatic equilibrium which is of the following form, correct to first order in $P_2(\\cos \\Theta )$ : $\\nabla P^{\\prime } = \\rho _o \\, \\nabla V^{\\prime } + \\rho _o^{\\prime } \\, \\nabla V^0 \\:.$ Therefore, the $\\Theta $ -projection of (REF ) leads to $P^{\\prime }(r,\\Theta ) = \\rho _o \\, V^{\\prime }(r,\\Theta ) \\:.$ Finally, since only the variable part of the pressure, i.e.",
"$P^{\\prime }$ , contributes to the normal effective traction $f^{T_N}_3$ that acts on the mean surface of core, this latter takes the following expression: $f^{T_N}_3(R_c) = P^{\\prime }(\\mathbf {r_c}) = \\rho _o \\, V^{\\prime }(\\mathbf {r_c}) = - \\rho _o \\, Z_c \\, R_c^2 \\, P_2 \\: .$ The sum of these three forces, represented on Fig.",
"REF , corresponds to the total normal effective traction ${f^{T_N} (R_c) = f^{T_N}_1 (R_c) + f^{T_N}_2 (R_c) + f^{T_N}_3 (R_c)}$ that deforms the mean surface of the core.",
"Using Eqs.",
"(REF -REF -REF ), we get: $f^{T_N} (R_c) = \\left( \\rho _c \\, Z - \\rho _o \\, Z_c \\right) \\, R_c^2 \\, P_2 - \\rho _c \\, g_c \\, R_c \\, S_2 \\, P_2 \\: ,$ where the expressions of $Z$ and $Z_c$ are given by equations (REF ) and (REF ) respectively, so that $f^{T_N} (R_c) = X \\, P_2 (\\cos \\Theta ) \\: ,$ where we have denoted by $X$ the following quantity: $X = \\frac{2}{5} \\, \\rho _c \\, g_c \\, R_c \\left( 1 - \\frac{\\rho _o}{\\rho _c} \\right) \\, \\left[ \\frac{5}{2} \\frac{{\\zeta }_c}{R_c} - S_2 + \\frac{3}{2} \\frac{\\rho _o}{\\rho _c} \\left( T_2 - S_2 \\right) \\right] \\: .$ Figure: Balance of forces that act on the mean surface of the core r=R c r=R_c: f 1 T N (R c )f^{T_N}_1 (R_c) are the gravitational forces, f 2 T N (R c )f^{T_N}_2 (R_c) the loading of the solid tide and f 3 T N (R c )f^{T_N}_3 (R_c) the hydrostatic pressure."
],
[
"Amplitude of the vertical deformation",
"According to Melchior (1966), a deforming potential $\\mathcal {U}_2$ of second order produces a deformation at each point $\\mathbf {r_c}$ of the surface of the core which radial component takes the form ${\\epsilon }_{rr} = \\frac{\\left(8 R_c^2 - 3 r_c^2\\right)}{19 \\mu } \\frac{\\rho _c \\,\\mathcal {U}_2}{r_c^2} \\: .$ Correct to first order in $P_2$ , recalling that ${r_c = R_c \\left( 1 + S_2 P_2 \\right)}$ , this reduces to: ${\\epsilon }_{rr} = \\frac{5}{19 \\mu } \\, \\rho _c \\mathcal {U}_2 \\: ,$ where $\\rho _c \\mathcal {U}_2$ is the deforming traction ${f^{T,N}(R_c) = X P_2}$ applied on the core.",
"Furthermore, since the amplitude of the displacement is also given by ${{\\epsilon }_{rr} = S_2 P_2}$ (Eq.",
"REF ), we have the following equality $S_2 P_2 = \\frac{5}{19 \\mu } \\, X P_2 \\: .$ Therefore the relation (REF ) beetween $\\mu $ and $\\bar{\\mu }$ and the expression (REF ) of $X$ enable us to relate the deformation of the surfaces of the core ($S_2$ ) and of the ocean ($T_2$ ) as $S_2 = \\frac{1}{\\bar{\\mu }} \\, \\left( 1 - \\frac{\\rho _o}{\\rho _c} \\right) \\left[ \\frac{5}{2} \\frac{{\\zeta }_c}{R_c} - S_2 + \\frac{3}{2} \\frac{\\rho _o}{\\rho _c} (T_2 - S_2) \\right] \\: .$ By definition, given in Eq.",
"(REF ), the impedance $F$ is of the form $F = \\frac{2}{5} \\, (1+\\bar{\\mu }) \\, \\frac{R_c}{{\\zeta }_c} \\, S_2 \\: .$ Since the surface of the planet is an equipotential, the total potential $V$ takes a constant value at any point $\\mathbf {r_p}$ of the surface of the ocean: $V(\\mathbf {r_p}) \\equiv U(\\mathbf {r_p}) + \\Phi (\\mathbf {r_p}) = {cst} \\: .$ As $V - V^{\\prime } = {cst}$ by definition, we get the simplier condition: $V^{\\prime }(\\mathbf {r_p}) \\equiv U(\\mathbf {r_p}) + \\Phi ^{\\prime }(\\mathbf {r_p}) = {cst} \\: .$ At a point $\\mathbf {r}$ of the ocean, $\\Phi _c (\\mathbf {r})$ corresponds to the external potential created by the core: $\\Phi _c (\\mathbf {r}) = - g_c \\, R_c^2 \\left( \\frac{1}{r} + \\frac{3}{5} \\, \\frac{R_c^2 \\, S_2}{r^3} \\, P_2 \\right) \\: .$ At the same point $\\mathbf {r}$ , $\\Phi _o (\\mathbf {r})$ is the internal potential created by the fluid shell of density $\\rho _o$ and of lower and upper surface boundaries $r_c$ and $r_p$ respectively: $\\begin{split}\\Phi _o (\\mathbf {r}) =& - g_c \\, \\frac{R_p^3}{R_c} \\, \\frac{\\rho _o}{\\rho _c} \\, \\left( \\frac{3 R_p^2 - r^2}{2 R_p^3} + \\frac{3}{5} \\, \\frac{r^2 T_2}{R_p^3} \\, P_2 \\right) \\\\& + g_c \\, R_c^2 \\, \\frac{\\rho _o}{\\rho _c} \\, \\left( \\frac{1}{r} + \\frac{3}{5} \\, \\frac{R_c^2 \\, S_2}{r^3} \\, P_2 \\right)\\: .\\end{split}$ Therefore ${V(\\mathbf {r}) = U(\\mathbf {r}) + \\Phi _c(\\mathbf {r}) + \\Phi _o(\\mathbf {r})}$ has the following expression, at any point $\\mathbf {r}$ inside the ocean: $V(\\mathbf {r}) = - g_c \\, R_c^2 \\, \\left[ \\left( 1 - \\frac{\\rho _o}{\\rho _c} \\right) \\, \\frac{1}{r} - \\frac{\\rho _o}{\\rho _c} \\frac{r^2}{2 R_c^3} + \\frac{3}{2} \\frac{\\rho _o}{\\rho _c} \\frac{R_p^2}{R_c^3} \\right] + V^{\\prime }(\\mathbf {r}) \\: ,$ where the effective deforming potential is expressed by $V^{\\prime }(\\mathbf {r}) = - g_c \\, R_c^2 \\, W(r) \\, P_2 \\: ,$ $W$ being a function of the distance $r$ to the center of the planet: $W(r) = \\frac{{\\zeta }_c}{R_c} \\, \\frac{r^2}{R_c^3} + \\frac{3}{5} \\, \\frac{\\rho _o}{\\rho _c} \\, \\frac{r^2}{R_c^3} \\, T_2 + \\frac{3}{5} \\, \\left( 1 - \\frac{\\rho _o}{\\rho _c} \\right) \\, \\frac{R_c^2}{r^3} \\, S_2 \\: .$ We then obtain its expression at a point ${\\mathbf {r_p} = r_p \\, \\mathbf {e_r}}$ of the surface of the planet: $V(\\mathbf {r_p}) = - g_c \\left[ \\left( 1 - \\frac{\\rho _o}{\\rho _c} \\right) \\, \\frac{R_c^2}{R_p} + \\frac{\\rho _o}{\\rho _c} \\frac{R_p^2}{R_c} \\right] + V^{\\prime }(\\mathbf {r_p})\\: ,$ where $V^{\\prime }(\\mathbf {r_p}) = - g_c \\, W_p \\, P_2 \\: ,$ $W_p$ being a constant that depends on the planet characteristics: $W_p = \\frac{R_p^2}{R_c^2} \\, {\\zeta }_c+ \\left[ \\frac{3}{5} \\, \\left( 1 - \\frac{\\rho _o}{\\rho _c} \\right) \\, \\frac{R_c^4}{R_p^3} \\right] \\, S_2 \\\\+ \\left[ - \\frac{2}{5} \\, \\frac{\\rho _o}{\\rho _c} \\, \\frac{R_p^2}{R_c} - \\left( 1 - \\frac{\\rho _o}{\\rho _c} \\right) \\, \\frac{R_c^2}{R_p} \\right] \\, T_2\\: .$ The condition (REF ) takes then the form: $\\frac{{\\zeta }_c}{R_c} =- \\frac{3}{5} \\, \\left( \\frac{R_c}{R_p} \\right)^5 \\, \\left( 1 - \\frac{\\rho _o}{\\rho _c} \\right) \\, S_2+ \\frac{2}{5} \\, \\frac{\\rho _o}{\\rho _c} \\alpha \\, T_2\\, ,$ where $\\alpha = 1 + \\frac{5}{2} \\, \\frac{\\rho _c}{\\rho _o} \\, \\left( \\frac{R_c}{R_p} \\right)^3 \\, \\left( 1 - \\frac{\\rho _o}{\\rho _c} \\right) \\: .$ We may eliminate the variable $T_2$ thanks to Eq.",
"(REF ): $\\frac{2}{5} \\, \\frac{\\rho _o}{\\rho _c} \\alpha \\, T_2 =\\frac{ \\frac{2}{5} \\, \\alpha }{ \\left( \\alpha + \\frac{3}{2} \\right) \\left( 1 - \\frac{\\rho _o}{\\rho _c} \\right) } \\\\\\times \\left[ 1 + \\bar{\\mu } - \\frac{\\rho _o}{\\rho _c}+ \\frac{3}{2} \\frac{\\rho _o}{\\rho _c} \\left( 1 - \\frac{\\rho _o}{\\rho _c} \\right)+ \\frac{3}{2} \\left( \\frac{R_c}{R_p} \\right)^5 \\left( 1 - \\frac{\\rho _o}{\\rho _c} \\right)^2\\right] \\: .$ Injecting this relation into the expression (REF ) of ${\\zeta }_c / R_c$ , and the resulting relation in the expression (REF ) of $F$ , we finally get: $F = \\frac{\\left( 1 - \\frac{\\rho _o}{\\rho _c} \\right) \\, (1 + \\bar{\\mu }) \\, \\left( 1 + \\frac{3}{2 \\alpha } \\right)}{1 + \\bar{\\mu } - \\frac{\\rho _o}{\\rho _c} + \\frac{3}{2} \\frac{\\rho _o}{\\rho _c} \\left( 1 - \\frac{\\rho _o}{\\rho _c} \\right) - \\frac{9}{4 \\alpha } \\left( \\frac{R_c}{R_p} \\right)^5 \\left( 1 - \\frac{\\rho _o}{\\rho _c} \\right)^2} \\: .$ In the case of a shallow oceanic envelope ($R_p \\simeq R_c$ ), the height of the oceanic tide is then given by $R_c (T_2 - S_2)$ at the surface of the core.",
"Using Eqs.",
"(REF -REF ), we obtain the classical expression of the height of oceanic tide $R_c (T_2 - S_2) = \\frac{ \\bar{\\mu } {\\zeta }_c }{ 1- \\frac{\\rho _o}{\\rho _c} + \\bar{\\mu } \\left( 1 - \\frac{3}{5} \\frac{\\rho _o}{\\rho _c} \\right) } \\:.$ The height of the solid tidal displacement is given by $R_c S_2$ .",
"Using Eqs.",
"(REF -REF -REF ), we obtain its classical expression: $R_c S_2 = \\frac{ \\frac{5}{2} {\\zeta }_c \\left( 1 - \\frac{\\rho _o}{\\rho _c} \\right) }{ 1- \\frac{\\rho _o}{\\rho _c} + \\bar{\\mu } \\left( 1 - \\frac{3}{5} \\frac{\\rho _o}{\\rho _c} \\right) } \\, ,$ which reduces to $R_c S_2 = \\frac{ \\frac{5}{2} {\\zeta }_p }{1 + \\bar{\\mu }}$ for an ocean-free planet ($\\rho _o = 0$ ), which corresponds to that given by Lord Kelvin (1863).",
"Thus, recalling Eq.",
"(REF ), we deduce that for an oceanless planet, $F$ is unity.",
"Fig.",
"REF displays the value of $F$ for three types of planets (i.e.",
"Earth-, Jupiter- and Saturn-like planets), with a given core (of fixed size, mass and shear modulus) and a fluid shell of fixed density but variable depth, so that the size and mass of the whole planet varies also.",
"The variation of $F$ is represented in function of $R_c/R_p$ : the smaller this ratio, the higher the ocean depth.",
"Table: Earth parameters.Table: Mass and mean radius of Jupiter and Saturn.For the Earth, all parameters are well known (see Table REF ).",
"Table: Mass and mean radius of Jupiter's and Saturn's cores.The values of the ocean density for the Jupiter- and Saturn-like planets correspond to the ones we may deduce from the well known values of their global size and mass (see Table REF ), and the much less constrained values of the core size and mass (see models A of Table REF ).",
"Concerning the shear modulus, we used the value taken as reference by Henning et al.",
"(2009) when studing the tidal heating of terrestrial exoplanets, i.e.",
"${\\mu = 5 \\times 10^{10} \\, \\mathrm {Pa}}$ .",
"These models of planets are used as starting points to compare the influence the ocean depth has on core deformation for different types of planets.",
"Since we do not try to estimate this deformation for realistic planets, we will not discuss in this section the validity of the values we use for the parameters.",
"Figure: Factor FF, accounting for the overloading exerted by the tidally deformed oceanic envelope on the solid core, in terms of the ocean depth throw the ratio R c /R p R_c/R_p for three types of planet.Parameters are given in Tables --: for Earth-, Jupiter- and Saturn-like planets, we assume respectively that: R p ={1,10.97,9.14}{R_p = \\lbrace 1, 10.97 , 9.14\\rbrace } (in units of R p Φ R_p^{\\mathcal {\\Phi }}), M p ={1,317.8,95.16}{M_p = \\lbrace 1, 317.8, 95.16\\rbrace } (in units of M p Φ M_p^{\\mathcal {\\Phi }}), R c ={0.99,0.126,0.219}×R p {R_c = \\lbrace 0.99, 0.126 , 0.219\\rbrace \\times R_p}, M c ={0.33,6.41,18.65}{M_c = \\lbrace 0.33, 6.41, 18.65\\rbrace } (in units of M p Φ M_p^{\\mathcal {\\Phi }}) and μ=5×10 10 ( Pa ){\\mu = 5 \\times 10^{10} \\:(\\textrm {Pa})} for all cores.Note that this figure is similar to that drawned by Dermott (1979), divergences coming from the use of different values of the parameters.Fig.",
"REF shows that for a planet with a shallow fluid shell (i.e.",
"when ${R_c \\gtrsim a \\, \\times \\, R_p}$ , where ${ a = \\lbrace 0.840 , 0.915 , 0.937 \\rbrace }$ for an Earth-, Jupiter- and Saturn-like planet respectively), $F$ is less than unity, which means that the ocean exerts a loading effect on the solid core which is stronger than the gravitational forces and opposed to it.",
"That is the case of the Earth where the depth of the oceanic envelope does not exceeds $1\\%$ of the size of the planet, but giant planets are supposed to have a solid core no bigger than the third of the planet size.",
"According to Fig.",
"REF , $F$ may reach values up to $2.3$ for this kind of planets.",
"That means that for a planet with a deep fluid envelope, the ocean tide has no loading effect on the core but exerts a gravitational force that amplifies the tidal deformation.",
"We refer the reader to Dermott (1979) for a complete discussion."
],
[
"Modified Love numbers",
"From Eq.",
"(REF ) we deduce the Love number $h^F_2$ (cf.",
"Eq.",
"REF ) which measures the surface deformation: $h^F_2 = \\frac{ \\frac{5}{2} \\,\\left( 1 - \\frac{\\rho _o}{\\rho _c} \\right) \\, \\left( 1 + \\frac{3}{2 \\alpha } \\right)}{1 + \\bar{\\mu } - \\frac{\\rho _o}{\\rho _c} + \\frac{3}{2} \\frac{\\rho _o}{\\rho _c} \\left( 1 - \\frac{\\rho _o}{\\rho _c} \\right) - \\frac{9}{4 \\alpha } \\left( \\frac{R_c}{R_p} \\right)^5 \\left( 1 - \\frac{\\rho _o}{\\rho _c} \\right)^2} \\: .$ Let us give here the expression of the second-order Love number (REF ) associated with the solid core.",
"First of all, let us recall its value for an ocean-free planet: According to Eq.",
"(REF ) with $r=R_c$ , we get $k_2 = \\frac{3}{5} \\, h_2 = \\frac{3}{2} \\, \\frac{1}{1+\\bar{\\mu }} \\: .$ In presence of an ocean on top of the solid core, we may also introduce the modified Love number $k^F_2 \\stackrel{\\text{def}}{=} \\frac{\\Phi ^{\\prime }(R_c)}{U(R_c)} = \\frac{V^{\\prime }(R_c)-U(R_c)}{U(R_c)} = \\frac{V^{\\prime }(R_c)}{U(R_c)} - 1 \\: ,$ where $V^{\\prime }(R_c)$ and $U(R_c)$ are obtained from Eqs.",
"(REF ) and (REF ) respectively, with $r=R_c$ .",
"Thus, expressing ${\\zeta _c}/{R_c}$ in function of the modified second-order Love number $h_2^F$ according to Eq.",
"(REF ), and using Eq.",
"(REF ), we obtain the expression of $k^F_2$ in terms of $h_2^F$ : $k^F_2 = \\left( 1 + \\frac{2}{5} \\, \\frac{ \\bar{\\mu } }{ 1-\\frac{\\rho _o}{\\rho _c} } \\right) \\, h^F_2 - 1 \\:.$ In this section we have studied the impact of the presence of a fluid envelope in the determination of the deformation imposed on an elastic core under tidal forcing.",
"In the following, we consider that the solid core also presents viscous properties so that its response to the tidal force exerced by the perturber is no more immediate, thus inducing dissipation.",
"The next section addresses the quantification of this conversion of energy, which drives the dynamical evolution of the whole system."
],
[
"Anelastic tidal dissipation: analytical results",
"Assuming that the anelasticity is linear, the correspondence principle established by Biot (1954) allows us to extend the formulation of the adiabatic elastic problem to the resolution of the equivalent dissipative anelastic problem.",
"For initial conditions taken as zero and similar geometries, the Laplace and Fourier transforms of the anelastic equations and boundary conditions are identical to the elastic equations, if the rheological parameters and radial functions are defined as complex numbers.",
"We will then denote $ {\\stackrel{\\approx }{\\sigma } } \\equiv {\\bar{ \\bar{\\sigma } }} _1 + i \\, {\\bar{ \\bar{\\sigma } }} _2$ the complex stress tensor, and $ {\\stackrel{\\approx }{\\epsilon } } \\equiv {\\bar{ \\bar{\\epsilon } }} _1 + i \\, {\\bar{ \\bar{\\epsilon } }} _2$ the complex strain tensor.",
"The perturbative strain is cyclic, with tidal pulsations $\\sigma _{2,m,p,q}$ .",
"For sake of clarity, we will use from now on the generic notation ${\\omega \\equiv \\sigma _{2,m,p,q}}$ , recalling that there is a large range of tidal frequencies for each term of the expansion of the tidal potential.",
"The stress and strain tensors take the following form: $ {\\stackrel{\\approx }{\\sigma } }(\\omega ) &= ( {\\bar{ \\bar{\\sigma } }} _1 + i {\\bar{ \\bar{\\sigma } }} _2) \\, e ^{i \\omega t} \\: ,\\\\ {\\stackrel{\\approx }{\\epsilon } }(\\omega ) &= {\\stackrel{\\approx }{\\epsilon } }_0 \\, e ^{i \\omega t} \\: .$ The complex rigidity $\\tilde{\\mu }(\\omega ) \\equiv {\\mu }_1(\\omega ) + i \\, {\\mu }_2(\\omega ) \\:,$ where ${\\mu }_1$ represents the energy storage and ${\\mu }_2$ the energy loss of the system, is defined by $\\tilde{\\mu }(\\omega ) \\equiv \\frac{ {\\stackrel{\\approx }{\\sigma } }(\\omega )}{ {\\stackrel{\\approx }{\\epsilon } }(\\omega )} \\: .$ We may also define the complex effective rigidity $\\hat{\\mu }(\\omega ) \\equiv \\bar{\\mu }_1(\\omega ) + i \\, \\bar{\\mu }_2(\\omega )$ by: $\\hat{\\mu }(\\omega ) = \\gamma \\, \\tilde{\\mu }(\\omega ) \\:,$ where (see Eq.",
"REF ) $\\gamma = \\frac{\\hat{\\mu }}{\\tilde{\\mu }} \\equiv \\frac{\\bar{\\mu }}{\\mu } = \\frac{19}{2 \\rho _c g_c R_c} \\:.$"
],
[
"Case of a fully-solid planet",
"The complex Love number $\\tilde{k}_2$ may be expressed in terms of the complex effective rigidity $\\hat{\\mu }$ , by: $\\begin{split}\\tilde{k}_2(\\omega )&= \\frac{3}{2} \\, \\frac{1}{1+ \\hat{\\mu }(\\omega )} \\\\&= \\frac{3}{2} \\, \\frac{1}{1+ \\gamma \\left[ \\mu _1(\\omega ) + i \\mu _2(\\omega ) \\right] } \\: ,\\end{split}$ in the case of a completely solid planet.",
"The real part of $\\tilde{k}_2$ characterizes the purely elastic deformation, since $- \\mathcal {I}\\!m \\ (\\tilde{k}_2)$ gives the phase lag due to the viscosity.",
"Therefore, we define the factor of tidal dissipation $Q$ , that represents the dissipation rate due to viscous friction, by $Q^{-1}(\\omega ) = - \\frac{\\mathcal {I}\\!m \\ {\\tilde{k}_2}(\\omega )}{\\left| \\tilde{k}_2(\\omega ) \\right|} \\: .$ Then, from equation (REF ), we deduce that $Q(\\omega ) = \\sqrt{ 1 + \\left[ \\frac{1}{\\bar{\\mu }_2(\\omega )} + \\frac{\\bar{\\mu }_1(\\omega )}{\\bar{\\mu }_2(\\omega )} \\right]^2 } \\: .$"
],
[
"Case of a two-layer planet",
"Let us introduce the quantities $A &= \\left( 1 - \\frac{\\rho _o}{\\rho _c} \\right) \\, \\left( 1 + \\frac{3}{2 \\alpha } \\right) \\:,\\:\\text{ and: } \\\\B &= 1 - \\frac{\\rho _o}{\\rho _c} + \\frac{3}{2} \\frac{\\rho _o}{\\rho _c} \\left( 1 - \\frac{\\rho _o}{\\rho _c} \\right) - \\frac{9}{4 \\alpha } \\left( \\frac{R_c}{R_p} \\right)^5 \\left( 1 - \\frac{\\rho _o}{\\rho _c} \\right)^2 \\:, $ in the expression (REF ) of F: $F = \\frac{ A (1+\\bar{\\mu }) }{B+\\bar{\\mu }} \\: ,$ and let us define its complex equivalent $\\widetilde{F} = \\frac{ A (1+\\hat{\\mu }) }{B+\\hat{\\mu }} \\:.$ The determination of how the presence of an oceanic envelope will modify the tidal dissipation consists in the determination of $\\tilde{k}^F_2$ , defined as the complex Love number $\\tilde{k}_2$ in presence of the fluid envelope.",
"According to the correspondence principle, this number is given by the complex Fourier transform of (REF ), i.e.",
": $\\tilde{k}^F_2(\\omega ) = \\left( 1 + \\frac{2}{5} \\, \\frac{ \\hat{\\mu }(\\omega ) }{ 1-\\frac{\\rho _o}{\\rho _c} } \\right) \\, \\tilde{h}^F_2(\\omega ) - 1 \\:.$ From (REF ) we get then $\\tilde{k}^F_2(\\omega ) = \\frac{1}{ \\left(B+\\bar{\\mu }_1 \\right)^2 + \\bar{\\mu }_2^2 } \\\\\\times \\left\\lbrace \\left[ \\left(B+\\bar{\\mu }_1 \\right) \\, \\left( C+\\frac{3}{2\\alpha } \\, \\bar{\\mu }_1 \\right) + \\frac{3}{2\\alpha } \\, \\bar{\\mu }_2^2 \\right]- i {A \\, D \\, \\bar{\\mu }_2} \\right\\rbrace \\:,$ where we made use of the dimensionless quantities $\\alpha $ , $A$ and $B$ previously defined (see respectively Eqs.",
"REF and REF ), $C$ and $D$ given by: $C &= \\frac{3}{2} \\, \\left( 1 - \\frac{\\rho _o}{\\rho _c} \\right) \\, \\left( 1 - \\frac{\\rho _o}{\\rho _c} + \\frac{5}{2\\alpha } \\right)+ \\frac{9}{4\\alpha } \\left(\\frac{R_c}{R_p}\\right)^5 \\, \\left( 1 - \\frac{\\rho _o}{\\rho _c} \\right)^2 \\:, \\\\D &= \\frac{3}{2} \\, \\left( 1 - \\frac{\\rho _o}{\\rho _c} \\right) \\, \\left[ 1 + \\frac{3}{2\\alpha } \\, \\left(\\frac{R_c}{R_p}\\right)^5 \\right] \\:.$ Finally, the dissipation factor $\\hat{Q}$ , defined here by $\\hat{Q}^{-1}(\\omega ) = - \\frac{\\mathcal {I}\\!m \\ {\\tilde{k}^F_2(\\omega )}}{ \\left| \\tilde{k}^F_2(\\omega ) \\right|} \\: ,$ is of the form: $\\hat{Q}(\\omega ) =\\sqrt{ 1 + \\frac{9}{4 \\alpha ^2 A^2 D^2} \\,\\left[ 1 + \\frac{\\left( B + \\bar{\\mu }_1(\\omega ) \\right) \\, \\left( \\frac{2 \\alpha C}{3} + \\bar{\\mu }_1(\\omega ) \\right) }{ \\bar{\\mu }_2(\\omega ) } \\right]^2 } \\: .$ Thanks to the correspondence principle, one is able to derive this general expression of the tidal dissipation valid for any rheology.",
"The obtained formulae explicitly reveal its dependence on the tidal frequency ${\\omega \\equiv \\sigma _{2,m,p,q}}$ , as shown, for example, by Remus et al.",
"(2012), Ogilvie & Lin (2004-2007) for fluid layers."
],
[
"Implementation of an anelastic model",
"Since the factor $Q$ depends on the real and imaginary components $\\bar{\\mu }_1$ and $\\bar{\\mu }_2$ of the complex effective shear modulus $\\hat{\\mu }$ , we need to define the rheology of the studied body to express it in terms of the constitutive parameters of the material.",
"The anelasticity of a material is evaluated by a quality factor $Q_a$ defined by $Q_a(\\omega ) = \\frac{\\mu _1(\\omega )}{\\mu _2(\\omega )} \\: .$ We may express the solid tidal dissipation, given by $Q$ (Eq.",
"REF ) for a fully-solid planet and $\\hat{Q}$ (Eq.",
"REF ) in the case of a two-layer planet, in terms of this factor: $Q(\\omega \\equiv \\sigma _{2,m,p,q}) = \\sqrt{ 1 + \\left[ \\frac{1}{\\bar{\\mu }_2(\\omega )} + Q_a(\\omega ) \\right]^2 } \\:,$ and $\\hat{Q}(\\omega \\equiv \\sigma _{2,m,p,q}) = \\\\\\sqrt{ 1 + \\frac{9}{4 \\alpha ^2 A^2 D^2}\\left[ 1 + \\left( \\frac{B}{\\bar{\\mu }_2(\\omega )} + Q_a(\\omega ) \\right)\\, \\left( \\frac{2}{3} \\frac{ \\alpha C}{\\bar{\\mu }_2(\\omega )} + Q_a(\\omega ) \\right) \\right]^2 } \\: .$ All previous results are independent of the viscoelastic rheological model.",
"We have now to apply these general expressions to a specific rheology which will depend on the physical properties of the material.",
"Considering our lack of knowledge on the internal structure of giant planets, we will implement the simplest model, namely the Maxwell model, which presents the advantage to involve only two parameters and thus to be easy to use (Tobie 2003, and Tobie et al.",
"2005).",
"A critical overview of the four main rheological models has also been done by Henning et al.",
"(2009), thus we refer the reader to the three above mentioned papers for a detailed comparison."
],
[
"The Maxwell model. ",
"This model considers a viscoelastic material as a spring-dashpot serie.",
"The instantaneous elastic response is characterized by a shear modulus $G$ , and the viscous yielding is represented by a viscous scalar modulus $\\eta $ (see Fig.",
"REF ).",
"Notice that the shear moduli $G$ and $\\mu $ (introduced in Sect.",
"REF ) designate the same quantity.",
"We change here the notation to avoid any confusion with the complex shear modulus $\\tilde{\\mu }$ used to study the anelastic tidal dissipation, and whose real and imaginary parts involve both $G$ and $\\eta $ .",
"Figure: Representation of the Maxwell model and its corresponding notations.The constitutive equation is given by Henning et al.",
"(2009): $G \\, {\\stackrel{\\approx }{\\sigma } }(\\omega ) + \\eta \\, \\dot{ {\\stackrel{\\approx }{\\sigma } }}(\\omega )= G \\, \\eta \\, \\dot{ {\\stackrel{\\approx }{\\epsilon } }}(\\omega ) \\: ,$ where the time derivative of a given quantity is denoted by a dot.",
"Recalling the relation (REF ) this equation becomes: $G \\, {\\stackrel{\\approx }{\\epsilon } }(\\omega ) \\, \\tilde{\\mu }(\\omega )+ \\eta \\, \\frac{\\mathrm {d}}{\\mathrm {d}t} \\left[ \\tilde{\\mu }(\\omega ) \\, {\\stackrel{\\approx }{\\epsilon } }(\\omega ) \\right]= G \\, \\eta \\, \\dot{ {\\stackrel{\\approx }{\\epsilon } }}(\\omega ) \\: .$ Therefore the real part $\\mu _1$ and the imaginary part $\\mu _2$ of the complex shear modulus $\\tilde{\\mu }$ have the following expressions: $& \\mu _1(\\omega ) = \\frac{{\\eta }^2 \\, G \\, \\omega ^2}{G^2 + {\\eta }^2 \\, \\omega ^2} \\:, \\\\& \\mu _2(\\omega ) = \\frac{\\eta \\, G^2 \\, \\omega }{G^2 + {\\eta }^2 \\, \\omega ^2} \\: .",
"$ The anelastic quality factor $Q_a$ is then given by $Q_a (\\omega ) = \\frac{\\mu _1(\\omega )}{\\mu _2(\\omega )} = \\frac{\\eta \\, \\omega }{G} \\equiv \\omega \\, \\tau _M \\:,$ where $\\tau _M = \\eta / G $ is the characteristic time of relaxation of the Maxwell model.",
"As confirmed by Fig.",
"REF , Eq.",
"(REF ) shows that $Q_a$ increases linearly with the frequency of the cyclic tidal strain: the shorter the oscillation period, the lower the dissipation due to intrinsic viscoelastic properties of the material.",
"Moreover, the anelastic quality factor is proportional to $\\tau _M = \\eta / G$ , so that it dissipates more if it is more rigid and less viscous.",
"Figure: Anelastic quality factor Q a Q_a of the Maxwell model in function of the tidal pulsation ω\\omega for different values of the viscosity η\\eta .",
"G is taken equal to 5×10 10 Pa {5 \\times 10^{10} ~\\mathrm {Pa}} (see Henning et.",
"al 2009).",
"Q a Q_a is represented on a logarithmic scale.Thus, we may express $\\mu _2$ (Eq. )",
"in terms of the anelastic quality factor $Q_a$ (Eq.",
"REF ): $\\mu _2(\\omega ) = \\frac{G}{\\left( \\omega \\, \\tau _M \\right)^{-1} + \\omega \\, \\tau _M} \\:.$ In the case of a fully-solid body, we get, from Eqs.",
"(REF ) and (REF ), the imaginary part of the complex Love number $\\tilde{k}_2$ (REF ): $\\mathcal {I}\\!m \\ \\left[ \\tilde{k}_2(\\omega ) \\right]= - \\frac{3 \\gamma }{2} \\, \\frac{G \\, \\omega \\, \\tau _M}{1 + \\left({\\omega \\, \\tau _M} \\right)^2 \\, ( 1 + \\gamma G )^2 } \\:.$ Therefore the dissipation factor $Q$ defined by (REF ) is of the form: $Q(\\omega \\equiv \\sigma _{2,m,p,q}) = \\\\\\sqrt{ 1 + \\left\\lbrace \\frac{1}{G \\gamma } \\, \\left[ \\left( \\omega \\, \\tau _M \\right)^{-1} + \\omega \\, \\tau _M \\right] + \\omega \\, \\tau _M \\right\\rbrace ^2 } \\:.$ In the more general case of a two-layer body, the imaginary part of the complex Love number $\\tilde{k}^F_2$ given by Eq.",
"(REF ), takes a different form than Eq.",
"(REF ) because of the presence of the fluid envelope, so does the two-layer dissipation factor $\\hat{Q}$ (REF ) with respect to its oceanless form (REF ).",
"To obtain them, one needs to replace the shear modulus $\\tilde{\\mu }$ and the anelastic quality factor $Q_a$ by their expression in the case of the Maxwell model (Eqs REF and REF respectively).",
"Fig.",
"REF compares the dissipation of the solid core with and without the presence of a fluid envelope of variable depth for a Saturn-like planet, using the parameters given by Tables REF -REF -REF : as expected, the difference between the two decreases with the size of the fluid envelope down to about ${0.34 \\times R_p}$ ; but for a thiner fluid shell, the dissipation get lower than it would be without it.",
"Figure: Relative difference between k ˜ 2 F /Q ^{{\\left|\\tilde{k}^F_2\\right|}/{\\hat{Q}}} and k ˜ 2 /Q{{\\left|\\tilde{k}_2\\right|}/{Q}}.We may distinguish two regimes: for 0<R c <0.661R p 0<R_c<0.661 \\, R_p the presence of the fluid envelope increases the tidal dissipation; over this value, tidal dissipation is lower than it would have been without fluid shell.Parameters are given in Tables -- for a Saturn-like planet perturbed at the tidal frequency of Enceladus (ω=2.25×10 -4 rad ·s -1 {\\omega = 2.25 \\times 10^{-4} \\, \\mathrm {rad}\\cdot \\mathrm {s}^{-1}}): R p =9.14R p Φ {R_p = 9.14 \\, R_p^{\\mathcal {\\Phi }}}, M p =95.16M p Φ {M_p = 95.16 \\, M_p^{\\mathcal {\\Phi }}}, R c =0.219R p Φ {R_c = 0.219 \\, R_p^{\\mathcal {\\Phi }}}, M c =18.65M p Φ {M_c = 18.65 \\, M_p^{\\mathcal {\\Phi }}}, G=5×10 10 ( Pa ){G = 5 \\times 10^{10} \\:(\\textrm {Pa})} and η=10 15 ( Pa ·s){\\eta = 10^{15}} \\, (\\mathrm {Pa}\\cdot \\mathrm {s}).Table: Tidal frequencies considered in numerical applications."
],
[
"Anelastic tidal dissipation: role of the structural and rheological parameters",
"With our choice of the Maxwell model to represent the rheology of the solid parts of the planet, the dissipation quality factor $\\hat{Q}$ depends on the tidal frequency $\\omega $ and on four structural and rheological parameters: the relative size of the core ($R_c / R_p$ ), the relative density of the envelope with respect to the core ($\\rho _o / \\rho _c$ ), the shear modulus ($G$ ) and the viscosity of the core ($\\eta $ ).",
"Our present knowledge of extrasolar giant planets, and also planets of our Solar System like Jupiter or Saturn, suffers some uncertainties on the values of these parameters, so that the range of values taken by core properties of giant planets presents poor constraints.",
"Moreover, even if the presence of a core in Jupiter is not yet confirmed (see Guillot 1999-2005), but new data coming from seismology may provide more constraints on giant planets internal structure (see Gaulme et al.",
"2011).",
"Nevertheless, we will explore the resulting tidal dissipation of such bodies around values of the structural and rheological parameters taken as reference and corresponding to those of the literature."
],
[
"Baseline structural and rheological parameters",
"As reference models, we chose Jupiter and Saturn although their core parameters are still uncertain.",
"The values of the global sizes and masses of these planets are those of Table REF .",
"Presently, two main types of models are available for Jupiter's interior.",
"The NHKFRB groupNettelmann, Holst, Kietzmann, French, Redmer and Blaschke (Nettelmann et al.",
"2008).",
"uses a three-layer model with a thin radiative zone, close to previous models by Saumon & Guillot (2004), whereas the MHVTB groupMilitzer, Hubbard, Vorberger, Tamblyn, and Bonev (Militzer et al.",
"2008).",
"proposes a new type of Jupiter model that possesses only two layers (see Militzer et al.",
"2008).",
"But, as explained in Militzer & Hubbard (2009), the crucial difference lies in the treatment of the molecular-to-metallic transition in dense fluid hydrogen, that leads to very different conclusions.",
"The first group predicts a small core of less than 10 $M_p^\\mathcal {\\Phi }$ (Saumon & Guillot 2004), while the second obtains a larger core of 14-18 $M_p^\\mathcal {\\Phi }$ (Militzer et al.",
"2008).",
"Among all these models of Jupiter's interior, we choose as reference the adiabatic model with Plasma Phase Transition (PPT) (http://www.oca.eu/guillot/jupsat.html)It is constructed with CEPAM, Code d'Evolution Planétaire Adaptatif et Modulaire (Guillot & Morel, 1995).",
"of Guillot (1998), which is of the first type.",
"It predicts a core of radius ${R_c = 0.126 \\times R_p}$ and mass ${M_c = 6.41 \\times M_p^\\mathcal {\\Phi }}$ .",
"Only the mass of the core of this reference model will be used in what follows.",
"The core radius will serve just as a first approximation, as a starting point in our study, since we will present our results for several core sizes.",
"Figure: Dissipation quality factor Q ^ eff \\hat{Q}_\\mathrm {eff} of the Maxwell model in function of the viscoelastic parameters GG and η\\eta .Top: for a Jupiter-like two-layer planet tidally perturbed at the Io's frequency ω≃2.79×10 -4 rad ·s -1 {\\omega \\simeq 2.79 \\times 10^{-4} ~\\mathrm {rad \\cdot s^{-1}}}.Bottom: for a Saturn-like two-layer planet tidally perturbed at the Enceladus' frequency ω≃2.25×10 -4 rad ·s -1 {\\omega \\simeq 2.25 \\times 10^{-4} ~\\mathrm {rad \\cdot s^{-1}}}.The red dashed line corresponds to the value of Q ^ eff ={(3.56±0.56)×10 4 ,(1.682±0.540)×10 3 }\\hat{Q}_\\mathrm {eff}=\\lbrace (3.56 \\pm 0.56) \\times 10^4,(1.682 \\pm 0.540) \\times 10^3\\rbrace (for Jupiter and Saturn respectively) determined by Lainey et al.",
"(2009-2012).The blue lines corresponds to the lower and upper limits of the reference values taken by the viscoelastic parameters GG and η\\eta for an unknown mix of ice and silicates.We assume the values of R p ={10.97,9.14}{R_p = \\lbrace 10.97 , 9.14\\rbrace } (in units of R p Φ R_p^{\\mathcal {\\Phi }}), M p ={317.8,95.16}{M_p = \\lbrace 317.8, 95.16\\rbrace } (in units of M p Φ M_p^{\\mathcal {\\Phi }}), M c ={6.41,18.65}×M p Φ {M_c = \\lbrace 6.41, 18.65\\rbrace \\times M_p^{\\mathcal {\\Phi }}} given in Tables -.There are also different models of Saturn's interior.",
"According to the model of Guillot with PPT (http://www.oca.eu/guillot/jupsat.html)[3], Saturn's core may have a mass of ${M_c = 6.55 \\times M_p^\\mathcal {\\Phi }}$ and a size of ${R_c = 0.174 \\times R_p}$ .",
"More recently, Hubbard et al.",
"(2009) infer, from Cassini-Huygens data, that Saturn has a larger core in the range $M_c = \\text{15-20} \\times M_p^{\\mathcal {\\Phi }}$ and a corresponding radius of more than $20\\%$ of the planet size.",
"We adopt this latter as reference model of Saturn, with ${M_c = 18.65 \\times M_p^{\\mathcal {\\Phi }}}$ and ${R_c = 0.219 \\times R_p}$ .",
"All these models support core accretion as the standard process for the formation of giant planets.",
"The corresponding parameters are listed in Table REF ."
],
[
"Rheological parameters of the core",
"The main uncertainties concern the viscoelastic properties of the core, namely its shear modulus $G$ and its viscosity $\\eta $ .",
"At high pressure and temperature, theoretical models and experiments show that $G$ and $\\eta $ values depend on temperature and pressure.",
"But no experiments are available at the very-high pressures and temperatures we may expect in Jupiter's and Saturn's cores (Guillot 2005).",
"Nevertheless, geophysical and experimental data have allowed to constrain the rheology of the icy satellites of Jupiter, since their ranges of pressure and temperature are similar to those of the outer mantle of the Earth (Tobie 2003).",
"Then, keeping in mind that these values may differ by several orders of magnitude in our case, we will adopt reference values based on these data, assuming that Jupiter's and Saturn's cores are made of ice and rock.",
"We will then explore, in the following Sect.",
"REF , the variation of the tidal dissipation for a large range of values of the rheological parameters around those taken as reference.",
"We thus assume that the shear modulus $G$ is in the range ${ \\left[ G_{ice} = 4\\times 10^9 \\,(\\mathrm {Pa}) \\,,\\, G_{silicate} = 10^{11} \\,(\\mathrm {Pa}) \\right]}$ (Henning et al.",
"2009).",
"Concerning the viscosity $\\eta $ , it takes its values in the range ${\\left[\\eta _{ice}=10^{14}\\mathrm {Pa}\\cdot \\mathrm {s},\\eta _{silicate}=10^{21}\\,\\mathrm {Pa}\\cdot \\mathrm {s}\\right]}$ for the icy satellites of Jupiter at high pressure (Tobie 2003).",
"We expand this range, reducing its lower boundary by two orders of magnitude, following the discussion of Karato (2011) which seems to indicate that, at the very high pressures, viscosity in deep interior of super-Earths may decrease by two or three orders of magnitude.",
"We refer the reader to the Karato's paper for an overview of all plausible mechanisms that may cause a change in the viscous-pressure relationship at very-high pressures.",
"Figure: Dissipation quality factor Q ^ eff {\\hat{Q}_\\mathrm {eff}} normalized to the size of the planet for Jupiter-like and Saturn-like giant planets.Note that all curves are represented with a logarithmic scale.Left: dependence to the perturbative strain pulsation ω\\omega , with R c ={0.20,0.34}×R p {R_c = \\lbrace 0.20 , 0.34\\rbrace \\times R_p}.Right: dependence to the size of the core, with ω≃2.25×10 -4 rad ·s -1 {\\omega \\simeq 2.25 \\times 10^{-4} ~\\mathrm {rad \\cdot s^{-1}}} (tidal frequency of Enceladus) for the blue curve associated to a Saturn-like planet, and ω≃2.79×10 -4 rad ·s -1 {\\omega \\simeq 2.79 \\times 10^{-4} ~\\mathrm {rad \\cdot s^{-1}}} (tidal frequency of Io) for the red curve associated to a Jupiter-like planet.The green curve corresponds to the prescription of Goodman & Lackner (2009) (see plain text for details).The red and blue dashed lines correspond to the value of Q ^ eff ={(3.56±0.56)×10 4 ,(1.682±0.540)×10 3 }\\hat{Q}_\\mathrm {eff}=\\lbrace (3.56 \\pm 0.56) \\times 10^4,(1.682 \\pm 0.540) \\times 10^3\\rbrace (for Jupiter and Saturn respectively) determined by Lainey et al.",
"(2009-2012).Their zone of uncertainty is also represented in the corresponding color.We assume the values of R p ={10.97,9.14}{R_p = \\lbrace 10.97 , 9.14\\rbrace } (in units of R p Φ R_p^{\\mathcal {\\Phi }}), M p ={317.8,95.16}{M_p = \\lbrace 317.8, 95.16\\rbrace } (in units of M p Φ M_p^{\\mathcal {\\Phi }}), M c ={6.41,18.65}×M p Φ {M_c = \\lbrace 6.41, 18.65\\rbrace \\times M_p^{\\mathcal {\\Phi }}} given in Tables -.We also take for the viscoelastic parameters G={4.85,4.45}×10 10 ( Pa ){G= \\lbrace 4.85 , 4.45\\rbrace \\times 10^{10} \\, \\textrm {(Pa)}}, and η={1.26,1.78}×10 14 ( Pa ·s -1 ){\\eta = \\lbrace 1.26 , 1.78\\rbrace \\times 10^{14} \\, (\\textrm {Pa}\\cdot {\\mathrm {s}^{-1}})} for Jupiter and Saturn respectively.Since tidal dissipation causes exchange of angular momentum in the system, it may be quantified by monitoring carefuly the orbital motion of the system.",
"Using astrometric data covering more than a century, Lainey et al.",
"(2009-2012) succeeded in determining from observations the tidal dissipation in Jupiter and Saturn: namely, ${Q_\\mathrm {Jupiter} = (3.56 \\pm 0.56) \\times 10^4}$ determined by Lainey et al.",
"(2009), and ${Q_\\mathrm {Saturn} = (1.682 \\pm 0.540) \\times 10^3}$ determined by Lainey et al.",
"(2012) and requireded by the formation scenario of Charnoz et al.",
"(2011).",
"However, with such a method, the different contributions to the global tidal dissipation, coming from each layer constituting the planet, are lumped together.",
"Equations of the dynamical evolution (Eqs.",
"REF to REF ) link the observed evolution rates of the rotational and orbital parameters to the observed tidal dissipation and system characteristics.",
"Since all these rates are proportional to $R_p^5$ , where $R_p$ is the planet radius, we introduce the associated dissipation factor This can also be demonstrated by calculating the perturbation of the gravific potential at the surface of the planet, adapting Zahn (1966) theory taking into account the boundary conditions at the core surface.",
": $\\hat{Q}_\\mathrm {eff} = \\left( \\frac{R_p}{R_c} \\right)^5 \\times \\frac{|\\tilde{k}_2^F(R_p)|}{|\\tilde{k}_2^F(R_c)|} \\times \\hat{Q} \\:,$ where $|\\tilde{k}_2^F(R_c)|$ can be deduced from Eq.",
"(REF ), and $|\\tilde{k}_2^F(R_p)|$ designates the modulus of the second order Love number of the planet's surface that is obtained from Eqs.",
"(REF ) and (REF ) $\\begin{split}\\tilde{k}_2^F(R_p) &= \\frac{V^{\\prime }(R_p)}{U(R_p)} - 1 \\\\&= \\frac{ \\frac{\\rho _o}{\\rho _c} - \\left( \\frac{R_c}{R_p} \\right)^3 \\, \\left( 1 - \\frac{\\rho _o}{\\rho _c} \\right) }{ \\frac{2}{5} \\frac{\\rho _o}{\\rho _c} \\alpha - \\frac{3}{5} \\left( \\frac{R_c}{R_p} \\right)^5 \\left( 1 - \\frac{\\rho _o}{\\rho _c} \\right)^2 \\frac{\\rho _o}{\\rho _c} \\left( \\alpha + \\frac{3}{2} \\right) \\frac{1}{H} }\\:,\\end{split}$ where $H$ acounts for the quantity $H = 1 + \\tilde{\\mu } - \\frac{\\rho _o}{\\rho _c}+ \\frac{3}{2} \\frac{\\rho _o}{\\rho _c} \\left( 1 - \\frac{\\rho _o}{\\rho _c} \\right)+ \\frac{3}{2} \\left( \\frac{R_c}{R_p} \\right)^5 \\left( 1 - \\frac{\\rho _o}{\\rho _c} \\right)^2 \\:.$ Since we have weak constraints on the viscoelastic parameters of giant planet cores (Guillot 2005), we thus have to explore a large range of values.",
"Fig.",
"REF shows the tidal dissipation factor $\\hat{Q}_\\mathrm {eff}$ around the reference values presented in Sect.",
"REF , expanding the range up to about $\\pm $ 2-4 orders of magnitude for $G$ and $\\eta $ .",
"In the middle region (inside the blue rectangle on Fig.",
"REF ), where $\\eta $ and $G$ correspond to the reference values, the dissipation factor $\\hat{Q}_\\mathrm {eff}$ of Saturn (resp.",
"Jupiter) may reach values of the order of $10^{3}$ (resp.",
"$10^{4}$ ), and in the whole field it varies up to $10^{20}$ .",
"From Fig.",
"REF , we deduce that the tidal dissipation of the core may reach the values observed for Jupiter (Lainey et al.",
"2009) and Saturn (Lainey et al.",
"2012) assuming that Jupiter's core (resp.",
"Saturn's core) has a radius $34.92$ % (resp.",
"$18.72$ %) larger than this of Guillot 1998 (resp.",
"Hubbard 2009).",
"Therefore, we can evaluate the real part of the second order Love numbers ${\\tilde{k}_2^F(R_c)}$ and ${\\tilde{k}_2^F(R_p)}$ , accounting respectively for the deformation of the core's and planet's surface, for parameters whose values are compatible with the tidal dissipation observations (Lainey et al., 2009-2012).",
"For Jupiter and Saturn, in this order, assuming that ${R_c = \\lbrace 0.170 , 0.260 \\rbrace \\times R_p}$ , ${G = \\lbrace 4.85 , 4.45 \\rbrace \\times 10^{10} \\, \\mathrm {Pa}}$ and ${\\eta = \\lbrace 1.26 , 1.78 \\rbrace \\times 10^{14} \\, \\mathrm {Pa}\\cdot \\mathrm {s} }$ , we obtain that ${\\mathcal {R} \\!e \\ \\left[\\tilde{k}_2^F(R_c) \\right] = \\lbrace 3.21 , 3.31 \\rbrace }$ and ${\\mathcal {R} \\!e \\ \\left[\\tilde{k}_2^F(R_p) \\right] = \\lbrace 1.37 , 0.24 \\rbrace }$ .",
"These estimations at the planet's surface can be compared to the value of Gavrilov & Zharkov (1977) of $k_2 = 0.379$ for Jupiter and $k_2 = 0.341$ for Saturn obtained for stratified models.",
"As discussed in the aforementioned paper, the differences between both evaluations are linked to the degree of stratification: the more the planet interior is stratified, the smaller the second order Love number (we recall that the second order Love number of a homogeneous fluid planet is ${3}/{2}$ )."
],
[
"Dependence of tidal dissipation on the size of the core and the tidal frequency",
"For Fig.",
"REF , the values of the viscoelastic parameters $G$ and $\\eta $ , and of the core size $R_c$ result from Fig.",
"REF : we chose them so as the tidal dissipation factor $\\hat{Q}$ reaches the observed values of Lainey et al.",
"(2009-2012) on the condition that the rheological parameters stand in the more realistic domain defined by the lowest and highest value of the ice and rock viscoelasticities taken as reference.",
"Taking into account the global dissipation values obtained by Lainey et al.",
"2009 & 2012 for Jupiter and Saturn, we may deduce some constraints on the viscoelastic parameters and also the size of the core (looking at the red dashed line in Fig.",
"REF ).",
"We thus will assume that they take values allowing such a dissipation: we first chose a core slightly larger than that assumed until now: ${R_c = 0.170 \\times R_p}$ for Jupiter, and ${R_c = 0.260 \\times R_p}$ for Saturn, we then fixed the value of the shear modulus $G$ to the lowest value needed to reach the observed tidal dissipation of Lainey et al.",
"(2009-2012), i.e.",
"${G = 4.85 \\times 10^{10} \\, \\mathrm {Pa}}$ for Jupiter, and ${G = 4.45 \\times 10^{10} \\, \\mathrm {Pa}}$ for Saturn, e finally searched the more realistic value of the viscosity which corresponds to the observed tidal dissipations of Jupiter and Saturn: ${\\eta = 1.26 \\times 10^{14} \\, \\mathrm {Pa}\\cdot \\mathrm {s}}$ for Jupiter, and ${\\eta = 1.78 \\times 10^{14} \\, \\mathrm {Pa}\\cdot \\mathrm {s}}$ for Saturn.",
"Fig.",
"REF explores, for the present model, the dependence of ${\\hat{Q}_\\mathrm {eff}}$ on the pulsation $\\omega $ and the size of the core $R_c$ normalized by the size of the planet $R_p$ .",
"With these parameters, the figure indicates that Saturn dissipates slightly more than ten times more than Jupiter, since ${\\left(R_c\\right)_\\mathrm {Saturn}>\\left(R_c\\right)_\\mathrm {Jupiter}}$ and ${\\left(\\rho _c\\right)_\\mathrm {Saturn}< \\left(\\rho _c\\right)_\\mathrm {Jupiter}}$ .",
"In the range of tidal frequencies of Jupiter's and Saturn's satellites (${2.25 \\times 10^{-4} ~\\mathrm {rad}\\cdot \\mathrm {s}^{-1} ~<~ \\omega ~<~2.95 \\times 10^{-4} ~\\mathrm {rad}\\cdot \\mathrm {s}^{-1}}$ , Lainey et al.",
"2009-2012), the effective dissipation factor ${\\hat{Q}_\\mathrm {eff}}$ keeps the same order of magnitude.",
"However, it strongly depends on the size of the core, as it may loose up to 6 orders of magnitude between a coreless planet and a fully-solid one.",
"Note that for a given core (so that $M_c$ , $R_c$ and then $\\rho _c$ are fixed) and a given mass of the planet $M_p$ , the density of the fluid envelope $\\rho _o$ , which varies with its height ${R_p - R_c}$ , can not exceed $\\rho _c$ .",
"Since: $\\rho _o \\left( R_c \\right) = \\frac{M_p - M_c}{4/3 \\, \\pi \\, \\left( R_p^3 - R_c^3 \\right)} \\:,$ this condition gives a limit for the core size: $\\left( \\frac{R_c}{R_p} \\right)_\\mathrm {sup}= \\left( \\frac{M_c}{M_p} \\right)^{1/3} \\:.$ In 2004, Ogilvie & Lin also studied tidal dissipation in giant planets, and particulary the tidal dissipation resulting from the excitation of inertial waves in the convective region by the tidal potential for rotating giant planets with an elastic solid core.",
"They obtained a decrease of the quality factor $Q$ of one order of magnitude considering the dynamical tide (due to inertial modes) compared to the equilibrium one: from ${Q = 10^6}$ to $Q=10^5$ , but it is not efficient enough to explain the observed tidal dissipations in Jupiter or Saturn which are of 1-2 orders of magnitude higher (Lainey et al.",
"2009-2012).",
"Moreover, they showed that the dissipation resulting from the resonance between fluid tide and inertial modes highly depends on the tidal frequency in the range of inertial waves, as do also the coreless models (Wu 2005).",
"This disagrees with the weak frequency-dependence obtained from astrometry (Lainey et al.",
"2012).",
"By discussing the size of the core, Goodman & Lackner (2009) got a higher value of the quality factor $Q$ , in the range $10^7-10^8 \\times \\left( 0.2 \\, R_p / R_c \\right)^5$ which disagrees with the observed value of the tidal dissipation of Saturn (see Fig.",
"REF ).",
"The present two-layer model proposes an alternative process to reach such a high dissipation with a smooth frequency-dependence of $\\hat{Q}$ .",
"But the uncertainties on the structural and rheological parameters does not allow us to firmly conclude that the tidal dissipation of the core can explain by its own the tidal dissipation observed in giant planets of our Solar System by Lainey et al.",
"(2009-2012).",
"Through these expressions of the tidal dissipation closely linked to the internal structure of the planet and its rheological properties, we are now able to derive the equations of the dynamical evolution of the system with explicit dependence on the tidal frequency."
],
[
"Comparison with previous work of Dermott",
"The difference beetwen our study and the work of Dermott (1979) lies in the treatment of the tidal dissipation.",
"Dermott draws his expression from an evolution scenario of Saturnian and Jovian systems.",
"He assumes that the satellites of Jupiter and Saturn were formed ${4.5 \\times 10^9}$ ago, and that their semi-major axis have changed by 10 % since their formation, with a stable resonance of the main satellites of Jupiter (Io, Europa and Ganymede) since their formation but a young resonance of the satellites Mimas and Thetys of Saturn.",
"He also assumes an average value of the tidal dissipation, independent of amplitude, frequency and time.",
"All these assumptions lead Dermott (1979) to a tidal dissipation factor $Q$ that only depends on the mass $M_c$ , the size $R_c$ , the elasticity $\\mu $ of the core and a dimensionless coefficient $K$ characteristic of the evolution scenario of the planet.",
"In particular, his tidal factor $Q$ is directly proportional to $R_c^5$ (Eq.",
"27 of Dermott 1979), so that the tidal dissipation gets lower as the core size increases (see Fig.",
"4 of Dermott 1979), while one should expect an opposite behaviour.",
"Instead, our model is constructed on physical considerations of the internal structure and properties of the core.",
"In particular, we derived our tidal dissipation factor $\\hat{Q}$ with no assumption on the evolution of the Jovian and Saturnian systems.",
"To do so, we used the correspondence principle of Biot (1954).",
"This allowed us to obtain an expression of the tidal dissipation factor valid for any rheological model of planets' cores.",
"Moreover, our expression (REF ) of $\\hat{Q}$ not only depends on the mass $M_c$ , the size $R_c$ and the elasticity $\\mu $ of the core, but also on the tidal frequency $\\omega $ and the viscosity $\\eta $ .",
"We notice, in particular, that tidal dissipation increases when the size of the core increases, contrary to Dermott's strange result."
],
[
"Equations of the dynamical evolution",
"Mass redistribution due to the tide generates a tidal torque of non-zero average which induces an exchange of angular momentum between the orbital motion and the rotation of each component.",
"As shown in MLP09 & Remus et al.",
"(2012), this tidal torque is proportional to the tidal dissipation ratio $\\displaystyle {{\\left|\\tilde{k}^F_2\\right|}/{Q}}$ (see also Correia & Laskar 2003, Correia et al.",
"2003, and Murray & Dermott 2000).",
"Note that for a perfectly elastic material, the core will be elongated in the direction of the line of centers, inducing a torque, $\\displaystyle {\\int _{\\mathcal {V}} \\, \\mathbf {r} \\wedge \\left( \\rho _c {\\nabla }U \\right) \\, \\rm {d} \\mathcal {V}}$ , with periodic variations of zero average, so that no secular exchanges of angular momentum will be possible (see Zahn 1966a and Remus et al.",
"2012).",
"On the other hand, if the core is anelastic, the deformation of the core resulting from the equilibrium adjustment presents a time delay $\\Delta t$ with respect to the tidal forcing, which may be measured also by the tidal lag angle ${2 \\delta _l}$ or equivalently by the quality factor $Q$ (see Ferraz-Mello et al.",
"2008 or Efroimsky & Williams 2009): $\\tan \\left[ \\Delta t \\times \\sigma _{2,m,p,q} \\right] =\\tan \\left[ 2 \\delta \\left( \\sigma _{2,m,p,q} \\right) \\right] =\\frac{1}{Q\\left( \\sigma _{2,m,p,q} \\right)} \\:.$ Thus, the tidal bulge is no more aligned with the line of centers, as shown in Fig.",
"REF .",
"Figure: Tidal interaction involving a solid body.Body B exerts a tidal force on body A, which adjusts itself with a phase lag 2δ2 \\, \\delta , because of internal friction in the anelastic core.This adjustment may be split in an adiabatic component, corresponding to the elastic deformation, in phase with the tide, and a dissipative one, resulting from the viscous internal frictions, which is in quadrature.The resulting tidal angle is at the origin of a torque of non-zero average which causes exchange of spin and orbital angular momentum between the components of the system.",
"The evolution of the semi-major axis $a$ , of the eccentricity $e$ , of the inclination $I$ , of the obliquity $\\varepsilon $ and of the angular velocity $\\Omega $ ($\\bar{I}_A$ denotes the moment of inertia of $A$ ), is governed by the following equations, established in MLP09 and Remus et al.",
"(2012): $\\frac{\\mathrm {d} \\left( \\bar{I}_A \\, \\Omega \\right) }{\\mathrm {d}t}= - \\frac{8\\pi }{5} \\frac{\\mathcal {G} M_{B}^{2} R_\\mathrm {eq}^5}{a^6} \\\\\\times \\sum _{(m,j,p,q) \\in \\mathbb {I}}\\left\\lbrace \\frac{\\left|\\tilde{k}^F_2(R_p,\\sigma _{2,m,p,q})\\right|}{\\hat{Q}_\\mathrm {eff}(\\sigma _{2,m,p,q})} \\left[ \\mathcal {H}_{2,m,j,p,q}(e,I,\\varepsilon ) \\right]^2 \\right\\rbrace \\:,$ $\\bar{I}_A \\, \\Omega \\, \\frac{\\mathrm {d} \\left( \\cos \\varepsilon \\right)}{\\mathrm {d}t}= \\frac{4\\pi }{5} \\frac{\\mathcal {G} M_{B}^{2} R_\\mathrm {eq}^5}{a^6} \\\\\\times \\sum _{(m,j,p,q) \\in \\mathbb {I}}\\left\\lbrace (j+2\\cos \\varepsilon ) \\, \\frac{\\left|\\tilde{k}^F_2(R_p,\\sigma _{2,m,p,q})\\right|}{\\hat{Q}_\\mathrm {eff}(\\sigma _{2,m,p,q})} \\left[ \\mathcal {H}_{2,m,j,p,q}(e,I,\\varepsilon ) \\right]^2 \\right\\rbrace \\:,$ $\\frac{1}{a} \\, \\frac{\\mathrm {d}a}{\\mathrm {d}t}= -\\frac{2}{n}\\frac{4\\pi }{5}\\frac{\\mathcal {G} M_{B} R_\\mathrm {eq}^5}{a^8} \\\\\\times \\sum _{(m,j,p,q) \\in \\mathbb {I}}\\left\\lbrace (2-2p+q) \\, \\frac{\\left|\\tilde{k}^F_2(R_p,\\sigma _{2,m,p,q})\\right|}{\\hat{Q}_\\mathrm {eff}(\\sigma _{2,m,p,q})} \\left[ \\mathcal {H}_{2,m,j,p,q}(e,I,\\varepsilon ) \\right]^2\\right\\rbrace \\:,$ $\\frac{1}{e} \\, \\frac{{\\rm d}e}{{\\rm d}t}= - \\frac{1}{n}\\frac{1-e^2}{e^2}\\frac{4\\pi }{5}\\frac{\\mathcal {G} M_{B} R_\\mathrm {eq}^5}{a^8} \\\\\\times \\sum _{(m,j,p,q) \\in \\mathbb {I}}\\left\\lbrace \\vphantom{\\frac{\\left|\\tilde{k}^F_2\\right|}{\\hat{Q}_e(\\sigma _q)}} \\left[ (2-2p) \\, \\left( 1 - \\frac{1}{\\sqrt{1-e^2}} \\right) + q \\right] \\right.",
"\\\\\\times \\left.",
"\\frac{\\left|\\tilde{k}^F_2(R_p,\\sigma _{2,m,p,q})\\right|}{\\hat{Q}_\\mathrm {eff}(\\sigma _{2,m,p,q})} \\left[\\mathcal {H}_{2,m,j,p,q}(e,I,\\varepsilon ) \\right] ^2 \\vphantom{\\left( \\frac{1}{\\sqrt{e^2}} \\right)} \\right\\rbrace \\:,$ $\\frac{\\mathrm {d} \\left( \\cos I \\right)}{\\mathrm {d}t}= \\frac{1}{n} \\, \\frac{1}{\\sqrt{1-e^2}} \\, \\frac{4\\pi }{5} \\frac{\\mathcal {G} M_{B}^{2} R_\\mathrm {eq}^5}{a^8} \\\\\\times \\sum _{(m,j,p,q) \\in \\mathbb {I}}\\left\\lbrace \\vphantom{\\frac{\\left|\\tilde{k}^F_2\\right|}{\\hat{Q}}} \\left[ j+(2q-2)\\cos I \\right] \\right.",
"\\\\\\times \\left.",
"\\frac{\\left|\\tilde{k}^F_2(R_p,\\sigma _{2,m,p,q})\\right|}{\\hat{Q}_\\mathrm {eff}(\\sigma _{2,m,p,q})} \\left[ \\mathcal {H}_{2,m,j,p,q}(e,I,\\varepsilon ) \\right]^2 \\right\\rbrace \\:,$ where the functions $\\mathcal {H}_{m,j,p,q}(e,I,\\varepsilon )$ are expressed in terms of $d_{j,m}^{2}(\\varepsilon )$ , $F_{2,j,p}(I)$ , and $G_{2,p,q}(e)$ , which are defined in Sect.",
"REF $\\mathcal {H}_{2,m,j,p,q}(e,I,\\varepsilon ) = \\sqrt{ \\frac{5}{4\\pi } \\frac{(2-|j|)!}{(2+|j|)!}",
"} \\\\\\times d_{j,m}^{2}(\\varepsilon ) \\, F_{2,j,p}(I) \\, G_{2,p,q}(e) \\:,$ and $R_\\mathrm {eq}$ designates the equatorial radius of body $A$ .",
"From these equations one may derive the characteristic times of synchronization, circularization and spin alignment: $&\\frac{1}{t_{\\rm sync}}= - \\frac{1}{\\Omega -n}\\frac{\\mathrm {d} \\Omega }{\\mathrm {d}t}= - \\frac{1}{\\bar{I}_A \\, \\left(\\Omega -n\\right)}\\frac{\\mathrm {d} \\left( \\bar{I}_A \\, \\Omega \\right) }{\\mathrm {d}t} \\:, \\\\&\\frac{1}{t_{\\rm circ}}= - \\frac{1}{e} \\, \\frac{{\\rm d}e}{{\\rm d}t} \\:, \\\\&\\frac{1}{t_{\\rm align_A}}= - \\frac{1}{\\varepsilon } \\, \\frac{{\\rm d}\\varepsilon }{{\\rm d}t}= \\frac{1}{\\varepsilon \\, \\sin \\varepsilon } \\, \\frac{{\\rm d} \\left(\\cos \\varepsilon \\right)}{{\\rm d}t} \\:, \\\\&\\frac{1}{t_{\\rm align_{Orb}}}= - \\frac{1}{I} \\, \\frac{{\\rm d}I}{{\\rm d}t}= \\frac{1}{I \\, \\sin I} \\, \\frac{{\\rm d} \\left(\\cos I \\right)}{{\\rm d}t} \\:.$"
],
[
"Conclusion",
"To conclude, the purpose of this work was to study the tidal dissipation in a two-layer planet consisting in a rocky/icy core and a fluid envelope, as one expects to be the case in Jupiter, Saturn, and many extrasolar planets.",
"We considered the most general configuration where the perturber (star or satellite) is moving on an elliptical and inclined orbit around the planet which rotates on an inclined axis.",
"We expanded the tidal displacement in Fourier series and spherical harmonics, each term of the expansion having a radial part proportional to that of the corresponding term of the tidal potential, which depends on the eccentricity, the inclination and the obliquity.",
"We followed the method by Dermott (1979) to derive the modified Love numbers $h _2^F$ and $k _2^F$ accounting for the tidal deformation at the boundary of the solid core.",
"Like Dermott, we made the simplifying assumption that core and envelope have a constant density.",
"Then, generalizing the results of his work invoking the correspondence principle, we obtained the tidal dissipation rate of the core expressed by $| k _2^F|/\\hat{Q}$ , $\\hat{Q}$ being the quality factor.",
"That ratio depends on the tidal frequency and on the rheological properties of the core; unlike Dermott we made no assumption on the formation history of the system.",
"As mentioned in Sect.",
"REF the rheological properties of planetary cores are still quite uncertain.",
"However, taking plausible values for the viscoelastic parameters $G$ and $\\eta $ , we obtain a tidal dissipation which may be much higher than for a fully fluid planet and weakly frequency-dependent.",
"Under these assumptions, we find that the low value of ${Q = (1.682 \\pm 0.540) \\times 10^3}$ , determined by Lainey et al.",
"(2012) and needed by Charnoz et al.",
"(2011) to explain the formation of all mid-size moons of Saturn from the rings, can be reached taking into account the tidal dissipation of Saturn's core.",
"In the same way, the dissipation in Jupiter's core may explain the value of the $Q$ -factor determined by Lainey et al.",
"(2009), i.e.",
"${Q = (3.56 \\pm 0.56) \\times 10^4}$ .",
"But to do so, we need to assume a core in Jupiter and Saturn slightly larger than those of Guillot (1998) and Hubbard et al.",
"(2009).",
"But we recall that in our model the density was assumed to be piecewise constant.",
"In the future, we shall consider a non constant density profile, to evaluate the impact on our results of a realistic density stratification.",
"Moreover, there are much uncertainties on the determination of the core size in giant planets as Jupiter and Saturn, so that we need more constraints on the system formation by core accretion (Pollack et al.",
"1996) and differenciation resulting from the internal structure evolution (Nettelmann 2011).",
"Furthermore, seismology seems to offer an interesting way to improve our knowledge of giant planets interiors (Gaulme et al.",
"2011).",
"To conclude, the purpose of this paper was to study tidal dissipation in the solid parts of a simple model of two-layer planet; we show how this mechanism may be powerful.",
"The results derived here are general in the sense that no specific rheological model has been assumed.",
"However, considering the lack of constraints on the rheology of giant planets cores, we have chosen the simplest Maxwell model to illustrate the tidal dissipation.",
"This work represents thus a first step for further numerical investigations in more realistic cases."
],
[
"Acknowledgments",
"The authors are grateful to the referee for his/her remarks and suggestions.",
"They also thank G. Tobie for fruitful discussions during this work and T. Guillot for providing numerical models of Jupiter and Saturn interiors.",
"This work was supported in part by the Programme National de Planétologie (CNRS/INSU), the EMERGENCE-UPMC project EME0911, and the CNRS programme Physique théorique et ses interfaces.",
"In Sect.",
"REF we gave the system of equations (REF ), with its boundary conditions (REF ), governing an elastic planet under tidal perturbation.",
"Considering that such perturbations are of small order of magnitude compared to the hydrostatic equilibrium, we proposed in Sect.",
"REF a method to linearize the system (REF ).",
"Thus, assuming the expansion (REF ) of all quantities in spherical harmonics, we obtain the following system of equations governing the scalar radial parts of these expansions (Alterman et al.",
"1959; Takeuchi & Saito 1972): $\\dot{y}^m_1 =& - \\frac{2 \\left(K-\\frac{2}{3} \\mu \\right)}{K+\\frac{4}{3}\\mu } \\, \\frac{y^m_1}{r} + \\frac{1}{K+\\frac{4}{3}\\mu } \\, y^m_2+ \\frac{6 \\left(K-\\frac{2}{3} \\mu \\right)}{K+\\frac{4}{3}\\mu } \\, \\frac{y^m_3}{r} \\:, \\\\\\dot{y}^m_2 =& \\left[ -4 \\, \\rho _0 \\, g_s \\, \\frac{r}{R} + \\frac{12 \\mu K}{\\left( K+\\frac{4}{3}\\mu \\right) r} \\right] \\, \\frac{y^m_1}{r}- \\frac{4 \\mu K}{\\left( K+\\frac{4}{3}\\mu \\right) r} \\, \\frac{y^m_2}{r} \\nonumber \\\\& + 6 \\left[ \\rho _0 \\, g_s \\, \\frac{r}{R} - \\frac{6 \\mu K}{\\left( K+\\frac{4}{3}\\mu \\right) r} \\right] \\, \\frac{y^m_3}{r}+6 \\, \\frac{y^m_4}{r}- \\rho _0 y^m_6 \\:, \\\\\\dot{y}^m_3 =& - \\frac{y^m_1}{r} + \\frac{y^m_3}{r} + \\frac{y^m_4}{\\mu } \\:, \\\\\\dot{y}_4^m =& \\left[ \\rho _0 \\, g_s \\, \\frac{r}{R} - \\frac{6 \\mu K}{\\left( K+\\frac{4}{3}\\mu \\right) r} \\right] \\, \\frac{y^m_1}{r} \\nonumber \\\\& + \\frac{2 \\mu \\left[ 11 \\, \\left(K-\\frac{2}{3} \\mu \\right) + 10 \\, \\mu \\right]}{\\left( K+\\frac{4}{3}\\mu \\right) r} \\, \\frac{y^m_3}{r} \\:, \\\\\\dot{y}^m_5 =& 4 \\pi \\mathcal {G} \\rho _0 y^m_1 +y^m_6 \\:, \\\\\\dot{y}^m_6 =& -24 \\pi \\mathcal {G} \\rho _0 \\frac{y^m_3}{r} + \\frac{6}{r} \\frac{y^m_5}{r} - 2 \\, \\frac{y^m_6}{r} \\:.$ Solutions of (REF ) are given by (REF )."
]
] | 1204.1468 |
[
[
"Spectral Duality in Integrable Systems from AGT Conjecture"
],
[
"Abstract We describe relationships between integrable systems with N degrees of freedom arising from the AGT conjecture.",
"Namely, we prove the equivalence (spectral duality) between the N-cite Heisenberg spin chain and a reduced gl(N) Gaudin model both at classical and quantum level.",
"The former one appears on the gauge theory side of the AGT relation in the Nekrasov-Shatashvili (and further the Seiberg-Witten) limit while the latter one is natural on the CFT side.",
"At the classical level, the duality transformation relates the Seiberg-Witten differentials and spectral curves via a bispectral involution.",
"The quantum duality extends this to the equivalence of the corresponding Baxter-Schrodinger equations (quantum spectral curves).",
"This equivalence generalizes both the spectral self-duality between the 2x2 and NxN representations of the Toda chain and the famous AHH duality."
],
[
"Introduction",
"In this paper we study the AGT correspondence [1] at the level of integrable systems [2], [3], [4]See also [5],[6],[7],[8],[9],[10],[11],[12],[13],[14].. More exactly, we deal with the AGT inspired models which emerge in the limiting case.",
"The full AGT correspondence associates the conformal block of the Virasoro or $W$ -algebra in two-dimensional conformal field theory with the LMNS integral [15] (Nekrasov functions [16])) describing the two-parametric deformation of Seiberg-Witten theory by $\\Omega $ -background.",
"The classical integrable systems emerge when both deformation parameters are brought to zero, while when only one of the parameters going to zero (the Nekrasov-Shatashvili limit [3]) the integrable system gets quantized.",
"We shall study here only the correspondence between AGT inspired integrable systems in these two limiting cases.",
"It is important that the two sides of the AGT correspond to a priori different types of integrable models which should actually coincide due to AGT.",
"This leads to non-trivial predictions of equivalence of different models and also illuminates what the equivalence exactly means.",
"Here we consider the simplest example of this kind: the equivalence of the four-point conformal block and the prepotential in the $SU(N)$ SUSY theory with vanishing $\\beta $ -function.",
"On the gauge theory side the (classical) integrable system is known [17] to be the Heisenberg chain [18] which is described by the spectral curve $\\Gamma ^{\\hbox{\\tiny {Heisen}}}(w,x):\\det (w-T(x))=0$ with ${\\rm GL}_2$ -valued $N$ -site transfer-matrix $T(x)$ and Seiberg-Witten [19] (SW) differential $\\hbox{d}S^{\\hbox{\\tiny {Heisen}}}(w,x)=x\\frac{\\hbox{d}w}{w}$ .",
"On the CFT side the corresponding integrable system was argued to be some special reduced Gaudin model [20] defined by its spectral curve $\\Gamma ^{\\hbox{\\tiny {Gaudin}}}(y,z):\\det (y-L(z))=0$ with ${\\rm gl}_N$ -valued Lax matrix $L(z)$ and the SW differential $\\hbox{d}S^{\\hbox{\\tiny {Gaudin}}}(y,z)=y\\hbox{d}z$ .",
"The original argument [1] dealt with the $SU(2)$ case and implied that on the conformal side of the AGT correspondence the counterpart of the SW differential is played by the average of the energy-momentum tensor, and this latter shows up a pole behaviour which is rather associated with the Gaudin model.",
"This argument was refined later by associating the SW differential with an insertion of the surface operator [6], [9], [11] or with the matrix model resolvent [7].",
"If considering the case of higher rank group $SU(N)$ , which on the gauge theory side is associated with the Heisenberg chain (on $N$ sites), one has to take into account that on the conformal side the AGT conjecture in this case deal with a four-point conformal block of the $W_N$ -algebra [5], however not an arbitrary one but that restricted with special conditions imposed onto two of the four external operators (states) of the block.",
"This means that there are two arbitrary operators parameterized by $N-1$ parameters each and two other operators parameterized by only one parameter each.",
"In integrable terms this means that one should expect for the associated integrable system, the reduced Gaudin model that it is described by two coadjoint orbits of the maximal dimensions inserted in two points, and by two coadjoint orbits of the minimal dimensions inserted in two other points.",
"As we shall see, this is, indeed, the case.",
"In this letter we show that the change of variables $z=w$ , $\\lambda =x/w$ relates the curves and SW differentials of the two integrable systems under discussion (the Heisenberg spin chain and the reduced Gaudin model).",
"It means that with this change of variables the following relations hold true: $\\begin{array}{c}\\Gamma ^{\\hbox{\\tiny {Gaudin}}}(y,z)=\\Gamma ^{\\hbox{\\tiny {Heisen}}}(z,zy)\\,,\\\\\\ \\\\\\end{array}\\hbox{d}S^{\\hbox{\\tiny {Gaudin}}}(y,z)=\\hbox{d}S^{\\hbox{\\tiny {Heisen}}}(z,zy)\\,.$ This type of relations between spectral curves appeared in [21]We call the duality between Gaudin models described in [21] as AHH duality.",
"See the comment in the end of the paper..",
"Following [22] we call it (classical) spectral duality.",
"The duality transformation acts by bispectral involution [24] which interchanges the roles of the eigenvalue-variable and spectral parameter.",
"A well-known simpler example is the periodic Toda chain.",
"It can be described by both the ${\\rm gl}(N)$ -valued Lax matrix: $ { L}^{Toda}_{N\\times N}(z) =\\left(\\begin{array}{ccccc}p_1 & e^{{1\\over 2}(q_2-q_1)} & 0 & & ze^{{1\\over 2}(q_1-q_{N})}\\\\e^{{1\\over 2}(q_2-q_1)} & p_2 & e^{{1\\over 2}(q_3 - q_2)} & \\ldots & 0\\\\0 & e^{{1\\over 2}(q_3-q_2)} & p_3 & & 0 \\\\& & \\ldots & & \\\\\\frac{1}{z}e^{{1\\over 2}(q_1-q_{N})} & 0 & 0 & & p_{N}\\end{array} \\right)$ and the ${\\rm GL}(2)$ -valued transfer-matrix: $T^{Toda}_{2\\times 2}(\\lambda )=L_N(\\lambda )...L_1(\\lambda ),\\ \\ \\ L_i(\\lambda ) = \\left(\\begin{array}{cc} \\lambda -p_i & e^{q_i} \\\\-e^{-q_i} & 0\\end{array}\\right), \\ \\ \\ \\ \\ i = 1,\\dots ,N$ The spectral curves defined by these representations are related by the bispectral involution, i.e.",
"$\\det (\\lambda -{ L}^{Toda}_{N\\times N}(z))=0\\ \\ \\ \\hbox{and}\\ \\ \\ \\ \\det (z-T^{Toda}_{2\\times 2}(\\lambda ))=0$ coincide.",
"The SW differential is the same in both cases $\\hbox{d}S=\\lambda \\frac{\\hbox{d}z}{z}$ .",
"Therefore, the periodic Toda chain is a self-dual model [25].",
"The quantum version of the duality appears from the exact quasi-classical quantization of the spectral curves.",
"Considering the SW differential as a symplectic 1-form [26] on ${\\mathbb {C}}^2$ -plane $(y,z)$ yields a pair of canonical variables $(p(y,z),q(z))$ which brings the SW differential to $\\hbox{d}S(y,z)=p\\hbox{d}q$ .",
"Then there is a natural quantization of the spectral curve defined by the rule $(p,q)\\rightarrow (\\hbar \\partial _q, q)$ .",
"For the above mentioned models one has: $\\begin{array}{c}{\\hat{\\Gamma }}^{\\hbox{\\tiny {Heisen}}}(z,\\hbar z\\partial _z)\\Psi ^{\\hbox{\\tiny {Heisen}}}(z)=0\\,,\\end{array}$ $\\begin{array}{c}\\hat{\\Gamma }^{\\hbox{\\tiny {Gaudin}}}(\\hbar \\partial _z,z)\\Psi ^{\\hbox{\\tiny {Gaudin}}}(z)=0\\end{array}$ with some choice of ordering.",
"The wave functions can be written in terms of the quantum deformation of the SW differential on the spectral curve, i.e.",
"$\\Psi (z)=\\exp \\left(-\\frac{1}{\\hbar }\\int ^q\\hbox{d}S(\\hbar )\\right)$ , where $\\hbox{d}S(\\hbar )=p(q,\\hbar )\\hbox{d}q$ and $p(q,0)=p(q)|_\\Gamma $ .",
"The monodromies of the wave function around $A$ - and $B$ - cycles of $\\Gamma $ are given by the quantum deformed action type variables [4]: $\\begin{array}{l}\\Psi (z+A_i)=\\exp \\left(-\\frac{1}{\\hbar }a_i^\\hbar \\right)\\Psi (z),\\ \\ a_i^\\hbar =\\oint \\limits _{A_i}\\hbox{d}S(\\hbar )\\,,\\\\\\ \\\\\\Psi (z+B_i)=\\exp \\left(-\\frac{1}{\\hbar }\\frac{\\partial \\mathcal {F}_{\\hbox{\\tiny {NS}}}}{\\partial a_i^\\hbar }\\right)\\Psi (z),\\ \\ \\frac{\\partial \\mathcal {F}_{\\hbox{\\tiny {NS}}}}{\\partial a_i^\\hbar }=\\oint \\limits _{B_i}\\hbox{d}S(\\hbar )\\,,\\end{array}$ where $\\mathcal {F}_{\\hbox{\\tiny {NS}}}$ is the Nekrasov-Shatashvili limit [3] of the LMNS integral [15].",
"The AGT conjecture predicts the following relations (quantum spectral duality): $\\begin{array}{l}a_i^\\hbar (\\Psi ^{\\hbox{\\tiny {Heisen}}})=a_i^\\hbar (\\Psi ^{\\hbox{\\tiny {Gaudin}}})\\,,\\\\\\ \\\\\\frac{\\partial \\mathcal {F}_{\\hbox{\\tiny {NS}}}}{\\partial a_i^\\hbar }(\\Psi ^{\\hbox{\\tiny {Heisen}}})=\\frac{\\partial \\mathcal {F}_{\\hbox{\\tiny {NS}}}}{\\partial a_i^\\hbar }(\\Psi ^{\\hbox{\\tiny {Gaudin}}})\\,.\\end{array}$ In this paper we deal with the known quantum equation (REF ) for the XXX chain - the Baxter equationIt arises as an equation for the Baxter Q-operator eigenvalues in the Quantum Inverse Scattering Method.",
"Originally, it was written in difference (Fourier-dual) form.",
"[27]: $\\left( {\\rm tr}\\, T(\\hbar z \\partial _z) - \\frac{z}{1+q} {K}_{+}(\\hbar z \\partial _z) - \\frac{q}{(1+q)z} {K}_{-}(\\hbar z \\partial _z) \\right)\\Psi ^{\\hbox{\\tiny {Heisen}}}(z)=0\\,.$ We verify that (REF ) can be re-written as the quantum spectral curve of the Gaudin model (REF ).",
"In this way we arrive to the quantum version of duality: $\\begin{array}{|c|}\\hline \\\\\\Psi ^{\\hbox{\\tiny {Heisen}}}(z)=\\Psi ^{\\hbox{\\tiny {Gaudin}}}(z)\\\\ \\ \\\\ \\hline \\end{array}$ In the next section we briefly describe the models and formulate the spectral duality.",
"Some comments are given at the end.",
"Most of details will be given in [28].",
"In that extended version we also plan to describe the Poisson map between models."
],
[
"Acknowledgments",
"The authors are grateful to A.Gorsky, A.Zabrodin and A.Zhedanov for useful comments and remarks.",
"The work was partially supported by the Federal Agency for Science and Innovations of Russian Federation under contract 14.740.11.0347 (A.Z., Y.Z.",
"), by NSh-3349.2012.2, by RFBR grants 10-02-00509 (A.Mir.",
"), 10-02-00499 (A.Mor., Y.Z.",
"), 12-01-00482 (A.Z.)",
"and by joint grants 11-02-90453-Ukr, 12-02-91000-ANF, 12-02-92108-Yaf-a, 11-01-92612-Royal Society.",
"The work of A.Zotov was also supported in part by the Russian President fund MK-1646.2011.1."
],
[
"Heisenberg Chain - Gaudin Model Duality",
"1.",
"$N$ -site ${\\rm GL}_2$ Heisenberg (XXX) chain.",
"It is classically defined by its spectral curve $\\Gamma ^{\\hbox{\\tiny {Heisen}}}(w,x):\\ \\ {\\rm tr}T(x)-\\frac{1}{1\\!+\\!q}wK^+(x)-\\frac{q}{1\\!+\\!q}w^{-1}K^-(x)=0,\\ K^\\pm (x)=\\prod \\limits _{i=1}^N(x-m_i^\\pm )$ and SW differential $\\hbox{d}S^{\\hbox{\\tiny {Heisen}}}(w,x)=x\\frac{\\hbox{d}w}{w}\\,.$ $T(x)$ in (REF ) is ${\\rm GL}_2$ -valued transfer-matrix: $\\begin{array}{c}T(x)= V L_N(x) \\ldots L_1(x),\\ \\ L_i(x)\\!=\\!x\\!-\\!x_i\\!+\\!S^i,\\ i = 1\\ldots N,\\ \\ \\end{array}V={{\\left(\\begin{array}{cc}{1}&{-\\frac{q}{(1+q)^2}}\\\\{1}&{0}\\end{array}\\right)}}\\,,\\\\S^i\\in {\\rm sl}_2:\\ \\hbox{Spec}(S^i)=(K_i,-K_i),\\ \\ m_i^\\pm =x_i\\pm K_i\\,.$ Function ${\\rm tr}T(x)=x^N + \\sum \\limits _{i=1}^N x^{i-1}H^{\\hbox{\\tiny {Heisen}}}_i$ provides commuting integrals of motion.",
"2.",
"Special (reduced) ${\\rm gl}_N$ Gaudin model on ${\\mathbb {CP}}^1\\backslash \\lbrace 0,1,q,\\infty \\rbrace $.",
"It is described by the spectral curve $\\Gamma ^{\\hbox{\\tiny {Gaudin}}}(y,z):\\ \\ \\det (y-L(z))=0,\\ \\ \\ L(z)=\\frac{A^0}{z}+\\frac{A^1}{z-1}+\\frac{A^q}{z-q}\\in {\\rm gl}_N$ with additional conditions including the reduction constraintsOne should also fix the action of the Cartan subgroup.",
"We do not discuss it here since it does not effect the curve.",
"$\\begin{array}{c}A^0+A^1+A^q+A^\\infty =0\\,,\\\\A^\\infty \\equiv \\Upsilon =\\hbox{diag}(\\upsilon _1,...,\\upsilon _N),\\ \\ \\hbox{Spec}(A^0)=(\\mu _1,...,\\mu _N)\\,,\\\\A^1=\\xi ^1\\times \\eta ^1,\\ \\ A^q=\\xi ^q\\times \\eta ^q\\,,\\end{array}$ i.e.",
"$A^1$ and $A^q$ are ${\\rm gl}_N$ matrices of rank 1 (this type of configuration was already discussed [29], [10]).",
"Using specification (REF ) the spectral curve can be find explicitly: $\\begin{array}{c} \\left(\\right.\\eta ^1(zy\\!+\\!\\Upsilon )^{\\!-1}\\xi ^1\\!+\\!q\\eta ^q(zy\\!+\\!\\Upsilon )^{\\!-1}\\xi ^q \\!+\\!q\\!+\\!1\\!\\prod \\limits _{i=1}^N(zy\\!+\\!\\upsilon _i)=z\\!\\prod \\limits _{i=1}^N(zy\\!+\\!\\upsilon _i)\\!+z^{\\!-1}\\!q\\!\\prod \\limits _{i=1}^N(zy\\!-\\!\\mu _i)\\end{array}$ or $\\begin{array}{c}\\prod \\limits _{i=1}^N(zy+\\upsilon _i)+ \\sum \\limits _{k=1}^N\\frac{\\eta ^1_k\\xi ^1_k+q\\eta ^q_k\\xi ^q_k}{q+1}\\prod \\limits _{i\\ne k}^N(zy+\\upsilon _i) = \\frac{z}{q+1}\\prod \\limits _{i=1}^N(zy+\\upsilon _i)\\!+\\!z^{-1}\\frac{q}{q+1}\\prod \\limits _{i=1}^N(zy-\\mu _i)\\,.\\end{array}$ The SW differential is $\\hbox{d}S^{\\hbox{\\tiny {Gaudin}}}(y,z)=y\\hbox{d}z\\,.$ The classical spectral duality.",
"First, notice that the both models (as classical mechanical systems) describe dynamics of $N-1$ degrees of freedom and depend on $2N+1$ parameters.",
"Indeed, the dynamical variables of the off-shell Gaudin model (REF ) are $A^{0,1,q,\\infty }$ .",
"Fixing the Casimir functions restricts $A^{0,1,q,\\infty }$ to the coadjoint orbits of maximum dimensions ($N^2-N$ ) at $z=0,\\ \\infty $ and of minimal dimensions ($2N-2$ ) at $z=1,\\ q$ .",
"Then the reduction by the coadjoint action of ${\\rm GL}_N$ gives the following dimension of the phase space: $2(N^2-N)+2(2N-2)-2(N^2-1)=2(N-1).$ The number of parameters is $2N+3:\\ \\lbrace \\upsilon _1,...,\\upsilon _N,\\mu _1,...,\\mu _N,{\\rm tr}A^1,{\\rm tr}A^q,q\\rbrace $ .",
"Two of them, (${\\rm tr}A^0,{\\rm tr}A^\\infty $ ) can be eliminated from the spectral curve by the shift of $y$ .",
"Therefore, the number of independent parameters is $2N+1$ .",
"For the Heisenberg chain, one initially has $N$ ${\\rm sl}_2$ -valued variables $S^i$ with the Casimir functions fixed at each site: $\\frac{1}{2}{\\rm tr}\\left(S^i\\right)^2=K_i^2$ .",
"The reduction by $\\hbox{Stab}(V(q))\\cong \\hbox{Cartan}({\\rm GL}_2)$ fixes two independent variables.",
"Therefore, for the dimension of the phase space one has $3N-N-2=2(N-1)$ and there are $2N+1$ parameters $\\lbrace x_1,...,x_N,K_1,...,K_N,q\\rbrace $ .",
"The duality between models is described by the following Theorem.",
"The N-site ${\\rm GL}_2$ Heisenberg XXX chain defined by (REF )-(REF ) and the ${\\rm gl}_N$ Gaudin model (REF )-(REF ) are spectrally dual at the classical level $\\begin{array}{|c|}\\hline \\\\\\Gamma ^{\\hbox{\\tiny {Gaudin}}}(y,z)=\\Gamma ^{\\hbox{\\tiny {Heisen}}}(w,x)\\\\\\ \\\\\\hbox{d}S^{\\hbox{\\tiny {Gaudin}}}(y,z)=\\hbox{d}S^{\\hbox{\\tiny {Heisen}}}(w,x)\\\\\\ \\\\\\hline \\end{array}$ with the following change of variables $\\begin{array}{c} z=w,\\ \\ y=\\frac{x}{w}\\,,\\end{array}$ identification of parameters $\\begin{array}{c}m^+_i=-\\upsilon _i,\\ \\ m^-_i=\\mu _i,\\ \\ 1\\le i\\le N\\,,\\end{array}$ and relation between generating functions of the Hamiltonians: $\\begin{array}{c}{\\rm tr}T^{\\hbox{\\tiny {Heisen}}}(y)= \\det (y\\!+\\!\\Upsilon )\\left(1\\!+\\!\\right.\\frac{1}{1+q}\\eta ^1(y\\!+\\!\\Upsilon )^{\\!-1}\\xi ^1+\\frac{q}{1+q}\\eta ^q(y\\!+\\!\\Upsilon )^{\\!-1}\\xi ^q\\,.\\end{array}$ The statement follows from the comparison of (REF ) and (REF ).",
"In particular, $H^{\\hbox{\\tiny {Heisen}}}_N=\\frac{1}{1+q}{\\rm tr}A^1+\\frac{q}{1+q}{\\rm tr}A^q+\\sum \\limits _{k=1}^N\\upsilon _k\\,.$ The quantum spectral duality.",
"The quantization of the XXX chain spectral curve (REF ) with the SW differential (REF ) means that $x$ should be simply replaced by $\\hbar w\\partial _w$ .",
"Then one gets the Baxter equation: $\\left( {\\rm tr}\\, T(\\hbar w \\partial _w) - \\frac{w}{1+q} {K}_{+}(\\hbar w \\partial _w) - \\frac{q}{(1+q)w} {K}_{-}(\\hbar w \\partial _w) \\right)\\Psi ^{\\hbox{\\tiny {Heisen}}}(w)=0\\,.$ Equivalently, for the Gaudin spectral curve (REF ) the quantization is given by the replacement $y\\rightarrow \\hbar \\partial _z$ : $\\begin{array}{l}\\left(\\prod \\limits _{i=1}^N(z\\hbar \\partial _z+\\upsilon _i)+\\sum \\limits _{k=1}^N\\frac{\\eta ^1_k\\xi ^1_k+q\\eta ^q_k\\xi ^q_k}{q+1}\\prod \\limits _{i\\ne k}^N(z\\hbar \\partial _z+\\upsilon _i) -\\right.\\\\\\left.-\\frac{z}{q+1}\\prod \\limits _{i=1}^N(z\\hbar \\partial _z+\\upsilon _i)\\right.-z^{-1}\\frac{q}{q+1}\\prod \\limits _{i=1}^N(z\\hbar \\partial _z-\\mu _i)\\Psi ^{\\hbox{\\tiny {Gaudin}}}(z)=0\\,.\\end{array}$ Obviously, the differential operators in the brackets of (REF ) and (REF ) can be identified in the same way as the classical spectral curves did."
],
[
"Comments",
" AHH duality.",
"In [21] (see also [23]) the authors considered the Gaudin model with $M$ marked points and the Lax matrix defined as follows: $L^G_{AHH}(z)=Y+\\sum \\limits _{c=1}^M \\frac{A^c}{z-z_c}\\,,\\ \\ Y= \\hbox{diag }(y_1,...,y_N)\\,,\\ \\ A^c\\in {\\rm gl}_N\\,.$ The later differs from ours.",
"The difference is significant since $Y\\ne 0$ leads to the second order pole at $\\infty $ for $L^G_{AHH}(z)\\hbox{d}z$ .",
"The phase space is also different.",
"It is a direct product of the coadjoint orbits (equipped with a natural Poisson-Lie structure) factorized by the stabilizer of $Y$ : ${\\mathcal {O}}^1\\times \\dots \\times {\\mathcal {O}}^M//\\hbox{Stab}(Y)$ .",
"In the case when all $A^c$ are of rank 1 the dual Lax matrix is the ${\\rm gl}_M$ -valued function with $\\tilde{Y}=\\hbox{diag}(z_1,...,z_M)$ and $N$ marked points at $y_1,...,y_N$ : ${\\tilde{L}}^G_{AHH}(z)=\\tilde{Y}+\\sum \\limits _{c=1}^N \\frac{\\tilde{A}^c}{z-y_c}\\,,\\ \\ \\tilde{Y}= \\hbox{diag }(z_1,...,z_M)\\,, \\ \\ \\tilde{A}^c\\in {\\rm gl}_M\\,.$ The duality implies the following relation between the spectral curves: $\\det (\\tilde{Y}-z)\\det (L^G_{AHH}(z)-\\lambda )=\\det (Y-\\lambda )\\det (\\tilde{L}^G_{AHH}(\\lambda )-z)\\,.$ The dimensions of the phase spaces of both models equal $2(N-1)(M-1)$ and the number of parameters is $N+M-1$ .",
"Sometimes ${\\rm sl}_N$ description of the Gaudin model is more convenient than the ${\\rm gl}_N$ one.",
"The transformation of the spectral curve from ${\\rm gl}_N$ to ${\\rm sl}_N$ is given by the simple shift: $y\\rightarrow y^{\\prime }=y-\\frac{1}{N}{\\rm tr}L(z)=y-\\frac{1}{zN}\\left(-{\\rm tr}\\Upsilon +\\frac{{\\rm tr}A^1}{z-1}+q\\frac{{\\rm tr}A^q}{z-q}\\right)\\,.$ In this case the change of variables (REF ) is modifiedIn this form the change of variables was found in [12] for ${\\rm sl}_2$ case.",
": $\\begin{array}{c} z=w,\\ \\ y^{\\prime }=\\frac{x-R(z)}{w},\\ \\ R(z)=\\frac{1}{N}\\left(-{\\rm tr}\\Upsilon +\\frac{{\\rm tr}A^1}{z-1}+q\\frac{{\\rm tr}A^q}{z-q}\\right)\\,.\\end{array}$ The equality of the wave functions (REF ) acquires the predictable multiple: $\\begin{array}{c}\\Psi ^{\\hbox{\\tiny {Heisen}}}(z)=\\Psi ^{\\hbox{\\tiny {Gaudin}}}(z)e^{\\frac{1}{N\\hbar }\\int ^z b_h(z)\\hbox{d}z}\\,,\\\\ \\ \\\\b_{\\hbar }(z) = \\frac{(1+q)}{(z-1)(z-q)} \\left( {H^{\\hbox{\\tiny {Heisen}}}_N} +\\frac{z \\sum _{k=1}^N m_k^{+}}{1+q} + \\frac{q \\sum _{k=1}^{N}m_k^{-}}{(1+q) z } \\right) - \\hbar \\frac{N(N-1)}{2z}\\,.\\end{array}$ It should be mentioned that we do not impose any boundary conditions which provide a valuable quantum problem, i.e.",
"we do not specify wave functions explicitly.",
"To compare the quantum problems one needs a construction of the Poisson (and then quantum) map between the phase spaces (Hilbert spaces) of the two models.",
"We are going to describe the Poisson map elsewhere [28].",
"Alternatively, one can specify the spaces of solutions initially and then verify their identification through the duality transformation.",
"This is the recipe of [30] where the authors considered very close problem in terms of the Bethe vectors.",
"The precise connection between the two approaches deserves further elucidation.",
"We will comment on it in [28].",
"Besides the approach proposed here, a quantization of the Gaudin model is known from [31] and [32].",
"We hope to shed light on relations between the quantizations in further publications.",
"At last, let us mention possible generalizations of the correspondence proposed in this letter.",
"First of all, one can naturally consider multi-point conformal blocks.",
"This provides one with the multi-point Gaudin model.",
"At the same time, the AGT predicts in this case on the other side of the correspondence the theory with gauge group being a product of a few gauge factors.",
"This latter is naturally embedded into the spin magnets with higher rank group [25].",
"Thus, one expects a correspondence between $GL(p)$ -magnets and multi-point Gaudin models.",
"Another interesting generalization is induced by the five-dimensional AGT [33] which implies a correspondence between the XXZ magnets (see [34]) and a Gaudin-like model with relativistic (difference) dynamics.",
"This latter would emerge, since on the conformal side one deal in this case with the q-Virasoro conformal block which implies a difference Schrödinger equation for the block with insertion of the degenerate field.",
"An extension to six dimensions (elliptic extension of the differential operator in the Schrödinger equation versus XYZ magnet) is also extremely interesting to construct.",
"As is well known, the ${\\rm sl}_2$ reduced Gaudin model with the configuration discussed above can be written in different elliptic forms [35] with $q$ be a function of the modular parameter.",
"Therefore, one can expect some elliptic parametrization for the ${\\rm sl}_N$ case as well.",
"We plan to return to these issues in further publications."
]
] | 1204.0913 |
[
[
"Modeling the polarization of radio-quiet AGN from the optical to the\n X-ray band"
],
[
"Abstract A thermal active galactic nucleus (AGN) consist of a powerful, broad-band continuum source that is surrounded by several reprocessing media with different geometries and compositions.",
"Here we investigate the expected spectropolarimetric signatures in the optical/UV and X-ray wavebands as they arise from the complex radiative coupling between different, axis-symmetric AGN media.",
"Using the latest version of the Monte-Carlo radiative transfer code STOKES, we obtain spectral fluxes, polarization percentages, and polarization position angles.",
"In the optical/UV, we assume unpolarized photons coming from a compact source that are reprocessed by an optically-thick, dusty torus and by equatorial and polar electron-scattering regions.",
"In the X-ray band, we additionally assume a lamp-post geometry with an X-ray source irradiating the accretion disk from above.",
"We compare our results for the two wavebands and thereby provide predictions for future X-ray polarimetric missions.",
"These predictions can be based on present-day optical/UV spectropolarimetric observations.",
"In particular, we conclude that the observed polarization dichotomy in the optical/UV band should extend into the X-ray range."
],
[
"Introduction",
"Since the early observations by [6], active galactic nuclei (AGN) have been intensively observed at all possible wavelengths using ground-based and space telescopes.",
"The core of an AGN cannot be resolved by current optical instruments.",
"In addition to that we find that in type-2 objects (those with narrow optical emission lines) the central engine is hidden by optically thick dust blocking most of the light.",
"According to the unified scheme of AGN [2], the obscuring dust is distributed anisotropically and in type-1 AGN, which show broad optical emission lines, the central region is directly visible.",
"This anisotropic distribution of absorbing and scattering media in AGN must induce a net polarization that we can exploit in order to investigate the complex radiative coupling between the innermost components of AGN.",
"In fact, spectropolarimetry is a unique tool to probe the unresolvable parts of AGN thanks to two more independent observables it adds: the percentage and the position angle of polarization.",
"So far, spectropolarimetry observations could be performed from the radio to the optical/UV band, but with the launch of the GEMS satellite [13] the first X-ray polarization data of bright AGN is soon going to be in reach.",
"To interpret the data, polarization modeling of the radiative interplay between different AGN components is necessary.",
"Such modeling has been conducted previously by a number of authors [10], [11], [16], [8].",
"For computational reasons, some models are restricted to a single-scattering approach following the suggestion by [9] about the predominance of first-order scattering in optically thick media.",
"But one should bear in mind that this argument does not necessarily hold for the multiple scattering between several non-absorbing, electron-scattering components.",
"Also, previous modeling is most often limited to a given waveband.",
"In this research note, we present a composite, multiple-scattering and reprocessing model of AGN from which spectropolarimetric fluxes are computed simultaneously in the optical/UV and in the X-ray band.",
"Our model setup is based on the classical, axis-symmetric unified scheme of AGN [15] and we are particularly interested in the polarization properties as a function of wavelength and viewing direction.",
"When observing the optical polarization of AGN, a dichotomy is found for the polarization angle [3]: at type-2 viewing angles, the position angle of the polarization is most often directed perpendicularly to the central radio structure; at type-1 viewing angles, the polarization vector favors a direction that is aligned with the (projected) radio axis.",
"Assuming that the radio structure is stretched along the symmetry axis of the torus, our modeling allows us to test if we can reproduce this observed dichotomy.",
"In the following, we define parallel (or perpendicular) polarization according to the preferentially observed polarization angle of type-1 (or type-2) AGN.",
"When plotting our results, we distinguish parallel polarization by adding a negative sign to the polarization percentage.",
"We investigate the radiative coupling between different axis-symmetric emission and reprocessing regions: the inner and outer parts of the accretion disk, the obscuring equatorial dust region, and double-conical outflows along the polar direction.",
"We consider a compact continuum source of unpolarized photons being emitted isotropically according to a power-law $F_{\\rm \\nu } \\propto \\nu ^{-\\alpha }$ with index $\\alpha = 1$ .",
"For the optical/UV part (1600 Å – 8000 Å) we assume the continuum source to be very compact and quasi point-like.",
"For the X-ray range (1 keV – 100 keV), we adopt a lamp-post geometry and include X-ray reprocessing of the primary radiation by the underlying disk.",
"The primary source is located at low height on the disk axis subtending a large solid angle with the disk.",
"The source region is surrounded by a geometrically and optically thin scattering annulus with a flared shape.",
"This radiation-supported wedge plays a major role as it produces a parallel polarization signature in type-1 view by electron scattering (see e.g.",
"[5], [1], [4], [12]).",
"At larger radius an optically thick, elliptical dusty torus surrounds the system.",
"It shares the same symmetry plane as the flared disk and is responsible for the optical obscuration at type-2 views.",
"The torus funnel supposedly collimates a mildly-ionized, optically thin outflow stretched along the symmetry axis of the system.",
"The polar wind has an hourglass shape and is centered on the photon source.",
"Parameters defining the shape and the composition of the three/four reprocessing regions are summarized in Table 1.",
"The dust model used for the torus at optical/UV wavelengths is based on a prescription for Galactic dust as described in [7].",
"In the X-ray band, we assume neutral reprocessing for the torus and for the accretion disk.",
"Details of this reprocessing model can be found in [8].",
"Table: Parameters of the different model components.",
"The accretion disk is only present when modeling the X-ray range.",
"The elevated primary X-ray source is located on the disk axis and subtends a half-angle of 76 ∘ ^\\circ with the disk.",
"Note that for the polar outflow, the half-opening angle is measured with respect to the vertical, symmetry axis of the torus, while for the flared-disk the half-opening angle is taken with respect to the equatorial plane."
],
[
"The radiative transfer code ",
"We apply the latest version of the Monte-Carlo code stokes (www.stokes-program.info).",
"For details on the code, consult [7] and [8].",
"It conducts radiative transfer in complex emission and reprocessing environments and includes the treatment of polarization.",
"The calculations include multiple scattering, an angle-dependent analysis in 3D, and different dust models.",
"The stokes code computes the total flux spectrum, the polarization angle, the percentage of polarization, and the polarized flux.",
"The reprocessing physics depends on the energy band considered.",
"Electron and Mie scattering are assumed in the optical/UV waveband; Compton scattering and neutral reprocessing predominate in the X-ray range.",
"We compute the spectropolarimetric flux as a function of wavelength or photon energy at a given polar viewing angle, $i$ , that is measured with respect to the symmetry axis of the system."
],
[
"Results",
"In the top panels of Fig.",
"REF , we present the total flux, $F$ , for the two wavebands considered.",
"The fluxes are normalized to the pure source flux, $F_*$ , that would emerge along the same line-of-sight if there were no scattering media.",
"The results for the polarization percentage, $P$ , and for the polarization angle are combined in the bottom panels of Fig.",
"REF .",
"We adopt a sign convention for the polarization percentage that is recalled in the figure caption.",
"Figure: Modeling the radiative coupling between the different axis-symmetric reprocessing regions.The normalized spectral flux is shown in the top panels, the polarization properties are shown below.A negative PP denotes a “type-1” polarization (parallel to the projected symmetry axis) and apositive PP stands for a “type-2” polarization (perpendicular to the axis).",
"The transition at P=0P = 0 isindicated by dashed lines.",
"Four different viewing angles, ii, are considered: a face-on view ati∼18 ∘ i \\sim 18^\\circ (black), a line-of-sight just below the torus horizon at cosi=0.63 ∘ \\cos i = 0.63^\\circ (red),an intermediate type-2 view at i∼76 ∘ i \\sim 76^\\circ (green), and an edge-on view at i∼87 ∘ i \\sim 87^\\circ (blue).Left: optical/UV energy band.",
"Right: X-ray band."
],
[
"Results for the optical/UV band",
"At face-on and edge-on view, the optical flux is wavelength-independent indicating that electron-scattering in the equatorial flared-disk and in the polar outflows dominates.",
"At intermediate viewing-angles, the impact of the wavelength-dependent dust scattering emerges, mostly around the $\\lambda _{2175}$ feature in the UV.",
"This little bump at 2175 Å in the flux spectrum is due to scattering by carbonaceous dust in the torus.",
"The model reproduces the observed polarization dichotomy.",
"The signature of the flared-disk is visible exceeding the effects of the polar outflow and producing low degrees of parallel polarization towards face-on viewing angles.",
"At higher inclination, the equatorial scattering is hidden by the dusty torus and polar scattering dominates causing perpendicular polarization.",
"The variations of $P$ with wavelength at intermediate viewing angles indicate that the dusty torus also has a significant impact on the polarization.",
"The rise in $P$ towards the UV at $i \\sim 73^\\circ $ is a combined effect of multiple scattering inside the torus funnel and of the wavelength-dependent polarization phase function that is associated with Mie scattering by Galactic dust."
],
[
"Results for the X-ray band",
"The X-ray spectrum shows typical features of neutral reprocessing – the iron K$\\alpha $ and K$\\beta $ fluorescence lines at 6.4 keV and 7.1 keV and their absorption edges, the Compton hump, and strong soft X-ray absorption at intermediate viewing angles are prominent spectral features.",
"The fluorescent line emission and the Compton reflection hump around 30 keV are present at every line of sight.",
"An important result of our modeling is that we predict a polarization dichotomy also for the X-ray band.",
"At all viewing directions below the torus horizon, $P$ is positive implying perpendicular polarization.",
"But towards a face-on view, the electron scattering in the equatorial flared disk predominates and produces a net parallel polarization.",
"A peculiar feature appears at $i \\sim 73^\\circ $ , where $P$ changes from positive to mildly negative values around the Compton hump.",
"This behavior is due to the competition between parallel and perpendicular polarization emerging from different reprocessing regions of the model.",
"Around 30 keV, the effect of the flared disk becomes less important than the Compton scattering in the other regions.",
"Explaining this behavior in detail is not trivial as several factors have to be taken into account, one of them being the angle-dependent scattering phase function.",
"But also,the energy-dependence of the electron scattering cross-section has an effect as it favors soft X-ray photons to scatter more than hard X-ray photons.",
"Higher energy photons therefore pass more easily through the optically thin, equatorial scattering region without interacting.",
"This partly explains the disappearance of the parallel polarization at higher photon energy.",
"A more detailed discussion about the X-ray polarization signature of isolated and coupled reprocessing regions is going to be provided elsewhere."
],
[
"Summary and conclusions",
"We have applied the latest version of the stokes radiative transfer code to examine the complex reprocessing between different, axis-symmetric media of an active nucleus.",
"We provide simultaneous results for the optical/UV and for the X-ray wavebands and we trace the spectral flux and the polarization as a function of photon wavelength.",
"The observed optical/UV polarization dichotomy is successfully reproduced and an analogous dichotomy is predicted for the X-ray range.",
"This work is carried out in anticipation of the forthcoming age of X-ray polarimetry.",
"The NASA space telescope GEMS [13] is planned to be launched in 2014 and will be entirely dedicated to X-ray polarimetry.",
"The satellite will observe X-ray sources in the 2–10 keV band allowing us to test the soft X-ray part of our modeling results for the brightest AGN.",
"Note that a next generation, broad-band X-ray polarimeter is technically feasible [14] and could even observe polarization up to 35 keV.",
"Such observations include the X-ray polarization of the Compton hump and thus put even stronger constraints on the validity of our modeling results.",
"The authors are grateful to Martin Gaskell at the University of Valparaíso in Chile for his great help."
]
] | 1204.0940 |
[
[
"Theory of ferromagnetic unconventional superconductors with spin-triplet\n electron pairing"
],
[
"Abstract A general phenomenological theory is presented for the phase behavior of ferromagnetic superconductors with spin-triplet electron Cooper pairing.",
"The theory describes in details the temperature-pressure phase diagrams of real inter-metallic compounds exhibiting the remarkable phenomenon of coexistence of spontaneous magnetic moment of the itinerant electrons and spin-triplet superconductivity.",
"The quantum phase transitions which may occur in these systems are also described.",
"The theory allows for a classification of these itinerant ferromagnetic superconductors in two types: type I and type II.",
"The classification is based on quantitative criteria.The comparison of theory and experiment is performed and outstanding problems are discussed."
],
[
"Introduction",
"In the beginning of this century the unconventional superconductivity of spin-triplet type had been experimentally discovered in several itinerant ferromagnets.",
"Since then much experimental and theoretical research on the properties of these systems has been accomplished.",
"Here we review the phenomenological theory of ferromagnetic unconventional superconductors with spin-triplet Cooper pairing of electrons.",
"Some theoretical aspects of the description of the phases and the phase transitions in these interesting systems, including the remarkable phenomenon of coexistence of superconductivity and ferromagnetism are discussed with an emphasis on the comparison of theoretical results with experimental data.",
"The spin-triplet or $p$ -wave pairing allows parallel spin orientation of the fermion Cooper pairs in superfluid $^3$ He and unconventional superconductors [1].",
"For this reason the resulting unconventional superconductivity is robust with respect to effects of external magnetic field and spontaneous ferromagnetic ordering, so it may coexist with the latter.",
"This general argument implies that there could be metallic compounds and alloys, for which the coexistence of spin-triplet superconductivity and ferromagnetism may be observed.",
"Particularly, both superconductivity and itinerant ferromagnetic orders can be created by the same band electrons in the metal, which means that spin-1 electron Cooper pairs participate in the formation of the itinerant ferromagnetic order.",
"Moreover, under certain conditions the superconductivity is enhanced rather than depressed by the uniform ferromagnetic order that can generate it, even in cases when the superconductivity does not appear in a pure form as a net result of indirect electron-electron coupling.",
"The coexistence of superconductivity and ferromagnetism as a result of collective behavior of $f$ -band electrons has been found experimentally for some Uranium-based intermetallic compounds as, UGe$_2$ [2], [3], [4], [5], URhGe [6], [7], [8], UCoGe [9], [10], and UIr [11], [12].",
"At low temperature $(T \\sim 1~\\mbox{K})$ all these compounds exhibit thermodynamically stable phase of coexistence of spin-triplet superconductivity and itinerant ($f$ -band) electron ferromagnetism (in short, FS phase).",
"In UGe$_2$ and UIr the FS phase appears at high pressure $(P \\sim 1~\\mbox{GPa})$ whereas in URhGe and UCoGe, the coexistence phase persists up to ambient pressure $(10^5 \\mbox{Pa} \\equiv 1\\mbox{bar})$ .",
"Experiments, carried out in ZrZn$_2$ [13], also indicated the appearance of FS phase at $ T < 1$ K in a wide range of pressures ($0<P \\sim 21~\\mbox{kbar}$ ).",
"In Zr-based compounds the ferromagnetism and the $p$ -wave superconductivity occur as a result of the collective behavior of the $d$ -band electrons.",
"Later experimental results [14], [15] had imposed the conclusion that bulk superconductivity is lacking in ZrZn$_2$ , but the occurrence of a surface FS phase at surfaces with higher Zr content than that in ZrZn$_2$ has been reliably demonstrated.",
"Thus the problem for the coexistence of bulk superconductivity with ferromagnetism in ZrZn$_2$ is still unresolved.",
"This raises the question whether the FS phase in ZrZn$_2$ should be studied by surface thermodynamics methods or should it be investigated by considering that bulk and surface thermodynamic phenomena can be treated on the same footing.",
"Taking into account the mentioned experimental results for ZrZn$_2$ and their interpretation by the experimentalists [13], [14], [15] we assume that the unified thermodynamic approach can be applied.",
"As an argument supporting this point of view let us mention that the spin-triplet superconductivity occurs not only in bulk materials but also in quasi-two-dimensional (2D) systems – thin films and surfaces and quasi-1D wires (see, e.g., Refs. [16]).",
"In ZrZn$_2$ and UGe$_2$ both ferromagnetic and superconducting orders vanish at the same critical pressure $P_c$ , a fact implying that the respective order parameter fields strongly depend on each other and should be studied on the same thermodynamic basis [17].",
"Fig.",
"REF illustrates the shape of the $T-P$ phase diagrams of real intermetallic compounds.",
"The phase transition from the normal (N) to the ferromagnetic phase (FM) (in short, N-FM transition) is shown by the line $T_F(P)$ .",
"The line $T_{FS}(P)$ of the phase transition from FM to FS (FM-FS transition) may have two or more distinct shapes.",
"Beginning from the maximal (critical) pressure $P_c$ , this line may extend, like in ZrZn$_2$ , to all pressures $P < P_c$ , including the ambient pressure $P_a$ ; see the almost straight line containing the point 3 in Fig.",
"REF .",
"A second possible form of this line, as known, for example, from UGe$_2$ experiments, is shown in Fig.",
"REF by the curve which begins at $P \\sim P_c$ , passes through the point 2, and terminates at some pressure $P_{1} > P_a$ , where the superconductivity vanishes.",
"These are two qualitatively different physical pictures: (a) when the superconductivity survives up to ambient pressure (type I), and (b) when the superconducting states are possible only at relatively high pressure (for UGe$_2$ , $P_1 \\sim 1$ GPa); type II.",
"At the tricritical points 1, 2 and 3 the order of the phase transitions changes from second order (solid lines) to first order (dashed lines).",
"It should be emphasized that in all compounds, mentioned above, $T_{FS}(P)$ is much lower than $T_{F}(P)$ when the pressure $P$ is considerably below the critical pressure $P_c$ (for experimental data, see Sec. 8).",
"Figure: An illustration of T-PT-P phase diagramof pp-wave ferromagnetic superconductors (details are omitted): N – normal phase,FM – ferromagnetic phase, FS – phase of coexistence of ferromagneticorder and superconductivity, T F (P)T_{F}(P) and T FS (P)T_{FS}(P) are therespective phase transition lines: solid lines correspond tosecond order phase transitions, dashed lines stand for firstorder phase transition; 1 and 2 are tricritical points;P c P_c is the critical pressure, and the circle CC surroundsa relatively small domain of high pressure andlow temperature, where the phase diagram may have several forms depending onthe particular substance.",
"The line of the FM-FS phase transition mayextend up to ambient pressure (type I ferromagnetic superconductors),or, may terminate at T=0T=0 at some high pressure P=P 1 P=P_1(type II ferromagnetic superconductors, as indicated in the figure).In Fig.",
"REF , the circle $C$ denotes a narrow domain around $P_c$ at relatively low temperatures ($T \\lesssim 300$ mK), where the experimental data are quite few and the predictions about the shape of the phase transition are not reliable.",
"It could be assumed, as in the most part of the experimental papers, that $(T=0,P=P_c)$ is the zero temperature point at which both lines $T_{F}(P)$ and $T_{FS}(P)$ terminate.",
"A second possibility is that these lines may join in a single (N-FS) phase transition line at some point $(T\\gtrsim 0,P^{\\prime }_c\\lesssim P_c)$ above the absolute zero.",
"In this second variant, a direct N-FS phase transition occurs, although this option exists in a very small domain of temperature and pressure variations: from point $(0,P_c)$ to point $(T\\gtrsim 0,P^{\\prime }_c\\lesssim P_c)$ .",
"A third variant is related with the possible splitting of the point $(0,P_c)$ , so that the N-FM line terminates at $(0,P_c)$ , whereas the FM-FS line terminates at another zero temperature point $(0, P_{0c})$ ; $P_{0c} \\lesssim P_c$ .",
"In this case, the $p$ -wave ferromagnetic superconductor has three points of quantum (zero temperature) phase transitions [18], [19].",
"These and other possible shapes of $T-P$ phase diagrams are described within the framework of the general theory of Ginzburg-Landau (GL) type [18], [19], [20] in a conformity with the experimental data; see also Ref. [21].",
"The same theory has been confirmed by a microscopic derivation based on a microscopic Hamiltonian including a spin-generalized BCS term and an additional Heisenberg exchange term [22].",
"For all compounds, cited above, the FS phase occurs only in the ferromagnetic phase domain of the $T-P$ diagram.",
"Particularly at equilibrium, and for given $P$ , the temperature $T_{F}(P)$ of the normal-to-ferromagnetic phase (or N-FM) transition is never lower than the temperature $T_{FS}(P)$ of the ferromagnetic-to-FS phase transition (FM-FS transition).",
"This confirms the point of view that the superconductivity in these compounds is triggered by the spontaneous magnetization $\\mbox{$M$}$ , in analogy with the well-known triggering of the superfluid phase A$_1$ in $^3$ He at mK temperatures by the external magnetic field $\\mbox{$H$}$ .",
"Such “helium analogy\" has been used in some theoretical studies (see, e.g., Ref.",
"[23], [24]), where Ginzburg-Landau (GL) free energy terms, describing the FS phase were derived by symmetry group arguments.",
"The non-unitary state, with a non-zero value of the Cooper pair magnetic moment, known from the theory of unconventional superconductors and superfluidity in $^3$ He [1], has been suggested firstly in Ref.",
"[23], and later confirmed in other studies [7], [24]; recently, the same topic was comprehensively discussed in Ref. [25].",
"For the spin-triplet ferromagnetic superconductors the trigger mechanism was recently examined in detail [20], [21].",
"The system main properties are specified by terms in the GL expansion of form $M_i\\psi _j\\psi _k$ , which represent the interaction of the magnetization $\\mbox{$M$} = \\lbrace M_j; j=1,2,3\\rbrace $ with the complex superconducting vector field $\\mbox{$\\psi $} =\\lbrace \\psi _j; j=1,2,3\\rbrace $ .",
"Particularly, these terms are responsible for the appearance of superconductivity ($|\\mbox{$\\psi $}| > 0$ ) for certain $T$ and $P$ values.",
"A similar trigger mechanism is familiar in the context of improper ferroelectrics [26].",
"A crucial feature of these systems is the nonzero magnetic moment of the spin-triplet Cooper pairs.",
"As mentioned above, the microscopic theory of magnetism and superconductivity in non-Fermi liquids of strongly interacting heavy electrons ($f$ and $d$ band electrons) is either too complex or insufficiently developed to describe the complicated behavior in itinerant ferromagnetic compounds.",
"Several authors (see [20], [21], [23], [24], [25]) have explored the phenomenological description by a self-consistent mean field theory, and here we will essentially use the thermodynamic results, in particular, results from the analysis in Refs.",
"[20], [21].",
"Mean-field microscopic theory of spin-mediated pairing leading to the mentioned non-unitary superconductivity state has been developed in Ref.",
"[17] that is in conformity with the phenomenological description that we have done.",
"The coexistence of $s$ -wave (conventional) superconductivity and ferromagnetic order is a long-standing problem in condensed matter physics [27], [28], [29].",
"While the $s$ -state Cooper pairs contain only opposite electron spins and can easily be destroyed by the spontaneous magnetic moment, the spin-triplet Cooper pairs possess quantum states with parallel orientation of the electron spins and therefore can survive in the presence of substantial magnetic moments.",
"This is the basic difference in the magnetic behavior of conventional ($s$ -state) and spin-triplet superconductivity phases.",
"In contrast to other superconducting materials, for example, ternaty and Chevrel phase compounds, where the effect of magnetic order on $s$ -wave superconductivity has been intensively studied in the seventies and eighties of last century (see, e.g., Refs.",
"[27], [28], [29]), in these ferromagnetic compounds the phase transition temperature $T_{F}$ to the ferromagnetic state is much higher than the phase transition temperature $T_{FS}$ from ferromagnetic to a (mixed) state of coexistence of ferromagnetism and superconductivity.",
"For example, in UGe$_2$ we have $T_{FS} \\sim 0.8$ K versus maximal $T_{F} = 52$ K [2], [3], [4], [5].",
"Another important difference between the ternary rare earth compounds and the intermetallic compounds (UGe$_2$ , UCoGe, etc.",
"), which are of interest in this paper, is that the experiments with the latter do no give any evidence for the existence of a standard normal-to-superconducting phase transition in zero external magnetic field.",
"This is an indication that the (generic) critical temperature $T_s$ of the pure superconductivity state in these intermetallic compounds is very low ($T_s \\ll T_{FS}$ ), if not zero or even negative.",
"In the reminder of this paper, we present general thermodynamic treatment of systems with itinerant ferromagnetic order and superconductivity due to spin-triplet Cooper pairing of the same band electrons, which are responsible for the spontaneous magnetic moment.",
"The usual Ginzburg-Landau (GL) theory of superconductors has been completed to include the complexity of the vector order parameter $\\mbox{$\\psi $}$ , the magnetization $\\mbox{$M$}$ and new relevant energy terms [20], [21].",
"We outline the $T-P$ phase diagrams of ferromagnetic spin-triplet superconductors and demonstrate that in these materials two contrasting types of thermodynamic behavior are possible.",
"The present phenomenological approach includes both mean-field and spin-fluctuation theory (SFT), as the arguments in Ref. [30].",
"We propose a simple, yet comprehensive, modeling of $P$ dependence of the free energy parameters, resulting in a very good compliance of our theoretical predictions for the shape the $T-P$ phase diagrams with the experimental data (for some preliminary results, see Ref.",
"[18], [19]).",
"The theoretical analysis is done by the standard methods of phase transition theory [31].",
"Treatment of fluctuation effects and quantum correlations [31], [32] is not included in this study.",
"But the parameters of the generalized GL free energy may be considered either in mean-field approximation as here, or as phenomenologically renormalized parameters which are affected by additional physical phenomena, as for example, spin fluctuations.",
"We demonstrate with the help of present theory that we can outline different possible topologies for the $T-P$ phase diagram, depending on the values of Landau parameters, derived from the existing experimental data.",
"We show that for spin-triplet ferromagnetic superconductors there exist two distinct types of behavior, which we denote as Zr-type (or, alternatively, type I) and U-type (or, type II); see Fig.",
"REF .",
"This classification of the FS, first mentioned in Ref.",
"[18], is based on the reliable interrelationship between a quantitative criterion derived by us and the thermodynamic properties of the ferromagnetic spin-triplet superconductors.",
"Our approach can be also applied to URhGe, UCoGe, and UIr.",
"The results shed light on the problems connected with the order of the quantum phase transitions at ultra-low and zero temperatures.",
"They also raise the question for further experimental investigations of the detailed structure of the phase diagrams in the high-$P$ /low-$T$ region."
],
[
"Theoretical framework",
"Consider the GL free energy functional of the form $F(\\mbox{$\\psi $}, \\mbox{$B$}) = \\int _V d \\mbox{$x$}\\left[ f_{\\mbox{\\scriptsize S}}(\\mbox{$\\psi $}) +f_{\\mbox{\\scriptsize F}}(\\mbox{$M$}) +f_{\\mbox{\\scriptsize I}}(\\psi ,\\mbox{$M$}) +\\frac{\\mbox{$B$}^2}{8\\pi } - \\mbox{$B.M$} \\right],$ where the fields $\\mbox{$\\psi $}$ , $\\mbox{$M$}$ , and $\\mbox{$B$}$ are supposed to depend on the spatial vector $\\mbox{$x$} \\in V$ in the volume $V$ of the superconductor.",
"In Eq.",
"(REF ), the free energy density generated by the generic superconducting subsystem ($\\mbox{$\\psi $}$ ) is given by $f_{\\mbox{\\scriptsize S}}(\\mbox{$\\psi $})= f_{grad}(\\mbox{$\\psi $})+ a_s|\\mbox{$\\psi $}|^2 +\\frac{b_s}{2}|\\mbox{$\\psi $}|^4 +\\frac{u_s}{2}|\\mbox{$\\psi $}^2|^2 +\\frac{v_s}{2}\\sum _{j=1}^{3}|\\psi _j|^4 \\;,$ with $f_{grad}(\\mbox{$\\psi $})& =& K_1(D_i\\psi _j)^{\\ast }(D_iD_j) +K_2\\left[(D_i\\psi _i)^{\\ast }(D_j\\psi _j) + (D_i\\psi _j)^{\\ast }(D_j\\psi _i)\\right] \\\\\\nonumber && + K_3(D_i\\psi _i)^{\\ast }(D_i\\psi _i),$ where a summation over the indices $(i,j)$ is assumed, the symbol $D_j= (\\hbar \\partial /i\\partial x_j + 2|e|A_j/c)$ of covariant differentiation is introduced, and $K_j$ are material parameters [1].",
"The free energy density $f_{\\mbox{\\scriptsize F}}(\\mbox{$M$})$ of a standard ferromagnetic phase transition of second order [31] is $f_{\\mbox{\\scriptsize F}}(\\mbox{$M$}) =c_f\\sum _{j=1}^{3}|\\nabla \\mbox{$M$}_j|^2 +a_f\\mbox{$M$}^2 + \\frac{b_f}{2}\\mbox{$M$}^4,$ with $c_f, b_f > 0$ , and $a_f = \\alpha (T-T_f)$ , where $\\alpha _f>0$ and $T_f$ is the critical temperature, corresponding of the generic ferromagnetic phase transition.",
"Finally, the energy $f_{\\mbox{\\scriptsize I}}(\\mbox{$\\psi $}, \\mbox{$M$})$ produced by the possible couplings of $\\mbox{$\\psi $}$ and $\\mbox{$M$}$ is given by $f_{\\mbox{\\scriptsize I}}(\\mbox{$\\psi $}, \\mbox{$M$}) = i\\gamma _0\\mbox{$M$}.",
"(\\mbox{$\\psi $}\\times \\mbox{$\\psi $}^*) +\\delta _0 \\mbox{$M$}^2|\\mbox{$\\psi $}|^2,$ where the coupling parameter $\\gamma _0 \\sim J$ depends on the ferromagnetic exchange parameter $J>0$ , [23], [24] and $\\delta _0$ is the standard $\\mbox{$M$}-\\mbox{$\\psi $}$ coupling parameter, known from the theory of multicritical phenomena [31] and from studies of coexistence of ferromagnetism and superconductivity in ternary compounds [27], [28].",
"As usual, in Eq.",
"(REF ), $a_s = (T-T_s)$ , where $T_s$ is the critical temperature of the generic superconducting transition, $b_s > 0$ .",
"The parameters $u_s$ and $v_s$ and $\\delta _0$ may take some negative values, provided the overall stability of the system is preserved.",
"The values of the material parameters $\\mu $ = ($T_s$ , $T_f$ , $\\alpha _s$ , $\\alpha _f$ , $b_s$ , $u_s$ , $v_s$ , $b_f$ , $K_j$ , $\\gamma _0$ and $\\delta _0$ ) depend on the choice of the substance and on intensive thermodynamic parameters, such as the temperature $T$ and the pressure $P$ .",
"From a microscopic point of view, the parameters $\\mu $ depend on the density of states $U_F(k_F)$ on the Fermi surface.",
"On the other hand $U_F$ varies with $T$ and $P$ .",
"Thus the relationships $(T,P) \\rightleftarrows U_F \\rightleftarrows \\mu $ , i.e., the functional relations $\\mu [U_F(T,P)]$ , are of essential interest.",
"While these relations are unknown, one may suppose some direct dependence $\\mu (T,P)$ .",
"The latter should correspond to the experimental data.",
"The free energy (REF ) is quite general.",
"It has been deduced by several reliable arguments.",
"In order to construct Eq.",
"(REF )–(REF ) we have used the standard GL theory of superconductors and the phase transition theory with an account of the relevant anisotropy of the $p$ -wave Cooper pairs and the crystal anisotropy, described by the $u_s$ - and $v_s$ -terms in Eq.",
"(REF ), respectively.",
"Besides, we have used the general case of cubic anisotropy, when all three components $\\psi _j$ of $\\mbox{$\\psi $}$ are relevant.",
"Note, that in certain real cases, for example, in UGe$_2$ , the crystal symmetry is tetragonal, $\\mbox{$\\psi $}$ effectively behaves as a two-component vector and this leads to a considerable simplification of the theory.",
"As shown in Ref.",
"[20], the mentioned anisotropy terms are not essential in the description of the main thermodynamic properties, including the shape of the $T-P$ phase diagram.",
"For this reason we shall often ignore the respective terms in Eq.",
"(REF ).",
"The $\\gamma _0$ -term triggers the superconductivity ($\\mbox{$M$}$ -triger effect [20], [21]) while the $\\delta _0\\mbox{$M$}^2 |\\psi |^2$ –term makes the model more realistic for large values of $\\mbox{$M$}$ .",
"This allows for an extension of the domain of the stable ferromagnetic order up to zero temperatures for a wide range of values of the material parameters and the pressure $P$ .",
"Such a picture corresponds to the real situation in ferromagnetic compounds [20].",
"The total free energy (REF ) is difficult for a theoretical investigation.",
"The various vortex and uniform phases described by this complex model cannot be investigated within a single calculation but rather one should focus on particular problems.",
"In Ref.",
"[24] the vortex phase was discussed with the help of the criterion [33] for a stability of this state near the phase transition line $T_{c2}(\\mbox{$B$})$ , ; see also, Ref. [34].",
"The phase transition line $T_{c2}(H)$ of a usual superconductor in external magnetic field $H = |\\mbox{$H$}|$ is located above the phase transition line $T_s$ of the uniform (Meissner) phase.",
"The reason is that $T_s$ is defined by the equation $a_s(T) = 0$ , whereas $T_{c2}(H)$ is a solution of the equation $|a_s| = \\mu _BH$ , where $\\mu _B = |e|\\hbar /2mc$ is the Bohr magneton [34].",
"For ferromagnetic superconductors, where $M > 0$ , one should use the magnetic induction $\\mbox{$B$}$ rather than $H$.",
"In case of $\\mbox{$H$} = 0$ one should apply the same criterion with respect to the magnetization $\\mbox{$M$}$ for small values of $|\\mbox{$\\psi $}|$ near the phase transition line $T_{c2}(M)$ ; $M = |\\mbox{$M$}|$ .",
"For this reason we should use the diagonal quadratic form [35] corresponding to the entire $\\mbox{$\\psi $}^2$ -part of the total free energy functional (REF ).",
"The lowest energy term in this diagonal quadratic part contains a coefficient $a$ of the form $a = (a_s - \\gamma _0 M - \\delta M^2)$ [35].",
"Now the equation $a(T) = 0$ defines the critical temperature of the Meissner phase and the equation $|a_s| = \\mu _BM$ stands for $T_{c2}(M)$ .",
"It is readily seen that these two equations can be written in the same form, provided the parameter $\\gamma _0$ in $a$ is substituted by $\\gamma _0^{\\prime } = (\\gamma _0- \\mu _B$ ).",
"Thus the phase transition line corresponding to the vortex phase, described by the model (REF ) at zero external magnetic field and generated by the magnetization $\\mbox{$M$}$ , can be obtained from the phase transition line corresponding to the uniform superconducting phase by an effective change of the value of the parameter $\\gamma _0$ .",
"Both lines have the same shape and this is a particular property of the present model.",
"The variation of the parameter $\\gamma _0$ generates a family of lines.",
"Now we propose a possible way of theoretical treatment of the $T_{FS}(P)$ line of the FM-FS phase transition, shown in Fig. (1).",
"This is a crucial point in our theory.",
"The phase transition line of the uniform superconducting phase can be calculated within the thermodynamic analysis of the uniform phases, described by the free energy (REF ).",
"This analysis is done in a simple variant of the free energy (REF ) in which the fields $\\mbox{$\\psi $}$ and $\\mbox{$M$}$ do not depend on the spatial vector $\\mbox{$x$}$ .",
"The accomplishment of such analysis will give a formula for the phase transition line $T_{FS}(P)$ which corresponds a Meissner phase coexisting with the ferromagnetic order.",
"The theoretical result for $T_{FS}(P)$ will contain a unspecified parameter $\\gamma _0$ .",
"If the theoretical line $T_{FS}(P)$ is fitted to the experimental data for the FM-FS transition line corresponding to a particular compound, the two curves will coincide for some value of $\\gamma _0$ , irrespectively on the structure of the FS phase.",
"If the FS phase contains a vortex superconductivity the fitting parameter $\\gamma _{0(eff)}$ should be interpreted as $\\gamma _0^{\\prime }$ but if the FS phase contains Meissner superconductivity, $\\gamma _{0(eff)}$ should be identified as $\\gamma _0$ .",
"These arguments justify our approach to the investigation of the experimental data for the phase diagrams of intermetallic compounds with FM and FS phases.",
"In the remainder of this paper, we shall investigate uniform phases."
],
[
"Model considerations",
"In the previous section we have justified a thermodynamic analysis of the free energy (REF ) in terms of uniform order parameters.",
"Neglecting the $\\mbox{$x$}$ -dependence of $\\mbox{$\\psi $}$ and $\\mbox{$M$}$ , the free energy per unit volume, $F/V =f(\\mbox{$\\psi $},\\mbox{$M$})$ in zero external magnetic field $(\\mbox{$H$}=0)$ , can be written in the form $ f(\\mbox{$\\psi $},\\mbox{$M$}) &= &a_s|\\mbox{$\\psi $}|^2 +\\frac{b_s}{2}|\\mbox{$\\psi $}|^4 +\\frac{u_s}{2}|\\mbox{$\\psi $}^2|^2 +\\frac{v_s}{2}\\sum _{j=1}^{3}|\\psi _j|^4 +a_f\\mbox{$M$}^2 +\\frac{b_f}{2}\\mbox{$M$}^4 \\\\ \\nonumber && + \\; i\\gamma _0 \\mbox{$M$}\\cdot (\\mbox{$\\psi $}\\times \\mbox{$\\psi $}^*) + \\delta _0\\mbox{$M$}^2 |\\mbox{$\\psi $}|^2.$ Here we slightly modify the parameter $a_f$ by choosing $a_f =\\alpha _{f}[T^n-T_{f}^n(P)]$ , where $n=1$ gives the standard form of $a_f$ , and $n=2$ applies for SFT [30] and the Stoner-Wohlfarth model [36].",
"Previous studies [20] have shown that the anisotropy represented by the $u_s$ and $v_s$ terms in Eq.",
"(REF ) slightly perturbs the size and shape of the stability domains of the phases, while similar effects can be achieved by varying the $b_s$ factor in the $b_s|\\mbox{$\\psi $}|^4$ term.",
"For these reasons, in the present analysis we ignore the anisotropy terms, setting $u_s =v_s = 0$ , and consider $b_s\\equiv b >0$ as an effective parameter.",
"Then, without loss of generality, we are free to choose the magnetization vector to have the form $\\mbox{$M$} =(0,0,M)$ .",
"According to the microscopic theory of band magnetism and superconductivity the macroscopic material parameters in Eq.",
"(REF ) depend in a quite complex way on the density of states at the Fermi level and related microscopic quantities [37].",
"That is why we can hardly use the microscopic characteristics of these complex metallic compounds in order to elucidate their thermodynamic properties, in particular, in outlining their phase diagrams in some details.",
"However, some microscopic simple microscopic models reveal useful results, for example, the zero temperature Stoner-type model employed in Ref. [38].",
"We redefine for convenience the free energy (REF ) in a dimensionless form by $\\tilde{f} = f/(b_f M_0^4)$ , where $M_0 = [\\alpha _fT_{f0}^n/b_f]^{1/2} >0$ is the value of the magnetization $M$ corresponding to the pure magnetic subsystem $(\\mbox{$\\psi $} \\equiv 0)$ at $T=P=0$ and $T_{f0}=T_f(0)$ .",
"The order parameters assume the scaling $m = M/M_0$ and $\\mbox{$\\varphi $} = \\mbox{$\\psi $}/[(b_f/b)^{1/4}M_0]$ , and as a result, the free energy becomes $ \\tilde{f}= r\\phi ^2 + \\frac{\\phi ^4}{2}+ tm^2+\\frac{m^4}{2} + 2\\gamma m\\phi _1\\phi _2\\mbox{sin}\\theta +\\gamma _1m^2\\phi ^2,$ where $\\phi _j =|\\varphi _j|$ , $\\phi =|\\mbox{$\\varphi $}|$ , and $\\theta = (\\theta _2 - \\theta _1)$ is the phase angle between the complex $\\varphi _1 = \\phi _1e^{i\\theta _1}$ and $\\varphi _2 = \\phi _2 e^{\\theta _2}$ .",
"Note that the phase angle $\\theta _3$ , corresponding to the third complex field component $\\varphi _3 = \\phi _3e^{i\\theta _3}$ does not enter explicitly in the free energy $\\tilde{f}$ , given by Eq.",
"(REF ), which is a natural result of the continuous space degeneration.",
"The dimensionless parameters $t$ , $r$ , $\\gamma $ and $\\gamma _1$ in Eq.",
"(REF ) are given by $ t = \\tilde{T}^n-\\tilde{T}_f^n(P),\\;\\;\\;\\; r = \\kappa (\\tilde{T}-\\tilde{T}_s),$ where $\\kappa = \\alpha _sb_f^{1/2}/\\alpha _fb^{1/2}T_{f0}^{n-1}$ , $\\gamma = \\gamma _0/ [\\alpha _fT_{f0}^nb]^{1/2}$ , and $\\gamma _1 =\\delta _0/(bb_f)^{1/2}$ .",
"The reduced temperatures are $\\tilde{T} =T/T_{f0}$ , $\\tilde{T}_f(P) = T_f(P)/T_{f0}$ , and $\\tilde{T}_s(P)=T_s(P)/T_{f0}$ .",
"The analysis involves making simple assumptions for the $P$ dependence of the $t$ , $r$ , $\\gamma $ , and $\\gamma _1$ parameters in Eq.",
"(REF ).",
"Specifically, we assume that only $T_f$ has a significant $P$ dependence, described by $ \\tilde{T}_f(P) = (1 - \\tilde{P})^{1/n},$ where $\\tilde{P} = P/P_0$ and $P_0$ is a characteristic pressure deduced later.",
"In ZrZn$_2$ and UGe$_2$ the $P_0$ values are very close to the critical pressure $P_c$ at which both the ferromagnetic and superconducting orders vanish, but in other systems this is not necessarily the case.",
"As we will discuss, the nonlinearity ($n=2$ ) of $T_f(P)$ in ZrZn$_2$ and UGe$_2$ is relevant at relatively high $P$ , at which the N-FM transition temperature $T_F(P)$ may not coincide with $T_f(P)$ ; $T_F(P)$ is the actual line of the N-FM phase transition, as shown in Fig. (1).",
"The form (REF ) of the model function $\\tilde{T}_f(P)$ is consistent with preceding experimental and theoretical investigations of the N-FM phase transition in ZrZn$_2$ and UGe$_2$ (see, e.g., Refs.",
"[4], [24], [39]).",
"Here we consider only non-negative values of the pressure $P$ (for effects at $P<0$ , see, e.g., Ref. [44]).",
"The model function (REF ) is defined for $P \\le P_0$ , in particular, for the case of $n >1$ , but we should have in mind that, in fact, the thermodynamic analysis of Eq.",
"(REF ) includes the parameter $t$ rather than $T_f(P)$ .",
"This parameter is given by $ t(T,P) = \\tilde{T}^n - 1 + \\tilde{P},$ and is well defined for any $\\tilde{P}$ .",
"This allows for the consideration of pressures $P > P_0$ within the free energy (REF ).",
"The model function $\\tilde{T}_f(P)$ can be naturally generalized to $\\tilde{T}_f(P) = (1-\\tilde{P}^{\\beta })^{1/\\alpha }$ but the present needs of interpretation of experimental data do not require such a complex consideration (hereafter we use Eq.",
"(REF ) which corresponds to $\\beta = 1$ and $\\alpha =n$ ).",
"Besides, other analytical forms of $\\tilde{T}_f(\\tilde{P})$ can also be tested in the free energy (REF ), in particular, expansion in powers of $\\tilde{P}$ , or, alternatively, in $(1 -\\tilde{P})$ which satisfy the conditions $\\tilde{T}_f(0) = 1$ and $\\tilde{T}_f(1) = 0$ .",
"Note, that in URhGe the slope of $T_{F}(P)\\sim T_f(P)$ is positive from $P=0$ up to high pressures [8] and for this compound the form (REF ) of $\\tilde{T}_f(P)$ is inconvenient.",
"Here we apply the simplest variants of $P$ -dependence, namely, Eqs.",
"(REF ) and (REF ).",
"In more general terms, all material parameters ($r$ , $t$ , $\\gamma $ , $\\dots $ ) may depend on the pressure.",
"We suppose that a suitable choice of the dependence of $t$ on $P$ is enough for describing the main thermodynamic properties and this supposition is supported by the final results, presented in the remainder of this paper.",
"But in some particular investigations one may need to introduce a suitable pressure dependence of other parameters."
],
[
"Stable phases",
"The simplified model (REF ) is capable of describing the main thermodynamic properties of spin-triplet ferromagnetic superconductors.",
"For $r >0$ , i.e., $T > T_s$ , there are three stable phases [20]: (i) the normal (N-) phase, given by $\\phi = m = 0$ (stability conditions: $t\\ge 0$ , $r \\ge 0$ ); (ii) the pure ferromagnetic phase (FM phase), given by $m = (-t)^{1/2}> 0$ , $\\phi =0$ , which exists for $t < 0$ and is stable provided $r\\ge 0$ and $r \\ge (\\gamma _1t + \\gamma |t|^{1/2})$ , and (iii) the already mentioned phase of coexistence of ferromagnetic order and superconductivity (FS phase), given by $\\mbox{sin}\\theta = \\mp 1$ , $\\phi _3 = 0$ , $\\phi _1=\\phi _2= \\phi /\\sqrt{2}$ , where $ \\phi ^2 = \\kappa (\\tilde{T}_s-\\tilde{T}) \\pm \\gamma m -\\gamma _1 m^2 \\ge 0.$ The magnetization $m$ satisfies the equation $ c_3 m^3 \\pm c_2 m^2 + c_1 m \\pm c_0 = 0$ with coefficients $c_0 = \\gamma \\kappa (\\tilde{T} - \\tilde{T}_s)$ , $ c_1 = 2\\left[\\tilde{T}^n + \\kappa \\gamma _1(\\tilde{T}_s-\\tilde{T}) +\\tilde{P} -1 -\\frac{\\gamma ^2}{2}\\right],$ $ c_2 = 3\\gamma \\gamma _1,\\;\\;\\;\\; c_3 = 2(1-\\gamma _1^2).$ Table 1.",
"Theoretical results for the location [$(\\tilde{T}, \\tilde{P})$ - reduced coordinates] of the tricritical points A $\\equiv (\\tilde{T}_A, \\tilde{P}_A)$ and B $\\equiv (\\tilde{T}_B, \\tilde{P}_B)$ , the critical-end point C $\\equiv (\\tilde{T}_C, \\tilde{P}_C)$ , and the point of temperature maximum, max =$(\\tilde{T}_m,\\tilde{P}_m)$ on the curve $\\tilde{T}_{FS}(\\tilde{P})$ of the FM-FS phase transitions of first and second orders (for details, see Sec. 5).",
"The first column shows $\\tilde{T}_N \\equiv \\tilde{T}_{(A,B,C,m)}$ .",
"The second column stands for $t_N =t_{(A,B,C,m)}$ .",
"The reduced pressure values $\\tilde{P}_{(A,B,C,m)}$ of points A, B, C, and max are denoted by $\\tilde{P}_N(n)$ : $n=1$ stands for the linear dependence $T_f(P)$ , and $n=2$ stands for the nonlinear $T_f(P)$ and $t(T)$ , corresponding to SFT.",
"Table: NO_CAPTIONThe FS phase contains two thermodynamically equivalent phase domains that can be distinguished by the upper and lower signs ($\\pm $ ) of some terms in Eqs.",
"(REF ) and (REF ).",
"The upper sign describes the domain (labelled bellow again by FS), where $m>0$ , $\\mbox{sin}\\theta = - 1$ , whereas the lower sign describes the conjunct domain FS$^{\\ast }$ , where $m < 0$ and $\\mbox{sin}\\theta = 1$ (for details, see, Ref. [20]).",
"Here we consider one of the two thermodynamically equivalent phase domains, namely, the domain FS, which is stable for $m >0$ (FS$^{\\ast }$ is stable for $m<0$ ).",
"This “one-domain approximation\" correctly presents the main thermodynamic properties described by the model (REF ), in particular, in the case of a lack of external symmetry breaking fields.",
"The stability conditions for the FS phase domain given by Eqs.",
"(REF ) and (REF ) are $\\gamma M \\ge 0$ , $ \\kappa (\\tilde{T}_s-\\tilde{T}) \\pm \\gamma m - 2\\gamma _1 m^2 \\ge 0,$ and $3(1-\\gamma _1^2)m^2 +3\\gamma \\gamma _1m + \\tilde{T}^n -1+\\tilde{P} +\\kappa \\gamma _1(\\tilde{T}_s-\\tilde{T})-\\frac{\\gamma ^2}{2} \\ge 0.$ These results are valid whenever $T_f(P) > T_s(P)$ , which excludes any pure superconducting phase ($\\mbox{$\\psi $} \\ne 0, m=0$ ) in accord with the available experimental data.",
"For $r <0$ , and $t > 0$ the models (REF ) and (REF ) exhibit a stable pure superconducting phase ($\\phi _1=\\phi _2=m=0$ , $\\phi _3^2 = -r$ ) [20].",
"This phase may occur in the temperature domain $T_f(P) < T < T_s$ .",
"For systems, where $T_f(0)\\gg T_s$ , this is a domain of pressure in a very close vicinity of $P_0\\sim P_c$ , where $T_{F}(P)\\sim T_f(P)$ decreases up to values lower than $T_s$ .",
"Of course, such a situation is described by the model (REF ) only if $T_s >0$ .",
"This case is interesting from the experimental point of view only when $T_s > 0$ is enough above zero to enter in the scope of experimentally measurable temperatures.",
"Up to date a pure superconducting phase has not been observed within the accuracy of experiments on the mentioned metallic compounds.",
"For this reason, in the reminder of this paper we shall often assume that the critical temperature $T_s$ of the generic superconducting phase transition is either non-positive $(T_s \\le 0)$ , or, has a small positive value which can be neglected in the analysis of the available experimental data.",
"The negative values of the critical temperature $T_s$ of the generic superconducting phase transition are generally possible and produce a variety of phase diagram topologies (Sec. 5).",
"Note, that the value of $T_s$ depends on the strength of the interaction mediating the formation of the spin-triplet Cooper pairs of electrons.",
"Therefore, for the sensitiveness of such electron couplings to the crystal lattice properties, the generic critical temperature $T_s$ depends on the pressure.",
"This is an effect which might be included in our theoretical scheme by introducing some convenient temperature dependence of $T_s$ .",
"To do this we need information either from experimental data or from a comprehensive microscopic theory.",
"Usually, $T_s\\le 0$ is interpreted as a lack of any superconductivity but here the same non-positive values of $T_s$ are effectively enhanced to positive values by the interaction parameter $\\gamma $ which triggers the superconductivity up to superconducting phase-transition temperatures $T_{FS}(P)> 0$ .",
"This is readily seen from Table 1, where we present the reduced critical temperatures on the FM-FS phase transition line $\\tilde{T}_{FS}(\\tilde{P})$ , calculated from the present theory, namely, $\\tilde{T}_m$ – the maximum of the curve $T_{FS}(P)$ (if available, see Sec.",
"5), the temperatures $\\tilde{T}_A$ and $\\tilde{T}_B$ , corresponding to the tricritical points A $\\equiv (\\tilde{T}_A, \\tilde{P}_A)$ and B $\\equiv (\\tilde{T}_B, \\tilde{P}_B)$ , and the temperature $\\tilde{T}_C$ , corresponding to the critical-end point C $\\equiv (\\tilde{T}_C, \\tilde{P}_C)$ .",
"The theoretical derivation of the dependence of the multicritical temperatures $\\tilde{T}_A$ , $\\tilde{T}_B$ and $\\tilde{T}_C$ on $\\gamma $ , $\\gamma _1$ , $\\kappa $ , and $\\tilde{T}_s$ , as well as the dependence of $\\tilde{T}_m$ on the same model parameters is outlined in Sec. 5.",
"All these temperatures as well as the whole phase transition line $T_{FS}(P)$ are considerably boosted above $T_s$ owing to positive terms of order $\\gamma ^2$ .",
"If $\\tilde{T}_s < 0$ , the superconductivity appears, provided $\\tilde{T}_m > 0$ , i.e., when $\\gamma ^2/4\\kappa \\gamma _1 > |\\tilde{T}_s|$ (see Table 1)."
],
[
"Temperature-pressure phase diagram",
"Although the structure of the FS phase is quite complicated, some of the results can be obtained in analytical form.",
"A more detailed outline of the phase domains, for example, in $T-P$ phase diagram, can be done by using suitable values of the material parameters in the free energy (REF ): $P_0$ , $T_{f0}$ , $T_s$ , $\\kappa $ , $\\gamma $ , and $\\gamma _1$ .",
"Here we present some of the analytical results for the phase transition lines and the multi-critical points.",
"Typical shapes of phase diagrams derived directly from Eq.",
"(REF ) are given in Figs. 2–7.",
"Figure 2 shows the phase diagram calculated from Eq.",
"(REF ) for parameters, corresponding to the experimental data [13] for ZrZn$_2$ .",
"Figures 3 and 4 show the low-temperature and the high-pressure parts of the same phase diagram (see Sec.",
"7 for details).",
"Figures 5–7 show the phase diagram calculated for the experimental data [2], [4] of UGe$_2$ (see Sec. 8).",
"In ZrZn$_2$ , UGe$_2$ , as well as in UCoGe and UIr, critical pressure $P_c$ exists, where both superconductivity and ferromagnetic orders vanish.",
"As in experiments, we find out from our calculation that in the vicinity of $P_0\\sim P_c$ the FM-FS phase transition is of fist order, denoted by the solid line BC in Figs.",
"3, 4, 6, and 7.",
"At lower pressure the same phase transition is of second orderq shown by the dotted lines in the same figures.",
"The second order phase transition line $\\tilde{T}_{FS}(P)$ separating the FM and FS phases is given by the solution of the equation $ \\tilde{T}_{FS}(\\tilde{P}) = \\tilde{T}_s +\\tilde{\\gamma _1}t_{FS}(\\tilde{P}) +\\tilde{\\gamma }[-t_{FS}(\\tilde{P})]^{1/2},$ where $t_{FS}(\\tilde{P}) = t(T_{FS}, \\tilde{P}) \\le 0$ , $\\tilde{\\gamma } = \\gamma /\\kappa $ , $\\tilde{\\gamma }_1 =\\gamma _1/\\kappa $ , and $0 < \\tilde{P} < \\tilde{P}_B$ ; $P_B$ is the pressure corresponding to the multi-critical point B, where the line $T_{FS}(P)$ terminates, as clearly shown in Figs.",
"4 and 7).",
"Note, that Eq.",
"(REF ) strictly coincides with the stability condition for the FM phase with respect to appearance of FS phase [20].",
"Additional information for the shape of this phase transition line can be obtained by the derivative $\\tilde{\\rho } =\\partial \\tilde{T}_{FS}(\\tilde{P})/\\partial \\tilde{P}$ , namely, $ \\tilde{\\rho } = \\frac{\\tilde{\\rho }_s +\\tilde{\\gamma _1}- \\tilde{\\gamma }/2(-t_{FS})^{1/2}}{1 -n\\tilde{T}_{FS}^{n-1}\\left[\\tilde{\\gamma _1} -\\tilde{\\gamma }/2[(-t_{FS})^{1/2} \\right]},$ where $\\tilde{\\rho }_s = \\partial \\tilde{T}_{s}(\\tilde{P})/\\partial \\tilde{P}$ .",
"Note, that Eq.",
"(REF ) is obtained from Eqs.",
"(REF ) and (REF ).",
"The shape of the line $\\tilde{T}_{FS}(P)$ can vary depending on the theory parameters (see, e.g., Figs.3 and 6).",
"For certain ratios of $\\tilde{\\gamma }$ , $\\tilde{\\gamma }_1$ , and values of $\\tilde{\\rho }_s$ , the curve $\\tilde{T}_{FS}(\\tilde{P})$ exhibits a maximum $\\tilde{T}_m =\\tilde{T}_{FS}(\\tilde{P}_m)$ , given by $\\tilde{\\rho }(\\tilde{\\rho }_s, T_m,P_m)=0$ .",
"This maximum is clearly seen in Figs.",
"6 and 7.",
"To locate the maximum we need to know $\\tilde{\\rho }_s$ .",
"We have already assumed $T_s$ does not depend on $P$ , as explained above, which from the physical point of view means that the function $T_s(P)$ is flat enough to allow the approximation $\\tilde{T}_{s}\\approx 0$ without a substantial error in the results.",
"From our choice of $P$ -dependence of the free energy [Eq.",
"(REF )] parameters, it follow that $\\tilde{\\rho }_s = 0 $ .",
"Setting $\\tilde{\\rho }_s =\\tilde{\\rho }=0 $ in Eq.",
"(REF ) we obtain $ t(T_m,P_m)=-\\frac{\\tilde{\\gamma }^2}{4\\tilde{\\gamma }_1^2},$ namely, the value $t_m(T,P) = t(T_m,P_m)$ at the maximum $T_m(P_m)$ of the curve $T_{FS}(P)$ .",
"Substituting $t_m$ back in Eq.",
"(REF ) we obtain $T_m$ , and with its help we also obtain the pressure $P_m$ , both given in Table 1, respectively.",
"We want to draw the attention to a particular feature of the present theory that the coordinates $T_m$ and $P_m$ of the maximum (point max) at the curve $T_{FS}(P)$ as well as the results from various calculations with the help of Eqs.",
"(REF ) and (REF ) are expressed in terms of the reduced interaction parameters $\\tilde{\\gamma }$ and $\\tilde{\\gamma }_1$ .",
"Thus, using certain experimental data for $T_m$ , $P_m$ , as well as Eqs.",
"(REF ) and (REF ) for $T_{FS}$ , $T_s$ , and the derivative $\\rho $ at particular values of the pressure $P$ , $\\tilde{\\gamma }$ and $\\tilde{\\gamma }_1$ can be calculated without any additional information, for example, for the parameter $\\kappa $ .",
"This property of the model (REF ) is quite useful in the practical work with the experimental data.",
"Figure: T-PT-P diagram of ZrZn 2 _2 calculated forT s =0T_s=0, T f0 =28.5T_{f0}=28.5 K, P 0 =21P_0 = 21 kbar, κ=10\\kappa = 10,γ ˜=2γ 1 ˜≈0.2\\tilde{\\gamma } = 2\\tilde{\\gamma _1} \\approx 0.2, and n=1n=1.",
"Thedotted line represents the FM-FS transition and the dashed linestands for the second order N-FM transition.",
"The dotted line has azero slope at P=0P=0.",
"The low-temperature and high-pressure domainsof the FS phase are seen more clearly in the following Figs.",
"3 and4.The conditions for existence of a maximum on the curve $T_{FS}(P)$ can be determined by requiring $\\tilde{P}_{m} > 0$ , and $\\tilde{T}_m > 0$ and using the respective formulae for these quantities, shown in Table 1.",
"This max always occurs in systems where $T_{FS}(0) \\le 0$ and the low-pressure part of the curve $T_{FS}(P)$ terminates at $T=0$ for some non-negative critical pressure $P_{0c}$ (see Sec. 6).",
"But the max may occur also for some sets of material parameters, when $T_{FS}(0)>0$ (see Fig.",
"3, where $P_m =0$ ).",
"All these shapes of the line $T_{FS}(P)$ are described by the model (REF ).",
"Irrespectively of the particular shape, the curve $T_{FS}(P)$ given by Eq.",
"(REF ) always terminates at the tricritical point (labeled B), with coordinates $(P_B,T_B)$ (see, e.g., Figs.",
"4 and 7).",
"Figure: Details of Fig.",
"2 with expanded temperaturescale.",
"The points A, B, C are located in the high-pressure part(P∼P c ∼21kbarP\\sim P_c\\sim 21~\\mbox{kbar}).",
"The maxmax point is at P≈0P\\approx 0 kbar.",
"The FS phase domain is shaded.",
"The dotted line shows thesecond order FM-FS phase transition with P m ≈0P_m \\approx 0.",
"Thesolid straight line BC shows the fist-order FM-FS transition forP>P B P > P_B.",
"The quite flat solid line AC shows the first order N-FStransition (the lines BC and AC are more clearly seen in Fig.",
"4.The dashed line stands for the second order N-FM transition.At pressure $P > P_B$ the FM-FS phase transition is of first order up to the critical-end point C. For $P_B < P < P_C$ the FM-FS phase transition is given by the straight line BC (see, e.g., Figs.",
"4 and 7).",
"The lines of all three phase transitions, N-FM, N-FS, and FM-FS, terminate at point C. For $P > P_C$ the FM-FS phase transition occurs on a rather flat smooth line of equilibrium transition of first order up to a second tricritical point A with $P_A \\sim P_0$ and $T_A \\sim 0$ .",
"Finally, the third transition line terminating at the point C describes the second order phase transition N-FM.",
"The reduced temperatures $\\tilde{T}_N$ and pressures $\\tilde{P}_N$ , $N$ = (A, B, C, max) at the three multi-critical points (A, B, and C), and the maximum $T_m(P_m)$ are given in Table 1.",
"Note that, for any set of material parameters, $T_A < T_C < T_B < T_m$ and $P_m<P_B<P_C<P_A$ .",
"There are other types of phase diagrams, resulting from model (REF ).",
"For negative values of the generic superconducting temperature $T_s$ , several other topologies of the $T-P$ diagram can be outlined.",
"The results for the multicritical points, presented in Table 1, shows that, when $T_s$ lowers below $T=0$ , $T_C$ also decreases, first to zero, and then to negative values.",
"When $T_C=0$ the direct N-FS phase transition of first order disappears and point C becomes a very special zero-temperature multicritical point.",
"As seen from Table 1, this happens for $T_s = - \\gamma ^2T_f(0)/4\\kappa (1+\\gamma _1)$ .",
"The further decrease of $T_s$ causes point C to fall below the zero temperature and then the zero-temperature phase transition of first order near $P_c$ splits into two zero-temperature phase transitions: a second order N-FM transition and a first order FM-FS transition, provided $T_B$ still remains positive.",
"At lower $T_s$ also point B falls below $T=0$ and the FM-FS phase transition becomes entirely of second order.",
"For very extreme negative values of $T_s$ , a very large pressure interval below $P_c$ may occur where the FM phase is stable up to $T=0$ .",
"Then the line $T_{FS}(P)$ will exist only for relatively small pressure values $(P \\ll P_c)$ .",
"This shape of the stability domain of the FS phase is also possible in real systems.",
"Figure: High-pressure part of the phase diagram ofZrZn 2 _2, shown in Fig. 1.",
"The thick solid lines AC and BC showthe first-order transitions N-FS, and FM-FS, respectively.",
"Othernotations are explained in Figs.",
"2 and 3."
],
[
"Quantum phase transitions",
"We have shown that the free energy (REF ) describes zero temperature phase transitions.",
"Usually, the properties of these phase transitions essentially depend on the quantum fluctuations of the order parameters.",
"For this reason the phase transitions at ultralow and zero temperature are called quantum phase transitions [31], [32].",
"The time-dependent quantum fluctuations (correlations) which describe the intrinsic quantum dynamics of spin-triplet ferromagnetic superconductors at ultralow temperatures are not included in our consideration but some basic properties of the quantum phase transitions can be outlines within the classical limit described by the free energy models (REF ) and (REF ).",
"Let we briefly clarify this point.",
"The classical fluctuations are entirely included in the general GL functional (REF )–(REF ) but the quantum fluctuations should be added in a further generalization of the theory.",
"Generally, both classical (thermal) and quantum fluctuations are investigated by the method of the renormalization group (RG) [31], which is specially intended to treat the generalized action of system, where the order parameter fields ($\\mbox{$\\varphi $}$ and $\\mbox{$M$}$ ) fluctuate in time $t$ and space $\\vec{x}$ [31], [32].",
"These effects, which are beyond the scope of the paper, lead either to a precise treatment of the narrow critical region in a very close vicinity of second order phase transition lines or to a fluctuation-driven change in the phase-transition order.",
"But the thermal fluctuations and quantum correlation effects on the thermodynamics of a given system can be unambiguously estimated only after the results from counterpart simpler theory, where these phenomena are not present, are known and, hence, the distinction in the thermodynamic properties predicted by the respective variants of the theory can be established.",
"Here we show that the basic low-temperature and ultralow-temperature properties of the spin-triplet ferromagnetic superconductors, as given by the preceding experiments, are derived from the model (REF ) without any account of fluctuation phenomena and quantum correlations.",
"The latter might be of use in a more detailed consideration of the close vicinity of quantum critical points in the phase diagrams of ferromagnetic spin-triplet superconductors.",
"Here we show that the theory predicts quantum critical phenomena only for quite particular physical conditions whereas the low-temperature and zero-temperature phase transitions of first order are favored by both symmetry arguments and detailed thermodynamic analysis.",
"There is a number of experimental [9], [40] and theoretical [17], [41], [42] investigations of the problem for quantum phase transitions in unconventional ferromagnetic superconductors, including the mentioned intermetallic compounds.",
"Some of them are based on different theoretical schemes and do not refer to the model (REF ).",
"Others, for example, those in Ref.",
"[41] reported results about the thermal and quantum fluctuations described by the model (REF ) before the comprehensive knowledge for the results from the basic treatment reported in the present investigation.",
"In such cases one could not be sure about the correct interpretation of the results from the RG and the possibilities for their application to particular zero-temperature phase transitions.",
"Here we present basic results for the zero-temperature phase transitions described by the model (REF ).",
"Figure: T-PT-P diagram of UGe 2 _2 calculated takingT s =0T_s=0, T f0 =52T_{f0}=52 K, P 0 =1.6P_0 = 1.6 GPa, κ=4\\kappa = 4,γ ˜=0.0984\\tilde{\\gamma } = 0.0984, γ 1 ˜=0.1678\\tilde{\\gamma _1} = 0.1678, and n=1n=1.The dotted line represents the FM-FS transition and the dashedline stands for the N-FM transition.",
"The low-temperature andhigh-pressure domains of the FS phase are seen more clearly in thefollowing Figs.",
"6 and 7.The RG investigation [41] has demonstrated up to two loop order of the theory that the thermal fluctuations of the order parameter fields rescale the model (REF ) in a way which corresponds to first order phase transitions in magnetically anisotropic systems.",
"This result is important for the metallic compounds we consider here because in all of them magnetic anisotropy is present.",
"The uniaxial magnetic anisotropy in ZrZn$_2$ is much weaker than in UGe$_2$ but cannot be neglected when fluctuation effects are accounted for.",
"Owing to the particular symmetry of model (REF ), for the case of magnetic isotropy (Heisenberg symmetry), the RG study reveals an entirely different class of (classical) critical behavior.",
"Besides, the different spatial dimensions of the superconducting and magnetic quantum fluctuations imply a lack of stable quantum critical behavior even when the system is completely magnetically isotropic.",
"The pointed arguments and preceding results lead to the reliable conclusion that the phase transitions, which have already been proven to be first order in the lowest-order approximation, where thermal and quantum fluctuations are neglected, will not undergo a fluctuation-driven change in the phase transition order from first to second.",
"Such picture is described below, in Sec.",
"8, and it corresponds to the behavior of real compounds.",
"Our results definitely show that the quantum phase transition near $P_c$ is of first order.",
"This is valid for the whole N-FS phase transition below the critical-end point C, as well as the straight line BC.",
"The simultaneous effect of thermal and quantum fluctuations do not change the order of the N-FS transition, and it is quite unlikely to suppose that thermal fluctuations of the superconductivity field $\\mbox{$\\psi $}$ can ensure a fluctuation-driven change in the order of the FM-FS transition along the line BC.",
"Usually, the fluctuations of $\\mbox{$\\psi $}$ in low temperature superconductors are small and slightly influence the phase transition in a very narrow critical region in the vicinity of the phase-transition point.",
"This effect is very weak and can hardly be observed in any experiment on low-temperature superconductors.",
"Besides, the fluctuations of the magnetic induction $\\mbox{$B$}$ always tend to a fluctuation-induced first-order phase transition rather than to the opposite effect - the generation of magnetic fluctuations with infinite correlation length at the equilibrium phase-transition point and, hence, a second order phase transition [31], [43].",
"Thus we can quire reliably conclude that the first-order phase transitions at low-temperatures, represented by the lines BC and AC in vicinity of $P_c$ do not change their order as a result of thermal and quantum fluctuation fluctuations.",
"Figure: Low-temperature part of the T-PT-P phasediagram of UGe 2 _2, shown in Fig. 5.",
"The points A, B, C arelocated in the high-pressure part (P∼P c ∼1.6P\\sim P_c\\sim 1.6 GPa).",
"TheFS phase domain is shaded.",
"The thick solid lines AC and BC showthe first-order transitions N-FS, and FM-FS, respectively.",
"Othernotations are explained in Figs.",
"2 and 3.Quantum critical behavior for continuous phase transitions in spin-triplet ferromagnetic superconductors with magnetic anisotropy can therefore be observed at other zero-temperature transitions, which may occur in these systems far from the critical pressure $P_c$ .",
"This is possible when $T_{FS}(0) = 0$ and the $T_{FS}(P)$ curve terminates at $T=0$ at one or two quantum (zero-temperature) critical points: $P_{0c} < P_m$ - “lower critical pressure\", and $P_{0c}^{\\prime }>P_m$ – “upper critical pressure.\"",
"In order to obtain these critical pressures one should solve Eq.",
"(REF ) with respect to $P$ , provided $T_{FS}(P) =0$ , $T_m > 0$ and $P_m>0$ , namely, when the continuous function $T_{FS}(P)$ exhibits a maximum.",
"The critical pressure $P_{0c}^{\\prime }$ is bounded in the relatively narrow interval ($P_m,P_B$ ) and can appear for some special sets of material parameters ($r,t,\\gamma ,\\gamma _1$ ).",
"In particular, as our calculations show, $P_{0c}^{\\prime }$ do not exists for $T_s \\ge 0$ ."
],
[
"Criteria for type I and type II spin-triplet ferromagnetic superconductors",
"The analytical calculation of the critical pressures $P_{0c}$ and $P_{0c}^{\\prime }$ for the general case of $T_s \\ne 0$ leads to quite complex conditions for appearance of the second critical field $P_{0c}^{\\prime }$ .",
"The correct treatment of the case $T_s\\ne 0$ can be performed within the entire two-domain picture for the phase FS (see, also, Ref. [20]).",
"The complete study of this case is beyond our aims but here we will illustrate our arguments by investigation of the conditions, under which the critical pressure $P_{oc}$ occurs in systems with $T_s \\approx 0$ .",
"Moreover, we will present the general result for $P_{0c} \\ge 0$ and $P_{0c}^{\\prime } \\ge 0$ in systems where $T_s \\ne 0$ .",
"Figure: High-pressure part of the phase diagram ofUGe 2 _2, shown in Fig.",
"4.",
"Notations are explained in Figs.",
"2, 3,5, and 6.Setting $T_{FS}(P_{0c}) = 0 $ in Eq.",
"(REF ) we obtain the following quadratic equation, $ \\tilde{\\gamma }_1 m^{2}_{0c} - \\tilde{\\gamma }m_{0c} -\\tilde{T}_s = 0,$ for the reduced magnetization, $ m_{0c} = [-t(0,\\tilde{P}_{oc})]^{1/2} =(1-\\tilde{P}_{0c})^{1/2}$ and, hence, for $\\tilde{P}_{0c}$ .",
"For $T_s \\ne 0$ , Eqs.",
"(REF ) and (REF ) have two solutions with respect to $\\tilde{P}_{0c}$ .",
"For some sets of material parameters these solutions satisfy the physical requirements for $P_{0c}$ and $P_{0c}^{\\prime }$ and can be identified with the critical pressures.",
"The conditions for existence of $P_{0c}$ and $P_{0c}^{\\prime }$ can be obtained either by analytical calculations or by numerical analysis for particular values of the material parameters.",
"For $T_s=0$ , the trivial solution $\\tilde{P}_{0c} = 1$ corresponds to $P_{0c} = P_0 > P_B$ and, hence, does not satisfy the physical requirements.",
"The second solution, $ \\tilde{P}_{0c} = 1 -\\frac{\\tilde{\\gamma }^2}{\\tilde{\\gamma }^2_1}$ is positive for $ \\frac{\\gamma _1}{\\gamma } \\ge 1$ and, as shown below, it gives the location of the quantum critical point $(T=0, P_{0c} < P_m)$ .",
"At this quantum critical point, the equilibrium magnetization $m_{0c}$ is given by $m_{0c} = \\gamma /\\gamma _1$ and is twice bigger that the magnetization $m_{m} =\\gamma /2\\gamma _1$ ([20]) at the maximum of the curve $T_{FS}(P)$ .",
"To complete the analysis we must show that the solution (REF ) satisfies the condition $P_{0c} < \\tilde{P}_m$ .",
"By taking $\\tilde{P}_m$ from Table 1, we can show that solution (REF ) satisfies the condition $P_{0c} < \\tilde{P}_m$ for $n=1$ , if $ \\gamma _1 < 3\\kappa ,$ and for $n=2$ (SFT case), when $ \\gamma < 2\\sqrt{3}\\kappa .$ Finally, we determine the conditions under which the maximum $T_m$ of the curve $T_{FS}(P)$ occurs at non-negative pressures.",
"For $n=1$ , we obtain that $P_m \\ge 0$ for $n=1$ , if $ \\frac{\\gamma _1}{\\gamma } \\ge \\frac{1}{2}\\left(1 +\\frac{\\gamma _1}{\\kappa }\\right)^{1/2},$ whereas for $n=2$ , the condition is $ \\frac{\\gamma _1}{\\gamma } \\ge \\frac{1}{2}\\left(1+\\frac{\\gamma ^2}{4\\kappa ^2}\\right)^{1/2}.$ Obviously, the conditions (REF )-(REF ) are compatible with one another.",
"The condition (REF ) is weaker than the condition Eq.",
"(REF ), provided the inequality (REF ) is satisfied.",
"The same is valid for the condition (REF ) if the inequality (REF ) is valid.",
"In Sec.",
"8 we will show that these theoretical predictions are confirmed by the experimental data.",
"Doing in the same way the analysis of Eq.",
"(REF ), some results may easily obtained for $T_s \\ne 0$ .",
"In this more general case the Eq.",
"(REF ) has two nontrivial solutions, which yield two possible values of the critical pressure $ \\tilde{P}_{0c(\\pm )} = 1 -\\frac{\\gamma ^2}{4\\gamma _1^2}\\left[1 \\pm \\left(1+\\frac{4\\tilde{T}_s\\kappa \\gamma _1}{\\gamma ^2}\\right)^{1/2}\\right]^2.$ The relation $\\tilde{P}_{0c(-)} \\ge \\tilde{P}_{0c(+)}$ is always true.",
"Therefore, to have both $\\tilde{P}_{0c(\\pm )} \\ge 0$ , it is enough to require $\\tilde{P}_{0c(+)} \\ge 0$ .",
"Having in mind that for the phase diagram shape, we study $\\tilde{T}_m > 0$ , and according to the result for $\\tilde{T}_m$ in Table 1, this leads to the inequality $\\tilde{T}_s > -\\gamma ^2/4\\kappa \\gamma _1$ .",
"So, we obtain that $\\tilde{P}_{0c(+)} \\ge 0$ will exist, if $ \\frac{\\gamma _1}{\\gamma } \\ge 1 +\\frac{\\kappa \\tilde{T}_s}{\\gamma },$ which generalizes the condition (REF ).",
"Now we can identify the pressure $P_{0c(+)}$ with the lower critical pressure $P_{0c}$ , and $P_{0c(-)}$ with the upper critical pressure $P_{0c}^{\\prime }$ .",
"Therefore, for wide variations in the parameters, theory (REF ) describes a quantum critical point $P_{oc}$ , that exists, provided the condition (REF ) is satisfied.",
"The quantum critical point $(T=0,P_{0c})$ exists in UGe$_2$ and, perhaps, in other $p$ -wave ferromagnetic superconductors, for example, in UIr.",
"Our results predict the appearance of second critical pressure – the upper critical pressure $P_{oc}^{\\prime }$ that exists under more restricted conditions and, hence, can be observed in more particular systems, where $T_s < 0$ .",
"As mentioned in Sec.",
"5, for very extreme negative values of $T_s$ , when $T_B < 0$ , the upper critical pressure $P_{0c}^{\\prime }>0$ occurs, whereas the lower critical pressure $P_{0c}>0$ does not appear.",
"Bue especially this situation should be investigated in a different way, namely, one should keep $T_{FS}(0)$ different from zero in Eq.",
"(REF ), and consider a form of the FS phase domain in which the curve $T_{FS}(P)$ terminates at $T=0$ for $P_{0c}^{\\prime }>0$ , irrespective of whether the maximum $T_m$ exists or not.",
"In such geometry of the FS phase domain, the maximum $T(P_m)$ may exist only in quite unusual cases, if it exists at all.",
"Using criteria like (REF ) in Sec.",
"8.4 we classify these superconductors in two types: (i) type I, when the condition (REF ) is satisfied, and (ii) type II, when the same condition does not hold.",
"As we show in Sec.",
"8.2, 8.3 and 8.4, the condition (REF ) is satisfied by UGe$_2$ but the same condition fails for ZrZn$_2$ .",
"For this reason the phase diagrams of UGe$_2$ and ZrZn$_2$ exhibit qualitatively different shapes of the curves $T_FS(P)$ .",
"For UGe$_2$ the line $T_FS(P)$ has a maximum at some pressure $P >0$ , whereas the line $T_FS(P)$ , corresponding to ZrZn$_2$ , does not exhibit such maximum (see also Sec.",
"8).",
"The quantum and thermal fluctuation phenomena in the vicinities of the two critical pressures $P_{0c}$ and $P_{0c}^{\\prime }$ need a nonstandard RG treatment because they are related with the fluctuation behavior of the superconducting field $\\mbox{$\\psi $}$ far below the ferromagnetic phase transitions, where the magnetization $\\mbox{$M$}$ does not undergo significant fluctuations and can be considered uniform.",
"The presence of uniform magnetization produces couplings of $\\mbox{$M$}$ and $\\mbox{$\\psi $}$ which are not present in previous RG studies and need a special analysis."
],
[
"Theoretical outline of the phase diagram",
"In order to apply the above displayed theoretical calculations, following from free energy (REF ), for the outline of $T-P$ diagram of any material, we need information about the values of $P_0$ , $T_{f0}$ , $T_s$ , $\\kappa $ $\\gamma $ , and $\\gamma _1$ .",
"The temperature $T_{f0}$ can be obtained directly from the experimental phase diagrams.",
"The pressure $P_0$ is either identical or very close to the critical pressure $P_c$ , for which the N-FM phase transition line terminates at $T \\sim 0$ .",
"The temperature $T_s$ of the generic superconducting transition is not available from the experiments because, as mentioned above, pure superconducting phase not coexisting with ferromagnetism has not been observed.",
"This can be considered as an indication that $T_s$ is very small and does not produce a measurable effect.",
"So the generic superconducting temperature will be estimated on the basis of the following arguments.",
"For $T_f(P)> T_s$ we must have $T_s(P) =0$ at $P \\ge P_c$ , where $T_f(P) \\le 0$ , and for $0 \\le P\\le P_0$ , $T_s < T_C$ .",
"Therefore for materials where $T_C$ is too small to be observed experimentally, $T_s$ can be ignored.",
"As far as the shape of FM-FS transition line is well described by Eq.",
"(REF ), we will make use of additional data from available experimental phase diagrams for ferroelectric superconductors.",
"For example, in ZrZn$_2$ these are the observed values of $T_{FS}(0)$ and the slope $\\rho _0 \\equiv [\\partial T_{FS}(P)/\\partial P]_0 =(T_{f0}/P_0)\\tilde{\\rho }_0 $ at $P=0$ ; see Eq.",
"(REF ).",
"For UGe$_2$ , where a maximum ($\\tilde{T}_m$ ) is observed on the phase-transition line, we can use the experimental values of $T_m$ , $P_m$ , and $P_{0c}$ .",
"The interaction parameters $\\tilde{\\gamma }$ and $\\tilde{\\gamma _1}$ are derived using Eq.",
"(REF ), and the expressions for $\\tilde{T}_m$ , $\\tilde{P}_m$ , and $\\tilde{\\rho }_0$ , see Table 1.",
"The parameter $\\kappa $ is chosen by fitting the expression for the critical-end point $T_C$ ."
],
[
"ZrZn$_2$",
"Experiments for ZrZn$_2$ [13] gives the following values: $T_{f0} =28.5$ K, $T_{FS}(0) = 0.29$ K, $P_0 \\sim P_c = 21$ kbar.",
"The curve $T_F(P)\\sim T_f(P)$ is almost a straight line, which directly indicates that $n=1$ is adequate in this case for the description of the $P$ -dependence.",
"The slope for $T_{FS}(P)$ at $P=0$ is estimated from the condition that its magnitude should not exceed $T_{f0}/P_c \\approx 0.014$ as we have assumed that is straight one, so as a result we have $-0.014 <\\rho \\le 0$ .",
"This ignores the presence of a maximum.",
"The available experimental data for ZrZn$_2$ do not give clear indication whether a maximum at ($T_m$ , $P_m$ ) exists.",
"If such a maximum were at $P=0$ we would have $\\rho _0= 0$ , whereas a maximum with $T_m \\sim T_{FS}(0)$ and $P_m \\ll P_0$ provides us with an estimated range $0\\le \\rho _0 < 0.005$ .",
"The choice $\\rho _0 = 0$ gives $\\tilde{\\gamma } \\approx 0.02$ and $\\tilde{\\gamma }_1 \\approx 0.01$ , but similar values hold for any $|\\rho _0| \\le 0.003$ .",
"The multicritical points A and C cannot be distinguished experimentally.",
"Since the experimental accuracy [13] is less than $\\sim 25$ mK in the high-$P$ domain ($P\\sim 20-21$ kbar), we suppose that $T_C \\sim 10$ mK, which corresponds to $\\kappa \\sim 10$ .",
"We employed these parameters to calculate the $T-P$ diagram using $\\rho _0 = 0$ and $0.003$ .",
"The differences obtained in these two cases are negligible, with both phase diagrams being in excellent agreement with experiment.",
"Phase diagram of ZrZn$_2$ calculated directly from the free energy (REF ) for $n=1$ , the above mentioned values of $T_s$ , $P_0$ , $T_{f0}$ , $\\kappa $ , and values of $\\tilde{\\gamma } \\approx 0.2$ and $\\tilde{\\gamma }_1 \\approx 0.1$ which ensure $\\rho _0 \\approx 0$ is shown in Fig.",
"2.",
"Note, that the experimental phase diagram [13] of ZrZn$_2$ looks almost exactly as the diagram in Fig.",
"2, which has been calculated directly from the model (REF ) without any approximations and simplifying assumptions.",
"The phase diagram in Fig.",
"2 has the following coordinates of characteristic points: $P_A\\sim P_c= 21.42$ kbar, $P_B =20.79$ kbar, $P_C = 20.98$ kbar, $T_A=T_F(P_c)=T_{FS}(P_c) = 0$ K, $T_B=0.0495$ K, $T_C =0.0259$ K, and $T_{FS}(0) =0.285$ K. The low-$T$ region is seen in more detail in Fig.",
"3, where the A, B, C points are shown and the order of the FM-FS phase transition changes from second to first order around the critical end-point C. The $T_{FS}(P)$ curve, shown by the dotted line in Fig.",
"3, has a maximum $T_m=0.290$ K at $P = 0.18$ kbar, which is slightly above $T_{FS}(0) = 0.285$ K. The straight solid line BC in Fig.",
"3 shows the first order FM-FS phase transition which occurs for $P_B < P <P_C$ .",
"The solid AC line shows the first order N-FS phase transition and the dashed line stands for the N-FM phase transition of second order.",
"Although the expanded temperature scale in Fig.",
"3, the difference $[T_m-T_{FS}(0)] = 5$ mK is hard to see.",
"To locate the point max exactly at $P=0$ one must work with values of $\\tilde{\\gamma }$ and $\\tilde{\\gamma }_1$ of accuracy up to $10^{-4}$ .",
"So, the location of the max for parameters corresponding to ZrZn$_2$ is very sensitive to small variations of $\\tilde{\\gamma }$ and $\\tilde{\\gamma }_1$ around the values $0.2$ and $0.1$ , respectively.",
"Our initial idea was to present a diagram with $T_m=T_{FS}(0) = 0.29$ K and $\\rho _0 = 0$ , namely, max exactly located at $P=0$ , but the final phase diagram slightly departs from this picture because of the mentioned sensitivity of the result on the values of the interaction parameters $\\gamma $ and $\\gamma _1$ .",
"The theoretical phase diagram of ZrZn$_2$ can be deduced in the same way for $\\rho _0 = 0.003$ and this yields $T_m= 0.301$ K at $P_m=6.915$ kbar for initial values of $\\tilde{\\gamma }$ and $\\tilde{\\gamma }_1$ which differs from $\\tilde{\\gamma } = 2\\tilde{\\gamma }_1 = 0.2$ only by numbers of order $10^{-3}-10^{-4}$ [18].",
"This result confirms the mentioned sensitivity of the location of the maximum $T_m$ towards slight variations of the material parameters.",
"Experimental investigations of this low-temperature/low-pressure region with higher accuracy may help in locating this maximum with better precision.",
"Fig.",
"4 shows the high-pressure part of the same phase diagram in more details.",
"In this figure the first order phase transitions (solid lines BC and AC) are clearly seen.",
"In fact the line AC is quite flat but not straight as the line BC.",
"The quite interesting topology of the phase diagram of ZrZn$_2$ in the high-pressure domain ($P_B < P < P_A$ ) is not seen in the experimental phase diagram [13] because of the restricted accuracy of the experiment in this range of temperatures and pressures.",
"These results account well for the main features of the experimental behavior [13], including the claimed change in the order of the FM-FS phase transition at relatively high $P$ .",
"Within the present model the N-FM transition is of second order up to $P_C \\sim P_c$ .",
"Moreover, if the experiments are reliable in their indication of a first order N-FM transition at much lower $P$ values, the theory can accommodate this by a change of sign of $b_f$ , leading to a new tricritical point located at a distinct $P_{tr} < P_C$ on the N-FM transition line.",
"Since $T_C>0$ a direct N-FS phase transition of first order is predicted in accord with conclusions from de Haas–van Alphen experiments [44] and some theoretical studies [40].",
"Such a transition may not occur in other cases where $T_C=0$ .",
"In SFT ($n=2$ ) the diagram topology remains the same but points B and C are slightly shifted to higher $P$ (typically by about $0.01--0.001$ kbar)."
],
[
"UGe$_2$",
"The experimental data for UGe$_2$ indicate $T_{f0} = 52$ K, $P_c=1.6$ GPa ($\\equiv 16$ kbar), $T_m = 0.75$ K, $P_m\\approx 1.15$ GPa, and $P_{0c} \\approx 1.05$ GPa [2], [3], [4], [5].",
"Using again the variant $n=1$ for $T_f(P)$ and the above values for $T_m$ and $P_{0c}$ we obtain $\\tilde{\\gamma } \\approx 0.0984$ and $\\tilde{\\gamma _1} \\approx 0.1678$ .",
"The temperature $T_C \\sim 0.1$ K corresponds to $\\kappa \\sim 4$ .",
"Using these initial parameters, together with $T_s=0$ , leads to the $T-P$ diagram of UGE$_2$ shown in Fig. 5.",
"We obtain $T_A=0$ K, $P_A = 1.723$ GPa, $T_B=0.481$ K, $P_B = 1.563$ GPa, $T_C=0.301$ K, and $P_C=1.591$ GPa.",
"Figs.",
"6 and 7 show the low-temperature and the high-pressure parts of this phase diagram, respectively.",
"There is agreement with the main experimental findings, although $P_m$ corresponding to the maximum (found at $\\sim 1.44$ GPa in Fig.",
"5) is about 0.3 GPa higher than suggested experimentally [4], [5].",
"If the experimental plots are accurate in this respect, this difference may be attributable to the so-called ($T_x$ ) meta-magnetic phase transition in UGe$_2$ , which is related to an abrupt change of the magnetization in the vicinity of $P_m$ .",
"Thus, one may suppose that the meta-magnetic effects, which are outside the scope of our current model, significantly affect the shape of the $T_{FS}(P)$ curve by lowering $P_m$ (along with $P_B$ and $P_C$ ).",
"It is possible to achieve a lower $P_m$ value (while leaving $T_m$ unchanged), but this has the undesirable effect of modifying $P_{c0}$ to a value that disagrees with experiment.",
"In SFT $(n=2)$ the multi-critical points are located at slightly higher $P$ (by about 0.01 GPa), as for ZrZn$_2$ .",
"Therefore, the results from the SFT theory are slightly worse than the results produced by the usual linear approximation ($n=1$ ) for the parameter $t$ ."
],
[
"Two types of ferromagnetic superconductors with\nspin-triplet electron pairing",
"The estimates for UGe$_2$ imply $\\gamma _1\\kappa \\approx 1.9$ , so the condition for $T_{FS}(P)$ to have a maximum found from Eq.",
"(REF ) is satisfied.",
"As we discussed for ZrZn$_2$ , the location of this maximum can be hard to fix accurately in experiments.",
"However, $P_{c0}$ can be more easily distinguished, as in the UGe$_2$ case.",
"Then we have a well-established quantum (zero-temperature) phase transition of second order, i.e., a quantum critical point at some critical pressure $P_{0c} \\ge 0$ .",
"As shown in Sec.",
"6, under special conditions the quantum critical points could be two: at the lower critical pressure $P_{0c} < P_m$ and the upper critical pressure $P_{0c}^{\\prime } <P_m$ .",
"This type of behavior in systems with $T_s=0$ (as UGe$_2$ ) occurs when the criterion (REF ) is satisfied.",
"Such systems (which we label as U-type) are essentially different from those such as ZrZn$_2$ where $\\gamma _1 < \\gamma $ and hence $T_{FS}(0) >0$ .",
"In this latter case (Zr-type compounds) a maximum $T_m> 0$ may sometimes occur, as discussed earlier.",
"We note that the ratio $\\gamma /\\gamma _1$ reflects a balance effect between the two $\\mbox{$\\psi $}$ -$\\mbox{$M$}$ interactions.",
"When the trigger interaction (typified by $\\gamma $ ) prevails, the Zr-type behavior is found where superconductivity exists at $P=0$ .",
"The same ratio can be expressed as $\\gamma _0/\\delta _0 M_0$ , which emphasizes that the ground state value of the magnetization at $P=0$ is also relevant.",
"Alternatively, one may refer to these two basic types of spin-triplet ferromagnetic superconductors as \"type I\" (for example, for the \"Zr-type compounds), and \"type II\" – for the U-type compounds.",
"As we see from this classification, the two types of spin-triplet ferromagnetic superconductors have quite different phase diagram topologies although some fragments have common features.",
"The same classification can include systems with $T_s\\ne 0$ but in this case one should use the more general criterion (REF )."
],
[
"Other compounds",
"In URhGe, $T_{f}(0) \\sim 9.5$ K and $T_{FS}(0) = 0.25$ K and, therefore, as in ZrZn$_2$ , here the spin-triplet superconductivity appears at ambient pressure deeply in the ferromagnetic phase domain [6], [7], [8].",
"Although some similar structural and magnetic features are found in UGe$_2$ the results in Ref.",
"[8] of measurements under high pressure show that, unlike the behavior of ZrZn$_2$ and UGe$_2$ , the ferromagnetic phase transition temperature $T_{F}(P)\\sim T_{f}(P)$ has a slow linear increase up to 140 kbar without any experimental indications that the N-FM transition line may change its behavior at higher pressures and show a negative slope in direction of low temperature up to a quantum critical point $T_{F}=0$ at some critical pressure $P_c$ .",
"Such a behavior of the generic ferromagnetic phase transition temperature cannot be explained by our initial assumption for the function $T_f(P)$ which was intended to explain phase diagrams where the ferromagnetic order is depressed by the pressure and vanishes at $T=0$ at some critical pressure $P_c$ .",
"The $T_{FS}(P)$ line of URhGe shows a clear monotonic negative slope to $T=0$ at pressures above 15 kbar and the extrapolation [8] of the experimental curve $T_{FS}(P)$ tends a quantum critical point $T_{FS}(P_{oc}^{\\prime })=0$ at $P_{0c} \\sim 25-30$ kbar.",
"Within the framework of the phenomenological theory (REF , this $T-P$ phase diagram can be explained after a modification on the $T_f(P)$ -dependence is made, and by introducing a convenient nontrivial pressure dependence of the interaction parameter $\\gamma $ .",
"Such modifications of the present theory are possible and follow from important physical requirements related with the behavior of the $f$ -band electrons in URhGe.",
"Unlike UGe$_2$ , where the pressure increases the hybridization of the $5f$ electrons with band states lading to a suppression of the spontaneous magnetic moment $M$ , in URhGe this effects is followed by a stronger effect of enhancement of the exchange coupling due to the same hybridization, and this effect leads to the slow but stable linear increase in the function $T_F(P)$[8].",
"These effects should be taken into account in the modeling the pressure dependence of the parameters of the theory (REF ) when applied to URhGe.",
"Another ambient pressure FS phase has been observed in experiments with UCoGe [9].",
"Here the experimentally derived slopes of the functions $T_{F}(P)$ and $T_{FS}(P)$ at relatively small pressures are opposite compared to those for URhGe and, hence, the $T-P$ phase diagram of this compound can be treated within the present theoretical scheme without substantial modifications.",
"Like in UGe$_2$ , the FS phase in UIr [12] is embedded in the high-pressure/low-temperature part of the ferromagnetic phase domain near the critical pressure $P_c$ which means that UIr is certainly a U-type compound.",
"In UGe$_2$ there is one metamagnetic phase transition between two ferromagnetic phases (FM1 and FM2), in UIr there are three ferromagnetic phases and the FS phase is located in the low-$T$ /high-$P$ domain of the third of them - the phase FM3.",
"There are two metamagnetic-like phase transitions: FM1-FM2 transition which is followed by a drastic decrease of the spontaneous magnetization when the the lower-pressure phase FM1 transforms to FM2, and a peak of the ac susceptibility but lack of observable jump of the magnetization at the second (higher pressure) “metamagnetic\" phase transition from FM2 to FM3.",
"Unlike the picture for UGe$_2$ , in UIr both transitions, FM1-FM2 and FM2-FM3 are far from the maximum $T_m(P_m)$ so in this case one can hardly speculate that the max is produced by the nearby jump of magnetization.",
"UIr seems to be a U-type spin-triplet ferromagnetic superconductor."
],
[
"Final remarks",
"Finally, even in its simplified form, this theory has been shown to be capable of accounting for a wide variety of experimental behavior.",
"A natural extension to the theory is to add a $\\mbox{$M$}^6$ term which provides a formalism to investigate possible metamagnetic phase transitions [45] and extend some first order phase transition lines.",
"Another modification of this theory, with regard to applications to other compounds, is to include a $P$ dependence for some of the other GL parameters.",
"The fluctuation and quantum correlation effects can be considered by the respective field-theoretical action of the system, where the order parameters $\\mbox{$\\psi $}$ and $\\mbox{$M$}$ are not uniform but rather space and time dependent.",
"The vortex (spatially non-uniform) phase due to the spontaneous magnetization $\\mbox{$M$}$ is another phenomenon which can be investigated by a generalization of the theory by considering nonuniform order parameter fields $\\mbox{$\\psi $}$ and $\\mbox{$M$}$ (see, e.g., Ref. [28]).",
"Note that such theoretical treatments are quite complex and require a number of approximations.",
"As already noted in this paper the magnetic fluctuations stimulate first order phase transitions for both finite and zero phase-transition temperatures."
]
] | 1204.1007 |
[
[
"Superconductivity Induced by Bond Breaking in the Triangular Lattice of\n IrTe2"
],
[
"Abstract IrTe2, a layered compound with a triangular iridium lattice, exhibits a structural phase transition at approximately 250 K. This transition is characterized by the formation of Ir-Ir bonds along the b-axis.",
"We found that the breaking of Ir-Ir bonds that occurs in Ir1-xPtxTe2 results in the appearance of a structural critical point in the T = 0 limit at xc = 0.035.",
"Although both IrTe2 and PtTe2 are paramagnetic metals, superconductivity at Tc = 3.1 K is induced by the bond breaking in a narrow range of x > xc in Ir1-xPtxTe2.",
"This result indicates that structural fluctuations can be involved in the emergence of superconductivity."
],
[
"Acknowledgment",
"We would like to thank S. Watanabe and T. C. Kobayashi for valuable discussions.",
"Part of this work was performed at the Advanced Science Research Center, Okayama University.",
"It was partially supported by a Grant-in-Aid for Young Scientists (B) (23740274) from Japan Society for the Promotion of Science (JSPS) and the Funding Program for World-Leading Innovative R&D on Science and Technology (FIRST Program) from JSPS.",
"Note added in proof - We noticed a paper by J. J. Yang et al.",
"[Phys.",
"Rev.",
"Lett.",
"108 (2012) 116402], reporting the superconductivity in Pd$_x$ IrTe$_2$ and Ir$_{1-x}$ Pd$_x$ Te$_2$ ."
]
] | 1204.1421 |
[
[
"Fast projections onto mixed-norm balls with applications"
],
[
"Abstract Joint sparsity offers powerful structural cues for feature selection, especially for variables that are expected to demonstrate a \"grouped\" behavior.",
"Such behavior is commonly modeled via group-lasso, multitask lasso, and related methods where feature selection is effected via mixed-norms.",
"Several mixed-norm based sparse models have received substantial attention, and for some cases efficient algorithms are also available.",
"Surprisingly, several constrained sparse models seem to be lacking scalable algorithms.",
"We address this deficiency by presenting batch and online (stochastic-gradient) optimization methods, both of which rely on efficient projections onto mixed-norm balls.",
"We illustrate our methods by applying them to the multitask lasso.",
"We conclude by mentioning some open problems."
],
[
"Introduction",
" Sparsity encodes key structural information about data and permits estimating unknown, high-dimensional vectors robustly.",
"No wonder, sparsity has been intensively studied in signal processing, machine learning, and statistics, and widely applied to many tasks therein.",
"But the associated literature has grown too large to be summarized here; so we refer the reader to [36], [2], [35], [28] as starting points.",
"Sparsity constrained problems are often cast as instances of the following high-level optimization problem $\\min \\nolimits _{x \\in \\mathbb {R}^d}\\quad L(x) + \\lambda f(x), $ where $L$ is a differentiable loss-function, $f$ is a convex (nonsmooth) regularizer, and $\\lambda > 0$ is a scalar.",
"Alternatively, one may prefer the constrained formulation $\\min \\nolimits _{x \\in \\mathbb {R}^d}\\quad L(x)\\quad \\text{s.t.",
"}\\quad f(x) \\le \\gamma .$ Both formulations (REF ) and (REF ) continue to be actively researched, the former perhaps more than the latter.",
"We focus on the latter, primarily because it often admits simple but effective first-order optimization algorithms.",
"Additional benefits that make this constrained formulation attractive include: Even when the loss $L$ is nonconvex, gradient-projection remains applicable; If the loss is separable, it is easy to derive highly scalable incremental or stochastic-gradient based optimization algorithms; If only inexact projections onto $f(x) \\le \\gamma $ are possible (a realistic case), convergence analysis of gradient-projection-type methods remains relatively simple.",
"In this paper, we study a particular subclass of (REF ) that has recently become important, namely, groupwise sparse regression.",
"Two leading examples are multitask learning [16], [15], [24], [30] and group-lasso [45], [44], [3].",
"A key component of these regression problems is the regularizer $f(x)$ , which is designed to enforce `groupwise variable selection'—for example, with $f(x)$ chosen to be a mixed-norm.",
"Definition 1 (Mixed-norm) Let $x \\in \\mathbb {R}^d$ be partitioned into subvectors $x^i \\in \\mathbb {R}^{d_i}$ , for $i \\in [m]$We use $[m]$ as a shorthand for the set $\\left\\lbrace {1,2,\\ldots ,m}\\right\\rbrace $ ..",
"The $\\ell _{p,q}$ -mixed-norm for $p$ , $q\\ge 1$ , is then defined as $f(x) = \\Vert {x} \\Vert _{p,q} := \\bigl (\\sum \\nolimits _{i=1}^m\\Vert {x^i} \\Vert _{q}^p\\bigr )^{1/p}.$ The most practical instances of (REF ) are $\\ell _{1,q}$ -norms, especially for $q \\in \\left\\lbrace {1,2,\\infty }\\right\\rbrace $ .",
"The choice $q=1$ yields the ordinary $\\ell _1$ -norm penalty; $q=2$ is used in group-lasso [45], while $q=\\infty $ arises in compressed sensing [43] and multitask lasso [24].",
"Less common, though potentially useful versions allow interpolating between these extremes by letting $q \\in (1,\\infty )$ ; see also [34], [46], [22].",
"Definition REF can be substantially generalized: we may allow the subvectors $x^i$ to overlap; or to even be normed differently [47].",
"But unless the overlapping has special structure [19], [28], [27], it leads to somewhat impractical mixed-norms, as the corresponding optimization problem (REF ) becomes much harder.",
"Since our chief aim is to develop fast, scalable algorithms for (REF ), we limit our discussion to $\\ell _{1,q}$ -norms—this choice is widely applicable, hence important [44], [24], [17], [20], [5], [40], [14], [30], [15].",
"Before moving onto the technical part, we briefly list the paper's main contents:Which also helps position this paper relative to its precursor at ECML 2011 [41].",
"Batch and online (stochastic-gradient based) algorithms for solving (REF ); Theory of and algorithms for fast projection onto $\\ell _{1,q}$ -norm balls; Application to $\\ell _{1,q}$ -norm based multitask lasso; both batch and online versions; Application to computing projections for matrix mixed-norms; A set of open problems."
],
[
"Basic theory",
" We begin by developing some basic theory.",
"Our aim is to efficiently implement a generic `first-order' algorithm: Generate a sequence $\\left\\lbrace {x_t}\\right\\rbrace $ by iterating $x_{t+1} = \\text{proj}_{f}(x_t-\\eta _t\\nabla _t),\\quad t=0,1,\\ldots ,$ where $\\eta > 0$ is a stepsize, $\\nabla _t$ is an estimate of the gradient, and $\\text{proj}_f$ is the projection operator that enforces the constraint $f(x) \\le \\gamma $ .",
"Below we expand on the most challenging component of iteration (REF ) when applied to mixed-norm regression, namely efficient computation of the projection operator $\\text{proj}_f$ ."
],
[
"Efficient projection via proximity",
"Formally, the (orthogonal) projection operator $\\text{proj}_f: \\mathbb {R}^d \\rightarrow \\mathbb {R}^d$ is defined as $\\text{proj}_{f}(y):= \\operatornamewithlimits{argmin}\\nolimits _{x}\\quad \\tfrac{1}{2}\\Vert {x-y} \\Vert _{2}^2\\quad \\text{s.t.",
"}\\quad f(x) \\le \\gamma .$ Closely tied to projection is the proximity operator $\\operatorname{prox}_h: \\mathbb {R}^d \\times \\mathbb {R}_+ \\rightarrow \\mathbb {R}^d$ $\\hspace{-34.14322pt}\\operatorname{prox}_h(y, \\theta ) := \\operatornamewithlimits{argmin}\\nolimits _{x}\\quad \\tfrac{1}{2}\\Vert {x-y} \\Vert _{2}^2 +\\theta h(x),$ where $h$ is a convex function on $\\mathbb {R}^d$ .",
"Operator (REF ) generalizes projections: if in (REF ) the function $h$ is chosen to be the indicator function for the set $\\left\\lbrace {x : f(x) \\le \\gamma }\\right\\rbrace $ , then the operator $\\operatorname{prox}_h$ reduces to the projection operator $\\text{proj}_f$ .",
"Alternatively, for convex $f$ and $h$ , operators $\\text{proj}_f$ and $\\operatorname{prox}_h$ are also intimately connected by duality.",
"Indeed, this connection proves key to computing a projection efficiently whenever its corresponding proximity operator is `easier'.",
"The idea is simple (see e.g., [31]), but exploiting it effectively requires some care; let us see how.",
"Let $\\mathcal {L}(x,\\theta )$ be the Lagrangian for (REF ); and let the optimal dual solution be denoted by $\\theta ^*$ .",
"Assuming strong-duality, the optimal primal solution is given by $x(\\theta ^*) := \\operatornamewithlimits{argmin}\\nolimits _{x}\\ \\mathcal {L}(x,\\theta ^*):= \\operatornamewithlimits{argmin}\\nolimits _{x}\\ \\tfrac{1}{2}\\Vert {x-y} \\Vert _{2}^2 + \\theta ^*(f(x) - \\gamma ).$ But to compute (REF ), we require the optimal $\\theta ^*$ —the key insight on obtaining $\\theta ^*$ is that it can be computed by solving a single nonlinear equation.",
"Here is how.",
"First, observe that if $f(y) \\le \\gamma $ , then $x(\\theta ^*)=y$ , and there is nothing to compute.",
"Thus, assume that $f(y) > \\gamma $ ; then, the optimal point $x(\\theta ^*)$ satisfies $f(x(\\theta ^*))=\\gamma .$ Next, observe from (REF ) that for a fixed $\\theta $ , the point $x(\\theta )$ equals the operator $\\operatorname{prox}_f(y, \\theta )$ .",
"Consider, therefore, the nonlinear function (residual) $g(\\theta ) := f(x(\\theta )) - \\gamma \\ =\\ f(\\operatorname{prox}_f(y, \\theta ))-\\gamma ,$ which measures how accurately equation (REF ) is satisfied.",
"The optimal $\\theta ^*$ can be then obtained by solving $g(\\theta )=0$ , for which the following lemma proves very useful.",
"Lemma 2 Let $f(x)$ be a gaugeThat is, $f$ is nonnegative, positively homogeneous, and disappears at the origin [37], and let $g(\\theta )$ be as defined in (REF ).",
"Then, there exists an interval $[0,\\theta _{\\max }]$ , on which $g(\\theta )$ is monotonically decreasing, and differs in sign at the endpoints.",
"By assumption on $f(y)$ , it holds that $g(0) = f(y)-\\gamma > 0$ .",
"We claim that for $\\theta \\ge {f}^{\\circ }(y)$ , where ${f}^{\\circ }$ denotes the polar of $f$ , the optimal point $x(\\theta ) = 0$ .",
"To see why, suppose that $\\theta \\ge {f}^{\\circ }(y)$ , but $x(\\theta ) \\ne 0$ .",
"Then, $\\tfrac{1}{2}\\Vert {x(\\theta )-y} \\Vert _{2}^2 +\\theta f(x(\\theta )) < \\tfrac{1}{2}\\Vert {y} \\Vert _{2}^2$ .",
"But since $\\Vert {\\cdot } \\Vert _{2}^2$ is strictly convex, the inequality $\\Vert {y} \\Vert _{2}^2 - \\Vert {x-y} \\Vert _{2}^2 < 2\\langle {y},\\, {x} \\rangle $ also holds for any $x$ .",
"Thus, it follows that $\\theta <\\langle {y},\\, {x(\\theta )} \\rangle /f(x(\\theta ))$ , whereby, for $\\theta \\ge \\sup _{x \\ne 0}{\\langle {y},\\, {x} \\rangle }/{f(x)}={f}^{\\circ }(y)$ , the optimal $x(\\theta )$ must equal 0.",
"Hence, we may select $\\theta _{\\max } = {f}^{\\circ }(y)$ .",
"Monotonicity of $g$ follows easily, as it is the derivative of the concave (dual) function $\\inf _{x}\\mathcal {L}(x,\\theta )$ .",
"Finally, $g(\\theta _{\\max }) = -\\gamma < 0$ , so it differs in sign.$\\Box $ Since $g(\\theta )$ is continuous, changes sign, and is monotonic in the interval $[0,\\theta _{\\max }]$ , it has a unique root therein.",
"This root can be computed to $\\epsilon $ -accuracy using bisection in $O(\\log (\\theta _{\\max }/\\epsilon ))$ iterations.",
"We recommend not to use mere bisection, but rather to invoke a more powerful root-finder that combines bisection, inverse quadratic interpolation, and the secant method (e.g., Matlab's fzero function).",
"Pseudocode encapsulating these ideas is given in Algorithm REF .",
"[tbp] Subroutine to compute $\\operatorname{prox}_f(y,\\theta )$ ; vector $y$ ; scalar $\\gamma > 0$ $x^* := \\text{proj}_f(y,\\gamma )$ $f(y) \\le \\gamma $$x^*=y$ Define $g(\\theta ) := \\operatorname{prox}_f(y,\\theta )-\\gamma $ Compute interval $[\\theta _{\\min },\\theta _{\\max }] = [0, {f}^{\\circ }(y)]$ Compute root $\\theta ^* = \\textsc {FindRoot}(g(\\theta ), \\theta _{\\min },\\theta _{\\max })$ $x^* = \\operatorname{prox}_f(y,\\theta ^*)$ Root-finding for projection via proximity"
],
[
"Projection onto $\\ell _{1,q}$ -norm balls",
"After the generic approach above, let us specialize to projections for the case of central interest to us, namely, $\\text{proj}_f$ with $f(x)=\\ell _{1,q}(x)$ .",
"Algorithm REF requires computing the upper bound $\\theta _{\\max } = {f}^{\\circ }(y)$ .",
"To that end Lemma REF , which actually proves much more, proves useful.",
"Lemma 3 (Dual-norm) Let $p,q \\ge 1$ ; and let $p^*,q^* \\ge 1$ be “conjugate” scalars, i.e., $1/p+1/p^*=1$ and $1/q+1/q^*=1$ .",
"The polar (dual-norm) of $\\Vert {\\cdot } \\Vert _{p,q}$ is $\\Vert {\\cdot } \\Vert _{p^*,q^*}$ .",
"By definition, the norm dual to an arbitrary norm $\\Vert {\\cdot } \\Vert _{}{}$ is given by $\\Vert {u} \\Vert _{*} := \\sup \\left\\lbrace {\\langle {x},\\, {u} \\rangle \\ |\\ \\ \\Vert {x} \\Vert _{}{} \\le 1}\\right\\rbrace .$ To prove the lemma, we prove two items: (i) for any two (conformally partitioned) vectors $x$ and $u$ , we have $|\\langle {x},\\, {u} \\rangle | \\le \\Vert {x} \\Vert _{p,q}\\Vert {u} \\Vert _{p^*,q^*}$ ; and (ii) for each $u$ , there exists an $x$ for which $\\langle {x},\\, {u} \\rangle =\\Vert {y} \\Vert _{p^*,q^*}$ .",
"Let $x$ be a vector partitioned conformally to $u$ , and consider the inequality $\\langle {x},\\, {u} \\rangle = \\sum \\nolimits _{i=1}^g \\langle {x^i},\\, {u^i} \\rangle \\le \\sum \\nolimits _{i=1}^g \\Vert {x^i} \\Vert _{q}\\Vert {u^g} \\Vert _{q^*},$ which follows from Hölder's inequality.",
"Define $\\psi =[\\Vert {x^i} \\Vert _{q}]$ and $\\xi =[\\Vert {u^i} \\Vert _{q^*}]$ , and invoke Hölder's inequality again to obtain $\\langle {\\psi },\\, {\\xi } \\rangle \\le \\Vert {\\psi } \\Vert _{p}\\Vert {\\xi } \\Vert _{p^*} =\\Vert {x} \\Vert _{p,q}\\Vert {u} \\Vert _{p^*,q^*}$ .",
"Thus, from definition (REF ) we conclude that $\\Vert {u} \\Vert _{*} \\le \\Vert {u} \\Vert _{p^*,q^*}$ .",
"To prove that the dual norm actually equals $\\Vert {u} \\Vert _{p^*,q^*}$ , we show that for each $u$ , we can find an $x$ that satisfies $\\Vert {x} \\Vert _{p,q}=1$ , for which the inner-product $\\langle {x},\\, {u} \\rangle = \\Vert {u} \\Vert _{p^*,q^*}$ .",
"Define therefore $\\beta = \\sum _i \\Vert {u^i} \\Vert _{q^*}^{p^*}$ —some juggling with indices suggests that we should set $x_j^i = \\frac{1}{\\beta ^{1/p}}\\frac{\\Vert {u^i} \\Vert _{q^*}^{p^*}}{\\Vert {u^i} \\Vert _{q^*}^{q^*}}\\operatorname{sgn}(u^i_j)|u^i_j|^{q^*-1},$ where $x^i_j$ denotes the $j$ -the element of the subvector $x^i$ (similarly $u^i_j$ ).",
"To see that (REF ) works, first consider the inner-product $\\langle {x},\\, {u} \\rangle = \\sum \\nolimits _i\\langle {x^i},\\, {u^i} \\rangle &= \\sum \\nolimits _i\\sum \\nolimits _j x^i_ju^i_j\\\\& = \\frac{1}{\\beta ^{1/p}}\\sum \\nolimits _i \\Vert {u^i} \\Vert _{q^*}^{p^*-q^*}\\sum \\nolimits _j |u^i_j|^{q*}\\qquad (\\text{since }\\operatorname{sgn}(u^i_j)u^i_j=|u^i_j|)\\\\& = \\frac{1}{\\beta ^{1/p}} \\sum \\nolimits _i \\Vert {u^i} \\Vert _{q^*}^{p^*} = \\frac{\\beta }{\\beta ^{1/p}} = \\beta ^{1-1/p} = \\beta ^{1/p^*}\\\\& = \\left(\\sum \\nolimits _i \\Vert {u^i} \\Vert _{q^*}^{p^*}\\right)^{1/p^*} = \\Vert {u} \\Vert _{p^*,q^*}.$ Next, we check that $\\Vert {x} \\Vert _{p,q}=\\left(\\sum _i \\Vert {x^i} \\Vert _{q}^p\\right)^{1/p}= 1$ .",
"Consider thus, the term $\\Vert {x^i} \\Vert _{q}^p = \\bigl ( \\sum _j |x^i_j|^q\\bigr )^{p/q}$ .",
"Using (REF ) we have $\\sum \\nolimits _j |x^i_j|^q &= \\frac{1}{\\beta ^{q/p}}\\Vert {u^i} \\Vert _{q^*}^{(p^*-q^*)q} \\sum \\nolimits _j|u^i_j|^{q(q^*-1)}\\\\&= \\frac{1}{\\beta ^{q/p}}\\Vert {u^i} \\Vert _{q^*}^{(p^*-q^*)q} \\sum \\nolimits _j |u^i_j|^{q^*}\\qquad &(\\text{since } q^*q^{-1} +1=q^*)\\\\&= \\frac{1}{\\beta ^{q/p}}\\Vert {u^i} \\Vert _{q^*}^{(p^*-q^*)q+q^*} = \\frac{1}{\\beta ^{q/p}}\\Vert {u^i} \\Vert _{q^*}^{p^*q-q^*(q-1)}\\\\& = \\frac{1}{\\beta ^{q/p}}\\Vert {u^i} \\Vert _{q^*}^{(p^*-1)q}&\\qquad (\\text{since } q(q^*)^{-1} +1=q).$ Thus, it follows that $\\Vert {x^i} \\Vert _{q}^p = \\Bigl ( \\sum \\nolimits _j |x^i_j|^q\\Bigr )^{p/q} = \\frac{1}{\\beta }\\Vert {u^i} \\Vert _{}{q^*}^{p(p^*-1)} = \\frac{1}{\\beta }\\Vert {u^i} \\Vert _{}{q^*}^{p^*},$ where the last equality holds because $1/p+1/p^*=1$ .",
"Finally, from (REF ) it follows that $\\Vert {x} \\Vert _{p,q} = \\bigl (\\sum \\nolimits _i \\Vert {x^i} \\Vert _{q}^p\\bigr )^{1/p} = \\bigl (\\tfrac{1}{\\beta }\\sum \\nolimits _i \\Vert {u^i} \\Vert _{q^*}^{p^*}\\bigr )^{1/p} = 1,$ since by definition $\\beta =\\sum _i \\Vert {u^i} \\Vert _{q^*}^{p^*}$ .",
"This concludes the proof.",
"The next key component for Algorithm REF is the proximity operator $\\operatorname{prox}_f$ .",
"For $f(x)=\\Vert {x} \\Vert _{1,q}$ , this operator requires solving $\\min _{x^1,\\ldots ,x^m}\\ \\sum \\nolimits _{i=1}^m\\tfrac{1}{2}\\Vert {x^i-y^i} \\Vert _{2}^2 + \\theta \\sum \\nolimits _{i=1}^m\\Vert {x^i} \\Vert _{q}.$ Fortunately, Problem (REF ) separates into a sum of $m$ independent, $\\ell _q$ -norm proximity operators.",
"It suffices, therefore, to only consider a subproblem of the form $\\min \\nolimits _u\\quad \\tfrac{1}{2}\\Vert {u-v} \\Vert _{2}^2 + \\theta \\Vert {u} \\Vert _{q}.$ For $q=1$ , the solution to (REF ) is given by the soft-thresholding operation [13]: $u(\\theta ) = \\operatorname{sgn}(v) \\odot \\max (|v|-\\theta , 0),$ where operator $\\odot $ performs elementwise multiplication.",
"For $q=2$ , we get $u(\\theta ) = \\max (1-\\theta \\Vert {v} \\Vert _{2}^{-1}, 0)v,$ while the case $q=\\infty $ is slightly more involved.",
"It can be solved via the Moreau decomposition [11], which, for a norm $f=\\Vert {\\cdot } \\Vert _{}$ implies that $\\operatorname{prox}_{f}(v, \\theta ) = v - \\text{proj}_{{f}^{\\circ }}(v,\\theta ).$ For $f=\\Vert {\\cdot } \\Vert _{\\infty }$ , the dual-norm (polar) is ${f}^{\\circ }=\\Vert {\\cdot } \\Vert _{1}$ ; but projection onto $\\ell _1$ -balls has been extremely well-studied—see e.g., [29], [21], [25].",
"For $q > 1$ (different from 2 and $\\infty $ ), problem (REF ) is much harder.",
"Fortunately, this problem was recently solved in [26], using nested root-finding subroutines.",
"But unlike the cases $q \\in \\left\\lbrace {1,2,\\infty }\\right\\rbrace $ , the proximity operator for general $q$ can be computed only approximately (i.e., in (REF ), each iteration generates only approximate $x(\\theta )$ )."
],
[
"Mixed norms for matrices: a brief digression",
"We now make a brief digression, which is afforded to us by the above results.",
"Our digression concerns mixed-norms for matrices, as well as their associated projection, proximity operators, which ultimately depend on the results of the previous section.",
"Our discussion is motivated by applications in [42], where the authors used mixed-norms on matrices to simultaneously.",
"We define mixed-norms on matrices by building upon the classic Schatten-$q$ matrix norms [7], defined as: $\\Vert {X} \\Vert _{q} := \\bigl (\\sum \\nolimits _i \\sigma _i^q(X)\\bigr )^{1/q},\\quad \\text{for}\\ q \\ge 1,$ where $X$ is an arbitrary complex matrix, and $\\sigma _i(X)$ is its $i$ th singular value.",
"Now, let $\\mathsf {X}= \\left\\lbrace {X^1,\\ldots ,X^m}\\right\\rbrace $ be an arbitrary set of matrices, and let $p, q \\ge 1$ .",
"We define the matrix $(p,q)$ -norm by the formula $\\Vert {\\mathsf {X}} \\Vert _{(p,q)} := \\bigl (\\sum \\nolimits _{i=1}^m \\Vert {X^i} \\Vert _{q}^p\\bigr )^{1/p}.$ As for the vector case, we have a similar lemma about norms dual to (REF ).",
"Lemma 4 (Matrix Hölder inequality) Let $X$ and $Y$ be matrices such that $\\operatorname{tr}(X^*Y)$ is well-defined.",
"Then, for $p \\ge 1$ , such that $1/p + 1/p^* = 1$ , it holds that $|\\langle {X},\\, {Y} \\rangle | = |\\operatorname{tr}(X^*Y)| \\le \\Vert {X} \\Vert _{p}\\Vert {Y} \\Vert _{p^*}.$ From the well-known von Neumann trace inequality [18] we know that $|\\operatorname{tr}(X^*Y)| \\le \\sum \\nolimits _i \\sigma _i(X)\\sigma _i(Y) = \\langle {\\sigma (X)},\\, {\\sigma (Y)} \\rangle .$ Now invoke the classical Hölder inequality and use definition (REF ) of matrix mixed-norms to conclude.",
"Lemma 5 (Dual norms) Let $p, q \\ge 1$ ; and let $p^*, q^*$ be their conjugate exponents.",
"The norm dual to $\\Vert {\\cdot } \\Vert _{(p,q)}$ is $\\Vert {\\cdot } \\Vert _{(p^*,q^*)}$ .",
"By the triangle-inequality and Lemma REF we have $|\\langle {\\mathsf {X}},\\, {\\mathsf {Y}} \\rangle |=\\left|\\sum \\nolimits _i \\langle {X^i},\\, {Y^i} \\rangle \\right| \\le \\sum \\nolimits _{i}|\\langle {X^i},\\, {Y^i} \\rangle | \\le \\sum \\nolimits _i\\Vert {X^i} \\Vert _{q}\\Vert {Y^i} \\Vert _{q^*}.$ Applying Hölder's inequality to the latter term we obtain $\\sum \\nolimits _i\\Vert {X^i} \\Vert _{q}\\Vert {Y^i} \\Vert _{q^*} \\le \\Vert {\\mathsf {X}} \\Vert _{(p,q)}\\Vert {\\mathsf {Y}} \\Vert _{(p^*,q^*)}.$ Now, we must show that for any $\\mathsf {Y}$ , we can find an $\\mathsf {X}$ such that (REF ) holds with equality.",
"To that end, let $Y^i = P_iS_iQ_i^*$ be the SVD of matrix $Y^i$ .",
"Setting $X^i = P_i\\Sigma _iQ_i^*$ , we see that $|\\langle {\\mathsf {X}},\\, {\\mathsf {Y}} \\rangle | = \\sum \\nolimits _i\\operatorname{tr}(\\Sigma _iS_i)$ ; since both $\\Sigma _i$ and $S_i$ are diagonal, this reduces to the vector case (REF ), completing the proof.",
"Projections onto $\\Vert {\\cdot } \\Vert _{(1,q)}$ -norm balls: As for vectors, we now consider the matrix $(1,q)$ -norm projection $\\min \\nolimits _{X^1,\\ldots ,X^m}\\quad \\sum \\nolimits _{i=1}^m\\tfrac{1}{2}\\Vert {X^i-Y^i}\\Vert _{\\text{F}}^2\\quad \\text{s.t.",
"}\\ \\sum \\nolimits _{i=1}^m\\Vert {X^i} \\Vert _{q} \\le \\gamma .$ Algorithm REF can be used to solve (REF ).",
"The upper bound $\\theta _{\\max }$ can be obtained via Lemma REF .",
"It only remains to solve proximity subproblems of the form $\\min \\nolimits _{X}\\quad \\tfrac{1}{2}\\Vert {X-Y}\\Vert _{\\text{F}}^2 + \\theta \\Vert {X} \\Vert _{q}\\ \\ .$ Since both $\\Vert {\\cdot }\\Vert _{\\text{F}}$ and $\\Vert {\\cdot } \\Vert _{q}$ are unitarily invariant, from Corollary 2.5 of [23] it follows that if $Y^i$ has the singular value decomposition $Y=U\\operatorname{Dg}(y)V^*$ , then (REF ) is solved by $X=U\\operatorname{Dg}(\\bar{x})V^*$ , where the vector $\\bar{x}$ is obtained by solving $\\bar{x} := \\operatorname{prox}_{\\Vert {\\cdot } \\Vert _{q}}(y) :=\\operatornamewithlimits{argmin}\\nolimits _x\\quad \\tfrac{1}{2}\\Vert {x-y} \\Vert _{2}^2+\\theta \\Vert {x} \\Vert _{q}.$ We note in passing that operator (REF ) generalizes the popular singular value thresholding operator [10], which corresponds to $q=1$ (trace norm)."
],
[
"Algorithms for solving (",
"We describe two realizations of the generic iteration (REF ) that can be particularly effective: (i) spectral projected gradients; and (ii) stochastic-gradient descent."
],
[
"Batch method: spectral projected gradient",
"The simplest method to solve (REF ) is perhaps gradient-projection [38], where starting with a suitable initial point $x_0$ , one iterates $x_{t+1} = \\text{proj}_{f}(x_t-\\eta _t\\nabla L(x_t)),\\quad t=0,1,\\ldots .$ We have already discussed $\\text{proj}_f$ ; the other two important parts of (REF ) are the stepsize $\\eta _t$ , and the gradient $\\nabla L$ .",
"Even when the loss $L$ is not convex, under fairly mild condition, we may still iterate (REF ) to obtain convergence to a stationary point—see [6] for a detailed discussion, including various strategies for computing stepsizes.",
"If, however, $L$ is convex, we may invoke a method that typically converges much faster: spectral projected gradient (SPG) [8].",
"SPG extends ordinary gradient-projection by using the famous (nonmonotonic) spectral stepsizes of Barzilai and Borwein [4] (BB).",
"Formally, these stepsizes are $\\eta _{BB1} := \\frac{\\langle {\\Delta x_t},\\, {\\Delta x_t} \\rangle }{\\langle {\\Delta g_t},\\, {\\Delta x_t} \\rangle },\\quad \\text{or}\\quad \\eta _{BB2} := \\frac{\\langle {\\Delta x_t},\\, {\\Delta g_t} \\rangle }{\\langle {\\Delta g_t},\\, {\\Delta g_t} \\rangle },$ where $\\Delta x_t = x_t - x_{t-1}$ , and $\\Delta g_t = \\nabla L(x_t)-\\nabla L(x_{t-1})$ .",
"SPG substitutes stepsizes (REF ) in (REF ) (using safeguards to ensure bounded steps).",
"Thereby, it leverages the strong empirical performance enjoyed by BB stepsizes [4], [8], [12], [39]; to ensure global convergence, SPG invokes a nonmontone line search strategy that allows the objective value to occasionally increase, while maintaining some information that allows extraction of a descending subsequence."
],
[
"Inexact projections:",
"Theoretically, the convergence analysis of SPG [8] depends on access to a subroutine that computes $\\text{proj}_f$ exactly.",
"Obviously, in general, this operator cannot be computed exactly (including for many of the mixed-norms).",
"To be correct, we must rely on an inexact SPG method such as [9].",
"In fact, due to roundoff error, even the so-called exact methods run inexactly.",
"So, to be fully correct, we must treat the entire iteration (REF ) as being inexact.",
"Such analysis can be done (see e.g., [32]); but it is not one of the main aims of this paper, so we omit it."
],
[
"Stochastic-gradient method",
"Suppose the loss-function $L$ in (REF ) is separable, that is, $L(x) = \\sum \\nolimits _{i=1}^r \\ell _i(x),\\qquad \\text{where}\\ x \\in \\mathbb {R}^d,$ for some large number $r$ of components (say $r \\gg d$ ).",
"In such a case, computing the entire gradient $\\nabla L$ at each iteration (REF ) may be too expensive, and it might be more preferable to use stochastic-gradient descent (SGD)This popular name is a misnomer because SGD does not necessarily lead to descent at each step.",
"instead.",
"In its simplest realization, at iteration $t$ , SGD picks a random index $s(t) \\in [r]$ , and replaces $\\nabla L(x)$ by a stochastic estimate $\\nabla \\ell _{s(t)}(x)$ .",
"This results in the iteration $x_{t+1} = \\text{proj}_{f}(x_t - \\eta _t\\nabla \\ell _{s(t)}(x_t)),\\quad t=0,1,\\ldots ,$ where $\\eta _t$ are suitable (e.g., $\\eta _t \\propto 1/t$ ) stepsizes.",
"Again, some additional analysis is also needed for (REF ) to account for the potential inexactness of the projections."
],
[
"Experimental results and applications",
"We present below numerical results that illustrate the computational performance of our methods.",
"In particular, we show the following main experiments: Running time behavior of our root-finding projection methods, including Comparisons against the method of [33] for $\\ell _{1,\\infty }$ projections Some results on $\\ell _{1,q}$ projections for a few different values of $q$ .",
"Application to the $\\ell _{1,\\infty }$ -norm multitask lasso [24], for which we show Running time behavior of SPG, both with our projection and that of [33]; Derivation of and numerical results with a SGD based method for MTL."
],
[
"Projection onto the $\\ell _{1,\\infty }$ -ball",
" For ease of comparison, we use the notation of [33], who seem to be the first to consider efficient projections onto the $\\ell _{1,\\infty }$ -norm ball.",
"The task is to solve $\\min \\nolimits _{W}\\quad \\tfrac{1}{2}\\Vert {W-V}\\Vert _{\\text{F}}^2,\\quad \\text{s.t.",
"}\\quad \\sum \\nolimits _{i=1}^d\\Vert {w^i} \\Vert _{\\infty } \\le \\gamma ,$ where $W$ is a $d \\times n$ matrix, and $w^i$ denotes its $i$ th row.",
"In our comparisons, we refer to the algorithm of [33] (C implementation)http://www.lsi.upc.edu/$\\sim $ aquattoni/CodeToShare/, as `QP',The runtimes for QP reported in this paper differ significantly from those in our previous paper [41].",
"This difference is due to an unfortunate bug in the previous implementation of [33], which got uncovered after the authors of [33] saw our experimental results in [41].",
"and to our method as `FP' (also C implementation).",
"The experiments were run on a single core of a quad-core AMD Opteron (2.6GHz), 64bit Linux machine with 16GB RAM.",
"We compute the optimal $W^*$ , as $\\gamma $ varies from $0.01\\Vert {V} \\Vert _{1,\\infty }$ (more sparse) to $0.6\\Vert {V} \\Vert _{1,\\infty }$ (less sparse) settings.",
"Tables REF –REF present running times, objective function values, and errors (as measured by the constraint violation: $|\\gamma -\\Vert {W^*} \\Vert _{1,\\infty }|$ , for an estimated $W^*$ ).",
"The tables also show the absolute difference in objective value between QP and FP.",
"While for small problems, QP is very competitive, for larger ones, FP consistently outperforms it.",
"Although on average FP is only about twice as fast as QP, it is noteworthy that despite FP being an “inexact” method (and QP an “exact” one), FP obtains solutions of accuracy many magnitudes of order better than QP.",
"Table: Runtime and accuracy for QP and FP on a 10,000×30010,000\\times 300 matrix VV.Table: Runtime and accuracy for QP and FP on a 50,000×100050,000\\times 1000 matrix VV.Table: Runtime and accuracy for QP and FP on a 50,000×10,00050,000\\times 10,000 matrix VV.",
"For this experiment, QP did not run on our machine with 16GB, so we performed this experiment on a machine with 32GB RAM."
],
[
"Projection onto $\\ell _{1,q}$ -balls",
"Next we show running time behavior displayed our method for projecting onto $\\ell _{1,q}$ balls; we show results for $q \\in \\left\\lbrace {1.5, 2.5, 3, 5}\\right\\rbrace $ , when solving $\\min \\nolimits _{W}\\quad \\Vert {W-V}\\Vert _{\\text{F}}^2,\\quad \\text{s.t.",
"}\\ \\ \\sum \\nolimits _{i=1}^d \\Vert {w^i} \\Vert _{q}.$ The plots (Figure REF ) also running time behavior as the parameter $\\gamma $ is varied.",
"These plots reveal four main points: (i) the runtimes seem to be largely independent of $\\gamma $ ; (ii) for smaller values of $q$ , the projection times are approximately same; and (iii) for larger values of, the projection times increase dramatically.",
"Moreover, from the actual running times it is apparent our projection code scales linearly with the data size.",
"For example, the matrix corresponding to the second bar plot has 25 times more parameters than the first plot, and the runtimes reported in the second plot are approximately 25–30 times higher.",
"Although the running times scale linearly, a single $\\ell _{1,q}$ -norm projection still takes nontrivial effort.",
"Thus, even though our $\\ell _{1,q}$ -projection method is relatively fast, currently we can recommend it only for small and medium-scale regression problems.",
"Figure: Running times for ℓ 1,q \\ell _{1,q}-norm projections as scalars qq and ratios γ/∥V∥ 1,q \\gamma /\\Vert {V} \\Vert _{1,q} vary.",
"The left plot is on a 1000×100{1000 \\times 100} matrix, while the right one is on a 5000×5005000 \\times 500 matrix."
],
[
"Application to Multitask Lasso",
"Multitask Lasso (Mtl) [44], [24] is a simple grouped feature selection problem, which separates important features from less important ones by using information shared across multiple tasks.",
"The feature selection is effected by a sparsity promoting mixed-norm, usually the $\\ell _{1,\\infty }$ -norm [24].",
"Formally, Mtl is setup as follows.",
"Let $\\mathbf {X}_j \\in \\mathbb {R}^{m_j \\times d}$ be the data matrix for task $j$ , where $1 \\le j \\le n$.",
"Mtl seeks a matrix $W\\in \\mathbb {R}^{d \\times n}$ , each column of which corresponds to parameters for a task; these parameters are regularized across features by applying a mixed-norm over the rows of $W$ .",
"This leads to a “grouped” feature selection, because if for a row, the norm $\\Vert {w^i} \\Vert _{\\infty }=0$ , then the entire row $w^i$ gets eliminated (i.e., feature $i$ is removed).",
"The standard Mtl optimization problem is $\\min _{w_1,\\ldots ,w_n}\\quad \\mathcal {L}(W) := \\sum \\nolimits _{j=1}^n\\tfrac{1}{2}\\Vert {y_j - X_jw_j} \\Vert _{2}^2,\\quad \\text{s.t.",
"}\\quad \\sum \\nolimits _{i=1}^d\\Vert {w^i} \\Vert _{\\infty } \\le \\gamma ,$ where the $y_j$ are the dependent variables, and $\\gamma > 0$ is a sparsity-tuning parameter.",
"Notice that the loss-function combines the different tasks (over columns of $W$ ), but the overall problem does not decompose into separable problems because the mixed-norm constrained is over the rows of $W$ ."
],
[
"Stochastic-gradient based MTL",
"We may rewrite the MTL problem as $\\begin{split}\\min \\ L(W) := &\\sum \\nolimits _{j=1}^n\\tfrac{1}{2}\\Vert {y_j-X_jw_j} \\Vert _{2}^2 = \\tfrac{1}{2}\\Vert {y-Xw} \\Vert _{2}^2,\\\\\\text{s.t.",
"}\\quad &\\sum \\nolimits _{i=1}^d\\Vert {w^i} \\Vert _{\\infty } \\le \\gamma ,\\end{split}$ where we have introduced the notation $y = \\operatorname{vec}(Y),\\quad X=X_1\\oplus \\cdots \\oplus X_n,\\quad \\text{and}\\ w = \\operatorname{vec}(W),$ in which $\\operatorname{vec}(\\cdot )$ is the operator that stacks columns of its argument to yield a long vector, and $\\oplus $ denotes the direct sum of two matrices.",
"Notice that if it were not for the $\\ell _{1,\\infty }$ -norm constraint, problem (REF ) would just reduce to ordinary least squares.",
"The form (REF ), however, makes it apparent how to derive a stochastic-gradient method.",
"In particular, suppose that we use a “mini-batch” of size $b$ , i.e., we choose $b$ rows of matrix $X$ , say $X_b$ .",
"Let $y_b$ denote the corresponding rows (components) of $y$ .",
"This subset of rows contributes $\\ell _b(w) := \\tfrac{1}{2}\\Vert {y_b-X_bw} \\Vert _{2}^2$ to the objective (REF ), whereby we have the stochastic-gradient $\\nabla \\ell _b(w) = X_b^T(X_bw-y_b).$ Then, upon instantiating iteration (REF ) with (REF ), we obtain Algorithm REF .",
"[tbp] Scalar $\\gamma > 0$ ; batchsize $b$ ; stepsize sequence: $\\eta _0,\\eta _1,\\ldots $ $W^* \\approx \\operatornamewithlimits{argmin}_{W}\\ L(W)$ , s.t.",
"$\\Vert {W^T} \\Vert _{1,\\infty } \\le \\gamma $ $W_0 \\leftarrow 0$ $\\lnot $ converged Pick $b$ different indices in $[mn]$ Obtain stochastic gradient using (REF ) $W_t \\leftarrow \\text{proj}(W_{t-1} - \\eta _t\\nabla \\ell _s(W_t))$ $t \\leftarrow t + 1$ $W^*$ MTL via stochastic-gradient descent"
],
[
"Implementation notes:",
"Despite our careful implementation, for large-scale problems the projection can become the bottleneck in Algorithm REF .",
"To counter this, we should perform projections only occasionally—the convergence analysis is unimpeded, as we may restrict our attention to the subsequence of iterates for which projection was performed.",
"Other implementation choices such as size of the mini-batch and the values of the stepsizes $\\eta _t$ are best determined empirically.",
"Although tuning $\\eta _t$ can be difficult, this drawback is offset by the gain in scalability."
],
[
"Simulation results",
"We illustrate running time results of SPG on two large-scale instances of Mtl (see Table REF ).",
"We report running time comparisons between two different invocations of an SPG-based method for solving (REF ), once with QP as the projection method and once with FP—we call the corresponding solvers SPG$_{\\text{QP}}$ , and SPG$_{\\text{FP}}$ .",
"We note in passing that other efficient Mtl algorithms (e.g., [20], [26]) solve the penalized version; our formulation is constrained, so we only show SPG.",
"Table: Sparse datasets used for MTL.",
"For simplicity, all matrices X j X_j (for each task 1≤j≤n1\\le j\\le n), were chosen to have the same size m×d{m \\times d}.Table: Running times (seconds) on datasets D1 and D2.",
"SPG was used to solve MTL, with stopping tolerance of 10 -5 10^{-5}.",
"Total number of projections required to reach this accuracy are reported as '#projs'.",
"The columns 'proj QP _{\\text{QP}}' and 'proj FP _{\\text{FP}}', report the total time spent by the SPG QP _{\\text{QP}} and SPG FP _{\\text{FP}} methods for the ℓ 1,∞ \\ell _{1,\\infty }-projections alone.",
"The last two columns report the overall time taken by SPG QP _{\\text{QP}} and SPG FP _{\\text{FP}}.The results in Table REF indicate that for large-scale problems, the savings accrued upon using our faster projections (in combination with SPG) can be substantial."
],
[
"MTL results on real-world data",
"We now show a running comparison between three methods: (i) SPG$_{\\text{QP}}$ , (ii) SPG$_{\\text{FP}}$ , and (iii) SGD (with projection step computed using FP).",
"For our comparison, we solve Mtl on a subset of the CMU Newsgroups datasetOriginal at: http://www.cs.cmu.edu/$\\sim $ textlearning/; we use the reduced version of [20]..",
"The dataset corresponds to 5 feature selection tasks based on data taken from the following newsgroups: computer, politics, science, recreation, and religion.",
"The feature selection tasks are spread over the matrices $\\mathbf {X}_1,\\ldots ,\\mathbf {X}_5$ , each of size $2907 \\times 53975$ , while the dependent variables $\\mathbf {y}_1,\\ldots ,\\mathbf {y}_5$ correspond to class labels.",
"Figure: Running time results on CMU Newsgroups subset (left: less sparse; right: more sparse problem).Figure REF reports running time results obtained by the three methods in question (all methods were initialized by the same $W_0$ ).",
"As expected, the stochastic-gradient based method rapidly achieves a low-accuracy solution, but start slowing down as time proceeds, and eventually gets overtaken by the SPG based methods.",
"Interestingly, in the first experiment, SPG$_{\\text{QP}}$ takes much longer than SPG$_{\\text{FP}}$ to convergence, while in the second experiment, it lags behind substantially before accelerating towards the end.",
"We attribute this difference to the difficulty of the projection subproblem: in the beginning, the sparsity pattern has not yet emerged, which drives SPG$_{\\text{QP}}$ to take more time.",
"In general, however, from the figure it seems that either SGD or SPG$_{\\text{FP}}$ yield an approximate solution more rapidly—so for problems of increasingly larger size, we might prefer them.Though some effort must always be spent to tune the batch and stepsizes for SGD."
],
[
"Discussion",
"We described mixed-norms for vectors, which we then naturally extended also to matrices.",
"We presented some duality theory, which enabled us to derive root-finding algorithms for efficiently computing projections onto mixed-norm balls, especially for the special class of $\\ell _{1,q}$ -mixed norms.",
"For solving an overall regression problem involving mixed-norms we suggested two main algorithms, spectral projected gradient and stochastic-gradient (for separable losses).",
"We presented a small but indicative set of experiments to illustrate the computational benefits of our ideas, in particular for the multitask lasso problem.",
"At this point, several directions of future work remain open—for instance: Designing fast projection methods for certain classes of non-separable mixed norms.",
"Some algorithms already exist for particular classes [1], [28].",
"Studying norm projections with additional simple constraints (e.g., bounds).",
"Extending the fast methods of this paper to non-Euclidean proximity operators.",
"Exploring applications of matrix mixed-norm regularizers."
]
] | 1204.1437 |
[
[
"Fundamentals of the Dwarf Fundamental Plane"
],
[
"Abstract Star-forming dwarfs are studied to elucidate the physical underpinnings of their fundamental plane.",
"It is confirmed that residuals in the Tully-Fisher relation are correlated with surface brightness, but that even after accommodating the surface brightness dependence through the dwarf fundamental plane, residuals in absolute magnitude are far larger than expected from observational errors.",
"Rather, a more fundamental plane is identified which connects the potential to HI line width and surface brightness.",
"Residuals correlate with the axis ratio in a way which can be accommodated by recognizing the galaxies to be oblate spheroids viewed at varying angles.",
"Correction of surface brightnesses to face-on leads to a correlation among the potential, line width, and surface brightness for which residuals are entirely attributable to observational uncertainties.",
"The mean mass-to-light ratio of the diffuse component of the galaxies is constrained to be 0.88 +/- 0.20 in Ks.",
"Blue compact dwarfs lie in the same plane as dwarf irregulars.",
"The dependence of the potential on line width is less strong than expected for virialized systems, but this may be because surface brightness is acting as a proxy for variations in the mass-to-light ratio from galaxy to galaxy.",
"Altogether, the observations suggest that gas motions are predominantly disordered and isotropic, that they are a consequence of gravity, not turbulence, and that the mass and scale of dark matter haloes scale with the amount and distribution of luminous matter.",
"The tight relationship between the potential and observables offers the promise of determining distances to unresolved star-forming dwarfs to an accuracy comparable to that provided by the Tully-Fisher relation for spirals."
],
[
"Introduction",
"$\\Lambda $ CDM cosmology leads to dwarf galaxies with dark matter haloes at the centre of which is a cusp in density.",
"However, dwarfs in which a significant portion of their internal energy is ordered show rotation curves which rise less steeply outward from their centres than expected.",
"Thus, their core density profiles must be quite flat [58], [19], [66], [12], [60].",
"Dwarfs over a wide range of absolute magnitudes display surface brightness profiles which are flat in the cores, also suggestive of a central dark matter framework whose density is slowly varying [73].",
"Hydrodynamic simulations taking into account star formation and its consequences suggest that a flat density profile is a response to the blowout of baryonic matter with low angular momentum by supernova explosions [24].",
"On the contrary, the chemical properties of star-forming dwarfs in low-density environments indicate that gas flows have not played a major role in their evolution [45], [75].",
"A clue as to whether or not the current state of dwarfs is a consequence of gas flows may come from velocity dispersions.",
"Galaxies whose evolution has been affected significantly by flows may well display internal motions which are not entirely explainable as a response to gravity.",
"Recently, rotating disk galaxies were discovered at low redshift in which the velocity dispersion is large [25].",
"Because line widths correlate with star formation rates but not masses or gas fractions, the unusual motions were attributed to turbulence resulting from star formation activity.",
"This may be relevant to understanding the Tully-Fisher relation for star-forming dwarfs, which is highly scattered [76].",
"Although some of the scatter can be explained through a connection to surface brightness (the fundamental plane for dwarfs), there remains a significant component which cannot be attributed to observational errors [76].",
"Within the context of evaluating the impact of star formation on dynamics, and by implication the evolution of both mass and chemistry, it is important to examine in more detail how closely the mass and distribution of visible matter in dwarfs are linked to kinematics.",
"This motivates, in particular, exploration of the baryonic Tully-Fisher relation [52], since a significant portion of the mass of star-forming dwarfs is in gaseous form.",
"From the standpoint of turbulence, it is of interest to compare blue compact dwarfs, in which there is evidence for a recent burst of star formation, with the more quiescent dwarf irregular galaxies [74].",
"A better understanding of the physics of star-forming dwarfs also has the potential to open up new avenues for determining distances.",
"At the moment, distances to unresolved systems are so poorly constrained that it is not possible to map peculiar motions on large scales independently from giants.",
"In this paper, star-forming dwarfs in the Local Volume whose structural properties are defined by near-infrared surface photometry are employed to study how the luminosity, baryonic mass, and baryonic potential are linked to kinematics.",
"Simultaneously, the mass-to-light ratio is constrained by optimizing linkages.",
"Section introduces new near-infrared observations of star-forming dwarfs, the surface photometry for which is presented in Section .",
"An expanded sample of galaxies suitable for study is assembled in Section , and then subjected to detailed analysis in Section .",
"This leads to the identification of a more fundamental plane for dwarfs.",
"Section follows with a discussion of results, especially examining how closely internal motions are tied to gravity.",
"As well, a new method for deriving distances to dwarfs is presented.",
"Finally, conclusions are presented in Section .",
"During 2008 Mar 10–13 and Aug 10–12, deep NIR images of 23 galaxies were acquired using the $4.1 \\, \\rm m$ Blanco telescope at Cerro Tololo Inter-American Observatory, Chile (Run IDs: 2008A-0913 and 2008B-0909).",
"All three nights of the first run were photometric, but only the second night of the August run was clear.",
"During both runs, the ISPI camera was used at the $f/8$ Cassegrain focus.",
"The detector was a Hawaii array with $2048 \\times 2048$ pixels.",
"The scale was $0\\,3 \\, \\rm pix^{-1}$ , yielding a field of view $1025\\times 1025$ .",
"Targets were imaged exclusively through the $K_s$ filter.",
"Table REF summarizes the observations.",
"To sample the sky, small objects were cycled through four quadrants of the array.",
"For large targets, the telescope was jogged to a sky field after every pair of dithered target images.",
"Data were reduced, calibrated, and analyzed in the manner described by [73].",
"Typically, 10 to 15 2MASS stars were employed to calibrate each field.",
"Imaging and surface photometry for the 13 dwarfs clearly detected with the Blanco telescope are presented in Figure REF .",
"For reference, Figure REF gives the reduced images of the fields of the 10 unexaminable dwarfs.",
"It is possible that galaxies imaged in August (HIPASS J1337$-$ 39, Sag DIG, and DDO 210) were obscured by thin clouds.",
"HIPASS J1351$-$ 47 and Sag DIG appear to have been detected, but not well enough to permit surface photometry.",
"The remaining galaxies were just too faint to detect with the chosen exposures.",
"Figure: Images and surface photometry of dIs observed at CTIO (Blanco) and La Silla (NTT).Left panels: K s K_s images (North is up, East to the left).",
"The field of view is about 5×55\\times 5(Blanco) or 25×2525 \\times 25 (NTT).Right panels: Surface brightness profiles in K s K_s for the unresolved components.The thick solid curves are fits of a sech function.",
"In a few cases, a Gaussian burst was fitted simultaneously,and is marked by a dashed curve.",
"In these cases, the sum of the sech and Gaussian components is shown as a thin solid line (sometimes hard to see due to overlap with the observations).Figure: (cont'd)Figure: (cont'd)Figure: (cont'd)Figure: (cont'd)Figure: Images in K s K_s of the fields of galaxies either marginally or not detected at CTIO (Blanco) and La Silla (NTT).",
"North is up, and East is to the left.The fields of view are about 5×55\\times 5 (Blanco) and 25×2525 \\times 25 (NTT)."
],
[
"NTT observations 2008",
"During 2008 Aug 13–17, deep NIR imaging of nine galaxies was undertaken with the 3.5m NTT telescope at ESO La Silla Observatory, Chile (Run ID: 081.B-0386(A)).",
"One night was clear, and the rest were clouded out.",
"The SOFI camera equipped with a Hawaii HgCdTe array was employed at the $f/11$ Nasmyth focus.",
"The array was composed of $1024 \\times 1024$ pixels.",
"The scale was $0\\,288 \\, \\rm pix^{-1}$ , so the field of view was $492\\times 492$ .",
"All targets were observed with the $K_s$ filter only.",
"Observations are summarized in Table REF .",
"Data were reduced, calibrated, and analyzed in the same manner as for the Blanco runs.",
"Around five 2MASS stars were employed to calibrate most of the fields.",
"Imaging and surface photometry for the six dIs solidly detected are included in Figure REF .",
"Figure REF includes the reduced images of the fields of the two unexaminable dwarfs not observed with the Blanco telescope.",
"In fact, ESO 540-30 appears to have been detected, but not well enough to carry out surface photometry."
],
[
"Other observations 2002–2007",
"As part of separate studies, deep $K_s$ images of 110 different dwarf galaxies were obtained from 2002 to 2007 over observing runs conducted with the 2.1 m telescope of OAN-SPM in Mexico (2002 and 2005), the 1.4m IRSF telescope of SAAO in South Africa (2005 and 2006), the 3.6m CFHT in Hawaii (2002, 2004, 2005, and 2006), and the Blanco Telescope at CTIO (2006 and 2007).",
"Images and surface photometry are presented in [73] (34 galaxies), [76] (17 galaxies, plus eight from 2MASS), and [17] (80 galaxies).",
"Of the newly-observed dwarfs, 15 out of 19 have flat cores and exponential wings.",
"[73] showed that a sech function provides a good fit to such profiles.",
"For this function, the apparent surface brightness $\\mu ^{app}$ in $\\rm mag \\, arcsec^{-2}$ at radius $r$ along the major axis is given by $\\mu ^{app} = \\mu _0^{app} - 2.5 \\log {2 \\over e^{r / r_0} +e^{-r / r_0}} $ where $\\mu _{0}^{app}$ is the apparent surface brightness at the centre and $r_0$ is the scale length.",
"Solutions for the parameters of the best fitting sech functions are given in Table REF .",
"In the right panels of Figure REF , fits are shown as thick solid lines.",
"Some dwarfs show an excess of light in their centres.",
"They can be interpreted as being normal dIs hosting a central starburst, i.e., blue compact dwarfs [74].",
"NGC 1311 and ESO 137-18 are two such objects.",
"Their surface brightness profiles were modeled by simultaneously fitting a Gaussian on top of the sech function describing the extended underlying light distribution.",
"In Figure REF , the Gaussian component is displayed as a dashed line, and the sum of the Gaussian and sech functions is marked by a thin solid line.",
"The brightness of the main body of each dwarf was estimated by integrating the sech function out to infinity.",
"The apparent total magnitude $m_{sech}^{app}$ , referred to here as the sech magnitude, was computed from $m_{sech}^{app} = -2.5 \\log \\left[ 11.51036 \\, r_0^2 \\, q \\, I_0^{app} \\right] $ where $I_0^{app}$ is the apparent central surface brightness in linear units and $q$ is the axis ratio ($b/a$ ) of the isophotes.",
"The magnitude within the outermost detected isophote, referred to here as the isophotal magnitude, was also estimated.",
"Sech and isophotal magnitudes for all 19 of the dwarfs detected at Blanco and the NTT are given in Table REF ."
],
[
"Amalgamated sample and data",
"The observations presented above and in [17] significantly expand the sample of dwarfs for which deep $K_s$ -band photometry is available.",
"Thus, it is appropriate to re-examine the scaling relations elucidated earlier, especially to seek deeper insights into the why scatter is so much greater than expected from observational errors.",
"A sample of star-forming dwarfs was compiled from galaxies with extant $K_s$ -band surface photometry to which a sech profile had been fitted.",
"In order to minimize scatter due to distance errors, the sample was restricted to objects for which the $I$ magnitude and $V-I$ colour of the tip of the red giant branch (TRGB) have been measured reliably.",
"A total of 66 galaxies satisfied the criteria for analysis, and are listed in Table .",
"The galaxies IC 10 and ESO 245-05 were not included due to obvious problems with their photometry.",
"In particular, IC 10 is a very large galaxy suffering from heavy extinction and severe crowding by foreground stars.",
"NGC 1560 was omitted because it is a late-type spiral.",
"To establish homogeneous distances, the absolute magnitude of the TRGB was estimated from $M_{I,TRGB} & = & (-3.935 \\pm 0.028) \\\\& & + (0.217 \\pm 0.020) \\left[ (V-I)_{TRGB} - 1.6 \\right] \\nonumber $ where $(V-I)_{TRGB}$ is the mean colour of the stars at the tip corrected for extinction and redshift [64].",
"The zero-point was determined from a pairwise analysis of 127 distances to 34 nearby galaxies derived from Cepheids, planetary nebulae, surface brightness fluctuations, and the TRGB (McCall, M. L., in preparation), and is anchored to the maser distance to NGC 4258 [27], [22], [47].",
"The uncertainty is that due to random errors only; it does not include the uncertainty in the distance to NGC 4258.",
"The rate of change of the absolute magnitude of the TRGB with colour was adopted from [64].",
"Apparent magnitudes and colours of TRGB stars were extracted from the literature.",
"In instances where colours were not recorded, they were estimated through inspection of colour-magnitude diagrams.",
"Where necessary, conversion of HST photometry to the Johnson-Cousins system was accomplished using the transformation equations of [67].",
"Apparent magnitudes and colours were corrected for extinction and redshift (i.e., K-corrections) using the York Extinction Solver [48].",
"Optical depths were computed from $B-V$ colour excesses tabulated by [65], and these were converted into extinctions and $V-I$ colour excesses assuming the spectral energy distribution of an M0 giant.",
"The adopted reddening law was that of [18], tuned to deliver a ratio of total-to-selective extinction of 3.07 for Vega [48].",
"Colour excesses, extinctions, corrected TRGB magnitudes and colours, and the resulting distance moduli for the 66 sample galaxies are recorded in Table .",
"Adopted heliocentric velocities are given in Table .",
"All K-corrections were less than $0.01 \\, \\rm mag$ .",
"Uncertainties in distance moduli are estimated to be $0.10 \\, \\rm mag$ typically.",
"This paper extends work on absolute magnitudes of dwarfs into the realm of masses, motivated by the fact that dIs and BCDs retain a large fraction of their mass in gaseous form.",
"It is reasonable to expect that any correlation of absolute magnitude with dynamics will have scatter enhanced by variations in gas fractions, because the stellar component is often a minority of the visible mass (see Section REF ).",
"The determination of a baryonic mass requires that both the stellar and gaseous masses be constrained.",
"The mass of stars in each galaxy was judged to be most reliably signified by the luminosity in $K_s$ , because the light from young stars is suppressed and the mass-to-light ratio is less sensitive to the star formation history than in bluer passbands [33], [61], [73].",
"The absolute magnitude of the diffuse stellar component was determined from the integrated magnitude of the sech function modeling the two-dimensional surface brightness profile.",
"The luminosity of any co-existing starburst was estimated by integrating the flux under the fitted Gaussian.",
"Corrections for extinction and redshift (i.e., K-corrections) were accomplished as for the TRGB, but using the Im spectral energy distribution of [48].",
"In computing luminosities from absolute magnitudes, the absolute magnitude of the Sun was adopted to be 3.315 in $K_s$ [28], [20].",
"Note that no corrections for redshift dimming were applied.",
"All galaxies in the sample are at low redshift, and the entire range spanned by dimming corrections is only $0.01 \\, \\rm mag$ .",
"Parameters describing the light distributions of the 66 sample galaxies are given in Table .",
"Listed are the corrected value $\\mu _0$ of the central surface brightness in $\\rm mag \\, arcsec^{-2}$ , the sech scale length $r_0$ converted to parsecs, the axis ratio $q$ , the limiting radius of the surface photometry in units of $r_0$ , and the source of the photometry.",
"The derived absolute magnitude $M_{Ks}$ of the sech component and the ratio of the luminosity of any burst relative to the luminosity of the sech component are given in Table .",
"The adopted value of the extinction is included in Table .",
"Generally, for galaxies observed on more than one occasion, parameters describing the fit to the deepest profile are presented.",
"However, parameters listed for ESO 381-18 and IC 4247 come from averages of fits to two independent observations.",
"The mass of gas was determined from the integrated flux of HI at $21 \\, \\rm cm$ .",
"Given the flux $F_{HI}$ in $\\, \\rm K \\, km \\, s^{-1}$ , the mass $M_{gas}$ in solar units was computed from $M_{gas} = k_{21} D_{Mpc}^2 F_{HI} / X $ where $D_{Mpc}$ is the distance in Mpc, $X$ is the mass fraction of hydrogen, and $k_{21} = 2.356 \\times 10^5 \\, \\rm M_\\odot K^{-1} km^{-1} s$ .",
"The value of $X$ was adopted to be 0.735 on the basis of measurements of the rate of change of the helium and metal fractions with the oxygen abundance in dwarfs [32], presuming a mean oxygen abundance of 8.25 for the current sample and a primordial helium mass fraction of 0.257 [32].",
"For galaxies for which multiple measurements of $F_{HI}$ were available, the single-dish measurement with the highest signal-to-noise ratio was normally adopted, unless there was evidence for confusion or peculiarities in the spectrum.",
"The adopted fluxes and sources are listed in Table , and corresponding gas masses are given in Table .",
"The width of the $21 \\, \\rm cm$ line at 20% of the peak, $W_{20}$ , was used to quantify internal motions.",
"The choice of $W_{20}$ over the width at 50% of the peak was motivated by a desire to measure kinematics representative of the broadest possible body of gas.",
"Earlier studies of the fundamental plane for dIs relied upon line widths with generally poor velocity resolution.",
"Most were taken from the Third Reference Catalogue of Bright Galaxies [14].",
"For this paper, a comprehensive survey of the literature was made to pinpoint $21 \\, \\rm cm$ line profiles with the highest resolution and least noise.",
"Where 20% line widths were not recorded, they were measured from the plotted profiles or, if justifiable, established mathematically from a fit used to determine the tabulated 50% line width.",
"Because line profiles for the galaxies in this sample were very close to being Gaussian in shape, the apparent line width $W_{20}^{app}$ was corrected for instrumental broadening by subtracting in quadrature the width of a Gaussian instrumental profile as defined by the full-width at half-maximum $R$ [77]: $W_{20} = {W_{20}^{app} \\over (1 + z)} \\sqrt{1 - {ln 5 \\over ln 2} \\left( R \\over W_{20}^{app} \\right)^2} $ Here, $R$ , $W_{20}^{app}$ , and $W_{20}$ are in $\\rm km \\, s^{-1}$ .",
"The factor of $1 + z$ corrects the width for redshift broadening.",
"The adopted values of the heliocentric velocity, $W_{20}^{app}$ , $R$ , and $\\log W_{20}$ , as well as the sources of the data, are given in Table ."
],
[
"Overview",
"Investigations below concentrate on elucidating how observed kinematics of dwarfs are tied to their scale and structure.",
"In the process, they lead to insights on what is driving gas motions, constraints on the mass-to-light ratio of stars, an evaluation of how close the galaxies are to being virialized, and the establishment of a method for determining reliable distances to unresolved objects.",
"The most important correlations between intrinsic galaxy properties (absolute magnitude, mass, and potential) and distance-independent observables (HI line width, central surface brightness, and axis ratio) are displayed along with their fits in Figure REF .",
"The ordinates of the panels have been configured to span identical ranges in magnitude units, so that the apparent vertical dispersions about the fits are inter-comparable.",
"The displayed correlations are founded upon the properties of the sech component of the light profiles alone.",
"Derived global properties of the galaxies are summarized in Table .",
"The relevance of the burst component and the distribution of the properties of bursting galaxies are discussed in Section REF .",
"Figure: Correlations among intrinsic and observed properties of star-forming dwarfs.",
"In all panels, line widths have been corrected for resolution but not for tilt.",
"In magnitude units, the range of ordinates is the same for all panels, so vertical dispersions are directly comparable.",
"Typical uncertainties in abscissae and ordinates are depicted by an error cross in the upper left corner of each panel.",
"Top left: Absolute magnitude in K s K_s versus the HI line width (the Tully-Fisher relation); Top right: Absolute magnitude in K s K_s versus the HI line width and the observed central surface brightness .",
"Correcting the surface brightnesses for tilt worsens the fit.",
"; Middle left: Baryonic mass versus the HI line width, based upon a mass-to-light ratio in K s K_s derived from the fit to the potential plane; Middle right: Baryonic mass versus the HI line width and the observed central surface brightness, based upon a mass-to-light ratio in K s K_s derived from the fit to the potential plane.",
"Correcting the surface brightnesses for tilt does not improve the fit; Bottom left: Baryonic potential versus the HI line width, based upon a mass-to-light ratio in K s K_s derived from the fit to the potential plane; Bottom right: Baryonic potential versus the HI line width and tilt-corrected central surface brightness (the potential plane), based upon a mass-to-light ratio in K s K_s derived during the fitting.",
"In all panels, every galaxy in the sample is plotted, but only galaxies marked with large circles were fitted.",
"Excluded from the fits were four galaxies observed exclusively by 2MASS (crosses), six galaxies in addition to two 2MASS galaxies for which photometry did not extend beyond 2.5 sech scale lengths (solid circles), one galaxy with an unusual morphology (small open circle), and seven extreme outliers identified while fitting the potential (small open circles).The correlations were defined without correcting line widths for projection.",
"Even though it is known that some of the more massive dwarfs are rotating [15], [71], there is considerable evidence that the kinetic energy of the gas in most of the galaxies in the sample is predominantly disordered [73].",
"For example, line profiles tend to be Gaussian in shape, and the sensitivity of line widths to the ellipticity of isophotes is weak at best, implying that motions are close to isotropic (see Section REF ).",
"All galaxies in the sample are displayed in Figure REF , but only the 48 galaxies marked by large open circles were included in fitting.",
"Photometry originating from 2MASS becomes suspect for galaxies whose surface brightnesses are as low as is typical of sample members [42], and galaxies whose surface brightness profiles do not extend beyond 2.5 sech scale lengths tend to display deviant properties.",
"Consequently, the following galaxies were excluded from the fits: four dwarfs observed by 2MASS only (Ho II, NGC 3077, NGC 4214, and NGC 6822, marked by crosses) in addition to two of the 2MASS galaxies (Ho II and NGC 6822), six dwarfs for which photometry did not extend beyond $2.5 r_0$ (Cam B, ESO444-84, KK98 230, KKH 86, Peg DIG, and UGCA 92, marked by solid circles) one object which displays a spiral-like morphology in HI and whose surface brightness profile is convex in the core (NGC 2915) seven extreme deviants identified during the course of analysis (DDO 47, DDO 168, ESO215-09, ESO223-09, KK98 17, KK98 182, and UGC 3755), all of which are signified by small open circles The last seven galaxies were revealed by a large gap in the histogram of residuals for the fit to the potential versus line width and surface brightness.",
"They lie $3.1 \\sigma $ or more away from the fit, whereas the most extreme of the retained galaxies lie within $1.8 \\sigma $ of that fit.",
"For maximum flexibility, fits were determined using the downhill simplex algorithm [59], [63].",
"For certain fits involving the stellar mass, it was postulated that any fundamental relationship is one for which there exists a mass-to-light ratio which minimizes the dispersion.",
"Stable solutions to the mass-to-light ratio proved to be possible by combining the simplex algorithm with a golden section search.",
"Uncertainties in derived parameters were ascertained through Monte Carlo simulations.",
"The starting point for these simulations were estimates for typical errors in the observables.",
"The adopted uncertainties were $0.1 \\, \\rm mag$ for distance moduli, $0.15 \\, \\rm mag$ for $\\mu _0$ , 5% for $r_0$ , 10% for $q$ , $0.23 \\, \\rm mag$ for $m_{sech}$ , 5% for $W_{20}$ , and 10% for $F_{HI}$ .",
"In each analysis, 1000 random deviates of these observational quantities were computed, from which deviates for the derived quantities were computed and fitted.",
"Below, any quoted uncertainty in a fitted parameter is the average of the standard deviations of the resulting solutions on either side of the solution obtained from the reference fit."
],
[
"Absolute magnitude",
"The upper left panel of Figure REF displays the Tully-Fisher relation for the dwarfs, albeit with no correction of line widths for tilt (as discussed above).",
"As expected, the dispersion is large.",
"The fitted relation is $M_{Ks,sech} & = & (-16.424 \\pm 0.040) \\\\& & - (5.066 \\pm 0.207) (\\log W_{20} - 1.8) \\nonumber $ with the standard deviation being $0.95 \\, \\rm mag$ .",
"By comparison, the expected vertical scatter due to observational errors alone is only $0.27 \\, \\rm mag$ (the quadrature sum of the uncertainties in the abscissa and ordinate).",
"Even if all motions were rotational, the dispersion in axis ratios is such that only about $0.5 \\, \\rm mag$ of the scatter would be attributable to projection.",
"The upper right panel of Figure REF displays the fundamental plane of [73], with the fit updated.",
"For the sample here, $M_{Ks, sech} & = & (-16.490 \\pm 0.041) \\\\& & - (2.789 \\pm 0.243) (\\log W_{20} - 1.8) \\nonumber \\\\& & + (0.721 \\pm 0.043) (\\mu _0 - 20) \\nonumber $ where $\\mu _0$ is the observed central surface brightness in $\\rm mag \\, arcsec^{-2}$ (i.e., without any correction for projection).",
"The standard deviation is $0.63 \\, \\rm mag$ .",
"The dispersion worsened when surface brightnesses were corrected for tilt (assuming an oblate spheroidal geometry: see Section REF ).",
"Efforts to improve the fit by attributing some of the motions to rotation, and correcting for tilt accordingly, met with failure.",
"The scatter is larger than suggested by [76], probably because of changes to rejection criteria.",
"Most importantly, as noted by [76], it is much larger than the dispersion to be expected on the basis of observational errors alone, which is $0.28 \\, \\rm mag$ ."
],
[
"Mass",
"The unexplained scatter in absolute magnitudes motivated the development of a corresponding plane for the baryonic mass $M_{bary}$ which would accommodate the often significant but highly variable proportion of matter in gaseous form.",
"There was reason to be optimistic that a well-defined relationship might be found because the baryonic Tully-Fisher relation for rotationally-supported systems is so tight [52], [50], [51].",
"Most of the galaxies in the sample appear to be pressure-supported, with random motions being close to isotropic (see Section REF ), so on energy grounds it is reasonable to construct the baryonic Tully-Fisher relation by substituting radial velocity dispersions for circular velocities [53], [81].",
"The computation of baryonic masses required the adoption of a mass-to-light ratio for the stars, since $M_{bary} = \\Upsilon L_{Ks} + M_{gas} $ Here, $L_{Ks}$ is the luminosity in $K_s$ and $\\Upsilon = M_{stars} / L_{Ks}$ is the mass-to-light ratio of the stars in $K_s$ .",
"Attention was restricted to the sech component of the light distribution.",
"The contribution of a burst component to the mass was assumed to be negligible.",
"This approximation is justified in Section REF , where the consequences of accommodating the light of a burst are discussed.",
"The best estimate of $\\Upsilon $ was gained from analysis of the gravitational potential (see Section REF ), which yielded a solution of $0.88 \\pm 0.20$ .",
"To establish the most credible relationship between the baryonic mass and observables, then, the mass-to-light ratio was fixed at $0.88$ .",
"The corresponding stellar and baryonic masses are summarized in Table .",
"Surprisingly, the baryonic Tully-Fisher relation proved to be as highly dispersed as the fundamental plane.",
"It is displayed in the middle left panel of Figure REF .",
"The fitted relation is given by $\\log M_{bary} & = & (8.138 \\pm 0.012) \\\\& & + (2.140 \\pm 0.064) (\\log W_{20} - 1.8) \\nonumber $ The dispersion is $0.24 \\, \\rm dex$ ($0.61 \\, \\rm mag$ ), versus the expected value of $0.08 \\, \\rm dex$ based upon observational uncertainties alone.",
"An improvement to the baryonic Tully-Fisher relation was realized by introducing surface brightness as a second parameter.",
"The resulting baryonic plane, which is displayed in the middle right panel of Figure REF , is described by $\\log M_{bary} & = & (8.152 \\pm 0.012) \\\\& & + (1.671 \\pm 0.074) (\\log W_{20} - 1.8) \\nonumber \\\\& & + (-0.148 \\pm 0.013) (\\mu _0 - 20) \\nonumber $ (for $\\Upsilon = 0.88$ ).",
"The standard deviation is $0.20 \\, \\rm dex$ ($0.49 \\, \\rm mag$ ).",
"Although lower than found for the Tully-Fisher relation, it is still much higher than the expected value of $0.08 \\, \\rm dex$ based upon observational uncertainties.",
"The fit did not improve when the surface brightnesses were corrected for projection (see Section REF ).",
"Allowing $\\Upsilon $ to be free, the dispersion dropped somewhat to $0.17 \\, \\rm dex$ .",
"However, the solution for $\\Upsilon $ was $0.15 \\pm 0.05$ , which is unreasonably low compared to expectations from population syntheses [61]."
],
[
"Potential",
"Thirty-two galaxies in the sample have line profiles which have been observed with a resolution of $2 \\, \\rm km \\, s^{-1}$ or less.",
"Profiles are close to being Gaussian in shape, which suggests that the dynamics of dIs may be simple.",
"As a starting point, it is reasonable to posit that the systems are close to being virialized.",
"Virialization requires that $2T + \\Omega = 0 $ where $T$ is the kinetic energy and $\\Omega $ is the potential energy.",
"If the line width is predominantly controlled by gravity, and if the potential defined by the baryonic mass scales with the potential setting the line width (which in large part must be controlled by the amount of dark matter), then one might surmise that $P \\equiv M_{bary} / r_0 \\propto \\left( W_{20} \\right)^2 $ Henceforth, $P$ will be referred to as the “baryonic potential”.",
"The baryonic potential (the stellar component of which being defined by the mass of the sech component) is plotted as a function of line width in the lower left panel of Figure REF .",
"The relationship is given by $\\log P & =& (5.559 \\pm 0.011) \\\\& & + (1.760 \\pm 0.058) (\\log W_{20} - 1.8) \\nonumber $ The standard deviation of the fit is $0.24 \\, \\rm dex$ ($0.59 \\, \\rm mag$ ), which, surprisingly, is comparable to that for the fits to the baryonic mass.",
"However, the introduction of surface brightness as a second parameter reduced the dispersion drastically.",
"With the mass-to-light ratio fixed at 0.88 (see below), the following relationship was found: $\\log P & =& (5.578 \\pm 0.011) \\\\& & + (1.101 \\pm 0.065) (\\log W_{20} - 1.8) \\nonumber \\\\& & + (-0.208 \\pm 0.012) (\\mu _0 - 20) \\nonumber $ The standard deviation is only $0.12 \\, \\rm dex$ ($0.29 \\, \\rm mag$ ).",
"The dispersion did not change significantly when the mass-to-light ratio was allowed to vary.",
"Figure REF shows the residuals in the fit to the potential as a function of the logarithm of the axis ratio $q$ .",
"Residuals become more negative as galaxies flatten.",
"To a significant extent, this is likely to be a consequence of the effect of projection on surface brightnesses.",
"For an oblate spheroid, the surface brightness varies with $q$ as $\\mu _0 = \\mu _{0}^{i=0} + 2.5 \\log q $ where $\\mu _{0}^{i=0}$ is the surface brightness that would be measured if the view were face-on (inclination $i$ equal to zero).",
"The dashed line in Figure REF displays the rate at which residuals in the potential should vary with $q$ if surface brightnesses are affected by projection in the way expected for oblate spheroids.",
"The observed trend is very close to that predicted.",
"Thus, a further refinement to the fit to the potential was possible by correcting surface brightnesses to a common viewing angle (face-on).",
"Figure: Residuals in the potential as a function of the axis ratio q=b/aq = b/a.",
"The dashed line displays the slope of the expected relationship for oblate spheroids (the y-intercept having been fixed at zero).It was unclear how line widths dominated by random motions might vary with tilt.",
"For the purpose of investigation, any sensitivity to tilt was approximated as a power law in $q$ .",
"Then, to convert measurements to face-on, $W_{20}^{i=0} = q^\\gamma W_{20} $ where $\\gamma $ is a constant.",
"It was expected that a relationship between the potential, line width, and surface brightness which is free of projection effects would have the form $\\log P & = & a + b \\log q^\\gamma W_{20} + c (\\mu _0 - 2.5 \\log q) \\\\& = & a + b \\log W_{20} + c \\mu _0 + (\\gamma b - 2.5 c) \\log q $ where $a$ , $b$ , and $c$ are constants.",
"By introducing $\\log q$ as a third variable, it was possible to constrain $\\gamma $ .",
"With $\\Upsilon $ fixed at 0.88, and with the geometry approximated to be oblate spheroidal, the solution for $\\gamma $ was $-0.10$ .",
"The sign is opposite to what would be expected if flattening is a consequence of anisotropic motions, be they ordered or disordered.",
"Also, $| \\gamma |$ is small, suggesting that motions are close to being isotropic.",
"It was concluded that the trend in the residuals of the potential with the axis ratio should be attributed primarily to variations in surface brightness expected for oblate spheroids viewed at different angles.",
"In the end, given how weakly line widths appeared to depend on tilt and uncertainty about precisely how they did, only surface brightnesses were corrected for projection.",
"The relationship between the potential, apparent line width, and the surface brightness corrected to face-on (via Equation REF ) is displayed in the lower right panel of Figure REF .",
"With the mass-to-light ratio free to vary, the fit was given by $\\log P & = & (5.697 \\pm 0.065) \\\\& & + (1.134 \\pm 0.080) (\\log W_{20} - 1.8) \\nonumber \\\\& & + (-0.198 \\pm 0.018) (\\mu _0^{i=0} - 20) \\nonumber $ The corresponding solution for $\\Upsilon $ was $0.883 \\pm 0.199$ .",
"Resulting values of $\\log M_{stars}$ , $\\log M_{bary}$ , and $\\log P$ are listed in Table .",
"Uncertainties in the coefficients are higher than for Equation REF because of the freedom in the mass-to-light ratio.",
"The standard deviation is only $0.096 \\, \\rm dex$ ($0.24 \\, \\rm mag$ ).",
"The vertical dispersion expected from random observational errors is $0.08 \\, \\rm dex$ .",
"Thus, a structural relationship has been identified for dwarfs for which observational errors overwhelm the cosmic dispersion.",
"This relationship can be regarded as a more fundamental plane for dwarfs, and henceforth will be referred to as the potential plane.",
"Although the solution for the mass-to-light ratio is twice as high as measured for the disk of the Milky Way from vertical kinematics of stars [62], it is nevertheless compatible with syntheses of exponential disks spanning a range of possible star formation histories [61].",
"It should be noted, though, that the estimate for $\\Upsilon $ is in part a dynamical estimate of the mass-to-light ratio, since it is tied to line widths.",
"The tightness of the correlation lends credence to the postulate that the mass and size of dark matter haloes scale straightforwardly with the mass and distribution of luminous matter.",
"To check the sensitivity of the fit to the selection of galaxies, random subsets of the original galaxy sample were formed by removing 10% of the objects (5 galaxies).",
"A total of 100 subsets were constructed and fitted.",
"The mean values of the free parameters agreed extremely well with those determined from the entire sample.",
"Most important, the mean value of $\\Upsilon $ was $0.895 \\pm 0.095$ , which is almost identical to the value derived by fitting the whole sample.",
"Because observational errors are predominant in setting the dispersion, it is not believed that the solution for the mass-to-light ratio suffers from biases which plague fits to less fundamental relations."
],
[
"Blue compact dwarfs and turbulence",
"Despite much research, the relationship between dIs and BCDs is not clear yet.",
"Structurally, they appear to be similar, because the near-infrared light profile of a BCD can be modeled well by superimposing a Gaussian starburst upon a sech function [74].",
"Thus, it is reasonable to consider any star-forming dwarf to be a blue compact dwarf if its light profile in $K_s$ displays an excess of light in the core over what is expected for a pure sech law.",
"That is the definition adopted here.",
"The left panel of Figure REF re-displays the correlation of the potential with line width and face-on central surface brightness (Equation REF ).",
"Those galaxies with an excess of light in the core, i.e., the BCDs, are marked with solid circles, and normal dIs are marked with open circles.",
"Starbursts span the entire range of galaxies in the sample.",
"Fits to the pure sech dwarfs and the BCDs separately were consistent within errors, proving that there is no segregation.",
"It appears that gravity, not turbulence, is the predominant determinant of gas kinematics in most of the star-forming dwarfs in the sample.",
"The mere existence of strong correlations of the baryonic mass and the potential with line width lends support to this conclusion.",
"Also, the dependence of the potential on surface brightness is opposite to what would be expected if line widths were inflated by gas flows stemming from recent star formation.",
"For a gas-dominated system at least, any concomitant enhancement in surface brightness would be expected to be incorporated in such a way as to oppose the change in line width and thereby preserve the potential.",
"Instead, the sign of the coefficient of $\\mu _0^{i=0}$ in Equation REF is such that brightening and broadening change the potential in the same way.",
"Figure: Two renditions of the potential for dwarfs.",
"Left: M stars /L Ks M_{stars} / L_{Ks} fixed.",
"Right: M stars /L Ks M_{stars} / L_{Ks} variable.",
"Individualized mass-to-light ratios for the plot on the right were estimated from the amount by which the central surface brightness of the sech component deviated from the mean for a given line width under the condition that the galaxies are virialized (see Figure ).",
"In both panels, dwarf irregulars (dIs) are marked with open circles, and blue compact dwarfs (BCDs) are flagged by solid circles.",
"No matter how the mass-to-light ratios are computed, BCDs and dIs populate the diagrams in the same way.To examine the influence of the Gaussian component on the fit to the potential, the mass-to-light ratio of stars in the burst relative to the mass-to-light ratio of stars in the sech was introduced as a free parameter.",
"Then, luminosities of both the burst and sech components were employed to compute baryonic masses.",
"The solution for the ratio of mass-to-light ratios was zero, meaning that the inclusion of a non-negligible burst mass degrades the correlation.",
"However, forcing the ratio of mass-to-light ratios to unity led to a fit which was still reasonable.",
"A larger sample of BCDs in which the burst light is a significant fraction of the total light will be required to constrain more reliably the appropriate mass-to-light ratio to use to accommodate the burst mass in the definition of the potential."
],
[
"Variable mass-to-light ratios and virialization",
"Equation REF reveals that $P \\propto \\left[ W_{20} \\right] ^{1.13} \\left[ I_0^{i=0} \\right]^{0.49} $ where $I_0^{i=0}$ is the face-on central surface brightness of the sech model in linear units.",
"Although the virial theorem motivated the quest for this relation, the exponent of $W_{20}$ is half what it should be.",
"The dependence on the surface brightness is puzzling, too.",
"Possibly, it is a reaction to variations in the mass-to-light ratio.",
"A dwarf which has undergone star formation more recently than is typical for galaxies of its kind in the sample may show enhancements in both surface brightness and luminosity relative to the norm for its potential.",
"If so, the mass-to-light ratio ought to be reduced by a factor which preserves the value of the potential.",
"In effect, such a reduction is happening through the dependence of $P$ on $I_0^{i=0}$ , although it is complicated by the presence of gas.",
"It is even possible that, in not accounting for variability in $\\Upsilon $ , the true dependence of the potential on the line width is obscured.",
"To individualize mass-to-light ratios, and thereby assess the impact on the relationship between the potential and internal motions, the premise was made that any deviation of the surface brightness from the mean at a given line width is a consequence of a different star formation history, and that the deviation is simultaneously incorporated in the luminosity of stars associated with the sech component.",
"Then, the mass-to-light ratio $\\Upsilon $ relative to some reference value $\\Upsilon _{ref}$ could be estimated from the face-on central surface brightness $\\mu _0^{i=0}$ relative to an appropriate norm $\\mu _{0,ref}^{i=0}$ as follows: $\\log (\\Upsilon / \\Upsilon _{ref}) = 0.4 (\\mu _0^{i=0} - \\mu _{0,ref}^{i=0}) $ In principle, the mean surface brightness $\\mu _{0,ref}^{i=0}$ could be tied to global properties.",
"Figure REF reveals the correlation of $\\mu _0^{i=0}$ with $\\log W_{20}$ .",
"There is a tendency for surface brightnesses to brighten with the line width, although the scatter is large.",
"The correlation must in part be responsible for weakening the $W_{20}$ -dependence of the potential relative to what is expected for virialized systems.",
"Figure: Tilt-corrected central surface brightness of the sech component as a function of the HI line width.",
"Dwarf irregulars (dIs) are marked with open circles, and blue compact dwarfs (BCDs) are flagged by solid circles.",
"The dashed line is the required relationship if the galaxies are virialized, based upon a reference mass-to-light ratio of 0.883.If it is hypothesized that star-forming dwarfs really are virialized, it is possible to predict how rapidly $\\mu _{0,ref}^{i=0}$ must vary with $\\log W_{20}$ , and thereby test compatibility with the observations.",
"To this end, it was approximated that $\\mu _{0,ref}^{i=0}$ varies linearly with $\\log W_{20}$ .",
"Based upon the earlier fit to the potential with $\\Upsilon $ held fixed, it was reasonable to approximate $\\Upsilon _{ref}$ to be 0.883.",
"Noting that $\\Upsilon $ for each dwarf is defined by Equation REF , it was possible to solve for the coefficients of the relation between $\\mu _{0,ref}^{i=0}$ and $\\log W_{20}$ by minimizing the dispersion in the relationship between the potential and $\\log W_{20}$ .",
"The dashed line in Figure REF displays the solution for the correlation between $\\mu _{0,ref}^{i=0}$ and $\\log W_{20}$ which arose when the galaxies were required to be virialized.",
"It is defined by $\\mu _{0,ref}^{i=0} & = & (20.778 \\pm 0.214) \\\\& & + (-4.371 \\pm 0.296) (\\log W_{20} - 1.8) \\nonumber $ The data admit the possibility of such a trend.",
"The right panel of Figure REF shows the simultaneous fit to the potential, with baryonic masses now determined using individualized mass-to-light ratios computed from Equations REF and REF (with $\\Upsilon _{ref}$ set to 0.883).",
"It is described by $\\log P = (5.526 \\pm 0.041) + 2 (\\log W_{20} - 1.8) $ Of course, the coefficient in front of $\\log W_{20}$ was forced by the requirement that the systems be virialized.",
"The standard deviation about the fit is $0.105 \\, \\rm dex$ ($0.26 \\, \\rm mag$ ), which is only a little worse than that for the potential plane (Equation REF ).",
"Resulting mass-to-light ratios are listed in Table and displayed as a function of $\\log W_{20}$ in Figure REF .",
"The range of variation is fairly large, in many cases beyond what is reasonable to expect in $K_s$ for typical star formation histories [61].",
"Observational errors are in part to blame, since mass-to-light ratios vary to compensate for errors in surface brightness.",
"Figure: Mass-to-light ratio of the sech component as a function of line width.",
"Dwarf irregulars (dIs) are marked with open circles, and blue compact dwarfs (BCDs) are flagged by solid circles.",
"Individualized mass-to-light ratios were determined from the deviations in surface brightness from the norms expected for virialized systems.",
"The horizontal dashed line is the value of the mass-to-light ratio which was computed for the entire sample in establishing the potential plane of Figure .Figure REF compares the gas fractions computed from a fixed value of the mass-to-light ratio ($\\Upsilon = 0.88$ ) with the gas fractions derived when the mass-to-light ratio is allowed to vary as above.",
"The spread in gas fractions at large $W_{20}$ is reduced when variable mass-to-light ratios are employed.",
"For many star-bursting dwarfs, the gas fraction rises.",
"This is because surface brightnesses for these galaxies are unusually high, leading to downward adjustments to mass-to-light ratios and consequent reductions in the stellar masses.",
"Gas fractions for many dwarf irregulars also change significantly, rising for the least massive galaxies and declining for the most massive.",
"Figure: Gas fractions as a function of line width.",
"Left: M stars /L Ks M_{stars} / L_{Ks} fixed.",
"Right: M stars /L Ks M_{stars} / L_{Ks} variable.",
"In both panels, dwarf irregulars (dIs) are marked with open circles, and blue compact dwarfs (BCDs) are flagged by solid circles.",
"With mass-to-light ratios adjusted for surface brightness deviations, gas fractions for many galaxies change significantly."
],
[
"Distances",
"The discovery of tight correlations between the gravitational potential and distance-independent observables opens up a way of mapping the spatial distribution of dwarf galaxies on large scales.",
"Locally, dwarfs are more dispersed than giants, so the relationship offers an avenue for exploring the distribution of matter in mass-poor regions of the universe.",
"A huge advantage over the Tully-Fisher relation for spirals is that no restriction need be placed on tilt.",
"In fact, the method for correcting surface brightness for the viewing angle does not even require that the tilt be evaluated (a complicated problem owing to the possible variability of the intrinsic axis ratio).",
"For any galaxy, the value of $P \\equiv M_{bary} / r_0$ in $\\rm M_\\odot \\, pc^{-1}$ can be estimated from observations of the $21 \\, \\rm cm$ line width $\\log W_{20}$ , the central surface brightness $\\mu _0$ , and the axis ratio $q$ using Equation REF or REF .",
"This provides an avenue to determining the distance.",
"Define $m_{sech}$ to be the apparent magnitude of the sech component of the light distribution corrected for extinction and redshift, $r_{0,arcsec}$ the sech scale length in arc seconds, $\\Upsilon $ the mass-to-light ratio in solar units for the stars constituting the sech component (for the passband defining $m_{sech}$ ), $F_{21}$ the $21 \\, \\rm cm$ line flux in $\\rm K \\, km \\, s^{-1}$ , $X$ the fraction of the gas mass which is hydrogen, and $k_{21}$ the factor required to convert the HI line flux to hydrogen mass.",
"The distance modulus $D_{mod}$ is given by $D_{mod} / 5 & = & \\log P + \\log r_{0,arcsec} \\\\& & - \\log \\left[ \\Upsilon 10^{-0.4 (m_{sech} - M_\\odot - 12.5)}+ 10^{-5} k_{21} F_{21} / X \\right] \\nonumber \\\\& & - \\log (0.648 / \\pi ) \\nonumber $ where $M_\\odot $ is the absolute magnitude of the Sun (3.315 in $K_s$ ) and $k_{21}$ has the value specified for Equation REF .",
"For $K_s$ , the mass-to-light ratio is either fixed at 0.883 if Equation REF is used to estimate $P$ , or computed from Equations REF and REF if Equation REF is employed to get $P$ (with $\\Upsilon _{ref}$ set to 0.883).",
"Figure REF shows for the galaxies defining the potential plane how the distances derived from Equation REF compare with those derived from the TRGB.",
"Typical uncertainties in observational quantities were given in Section REF .",
"Because observational errors are responsible for the bulk of the dispersion in the potential, it is reasonable to examine the accuracy with which a distance can be determined considering observational errors alone.",
"Based upon the uncertainties in $W_{20}$ , $\\mu _0$ , and $q$ , $\\log P$ can be estimated to an accuracy of about $0.04 \\, \\rm dex$ (Equation REF ) if the cosmic dispersion is smaller.",
"Accounting for the errors in $m_{sech}$ , $r_{0,arcsec}$ and $F_{21}$ , then the uncertainty in the derived distance modulus comes out to be $0.38 \\, \\rm mag$ for $\\Upsilon = 0.88$ .",
"Figure: Difference between the potential plane distance modulus and TRGB distance modulus as a function of the TRGB distance modulus for dwarfs defining the potential plane.",
"The horizontal dashed line marks perfect agreement.",
"Dwarf irregulars (dIs) are marked with open circles, and blue compact dwarfs (BCDs) are flagged by solid circles.",
"The vertical component of the error bar shows the mean uncertainty expected for potential plane distances from observational errors if the cosmic scatter about the potential plane is zero.The most important contributors to the uncertainty are the errors in $m_{sech}$ and $\\mu _0$ .",
"In the sample studied here, these errors were quite large due to limitations in the field of view, which restricted the number of 2MASS stars available to calibrate the photometry.",
"Significant improvements are possible using more modern detectors with wider fields of view.",
"If the uncertainty in $m_{sech}$ can be reduced to $0.10 \\, \\rm mag$ and the uncertainty in $\\mu _0$ to $0.05 \\, \\rm mag$ , then the error in the distance modulus will come down to $0.26 \\, \\rm mag$ .",
"A further reduction is possible with deeper photometry, which would enable improvement in the accuracy with which $r_0$ and $q$ are measured.",
"Overall, it appears that the potential plane for dwarfs offers a means to determine distances to dwarfs as good as the Tully-Fisher relation yields for spirals."
],
[
"Conclusions",
"Deep imaging in $K_s$ has been presented for 19 star-forming dwarf galaxies.",
"Structural properties were measured by fitting a sech function to surface brightness profiles, and additionally a Gaussian for those objects displaying evidence for a central starburst (interpreted thereby as being blue compact dwarfs).",
"Results for these galaxies were combined with photometry for others published previously to examine how kinematics, as conveyed by HI line widths, are connected to global properties.",
"Most dwarfs in the sample displayed HI line profiles close to Gaussian in shape.",
"Also, in optimizing relationships among global properties and kinematics, no strong tie between apparent line widths and isophotal axis ratios was evident.",
"Thus, in the majority of sample galaxies, most of the kinetic energy of the gas appeared to be disordered, and internal motions appeared to be close to isotropic.",
"Consequently, it proved to be possible to establish relationships without correcting line widths for tilt.",
"It was confirmed that much of the scatter in the Tully-Fisher relation for dwarfs is correlated with surface brightness.",
"The “fundamental plane” defined by the correlation of absolute magnitude with line width and surface brightness still displayed a dispersion which exceeded what was expected from observational errors alone.",
"Conjecturing that some of the scatter might be a consequence of not accommodating the highly variable gaseous component, the baryonic Tully-Fisher relation was constructed.",
"It proved to be as dispersed as the fundamental plane, although some improvement was possible by adding surface brightness as a second parameter.",
"Motivated by the possibility that the galaxies may be virialized, the correlation between the potential and the line width was examined.",
"The potential defined by the ratio of the baryonic mass to the sech scale length was hypothesized to be proportional to the potential setting internal motions, but the mass-to-light ratio required to compute the stellar component of the mass was left as a free parameter.",
"The derived relationship between the potential and the line width displayed large residuals correlated with surface brightness.",
"When surface brightness variations were accommodated, an extremely tight relationship between the potential, line width, and surface brightness resulted.",
"Remaining residuals were found to correlate with tilt, with the bulk of the trend explainable by variations in surface brightness arising from viewing oblate spheroids at different angles.",
"Once surface brightnesses were corrected for tilt, the remaining dispersion of this more fundamental plane, referred to as the potential plane, could be almost entirely attributable to observational errors.",
"The solution for the mean mass-to-light ratio for the stars (i.e., the sech component of the light distribution) was $0.88 \\pm 0.20$ .",
"BCDs lay precisely in the plane described by dIs.",
"The placement of BCDs, the strength of the correlation, and the direction of the sensitivity to surface brightness all point to gravity, not turbulence, being primarily responsible for determining gas motions in star-forming dwarfs.",
"The potential plane suggests a strong linkage between the mass and distribution of luminous matter and the mass and scale of dark matter haloes.",
"It also offers a new avenue for determining the distances to unresolvable star-forming dwarfs which may be as good as the Tully-Fisher relation for spirals.",
"The potential plane described a potential varying as $W_{20}^{1.13}$ , i.e., less steeply than expected for virialized systems (for which the exponent should be 2).",
"However, the dependence of the potential on surface brightness was such that surface brightness might be considered to be acting as a proxy for variations in the mass-to-light ratio of stars.",
"To explore this possibility, it was hypothesized that star-forming dwarfs are in fact virialized, and the required dependence of mean surface brightness on line width was derived.",
"Deviations in surface brightness from the norm for a particular line width were used to adjust mass-to-light ratios to compensate for differing star formation histories (and observational errors).",
"Residuals in the correlation of the modified potential with line width were only slightly degraded with respect to those for the potential plane.",
"Computed mass-to-light ratios covered a range greater than expected from theoretical expectations, but some of the variation was attributable to the uncertainties in surface brightnesses.",
"Further studies of star formation rates would be productive in evaluating whether the predicted range of mass-to-light ratios is justified.",
"Work is in progress to evaluate implications for chemical evolution.",
"MLM is grateful to M. De Robertis for advice on testing the robustness of the fit to the potential plane, and to the Natural Sciences and Engineering Council of Canada for its continuing support.",
"OV, EUS, MA, FPN and ABD acknowledge the Chilean TACs for the time allocation at ESO (Run ID: 081.B-0386(A)) and CTIO (2008A-0913 and 2008B-0909).",
"This research has made use of the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration.",
"We acknowledge the usage of the HyperLeda database (http://leda.univ-lyon1.fr - Paturel et al.",
"2003).",
"Our work used IRAF, a software package distributed by the National Optical Astronomy Observatory, which is operated by the Association of Universities for Research in Astronomy (AURA) under cooperative agreement with the National Science Foundation.",
"This research has made use of SAOImage DS9, developed by the Smithsonian Astrophysical Observatory.",
"3 lccccccccc Sample for analysis: Extinctions and Distances Galaxy $E(B-V)$ $A_V^{gal}$ $A_I^{gal}$ $A_{Ks}^{gal}$ $I_{TRGB}$ $(V - I)_{TRGB}$ $M_{I,TRGB}$ $D_{mod}$ Source (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) continued.",
"Galaxy $E(B-V)$ $A_V^{gal}$ $A_I^{gal}$ $A_{Ks}^{gal}$ $I_{TRGB}$ $(V - I)_{TRGB}$ $M_{I,TRGB}$ $D_{mod}$ Source (mag) (mag) (mag) (mag) (mag) (mag) (mag) (mag) (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) Cam B 0.219 0.695 0.394 0.082 23.60 1.65 $-$ 3.92 27.52 1 CGCG 087-33 0.032 0.102 0.058 0.012 25.47 1.15 $-$ 4.03 29.50 2 DDO 006 0.017 0.054 0.031 0.006 23.57 1.48 $-$ 3.96 27.53 3 DDO 047 0.033 0.105 0.059 0.012 25.48 1.13 $-$ 4.04 29.52 2 DDO 099 0.026 0.083 0.047 0.010 23.11 1.29 $-$ 4.00 27.11 4 DDO 167 0.010 0.032 0.018 0.004 24.06 1.49 $-$ 3.96 28.02 5 DDO 168 0.015 0.048 0.027 0.006 24.13 1.40 $-$ 3.98 28.11 5 DDO 181 0.006 0.019 0.011 0.002 23.46 1.35 $-$ 3.99 27.45 4 DDO 187 0.023 0.073 0.041 0.009 22.69 1.34 $-$ 3.99 26.68 4 DDO 190 0.012 0.039 0.022 0.005 23.19 1.38 $-$ 3.98 27.17 4 DDO 226 0.015 0.049 0.028 0.006 24.41 1.47 $-$ 3.96 28.37 3 ESO 059-01 0.147 0.467 0.265 0.055 24.26 1.54 $-$ 3.95 28.21 6 ESO 121-20 0.042 0.132 0.075 0.016 24.86 1.33 $-$ 3.99 28.86 6 ESO 137-18 0.245 0.777 0.441 0.092 25.01 1.55 $-$ 3.95 28.95 7 ESO 215-09 0.221 0.700 0.397 0.083 24.58 1.43 $-$ 3.97 28.55 7 ESO 223-09 0.260 0.824 0.467 0.097 25.04 1.59 $-$ 3.94 28.98 7 ESO 269-58 0.109 0.345 0.196 0.041 23.86 1.61 $-$ 3.93 27.80 8 ESO 320-14 0.143 0.454 0.257 0.053 24.89 1.45 $-$ 3.97 28.86 7 ESO 321-14 0.094 0.299 0.170 0.035 23.51 1.34 $-$ 3.99 27.50 4 ESO 324-24 0.113 0.358 0.203 0.042 23.81 1.43 $-$ 3.97 27.79 9 ESO 325-11 0.088 0.279 0.158 0.033 23.62 1.36 $-$ 3.99 27.61 9 ESO 349-31 0.012 0.038 0.022 0.004 23.48 1.43 $-$ 3.97 27.45 6 ESO 379-07 0.074 0.236 0.134 0.028 24.54 1.59 $-$ 3.94 28.48 9 ESO 381-18 0.063 0.199 0.113 0.023 24.58 1.46 $-$ 3.97 28.55 7 ESO 381-20 0.066 0.208 0.118 0.024 24.64 1.42 $-$ 3.98 28.61 7 ESO 384-16 0.074 0.235 0.133 0.028 24.23 1.55 $-$ 3.95 28.18 7 ESO 444-78 0.053 0.167 0.095 0.020 24.55 1.42 $-$ 3.97 28.53 7 ESO 444-84 0.069 0.218 0.123 0.026 24.27 1.29 $-$ 4.00 28.28 9 ESO 461-36 0.303 0.962 0.546 0.114 25.45 1.32 $-$ 4.00 29.45 6 GR 8 0.026 0.082 0.046 0.010 22.63 2.21 $-$ 3.80 26.43 2 Ho II 0.032 0.101 0.057 0.012 23.61 1.41 $-$ 3.98 27.59 4 IC 3104 0.410 1.301 0.739 0.154 22.75 1.29 $-$ 4.00 26.75 10 IC 4247 0.065 0.205 0.116 0.024 24.43 1.39 $-$ 3.98 28.41 7 IC 4316 0.055 0.173 0.098 0.020 24.10 1.72 $-$ 3.91 28.01 9 IC 4662 0.070 0.222 0.126 0.026 22.89 1.64 $-$ 3.93 26.82 6 IC 5152 0.025 0.079 0.045 0.009 22.42 1.58 $-$ 3.94 26.36 2 KK98 17 0.055 0.173 0.098 0.020 24.41 1.07 $-$ 4.05 28.46 2 KK98 182 0.102 0.325 0.184 0.038 24.77 1.29 $-$ 4.00 28.77 7 KK98 200 0.069 0.219 0.124 0.026 24.20 1.60 $-$ 3.93 28.14 9 KK98 230 0.014 0.045 0.025 0.005 22.47 1.31 $-$ 4.00 26.47 4 KKH 086 0.027 0.085 0.048 0.010 23.09 1.32 $-$ 4.00 27.08 4 KKH 098 0.123 0.389 0.221 0.046 22.92 1.49 $-$ 3.96 26.88 10 Mrk 178 0.019 0.060 0.034 0.007 23.90 1.49 $-$ 3.96 27.86 5 NGC 1311 0.022 0.068 0.039 0.008 24.63 1.15 $-$ 4.03 28.66 2 NGC 1569 0.694 2.206 1.254 0.262 23.12 1.36 $-$ 3.99 27.10 11 NGC 2915 0.275 0.872 0.495 0.103 23.87 1.76 $-$ 3.90 27.77 12 NGC 3077 0.067 0.212 0.120 0.025 23.93 1.77 $-$ 3.90 27.82 4 NGC 3738 0.010 0.033 0.019 0.004 24.40 1.75 $-$ 3.90 28.30 5 NGC 4163 0.020 0.063 0.036 0.007 23.27 1.44 $-$ 3.97 27.24 4 NGC 4214 0.022 0.069 0.039 0.008 23.43 1.47 $-$ 3.96 27.39 4 NGC 5408 0.068 0.216 0.123 0.025 24.36 0.90 $-$ 4.09 28.45 9 NGC 6822 0.231 0.732 0.415 0.086 19.39 1.58 $-$ 3.94 23.32 13 Peg DIG 0.068 0.216 0.122 0.025 20.75 1.52 $-$ 3.95 24.70 14 Sex A 0.045 0.141 0.080 0.017 21.76 1.23 $-$ 4.02 25.77 4 Sex B 0.031 0.099 0.056 0.012 21.77 1.35 $-$ 3.99 25.76 4 UGC 0685 0.057 0.182 0.103 0.021 24.32 1.12 $-$ 4.04 28.36 2 UGC 3755 0.088 0.280 0.159 0.033 25.31 1.08 $-$ 4.05 29.36 2 UGC 4115 0.028 0.090 0.051 0.011 25.39 1.04 $-$ 4.06 29.44 2 UGC 4483 0.034 0.108 0.061 0.013 23.72 1.23 $-$ 4.02 27.74 4 UGC 6456 0.037 0.119 0.067 0.014 24.20 1.38 $-$ 3.98 28.19 15 UGC 7605 0.015 0.046 0.026 0.005 24.18 1.34 $-$ 3.99 28.17 5 UGC 8508 0.015 0.048 0.027 0.006 22.99 1.38 $-$ 3.98 26.98 10 UGC 8833 0.012 0.037 0.021 0.004 23.41 1.36 $-$ 3.99 27.40 4 UGCA 092 0.785 2.496 1.420 0.296 23.46 1.18 $-$ 4.03 27.49 6 UGCA 438 0.015 0.046 0.026 0.005 22.69 1.37 $-$ 3.98 26.67 4 WLM 0.038 0.120 0.068 0.014 20.85 1.47 $-$ 3.96 24.82 16 (1) Name of galaxy; (2) Galactic colour excess from [65]; (3) Galactic extinction of M0 III star in V; (4) Galactic extinction of M0 III star in I; (5) Galactic extinction of dwarf irregular galaxy in $K_s$ ; (6) $I$ magnitude of stars at tip of red giant branch, corrected for extinction and redshift; (7) $(V-I)$ colour of stars at tip of red giant branch, corrected for extinction and redshift; (8) Absolute magnitude of TRGB stars in $I$ ; (9) Distance modulus; (10) Source of colour-magnitude diagram.",
"(1) [39]; (2) [72]; (3) [38]; (4) [9]; (5) [36]; (6) [40]; (7) [41]; (8) [10]; (9) [34]; (10) [35]; (11) [26]; (12) [37]; (13) [21]; (14) [49]; (15) [54]; (16) [64].",
"4 lccccccccccc Sample for Analysis: Radio and Optical Parameters Galaxy $V_\\odot $ $F_{HI}$ $W_{20}^{app}$ $R$ $\\log W_{20}$ HI $\\mu _0$ $r_{0,pc}$ $q$ $r_{lim} / r_0$ Photometry ($\\rm km \\, s^{-1}$ ) ($\\rm Jy \\, km \\, s^{-1}$ ) ($\\rm km \\, s^{-1}$ ) ($\\rm km \\, s^{-1}$ ) ($\\rm km \\, s^{-1}$ ) Source ($\\rm mag \\, arcsec^{-2}$ ) (pc) Source (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) continued.",
"Galaxy $V_\\odot $ $F_{HI}$ $W_{20}^{app}$ $R$ $\\log W_{20}$ HI $\\mu _0$ $r_{0,pc}$ $q$ $r_{lim} / r_0$ Photometry ($\\rm km \\, s^{-1}$ ) ($\\rm Jy \\, km \\, s^{-1}$ ) ($\\rm km \\, s^{-1}$ ) ($\\rm km \\, s^{-1}$ ) ($\\rm km \\, s^{-1}$ ) Source ($\\rm mag \\, arcsec^{-2}$ ) (pc) Source (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) Cam B 78 5.0 32.6 1.65 1.512 H1, H2, H3 21.87 407 0.44 2.40 P1 CGCG 087-33 279 2.6 55.8 1.4 1.746 H1 19.74 532 0.41 4.13 P2 DDO 006 295 3.4 32.9 1.4 1.516 H1 22.04 570 0.29 2.53 P3 DDO 047 272 61.4 111.0 2.06 2.045 H4 21.78 555 1.00 4.20 P4 DDO 099 243 46.0 91.0 2.06 1.958 H4 21.30 393 0.71 2.67 P1 DDO 167 163 4.6 40.2 1.4 1.604 H1 21.65 337 0.60 3.47 P4 DDO 168 191 74.4 88.0 8.24 1.940 H5, H6 20.36 733 0.38 3.96 P1 DDO 181 202 11.4 56.8 1.4 1.754 H1 20.77 356 0.70 2.87 P1 DDO 187 153 12.0 51.4 1.4 1.710 H1 21.71 287 0.60 3.68 P4 DDO 190 151 27.1 64.0 8.24 1.797 H5, H6 19.62 257 0.89 4.21 P1 DDO 226 359 6.1 56.4 1.65 1.750 H7, H3 20.58 982 0.15 3.05 P3 ESO 059-01 530 17.7 104.0 18 2.001 H8 19.84 389 0.80 4.92 P5 ESO 121-20 577 14.1 96.0 18 1.963 H8 20.47 384 0.70 4.48 P5 ESO 137-18 605 37.4 155.0 18 2.183 H8 18.22 727 0.50 6.17 P5 ESO 215-09 597 122.0 93.0 4 1.967 H9 20.90 430 0.83 3.49 P6 ESO 223-09 593 96.2 103.0 8.2 2.009 H10 19.21 1044 0.70 4.35 P7 ESO 269-58 400 7.2 84.0 18 1.899 H11 19.10 636 0.63 4.72 P3 ESO 320-14 654 2.5 61.3 18 1.738 H7 20.26 304 0.70 4.25 P7 ESO 321-14 609 6.4 29.0 1.65 1.459 H8, H3 21.18 522 0.27 2.59 P6 ESO 324-24 526 52.1 113.0 8.2 2.050 H10 20.43 630 0.93 3.13 P6 ESO 325-11 550 25.4 77.0 8.2 1.880 H10 20.93 775 0.35 3.13 P3 ESO 349-31 229 2.7 31.0 8.2 1.453 H10 21.51 323 0.58 3.21 P3 ESO 379-07 644 5.2 40.0 1.65 1.600 H8, H3 22.10 443 0.85 2.81 P6 ESO 381-18 625 3.3 61.6 18 1.741 H7 20.29 224 0.70 4.77 P6, P5 ESO 381-20 596 31.9 103.0 8.2 2.009 H10 20.99 891 0.32 2.89 P8 ESO 384-16 504 1.5 41.0 1.2 1.612 H12 19.45 214 0.92 3.00 P6 ESO 444-78 573 4.0 52.1 1.4 1.716 H1 20.46 475 0.41 3.00 P1 ESO 444-84 583 21.1 75.0 4 1.873 H8, H13 20.60 318 0.89 2.46 P1 ESO 461-36 427 7.5 84.0 10.2 1.916 H7, H14 20.51 492 0.50 4.58 P7 GR 8 214 7.8 39.2 1.4 1.592 H1 21.36 152 0.80 3.70 P4 Ho II 156 267.0 72.0 5.2 1.854 H15 19.66 1348 1.00 2.25 P9 IC 3104 429 10.3 63.0 18 1.753 H8 18.85 468 0.45 3.42 P3 IC 4247 419 3.4 49.0 18 1.608 H16 18.77 370 0.33 5.30 P2, P5 IC 4316 576 2.1 32.8 1.65 1.513 H16, H3 20.31 520 0.60 4.47 P5 IC 4662 302 130.0 133.0 18 2.114 H8 17.40 242 0.73 5.69 P3 IC 5152 122 97.2 100.0 18 1.983 H8 18.08 345 0.66 5.22 P3 KK98 17 156 1.0 53.0 10.2 1.705 H14 21.67 470 0.31 2.98 P10 KK98 182 613 2.1 24.0 7.9 1.316 H14 21.28 400 0.68 3.07 P3 KK98 200 494 1.7 26.5 1.65 1.421 H1, H3 20.51 179 0.70 5.17 P7 KK98 230 63 2.6 25.9 1.65 1.411 H1, H17 22.57 140 0.95 1.76 P11 KKH 086 287 0.5 20.6 1.4 1.310 H1 21.56 181 0.61 1.98 P3 KKH 098 $-$ 132 4.1 31.5 1.65 1.498 H1, H3 21.39 184 0.59 2.68 P10 Mrk 178 250 3.0 44.8 1.4 1.650 H1 19.15 228 0.50 5.56 P4 NGC 1311 568 14.6 105.0 18 2.005 H8 19.15 586 0.40 5.36 P12 NGC 1569 $-$ 86 84.0 123.8 5.2 2.092 H18 16.83 271 0.55 6.13 P4 NGC 2915 468 145.0 163.0 6.6 2.211 H19 17.49 306 0.50 7.67 P5 NGC 3077 $-$ 20 256.0 157.3 5.2 2.196 H18 17.28 593 0.70 4.50 P9 NGC 3738 225 22.0 122.0 8.24 2.084 H6 18.41 437 0.70 5.08 P4 NGC 4163 164 9.6 38.0 4.1 1.574 H5, H20 19.29 223 0.70 5.18 P4 NGC 4214 293 319.8 89.8 2.6 1.952 H5, H18 17.63 491 0.50 4.46 P9 NGC 5408 506 65.5 123.0 8.2 2.087 H10 18.88 456 0.50 7.03 P5 NGC 6822 $-$ 55 2266.0 115.0 1.9 2.061 H21, H22 19.55 239 0.80 1.87 P9 Peg DIG $-$ 183 28.1 38.6 5.3 1.577 H23 20.93 333 0.55 2.08 P10 Sex A 324 168.0 64.0 1.12 1.806 H24 21.09 362 0.95 3.03 P3 Sex B 301 72.9 56.0 1.4 1.747 H1 20.57 255 0.87 3.53 P6 UGC 0685 156 13.4 83.0 1.65 1.919 H25, H3 19.72 336 0.70 5.07 P12 UGC 3755 315 6.8 50.6 1.4 1.703 H1 19.81 808 0.50 3.72 P2 UGC 4115 343 21.0 106.0 1.65 2.025 H5, H3 20.24 803 0.40 3.74 P4 UGC 4483 156 13.6 50.6 1.4 1.703 H1 20.70 306 0.55 2.81 P1 UGC 6456 $-$ 94 10.1 52.0 1.65 1.716 H3 20.16 248 0.70 5.08 P4 UGC 7605 310 5.7 43.7 1.4 1.640 H1 20.75 402 0.67 3.91 P11 UGC 8508 56 18.3 65.0 1.65 1.813 H3 19.95 234 0.55 4.64 P13 UGC 8833 227 6.0 42.8 1.4 1.631 H1 20.94 274 0.77 4.05 P2 UGCA 092 $-$ 95 104.7 73.0 1.65 1.863 H5, H3 20.53 635 0.50 2.40 P4 UGCA 438 62 15.0 35.0 8.2 1.514 H26 20.48 283 0.90 3.97 P3 WLM $-$ 122 292.0 81.0 1.12 1.909 H24 21.28 437 0.40 3.06 P7 (1) Name of galaxy; (2) Heliocentric radial velocity defined by HI; (3) Integrated HI flux; (4) Apparent full width of 21 cm line at 20% of peak; (5) FWHM of instrumental profile; (6) Logarithm of the 21 cm line width at 20% of peak, corrected for instrumental broadening and redshift; (7) Source of HI data; (8) Central surface brightness of sech model, corrected for extinction and redshift; (9) Scale length of sech model in parsecs; (10) Adopted ratio of minor to major axes of isophotes; (11) Ratio of radius of limiting isophote to scale length of sech; (12) Source of surface photometry.",
"(H1) [30]; (H2) [4]; (H3) [6]; (H4) [68]; (H5) [31]; (H6) [69]; (H7) [56]; (H8) [44]; (H9) [79]; (H10) [8]; (H11) [1]; (H12) [3]; (H13) [13]; (H14) [29]; (H15) [7]; (H16) [57]; (H17) [5]; (H18) [78]; (H19) [55]; (H20) [70]; (H21) [11]; (H22) [80]; (H23) [43]; (H24) [2]; (H25) [23] (H26) [46]; (P1) CFHT 2004: [17]; (P2) SPM 2005: [17]; (P3) IRSF 2006: [17]; (P4) CFHT 2002: [73]; (P5) CTIO 2008: this paper; (P6) IRSF 2005: [17]; (P7) CTIO 2007: [76]; (P8) CTIO 2006: [17]; (P9) 2MASS: [76]; (P10) CFHT 2005: [17]; (P11) CFHT 2006: [17]; (P12) ESO 2008: this paper; (P13) SPM 2002: [73].",
"5 lccccccccc Sample for Analysis: Derived Global Properties Galaxy Weight $M_{Ks}$ $L_{burst} / L_{sech}$ $\\log M_{gas}$ $\\log M_{stars}$ $\\log M_{bary}$ Gas Fraction $\\log P$ $M / L \\, \\rm (virial)$ (mag) ($M_\\odot $ ) ($M_\\odot $ ) ($M_\\odot $ ) ($M_\\odot \\, \\rm pc^{-1}$ ) ($M_\\odot / L_\\odot $ ) (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) continued.",
"Galaxy Weight $M_{Ks}$ $L_{burst} / L_{sech}$ $\\log M_{gas}$ $\\log M_{stars}$ $\\log M_{bary}$ Gas Fraction $\\log P$ $M / L \\, \\rm (virial)$ (mag) ($M_\\odot $ ) ($M_\\odot $ ) ($M_\\odot $ ) ($M_\\odot \\, \\rm pc^{-1}$ ) ($M_\\odot / L_\\odot $ ) (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) Cam B 0 $-$ 14.51 0.010 7.213 7.077 7.451 0.578 4.842 2.95 CGCG 087-33 1 $-$ 17.15 0.000 7.722 8.131 8.274 0.281 5.548 0.88 DDO 006 1 $-$ 14.62 0.000 7.049 7.122 7.388 0.458 4.632 5.28 DDO 047 0 $-$ 16.17 0.000 9.100 7.738 9.119 0.958 6.374 5.65 DDO 099 1 $-$ 15.52 0.017 8.012 7.481 8.124 0.772 5.530 3.98 DDO 167 1 $-$ 14.66 0.000 7.377 7.137 7.575 0.635 5.047 2.30 DDO 168 0 $-$ 17.14 0.000 8.622 8.130 8.743 0.756 5.878 2.95 DDO 181 1 $-$ 15.82 0.003 7.544 7.602 7.875 0.467 5.324 1.36 DDO 187 1 $-$ 14.24 0.000 7.259 6.969 7.439 0.661 4.982 3.33 DDO 190 1 $-$ 16.53 0.000 7.807 7.884 8.148 0.456 5.738 0.42 DDO 226 1 $-$ 16.55 0.000 7.641 7.891 8.085 0.360 5.093 5.27 ESO 059-01 1 $-$ 17.09 0.000 8.038 8.109 8.376 0.459 5.786 1.04 ESO 121-20 1 $-$ 16.29 0.000 8.198 7.787 8.340 0.720 5.756 1.90 ESO 137-18 1 $-$ 19.56 0.059 8.660 9.095 9.231 0.269 6.369 0.64 ESO 215-09 0 $-$ 16.29 0.000 9.013 7.788 9.038 0.944 6.404 2.42 ESO 223-09 0 $-$ 19.72 0.000 9.080 9.162 9.424 0.453 6.405 0.68 ESO 269-58 1 $-$ 18.64 0.258 7.482 8.727 8.751 0.054 5.948 0.50 ESO 320-14 1 $-$ 15.99 0.000 7.447 7.667 7.872 0.376 5.389 0.81 ESO 321-14 1 $-$ 15.21 0.012 7.311 7.357 7.636 0.474 4.918 2.18 ESO 324-24 1 $-$ 17.71 0.000 8.337 8.356 8.648 0.489 5.849 1.78 ESO 325-11 1 $-$ 16.60 0.000 7.954 7.912 8.234 0.524 5.345 4.57 ESO 349-31 1 $-$ 14.68 0.000 6.917 7.142 7.345 0.373 4.835 1.35 ESO 379-07 1 $-$ 15.18 0.014 7.614 7.345 7.801 0.650 5.155 2.44 ESO 381-18 1 $-$ 15.29 0.000 7.444 7.390 7.719 0.531 5.369 0.84 ESO 381-20 1 $-$ 16.75 0.000 8.455 7.971 8.579 0.753 5.629 7.68 ESO 384-16 1 $-$ 16.34 0.045 6.962 7.807 7.865 0.125 5.535 0.20 ESO 444-78 1 $-$ 16.18 0.000 7.518 7.745 7.948 0.372 5.271 1.56 ESO 444-84 0 $-$ 16.01 0.000 8.140 7.676 8.269 0.745 5.767 1.30 ESO 461-36 1 $-$ 16.42 0.000 8.160 7.842 8.330 0.675 5.638 2.41 GR 8 1 $-$ 13.53 0.000 6.972 6.684 7.153 0.660 4.970 1.28 Ho II 0 $-$ 20.21 0.000 8.969 9.358 9.507 0.290 6.377 0.46 IC 3104 1 $-$ 17.86 0.000 7.219 8.416 8.442 0.060 5.772 0.36 IC 4247 1 $-$ 17.06 0.000 7.403 8.097 8.177 0.168 5.608 0.30 IC 4316 1 $-$ 16.94 0.109 7.031 8.050 8.089 0.087 5.374 0.51 IC 4662 1 $-$ 18.41 0.000 8.347 8.634 8.815 0.341 6.431 0.17 IC 5152 1 $-$ 18.38 0.077 8.039 8.625 8.725 0.206 6.187 0.24 KK98 17 0 $-$ 14.64 0.005 6.868 7.129 7.319 0.354 4.647 6.11 KK98 182 0 $-$ 15.54 0.000 7.342 7.487 7.722 0.417 5.120 0.62 KK98 200 1 $-$ 14.59 0.000 6.992 7.109 7.356 0.433 5.103 0.41 KK98 230 0 $-$ 12.34 0.015 6.508 6.207 6.684 0.667 4.537 1.93 KKH 086 0 $-$ 13.42 0.000 6.039 6.638 6.736 0.201 4.478 0.89 KKH 098 1 $-$ 13.59 0.000 6.870 6.706 7.097 0.593 4.832 1.36 Mrk 178 1 $-$ 16.12 0.000 7.128 7.720 7.819 0.204 5.461 0.32 NGC 1311 1 $-$ 17.92 0.194 8.135 8.442 8.616 0.330 5.848 1.11 NGC 1569 1 $-$ 18.91 0.348 8.272 8.837 8.941 0.214 6.509 0.12 NGC 2915 0 $-$ 18.41 0.000 8.777 8.636 9.013 0.580 6.527 0.35 NGC 3077 0 $-$ 20.43 0.107 9.044 9.443 9.589 0.285 6.816 0.20 NGC 3738 1 $-$ 18.63 0.204 8.169 8.725 8.831 0.218 6.191 0.40 NGC 4163 1 $-$ 16.28 0.000 7.383 7.785 7.930 0.284 5.582 0.21 NGC 4214 0 $-$ 19.29 0.016 8.969 8.989 9.280 0.488 6.589 0.19 NGC 5408 1 $-$ 17.89 0.000 8.703 8.428 8.888 0.653 6.229 0.88 NGC 6822 0 $-$ 16.32 0.000 8.191 7.802 8.339 0.710 5.960 0.95 Peg DIG 0 $-$ 15.25 0.000 6.835 7.374 7.484 0.224 4.962 1.20 Sex A 1 $-$ 15.87 0.000 8.040 7.622 8.180 0.723 5.622 1.56 Sex B 1 $-$ 15.53 0.000 7.673 7.485 7.890 0.606 5.484 0.90 UGC 0685 1 $-$ 16.75 0.000 7.975 7.973 8.275 0.501 5.748 0.84 UGC 3755 0 $-$ 18.20 0.000 8.082 8.552 8.679 0.253 5.771 0.68 UGC 4115 1 $-$ 17.51 0.000 8.606 8.277 8.773 0.681 5.868 3.23 UGC 4483 1 $-$ 15.31 0.000 7.734 7.395 7.898 0.686 5.412 1.40 UGC 6456 1 $-$ 15.66 0.000 7.785 7.535 7.979 0.640 5.584 0.69 UGC 7605 1 $-$ 16.06 0.000 7.532 7.697 7.923 0.406 5.319 1.00 UGC 8508 1 $-$ 15.46 0.000 7.559 7.457 7.812 0.558 5.443 0.97 UGC 8833 1 $-$ 15.19 0.000 7.244 7.349 7.601 0.440 5.163 1.01 UGCA 092 0 $-$ 16.95 0.000 8.520 8.054 8.648 0.745 5.845 2.11 UGCA 438 1 $-$ 15.89 0.000 7.350 7.629 7.812 0.345 5.361 0.40 WLM 1 $-$ 15.15 0.000 7.898 7.331 8.002 0.787 5.361 6.02 (1) Name of galaxy; (2) Weight during fitting; (3) Absolute magnitude of sech model in $K_s$ ; (4) Luminosity of burst relative to luminosity of sech model; (5) Logarithm of the gas mass; (6) Logarithm of the mass of stars in the sech model, computed assuming a fixed mass-to-light ratio of 0.883 in $K_s$ ; (7) Logarithm of the baryonic mass (sum of gaseous and stellar masses); (8) Gas fraction (for a fixed mass-to-light ratio of 0.883 in $K_s$ ); (9) Logarithm of the potential defined by the ratio of the baryonic mass to the scale length of the sech model; (10) Mass-to-light ratio of the stars in $K_s$ as indicated by the deviation of the surface brightness from the norm for a virialized system."
]
] | 1204.1074 |
[
[
"How to Classify Reflexive Gorenstein Cones"
],
[
"Abstract Two of my collaborations with Max Kreuzer involved classification problems related to string vacua.",
"In 1992 we found all 10,839 classes of polynomials that lead to Landau-Ginzburg models with c=9 (Klemm and Schimmrigk also did this); 7,555 of them are related to Calabi-Yau hypersurfaces.",
"Later we found all 473,800,776 reflexive polytopes in four dimensions; these give rise to Calabi-Yau hypersurfaces in toric varieties.",
"The missing piece - toric constructions that need not be hypersurfaces - are the reflexive Gorenstein cones introduced by Batyrev and Borisov.",
"I explain what they are, how they define the data for Witten's gauged linear sigma model, and how one can modify our classification ideas to apply to them.",
"I also present results on the first and possibly most interesting step, the classification of certain basic weights systems, and discuss limitations to a complete classification."
],
[
"Landau–Ginzburg models",
"I first met Max in 1987, when he was in the last stages of his doctorate studies at the Institute for Theoretical Physics of the TU Vienna, and I was just starting mine.",
"We were then both working in quantum field theory, and Max already displayed his well known capacity for compressing essential information; in particular, he had one piece of paper containing all the formulas one would ever need for computing certain Feynman diagrams.",
"Max then went on to postdoctoral positions at Hanover and Santa Barbara.",
"When he came back to Vienna, I had finished my thesis and was looking for something new to work on.",
"Max, who had started to collaborate with Rolf Schimmrigk in Santa Barbara, invited me to join a project related to string compactifications.",
"Since neither of us knew much string theory at the time (but both of us had math diplomas), our strategy was to isolate a mathematical nucleus from an important topic, namely orbifolds of N=2 superconformal field theories (SCFTs) of Landau–Ginzburg type that can be used for string compactifications.",
"Such a model requires a potential that is a quasihomogeneous function $f({\\phi _i})$ of the fields with an isolated singularity at the origin: $f({\\lambda ^{q_i}\\phi _i})= \\lambda f({\\phi _i}),~~~~~~\\frac{\\partial f}{\\partial \\phi _i}=0~\\forall i~~~\\Rightarrow ~~~\\phi _i=0~\\forall i.$ This gives rise to a superconformal field theory whose anomaly $c$ is determined by the positive rational numbers $q_i$ (the `weights') via $c/3 = \\sum _{i=1}^N(1-2q_i)$ .",
"In a first paper with Rolf [1] we considered symmetries of some known models of that type, but then we turned to the classification problem.",
"This had been treated for very simple cases in a book by Arnold et al.",
"[2].",
"For realistic string compactifications one requires $c=9$ , i.e.",
"$\\sum _{i=1}^N(1-2q_i)=3$ .",
"Well known examples include $f=\\phi _1^5+\\ldots +\\phi _5^5 ~~~\\hbox{ and }~~~ f=\\phi _1^3+\\ldots +\\phi _9^3.$ At that time it was known that the cases $N=4,5$ were related to Calabi–Yau manifolds — the functions could be used to define hypersurfaces in weighted projective spaces — whereas the cases $N=6,7,8,9$ were not (at least, not directly); however, the precise form of the relationship between Landau–Ginzburg models and Calabi–Yaus was unclear.",
"Max and I extended the approach of [2] and used the resulting algorithm [3] to find 10,838 sets of $q_i$ that fit the criteria.",
"In the meantime Klemm and Schimmrigk had done a more thorough search of the literature which allowed them to attack the problem directly, and while we were in the last stages of writing up our results they published theirs [4] which contained precisely one model more.",
"By taking another look at some candidates that we had rejected we managed to find that model and arrived at the same set of 10,839 [5].",
"One thing that was noticeable from these results was the fact that mirror symmetry was not complete within this class of models."
],
[
"The gauged linear sigma model",
"The question about the precise relationship between Landau–Ginzburg models and Calabi–Yaus was settled quite beautifully by Witten [6].",
"He introduced a (2,2)–supersymmetric gauged linear sigma model in two dimensions which contained chiral superfields $\\Phi _i$ (with component fields $\\phi _i$ , $F_i$ ) and vector superfields $V_a$ (with components $\\sigma _a$ , $D_a$ ).",
"The theory's superpotential $W(\\Phi )$ is invariant under $\\Phi _i\\rightarrow e^{-iQ_{i,a}\\lambda _a}\\Phi _i$ , and there are real parameters $r_a$ coming from the Fayet-Iliopoulos terms.",
"It turns out that the model determines the $D$ - and $F$ -terms as $D_a = -e_a^2(\\sum _iQ_{i,a}|\\phi _i|^2-r_a), ~~~F_i =\\frac{\\partial W}{\\partial \\phi _i},$ and requires minimization of the potential $U = \\sum _a\\frac{1}{ 2e_a^2}D_a^2 +\\sum _i|F_i|^2+2\\sum _{i,a}|\\sigma _a|^2|\\phi _i|^2Q_{i,a}^2.$ The behaviour of the theory depends crucially on the values of the $r_a$ as the following classic example demonstrates.",
"Example 1.",
"Consider the case of just one vector field and six chiral superfields with charges $Q_0=-5$ , $Q_1=\\ldots =Q_5=1$ , and a superpotential $W=\\Phi _0P^5(\\Phi _1,\\ldots ,\\phi _5)$ , where $P^5$ is a polynomial of degree 5 that is non-degenerate, i.e.",
"obeys eq.",
"(REF ) .",
"Then $ D \\sim r+5|\\phi _0|^2-\\sum _{i=1}^5|\\phi _i|^2, ~~~|F_0|^2=|P^5|^2,~~~\\sum _{i=1}^5|F_i|^2 = |\\phi _0|^2\\sum _{i=1}^5|\\frac{\\partial P^5}{\\partial \\phi _i}|^2,$ and we can distinguish the following cases.",
"$r >> 0$ : Then $D^2\\rightarrow $ min implies $(\\phi _1,\\ldots ,\\phi _5)\\ne (0,\\ldots ,0)$ , so by non-degeneracy of $P^5$ we need $\\phi _0=0$ to minimize $\\sum _{i=1}^5|F_i|^2$ .",
"The ground state is located at $ \\lbrace (\\phi _1,\\ldots ,\\phi _5): \\sum _{i=1}^5|\\phi _i|^2 = \\sqrt{r},P^5(\\phi _i)=0\\rbrace /U(1),$ which is just the symplectic quotient description of a quintic hypersurface in ${\\mathbb {P}}^4$ , i.e.",
"the standard Calabi–Yau example.",
"$r << 0$ : Then $D^2\\rightarrow $ min requires $\\phi _0\\ne 0$ , implying $\\frac{\\partial P^5}{\\partial \\phi _i}=0$ and therefore $\\phi _1=\\ldots =\\phi _5=0$ .",
"The $U(1)$ gauge symmetry may be used to fix $\\phi _0=\\sqrt{-r/5}$ , leaving a residual ${\\mathbb {Z}}_5$ symmetry.",
"The resulting model is just a ${\\mathbb {Z}}_5$ orbifold of a Landau–Ginzburg model with potential $P^5$ , i.e.",
"one of the 10,839 models we had classified.",
"By introducing more than one gauge field and more than one analogue of $\\phi _0$ one can easily build models that correspond to complete intersection Calabi–Yaus."
],
[
"Toric constructions",
"Around that time Victor Batyrev [7] introduced a construction that was manifestly mirror symmetric in the following sense.",
"Given a dual pair of lattices $M\\simeq {\\mathbb {Z}}^d$ , $N=\\mathrm {Hom}(M,{\\mathbb {Z}})$ and their real extensions $M_{\\mathbb {R}}\\simeq {\\mathbb {R}}^d$ , $N_{\\mathbb {R}}\\simeq {\\mathbb {R}}^d$ , one defines a lattice polytope $\\Delta \\subset M_{\\mathbb {R}}$ as a polytope with vertices in $M$ , and a reflexive polytope as a lattice polytope $\\Delta \\ni 0_M$ whose dual $\\Delta ^* = \\lbrace y\\in N_{\\mathbb {R}}: \\langle y,x \\rangle +1\\ge 0 ~~\\forall x\\in \\Delta \\rbrace $ is also a lattice polytope.",
"To any triangulation of the surface of $\\Delta ^*$ one can assign the toric variety $\\cal V$ that is determined by the corresponding fan, with homogeneous coordinates $z_i$ that are in one to one correspondence with the nonzero lattice points $y_i$ of $\\Delta ^*$ .",
"Then every lattice point $x_j$ of $\\Delta $ determines a monomial $M_j=\\prod _i z_i^{\\langle y_i,x_j \\rangle +1}$ , and the generic polynomial consisting of these monomials defines a Calabi–Yau hypersurface $\\cal C\\subset \\cal V$ .",
"The Hodge numbers of $\\cal C$ can be computed directly from the structure of $\\Delta $ and it turns out that the exchange $(M, \\Delta )\\leftrightarrow (N, \\Delta ^*)$ effects precisely the flip of the Hodge diamond that is associated with mirror symmetry.",
"Borisov [8] generalized this construction to complete intersections $\\cal C$ in toric varieties.",
"The main idea is to generalize the duality of eq.",
"(REF ) to sets of polytopes $\\nabla _i\\subset N_{\\mathbb {R}}$ , $\\Delta _j\\subset M_{\\mathbb {R}}$ for $i, j\\in \\lbrace 1,\\ldots , {\\rm codim }~ \\cal C\\rbrace $ via $\\langle y,x \\rangle +\\delta _{ij}\\ge 0 ~~\\forall ~y\\in \\nabla _i, ~x\\in \\Delta _j; $ the fan for $\\cal V$ is given by a triangulation of Conv($\\lbrace \\nabla _i\\rbrace $ ) which turns out to be reflexive.",
"At this point it is not clear how this is related to the gauged linear sigma model; in particular, fields like $\\phi _0$ have no analogues in the toric coordinates, and Landau–Ginzburg models without Calabi–Yau interpretation are missing.",
"This situation changed with two papers by Batyrev and Borisov who introduced reflexive Gorenstein cones [9] and a formula for the corresponding `stringy Hodge numbers' [10] that displays exactly the type of combinatorial duality required by mirror symmetry.",
"We postpone precise definitions to the next section and just mention here that the data of these cones can be used to define gauged linear sigma models."
],
[
"The classification of reflexive polyhedra",
"Given all these developments it was clear that the answer to the `missing mirror problem' lay in the realm of gauged linear sigma models and toric geometry.",
"At that time reflexive polytopes were classified only in dimensions up to two (there are 16 reflexive polygons), and no algorithm for a classification in higher dimensions was known.",
"In the autumn of 1995 Max in I were both in Vienna again and started to work on a general algorithm.",
"We realized that the inversion of inclusion relations by duality, $\\Delta \\subseteq \\tilde{\\Delta }\\Leftrightarrow \\Delta ^*\\supseteq \\tilde{\\Delta }^*$ , has the following implication.",
"If we find a set $S$ of polytopes such that every reflexive polytope contains at least one member of $S$ , then every reflexive polytope must be contained in one of the duals of the members of $S$ .",
"One can choose $S$ as a set of polytopes $\\nabla _{\\rm min}$ that are minimal in the sense that 0 is in the interior of $\\nabla _{\\rm min}$ but not in the interior of the convex hull of any subset of the set of vertices of $\\nabla _{\\rm min}$ .",
"We proved that every such $\\nabla _{\\rm min}$ is either a simplex or the convex hull of lower dimensional simplices in a specific way [11].",
"In two dimensions the only possibilities are triangles and parallelograms; in 3d examples would include tetrahedra, octahedra, egyptian pyramids, etc.",
"Any simplex involved in such a construction determines a weight system $\\lbrace q_i\\rbrace $ via $0=\\sum _iq_iV_i$ where the $V_i$ are the vertices of the simplex.",
"In order to play a role for the classification of reflexive polytopes a minimal polytope must satisfy $0 \\in {\\rm int}({\\rm conv}(\\nabla _{\\rm min}^*\\cap M))$ .",
"This condition restricts the admissible weight systems to a finite set; a procedure for obtaining them in arbitrary dimensions and the results in up to four dimensions were presented in [12].",
"Combining these with all possible combinatorial structures of minimal polytopes led to a complete list of minimal polytopes.",
"By considering all subpolytopes of polytopes in the dual list of maximal objects we could find all 4,319 reflexive polytopes in three dimensions [13] and all 473,800,776 in four dimensions [14].",
"A thorough description of the complete algorithm in its final form can be found in [15].",
"This project was at the limit of what could be achieved with the computers that were available to us, so we required extremely efficient routines for handling lattice polytopes.",
"After some polishing these routines were published as the package PALP [16] which is still being updated every now and then.",
"An up-to-date manual of the current version can be found in this volume [17]."
],
[
"Structure of the paper",
"From everything discussed so far it is clear that the piece that is missing from our classification results is the case of toric constructions that need not correspond to Calabi–Yau hypersurfaces; in other words it is the Batyrev/Borisov construction of reflexive Gorenstein cones.",
"This is what the rest of this paper will be about.",
"In the following section some of the essential definitions are given.",
"In section 3 the classification problem is analysed in the spirit of [11], [12], [15].",
"Section 4 describes the classification of the relevant new weight systems, and section 5 discusses the further steps that could be taken."
],
[
"Some definitions",
"A Gorenstein cone $\\sigma $ is a cone in $M_{\\mathbb {R}}$ with generators $V_1,\\ldots ,V_k\\in M$ satisfying $\\langle V_i,n_\\sigma \\rangle =1$ for some element $n_\\sigma \\in N$ .",
"The support $\\Delta _\\sigma $ of $\\sigma $ is the polytope Conv($\\lbrace V_1,\\ldots ,V_k\\rbrace $ ) in the hyperplane $\\langle x,n_\\sigma \\rangle =1$ in $M_{\\mathbb {R}}$ .",
"A reflexive Gorenstein cone $\\sigma $ is a Gorenstein cone whose dual $\\sigma ^{\\vee }=\\lbrace y\\in N_{\\mathbb {R}}:\\langle y,x \\rangle \\ge 0 ~~\\forall x\\in \\sigma \\rbrace $ is also Gorenstein, i.e.",
"there exists an $m_\\sigma \\in M$ such that $\\langle m_\\sigma ,W_i \\rangle =1$ for all generators $W_i$ of $\\sigma ^{\\vee }$ ; the integer $r=\\langle m_\\sigma ,n_\\sigma \\rangle $ is called the index of $\\sigma $ .",
"If $\\sigma $ is reflexive with index $r$ then $r\\Delta _\\sigma $ is a reflexive polytope [9].",
"A reflexive Gorenstein cone $\\sigma $ of index $r$ is called split if $M\\simeq {\\mathbb {Z}}^k\\oplus \\tilde{M}$ and $\\sigma $ is generated by $(e_1,\\Delta _1),\\ldots ,(e_k,\\Delta _k)$ where the $e_i$ form a basis of ${\\mathbb {Z}}^k$ and the $\\Delta _i$ are lattice polytopes in $\\tilde{M}_{\\mathbb {R}}$ .",
"This implies $k\\le r$ ; $\\sigma $ is called completely split if $k=r$ .",
"If both $\\sigma $ and $\\sigma ^\\vee $ are completely split (the latter with a basis $ \\lbrace f_i\\rbrace $ for ${\\mathbb {Z}}^r$ and polytopes $\\nabla _i\\subset \\tilde{N}_{\\mathbb {R}}$ ) it can be shown [18] that one can choose the bases $\\lbrace e_i\\rbrace $ and $\\lbrace f_i\\rbrace $ dual to each other.",
"Then the duality of the cones is equivalent to eq.",
"(REF ) which is the defining property of a nef-partition [8].",
"The cartesian product $\\sigma _1\\times \\sigma _2\\subset M_{1,{\\mathbb {R}}}\\oplus M_{2,{\\mathbb {R}}}$ of two reflexive Gorenstein cones is again a reflexive Gorenstein cone, with dimension $d=d_1+d_2$ , index $r=r_1+r_2$ and dual cone $\\sigma _1^\\vee \\times \\sigma _2^\\vee \\subset N_{1,{\\mathbb {R}}}\\oplus N_{2,{\\mathbb {R}}}$ .",
"Given a reflexive pair $(\\sigma ,\\sigma ^\\vee )$ of Gorenstein cones and denoting by $\\lbrace x_j\\rbrace $ ($\\lbrace y_i\\rbrace $ ) the set of lattice points in the support of $\\sigma $ ($\\sigma ^\\vee $ ) and by $l_a: \\sum _i Q_{i,a}y_i=0$ a basis for the set of linear relations among the $y_i$ , one can define a gauged linear sigma model by introducing a chiral superfield $\\Phi _i$ for every $y_i$ , a gauge field $V_a$ for every $l_a$ , the charges $Q_{i,a}$ as the coefficients of the $l_a$ , a monomial $M_j=\\prod _i\\Phi _i^{\\langle y_i,x_j \\rangle }$ for every $x_j$ ."
],
[
"Analysis of the classification problem",
"Let us fix $n\\in N$ and $m\\in M$ with $\\langle m,n \\rangle =r$ .",
"The main ideas of [11], [12], [15] can be adapted as follows.",
"We say that a Gorenstein cone $\\sigma \\subset M_{\\mathbb {R}}$ with $n_\\sigma =n$ has the IP (for `interior point') property if $m$ is in the interior of $\\sigma $ .",
"This is equivalent to $m/r$ being in the interior of the support $\\Delta _\\sigma $ .",
"With an analogous definition of the IP property for a cone $\\rho \\subset N_{\\mathbb {R}}$ we call $\\rho $ minimal if it has the IP property, but if no cone generated by a proper subset of the set of generators of $\\rho $ has it.",
"The support $\\nabla _\\rho $ of $\\rho $ is a minimal polytope in the sense of [11], characterized by the fact that the set $\\lbrace V_1,\\ldots ,V_{d+k-1}\\rbrace $ of its vertices is the union of $k\\ge 1$ subsets (possibly overlapping) such that each of them determines a simplex (of lower dimension unless $k=1$ ) with the interior point (here, $n/r$ ) in its relative interior.",
"This implies that $n$ lies in the interior of the cone generated by the vertices $V_i$ of such a simplex, so there exists a uniquely defined set of positive rational numbers $q_i$ such that $\\sum q_i V_i = n$ ; acting with $m$ on this equation we see that $\\sum q_i = r$ .",
"We call the $q_i$ the weight system associated with the simplex; if $k>1$ the collection of weight systems is referred to as a combined weight system or CWS.",
"In the case $k=1$ where $\\rho $ itself is simplicial we have an identification of $\\rho \\subset N_{\\mathbb {R}}$ with ${\\mathbb {R}}_{\\ge 0}^d\\subset {\\mathbb {R}}^d$ via $V_i\\leftrightarrow e_i$ .",
"The corresponding identification of dual spaces implies $m \\leftrightarrow (1,\\ldots ,1)$ .",
"Up to now we have not specified the lattice $N$ .",
"Given $n$ and the generators $V_i$ , clearly the coarsest possible lattice $N_\\mathrm {coarsest}$ is the one generated by these vectors, corresponding to the lattice in ${\\mathbb {R}}^d$ generated by $e_1,\\ldots ,e_d$ and ${\\mathbf {q}} = (q_1,\\ldots ,q_d)$ .",
"The lattice $M_\\mathrm {finest}$ dual to $N_\\mathrm {coarsest}$ is then determined by the isomorphism $ M_\\mathrm {finest}\\simeq \\lbrace (x_1,\\ldots ,x_d): x_i\\in {\\mathbb {Z}}, \\sum x_i q_i \\in {\\mathbb {Z}}\\rbrace .",
"$ Let us now define $\\sigma (\\rho )$ as the cone over $\\mathrm {Conv}(\\rho ^\\vee \\cap \\lbrace x\\in M: \\langle x,n \\rangle = 1\\rbrace ),$ and $\\sigma (\\mathbf {q})$ as $\\sigma (\\rho )$ for the case $ M = M_\\mathrm {finest}$ .",
"We say that $\\mathbf {q}$ has the IP property if $\\sigma (\\mathbf {q})$ has it, i.e.",
"if $(1,\\ldots ,1)$ is interior to the cone over $\\lbrace (x_1,\\ldots ,x_d):x_i \\in {\\mathbb {Z}}_{\\ge 0}, \\sum x_iq_i=1\\rbrace $ .",
"This is equivalent to $(1/r,\\ldots ,1/r) \\in \\mathrm {int}(\\Delta _{\\mathbf {q}})$ with $\\Delta _{\\mathbf {q}}=\\mathrm {Conv}(\\lbrace (x_1,\\ldots ,x_d):x_i \\in {\\mathbb {Z}}_{\\ge 0}, \\sum x_iq_i=1\\rbrace ).$ We note that this does not rely on $r$ being integer, allowing us to talk about IPWSs (`IP weight systems') for $(d,r)$ with rational $r$ .",
"The cartesian product of cones has an analogue in the fact that if ${\\mathbf {q}^{(1)}}$ , ${\\mathbf {q}^{(2)}}$ are IPWSs for $(d_1,r_1)$ and $(d_2,r_2)$ , respectively, then $\\mathbf {q} = ({\\mathbf {q}^{(1)}}, {\\mathbf {q}^{(2)}})$ is an IPWS for $(d_1+d_2,r_1+r_2)$ .",
"Note, however, that generically $M_\\mathrm {finest}(\\mathbf {q})$ is finer than $M_\\mathrm {finest}(\\mathbf {q}^{(1)})\\oplus M_\\mathrm {finest}(\\mathbf {q}^{(2)})$ .",
"Lemma 1.",
"Assume that $(q_1,\\ldots ,q_{d})$ form a $(d,r)$ –IPWS.",
"Then a) every $q_i$ obeys $q_i\\le 1$ ; b) if $q_d=1$ then $(q_1,\\ldots ,q_{d-1})$ form a $(d-1,r-1)$ –IPWS; c) if $q_d=1/2$ then $(q_1,\\ldots ,q_{d-1})$ form a $(d-1,r-1/2)$ –IPWS; d) if $q_{d-1}+q_d=1$ then $(q_1,\\ldots ,q_{d-2})$ form a $(d-2,r-1)$ –IPWS, and $q_{d-1}=q_d=1/2$ or $q_{d-1}$ and $q_d$ can be written as nonnegative integer linear combinations of $q_1,\\ldots ,q_{d-2}$ .",
"Proof.",
"a) If $q_i > 1$ then $x_i=0$ in $\\Delta _{\\mathbf {q}}$ , so $(1/r,\\ldots ,1/r)$ is not in the interior.",
"b), c) Here $\\Delta _{\\mathbf {q}}$ is the pyramid over $\\Delta _{(q_1,\\ldots ,q_{d-1})}$ with apex the point $e_d$ or $2e_d$ which has the IP property if and only if $\\Delta _{(q_1,\\ldots ,q_{d-1})}$ has it.",
"d) The case $q_{d-1}=q_d=1/2$ can be reduced to case c), so let us assume $q_{d-1}=1-q_d > 1/2$ , $q_d < 1/2$ .",
"If we denote by $\\lambda $ the largest integer satisfying $\\lambda q_d\\le 1$ , then $\\Delta _{\\mathbf {q}}$ is the convex hull of $\\Delta _1\\cup \\lbrace e_{d-1}+e_d\\rbrace \\cup \\lbrace e_{d-1}+\\Delta _{q_d}\\rbrace \\cup \\bigcup _{\\mu =1}^\\lambda \\lbrace \\mu e_d+\\Delta _{1-\\mu q_d}\\rbrace $ where we have written $\\Delta _y$ for $\\mathrm {Conv}(\\lbrace (x_1,\\ldots ,x_{d-2},0,0):x_i \\in {\\mathbb {Z}}_{\\ge 0},x_1q_1+\\ldots x_{d-2}q_{d-2}=y\\rbrace )$ .",
"If $q_d$ could not be written as a nonnegative integer linear combination of $q_1,\\ldots ,q_{d-2}$ then $\\Delta _{q_d}$ would be empty and every point of $\\Delta _{\\mathbf {q}}$ would satisfy $x_d\\ge x_{d-1}$ .",
"But then $(1/r,\\ldots ,1/r)$ would lie at the boundary of $\\Delta _{\\mathbf {q}}$ , thus violating the IP assumption.",
"Similarly, if all $\\Delta _{1-\\mu q_d}$ were empty we would have the same type of contradiction via $x_{d-1}\\ge x_d$ , so at least one of the $\\Delta _{1-\\mu q_d}$ must be non-empty, but then $\\Delta _{1-q_d}\\supseteq \\Delta _{1-\\mu q_d}+(\\mu -1)\\Delta _{q_d}$ implies that $\\Delta _{1-q_d}$ must also be non-empty, hence $1-q_d=q_{d-1}$ is a nonnegative linear combination of $q_1,\\ldots ,q_{d-2}$ .",
"Finally, let us assume that $(q_1,\\ldots ,q_{d-2})$ does not form a $(d-2,r-1)$ –IPWS.",
"Then there is some hyperplane through 0 and $(1/(r-1),\\ldots ,1/(r-1))$ such that all of $\\Delta _1$ lies on the same side of it: $a_1x_1+\\ldots +a_{d-2}x_{d-2}\\ge 0 $ for all $x\\in \\Delta _1$ , with $a_i$ satisfying $a_1+\\ldots +a_{d-2}=0$ .",
"As the point $x_1=\\ldots =x_d=1/r$ lies in the same hyperplane, the IP property can hold for $\\Delta _{\\mathbf {q}}$ only if there is at least one point with $a_1x_1+\\ldots +a_{d-2}x_{d-2}<0$ .",
"If this point pertains to $\\Delta _{q_d}$ , denote by $c$ the maximal value for which $a_1x_1+\\ldots +a_{d-2}x_{d-2}=-c$ .",
"Then $\\Delta _1\\supseteq \\Delta _{1-\\mu q_d}+\\mu \\Delta _{q_d}$ implies $a_1x_1+\\ldots +a_{d-2}x_{d-2}\\ge \\mu c$ for all $x\\in \\Delta _{1-\\mu q_d}$ and inspection of the components of $\\Delta _{\\mathbf {q}}$ shows that they all obey $a_1x_1+\\ldots +a_{d-2}x_{d-2}+cx_{d-1}-cx_d\\ge 0,$ thereby violating the IP condition for $\\Delta _{\\mathbf {q}}$ .",
"Similarly, if one or more of the $\\Delta _{1-\\mu q_d}$ contain points with $a_1x_1+\\ldots +a_{d-2}x_{d-2}<0$ , we choose $c$ to be the maximal value for which $a_1x_1+\\ldots +a_{d-2}x_{d-2}=-\\mu c$ .",
"Then $\\Delta _1\\supseteq \\Delta _{1-\\mu q_d}+\\mu \\Delta _{q_d}$ implies $a_1x_1+\\ldots +a_{d-2}x_{d-2}\\ge c$ for all $x\\in \\Delta _{q_d}$ and all components of $\\Delta _{\\mathbf {q}}$ obey $a_1x_1+\\ldots +a_{d-2}x_{d-2}-cx_{d-1}+cx_d\\ge 0,$ again violating the IP condition for $\\Delta _{\\mathbf {q}}$ .",
"$\\Box $ Note, however: $q_{d-1}+q_d=1$ does not imply that one of these two repeats one of the other weights as the IPWS $(111126)[8]$ shows (the notation $(n_1\\ldots n_d)[k]$ means $q_i=n_i/k$ ); $q_d>1/2$ does not imply $1-q_d\\in \\lbrace q_1,\\ldots ,q_{d-1}\\rbrace $ as demonstrated by the IPWS $(111114)[6]$ .",
"Motivated by the lemma we shall refer to a weight system as basic if it contains no weights $q_i \\in \\lbrace 1/2,1\\rbrace $ and no $q_i$ , $q_j$ with $q_i+q_j=1$ .",
"For such a weight system, any $\\mathbf {x}$ satisfying $\\sum x_i q_i =1$ must obey $\\sum x_i > 2$ .",
"What happens if $\\rho $ is not simplicial, i.e.",
"$\\nabla _\\rho $ consists of more than one simplex?",
"Then one embeds each of the $k>1$ simplices $S_i$ into ${\\mathbb {R}}^{d_i}\\subset {\\mathbb {R}}^{d+k-1}$ , where ${\\mathbb {R}}^{d_i}$ is the subspace spanned by the $e_j$ corresponding to the $d_i$ vertices of $S_i$ ; the interior points ${\\mathbf {q}}^{(i)}$ of the resulting simplicial cones are identified and on gets $ N\\simeq ({\\mathbb {Z}}^{d+k-1}\\oplus {\\mathbb {Z}}{\\mathbf {q}}^{(1)}\\cdots \\oplus {\\mathbb {Z}}{\\mathbf {q}}^{(k)})/\\lbrace a_{ij}({\\mathbf {q}}^{(i)}-{\\mathbf {q}}^{(j)}):a_{ij}\\in {\\mathbb {Z}}\\rbrace .",
"$ On the $M$ lattice side one now has $k$ equations of the type $\\sum x_iq_i=1$ in ${\\mathbb {Z}}_{\\ge 0}^{d+k-1}$ .",
"In particular, if the simplices all have distinct vertices, one starts with the cartesian product of cones in $N$ and projects along the differences of the $n_i$ , $i\\in \\lbrace 1, \\ldots ,k\\rbrace $ ; in $M$ this results in the support $\\Delta _\\sigma $ being the product of the supports $\\Delta _{\\sigma _i}$ , $i\\in \\lbrace 1, \\ldots ,k\\rbrace $ .",
"Given these preparations the following algorithm for the classification of reflexive Gorenstein cones in dimension $d$ with index $r$ emerges.",
"Find all basic IPWSs for $d^{\\prime }\\in \\lbrace 0,1,\\ldots ,d\\rbrace $ , $r^{\\prime }\\in \\lbrace 0,1/2,1,\\ldots ,r\\rbrace $ with $r-r^{\\prime }\\le d-d^{\\prime }$ .",
"Extend the results of the first step by weights 1, 1/2 and $(q, 1-q)$ to get all IPWSs with index $r$ and dimension $d^{\\prime }\\le d$ .",
"Determine all possible structures of minimal polytopes in dimension $d-1$ .",
"Combine the last two steps to get all $d$ -dimensional minimal cones.",
"Determine all subcones on all sublattices of $M_\\mathrm {finest}$ ."
],
[
"Classification of basic weight systems",
"The classification of basic IPWSs relies on the algorithm of [12].",
"In order to find all $\\mathbf {q}$ 's satisfying $(1/r,\\ldots ,1/r) \\in \\mathrm {int}(\\Delta _{\\mathbf {q}})$ with $\\Delta _{\\mathbf {q}}$ determined by eq.",
"(REF ) one uses the fact that $\\mathbf {q}$ is determined by a set of linearly independent $\\mathbf {x}$ 's satisfying $\\sum x_iq_i=1$ .",
"The classification proceeds by successively choosing such ${\\mathbf {x}}^{(i)}$ , starting with ${\\mathbf {x}}^{(0)}=(1/r,\\ldots ,1/r)$ and continuing with lattice points ${\\mathbf {x}}^{(1)},\\ldots {\\mathbf {x}}^{(k)}$ .",
"Every choice of a new $\\mathbf {x}$ restricts the set of allowed $\\mathbf {q}$ 's.",
"Given ${\\mathbf {x}}^{(0)},\\ldots , {\\mathbf {x}}^{(k)}$ one can choose any $\\tilde{\\mathbf {q}}$ compatible with them and check whether it has the IP property.",
"A further $\\mathbf {q} \\ne \\tilde{\\mathbf {q}}$ can have the IP property only if $k+1<d$ and $\\Delta _{\\mathbf {q}}$ contains points on both sides of the hyperplane $\\sum x_i\\tilde{q}_i=1$ .",
"In particular, such a $\\mathbf {q}$ must be compatible with one of the finitely many lattice points obeying $x_i\\ge 0$ for all $i$ and $\\sum x_i\\tilde{q}_i<1$ .",
"For every choice of ${\\mathbf {x}}^{(k+1)}$ among these one should then continue in the same way.",
"Example 2.",
"$d=2$ , $r=1/2$ : ${\\mathbf {x}}^{(0)}=(2,2)$ is compatible with $\\tilde{\\mathbf {q}} = (1/4,1/4)$ , which has the IP property.",
"Any further $\\mathbf {q}$ must allow at least one integer point with $x_1+x_2<4$ .",
"Up to permutation of coordinates the only possiblities are ${\\mathbf {x}}^{(1)}=(3,0)$ which leads to ${\\mathbf {q}} = (1/6,1/3)$ , and ${\\mathbf {x}}^{(1)}=(2,1)$ which does not result in a positive weight system.",
"Lemma 2.",
"If $d=3r$ there is precisely one basic IPWS $(1/3,\\ldots ,1/3)$ , and for $d<3r$ there is no basic IPWS.",
"Proof.",
"Let us assume $d\\le 3r$ .",
"The point $(1/r,\\ldots ,1/r)$ is compatible with $\\tilde{\\mathbf {q}} = (r/d,\\ldots ,r/d)$ , which has the IP property if $d=3r$ .",
"Any other $\\mathbf {q}$ must admit at least one point $\\mathbf {x}$ such that $1 > \\sum x_i \\tilde{\\mathbf {q}}_i \\ge (\\sum x_i)/3$ , i.e.",
"$\\sum x_i \\le 2$ , which is not consistent with a basic IPWS.",
"$\\Box $ The cases covered neither by example 2 nor by lemma 2 require the use of a computer.",
"PALP [16] contains an implementation of the algorithm of [12] that works reasonably well for $r\\le 1$ and $d\\le 5$ .",
"In order to get a program that is fast enough even for the case $r=3$ , $d=8$ the corresponding routines had to be rewritten completely.",
"In particular, the present implementation takes into account some of the symmetry coming from permutations of the coordinates.",
"At every choice of ${\\mathbf {x}}^{(k)}$ in the recursive construction the program computes the vertices of the $(d-k-1)$ –dimensional polytope in $\\mathbf {q}$ –space that is determined by $q_i\\ge 0$ and $\\sum _i x_i^{(j)}q_i=1$ for $j\\in \\lbrace 0,\\ldots ,k\\rbrace $ .",
"This can be done efficiently by using the $(d-k)$ –dimensional polytope of the previous recursive step.",
"$\\tilde{\\mathbf {q}}$ is chosen as the average of the vertices of the $\\mathbf {q}$ –space polytope.",
"This program was used to determine all basic IPWSs for $r\\le 3$ and $d\\le 9$ .",
"The complete lists can be found at the website [19].",
"The results are summarized in table REF Table: Numbers of basic IPWSs for given values of rr vs. d CY =d-2rd_{\\rm CY}=d-2rwhich shows the numbers of basic IPWSs for given index $r$ and $d-2r$ .",
"Following [18] we call the latter `Calabi–Yau dimension' $d_{\\rm CY}$ ; in the case of a complete splitting of the cone it is indeed the dimension of a complete intersection Calabi–Yau variety defined by the corresponding nef-partition, and for any cone leading to a sensible superconformal field theory it is $c/3$ where $c$ is the conformal anomaly.",
"The first entry is the empty IPWS for $d=r=0$ which is required as a starting point for the construction of IPWSs containing only weights $1/2$ or 1.",
"For $d_{\\rm CY}=1$ there are the three basic weight systems $(1/4,1/4)$ , $(1/6,1/3)$ and $(1/3,1/3,1/3)$ from example 2 and lemma 2.",
"For $d_{\\rm CY}=2$ there are 48 basic IPWSs with $r=1/2$ and 47 with $r=1$ .",
"Together they determine precisely the well known 95 weight systems for weighted ${\\mathbb {P}}^4$ 's that have K3 hypersurfaces [20], [21]; as weight systems for reflexive polytopes they were determined in [12].",
"In addition there are 28 basic IPWSs with $r=3/2$ as well as the IPWS $(1/3,\\ldots ,1/3)$ for $r=2$ .",
"These 29 additional basic IPWSs are again identical with the ones relevant to Landau–Ginzburg type SCFTs as determined in [5]; each of them gives rise to a reflexive Gorenstein cone.",
"Finally, for $d_{\\rm CY}=3$ there are the 184,026 weight systems with $r\\le 1$ relevant to Calabi–Yau hypersurfaces in toric varieties [12], which contain the 7,555 weight systems relevant to weighted projective spaces [4], [5] as a small subset.",
"In addition there are $168,107+34,256+6,066+1=208,430$ IPWSs with $r>1$ which are new (except for $3,284$ Landau–Ginzburg weights [4], [5]).",
"These weight systems can be the starting points for constructing codimension 2 and 3 Calabi–Yau threefolds in toric varieties as well as N=2 SCFTs with $c=9$ .",
"While each of the 184,026 weight systems with $r\\le 1$ determines a reflexive polytope (hence a reflexive Gorenstein cone) as shown already in [12], among the Gorenstein cones determined by IPWSs with $r>1$ only $112,817+18,962+1,321+1=133,101$ of $208,430$ are reflexive; nevertheless the others are relevant to the classification because they may contain reflexive subcones.",
"For the reflexive cases the `stringy Hodge numbers' of [10] as computed by PALP 2.1 [17] are also listed at the website [19].",
"The pairs of Hodge numbers all seem to be in the range that is well known from the earlier classifications."
],
[
"Further steps of the algorithm",
"We shall now illustrate further steps of the algorithm presented at the end of section for some of the smallest $(d,r)$ –pairs.",
"The case of $d_{\\rm CY}=1$ corresponds to $(d,r)\\in \\lbrace (3,1), (5,2), (7,3),\\ldots \\rbrace $ ."
],
[
"$d=3,~r=1$",
" According to the previous section, the relevant basic IPWSs are $d^{\\prime }=0,~r^{\\prime }=0$ : $()$ ; $d^{\\prime }=2,~r^{\\prime }=1/2$ : $(1/4,~1/4),~ (1/6,~1/3)$ ; $d^{\\prime }=3,~r^{\\prime }=1$ : $(1/3,~1/3,~1/3)$ .",
"These give rise to the $r=1$ IPWSs $d^{\\prime }=2$ : $(1/2,~1/2)$ ; $d^{\\prime }=3$ : $(1/3,~1/3,~1/3),~ (1/4,~1/4,~1/2),~ (1/6,~1/3,~1/2)$ .",
"A 2–dimensional minimal polytope is a triangle or a rhomboid [11].",
"A minimal cone is determined by one of the weight systems (1/3, 1/3, 1/3), (1/4, 1/4, 1/2), (1/6, 1/3, 1/2) or the CWS (1/2, 1/2, 0, 0; 0, 0, 1/2, 1/2).",
"All reflexive subcones correspond to all reflexive subpolytopes of the corresponding support polytopes (3 triangles and a square); these are the well known 16 reflexive polygons."
],
[
"$d=5,~r=2$",
" The relevant basic IPWSs are $d^{\\prime }=0,~r^{\\prime }=0$ : $()$ ; $d^{\\prime }=2,~r^{\\prime }=1/2$ : $(1/4,~1/4),~ (1/6,~1/3)$ ; $d^{\\prime }=3,~r^{\\prime }=1/2$ : 48 basic IPWSs (cf.",
"table REF ); $d^{\\prime }=3,~r^{\\prime }=1$ : $(1/3,~1/3,~1/3)$ ; $d^{\\prime }=4,~r^{\\prime }=1$ : 47 basic IPWSs (cf.",
"table REF ).",
"These give rise to the $r=2$ IPWSs $d^{\\prime }=2$ : $(1,~1)$ ; $d^{\\prime }=3$ : $(1/2,~1/2,~1)$ ; $d^{\\prime }=4$ : $(1/2,~1/2~,~1/2,~1/2)$ , ${}$ $(1/3,~1/3,~1/3,~1)$ , $(1/4,~1/4,~1/2,~1)$ , $(1/6,~1/3,~1/2,~1)$ ; $d^{\\prime }=5$ : $(1/4,~1/4,~1/2,~1/2,~1/2)$ , $(1/6,~1/3,~1/2,~1/2,~1/2)$ , ${}$ $(1/3,~1/3,~1/3,~1/2,~1/2)$ , ${}$ $(1/4,~1/4,~1/4,~1/2,~3/4)$ , $(1/6,~1/6,~1/3,~1/2,~5/6)$ , ${}$ $(1/6,~1/3,~1/3,~1/2,~2/3)$ , $(1/3,~1/3,~1/3,~1/3,~2/3)$ , ${}$ 48 IPWSs of the type $(q_1,~q_2,~q_3,~1/2,~1)$ , ${}$ 47 IPWSs of the type $(q_1,~q_2,~q_3,~q_4,~1)$ .",
"The 4–dimensional minimal polytopes were classified in[11].",
"5.",
"These steps would require the use of a computer and have not yet been performed."
],
[
"Other cases",
"The next case with $d_{\\rm CY}=1$ is $d=7$ , $r=3$ .",
"Here already the first step of the algorithm involves the 184,026 weight systems that were used in the classification of reflexive polytopes in four dimensions, as well as as the 28 basic IPWSs for $d^{\\prime }=5$ , $r^{\\prime }=3/2$ .",
"In addition it requires an analysis of the possible structures of minimal polytopes in dimensions up to 6.",
"This should not be too hard, but one should be aware of the fact that a description of a minimal polytope in terms of IP simplices need not be unique, as pointed out already in [11].",
"From what we have seen it is clear that for any fixed value of $d_{\\rm CY}$ the complete classification problem gets harder for rising $r$ .",
"In particular the lists for $d_{\\rm CY}=3$ contain weight systems of the type $(q_1,\\ldots ,q_6,1)$ for $r=2$ and of the type $(q_1,\\ldots ,q_7,1,1)$ for $r=3$ .",
"In the cases where classifications have been completed it turns out that there are more weight systems for reflexive ($d-1$ )–polytopes than there are reflexive $d$ –polytopes, so while it is conceivable that $(d=5,r=1/2)$ and $(d=6,r=1)$ might be within the range of present computer power, $(d=7,r=1)$ is definitely impossible.",
"However, one would not expect all reflexive Gorenstein cones to lead to sensible SCFTs.",
"For example, consider a cone $\\sigma $ whose support is a height one pyramid, which is equivalent to $\\sigma = \\sigma _b\\times \\sigma _1$ where $\\sigma _b$ is the cone over the base of the pyramid and $\\sigma _1$ is the unique one dimensional cone; this case leads to trivial $E_{\\rm string}$ [18].",
"Now $\\sigma ({\\mathbf {q}})$ with ${\\mathbf {q}}=(\\tilde{\\mathbf {q}},1)$ is of this type, and any of its subcones with the IP property is also of this type because all lattice points are in the base or the apex; hence the apex of the pyramid cannot be dropped without violating the IP property.",
"Therefore one can omit cones defined by single weight systems containing a weight of 1 from the list of cones serving as starting points for step (5) of the classification procedure.",
"This implies that in addition to the basic weight systems of table REF only $(d=5,r=1/2)$ and $(d=6,r=1)$ are required for a classification of relevant CWS for $d_{\\rm CY}\\le 3$ , $r\\le 3$ .",
"More generally one might use the fact that $E_{\\rm string}$ is multiplicative under taking cartesian products of cones [22]; the case above is a special case of this since $E_{\\rm string}=0$ for the one dimensional cone (actually $E_{\\rm string}=0$ whenever $d_{\\rm CY}< 0$ [22]).",
"A further reduction of the number of relevant (C)WS may come from the following consideration related to the gauged linear sigma model.",
"If the superpotential contains quadratic terms then its derivatives $F_i$ (cf.",
"eq.",
"REF ) have linear terms that can be used to eliminate (`integrate out' in physicists' language) fields by replacing them by the expressions determined by $F_i=0$ .",
"In this way one can argue for the following simplifications: a support polytope that is a height 2 pyramid over a height 2 pyramid can be reduced to the base, implying that a weight system $({\\mathbf {q}},1/2,1/2)$ is equivalent to just $({\\mathbf {q}})$ ; the product of two height one pyramids can be reduced to the product of the bases, implying the equivalence $({\\mathbf {q}},1,{\\mathbf {0}},0;{\\mathbf {0}},0,\\tilde{\\mathbf {q}},1)\\sim ({\\mathbf {q}},{\\mathbf {0}};{\\mathbf {0}},\\tilde{\\mathbf {q}})$ of CWS; a weight system $(\\tilde{\\mathbf {q}},q,1-q)$ should be equivalent to $(\\tilde{\\mathbf {q}})$ .",
"While these considerations certainly need to be put on a firmer footing, they seem to be confirmed `experimentally' as the following lines of PALP output (version 2.1 is required, see [17]) indicate.",
"4 1 1 1 1 0 0 2 0 0 0 0 1 1 M:105 8 N:7 6 H:2,86 [-168] 4 1 1 1 1 4 0 0 0 2 0 0 0 0 0 1 1 2 M:144 15 N:10 8 H:2 86 [-168] 3 1 1 1 0 0 0 3 0 0 0 1 1 1 M:100 9 N:7 6 H:2,83 [-162] 3 1 1 1 3 0 0 0 0 3 0 0 0 0 1 1 1 3 M:121 16 N:10 8 H:2 83 [-162] 6 1 1 1 1 2 3 3 M:181 7 N:7 7 H:1 103 [-204] 5 1 1 1 1 1 1 4 M:258 12 N:8 8 H:1 101 [-200] However, one should not draw the conclusion that only basic weight systems are relevant: for example, the CWS $(1, 1, 0, 0, 0, 0; 0, 0, 1/2, 1/2, 1/2, 1/2)$ corresponds to the perfectly sensible case of two quadrics in ${\\mathbb {P}}^3$ .",
"Finally let us discuss what can be done in the future.",
"Extending the basic weight systems with $q\\ge 1/2$ –weights is completely straightforward but only interesting once we also combine several weight systems into CWS, which should not be too hard, either.",
"The classification of $(d=5,r=1/2)$ and $(d=6,r=1)$ basic weight systems probably is the most interesting step that may still be achieved, in particular since these same weight systems also give rise to Calabi–Yau fourfolds.",
"In principle this could be done with the existing algorithm.",
"In practice it is very unlikely that it would produce results within a reasonable computation time.",
"One would probably need to work very hard on further elimination of redundancies, on parallelizing the computation and on obtaining the necessary computer power.",
"This would require someone with great skills in understanding the problem, programming, and organizing resources; in other words, someone like Max Kreuzer."
]
] | 1204.1181 |
[
[
"On the Extension of the Erdos-Mordell Type Inequalities"
],
[
"Abstract We discuss the extension of inequality R_A >= c/a * r_b + b/a * r_c to the plane of triangle ABC.",
"Based on the obtained extension, in regard to all three vertices of the triangle, we get the extension of Erdos-Mordell inequality, and some inequalities of Erdos-Mordell type."
],
[
"Introduction",
"Let triangle $\\triangle ABC$ be given in Euclidean plane.",
"Denote by $R_A, \\, R_B$ and $R_C$ the distances from the arbitrary point $M$ in the interior of $\\triangle ABC$ to the vertices $A, \\, B$ and $C$ respectively, and denote by $r_a, \\, r_b$ and $r_c$ the distances from the point $M$ to the sides $BC, \\, CA$ and $AB$ respectively (Figure 1).",
"Figure: NO_CAPTION Figure 1: Erdös-Mordell inequality Then Erdös-Mordell inequality is true: $R_A + R_B + R_C\\ge 2\\left( r_a + r_b + r_c \\right)$ whereat equality holds if and only if triangle $ABC$ is equilateral and $M$ is its center.",
"This inequality was conjectured by P. Erdös as Amer.",
"Math.",
"Monthly Problem 3740 in 1935.",
"[9], after his experimental conjecture in 1932.",
"[13].",
"It was proved by L.J.",
"Mordell in 1935.",
"(in Hungarian, according to [13]), and as the solution of the Problem 3740 in 1937.",
"[22].",
"Considering the Erdös-Mordell inequality (REF ) the goal of this research is to determine areas in the plane of the triangle, where the following three inequalities are valid: $R_A \\ge \\frac{c}{a}r_b + \\frac{b}{a}r_c$ $R_B \\ge \\frac{c}{b}r_a + \\frac{a}{b}r_c$ $R_C\\ge \\frac{b}{c}r_a + \\frac{a}{c}r_b$ where $a=\\left|BC\\right|, \\, b=\\left|CA\\right|, \\, c=\\left|AB\\right|$ .",
"In this paper we determine a set of points E for which $R_A + R_B + R_C\\ge \\left(\\frac{c}{b}+\\frac{b}{c}\\right)r_a+\\left(\\ \\frac{c}{a}+\\frac{a}{c}\\right)r_b+\\left(\\frac{a}{b}+\\frac{b}{a}\\right)r_c$ is valid.",
"It is known that the triangular area of $\\triangle ABC$ is contained in the set E [3], [4], [11], [13], [14], [26].",
"Here we show that the set E is greater than the triangle $\\triangle ABC$ , and we give a geometric interpretation of the set E. The proofs of Erdös-Mordell inequality are often based on different proofs of inequality (REF ), as given in [4], [6], [7], [11], [12], [23], [26].",
"N. Derigades in [8] proved the inequality (REF ) valid in the whole plane of the triangle, where $r_a, \\, r_b$ and $r_c$ , are signed distances.",
"A similar result was given by B. Malešević [20], [21].",
"Note that V. Pambuccian [24] recently proved that the Erdös-Mordell inequality is equivalent to non-positive curvature.",
"Overview of recent results on Erdös-Mordell inequalities and related inequalities is given in [1] - [3], [5], [8], [10], [13] - [21], [24], [25], [27] - [30] ."
],
[
"The Main Results",
"In this section we analyze only the inequality (REF ).",
"Let $\\triangle ABC$ be a triangle with vertices $A\\left(0,r\\right), \\, B\\left(p,0\\right), \\, C\\left(q,0\\right), \\, p \\ne q, \\, r\\ne 0$ .",
"Without diminishing generality, let $p<q$ .",
"We denote by $M\\left(x,y\\right)$ an arbitrary point in the plane of the triangle $\\triangle ABC$ .",
"The distance from the point $M$ to the point $A$ , and the distance from the point $M$ to the straight lines b and c are given by functions: $R_A = \\displaystyle \\sqrt{\\strut x^2+{\\left(y-r\\right)}^2}$ $r_b = \\mbox{\\small $\\displaystyle \\frac{\\left|-qy-rx+qr\\right|}{\\sqrt{\\strut r^2+q^2}}$}$ $r_c = \\mbox{\\small $\\displaystyle \\frac{\\left|py+rx-pr\\right|}{\\sqrt{r^2+p^2}}$}$ respectively.",
"Consider the inequality (REF ) related to the vertex $A$ .",
"The analytical notation of this inequality is: $\\sqrt{x^2+{\\left(y-r\\right)}^2}\\ge \\mbox{\\small $\\displaystyle \\frac{\\sqrt{r^2+p^2}}{\\left|q-p\\right|}\\frac{\\left|-qy-rx+qr\\right|}{\\sqrt{r^2+q^2}}$}+\\mbox{\\small $\\displaystyle \\frac{\\sqrt{r^2+q^2}}{\\left|q-p\\right|}\\frac{\\left|py+rx-pr\\right|}{\\sqrt{r^2+p^2}}$},$ i.e.",
"$\\begin{array}{rcc}\\left|q-p\\right| \\displaystyle \\sqrt{\\strut r^2+p^2} \\displaystyle \\sqrt{\\strut r^2+q^2} \\displaystyle \\sqrt{\\strut x^2+{\\left(y-r\\right)}^2}&\\;\\ge &\\left(r^2+p^2\\right)\\left|-qy-rx+qr\\right| \\\\[1.0 ex]&\\; &+\\left(r^2+q^2\\right)\\left|py+rx-pr\\right|.\\end{array}$ Let $y=kx+r, \\, k \\in \\overline{\\mathbb {R}}$ , then the inequality (REF ) reads as follows: $\\left|x\\right|\\left|q\\!-\\!p\\right|\\displaystyle \\sqrt{\\strut r^2\\!+\\!p^2}\\displaystyle \\sqrt{\\strut r^2\\!+\\!q^2}\\displaystyle \\sqrt{\\strut 1\\!+\\!k^2}\\ge \\left|x\\right|{\\Big (}\\!\\left(r^2\\!+\\!p^2\\right)\\!\\left|-\\!qk\\!-\\!r\\right|+\\left(r^2\\!+\\!q^2\\right)\\!\\left|pk\\!+\\!r\\right|\\!",
"{\\Big )}$ For $x = 0$ , the previous inequality is reduced to an equality which solution is the point $A\\left(0,r\\right)$ .",
"For $x \\ne 0$ we obtain inequality by a single variable $k$ : $\\left|q\\!-\\!p\\right|\\displaystyle \\sqrt{\\strut r^2\\!+\\!p^2}\\displaystyle \\sqrt{\\strut r^2\\!+\\!q^2}\\displaystyle \\sqrt{\\strut 1\\!+\\!k^2}\\ge \\left(r^2+p^2\\right)\\!\\left|\\!-qk\\!-\\!r\\right|+\\left(r^2+q^2\\right)\\!\\left|pk\\!+\\!r\\right|.$ Solution of the inequality (REF ) reduces to four cases per parameter $k$ : $\\left(\\alpha _1\\right) \\, : \\;\\left\\lbrace \\!\\begin{array}{c}\\;\\; pk+r\\ge 0 \\\\-qk-r\\ge 0,\\end{array}\\right.$ $\\left(\\alpha _2\\right) \\, : \\;\\left\\lbrace \\!\\begin{array}{c}\\;\\; pk+r<0 \\\\-qk-r\\ge 0,\\end{array}\\right.$ $\\left(\\alpha _3\\right) \\, : \\;\\left\\lbrace \\!\\begin{array}{c}\\;\\; pk+r\\ge 0 \\\\-qk-r<0,\\end{array}\\right.$ $\\left(\\alpha _4\\right) \\, : \\;\\left\\lbrace \\!\\begin{array}{c}\\;\\; pk+r<0 \\\\-qk-r<0.\\end{array}\\right.$ Note that the value $k$ corresponds to the points $(x,y) \\in {\\mathbb {R}}^2$ located on the straight line $y=kx+r$ .",
"With its values, the mentioned parameter of the line $y = kx + r$ decomposes ${\\mathbb {R}}^2$ on four corner areas.",
"Inquiring the existence of parameter $k$ (i.e.",
"the pencil of lines $y=kx+r$ through the vertex $A$ ) depending on the signs of parameters $p, \\, q$ and $r$ , we provide the following table of existing corner areas $\\left(\\alpha _1\\right)-\\left(\\alpha _4\\right)$ : Table: NO_CAPTIONTable 1: The existence of the corner area depending on the parameters p, q and r The corner areas $\\left(\\alpha _1\\right)$ and $\\left(\\alpha _4\\right)$ are always in the interior of $\\sphericalangle BAC$ and its cross angle, while the areas $\\left(\\alpha _2\\right)$ and $\\left(\\alpha _3\\right)$ are in the interior of its supplementary angle (Figure 2).",
"Figure: NO_CAPTION Figure 2: Existence of the corner area for the vertex A (Cases 1. to 6. in the Table 1) Let us consider the equation: $\\left(q-p\\right)\\displaystyle \\sqrt{\\strut r^2\\!+\\!p^2}\\displaystyle \\sqrt{\\strut r^2\\!+\\!q^2}\\displaystyle \\sqrt{\\strut 1\\!+\\!k^2}=\\left(r^2\\!+\\!p^2\\right)\\left|-qk-r\\right|+\\left(r^2\\!+\\!q^2\\right)\\left|pk+r\\right|.$ I)$\\;$ Let $k$ fulfill $\\left(\\alpha _1\\right)$ or $\\left(\\alpha _4\\right)$ .",
"Then the previous equation can be rewritten in a way that follows, with positive sign (+) in the case of area $\\left(\\alpha _1\\right)$ and negative sign (-) in the case of area $\\left(\\alpha _4\\right)$ $\\left(q-p\\right)\\displaystyle \\sqrt{\\strut r^2\\!+\\!p^2}\\displaystyle \\sqrt{\\strut r^2\\!+\\!q^2}\\displaystyle \\sqrt{\\strut 1\\!+\\!k^2}=\\pm {\\big (}\\!\\left(-qk-r\\right)\\left(r^2\\!+\\!p^2\\right)+\\left(pk+r\\right)\\left(r^2\\!+\\!q^2\\right)\\!",
"{\\big )}$ i.e.",
"$\\left(q-p\\right)\\displaystyle \\sqrt{\\strut r^2\\!+\\!p^2}\\displaystyle \\sqrt{\\strut r^2\\!+\\!q^2}\\displaystyle \\sqrt{\\strut 1\\!+\\!k^2}=\\pm \\left(q-p\\right){\\big (}r\\left(q\\!+\\!p\\right)+k\\left(pq-r^2\\right)\\!",
"{\\big )}$ abbreviated written as $\\lambda \\displaystyle \\sqrt{\\strut 1+k^2}=\\pm \\beta k \\pm \\gamma =\\left\\lbrace \\begin{array}{c}\\;\\;\\;\\;\\beta k+\\gamma ,\\ \\ \\ \\ k \\in \\left(\\alpha _1\\right) \\\\[1.0 ex]-\\beta k-\\gamma ,\\ \\ \\ k \\in \\left(\\alpha _4\\right)\\end{array}\\right.$ where at: $\\lambda =\\left(q-p\\right)\\displaystyle \\sqrt{\\strut r^2+p^2}\\displaystyle \\sqrt{\\strut r^2+q^2}\\;\\;\\;\\;\\mbox{\\rm and}\\;\\;\\;\\;\\lambda >0$ $\\beta =\\left(pq-r^2\\right)\\left(q-p\\right)$ $\\gamma =r\\left(q^2-p^2\\right).$ As $p<q$ , the equation (REF ) can be divided by $ q - p \\ne 0$ and then squared: $\\left(r^2\\!+\\!p^2\\right)\\left(r^2\\!+\\!q^2\\right)\\left(1+k^2\\right)={\\big (} \\!\\, r \\left(q+p\\right) + k \\left( pq-r^2 \\right) \\!",
"{\\big )}^2$ which transforms into $ {\\big (} \\!\\, r\\left(p+q\\right)k-\\left(pq-r^2\\right) \\!",
"{\\big )}^2 = 0.$ Based on the above equation, we conclude that there exists the unique solution: $ k_1=\\frac{pq-r^2}{r\\left(p+q\\right)}$ only if, for $k=k_1$ : $ \\pm \\beta k \\pm \\gamma \\ge 0$ is valid.",
"Hence, the straight line $y=k_1x+r$ is in the interior of $\\sphericalangle BAC$ and its cross angle, or it doesn't exist.",
"The cases where values $k_1$ from the formula (REF ) does not meet the condition (REF ) are presented in the Table 1 with: in the case 1: $k_{{\\rm 1}}{\\rm >}-{r}/{q}\\Longleftrightarrow p\\left(q^{{\\rm 2}}{\\rm +}r^{{\\rm 2}}\\right){\\rm >}0\\ $ ; in the case 3: $k_{{\\rm 1}}{\\rm >}-{r}/{p}\\Longleftrightarrow \\left({\\rm -}q\\right)\\left(p^{{\\rm 2}}{\\rm +}r^{{\\rm 2}}\\right){\\rm >}0$ ; in the case 4: $k_{{\\rm 1}}{\\rm <}-{r}/{q}\\Longleftrightarrow p\\left(q^{{\\rm 2}}{\\rm +}r^{{\\rm 2}}\\right){\\rm >}0\\ $ ; in the case 6: $k_{{\\rm 1}}{\\rm <}-{r}/{p}\\Longleftrightarrow \\left({\\rm -}q\\right)\\left(p^{{\\rm 2}}{\\rm +}r^{{\\rm 2}}\\right){\\rm >}0$ .",
"Lemma 1 For $k\\in \\left(\\alpha _1\\right)\\ \\cup \\left(\\alpha _4\\right)$ inequality (REF ) is valid, where equality holds for $k=k_1$ if (REF ) is fulfilled.",
"$\\mbox{(\\ref {GrindEQ__12_})} \\Longleftrightarrow {\\big (} r\\left(p+q\\right)k - \\left(pq-r^2\\right){\\big )}^2 \\ge 0$ .",
"$\\Box $ Corollary 1 Inequality (REF ) is valid for lines b and c. II)$\\;$ Let $k$ fulfill $\\left(\\alpha _2\\right)$ or $\\left(\\alpha _3\\right)$ .",
"Then equation (REF ) can be rewritten in a way that follows, with negative sign (-) in the case of area $\\left(\\alpha _2\\right)$ and positive sign (+) in the case of area $\\left(\\alpha _3\\right)$ $\\left(q-p\\right)\\displaystyle \\sqrt{\\strut r^2\\!+\\!p^2}\\displaystyle \\sqrt{\\strut r^2\\!+\\!q^2}\\displaystyle \\sqrt{\\strut 1\\!+\\!k^2}=\\pm {\\big (}\\!\\left(qk+r\\right)\\left(r^2\\!+\\!p^2\\right) + \\left(pk+r\\right)\\left(r^2\\!+\\!q^2\\right)\\!",
"{\\big )}$ or abbreviated written as $\\lambda \\displaystyle \\sqrt{\\strut 1 + k^2}=\\pm \\delta k \\pm \\varepsilon =\\left\\lbrace \\begin{array}{c}\\;\\;\\; \\delta k + \\varepsilon ,\\ \\ \\ \\ k\\in \\left(\\alpha _3\\right) \\\\[1.0 ex]-\\delta k - \\varepsilon ,\\ \\ \\ k\\in \\left(\\alpha _2\\right)\\end{array}\\right.$ with parameters: $\\lambda =\\left(q-p\\right)\\displaystyle \\sqrt{\\strut r^2+p^2}\\displaystyle \\sqrt{\\strut r^2+q^2} \\quad \\mbox{and} \\quad \\lambda >0$ $\\delta =\\left(r^2+pq\\right)\\left(q+p\\right)$ $\\varepsilon =r\\left(2r^2+q^2+p^2\\right) \\!",
".$ The equation (REF ) is considered under the following condition: $\\pm \\delta k\\pm \\varepsilon \\ge 0.$ By squaring the equation (REF ) we obtain $P(k)=\\lambda ^2\\left(1+k^2\\right) - \\left(\\pm \\delta k \\pm \\varepsilon \\right)^2=\\left(\\lambda ^2-\\delta ^2\\right) k^2 - 2 \\delta \\varepsilon k + \\left(\\lambda ^2-\\varepsilon ^2\\right)=0.$ For the square trinomial $P(k)=\\widehat{{\\rm A}} \\, k^2 + \\widehat{{\\rm B}} \\, k + \\widehat{{\\rm C}}$ coefficients $\\widehat{{\\rm A}}, \\, \\widehat{{\\rm B}}, \\, \\widehat{{\\rm C}}$ are determined by: $\\widehat{{\\rm A}}=\\lambda ^2 - \\delta ^2=\\left(q-p\\right)^2 \\!",
"\\left(r^2\\!+\\!p^2\\right) \\!",
"\\left(r^2\\!+\\!q^2\\right) - \\left(r^2+pq\\right)^2 \\!",
"\\left(q+p\\right)^2$ $\\widehat{{\\rm B}}=-2 \\delta \\varepsilon =-2 r\\left(r^2 + pq\\right) \\!",
"{\\big (}q + p{\\big )} \\!",
"\\left(2 r^2 + q^2 + p^2 \\right)$ $\\widehat{{\\rm C}}=\\lambda ^2 - \\varepsilon ^2=\\left(r^2+pq\\right){\\big (}\\!\\left(pq-r^2\\right) \\left(q-p\\right)^2 - 2r^2(2r^2+q^2+p^2){\\big )}.$ Let us consider the equation: $\\widehat{{\\rm A}}=-4pqr^4 + \\left(p^4\\!+\\!q^4\\!-\\!4pq^3\\!-\\!4p^3q\\!-\\!2p^2q^2\\right) \\!",
"r^2 - 4p^3q^3=0.$ It has real solutions for $r$ in the following form: $\\left\\lbrace \\begin{array}{c}r_{1,2}=\\mbox{\\small \\mbox{$\\displaystyle \\frac{1}{4\\displaystyle \\sqrt{\\strut pq}}$}}\\left((q-p)^2\\ \\pm \\ \\displaystyle \\sqrt{\\strut {\\left(q-p\\right)}^4-16{p^2q}^2} \\;\\; \\right) > 0 \\\\[1.75 ex]r_{3,4}=-\\mbox{\\small \\mbox{$\\displaystyle \\frac{1}{4\\displaystyle \\sqrt{\\strut pq}}$}}\\left((q-p)^2\\ \\pm \\ \\displaystyle \\sqrt{\\strut {\\left(q-p\\right)}^4-16{p^2q}^2} \\;\\; \\right) < 0\\end{array}\\right.$ iff ${\\Big (} p \\ge 0 \\; \\wedge \\; q \\ge (3+2\\displaystyle \\sqrt{2})p {\\Big )}\\;\\; \\vee \\;\\;{\\Big (} p < 0 \\; \\wedge \\; q \\le (3-2\\displaystyle \\sqrt{2})p {\\Big )}.$ Remark 1 When $p < 0$ and $q > 0$ then $\\widehat{{\\rm A}}=4\\left|p\\right|qr^4 + \\left(q^2{-p}^2\\right)^2 \\!",
"r^2 + 4\\left|p\\right|q$ $\\left(p^2+q^2\\right) \\!",
"r^2 + 4\\left|p\\right|^3q^3 > 0$ is valid.",
"Note that the equation $\\widehat{{\\rm A}} =0$ is not considered for $p=0$ or $q=0$ (because we obtain the contradictions: $p = 0, \\; q \\ne 0$ : $\\widehat{{\\rm A}} = r^2 q^4 = 0 \\, \\Longrightarrow \\, r=0$ ; i.e.",
"$p \\ne 0, \\; q=0$ : $\\widehat{{\\rm A}} = r^2 p^4 = 0 \\, \\Longrightarrow \\, r = 0$ ).",
"We distinguish the cases: a) Let $r = r_j$ for some $j = 1, \\, 2, \\, 3, \\, 4$ , then $\\widehat{{\\rm A}} = 0$ .",
"In this case, $\\widehat{{\\rm B}} \\ne 0$ , because $r^2 + pq \\ne 0$ and $q + p \\ne 0$ (in the case of equilateral triangle, there will be valid $q + p = 0$ and then $r = \\pm pi, \\; i = \\displaystyle \\sqrt{\\!-\\!1}\\,$ ).",
"Therefore, by solving the linear equation $\\widehat{{\\rm B}} \\, k + \\widehat{{\\rm C}} = 0$ we find that: $k_2=-\\displaystyle \\frac{\\widehat{{\\rm C}}}{\\widehat{{\\rm B}}}=\\displaystyle \\frac{\\lambda ^2 - \\varepsilon ^2}{2 \\delta \\varepsilon }=\\displaystyle \\frac{\\left(q-p\\right)^2\\left(r^2+p^2\\right)\\left(r^2+q^2\\right) - r^2\\left(2r^2+q^2+p^2\\right)^2}{2r\\left(q+p\\right)\\left(2r^2+q^2+p^2\\right)}.$ For $p < 0$ and $q > 0$ the case a) is not considered (because $\\widehat{{\\rm A}} > 0$ ).",
"Let us examine when the value $k_2$ meet the condition (REF ).",
"It is valid that: $\\pm \\delta k_2 \\pm \\varepsilon \\ge 0\\; \\Longleftrightarrow \\;\\pm \\left(\\delta k_2+\\varepsilon \\right)=\\pm \\left(\\delta \\displaystyle \\frac{\\,\\lambda ^2 - \\varepsilon ^2}{2 \\delta \\varepsilon } + \\varepsilon \\right)=\\pm \\left(\\displaystyle \\frac{\\,\\lambda ^2+\\varepsilon ^2}{2 \\varepsilon }\\right)\\ge 0.$ Based on $\\varepsilon = r\\left(2r^2+q^2+p^2\\right)$ we conclude: if $r > 0$ then $\\delta k_2 + \\varepsilon \\ge 0$ is fulfilled, whereby $k_2$ fulfills condition (REF ) and $k_2 \\in \\left(\\alpha _3\\right)$ ; if $r < 0$ then $-\\delta k_2-\\varepsilon \\ge 0\\ $ is fulfilled, whereby $k_2$ fulfills condition (REF ) and $k_2 \\in \\left(\\alpha _2\\right)$ .",
"In this case, the line $y = k_2 x + r$ is in the exterior of $\\sphericalangle BAC$ and its cross angle.",
"b) Let $r \\ne r_j$ for each $j = 1, \\, 2, \\, 3, \\, 4$ , then $\\widehat{{\\rm A}} \\ne 0$ and in this case, by solving the quadratic equation (REF ), we find the values: $\\begin{array}{rcl}k_{2,3}& \\!\\!=\\!\\!",
"&\\displaystyle \\frac{-\\delta \\varepsilon \\pm \\displaystyle \\sqrt{\\strut \\lambda ^2 \\left(\\delta ^2 \\!+\\!",
"\\varepsilon ^2 \\!-\\!",
"\\lambda ^2\\right)}}{\\delta ^2\\!-\\!\\lambda ^2} \\\\[1.0 ex]& \\!\\!=\\!\\!",
"&\\displaystyle \\frac{r(p+q)(r^2\\!+pq)(q^2\\!+\\!p^2\\!+\\!2r^2)\\pm 2\\left(r^2\\!+\\!p^2\\right)\\left(r^2\\!+\\!q^2\\right)\\left(q\\!-\\!p\\right)\\displaystyle \\sqrt{\\strut r^2\\!+pq}}{\\left(q\\!-\\!p\\right)^2\\left(r^2\\!+\\!p^2\\right)\\left(r^2\\!+\\!q^2\\right)-\\left(r^2\\!+pq\\right)^2\\left(q\\!+\\!p\\right)^2}.\\end{array}$ If $r^2 + pq \\ge 0$ then exists $k_{2,3}\\in {\\mathbb {R}}$ .",
"Incidence of $k_{2,3} \\in {\\mathbb {R}}$ to the area $\\left(\\alpha _3\\right)$ , as to the area $\\left(\\alpha _2\\right)$ is determined by the inequality (REF ).",
"The expression $\\delta k_{2,3} + \\varepsilon $ exists for $\\delta \\ne \\pm \\lambda $ , whereby the expression $\\delta k_{2,3} + \\varepsilon $ is either positive or negative (because $\\delta k_{2,3} + \\varepsilon = 0 \\Longrightarrow \\delta = \\pm \\lambda $ ).",
"Based on the Corollary 1, the straight lines $y=k_sx+r, (s=2,\\ 3)$ are in the exterior of $\\sphericalangle BAC$ and its cross angle (Figure 3).",
"Consider the limiting case for $k_{2,3}$ when $r \\rightarrow r_{j}$ .",
"Note that $\\widehat{{\\rm A}} = \\lambda ^2 - \\delta ^2{{\\underset{r\\rightarrow r_{j}}{\\longrightarrow }}} \\,0$ is valid, whereat from $k_{2,3}=\\displaystyle \\frac{-\\varepsilon }{\\left(\\delta -\\lambda \\right)\\left(\\delta +\\lambda \\right)}\\cdot \\left(\\delta \\mp \\left|\\lambda \\right|\\displaystyle \\sqrt{\\strut 1 + \\frac{\\delta ^2-\\lambda ^2}{\\varepsilon ^2}}\\;\\;\\right)$ follows $\\mathop {\\mbox{\\rm lim}}_{r\\rightarrow r_j}{k}_2=\\displaystyle \\frac{-\\varepsilon }{\\left(\\delta + \\lambda \\right)}\\;\\;\\; \\wedge \\;\\;\\;\\mathop {\\mbox{\\rm lim}}_{r\\rightarrow r_j}{k}_3 = \\infty .$ Figure: NO_CAPTION Figure 3: The existence of the lines $y = k_s x \\!+\\!",
"r,\\; (\\!\\mbox{\\small $s\\!=\\!2,3$}\\!",
")$ depending on the parameter $\\widehat{{\\rm A}}$ Related to the $\\sphericalangle BAC$ we distinguish the cases: 1.",
"$\\sphericalangle BAC < \\pi /2 \\Longleftrightarrow r^2 + pq > 0$ and if $\\widehat{{\\rm A}} \\ne 0$ then there are two real and different values of $k_2$ and $k_3$ .",
"In this case, the following lemma is valid: Lemma 2 For $\\sphericalangle BAC < \\pi /2$ , $k \\in \\left(\\alpha _2\\right) \\cup \\left(\\alpha _3\\right)$ the inequality (REF ) is valid, just in the cases: $\\widehat{{\\rm A}} > 0\\; \\wedge \\;k \\in \\left[-\\infty , \\, k_2\\right] \\cup \\left[k_3, \\, +\\infty \\right] \\, \\backslash \\, {\\big (}\\!\\left(\\alpha _1\\right) \\cup \\left(\\alpha _4\\right)\\!",
"{\\big )}$ ; $\\widehat{{\\rm A}} = 0\\; \\wedge \\;k \\in \\left[-\\infty , \\, k_2\\right] \\backslash {\\big (}\\left(\\alpha _1\\right) \\cup \\left(\\alpha _4\\right){\\big )}$ ; $\\widehat{{\\rm A}} < 0\\; \\wedge \\;k \\in \\left[k_2, \\, k_3\\right] \\backslash {\\big (}\\left(\\alpha _1\\right) \\cup \\left(\\alpha _4\\right){\\big )}$ ; where the equality holds for $k = k_2$ or $k = k_3$ .",
"2.",
"If $\\sphericalangle BAC = \\pi /2 \\Longleftrightarrow r^2+pq=0$ then $\\widehat{{\\rm A}} = -qp\\left(q-p\\right)^4$ , $\\widehat{{\\rm B}} = 0 $ and $\\widehat{{\\rm C}} = 0$ , according to the equation (REF ) that $k_{2,3}=0.$ Hence is valid: Lemma 3 For $\\sphericalangle BAC=\\pi /2$ and $k \\in \\left(\\alpha _2\\right) \\cup \\left(\\alpha _3\\right)$ the inequality (REF ) is valid.",
"The equality is valid only for $k = 0$ .",
"$\\mbox{(\\ref {GrindEQ__12_})}\\Longleftrightarrow \\widehat{{\\rm A}} \\, k^2 + \\widehat{{\\rm B}} \\, k + \\widehat{{\\rm C}} \\ge 0\\Longleftrightarrow -qp\\left(q-p\\right)^4 k^2 \\ge 0$ .",
"$\\Box $ 3.",
"$\\sphericalangle BAC > \\pi /2 \\Longleftrightarrow r^2+pq < 0$ .",
"In this case, for: $r^2 < -pq$ and for the coefficient $\\widehat{{\\rm A}}$ : $\\begin{array}{rcl}\\widehat{{\\rm A}}& \\!>\\!",
"&4r^6 + \\left(p^4+q^4\\right)r^2 + 4\\left(p^2+q^2\\right)r^4 - 2r^6 + 4p^2q^2r^2 \\\\[1.25 ex]& \\!=\\!",
"&2r^6 + 4\\left(p^2+q^2\\right)r^4 + \\left(p^4+q^4+4p^2q^2\\right)r^2>0\\end{array}$ is valid.",
"Since $k_{2,3} \\in \\mathbb {C}$ and $\\widehat{{\\rm A}} > 0$ the inequality (REF ) is valid, which proves the claim: Lemma 4 For $\\sphericalangle BAC > \\pi /2$ and $k \\in \\left(\\alpha _2\\right) \\cup \\left(\\alpha _3\\right)$ the inequality (REF ) is valid in the strict form.",
"Based on the previous considerations in I) and II), follows: Statement 1 The inequality (REF ) holds in following cases: $\\begin{array}{lcl}& \\hspace*{56.9055pt} & k \\in \\left(\\alpha _1\\right) \\cup \\left(\\alpha _4\\right) \\\\[1.0 ex]\\hspace*{-44.81308pt}\\mbox{\\it or} & \\hspace*{56.9055pt} & \\\\[1.0 ex]& \\hspace*{56.9055pt} & k \\in \\left(\\alpha _2\\right) \\cup \\left(\\alpha _3\\right)\\;\\;\\mbox{\\it for}\\;\\; \\sphericalangle BAC \\ge \\pi /2 \\\\[1.0 ex]\\hspace*{-44.81308pt}\\mbox{\\it i.e.}",
"& \\hspace*{56.9055pt} & \\\\[1.0 ex]& \\hspace*{56.9055pt} & k \\in \\left[-\\infty , \\, k_2\\right] \\cup \\left[k_3, \\, +\\infty \\right]\\backslash {\\big (}\\left(\\alpha _1\\right) \\cup \\left(\\alpha _4\\right){\\big )}\\;\\wedge \\; \\widehat{{\\rm A}} > 0 \\\\[1.0 ex]& \\hspace*{56.9055pt} & k \\in \\left[-\\infty , \\, k_2\\right]\\backslash {\\big (}\\left(\\alpha _1\\right) \\cup \\left(\\alpha _4\\right){\\big )}\\;\\wedge \\; \\widehat{{\\rm A}} = 0 \\\\[1.0 ex]& \\hspace*{56.9055pt} & k \\in \\left[k_2, \\, k_3\\right]\\backslash {\\big (}\\left(\\alpha _1\\right) \\cup \\left(\\alpha _4\\right){\\big )}\\;\\wedge \\; \\widehat{{\\rm A}} <0,\\end{array}$ for $\\sphericalangle BAC < \\pi /2$ ."
],
[
"Conclusion",
"For the vertex $A$ , let us define $\\mbox{\\textbf {\\textit {E}}}_A=\\left\\lbrace \\left( x, y \\right) \\mbox{\\large $|$}R_A\\ge \\mbox{\\small $\\displaystyle \\frac{c}{a}$} r_b + \\mbox{\\small $\\displaystyle \\frac{b}{a}$} r_c\\right\\rbrace ,$ and for the vertices $B$ and $C$ , let us define $\\mbox{\\textbf {\\textit {E}}}_B=\\left\\lbrace \\left( x, y \\right) \\mbox{\\large $|$}R_B\\ge \\mbox{\\small $\\displaystyle \\frac{c}{b}$} r_a + \\mbox{\\small $\\displaystyle \\frac{a}{b}$} r_c\\right\\rbrace ,$ $\\mbox{\\textbf {\\textit {E}}}_C=\\left\\lbrace \\left( x, y \\right) \\mbox{\\large $|$}R_C\\ge \\mbox{\\small $\\displaystyle \\frac{b}{c}$} r_a + \\mbox{\\small $\\displaystyle \\frac{a}{c}$} r_b\\right\\rbrace ,$ respectively.",
"Based on the analysis of the inequalities (REF ), (REF ) and (REF ), the inequality (REF ) is valid in the intersection of the areas: $\\mbox{\\textbf {\\textit {E}}}=\\mbox{\\textbf {\\textit {E}}}_A\\cap \\mbox{\\textbf {\\textit {E}}}_B\\cap \\mbox{\\textbf {\\textit {E}}}_C.$ Therefore follows Statement 2 Erdös-Mordell inequality is valid in the area $\\mbox{\\textbf {\\textit {E}}}\\,$ .",
"Let us define the set $\\textbf {\\textit {M}}$ by the intersection of the corner areas formed from $\\mbox{\\textbf {\\textit {E}}}_A$ , $\\mbox{\\textbf {\\textit {E}}}_B$ and $\\mbox{\\textbf {\\textit {E}}}_C$ , containing the initial triangle.",
"Then the set of points $\\textbf {\\textit {M}}$ is quadrilateral or hexagonal shape, and is contained the area $\\mbox{\\textbf {\\textit {E}}}$ (Figure 4).",
"Figure: NO_CAPTION Figure 4: Extension of the triangle ABC to the area ${\\mathbf {M}} \\subset {\\mathbf {E}}$ Let us define Erdös-Mordell curve in the plane of triangle, by the following equation: $R_A + R_B + R_C=2\\left(r_a + r_b + r_c\\right),$ where $\\begin{array}{ccccc}R_A=\\displaystyle \\sqrt{\\strut x^2+{\\left(y-r\\right)}^2} \\, ,& \\;\\;\\; &R_B=\\displaystyle \\sqrt{\\strut {\\left(x-p\\right)}^2+y^2} \\, ,& \\;\\;\\; &R_C=\\displaystyle \\sqrt{\\strut {\\left(x-q\\right)}^2+y^2} \\, , \\\\[1.5 ex]r_a=\\mbox{\\small $\\displaystyle \\frac{\\left|y\\left(q-p\\right)\\right|}{\\displaystyle \\sqrt{\\strut {\\left(q-p\\right)}^2}}$}=\\left|y\\right| \\, ,& \\;\\;\\; &r_b=\\mbox{\\small $\\displaystyle \\frac{\\left|-q\\left(y-r\\right)-rx\\right|}{\\displaystyle \\sqrt{\\strut r^2+q^2}}$} \\, ,& \\;\\;\\; &r_c=\\mbox{\\small $\\displaystyle \\frac{\\left|-p\\left(y-r\\right)-rx\\right|}{\\displaystyle \\sqrt{\\strut r^2+p^2}}$} \\, .\\end{array}$ The curve (REF ) is a union of parts of algebraic curves of order eight (Figure 5).",
"Figure: NO_CAPTION Figure 5: Erdös-Mordell curve and the area ${\\mathbf {E}}$ Let us denote by E' the part of the plane ${\\mathbb {R}}^2$ bounded by the Erdös-Mordell's curve and consisting the triangle $\\triangle ABC$ .",
"Thus, according to the fact that inequality (REF ) is valid in the area of the triangle $\\triangle ABC$ , and based on continuity, it follows that inequality (REF ) is valid in the area E'.",
"Remark that the area E' allows us to precise when, except for the inequality (REF ), some of the inequalities (REF ), (REF ) and/or (REF ) are true.",
"For example, in the area $\\left(\\mbox{\\textbf {\\textit {E'}}} \\backslash \\mbox{\\textbf {\\textit {E}}}_A\\right)\\cap \\mbox{\\textbf {\\textit {E}}}_B \\cap \\mbox{\\textbf {\\textit {E}}}_C$ the inequalities (REF ), (REF ), (REF ) are true and (REF ) is not true.",
"At end of this section let us emphasize that the following statement is true.",
"Statement 3 All geometric inequalities based on the inequalities (REF ), (REF ) and (REF ) can be extended from the triangle interior to the area $\\mbox{\\textbf {\\textit {E}}}\\,$ .",
"Example 1 In the area $\\mbox{\\textbf {\\textit {E}}}$ , the inequality of Child [7] is valid: $R_A \\cdot R_B \\cdot R_C\\ge 8 \\cdot r_a \\cdot r_b \\cdot r_c$ because, based on inequality between arithmetic and geometric mean, follows: $a \\cdot R_A \\ge b \\cdot r_c + c \\cdot r_b \\ge 2\\displaystyle \\sqrt{\\strut b \\cdot c \\cdot r_b \\cdot r_c}$ $b \\cdot R_B \\ge c \\cdot r_a + a \\cdot r_c \\ge 2\\displaystyle \\sqrt{\\strut c \\cdot a \\cdot r_c \\cdot r_a}$ $c \\cdot R_C \\ge a \\cdot r_b + b \\cdot r_a \\ge 2\\displaystyle \\sqrt{\\strut a \\cdot b \\cdot r_a \\cdot r_b}.$ Hence, by multiplying the left and right sides of inequalities (REF ) - (REF ), we get the inequality (REF ) in the area $\\mbox{\\textbf {\\textit {E}}}$ .",
"$\\Box $ At the end of this paper, let us set up an open problem (proposed by anonymous reviewer): prove or disprove that there exist a positive number $\\varepsilon $ such that the area of $\\mbox{\\textbf {\\textit {E'}}}$ is bigger than 1+$\\varepsilon $ times the area of the triangle for every triangle.",
"Thus, we set a conjecture: for the finite area of $\\mbox{\\textbf {\\textit {E'}}}$ the value $\\varepsilon $ is determined in the case of equilateral triangle.",
"Acknowledgment.",
"The authors would like to thank anonymous reviewer for his/her valuable comments and suggestions, which were helpful in improving the paper.",
"Branko Malešević $($CORRESPONDING AUTHOR$)$ Faculty of Electrical Engineering, University of Belgrade, Bulevar Kralja Aleksandra 73, 11000 Belgrade, Serbia [email protected] Maja Petrović $($CORRESPONDING AUTHOR$)$ Faculty of Transport and Traffic Engineering, University of Belgrade, Vojvode Stepe 305, 11000 Belgrade, Serbia [email protected] Marija Obradović, Faculty of Civil Engineering, University of Belgrade, Bulevar Kralja Aleksandra 73, 11000 Belgrade, Serbia [email protected] Branislav Popkonstantinović, Faculty of Mechanical Engineering, University of Belgrade, Kraljice Marije 16, 11000 Belgrade, Serbia Faculty of Technical Sciences, University of Novi Sad, Trg D. Obradovića 16, 21000 Novi Sad, Serbia [email protected]"
]
] | 1204.1003 |
[
[
"Diagonal and Low-Rank Matrix Decompositions, Correlation Matrices, and\n Ellipsoid Fitting"
],
[
"Abstract In this paper we establish links between, and new results for, three problems that are not usually considered together.",
"The first is a matrix decomposition problem that arises in areas such as statistical modeling and signal processing: given a matrix $X$ formed as the sum of an unknown diagonal matrix and an unknown low rank positive semidefinite matrix, decompose $X$ into these constituents.",
"The second problem we consider is to determine the facial structure of the set of correlation matrices, a convex set also known as the elliptope.",
"This convex body, and particularly its facial structure, plays a role in applications from combinatorial optimization to mathematical finance.",
"The third problem is a basic geometric question: given points $v_1,v_2,...,v_n\\in \\R^k$ (where $n > k$) determine whether there is a centered ellipsoid passing \\emph{exactly} through all of the points.",
"We show that in a precise sense these three problems are equivalent.",
"Furthermore we establish a simple sufficient condition on a subspace $U$ that ensures any positive semidefinite matrix $L$ with column space $U$ can be recovered from $D+L$ for any diagonal matrix $D$ using a convex optimization-based heuristic known as minimum trace factor analysis.",
"This result leads to a new understanding of the structure of rank-deficient correlation matrices and a simple condition on a set of points that ensures there is a centered ellipsoid passing through them."
],
[
"Introduction",
"Decomposing a matrix as a sum of matrices with simple structure is a fundamental operation with numerous applications.",
"A matrix decomposition may provide computational benefits, such as allowing the efficient solution of the associated linear system in the square case.",
"Furthermore, if the matrix arises from measurements of a physical process (such as a sample covariance matrix), decomposing that matrix can provide valuable insight about the structure of the physical process.",
"Among the most basic and well-studied additive matrix decompositions is the decomposition of a matrix as the sum of a diagonal matrix and a low-rank matrix.",
"This decomposition problem arises in the factor analysis model in statistics, which has been studied extensively since Spearman's original work of 1904 [29].",
"The same decomposition problem is known as the Frisch scheme in the system identification literature [17].",
"For concreteness, in Section REF we briefly discuss a stylized version of a problem in signal processing that under various assumptions can be modeled as a (block) diagonal and low-rank decomposition problem.",
"Much of the literature on diagonal and low-rank matrix decompositions is in one of two veins.",
"An early approach [1] that has seen recent renewed interest [11] is an algebraic one, where the principal aim is to give a characterization of the vanishing ideal of the set of symmetric $n\\times n$ matrices that decompose as the sum of a diagonal matrix and a rank $k$ matrix.",
"Such a characterization has only been obtained for the border cases $k=1$ , $k=n-1$ (due to Kalman [17]), and the recently resolved $k=2$ case (due to Brouwer and Draisma [3] following a conjecture by Drton et al. [11]).",
"This approach does not (yet) offer scalable algorithms for performing decompositions, rendering it unsuitable for many applications including those in high-dimensional statistics, optics [12], and signal processing [24].",
"The other main approach to factor analysis is via heuristic local optimization techniques, often based on the expectation maximization (EM) algorithm [9].",
"This approach, while computationally tractable, typically offers no provable performance guarantees.",
"A third way is offered by convex optimization-based methods for diagonal and low-rank decompositions such as minimum trace factor analysis (MTFA), the idea and initial analysis of which dates at least to Ledermann's 1940 work [21].",
"MTFA is computationally tractable, being based on a semidefinite program (see Section ), and yet offers the possibility of provable performance guarantees.",
"In this paper we provide a new analysis of MTFA that is particularly suitable for high-dimensional problems.",
"Semidefinite programming duality theory provides a link between this matrix decomposition heuristic and the facial structure of the set of correlation matrices— positive semidefinite matrices with unit diagonal—also known as the elliptope [19].",
"This set is one of the simplest of spectrahedra—affine sections of the positive semidefinite cone.",
"Spectrahedra are of particular interest for two reasons.",
"First, spectrahedra are a rich class of convex sets that have many nice properties (such as being facially exposed).",
"Second, there are well-developed algorithms, efficient both in theory and in practice, for optimizing linear functionals over spectrahedra.",
"These optimization problems are known as semidefinite programs [30].",
"The elliptope arises in semidefinite programming-based relaxations of problems in areas such as combinatorial optimization (e.g.",
"the max-cut problem [14]) and statistical mechanics (e.g.",
"the $k$ -vector spin glass problem [2]).",
"In addition, the problem of projecting onto the set of (possibly low-rank) correlation matrices has enjoyed considerable interest in mathematical finance and numerical analysis in recent years [16].",
"In each of these applications the structure of the set of low-rank correlation matrices, i.e.",
"the facial structure of this convex body, plays an important role.",
"Understanding the faces of the elliptope turns out to be related to the following ellipsoid fitting problem: given $n$ points in $\\mathbb {R}^k$ (with $n > k$ ), under what conditions on the points is there an ellipsoid centered at the origin that passes exactly through these points?",
"While there is considerable literature on many ellipsoid-related problems, we are not aware of any previous systematic investigation of this particular problem.",
"Direction of arrival estimation is a classical problem in signal processing where (block) diagonal and low-rank decomposition problems arise naturally.",
"In this section we briefly discuss some stylized models of the direction of arrival estimation problem that can be reduced to matrix decomposition problems of the type considered in this paper.",
"Suppose we have $n$ sensors at locations $(x_1,y_1),(x_2,y_2),\\ldots ,(x_n,y_n)\\in \\mathbb {R}^2$ that are passively `listening' for waves (electromagnetic or acoustic) at a known frequency from $r \\ll n$ sources in the far field (so that the waves are approximately plane waves when they reach the sensors).",
"The aim is to estimate the number of sources $r$ and their directions of arrival $\\theta = (\\theta _1,\\theta _2,\\ldots ,\\theta _r)$ given sensor measurements and knowledge of the sensor locations (see Figure REF ).",
"Figure: Plane waves from directions θ 1 \\theta _1 and θ 2 \\theta _2 arriving at anarray of sensors equally spaced on a circle (a uniform circular array).A standard mathematical model for this problem (see [18] for a derivation) is to model the vector of sensor measurements $z(t)\\in n$ at time $t$ as $z(t) = A(\\theta )s(t) + n(t)$ where $s(t)\\in r$ is the vector of baseband signal waveforms from the sources, $n(t)\\in n$ is the vector of sensor measurement noise, and $A(\\theta )$ is the $n\\times r$ matrix with complex entries $[A(\\theta )]_{ij} = e^{-k\\sqrt{-1}(x_i\\cos (\\theta _j)+y_i\\sin (\\theta _j))}$ , with $k$ a positive constant related to the frequency of the waves being sensed.",
"The column space of $A(\\theta )$ contains all the information about the directions of arrival $\\theta $ .",
"As such, subspace-based approaches to direction of arrival estimation aim to estimate the column space of $A(\\theta )$ (from which a number of standard techniques can be employed to estimate $\\theta $ ).",
"Typically $s(t)$ and $n(t)$ are modeled as zero-mean stationary white Gaussian processes with covariances $\\mathbb {E}[s(t)s(t)^H] = P$ and $\\mathbb {E}[n(t)n(t)^H] = Q$ respectively (where $A^H$ denotes the Hermitian transpose of $A$ and $\\mathbb {E}[\\cdot ]$ the expectation).",
"In the simplest setting, $s(t)$ and $n(t)$ are assumed to be uncorrelated so that the covariance of the sensor measurements at any time is $ \\Sigma = A(\\theta )PA(\\theta )^H + Q.$ The first term is Hermitian positive semidefinite with rank $r$ , i.e.",
"the number of sources.",
"Under the assumption that spatially well-separated sensors (such as in a sensor network) have uncorrelated measurement noise $Q$ is diagonal.",
"In this case the covariance $\\Sigma $ of the sensor measurements decomposes as a sum of a positive semidefinite matrix of rank $r\\ll n$ and a diagonal matrix.",
"Given an approximation of $\\Sigma $ (e.g.",
"a sample covariance) approximately performing this diagonal and low-rank matrix decomposition allows the estimation of the column space of $A(\\theta )$ and in turn the directions of arrival.",
"A variation on this problem occurs if there are multiple sensors at each location, sensing, for example, waves at different frequencies.",
"Again under the assumption that well-separated sensors have uncorrelated measurement noise, and sensors at the same location have correlated measurement noise, the sensor noise covariance matrix $Q$ would be block-diagonal.",
"As such the covariance of all of the sensor measurements would decompose as the sum of a low-rank matrix (with rank equal to the total number of sources over all measured frequencies) and a block-diagonal matrix.",
"A block-diagonal and low-rank decomposition problem also arises if the second-order statistics of the noise have certain symmetries.",
"This might occur in cases where the sensors themselves are arranged in a symmetric way (such as in the uniform circular array shown in Figure REF ).",
"In this case there is a unitary matrix $T$ (depending only on the symmetry group of the array) such that $TQT^H$ is block-diagonal [25].",
"Then the covariance of the sensor measurements, when written in coordinates with respect to $T$ , is $T\\Sigma T^H = TA(\\theta )PA(\\theta )^HT^H + TQT^H$ which has a decomposition as the sum of a block diagonal matrix and a rank $r$ Hermitian positive semidefinite matrix (as conjugation by $T$ does not change the rank of this term).",
"Note that the matrix decomposition problems discussed in this section involve Hermitian matrices with complex entries, rather than the symmetric matrices with real entries considered elsewhere in this paper.",
"It is straightforward to generalize the main problems and results throughout the paper to the complex setting.",
"We introduce and make explicit the links between the analysis of MTFA, the facial structure of the elliptope, and the ellipsoid fitting problem, showing that these problems are, in a precise sense, equivalent (see Proposition ).",
"As such, we relate a basic problem in statistical modeling (tractable diagonal and low-rank matrix decompositions), a basic problem in convex algebraic geometry (understanding the facial structure of perhaps the simplest of spectrahedra), and a basic geometric problem."
],
[
"A sufficient condition for the three problems",
"The main result of the paper is to establish a new, simple, sufficient condition on a subspace $\\mathcal {U}$ of $\\mathbb {R}^n$ that ensures that MTFA correctly decomposes matrices of the form $D^\\star + L^\\star $ where $\\mathcal {U}$ is the column space of $L^\\star $ .",
"The condition is stated in terms of a measure of coherence of a subspace (made precise in Definition ).",
"Informally, the coherence of a subspace is a real number between zero and one that measures how close the subspace is to containing any of the elementary unit vectors.",
"This result can be translated into new results for the other two problems under consideration based on the relationship between the analysis of MTFA, the faces of the elliptope, and ellipsoid fitting."
],
[
"Block-diagonal and low-rank decompositions",
"In Section we turn our attention to the block-diagonal and low-rank decomposition problem, showing how our results generalize to that setting.",
"Our arguments combine our results for the diagonal and low-rank decomposition case with an understanding of the symmetries of the block-diagonal and low-rank decomposition problem.",
"The remainder of the paper is organized as follows.",
"We describe notation, give some background on semidefinite programming, and provide precise problem statements in Section .",
"In Section we present our first contribution by establishing relationships between the success of MTFA, the faces of the elliptope, and ellipsoid fitting.",
"We then illustrate these connections by noting the equivalence of a known result about the faces of the elliptope, and a known result about MTFA, and translating these into the context of ellipsoid fitting.",
"Section is focused on establishing and interpreting our main result: a sufficient condition for the three problems based on a coherence inequality.",
"Finally in Section we generalize our results to the analogous tractable block-diagonal and low-rank decomposition problem."
],
[
"Notation",
"If $x,y\\in \\mathbb {R}^n$ we denote by $\\langle x,y\\rangle = \\sum _{i=1}^{n} x_iy_i$ the standard Euclidean inner product and by $\\Vert x\\Vert _2 = \\langle x,x\\rangle ^{1/2}$ the corresponding Euclidean norm.",
"We write $x \\ge 0$ and $x > 0$ to indicate that $x$ is entry-wise non-negative and strictly positive, respectively.",
"Correspondingly, if $X,Y\\in \\mathcal {S}^n$ , the set of $n \\times n$ symmetric matrices, then we denote by $\\langle X,Y\\rangle = \\operatornamewithlimits{\\text{tr}}(XY)$ the trace inner product and by $\\Vert X\\Vert _F =\\langle X,X\\rangle ^{1/2}$ the Frobenius norm.",
"We write $X \\succeq 0$ and $X \\succ 0$ to indicate that $X$ is positive semidefinite and strictly positive definite, respectively.",
"We write $\\mathcal {S}_+^n$ for the cone of $n\\times n$ positive semidefinite matrices.",
"The column space of a matrix $X$ is denoted $\\mathcal {R}(X)$ and the nullspace is denoted $\\mathcal {N}(X)$ .",
"If $X$ is an $n\\times n$ matrix then $(X) \\in \\mathbb {R}^n$ is the diagonal of $X$ .",
"If $x\\in \\mathbb {R}^n$ then $^*(x)\\in \\mathcal {S}^n$ is the diagonal matrix with $[^*(x)]_{ii} =x_i$ for $i=1,2,\\ldots ,n$ .",
"If $\\mathcal {U}$ is a subspace of $\\mathbb {R}^n$ then $P_{\\mathcal {U}}:\\mathbb {R}^n\\rightarrow \\mathbb {R}^n$ denotes the orthogonal projector onto $\\mathcal {U}$ , that is the self-adjoint linear map such that $\\mathcal {R}(P_{\\mathcal {U}}) = \\mathcal {U}$ , $P_{\\mathcal {U}}^2 = P_{\\mathcal {U}}$ and $\\operatornamewithlimits{\\text{tr}}(P_{\\mathcal {U}}) =\\dim (\\mathcal {U})$ .",
"We use the notation $e_i$ for the vector with a one in the $i$ th position and zeros elsewhere and the notation $\\mathbf {1}$ to denote the vector all entries of which are one.",
"We use the shorthand $[n]$ for the set $\\lbrace 1,2,\\ldots ,n\\rbrace $ .",
"The set of $n\\times n$ correlation matrices, i.e.",
"positive semidefinite matrices with unit diagonal, is denoted $\\mathcal {E}_n$ .",
"For brevity we typically refer to $\\mathcal {E}_n$ as the elliptope, and the elements of $\\mathcal {E}_n$ as correlation matrices."
],
[
"Semidefinite programming",
"The term semidefinite programming [30] refers to convex optimization problems of the form $\\operatornamewithlimits{\\text{minimize}}_X\\; \\langle C,X \\rangle \\quad \\text{subject to}\\quad \\left\\lbrace \\begin{array}{rcl}\\mathcal {A}(X)\\!\\!\\!\\!& = &\\!\\!\\!\\!",
"b\\\\X\\!\\!\\!\\!",
"& \\succeq &\\!\\!\\!\\!",
"0\\end{array}\\right.$ where $X$ and $C$ are $ n\\times n$ symmetric matrices, $b\\in \\mathbb {R}^m$ , and $\\mathcal {A}: \\mathcal {S}^{n} \\rightarrow \\mathbb {R}^m$ is a linear map.",
"The dual semidefinite program is $\\operatornamewithlimits{\\text{maximize}}_{y,S}\\; \\langle b,y\\rangle \\quad \\text{subject to}\\quad \\left\\lbrace \\begin{array}{rcl}C - \\mathcal {A}^*(y) \\!\\!\\!\\!",
"& = & \\!\\!\\!\\!",
"S\\\\S \\!\\!\\!\\!",
"& \\succeq & \\!\\!\\!\\!",
"0\\end{array}\\right.$ where $\\mathcal {A}^*:\\mathbb {R}^m\\rightarrow \\mathcal {S}^n$ is the adjoint of $\\mathcal {A}$ .",
"General semidefinite programs can be solved in polynomial time using interior point methods [30].",
"While our focus in this paper is not on algorithms, we remark that for the structured semidefinite programs discussed in this paper, many different special-purpose methods have been devised.",
"The main result about semidefinite programming that we use is the following optimality condition (see [30] for example).",
"Suppose (REF ) and (REF ) are strictly feasible.",
"Then $X^\\star $ and $(y^\\star ,S^\\star )$ are optimal for the primal (REF ) and dual (REF ) respectively if and only if $X^\\star $ is primal feasible, $(y^\\star ,S^\\star )$ is dual feasible and $X^\\star S^\\star = 0$ ."
],
[
"Tractable diagonal and low-rank matrix decompositions",
"To decompose $X$ into a diagonal part and a positive semidefinite low-rank part, we may try to solve the following rank minimization problem $\\operatornamewithlimits{\\text{minimize}}_{D,L} \\;(L)\\quad \\text{subject to}\\quad \\left\\lbrace \\begin{array}{rcl}X \\!\\!\\!\\!",
"& = & \\!\\!\\!\\!",
"D + L\\\\L \\!\\!\\!\\!",
"& \\succeq & \\!\\!\\!\\!",
"0\\\\D \\!\\!\\!\\!",
"& & \\!\\!\\!\\!\\!\\!\\!\\!\\text{diagonal.",
"}\\end{array}\\right.$ Since the rank function is non-convex and non-differentiable, it is not clear how to solve this optimization problem directly.",
"One approach that has been successful for other rank minimization problems (for example those in [22], [23]), is to replace the rank function with the trace function in the objective.",
"This can be viewed as a convexification of the problem as the trace function is the convex envelope of the rank function when restricted to positive semidefinite matrices with spectral norm at most one.",
"Performing this convexification leads to the semidefinite program we refer to as minimum trace factor analysis (MTFA): $\\operatornamewithlimits{\\text{minimize}}_{D,L} \\;\\operatornamewithlimits{\\text{tr}}(L)\\quad \\text{subject to}\\quad \\left\\lbrace \\begin{array}{rcl}X \\!\\!\\!\\!",
"& = & \\!\\!\\!\\!",
"D + L\\\\L \\!\\!\\!\\!",
"& \\succeq & \\!\\!\\!\\!",
"0\\\\D \\!\\!\\!\\!",
"& & \\!\\!\\!\\!\\!\\!\\!\\!\\text{diagonal.",
"}\\end{array}\\right.$ It has been shown by Della Riccia and Shapiro [7] that if MTFA is feasible it has a unique optimal solution.",
"One central concern of this paper is to understand when the diagonal and low-rank decomposition of a matrix given by MTFA is `correct' in the following sense."
],
[
"Recovery problem I",
"Suppose $X$ is a matrix of the form $X = D^\\star + L^\\star $ where $D^\\star $ is diagonal and $L^\\star $ is positive semidefinite.",
"What conditions on $(D^\\star ,L^\\star )$ ensure that $(D^\\star ,L^\\star )$ is the unique optimum of MTFA with input $X$ ?",
"We establish in Section that whether $(D^\\star ,L^\\star )$ is the unique optimum of MTFA with input $X = D^\\star + L^\\star $ depends only on the column space of $L^\\star $ , motivating the following definition.",
"A subspace $\\mathcal {U}$ of $\\mathbb {R}^n$ is recoverable by MTFA if for every diagonal $D^\\star $ and every positive semidefinite $L^\\star $ with column space $\\mathcal {U}$ , $(D^\\star ,L^\\star )$ is the unique optimum of MTFA with input $X = D^\\star + L^\\star $ .",
"In these terms, we can restate the recovery problem succinctly as follows."
],
[
"Recovery problem II",
"Determine which subspaces of $\\mathbb {R}^n$ are recoverable by MTFA.",
"Much of the basic analysis of MTFA, including optimality conditions and relations between minimum rank and minimum trace factor analysis, was carried out in a sequence of papers by Shapiro [26], [27], [28] and Della Riccia and Shapiro [7].",
"More recently, Chandrasekaran et al.",
"[6] and Candès et al.",
"[4] considered convex optimization heuristics for decomposing a matrix as a sum of a sparse and low-rank matrix.",
"Since a diagonal matrix is certainly sparse, the analysis in [6] can be specialized to give fairly conservative sufficient conditions for the success of MTFA.",
"The diagonal and low-rank decomposition problem can also be interpreted as a low-rank matrix completion problem, where we are given all the entries of a low-rank matrix except the diagonal, and aim to correctly reconstruct the diagonal entries.",
"As such, this paper is closely related to the ideas and techniques used in the work of Candès and Recht [5] and a number of subsequent papers on this topic.",
"We would like to emphasize a key point of distinction between that line of work and the present paper.",
"The recent low-rank matrix completion literature largely focuses on determining the proportion of randomly selected entries of a low-rank matrix that need to be revealed to be able to reconstruct that low-rank matrix using a tractable algorithm.",
"The results of this paper, on the other hand, can be interpreted as attempting to understand which low-rank matrices can be reconstructed from a fixed and quite canonical pattern of revealed entries."
],
[
"Faces of the elliptope",
"The faces of the cone of $n\\times n$ positive semidefinite matrices are all of the form $\\mathcal {F}_{\\mathcal {U}} = \\lbrace X \\succeq 0: \\mathcal {N}(X) \\supseteq \\mathcal {U}\\rbrace $ where $\\mathcal {U}$ is a subspace of $\\mathbb {R}^n$ [19].",
"Conversely given any subspace $\\mathcal {U}$ of $\\mathbb {R}^n$ , $\\mathcal {F}_{\\mathcal {U}}$ is a face of $\\mathcal {S}_+^n$ .",
"As a consequence, the faces of $\\mathcal {E}_n$ are all of the form $\\mathcal {E}_n \\cap \\mathcal {F}_{\\mathcal {U}} = \\lbrace X \\succeq 0: \\mathcal {N}(X) \\supseteq \\mathcal {U},\\; (X) = \\mathbf {1}\\rbrace $ where $\\mathcal {U}$ is a subspace of $\\mathbb {R}^n$ [19].",
"It is not the case, however, that for every subspace $\\mathcal {U}$ of $\\mathbb {R}^n$ there is a correlation matrix with nullspace containing $\\mathcal {U}$ , motivating the following definition.",
"[[19]] A subspace $\\mathcal {U}$ of $\\mathbb {R}^n$ is realizable if there is an $n\\times n$ correlation matrix $Q$ such that $\\mathcal {N}(Q) \\supseteq \\mathcal {U}$ .",
"The problem of understanding the facial structure of the set of correlation matrices can be restated as follows."
],
[
"Facial structure problem",
"Determine which subspaces of $\\mathbb {R}^n$ are realizable.",
"Much is already known about the faces of the elliptope.",
"For example, all possible dimensions of faces as well as polyhedral faces, are known [20].",
"Characterizations of the realizable subspaces of $\\mathbb {R}^n$ of dimension 1, $n-2$ , and $n-1$ are given in [8] and implicitly in [19] and [20].",
"Nevertheless, little is known about which $k$ dimensional subspaces of $\\mathbb {R}^n$ are realizable for general $n$ and $k$ ."
],
[
"Ellipsoid fitting problem I",
"What conditions on a collection of $n$ points in $\\mathbb {R}^k$ ensure that there is a centered ellipsoid passing exactly through all those points?",
"Let us consider some basic properties of this problem."
],
[
"Number of points",
"If $n\\le k$ we can always fit an ellipsoid to the points.",
"Indeed if $V$ is the matrix with columns $v_1,v_2,\\ldots ,v_n$ then the image of the unit sphere in $\\mathbb {R}^n$ under $V$ is a centered ellipsoid passing through $v_1,v_2,\\ldots ,v_n$ .",
"If $n > \\binom{k+1}{2}$ and the points are `generic' then we cannot fit a centered ellipsoid to them.",
"This is because if we represent the ellipsoid by a symmetric $k\\times k$ matrix $M$ , the condition that it passes through the points (ignoring the positivity condition on $M$ ) means that $M$ must satisfy $n$ linearly independent equations."
],
[
"Invariances",
"If $T\\in GL(k)$ is an invertible linear map then there is an ellipsoid passing through $v_1,v_2,\\ldots ,v_n$ if and only if there is an ellipsoid passing through $Tv_1,Tv_2,\\ldots ,Tv_n$ .",
"This means that whether there is an ellipsoid passing through $n$ points in $\\mathbb {R}^k$ does not depend on the actual set of $n$ points, but on a subspace of $\\mathbb {R}^n$ related to the points.",
"We summarize this observation in the following lemma.",
"Suppose $V$ is a $k\\times n$ matrix with row space $\\mathcal {V}$ .",
"If there is a centered ellipsoid in $\\mathbb {R}^k$ passing through the columns of $V$ then there is a centered ellipsoid passing through the columns of any matrix $\\tilde{V}$ with row space $\\mathcal {V}$ .",
"Lemma REF asserts that whether it is possible to fit an ellipsoid to $v_1,v_2,\\ldots ,v_n$ depends only on the row space of the matrix with columns given by the $v_i$ , motivating the following definition.",
"A subspace $\\mathcal {V}$ of $\\mathbb {R}^n$ has the ellipsoid fitting property if there is a $k\\times n$ matrix $V$ with row space $\\mathcal {V}$ and a centered ellipsoid in $\\mathbb {R}^k$ that passes through each column of $V$ .",
"As such we can restate the ellipsoid fitting problem as follows."
],
[
"Ellipsoid fitting problem II",
"Determine which subspaces of $\\mathbb {R}^n$ have the ellipsoid fitting property."
],
[
"Relating ellipsoid fitting, diagonal and low-rank decompositions, and correlation matrices",
"In this section we show that the ellipsoid fitting problem, the recovery problem, and the facial structure problem are equivalent in the following sense.",
"Let $\\mathcal {U}$ be a subspace of $\\mathbb {R}^n$ .",
"Then the following are equivalent: $\\mathcal {U}$ is recoverable by MTFA.",
"$\\mathcal {U}$ is realizable.",
"$\\mathcal {U}^\\perp $ has the ellipsoid fitting property.",
"To see that REF implies REF , let $V$ be a $k\\times n$ matrix with nullspace $\\mathcal {U}$ and let $v_i$ denote the $i$ th column of $V$ .",
"If $\\mathcal {U}$ is realizable there is a correlation matrix $Y$ with nullspace containing $\\mathcal {U}$ .",
"Hence there is some $M\\succeq 0$ such that $Y = V^T M V$ and $v_i^TMv_i = 1$ for $i\\in [n]$ .",
"Since $V$ has nullspace $\\mathcal {U}$ , it has row space $\\mathcal {U}^\\perp $ .",
"Hence the subspace $\\mathcal {U}^\\perp $ has the ellipsoid fitting property.",
"By reversing the argument we see that the converse also holds.",
"The equivalence of REF and REF arises from semidefinite programming duality.",
"Following a slight reformulation, MTFA (REF ) can be expressed as $\\operatornamewithlimits{\\text{maximize}}_{d,L} \\; \\langle \\mathbf {1},d\\rangle \\quad \\text{subject to}\\quad \\left\\lbrace \\begin{array}{rcl}X\\!\\!\\!\\!",
"& = & \\!\\!\\!\\!^*(d) + L\\\\L \\!\\!\\!\\!",
"& \\succeq & \\!\\!\\!\\!0\\end{array}\\right.$ and its dual as $\\operatornamewithlimits{\\text{minimize}}_{Y} \\;\\langle X,Y\\rangle \\quad \\text{subject to}\\quad \\left\\lbrace \\begin{array}{rcl}(Y)\\!\\!\\!\\!",
"& = & \\!\\!\\!\\!\\mathbf {1}\\\\Y \\!\\!\\!\\!",
"& \\succeq & \\!\\!\\!\\!",
"0\\end{array}\\right.$ which is clearly just the optimization of the linear functional defined by $X$ over the elliptope.",
"We note that (REF ) is exactly in the standard dual form (REF ) for semidefinite programming and correspondingly that (REF ) is in the standard primal form (REF ) for semidefinite programming.",
"Suppose $\\mathcal {U}$ is recoverable by MTFA.",
"Fix a diagonal matrix $D^\\star $ and a positive semidefinite matrix $L^\\star $ with column space $\\mathcal {U}$ and let $X = D^\\star + L^\\star $ .",
"Since (REF ) and (REF ) are strictly feasible, by Theorem REF (optimality conditions for semidefinite programming), the pair $((D^\\star ),L^\\star )$ is an optimum of (REF ) if and only if there is some correlation matrix $Y^\\star $ such that $Y^\\star L^\\star = 0$ .",
"Since $\\mathcal {R}(L^\\star ) = \\mathcal {U}$ this implies that $\\mathcal {U}$ is realizable.",
"Conversely, if $\\mathcal {U}$ is realizable, there is some $Y^\\star $ such that $Y^\\star L^\\star = 0$ for every $L^\\star $ with column space $\\mathcal {U}$ , showing that $\\mathcal {U}$ is recoverable by MTFA.",
"Remark We note that in the proof of Proposition we established that the two versions of the recovery problem stated in Section REF are actually equivalent.",
"In particular, whether $(D^\\star ,L^\\star )$ is the optimum of MTFA with input $X = D^\\star + L^\\star $ depends only on the column space of $L^\\star $ .",
"Certificates of failure We can prove that a subspace $\\mathcal {U}$ is realizable by constructing a correlation matrix with nullspace containing $\\mathcal {U}$ .",
"We can prove that a subspace is not realizable by constructing a matrix that certifies this fact.",
"Geometrically, a subspace $\\mathcal {U}$ is realizable if and only if the subspace $\\mathcal {L}_{\\mathcal {U}} = \\lbrace X\\in \\mathcal {S}^n: \\mathcal {N}(X) \\supseteq \\mathcal {U}\\rbrace $ of symmetric matrices intersects with the elliptope.",
"So a certificate that $\\mathcal {U}$ is not realizable is a hyperplane in the space of symmetric matrices that strictly separates the elliptope from $\\mathcal {L}_{\\mathcal {U}}$ .",
"The following lemma describes the structure of these separating hyperplanes.",
"A subspace $\\mathcal {U}$ of $\\mathbb {R}^n$ is not realizable if and only if there is a diagonal matrix $D$ such that $\\operatornamewithlimits{\\text{tr}}(D) > 0$ and $v^TDv \\le 0$ for all $v\\in \\mathcal {U}^\\perp $ .",
"By Proposition , $\\mathcal {U}$ is not realizable if and only if $\\mathcal {U}^\\perp $ does not have the ellipsoid fitting property.",
"Let $V$ be a $k \\times n$ matrix with row space $\\mathcal {U}^\\perp $ .",
"Then $\\mathcal {U}^\\perp $ does not have the ellipsoid fitting property if and only if we cannot find an ellipsoid passing through the columns of $V$ , i.e.",
"the semidefinite program $\\operatornamewithlimits{\\text{minimize}}_{M} \\; \\langle 0,M\\rangle \\quad \\text{subject to}\\quad \\left\\lbrace \\begin{array}{rcl} (V^TMV)\\!\\!\\!\\!",
"& = & \\!\\!\\!\\!\\mathbf {1}\\\\M \\!\\!\\!\\!",
"& \\succeq & \\!\\!\\!\\!",
"0 \\end{array}\\right.$ is infeasible.",
"The semidefinite programming dual of (REF ) is $\\operatornamewithlimits{\\text{maximize}}_d \\; \\langle d,\\mathbf {1}\\rangle \\quad \\text{subject to}\\quad \\left\\lbrace \\begin{array}{rcl} V^*(d)V^T\\!\\!\\!\\!",
"& \\preceq & \\!\\!\\!\\!",
"0.\\end{array}\\right.$ Since (REF ) is clearly always feasible, by strong duality (which holds because both primal and dual problems are strictly feasible) (REF ) is infeasible if and only if (REF ) is unbounded.",
"This occurs if and only if there is some $d$ with $\\sum _{i\\in [n]} d_i > 0$ and yet $V^*(d)V^T \\preceq 0$ .",
"Then $D = ^*(d)$ has the properties in the statement of the lemma.",
"Exploiting connections: results for one dimensional subspaces In 1940, Ledermann [21] characterized the one dimensional subspaces that are recoverable by MTFA.",
"In 1990, Grone et al.",
"[15] gave a necessary condition for a subspace to be realizable.",
"In 1993, independently of Ledermann's work, Delorme and Poljak [8] showed that this condition is also sufficient for one dimensional subspaces.",
"Since we have established that a subspace is recoverable by MTFA if and only if it is realizable, Ledermann's result and Delorme and Poljak's results are equivalent.",
"In this section we translate these equivalent results into the context of the ellipsoid fitting problem, giving a geometric characterization of when it is possible to fit a centered ellipsoid to $k+1$ points in $\\mathbb {R}^k$ .",
"Delorme and Poljak state their result in terms of the following definition.",
"[[8]] A vector $u\\in \\mathbb {R}^n$ is balanced if, for all $i\\in [n]$ , $|u_i| \\le \\sum _{j\\ne i} |u_j|.$ If the inequality is strict we say that $u$ is strictly balanced.",
"In the following, the necessary condition is due to Grone et al.",
"[15] and the sufficient condition is due to Ledermann [21] (in the context of the analysis of MTFA) and Delorme and Poljak [8] (in the context of the facial structure of the elliptope).",
"We state the result only in terms of realizability of a subspace.",
"If a subspace $\\mathcal {U}$ of $\\mathbb {R}^n$ is realizable then every $u\\in \\mathcal {U}$ is balanced.",
"If $\\mathcal {U} = \\operatornamewithlimits{\\text{span}}\\lbrace u\\rbrace $ is one-dimensional then $\\mathcal {U}$ is realizable if and only if $u$ is balanced.",
"The balance condition has a particularly natural geometric interpretation in the ellipsoid fitting setting (Lemma REF , below).",
"The proof is a fairly straightforward application of linear programming duality, which we defer to Appendix .",
"Suppose $V$ is any $k \\times n$ matrix with $\\mathcal {N}(V) = \\mathcal {U}$ .",
"Denote the columns of $V$ by $v_1,v_2,\\ldots ,v_n \\in \\mathbb {R}^{k}$ .",
"Then every $u\\in \\mathcal {U}$ is balanced if and only if for each $i\\in [n]$ , $v_i$ lies on the boundary of the convex hull of $\\pm v_1,\\pm v_2,\\ldots ,\\pm v_n$ .",
"By combining Theorem REF with Lemma REF , we are in a position to interpret Theorem REF purely in terms of ellipsoid fitting.",
"If there is an ellipsoid passing through $\\pm v_1,\\pm v_2,\\ldots ,\\pm v_n\\in \\mathbb {R}^k$ then $\\pm v_1,\\pm v_2,\\ldots ,\\pm v_n$ lie on the boundary of their convex hull.",
"If, in addition, $k=n-1$ the converse also holds.",
"We note that $\\pm v_1,\\pm v_2,\\ldots ,\\pm v_n$ lie on the boundary of their convex hull if and only if there exists some convex set with boundary containing $\\pm v_1,\\pm v_2,\\ldots ,\\pm v_n$ .",
"In this geometric setting, it is clear that this is a necessary condition to be able to find a centered ellipsoid passing through the points, but not so obvious that it is sufficient if $k=n-1$ .",
"A sufficient condition for the three problems In this section we establish a new sufficient condition for a subspace $\\mathcal {U}$ of $\\mathbb {R}^n$ to be realizable and consequently a sufficient condition for $\\mathcal {U}$ to be recoverable by MTFA and $\\mathcal {U}^\\perp $ to have the ellipsoid fitting property.",
"Our condition is based on a simple property of a subspace known as coherence.",
"Given a subspace $\\mathcal {U}$ of $\\mathbb {R}^n$ , the coherence of $\\mathcal {U}$ is a measure of how close the subspace is to containing any of the elementary unit vectors.",
"This notion was introduced (with a different scaling) by Candès and Recht in their work on low-rank matrix completion [5], although related quantities have played an important role in the analysis of sparse reconstruction problems since the work of Donoho and Huo [10].",
"If $\\mathcal {U}$ is a subspace of $\\mathbb {R}^n$ then the coherence of $\\mathcal {U}$ is $ \\mu (\\mathcal {U}) = \\max _{i\\in [n]}\\Vert P_\\mathcal {U}e_i\\Vert _2^2.$ A basic property of coherence is that it satisfies the inequality $\\frac{\\dim (\\mathcal {U})}{n} \\le \\mu (\\mathcal {U}) \\le 1$ for any subspace $\\mathcal {U}$ of $\\mathbb {R}^n$ [5].",
"This inequality, together with the definition of coherence, provides useful intuition about the properties of subspaces with low coherence, that is incoherence.",
"Any subspace with low coherence is necessarily of low dimension and far from containing any of the elementary unit vectors $e_i$ .",
"As such, any symmetric matrix with incoherent row/column spaces is necessarily of low-rank and quite different from being a diagonal matrix.",
"Coherence-threshold-type sufficient conditions In this section we focus on finding the largest possible $\\alpha $ such that $ \\text{$\\mu (\\mathcal {U}) < \\alpha \\Rightarrow \\mathcal {U}$ is realizable,}$ that is finding the best possible coherence-threshold-type sufficient condition for a subspace to be realizable.",
"Such conditions are of particular interest because the dependence they have on the ambient dimension and the dimension of the subspace is only the mild dependence implied by (REF ).",
"In contrast, existing results (e.g.",
"[8], [20], [19]) about realizability of subspaces hold only for specific combinations of the ambient dimension and the dimension of the subspace.",
"The following theorem, our main result, gives a sufficient condition for realizability based on a coherence-threshold condition.",
"Furthermore, it establishes that this is the best possible coherence-threshold-type sufficient condition.",
"If $\\mathcal {U}$ is a subspace of $\\mathbb {R}^n$ and $\\mu (\\mathcal {U}) < 1/2$ then $\\mathcal {U}$ is realizable.",
"On the other hand, given any $\\alpha > 1/2$ , there is a subspace $\\mathcal {U}$ with $\\mu (\\mathcal {U}) = \\alpha $ that is not realizable.",
"We give the main idea of the proof, deferring some details to Appendix .",
"Instead of proving that there is some $Y\\in \\mathcal {F}_{\\mathcal {U}} = \\lbrace Y \\succeq 0: \\mathcal {N}(Y) \\supseteq \\mathcal {U}\\rbrace $ such that $Y_{ii} = 1$ for $i\\in [n]$ , it suffices to choose a convex cone $\\mathcal {K}$ that is an inner approximation to $\\mathcal {F}_{\\mathcal {U}}$ and establish that there is some $Y\\in \\mathcal {K}$ such that $Y_{ii} = 1$ for $i\\in [n]$ .",
"One natural choice is to take $\\mathcal {K} = \\lbrace P_{\\mathcal {U}^\\perp }^*(\\lambda )P_{\\mathcal {U}^\\perp }: \\lambda \\ge 0\\rbrace $ , which is clearly contained in $\\mathcal {F}_{\\mathcal {U}}$ .",
"Note that there is some $Y \\in \\mathcal {K}$ such that $Y_{ii} = 1$ for all $i\\in [n]$ if and only if there is $\\lambda \\ge 0$ such that $\\left(P_{\\mathcal {U}^\\perp }^*(\\lambda )P_{\\mathcal {U}^\\perp }\\right) = \\mathbf {1}.$ The rest of the proof of the sufficient condition involves showing that if $\\mu (\\mathcal {U}) < 1/2$ then such a non-negative $\\lambda $ exists.",
"We establish this in Lemma REF .",
"Now let us construct, for any $\\alpha > 1/2$ , a subspace with coherence $\\alpha $ that is not realizable.",
"Let $\\mathcal {U}$ to be the subspace of $\\mathbb {R}^2$ spanned by $u = (\\sqrt{\\alpha },\\sqrt{1-\\alpha })$ .",
"Then $\\mu (\\mathcal {U}) = \\max \\lbrace \\alpha ,1-\\alpha \\rbrace = \\alpha $ and yet by Theorem REF , $\\mathcal {U}$ is not realizable because $u$ is not balanced.",
"Remarks Theorem REF illustrates both the power and limitations of coherence-threshold-type conditions.",
"On the one hand, since coherence is quite a coarse property of a subspace, the result applies to `many' subspaces (see Proposition REF in Section REF ).",
"On the other hand, since coherence has very mild dimension dependence, the power of coherence-threshold-type conditions is limited to their specialization to low-dimensional situations, such as one dimensional subspaces of $\\mathbb {R}^2$ .",
"Interpretations of Theorem REF We now establish two corollaries of our coherence-threshold-type sufficient condition for realizability.",
"These corollaries can be thought of as re-interpretations of the coherence inequality $\\mu (\\mathcal {U}) < 1/2$ in terms of other natural quantities.",
"An ellipsoid-fitting interpretation With the aid of Proposition we reinterpret our coherence-threshold-type sufficient condition as a sufficient condition on a set of points in $\\mathbb {R}^k$ that ensures there is a centered ellipsoid passing through them.",
"The condition involves `sandwiching' the points between two ellipsoids (that depend on the points).",
"Indeed, given $v_1,v_2,\\ldots ,v_n\\in \\mathbb {R}^k$ and $0 < \\beta < 1$ we define the ellipsoid $ \\mathcal {E}_{\\beta }(v_1,\\ldots ,v_n) = \\text{$\\lbrace x\\in \\mathbb {R}^k: x^T(\\textstyle \\sum _{j=1}^n v_jv_j^T)^{-1}x \\le \\beta \\rbrace $}.$ Given $0< \\beta < 1$ the points $v_1,v_2,\\ldots ,v_n$ satisfy the $\\beta $ -sandwich condition if $\\lbrace v_1,v_2,\\ldots ,v_n\\rbrace \\subset \\mathcal {E}_1(v_1,\\ldots ,v_n) \\setminus \\mathcal {E}_\\beta (v_1,\\ldots ,v_n).$ The intuition behind this definition (illustrated in Figure REF ) is that if the points satisfy the $\\beta $ -sandwich condition for $\\beta $ close to one, then they are confined to a thin elliptical shell that is adapted to their position.",
"One might expect that it is `easier' to fit an ellipsoid to points that are confined in this way.",
"Indeed this is the case.",
"If $v_1,v_2,\\ldots ,v_n\\in \\mathbb {R}^k$ satisfy the $1/2$ -sandwich condition then then there is a centered ellipsoid passing through $v_1,v_2,\\ldots ,v_n$ .",
"Let $V$ be the $k\\times n$ matrix with columns given by the $v_i$ , and let $\\mathcal {U}$ be the nullspace of $V$ .",
"Then the orthogonal projection onto the row space of $V$ is $P_{\\mathcal {U}^\\perp }$ , and can be written as $ P_{\\mathcal {U}^\\perp } = V^T(VV^T)^{-1}V.$ Our assumption that the points satisfy the $1/2$ -sandwich condition is equivalent to assuming that $1/2 < [P_{\\mathcal {U}^\\perp }]_{ii} \\le 1$ for all $i\\in [n]$ or alternatively that $\\mu (\\mathcal {U}) = \\max _{i\\in [n]} [P_{\\mathcal {U}}]_{ii} = 1-\\min _{i\\in [n]}[P_{\\mathcal {U}^\\perp }]_{ii} < 1/2.$ From Theorem REF we know that $\\mu (\\mathcal {U}) < 1/2$ implies that $\\mathcal {U}$ is realizable.",
"Invoking Proposition we then conclude that there is a centered ellipsoid passing through $v_1,v_2,\\ldots ,v_n$ .",
"Figure: The ellipsoidsshown are ℰ=ℰ 1 (v 1 ,v 2 ,v 3 )\\mathcal {E} = \\mathcal {E}_1(v_1,v_2,v_3) and ℰ ' =ℰ 1/2 (v 1 ,v 2 ,v 3 )\\mathcal {E}^{\\prime } = \\mathcal {E}_{1/2}(v_1,v_2,v_3).There is an ellipsoid passing through v 1 ,v 2 v_1,v_2 and v 3 v_3 because the points aresandwiched between ℰ\\mathcal {E} and ℰ ' \\mathcal {E}^{\\prime }.",
"A balance interpretation In Section REF we saw that if a subspace $\\mathcal {U}$ is realizable, every $u\\in \\mathcal {U}$ is balanced.",
"The sufficient condition of Theorem REF can be expressed in terms of a balance condition on the element-wise square of the elements of a subspace.",
"(In what follows $u\\circ u$ denotes the element-wise square of a vector in $\\mathbb {R}^n$ .)",
"Suppose $\\mathcal {U}$ is a subspace of $\\mathbb {R}^n$ .",
"If $u\\circ u$ is strictly balanced for every $u\\in \\mathcal {U}$ then $\\mathcal {U}$ is realizable.",
"It suffices to show that if for every $u\\in \\mathcal {U}$ , $u\\circ u$ is strictly balanced, then $\\mu (\\mathcal {U}) <1/2$ (although we could reverse the argument to establish the equivalence of these conditions).",
"If $u\\circ u$ is strictly balanced for all $u\\in \\mathcal {U}$ then for all $i\\in [n]$ and all $u\\in \\mathcal {U}$ $2\\langle e_i,u\\rangle ^2 < \\sum _{j=1}^{n}\\langle e_j,u\\rangle ^2 = \\Vert u\\Vert _2^2.$ Since $\\Vert P_{\\mathcal {U}}e_i\\Vert _2 = \\max _{u\\in \\mathcal {U}\\setminus \\lbrace 0\\rbrace } \\langle e_i,u\\rangle /\\Vert u\\Vert _2$ , it follows from (REF ) that $2\\Vert P_{\\mathcal {U}}e_i\\Vert _2^2 < 1$ .",
"Since this holds for all $i\\in [n]$ it follows that $\\mu (\\mathcal {U}) < 1/2$ .",
"Remark Suppose $\\mathcal {U} = \\operatornamewithlimits{\\text{span}}\\lbrace u\\rbrace $ is a one-dimensional subspace of $\\mathbb {R}^n$ .",
"We have just established that if $u\\circ u$ is strictly balanced then $\\mathcal {U}$ is realizable and so (by Theorem REF ) $u$ must be balanced.",
"We note that it is straightforward to establish directly that if $u\\circ u$ is balanced then $u$ is balanced by using the definition of balance and the fact that $\\Vert x\\Vert _1 \\ge \\Vert x\\Vert _2$ for any $x\\in \\mathbb {R}^n$ .",
"Examples To gain more intuition for what Theorem REF means, we consider its implications in two particular cases.",
"First, we compare the characterization of when it is possible to fit an ellipsoid to $k+1$ points in $\\mathbb {R}^k$ (Corollary REF ) with the specialization of our sufficient condition to this case (Corollary REF ).",
"This comparison provides some insight into how conservative our sufficient condition is.",
"Second, we investigate the coherence properties of suitably `random' subspaces.",
"This provides intuition about whether or not $\\mu (\\mathcal {U}) < 1/2$ is a very restrictive condition.",
"In particular, we establish that `most' subspaces of $\\mathbb {R}^n$ with dimension bounded above by $(1/2 - \\epsilon )n$ are realizable.",
"Fitting an ellipsoid to $k+1$ points in $\\mathbb {R}^k$ Recall that Ledermann and Delorme and Poljak's result, interpreted in terms of ellipsoid fitting, tells us that we can fit an ellipsoid to $k+1$ points $v_1,\\ldots ,v_{k+1}\\in \\mathbb {R}^k$ if and only if those points are on the boundary of the convex hull of $\\lbrace \\pm v_1,\\ldots ,\\pm v_{k+1}\\rbrace $ (see Corollary REF ).",
"We now compare this characterization with the $1/2$ -sandwich condition, which is sufficient by Corollary REF .",
"Without loss of generality we assume that $k$ of the points are $e_1,\\ldots ,e_k$ , the standard basis vectors, and compare the conditions by considering the set of locations of the $k+1$ st point $v\\in \\mathbb {R}^k$ for which we can fit an ellipsoid through all $k+1$ points.",
"Corollary REF gives a characterization of this region as $ R = \\lbrace v\\in \\mathbb {R}^k: \\sum _{j=1}^k|v_j| \\ge 1, \\; |v_i| - \\sum _{j\\ne i}|v_j| \\le 1\\quad \\text{for $i\\in [k]$}\\rbrace $ which is shown in Figure REF in the case $k=2$ .",
"The set of $v$ such that $v,e_1,\\ldots ,e_n$ satisfy the $1/2$ -sandwich condition can be written as $R^{\\prime } & = \\lbrace v\\in \\mathbb {R}^k: v^T(I + vv^T)^{-1}v > 1/2,\\; e_i^T(I+vv^T)^{-1}e_i > 1/2 \\quad \\text{for $i\\in [k]$}\\rbrace \\\\& = \\lbrace v\\in \\mathbb {R}^k: \\sum _{j=1}^{k}v_j^2 > 1,\\; v_i^2 - \\sum _{j\\ne i}v_j^2 < 1 \\quad \\text{for $i\\in [k]$}\\rbrace $ which is shown in Figure REF .",
"It is clear that $R^{\\prime }\\subseteq R$ .",
"Figure: Comparing our sufficient condition for ellipsoid fitting (Corollary ) withthe characterization (Corollary ) in the case of fitting an ellipsoid to k+1k+1 points in ℝ k \\mathbb {R}^k.",
"Realizability of random subspaces Suppose $\\mathcal {U}$ is a subspace generated by taking the column space of an $n\\times r$ matrix with i.i.d.",
"standard Gaussian entries.",
"For what values of $r$ and $n$ does such a subspace have $\\mu (\\mathcal {U}) < 1/2$ with high probability, i.e.",
"satisfy our sufficient condition for being realizable?",
"The following result essentially shows that for large $n$ , `most' subspaces of dimension at most $(1/2-\\epsilon )n$ are realizable.",
"This suggests that MTFA is a very good heuristic for diagonal and low-rank decomposition problems in the high-dimensional setting.",
"Indeed `most' subspaces of dimension up to one half the ambient dimension—hardly just low-dimensional subspaces—are recoverable by MTFA.",
"Let $0 < \\epsilon < 1/2$ be a constant and suppose $n > 6/(\\epsilon ^2-2\\epsilon ^3)$ .",
"There are positive constants $\\bar{c}$ , $\\tilde{c}$ , (depending only on $\\epsilon $ ) such that if $\\mathcal {U}$ is a random $(1/2 - \\epsilon )n$ dimensional subspace of $\\mathbb {R}^n$ then $ \\Pr [\\text{$\\mathcal {U}$ is realizable}] \\ge 1-\\bar{c}\\sqrt{n}e^{-\\tilde{c} n}.$ We provide a proof of this result in Appendix .",
"The main idea is that the coherence of a random $r$ dimensional subspace of $\\mathbb {R}^n$ is the maximum of $n$ random variables that concentrate around their mean of $r/n$ for large $n$ .",
"To illustrate the result, we consider the case where $\\epsilon = 1/4$ and $n>192$ .",
"Then (by examining the proof in Appendix ) we see that we can take $\\tilde{c} = 1/24$ and $\\bar{c} = 24/\\sqrt{3\\pi } \\approx 7.8$ .",
"Hence if $n > 192$ and $\\mathcal {U}$ is a random $n/4$ dimensional subspace of $\\mathbb {R}^n$ we have that $ \\Pr [\\text{$\\mathcal {U}$ is realizable}] \\ge 1 - 7.8\\sqrt{n}e^{-n/24}.$ Tractable block diagonal and low-rank decompositions and related problems In this section we generalize our results to the analogue of MTFA for block-diagonal and low-rank decompositions.",
"Mimicking our earlier development, we relate the analysis of this variant of MTFA to the facial structure of a variant of the elliptope and a generalization of the ellipsoid fitting problem.",
"The key point is that these problems all possess additional symmetries that, once taken into account, essentially allow us to reduce our analysis to cases already considered in Sections and .",
"Throughout this section, let $\\mathcal {P}$ be a fixed partition of $\\lbrace 1,2,\\ldots ,n\\rbrace $ .",
"We say a matrix is $\\mathcal {P}$ -block-diagonal if it is zero except for the principal submatrices indexed by the elements of $\\mathcal {P}$ .",
"We denote by $\\text{blkdiag}_{\\mathcal {P}}$ the map that takes an $n\\times n$ matrix and maps it to the principal submatrices indexed by $\\mathcal {P}$ .",
"Its adjoint, denoted $\\text{blkdiag}_{\\mathcal {P}}^*$ , takes a tuple of symmetric matrices $(X_I)_{I\\in \\mathcal {P}}$ and produces an $n\\times n$ matrix that is $\\mathcal {P}$ -block diagonal with blocks given by the $X_{I}$ .",
"We now describe the analogues of MTFA, ellipsoid fitting, and the problem of determining the facial structure of the elliptope.",
"Block minimum trace factor analysis If $X = B^\\star + L^\\star $ where $B^\\star $ is $\\mathcal {P}$ -block-diagonal and $L^\\star \\succeq 0$ is low rank, the obvious analogue of MTFA is the semidefinite program $\\operatornamewithlimits{\\text{minimize}}_{B,L} \\; \\operatornamewithlimits{\\text{tr}}(L) \\quad \\text{subject to}\\quad \\left\\lbrace \\begin{array}{rcl} X \\!\\!\\!\\!",
"& = & \\!\\!\\!\\!",
"B + L\\\\L \\!\\!\\!\\!",
"& \\succeq \\!\\!\\!\\!",
"& 0\\\\B\\!\\!\\!\\!",
"& \\text{is} &\\!\\!\\!\\!",
"\\text{$\\mathcal {P}$-block-diagonal}\\end{array}\\right.$ which we call block minimum trace factor analysis (BMTFA).",
"A subspace $\\mathcal {U}$ of $\\mathbb {R}^n$ is recoverable by BMTFA if for every $B^\\star $ that is $\\mathcal {P}$ -block-diagonal and every positive semidefinite $L^\\star $ with column space $\\mathcal {U}$ , $(B^\\star ,L^\\star )$ is the unique optimum of BMTFA with input $X = B^\\star + L^\\star $ .",
"Faces of the $\\mathcal {P}$ -elliptope Just as MTFA is related to the facial structure of the elliptope, BMTFA is related to the facial structure of the spectrahedron $ \\mathcal {E}_{\\mathcal {P}} = \\lbrace Y \\succeq 0: \\text{blkdiag}_{\\mathcal {P}}(Y) = (I,I,\\ldots ,I)\\rbrace .$ We refer to $\\mathcal {E}_\\mathcal {P}$ as the $\\mathcal {P}$ -elliptope.",
"We extend the definition of a realizable subspace to this context.",
"A subspace $\\mathcal {U}$ of $\\mathbb {R}^n$ is $\\mathcal {P}$ -realizable if there is some $Y\\in \\mathcal {E}_{\\mathcal {P}}$ such that $\\mathcal {N}(Y) \\supseteq \\mathcal {U}$ .",
"Generalized ellipsoid fitting To describe the $\\mathcal {P}$ -ellipsoid fitting problem we first introduce some convenient notation.",
"If $I\\subset [n]$ we write $S^{I} = \\lbrace x\\in \\mathbb {R}^n: \\Vert x\\Vert _2 = 1,\\;\\text{$x_j = 0$ if $j\\notin I$}\\rbrace $ for the intersection of the unit sphere with the coordinate subspace indexed by $I$ .",
"Suppose $v_1,v_2,\\ldots ,v_n\\in \\mathbb {R}^k$ is a collection of points and $V$ is the $k\\times n$ matrix with columns given by the $v_i$ .",
"Noting that $S^{\\lbrace i\\rbrace } = \\lbrace -e_i,e_i\\rbrace $ , and thinking of $V$ as a linear map from $\\mathbb {R}^n$ to $\\mathbb {R}^k$ , we see that the ellipsoid fitting problem is to find an ellipsoid in $\\mathbb {R}^k$ with boundary containing $\\cup _{i\\in [n]} V(S^{\\lbrace i\\rbrace })$ , i.e.",
"the collection of points $\\pm v_1,\\ldots ,\\pm v_n$ .",
"The $\\mathcal {P}$ -ellipsoid fitting problem is then to find an ellipsoid in $\\mathbb {R}^k$ with boundary containing $\\cup _{I\\in \\mathcal {P}} V(S^{I})$ , i.e.",
"the collection of ellipsoids $V(S^{I})$ .",
"The generalization of the ellipsoid fitting property of a subspace is as follows.",
"A subspace $\\mathcal {V}$ of $\\mathbb {R}^n$ has the $\\mathcal {P}$ -ellipsoid fitting property if there is a $k\\times n$ matrix $V$ with row space $\\mathcal {V}$ such that there is a centered ellipsoid in $\\mathbb {R}^k$ with boundary containing $\\cup _{I\\in \\mathcal {P}} V(S^{I})$ .",
"Relating the generalized problems The facial structure of the $\\mathcal {P}$ -elliptope, BMTFA, and the $\\mathcal {P}$ -ellipsoid fitting problem are related by the following result, the proof of which is omitted as it is almost identical to that of Proposition .",
"Let $\\mathcal {U}$ be a subspace of $\\mathbb {R}^n$ .",
"Then the following are equivalent: $\\mathcal {U}$ is recoverable by BMTFA.",
"$\\mathcal {U}$ is $\\mathcal {P}$ -realizable.",
"$\\mathcal {U}^\\perp $ has the $\\mathcal {P}$ -ellipsoid fitting property.",
"The following lemma is the analogue of Lemma REF .",
"It describes certificates that a subspace $\\mathcal {U}$ is not $\\mathcal {P}$ -realizable.",
"Again the proof is almost identical to that of Lemma REF so we omit it.",
"A subspace $\\mathcal {U}$ of $\\mathbb {R}^n$ is not $\\mathcal {P}$ -realizable if and only if there is a $\\mathcal {P}$ -block-diagonal matrix $B$ such that $\\operatornamewithlimits{\\text{tr}}(B) > 0$ and $v^TBv \\le 0$ for all $v\\in \\mathcal {U}^\\perp $ .",
"For the sake of brevity, in what follows we only discuss the problem of whether $\\mathcal {U}$ is $\\mathcal {P}$ -realizable without explicitly translating the results into the context of the other two problems.",
"Symmetries of the $\\mathcal {P}$ -elliptope We now consider the symmetries of the $\\mathcal {P}$ -elliptope.",
"Our motivation for doing so is that it allows us to partition subspaces into classes for which either all elements are $\\mathcal {P}$ -realizable or none of the elements are $\\mathcal {P}$ -realizable.",
"It is clear that the $\\mathcal {P}$ -elliptope is invariant under conjugation by $\\mathcal {P}$ -block-diagonal orthogonal matrices.",
"Let $G_{\\mathcal {P}}$ denote this subgroup of the group of $n\\times n$ orthogonal matrices.",
"There is a natural action of $G_{\\mathcal {P}}$ on subspaces of $\\mathbb {R}^n$ defined as follows.",
"If $P\\in G_{\\mathcal {P}}$ and $\\mathcal {U}$ is a subspace of $\\mathbb {R}^n$ then $P\\cdot \\mathcal {U}$ is the image of the subspace $\\mathcal {U}$ under the map $P$ .",
"(It is straightforward to check that this is a well defined group action.)",
"If there exists some $P\\in G_{\\mathcal {P}}$ such that $P\\cdot \\mathcal {U} = \\mathcal {U}^{\\prime }$ then we write $\\mathcal {U} \\sim \\mathcal {U}^{\\prime }$ and say that $\\mathcal {U}$ and $\\mathcal {U}^{\\prime }$ are equivalent.",
"We care about this equivalence relation on subspaces because the property of being $\\mathcal {P}$ -realizable is really a property of the corresponding equivalence classes.",
"Suppose $\\mathcal {U}$ and $\\mathcal {U}^{\\prime }$ are subspaces of $\\mathbb {R}^n$ .",
"If $\\mathcal {U}\\sim \\mathcal {U}^{\\prime }$ then $\\mathcal {U}$ is $\\mathcal {P}$ -realizable if and only if $\\mathcal {U}^{\\prime }$ is $\\mathcal {P}$ -realizable.",
"If $\\mathcal {U}$ is $\\mathcal {P}$ -realizable there is $Y\\in \\mathcal {E}_{\\mathcal {P}}$ such that $Yu = 0$ for all $u\\in \\mathcal {U}$ .",
"Suppose $\\mathcal {U}^{\\prime } = P\\cdot \\mathcal {U}$ for some $P\\in G_{\\mathcal {P}}$ and let $Y^{\\prime } = PYP^T$ .",
"Then $Y^{\\prime }\\in \\mathcal {E}_{\\mathcal {P}}$ and $Y^{\\prime }(Pu) =(PYP^T)(Pu) = 0$ for all $u\\in \\mathcal {U}$ .",
"By the definition of $\\mathcal {U}^{\\prime }$ it is then the case that $Y^{\\prime }u^{\\prime } = 0$ for all $u^{\\prime }\\in \\mathcal {U}^{\\prime }$ .",
"Hence $\\mathcal {U}^{\\prime }$ is $\\mathcal {P}$ -realizable.",
"The converse clearly also holds.",
"Exploiting symmetries: relating realizability and $\\mathcal {P}$ -realizability For a subspace of $\\mathbb {R}^n$ , we now consider how the notions of $\\mathcal {P}$ -realizability and realizability (i.e.",
"$[n]$ -realizability) relate to each other.",
"Since $\\mathcal {E}_{\\mathcal {P}} \\subset \\mathcal {E}_n$ , if $\\mathcal {U}$ is $\\mathcal {P}$ -realizable, it is certainly also realizable.",
"While the converse does not hold, we can establish the following partial converse, which we subsequently use to extend our analysis from Sections and to the present setting.",
"A subspace $\\mathcal {U}$ of $\\mathbb {R}^n$ is $\\mathcal {P}$ -realizable if and only if $\\mathcal {U}^{\\prime }$ is realizable for every $\\mathcal {U}^{\\prime }$ such that $\\mathcal {U}^{\\prime } \\sim \\mathcal {U}$ .",
"We note that one direction of the proof is obvious since $\\mathcal {P}$ -realizability implies realizability.",
"It remains to show that if $\\mathcal {U}$ is not $\\mathcal {P}$ -realizable then there is some $\\mathcal {U}^{\\prime }$ equivalent to $\\mathcal {U}$ that is not realizable.",
"Recall from Lemma REF that if $\\mathcal {U}$ is not $\\mathcal {P}$ -realizable there is some $\\mathcal {P}$ -block-diagonal $X$ with positive trace such that $v^T X v \\le 0$ for all $v\\in \\mathcal {U}^\\perp $ .",
"Since $X$ is $\\mathcal {P}$ -block-diagonal there is some $P\\in G_{\\mathcal {P}}$ such that $PXP^T$ is diagonal.",
"Since conjugation by orthogonal matrices preserves eigenvalues, $\\operatornamewithlimits{\\text{tr}}(PXP^T) = \\operatornamewithlimits{\\text{tr}}(X) > 0$ .",
"Furthermore $v^T(PXP^T)v = (P^Tv)^TX(P^Tv) \\le 0$ for all $P^Tv \\in \\mathcal {U}^\\perp $ .",
"Hence $w^T(PXP^T)w \\ge 0$ for all $w\\in P\\cdot \\mathcal {U}^\\perp = (P\\cdot \\mathcal {U})^{\\perp }$ .",
"By Lemma REF , $PXP^T$ is a certificate that $P\\cdot \\mathcal {U}$ is not realizable, completing the proof.",
"The power of Theorem REF lies in its ability to turn any condition for a subspace to be realizable into a condition for the subspace to be $\\mathcal {P}$ -realizable by appropriately symmetrizing the condition with respect to the action of $G_{\\mathcal {P}}$ .",
"We now illustrate this approach by generalizing Theorem REF and our coherence based condition (Theorem REF ) for a subspace to be $\\mathcal {P}$ -realizable.",
"In each case we first define an appropriately symmetrized version of the original condition.",
"The natural symmetrized version of the notion of balance is as follows.",
"A vector $u\\in \\mathbb {R}^n$ is $\\mathcal {P}$ -balanced if for all $I\\in \\mathcal {P}$ $ \\Vert u_{I}\\Vert _2 \\le \\sum _{J\\in \\mathcal {P}\\setminus \\lbrace I\\rbrace } \\Vert u_{J}\\Vert _2.$ We next define the appropriately symmetrized analogue of coherence.",
"Just as coherence measures how far a subspace is from any one-dimensional coordinate subspace, $\\mathcal {P}$ -coherence measures how far a subspace is from any of the coordinate subspaces indexed by elements of $\\mathcal {P}$ .",
"The $\\mathcal {P}$ -coherence of a subspace $\\mathcal {U}$ of $\\mathbb {R}^n$ is $ \\mu _{\\mathcal {P}}(\\mathcal {U}) = \\max _{I\\in \\mathcal {P}}\\max _{x\\in S^{I}} \\Vert P_{\\mathcal {U}}x\\Vert _2^2.$ Just as the coherence of $\\mathcal {U}$ can be computed by taking the maximum diagonal element of $P_{\\mathcal {U}}$ , it is straightforward to veify that the $\\mathcal {P}$ -coherence of $\\mathcal {U}$ can be computed by taking the maximum of the spectral norms of the principal submatrices $[P_{\\mathcal {U}}]_{I}$ indexed by $I\\in \\mathcal {P}$ .",
"We now use Theorem REF to establish the natural generalization of Theorem REF .",
"If a subspace $\\mathcal {U}$ of $\\mathbb {R}^n$ is $\\mathcal {P}$ -realizable then every element of $\\mathcal {U}$ is $\\mathcal {P}$ -balanced.",
"If $\\mathcal {U} = \\operatornamewithlimits{\\text{span}}\\lbrace u\\rbrace $ is one dimensional then $\\mathcal {U}$ is $\\mathcal {P}$ -realizable if and only if $u$ is $\\mathcal {P}$ -balanced.",
"If there is $u\\in \\mathcal {U}$ that is not $\\mathcal {P}$ -balanced then there is $P\\in G_{\\mathcal {P}}$ such that $Pu$ is not balanced (choose $P$ so that it rotates each $u_{I}$ until it has only one non-zero entry).",
"But then $P\\cdot \\mathcal {U}$ is not realizable and so $\\mathcal {U}$ is not $\\mathcal {P}$ -realizable.",
"For the converse, we first show that if a vector is $\\mathcal {P}$ -balanced then it is balanced.",
"Let $I\\in \\mathcal {P}$ , and consider $i\\in I$ .",
"Then since $u$ is $\\mathcal {P}$ -balanced, $ 2|u_i| \\le 2 \\Vert u_{I}\\Vert _2 \\le \\sum _{J\\in \\mathcal {P}} \\Vert u_{J}\\Vert _2 \\le \\sum _{i=1}^{n}|u_i|$ and so $u$ is balanced.",
"Now suppose $\\mathcal {U} = \\operatornamewithlimits{\\text{span}}\\lbrace u\\rbrace $ is one dimensional and $u$ is $\\mathcal {P}$ -balanced.",
"Since $u$ is $\\mathcal {P}$ -balanced it follows that $Pu$ is $\\mathcal {P}$ -balanced (and hence balanced) every $P\\in G_{\\mathcal {P}}$ .",
"Then by Theorem REF $\\operatornamewithlimits{\\text{span}}\\lbrace Pu\\rbrace $ is realizable for every $P\\in G_{\\mathcal {P}}$ .",
"Hence by Theorem REF , $\\mathcal {U}$ is $\\mathcal {P}$ -realizable.",
"Similarly, with the aid of Theorem REF we can write down a $\\mathcal {P}$ -coherence-threshold condition that is a sufficient condition for a subspace to be $\\mathcal {P}$ -realizable.",
"The following is a natural generalization of Theorem REF .",
"If $\\mu _{\\mathcal {P}}(\\mathcal {U}) < 1/2$ then $\\mathcal {U}$ is $\\mathcal {P}$ -realizable.",
"By examining the constraints in the variational definitions of $\\mu (\\mathcal {U})$ and $\\mu _{\\mathcal {P}}(\\mathcal {U})$ we see that $\\mu (\\mathcal {U}) \\le \\mu _{\\mathcal {P}}(\\mathcal {U})$ .",
"Consequently if $\\mu _{\\mathcal {P}}(\\mathcal {U}) < 1/2$ it follows from Theorem REF that $\\mathcal {U}$ is realizable.",
"Since $\\mu _{\\mathcal {P}}$ is invariant under the action of $G_{\\mathcal {P}}$ on subspaces we can apply Theorem REF to complete the proof.",
"Conclusions We established a link between three problems of independent interest: deciding whether there is a centered ellipsoid passing through a collection of points, understanding the structure of the faces of the elliptope, and deciding which pairs of diagonal and low rank-matrices can be recovered from their sum using a tractable semidefinite-programming-based heuristic, namely minimum trace factor analysis.",
"We provided a simple sufficient condition, based on the notion of the coherence of a subspace, which ensures the success of minimum trace factor analysis, and showed that this is the best possible coherence-threshold-type sufficient condition for this problem.",
"We provided natural generalizations of our results to the problem of analyzing tractable block-diagonal and low-rank decompositions, showing how the symmetries of this problem allow us to reduce much of the analysis to the original diagonal and low-rank case.",
"Our results suggest both the power and the limitations of using `coarse' properties of a subspace such as coherence to gain understanding of the faces of the elliptope (and related problems).",
"The power of results based on such properties is that they do not have explicit dimension-dependence, unlike previous results on the faces of the elliptope.",
"At the same time, the lack of explicit dimension dependence typically yields conservative sufficient conditions for high-dimensional problems.",
"It would be interesting to find a hierarchy of coherence-like conditions that provide less conservative sufficient conditions for higher dimensional problem instances.",
"Additional proofs Proof of Lemma REF We first establish Lemma REF which gives an interpretation of the balance condition in terms of ellipsoid fitting.",
"The proof is a fairly straightforward application of linear programming duality.",
"Throughout let $V$ be the $k\\times n$ matrix with columns given by the $v_i$ .",
"The point $v_i\\in \\mathbb {R}^k$ is on the boundary of the convex hull of $\\pm v_1,\\ldots ,\\pm v_n$ if and only if there exists $x\\in \\mathbb {R}^k$ such that $\\langle x,v_i\\rangle = 1$ and $|\\langle x,v_j\\rangle | \\le 1$ for all $j\\ne i$ .",
"Equivalently, the following linear program (which depends on $i$ ) is feasible $\\operatornamewithlimits{\\text{minimize}}_x \\; \\langle 0,x\\rangle \\quad \\text{subject to}\\quad \\left\\lbrace \\begin{array}{rcl} v_i^Tx \\!\\!\\!\\!",
"& = & \\!\\!\\!\\!",
"1\\\\|v_j^Tx| \\!\\!\\!\\!",
"& \\le & \\!\\!\\!\\!",
"1\\;\\;\\text{for all $j\\ne i$.",
"}\\end{array}\\right.$ Suppose there is some $i$ such that $v_i$ is in the interior of $\\text{conv}\\lbrace \\pm v_1,\\ldots ,\\pm v_n\\rbrace $ .",
"Then (REF ) is not feasible so the dual linear program (which depends on $i$ ) $\\operatornamewithlimits{\\text{maximize}}_u \\; u_i - \\sum _{j\\ne i} |u_j| \\quad \\text{subject to}\\quad \\begin{array}{rcl} V u \\!\\!\\!\\!",
"& = & \\!\\!\\!\\!",
"0\\end{array}$ is unbounded.",
"This is the case if and only if there is some $u$ in the nullspace of $V$ such that $u_i > \\sum _{j\\ne i} |u_j|$ .",
"If such a $u$ exists, then it is certainly the case that $|u_i| \\ge u_i > \\sum _{j\\ne i} |u_j|$ and so $u$ is not balanced.",
"Conversely if $u$ is in the nullspace of $V$ and $u$ is not balanced then either $u$ or $-u$ satisfies $u_i > \\sum _{j\\ne i} |u_j|$ for some $i$ .",
"Hence the linear program (REF ) associated with the index $i$ is unbounded and so the corresponding linear program (REF ) is infeasible.",
"It follows that $v_i$ is in the interior of the convex hull of $\\pm v_1,\\ldots ,\\pm v_n$ .",
"Completing the proof of Theorem REF We now complete the proof of Theorem REF by establishing the following result about the existence of a non-negative solution to the linear system (REF ).",
"If $\\mu (\\mathcal {U}) < 1/2$ then there is $\\lambda \\ge 0$ such that $\\left(P_{\\mathcal {U}^\\perp }^*(\\lambda )P_{\\mathcal {U}^\\perp }\\right) = \\mathbf {1}.$ We note that the linear system (REF ) can be written as $P_{\\mathcal {U}^\\perp }\\circ P_{\\mathcal {U}^\\perp }\\lambda = \\mathbf {1}$ where $\\circ $ denotes the entry-wise product of matrices.",
"As such, we need to show that $P_{\\mathcal {U}^\\perp }\\circ P_{\\mathcal {U}^\\perp }$ is invertible and $(P_{\\mathcal {U}^\\perp }\\circ P_{\\mathcal {U}^\\perp })^{-1}\\mathbf {1}\\ge 0$ .",
"To do so, we appeal to the following (slight restatement) of a theorem of Walters [31] regarding positive solutions to certain linear systems.",
"[Walters [31]] Suppose $A$ is a square matrix with non-negative entries and positive diagonal entries.",
"Let $D$ be a diagonal matrix with $D_{ii} = A_{ii}$ for all $i$ .",
"If $y>0$ and $2y - AD^{-1}y > 0$ then $A$ is invertible and $A^{-1}y > 0$ .",
"In our case we take $A = P_{\\mathcal {U}^\\perp }\\circ P_{\\mathcal {U}^\\perp }$ and $y = \\mathbf {1}$ in Theorem REF .",
"It is clear that $P_{\\mathcal {U}^\\perp } \\circ P_{\\mathcal {U}^\\perp }$ is entry-wise non-negative.",
"Furthermore $[P_{\\mathcal {U}^\\perp }]_{ii} = 1-[P_\\mathcal {U}]_{ii} > 1-\\mu (\\mathcal {U}) > 1/2$ and so $D_{ii} = [P_{\\mathcal {U}^\\perp }\\circ P_{\\mathcal {U}^\\perp }]_{ii} > 1/4$ .",
"It then remains to show that $P_{\\mathcal {U}^\\perp }\\circ ~P_{\\mathcal {U}^\\perp }~D^{-1}\\mathbf {1}<2\\mathbf {1}$ .",
"Consider the $i$ th such inequality, and observe that $[P_{\\mathcal {U}^\\perp }\\circ P_{\\mathcal {U}^\\perp } D^{-1}\\mathbf {1}]_i& = \\left(P_{\\mathcal {U}^\\perp }D^{-1}P_{\\mathcal {U}^\\perp }\\right)_{ii}\\\\& = \\left(P_{\\mathcal {U}^\\perp }D_{ii}^{-1}e_ie_i^TP_{\\mathcal {U}^\\perp }\\right)_{ii} +\\left(P_{\\mathcal {U}^\\perp }(D^{-1} - D_{ii}^{-1}e_ie_i^T)P_{\\mathcal {U}^\\perp }\\right)_{ii}\\\\& \\le 1 + \\max _{j\\in [n]}D_{jj}^{-1} \\left(P_{\\mathcal {U}^\\perp }(I-e_ie_i^T)P_{\\mathcal {U}^\\perp }\\right)_{ii}\\\\& < 1 + 4[P_{\\mathcal {U}^\\perp }]_{ii} - 4[P_{\\mathcal {U}^\\perp }]_{ii}^2\\\\& = 2 - 4([P_{\\mathcal {U}^\\perp }]_{ii} - 1/2)^{2}\\\\& \\le 2$ where we have used the assumption that $[P_{\\mathcal {U}^\\perp }]_{ii}>1/2$ for all $i$ and the fact that $P_{\\mathcal {U}^\\perp }^{2} = P_{\\mathcal {U}^\\perp }$ .",
"Applying Walters's theorem completes the proof.",
"Proof of Proposition REF We now establish Proposition REF , giving a bound on the probability that a suitably random subspace is realizable by bounding the probability that it has coherence strictly bounded above by $1/2$ .",
"It suffices to show that $\\Vert P_{\\mathcal {U}} e_i\\Vert ^2 \\le (1-2\\epsilon )(1/2-\\epsilon ) = 1/2 - 2\\epsilon ^2 < 1/2$ for all $i$ with high probability.",
"The main observation we use is that if $\\mathcal {U}$ is a random $r$ dimensional subspace of $\\mathbb {R}^n$ and $x$ is any fixed vector with $\\Vert x\\Vert =1$ then $\\Vert P_{\\mathcal {U}} x\\Vert ^2 \\sim \\beta (r/2,(n-r)/2)$ where $\\beta (p,q)$ denotes the beta distribution [13].",
"In the case where $r = (1/2 - \\epsilon )n$ , using a tail bound for $\\beta $ random variables [13] we see that if $x\\in \\mathbb {R}^n$ is fixed and $r > 3/\\epsilon ^2$ then $ \\Pr [ \\Vert P_\\mathcal {U} x\\Vert ^2 \\ge (1+2\\epsilon )(1/2 -\\epsilon )] <\\frac{1}{a_\\epsilon }\\frac{1}{(\\pi (1/4 - \\epsilon ^2))^{1/2}}n^{-1/2}e^{-a_\\epsilon k}$ where $a_\\epsilon = \\epsilon - 4\\epsilon ^2/3$ .",
"Taking a union bound over $n$ events, as long as $r > 3/\\epsilon ^2$ $\\Pr \\left[\\mu (\\mathcal {U}) \\ge 1/2\\right] & \\le \\Pr \\left[\\Vert P_{\\mathcal {U}} e_i\\Vert ^2 \\ge (1-2\\epsilon )(1/2-\\epsilon )\\;\\; \\text{for some $i\\in [n]$}\\right]\\\\& \\le n\\cdot \\frac{1}{a_\\epsilon (\\pi (1/4 - \\epsilon ^2))^{1/2}} n^{-1/2} e^{- a_\\epsilon k} =\\bar{c}n^{1/2}e^{-\\tilde{c}n}$ for appropriate positive constants $\\bar{c}$ and $\\tilde{c}$ .",
"Acknowledgements The authors would like to thank Prof. Sanjoy Mitter for helpful discussions."
],
[
"A sufficient condition for the three problems",
"In this section we establish a new sufficient condition for a subspace $\\mathcal {U}$ of $\\mathbb {R}^n$ to be realizable and consequently a sufficient condition for $\\mathcal {U}$ to be recoverable by MTFA and $\\mathcal {U}^\\perp $ to have the ellipsoid fitting property.",
"Our condition is based on a simple property of a subspace known as coherence.",
"Given a subspace $\\mathcal {U}$ of $\\mathbb {R}^n$ , the coherence of $\\mathcal {U}$ is a measure of how close the subspace is to containing any of the elementary unit vectors.",
"This notion was introduced (with a different scaling) by Candès and Recht in their work on low-rank matrix completion [5], although related quantities have played an important role in the analysis of sparse reconstruction problems since the work of Donoho and Huo [10].",
"If $\\mathcal {U}$ is a subspace of $\\mathbb {R}^n$ then the coherence of $\\mathcal {U}$ is $ \\mu (\\mathcal {U}) = \\max _{i\\in [n]}\\Vert P_\\mathcal {U}e_i\\Vert _2^2.$ A basic property of coherence is that it satisfies the inequality $\\frac{\\dim (\\mathcal {U})}{n} \\le \\mu (\\mathcal {U}) \\le 1$ for any subspace $\\mathcal {U}$ of $\\mathbb {R}^n$ [5].",
"This inequality, together with the definition of coherence, provides useful intuition about the properties of subspaces with low coherence, that is incoherence.",
"Any subspace with low coherence is necessarily of low dimension and far from containing any of the elementary unit vectors $e_i$ .",
"As such, any symmetric matrix with incoherent row/column spaces is necessarily of low-rank and quite different from being a diagonal matrix."
],
[
"Coherence-threshold-type sufficient conditions",
"In this section we focus on finding the largest possible $\\alpha $ such that $ \\text{$\\mu (\\mathcal {U}) < \\alpha \\Rightarrow \\mathcal {U}$ is realizable,}$ that is finding the best possible coherence-threshold-type sufficient condition for a subspace to be realizable.",
"Such conditions are of particular interest because the dependence they have on the ambient dimension and the dimension of the subspace is only the mild dependence implied by (REF ).",
"In contrast, existing results (e.g.",
"[8], [20], [19]) about realizability of subspaces hold only for specific combinations of the ambient dimension and the dimension of the subspace.",
"The following theorem, our main result, gives a sufficient condition for realizability based on a coherence-threshold condition.",
"Furthermore, it establishes that this is the best possible coherence-threshold-type sufficient condition.",
"If $\\mathcal {U}$ is a subspace of $\\mathbb {R}^n$ and $\\mu (\\mathcal {U}) < 1/2$ then $\\mathcal {U}$ is realizable.",
"On the other hand, given any $\\alpha > 1/2$ , there is a subspace $\\mathcal {U}$ with $\\mu (\\mathcal {U}) = \\alpha $ that is not realizable.",
"We give the main idea of the proof, deferring some details to Appendix .",
"Instead of proving that there is some $Y\\in \\mathcal {F}_{\\mathcal {U}} = \\lbrace Y \\succeq 0: \\mathcal {N}(Y) \\supseteq \\mathcal {U}\\rbrace $ such that $Y_{ii} = 1$ for $i\\in [n]$ , it suffices to choose a convex cone $\\mathcal {K}$ that is an inner approximation to $\\mathcal {F}_{\\mathcal {U}}$ and establish that there is some $Y\\in \\mathcal {K}$ such that $Y_{ii} = 1$ for $i\\in [n]$ .",
"One natural choice is to take $\\mathcal {K} = \\lbrace P_{\\mathcal {U}^\\perp }^*(\\lambda )P_{\\mathcal {U}^\\perp }: \\lambda \\ge 0\\rbrace $ , which is clearly contained in $\\mathcal {F}_{\\mathcal {U}}$ .",
"Note that there is some $Y \\in \\mathcal {K}$ such that $Y_{ii} = 1$ for all $i\\in [n]$ if and only if there is $\\lambda \\ge 0$ such that $\\left(P_{\\mathcal {U}^\\perp }^*(\\lambda )P_{\\mathcal {U}^\\perp }\\right) = \\mathbf {1}.$ The rest of the proof of the sufficient condition involves showing that if $\\mu (\\mathcal {U}) < 1/2$ then such a non-negative $\\lambda $ exists.",
"We establish this in Lemma REF .",
"Now let us construct, for any $\\alpha > 1/2$ , a subspace with coherence $\\alpha $ that is not realizable.",
"Let $\\mathcal {U}$ to be the subspace of $\\mathbb {R}^2$ spanned by $u = (\\sqrt{\\alpha },\\sqrt{1-\\alpha })$ .",
"Then $\\mu (\\mathcal {U}) = \\max \\lbrace \\alpha ,1-\\alpha \\rbrace = \\alpha $ and yet by Theorem REF , $\\mathcal {U}$ is not realizable because $u$ is not balanced."
],
[
"Remarks",
"Theorem REF illustrates both the power and limitations of coherence-threshold-type conditions.",
"On the one hand, since coherence is quite a coarse property of a subspace, the result applies to `many' subspaces (see Proposition REF in Section REF ).",
"On the other hand, since coherence has very mild dimension dependence, the power of coherence-threshold-type conditions is limited to their specialization to low-dimensional situations, such as one dimensional subspaces of $\\mathbb {R}^2$ ."
],
[
"Interpretations of Theorem ",
"We now establish two corollaries of our coherence-threshold-type sufficient condition for realizability.",
"These corollaries can be thought of as re-interpretations of the coherence inequality $\\mu (\\mathcal {U}) < 1/2$ in terms of other natural quantities."
],
[
"An ellipsoid-fitting interpretation",
"With the aid of Proposition we reinterpret our coherence-threshold-type sufficient condition as a sufficient condition on a set of points in $\\mathbb {R}^k$ that ensures there is a centered ellipsoid passing through them.",
"The condition involves `sandwiching' the points between two ellipsoids (that depend on the points).",
"Indeed, given $v_1,v_2,\\ldots ,v_n\\in \\mathbb {R}^k$ and $0 < \\beta < 1$ we define the ellipsoid $ \\mathcal {E}_{\\beta }(v_1,\\ldots ,v_n) = \\text{$\\lbrace x\\in \\mathbb {R}^k: x^T(\\textstyle \\sum _{j=1}^n v_jv_j^T)^{-1}x \\le \\beta \\rbrace $}.$ Given $0< \\beta < 1$ the points $v_1,v_2,\\ldots ,v_n$ satisfy the $\\beta $ -sandwich condition if $\\lbrace v_1,v_2,\\ldots ,v_n\\rbrace \\subset \\mathcal {E}_1(v_1,\\ldots ,v_n) \\setminus \\mathcal {E}_\\beta (v_1,\\ldots ,v_n).$ The intuition behind this definition (illustrated in Figure REF ) is that if the points satisfy the $\\beta $ -sandwich condition for $\\beta $ close to one, then they are confined to a thin elliptical shell that is adapted to their position.",
"One might expect that it is `easier' to fit an ellipsoid to points that are confined in this way.",
"Indeed this is the case.",
"If $v_1,v_2,\\ldots ,v_n\\in \\mathbb {R}^k$ satisfy the $1/2$ -sandwich condition then then there is a centered ellipsoid passing through $v_1,v_2,\\ldots ,v_n$ .",
"Let $V$ be the $k\\times n$ matrix with columns given by the $v_i$ , and let $\\mathcal {U}$ be the nullspace of $V$ .",
"Then the orthogonal projection onto the row space of $V$ is $P_{\\mathcal {U}^\\perp }$ , and can be written as $ P_{\\mathcal {U}^\\perp } = V^T(VV^T)^{-1}V.$ Our assumption that the points satisfy the $1/2$ -sandwich condition is equivalent to assuming that $1/2 < [P_{\\mathcal {U}^\\perp }]_{ii} \\le 1$ for all $i\\in [n]$ or alternatively that $\\mu (\\mathcal {U}) = \\max _{i\\in [n]} [P_{\\mathcal {U}}]_{ii} = 1-\\min _{i\\in [n]}[P_{\\mathcal {U}^\\perp }]_{ii} < 1/2.$ From Theorem REF we know that $\\mu (\\mathcal {U}) < 1/2$ implies that $\\mathcal {U}$ is realizable.",
"Invoking Proposition we then conclude that there is a centered ellipsoid passing through $v_1,v_2,\\ldots ,v_n$ .",
"Figure: The ellipsoidsshown are ℰ=ℰ 1 (v 1 ,v 2 ,v 3 )\\mathcal {E} = \\mathcal {E}_1(v_1,v_2,v_3) and ℰ ' =ℰ 1/2 (v 1 ,v 2 ,v 3 )\\mathcal {E}^{\\prime } = \\mathcal {E}_{1/2}(v_1,v_2,v_3).There is an ellipsoid passing through v 1 ,v 2 v_1,v_2 and v 3 v_3 because the points aresandwiched between ℰ\\mathcal {E} and ℰ ' \\mathcal {E}^{\\prime }."
],
[
"A balance interpretation",
"In Section REF we saw that if a subspace $\\mathcal {U}$ is realizable, every $u\\in \\mathcal {U}$ is balanced.",
"The sufficient condition of Theorem REF can be expressed in terms of a balance condition on the element-wise square of the elements of a subspace.",
"(In what follows $u\\circ u$ denotes the element-wise square of a vector in $\\mathbb {R}^n$ .)",
"Suppose $\\mathcal {U}$ is a subspace of $\\mathbb {R}^n$ .",
"If $u\\circ u$ is strictly balanced for every $u\\in \\mathcal {U}$ then $\\mathcal {U}$ is realizable.",
"It suffices to show that if for every $u\\in \\mathcal {U}$ , $u\\circ u$ is strictly balanced, then $\\mu (\\mathcal {U}) <1/2$ (although we could reverse the argument to establish the equivalence of these conditions).",
"If $u\\circ u$ is strictly balanced for all $u\\in \\mathcal {U}$ then for all $i\\in [n]$ and all $u\\in \\mathcal {U}$ $2\\langle e_i,u\\rangle ^2 < \\sum _{j=1}^{n}\\langle e_j,u\\rangle ^2 = \\Vert u\\Vert _2^2.$ Since $\\Vert P_{\\mathcal {U}}e_i\\Vert _2 = \\max _{u\\in \\mathcal {U}\\setminus \\lbrace 0\\rbrace } \\langle e_i,u\\rangle /\\Vert u\\Vert _2$ , it follows from (REF ) that $2\\Vert P_{\\mathcal {U}}e_i\\Vert _2^2 < 1$ .",
"Since this holds for all $i\\in [n]$ it follows that $\\mu (\\mathcal {U}) < 1/2$ ."
],
[
"Remark",
"Suppose $\\mathcal {U} = \\operatornamewithlimits{\\text{span}}\\lbrace u\\rbrace $ is a one-dimensional subspace of $\\mathbb {R}^n$ .",
"We have just established that if $u\\circ u$ is strictly balanced then $\\mathcal {U}$ is realizable and so (by Theorem REF ) $u$ must be balanced.",
"We note that it is straightforward to establish directly that if $u\\circ u$ is balanced then $u$ is balanced by using the definition of balance and the fact that $\\Vert x\\Vert _1 \\ge \\Vert x\\Vert _2$ for any $x\\in \\mathbb {R}^n$ ."
],
[
"Examples",
"To gain more intuition for what Theorem REF means, we consider its implications in two particular cases.",
"First, we compare the characterization of when it is possible to fit an ellipsoid to $k+1$ points in $\\mathbb {R}^k$ (Corollary REF ) with the specialization of our sufficient condition to this case (Corollary REF ).",
"This comparison provides some insight into how conservative our sufficient condition is.",
"Second, we investigate the coherence properties of suitably `random' subspaces.",
"This provides intuition about whether or not $\\mu (\\mathcal {U}) < 1/2$ is a very restrictive condition.",
"In particular, we establish that `most' subspaces of $\\mathbb {R}^n$ with dimension bounded above by $(1/2 - \\epsilon )n$ are realizable."
],
[
"Fitting an ellipsoid to $k+1$ points in {{formula:0d9e0bc9-0619-4678-9fa2-35f715d9dec5}}",
"Recall that Ledermann and Delorme and Poljak's result, interpreted in terms of ellipsoid fitting, tells us that we can fit an ellipsoid to $k+1$ points $v_1,\\ldots ,v_{k+1}\\in \\mathbb {R}^k$ if and only if those points are on the boundary of the convex hull of $\\lbrace \\pm v_1,\\ldots ,\\pm v_{k+1}\\rbrace $ (see Corollary REF ).",
"We now compare this characterization with the $1/2$ -sandwich condition, which is sufficient by Corollary REF .",
"Without loss of generality we assume that $k$ of the points are $e_1,\\ldots ,e_k$ , the standard basis vectors, and compare the conditions by considering the set of locations of the $k+1$ st point $v\\in \\mathbb {R}^k$ for which we can fit an ellipsoid through all $k+1$ points.",
"Corollary REF gives a characterization of this region as $ R = \\lbrace v\\in \\mathbb {R}^k: \\sum _{j=1}^k|v_j| \\ge 1, \\; |v_i| - \\sum _{j\\ne i}|v_j| \\le 1\\quad \\text{for $i\\in [k]$}\\rbrace $ which is shown in Figure REF in the case $k=2$ .",
"The set of $v$ such that $v,e_1,\\ldots ,e_n$ satisfy the $1/2$ -sandwich condition can be written as $R^{\\prime } & = \\lbrace v\\in \\mathbb {R}^k: v^T(I + vv^T)^{-1}v > 1/2,\\; e_i^T(I+vv^T)^{-1}e_i > 1/2 \\quad \\text{for $i\\in [k]$}\\rbrace \\\\& = \\lbrace v\\in \\mathbb {R}^k: \\sum _{j=1}^{k}v_j^2 > 1,\\; v_i^2 - \\sum _{j\\ne i}v_j^2 < 1 \\quad \\text{for $i\\in [k]$}\\rbrace $ which is shown in Figure REF .",
"It is clear that $R^{\\prime }\\subseteq R$ .",
"Figure: Comparing our sufficient condition for ellipsoid fitting (Corollary ) withthe characterization (Corollary ) in the case of fitting an ellipsoid to k+1k+1 points in ℝ k \\mathbb {R}^k."
],
[
"Realizability of random subspaces",
"Suppose $\\mathcal {U}$ is a subspace generated by taking the column space of an $n\\times r$ matrix with i.i.d.",
"standard Gaussian entries.",
"For what values of $r$ and $n$ does such a subspace have $\\mu (\\mathcal {U}) < 1/2$ with high probability, i.e.",
"satisfy our sufficient condition for being realizable?",
"The following result essentially shows that for large $n$ , `most' subspaces of dimension at most $(1/2-\\epsilon )n$ are realizable.",
"This suggests that MTFA is a very good heuristic for diagonal and low-rank decomposition problems in the high-dimensional setting.",
"Indeed `most' subspaces of dimension up to one half the ambient dimension—hardly just low-dimensional subspaces—are recoverable by MTFA.",
"Let $0 < \\epsilon < 1/2$ be a constant and suppose $n > 6/(\\epsilon ^2-2\\epsilon ^3)$ .",
"There are positive constants $\\bar{c}$ , $\\tilde{c}$ , (depending only on $\\epsilon $ ) such that if $\\mathcal {U}$ is a random $(1/2 - \\epsilon )n$ dimensional subspace of $\\mathbb {R}^n$ then $ \\Pr [\\text{$\\mathcal {U}$ is realizable}] \\ge 1-\\bar{c}\\sqrt{n}e^{-\\tilde{c} n}.$ We provide a proof of this result in Appendix .",
"The main idea is that the coherence of a random $r$ dimensional subspace of $\\mathbb {R}^n$ is the maximum of $n$ random variables that concentrate around their mean of $r/n$ for large $n$ .",
"To illustrate the result, we consider the case where $\\epsilon = 1/4$ and $n>192$ .",
"Then (by examining the proof in Appendix ) we see that we can take $\\tilde{c} = 1/24$ and $\\bar{c} = 24/\\sqrt{3\\pi } \\approx 7.8$ .",
"Hence if $n > 192$ and $\\mathcal {U}$ is a random $n/4$ dimensional subspace of $\\mathbb {R}^n$ we have that $ \\Pr [\\text{$\\mathcal {U}$ is realizable}] \\ge 1 - 7.8\\sqrt{n}e^{-n/24}.$"
],
[
"Tractable block diagonal and low-rank decompositions and related problems",
"In this section we generalize our results to the analogue of MTFA for block-diagonal and low-rank decompositions.",
"Mimicking our earlier development, we relate the analysis of this variant of MTFA to the facial structure of a variant of the elliptope and a generalization of the ellipsoid fitting problem.",
"The key point is that these problems all possess additional symmetries that, once taken into account, essentially allow us to reduce our analysis to cases already considered in Sections and .",
"Throughout this section, let $\\mathcal {P}$ be a fixed partition of $\\lbrace 1,2,\\ldots ,n\\rbrace $ .",
"We say a matrix is $\\mathcal {P}$ -block-diagonal if it is zero except for the principal submatrices indexed by the elements of $\\mathcal {P}$ .",
"We denote by $\\text{blkdiag}_{\\mathcal {P}}$ the map that takes an $n\\times n$ matrix and maps it to the principal submatrices indexed by $\\mathcal {P}$ .",
"Its adjoint, denoted $\\text{blkdiag}_{\\mathcal {P}}^*$ , takes a tuple of symmetric matrices $(X_I)_{I\\in \\mathcal {P}}$ and produces an $n\\times n$ matrix that is $\\mathcal {P}$ -block diagonal with blocks given by the $X_{I}$ .",
"We now describe the analogues of MTFA, ellipsoid fitting, and the problem of determining the facial structure of the elliptope."
],
[
"Block minimum trace factor analysis",
"If $X = B^\\star + L^\\star $ where $B^\\star $ is $\\mathcal {P}$ -block-diagonal and $L^\\star \\succeq 0$ is low rank, the obvious analogue of MTFA is the semidefinite program $\\operatornamewithlimits{\\text{minimize}}_{B,L} \\; \\operatornamewithlimits{\\text{tr}}(L) \\quad \\text{subject to}\\quad \\left\\lbrace \\begin{array}{rcl} X \\!\\!\\!\\!",
"& = & \\!\\!\\!\\!",
"B + L\\\\L \\!\\!\\!\\!",
"& \\succeq \\!\\!\\!\\!",
"& 0\\\\B\\!\\!\\!\\!",
"& \\text{is} &\\!\\!\\!\\!",
"\\text{$\\mathcal {P}$-block-diagonal}\\end{array}\\right.$ which we call block minimum trace factor analysis (BMTFA).",
"A subspace $\\mathcal {U}$ of $\\mathbb {R}^n$ is recoverable by BMTFA if for every $B^\\star $ that is $\\mathcal {P}$ -block-diagonal and every positive semidefinite $L^\\star $ with column space $\\mathcal {U}$ , $(B^\\star ,L^\\star )$ is the unique optimum of BMTFA with input $X = B^\\star + L^\\star $ ."
],
[
"Faces of the $\\mathcal {P}$ -elliptope",
"Just as MTFA is related to the facial structure of the elliptope, BMTFA is related to the facial structure of the spectrahedron $ \\mathcal {E}_{\\mathcal {P}} = \\lbrace Y \\succeq 0: \\text{blkdiag}_{\\mathcal {P}}(Y) = (I,I,\\ldots ,I)\\rbrace .$ We refer to $\\mathcal {E}_\\mathcal {P}$ as the $\\mathcal {P}$ -elliptope.",
"We extend the definition of a realizable subspace to this context.",
"A subspace $\\mathcal {U}$ of $\\mathbb {R}^n$ is $\\mathcal {P}$ -realizable if there is some $Y\\in \\mathcal {E}_{\\mathcal {P}}$ such that $\\mathcal {N}(Y) \\supseteq \\mathcal {U}$ ."
],
[
"Generalized ellipsoid fitting",
"To describe the $\\mathcal {P}$ -ellipsoid fitting problem we first introduce some convenient notation.",
"If $I\\subset [n]$ we write $S^{I} = \\lbrace x\\in \\mathbb {R}^n: \\Vert x\\Vert _2 = 1,\\;\\text{$x_j = 0$ if $j\\notin I$}\\rbrace $ for the intersection of the unit sphere with the coordinate subspace indexed by $I$ .",
"Suppose $v_1,v_2,\\ldots ,v_n\\in \\mathbb {R}^k$ is a collection of points and $V$ is the $k\\times n$ matrix with columns given by the $v_i$ .",
"Noting that $S^{\\lbrace i\\rbrace } = \\lbrace -e_i,e_i\\rbrace $ , and thinking of $V$ as a linear map from $\\mathbb {R}^n$ to $\\mathbb {R}^k$ , we see that the ellipsoid fitting problem is to find an ellipsoid in $\\mathbb {R}^k$ with boundary containing $\\cup _{i\\in [n]} V(S^{\\lbrace i\\rbrace })$ , i.e.",
"the collection of points $\\pm v_1,\\ldots ,\\pm v_n$ .",
"The $\\mathcal {P}$ -ellipsoid fitting problem is then to find an ellipsoid in $\\mathbb {R}^k$ with boundary containing $\\cup _{I\\in \\mathcal {P}} V(S^{I})$ , i.e.",
"the collection of ellipsoids $V(S^{I})$ .",
"The generalization of the ellipsoid fitting property of a subspace is as follows.",
"A subspace $\\mathcal {V}$ of $\\mathbb {R}^n$ has the $\\mathcal {P}$ -ellipsoid fitting property if there is a $k\\times n$ matrix $V$ with row space $\\mathcal {V}$ such that there is a centered ellipsoid in $\\mathbb {R}^k$ with boundary containing $\\cup _{I\\in \\mathcal {P}} V(S^{I})$ ."
],
[
"Relating the generalized problems",
"The facial structure of the $\\mathcal {P}$ -elliptope, BMTFA, and the $\\mathcal {P}$ -ellipsoid fitting problem are related by the following result, the proof of which is omitted as it is almost identical to that of Proposition .",
"Let $\\mathcal {U}$ be a subspace of $\\mathbb {R}^n$ .",
"Then the following are equivalent: $\\mathcal {U}$ is recoverable by BMTFA.",
"$\\mathcal {U}$ is $\\mathcal {P}$ -realizable.",
"$\\mathcal {U}^\\perp $ has the $\\mathcal {P}$ -ellipsoid fitting property.",
"The following lemma is the analogue of Lemma REF .",
"It describes certificates that a subspace $\\mathcal {U}$ is not $\\mathcal {P}$ -realizable.",
"Again the proof is almost identical to that of Lemma REF so we omit it.",
"A subspace $\\mathcal {U}$ of $\\mathbb {R}^n$ is not $\\mathcal {P}$ -realizable if and only if there is a $\\mathcal {P}$ -block-diagonal matrix $B$ such that $\\operatornamewithlimits{\\text{tr}}(B) > 0$ and $v^TBv \\le 0$ for all $v\\in \\mathcal {U}^\\perp $ .",
"For the sake of brevity, in what follows we only discuss the problem of whether $\\mathcal {U}$ is $\\mathcal {P}$ -realizable without explicitly translating the results into the context of the other two problems.",
"Symmetries of the $\\mathcal {P}$ -elliptope We now consider the symmetries of the $\\mathcal {P}$ -elliptope.",
"Our motivation for doing so is that it allows us to partition subspaces into classes for which either all elements are $\\mathcal {P}$ -realizable or none of the elements are $\\mathcal {P}$ -realizable.",
"It is clear that the $\\mathcal {P}$ -elliptope is invariant under conjugation by $\\mathcal {P}$ -block-diagonal orthogonal matrices.",
"Let $G_{\\mathcal {P}}$ denote this subgroup of the group of $n\\times n$ orthogonal matrices.",
"There is a natural action of $G_{\\mathcal {P}}$ on subspaces of $\\mathbb {R}^n$ defined as follows.",
"If $P\\in G_{\\mathcal {P}}$ and $\\mathcal {U}$ is a subspace of $\\mathbb {R}^n$ then $P\\cdot \\mathcal {U}$ is the image of the subspace $\\mathcal {U}$ under the map $P$ .",
"(It is straightforward to check that this is a well defined group action.)",
"If there exists some $P\\in G_{\\mathcal {P}}$ such that $P\\cdot \\mathcal {U} = \\mathcal {U}^{\\prime }$ then we write $\\mathcal {U} \\sim \\mathcal {U}^{\\prime }$ and say that $\\mathcal {U}$ and $\\mathcal {U}^{\\prime }$ are equivalent.",
"We care about this equivalence relation on subspaces because the property of being $\\mathcal {P}$ -realizable is really a property of the corresponding equivalence classes.",
"Suppose $\\mathcal {U}$ and $\\mathcal {U}^{\\prime }$ are subspaces of $\\mathbb {R}^n$ .",
"If $\\mathcal {U}\\sim \\mathcal {U}^{\\prime }$ then $\\mathcal {U}$ is $\\mathcal {P}$ -realizable if and only if $\\mathcal {U}^{\\prime }$ is $\\mathcal {P}$ -realizable.",
"If $\\mathcal {U}$ is $\\mathcal {P}$ -realizable there is $Y\\in \\mathcal {E}_{\\mathcal {P}}$ such that $Yu = 0$ for all $u\\in \\mathcal {U}$ .",
"Suppose $\\mathcal {U}^{\\prime } = P\\cdot \\mathcal {U}$ for some $P\\in G_{\\mathcal {P}}$ and let $Y^{\\prime } = PYP^T$ .",
"Then $Y^{\\prime }\\in \\mathcal {E}_{\\mathcal {P}}$ and $Y^{\\prime }(Pu) =(PYP^T)(Pu) = 0$ for all $u\\in \\mathcal {U}$ .",
"By the definition of $\\mathcal {U}^{\\prime }$ it is then the case that $Y^{\\prime }u^{\\prime } = 0$ for all $u^{\\prime }\\in \\mathcal {U}^{\\prime }$ .",
"Hence $\\mathcal {U}^{\\prime }$ is $\\mathcal {P}$ -realizable.",
"The converse clearly also holds.",
"Exploiting symmetries: relating realizability and $\\mathcal {P}$ -realizability For a subspace of $\\mathbb {R}^n$ , we now consider how the notions of $\\mathcal {P}$ -realizability and realizability (i.e.",
"$[n]$ -realizability) relate to each other.",
"Since $\\mathcal {E}_{\\mathcal {P}} \\subset \\mathcal {E}_n$ , if $\\mathcal {U}$ is $\\mathcal {P}$ -realizable, it is certainly also realizable.",
"While the converse does not hold, we can establish the following partial converse, which we subsequently use to extend our analysis from Sections and to the present setting.",
"A subspace $\\mathcal {U}$ of $\\mathbb {R}^n$ is $\\mathcal {P}$ -realizable if and only if $\\mathcal {U}^{\\prime }$ is realizable for every $\\mathcal {U}^{\\prime }$ such that $\\mathcal {U}^{\\prime } \\sim \\mathcal {U}$ .",
"We note that one direction of the proof is obvious since $\\mathcal {P}$ -realizability implies realizability.",
"It remains to show that if $\\mathcal {U}$ is not $\\mathcal {P}$ -realizable then there is some $\\mathcal {U}^{\\prime }$ equivalent to $\\mathcal {U}$ that is not realizable.",
"Recall from Lemma REF that if $\\mathcal {U}$ is not $\\mathcal {P}$ -realizable there is some $\\mathcal {P}$ -block-diagonal $X$ with positive trace such that $v^T X v \\le 0$ for all $v\\in \\mathcal {U}^\\perp $ .",
"Since $X$ is $\\mathcal {P}$ -block-diagonal there is some $P\\in G_{\\mathcal {P}}$ such that $PXP^T$ is diagonal.",
"Since conjugation by orthogonal matrices preserves eigenvalues, $\\operatornamewithlimits{\\text{tr}}(PXP^T) = \\operatornamewithlimits{\\text{tr}}(X) > 0$ .",
"Furthermore $v^T(PXP^T)v = (P^Tv)^TX(P^Tv) \\le 0$ for all $P^Tv \\in \\mathcal {U}^\\perp $ .",
"Hence $w^T(PXP^T)w \\ge 0$ for all $w\\in P\\cdot \\mathcal {U}^\\perp = (P\\cdot \\mathcal {U})^{\\perp }$ .",
"By Lemma REF , $PXP^T$ is a certificate that $P\\cdot \\mathcal {U}$ is not realizable, completing the proof.",
"The power of Theorem REF lies in its ability to turn any condition for a subspace to be realizable into a condition for the subspace to be $\\mathcal {P}$ -realizable by appropriately symmetrizing the condition with respect to the action of $G_{\\mathcal {P}}$ .",
"We now illustrate this approach by generalizing Theorem REF and our coherence based condition (Theorem REF ) for a subspace to be $\\mathcal {P}$ -realizable.",
"In each case we first define an appropriately symmetrized version of the original condition.",
"The natural symmetrized version of the notion of balance is as follows.",
"A vector $u\\in \\mathbb {R}^n$ is $\\mathcal {P}$ -balanced if for all $I\\in \\mathcal {P}$ $ \\Vert u_{I}\\Vert _2 \\le \\sum _{J\\in \\mathcal {P}\\setminus \\lbrace I\\rbrace } \\Vert u_{J}\\Vert _2.$ We next define the appropriately symmetrized analogue of coherence.",
"Just as coherence measures how far a subspace is from any one-dimensional coordinate subspace, $\\mathcal {P}$ -coherence measures how far a subspace is from any of the coordinate subspaces indexed by elements of $\\mathcal {P}$ .",
"The $\\mathcal {P}$ -coherence of a subspace $\\mathcal {U}$ of $\\mathbb {R}^n$ is $ \\mu _{\\mathcal {P}}(\\mathcal {U}) = \\max _{I\\in \\mathcal {P}}\\max _{x\\in S^{I}} \\Vert P_{\\mathcal {U}}x\\Vert _2^2.$ Just as the coherence of $\\mathcal {U}$ can be computed by taking the maximum diagonal element of $P_{\\mathcal {U}}$ , it is straightforward to veify that the $\\mathcal {P}$ -coherence of $\\mathcal {U}$ can be computed by taking the maximum of the spectral norms of the principal submatrices $[P_{\\mathcal {U}}]_{I}$ indexed by $I\\in \\mathcal {P}$ .",
"We now use Theorem REF to establish the natural generalization of Theorem REF .",
"If a subspace $\\mathcal {U}$ of $\\mathbb {R}^n$ is $\\mathcal {P}$ -realizable then every element of $\\mathcal {U}$ is $\\mathcal {P}$ -balanced.",
"If $\\mathcal {U} = \\operatornamewithlimits{\\text{span}}\\lbrace u\\rbrace $ is one dimensional then $\\mathcal {U}$ is $\\mathcal {P}$ -realizable if and only if $u$ is $\\mathcal {P}$ -balanced.",
"If there is $u\\in \\mathcal {U}$ that is not $\\mathcal {P}$ -balanced then there is $P\\in G_{\\mathcal {P}}$ such that $Pu$ is not balanced (choose $P$ so that it rotates each $u_{I}$ until it has only one non-zero entry).",
"But then $P\\cdot \\mathcal {U}$ is not realizable and so $\\mathcal {U}$ is not $\\mathcal {P}$ -realizable.",
"For the converse, we first show that if a vector is $\\mathcal {P}$ -balanced then it is balanced.",
"Let $I\\in \\mathcal {P}$ , and consider $i\\in I$ .",
"Then since $u$ is $\\mathcal {P}$ -balanced, $ 2|u_i| \\le 2 \\Vert u_{I}\\Vert _2 \\le \\sum _{J\\in \\mathcal {P}} \\Vert u_{J}\\Vert _2 \\le \\sum _{i=1}^{n}|u_i|$ and so $u$ is balanced.",
"Now suppose $\\mathcal {U} = \\operatornamewithlimits{\\text{span}}\\lbrace u\\rbrace $ is one dimensional and $u$ is $\\mathcal {P}$ -balanced.",
"Since $u$ is $\\mathcal {P}$ -balanced it follows that $Pu$ is $\\mathcal {P}$ -balanced (and hence balanced) every $P\\in G_{\\mathcal {P}}$ .",
"Then by Theorem REF $\\operatornamewithlimits{\\text{span}}\\lbrace Pu\\rbrace $ is realizable for every $P\\in G_{\\mathcal {P}}$ .",
"Hence by Theorem REF , $\\mathcal {U}$ is $\\mathcal {P}$ -realizable.",
"Similarly, with the aid of Theorem REF we can write down a $\\mathcal {P}$ -coherence-threshold condition that is a sufficient condition for a subspace to be $\\mathcal {P}$ -realizable.",
"The following is a natural generalization of Theorem REF .",
"If $\\mu _{\\mathcal {P}}(\\mathcal {U}) < 1/2$ then $\\mathcal {U}$ is $\\mathcal {P}$ -realizable.",
"By examining the constraints in the variational definitions of $\\mu (\\mathcal {U})$ and $\\mu _{\\mathcal {P}}(\\mathcal {U})$ we see that $\\mu (\\mathcal {U}) \\le \\mu _{\\mathcal {P}}(\\mathcal {U})$ .",
"Consequently if $\\mu _{\\mathcal {P}}(\\mathcal {U}) < 1/2$ it follows from Theorem REF that $\\mathcal {U}$ is realizable.",
"Since $\\mu _{\\mathcal {P}}$ is invariant under the action of $G_{\\mathcal {P}}$ on subspaces we can apply Theorem REF to complete the proof.",
"Conclusions We established a link between three problems of independent interest: deciding whether there is a centered ellipsoid passing through a collection of points, understanding the structure of the faces of the elliptope, and deciding which pairs of diagonal and low rank-matrices can be recovered from their sum using a tractable semidefinite-programming-based heuristic, namely minimum trace factor analysis.",
"We provided a simple sufficient condition, based on the notion of the coherence of a subspace, which ensures the success of minimum trace factor analysis, and showed that this is the best possible coherence-threshold-type sufficient condition for this problem.",
"We provided natural generalizations of our results to the problem of analyzing tractable block-diagonal and low-rank decompositions, showing how the symmetries of this problem allow us to reduce much of the analysis to the original diagonal and low-rank case.",
"Our results suggest both the power and the limitations of using `coarse' properties of a subspace such as coherence to gain understanding of the faces of the elliptope (and related problems).",
"The power of results based on such properties is that they do not have explicit dimension-dependence, unlike previous results on the faces of the elliptope.",
"At the same time, the lack of explicit dimension dependence typically yields conservative sufficient conditions for high-dimensional problems.",
"It would be interesting to find a hierarchy of coherence-like conditions that provide less conservative sufficient conditions for higher dimensional problem instances.",
"Additional proofs Proof of Lemma REF We first establish Lemma REF which gives an interpretation of the balance condition in terms of ellipsoid fitting.",
"The proof is a fairly straightforward application of linear programming duality.",
"Throughout let $V$ be the $k\\times n$ matrix with columns given by the $v_i$ .",
"The point $v_i\\in \\mathbb {R}^k$ is on the boundary of the convex hull of $\\pm v_1,\\ldots ,\\pm v_n$ if and only if there exists $x\\in \\mathbb {R}^k$ such that $\\langle x,v_i\\rangle = 1$ and $|\\langle x,v_j\\rangle | \\le 1$ for all $j\\ne i$ .",
"Equivalently, the following linear program (which depends on $i$ ) is feasible $\\operatornamewithlimits{\\text{minimize}}_x \\; \\langle 0,x\\rangle \\quad \\text{subject to}\\quad \\left\\lbrace \\begin{array}{rcl} v_i^Tx \\!\\!\\!\\!",
"& = & \\!\\!\\!\\!",
"1\\\\|v_j^Tx| \\!\\!\\!\\!",
"& \\le & \\!\\!\\!\\!",
"1\\;\\;\\text{for all $j\\ne i$.",
"}\\end{array}\\right.$ Suppose there is some $i$ such that $v_i$ is in the interior of $\\text{conv}\\lbrace \\pm v_1,\\ldots ,\\pm v_n\\rbrace $ .",
"Then (REF ) is not feasible so the dual linear program (which depends on $i$ ) $\\operatornamewithlimits{\\text{maximize}}_u \\; u_i - \\sum _{j\\ne i} |u_j| \\quad \\text{subject to}\\quad \\begin{array}{rcl} V u \\!\\!\\!\\!",
"& = & \\!\\!\\!\\!",
"0\\end{array}$ is unbounded.",
"This is the case if and only if there is some $u$ in the nullspace of $V$ such that $u_i > \\sum _{j\\ne i} |u_j|$ .",
"If such a $u$ exists, then it is certainly the case that $|u_i| \\ge u_i > \\sum _{j\\ne i} |u_j|$ and so $u$ is not balanced.",
"Conversely if $u$ is in the nullspace of $V$ and $u$ is not balanced then either $u$ or $-u$ satisfies $u_i > \\sum _{j\\ne i} |u_j|$ for some $i$ .",
"Hence the linear program (REF ) associated with the index $i$ is unbounded and so the corresponding linear program (REF ) is infeasible.",
"It follows that $v_i$ is in the interior of the convex hull of $\\pm v_1,\\ldots ,\\pm v_n$ .",
"Completing the proof of Theorem REF We now complete the proof of Theorem REF by establishing the following result about the existence of a non-negative solution to the linear system (REF ).",
"If $\\mu (\\mathcal {U}) < 1/2$ then there is $\\lambda \\ge 0$ such that $\\left(P_{\\mathcal {U}^\\perp }^*(\\lambda )P_{\\mathcal {U}^\\perp }\\right) = \\mathbf {1}.$ We note that the linear system (REF ) can be written as $P_{\\mathcal {U}^\\perp }\\circ P_{\\mathcal {U}^\\perp }\\lambda = \\mathbf {1}$ where $\\circ $ denotes the entry-wise product of matrices.",
"As such, we need to show that $P_{\\mathcal {U}^\\perp }\\circ P_{\\mathcal {U}^\\perp }$ is invertible and $(P_{\\mathcal {U}^\\perp }\\circ P_{\\mathcal {U}^\\perp })^{-1}\\mathbf {1}\\ge 0$ .",
"To do so, we appeal to the following (slight restatement) of a theorem of Walters [31] regarding positive solutions to certain linear systems.",
"[Walters [31]] Suppose $A$ is a square matrix with non-negative entries and positive diagonal entries.",
"Let $D$ be a diagonal matrix with $D_{ii} = A_{ii}$ for all $i$ .",
"If $y>0$ and $2y - AD^{-1}y > 0$ then $A$ is invertible and $A^{-1}y > 0$ .",
"In our case we take $A = P_{\\mathcal {U}^\\perp }\\circ P_{\\mathcal {U}^\\perp }$ and $y = \\mathbf {1}$ in Theorem REF .",
"It is clear that $P_{\\mathcal {U}^\\perp } \\circ P_{\\mathcal {U}^\\perp }$ is entry-wise non-negative.",
"Furthermore $[P_{\\mathcal {U}^\\perp }]_{ii} = 1-[P_\\mathcal {U}]_{ii} > 1-\\mu (\\mathcal {U}) > 1/2$ and so $D_{ii} = [P_{\\mathcal {U}^\\perp }\\circ P_{\\mathcal {U}^\\perp }]_{ii} > 1/4$ .",
"It then remains to show that $P_{\\mathcal {U}^\\perp }\\circ ~P_{\\mathcal {U}^\\perp }~D^{-1}\\mathbf {1}<2\\mathbf {1}$ .",
"Consider the $i$ th such inequality, and observe that $[P_{\\mathcal {U}^\\perp }\\circ P_{\\mathcal {U}^\\perp } D^{-1}\\mathbf {1}]_i& = \\left(P_{\\mathcal {U}^\\perp }D^{-1}P_{\\mathcal {U}^\\perp }\\right)_{ii}\\\\& = \\left(P_{\\mathcal {U}^\\perp }D_{ii}^{-1}e_ie_i^TP_{\\mathcal {U}^\\perp }\\right)_{ii} +\\left(P_{\\mathcal {U}^\\perp }(D^{-1} - D_{ii}^{-1}e_ie_i^T)P_{\\mathcal {U}^\\perp }\\right)_{ii}\\\\& \\le 1 + \\max _{j\\in [n]}D_{jj}^{-1} \\left(P_{\\mathcal {U}^\\perp }(I-e_ie_i^T)P_{\\mathcal {U}^\\perp }\\right)_{ii}\\\\& < 1 + 4[P_{\\mathcal {U}^\\perp }]_{ii} - 4[P_{\\mathcal {U}^\\perp }]_{ii}^2\\\\& = 2 - 4([P_{\\mathcal {U}^\\perp }]_{ii} - 1/2)^{2}\\\\& \\le 2$ where we have used the assumption that $[P_{\\mathcal {U}^\\perp }]_{ii}>1/2$ for all $i$ and the fact that $P_{\\mathcal {U}^\\perp }^{2} = P_{\\mathcal {U}^\\perp }$ .",
"Applying Walters's theorem completes the proof.",
"Proof of Proposition REF We now establish Proposition REF , giving a bound on the probability that a suitably random subspace is realizable by bounding the probability that it has coherence strictly bounded above by $1/2$ .",
"It suffices to show that $\\Vert P_{\\mathcal {U}} e_i\\Vert ^2 \\le (1-2\\epsilon )(1/2-\\epsilon ) = 1/2 - 2\\epsilon ^2 < 1/2$ for all $i$ with high probability.",
"The main observation we use is that if $\\mathcal {U}$ is a random $r$ dimensional subspace of $\\mathbb {R}^n$ and $x$ is any fixed vector with $\\Vert x\\Vert =1$ then $\\Vert P_{\\mathcal {U}} x\\Vert ^2 \\sim \\beta (r/2,(n-r)/2)$ where $\\beta (p,q)$ denotes the beta distribution [13].",
"In the case where $r = (1/2 - \\epsilon )n$ , using a tail bound for $\\beta $ random variables [13] we see that if $x\\in \\mathbb {R}^n$ is fixed and $r > 3/\\epsilon ^2$ then $ \\Pr [ \\Vert P_\\mathcal {U} x\\Vert ^2 \\ge (1+2\\epsilon )(1/2 -\\epsilon )] <\\frac{1}{a_\\epsilon }\\frac{1}{(\\pi (1/4 - \\epsilon ^2))^{1/2}}n^{-1/2}e^{-a_\\epsilon k}$ where $a_\\epsilon = \\epsilon - 4\\epsilon ^2/3$ .",
"Taking a union bound over $n$ events, as long as $r > 3/\\epsilon ^2$ $\\Pr \\left[\\mu (\\mathcal {U}) \\ge 1/2\\right] & \\le \\Pr \\left[\\Vert P_{\\mathcal {U}} e_i\\Vert ^2 \\ge (1-2\\epsilon )(1/2-\\epsilon )\\;\\; \\text{for some $i\\in [n]$}\\right]\\\\& \\le n\\cdot \\frac{1}{a_\\epsilon (\\pi (1/4 - \\epsilon ^2))^{1/2}} n^{-1/2} e^{- a_\\epsilon k} =\\bar{c}n^{1/2}e^{-\\tilde{c}n}$ for appropriate positive constants $\\bar{c}$ and $\\tilde{c}$ .",
"Acknowledgements The authors would like to thank Prof. Sanjoy Mitter for helpful discussions."
],
[
"Conclusions",
"We established a link between three problems of independent interest: deciding whether there is a centered ellipsoid passing through a collection of points, understanding the structure of the faces of the elliptope, and deciding which pairs of diagonal and low rank-matrices can be recovered from their sum using a tractable semidefinite-programming-based heuristic, namely minimum trace factor analysis.",
"We provided a simple sufficient condition, based on the notion of the coherence of a subspace, which ensures the success of minimum trace factor analysis, and showed that this is the best possible coherence-threshold-type sufficient condition for this problem.",
"We provided natural generalizations of our results to the problem of analyzing tractable block-diagonal and low-rank decompositions, showing how the symmetries of this problem allow us to reduce much of the analysis to the original diagonal and low-rank case.",
"Our results suggest both the power and the limitations of using `coarse' properties of a subspace such as coherence to gain understanding of the faces of the elliptope (and related problems).",
"The power of results based on such properties is that they do not have explicit dimension-dependence, unlike previous results on the faces of the elliptope.",
"At the same time, the lack of explicit dimension dependence typically yields conservative sufficient conditions for high-dimensional problems.",
"It would be interesting to find a hierarchy of coherence-like conditions that provide less conservative sufficient conditions for higher dimensional problem instances."
],
[
"Proof of Lemma ",
"We first establish Lemma REF which gives an interpretation of the balance condition in terms of ellipsoid fitting.",
"The proof is a fairly straightforward application of linear programming duality.",
"Throughout let $V$ be the $k\\times n$ matrix with columns given by the $v_i$ .",
"The point $v_i\\in \\mathbb {R}^k$ is on the boundary of the convex hull of $\\pm v_1,\\ldots ,\\pm v_n$ if and only if there exists $x\\in \\mathbb {R}^k$ such that $\\langle x,v_i\\rangle = 1$ and $|\\langle x,v_j\\rangle | \\le 1$ for all $j\\ne i$ .",
"Equivalently, the following linear program (which depends on $i$ ) is feasible $\\operatornamewithlimits{\\text{minimize}}_x \\; \\langle 0,x\\rangle \\quad \\text{subject to}\\quad \\left\\lbrace \\begin{array}{rcl} v_i^Tx \\!\\!\\!\\!",
"& = & \\!\\!\\!\\!",
"1\\\\|v_j^Tx| \\!\\!\\!\\!",
"& \\le & \\!\\!\\!\\!",
"1\\;\\;\\text{for all $j\\ne i$.",
"}\\end{array}\\right.$ Suppose there is some $i$ such that $v_i$ is in the interior of $\\text{conv}\\lbrace \\pm v_1,\\ldots ,\\pm v_n\\rbrace $ .",
"Then (REF ) is not feasible so the dual linear program (which depends on $i$ ) $\\operatornamewithlimits{\\text{maximize}}_u \\; u_i - \\sum _{j\\ne i} |u_j| \\quad \\text{subject to}\\quad \\begin{array}{rcl} V u \\!\\!\\!\\!",
"& = & \\!\\!\\!\\!",
"0\\end{array}$ is unbounded.",
"This is the case if and only if there is some $u$ in the nullspace of $V$ such that $u_i > \\sum _{j\\ne i} |u_j|$ .",
"If such a $u$ exists, then it is certainly the case that $|u_i| \\ge u_i > \\sum _{j\\ne i} |u_j|$ and so $u$ is not balanced.",
"Conversely if $u$ is in the nullspace of $V$ and $u$ is not balanced then either $u$ or $-u$ satisfies $u_i > \\sum _{j\\ne i} |u_j|$ for some $i$ .",
"Hence the linear program (REF ) associated with the index $i$ is unbounded and so the corresponding linear program (REF ) is infeasible.",
"It follows that $v_i$ is in the interior of the convex hull of $\\pm v_1,\\ldots ,\\pm v_n$ ."
],
[
"Completing the proof of Theorem ",
"We now complete the proof of Theorem REF by establishing the following result about the existence of a non-negative solution to the linear system (REF ).",
"If $\\mu (\\mathcal {U}) < 1/2$ then there is $\\lambda \\ge 0$ such that $\\left(P_{\\mathcal {U}^\\perp }^*(\\lambda )P_{\\mathcal {U}^\\perp }\\right) = \\mathbf {1}.$ We note that the linear system (REF ) can be written as $P_{\\mathcal {U}^\\perp }\\circ P_{\\mathcal {U}^\\perp }\\lambda = \\mathbf {1}$ where $\\circ $ denotes the entry-wise product of matrices.",
"As such, we need to show that $P_{\\mathcal {U}^\\perp }\\circ P_{\\mathcal {U}^\\perp }$ is invertible and $(P_{\\mathcal {U}^\\perp }\\circ P_{\\mathcal {U}^\\perp })^{-1}\\mathbf {1}\\ge 0$ .",
"To do so, we appeal to the following (slight restatement) of a theorem of Walters [31] regarding positive solutions to certain linear systems.",
"[Walters [31]] Suppose $A$ is a square matrix with non-negative entries and positive diagonal entries.",
"Let $D$ be a diagonal matrix with $D_{ii} = A_{ii}$ for all $i$ .",
"If $y>0$ and $2y - AD^{-1}y > 0$ then $A$ is invertible and $A^{-1}y > 0$ .",
"In our case we take $A = P_{\\mathcal {U}^\\perp }\\circ P_{\\mathcal {U}^\\perp }$ and $y = \\mathbf {1}$ in Theorem REF .",
"It is clear that $P_{\\mathcal {U}^\\perp } \\circ P_{\\mathcal {U}^\\perp }$ is entry-wise non-negative.",
"Furthermore $[P_{\\mathcal {U}^\\perp }]_{ii} = 1-[P_\\mathcal {U}]_{ii} > 1-\\mu (\\mathcal {U}) > 1/2$ and so $D_{ii} = [P_{\\mathcal {U}^\\perp }\\circ P_{\\mathcal {U}^\\perp }]_{ii} > 1/4$ .",
"It then remains to show that $P_{\\mathcal {U}^\\perp }\\circ ~P_{\\mathcal {U}^\\perp }~D^{-1}\\mathbf {1}<2\\mathbf {1}$ .",
"Consider the $i$ th such inequality, and observe that $[P_{\\mathcal {U}^\\perp }\\circ P_{\\mathcal {U}^\\perp } D^{-1}\\mathbf {1}]_i& = \\left(P_{\\mathcal {U}^\\perp }D^{-1}P_{\\mathcal {U}^\\perp }\\right)_{ii}\\\\& = \\left(P_{\\mathcal {U}^\\perp }D_{ii}^{-1}e_ie_i^TP_{\\mathcal {U}^\\perp }\\right)_{ii} +\\left(P_{\\mathcal {U}^\\perp }(D^{-1} - D_{ii}^{-1}e_ie_i^T)P_{\\mathcal {U}^\\perp }\\right)_{ii}\\\\& \\le 1 + \\max _{j\\in [n]}D_{jj}^{-1} \\left(P_{\\mathcal {U}^\\perp }(I-e_ie_i^T)P_{\\mathcal {U}^\\perp }\\right)_{ii}\\\\& < 1 + 4[P_{\\mathcal {U}^\\perp }]_{ii} - 4[P_{\\mathcal {U}^\\perp }]_{ii}^2\\\\& = 2 - 4([P_{\\mathcal {U}^\\perp }]_{ii} - 1/2)^{2}\\\\& \\le 2$ where we have used the assumption that $[P_{\\mathcal {U}^\\perp }]_{ii}>1/2$ for all $i$ and the fact that $P_{\\mathcal {U}^\\perp }^{2} = P_{\\mathcal {U}^\\perp }$ .",
"Applying Walters's theorem completes the proof."
],
[
"Proof of Proposition ",
"We now establish Proposition REF , giving a bound on the probability that a suitably random subspace is realizable by bounding the probability that it has coherence strictly bounded above by $1/2$ .",
"It suffices to show that $\\Vert P_{\\mathcal {U}} e_i\\Vert ^2 \\le (1-2\\epsilon )(1/2-\\epsilon ) = 1/2 - 2\\epsilon ^2 < 1/2$ for all $i$ with high probability.",
"The main observation we use is that if $\\mathcal {U}$ is a random $r$ dimensional subspace of $\\mathbb {R}^n$ and $x$ is any fixed vector with $\\Vert x\\Vert =1$ then $\\Vert P_{\\mathcal {U}} x\\Vert ^2 \\sim \\beta (r/2,(n-r)/2)$ where $\\beta (p,q)$ denotes the beta distribution [13].",
"In the case where $r = (1/2 - \\epsilon )n$ , using a tail bound for $\\beta $ random variables [13] we see that if $x\\in \\mathbb {R}^n$ is fixed and $r > 3/\\epsilon ^2$ then $ \\Pr [ \\Vert P_\\mathcal {U} x\\Vert ^2 \\ge (1+2\\epsilon )(1/2 -\\epsilon )] <\\frac{1}{a_\\epsilon }\\frac{1}{(\\pi (1/4 - \\epsilon ^2))^{1/2}}n^{-1/2}e^{-a_\\epsilon k}$ where $a_\\epsilon = \\epsilon - 4\\epsilon ^2/3$ .",
"Taking a union bound over $n$ events, as long as $r > 3/\\epsilon ^2$ $\\Pr \\left[\\mu (\\mathcal {U}) \\ge 1/2\\right] & \\le \\Pr \\left[\\Vert P_{\\mathcal {U}} e_i\\Vert ^2 \\ge (1-2\\epsilon )(1/2-\\epsilon )\\;\\; \\text{for some $i\\in [n]$}\\right]\\\\& \\le n\\cdot \\frac{1}{a_\\epsilon (\\pi (1/4 - \\epsilon ^2))^{1/2}} n^{-1/2} e^{- a_\\epsilon k} =\\bar{c}n^{1/2}e^{-\\tilde{c}n}$ for appropriate positive constants $\\bar{c}$ and $\\tilde{c}$ ."
],
[
"Acknowledgements",
"The authors would like to thank Prof. Sanjoy Mitter for helpful discussions."
]
] | 1204.1220 |
[
[
"NIKEL: Electronics and data acquisition for kilopixels kinetic\n inductance camera"
],
[
"Abstract A prototype of digital frequency multiplexing electronics allowing the real time monitoring of microwave kinetic inductance detector (MKIDs) arrays for mm-wave astronomy has been developed.",
"Thanks to the frequency multiplexing, it can monitor simultaneously 400 pixels over a 500 MHz bandwidth and requires only two coaxial cables for instrumenting such a large array.",
"The chosen solution and the performances achieved are presented in this paper."
],
[
"Introduction",
"Microwave kinetic inductance detectors (MKIDs) have proven to be a solid working alternative to traditional bolometers for millimeter and sub-millimeter astronomy [1], [2], [3], [4].",
"MKIDs are composed of high-quality superconducting resonant circuits electromagnetically coupled to a transmission line.",
"They are designed to resonate in the microwave domain [5], [6], [7].",
"For astronomical applications, the resonances typically lie between 1 to 10 GHz and have loaded quality factors around $\\rm Q_{L}=10^5$ , corresponding to a typical bandwidth of $\\rm \\Delta f = f/Q_{L} \\sim 10-100$ kHz.",
"Provided that the MKID resonant frequencies can be easily adjusted by layout design, it is possible to couple a large number of MKIDs with different resonance frequencies to a single transmission line [8].",
"Indeed, a large number of MKIDs can naturally be read out by a frequency-based multiplexing system with no loss of performance [9].",
"In practice, the average frequency spacing between resonators is between 1 and 2 MHz [1].",
"Thus, in order to ensure the largest sky coverage and overall signal to noise per unit of time with a reduce number of cables (few) feedthrough to the cryostat, the analog bandwidth and the number of detectors (resonators) managed by the electronics must be maximized.",
"At this respect, we present here a building block for the NIKA camera [1], [2] that is able to monitor simultaneously 400 pixels over a 500 MHz bandwidth."
],
[
"Instrumentation methodology",
"The instrumentation setup used for NIKA and its associated electronics is extensively described in [9].",
"In summary, the excitation frequency comb is generated at baseband in the electronics using coordinate rotation digital computer (CORDIC), up-converted with an IQ mixer to the 1 to 10 GHz frequency and injected in the resonator line.",
"The returning and thus modified frequency comb is down-converted and analyzed by channelized Digital Down Converters (DDC) to determine each tone amplitude and phase.",
"Aside from good signal to noise ratio (SNR) on the whole chain, the first limitation on the number of MKIDs managed by this solution is given by the digital to analog converter (DAC) and the analog to digital converter (ADC) bandwidths.",
"The second constraint comes from the computing power limitation.",
"For a FPGA (Field Programmable Gate Array), the computational power is determined by the available amount user logic and multiplier block times their maximum running frequency.",
"Indeed, thanks to the inherently achievable parallelization in FPGAs, this figure is much larger compared to DSPs that have only a few Multiplier Accumulators.",
"Starting from the previous version, which was able to manage a line of 128 tones over a bandwidth of 125 MHz, three solutions are possible to increase the multiplexing factor per line.",
"The first solution would be to juxtapose several of the previous electronic boards, each one managing its share of bandwidth, see figure REF .",
"Unfortunately, the analog filters required to separate each share of bandwidth before down-converting have such a stringent separation requirement to avoid crosstalk due to image frequencies that they cannot be constructed.",
"Figure: Overview of the setup using the juxtaposition of several electronics to monitor a MKID array.",
"Each electronics generating the two frequency combs (each tone phase shifted by 90° between I and Q) is followed by an IQ up-mixer.",
"The excitation combs up-converted at high frequencies are summed and the resulting signal is fed to a programmable attenuator for power adjustment.",
"After passing through the cryostat and the low noise boost amplifier each share of bandwidth is separated by highly selective filters before passing through the down-mixers and returned to the corresponding electronics.The second option is to use faster ADCs an DACs combined to a larger computing power (FPGA) in order to directly cover a larger bandwidth.",
"Following this path, two concurrent approaches still remain.",
"The first “obvious” solution is to directly generate the frequency comb at twice the desired bandwidth and to perform channelized DDC with the ADC signal.",
"Unfortunately, due to the frequency limitation of state of the art FPGAs this can only be achieved by performing massive design pipelining on both sides, excitation and analysis, and therefore makes it extremely complicated.",
"The third option, which we have chosen, is to use modern DACs featuring digital modulator and interpolator followed by very steep half-band filters for generating the excitation comb.",
"With these, the total frequency bandwidth to cover is split into smaller bands where the frequency combs can be computed at a moderate frequency, digitally up-converted and filtered to avoid unwanted spurious frequencies.",
"Finally, each band contribution is then summed before being up-converted to the frequency band of interest by an IQ up-mixer.",
"At reception side, the returning signal is down converted to baseband and is digitized by a fast ADC.",
"Then, the digitized signal goes through a polyphase filter bank with equal bandwidth overlapping bands.",
"This filter, has the ability to separate the total bandwidth in five smaller frequency bands and to down convert each of them to baseband.",
"The sub frequency bands are chosen such as to match the excitation bands.",
"The filter outputs are fed to the corresponding channelized DDC in order to be analyzed.",
"The benefit of this architecture is to limit the massive pipelining to the polyphase filter part, and thus, to dramatically reduce the required amount of user logic for the frequency comb generation and the channelized DDC."
],
[
"Hardware development",
"Following section , a dedicated hardware, the New Iram KID ELectronics (NIKEL), able to manage 400 resonators over a bandwidth of 500 MHz was developed.",
"NIKEL is designed such as to manage five adjacent bands of 100 MHz.",
"This choice was driven by the chosen DAC capabilities (AD9125 from analog devices).",
"As shown in figure REF , the NIKEL electronic board is composed of a central FPGA (labeled `split') which receives the 12 bit ADC (ADS5400 from Texas Instruments) output data flow at 1 GSPS and of five processing FPGAs (labeled `proc').",
"Figure: The electronic board is composed of a central FPGA (labeled `split') which receives the 12 bit ADC (ADS5400 from Texas Instruments) output data flow at 1 GSPS and of five processing FPGA (labeled `proc').Each of the latter is driving its associated DAC with the adequate frequency comb which can feature up to 80 tones.",
"The `proc' FPGA is connected to the `split' FPGA with two links.",
"The first of these, labeled `fake ADC', is a 12 bit parallel LVDS link running at 250 MSPS that is carrying the part of bandwidth corresponding to the excitation signal.The second link, labeled `GTX link', is periodically (at ∼\\sim 953 Hz) conveying the 80 DDC results over a 2 Gb/s serial link.An additional slow speed DAC, driven by the `split' FPGA, is implemented to be able to provide a 500 Hz modulation signal.",
"The communication with the hardware is ensured via a USB2 capable micro-controller and an interface FPGA that accommodates different voltage levels.Each of the latter is driving its associated DAC with the adequate frequency comb which can feature up to 80 tones.",
"The `proc' FPGA is connected to the `split' FPGA with two links.",
"The first of these, labeled `fake ADC', is a 12 bit parallel LVDS link running at 250 MSPS that is carrying the part of bandwidth corresponding to the excitation signal.",
"The second link, labeled `GTX link', is periodically (at $\\sim $ 953 Hz) conveying the 80 DDC results over a 2 Gb/s serial link.",
"The six FPGAs are from the same founder (Xilinx XC6VLX75T-2FFG484C).",
"They provide a satisfactory amount of available user logic, coupled to a sufficiently large Multiplier Accumulator block (MAC) count.",
"They also feature eight high speed serial links.",
"An additional slow speed DAC, driven by the `split' FPGA, is implemented in order to be able to provide a $\\sim $ 500 Hz modulation signal.",
"Provided that the board can be clocked with a reference clock, a bidirectional port was provided to allow synchronization between several boards performing acquisition on the same kilo-pixel camera.",
"When using several NIKEL electronics, one board must be configured as master and provide the synchronization signal, while the others are configured as slaves and should start their acquisition upon reception of this synchronization signal.",
"The communication with the hardware is ensured via a USB2 capable micro-controller and an interface FPGA that accommodates the different voltage levels.",
"It allows the dynamic FPGA reconfiguration, the tone frequencies adjustments and the data readout.",
"A picture of the board can be seen in figure REF .",
"It is a 14 layers PCB having a dimension of $\\rm 184\\ mm \\times 153\\ mm$ .",
"The inner dielectric layers are made of traditional FR4 epoxy while the outside layers consist of RO4350 high frequency circuit [10] that have lower dielectric losses and therefore are well suited to accommodate the 2 Gb/s serial links, the DAC outputs that provides samples at 1 GSPS and the ADC input signal.",
"Figure: Picture of the NIKEL board.",
"It is a 14 layers PCB having a dimension of 184 mm ×153 mm \\rm 184\\ mm \\times 153\\ mm.Due to to the extensive FPGA resource usage and their running frequency (250 MHz) special care was taken in designing the electronic board power supply.",
"Indeed, each FPGA core supply draws a current of about 5 A when all tones are activated.",
"Thanks to the usage of DC/DC converters the total current drawn on the input power supply is below 20 A, thus a maximum required total power of 100 W (or 0.25 W per channel)."
],
[
"Polyphase filter design and full chain simulation",
"As introduced in sections and , the received signal, sampled at 1 GS/s, must be decomposed in five 250 MS/s data streams, each stream having a useful bandwidth of 100 MHz in order to cover the 0-500 MHz full bandwidth.",
"Frequency modulation/demodulation is a well documented digital signal processing technique [11] for data transmission (channelization) or audio/image coding application.",
"Those classical techniques, based on Discrete Fourier Transform (DFT), Modified DFT (MDFT) or Cosine Modulated filter banks, suffer from aliasing and data distortion mainly due to the critical sampling of created sub-bands.",
"Overlapping polyphase filter banks as described in [12], offer a computationally efficient solution and have only two drawbacks for our application: a first sub-band which is half the bandwidth of the others and the last sub-band (also half bandwidth) is not usable.",
"The same technique can however be adapted to the wanted filter bank specificity, with an acceptable increase of complexity."
],
[
"Theoretical formulation",
"Digital filter banks implementations are often non-intuitive, but are however composed of simple successive digital signal processing blocks, re-arranged in different form to increase computing efficiency.",
"The simplified processing for each band of the filter bank is described hereafter.",
"At first, a frequency shift is performed to translate the band of interest around 0 Hz.",
"Then a low pass filtering followed by decimation is applied to select the frequencies of interest.",
"This paragraph described the basic blocks arrangement involved in the specific processing used here.",
"The input data stream is a real signal, sampled at $\\rm F_{si}=1\\, GS/s$ where four consecutive samples are presented at the filter bank input at each system clock cycle (250 MHz) while the filter bank outputs five different samples, one for each output band.",
"The signal processing for each band k (k=0..4) is done in five consecutive steps.",
"An illustration is provided for band k=2 in figure REF and the operations are described hereafter: Perform an input signal frequency shift of $\\rm -(2k+1)\\cdot F_{si}/20$ where $\\rm F_{si}/20=$ 50 MHz.",
"This is obtained by multiplying the input signal by the complex exponential $\\rm e^{-j\\pi (2k+1)n/10}$ where n is the input signal sample index.",
"Filter the complex signal by a low pass Finite Impulse Response (FIR) filter having a passband of $\\rm F_{si}/20$ and a maximum rejection after $\\rm F_{si}/10+ F_{si}/80=F_{si}/16$ .",
"Decimate the result by a factor of 4.",
"The new data rate then becoming $\\rm F_{so} = F_{si}/4$ and the resulting filtered signal bandwidth $\\rm [-F_{so}/4,F_{so}/4]$ .",
"Up convert by $\\rm F_{so}/4$ .",
"In practice, realized by multiplying the previous signal by $e^{-j\\pi m/2}$ where $m=n/4$ is the sample index of the decimated data stream.",
"The resulting complex signal covers the frequency band $\\rm [0, F_{so}/2]$ Finally, keep only the complex signal real part.",
"This will add the complex conjugate negative frequency image in the frequency plane.",
"The output real signal is then correctly sampled at $\\rm F_{so}$ without aliasing.",
"Figure: Illustration of the polyphase filtering algorithm detailed for band k=2.For a given tone c (frequency $\\rm f_{ck}$ ), located in the kth band of the input data stream can have its frequency expressed as $\\rm f_{ck} = F_{si}/10 \\cdot k + f_c$ .",
"Due to the whole processing, it should be noted this tone will not appear in the kth filter bank output at the frequency $\\rm f_{c}$ , but at $\\rm f_c+F_{so}/20$ .",
"Consequently, the tones used for KID excitation must present a frequency shift of $\\rm -F_{so}/20$ with respect to the one used for performing DDC on the returning signal provided by the filter.",
"In practice, the FIR filter do not have to be as steep as noted in step 2 given the fact that any aliasing causing frequency folding in the useless sidebands causes no harm.",
"Consequently, a poorer rejection up to $\\rm F_{si}/16+ F_{si}/80=3F_{si}/40$ can be tolerated and greatly eases the FIR filter design.",
"Unfortunately, this processing is very inefficient in this direct form for several reasons.",
"For instance, the filtering is done for each band and on complex data.",
"Furthermore, resource consuming FIR filtering is performed on the frequency shifted data, but it is followed by decimation.",
"In other words, samples are computed needlessly.",
"These computing inefficiencies can be considerably improved by grouping the different frequency shifts and by using polyphase filters."
],
[
"Polyphase filters",
"If $x(n)$ is the input sample signal, $x^{^{\\prime }}_k(n)$ , the frequency shifted data stream for the band k, k=0..4, is expressed by equation REF .",
"$x^{^{\\prime }}_k(n)=x(n)\\cdot e^{-2j\\pi \\frac{(2k+1)n}{20}}$ The output, $ x^{^{\\prime \\prime }}_k(n)$ , of the low pass FIR filter with coefficients $a(p)$ is then $x^{^{\\prime \\prime }}_k(n)=\\sum _p a(p) \\cdot x^{^{\\prime }}_k(n-p)=e^{-2j\\pi \\frac{(2k+1)n}{20}} \\sum _p a(p)\\cdot x(n-p) e^{2j\\pi \\frac{(2k+1)p}{20}}$ Provided that the filtered signal is down-sampled by a factor of 4, $ x^{^{\\prime \\prime }}_k(n)$ can be only computed for $n=4 m$ .",
"By decomposing the filter into a 20 phases polyphase filters, where the coefficients index p is given by $p = q+20r$ with q=0..19, equation REF can be written in the following form $x^{^{\\prime \\prime }}_k(m)=e^{-2j\\pi \\frac{(2k+1)m}{5}} \\sum _{q=0}^{19} e^{2j\\pi \\frac{(2k+1)q}{20}} \\cdot w_q(m)$ where $w_q(m)$ is the output of the qth phase polyphase filter.",
"$w_q(m)=\\sum _r a(q+20r)\\cdot x(4m-q-20r)$ The final step is to up convert the signal by $\\rm F_{so}/4$ , which is equivalent to a complex multiplication by $\\rm j^m$ and taking the real part of the resulting complex number $y_k(m)= Re \\left[e^{j\\frac{\\pi }{2}m} \\cdot e^{-2j\\pi \\frac{(2k+1)m}{5}} \\cdot \\sum _{q=0}^{19} e^{j\\pi \\frac{(2k+1)q}{10}} \\cdot w_q(m)\\right]$ The use of the polyphase decomposition of the FIR filter considerably reduces the computation cost.",
"However, it can be seen in equation REF that a lot of calculation still need to be done on complex numbers before keeping only the real part.",
"This leaves some margins for optimization."
],
[
"Optimized reconstruction",
"Since input and outputs of the polyphase filter banks are real signals, it is possible to perform all computation only on real numbers.",
"Equation REF can be re-written as $y_k(m)= Re \\left[\\sum _{q=0}^{19} e^{j\\frac{\\pi }{10}[5m+(2k+1)(q-4m)]} \\cdot w_q(m)\\right]$ We can change the order of the polyphase filter outputs $w_q(m)$ in the sum by introducing new data streams $w^{^{\\prime }}_l(m) =w_q(m)$ with $l = q +( m\\ mod\\ 20)$ .",
"Due to the $2j\\pi $ periodicity of the complex exponential function, the output of the filter bank can be expressed by the following formula: $y_k(m)= Re \\left[(-1)^{mk} \\cdot \\sum _{l=0}^{19} e^{j\\frac{\\pi (2k+1)l}{10}} \\cdot w^{^{\\prime }}_l(m)\\right]= (-1)^{mk} \\cdot \\sum _{l=0}^{19} \\cos \\left(\\frac{(2k+1)l\\pi }{10}\\right) \\cdot w^{^{\\prime }}_l(m)$ This simple rotation of the polyphase filter outputs orders, greatly simplify the formula.",
"Moreover, each filter bank output can now be computed without complex arithmetics."
],
[
"Excitation DAC",
"In order to validate the DAC choice and to select its best configuration for each band, that are the digital modulator frequency and the half-band filters to engage, the DAC behavior was simulated.",
"Indeed, the Frequency Tuning Word (FTW) allowing the configuration of the modulator frequency is given by the following formula $FTW=\\dfrac{f_{carrier}}{f_{nco}} \\times 2^{32}$ , where $\\rm f_{nco}$ is 500 MHz.",
"Ideally, it is desired to have five frequency bands and thus five different carrier frequencies going from 0 to 400 Mhz in steps of 100 MHz.",
"Consequently, the first approach would be to select these exact values that are perfectly suited to fit with the half-band filters.",
"Unfortunately, these carrier frequencies would yield real FTW instead of integer FTW.",
"Using these rounded values would induce a small offset in frequency which would be observed as a $2 \\pi /2^{32}$ phase shift every $2^{32}$ clock cycles.",
"Consequently, the carrier frequencies were adequately chosen to obtain a $\\dfrac{f_{carrier}}{500\\ MHz}$ ratio of 0, 7/32, 13/32, 19/32 and 26/32 yielding integer FTW values.",
"Given the fact that the manufacturer provided the DAC half-band filter coefficients, a thorough simulation of the DAC, having the appropriate filters selected, was conducted for the five excitation bands.",
"The results confirm that non optimal carrier frequencies are acceptable.",
"In particular, the flatness is slightly degraded while the ripple remains below 0.06 dB.",
"This mandatory carrier frequency shift with respect to the ideal value, must be pre-compensated in the FPGA `proc' building the excitation frequency comb by a digital modulator that apply a frequency shift in the opposite direction to virtually obtain carrier frequencies at the requested values (from 0 to 400 MHz).",
"The required compensations, expressed as a ratio of $\\rm F_{so}$ , are respectively: 0, -3/80, -2/80, +2/80 and -4/80.",
"This shift is accomplished in the meantime as the frequency shift of $\\rm -F_{so}/20$ needed to compensate the polyphase filter bank induced shift (see section REF ).",
"Consequently, the final required compensations, again expressed in ratio of $\\rm F_{so}$ are : -1/20, -7/80, -4/80, -3/80 and -3/40.",
"In practice, these are implemented with 80 values sine and cosine table feeding digital modulator."
],
[
"Polyphase filter",
"Likewise to the excitation DAC, the polyphase filter was simulated in order to assess its performances and to find the best implementation options matching the FPGA available resources.",
"During the firmware design, the mathematical simulation tool was used to build stimulus files and reference filter output that were used by the VHDL simulation tool to speed up the design and validate the firmware implementation of the filter.",
"Figure: Simulation of the selected FIR filter.",
"Top left figure shows the global filter response.",
"Top right figure, shows the gain fluctuation in the passband and bottom figure shows the steep rejection after the passband.The simulation was also an asset in designing the FIR filter used.",
"As shown in figure REF , the selected FIR has a good flatness over the useful bandwidth ($<$ 0.01 dB).",
"The choice was made to concede a larger than specified transition band [50-75 MHz] while having an excellent rejection (-170 dB) in the stopband.",
"As explained in section REF , possible resulting aliasing does not impact DDC performances in the useful bandwidth.",
"Additionally, the quantization noise due to the use of the fixed point Multiplier ACcumulator (MAC) was evaluated and confirmed to be negligible with respect to the quantization noise of the ADC.",
"A full polyphase filter simulation, where three tones (205 MHz, 250 MHz, 299 MHz) are injected at the filter input, is shown in figure REF .",
"The top left figure shows the input signal spectrum, and the other plots show the frequency content of each output of the polyphase filter.",
"It can be observed that the expected tones lie in the expected band k=2, while the spurious appearing in band k=1 and k=3 are in their rejected side bands, i.e above 200 MHz for band k=1 and below 300 MHz for band k=3.",
"Figure: full polyphase filter simulation, where three tones (205 MHz, 250 MHz, 299 MHz) are injected at the filter input.",
"The top left figure shows the input signal spectrum, and the other plots show the frequency content of each output of the polyphase filter.",
"It can be observed that the expected tones lie in the expected band k=2, while the spurious appearing in band k=1 and k=3 are in their rejected side bands, i.e above 200 MHz for band k=1 and below 300 MHz for band k=3."
],
[
"FPGA `split' description",
"The `split' FPGA, shown in figure REF , contains two main parts.",
"The first part, which is the key-point of the overall design, is composed of the ADC interface, the polyphase filter bank and the five `fakeADC' outputs, each carrying its share of the bandwidth to the dedicated `proc' FPGA.",
"The second part consists of five GTX receivers that collect the I/Q data provided by each `proc' FPGA, a data concentrator, a large FIFO and a USB interface.",
"The GTX2IQ receiver blocks are designed to operate at a speed of 2 Gb/s.",
"This is the speed required to carry 32 bit at 50 MHz with an 8b/10b encoding.",
"Every $\\sim $ 1.05 ms (218 clock cycles at 250 MHz) a 644 bytes data frame is received (see section REF ) and stored in a small reception buffer (1 k word deep).",
"Once all GTX2IQ received its data frame, the `data concentrator' transfers each link data into the global data buffer labeled `USB interface FIFO' (32 k word deep) to make the complete data frame available for data acquisition via the USB interface.",
"The USB interface is mostly in charge of reading out the `USB interface FIFO' and thus of performing data acquisition.",
"The required data throughput is $\\rm 644 \\times 5 \\times 953=3\\ MB/s$ .",
"The interface is also used to set the master/slave mode, to arm the acquisition, to select the modulation mode and to configure and recover the status of the GTX transceiver links.",
"The DAC modulation block is used to generate an optional modulation signal which can be a 2 or 4 values modulation signal, depending whether it is desired or not to compute the sensitivity (first derivative) and the sensitivity variation (second derivative) of the I/Q measurement.",
"When this block is activated, the modulation signal is modified every integration cycle.",
"To ensure the modulation synchronousness with the integration performed in the DDC, the initial start of the modulation is adjustable with a resolution of 4 ns and up to one full integration cycle.",
"The polyphase filter bank implementation (shown in figure REF ) is composed of five successive stages.",
"During the design, several stratagems where used to minimize the number of DSP48 blocks used and hence to allow the filter to fit in the chosen FPGA.",
"The input stage is composed of a shift register bank featuring 20 registers of 12 bit.",
"It receives four new ADC samples every clock cycle and at the same time performs a four data samples shifting from the newest data to the oldest.",
"At the output of this stage the $\\rm n$ to $\\rm n-19$ samples are provided to the following stage.",
"The following stage is composed of 20 FIR filters, each processing one of the `input stage' output.",
"The FIR filters feature 45 taps and are implemented in the transposed direct form which suits perfectly the possibilities offered by the DSP48 blocks inside the Virtex 6 FPGA.",
"Given the fact that for each FIR filter only 9 taps out of 45 are non zero, the zero coefficient taps are replaced by simple registers.",
"This artifice alone allows to use only 180 DSP48 blocks for the whole filter bank instead of 450.",
"The third stage, named `rotation block', is used to rotate the vector composed of the 20 FIR filter outputs and to provide it to the `optimized reconstruction block'.",
"The rotation consist in routing the data according to the following equation: $W^{^{\\prime }}_l(m) =W_{(l- m)\\ mod\\ 20}(m)$ where l=0..19 and m the sample index.",
"In practice, this is implemented with 20 high performance multiplexers having 20 inputs and one output.",
"Each of these multiplexers is controlled by a counter having a 0 to 19 range and is initialized with a value according to the `optimized reconstruction block' input it is connected to.",
"Figure: Polyphase filter bank implementation overview.According to equation REF given in section REF , having the vectors Y(m), $\\rm W^{^{\\prime \\prime }}(m)$ and $\\rm W^{^{\\prime }}(m)$ being respectively composed of $y_k(m)$ , $w^{^{\\prime \\prime }}_k(m)$ and $w^{^{\\prime }}_l(m)$ for k=0..4 and l=0..19, the optimized reconstruction can be computed by $\\rm Y(m)=J^{m} \\cdot W^{^{\\prime \\prime }}(m)=J^{m} \\cdot A \\cdot W^{^{\\prime }}(m)$ , where: $ J=\\begin{bmatrix}1 & 0 & 0 & 0 & 0 \\\\0 & -1 & 0 & 0 & 0 \\\\0 & 0 & 1 & 0 & 0 \\\\0 & 0 & 0 & -1 & 0 \\\\0 & 0 & 0 & 0 & 1 \\\\\\end{bmatrix}$ and $ A=\\begin{bmatrix}1 & a & b & c & d & 0 & -d & -c & -b & -a & -1 & -a & -b & -c & -d & 0 & d & c & b & a \\\\1 & c & -d & -a & -b & 0 & b & a & d & -c & -1 & -c & d & a & b & 0 & -b & -a & -d & c \\\\1 & 0 & -1 & 0 & 1 & 0 & -1 & 0 & 1 & 0 & -1 & 0 & 1 & 0 & -1 & 0 & 1 & 0 & -1 & 0 \\\\1 & -c & -d & a & -b & 0 & b & -a & d & c & -1 & c & d & -a & b & 0 & -b & a & -d & -c \\\\1 & -a & b & -c & d & 0 & -d & c & -b & a & -1 & a & -b & c & -d & 0 & d & -c & b & -a\\end{bmatrix}$ with: $ a=\\cos \\left(\\frac{\\pi }{10}\\right), b=\\cos \\left(\\frac{\\pi }{5}\\right),c=\\cos \\left(\\frac{3\\pi }{10}\\right), d=\\cos \\left(\\frac{2\\pi }{5}\\right)$ It can be seen that computing $\\rm W^{^{\\prime \\prime }}(2)$ does not need any multiplier since the sign inversion can be simply obtained by computing the two complement of the input value.",
"Moreover, by using $2 \\times 16$ DSP48 slices for computing the non zero and non one values multiplication of the two first row of the A matrix, the last two rows can be obtained by sign inversion only.",
"The sign inversion is applied on one out of two multiplications only (when l is odd).",
"Figure REF provides a visual summary of the block implementation scheme.",
"The whole processing must be pipelined and as opposed to a FIR filter, each single sample is multiplied by the 20 coefficients of each row and then the operation results are all summed together.",
"This requires the use of two pipelined adders types: one having ten inputs for $\\rm W^{^{\\prime \\prime }}(2)$ (with four pipeline stages) and another having 18 inputs for the others (with five pipeline levels).",
"Finally, for a $[5,20] \\times [20,1]$ matrix multiplication, only 16 DSP48 slices are used.",
"Figure: Optimized operator taking benefit of the sign symmetry in A matrix between row k=0 and k=4 and row k=1 and k=3.",
"For row k=2 no multiplier is needed and since there is no phase requirement between the different frequency bands, the delay adjustment needed to compensate the DSP48 latency are in fact unnecessary.The last stage is actually associated with the previous stage (`optimized reconstruction output stage'), but for the sake clarity it is shown as separate block.",
"It corresponds to the first term of equation REF , which performs an alternate sign inversion for the odd bands resulting in a frequency shift by half the sampling frequency and in a frequency scale reversion.",
"The whole design uses 216 out of 288 DSP48 blocks, 18442 out of 93120 slice registers and 14879 out of 46560 slice LUT."
],
[
"FPGA `proc' description",
"As explained before, the processing FPGA, whose block diagram is shown in figure REF , is in charge of generating the frequency comb in its share of bandwidth and to perform the channelized DDC for each considered tone.",
"Figure: Overview of the `split' FPGA firmware.The communication between the USB interface and the FPGA is ensured via a serial link running at 50 MHz.",
"The various commands received are interpreted by the `proc_cmd' state machine.",
"Commands are of two kinds: the write commands and the read commands.",
"The write commands are used, for instance, to set the individual phase increment values and tone attenuation, the digital gain and the mixing table to use for performing a frequency shift.",
"Configuration and test modes can also be set via this interface.",
"Among the provided test modes, it may be noted that it is possible to record a `fakeADC' signal snapshot of 32 k samples in the `fakeADC_mem'.",
"The read command are used to request data from the FPGA like the GTX link status, the `fakeADC' link synchronization status.",
"Moreover the DAC internal registers values can be accessed.",
"Given the fact that the `fakeADC' data emitted by the `split' FPGA are synchronized by the system wide reference clock, a dedicated interface (fakeADC_input) is used to adjust the `fakeADC' bus delay in order to compensate the data sampling phase misalignment and thus to guarantee stable information sampling.",
"The locally synchronized data are provided to the tone managers.",
"The 80 tone manager outputs are fed to two pipelined adder in order to construct the in-phase and quadrature versions of the frequency comb.",
"Each comb version is then frequency shifted by an IQ mixer in order to compensate the residual up converting due to the polyphase filtering and the frequency shift due to the non optimal selection of the DAC internal modulator frequency (see section ).",
"The digital gain is used to numerically amplify the resulting signal by 0 dB up to 36 dB in steps of 6 dB before driving the DAC.",
"This feature is useful to adapt the signal to the ADC input range when less than 80 tones are used.",
"The IQ2GTX block is used to transmit the DDC results through the high speed link to the `split' FPGA for data concentration.",
"Along with these data, the detected peak amplitude, in absolute value, is transmitted for monitoring and to avoid DAC clipping.",
"Hence, the data frame is composed of $\\rm 2 \\times 80$ 32 bit words representing the in-phase/quadrature information.",
"Figure: Overview of a tone manager.",
"The block comprises a CORDIC generator, two digital attenuators for individual tone power adjustment and a DDC implemented with DSP48 blocks.The tone manager, which is depicted in figure REF , features a COordinate Rotation DIgital Computer (CORDIC) [13] block and a DDC that is composed of an I&Q demodulator followed by a Low Pass Filter (LPF).",
"The LPF, which is primarily used to remove the summed frequencies component from the spectrum, also provides unwanted frequencies rejection (e.g.",
"frequencies tuned to other pixels, white noise, ...).",
"Each CORDIC, implemented in a pipelined fashion and composed only of adders and subtracters, was designed to provide a 10 bit precision on the sine and cosine values calculated.",
"It uses 10 precalculated arc tangent values with 20 bit resolution.",
"The phase accumulator that feeds the CORDIC is used to adjust the frequency with a precision of $\\rm 250\\ MHz/2^{18} \\sim 953\\ Hz$ .",
"In order to avoid in phase startup at the maximum cosine or sine amplitude of all CORDIC, the phase accumulator is initialized at a quarter of its full scale, i.e each phase accumulator is reseted at $\\pi /4$ .",
"The I&Q demodulation is performed by multiplying a copy of the ADC output by replicas of the generated sine and cosine values.",
"For practical reasons (FPGA logic resources), the Low pass Filter (LPF) is obtained by averaging $2^{18}$ data samples and it is thus in the order of the kHz of bandwidth.",
"It must be noted, that the accumulator period must be chosen as a multiple of the phase accumulator period in order to avoid beat frequency phenomena.",
"At the end of the accumulation cycle, each tone manager transfers its I&Q data to the IQ2GTX interface for transmission to the `split' FPGA.",
"To allow individual tone power adjustment, the sine and cosine wave are passed through digital attenuators before being provided to the block output.",
"Tones can be tuned in the range 0 to 8/8 and have a resolution of 1/8th of the input power.",
"The whole design uses 164 out of 288 DSP48 blocks, 60412 out of 93120 slice registers and 43508 out of 46560 slice LUT."
],
[
"System frequency response",
"The frequency response of the system was measured for In phase and Quadrature output of the board in loop-back mode, i.e one of the board output connected directly to the board ADC input.",
"For each measurement, 400 tones uniformly distributed over the system bandwidth were generated and analyzed by the embedded DDC.",
"The amplitude of each tone is plotted in figure REF .",
"Figure: Plot of the system frequency response measured for In phase and Quadrature output of the board in loop-back mode, i.e one of the board output connected directly to the board ADC input.",
"For each measurement, 400 tones uniformly distributed over the system bandwidth were generated and analyzed by the embedded DDC.The expected juxtaposition of the five frequency bands of 100 MHz, corresponding to each DAC contribution, can be observed on the plot.",
"The maximum amplitude variation observed over the full bandwidth is less than 6 dB.",
"We explain the amplitude variation by several factors.",
"A part of the dispersion is due to the active and passive electronic components that display a certain amount of dispersion.",
"For instance, the DAC gain has a worst case dispersion of $\\pm $ 3.6 %, while the DAC full scale current resistor has a dispersion of $\\pm $ 1 %.",
"Then, there are also the dispersion of the resistor in the passive combiner and the balun transformer loss dispersion (not documented).",
"Additionally, the balun transformers have a frequency dependent loss (-2 dB at 500 MHz) which partly explains the decreasing tendency of the curve.",
"It may also be noted, that the original board design was foreseen to use sum amplifiers to sum the five DAC signals (I and Q).",
"Unfortunately they were causing distortion and picking noise from the power supplies.",
"In consequence, they were replaced by passive combiners.",
"This modification required the implementation of wire straps to bypass the amplifiers that certainly induces attenuation as the frequency increase.",
"Besides, the DAC output of the bands 100-300 MHz and 400-500 MHz were not routed on the outer PCB layers (as striplines), but in the inner FR4 layers (as microstrips) and thus they have higher dielectric losses.",
"From the dielectric manufacturer specification, a loss difference of 0.2 dB can be observed between FR4 and RO4350 microstrips.",
"Finally, some routing choices were not optimal (bends, stubs, ...) and certainly, they cause small impedance variations over the lines which induce transmission losses as well.",
"Even though the fluctuation is not fully explained, it remains totally acceptable for such a bandwidth.",
"Moreover, this can be corrected by applying tone per tone power adjustment."
],
[
"System noise",
"As shown in [9], the main system noise contributors in the KID readout electronic chain are the RF mixing electronics and the cold amplifier.",
"Consequently, this prototype was also tested in loop-back to measure its noise power spectrum distribution.",
"The measurements were performed for one tone generated in the middle of each frequency band and at different output power level.",
"The output level was digitally adjusted with the digital gain module available in each FPGA `proc' (see figure REF ).",
"The highest signal level reached by this method was just slightly above midscale for the 25 gain.",
"For each tone and in each digital gain conditions, 6000 points were recorded at 23.84 Hz and were windowed with a Hann function.",
"The resulting data were used for computing the Fast Fourier Transform (FFT) and the 6 dB loss due to the windowing function was compensated.",
"Finally the resulting FFT was smoothed by FFT filtering (20 bins kept).",
"Figure REF shows the system noise Power Spectrum Distribution (PSD) for one tone in each frequency band.",
"With the exception of tone 4, all tones have a similar Signal to Noise Ratio (SNR).",
"This is compatible with the board losses mentioned previously that reduce the signal amplitude by about 6 dB.",
"Figure: Power spectrum distribution plot showing system noise for one tone in each band.",
"At the exception of tone 4, all tone have a similar Signal to Noise Ratio (SNR).Figure REF shows the system noise PSD for a given tone but for different excitation signal amplitudes.",
"The noise floor (relative to carrier) is seen to increase accordingly with each amplitude decrease.",
"It may be noticed that when all tones are activated in a single band, it is possible to keep a digital gain between 21 and 22 without DAC clipping because of the frequency values random distribution which minimizes the risk to sum all tones at their maximum amplitude at the same time.",
"Therefore, the 21 and 22 gain curves, provide the achievable performance when the full capabilities of the board are used.",
"Figure: Power spectrum distribution for a given tone but for different signal amplitude.",
"The noise floor (relative to carrier) can be seen to be increased accordingly to each amplitude decrease."
],
[
"Conclusion",
"We have presented in this paper a first prototype of the NIKEL electronic board which was specifically designed for the NIKA camera to be installed at the IRAM 30 m telescope at Pico Veleta, Spain.",
"We have proved that NIKEL is able to perform real-time frequency multiplexing of an array of up to 400 MKIDs over a bandwidth of 500 MHz with outstanding performances in terms of noise.",
"This is due to an innovative solution based on the splitting of the original 500 MHz band into five bands of 100 MHz each, thanks to state of art electronic components and sophisticated numerical filtering algorithms.",
"The NIKEL multiplexing factor is three times larger compared to previous single board systems and it opens a clear path towards the exploitation and monitoring of future kilo-pixel arrays of MKIDs.",
"Consequently, the resulting minimization of the cable count towards the cryogenic system makes it an asset.",
"Such large arrays will be with no doubt a serious alternative to standard bolometric techniques for millimeter astronomy both because of the intrinsic quality of MKIDs (low noise and fast response) and because of the large multiplexing capabilities."
]
] | 1204.1415 |
[
[
"On Entire Solutions of an Elliptic System Modeling Phase Separations"
],
[
"Abstract We study the qualitative properties of a limiting elliptic system arising in phase separation for Bose-Einstein condensates with multiple states: \\Delta u=u v^2 in R^n, \\Delta v= v u^2 in R^n, u, v>0\\quad in R^n.",
"When n=1, we prove uniqueness of the one-dimensional profile.",
"In dimension 2, we prove that stable solutions with linear growth must be one-dimensional.",
"Then we construct entire solutions in $\\R^2$ with polynomial growth $|x|^d$ for any positive integer $d \\geq 1$.",
"For $d\\geq 2$, these solutions are not one-dimensional.",
"The construction is also extended to multi-component elliptic systems."
],
[
"Introduction and Main Results",
"Consider the following two-component Gross-Pitaevskii system $& -\\Delta u + \\alpha u^3 + \\Lambda v^2 u = \\lambda _1 u &&\\text{in }\\Omega , \\\\& -\\Delta v +\\beta v^3 + \\Lambda u^2 v = \\lambda _2 v &&\\text{in }\\Omega , \\\\& u>0,\\quad v>0 && \\text{in }\\Omega , \\\\& u=0,\\quad v=0 && \\text{on }\\partial \\Omega \\,, \\\\& \\int _\\Omega u^2=N_1,\\quad \\int _\\Omega v^2=N_2\\, , &&$ where $\\alpha , \\beta , \\Lambda >0$ and $\\Omega $ is a bounded smooth domain in ${\\mathbb {R}}^n$ .",
"Solutions of (REF )-() can be regarded as critical points of the energy functional $E_\\Lambda (u,v)=\\int _\\Omega \\,\\left(|\\nabla u|^2+|\\nabla v|^2\\right)+\\frac{\\alpha }{2}u^4+\\frac{\\beta }{2}v^4+\\frac{\\Lambda }{2}u^2v^2\\,,$ on the space $(u,v)\\in H^1_0(\\Omega )\\times H^1_0(\\Omega )$ with constraints $\\int _\\Omega u^2 dx=N_1, \\int _\\Omega v^2 dx=N_2.$ The eigenvalues $\\lambda _j$ 's are Lagrange multipliers with respect to (REF ).",
"Both eigenvalues $\\lambda _j=\\lambda _{j,\\Lambda }, j=1,2$ , and eigenfunctions $u=u_\\Lambda ,v=v_\\Lambda $ depend on the parameter $\\Lambda $ .",
"As the parameter $\\Lambda $ tends to infinity, the two components tend to separate their supports.",
"In order to investigate the basic rules of phase separations in this system one needs to understand the asymptotic behavior of $(u_\\Lambda , v_\\Lambda )$ as $ \\Lambda \\rightarrow +\\infty $ .",
"We shall assume that the solutions $(u_\\Lambda ,v_\\Lambda )$ of (REF )-() are such that the associated eigenvalues $\\lambda _{j,\\Lambda }$ 's are uniformly bounded, together with their energies $ E_\\Lambda (u_\\Lambda , v_\\Lambda )$ .",
"Then, as $\\Lambda \\rightarrow +\\infty $ , there is weak convergence (up to a subsequence) to a limiting profile $(u_\\infty , v_\\infty )$ which formally satisfies ${\\left\\lbrace \\begin{array}{ll}-\\Delta u_{\\infty } +\\alpha u_{\\infty }^3 =\\lambda _{1,\\infty }u_{\\infty } \\qquad & \\text{in $\\Omega _u$}\\,,\\\\-\\Delta v_{\\infty } +\\beta v_{\\infty }^3 =\\lambda _{2,\\infty }v_{\\infty } \\qquad &\\text{in $\\Omega _v$}\\,,\\\\\\end{array}\\right.",
"}$ where $\\Omega _u=\\lbrace x\\in \\Omega : u_\\infty (x)>0\\rbrace $ and $\\Omega _v=\\lbrace x\\in \\Omega : v_\\infty (x)>0\\rbrace $ are positivity domains composed of finitely disjoint components with positive Lebesgue measure, and each $\\lambda _{j,\\infty }$ is the limit of $\\lambda _{j,\\Lambda }$ 's as $\\Lambda \\rightarrow \\infty $ (up to a subsequence).",
"There is a large literature about this type of questions.",
"Effective numerical simulations for (REF ) can be found in [5], [6] and [13].",
"Chang-Lin-Lin-Lin [13] proved pointwise convergence of $(u_\\Lambda , v_\\Lambda )$ away from the interface $\\Gamma \\equiv \\lbrace x\\in \\Omega : u_\\infty (x)=v_\\infty (x)=0\\rbrace $ .",
"In Wei-Weth [27] the uniform equicontinuity of $(u_\\Lambda ,v_\\Lambda )$ is established, while Noris-Tavares-Terracini-Verzini [24] proved the uniform-in-$\\Lambda $ Hölder continuity of $(u_\\Lambda , v_\\Lambda )$ .",
"The regularity of the nodal set of the limiting profile has been investigated in [12], [26] and in [16]: it turns out that the limiting pair $(u_\\infty (x),v_\\infty (x))$ is the positive and negative pair $(w^+,w^-)$ of a solution of the equation $-\\Delta w+\\alpha (w^{+})^3-\\beta (w^{-})^3 =\\lambda _{1,\\infty }w^+-\\lambda _{2,\\infty }w^-$ .",
"To derive the asymptotic behavior of $(u_\\Lambda , v_\\Lambda )$ near the interface $\\Gamma =\\lbrace x\\in \\Omega :u_\\infty (x)=v_\\infty (x)=0\\rbrace $ , one is led to considering the points $x_\\Lambda \\in \\Omega $ such that $ u_\\Lambda (x_\\Lambda )=v_\\Lambda (x_\\Lambda )= m_\\Lambda \\rightarrow 0$ and $x_\\Lambda \\rightarrow x_\\infty \\in \\gamma \\subset \\Omega $ as $\\Lambda \\rightarrow +\\infty $ (up to a subsequence).",
"Assuming that $m_\\Lambda ^4 \\Lambda \\rightarrow C_0>0,$ (without loss of generality we may assume that $ C_0=1$ ), then, by blowing up, we find the following nonlinear elliptic system $\\Delta u= u v^2\\,, \\quad \\Delta v= v u^2\\,, \\quad u,v > 0 \\quad \\mbox{in} \\quad {\\mathbb {R}}^n\\,.$ Problem (REF ) has been studied in Berestycki-Lin-Wei-Zhao [8], and Noris-Tavares-Terracini-Verzini [24].",
"It has been proved in [8] that, in the one-dimensional case, (REF ) always holds.",
"In addition, the authors showed the existence, symmetry and nondegeneracy of the solution to one-dimensional limiting system $u^{^{\\prime \\prime }}= uv^2, v^{^{\\prime \\prime }}=v u^2, u, v>0 \\ \\mbox{in} \\ {\\mathbb {R}}.$ In particular they showed that entire solutions are reflectionally symmetric, i.e., there exists $x_0$ such that $ u(x-x_0)= v(x_0-x)$ .",
"They also established a two-dimensional version of the De Giorgi Conjecture in this framework.",
"Namely, under the growth condition $u(x)+v(x)\\le C (1+|x|),$ all monotone solution is one dimensional.",
"On the other hand, in [24], it was proved that the linear growth is the lowest possible for solutions to (REF ).",
"In other words, if there exists $\\alpha \\in (0,1)$ such that $u(x)+v(x)\\le C (1+|x|)^{\\alpha },$ then $u, v \\equiv 0$ .",
"In this paper we address three problems left open in [8].",
"First, we prove the uniqueness of (REF ) (up to translations and scaling).",
"This answers the question stated in Remark 1.4 of [8].",
"Second, we prove that the De Giorgi conjecture still holds in the two dimensional case, when we replace the monotonicity assumption by the stability condition.",
"A third open question of (REF ) is whether all solutions to (REF ) necessarily satisfy the growth bound (REF ).",
"We shall answer this question negatively in this paper.",
"We first study the one-dimensional problem (REF ).",
"Observe that problem (REF ) is invariant under the translations $ (u(x), v(x)) \\rightarrow ( u(x+t), v(x+t)), \\forall t \\in {\\mathbb {R}}$ and scalings $ (u(x), v(x)) \\rightarrow ( \\lambda u(\\lambda x), \\lambda v(\\lambda x)), \\forall \\lambda >0$ .",
"The following theorem classifies all entire solutions to (REF ).",
"Theorem 1.1 The solution to (REF ) is unique, up to translations and scaling.",
"Next,we want to classify the stable solutions in ${\\mathbb {R}}^2$ .",
"We recall that a stable solution $(u, v)$ to (REF ) is such that the linearization is weakly positive definite.",
"That is, it satisfies $\\int _{{\\mathbb {R}}^n} [\\nabla \\varphi |^2+|\\nabla \\psi |^2 + v^2 \\varphi ^2+u^2 \\psi ^2 +4 uv \\varphi \\psi ] \\ge 0, \\qquad \\forall \\varphi , \\psi \\in C_0^\\infty ({\\mathbb {R}}^n).$ In [8], it was proved that the one-dimensional solution is stable in ${\\mathbb {R}}^n$ .",
"Our first result states that the only stable solution in ${\\mathbb {R}}^2$ , among those growing at most linearly, is the one-dimensional family.",
"Theorem 1.2 Let $(u,v)$ be a stable solution to (REF ) in ${\\mathbb {R}}^2$ .",
"Furthermore, we assume that the growth bound (REF ) holds.",
"Then $(u, v)$ is one-dimensional, i.e., there exists $a \\in {\\mathbb {R}}^2, |a|=1$ such that $(u, v)= (U (a \\cdot x), V (a \\cdot x))$ where $(U, V)$ are functions of one variable and satisfies (REF ).",
"Our third result shows that there are solutions to (REF ) with polynomial growth $|x|^d$ that are not one dimensional.",
"The construction depends on the following harmonic polynomial $\\Phi $ of degree $d$ : $\\Phi :=\\mbox{Re}(z^d).$ Note that $\\Phi $ has some dihedral symmetry; indeed, let us take its $d$ nodal lines $L_1, \\cdots , L_d$ and denote the corresponding reflection with respect to these lines by $T_1,\\cdots , T_d$ .",
"Then there holds $\\Phi (T_i z)=-\\Phi (z).$ The third result of this paper is the following one.",
"Theorem 1.3 For each positive integer $d \\ge 1$ , there exists a solution $(u,v)$ to problem (REF ), satisfying $u-v>0$ in $\\lbrace \\Phi >0\\rbrace $ and $u-v<0$ in $\\lbrace \\Phi <0\\rbrace $ ; $u \\ge \\Phi ^+$ and $v\\ge \\Phi ^-$ ; $\\forall i=1,\\cdots , d$ , $u(T_iz)=v(z)$ ; $\\forall r>0$ , the Almgren frequency function satisfies $ N(r):=\\frac{r\\int _{B_r(0)}|\\nabla u|^2+|\\nabla v|^2+u^2v^2}{\\int _{\\partial B_r(0)}u^2+v^2}\\le d;$ $\\lim _{r \\rightarrow +\\infty } N(r) =d.$ Note that the one-dimensional solution constructed in [8] can be viewed as corresponding to the case $d=1$ .",
"For $d\\ge 2$ , the solutions of Theorem REF will be obtained by a minimization argument under symmetric variations $(\\varphi ,\\psi )$ (i.e.",
"satisfying $\\varphi \\circ T_i=\\psi $ for every reflection $T_i$ ).",
"The first four claims will be derived from the construction.",
"See Theorem REF .",
"Regarding the claim 5, we note that by Almgren's monotonicity formula, (see Proposition REF below), the Almgren frequency quotient $N(r)$ is increasing in $r$ .",
"Hence $ \\lim _{r \\rightarrow +\\infty } N(r)$ exists.",
"To understand the asymptotics at infinity of the solutions, one way is to study the blow-down sequence defined by: $(u_R(x), v_R(x)):=(\\frac{1}{L(R)}u(Rx)\\frac{1}{L(R)}v(Rx)),$ where $L(R)$ is chosen so that $\\int _{\\partial B_1(0)}u_R^2+v_R^2=1.$ In Section 6, we will prove Theorem 1.4 Let $(u,v)$ be a solution of (REF ) such that $d:=\\lim \\limits _{r\\rightarrow +\\infty }N(r)<+\\infty .$ Then $d$ is a positive integer.",
"As $R\\rightarrow \\infty $ , $(u_R, v_R)$ defined above (up to a subsequence) converges to $(\\Psi ^+,\\Psi ^-)$ uniformly on any compact set of $\\mathbb {R}^N$ where $\\Psi $ is a homogeneous harmonic polynomial of degree $d$ .",
"If $d=1$ then $(u,v)$ is asymptotically flat at infinity.",
"In particular this applies to the solutions found by Theorem REF to yield the following property Corollary 1.5 Let $(u,v)$ be a solution of (REF ) given by Theorem REF .",
"Then $(u_R(x), v_R(x)):=(\\frac{1}{R^d}u(Rx)\\frac{1}{R^d}v(Rx))$ converges uniformly on compact subsets of $\\mathbb {R}^2$ to a multiple of $(\\Phi ^+,\\Phi ^-)$ , where $\\Phi :=\\mbox{Re}(z^d)$ .",
"Theorem REF roughly says that $(u,v)$ is asymptotic to $(\\Psi ^+,\\Psi ^-)$ at infinity for some homogeneous harmonic polynomial.",
"The extra information we have in the setting of Theorem REF is that $\\Psi \\equiv \\Phi =\\mbox{Re}(z^d)$ .",
"This can be inferred from the symmetries of the solution (property 3 in Theorem REF ).",
"For another elliptic system with a similar form, $\\left\\lbrace \\begin{aligned}&\\Delta u=uv, u>0 \\ \\mbox{in} \\ {\\mathbb {R}}^n,\\\\&\\Delta v=vu, v>0 \\ \\mbox{in} \\ {\\mathbb {R}}^n \\end{aligned} \\right.$ the same result has been proved by Conti-Terracini-Verzini in [15].",
"In fact, their result hold for any dimension $n\\ge 1$ and any harmonic polynomial function on $\\mathbb {R}^n$ .",
"Note however that the problem here is different from (REF ).",
"Actually, equation (REF ) can be reduced to a single equation: indeed, the difference $u-v$ is a harmonic function ($\\Delta (u-v)=0$ ) and thus we can write $v= u-\\Phi $ where $\\Phi $ is a harmonic function.",
"By restricting to certain symmetry classes, then (REF ) can be solved by sub-super solution method.",
"However, this reduction does not work for system (REF ) that we study here.",
"For the proof of Theorem REF , we first construct solutions to (REF ) in any bounded ball $B_R(0)$ satisfying appropriate boundary conditions: $\\left\\lbrace \\begin{aligned}&\\Delta u=uv^2, ~~\\mbox{in}~~B_R(0),\\\\&\\Delta v=vu^2,~~\\mbox{in}~~B_R(0), \\\\& u=\\Phi ^+, v=\\Phi ^- \\ \\mbox{ on} \\ \\partial B_R(0).\\end{aligned} \\right.$ This is done by variational method and using heat flow.",
"The next natural step is to let $R\\rightarrow +\\infty $ and obtain some convergence result.",
"This requires some uniform (in $R$ ) upper bound for solutions to (REF ).",
"In order to prove this, we will exploit a new monotonicity formula for symmetric functions (Proposition REF ).",
"We also need to exclude the possibility of degeneracy, that is that the limit could be 0 or a solution with lower degree such as a one dimensional solution.",
"To this end, we will give some lower bound using the Almgren monotonicity formula.",
"Lastly, we observe that the same construction works also for a system with many components.",
"Let $d$ be an integer or a half-integer and $2d=hk$ be a multiple of the number of components $k$ , and $G$ denote the rotation of order $2d$ .",
"In this way we prove the following result Theorem 1.6 There exists a positive solution to the system $\\left\\lbrace \\begin{aligned}&\\Delta u_i=u_i\\sum _{j\\ne i,j=1}^ku_j^2, ~~\\mbox{in}~~\\mathbb {C}={\\mathbb {R}}^2, i=1,\\dots , k,\\\\& u_i>0, i=1,\\ldots , k,\\end{aligned} \\right.$ having the following symmetries (here $\\overline{z}$ is the complex conjugate of $z$ ) $\\begin{aligned}u_{i}(z)&=u_i(G^hz), \\qquad \\ &\\mbox{ on} \\ &\\mathbb {C}\\,,i=1,\\dots ,k,\\\\u_i(z)&=u_{i+1}(Gz), \\qquad \\ &\\mbox{ on} \\ &\\mathbb {C}\\,,i=1,\\dots ,k,\\\\u_{k+1}(z)&=u_1(z), \\ &\\mbox{ on} \\ &\\mathbb {C}\\\\u_{k+2-i}(z)&=u_i(\\overline{z}), \\qquad \\ &\\mbox{ on} \\ &\\mathbb {C}\\,,i=1,\\dots ,k.\\\\\\end{aligned}$ Furthermore, $\\lim _{r\\rightarrow \\infty } \\dfrac{1}{r^{1+2d}}\\int _{\\partial B_r(0)}\\sum _1^k u_{i}^2=b\\in (0,+\\infty )\\;;$ and $\\lim _{r\\rightarrow \\infty } \\frac{r\\int _{B_r(0)}\\sum _1^k |\\nabla u_{i}|^2+\\sum _{i<j}u_i^2u_j^2}{\\int _{\\partial B_r(0)}\\sum _1^k u_{i}^2}=d\\;.$ The problem of the full classification of solutions to (REF ) is largely open.",
"In view of our results, one can formulate several open questions.",
"Open problem 1.",
"We recall from [8] that it is still an open problem to know in which dimension it is true that all monotone solution is one-dimensional.",
"A similar open question is in which dimension it is true that all stable solution is one-dimensional.",
"We refer to [2], [20], [18], [23], and [25] for results of this kind for Allen-Cahn equation.",
"Open problem 2.",
"Let us recall that in one space dimension, there exists a unique solution to (REF ) (up to translations and scalings).",
"Such solutions have linear growth at infinity and, in the Almgren monotonicity formula, they satisfy $\\lim \\limits _{r\\rightarrow +\\infty }N(r)=1.$ It is natural to conjecture that, in any space dimension, a solution of (REF ) satisfying (REF ) is actually one dimensional, that is, there is a unit vector $a$ such that $(u(x),v(x))=(U (a \\cdot x ), V (a \\cdot x))$ for $x \\in \\mathbb {R}^n$ , where $(U,V)$ solves (REF ).",
"However this result seems to be difficult to obtain at this stage.",
"Open problem 3.",
"A further step would be to prove uniqueness of the (family of) solutions having polynomial asymptotics given by Theorem REF in two space dimension.",
"A more challenging question is to classify all solutions with $\\lim \\limits _{r\\rightarrow +\\infty }N(r)=d.$ Open problem 4.",
"For the Allen-Cahn equation $ \\Delta u+u-u^3=0$ in ${\\mathbb {R}}^2$ , solutions similar to Theorem REF was first constructed in [17] for $d=2$ and in [1] for $ d\\ge 3$ .",
"(However all solutions to Allen-Cahn equation are bounded.)",
"On the other hand, it was also proved in [19] that Allen-Can equation in ${\\mathbb {R}}^2$ admits solutions with multiple fronts.",
"An open question is whether similar result holds for (REF ).",
"Namely, are there solutions to (REF ) such that the set $\\lbrace u=v\\rbrace $ contains disjoint multiple curves?",
"Open problem 5.",
"This question is related to extension of Theorem REF to higher dimensions.",
"We recall that for the Allen-Cahn equation $ \\Delta u+u-u^3=0$ in ${\\mathbb {R}}^{2m}$ with $m\\ge 2$ , saddle-like solutions were constructed in [10] by employing properties of Simons cone.",
"Stable solutions to Allen-Cahn equation in ${\\mathbb {R}}^8$ with non planar level set were found in [23], using minimal cones.",
"We conjecture that all these results should have analogues for (REF )."
],
[
"Uniqueness of solutions in ${\\mathbb {R}}$ : Proof of Theorem ",
"In this section we prove Theorem REF .",
"Without loss of generality, we assume that $\\lim _{x \\rightarrow +\\infty } u(x)= +\\infty , \\lim _{x \\rightarrow +\\infty } v(x)=0.$ The existence of such entire solutions has been proved in [8].",
"By symmetry property of solutions to (REF ) (Theorem 1.3 of [8]), we may consider the following problem $\\left\\lbrace \\begin{aligned}&u^{^{\\prime \\prime }}=uv^2,v^{^{\\prime \\prime }}=vu^2, u,v>0~~\\text{in}~~\\mathbb {R},\\\\&\\lim \\limits _{x\\rightarrow +\\infty }u^{^{\\prime }}(x)=-\\lim \\limits _{x\\rightarrow -\\infty }v^{^{\\prime }}(x)=a\\end{aligned} \\right.$ where $a>0$ is a constant.",
"We now prove that there exists a unique solution $(u, v)$ to (REF ), up to translations.",
"We will prove it using the method of moving planes.",
"First we observe that for any solution $(u,v)$ of (REF ), $u^{^{\\prime \\prime }}$ and $v^{^{\\prime \\prime }}$ decay exponentially at infinity.",
"Integration shows that as $x\\rightarrow +\\infty $ , $|u^{^{\\prime }}(x)-a|$ decays exponentially.",
"(See also [8].)",
"This implies the existence of a positive constant $A$ such that $|u(x)-ax^+|+|v(x)-ax^-|\\le A.$ Moreover, the limits $\\lim \\limits _{x\\rightarrow +\\infty }(u(x)-ax^+),\\lim \\limits _{x\\rightarrow -\\infty }(v(x)-ax^-)$ exist.",
"Now assume $(u_1,v_1)$ and $(u_2,v_2)$ are two solutions of (REF ).",
"For $t>0$ , denote $u_{1,t}(x):=u_1(x+t),v_{1,t}(x):=v_1(x+t).$ We want to prove that there exists an optimal $t_0$ such that for all $t\\ge t_0$ , $u_{1,t}(x)\\ge u_2(x),v_{1,t}(x)\\le v_2(x)~~\\text{in}~~\\mathbb {R}.$ Then we will show that when $t=t_0$ these inequalities are identities.",
"This will imply the uniqueness result.",
"Without loss of generality, assume $(u_1,v_1)$ and $(u_2,v_2)$ satisfy the estimate (REF ) with the same constant $A$ .",
"Step 1.",
"For $t\\ge \\frac{16A}{a}$ ($A$ as in (REF )), (REF ) holds.",
"Firstly, in the region $\\lbrace x\\ge -t+\\frac{2A}{a}\\rbrace $ , by (REF ) we have $u_{1,t}(x)\\ge a(x+t)-A\\ge ax^++A\\ge u_2(x);$ while in the region $\\lbrace x\\le -t+\\frac{2A}{a}\\rbrace $ , we have $v_{1,t}(x)\\le a(x+t)^-+A\\le ax^--A\\le v_2(x).$ On the interval $\\lbrace x<-t+\\frac{2A}{a}\\rbrace $ , we have $\\left\\lbrace \\begin{aligned}&u_{1,t}^{^{\\prime \\prime }}=u_{1,t}v_{1,t}^2\\le u_{1,t}v_2^2,\\\\&u_2^{^{\\prime \\prime }}=u_2v_2^2.\\\\\\end{aligned} \\right.$ With the right boundary conditions $u_{1,t}(-t+\\frac{2A}{a})\\ge u_2(-t+\\frac{2A}{a}),\\lim \\limits _{x\\rightarrow -\\infty }u_{1,t}(x)=\\lim \\limits _{x\\rightarrow -\\infty }u_2(x)=0,$ a direct application of the maximum principle implies $\\inf _{\\lbrace x<-t+\\frac{2A}{a}\\rbrace }(u_{1,t}-u_2)\\ge 0.$ By the same type of argument also show that $\\sup _{\\lbrace x>-t+\\frac{2A}{a} \\rbrace }(v_{1,t}-v_2)\\le 0.$ Therefore, we have shown that for $ t\\ge \\frac{16A}{a}, u_{1,t}\\ge u_2$ and $ v_{1,t}\\le v_2 $ .",
"Step 2.",
"We now decrease the $t$ to an optimal value when (REF ) holds $t_0=\\inf \\lbrace t^{^{\\prime }} | \\ \\text{such that}~~(\\ref {sliding})~~\\text{holds} ~~\\text{for all} ~~ t \\ge t^{^{\\prime }} \\rbrace .$ Thus $t_0$ is well defined by Step 1.",
"Since $ -(u_{1,t_0}-u_2)^{^{\\prime \\prime }} +v_{1,t_0}^2 (u_{1,t_0}-u_2) \\ge 0,\\ -(v_{2}-v_{1,t_0})^{^{\\prime \\prime }} +u_{1,t_0}^2 (v_2- v_{1,t_0}) \\ge 0,$ by the strong maximum principle, either $u_{1,t_0}(x)\\equiv u_2(x),v_{1,t_0}(x)\\equiv v_2(x)~~\\text{in}~~\\mathbb {R},$ or $u_{1,t_0}(x)> u_2(x),v_{1,t_0}(x)<v_2(x)~~\\text{in}~~\\mathbb {R}.$ Let us argue by contradiction that (REF ) holds.",
"By the definition of $t_0$ , there exists a sequence of $t_k<t_0$ such that $\\lim \\limits _{k\\rightarrow +\\infty }t_k=t_0$ and either $\\inf _{\\mathbb {R}}(u_{1,t_k}-u_2)<0,$ or $\\sup _{\\mathbb {R}}(v_{1,t_k}-v_2)>0.$ Let us only consider the first case.",
"Define $w_{1,k}:=u_{1,t_k}-u_2$ and $w_{2,k}:=v_2-v_{1,t_k}$ .",
"Direct calculations show that they satisfy $\\left\\lbrace \\begin{aligned}&-w_{1,k}^{^{\\prime \\prime }}+v_{1,t_k}^2w_{1,k}=u_2(v_2+v_{1,t_k})w_{2,k}~~\\mbox{in}~~\\mathbb {R}, \\\\&-w_{2,k}^{^{\\prime \\prime }}+u_{1,t_k}^2w_{2,k}=v_2(u_2+u_{1,t_k})w_{1,k}~~\\mbox{in}~~\\mathbb {R}.\\end{aligned} \\right.$ We use the auxiliary function $g(x)=\\log (|x|+3)$ as in [14].",
"Note that $g\\ge 1,~~g^{^{\\prime \\prime }}<0~~\\mbox{in}~~\\lbrace x\\ne 0\\rbrace .$ Define $\\widetilde{w}_{1,k}:=w_{1,k}/g$ and $\\widetilde{w}_{2,k}:=w_{2,k}/g$ .",
"For $x \\ne 0$ we have $\\left\\lbrace \\begin{aligned}&-\\widetilde{w}_{1,k}^{^{\\prime \\prime }}-2\\frac{g^{^{\\prime }}}{g}\\widetilde{w}_{1,k}^{^{\\prime }}+[v_{1,t_k}^2-\\frac{g^{^{\\prime }}}{g}]\\widetilde{w}_{1,k}=u_2(v_2+v_{1,t_k})\\widetilde{w}_{2,k},~~\\mbox{in}~~\\mathbb {R}, \\\\&-\\widetilde{w}_{2,k}^{^{\\prime \\prime }}-2\\frac{g^{^{\\prime }}}{g}\\widetilde{w}_{2,k}^{^{\\prime }}+[u_{1,t_k}^2-\\frac{g^{^{\\prime }}}{g}]\\widetilde{w}_{2,k}=v_2(u_2+u_{1,t_k})\\widetilde{w}_{1,k},~~\\mbox{in}~~\\mathbb {R}.\\end{aligned} \\right.$ By definition, $w_{1,k}$ and $w_{2,k}$ are bounded in $\\mathbb {R}$ , and hence $\\widetilde{w}_{1,k},\\widetilde{w}_{2,k}\\rightarrow 0~~\\text{as}~~|x|\\rightarrow \\infty .$ In particular, in view of (REF ), we know that $\\inf _{{\\mathbb {R}}} (\\widetilde{w}_{1,k})<0$ is attained at some point $x_{k,1}$ .",
"Note that $|x_{k,1}|$ must be unbounded, for if $ x_{k,1} \\rightarrow x_{\\infty }, t_k\\rightarrow t_0$ , then $ w_{1,k} (x_{k,1}) \\rightarrow u_{1,t_0} (x_\\infty )- u_2 (x_\\infty ) =0$ .",
"But this violates the assumption (REF ).",
"Since $|x_{k,1}|$ is unbounded, at $x=x_{k,1}$ there holds $\\widetilde{w}_{1,k}^{^{\\prime \\prime }}\\ge 0~~\\mbox{and}~~\\widetilde{w}_{1,k}^{^{\\prime }}=0.$ Substituting this into the first equation of (REF ), we get $[v_{1,t_k}(x_{k,1})^2-\\frac{ g^{^{\\prime \\prime }}(x_{k,1})}{g(x_{k,1})}]\\widetilde{w}_{1,k}(x_{k,1})\\ge u_2(x_{k,1})(v_2(x_{k,1})+v_{1,t_k}(x_{k,1}))\\widetilde{w}_{2,k}(x_{k,1})$ which implies that $\\widetilde{w}_{2,k}(x_{k,1})<0$ .",
"Thus we also have $\\inf \\limits _{\\mathbb {R}}\\widetilde{w}_{2,k}<0$ .",
"Assume it is attained at $x_{k,2}$ .",
"Same argument as before shows that $|x_{k,2}|$ must also be unbounded.",
"Similar to (REF ), we have $[u_{1,t_k}(x_{k,2})^2-\\frac{g^{^{\\prime \\prime }}(x_{k,2})}{g(x_{k,2})}]\\widetilde{w}_{2,k} (x_{k,2})\\ge v_2(x_{k,2})(u_2(x_{k,2})+u_{1,t_k}(x_{k,2}))\\widetilde{w}_{1,k} (x_{k,2}).$ Observe that $\\widetilde{w}_{2,k} (x_{k,2})=\\inf \\limits _{\\mathbb {R}}\\widetilde{w}_{2,k} \\le \\widetilde{w}_{2, k} (x_{k,1}),$ $\\widetilde{w}_{1, k} (x_{k,1})=\\inf \\limits _{\\mathbb {R} }\\widetilde{w}_{1,k} \\le \\widetilde{w}_{1, k} (x_{k,2}).$ Substituting these into (REF ) and (REF ), we obtain $\\widetilde{w}_{1, k} (x_{k,1})\\ge \\frac{u_2(x_{k,1})[v_2(x_{k,1})+v_{1,t_k}(x_{k,1})]}{v_{1,t_k}(x_{k,1})^2-\\frac{g^{^{\\prime \\prime }}(x_{k,1})}{g(x_{k,1})}}\\frac{v_2(x_{k,2})[u_2(x_{k,2})+u_{1,t_k}(x_{k,2})]}{u_{1,t_k}(x_{k,2})^2-\\frac{g^{^{\\prime \\prime }}(x_{k,2})}{g(x_{k,2})}}\\widetilde{w}_{1,k}(x_{k,1}).$ Since $ \\tilde{w}_{1,k} (x_{k,1}) <0$ , we conclude from (REF ) that $\\frac{u_2(x_{k,1})[v_2(x_{k,1})+v_{1,t_k}(x_{k,1})]}{v_{1,t_k}(x_{k,1})^2-\\frac{g^{^{\\prime \\prime }}(x_{k,1})}{g(x_{k,1})}}\\frac{v_2(x_{k,2})[u_2(x_{k,2})+u_{1,t_k}(x_{k,2})]}{u_{1,t_k}(x_{k,2})^2-\\frac{g^{^{\\prime \\prime }}(x_{k,2})}{g(x_{k,2})}} \\ge 1$ where $|x_{k,1}|\\rightarrow +\\infty , |x_{k,2}| \\rightarrow +\\infty $ .",
"This is impossible since $ \\frac{ g^{^{\\prime \\prime }} (x)}{g (x)} \\sim -\\frac{1}{|x|^2 \\log (|x|+3)}$ as $|x| \\rightarrow +\\infty $ , and we also use the decaying as well as the linear growth properties of $u$ and $v$ at $\\infty $ .",
"We have thus reached a contradiction, and the proof of Theorem REF is thereby completed."
],
[
"Stable solutions: Proof of Theorem ",
"In this section, we prove Theorem REF .",
"The proof follows an idea from Berestycki-Caffarelli-Nirenberg [7]-see also Ambrosio-Cabré [2] and Ghoussoub-Gui [20].",
"First, by the stability, we have the following Lemma 3.1 There exist a constant $\\lambda \\ge 0$ and two functions $\\varphi >0$ and $\\psi <0$ , smoothly defined in $\\mathbb {R}^2$ such that $\\left\\lbrace \\begin{aligned}&\\Delta \\varphi =v^2\\varphi +2uv\\psi -\\lambda \\varphi ,\\\\&\\Delta \\psi = 2uv\\varphi +v^2\\psi -\\lambda \\psi .\\end{aligned} \\right.$ For any $R<+\\infty $ the stability assumption reads $\\lambda (R):=\\min \\limits _{\\varphi ,\\psi \\in H_0^1(B_R(0))\\setminus \\lbrace 0\\rbrace }\\frac{\\int _{B_R(0)}|\\nabla \\varphi |^2+|\\nabla \\psi |^2+v^2\\varphi ^2+u^2\\psi ^2+4uv\\varphi \\psi }{\\int _{B_R(0)}\\varphi ^2+\\psi ^2}\\ge 0.$ It's well known that the corresponding minimizer is the first eigenfunction.",
"That is, let $(\\varphi _R,\\psi _R)$ realizing $\\lambda (R)$ , then $\\left\\lbrace \\begin{aligned}&\\Delta \\varphi _R= v^2\\varphi _R+2uv\\psi _R-\\lambda (R)\\varphi _R, \\ \\mbox{in} \\ B_R (0),\\\\&\\Delta \\psi _R= 2uv\\varphi _R+v^2\\psi _R-\\lambda (R)\\psi _R, \\ \\mbox{in} \\ B_R (0), \\\\& \\varphi _R =\\psi _R=0 \\ \\mbox{on} \\ \\partial B_R (0).\\end{aligned} \\right.$ By possibly replacing $(\\varphi _R,\\psi _R)$ with $(|\\varphi _R|,-|\\psi _R|)$ , we can assume $\\varphi _R\\ge 0$ and $\\psi _R\\le 0$ .",
"After a normalization, we also assume $|\\varphi _R(0)|+|\\psi _R(0)|=1.$ $\\lambda (R)$ is decreasing in $R$ , thus uniformly bounded as $R\\rightarrow +\\infty $ .",
"Let $\\lambda :=\\lim \\limits _{R\\rightarrow +\\infty }\\lambda (R).$ The equation for $\\varphi _R$ and $-\\psi _R$ (both of them are nonnegative functions) forms a cooperative system, thus by the Harnack inequality ([3] or [9]), $\\varphi _R$ and $\\psi _R$ are uniformly bounded on any compact set of $\\mathbb {R}^2$ .",
"By letting $R\\rightarrow +\\infty $ , we can obtain a converging subsequence and the limit $(\\varphi ,\\psi )$ satisfies (REF ).",
"We also have $\\varphi \\ge 0$ and $\\psi \\le 0$ by passing to the limit.",
"Hence $-\\Delta \\varphi +(v^2-\\lambda )\\varphi \\ge 0.$ Applying the strong maximum principle, either $\\varphi >0$ strictly or $\\varphi \\equiv 0$ .",
"If $\\varphi \\equiv 0$ , substituting this into the first equation in (REF ), we obtain $\\psi \\equiv 0$ .",
"This contradicts the normalization condition (REF ).",
"Thus, it holds true that $\\varphi >0$ and similarly $\\psi <0$ .",
"Fix a unit vector $\\xi $ .",
"Differentiating the equation (REF ) yields the following equation for $(u_{\\xi },v_{\\xi })$ $\\left\\lbrace \\begin{aligned}&\\Delta u_{\\xi }=v^2u_{\\xi }+2uvv_{\\xi },\\\\&\\Delta v_{\\xi }=2uvu_{\\xi }+v^2v_{\\xi }.\\end{aligned} \\right.$ Let $w_1=\\frac{u_{\\xi }}{\\varphi },w_2=\\frac{v_{\\xi }}{\\psi }.$ Direct calculations using (REF ) and (REF ) show $\\left\\lbrace \\begin{aligned}&\\text{div}(\\varphi ^2\\nabla w_1)=2uv\\varphi \\psi (w_2-w_1)+\\lambda \\varphi ^2w_1,\\\\\\nonumber &\\text{div}(\\varphi ^2\\nabla w_2)=2uv\\varphi \\psi (w_1-w_2)+\\lambda \\psi ^2w_2.\\end{aligned} \\right.$ For any $\\eta \\in C_0^{\\infty }(\\mathbb {R}^2)$ , testing these two equations with $w_1\\eta ^2$ and $w_2\\eta ^2$ respectively, we obtain $\\left\\lbrace \\begin{aligned}&-\\int \\varphi ^2|\\nabla w_1|^2\\eta ^2-2\\varphi ^2w_1\\eta \\nabla w_1\\nabla \\eta =\\int 2uv\\varphi \\psi (w_2-w_1)w_1\\eta ^2+\\lambda \\varphi ^2w_1\\eta ^2,\\\\\\nonumber &-\\int \\psi ^2|\\nabla w_2|^2\\eta ^2-2\\psi ^2w_2\\eta \\nabla w_2\\nabla \\eta =\\int 2uv\\varphi \\psi (w_1-w_2)w_2\\eta ^2+\\lambda \\psi ^2w_2\\eta ^2.\\end{aligned} \\right.$ Adding these two and applying the Cauchy-Schwarz inequality, we infer that $\\int \\varphi ^2|\\nabla w_1|^2\\eta ^2+\\psi ^2|\\nabla w_2|^2\\eta ^2\\le 16\\int \\varphi ^2w_1^2|\\nabla \\eta |^2+\\psi ^2w_2^2|\\nabla \\eta |^2\\le 16\\int (u_{\\xi }^2+v_{\\xi }^2)|\\nabla \\eta |^2.$ Here we have taken away the positive term in the right hand side and used the fact that $2uv\\varphi \\psi (w_2-w_1)w_1\\eta ^2+2uv\\varphi \\psi (w_1-w_2)w_2\\eta ^2=-2uv\\varphi \\psi (w_1-w_2)^2\\eta ^2\\ge 0,$ because $\\varphi >0$ and $\\psi <0$ .",
"On the other hand, testing the equation $\\Delta u \\ge 0$ with $u\\eta ^2$ ($\\eta $ as above) and integrating by parts, we get $\\int |\\nabla u|^2\\eta ^2\\le 16\\int u^2|\\nabla \\eta |^2.$ The same estimate also holds for $v$ .",
"For any $r>0$ , take $\\eta \\equiv 1$ in $B_r(0)$ , $\\eta \\equiv 0$ outside $B_{2r}(0)$ and $|\\nabla \\eta |\\le 2/r$ .",
"By the linear growth of $u$ and $v$ , we obtain a constant $C$ such that $\\int _{B_r(0)}|\\nabla u|^2+|\\nabla v|^2\\le Cr^2.$ Now for any $R>0$ , in (REF ), we take $\\eta $ to be $\\eta (z)= \\left\\lbrace \\begin{array}{ll}1, & x\\in B_R(0), \\\\\\nonumber 0, & x\\in B_{R^2}(0)^c,\\\\\\nonumber 1-\\frac{\\log (|z|/R)}{\\log R} & x\\in B_{R^2}(0)\\setminus B_R(0).\\end{array}\\right.",
"$ With this $\\eta $ , we infer from (REF ) $&&\\int _{B_R(0)}\\varphi ^2|\\nabla w_1|^2+\\psi ^2|\\nabla w_2|^2\\\\&\\le &\\frac{C}{(\\log R)^2}\\int _{B_{R^2}(0)\\setminus B_R(0)}\\frac{1}{|z|^2}(|\\nabla u|^2+|\\nabla v|^2)\\\\&\\le &\\frac{C}{(\\log R)^2}\\int _R^{R^2}r^{-2}(\\int _{\\partial B_r(0)}|\\nabla u|^2+|\\nabla v|^2)dr\\\\&=&\\frac{C}{(\\log R)^2}\\int _R^{R^2}r^{-2}(\\frac{d}{dr}\\int _{B_r(0)}|\\nabla u|^2+|\\nabla v|^2)dr\\\\&=&\\frac{C}{(\\log R)^2}[r^{-2}\\int _{\\partial B_r(0)}|\\nabla u|^2+|\\nabla v|^2)|_R^{R^2}+2\\int _R^{R^2}r^{-3}(\\int _{B_r(0)}|\\nabla u|^2+|\\nabla v|^2)dr]\\\\&\\le & \\frac{C}{\\log R}.$ By letting $R\\rightarrow +\\infty $ , we see $\\nabla w_1\\equiv 0$ and $\\nabla w_2\\equiv 0$ in $\\mathbb {R}^2$ .",
"Thus, there is a constant $c$ such that $(u_{\\xi },v_{\\xi })=c(\\varphi ,\\psi ).$ Because $\\xi $ is an arbitrary unit vector, from this we actually know that after changing the coordinates suitably, $u_y\\equiv 0,v_y\\equiv 0\\ \\ \\text{in}~\\mathbb {R}^2.$ That is, $u$ and $v$ depend on $x$ only and they are one dimensional."
],
[
"Existence in bounded balls",
"In this section we first construct a solution $(u,v)$ to the problem $\\left\\lbrace \\begin{aligned}&\\Delta u=uv^2 ~~\\mbox{in}~~B_R(0),\\\\&\\Delta v=vu^2 ~~\\mbox{in}~~B_R(0),\\end{aligned} \\right.$ satisfying the boundary condition $u=\\Phi ^+, v=\\Phi ^- \\ \\mbox{ on} \\ \\partial B_R(0)\\subset \\mathbb {R}^2.$ More precisely, we prove Theorem 4.1 There exists a solution $(u_R,v_R)$ to problem (REF ), satisfying $u_R-v_R>0$ in $\\lbrace \\Phi >0\\rbrace $ and $u_R-v_R<0$ in $\\lbrace \\Phi <0\\rbrace $ ; $u_R\\ge \\Phi ^+$ and $v_R\\ge \\Phi ^-$ ; $\\forall i=1,\\cdots , d$ , $u_R(T_iz)=v_R(z)$ ; $\\forall r\\in (0,R)$ , $N(r;u_R,v_R):=\\frac{r\\int _{B_r(0)}|\\nabla u_R|^2+|\\nabla v_R|^2+u_R^2v_R^2}{\\int _{\\partial B_r(0)}u_R^2+v_R^2}\\le d.$ Let us denote $\\mathcal {U}\\subset H^1(B_R(0))^2$ the set of pairs satisfying the boundary condition (REF ), together with conditions $(1,2,3)$ of the statement of the Theorem (with the strict inequality $<$ replaced by $\\le $ , and so now $\\mathcal {U}$ is a closed set).",
"The desired solution will be a minimizer of the energy functional $E_R(u,v):=\\int _{B_R(0)}|\\nabla u|^2+|\\nabla v|^2+u^2v^2$ over $\\mathcal {U}$ .",
"Existence of at least one minimizer follows easily from the direct method of the Calculus of Variations.",
"To prove that the minimizer also satisfies equation (REF ), we use the heat flow method.",
"More precisely, we consider the following parabolic problem $\\left\\lbrace \\begin{aligned}&U_t-\\Delta U=-UV^2, ~~\\mbox{in}~~[0,+\\infty )\\times B_R(0),\\\\&V_t-\\Delta V=-VU^2,~~\\mbox{in}~~[0,+\\infty )\\times B_R(0),\\end{aligned} \\right.$ with the boundary conditions $U=\\Phi ^+$ and $V=\\Phi ^{-}$ on $(0,+\\infty )\\times \\partial B_R(0)$ and initial conditions in $\\mathcal {U}$ .",
"By the standard parabolic theory, there exists a unique local solution $(U,V)$ .",
"Then by the maximum principle, $0\\le U\\le \\sup _{B_R(0)}\\Phi ^+, \\ \\ 0\\le V \\le \\sup _{B_R(0)} \\Phi ^{-}$ , hence the solution can be extended to a global one, for all $t\\in (0,+\\infty )$ .",
"By noting the energy inequality $\\frac{d}{dt}E_R(U(t),V(t))=-\\int _{B_R(0)}|\\frac{\\partial U}{\\partial t}|^2+|\\frac{\\partial V}{\\partial t}|^2$ and the fact that $E_R\\ge 0$ , standard parabolic theory implies that for any sequence $t_i\\rightarrow +\\infty $ , there exists a subsequence of $t_i$ such that $(U(t_i),V(t_i))$ converges to a solution $(u,v)$ of (REF ).",
"Next we show that $\\mathcal {U}$ is positively invariant by the parabolic flow.",
"First of all, by the symmetry of initial and boundary data, $(V(t,T_iz),U(t,T_iz))$ is also a solution to the problem (REF ).",
"By the uniqueness of solutions to the parabolic system (REF ), $(U,V)$ inherits the symmetry of $(\\Phi ^+,\\Phi ^-)$ .",
"That is, for all $t\\in [0,+\\infty )$ and $i=1,\\cdots , d$ , $U(t,z)=V(t,T_iz).$ This implies $U-V=0~~\\mbox{on}~~\\lbrace \\Phi =0\\rbrace .$ Thus, in the open set $D_R:=B_R(0)\\cap \\lbrace \\Phi >0\\rbrace $ , we have, for any initial datum $(u_0,v_0)\\in \\mathcal {U}$ , $\\left\\lbrace \\begin{aligned}&(U-V)_t-\\Delta (U-V)=UV(U-V), ~~\\mbox{in}~~[0,+\\infty )\\times D_R(0),\\\\&U-V\\ge 0,~~\\mbox{on}~~[0,+\\infty )\\times \\partial D_R(0),\\\\&U-V\\ge 0,~~\\mbox{on}~~\\lbrace 0\\rbrace \\times D_R(0).\\end{aligned} \\right.$ The strong maximum principle implies $U-V>0$ in $(0,+\\infty )\\times D_R(0)$ .",
"By letting $t\\rightarrow +\\infty $ , we obtain that the limit satisfies $u-v\\ge 0~~\\mbox{in}~~D_R(0).$ $(u,v)$ also has the symmetry, $\\forall i=1,\\cdots , d$ $u (T_i z)=v (z).$ Similar to (REF ), noting (REF ), we have $\\left\\lbrace \\begin{aligned}&-\\Delta (u-v)\\ge 0, ~~\\mbox{in}~~D_R(0),\\\\&u-v=\\Phi ^+,~~\\mbox{on}~~\\partial D_R(0).\\end{aligned} \\right.$ Comparing with $\\Phi ^+$ on $D_R(0)$ , we obtain $u-v>\\Phi ^+>0,~~\\mbox{in}~~D_R(0).$ Because $u>0$ and $v>0$ in $B_R(0)$ , we in fact have $u>\\Phi ^+,~~\\mbox{in}~~B_R(0).$ In conclusion, $(u,v)$ satisfies conditions $(1,2,3)$ in the statement of the theorem.",
"Let $(u_R, v_R)$ be a minimizer of $E_{R}$ over ${\\mathcal {U}}$ .",
"Now we consider the parabolic equation (REF ) with the initial condition $U(x, t)= u_R (x), V (x, t) = v_R (x).$ By (REF ), we deduce that $ E_R (u_R, v_R) \\le E_{R} (U, V) \\le E_R (u_R, v_R) $ and hence $ (U(x, t), V(x, t) )\\equiv (u_R(x), v_R (x))$ for all $ t \\ge 0$ .",
"By the arguments above, we see that $ (u_R, v_R)$ satisfies (REF )and conditions $(1,2,3)$ in the statement of the theorem.",
"In order to prove (4), we firstly note that, as $(u_R,v_R)$ minimizes the energy and $(\\Phi ^+,\\Phi ^-)\\in \\mathcal {U}$ , there holds $\\int _{B_R(0)}|\\nabla u_R|^2+|\\nabla v_R|^2+u_R^2v_R^2\\le \\int _{B_R(0)}|\\nabla \\Phi |^2.$ Now by the Almgren monotonicity formula (Proposition REF below) and the boundary conditions, $\\forall r\\in (0,R)$ , we derive $N(r;u_R,v_R)\\le N(R;u_R,v_R)\\le \\frac{R\\int _{B_R(0)}|\\nabla \\Phi |^2}{\\int _{\\partial B_R(0)}|\\Phi |^2}=d.$ This completes the proof of Theorem REF .",
"Let us now turn to the system with many components.",
"In a similar way we shall prove the existence on bounded sets.",
"Let $d$ be an integer or a half-integer and $2d=hk$ be a multiple of the number of components $k$ , and $G$ denote the rotation of order $2d$ .",
"Take the fundamental domain $F$ of the rotations group of degree $2d$ , that is $F=\\lbrace z\\in \\mathbb {C}\\;:\\;\\theta =\\mbox{arg}(z)\\in (-\\pi /{2d},\\pi /{2d})\\rbrace $ .",
"$\\Psi (z)={\\left\\lbrace \\begin{array}{ll}r^{d}\\cos (d\\theta )\\qquad &\\text{if $z\\in \\cup _{i=0}^{h-1}G^{ik}(F)$,}\\\\0 & \\text{otherwise in $\\mathbb {C}$.}\\end{array}\\right.",
"}$ Note that $\\Psi (z)$ is positive whenever it is not zero.",
"Next we construct a solution $(u_1,\\dots ,u_k)$ to the system $\\Delta u_i=u_i\\sum _{j\\ne i,j=1}^ku_j^2, ~~\\mbox{in}~~B_R(0), i=1,\\dots , k$ satisfying the symmetry and boundary condition (here $\\overline{z}$ is the complex conjugate of $z$ ) $\\left\\lbrace \\begin{array}{l}u_{i}(z)= u_i(G^hz), \\qquad \\ \\ \\ \\ \\mbox{ on} \\ B_R(0)\\,,i=1,\\dots ,k,\\\\u_i (z)= u_{i+1}(Gz), \\qquad \\ \\ \\mbox{ on} \\ B_R(0)\\,,i=1,\\dots ,k,\\\\u_{k+2-i}(z)= u_i(\\overline{z}), \\qquad \\ \\mbox{ on} \\ B_R(0)\\,,i=1,\\dots ,k,\\\\u_{k+1}(z)= u_1(z), \\ \\ \\ \\ \\ \\ \\ \\ \\ \\mbox{ on} \\ B_R(0),\\end{array}\\right.$ $u_{i+1}(z)=\\Psi (G^i(z)), \\qquad \\mbox{ on} \\ \\partial B_R(0)\\,,i=0,\\dots ,k-1.$ More precisely, we prove the following.",
"Theorem 4.2 For every $R>0$ , there exists a solution $(u_{1,R},\\dots ,u_{k,R})$ to the system (REF ) with symmetries (REF ) and boundary conditions (REF ), satisfying, $N(r):=\\frac{r\\int _{B_r(0)}\\sum _1^k|\\nabla u_{i,R}|^2+\\sum _{i<j}u_{i,R}^2u_{j,R}^2}{\\int _{\\partial B_r(0)}\\sum _1^k u_{i,R}^2}\\le d, \\ \\forall r\\in (0,R).$ Let us denote by $\\mathcal {U}\\subset H^1(B_R(0))^k$ the set of pairs satisfying the symmetry and boundary condition (REF ), (REF ).",
"The desired solution will be the minimizer of the energy functional $\\int _{B_r(0)}\\sum _1^k|\\nabla u_{i,R}|^2+\\sum _{i<j}u_{i,R}^2u_{j,R}^2$ over $\\mathcal {U}$ .",
"Once more, to deal with the constraints, we may take advantage of the positive invariance of the associated heat flow: $\\left\\lbrace \\begin{aligned}&\\dfrac{\\partial U_i}{\\partial t}-\\Delta U_i=-U_i\\sum _{j\\ne i}U_j^2, ~~\\mbox{in}~~[0,+\\infty )\\times B_R(0),\\\\\\end{aligned} \\right.$ which can be solved under conditions (REF ), (REF ) and initial conditions in $\\mathcal {U}$ .",
"Thus, the minimizer of the energy $(u_{1,R},\\dots ,u_{k,R})$ solves the differential system.",
"In addition, using the test function $(\\Psi _1,\\dots ,\\Psi _k)$ , where $\\Psi _i=\\Psi \\circ G^{i-1}$ , $i=1,\\dots ,k$ , we have $\\int _{B_R(0)}\\sum _1^k|\\nabla u_{i,R}|^2+\\sum _{i<j}u_{i,R}^2u_{j,R}^2\\le k\\int _{B_R(0)}|\\nabla \\Psi |^2.$ Now by the Almgren monotonicity formula below (Proposition REF ) and the boundary conditions, we get $N(r)\\le N(R)\\le \\frac{R\\int _{B_R(0)}|\\nabla \\Psi |^2}{\\int _{\\partial B_R(0)}|\\Psi |^2}=d, \\ \\forall r \\in (0, R).$ In order to conclude the proof of Theorems REF and REF , we need to find upper and lower bounds for the solutions, uniform with respect to $R$ on bounded subsets of $\\mathbb {C}$ .",
"That is, we will prove that for any $ r>0$ , there exists positive constants $0<c(r)<C(r)$ (independent of $R$ ) such that $c(r)<\\sup \\limits _{B_r(0)}u_R\\le C(r).$ Once we have this estimate, then by letting $R\\rightarrow +\\infty $ , a subsequence of $(u_R,v_R)$ will converge to a solution $(u,v)$ of problem (REF ), uniformly on any compact set of $\\mathbb {R}^2$ .",
"It is easily seen that properties (1), (2), (3) and (4) in Theorem REF can be derived by passing to the limit, and we obtain the main results stated in Theorem REF and REF .",
"It then remains to establish the bound (REF ).",
"In the next section, we shall obtain this estimate by using the monotonicity formula."
],
[
"Monotonicity formula",
"Let us start by stating some monotonicity formulae for solutions to (REF ), for any dimension $n\\ge 2$ .",
"The first two are well-known and we include them here for completeness.",
"But we will also require some refinements.",
"Proposition 5.1 For $r>0$ and $x\\in \\mathbb {R}^n$ , $E(r)=r^{2-n}\\int _{B_r(x)}\\sum _1^k|\\nabla u_i|^2+\\sum _{i<j}u_i^2u_j^2$ is nondecreasing in $r$ .",
"For a proof, see [12].",
"The next statement is an Almgren-type monotonicity formula with remainder.",
"Proposition 5.2 For $r>0$ and $x\\in \\mathbb {R}^n$ , let us define $H(r)=r^{1-n}\\int _{\\partial B_r(x)}\\sum _1^k u_i^2.$ Then $N(r;x):=\\frac{E(r)}{H(r)}$ is nondecreasing in $r$ .",
"In addition there holds $\\int _{0}^r \\dfrac{2\\int _{B_s}\\sum _{i<j}^ku_i^2u_j^2}{\\int _{\\partial B_s}\\sum _1^ku_i^2}ds\\le N(r)\\;.$ For simplicity, take $x$ to be the origin 0 and let $k=2$ .",
"We have $H(r)= r^{1-n}\\int _{\\partial B_r}u^2+v^2\\;,\\qquad E(r)=r^{2-n}\\int _{B_r}|\\nabla u|^2+|\\nabla v|^2+u^2v^2\\;.$ Then, direct calculations show that $\\frac{d}{dr} H(r)=2r^{1-n}\\int _{B_r}|\\nabla u|^2+|\\nabla v|^2+2u^2v^2.$ By the proof of Proposition REF , we have $\\frac{d}{dr} E(r)=2r^{2-n}\\int _{\\partial B_r}[u_r^2+v_r^2] +2r^{1-n}\\int _{B_r}u^2v^2.$ With these two identities, we obtain $\\frac{d}{dr}\\frac{E}{H}(r)=\\dfrac{H [2r^{2-n}\\int _{\\partial B_r}(u_r^2+v_r^2)+2r^{1-n}\\int _{B_r}u^2v^2]-E[2r^{1-n}\\int _{\\partial B_r} uu_r+vv_r]}{H^2}\\\\\\ge \\dfrac{2r^{3-2n}\\int _{\\partial B_r}(u^2+v^2)\\int _{\\partial B_r}(u_r^2+v_r^2)-2r^{3-2n}\\left[\\int _{\\partial B_r} uu_r+vr_r\\right]^2}{H^{2}}+\\\\+\\dfrac{2r^{1-n}\\int _{B_r}u^2v^2}{H}\\ge \\dfrac{2r^{1-n}\\int _{B_r}u^2v^2}{H}.$ Here we have used the following inequality $E(r)\\le \\int _{B_r}|\\nabla u|^2+|\\nabla v|^2+2u^2v^2=\\int _{\\partial B_r} uu_r+vr_r.$ Hence this yields monotonicity of the Almgren quotient.",
"In addition, by integrating the above inequality we obtain $\\int _{r_0}^r \\dfrac{2\\int _{B_s}u^2v^2}{\\int _{\\partial B_s}u^2+v^2}ds\\le N(r)\\;.$ If $x=0$ , we simply denote $N(r;x)$ as $N(r)$ .",
"Assuming an upper bound on $N(r)$ , we establish a doubling property by the Almgren monotonicity formula.",
"Proposition 5.3 Let $R>1$ and let $(u_1,\\dots ,u_k)$ be a solution of (REF ) on $B_R$ .",
"If $N(R)\\le d$ , then for any $1<r_1\\le r_2\\le R$ $\\dfrac{H(r_2)}{H(r_1)}\\le e^{d}\\dfrac{r_2^{2d}}{r_1^{2d}}.$ For simplicity of notation, we expose the proof for the case of two components.",
"By direct calculation using (REF ), we obtain $\\frac{d}{dr}\\log \\Bigg [r^{1-n} (\\int _{\\partial B_r(0)}u^2+v^2)\\Bigg ]&=&\\frac{2\\int _{B_r}|\\nabla u|^2+|\\nabla v|^2+2u^2v^2}{\\int _{\\partial B_r(0)}u^2+v^2}\\\\&\\le & \\frac{2N(r)}{r}+\\frac{2\\int _{B_r}u^2v^2}{\\int _{\\partial B_r(0)}u^2+v^2}\\\\&\\le & \\frac{2d}{r}+\\frac{2\\int _{B_r}u^2v^2}{\\int _{\\partial B_r(0)}u^2+v^2}\\\\$ Thanks to (REF ), by integrating, we find that, if $r_1\\le r_2\\le 2r_0$ then $\\dfrac{H(r_2)}{H(r_1)}\\le e^{d}\\dfrac{r_2^{2d}}{r_1^{2d}}.$ An immediate consequence of Proposition REF is the lower bound on bounded sets for the solutions found in Theorems REF and REF .",
"Proposition 5.4 Ler $(u_{1,R},\\dots ,u_{k,R})$ be a family of solutions to (REF ) such that $N(R)\\le d$ and $H(R)= CR^{2d}$ .",
"Then, for every fixed $r<R$ , there holds $H(r)\\ge Ce^{-d}r^{2d}.$ Another byproduct of the monotonicity formula with the remainder (REF ) is the existence of the limit of $H(r)/r^{2d}$ .",
"Corollary 5.5 Let $R>1$ and let $(u_1,\\dots ,u_k)$ be a solution of (REF ) on $\\mathbb {C}$ such that $\\lim _{r\\rightarrow +\\infty }N(r)\\le d$ , then there exists $\\lim _{r\\rightarrow +\\infty }\\dfrac{H(r)}{r^{2d}}<+\\infty \\;.$ Now we prove the optimal lower bound on the growth of the solution.",
"To this aim, we need a fine estimate on the asymptotics of the lowest eigenvalue as the competition term diverges.",
"The following result is an extension of Theorem 1.6 in [8], where the estimate was proved in case of two components.",
"Theorem 5.6 Let $d$ be a fixed integer and let us consider $\\mathcal {L}(d,\\Lambda )=\\min \\left\\lbrace \\int _0^{2\\pi }\\sum _i^d|u^\\prime _i|^2+\\Lambda \\sum _{i<j}^d u_i^2u_j^2\\; \\Bigg | \\; \\begin{array}{l} \\int _0^{2\\pi }\\sum _i u_i^2=1, \\ u_{i+1}(x)=u_i(x-2\\pi /d),\\\\u_1(-x)=u_1(x)\\;,u_{d+1}=u_1\\;\\end{array}\\right\\rbrace .$ Then, there exists a constant $C$ such that for all $\\Lambda >1$ we have $d^2-C \\Lambda ^{-1/4}\\le \\mathcal {L}(d,\\Lambda )\\le d^2\\;.$ Any minimizer $(u_{1,\\Lambda },\\dots ,u_ {d,\\Lambda })$ solves the system of ordinary differential equations $u_i^{^{\\prime \\prime }}=\\Lambda u_i\\sum _{j\\ne i}u_j^2-\\lambda u_i\\;,\\qquad i=1,\\dots ,d,$ together with the associated energy conservation law $\\sum _1^d(u_i^{^{\\prime }})^2+\\lambda u_i^2-\\Lambda \\sum _{i<j}^du_i^2u_j^2=h\\;.$ Note that the Lagrange multiplier satisfies $\\lambda =\\int _0^{2\\pi }\\sum _i^d|u^\\prime _i|^2+2\\Lambda \\sum _{i<j}^d u_i^2u_j^2=\\mathcal {L}(d,\\Lambda )+\\int _0^{2\\pi }\\Lambda \\sum _{i<j}^d u_i^2u_j^2\\;.$ As $\\Lambda \\rightarrow \\infty $ , we see convergence of the eigenvalues $\\lambda \\simeq \\mathcal {L}(d,\\Lambda )\\rightarrow d^2$ , together with the energies $h\\rightarrow 2d^2$ .",
"Moreover, the solutions remain bounded in Lipschitz norm and converge in Sobolev and Hölder spaces (see [8] for more details).",
"Now, let us focus on the interval $I=(a,a+2\\pi /d)$ where the $i$ -th component is active.",
"The symmetry constraints imply $u_{i-1}(a)=u_i(a)\\;,u_{i-1}^{^{\\prime }}(a)=-u_i^{^{\\prime }}(a)\\;,\\\\u_{i+1}(a+2\\pi /d)=u_i(a+2\\pi /d)\\;,u_{i+1}^{^{\\prime }}(a+2\\pi /d)=-u_i^{^{\\prime }}(a+2\\pi /d)$ We observe that there is interaction only with the two prime neighboring components, while the others are exponentially small (in $\\Lambda $ ) on $I$ .",
"Close to the endpoint $a$ , the component $u_i$ is increasing and convex, while $u_{i-1}$ is decreasing and again convex.",
"Similarly to [8] we have that $u_i(a)= u_{i-1}(a)\\simeq K\\Lambda ^{-1/4}\\;, u^{\\prime }_i(a)= -u^{^{\\prime }}_{i-1}(a)\\simeq H=(h+K)/2\\;.$ Hence, in a right neighborhood of $a$ , there holds $u_i(x)\\ge u_i(a)$ , and therefore, as $u_{i-1}^{^{\\prime \\prime }}\\ge \\Lambda u_i^2(a)u_{i-1}$ , from the initial value problem (REF ) we infer $u_{i-1}(x)\\le C u_i(a)e^{-\\Lambda ^{1/2}u_i(a)(x-a)}\\;,\\forall x\\in [a,b].$ On the other hand, on the same interval we have $u_{i}(x)\\le u_i(a)+C(x-a)\\;,\\forall x\\in [a,b].$ (here and below $C$ denotes a constant independent of $\\Lambda $ ).",
"Consequently, there holds $\\Lambda \\int _I u_{i-1}^2 u_{i}^2+u_{i-1}^3 u_{i}+u_{i-1} u_{i}^2\\le C\\Lambda ^{-1/2}u_i(a)^{-1}\\simeq C\\Lambda ^{-1/4}\\;.$ In particular, this yields $\\mathcal {L}(d,\\Lambda )\\ge \\lambda -C\\Lambda ^{-1/4}\\;.$ In order to estimate $\\lambda $ , let us consider $\\widehat{u}_i=\\left(u_i-\\sum _{j=i\\pm 1}u_j\\right)^+$ .",
"Then, as $u_i(a)=u_{i-1}(a)$ and $u_i(a+2\\pi /d)=u_{i+1}(a+2\\pi /d)$ , $\\widehat{u}_i\\in H^1_0(I)$ .",
"By testing the differential equation for $u_i-\\sum _{j=i\\pm 1}u_j$ with $\\widehat{u}_i$ on $I$ we find $\\int _I |\\widehat{u}_i^{^{\\prime }}|^2\\le \\lambda \\int _I |\\widehat{u}_i|^2+C\\Lambda ^{-1/4}\\;,$ where in the last term we have majorized all the integrals of mixed fourth order monomials with (REF ).",
"As $|I|=2\\pi /d$ , using Poincaré inequality and (REF ) we obtain the desired estimate on $\\mathcal {L}(d,\\Lambda )$ .",
"We are now ready to apply the estimate from below on $\\mathcal {L}$ to derive a lower bound on the energy growth.",
"We recall that there holds $\\widehat{E}(r):=\\int _{B_r(x)}\\sum _1^k|\\nabla u_i|^2+2\\sum _{i<j}u_i^2u_j^2=\\int _{\\partial B_r(x)}\\sum _1^k u_i\\dfrac{\\partial u_i}{\\partial r}$ Proposition 5.7 Let $(u_{1,R},\\dots ,u_{k,R})$ be a solution of (REF ) having the symmetries (REF ) on $B_R$ .",
"There exists a constant $C$ (independent of $R$ ) such that for all $\\;1\\le r_1\\le r_2\\le R$ there holds $\\dfrac{\\widehat{E}(r_2)}{\\widehat{E}(r_1)}\\ge C\\dfrac{r_2^{2d}}{r_1^{2d}}$ Let us compute, $\\dfrac{d}{dr}\\log \\left(r^{-2d}\\widehat{E}(r)\\right)=-\\dfrac{2d}{r}+\\dfrac{\\int _{\\partial B_r(x)}{\\sum _1^k|\\nabla u_i|^2}+2\\sum _{i<j}u_i^2u_j^2}{\\int _{\\partial B_r(x)}\\sum _1^k u_i\\dfrac{\\partial u_i}{\\partial r}}\\\\=-\\dfrac{2d}{r}+\\dfrac{\\int _{\\partial B_r(x)}\\sum _1^k\\left(\\dfrac{\\partial u_i}{\\partial r}\\right)^2+\\dfrac{1}{r^2}\\left[\\sum _1^k\\left(\\dfrac{\\partial u_i}{\\partial \\theta }\\right)^2+2r^2\\sum _{i<j}u_i^2u_j^2\\right]}{\\int _{\\partial B_r(x)}\\sum _1^k u_i\\dfrac{\\partial u_i}{\\partial r}}\\\\=-\\dfrac{2d}{r}+\\dfrac{\\int _{0}^{2\\pi }\\sum _1^k\\left(\\dfrac{\\partial u_i}{\\partial r}\\right)^2+\\dfrac{1}{r^2}\\left[\\sum _1^k\\left(\\dfrac{\\partial u_i}{\\partial \\theta }\\right)^2+2r^2\\sum _{i<j}u_i^2u_j^2\\right]}{\\int _{0}^{2\\pi }\\sum _1^k u_i\\dfrac{\\partial u_i}{\\partial r}}$ Now we use Theorem REF and we continue the chain of inequalities: $\\dfrac{d}{dr}\\log \\left(r^{-2d}\\widehat{E}(r)\\right)\\ge -\\dfrac{2d}{r}+\\dfrac{\\int _{0}^{2\\pi }\\sum _1^k\\left(\\dfrac{\\partial u_i}{\\partial r}\\right)^2+\\dfrac{\\mathcal {L}(d,2r^2)}{r^2}\\int _0^{2\\pi }\\sum _1^k u_i^2}{\\int _{0}^{2\\pi }\\sum _1^k u_i\\dfrac{\\partial u_i}{\\partial r}}\\\\\\ge -\\dfrac{2d-2 \\sqrt{\\mathcal {L}(d,2r^2)}}{r}\\ge -\\dfrac{C}{r^{3/2}}\\;,$ where in the last line we have used Hölder inequality.",
"By integration we easily obtain the assertion.",
"A direct consequence of the above inequalities is the non vanishing of the quotient $E/r^{2d}$ : Corollary 5.8 Let $R>1$ and let $(u_1,\\dots ,u_k)$ be a solution of (REF ) on $\\mathbb {C}$ satisfying REF : then there exists $\\lim _{r\\rightarrow +\\infty }\\dfrac{\\widehat{E}(r)}{r^{2d}}=b\\in (0,+\\infty ]\\;.$ If, in addition, $\\lim _{r\\rightarrow +\\infty }N(r)\\le d$ , then we have that $b<+\\infty $ and $\\lim _{r\\rightarrow +\\infty }N(r)=d,\\quad \\text{and}\\quad \\lim _{r\\rightarrow +\\infty }\\dfrac{E(r)}{r^{2d}}=b\\;.$ Note that (REF ) is a straightforward consequence of the monotonicity formula (REF ).",
"To prove (REF ), we first notice that $\\lim _{r\\rightarrow +\\infty }\\dfrac{E(r)}{r^{2d}}=\\lim _{r\\rightarrow +\\infty }N(r)\\dfrac{H(r)}{r^{2d}}.$ So the limit of $E(r)/r^{2d}$ exists finite.",
"Now we use (REF ) $\\int _{0}^{+\\infty } \\dfrac{2\\int _{B_s}\\sum _{i<j}^ku_i^2u_j^2}{\\int _{\\partial B_s}\\sum _1^ku_i^2}ds<+\\infty $ and we infer $\\liminf _{r\\rightarrow +\\infty }\\dfrac{r\\int _{B_{r}}\\sum _{i<j}^ku_i^2u_j^2}{\\int _{\\partial B_{r}}\\sum _1^ku_i^2}=0.$ Next, using Corollary REF we can compute $\\liminf _{r\\rightarrow +\\infty }\\dfrac{\\int _{B_{r}}\\sum _{i<j}^ku_i^2u_j^2}{r^{2d}}=\\liminf _{r\\rightarrow +\\infty }\\dfrac{\\int _{B_{r}}\\sum _{i<j}^ku_i^2u_j^2}{H(r)} \\dfrac{H(r)}{r^{2d}}=0,$ and finally $\\liminf _{r\\rightarrow +\\infty }\\dfrac{\\widehat{E}(r)-E(r)}{r^{2d}}=0;.$ Was the limit of $N(r)$ strictly less that $d$ , the growth of $H(r)$ would be in contradiction with that of $E(r)$ .",
"Now we can combine the upper and lower estimates to obtain convergence of the approximating solutions on compact sets and complete the proof of Theorems REF [Proof of Theorem REF .]",
"Let $(u_{1,R},\\dots ,u_{k,R})$ be a family of solutions to (REF ) such that $N_R(R)\\le d$ and $H_R(R)= CR^{2d}$ .",
"Since $H_R(R)=CR^{2d}$ , then, by Proposition REF we deduce that, for every fixed $1<r<R$ , there holds $H_R(r)\\ge Ce^{-d}r^{2d}\\;.",
"$ Assume first that there holds a uniform bound for some $r>1$ , $H_R(r)\\le C\\;.$ Then $H_R(r)$ and $E_R(r)$ are uniformly bounded on $R$ .",
"This implies a uniform bound on the $H^1(B_{r})$ norm.",
"As the components are subharmonic, standard elliptic estimates (Harnack inequality) yield actually a $\\mathcal {C}^2$ bound on $B_{r/2}$ , which is independent on $R$ .",
"Note that, by Proposition REF , $H_R(r)$ is bounded away from zero, so the weak limit cannot be zero.",
"By the doubling Property REF the uniform bound on $H_R(r_2)\\le C r_2^{2d}$ holds for every $r_2\\in \\mathbb {R}$ larger than $r$ .",
"Thus, a diagonal procedure yields existence of a nontrivial limit solution of the differential system, defined on the whole of $\\mathbb {C}$ .",
"It is worthwhile noticing that this solution inherits all the symmetries of the approximating solutions together with the upper bound on the Almgren's quotient.",
"Finally, from Corollary REF and REF infer the limit $\\lim _{r\\rightarrow +\\infty }\\dfrac{H(r)}{r^{2d}}=\\lim _{r\\rightarrow +\\infty }\\dfrac{1}{N(r)},\\lim _{r\\rightarrow +\\infty }\\dfrac{E(r)}{r^{2d}}=\\dfrac{b}{d}\\in (0,+\\infty )\\:.$ Let us now show that $H_R (r)$ is uniformly bounded with respect to $R$ for fixed $r$ .",
"We argue by contradiction and assume that, for a sequence $R_n\\rightarrow +\\infty $ , there holds $\\lim _{n\\rightarrow +\\infty }H_{R_n}(r)=+\\infty \\;.$ Denote $u_{i,n}=u_{i,R_n}$ and $H_n$ , $E_n$ , $N_n$ the corresponding functions.",
"Note that, as $E_n$ is bounded, we must have $N_n(r)\\rightarrow 0$ .",
"For each $n$ , let $\\lambda _n\\in (0,r)$ such that $\\lambda ^2_nH_n(\\lambda _n)=1\\;$ (such $\\lambda _n$ exist right because of (REF )) and scale $\\tilde{u}_{i,n}(z)=\\lambda _n u_{i,n}(\\lambda _n z)\\;, \\quad |z|<R_n/\\lambda _n\\;.$ Note that the $(\\tilde{u}_{i,n})_i$ still solve system (REF ) on the disk $B(0,R_n/\\lambda _n)$ and enjoy all the symmetries (REF ).",
"Let us denote $\\tilde{H}_n$ , $\\tilde{E}_n$ , $\\tilde{N}_n$ the corresponding quantities.",
"We have $\\begin{aligned}\\tilde{H}_n(1)&=\\lambda ^2_nH_n(\\lambda _n)=1, \\\\\\tilde{E}_n(1)&=\\lambda ^2_nE_n(\\lambda _n) \\rightarrow 0 \\\\\\tilde{N}_n(1)&=N_n(\\lambda _n)\\rightarrow 0\\end{aligned}$ In addition there holds $\\tilde{N}_n(s)\\le d$ for $s<R_n/\\lambda _n$ .",
"By the compactness argument exposed above, we can extract a subsequence converging in the compact-open topology of $\\mathcal {C}^2$ to a nontrivial symmetric solution of (REF ) with Almgren quotient vanishing constantly.",
"Thus, such solution should be a nonzero constant in each component, but constant solution are not compatible with the system of PDE's (REF ) ."
],
[
"Asymptotics at infinity",
"We now come to the proof of Theorem .",
"Note that by Proposition REF , the condition on $N(r)$ implies that $u$ and $v$ have a polynomial growth.",
"(In fact, with more effort we can show the reverse also holds.",
"Namely, if $u$ and $v$ have polynomial growth, then $N(r)$ approaches a positive integer as $r\\rightarrow +\\infty $ .",
"We leave out the proof.)",
"Recall the blow down sequence is defined by $(u_R(x), v_R(x)):=(\\frac{1}{L(R)}u(Rx),\\frac{1}{L(R)}v(Rx)),$ where $L(R)$ is chosen so that $\\int _{\\partial B_1(0)}u_R^2+v_R^2=\\int _{\\partial B_1(0)}\\Phi ^2.$ For the solutions in Theorem REF , by (), we have $L(R)\\sim R^d.$ We will now analyze the limit of $(u_R,v_R)$ as $R\\rightarrow +\\infty $ .",
"Because for any $r\\in (0,+\\infty )$ , $N(r)\\le d$ , $(u, v)$ satisfies Proposition REF for any $r\\in (1,+\\infty )$ .",
"After rescaling, we see that Proposition REF holds for $(u_R,v_R)$ as well.",
"Hence, there exists a constant $C>0$ , such that for any $R$ and $r\\in (1,+\\infty )$ , $\\int _{\\partial B_r(0)}u_R^2+v_R^2\\le C e^{d}r^d.$ Next, $(u_R,v_R)$ satisfies the equation $\\left\\lbrace \\begin{aligned}&\\Delta u_R=L(R)^2R^2u_Rv_R^2,\\\\&\\Delta v_R=L(R)^2R^2v_Ru_R^2,\\\\&u_R,v_R>0~~\\mbox{in}~~\\mathbb {R}^2.\\end{aligned} \\right.$ Here we need to observe that, by (REF ), $\\lim \\limits _{R\\rightarrow +\\infty }L(R)^2R^2=+\\infty .$ By (REF ), as $R\\rightarrow +\\infty $ , $u_R$ and $v_R$ are uniformly bounded on any compact set of $\\mathbb {R}^2$ .",
"Then by the main result in [16], [24] and [26], there is a harmonic function $\\Psi $ defined in $\\mathbb {R}^2$ , such that (a subsequence of) $(u_R,v_R)\\rightarrow (\\Psi ^+,\\Psi ^-)$ in $H^1$ and in Hölder spaces on any compact set of $\\mathbb {R}^2$ .",
"By (REF ), $\\int _{\\partial B_1(0)}\\Psi ^2=\\int _{\\partial B_1(0)}\\Phi ^2,$ so $\\Psi $ is nonzero.",
"Because $L(R)\\rightarrow +\\infty $ , $u_R(0)$ and $v_R(0)$ goes to 0, hence $\\Psi (0)=0.$ After rescaling in Proposition REF , we obtain a corresponding monotonicity formula for $(u_R,v_R)$ , $N(r;u_R,v_R):=\\frac{r\\int _{B_r(0)}|\\nabla u_R|^2+|\\nabla v_R|^2+L(R)^2R^2u_R^2v_R^2}{\\int _{\\partial B_r(0)}u_R^2+v_R^2}=N(Rr)$ is nondecreasing in $r$ .",
"By (4) in Theorem REF and from Corollary REF , $N(r;u_R,v_R)\\le d=\\lim _{r\\rightarrow +\\infty } N(r;u_R,v_R)\\;\\;, \\forall \\; r\\in (0,+\\infty ).$ In [16], it's also proved that $(u_R,v_R)\\rightarrow (\\Psi ^+,\\Psi ^-)$ in $H^1_{loc}$ and for any $r<+\\infty $ , $\\lim \\limits _{R\\rightarrow +\\infty }\\int _{B_r(0)}L(R)^2R^2u_R^2v_R^2=0.$ After letting $R\\rightarrow +\\infty $ in (REF ), we get $N(r;\\Psi ):=\\frac{r\\int _{B_r(0)}|\\nabla \\Psi |^2}{\\int _{\\partial B_r(0)}\\Psi ^2}=\\lim \\limits _{R\\rightarrow +\\infty }N(r;u_R,v_R)=\\lim \\limits _{R\\rightarrow +\\infty }N(Rr)=d.$ In particular, $N(r;\\Psi )$ is a constant for all $r\\in (0,+\\infty )$ .",
"So $\\Psi $ is a homogeneous polynomial of degree $d$ .",
"Actually the number $d$ is the vanishing order of $\\Psi $ at 0, which must therefore be a positive integer.",
"Now it remains to prove that $\\Psi \\equiv \\Phi $ : this is easily done by exploiting the symmetry conditions on $\\Psi $ (point $(3)$ of Theorem REF ).",
"Acknowledgment.",
"Part of this work was carried out while Henri Berestycki was visiting the Department of Mathematics at the University of Chicago.",
"Heá was supported by an NSF FRG grant DMS-1065979 and by the French \"Agence Nationale de la Recherche\" within the project PREFERED (ANR 08-BLAN-0313).",
"Juncheng Wei was supported by a GRF grant from RGC of Hong Kong.",
"Susanna Terracini was partially supported by the Italian PRIN2009 grant “Critical Point Theory and Perturbative Methods for Nonlinear Differential Equations\".",
"Kelei Wang was supported by the Australian Research Council.",
"tocsectionReferences"
]
] | 1204.1038 |
[
[
"A new graph parameter related to bounded rank positive semidefinite\n matrix completions"
],
[
"Abstract The Gram dimension $\\gd(G)$ of a graph $G$ is the smallest integer $k\\ge 1$ such that any partial real symmetric matrix, whose entries are specified on the diagonal and at the off-diagonal positions corresponding to edges of $G$, can be completed to a positive semidefinite matrix of rank at most $k$ (assuming a positive semidefinite completion exists).",
"For any fixed $k$ the class of graphs satisfying $\\gd(G) \\le k$ is minor closed, hence it can characterized by a finite list of forbidden minors.",
"We show that the only minimal forbidden minor is $K_{k+1}$ for $k\\le 3$ and that there are two minimal forbidden minors: $K_5$ and $K_{2,2,2}$ for $k=4$.",
"We also show some close connections to Euclidean realizations of graphs and to the graph parameter $\\nu^=(G)$ of \\cite{H03}.",
"In particular, our characterization of the graphs with $\\gd(G)\\le 4$ implies the forbidden minor characterization of the 3-realizable graphs of Belk and Connelly \\cite{Belk,BC} and of the graphs with $\\nu^=(G) \\le 4$ of van der Holst \\cite{H03}."
],
[
"Introduction",
"Given a graph $G=(V=[n],E)$ , a $G$ -partial matrix is a real symmetrix $n\\times n$ matrix whose entries are specified on the diagonal and at the off-diagonal positions corresponding to the edges of $G$ .",
"The problem of completing a partial matrix to a full positive semidefinite (psd) matrix is one of the most extensively studied matrix completion problems.",
"A particular instance is the completion problem for correlation matrices (where all diagonal entries are equal to 1) arising in probability and statistics, and it is also closely related to the completion problem for Euclidean distance matrices with applications, e.g., to sensor network localization and molecular conformation in chemistry.",
"We give definitions below and refer, e.g., to [12], [24] and further references therein for additional details.",
"Among all psd completions of a partial matrix, the ones with the lowest possible rank are of particular importance.",
"Indeed the rank of a matrix is often a good measure of the complexity of the data it represents.",
"As an example, it is well known that the minimum dimension of a Euclidean embedding of a finite metric space can be expressed as the rank of an appropriate psd matrix (see e.g.",
"[12]).",
"Moreover, in applications, one is often interested in embeddings in low dimension, say 2 or 3.",
"The problem of computing (approximate) low rank psd (or Euclidean) completions of a partial matrix is a challenging non-continuous, non-convex problem which, due to its great importance, has been extensively studied (see, e.g., [1], [2], [33], the recent survey [22] and further references therein).",
"The following basic questions arise about psd matrix completions: Decide whether a given partial rational matrix has a psd completion, what is the smallest rank of a completion, and if so find an (approximate) one (of smallest rank).",
"This leads to hard problems and of course the answer depends on the actual values of the entries of the partial matrix.",
"However, taking a combinatorial approach to the problem and looking at the structure of the graph $G$ of the specified entries, one can sometimes get tractable instances.",
"For instance, when the graph $G$ is chordal (i.e., has no induced circuit of length at least 4), the above questions are fully answered in [18], [25] (see also the proof of Lemma REF below): There is a psd completion if and only if each fully specified principal submatrix is psd, the minimum possible rank is equal to the largest rank of the fully specified principal submatrices, and such a psd completion can be found in polynomial time (in the bit number model).",
"Further combinatorial characterizations (and some efficient algorithms for completions – in the real number model) exist for graphs with no $K_4$ -minor (more generallly when excluding certain splittings of wheels), see [6], [23], [25].",
"In the present paper we focus on the question of existence of low rank psd completions.",
"Our approach is combinatorial, so we look for conditions on the graph $G$ of specified entries permitting to guarantee the existence of low rank completions.",
"This is captured by the notion of Gram dimension of a graph which we introduce in Definition below.",
"We use the following notation: ${\\mathcal {S}}^n$ denotes the set of symmetric $n\\times n$ matrices and ${\\mathcal {S}}^n_+$ (resp., ${\\mathcal {S}}^n_{++}$ ) is the subset of all positive semidefinite (psd) (resp., positive definite) matrices.",
"For a matrix $X\\in {\\mathcal {S}}^n$ , the notation $X\\succeq 0$ means that $X$ is psd.",
"Given a graph $G=(V=[n],E)$ , it will be convenient to identify $V$ with the set of diagonal pairs, i.e., to set $V=\\lbrace (i,i)\\mid i\\in [n]\\rbrace $ .",
"Then, a $G$ -partial matrix corresponds to a vector $a\\in {\\mathbb {R}}^{V\\cup E}$ and $\\pi _{V E}$ denotes the projection from ${\\mathcal {S}}^n$ onto the subspace ${\\mathbb {R}}^{V\\cup E}$ indexed by the diagonal entries and the edges of $G$ .",
"The Gram dimension $\\text{\\rm gd}(G)$ of a graph $G=([n],E)$ is the smallest integer $k\\ge 1$ such that, for any matrix $X\\in {\\mathcal {S}}^n_+$ , there exists another matrix $X^{\\prime } \\in {\\mathcal {S}}^n_+$ with rank at most $k$ and such that $\\pi _{VE}(X)=\\pi _{VE}(X^{^{\\prime }})$ .",
"Hence, if a $G$ -partial matrix admits a psd completion, it also has one of rank at most $\\text{\\rm gd}(G)$ .",
"This motivates the study of bounds for the graph parameter $\\text{\\rm gd}(G)$ .",
"As we will see in Section REF , for any fixed $k$ the class of graphs with $\\text{\\rm gd}(G)\\le k$ is closed under taking minors, hence it can be characterized by a finite list of forbidden minors.",
"Our main result is such a characterization for each integer $k\\le 4$ .",
"Main Theorem.",
"For $k\\le 3$ , $\\text{\\rm gd}(G)\\le k$ if and only if $G$ has no $K_{k+1}$ minor.",
"For $k=4$ , $\\text{\\rm gd}(G)\\le 4$ if and only if $G$ has no $K_5$ and $K_{2,2,2}$ minors.",
"An equivalent way of rephrasing the notion of Gram dimension is in terms of ranks of feasible solutions to semidefinite programs.",
"Indeed, the Gram dimension of a graph $G=(V,E)$ is at most $k$ if and only if the set $S(G,a)=\\lbrace X \\succeq 0 \\mid X_{ij}=a_{ij} \\text{ for } ij \\in V \\cup E\\rbrace $ contains a matrix of rank at most $k$ for all $a\\in {\\mathbb {R}}^{V\\cup E}$ for which $S(G,a)$ is not empty.",
"The set $S(G,a)$ is a typical instance of spectrahedron.",
"Recall that a spectrahedron is the convex region defined as the intersection of the positive semidefinite cone with a finite set of affine hyperplanes, i.e., the feasibility region of a semidefinite program in canonical form: $\\max \\langle A_0,X \\rangle \\text{ subject to } \\langle A_j,X\\rangle =b_j,\\ (j=1,\\ldots ,m), \\qquad X \\succeq 0.$ If the feasibility region of (REF ) is not empty, it follows from well known geometric results that it contains a matrix $X$ of rank $k$ satisfying ${k+1\\atopwithdelims ()2}\\le m$ (see e.g.",
"[7]).",
"Applying this to the spectahedron $S(G,a)$ , we obtain the bound $\\text{\\rm gd}(G)\\le \\left\\lfloor \\frac{ \\sqrt{1+8(|V|+|E|)}-1}{2}\\right\\rfloor .$ For the complete graph $G=K_n$ the upper bound is equal to $n$ , so it is tight.",
"As we will see one can get other bounds depending on the structure of $G$ ; for instance, $\\text{\\rm gd}(G)$ is at most the tree-width plus 1 (cf.",
"Lemma REF ).",
"As an application, the Gram dimension can be used to bound the rank of optimal solutions to semidefinite programs.",
"Namely, consider a semidefinite program in canonical form (REF ).",
"Its aggregated sparsity pattern is the graph $G$ with node set $[n]$ and whose edges are the pairs corresponding to the positions where at least one of the matrices $A_j$ ($j\\ge 0$ ) has a nonzero entry.",
"Then, whenever (REF ) attains its maximum, it has an optimal solution of rank at most $\\text{\\rm gd}(G)$ .",
"Results ensuring existence of low rank solutions are important, in particular, for approximation algorithms.",
"Indeed semidefinite programs are widely used as convex tractable relaxations to hard combinatorial problems.",
"Then the rank one solutions typically correspond to the desired optimal solutions of the discrete problem and low rank solutions can sometimes lead to improved performance guarantees (see, e.g., the result of [4] for max-cut and the result of [10] for maximum stable sets).",
"As an illustration, consider the max-cut problem for graph $G$ and its standard semidefinite programming relaxation: $\\max \\frac{1}{4}\\langle L_G,X\\rangle \\text{ subject to } X_{ii} =1\\ (i=1,\\ldots ,n), \\ \\ X\\succeq 0,$ where $L_G$ denotes the Laplacian matrix of $G$ .",
"Clearly, $G$ is the aggregated sparsity pattern of the program (REF ).",
"In particular, our main Theorem implies that if $G$ is $K_5$ and $K_{2,2,2}$ minor free, then (REF ) has an optimal solution of rank at most four.",
"(On the other hand recall that the max-cut problem can be solved in polynomial time for $K_5$ minor free graphs [5]).",
"In a similar flavor, for a graph $G=([n],E)$ with weights $w\\in {\\mathbb {R}}^{V\\cup E}_+$ , the authors of [17] study semidefinite programs of the form $\\max \\sum _{i=1}^n w_{i}X_{ii}\\ \\text{ s.t. }",
"\\sum _{i,j=1}^nw_{i} w_{j}X_{ij}=0,\\ X_{ii}+X_{ij}-2X_{ij}\\le w_{ij}\\ (ij\\in E),\\ X\\succeq 0,$ and show the existence of an optimal solution of rank at most the tree-width of $G$ plus 1.",
"There is a large literature on dimensionality questions for various geometric representations of graphs.",
"We refer, e.g., to [15], [16], [19], [27], [29] for results and further references.",
"We will point out links to the parameter $\\nu ^=(G)$ of [20], [21] in Section REF .",
"Yet another, more geometrical, way of interpreting the Gram dimension is in terms of isometric embeddings in the spherical metric space [12].",
"For this, consider the unit sphere $\\mathbf {S}^{k-1}=\\lbrace x \\in {\\mathbb {R}}^k: \\Vert x\\Vert =1\\rbrace $ , equipped with the distance $d_{\\mathbf {S}}(x,y)=\\arccos (x^Ty) \\ \\text{ for } x,y\\in \\mathbf {S}^{k-1}.$ Here, $\\Vert x\\Vert $ denotes the usual Euclidean norm.",
"Then $(\\mathbf {S}^{k-1},d_{\\mathbf {S}})$ is a metric space, known as the spherical metric space.",
"A graph $G=([n],E)$ has Gram dimension at most $k$ if and only if, for any assignment of vectors $p_1,\\ldots ,p_n \\in \\mathbf {S}^h$ (for some $h\\ge 1$ ), there exists another assignment $q_1,\\ldots ,q_n \\in \\mathbf {S}^{k-1}$ such that $d_{\\mathbf {S}}(p_i,p_j)=d_{\\mathbf {S}}(q_i,q_j), \\text{ for } ij \\in E.$ In other words, this is the question of deciding whether a partial matrix can be realized in the $(k-1)$ -dimensional spherical space.",
"The analogous question for the Euclidean metric space $({\\mathbb {R}}^k,\\Vert \\cdot \\Vert )$ has been extensively studied.",
"In Section REF we will establish close connections with the notion of $k$ -realizability of graphs introduced in [8], [9] and to the corresponding graph parameter ${\\text{\\rm ed}}(G)$ .",
"Complexity issues concerning the parameter $\\text{\\rm gd}(G,x)$ are discussed in [14].",
"Specifically, given a graph $G$ and a rational vector in $\\mathcal {E}(G)$ , the problem of deciding whether $\\text{\\rm gd}(G,x)\\le k$ is proven to be NP-hard for every fixed $k\\ge 2$ [14]."
],
[
"Contents of the paper.",
"In Section REF we determine basic properties of the graph parameter $\\text{\\rm gd}(G)$ and in Section REF we reduce the proof of our main Theorem to the problem of computing the Gram dimension of the two graphs $V_8$ and $C_5 \\times C_2$ .",
"In Sections REF and REF we investigate the links of $\\text{\\rm gd}(G)$ with the graph parameters ${\\text{\\rm ed}}(G)$ and $\\nu ^=(G)$ , respectively.",
"Section introduces the main ingredients for our proof: In Section REF we discuss some genericity assumptions we can make, in Section REF we show how to use semidefinite programming, in Section REF we establish a number of useful lemmas, and in Section REF we show that $\\text{\\rm gd}(V_8)=4$ .",
"Section is dedicated to proving that $\\text{\\rm gd}(C_5\\times C_2)=4$ – this is the most technical part of the paper.",
"Lastly, in Section we conclude with some comments and open problems.",
"Part of this work will appear as an extended abstract in the proceedings of ISCO 2012 [26]."
],
[
"Basic definitions and properties",
"For a graph $G=(V=[n],E)$ let $ \\mathcal {S}_{+}(G)=\\pi _{VE}({\\mathcal {S}}^n_+)\\subseteq {\\mathbb {R}}^{V\\cup E}$ denote the projection of the positive semidefinite cone onto ${\\mathbb {R}}^{V\\cup E}$ , whose elements can be seen as the $G$ -partial matrices that can be completed to a psd matrix.",
"Let $\\mathcal {E}_n$ denote the set of matrices in ${\\mathcal {S}}^n_+$ with an all-ones diagonal (aka the correlation matrices), and let $\\mathcal {E}(G)=\\pi _{E}(\\mathcal {E}_n)\\subseteq {\\mathbb {R}}^E$ denote its projection onto the edge subspace ${\\mathbb {R}}^E$ , known as the elliptope of $G$ ; we only project on the edge set since all diagonal entries are implicitly known and equal to 1 for matrices in $\\mathcal {E}_n$ .",
"Given a graph $G=(V,E)$ and a vector $a\\in {\\mathbb {R}}^{V \\cup E}$ , a Gram representation of $a$ in ${\\mathbb {R}}^k$ consists of a set of vectors $p_1,\\ldots ,p_n\\in {\\mathbb {R}}^k$ such that $p_i^Tp_j=a_{ij}\\ \\forall ij \\in V \\cup E.$ The Gram dimension of $a\\in \\mathcal {S}_{+}(G)$ , denoted as $\\text{\\rm gd}(G,a)$ , is the smallest integer $k$ for which $a$ has a Gram representation in ${\\mathbb {R}}^k$ .",
"The Gram dimension of a graph $G=(V,E)$ is defined as $\\text{\\rm gd}(G)=\\underset{a \\in \\mathcal {S}_{+}(G)}{\\max } \\text{\\rm gd}(G,a).$ Clearly, the maximization in (REF ) can be restricted to be taken over all vectors $a \\in \\mathcal {E}(G)$ (where all diagonal entries are implicitly taken to be equal to 1).",
"We denote by ${\\mathcal {G}}_k$ the class of graphs $G$ for which $\\text{\\rm gd}(G)\\le k$ .",
"As a warm-up example, $\\text{\\rm gd}(K_n)=n$ : The upper bound is clear as $|V(K_n)|=n$ and the lower bound follows by considering, e.g., $a=\\pi _{V\\cup E}(I_n)$ .",
"We now investigate the behavior of the graph parameter $\\text{\\rm gd}(G)$ under some simple graph operations.",
"Recall that $G\\backslash e$ (resp., $G\\slash e$ ) denotes the graph obtained from $G$ by deleting (resp., contracting) the edge $e$ .",
"A graph $H$ is a minor of $G$ (denoted as $H\\preceq G$ ) if $H$ can be obtained from $G$ by successively deleting and contracting edges and deleting nodes.",
"The graph parameter $\\text{\\rm gd}(G)$ is monotone nonincreasing with respect to edge deletion and contraction: $\\text{\\rm gd}(G\\backslash e), \\text{\\rm gd}(G\\slash e)\\le \\text{\\rm gd}(G) $ for any edge $e\\in E$ .",
"Let $G=([n],E)$ and $e\\in E$ .",
"It follows directly from the definition that $\\text{\\rm gd}(G\\backslash e)\\le \\text{\\rm gd}(G)$ .",
"We show that $\\text{\\rm gd}(G\\slash e)\\le \\text{\\rm gd}(G)$ .",
"Say $e$ is the edge $(1,n)$ and $G\\slash e=([n-1],E^{\\prime })$ .",
"Consider $ X \\in {\\mathcal {S}}_{+}^{n-1}$ ; we show that there exists $X^{\\prime }\\in {\\mathcal {S}}^{n-1}_+$ with rank at most $k=\\text{\\rm gd}(G)$ and such that $\\pi _{E^{\\prime }}(X)=\\pi _{E^{\\prime }}(X^{\\prime })$ .",
"For this, extend $X$ to the matrix $Y\\in {\\mathcal {S}}^n_+$ defined by $Y_{nn}=X_{11}$ and $Y_{in}=X_{1i}$ for $i\\in [n-1]$ .",
"By assumption, there exists $Y^{\\prime }\\in {\\mathcal {S}}^n_+$ with rank at most $k$ such that $\\pi _E(Y)=\\pi _E(Y^{\\prime })$ .",
"Hence $Y^{\\prime }_{1i}=Y^{\\prime }_{ni}$ for all $i\\in [n]$ , so that the principal submatrix $X^{\\prime }$ of $Y^{\\prime }$ indexed by $[n-1]$ has rank at most $k$ and satisfies $\\pi _{E^{\\prime }}(X^{\\prime })=\\pi _{E^{\\prime }}(X)$ .",
"$\\Box $ Let $G_1=(V_1,E_1)$ , $G_2=(V_2,E_2)$ be two graphs, where $V_1\\cap V_2$ is a clique in both $G_1$ and $G_2$ .",
"Their clique sum is the graph $G=(V_1\\cup V_2, E_1\\cup E_2)$ , also called their clique $k$ -sum when $|V_1\\cap V_2|=k$ .",
"The following result follows from well known arguments (cf.",
"e.g.",
"[18]; a proof is included for completeness).",
"For a matrix $X$ indexed by $V$ and a subset $U\\subseteq V$ , $X[U]$ denotes the principal submatrix of $X$ indexed by $U$ .",
"If $G$ is the clique sum of two graphs $G_1$ and $G_2$ , then $\\text{\\rm gd}(G)=\\max \\lbrace \\text{\\rm gd}(G_1),\\text{\\rm gd}(G_2)\\rbrace .$ The proof relies on the following fact: Two psd matrices $X_i$ indexed by $V_i$ ($i=1,2$ ) such that $X_1[V_1\\cap V_2]=X_2[V_1\\cap V_2]$ admit a common psd completion $X$ indexed by $V_1\\cup V_2$ with rank $\\max \\lbrace {\\text{\\rm dim}}(X_1),{\\text{\\rm dim}}(X_2)\\rbrace $ .",
"Indeed, let $u^{(i)}_j$ ($j\\in V_i$ ) be a Gram representation of $X_i$ and let $U$ an orthogonal matrix mapping $u^{(1)}_j$ to $u^{(2)}_j$ for $j\\in V_1\\cap V_2$ , then the Gram representation of $Uu^{(1)}_j$ ($j\\in V_1$ ) together with $u^{(2)}_j$ ($j\\in V_2\\setminus V_1$ ) is such a common completion.",
"$\\Box $ Recall that the tree-width of a graph $G$ , denoted by ${\\rm tw}(G)$ , is the minimum integer $k$ for which $G$ is contained (as a subgraph) in a clique sum of copies of ${K_{k+1}}.$ As a direct application of Lemmas REF and REF we obtain the following bound: For any graph $G$ , $\\text{\\rm gd}(G)\\le {\\rm tw}(G)+1$ .",
"In view of Lemma REF , the class ${\\mathcal {G}}_k$ of graphs with Gram dimension at most $k$ is closed under taking minors.",
"Hence, by the celebrated graph minor theorem of [34], it can be characterized by finitely many minimal forbidden minors.",
"Clearly, $K_n$ is a minimal forbidden minor for ${\\mathcal {G}}_{n-1}$ for all $n$ , since contracting an edge yields a graph with $n-1$ nodes and deleting an edge yields a graph with tree-width at most $n-2$ .",
"It follows by its definition that the tree-width of a graph is a minor-monotone graph parameter.",
"One can easily verify that ${\\rm tw}(G) \\le 1 \\Longleftrightarrow K_3 \\lnot \\preceq G$ and it is known that ${\\rm tw}(G) \\le 2 \\Longleftrightarrow K_4 \\lnot \\preceq G$ [13].",
"Combining these two facts with Lemma REF yields the full list of forbidden minors for the class ${\\mathcal {G}}_k$ when $k\\le 3$ .",
"For $k\\le 3$ , $\\text{\\rm gd}(G)\\le k$ if and only if $G$ has no minor $K_{k+1}$ ."
],
[
"Characterizing graphs with Gram dimension at most 4",
"The next natural question is to characterize the class ${\\mathcal {G}}_4$ .",
"Clearly, $K_5$ is a minimal forbidden minor for ${\\mathcal {G}}_4$ .",
"We now show that this is also the case for the complete tripartite graph $K_{2,2,2}$ .",
"The graph $K_{2,2,2}$ is a minimal forbidden minor for ${\\mathcal {G}}_4$ .",
"First we construct $a\\in \\mathcal {E}(K_{2,2,2})$ with $\\text{\\rm gd}(K_{2,2,2},a)\\ge 5$ , thus implying $\\text{\\rm gd}(K_{2,2,2})\\ge 5$ .",
"For this, let $K_{2,2,2}$ be obtained from $K_6$ by deleting the edges $(1,4)$ , $(2,5)$ and $(3,6)$ .",
"Let $e_1,\\ldots ,e_5$ denote the standard unit vectors in ${\\mathbb {R}}^5$ , let $X$ be the Gram matrix of the vectors $e_1,e_2,e_3,e_4,e_5$ and $(e_1+e_2)/\\sqrt{2}$ labeling the nodes $1,\\ldots ,6$ , respectively, and let $a\\in \\mathcal {E}(K_{2,2,2})$ be the projection of $X$ .",
"We now verify that $X$ is the unique psd completion of $a$ which shows that $ \\text{\\rm gd}(K_{2,2,2},a)\\ge 5$ .",
"Indeed the chosen Gram labeling of the matrix $X$ implies the following linear dependency: $X[\\cdot ,6]=(X[\\cdot ,4]+X[\\cdot ,5])/\\sqrt{2}$ among its columns $X[\\cdot ,i]$ indexed respectively by $i=4,5,6$ ; this implies that the unspecified entries $X_{14}, X_{25}, X_{36}$ are uniquely determined in terms of the specified entries of $X$ .",
"On the other hand, one can easily verify that $K_{2,2,2}$ is a partial 4-tree, therefore $\\text{\\rm gd}(K_{2,2,2})\\le 5$ .",
"Moreover, deleting or contracting an edge in $K_{2,2,2}$ yields a partial 3-tree, thus with Gram dimension at most 4.",
"$\\Box $ By Lemma REF , all graphs with tree-width at most three belong to ${\\mathcal {G}}_4$ .",
"Moreover, these graphs can be characterized in terms of forbidden minors as follows: [3] A graph $G$ has ${\\rm tw}(G)\\le 3$ if and only if $G$ does not have $K_5,K_{2,2,2}, V_8$ and $C_5 \\times C_2$ as a minor.",
"The graphs $V_8$ and $C_5\\times C_2$ are shown in Figures REF and REF below, respectively.",
"These four graphs are natural candidates for being forbidden minors for the class ${\\mathcal {G}}_4$ .",
"We have already seen that for $K_5$ and $K_{2,2,2}$ this is indeed the case.",
"However, this is not true for $V_8$ and $C_5 \\times C_2$ .",
"Both belong to $ {\\mathcal {G}}_4$ , this will be proved in Section REF for $V_8$ (Theorem REF ) and in Section for $C_5\\times C_2$ (Theorem ).",
"These two results form the main technical part of the paper.",
"Using them, we can complete our characterization of the class ${\\mathcal {G}}_4$ .",
"For a graph $G$ , $\\text{\\rm gd}(G)\\le 4$ if and only if $G$ does not have $K_5$ or $K_{2,2,2}$ as a minor.",
"Necessity follows from Lemmas REF and REF .",
"Sufficiency follows from the following graph theoretical result, obtained by combining Theorem REF with Seymour's splitter theorem (for a self-contained proof see [20]): every graph with no $K_5$ and $K_{2,2,2}$ minors can be obtained as a subgraph of a clique $k$ -sum ($k\\le 2$ ) of copies of graphs with tree-width at most 3, $V_8$ and $C_5 \\times C_2$ .",
"Combining this fact with Theorems REF , and Lemmas REF , REF the claim follows.",
"$\\Box $"
],
[
"Links to Euclidean graph realizations",
"In this section we investigate the links between the Gram dimension and the notion of $k$ -realizability of graphs introduced in [8], [9].",
"We start the discussion with some necessary definitions.",
"Recall that a matrix $D=(d_{ij})\\in {\\mathcal {S}}^n$ is a Euclidean distance matrix (EDM) if there exist vectors $p_1,\\ldots ,p_n\\in {\\mathbb {R}}^k$ (for some $k\\ge 1$ ) such that $d_{ij}=\\Vert p_i-p_j\\Vert ^2$ for all $i,j\\in [n]$ .",
"Then $\\text{\\rm EDM}_n$ denotes the cone of all $n \\times n$ Euclidean distance matrices and, for a graph $G=([n],E)$ , $\\text{\\rm EDM}(G)=\\pi _E(\\text{\\rm EDM}_n)$ is the set of $G$ -partial matrices that can be completed to a Euclidean distance matrix.",
"Given a graph $G=([n],E)$ and $d\\in {\\mathbb {R}}_{+}^{E}$ , a Euclidean (distance) representation of $d$ in ${\\mathbb {R}}^k$ consists of a set of vectors $p_1,\\ldots ,p_n\\in {\\mathbb {R}}^k$ such that $\\Vert p_i-p_j\\Vert ^2=d_{ij}\\ \\forall ij \\in E.$ Then, ${\\text{\\rm ed}}(G,d)$ is the smallest integer $k$ for which $d$ has a Euclidean representation in ${\\mathbb {R}}^k$ and the graph parameter ${\\text{\\rm ed}}(G)$ is defined as ${\\text{\\rm ed}}(G)=\\underset{d \\in \\text{\\rm EDM}(G)}{\\max } {\\text{\\rm ed}}(G,d).$ In the terminology of [8], [9] a graph $G$ satisfying ${\\text{\\rm ed}}(G)\\le k$ is called $k$ -realizable.",
"It is easy to verify that the graph parameter ${\\text{\\rm ed}}(G)$ is minor monotone.",
"Hence for any fixed $k\\ge 1$ the class of graphs satisfying ${\\text{\\rm ed}}(G)\\le k$ can be characterized by a finite list of minimal forbidden minors.",
"For $k\\le 2$ the only forbidden minor is $K_{k+2}$ .",
"Belk and Connelly [8], [9] have determined the list of forbidden minors for $k=3$ .",
"[8], [9] For a graph $G$ , ${\\text{\\rm ed}}(G)\\le 3$ if and only if $G$ does not have $K_5$ and $K_{2,2,2}$ as minors.",
"The hard part of the proof of [8], [9] is to prove sufficiency, i.e., that if a graph $G$ has no $K_5$ and $K_{2,2,2}$ minors then ${\\text{\\rm ed}}(G)\\le 3$ .",
"We will obtain this result as a corollary of our main theorem (cf.",
"Corollary REF ).",
"To this end, we have to establish some connections between the graphs parameters ${\\text{\\rm ed}}(G)$ and $\\text{\\rm gd}(G)$ .",
"There is a well known correspondence between psd and EDM completions (for details and references see, e.g., [12]).",
"Namely, for a graph $G$ , let $\\nabla G$ denote its suspension graph, obtained by adding a new node (the apex node, denoted by 0), adjacent to all nodes of $G$ .",
"Consider the one-to-one map $\\phi : {\\mathbb {R}}^{V \\cup E(G)} \\mapsto {\\mathbb {R}}_{+}^{E(\\nabla G)}$ , which maps $x\\in {\\mathbb {R}}^{V \\cup E(G)}$ to $d=\\phi (x )\\in {\\mathbb {R}}_{+}^{E(\\nabla G)}$ defined by $ d_{0i}= x_{ii} \\ (i\\in [n]),\\ \\ \\ d_{ij}=x_{ii}+x_{jj}-2x_{ij} \\ (ij\\in E(G)).$ Then the vectors $u_1,\\ldots ,u_n\\in {\\mathbb {R}}^k$ form a Gram representation of $x$ if and only if the vectors $u_0=0,u_1,\\ldots ,u_n$ form a Euclidean representation of $d=\\phi (x)$ in ${\\mathbb {R}}^k$ .",
"This shows: Let $G=(V,E)$ be a graph.",
"Then, $\\text{\\rm gd}(G,x)={\\text{\\rm ed}}(\\nabla G,\\phi (x))$ for any $x\\in {\\mathbb {R}}^{V\\cup E}$ and thus $\\text{\\rm gd}(G)={\\text{\\rm ed}}(\\nabla G)$ .",
"For the Gram dimension of a graph one can show the following property: Consider a graph $G=(V=[n],E)$ and its suspension graph $\\nabla G=([n]\\cup \\lbrace 0\\rbrace ,E\\cup F)$ , where $F=\\lbrace (0,i)\\mid i\\in [n]\\rbrace $ .",
"Given $x\\in {\\mathbb {R}}^E$ , its 0-extension is the vector $y=(x,0)\\in {\\mathbb {R}}^{E\\cup F}$ .",
"If $x\\in {\\mathcal {S}}_+(G)$ , then $y\\in {\\mathcal {S}}_+(\\nabla G)$ and $\\text{\\rm gd}(\\nabla G, y)=\\text{\\rm gd}(G,x)+1$ .",
"Moreover, $\\text{\\rm gd}(\\nabla G)= \\text{\\rm gd}(G)+1$ .",
"The first part is clear and implies $\\text{\\rm gd}(\\nabla G)\\ge \\text{\\rm gd}(G)+1$ .",
"Set $k=\\text{\\rm gd}(G)$ ; we show the reverse inequality $\\text{\\rm gd}(\\nabla G)\\le k+1$ .",
"For this, let $X\\in {\\mathcal {S}}^{n+1}_+$ , written in block-form as $X=\\left(\\begin{matrix} \\alpha & a^T \\cr a & A\\end{matrix}\\right)$ , where $A\\in {\\mathcal {S}}^n_+$ and the first row/column is indexed by the apex node 0 of $\\nabla G$ .",
"If $\\alpha =0$ then $a=0$ , $\\pi _{VE}(A)$ has a Gram representation in ${\\mathbb {R}}^k$ and thus $\\pi _{ V(\\nabla G) E(\\nabla G)}(X)$ too.",
"Assume now $\\alpha > 0$ and without loss of generality $\\alpha =1$ .",
"Consider the Schur complement $Y$ of $X$ with respect to the entry $\\alpha =1$ , given by $Y=A-aa^T$ .",
"As $Y\\in {\\mathcal {S}}^n_+$ , there exists $Z\\in {\\mathcal {S}}^n_+$ such that $\\text{rank}(Z) \\le k$ and $\\pi _{V E}(Z)=\\pi _{V E}(Y)$ .",
"Define the matrix $X^{\\prime }:= \\left(\\begin{matrix} 1 & a^T \\cr a & aa^T\\end{matrix}\\right)+\\left(\\begin{matrix} 0 & 0 \\cr 0 & Z\\end{matrix}\\right).$ Then, ${\\rm rank}( X^{\\prime }) ={\\rm rank} ( Z)+1\\le k+1$ .",
"Moreover, $X^{\\prime }$ and $X$ coincide at all diagonal entries as well as at all entries corresponding to edges of $\\nabla G$ .",
"This concludes the proof that $\\text{\\rm gd}(\\nabla G)\\le k+1$ .",
"$\\Box $ We do not know whether the analogous property is true for the graph parameter ${\\text{\\rm ed}}(G)$ .",
"On the other hand, the following partial result holds, whose proof was communicated to us by A. Schrijver.",
"For a graph $G$ , ${\\text{\\rm ed}}(\\nabla G)\\ge {\\text{\\rm ed}}(G)+1$ .",
"Set ${\\text{\\rm ed}}(\\nabla G)=k$ ; we show ${\\text{\\rm ed}}(G)\\le k-1.$ We may assume that $G$ is connected (else deal with each connected component separately).",
"Let $d \\in \\text{\\rm EDM}(G)$ and let $p_1=0,p_2, \\ldots , p_n$ be a Euclidean representation of $d$ in ${\\mathbb {R}}^h$ ($h\\ge 1$ ).",
"Extend the $p_i$ 's to vectors $\\widehat{p_i}=(p_i,0)\\in {\\mathbb {R}}^{h+1}$ by appending an extra coordinate equal to zero, and set $\\widehat{p}_0(t)=(0,t)\\in {\\mathbb {R}}^{h+1}$ where $t$ is any positive real scalar.",
"Now consider the distance $\\widehat{d}(t) \\in \\text{\\rm EDM}(\\nabla G)$ with Euclidean representation $\\widehat{p_0}(t), \\widehat{p_1},\\ldots ,\\widehat{p_n}$ .",
"As ${\\text{\\rm ed}}(\\nabla G)=k$ , there exists another Euclidean representation of $\\widehat{d}(t)$ by vectors $q_0(t), q_1(t),\\ldots ,q_n(t)$ lying in ${\\mathbb {R}}^k$ .",
"Without loss of generality, we can assume that $q_0(t)=\\widehat{p_0}(t)=(0,t)$ and $q_1(t)$ is the zero vector; for $i\\in [n]$ , write $q_i(t)=(u_i(t),a_i(t))$ , where $u_i(t)\\in {\\mathbb {R}}^{k-1}$ and $a_i(t) \\in {\\mathbb {R}}$ .",
"Then $\\Vert q_i(t)\\Vert = \\Vert \\widehat{p_i}\\Vert =\\Vert p_i\\Vert $ whenever node $i$ is adjacent to node 1 in $G$ .",
"As the graph $G$ is connected, this implies that, for any $i\\in [n]$ , the scalars $\\Vert q_i(t)\\Vert $ ($t \\in {\\mathbb {R}}_+$ ) are bounded.",
"Therefore there exists a sequence $t_m \\in {\\mathbb {R}}_+$ ($m\\in {\\mathbb {N}}$ ) converging to $+\\infty $ and for which the sequence $(q_i(t_m))_m$ has a limit.",
"Say $q_i(t_m)=(a_i(t_m),u_i(t_m))$ converges to $(u_i,a_i)\\in {\\mathbb {R}}^k$ as $m \\rightarrow +\\infty $ , where $u_i\\in {\\mathbb {R}}^{k-1}$ and $a_i\\in {\\mathbb {R}}$ .",
"The condition $\\Vert q_0(t)-q_i(t)\\Vert ^2=\\widehat{d}(t)_{0i}$ implies that $\\Vert p_i\\Vert ^2+t^2=\\Vert u_i(t)\\Vert ^2+(a_i(t)-t)^2$ and thus $ a_i(t_m)=\\frac{a_i^2(t_m)+\\Vert u_i(t_m)\\Vert ^2-\\Vert p_i\\Vert ^2}{2t_m} \\hspace{5.69046pt} \\forall m\\in {\\mathbb {N}}.$ Taking the limit as $m \\rightarrow \\infty $ we obtain that $\\underset{m \\rightarrow \\infty }{\\lim } a_i(t_m)=0$ and thus $a_i=0$ .",
"Then, for $i,j\\in [n]$ , $d_{ij}=\\widehat{d}(t_m)_{ij}=\\Vert (a_i(t_m),u_i(t_m))-(a_j(t_m),u_j(t_m))\\Vert ^2$ and taking the limit as $m \\rightarrow +\\infty $ we obtain that $d_{ij}=\\Vert u_i-u_j\\Vert ^2$ .",
"This shows that the vectors $u_1,\\ldots ,u_n$ form a Euclidean representation of $d$ in ${\\mathbb {R}}^{k-1}$ .",
"$\\Box $ Combining Lemma REF with Theorem REF we obtain the following inequality relating the parameters ${\\text{\\rm ed}}(G)$ and $\\text{\\rm gd}(G)$ .",
"For any graph $G$ we have that ${\\text{\\rm ed}}(G)\\le \\text{\\rm gd}(G)-1.$ Combining Theorem REF with our main theorem we can recover sufficiency in Theorem REF .",
"For a graph $G$ , if $G$ has no $K_5$ and $K_{2,2,2}$ minors then ${\\text{\\rm ed}}(G)\\le 3$ ."
],
[
"Relation with the graph parameter $\\nu ^=(G)$",
"In this section we investigate the relation between the Gram dimension of a graph and the graph parameter $\\nu ^=(G)$ introduced in [20], [21].",
"Recall that the corank of a matrix $M\\in {\\mathbb {R}}^{n \\times n}$ is the dimension of its kernel.",
"Consider the cone ${\\mathcal {C}}(G)=\\lbrace M\\in {\\mathcal {S}}^n_+: M_{ij}=0\\ \\text{ for all distinct } i,j\\in V \\text{ with } (i,j)\\notin E\\rbrace $ which, as is well known, can be seen as the dual cone of the cone ${\\mathcal {S}}_+(G)$ .",
"We now introduce the graph parameter $\\nu ^=(G)$ .",
"Given a graph $G=([n],E)$ the parameter $\\nu ^=(G)$ is defined as the maximum corank of a matrix $M\\in {\\mathcal {C}}(G)$ satisfying the following property: $\\forall X\\in {\\mathcal {S}}^n \\ \\ \\ MX=0, \\ X_{ii}=0 \\ \\forall i\\in V, \\ X_{ij}=0\\ \\forall (i,j)\\in E \\ \\Longrightarrow X=0,$ known as the strong Arnold property.",
"It is proven in [20], [21] that $\\nu ^=(G)$ is a minor monotone graph parameter.",
"Hence for any fixed integer $k \\ge 1$ the class of graphs with $\\nu ^=(G)\\le k$ can be characterized by a finite family of minimal forbidden minors.",
"For $k\\le 3$ the only forbidden minor is $K_{k+1}$ .",
"Van der Holst [20], [21] has determined the list of forbidden minors for $k=4$ .",
"[20], [21] For a graph $G$ , $\\nu ^=(G)\\le 4$ if and only if $G$ does not have $K_5$ and $K_{2,2,2}$ as minors.",
"By relating the two parameters $\\text{\\rm gd}(G)$ and $\\nu ^=(G)$ we can derive sufficiency in Theorem REF from our main Theorem.",
"For any graph $G$ , $\\text{\\rm gd}(G) \\ge \\nu ^=(G)$ .",
"Let $k=\\nu ^=(G)$ be attained by some matrix $M\\in {\\mathcal {S}}^n_+$ .",
"Write $M=\\sum _{i=1}^n \\lambda _iv_iv_i^T$ , where $\\lambda _i\\ge 0$ , $\\lbrace v_1,\\ldots ,v_n\\rbrace $ is an orthonormal base of eigenvectors of $M$ , and $\\lbrace v_1,\\ldots ,v_k\\rbrace $ spans the kernel of $M$ .",
"Consider the matrix $X=\\sum _{i=1}^kv_iv_i^T$ and its projection $a=\\pi _{E\\cup V}(X)\\in {\\mathcal {S}}_+(G)$ .",
"By construction, ${\\rm rank} (X)=k$ .",
"Hence it is enough to show that $a$ has a unique psd completion, which will imply $\\text{\\rm gd}(G)\\ge \\text{\\rm gd}(G,a)=k$ .",
"For this let $Y\\in {\\mathcal {S}}^n_+$ be another psd completion of $a$ .",
"Hence the matrix $X-Y$ has zero entries at all positions $(i,j)\\in V\\cup E$ .",
"Since the matrix $M$ has zero entries at all off-diagonal positions corresponding to non-edges of $G$ , we deduce that $\\langle M,X-Y \\rangle =0$ .",
"On the other hand, $\\langle M,X\\rangle =\\sum _{i=1}^k\\lambda _iv_i^TMv_i=0$ .",
"Therefore, $\\langle M,Y\\rangle =0$ .",
"As $M,X,Y$ are psd, the conditions $\\langle M,X\\rangle =\\langle M,Y\\rangle =0$ imply that $MX=MY=0$ and thus $M(X-Y)=0$ .",
"Now we can apply the assumption that the matrix $M$ satisfies the strong Arnold property and deduce that $X=Y$ .",
"$\\Box $ Combining Theorem REF with our main theorem we can recover sufficiency in Theorem REF .",
"For a graph $G$ , if $G$ does not have $K_5$ and $K_{2,2,2}$ as minors then $\\nu ^=(G)\\le 4$ .",
"Colin de Verdière [11] studies the graph parameter $\\nu (G)$ , defined as the maximum corank of a matrix $M$ satisfying the strong Arnold property and such that, for any $i, j\\in V$ , $M_{ij}=0 \\Longleftrightarrow (i,j)\\notin E$ .",
"In particular he shows that $\\nu (G)$ is unbounded for the class of planar graphs.",
"As $\\nu (G)\\le \\nu ^=(G)\\le \\text{\\rm gd}(G)$ , we obtain as a direct application: The graph parameter $\\text{\\rm gd}(G)$ is unbounded for the class of planar graphs."
],
[
"Bounding the Gram dimension",
"In this section we sketch our approach to show that $\\text{\\rm gd}(V_8)=\\text{\\rm gd}(C_5\\times C_2)=4$ .",
"Given a graph $G=(V=[n],E)$ , a configuration of $G$ is an assignment of vectors $p_1,\\ldots ,p_n $ (in some space) to the nodes of $G$ ; the pair $(G,{p})$ is called a framework.",
"We use the notation ${p}=\\lbrace p_1,\\ldots ,p_n\\rbrace $ and, for a subset $T\\subseteq V$ , ${p}_T=\\lbrace p_i\\mid i\\in T\\rbrace $ .",
"Thus ${p}={p}_V$ and we also set ${p}_{-i}={p}_{V\\setminus \\lbrace i\\rbrace }$ .",
"Two configurations ${p},{q}$ of $G$ (not necessarily lying in the same space) are said to be equivalent if $p_i^Tp_j=q^T_i q_j$ for all $ij \\in V\\cup E$ .",
"Our objective is to show that the two graphs $G=V_8$ , $C_5 \\times C_2$ belong to ${\\mathcal {G}}_4$ .",
"That is, we must show that, given any $a\\in {\\mathcal {S}}_+(G)$ , one can construct a Gram representation ${q}$ of $(G,a)$ lying in the space ${\\mathbb {R}}^4$ .",
"Along the lines of [8] (which deals with Euclidean distance realizations), our strategy to achieve this is as follows: First, we construct a `flat' Gram representation ${p}$ of $(G,a)$ obtained by maximizing the inner product $p_{i_0}^Tp_{j_0}$ along a given pair $(i_0,j_0)$ which is not an edge of $G$ .",
"As suggested in [31] (in the context of Euclidean distance realizations), this configuration ${p}$ can be obtained by solving a semidefinite program; then ${p}$ corresponds to the Gram representation of an optimal solution $X$ to this program.",
"In general we cannot yet claim that ${p}$ lies in ${\\mathbb {R}}^4$ .",
"However, we can derive useful information about ${p}$ by using an optimal solution $\\Omega $ (which will correspond to a `stress matrix') to the dual semidefinite program.",
"Indeed, the optimality condition $X\\Omega =0$ will imply some linear dependencies among the $p_i$ 's that can be used to show the existence of an equivalent representation ${q}$ of $(G,a)$ in low dimension.",
"Roughly speaking, most often, these dependencies will force the majority of the $p_i$ 's to lie in ${\\mathbb {R}}^4$ , and one will be able to rotate each remaining vector $p_j$ about the space spanned by the vectors labeling the neighbors of $j$ into ${\\mathbb {R}}^4$ .",
"Showing that the initial representation ${p}$ can indeed be `folded' into ${\\mathbb {R}}^4$ as just described makes up the main body of the proof.",
"Before going into the details of the proof, we indicate some additional genericity assumptions that can be made w.l.o.g.",
"on the vector $a\\in {\\mathcal {S}}_+(G)$ .",
"This will be particularly useful when treating the graph $C_5\\times C_2$ ."
],
[
"Genericity assumptions",
"By definition, $\\text{\\rm gd}(G)$ is the maximum value of $\\text{\\rm gd}(G,a)$ taken over all $a\\in \\mathcal {E}(G)$ .",
"Clearly we can restrict the maximum to be taken over all $a$ lying in a dense subset of $\\mathcal {E}(G)$ .",
"For instance, the set ${\\mathcal {D}}$ consisting of all $x\\in \\mathcal {E}(G)$ that admit a positive definite completion in $\\mathcal {E}_n$ is dense in $\\mathcal {E}(G)$ .",
"We next identify a smaller dense subset ${\\mathcal {D}}^*$ of ${\\mathcal {D}}$ which will we use in our study of the Gram dimension of $C_5\\times C_2$ .",
"We start with a useful lemma, which characterizes the vectors $a\\in \\mathcal {E}(C_n)$ admitting a Gram realization in ${\\mathbb {R}}^2$ .",
"Here $C_n$ denotes the cycle on $n$ nodes.",
"Consider the vector $a=(\\cos \\vartheta _1, \\cos \\vartheta _2,\\ldots ,\\cos \\vartheta _n)\\in {\\mathbb {R}}^{E(C_n)}$ , where $\\vartheta _1,\\ldots ,\\vartheta _n\\in [0,\\pi ]$ .",
"Then $\\text{\\rm gd}(C_n,a)\\le 2$ if and only if there exist $\\epsilon \\in \\lbrace \\pm 1\\rbrace ^n$ and $k\\in {\\mathbb {Z}}$ such that $\\sum _{i=1}^n\\epsilon _i \\vartheta _i=2k\\pi $ .",
"We prove the `only if' part.",
"Assume that $u_1,\\ldots ,u_n\\in {\\mathbb {R}}^2$ are unit vectors such that $u_i^Tu_{i+1}= \\cos \\vartheta _{i}$ for all $i\\in [n]$ (setting $u_{n+1}=u_1$ ).",
"We may assume that $u_1=(1, 0)^T$ .",
"Then, $u_1^Tu_2=\\cos \\vartheta _1$ implies that $u_2=(\\cos (\\epsilon _1 \\vartheta _1),\\sin (\\epsilon _1 \\vartheta _1))^T$ for some $\\epsilon _1\\in \\lbrace \\pm 1\\rbrace $ .",
"Analogously, $u_2^Tu_3=\\cos \\vartheta _2$ implies $u_3=(\\cos (\\epsilon _1\\vartheta _1+\\epsilon _2\\vartheta _2), \\sin (\\epsilon _1\\vartheta _1+\\epsilon _2\\vartheta _2))^T$ for some $\\epsilon _2\\in \\lbrace \\pm 1\\rbrace $ .",
"Iterating, we find that there exists $\\epsilon \\in \\lbrace \\pm 1\\rbrace ^n$ such that $u_{i}=(\\cos (\\sum _{j=1}^{i-1}\\epsilon _i \\vartheta _i), \\sin (\\sum _{j=1}^{i-1}\\epsilon _i \\vartheta _i))^T$ for $i=1,\\ldots ,n$ .",
"Finally, the condition $u_n^Tu_1=\\cos \\vartheta _n= \\cos (\\sum _{i=1}^{n-1} \\epsilon _i \\vartheta _i)$ implies $\\sum _{i=1}^n\\epsilon _i\\vartheta _i\\in 2\\pi {\\mathbb {Z}}$ .",
"The arguments can be reversed to show the `if part'.",
"$\\Box $ Let ${\\mathcal {D}}^*$ be the set of all $a\\in \\mathcal {E}(G)$ that admit a positive definite completion in $\\mathcal {E}_n$ satisfying the following condition: For any circuit $C$ in $G$ , the restriction $a_C=(a_e)_{e\\in C}$ of $a$ to $C$ does not admit a Gram representation in ${\\mathbb {R}}^2$ .",
"Then the set ${\\mathcal {D}}^*$ is dense in $\\mathcal {E}(G)$ .",
"We show that ${\\mathcal {D}}^*$ is dense in ${\\mathcal {D}}$ .",
"Let $a\\in {\\mathcal {D}}$ and set $a=\\cos \\vartheta $ , where $\\vartheta \\in [0,\\pi ]^E$ .",
"Given a circuit $C$ in $G$ (say of length $p$ ), it follows from Lemma REF that $a_C$ has a Gram realization in ${\\mathbb {R}}^2$ if and only if $\\sum _{i=1}^p\\epsilon _i\\vartheta _i=2k\\pi $ for some $\\epsilon \\in \\lbrace \\pm 1\\rbrace ^p$ and $k\\in {\\mathbb {Z}}$ with $|k|\\le p/2$ .",
"Let ${\\mathcal {H}}_C$ denote the union of the hyperplanes in ${\\mathbb {R}}^{E(C)}$ defined by these equations.",
"Therefore, $a\\notin {\\mathcal {D}}^*$ if and only if $\\vartheta \\in \\cup _C {\\mathcal {H}}_C$ , where the union is taken over all circuits $C$ of $G$ .",
"Clearly we can find a sequence $\\vartheta ^{(i)} \\in [0,\\pi ]^E \\setminus \\cup _C {\\mathcal {H}}_C$ converging to $\\vartheta $ as $i\\rightarrow \\infty $ .",
"Then the sequence $a^{(i)}:=\\cos \\vartheta ^{(i)}$ tends to $a$ as $i\\rightarrow \\infty $ and, for all $i$ large enough, $a^{(i)}\\in {\\mathcal {D}}^*$ .",
"This shows that ${\\mathcal {D}}^*$ is a dense subset of ${\\mathcal {D}}$ and thus of $\\mathcal {E}(G)$ .",
"$\\Box $ For any graph $G=([n],E)$ , $\\text{\\rm gd}(G)=\\max \\text{\\rm gd}(G,a)$ , where the maximum is over all $a\\in \\mathcal {E}(G)$ admitting a positive definite completion and whose restriction to any circuit of $G$ has no Gram representation in the plane."
],
[
"Semidefinite programming formulation",
"We now describe how to model the `flattening' procedure using semidefinite programming (sdp) and how to obtain a `stress matrix' using sdp duality.",
"Let $G=(V=[n],E)$ be a graph and let $e_0=(i_0,j_0)$ be a non-edge of $G$ (i.e., $i_0\\ne j_0$ and $e_0\\notin E$ ).",
"Let $a\\in {\\mathcal {S}}_+(G)$ be a partial positive semidefinite matrix for which we want to show the existence of a Gram representation in a small dimensional space.",
"For this consider the semidefinite program: $\\max \\ \\langle E_{i_0j_0},X\\rangle \\ \\ \\text{\\rm s.t. }",
"\\langle E_{ij},X\\rangle =a_{ij} \\ (ij\\in V\\cup E),\\ \\ X\\succeq 0,$ where $E_{ij}=(e_ie_j^T+e_je_i^T)/2$ and $e_1,\\ldots ,e_n$ are the standard unit vectors in ${\\mathbb {R}}^n$ .",
"The dual semidefinite program of (REF ) reads: $\\min \\sum _{ij\\in V\\cup E} w_{ij} a_{ij}\\text{ \\rm s.t. }",
"\\Omega = \\sum _{ij\\in V\\cup E} w_{ij} E_{ij}-E_{i_0j_0}\\succeq 0.$ Consider a graph $G=([n],E)$ , a pair $e_0=(i_0,j_0)\\notin E$ , and let $a\\in {\\mathcal {S}}_{++}(G)$ .",
"Then there exists a Gram realization ${p}=(p_1,\\ldots ,p_n)$ in ${\\mathbb {R}}^k$ (for some $k\\ge 1$ ) of $(G,a)$ and a matrix $\\Omega =(w_{ij})\\in {\\mathcal {S}}^n_+$ satisfying $w_{i_0j_0}\\ne 0,$ $w_{ij}=0\\ \\text{ for all } ij\\notin V\\cup E\\cup \\lbrace e_0\\rbrace ,$ $w_{ii}p_i+\\sum _{j|ij \\in E \\cup \\lbrace e_0\\rbrace } w_{ij}p_j=0\\ \\text{for all } i\\in [n],$ $\\dim \\langle p_i,p_j\\rangle =2 \\ \\text{ for all } ij \\in E.$ We refer to equation (REF ) as the equilibrium condition at vertex $i$ .",
"Consider the sdp (REF ) and its dual program (REF ).",
"By assumption, $a$ has a positive definite completion, hence the program (REF ) is strictly feasible.",
"Clearly, the dual program (REF ) is also strictly feasible.",
"Hence there is no duality gap and the optimal values are attained in both programs.",
"Let $(X,\\Omega )$ be a pair of primal-dual optimal solutions.",
"Then $(X,\\Omega )$ satisfies the optimality condition: $\\langle X,\\Omega \\rangle =0$ or, equivalently, $X\\Omega =0$ .",
"Say $X$ has rank $k$ and let ${p}=\\lbrace p_1,\\ldots ,p_n\\rbrace \\subseteq {\\mathbb {R}}^k$ be a Gram realization of $X$ .",
"Now it suffices to observe that the condition $X\\Omega =0$ can be reformulated as the equilibrium conditions (REF ).",
"The conditions (REF ) and (REF ) follow from the form of the dual program (REF ), and (REF ) follows from the assumption $a\\in {\\mathcal {S}}_{++}(G)$ .",
"$\\Box $ Note that, using the following variant of Farkas' lemma for semidefinite programming, one can show the existence of a nonzero positive semidefinite matrix $\\Omega =(w_{ij})$ satisfying (REF ) and the equilibrium conditions (REF ) also in the case when the sdp (REF ) is not strictly feasible, however now with $w_{i_0j_0}=0$ .",
"This remark will be useful in the exceptional case considered in Section REF where we will have to solve again a semidefinite program of the form (REF ); however this program will have additional conditions imposing that some of the $p_i$ 's are pinned so that one cannot anymore assume strict feasibility (see the proof of Lemma REF ).",
"(Farkas' lemma for semidefinite programming) (see [28]) Let $b\\in {\\mathbb {R}}^m$ and let $A_1,\\ldots ,A_m\\in {\\mathcal {S}}^n$ be given.",
"Then exactly one of the following two assertions holds: (i) Either there exists $X\\in {\\mathcal {S}}^n_{++}$ such that $\\langle A_j,X\\rangle =b_j$ for $j=1,\\ldots ,m$ .",
"(ii) Or there exists a vector $y\\in {\\mathbb {R}}^m$ such that $\\Omega :=\\sum _{j=1}^m y_j A_j\\succeq 0$ , $\\Omega \\ne 0$ and $b^Ty \\le 0$ .",
"Moreover, for any $X\\succeq 0$ satisfying $\\langle A_j,X\\rangle =b_j$ ($j=1,\\ldots ,m$ ), we have in (ii) $\\langle X,\\Omega \\rangle = b^Ty=0$ and thus $X\\Omega =0$ .",
"Clearly, if (i) holds then (ii) does not hold.",
"Conversely, assume (i) does not hold, i.e., ${\\mathcal {S}}^n_{++}\\cap {\\mathcal {L}}=\\emptyset $ , where $\\mathcal {L}=\\lbrace X\\in {\\mathcal {S}}^n\\mid \\langle A_j,X\\rangle =b_j\\ \\forall j\\rbrace $ .",
"Then there exists a separating hyperplane, i.e., there exists a nonzero matrix $\\Omega \\in {\\mathcal {S}}^n$ and $\\alpha \\in {\\mathbb {R}}$ such that $\\langle \\Omega ,X\\rangle \\ge \\alpha $ for all $X\\in {\\mathcal {S}}^n_{++}$ and $\\langle \\Omega ,X\\rangle \\le \\alpha $ for all $X\\in \\mathcal {L}$ .",
"This implies $\\Omega \\succeq 0$ , $\\Omega \\in {\\mathcal {L}}^\\perp $ , and $\\alpha \\le 0$ , so that (ii) holds and the lemma follows.",
"$\\Box $"
],
[
"Useful lemmas",
"We start with some definitions about stressed frameworks and then we establish some basic tools that we will repeatedly use later in our proof for $V_8$ and $C_5\\times C_2$ .",
"For a matrix $\\Omega \\in {\\mathcal {S}}^n$ its support graph is the graph ${\\mathcal {S}}(\\Omega )$ is the graph with node set $[n]$ and with edges the pairs $(i,j)$ with $\\Omega _{ij}\\ne 0$ .",
"(Stressed framework $(H,{p},\\Omega )$ ) Consider a framework $(H=(V=[n],F), {p})$ .",
"A nonzero matrix $\\Omega =(w_{ij})\\in {\\mathcal {S}}^n$ is called a stress matrix for the framework $(H,{p}) $ if its support graph ${\\mathcal {S}}(\\Omega )$ is contained in $H$ (i.e., $w_{ij}=0$ for all $ij\\notin V\\cup F$ ) and $\\Omega $ satisfies the equilibrium condition $w_{ii}p_i+\\sum _{j: (i,j) \\in F}w_{ij} p_j=0 \\ \\ \\forall i\\in V.$ Then the triple $(H,{p},\\Omega )$ is called a stressed framework, and a psd stressed framework if moreover $\\Omega \\succeq 0$ .",
"We let $V_\\Omega $ denote the set of nodes $i\\in V$ for which $\\Omega _{ij}\\ne 0$ for some $j\\in V$ .",
"A node $i\\in V$ is said to be a 0-node when $w_{ij}=0$ for all $j\\in V$ .",
"Hence, $V\\setminus V_\\Omega $ is the set of all 0-nodes and, when $\\Omega \\succeq 0$ , $i$ is a 0-node if and only if $w_{ii}=0$ .",
"The support graph ${\\mathcal {S}}(\\Omega )$ of $\\Omega $ is called the stressed graph; its edges are called the stressed edges of $H$ and the nodes $i\\in V_\\Omega $ are called the stressed nodes.",
"Given an integer $t\\ge 1$ , a node $i\\in V$ is said to be a $t$ -node if its degree in the stressed graph ${\\mathcal {S}}(\\Omega )$ is equal to $t$ .",
"Throughout we will deal with stressed frameworks $(H,{p},\\Omega )$ obtained by applying Theorem REF .",
"Hence the graph $H$ arises by adding a new edge $e_0$ to a given graph $G$ , which we then denote as $H=\\widehat{G}$ , as indicated below.",
"(Extended graph $\\widehat{G}$ ) Given a graph $G=(V=[n],E)$ and a fixed pair $e_0=(i_0,j_0)$ not belonging to $E$ , we set $\\widehat{G}=(V,\\widehat{E}=E\\cup \\lbrace e_0\\rbrace )$ .",
"We now group some useful lemmas which we will use in order to show that a given framework $(H,{p})$ admits an equivalent configuration in lower dimension.",
"Clearly, the stress matrix provides some linear dependencies among the vectors $p_i$ labeling the stressed nodes, but it gives no information about the vectors labeling the 0-nodes.",
"However, if we have a set $S$ of 0-nodes forming a stable set, then we can use the following lemma in order to `fold' the corresponding vectors $p_i$ ($i\\in S$ ) in a lower dimensional space.",
"(Folding the vectors labeling a stable set) Let $(H=(V,F),{p})$ be a framework and let $T\\subseteq V$ .",
"Assume that $S=V\\setminus T$ is a stable set in $H$ , that each node $i\\in S$ has degree at most $k-1$ in $H$ , and that ${\\text{\\rm dim}}\\langle {p}_T\\rangle \\le k$ .",
"Then there exists a configuration ${q}$ of $H$ in $ {\\mathbb {R}}^k$ which is equivalent to $(H,{p})$ .",
"Fix a node $i\\in S$ .",
"Let $N[i]$ denote the closed neighborhood of $i$ in $H$ consisting of $i$ and the nodes adjacent to $i$ .",
"By assumption, $|N[i]| \\le k$ and both sets of vectors ${p}_T$ and ${p}_{N[i]}$ have rank at most $k$ .",
"Hence one can find an orthogonal matrix $P$ mapping all vectors $p_j$ ($j\\in T\\cup N[i]$ ) into the space ${\\mathbb {R}}^k$ .",
"Repeat this construction with every other node of $S$ .",
"As no two nodes of $S$ are adjacent, this produces a configuration ${q}$ in ${\\mathbb {R}}^k$ which is equivalent to $(H,{p})$ .",
"$\\Box $ The next lemma uses the stress matrix to upper bound the dimension of a given stressed configuration.",
"(Bounding the dimension) Let $(H=(V=[n],F),{p},\\Omega )$ be a psd stressed framework.",
"Then ${\\text{\\rm dim}}\\langle {p}_V\\rangle \\le n-2$ , except ${\\text{\\rm dim}}\\langle {p}_V\\rangle \\le n-1$ if ${\\mathcal {S}}(\\Omega )$ is a clique.",
"Let $X$ denote the Gram matrix of the $p_i$ 's, so that $\\text{rank} (X)={\\text{\\rm dim}}\\langle {p}_V\\rangle $ .",
"By assumption, $X\\Omega =0$ .",
"This implies that $\\text{rank} (X)\\le n-1$ .",
"Moreover, if ${\\mathcal {S}}(\\Omega )$ is not a clique, then $\\text{rank} (\\Omega )\\ge 2$ and thus $\\text{rank} (X)\\le n-2$ .",
"$\\Box $ The next lemma indicates how 1-nodes can occur in a stressed framework.",
"Let $(H=(V,F),{p},\\Omega )$ be a stressed framework.",
"If node $i$ is a 1-node in the stressed graph ${\\mathcal {S}}(\\Omega )$ , i.e., there is a unique edge $ij\\in F$ such that $w_{ij}\\ne 0$ , then ${\\text{\\rm dim}}\\langle p_i, p_j\\rangle \\le 1$ .",
"Directly, using the equilibrium condition (REF ) at node $i$ .",
"$\\Box $ We now consider 2-nodes in a stressed framework.",
"First recall the notion of Schur complement.",
"For a matrix $\\Omega =(w_{ij})\\in {\\mathcal {S}}^n$ and $i\\in [n]$ with $w_{ii}\\ne 0$ , the Schur complement of $\\Omega $ with respect to its $(i,i)$ -entry is the matrix, denoted as $\\Omega _{-i}=(w^{\\prime }_{jk})_{j,k\\in [n]\\setminus \\lbrace i\\rbrace }\\in {\\mathcal {S}}^{n-1}$ , with entries $w^{\\prime }_{jk}= w_{jk} -w_{ik}w_{jk}/w_{ii}$ for $i,j\\in [n]\\setminus \\lbrace i\\rbrace $ .",
"As is well known, $\\Omega \\succeq 0$ if and only if $w_{ii}>0$ and $\\Omega _{-i}\\succeq 0$ .",
"We also need the following notion of `contracting a degree two node' in a graph.",
"Let $H=(V,F)$ be a graph, let $i\\in V$ be a node of degree two in $H$ which is adjacent to nodes $i_1,i_2\\in V$ .",
"The graph obtained by contracting node $i$ in $H$ is the graph $H/i$ with node set $V\\setminus \\lbrace i\\rbrace $ and with edge set $F/i= F\\setminus \\lbrace (i,i_1),(i,i_2)\\rbrace \\cup \\lbrace (i_1,i_2)\\rbrace $ (ignoring multiple edges).",
"(Contracting a 2-node) Let $(H=(V,F),{p}, \\Omega )$ be a psd stressed framework, let $i\\in V$ be a 2-node in the stressed graph ${\\mathcal {S}}(\\Omega )$ and set $N(i)=\\lbrace i_1,i_2\\rbrace $ .",
"Then $p_i\\in \\langle p_{i_1},p_{i_2}\\rangle $ and thus ${\\text{\\rm dim}}\\langle {p}\\rangle = {\\text{\\rm dim}}\\langle {p}_{-i}\\rangle .$ Moreover, if the stressed graph ${\\mathcal {S}}(\\Omega )$ is not the complete graph on $N[i]=\\lbrace i,i_1,i_2\\rbrace $ , then $(H/i,{p}_{-i},\\Omega _{-i})$ is a psd stressed framework.",
"The equilibrium condition at node $i$ implies $p_i \\in \\langle p_{i_1},p_{i_2}\\rangle $ .",
"Note that the Schur complement $\\Omega _{-i}$ of $\\Omega $ with respect to the $(i,i)$ -entry $w_{ii}$ has entries $w^{\\prime }_{i_1i_2}=w_{i_1i_2}- w_{ii_1}w_{ii_2}/w_{ii}$ , $w_{i_ri_r}^{\\prime }=w_{i_ri_r}- w_{ii_r}^2/w_{ii}$ for $r=1,2$ , and $w^{\\prime }_{jk}=w_{jk}$ for all other edges $jk$ of $H/i$ .",
"As $\\Omega \\succeq 0$ we also have $\\Omega _{-i}\\succeq 0$ .",
"Moreover, $\\Omega _{-i}\\ne 0$ .",
"Indeed, $w_{i_1i_2}^{\\prime }\\ne 0$ if $(i_1,i_2)\\notin F$ ; otherwise, as ${\\mathcal {S}}(\\Omega )$ is not the clique on $N[i]$ , there is another edge $jk$ of $H/i$ in the support of $\\Omega $ so that $w_{jk}^{\\prime }=w_{jk}\\ne 0$ .",
"In order to show that $\\Omega _{-i}$ is a stress matrix for $(H/i,{p}_{-i})$ , it suffices to check the stress equilibrium at the nodes $i_1$ and $i_2$ .",
"To fix ideas consider node $i_1$ .",
"Then we can rewrite $w^{\\prime }_{i_1i_1}p_{i_1}+w^{\\prime }_{i_1i_2}p_{i_2}+\\sum _{j\\in N(i_1)\\setminus \\lbrace i_2\\rbrace }w^{\\prime }_{i_1j}p_j$ as $( \\sum _{j} w_{i_1j}p_j) - \\left(w_{ii}p_i+ w_{ii_1}p_{i_1}+w_{ii_2}p_{i_2}\\right) w_{ii_1}/w_{ii},$ where both terms are equal to 0 using the equilibrium conditions of $(\\Omega ,{p})$ at nodes $i_1$ and $i$ .",
"$\\Box $ We will apply the above lemma iteratively to contract a set $I$ containing several 2-nodes.",
"Of course, in order to obtain useful information, we want to be able to claim that, after contraction, we obtain a stressed framework $(H/I, {p}_{V\\setminus I}, \\Omega _{-I})$ , i.e., with $\\Omega _{-I}\\ne 0$ .",
"Problems might occur if at some step we get a stressed graph which is a clique on 3 nodes.",
"Note that this can happen only when a connected component of the stressed graph is a circuit.",
"However, when we will apply this operation of contracting 2-nodes to the case of $G=C_5\\times C_2$ , we will make sure that this situation cannot happen; that is, we will show that we may assume that the stressed graph does not have a connected component which is a circuit (see Remark REF in Section REF )."
],
[
"The graph $V_8$ has Gram dimension 4",
"Let $V_8=(V=[8],E)$ be the graph shown in Figure REF .",
"In this section we use the tools developed above to show that $V_8$ has Gram dimension 4.",
"Figure: The graph V 8 V_8.The graph $V_8$ has Gram dimension 4.",
"Set $G=V_8=([8],E)$ .",
"Clearly $\\text{\\rm gd}(G)\\ge 4$ since $K_4$ is a minor of $ G$ .",
"Fix $a\\in {\\mathcal {S}}_{++}(G)$ ; we show that $(G,a)$ has a Gram realization in ${\\mathbb {R}}^4$ .",
"For this we first apply Theorem REF .",
"As stretched edge $e_0$ , we choose the pair $e_0=(1,4)$ and we denote by $\\widehat{G}=([8], \\widehat{E}=E\\cup \\lbrace (1,4)\\rbrace )$ the extended graph obtained by adding the stretched pair $(1,4)$ to $G$ .",
"Let ${p}$ be the initial Gram realization of $(G,a)$ and let $\\Omega =(w_{ij})$ be the corresponding stress matrix obtained by applying Theorem REF .",
"We now show how to construct from ${p}$ an equivalent realization $q$ of $(G,a)$ lying in ${\\mathbb {R}}^4$ .",
"In view of Lemma REF , we know that we are done if we can find a subset $S\\subseteq V$ which is stable in the graph $G$ and satisfies ${\\text{\\rm dim}}\\langle {p}_{V\\setminus S}\\rangle \\le 4$ .",
"This permits to deal with 1-nodes.",
"Indeed suppose that there is a 1-node in the stressed graph ${\\mathcal {S}}(\\Omega )$ .",
"In view of Lemma REF and (REF ), this can only be node 1 (or node 4) (i.e., the end points of the stretched pair) and ${\\text{\\rm dim}}\\langle p_1,p_4\\rangle \\le 1$ .",
"Then, choosing the stable set $S=\\lbrace 2,5,7\\rbrace $ , we have ${\\text{\\rm dim}}\\langle {p}_{V\\setminus S}\\rangle \\le 4$ and we can conclude using Lemma REF .",
"Hence we can now assume that there is no 1-node in the stressed graph ${\\mathcal {S}}(\\Omega )$ .",
"Next, observe that we are done in any of the following two cases: (i) There exists a set $T\\subseteq V$ with $|T|=4$ and ${\\text{\\rm dim}}\\langle {p}_T\\rangle \\le 2$ .",
"(ii) There exists a set $T\\subseteq V$ of cardinality $|T|=3$ such that $T$ does not consist of three consecutive nodes on the circuit $(1,2,\\ldots ,8)$ and ${\\text{\\rm dim}}\\langle {p}_T\\rangle \\le 2$ .",
"Indeed, in case (i) (resp., case (ii)), there is a stable set $S \\subseteq V\\setminus T$ of cardinality $|S|=2$ (resp., $|S|=3$ ), so that $|V\\setminus (S\\cup T)|=2$ and thus ${\\text{\\rm dim}}\\langle {p}_{V\\setminus S} \\rangle \\le {\\text{\\rm dim}}\\langle {p}_T\\rangle + {\\text{\\rm dim}}\\langle {p}_{V\\setminus (S\\cup T)} \\rangle \\le 2+2 =4$ .",
"Hence we may assume that we are not in the situation of cases (i) and (ii).",
"Assume first that one of the nodes in $\\lbrace 5,6,7,8\\rbrace $ is a 0-node.",
"Then all of them are 0-nodes.",
"Indeed, if (say) 5 is a 0-node and 6 is not a 0-node then the equilibrium equation at node 6 implies that ${\\text{\\rm dim}}\\langle p_6,p_7,p_2\\rangle \\le 2$ so that we are in the situation of case (ii).",
"As nodes 1, 4 are not 1-nodes, the stressed graph ${\\mathcal {S}}(\\Omega )$ is the circuit $(1,2,3,4)$ .",
"Using Lemma REF , we deduce that ${\\text{\\rm dim}}\\langle p_1,p_2,p_3,p_4\\rangle \\le 2$ and thus we are in the situation of case (i) above.",
"Assume now that none of the nodes in $\\lbrace 5,6,7,8\\rbrace $ is a 0-node but one of the nodes in $\\lbrace 2,3\\rbrace $ is a 0-node.",
"Then both nodes 2 and 3 are 0-nodes (else we are in the situation of case (ii)).",
"Therefore, both nodes 6 and 7 are 2-nodes.",
"Applying Lemma REF , after contracting both nodes 6,7, we obtain a stressed framework on $\\lbrace 1,4,5,8\\rbrace $ and thus ${\\text{\\rm dim}}\\langle {p}_{V\\setminus \\lbrace 2,3\\rbrace }\\rangle = {\\text{\\rm dim}}\\langle p_1,p_4,p_5,p_8 \\rangle $ .",
"Using Lemma REF , we deduce that ${\\text{\\rm dim}}\\langle p_1,p_4,p_5,p_8\\rangle \\le 3$ .",
"Therefore, ${\\text{\\rm dim}}\\langle {p}_{V\\setminus \\lbrace 3\\rbrace }\\rangle \\le 4$ and one can find a new realization ${q}$ in ${\\mathbb {R}}^4$ equivalent to $(G,{p})$ using Lemma REF .",
"Finally assume that none of the nodes in $\\lbrace 2,3,5,6,7,8\\rbrace $ is a 0-node.",
"We show that $\\langle {\\bf p}\\rangle = \\langle p_2,p_3,p_6,p_7\\rangle $ .",
"The equilibrium equation at node 6 implies that ${\\text{\\rm dim}}\\langle p_2,p_5,p_6,p_7 \\rangle \\le 3$ .",
"Moreover, ${\\text{\\rm dim}}\\langle p_2,p_6,p_7\\rangle =3$ (else we are in case (ii) above).",
"Hence $p_5\\in \\langle p_2, p_6,p_7\\rangle $ .",
"Analogously, the equilibrium equations at nodes 7,2,3 give that $p_8,p_1,p_4 \\in \\langle p_2,p_3, p_6,p_7\\rangle $ , respectively.",
"$\\Box $ This section is devoted to proving that the graph $C_5\\times C_2$ has Gram dimension 4.",
"The analysis is considerably more involved than the analysis for $V_8$ .",
"Figure REF shows two drawings of $C_5\\times C_2$ , the second one making its symmetries more apparent.",
"The graph $C_5 \\times C_2$ has Gram dimension 4.",
"Throughout this section we set $G= C_5\\times C_2=(V=[10],E)$ .",
"Clearly, $\\text{\\rm gd}(G)\\ge 4$ since $K_4$ is a minor of $G$ .",
"In order to show that $\\text{\\rm gd}(G)\\le 4$ , we must show that $\\text{\\rm gd}(G,a)\\le 4$ for any $a\\in {\\mathcal {S}}_{++}(G)$ .",
"Moreover, in view of Corollary REF , it suffices to show this for all $a\\in {\\mathcal {S}}_{++}(G)$ satisfying the following `genericity' property: For any Gram realization ${p}$ of $(G,a)$ , ${\\text{\\rm dim}}\\langle {p}_C\\rangle \\ge 3 \\ \\text{ for any circuit } C \\text{ in } G. $ From now on, we fix $a\\in {\\mathcal {S}}_{++}(G)$ satisfying this genericity property.",
"Our objective is to show that there exists a Gram realization of $(G,a)$ in ${\\mathbb {R}}^4$ .",
"Again we use Theorem REF to construct an initial Gram realization ${p}$ of $(G,a)$ .",
"As stretched edge $e_0$ , we choose the pair $e_0=(3,8)$ and we denote by $\\widehat{G}=([8], \\widehat{E}=E\\cup \\lbrace (3,8)\\rbrace )$ the extended graph obtained by adding the stretched pair $(3,8)$ to $G$ .",
"By Theorem REF , we also have a stress matrix $\\Omega $ so that $(\\widehat{G},{p},\\Omega )$ is a psd stressed framework.",
"Our objective is now to construct from ${p}$ another Gram realization ${q}$ of $(G,a)$ lying in ${\\mathbb {R}}^4$ .",
"Figure: Two drawings of the graph C 5 ×C 2 C_5\\times C_2."
],
[
"Additional useful lemmas",
"First we deal with the case when ${\\text{\\rm dim}}\\langle p_i,p_j\\rangle =1$ for some pair $(i,j)$ of distinct nodes.",
"As $a\\in {\\mathcal {S}}_{++}(G)$ , this can only happen when $(i,j)\\notin E$ .",
"If ${\\text{\\rm dim}}\\langle p_i,p_j\\rangle =1$ for some pair $(i,j)\\notin E$ , then there is a configuration in ${\\mathbb {R}}^4$ equivalent to $(G,{p})$ .",
"By assumption, $p_i=\\epsilon p_j$ for some scalar $\\epsilon \\ne 0$ .",
"Up to symmetry there are two cases to consider: (i) $(i,j)=(1,5)$ (two nodes at distance 2 in $G$ ), or (ii) $(i,j)=(1,6)$ (two nodes at distance 3).",
"Consider first case (i) when $(i,j)=(1,5)$ , so $p_1=\\epsilon p_5$ .",
"Set $V^{\\prime }=V\\setminus \\lbrace 1\\rbrace $ .",
"Let $G^{\\prime }=(V^{\\prime },E^{\\prime })$ be the graph on $V^{\\prime }$ obtained from $G$ by deleting node 1 and adding the edges $(2,5)$ and $ (5,9)$ (in other words, get $G^{\\prime }$ by identifying nodes 1 and 5 in $G$ ).",
"Let $X^{\\prime }$ be the Gram matrix of the vectors $p_i$ ($i\\in V^{\\prime }$ ) and define $a^{\\prime }=(X^{\\prime }_{jk})_{jk\\in V^{\\prime }\\cup E^{\\prime }} \\in {\\mathcal {S}}_+(G^{\\prime })$ .",
"First we show that $(G^{\\prime },a^{\\prime })$ has a Gram realization in ${\\mathbb {R}}^4$ .",
"For this, consider the graph $H$ obtained from $G$ by deleting both nodes 1 and 5.",
"Then $G^{\\prime }$ is a subgraph of $\\nabla H$ and thus $\\text{\\rm gd}(G^{\\prime })\\le \\text{\\rm gd}(\\nabla H)=\\text{\\rm gd}(H)+1$ .",
"As ${\\rm tw}(H)\\le 2$ it follows that $\\text{\\rm gd}(H)\\le 3$ and thus $\\text{\\rm gd}(G^{\\prime })\\le 4$ .",
"Finally, if ${\\bf q}_{V^{\\prime }}$ is a Gram realization in ${\\mathbb {R}}^4$ of $(G^{\\prime },a^{\\prime })$ then, setting $q_1=\\epsilon q_5$ , we obtain a Gram realization ${\\bf q}$ of $(G,a)$ in ${\\mathbb {R}}^4$ .",
"Case (ii) is analogous, based on the fact that the graph $H$ obtained from $G$ by deleting nodes 1 and 6 is a partial 2-tree.",
"$\\Box $ We now consider the case when the stressed graph might have a circuit as a connected component.",
"Let $C$ be a circuit in $\\widehat{G}$ .",
"If $C$ is a connected component of ${\\mathcal {S}}(\\Omega )$ , then ${\\text{\\rm dim}}\\langle {p}_C\\rangle \\le 2$ .",
"Directy, using Lemma REF combined with Lemma REF .",
"$\\Box $ Therefore, in view of the genericity assumption (REF ), if a circuit $C$ is a connected component of the stressed graph, then $C$ cannot be a circuit in $G$ and thus $C$ must contain the stretched pair $e_0=(3,8)$ .",
"The next result is useful to handle this case, treated in Corollary REF below.",
"Let $N_2(i)$ be the set of nodes at distance 2 from a given node $i$ in $G$ .",
"If ${\\text{\\rm dim}}\\langle {p}_{N_2(i)}\\rangle \\le 3$ , then there is a configuration equivalent to $(G,{p})$ in ${\\mathbb {R}}^4$ .",
"Say, $i=1$ so that $N_2(1)= \\lbrace 4,5,7,10\\rbrace $ , cf.",
"Figure REF .",
"Consider the set $S=\\lbrace 2,3,6,9\\rbrace $ which is stable in $G$ .",
"Let $H$ denote the graph obtained from $G$ in the following way: For each node $i\\in S$ , delete $i$ and add the clique on $N(i)$ .",
"One can verify that $H$ is contained in the clique 4-sum of the two cliques $H_1$ and $H_2$ on the node sets $V_1=\\lbrace 1,4,5,7,10\\rbrace $ and $V_2=\\lbrace 4,5,7,8,10\\rbrace $ , respectively.",
"By assumption, ${\\text{\\rm dim}}\\langle {p}_{V_1}\\rangle \\le 4$ and ${\\text{\\rm dim}}\\langle {p}_{V_2}\\rangle \\le 4$ .",
"Therefore, one can apply an orthogonal transformation and find vectors $q_i\\in {\\mathbb {R}}^4$ ($i\\in V_1\\cup V_2$ ) such that ${p}_{V_r}$ and ${q}_{V_r}$ have the same Gram matrix, for $r=1,2$ .",
"Finally, as $V_1\\cup V_2=V\\setminus S$ and the set $S$ is stable in $G$ , one can extend to a configuration ${\\bf q}_V$ equivalent to ${p}_V$ by applying Lemma REF .",
"$\\Box $ If there is a circuit $C$ in $\\widehat{G}$ containing the (stretched) edge $(3,8)$ such that ${\\text{\\rm dim}}\\langle {p}_C\\rangle \\le 2$ , then there is a configuration equivalent to $(G,{p})$ in ${\\mathbb {R}}^4$ .",
"If $|C|\\ge 7$ , pick $i\\in V\\setminus C$ and note that ${\\text{\\rm dim}}\\langle {p}_{-i}\\rangle \\le 4$ .",
"If $|C|=6$ , pick a subset $S\\subseteq V\\setminus C$ of cardinality 2 which is stable in $G$ , so that ${\\text{\\rm dim}}\\langle {p}_{V\\setminus S}\\rangle \\le 4$ .",
"In both cases we can conclude using Lemma REF .",
"Assume now that $|C|= 4$ or 5.",
"In view of Lemma REF , it suffices to check that there exists a node $i$ for which $|C\\cap N_2(i)| =3$ .",
"For instance, for $C=(3,8,7,5)$ , this holds for node $i= 9$ , and for $C=(3,8,10,9,1)$ this holds for $i=2$ .",
"Then, Lemma REF implies that ${\\text{\\rm dim}}\\langle {p}_{N_2(i)}\\rangle \\le 3$ .",
"$\\Box $ From now on, we will assume that ${\\text{\\rm dim}}\\langle p_i,p_j\\rangle =2$ for all $i\\ne j\\in V$ (by Lemma REF ).",
"Hence there is no 1-node in the stressed graph.",
"Moreover, we will assume that no circuit $C$ of $\\widehat{G}$ satisfies ${\\text{\\rm dim}}\\langle {p}_C\\rangle \\le 2$ .",
"Therefore, the stressed graph does not have a connected component which is a circuit (by (REF ), Lemma REF and Corollary REF ).",
"Hence we are guaranteed that after contracting several 2-nodes we do obtain a stressed framework (i.e, with a nonzero stress matrix).",
"The next two lemmas settle the case when there are sufficiently many 2-nodes.",
"If there are at least four 2-nodes in the stressed graph ${\\mathcal {S}}(\\Omega )$ , then there is a configuration equivalent to $(G,{p})$ in ${\\mathbb {R}}^4$ .",
"Let $I$ be a set of four 2-nodes in ${\\mathcal {S}}(\\Omega )$ .",
"Hence, ${p}_I\\subseteq \\langle {p}_{V\\setminus I}\\rangle $ and thus it suffices to show that ${\\text{\\rm dim}}\\langle {p}_{V\\setminus I}\\rangle \\le 4$ .",
"After contracting each of the four 2-nodes of $I$ , we obtain a psd stressed framework $(\\widehat{G}/ I, {p}_{V\\setminus I }, \\Omega ^{\\prime })$ .",
"Indeed, we can apply Lemma REF and obtain a nonzero psd stress matrix $\\Omega ^{\\prime }$ in the contracted graph (recall Remark REF ).",
"If the support graph of $\\Omega ^{\\prime }$ is not a clique, Lemma REF implies that ${\\text{\\rm dim}}\\langle {p}_{V\\setminus I}\\rangle \\le |V\\setminus I|-2 = 4$ .",
"Assume now that ${\\mathcal {S}}(\\Omega ^{\\prime })$ is a clique on $T\\subseteq V\\setminus I$ .",
"Then ${\\text{\\rm dim}}\\langle {p}_T\\rangle \\le t-1$ , $|V\\setminus (I\\cup T)|=6-t$ , and $t=|T| \\in \\lbrace 3,4,5\\rbrace $ .",
"Indeed one cannot have $t=6$ since, after contracting the four 2-nodes, at least 4 edges are lost so that there remains at most $16- 4=12<15$ edges.",
"It suffices now to show that we can partition $V\\setminus (I\\cup T)$ as $S\\cup S^{\\prime }$ , where $S$ is stable in $G$ and $|S^{\\prime }|+t-1 \\le 4$ .",
"Indeed, we then have ${\\text{\\rm dim}}\\langle {p}_{V\\setminus S} \\rangle = {\\text{\\rm dim}}\\langle {p}_{T\\cup S^{\\prime }}\\rangle \\le t-1+|S^{\\prime }|\\le 4$ and we can conclude using Lemma REF .",
"If $t=5$ , then $|V\\setminus (I\\cup T)|=1$ and choose $S^{\\prime }=\\emptyset $ .",
"If $t=4$ , then choose $S^{\\prime }\\subseteq V\\setminus (I\\cup T)$ of cardinality 1.",
"If $t=3$ , then one can choose a stable set of cardinality 2 in $V\\setminus (I\\cup T)$ and $|S^{\\prime }|=1$ .",
"$\\Box $ If there is at least one 0-node and at least three 2-nodes in the stressed graph ${\\mathcal {S}}(\\Omega )$ , then there is a configuration equivalent to $(G,{p})$ in ${\\mathbb {R}}^4$ .",
"For $r=0,2$ , let $V_r$ denote the set of $r$ -nodes and set $n_r=|V_r|$ .",
"By assumption, $n_0\\ge 1$ and we can assume $n_2=3$ (else apply Lemma REF ).",
"Set $W=V\\setminus (V_0\\cup V_2)$ .",
"After contracting the three 2-nodes in the stressed framework $(\\widehat{G},{p},\\Omega )$ , we get a stressed framework $(H,{p}_W,\\Omega ^{\\prime })$ on $|W|=7-n_0$ nodes.",
"Hence $n_0\\le 4$ and ${p}_{V_2}\\subseteq \\langle {p}_W\\rangle $ .",
"Assume first that ${\\mathcal {S}}(\\Omega ^{\\prime })$ is not a clique.",
"Then ${\\text{\\rm dim}}\\langle {p}_W\\rangle \\le |W|-2 =5-n_0$ by Lemma REF .",
"Now we can conclude using Lemma REF since in each of the cases: $n_0=1,2,3,4$ , one can find a stable set $S\\subseteq V_0$ such that ${\\text{\\rm dim}}\\langle {p}_{W\\cup (V_0\\setminus S)}\\rangle \\le 4$ .",
"Assume now that ${\\mathcal {S}}(\\Omega ^{\\prime })$ is a clique.",
"Then ${\\text{\\rm dim}}\\langle {p}_W\\rangle \\le |W|-1=6-n_0$ by Lemma REF .",
"Note first that $n_0\\ne 1,2$ .",
"Indeed, if $n_0=1$ then, after deleting the 0-node and contracting the three 2-nodes, we have lost at least $3+3=6$ edges.",
"Hence there remains at most $16-6=10$ edges in the stressed graph ${\\mathcal {S}}(\\Omega ^{\\prime })$ , which therefore cannot be a clique on six nodes.",
"If $n_0=2$ then, after deleting the two 0-nodes and contracting the three 2-nodes, we have lost at least $5 + 3=8$ edges.",
"Hence there remain at most $16-8=8$ edges in the stressed graph ${\\mathcal {S}}(\\Omega ^{\\prime })$ , which therefore cannot be a clique on five nodes.",
"In each of the two cases $n_0=3,4$ , one can find a stable set $S\\subseteq V_0$ of cardinality 2 and thus ${\\text{\\rm dim}}\\langle {p}_{W\\cup (V_0\\setminus S)}\\rangle \\le (6-n_0)+(n_0-2)=4$ .",
"Again conclude using Lemma REF .",
"$\\Box $"
],
[
"Sketch of the proof",
"In the proof we distinguish two cases: (i) when there is no 0-node, and (ii) when there is at least one 0-node, which are considered, respectively, in Sections REF and REF .",
"In both cases the tools developed in the preceding section permit to conclude, except in one exceptional situation, occurring in case (ii).",
"This execptional situation is when nodes 1,2,9 and 10 are 0-nodes and all edges of $\\widehat{G}\\setminus \\lbrace 1,2,9,10\\rbrace $ are stressed.",
"This situation needs a specific treatment which is done in Section REF ."
],
[
"There is no 0-node in the stressed graph",
"In this section we consider the case when each node is stressed in ${\\mathcal {S}}(\\Omega )$ , i.e., $w_{ii}\\ne 0$ for all $i\\in [n]$ .",
"Assume that all vertices are stressed in the stressed graph ${\\mathcal {S}}(\\Omega )$ and that there exists a circuit $C$ of length 4 in $G$ such that all edges in the cut $\\delta (C)$ are stressed, i.e., $w_{ij}\\ne 0$ for all edges $ij\\in \\widehat{E}$ with $i\\in C$ and $j\\in V\\setminus C$ .",
"Then $\\dim \\langle {p}_V \\rangle \\le 4$ .",
"Up to symmetry, there are three types of circuits $C$ of length 4 to consider: (i) $C$ does not meet $\\lbrace 3,8\\rbrace $ , i.e., $C=(1,2,10,9)$ ; or (ii) $C$ contains one of the two nodes 3,8, say node 8, and it contains a node adjacent to the other one, i.e., node 3, like $C=(5,6,8,7)$ ; or (iii) $C$ contains one of 3,8 but has no node adjacent to the other one, like $C=(7,8,10,9)$ .",
"Consider first the case (i), when $C=(1,2,10,9)$ .",
"We show that the set ${p}_C$ spans ${p}_V$ .",
"Using the equilibrium conditions at the nodes 1,2,9,10, we find that $p_3,p_4,p_7,p_8 \\in \\langle {p}_C \\rangle $ .",
"As 6 is not a 0-node, $w_{6i}\\ne 0$ for some $i\\in \\lbrace 4,8\\rbrace $ .",
"Then, the equilibrium condition at node $i$ implies that $p_6\\in \\langle {p}_C\\rangle $ .",
"Analogously for node 5.",
"Case (ii) when $C=(5,6,8,7)$ can be treated in analogous manner.",
"Just note that the equilibrium conditions applied to nodes 7,5,6 and 8 respectively, imply that $p_9,p_3,p_4,p_{10}\\in \\langle {p}_C\\rangle $ .",
"We now consider case (iii) when $C=(7,8,10,9)$ .",
"Then one sees directly that $p_1,p_2,p_5 \\in \\langle {p}_C\\rangle $ .",
"If $w_{24}\\ne 0$ , then the equilibrium conditions at nodes 2,3,6 imply that $p_4,p_3,p_6 \\in \\langle {p}_C\\rangle $ and thus $\\langle {p}_C\\rangle =\\langle {p}_V\\rangle $ .",
"Assume now that $w_{24}=0$ , which implies $w_{34},w_{46}\\ne 0$ .",
"If $w_{13}\\ne 0$ , then the equilibrium conditions at nodes 1,3,4 imply that $p_C$ spans $p_3,p_4,p_6$ and we are done.",
"Assume now that $w_{24}=w_{13}=0$ , so that 1,2,4 are 2-nodes.",
"If there is one more 2-node then we are done by Lemma REF .",
"Hence we can now assume that $w_{ij}\\ne 0$ whenever $(i,j)\\ne (2,4)$ or $(1,3)$ .",
"After contracting the three 2-nodes 1,2,4 in the psd stressed framework $(\\widehat{G}, {p}, \\Omega )$ , we obtain a new psd stressed framework on $V\\setminus \\lbrace 1,2,4\\rbrace $ where nodes 9, 10 have again degree 2.",
"So contract these two nodes and get another psd stressed framework on $V\\setminus \\lbrace 1,2,4,9,10\\rbrace $ .",
"Finally this implies ${\\text{\\rm dim}}\\langle {p}_V\\rangle ={\\text{\\rm dim}}\\langle {p}_{V\\setminus \\lbrace 1,2,4,9,10\\rbrace }\\rangle \\le 4.$ $\\Box $ In view of Lemma REF , we can now assume that, for each circuit $C$ of length 4 in $G$ , there is at least one edge $ij\\in \\delta (C)$ which is not stressed, i.e., $w_{ij}=0$ .",
"It suffices now to show that this implies the existence of at least four 2-nodes, as we can then conclude using Lemma REF .",
"For this let us enumerate the cuts $\\delta (C)$ of the 4-circuits $C$ in $G$ : $\\bullet $ For $C=(1,2,10,9)$ , $\\delta (C)=\\lbrace (1,3),(2,4),(7,9),(8,10)\\rbrace $ .",
"$\\bullet $ For $C=(7,9,10,8)$ , $\\delta (C)=\\lbrace (1,9), (2,10), (5,7), (6,8)\\rbrace .$ $\\bullet $ For $C=(5,6,8,7)$ , $\\delta (C)=\\lbrace (7,9), (8,10), (3,5), (4,6)\\rbrace $ .",
"$\\bullet $ For $C=(3,5,6,4)$ , $\\delta (C)=\\lbrace (1,3), (2,4), (5,7),(6,8)\\rbrace $ .",
"$\\bullet $ For $C=(1,3,4,2)$ , $\\delta (C)=\\lbrace (3,5), (4,6), (1,9), (2, 10)\\rbrace $ .",
"For instance, $w_{24}=0$ implies that both 2 and 4 are 2-nodes, while $w_{13}=0$ implies that 1 is a 2-node.",
"One can easily verify that there are at least four 2-nodes in ${\\mathcal {S}}(\\Omega )$ ."
],
[
"There is at least one 0-node in the stressed graph",
"Note that the mapping $\\sigma : V\\rightarrow V$ that permutes each of the pairs $(1,10)$ , $(4,7)$ , $(5,6)$ , $(2,9)$ and $(3,8)$ is an automorphism of $G$ .",
"This can be easily seen using the second drawing of $C_5 \\times C_2$ in Figure REF .",
"Hence, as nodes 3 and 8 are not 0-nodes, up to symmetry, it suffices to consider the following three cases: $\\bullet $ Node 1 is a 0-node.",
"$\\bullet $ Nodes 1, 10 are not 0-nodes and node 4 is a 0-node.",
"$\\bullet $ Nodes 1, 10, 4, 7 are not 0-nodes and one of 5 or 2 is a 0-node."
],
[
"Node 1 is a 0-node.",
"It will be useful to use the drawing of $\\widehat{G}$ from Figure REF .",
"There, the thick edge (3,8) is known to be stressed, the dotted edges are known to be non-stressed (i.e., $w_{ij}=0$ ), while the other edges could be stressed or not.",
"In view of Lemma REF , we can assume that there are at most two 2-nodes (else we are done).",
"Figure: A drawing of C 5 ×C 2 ^\\widehat{C_5 \\times C_2} with 1 as the root node.Assume first that both nodes 2 and 9 are 0-nodes.",
"Then node 10 too is a 0-node and each of nodes 4 and 7 is a 0- or 2-node.",
"If both 4,7 are 2-nodes, then all edges in the graph $G\\backslash \\lbrace 1,2,9,10\\rbrace $ are stressed.",
"Hence we are in the exceptional case, which we will consider in Section REF below.",
"If 4 is a 0-node and 7 is a 2-node, then 3,7 must be the only 2-nodes and thus 6 is a 0-node.",
"Hence, the stressed graph is the circuit $C=(3,8,5,7)$ , which implies ${\\text{\\rm dim}}\\langle {\\bf p}_C\\rangle \\le 2$ and thus we can conclude using Corollary REF .",
"If 4 is a 2-node and 7 is a 0-node, then we find at least two more 2-nodes.",
"Finally, if both 4,7 are 0-nodes, then the stressed graph is the circuit $C=(3,8,6,5)$ and thus we can again conclude using Corollary REF .",
"We can now assume that at least one of the two nodes 2,9 is a 2-node.",
"Then, node 3 has degree 3 in the stressed graph.",
"(Indeed, if 3 is a 2-node, then 10 must be a 0-node (else we have three 2-nodes), which implies that 2,9 are 0-nodes, a contradiction.)",
"If exactly one of nodes 2,9 is stressed, one can easily see that there should be at least three 2-nodes.",
"Finally consider the case when both nodes 2,9 are stressed.",
"Then they are the only 2-nodes which implies that all edges of $G\\backslash 1$ are stressed.",
"Set $I=\\lbrace 4,5,8\\rbrace $ .",
"We show that ${p}_I$ spans ${p}_{V\\setminus \\lbrace 1\\rbrace }$ , so that ${p}_{\\lbrace 1,4,5,8\\rbrace }$ spans ${p}_V$ .",
"Indeed, the equilibrium conditions at 3 and 6 imply that $p_3,p_6\\in \\langle {p}_I\\rangle $ .",
"Next, the equilibrium conditions at $4,5,2,9$ imply, respectively, that $p_2\\in \\langle p_3,p_4,p_6\\rangle \\subseteq \\langle {p}_I\\rangle $ , $p_7\\in \\langle p_3,p_5,p_6\\rangle \\subseteq \\langle {p}_I\\rangle $ , $p_{10}\\in \\langle p_2,p_4\\rangle \\subseteq \\langle {p}_I\\rangle $ , and $p_9\\in \\langle p_7,p_{10}\\rangle \\subseteq \\langle {p}_I\\rangle $ .",
"This concludes the proof."
],
[
"Nodes 1, 10 are not 0-nodes and node 4 is a 0-node.",
"It will be useful to use the drawing of $\\widehat{G}$ from Figure REF .",
"We can assume that node 2 is a 2-node and that node 3 has degree 3 in the stressed graph, since otherwise one would find at least three 2-nodes.",
"Consider first the case when 6 is a 2-node.",
"Figure: A drawing of C 5 ×C 2 ^\\widehat{C_5 \\times C_2} with 2 as the root node.Then nodes 2 and 6 are the only 2-nodes which implies that all edges in the graph $G\\backslash 4$ are stressed.",
"Set $I=\\lbrace 3,5,7,10\\rbrace $ .",
"We show that $p_I$ spans $p_{V\\setminus \\lbrace 4\\rbrace }$ , and then we can conclude using Lemma REF .",
"Indeed, the equilibrium conditions applied, respectively, to nodes 5,6,3,1,2 imply that $p_6\\in \\langle {p}_I\\rangle $ , $p_8\\in \\langle p_5,p_6\\rangle \\subseteq \\langle {p}_I\\rangle $ , $p_1\\in \\langle p_3,p_5,p_8\\rangle \\subseteq \\langle {p}_I\\rangle $ , $p_9\\in \\langle p_1,p_7,p_{10}\\rangle \\subseteq \\langle {p}_I\\rangle $ , $p_2\\in \\langle p_1,p_{10}\\rangle \\subseteq \\langle {p}_I\\rangle $ .",
"Consider now the case when 6 is a 0-node.",
"Then 2 and 5 are the only 2-nodes so that all edges in the graph $G\\backslash \\lbrace 4,6\\rbrace $ are stressed.",
"Set $I=\\lbrace 3,7,10\\rbrace $ .",
"We show that ${p}_I$ spans ${p}_{V\\setminus \\lbrace 4,6\\rbrace }$ , and then we can again conclude using Lemma REF .",
"Indeed the equilibrium conditions applied, respectively, at nodes 5,8,3,2,1 imply that $p_5, p_8 \\in \\langle {p}_I\\rangle $ , $p_1\\in \\langle p_3,p_5,p_8\\rangle \\subseteq \\langle {p}_I\\rangle $ , $p_2\\in \\langle p_1,p_{10}\\rangle \\subseteq \\langle {p}_I\\rangle $ , $p_9\\in \\langle p_2,p_8,p_{10}\\rangle \\subseteq \\langle {p}_I\\rangle $ ."
],
[
"Nodes 1, 4, 7, 10 are not 0-nodes and node 5 or 2 is a 0-node.",
"It will be useful to use the drawing of $\\widehat{G}$ from Figure REF .",
"We assume that nodes 1,4,7,10 are not 0-nodes.",
"Consider first the case when node 5 is a 0-node.",
"Then node 7 is a 2-node.",
"Figure: A drawing of C 5 ×C 2 ^\\widehat{C_5 \\times C_2} with 3 as the root node.If node 6 is a 2-node, then 6 and 7 are the only 2-nodes and thus all edges of the graph $G\\backslash 5$ are stressed.",
"Setting $I=\\lbrace 1,2,4,8\\rbrace $ , one can verify that ${p}_I$ spans ${p}_{V\\setminus \\lbrace 5\\rbrace }$ and then one can conclude using Lemma REF .",
"If node 6 is a 0-node, then nodes 4 and 7 are the only 2-nodes and thus all edges in the graph $G\\backslash \\lbrace 5,6\\rbrace $ are stressed.",
"Setting $I=\\lbrace 2,3,9\\rbrace $ , one can verify that ${p}_I$ spans ${p}_{V\\setminus \\lbrace 5,6\\rbrace }$ .",
"Thus ${p}_{\\lbrace 2,3,9,6\\rbrace }$ spans ${p}_{V\\setminus \\lbrace 5\\rbrace }$ and one can again conclude using Lemma REF .",
"Consider finally the case when nodes 1,4,7,10, 5 and 6 are not 0-nodes and node 2 is a 0-node.",
"As node 2 is adjacent to nodes 1, 4 and 10 in $G$ , we find three 2-nodes and thus we are done."
],
[
"The exceptional case",
"In this section we consider the following case which was left open in the first case considered in Section REF : Nodes 1, 2, 9 and 10 are 0-nodes and all edges of the graph $\\widehat{G}\\backslash \\lbrace 1,2,9,10\\rbrace $ are stressed.",
"Then, nodes 4 and 7 are 2-nodes in the stressed graph.",
"After contracting both nodes 4,7, we obtain a stressed graph which is the complete graph on 4 nodes.",
"Hence, using Lemma REF , we can conclude that $\\dim \\langle {p}_{V_1}\\rangle \\le 3$ , where $V_1= V\\setminus \\lbrace 1,2,9,10\\rbrace $ .",
"Among the nodes 1, 2, 9 and 10, we can find a stable set of size 2.",
"Hence, if $\\dim \\langle {p}_{V_1}\\rangle \\le 2$ then, using Lemma REF , we can find an equivalent configuration in dimension $2+2 =4$ and we are done.",
"From now on we assume that $ \\dim \\langle {p}_{V_1}\\rangle =3.$ In this case it is not clear how to fold ${p}$ in ${\\mathbb {R}}^4$ .",
"In order to settle this case, we proceed as in Belk [8]: We fix (or pin) the vectors $p_i$ labeling the nodes $i\\in V_1$ and we search for another set of vectors $p^{\\prime }_i$ labeling the nodes $i\\in V_2=V\\setminus V_1=\\lbrace 1,2,9,10\\rbrace $ so that ${p}_{V_1}\\cup {p}^{\\prime }_{V_2}$ can be folded into ${\\mathbb {R}}^4$ .",
"Again, our starting point is to get such new vectors $p^{\\prime }_i$ ($i\\in V_2$ ) which, together with ${p}_{V_1}$ , provide a Gram realization of $(G,a)$ , by stretching along a second pair $e^{\\prime }$ ; namely we stretch the pair $e^{\\prime }=(4,9)\\in V_1\\times V_2$ .",
"As in So and Ye [31], this configuration ${p}^{\\prime }_{V_2}$ is again obtained by solving a semidefinite program; details are given below."
],
[
"Computing ${p}^{\\prime }_{V_2}$ via semidefinite programming.",
"Let $E[V_2]$ denote the set of edges of $G$ contained in $V_2$ and let $E[V_1,V_2]$ denote the set of edges $(i,j)\\in E$ with $i\\in V_1$ , $j\\in V_2$ .",
"Moreover, set $|V_1|=n_1 \\ge |V_2|=n_2$ , so the configuration ${p}_{V_1}$ lies in $ {\\mathbb {R}}^{n_1}$ .",
"(Here $n_1=6$ , $n_2=4$ ).",
"We now search for a new configuration ${p}^{\\prime }_{V_2}$ by stretching along the pair $(4,9)$ .",
"For this we use the following semidefinite program: $\\begin{array}{lll}\\max \\ \\langle F_{49},Z\\rangle \\ \\text{\\rm such that } & \\langle F_{ij},Z\\rangle =a_{ij} & \\forall ij \\in E[V_1,V_2]\\\\& \\langle E_{ij}, Z\\rangle =a_{ij} & \\forall ij \\in V_2\\cup E[V_2]\\\\& \\langle E_{ij},Z\\rangle =0 & \\forall i< j, i,j\\in V_1\\\\& \\langle E_{ii},Z\\rangle =1& \\forall i\\in V_1\\\\&Z\\succeq 0.\\end{array}$ Here, $E_{ij}=(e_ie_j^T+e_je_i^T)/2 \\in {\\mathcal {S}}^{n_1+n_2} $ , where $e_i$ ($ i\\in [n_1+ n_2]$ ) are the standard unit vectors in ${\\mathbb {R}}^{n_1+n_2}$ .",
"Moreover, for $i\\in V_1$ , $j\\in V_2$ , $F_{ij}= (p_i^{\\prime }e_j^T+e_j(p_i^{\\prime })^T)/2$ , after setting $p^{\\prime }_i=(p_i,0)\\in {\\mathbb {R}}^{n_1+n_2}$ .",
"Consider now a matrix $Z$ feasible for (REF ).",
"Then $Z$ can be written in the block form $Z=\\left(\\begin{matrix} I_{n_1} & Y \\cr Y^T & X\\end{matrix}\\right)$ , and let $y_i\\in {\\mathbb {R}}^{n_1}$ ($i\\in V_2$ ) denote the columns of $Y$ .",
"The condition $Z\\succeq 0$ is equivalent to $X-Y^TY\\succeq 0$ .",
"Say, $X-Y^TY$ is the Gram matrix of the vectors $z_i\\in {\\mathbb {R}}^{n_2}$ ($i\\in V_2$ ).",
"For $i\\in V_2$ , set $p^{\\prime }_i=(y_i,z_i)\\in {\\mathbb {R}}^{n_1+n_2}$ .",
"Then $X$ is the Gram matrix of the vectors $p^{\\prime }_i \\ (i \\in V_2)$ .",
"For $i\\in V_1$ , $j\\in V_2$ we have that $\\langle F_{ij},Z\\rangle =(p_i,0)^T(y_j,z_j)=(p^{\\prime }_i)^T p^{\\prime }_j.$ Moreover, for $i,j\\in V_2$ , we have that $\\langle {E}_{ij},Z\\rangle =X_{ij}=(p^{\\prime }_i)^Tp^{\\prime }_j$ .",
"Therefore, the linear conditions $\\langle F_{ij},Z\\rangle =a_{ij}$ for $ij \\in E[V_1,V_2] $ and $\\langle E_{ij},Z\\rangle =a_{ij} $ for $ij\\in V_2\\cup E[V_2]$ imply that the vectors $p^{\\prime }_i$ ($i\\in V_1\\cup V_2$ ) form a Gram realization of $(G,a)$ .",
"We now consider the dual semidefinite program of (REF ) which, as we see in Lemma REF below, will give us some equilibrium conditions on the new vectors $p^{\\prime }_i$ ($i\\in V_2$ ).",
"The dual program involves scalar variables $w^{\\prime }_{ij}$ (for $ij\\in E[V_1,V_2]\\cup V_2\\cup E[V_2]$ ) and a matrix $U^{\\prime }=\\left(\\begin{matrix} U & 0 \\cr 0 & 0\\end{matrix}\\right)$ , and it reads: $\\begin{array}{ll}\\min & \\displaystyle \\langle I_{n_1}, U\\rangle + \\sum _{ij \\in E[V_1,V_2]} w^{\\prime }_{ij} a_{ij} +\\sum _{ij \\in V_2\\cup E[V_2]} w^{\\prime }_{ij} a_{ij} \\\\\\text{such that }& \\displaystyle \\Omega ^{\\prime } = -F_{49} + U^{\\prime } +\\sum _{ij \\in E[V_1,V_2] } w^{\\prime }_{ij}F_{ij} +\\sum _{ij\\in V_2\\cup E[V_2]} w^{\\prime }_{ij} E_{ij}\\succeq 0.\\end{array}$ Since the primal program (REF ) is bounded and the dual program (REF ) is strictly feasible it follows that program (REF ) has an optimal solution $Z$ .",
"Let $p^{\\prime }_i \\in {\\mathbb {R}}^{n_1+n_2}$ ($i\\in V_1\\cup V_2$ ) be the vectors as defined above, which thus form a Gram realization of $(G,a)$ .",
"There exists a nonzero matrix $\\Omega ^{\\prime }=(w^{\\prime }_{ij}) \\succeq 0$ satisfying the optimality condition $Z\\Omega ^{\\prime }=0$ and the following conditions on its support: $\\begin{array}{l}w^{\\prime }_{ij}=0 \\ \\ \\forall (i,j)\\in (V_1\\times V_2) \\setminus (E[V_1,V_2] \\cup \\lbrace (4,9)\\rbrace ), \\\\w^{\\prime }_{ij}=0 \\ \\ \\forall i\\ne j\\in V_2, (i,j) \\notin E[V_2].\\end{array}$ Moreover, the following equilibrium conditions hold: $w^{\\prime }_{ii}p^{\\prime }_i+\\sum _{j\\in V_1\\cup V_2\\mid ij\\in E\\cup \\lbrace (4,9)\\rbrace } w^{\\prime }_{ij}p^{\\prime }_j=0\\ \\ \\forall i\\in V_2$ and $w^{\\prime }_{ij}\\ne 0$ for some $ij\\in V_2\\cup E[V_2]$ .",
"Furthermore, a node $i\\in V_2$ is a 0-node, i.e., $w^{\\prime }_{ij}=0$ for all $j\\in V_1\\cup V_2$ , if and only if $w^{\\prime }_{ii}=0$ .",
"If the primal program (REF ) is strictly feasible, then (REF ) has an optimal solution $\\Omega ^{\\prime }$ which satisfies $Z\\Omega ^{\\prime }=0$ and (REF ) (with $w^{\\prime }_{49}=-1$ ).",
"Otherwise, if (REF ) is feasible but not strictly feasible then, using Farkas' lemma (Lemma REF ), we again find a matrix $\\Omega ^{\\prime }\\succeq 0$ satisfying $Z\\Omega ^{\\prime }=0$ and (REF ) (now with $w^{\\prime }_{49}=0$ ).",
"We now indicate how to derive (REF ) from the condition $Z\\Omega ^{\\prime }=0$ .",
"For this write the matrices $Z$ and $\\Omega ^{\\prime }$ in block form $Z=\\left(\\begin{matrix} I_{n_1} & Y\\cr Y^T & X \\end{matrix}\\right),\\ \\ \\Omega ^{\\prime }=\\left(\\begin{matrix} \\Omega ^{\\prime }_{1}& \\Omega ^{\\prime }_{12}\\cr (\\Omega ^{\\prime }_{12})^T & \\Omega ^{\\prime }_2\\end{matrix}\\right).$ From $Z\\Omega ^{\\prime }=0$ , we derive $Y^T\\Omega ^{\\prime }_{12}+X\\Omega ^{\\prime }_2=0$ and $\\Omega ^{\\prime }_{12}+Y\\Omega ^{\\prime }_2=0$ .",
"First this implies $(X-Y^TY)\\Omega ^{\\prime }_2=0$ which in turn implies that the $V_2$ -coordinates of the vectors on the left hand side in (REF ) are equal to 0.",
"Second, the condition $\\Omega ^{\\prime }_{12}+Y\\Omega ^{\\prime }_2=0$ together with expressing $\\Omega ^{\\prime }_{12}=\\sum _{ij\\in E[V_1,V_2]\\cup \\lbrace (4,9)\\rbrace } w^{\\prime }_{ij} p^{\\prime }_i e_j^T $ , implies that the $V_1$ -coordinates of the vectors on the left hand side in (REF ) are 0.",
"Thus (REF ) holds.",
"Finally, we verify that $\\Omega ^{\\prime }_2\\ne 0$ .",
"Indeed, $\\Omega ^{\\prime }_2=0$ implies $\\Omega ^{\\prime }_{12}=0$ and thus $\\Omega ^{\\prime }=0$ since $0=\\langle Z,\\Omega ^{\\prime }\\rangle = \\langle I_{n_1}, \\Omega ^{\\prime }_1\\rangle $ .",
"$\\Box $"
],
[
"Folding ${p}^{\\prime }$ into {{formula:09006d39-ac29-4fb7-b37e-5d59d1fc7847}} .",
"We now use the above configuration $p^{\\prime }$ and the equilibrium conditions (REF ) at the nodes of $V_2$ to construct a Gram realization of $(G,a)$ in ${\\mathbb {R}}^4$ .",
"By construction, $p^{\\prime }_i=(p_i,0)$ for $i\\in V_1$ .",
"Note that no node $i\\in V_2$ is a 1-node with respect to the new stress $\\Omega ^{\\prime }$ (recall Lemma REF ).",
"Let us point out again that Lemma REF does not guarantee that $w^{\\prime }_{49}\\ne 0$ (as opposed to relation (REF ) in Theorem REF ).",
"By assumption nodes 1,2,9 and 10 are 0-nodes and all other edges of the graph $\\widehat{G}\\setminus \\lbrace 1,2,9,10\\rbrace $ are stressed w.r.t.",
"the old stress matrix $\\Omega $ .",
"We begin with the following easy observation about ${p}^{\\prime }_{V_1}$ .",
"$\\dim \\langle p^{\\prime }_4,p^{\\prime }_7,p^{\\prime }_8\\rangle = \\dim \\langle p^{\\prime }_3,p^{\\prime }_4,p^{\\prime }_8\\rangle =3$ .",
"It is easy to see that each of these sets spans ${p}^{\\prime }_{V_1}$ and ${\\text{\\rm dim}}\\langle {p}^{\\prime }_{V_1}\\rangle ={\\text{\\rm dim}}\\langle {p}_{V_1}\\rangle =3$ by (REF ).",
"$\\Box $ As an immediate corollary we may assume that $p^{\\prime }_i\\notin \\langle {p}^{\\prime }_{V_1}\\rangle \\ \\forall i\\in V_2$ Indeed, if there exists $i\\in V_2$ satisfying $p^{\\prime }_i \\in \\langle {p}^{\\prime }_{V_1}\\rangle $ then we can find a stable set of size two in $V_2\\setminus \\lbrace i\\rbrace $ and using Lemma REF we can construct an equivalent configuration in ${\\mathbb {R}}^4$ .",
"Therefore, we can assume that at most two nodes in $V_2$ are 0-nodes in ${\\mathcal {S}}(\\Omega {^{\\prime }})$ since, by construction, for the new stress matrix $\\Omega ^{\\prime }$ there exists $ij \\in V_2\\cup E[V_2]$ such that $w^{\\prime }_{ij}\\ne 0$ .",
"This guides our discussion below.",
"Figure REF shows the graph containing the support of the new stress matrix $\\Omega ^{\\prime }$ .",
"Figure: The graph containing the support of the new stress matrix Ω ' \\Omega ^{\\prime }"
],
[
"There are two 0-nodes in $V_2$ .",
"The cases when either 2,9, or 1,10, are 0-nodes are excluded (since then one would have a 1-node).",
"If nodes 1 and 9 are 0-nodes, then the equilibrium conditions at nodes 2 and 10 imply that $p^{\\prime }_4,p^{\\prime }_8\\in \\langle p^{\\prime }_2,p^{\\prime }_{10}\\rangle $ and by Lemma REF we have that $\\langle p^{\\prime }_2,p^{\\prime }_{10}\\rangle =\\langle p^{\\prime }_4,p^{\\prime }_8\\rangle \\subseteq \\langle {p}^{\\prime }_{V_1}\\rangle $ , contradicting (REF ).",
"The case when nodes 9,10 are 0-nodes is similar.",
"Finally assume that nodes 1,2 are 0-nodes (the case when 2,10 are 0-nodes is analogous).",
"As $w^{\\prime }_{8, 10}\\ne 0$ , the equilibrium condition at node 10 implies that $p^{\\prime }_8\\in \\langle p^{\\prime }_9,p^{\\prime }_{10}\\rangle $ .",
"If $w^{\\prime }_{49}=0$ then the equilibrium condition at node 9 implies that $p^{\\prime }_7\\in \\langle p^{\\prime }_9,p^{\\prime }_{10}\\rangle $ .",
"Hence $\\langle p^{\\prime }_7,p^{\\prime }_8\\rangle \\subseteq \\langle p^{\\prime }_9,p^{\\prime }_{10}\\rangle $ , thus equality holds, contradicting (REF ).",
"If $w^{\\prime }_{49}\\ne 0$ , then $p^{\\prime }_4\\in \\langle p^{\\prime }_7,p^{\\prime }_9,p^{\\prime }_{10}\\rangle $ and thus $\\langle p^{\\prime }_4, p^{\\prime }_7,p^{\\prime }_8\\rangle \\subseteq \\langle p^{\\prime }_7,p^{\\prime }_9,p^{\\prime }_{10}\\rangle $ .",
"Hence equality holds (by Lemma REF ), contradicting again (REF ).",
"Suppose first 9 is the only 0-node in $V_2$ .",
"The equilibrium conditions at nodes 1 and 10 imply that $p^{\\prime }_1\\in \\langle p^{\\prime }_3,p^{\\prime }_2\\rangle $ and $p^{\\prime }_{10}\\in \\langle p^{\\prime }_2, p^{\\prime }_8\\rangle $ .",
"Hence $\\langle p^{\\prime }_1,p^{\\prime }_{10}\\rangle \\subseteq \\langle p^{\\prime }_{V_1}, p^{\\prime }_2\\rangle $ and thus ${\\text{\\rm dim}}\\langle {p}^{\\prime }_{V\\setminus \\lbrace 9\\rbrace }\\rangle = 4$ .",
"Then conclude using Lemma REF .",
"Suppose now that node 1 is the only 0-node (the cases when 2 or 10 is 0-node are analogous).",
"The equilibrium conditions at nodes 2 and 9 imply that $p^{\\prime }_2 \\in \\langle p^{\\prime }_4,p^{\\prime }_{10}\\rangle $ and $p^{\\prime }_9\\in \\langle p^{\\prime }_4,p^{\\prime }_7,p^{\\prime }_{10}\\rangle $ .",
"Hence, $\\langle p^{\\prime }_2,p^{\\prime }_9 \\rangle \\subseteq \\langle p^{\\prime }_{V_1},p^{\\prime }_{10}\\rangle $ and we can conclude using Lemma REF .",
"We can assume that $w^{\\prime }_{ij}\\ne 0$ for some $(i,j)\\in V_1\\times V_2$ for otherwise we get the stressed circuit $C=(1,2,10,9)$ , thus with ${\\text{\\rm dim}}\\langle {p}^{\\prime }_C\\rangle =2$ , contradicting Corollary REF .",
"We show that ${\\text{\\rm dim}}\\langle {p}^{\\prime }_V\\rangle =4$ .",
"For this we discuss according to how many parameters are equal to zero among $w^{\\prime }_{13},w^{\\prime }_{24}, w^{\\prime }_{8,10}$ .",
"If none is zero, then the equilibrium conditions at nodes 1,2 and 10 imply that $p^{\\prime }_3,p^{\\prime }_4, p^{\\prime }_8\\in \\langle {p}^{\\prime }_{V_2}\\rangle $ and thus Lemma REF implies that $\\dim ( \\langle {p}^{\\prime }_{V_1}\\rangle \\cap \\langle {p}^{\\prime }_{V_2}\\rangle ) \\ge 3$ .",
"Therefore, $\\dim \\langle {p}^{\\prime }_{V_1}, {p}^{\\prime }_{V_2}\\rangle = \\dim \\langle {p}^{\\prime }_{V_1}\\rangle + \\dim \\langle {p}^{\\prime }_{V_2}\\rangle - \\dim (\\langle {p}^{\\prime }_{V_1}\\rangle \\cap \\langle {p}^{\\prime }_{V_2}\\rangle )\\le 3 + 4 -3 =4$ .",
"Assume now that (say) $w^{\\prime }_{13}=0$ , $w^{\\prime }_{24},w^{\\prime }_{8,10}\\ne 0$ .",
"Then ${\\text{\\rm dim}}\\langle {p}^{\\prime }_{V_2}\\rangle \\le 3$ (using the equilibrium condition at node 1).",
"As $w^{\\prime }_{24},w^{\\prime }_{8,10}\\ne 0$ , we know that $p^{\\prime }_4,p^{\\prime }_8\\in \\langle {p}^{\\prime }_{V_2}\\rangle $ .",
"Hence $ \\dim (\\langle {p}^{\\prime }_{V_1}\\rangle \\cap \\langle {p}^{\\prime }_{V_2}\\rangle ) \\ge 2$ and thus $\\dim \\langle {p}^{\\prime }_{V_1}, {p}^{\\prime }_{V_2}\\rangle \\le 3 + 3 - 2=4$ .",
"Assume now (say) that $w^{\\prime }_{13}=w^{\\prime }_{24}=0$ , $w^{\\prime }_{8,10}\\ne 0$ .",
"Then the equilbrium conditions at nodes 1 and 2 imply that ${\\text{\\rm dim}}\\langle {p}^{\\prime }_{V_2}\\rangle \\le 2$ .",
"Moreover, $p^{\\prime }_8\\in \\langle {p}^{\\prime }_{V_2}\\rangle $ .",
"Hence $ \\dim (\\langle {p}^{\\prime }_{V_1}\\rangle \\cap \\langle {p}^{\\prime }_{V_2}\\rangle ) \\ge 1$ and thus $\\dim \\langle {p}^{\\prime }_{V_1}, {p}^{\\prime }_{V_2}\\rangle \\le 3 + 2 -1 =4$ .",
"Finally assume now that $w^{\\prime }_{13}=w^{\\prime }_{24}=w^{\\prime }_{8,10}=0$ .",
"Then ${\\text{\\rm dim}}\\langle {p}^{\\prime }_{V_2}\\rangle = 2$ .",
"Moreover, at least one of $w^{\\prime }_{49},w^{\\prime }_{79}$ is nonzero.",
"Hence $ \\dim (\\langle {p}^{\\prime }_{V_1}\\rangle \\cap \\langle {p}^{\\prime }_{V_2}\\rangle ) \\ge 1$ and thus $\\dim \\langle {p}^{\\prime }_{V_1}, {p}_{V_2}\\rangle \\le 3 + 2 -1 =4$ ."
],
[
"Concluding remarks",
"One of the main contributions of this paper is the proof that $\\text{\\rm gd}(C_5 \\times C_2)\\le ~4$ , an inequality which underlies the characterization of graphs with Gram dimension at most four.",
"As already explained we obtain as corollaries the inequalities ${\\text{\\rm ed}}(C_5\\times C_2)\\le 3$ of [8] and $\\nu ^=(C_5\\times C_2)\\le 4$ of [21].",
"Although our proof of the inequality $\\text{\\rm gd}(C_5 \\times C_2)\\le 4$ goes roughly along the same lines as the proof of the inequality ${\\text{\\rm ed}}(C_5 \\times C_2) \\le 3$ given in [8], there are important differences and we believe that our proof is simpler.",
"This is due in particular to the fact that we introduce a number of new auxiliary lemmas (cf.",
"Sections REF and REF ) that enable us to deal more efficiently with the case checking which constitutes the most tedious part of the proof.",
"Furthermore, the use of semidefinite programming to construct a stress matrix permits to eliminate some case checking since, as was already noted in [31] (in the context of Euclidean realizations), the stress is nonzero along the stretched pair of vertices.",
"Additionally, our analysis complements and at several occasions even corrects the proof in [8].",
"As an example, the case when two vectors labeling two non-adjacent nodes are parallel in not discussed in [8]; this leads to some additional case checking which we address in Lemma REF .",
"As the class of graphs with $\\text{\\rm gd}(G)\\le k$ is closed under taking minors, it follows from the general theory of Robertson and Seymour [34] that there exists a polynomial time algorithm for testing $\\text{\\rm gd}(G)\\le k$ .",
"For $k\\le 4$ the forbidden minors for $\\text{\\rm gd}(G)\\le k$ are known, hence one can make this polynomial time algorithm explicit.",
"We refer to [30] for details on how to check $\\text{\\rm gd}(G)\\le 4$ or, equivalently, ${\\text{\\rm ed}}(G)\\le 3$ .",
"The next algorithmic question is how to construct a Gram representation in ${\\mathbb {R}}^4$ of a given partial matrix $a\\in {\\mathcal {S}}_+(G)$ when $G$ has Gram dimension 4.",
"As explained in [30], the first step consists of finding a graph $G^{\\prime }$ containing $G$ as a subgraph and such that $G^{\\prime }$ is a clique sum of copies of $V_8$ , $C_5\\times C_2$ and chordal graphs with tree-width at most 3.",
"Then, if a psd completion $A$ of $a$ is available, it suffices to deal with each of these components separately.",
"Such a psd completion can be computed approximately by solving a semidefinite program.",
"Chordal components are easy to deal with in view of the general results on psd completions in the chordal case.",
"For the components of the form $V_8$ or $C_5\\times C_2$ one has to go through the steps of the proof to get a new Gram representation in ${\\mathbb {R}}^4$ .",
"The basic ingredient of our proof is the existence of a primal-dual pair of optimal solutions to the programs (REF ) and (REF ).",
"Under appropriate genericity assumptions, the existence of such a pair of optimal solutions is guaranteed by standard results of semidefinite programming duality theory (cf.",
"Section REF ).",
"Also in the case when the primal program is not strictly feasible, we can still guarantee the existence of a psd stress matrix; our proof uses Farkas' lemma and is simpler than the proof in [30] of the analogous result in the context of Euclidean realizations.",
"However, in the case of the graph $C_5\\times C_2$ , we must make an additional genericity assumption on the vector $a\\in {\\mathcal {S}}_{++}(G)$ (namely, that the configuration restricted to any circuit is not coplanar).",
"This is problematic since the folding procedure apparently breaks down for non-generic configurations; note that this issue also arises in the case of Euclidean embeddings since an analogous genericity assumption is made in [8], although it is not discussed in the algorithmic approach of [30], [31].",
"Moreover, the above procedure relies on solving several semidefinite programs, which however cannot be solved exactly in general, but only to some given precision.",
"This thus excludes the possibility of turning the proof into an efficient algorithm for computing exact Gram representations in ${\\mathbb {R}}^4$ .",
"We conclude with the following question about the relation between the two parameters $\\text{\\rm gd}(G)$ and ${\\text{\\rm ed}}(G)$ , which has been left open: Prove or disprove the inequality: ${\\text{\\rm ed}}(\\nabla G)\\le {\\text{\\rm ed}}(G)+1.$ Acknowledgements.",
"We thank M. E.-Nagy for useful discussions and A. Schrijver for his suggestions for the proof of Theorem REF ."
]
] | 1204.0734 |
[
[
"Generalization of the Correspondence about DTr-selfinjective algebras"
],
[
"Abstract We give a correspondence between (n-1)-DTr-selfinjective algebras and algebras with dominant dimension and selinjective dimension being both n for any n bigger than 1 .",
"Furthermore, we show the relation between the module categories of the two kinds of algebras.",
"At last we show the sepcial condition n = 2."
],
[
"Introduction",
"The relation between the dominant dimension and the representation property of an algebra is a very hot topic since 60th in the last century.",
"The papers about this topic in that time are [11], [12], [14] and so on.",
"The main interest on this topic is this fact.",
"For any artin algebra, we always can construct algebras with dominant dimension more than or equal to 2 by their generator-cogenerators, and these algebras constructed are invariants of the original algebras to some extent [13].",
"So those particular algebras with dominant dimension more than or equal to 2 will reflect the properties of all algebras, for example, [2], [9].",
"On the other hand, the dominant dimension is also associated with the injective resolution of the regular module.",
"So it has some relation with Gorenstein(or Cohen-Maucauley) theory, for example, [4].",
"In [4], Auslander and Solberg found a correspondence between those algebras with dominant dimension and selinjective dimension being both 2 and $\\operatorname{DTr}$ -selfinjective algebras.",
"What is surprising is that the Gorenstein projective module categories of those algebras with dominant dimension and selinjective dimension being both 2 are always module categories of algebras whose $\\operatorname{DTr}$ -orbits has some periodic property.",
"This implies there is a close relation between Auslander-Reiten theory and Cohen-Maucauley thory.",
"In 2007, Iyama developed Auslander-Reiten theory.",
"He demonstrated higher dimensional Auslander-Reiten theory as a generalization of classic Auslander-Reiten theory in [10].",
"In that article, he developed a lot of useful tools such as higher dimensional Auslander-Reiten translation, maximal orthogonal subcategories and so on.",
"As an application, in [9] he showed the higher dimensional Auslander correspondence which is a generalization of theories in [2].",
"If we associate [4] with [9] and [10], we can find that the higher dimensional Auslander-Reiten theory should be useful to characterize the Goreinsten projective module category at least in some particular algebras.",
"In this article, we will show it.",
"We will show the generalization of the correspondence in [4].",
"And we will find that the periodic property of higher dimensional $\\operatorname{DTr}$ -orbits appears again in our background.",
"We always assume $R$ is a commutative artin ring, $\\operatorname{D}$ is the duality functor, all agebras are artin $R$ -algebras.",
"If there is no special instruction, we always assume all modules are left finitely generated modules."
],
[
"Main theory",
"Before describing our main theory, we need the following definitions and notations.",
"Definition 2.1 Let $\\Lambda $ be a basic artin algebra with $dom.\\Lambda \\ge 1$ .",
"Then there exists a uniquely basic $\\Lambda $ module $I$ such that $\\operatorname{add}I = \\lbrace M \\mid M \\text{ is a projective-injective $\\Lambda $ mod-} $ ule}.",
"We denote $I$ by $I^{\\Lambda }$ .",
"And it is called the minimal faithful $\\Lambda $ -module just as in $\\cite {12}$ .",
"We follow the notations in [9] and [10].",
"Suppose $ is an artin algebra.",
"Let $$ be the Auslander Reinten translation of $ -mod, $\\tau ^-$ be the quasi-inverse Auslander Reinten translation of $-mod, $$ be the syzygy functor, $ - 1$ be the cosyzygy functor.",
"Then just as in \\cite {8} and \\cite {9}, for any $ m 1$, let $ m = m - 1$ and $ - m = - -(m - 1)$.$ Also as in [9] and [10], suppose $ n \\ge 1 $ and $\\mathcal {D}$ is a full subcategory of $-mod.",
"Then $ nD = {M Exti(M, X) = 0, X D and 1 i n }, Dn = {M Exti(X, M) = 0, X D and 1 i n }$.",
"Especially, for a module $ M, n M = n (addM) , Mn = (addM)n$.",
"For modules M and N, we say $ M n N$ if $ Ext(M,N)= 0, 1 i n$.",
"We say M is n-self-orthogonal if $ M n M$.$ Definition 2.2 Let $ be a basic artin algebra and $ n 2$.",
"If there exists a basic $ module ${_Q which satisfies the following conditions: (1) it is a generator-cogenerator of -mod; (2) it is (n - 2)-self-orthogonal; (3) \\tau ^{n - 1}Q \\oplus \\tau ^{-(n - 1)}Q \\in \\operatorname{add}Q, then we call is a (n - 1)-\\operatorname{DTr}-selfinjective algebra, Q is a (n-2)-self-orthogonal (n - 1)-\\operatorname{DTr}-closed generator-cogenerator.",
"1-\\operatorname{DTr}-selfinjective algebra is also called \\operatorname{DTr}-selfinjective algebra as in \\cite {3}}$ Suppose $n \\ge 2$ , 1, 2 are two n-$\\operatorname{DTr}$ -selfinjective modules , ${_{1}}Q_1$ and ${_{2}}Q_2$ are respectively (n-2)-self-orthogonal $(n - 1)$ -$\\operatorname{DTr}$ -closed generator-cogenerator of 1 and 2.",
"Then we say that the pair $(1, {_{1}}Q_1 )$ is equivalent to $(2, {_{2}}Q_2 )$ if $\\operatorname{End}{_{1}}Q_1$ is Morita equivalent to $\\operatorname{End}{_{2}}Q_2$ (or equivalently, $\\operatorname{End}{_{1}}Q_1 \\cong \\operatorname{End}{_{2}}Q_2$ since both are basic modules).",
"We denote the equivalent class by $[1, {_{1}}Q_1]$ .",
"For a basic artin algebra $\\Lambda $ , we denote the equivalent class of $\\Lambda $ under algebraic isomorphism by $[\\Lambda ]$ (we don't use Morita equivalent class in order to ensure all algebras are basic).",
"Then we have the following notations: $\\mathfrak {U}_n = \\lbrace [\\Lambda ] \\mid dom.dim\\Lambda = inj.dim\\Lambda = n\\rbrace $ ; $\\mathfrak {B}_n =\\lbrace [ Q] \\mid \\text{$ is an (n - 1)-$\\operatorname{DTr}$-selfinjective algbra, $Q$ is an $(n - 2)$-self-orthogonal$$ $(n - 1)$-$\\operatorname{DTr}$-closed generator-cogenerator\\rbrace .",
"Now we can describing the main theorem.$\\begin{thm}Suppose n \\ge 2.",
"Then there is a one to one correspondence \\mathfrak {U}_n {F}{G} \\mathfrak {B}_n such that \\forall [\\Lambda ] \\in \\mathfrak {U}_n , F([\\Lambda ]) = [\\operatorname{End}^{op} I^\\Lambda , {_{(\\operatorname{End}I^\\Lambda )^{op}}} (\\operatorname{D}(I^\\Lambda ))]; \\forall [ Q] \\in \\mathfrak {B}_n, G([ Q]) = [\\operatorname{End}^{op} Q].\\end{thm}}$ Now suppose $[\\Lambda ] \\in \\mathfrak {U}_n, \\operatorname{End}^{op} I^\\Lambda , {_Q_\\Lambda = {_(\\operatorname{D}(I^\\Lambda ))_\\Lambda .",
"We denote \\lbrace {_X \\mid \\text{there is an exact sequence $0 \\rightarrow X \\rightarrow I_0 \\rightarrow I_1 \\rightarrow \\dots I_{m - 1}$ such that $I_0, I_1, \\dots I_{m - 1} \\in $ } \\linebreak \\operatorname{add}I^\\Lambda \\rbrace by \\mathcal {C}^m_\\Lambda for any m \\ge 1 .",
"We have the following lemma.",
"}}\\begin{lem}The exact functor \\operatorname{D}\\operatorname{Hom}_\\Lambda (-, I^\\Lambda ) = Q\\otimes _\\Lambda -: \\mathcal {C}^2_\\Lambda \\rightarrow -mod is an equivalence between categories.\\end{lem}}{\\bf {Proof.}}",
"For any $$-module M, $ DHom(M, I) DHom(M, DD(I)) DD (D(I) M) D(I) M$.",
"So $ DHom(-, I) $ and $ Q-$ are naturally isomorphic.$ The equivalence between categories is proved for the similar reason as Proposition 2.5 in chapter 2 of [3] .",
"Using the above lemma we prove the following two corollaries which is also proved in [13] in a different way.",
"Corollary 2.3 $Q$ is a generator-cogenerator of $-mod.$ Proof.",
"${_Q = {_(\\operatorname{D}(I^\\Lambda )) = \\operatorname{D}\\operatorname{Hom}_\\Lambda (\\Lambda , I^\\Lambda ) = \\operatorname{Hom}_\\Lambda (I^\\Lambda , \\operatorname{D}\\Lambda ).",
"Since I^\\Lambda \\in \\operatorname{add}\\Lambda \\bigcap \\linebreak \\operatorname{add}\\operatorname{D}\\Lambda , \\operatorname{D}( \\oplus {_ \\operatorname{D}\\operatorname{Hom}_\\Lambda (I^\\Lambda , I^\\Lambda ) \\oplus \\operatorname{Hom}_\\Lambda (I^\\Lambda , I^\\Lambda ) \\in \\operatorname{add}{_Q.",
"So Q is a generator-cogenerator of -mod.",
"}}}\\begin{cor}{_Q_\\Lambda is faithful balanced.",
"}{\\bf {Proof.}}",
"The canonical map \\operatorname{End}(\\operatorname{D}(I^\\Lambda ))_\\Lambda is an isomorphism since the canonical map \\operatorname{End}^{op}{_\\Lambda }I^\\Lambda is an isomorphism.\\end{cor}On the other hand since \\operatorname{D}\\operatorname{Hom}_\\Lambda (-, I^\\Lambda ) = Q\\otimes _\\Lambda -: \\mathcal {C}^2_\\Lambda \\rightarrow -mod is an equivalence between categories, \\operatorname{End}^{op}{_(\\operatorname{D}(I^\\Lambda )) = \\operatorname{End}^{op}{_(\\operatorname{D}\\operatorname{Hom}_\\Lambda (\\Lambda , I^\\Lambda )) = \\operatorname{End}^{op}{_\\Lambda }\\Lambda = \\Lambda .",
"Since {_\\Lambda }I^\\Lambda is a faithful \\Lambda -module, we know the canonical map \\Lambda \\rightarrow \\operatorname{End}^{op}{_(\\operatorname{D}(I^\\Lambda )) is a monomorphism.",
"So it is an isomorphism.",
"}}\\begin{lem}There is an exact sequence: 0 \\rightarrow \\Lambda \\rightarrow I_0 \\rightarrow I_1 \\rightarrow \\dots \\rightarrow I_{n - 1} \\rightarrow \\operatorname{D}\\Lambda \\rightarrow 0 such that I_0, I_1, \\dots , I_{n - 1} \\in \\operatorname{add}I^\\Lambda .",
"Especially, \\Lambda is an n-Gorenstein algebra.\\end{lem}{\\bf {Proof.}}",
"Since dom.dim.\\Lambda = inj.dim.",
"{_\\Lambda }{\\Lambda }, for any indecomposable projective module P,inj.dim.P = 0 or n.If inj.dim.P = 0, P is a projective-injective module.",
"If not, P hasa minimal injective resolution: {\\begin{matrix}0 \\rightarrow P \\rightarrow I_0 \\rightarrow I_1 \\rightarrow \\dots \\rightarrow I_{n - 1} \\rightarrow \\Omega ^{-n}(P) \\rightarrow 0\\end{matrix}}\\\\such that I_0, I_1, \\dots , I_{n - 1} are projective-injective modules.Since this is also a projective resolution of \\Omega ^{-n}(P),\\Omega ^{-n}(P) is an indecomposable module.",
"If not, the injective resolution of P is not minimal.",
"So \\Omega ^{-n}(P) is an indecomposable injective nonprojective module.",
"}On the other hand, the number of non-injective projective modules is equalto the number of non-projective injective modules.",
"So \\Omega ^{-n} constructs a one to one correspondence between non-injective projective modules and non-projective injective modules.",
"So \\Omega ^{-n}(A) is a basic module which is the direct sum of all mutually nonisomorphic nonprojective injective modules.",
"So the exact sequence in the lemma exists.",
"By duality \\operatorname{D}, we know \\Lambda is n-Gorenstein algebra.",
"}$ Proposition 2.4 ${_Q \\ \\bot _{n - 2} \\ {_Q}{\\bf {Proof.",
"}}If n = 2, it is clear.",
"Now suppose n > 2.",
"There exists an injective resolution of {_\\Lambda }\\Lambda :0 \\rightarrow \\Lambda \\rightarrow I_0 \\rightarrow I_1 \\rightarrow \\dots \\rightarrow I_{n - 2} \\rightarrow I_{n - 1}such that I_i \\in \\operatorname{add}I^\\Lambda for all i.",
"}Applying $ Q-$ to the exact sequence, we obtain the following exact sequence since it is an exact functor:$$0 \\rightarrow {_Q \\rightarrow Q \\otimes I_1 \\rightarrow Q \\otimes I_2 \\rightarrow \\dots \\rightarrow Q \\otimes I_{n - 2}\\rightarrow Q \\otimes I_{n - 1}}Since $ Q Ii = DHom(Ii, I) addD($, the above sequence is an injective resolution of $ Q$.$ By Lemma 2.4 we have the following commutative diagram: ${0 @{->}[r] & \\operatorname{Hom}_\\Lambda ({_\\Lambda }\\Lambda , {_\\Lambda }\\Lambda ) @{->}[r] @{->}[d] &\\operatorname{Hom}_\\Lambda ({_\\Lambda }\\Lambda , I_0) @{->}[r] @{->}[d] &\\dots @{->}[r] &\\operatorname{Hom}_\\Lambda ({_\\Lambda }\\Lambda , I_{n - 1}) @{->}[d]\\\\0 @{->}[r] & \\operatorname{Hom}_Q, Q) @{->}[r] & \\operatorname{Hom}_Q, Q\\otimes I_0) @{->}[r] &\\dots @{->}[r] &\\operatorname{Hom}_Q, Q \\otimes I_{n - 1})}$ Since the above is an exact sequence, so is the bellow one.",
"Therefore, ${_Q \\ \\bot _{n - 2} \\ {_Q.\\\\}Now we suppose \\mathcal {N} = \\operatorname{D}\\operatorname{Hom}_-, is the Nakayama functor and \\mathcal {N}^- = \\operatorname{Hom}_\\operatorname{D}(-), is the quasi-inverse Nakayama functor.",
"We denote the sable module category of -mod by {\\text{mod}}.",
"Given M \\in -mod, the corresponding module in {\\text{mod}} by {M}.",
"Dually, we have the notations \\overline{\\text{mod}} and \\overline{M}.\\\\}\\begin{prop}{_Q = \\operatorname{D}( \\oplus \\tau ^{n - 1}Q = \\tau ^{-(n - 1)}Q}{\\bf {Proof.}}Step1.",
"{_Q = \\operatorname{D}( \\oplus \\tau ^{n - 1}Q .",
"}By Lemma 2.7, there is an exact sequence:{\\begin{matrix} \\unknown.",
"{0 \\rightarrow {_\\Lambda }\\Lambda \\xrightarrow{} I_0 \\xrightarrow{} I_1\\xrightarrow{} I_2\\xrightarrow{} \\dots \\xrightarrow{} I_{n - 1} \\xrightarrow{} \\operatorname{D}(\\Lambda _\\Lambda ) \\rightarrow 0 \\hspace{28.45274pt} \\left( * \\right)} \\end{matrix}}such that I_i \\in \\operatorname{add}I^\\Lambda for all i.\\end{prop}Applying $ Hom(I, -)$ to it, we obtain the exact sequence since it is an exact functor:$${\\begin{matrix} 0 \\rightarrow \\operatorname{Hom}_\\Lambda (I^\\Lambda , \\Lambda ) \\xrightarrow{} \\operatorname{Hom}_\\Lambda (I^\\Lambda , I_0) \\xrightarrow{} \\operatorname{Hom}_\\Lambda (I^\\Lambda , I_1) \\xrightarrow{} \\\\ \\unknown.",
"{ \\dots \\xrightarrow{} \\operatorname{Hom}_\\Lambda (I^\\Lambda , I_{n - 1}) \\xrightarrow{} \\operatorname{Hom}_\\Lambda (I^\\Lambda , \\operatorname{D}(\\Lambda _\\Lambda )) \\rightarrow 0 \\hspace{65.44133pt}} \\end{matrix}}$$$ Since $\\operatorname{Hom}_\\Lambda (I^\\Lambda , I_i) $ is a projective $ module for all i, $ n - 2${_Q}$ = KerHom(I, d2) = Hom(I, Kerd2) $.$ We have the projective resolution of $\\operatorname{Hom}_\\Lambda (I^\\Lambda , \\operatorname{Ker}d_2)$ : $ \\operatorname{Hom}_\\Lambda (I^\\Lambda , I_0) \\xrightarrow{} \\operatorname{Hom}_\\Lambda (I^\\Lambda , I_1) \\rightarrow \\operatorname{Hom}_\\Lambda (I^\\Lambda , \\operatorname{Ker}d_2) \\rightarrow 0$ Since $\\operatorname{Hom}_\\Lambda (I^\\Lambda , -): \\operatorname{add}I^\\Lambda \\rightarrow \\operatorname{add}{_ is an equivalence between categories.",
"We have the following commutative diagram:}$$@C=4 em{&&\\mathcal {N}(\\operatorname{Hom}_\\Lambda (I^\\Lambda , I_0))[r]^{\\mathcal {N}\\operatorname{Hom}_\\Lambda (I^\\Lambda , d_0)} [d]&\\mathcal {N}(\\operatorname{Hom}_\\Lambda (I^\\Lambda , I_1))[d]\\\\0 [r] &\\operatorname{D}\\operatorname{Hom}_\\Lambda (\\Lambda , I^\\Lambda ) [r]^{\\operatorname{D}\\operatorname{Hom}_\\Lambda (d_{- 1}, I^\\Lambda )} &\\operatorname{D}\\operatorname{Hom}_\\Lambda (I_0, I^\\Lambda ) [r]^{\\operatorname{D}\\operatorname{Hom}_\\Lambda (d_0, I^\\Lambda )} &\\operatorname{D}\\operatorname{Hom}_\\Lambda (I_1, I^\\Lambda )}$$$ The vertical morphisms are morphisms.",
"Since the bellow sequence is exact, we have $\\overline{\\tau ^{n - 1} Q} = \\overline{\\tau \\operatorname{Hom}_\\Lambda (I^\\Lambda , \\operatorname{Ker}d_2)} = \\overline{\\operatorname{D}\\operatorname{Hom}_\\Lambda (\\Lambda , I^\\Lambda )} = \\overline{\\text{${_Q}$$.$Thus $\\tau ^{n - 1}Q \\in \\operatorname{add}Q$ since ${_Q is a cogenerator.",
"For the same reason in Lemma 2.7, \\tau ^{n - 1}Q is a basic module.",
"So we have {_Q = \\operatorname{D}( \\oplus \\tau ^{n - 1}Q since \\tau ^{n - 1}\\operatorname{D}( = 0 and {_Q is a cogenerator.",
"}Step 2.",
"{_Q = \\tau ^{-(n - 1)}Q.",
"}Applying the exact functor Q\\otimes _\\Lambda - to (*), we obtain the folllowing exact sequence:0 \\rightarrow Q\\otimes \\Lambda \\xrightarrow{} Q\\otimes I_0 \\xrightarrow{} \\dots \\xrightarrow{} Q\\otimes I_{n -1} \\xrightarrow{} Q\\otimes \\operatorname{D}\\Lambda \\rightarrow 0}Since Q\\otimes I_i is an injective -module for all i, \\overline{\\Omega ^{-(n - 2)}\\text{${_Q}$ = \\overline{\\operatorname{Ker}(Q\\otimes d_{n - 2})} = \\overline{ Q\\otimes \\operatorname{Ker}d_{n - 2}}$.$We have injective resolution of $Q\\otimes \\operatorname{Ker}d_{n - 2}$:$$ 0 \\rightarrow Q\\otimes \\operatorname{Ker}d_{n - 2} \\rightarrow Q\\otimes I_{n -2} \\xrightarrow{} Q\\otimes I_{n -1} $$}}By Lemma 2.4, we have the following commutative diagram:}$${\\begin{matrix}\\mathcal {N}^-(Q\\otimes I_{n -2}) &\\xrightarrow{}& \\mathcal {N}^-(Q\\otimes I_{n -1})\\\\\\uparrow && \\uparrow &&\\\\\\operatorname{Hom}_\\Lambda (I^\\Lambda , I_{n -2}) &\\xrightarrow{}& \\operatorname{Hom}_\\Lambda (I^\\Lambda , I_{n - 1}) &\\xrightarrow{}& \\operatorname{Hom}_\\Lambda (I^\\Lambda , \\operatorname{D}\\Lambda ) &\\xrightarrow{}& 0\\end{matrix}}$$$}The vertical morphisms are morphisms.",
"Since the bellow sequence is exact, we have {\\tau ^{-(n - 1)}\\text{${_Q}$ = {\\tau ^- (Q\\otimes \\operatorname{Ker}d_{n - 2})} = {\\operatorname{Hom}_\\Lambda (I^\\Lambda , D\\Lambda )} = {\\text{${_Q}$$$Thus $\\tau ^{-(n - 1)}Q \\in \\operatorname{add}Q$ since ${_Q is a generator.",
"For the same reason in Lemma 2.7, \\tau ^{-(n - 1)}Q is a basic module.",
"So we have {_Q = \\tau ^{-(n - 1)}Q since \\tau ^{-(n - 1)} 0 and {_Q is a generator.\\\\}}}$}}The following lemma is from $\\cite {9}$.",
"}\\begin{lem}Suppose n \\ge 2, \\Sigma is an artin algebra.",
"Let X, Y \\in \\Sigma -mod.",
"\\\\\\left( 1 \\right) If X \\in {^{\\bot _{n - 2}}}\\Sigma .",
"Then we have the following isomorphism for any 1 \\le i \\le n - 2: \\operatorname{Ext}^i(X, Y) \\cong \\operatorname{Ext}^{n - 1 - i}(Y, \\tau ^{n - 1}X).\\\\\\left( 2 \\right) If Y \\in D(\\Sigma _\\Sigma )^{\\bot _{n - 2}}.",
"Then we have the following isomorphism for any 1 \\le i \\le n - 2: \\operatorname{Ext}^i(X, Y) \\cong \\operatorname{Ext}^{n - 1 - i}(\\tau ^{-(n - 1)}Y, X).\\end{lem}}}The following lemma may be well known.",
"But we give a proof there.$ Lemma 2.5 Suppose $\\Sigma $ is an artin algebra.",
"Let ${_\\Sigma }M$ be a generator of $\\Sigma $ -mod.",
"Then the functor $\\operatorname{Hom}_\\Sigma (M, -): \\Sigma $ -mod $\\rightarrow \\operatorname{End}^{op} M$ -mod is fully faithful.",
"Dually, if ${_\\Sigma }M$ is a cogenerator of $\\Sigma $ -mod, then the functor $\\operatorname{Hom}_\\Sigma (-, M): \\Sigma $ -mod $\\rightarrow \\operatorname{End}M$ -mod is fully faithful.",
"Proof.",
"We just prove the first assertion.",
"Since $M$ is a generator it is faithful.",
"Suppose $X, Y \\in \\Gamma $ -Mod, $f: \\operatorname{Hom}(M, X) \\rightarrow \\operatorname{Hom}(M, Y)$ is a $\\operatorname{End}^{op} M$ -morphism.",
"Suppose $\\pi : T \\rightarrow X \\rightarrow 0$ is a right $\\operatorname{add}M$ approximation.",
"Then there exists an exact sequence: $ {\\begin{matrix}0 &\\xrightarrow{}& \\operatorname{Hom}(M, \\operatorname{ker}\\pi ) &\\xrightarrow{}&\\operatorname{Hom}(M, T) &\\xrightarrow{}&\\operatorname{Hom}(M, X) &\\xrightarrow{}& 0\\end{matrix}}$ Since $\\operatorname{Hom}(T,Y) \\rightarrow \\operatorname{Hom}(\\operatorname{Hom}(M, T), \\operatorname{Hom}(M,Y))$ is an isomorphism, there exists $g: T \\rightarrow Y$ such that $f \\cdot \\operatorname{Hom}(M,\\pi ) = \\operatorname{Hom}(M,g)$ .",
"So $\\operatorname{Hom}(M, g \\cdot i) = f \\cdot \\operatorname{Hom}(M,\\pi ) \\cdot \\operatorname{Hom}(M,i) = 0\\Rightarrow g \\cdot i = 0$ $\\Rightarrow \\exists f^{\\prime }: X \\rightarrow Y$ such that $g = f^{\\prime } \\cdot \\pi $ $\\Rightarrow \\operatorname{Hom}(M, f^{\\prime }) = f$ .",
"Now we can give the proof of the main theorem.",
"Proof of the Theorem 2.3.",
"Give $[\\Lambda ] \\in \\mathfrak {U}_n$ , by Corollary 2.5, Proposition 2.8 and Proposition 2.9, $F([\\Lambda ]) \\in \\mathfrak {B}_n$ .",
"By Corollary 2.6, $GF([\\Lambda ]) = [\\Lambda ]$ .",
"Now suppose $[ {_M] \\in \\mathfrak {B}_n.",
"Let \\Sigma = \\operatorname{End}^{op}{M}.",
"We prove the another part of the theorem by 3 steps.}Step1.",
"M has an injective resolution:$$0 \\rightarrow {_M \\rightarrow J_0 \\rightarrow J_1 \\rightarrow \\dots \\rightarrow J_{n - 2}\\rightarrow J_{n - 1}}Since $ M n -2 M$, applying $ HomM, -)$ to it, we have the following exact sequence:$$0 \\rightarrow \\operatorname{Hom}_{_M, {_M) \\rightarrow \\operatorname{Hom}_M, J_0) \\rightarrow \\operatorname{Hom}_M, J_1) \\rightarrow \\dots \\rightarrow \\operatorname{Hom}_M, J_{n - 1})}Since M is a generator-cogenerator, by \\cite {12}, \\operatorname{Hom}(M, \\operatorname{D}() is a projective-injective \\Sigma -module.",
"So the above exact sequence is the injective resolution of {_\\Sigma }\\Sigma .",
"And \\operatorname{Hom}_M, J_i) is a projective-injective \\Sigma -module.",
"So dom.\\Sigma \\ge n.}$ Step2.",
"Now suppose $Z \\in -mod, and the following is an exact sequence: $ 0 Y Mf Z 0 $ such that $ f$ is a right $ addM$-approximation of $ X$.",
"Then $ Ext1(M, Y) = 0$.",
"So by Lemma Proposition 2.9 and 2.10 .",
"$ Extn - 2(Y, M) = Extn - 2(Y, n - 2M) = Ext1(M, Y) = 0$.$ Suppose $X \\in -mod and the following is an exact sequence:\\begin{center}\\unknown.",
"{0 \\rightarrow X_{n - 1 } \\xrightarrow{} M_{n - 1} \\xrightarrow{} \\dots \\xrightarrow{} M_2 \\xrightarrow{} M_1 \\xrightarrow{} X \\rightarrow 0 \\hspace{28.45274pt}(**)}\\end{center}such that $ fi: Mi Imfi$ is a right $ addM$-approximation for all $ i$.$ If $n > 2$ , since $M \\bot _{n -2} M$ , $\\operatorname{Ext}^{1}(\\operatorname{Ker}f_{n - 2}, M) = \\operatorname{Ext}^{n - 2}(\\operatorname{Ker}f_1, M) = \\operatorname{Ext}^{1}(M, \\operatorname{Ker}f_1) = 0$ .",
"Thus, $\\operatorname{Hom}(h, M)$ is an epic morphism.",
"If $n = 2$ , since $f_1$ is a right $\\operatorname{add}M$ -approximation, $h$ is a left $\\operatorname{add}M$ -approximation since $M$ is $\\operatorname{DTr}$ -closed and a cogenerator.",
"Thus, $\\operatorname{Hom}(h, M)$ is an epic morphism.",
"Applying $\\operatorname{Hom}_M, -)$ to $(**)$ , we have the following exact diagram: $0 \\rightarrow \\operatorname{Hom}(M, X_{n - 1}) \\xrightarrow{} \\dots \\xrightarrow{} \\operatorname{Hom}(M, X) \\rightarrow 0$ By Lemma 2.11, we know that $\\operatorname{Hom}_\\Sigma (\\operatorname{Hom}_M, \\text{$h$}), \\text{${_\\Sigma }\\Sigma $})$ is isomorphic to $\\operatorname{Hom}(h, \\linebreak M)$ .",
"So $\\operatorname{Hom}_\\Sigma (\\operatorname{Hom}_M, \\text{$h$}), \\text{${_\\Sigma }\\Sigma $})$ is an epic morphism.",
"Since $\\operatorname{Hom}(M, M_i)$ is a projective $\\Sigma $ -module for all $i$ , $\\operatorname{Ext}^{n - 2}_\\Sigma (\\operatorname{Hom}(M, X), \\Sigma ) = 0$ .",
"Suppose $V \\in \\Sigma $ module.",
"Then there is a morphism $f: M_1 \\rightarrow M_2$ such that there exists a projective resolution of $V$ : $ \\operatorname{Hom}(M, M_1) \\xrightarrow{}\\operatorname{Hom}(M, M_2) \\rightarrow V \\rightarrow 0$ Therefore, $\\operatorname{Ext}^{n + 1}_\\Sigma (V, \\Sigma )= \\operatorname{Ext}^{n - 2}_\\Sigma (\\operatorname{Ker}(\\operatorname{Hom}(M, f)), \\Sigma ) = \\operatorname{Ext}^{n - 2}_\\Sigma (\\operatorname{Hom}(M, \\operatorname{Ker}f), \\linebreak \\Sigma ) = 0$ .",
"So $inj.dim.",
"{_\\Sigma }\\Sigma \\le n$ Step3.",
"Also we know $\\operatorname{Hom}(M, \\operatorname{D}()$ is the minimal faithful $\\Sigma $ -module by [2].",
"So $\\Sigma $ is not a selfinjective algebra.",
"So $inj.dim.",
"{_\\Sigma }\\Sigma = n = dom.dim.\\Sigma $ .",
"Therefore, $G([ Q]) \\in \\mathfrak {U}_n$ .",
"By Lemma 2.11 $ \\operatorname{End}^{op}\\operatorname{Hom}(M, \\operatorname{D}() = \\operatorname{End}^{op} \\operatorname{D}(= \\operatorname{End}( = .",
"And $ (DHom(M, D()) = (DD( M)) = M.$ So $ GF([ M]) = [ M].$\\\\$ For an artin algebra $\\Sigma $ , we denote its finitely generated Gorenstein projective module category by $Gproj(\\Sigma )$ .",
"Lemma 2.6 Suppose $n \\ge 2, [\\Lambda ] \\in \\mathfrak {U}_n$ .",
"Then $Gproj(\\Lambda ) = \\mathcal {C}_\\Lambda ^{n}$ Proof.",
"Suppose $X \\in \\mathcal {C}_\\Lambda ^{n}$ .",
"Then $X$ has an injective resolution: $0 \\rightarrow X \\rightarrow I_0 \\rightarrow I_1 \\rightarrow \\dots \\rightarrow I_{n - 1}$ such that $I_i$ is a projective module for all $i$ .",
"So $\\operatorname{Ext}^i(X, \\Lambda )= \\operatorname{Ext}^{n + i}(\\Omega ^{-n }X, \\Lambda ) = 0, \\forall i > 0$ .",
"So $X \\in Gproj(\\Lambda )$ .",
"Suppose $\\operatorname{Ext}^{n}(Z, \\Lambda ) = 0$ .",
"Applying $\\operatorname{Hom}(Z, -)$ to $(*)$ in Proposition 2.9, we have an epic morphism $\\operatorname{Hom}(Z, I_{n - 1}) \\rightarrow \\operatorname{Hom}(Z, \\operatorname{D}\\Lambda )$ .",
"So $Z$ is cogenerated by $\\operatorname{add}I_{n - 1}$ .",
"Thus it is cogenerated by $\\operatorname{add}I^{\\Lambda }$ .",
"Suppose $Y \\in Gproj(\\Lambda )$ .",
"Then $\\operatorname{Ext}^i(Y, \\Lambda ) = 0, \\forall i \\ge 1$ .",
"Using the above assertion by induction on $i$ .",
"We know $\\operatorname{Ext}^{n}(\\Omega ^{-i}Y, \\Lambda ) = 0, $ and $\\Omega ^{-i}X$ is cogenerated by $\\operatorname{add}I^{\\Lambda }$ for all $ i \\le n - 1$ .",
"Then we have an injective resolution of $Y : 0 \\rightarrow Y \\rightarrow I^\\prime _ 0 \\rightarrow I^\\prime _1 \\rightarrow \\dots \\rightarrow I^\\prime _{n - 1}$ such that $I_i \\in \\operatorname{add}I^\\Lambda $ for all $i$ .",
"So $Y \\in \\mathcal {C}_\\Lambda ^{n}$ .",
"Theorem 2.7 Suppose $n \\ge 2, [ Q] \\in \\mathfrak {B}_n$ .",
"Let $\\Sigma = \\operatorname{End}^{op} Q$ .",
"Then ${_Q^{\\bot _{n - 2}} = {^{\\bot _{n - 2}}}({_Q), and the functor \\operatorname{Hom}_Q, -) gives an equivalence between Q^{\\bot _{n - 2}} and Gproj(\\Sigma ).",
"}{\\bf {Proof.}}",
"Suppose X \\in -mod.",
"By Lemma 2.10 \\operatorname{Ext}^i(Q, X) = \\operatorname{Ext}^{n - 1 - i}(X, \\tau ^{n - 1}Q), \\forall 1 \\le i \\le n - 2.",
"However, since we have the correspondence as in Theorem 2.3, Q = \\operatorname{D}( \\oplus \\tau ^{n - 1}Q.",
"So \\operatorname{Ext}^i(Q, X) = \\operatorname{Ext}^{n - 1 - i}(X, Q).",
"So the first assertion is proved.",
"}Now suppose $ X Qn - 2$ and the following is a injective resolution of $ X: 0 X J0 J1 ...Jn - 1$.",
"Applying $ Hom(Q, -)$ to it we get an exact sequence: $ 0 Hom(Q, X) Hom(Q, J0) Hom(Q, J1) ...Hom(Q, Jn - 1)$ since $ X Qn - 2$.",
"$ Hom(Q, Ji)$ is a projective-injective module for all $ i$ since $ Hom(Q, D()$ is the minimal faithful module of $$.",
"So $ Hom(Q, X) Cn$.",
"Thus$ Hom(Q, X) Gproj()$ by the above lemma.$ Conversely, suppose that $M \\in Gproj(\\Sigma )$ .",
"Then there is an injective resolution of $M: 0 \\rightarrow M \\rightarrow I_0 \\xrightarrow{} I_1 \\xrightarrow{} \\dots \\xrightarrow{} I_{n - 1}$ such that $I_i \\in I^{\\Sigma }$ .",
"By Lemma 2.11, we know that there exists $J_0, J_1, \\dots , J_{n - 1} \\in \\operatorname{add}\\operatorname{D}($ and morphisms $d_i: J_i \\rightarrow J_{i + 1}$ such that $\\operatorname{Hom}(Q, d_i)$ is isomorphic to $f_i$ for all $i$ .",
"So there is a commutative diagram.",
"${\\begin{matrix}&& &&\\operatorname{Hom}(Q, J_0) &\\xrightarrow{}& \\dots &\\xrightarrow{}& \\operatorname{Hom}(Q, J_{n - 1}) \\\\&&&& \\downarrow && && \\downarrow &&\\\\0 &\\xrightarrow{}& M &\\xrightarrow{}& I_0 &\\xrightarrow{}& \\dots &\\xrightarrow{}& I_{n - 1}\\end{matrix}}$ The vertical morphisms are morphisms.",
"Since the bellow sequence is exact, so is the above and $\\operatorname{Ker}\\operatorname{Hom}(Q, d_0) = M$ .",
"Therefore, since ${_Q is a generator, the sequence I_0 \\xrightarrow{} I_1\\xrightarrow{} \\dots \\xrightarrow{} I_{n - 1} is exact and \\operatorname{Ker}d_0 \\in Q^{\\bot _{n - 2}}.",
"On the other hand, \\operatorname{Ker}\\operatorname{Hom}(Q, d_0) = \\operatorname{Hom}(Q, \\operatorname{Ker}d_0).",
"So M = \\operatorname{Hom}(Q, \\operatorname{Ker}d_0).",
"Thus \\operatorname{Hom}_Q, -): Q^{\\bot _{n - 2}} \\rightarrow Gproj(\\Sigma ) is dense.",
"It is also faithful by Lemma 2.11.",
"So \\operatorname{Hom}_Q, -)gives an equivalence between Q^{\\bot _{n - 2}} and Gproj(\\Sigma ).\\\\}$ Qn - 2$ has a very interesting property\\begin{cor}Suppose n \\ge 2, [ Q] \\in \\mathfrak {B}_n.",
"Then {_Q^{\\bot _{n - 2}} is closed under \\tau ^{n - 1} and \\tau ^{-(n - 1)}}\\end{cor}{\\bf {Proof.}}",
"Since $ Qn - 2 = n - 2(Q)$, by Lemma 2.10, it^{\\prime }s obvious.\\\\$ Now we give a homological characterization for $(n - 1)$ -$\\operatorname{DTr}$ -selfinjective algebras.",
"First, we give a lemma.",
"Lemma 2.8 Suppose $n \\ge 2$ .",
"If $ is an $ (n - 1)$-$ DTr$-Selfinjective algebra, so is $ op$$ Proof.",
"Suppose $Q$ is an $(n -2)$ -self-orthogonal $(n - 1)$ -$\\operatorname{DTr}$ -closed generator-cogenerator.",
"Then $\\operatorname{D}Q$ is an (n - 2)-self-orthogonal $ {op}$ -module.",
"It is also a generator-cogenerator of ${op}$ -mod.",
"Given $X \\in -mod, then $ n - 1(DX) = n - 2(DX) =D -(n - 2)X = D--(n - 2)X = D(-(n - 1)X)$.",
"For the same reason, $ -(n - 1)(DX) = D(n - 1X)$.$ Thus $\\operatorname{D}Q$ is a (n - 2)-self-orthogonal $(n - 1)$ -$\\operatorname{DTr}$ -closed generator-cogenerator of $ {op}$ -mod.",
"So ${op}$ is an $(n - 1)$ -$\\operatorname{DTr}$ -Selfinjective algebra.",
"Theorem 2.9 Suppose $n \\ge 2$ , and $ is a basic artin algebra such that $ Dn - 2 .",
"Then the following is equivalent.",
"$\\left( 1 \\right)$ $ is an $ (n - 1)$-$ DTr$-selfinjective algebra.",
"\\\\$ ( 2 )$ $ Inf {inj.dim.",
"= EndopM, M is a basic generator-cogenerator of $-mod$$ \\linebreak such that $ M n - 2 M} = n$.\\\\$ ( 3 )$ $ Inf {inj.dim.",
"= EndopM, M is a basic generator-cogenerator of $-mod$$ \\linebreak such that $ M n - 2 M} = n$.\\\\$ ( 4 )$ $ Inf {max(inj.dim.",
", inj.dim. )",
"= EndopM, M is a basic generator$ \\linebreak -cogenerator$of $-mod such that $ M n - 2 M} = n$.$ Proof.",
"$(1) \\Rightarrow (2), (3), (4)$ is obvious since we can choose $M$ is a $(n - 2)$ -self-orthogonal $(n - 1)$ -$\\operatorname{DTr}$ -closed generator-cogenerator.",
"$(2) \\Rightarrow (1)$ .",
"Suppose $M \\text{ is a basic generator-cogenerator of $-mod such that $ M \\linebreak \\bot _{n - 2} \\ M$ and $inj.dim.",
"{_\\Sigma }\\Sigma = n$ for $\\Sigma =\\operatorname{End}^{op}M$.",
"For the same reason in the proof of Theorem 2.3, $dom.dim.\\Sigma = n$.",
"So $\\Sigma \\in \\mathfrak {U}_n$.",
"Since $\\operatorname{Hom}_M, -): -mod $\\rightarrow \\Sigma $-mod is fully faithful By Lemma 2.11.",
"So $\\operatorname{End}^{op} {_\\Sigma }\\operatorname{Hom}_M, \\operatorname{D}() = \\operatorname{End}^{op} \\operatorname{D}( =\\operatorname{End} .",
"On the other hand, $\\operatorname{Hom}_M, \\operatorname{D}()$ is a minimal faithful $\\Sigma $-module (by \\cite {2},\\cite {12}), So we know $ is an $(n-1)$-$\\operatorname{DTr}$-selfinjective algebra by Theorem 2.3.",
"}$ (3) (1)$.",
"If $ (3)$ is true, then there exists a basic generator-cogenerator of $ op$-module N such that $ N n - 2 N$ and $ inj.dim.",
"= n$ for $ = EndopDN$.",
"However, $ EndopDN = EndN$.",
"So $ op$ satisfies $ (2)$.",
"By $ (2) (1)$, $ op$ is an$ (n - 1)$-$ DTr$-selfinjective algebra.",
"By Lemma 2.14, so is $ .",
"$(4) \\Rightarrow (2)$ .",
"Obvious."
],
[
"The case n = 2",
"When n = 2, 1-$\\operatorname{DTr}$ -selfinjective algebras are called $\\operatorname{DTr}$ -selfinjective algebras just as in[4].",
"The correspondence in Theorem 2.3 about it ($n = 2$ ) is the analogy of representation-finite algebras which is obtained in [2].",
"So we think $\\operatorname{DTr}$ -selfinjective algebras have some similar properties as representation-finite algebras.",
"For the same reason, the algebras with diminant dimension and selfinjective dimension being both 2 should have some similar properties of Auslander algebras.",
"The homological characterization of $\\operatorname{DTr}$ -selfinjective algebras which is demonstrated in Theorem 2.16 when $n= 2$ is the analogy of the representation dimension characterization of representation-finite algebras.",
"In this section we will give another two similar properties as representation-finite algebras.",
"Firs , we will prove the following theorem.",
"The similar property about Auslander-algebras is placed in the appendix.",
"Theorem 3.1 Let $\\Gamma $ be an artin algebra.",
"Then the following are equivalent.",
"$\\left( 1 \\right)$ $Gproj(\\Gamma )$ is an abelian category $\\left(\\text{Notice: not necessary an abelian subcategory}\\right)$ .",
"$\\left( 2 \\right)$ $\\operatorname{dom.dim} \\Gamma \\ge 2,\\operatorname{id}_{\\Gamma }\\Gamma \\le 2$ As a corollary, we can know the form of the Gorenstein projective module category of an artin algebra when its Gorenstein projective module category is an abelian category by Theorem 2.3 and Theorem 2.13.",
"They are precisely the module category of all DTr-selfinjective algebras.",
"We denote $\\lbrace M \\in \\Gamma \\text{-mod } \\mid \\operatorname{Ext}^i(M, \\Gamma )=0\\rbrace , i=0,1, 2$ by ${^{\\bot _i}}{\\Gamma }$ , the submodule category of $\\operatorname{add}\\Gamma $ by $Sub\\Gamma $ , the Gorenstein Projective dimension of $X$ by $ \\operatorname{Gproj.dim}X$ for every $X \\in \\Gamma $ -mod, $\\bigcap \\lbrace \\operatorname{ker}f \\mid f \\in \\operatorname{Hom}(X, Y)\\rbrace $ by $\\operatorname{Rej}_X(Y)$ for all $X, Y \\in \\Lambda $ -mod.",
"We have the following lemma.",
"Lemma 3.2 Let $\\Gamma $ be an artin algebra, and $Gproj(\\Gamma )$ be an abelian category.",
"Then $\\left( 1 \\right)$ $\\Gamma $ is 2-Gorenstein algebra.",
"$\\left( 2 \\right)$ If $\\operatorname{Gproj.dim}X \\le 1$ , then $X \\in Sub\\Gamma $ for every $X \\in \\Gamma $ -mod.",
"$\\left( 3 \\right)$ ${^{\\bot _0}}{\\Gamma } \\subseteq {^{\\bot _1}}{\\Gamma }$ .",
"Proof.",
"For every morphism $f: X_1 \\rightarrow X_2$ where $X_1, X_2 \\in Gproj(\\Gamma )$ , we denote the kernel and cokernel of $f$ in $Gproj(\\Gamma )$ by $\\operatorname{ker}_{Gproj(\\Gamma )}f,\\operatorname{cok}_{Gproj(\\Gamma )}f$ since $Gproj(\\Gamma )$ is an abelian category.",
"(1) Given a morphism $f: X_1 \\rightarrow X_2$ where $X_1, X_2 \\in Gproj(\\Gamma )$ , since $Gproj(\\Gamma )$ is an abelian category and $\\operatorname{add}\\Gamma \\subseteq Gproj(\\Gamma )$ , $\\operatorname{ker}f = \\operatorname{ker}_{Gproj(\\Gamma )}f\\in Gproj(\\Gamma )$ .",
"$\\Rightarrow $ For every module $X$ , there exists an exact sequence $ {\\begin{matrix}0 \\rightarrow G \\rightarrow P_1 \\xrightarrow{} P_0 \\rightarrow X \\rightarrow 0\\end{matrix}}$ such that $P_1, P_0 \\in \\operatorname{add}\\Gamma $ .",
"Then $G \\cong \\operatorname{ker}f_X\\cong \\operatorname{ker}_{Gproj(\\Gamma )}f_X \\in Gproj(\\Gamma )$ .",
"$\\Rightarrow \\operatorname{Ext}^i_{\\Gamma }(X, \\Gamma )=0$ , for $i \\ge 3$ $\\Rightarrow \\operatorname{\\operatorname{id}} {_\\Gamma }{\\Gamma } \\le 2$ .",
"Since the left and right Gorenstein projective category are dual, the right Gorenstein projective module category is also an abelian category.",
"So $\\operatorname{id} \\Gamma _{\\Gamma } \\le 2$ .",
"(2) Suppose $X \\in \\Gamma $ -mod such that $\\operatorname{Gproj.dim}X \\le 1$ .",
"Then there is an exact sequence: $0 \\rightarrow X_1 \\xrightarrow{} X_2 \\rightarrow X \\rightarrow 0$ such that $X_1, X_2 \\in Gproj(\\Gamma )$ .",
"Suppose $g: X_2 \\rightarrow X_3$ is the cokernal of $f$ in $Gproj(\\Gamma )$ .",
"So $f = \\operatorname{ker}_{Gproj(\\Gamma )}g$ by abelian categories's axioms.",
"There exists a commutative diagram: ${0 @{>}[r] & X_1 [r]^{f} & X_2 [r]^{g} @{>>}[d]_{i} & X_3\\\\& & X @{->}_{\\pi }[ur]}$ By (1) $\\operatorname{ker}_{Gproj(\\Gamma )}g = \\operatorname{ker}g$ $\\Rightarrow \\operatorname{ker}g = f \\Rightarrow \\pi $ is an injective map.",
"$\\Rightarrow $ $X \\in Sub\\Gamma $ since $Gproj(\\Gamma ) \\subseteq Sub\\Gamma $ .",
"(3) Suppose $X \\in {^{\\perp _0}}{\\Gamma }$ .",
"There is an exact sequence: $ {\\begin{matrix}0 \\rightarrow K \\xrightarrow{} P_1 \\xrightarrow{} P_0 \\rightarrow X \\rightarrow 0\\end{matrix}}$ such that $P_1, P_0 \\in \\operatorname{add}\\Gamma $ .",
"Since $X \\in {^{\\perp _0}}{\\Gamma }$ , $f$ is a surjective map in $Gproj(\\Gamma )$ .",
"On the other hand, $i = \\operatorname{ker}f = \\operatorname{ker}_{\\mathcal {G}P}f$ by (1).",
"So $f$ is the cokernel of i in $Gproj(\\Gamma )$ by abelian categories's axioms.",
"$\\Rightarrow $ ${\\begin{matrix}0 \\rightarrow \\operatorname{Hom}(P_0 , \\Gamma ) \\rightarrow \\operatorname{Hom}(P_1, \\Gamma ) \\rightarrow \\operatorname{Hom}(K, \\Gamma )\\end{matrix}}$ is an exact sequence.",
"$\\Rightarrow \\operatorname{Ext}^1_{\\Gamma }(X, \\Gamma ) = 0$ $\\Rightarrow X \\in {^{\\perp _1}}{\\Gamma }$ From now on we can abandon the abstract abelian category structure to prove the property of $\\Gamma $ .",
"What is surprising is that we didn't use the whole abelian categories'axioms in the above lemma.",
"Corollary 3.3 $Sub(\\Gamma )$ is extension closed.",
"Moreover, $\\left({^{\\perp _0}}{\\Gamma }, Sub(\\Gamma )\\right)$ is a torsion pair on $\\Gamma $ -mod.",
"Proof.",
"If $X \\in {^{\\perp _2}}{\\Gamma }$ , then $\\operatorname{Gproj.dim}X \\le 1$ .",
"By Lemma 3.2(2), $X \\in Sub\\Gamma $ .",
"On the other hand, if $X \\in Sub\\Gamma $ , since $\\operatorname{id}{_{\\Gamma }}{\\Gamma } \\le 2$ , then $X \\in {^{\\perp _2}}{\\Gamma }$ .",
"So ${^{\\perp _2}}{\\Gamma } =Sub\\Gamma $ .",
"$\\Rightarrow Sub\\Gamma $ is extension closed.",
"It is also closed under submodules.",
"So (${^{\\perp _0}}{\\Gamma },Sub\\Gamma $ ) is a torsion pair on $\\Gamma $ -mod.",
"$\\forall M\\in \\Gamma $ -mod.",
"$0 \\rightarrow Rej_M(\\Gamma ) \\rightarrow M \\rightarrow M/Rej_M(\\Gamma ) \\rightarrow 0$ is the decomposition of $M$ by the torsion pair.",
"Proof of Theorem 3.1. we just need to prove $\\left(1 \\right) \\Longrightarrow \\left( 2 \\right)$ Step1.",
"Suppose $X \\in Sub\\Gamma $ .",
"$f: X\\hookrightarrow I$ is the injective envelope of $X$ .",
"Suppose $K =\\operatorname{Rej}_I(\\Gamma )$ .",
"By (${^{\\perp _0}}{\\Gamma },Sub\\Gamma $ ), there is an exact sequence: ${\\begin{matrix}0 \\rightarrow K \\xrightarrow{} I \\rightarrow L \\rightarrow 0\\end{matrix}}$ where $L \\in Sub(\\Gamma ), K \\in {^{\\perp _0}}{\\Gamma }$ .",
"By the pull back of $i$ and $f$ , there exists a commutative diagram: $ {\\begin{matrix}&& 0 && 0\\\\&& \\downarrow && \\downarrow &&\\\\0 &\\xrightarrow{}& K^{\\prime } &\\xrightarrow{}& K &\\xrightarrow{}& K/K^{\\prime } &\\xrightarrow{}& 0\\\\&& \\downarrow && {\\scriptstyle i}\\downarrow \\mathbox{mphantom}{\\scriptstyle i}&& \\downarrow &&\\\\0 &\\xrightarrow{}& X &\\xrightarrow{}& I &\\xrightarrow{}& I/X &\\xrightarrow{}& 0\\\\&& \\downarrow && \\downarrow &&\\\\0 &\\xrightarrow{}& L^{\\prime } &\\xrightarrow{}& L\\\\&& \\downarrow && \\downarrow &&\\\\&& 0 && 0\\end{matrix}}$ Since ${^{\\perp _0}}{\\Gamma } \\subseteq {^{\\perp _1}}{\\Gamma }$ and $K/K^{\\prime } \\in {^{\\perp _0}}{\\Gamma }$ , $K/K^{\\prime } \\in {^{\\perp _1}}{\\Gamma }$ .",
"$\\Rightarrow K^{\\prime } \\in {^{\\perp _0}}{\\Gamma }$ .",
"$\\Rightarrow K^{\\prime } \\in {^{\\perp _0}}{\\Gamma } \\bigcap Sub\\Gamma .",
"\\Rightarrow K^{\\prime } = 0.",
"\\Rightarrow X \\cong L^{\\prime }$ .",
"So there exists a commutative diagram: $ {X @{>}[r]^f @{>}[d]_f @{_{(}->}[dr]^g& I @{>}[d]^{\\pi }\\\\I & L @{-->}[l]_h}$ Since $g$ is an injective map, there exists $h: L \\rightarrow I$ such that $f = hg$ .",
"$f$ is left minimal, so $h\\pi $ is an isomorphism.",
"$\\Rightarrow \\pi $ is an isomorphism.",
"$\\Rightarrow I \\in Sub\\Gamma .\\Rightarrow I$ is a projective module.",
"Step2.",
"Suppose $X \\in {^{\\perp _0}}{\\Gamma } \\bigcap {^{\\perp _2}}{\\Gamma }$ .",
"Then $X \\in {^{\\perp _i}}{\\Gamma }$ for $i = 0, 1, 2$ .",
"So $X \\in Gproj(\\Gamma )$ .",
"$\\Rightarrow X \\in Sub\\Gamma $ .",
"However, $X\\in {^{\\perp _0}}{\\Gamma }.$ So $X = 0$ .",
"Suppose $f: X_1 \\rightarrow X_2$ is an injective morphism such that $X_1,X_2 \\in Gproj(\\Gamma ), X = \\operatorname{cok}f$ .",
"Then $X \\in {^{\\perp _2}}{\\Gamma }$ .",
"By (${^{\\perp _0}}{\\Gamma }, Sub\\Gamma $ ), there is an exact sequence: $ {\\begin{matrix}0 \\rightarrow K \\rightarrow X \\rightarrow L \\rightarrow 0\\end{matrix}}$ such that $K \\in {^{\\perp _0}}{\\Gamma }, L \\in Sub\\Gamma .$ $\\Rightarrow \\operatorname{Ext}^2_{\\Gamma }(K, \\Gamma ) \\ne 0$ if $K \\ne 0$ .",
"But $\\operatorname{Ext}^2_{\\Gamma }(X, \\Gamma ) = 0$ .",
"So $\\operatorname{Ext}^2_{\\Gamma }(K, \\Gamma ) = 0$ .",
"That is contradictive.",
"So $K = 0$ .",
"$\\Rightarrow X \\in Sub\\Gamma $ Step3.",
"By step1, there is an exact sequence: ${\\begin{matrix}0 \\rightarrow {_{\\Gamma }}{\\Gamma } \\rightarrow I_0 \\rightarrow K \\rightarrow 0\\end{matrix}}$ such that $I_0$ is a projective-injective module.",
"By step2, $K \\in Sub\\Gamma $ .",
"So by step1, there exists an exact sequence: ${\\begin{matrix}0 \\rightarrow K \\rightarrow I_1 \\rightarrow I_2 \\rightarrow 0\\end{matrix}}$ such that $I_1$ is a projective-injective module.",
"So there is an exact sequence: ${\\begin{matrix}0 \\rightarrow {_{\\Gamma }}{\\Gamma } \\rightarrow I_0 \\rightarrow I_1 \\rightarrow I_2 \\rightarrow 0\\end{matrix}}.$ Since $\\operatorname{id}{_{\\Gamma }}{\\Gamma } \\le 2$ , $I_2$ is an injective module.",
"Now we suppose $k$ be a field, denote $\\bigotimes _k$ by $\\bigotimes $ .",
"We will prove the following theorem which is similar as representation- finite algebras.",
"And it is also an example of $\\operatorname{DTr}$ -selfinjective algebras.",
"Theorem 3.4 Suppose Q is a acyclic quiver, $\\Lambda $ is a finitely dimensional self-injective $k$ algebra.",
"Let $\\Gamma = kQ \\bigotimes \\Lambda $ .",
"Then $\\Gamma $ is a $\\operatorname{DTr}$ -selfinjective algebra if and only if Q is a Dykin quiver.",
"For this, we need some lemmas.",
"Lemma 3.5 Suppose k is a field, A and B are two finitely dimensional algebra over k. Let $M_A$ a right finitely generated A module and $N_B$ a right finitely generated B module.",
"Then $\\operatorname{D}(M \\bigotimes N) = \\operatorname{D}M\\bigotimes \\operatorname{D}N$ as $A \\bigotimes B$ modules.",
"Proof.",
"There is an $A \\bigotimes B$ homomorphism $\\sigma : \\operatorname{D}M \\bigotimes \\operatorname{D}N \\rightarrow \\operatorname{D}(M \\bigotimes N)$ such that $\\forall f \\in \\operatorname{D}M, g \\in \\operatorname{D}N,m \\in M, n \\in N, \\sigma (f \\bigotimes g)(m \\bigotimes n)= f(m)g(n)$ .",
"We choose the bases and the dual bases of M and N as $k$ linear spaces.",
"Then it is easy to check $\\sigma $ is an isomorphism.",
"Corollary 3.6 Suppose k is a field, A and B are two finitely dimensional algebra over k, B is self-injective.",
"Then $\\operatorname{D}(A_A) \\bigotimes {_B}B$ is an injective cogenerator of left $A \\bigotimes B$ module category.",
"Proof.",
"$\\operatorname{D}(A_A \\bigotimes B_B) =\\operatorname{D}(A_A) \\bigotimes \\operatorname{D}(B_B)$ by the above lemma.",
"since ${_B}B \\in \\operatorname{add}\\operatorname{D}(B_B)$ , then $\\operatorname{D}(A_A) \\bigotimes {_B}B \\in \\operatorname{add} \\operatorname{D}(A_A)\\bigotimes \\operatorname{D}(B_B)$ .",
"So $\\operatorname{D}(A_A)\\bigotimes {_B}B$ is an injective module.",
"On the other hand, since $ \\operatorname{D}(B_B) \\in \\operatorname{add} {_B}B$ , then $\\operatorname{D}(A_A) \\bigotimes \\operatorname{D}(B_B) \\in \\operatorname{add} \\operatorname{D}(A_A) \\bigotimes {_B}B$ .",
"So $\\operatorname{D}(A_A) \\bigotimes {_B}B$ is a cogenerator.",
"Now, we give the proof of the theorem.",
"Proof of Proposition 3.4.",
"Suppose $\\lbrace e_1, e_2, \\dots ,e_n \\rbrace $ is the set of all vertices of Q, $\\lbrace \\varepsilon _1,\\varepsilon _2, \\dots , \\varepsilon _m\\rbrace $ is a complete set of primitive idempotents of $\\Lambda , M \\in kQ \\text{-mod}$ .",
"Then there exists the minimal projective resolution of $M$ : ${\\begin{matrix}\\unknown.",
"{\\bigoplus (kQ)e^i \\xrightarrow{} \\bigoplus (kQ)e^j \\rightarrow M \\rightarrow 0 \\hspace{170.71652pt} \\left( *\\right)}\\end{matrix}}$ where $e^i, e^j \\in \\lbrace e_1, \\dots , e_n\\rbrace $ , $f = \\lbrace f_{ij} \\mid f_{ij} \\in \\operatorname{Hom}_{kQ}((kQ)e^i, (kQ)e^j)\\rbrace $ .",
"So $f$ can be represented as a matrix $A = \\lbrace a_{ij} \\mid a_{ij} \\in e^i(kQ)e^j\\rbrace $ .",
"Suppose $\\varepsilon \\in \\lbrace \\varepsilon _1, \\varepsilon _2, \\dots ,\\varepsilon _m\\rbrace $ .",
"$- \\bigotimes \\Lambda \\varepsilon $ acts to ($\\ast $ ).",
"Then we get the following exact sequence: ${\\begin{matrix}\\bigoplus (kQ)e^i \\bigotimes \\Lambda \\varepsilon \\xrightarrow{} \\bigoplus (kQ)e^j \\bigotimes \\Lambda \\varepsilon \\rightarrow M \\bigotimes \\Lambda \\varepsilon \\rightarrow 0\\end{matrix}}$ So the following exact sequence is the projective resolution of $M \\bigotimes \\Lambda \\varepsilon $ : ${\\begin{matrix}\\unknown.",
"{\\bigoplus \\Gamma (e^i \\bigotimes \\varepsilon ) \\xrightarrow{} \\bigoplus \\Gamma (e^j \\bigotimes \\varepsilon ) \\rightarrow M \\bigotimes \\Lambda \\varepsilon \\rightarrow 0 \\hspace{76.82243pt} \\left( **\\right)}\\end{matrix}}$ Where $f \\bigotimes \\Lambda \\varepsilon = \\lbrace f_{ij}\\bigotimes \\Lambda \\varepsilon \\mid f_{ij} \\bigotimes \\Lambda \\varepsilon \\in \\operatorname{Hom}_{\\Gamma }(\\Gamma (e^i \\bigotimes \\varepsilon ), \\Gamma (e^j\\bigotimes \\varepsilon ))\\rbrace $ .",
"By ($\\ast $ ), $f \\bigotimes \\Lambda \\varepsilon $ can be represented by the matrix $B =\\lbrace a_{ij}\\bigotimes \\varepsilon \\rbrace $ .",
"$\\operatorname{Hom}_{\\Gamma }(-, \\Gamma )$ acts to ($**$ ).",
"Then we get an exact sequence: ${\\begin{matrix}\\unknown.",
"{\\bigoplus (e^j \\bigotimes \\varepsilon ) \\Gamma \\xrightarrow{} \\bigoplus (e^j \\bigotimes \\varepsilon ) \\Gamma \\rightarrow N \\rightarrow 0 \\hspace{71.13188pt} \\left( ***\\right)}\\end{matrix}}$ Where $(f \\bigotimes \\Lambda \\varepsilon )^* = \\lbrace g_{ji} =(f_{ij}\\bigotimes \\Lambda \\varepsilon )^* \\mid g_{ji} \\in \\operatorname{Hom}_{\\Gamma }((e^j \\bigotimes \\varepsilon ) \\Gamma , (e^i \\bigotimes \\varepsilon )\\Gamma )\\rbrace $ .",
"By ($**$ ), $(f \\bigotimes \\Lambda \\varepsilon )^*$ can be represented by the matrix $C =\\lbrace c_{ji} = a_{ij} \\bigotimes \\varepsilon \\rbrace $ , and ${N} = {\\operatorname{Tr} (M\\bigotimes \\Lambda \\varepsilon )}$ .",
"So we have the following commutative diagram: ${\\begin{matrix}\\bigoplus (e^j \\bigotimes \\varepsilon ) \\Gamma &\\xrightarrow{}& \\bigoplus (e^j \\bigotimes \\varepsilon ) \\Gamma &\\xrightarrow{}& N&\\xrightarrow{}&0\\\\{\\scriptstyle \\alpha _1}\\downarrow \\mathbox{mphantom}{\\scriptstyle \\alpha _1}&& {\\scriptstyle \\alpha _2}\\downarrow \\mathbox{mphantom}{\\scriptstyle \\alpha _2}&& {\\scriptstyle \\alpha _3}\\downarrow \\mathbox{mphantom}{\\scriptstyle \\alpha _3}&&\\\\\\bigoplus ((kQ)e^j)^* \\bigotimes (\\Lambda \\varepsilon )^* &\\xrightarrow{}& \\bigoplus ((kQ)e^i)^* \\bigotimes (\\Lambda \\varepsilon )^* &\\xrightarrow{}& \\operatorname{Tr}M \\bigotimes (\\Lambda \\varepsilon )^* &\\xrightarrow{}&0\\end{matrix}}$ such that $\\alpha _1, \\alpha _2$ are isomorphisms.",
"So $\\alpha _3$ is an isomorphism.",
"$\\Rightarrow {\\operatorname{Tr} (M \\bigotimes \\Lambda \\varepsilon )} \\cong {\\operatorname{Tr}M \\bigotimes (\\Lambda \\varepsilon )^*}$ $\\Rightarrow \\overline{\\operatorname{DTr} (M \\bigotimes \\Lambda \\varepsilon )} \\cong \\operatorname{D}{\\operatorname{Tr} (M \\bigotimes \\Lambda \\varepsilon )} \\cong \\operatorname{D}{\\operatorname{Tr}M \\bigotimes \\operatorname{D}(\\Lambda \\varepsilon )^*} \\cong \\overline{\\operatorname{DTr}M \\bigotimes \\operatorname{D}(\\Lambda \\varepsilon )^*}$ by Lemma 5.6 Now we can start to calculate the $\\operatorname{DTr}$ -obit of the injective $\\Gamma $ module.",
"Since $\\operatorname{D}(kQ) \\bigotimes \\Lambda $ is an injective cogenerator of $\\Gamma $ -mod, and it is a direct sum of the modules with the form $I \\bigotimes \\Lambda \\varepsilon $ where $I$ is an injective $kQ$ module and $\\varepsilon \\in \\lbrace \\varepsilon _1, \\varepsilon _2, \\dots , \\varepsilon _m\\rbrace $ , then we only have to check the length of $I \\bigotimes \\Lambda \\varepsilon $ .",
"Define $\\mathcal {N}(-) = \\operatorname{D}\\operatorname{Hom}_{\\Lambda }(-, \\Lambda )$ , and $\\mathcal {N}^{n + 1}(-) = \\mathcal {N}(\\mathcal {N}^n(-)),\\operatorname{DTr}^{n +1}(-) =\\operatorname{DTr}(\\operatorname{DTr}^n(-))$ .",
"Then $\\exists \\varepsilon _k \\in \\lbrace \\varepsilon _1, \\varepsilon _2, \\dots ,\\varepsilon _m\\rbrace $ such that $\\Lambda \\varepsilon _k = \\mathcal {N}^k(\\Lambda \\varepsilon )$ .",
"So we have ${\\begin{matrix}\\overline{\\operatorname{DTr}^n (I \\bigotimes \\Lambda \\varepsilon )} =\\overline{\\operatorname{DTr}^nI \\bigotimes \\Lambda \\varepsilon ^n}.\\end{matrix}}$ This is easy to be proved by induction.",
"So the length of $\\operatorname{DTr}$ -obit of $I \\bigotimes \\Lambda \\varepsilon $ is equal to that of $I$ .",
"The theorem is proved."
],
[
"In this section we will prove the following theorem where k can be a field or commutative artin ring.",
"Although it can be proved by the way in section 3, we decide to introduce a way which is more combinatory.",
"Theorem A.1 If $\\mathcal {A}$ is an hom-finite k abeliean category with a finite number of nonisomorphic indecomposable objects, then $\\mathcal {A}$ is equivalent to the finitely generated module category of a finite dimensional k algebra of Representation-finite type .",
"As a corollary,we have Corollary A.2 Suppose $\\Lambda $ is an artin algebra.",
"If the projective module category is an abelian category, then it is equivalent to the finitely generated module category of a representation-finite artin algebra.",
"So $\\Lambda $ is a Auslander algebra.",
"The corollary is a analogy of Theorem 3.1.",
"The above theorem needs several lemmas.",
"From now on we, we suppose $\\mathcal {A}$ is an home-finite k abeliean category with a finite number of nonisomorphic indecomposable objects $A_1, A_2, \\dots , A_n$ .",
"Lemma A.3 If $M \\in \\mathcal {A}$ , then $M$ is of finite length.",
"Proof.",
"We have to prove M satisfies artin conditions and norther conditions.",
"Step1 $\\forall X \\in \\mathcal {A} $ , if $f:X \\rightarrow X$ is an injective morphism(or epicmorphism), then f is an isomorphism.",
"Suppose $f: X \\in \\mathcal {A} $ is an injective morphism but not epic and $\\forall i >0, g_i = \\operatorname{cok}f^i $ where$f^i = f \\dots f, f^1 = f$ .",
"Then $ \\forall j$ , $g_if^j = 0$ if and only if $ i \\le j$ .",
"Now suppose $h = k_1f_1 + k_2f_2 + \\dots +k_mf_m = 0, m >0$ .",
"Then $g_2h = k_1(g_2f_1) + k_2(g_2f_2) + \\dots +k_m(g_2f_m) = k_1(g_2f_1) = 0$ .",
"So $k_1 = 0$ .",
"By induction, $k_1 = k_2 = \\dots = k_m = o$ .",
"So {$f,f^2,f^3,\\dots $ } is linear independent in $Hom(X,X)$ which is an contradiction with the hom-finite property of $\\mathcal {A}$ .",
"Step2 M satisfies artin conditions.",
"Because the object in $\\mathcal {A}$ is of a Krull-Schmidt category, for each $X\\in \\mathcal {A},\\exists x^1,x^2\\dots $ $x^n,X \\cong A_1^{x^1}\\oplus A_2^{x^2}\\dots \\oplus A_n^{x^n}$ , we denote $x=(x^1,x^2\\dots x^n)$ as this decomposition.",
"Suppose $\\exists $ an infinite chain: $\\dots \\stackrel{f_3}{\\rightarrow }X_2\\stackrel{f_2}{\\rightarrow }X_1\\stackrel{f_1}{\\rightarrow }X$ such that $f_i$ is a injective morphism but not an isomorphism.",
"We denote $x_i=(x_1^1,x_i^2\\dots x_i^n)$ if $X_i=A_1^{x_i^1}\\oplus A_2^{x_i^2}\\dots \\oplus A_n^{x_i^n}$ .",
"Then we get a sequence in $N^n$ .",
"There exists $i > 0$ such that $\\forall j > i, 1\\le k\\le n, x_i^k \\le x_j^k$ .",
"Thus there is an injective morphism: $g: X_i \\rightarrow X_{i+1}$ .",
"So $f_{i + 1}g : X_i \\rightarrow X_i$ is an injective morphism.",
"By (1), it is an isomorphism.",
"So $f_{i + 1}$ is also is an injective morphism.",
"That is contradictive.",
"Step3 M satisfies noetherian conditions.",
"Suppose $\\exists $ an infinite subobject chain of $X$ : $X_1\\stackrel{f_1}{\\rightarrow }X_2\\stackrel{f_2}{\\rightarrow }X_3\\stackrel{f_3}{\\rightarrow }\\dots $ such that $f_i$ is a injective morphism but not an isomorphism.",
"Then we also get a sequence $\\lbrace x_1,x_2\\dots \\rbrace $ in $N^n$ .",
"Denote $S(x_i)=\\sum _{k=1}^nx_i^k$ .",
"By step 1, we know $sup\\lbrace S(x_1),S(x_2)\\dots \\rbrace $ $=\\infty $ $\\Rightarrow \\exists i, sup\\lbrace x_1^i,x_2^i\\dots \\rbrace =\\infty $ $\\Rightarrow sup\\lbrace dim_kHom(A_i,X_1), dim_kHom(A_i,X_2)\\dots \\rbrace =\\infty $ .",
"But we know $dim_kHom(A_i,X_1)\\le dim_kHom(A_i,X)$ .",
"So $dim_kHom(A_i,X)=\\infty $ that's contradicted with the hom-finite property.",
"The following lemma can be proved similarly by the way in [1, chapter 6].",
"Lemma A.4 $\\exists m \\in N$ , for every chain $X_1\\stackrel{f_1}{\\rightarrow }X_2\\stackrel{f_2}{\\rightarrow }X_3\\stackrel{f_3}{\\rightarrow }\\dots \\stackrel{f_m}{\\rightarrow }X_{m+1}$ with $X_i\\in \\lbrace A_1, A_2, \\dots , A_n\\rbrace $ , if $f_j$ is not an isomorphism for every $j=1, 2, \\dots , m+1$ , then $f_mf_{m-1}\\dots f_1=0$ .",
"Lemma A.5 Suppose$X\\in \\mathcal {A}$ .",
"The following are equivalent.",
"$\\left( 1 \\right)$ $X$ is a projective object.",
"$\\left( 2 \\right)$ if $f: Y\\rightarrow X$ is a right minimal epic morphism, then $f$ is an isomorphism.",
"Proof.",
"$\\left( 1 \\right) \\Rightarrow \\left( 2\\right)$ : clear.",
"$\\left( 2 \\right) \\Rightarrow \\left( 1 \\right)$ : Suppose $X$ has the property in $\\left( 2 \\right)$ .",
"And $f: Y\\rightarrow X$ is an epic morphism.",
"Then $f = (f_1, f_2)$ where $f_1 \\in \\operatorname{Hom}(Y_1,X), f_2 \\in \\operatorname{Hom}(Y_2, X)$ , $Y = Y_1 \\bigoplus Y_2$ such that $f_1$ is right minimal and $f_2 = 0$ .",
"So $f_1$ is an isomorphism.",
"$f$ is a split epic morphism.",
"So $X$ is a projective object.",
"Lemma A.6 $\\mathcal {A}$ has enough projective objects Proof.",
"Suppose $X \\in \\mathcal {A}$ such that $X$ has no projective cover and $X$ is an indecomposable object.",
"So there exists a right minimal epic morphism $f_1: Y_1 \\rightarrow X$ such that $f_1$ is not an isomorphism by the above lemma.",
"So there exists $Y_1= Q_1 \\bigoplus X_1$ such that $Q_1$ is a projective object, $X_1$ has no projective direct summand, $X_1 \\ne 0$ , and $f_1 = (g_1,h_1)$ where $g_1 \\in \\operatorname{Hom}(Q_1, X), h_1 \\in \\operatorname{Radical}\\operatorname{Hom}(X_1, X)$ .",
"By the way above, we consider the indecomposable direct summand of $X_1$ .",
"Then there exists an epic morphism $f_2: Y_2 \\rightarrow X_1$ such that $f_2 \\in \\operatorname{Radical} \\operatorname{Hom}(Y_2, X_1).",
"$ So there exists $Y_2 = Q_2 \\bigoplus X_2$ such that $Q_2$ is a projective object, $X_2$ has no projective direct summand, $X_2 \\ne 0$ since $X$ has no projective cover, and$f_2 = (g_2, h_2)$ where $g_2 \\in \\operatorname{Hom}(Q_2, X_1), h_2 \\in \\operatorname{Radical} \\operatorname{Hom}(X_2,X_1)$ .",
"By induction, for $k > 0$ , we get $Y_k = Q_k \\bigoplus X_k$ such that $Q_k$ is a projective object, $X_k$ has no projective direct summand, $X_k \\ne 0$ since $X$ has no projective cover, and$f_k = (g_k, h_k)$ where $g_k \\in \\operatorname{Hom}(Q_k,X_{k - 1}), h_k \\in \\operatorname{Radical} \\operatorname{Hom}(X_k, X_{k - 1})$ .",
"We have the following diagram to explain the operation: ${\\dots &X_2 ^{h_2}[rd]\\\\&&X_1 [rd]^{h_1}\\\\&Q_2 [ru]^{g_2} & & X\\\\&&Q_1 [ru]^{g_1}}$ So there exists an epic morphism $(h_m \\dots h_1, \\phi _m) : X_m\\bigoplus (Q_1 \\bigoplus \\dots \\bigoplus Q_m) \\rightarrow X$ .",
"Since $X$ has no projective cover, $h_m \\dots h_1 \\ne 0$ .",
"That is contradicted with the property of $m$ .",
"Thus the above lemma tells us the abelian category has a projective generator.",
"So by the following well known lemma.",
"The theorem is proved.",
"Lemma A.7 If an abelian categoryis a hom-finite k category with a projective generator, then it is equivalent to the left finitely generated module category of the opposite endomorphism ring of the projective generator.",
"Acknowledgement.",
"This article is part of the author's Ph.D. thesis under the supervision of Pu Zhang.",
"The author is deeply grateful to him for his guidance and encouragement.",
"The author also thanks Professor Ringel for providing him the references [11], [12], [13], [14] and his excellent lectures in Shanghai Jiao Tong University in 2011.",
"The author also deeply thanks Baolin Xiong for his helpful discussions and constant encouragement.",
"Fan Kong, Department of Mathematics, Shanghai Jiao Tong University, 200240 Shanghai, People's Republic of China.",
"Email: [email protected]"
],
[
"The case n = 2",
"When n = 2, 1-$\\operatorname{DTr}$ -selfinjective algebras are called $\\operatorname{DTr}$ -selfinjective algebras just as in[4].",
"The correspondence in Theorem 2.3 about it ($n = 2$ ) is the analogy of representation-finite algebras which is obtained in [2].",
"So we think $\\operatorname{DTr}$ -selfinjective algebras have some similar properties as representation-finite algebras.",
"For the same reason, the algebras with diminant dimension and selfinjective dimension being both 2 should have some similar properties of Auslander algebras.",
"The homological characterization of $\\operatorname{DTr}$ -selfinjective algebras which is demonstrated in Theorem 2.16 when $n= 2$ is the analogy of the representation dimension characterization of representation-finite algebras.",
"In this section we will give another two similar properties as representation-finite algebras.",
"Firs , we will prove the following theorem.",
"The similar property about Auslander-algebras is placed in the appendix.",
"Theorem 3.1 Let $\\Gamma $ be an artin algebra.",
"Then the following are equivalent.",
"$\\left( 1 \\right)$ $Gproj(\\Gamma )$ is an abelian category $\\left(\\text{Notice: not necessary an abelian subcategory}\\right)$ .",
"$\\left( 2 \\right)$ $\\operatorname{dom.dim} \\Gamma \\ge 2,\\operatorname{id}_{\\Gamma }\\Gamma \\le 2$ As a corollary, we can know the form of the Gorenstein projective module category of an artin algebra when its Gorenstein projective module category is an abelian category by Theorem 2.3 and Theorem 2.13.",
"They are precisely the module category of all DTr-selfinjective algebras.",
"We denote $\\lbrace M \\in \\Gamma \\text{-mod } \\mid \\operatorname{Ext}^i(M, \\Gamma )=0\\rbrace , i=0,1, 2$ by ${^{\\bot _i}}{\\Gamma }$ , the submodule category of $\\operatorname{add}\\Gamma $ by $Sub\\Gamma $ , the Gorenstein Projective dimension of $X$ by $ \\operatorname{Gproj.dim}X$ for every $X \\in \\Gamma $ -mod, $\\bigcap \\lbrace \\operatorname{ker}f \\mid f \\in \\operatorname{Hom}(X, Y)\\rbrace $ by $\\operatorname{Rej}_X(Y)$ for all $X, Y \\in \\Lambda $ -mod.",
"We have the following lemma.",
"Lemma 3.2 Let $\\Gamma $ be an artin algebra, and $Gproj(\\Gamma )$ be an abelian category.",
"Then $\\left( 1 \\right)$ $\\Gamma $ is 2-Gorenstein algebra.",
"$\\left( 2 \\right)$ If $\\operatorname{Gproj.dim}X \\le 1$ , then $X \\in Sub\\Gamma $ for every $X \\in \\Gamma $ -mod.",
"$\\left( 3 \\right)$ ${^{\\bot _0}}{\\Gamma } \\subseteq {^{\\bot _1}}{\\Gamma }$ .",
"Proof.",
"For every morphism $f: X_1 \\rightarrow X_2$ where $X_1, X_2 \\in Gproj(\\Gamma )$ , we denote the kernel and cokernel of $f$ in $Gproj(\\Gamma )$ by $\\operatorname{ker}_{Gproj(\\Gamma )}f,\\operatorname{cok}_{Gproj(\\Gamma )}f$ since $Gproj(\\Gamma )$ is an abelian category.",
"(1) Given a morphism $f: X_1 \\rightarrow X_2$ where $X_1, X_2 \\in Gproj(\\Gamma )$ , since $Gproj(\\Gamma )$ is an abelian category and $\\operatorname{add}\\Gamma \\subseteq Gproj(\\Gamma )$ , $\\operatorname{ker}f = \\operatorname{ker}_{Gproj(\\Gamma )}f\\in Gproj(\\Gamma )$ .",
"$\\Rightarrow $ For every module $X$ , there exists an exact sequence $ {\\begin{matrix}0 \\rightarrow G \\rightarrow P_1 \\xrightarrow{} P_0 \\rightarrow X \\rightarrow 0\\end{matrix}}$ such that $P_1, P_0 \\in \\operatorname{add}\\Gamma $ .",
"Then $G \\cong \\operatorname{ker}f_X\\cong \\operatorname{ker}_{Gproj(\\Gamma )}f_X \\in Gproj(\\Gamma )$ .",
"$\\Rightarrow \\operatorname{Ext}^i_{\\Gamma }(X, \\Gamma )=0$ , for $i \\ge 3$ $\\Rightarrow \\operatorname{\\operatorname{id}} {_\\Gamma }{\\Gamma } \\le 2$ .",
"Since the left and right Gorenstein projective category are dual, the right Gorenstein projective module category is also an abelian category.",
"So $\\operatorname{id} \\Gamma _{\\Gamma } \\le 2$ .",
"(2) Suppose $X \\in \\Gamma $ -mod such that $\\operatorname{Gproj.dim}X \\le 1$ .",
"Then there is an exact sequence: $0 \\rightarrow X_1 \\xrightarrow{} X_2 \\rightarrow X \\rightarrow 0$ such that $X_1, X_2 \\in Gproj(\\Gamma )$ .",
"Suppose $g: X_2 \\rightarrow X_3$ is the cokernal of $f$ in $Gproj(\\Gamma )$ .",
"So $f = \\operatorname{ker}_{Gproj(\\Gamma )}g$ by abelian categories's axioms.",
"There exists a commutative diagram: ${0 @{>}[r] & X_1 [r]^{f} & X_2 [r]^{g} @{>>}[d]_{i} & X_3\\\\& & X @{->}_{\\pi }[ur]}$ By (1) $\\operatorname{ker}_{Gproj(\\Gamma )}g = \\operatorname{ker}g$ $\\Rightarrow \\operatorname{ker}g = f \\Rightarrow \\pi $ is an injective map.",
"$\\Rightarrow $ $X \\in Sub\\Gamma $ since $Gproj(\\Gamma ) \\subseteq Sub\\Gamma $ .",
"(3) Suppose $X \\in {^{\\perp _0}}{\\Gamma }$ .",
"There is an exact sequence: $ {\\begin{matrix}0 \\rightarrow K \\xrightarrow{} P_1 \\xrightarrow{} P_0 \\rightarrow X \\rightarrow 0\\end{matrix}}$ such that $P_1, P_0 \\in \\operatorname{add}\\Gamma $ .",
"Since $X \\in {^{\\perp _0}}{\\Gamma }$ , $f$ is a surjective map in $Gproj(\\Gamma )$ .",
"On the other hand, $i = \\operatorname{ker}f = \\operatorname{ker}_{\\mathcal {G}P}f$ by (1).",
"So $f$ is the cokernel of i in $Gproj(\\Gamma )$ by abelian categories's axioms.",
"$\\Rightarrow $ ${\\begin{matrix}0 \\rightarrow \\operatorname{Hom}(P_0 , \\Gamma ) \\rightarrow \\operatorname{Hom}(P_1, \\Gamma ) \\rightarrow \\operatorname{Hom}(K, \\Gamma )\\end{matrix}}$ is an exact sequence.",
"$\\Rightarrow \\operatorname{Ext}^1_{\\Gamma }(X, \\Gamma ) = 0$ $\\Rightarrow X \\in {^{\\perp _1}}{\\Gamma }$ From now on we can abandon the abstract abelian category structure to prove the property of $\\Gamma $ .",
"What is surprising is that we didn't use the whole abelian categories'axioms in the above lemma.",
"Corollary 3.3 $Sub(\\Gamma )$ is extension closed.",
"Moreover, $\\left({^{\\perp _0}}{\\Gamma }, Sub(\\Gamma )\\right)$ is a torsion pair on $\\Gamma $ -mod.",
"Proof.",
"If $X \\in {^{\\perp _2}}{\\Gamma }$ , then $\\operatorname{Gproj.dim}X \\le 1$ .",
"By Lemma 3.2(2), $X \\in Sub\\Gamma $ .",
"On the other hand, if $X \\in Sub\\Gamma $ , since $\\operatorname{id}{_{\\Gamma }}{\\Gamma } \\le 2$ , then $X \\in {^{\\perp _2}}{\\Gamma }$ .",
"So ${^{\\perp _2}}{\\Gamma } =Sub\\Gamma $ .",
"$\\Rightarrow Sub\\Gamma $ is extension closed.",
"It is also closed under submodules.",
"So (${^{\\perp _0}}{\\Gamma },Sub\\Gamma $ ) is a torsion pair on $\\Gamma $ -mod.",
"$\\forall M\\in \\Gamma $ -mod.",
"$0 \\rightarrow Rej_M(\\Gamma ) \\rightarrow M \\rightarrow M/Rej_M(\\Gamma ) \\rightarrow 0$ is the decomposition of $M$ by the torsion pair.",
"Proof of Theorem 3.1. we just need to prove $\\left(1 \\right) \\Longrightarrow \\left( 2 \\right)$ Step1.",
"Suppose $X \\in Sub\\Gamma $ .",
"$f: X\\hookrightarrow I$ is the injective envelope of $X$ .",
"Suppose $K =\\operatorname{Rej}_I(\\Gamma )$ .",
"By (${^{\\perp _0}}{\\Gamma },Sub\\Gamma $ ), there is an exact sequence: ${\\begin{matrix}0 \\rightarrow K \\xrightarrow{} I \\rightarrow L \\rightarrow 0\\end{matrix}}$ where $L \\in Sub(\\Gamma ), K \\in {^{\\perp _0}}{\\Gamma }$ .",
"By the pull back of $i$ and $f$ , there exists a commutative diagram: $ {\\begin{matrix}&& 0 && 0\\\\&& \\downarrow && \\downarrow &&\\\\0 &\\xrightarrow{}& K^{\\prime } &\\xrightarrow{}& K &\\xrightarrow{}& K/K^{\\prime } &\\xrightarrow{}& 0\\\\&& \\downarrow && {\\scriptstyle i}\\downarrow \\mathbox{mphantom}{\\scriptstyle i}&& \\downarrow &&\\\\0 &\\xrightarrow{}& X &\\xrightarrow{}& I &\\xrightarrow{}& I/X &\\xrightarrow{}& 0\\\\&& \\downarrow && \\downarrow &&\\\\0 &\\xrightarrow{}& L^{\\prime } &\\xrightarrow{}& L\\\\&& \\downarrow && \\downarrow &&\\\\&& 0 && 0\\end{matrix}}$ Since ${^{\\perp _0}}{\\Gamma } \\subseteq {^{\\perp _1}}{\\Gamma }$ and $K/K^{\\prime } \\in {^{\\perp _0}}{\\Gamma }$ , $K/K^{\\prime } \\in {^{\\perp _1}}{\\Gamma }$ .",
"$\\Rightarrow K^{\\prime } \\in {^{\\perp _0}}{\\Gamma }$ .",
"$\\Rightarrow K^{\\prime } \\in {^{\\perp _0}}{\\Gamma } \\bigcap Sub\\Gamma .",
"\\Rightarrow K^{\\prime } = 0.",
"\\Rightarrow X \\cong L^{\\prime }$ .",
"So there exists a commutative diagram: $ {X @{>}[r]^f @{>}[d]_f @{_{(}->}[dr]^g& I @{>}[d]^{\\pi }\\\\I & L @{-->}[l]_h}$ Since $g$ is an injective map, there exists $h: L \\rightarrow I$ such that $f = hg$ .",
"$f$ is left minimal, so $h\\pi $ is an isomorphism.",
"$\\Rightarrow \\pi $ is an isomorphism.",
"$\\Rightarrow I \\in Sub\\Gamma .\\Rightarrow I$ is a projective module.",
"Step2.",
"Suppose $X \\in {^{\\perp _0}}{\\Gamma } \\bigcap {^{\\perp _2}}{\\Gamma }$ .",
"Then $X \\in {^{\\perp _i}}{\\Gamma }$ for $i = 0, 1, 2$ .",
"So $X \\in Gproj(\\Gamma )$ .",
"$\\Rightarrow X \\in Sub\\Gamma $ .",
"However, $X\\in {^{\\perp _0}}{\\Gamma }.$ So $X = 0$ .",
"Suppose $f: X_1 \\rightarrow X_2$ is an injective morphism such that $X_1,X_2 \\in Gproj(\\Gamma ), X = \\operatorname{cok}f$ .",
"Then $X \\in {^{\\perp _2}}{\\Gamma }$ .",
"By (${^{\\perp _0}}{\\Gamma }, Sub\\Gamma $ ), there is an exact sequence: $ {\\begin{matrix}0 \\rightarrow K \\rightarrow X \\rightarrow L \\rightarrow 0\\end{matrix}}$ such that $K \\in {^{\\perp _0}}{\\Gamma }, L \\in Sub\\Gamma .$ $\\Rightarrow \\operatorname{Ext}^2_{\\Gamma }(K, \\Gamma ) \\ne 0$ if $K \\ne 0$ .",
"But $\\operatorname{Ext}^2_{\\Gamma }(X, \\Gamma ) = 0$ .",
"So $\\operatorname{Ext}^2_{\\Gamma }(K, \\Gamma ) = 0$ .",
"That is contradictive.",
"So $K = 0$ .",
"$\\Rightarrow X \\in Sub\\Gamma $ Step3.",
"By step1, there is an exact sequence: ${\\begin{matrix}0 \\rightarrow {_{\\Gamma }}{\\Gamma } \\rightarrow I_0 \\rightarrow K \\rightarrow 0\\end{matrix}}$ such that $I_0$ is a projective-injective module.",
"By step2, $K \\in Sub\\Gamma $ .",
"So by step1, there exists an exact sequence: ${\\begin{matrix}0 \\rightarrow K \\rightarrow I_1 \\rightarrow I_2 \\rightarrow 0\\end{matrix}}$ such that $I_1$ is a projective-injective module.",
"So there is an exact sequence: ${\\begin{matrix}0 \\rightarrow {_{\\Gamma }}{\\Gamma } \\rightarrow I_0 \\rightarrow I_1 \\rightarrow I_2 \\rightarrow 0\\end{matrix}}.$ Since $\\operatorname{id}{_{\\Gamma }}{\\Gamma } \\le 2$ , $I_2$ is an injective module.",
"Now we suppose $k$ be a field, denote $\\bigotimes _k$ by $\\bigotimes $ .",
"We will prove the following theorem which is similar as representation- finite algebras.",
"And it is also an example of $\\operatorname{DTr}$ -selfinjective algebras.",
"Theorem 3.4 Suppose Q is a acyclic quiver, $\\Lambda $ is a finitely dimensional self-injective $k$ algebra.",
"Let $\\Gamma = kQ \\bigotimes \\Lambda $ .",
"Then $\\Gamma $ is a $\\operatorname{DTr}$ -selfinjective algebra if and only if Q is a Dykin quiver.",
"For this, we need some lemmas.",
"Lemma 3.5 Suppose k is a field, A and B are two finitely dimensional algebra over k. Let $M_A$ a right finitely generated A module and $N_B$ a right finitely generated B module.",
"Then $\\operatorname{D}(M \\bigotimes N) = \\operatorname{D}M\\bigotimes \\operatorname{D}N$ as $A \\bigotimes B$ modules.",
"Proof.",
"There is an $A \\bigotimes B$ homomorphism $\\sigma : \\operatorname{D}M \\bigotimes \\operatorname{D}N \\rightarrow \\operatorname{D}(M \\bigotimes N)$ such that $\\forall f \\in \\operatorname{D}M, g \\in \\operatorname{D}N,m \\in M, n \\in N, \\sigma (f \\bigotimes g)(m \\bigotimes n)= f(m)g(n)$ .",
"We choose the bases and the dual bases of M and N as $k$ linear spaces.",
"Then it is easy to check $\\sigma $ is an isomorphism.",
"Corollary 3.6 Suppose k is a field, A and B are two finitely dimensional algebra over k, B is self-injective.",
"Then $\\operatorname{D}(A_A) \\bigotimes {_B}B$ is an injective cogenerator of left $A \\bigotimes B$ module category.",
"Proof.",
"$\\operatorname{D}(A_A \\bigotimes B_B) =\\operatorname{D}(A_A) \\bigotimes \\operatorname{D}(B_B)$ by the above lemma.",
"since ${_B}B \\in \\operatorname{add}\\operatorname{D}(B_B)$ , then $\\operatorname{D}(A_A) \\bigotimes {_B}B \\in \\operatorname{add} \\operatorname{D}(A_A)\\bigotimes \\operatorname{D}(B_B)$ .",
"So $\\operatorname{D}(A_A)\\bigotimes {_B}B$ is an injective module.",
"On the other hand, since $ \\operatorname{D}(B_B) \\in \\operatorname{add} {_B}B$ , then $\\operatorname{D}(A_A) \\bigotimes \\operatorname{D}(B_B) \\in \\operatorname{add} \\operatorname{D}(A_A) \\bigotimes {_B}B$ .",
"So $\\operatorname{D}(A_A) \\bigotimes {_B}B$ is a cogenerator.",
"Now, we give the proof of the theorem.",
"Proof of Proposition 3.4.",
"Suppose $\\lbrace e_1, e_2, \\dots ,e_n \\rbrace $ is the set of all vertices of Q, $\\lbrace \\varepsilon _1,\\varepsilon _2, \\dots , \\varepsilon _m\\rbrace $ is a complete set of primitive idempotents of $\\Lambda , M \\in kQ \\text{-mod}$ .",
"Then there exists the minimal projective resolution of $M$ : ${\\begin{matrix}\\unknown.",
"{\\bigoplus (kQ)e^i \\xrightarrow{} \\bigoplus (kQ)e^j \\rightarrow M \\rightarrow 0 \\hspace{170.71652pt} \\left( *\\right)}\\end{matrix}}$ where $e^i, e^j \\in \\lbrace e_1, \\dots , e_n\\rbrace $ , $f = \\lbrace f_{ij} \\mid f_{ij} \\in \\operatorname{Hom}_{kQ}((kQ)e^i, (kQ)e^j)\\rbrace $ .",
"So $f$ can be represented as a matrix $A = \\lbrace a_{ij} \\mid a_{ij} \\in e^i(kQ)e^j\\rbrace $ .",
"Suppose $\\varepsilon \\in \\lbrace \\varepsilon _1, \\varepsilon _2, \\dots ,\\varepsilon _m\\rbrace $ .",
"$- \\bigotimes \\Lambda \\varepsilon $ acts to ($\\ast $ ).",
"Then we get the following exact sequence: ${\\begin{matrix}\\bigoplus (kQ)e^i \\bigotimes \\Lambda \\varepsilon \\xrightarrow{} \\bigoplus (kQ)e^j \\bigotimes \\Lambda \\varepsilon \\rightarrow M \\bigotimes \\Lambda \\varepsilon \\rightarrow 0\\end{matrix}}$ So the following exact sequence is the projective resolution of $M \\bigotimes \\Lambda \\varepsilon $ : ${\\begin{matrix}\\unknown.",
"{\\bigoplus \\Gamma (e^i \\bigotimes \\varepsilon ) \\xrightarrow{} \\bigoplus \\Gamma (e^j \\bigotimes \\varepsilon ) \\rightarrow M \\bigotimes \\Lambda \\varepsilon \\rightarrow 0 \\hspace{76.82243pt} \\left( **\\right)}\\end{matrix}}$ Where $f \\bigotimes \\Lambda \\varepsilon = \\lbrace f_{ij}\\bigotimes \\Lambda \\varepsilon \\mid f_{ij} \\bigotimes \\Lambda \\varepsilon \\in \\operatorname{Hom}_{\\Gamma }(\\Gamma (e^i \\bigotimes \\varepsilon ), \\Gamma (e^j\\bigotimes \\varepsilon ))\\rbrace $ .",
"By ($\\ast $ ), $f \\bigotimes \\Lambda \\varepsilon $ can be represented by the matrix $B =\\lbrace a_{ij}\\bigotimes \\varepsilon \\rbrace $ .",
"$\\operatorname{Hom}_{\\Gamma }(-, \\Gamma )$ acts to ($**$ ).",
"Then we get an exact sequence: ${\\begin{matrix}\\unknown.",
"{\\bigoplus (e^j \\bigotimes \\varepsilon ) \\Gamma \\xrightarrow{} \\bigoplus (e^j \\bigotimes \\varepsilon ) \\Gamma \\rightarrow N \\rightarrow 0 \\hspace{71.13188pt} \\left( ***\\right)}\\end{matrix}}$ Where $(f \\bigotimes \\Lambda \\varepsilon )^* = \\lbrace g_{ji} =(f_{ij}\\bigotimes \\Lambda \\varepsilon )^* \\mid g_{ji} \\in \\operatorname{Hom}_{\\Gamma }((e^j \\bigotimes \\varepsilon ) \\Gamma , (e^i \\bigotimes \\varepsilon )\\Gamma )\\rbrace $ .",
"By ($**$ ), $(f \\bigotimes \\Lambda \\varepsilon )^*$ can be represented by the matrix $C =\\lbrace c_{ji} = a_{ij} \\bigotimes \\varepsilon \\rbrace $ , and ${N} = {\\operatorname{Tr} (M\\bigotimes \\Lambda \\varepsilon )}$ .",
"So we have the following commutative diagram: ${\\begin{matrix}\\bigoplus (e^j \\bigotimes \\varepsilon ) \\Gamma &\\xrightarrow{}& \\bigoplus (e^j \\bigotimes \\varepsilon ) \\Gamma &\\xrightarrow{}& N&\\xrightarrow{}&0\\\\{\\scriptstyle \\alpha _1}\\downarrow \\mathbox{mphantom}{\\scriptstyle \\alpha _1}&& {\\scriptstyle \\alpha _2}\\downarrow \\mathbox{mphantom}{\\scriptstyle \\alpha _2}&& {\\scriptstyle \\alpha _3}\\downarrow \\mathbox{mphantom}{\\scriptstyle \\alpha _3}&&\\\\\\bigoplus ((kQ)e^j)^* \\bigotimes (\\Lambda \\varepsilon )^* &\\xrightarrow{}& \\bigoplus ((kQ)e^i)^* \\bigotimes (\\Lambda \\varepsilon )^* &\\xrightarrow{}& \\operatorname{Tr}M \\bigotimes (\\Lambda \\varepsilon )^* &\\xrightarrow{}&0\\end{matrix}}$ such that $\\alpha _1, \\alpha _2$ are isomorphisms.",
"So $\\alpha _3$ is an isomorphism.",
"$\\Rightarrow {\\operatorname{Tr} (M \\bigotimes \\Lambda \\varepsilon )} \\cong {\\operatorname{Tr}M \\bigotimes (\\Lambda \\varepsilon )^*}$ $\\Rightarrow \\overline{\\operatorname{DTr} (M \\bigotimes \\Lambda \\varepsilon )} \\cong \\operatorname{D}{\\operatorname{Tr} (M \\bigotimes \\Lambda \\varepsilon )} \\cong \\operatorname{D}{\\operatorname{Tr}M \\bigotimes \\operatorname{D}(\\Lambda \\varepsilon )^*} \\cong \\overline{\\operatorname{DTr}M \\bigotimes \\operatorname{D}(\\Lambda \\varepsilon )^*}$ by Lemma 5.6 Now we can start to calculate the $\\operatorname{DTr}$ -obit of the injective $\\Gamma $ module.",
"Since $\\operatorname{D}(kQ) \\bigotimes \\Lambda $ is an injective cogenerator of $\\Gamma $ -mod, and it is a direct sum of the modules with the form $I \\bigotimes \\Lambda \\varepsilon $ where $I$ is an injective $kQ$ module and $\\varepsilon \\in \\lbrace \\varepsilon _1, \\varepsilon _2, \\dots , \\varepsilon _m\\rbrace $ , then we only have to check the length of $I \\bigotimes \\Lambda \\varepsilon $ .",
"Define $\\mathcal {N}(-) = \\operatorname{D}\\operatorname{Hom}_{\\Lambda }(-, \\Lambda )$ , and $\\mathcal {N}^{n + 1}(-) = \\mathcal {N}(\\mathcal {N}^n(-)),\\operatorname{DTr}^{n +1}(-) =\\operatorname{DTr}(\\operatorname{DTr}^n(-))$ .",
"Then $\\exists \\varepsilon _k \\in \\lbrace \\varepsilon _1, \\varepsilon _2, \\dots ,\\varepsilon _m\\rbrace $ such that $\\Lambda \\varepsilon _k = \\mathcal {N}^k(\\Lambda \\varepsilon )$ .",
"So we have ${\\begin{matrix}\\overline{\\operatorname{DTr}^n (I \\bigotimes \\Lambda \\varepsilon )} =\\overline{\\operatorname{DTr}^nI \\bigotimes \\Lambda \\varepsilon ^n}.\\end{matrix}}$ This is easy to be proved by induction.",
"So the length of $\\operatorname{DTr}$ -obit of $I \\bigotimes \\Lambda \\varepsilon $ is equal to that of $I$ .",
"The theorem is proved."
],
[
"In this section we will prove the following theorem where k can be a field or commutative artin ring.",
"Although it can be proved by the way in section 3, we decide to introduce a way which is more combinatory.",
"Theorem A.1 If $\\mathcal {A}$ is an hom-finite k abeliean category with a finite number of nonisomorphic indecomposable objects, then $\\mathcal {A}$ is equivalent to the finitely generated module category of a finite dimensional k algebra of Representation-finite type .",
"As a corollary,we have Corollary A.2 Suppose $\\Lambda $ is an artin algebra.",
"If the projective module category is an abelian category, then it is equivalent to the finitely generated module category of a representation-finite artin algebra.",
"So $\\Lambda $ is a Auslander algebra.",
"The corollary is a analogy of Theorem 3.1.",
"The above theorem needs several lemmas.",
"From now on we, we suppose $\\mathcal {A}$ is an home-finite k abeliean category with a finite number of nonisomorphic indecomposable objects $A_1, A_2, \\dots , A_n$ .",
"Lemma A.3 If $M \\in \\mathcal {A}$ , then $M$ is of finite length.",
"Proof.",
"We have to prove M satisfies artin conditions and norther conditions.",
"Step1 $\\forall X \\in \\mathcal {A} $ , if $f:X \\rightarrow X$ is an injective morphism(or epicmorphism), then f is an isomorphism.",
"Suppose $f: X \\in \\mathcal {A} $ is an injective morphism but not epic and $\\forall i >0, g_i = \\operatorname{cok}f^i $ where$f^i = f \\dots f, f^1 = f$ .",
"Then $ \\forall j$ , $g_if^j = 0$ if and only if $ i \\le j$ .",
"Now suppose $h = k_1f_1 + k_2f_2 + \\dots +k_mf_m = 0, m >0$ .",
"Then $g_2h = k_1(g_2f_1) + k_2(g_2f_2) + \\dots +k_m(g_2f_m) = k_1(g_2f_1) = 0$ .",
"So $k_1 = 0$ .",
"By induction, $k_1 = k_2 = \\dots = k_m = o$ .",
"So {$f,f^2,f^3,\\dots $ } is linear independent in $Hom(X,X)$ which is an contradiction with the hom-finite property of $\\mathcal {A}$ .",
"Step2 M satisfies artin conditions.",
"Because the object in $\\mathcal {A}$ is of a Krull-Schmidt category, for each $X\\in \\mathcal {A},\\exists x^1,x^2\\dots $ $x^n,X \\cong A_1^{x^1}\\oplus A_2^{x^2}\\dots \\oplus A_n^{x^n}$ , we denote $x=(x^1,x^2\\dots x^n)$ as this decomposition.",
"Suppose $\\exists $ an infinite chain: $\\dots \\stackrel{f_3}{\\rightarrow }X_2\\stackrel{f_2}{\\rightarrow }X_1\\stackrel{f_1}{\\rightarrow }X$ such that $f_i$ is a injective morphism but not an isomorphism.",
"We denote $x_i=(x_1^1,x_i^2\\dots x_i^n)$ if $X_i=A_1^{x_i^1}\\oplus A_2^{x_i^2}\\dots \\oplus A_n^{x_i^n}$ .",
"Then we get a sequence in $N^n$ .",
"There exists $i > 0$ such that $\\forall j > i, 1\\le k\\le n, x_i^k \\le x_j^k$ .",
"Thus there is an injective morphism: $g: X_i \\rightarrow X_{i+1}$ .",
"So $f_{i + 1}g : X_i \\rightarrow X_i$ is an injective morphism.",
"By (1), it is an isomorphism.",
"So $f_{i + 1}$ is also is an injective morphism.",
"That is contradictive.",
"Step3 M satisfies noetherian conditions.",
"Suppose $\\exists $ an infinite subobject chain of $X$ : $X_1\\stackrel{f_1}{\\rightarrow }X_2\\stackrel{f_2}{\\rightarrow }X_3\\stackrel{f_3}{\\rightarrow }\\dots $ such that $f_i$ is a injective morphism but not an isomorphism.",
"Then we also get a sequence $\\lbrace x_1,x_2\\dots \\rbrace $ in $N^n$ .",
"Denote $S(x_i)=\\sum _{k=1}^nx_i^k$ .",
"By step 1, we know $sup\\lbrace S(x_1),S(x_2)\\dots \\rbrace $ $=\\infty $ $\\Rightarrow \\exists i, sup\\lbrace x_1^i,x_2^i\\dots \\rbrace =\\infty $ $\\Rightarrow sup\\lbrace dim_kHom(A_i,X_1), dim_kHom(A_i,X_2)\\dots \\rbrace =\\infty $ .",
"But we know $dim_kHom(A_i,X_1)\\le dim_kHom(A_i,X)$ .",
"So $dim_kHom(A_i,X)=\\infty $ that's contradicted with the hom-finite property.",
"The following lemma can be proved similarly by the way in [1, chapter 6].",
"Lemma A.4 $\\exists m \\in N$ , for every chain $X_1\\stackrel{f_1}{\\rightarrow }X_2\\stackrel{f_2}{\\rightarrow }X_3\\stackrel{f_3}{\\rightarrow }\\dots \\stackrel{f_m}{\\rightarrow }X_{m+1}$ with $X_i\\in \\lbrace A_1, A_2, \\dots , A_n\\rbrace $ , if $f_j$ is not an isomorphism for every $j=1, 2, \\dots , m+1$ , then $f_mf_{m-1}\\dots f_1=0$ .",
"Lemma A.5 Suppose$X\\in \\mathcal {A}$ .",
"The following are equivalent.",
"$\\left( 1 \\right)$ $X$ is a projective object.",
"$\\left( 2 \\right)$ if $f: Y\\rightarrow X$ is a right minimal epic morphism, then $f$ is an isomorphism.",
"Proof.",
"$\\left( 1 \\right) \\Rightarrow \\left( 2\\right)$ : clear.",
"$\\left( 2 \\right) \\Rightarrow \\left( 1 \\right)$ : Suppose $X$ has the property in $\\left( 2 \\right)$ .",
"And $f: Y\\rightarrow X$ is an epic morphism.",
"Then $f = (f_1, f_2)$ where $f_1 \\in \\operatorname{Hom}(Y_1,X), f_2 \\in \\operatorname{Hom}(Y_2, X)$ , $Y = Y_1 \\bigoplus Y_2$ such that $f_1$ is right minimal and $f_2 = 0$ .",
"So $f_1$ is an isomorphism.",
"$f$ is a split epic morphism.",
"So $X$ is a projective object.",
"Lemma A.6 $\\mathcal {A}$ has enough projective objects Proof.",
"Suppose $X \\in \\mathcal {A}$ such that $X$ has no projective cover and $X$ is an indecomposable object.",
"So there exists a right minimal epic morphism $f_1: Y_1 \\rightarrow X$ such that $f_1$ is not an isomorphism by the above lemma.",
"So there exists $Y_1= Q_1 \\bigoplus X_1$ such that $Q_1$ is a projective object, $X_1$ has no projective direct summand, $X_1 \\ne 0$ , and $f_1 = (g_1,h_1)$ where $g_1 \\in \\operatorname{Hom}(Q_1, X), h_1 \\in \\operatorname{Radical}\\operatorname{Hom}(X_1, X)$ .",
"By the way above, we consider the indecomposable direct summand of $X_1$ .",
"Then there exists an epic morphism $f_2: Y_2 \\rightarrow X_1$ such that $f_2 \\in \\operatorname{Radical} \\operatorname{Hom}(Y_2, X_1).",
"$ So there exists $Y_2 = Q_2 \\bigoplus X_2$ such that $Q_2$ is a projective object, $X_2$ has no projective direct summand, $X_2 \\ne 0$ since $X$ has no projective cover, and$f_2 = (g_2, h_2)$ where $g_2 \\in \\operatorname{Hom}(Q_2, X_1), h_2 \\in \\operatorname{Radical} \\operatorname{Hom}(X_2,X_1)$ .",
"By induction, for $k > 0$ , we get $Y_k = Q_k \\bigoplus X_k$ such that $Q_k$ is a projective object, $X_k$ has no projective direct summand, $X_k \\ne 0$ since $X$ has no projective cover, and$f_k = (g_k, h_k)$ where $g_k \\in \\operatorname{Hom}(Q_k,X_{k - 1}), h_k \\in \\operatorname{Radical} \\operatorname{Hom}(X_k, X_{k - 1})$ .",
"We have the following diagram to explain the operation: ${\\dots &X_2 ^{h_2}[rd]\\\\&&X_1 [rd]^{h_1}\\\\&Q_2 [ru]^{g_2} & & X\\\\&&Q_1 [ru]^{g_1}}$ So there exists an epic morphism $(h_m \\dots h_1, \\phi _m) : X_m\\bigoplus (Q_1 \\bigoplus \\dots \\bigoplus Q_m) \\rightarrow X$ .",
"Since $X$ has no projective cover, $h_m \\dots h_1 \\ne 0$ .",
"That is contradicted with the property of $m$ .",
"Thus the above lemma tells us the abelian category has a projective generator.",
"So by the following well known lemma.",
"The theorem is proved.",
"Lemma A.7 If an abelian categoryis a hom-finite k category with a projective generator, then it is equivalent to the left finitely generated module category of the opposite endomorphism ring of the projective generator.",
"Acknowledgement.",
"This article is part of the author's Ph.D. thesis under the supervision of Pu Zhang.",
"The author is deeply grateful to him for his guidance and encouragement.",
"The author also thanks Professor Ringel for providing him the references [11], [12], [13], [14] and his excellent lectures in Shanghai Jiao Tong University in 2011.",
"The author also deeply thanks Baolin Xiong for his helpful discussions and constant encouragement.",
"Fan Kong, Department of Mathematics, Shanghai Jiao Tong University, 200240 Shanghai, People's Republic of China.",
"Email: [email protected]"
]
] | 1204.0967 |
[
[
"Properties of canonical determinants and a test of fugacity expansion\n for finite density lattice QCD with Wilson fermions"
],
[
"Abstract We analyze canonical determinants, i.e., grand canonical determinants projected to a fixed net quark number.",
"The canonical determinants are the coefficients in a fugacity expansion of the grand canonical determinant and we evaluate them as the Fourier moments of the grand canonical determinant with respect to imaginary chemical potential, using a dimensional reduction technique.",
"The analysis is done for two mass-degenerate flavors of Wilson fermions at several temperatures below and above the confinement/deconfinement crossover.",
"We discuss various properties of the canonical determinants and analyse the convergence of the fugacity series for different temperatures."
],
[
"Introduction",
"QCD with finite density is an important topic of both, experimental and theoretical studies.",
"In particular one would like to understand the various transitions to different states of matter that are conjectured for the QCD phase diagram.",
"Exploring these phase transitions is clearly a non-perturbative problem and non-perturbative methods must be applied for such studies.",
"In principle the lattice QCD formulation provides a suitable non-perturbative framework, and as long as the chemical potential is zero this approach may be used to obtain reliable quantitative results.",
"However, if one works at finite density a severe technical problem emerges: The fermion determinant is complex at non-zero chemical potential $\\mu $ and cannot be used as a probability weight in a Monte Carlo calculation.",
"Various reweighting and expansion techniques have been explored in recent years and the reviews at the annual Lattice conference provide a glimpse at the corresponding developments [1].",
"An interesting option is to work not in the grand canonical approach with a chemical potential $\\mu $ , but instead to use canonical partition sums with fixed net quark number $q$ .",
"Several studies of this alternative perspective can be found in the literature [2] – [10].",
"So far the focus was mainly either on reweighting techniques or full canonical simulations at a fixed quark number.",
"Here we explore the perspectives of a fugacity expansion of the grand canonical ensemble.",
"The expansion coefficients are fermion determinants projected to a fixed net quark number, so-called canonical determinants.",
"The expansion is a Laurent series in the fugacity parameter $e^{\\mu \\beta }$ ($\\beta $ is the inverse temperature) which has properties different from the more conventional Taylor expansion in $\\mu \\beta $ .",
"However, the possible application to lattice QCD at finite density is the same for both expansions: One computes the expansion coefficients in simulations at $\\mu = 0$ and uses the series to explore QCD at $\\mu > 0$ .",
"In this article we discuss the evaluation and the properties of canonical determinants.",
"They are computed as Fourier moments of the fermion determinant at imaginary chemical potential $\\mu \\beta = i \\varphi $ , which we evaluate efficiently using a recently proposed dimensional reduction formula [8].",
"We study the distribution properties of the canonical determinants and analyze the convergence of the fugacity expansion for various temperatures in the confined and deconfined phases.",
"This analysis is done for a fixed numerical effort, i.e., a fixed number of 256 sampling points of $\\varphi $ in the interval $(-\\pi ,\\pi ]$ used for the evaluation of the Fourier transformation to obtain the canonical determinants."
],
[
"Canonical determinants and their evaluation",
"The starting point for our study of canonical determinants and the fugacity expansion is Wilson's lattice Dirac operator $D(\\mu )$ with chemical potential $\\mu $ on lattices of size $N_s^3 \\times N_t$ .",
"The lattice spacing will be denoted by $a$ , such that $\\beta = a N_t$ .",
"For two mass-degenerate flavors the corresponding grand canonical partition function is given by $Z(\\mu ) \\; = \\; \\int {\\cal D}[U] \\, e^{-S_G[U]} \\, \\det [D(\\mu )]^2 \\; ,$ where $S_G[U]$ is the action for the gauge fields $U$ , ${\\cal D}[U]$ the path integral measure and $\\det [D(\\mu )]$ the fermion determinant for chemical potential $\\mu $ , which in the following will be referred to as grand canonical determinant.",
"The grand canonical determinant can be expanded in a finite fugacity series, $\\det [D(\\mu )] \\; = \\; \\sum _q e^{\\mu \\beta q} \\, D^{(q)} \\; ,$ where the sum runs over integer valued quark numbers $q \\in [-6N_s^3,+6N_s^3]$ .",
"The expansion coefficients $D^{(q)}$ are the canonical determinants and may be obtained using Fourier transformation with respect to an imaginary chemical potential, $D^{(q)} \\; = \\; \\frac{1}{2\\pi } \\int _{-\\pi }^\\pi \\!\\!\\!\\!",
"d \\varphi \\;e^{-i q \\varphi }\\, \\det [D(\\mu \\beta = i\\varphi )] \\;\\; .$ While for real chemical potential $\\mu $ the grand canonical determinant is complex, for imaginary chemical potential the determinant $\\det [D(\\mu \\beta = i\\varphi )]$ is real.",
"One may explore the generalized $\\gamma _5$ -hermiticity relation for Wilson fermions, $\\gamma _5 D(\\mu ) \\gamma _5 = D(-\\mu )^\\dagger $ , to establish the relation $D^{(-q)} \\, = \\, ( D^{(q)} )^*$ between canonical determinants with positive and negative quark numbers $q$ .",
"For vanishing quark number $q = 0$ the canonical determinant is real, i.e., $D^{(0)} \\in {R}$ .",
"Table: Parameters of our numerical simulation.In principle Eq.",
"(REF ) is a very elegant expression for the canonical determinants $D^{(q)}$ .",
"However, the Fourier integral has to be evaluated numerically and the evaluation of the grand canonical determinant $\\det [D(\\mu \\beta = i\\varphi )]$ for the necessary $\\varphi $ -values is computationally rather costly.",
"To alleviate the problem we use a domain decomposition technique [8] where the grand canonical fermion determinant may be rewritten exactly in a dimensionally reduced form, $\\det [D(\\mu )] = A_0 \\det \\big [ {1} - H_0 - e^{\\mu \\beta } H - e^{-\\mu \\beta } H^{\\, \\dagger } \\big ] \\;,$ which holds both for real and imaginary chemical potential.",
"Here $A_0$ is a real factor which depends only on the background gauge field configuration but is independent of the chemical potential $\\mu $ .",
"$H_0 = H_0^\\dagger $ , and $H$ are matrices that are built from propagators on sub-domains of the lattice and live on only a single time slice (see [8] for details).",
"Thus the determinant in (REF ) is dimensionally reduced, i.e., the determinant is taken over a matrix with $12 N_s^3$ rows and columns, where the factor 12 comes from the color and Dirac indices.",
"The terms $H_0$ and $H$ can be completely stored in memory and are then used many times for the evaluation of $\\det [D(\\mu \\beta = i\\varphi )]$ at the necessary values of $\\varphi $ .",
"Due to the reduced size of the matrix where the determinant has to be evaluated, we gain a factor of ${\\cal O} (N_t^3)$ , which here, at $N_t = 4$ , corresponds to a speedup by a factor of 64.",
"The Fourier integral (REF ) is evaluated numerically with standard techniques [11] using 256 values of $\\varphi $ in the interval $(-\\pi ,\\pi ]$ .",
"We remark at this point that also alternative dimensional reduction formulas with different properties were proposed [2], [12], [13], [14].",
"Our numerical tests are done on dynamical configurations with two flavors of Wilson fermions at $\\mu = 0$ generated with the MILC code [15].",
"In this exploratory study of the canonical determinants and the fugacity expansion we are here limited to a single lattice volume, $8^3 \\times 4$ .",
"The parameters (inverse gauge coupling $\\beta _{gauge}$ and hopping parameter $\\kappa $ ) and the values for the corresponding lattice spacing $a$ and the temperature $T$ are taken from [5] and listed in Table REF .",
"Due to the small lattice the pion mass has to be kept rather large – it is close to 950 MeV for all our ensembles.",
"For each set of parameters we use a statistics of 200 configurations.",
"All errors we show are statistical errors determined with the jackknife method.",
"In addition to the parameters of the simulation, in Table REF we also list the cutoff parameter $q_{cut}$ used in the fugacity expansion which we discuss in more detail in Section IV.",
"In the range of temperatures we consider, the lowest three values of $T$ are in the confined phase, the crossover into the deconfined phase is at $T \\sim 165$ MeV for our simulation parameters, and the two largest temperatures are in the deconfined phase.",
"As a first result we now have a look at the average $\\langle \\log _{10} \\det [D(\\mu \\beta = i\\varphi )] \\rangle $ of the logarithm of the grand canonical determinant as a function of the imaginary chemical potential parameter $\\varphi $ .",
"This is interesting because the Fourier moments with respect to $\\varphi $ are the canonical determinants we want to study and the $\\varphi $ -dependence of $ \\det [D(\\mu \\beta = i\\varphi )]$ already provides a first insight into their properties.",
"$\\langle \\, .. \\,\\rangle $ is the average over gauge configurations generated for two mass degenerate flavors of Wilson fermions at $\\mu = 0$ as detailed above.",
"We study the $\\varphi $ -dependence of $\\langle \\log _{10}\\det [D(\\mu \\beta = i\\varphi )] \\rangle $ using all 256 values of $\\varphi $ in the interval $(-\\pi ,\\pi ]$ .",
"The integrands $\\det [D(\\mu \\beta = i\\varphi )]$ are computed with the dimensional reduction formula (REF ) as outlined.",
"We show the corresponding results for all our temperature values in Fig.",
"REF .",
"The figure clearly demonstrates that in the confined phase the grand canonical determinant shows only a weak dependence on $\\varphi $ , while above the crossover a pronounced variation with $\\varphi $ develops.",
"This behavior has a simple and well known interpretation: An imaginary chemical potential $\\mu \\beta = i \\varphi $ is equivalent to an additional temporal boundary condition for the fermion fields, $\\psi (\\vec{x},N_t+1) = - e^{i\\varphi } \\psi (\\vec{x},1)$ , where the minus sign is the usual anti-periodic temporal boundary condition for fermions.",
"While in the confined phase the correlations are short ranged and do not give rise to a dependence on the boundary angle $\\varphi $ , in the deconfined phase correlations over distances larger than the temporal extent of the lattice generate the non-trivial response to changing $\\varphi $ .",
"The canonical determinants $D^{(q)}$ are the Fourier moments of the $\\varphi $ -dependence and the first analysis in Fig.",
"REF already demonstrates that in the deconfined phase the higher modes, i.e., $D^{(q)}$ with larger values of $|q|$ will play a more prominent role.",
"Figure: Average of the logarithm of the grand canonicalfermion determinant as a function of the boundary phaseϕ\\varphi .",
"We compare the results for different temperatures from 144 MeV (bottom curve) to 211 MeV (top).",
"Figure: Scatter plot of the canonical determinants D (q) /det[D(μ=0)]D^{(q)} / \\det [D(\\mu = 0)] in thecomplex plane at T=144T = 144 MeV (top row of plots), T=164T = 164 MeV (middle) and T=211T = 211 MeV (bottom) for q=0,1,2,3q = 0,1,2,3.",
"The canonical determinants arenormalized by the grand canonical determinant at μ=0\\mu = 0.",
"Note that we use differentscales for the different temperatures."
],
[
"Properties of the canonical determinants",
"Having discussed the canonical determinants $D^{(q)}$ and the setting of our calculation we now come to analyzing properties of the $D^{(q)}$ .",
"In Fig.",
"REF we show scatter plots in the complex plane for $D^{(q)} / \\det [D(\\mu = 0)]$ , i.e., the canonical determinants normalized by the grand canonical determinant at $\\mu = 0$ .",
"This normalization is such that when one sums over $q$ all data points $D^{(q)} / \\det [D(\\mu = 0)]$ in the complex plane for a single configuration the result is 1.",
"In the plot we show results for $q = 0,1,2,3$ and compare $T = 144$ MeV (top row of plots in Fig.",
"REF ), $T = 164$ MeV (middle) and $T = 211$ MeV (bottom).",
"Note that different axis scales are used.",
"As already discussed, for $q=0$ the canonical determinants $D^{(q=0)}$ must be real, a fact that is obvious in the figures.",
"The corresponding values scatter on the real axis for all temperatures.",
"For $q > 0$ the points spread out also in the imaginary direction.",
"In the confined phase ($T = 144$ MeV) the points for the higher Fourier modes $D^{(q)}$ then quickly approach the origin of the complex plane when increasing $q$ .",
"The situation is different in the deconfined phase ($T = 211$ MeV), where we observe that for increasing $q$ the values of $D^{(q)}$ do not move toward the origin as quickly as for $T = 144$ MeV.",
"For $T = 164$ MeV (middle row of plots in Fig.",
"REF ) one finds an intermediate behavior.",
"The relative size of the canonical determinants $D^{(q)}$ at the different temperatures reflects the observation we have already made in the discussion of Fig.",
"REF : For temperatures above the crossover transition higher Fourier components $D^{(q)}$ are necessary to resolve the strong dependence of the grand canonical determinant $\\det [D(\\mu \\beta = i\\varphi )]$ on the boundary phase $\\varphi $ .",
"We conclude the discussion of Fig.",
"REF with noting, that the scatter plots of $D^{(q)}$ for the smaller values of $q$ show a curious pattern, in particular the $q = 1$ data for $T = 144$ and 164 MeV: They arrange in two oblong structures that seem to have a preferred angle relative to the real axis.",
"A similar pattern at temperatures near the crossover was observed also in canonical determinants for staggered fermions [16].",
"At the moment we do not understand the origin of this pattern, but it might be related to the properties of the $D^{(q)}$ under center transformations which rotate them according to $D^{(q)} \\, \\rightarrow \\, z^{q\\!\\!\\mod {\\!",
"}3} D^{(q)}$ , where $z \\in \\lbrace 1, e^{i2\\pi /3}, e^{-i2\\pi /3}\\rbrace $ is the center element multiplied to all temporal gauge links in a fixed time slice.",
"For pure gauge theory the spontaneous breaking of center symmetry is intimately related to the deconfinement transition, and it could be that the pattern we observe is a remnant of this mechanism.",
"This speculation is further supported by the observation, that the phase of the Polyakov loop (the order parameter for confinement in pure gauge theory) is strongly correlated with the phases of the $D^{(q)}$ [16].",
"Also in an analysis of the canonical determinants of the quenched case [9], [10] it was demonstrated that the $D^{(q)}$ in the broken phase very cleanly map the center orientation of the underlying center sector.",
"Figure: Distribution of 〈|D (q) |/D (0) 〉\\langle | D^{(q)} | /D^{(0)} \\rangle as a function of the quark number qq.",
"We compare three temperatures: T=144T =144 MeV (top plot), T=164T = 164 MeV (middle) and T=211T = 211 MeV (bottom).Figure: Expectation value of the phase of the canonical determinants,〈e i2θ (q) 〉=〈D (q) /D (-q) 〉\\langle e^{i2\\theta ^{(q)}}\\rangle = \\langle D^{(q)}/D^{(-q)}\\rangle as a function of qq.",
"We compare the results for different temperatures.",
"The change of the relative weight of the $D^{(q)}$ with temperature is manifest also in Fig.",
"REF , where we study the size distribution $\\langle |D^{(q)} | / D^{(0)} \\rangle $ of the canonical determinants normalized to the trivial $q = 0$ sector as a function of $q$ and again compare results for $T = 144$ MeV (top), $T = 164$ MeV (middle) and $T = 211$ MeV (bottom).",
"The plots show that the distribution roughly follows a Gaussian centered around the dominant $q = 0$ sector.",
"While in the confined phase the distribution is rather narrow, in the deconfined phase it widens and reflects the fact that at higher temperatures also sectors with larger quark numbers become accessible.",
"When one turns on a chemical potential $\\mu > 0$ the canonical determinants are multiplied with powers of the fugacity factor, i.e., $D^{(q)} \\rightarrow e^{\\mu \\beta q} D^{(q)}$ , which has the effect of shifting the center of the distribution towards larger quark numbers.",
"Finally, in Fig.",
"REF we have a look at the phase $\\theta ^{(q)}$ of the canonical determinants $D^{(q)}$ .",
"More precisely we look at $\\langle e^{i 2 \\theta ^{(q)}} \\rangle = \\langle D^{(q)}/ D^{(-q)} \\rangle $ (remember that $D^{(-q)} = {D^{(q)}}^\\star $ ).",
"In Fig.",
"REF we show $\\langle D^{(q)}/ D^{(-q)} \\rangle $ as a function of $q$ for different values of the temperature.",
"Below the crossover temperature the results for $\\langle D^{(q)}/ D^{(-q)} \\rangle $ drop to 0 rather quickly with increasing $q$ , i.e., the phases of the $D^{(q)}$ for $q$ above roughly $q = 5$ fluctuate strongly and average to a very small number.",
"For the largest temperatures the decrease of $\\langle D^{(q)}/ D^{(-q)} \\rangle $ with $q$ is slower and the phases have a sizable expectation value up to $q \\sim 25$ (we stress that these are statements specific to the parameters of our calculation, in particular the volume used).",
"The decrease of the $\\langle D^{(q)}/ D^{(-q)} \\rangle $ with increasing $q$ also sheds light on how the fermion sign problem may be viewed in the fugacity expansion: When increasing $\\mu $ the powers $e^{\\mu \\beta q}$ of the fugacity factors put a larger weight on the canonical determinants $D^{(q)}$ with higher quark numbers, which in turn are the ones that are suppressed due to the fluctuations of their phase.",
"The fact that for higher temperatures the sign problem is milder is clearly visible from the slower decrease of $\\langle D^{(q)}/ D^{(-q)} \\rangle $ for larger temperatures.",
"Figure: Relative error Δ\\Delta (see the text for its definition) of thefugacity expansion as a function of μβ\\mu \\beta at different temperatures.",
"We showthe error of the best configuration, the worst configuration, the average over allconfiguration as well as the median."
],
[
"Convergence properties of the fugacity series",
"After the study of the canonical determinants $D^{(q)}$ , we now can start to analyze the fugacity series (REF ).",
"In a numerical study it is necessary to truncate the series and we denote the truncated fugacity expansion by $S(\\mu )_{q_{cut}} \\; = \\; \\sum _{q = -q_{cut}}^{q_{cut}} \\, e^{\\mu \\beta q} \\, D^{(q)} \\; ,$ where $q_{cut}$ denotes the highest (anti-) quark sector we take into account in the truncated series.",
"The central question we want to study in this section is how well the truncated series $S(\\mu )_{q_{cut}}$ approximates the full grand canonical determinant $\\det [D(\\mu )]$ and how the quality of the approximation depends on the parameters $\\mu $ and $T$ .",
"This is a question which we address both for individual configurations and for the gauge average.",
"We stress at this point, that for a finite volume and non-zero temperature, the representation of $\\det [D(\\mu )]$ by $S(\\mu )_{q_{cut}}$ would be exact with $q_{cut} = 6N_s^3$ if all coefficients $D^{(q)}$ were known with arbitrary precision.",
"Thus as long as the lattice is finite such that no singularities can emerge, an insufficient representation of the grand canonical determinant is only due to limited accuracy in the numerical evaluation of the Fourier integrals (REF ).",
"We assess the quality of the approximation by considering the relative error $\\Delta \\; = \\; \\left| \\frac{ S(\\mu )_{q_{cut}} - \\det [D(\\mu )] }{ S(\\mu )_{q_{cut}}\\, } \\right| \\; \\; .$ For the study of $\\Delta $ we also evaluated the grand canonical determinant $\\det [D(\\mu )]$ for a few values of real chemical potential, again applying the dimensional reduction formula (REF ).",
"These values are then used for the assessment of the fugacity expansion using $\\Delta $ .",
"Inspecting the fugacity series (REF ) or (REF ) one observes that for increasing $q$ there is competition of two terms: The powers of the fugacity, i.e., the factors $e^{\\mu \\beta q}$ , are terms that grow exponentially with $q$ and their rate of growth is determined by the product $\\mu \\beta $ .",
"The exponential growth with $q$ is compensated by the quick decrease of the canonical determinants $D^{(q)}$ .",
"This decrease is roughly Gaussian as we demonstrated in Fig.",
"REF and thus dominates the increasing function $e^{\\mu \\beta q}$ .",
"However, this mechanism can be spoiled by numerical instabilities: If the higher coefficients $D^{(q)}$ are not known with perfect accuracy (as will always be the case when they are evaluated via numerical Fourier transformation), their fluctuations will be amplified by the factors $e^{\\mu \\beta q}$ .",
"This effect is manifest in the behavior of $S(\\mu )_{q_{cut}}$ when $q_{cut}$ is varied: First $S(\\mu )_{q_{cut}}$ quickly saturates as a function of $q_{cut}$ , but for too large values, e.g., already $q_{cut} \\sim 30$ for some of our parameter values, the series $S(\\mu )_{q_{cut}}$ starts to diverge.",
"Fortunately the $D^{(q)}$ decrease very rapidly: Fig.",
"REF shows that also for our highest temperature ensemble all $D^{(q)}$ with $|q| > 20$ essentially vanish, and the series $S(\\mu )_{q_{cut}}$ can be truncated at small $q_{cut}$ .",
"The optimal values for $q_{cut}$ were determined using the relative error $\\Delta $ and we list them in Table REF .",
"In Fig.",
"REF we show how the relative error $\\Delta $ of the fugacity expansion depends on the parameters $\\mu \\beta $ and $T$ .",
"For all 200 configurations in each of our ensembles we compute the canonical determinants $D^{(q)}$ , as well as the canonical determinant $\\det [D(\\mu )]$ for several real values of $\\mu $ .",
"For these values of $\\mu $ we can evaluate $\\Delta $ .",
"Fig.",
"REF shows $\\Delta $ as a function of the dimensionless combination $\\mu \\beta $ , and we compare the results for all available temperatures.",
"For each ensemble we show $\\Delta $ for the configuration where $\\Delta $ is largest (\"worst\"), for the configuration where $\\Delta $ is smallest (\"best\"), for the median of the 200 configurations, as well as the average of $\\Delta $ over all configurations.",
"From Fig.",
"REF it is obvious that for the lower values of $T$ the fugacity expansion has better convergence properties: At $T = 144$ MeV all four categories (best, worst, median and average) have a relative error $\\Delta $ smaller than 1 % for values of the chemical potential up to $\\mu \\beta = 0.8$ and the average error over all configurations remains below 1% even up to $\\mu \\beta = 1.1$ .",
"For $T = 211$ MeV an error smaller than 1 % can be maintained only up to $\\mu \\beta = 0.4$ .",
"The reason is that here the distribution of the $D^{(q)}$ is much wider (compare Fig.",
"REF ), and the $D^{(q)}$ for larger values of $|q|$ contribute significantly in the fugacity sum already at not too large $\\mu \\beta $ .",
"Since these higher Fourier modes are only known with less relative accuracy, the fugacity expansion at the highest temperatures breaks down already at smaller chemical potentials.",
"We stress that our convergence analysis is specific for the setting we use here, i.e., lattice size $8^3 \\times 4$ , parameters as listed in Table REF and 256 values of $\\varphi $ in the evaluation of the Fourier integrals (REF ).",
"In particular increasing the number of sampling points for $\\varphi $ in the Fourier integral will allow for a higher precision of the $D^{(q)}$ which leads to better convergence properties of the fugacity sum, such that higher values of $\\mu \\beta $ can be reached.",
"An interesting question is how the number of $D^{(q)}$ that need to be evaluated depends on the volume of the box.",
"This number is roughly proportional to the width of the distribution of the $D^{(q)}$ as displayed in Fig.",
"REF .",
"In the next section we will show that this width is related to the quark number susceptibility $\\chi _q$ .",
"This is an extensive quantity, i.e., it grows with the spatial volume $V = (aN_s)^3$ .",
"We thus conclude that the number of $D^{(q)}$ that contribute significantly to the fugacity sum is proportional to the spatial volume $V$ .",
"In turn this means that also $q_{cut}$ needs to be increased linearly in $V$ ."
],
[
"Quark number density and quark number susceptibility",
"Let us finally discuss an exploratory calculation of the quark number density and the quark number susceptibility based on canonical determinants and the fugacity expansion.",
"Combining the general expression for the grand canonical partition sum $Z(\\mu )$ from Eq.",
"(REF ) with the fugacity expansion (REF ) we find $Z(\\mu ) & = & \\int \\!\\!",
"{\\cal D}[U] \\, e^{-S_G[U]} \\!\\left(\\sum _q e^{q \\mu \\beta } \\, D^{(q)} \\right)^2\\\\& = & \\int \\!\\!",
"{\\cal D}[U] \\, e^{-S_G[U]} \\, \\det [D(0)]^2\\left(\\!\\sum _q \\,\\frac{e^{q \\mu \\beta } \\, D^{(q)}}{\\det [D(0)]} \\right)^{\\!\\!2} \\!,\\nonumber $ where in the second step we inserted $1 = \\det [D(0)]^2/\\det [D(0)]^2$ to write the whole expression as a vacuum expectation value (up to normalization) in the grand canonical ensemble at $\\mu = 0$ where we perform our simulation.",
"The definitions of the quark number density $n_q$ and the quark number susceptibility $\\chi _q$ are $\\frac{n_q}{T^3} \\, = \\, \\frac{\\beta ^3}{V} \\frac{\\partial \\ln Z(\\mu )}{\\partial \\mu \\beta }\\; \\; , \\; \\; \\; \\frac{\\chi _q}{T^2} \\, = \\, \\frac{\\beta ^3}{V} \\frac{\\partial ^2 \\ln Z(\\mu )}{\\partial (\\mu \\beta )^2}\\; .$ Both observables are intensive quantities after the normalization with the 3-volume $V$ and are made dimensionless using suitable powers of $T$ .",
"For the necessary first and second derivatives one finds $\\frac{\\partial \\ln Z(\\mu )}{\\partial \\mu \\beta } &\\!\\!=\\!\\!& \\frac{2}{Z(\\mu )}\\!",
"\\int \\!\\!",
"{\\cal D}[U] \\,e^{-S_G[U]}\\det [D(0)]^2 \\; M_0 M_1 \\; ,\\\\\\frac{\\partial ^2\\!\\ln \\!Z(\\mu )}{\\partial (\\mu \\beta )^2} &\\!\\!=\\!\\!& \\frac{2}{Z(\\mu )} \\!\\int \\!\\!",
"{\\cal D}[U] \\, e^{-S_G[U]}\\det [D(0)]^2\\!\\Big [\\!M_2 M_0\\!+\\!M_1^2\\!",
"\\Big ] \\nonumber \\\\& & \\hspace{14.22636pt}- \\, \\left( \\frac{\\partial \\ln Z(\\mu )}{\\partial \\mu \\beta } \\right)^2 ,\\nonumber $ where we introduced the moments $M_n, n = 0,1,2$ of the fugacity series as $M_n \\; = \\; \\sum _{q= -q_{cut}}^{q_{cut}} \\, q^n \\, e^{q\\mu \\beta } \\, \\frac{D^{(q)}}{\\det [D(0)]} \\; .$ Finally we express the normalization factor $1/Z(\\mu )$ with the moment $M_0$ and obtain (use $\\beta ^3 / V = (N_t/N_s)^3$ ) $n_q &\\!",
"= \\!",
"& 2\\!",
"\\left(\\frac{N_t}{N_s}\\right)^{\\!3} \\frac{\\langle \\, M_0 M_1 \\, \\rangle }{\\langle \\, (M_0)^2 \\,\\rangle } \\; ,\\\\\\chi _q &\\!",
"= \\!& 2\\!",
"\\left(\\frac{N_t}{N_s}\\right)^{\\!3} \\left[ \\frac{\\langle \\, M_0 M_2 + (M_1)^2 \\, \\rangle }{\\langle \\, (M_0)^2 \\, \\rangle } \\, - \\, 2 \\left(\\frac{\\langle \\, M_0 M_1 \\, \\rangle }{\\langle \\, (M_0)^2 \\,\\rangle } \\right)^{\\!2\\,} \\right]\\!",
".\\nonumber $ In the final result (REF ) both observables are expressed in terms of vacuum expectation values of moments of the fugacity series.",
"These vacuum expectation values $\\langle .. \\rangle $ are computed with two flavors of Wilson fermions at $\\mu = 0$ .",
"In Fig.",
"REF we show our results (symbols) for $n_q/T^3$ as function of the dimensionless combination $\\mu \\beta $ for three different temperatures.",
"We compare the results to the outcome of the Taylor expansion presented in [17], [18].",
"More precisely we used the terms up to fourth order from [18] and interpolated the Taylor coefficients to the temperatures used in our study.",
"However, this comparison should be viewed with caution: The calculation in [18] is done on considerably larger lattices, has a statistics that is by a factor 20 larger than the statistics available in our exploratory study, and also is based on the staggered formulation.",
"Nevertheless we find reasonable agreement for our results at these temperatures and range of $\\mu \\beta $ .",
"Figure: Quark number density as a function of μβ\\mu \\beta .",
"We compare our results at three different temperatures(symbols) to the results from Taylor expansion (curves).",
"Table: Comparison of results for χ q /T 2 \\chi _q/T^2 at μ=0\\mu = 0.We also attempt a comparison of our results for the quark number susceptibility $\\chi _q$ at $\\mu = 0$ to the results from the Taylor expansion.",
"The corresponding numbers are listed in Table REF (again interpolated to our values of $T$ ).",
"We find that up to temperatures near the crossover the agreement of the two approaches is reasonable.",
"Given the fact that the calculation [18] and our exploratory study differ considerable in statistics and volume, this agreement is satisfactory.",
"Above the crossover we see, however, a rather strong discrepancy.",
"We suspect two main reasons for this discrepancy: 1) The rather small volumes that are available for our calculation, and 2) the fact that for the larger values of $T$ higher Fourier modes contribute significantly (the factor $q^2$ in the second moment $M_2$ of Eq.",
"(REF ) enhances them further) which in the current setting of 256 sampling points for the numerical evaluation of the Fourier integral are not sufficiently accurate."
],
[
"Summary and discussion",
"In this exploratory study we analyzed the canonical determinants of 2-flavor lattice QCD with Wilson fermions.",
"The canonical determinants are the coefficients in the fugacity expansion of the fermion determinant and may be calculated as the Fourier modes with respect to imaginary chemical potential.",
"We speed up this evaluation by using a dimensional reduction formula for the grand canonical determinant.",
"We illustrate that a sizable dependence of the fermion determinant on the imaginary chemical potential sets in only at temperatures near the crossover temperature, which already shows that at the crossover an enhancement of the higher quark numbers can be expected.",
"Studying the size distribution of the canonical determinants we find that their distribution is roughly Gaussian in the net quark number $q$ with a width that starts to increase at the crossover.",
"The average phases of the canonical determinants drop quickly with $q$ , but the drop is slowed above the crossover temperature.",
"This analysis sheds light on the complex phase problem from the point of view of the fugacity series.",
"We continue with a systematical analysis of the convergence properties of the fugacity expansion by comparing the truncated fugacity series to an exact evaluation of the grand canonical determinant at several values of the chemical potential.",
"It is shown that for lower temperatures we can obtain very good accuracy up to $\\mu \\beta \\sim 0.8$ , while above the crossover we only reach $\\mu \\beta \\sim 0.4$ .",
"We stress that these results are specific for the numerical effort we invest here, in particular 256 sampling points for the evaluation of the Fourier moments.",
"On a finite lattice the fugacity expansion is a finite Laurent series, and in principle it is possible to compute all coefficients such that the series representation of the grand canonical determinant becomes exact.",
"Finally we use the fugacity series to explore the evaluation of the quark number density and the quark number susceptibility through first and second moments of the truncated fugacity series.",
"This is an interesting alternative to a standard calculation where the quark number $q$ is a binomial in the quark fields, $q\\propto \\int d^4 x \\overline{\\psi }(x) \\gamma _4 \\psi (x)$ which after integrating out the fermions is related to the trace of the quark propagator, $q \\propto $ Tr $ \\gamma _4 D^{-1}$ .",
"In this form $q$ is difficult to evaluate numerically using, e.g., stochastic estimators.",
"Once the canonical determinants $D^{(q)}$ are available the expressions in terms of the moments of the fugacity expansion are considerably simpler and we demonstrate that the approach is compatible with the conventional method up to temperatures where we have sufficiently control over the fugacity series.",
"Conceptually the fugacity expansion falls in the same category as the Taylor expansion: A series is used to extrapolate the information from a Monte Carlo simulation at $\\mu = 0$ to finite values of the chemical potential.",
"Thus both, Taylor- and fugacity expansion face the same overlap problem, i.e., the $\\mu = 0$ configurations used for evaluating the expansion coefficients may have rather little overlap with the configurations that dominate physics at finite $\\mu $ .",
"This limits both expansion approaches to small values of $\\mu $ .",
"Concerning other properties, Taylor- and fugacity expansion are different: At finite volume the fugacity series is a finite Laurent series in the fugacity parameter $e^{\\mu \\beta }$ , while the Taylor series is an infinite series even at finite volume.",
"Thus the two series will have different properties and one should explore which of the two approaches provides a better expansion around $\\mu = 0$ .",
"To study this question we currently compare the two series in an effective theory for QCD, where a flux representation free of the complex phase problem allows one to obtain high precision reference data from a Monte Carlo simulation [19].",
"These will be used to study the quality of the approximation from Taylor- and fugacity series.",
"We conclude with stressing that this exploratory work should be considered only as a first step towards a full development of fugacity expansion as a reliable tool in finite density lattice QCD.",
"Certainly further improvement of the numerical techniques and more tests will be necessary to assess the full potential of the approach.",
"Acknowledgments: We thank Christian Lang, Anyi Li and Bernd-Jochen Schaefer for interesting discussions and remarks.",
"We also would like to thank Christian Lang and Ludovit Liptak for help with parts of the computer code.",
"This work has been supported by the Austrian Science Fund FWF: Doctoral School ”Hadrons in Vacuum, Nuclei and Stars”, DK W1203.",
"Christof Gattringer thanks the members of the Institute for Nuclear Theory at the University of Washington, Seattle, where part of this work was done, for their hospitality and the inspiring atmosphere.",
"Christof Gattringer also acknowledges support by the Dr. Heinrich-Jörg foundation of the University of Graz.",
"The simulations were done on the Linux clusters of the computer center at the University of Graz, and we thank their staff for service and support.",
"This work was in part based on the MILC collaboration's public lattice gauge theory code."
]
] | 1204.1020 |
[
[
"Worldline approach to noncommutative field theory"
],
[
"Abstract The study of the heat-trace expansion in noncommutative field theory has shown the existence of Moyal nonlocal Seeley-DeWitt coefficients which are related to the UV/IR mixing and manifest, in some cases, the non-renormalizability of the theory.",
"We show that these models can be studied in a worldline approach implemented in phase space and arrive to a master formula for the $n$-point contribution to the heat-trace expansion.",
"This formulation could be useful in understanding some open problems in this area, as the heat-trace expansion for the noncommutative torus or the introduction of renormalizing terms in the action, as well as for generalizations to other nonlocal operators."
],
[
"Introduction",
"Noncommutative field theory [1], [2] involves two concepts which could be considered as fundamental ingredients of a theory of quantum gravity: nonlocality and a minimal length scale.",
"As a consequence, the theory presents interesting renormalization properties, as the so-called UV/IR mixing [3].",
"Renormalization in quantum field theory can be implemented with heat-kernel techniques [4] since the one-loop counterterms that regularize the high energy divergencies of the effective action can be obtained in terms of the Seeley-DeWitt (SDW) coefficients, that are determined by the asymptotic expansion, for small values of the proper time, of the heat-trace of a relevant operator.",
"Such an operator defines the spectrum of quantum fluctuations and is obtained as the second order correction of the classical action in a background field expansion.",
"In a commutative field theory, this operator of quantum fluctuations is, in general, a (local) differential operator whose SDW coefficients have been extensively studied; in particular the worldline formalism (WF) has proved an efficient technique for the computation of these coefficients [5].",
"On the other hand, in a noncommutative field theory the operator of quantum fluctuations is, in general, a nonlocal operator.",
"Lately it has been shown that the SDW coefficients corresponding to these types of nonlocal operators have peculiar contributions –which are related to the non-planar diagrams of the UV/IR mixing– and are linked to the (non)renormalizability of the theory (see the short review [6].)",
"Let us also mention that in the context of the spectral action principle [7] the SDW coefficients for models defined on noncommutative spaces have an essential role in the determination of the corresponding classical actions.",
"In this article we use WF techniques to obtain a systematic description of the SDW coefficients of nonlocal operators relevant for the quantization of noncommutative self-interacting scalar fields on Moyal Euclidean spacetime.",
"In order to do that we implement the WF in phase space and we derive a master formula (eq.",
"(REF )) that can be applied to different settings.",
"We consider this formula to be a step towards a more systematic understanding of the heat-trace expansion of nonlocal operators as well as a potentially useful approach to some open problems in this topic, as those one encounters for example on the NC torus [8].",
"Further applications to other nonlocal operators could also be considered.",
"Although the main motivation of the present article is to develop a new tool for performing one-loop calculations in noncommutative field theories, let us point out that we also apply a path integral representation of quantum mechanical transition amplitudes in Moyal spaces which, as far as we know, has not been used in previous works on noncommutative quantum mechanics.",
"The path integral expressions presented in [9], [10] are based on the symplectic form that defines noncommutativity on phase space whereas our worldline formulation depends on the particular representation of the noncommutative algebra given by the Moyal product.",
"Other path integral representations of transition amplitudes have been derived in [11], [12] in the coherent states approach to noncommutative quantum mechanics (in [12] coherent states are defined in the Hilbert space consisting on a certain class of Hilbert-Schmidt operators).",
"However, the path integral formulation derived from coherent states is connected instead to the representation of noncommutativity given by the Voros product.",
"There exist applications of path integrals to quantum mechanics in the Moyal plane [13], [14] but in these models the Hamiltonians are quadratic in momenta and coordinates so that, after a Bopp shift, they can be reduced to analogue models in the commutative plane.",
"Let us mention that the application of path integrals to the Aharonov-Bohm problem in the Moyal plane [15] is performed to leading order in the parameter which defines noncommutativity."
],
[
"Effective action in Euclidean Moyal spacetime",
"A field theory on Euclidean Moyal spacetime can be formulated in terms of the nonlocal Moyal product usually defined as $(\\phi \\star \\psi )(x):=e^{i\\,\\partial ^{^\\phi }\\Theta \\partial ^{^\\psi }}\\,\\phi (x)\\psi (x)\\,.$ The conditions under which the exponential in this expression is well-defined are studied in [16].",
"The scalar functions $\\phi $ and $\\psi $ depend on $x\\in \\mathbb {R}^d$ ; $\\partial ^{^\\phi }$ and $\\partial ^{^\\psi }$ denote their gradients, respectively.",
"The expression $\\partial ^{^\\phi }\\Theta \\partial ^{^\\psi }$ represents $\\Theta ^{ab} \\partial ^{^\\phi }_{x^a} \\partial ^{^\\psi }_{x^b}$ , where $\\Theta ^{ab}$ are the components of an antisymmetric matrix $\\Theta $ independent of $x$ .",
"With respect to this $\\star $ - product, the coordinates do not commute: $[x^a,x^b]_{\\star }:=x^a\\star x^b-x^b\\star x^a=2i\\,\\Theta ^{ab}\\,.$ Thus, the matrix $\\Theta $ characterizes the noncommutativity of the base space.",
"Throughout this article we will consider a possibly degenerate matrix $\\Theta $ .",
"Assuming $\\Theta $ of rank $2b$ we split $\\mathbb {R}^d$ in a commutative $\\mathbb {R}^c$ and a noncommutative $\\mathbb {R}^{2b}$ , with $d=c+2b$ , by choosing coordinates $x=(\\tilde{x},\\hat{x})$ with commuting $\\tilde{x}\\in \\mathbb {R}^{c}$ and noncommuting $\\hat{x}\\in \\mathbb {R}^{2b}$ .",
"Consequently, the matrix $\\Theta $ can be written as $\\Theta =\\mathbf {0}_c\\oplus \\Xi \\,,$ where $\\mathbf {0}_c$ is the null matrix in $\\mathbb {R}^c$ and $\\Xi $ is a nondegenerate antisymmetric matrix in $\\mathbb {R}^{2b}$ .",
"In terms of the Fourier transform $\\tilde{\\phi }(p)=\\int \\frac{dx}{(2\\pi )^{d}}e^{-ipx}\\phi (x)$ the Moyal product reads $\\widetilde{\\phi \\star \\psi }(p)=\\int dq\\,e^{-i p\\Theta q}\\,\\tilde{\\phi }(p-q)\\tilde{\\psi }(q)\\,.$ A simple model in noncommutative field theory in Euclidean Moyal spacetime is given by a scalar field with a $\\star $ - cubic self-interaction, described by the Langrangian $\\mathcal {L}=\\frac{1}{2}(\\partial \\phi )^2+\\frac{m^2}{2}\\phi ^2+\\frac{\\lambda }{3!",
"}\\phi ^3_\\star \\,,$ where $\\phi ^3_\\star :=\\phi \\star \\phi \\star \\phi $ .",
"In the ordinary commutative case (i.e.",
"$\\Theta =0$ ) the one-loop effective action $\\Gamma _{C}$ can be represented as $\\Gamma _C=\\frac{1}{2}\\log {\\rm Det}\\left\\lbrace -\\partial ^2+m^2+\\lambda \\,\\phi (x)\\right\\rbrace \\,,$ where the Schrödinger differential operator between brackets, which determines the spectrum of quantum fluctuations, arises from the second functional derivative of the action with respect to the background field $\\phi $ .",
"As we have already mentioned, the regularization of the effective action can be implemented in terms of the heat-trace of this Schrödinger operator.",
"The spectral theory of Schrödinger operators of the type $-\\partial ^2+V(x)$ on $\\mathbb {R}^d$ shows that for regular potentials $V(x)$ the heat-trace admits the following asymptotic expansion in powers of the proper time $\\beta >0$ [17] ${\\rm Tr}\\left(f(x)\\cdot e^{-\\beta \\left\\lbrace -\\partial ^2+V(x)\\right\\rbrace }\\right)\\sim \\frac{1}{(4\\pi \\beta )^{d/2}}\\sum _{n=0}^\\infty a_n\\beta ^n\\,.$ In this expression we have introduced a smearing function $f(x)$ .",
"The coefficients $a_n$ are called Seeley-DeWitt (SDW) coefficients and are given by integrals on $\\mathbb {R}^d$ of products of the smearing function $f(x)$ , the potential $V(x)$ and derivatives thereof (the reader can find some proof of this statement in subsection REF .)",
"In the noncommutative case ($\\Theta \\ne 0$ ) the one-loop effective action $\\Gamma _{NC}$ corresponding to the Lagrangian (REF ) instead results in $\\Gamma _{NC}=\\frac{1}{2}\\log {\\rm Det}\\left\\lbrace -\\partial ^2+m^2+\\frac{\\lambda }{2}\\,L(\\phi )+\\frac{\\lambda }{2}R(\\phi )\\right\\rbrace \\,,$ where $L(\\phi )$ is an operator whose action on a function $\\psi (x)$ is defined as $L(\\phi )\\psi (x):=(\\phi \\star \\psi )(x)$ whereas $R(\\phi )\\psi (x):=(\\psi \\star \\phi )(x)$ .",
"These nonlocal operators represent the left- respectively right-Moyal multiplication by $\\phi $ and can also be expressed as $L(\\phi )\\psi (x)=\\phi (x+i\\Theta \\partial )\\psi (x)\\,,\\\\R(\\phi )\\psi (x)=\\phi (x-i\\Theta \\partial )\\psi (x)\\,,\\nonumber $ where $x\\pm i\\Theta \\partial $ have components $x^a\\pm i\\Theta ^{ab} \\partial _b$ .",
"Thus, in order to regularize the noncommutative effective action $\\Gamma _{NC}$ we must study the heat-trace of the nonlocal operator $-\\partial ^2+m^2+\\frac{\\lambda }{2}\\,\\phi (x+i\\Theta \\partial )+\\frac{\\lambda }{2}\\,\\phi (x-i\\Theta \\partial )\\,.$ Notice that in the case of a quartic self-interaction $\\phi _\\star ^4$ the operator corresponding to the second functional derivative of the action contains the following terms: $L(\\phi _\\star ^2)$ , $R(\\phi _\\star ^2)$ , $L(\\phi )R(\\phi )$ .",
"It has been shown that the heat-trace of an operator of the type $-\\partial ^2+L(l_1(x))+R(r_1(x))+L(l_2(x))R(r_2(x))$ also admits an asymptotic expansion in powers of the proper time $\\beta $ [18] .",
"However, the SDW coefficients are not given, in general, by integrals on $\\mathbb {R}^d$ of local expressions depending on the smearing function and the potential functions $l_i(x),r_i(x)$ .",
"Nevertheless, if the operator (REF ) contains only a left-Moyal product (i.e., $r_1(x)=r_2(x)=0$ ) and the smearing function acts by left-Moyal multiplication, then the SDW coefficients can be obtained from the commutative ones by replacing every commutative pointwise product by the noncommutative Moyal productAs we will see, to avoid ordering ambiguities, one should consider a commutative operator with matrix-valued coefficients.",
"[19], [20].",
"The same holds if the operator contains only a right-Moyal product (i.e., $l_1(x)=l_2(x)=0$ ) and the smearing function acts by right-Moyal multiplication.",
"Therefore, when there is no mixing between left- and right-Moyal multiplications, the SDW coefficients are still integrals of “local products” (but in a Moyal sense) of the smearing function, the potential functions and their derivatives; we will refer to these as “Moyal local” coefficients.",
"In consequence, they are likely to be introduced as counterterms in the Lagrangian.",
"On the contrary, for the general case given by expression (REF ) some SDW coefficients are not even Moyal local, i.e.",
"they cannot be written as integrals of Moyal products of the potential functions and the smearing function.",
"In some field theories, these coefficients manifest the non-renormalizability of the corresponding effective actionLet us mention that in some noncommutative models in emergent gravity a low-energy regime is to be considered and this UV/IR mixing problem is avoided [22].",
"[21].",
"In this article we develop a worldline approach to the computation of the SDW coefficients of operator (REF ).",
"For simplicity, we will consider the case $l_1(x)=r_1(x)=0$ since the mixing term $L(l_2)R(r_2)$ will suffice to show the appearance of the (Moyal) nonlocal coefficients.",
"In order to do that, we write a representation of the heat-kernel in terms of a path integral in phase space; this is done in Section .",
"This path integral is solved in Section , where we compute the corresponding generating functional.",
"Our main result, the master formula for the noncommutative heat-trace, is given in Section , where we also show how it works for some particular settings and we confirm existing results [6].",
"In Section we show a simple application of formula (REF ) to noncommutative field theroy: we consider a scalar field with a quartic self-interaction in Euclidean noncommutative spacetime to show how (Moyal) nonlocal SDW coefficients manifest two different phenomena that could lead to non-renormalizability: the so called UV/IR mixing and the existence of more than one commuting direction in the noncommutative spacetime.",
"Finally, in Section we draw our conclusions."
],
[
"Path Integral in phase space",
"Let us consider an operator which contains a nonlocal potential mixing left- and right-Moyal multiplication $H=-\\partial ^2+L(l)\\,R(r)\\,,$ where $l(x)$ and $r(x)$ are functions of $x\\in \\mathbb {R}^d$ .",
"Notice that, due to the associativity of Moyal product, $L(l)$ and $R(r)$ commute with each other.",
"In accordance with representation (REF ) we write $H=p^2+l(x-\\Theta p)\\,r(x+\\Theta p)\\,,$ where $p=-i\\partial $ .",
"Hamiltonian operator (REF ) has a fixed ordering given by replacing coordinate $x$ with operator $x-\\Theta p$ in $l$ , and with operator $x+\\Theta p$ in $r$ .",
"In order to obtain the path integral representation of the transition amplitude associated to (REF ) it is convenient to rearrange the hamiltonian operator in a different form, for example the Weyl ordered form.",
"A phase space operator $A(x,p)$ is written in Weyl-ordered form when it is arranged in such a way that $A(x,p) = A_S(x,p) + \\Delta A\\equiv A_W(x,p)$ , where $A_S(x,p)$ involves symmetric products of $x$ 's and $p$ 's and $\\Delta A$ includes all terms resulting from eventual commutators between $x$ 's and $p$ 's, necessary to rearrange $A(x,p)$ in its symmetric form; for example $xp =(xp)_S +\\frac{1}{2}[x,p] = (xp)_S +\\frac{i\\hbar }{2}\\equiv (xp)_W$ (for details see e.g.",
"Appendices B and C in [5]).",
"Above, since the phase-space operator $l(x-\\Theta p)\\,r(x+\\Theta p)$ mixes coordinates and momenta it is not, a priori, written in symmetrized form: in other words $l(x-\\Theta p)\\,r(x+\\Theta p) = (l(x-\\Theta p)\\,r(x+\\Theta p))_S +\\Delta V$ .",
"However, one can show by Taylor expanding the functions $l$ and $r$ , that the operator (REF ) only involves symmetric products; in other words products of $x$ 's and $p$ 's in (REF ) can be cast in their symmetrized form (for example $(xp)_S=\\frac{1}{2} (xp+px)$ and $(x^2 p)_S= \\frac{1}{3} (x^2 p +xpx +px^2) =\\frac{1}{2}(x^2 p +p x^2)$ ) without introducing extra terms, i.e.",
"$\\Delta V =0$ .",
"Let us, for example, consider the product of the linear contributions in the Taylor expansions of $l$ and $r$ : it involves the product $x_-^a x_+^b \\equiv (x^a-\\Theta ^{ac}p_c)(x^b+\\Theta ^{bd}p_d)$ whose Weyl-ordering reads $x_-^a x_+^b &=& (x^ax^b)_S-\\Theta ^{ac}\\Theta ^{bd} (p_c p_d)_S -\\Theta ^{ac} p_cx^b+\\Theta ^{bd} x^a p_d\\\\&=&\\nonumber (x_-^a x_+^b)_S -\\frac{1}{2} \\Theta ^{ac} [p_c, x^b]+\\frac{1}{2}\\Theta ^{bd} [x^a,p_d] = (x_-^a x_+^b)_S$ thanks to the antisymmetry of the $\\Theta ^{ab}$ symbol.",
"It is thus easy to convince oneself that the latter antisymmetry, along with the commutativity property $[x_-^a,x_+^b]=0$ and the total symmetry of the coefficients of Taylor expansions of functions $l(x)$ and $r(x)$ , allows to easily prove that all the contributions to the product of Taylor expansions of operators $l(x-\\Theta p)$ and $r(x+\\Theta p)$ are symmetric.",
"Therefore –using the midpoint rule [23], [5]– one can write the following path integral representation for the heat-kernel of operator (REF ) $\\langle x+z\\vert e^{-\\beta H}\\vert x \\rangle = \\int \\mathcal {D}x(t)\\mathcal {D}p(t)\\,e^{-\\int _0^{\\beta }dt\\,\\left\\lbrace p^2(t)-ip(t)\\dot{x}(t)\\right\\rbrace } e^{-\\int _0^{\\beta }dt\\,l(x(t)-\\Theta p(t))\\,r(x(t)+\\Theta p(t))}\\,,$ where $\\beta >0$ and $x(t),p(t)$ represent trajectories in phase space $\\mathbb {R}^{2d}$ .",
"The path integral is performed on trajectories $x(t)$ that satisfy the boundary conditions $x(0)=x$ and $x(\\beta )=x+z$ and on trajectories $p(t)$ that do not satisfy any boundary condition.",
"It is convenient to replace the integral on the trajectories $x(t)$ by an integral on perturbations $q(t)$ about the free classical path $x_{cl}(t)=z~t/\\beta +x$ .",
"We also make the following rescaling: $t\\rightarrow \\beta t$ , $q\\rightarrow \\sqrt{\\beta }q$ , $p\\rightarrow p/\\sqrt{\\beta }$ .",
"Expression (REF ) then takes the form ${}\\langle x+z\\vert e^{-\\beta H}\\vert x \\rangle = \\beta ^{-d/2}\\int \\mathcal {D}q\\mathcal {D}p\\,e^{-\\int _0^{1} dt\\,\\left\\lbrace p^2-ip\\dot{q}\\right\\rbrace }\\times \\\\\\times \\,e^{i\\frac{z}{\\sqrt{\\beta }}\\int _0^1dt\\,p}e^{-\\beta \\int _0^{1}dt\\,l(x+tz+\\sqrt{\\beta }q-\\Theta p/\\sqrt{\\beta })\\,r(x+tz+\\sqrt{\\beta }q+\\Theta p/\\sqrt{\\beta })}\\,,\\nonumber $ where the perturbations $q$ in configuration space satisfy $q(0)=q(1)=0$ .",
"If we define the mean value of a functional $f[q(t),p(t)]$ as $\\left\\langle f[q(t),p(t)]\\right\\rangle :=\\frac{\\int \\mathcal {D}q\\mathcal {D}p\\,e^{-\\int _0^{1}dt\\,\\left\\lbrace p^2-ip\\dot{q}\\right\\rbrace } f[q(t),p(t)]}{\\int \\mathcal {D}q\\mathcal {D}p\\,e^{-\\int _0^{1}dt\\,\\left\\lbrace p^2-ip\\dot{q}\\right\\rbrace }}$ then the transition amplitude (REF ) reads $\\langle x+z\\vert e^{-\\beta H}\\vert x \\rangle = \\frac{1}{(4\\pi \\beta )^{d/2}}\\left\\langle e^{i\\frac{z}{\\sqrt{\\beta }}\\int _0^1dt\\,p}e^{-\\beta \\int _0^{1}dt\\,l(x+tz+\\sqrt{\\beta }q-\\Theta p/\\sqrt{\\beta })\\,r(x+tz+\\sqrt{\\beta }q+\\Theta p/\\sqrt{\\beta })}\\right\\rangle \\,.$ Next, we make a small $\\beta $ expansion of the second exponential and a Taylor expansion of $l$ and $r$ around $x$ ${}\\langle x+z\\vert e^{-\\beta H}\\vert x \\rangle =\\frac{1}{(4\\pi \\beta )^{d/2}}\\sum _{n=0}^\\infty \\frac{(-\\beta )^n}{n!",
"}\\int _0^1dt_1\\ldots \\int _0^1dt_n\\times \\\\\\nonumber \\times \\,\\left\\langle e^{i\\frac{z}{\\sqrt{\\beta }}\\int _0^1dt\\,p}e^{\\sum _{i=1}^n[t_iz+\\sqrt{\\beta }q(t_i)-\\Theta p(t_i)/\\sqrt{\\beta }]\\partial ^l_i+[t_iz+\\sqrt{\\beta }q(t_i)+\\Theta p(t_i)/\\sqrt{\\beta }]\\partial ^r_i}\\right\\rangle \\times \\\\\\nonumber \\left.\\times \\,l(x_1)\\ldots l(x_n)\\,r(x_1)\\ldots r(x_n)\\right|_{x}\\,,$ where $\\partial ^l_i,\\partial ^r_i$ are the gradientsNotice that the subindex $i$ does not denote spacetime components but refers to the $i$ -th spacetime point $x_i$ .",
"of $l(x_i),r(x_i)$ .",
"As indicated, all $x_i$ must be evaluated at $x$ after performing these derivatives.",
"Expression (REF ) can be written as ${}\\langle x+z\\vert e^{-\\beta H}\\vert x \\rangle =\\frac{1}{(4\\pi \\beta )^{d/2}}\\sum _{n=0}^\\infty \\frac{(-\\beta )^n}{n!",
"}\\int _0^1dt_1\\ldots \\int _0^1dt_n\\times \\\\\\nonumber \\left.\\times \\,e^{\\sum _{i=1}^n t_iz(\\partial ^l_i+\\partial ^r_i)}\\left\\langle e^{\\int _0^1 dt \\left(p\\, k_n +q\\, j_n\\right)}\\right\\rangle l(x_1)\\ldots l(x_n)\\,r(x_1)\\ldots r(x_n)\\right|_{x}\\nonumber $ if we define the sources $k_n(t),j_n(t)$ as $k_n(t)=\\frac{iz}{\\sqrt{\\beta }}+\\frac{\\Theta }{\\sqrt{\\beta }}\\sum _{i=1}^{n}\\delta (t-t_i)(\\partial ^l_i-\\partial ^r_i)\\,,\\\\j_n(t)=\\sqrt{\\beta }\\,\\sum _{i=1}^{n}\\delta (t-t_i)(\\partial ^l_i+\\partial ^r_i)\\,.\\nonumber $ In the following section we will compute the expectation value $\\left\\langle e^{\\int _0^1 dt\\left(p\\, k +q\\, j\\right)}\\right\\rangle $ for arbitrary sources $k(t),j(t)$ ."
],
[
"The generating functional in phase space",
"Let us compute the generating functional $Z[k,j]:=\\left\\langle e^{\\int _0^1 dt \\left(p\\, k +q\\, j \\right)}\\right\\rangle =\\frac{\\int \\mathcal {D}q\\mathcal {D}p\\,e^{-\\int _0^{1} dt\\,\\left(p^2-ip\\,\\dot{q}\\right)}e^{\\int _0^{1} dt\\,(p\\, k +q\\, j)}}{\\int \\mathcal {D}q\\mathcal {D}p\\,e^{-\\int _0^{1} dt\\,\\left(p^2-ip\\,\\dot{q}\\right)}}\\\\[1mm]\\nonumber =\\frac{\\int \\mathcal {D}P\\ e^{-\\frac{1}{2}\\int _0^{1} dt\\,P^t A P+\\int _0^1dt\\,P^tK}}{\\int \\mathcal {D}P\\ e^{-\\int _0^{1} dt\\,P^t A P}}$ for arbitrary sources $k(t), j(t)$ .",
"In this last expression we have defined the vectors $P:=\\left(\\begin{array}{c}p(t)\\\\q(t)\\end{array}\\right)&K:=\\left(\\begin{array}{c}k(t)\\\\j(t)\\end{array}\\right)$ and the operator $A:=\\begin{pmatrix}2&-i\\partial _t\\\\i\\partial _t&0\\end{pmatrix}$ Completing squares and inverting the operator $A$ –taking into account the boundary conditions $q(0)=q(1)=0$ – we obtain the generating functional in phase space $Z[k,j]=e^{\\frac{1}{2}\\int _0^1 dt\\, K^t A^{-1} K}\\,.$ The kernel of the operator $A^{-1}$ is given by $A^{-1}(t,t^{\\prime })=\\begin{pmatrix}\\frac{1}{2}&\\frac{i}{2}\\left[h(t,t^{\\prime })+f(t,t^{\\prime })\\right]\\\\\\frac{i}{2}\\left[h(t,t^{\\prime })-f(t,t^{\\prime })\\right]&2g(t,t^{\\prime })\\end{pmatrix}\\,,$ where $h(t,t^{\\prime }):=1-t-t^{\\prime }\\,,\\\\f(t,t^{\\prime }):=t-t^{\\prime }-\\epsilon (t-t^{\\prime })\\,,\\nonumber \\\\g(t,t^{\\prime }):=t(1-t^{\\prime })H(t^{\\prime }-t)+t^{\\prime }(1-t)H(t-t^{\\prime })\\,.\\nonumber $ The sign function $\\epsilon (\\cdot )$ is $\\pm 1$ if its argument is positive or negative, respectively; $H(\\cdot )$ represents the Heaviside function.",
"To obtain the expectation value of expression (REF ) we replace the sources given by eqs.",
"(REF ) into expression (REF ) $\\left\\langle e^{\\int _0^1 dt \\left(p\\, k_n +q\\, j_n \\right)}\\right\\rangle =e^{-\\frac{z^2}{4\\beta }+\\frac{iz}{2\\beta }\\Theta \\sum _{i=1}^n(\\partial ^l_i-\\partial ^r_i)}e^{\\triangle _n}\\,,$ where the operator $\\triangle _n$ is defined as $\\\\\\nonumber \\triangle _n:=\\sum _{i,j=1}^n\\left[\\beta g(t_i,t_j)(\\partial ^l_i+\\partial ^r_i)(\\partial _j^l+\\partial _j^r)-\\frac{1}{4\\beta }(\\partial ^l_i-\\partial ^r_i)\\Theta ^2(\\partial _j^l-\\partial _j^r)\\right.\\\\\\nonumber \\left.\\mbox{}-\\frac{i}{2}f(t_i,t_j)(\\partial _i^l\\Theta \\partial _j^l-\\partial _i^r\\Theta \\partial _j^r)-ih(t_i,t_j)\\partial _i^l\\Theta \\partial _j^r\\right]\\,.$"
],
[
"The noncommutative heat-kernel",
"The smeared heat-trace of the nonlocal operator $H$ , defined in eq.",
"(REF ), can be written as ${\\rm Tr}\\left(f(x)\\,\\bar{\\star }\\, e^{-\\beta H}\\right)=\\int _{\\mathbb {R}^d}dx\\,\\left.\\langle x+z\\vert e^{-\\beta H}\\vert x \\rangle \\right|_{z=-i\\bar{\\Theta }\\partial ^f}f(x)\\,,$ where $\\bar{\\star }$ represents a Moyal product as defined in (REF ) but in terms of another antisymmetric matrix $\\bar{\\Theta }$ which, in principle, differs from $\\Theta $ .",
"Expression (REF ) can be easily obtained by inserting in the l.h.s a spectral decomposition of unity in terms of position eigenstates and using that $e^{-i\\bar{\\Theta }\\partial ^{f}\\partial } \\langle x| =\\langle x-i\\bar{\\Theta }\\partial ^{f} |$ .",
"The matrix $\\bar{\\Theta }$ allows us to consider at the same time the cases where the smearing function $f$ acts by commutative pointwise multiplication ($\\bar{\\Theta }=0$ ), by left-Moyal multiplication ($\\bar{\\Theta }=\\Theta $ ) or right-Moyal multiplication ($\\bar{\\Theta }=-\\Theta $ ).",
"As indicated, the variable $z$ must be replaced by the operator $-i\\bar{\\Theta }\\partial ^f$ , where $\\partial ^f$ is the gradient acting on the smearing function $f$ only.",
"The heat-kernel $\\langle x+z\\vert e^{-\\beta H}\\vert x \\rangle $ is obtained by replacing eq.",
"(REF ) into expression (REF ).",
"Inserting the result into (REF ) we get our master formula ${}{\\rm Tr}\\left(f(x)\\,\\bar{\\star }\\, e^{-\\beta H}\\right)=\\frac{1}{(4\\pi \\beta )^{d/2}}\\sum _{n=0}^\\infty (-\\beta )^n\\int _{\\mathbb {R}^d}dx\\,f(x)\\,e^{\\frac{1}{4\\beta }\\sum _{i,j=1}^n D_iD_j}\\times \\\\\\nonumber \\left.\\times \\,\\int _0^1 dt_1\\int _0^{t_1}dt_2\\ldots \\int _0^{t_{n-1}}dt_n\\ e^{i\\triangle ^{NC}_n+\\beta \\triangle ^{C}_n}l(x_1)\\ldots l(x_n)\\,r(x_1)\\ldots r(x_n)\\right|_{x}\\,,$ where we have defined the following differential operators $\\triangle ^{C}_n:=\\sum _{i,j=1}^n g(t_i,t_j)(\\partial ^l_i+\\partial ^r_i)(\\partial ^l_j+\\partial ^r_j)\\,,\\\\[1mm]D_i:=\\left(\\Theta -\\bar{\\Theta }\\right)\\partial ^l_i-\\left(\\Theta +\\bar{\\Theta }\\right)\\partial ^r_i\\,,\\nonumber \\\\[1mm]\\triangle ^{NC}_n:=\\sum _{i< j=1}^n\\Big \\lbrace \\left[\\partial ^l_i\\Theta \\partial ^l_j-\\partial ^r_i\\Theta \\partial ^r_j\\right]-(1-t_i-t_j)(\\partial ^l_i\\Theta \\partial ^r_j-\\partial ^r_i\\Theta \\partial ^l_j)\\nonumber \\\\\\mbox{} + (t_i-t_j)\\left[\\partial ^l_i(-\\Theta +\\bar{\\Theta })\\partial ^l_j+\\partial ^r_i(\\Theta +\\bar{\\Theta })\\partial ^r_j+\\partial ^l_i\\bar{\\Theta }\\partial ^r_j+\\partial ^r_i\\bar{\\Theta }\\partial ^l_j\\right]\\Big \\rbrace \\nonumber \\\\-\\sum _{i=1}^n (1-2t_i)\\partial ^l_i\\Theta \\partial ^r_i\\,.\\nonumber $ In the derivation of (REF ) we have used the symmetry of the integrand with respect to permutations of the variables $t_i$ and we have integrated by parts to make the replacement $\\partial ^f\\rightarrow -\\sum _{i=1}^n(\\partial ^l_i+\\partial ^r_i)$ .",
"A few remarks are now in order.",
"As we will see next, the SDW coefficients for the commutative case are fully determined by the action of the operator $\\triangle _n^C$ since $\\triangle _n^{NC}$ and $D_i$ vanish for $\\Theta =\\bar{\\Theta }=0$ .",
"On the other hand, if $r(x)\\equiv 1$ (or $l(x)\\equiv 1$ ) and the smearing function acts by left- (respectively right-) Moyal multiplication, then $D_i$ vanishes and the only non-vanishing term in $\\triangle _n^{NC}$ is the first one $\\partial ^l_i\\Theta \\partial ^l_j$ (respectively $-\\partial ^r_i\\Theta \\partial ^r_j$ ) which replaces any pointwise product by a left- (respectively right-) Moyal product.",
"Let us also mention that the heat-trace with no smearing function corresponds to $f(x)\\equiv 1$ and $\\bar{\\Theta }=0$ .",
"Finally, we will show that the $1/\\beta $ coefficient of $\\sum _{i,j} D_iD_j$ in expression (REF ) is responsible for Moyal nonlocal SDW coefficients, which can be shown to correspond to contributions of non-planar diagrams, leading to the well-known UV/IR mixing.",
"In the rest of this section, we will apply master formula (REF ) to these different settings in order to describe the SDW coefficients."
],
[
"Commutative limit",
"First of all, we apply formula (REF ) to the heat-trace in the commutative case for future comparison with the subsequent noncommutative expressions.",
"For $\\bar{\\Theta }=\\Theta =0$ it is sufficient to consider the case $r(x)\\equiv 1$ .",
"As already mentioned, the operators $D_i$ and $\\triangle _n^{NC}$ vanish so that formula (REF ) reads ${\\rm Tr}\\left(f(x)\\,\\cdot \\, e^{-\\beta H}\\right)=\\frac{1}{(4\\pi \\beta )^{d/2}}\\sum _{n=0}^\\infty (-\\beta )^n\\int _{\\mathbb {R}^d}dx\\,f(x)\\times \\\\\\nonumber \\left.\\times \\,\\int _0^1 dt_1\\int _0^{t_1}dt_2\\ldots \\int _0^{t_{n-1}}dt_n\\ e^{\\beta \\sum _{i,j=1}^n g(t_i,t_j)\\partial _i\\partial _j}\\ l(x_1)\\ldots l(x_n)\\right|_{x}\\,.$ This expression shows that the SDW coefficients $a_n$ are integrals of products of the smearing function, the potential and derivatives thereof (see eq.",
"(REF ).)",
"For later use, in expression (REF ) we keep the time-ordering explicit in the multi worldline integral so that (REF ) gives the correct SDW coefficients even if the potential $l(x)$ is a matrix potential.",
"In such a case the product of two adjacent $l(x)$ 's has to be understood as a spatially-pointwise matricial product."
],
[
"Pointwise multiplication by a smearing function",
"In this subsection we consider the noncommutative operator (REF ) but for the case in which the smearing function $f(x)$ acts by pointwise multiplication, i.e.",
"$\\bar{\\Theta }=0$ .",
"We will show that some SDW coefficients are Moyal nonlocal [19].",
"For $\\bar{\\Theta }=0$ formula (REF ) reads ${}{\\rm Tr}\\left(f(x)\\,\\cdot \\, e^{-\\beta H}\\right)=\\frac{1}{(4\\pi \\beta )^{d/2}}\\sum _{n=0}^\\infty (-\\beta )^n\\int _{\\mathbb {R}^d}dx\\,f(x)\\,\\times \\\\\\nonumber \\left.\\times \\,\\int _0^1 dt_1\\int _0^{t_1}dt_2\\ldots \\int _0^{t_{n-1}}dt_n\\ e^{\\triangle _n}l(x_1)\\ldots l(x_n)\\,r(x_1)\\ldots r(x_n)\\right|_{x}\\,.$ For the purposes of this section it will suffice to consider a potential which contains only a left-Moyal product so we will consider the case $r(x)\\equiv 1$ .",
"Thus, the leading terms in (REF ) read ${\\rm Tr}\\left(f(x)\\,\\cdot \\, e^{-\\beta H}\\right)=\\frac{1}{(4\\pi \\beta )^{d/2}}\\int _{\\mathbb {R}^d}dx\\,f(x)\\left(1-\\beta \\,e^{-\\frac{1}{4\\beta }\\partial ^l\\Theta ^2\\partial ^l}l(x)+\\ldots \\right)\\,.$ The first term coincides with the leading term –the volume contribution– in the commutative case, whereas the second one can be written as (see the discussion below eq.",
"(REF )) $\\\\\\nonumber -\\frac{\\beta }{(4\\pi \\beta )^{d/2}}\\int _{\\mathbb {R}^c}d\\tilde{x}\\int _{\\mathbb {R}^{2b}}d\\hat{x}\\,f(\\tilde{x},\\hat{x})\\int _{\\mathbb {R}^{2b}}d\\hat{y}\\frac{\\beta ^{b}}{\\pi ^{b}\\ {\\rm det}\\,\\Xi }\\ e^{\\beta (\\hat{x}-\\hat{y})\\Xi ^{-2}(\\hat{x}-\\hat{y})}\\ l(\\tilde{x},\\hat{y})\\sim \\\\\\nonumber \\sim -\\frac{1}{(4\\pi \\beta )^{d/2}}\\cdot \\frac{\\beta ^{b+1}}{\\pi ^b\\ {\\rm det}\\,\\Xi }\\int _{\\mathbb {R}^c}d\\tilde{x}\\left(\\int _{\\mathbb {R}^{2b}}d\\hat{x}\\,f(\\tilde{x},\\hat{x})\\right)\\left(\\int _{\\mathbb {R}^{2b}}d\\hat{y}\\,l(\\tilde{x},\\hat{y})\\right)+\\ldots $ As can be seen from this expression, the presence of a smearing function that acts by pointwise multiplication originates a contribution to the SDW coefficient $a_{b+1}$ which is nonlocal in the generalized Moyal sense.",
"Notice that this Moyal nonlocal coefficient was obtained when considering the case $r(x)\\equiv 1$ , i.e.",
"even when the potential in $H$ does not mix left- and right-Moyal actions."
],
[
"No mixing between left- and right-Moyal multiplication",
"We will now consider the case in which the heat-trace involves only left-Moyal multiplication ($r(x)\\equiv 1$ and $\\bar{\\Theta }=\\Theta $ ) or only right-Moyal multiplication ($l(x)\\equiv 1$ and $\\bar{\\Theta }=-\\Theta $ ).",
"Under any of these alternative assumptions the operators $D_i$ vanish, whereas $\\triangle _n^{NC}$ takes the form $\\triangle ^{NC}_n:=\\pm \\sum _{i< j=1}^n\\partial _i\\Theta \\partial _j\\,,$ where the upper (lower) sign corresponds to the case where only left- (right-) Moyal multiplication is considered; the derivatives $\\partial _i$ act consequently on $l(x_i)$ ($r(x_i)$ ).",
"Therefore, according to formula (REF ), the action of the operator $e^{i\\triangle _{n}^{NC}}$ is the only effect of noncommutativity on the SDW coefficients.",
"Notice that the action of these operator defines Moyal multiplication (see eq.",
"(REF )), so that the heat-trace (for the left-Moyal case) reduces to ${\\rm Tr}\\left(f(x)\\star e^{-\\beta H}\\right)=\\frac{1}{(4\\pi \\beta )^{d/2}}\\sum _{n=0}^\\infty (-\\beta )^n\\int _{\\mathbb {R}^d}dx\\,f(x)\\times \\\\\\nonumber \\left.\\times \\,\\int _0^1 dt_1\\int _0^{t_1}dt_2\\ldots \\int _0^{t_{n-1}}dt_n\\ e^{\\beta \\sum _{i,j=1}^n g(t_i,t_j)\\partial _i\\partial _j}\\ l(x_1)\\star \\cdots \\star l(x_n)\\right|_{x}\\,.$ In this case, we conclude that the noncommutative SDW coefficients can be obtained from the commutative coefficients for a matrix potential $l(x)$ (cfr.",
"(REF )), by replacing every spatially-pointwise matrix product of adjacent potentials with left- (right-) Moyal products.",
"For example, up to $\\beta ^3$ one gets ${\\rm Tr}\\left(f(x)\\star e^{-\\beta H}\\right)=\\frac{1}{(4\\pi \\beta )^{d/2}} \\int _{\\mathbb {R}^d}dx\\,f(x) \\times \\\\ \\nonumber \\times \\Biggl [1-\\beta l(x) +\\beta ^2 \\left( \\frac{1}{2} l_\\star ^2(x) -\\frac{1}{6}\\partial ^2 l(x)\\right)+\\beta ^3 \\Biggl ( -\\frac{1}{60}\\partial ^4 l(x)\\\\ \\nonumber +\\frac{1}{12} \\Big (\\partial ^2l \\star l(x) +l\\star \\partial ^2 l(x) +\\partial l\\star \\partial l(x)\\Big )-\\frac{1}{3!}",
"l_\\star ^3(x)\\Biggr ) +\\cdots \\Biggr ]$ In summary, when there is only left- (right-) Moyal multiplication the SDW coefficients are Moyal local [19], [20]."
],
[
"Mixing between left- and right-Moyal multiplication",
"In this last subsection we will consider the presence of both functions $l(x)$ and $r(x)$ in the potential of the operator (REF ).",
"The calculation goes along the same line followed in subsection REF and we will arrive to a similar conclusion, namely, the existence of Moyal nonlocal SDW coefficients.",
"For simplicity, we will consider $f(x)\\equiv 1$ .",
"Formula (REF ) then reads ${\\rm Tr}\\left(e^{-\\beta H}\\right)=\\frac{1}{(4\\pi \\beta )^{d/2}}\\sum _{n=0}^\\infty (-\\beta )^n\\int _{\\mathbb {R}^d}dx\\,\\times \\\\\\nonumber \\left.\\times \\,\\int _0^1 dt_1\\int _0^{t_1}dt_2\\ldots \\int _0^{t_{n-1}}dt_n\\ e^{\\triangle _n}l(x_1)\\ldots l(x_n)\\,r(x_1)\\ldots r(x_n)\\right|_{x}\\,,$ Let us consider the leading terms in expansion (REF ) corresponding to an increasing number $n$ of insertions of $l(x)$ and $r(x)$ .",
"For $n=0$ we get the commutative volume contribution.",
"For $n=1$ we get the leading contribution $-\\frac{\\beta }{(4\\pi \\beta )^{d/2}}\\int _{\\mathbb {R}^d}dx\\,r(x)e^{-\\frac{1}{\\beta }\\partial ^l\\Theta ^2\\partial ^l}l(x)\\,,$ where upon integration by parts we have replaced $\\partial ^r\\rightarrow -\\partial ^l$ .",
"Expression (REF ) can be obtained from the subleading term of (REF ) by identifying $r(x)$ with $f(x)$ .",
"Proceeding analogously we get for (REF ) $\\mbox{}\\\\\\nonumber -\\frac{1}{(4\\pi \\beta )^{d/2}}\\frac{\\beta ^{b+1}}{(4\\pi )^{b}\\ {\\rm det}\\,\\Xi }\\int _{\\mathbb {R}^c}d\\tilde{x}\\int _{\\mathbb {R}^{2b}}d\\hat{x}\\,r(\\tilde{x},\\hat{x})\\int _{\\mathbb {R}^{2b}}d\\hat{y}\\ e^{\\frac{\\beta }{4}(\\hat{x}-\\hat{y})\\Xi ^{-2}(\\hat{x}-\\hat{y})}\\ l(\\tilde{x},\\hat{y})\\sim \\\\\\nonumber \\sim -\\frac{1}{(4\\pi \\beta )^{d/2}}\\cdot \\frac{\\beta ^{b+1}}{(4\\pi )^b\\ {\\rm det}\\,\\Xi }\\int _{\\mathbb {R}^c}d\\tilde{x}\\left\\lbrace \\left(\\int _{\\mathbb {R}^{2b}}d\\hat{x}\\,r(\\tilde{x},\\hat{x})\\right)\\left(\\int _{\\mathbb {R}^{2b}}d\\hat{y}\\,l(\\tilde{x},\\hat{y})\\right)\\right.\\\\\\nonumber \\mbox{}+\\frac{\\beta }{4}(\\Xi ^{-2})_{ij}\\left[\\int _{\\mathbb {R}^{2b}}d\\hat{x}\\,\\hat{x}^i\\hat{x}^jr(\\tilde{x},\\hat{x})\\cdot \\int _{\\mathbb {R}^{2b}}d\\hat{y}\\,l(\\tilde{x},\\hat{y})\\right.",
"+ \\int _{\\mathbb {R}^{2b}}d\\hat{x}\\,r(\\tilde{x},\\hat{x})\\cdot \\int _{\\mathbb {R}^{2b}}d\\hat{y}\\,\\hat{y}^i\\hat{y}^jl(\\tilde{x},\\hat{y})\\\\\\nonumber \\left.\\left.\\mbox{}-2\\int _{\\mathbb {R}^{2b}}d\\hat{x}\\,\\hat{x}^ir(\\tilde{x},\\hat{x})\\int _{\\mathbb {R}^{2b}}d\\hat{y}\\,\\hat{y}^jl(\\tilde{x},\\hat{y})\\right]+\\ldots \\right\\rbrace \\,.$ In this expression we have explicitly written the splitting of the coordinates $x\\in \\mathbb {R}^d$ in commuting $\\tilde{x}\\in \\mathbb {R}^c$ and noncommuting $\\hat{x}\\in \\mathbb {R}^{2b}$ coordinates (see the discussion below eq.",
"(REF )).",
"Expansion (REF ) shows the Moyal nonlocal contributions, which are linear in the product $r(x)l(y)$ , to the coefficients $a_{b+1}$ and $a_{b+2}$ [18] that could lead to the non-renormalizability of the corresponding theory [21]."
],
[
"The noncommutative torus",
"As our last example, we will consider the operator (REF ) on the $d$ -dimensional noncommutative torus $T^d$ , as defined in [8].",
"The coordinates $x=(\\tilde{x},\\hat{x})$ on $T^d$ can be split in commuting $\\tilde{x}\\in T^{c}$ and noncommuting $\\hat{x}\\in T^{2b}$ components, with $d=c+2b$ , as discussed after eq.",
"(REF ).",
"We also define $0\\le x_i\\le L_i$ .",
"The heat-kernel $\\langle y|e^{-\\beta H}|x\\rangle _{T^d}$ subject to the corresponding periodic boundary conditions can be obtained as the sum of infinitely many transition amplitudes $\\langle y+t_k|e^{-\\beta H}|x\\rangle _{\\mathbb {R}^d}$ computed on the whole space $\\mathbb {R}^d$ , where $t_k=(L_1 k_1,\\ldots ,L_dk_d)$ and $k=(k_1,\\ldots ,k_d)\\in \\mathbb {Z}^d$ : $\\langle y|e^{-\\beta H}|x\\rangle _{T^d}=\\sum _{k\\in \\mathbb {Z}^d}\\langle y+t_k|e^{-\\beta H}|x\\rangle _{\\mathbb {R}^d}$ The functions $l,r$ in the operator $H$ of the r.h.s.",
"of this equation are the periodic extensions to the whole space $\\mathbb {R}^d$ of the functions $l,r$ in the operator $H$ of the l.h.s.",
"Correspondingly, the heat-trace on the torus can be computed as ${\\rm Tr}\\left(e^{-\\beta H}\\right)=\\int _{T^d}dx\\, \\langle x|e^{-\\beta H}|x\\rangle _{T^d}=\\int _{T^d}dx\\,\\sum _{k\\in \\mathbb {Z}^d}\\langle x+t_k|e^{-\\beta H}|x\\rangle _{\\mathbb {R}^d}$ The transition amplitudes on the whole $\\mathbb {R}^d$ were already computed and can be read from eqs.",
"(REF ) and (REF ).",
"The first contribution, corresponding to $n=0$ in eq.",
"(REF ), gives the volume term $\\frac{1}{(4\\pi \\beta )^{d/2}}\\,\\prod _{i=1}^d L_i\\sum _{k\\in \\mathbb {Z}}e^{-\\frac{L_i^2}{4\\beta } k^2}\\sim \\frac{1}{(4\\pi \\beta )^{d/2}}\\,L_1\\ldots L_d\\,.$ Notice that the only contribution to the asymptotic expansion for small $\\beta $ comes from the term corresponding to $k=0$ .",
"On the other hand, the term corresponding to $n=1$ in eq.",
"(REF ) reads $-\\frac{\\beta }{(4\\pi \\beta )^{d/2}}\\sum _{k\\in \\mathbb {Z}^d}\\int _{T^d}dx\\,r(x)e^{-\\frac{1}{\\beta }\\left(-i\\Theta \\partial +\\frac{1}{2}t_k\\right)^2}l(x)\\,,$ where we have used the periodicity of $l(x)$ and $r(x)$ to replace $\\partial ^r$ by $-\\partial ^l$ .",
"This contribution has been analyzed in [8] for the case in which the matrix elements of $\\Theta $ satisfy a Diophantine condition."
],
[
"Application to $\\lambda \\, \\phi \\star ^4$",
"In this section we will apply our formula (REF ) to a simple noncommutative model.",
"In particular, we will compute some SDW coefficients to study the one-loop corrections to the propagator of a real scalar field with a quartic self-interaction defined on the Euclidean spacetime $\\mathbb {R}^d$ .",
"We therefore consider the following action $\\mathcal {L}=\\frac{1}{2}(\\partial \\phi )^2+\\frac{m^2}{2}\\phi ^2+\\frac{\\lambda }{4!",
"}\\phi ^4_\\star \\,,$ where $\\phi ^4_\\star :=\\phi \\star \\phi \\star \\phi \\star \\phi $ .",
"The one-loop effective action $\\Gamma $ can be written as [18] $\\Gamma =\\frac{1}{2}\\log {\\rm Det}\\left\\lbrace -\\partial ^2+m^2+\\frac{\\lambda }{3}\\,[L(\\phi _\\star ^2)+R(\\phi _\\star ^2)+L(\\phi )R(\\phi )]\\right\\rbrace \\\\ \\nonumber =-\\frac{1}{2}\\int _{\\Lambda ^{-2}}^\\infty \\frac{d\\beta }{\\beta }\\,e^{-\\beta m^2}\\,{\\rm Tr}\\,e^{-\\beta \\left\\lbrace -\\partial ^2+\\frac{\\lambda }{3}\\,\\left[ L(\\phi _\\star ^2)+R(\\phi _\\star ^2)+L(\\phi )R(\\phi )\\right] \\right\\rbrace }\\,,$ where we have used the Schwinger proper time approach to represent the functional determinant and we have introduced an UV-cutoff $\\Lambda $ .",
"Since the propagator is obtained from the quadratic terms in the effective action, we only need to consider the terms in expression (REF ) which are linear in $\\lambda $ .",
"Therefore, we can use formula $(\\ref {mf})$ to compute the contributions of the three terms $L(\\phi _\\star ^2)$ , $R(\\phi _\\star ^2)$ and $L(\\phi )R(\\phi )$ separately.",
"By replacing $f(x)\\equiv 1$ , $\\bar{\\Theta }=0$ and $r(x)\\equiv 1$ in the term corresponding to $n=1$ in eq.",
"(REF ) we obtain the contribution of $L(\\phi _\\star ^2)$ to the effective action.",
"Analogously, the contribution corresponding to $R(\\phi _\\star ^2)$ is obtained by replacing instead $l(x)\\equiv 1$ .",
"Both contributions are equal and the sum of them reads $\\int _{\\Lambda ^{-2}}^\\infty d\\beta \\,e^{-\\beta m^2}\\,\\frac{1}{(4\\pi \\beta )^{d/2}}\\int _{\\mathbb {R}^d}dx\\,\\frac{\\lambda }{3}\\phi _\\star ^2(x)=\\frac{\\lambda }{3}\\frac{m^{d-2}}{(2\\pi )^{d/2}}\\,\\Gamma (1-d/2,m^2/\\Lambda ^2)\\int _{\\mathbb {R}^d}\\phi ^2\\,,$ where $\\Gamma (\\cdot ,\\cdot )$ represents the incomplete gamma function [26].",
"In the limit $\\Lambda \\rightarrow \\infty $ , this contribution diverges as $\\log {\\Lambda }$ , for $d=2$ , and as $\\Lambda ^{d-2}$ , for $d>2$ .",
"This divergence is eliminated by a mass redefinition; in this way, the dependence of the mass with the cutoff $\\Lambda $ is determined.",
"Notice that result (REF ) –which corresponds to the contribution of one-loop planar diagrams– does not depend on the noncommutativity parameters and holds also in the commutative case.",
"The remaining contribution, corresponding to the term $L(\\phi )R(\\phi )$ , is obtained by replacing $f(x)\\equiv 1$ , $\\bar{\\Theta }=0$ and $\\partial ^l=-\\partial ^r$ in the term corresponding to $n=1$ in eq.",
"(REF ); the result reads, $\\frac{1}{2}\\int _{\\Lambda ^{-2}}^\\infty d\\beta \\,e^{-\\beta m^2}\\,\\frac{1}{(4\\pi \\beta )^{d/2}}\\int _{\\mathbb {R}^d}dx\\,\\frac{\\lambda }{3}\\phi (x)e^{-\\frac{1}{\\beta }\\partial \\Theta ^2\\partial }\\phi (x)=\\\\\\nonumber =\\frac{\\lambda }{6}(2\\pi )^{3d/2}m^{d-2}\\,\\int d^c\\tilde{p}\\,d^{2b}\\hat{p}\\ \\tilde{\\phi }^*(\\tilde{p},\\hat{p})\\tilde{\\phi }(\\tilde{p},\\hat{p})\\cdot \\Sigma _{NP}(\\hat{p})\\,,$ where $\\Sigma _{NP}(\\hat{p}):=\\int _0^\\infty \\frac{d\\beta }{\\beta ^{d/2}}\\,e^{-\\beta -\\frac{m^2}{\\beta }|\\Xi \\hat{p}|^2}=2(m|\\Xi \\hat{p}|)^{1-d/2}K_{d/2-1}(2m|\\Xi \\hat{p}|)\\,,$ being $K_{d/2-1}(\\cdot )$ the modified Bessel function.",
"In eq.",
"(REF ) we have made use of the splitting defined in eq.",
"(REF ) to separate spacetime $\\mathbb {R}^d$ into $\\mathbb {R}^c$ , which is described by commuting coordinates, and $\\mathbb {R}^{2b}$ , where noncommutativity is defined by the non-degenerate matrix $\\Xi $ .",
"We have also written this contribution to the effective action in terms of the Fourier transform $\\tilde{\\phi }(\\tilde{p},\\hat{p})$ of the field, where $\\tilde{p}\\in \\mathbb {R}^c$ and $\\hat{p}\\in \\mathbb {R}^{2b}$ .",
"Notice also that in expression (REF ) the term depending on $\\Xi $ makes the integral convergent at $\\beta \\rightarrow 0$ and, in consequence, the cutoff $\\Lambda $ has been removed.",
"In other words, noncommutativity regularizes UV-divergence at the one-loop level.",
"It can be shown that expression (REF ) corresponds to the contribution of one-loop non-planar diagrams.",
"Since $\\Sigma _{NP}(\\hat{p})\\sim |\\hat{p}|^{-d+2}$ , for small $|\\hat{p}|$ and $d>2$ , the integrand in (REF ) grows as $|\\hat{p}|^{1-c}$ for $|\\hat{p}|\\rightarrow 0$ , where $c=d-2b$ is the number of commuting coordinates.",
"Therefore, this contribution to the one-loop effective action is divergent if the number of commuting coordinates is greater or equal than two.",
"The same result can be obtained from expression (REF ), whose leading term reads $-\\frac{1}{(4\\pi \\beta )^{d/2}}\\cdot \\frac{\\beta ^{b+1}}{(4\\pi )^b\\ {\\rm det}\\,\\Xi }\\int _{\\mathbb {R}^c}d\\tilde{x}\\left(\\int _{\\mathbb {R}^{2b}}d\\hat{x}\\,\\phi (\\tilde{x},\\hat{x})\\right)\\left(\\int _{\\mathbb {R}^{2b}}d\\hat{y}\\,\\phi (\\tilde{x},\\hat{y})\\right)\\,.$ If this contribution is inserted in the last line of expression (REF ) one can see that the integrand behaves as $\\beta ^{-c/2}$ for small $\\beta $ and then the integral diverges as $\\Lambda \\rightarrow \\infty $ if $c\\ge 2$ .",
"This UV-divergence cannot be eliminated by a counterterm in the Lagrangian due to the high non-locality in the fields of the contribution (REF ).",
"As is well-known, the $S$ -matrix is not unitary if time is a noncommutative coordinate [27], [28], [29].",
"Therefore, for an odd number of spacelike coordinates, unitarity imposes that at least two coordinates are commutative, i.e.",
"$c\\ge 2$ .",
"In this case, as we have seen, some one-loop non-planar diagrams generate divergencies that cannot be regularized by a redefinition of the parameters of the theory.",
"In consequence, as has been shown by V. Gayral et al.",
"[21], a real scalar field in four dimensions with a quartic self-interaction does not have a well-defined effective action.",
"On the other hand, notice that the function $\\Sigma _{NP}(\\hat{p})$ behaves as $\\Sigma _{NP}(\\hat{p})=\\left[(d/2-2)!",
"(m|\\Xi \\hat{p}|)^{2-d}+\\frac{2(-1)^{d/2}}{(d/2-1)!",
"}\\log {(m|\\Xi \\hat{p}|)}\\right](1+O(|\\hat{p}|^2))\\,,$ for small $|\\hat{p}|$ and $d>2$ .",
"The corresponding result for $d=2$ reads $\\Sigma _{NP}(\\hat{p})=-2\\left[\\log {m|\\Xi \\hat{p}|+\\gamma }\\right](1+O(|\\hat{p}|^2))\\,,$ where $\\gamma $ is Euler's constant.",
"The result of expression (REF ) evaluated at $d=4$ and $c=0$ corresponds to the contribution to the effective action computed by S. Minwalla et al.",
"[3] by considering one-loop non-planar diagrams.",
"The divergent behavior at small $\\hat{p}$ shown in expressions (REF ) and (REF ) implies that the propagator receives one-loop corrections which are divergent for small values of the momentum in the noncommutative directions.",
"This is the well-known UV/IR-mixing, which shows that in some noncommutative theories the integration of internal momenta can generate divergencies at small values of the external momenta, even for massive fields.",
"As a consequence, these noncommutative theories are non-renormalizable."
],
[
"Conclusions",
"We have determined the phase space propagator and the phase space generating functional and written a path integral formulation for non-local operators which are relevant in field theories on noncommutative spacetimes.",
"In this formulation, we have derived a master formula (cfr.",
"(REF )) for the heat-trace expansion which can be applied to different noncommutative settings.",
"In particular, we considered a non-local operator involving the product of left-Moyal and right-Moyal multiplications.",
"We have shown that the natural rescaling $x\\rightarrow \\sqrt{\\beta }x$ and $p\\rightarrow p/\\sqrt{\\beta }$ in phase space introduces $O(1/\\beta )$ differential operators which act on the potentials.",
"These operators are given by $D_i$ , defined in eqs.",
"(REF ).",
"Notice that when the heat operator involves only left-Moyal multiplication ($\\partial ^r=0$ and $\\bar{\\Theta }=\\Theta $ ) or only right-Moyal multiplication ($\\partial ^l=0$ and $\\bar{\\Theta }=-\\Theta $ ) the operators $D_i$ vanish.",
"However, in the case where both left- and right-Moyal multiplications are present, the operators $D_i$ generate SDW coefficients which are non-local even in the generalized Moyal sense.",
"These non-local SDW coefficients are equivalent to the non-planar contributions in a perturbative calculation of the effective action in terms of Feynman diagrams.",
"Our phase space formulation provides a simple derivation of these results and is suitable for further generalizations in noncommutative models.",
"In particular, the introduction of a Grosse-Wulkenhaar term [24], [25] can be straightforwardly implemented in our phase space approach by replacing the vanishing matrix element of the propagator given by eq.",
"(REF ) by a constant matrix element, proportional to the squared frequency of the harmonic oscillator term.",
"Finally, we consider that this formalism could be a useful tool in the study of other models involving more general non-local operators.",
"Research along these lines is currently in progress."
],
[
"Acknowledgments",
"The authors thank F. Bastianelli for help and suggestions and for participating in the earlier stages of this project.",
"P.A.G.P.",
"and S.A.F.V.",
"acknowledge D.V.",
"Vassilevich for calling their attention on the results of [8] and for a discussion on the open problems of heat-trace expansion on the NC torus.",
"O.C.",
"thanks the Dipartimento di Fisica and INFN Bologna for hospitality and support while parts of this work were completed.",
"The work of O.C.",
"is partly funded by SEP-PROMEP/103.5/11/6653.",
"The work of P.A.G.P.",
"and S.A.F.V.",
"is partly funded by CONICET (PIP 01787) and UNLP (proj.",
"11/X492).",
"S.A.F.V.",
"'s stay at the Università di Roma “La Sapienza” was partly financed by the ERASMUS MUNDUS Action 2 programme."
]
] | 1204.1013 |
[
[
"\"Frobenius twists\" in the representation theory of the symmetric group"
],
[
"Abstract For the general linear group $GL_n(k)$ over an algebraically closed field $k$ of characteristic $p$, there are two types of \"twisting\" operations that arise naturally on partitions.",
"These are of the form $\\lambda \\rightarrow p\\lambda$ and $\\lambda \\rightarrow \\lambda + p^r\\tau$ The first comes from the Frobenius twist, and the second arises in various tensor product situations, often from tensoring with the Steinberg module.",
"This paper surveys and adds to an intriguing series of seemingly unrelated symmetric group results where this partition combinatorics arises, but with no structural explanation for it.",
"This includes cohomology of simple, Specht and Young modules, support varieties for Specht modules, homomorphisms between Specht modules, the Mullineux map, $p$-Kostka numbers and tensor products of Young modules."
],
[
"Introduction",
"Let $k$ be an algebraically closed field of characteristic $p$ .",
"An important construction in the representation theory of the general linear group $G:=GL_n(k)$ is the Frobenius twist, which takes a $G$ module $M$ to the module $M^{(1)}$ .",
"The action of $G$ on $M^{(1)}$ is as on $M$ except twisted by the Frobenius endomorphism $F: G \\rightarrow G$ , which raises each matrix entry to the $p$ th power.",
"Probably the most important $G$ modules are the Steinberg modules $\\operatorname{St}_r=L((p^r-1)\\rho )$ .",
"For example the operation of “twist then tensor with $\\operatorname{St}_r$ \" plays a key role in the proof of Kempf's vanishing theorem.",
"In the last decade or so there have been a great variety of results and conjectures on the symmetric group $\\Sigma _d$ that “look like\" they should come from doing a Frobenius twist or taking a tensor product with a Steinberg module, even though neither construction has any reasonable analogue in the world of $k\\Sigma _d$ modules.",
"Regular twisting results involve partitions $\\lambda , p\\lambda , p^2\\lambda $ , etc...",
"Results reminiscent of twisting then tensor with $\\operatorname{St}_r$ could relate $\\lambda $ with $\\lambda + p^r\\tau .$ Both results we informally think of as twisting type theorems, keeping in mind again that there is no Frobenius twist for $k\\Sigma _d$ -modules.",
"In this paper we survey the known results of this type, and add a couple more new results together with new examples, conjectures and a variety of open problems that remain.",
"Particularly striking is the array of techniques that arise in the proofs of the various results.",
"Twisting behavior seems to arise in many different ways.",
"It is probably naive to expect some kind of uniform “Frobenius twist\" for symmetric groups that captures these diverse results.",
"We will assume background information for $k\\Sigma _d$ representation theory as found in [22], and use the same notation.",
"Let $\\lambda =(\\lambda _1, \\lambda _2, \\ldots , \\lambda _n)$ be a partition of $d$ with at most $n$ parts.",
"These partitions correspond to dominant polynomial weights for $G$ and many natural $G$ modules are labeled by them; for example, irreducible modules $L(\\lambda )$ , Weyl modules $V(\\lambda )$ and induced modules $\\operatorname{H}^0(\\lambda )$ .",
"The interested reader will find Jantzen's book [23] a definitive although likely unnecessary reference, as this paper will focus on the symmetric group.",
"For a $G$ module $M$ , let $M^{(r)}$ denote the $r$ th Frobenius twist [23] of $M$ .",
"A special case of the Steinberg Tensor Product Theorem [23] gives that: $L(\\lambda )^{(1)} \\cong L(p\\lambda ),$ where $p\\lambda :=(p\\lambda _1, \\ldots , p\\lambda _n) \\vdash pd$ .",
"Equation REF suggests the operation $\\lambda \\rightarrow p\\lambda $ is quite natural for $G$ , and it is not surprising to encounter theorems involving these “twisted\" partitions.",
"For example the isomorphism $ \\operatorname{H}^1(G, L(\\lambda )) \\cong \\operatorname{H}^1(G, L(p\\lambda ))$ is a special case of [4] and is realized explicitly on the level of short exact sequences by applying the Frobenius twist.",
"Let $\\rho =\\rho _n$ denote the partition $(n-1,n-2, \\ldots , 2, 1,0)$ .",
"Another operation that arises frequently in the representation theory of $G$ takes a partition $\\lambda $ to $\\lambda +p^r\\tau $ for some other partition $\\tau $ , where $\\lambda $ often involves the so-called Steinberg weight $(p^r-1)\\rho $ .",
"For example [23]: $\\operatorname{H}^i((p^r-1)\\rho ) \\otimes \\operatorname{H}^i(\\tau )^{(r)} \\cong \\operatorname{H}^i((p^r-1)\\rho + p^r\\tau ).$ These results are also quite natural as $\\operatorname{H}^i((p^r-1)\\rho )$ is the ubiquitous Steinberg module $\\operatorname{St}_r$ , which is simple and both projective and injective as a module for the Frobenius kernel $G_r$ .",
"Turning our attention to $k\\Sigma _d$ , we again find modules labeled by partitions, this time by all partitions of $d$ , not just those with at most $n$ parts.",
"For example we have Specht modules $S^\\lambda $ , Young modules $Y^\\lambda $ , irreducible modules $D^\\lambda $ for $\\lambda $ $p$ -regular, etc.",
"However there is no analogue of the Frobenius twist.",
"Moreover $p\\lambda $ is a partition of $pd$ , and so, for example, $S^\\lambda $ and $S^{p\\lambda }$ are modules for different groups with no apparent connection.",
"Nevertheless over the last ten years or so there have been numerous symmetric group results involving this kind of “twisting\" of partitions, and the proofs use an impressive variety of different techniques.",
"Other results are reminiscent of (REF ), even though there is no natural analogue of the Steinberg module for $\\Sigma _d$ ."
],
[
"Schur subalgebras and the original “twist\".",
"We believe the first appearance of “twisting\" type results for the symmetric group arose in the thesis of Henke, published in part in the paper [20].",
"(Although James' computation of decomposition numbers for two-part partitions [21] can be put in similar form).",
"For example Henke proved: Theorem 2.1 Fix $d$ and let $\\lambda =(d-k, k)$ be a $p$ -regular partition.",
"Then there is an $a \\ge 1$ such that there exists a strong submodule lattice isomorphism between $S^{(d-k, k)}$ and $S^{(d-k+cp^a, k)}$ for any $c \\ge 1$ such that $cp^a$ is even.",
"Similar lattice isomorphisms exist for Young modules and permutation modules.",
"More general results along the same line were obtain in [19], involving adding a large power of $p$ to the first part of a partition and obtaining equality of decomposition numbers and $p$ -Kostka numbers.",
"We should warn though that Section 4 of [19] has a gap.",
"For example $p=2, \\lambda =(4,3,1), \\mu =(8), p^d=4$ gives a counterexample to Cor.",
"4.5 The problem is that the set of weights considered, for example in Cor.",
"4.1 and following, is not an ideal or coideal.",
"The proofs of these results involve constructing explicit isomorphisms between generalized Schur algebras in different degrees.",
"The numerical equalities obtained are then translated to the symmetric group setting."
],
[
"Generic cohomology",
"Only more recently have symmetric group results relating $\\lambda $ and $p\\lambda $ appeared.",
"Many of these results are motivated by or suggestive of the famous generic cohomology theorem from [4], which we describe briefly now.",
"For $G$ modules $M_1$ and $M_2$ , there is a natural map $\\operatorname{Ext}^i_G(M_1,M_2) \\rightarrow \\operatorname{Ext}^i_G(M_1^{(r)}, M_2^{(r)})$ induced by applying the Frobenius twist to the corresponding exact sequences of $G$ modules.",
"The map (REF ) is always an injection [23].",
"Thus, for fixed $i$ , we have a sequence of injective maps $\\operatorname{H}^i(G,M) \\rightarrow \\operatorname{H}^i(G,M^{(1)}) \\rightarrow \\operatorname{H}^i(G, M^{(2)}) \\rightarrow \\cdots .$ When $M$ is finite-dimensional, the sequence (REF ) is known [4] to stabilize, and the limit is called the generic cohomology $\\operatorname{H}^i_{gen}(G,M)$ of $M$ .",
"Since $G=GL_n(k)$ , the sequence (REF ) is known to stabilize immediately for $i=1$ , [4] i.e.",
"$\\operatorname{H}^1_{\\operatorname{gen}}(G,M) \\cong \\operatorname{H}^1(G,M).$ Note that (REF ) is a special case.",
"We call theorems relating cohomology of symmetric group modules $U^\\lambda $ and $U^{p\\lambda }$ “generic cohomology\" or “stability\" type theorems, where $U$ can be $S, M, D, Y$ .",
"For example, a cohomology result relating $S^{p\\lambda }$ and $S^{p^2\\lambda }$ would be called a generic cohomology type theorem, although we should warn that (REF ) does not hold for other natural $G$ modules.",
"For example $\\operatorname{H}^0(\\lambda )^{(1)}$ is always a proper submodule of $\\operatorname{H}^0(p\\lambda ).$"
],
[
"Young modules",
"For a partition $\\lambda \\vdash d$ there is a corresponding Young subgroup $\\Sigma _\\lambda \\le \\Sigma _d$ and the permutation module on the cosets is denoted $M^\\lambda $ .",
"The isomorphism classes of indecomposable summands of these permutation modules are also indexed by partitions of $d$ and are called Young modules, denoted $Y^\\lambda $ .",
"These are important and well-studied modules.",
"For example the set $\\lbrace Y^\\lambda \\mid \\lambda \\text{ is $p$-restricted}\\rbrace $ is a complete set of projective indecomposable $k\\Sigma _d$ modules.",
"Each $Y^\\lambda $ is self-dual.",
"Section 4.6 of [31] is a good basic reference for Young modules."
],
[
"$p$ -Kostka numbers and tensor products",
"We have: $M^\\lambda \\cong Y^\\lambda \\oplus \\bigoplus _{\\mu > \\lambda } [M^\\lambda : Y^\\mu ] Y^\\mu $ where $>$ is the lexicographic order on partitions.",
"The multiplicities $[M^\\lambda : Y^\\mu ]$ in (REF ) are known as $p$ -Kostka numbers.",
"As we will see there are results for Young modules of both types, relating $Y^\\lambda $ with $Y^{p\\lambda }$ and with $Y^{\\lambda + p^r \\tau }.$ In fact just considering $p$ -Kostka numbers, both types of results arise.",
"In [18], Henke determines completely the $p$ -Kostka numbers when $\\lambda $ has two parts.",
"She also obtains: Theorem 4.1 [18] Let $\\lambda \\vdash d$ and suppose $\\lambda _1 \\ge d/2$ and $\\lambda _2 < p^r$ .",
"Then: $[M^{\\lambda +(ap^r)} : Y^{\\mu + (ap^r )}] = [M^{\\lambda } : Y^{\\mu }]$ for every $a \\ge 1$ and $\\mu \\vdash d$ .",
"Her proof uses the well-known multiplicity formula of Klyachko which gives a kind of recursion for $p$ -Kostka numbers in terms of those for smaller partitions, where the assumptions in Theorem REF ensures those decompositions are closely related.",
"More recently in his 2011 thesis Gill proved a strengthened result: Theorem 4.2 [15] Let $\\lambda , \\mu \\vdash d$ and $a \\ge 1$ .",
"Suppose $\\mu $ has $p$ -adic expansion $\\mu =\\sum _{i=0}^s\\mu (i) p^i.$ If $p^r > \\operatorname{max} (p^s, \\lambda _2)$ , then: $[M^{\\lambda +(ap^r)} : Y^{\\mu + (ap^r )}] = [M^{\\lambda } : Y^{\\mu }].$ Gill's techniques include an extensive analysis of Young vertices and the Broué correspondence for $p$ -permutation modules.",
"This approach to studying Young modules was pioneered by Erdmann in [10].",
"The twisting $\\lambda \\rightarrow p\\lambda $ behaves very well with respect to the Young vertices, which Gill used to prove the following stability result on $p$ -Kostka numbers under twisting: Theorem 4.3 [15] Let $\\lambda , \\mu \\vdash d$ .",
"Then: $[M^\\lambda : Y^\\mu ]= [M^{p\\lambda } : Y^{p\\mu }].$ We remark that an alternative proof of Theorem REF could be given using the general linear group and an actual Frobenius twist.",
"This is because $p$ -Kostka numbers are equal to weight space multiplicities in simple $GL_n(k)$ modules.",
"Namely: Proposition 4.4 [8] The $p$ -Kostka number $[M^\\lambda : Y^\\mu ]$ is the dimension of $L(\\mu )^\\lambda $ , the $\\lambda $ weight space in the simple module $L(\\mu )$ .",
"But there is no obvious way to use transfer the proof of Proposition REF to give a short symmetric group proof of Theorem REF ."
],
[
"Young module cohomology",
"If one thinks of $Y^{p\\lambda }$ as a twist of $Y^\\lambda $ then the following theorem can be interpreted as a generic cohomology theorem for Young modules, valid in arbitrary degree: Theorem 4.5 [5] Fix $i>0$ and let $p$ be arbitrary.",
"Then there exists $s(i)>0$ such that for any $d$ and $\\lambda \\vdash d$ we have $\\operatorname{H}^i(\\Sigma _{p^{a}d},Y^{p^{a}\\lambda }) \\cong \\operatorname{H}^i(\\Sigma _{p^{a+1}d},Y^{p^{a+1}\\lambda }) $ whenever $a \\ge s(i)$ .",
"Here we have our first example of what will be a common theme, a vector space isomorphism of cohomology but without any map realizing the isomorphism.",
"Indeed the proof of Theorem REF proceeds by using Schur functor techniques to translate the symmetric group cohomology problem to a representation theory problem for $G$ .",
"That in turn is solved using powerful algebraic topology techniques, and “reverse engineering\" the answer gives the isomorphism; but one cannot trace back to find a symmetric group proof or explicit realization of the isomorphism.",
"For example if $p=2$ and $i=1$ we can compute $s(i)=1$ , i.e.",
"[5] gives: $\\operatorname{Ext}^1_{\\Sigma _{2d}}(k, Y^{2\\lambda }) \\cong \\operatorname{Ext}^1_{\\Sigma _{4d}}(k, Y^{4\\lambda }).$ This leads us to ask: Problem 4.6 Can you prove (REF ) purely using symmetric group representation theory?",
"Can you give an explicit map that takes a short exact sequence $0 \\rightarrow Y^{2\\lambda } \\rightarrow U \\rightarrow k \\rightarrow 0$ and produces the corresponding one for $Y^{4\\lambda }$ ?",
"Recall that $\\operatorname{H}^i(G, M) \\cong \\operatorname{Ext}^i_G(k, M)$ .",
"This suggests a natural generalization of Theorem REF would be to consider $\\operatorname{Ext}^i_{\\Sigma _d}(Y^\\lambda , Y^\\mu ) \\cong \\operatorname{H}^i(\\Sigma _d, Y^\\mu \\otimes Y^\\lambda )$ .",
"Already the $i=0$ case of this problem is extremely difficult.",
"Indeed knowing the dimension of $\\operatorname{Hom}_{k\\Sigma _d}(Y^\\lambda , Y^\\mu )$ for all $\\lambda , \\mu \\vdash d$ is equivalent [5] to knowing the decomposition matrix for the Schur algebra $S(d,d)$ , and thus contains more information than computing the decomposition matrix for the symmetric group, a notoriously intractable problem.",
"While computing the actual dimensions is beyond reach, Gill managed to prove: Theorem 4.7 [14] Let $\\lambda , \\mu \\vdash d$ .",
"Then: $\\dim _k\\operatorname{Hom}_{k\\Sigma _{d}}(Y^{\\lambda }, Y^{\\mu }) \\le \\dim _k\\operatorname{Hom}_{k\\Sigma _{pd}}(Y^{p\\lambda }, Y^{p\\mu }).$ Mackey's theorem easily implies that $Y^\\lambda \\otimes Y^\\mu $ is a direct sum of Young modules.",
"The proof of Theorem REF uses fact that $\\dim _k\\operatorname{Hom}_{k\\Sigma _{d}}(Y^{\\lambda }, Y^{\\mu })$ is the number of summands in $Y^\\lambda \\otimes Y^\\mu $ which have a trivial submodule.",
"Since $Y^\\lambda \\subseteq M^\\lambda $ it is clear that $\\operatorname{Hom}_{k\\Sigma _d}(k, Y^\\lambda )$ is at most one-dimensional.",
"The $\\lambda $ for which it is nonzero are known (see [5] for example), and these partitions are preserved under twisting.",
"Then the following key theorem from [14] is used to complete the proof.",
"Theorem 4.8 [14] Let $\\lambda , \\mu , \\tau \\vdash d$ .",
"Then: $[Y^\\lambda \\otimes Y^\\mu : Y^\\tau ] = [Y^{p\\lambda } \\otimes Y^{p\\mu } : Y^{p\\tau }].$ This theorem is proved numerically by counting multiplicities, but looks like it should come from some explicit twist map!",
"Several obvious questions arise: Problem 4.9 Theorem REF implies a sequence of injections $0 \\rightarrow \\operatorname{Hom}_{k\\Sigma _d}(Y^\\lambda , Y^\\mu ) \\rightarrow \\operatorname{Hom}_{k\\Sigma _{pd}}(Y^{p\\lambda }, Y^{p\\mu }) \\rightarrow \\operatorname{Hom}_{k\\Sigma _{p^2d}}(Y^{p^2\\lambda }, Y^{p^2\\mu }) \\rightarrow \\cdots .$ Does this sequence stabilize?",
"How many twists does it take to do so?",
"Problem 4.10 Can one find an explicit embedding of $\\operatorname{Hom}_{k\\Sigma _d}(Y^\\lambda , Y^\\mu )$ into $ \\operatorname{Hom}_{k\\Sigma _{pd}}(Y^{p\\lambda }, Y^{p\\mu })$ ?",
"Problem 4.11 Can Theorem REF be extended to $\\operatorname{Ext}^i_{k\\Sigma _d}(Y^\\lambda , Y^\\mu )$ for $i>0$ , perhaps with more twists required as $i$ grows in the spirit of Theorem REF ?"
],
[
"Specht modules",
"The Specht modules $S^\\lambda $ are perhaps the most well-studied among all $k\\Sigma _d$ modules.",
"They are a complete set of irreducible ${\\mathbb {C}}\\Sigma _d$ modules, but they are defined over any field and are not well understood over $k$ .",
"For example only quite recently was it proven which remain irreducible over $k$ [11].",
"Computing the homomorphism space $\\operatorname{Hom}_{k\\Sigma _d}(S^\\lambda , S^\\mu )$ is an active area of research.",
"Cohomology $\\operatorname{H}^i(\\Sigma _d, S^\\lambda )$ was worked out in degree $i=0$ more than thirty years ago in [21], but the $i=1$ case remains open.",
"Recent results for Specht modules involve both twisting $\\lambda \\rightarrow p\\lambda $ and $\\lambda \\rightarrow \\lambda +p^a\\tau $ ."
],
[
"Homomorphisms between Specht modules and decomposable Specht modules",
"There is quite a large literature on homomorphisms between Specht modules, for example Carter-Payne maps, row removal theorems, etc.",
"When $p>2$ it is known that $S^\\lambda $ is indecomposable and $\\operatorname{Hom}_{k\\Sigma _d}(S^\\lambda , S^\\lambda ) \\cong k$ .",
"In 1980 Murphy [33] analyzed the hook Specht modules $S^{(d-r, 1^r)}$ in characteristic $p=2$ and discovered they can have arbitrarily many indecomposable summands, so the dimension of $\\operatorname{Hom}_{k\\Sigma _d}(S^\\lambda , S^\\lambda )$ can be arbitrarily large.",
"Only in 2011 were such homomorphism spaces with dimension larger than one discovered in odd characteristic [6], [30].",
"Dodge's examples are found in Rouquier blocks while Lyle finds explicitly examples with dimension two, then uses row and column removal theorems to get arbitrary dimension.",
"In line with the theme of this paper we observe that the examples from [30] are of the form: $\\operatorname{Hom}_{k\\Sigma _{d+3ap}}(S^{\\lambda + a(p,p,p)}, S^{\\mu + a(p,p,p)}).$ Lyle suspects the spaces in REF are all two-dimensional but does not prove this.",
"If so it would give another example of the $\\lambda \\rightarrow \\lambda + p^r\\tau $ twisting.",
"However she constructs the maps individually for each choice of $a$ rather than, for example, by “twisting\" the $a=1$ case, so this is somewhat speculative at this point.",
"We collect Murphy's results below.",
"Recall from [21] that $S^\\lambda \\otimes \\operatorname{sgn}\\cong (S^{\\lambda ^{\\prime }})^*$ .",
"Since the sign representation is trivial in characteristic two, the assumption $d \\ge 2r$ below does not impose a real restriction, all possible hooks are handled.",
"Theorem 5.1 Let $p=2$ and assume $d \\ge 2r$ .",
"If $d$ is even then $S^{(d-r, 1^r)}$ is indecomposable and $\\dim \\operatorname{Hom}_{k\\Sigma _d}(S^{(d-r, 1^r)}, S^{(d-r, 1^r)})=1.$ If $d$ is odd and $r$ is even then $\\dim \\operatorname{Hom}_{k\\Sigma _d}(S^{(d-r, 1^r)}, S^{(d-r, 1^r)})=r/2.$ If $d$ is odd and $r$ is odd then $\\dim \\operatorname{Hom}_{k\\Sigma _d}(S^{(d-r, 1^r)}, S^{(d-r, 1^r)})=(r+1)/2.$ If $d$ is odd then $S^{(d-r, 1^r)}$ is indecomposable if and only if $d-r-1 \\equiv 0 \\operatorname{mod} 2^L$ where $2^{L-1} \\le r < 2^L.$ From Theorem REF we can extract the following twisting result: Proposition 5.2 Let $p=2$ and $d \\ge 2r$ .",
"Then: $\\dim \\operatorname{Hom}_{k\\Sigma _d}(S^{(d-r, 1^r)}, S^{(d-r, 1^r)})= \\dim \\operatorname{Hom}_{k\\Sigma _d}(S^{(d-r+2, 1^r)}, S^{(d-r+2, 1^r)}).$ Suppose $2^{L-1} \\le r < 2^L$ .",
"Then $S^{(d-r, 1^r)}$ is indecomposable if and only if $S^{(d-r+2^L, 1^r)}$ is.",
"Since a module is indecomposable if and only if its endomorphism algebra is local, we conclude from Proposition REF (2) that the vector space isomorphisms in Proposition REF (1) are not, in general, algebra isomorphisms.",
"In [7], Dodge and Fayers discovered new infinite series of decomposable Specht modules in characteristic two, the first new examples since Murphy's 1980 paper.",
"Again in their series we see the twisting $\\lambda \\rightarrow \\lambda +p^r\\tau $ occurring.",
"For example a special case of Theorem 3.1 in [7] is: Proposition 5.3 Let $p=2$ .",
"Then $S^{(4+4n,3,1,1)}$ is decomposable for $n \\ge 0$ .",
"We can ask much more generally: Problem 5.4 Find general theorems relating $\\operatorname{Hom}_{k\\Sigma _d}(S^\\lambda , S^\\mu )$ with the space $\\operatorname{Hom}_{k\\Sigma _d}(S^{\\lambda +(p^a)}, S^{\\mu +(p^a)})$ .",
"More generally, inspired by Lyle's work, we could ask for homomorphism results relating $\\lambda + p^r\\tau $ and $\\mu + p^r\\tau $ for more general $\\tau .$ The work in [19] mentioned in Section may be relevant here for Problem REF As for comparing $\\operatorname{Hom}_{k\\Sigma _d}(S^\\lambda , S^\\mu )$ with $\\operatorname{Hom}_{k\\Sigma _d}(S^{p\\lambda }, S^{p\\mu })$ in hopes of a result like Theorem REF , the following example suggests some caution.",
"The computations were done in GAP4 [13] using code written by Matthew Fayers.",
"Example 5.5 Let $p=3$ .",
"Then: $\\dim \\operatorname{Hom}_{k\\Sigma _9}(S^{(7,1,1)}, S^{(3,1^6)}) &= & 0 \\\\\\dim \\operatorname{Hom}_{k\\Sigma _{27}}(S^{(21,3,3)}, S^{(9,3^6)}) &= & 1 \\\\\\dim \\operatorname{Hom}_{k\\Sigma _{81}}(S^{(63,9,9)}, S^{(27,9^6)}) &= & 0.",
"$"
],
[
"Generic cohomology for Specht modules",
"We recently proved a generic cohomology type result for Specht modules.",
"The proof proceeds by translating the problem to $GL_n(k)$ using the result of Kleshchev and Nakano [26]that: $\\operatorname{H}^i(\\Sigma _d, S^\\lambda ) \\cong \\operatorname{Ext}^i_{GL_d(k)}(\\operatorname{H}^0(d), \\operatorname{H}^0(\\lambda ), \\,\\, 0 \\le i \\le 2p-4.$ Using extensive knowledge on the structure of $\\operatorname{H}^0(d)$ worked out by Doty [9] together with knowledge of cohomology for the Borel subgroup $B$ and its Frobenius kernel $B_r$ , we applied the Lyndon-Hochschild-Serre spectral sequence to obtain: Theorem 5.6 Let $p \\ge 3$ and $\\lambda \\vdash d$ .",
"Then $\\operatorname{H}^1(\\Sigma _{pd}, S^{p\\lambda }) \\cong \\operatorname{H}^1(\\Sigma _{p^2d}, S^{p^2\\lambda }).$ Theorem REF can be interpreted as a generic cohomology theorem for Specht modules in degree one.",
"We remark that the same result holds for $\\operatorname{H}^0(\\Sigma _d, S^\\lambda )$ , although somewhat trivially as $\\operatorname{Hom}_{k\\Sigma _{pd}}(k, S^{p\\lambda })$ is always zero unless $\\lambda =(d).$ It is natural to ask if this extends to a more general theorem like Theorem REF , that is: Problem 5.7 Fix $i>0$ .",
"Is there a constant $c(i)$ such that for any $d$ with $\\lambda \\vdash d$ and any $a \\ge c(i)$ that $\\operatorname{H}^i(\\Sigma _{p^ad}, S^{p^a\\lambda }) \\cong \\operatorname{H}^i(\\Sigma _{p^{a+1}d}, S^{p^{a+1}\\lambda })?$ As in the case of the Young module cohomology, the proof of Theorem REF leaves one unable to produce an explicit map, so we can ask: Problem 5.8 Given an element $0 \\rightarrow S^{p\\lambda } \\rightarrow M \\rightarrow k \\rightarrow 0$ in $\\operatorname{H}^1(\\Sigma _{pd}, S^{p\\lambda })$ , can one explicitly construct an extension of $S^{p^2\\lambda }$ by $k$ realizing the isomorphism in Theorem REF ?",
"We also proved a generic cohomology result of the other variety.",
"Namely: Theorem 5.9 Let $\\lambda \\vdash d$ and $p^r>d$ .",
"Then: $\\operatorname{H}^1(\\Sigma _d, S^\\lambda ) = \\operatorname{H}^1(\\Sigma _{d+p^r}, S^{\\lambda +(p^r)}).$ Finally we ask for stronger results like that of Theorem REF .",
"Problem 5.10 Let $\\lambda \\vdash d$ and $\\mu \\vdash c$ .",
"Can one find more results that relate the cohomology $\\operatorname{H}^i(\\Sigma _d, S^\\lambda )$ and $\\operatorname{H}^i(\\Sigma _{d+cp^r}, S^{\\lambda + p^r\\mu })$ ?",
"We end this section by mentioning a different sort of stability result that holds for $\\operatorname{H}^0(\\Sigma _d, S^\\lambda )$ and, in all examples we have computed, also for $\\operatorname{H}^1(\\Sigma _d, S^\\lambda )$ .",
"For an integer $t$ let $l_p(t)$ be the least nonnegative integer satisfying $t<p\\,^{l_p(t)}$ .",
"The following is an easy consequence of Theorem 24.4 in [21]: Lemma 5.11 Suppose $\\lambda =(\\lambda _1, \\lambda _2, \\ldots , \\lambda _s)\\vdash d$ and suppose $a \\equiv -1 \\operatorname{mod } p^{l_p(\\lambda _1)}$ .",
"Then $\\operatorname{H}^0(\\Sigma _d, S^\\lambda ) \\cong \\operatorname{H}^0(\\Sigma _{d+a}, S^{(a,\\lambda _1, \\lambda _2, \\ldots , \\lambda _s)}).$ This leads to the following Problem 5.12 Does the isomorphism in (REF ) hold for $\\operatorname{H}^i$ for any other $i>0$ ?"
],
[
"Complexity of symmetric group modules.",
"Some computer calculations done by the VIGRE algebra group at the University of Georgia suggest that twisting of partitions may arise in determining the complexity of Specht modules.",
"A thorough discussion of the complexity of modules can be found in [1] Recall that an indecomposable module $M$ has complexity the smallest $c=c(M)$ such that the dimensions in a minimal projective resolution are bounded by a polynomial of degree $c-1$ .",
"The maximum possible complexity for $M$ is the $p$ -rank of the defect group of its block, which for the symmetric group is just the $p$ -weight $w$ of the block.",
"Determining the complexity of various $k\\Sigma _d$ modules is an active area of research.",
"The complexity $c(Y^\\lambda )$ was determined in [17].",
"It is worth remarking that for $\\lambda \\vdash d$ it follows immediately from that result that $c(Y^{p\\lambda })=d$ , the maximum possible.",
"Very little is known on the complexity of simple modules $D^\\lambda $ .",
"The paper [17] gives an answer when $D^\\lambda $ is completely splittable.",
"The preprint [29] show that simple modules in Rouquier blocks of weight $w<p$ all have complexity $w$ .",
"In contrast to the “twisted\" Young modules $Y^{p\\lambda }$ having maximal possible complexity, it seems the situation for Specht modules $S^\\lambda $ may be somewhat reversed.",
"Say a partition $\\lambda $ is $p \\times p$ if both $\\lambda $ and $\\lambda ^{\\prime }$ are of the form $p\\tau $ .",
"Equivalently if the Young diagram of $\\lambda $ is made up of $p \\times p$ blocks.",
"The UGA VIGRE Algebra Group made the following conjecture: Conjecture 6.1 (UGA VIGRE This conjecture and some discussion can be found at http://www.math.uga.edu/$\\sim $ nakano/vigre/vigre.html) Let $S^\\lambda $ be in a block $B$ of weight $w$ .",
"Then the complexity of $S^\\lambda $ is $w$ if and only if $\\lambda $ is not $p \\times p$ .",
"In [16] we proved that when $\\lambda $ is $p \\times p$ then its complexity is not maximal, by finding a natural equivalent condition for $p \\times p$ in terms of the abacus display for $\\lambda $ , and then looking at the branching behavior of $S^\\lambda $ .",
"The other (surely more difficult!)",
"direction of the conjecture remains open: Problem 6.2 Resolve the other direction of Conjecture REF .",
"Problem 6.3 Suppose $\\lambda $ is $p \\times p$ of weight $w$ .",
"Is the complexity of $S^\\lambda $ equal to $w-1$ , or can it be less than $w-1$ ?",
"Problem REF has been resolved only in the case $\\lambda =(p, p, \\ldots , p) \\vdash p^2$ .",
"In this case the support variety was computed explicitly by Lim [28], and its dimension (which equals the complexity) is indeed $p-1$ .",
"The support variety for the Specht module $S^{(3,3,3)}$ in characteristic three provides a motivating example in Chapter 7 of the book [2] as a small-dimensional module with a very interesting support variety.",
"Perhaps further twisting might lower the complexity even more: Problem 6.4 One can generalize the definition of $p \\times p$ in several ways.",
"For example one obvious generalization would be to require $\\lambda $ be $p^2 \\times p^2$ .",
"Can one say anything interesting about these situations?",
"Perhaps the complexity drops by even more in this case?",
"For example can one determine the complexity of $S^{(9^9)}$ in characteristic three?"
],
[
"Generic cohomology for simple modules and twists of the Mullineux map",
"Recall that the irreducible modules for $k\\Sigma _d$ are labeled by $p$ -regular partitions and are denoted $D^\\lambda $ .",
"However there is another indexing, by $p$ -restricted partitions and denoted $D_\\mu $ .",
"The latter is more natural in some ways, as $D_\\mu $ is the image under the Schur functor of the irreducible $G$ module $L(\\mu )$ .",
"The labellings are related by: $D^\\lambda \\otimes \\operatorname{sgn}\\cong D_{\\lambda ^{\\prime }}.$ It was a longstanding open problem to determine the partition $m(\\lambda )$ so that $D^\\lambda \\otimes \\operatorname{sgn}\\cong D^{m(\\lambda )}$ .",
"This problem was finally solved by Kleshchev in [24].",
"A short time later Ford and Kleshchev [12] confirmed that Kleshchev's answer agreed with the original conjecture made by Mullineux in [32].",
"We will find it useful to use Mullineux's original algorithm and also a different (but of course equivalent) description given later by Xu in [34].",
"Both are nicely described in [3].",
"It follow from (REF ) that: $D^\\lambda \\cong D_{m(\\lambda )^{\\prime }}$ so one can easily arrive at a version of the Mullineux bijection, except on $p$ -restricted partitions; namely $\\lambda \\rightarrow m(\\lambda ^{\\prime }) ^{\\prime }.$"
],
[
"Generic cohomology for two-part irreducibles",
"Suppose one wanted a generic cohomology theorem for extensions between simple $k\\Sigma _d$ modules.",
"The simple module $L(\\lambda )$ corresponds to $D_\\lambda $ , but $p\\lambda $ is never $p$ -restricted so $D_{p\\lambda }$ does not exist.",
"Strangely though something seems to be going on with the “wrong\" upper notation.",
"For example: Proposition 7.1 Assume $p>2$ .",
"Let $\\lambda =(v,u)$ and $\\mu =(s,r)$ be partitions of $d$ .",
"Then $\\operatorname{Ext}^1_{k\\Sigma _{pd}}(D^{p\\lambda }, D^{p\\mu }) \\cong \\operatorname{Ext}^1_{k\\Sigma _{p^2d}}(D^{p^2\\lambda }, D^{p^2\\mu }).$ Assume $u \\ge r$ without loss of generality.",
"All the extensions between two-part simple modules were worked out by Kleshchev and Sheth in [27] (but see the Corrigendum [25]).",
"In their notation we have $pv-pu+1 = 1 + \\sum _{i \\ge 1}a_ip^i$ and the condition for the $\\operatorname{Ext}$ group to be nonzero is that $pu-pr=(p-a_i)p^i$ for some $i$ such that $a_i >0$ and either $a_{i+1} < p-1$ or $u < p^{i+1}$ .",
"This condition is clearly equivalent to the corresponding one for $p^2v-p^2u+1$ and $p^2u-p^2r.$ We remark that Proposition REF requires the additional twist before the cohomology stabilizes.",
"For example when $p=3$ one can use Kleshchev-Sheth's result to compute: $\\operatorname{Ext}^1_{\\Sigma _{29}}(D^{(20,9)}, D^{(26,3)}) &\\cong & k\\\\\\operatorname{Ext}^1_{\\Sigma _{87}}(D^{(60,27)}, D^{(78,9)}) &\\cong & 0.$ Of course this suggests the following: Conjecture 7.2 Let $\\lambda , \\mu \\vdash d$ .",
"Then: $\\operatorname{Ext}^1_{k\\Sigma _{pd}}(D^{p\\lambda }, D^{p\\mu }) \\cong \\operatorname{Ext}^1_{k\\Sigma _{p^2d}}(D^{p^2\\lambda }, D^{p^2\\mu }).$ Once again we have a vector space isomorphism in cohomology (Proposition REF ) with no module homomorphisms realizing it!"
],
[
"Mullineux map and twists",
"The labelling of irreducibles $D_\\mu $ by $p$ -restricted partitions seems more natural when comparing with $GL_n(k)$ (where actual Frobenius twists can occur).",
"But $p\\lambda $ is never $p$ -restricted, and we observed above a relationship between $D^{p\\lambda }$ and $D^{p^2\\lambda }$ .",
"Using (REF ) suggests some relationship between $m(p\\lambda )^{\\prime }$ and $m(p^2\\lambda )^{\\prime }$ .",
"This led us to a strictly combinatorial question, namely is there any relation between twisting and the Mullineux map?",
"And then of course given any such relation, is there a representation-theoretic interpretation?",
"We have found a large class of partitions that have interesting behavior here.",
"For example if $\\lambda =(\\lambda _1, \\lambda _2, \\ldots , \\lambda _s) \\vdash d$ is a partition with distinct parts, define $\\hat{\\lambda }=(\\lambda _1^{p-1}, \\lambda _2^{p-1}, \\ldots , \\lambda _s^{p-1}) \\vdash (p-1)d.$ Proposition 7.3 Suppose $\\lambda =(\\lambda _1, \\lambda _2, \\ldots , \\lambda _s) \\vdash d$ has distinct parts.",
"Then $m(\\hat{\\lambda })=(p-1)\\lambda $ .",
"Consider the $p$ -regular version of Xu's algorithm from [34] (nicely described in [3]).",
"If we apply it to calculate ${\\hat{\\lambda }}^X$ using [3] we obtain $j_1=j_2= \\cdots = j_{p-1}=s$ .",
"At this point in the algorithm the first column (consisting of $s(p-1)$ nodes) will have been removed from $\\hat{\\lambda }$ , and what remains is $\\widehat{\\overline{\\lambda }}$ where $\\overline{\\lambda }$ denotes $\\lambda $ with its first column removed.",
"One can now apply induction using [3] or just continue with the algorithm to obtain $(p-1)\\lambda .$ Corollary 7.4 Let $\\lambda \\vdash d$ have distinct parts.",
"Then $m(p\\mu )=pm(\\mu )$ for both $\\mu =(p-1)\\lambda $ and $\\mu =\\hat{\\lambda }.$ By Proposition REF we have: $m((p)(p-1)\\lambda ) &=& \\widehat{p\\lambda } \\\\&=& p\\hat{\\lambda } \\\\&=&pm((p-1)\\lambda )$ and $m(p\\hat{\\lambda }) &=& m(\\widehat{p\\lambda }) \\\\&=& p(p-1)\\lambda \\\\&=&pm(\\hat{\\lambda }).$ The appearance above of $p(p-1)\\lambda $ for $\\lambda $ having distinct parts is reminiscent of the twist of the Steinberg weight $p(p-1)\\rho $ from the $GL_n(k)$ theory, although we have no representation-theoretic interpretation at this time.",
"The examples in Corollary REF are not the only ones where $m(p\\mu )=pm(\\mu )$ although other examples seem to be rare.",
"For example if $p=5$ and $d=20$ then Corollary REF yields all $\\mu \\vdash 20$ where $m(5\\mu )=5m(\\mu )$ .",
"On the other hand for $p=5$ and $\\lambda =(3,3)$ we have $m(15,15)=(10,10,10)=5m(3,3)$ , although $\\lambda $ does not have distinct parts..",
"This brings us to: Problem 7.5 Classify all $\\lambda $ such that $m(p\\lambda )=pm(\\lambda )$ and give a representation-theoretic interpretation of the answer.",
"The subset of such $\\lambda $ arising in Corollary REF is certainly closed under twisting, as are all other examples we have computed, so we conjecture: Conjecture 7.6 Suppose $m(p\\lambda )=pm(\\lambda )$ .",
"Then $m(p^2\\lambda )=pm(p\\lambda )$ .",
"More generally we can ask: Problem 7.7 Classify all $\\lambda $ such that $m(p\\lambda )=p\\tau $ for some $\\tau .$ For example when $p=5$ and $\\lambda =(6,6,4)$ we have: $m(30,30,20)=(20^3,5^4)=5m(10,3,3).$ Even for partitions without such a nice relationship, there still seems to be some intriguing behavior among the $m(p^a\\lambda )$ for various $a$ .",
"For example we will now derive a sort of Steinberg tensor product theorem for Mullineux maps.",
"First define $\\tau _n=m(1^n)$ , so the trivial $k\\Sigma _n$ module is $D_{\\tau _n}$ .",
"It is well known that $\\tau _n=(p-1,p-1, p-1, \\cdots , p-1, a)$ .",
"Lemma 7.8 Suppose $\\lambda =(\\lambda _1, \\lambda _2, \\ldots , \\lambda _s)$ is a partition with distinct parts.",
"Then: $m(p\\lambda )^{\\prime }= \\tau _{p\\lambda _1} + \\tau _{p\\lambda _2} + \\cdots + \\tau _{p\\lambda _s}.$ For example if $\\lambda =(4,2,1)$ and $p=5$ then $m(5\\lambda )^{\\prime } = (12, 9, 6, 4, 4)=(4^5)+(4,4,2)+(4,1)=\\tau _{20} + \\tau _{10} + \\tau _5.$ The proof is by induction on $s$ where the case $s=1$ is just the definition of $\\tau _n$ .",
"We will use the original algorithm of Mullineux, described and proved in [12].",
"Since the parts of $\\lambda $ are distinct, then all the rim $p$ -hooks removed in determining the Mullineux symbol $G_p(p\\lambda )$ are horizontal.",
"Thus we determine the Mullineux symbol $G_p(m(p\\lambda ))= $ $\\left(\\begin{array}{llllllllll}sp & \\cdots & sp & (s-1)p & \\cdots & (s-1)p & \\cdots & p & \\cdots & p \\\\s(p-1) & \\cdots & s(p-1) & (s-1)(p-1) & \\cdots & (s-1)(p-1) & \\cdots & p-1 & \\cdots & p-1 \\\\\\end{array}\\right)$ where the first column in (REF ) occurs $\\lambda _s$ times, the second occurs $\\lambda _{s-1}-\\lambda _s$ times, the third $\\lambda _{s-2}-\\lambda _{s-1}$ , etc.",
"So the last column occurs $\\lambda _1-\\lambda _2$ times.",
"Notice then that removing the columns $\\begin{array}{l}sp \\\\s(p-1)\\end{array}$ we obtain the Mullineux symbol for $p(\\lambda _1-\\lambda _s, \\lambda _2-\\lambda _s, \\ldots , \\lambda _{s-1}-\\lambda _s)$ .",
"So we apply induction and the result follows.",
"Using the result above and drawing a diagram with the appropriate $\\tau _{p\\lambda _i}$ and $\\tau _{p^2\\lambda _i}$ , the following corollary is immediate: Corollary 7.9 Suppose $\\lambda $ has distinct parts.",
"Then: $m(p^2\\lambda )-m(p\\lambda )=p\\hat{\\lambda }.$ Using Proposition REF one can rewrite this as: $m(p^2\\lambda )-m(p\\lambda )=pm((p-1)\\lambda ).$ For arbitrary $\\lambda $ with repeated parts we conjecture a weaker form of stability after multiple “twists:\" Conjecture 7.10 Let $\\lambda \\vdash d$ .",
"Then there exist $1 \\le a<b$ such that: $m(p^b\\lambda )=m(p^a\\lambda ) + p^a\\tau .$ As an example of Conjecture REF needing multiple twists consider the following: Example 7.11 Let $p=7$ and $\\lambda =(29^2, 24, 4^2, 3^3, 2, 1).$ Then: $m(7^5\\lambda )-m(7\\lambda )=7(123840^5, 9600^5, 5400^4, 3840^5, 800^6, 400^6).$ If $1\\le x<y <5$ then $m(7^y\\lambda )-m(7^x\\lambda )$ is not of the form $7\\tau $ , i.e.",
"this stability really requires at least five twists.",
"Finally we remark that the results above on the Mullineux map did not depend on $p$ being prime, and any possible representation-theoretic interpretations might hold true for the Hecke algebra of type $A$ at an $e$ th root of unity.",
"It seems fitting to close with a completely general (and quite possibly absurd) question: Problem 7.12 Can one say anything interesting about how the structure of the principal block $B_0(k\\Sigma _{pd})$ is reflected inside $B_0(k\\Sigma _{p^2d})$ ?",
"For example Theorem REF is a statement about Cartan invariants."
]
] | 1204.1045 |
[
[
"Asymptotic behavior of solutions to the Helmholtz equations with sign\n changing coefficients"
],
[
"Abstract This paper is devoted to the study of the behavior of the unique solution $u_\\delta \\in H^{1}_{0}(\\Omega)$, as $\\delta \\to 0$, to the equation \\begin{equation*} \\dive(\\epss_\\delta A \\nabla u_{\\delta}) + k^2 \\epss_0 \\Sigma u_{\\delta} = \\epss_0 f \\mbox{in} \\Omega, \\end{equation*} where $\\Omega$ is a smooth connected bounded open subset of $\\mR^d$ with $d=2$ or 3, $f \\in L^2(\\Omega)$, $k$ is a non-negative constant, $A$ is a uniformly elliptic matrix-valued function, $\\Sigma$ is a real function bounded above and below by positive constants, and $\\epss_\\delta$ is a complex function whose {\\bf the real part takes the value 1 and -1}, and the imaginary part is positive and converges to 0 as $\\delta$ goes to 0.",
"This is motivated from a result in \\cite{NicoroviciMcPhedranMilton94} and the concept of complementary suggested in \\cite{LaiChenZhangChanComplementary, PendryNegative, PendryRamakrishna}.",
"After introducing the reflecting complementary media, complementary media generated by reflections, we characterize $f$ for which $\\|u_\\delta\\|_{H^1(\\Omega)}$ remains bounded as $\\delta$ goes to 0.",
"For such an $f$, we also show that $u_\\delta$ converges weakly in $H^1(\\Omega)$ and provide a formula to compute the limit."
],
[
"Introduction",
"Negative index materials (NIMs) were first investigated theoretically by Veselago in [22] and were innovated by Pendry in [15].",
"The existence of such materials was confirmed by Shelby, Smith, and Schultz in [20] (see also [21]).",
"Cloaking space and illusion optics using NIMs were discussed in [7], [8] (see also [14]) based on the concept of complementary and in [2], [3], [5], [6], [11], [12], [13], [14] based on the anomalous localized resonance.",
"Perfect lens using NIMs was studied in [13], [14], [16], [19].",
"The first motivation of this work comes from the following two dimensional result of Nicorovici et.",
"al.",
"in [14].",
"Let $0 < r_{1} < r_{2} < R$ , and $f \\in L^2(B_R)$ .",
"Here and in what follows, for $r>0$ , $B_{r}$ denotes the ball centered at 0 of radius $r$ .",
"Set $r_3 = r_2^2/ r_1$ .",
"Assume that $R > r_3$ and $\\operatorname{supp}f \\cap \\lbrace x \\; |x| < r_3\\rbrace = Ø$ .",
"Let $u_\\delta \\in H^1_0(B_R)$ be the unique solution to the equation $\\operatorname{div}(\\varepsilon _\\delta \\nabla u_\\delta ) = f \\mbox{ in } B_R .$ Here $\\varepsilon _\\delta = \\varepsilon + i 1_{r_1 < |x| < r_2},$ where $\\varepsilon (x) = \\left\\lbrace \\begin{array}{cl}-1 & \\mbox{ if } r_{1} < |x| < r_{2}, \\\\[6pt]1 & \\mbox{ otherwise}.\\end{array} \\right.$ Physically, the imaginary part of $\\varepsilon _{\\delta }$ is the loss of the medium.",
"It is showed in [14] by separation of variables that $u_\\delta \\rightarrow {\\cal U} \\mbox{ for } |x| > r_3,$ where ${\\cal U} \\in H^1_0(B_R)$ is the unique solution to the equation $\\Delta {\\cal U}= f \\mbox{ in } B_R.$ The surprising fact on this result is that (REF ) holds for any $f$ with $\\operatorname{supp}f \\cap B_{r_3} = Ø$ .",
"From (REF ), one might say that the region $\\lbrace r_{2} < |x| < r_{3}\\rbrace $ is canceled by the one in $\\lbrace r_{1} < |x| < r_{2}\\rbrace $ and the total system is effectively equal to the free space; invisibility is achieved.",
"The following questions naturally arise: What happens if $\\operatorname{supp}f \\cap B_{r_3} \\ne Ø$ ?",
"Is the radial symmetry necessary?",
"If not, what are conditions on $\\varepsilon $ ?",
"What happens in the finite frequency regime?",
"Do similar phenomena hold in three dimensions?",
"If yes, under which conditions?",
"Another motivation of this work is the concept of complementary media which was suggested in [7], [18], [19] (see also [14], [15]).",
"This concept has played an important role in the study of NIMs and its applications such as cloaking, perfect lens, and illusion optics see [7], [8], [12], [15], [19].",
"Although many examples have been suggested, this concept has not be defined in a precise manner.",
"A common point in examples studied is $F_{*} a = - b \\mbox{ in } D_{2},$ for some differomorphism $F: D_{1} \\rightarrow D_{2}$ if a matrix $a$ defined in a region $D_{1}$ is complementary to a matrix $b$ defined in a region $D_{2}$ .",
"Here $F_*a(y) = \\frac{D F (x) a(x) D F ^{T}(x)}{J(x)} \\mbox{ where } x =F^{-1}(y) \\mbox{ and } J(x) = |\\det D F(x)|.$ It is easy to verify in two dimensions that if $F : \\lbrace r_{1} < |x| < r_{2}\\rbrace \\rightarrow \\lbrace r_{2} < |x| < r_{3}\\rbrace $ with $r_{3} = r_{2}^{2}/ r_{1}$ defined by $F(x) = r_{2}^{2} x /|x|^{2}$ then $F$ is a diffeomorphism and $F*(-I) = - I \\mbox{ in } \\lbrace r_{2} < |x| < r_{3}\\rbrace .$ In other words, the medium $-I$ in $\\lbrace r_{1} < |x| < r_{2}\\rbrace $ is complementary to the medium $I$ in $\\lbrace r_{2} < |x| < r_{3}\\rbrace $ : complementary media appears in the setting of Nocorovici et.",
"al.'s.",
"In this paper, we address the above questions.",
"For this end, we first introduce the notion of reflecting complementary media.",
"Similar phenomena as in (REF ) take place for media inherits this property in the quasistatic and the finite frequency regimes.",
"Two media (two matrices in two regions in the quasistatic regime) are called reflecting complementary if they are complementary and the complementary is generated by a reflection which satisfies some mild conditions.",
"The motivation for the definition of this notion comes from the reflection technique used in an heuristic argument for (REF ) in Section REF .",
"We then establish results in the spirit of (REF ) for media of this property.",
"The analysis again has root from the heuristic argument.",
"The key of the analysis is the derivation of two Cauchy problems for elliptic equations by the relection technique using the reflecting complementary property.",
"Concerning the analysis, we characterize $f$ (based on the compatibility condition in Definition REF ) for which $\\Vert u_\\delta \\Vert _{H^1}$ remains bounded as $\\delta \\rightarrow 0_+$ ; moreover, we show that for such a function $f$ , the limit of $u_\\delta $ exists as $\\delta \\rightarrow 0_{+}$ (Theorem REF ).",
"We also provide a formula to compute the limit which involves only the solutions of standard elliptic equations (no sign changing coefficients), and show that the limit has properties in the spirit of (REF ) (Theorem REF ).",
"To our knowledge, the results presented in this paper are new even in the 2d-quasistatic regime.",
"The use of reflections to study NIMs has been considered previously in [12].",
"However, there is a big difference between the use of reflections in [12] and in this paper.",
"In [12], the authors used reflections as a change of variables to obtain a new simple setting from an old more complicated one and hence the analysis of the old problem becomes simpler.",
"In this paper, we use reflections to derive the two Cauchy problems.",
"This derivation makes use essentially the complementary property of media.",
"The global or non-glocal existence of the solutions to these Cauchy problems will determine the boundedness or unboundedness of the $H^{1}$ -norm of the solutions.",
"The goals and the setting of this paper are different from the ones of Ammari et al.",
"'s in [2] and Kohn et.",
"al.",
"'s in [6].",
"In this paper, we introduce the concept of reflecting complementary media and we study the boundedness of $\\Vert u_{\\delta }\\Vert _{H^{1}}$ and the limit of $u_{\\delta }$ in the whole domain as $\\delta \\rightarrow 0$ in the quasistatic and the finite frequency regimes.",
"In [2], [6], the authors investigate the unboundedness of $\\delta ^{1/2} \\Vert u_{\\delta }\\Vert _{H^{1}}$ for piecewise constant media (up to a diffeomorphism in [6]) in the quasistatic case.",
"It is clear that the boundedness of $\\Vert u_\\delta \\Vert _{H^1}$ implies the boundedness of $\\delta ^{1/2}\\Vert u_\\delta \\Vert _{H^1}$ and the unboundedness of $\\delta ^{1/2}\\Vert u_\\delta \\Vert _{H^1}$ implies the unboundedness of $\\Vert u_\\delta \\Vert _{H^1}$ .",
"The media considered in [6] are of the reflecting complimentary property; however the ones in [2] are, in general, not (the radial setting considered in [2] is an exclusion).",
"In [2], the authors also deal with the boundedness of $u_{\\delta }$ in some region.",
"To make use of their results mentioned above, one needs detailed information on the spectral properties of certain boundary integral operators.",
"This information is difficult to come by in general.",
"The method in this paper is different from and more elementary than the spectral one in [2] and the variational one in [6].",
"The method in this paper is used in [9] and developed in [10].",
"In [9] the authors study the complete resonance and localized resonance in plasmonic structures whileas in [10] the author investigate the cloaking via complementary media.",
"Our paper is organized as follows.",
"Section contains two subsections devotes to the concept of reflecting complementary media.",
"In the first subsection, we present the heuristic argument to obtain (REF ) and to motivate the definition of this concept.",
"The second subsection devotes to the definition.",
"In Section , we state and prove properties on the reflecting complementary media.",
"More precisely, we state and prove Theorems REF and REF and present their two corollaries there."
],
[
"Reflecting complementary media",
"In this section, we introduce the notion of reflecting complementary media.",
"To motivate the definition, we first present an heuristic argument, in Section REF , to obtain (REF ) based on the reflecting technique.",
"The definition of reflecting complementary media is introduced in Section ."
],
[
"Motivation - an heuristic argument for (",
"In this section, we present an heuristic (elementary) argument to obtain (REF ).",
"This argument motivates not only the notion of the reflecting complementary media but also the analysis in Section .",
"In this section, we assume that $u_\\delta \\rightarrow u \\in H^1(B_{R})$ .",
"It follows that $u \\in H^1_{0}(B_{R})$ is a solution to the equation $\\operatorname{div}(\\varepsilon \\nabla u ) = f \\mbox{ in } B_{R}.$ Let $F$ defined in $\\lbrace |x| < r_{2} \\rbrace $ be the Kelvin transform w.r.t.",
"$\\partial B_{r_{2}}$ , i.e., $F(x) = \\frac{r_{2}^2}{|x|^{2}} x \\mbox{ for } |x| < r_{2}.$ Let $u_{1}$ defined in $\\lbrace |x| > r_{2}\\rbrace $ be the Kelvin transform $F$ of $u$ , i.e., $u_{1}(x) = u \\circ F^{-1} (x) \\quad \\mbox{ for } |x| > r_{2}.$ Then, by the transmission condition on $\\partial B_{r_2}$ , we have $u_{1} = u \\quad \\mbox{ and } \\quad \\partial _{r} u_{1} \\Big |_{r \\rightarrow {r_{2}}_{+}}= \\partial _{r} u \\Big |_{r \\rightarrow {r_{2}}_{+}}\\quad \\mbox{ for } |x| = r_{2}.$ Since $F$ is a Kelvin transform and $\\operatorname{supp}f \\cap \\lbrace |x| < r_{3} \\rbrace = Ø$ , it follows that $\\operatorname{div}(\\hat{\\varepsilon }\\nabla u_{1}) = 0 \\mbox{ for } |x| > r_{2},$ where $\\hat{\\varepsilon }(x) = \\left\\lbrace \\begin{array}{cl} 1 & \\mbox{ if } r_{2 } < |x| < r_3, \\\\[6pt]- 1& \\mbox{ if } |x| > r_3.\\end{array} \\right.$ (Note that $F$ transforms $\\partial B_{r_{1}}$ into $\\partial B_{r_3}$ .)",
"By the unique continuation principle, we have $u_{1} = u \\mbox{ in } \\lbrace r_{2} < |x| < r_{3}\\rbrace .$ Let $G$ defined in $\\lbrace |x| > r_{3} \\rbrace $ be the Kelvin transform w.r.t.",
"$\\partial B_{r_3}$ , i.e., $G(x) = \\frac{r_{3}^2}{|x|^{2}} x \\mbox{ for } |x| > r_{3}.$ Define $u_{2}$ in $\\lbrace |x| < r_{3}\\rbrace $ , the Kelvin transform $G$ of $u_{1}$ , as follows $u_{2} (x)= u_{1} \\circ G^{-1} (x) \\quad \\mbox{ for } |x| < r_{3}.$ Similar to (REF ), we have $u_{2} = u_{1} \\quad \\mbox{ and } \\quad \\partial _{r} u_{2} \\Big |_{r \\rightarrow {r_{3}}_{-}}= \\partial _{r} u_{1} \\Big |_{r \\rightarrow {r_{3}}_{-}}\\quad \\mbox{ for } |x| = r_{3}.$ It follows from (REF ) that $u_{2} = u \\quad \\mbox{ and } \\quad \\partial _{r} u_{2} \\Big |_{r \\rightarrow {r_{3}}_{-}} = \\partial _{r} u \\quad \\mbox{ for } |x| = r_{3}.$ We also have $\\Delta u_{2} = 0 \\mbox{ in } |x| < r_{3},$ by the property of the Kelvin transforms.",
"Define ${\\cal U}$ by ${\\cal U} (x)= \\left\\lbrace \\begin{array}{cl} u (x) & \\mbox{ if } |x| > r_{3}, \\\\[6pt]u_{2} (x), & \\mbox{ if } |x| < r_{3}.\\end{array} \\right.$ Since $\\Delta u = f$ for $|x| > r_{3}$ , it follows from (REF ) and (REF ) that $\\Delta {\\cal U} = f \\mbox{ in } B_{R}.$ Therefore, we obtain (REF )."
],
[
"Reflecting complementary media",
"In this section, we introduce the notion of reflecting complementary media.",
"Let $\\Omega _{1} \\subset \\subset \\Omega _{2} \\subset \\subset \\Omega _{3} \\subset \\subset \\Omega $ be smooth connected bounded open subsets of $\\mathbb {R}^{d}$ ($d = 2, \\, 3$ ).",
"Let $A$ be a measurable matrix function and $\\Sigma $ be a measurable real function defined in $\\Omega $ such that $\\frac{1}{\\Lambda } |\\xi |^2 \\le \\langle A(x) \\xi , \\xi \\rangle \\le \\Lambda |\\xi |^2 \\quad \\forall \\, \\xi \\in \\mathbb {R}^d,$ for a.e.",
"$x \\in \\Omega $ and for some $0< \\Lambda < + \\infty $ and $0 < \\mathop {\\mathrm {ess} \\,\\inf }_{\\Omega } \\Sigma \\le \\mathop {\\mathrm {ess} \\,sup}_{\\Omega } \\Sigma < +\\infty .$ Set $s_\\delta (x) = \\left\\lbrace \\begin{array}{cl} -1 + i \\delta & \\mbox{ if } x \\in \\Omega _2 \\setminus \\Omega _1, \\\\[6pt]1 & \\mbox{ otherwise}.\\end{array}\\right.$ We are later interested in the behavior of the unique solution $u_\\delta \\in H^1_0(\\Omega )$ to the equation $\\operatorname{div}(s_\\delta A \\nabla u_\\delta ) + k^{2} s_{0} \\Sigma u_{\\delta } = s_0 f \\mbox{ in } \\Omega ,$ as $\\delta \\rightarrow 0$ .",
"We are ready to give Definition 1 (Reflecting complementary media) The media $(A, \\Sigma )$ in $\\Omega _{3} \\setminus \\Omega _{2}$ and $(-A, - \\Sigma )$ in $\\Omega _{2} \\setminus \\Omega _{1}$ are said to be reflecting complementary if there exists a diffeomorphism $F: \\Omega _{2} \\setminus \\bar{\\Omega }_{1} \\rightarrow \\Omega _{3} \\setminus \\bar{\\Omega }_{2}$ such that $F_*A (x) = A(x), \\quad F_{*} \\Sigma (x) = \\Sigma (x) \\mbox{ for } x \\in \\Omega _3 \\setminus \\bar{\\Omega }_2,$ $F(x) = x \\mbox{ on } \\partial \\Omega _2.$ and the following two conditions hold: There exists an diffeomorphism extension of $F$ , which is still denoted by $F$ , from $\\Omega _{2} \\setminus \\lbrace x_{1}\\rbrace \\rightarrow \\Omega _{4} \\setminus \\bar{\\Omega }_{2}$ for some $x_{1} \\in \\Omega _{1}$ and some smooth open subset $\\Omega _{4}$ of $\\mathbb {R}^{d}$ with $\\Omega _{3} \\subset \\Omega _{4}$ .",
"There exists a diffeomorphism $G: \\Omega _{4} \\setminus \\bar{\\Omega }_{3} \\rightarrow \\Omega _{3} \\setminus x_{2}$ for some $x_{2} \\in \\Omega _{3}$ such that In (REF ) and (REF ), $F$ and $G$ denote some diffeomorphism extensions of $F$ and $G$ in a neighborhood of $\\partial \\Omega _2$ and of $\\partial \\Omega _3$ .",
"$\\ \\quad G(x) = x \\mbox{ on } \\partial \\Omega _3,$ and $G \\circ F : \\Omega _1 \\rightarrow \\Omega _3 \\mbox{ is a diffeomorphism if one sets } G\\circ F(x_1) = x_2.$ Here and in what follows, we use the standard notations: ${\\cal F}_*{\\cal A}(y) = \\frac{D {\\cal F} (x) {\\cal A}(x) D {\\cal F}^{T}(x)}{J(x)}, \\;\\;\\; {\\cal F}_* {\\it \\Sigma }(y) = \\frac{{\\it \\Sigma }(x)}{J(x)}, \\;\\;\\; \\mbox{and} \\;\\;\\; {\\cal F}_* \\textsl {f}(y) = \\frac{ \\textsl {f} (x)}{J(x)},$ where $x = {\\cal F}^{-1}(y)$ and $J(x) = |\\det D {\\cal F}(x)|$ .",
"Some comments on the definition are: If $k=0$ , then the condition on $\\Sigma $ is irrelevant in Definition REF .",
"Condition (REF ) implies that $(A, \\Sigma )$ in $\\Omega _{3} \\setminus \\Omega _{2}$ and $(-A, - \\Sigma )$ in $\\Omega _{2} \\setminus \\Omega _{1}$ are complementary in the usual sense.",
"The term “reflecting” in the definition comes from (REF ) and the assumption $\\Omega _{1} \\subset \\Omega _{2} \\subset \\Omega _{3}$ .",
"Conditions (REF ) and (REF ) are the main assumptions in the definition.",
"Condition (REF ) makes sure that $u$ (the “solution” for $\\delta =0$ ) and $u_{1} : = u \\circ F$ satisfy the same equation in $\\Omega _{3} \\setminus \\Omega _{2}$ ; hence the reflecting technique in Section REF can be used.",
"Condition (REF ) and (REF ) assure that $u = u_1$ on $\\partial \\Omega _2$ and $u_2 = u_1$ on $\\partial \\Omega _3$ where $u_2 = u_1 \\circ G^{-1}$ .",
"Condition (REF ) is a technical one which is required by the proof.",
"Conditions 1) and 2) in the definition are mild assumptions.",
"Introducing $G$ in the definition makes the analysis more accessible as in Section REF (see Remark for other comments on $G$ ).",
"In general, it is difficult to verify whether (REF ) holds for some $F$ .",
"In practice, to obtain the reflecting complementary in $\\Omega _{2} \\setminus \\Omega _{1}$ for $(A, \\Sigma )$ in $\\Omega _{3} \\setminus \\Omega _{2}$ , it suffices to choose a diffeomorphism $F: \\Omega _2 \\setminus \\lbrace x_1\\rbrace \\rightarrow \\Omega _4 \\setminus \\bar{\\Omega }_2$ for some $x_{1} \\in \\Omega _{1}$ and for some smooth bounded open subset $\\Omega _{4}$ containing $\\Omega _{3}$ , and define $(-A, -\\Sigma )$ in $\\Omega _{2} \\setminus \\Omega _{1}$ by $(-F^{-1}_{*} A, -F^{-1}_{*} \\Sigma )$ .",
"It is clear that the medium $\\varepsilon $ given in (REF ) has the reflecting complementary property with $\\Omega _i = B_{r_i}$ (for $i=1, 2, 3$ ) and $\\Omega = B_R$ where $r_4 : = +\\infty $ , and $F$ and $G$ are the Kelvin transforms w.r.t.",
"$\\partial B_{r_{2}}$ and $\\partial B_{r_{3}}$ resp.",
"Remark 1 Concerning reflecting complementary media, the 2d quasistatic case is quite special in the sense that two constant media (two constant matrices) can be complementary (see the example in the introduction).",
"In fact, in the 2d finite frequency case, it seems that there do not exist two constant media (two constant matrices and two constant functions) which are complementary and in the 3d case there do not exist two constant media (two constant matrices) which are complementary."
],
[
"Statement of the results",
"Let $\\Omega _{1} \\subset \\subset \\Omega _{2} \\subset \\subset \\Omega _{3} \\subset \\subset \\Omega $ be smooth connected bounded open subsets of $\\mathbb {R}^{d}$ ($d = 2, \\, 3$ ).",
"Let $A$ be a measurable matrix function and $\\Sigma $ be a measurable real function defined in $\\Omega $ such that $\\frac{1}{\\Lambda } |\\xi |^2 \\le \\langle A(x) \\xi , \\xi \\rangle \\le \\Lambda |\\xi |^2 \\quad \\forall \\, \\xi \\in \\mathbb {R}^d,$ for a.e.",
"$x \\in \\Omega $ and for some $0< \\Lambda < + \\infty $ , and $0 < \\mathop {\\mathrm {ess} \\,\\inf }_{\\Omega } \\Sigma \\le \\mathop {\\mathrm {ess} \\,sup}_{\\Omega } \\Sigma < +\\infty .$ We will assume that $A$ is piecewise differentiable in $\\Omega $ in three dimensions This condition is necessary to obtain the uniqueness for the Cauchy problems.. Set $s_\\delta (x) = \\left\\lbrace \\begin{array}{cl} -1 + i \\delta & \\mbox{ if } x \\in \\Omega _2 \\setminus \\Omega _1, \\\\[6pt]1 & \\mbox{ otherwise}.\\end{array}\\right.$ We are interested in the behavior of the unique solution $u_\\delta \\in H^1_0(\\Omega )$ to the equation $\\operatorname{div}(s_\\delta A \\nabla u_\\delta ) + k^{2} s_{0} \\Sigma u_{\\delta } = s_0 f \\mbox{ in } \\Omega ,$ as $\\delta \\rightarrow 0$ .",
"Throughout this section, we will assume that: $\\mbox{Systems (\\ref {frequency-assumption}) and (\\ref {frequency-assumption-2}) have only zero solutions in $H^1(\\Omega \\setminus \\bar{\\Omega }_2)$ and $H^1(\\Omega )$ resp.",
"},$ where $\\left\\lbrace \\begin{array}{cl}\\operatorname{div}(A \\nabla v) + k^2 \\Sigma v = 0 & \\mbox{ in } \\Omega \\setminus \\bar{\\Omega }_2,\\\\[6pt]v = 0 & \\mbox{ on } \\partial \\Omega \\cup \\partial \\Omega _{2},\\end{array}\\right.$ and $\\left\\lbrace \\begin{array}{cl}\\operatorname{div}(\\hat{A} \\nabla U) + k^2 \\hat{\\Sigma }U = 0 & \\mbox{ in } \\Omega , \\\\[6pt]U = 0 & \\mbox{ on } \\partial \\Omega .\\end{array} \\right.$ Here $\\hat{A}$ and $\\hat{\\Sigma }$ are defined as follows $\\hat{A} := \\left\\lbrace \\begin{array}{cl}A & \\mbox{ if } x \\in \\Omega \\setminus \\Omega _3,\\\\[6pt]G_*F_*A & \\mbox{ if } x \\in \\Omega _3,\\end{array}\\right.\\quad \\mbox{ and } \\quad \\hat{\\Sigma }:= \\left\\lbrace \\begin{array}{cl}\\Sigma & \\mbox{ if } x \\in \\Omega \\setminus \\Omega _3,\\\\[6pt]G_*F_*\\Sigma & \\mbox{ if } x \\in \\Omega _3.\\end{array}\\right.$ We also define $\\hat{f} : = \\left\\lbrace \\begin{array}{cl}f & \\mbox{ if } x \\in \\Omega \\setminus \\Omega _3,\\\\[6pt]G_*F_*f & \\mbox{ if } x \\in \\Omega _3.\\end{array}\\right.$ Remark 2 The well-posedness of (REF ) and (REF ) always hold for $k=0$ .",
"In the case $k>0$ , if one is interested the corresponding problem on the whole space in which outgoing solutions are considered, the well-posedness assumption is not necessary.",
"In what follows we assume that $k > 0$ .",
"The case $k=0$ is similar and even easier to obtain.",
"The first main result in this section is Theorem 1 Let $d=2, \\, 3$ , $\\delta >0$ , $f \\in L^2(\\Omega )$ and let $u_\\delta \\in H^1_0(\\Omega )$ be the unique solution to equation (REF ): $\\operatorname{div}(s_\\delta A \\nabla u_\\delta ) + k^{2} s_{0} \\Sigma u_{\\delta } = s_0 f \\mbox{ in } \\Omega .$ Assume that the media $(A, \\Sigma )$ in $\\Omega _{3} \\setminus \\Omega _{2}$ and $(-A, - \\Sigma )$ in $\\Omega _{2} \\setminus \\Omega _{1}$ are reflecting complementary.",
"We have Case 1: $f$ is compatible with the medium.",
"Then $(u_\\delta )$ converges weakly in $H^1(\\Omega )$ and strongly in $L^2(\\Omega )$ to $u_0 \\in H^1_0(\\Omega )$ , the unique solution to the equation $\\operatorname{div}(s_0 \\nabla u_0) + k^2 s_0 \\Sigma u_0 = s_0 f \\mbox{ in } \\Omega .$ as $\\delta \\rightarrow 0$ .",
"Case 2: $f$ is not compatible with the medium.",
"We have $\\lim _{\\delta \\rightarrow 0}\\Vert u_\\delta \\Vert _{H^1(\\Omega )} = + \\infty .$ In the statement of Theorem REF , we use the following Definition 2 (Compatibility condition) Assume that the media $(A, \\Sigma )$ in $\\Omega _{3} \\setminus \\Omega _{2}$ and $(-A, - \\Sigma )$ in $\\Omega _{2} \\setminus \\Omega _{1}$ are reflecting complementary.",
"Then $f \\in L^2(\\Omega )$ is said to be compatible with the system if and only if there exist $U \\in H^1(\\Omega _3 \\setminus \\Omega _2)$ and $V \\in H^1(\\Omega _3 \\setminus \\Omega _2)$ such that $\\left\\lbrace \\begin{array}{cl}\\operatorname{div}(A \\nabla U) + k^2 \\Sigma U = F_*f - f & \\mbox{ in } \\Omega _3 \\setminus \\bar{\\Omega }_2, \\\\[6pt]U = 0 & \\mbox{ on } \\partial \\Omega _2, \\\\[6pt]A \\nabla U \\cdot \\eta = 0 & \\mbox{ on } \\partial \\Omega _2,\\end{array} \\right.$ and $\\left\\lbrace \\begin{array}{cl}\\operatorname{div}(A \\nabla V) + k^2 \\Sigma V = f & \\mbox{ in } \\Omega _3 \\setminus \\bar{\\Omega }_2, \\\\[6pt]V = W \\Big |_{\\mathrm {ext}}& \\mbox{ on } \\partial \\Omega _3, \\\\[6pt]A \\nabla V \\cdot \\eta = A \\nabla W \\cdot \\eta \\Big |_{\\mathrm {ext}} & \\mbox{ on } \\partial \\Omega _3.\\end{array} \\right.$ Here $W \\in H^1(\\Omega \\setminus \\partial \\Omega _3)$ is the unique solution to the system $\\left\\lbrace \\begin{array}{cl}\\operatorname{div}(\\hat{A} \\nabla W) + k^2 \\hat{\\Sigma }W = \\hat{f} & \\mbox{ in } \\Omega \\setminus \\partial \\Omega _3, \\\\[6pt]W = 0 & \\mbox{ on } \\partial \\Omega , \\\\[6pt][W] = - U & \\mbox{ on } \\partial \\Omega _3, \\\\[6pt][\\hat{A} \\nabla W \\cdot \\eta ] = - A \\nabla U \\cdot \\eta & \\mbox{ on } \\partial \\Omega _3.\\end{array} \\right.$ The compatibility condition is an intrinsic one, i.e., it does not depend on the choice of $F$ and $G$ .",
"In fact, it is equivalent to the boundedness of $\\Vert u_{\\delta }\\Vert _{H^{1}(\\Omega )}$ as $\\delta \\rightarrow 0$ .",
"Given $F$ , there are infinitely many choices of $G$ .",
"A choice of $G$ that would make the compatibility condition more accessible is preferred.",
"Problems (REF ) and (REF ) are the two Cauchy problems obtained from the reflecting technique.",
"The reflecting complementary condition is necessary so that $(\\Vert u_{\\delta \\Vert _{H^{1}(\\Omega )}})$ explodes for some $f$ (see [6]).",
"The uniqueness of $U$ and $V$ follow from the unique continuation principle (see, e.g., [1], [17]).",
"The existence and uniqueness of $W$ are standard and follow from Fredholm's theory (see, e.g., [4]) since system (REF ) is well-posed.",
"The next result is in the spirit of (REF ).",
"Theorem 2 Let $d=2, \\, 3$ , $\\delta >0$ , $f \\in L^2(\\Omega )$ and let $u_\\delta \\in H^1_0(\\Omega )$ be the unique solution to equation (REF ): $\\operatorname{div}(s_\\delta A \\nabla u_\\delta ) + k^{2} s_{0} \\Sigma u_{\\delta } = s_0 f \\mbox{ in } \\Omega .$ Assume that the media $(A, \\Sigma )$ in $\\Omega _{3} \\setminus \\Omega _{2}$ and $(-A, - \\Sigma )$ in $\\Omega _{2} \\setminus \\Omega _{1}$ are reflecting complementary and $f$ is compatible with the system.",
"Then $(u_\\delta )$ converges weakly to $NI(f)$ in $H^{1}(\\Omega )$ where $NI(f)=\\left\\lbrace \\begin{array}{cl}W & \\mbox{ if } x \\in \\Omega \\setminus \\Omega _3, \\\\[6pt]V & \\mbox{ if } x \\in \\Omega _3 \\setminus \\Omega _2, \\\\[6pt](U + V) \\circ F & \\mbox{ if } x \\in \\Omega _2 \\setminus \\Omega _1, \\\\[6pt]W \\circ G \\circ F & \\mbox{ if } x \\in \\Omega _1.\\end{array}\\right.$ Here $U, V$ , and $W$ are given in Definition REF .",
"If $f = 0$ in $\\Omega _{3}$ then $U = 0$ .",
"In this case, the compatibility condition is equivalent to the existence of $V \\in H^{1}(\\Omega _{3} \\setminus \\bar{\\Omega }_{2})$ to the Cauchy problem $\\left\\lbrace \\begin{array}{cl}\\operatorname{div}(A \\nabla V) + k^{2} \\Sigma V = 0 & \\mbox{ in } \\Omega _3 \\setminus \\bar{\\Omega }_2, \\\\[6pt]V = W \\Big |_{\\mathrm {ext}} & \\mbox{ on } \\partial \\Omega _3, \\\\[6pt]A \\nabla V \\cdot \\eta = A \\nabla W \\cdot \\eta \\Big |_{\\mathrm {ext}} & \\mbox{ on } \\partial \\Omega _3,\\end{array} \\right.$ where $W \\in H^{1}_{0}(\\Omega )$ is the unique solution to $\\operatorname{div}(\\hat{A} \\nabla W) + k^{2} \\hat{\\Sigma }W= f \\mbox{ in } \\Omega .$ We have Corollary 1 Let $d=2, \\, 3$ , $f \\in L^2(\\Omega )$ with $\\operatorname{supp}f \\cap \\Omega _{3} = Ø$ .",
"Assume that the media $(A, \\Sigma )$ in $\\Omega _{3} \\setminus \\Omega _{2}$ and $(-A, - \\Sigma )$ in $\\Omega _{2} \\setminus \\Omega _{1}$ are reflecting complementary and there exists a solution $V \\in H^{1}(\\Omega _{3} \\setminus \\bar{\\Omega }_{2})$ to (REF ).",
"Then $f$ is compatible to the system.",
"It follows from Corollary REF that if $\\operatorname{supp}f \\cap \\Omega _{3} = Ø$ and $G_{*} F_{*} A(x) = A(x)$ and $G_{*} F_{*} \\Sigma (x) = \\Sigma (x)$ for $x \\in \\Omega _{3} \\setminus \\bar{\\Omega }_{2}$ then $V = W$ .",
"We obtain Corollary 2 Let $d=2, \\, 3$ , $f \\in L^2(\\Omega )$ with $\\operatorname{supp}f \\cap \\Omega _{3} = Ø$ .",
"Assume that the media $(A, \\Sigma )$ in $\\Omega _{3} \\setminus \\Omega _{2}$ and $(-A, - \\Sigma )$ in $\\Omega _{2} \\setminus \\Omega _{1}$ are reflecting complementary and $G_{*} F_{*} A(x) = A(x) \\quad \\mbox{ and } \\quad G_{*} F_{*} \\Sigma (x) = \\Sigma (x) \\quad \\mbox{ for } \\quad x \\in \\Omega _{3} \\setminus \\bar{\\Omega }_{2}.$ Then $f$ is compatible to the system.",
"Remark 3 It is easy to verify that the $2d$ setting considered in the introduction satisfies the assumptions of Corollary REF .",
"Remark 4 In this paper, we characterize the behavior of $u_{\\delta }$ as $\\delta \\rightarrow 0$ for compatible $f$ .",
"The paper [10] develops the method introduced here to deal with this problem without the assumption on the compatibility in the cloaking setting."
],
[
" Proofs of Theorems ",
"This section containing two subsections is devoted to the proof of Theorems REF and REF .",
"In the first subsection, we establish basis properties of solutions to equation (REF ) such as the existence, uniqueness, and stability, and establish a result on the change of variables concerning reflections.",
"The proof of Theorem REF and REF are given in the second subsection."
],
[
"Preliminaries",
"This section contains two lemmas.",
"The first one is on the wellposedness of (REF ).",
"Lemma 1 Let $d=2, \\, 3$ , $k>0$ , $0 < \\delta < 1$ , $g \\in H^{-1}(\\Omega )$ $($ the duality of $H^1_0(\\Omega )$$)$ and let $s_{\\delta }$ be defined in (REF ).",
"Assume that $A$ and $\\Sigma $ satisfy (REF ) and (REF ), and (REF ) holds.",
"Then there exists a unique solution $v_\\delta \\in H^1_0(\\Omega )$ to the equation $\\operatorname{div}(s_\\delta A \\nabla v_\\delta ) + k^2 s_0 \\Sigma v_\\delta = g \\mbox{ in } \\Omega .$ Moreover, $\\Vert v_\\delta \\Vert _{H^1(\\Omega )} \\le C \\Big (\\frac{1}{\\delta } \\Vert g \\Vert _{H^{-1}(\\Omega )}+ \\Vert g\\Vert _{L^2(\\Omega _1)} + \\Vert g\\Vert _{L^2(\\Omega _2 \\setminus \\bar{\\Omega }_1)} + \\Vert g\\Vert _{L^2(\\Omega \\setminus \\bar{\\Omega }_2)} \\Big ),$ for some positive constant $C$ independent of $g$ and $\\delta $ , as $\\delta $ is small.",
"Proof.",
"The existence of $v_\\delta $ follows from the uniqueness of $v_{\\delta }$ by Fredholm's theorem (see, e.g., [4]).",
"We now establish the uniqueness of $v_{\\delta }$ by showing that $v_\\delta = 0$ if $v_\\delta \\in H^1_0(\\Omega )$ is a solution to the equation $\\operatorname{div}(s_\\delta A \\nabla v_\\delta ) + k^2 s_0 \\Sigma v_\\delta = 0 \\mbox{ in } \\Omega .$ Multiplying the above equation by $\\bar{v}_\\delta $ (the conjugate of $v_{\\delta }$ ) and integrating the obtained expression on $\\Omega $ , we have $\\int _{\\Omega } s_\\delta \\langle A \\nabla v_\\delta , \\nabla v_\\delta \\rangle \\, dx - \\int _{\\Omega } k^2 s_0 \\Sigma |v_\\delta |^2 \\, dx = 0.$ This implies, by considering the imaginary part, $\\int _{\\Omega _2 \\setminus \\Omega _1} \\langle A \\nabla v_\\delta , \\nabla v_\\delta \\rangle \\, dx= 0.$ It follows from (REF ) that $v_\\delta $ is constant in $\\Omega _2 \\setminus \\Omega _1$ .",
"Thus $v_\\delta = 0$ in $\\Omega _2 \\setminus \\Omega _1$ since $\\operatorname{div}(s_\\delta A \\nabla v_\\delta ) + k^2 s_0 \\Sigma v_\\delta = 0$ in $\\Omega _2 \\setminus \\Omega _1$ .",
"This implies $v_\\delta = 0 $ in $\\Omega \\setminus \\Omega _{2}$ and in $\\Omega _{1}$ by (REF ).",
"Here to establish $v_{\\delta } = 0 $ in $\\Omega _{1}$ , we considered the function $V_{\\delta }$ defined in $\\Omega $ as follows $V_{\\delta } = v_{\\delta } \\circ F^{-1} \\circ G^{-1}$ in $\\Omega _{3}$ and $V_{\\delta } = 0$ in $\\Omega \\setminus \\Omega _{3}$ and used the well-posedness of (REF ).",
"The proof of the uniqueness is complete.",
"We next establish (REF ) by a contradiction argument.",
"Assume that (REF ) is not true.",
"Then there exists $(g_\\delta ) \\subset H^{-1}(\\Omega )$ such that $\\Vert v_\\delta \\Vert _{H^1(\\Omega )} =1 \\mbox{ and } \\frac{1}{\\delta } \\Vert g_\\delta \\Vert _{H^{-1}} + \\Vert g_\\delta \\Vert _{L^2(\\Omega _1)} + \\Vert g_\\delta \\Vert _{L^2(\\Omega _2 \\setminus \\bar{\\Omega }_1)} + \\Vert g\\Vert _{L^2(\\Omega \\setminus \\bar{\\Omega }_2)} \\rightarrow 0,$ as $\\delta \\rightarrow 0$ , where $v_\\delta \\in H^1_0(\\Omega )$ is the unique solution to the equation $\\operatorname{div}(s_\\delta A \\nabla v_\\delta ) + k^2 s_0 \\Sigma v_\\delta = g_\\delta \\mbox{ in } \\Omega .$ Multiplying this equation by $\\bar{v}_\\delta $ and integrating the obtained expression on $\\Omega $ , we have $\\int _{\\Omega } s_\\delta \\langle A \\nabla v_\\delta , \\nabla v_\\delta \\rangle \\, dx - \\int _{\\Omega } k^2 s_0 \\Sigma |v_\\delta |^2 \\, dx = - \\int _\\Omega g_\\delta \\bar{v}_\\delta \\, dx.$ Considering the imaginary part and using the fact that $\\frac{1}{\\delta }\\Big | \\int _\\Omega g_\\delta \\bar{v}_\\delta \\Big | \\le \\frac{1}{\\delta }\\Vert g_\\delta \\Vert _{H^{-1}} \\Vert v_\\delta \\Vert _{H^1(\\Omega )} \\rightarrow 0 \\mbox{ as $\\delta \\rightarrow 0$ by }(\\ref {contradict-assumption}),$ we obtain, by (REF ), $\\Vert \\nabla v_\\delta \\Vert _{L^2(\\Omega _2 \\setminus \\Omega _1)} \\rightarrow 0 \\mbox{ as } \\delta \\rightarrow 0.$ Since $\\operatorname{div}(A \\nabla v_{\\delta }) + k^{2} \\Sigma v_{\\delta } = g_{\\delta }$ in $\\Omega _{2} \\setminus \\bar{\\Omega }_{1}$ , it follows from (REF ) and a standard compactness argument that $\\Vert v_{\\delta }\\Vert _{L^{2}(\\Omega _{2} \\setminus \\Omega _{1})} \\rightarrow 0 \\mbox{ as } \\delta \\rightarrow 0.$ A combination of (REF ) and (REF ) yields $\\Vert v_{\\delta }\\Vert _{H^{1}(\\Omega _{2} \\setminus \\Omega _{1})} \\rightarrow 0 \\mbox{ as } \\delta \\rightarrow 0.$ In particular, $\\Vert v_{\\delta }\\Vert _{H^{1/2}(\\partial \\Omega _{2})} + \\Vert v_{\\delta }\\Vert _{H^{1/2}(\\partial \\Omega _{1})} \\rightarrow 0 \\mbox{ as } \\delta \\rightarrow 0.$ We derive from the well-posedness of (REF ) and (REF ) that $\\Vert v_{\\delta }\\Vert _{H^{1}(\\Omega \\setminus \\Omega _{2})} \\rightarrow 0,$ and $\\Vert v_{\\delta }\\Vert _{H^{1}(\\Omega \\setminus \\Omega _{1})} \\rightarrow 0.$ A combination of (REF ), (REF ), and (REF ) yields $\\Vert v_{\\delta }\\Vert _{H^{1}(\\Omega )} \\rightarrow 0.$ We have a contradiction by (REF ).",
"The proof of (REF ) completes.",
"$\\Box $ The second lemma is on the change of variables for reflections.",
"Lemma 2 Let $k \\ge 0$ , $D_1$ and $D_2$ be two smooth open subsets of $\\mathbb {R}^d$ , $T$ be a diffeomorphism from $D_1$ onto $D_2$ , $a \\in [L^\\infty (D_1)]^{d \\times d}$ be a matrix function, and $\\sigma \\in L^\\infty (D_1)$ be a complex function.",
"Fix $u \\in H^1(D_1)$ and set $v = u \\circ T^{-1}$ .",
"Then $\\operatorname{div}(a \\nabla u) + k^2 \\sigma u = f \\mbox{ in } D_{1}$ iff $\\operatorname{div}(T_*a \\nabla v) + k^2 T_*\\sigma v = T_* f \\mbox{ in } D_{2}.$ Assume that $\\Gamma _1$ and $\\Gamma _2$ are open subsets of $\\partial D_1$ and $\\partial D_2$ such that $\\Gamma _1$ and $\\Gamma _2$ are smooth, $\\Gamma _2 = T(\\Gamma _1)$ , and ${\\bf T}: = T\\Big |_{ \\Gamma _1} \\Gamma _1 \\rightarrow \\Gamma _2$ is a diffeomorphism We assume here that there is an extension of $T$ in a neighborhood of $\\partial D_{1}$ (which is also called $T$ ) such that it is a diffeomorphism.. We have $a \\nabla u \\cdot \\eta _1 = g_1 \\mbox{ on } \\Gamma _1$ iff $T_*a \\nabla v \\cdot \\eta _2 = g_2 \\mbox{ on } \\Gamma _2,$ where In the identity below, $\\nabla {\\bf T}$ stands for the gradient of a transformation from a $(d-1)$ -manifold into a $(d-1)$ -manifold, and $\\det \\nabla {\\bf T}$ denotes the determinant of $(d-1) \\times (d-1)$ matrix.",
"$g_2(y) = g_1(x)/ |\\det \\nabla {\\bf T}(x)| \\mbox{ with } x = {\\bf T}^{-1}(y).$ Here $\\eta _1$ and $\\eta _2$ are the normal unit vectors on $\\Gamma _1$ and $\\Gamma _2$ directed to the exterior of $D_1$ and $D_2$ .",
"In particular, if $\\Gamma _1 = \\Gamma _2$ , ${\\bf T}(x) = x$ on $\\Gamma _1$ , $D_2 \\cap D_1 = Ø$ .",
"We have $T_*a \\nabla v \\cdot \\eta _1 = - a \\nabla u \\cdot \\eta _1 \\mbox{ on } \\Gamma _1 = \\Gamma _2.$ Proof.",
"Lemma REF is a consequence of the change of variables.",
"The first equivalence relation is known and can be proved by using the weak formula for $u$ and $v$ .",
"The second equivalence follows similarly.",
"The details are left to the reader.",
"$\\Box $"
],
[
"Proof of the first statement of Theorem ",
"In this section, $f$ is compatible.",
"The proof is derived from the following steps: Step 1: Let $v \\in H^1_0(\\Omega )$ be a solution to the equation $\\operatorname{div}(s_0 A \\nabla v) + k^{2} s_{0} \\Sigma v = s_0 f \\mbox{ in } \\Omega .$ We prove that $v = NI(f)$ .",
"Step 2: Define $u_0: = NI(f)$ .",
"We prove that $u_0 \\in H^1_0(\\Omega )$ is a solution to the equation $\\operatorname{div}(s_0 A \\nabla u_0) + k^{2} s_{0} \\Sigma u_{0} = s_0 f \\mbox{ in } \\Omega .$ Step 3: We prove that $(u_\\delta )_{0 < \\delta < 1}$ is bounded in $H^1(\\Omega )$ .",
"Step 4: We prove that $(u_\\delta )$ converges weakly in $H^1(\\Omega )$ and strongly in $L^2(\\Omega )$ to $u_0$ as $\\delta $ goes to 0.",
"It is clear that the proof of the first statement of Theorem REF and Theorem REF is complete after these four steps.",
"We now process these steps.",
"Step 1: Assume that $v \\in H^1_0(\\Omega )$ is a solution to the equation $\\operatorname{div}(s_0 A \\nabla v) + k^{2} s_{0} \\Sigma v= s_0 f \\mbox{ in } \\Omega .$ Set $v_1 = v \\circ F^{-1} \\mbox{ in } \\Omega _4 \\setminus \\bar{\\Omega }_2$ and $\\hat{s}_0 = \\left\\lbrace \\begin{array}{cl} 1 & \\mbox{ if } x \\in \\Omega _3 \\setminus \\Omega _2, \\\\[6pt]-1 & \\mbox{ if } x \\in \\Omega _4 \\setminus \\Omega _3.\\end{array}\\right.$ Then $v_1 \\in H^1(\\Omega _3 \\setminus \\bar{\\Omega }_2) \\cap H^1_{_{loc}}(\\Omega _4 \\setminus \\bar{\\Omega }_2)$ and, by Lemma REF , $v_1$ satisfies the equation $\\operatorname{div}(\\hat{s}_0 F_*A \\nabla v_1) + k^{2} \\hat{s}_{0} F_{*}\\Sigma v_{1} = \\hat{s}_0 F_*f \\mbox{ in } \\Omega _4 \\setminus \\bar{\\Omega }_2,$ and $v_1 = v \\mbox{ on } \\partial \\Omega _2 \\quad \\mbox{ and } \\quad F_*A \\nabla v_1 \\cdot \\eta = A \\nabla v_ \\cdot \\eta \\Big |_{\\mathrm {ext}} \\mbox{ on } \\partial \\Omega _2.$ In the last identity, we use the fact that $F_*A \\nabla v_1 \\cdot \\eta = - A \\nabla v \\cdot \\eta \\Big |_{\\mathrm {int}}$ on $\\partial \\Omega _2$ by (REF ), and $A \\nabla v \\cdot \\eta \\Big |_{\\mathrm {ext}} = - A \\nabla v \\cdot \\eta \\Big |_{\\mathrm {int}} $ on $\\partial \\Omega _2$ by the transmission condition on $\\partial \\Omega _2$ .",
"Define ${\\bf U} = v_1 - v \\mbox{ in } \\Omega _3 \\setminus \\bar{\\Omega }_2.$ Since $F_*A = A$ and $F_{*} \\Sigma = \\Sigma $ in $\\Omega _3 \\setminus \\bar{\\Omega }_2$ , it follows that $\\left\\lbrace \\begin{array}{cl}\\operatorname{div}(A \\nabla {\\bf U}) + k^{2} \\Sigma {\\bf U} = F_*f - f & \\mbox{ in } \\Omega _3 \\setminus \\bar{\\Omega }_2, \\\\[6pt]{\\bf U} = 0 & \\mbox{ on } \\partial \\Omega _2, \\\\[6pt]A \\nabla {\\bf U} \\cdot \\eta = 0 & \\mbox{ on } \\partial \\Omega _2.\\end{array} \\right.$ Applying the unique continuation principle (see, e.g., [1], [17]), from (REF ), we have ${\\bf U} = U \\mbox{ in } \\Omega _3 \\setminus \\bar{\\Omega }_2.$ Define $v_2$ in $\\Omega $ as follows $v_2(x) =\\left\\lbrace \\begin{array}{cl}v_1 \\circ G^{-1} (x) & \\mbox{ if } x \\in \\Omega _3, \\\\[6pt]v (x) & \\mbox{ if } x \\in \\Omega \\setminus \\Omega _3.\\end{array}\\right.$ Using (REF ) and applying Lemma REF , we have $\\operatorname{div}(\\hat{A} \\nabla v_2) + k^{2 } \\hat{\\Sigma }v_{2} = \\hat{f} \\mbox{ in } \\Omega \\setminus \\partial \\Omega _3,$ and, on $\\partial \\Omega _3$ , $\\hat{A} \\nabla v_2 \\cdot \\eta \\Big |_\\mathrm {ext} - \\hat{A} \\nabla v_2 \\cdot \\eta \\Big |_\\mathrm {int} &= A \\nabla v \\cdot \\eta \\Big |_\\mathrm {ext} + F_* A \\nabla v_1 \\cdot \\eta \\Big |_\\mathrm {ext} \\quad ( \\mbox{by } (\\ref {reflextion})) \\\\[6pt]& = A \\nabla v \\cdot \\eta \\Big |_\\mathrm {ext} - F_* A \\nabla v_1 \\cdot \\eta \\Big |_\\mathrm {int} \\quad (\\mbox{by } (\\ref {eq-v1}))\\\\[6pt]& = A \\nabla v \\cdot \\eta \\Big |_\\mathrm {ext} - F_* A \\nabla (v + {\\bf U}) \\cdot \\eta \\Big |_\\mathrm {int} \\quad ( \\mbox{by } (\\ref {def-bfU})).$ Since $F_{*} A = A$ in $\\Omega _{3} \\setminus \\bar{\\Omega }_{2}$ , it follows from (REF ) that $\\hat{A} \\nabla v_2 \\cdot \\eta \\Big |_\\mathrm {ext} - \\hat{A} \\nabla v_2 \\cdot \\eta \\Big |_\\mathrm {int}= - A \\nabla U \\cdot \\eta \\Big |_\\mathrm {int} \\mbox{ on } \\partial \\Omega _{3}.$ Since $G(x) = x$ on $\\partial \\Omega _{3}$ , we obtain, on $\\partial \\Omega _3$ , $v_2\\Big |_\\mathrm {ext} - v_2\\Big |_\\mathrm {int} = v\\Big |_\\mathrm {ext} - v_1\\Big |_\\mathrm {ext} = v\\Big |_\\mathrm {ext} - v_1\\Big |_\\mathrm {int} = v\\Big |_\\mathrm {ext} - ({\\bf U} + v)\\Big |_\\mathrm {int} = - U,$ by (REF ).",
"Combining (REF ), (REF ), (REF ), and (REF ), and applying the unique continuation principle, we have $v_2 = W \\mbox{ in } \\Omega .$ Since $\\operatorname{div}(A \\nabla v) + k^{2} \\Sigma v = f$ in $\\Omega _3 \\setminus \\bar{\\Omega }_2$ , it follows from (REF ) and (REF ) that $\\left\\lbrace \\begin{array}{lc}\\operatorname{div}(A \\nabla v) + k^{2} \\Sigma v= f & \\mbox{ in } \\Omega _3 \\setminus \\bar{\\Omega }_2, \\\\[6pt]v= W \\Big |_{\\mathrm {ext}} & \\mbox{ on } \\partial \\Omega _3, \\\\[6pt]A \\nabla v \\cdot \\eta \\Big |_\\mathrm {int} = A \\nabla W \\cdot \\eta \\Big |_{\\mathrm {ext}} & \\mbox{ on } \\partial \\Omega _3.\\end{array}\\right.$ By the unique continuation principle, it follows from (REF ) that $v = V \\mbox{ in } \\Omega _3 \\setminus \\bar{\\Omega }_2.$ We claim that $ v =\\left\\lbrace \\begin{array}{cl}W & \\mbox{ in } \\Omega \\setminus \\Omega _3, \\\\[6pt]V &\\mbox{ in } \\Omega _3 \\setminus \\Omega _2, \\\\[6pt](V+U) \\circ F & \\mbox{ in } \\Omega _2 \\setminus \\Omega _1, \\\\[6pt]W \\circ G \\circ F & \\mbox{ in } \\Omega _1.\\end{array}\\right.$ In fact, the statement $v = W$ in $\\Omega \\setminus \\Omega _{3}$ is a consequence of (REF ) and (REF ); the statement $v = V$ in $\\Omega _{3} \\setminus \\Omega _{2}$ follows from (REF ); the statement $v = (V+U) \\circ F$ in $\\Omega _{2} \\setminus \\Omega _{1}$ is a consequence of the fact $v_{1} = v + U = V + U$ in $\\Omega _{3} \\setminus \\Omega _{2}$ , and the statement $v = v_{1} \\circ F $ in $\\Omega _{2} \\setminus \\Omega _{1}$ ; $v =W \\circ G \\circ F $ in $\\Omega _{1}$ is a consequence of the definition of $v_{1}$ and $v_{2}$ , and $v_{2} = W$ in $\\Omega _{3}$ .",
"The claim is proved.",
"Therefore, $v = NI(f) \\mbox{ in } \\Omega .$ The proof of Step 1 is complete.",
"Step 2: We claim that $\\operatorname{div}(A \\nabla u_0) + k^{2} \\Sigma u_{0}= f \\mbox{ in } \\Omega \\setminus (\\partial \\Omega _3 \\cup \\partial \\Omega _2 \\cup \\partial \\Omega _{1}).$ where $u_0 : = NI(f)$ .",
"Indeed, it is just a consequence of the definition of $U, \\, V$ and $W$ and the fact that $F_{*} A = A$ and $F_{*} \\Sigma = \\Sigma $ in $\\Omega _{3} \\setminus \\Omega _{2}$ .",
"It remains to verify $[A \\nabla u_0] = [u_0] = 0 \\mbox{ on } \\partial \\Omega _3, \\quad [s_0 A \\nabla u_0] = [u_0] = 0 \\mbox{ on } \\partial \\Omega _2,$ and $[A \\nabla u_0] = [u_0] = 0 \\mbox{ on } \\partial \\Omega _1.$ From the definitions of $V$ in (REF ) and $NI(f)$ in (REF ), we have $[A \\nabla u_0] = [u_0] = 0 \\mbox{ on } \\partial \\Omega _3.$ Since $U = 0$ and $A \\nabla U \\cdot \\eta = 0$ on $\\partial \\Omega _2$ , it follows from (REF ) that $[s_0 A \\nabla u_0] = [u_0] = 0 \\mbox{ on } \\partial \\Omega _2.$ From (REF ) and (REF ), we have $[s_0 A \\nabla u_0] = [u_0] = 0 \\mbox{ on } \\partial \\Omega _1.$ The proof of Step 2 is complete.",
"Step 3: Set $v_\\delta = u_\\delta - u_0 \\mbox{ in } \\Omega .$ We have, in $\\Omega $ , $\\operatorname{div}(s_\\delta A \\nabla v_\\delta ) + k^{2} s_{0} \\Sigma v_{\\delta } = & \\operatorname{div}(s_\\delta A \\nabla u_\\delta ) - \\operatorname{div}(s_\\delta A \\nabla u_0) + k^{2} s_{0} \\Sigma u_{\\delta } - k^{2} s_{0} \\Sigma u_{0}$ This implies $\\operatorname{div}(s_\\delta A \\nabla v_\\delta ) + k^{2} s_{0} \\Sigma v_{\\delta } = \\operatorname{div}\\big [(s_0 - s_\\delta ) A \\nabla u_0\\big ] \\mbox{ in } \\Omega .$ By Lemma REF , we have $\\Vert \\nabla v_\\delta \\Vert _{L^2(\\Omega )} \\le C \\Vert \\nabla u_0 \\Vert _{L^2(\\Omega )},$ which yields, since $u_\\delta = v_\\delta + u_0$ , $\\Vert \\nabla u_\\delta \\Vert _{L^2(\\Omega )} \\le C \\Vert \\nabla u_0 \\Vert _{L^2(\\Omega )}.$ Since $u_\\delta \\in H^1_0(\\Omega )$ , by Poincaré's inequality, it follows that $\\Vert u_\\delta \\Vert _{H^1(\\Omega )} \\le C \\Vert \\nabla u_0 \\Vert _{L^2(\\Omega )}.$ Step 3 completes.",
"Step 4: The conclusion of Step 4 follows from Step 3 and the fact that the limit of $u_\\delta $ (up to a subsequence) satisfies (REF ) and (REF ) has a unique solution in $H^1_0(\\Omega )$ by Steps 1 and 2.",
"$\\Box $"
],
[
"Proof of the second statement of Theorem ",
"In this section $f$ is not compatible.",
"We prove the second statement of Theorem REF by contradiction.",
"Assume that (REF ) is not true.",
"Without loss of generality, there exists a bounded sequence $(u_\\delta )$ in $H_{0}^1(\\Omega )$ such that $u_{\\delta }$ is the unique solution to the equation $\\operatorname{div}(s_{\\delta } A \\nabla u_{\\delta }) + k^{2} s_{0} \\Sigma u_{\\delta } = s_{0} f \\mbox{ in } \\Omega ,$ and $u_\\delta $ converges weakly to (some) $u \\in H^1(\\Omega )$ as $\\delta \\rightarrow 0$ .",
"It follows that $u \\in H^1_0(\\Omega )$ is a solution to the equation $\\operatorname{div}(s_0 A \\nabla u) + k^{2} s_{0} u= s_0 f \\mbox{ in } \\Omega .$ Define $U= u \\circ F^{-1} - u \\mbox{ in } \\Omega _3 \\setminus \\bar{\\Omega }_2 \\quad \\mbox{ and } \\quad V = u \\mbox{ in } \\Omega _3 \\setminus \\bar{\\Omega }_2.$ As in Step 1 of Section REF , $U$ and $V$ satisfy (REF ) and (REF ) respectively.",
"We have a contradiction since $f$ is not compatible with the system.",
"$\\Box $ Acknowledgment.",
"The author would like to thank Bob Kohn and Grame Milton for interesting discussions."
]
] | 1204.1518 |
[
[
"Assessing the Significance of Apparent Correlations Between Radio and\n Gamma-ray Blazar Fluxes"
],
[
"Abstract Whether a correlation exists between the radio and gamma-ray flux densities of blazars is a long-standing question, and one that is difficult to answer confidently because of various observational biases which may either dilute or apparently enhance any intrinsic correlation between radio and gamma-ray luminosities.",
"We introduce a novel method of data randomization to evaluate quantitatively the effect of these biases and to assess the intrinsic significance of an apparent correlation between radio and gamma-ray flux densities of blazars.",
"The novelty of the method lies in a combination of data randomization in luminosity space (to ensure that the randomized data are intrinsically, and not just apparently, uncorrelated) and significance assessment in flux space (to explicitly avoid Malmquist bias and automatically account for the limited dynamical range in both frequencies).",
"The method is applicable even to small samples that are not selected with strict statistical criteria.",
"For larger samples we describe a variation of the method in which the sample is split in redshift bins, and the randomization is applied in each bin individually; this variation is designed to yield the equivalent to luminosity-function sampling of the underlying population in the limit of very large, statistically complete samples.",
"We show that for a smaller number of redshift bins, the method yields a worse significance, and in this way it is conservative in that it does not assign a stronger, artificially enhanced significance.",
"We demonstrate how our test performs as a function of number of sources, strength of correlation, and number of redshift bins used, and we show that while our test is robust against common-distance biases and associated false positives for uncorrelated data, it retains the power of other methods in rejecting the null hypothesis of no correlation for correlated data."
],
[
"Introduction",
"Whether the radio and the gamma-ray luminosities of blazars are intrinsically correlated is a long-standing debate.",
"The presence or absence of such a correlation could provide insight into blazar emission physics.",
"At radio frequencies low enough that synchrotron emission is self-absorbed on physical scales likely to be associated with gamma-ray emission, measurements of the gamma-ray and radio flux densities typically probe different parts of the blazar jet.",
"If concurrently-measured, time-averaged flux densities at self-absorbed radio frequencies and high-energy ($\\ge 100$ MeV) gamma-rays are intrinsically correlated, the implication would be that emission and flaring in different parts of blazar jets are driven by the same disturbances.",
"In this case, further progress on the sequence of events that produce blazar flares can be made through high-cadence monitoring in both wavebands.",
"If on the other hand radio and gamma-ray flux densities can be shown to be uncorrelated (a statement that needs to be carefully distinguished from the absence of evidence for correlation) then it is more likely that, over the timescales used for the flux-averaging, emission regions probed by radio and gamma-ray observations evolve and radiate independently.",
"Furthermore, should an intrinsic correlation between gamma-ray and radio flux densities be unambiguously demonstrated, radio blazar luminosity functions could be used to establish the shape and normalization of gamma-ray luminosity functions or $\\log N-\\log S$ distributions (however, proper care should be exercised to account for any significant scatter in the correlation, see, e.g., the discussion in Ackermann et al. 2011).",
"From there, the unresolved blazar contribution to the diffuse gamma-ray background could be estimated (e.g., Stecker & Salamon 1996; Kazanas & Perlman 1997; Stecker & Venters 2010).",
"This is particularly important as blazars constitute a guaranteed background for any search in the diffuse gamma-ray emission for yet-undetected classes of sources such as galaxy clusters, and for signatures of exotic physics.",
"Strong correlations between radio and gamma-ray luminosities have been claimed based on EGRET data (e.g., Stecker et al.",
"1993; Padovani et al.",
"1993; Stecker & Salamon 1996).",
"However, these findings have been disputed (e.g., Mücke et al.",
"1997; Chiang & Mukherjee 1998) based on more detailed statistical analyses.",
"The objections against the claimed correlations can be summarized as follows.",
"First, artificial flux-flux correlations can be induced due to the effect of a common distance modulation of gamma-ray and radio luminosities.",
"Feigelson & Berg (1983) have argued that in statistically complete surveys of relatively small depth, apparent flux-flux correlations do not appear unless the corresponding luminosities are intrinsically correlated: if luminosities are intrinsically uncorrelated most objects will only have an upper limit rather than a detection in one of the wavebands.",
"However this is not the case in samples that are selected with complex or subjective criteria, samples in which there is clustering around a preferred luminosity value, samples in which detection in both wavebands is one of the selection criteria, or samples in which the luminosity dynamical range is, for any reason, small compared to the distance modulation range.",
"In such cases, the application of a common distance-squared factor to both radio and gamma-ray luminosity will automatically induce an artificial flux-flux correlation.",
"This effect cannot be avoided simply by searching for correlation in luminosity space, as the danger of inducing an artificial apparent correlation is even greater in this case due to Malmquist bias: in flux-limited (or approximately flux-limited) surveys, most objects are concentrated close to the survey sensitivity at each wavelength.",
"By modulating these limiting fluxes by a common distance factor to return to luminosity space, artificial correlations arise.",
"Finally, the data used to obtain the claimed correlations were not synchronous.",
"The direction in which non-simultaneity affects any intrinsic correlation is unclear.",
"On the one hand, non-simultaneous data may wash out an intrinsic correlation which might otherwise be found in concurrently measured data.",
"On the other hand, the tendency to detect more flaring objects than objects in a quiescent state in surveys may lead to enhanced correlations, essentially representing peak flux / peak flux correlations of different flares, which although they may be indicative of the overall energetics of flares in a single object, they do not convey any detailed information regarding the time-averaged behavior of the object.",
"In the Fermi era, the possibility of a correlation between gamma-ray and radio fluxes of blazars has generated a lot of interest, and the question has been explored using Fermi Large Area Telescope (LAT) fluxes in combination with archival (Ghirlanda et al.",
"2010, Mahony et al.",
"2010, Giroletti et al.",
"2010; Ackermann et al.",
"2011), quasi-concurrent (Kovalev et al.",
"2009) and concurrent (Giroletti et al.",
"2010; Ackermann et al.",
"2011, Angelakis et al.",
"2010; Fuhrmann et al.",
"2012, in preparation) radio data.",
"The intrinsic significance of an apparent correlation between radio and gamma-ray flux densities in strictly flux-limited, large datasets is relatively straight-forward to assess, by Monte-Carlo draws from the underlying luminosity functions in both datasets, obeying the same selection criteria as the observed sample of sources (e.g., Bloom 2008).",
"In practice however we frequently encounter the case where a sample of monitored sources has been selected to optimize the likelihood of high-impact observations in individual objects using complex and often subjective criteria, which are difficult to reproduce in a simulation.",
"Although such samples are not ideally configured for unbiased population studies, they may present significant advantages in other respects, such as multi-band coverage, high cadence of observations, and simultaneity between different waveband data.",
"It is thus important to be able to assess as robustly as possible the intrinsic significance of any apparent correlations observed in such samples.",
"Here, we introduce a method for the quantitative assessment of the significance of a correlation in such cases, based on permutations of observed flux densities, while ensuring that the dynamical ranges in luminosity and flux density are kept fixed.",
"When this method is applied in large, statistically complete samples that are split in redshift bins, it asymptotically approaches luminosity-function sampling.",
"For smaller samples, the significances it returns are conservative: existing intrinsic correlations may not be verified, but exaggerated significances are avoided.",
"Our method has been recently used by the Fermi-LAT collaboration (Ackermann et al.",
"2011) to study the correlation between GeV and cm radio fluxes (both archival and concurrent, the latter from the Owens Valley Radio Observatory 15 GHz monitoring program, Richards et al.",
"2011Program description and data also available online, at: http://www.astro.caltech.edu/ovroblazars/ ).",
"They have established, at a very high significance level, the existence of a positive correlation ($<10^{-7}$ probability of the correlation arising by chance).",
"Our method is also currently used in studies of multi-frequency concurrent radio observations by the F-GAMMA program (Angelakis et al.",
"2010; Fuhrmann et al.",
"2012, in preparation).",
"Here, we discuss in detail the method and its implementation, and we evaluate its performance using both simulated and real (Fermi and OVRO) data.",
"We caution the reader that our proposed algorithm assumes perfectly concurrent data and thus does not address any possible effects of non-simultaneity.",
"In addition, we stress that our method cannot compensate for sample selection effects or incompleteness relative to a parent population.",
"For example, if the objects in the examined sample do not constitute a representative sample of the blazar population, even when a statistically significant correlation between radio and gamma-ray flux densities can be established in the objects of the observed sample, it is not possible to generalize this result to the blazar population as a whole.",
"This limitation can only be addressed by more careful sample selection.",
"This paper is organized as follows.",
"In §2 we discuss our method, and in §3 we present in detail the implementation of the statistical test we have adopted.",
"Demonstrations of the test and evaluations of its performance are presented in §4.",
"We summarize and discuss our conclusions in §5.",
"The purpose of the test is to quantitatively assess the significance of an apparent correlation between concurrent radio and gamma-ray flux densities of blazars in the presence of distance effects and subjective sample selection criteria.",
"We will do so by testing the hypothesis that emission in the two wavebands is intrinsically uncorrelated: we will calculate how frequently a sample of objects similar to the sample at hand, with intrinsically uncorrelated gamma/radio luminosities, will yield an apparent correlation as strong as the one seen in the data, when subjected to the same distance and dynamical-range effects as our actual sample.",
"In our implementation of the test, the strength of the apparent correlation is quantified by the Pearson product-moment correlation coefficient $r$ (Fisher 1944), defined as $ r = \\frac{\\sum _{i=1}^N(X_i-\\bar{X})(Y_i-\\bar{Y})}{\\sqrt{\\sum _{i=1}^N(X_i-\\bar{X})^2\\sum _{i=1}^N(Y_i-\\bar{Y})^2}}\\,,$ with $(X_i,Y_i)$ in our case being a pair of the logarithms of the flux densities in each frequency for a single object.",
"The reason for taking the logarithm is two-fold.",
"First, it ensures that, for sources with a power-law distribution of fluxes, there will not be a clustering of most measurements around the low-flux corner of the flux-flux plane, which would then allow single high-flux outliers to induce an artificially high $r$ value.",
"Second, it linearizes any power-law relation between the variables, which improves the behavior of correlation measures that target specifically the linear correlation between variables (such as the Pearson $r$ ).",
"This test can also be used with any statistic quantifying correlation strength instead of the Pearson product-moment coefficient, including non-parametric correlation measures (e.g., Siegel & Castellan 1988; Conover 1999).",
"Since the sample selection criteria are assumed to be subjective, the challenge in defining our test lies in constructing simulated object samples of intrinsically uncorrelated gamma/radio flux densities, similar in other respects to our actual object sample.",
"In order to overcome this difficulty, we use only permutations of measured quantities.",
"Our method is a variation of a classical permutation test for the assessment of a correlation (e.g., Wall & Jenkins 2003, §4.2.3; see also Efron & Petrosian 1998 for permutation methods for doubly truncated datasets).",
"Its novelty lies in the fact that while we are trying to establish a correlation between flux densities and calculate a distribution of correlation measures in simulated sets of flux density logarithms, we perform permutations in luminosity space (see also Fender & Hendry 2000 for a similar Monte Carlo approach of evaluating an apparent distance-squared effect and the possible effect of Doppler beaming in the case of radio data of persistent X-ray binaries).",
"In this way, we can simulate the effect of a common distance on intrinsically uncorrelated luminosities, by applying a common redshift to permuted luminosity pairs to return to flux space.",
"By assessing the significance in flux space we avoid Malmquist bias, and we automatically account for the limited flux dynamical ranges in the two frequencies under consideration.",
"We do so as follows: From the measured radio and gamma-ray flux densities, we calculate radio and gamma-ray luminosities at a common rest-frame radio frequency and rest-frame gamma-ray energy.",
"We permute the evaluated luminosities, to simulate objects with intrinsically uncorrelated radio/gamma luminosities.",
"We assign a common redshift (one of the redshifts of the objects in our sample, randomly selected) to each luminosity pair, and return to flux-density space.",
"Assigning a common redshift allows us to simulate the common-distance effect on uncorrelated luminosities.",
"Using measured redshifts and luminosities guarantees that the distance and luminosity dynamical range in our simulated samples is also identical to that of our actual sample.",
"To avoid apparent correlations induced by a single very bright or very faint object much brighter or fainter than the objects in our actual sample, we reject any flux-density pairs where one of the flux densities is outside the flux-density dynamical range in our original sample.",
"Using a randomly selected set of flux density pairs, with number equal to the number of objects in our actual sample, we calculate a value for $r$ .",
"We repeat the process a large number of times, and calculate a distribution of $r-$ values for intrinsically uncorrelated flux densities.",
"The fraction of $|r|\\ge r_{\\rm data}$ , where $r_{\\rm data}$ is the $r-$ value for the observed flux densities, is the probability to have obtained an apparent correlation at least as strong as the one seen in the data from a sample with intrinsically uncorrelated gamma-ray/radio emission.",
"This quantifies the statistical significance of the observed correlation.",
"Formally, the null hypothesis tested with this procedure is $H_0:$ The radio and gamma-ray luminosity of blazars are independent, and redshift is independent of both luminosities.",
"We note that in many cases, this is not the hypothesis we would like to be testing, as luminosities depend on redshift in most population models of active galactic nuclei.",
"Ideally, we would like to test for independence between radio and gamma-ray luminosities conditioned on redshift.",
"However, this is not always practically possible due to sample size and redshift span of the sources.",
"For the cases when the sample size is large enough and the sources included in the sample are adequately spread over redshifts, the test discussed in the next subsection will fulfill this requirement.",
"For cases however when sample limitations are prohibitive for such a study, we show that testing $H_0$ with the implementation presented in this work can provide a conservative alternative to the full problem: if $H_0$ is rejected with high significance, then it is safe to assume that radio and gamma-ray luminosities are also not independent conditioned on redshift.",
"However, if $H_0$ cannot be rejected, no conclusion can be reached for either hypothesis, as absence of evidence for a correlation is not equivalent with evidence for absence of a correlation."
],
[
"Larger samples: splitting the sample in redshift bins",
"The process of pair rejection discussed in step 4 above may alter the distribution of luminosities, fluxes, and redshifts of the randomized data and introduce substantial differences from the corresponding distributions of the original dataset.",
"The cause of this effect is the randomization of redshifts among all sources, and it is straight-forward to understand.",
"Low-luminosity nearby objects, when combined with large redshifts, will result in very faint fluxes which are outside the original flux dynamical range and thus rejected.",
"For this reason, the simulated datasets will have fewer very-low–luminosity objects compared to the original dataset.",
"In addition, rare, high-luminosity, high-redshift objects, when combined with low redshifts, will result in very high fluxes, also outside the original flux dynamical range and thus rejected.",
"For this reason, the simulated datasets will also have fewer very-high–luminosity objects compared to the original dataset.",
"In contrast, the number of intermediate-luminosity objects will be relatively enhanced in simulated datasets.",
"The distributions of redshifts and fluxes of the simulated datasets will also be altered for similar reasons.",
"If the pair rejection rate is high, the properties of the simulated datasets could be different than the properties of the original dataset, and these biases could affect our estimation of a correlation significance.",
"In small and subjectively selected datasets, this problem is a necessary evil.",
"The effect of these biases is, as we will show below, to worsen the estimated significance of a correlation, rather than induce false positives of enhanced significance.",
"However, in the case of larger samples, there is a simple alteration in the methodology described in §REF that can significantly alleviate these biases: splitting the sample in redshift bins.",
"In this variation of the test, the original sample is split into a number of bins dependent on the available number of objects (as we discuss below, we need about 10 objects or more per bin, and in any case no fewer than 8).",
"We then generate randomized flux-density pairs in each redshift bin with the process described above.",
"Because the range of redshifts that are permuted between objects of different luminosities is much smaller, the likelihood that one of the resulting randomized flux densities will exceed the flux-density dynamical range of the original dataset is much smaller.",
"As a result, the pair rejection rate is decreased, and the luminosity, redshift, and flux distributions of the randomized data pairs resemble more closely those of the original dataset.",
"The similarity between distributions of the randomized and the original data increases as the size of the sample increases and the width of each redshift bin decreases.",
"If the original dataset is also a statistically complete and flux-limited sample, then the test asymptotically approaches the luminosity-function–sampling test as the size of the original dataset approaches infinity and the size of each redshift bin used approaches zero.",
"This can be understood as follows.",
"In the limit of zero-size redshift bin, all objects within a single redshift bin are at the same distance.",
"Therefore, permuting the luminosities of objects at that distance is equivalent to forming luminosity pairs by randomly sampling each frequency's luminosity function at a specific redshift and with a specific flux-density limit (the limit of the original sample).",
"Repeating the process at all redshift bins is equivalent to sampling the luminosity functions at all redshifts.",
"Then, the “pool” of randomized data pairs, from which we draw the mock datasets, could have been equivalently generated through luminosity function sampling.",
"Formally, the null hypothesis tested with this procedure is $H_0:$ Conditional on redshift, the radio and gamma-ray luminosity of blazars are independent, which is the hypothesis that one would generally wish to test.",
"For this reason, this version of the test should be preferred whenever possible."
],
[
"Implementation",
"In this section we describe how the method discussed above can be implemented in practice for small and large datasets."
],
[
"Small, subjectively selected samples",
"The first step is to convert the blazar gamma-ray fluxes (which are usually reported as integrated photon fluxes $F$ above some fiducial energy $E_0$ , usually 100 MeV), to energy flux densities, so that the comparison with radio flux densities can be done on an equal footingOther possible choices is to correlate radio flux densities with gamma-ray photon fluxes at some particular energy bin, or with the integrated photon fluxes themselves (see, for example, Abdo et al. 2011).",
"In these cases, Eq.",
"REF should be changed accordingly.. We do so by assuming that the photon fluxes are power laws, so that the flux (number of photons per unit area-time-energy bin) is $\\frac{dN_{\\rm photon}}{dE\\, dA\\, dt} = F_0\\left(\\frac{E}{E_0}\\right)^{-\\Gamma }\\,.$ In this case, the gamma-ray energy flux density $S_\\gamma \\equiv dE/dE\\, dA\\, dt $ at $E_0$ is given by $S_\\gamma (E_0) =F_0E_0 = F(\\Gamma -1)$ and its energy dependence is $S_\\gamma (E) = (\\Gamma -1)F\\left(\\frac{E}{E_0} \\right)^{-\\Gamma +1}\\,.$ The relation between monochromatic flux density $S(\\nu )$ and monochromatic luminosity $L(\\nu )$ for a source at redshift $z$ is $S(\\nu )= \\frac{L[\\nu (1+z)]}{4\\pi d^2(1+z)}$ where $d=(c/H_0)\\int _0^z dz/\\sqrt{\\Omega _\\Lambda +\\Omega _m(1+z)^3}$ .",
"Here $H_0$ is the present-day value of the Hubble parameter, and $\\Omega _\\Lambda $ and $\\Omega _m$ are the vacuum energy and matter density parameters.",
"In this work, we have used $\\Omega _m = 0.26$ and $\\Omega _\\Lambda = 1-\\Omega _m$ , consistent with, e.g., Larson et al.",
"(2011).",
"Note that the value of $H_0$ drops out of the calculation as $d$ in the formalism we describe below appears only in ratios.",
"If the source has a spectral index $\\alpha $ so that $S(\\nu ) \\propto \\nu ^\\alpha $ at the frequency of interest, Eq.",
"(REF ) implies that the relation between $S(\\nu )$ at observer-frame $\\nu $ and $L(\\nu )$ at rest-frame $\\nu $ (the K-correction) is $L(\\nu ) = S(\\nu )4\\pi d^2(1+z)^{1-\\alpha }.$ So if a radio flux density $S_r(\\nu )$ (at observer-frame $\\nu $ ) is turned into a luminosity density (at rest-frame $\\nu $ ) using a redshift $z_1$ and a spectral index $\\alpha _r$ , and this luminosity density is then returned to flux-density–space (at observer-frame $\\nu $ ) using a different redshift $z^{\\prime }$ but the same spectral index $\\alpha _r$ , we can write $S^{\\prime }_r(\\nu ) = S_r(\\nu ) \\left(\\frac{d_1}{d^{\\prime }}\\right)^2\\left(\\frac{1+z_1}{1+z^{\\prime }}\\right)^{1-\\alpha _r}\\,,$ where $d_1=d(z_1)$ and $d^{\\prime }=d(z^{\\prime })$ .",
"For the same procedure with gamma-ray flux densities and a source at a redshift $z_2$ we can write $S^{\\prime }_\\gamma (E_0) = (\\Gamma -1)F\\left(\\frac{d_2}{d^{\\prime }}\\right)^2\\left(\\frac{1+z_2}{1+z^{\\prime }}\\right)^\\Gamma \\,.$ In practice, we perform the following steps.",
"(i) For each blazar, we use the flux density in radio and gamma-ray frequency to produce monochromatic luminosities at the same (now rest-frame) frequency in the two bands.",
"(ii) We construct all possible pairings (excluding the original ones) of radio and gamma-ray luminosities from our observed sample.",
"(iii) We assign a common redshift $z^{\\prime }$ to each permuted pair (one of the available redshifts in our sample).",
"(iv) We calculate “mock” radio and gamma-ray flux densities $S^{\\prime }_r, S^{\\prime }_\\gamma $ for each pair using Eq.",
"(REF )Equivalently, we can use directly Eqs.",
"(REF ) and (REF ), without explicitly calculating luminosities first.. (v) We accept the pair if both flux densities are within our original flux-density dynamical range in each band, or reject it otherwise.",
"(vi) We randomly select $N$ pairs out of all the possible combinations, where $N$ is equal to the number of our original observations.",
"Each set of $N$ pairs is now a simulated dataset of intrinsically uncorrelated flux/flux observations.",
"(vii) For each simulated dataset, we compute $r$ using Eq.",
"(REF ), where $X_i = \\log (S^{\\prime }_{r,i})$ and $Y_i = \\log (S^{\\prime }_{\\gamma ,i})$ , with $i$ running from 1 to N. (viii) We repeat steps (vi-vii) $m$ times, where $m$ is a sufficiently large number to sample the underlying $|r|$ distribution.",
"In our tests below $m$ is between $10^6 - 10^7$ .",
"(ix) We calculate the probability for the observed $|r|$ to have occurred through uncorrelated flux densities from the $|r|-$ values obtained in step (viii).",
"Our technique can be applied to samples that are very small and still yield a reliable estimate of the distribution of $|r|$ .",
"The total number of simulated pairs that we can construct through our permutation technique from $N$ objects is $N_{\\rm pairs}=N^2(N-1)$ (where we permute both flux densities as well as redshifts).",
"Only a fraction $N_{\\rm surv}$ will survive the low- and high- flux-density cuts that ensure that the flux-density dynamical range remains the same as in the original sample.",
"Assuming a reduction no larger than a factor of 5 (i.e.",
"$N_{\\rm surv} \\gtrsim N_{\\rm pairs}/5$ , shown in practice to be a conservative assumption), the total number of combinations of $N$ pairings different from each other by one or more pairs out of a population of $N_{\\rm surv}$ objects then is ${\\rm pair \\,\\,\\, combinations} = \\frac{N_{\\rm surv}!}{N!",
"(N_{\\rm surv}-N)!}",
"\\,,$ which is $\\gg 10^7$ for samples with $N \\gtrsim 8$ .",
"However, in small datasets a statistically significant correlation is harder to establish, even if the distribution of $|r|$ can be estimated with sufficient statistics.",
"In addition, as we will also show in §, the biases in the luminosity, redshift, and flux distributions of the simulated datasets introduced due to pair rejections (see discussion in §REF ) tend to worsen the significance that can be established through this test."
],
[
"Splitting larger samples in redshift bins",
"Whenever the size of the source sample is large enough to allow splitting in more than one redshift bins, this variation of the test is recommended, as the effect of biases introduced through pair rejection decreases with increasing number of redshift bins (decreasing redshift bin size).",
"To implement this variation of the test, we split the sample in $N_z$ redshift bins.",
"Our choice for the test implementation is to use variable redshift bin size, selected in such a way that the number of sources in each bin is as close to equal as possible, but never fewer than 8.",
"However, other choices are also possible (for example, keeping the redshift bin size approximately equal; or splitting by luminosity distance rather than redshift, and keeping the luminosity distance bin size approximately equal).",
"For the sources in each one of the $N_z$ bins, we apply steps (i)-(v) of §REF .",
"We then combine all accepted simulated data pairs from all redshift bins to generate the “pool” of all possible pair combinations.",
"Finally, we apply steps (vi) - (ix) to this combined randomized pair “pool”."
],
[
"Demonstrations of the test",
"In this section we present example applications of our tests, using both real and simulated data, to evaluate the performance of our proposed test and demonstrate several aspects of its implementation.",
"For the applications on real data, we will use gamma-ray flux measurements from Fermi LAT and radio flux-density measurements from the OVRO 40 M Monitoring Program (Richards et al.",
"2011).",
"In addition, we will use simulated data to evaluate the the performance of the method: its effectiveness in rejecting false positives due to common-distance biases in correlation assessments, and its power in establishing significant correlations when such correlations do exist.",
"As a benchmark we will use the face-value estimate of the significance for the Pearson correlation coefficient $r$ , which evaluates the probability of a certain (or bigger) value of $r$ to occur by chance in the “dart-throwing” scenario (i.e., when pairs are randomly drawn from uncorrelated Gaussian distributions, assuming that no biases exist).",
"In the latter scenario, the significance only depends on the value of $r$ and the sample size $N$ .",
"In the null hypothesis (uncorrelated data), the variable $ t = \\frac{r\\sqrt{N-2}}{1-r^2}$ follows a Student's t-distribution with $N-2$ degrees of freedom.",
"Using Eq.",
"(REF ) significances (p-values) can be estimated for any given values of $r$ and $N$ by taking the two-tail integral of the appropriate t-distribution.",
"In general, the variation of the test with redshift binning is the one which we recommend whenever possible (whenever sample restrictions allow its use), and it is the one which we have used in our simulated datasets."
],
[
"Small sample, no redshift bin splitting",
"As an example of a relatively small dataset, we use the set of blazars that are included both in the LAT bright AGN source list (Abdo et al.",
"2009, produced using three months of LAT observations), as well as in the “complete sample” of the OVRO 40 M Monitoring Program (Richards et al.",
"2011).",
"The latter consists of the 1158 sources north of $-20^\\circ $ declination in the Candidate Gamma-Ray Blazar Survey (CGRaBS) sample, which is a sample of 1625 sources, mostly blazars, selected by their flux and spectral index in radio, and flux in X-rays, to resemble the blazars detected by EGRET (Healey et al.",
"2008).",
"The 1158 of the “complete sample” are observed approximately twice a week at 15 GHz with the Owens Valley Radio Observatory (OVRO) 40 M Telescope.",
"For this study, we only use sources with known redshifts, and for which a sufficient number of high-quality 15 GHz observations were taken in the same three-month time interval of LAT observations so as to produce a meaningful concurrent 15 GHz average flux density (see Richards et al.",
"2011).",
"This sample contains 38 sources.",
"Figure: 3-month averaged concurrent 15 GHz versus 100 MeV observer-frame flux densities for the 38 blazars in our sample.Figure: Distribution of |r|-|r|-values for 38 blazars of the same dynamical range in redshift and radio and gamma-ray flux densities and luminosities as blazars in our sample.",
"The vertical arrow indicates the r-r-value for the actual observations (r=0.62r=0.62).",
"The significance of the correlation is 1.5×10 -4 .1.5\\times 10^{-4}.Figure REF shows 3-month averaged 15 GHz flux densities plotted against 100 MeV observer-frame flux densities obtained by integration over the same time interval for the 38 blazars in our sample.",
"The error bars in this plot are substantially smaller than the scatter of points (see, e.g., Ackermann et al.",
"2011) and have been omitted for clarity.",
"An apparent correlation between the radio and gamma-ray time-averaged flux densities is obvious, however the statistical significance of an intrinsic correlation between the radio and gamma-ray emission of these objects needs to be quantitatively assessed.",
"To this end, we apply the data randomization analysis we have introduced in §REF .",
"Figure: 11-month averaged concurrent 15 GHz versus 100 MeV observer-frame flux densities for the 160 blazars in the larger sample.The probability distribution of the values of $|r|$ in our simulated samples with intrinsically uncorrelated radio/gamma luminosities is shown in Fig.",
"REF .",
"The vertical arrow in this figure indicates the $r-$ value for the observed data, equal to $0.62$ .",
"From the 38 objects in our sample a total number of $38^2\\times 37 = 53,428$ permuted pairs were generated.",
"Of those, $13,003$ pairs had both gamma-ray and radio flux densities within the dynamical range of the original dataset.",
"The accepted pairs were used (in $10^7$ randomly drawn sets of 38) to generate the distribution shown in Fig.",
"REF .",
"The probability to obtain $|r|\\ge 0.62$ from intrinsically uncorrelated flux-density measurements due to the effect of a common distance is $1.5\\times 10^{-4}$ .",
"For comparison, the significance estimate ignoring any biases and using only Eq.",
"(REF ) is $3.3\\times 10^{-5}$ : without a careful analysis, we would evaluate the observed correlation as more significant than we do when accounting for common-distance and flux biases, as these effects are likely to contribute at least part of the observed correlation strength.",
"We will elaborate on the origin and quantitative behavior of this discrepancy in the following sections.",
"Note that the pair rejection rate is high - only 24% of the permuted pairs are within the original flux density dynamical range and were accepted; biases introduced in the luminosity, flux, and redshift distributions of the simulated data are therefore a concern.",
"However, as we will show below, were these biases absent, the significance of the correlation would improve."
],
[
"Larger sample, behavior of test with increasing number of redshift bins",
"We now turn to a demonstration of the second variation of our test, where the sample is split in redshift bins, and we discuss the alleviation of biases induced through pair rejection, and the improvement of the correlation significance with increasing number of bins.",
"To allow splitting in enough redshift bins to adequately demonstrate the behavior of the test in the many-bins limit we use the significantly larger sample of 160 sources that: (a) are included in the first year LAT catalog (Abdo et al.",
"2010); (b) are part of the OVRO 40 M telescope monitored sample; (c) have known redshifts.",
"This same sample has been examined in detail for intrinsic correlations between 15 GHz flux density and LAT gamma-ray fluxes at various energy ranges by Ackermann et al.",
"(2011), using the test discussed hereHere, we use for the gamma-ray band data the 100 MeV flux density calculated according to Eq.",
"REF from integrated photon fluxes for $E>100$ MeV and using the photon index provided in 1LAC, which is different that any of the flux densities or integrated fluxes examined by Ackermann et al.",
"2011; this is the origin of the small differences in the value of $r$ obtained here for the data..",
"Figure REF shows 11-month–averaged 15 GHz flux densities plotted against 100 MeV observer-frame flux densities obtained by integration over the same time interval for the 160 blazars in the sample described above.",
"The error bars in this plot are again substantially smaller than the scatter of points (see, e.g., Ackermann et al.",
"2011) and have been omitted for clarity.",
"Through visual inspection, this sample also appears to feature an apparent correlation between radio and gamma-ray flux densities, with scatter comparable to that of the smaller sample of §REF .",
"The correlation coefficient of the data in this case is $r=0.48$ .",
"The biases introduced through pair rejection in our first variation of the test (where the sample is not split in redshift bins) are demonstrated in Figs.",
"REF -REF .",
"The luminosities in these figures are in units of $4\\pi (c/H_0)^2S_0$ , where $H_0$ is the Hubble parameter, and $S_0 = 1$ Jy for 15 GHz source-frame luminosities and $S_0=10^{-8} {\\rm GeV/s-cm^2-GeV}$ for 100 MeV source-frame luminosities.",
"Figure REF shows the fraction of objects in each logarithmic radio luminosity bin for the data (thick black line) and the accepted scrambled pairs (thin lines).",
"Different line colors correspond to different numbers of redshift bins, as in the figure legend.",
"When only one redshift bin is used (thin black line, equivalent to the first variation of our test), the shape of the luminosity distribution of the accepted scrambled pairs has a qualitatively different shape than that of the data: objects in the bins corresponding to the $\\sim $ 3 lowest orders of magnitude in luminosity are significantly underrepresented compared to the original sample, because these low luminosities, corresponding to nearby objects in the data, are frequently rejected when they are combined with high redshifts and produce very low simulated flux densities outside the original flux density dynamical range.",
"When we split the sample in a larger number of bins the effect is alleviated.",
"At 16 redshift bins the radio luminosity distribution of simulated data is very close to that of the original data, and it is essentially converged, as it does not change appreciably when the number of redshift bins is increased to 20.",
"A very similar behavior for the gamma-ray luminosity distribution is shown in Fig.",
"REF .",
"In the case of the redshift distribution, shown in Fig.",
"REF , both the very low and the very high redshift bins are underestimated when no data splitting is applied (thin black line).",
"However, at 16 redshift bins the real and simulated data distributions are very similar, and the simulated data distribution is, again, converged.",
"In all distributions, as the number of redshift bins increases, the difference between data and simulated distributions decreases, as a result of the decreasing pair rejection rate which, at 16 redshift bins, is $\\lesssim 20\\%$ for all bins.",
"Figure: Fraction of objects in each logarithm-in radio luminosity bin for the data (thick black line) and the accepted scrambled pairs (thin lines; different colors correspond to different numbers of redshift bins, as in legend).",
"The radio luminosities are in units of L 0, radio =4π(c/H 0 ) 2 S 0 L_{0,\\rm radio} = 4\\pi (c/H_0)^2S_0 where S 0 =1S_0=1Jy.Figure: Fraction of objects in each logarithm-in gamma-ray luminosity bin for the data (thick black line) and the accepted scrambled pairs (thin lines; different colors correspond to different numbers of redshift bins, as in legend).",
"The gamma-ray luminosities are in units of L 0,γ =4π(c/H 0 ) 2 S 0 L_{0,\\rm \\gamma }= 4\\pi (c/H_0)^2S_0 where S 0 =10 -8 GeV /s- cm 2 - GeV S_0=10^{-8} {\\rm GeV/s-cm^2-GeV}.Figure: Fraction of objects in each redshift bin for the data (thick black line) and the accepted scrambled pairs (thin lines; different colors correspond to different numbers of redshift bins, as in legend).Figure: Distribution of |r|-|r|-values for randomly selected 160-blazar sets picked from the ensemble of accepted pairs generated through data scrambling.",
"The vertical arrow indicates the r-r-value (r=0.48r=0.48) for the actual observations.",
"Different colors correspond to different numbers of redshift bins, as in legend.The behavior of the estimated significance as a function of the number of redshift bins is shown in Fig.",
"REF , where we have plotted the distribution of the absolute values of the correlation coefficients $|r|$ for each test implementation.",
"Again, different colors correspond to different numbers of redshift bins used as in Figs.",
"REF -REF .",
"$10^6$ simulations were used to produce each curve.",
"The $r-$ value for the data is shown with the arrow.",
"The significance of the correlation as evaluated with 16 redshift bins is $<10^{-6}$ (if we fit the distribution shown with the blue line in Fig.",
"REF with a Gaussian, we obtain a significance of $\\sim 10^{-7}$ ).",
"Again, using Eq.",
"(REF ) to compare with the simple significance estimate based only on $r$ and $N$ , we find that $t=6.88$ , for which the two-tailed t-distribution with 158 degrees of freedom yields a much stronger significance of $1.3\\times 10^{-10}$ .",
"The reason for this substantial difference can be immediately understood qualitatively through inspection of Figs.",
"REF -REF .",
"Both the radio and the gamma-ray luminosity distributions of the data show broad peaks, which means that even if there was no intrinsic correlation between radio and gamma-ray emission and radio and gamma-ray luminosities were simply randomly drawn from these distributions, values of radio and gamma-ray luminosity around the peaks would appear frequently.",
"As a result, pairs of radio/gamma-ray fluxes corresponding to underlying luminosities clustered around likely values would be common.",
"Such pairs, when modulated with a common distance factor, would yield an apparent correlation in flux-flux space by chance, much more frequently than if there was no peak in the luminosity distributions.",
"The unsophisticated significance estimate contains no information about common-distance effects and the behavior of the underlying luminosity distributions, and for this reason overestimates the significance of the apparent correlation.",
"However, even when these effects are accounted for using our method, the data show significant intrinsic correlation between radio and gamma-ray fluxes.",
"We can see that the significance of the correlation monotonically improves with increasing number of binsThe exact statement is that the significance monotonically improves with decreasing fraction of rejected pairs.",
"Should a particular choice in redshift binning result in increased rejection fraction, the significance would worsen, even if the number of bins was larger..",
"The reason for this behavior can be understood from Figs.",
"REF and REF .",
"The more frequent rejection of pairs at the edges of the luminosity distributions results in the artificial enhancement of the peaks in the luminosity distributions at intermediate luminosities.",
"This stronger peak results in an enhanced incidence of artificial correlations.",
"As a result, the significance of the apparent correlation of the data drops.",
"This is also the reason for the appearance of a peak at positive values of $|r|$ in the $|r|$ distribution of simulated datasets in Fig.",
"REF when the number of redshift bins is low.",
"However, it is not guaranteed that a small number of redshift bins will generate such a peak at positive $|r|$ - this depends on the details of the luminosity distribution of the original dataset and of the pair rejection.",
"For example, such a peak does not appear in our smaller dataset example in Fig.",
"REF .",
"Conversely, a large number of redshift bins does not guarantee that such a peak will not appear.",
"If our original dataset is selected in such a way that a certain narrow range of luminosities is over-represented, then such a peak is intrinsic to the dataset and it will appear regardless of number of bins used.",
"It is also interesting to consider the behavior of the test in the limit of a very large number of redshift bins that could be in principle used if we had a very large sample available for study, and in the case that our sample was a statistically complete, flux-limited set of sources.",
"In this case, each redshift bin could be made very narrow, and all sources within the bin would be located essentially at the same distance.",
"The set of radio luminosities within each bin would then be a representation of the radio luminosity function at a fixed redshift, with a limiting luminosity set by the limiting flux and the bin redshift.",
"The set of gamma-ray luminosities within the same bin would similarly be a representation of the gamma-ray luminosity function.",
"Since all sources would be located at the same distance, data randomization within the bin would never produce fluxes outside the original dynamical range, and no pairs would be rejected.",
"The simulated pairs would then have exactly the same luminosity distribution as the data, and they would continue to be a representation of the luminosity functions at the two frequencies under consideration, as “fair” as the original data.",
"As a result, in the limit of the “perfect sample” and a large number of redshift bins, our test would yield exactly the same result as a statistical test sampling random radio and gamma-ray fluxes from known luminosity functions.",
"Our test deviates increasingly from this result as the statistical properties and the size of the sample deteriorate.",
"As shown in Fig.",
"REF , our proposed test is conservative: a smaller number of redshift bins will generally result in an increased rate of pair rejection and a worse correlation significance.",
"In this way, it is possible that a real, intrinsic correlation cannot be confirmed by this test if a poor sample is used.",
"However, the test will not yield artificially enhanced significances.",
"In this section we discuss the performance of the test when applied to datasets drawn from intrinsically uncorrelated populations.",
"In particular, we evaluate the effectiveness of our test in rejecting false positives that we might have obtained due to common-distance biases had we used the estimate of the significance given by Eq.",
"(REF ).",
"In §REF we describe how we generate the intrinsically uncorrelated simulated datasets that we use to test the performance of our method, and in §REF we examine this performance and the robustness of the evaluated significances against common-distance biases."
],
[
"Generation of uncorrelated simulated datasets",
"To test the performance of our method in the case of intrinsically uncorrelated data, we produce simulated datasets in the following way.",
"We draw a gamma-ray luminosity from a log-normal distribution, with probability density function $p(L_\\gamma ) = \\frac{1}{L_\\gamma \\sqrt{2\\pi \\sigma _1^2}}\\exp \\left[-\\frac{\\left(\\ln L_\\gamma -\\mu _1\\right)^2}{2\\sigma _1^2}\\right]\\,.$ We draw a radio luminosity from a log-normal distribution, with probability density function $p(L_r) = \\frac{1}{L_r\\sqrt{2\\pi \\sigma _2^2}}\\exp \\left[-\\frac{\\left(\\ln L_r -\\mu _2\\right)^2}{2\\sigma _2^2}\\right]\\,.$ For this pair, we also draw a common redshift from a uniform distribution with lower limit $z_{\\rm low}$ and upper limit $z_{\\rm up}$ .",
"We evaluate the resulting gamma-ray and radio fluxes, and check whether they reside within an allowed flux dynamical range of three orders of magnitude.",
"If either one does not, we reject the pair and repeat the draw.",
"We repeat the process above until we have 30 pairs within our desired flux dynamical range.",
"This then is our simulated, intrinsically uncorrelated dataset, to which both a common distance factor and a limit in the flux dynamical range have been applied.",
"We anticipate that the effect of the common-distance biases will increase as the luminosity dynamical range decreases and the redshift dynamical range increases.",
"This can be easily understood by considering the extreme limits.",
"Datasets drawn from luminosity delta-functions will always appear perfectly correlated within errors: the spread in fluxes in each waveband is only due to the distance factor, which is the same in each pair, and errors.",
"Conversely, if all sources are at the same redshift, there will be no common-distance effect: the distance factor is always the same, and any observed correlation has to be intrinsic.",
"To assess when common-distance biases become important, we will use the coefficient of variation (eg., Frank & Althoen 1995) of the redshift and luminosity distributions (standard deviation in units of the mean, $c_z$ and $c_L$ respectively) to quantify the dynamical range of each distribution.",
"As we will see in the next section, the importance of common-distance biases is generally dependent on the ratio of the luminosity coefficient of variation to the redshift coefficient variation, $c_L/c_z$ , and decreases as this ratio increases.",
"In our simulated datasets we have used radio and gamma-ray luminosity distributionsSince the flux/flux correlation coefficient is evaluated in logarithmic space, changing the units of the luminosity, or, equivalently, the mean of the luminosity distribution, will only uniformly slide the points along the flux axes and will not affect the apparent correlation strength, as long as the flux limits are also shifted accordingly.",
"with $\\mu _1=\\mu _2 = \\mu _0$ and $\\sigma _1=\\sigma _2=\\sigma _0$ and, as a result, the same value of $c_L$ , but in practice the relevant value of $c_L$ is the one of the more extended of the two distributions.",
"For the distributions we have used here, $c_L = \\left[\\exp \\left(\\sigma _0^2\\right)-1\\right]^{1/2}$ and $c_z = \\frac{z_{\\rm up}-z_{\\rm low}}{\\sqrt{3}(z_{\\rm up}+z_{\\rm low})}\\,.$"
],
[
"Robustness of the test against common-distance biases",
"To evaluate the robustness of our test against common-distance biases and its ability to reject false positives, we generate, using the procedure described in §REF , simulated datasets with varying values of the ratio $c_L/c_z$ of 30 objects each, and we calculate the significance of the apparent correlation using our method and the simple estimate of Eq.",
"(REF ).",
"In practice, we implement the simulated dataset generation for a specific value of $c_L/c_z$ in two distinct ways, and we compare the results as shown in Fig.",
"REF .",
"First, we keep the redshift distribution fixed to a uniform distribution with lower limit $z_{\\rm low} = 0$ and an upper limit $z_{\\rm up} = 2$ , and we draw the radio and gamma-ray luminosities from identical distributions with $\\mu _0=0$ and a varying value of $\\sigma _0$ .",
"In this way, we derive the black points in Fig.",
"REF .",
"Next, we keep the luminosity distributions fixed at $\\mu _0=0$ , $\\sigma _0 =1$ , and we draw redshifts from uniform distributions with varying upper and lower limits, always symmetric about $z=1$ .",
"In this way, we derive the red points in Fig.",
"REF .",
"For each dataset, we then evaluate the significance of the apparent correlation using the variation of our test that utilizes redshift binning; these results are shown with the circles/solid lines in Fig.",
"REF .",
"We compare these values with the significance estimate of Eq.",
"(REF ) which does not account for any common-distance bias; these results are shown with the diamonds/dashed lines in Fig.",
"REF .",
"For low values of $c_L/c_z$ , the simple estimate of Eq.",
"(REF ) returns false positives with high significance for these intrinsically uncorrelated datasets.",
"Our method however correctly identifies these apparent correlations as artifacts of common-distance biases, and returns a significance value always consistent with no correlation.",
"For higher values of $c_L/c_z$ , common-distance biases are less important, and both significance estimates agree, returning a result consistent with no correlation.",
"The roughly consistent, within noise, behavior of the black and red lines, despite the different method of implementation of the same value of $c_L/c_z$ , implies that the $c_L/c_z$ ratio is a good way to quantify the way in which the dynamical ranges in the luminosity and redshift distributions induce common-distance biases in correlations between different wavebands evaluated in flux space.",
"As a rule of thumb, a value of $c_L/c_Z $ smaller than about 5 indicates that common-distance biases may be important, and the simple estimate of Eq.",
"(REF ) (or, equivalently, permutation methods in flux space alone which do not account for the common distance modulation in each flux pair) should not be trusted as they might yield false positives.",
"Figure: Significance (probability to obtain an rr as big or bigger than the data by chance) returned by our method (circles, solid lines) compared to significance returned by the simple estimate of Eq.",
"() which does not account for common distance biases, as a function of the ratio of coefficients of variation of the luminosity and redshift distributions.",
"Black points were generated by varying the width of the luminosity distribution while keeping the redshift distribution fixed.",
"Red points were generated by varying the width of the redshift distribution while keeping the luminosity distribution fixed.",
"Our method always succeeds in rejecting artificial correlations induced by common-distance biases."
],
[
"Demonstrations on simulated data: Correlated datasets",
"In the previous section we have shown that our proposed method successfully accounts for common-distance biases and returns results consistent with no correlation even when the simple estimate of Eq.",
"(REF ) yields very significant false positives.",
"Here we wish to examine whether this robustness against false positives comes at the expense of the power of the test in rejecting the null hypothesis of no correlation when the data are intrinsically correlated.",
"In §REF we discuss how we generate intrinsically correlated datasets with minimal common-distance biases, and in §REF we discuss how the power of the test depends on the number of objects in the dataset, $N$ , and on the apparent correlation strength, $r$ , as well as how these dependencies compare with the simple formula of Eq.",
"REF ."
],
[
"Generation of correlated datasets",
"To generate mock datasets with known intrinsic correlation signals, we assume that the radio and gamma-ray monochromatic luminosities at the frequencies of interest are linearly correlatedWe adopt this assumption in the interest of simplicity for these demonstrations; this does not have to be the case in nature.",
"Nonlinear correlations between the luminosities in the two wavebands will further complicate the relation between intrinsic and apparent correlation strength., with some scatter obeying a log-normal distribution: $\\log L _r = C + \\log L_\\gamma + \\Delta \\log L_r$ where $C$ is a normalization constant, and $\\Delta \\log L_r$ is normally distributed with mean 0 and standard deviation $\\sigma $ , i.e.",
"if $\\Delta \\log L_r =x$ then the probability density of $x$ is given by $p(x) = \\frac{1}{\\sqrt{2\\pi } \\sigma }\\exp \\left[-\\frac{x^2}{2\\sigma ^2}\\right]\\,.$ Using Eq.",
"REF , this yields a relationship between radio and gamma-ray flux densities: $\\frac{S_r}{S_{r,0}} = \\frac{S_\\gamma }{S_{\\gamma ,0}} (1+z)^{\\alpha _r+\\Gamma - 1} \\times 10^{\\Delta \\log r}\\,.$ The scatter in this intrinsic correlation is quantified by $\\sigma $ .",
"We normalize the relation assuming that, for $z=\\Delta \\log r=0$ , a 15 GHz radio flux density of $1 {\\rm \\, Jy}$ corresponds to a gamma-ray flux density of $10^{-8} {\\rm GeV cm^{-2} s^{-1} GeV^{-1}}$ at 300 MeV.",
"We generate mock datasets by starting from the set of 136 sources which (a) are detected by Fermi LAT at energies between 300 MeV and 1 GeV and are included in the First Fermi Catalog (1LAC, Abdo et al.",
"2010); (b) are included in the OVRO 40 M monitoring sample; (c) have known redshifts (see Ackermann et al.",
"2011 for the details of this sample).",
"We use this set to obtain redshifts, gamma-ray fluxes, and gamma-ray spectral indices for our sources; for radio spectral indices, we use the historical values quoted in Ackermann et al. 2011.",
"We then use Eq.",
"(REF ) to obtain radio fluxes with a known correlation signal, by using the desired value of $\\sigma $ .",
"The value of the $c_L/c_z$ ratio in the sample we use is $\\sim 4$ , so, according to the findings of §REF , the effect of common-distance biases should be limited, and any apparent correlation between gamma-ray and radio emission should be primarily due to the intrinsic correlations we have imposed in the simulated datasets.",
"In this case, we would expect a well-behaved test to return results that are close to the simple significance estimates of Eq.",
"(REF ).",
"Figure: Distribution of Pearson product-moment correlation coefficients rr that arise from random realizations of 80 objects obtained using 80 randomly chosen Fermi sources from 1LAC and Eq.",
", for various values of σ\\sigma as in legend.Figure REF shows the distribution of Pearson product-moment correlation coefficients $r$ that arise from random realizations of 80 objects obtained in the manner described above, for various values of the intrinsic correlation scatter $\\sigma $ .",
"The striking feature of this plot is that the distributions of possible $r$ values of “observed” flux/flux correlations arising from different random realizations of the same intrinsic luminosity/luminosity correlation sampled with the same number of points can be quite extended, with its width increasing with increasing $\\sigma $ .",
"Even if we assumed that we knew the form of the underlying intrinsic luminosity/luminosity correlation, i.e.",
"if, in our case, we assumed that Eq.",
"REF holds exactly, and even with a relatively large sample (80 objects in this case), the observed value of the flux/flux correlation coefficient would only yield a rough and uncertain estimate of the scatter $\\sigma $ of the underlying correlation, although the uncertainty of the estimate would improve for increasing values of $r$ (decreasing values of $\\sigma $ )."
],
[
"Dependence of significance on number of observations and apparent correlation strength",
"Figure REF shows the dependence of the calculated significance of a correlation of fixed apparent and intrinsic strength (i.e, fixed values of $r$ and $\\sigma $ ) on the number of objects in the sample.",
"In the example presented here, mock datasets of $N$ objects were generated as described in §REF , using a fixed intrinsic correlation scatter $\\sigma = 0.4$ , and requiring an apparent correlation strength of $r=0.55\\pm 0.001$ .",
"For every value of $N$ plotted in Fig.",
"REF , 10 such mock datasets were produced, and for each dataset the significance was evaluated using the redshift-bin-splitting variation of the test (with the number of redshift bins chosen, for each value of $N$ , as discussed in §REF .)",
"The datapoints in Fig.",
"REF represent the the mean of $\\log _{10}$ (Significance) for these 10 realizations, and the error bars indicate the standard deviation of the 10 values of $\\log _{10}$ (Significance).",
"The solid line shows the result of Eq.",
"(REF ) for $r=0.55$ and varying $N$ .",
"Even this modest correlation with appreciable scatter can be established at high significance (better than $\\sim 10^{-5}$ ) with 60 or more objects.",
"Figure: Significance of an intrinsic correlation with σ=0.4\\sigma = 0.4 sampled with NN objects and resulting in an apparent correlation strength of r=0.55±0.001r=0.55\\pm 0.001, as a function of NN.",
"The points indicate the mean and error bars indicate the standard deviation of the calculated significances in 10 random implementations of the correlation.",
"The downwards triangle indicates an upper limit for NN=100, where the probability of the correlation to arise by chance was always found to be <10 -6 <10^{-6} (none out of 10 7 10^7 scrambled datasets had an |r||r| at least as big as the “data”).",
"The solid line shows the result of Eq.",
"() for r=0.55r=0.55 and varying NN.Figure: Significance of an intrinsic correlation with σ=0.6\\sigma = 0.6 sampled with 20 objects and resulting in an apparent correlation strength of varying rr, as a function of rr.",
"The points indicate the mean and error bars indicate the 1σ\\sigma variation of the calculated significances in 10 random implementations of the correlation.",
"The solid line shows the result of Eq.",
"() for N=20N=20 and varying rr.Figure REF shows the dependence of the significance that the redshift-bin-splitting variation of our method yields as a function of the apparent correlation strength (as quantified by $r$ ), when the underlying, intrinsic correlation and the number of objects are fixed.",
"We have used an intrinsic correlation with relatively large scatter ($\\sigma =0.6$ ), sampled with a relatively small number of objects ($N=20$ ).",
"As it is obvious from Fig.",
"REF (red line), this large scatter can result in a variety of apparent correlation values.",
"Again, for each value of $r$ plotted in Fig.",
"REF , we have generated 10 mock datasets as described in §REF , demanding that their apparent correlation strength is within $0.001$ of the plotted $r$ value.",
"For each dataset the significance was evaluated using the redshift-bin-splitting variation of the test (with the number of redshift bins chosen, for each value of $N$ , as discussed in §REF .)",
"The datapoints in Fig.",
"REF represent the the mean of $\\log _{10}$ (Significance) for these 10 realizations, and the error bars indicate the standard deviation of the 10 values of $\\log _{10}$ (Significance).",
"The solid line shows the result of Eq.",
"(REF ) for $N=20$ and varying $r$ .",
"As we can see in both Fig.",
"REF and Fig.",
"REF , the significance returned by our method in the absence of strong common-distance biases and for correlated datasets is consistent with, or only marginally worse than the results of the simple estimate of Eq.",
"(REF ).",
"We therefore conclude that the robustness of our method against common-distance biases in uncorrelated datasets does not come at the expense of the power of the method in the case of correlated datasets."
],
[
"Discussion and Conclusions",
"In this paper, we have introduced a data-randomization method for assessing the significance of apparent correlations between radio and gamma-ray emission in blazar jets, accounting explicitly for biases introduced through a common distance, small sample size, and complex or subjective sample selection criteria.",
"Our method is designed to be conservative and applicable even to small samples selected with subjective criteria.",
"An application of this technique to the first Fermi catalog of point sources (Abdo et al.",
"2010) has been discussed in Ackermann et al.",
"(2011), which also discusses the dependency of the strength and significance of the radio/gamma flux correlation on gamma-ray photon energy, and on the concurrency of the two datasets.",
"A study of the dependency of correlation strength and significance on radio frequency can be assessed by application of this method to multi-frequency radio monitoring data, such as the results of the F-GAMMA Program (Angelakis et al.",
"2010; Fuhrmann et al.",
"2012, in preparation).",
"Using simulated datasets of intrinsically uncorrelated data, we have demonstrated that our proposed method is robust against artificial correlations induced by common-distance biases, and returns results consistent with no correlation even when simple face-value significance estimates that do not account for these biases would have incorrectly claimed highly significant correlations.",
"We have shown that the effect of these biases can be quantified by the ratio of the coefficients of variation of the luminosity distribution (in the waveband which has the widest luminosity distribution) over the redshift distribution.",
"When this ratio is lower than $\\sim 5$ false positives are possible when the biases are not accounted for, with their significance increasing with decreasing value of the ratio.",
"In addition, using simulated datasets of intrinsically correlated data, we have shown that our method can establish existing correlations with significance comparable to that of other tests, and thus its robustness against false positives does not come at the expense of its power in rejecting the null hypothesis.",
"As our method is designed to be applied to astronomical datasets, we have implemented it in such a way to directly address the limited flux dynamical range that is generally encountered in such data.",
"Astronomical datasets are generally expected to have a low-flux limit in each frequency due to the limited sensitivity of any given observing instrument, and a high-flux limit corresponding to the most favorable combination of luminosity/distance that happened to occur given our position in the universe, which is generally determined by chance, and fixed by the observed dataset.",
"Simulated data with one or two of the fluxes outside these limits represent situations possible in nature but impossible to observe in the specific experiment.",
"Such simulated data coud result in apparent correlations in our simulated samples induced by a single very bright or very faint object much brighter or fainter than the objects in our actual sample, while most of the other pairs would be scattered in a limited area of the flux/flux space (the classical case of an artificial correlation seen when an uncorrelated scattered cluster of points is combined with a single point far away from the main cluster).",
"Such configurations would be impossible in our actual observed datasets, but could be frequent in our simulated datasets, thus biasing the simulated datasets toward much higher correlation coefficients, and artificially reducing the significance of any correlation seen in the observed datasets.",
"We exercise care to avoid this bias by limiting the flux dynamical range of our randomized data to that of the observed sample and by rejecting simulated flux pairs outside this range.",
"We caution the reader that our test does not in any way account for the effects of non-simultaneity.",
"Ideally the test should be applied to data in different frequencies averaged over the same time interval.",
"The spectral indices used in each band to implement the K-correction should also be concurrently measured with the flux averages.",
"Such concurrent spectral indices are straight-forward to obtain in gamma rays, however radio monitoring is routinely performed at a single waveband (as is the case for the OVRO 40 M Monitoring Program), and in practice only archival radio spectral indices are available.",
"It is thus fortunate that our test is robust against small changes in the value of the radio spectral index used in the K-correction.",
"This property of the test can be understood by taking into account that blazars are spectrally flat at radio frequencies so the effect of the K-correction in radio is small to begin with.",
"We have confirmed this by alternatively using archival radio spectral indices measured for each source, or a uniform value of $\\alpha _r=-0.5$ across all sources; the evaluated significance in the two cases did not change appreciably (fractional change in the quoted significance less than $0.01$ ).",
"This result can be explicitly confirmed by using multi-frequency simultaneous data from the F-GAMMA Program where simultaneous radio spectral indices can be obtained.",
"These tests are described in detail in Fuhrmann et al.",
"2012 (in preparation).",
"In contrast, the test is quite sensitive to the redshift of the sources included in the sample under consideration, as shown in Ackermann et al.",
"(2011): as the main purpose of the test is to assess the effect of distance biases, the calculations involved are sensitively dependent on said distances.",
"For this reason the test should only be used on samples with known redshifts for all members.",
"Finally, we stress that this test in itself does not assess the strength of the intrinsic correlation between flux densities of different frequencies.",
"It only addresses its statistical significance, i.e.",
"the probability that an apparent correlation as strong or stronger than the observed one can be obtained from intrinsically uncorrelated data due to observational biases.",
"A correlation may be very weak but picked up at high significance if the dataset is large and the data quality is high; in contrast, a strong correlation in a very small sample may not be very significant.",
"It is similarly important when the test returns a low statistical significance to carefully distinguish between lack of evidence for correlation and evidence for intrinsically uncorrelated data.",
"Intrinsic lack of correlation cannot generally be established.",
"However it is possible to show that a correlation with strength above a certain threshold would have been picked up at a given level of significance by a particular test.",
"We thank Andy Strong and the anonymous referees for insightful comments which improved this manuscript.",
"The OVRO 40 M program is supported in part by NASA grants NNX08AW31G and NNG06GG1G, and NSF grant AST-0808050.",
"Support from the Max-Planck Institut f Radioastronomie for upgrading the OVRO 40 M telescope receiver is also acknowledged.",
"We are grateful to Russ Keeney for his tireless efforts in support of observations at the Owens Valley Radio Observatory.",
"VP acknowledges support for this work provided by NASA through Einstein Postdoctoral Fellowship grant number PF8-90060 awarded by the Chandra X-ray Center, which is operated by the Smithsonian Astrophysical Observatory for NASA under contract NAS8-03060, and thanks the Department of Physics at the University of Crete for their hospitality during the completion of part of this work.",
"WM acknowledges support from the U.S. Department of State and the Comisión Nacional de Investigación Científica y Tecnológica (CONICYT) in Chile for a Fulbright-CONICYT scholarship."
]
] | 1204.0790 |
[
[
"Fractal powers in Serrin's swirling vortex solutions"
],
[
"Abstract We consider a modification of the fluid flow model for a tornado-like swirling vortex developed by J. Serrin, where velocity decreases as the reciprocal of the distance from the vortex axis.",
"Recent studies, based on radar data of selected severe weather events, indicate that the angular momentum in a tornado may not be constant with the radius, and thus suggest a different scaling of the velocity/radial distance dependence.",
"Motivated by this suggestion, we consider Serrin's approach with the assumption that the velocity decreases as the reciprocal of the distance from the vortex axis to the power $b$ with a general $b>0$.",
"This leads to a boundary-value problem for a system of nonlinear differential equations.",
"We analyze this problem for particular cases, both with nonzero and zero viscosity, discuss the question of existence of solutions, and use numerical techniques to describe those solutions that we cannot obtain analytically."
],
[
"Introduction",
"Rotating thunderstorms, also known as supercells, and tornadoes generated from them have been modeled using axisymmetric flows.",
"A variety of approaches to investigate axisymmetric flows has led to various models of vortex dynamics [25], [31].",
"Among the most prominent ones are Rankine combined, Burgers–Rott, Lamb–Oseen, and Sullivan vortex models.",
"Some of these models (e.g., Burgers–Rott) balance vorticity diffusion and advection mechanisms that are important to modeling the inner core of tornadoes and other intense vortices.",
"Most of the models describe rotation in the whole space and therefore they do not take into account friction resulting from contact with the ground.",
"See [31] for a detailed list of various axisymmetric models, some of which are exact solutions to Navier–Stokes equations.",
"In 1972, J. Serrin, following the works of Long [22], [23] and Goľdshtik [14], discovered a special class of tornado-like swirling vortex solutions to the Navier–Stokes equations in half-space [30], in which the velocity decreases as the reciprocal of the radial distance, $r$ , from the vortex axis, a phenomenon observed in real tornadoes [39], [31].",
"Serrin's solutions, unlike Long's, model the interaction of a swirling vortex with the horizontal boundary, and they are some of the few exact solutions of Navier–Stokes equations in half-space, in which both the impermeability and the no-slip condition are enforced on a rigid horizontal boundary representing the ground.",
"This should be contrasted with, for example, the popular Burgers–Rott or Sullivan models, in which the no-slip condition is violated.",
"Serrin described three types of solutions depending on the values of kinematic viscosity and a “pressure” parameter: downdraft core with radial outflow, updraft core with radial inflow (single-cell vortices), and downdraft core with a compensating radial inflow (double-cell vortex).",
"See Fig.",
"REF for a sketch of a single-cell and a double-cell vortex.",
"While these solutions may not be accurate near the vortex core due to the singularity along the vortex axis, outside the region of the most intense winds they seem to provide a reasonable description of a tornado [30], [31], [39].",
"In fact, in [18], the authors note that their solution was similar to a similarity solution of Long [22], [23], and Serrin's computations are analogous to Long's.",
"Also, as stated in [31], “The near-surface flow of Serrin's vortex beyond the core region may be a useful analog for the frictional boundary layer in the region of tornadoes beyond the radius of maximum wind [speed].” Regarding the inner core, the singularity near the vortex axis present in Serrin's model is not present in the Burgers–Rott, Sullivan, or Long's models.",
"On the other hand, some numerical, radar, and ground velocity tracking studies suggest that updraft wind speeds near the tornado axis can achieve large values, approaching and possibly exceeding the speed of sound [1], [9], [10], [11], [12], [19], [21], [40], [38].",
"Figure: An illustration of a vortex breakdown process.",
"Viewed from left to right, the flow undergoes several bifurcations from a single-cell vortex on the left to multiple vortices on the right.",
"The middle two images show a single-cell vortex below and a double-cell vortex above (left) and a double-cell vortex (right).",
"Modified with permission from .The search for axisymmetric flow solutions has continued through the last few decades because of their importance in modeling a wide range of phenomena.",
"Particular types of tornado-like conical solutions influenced by Serrin's work can be found in [35], [41], [15].",
"Relevant reviews are presented in [32], [33].",
"These solutions all exhibit velocity decay reciprocal to the distance from the vortex axis.",
"However, more recent high-resolution mobile Doppler radar studies [37], [36] provide evidence that the velocities decay as the reciprocal of a different power of the radial distance from the vortex axis.",
"In these papers, devoted to analyzing data associated to strong or violent tornadoes, an attempt is made to calculate the value of the exponent in the “velocity power law” $v\\propto r^b$ .",
"Wurman and Alexander [36] calculated the exponent $b$ from radar data obtained in an intercept of the May 31, 1998 Spencer, South Dakota, tornado and obtained the value $b=-0.67$ for the velocity field away from the core-flow region, in which data indicated a solid-body rotation.",
"The tornado was rated EF4.",
"These values were calculated from the data taken at one instant during the tornado's existence.",
"In the case of the June 2, 1995 Dimmit, Texas, tornado, Wurman and Gill [37] observed exponent values in the range $-0.5$ to $-0.7$ , concluding that “it implies that the angular momentum in the tornado was not constant with radius, but decays toward the center.” In an attempt to distinguish between tornadic and non-tornadic storms, Cai observed [2] that tornado-related vorticity fields might have a fractal nature with respect to the grid size; more specifically, natural log of the vorticity, $\\zeta $ , and natural log of the grid spacing seem to have a linear relationship, with a constant negative ratio.",
"Larger absolute values of this ratio correspond to stronger storms.",
"In some cases of tornadic storms, the ratio is found close to $-1.6$ .",
"For tornadic mesocyclones, this suggests a power law of the form $\\zeta \\propto r^\\beta $ .",
"Since $\\zeta =\\nabla \\times {v}$ , the results of Wurman et al.",
"and Cai appear to be consistent, even though they consider different scales.",
"We further explore the potential scale invariance between the mesocyclone and tornado scales in a related work [7].",
"Cai also noted that the exponent in the power law for vorticity can be thought of as measuring a fractal dimension associated with the vortex.",
"The possibility of fractalization of a vortex undergoing stretching was pointed out by Chorin [4].",
"In [7], we also explore the relationship between vortex stretching and a vortex breakdown.",
"Additional discussions of a vortex breakdown can be found in [1], [9], [12], [18], [19], [20], [40].",
"We briefly comment on how our results relate to a vortex breakdown (as illustrated in Fig.",
"REF ) in the conclusions section.",
"We therefore find it natural to ask whether there are Serrin-type similarity solutions to the Navier–Stokes equations of the form described in (REF ), in which the velocity field is proportional to $r^{-b}$ , where $b>0$ and $b\\ne 1$ , with the most interesting case being $0<b<1$ .",
"This work attempts to answer this question.",
"Although other models could conceivably be used as well to try to derive a first model with a velocity decay different from $r^{-1}$ , we use Serrin's model as a starting point for its mathematical simplicity and for being an exact solution to the Navier–Stokes equations satisfying the boundary conditions at the ground.",
"We show that under the assumption of constant nonzero viscosity and suitable assumptions on the form of the velocity field, similar to that in [30], the Navier–Stokes equations do not admit any nontrivial solutions except when $b=1$ , the case studied by Serrin.",
"However, in the relaxed case of zero viscosity, the Euler equations always admit a simple, purely rotational solution with azimuthal velocity of the form $Cr^{-b}$ .",
"In addition, when $b=1$ , another set of nontrivial solutions is found analytically.",
"When $b\\ge 2$ , we show that no other solutions exist.",
"The most intriguing cases are when $0<b<1$ and $1<b<2$ , for which we have not found analytic solutions; for the former case we present numerically computed solutions for various values of the parameter $b$ , while for the latter case we show that any solution would have to be unstable in the sense of Rayleigh's circulation criterion [8].",
"We summarize the main results below.",
"Table: Summary of the main results for various values of bb and viscosity ν\\nu The paper is organized as follows.",
"In section , we describe the basic geometry of the problem, the governing equations, and the form of solutions we are interested in finding.",
"In section , we analyze the Navier–Stokes equations in the case of constant nonzero viscosity.",
"In section , we focus on the case of zero viscosity, governed by the Euler equations.",
"Finally, in section we discuss the conclusions and implications of our findings for tornadogenesis.",
"The appendix contains some auxiliary equations needed for our work that would unnecessarily clutter the presentation in the paper."
],
[
"Governing equations, basic geometry, and modified Serrin's variables",
"In this section, we discuss the relevant equations, introduce a change of variables in the spirit of [30], and also introduce a special form of solutions we seek, which eventually allows us to reformulate the problem in terms of ordinary differential equations.",
"Finally, we discuss the continuity equation and its implications in terms of boundary conditions."
],
[
"Governing equations",
"The equations governing fluid flow are the Navier–Stokes equations, $\\rho \\,\\frac{D{\\bf v}}{Dt}\\equiv \\rho \\left(\\frac{\\partial {\\bf v}}{\\partial t}+({\\bf v}\\cdot \\nabla ){\\bf v}\\right)=-\\nabla P+(\\lambda +\\mu )\\nabla (\\nabla \\cdot {\\bf v})+\\mu \\,\\Delta {\\bf v},$ where $\\bf v$ , $P$ , and $\\rho $ are velocity, pressure, and density fields, respectively, and $\\mu $ and $\\lambda $ are dynamic viscosity coefficients.",
"The conservation of mass, or continuity, equation is $\\frac{D\\rho }{Dt}+\\rho \\nabla \\cdot {\\bf v}\\equiv \\frac{\\partial \\rho }{\\partial t}+{\\bf v}\\cdot \\nabla \\rho +\\rho \\,\\nabla \\cdot {\\bf v}=0.$ We will consider the case of an incompressible and homogeneous flow, so that $D\\rho /Dt=\\nabla \\cdot {\\bf v}=0$ and $\\nabla \\rho =0$ , respectively.",
"We will seek steady-state solutions, i.e., those that satisfy ${\\partial {\\bf v}}/{\\partial t}={\\partial \\rho }/{\\partial t}=0$ .",
"Under these assumptions equation (REF ) is automatically satisfied.",
"As a consequence, steady-state solutions for an incompressible, homogeneous flow satisfy the following simplified versions of (REF ) and (REF ), $({\\bf v}\\cdot \\nabla ){\\bf v}=-\\nabla p+\\nu \\,\\Delta {\\bf v}$ and $\\nabla \\cdot {\\bf v}=0,$ where $p = P/\\rho $ is a (scaled) pressure field and $\\nu =\\mu /\\rho $ is a (constant) kinematic viscosity.",
"The relevant boundary conditions are ${\\bf v} = {\\bf 0} & \\qquad \\text{when }\\nu >0 \\quad \\text{(no source/sink and no slip)},\\\\\\dfrac{\\partial \\bf v}{\\partial n} =0 & \\qquad \\text{when }\\nu =0 \\quad \\text{(no source/sink, but slip allowed)}.$"
],
[
"Spherical coordinate system and components of the velocity field",
"Following Serrin [30], we use the (right) spherical coordinates $(R,\\alpha ,\\theta )$ , where $R$ is radial distance from the origin, $\\alpha $ is the angle between the radius vector and the positive $z$ -axis, and $\\theta $ is the meridian angle about the $z$ -axis.",
"The positive $z$ -axis then corresponds to $\\alpha =0$ and the boundary (ground) plane to $\\alpha =\\pi /2$ .",
"We are interested in solutions in the upper half space, $z>0$ , which corresponds to $R>0$ and $0\\le \\alpha <\\pi /2$ in our coordinate system.",
"We denote the components of the velocity vector ${\\bf v}(R,\\alpha ,\\theta )$ in the spherical coordinate system by $v_R$ , $v_\\alpha $ , and $v_\\theta $ , and write ${\\bf v}(R,\\alpha ,\\theta )=\\left(v_R(R,\\alpha ,\\theta ),\\,v_\\alpha (R,\\alpha ,\\theta ),\\,v_\\theta (R,\\alpha ,\\theta )\\right).$ We will refer to the individual components as radial ($v_R$ ), meridional ($v_\\alpha $ ), and azimuthal ($v_\\theta $ ).",
"The scaled pressure field will be denoted by $p(R,\\alpha ,\\theta )$ .",
"The three components of the Navier–Stokes equations (REF ) in this coordinate system and in the spherical velocity components are given in the appendix in (REF )–(REF ), and the continuity equation (REF ) is given in (REF ).",
"We will follow Serrin's approach [30] and consider velocities of the form $v_R(R,\\alpha ,\\theta )=\\frac{G(x)}{r^b},\\qquad v_\\alpha (R,\\alpha ,\\theta )=\\frac{F(x)}{r^b},\\qquad v_\\theta (R,\\alpha ,\\theta )=\\frac{\\Omega (x)}{r^b},$ where $r=R\\sin \\alpha $ is the horizontal distance to the $z$ -axis, $x=\\cos \\alpha $ , and $b>0$ .",
"We remark that the case studied by Serrin [30] corresponds to $b=1$ .",
"Since $\\sin \\alpha =\\sqrt{1-x^2}$ , we can use the change of variables $f(x)=F(x)(1-x^2)^{(1-b)/2},\\qquad g(x)=G(x)(1-x^2)^{(1-b)/2},\\qquad \\omega (x)=\\Omega (x)(1-x^2)^{(1-b)/2},$ and rewrite (REF ) as $v_R(R,\\alpha ,\\theta )=\\frac{g(x)}{R^b\\sin \\alpha },\\qquad v_\\alpha (R,\\alpha ,\\theta )=\\frac{f(x)}{R^b\\sin \\alpha },\\qquad v_\\theta (R,\\alpha ,\\theta )=\\frac{\\omega (x)}{R^b\\sin \\alpha }.$ When $b=1$ , the upper-case functions, $F$ , $G$ , and $\\Omega $ , agree with the lower-case functions, $f$ , $g$ , and $\\omega $ .",
"The continuity equation (REF ) written in terms of the newly introduced functions is given below in (REF ), while the Navier–Stokes equations (REF ) are discussed in section REF .",
"We also remark that, as a consequence of (REF ) below, the function $\\Psi (R,\\alpha )=R^{2-b}f(x)=R^{2-b}f(\\cos \\alpha )$ satisfies $\\nabla \\Psi \\cdot {\\bf v}=0,$ and therefore the surfaces $\\Psi =\\text{constant}$ contain the streamlines of the fluid motion.",
"Notice that this is a direct generalization of Serrin's $\\Psi =RF(x)$ , since when $b=1$ we have $F(x)=f(x)$ ."
],
[
"The continuity equation",
"We now consider the continuity equation (REF ) and its consequences for solutions to (REF ).",
"We first observe that direct substitution of (REF ) into the continuity equation (REF ) yields (see (REF ) in the appendix) $\\begin{split}(2-b)G(x)=\\sqrt{1-x^2}\\,&F^{\\prime }(x)-(1-b)\\frac{x}{\\sqrt{1-x^2}}F(x),\\\\(2-b)g(x)=\\ &\\sqrt{1-x^2}\\,f^{\\prime }(x).\\end{split}$ The prime symbol will denote differentiation with respect to $x$ throughout the paper.",
"From (REF ) we see that if $b\\ne 2$ , then $G$ can be expressed in terms of $F$ and $g$ in terms of $f$ .",
"We next derive an integral version of the continuity equation that will lead to naturally arising boundary conditions needed later in addition to the natural boundary condition that the ground contains no source or sink.",
"Let $R_0>0$ and $E\\subset \\mathbb {R}^3$ be the upper half ball bounded below by the horizontal disk $D=\\lbrace (R,\\alpha ,\\theta ):\\ 0\\le R<R_0,\\ \\alpha =\\pi /2,\\ 0\\le \\theta <2\\pi \\rbrace $ and above by the hemisphere $S=\\lbrace (R,\\alpha ,\\theta ):\\ R=R_0,\\ 0\\le \\alpha <\\pi /2,\\ 0\\le \\theta <2\\pi \\rbrace $ , i.e., $E=\\lbrace (R,\\alpha ,\\theta ):\\ 0<R<R_0,\\ 0\\le \\alpha <\\pi /2,\\ 0\\le \\theta <2\\pi \\rbrace .$ Applying (REF ) and the divergence theorem, we obtain $0=\\iiint _E\\nabla \\cdot {\\bf v}\\,dV=\\iint _{\\partial E}{\\bf v}\\cdot {\\bf n}\\,dA=\\iint _D v_\\alpha \\,dA+\\iint _S v_R\\,dA.$ However, since there can be no source or sink at the ground ($\\alpha =\\pi /2$ ), we have that $v_\\alpha (R,\\pi /2,\\theta )=F(0)/R^b=0$ for all $R>0$ , or $F(0)=f(0)=0,$ and thus the integral over the disk $D$ vanishes.",
"Substituting (REF ) into the integral over $S$ , we obtain $\\iint _S v_R\\,dA=\\int _0^{2\\pi }\\int _0^{\\pi /2}\\frac{G(x)}{r^b}R_0^2\\sin {\\alpha }\\,d\\alpha \\,d\\theta =2\\pi R_0^{2-b}\\int _0^{\\pi /2}G(x)(\\sin {\\alpha })^{1-b}\\,d\\alpha =0,$ or, in terms of $x$ , $\\int _0^1\\frac{G(x)}{(1-x^2)^{b/2}}\\,dx=\\int _0^1\\frac{g(x)}{\\sqrt{1-x^2}}\\,dx=0.$ Substituting (REF ) into (REF ), integrating, and using the boundary value from (REF ), we obtain $\\lim _{x\\rightarrow 1}F(x)(1-x^2)^{(1-b)/2}=\\lim _{x\\rightarrow 1}f(x)=0.$ Remark 2.1 (Consequences of the continuity equation) We can summarize the consequences of the continuity equation (REF ) and the no source/sink boundary condition (REF ) as follows.",
"If $b=2$ , then, using (REF )–(REF ), we have $F=f\\equiv 0,\\qquad \\int _0^1\\frac{G(x)}{1-x^2}\\,dx=0.$ If $b\\ne 2$ , then, using (REF ), (REF ), and (REF ), we have $F(0)=\\lim _{x\\rightarrow 1}F(x)(1-x^2)^{(1-b)/2}=0,\\qquad (2-b)G(x)=\\sqrt{1-x^2}\\,F^{\\prime }(x)-(1-b)\\frac{x}{\\sqrt{1-x^2}}\\,F(x),$ or, in terms of the lower-case functions, $f(0)=\\lim _{x\\rightarrow 1}f(x)=0,\\qquad (2-b)g(x)=\\sqrt{1-x^2}\\,f^{\\prime }(x).$ Remark 2.2 Note that the special case $b=1$ in the previous remark gives rise to $F(0)=0$ , $\\lim _{x\\rightarrow 1}F(x)=0$ , and $G(x)=\\sqrt{1-x^2}\\,F^{\\prime }(x)$ , which is consistent with [30]."
],
[
"Simplification of the Navier–Stokes equations",
"In this section, we consider the Navier–Stokes equations (REF ) in terms of the velocity expressions (REF ) and (REF ).",
"We first observe that all of the velocity components have the form $v(R,\\alpha ,\\theta )=K(\\alpha )/R^b$ , so their partial derivatives are of the form $\\frac{\\partial v}{\\partial R}=-b\\,\\frac{K(\\alpha )}{R^{b+1}},\\qquad \\qquad \\frac{\\partial v}{\\partial \\alpha }=\\frac{\\dot{K}(\\alpha )}{R^b},\\qquad \\qquad \\frac{\\partial v}{\\partial \\theta }=0.$ The dot symbol will denote differentiation with respect to $\\alpha $ throughout the paper.",
"Note that all terms in the left-hand sides of (REF )–(REF ), arising from the convective term in (REF ), are of the form $C(\\alpha )/R^{1+2b}$ , while all terms in the right-hand sides, arising from the diffusive term in (REF ), (i.e., all the terms multiplied by the viscosity coefficient, $\\nu $ ) are of the form $D(\\alpha )/R^{2+b}$ .",
"Therefore, the governing equations (REF )–(REF ) have the general form $\\frac{C_1(\\alpha )}{R^{1+2b}}=-\\frac{\\partial p}{\\partial R}+\\nu \\frac{D_1(\\alpha )}{R^{2+b}},$ $\\frac{C_2(\\alpha )}{R^{1+2b}}=-\\frac{1}{R}\\frac{\\partial p}{\\partial \\alpha }+\\nu \\frac{D_2(\\alpha )}{R^{2+b}},$ $\\frac{C_3(\\alpha )}{R^{1+2b}}=-\\frac{1}{R\\sin \\alpha }\\frac{\\partial p}{\\partial \\theta }+\\nu \\frac{D_3(\\alpha )}{R^{2+b}},$ where the expressions for $C_i(\\alpha )$ and $D_i(\\alpha )$ are given, in their various forms, in the appendix.",
"Like in [30], we argue that (REF ) yields $\\partial p/\\partial \\theta $ independent of $\\theta $ , so $p$ must be linear in $\\theta $ .",
"Together with periodicity in $\\theta $ , this implies that $\\partial p/\\partial \\theta =0$ , so $p(R,\\alpha ,\\theta )=p(R,\\alpha )$ .",
"Consequently, (REF ) reduces to $C_3(\\alpha )=\\nu R^{b-1}D_3(\\alpha ).$ The scaled pressure function has to satisfy (REF ), so by integrating it with respect to $R$ we obtain $p(R,\\alpha )=\\frac{C_1(\\alpha )}{2bR^{2b}}-\\nu \\frac{D_1(\\alpha )}{(1+b)R^{1+b}}+T(\\alpha ).$ Substituting this expression into (REF ), we conclude that $\\dot{T}(\\alpha )=0$ , and thus $T(\\alpha )\\equiv T$ is a constant and $p(R,\\alpha )=\\frac{C_1(\\alpha )}{2bR^{2b}}-\\nu \\frac{D_1(\\alpha )}{(1+b)R^{1+b}}+T.$ In addition, from (REF ) and (REF ) we obtain a compatibility condition for the existence of the pressure, $\\dot{C}_1(\\alpha )+2b\\,C_2(\\alpha )=\\nu R^{b-1}\\frac{2b}{1+b}\\left(\\dot{D}_1(\\alpha )+(1+b)D_2(\\alpha )\\right).$ We now have the following equivalence lemma.",
"Lemma 2.1 The system of Navier–Stokes equations (REF )–(REF ) is equivalent to the system (REF )–(REF ).",
"First, note that (REF )–(REF ) follow directly from the Navier–Stokes equations (REF )–(REF ).",
"Vice versa, if $C_i(\\alpha )$ and $D_i(\\alpha )$ are such that (REF ) and (REF ) are satisfied, then (REF ) gives an expression for the scaled pressure so that (REF ) is immediately satisfied, (REF ) is satisfied due to the compatibility equation (REF ), and (REF ) follows immediately from (REF )."
],
[
"Analysis of the viscous case: $\\nu >0$",
"In this section we discuss the existence of classical solutions to our problem with constant nonzero viscosity.",
"We show that nontrivial solutions of the form (REF ) exist only for the case $b=1$ discussed by Serrin [30].",
"We start by discussing the boundary conditions and then analyze the various cases that arise for various values of $b$ ."
],
[
"Boundary conditions",
"In the case of nonzero viscosity, the no-slip requirement at the ground implies $v_R(R,\\pi /2,\\theta )=v_\\theta (R,\\pi /2,\\theta )=0$ for all $R>0$ and $0\\le \\theta <2\\pi $ , so that, using (REF ), $G(0)=\\Omega (0)=0$ .",
"The no-sink/source requirement gives $v_\\alpha (R,\\pi /2,\\theta )=0$ , or (REF ), $F(0)=0$ .",
"In addition, as a consequence of incompressibility, in particular (REF ), we have $F^{\\prime }(0)=0$ .",
"(Note that as discussed in Remark REF , if $b=2$ , then $F\\equiv 0$ and all boundary conditions concerning $F$ are automatically satisfied.)",
"Near the vortex axis, we have (REF ) if $b\\ne 2$ , while if $b=2$ , then $F\\equiv 0$ , and there are no physical restrictions on the behavior of $G$ and $\\Omega $ as $x\\rightarrow 1$ .",
"However, we will assume, similarly as in [30], that near the vortex axis the azimuthal velocity, $v_\\theta $ , behaves like $C/r^b$ , i.e., we will assume that $\\lim _{x\\rightarrow 1}\\Omega (x)=\\lim _{x\\rightarrow 1}\\omega (x)(1-x^2)^{-(1-b)/2}=C_\\omega \\ne 0.$ Similarly as in [30], this boundary condition is based on the observation that ${\\bf v}=\\left(0,0,C_\\omega /r^b\\right)$ is a solution to our problem for any $b>0$ (see Sections REF and REF below), which can also be easily verified to be a solution in the full 3D space.",
"Remark 3.1 (Boundary conditions in the viscous case) In the case of constant nonzero viscosity, sought solutions $F$ , $G$ , and $\\Omega $ are subject to the following requirements: If $b=2$ , then $F=f\\equiv 0,\\qquad G(0)=\\Omega (0)=g(0)=\\omega (0)=0,\\qquad \\lim _{x\\rightarrow 1}\\Omega (x)=C_\\omega .$ If $b\\ne 2$ , then $F(0)=F^{\\prime }(0)=G(0)=\\Omega (0)=0,\\qquad \\lim _{x\\rightarrow 1}F(x)(1-x^2)^{(1-b)/2}=0,\\qquad \\lim _{x\\rightarrow 1}\\Omega (x)=C_\\omega .$ or $f(0)=f^{\\prime }(0)=g(0)=\\omega (0)=0,\\qquad \\lim _{x\\rightarrow 1}f(x)=0,\\qquad \\lim _{x\\rightarrow 1}\\omega (x)(1-x^2)^{-(1-b)/2}=C_\\omega .$ Remark 3.2 Note that the boundary conditions studied by Serrin are consistent with ours when $b=1$ , since then $\\lim _{x\\rightarrow 1}F(x)(1-x^2)^{(1-b)/2}=\\lim _{x\\rightarrow 1}F(x)=0$ ."
],
[
"Governing equations",
"The governing equations are (REF ) and (REF ), together with the continuity equations (REF ).",
"We will need to distinguish between the special case $b=1$ studied by Serrin and the remaining cases when $b\\ne 1$ ."
],
[
"Case $b=1$",
"In this case, (REF ) and (REF ) reduce to $C_3(\\alpha )=\\nu D_3(\\alpha ),\\qquad \\dot{C}_1(\\alpha )+2\\,C_2(\\alpha )=\\nu (\\dot{D}_1(\\alpha )+2D_2(\\alpha )).$ Using (REF )–(REF ), these equations can be rewritten as $\\begin{split}\\nu (1-x^2)F^{(4)}(x)-4\\nu xF^{\\prime \\prime \\prime }(x)+F(x)F^{\\prime \\prime \\prime }(x)+3F^{\\prime }(x)F^{\\prime \\prime }(x)&=-\\frac{2\\Omega (x)\\Omega ^{\\prime }(x)}{1-x^2},\\\\\\nu (1-x^2)\\Omega ^{\\prime \\prime }(x)+F(x)\\Omega ^{\\prime }(x)&=0.\\end{split}$ We note that system (REF ) is identical to system (5) in [30] and is analyzed there.",
"In what follows, we focus on the case with $b\\ne 1$ ."
],
[
"Case $b\\ne 1$",
"In this case, the only way to satisfy (REF ) and (REF ) for all $R>0$ is to satisfy $C_3(\\alpha )=0,\\qquad D_3(\\alpha )=0,\\qquad \\dot{C}_1(\\alpha )+2b\\,C_2(\\alpha )=0,\\qquad \\dot{D}_1(\\alpha )+(1+b)D_2(\\alpha )=0.$ The relevant expressions for the quantities in (REF ) are given in the appendix and we will recall them as needed.",
"The last equation that needs to be satisfied is (REF ), the consequence of the continuity equation, restated here in both forms for completeness, $(2-b)G(x)=\\sqrt{1-x^2}\\,F^{\\prime }(x)-(1-b)\\frac{x}{\\sqrt{1-x^2}}\\,F(x)\\qquad \\text{ or }\\qquad (2-b)g(x)=\\sqrt{1-x^2}f^{\\prime }(x).$"
],
[
"Case $b=2$ (no solutions)",
"We first address the special case with $b=2$ .",
"In this case, $F=f\\equiv 0$ and (REF ) provides no information.",
"Using (REF )–(REF ), the first three equations in (REF ) reduce to $G(x)\\Omega (x)=0,\\\\(1-x^2)^2\\Omega ^{\\prime \\prime }(x)+2x(1-x^2)\\Omega ^{\\prime }(x)+3\\Omega (x)=0,\\\\\\frac{8x}{1-x^2}\\,G^2(x)+2(G^2(x))^{\\prime }+(\\Omega ^2(x))^{\\prime }=0.$ Because of the boundary condition (REF ), $\\Omega \\ne 0$ on some interval $(x_0,1)$ by continuity.",
"Equation (REF ) then implies that $G\\equiv 0$ in $(x_0,1)$ , and () reduces to $(\\Omega ^2)^{\\prime }=0$ in $(x_0,1)$ .",
"Thus $\\Omega \\equiv C_\\omega $ in $(x_0,1)$ .",
"This, in turn, implies that $\\Omega \\equiv C_\\omega $ and $G\\equiv 0$ in $(0,1)$ .",
"However, neither (), nor the boundary condition $\\Omega (0)=0$ are then satisfied, so no solution exists when $b=2$ ."
],
[
"Case $b\\ne 1,2$ (no solutions)",
"In this case, (REF ) can be substituted into (REF ) to yield the set of equations given in (REF )–(REF ) in the appendix.",
"In order to conclude that no solutions exist, it suffices to analyze (REF ), which reads $(1-x^2)^2\\Omega ^{\\prime \\prime }(x)-2(1-b)x(1-x^2)\\Omega ^{\\prime }(x)-(1-b^2)\\Omega (x)=0.$ When equipped with the initial conditions $\\Omega (0)=0$ and $\\Omega ^{\\prime }(0)=C$ , its solution is $\\Omega (x)=Cx(1-x^2)^{(b-1)/2}\\,{}_2{\\mathcal {F}}_1\\left(\\frac{1-b}{2},\\frac{b}{2};\\frac{3}{2};x^2\\right),$ where ${}_2 \\mathcal {F}_1$ is the Gaussian hypergeometric function given by (see [27]) ${}_2{\\mathcal {F}}_1(\\alpha ,\\beta ;\\gamma ;z)=\\sum _{n=0}^{\\infty }\\frac{(\\alpha )_n(\\beta )_n}{(\\gamma )_n}\\frac{z^n}{n!",
"},$ and $(x)_n$ with $n\\in \\mathbb {N}\\cup \\lbrace 0\\rbrace $ is the Pochhammer symbol given by $(x)_n={\\left\\lbrace \\begin{array}{ll}1 & \\text{if }n=0,\\\\x(x+1)\\dots (x+n-1) & \\text{if }n>0.\\end{array}\\right.",
"}$ Since the case $C=0$ in (REF ) would yield $\\Omega \\equiv 0$ , which does not satisfy the boundary condition (REF ), we only consider the case with $C\\ne 0$ and show that (REF ) cannot be satisfied for any $b>0$ .",
"We first have [27] ${}_2{\\mathcal {F}}_1\\left(\\frac{1-b}{2},\\frac{b}{2};\\frac{3}{2};1\\right)=\\frac{\\Gamma \\left(\\frac{3}{2}\\right)\\Gamma (1)}{\\Gamma \\left(\\frac{3-b}{2}\\right)\\Gamma \\left(\\frac{2+b}{2}\\right)}=\\frac{\\sqrt{\\pi }}{2\\,\\Gamma \\left(\\frac{3-b}{2}\\right)\\Gamma \\left(\\frac{2+b}{2}\\right)}.$ Since $1/\\Gamma (z)$ is an entire function vanishing only for $z=0,-1,-2,\\dots $ , the value above is finite, and it is zero only for $b=3,5,7,\\dots $ (or $b=-2,-4,-6,\\dots $ , but this case is excluded from our consideration).",
"This observation, together with the behavior of $(1-x^2)^{(b-1)/2}$ , allows us to conclude that $\\lim _{x\\rightarrow 1-}\\Omega (x)={\\left\\lbrace \\begin{array}{ll}\\infty & \\text{if }0<b<1,\\\\0 & \\text{if }b>1,\\end{array}\\right.",
"}$ and the boundary condition (REF ) cannot be satisfied for any choice of $C$ .",
"We can therefore conclude that no solutions with $b\\ne 1,2$ exist."
],
[
"Analysis of the inviscid case: $\\nu =0$",
"In this section we discuss the existence of classical solutions in case of zero viscosity.",
"In tornadic thunderstorms one can expect a large Reynolds number on the order of $10^{10}$ , and thus very small viscosity [11].",
"When studying the case with $\\nu =0$ , we have to modify the boundary conditions at the ground and allow slip, and also tacitly assume that the solutions with $\\nu =0$ are “close” to the physical solutions with large Reynolds numbers (see, e.g., [6]).",
"We start this section by showing that the purely rotational trivial solution $F=G\\equiv 0$ and $\\Omega \\equiv C_\\omega $ exists for all $b>0$ , but that no nontrivial solutions of the form (REF ) exist if $b\\ge 2$ .",
"For $b=1$ , we present and discuss analytic solutions and provide a numerical and graphical comparison with some of Serrin's solutions.",
"We observe that our solutions with $b=1$ appear to be the limiting cases as viscosity tends to 0, and, compared to our solutions, Serrin's solutions exhibit a boundary layer near the physical ground whose thickness tends to 0.",
"In particular, we observe that the size of the boundary layer tends to 0 at the same rate as theoretically established in [30].",
"For the cases $0<b<1$ and $1<b<2$ , we present the governing equations that apparently admit nontrivial solutions.",
"While we have not completed the existence and uniqueness analysis, we present our insights and numerical results in the case $0<b<1$ , and we show that all potential solutions with $1<b<2$ could not satisfy Rayleigh's circulation criterion and would thus be unstable with respect to axisymmetric perturbations.",
"We again start by discussing the boundary conditions and then analyze the various cases that arise for various values of $b$ ."
],
[
"Boundary conditions",
"Since slip is of no concern in the case of zero viscosity, we only need to address the no-source/sink requirements at the ground ($\\alpha =\\pi /2$ ) and at the center of the vortex ($\\alpha =0$ ).",
"From the analysis in the previous sections, it is clear that there will be no a priori restrictions on $G$ or $\\Omega $ .",
"Regarding restrictions on $F$ , we have $F=f\\equiv 0\\qquad \\text{ if } b=2,\\\\F(0)=f(0)=0\\quad \\text{ and }\\quad \\lim _{x\\rightarrow 1}F(x)(1-x^2)^{(1-b)/2}=\\lim _{x\\rightarrow 1}f(x)=0\\qquad \\text{ if } b\\ne 2.$ We still assume that near the vortex axis the azimuthal velocity, $v_\\theta $ , behaves like $C/r^b$ , i.e., we still assume that (REF ) holds.",
"We restate it here for completeness, $\\lim _{x\\rightarrow 1}\\Omega (x)=\\lim _{x\\rightarrow 1}\\omega (x)(1-x^2)^{-(1-b)/2}=C_\\omega \\ne 0.$"
],
[
"Governing equations",
"Since the viscosity coefficient, $\\nu $ , is zero, (REF ) and (REF ) reduce to $C_3(\\alpha )=0,\\\\\\dot{C}_1(\\alpha )+2b\\,C_2(\\alpha )=0,$ which can be rewritten using (REF ) and (REF ) in the appendix.",
"The third equation is (REF ) (restated later as (REF )).",
"Remark 4.1 Note that the governing equations (REF ) and (REF ) in the case of zero viscosity are invariant under the transformation $\\bf {v}\\mapsto -\\bf {v}$ , and so any obtained solution can be also “reversed” by changing its sign.",
"In addition, we will see in some cases below that some of the equations are invariant under sign changes of some of the functions $F$ , $G$ , $\\Omega $ , etc., individually."
],
[
"Case $b=2$ (no nontrivial solutions)",
"In this case, $F\\equiv 0$ by (REF ), and (REF ) provides no information.",
"Using (REF ) and (REF ), equations (REF ) and () reduce to $G(x)\\Omega (x)=0,\\\\2\\left[(G^2(x))^{\\prime }+\\frac{4x}{1-x^2}\\,G^2(x)\\right]+(\\Omega ^2(x))^{\\prime }=0.$ Because of the boundary condition (REF ), we have that $\\Omega \\ne 0$ in some interval $(x_0,1)$ by continuity.",
"Equation (REF ) implies that $G\\equiv 0$ in $(x_0,1)$ , and () reduces to $(\\Omega ^2)^{\\prime }=0$ in $(x_0,1)$ .",
"Thus $\\Omega \\equiv C_\\omega $ in $(x_0,1)$ .",
"This then implies that $\\Omega \\equiv C_\\omega $ and $G\\equiv 0$ in $(0,1)$ .",
"Thus we obtain the “trivial” solution $F=G\\equiv 0,\\qquad \\Omega \\equiv C_\\omega .$"
],
[
"Case $b\\ne 2$ (existence of the trivial solution)",
"In this case, we can use relationship (REF ) between $G(x)$ and $F(x)$ and substitute it into (REF ) and ().",
"The resulting equations are given in the appendix in (REF ) and (REF ).",
"Notice that we still have the trivial solution (REF ), since if $\\Omega \\equiv C_\\omega $ , then (REF ) implies $F(x)=c\\sqrt{1-x^2}$ , and the initial condition $F(0)=0$ gives $c=0$ and $F\\equiv 0$ ; equation (REF ) then gives $G\\equiv 0$ .",
"There remains to be seen if there exist other, nontrivial solutions.",
"We first address the simple case $b=1$ and then turn to the more complicated case $b\\ne 1,2$ ."
],
[
"Case $b=1$ (existence of nontrivial solutions)",
"If $b=1$ , then (REF ) and (REF ) reduce to $F(x)\\Omega ^{\\prime }(x)=0,\\\\(\\Omega ^2(x))^{\\prime }+\\frac{1}{2}(1-x^2)\\left(F^2(x)\\right)^{\\prime \\prime \\prime }=0.$ Recall that in this case $F$ vanishes at both $x=0$ and $x=1$ by ().",
"We can consider two cases.",
"Either $F\\equiv 0$ , in which case () implies $\\Omega \\equiv C_\\omega $ , (REF ) is trivially satisfied, and from (REF ) we have $G\\equiv 0$ .",
"This case corresponds to the trivial solution (REF ).",
"In the second case, if $F(x_0)\\ne 0$ for some $x_0\\in (0,1)$ , then consider the largest interval $(x_1,x_2)\\subset (0,1)$ containing $x_0$ such that $F(x)\\ne 0$ in $(x_1,x_2)$ and $F(x_1)=F(x_2)=0$ .",
"In $(x_1,x_2)$ , $\\Omega $ has to be constant in view of (REF ) and $F^2(x)=c(x-x_1)(x_2-x)$ with $c>0$ in view of ().",
"However, in this case all (one-sided) derivatives of $F$ become infinite at $x_1$ and $x_2$ , and therefore the only possibility is that $x_1=0$ and $x_2=1$ , in which case we have $\\Omega \\equiv C_\\omega $ , $F(x)=C_1\\sqrt{x(1-x)}$ with $C_1\\ne 0$ , and, from (REF ), $G(x)=C_1\\dfrac{(1-2x)\\sqrt{1+x}}{2\\sqrt{x}}$ .",
"In summary, when $b=1$ , we have a set of solutions of the form $\\Omega \\equiv C_\\omega ,\\qquad F(x)=C_1\\sqrt{x(1-x)},\\qquad G(x)=C_1\\frac{(1-2x)\\sqrt{1+x}}{2\\sqrt{x}}\\quad \\text{ for }C_1\\in \\mathbb {R},$ which also includes the trivial solution (REF ) when $C_1=0$ .",
"We see that the solutions with $C_1\\ne 0$ will have infinite flow speeds both near the ground and near the vortex axis.",
"The velocity becomes infinite near the ground in the radial direction (inflow for updraft solutions with $C_1<0$ and outflow for downdraft solutions with $C_1>0$ ) due to $G$ having an asymptote at $x=0$ .",
"Near the vortex axis ($x=1$ ) both $\\Omega $ and $G$ have a finite limit and therefore the flow speed becomes infinite due to the $r$ term in the denominators of the velocity components (REF ).",
"Both of these phenomena are observed in Fig.",
"REF in the middle plot.",
"Remark 4.2 Note that we are only looking for classical solutions for $x\\in (0,1)$ that lead to the solution (REF ), resulting in only updraft or only downdraft flows.",
"If we allowed more general solutions, we could generate flows with an arbitrary number of cells, $n$ , with alternating updraft and downdraft flows by considering a partition of the interval $(0,1)$ , say, $0=a_0<a_1<\\dots <a_n=0$ , and on each $(a_i,a_{i+1})$ have $F^2(x)=c_i(x-a_i)(a_{i+1}-x)$ with $c_i>0$ .",
"By alternating the signs of $F$ from interval to interval, we could obtain $\\lim _{x\\rightarrow a_i}F^{\\prime }(x)=\\lim _{x\\rightarrow a_i}G(x)=+\\infty $ or $-\\infty $ for each $0<i<n$ , and thus obtain a collection of conical flows with infinite inflows or outflows along every cone.",
"To assess how reasonable solution (REF ) is, we have implemented the iterative procedure described by Serrin [30] to compute solutions to (REF ) with small nonzero viscosity.",
"In [30], solutions depend on two parameters, $k$ and $P$ ; viscosity is related to $k$ via $\\nu =1/(2k)$ , so small values of viscosity correspond to large values of $k$ .",
"In Fig.",
"REF we present two solutions, one for $k=100$ and $P=0$ (left) and one for $k=1000$ and $P=0$ (right), and we compare them to (REF ) with $C_1=C_\\omega =1$ .",
"We observe very good agreement of the two solutions in the interval $(0,1)$ except near $x=0$ , where the nonzero-viscosity solution exhibits a thin boundary layer due to the no-slip boundary conditions, while $F^{\\prime }$ and $G$ from (REF ) both tend to infinity as $x\\rightarrow 0$ .",
"Increasing $k$ (i.e., decreasing the viscosity, $\\nu $ ) results in shrinking of the size of the boundary layer.",
"More specifically, to numerically estimate the size of this boundary layer, we focus on the $x$ -value at which Serrin's $\\Omega (x)$ (blue curve in Fig.",
"REF ) starts to deviate from the inviscid solution $\\Omega (x)\\equiv 1$ (dashed).",
"For several decreasing values of $\\nu $ we estimate the layer size and plot the results on a log-log scale in Fig.",
"REF .",
"We observe a linear relationship between the logarithm of the layer size and the logarithm of the viscosity with a slope estimated by linear regression to be approximately $0.669$ .",
"In [30], Serrin defines the boundary layer independently of the solutions of Euler equations, and analytically estimates its size to be on the order of $\\nu ^{2/3}$ , which very well agrees with our result.",
"We, therefore, conclude that outside this boundary layer the solutions of Navier–Stokes equations are in good agreement with the solutions of Euler equations.",
"This provides numerical evidence that downdraft solutions ($C_1>0$ ) in (REF ) are limits of downdraft solutions of (REF ) as $\\nu \\rightarrow 0$ .",
"On the other hand, it follows from Serrin's results that updraft solutions ($C_1<0$ ) in (REF ) cannot be limits as viscosity tends to zero of any of the solutions presented in [30].",
"This leaves open the question whether solutions other than those described by Serrin exist that tend to solution (REF ) with $C_1<0$ as viscosity tends to zero.",
"Figure: Comparison of the inviscid flow solution () with C 1 =C ω =1C_1=C_\\omega =1 with the solutions with small nonzero viscosity corresponding to k=100k=100 and P=0P=0 (left) and k=1000k=1000 and P=0P=0 (right) in .",
"The plot of FF is shown in red, plot of GG in green, and plot of Ω\\Omega in blue, with the plots for solution () dashed.Figure: Boundary layer analysis for the case b=1b=1.",
"A linear relationship between the logarithm of the kinematic viscosity (horizontal axis) and the logarithm of the estimated size of the boundary layer (vertical axis) is observed with a slope of 0.6690.669 obtained by linear regression.To visualize solution (REF ) in other ways, in Fig.",
"REF we show a streamlines plot that represents particle trajectories without the azimuthal component, a contour plot of the speed, $\\Vert {\\bf v}\\Vert =\\sqrt{v_R^2+v_\\alpha ^2+v_\\theta ^2}$ , and a contour plot of the pressure obtained from (REF ) with $T=0$ .",
"The shown ranges are $0<r<1$ , $0<z<1$ with $r=R\\sin \\alpha $ and $z=R\\cos \\alpha $ .",
"We choose $C_1=4\\sqrt{2}$ and $C_\\omega =1$ since the corresponding solution is also displayed later in red in Fig.",
"REF (as a limit of numerically computed solutions corresponding to $b\\nearrow 1$ ).",
"All of the contour plots in this paper have been generated with fifty, automatically chosen and uniformly spaced contour levels.",
"Due to the singularities in the speed and the pressure near the axis of the vortex or near the ground, the holes that appear in the isospeed and isobar plots correspond to values that are out of the automatically chosen range.",
"Figure: Plots of the streamlines (left), the corresponding isocurves for speed (middle), and isocurves for pressure (right) for the solution in () with C 1 =42C_1=4\\sqrt{2} and C ω =1C_\\omega =1.",
"The horizontal axes correspond to r=Rsinαr=R\\sin \\alpha and the vertical axes to z=Rcosαz=R\\cos \\alpha .",
"The values for speed range from 4 to 50 with a step of 1, increasing towards the vortex axis.",
"The values for pressure range from -36.8-36.8 to 75.075.0 with a step of 2.32.3 with lowest values near the vortex axis.",
"The straight red line corresponds to the level set p(R,α)=0p(R,\\alpha )=0, which is the line cosα=1-(C ω /C 1 ) 2 \\cos \\alpha =1-(C_\\omega /C_1)^2.We remark that the solution in (REF ) is self-similar, and the self-similarity is clearly observed when one zooms out of the plots in Fig.",
"REF .",
"The zoomed out figures look identical to Fig.",
"REF with the contours only corresponding to different values for each level set.",
"This is clear from the definition of the velocities in (REF ), since the functions $F$ , $G$ , and $\\Omega $ only depend on $x=\\cos \\alpha $ .",
"As an illustration of Cai's power law method, we computed the exponent for the velocity-radius power law from the data in the middle plot in Fig.",
"REF by computing the slope of the logarithm of the speed against the logarithm of the scale for several different pairs of points and obtained $-1$ for the slope.",
"This computation was done using the values at the height of $1.0$ unit in the middle plot, where the speed values can be easily read off.",
"While this height was chosen for convenience, the results would be the same at any height due to the assumption on the structure of the solution (REF ).",
"The pressure plot in Fig.",
"REF shows low values near the vortex axis and finite values along the ground, which increase as $r\\rightarrow 0$ .",
"In fact, from the solution (REF ) and the expression for pressure (REF ), one quickly obtains (taking $T=0$ ) that $p(R,\\alpha )=-\\frac{C_\\omega ^2-C_1^2(1-x)}{2r^2}=-\\frac{C_\\omega ^2-C_1^2(1-\\cos \\alpha )}{2R^2\\sin ^2\\alpha },$ so we see that as one approaches the vortex axis, i.e., as $x\\rightarrow 1$ , the pressure behaves like $-1/r^2$ .",
"(Note that the physical pressure has the form $p(R,\\alpha )+T$ and thus has a singularity near the vortex axis no matter what the value of $T$ is.",
"As in Serrin's approach, this is a consequence of the assumption on the velocity (REF ).)",
"It is also immediate to observe that the pressure is zero along the line $x=1-(C_\\omega /C_1)^2$ ; this line is indicated by the bolder red line in Fig.",
"REF .",
"Notice that near the corner, where the vortex axis meets the ground, all of the other level curves are tangent to this line, and our model in this case would formally indicate a large pressure gradient (singular at the origin as well as along the vortex axis) as the pressure undergoes a sudden change from positive to negative values when crossing the red line and approaching the vortex axis.",
"Clearly, such a behavior will be observed if $C_1^2>C_\\omega ^2$ , since then the level line $p(R,\\alpha )=0$ have positive slope.",
"In Fig.",
"REF , we show the cases that correspond to the line $p(R,\\alpha )=0$ having angles $\\pi /4$ , 0, and $-\\pi /4$ with the horizontal, respectively.",
"The corresponding values of $C_1^2$ are then $2+\\sqrt{2}$ , 1, and $2-\\sqrt{2}$ , respectively (with $C_\\omega =1$ ).",
"We note that the apparent singularity of the pressure gradient near the vortex axis might be due to the original assumption on the velocity field (REF ) and therefore not be physically reasonable.",
"Figure: Plots of the streamlines (left), the corresponding isocurves for speed (middle), and isocurves for pressure (right) for the solution in () with C ω =1C_\\omega =1 and C 1 =2+2C_1=\\sqrt{2+\\sqrt{2}} (top row), C 1 =1C_1=1 (middle row), and C 1 =2-2C_1=\\sqrt{2-\\sqrt{2}} (bottom row).",
"The values for speed range from 1.681.68 to 21.0021.00 with a step of 0.420.42 (top), from 1.241.24 to 15.5015.50 with a step of 0.310.31 (middle), and from 1.301.30 to 13.7813.78 with a step of 0.260.26 (bottom), increasing towards the vortex axis.",
"The values for pressure range from -24.94-24.94 to 17.2017.20 with a step of 0.860.86 (top), from -32.50-32.50 to 0 (red line) with a step of 0.650.65 (middle), and from -0.7-0.7 to -35.0-35.0 with a step of 0.70.7 (bottom), with lowest values near the vortex axis.The pressure plots in Fig.",
"REF and Fig.",
"REF also provide an interesting characterization of possible shapes of a visible tornado funnel.",
"Since the funnel outline should approximately follow the isobars, we see three distinct possible shapes: one that is conical near the ground (Fig.",
"REF and Fig.",
"REF (top row)); one that can be viewed as a degenerate, fully open cone, yet still with a single point touching the ground (Fig.",
"REF (middle row)); and one with the funnel having a nonzero width at the ground (Fig.",
"REF (bottom row)).",
"Note that changing $C_1$ while keeping $C_\\omega $ fixed changes the relative magnitudes of the azimuthal component of the velocity with respect to the non-azimuthal ones.",
"We could view a large value of $C_\\omega /C_1$ as corresponding to large amount of swirl, and a small value corresponding to a small amount of swirl [34], [5], [20].",
"In this sense, we can say that in Fig.",
"REF swirl increases from top to bottom, and wider funnels correspond to more swirl.",
"We illustrate this behavior in Fig.",
"REF , in which a few streamlines are shown for three cases, $C_1=-10$ (left), $C_1=-1$ , and $C_1=-1/10$ in (REF ).",
"It is believed that increasing the swirl ratio in a single-cell vortex can lead to a vortex breakdown into multiple vortices as shown in Fig.",
"REF [28].",
"Our model only captures an updraft or a downdraft flow, so even though the swirl increases from top to bottom in Fig.",
"REF , our model cannot capture the whole dynamics of a vortex breakdown.",
"See the conclusions section for more discussion of a vortex breakdown.",
"Figure: Illustration of the effect of the ratio C ω /C 1 C_\\omega /C_1 on the swirl of solutions ().",
"In all cases C ω =1C_\\omega =1, but C 1 =-10C_1=-10 in the left plot, C 1 =-1C_1=-1 in the middle plot, and C 1 =-1/10C_1=-1/10 in the right plot.",
"Four streamlines are plotted in each case, with initial points (20,20,0.02)(20,20,0.02), (20,20,0.5)(20,20,0.5), (20,20,2)(20,20,2), and (20,20,5)(20,20,5)."
],
[
"Case $b\\ne 1,2$ (general observations)",
"In this case, the governing equations (REF ) and (), using the lower-case functions and expressing $g$ using (REF ), reduce to $f\\omega ^{\\prime }=\\frac{1-b}{2-b}\\,f^{\\prime }\\omega ,$ $(1-x^2)\\left[\\frac{2+b}{2-b}\\,f^{\\prime }f^{\\prime \\prime }+ff^{\\prime \\prime \\prime }\\right]+4(1-b)ff^{\\prime }+2(1-b)(2-b)\\frac{x}{1-x^2}\\left(f^2+\\omega ^2\\right)+2(2-b)\\omega \\omega ^{\\prime }=0.$ Remark 4.3 The last two terms containing $\\omega $ in (REF ) can be written in terms of $\\Omega $ in the form $2(2-b)\\left[\\omega \\omega ^{\\prime }+(1-b)\\frac{x}{1-x^2}\\,\\omega ^2\\right]=2(2-b)(1-x^2)^{1-b}\\,\\Omega \\Omega ^{\\prime },$ and hence we see from (REF ) that if $F=f\\equiv 0$ , then we recover the trivial solution (REF ), $F=G\\equiv 0$ and $\\Omega \\equiv C_\\omega $ .",
"Similarly, if $\\Omega \\equiv C_\\omega $ , then it follows from (REF ) and the boundary condition $F(0)=f(0)=0$ that $F=f\\equiv 0$ .",
"We thus seek solutions with $f\\lnot \\equiv 0$ (or $F\\lnot \\equiv 0$ ) and $\\Omega \\lnot \\equiv C_\\omega $ .",
"It will be more convenient to work in terms of $f$ and $\\Omega $ , so we rewrite equations (REF ) and (REF ) as $f\\Omega ^{\\prime }=\\frac{1-b}{2-b}\\left[f^{\\prime }+(2-b)\\frac{x}{1-x^2}\\,f\\right]\\Omega ,$ $(1-x^2)\\left[\\frac{2+b}{2-b}\\,f^{\\prime }f^{\\prime \\prime }+ff^{\\prime \\prime \\prime }\\right]+4(1-b)ff^{\\prime }+2(1-b)(2-b)\\frac{x}{1-x^2}f^2+2(2-b)(1-x^2)^{1-b}\\,\\Omega \\Omega ^{\\prime }=0.$ Remark 4.4 Notice that both (REF ) and (REF ) are invariant under sign changes $f\\mapsto -f$ and $\\Omega \\mapsto -\\Omega $ ; as one consequence, we can assume that $C_\\omega >0$ .",
"On any interval where $f\\ne 0$ , we can solve (REF ) for $\\omega $ and obtain $\\omega (x)=c|f(x)|^{(1-b)/(2-b)}\\quad \\text{ for }c>0.$ Vice versa, on any interval where $\\omega \\ne 0$ (and in particular on some interval $(x_1,1)$ due to the boundary condition (REF )) we can solve for $f$ and obtain $f(x)=c|\\omega (x)|^{(2-b)/(1-b)}=c(1-x^2)^{(2-b)/2}|\\Omega (x)|^{(2-b)/(1-b)}\\quad \\text{ for }c>0.$ We thus have that $f(x)=\\mathcal {O}\\left((1-x^2)^{(2-b)/2}\\right)\\qquad \\text{and}\\qquad F(x)=\\mathcal {O}\\left(\\sqrt{1-x^2}\\right)\\qquad \\text{ as }x\\rightarrow 1.$"
],
[
"Case $2<b<\\infty $ (no nontrivial solutions)",
"We now show that no nontrivial solutions of (REF ) and (REF ) in the classical sense can exist for $b>2$ .",
"First, from the definition of $\\omega (x)=\\Omega (x)(1-x^2)^{(1-b)/2}$ and the boundary condition (REF ) we see that $\\omega (x)\\rightarrow +\\infty $ as $x\\rightarrow 1$ .",
"Since in this case $\\frac{2-b}{1-b}>0$ , then, in view of (REF ), the only way to not violate the boundary condition $f(x)\\rightarrow 0$ as $x\\rightarrow 1$ is if $f$ is identically zero on some interval $[x_1,1)$ .",
"Since we are looking for solutions with $f\\lnot \\equiv 0$ , let us assume, without loss of generality, that $f>0$ on some interval $(x_0,x_1)$ .",
"We now compare the limiting behavior of $\\omega $ on either side of $x_1$ .",
"On $(x_1,1)$ , where $f\\equiv 0$ , we have from (REF ) that $\\Omega \\equiv C_\\omega $ , so $\\omega $ has a nonzero limit as $x\\rightarrow x_1$ from the right.",
"On the other hand, on $(x_0,x_1)$ , where $f>0$ , we have (REF ), and hence $\\lim _{x\\rightarrow x_1-}\\omega (x)=0$ .",
"Therefore $\\omega $ cannot be continuous and there are no nontrivial solutions in the case $2<b<\\infty $ ."
],
[
"Case $1<b<2$ (behavior of potential solutions)",
"In this case we have $\\frac{2-b}{1-b}<0$ .",
"Since again $\\omega (x)\\rightarrow +\\infty $ as $x\\rightarrow 1$ , we can use (REF ) to express $f$ in terms of $\\omega $ and observe that this time the boundary condition $f(x)\\rightarrow 0$ as $x\\rightarrow 1$ is satisfied independently of the value of $c$ in (REF ).",
"Note that in this case $\\omega $ has to be positive in $(0,1)$ , since if $\\omega (x_0)=0$ for some $x_0\\in (0,1)$ , then, in view of (REF ), $f$ would have an asymptote at $x_0$ and thus be discontinuous there.",
"It follows from (REF ) that $f$ cannot change sign in $(0,1)$ either, and (REF ) implies that $\\omega (x)\\rightarrow +\\infty $ as $x\\rightarrow 0$ .",
"Consequently, we also have $\\Omega (x)\\rightarrow +\\infty $ as $x\\rightarrow 0$ and the azimuthal velocity becomes infinite near the ground.",
"We have been unable to find analytic expressions for such solutions and also encountered difficulties when approximating them numerically.",
"However, in the next section, we show that such solutions would be unstable with respect to axisymmetric perturbations."
],
[
"Instability of potential solutions for $1<b<2$",
"In this section we will assume $0<b<2$ and address the centrifugal stability of solutions to (REF ) and (REF ) with respect to axisymmetric perturbations.",
"We will use Rayleigh's circulation criterion [8], which can be stated as the requirement that the Rayleigh discriminant $\\Phi $ is nonnegative, where $\\Phi (r)=\\frac{1}{r^3}\\frac{\\partial }{\\partial r}(rv_\\theta )^2.$ Substituting in the expression $v_\\theta =\\Omega (x)/r^b$ and using the relationship between $x$ and the cylindrical coordinates $r$ and $z$ , $x=\\cos \\alpha =z/\\sqrt{r^2+z^2}$ , we obtain $\\Phi (r)=\\frac{2}{r^{2(1+b)}}\\,\\Omega (x)\\left[(1-b)\\Omega (x)-x(1-x^2)\\Omega ^{\\prime }(x)\\right].$ The particular case with $b=1$ gives $\\Phi \\equiv 0$ since from (REF ) we have $\\Omega \\equiv C_\\omega $ .",
"Also, the trivial solution with $F=G\\equiv 0$ and $\\Omega \\equiv C_\\omega $ is clearly stable only for $0<b\\le 1$ .",
"We will show that $b=1$ is the largest value of $b$ that permits stable nontrivial solutions.",
"For $1<b<2$ , the stability requirement $\\Phi (r)\\ge 0$ can be replaced by an equivalent statement $f^2(x)\\Phi (r)\\ge 0$ since $f\\ne 0$ in $(0,1)$ .",
"We can then use (REF ) to get $f^2(x)\\Phi (r)=\\frac{2(1-x^2)\\Omega ^2(x)}{r^{2(1+b)}}\\frac{1-b}{2-b}f(x)\\left[(2-b)f(x)-xf^{\\prime }(x)\\right],$ which is invariant under the sign change of $f$ , so we can assume, without loss of generality, that $f>0$ in $(0,1)$ .",
"The stability requirement $f^2(x)\\Phi (r)\\ge 0$ now implies $(2-b)f(x)-xf^{\\prime }(x)\\le 0$ , and, in particular, $f$ is nondecreasing in $(0,1)$ .",
"However, this contradicts the assumptions $f>0$ in $(0,1)$ and $\\lim _{x\\rightarrow 1}f(x)=0$ , so no solutions corresponding to $1<b<2$ can be stable with respect to axisymmetric perturbations."
],
[
"Case $0<b<1$ (numerical solutions)",
"Lack of analytic solutions for $b=1$ and constant nonzero viscosity led to numerical approaches presented, e.g., in [30], [17], [24].",
"We have not been able to find analytic expressions for any nontrivial solutions in the case $0<b<1$ either, but we used a numerical approach to generate their approximations for various values of $b$ between 0 and 1.",
"Once our solutions are computed, it can then be numerically or graphically verified that they satisfy the stability criterion $\\Phi \\ge 0$ with $\\Phi $ given in (REF ).",
"All of our numerical solutions for $0<b<1$ were graphically checked to satisfy the correct inequality and thus were stable with respect to axisymmetric perturbations.",
"We now describe our numerical approach to obtain approximations to solutions to (REF ) and (REF ).",
"We first note that we can rescale the functions in consideration using the boundary condition (REF ), $f(x)=C_\\omega \\tilde{f}(x)\\qquad \\text{ and }\\qquad \\Omega (x)=C_\\omega \\tilde{\\Omega }(x),$ so that the boundary condition (REF ) becomes $\\lim _{x\\rightarrow 1}\\tilde{\\Omega }(x)=1.$ Note that we can simply replace $f$ and $\\Omega $ in (REF ) and (REF ) by $\\tilde{f}$ and $\\tilde{\\Omega }$ , since the scaling constants cancel out.",
"We will thus work with (REF ) and (REF ) in their original form and only replace (REF ) with (REF ), keeping in mind that any solutions will correspond to the rescaled functions.",
"Disregarding the boundary condition on $\\Omega $ , it is clear that if the pair $(f,\\Omega )$ solves (REF ) and (REF ), then so does any pair $(\\pm \\tilde{c}f,\\pm \\tilde{c}\\Omega )$ with $\\tilde{c}\\in \\mathbb {R}$ .",
"In our numerical approach we will seek solutions with $f>0$ and $\\Omega >0$ in $(0,1)$ .",
"Note that if $f$ and $\\Omega $ satisfy (REF ), then equation (REF ) will be automatically satisfied.",
"It is not difficult to check that the pair $f_0(x)=2^{(1-b)/2}\\left(x(1-x)\\right)^{(2-b)/2}\\qquad \\text{ and }\\qquad \\Omega _0(x)=\\left(\\frac{2x}{1+x}\\right)^{(1-b)/2}$ satisfies (REF ) and the left-hand side of (REF ) evaluates to a well-behaved expression $2^{1-b}(2-b)(1-b)\\frac{2+x}{1+x}\\left(x(1-x)\\right)^{1-b}$ that vanishes at both endpoints for all $0<b<1$ and converges uniformly to 0 as $b\\rightarrow 1$ .",
"The expressions in (REF ) can therefore serve as a basis for initial guesses in a numerical scheme.",
"However, we observe that their derivatives behave singularly near the endpoints.",
"To bypass this difficulty, we recall (REF ) and define a function $\\gamma (x)$ by $f(x)=\\gamma (x) (1-x^2)^{(2-b)/2},$ so that, using (REF ), $\\Omega (x)=c(1-x^2)^{-(1-b)/2}f(x)^{(1-b)/(2-b)}=c\\gamma (x)^{(1-b)/(2-b)}\\quad \\text{ for some }c>0.$ We can then substitute (REF ) and (REF ) into (REF ).",
"Since $\\gamma $ is expected to have an infinite slope at $x=0$ , we also reformulate the newly obtained version of (REF ) in terms of the square of $\\gamma $ , $p(x)=\\gamma ^2(x),$ and, after factoring out and discarding some positive terms, get the equation $\\begin{split}p^2\\biggr [(1-x^2)\\big ((1-x^2)p^{\\prime \\prime \\prime }&-2(4-b)xp^{\\prime \\prime }\\big )-2(2+b-3(2-b)x^2)p^{\\prime }\\biggr ]+2c^2(1-b)p^{(3-2b)/(2-b)}p^{\\prime }\\\\&+\\frac{1-b}{2-b}(1-x^2)p^{\\prime }\\biggr [(1-x^2)(p^{\\prime })^2-2p\\left((1-x^2)p^{\\prime \\prime }-(2-b)xp^{\\prime }\\right)\\biggr ]=0.\\end{split}$ The boundary conditions arising immediately from those for $f$ are $\\gamma (0)=0$ and, in view of the asymptotic behavior of $f$ given in (REF ), $\\gamma $ having a finite limit as $x\\rightarrow 1$ .",
"Since the solutions can be rescaled as discussed above, we can assume $\\gamma (x)\\rightarrow 1$ as $x\\rightarrow 1$ .",
"We approximate the solution to (REF ) by discretizing it using a uniform mesh with stepsize $h$ and solving the discretized system by Newton's method, using (REF ) and (REF ) to assemble an initial guess for $p(x)$ .",
"The results presented in this section correspond to $h=10^{-3}$ .",
"While it is not clear to us whether solutions to (REF ) exist for any combination of the constants $b$ and $c$ , we have found that for a given value of $b$ , increasing the value of $c$ eventually creates instability in the numerical code, suggesting a potential restriction on (a combination of) these values.",
"Note from, e.g., (REF ) and (REF ) that the constant $c$ can be viewed as a scaling constant between $\\Omega $ and $f$ (or, more generally, between the azimuthal component of the flow and the non-azimuthal ones); a large value of $c$ corresponds to a relatively large azimuthal component of the velocity with respect to the other two components, while a small value of $c$ corresponds to the azimuthal component being relatively small.",
"In other words, increasing $c$ can be viewed as increasing swirl in the flow.",
"Notice the similarity to changing $C_1$ in the case $b=1$ above.",
"In Fig.",
"REF , we present computed solutions for $c=0.25$ and $b=0.1,\\dots ,0.9$ with increments of $0.1$ .",
"Notice that the results demonstrate continuous dependence on the parameter $b$ , and the solution (REF ) for $b=1$ can be viewed as their limit as $b\\rightarrow 1$ .",
"This solution, with constants $C_\\omega =1$ and $C_1=4\\sqrt{2}$ , is plotted in Fig.",
"REF in red for comparison.",
"Figure: Graphs of F(x)F(x) (left), Ω(x)\\Omega (x) (middle), and G(x)G(x) (right) as numerical solutions obtained from () with c=0.25c=0.25 for b=0.1,0.2,⋯,0.9b=0.1,0.2,\\dots ,0.9 (in blue).",
"The functions FF and Ω\\Omega are increasing with increasing bb at all x∈(0,1)x\\in (0,1), while the magnitude of GG increases in most of the interval (0,1)(0,1) as bb increases.",
"The red plot corresponds to the solution () with C ω =1C_\\omega =1 and C 1 =42C_1=4\\sqrt{2}, which can be viewed as a limiting case as b→1b\\rightarrow 1.We can observe that in most of the interval $(0,1)$ the magnitudes of $F$ , $\\Omega $ , and $G$ decrease as $b$ decreases, although the results do not suggest that these functions would vanish if $b$ approached 0.",
"Since the expression $1/r^b$ also decreases with decreasing $b$ for $0<r<1$ , we see that if $c$ is fixed, then the flow speed decreases with decreasing $b$ near the vortex axis.",
"This is consistent with observations of Cai [2] and Wurman [37], [36] that larger values of $b$ correspond to stronger storms.",
"The azimuthal velocity exhibits an interesting feature for $0<b<1$ .",
"Notice in Fig.",
"REF that $\\Omega (0)=0$ for every $0<b<1$ , and therefore the azimuthal velocity vanishes at the ground.",
"This behavior of $\\Omega $ is not enforced by an a priori boundary condition, rather is it a consequence of the Euler equations and the boundary condition $F(0)=0$ .",
"It means that nontrivial solutions with $0<b<1$ exhibit purely radial inflow or outflow at the ground.",
"To see how the choice of $c$ affects the results, we also present results with a fixed value of $b$ and varying values of $c$ for which we were able to generate results.",
"In Fig.",
"REF , we present results with $b=0.6$ and $c=0.1,\\dots ,1.0$ with increments of $0.1$ .",
"(The value $b=0.6$ is chosen since it corresponds to the midpoint of the interval $(-0.7,-0.5)$ found in [36].)",
"For comparison, we also display the graphs corresponding to $c=0.25$ in red; these same graphs are also shown in Fig.",
"REF .",
"Figure: Graphs of F(x)F(x) (left), Ω(x)\\Omega (x) (middle), and G(x)G(x) (right) as numerical solutions obtained from () with b=0.6b=0.6 for c=0.1,0.2,⋯,1.0c=0.1,0.2,\\dots ,1.0 (in blue).",
"The functions FF and Ω\\Omega are decreasing with increasing cc at all x∈(0,1)x\\in (0,1), while the magnitude of GG decreases in most of the interval (0,1)(0,1) as cc increases.",
"The red plots correspond to the solutions with c=0.25c=0.25, which are also shown in Fig.",
".We see that as $c$ increases, the magnitudes of $F$ , $\\Omega $ , and $G$ decrease in most of the interval $(0,1)$ , although the change in $\\Omega $ is fairly small.",
"This behavior is in agreement with the meaning of the constant $c$ discussed earlier, i.e., that $c$ reflects the relative importance of the azimuthal component of the velocity with respect to the other two components.",
"To compare the flows, speeds, and pressure fields, we present in Fig.",
"REF the analogs of Fig.",
"REF for the cases $c=0.25$ and $b=0.8$ and $0.2$ .",
"The plots corresponding to the intermediate values of $b$ showed continuous dependence on the parameter $b$ and consequently we do not display them.",
"We observe that while the streamlines remain relatively the same for various values of $b$ , the speeds of the flow and the pressure fields exhibit discernible changes.",
"The speeds are significantly larger in the plot with the larger value of $b$ (top row), and the figures also suggest a wider funnel for the larger $b$ .",
"Both of these observations are consistent with larger values of $b$ being associated with more violent storms.",
"Figure: Plots of the streamlines (left), the corresponding isocurves for speed (middle), and isocurves for pressure (right) for the numerically computed solution with c=0.25c=0.25, b=0.8b=0.8 (top row) and c=0.25c=0.25, b=0.2b=0.2 (bottom row).",
"The values for speed range from 3.203.20 to 34.5634.56 with a step of 0.640.64 (top), and from 1.71.7 to 6.66.6 with a step of 0.10.1 (bottom), both increasing towards the vortex axis.",
"The values for pressure range from -24.0-24.0 to 54.454.4 with a step of 1.61.6 (top), and from 1.821.82 to 14.5614.56 with a step of 0.260.26 (bottom), with lowest values near the vortex axis.",
"(The contours near the vortex axis for b=0.2b=0.2 are significantly affected by numerical errors in the contour plot routine.)",
"The straight red lines correspond to the level sets p(R,α)=0p(R,\\alpha )=0.Our numerical results in this section were obtained under the assumption that $\\Omega $ does not change sign in $(0,1)$ .",
"We do not yet know whether our equations (REF ) and (REF ) admit solutions that change sign in the interval $(0,1)$ .",
"Such cases would be interesting to study, since, unlike in [30], equation (REF ) suggests that if $f$ changes sign, so would $\\Omega $ and vice versa.",
"This could give rise to very interesting types of flows."
],
[
"Discussion, Conclusions, and Implications",
"In this work, we focused on finding solutions of the form (REF ) to the Navier–Stokes and Euler equations in the upper half-space.",
"We were motivated by the 1972 work of J. Serrin [30], in which he studies the viscous case with $b=1$ , and by later studies [2], [37], [36], in which it is suggested that the velocity may decay with different powers of the radial distance from the vortex axis than 1.",
"We found that no solutions of the form (REF ) exist in the case of constant nonzero viscosity when $b\\ne 1$ , primarily due to the boundary conditions at the ground, i.e., the plane bounding our half-space.",
"The situation is different in the case with zero viscosity and Euler equations.",
"In this case the boundary conditions are relaxed since slip is allowed.",
"Assuming that the azimuthal velocity behaves like $C_\\omega /r^b$ with $C_\\omega \\ne 0$ near the vortex axis, we were able to show that the trivial solution $\\Omega \\equiv C_\\omega $ and $F=G\\equiv 0$ works for any $b>0$ , although it is stable with respect to axisymmetric perturbations only if $0<b\\le 1$ .",
"Nontrivial solutions are harder to find.",
"We showed that if $b\\ge 2$ , no nontrivial solutions exist.",
"We also showed that any potential solutions for $1<b<2$ would be unstable with respect to axisymmetric perturbations.",
"The case with $b=1$ was fully analytically resolved and its solution is given in (REF ).",
"We discussed its characteristics and showed that the downdraft ($C_1>0$ ) solution in (REF ) can be viewed as a limit as viscosity goes to zero of the downdraft solutions found in [30].",
"The case with $0<b<1$ proved analytically difficult, and we only presented some numerical results indicating the existence of solutions that have similar characteristics to those with $b=1$ .",
"This case is most interesting, as it allows the coefficient $b$ to fall into the ranges discussed by Cai and Wurman et al.",
"[2], [37], [36].",
"We were able to numerically find solutions with $F$ and $\\Omega $ that do not change sign, which corresponds to either an updraft or a downdraft solution.",
"It would be interesting to see whether solutions with sign changes are possible.",
"The numerically found solutions exhibit continuous dependence on the parameter $b$ and tend to the solution (REF ) as $b\\rightarrow 1$ as demonstrated in Fig.",
"REF .",
"We have shown how the value of $b$ affects the intensity of the modeled vortex.",
"As in Serrin's model, our model exhibits a singularity near the vortex axis; in particular, the updraft/downdraft and azimuthal speeds tend to infinity, although at a rate of $1/r^b$ , rather than $1/r$ .",
"As we discussed briefly in the introduction, updraft wind speeds may exceed the speed of sound, which may occur during the process of a vortex breakdown, illustrated in Fig.",
"REF .",
"During this process, a single-cell flow bifurcates into a double-cell flow, and then further bifurcates into a flow with multiple vortices.",
"The portion of the vortex near the axis where the horizontal flow turns into vertical (updraft case) is called the corner flow region.",
"If we assume that this part of the flow with large vertical updraft speeds is quasi-steady, our and Serrin's models can be viewed as describing the lower portion of the flow.",
"Additionally, extremely intense vortices with large updraft speeds can develop inside larger tornadoes as evidenced by the formation of “suction spot” paths in crop fields, paths less than 1 meter in diameter where corn crops have been ripped out of the ground by the roots [13].",
"Such vortices could potentially be described using our and Serrin's models as well.",
"We believe that current radar research in [2], [37], [36], giving the power-law drop for velocity where $b\\ne 1$ (2 for vorticity) in tornadoes but varies, justifies our approach to modify Serrin's model.",
"We have also provided numerical evidence that some of our solutions are viscosity solutions and hence remove the singular behavior near the ground, where velocity should tend to 0.",
"Finally, we remark that in the case of a turbulent motion the functions of the form (REF ) correspond to the mean velocity, and thus $\\nu $ could play a role of eddy viscosity to maintain the delicate balance between the mean and turbulent components of the flow.",
"In recent studies, testing eddy viscosity assumptions with direct numerical simulations showed varied success [3], but has remained an important tool for understanding the connection between the scale of the model and dissipation of energy [16], [26].",
"Following [29], Serrin suggests an experimentally motivated relationship $\\nu \\approx \\sigma \\Vert {\\bf v}\\Vert r$ , where $\\sigma $ is a small dimensionless constant [30].",
"Taking into account (REF ), we thus obtain $\\nu \\approx \\tau (x)R^{1-b}$ , with the simplest case being $\\tau \\equiv \\text{const}$ .",
"Such an assumption on viscosity then leads to a modification of equations (REF ) and (REF ) that will be investigated in the future."
],
[
"Navier–Stokes equations in spherical coordinates and incompressibility",
"The three components of the Navier–Stokes equations (REF ) expressed in spherical coordinates and in terms of the velocity components (REF ) have the form $\\begin{split}v_R\\frac{\\partial v_R}{\\partial R}+\\frac{v_\\alpha }{R}\\frac{\\partial v_R}{\\partial \\alpha }+\\frac{v_\\theta }{R\\sin \\alpha }\\frac{\\partial v_R}{\\partial \\theta }&-\\frac{v^2_\\alpha +v^2_\\theta }{R}=\\\\-\\frac{\\partial p}{\\partial R}+\\frac{\\nu }{R^2}&\\Biggr [\\frac{\\partial }{\\partial R}\\left(R^2\\frac{\\partial v_R}{\\partial R}\\right)+\\frac{1}{\\sin \\alpha }\\frac{\\partial }{\\partial \\alpha }\\left(\\sin \\alpha \\frac{\\partial v_R}{\\partial \\alpha }\\right)\\\\&\\quad +\\frac{1}{\\sin ^2\\alpha }\\frac{\\partial ^2v_R}{\\partial \\theta ^2}-2\\left(v_R+\\frac{\\partial v_\\alpha }{\\partial \\alpha }+v_\\alpha \\cot \\alpha +\\frac{1}{\\sin \\alpha }\\frac{\\partial v_\\theta }{\\partial \\theta }\\right)\\Biggr ],\\end{split}$ $\\begin{split}v_R\\frac{\\partial v_\\alpha }{\\partial R}+\\frac{v_\\alpha }{R}\\frac{\\partial v_\\alpha }{\\partial \\alpha }+\\frac{v_\\theta }{R\\sin \\alpha }\\frac{\\partial v_\\alpha }{\\partial \\theta }&+\\frac{v_Rv_\\alpha -v^2_\\theta \\cot \\alpha }{R}=\\\\-\\frac{1}{R}\\frac{\\partial p}{\\partial \\alpha }&+\\frac{\\nu }{R^2}\\Biggr [\\frac{\\partial }{\\partial R}\\left(R^2\\frac{\\partial v_\\alpha }{\\partial R}\\right)+\\frac{1}{\\sin \\alpha }\\frac{\\partial }{\\partial \\alpha }\\left(\\sin \\alpha \\frac{\\partial v_\\alpha }{\\partial \\alpha }\\right)\\\\&\\qquad \\qquad +\\frac{1}{\\sin ^2\\alpha }\\frac{\\partial ^2v_\\alpha }{\\partial \\theta ^2}+2\\frac{\\partial v_R}{\\partial \\alpha }-\\frac{1}{\\sin ^2\\alpha }\\left(v_\\alpha +2\\cos \\alpha \\frac{\\partial v_\\theta }{\\partial \\theta }\\right)\\Biggr ],\\end{split}$ $\\begin{split}v_R\\frac{\\partial v_\\theta }{\\partial R}+\\frac{v_\\alpha }{R}\\frac{\\partial v_\\theta }{\\partial \\alpha }+\\frac{v_\\theta }{R\\sin \\alpha }\\frac{\\partial v_\\theta }{\\partial \\theta }&+\\frac{v_Rv_\\theta +v_\\alpha v_\\theta \\cot \\alpha }{R}=\\\\-\\frac{1}{R\\sin \\alpha }\\frac{\\partial p}{\\partial \\theta }+\\frac{\\nu }{R^2}&\\Biggr [\\frac{\\partial }{\\partial R}\\left(R^2\\frac{\\partial v_\\theta }{\\partial R}\\right)+\\frac{1}{\\sin \\alpha }\\frac{\\partial }{\\partial \\alpha }\\left(\\sin \\alpha \\frac{\\partial v_\\theta }{\\partial \\alpha }\\right)\\\\&\\quad +\\frac{1}{\\sin ^2\\alpha }\\frac{\\partial ^2v_\\theta }{\\partial \\theta ^2}+\\frac{1}{\\sin ^2\\alpha }\\left(2\\sin \\alpha \\frac{\\partial v_R}{\\partial \\theta }+2\\cos \\alpha \\frac{\\partial v_\\alpha }{\\partial \\theta }-v_\\theta \\right)\\Biggr ].\\end{split}$ Similarly, the continuity equation (REF ) has the form $\\frac{1}{R^2}\\frac{\\partial }{\\partial R}\\left(R^2v_R\\right)+\\frac{1}{R\\sin \\alpha }\\left[\\frac{\\partial }{\\partial \\alpha }\\left(v_\\alpha \\sin \\alpha \\right)+\\frac{\\partial v_\\theta }{\\partial \\theta }\\right]=0.$"
],
[
"Expressions $C_i$ and {{formula:3bf93d92-aa93-4a05-a064-776f538d1d9d}} for general {{formula:563ed847-4a30-4e19-8d67-208975c3da2f}} ; the continuity equation",
"Substituting variables (REF ) into the Navier–Stokes equations (REF )–(REF ) and comparing with the forms in (REF )–(REF ), we obtain the expressions for $C_i(\\alpha )$ and $D_i(\\alpha )$ (we omit the argument $x=\\cos \\alpha $ in the functions $F$ , $G$ , $\\Omega $ , $f$ , $g$ , and $\\omega $ , and, for example, we write $G^{\\prime }$ instead of $G^{\\prime }(\\cos \\alpha )$ ) $C_1(\\alpha )&=-\\frac{F^2+b\\,G^2+\\Omega ^2+b\\,FG\\cot \\alpha +FG^{\\prime }\\sin \\alpha }{(\\sin \\alpha )^{2b}},\\\\C_2(\\alpha )&=\\frac{-(b\\,F^2+\\Omega ^2)\\cot \\alpha +(1-b)FG-FF^{\\prime }\\sin \\alpha }{(\\sin \\alpha )^{2b}},\\\\C_3(\\alpha )&=\\frac{(1-b)G\\Omega +F\\left[(1-b)\\Omega \\cot \\alpha -\\Omega ^{\\prime }\\sin \\alpha \\right]}{(\\sin \\alpha )^{2b}},$ and $D_1(\\alpha )&=\\frac{G^{\\prime \\prime }\\sin ^2\\alpha -2(1-b)G^{\\prime }\\cos \\alpha -(1-b^2-\\cos {2\\alpha })G\\csc ^2\\alpha -2(1-b)F\\cot \\alpha +2F^{\\prime }\\sin \\alpha }{(\\sin \\alpha )^b},\\\\D_2(\\alpha )&=\\frac{F^{\\prime \\prime }\\sin ^2\\alpha -2(1-b)F^{\\prime }\\cos \\alpha -(1-b^2)F\\csc ^2\\alpha -2b\\,G\\cot \\alpha -2G^{\\prime }\\sin \\alpha }{(\\sin \\alpha )^b},\\\\D_3(\\alpha )&=\\frac{\\Omega ^{\\prime \\prime }\\sin ^2\\alpha -2(1-b)\\Omega ^{\\prime }\\cos \\alpha -(1-b^2)\\Omega \\csc ^2\\alpha }{(\\sin \\alpha )^b}.$ In terms of $x$ , and the functions $F$ , $G$ , $\\Omega $ , and $f$ , $g$ , $\\omega $ , these expressions can be written as $\\begin{split}C_1(x)&=(1-x^2)^{-b}\\left[-F^2-b\\,G^2-\\Omega ^2-F\\left(\\sqrt{1-x^2}\\,G^{\\prime }+b\\dfrac{x}{\\sqrt{1-x^2}}\\,G\\right)\\right]\\\\&=(1-x^2)^{-1}\\left[-f^2-b\\,g^2-\\omega ^2-f\\left(\\sqrt{1-x^2}\\,g^{\\prime }+\\dfrac{x}{\\sqrt{1-x^2}}\\,g\\right)\\right],\\\\\\end{split}$ $\\begin{split}C_2(x)&=(1-x^2)^{-b}\\left[-\\dfrac{x}{\\sqrt{1-x^2}}(b\\,F^2+\\Omega ^2)-F\\left(\\sqrt{1-x^2}\\,F^{\\prime }-(1-b)G\\right)\\right]\\\\&=(1-x^2)^{-1}\\left[-\\dfrac{x}{\\sqrt{1-x^2}}(f^2+\\omega ^2)-f\\left(\\sqrt{1-x^2}\\,f^{\\prime }-(1-b)g\\right)\\right],\\\\\\end{split}$ $\\begin{split}C_3(x)&=(1-x^2)^{-b}\\left[(1-b)G\\Omega -F\\left(\\sqrt{1-x^2}\\,\\Omega ^{\\prime }-(1-b)\\dfrac{x}{\\sqrt{1-x^2}}\\,\\Omega \\right)\\right]\\\\&=(1-x^2)^{-1}\\left[(1-b)g\\omega -\\sqrt{1-x^2}\\,f\\omega ^{\\prime }\\right],\\end{split}$ and $\\begin{split}D_1(x)&=(1-x^2)^{-b/2}\\left[(1-x^2)G^{\\prime \\prime }-2(1-b)x\\,G^{\\prime }-\\dfrac{2(1-x^2)-b^2}{1-x^2}\\,G-2(1-b)\\dfrac{x}{\\sqrt{1-x^2}}\\,F+2\\sqrt{1-x^2}\\,F^{\\prime }\\right]\\\\&=(1-x^2)^{-b/2}\\left[(1-x^2)g^{\\prime \\prime }+\\left(\\dfrac{1}{1-x^2}-(2-b)(1+b)\\right)g+2\\sqrt{1-x^2}\\,f^{\\prime }\\right],\\\\\\end{split}$ $\\begin{split}D_2(x)&=(1-x^2)^{-b/2}\\left[(1-x^2)F^{\\prime \\prime }-2(1-b)x\\,F^{\\prime }-\\dfrac{1-b^2}{1-x^2}\\,F-2b\\dfrac{x}{\\sqrt{1-x^2}}\\,G-2\\sqrt{1-x^2}\\,G^{\\prime }\\right]\\\\&=(1-x^2)^{-b/2}\\left[(1-x^2)f^{\\prime \\prime }-b(1-b)f-2\\dfrac{x}{\\sqrt{1-x^2}}\\,g-2\\sqrt{1-x^2}\\,g^{\\prime }\\right],\\\\\\end{split}$ $\\begin{split}D_3(x)&=(1-x^2)^{-b/2}\\left[(1-x^2)\\Omega ^{\\prime \\prime }-2(1-b)x\\,\\Omega ^{\\prime }-\\dfrac{1-b^2}{1-x^2}\\,\\Omega \\right]\\\\&=(1-x^2)^{-b/2}\\left[(1-x^2)\\omega ^{\\prime \\prime }-b(1-b)\\,\\omega \\right].\\end{split}$ The expressions $\\dot{C}_1+2b\\,C_2$ and $\\dot{D}_1+(1+b)D_2$ are then $\\begin{split}\\dot{C}_1+2b\\,C_2&=(1-x^2)^{-b}\\Big [2b\\dfrac{x}{\\sqrt{1-x^2}}\\left((1-b)F^2+b\\,G^2\\right)+2\\sqrt{1-x^2}\\left((1-b)FF^{\\prime }+b\\,GG^{\\prime }+\\Omega \\Omega ^{\\prime }\\right)\\\\&\\qquad +(1-x^2)\\left(F^{\\prime }G^{\\prime }+FG^{\\prime \\prime }\\right)+b\\dfrac{3-2b-2(1-2b)x^2}{1-x^2}FG+b\\,x\\,F^{\\prime }G-(1-3b)x\\,FG^{\\prime }\\Big ],\\\\=(1-&x^2)^{-1}\\Big [\\dfrac{2x}{\\sqrt{1-x^2}}\\left((1-b)f^2+b\\,g^2+(1-b)\\omega ^2\\right)+2\\sqrt{1-x^2}\\left((1-b)ff^{\\prime }+b\\,gg^{\\prime }+\\omega \\omega ^{\\prime }\\right)\\\\&\\qquad \\qquad +(1-x^2)\\left(f^{\\prime }g^{\\prime }+fg^{\\prime \\prime }\\right)+\\dfrac{1+2x^2+2b(1-b)(1-x^2)}{1-x^2}fg+x\\,f^{\\prime }g+2x\\,fg^{\\prime }\\Big ],\\end{split}$ and $\\begin{split}\\dot{D}_1+(1+b)\\,D_2&=(1-x^2)^{-(2+b)/2}\\Biggr [(1-b)(1-b^2-2b(1-x^2))F-b^2(4+b-2x^2)\\frac{x}{\\sqrt{1-x^2}}\\,G\\\\&\\qquad \\qquad \\qquad +2(1-b)^2x(1-x^2)F^{\\prime }+(2-4b-b^2-2(1-3b+b^2)x^2)\\sqrt{1-x^2}\\,G^{\\prime }\\\\&\\qquad \\qquad \\qquad -(1-x^2)^2\\left[(1-b)F^{\\prime \\prime }-(4-3b)\\frac{x}{\\sqrt{1-x^2}}\\,G^{\\prime \\prime }+\\sqrt{1-x^2}\\,G^{\\prime \\prime \\prime }\\right]\\Biggr ]\\\\&=(1-x^2)^{-3/2}\\Biggr [-b(1-b^2)(1-x^2)f+(-3-b(1+b)(1-x^2))\\frac{x}{\\sqrt{1-x^2}}\\,g\\\\&\\qquad \\qquad \\qquad \\qquad -(1+b(1+b)(1-x^2))\\sqrt{1-x^2}\\,g^{\\prime }\\\\&\\qquad \\qquad \\qquad \\qquad -(1-x^2)^2\\left[(1-b)f^{\\prime \\prime }-\\frac{x}{\\sqrt{1-x^2}}\\,g^{\\prime \\prime }+\\sqrt{1-x^2}\\,g^{\\prime \\prime \\prime }\\right]\\Biggr ].\\end{split}$ Finally, substituting (REF ) and (REF ) into the continuity equation (REF ), we get $\\begin{split}R^{-(1+b)}(1-x^2)^{-b/2}\\left[(2-b)G-\\left(\\sqrt{1-x^2}\\,F^{\\prime }-(1-b)\\dfrac{x}{\\sqrt{1-x^2}}\\,F\\right)\\right]=0,\\\\R^{-(1+b)}(1-x^2)^{-1/2}\\left[(2-b)g-\\sqrt{1-x^2}\\,f^{\\prime }\\right]=0.\\end{split}$"
],
[
"Expressions $C_i$ and {{formula:34905f94-8940-47c8-9447-41a21f29f061}} for {{formula:07138389-9783-42dd-b5c8-6c021fc98069}}",
"In this case, we can use equations (REF ) and eliminate $G(x)$ and $g(x)$ from the expressions in the previous section.",
"We have $\\begin{split}C_3&=(1-x^2)^{1/2-b}\\left[\\frac{1-b}{2-b}\\,\\Omega (x)\\left(F^{\\prime }(x)+\\frac{x}{1-x^2}\\,F(x)\\right)-F(x)\\Omega ^{\\prime }(x)\\right]\\\\&=(1-x^2)^{-1/2}\\left[\\frac{1-b}{2-b}f^{\\prime }(x)\\omega (x)-f(x)\\omega ^{\\prime }(x)\\right],\\end{split}$ $\\begin{split}D_3&=(1-x^2)^{-1-b/2}\\left[(1-x^2)\\Omega ^{\\prime \\prime }(x)-2(1-b)x(1-x^2)\\Omega ^{\\prime }(x)-(1-b^2)\\Omega (x)\\right]\\\\&=(1-x^2)^{-1/2}\\left[(1-x^2)\\omega ^{\\prime \\prime }(x)-b(1-b)\\omega (x)\\right],\\end{split}$ $\\begin{split}\\dot{C}_1+2b\\,C_2=\\frac{(1-x^2)^{1/2-b}}{(2-b)}&\\Biggr [(1-x^2)\\left(\\frac{2+b}{2-b}F^{\\prime }(x)F^{\\prime \\prime }(x)+F(x)F^{\\prime \\prime \\prime }(x)\\right)+2(2-b)\\Omega (x)\\Omega ^{\\prime }(x)\\\\&\\quad -2\\,\\frac{1-b}{2-b}\\biggr [\\frac{2x(1+bx^2)}{(1-x^2)^2}F^2(x)+\\frac{b+(2+3b)x^2}{1-x^2}F(x)F^{\\prime }(x)\\\\&\\qquad \\qquad \\qquad +(2+b)x(F^{\\prime }(x))^2+(4-b)xF(x)F^{\\prime \\prime }(x)\\biggr ]\\Biggr ]\\\\=\\frac{(1-x^2)^{-1/2}}{2-b}&\\biggr [(1-x^2)\\left(\\frac{2+b}{2-b}f^{\\prime }(x)f^{\\prime \\prime }(x)+f(x)f^{\\prime \\prime \\prime }(x)\\right)+2(2-b)\\omega (x)\\omega ^{\\prime }(x)\\\\&\\quad +2(1-b)\\left[(2-b)\\frac{x}{1-x^2}\\left(f^2(x)+\\omega ^2(x)\\right)+2f(x)f^{\\prime }(x)\\right]\\biggr ],\\end{split}$ and $\\begin{split}\\dot{D}_1+(1+b)D_2=-\\frac{(1-x^2)^{-2-b/2}}{2-b}&\\biggr [(1-x^2)^4F^{(4)}(x)-4(2-b)x(1-x^2)^3F^{\\prime \\prime \\prime }(x)\\\\&\\quad -2(1-b)(3+b-2(3-b)x^2)(1-x^2)^2F^{\\prime \\prime }(x)\\\\&\\quad -4b(1-b)(2+b-x^2)x(1-x^2)F^{\\prime }(x)\\\\&\\quad -(1-b)(3-b(1-b-b^2-4(3+b)x^2+4x^4))F(x)\\biggr ]\\\\=-\\frac{(1-x^2)^{-1/2}}{2-b}&\\biggr [(1-x^2)^2f^{(4)}(x)-4x(1-x^2)f^{\\prime \\prime \\prime }(x)\\\\&-2b(1-b)(1-x^2)f^{\\prime \\prime }(x)+b(1-b^2)(2-b)f(x)\\biggr ].\\end{split}$"
],
[
"Equations (",
"When $b\\ne 2$ , we can use expressions (REF ) and substitute them into equations (REF ) to get, in terms of $F$ and $\\Omega $ , $F\\Omega ^{\\prime }=\\frac{1-b}{2-b}\\left[F^{\\prime }+\\frac{x}{1-x^2}\\,F\\right]\\Omega ,$ $(1-x^2)^2\\Omega ^{\\prime \\prime }-2(1-b)x(1-x^2)\\Omega ^{\\prime }-(1-b^2)\\Omega =0,$ $\\begin{split}2(2-b)&\\Omega \\Omega ^{\\prime }+(1-x^2)\\left[\\frac{2+b}{2-b}\\,F^{\\prime }F^{\\prime \\prime }+FF^{\\prime \\prime \\prime }\\right]\\\\&\\quad =2\\,\\frac{1-b}{2-b}\\left[2\\frac{x(1+bx^2)}{(1-x^2)^2}\\,F^2+\\frac{b+(2+3b)x^2}{1-x^2}\\,FF^{\\prime }+(2+b)x(F^{\\prime })^2+(4-b)xF(x)F^{\\prime \\prime }\\right],\\end{split}$ $\\begin{split}(1-x^2)^4&F^{(4)}-4(2-b)x(1-x^2)^3F^{\\prime \\prime \\prime }-(1-b)\\Big [2(3+b-2(3-b)x^2)(1-x^2)^2F^{\\prime \\prime }\\\\&\\quad +4bx(2+b-x^2)(1-x^2)F^{\\prime }+(3-b+b^2+b^3+4b(3+b)x^2-4bx^4)F\\Big ]=0,\\end{split}$ or, in terms of $f$ and $\\omega $ , $f(x)\\omega ^{\\prime }(x)=\\frac{1-b}{2-b}f^{\\prime }(x)\\omega (x),$ $(1-x^2)\\omega ^{\\prime \\prime }(x)-b(1-b)\\omega (x)=0,$ $(1-x^2)\\left[\\frac{2+b}{2-b}\\,f^{\\prime }f^{\\prime \\prime }+ff^{\\prime \\prime \\prime }\\right]+4(1-b)ff^{\\prime }+2(1-b)(2-b)\\frac{x}{1-x^2}\\left(f^2+\\omega ^2\\right)+2(2-b)\\omega \\omega ^{\\prime }=0,$ $(1-x^2)^2f^{(4)}-4x(1-x^2)f^{\\prime \\prime \\prime }-2b(1-b)(1-x^2)f^{\\prime \\prime }+b(1-b^2)(2-b)f=0.$"
]
] | 1204.0835 |
[
[
"The Horizontal Component of Photospheric Plasma Flows During the\n Emergence of Active Regions on the Sun"
],
[
"Abstract The dynamics of horizontal plasma flows during the first hours of the emergence of active region magnetic flux in the solar photosphere have been analyzed using SOHO/MDI data.",
"Four active regions emerging near the solar limb have been considered.",
"It has been found that extended regions of Doppler velocities with different signs are formed in the first hours of the magnetic flux emergence in the horizontal velocity field.",
"The flows observed are directly connected with the emerging magnetic flux; they form at the beginning of the emergence of active regions and are present for a few hours.",
"The Doppler velocities of flows observed increase gradually and reach their peak values 4-12 hours after the start of the magnetic flux emergence.",
"The peak values of the mean (inside the +/-500 m/s isolines) and maximum Doppler velocities are 800-970 m/s and 1410-1700 m/s, respectively.",
"The Doppler velocities observed substantially exceed the separation velocities of the photospheric magnetic flux outer boundaries.",
"The asymmetry was detected between velocity structures of leading and following polarities.",
"Doppler velocity structures located in a region of leading magnetic polarity are more powerful and exist longer than those in regions of following polarity.",
"The Doppler velocity asymmetry between the velocity structures of opposite sign reaches its peak values soon after the emergence begins and then gradually drops within 7-12 hours.",
"The peak values of asymmetry for the mean and maximal Doppler velocities reach 240-460 m/s and 710-940 m/s, respectively.",
"An interpretation of the observable flow of photospheric plasma is given."
],
[
"Introduction",
"According to measurements of Doppler velocities during the emergence of active regions in the solar photosphere, the presence of upflows at the tops [5], [6], [42], [29], [40], [27], [19], [16], [17] and downflows of plasma at the footpoints [14], [15], [22], [2], [47], [5], [6], [7], [29], [39], [28], [46] of the emerging magnetic loops is well-established.",
"Uptil now horizontal photospheric velocities accompanying the emergence of active regions have only been measured indirectly.",
"fra72 studied a young active region and found that the photospheric magnetic field knots (21 events) moved along the arch filament system (AFS) at velocities of 0.1 – 0.4 .",
"Using observations of the young active region for 6.5 hours, sch73 found that magnetic elements moved in random directions with velocities of 0.4 – 1.0 .",
"bar90 measured the separation velocities of the opposite polarities for 45 bipolar pairs in the young active region and obtained velocity values up to 0.5 – 3.5 , decreasing with time.",
"str99 considered the emerging magnetic flux in a growing active region and found that the footpoints of the magnetic loops separated at an average velocity of 1.4 .",
"gri09 estimated the separation velocities of the photospheric magnetic flux outer boundaries in the NOAA 10488 active region.",
"The velocities decreased as the magnetic flux emerged: they were 2 – 2.5 at the end of the first hour and 0.3 in two and a half hours.",
"High values of horizontal photospheric velocities were obtained during the emergence of ephemeral active regions, except for the work of cho87.",
"har73 found that during the first 30 minutes of the emergence of ephemeral active regions the footpoints of magnetic loops separated at 5 ; the divergence velocity decreased to 0.7 – 1.3 over the next six hours and continued to decrease later on.",
"cho87 obtained low separation velocities of opposite polarities of 0.2 – 1 for 24 emerging ephemeral active regions.",
"hag01 found that the outer boundaries of the ephemeral regions expand with velocities up to 5.5 ; there is a tendency for the velocities to decrease with time.",
"ots11 studied 101 emerging flux regions of different spatial scales.",
"They found that the separation velocities of opposite polarities are lower than 1 for large emerging magnetic fluxes, whereas they reach 4 in small-scale ones.",
"Interesting results were obtained from the analysis of horizontal flows in the emerging active regions by granular motion.",
"str96 found large-scale horizontal divergent granular flows in a growing active region which were comparable with the common drift of magnetic polarities.",
"The authors interpreted this fact as a close interaction between granulation and magnetic fields.",
"koz05 and koz06 also found divergent flows located between the footpoints of emerging flux loops.",
"These authors consider these flows to be either convective flows which may trigger magnetic flux emergence from deep layers.",
"They also suggest that the observed flows are formed by the emergence of magnetic flux.",
"Note that the articles listed above concerned developing active regions already containing pores.",
"The present investigation involves an analysis of photospheric Doppler velocities for active regions emerging near the limb.",
"This subject was considered earlier by khl11."
],
[
"Data Analysis",
"We used full-disk solar magnetograms and Dopplergrams in the photospheric line Ni i 6768 Å and continuum images obtained on board the Solar and Heliospheric Observatory (SOHO) using the Michelson Doppler Imager (MDI) [34].",
"The temporal resolution of the magnetograms and Dopplergrams is 1 minute, while that of the continuum is 96 minutes.",
"The spatial resolution of the data is 4$^{\\prime \\prime }$ , and the pixel size is approximately 2$^{\\prime \\prime }$ .",
"We have cropped a region of emerging magnetic flux from a time sequence of data, taking into account its displacement caused by solar rotation.",
"The approximate displacement of the region was calculated by the differential rotation law for photospheric magnetic fields [38].",
"The exact tracking was performed by applying a cross-correlation analysis to two magnetograms adjacent in time.",
"A precise spatial superposition of the data was achieved by cropping fragments with identical coordinates from simultaneously acquired magnetograms, Dopplergrams, and continuum images.",
"For the Dopplergrams we applied a moving average over five images to reduce a contribution of five-minute oscillations to the velocity signal.",
"The solar differential rotation and other factors distorting the Doppler velocity signal were removed by using the following technique.",
"We averaged three upper and three lower rows of the cropped region and linearly smoothed the averaged rows, thus obtaining upper and lower rows of the array of solar rotation velocities.",
"To obtain the internal field of the array, we performed a linear interpolation between the upper and lower pixels of the columns belonging to the smoothed rows.",
"The obtained array of solar rotation velocities was subtracted from the original one.",
"The parameters under study were calculated in the region of the emerging magnetic flux.",
"The boundary of the emerging flux region was visually inspected.",
"For active regions we determined the following.",
"$\\theta $ is the heliocentric angle characterizing the distance from the solar disk center to the location of active region emergence; it is also approximately the angle between the normal to the surface and the line of sight to the emerging magnetic flux: $\\theta = \\arcsin (r/R),$ where $r$ is the distance from the solar disk center to the location of active region emergence and $R$ is the solar radius.",
"$\\Phi _{max}$ is the total unsigned magnetic flux at the maximum development of the active region which was measured inside isolines $\\pm $ 60 G, taking into account the projection effect and supposing that the magnetic field vector is perpendicular to the solar surface: $\\Phi _{max} = S_{0} \\sum _{i=1}^N \\frac{|B_{i}|}{cos \\theta _{i}},\\ $ where $N$ is the number of pixels with $|B_{i}|$ $>$ 60 G, $S_{0}$ is the area of the solar surface of the pixel in the center of the solar disk in $cm^{2}$ , $B_{i}$ is the line of sight magnetic field strength of the $i-th$ pixel in $G$ , and $\\theta _{i}$ is the heliocentric angle of the $i-th$ pixel.",
"d$\\Phi $ /dt is the total unsigned magnetic flux growth rate in the first 12 hours of the active region emergence.",
"$V_{mean-}$ and $V_{mean+}$ are the peak values of the mean negative and positive Doppler velocities inside the isoline, $-$ 500 and $+$ 500 , during the period considered.",
"The isoline of 500 was selected because it outlines the observable Doppler velocity structures well and the main contribution of convection flow is below this level.",
"The Doppler velocity values did not always exceed 500 in the calculation region, so the mean velocities were not determined for these instants.",
"$V_{max-}$ and $V_{max+}$ are the peak values of absolute maximum negative and positive Doppler velocities during the period considered.",
"$V_{sep}$ is a mean relative separation velocity between the two outer boundaries of the photospheric magnetic flux of opposite polarities.",
"It is calculated over a two-hour period at the peak values of the Doppler velocities, taking into account the projection effect: $\\overrightarrow{V}_{sep} = \\frac{1}{2 (T_{2}-T_{1})} \\left( \\frac{L_{2}}{cos \\theta _{2}} - \\frac{L_{1}}{cos \\theta _{1}} \\right),$ where $T_{2}-T_{1}$ is period of time under consideration in $s$ , $L_{1}$ and $L_{2}$ are the distances between the two outer boundaries of the photospheric magnetic flux in the image plane for points of time $T_{1}$ and $T_{2}$ in $m$ , and $\\theta _{1}$ and $\\theta _{2}$ are the heliocentric angles corresponding to the active region position in points of time $T_{1}$ and $T_{2}$ .",
"$V_{sep}$ contains the contribution of the horizontal expansion velocities of the emerging magnetic flux $V_{exp}$ and the magnetic polarity displacement due to the geometry of the rising magnetic loop $V_{foot}$ .",
"The contribution of the magnetic flux expansion $V_{exp}$ will essentially exceed the displacement of the magnetic polarities $V_{foot}$ at the very beginning of the active region appearance.",
"The displacement of the magnetic polarities resulting from vertical emergence of the magnetic loop $V_{foot}$ will not give a Doppler shift.",
"Thus, the contribution of the magnetic flux expansion $V_{exp}$ to the Doppler velocity signal will be calculated as $V_{exp} \\lesssim \\overrightarrow{V}_{sep} sin \\theta ,$ Table: Active regions studied"
],
[
"The Investigated Active Regions",
"Four active regions emerging near the solar limb have been considered (see Figures , , , and ).",
"The continuum images show that in the first hours of the emergence of active regions there are only pores.",
"In the first hours of appearance, active regions have a high magnetic flux growth rate (Table REF ).",
"Because of the projection effect of the magnetic field vector to the line of sight, the emergence of active regions begins with the appearance of one, then another magnetic polarity.",
"The boundary where the magnetic field changes sign is not the polarity inversion line (the upper-row panel in Figures , , , and ).",
"The polarity inversion line location can be indirectly estimated by the pore positions.",
"They arise in both polarities of the studied active regions (the bottom-row panel in Figures , , , and ).",
"The polarity inversion line will pass in the middle between the leading and following pores with a small shift towards the following pore at the beginning of the magnetic flux emergence due to the geometrical asymmetry of the emerging magnetic flux (Figure 4a of dri90).",
"Thus, the observable boundary where the magnetic field changes sign lies in the polarity located closer to the solar disk center.",
"In the active regions, the axis connecting opposite magnetic polarities rotates as the magnetic flux emerges (for an in-depth analysis see luo11).",
"Figure: NO_CAPTION The active region NOAA 9037 emerges on 10 Jun 2000 at N21 E59.",
"On the magnetograms (isolines $\\pm $ 60, 100, 150, 300 G), Dopplergrams, and continuum the isolines of Doppler velocities are superimposed.",
"The blue isoline corresponds to $-$ 500,$-$ 1000 – plasma motion towards the observer; the red isoline corresponds to $+$ 500,$+$ 1000 – plasma motion away from the observer.",
"Forming Doppler velocity structures are marked by an ellipse.",
"The orientation of the images is shown in the upper left corner; the direction from the solar disk center to the emerging magnetic flux is marked by a white arrow.",
"The black straight line on the upper-row panels marks the location of the slice of the time slice diagrams.",
"Figure: NO_CAPTION Active region NOAA 8536 emerges on 6 May 1999 at S24 E65.",
"The marking convention is the same as in Figure .",
"Figure: NO_CAPTION Active region NOAA 8635 emerges on 14 Jul 1999 at N42 W47.",
"The marking convention is the same as in Figure .",
"Figure: NO_CAPTION Active region NOAA 9064 emerges on 26 Jun 2000 at S21 W46.",
"The marking convention is the same as in Figure ."
],
[
"Doppler Velocity Structures",
"The SOHO/MDI data have a low spatial resolution which enables one to observe large-scale flows of plasma.",
"The active regions under consideration emerge in different sectors of the solar disk, but the morphology of Doppler velocity structures during the first hours of magnetic flux emergence is similar.",
"An amplification of negative Doppler velocities (plasma motion towards the observer) is observed on the boundary where the magnetic field changes sign located at the disk-side polarity (closer to the solar disk center) because of the projection effect of the magnetic field vector to the line of sight and of positive Doppler velocities (plasma motion away from the observer) in the limb-side polarity (closer to the solar limb) (Figures , , , and ).",
"High Doppler velocities inside the $\\pm $ 1000 isolines occupy significant areas (up to 50% of the Doppler velocity structure area inside the 500 isoline within individual time intervals) that are localized in the central part of Doppler velocity structure and exist for a long time: for NOAA 9037 on June 10 at 17:36 and 19:12 UT (Figure ); for NOAA 8536 on May 6 at 11:09 UT (Figure ); for NOAA 8635 on July 14 at 14:24 and 17:36 UT (Figure ); for NOAA 9064 on June 26 at 13:55 and 14:24 UT (Figure ).",
"Time slice diagrams for magnetic field strength and Doppler velocities were constructed in the first hours of emergence of active regions (panels (a) and (b) in Figures REF , REF , REF , and REF ).",
"The slices are orientated along the axis of the emerging bipolar pairs, and their position is marked by a black line on the magnetograms in Figures , , , and .",
"During the emergence of active regions a slight rotation of the axis of dipoles is observed; therefore, at the very beginning of magnetic flux emergence both polarities with Doppler velocity structures do not always fall into the slices.",
"The time slice diagrams clearly show that the Doppler velocity structures are located within the limits of emerging magnetic flux, and that they are not present in the surrounding regions (Figures REF a, REF a, REF a, REF a).",
"The Doppler velocity structures form at the beginning of magnetic flux emergence, occupy an extensive region, and persist for a few hours (Figures REF b, REF b, REF b, REF b and Table REF ).",
"Regions of Doppler velocities of different signs do not appear at the same time.",
"In the first hours of magnetic flux emergence they are adjacent to each other; they then separate together with the opposite polarities.",
"The Doppler velocity values increase gradually and reach their peak values 4 – 12 hours after the start of the emergence of active regions, approximately at half of the velocity structure's life time (Figures REF d, REF d, REF d, REF d).",
"The peak values of mean Doppler velocity inside the $\\pm $ 500 isolines are 800 – 970 , and the peak values of maximum Doppler velocities reach 1410 – 1700 (Table REF ).",
"The mean Doppler velocities show the presence of velocity structures rather weakly.",
"At the same time, as noted above, the Doppler velocities inside the $\\pm $ 1000 isolines, within individual time intervals, occupy significant areas which exist for a long time.",
"The peak values of the Doppler velocities accompanying the emergence of active regions (Table REF ) substantially exceed the maximum Doppler velocities of the convective flow of the quiet Sun, which, in the SOHO/MDI data, do not exceed 1200 (the points with $\\theta >50^{\\circ }$ in Figure 3 b of khl11).",
"We calculated the separation velocities of the photospheric magnetic flux outer boundaries $V_{sep}$ , and estimated the contribution of magnetic flux expansion to the Doppler velocity signal $V_{exp}$ (Table REF ).",
"From the comparison in Table REF and Table REF one can see that the Doppler velocity values observed substantially exceed the magnetic flux expansion velocity $V_{exp}$ .",
"Previews of other active regions emerging near the limb showed that these powerful and long-lived Doppler velocity structures do not always appear.",
"They seem to be present only in events with a high flux growth rate.",
"khl12 has carried out a statistical investigation of 54 active regions with different spatial scales (total unsigned magnetic flux $8\\times 10^{19}$ – $5\\times 10^{22}$ Mx) emerging near the limb.",
"It was found that the peak values of negative and positive Doppler velocities are related quadratically to the magnetic flux growth rate in the first hours of the emergence of active regions (Figure 6a of khl12).",
"Figure: Active region NOAA 9037: time slice diagrams of (a) magnetic field strength and(b) Doppler velocities.",
"The location of the slice is marked on the magnetogramsin Figure by a black line.",
"The blue and red isolines correspond to--500, --1000 and ++500, ++1000 , and the Doppler velocity structuresanalyzed are marked by arrows; (c) time variation of the total unsigned magneticflux; (d) time variation of mean (thick line) and absolut maximum (thin line)values of negative and positive Doppler velocities in the region of emergingmagnetic flux.",
"The vertical dotted line marks the time of the beginning ofmagnetic flux emergence.Figure: Active region NOAA 8536: time slice diagrams of (a) magnetic field strength and(b) Doppler velocities.",
"The location of the slice is marked on the magnetogramsin Figure by a black line.",
"The marking convention is the same asin Figure .Figure: Active region NOAA 8635: time slice diagrams of (a) magnetic field strength and(b) Doppler velocities.",
"The location of the slice is marked on the magnetogramsin Figure by a black line.",
"The marking convention is the same asin Figure .Figure: Active region NOAA 9064: time slice diagrams of (a) magnetic field strength and(b) Doppler velocities.",
"The location of the slice is marked on the magnetogramsin Figure by a black line.",
"The marking convention is the same asin Figure .Table: Peak values of mean and absolut maximum Doppler velocities(V mean- V_{mean-}, V max- V_{max-}, V mean+ V_{mean+}, V max+ V_{max+}), size (dd), and life time(tt) of the velocity structures during the first hours of the emergence ofactive regionsTable: The mean relative separation velocities of the photospheric magnetic flux outer boundaries (V → sep \\overrightarrow{V}_{sep}), the contribution of the magnetic flux expansion to the line of sight signal of Doppler velocity (V exp V_{exp}), and the periods of time (Time intervals) for which V → sep \\overrightarrow{V}_{sep} and V exp V_{exp} were determined"
],
[
"Geometrical Asymmetry of Emerging Magnetic Flux",
"During the appearance of active regions it is possible to see the well-known geometrical asymmetry of emerging magnetic flux (dri90 and references therein).",
"It appears that sunspots of leading polarity move away from the location of the emergence much faster than those of following polarity do.",
"In the time slice diagrams of the magnetic field of the active regions under consideration, the following polarity is situated at the bottom, and the leading one at the top (Figures REF a, REF a, REF a, REF a).",
"In NOAA 9037, soon after emergence, the following (negative) polarity is set against the existing concentration of the positive magnetic field and stops drifting, while the leading polarity moves along the slice at high velocity (Figure REF a).",
"NOAA 8536 emerges before an existing concentration of negative magnetic field.",
"Therefore, the following polarity moves faster than the leading one in the first hours of emergence, but then the leading polarity begins to move with greater velocity (Figure REF a).",
"In NOAA 8635 and 9064, emerging at the W-limb, it is clearly seen that the leading polarity moves considerably faster than the following one (Figures REF a and REF a).",
"Figure: The time variation for the difference between the positive and negative Dopplervelocities for mean (thick line) and maximal (thin line) values in the studiedactive regions.",
"The polarities of the locations of the positive and negativeDoppler velocities are marked on the plots."
],
[
"Asymmetry in the Doppler Velocity Fields",
"Asymmetry is also observed in the Doppler velocity fields.",
"In the time slice diagrams in Figures REF a, REF a, REF a, REF a, one can clearly see that the Doppler velocity structures located at the region of the leading magnetic polarity are more powerful and exist longer than in the following one.",
"We performed a quantitative analysis for the Doppler velocity asymmetry.",
"Figures REF presents the plots of time variation for the difference between the positive and negative Doppler velocities for mean and maximal values.",
"There are some blanks in the mean Doppler velocity asymmetry for those time instants when the velocities in the region of emerging magnetic flux were less than 500 .",
"For active regions NOAA 8536 and 8635 emerging near different limbs, a well-defined dominance of the Doppler velocities located in the leading polarity is observed during the first 12 hours of the emergence.",
"In two other active regions, NOAA 9037 and NOAA 9064, also emerging near different limbs, the asymmetry value changes its sign.",
"The NOAA 9037 appearance starts with the emergence of a magnetic loop whose axis is oriented almost perpendicularly to the line of sight.",
"Its rise is accompanied by a dominance of the negative Doppler velocity.",
"Approximately from 09:00 UT, a magnetic flux emergence appears whose axis is oriented along the line of sight.",
"Against this background, one observes a significant dominance of the positive Doppler velocities localized in the following polarity.",
"In the NOAA 9064 early emergence, the mean Doppler velocity values show practically no asymmetry, but one observes a dominance of the maximal negative Doppler velocities corresponding to the following polarity.",
"Approximately three hours after the start of emergence, the asymmetry value changes sign, and a noticeable dominance of the positive Doppler velocity localized in the leading polarity begins.",
"In all the active regions under consideration, the Doppler velocity asymmetry reaches its peak value soon after the start of magnetic flux emergence, and then gradually drops within 7 – 12 hours.",
"The Doppler velocity asymmetry decreasing time is comparable to the lifetime of the velocity structures.",
"The peak values of asymmetry for the mean and maximal Doppler velocities reach 240 – 460 and 710 – 940 , respectively."
],
[
"Interpretation of the Results",
"Let us consider the contribution of possible motions to the line of sight Doppler velocity signal in data with low spatial resolution (Figure REF a).",
"i) The magnetic flux rise velocity $\\overrightarrow{V}_{up}$ .",
"$\\overrightarrow{V}_{up}$ should have a similar contribution in the leading and following polarities.",
"When active regions emerge near the solar disk center, the maximum values of the upflow Doppler velocities reach $\\overrightarrow{V}_{up}\\sim $ 300 – 1000 (Figure 4 of khl12 and references in Section 1 of this paper).",
"The projection of these velocities to the line of sight at the heliocentric angle $\\theta $ = $60^\\circ $ will be $V_{up} \\lesssim $ 150 – 500 .",
"ii) The plasma downflow velocity $\\overrightarrow{V}_{down}$ .",
"In the active regions emerging near the solar disk center, at the footpoints of the magnetic loops, one observes positive Doppler velocities interpreted as a downflow of plasma being carried out into the solar atmosphere (see references in Section 1 of this paper).",
"The plasma downflow takes place along the magnetic field lines.",
"Theoretical models show that the velocities of draining plasma have a significant horizontal component at the beginning of the magnetic flux emergence (e.g., shi90, arch04, tor10, tor11).",
"iii) The magnetic flux horizontal expansion velocity $\\overrightarrow{V}_{exp}$ .",
"$\\overrightarrow{V}_{exp}$ should have a similar contribution in the leading and the following polarities under the condition of equality of their gas pressure.",
"From the theory of magnetic flux emergence into the solar atmosphere, an inverse relation exists between the expansion velocities and degree of twist of the emerging magnetic flux (see, for example, mur06).",
"che10 showed that an emerging tube with a total toroidal flux content of $7.6\\times 10^{21}$ Mx creates a transient pressure excess that results in diverging horizontal flows with velocities over 3 for approximately five hours.",
"In this case, the plasma flow velocities are comparable to the separation velocities of the photospheric magnetic flux outer boundaries.",
"The observations show that at the very beginning of the emergence the magnetic flux expansion has a maximal velocity that then drops rapidly with time.",
"Even at the emergence of the powerful active region NOAA 10488, with a total unsigned magnetic flux at the maximum evolution of $>6\\times 10^{22}$ Mx and a total unsigned magnetic flux growth rate during the first hours of $4.1\\times 10^{20}$ Mx h$^{-1}$ , the separation velocities of the photospheric magnetic flux outer boundaries had already dropped to 0.3 two hours after the beginning of the emergence [17].",
"Figure: (a) The simple scheme of the plasma flows accompanying the emergence ofthe magnetic flux near the W-limb.",
"The velocity vectors of the possibleflows and their projection to the line of sight in the leading and followingpolarities are marked by arrows.",
"V up V_{up} is the magnetic fluxrise velocity; V down V_{down} is the plasma downflow velocity being carriedout into the solar atmosphere by the emerging magnetic flux; V exp V_{exp}is the magnetic flux horizontal expansion velocity; V dir V_{dir} is the plasmadirectional flow velocity inside the emerging magnetic structure (in this casefrom the following polarity to the leading one).",
"(b) The compositionalresult of the velocity components along the line of sight without the plasmadirectional flow V dir V_{dir}.",
"(c) The compositional result of the velocitycomponents along the line of sight taking into account the plasma directionalflow V dir V_{dir}.iv) It is possible that plasma directional flows exist inside the emerging magnetic structure $\\overrightarrow{V}_{dir}$ .",
"Directional flows, as well as the plasma downflow $\\overrightarrow{V}_{down}$ , take place along the magnetic field lines.",
"Searches for such flows in the Doppler velocity asymmetry between the leading and following polarities in the active regions emerging near the solar disk center did not provide consistent results [8], [37], [9], [33], [4], [18].",
"Plasma flows are theoretically expected to be directed from the leading polarity into the following one.",
"The thin flux tube theory shows that these flows are due to the Coriolis force acting on the emerging magnetic structure, and the velocities of these flows reach some hundreds of meters per second (for a detailed analysis, see fan09).",
"However, when entering the convection zone high layers (20 – 30 Mm under the photosphere), the magnetic flux undergoes a strong expansion and fragmentation.",
"Therefore, the probability of conservation of the flows caused by the Coriolis force is not known.",
"The models of the magnetic flux emergence from the near-surface layers into the solar atmosphere do not consider the existence of these flows.",
"It is also possible that plasma flows occur from the following polarity into the leading one.",
"A discussion of the mechanisms leading to their origin may be found in cau96.",
"Thus, the line of sight Doppler velocity signal in data with low spatial resolution contains contributions from the following flows (Figure REF a): $\\overrightarrow{V} = \\overrightarrow{V}_{up} + \\overrightarrow{V}_{down} + \\overrightarrow{V}_{exp} + \\overrightarrow{V}_{dir},$ Their projections onto the line of sight in the positive and negative Doppler velocity structures are determined as: $V_{-} = +\\overrightarrow{V}_{up} cos \\theta + \\overrightarrow{V}_{down} cos X_{1} + \\overrightarrow{V}_{exp} sin \\theta \\pm \\overrightarrow{V}_{dir} cos X_{1},$ $V_{+} = -\\overrightarrow{V}_{up} cos \\theta + \\overrightarrow{V}_{down} cos X_{2} + \\overrightarrow{V}_{exp} sin \\theta \\pm \\overrightarrow{V}_{dir} cos X_{2},$ where $\\theta $ is the heliocentric angle, and $X_{1}$ and $X_{2}$ are the angles between the line of sight and the magnetic field lines in the polarities where the negative and positive velocity structures are localized.",
"In the velocity structures accompanying the emergence of active regions under consideration, one observes high Doppler velocities that reach their peak values 4 – 12 hours after the magnetic flux emergence begins (Figures REF d, REF d, REF d, REF d).",
"Doppler velocities more than 1000 concentrate in the central part of the velocity structures, occupy significant areas, and exist over long time intervals (Figures , , , and ).",
"Let us discuss what the observed velocities are associated with, having considered the contribution of possible motions to the line of sight Doppler velocity signal at the heliocentric angle $\\theta =60^\\circ $ .",
"The maximal contribution of the magnetic flux rise velocities should be $V_{up}$ $\\sim $ 150 – 500 , with a mean contribution of 300 .",
"The contribution of the magnetic flux horizontal expansion velocities to our active regions is $V_{exp}$ $\\sim $ 130 – 660 (Table REF ).",
"The velocity of directional flow $\\overrightarrow{V}_{dir}$ will determine the value of the Doppler velocity asymmetry between the velocity structures of opposite sign and, for active regions NOAA 9037 and 8536, 12 hours after the start of emergence it has no contribution.",
"Thus, one can see that the high Doppler velocities are caused by the significant component of the plasma downflow velocities $\\overrightarrow{V}_{down}$ (Figure REF b).",
"The Doppler velocity asymmetry between velocity structures of the leading and following polarities reaches its peak values soon after the emergence begins, and then it gradually drops (Figure REF ).",
"In two active regions, NOAA 8536 and 8635, emerging near different limbs, one observes an explicit dominance of the Doppler velocities localized in the leading polarity.",
"In two other active regions, NOAA 9037 and 9064, also emerging near different limbs, the Doppler velocity asymmetry value changes sign, but, at individual time intervals, there is a Doppler velocity dominance in the following polarity.",
"There is probably a contribution from the plasma directional flow $V_{dir}$ (Figure REF c).",
"The Doppler velocity asymmetry may be caused by a morphological asymmetry of active regions that becomes apparent because the magnetic flux of the leading polarity is more compact than of the following one.",
"However, to better understand the causes for the asymmetry between Doppler velocity structures, one should search for some additional relations."
],
[
"Conclusion",
"Analysis of photospheric flows accompanying the emergence of active regions near the limb showed the existence of an extensive region of enhanced negative Doppler velocities on the boundary where the magnetic field changes sign, which is located in the polarity closer to the solar disk center because of the projection effect.",
"The region of positive Doppler velocities is adjacent to it in the magnetic polarity closer to the solar limb.",
"The observed flows form at the start of active region emergence and are present for a few hours.",
"The Doppler velocity values increase gradually and reach their peak values 4 – 12 hours after the start of the magnetic flux emergence.",
"The peak values of the mean (inside the $\\pm $ 500 isolines) and maximum Doppler velocities are 800 – 970 and 1410 – 1700 , respectively.",
"The Doppler velocity values observed substantially exceed the separation velocities of the photospheric magnetic flux outer boundaries and most likely are caused by a significant component of the plasma downflow velocities being carried out into the solar atmosphere by emerging magnetic flux.",
"An asymmetry was detected between the velocity structures of the leading and following polarities.",
"Doppler velocity structures located in a region of leading magnetic polarity are more powerful and exist longer than those in a region of following magnetic polarity.",
"The Doppler velocity asymmetry between velocity structures of the leading and following polarities reaches peak values soon after emergence begins and then gradually drops within 7 – 12 hours.",
"The peak values of asymmetry for the mean and maximal Doppler velocities reach 240 – 460 and 710 – 940 , respectively.",
"Additional investigations are necessary to understand the reasons for the Doppler velocity asymmetry.",
"However, it could be caused by the plasma directional flows inside the emerging magnetic structure or morphological asymmetry between the leading and following polarities.",
"The author expresses sincere gratitude to the Geuest Editor for useful comments.",
"The author is grateful to Prof. V.M.",
"Grigoriev, L.V.",
"Ermakova, and V.G.",
"Eselevich for important suggestions and help in understanding the obtained results.",
"This work used data obtained by the SOHO/MDI instrument.",
"SOHO is a mission of international cooperation between ESA and NASA.",
"The MDI is a project of the Stanford-Lockheed Institute for Space Research.",
"This study was supported by RFBR grants 10-02-00607-a, 10-02-00960-a, 11-02-00333-a, 12-02-00170-a, state contracts of the Ministry of Education and Science of the Russian Federation Nos.",
"02.740.11.0576 and 16.518.11.7065, the program of the Division of Physical Sciences of the Russian Academy of Sciences No.",
"16, the Integration Project of SB RAS No.",
"13, and the program of the Presidium of Russian Academy of Sciences No.",
"22."
]
] | 1204.1189 |
[
[
"The Role of Temperature in the occurrence of some Zeno Phenomena"
],
[
"Abstract Temperature can be responsible for strengthening effective couplings between quantum states, determining a hierarchy of interactions, and making it possible to establish such dynamical regimes known as Zeno dynamics, wherein a strong coupling can hinder the effects of a weak one.",
"The relevant physical mechanisms which connect the structure of a thermal state with the appearance of special dynamical regimes are analyzed in depth."
],
[
"Introduction",
"In the last decades a lot of attention has been devoted to the role of temperature in quantum information and in quantum mechanics in general.",
"In particular, the detrimental effects of an environment at finite or even high temperature have been studied in connection with many aspects of quantum mechanics.",
"Nevertheless, many studies aimed at singling out robustness of quantum features even in the presence of high temperature, which for instance is the case of thermal entanglement.",
"Recently, the connection between temperature and Quantum Zeno phenomena has also been explored.",
"The concept of quantum Zeno effect (QZE) — according to its original definition QZE consists in the inhibition of the dynamics of a physical system due to repeated “pulsed” measurements [1], which has been demonstrated in various systems [2], [3] — has been progressively generalized.",
"The first generalization consists in considering continuous measurements meant as dissipative processes [4], showing that a strong decay can be responsible for a dynamical decoupling that weakens the effects of other interactions [5], [6], [8], [7].",
"Another important generalization is related to the knowledge that even an additional coherent coupling can be responsible for diminishing the effects of other interactions, hence realizing the so called Zeno Dynamics [9], [10], [11] and eventually leading to the concept of Zeno subspaces [12].",
"Also pulsed interactions have been proven to be responsible for the formation of dynamically invariant subspaces [13], [14], [15].",
"Quantum Zeno effect has been exploited to develop applications in various fields, such as quantum information and nanotechnologies [16], [17], [18].",
"Moreover, its importance in connection with fundamental concepts of quantum mechanics has been widely discussed [19].",
"Nevertheless, Zeno phenomena are not confined to the realm of quantum mechanics, but, according to Peres, similar occurrences are possible even in classical mechanics [20].",
"The connection between Zeno phenomena and temperature has been investigated by various authors.",
"In 2002 Ruseckas analyzed the effects of thermalization of a measurement apparatus on the appearance and intensity of the quantum Zeno effect, showing that an higher temperature can amplify the power of pulsed measurements in inducing Zeno phenomena [21].",
"In 2006 Maniscalco et al have studied the crossover between QZE and AZE — Anti-Zeno effect is the acceleration of the dynamics induced by measurements, usually incoming for short-but-not-too-short time intervals between subsequent measurements — in connection with the temperature of the bath the system under scrutiny is interacting with, showing that the QZE-AZE border can be significantly affected by temperature [22].",
"Recently, Scala et al have analyzed Landau-Zener (LZ) transitions in the presence of environmental effects, even in the case of a bath at finite or high temperature [23].",
"The analysis, developed through a master equation approach beyond the secular approximation, has shown that temperature can be responsible for a dynamical decoupling, which eventually leads to a Zeno effect.",
"A subsequent study [24] developed in connection with a different physical situation — a simplified version of the previous problem, wherein a time-independent Hamiltonian has been considered in place of the LZ time-dependent coupling scheme — has shown that in some situations temperature can clearly contribute to induce a Zeno dynamics, weakening the effects of coherent couplings.",
"Starting from these results, one could wonder whether in those cases wherein temperature can influence more than one interactions it is possible to have temperature-dependent hierarchies of couplings, driving the appearance of Zeno behaviors.",
"Indeed, for instance, if two or more interactions are strengthened by temperature, occurrence of Zeno phenomena could be obstructed, because of the impossibility of finding a coupling which is much stronger than others; in other cases temperature can enhance inhibition of the dynamics.",
"In this paper we analyze the role of temperature when it affects more than one interactions, discussing appearance and disappearance of temperature-induced Zeno dynamics, and studying in depth the underlying physical mechanisms.",
"The paper is organized as follows.",
"In section we analyze the dynamics of a three-level system resonantly interacting with two harmonic oscillators initially prepared in a thermal state.",
"In section we extend the scheme by considering some detuning effects and show an interesting competition between detuning and temperature in determining Zeno regimes.",
"In section we give some preliminary results that pave the way to the analysis of the case where many oscillators are involved in the coupling scheme.",
"Finally, in section we give some conclusive remarks."
],
[
"Zeno dynamics induced by Temperature",
"Let us consider a three-state system interacting with two harmonic oscillators initially prepared in a thermal state.",
"The interaction with the oscillators can induce transitions from the upper state to the intermediate one and from the intermediate to the lowest.",
"Both interactions are assumed to be resonant (see Fig.",
"REF ).",
"From general statements for the appearance of Zeno dynamics [12], [13], [11], if the strength of the coupling between $\\left|2\\right\\rangle $ and $\\left|3\\right\\rangle $ is much greater than the strength of the other coupling, i.e.",
"if $|g_{23}| \\gg |g_{12}|$ , then transitions from $\\left|1\\right\\rangle $ to $\\left|2\\right\\rangle $ — and vice versa — are hindered.",
"Therefore, if the atomic system is initially prepared in the state $\\left|1\\right\\rangle $ then the system does not evolve significantly.",
"We will show that this behavior occurs when temperature is high even if the condition $|g_{23}| \\gg |g_{12}|$ is not fulfilled, provided the frequency of the oscillator assisting $2-3$ transitions is much smaller than the frequency of the oscillator which couples $\\left|1\\right\\rangle $ and $\\left|2\\right\\rangle $ ."
],
[
"The Hamiltonian Model",
"Consider the following Hamiltonian model ($\\hbar =1$ ): $\\nonumber && \\hat{H} = \\sum _k \\omega _k \\left|k\\right\\rangle \\left\\langle k\\right| + \\omega _a \\hat{a}^\\dag \\hat{a} + \\omega _b\\hat{b}^\\dag \\hat{b} \\\\\\nonumber &+& g_{12} (\\left|1\\right\\rangle \\left\\langle 2\\right|\\hat{a}+\\left|2\\right\\rangle \\left\\langle 1\\right|\\hat{a}^\\dag )+ g_{23} (\\left|2\\right\\rangle \\left\\langle 3\\right|\\hat{b}+\\left|3\\right\\rangle \\left\\langle 2\\right|\\hat{b}^\\dag )\\,.",
"\\\\$ Assuming complete resonance ($\\omega _1-\\omega _2=\\omega _a$ and $\\omega _2-\\omega _3=\\omega _b$ ), it clearly corresponds to the coupling scheme represented in Fig.",
"REF .",
"Figure: (Color online) The upper atomic level (corresponding tothe state 1\\left|1\\right\\rangle ) is resonantly coupled to the intermediatelevel (2\\left|2\\right\\rangle ), which in turn is resonantly coupled to thelowest level (3\\left|3\\right\\rangle ).The structure of the Hamiltonian is such that the total number of excitations, $\\hat{N} = 2\\left|1\\right\\rangle \\left\\langle 1\\right| + \\left|2\\right\\rangle \\left\\langle 2\\right| + \\hat{a}^\\dag \\hat{a} +\\hat{b}^\\dag \\hat{b}\\,,$ is a constant of the motion.",
"Therefore the Hilbert space is partitioned into invariant three-dimensional subspaces, each spanned by a set $\\left|1, n_a, n_b\\right\\rangle $ , $\\left|2, n_a+1, n_b\\right\\rangle $ and $\\left|3,n_a+1, n_b+1\\right\\rangle $ , and each one corresponding to a restriction of the Hamiltonian operator which has the following form: $\\nonumber H_{n_a n_b} &=& (n_a\\omega _a+n_b\\omega _b+\\omega _1)\\,\\mathbb {I}_3\\\\\\nonumber &\\!\\!\\!\\!\\!\\!+&\\left(\\begin{array}{ccc}0 & g_{12} \\sqrt{n_a+1} & 0 \\cr \\,\\, g_{12} \\sqrt{n_a+1} \\,\\, & 0 & \\,\\, g_{23} \\sqrt{n_b+1} \\,\\, \\cr 0 & g_{23} \\sqrt{n_b+1} & 0\\end{array}\\right)\\,.",
"\\\\$ where $\\mathbb {I}_3$ is the identity of the $3\\times 3$ matrix space.",
"The `restricted' operator $H_{n_a n_b}$ generates the `restricted' unitary evolution $U_{n_a n_b} = \\exp (-i tH_{n_a n_b})$ .",
"We now prove that when $g_{23}\\sqrt{n_b+1}\\gg g_{12}\\sqrt{n_a+1}$ the survival probability of the state $\\left|1, n_a, n_b\\right\\rangle $ is preserved.",
"Preliminarily, consider that one of the three eigenenergies associated to $H_{n_a n_b}$ is $n_a\\omega _a+n_b\\omega _b+\\omega _1$ — according to the analysis in the appendix , the matrix in the second line has a vanishing eigenvalue — and that the corresponding eigenstate is $\\nonumber \\left|n_a\\omega _a+n_b\\omega _b+\\omega _1\\right\\rangle &=& \\frac{g_{23}\\sqrt{n_b+1}}{g}\\left|1, n_a, n_b\\right\\rangle \\\\&& \\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!- \\frac{g_{12}\\sqrt{n_a+1}}{g}\\left|3, n_a+1, n_b+1\\right\\rangle \\,,$ with $g=\\left[g_{12}^2(n_a+1)+g_{23}^2(n_b+1)\\right]^{1/2}$ .",
"Now, given a positive number $\\varepsilon $ , provided $\\frac{g_{23}\\sqrt{n_b+1}}{g_{12}\\sqrt{n_a+1}} \\,\\ge \\,\\chi _\\varepsilon \\,\\equiv \\,\\sqrt{\\frac{1+(1-\\varepsilon )^{1/4}}{1-(1-\\varepsilon )^{1/4}}} \\,,$ one easily gets that if the initial state is $\\left|\\psi (0)\\right\\rangle =\\left|1, n_a, n_b\\right\\rangle $ then the population of such initial state is kept close to unity through all the evolution: $P_{1 n_a n_b}(t)&\\ge & \\sqrt{1-\\varepsilon }\\,,\\qquad \\forall t \\ge 0\\,,$ which can be proven on the basis of the analysis in appendix .",
"This result can be thought of as the occurrence of a Zeno dynamics, since the $2-3$ coupling hinders the dynamics induced by the $1-2$ interaction."
],
[
"Inhibition of Time Evolution",
"Let us now consider the system in the following initial state: $\\rho (0) = \\left|1\\right\\rangle \\left\\langle 1\\right| \\otimes \\rho _a \\otimes \\rho _b\\,,$ where $\\rho _k &=& \\mathcal {N}_k\\sum _n \\exp \\left({-\\,\\omega _k n /k_BT}\\right) \\left|n\\right\\rangle \\left\\langle n\\right|\\\\\\mathcal {N}_k &=& \\left[1-\\exp \\left({-\\,\\omega _k/k_B T}\\right)\\right]\\,,$ with $k=a,b$ , are the thermal states of the two oscillators.",
"The probability to find the three-state system in the initial state $\\left|1\\right\\rangle $ after a time $t$ , irrespectively of the number of excitations of the harmonic oscillators, is given by: $P_1(t)\\nonumber &=& \\mathcal {N}_a \\mathcal {N}_b\\sum _{n_a} \\sum _{n_b} \\,\\exp \\left(-\\frac{\\omega _a n_a + \\omega _b n_b}{k_B T} \\right)\\\\&\\times &\\left|\\left\\langle 1, n_a, n_b\\right| \\hat{U}_{n_a n_b}(t) \\left|1, n_a, n_b\\right\\rangle \\right|^2\\,.$ Condition in Eq.",
"(REF ) can be rewritten as $n_b \\ge \\tilde{n}_b(n_a, \\varepsilon )\\,,$ with $\\nonumber \\tilde{n}_b(n_a, \\varepsilon ) &=& \\frac{g_{12}^2}{g_{23}^2}\\, \\chi _\\varepsilon ^2 \\,n_a + \\frac{\\chi _\\varepsilon ^2 \\, g_{12}^2 - g_{23}^2}{g_{23}^2}\\\\ &\\equiv & \\alpha _\\varepsilon n_a +\\beta _\\varepsilon \\,.$ Since all the terms in the right-hand side of Eq.",
"(REF ) are nonnegative, one has $P_1(t)\\nonumber &\\ge & \\mathcal {N}_a \\mathcal {N}_b\\sum _{n_a} \\sum _{n_b^*} \\,\\exp \\left(-\\frac{\\omega _a n_a + \\omega _b n_b}{k_B T}\\right) \\\\&\\times &\\left|\\left\\langle 1, n_a, n_b\\right| \\hat{U}_{n_a n_b}(t) \\left|1, n_a, n_b\\right\\rangle \\right|^2\\,,$ where $n_b^*$ stands for $n_b \\ge \\tilde{n}_b(n_a,\\,\\varepsilon )$ .",
"By definition, $|\\left\\langle 1, n_a, n_b\\right| \\hat{U}_{n_an_b}(t) \\left|1, n_a, n_b\\right\\rangle |^2 \\ge \\sqrt{1-\\varepsilon }$ for any $n_b^*$ , and then one has: $P_1(t)\\nonumber &\\ge & \\mathcal {N}_a \\mathcal {N}_b\\sum _{n_a} \\sum _{n_b^*} \\,\\exp \\left(-\\frac{\\omega _a n_a + \\omega _b n_b}{k_B T}\\right)\\sqrt{1-\\varepsilon }\\nonumber \\,\\, = \\,\\, \\mathcal {N}_a \\sum _{n_a}\\, \\exp \\left[-\\frac{\\omega _a n_a + \\omega _b [[\\tilde{n}_b(n_a,\\,\\varepsilon )]]}{k_B T}\\right] \\, \\sqrt{1-\\varepsilon } \\\\\\nonumber &\\ge & \\mathcal {N}_a \\sum _{n_a}\\, \\exp \\left[-\\frac{\\omega _a n_a + \\omega _b [\\tilde{n}_b(n_a,\\,\\varepsilon )+1]}{k_B T}\\right] \\, \\sqrt{1-\\varepsilon } \\\\&=& \\exp \\left[-(\\beta _\\varepsilon +1)\\omega _b/(k_B T)\\right]\\frac{ 1 - \\exp \\left[-\\,\\omega _a/(k_B T)\\right] }{ 1 - \\exp \\left[-\\left(\\omega _a+\\alpha _\\varepsilon \\omega _b\\right) / (k_B T)\\right] }\\, \\sqrt{1-\\varepsilon } \\,,$ where the symbol $[[x]]$ indicates the first integer number larger than or equal to $x$ , and the relations $x+1\\ge [[x]]$ and $\\sum _{n=s}^{\\infty } e^{-nx}=e^{-sx}/(1-e^{-x})$ have been used.",
"For any given value of $\\varepsilon $ , positive and smaller than unity, in the limit of very high temperature, one can assume $\\exp [-(\\beta _\\varepsilon +1)\\omega _b/(k_B T)] \\approx 1$ and $1-\\exp [x/(k_B T)] \\approx x/(k_B T)$ , so that one gets $P_1(t) &\\gtrsim &\\frac{\\sqrt{1-\\varepsilon }}{1+\\eta \\,\\chi ^2_\\varepsilon }\\,,$ with $\\eta =\\frac{g_{12}^2}{g_{23}^2}\\frac{\\omega _b}{\\omega _a}\\,.$ Therefore, in the limit of high temperature the survival probability becomes higher and higher for any $t$ , approaching unity in the limit $\\eta \\rightarrow 0$ .",
"In fact, for $\\eta \\ll 1$ , by choosing $\\varepsilon = \\sqrt{\\eta }$ , since $\\chi ^2_\\varepsilon \\approx 8/\\varepsilon $ for very small $\\varepsilon $ , one gets $P_1(t) \\gtrsim 1-(17/2)\\sqrt{\\eta } \\rightarrow 1$ .",
"The limit $\\eta \\rightarrow 0$ can be reached in different ways: two possibilities are assuming $|g_{12}| \\ll |g_{23}|$ or assuming $\\omega _b/\\omega _a \\ll 1$ .",
"On this basis, one can assert that in each of the two limits, $|g_{12}/g_{23}| \\rightarrow 0$ or $\\omega _b/\\omega _a \\rightarrow 0$ , it turns out that $P_1(t)\\rightarrow 1$ for any $t$ , provided the temperature is high enough.",
"It is worth emphasizing that in order to achieve the limit in (REF ) one must legitimate truncation of the series expansions of the exponentials involved, which is possible under the assumption that the following conditions are fulfilled: $(\\beta _\\varepsilon +1)\\omega _b / (k_B T) \\ll 1\\,, \\\\(\\omega _a+\\alpha _\\varepsilon \\omega _b) / (k_B T) \\ll 1\\,.$ These two inequalities define the limit of high temperature for any fixed value of $\\varepsilon $ .",
"Figure: (Color online) Thesurvival probability of the atomic state 1\\left|1\\right\\rangle as a functionof time, for different values of the relevant parameters.",
"(a)ω a /ω b =10\\omega _a/\\omega _b=10, g 12 /ω b =g 23 /ω b =1g_{12}/\\omega _b=g_{23}/\\omega _b = 1, fordifferent values of the temperature: T/ω b =0.1T/\\omega _b=0.1 (red solidline), T/ω b =1T/\\omega _b=1 (green dotted line), T/ω b =50T/\\omega _b=50 (bluedashed line), T/ω b =250T/\\omega _b=250 (black bold line).",
"(b)ω a /ω b =50\\omega _a/\\omega _b=50, g 12 /ω b =g 23 /ω b =1g_{12}/\\omega _b=g_{23}/\\omega _b = 1, fordifferent values of the temperature: T/ω b =0.1T/\\omega _b=0.1 (red solidline), T/ω b =1T/\\omega _b=1 (green dotted line), T/ω b =250T/\\omega _b=250 (bluedashed line), T/ω b =1250T/\\omega _b=1250 (black bold line).In Fig.",
"REF are shown the evolutions of the survival probability for different temperatures and for different ratios of the frequencies of the two oscillators.",
"It is evident that the higher the temperature the higher the survival probability at any time.",
"Moreover, the smaller the ratio $\\omega _b/\\omega _a$ the higher the asymptotic value of $P_1(t)$ .",
"It is worth noting the difference with the case analyzed in the previous paper on Zeno subspaces induced by temperature [24].",
"In fact, in that case the limit of the survival probability for $T\\rightarrow \\infty $ is unity, while in the present case the survival probability does not tend toward unity, unless assuming $\\omega _b/\\omega _a\\rightarrow 0$ (or $|g_{12}| / |g_{23}| \\rightarrow 0$ ).",
"This difference can be explained analyzing the physical mechanism on the basis of these Zeno-like behaviors induced by temperature.",
"Indeed, the key points are two: first, in each invariant subspace the strength of the coupling with an oscillator grows up as the number of excitations increases; second, the higher the temperature the more the subspaces with high excitation numbers are populated.",
"Now, in the case of Ref.",
"[24] the interaction $1-2$ is not assisted by an harmonic oscillator and therefore it is not strengthened by temperature as the interaction $2-3$ is.",
"As a consequence, an higher temperature makes the ratio between the two coupling strengths tend toward zero in most of the subspaces.",
"On the contrary, in the present case, both couplings, $1-2$ and $2-3$ , are effectively enhanced by temperature, and one needs $\\omega _b \\ll \\omega _a$ in order to make the coupling $2-3$ prevail in the majority of the subspaces.",
"Indeed, $\\omega _b \\ll \\omega _a$ means that excited states of the oscillator $b$ are more populated than the corresponding states of oscillator $a$ , at the same temperature.",
"If this condition (or the alternative one, i.e., $|g_{12}| / |g_{23}|\\ll 1$ ) is not properly fulfilled, occurrence of Zeno dynamics is attenuated."
],
[
"Low vs High Temperature",
"In order to better understand the role of temperature, it could be useful to compare the behaviors of the survival probability at low and high temperature.",
"In fact, in the zero-temperature case, the only block involved in the dynamics is the one related to $n_a=n_b=0$ , and the inhibition of the dynamics of $\\left|1,0,0\\right\\rangle $ is determined by the condition $|g_{23}|\\gg |g_{12}|$ , so that it is essentially due to a significant difference between the strengths of the two couplings.",
"On the contrary, when the high-temperature limit is considered, the parameter determining the minimum value of the survival probability is $\\eta $ , as defined in Eq.",
"(REF ), which involves both the coupling strengths and the frequencies of the oscillators.",
"As already discussed before, a smaller frequency implies an higher degree of population of the excited levels of the oscillator, and eventually a stronger coupling in most of the Hilbert space.",
"On this basis, one finds that Zeno dynamics is possible even in the case $|g_{23}|<|g_{12}|$ , provided $\\omega _b \\ll \\omega _a$ and temperature high enough.",
"On the contrary, temperature can produce the opposite effect, determining disappearance of Zeno dynamics.",
"As an example, consider the case $|g_{23}|\\gg |g_{12}|$ , which implies Zeno dynamics at zero temperature, and $\\omega _b >\\omega _a |g_{23}/g_{12}|^2 \\gg \\omega _a$ , which renders $\\eta $ large enough to invalidate inhibition of the dynamics.",
"In Fig.",
"REF it is shown a case wherein by increasing temperature one lowers the survival probability at any time.",
"Figure: (Color online) Survival probability of the atomic state1\\left|1\\right\\rangle at low and high temperature.",
"The relevant quantities areω b /ω a =10\\omega _b/\\omega _a=10, g 12 /ω a =1g_{12}/\\omega _a=1, g 23 /g 12 =6g_{23}/g_{12} = 6;temperature: T/ω a =0.1T/\\omega _a=0.1 (red solid line), T/ω a =250T/\\omega _a=250(black dashed line).Let us now consider again the Hamiltonian in Eq.",
"(REF ) but without the resonance conditions (see Fig.",
"REF ).",
"Generally speaking the presence of a detuning causes a diminishing of the interaction strength, so that the presence of a detuning in the $2-3$ transition will make us analyze the competition between detuning and temperature in connection with the occurrence of Zeno dynamics.",
"On the contrary, the presence of a detuning in the $1-2$ transition would simply hinder the evolution of the initial atomic state ($\\left|1\\right\\rangle $ ) even in the absence of the second oscillator and relevant coupling.",
"Therefore this analysis would be less interesting.",
"Hence, for the sake of simplicity, let us assume that the transition $1-2$ is resonant with the relevant oscillator ($\\omega _a=\\omega _1-\\omega _2$ ), while the transition $2-3$ is assumed not to be resonant with the second oscillator: $\\Delta =\\omega _b-(\\omega _2-\\omega _3)\\ne 0$ .",
"The restriction of the Hamiltonian in the generic invariant subspace corresponding to $n_a$ and $n_b$ is given by the following matrix: $\\nonumber H_{n_a n_b} &=& (n_a\\omega _a+n_b\\omega _b+\\omega _1)\\mathbb {I}_3\\\\\\nonumber &\\!\\!\\!\\!\\!\\!+&\\left(\\begin{array}{ccc}0 & g_{12} \\sqrt{n_a+1} & 0 \\cr \\,\\, g_{12} \\sqrt{n_a+1} \\,\\, & 0 & \\,\\, g_{23} \\sqrt{n_b+1} \\,\\, \\cr 0 & g_{23} \\sqrt{n_b+1} & \\Delta \\end{array}\\right)\\,.",
"\\\\$ Figure: (Color online) The upper atomic level (1\\left|1\\right\\rangle ) isresonantly coupled to the intermediate level (2\\left|2\\right\\rangle ), which inturn is non resonantly coupled to the lowest level (3\\left|3\\right\\rangle ).According with the analysis in appendix , in order to have inhibition of the dynamics of the state $\\left|1,n_a,n_b\\right\\rangle $ — making it a quasi-eigenstate of the Hamiltonian — one needs that $g_{23}\\sqrt{n_b+1}$ is much grater than $g_{12}\\sqrt{n_a+1}$ and greater than $\\Delta $ .",
"A large value of detuning can obstacle the appearance of Zeno dynamics, but larger values of temperature can establish again such a regime by populating more and more those subspaces wherein $g_{23}\\sqrt{n_b+1}$ is larger (and even much larger) than $\\Delta $ .",
"In Fig.",
"REF it is shown the behavior of the population of the atomic state $\\left|1\\right\\rangle $ as a function of time but for different values of temperature and detuning.",
"From Fig.",
"REF a we learn that even in the presence of non vanishing detuning there is an increase of the survival probability when temperature becomes higher.",
"Nevertheless, comparing the bold line in this figure with the one in Fig.",
"REF a, we see that the asymptotic value of the population is slightly smaller in the presence of detuning, in spite of the fact that the temperature is the same.",
"Figure: (Color online) Thesurvival probability of the atomic state 1\\left|1\\right\\rangle as a functionof time, for different values of the relevant parameters.",
"(a)ω a /ω b =10\\omega _a/\\omega _b=10, g 12 /ω b =g 23 /ω b =1g_{12}/\\omega _b=g_{23}/\\omega _b = 1,Δ/ω b =1\\Delta /\\omega _b=1, for different values of the temperature:T/ω b =0.1T/\\omega _b=0.1 (red solid line), T/ω b =10T/\\omega _b=10 (green dottedline), T/ω b =100T/\\omega _b=100 (blue dashed line), T/ω b =250T/\\omega _b=250(black bold line).",
"(b) ω a /ω b =10\\omega _a/\\omega _b=10,g 12 /ω b =g 23 /ω b =1g_{12}/\\omega _b=g_{23}/\\omega _b = 1, k B T/ω b =250k_B T/\\omega _b = 250, fordifferent values of the detuning: Δ/ω b =4\\Delta /\\omega _b=4 (red solidline), Δ/ω b =2\\Delta /\\omega _b=2 (green dotted line),Δ/ω b =1\\Delta /\\omega _b=1 (blue dashed line), Δ=0\\Delta =0 (black boldline).In Fig.",
"REF b it is shown that, for a fixed value of temperature, a diminishing of the detuning produces an increase of the asymptotic population, as expected."
],
[
"Attempt of Generalization",
"In view of a possible study of interactions between a few-level system and its environment, the previous analysis could deserve a generalization to the case wherein many harmonic oscillators couple both transitions, $1-2$ and $2-3$ .",
"Since we are interested to the case of quite different Bohr frequencies, we can consider two independent sets of oscillators.",
"The relevant Hamiltonian is given by, $\\nonumber \\hat{H} &=& \\sum _k \\omega _n \\left|n\\right\\rangle \\left\\langle n\\right| + \\sum _{k=1}^p \\nu _k \\hat{a}_k^\\dag \\hat{a}_k +\\sum _{l=1}^q\\mu _l \\hat{b}_l^\\dag \\hat{b}_l \\\\\\nonumber &+& \\sum _{k=1}^p g^{(12)}_k(\\left|1\\right\\rangle \\left\\langle 2\\right|\\hat{a}_k+\\left|2\\right\\rangle \\left\\langle 1\\right|\\hat{a}_k^\\dag ) \\\\\\nonumber &+& \\sum _{l=1}^q g^{(23)}_l (\\left|2\\right\\rangle \\left\\langle 3\\right|\\hat{b}_l+\\left|3\\right\\rangle \\left\\langle 2\\right|\\hat{b}_l^\\dag )\\,, \\\\$ which conserves the following quantity: $\\hat{N}=2\\left|1\\right\\rangle \\left\\langle 1\\right|+\\left|2\\right\\rangle \\left\\langle 2\\right|+\\sum _{k=1}^p\\hat{a}_k^\\dag \\hat{a}_k +\\sum _{l=1}^q\\hat{b}_l^\\dag \\hat{b}_l\\,.$ Therefore the Hilbert space is partitioned into invariant subspaces ${\\cal H}_{\\vec{n}_p\\vec{m}_q}$ of dimension $\\dim {\\cal H}_{\\vec{n}_p\\vec{m}_q}=1+p+pq$ and spanned by the following vectors: $\\left|1,\\vec{n}_p,\\vec{m}_q\\right\\rangle \\equiv \\left|1,n_1,...n_p,m_1,...m_q\\right\\rangle $ ; $(n_k+1)^{-1/2}\\hat{a}_k^\\dag \\left|2,\\vec{n}_p, \\vec{m}_q\\right\\rangle $ with $k=1,...p$ ; $(n_k+1)^{-1/2}(m_l+1)^{-1/2}\\hat{a}_k^\\dag \\hat{b}_l^\\dag \\left|3,\\vec{n}_p,\\vec{m}_q\\right\\rangle $ , with $k=1,...p$ and $l=1,...q$ .",
"The restriction of the Hamiltonian to each subspace has the following structure: $B_{p\\times q} = \\left(\\begin{array}{ccccccccccccccc}\\delta _0 & \\alpha _1 & \\alpha _2 & ... & \\alpha _p & 0 & 0 & 0 & 0 & ... & 0 & ... & 0 & ... & 0 \\cr \\alpha _1 & \\delta _1 & 0 & ... & 0 & \\beta _{11} & ... & \\beta _{1q} & 0 & ... & 0 & ... & 0 & ... & 0 \\cr \\alpha _2 & 0 & \\delta _2 & ... & 0 & 0 & ... & 0 & \\beta _{21} & ... & \\beta _{2q} & ... & 0 & ... & 0 \\cr \\vdots & \\vdots & \\vdots & \\ddots & & \\vdots & & & & & & & & & \\cr \\alpha _p & 0 & 0 & & \\delta _p & 0 & ... & 0 & 0 & ... & 0 & ... & \\beta _{p1} & ... & \\beta _{pq} \\cr 0 & \\beta _{11} & 0 & ... & 0 & \\delta _{11}& & 0 & 0 & ... & 0 & ... & 0 & ... & 0 \\cr \\vdots & \\vdots & \\vdots & & \\vdots & ... & \\ddots & \\vdots & 0 & ... & 0 & ... & 0 & ... & 0 \\cr 0 & \\beta _{1q} & 0 & ... & 0 & 0 & ... & \\delta _{1q}& 0 & ... & 0 & ... & 0 & ... & 0 \\cr 0 & 0 & \\beta _{21} & ... & 0 & 0 & ... & 0 & \\delta _{21}& ... & 0 & ... & 0 & ... & 0 \\cr \\vdots & \\vdots & \\vdots & & \\vdots & \\vdots & & 0 & 0 & \\ddots & 0 & ... & 0 & ... & 0 \\cr 0 & 0 & \\beta _{2q} & ... & 0 & 0 & ... & 0 & 0 & ... & \\delta _{2q}& ... & 0 & ... & 0 \\cr \\vdots & \\vdots & \\vdots & & \\vdots & \\vdots & & \\vdots & \\vdots & ... & \\vdots & \\ddots & \\vdots & ... & \\vdots \\cr 0 & 0 & 0 & ... & \\beta _{p1} & 0 & ... & 0 & 0 & ... & 0 & ... & \\delta _{p1}& ... & 0 \\cr \\vdots & \\vdots & \\vdots & & \\vdots & \\vdots & & 0 & 0 & & 0 & ... & 0 & \\ddots & 0 \\cr 0 & 0 & 0 & ... & \\beta _{pq} & 0 & ... & 0 & 0 & ... & 0 & ... & 0 & ... & \\delta _{pq}\\cr \\end{array}\\right)\\,,$ where the only non vanishing terms in a line or column are between two symbols different from ` 0' in that line or column.",
"In spite of this fact, the matrix is rather involved and difficult to study.",
"In the very special case of $p=1$ , such structure becomes quite simple: $B_{1\\times q} = \\left(\\begin{array}{cccccc}\\delta & \\alpha & 0 & 0 & ... & 0 \\cr \\alpha & \\delta _1 & \\beta _1 & \\beta _2 & ... & \\beta _M \\cr 0 & \\beta _1 & \\delta _{11} & 0 & ... & 0 \\cr 0 & \\beta _2 & 0 & \\delta _{12} & ... & 0 \\cr \\vdots & \\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\cr 0 & \\beta _M & 0 & 0 & ... & \\delta _{1M} \\cr \\end{array}\\right)\\,,$ where $\\alpha =\\alpha _1$ and $\\beta _l=\\beta _{1l}$ .",
"In the generic subspace ${\\cal H}_{\\vec{n}_p\\vec{m}_q}$ one has $\\alpha =g_1^{(12)}(n_1+1)^{1/2}$ and $\\beta _l=g_l^{(23)}(m_l+1)^{1/2}$ .",
"The matrix in Eq.",
"(REF ) has the same structure of the matrix analyzed in the Appendix B of Ref. [24].",
"On the basis of that analysis, we claim that when the system starts form $\\left|1,n_1,\\vec{m}_q\\right\\rangle $ and $(\\sum _l \\beta _l^2)^{1/2} \\gg |\\alpha |$ then the dynamics is hindered.",
"Therefore, focusing our attention on the population of the atomic state $\\left|1\\right\\rangle $ irrespectively of the number of excitations of the oscillators, by reasoning as in the previous sections, we can say that the evolution of such population turns out to be hindered when the frequencies $\\mu _l$ are smaller — preferably much smaller — than $\\nu \\equiv \\nu _1$ .",
"Such a condition on the frequencies makes the previous assertion true even if, for example, $|g_1^{(12)}|^2 \\sim \\sum _l|g_l^{(23)}|^2$ ."
],
[
"Discussion",
"The analysis developed in this paper shows in a clear way the role of the temperature in determining the occurrence of Zeno dynamics.",
"In particular, we have considered a few-level system prepared in an excited state and interacting with a set of harmonic oscillators initially prepared in thermal states at the same temperature.",
"We have pointed out that an higher temperature implies a more significant role of the subspaces of the Hilbert space characterized by higher numbers of excitations, which in turn imply stronger effective couplings.",
"The smaller the frequency of an oscillator the more its excited states are involved in the dynamics, for a given temperature.",
"Therefore, the presence of a coupling between the lowest state $\\left|3\\right\\rangle $ and the intermediate one $\\left|2\\right\\rangle $ can significantly weaken the effects of the $1-2$ coupling, especially when the Bohr frequency of the $2-3$ transition is sufficiently small.",
"This is a significant difference between high-temperature and low-temperature limit, since in the latter case the role of the frequencies is absent and only coupling strengths are important.",
"This analysis is the prosecution of the study developed in Ref.",
"[24] where the $1-2$ coupling was induced by an external field assumed not to be affected by temperature.",
"In the present work, instead, we have considered the competition between the growing of both the couplings, $1-2$ and $2-3$ , and the role of temperature and Bohr frequencies in making one of them prevail.",
"A further analysis aimed at considering a set of harmonic oscillators driving each atomic transition — similar to the one performed in the previous paper — would be appropriate.",
"Some preliminary results have been reported in section .",
"Another point that could deserve attention is the role of possible counter rotating terms in the interaction between the few-level system and the bosonic part of the total system."
],
[
"Acknowledgements",
"The Author wishes to thank Antonino Messina and Matteo Scala for carefully reading the manuscript."
],
[
"In this appendix we prove that a matrix of the following form (written in the basis $\\left|A\\right\\rangle $ , $\\left|B\\right\\rangle $ , $\\left|C\\right\\rangle $ ): $M=\\left(\\begin{array}{ccc}0 & \\alpha _{1} & 0 \\cr \\alpha _{1}^* & 0 & \\,\\, \\alpha _{2} \\,\\, \\cr 0 & \\alpha _{2}^* & \\Delta \\end{array}\\right)\\,,$ has an eigenvalue whose corresponding eigenstate becomes closer to $\\left|A\\right\\rangle $ as the modulus of the coupling constant $\\alpha _2$ grows up.",
"Let us first consider the special situation $\\Delta =0$ , in which case we can directly check that $\\lambda =0$ is an eigenvalue and that the corresponding eigenstate is $\\left|0\\right\\rangle = \\frac{\\alpha _2}{\\alpha }\\left|A\\right\\rangle - \\frac{\\alpha _1^*}{\\alpha }\\left|C\\right\\rangle \\,,\\qquad \\alpha =\\sqrt{|\\alpha _1|^2+|\\alpha _2|^2}\\,\\,,$ from which one deduces that $|\\left\\langle A\\vert 0\\right\\rangle |=|\\alpha _2/\\alpha |$ tends toward unity in the limit $|\\alpha _1|\\ll |\\alpha _2|$ .",
"Concerning the general case $\\Delta \\ne 0$ , the secular polynomial $P(\\lambda )=\\det (M-\\lambda \\mathbb {I}_3)$ is: $P(\\lambda ) = -\\lambda ^3 + \\Delta \\lambda ^2 +(|\\alpha _1|^2+|\\alpha _2|^2)\\lambda - |\\alpha _1|^2\\Delta \\,.$ It turns out that $&& P(0) = - |\\alpha _1|^2\\Delta \\,,$ and $\\nonumber && P\\left(\\frac{|\\alpha _1^2|}{|\\alpha _2^2|}\\Delta \\right) = \\frac{|\\alpha _1|^4}{|\\alpha _2|^2}\\Delta +\\left(\\left|\\frac{\\alpha _1}{\\alpha _2}\\right|^4-\\left|\\frac{\\alpha _1}{\\alpha _2}\\right|^6\\right)\\Delta ^3\\,,\\\\$ which for $\\Delta \\ne 0$ and $|\\alpha _1|<|\\alpha _2|$ implies a change of sign.",
"Therefore there is an eigenvalue $\\eta $ of modulus $|\\eta | < |\\alpha _1^2\\Delta /\\alpha _2^2|$ .",
"The relevant eigenvector is: $\\left|\\eta \\right\\rangle = \\mathcal {P}_\\eta \\left(\\left|A\\right\\rangle +\\frac{\\eta }{\\alpha _1}\\left|B\\right\\rangle -\\frac{\\eta \\,\\alpha _2^*}{(\\Delta -\\eta )\\,\\alpha _1}\\left|C\\right\\rangle \\right)\\,,$ with $\\mathcal {P}_\\eta = \\left[1 + \\frac{|\\eta |^2}{|\\alpha _1|^2} + \\frac{|\\eta |^2 |\\alpha _2|^2}{|\\Delta -\\eta |^2\\,|\\alpha _1|^2}\\right]^{-1/2}\\,.$ Now, assume $|\\Delta | < |\\alpha _2|$ and $|\\alpha _1|\\ll |\\alpha _2|$ (which in turn implies $|\\eta | \\ll |\\Delta |$ and then $|\\Delta /(\\Delta -\\eta )|<2$ ), one finds that $|\\eta /\\alpha _1| <|\\alpha _1/\\alpha _2|\\times |\\Delta /\\alpha _2|<|\\alpha _1/\\alpha _2|$ and $\\left|\\eta ^2 \\alpha _2^2/[(\\Delta -\\eta )^2\\,\\alpha _1^2]\\right|< |\\alpha _1/\\alpha _2|^2\\times |\\Delta /(\\Delta -\\eta )|^2 <2|\\alpha _1/\\alpha _2|^2$ .",
"Then, under the previous hypotheses one has $|\\left\\langle A\\vert \\eta \\right\\rangle |=\\mathcal {P}_\\eta \\rightarrow 1$ .",
"We conclude reminding that (see appendix A of Ref.",
"[24]) if the system is prepared in the initial state $\\left|\\psi (0)\\right\\rangle =\\left|A\\right\\rangle $ , then the survival probability has a non vanishing lower bound: $P_A(t)=|\\left\\langle A\\vert \\psi (t)\\right\\rangle |^2 \\ge (2|\\left\\langle A\\vert \\eta \\right\\rangle |^2-1)^2$ , which of course is close to unity when $|\\left\\langle A\\vert \\eta \\right\\rangle | \\approx 1$ ."
]
] | 1204.0889 |
[
[
"A quantum query algorithm for the graph collision problem"
],
[
"Abstract We construct a new quantum algorithm for the graph collision problem; that is, the problem of deciding whether the set of marked vertices contains a pair of adjacent vertices in a known graph G. The query complexity of our algorithm is O(sqrt(n)+sqrt(alpha*(G))), where n is the number of vertices and alpha*(G) is the maximum total degree of the vertices in an independent set of G. Notably, if G is a random graph where every edge is present with a fixed probability independently of other edges, then our algorithm requires O(sqrt(n log n)) queries on most graphs, which is optimal up to the sqrt(log n) factor on most graphs."
],
[
"Introduction",
"The quantum query complexity of a function is a natural counterpart of its classical query complexity; namely, it is the number of quantum oracle calls that an algorithm has to make in order to compute its value.",
"Grover's search algorithm [9] gave the first example of a function whose quantum query complexity is significantly smaller than classical: computing the value of the OR function $\\bigvee _{i=1}^nx_i$ requires $\\Omega (n)$ classical queries, but only $O(\\sqrt{n})$ quantum queries.",
"It has been shown by Bennett et al.",
"[3] that Grover's algorithm is optimal for computing the OR function.",
"Later, Beals et al.",
"[4] proved that no total function can have the gap between its quantum and classical query complexities larger than polynomial.",
"Another important problem where a quantum query algorithm can be much faster than a classical one is the element distinctness problem, where $n$ elements are “colored” by an oracle and the goal is to decide whether there are at least two elements of the same color.",
"In 2001 Buhrman et al.",
"[5] constructed an algorithm that required $O(n^{3/4})$ quantum queries, later a lower bound of $\\Omega (n^{2/3})$ was shown by Aaronson and Shi [2].",
"Finally, in 2003 Ambainis [1] gave a new algorithm that had query complexity $O(n^{2/3})$ , thus matching the lower bound.",
"The graph collision problem was first considered by Magniez et al.",
"[12], where it was shown to have quantum query complexity $O(n^{2/3})$ .",
"The algorithm used in [12] can be viewed as a natural adaptation of Ambainis' algorithm for the element distinctness problem.",
"On the other hand, the lower bound techniques used in [2] don't seem to be applicable to the graph collision problem, and the actual quantum query complexity of it is still an open question."
],
[
"Our results and techniques",
"We present a new quantum algorithm for the graph collision problem.",
"The complexity of our algorithm depends on the properties of the given graph $G$ .",
"Throughout the paper, the quantum query complexity of a decision problem refers to that with a constant two-sided error.",
"Theorem 1 For a graph $G$ on $n$ vertices, the quantum query complexity of the graph collision problem on $G$ is $O(\\sqrt{n}+\\sqrt{\\alpha ^*(G)})$ , where $\\alpha ^*(G)$ is the maximum total degree of the vertices in an independent set of $G$ .",
"Notably, this implies that the graph collision problem requires only $\\tilde{O}(\\sqrt{n})$ quantum queries for most graphs in the following sense.",
"Let $\\mu _{n,p}$ be the distribution corresponding to choosing a graph on $n$ vertices, where every edge is present with probability $p$ independently of other edges.",
"Corollary 2 For arbitrary function $p\\colon \\mathbb {N}\\rightarrow [0,1]$ , let $G\\sim \\mu _{n,p(n)}$ .",
"Then the (worst-case) quantum query complexity of the graph collision problem on $G$ is almost alwaysCf.",
"Theorem REF for the corresponding quantitative statement.",
"$O(\\sqrt{n\\log n})$ .",
"The above result is optimal up to the $\\sqrt{\\log n}$ factor for most random graphs, as computing the OR of $n$ variables can be reduced to solving the graph collision problem on any graph $G$ that contains $\\Omega (n)$ non-isolated vertices.",
"Our algorithm for Theorem REF works as follows.",
"As a preprocessing, we estimate the sum of the degrees of the vertices in $S$ .",
"If this sum is much larger than $\\max \\lbrace \\alpha ^*(G),n\\rbrace $ , then we answer “$S$ is not an independent set” and halt.",
"This requires $O(\\sqrt{n})$ queries, due to the approximate counting algorithm by Brassard, Høyer, and Tapp [6].",
"To handle the remaining (main) case, we construct a span program with witness size $O(\\sqrt{n}+\\sqrt{\\alpha ^*(G)})$ .",
"It was shown by Reichardt [13], [14] that the quantum query complexity of a promise decision problem is at most a constant factor away from the witness size of a span program computing it."
],
[
"Related work",
"Magniez, Santha, and Szegedy [12] introduced the graph collision problem and gave a quantum algorithm with $O(n^{2/3})$ queries.",
"They used it as a subroutine used in their $O(n^{13/10})$ -query algorithm for the triangle finding problem.",
"This $O(n^{2/3})$ is the best known upper bound on the quantum query complexity of the graph collision problem.",
"The best known lower bound for the graph collision problem is $\\Omega (\\sqrt{n})$ , which follows easily from the lower bound for the search problem [3].",
"Jeffery, Kothari, and Magniez [10] recently gave a quantum algorithm for the graph collision problem on a bipartite graph which is useful when the given bipartite graph is close to the complete bipartite graph: the query complexity of their algorithm is $\\tilde{O}(\\sqrt{n}+\\sqrt{m})$ , where $m$ is the number of missing edges compared to the complete bipartite graph.",
"Improving the query complexity of the graph collision problem has important consequences.",
"First, improving it is likely to give a better algorithm for the triangle finding problem by applying the same technique as the one used in Ref. [12].",
"Second, the graph collision problem is equivalent to the evaluation of a 2-DNF formula, and the techniques used in the graph collision problem may be also applicable to the more general $k$ -DNF evaluation.",
"Our algorithm for the main case of the graph collision problem, including its use of span programs, is inspired by the recent result by Belovs and Lee [7]."
],
[
"Preliminaries",
"We will consider the quantum query complexity of the following problem.",
"Definition 1 (Graph collision problem) Let $G=(V,E)$ be a graph.",
"The graph collision problem on $G$ asks, given oracle access to a string $x\\in \\lbrace 0,1\\rbrace ^V$ , whether there exists an edge $(i,j)\\in E$ such that $x_i=x_j=1$ .",
"Note that graph $G$ is given explicitly to the algorithm, and the only part of the input which needs to be queried is the string $x\\in \\lbrace 0,1\\rbrace ^V$ .",
"We call a vertex $i$ marked if $x_i=1$ .",
"Note that the graph collision problem is equivalent to deciding whether the marked vertices form an independent set in $G$ , with the answers “yes” and “no” swapped.",
"In the rest of the paper, we let $V=[n]$ , where $[n]$ denotes the set $\\lbrace 1,\\dots ,n\\rbrace $ .",
"For a graph $G=(V,E)$ and a set $S\\subseteq V$ of vertices, we denote by $\\deg (S)$ the sum of degrees of vertices in $S$ .",
"For any graph $G$ , we denote by $\\alpha ^*(G)$ the maximum total degree of the vertices in an independent set of $G$ ; that is, $\\alpha ^*(G) = \\max \\lbrace \\deg (S) \\colon \\text{$S$ is an independent set in~$G$}\\rbrace .$ We will use the following form of Chernoff bound, as stated by Drukh and Mansour [8].",
"Lemma 3 (Chernoff bound) Let $X_1,\\dots ,X_n$ be mutually independent random variables taking values in $[0,1]$ , such that $\\mathop {\\mathbf {E}}_{}{\\left[{X_i}\\right]}=\\mu $ for all $i\\in [n]$ .",
"Then for any $\\lambda >1$ , $ \\mathop {\\mathbf {Pr}}_{}{\\left[{\\sum _{i\\in [n]}X_i\\ge \\lambda n\\mu }\\right]}\\le \\exp \\left(-\\frac{n(\\lambda -1)^2\\mu }{\\lambda +1}\\right)\\le \\exp \\bigl ((3-\\lambda )n\\mu \\bigr ).",
"$ All logarithms in this paper are natural."
],
[
"Span programs",
"Span program is a linear-algebraic model of computation introduced by Karchmer and Wigderson [11] to study the computational power of counting in branching programs and space-bounded computation.",
"In our context, the relevant complexity measure is its witness size introduced by Reichardt and Špalek [15].",
"We use a formulation closer to that used by Reichardt [13].",
"Definition 2 (Span program) A span program $P=(\\mathcal {H},{t};V_{10},V_{11},\\dots ,V_{n0},V_{n1})$ with $n$ -bit input is defined by a finite-dimensional Hilbert space $\\mathcal {H}$ over $\\mathbb {C}$ , a vector ${t}\\in \\mathcal {H}$ , and a finite set $V_{jb}\\subseteq \\mathcal {H}$ for each $j\\in [n]$ and each $b\\in \\lbrace 0,1\\rbrace $ .",
"This span program is said to compute a function $f\\colon D\\rightarrow \\lbrace 0,1\\rbrace $ , where $D\\subseteq \\lbrace 0,1\\rbrace ^n$ , when for $x\\in D$ , we have $f(x)=1$ if and only if ${t}$ lies in the subspace of $\\mathcal {H}$ spanned by $\\bigcup _{j\\in [n]}V_{jx_j}$ .",
"The vector ${t}$ is called the target vector of this span program $P$ .",
"Definition 3 (Witness size of a span program) Let $P$ be a span program as in Definition REF .",
"For an input $x\\in f^{-1}(1)$ , a witness for $x$ is an $n$ -tuple of mappings $w_1,\\dots ,w_n$ , where $w_j\\colon V_{jx_j}\\rightarrow \\mathbb {C}$ , such that ${t}=\\sum _{j\\in [n]}\\sum _{{v}\\in V_{jx_j}}w_j({v}){v}$ .",
"The witness size on input $x\\in f^{-1}(1)$ , denoted by $\\operatorname{wsize}(P,x)$ , is defined as $\\operatorname{wsize}(P,x)=\\min _{(w_1,\\dots ,w_n):\\text{witness for~$x$ in~$P$}}\\sum _{j\\in [n]}\\sum _{{v}\\in V_{jx_j}}{w_j({v})}^2.$ For an input $x\\in f^{-1}(0)$ , a witness for $x$ is a vector ${w^{\\prime }}\\in \\mathcal {H}$ such that $\\langle t|w^{\\prime }\\rangle =1$ and $\\langle v|w^{\\prime }\\rangle =0$ for every ${v}\\in \\bigcup _{j\\in [n]}V_{jx_j}$ .",
"This time, the witness size on input $x\\in f^{-1}(0)$ , again denoted by $\\operatorname{wsize}(P,x)$ , is defined as $\\operatorname{wsize}(P,x)=\\min _{{w^{\\prime }}:\\text{witness for~$x$ in~$P$}}\\sum _{j\\in [n],b\\in \\lbrace 0,1\\rbrace }\\sum _{{v}\\in V_{jb}}{\\langle v|w^{\\prime }\\rangle }^2.$ The witness size of this span program is $\\operatorname{wsize}(P)=\\max _{x\\in D}\\operatorname{wsize}(P,x)$ .",
"Finally, we denote by $\\operatorname{wsize}(f)$ the minimum witness size of a span program which computes $f$ .",
"For a function $f\\colon D\\rightarrow \\lbrace 0,1\\rbrace $ , where $D\\subseteq \\lbrace 0,1\\rbrace ^n$ , we denote by $Q(f)$ the quantum query complexity of $f$ with two-sided error probability at most $1/3$ .",
"As is well known, changing the error probability to other constants less than $1/2$ affects the query complexity only within a constant factor.",
"Theorem 4 (Reichardt [13], [14]) Let $f\\colon D\\rightarrow \\lbrace 0,1\\rbrace $ , where $D\\subseteq \\lbrace 0,1\\rbrace ^n$ .",
"Then $Q(f)$ and $\\operatorname{wsize}(f)$ coincide up to a constant factor.",
"That is, there exists a constant $c>1$ which does not depend on $n$ or $f$ such that $(1/c)\\operatorname{wsize}(f)\\le Q(f)\\le c\\cdot \\operatorname{wsize}(f)$ .",
"Ref.",
"[13] showed that $\\operatorname{wsize}(f)$ is equal to the general adversary bound for $f$ , and Ref.",
"[14] showed that the general adversary bound for $f$ and $Q(f)$ coincide up to a constant factor."
],
[
"Quantum algorithm for approximate counting",
"To detect the case where marked vertices have too many edges, we will use the following result by Brassard, Høyer, and Tapp [6].",
"Theorem 5 (Approximate counting [6]) There exists a quantum algorithm which, given integers $N\\ge 1$ (domain size) and $P\\ge 4$ (precision) and oracle access to a function $F\\colon [N]\\rightarrow \\lbrace 0,1\\rbrace $ satisfying $t={F^{-1}(1)}\\le N/2$ , makes $P$ queries to the oracle and outputs an integer $\\tilde{t}$ , such that ${t-\\tilde{t}}<\\frac{2\\pi }{P}\\sqrt{tN}+\\frac{\\pi ^2}{P^2}N$ with probability at least $8/\\pi ^2$ .",
"We can remove the assumption that $t\\le N/2$ by doubling the size of the domain, and we can reduce the error probability to an arbitrarily small constant by repeating the algorithm constantly many times and taking the majority vote: Corollary 6 Let $\\varepsilon >0$ be a constant.",
"Then there exists a quantum algorithm which, given integers $N\\ge 1$ (domain size) and $P\\ge 4$ (precision) and oracle access to a function $F\\colon [N]\\rightarrow \\lbrace 0,1\\rbrace $ , makes $O(P)$ queries to the oracle and outputs an integer $\\tilde{t}$ satisfying the following.",
"Let $t={F^{-1}(1)}$ .",
"Then it holds that ${t-\\tilde{t}}<\\frac{2\\sqrt{2}\\,\\pi }{P}\\sqrt{tN}+\\frac{2\\pi ^2}{P^2}N$ with probability at least $1-\\varepsilon $ .",
"(The constant factor hidden in the $O$ -notation of the number of queries depends only on $\\varepsilon $ and not on $N$ , $P$ , or $F$ .)"
],
[
"Algorithm for the main case",
"The following lemma is useful in the case where not too many edges are incident to marked vertices.",
"In the next section, we will use it with $k=2\\max \\lbrace n,\\alpha ^*(G)\\rbrace $ to prove Theorem REF .",
"Lemma 7 Let $G$ be a graph on $n$ vertices, and let $k\\in \\mathbb {N}$ .",
"Consider the special case of the graph collision problem on $G$ where it is promised that the set $S$ of marked vertices satisfies $\\deg (S)\\le k$ .",
"There exists a span program for this promise problem whose witness size is at most $\\sqrt{2(n+k)}$ .",
"There exists a quantum algorithm for this promise problem with two-sided error probability at most $1/6$ whose query complexity is $O(\\sqrt{n}+\\sqrt{k})$ .",
"Item (ii) follows immediately from item (i) and Theorem REF .",
"In the rest of the proof, we will prove item (i) by constructing a span program explicitly.",
"Let $\\mathcal {H}=\\mathbb {C}^{\\lbrace 0,1\\rbrace ^n}$ , and let ${t}=\\gamma \\sum _{z\\in \\lbrace 0,1\\rbrace ^n}{z},\\qquad \\gamma =\\biggl (\\frac{n+k}{2}\\biggr )^{1/4}.$ For $j\\in [n]$ and $b\\in \\lbrace 0,1\\rbrace $ , let ${s_{jb}}=\\sum _{z\\in \\lbrace 0,1\\rbrace ^n:z_j=b}{z}$ .",
"Let $V_{j0}=\\varnothing $ and $V_{j1}=\\lbrace {s_{j0}}\\rbrace \\cup \\lbrace {s_{i1}}\\colon i\\in \\mathrm {N}(j)\\rbrace $ , where $\\mathrm {N}(j)$ is the set of neighbors of vertex $j$ in graph $G$ .",
"Define a span program $P$ as $P=(\\mathcal {H},{t};V_{10},V_{11},\\dots ,V_{n0},V_{n1})$ .",
"It is easy to see that $P$ computes the promise problem stated in the lemma.",
"Indeed, if $x\\in f^{-1}(1)$ , then there exists an edge $ij\\in E$ such that $x_i=x_j=1$ .",
"Therefore, ${s_{j0}}\\in V_{j1}$ and ${s_{j1}}\\in V_{i1}$ , which implies that ${t}=\\gamma {s_{j0}}+\\gamma {s_{j1}}\\in \\operatorname{span}(V_{j1}\\cup V_{i1})$ .",
"On the other hand, if $x\\in f^{-1}(0)$ , then ${w^{\\prime }}={x}/\\gamma $ is a witness for $x$ .",
"Indeed, $\\langle t|w^{\\prime }\\rangle =1$ , $\\langle s_{j0}|w^{\\prime }\\rangle =0$ if $x_j=1$ , and $\\langle s_{i1}|w^{\\prime }\\rangle =0$ if $x_j=1$ and $i\\in \\mathrm {N}(j)$ .",
"From these witnesses, the witness size of $P$ can be bounded easily.",
"If $x\\in f^{-1}(1)$ , then the witness stated above shows that the witness size for $x$ is at most $2\\gamma ^2=\\sqrt{2(n+k)}$ .",
"If $x\\in f^{-1}(0)$ , then the witness stated above shows that the witness size for $x$ is at most $(n+k)/\\gamma ^2=\\sqrt{2(n+k)}$ ."
],
[
"Preprocessing and overall algorithm",
"In this section, we will prove Theorem REF .",
"Consider the following quantum algorithm.",
"Compute $\\alpha ^*(G)$ .",
"Let $s=\\max \\lbrace \\alpha ^*(G),n\\rbrace $ .",
"(Because this step does not use any queries to $x$ , how $\\alpha ^*(G)$ is computed does not matter as long as the query complexity is concerned.)",
"(Preprocessing.)",
"Estimate the number $t$ of pairs $(i,j)\\in [n]^2$ such that $ij\\in E$ and $x_i=1$ by running the approximate counting algorithm in Corollary REF with error probability $\\varepsilon =1/6$ and precision parameter $P=\\max \\lbrace 4,{7\\pi \\sqrt{n}}\\rbrace $ .",
"If the result $\\tilde{t}$ of counting satisfies $\\tilde{t}>3s/2$ , then answer “yes” and halt.",
"Otherwise, proceed to the next step.",
"(Main case.)",
"Run the algorithm in Lemma REF (ii) using error probability $1/6$ and parameter $k=2s$ , and answer “yes” or “no” accordingly."
],
[
"Query complexity.",
"Step 1 does not make any queries to $x$ .",
"Step 2 makes $O(P)=O(\\sqrt{n})$ queries.",
"Step 3 makes $O(\\sqrt{n}+\\sqrt{2s})=O(\\sqrt{n}+\\sqrt{\\alpha ^*(G)})$ queries.",
"Therefore, the query complexity of the whole algorithm is $O(\\sqrt{n}+\\sqrt{\\alpha ^*(G)})$ ."
],
[
"Correctness.",
"In the rest of this section, we will show that this algorithm reports an incorrect answer with probability at most $1/3$ by considering the following three cases: (a) the correct answer is “yes” and $t\\le 2s$ , (b) the correct answer is “yes” and $t>2s$ , (c) the correct answer is “no.” If the correct answer is “yes” and $t\\le 2s$ , the only step where the algorithm reports an incorrect answer is step 3, and the promise of the algorithm in Lemma REF (ii) is satisfied.",
"Therefore, the error probability is at most $1/6$ .",
"If the correct answer is “yes” and $t>2s$ , then with probability at least $5/6$ , $\\tilde{t}$ satisfies that $\\tilde{t}&\\ge t-{t-\\tilde{t}}\\\\&>t-\\frac{2\\sqrt{2}\\,\\pi }{P}\\sqrt{tn^2}-\\frac{2\\pi ^2}{P^2}n^2[0] \\\\&\\ge t-\\frac{2\\sqrt{2}}{7}\\sqrt{tn}-\\frac{2}{49}n[0] \\\\&\\ge \\sqrt{t}\\Bigl (\\sqrt{t}-\\frac{2\\sqrt{2}}{7}\\sqrt{n}\\Bigr )-\\frac{2}{49}n\\\\&\\ge \\sqrt{2s}\\Bigl (\\sqrt{2s}-\\frac{2\\sqrt{2}}{7}\\sqrt{s}\\Bigr )-\\frac{2}{49}s>\\frac{3}{2} s.$ Therefore, the algorithm reports “yes” in step 2 alone with probability at least $5/6$ .",
"In this case, the promise of the algorithm in Lemma REF (ii) is not satisfied, but it does not matter what step 3 reports.",
"Finally, consider the case where the correct answer is “no.” In this case, both steps 2 and 3 can report an incorrect answer, and we will bound each of these probability by $1/6$ .",
"The correct answer being “no” means that the set of marked vertices is an independent set of $G$ , and therefore it holds that $t\\le \\alpha ^*(G)\\le s$ by the definitions of $\\alpha ^*(G)$ and $s$ .",
"This implies that with probability at least $5/6$ , $\\tilde{t}$ satisfies that $\\tilde{t}&\\le t+{t-\\tilde{t}}\\\\&<t+\\frac{2\\sqrt{2}\\,\\pi }{P}\\sqrt{tn^2}+\\frac{2\\pi ^2}{P^2}n^2[0] \\\\&\\le t+\\frac{2\\sqrt{2}}{7}\\sqrt{tn}+\\frac{2}{49}n\\\\&\\le s+\\frac{2\\sqrt{2}}{7}\\sqrt{s\\cdot s}+\\frac{2}{49}s<\\frac{3}{2} s.$ Therefore, step 2 reports in an incorrect answer with probability at most $1/6$ .",
"Moreover, because the promise of the algorithm in Lemma REF (ii) is satisfied, step 3 reports an incorrect answer with probability at most $1/6$ .",
"By union bound, the overall error probability is at most $1/6+1/6=1/3$ ."
],
[
"The case of random graphs",
"In this section we analyze the query complexity of the graph collision problem defined over random graphs.",
"Recall that we denote by $\\mu _{n,p}$ the distribution of random graphs on $n$ vertices, where every edge is present with probability $p$ , independently of other edges.",
"We need the following combinatorial lemma.",
"Lemma 8 For arbitrary $p\\in (0,1]$ and $t\\ge 40n\\log n$ , $ \\mathop {\\mathbf {Pr}}_{G\\sim \\mu _{n,p}}{\\left[{\\alpha ^*(G)\\ge t}\\right]}\\le n^{-14n} + 2\\exp \\left(\\frac{-t^2}{200n^2p}\\right).",
"$ Assume $G\\sim \\mu _{n,p}$ .",
"For any $t\\in \\mathbb {N}$ , let $Y_t$ be the expected number of independent sets $S\\subseteq [n]$ that satisfy $\\deg (S)\\ge t$ .",
"Clearly, $ \\mathop {\\mathbf {Pr}}_{}{\\left[{\\alpha ^*(G)\\ge t}\\right]}\\le \\mathop {\\mathbf {E}}_{}{\\left[{Y_t}\\right]}, $ and therefore we want an upper bound on $\\mathop {\\mathbf {E}}_{}{\\left[{Y_t}\\right]}$ .",
"For any $i\\ge 2$ it holds that $ \\genfrac(){0.0pt}{}{n}{i}\\cdot (1-p)^{\\genfrac(){0.0pt}{}{i}{2}}\\le \\exp \\left(i\\log n-\\frac{pi^2}{4}\\right).",
"$ Let $x_0\\in [n]$ , to be fixed later.",
"Then $ \\begin{aligned} \\mathop {\\mathbf {E}}_{}{\\left[{Y_t}\\right]}&\\le \\sum _{i=1}^{x_0}\\genfrac(){0.0pt}{}{n}{i}\\cdot (1-p)^{\\genfrac(){0.0pt}{}{i}{2}}\\cdot \\mathop {\\mathbf {Pr}}_{}{\\left[{\\deg (S)\\ge t~:~|S|=i}\\right]}\\\\&~~+\\sum _{i=x_0+1}^n\\genfrac(){0.0pt}{}{n}{i}\\cdot (1-p)^{\\genfrac(){0.0pt}{}{i}{2}}\\\\&\\le \\sum _{i=1}^{x_0}\\exp (i\\log n + 3nip-t)+\\sum _{i=x_0+1}^n\\exp \\left(i\\log n-\\frac{pi^2}{4}\\right), \\end{aligned} $ where the last inequality follows from Lemma REF .",
"Fix $x_0=\\min \\bigl \\lbrace [\\big ]{\\frac{t}{5np}},n\\bigr \\rbrace $ .",
"Then, noting $t\\ge 40n\\log n$ , it holds that $ \\sum _{i=x_0+1}^n\\exp \\left(i\\log n-\\frac{pi^2}{4}\\right)\\le 2\\exp \\left(-\\frac{px_0^2}{8}\\right), $ and we continue: $ \\begin{aligned} \\mathop {\\mathbf {E}}_{}{\\left[{Y_t}\\right]} &\\le \\exp (\\log x_0+x_0\\log n+3nx_0p-t)+2\\exp \\left(-\\frac{px_0^2}{8}\\right)\\\\&\\le \\exp \\left(-\\frac{7t}{20}\\right)+2\\exp \\left(-\\frac{t^2}{200n^2p}\\right).",
"\\end{aligned} $ The result follows.",
"The theorem below and Corollary REF follow immediately from Theorem REF and Lemma REF .",
"Theorem 9 There exists a universal constant $C$ such that for any $p\\in (0,1]$ , $n\\in \\mathbb {N}$ and $t\\ge 40n\\log n$ the following holds.",
"For $G\\sim \\mu _{n,p}$ , the probability that the (worst-case) quantum query complexity of the graph collision problem on $G$ is greater than $C(\\sqrt{n}+\\sqrt{t})$ is at most $n^{-14n} + 2\\exp \\bigl (\\frac{-t^2}{200n^2p}\\bigr )$ ."
],
[
"Concluding remarks",
"We gave a quantum algorithm for the graph collision problem on graph $G$ on $n$ vertices whose query complexity is bounded as $O(\\sqrt{n}+\\sqrt{\\alpha ^*(G)})$ in terms of the maximum sum of degrees of the vertices in an independent set of $G$ .",
"We used this to show that the graph collision problem has quantum query complexity $\\tilde{O}(\\sqrt{n})$ for almost all graphs if a graph is chosen at random so that each edge is present with a fixed probability independently of other edges.",
"We conclude by stating a few open problems.",
"Clearly improving the algorithm so that its query complexity becomes $\\tilde{O}(\\sqrt{n})$ for all graphs is an important open problem.",
"As another direction, the graph collision problem can be defined also for hypergraphs, and it is used in an algorithm for the subgraph finding problem [12], a natural generalization of the triangle finding problem.",
"Extending the present algorithm to the case of hypergraphs is another open problem."
],
[
"Acknowledgments",
"Dmitry Gavinsky is grateful to Ryan O'Donnell, Rocco Servedio, Srikanth Srinivasan and Li-Yang Tan for helpful discussions.",
"The authors acknowledge support by ARO/NSA under grant W911NF-09-1-0569."
]
] | 1204.1527 |
[
[
"Compton Scattering of Self-Absorbed Synchrotron Emission"
],
[
"Abstract Synchrotron self-Compton (SSC) scattering is an important emission mechanism in many astronomical sources, such as gamma-ray bursts (GRBs) and active galactic nuclei (AGNs).",
"We give a complete presentation of the analytical approximations for the Compton scattering of synchrotron emission with both weak and strong synchrotron self-absorption.",
"All possible orders of the characteristic synchrotron spectral breaks ($\\nu_{\\rm a}$, $\\nu_{\\rm m}$, and $\\nu_{\\rm c}$) are studied.",
"In the weak self-absorption regime, i.e., $\\nu_{\\rm a} < \\nu_c$, the electron energy distribution is not modified by the self-absorption process.",
"The shape of the SSC component broadly resembles that of synchrotron, but with the following features: The SSC flux increases linearly with frequency up to the SSC break frequency corresponding to the self-absorption frequency $\\nu_{\\rm a}$; and the presence of a logarithmic term in the high-frequency range of the SSC spectra makes it harder than the power-law approximation.",
"In the strong absorption regime, i.e.",
"$\\nu_{\\rm a} > \\nu_{\\rm c}$, heating of low energy electrons due to synchrotron absorption leads to pile-up of electrons, and form a thermal component besides the broken power-law component.",
"This leads to two-component (thermal + non-thermal) spectra for both the synchrotron and SSC spectral components.",
"For $\\nu_{\\rm c} < \\nu_{\\rm a} < \\nu_{\\rm m}$, the spectrum is thermal (non-thermal) -dominated if $\\nu_a > \\sqrt{\\nu_m \\nu_c}$ ($\\nu_a < \\sqrt{\\nu_m \\nu_c}$).",
"Similar to the weak-absorption regime, the SSC spectral component is broader than the simple broken power law approximation.",
"We derive the critical condition for strong absorption (electron pile-up), and discuss a case of GRB reverse shock emission in a wind medium, which invokes $\\nu_{\\rm a} > {\\rm max} (\\nu_{\\rm m}, \\nu_{\\rm c})$."
],
[
"Introduction",
"Astrophysical sources powered by synchrotron radiation should have a synchrotron self-Compton (SSC) scattering component.",
"The same electrons that radiate synchrotron photons would scatter these synchrotron seed photons to high energies, forming a distinct spectral component.",
"The SSC mechanism has been invoked to account for the observed high energy emission in many astrophysical sources, such as gamma-ray bursts (GRBs) [12], [23], [4], [14], [19], [25], [22], [24], [27] and active galactic nuclei (AGNs) [8], [3], [26].",
"SSC is a complex process.",
"The flux at each observed frequency includes the contributions from electrons in a wide range of energies, which scatter seed photons in a wide range of frequencies.",
"Therefore, a precise description of the SSC spectrum invokes a complex convolution of the seed photon spectrum and electron energy distribution, which requires numerical calculations.",
"However, for a synchrotron source with shock-accelerated electrons, the injected electron spectrum is usually assumed to be a simple power-law function, the corresponding electron energy distribution and seed synchrotron spectrum thus have simple patterns.",
"Some analytical approximations for the SSC spectrum can be then made if Compton scattering is in the Thomson regime.",
"Besides the injected electron spectrum, two other factors are essential to define the shape of the final electron energy distribution in a synchrotron source: radiation cooling and self-absorption heating.",
"There are three characteristic synchrotron frequencies in the spectrum: the minimum injection frequency ($\\nu _{\\rm m}$ ), the cooling frequency ($\\nu _c$ ), and the self-absorption frequency ($\\nu _{\\rm a}$ ).",
"When $\\nu _{\\rm a} <\\nu _{\\rm c}$ , the heating effect due to self-absorption is not important in modifying the electron energy spectrum.",
"For a continuous injection of a power-law electron spectrum, the final electron energy distribution is a broken power law.",
"The seed synchrotron spectrum for SSC is characterized by a multi-segment broken power law, separated by $\\nu _m$ , $\\nu _c$ , and $\\nu _a$ .",
"Different ordering of the three characteristic frequencies leads to different shapes of the seed synchrotron spectrum.",
"In the literature, usually $\\nu _{\\rm a} < {\\rm min}(\\nu _{\\rm m}, \\nu _{\\rm c})$ is assumed.",
"[19] have derived the approximated expressions of the SSC spectrum in the $\\nu _{\\rm a} < \\nu _{\\rm m} <\\nu _{\\rm c}$ and $\\nu _{\\rm a} < \\nu _{\\rm c} < \\nu _{\\rm m}$ regimes, respectivelyAssuming weak self-absorption, Gou et al.",
"(2007) derived analytical approximations of the SSC component for several other spectral regimes..",
"When $\\nu _{\\rm a}>\\nu _{\\rm c}$ , synchrotron self-absorption becomes an important heating source for the low-energy electrons.",
"Consequently, the electrons are dominated by a quasi-thermal component until a “transition” Lorentz factor $\\gamma _{\\rm t}$ , above which the electrons are no longer affected by the self-absorption heating and keep the normal power law distribution [5], [6], [7].",
"For these strong absorption cases, a thermal peak due to pile-up electrons would appear around $\\nu _{\\rm a}$ in the synchrotron spectrum [10], which would also result in some new features in the SSC spectrum.",
"In this paper, we extend the analysis of [19] and present the full analytical approximated expressions of the SSC spectrum in all six possible cases of $\\nu _{\\rm a}$ , $\\nu _{\\rm m}$ , $\\nu _{\\rm c}$ ordering.",
"In Section 2, three weak synchrotron self-absorption cases ($\\nu _{\\rm a}<\\nu _{\\rm c}$ ) are discussed.",
"In Section 3, we focus on the strong synchrotron self-absorption regime ($\\nu _{\\rm a}>\\nu _{\\rm c}$ ), where synchrotron self-absorption significantly affects the electron energy distribution.",
"By adopting a simplified prescription of the pile-up electron distribution, we derive the expressions of both synchrotron and SSC spectral components.",
"All the expressions in this work are valid in the Thomson regime, so that the Klein-Nishina correction effect [16], [13] is not important in the first order SSC component.",
"We also limit our treatment to the first-order SSC, and assume that the higher-order SSC components [11], [15] are suppressed by the Klein-Nishina effect.",
"Such an assumption is usually valid for most problems.",
"In order to make a simple analytical treatment, we have applied a simplified approximation for the synchrotron spectra, and adopted the simplification that the inverse Compton scattering of mono-energetic electrons off mono-energetic seed photons is also mono-energetic [19].",
"This would not significantly deteriorate precision of the analysis, while making it much simpler."
],
[
"Weak Synchrotron Self-absorption Cases",
"In the single scattering regime, the inverse Compton volume emissivity for a power-law distribution of electrons is [17], [19] $ j^{IC}_{\\nu } = 3 \\sigma _T \\int _{\\gamma _{\\rm m}}^{\\infty }{d \\gamma N(\\gamma )\\int _0^1{d x\\,g(x) \\tilde{f}_{\\nu _s}(x)}},$ where $x \\equiv \\nu /4 \\gamma ^2 \\nu _s$ (an angle-dependent parameter), $\\tilde{f}_{\\nu _s}$ is the incident specific flux in the shock front, $\\sigma _T$ is Thomson scattering cross section, and $g(x) = 1+x+2 x\\ln {x}-2 x^2$ takes care of the angular dependence of the scattering cross section in the limit of $\\gamma \\gg 1$ [1].",
"One can approximate $g(x)= 1$ for $0<x<x_0$ to simplify the integration, which would yield a correct behavior for $x\\ll 1$ (Sari & Esin 2001).",
"With such a simplification, the SSC spectrum is given by (Sari & Esin 2001), $ f^{\\rm {IC}}_{\\nu } = R \\sigma _T \\int _{\\gamma _{\\rm m}}^{\\infty }{d \\gamma N(\\gamma ) \\int _0^{x_0}{d x\\, f_{\\nu _{\\rm s}}(x)}},$ where $f_{\\nu _{\\rm s}}(x)$ is the synchrotron flux, $R$ is the co-moving size of the emission region, and the value of the parameter $x_0$ is set by ensuring energy conservation, i.e.",
"$\\int _0^1{x\\, g(x) d x} = \\int _0^{x_0}{x\\, d x}$ .",
"When $\\nu _{\\rm a}<\\nu _{\\rm c}$ , in the slow cooling regime ($\\gamma _{\\rm m} < \\gamma _{\\rm c}$ ), the electron energy distribution is $ N(\\gamma ) = \\left\\lbrace \\begin{array}{ll}n(p-1)\\gamma _{\\rm m}^{p-1}\\gamma ^{-p}, & \\gamma _{\\rm m} \\le \\gamma \\le \\gamma _{\\rm c}, \\\\n(p-1)\\gamma _{\\rm m}^{p-1}\\gamma _{\\rm c}\\gamma ^{-p-1}, & \\gamma > \\gamma _{\\rm c}.\\end{array} \\right.$ Here $\\gamma _{\\rm m}$ is the minimum Lorentz factor of the injected electrons, and $p$ is electron spectral index.",
"Cooling is efficient for electrons with Lorentz factor above the critical value $\\gamma _{\\rm c}$ .",
"Notice that Eq.REF is only valid for $p>1$ .",
"In the fast cooling regime ($\\gamma _{\\rm c} < \\gamma _{\\rm m}$ ), the electron energy distribution isThis is valid only in the deep fast cooling regime.",
"For a non-steady state with not too deep fast cooling, the electron spectrum can be harder than -2 [21].",
"$ N(\\gamma ) = \\left\\lbrace \\begin{array}{ll}n \\gamma _{\\rm c} \\gamma ^{-2}, & \\gamma _{\\rm c} \\le \\gamma \\le \\gamma _{\\rm m}, \\\\n \\gamma _{\\rm m}^{p-1}\\gamma _{\\rm c} \\gamma ^{-p-1}, & \\gamma >\\gamma _{\\rm m}.\\end{array} \\right.$ In this regime, all the injected electrons are able to cool on the dynamical timescale.",
"Therefore, there is a population of electrons with Lorentz factor below the injection minimum Lorentz factor $\\gamma _{\\rm m}$ .",
"The seed synchrotron spectrum $f_{\\nu _{\\rm s}}$ has spectral beaks at $\\nu _{\\rm a}$ , $\\nu _{\\rm m}$ and $\\nu _{\\rm c}$ , where $\\nu _{\\rm a}$ is the self-absorption frequency, below which the system becomes optically thick, and $\\nu _{\\rm m}$ and $\\nu _{\\rm c}$ are the characteristic synchrotron frequencies for the electrons with Lorentz factors $\\gamma _{\\rm m}$ and $\\gamma _{\\rm c}$ , respectively.",
"As shown in [19], the critical frequencies in the SSC component are defined by different combination of $\\gamma _{\\rm a}, \\gamma _{\\rm m},\\gamma _{\\rm c}$ and $\\nu _{\\rm a},\\nu _{\\rm m}, \\nu _{\\rm c}$ .",
"For convenience, we use a new notation in this paper $\\nu _{ij}^{\\rm {IC}}=4 \\gamma _i^2 \\nu _jx_0,~~~~~~~~~~~~~~~~~~~~~i,j={a,c,m}.",
"$ The physical meaning is the characteristic upscattered frequency for mono-energetic electrons with Lorentz factor $\\gamma _i$ scattering off mono-energetic photons with frequency $\\nu _j$ ."
],
[
"Case I: $\\nu _{\\rm a} < \\nu _{\\rm m} < \\nu _{\\rm c}$",
"This case has been studied by [19].",
"The synchrotron spectrum readsHereafter, the synchrotron spectra are denoted as $f_\\nu (\\nu )$ for simple presentation.",
"Notice that when they are taken as seed spectrum, one should consider them as $f_{\\nu _{\\rm s}}(\\nu _{\\rm s})$ and apply equation (REF ) to calculate the SSC spectra.",
"$ f_{\\nu } = \\left\\lbrace \\begin{array}{ll}f_{\\rm {max}}\\left(\\frac{\\nu _{\\rm a}}{\\nu _{\\rm m}}\\right)^{\\frac{1}{3}}\\left(\\frac{\\nu }{\\nu _{\\rm a}}\\right)^{2}, &\\nu \\le \\nu _{\\rm a}; \\\\f_{\\rm {max}}\\left(\\frac{\\nu }{\\nu _{\\rm m}}\\right)^{\\frac{1}{3}}, &\\nu _{\\rm a} < \\nu \\le \\nu _{\\rm m}; \\\\f_{\\rm {max}}\\left(\\frac{\\nu }{\\nu _{\\rm m}}\\right)^{\\frac{1-p}{2}}, &\\nu _{\\rm m} < \\nu \\le \\nu _{\\rm c}; \\\\f_{\\rm {max}}\\left(\\frac{\\nu _{\\rm c}}{\\nu _{\\rm m}}\\right)^{\\frac{1-p}{2}}\\left(\\frac{\\nu }{\\nu _{\\rm c}}\\right)^{-\\frac{p}{2}}, &\\nu > \\nu _{\\rm c}, \\\\\\end{array} \\right.$ where $f_{\\rm {max}} = f_{\\nu } (\\nu _{\\rm m})$ is the peak flux density of the synchrotron component, which is taken as a constant.",
"Substituting this seed photon spectrum into equation (REF ), the inner integral reads [19] $ I = \\left\\lbrace \\begin{array}{ll} I_1 \\simeq \\frac{5}{2}f_{\\rm {max}} x_0 \\left(\\frac{\\nu _{\\rm a}}{\\nu _{\\rm m}}\\right)^{\\frac{1}{3}}\\left(\\frac{\\nu }{4 \\gamma ^2 \\nu _{\\rm a} x_0}\\right), & \\nu < 4 \\gamma ^2 \\nu _{\\rm a} x_0 \\\\I_2 \\simeq \\frac{3}{2} f_{\\rm {max}} x_0 \\left(\\frac{\\nu }{4 \\gamma ^2 \\nu _{\\rm m}x_0}\\right)^{\\frac{1}{3}}, &4 \\gamma ^2 \\nu _{\\rm a} x_0 < \\nu < 4 \\gamma ^2 \\nu _{\\rm m} x_0 \\\\I_3 \\simeq \\frac{2}{(p+1)} f_{\\rm {max}} x_0 \\left(\\frac{\\nu }{4 \\gamma ^2 \\nu _{\\rm m}x_0}\\right)^{\\frac{1-p}{2}}, &4 \\gamma ^2 \\nu _{\\rm m} x_0 < \\nu < 4 \\gamma ^2 \\nu _{\\rm c} x_0 \\\\I_4 \\simeq \\frac{2}{(p+2)} f_{\\rm {max}} x_0\\left(\\frac{\\nu _{\\rm c}}{\\nu _{\\rm m}}\\right)^{\\frac{1-p}{2}} \\left(\\frac{\\nu }{4\\gamma ^2 \\nu _{\\rm c} x_0}\\right)^{-\\frac{p}{2}}, & \\nu > 4 \\gamma ^2 \\nu _{\\rm c} x_0.\\end{array} \\right.$ Similar to [19], only the leading order of $\\nu $ and zeroth order of $\\nu _{\\rm a}/\\nu _{\\rm m}$ and $\\nu _{\\rm m}/\\nu _{\\rm c}$ are shown.",
"However, we note that higher order small terms are needed to derive the following SSC spectrum (REF ) through integrating the outer integral of equation (REF ).",
"After integration, $f^{\\rm {IC}}_{\\nu }$ is very complex.",
"Keeping only the dominant terms, one gets the analytical approximation $ f_{\\nu }^{\\rm {IC}} &\\simeq & R \\sigma _T n f_{\\rm {max}} x_0 \\\\\\nonumber &\\times &\\left\\lbrace \\begin{array}{ll}\\end{array}\\frac{5}{2} \\frac{(p-1)}{(p+1)}\\left(\\frac{\\nu _{\\rm a}}{\\nu _{\\rm m}}\\right)^{\\frac{1}{3}}\\left(\\frac{\\nu }{\\nu _{\\rm ma}^{\\rm {IC}}}\\right), &\\nu < \\nu _{\\rm ma}^{\\rm {IC}}; \\\\\\right.\\frac{3}{2} \\frac{(p-1)}{(p-1/3)}\\left(\\frac{\\nu }{\\nu _{\\rm mm}^{\\rm {IC}}}\\right)^{\\frac{1}{3}}, &\\nu _{\\rm ma}^{\\rm {IC}} < \\nu < \\nu _{\\rm mm}^{\\rm {IC}}; \\\\$ (p-1)(p+1) (mmIC)1-p2 [4 (p+1/3)(p+1)(p-1/3) + (mmIC)], mmIC < < mcIC; (p-1)(p+1) (mmIC)1-p2 [2(2 p+3)(p+2) - 2(p+1)(p+2) + (ccIC)], mcIC < < ccIC; (p-1)(p+1) (mmIC)-p2 (cm) [2(2p+3)(p+2) - 2(p+2)2 + (p+1)(p+2) (ccIC)], > ccIC.",
".",
"Notice that [19] presented an opposite sign for the term $\\frac{2}{(p+2)^2}$ in the last segment, which might be a typo in that paper.",
"The normalized synchrotron + SSC spectra for this and other two weak self-absorption cases are presented in Figure REF .",
"We note that these analytical expressions are not continuous around the breaks because of dropping the small order terms (see also Sari & Esin 2001), but the mis-match is small.",
"When plotting the SSC curve in Figure REF , we have used the analytical approximations, but added back some smaller order terms to remove the discontinuity."
],
[
"Case II: $\\nu _{\\rm m} < \\nu _{\\rm a} < \\nu _{\\rm c}$",
"The synchrotron photons spectrum reads $ f_{\\nu } = \\left\\lbrace \\begin{array}{ll}f_{\\rm {max}}\\left(\\frac{\\nu _{\\rm m}}{\\nu _{\\rm a}}\\right)^{\\frac{p+4}{2}}\\left(\\frac{\\nu }{\\nu _{\\rm m}}\\right)^{2}, &\\nu \\le \\nu _{\\rm m}; \\\\f_{\\rm {max}}\\left(\\frac{\\nu _{\\rm a}}{\\nu _{\\rm m}}\\right)^{\\frac{1-p}{2}}\\left(\\frac{\\nu }{\\nu _{\\rm a}}\\right)^{\\frac{5}{2}},&\\nu _{\\rm m} < \\nu \\le \\nu _{\\rm a}; \\\\f_{\\rm {max}}\\left(\\frac{\\nu }{\\nu _{\\rm m}}\\right)^{\\frac{1-p}{2}}, &\\nu _{\\rm a} < \\nu \\le \\nu _{\\rm c}; \\\\f_{\\rm {max}}\\left(\\frac{\\nu _{\\rm c}}{\\nu _{\\rm m}}\\right)^{\\frac{1-p}{2}}\\left(\\frac{\\nu }{\\nu _{\\rm c}}\\right)^{-\\frac{p}{2}}, &\\nu > \\nu _{\\rm c}; \\\\\\end{array} \\right.$ Evaluating the inner integral in equation (REF ), we obtain $ I = \\left\\lbrace \\begin{array}{ll} I_1 \\simeq \\frac{2(p+4)}{3(p+1)} f_{\\rm {max}} x_0\\left(\\frac{\\nu _{\\rm m}}{\\nu _{\\rm a}}\\right)^{\\frac{p+1}{2}}\\frac{\\nu }{4 \\gamma ^2 \\nu _{\\rm m} x_0}, & \\nu < 4 \\gamma ^2 \\nu _{\\rm a} x_0 \\\\I_2 \\simeq \\frac{2}{p+1} f_{\\rm {max}} x_0 \\left(\\frac{\\nu }{4 \\gamma ^2 \\nu _{\\rm m}x_0}\\right)^{\\frac{1-p}{2}}, &4 \\gamma ^2 \\nu _{\\rm a} x_0 < \\nu < 4 \\gamma ^2 \\nu _{\\rm c} x_0 \\\\I_3 \\simeq \\frac{2}{(p+2)} f_{\\rm {max}} x_0\\left(\\frac{\\nu _{\\rm c}}{\\nu _{\\rm m}}\\right)^{\\frac{1}{2}}\\left(\\frac{\\nu }{4\\gamma ^2 \\nu _{\\rm m} x_0}\\right)^{-\\frac{p}{2}}, &\\nu > 4 \\gamma ^2 \\nu _{\\rm c} x_0 \\\\\\end{array} \\right.$ An interesting feature of this result is that $I_1$ is linear with $\\nu $ all the way to $\\nu = 4 \\gamma ^2 \\nu _{\\rm a} x_0$ , indicating that a break corresponding to the break in the synchrotron spectrum at $\\nu _{\\rm m}$ does not show up in the SSC spectrum for monoenergetic electron scattering.",
"When $\\nu > 4 \\gamma ^2 \\nu _{\\rm a} x_0$ , the SSC spectrum follows the same frequency dependence as the corresponding seed synchrotron spectrum.",
"After second integration, we get the analytical approximation in this regime: $ f_{\\nu }^{\\rm {IC}} &\\simeq & R \\sigma _T n f_{\\rm {max}} x_0 \\\\\\nonumber &\\times &\\left\\lbrace \\begin{array}{ll}\\end{array}\\frac{2(p+4)(p-1)}{3(p+1)^2}\\left(\\frac{\\nu _{\\rm m}}{\\nu _{\\rm a}}\\right)^{\\frac{p+1}{2}}\\left(\\frac{\\nu }{\\nu _{\\rm mm}^{\\rm {IC}}}\\right), &\\nu < \\nu _{\\rm ma}^{\\rm {IC}}; \\\\\\right.\\frac{(p-1)}{(p+1)}\\left(\\frac{\\nu }{\\nu _{\\rm mm}^{\\rm {IC}}}\\right)^{\\frac{1-p}{2}}\\left[\\frac{ 2(2p+5)}{(p+1)(p+4)} +\\ln {\\left(\\frac{\\nu }{\\nu _{\\rm ma}^{\\rm {IC}}}\\right)}\\right], &\\nu _{\\rm ma}^{\\rm {IC}} < \\nu < \\nu _{\\rm mc}^{\\rm {IC}}; \\\\$ (p-1)(p+1) (mmIC)1-p2 [2 + 2p+4 + (ca)], mcIC < < caIC; (p-1)(p+1) (mmIC)1-p2 [2(2 p+1)(p+1) + (ccIC)], caIC < < ccIC; (p-1)(p+2) (cm)(mmIC)-p2 [2(2p+5)(p+2)+ (ccIC)], > ccIC.",
".",
"Similar to the $I$ result, there is no spectral break around $\\nu _{\\rm mm}^{\\rm {IC}}$ .",
"Another comment is that the logarithmic terms make the SSC spectrum harder than the simple broken power-law approximation above the $\\nu F_\\nu $ peak frequency.",
"At high frequencies, the simple broken power-law approximation may not be adequate to represent the true SSC spectrum."
],
[
"Case III: $\\nu _{\\rm a} < \\nu _{\\rm c} < \\nu _{\\rm m}$",
"This case was also studied by [19].",
"The seed synchrotron spectrum reads $ f_{\\nu } = \\left\\lbrace \\begin{array}{ll}f_{\\rm {max}}\\left(\\frac{\\nu _{\\rm a}}{\\nu _{\\rm c}}\\right)^{\\frac{1}{3}}\\left(\\frac{\\nu }{\\nu _{\\rm a}}\\right)^{2}, &\\nu \\le \\nu _{\\rm a}; \\\\f_{\\rm {max}}\\left(\\frac{\\nu }{\\nu _{\\rm c}}\\right)^{\\frac{1}{3}}, &\\nu _{\\rm a} < \\nu \\le \\nu _{\\rm c}; \\\\f_{\\rm {max}}\\left(\\frac{\\nu }{\\nu _{\\rm c}}\\right)^{-\\frac{1}{2}}, &\\nu _{\\rm c} < \\nu \\le \\nu _{\\rm m}; \\\\f_{\\rm {max}}\\left(\\frac{\\nu _{\\rm c}}{\\nu _{\\rm m}}\\right)^{\\frac{1}{2}}\\left(\\frac{\\nu }{\\nu _{\\rm m}}\\right)^{-\\frac{p}{2}}, &\\nu > \\nu _{\\rm m}; \\\\\\end{array} \\right.$ This gives $ I = \\left\\lbrace \\begin{array}{ll} I_1 \\simeq \\frac{5}{2}f_{\\rm {max}} x_0 \\left(\\frac{\\nu _{\\rm a}}{\\nu _{\\rm c}}\\right)^{\\frac{1}{3}}\\left(\\frac{\\nu }{4 \\gamma ^2 \\nu _{\\rm a} x_0}\\right), & \\nu < 4 \\gamma ^2 \\nu _{\\rm a} x_0 \\\\I_2 \\simeq \\frac{3}{2} f_{\\rm {max}} x_0 \\left(\\frac{\\nu }{4 \\gamma ^2\\nu _{\\rm c} x_0}\\right)^{\\frac{1}{3}}, &4 \\gamma ^2 \\nu _{\\rm a} x_0 < \\nu < 4 \\gamma ^2 \\nu _{\\rm c} x_0 \\\\I_3 \\simeq \\frac{2}{3} f_{\\rm {max}} x_0 \\left(\\frac{\\nu }{4 \\gamma ^2\\nu _{\\rm c} x_0}\\right)^{-\\frac{1}{2}}, &4 \\gamma ^2 \\nu _{\\rm c} x_0 < \\nu < 4 \\gamma ^2 \\nu _{\\rm m} x_0 \\\\I_4 \\simeq \\frac{2}{(p+2)} f_{\\rm {max}} x_0\\left(\\frac{\\nu _{\\rm c}}{\\nu _{\\rm m}}\\right)^{\\frac{1}{2}} \\left(\\frac{\\nu }{4\\gamma ^2 \\nu _{\\rm m} x_0}\\right)^{-\\frac{p}{2}}, & \\nu > 4 \\gamma ^2 \\nu _{\\rm m} x_0\\end{array} \\right.$ and the final SSC spectrum $ f_{\\nu }^{\\rm {IC}} &\\simeq & R \\sigma _T n f_{\\rm {max}} x_0 \\\\\\nonumber &\\times &\\left\\lbrace \\begin{array}{ll}\\end{array}\\frac{5}{6} \\left(\\frac{\\nu _{\\rm a}}{\\nu _{\\rm c}}\\right)^{\\frac{1}{3}}\\left(\\frac{\\nu }{\\nu _{\\rm ca}^{\\rm {IC}}}\\right), &\\nu < \\nu _{\\rm ca}^{\\rm {IC}}; \\\\\\right.\\frac{9}{10}\\left(\\frac{\\nu }{\\nu _{\\rm cc}^{\\rm {IC}}}\\right)^{\\frac{1}{3}}, &\\nu _{\\rm ca}^{\\rm {IC}} < \\nu < \\nu _{\\rm cc}^{\\rm {IC}}; \\\\$ 13 (ccIC)-12 [2815 + (ccIC)], ccIC < < cmIC; 13 (ccIC)-12 [2(p+5)(p+2)(p-1) - 2 (p-1)3 (p+2) + (mmIC)], cmIC < < mmIC; 1(p+2) (cm)(mmIC)-p2 [23 (p+5)(p-1) - 23(p-1)(p+2) + (mmIC)], > mmIC.",
".",
"We note that [19] has an opposite sign in the term $\\ln {\\left(\\frac{\\nu }{\\nu _{\\rm cc}^{\\rm {IC}}}\\right)}$ in the third segment, which might be another typo in that paper.",
"Figure: Total synchrotron ++ SSC spectra for weak synchrotron reabsorption cases (ν a <ν c \\nu _{\\rm a}<\\nu _{\\rm c}).The top panel is for ν a <ν m <ν c \\nu _{\\rm a} <\\nu _{\\rm m} <\\nu _{\\rm c} case; the middle panel is for ν m <ν a <ν c \\nu _{\\rm m} <\\nu _{\\rm a} <\\nu _{\\rm c} case; and the bottom panel is for ν a <ν c <ν m \\nu _{\\rm a} <\\nu _{\\rm c} <\\nu _{\\rm m} case.",
"The thin solid line issynchrotron component.",
"The thick solid line in the SSC component is drawn using theanalytical approximations, while the dashed lines are the brokenpower-law approximation for comparison.",
"In all the cases, the νF ν \\nu F_\\nu peaksfor both the synchrotron and the SSC components are normalized to unity.We define ratio between the SSC luminosity and the synchrotron luminosity as the $X$ parameter similar to Sari & Esin (2001), i.e., $X \\equiv \\frac{L_{\\rm IC}}{L_{\\rm syn}}=\\frac{U_{\\rm ph}}{U_{\\rm B}},$ where $U_{\\rm ph}$ and $U_{\\rm B}$ are the synchrotron photon energy density and magnetic field energy density, respectively.",
"For $\\nu _{\\rm a} < \\nu _{\\rm m} <\\nu _{\\rm c}$ (case I) and $\\nu _{\\rm m} < \\nu _{\\rm a} <\\nu _{\\rm c}$ (case II), the $\\nu f_{\\nu }$ peaks of the synchrotron and the SSC components are at $\\nu _{\\rm c}$ and $\\nu _{\\rm cc}^{IC}$ , respectively (see Figure 1).",
"One can estimate $X = \\frac{L_{\\rm IC}}{L_{\\rm syn}}&\\sim &\\frac{\\nu _{\\rm cc}^{\\rm {IC}}f_{\\nu }^{\\rm {IC}}(\\nu _{\\rm cc}^{\\rm {IC}})}{\\nu _{\\rm c}f_{\\nu }(\\nu _{\\rm c})}\\nonumber \\\\&\\sim &\\frac{\\nu _{\\rm cc}^{\\rm {IC}}R \\sigma _T n f_{\\rm {max}}x_0\\left(\\frac{\\nu _{\\rm cc}^{\\rm {IC}}}{\\nu _{\\rm mm}^{\\rm {IC}}}\\right)^{\\frac{1-p}{2}}}{\\nu _{\\rm c}f_{\\rm {max}}\\left(\\frac{\\nu _{\\rm c}}{\\nu _{\\rm m}}\\right)^{\\frac{1-p}{2}}}\\nonumber \\\\&\\sim &4x_0^2\\sigma _T n R \\gamma _{\\rm c}^2\\left( \\frac{\\gamma _{\\rm c}}{\\gamma _{\\rm m}}\\right)^{1-p}, $ which is consistent with Sari & Esin (2001).",
"Note that when calculating $X$ , we did not include the coefficients in the analytical approximations of the SSC component, which is of order unity.",
"For $\\nu _{\\rm a} < \\nu _{\\rm c} <\\nu _{\\rm m}$ (case III), the $\\nu f_{\\nu }$ peaks of the synchrotron and SSC components are at $\\nu _{\\rm m}$ , and $\\nu _{\\rm mm}^{\\rm IC}$ , respectively.",
"One therefore has $X=\\frac{L_{\\rm IC}}{L_{\\rm syn}}&\\sim &\\frac{\\nu _{\\rm mm}^{\\rm {IC}}f_{\\nu }^{\\rm {IC}}(\\nu _{\\rm mm}^{\\rm {IC}})}{\\nu _{\\rm m}f_{\\nu }(\\nu _{\\rm m})}\\nonumber \\\\&\\sim &\\frac{\\nu _{\\rm mm}^{\\rm {IC}}R \\sigma _T n f_{\\rm {max}}x_0\\left(\\frac{\\nu _{\\rm mm}^{\\rm {IC}}}{\\nu _{\\rm cc}^{\\rm {IC}}}\\right)^{-\\frac{1}{2}}}{\\nu _{\\rm m}f_{\\rm {max}}\\left(\\frac{\\nu _{\\rm m}}{\\nu _{\\rm c}}\\right)^{-\\frac{1}{2}}}\\nonumber \\\\&\\sim &4x_0^2\\sigma _T n R \\gamma _{\\rm c}\\gamma _{\\rm m},$ which is also consistent with Sari & Esin (2001).",
"When $\\nu _{\\rm a}>\\nu _{\\rm c}$ , synchrotron/SSC cooling and self-absorption heating would reach a balance around a specific electron energy under certain conditions (see details in Appendix A).",
"For such cases, the electron energy distribution and the photon spectrum are coupled, a numerical iterative procedure is needed to obtain the self-consistent solution.",
"[5] solved the kinetic equation and found that the electron energy distribution would include two components: a thermal component shaped by synchrotron self-absorption heating, and a non-thermal power-law component.",
"Based on their results [5], the electron distribution is close but not strictly Maxwellian.",
"Strictly, one needs to use equation (REF ) to calculate the SSC spectral component numerically.",
"In the following, we make an approximation to derive analytical results.",
"For the quasi-thermal component, we take $N(\\gamma )\\propto \\gamma ^{2}$ for $\\gamma <\\gamma _{\\rm a}$ to denote the thermal component, and take a sharp cutoff at $\\gamma _{\\rm a}$ .",
"Above this energy, the electron energy distribution is taken as the standard (broken) power law distribution.",
"In particular, for $\\nu _{\\rm c} < \\nu _{\\rm a} < \\nu _{\\rm m}$ , the electron distribution becomes $ N(\\gamma ) = \\left\\lbrace \\begin{array}{ll}n\\frac{3\\gamma ^2}{\\gamma _{\\rm a}^3}, & \\gamma \\le \\gamma _{\\rm a}, \\\\n\\gamma _{\\rm c}\\gamma ^{-2}, & \\gamma _{\\rm a}<\\gamma \\le \\gamma _{\\rm m}.\\\\n\\gamma _{\\rm m}^{p-1}\\gamma _{\\rm c}\\gamma ^{-p-1}, & \\gamma >\\gamma _{\\rm m}.\\end{array} \\right.$ For $\\nu _{\\rm m} < \\nu _{\\rm c} < \\nu _{\\rm a}$ , one has $ N(\\gamma ) = \\left\\lbrace \\begin{array}{ll}n\\frac{3\\gamma ^2}{\\gamma _{\\rm a}^3}, & \\gamma \\le \\gamma _{\\rm a}, \\\\n(p-1)\\gamma _{\\rm m}^{p-1}\\gamma _{\\rm c}\\gamma ^{-p-1}, & \\gamma >\\gamma _{\\rm a}.\\end{array} \\right.$ For $\\nu _{\\rm c} < \\nu _{\\rm m} < \\nu _{\\rm a}$ , one has $ N(\\gamma ) = \\left\\lbrace \\begin{array}{ll}n\\frac{3\\gamma ^2}{\\gamma _{\\rm a}^3}, & \\gamma \\le \\gamma _{\\rm a}, \\\\n\\gamma _{\\rm m}^{p-1}\\gamma _{\\rm c}\\gamma ^{-p-1}, & \\gamma >\\gamma _{\\rm a}.\\end{array} \\right.$ Following these new shapes of the electron distribution, the synchrotron photon spectra can be calculated, which also contain a thermal component and a (broken) power-law component.",
"Still applying equation (REF ), one can analytically calculate the SSC spectral component for another three cases in this regime.",
"We note that due to the simple approximation to the complicated electron pile-up process, the analytical results presented below are not as precise as those in the weak absorption cases."
],
[
"Case IV: $\\nu _{\\rm c} < \\nu _{\\rm a} < \\nu _{\\rm m}$",
"In this case, the synchrotron photon spectrum reads $ f_{\\nu } = \\left\\lbrace \\begin{array}{ll}f_{\\rm {max}}\\left(\\frac{\\nu }{\\nu _{\\rm a}}\\right)^{2}, &\\nu \\le \\nu _{\\rm a}; \\\\f_{\\rm {max}}\\mathfrak {R}\\left(\\frac{\\nu }{\\nu _{\\rm a}}\\right)^{-\\frac{1}{2}}, &\\nu _{\\rm a}<\\nu \\le \\nu _{\\rm m}; \\\\f_{\\rm {max}}\\mathfrak {R}\\left(\\frac{\\nu _{\\rm m}}{\\nu _{\\rm a}}\\right)^{-\\frac{1}{2}}\\left(\\frac{\\nu }{\\nu _{\\rm m}}\\right)^{-\\frac{p}{2}}, & \\nu > \\nu _{\\rm m};\\end{array} \\right.$ where $\\mathfrak {R}$ is the discontinuity ratio in the electron distribution at $\\gamma _{\\rm a}$ , $\\mathfrak {R}=\\frac{\\gamma _{\\rm c}}{3\\gamma _{\\rm a}}~.$ One can then derive $ I = \\left\\lbrace \\begin{array}{ll} I_1 \\simeq f_{\\rm {max}}x_0 \\left(\\frac{1}{2}\\mathfrak {R}+1\\right)\\left(\\frac{\\nu }{4 \\gamma ^2 \\nu _{\\rm a} x_0}\\right), & \\nu < 4 \\gamma ^2 \\nu _{\\rm a} x_0 \\\\I_2 \\simeq \\frac{1}{2} f_{\\rm {max}} x_0\\mathfrak {R}\\left(\\frac{\\nu }{4 \\gamma ^2 \\nu _{\\rm a}x_0}\\right)^{-\\frac{1}{2}}, &4 \\gamma ^2 \\nu _{\\rm a} x_0 < \\nu < 4 \\gamma ^2 \\nu _{\\rm m} x_0 \\\\I_3 \\simeq \\frac{3}{2(p+2)} f_{\\rm {max}} x_0\\mathfrak {R}\\left(\\frac{\\nu _{\\rm a}}{\\nu _{\\rm m}}\\right)^{\\frac{1}{2}}\\left(\\frac{\\nu }{4 \\gamma ^2 \\nu _{\\rm m}x_0}\\right)^{-\\frac{p}{2}}, &\\nu > 4 \\gamma ^2 \\nu _{\\rm m} x_0 \\\\\\end{array} \\right.$ and $ f_{\\nu }^{\\rm {IC}} &\\simeq & R \\sigma _{\\rm T} n f_{\\rm {max}} x_0 \\\\\\nonumber &\\times &\\left\\lbrace \\begin{array}{ll}\\end{array}\\left(\\frac{1}{2}\\mathfrak {R}+1\\right)\\left(\\mathfrak {R}+4\\right)\\left(\\frac{\\nu }{\\nu _{\\rm aa}^{\\rm {IC}}}\\right), &\\nu < \\nu _{\\rm aa}^{\\rm {IC}}; \\\\\\right.\\mathfrak {R}\\left(\\frac{\\nu }{\\nu _{\\rm aa}^{\\rm {IC}}}\\right)^{-\\frac{1}{2}}\\left[\\frac{1}{6}\\mathfrak {R}+\\frac{9}{10}+\\frac{1}{4}\\mathfrak {R}\\ln {\\left(\\frac{\\nu }{\\nu _{\\rm aa}^{\\rm {IC}}}\\right)}\\right], &\\nu _{\\rm aa}^{\\rm {IC}} < \\nu < \\nu _{\\rm am}^{\\rm {IC}}; \\\\$ R2 (aaIC)-12[3p-1 -12 + 34(mmIC)], amIC < < mmIC; 92(p+2)R2 (am)(mmIC)-p2 [4p+3(am)p-1ac+3(p+1)(p-1)(p+2) +12mmIC], > mmIC.",
".",
"In this case, there are two peaks in the $\\nu F_\\nu $ spectrum for the synchrotron and SSC components, respectively.",
"For the synchrotron component, the thermal peak is at $(25/9) \\nu _{\\rm a} \\simeq 2.8\\nu _a$ , and the non-thermal peak is at $\\nu _{\\rm m}$ .",
"For the SSC component, the thermal peak at $\\nu _{\\rm aa}^{IC}$ , and the non-thermal peak at $\\nu _{\\rm mm}^{\\rm IC}$ .",
"The relative importance of the two peaks depend on the relative location of $\\nu _{\\rm a}$ with respect to $\\nu _{\\rm c}$ and $\\nu _{\\rm m}$ .",
"More specifically, the spectrum is non-thermal-dominated when $\\nu _{\\rm a} <\\sqrt{\\nu _{\\rm m} \\nu _{\\rm c}}$ , and is thermal-dominated when $\\nu _{\\rm a} > \\sqrt{\\nu _{\\rm m} \\nu _{\\rm c}}$ .",
"In Figure 2, we compare the above simplified analytical approximation (solid) with a simplest power law prescription (dashed) of the SSC component.",
"The non-thermal-dominated and the thermal-dominated cases are presented in Figures 2a and 2b, respectively.",
"Below $\\nu _{\\rm mm}^{\\rm IC}$ , similar to the weak self-absorption regime (cases I-III), the logarithmic terms make the analytical spectrum harder than the simple broken power-law approximation above the non-thermal $\\nu F_\\nu $ peak frequency.",
"At high frequencies, the simple broken power-law approximation is not adequate to represent the true SSC spectrum."
],
[
"Case V and VI: $\\nu _{\\rm a} > {\\rm max} (\\nu _{\\rm m}, \\nu _{\\rm c})$",
"For these two cases ($\\nu _{\\rm m} < \\nu _{\\rm c} < \\nu _{\\rm a}$ and $\\nu _{\\rm c} < \\nu _{\\rm m} < \\nu _{\\rm a}$ ), the treatments and results are rather similar to each other.",
"we take $\\nu _{\\rm m} < \\nu _{\\rm c} < \\nu _{\\rm a}$ as an example.",
"In this case, the synchrotron spectrum reads $ f_{\\nu } = \\left\\lbrace \\begin{array}{ll}f_{\\rm {max}}\\left(\\frac{\\nu }{\\nu _{\\rm a}}\\right)^{2}, &\\nu \\le \\nu _{\\rm a}; \\\\f_{\\rm {max}}\\mathfrak {R}\\left(\\frac{\\nu }{\\nu _{\\rm a}}\\right)^{-\\frac{p}{2}}, &\\nu > \\nu _{\\rm a}; \\\\\\end{array} \\right.$ where $\\mathfrak {R}=(p-1)\\frac{\\gamma _{\\rm c}}{3\\gamma _{\\rm a}}\\left(\\frac{\\gamma _{\\rm m}}{\\gamma _{\\rm a}}\\right)^{p-1}.$ Applying equation (REF ), the inner integral $I$ can be then approximated as $ I = \\left\\lbrace \\begin{array}{ll} I_1 \\simeq f_{\\rm {max}}x_0\\left(\\frac{3\\mathfrak {R}}{2(p+2)}+1\\right)\\left(\\frac{\\nu }{4 \\gamma ^2 \\nu _{\\rm a} x_0}\\right), & \\nu < 4 \\gamma ^2 \\nu _{\\rm a} x_0 \\\\I_2 \\simeq \\frac{3}{2(p+2)} f_{\\rm {max}} x_0\\mathfrak {R}\\left(\\frac{\\nu }{4 \\gamma ^2 \\nu _{\\rm a}x_0}\\right)^{-\\frac{p}{2}}, & \\nu > 4 \\gamma ^2 \\nu _{\\rm a} x_0.\\end{array} \\right.$ Integrating over the outer integral, one gets $ f_{\\nu }^{\\rm {IC}} &\\simeq & R \\sigma _{\\rm T} n f_{\\rm {max}} x_0 \\\\\\nonumber &\\times &\\left\\lbrace \\begin{array}{ll}\\left(\\frac{\\mathfrak {3R}}{2(p+2)}+1\\right)\\left(\\frac{\\mathfrak {3R}}{p+2}+4\\right)\\left(\\frac{\\nu }{\\nu _{\\rm aa}^{\\rm {IC}}}\\right), &\\nu < \\nu _{\\rm aa}^{\\rm {IC}}; \\\\\\frac{1}{p+2}\\left[\\frac{6\\mathfrak {R}}{p+3}+\\mathfrak {R}\\left(\\frac{9\\mathfrak {R}}{2(p+2)}+1\\right)+\\frac{9\\mathfrak {R}^2}{4}\\ln {\\left(\\frac{\\nu }{\\nu _{\\rm aa}^{\\rm {IC}}}\\right)}\\right]\\left(\\frac{\\nu }{\\nu _{\\rm aa}^{\\rm {IC}}}\\right)^{-\\frac{p}{2}}, & \\nu > \\nu _{\\rm aa}^{\\rm {IC}}; \\\\\\end{array} \\right.$ The case of $\\nu _{\\rm c} < \\nu _{\\rm m} < \\nu _{\\rm a}$ is almost identical to the above $\\nu _{\\rm m} < \\nu _{\\rm c} < \\nu _{\\rm a}$ .",
"The only difference is that the expression of $\\mathfrak {R}$ is modified to $\\mathfrak {R}=\\frac{\\gamma _{\\rm c}}{3\\gamma _{\\rm a}}\\left(\\frac{\\gamma _{\\rm m}}{\\gamma _{\\rm a}}\\right)^{p-1}.$ This is reasonable, since in the fast cooling case, the electron energy spectral index is $p=2$ , so that the factor $(p-1)$ can be reduced to 1.",
"The analytical results and simple broken power-law approximation in this regime is identical to Figure 3c.",
"We note again that full numerical calculations are needed to obtain more accurate results.",
"Figure: Same as Figure , but for strong synchrotron reabsorption cases .The top panel is for ν c <ν a <ν m \\nu _{\\rm c} < \\nu _{\\rm a} <\\nu _{\\rm m} andν a <ν m ν c \\nu _{\\rm a} < \\sqrt{\\nu _{\\rm m} \\nu _{\\rm c}} case; the middlepanel is for ν c <ν a <ν m \\nu _{\\rm c} < \\nu _{\\rm a} <\\nu _{\\rm m} and ν a >ν m ν c \\nu _{\\rm a} > \\sqrt{\\nu _{\\rm m} \\nu _{\\rm c}} case; and the bottom panel isfor ν a > max (ν m ,ν c )\\nu _{\\rm a} > {\\rm max} (\\nu _{\\rm m}, \\nu _{\\rm c}) case.",
"Allthe solid lines are analytical approximations and the dashed linesare broken power-law approximations.Finally, we investigate the $X$ parameter in the strong synchrotron self-absorption regime.",
"For $\\nu _{\\rm c} < \\nu _{\\rm a} <\\nu _{\\rm m}$ (case IV), if the spectrum is non-thermal-dominated, the synchrotron and SSC emission components peak at $\\nu _{\\rm m}$ and $\\nu _{\\rm mm}^{\\rm IC}$ , respectively.",
"One thus has $X=\\frac{L_{\\rm IC}}{L_{\\rm syn}}&\\sim &\\frac{\\nu _{\\rm mm}^{\\rm {IC}}f_{\\nu }^{\\rm {IC}}(\\nu _{\\rm mm}^{\\rm {IC}})}{\\nu _{\\rm m}f_{\\nu }(\\nu _{\\rm m})}\\nonumber \\\\&\\sim &\\frac{\\nu _{\\rm mm}^{\\rm {IC}}R \\sigma _{\\rm T} n f_{\\rm {max}}x_0\\mathfrak {R}^2\\left(\\frac{\\nu _{\\rm mm}^{\\rm {IC}}}{\\nu _{\\rm aa}^{\\rm {IC}}}\\right)^{-\\frac{1}{2}}}{\\nu _{\\rm m}\\mathfrak {R}f_{\\rm {max}}\\left(\\frac{\\nu _m}{\\nu _a}\\right)^{-\\frac{1}{2}}}\\nonumber \\\\&\\sim &4x_0^2\\sigma _{\\rm T} n R \\gamma _{\\rm c}\\gamma _{\\rm m}.$ If the spectrum is thermal-dominated, the synchrotron and SSC emission components peak at $\\nu _{\\rm a}$ and $\\nu _{\\rm aa}^{\\rm IC}$ , respectively.",
"One has $X=\\frac{L_{\\rm IC}}{L_{\\rm syn}}&\\sim &\\frac{\\nu _{\\rm aa}^{\\rm {IC}}f_{\\nu }^{\\rm {IC}}(\\nu _{\\rm aa}^{\\rm {IC}})}{\\nu _{\\rm a}f_{\\nu }(\\nu _{\\rm a})}\\nonumber \\\\&\\sim &\\frac{\\nu _{\\rm aa}^{\\rm {IC}}R \\sigma _{\\rm T} n f_{\\rm {max}} x_0}{\\nu _{\\rm a}f_{\\rm {max}}}\\nonumber \\\\&\\sim &4x_0^2\\sigma _{\\rm T} n R \\gamma _{\\rm a}^2.$ In general, the $X$ parameter for $\\nu _{\\rm c} < \\nu _{\\rm a}<\\nu _{\\rm m}$ (case IV) is $4x_0^2\\sigma _{\\rm T} n R \\cdot {\\rm max}(\\gamma _{\\rm a}^2,\\gamma _{\\rm c}\\gamma _{\\rm m})$ .",
"For $\\nu _{\\rm m} < \\nu _{\\rm c} <\\nu _{\\rm a}$ (case V) and $\\nu _{\\rm c} < \\nu _{\\rm m} <\\nu _{\\rm a}$ (case VI), the synchrotron and SSC emission components peak at $\\nu _{\\rm a}$ and $\\nu _{\\rm aa}^{IC}$ , respectively.",
"In this case, one has $ X=\\frac{L_{\\rm IC}}{L_{\\rm syn}}&\\sim &\\frac{\\nu _{\\rm aa}^{\\rm {IC}}f_{\\nu }^{\\rm {IC}}(\\nu _{\\rm aa}^{\\rm {IC}})}{\\nu _{\\rm a}f_{\\nu }(\\nu _{\\rm a})}\\nonumber \\\\&\\sim &\\frac{\\nu _{\\rm aa}^{\\rm {IC}}R \\sigma _{\\rm T} n f_{\\rm {max}} x_0}{\\nu _{\\rm a}f_{\\rm {max}}}\\nonumber \\\\&\\sim &4x_0^2\\sigma _{\\rm T} n R \\gamma _{\\rm a}^2.$ which is same as the thermal-dominated case for $\\nu _{\\rm c} <\\nu _{\\rm a} <\\nu _{\\rm m}$ (case IV).",
"So in general the expression of $X$ is equation (REF ) only if the spectrum is thermal-dominated."
],
[
"Conclusion and discussion",
"We have extended the analysis of [19] and derived the analytical approximations of the SSC spectra of all possible orders of the three synchrotron characteristic frequencies $\\nu _{\\rm a}$ , $\\nu _{\\rm m}$ , and $\\nu _{\\rm c}$ .",
"Based on the relative order between $\\nu _{\\rm a}$ and $\\nu _{\\rm c}$ , we divide the six possible orders into two regimes.",
"In the weak self-absorption regime $\\nu _{\\rm a}<\\nu _{\\rm c}$ , self-absorption does not affect the electron energy distribution.",
"Two cases in this regime have been studied by [19].",
"Our results are consistent with theirs (except the two typos in their paper).",
"For the other regime $\\nu _{\\rm m}<\\nu _{\\rm a}<\\nu _{\\rm c}$ , we find that the SSC spectrum is linear to $\\nu $ all the way to $\\nu _{\\rm ma}^{\\rm {IC}}$ , and there is no break corresponding to $\\nu _{\\rm mm}^{\\rm {IC}}$ .",
"In the strong self-absorption $\\nu _{\\rm a}>\\nu _{\\rm c}$ regime, synchrotron self-absorption heating balances synchrotron/SSC cooling, leading to pile-up of electrons at a certain energy, so that the electron energy distribution is significantly altered, with an additional thermal component besides the non-thermal power law component.",
"Both the synchrotron and the SSC spectral components become two-hump shaped.",
"To get an analytical approximation of the SSC spectrum, we simplified the quasi-thermal electron energy distribution as a power law with a sharp cutoff above the piling up energy, and derived the analytical approximation results of the synchrotron and SSC spectral components.",
"We suggest that for the thermal-dominated cases, i.e.",
"$\\nu _{\\rm a} > \\sqrt{\\nu _{\\rm m}\\nu _{\\rm c}}$ in the $\\nu _{\\rm c} < \\nu _{\\rm a} < \\nu _{\\rm m}$ regime or the $\\nu _{\\rm a}> \\max (\\nu _{\\rm m}, \\nu _{\\rm c})$ regime, full numerical calculations are needed to get accurate results.",
"In general, the SSC component roughly tracks the shape of the seed synchrotron component, but is smoother and harder at high energies.",
"For all the cases, we compare our analytical approximation results of SSC component with the simplest broken power-law prescription.",
"We find that in general the presence of the logarithmic terms in the high energy range makes the SSC spectrum harder than the broken power-law approximation.",
"One should consider these terms when studying high energy emission.",
"The only exceptions are the $\\nu _{\\rm a} > \\max (\\nu _{\\rm m}, \\nu _{\\rm c})$ regimes.",
"However, in these regimes the analytical approximations may be no longer good, and one should appeal to full numerical calculations.",
"Our newly derived spectral regimes may find applications in astrophysical objects with high “compactness” (i.e.",
"high luminosity, and small size).",
"In these cases, $\\nu _{\\rm a}$ can be higher than $\\nu _{\\rm c}$ or $\\nu _{\\rm m}$ , or even both (see Appendix A for the critical condition).",
"For example, in the early afterglow phase of GRBs, slow cooling may be relevant, and the radio afterglow is self-absorbed with $\\nu _{\\rm a}$ above $\\nu _{\\rm m}$ [2].",
"In the prompt emission phase when fast cooling is more relevant, the self-absorption frequency can be above $\\nu _{\\rm c}$ [20].",
"An example of the extreme case $\\nu _{\\rm a} > {\\rm max} (\\nu _{\\rm m}, \\nu _{\\rm c})$ can be identified for a GRB problem.",
"For a dense circumburst medium with a wind-like ($n \\propto r^{-2}$ ) structure, in the reverse shock region, the condition $\\nu _{\\rm a} > \\max (\\nu _{\\rm m}, \\nu _{\\rm c})$ can be satisfied [10].",
"For a GRB with isotropic energy $E=10^{52} E_{52}$ , initial Lorentz factor $\\Gamma _0=100\\Gamma _{0,2}$ , initial shell width $\\Delta = 10^{12} \\Delta _{12}$ running into stellar wind with density $\\rho = (5\\times 10^{11} {\\rm g~cm^{-1}}) A_* r^{-2}$ , one can derive following parameters at the shock crossing radius $r_\\times $ : The blastwave Lorentz factor $\\Gamma _\\times = 25.8 A_*^{-1/4} \\Delta _{12}^{-1/4} E_{52}^{1/4}$ , $\\nu _{\\rm m} = 3.1 \\times 10^{14} ~{\\rm Hz}~ [g(p)/g(2.3)] A_*\\Delta _{12}^{-1/2} E_{52}^{-1/2} \\epsilon _{e,-1}^2\\epsilon _{B,-2}^{1/2} \\Gamma _{0,2}^2$ , $\\nu _c = 1.2\\times 10^{12}~{\\rm Hz}~ A_*^{-2} \\Delta _{12}^{1/2} E_{52}^{1/2}\\epsilon _{B,-2}^{-3/2}$ , $\\nu _a = 4.6\\times 10^{14}~{\\rm Hz}~A_*^{3/5} \\Delta _{12}^{-11/10} E_{52}^{1/10} \\epsilon _{B.-2}^{3/10}\\Gamma _{0,2}^{-2/5}$ .",
"Here $\\epsilon _e = 0.1 \\epsilon _{e,-1}$ and $\\epsilon _B = 0.01 \\epsilon _{B,-2}$ are microphysics shock parameters for the internal energy fraction that goes to electrons and magnetic fields, $p$ is the electron spectral index, and $g(p) =(p-2)/(p-1)$ .",
"We can see that for typical parameters, $\\nu _a > {\\rm max} (\\nu _c, \\nu _m)$ is satisfied.",
"In this regime, one should check whether the “Razin” plasma effect is important.",
"At shock crossing time, the comoving number density of the shocked ejecta region is $n^{\\prime } = 2.3 \\times 10^8 ~{\\rm cm^{-3}}~ A_*^{5/4} \\Delta _{12}^{-7/4}E_{52}^{-1/4} \\Gamma _{0,1}^{-1}$ .",
"Noticing that the comoving plasma angular frequency is $\\omega ^{\\prime }_p = 5.63 \\times 10^4 ~{\\rm s^{-1}}{n^{\\prime }}^{1/2}$ , one can write the plasma frequency in the observer frame as $\\nu _p = 1.4\\times 10^{11}~{\\rm Hz}~ A_*^{3/8}\\Delta _{12}^{-9/8} E_{52}^{1/8} \\Gamma _{0,2}^{-1/2}$ .",
"Multiplying by $\\gamma _{\\rm a} =102 A_*^{1/20} \\Delta _{12}^{-1/20} E_{52}^{1/20}\\epsilon _{B.-2}^{-1/10} \\Gamma _{0,2}^{-1/5}$ , one gets $\\gamma _{\\rm a}\\nu _p = 1.4 \\times 10^{13}~{\\rm Hz}~ A_*^{17/40}\\Delta _{12}^{-47/40} E_{52}^{7/40} \\epsilon _{B,-2}^{-1/10}\\Gamma _{0,2}^{-7/10}$ , which is much smaller than $\\nu _{\\rm a}$ .",
"This suggests that the Razin effect is not important [17], and the dominant mechanism to suppress synchrotron emission at low energies is synchrotron self-absorption.",
"Notice that for this particular problem, the second order Comptonization may not be suppressed by the Klein-Nishina effect, and one has to introduce it for a fully self-consistent treatment."
],
[
"Acknowledgements",
"We thank the referee for constructive comments, Martin Rees for an important remark, and Zhuo Li, Resmi Lekshmi and Yuan-Chuan Zou for helpful discussions.",
"We acknowledge the National Basic Research Program (“973\" Program) of China under Grant No.",
"2009CB824800.",
"This work is supported by NSF under Grant No.",
"AST-0908362.",
"WHL acknowledges support by National Natural Science Foundation of China (grants 11003004, 11173011 and U1231101), HG and WHL acknowledge Fellowship support from China Scholarship Program, and XFW acknowledges support by the One-Hundred-Talents Program."
],
[
"Condition of electron pile-up and strong absorption",
"By applying the Einstein coefficients and their relations to a system with three energy levels, [6] have derived one useful analytical expression of the cross section for synchrotron self-absorption: $ \\sigma _{\\rm S}(\\gamma ,\\nu ) = \\left\\lbrace \\begin{array}{ll}\\frac{2^{2/3}\\sqrt{3}\\pi \\Gamma ^2(4/3)}{5}\\frac{\\sigma _{\\rm T}}{\\alpha _{\\rm f}}\\frac{B_{\\rm cr}}{B}\\left(\\frac{\\gamma \\nu }{3\\nu _{\\rm L}}\\right)^{-5/3}, & \\frac{\\nu _{\\rm L}}{\\gamma } < \\nu \\ll \\frac{3}{2}\\gamma ^2\\nu _{\\rm L}, \\\\\\frac{\\sqrt{3}}{2}\\pi ^2\\frac{\\sigma _{\\rm T}}{\\alpha _{\\rm f}}\\frac{B_{\\rm cr}}{B}\\frac{1}{\\gamma ^3}\\left(\\frac{\\nu _{\\rm L}}{\\nu }\\right)\\rm {exp}\\left(\\frac{-2\\nu }{3\\gamma ^2\\nu _{\\rm L}}\\right), & \\nu \\gg \\frac{3}{2}\\gamma ^2\\nu _{\\rm L}.\\end{array} \\right.$ where $\\gamma $ is the relevant electron Lorentz factor, $\\nu $ is photon frequency being absorbed, $\\alpha _{\\rm f}$ is the fine structure constant, $B_{\\rm cr}=\\alpha _{\\rm f}(m_ec^2/r_e^3)^{1/2}\\approx 4.4\\times 10^{13} \\rm G$ is the critical magnetic field strength, $r_e$ is the classical electron radius, and $\\nu _{\\rm L}=eB/2\\pi m_ec$ is the electron cyclotron frequency.",
"All the parameters introduced in this section are in the comoving frame.",
"For a simple derivation of the electron pile-up condition, we take an approximate form $ \\sigma _{\\rm S}(\\gamma ,\\nu ) = \\left\\lbrace \\begin{array}{ll}\\frac{2^{2/3}\\sqrt{3}\\pi \\Gamma ^2(4/3)}{5}\\frac{\\sigma _{\\rm T}}{\\alpha _{\\rm f}}\\frac{B_{\\rm cr}}{B}\\left(\\frac{\\gamma \\nu }{3\\nu _{\\rm L}}\\right)^{-5/3}, & \\frac{\\nu _{\\rm L}}{\\gamma } < \\nu \\le \\frac{3}{2}\\gamma ^2\\nu _{\\rm L}, \\\\0, & \\nu > \\frac{3}{2}\\gamma ^2\\nu _{\\rm L}.\\end{array} \\right.$ For electrons with Lorenz factor $\\gamma $ , the heating rate due to synchrotron self-absorption can be estimated as $\\dot{\\gamma }^+(\\gamma )=\\int _{0}^{\\infty }c\\cdot n_{\\nu }\\cdot h\\nu \\cdot \\sigma _{\\rm S}(\\gamma ,\\nu ) \\cdot d\\nu $ where $n_{\\nu }$ is the specific photon number density at frequency $\\nu $ contributed by all the electrons.",
"The cooling rate for electrons with Lorentz factor of $\\gamma $ is $\\dot{\\gamma }^-(\\gamma )&=&(1+Y)\\cdot P_{\\rm syn}\\nonumber \\\\&=&(1+Y)\\times \\frac{4}{3}\\sigma _{\\rm T}c\\gamma ^2\\frac{B^2}{8\\pi },$ where $Y\\equiv \\frac{P_{\\rm ssc}}{P_{\\rm syn}}$ is the Compton parameter.",
"By balancing the heating and cooing rate, one can easily obtain the critical electron Lorentz factor $\\gamma _{\\rm cr}$ , which satisfies $ \\dot{\\gamma }^+(\\gamma _{\\rm cr})=\\dot{\\gamma }^-(\\gamma _{\\rm cr})$ Initially, the photon spectrum has not been revised through self-absorption, i.e., $n_{\\nu }\\propto \\nu ^{1/3}$ .",
"One therefore has $\\gamma _{\\rm cr} =2.1\\times 10^4B^{-3/5}{\\cal F}_{\\rm \\nu ,max}^{3/10}\\gamma _{c}^{-1/5}(1+Y)^{-3/10}$ The electron pile-up (strong absorption) condition can be expressed as $ \\gamma _{\\rm cr} > \\gamma _{c}=\\frac{6\\pi m_ec}{\\sigma _{\\rm T} B^2 t(1+Y)}~.$ With equations REF - REF , the pile-up condition can be expressed as $\\left(\\frac{B}{100\\rm G}\\right)^2\\times \\left(\\frac{t}{100\\rm s}\\right)^{4/3}\\times {\\cal F}_{\\rm \\nu ,max}^{1/3}\\times \\left(\\frac{1+Y}{2}\\right)>1 $ where ${\\cal F}_{\\rm \\nu , max}=\\frac{f_{\\rm max}}{\\Gamma (1+z)}\\left(\\frac{d_{\\rm L}}{R}\\right)^2=1~{\\rm erg~cm^{-2}~s^{-1}~Hz^{-1}}~\\frac{f_{\\rm max,mJy}}{\\Gamma _{2}(1+z)}\\left(\\frac{d_{\\rm L,28}}{R_{14}}\\right)^2$ is the synchrotron peak flux in the emission region.",
"Here $d_{\\rm L}$ is the luminosity distance of the source, and $R$ is the distance of the emission region from the central engine.",
"One can immediately see that this condition is very difficult to satisfy.",
"It requires a strong magnetic field, long dynamical time scale and high synchrotron flux.",
"In the GRB afterglow problem, for forward shock emission, $B$ decreases with $t$ rapidly, and there is essentially no parameter space to satisfy the condition.",
"This condition may be realized in extreme conditions, e.g.",
"the reverse shock emission during shock crossing phase for a wind medium, as discussed in Sect.4 in the main text.",
"One interesting note is that SSC cooling only enhances the pile-up condition.",
"Once the pile-up condition is satisfied for synchrotron cooling only, adding SSC cooling only makes the condition more easily satisfied (as shown in equation REF ).",
"Once the electron pile-up process is triggered, both electron distribution and photon spectrum would be modified, so that the value of $\\gamma _{\\rm cr}$ is modified correspondingly.",
"According to the numerical calculation results [5], [6], [7], the electron distribution is dominated by a quasi-thermal component until a “transition” Lorentz factor $\\gamma _{\\rm t}$ , above which the electrons go back to the optically-thin normal power law.",
"In this case, $\\gamma _{\\rm cr}$ should be around the thermal peak, and $\\gamma _a$ should be around the “transition” Lorentz factor $\\gamma _{\\rm t}$ , which is slightly larger than $\\gamma _{\\rm cr}$ .",
"Consequently, one would roughly have $\\gamma _a \\sim \\gamma _{\\rm cr}\\sim \\gamma _{t}$ , so that the assumption of a sharp cutoff in the electron distribution around this energy is justified.",
"In the main text, we did not differentiate the three Lorentz factors, and only adopt $\\gamma _a$ in the expressions."
]
] | 1204.1386 |
[
[
"A class of Fejer convergent algorithms, approximate resolvents and the\n Hybrid Proximal-Extragradient method"
],
[
"Abstract A new framework for analyzing Fejer convergent algorithms is presented.",
"Using this framework we define a very general class of Fejer convergent algorithms and establish its convergence properties.",
"We also introduce a new definition of approximations of resolvents which preserve some useful features of the exact resolvent, and use this concept to present an unifying view of the Forward-Backward splitting method, Tseng's Modified Forward-Backward splitting method and Korpelevich's method.",
"We show that methods based on families of approximate resolvents fall within the aforementioned class of Fejer convergent methods.",
"We prove that such approximate resolvents are the iteration maps of the Hybrid Proximal-Extragradient method."
],
[
"Introduction",
"In this work we introduce a new framework for analysing Fejér convergent algorithms in Hilbert spaces, by means of recursive inclusions and sequences of point-to-set maps.",
"This framework defines a new class of Fejér convergent methods, which is general enough to encompass, for example, the classical Forward-Backward splitting method, Tseng's Modified Forward-Backward splitting method and Korpelevich's method.",
"Using this framework, we prove for good that convergence with summable errors is a generic property of a large class of Fejér convergent algorithms.",
"Therefore, we regard this convergence result (with summable errors) as a rather negative result, in the sense that it is too generic to convey useful information on Fejér convergent methods.",
"For sure, this kind of convergence for particular Fejér convergent algorithms lacks particular value.",
"Another original contribution of this work is the concept of approximate resolvents of maximal monotone operators.",
"Approximate resolvents retain the relevant features of exact resolvents as iteration maps for finding zeros of maximal monotone operators, their elements are more easily computable and are indeed calculated in, for instance, the classical Forward-Backward splitting method, Tseng's Modified Forward-Backward splitting method and Korpelevich's method.",
"We prove that any algorithm based on approximate resolvents fall within the above mentioned class of Fejér convergent methods, providing an unifying framework for establishing their convergence properties.",
"We present a new transportation formula for cocoercive operators and use it for establishing that the Forward-Backward splitting method is a particular case of the Hybrid Proximal-Extragradient method.",
"The relationship between the Hybrid Proximal-Extragradient method and approximate resolvents is also discussed.",
"This work is organized as follows.",
"In Section , we introduce some basic definitions and results.",
"In Section , we define a very general class of Fejér convergent by means of recursive inclusions and sequences of point-to-set maps satisfying two basic properties.",
"In Section , we define approximate resolvents, show that they are the iteration maps of the Hybrid Proximal-Extragradient method and prove that methods based on approximate resolvents fall within the aforementioned class of Fejér convergent methods.",
"In Section , we show that the Forward-Backward method is based on approximate resolvents, which is to say that it is a particular case of the Hybrid Proximal-Extragradient method.",
"In Section , we recall that Tseng's Modified Forward-Backward method is a particular case of the Hybrid Proximal-Extragradient method, which is to say that it is based on approximate resolvents.",
"In Section , we recall that Korpelevich's method is a particular case of the Hybrid Proximal-Extragradient method, which is to say that it is based on approximate resolvents.",
"In Section , we make some comments."
],
[
"Basic definitions and results",
"In the first part of this section, we review the concept and properties of Quasi-Fejér convergence, which will be used in our analysis of a class of Fejér convergent methods.",
"In the second part, we establish the notation concerning point-to-set maps, which will be used for defining the aforementioned class.",
"The last part of this section contains the material needed to define approximate resolvents and the Hybrid Proximal-Extragradient (HPE) method, to prove that methods based on approximate resolvents/HPE method belong to the aforementioned class, and that some well known decomposition methods are based on approximate resolvents/HPE method.",
"As far as we know, this section contains just one original result, namely, Lemma REF , which is a transportation formula for cocoercive operators."
],
[
"Quasi-Fejér convergence",
"The concept of Quasi-Fejér convergence was introduce by Ermol$^{\\prime }$ ev [9] in the context of sequences of random variables (see also [10] and its translation [11]).",
"We will use a deterministic version of this notion, considered first in metric spaces by Isuem, Svaiter and Teboulle [12], [13], in Euclidean spaces [4], in Hilbert spaces in [2], in reflexive Banach spaces in [1].",
"All this material is now standard knowledge and is included for the sake of completeness.",
"We do not claim to give here any original contribution to this over-studied concept.",
"We will use an arbitrary exponent $p$ just to unify the results related in the references; its particular value seems of little importance as indicated by the next results.",
"Definition 2.1 Let $X$ be a metric space and $0<p<\\infty $ .",
"A sequence $(x_n)$ in $X$ is $p$ -Quasi-Fejér convergent to $\\Omega \\subset X$ if, for each $x^*\\in \\Omega $ , there exists a non-negative, summable sequence $(\\rho _n)$ such that $d(x^*,x_{n})^p\\le d(x^*,x_{n-1})^p+\\rho _n\\qquad n=1,2,\\dots $ Note that if $\\rho _1=\\rho _2=\\cdots =0$ in the above definition, we retrieve the classical definition of Fejér convergence and the exponent $p$ becomes immaterial.",
"Ermol$^{\\prime }$ lev considered the stochastic case with $p=2$ and the deterministic case was considered in [12], [13] with $p=1$ and in [4], [2] with $p=2$ in Euclidean and Hilbert spaces respectively.",
"The next proposition summarizes the main properties of Quasi-Fejér convergent sequences in metric spaces.",
"Proposition 2.2 Let $X$ be a metric space, $p\\in (0,\\infty )$ and $(x_n)$ be a sequence in $X$ which is $p$ -Quasi-Fejér convergent to $\\Omega \\subset X$ then, if $\\Omega $ is non-empty, then $(x_n)$ is bounded; for any $x^*\\in \\Omega $ there exists $\\lim _{n\\rightarrow \\infty }d(x^*,x_n)<\\infty $ ; if the sequence $(x_n)$ has a cluster point $x^*\\in \\Omega $ , then it converges to such a point.",
"Take $x^*\\in \\Omega $ and let $(\\rho _n)$ be as in Definition REF .",
"Then for $n<m$ $d(x^*,x_m)^p\\le d(x^*,x_n)^p+\\sum _{i=n+1}^m\\rho _i.$ Hence $\\lim \\sup _{m\\rightarrow \\infty } d(x^*,x_m)^p\\le d(x^*,x_n)^p+\\sum _{i=n+1}^\\infty \\rho _i<\\infty ,$ which proves item 1.",
"To prove item 2, note that $(\\rho _n)$ is summable and take the $\\lim \\inf _{n\\rightarrow \\infty }$ at the right hand side of the first inequality in the above equation.",
"Item 3 follows trivially from item 2.",
"Now we recall Opial's Lemma [15], which is useful for analyzing Quasi-Fejér convergence in Hilbert spaces: Lemma 2.3 (Opial) If, in a Hilbert space $X$ , the sequence $(x_n)$ is weakly convergent to $x^*$ , then for any $x\\ne x^*$ $\\lim \\inf _{n\\rightarrow \\infty }\\Vert x_n-x\\Vert > \\lim \\inf _{n\\rightarrow \\infty }\\Vert x_n-x^*\\Vert .$ The next result was proved in [16], for the case of a specific sequence generated by an inexact proximal point method with $p=1$ , but the proof presented there is quite general, and we provide it here for the sake of completeness.",
"The idea of using Opial's Lemma seems to be due to H. Brezis.",
"Latter on this result was explicitly proved for Quasi-Fejér convergent sequences in Hilbert and Banach spaces with $p=2$ in [2], [1] respectively.",
"Proposition 2.4 If, in a Hilbert space $X$ , the sequence $(x_n)$ is $p$ -Quasi-Fejér convergent to $\\Omega \\subset X$ , then it has at most one weak cluster point in $\\Omega $ .",
"If $x^*\\in \\Omega $ is a weak cluster point of $(x_n)$ , then there exists a subsequence $(x_{n_k})$ weakly convergent to $x^*$ .",
"Therefore, using item 2 of Proposition REF and Opial's Lemma, we conclude that for any $x^{\\prime }\\in \\Omega $ , $x^{\\prime }\\ne x^*$ $\\lim _{n\\rightarrow \\infty }\\Vert x_n-x^{\\prime }\\Vert =\\lim \\inf _{k\\rightarrow \\infty }\\Vert x_{n_k}-x^{\\prime }\\Vert > \\lim \\inf _{k\\rightarrow \\infty }\\Vert x_n-x^*\\Vert = \\lim _{n\\rightarrow \\infty }\\Vert x_n-x^*\\Vert $ which trivially implies the desired result.",
"It is trivial that the specific value of $p\\in (0,\\infty )$ is immaterial in the above proofs.",
"It would be preposterous to claim that for each $p$ one has a “specific kind” of Quasi-Fejér convergence.",
"We hope to reinforce this point of view with the next remark.",
"Remark 2.5 Let $X$ be a metric space, $p\\in (0,\\infty )$ and $(x_n)$ be a sequence in $X$ which is $p$ -Quasi-Fejér convergent to $\\Omega \\subset X$ then either, $d(x_n,x^*)\\rightarrow 0$ for some $x^*\\in \\Omega $ ; $(x_n)$ is $q$ -Quasi-Fejér convergent to $\\Omega $ for any $q\\in (0,\\infty )$ .",
"From now on, $p$ -Quasi-Fejér convergence will be called simply Quasi-Fejér convergence, the exponent $p$ being 1 unless otherwise stated.",
"Let $X$ , $Y$ be arbitrary sets.",
"A point-to-set map $F:X\\rightrightarrows Y$ is a function $F:X\\rightarrow \\wp (Y)$ , where $\\wp (Y)$ is the power set of $Y$ , that is, the family of all subsets of $Y$ .",
"If $F(x)$ is a singleton for all $x$ , that is, a set with just one element, one says that $F$ is point-to-point.",
"Whenever necessary, we will identify a point-to-point map $F:X\\rightrightarrows Y$ with the unique function $f:X\\rightarrow Y$ such that $F(x)=\\lbrace f(x)\\rbrace $ for all $x\\in X$ , A point-to-set map $F:X\\rightrightarrows Y$ is $L$ -Lipschitz if $X$ and $Y$ are normed vector spaces and, $\\emptyset \\ne F(x^{\\prime })\\subset \\lbrace y+u\\;|\\; y\\in F(x),\\;u\\in Y,\\;\\Vert u\\Vert \\le L\\Vert x-x^{\\prime }\\Vert \\rbrace ,\\qquad \\forall x,x^{\\prime }\\in X.$ Note that if $F$ is point-to-point and it is identified with a function, then in the above definition we retrieve the classical notion of a $L$ -Lipschitz continuous function.",
"The $\\varepsilon $ -enlargement of a maximal monotone operators will be used to define approximate resolvents in Section .",
"In this section we review the definition of the $\\varepsilon $ -enlargement and discuss those of its properties which will be used in the analysis and applications of approximate resolvents.",
"From now on, $X$ is a real Hilbert space.",
"Recall that a point-to-set operator $T:X\\rightrightarrows X$ is monotone if $\\langle {x-y},{u-v}\\rangle \\ge 0\\qquad \\forall x,y\\in X,\\;u\\in T(x),v\\in T(y),$ and it is maximal monotone if it is monotone and maximal in the family of monotone operators in $X$ with respect to the partial order of the inclusion.",
"Let $T:X\\rightrightarrows X$ be a maximal monotone operator.",
"Recall that the $\\varepsilon $ -enlargement [5] of $T$ is defined $T^{[\\varepsilon ]}(x)=\\lbrace v\\;|\\; \\langle {x-y},{v-u}\\rangle \\ge -\\varepsilon \\rbrace ,\\qquad x\\in X,\\varepsilon \\ge 0.$ Now we state some elementary properties of the $\\varepsilon $ -enlargement which follow trivially from the above definition and the basic properties of maximal monotone operators.",
"Their proofs can be found in [5], [7], [20].",
"Proposition 2.6 Let $T:X\\rightrightarrows X$ be maximal monotone.",
"Then $T=T^{[0]}$ ; if $0\\le \\varepsilon _1\\le \\varepsilon _2$ , then $T^{[\\varepsilon _1]}(x)\\subset T^{[\\varepsilon _2]}(x)$ for any $x\\in X$ ; $\\lambda \\left(T^{[\\varepsilon ]}(x)\\right)=(\\lambda T)^{[\\lambda \\varepsilon ]}(x)$ for any $x\\in X$ , $\\varepsilon \\ge 0$ and $\\lambda >0$ ; if $v_k\\in T^{[\\varepsilon _k]}(x_k)$ for $k=1,2,\\dots $ , $(x_k)$ converges weakly to $ x$ , $(v_k)$ converges strongly to $v$ and $(\\varepsilon _k)$ converges to $\\varepsilon $ , then $v\\in T^{[\\varepsilon ]}(x)$ ; if $T=\\partial f$ , where $f$ is a proper closed convex function in $X$ , then $\\partial _\\varepsilon f(x)\\subset T^{[\\varepsilon ]}(x)=(\\partial f)^{[\\varepsilon ]}(x)$ for any $x\\in X$ , $\\varepsilon \\ge 0$ .",
"The $\\varepsilon $ -enlargements of two operators can be “added” as follows.",
"This fact was proved in [5] in a finite dimensional setting, but its extension to Hilbert and Banach spaces are straightforward.",
"Proposition 2.7 It $T_1,T_2:X\\rightrightarrows X$ are maximal monotone and $T_1+T_2$ is also maximal monotone then, for any $\\varepsilon _1,\\varepsilon _2\\ge 0$ and $x\\in X$ $T_1^{[\\varepsilon _1]}(x)+ T_2^{[\\varepsilon _2]}(x)\\subset (T_1+T_2)^{[\\varepsilon _1+\\varepsilon _2]}(x).$ Recall that a (maximal) monotone operator $A:X\\rightarrow X$ is $\\alpha $ -cocoercive (for $\\alpha >0$ ) if $\\langle {x-y},{Ax-Ay}\\rangle \\ge \\alpha \\Vert Ax-Ay\\Vert ^2,\\qquad \\forall x,y\\in X.$ There is an interesting “transportation formula” for cocoercive operators.",
"This result was proved by R.D.C Monteiro and myself.",
"Lemma 2.8 If ${A}:X\\rightarrow X$ is $\\alpha $ -cocoercive, then for any $x,z\\in X$ , ${A}(z)\\in {A}^{[\\varepsilon ]}(x), \\qquad \\mbox{with } \\varepsilon =\\frac{\\Vert x-z\\Vert ^2}{4\\alpha }.$ Take $y\\in X$ .",
"Then $\\langle {x-y},{{A}z-{A}y}\\rangle &=\\langle {x-z},{{A}z-{A}y}\\rangle +\\langle {z-y},{{A}z-{A}y}\\rangle \\\\&\\ge \\langle {x-z},{{A}z-{A}y}\\rangle +\\alpha \\Vert {A}z-{A}y\\Vert ^2\\\\&\\ge -\\Vert x-z\\Vert \\Vert {A}z-{A}y\\Vert +\\alpha \\Vert {A}z-{A}y\\Vert ^2,$ where the first inequality follows form the cocoercivity of ${A}$ and the second one from Cauchy-Schwarz inequality.",
"To end the proof, note that $-\\Vert x-z\\Vert \\Vert {A}z-{A}y\\Vert +\\alpha \\Vert {A}z-{A}y\\Vert ^2\\ge \\inf _{t\\in \\mathbb {R}}\\alpha t^2-\\Vert x-z\\Vert t$ and compute the value of the left hand-side of this inequality.",
"The usefulness of the $\\sigma $ -approximate resolvent (to be defined in Section ) follows from the next elementary result, essentially proved in [17].",
"Lemma 2.9 Suppose that $T:X\\rightrightarrows X$ is maximal monotone, $x\\in X$ , $\\lambda >0$ and $\\sigma \\ge 0$ .",
"If $\\left\\lbrace \\begin{array}{l}v\\in T^{[\\varepsilon ]}(y),\\\\ \\Vert \\lambda v+y-x\\Vert ^2+2\\lambda \\varepsilon \\le \\sigma ^2\\Vert y-x\\Vert ^2,\\end{array}\\right.\\qquad \\mbox{ and }\\qquad z=x-\\lambda v,&$ then $\\Vert \\lambda v\\Vert \\le (1+\\sigma )\\Vert y-x\\Vert $ , $\\Vert z-y\\Vert \\le \\sigma \\Vert y-x\\Vert $ and for any $x^*\\in T^{-1}(0)$ $\\Vert x^*-x\\Vert ^2&\\ge \\Vert x^*-z\\Vert ^2+\\Vert y-x\\Vert ^2-\\bigg [\\Vert \\lambda v+y-x\\Vert ^2+2\\varepsilon \\bigg ]\\\\&\\ge \\Vert x^*-z\\Vert ^2+(1-\\sigma ^2)\\Vert y-x\\Vert ^2.$ Since $\\varepsilon \\ge 0$ we have $\\Vert \\lambda v + y - x\\Vert \\le \\sigma \\Vert y-x\\Vert $ .",
"The two first inequalities of the lemma follows trivially from this inequality, triangle inequality and the definition of $z$ .",
"To prove the third inequality of the lemma, take $x^*\\in T^{-1}(0)$ .",
"Direct combination of the algebraic identities $\\Vert x^*-x\\Vert ^2&=\\Vert x^*-z\\Vert ^2+2\\left\\langle {x^*-y},{z-x}\\right\\rangle +2\\left\\langle {y-z},{z-x}\\right\\rangle +\\Vert z-x\\Vert ^2\\\\&=\\Vert x^*-z\\Vert ^2+2\\left\\langle {x^*-y},{z-x}\\right\\rangle +\\Vert y-x\\Vert ^2-\\Vert y-z\\Vert ^2$ with the definition of $z$ yields $\\Vert x^*-x\\Vert ^2=\\Vert x^*-z\\Vert ^2+2\\lambda \\left\\langle {x^*-y},{-v}\\right\\rangle +\\Vert y-x\\Vert ^2-\\Vert \\lambda v+y-x\\Vert ^2.$ Using the inclusions $0\\in T(x^*)$ , $v\\in T^{[\\varepsilon ]}(y)$ and the definition in (REF ), we conclude that $ \\langle {x^*-y},{0-v}\\rangle \\ge -\\varepsilon $ .",
"To end the proof, of the third inequality, combine this inequality with the above equations.",
"The last inequality follows trivially from the third one and the assumptions of the lemma."
],
[
"A class of Fejér convergent methods",
"Let $X$ be a Hilbert space and $\\Omega \\subset X$ .",
"We are concerned with iterative methods for solving problem $x\\in \\Omega .$ These methods, in their exact or inexact form, generate sequences $(x_n)$ by means of the recursive inclusions $x_n\\in F_n(x_{n-1})\\mbox{ or }x_n\\in F_n(x_{n-1})+r_n, \\qquad n=1,2,\\dots ,$ respectively, where $F_1:X\\rightrightarrows X,F_2:X\\rightrightarrows X,\\dots $ are point-to-set maps, $r_1,r_2\\dots $ are errors and $x_0\\in X$ is a starting point.",
"The basic elements here are the set $\\Omega $ and the family of point-to-set maps $(F_n)$ , which we will call the family of iteration-maps.",
"We will consider two properties of a general family of point-to-set maps $(F_n:X\\rightrightarrows X)_{n\\in \\mathbb {N}}$ with respect to $\\Omega \\subset X$ : P1: if $\\hat{x}\\in F_n(x)$ and $x^*\\in \\Omega $ then $\\Vert x^*-\\hat{x}\\Vert \\le \\Vert x^*-x\\Vert ;$ P2: if $(z_k)_{k\\in \\mathbb {N}}$ converges weakly to $\\bar{z}$ , $\\hat{z}_k\\in F_{n_k}(z_k)$ for $n_1<n_2<\\cdots $ and for some $x^*\\in \\Omega $ $\\lim _{k\\rightarrow \\infty } \\Vert x^*-z_k\\Vert -\\Vert x^*-\\hat{z}_k\\Vert =0,$ then $\\bar{z}\\in \\Omega $ .",
"Property P1 ensures that points in the image of $F_n(x)$ are closer (or no more distant) to $\\Omega $ than $x$ .",
"Regarding property P2, note that (using property P1) we have $\\Vert x^*-z_k\\Vert -\\Vert x^*-\\hat{z}_k\\Vert \\ge 0.$ The left hand-side of the above inequality measures the progress of $\\hat{z}_k$ toward the solution $x^*$ , as compared to $z_k$ .",
"Hence, property P2 ensures that if the progress becomes “negligible”, then the weak limit point of $(z_k)$ belongs to $\\Omega $ .",
"Theorem 3.1 Suppose that $\\Omega \\subset X$ is non-empty and $\\left( F_n:X\\rightrightarrows X\\right)$ is a sequence of point-to-set maps which satisfies conditions P1, P2 with respect to $\\Omega $ .",
"If $x_n\\in F_n(x_{n-1})+r_n,\\qquad \\sum \\Vert r_n\\Vert <\\infty $ then $(x_n)$ is Quasi-Fejér convergent to $\\Omega $ , it converges weakly to some $\\bar{x}\\in \\Omega $ and for any $w\\in \\Omega $ there exists $\\lim _{n\\rightarrow \\infty } \\Vert x^*-x_n\\Vert $ .",
"Moreover, if $r_n=0$ for all $n$ , then $(x_n)$ is Fejér convergent to $\\Omega $ .",
"To simplify the proof, define $\\hat{x}_n=x_n-r_n.$ Take an arbitrary $x^*\\in \\Omega $ .",
"Since $\\hat{x}_n\\in F_n(x_{n-1})$ , $\\Vert x^*-\\hat{x}_n\\Vert \\le \\Vert x^*-x_{n-1}\\Vert $ , $\\Vert x^*-x_n\\Vert \\le \\Vert x^*-\\hat{x}_n\\Vert +\\Vert r_n\\Vert \\le \\Vert x^*-x_{n-1}\\Vert +\\Vert r_n\\Vert $ and $(x_n)_{n\\in \\mathbb {N}}$ is Quasi-Fejér convergent to $\\Omega $ .",
"Therefore, by Proposition REF , this sequence is bounded and there exists $\\lim _{n\\rightarrow \\infty } \\Vert x^*-x_n\\Vert <\\infty $ .",
"Using this fact, the above equation and the assumption of $(r_n)$ being summable we conclude that $\\lim _{n\\rightarrow \\infty }\\Vert x^*-x_{n-1}\\Vert -\\Vert x^*-\\hat{x}_n\\Vert =0.$ Since $(x_n)_{n\\in \\mathbb {N}}$ is bounded it has a weak cluster point, say $\\bar{x}$ and there exists a subsequence $(x_{n_k})_{k\\in \\mathbb {N}}$ which converges weakly to $\\bar{x}$ .",
"The above equation shows that, in particular $\\lim _{k\\rightarrow \\infty }\\Vert x^*-x_{n_k-1}\\Vert -\\Vert x^*-\\hat{x}_{n_k}\\Vert =0$ Therefore, using P2, the two above equations and the inclusion $\\hat{x}_{n_k}\\in F_{n_k}(x_{n_k-1})$ , we conclude that $\\bar{x}\\in \\Omega $ .",
"Hence, all weak cluster points of $(x_n)$ belong to $\\Omega $ .",
"To end the proof, use Proposition REF Note that properties P1, P2 are “inherited” by specializations.",
"We state formally this result and the proof, being quite trivial, is be omitted Proposition 3.2 If $\\left( F_n:X\\rightrightarrows X\\right)$ is a sequence of point-to-set maps which satisfies conditions P1, P2 with respect to $\\Omega \\subset X$ and $\\left( G_n:X\\rightrightarrows X\\right)$ is a sequence of point-to-set maps such that, for any $x\\in X$ $G_n(x)\\subset F_n(x),\\qquad n=1,2,\\dots ,$ then $\\left( G_n:X\\rightrightarrows X\\right)$ also satisfies conditions P1, P2 with respect to $\\Omega $ .",
"What about compositions?",
"Suppose that $(F_n)$ is a sequence satisfying P1, P2 with respect to $\\Omega \\subset X$ , and that $F_n=G_n\\circ H_n$ where $G_n:X\\rightrightarrows Y$ , $H_n:Y\\rightrightarrows X$ and all $G_n$ 's are $L$ -Lipschitz continuous ($Y$ is Hilbert).",
"One may consider sequences ${y}_n\\in H_n(x_{n-1})+{u}_n, \\qquad x_n\\in G_n({y}_n)+{r}_n,$ where $({u}_n)$ and $({r}_n)$ are summable.",
"Since $x_n-{r}_n\\in G_n({y}_n)$ , using also (REF ), we conclude that there exists $\\hat{x}_n\\in G_n({y}_n-{u}_n)\\subset G_n\\circ H_n(x_{n-1})$ $\\Vert \\hat{x}_n-(x_n-{r}_n)\\Vert \\le L\\;\\Vert {u}_n\\Vert .$ Therefore, defining $s_n=x_n-\\hat{x}_n$ we conclude that $x_n\\in F_n(x_{n-1})+s_n,\\qquad \\sum \\Vert s_n\\Vert \\le \\sum L\\Vert {u}_n\\Vert +\\Vert {r}_n\\Vert <\\infty .$ Therefore, if $\\Omega \\ne \\emptyset $ , a sequence $(x_n)$ generated as in (REF ) converges weakly to some point $x^*\\in \\Omega $ .",
"On may also consider compositions of $m+1$ maps $F= G_{1,n}\\circ G_{2,n}\\dots \\circ G_{m,n}\\circ H_n$ adding summable errors in each stage, assuming each $G_{i,n}:Y_i\\rightrightarrows Y_{i-1}$ to be $L$ -Lipschitz continuous, $H_n:X\\rightrightarrows Y_m$ , $Y_0=X$ etc."
],
[
"Approximate resolvents and the Hybrid Proximal-Extragradient\nMethod",
"In this section, first we define $\\sigma $ -approximate resolvents, analyze some of their properties and study conditions under which sequences of $\\sigma $ -approximate resolvents satisfy properties P1, P2.",
"After that, we recall the definition of the Hybrid Proximal-Extragradient method and show that $\\sigma $ -approximate resolvents are the iteration maps of such method.",
"At the end of the section we discuss the incorporation of summable errors to sequences of $\\sigma $ -approximate resolvents and to the Hybrid Proximal-Extragradient method.",
"Recall that the resolvent of a maximal monotone operator $T:X\\rightrightarrows X$ is defined as $J_T(x)=(I+T)^{-1}(x),\\qquad x\\in X.$ We shall consider approximations of the resolvent in the following sense.",
"Definition 4.1 The $\\sigma $ -approximate resolvent of a maximal monotone operator $T:X\\rightrightarrows X$ is the point-to-set operator $J_{T,\\sigma }:X\\rightrightarrows X$ $J_{T,\\sigma }(x)=\\left\\lbrace x-v\\;\\left|\\begin{array}{l} \\exists \\varepsilon \\ge 0, y\\in X,\\\\v\\in T^{[\\varepsilon ]}(y)\\\\\\Vert v+y-x\\Vert ^2+2\\varepsilon \\le \\sigma ^2\\Vert y-x\\Vert ^2\\end{array}\\right\\rbrace \\right.$ where $\\sigma \\ge 0$ .",
"First, we analyze some elementary properties of approximate resolvents and find a convenient expression for $J_{\\lambda T,\\sigma }$ .",
"In particular, we show that the $\\sigma $ -approximate resolvent is indeed and extension (in the sense of point-to-set maps) of the classical resolvent.",
"Proposition 4.2 Let $T:X\\rightrightarrows X$ be maximal monotone.",
"Then, for any $x\\in X$ , $J_{T,\\sigma =0}(x)=\\lbrace J_T(x)\\rbrace $ ; if $0\\le \\sigma _1\\le \\sigma _2$ then $J_{T,\\sigma _1}(x) \\subset J_{T,\\sigma _2}(x)$ ; for any $\\lambda >0$ and $\\sigma \\ge 0$ , $J_{\\lambda T,\\sigma }(x)&=\\left\\lbrace x-\\lambda v\\;\\left|\\begin{array}{l}\\exists \\varepsilon \\ge 0, y\\in X,\\\\v\\in T^{[\\varepsilon ]}(y)\\\\\\Vert \\lambda v+y-x\\Vert ^2+2\\lambda \\varepsilon \\le \\sigma ^2\\Vert y-x\\Vert ^2\\end{array}\\right\\rbrace \\right.$ Items 1, 2 and 3 follow trivially from Definition REF and Proposition REF , items REF , REF and REF .",
"Note that in view of item 1 of the above proposition, if point-to-set operators which are point-to-point are identified with functions, we have $J_{T,0}=J_T\\,.$ In view of item 3, $J_{\\lambda T,\\sigma }=\\left\\lbrace z\\in X;\\left|\\begin{array}{l}\\exists \\varepsilon \\ge 0, y\\in X,\\\\\\displaystyle \\frac{x-z}{\\lambda }\\in T^{[\\varepsilon ]}(y)\\\\\\Vert y-z\\Vert ^2+2\\lambda \\varepsilon \\le \\sigma ^2\\Vert y-x\\Vert ^2\\end{array}\\right\\rbrace \\right.$ The next theorem is the main result of this section and states that, in some sense, approximate resolvents are “almost as good” as resolvents for finding zeros of maximal monotone operators, that is, for solving problem (REF ) with $\\Omega =\\lbrace x\\:|\\; 0\\in T(x)\\rbrace =T^{-1}(0)$ .",
"Theorem 4.3 Suppose that $T:X\\rightrightarrows X$ is maximal monotone, $\\sigma \\in [0,1)$ , $\\underline{\\lambda }>0$ and $(\\lambda _k)$ is a sequence in $[\\underline{\\lambda }, \\infty )$ .",
"Then, the sequence of point-to-set maps $\\left(J_{\\lambda _kT,\\sigma }\\right)_{k\\in \\mathbb {N}}$ satisfies properties P1, P2 with respect to $\\Omega =\\lbrace x\\in X\\;|\\; 0\\in T(x)\\rbrace =T^{-1}(0)$ .",
"Suppose that $\\hat{x}\\in J_{\\lambda _kT,\\sigma }(x)$ .",
"This means that there exists $y,v\\in X$ , $\\varepsilon \\ge 0$ such that $\\hat{x}=x-\\lambda _kv,\\quad v\\in T^{[\\varepsilon ]}(x),\\quad \\Vert \\lambda _kv+y-x\\Vert ^2+2\\lambda \\varepsilon \\le \\sigma ^2\\Vert y-x\\Vert ^2.$ Therefore, using Lemma REF we conclude that for any $x^*\\in =T^{-1}(0)$ , $\\Vert x^*-x\\Vert ^2\\ge \\Vert x^*-\\hat{x}\\Vert ^2+(1-\\sigma ^2)\\Vert y-x\\Vert ^2\\ge \\Vert x^*-\\hat{x}\\Vert ^2$ which proves that the family $(J_{\\lambda _kT,\\sigma })$ satisfies P1.",
"Now we prove P2.",
"Suppose that $(z_k)$ converges weakly to $\\bar{z}$ , $\\hat{z}_k\\in J_{\\lambda _{n_k}T,\\sigma }(z_k)$ , $0\\in T(x^*)$ and $\\lim _{k\\rightarrow \\infty }\\Vert x^*-z_k\\Vert -\\Vert x^*-\\hat{z}_k\\Vert =0.$ To simplify the proof, let $\\mu _k=\\lambda _{n_k}\\ge \\underline{\\lambda }$ .",
"For each $k$ there exists $v_k,y_k\\in X$ , $\\varepsilon _k\\ge 0$ such that $\\hat{z}_k=z_k-\\mu _kv,\\quad v_k\\in T^{\\varepsilon _k}(z_k),\\quad \\Vert \\mu _kv_k+y_k-z_k\\Vert ^2+2\\mu _k\\varepsilon \\le \\sigma ^2\\Vert y_k-z_k\\Vert ^2.$ Using again Lemma REF , we conclude that $\\Vert x^*-z_k\\Vert ^2\\ge \\Vert x^*-\\hat{z}_k\\Vert ^2+(1-\\sigma ^2)\\Vert y_k-z_k\\Vert ^2.$ Therefore $(1-\\sigma ^2)\\Vert y_k-z_k\\Vert ^2&\\le \\Vert x^*-z_k\\Vert ^2-\\Vert x^*-\\hat{z}_k\\Vert ^2\\\\&=(\\Vert x^*-z_k\\Vert -\\Vert x^*-\\hat{z}_k\\Vert )(\\Vert x^*-z_k\\Vert + \\Vert x^*-\\hat{z}_k\\Vert ).$ Since $(z^k)$ is weakly convergent, it is also bounded.",
"Taking this fact into account and using the above equation and (REF ) we conclude that $\\lim _{k\\rightarrow \\infty }\\Vert y_k-z_k\\Vert =0.$ So, $(y_k)$ also converges weakly to $\\bar{z}$ .",
"Since $\\varepsilon _k\\ge 0$ , using the last relation in (REF ) we conclude that $\\mu _k\\varepsilon _k\\le \\frac{\\sigma ^2}{2}\\Vert y_k-z_k\\Vert ^2, \\qquad \\Vert \\mu _kv_k\\Vert \\le (1+\\sigma )\\Vert y_k-z_k\\Vert .$ Therefore, since $(\\mu _k)$ is bounded away from 0, $ \\lim _{k\\rightarrow \\infty }\\varepsilon _k=0, \\quad \\lim _{k\\rightarrow \\infty }v_k=0$ and $0\\in T^{[0]}(\\bar{z})=T(\\bar{z})$ .",
"The Hybrid Proximal-Extragradient/Projection methods were introduced in [18], [17], [19].",
"These methods are variants of the Proximal Point method which use relative error tolerances for accepting inexact solutions of the proximal sub-problems.",
"Here we are concerned with the variant introduced in [17], which will be called, from now on, the Hybrid Proximal-Extragradient (HPE) method.",
"It solves iteratively the problem $0\\in T(x),$ where $T:X\\rightrightarrows X$ is maximal monotone.",
"This method proceeds as follows.",
"Algorithm: (projection free) HPE method [17]: Choose $x_0\\in X$ , $\\sigma \\in [0,1)$ , $\\underline{\\lambda }>0$ and for $k=1,2,\\dots $ a) Choose $\\lambda _k\\ge \\underline{\\lambda }$ and find/compute $v_k,y_k\\in X$ , $\\varepsilon \\ge 0$ such that $v_k\\in T^{\\varepsilon _k}( y_k),\\qquad \\Vert \\lambda _k v_k+y_k-x_{k-1}\\Vert ^2+2\\lambda _k\\varepsilon _k\\le \\sigma ^2\\Vert y_k-x_{k-1}\\Vert ^2$ b) Set $x_k=x_{k-1}-\\lambda _kv_k$ To generate iteratively sequences by means of approximate resolvents is equivalent to apply the HPE method in the following sense.",
"Proposition 4.4 Let $T:X\\rightrightarrows X$ be maximal monotone, $\\sigma \\ge 0$ , $\\underline{\\lambda }>0$ and $(\\lambda _k)$ be sequence in $[\\underline{\\lambda },\\infty )$ .",
"A sequence $(x_k)$ satisfies the recurrent inclusion $x_k\\in J_{\\lambda _k T,\\sigma }(x_{k-1}),\\qquad k=1,2,\\dots $ if and only if there exists sequences $(y_k)$ , $(v_k)$ , $(\\varepsilon _k)$ which, together with the sequences $(x_k)$ , $(\\lambda _k)$ satisfy steps a) and b) of the HPE method.",
"Use Definition REF and Proposition REF item REF .",
"Convergence of the HPE method perturbed by a summable sequence of errors was proved directly in [6].",
"Here we see that it can be easily and effortlessly derived as a particular case of a generic convergence result, combining Proposition REF with Theorem REF .",
"Corollary 4.5 If $T:X\\rightrightarrows X$ is maximal monotone, $T^{-1}(0)\\ne \\emptyset $ , $\\underline{\\lambda }>0$ , $\\sigma \\in [0,1)$ , for $k=1,2,\\dots $ $&\\lambda _k\\ge \\underline{\\lambda }\\\\& v_k\\in T^{[\\varepsilon _k]}(\\tilde{x}_k),\\;\\Vert \\lambda _kv_k+\\tilde{x}_k-x_{k-1}\\Vert ^2+2\\lambda _k\\varepsilon _k\\le \\sigma \\Vert \\tilde{x}_k-x_{k-1}\\Vert ^2\\\\&x_k=x_{k-1}-\\lambda _kv_k+r_k$ and $\\sum \\Vert r_k\\Vert <\\infty $ , then $(x_k)$ (and $(\\tilde{x}_k)$ ) converges weakly to a point $\\bar{x}\\in T^{-1}(0)$ .",
"Corollary 4.6 If $T:X\\rightrightarrows X$ is maximal monotone, $T^{-1}(0)\\ne \\emptyset $ , $\\overline{\\lambda }\\ge \\underline{\\lambda }>0$ , $\\sigma \\in [0,1)$ , for $k=1,2,\\dots $ $&\\overline{\\lambda }\\ge \\lambda _k\\ge \\underline{\\lambda }\\\\& v_k\\in T^{[\\varepsilon _k]}(\\tilde{x}_k),\\;\\Vert \\lambda _kv_k+\\tilde{x}_k-x_{k-1}\\Vert ^2+2\\lambda _k\\varepsilon _k\\le \\sigma \\Vert \\tilde{x}_k-x_{k-1}\\Vert ^2\\\\&x_k=x_{k-1}-\\lambda _k(v_k+r_k)$ and $\\sum \\Vert r_k\\Vert <\\infty $ , then $(x_k)$ (and $(\\tilde{x}_k)$ ) converges weakly to a point $\\bar{x}\\in T^{-1}(0)$ ."
],
[
"The Forward-Backward splitting method",
"We will prove in this section that the iteration maps of the Forward-Backward splitting method are specializations or selections of $\\sigma $ -approximate resolvents and the sequence of iteration maps satisfies properties P1, P2.",
"Equivalently, the Forward-Backward splitting method is a particular instance of the HPE method.",
"Observe that, as a consequence, sequences generated by the inexact Forward-Backward splitting methods with summable errors still converge weakly to solutions of the inclusion problem, if any.",
"This convergence result was previously obtained in [8] by a detailed analysis of the Forward-Backward splitting method.",
"Here we see that it can be easily and effortlessly derived as a particular case of a generic convergence result.",
"The Forward-Backward Splitting method solves the inclusion problem $0\\in (A+B)x$ where f1) ${A}:X\\rightarrow X$ is $\\alpha $ -cocoercive, $\\alpha >0$ ; f2) ${B}:X\\rightrightarrows X$ is maximal monotone.",
"This method proceeds as follows: Forward-Backward Splitting method 0) Initialization: Choose $0<\\underline{\\lambda }\\le \\bar{\\lambda }< 2 \\alpha $ and $x_0\\in X$ ; 1) for $k=1,2,\\dots $ a) choose $\\lambda _k\\in [\\underline{\\lambda },\\bar{\\lambda }]$ and define $\\nonumber x_k&=(I+\\lambda _k{B})^{-1}(x_k-\\lambda _k{A}(x_{k-1}))\\\\&=J_{\\lambda _k{B}}\\circ (I-\\lambda _k{A})\\;\\,( x_{k-1}).$ Note that the generic iteration map of the Forward-Backward method is $J_{\\lambda {B}}\\circ (I-\\lambda {A})$ with $\\lambda =\\lambda _k$ in the $k$ -th iteration.",
"Lemma 5.1 If $A,B$ satisfy f1 and f2 then, for any $\\lambda >0$ and $x\\in X$ , $J_{\\lambda {B}}\\circ (I-\\lambda {A})(x)\\in J_{\\lambda (A+B),\\sigma }(x),$ with $\\sigma =\\sqrt{\\lambda /(2\\alpha )}$ .",
"Take $x\\in X$ and let $z=J_{\\lambda {B}}\\circ (I-\\lambda {A})(x)$ .",
"This means that $b:=\\lambda ^{-1}(x-\\lambda {A}(x)-z)\\in {B}(z).$ Define $\\varepsilon =\\Vert x-z\\Vert ^2/(4\\alpha )$ , $v=A(x)+b$ .",
"Using Lemma REF we conclude that ${A}(x)\\in {A}^{[\\varepsilon ]}(z)$ .",
"Therefore, combining this result with these two definitions, the above equation, Proposition REF and Proposition REF item REF , we conclude that $& v\\in (A^{[\\varepsilon ]}+b)(z)\\subset (A+B)^{[\\varepsilon ]}(z),\\quad \\Vert \\lambda v+z-x\\Vert ^2+2\\lambda \\varepsilon =\\sigma ^2\\Vert z-x\\Vert ^2\\\\& z=x-\\lambda v,$ which, together with Proposition REF item 3, proves the lemma.",
"Corollary 5.2 Let $A,B$ be as in f1, f2 and $\\underline{\\lambda }$ , $\\bar{\\lambda }$ and $(\\lambda _k)$ , $(x_k)$ be as in the Forward Backward method.",
"Define $\\sigma =\\sqrt{\\bar{\\lambda }/(2\\alpha )}.$ Then $0<\\sigma <1$ and for any $x\\in X$ $J_{\\lambda _k{B}}\\circ (I-\\lambda _k{A})(x)\\in J_{\\lambda _k(A+B),\\sigma \\,}(x),\\qquad k=1,2,\\dots $ In particular $x_k= J_{\\lambda _k{B}}\\circ (I-\\lambda _k{A})(x_k)\\in J_{\\lambda _k(A+B),\\sigma \\,}(x_k),\\qquad k=1,2,\\dots $ and the sequence of maps $(J_{\\lambda _k{B}}\\circ (I-\\lambda _k{A}))$ satisfies properties P1, P2 with respect to $(A+B)^{-1}(0)$ .",
"The bounds for $\\sigma $ follow trivially from its definition and the choices for $\\underline{\\lambda }$ , $\\bar{\\lambda }$ in the Forward-Backward method.",
"Define $ \\sigma _k=\\sqrt{\\lambda _k/(2\\alpha )}$ for $k=1,2,\\dots $ Since $\\lambda _k\\in [\\underline{\\lambda },\\bar{\\lambda }]$ , $0<\\sigma _k\\le \\sigma $ for all $k$ .",
"Therefore, using also Lemma REF and Proposition REF item REF , we conclude that for any $x\\in X$ $J_{\\lambda _k{B}}\\circ (I-\\lambda _k{A})(x)\\in J_{\\lambda _k(A+B),\\sigma _k}(x)\\subset J_{\\lambda _k(A+B),\\sigma }(x),\\quad k=1,2,\\dots $ The equality in (REF ) follows trivially from the definition of the Forward-Backward method, while the inclusion follows from the above equation.",
"To end the proof, note that $0<\\underline{\\lambda }<\\lambda _k$ for all $k$ , and use Theorem REF , Proposition REF and the above equation.",
"Proposition 5.3 Let $(\\lambda _k)$ , $(x_k)$ be sequences generated by the Forward-Backward Splitting method.",
"Define $\\sigma =\\sqrt{\\frac{\\overline{\\lambda }}{2\\alpha }},\\;\\;v_k=\\lambda _k^{-1}(x_{k-1}-x_k),\\;\\;\\varepsilon _k=\\frac{\\Vert x_k-x_{k-1}\\Vert ^2}{4\\alpha },\\; k=1,2,\\dots $ Then $0<\\sigma <1$ and for $k=1,2,\\dots $ $& v^k\\in ({B}+{A})^{[\\varepsilon _k]}(x_k), \\quad \\Vert \\lambda _kv_k+x_k-x_{k-1}\\Vert ^2+2\\lambda _k\\varepsilon _k\\le \\sigma \\Vert x_k-x_{k-1}\\Vert ^2\\\\&x_k=x_{k-1}-\\lambda _kv_k.$ In particular, the Forward-Backward splitting method above defined is a particular case of the HPE method with $\\sigma \\in (0,1)$ .",
"See the proofs of Lemma REF and Corollary REF ."
],
[
"Tseng's Modified Forward-Backward splitting method",
"In [17] it was proved that Tseng's Modified Forward-Backward splitting method [21] is a particular case of the HPE method.",
"Here we will cast this result in the framework of approximate resolvents, prove that the iteration maps of the Tseng's Modified Forward-Backward splitting method are specializations or selections of $\\sigma $ -approximate resolvents and that the sequence of its iteration maps satisfies properties P1, P2.",
"Observe that, as a consequence, sequences generated by inexact Tseng's Modified Forward-Backward splitting method with summable errors still converge weakly to solutions of the inclusion problem, if any.",
"This result follows also from the fact that Tseng's method is a particular case of the HPE (as proved in [17]) and that the HPE with summable errors converges (as proved in [6]).",
"Convergence of Tseng's Modified Forward-Backward splitting method with summable errors was obtained in [3] by a detailed analysis of the Tseng's Modified Forward-Backward splitting method.",
"Here we see that this result can be easily and effortlessly derived as a particular case of a generic convergence result.",
"In this section we consider the inclusion problem $0\\in (A+B)\\,x$ where t1) ${A}:X\\rightarrow X$ is monotone and $L$ -Lipschitz continuous ($L>0$ ); t2) ${B}:X\\rightrightarrows X$ is maximal monotone.",
"The exact Tseng's Modified Forward-Backward Splitting method (without the auxiliary projection step) proceeds as follows: Tseng's Modified Forward-Backward method [21] Choose $0<\\underline{\\lambda }\\le \\bar{\\lambda }< 1/L$ and $x_0\\in X$ ; for $k=1,2,\\dots $ a) choose $\\lambda _k\\in [\\underline{\\lambda },\\bar{\\lambda }]$ and compute $y_k=( I+\\lambda _k {B})^{-1}(x_{k-1}-\\lambda _k {A}(x_{k-1})),\\qquad x_k=y_{k-1}-\\lambda _k({A}(y_k)-{A}(x_{k-1})).$ In order to cast this method in the formalism of Section , define for $\\lambda >0$ $&H_\\lambda :X\\rightarrow X\\times X,&&H_\\lambda (x)=(x,J_{\\lambda {B}}(x-\\lambda {A}(x)))\\\\&G_\\lambda :X\\times X\\rightarrow X,&&G_\\lambda (x,y)=y-\\lambda ({A}(y)-{A}(x)))$ Note that the second component of the (generic) operator $H_{\\lambda }$ is $J_{\\lambda {B}}\\circ (I-\\lambda {A})$ which is the generic iteration map of the Forward-Backward method in (REF ).",
"Trivially, $x_k=G_{\\lambda _k}\\circ H_{\\lambda _k}\\;(x_{k-1}).$ The next two result were essentially proved in [17], in the context of the Hybrid Proximal-Extragradient Method.",
"We will state and prove it in the context of $\\sigma $ -approximate resolvents.",
"Lemma 6.1 If $A,B$ satisfy assumptions t1, t2, then, for any $\\lambda >0$ and $x\\in X$ $G_\\lambda \\circ H_\\lambda (x)\\in J_{A+B,\\sigma \\;}(x)$ for $\\sigma =\\lambda L$ .",
"Take $x\\in X$ and let $y=J_{\\lambda {B}}(x-\\lambda {A}(x)),\\qquad z=y+\\lambda ({A}(y)-{A}(x)).$ Note that $z=G_\\lambda \\circ H_\\lambda (x)$ .",
"Using the definition of $y$ we have $a:=\\lambda ^{-1}(x-\\lambda {A}(x)-y)\\in {B}(y).$ Therefore, $v:=a+{A}(y)\\in (A+B)(y),\\qquad \\Vert \\lambda v+x-y\\Vert ^2&=\\Vert \\lambda ({A}(y)-{A}(x))\\Vert ^2\\\\&\\le (\\lambda L)^2\\Vert y-x\\Vert ^2,$ where the inequality follows from assumption t2).",
"To end the proof, note that $z=x-\\lambda v$ .",
"Corollary 6.2 Let $A,B$ be as in t1, t2 and $0<\\underline{\\lambda }<\\bar{\\lambda }<2\\alpha $ and $(\\lambda _k)$ , $(x_k)$ be as in Tseng's Modified Forward-Backward method.",
"Define $\\sigma =\\bar{\\lambda }L.$ Then $0<\\sigma <1$ and for any $x\\in X$ $G_{\\lambda _k}\\circ H_{\\lambda _k}(x)\\in J_{\\lambda _k(A+B),\\sigma \\,}(x),\\qquad k=1,2,\\dots $ In particular $x_k= G_{\\lambda _k}\\circ H_{\\lambda _k} (x_{k-1})\\in J_{\\lambda _k(A+B),\\sigma \\,}(x_{k-1}),\\qquad k=1,2,\\dots $ and the sequence of maps $( G_{\\lambda _k}\\circ H_{\\lambda _k})$ satisfies properties P1, P2 with respect to $(A+B)^{-1}(0)$ .",
"The bounds for $\\sigma $ follow trivially from its definition and the choices for $\\underline{\\lambda }$ and $\\bar{\\lambda }$ in Tseng's Forward-Backward method.",
"Define $ \\sigma _k=\\lambda _kL$ for $k=1,2,\\dots $ Since $\\lambda _k\\in [\\underline{\\lambda },\\bar{\\lambda }]$ we have $0<\\sigma _k\\le \\sigma $ for all $k$ .",
"Therefore, using also Lemma REF and Proposition REF item REF , we conclude that for any $x\\in X$ $J_{\\lambda _k{B}}\\circ (I-\\lambda _k{A})(x)\\in J_{\\lambda _k(A+B),\\sigma _k}(x)\\subset J_{\\lambda _k(A+B),\\sigma }(x),\\quad k=1,2,\\dots $ The equality in (REF ) follows trivially from the definition of the Tseng's Modified Forward-Backward method, while the inclusion follows from the above equation.",
"To end the proof, note that $0<\\underline{\\lambda }<\\lambda _k$ for all $k$ , and use Theorem REF , Proposition REF and the above equation.",
"Note that for $0<\\lambda \\le \\bar{\\lambda }$ , the maps $H_\\lambda $ , $G_\\lambda $ are Lipschitz continuous with constant $2+\\bar{\\lambda }L,\\qquad 1+2\\bar{\\lambda }L,$ respectively.",
"Hence, this method can be perturbed by summable sequences of errors in the evaluations of the resolvents $J_{\\lambda _k {B}}$ and/or in the evaluation of ${A}(x_k)$ , ${A}(y_k)$ etc, and will still converge weakly to a solution, if any exists."
],
[
"Korpelevich's method",
"In [14] it was proved that Korpelevich's method, with fixed stepsize, is a particular case of the HPE method.",
"The extension of this result for variable stepsizes is trivial, and here we will analyze such an extension in the framework of approximate resolvents.",
"Observe that, as a consequence, sequences generated by inexact Korpelevich's method with summable errors still converges weakly to solutions of the inclusion problem, if any.",
"In this section we consider the inclusion problem $0\\in A(x)+N_C(x)$ where k1) ${A}:X\\rightarrow X$ is monotone and $L$ -Lipschitz continuous ($L>0$ ); k2) $N_C$ is the normal cone operator of $C\\subset X$ , a non-empty closed convex set.",
"Korpelevich's method Choose $0<\\underline{\\lambda }\\le \\bar{\\lambda }< 1/L$ and $x_0\\in X$ ; for $k=1,2,\\dots $ a) choose $\\lambda _k\\in [\\underline{\\lambda },\\bar{\\lambda }]$ and define $y_k=P_C(x_{k-1}-\\lambda _kF(x_{k-1})),\\qquad x_k=P_C(x_{k-1}-\\lambda _kF(y_k)),$ where $P_C$ stands for the orthogonal projection onto $C$ .",
"In order to cast this method in the formalism of Section , define for $\\lambda >0$ $&H_\\lambda :X\\rightarrow X\\times X,&&H_\\lambda (x)=(x,P_C(x-\\lambda {A}(x)),\\\\&G_\\lambda :X\\times X\\rightarrow X,&&G_\\lambda (x,y)=P_C(x-\\lambda {A}(y)).$ Observe that since $P_C=J_{\\lambda N_C}$ , the second component of the (generic) operator $H_{\\lambda }$ is $J_{\\lambda {B}}\\circ (I-\\lambda {A})$ with ${B}=N_C$ which is the generic iteration map of the forward backward method in (REF ) (with ${B}=N_C$ ).",
"Note also that the map $H_\\lambda $ above defined has an equivalent expression $H_\\lambda (x)=(x,J_{\\lambda N_C}(x-\\lambda {A}(x)))$ which can be obtained by setting ${B}=N_C$ in (REF ) ${B}=N_C$ .",
"Trivially, $x_k=G_{\\lambda _k}\\circ H_{\\lambda _k}(x_{k-1}),\\qquad k=1,2,\\dots $ The next two result were essentially proved in [14], in the context of the Hybrid Proximal-Extragradient Method.",
"We will state and prove them in the context of $\\sigma $ -approximate resolvents.",
"Lemma 7.1 If ${A}$ and $C$ satisfy assumptions k1 and k2, then, for any $\\lambda >0$ and $x\\in X$ $G_\\lambda \\circ H_\\lambda (x)\\in J_{{A}+N_C,\\sigma \\;}(x)$ for $\\sigma =\\lambda L$ .",
"Take $x\\in X$ and let $y&=P_C(x-\\lambda {A}(x)),\\quad z=P_C(x-\\lambda {A}(y)).$ Note that $z= G_\\lambda \\circ H_\\lambda (x)$ .",
"Define $\\eta &=\\frac{1}{\\lambda }(x-\\lambda {A}(x)-y),\\\\\\nu &=\\frac{1}{\\lambda }(x-\\lambda {A}(y)-z),\\quad \\varepsilon =\\langle {\\nu },{z-y}\\rangle ,\\quad v=\\nu +{A}(y).$ Trivially, $\\eta \\in N_C(y)$ and $\\nu \\in N_C(z)=\\partial \\delta _C (z)$ .",
"Therefore, $\\nu \\in \\partial _\\varepsilon \\delta _C(y)\\subset (\\partial \\delta _C)^{[\\varepsilon ]}(y)= (N_C)^{[\\varepsilon ]}(y)$ and $v\\in ({A}+N_C)^{[\\varepsilon ]}(y),\\qquad z=x-\\lambda v.$ Therefore $\\Vert \\lambda v+y-x\\Vert ^2+2\\lambda \\varepsilon &=\\Vert y-z\\Vert ^2+2\\lambda \\langle {\\nu },{z-y}\\rangle \\\\&=\\Vert y-z\\Vert ^2+2\\lambda \\langle {\\nu -\\eta },{z-y}\\rangle +2\\lambda \\langle {\\eta },{z-y}\\rangle \\\\&\\le \\Vert y-z\\Vert ^2+2\\lambda \\langle {\\nu -\\eta },{z-y}\\rangle ,$ where the inequality follows from the inclusions $\\eta \\in N_C(y)$ , $z\\in C$ .",
"Direct algebraic manipulations yield $\\Vert y-z\\Vert ^2+2\\lambda \\langle {\\nu -\\eta },{z-y}\\rangle &=\\Vert \\lambda (\\nu -\\eta )+z-y\\Vert ^2-\\Vert \\lambda (\\nu -\\eta )\\Vert ^2\\\\&\\le \\Vert \\lambda (\\nu -\\eta )+z-y\\Vert ^2\\\\&=\\Vert \\lambda ({A}(x)-{A}(y))\\Vert ^2.$ Combining the two above equations, and using assumption k1, we conclude that $\\Vert \\lambda v+y-x\\Vert ^2+2\\lambda \\varepsilon \\le (\\lambda L)^2\\Vert y-x\\Vert ^2.$ The conclusion follows combining this inequality with (REF ).",
"Corollary 7.2 Let ${A}, C$ be as in k1, k2, and $0<\\underline{\\lambda }<\\bar{\\lambda }<2\\alpha $ and $(\\lambda _k)$ , $(x_k)$ be as in Korpelevich's method.",
"Define $\\sigma =\\bar{\\lambda }L.$ Then $0<\\sigma <1$ and for any $x\\in X$ $G_{\\lambda _k}\\circ H_{\\lambda _k}(x)\\in J_{\\lambda _k(A+B),\\sigma \\,}(x),\\qquad k=1,2,\\dots $ In particular $x_k= G_{\\lambda _k}\\circ H_{\\lambda _k} (x_{k-1})\\in J_{\\lambda _k(A+B),\\sigma \\,}(x_{k-1}),\\qquad k=1,2,\\dots $ and the sequence of maps $( G_{\\lambda _k}\\circ H_{\\lambda _k})$ satisfies properties P1, P2 with respect to $({A}+N_C)^{-1}(0)$ .",
"Use Lemma REF and the same reasoning as in corollaries REF and REF .",
"Endowing $X\\times X$ with the canonical inner product of Hilbert space products $\\langle {(x,y)},{(x^{\\prime },y^{\\prime })}\\rangle =\\langle {x},{x^{\\prime }}\\rangle +\\langle {y},{y^{\\prime }}\\rangle ,$ it is trivial to check that for $0<\\lambda \\le \\bar{\\lambda }$ , the maps $H_\\lambda $ and $G_\\lambda $ are Lipschitz continuous with constants $2+\\bar{\\lambda }L,\\qquad 1+\\bar{\\lambda }L,$ respectively.",
"Hence, one can analyze Korpelevich's method with (summable) errors in the projections and/or evaluations of ${A}$ etc."
],
[
"Discussion",
"We provided a general definition of generic methods by means of recursive inclusions and sequences of point-to-set maps.",
"Using this formulation, we defined two properties of those maps which guarantee that the associated method is Fejér convergent and generates sequences which converge to a solution, if any, even when perturbed by summable errors.",
"We think these results obviate the summable error convergence analysis of a number of convergent Fejér methods.",
"The framework for the analysis of Fejér convergent methods introduced here is, of course, not general enough to encompasses all of these methods.",
"Indeed, if $X=\\mathbb {R}$ , $\\Omega =\\lbrace 0\\rbrace $ and $F(x)={\\left\\lbrace \\begin{array}{ll}-x& x>0,\\\\x/2,& x\\le 0\\end{array}\\right.",
"}$ then any sequence $(x_n)$ satisfying $x_n=F(x_{n-1})$ is Fejér convergent to $\\lbrace 0\\rbrace $ and converges to 0.",
"However, the sequence $(F_n=F)$ does not satisfies P2.",
"It has been since long recognized that Korpelevich's method (and may be even the Forward-Backward method) was an “inexact” version of the proximal point method.",
"However, the nature and degree of this “inexactness” were not known.",
"We provided a formal definition of approximate solutions of the prox by means of the $\\sigma $ -approximate resolvent which, while encompassing many classical decomposition schemes, also guarantees weak convergence of sequences generated by such approximate resolvents (even in the presence of additional summable errors)."
]
] | 1204.1353 |
[
[
"Coherent states quantization of generalized bergman spaces on the unit\n ball of cn with a new formula for their associated berezin transforms"
],
[
"Abstract While dealing with a class of generalized Bergman spaces on the unit ball, we construct for each of these spaces a set of coherent states to apply a coherent states quantization method.",
"This provides us with another way to recover the Berezin transforms attached to these spaces.",
"Finally, a new formula representing these transforms a functions of the Laplace-Beltrami operator is established in terms ofWilson polynomials by using the Fourier-Helgason transform."
],
[
"Introduction",
"The Berezin transform introduced in [3] for certain classical bounded symmetric domains in $\\mathbb {C}^{n}$ is a transform linking the Berezin symbols and symbols for Toeplitz operators.",
"It is present in the study of the correspondence principle.",
"The formula representing the Berezin transform as a function of the Laplace operators $\\Delta _{1},...,\\Delta _{r}$ ( $r$ being the rank of the domain) plays a key role in the Berezin quantization [4].",
"In this paper, we deal with the rank one symmetric domains.",
"Namely the unit ball $\\mathbb {B}^{n}$ in $\\left(\\mathbb {C}^{n},\\left\\langle ,\\right\\rangle \\right) \\mathbb {\\ }$ endowed with its Bergman metric.",
"We are precisely concerned with the $L^{2}$ -eigenspaces $\\mathcal {A}_{m}^{2,\\nu }\\left( \\mathbb {B}^{n}\\right) =\\left\\lbrace \\varphi \\in L^{2}(\\mathbb {B}^{n},(1-\\left| \\xi \\right| ^{2})^{-n-1}d\\mu ),H_{\\nu }\\varphi =\\epsilon _{m}^{\\nu ,n}\\varphi \\right\\rbrace $ associated to the discrete spectrum $\\epsilon _{m}^{\\nu ,n}=4\\nu (2m+n)-4m(m+n),m=0,1,2,...,\\left[ \\nu -n/2\\right]$ of the Schrödinger operator with uniform magnetic field on $\\mathbb {B}^{n}$ given by $H_{\\nu }=-4(1-\\left| z\\right| ^{2})\\left(\\sum \\limits _{i,j=1}^{n}\\left( \\delta _{ij-}z_{i}\\overline{z_{j}}\\right) \\frac{\\partial ^{2}}{\\partial z_{i}\\partial \\overline{ z}_{j}}+\\nu \\sum \\limits _{j=1}^{n}(z_{j}\\frac{\\partial }{\\partial z _{j}}-\\overline{z}_{j}\\frac{\\partial }{\\partial \\overline{z} _{j}})+\\nu ^{2}\\right)+4\\nu ^{2}$ provided that $\\nu >n/2$ .",
"Above $\\left[ x\\right] $ denotes the greatest integer not exceeding $x.$ For $m\\in \\mathbb {Z}_{+},$ the Berezin transform associated with the space in $\\left( 1.1\\right) $ was obtained in [13] via the well known formalism of Toeplitz operators as $\\mathfrak {B}_{m}^{\\nu ,n}\\left[ \\varphi \\right] \\left( z\\right) &=&\\frac{m!\\left( 2\\nu -2m-n\\right) \\Gamma \\left( 2\\nu -m\\right)\\Gamma (n) }{n!\\Gamma \\left( 2\\nu -m-n+1\\right)\\Gamma (n+m) }\\int \\limits _{\\mathbb {B}}\\left( \\frac{(1-\\left| z\\right| ^{2}(1-\\left| \\xi \\right| ^{2})}{\\left|1-\\left\\langle z,\\xi \\right\\rangle \\right| ^{2}}\\right) ^{2\\left( \\nu -m\\right) } \\\\&&\\times \\left( P_{m}^{\\left( n-1,2(\\nu -m)-n\\right) }\\left( 1-2\\left| \\xi \\right| ^{2}\\right) \\right) ^{2}\\frac{\\varphi \\left( \\xi \\right) }{\\left(1-\\left| \\xi \\right| ^{2}\\right) ^{n+1}}d\\mu \\left( \\xi \\right)$ where $P_{m}^{\\left( \\alpha ,\\beta \\right) }\\left( .\\right) $ denotes the Jacobi polynomial [16].",
"Moreover this transform have been expressed as a function $f\\left( \\Delta _{\\mathbb {B}^{n}}\\right) $ of the Laplace-Beltrami operator $\\Delta _{\\mathbb {B}^{n}}$ in terms of an $_{3}\\digamma _{2}$ -sum, see $\\left( 5.24\\right) $ below.",
"Our aim here is to construct for each of the eigenspaces in $\\left(1.1\\right) $ a set of coherent states by following a generalized formalism [11] in order to apply a coherent states quantization method.",
"This provides us with another way to recover the Berezin transforms in $\\left( 1.4\\right) $ attached to the $L^{2}$ -eigenspace spaces in (1.1).",
"Finally, we add a new formula expressing the transform $\\left( 1.4\\right) $ as a function of the Laplace-Beltrami operator.",
"The idea is to make the integral $\\left( 1.4\\right) $ appear as ”convolution product” of the function $\\varphi $ with a specific radial function given in terms of the square of a Jacobi polynomial$.$ Next, a straightforward computation of the spherical transform of this radial function with the use of a Clebsh-Gordon type linearisation [8] for the square of a Jacobi polynomial amounts to a finite sum containing some integrals whose general form was given by Koornwinder [17] in terms of Wilson polynomials.",
"This paper is summarized as follows.",
"In Section 2, we recall briefly the formalism of coherent states quantization we will be using.",
"Section 3 deals with some needed facts on the generalized Bergman spaces.",
"In Section 4, we construct for each of these spaces a set of coherent states and we apply the corresponding quantization scheme in order to recover their associated Berezin transforms.",
"In Section 5, we present the formula expressing these Berezin transforms as functions of the Laplace-Beltrami operator by a different way and in a new form."
],
[
"Coherent states quantization",
"Coherent states are mathematical tools which provide a close connection between classical and quantum formalism.",
"In general, they are a specific overcomplete set of vectors in a Hilbert space satisfying a certain resolution of the identity condition.",
"Here, we review a coherent states formalism starting from a measure space ”as a set of data” as presented in [11].",
"Let $X=\\left\\lbrace x\\mid x\\in X\\right\\rbrace $ be a set equipped with a measure $d\\mu $ and $L^{2}(X,d\\mu )$ the space of $d\\mu -$ square integrable functions on $X.$ Let $\\mathcal {A}^{2}$ be a subspace of $L^{2}(X,d\\mu )$ with an orthonormal basis $\\left\\lbrace \\Phi _{j}\\right\\rbrace _{j=0}^{+\\infty }$ .",
"Let $\\mathcal {H}$ be another (functional) space with a given orthonormal basis $\\left\\lbrace \\phi _{j}\\right\\rbrace _{j=0}^{+\\infty }$ .",
"Then consider the family of states $\\left\\lbrace \\mid x>\\right\\rbrace _{x\\in X}$ in $\\mathcal {H}$ , through the following linear superposition: $\\mid x>:=\\left( \\mathcal {N}\\left( x\\right) \\right) ^{-\\frac{1}{2}}\\sum _{j=0}^{+\\infty }\\Phi _{j}\\left( x\\right) \\mid \\phi _{j}>,$ where $\\mathcal {N}\\left( x\\right) =\\sum _{j=0}^{+\\infty }\\Phi _{j}\\left( x\\right) \\overline{\\Phi _{j}\\left( x\\right) }.$ These coherent states obey the normalization condition $\\left\\langle x\\mid x\\right\\rangle _{\\mathcal {H}}=1$ and the following resolution of the identity of $\\mathcal {H}$ $\\mathbf {1}_{\\mathcal {H}}=\\int \\limits _{X}\\mid x><x\\mid N\\left( x\\right) d\\mu \\left( x\\right)$ which is expressed in terms of Dirac's bra-ket notation $\\mid x><x\\mid $ meaning the rank-one -operator $\\varphi \\mapsto \\left\\langle \\varphi \\mid x\\right\\rangle _{\\mathcal {H}}.\\mid x>.$ The choice of the Hilbert space $\\mathcal {H}$ define in fact a quantization of the space $X$ by the coherent states in $\\left( 2.1\\right) $ , via the inclusion map $x\\mapsto \\mid x>\\in \\mathcal {H}$ and the property $\\left( 2.4\\right) $ is crucial in setting the bridge between the classical and the quantum mechanics$.$ The Klauder-Berezin coherent states quantization consists in associating to a classical observable that is a function $f\\left( x\\right) $ on $X$ having specific properties the operator-valued integral $A_{f}:=\\int \\limits _{X}\\mid x><x\\mid f\\left( x\\right) \\mathcal {N}\\left(x\\right) d\\mu \\left( x\\right)$ The function $f\\left( x\\right) \\equiv \\widehat{A}_{f}\\left( x\\right) $ is called upper (or contravariant) symbol of the operator $A_{f}$ and is nonunique in general.",
"On the other hand, the expectation value $\\left\\langle x\\mid A_{f}\\mid x\\right\\rangle $ of $A_{f}$ with respect to the set of coherent states $\\left\\lbrace \\mid x>\\right\\rbrace _{x\\in X}$ is called lower ( or covariant) symbol of $A_{f}.$ Finally, associating to the classical observable $f\\left( x\\right) $ the obtained mean value $\\left\\langle x\\mid A_{f}\\mid x\\right\\rangle ,$ we get the Berezin transform of this observable.",
"That is, $B\\left[ f\\right] \\left( x\\right) :=\\left\\langle x\\mid A_{f}\\mid x\\right\\rangle ,\\text{ }x\\in X.$ For all aspect of the theory of coherent states and their genesis, we refer to the survey [9] by Dodonov or to the book by Gazeau [11]."
],
[
"The spaces $\\mathcal {A}_{m}^{2,\\protect \\nu }\\left( \\mathbb {B}^{n}\\right) $",
"In this section, we review some results on the $L^{2}$ -concrete spectral analysis of the Schrödinger operator $H_{\\nu }$ in $\\left( 1.3\\right) $ and acting in the Hilbert space $L^{2}(\\mathbb {B}^{n},d\\mu _{n})$ ,see [7], for more details.",
"Let $\\mathbb {B}^{n}=\\lbrace z\\in \\mathbb {C}^{n};\\mid z\\mid <1\\rbrace $ be the unit ball in ${^{n} with the Lebesgue measure d\\mu normalized so that \\mu (\\mathbb {B}^{n}) and let \\partial {\\mathbb {B}^{n}}=\\lbrace \\omega \\in \\mathbb {C}^{n},\\mid \\omega \\mid =1\\rbrace be the unit sphere with d\\sigma the normalized measure on it.Let G=SU(n,1) be the group of all \\mathbb {C}-lineartransforms g on \\mathbb {C}^{n+1} that preserve the indefinite hermitianform \\sum _{j=1}^{n}\\mid z_{j}\\mid ^{2}-\\mid z_{n+1}\\mid ^{2}, with \\det g=1.",
"Then G acts transitively on the unit ball by\\begin{equation}G\\ni g=\\left(\\begin{matrix}a & b \\\\c & d\\end{matrix}\\right):z\\rightarrow g.z=(az+b)(cz+d)^{-1}.\\end{equation}As a homogeneous space we have the identification \\mathbb {B}^{n}=G/K whereK=S(U(n)\\times U(1)) is the stabilizer of 0.It is endowed with itsusual Khaler-Bergman metric ds^{2}=-\\sum _{i,j}^{n}\\partial _{j}\\overline{\\partial }_{j}(Log(1-\\left| z\\right| ^{2}))dz_{i}\\otimes \\overline{dz_{j}}.The Bergman distance and the volume element on \\mathbb {B}^{n} are givenrespectively by\\begin{equation}\\cosh ^{2}d\\left( z,w\\right) =\\frac{\\left| 1-\\left\\langle z,w\\right\\rangle \\right| ^{2}}{(1-\\mid z\\mid ^{2} )(1-\\mid w\\mid ^{2} )}\\end{equation}and d\\mu _{n}\\left( z\\right) =(1-\\mid z\\mid ^{2} )^{-\\left(n+1\\right) }d\\mu \\left( z\\right).\\\\The group G acts unitarily on the space L^{2}(\\mathbb {B}^{n},d\\mu _{n}), viaU(g)F(z)=F(g^{-1}.z).Let consider the magnetic gauge vector potential given through the canonical1-form on \\mathbb {B}^{n}: \\theta =-i(\\partial -\\overline{\\partial })Log(1-\\mid z\\mid ^{2} ), to which the Schrodinger operator\\begin{equation}H_{\\nu }=-\\left( d+i\\nu \\text{ext}\\left( \\theta \\right) \\right) ^{\\ast }\\left( d+i\\text{ext}\\left( \\theta \\right) \\right) +4\\nu ^{2}\\end{equation}can be associated.",
"Here \\nu \\ge 0 is a fixed number d \\ denotes theusual exterior derivative on differential forms on \\mathbb {B}^{n} and ext\\left( \\theta \\right) is the exterior multiplication by \\theta whilethe symbol \\ast stands for the adjoint operation with respect to theHermitian scalar product induced by the Bergman metric ds^{2} ondifferential forms.",
"Note that when \\nu =0, the operator in \\left(3.3\\right) reduces to\\begin{equation}H_{0}\\equiv \\Delta _{\\mathbb {B}^{n}}=4(1-\\mid z\\mid ^{2})\\sum _{i,j=1}^{n}\\left( \\delta _{ij}-z_{i}\\overline{z_{j}}\\right) \\frac{\\partial ^{2}}{\\partial z_{i}\\partial \\overline{ z}_{j}}\\end{equation}which is the Laplace-Beltrami operator of the Bergman ball \\mathbb {B}^{n}.",
"For general \\nu \\ge 0, the Schrodinger operator H_{\\nu } in \\left( 3.3\\right) can beexpressed in the complex coordinates \\left( z_{1},...,z_{n}\\right) by theformula \\left( 1.3\\right) see \\cite {Ay},\\cite {Bo} and \\cite {Ge}.",
"}Now, for an arbitrary complex number $$, a fundamental family ofeigenfunctions of $ H$ with eigenvalue $ 2+42+n2$is given by the Poisson kernels :\\begin{equation}z\\mapsto P_{\\lambda }^{\\nu }(z,\\theta )=\\left( \\frac{1-\\mid z\\mid ^{2}}{\\mid 1-<z,\\theta >\\mid ^{2}}\\right) ^{\\frac{1}{2}\\left( i\\lambda +1\\right)}\\left( \\frac{1-\\overline{<z,\\theta >}}{1-<z,\\theta >}\\right) ^{\\nu },z\\in \\mathbb {B}^{n}.\\end{equation}Moreover, a complete description of theexpansion of an eigenfunction $ f$ of $ H$ with eigenvalue $ 2+42+n2$, in terms of the appropriate Fourier series in $ Bn$ have been given in \\cite [Proposition 2.2]{B}.",
"Precisely,\\begin{equation}f(z)=(1-\\rho ^{2})^{\\frac{i\\lambda +n}{2}}\\sum \\limits _{p,q=0}^{+\\infty }\\rho ^{p+q} \\cdot _{2}\\digamma _{1}\\left( \\frac{i\\lambda +n}{2}+\\nu +p,\\frac{i\\lambda +n}{2}-\\nu +q,p+q+n;\\rho ^{2}\\right) a_{p,q}^{\\lambda ,\\nu }.h_{p,q}(\\theta ),\\end{equation}in $ C([0,1[Bn)$, $ z=$, $ [0,1[$and $ =1$.",
"Above $ 2F1$ denotes the Gausshypergeometric function \\cite {Gr} and $ ap,q,=(ap,q,j,)d(n,p,q)$ are complex numbers,where\\begin{equation}d(n,p,q):=\\frac{(p+q+n-1)(p+n-2)!(q+n-2)!}{p!q!(n-1)!(n-2)!",
"}\\end{equation}is the dimension of the space $ H(p,q)$ of restrictions to the unit sphere $ Bn$ of harmonic polynomials $ h(z)$ on $ n$, whichare homogeneous of degree $ p$ in $ z$ and degree $ q$ in $z$, see \\cite {F} or \\cite {R} for more details.The notation ^{\\prime \\prime }.^{\\prime \\prime } in (3.6) means the following finite sum\\begin{equation}a_{p,q}^{\\lambda ,\\nu }.h_{p,q}(\\theta )=\\sum \\limits _{j=1}^{d(n,p,q)}a_{p,q,j}^{\\lambda ,\\nu }h_{p,q}^{j}(\\theta ),\\end{equation}where $ {hp,qj}1jd(n,p,q)$ is an orthonormal basis of$ H(p,q)$.",
"The spectral analysis of $ H$ have been studied by manyauthors, see \\cite {B} and references therein.Actually, $ H$ is an elliptic densely defined operator on the Hilbertspace $ L2(Bn,(1-z,z)-( n+1) d)$ admitting a unique self-adjoint realization also denotedby $ H$.",
"Its spectrum consists of a continuous part given by $ [ n2,+[ $ (corresponding to scattering states) and a finitenumber of infinitely degenerate eigenvalues $ m,n$ givenby $ ( 1.2) $ (characterizing bound states) provided that $ 2>n.",
"$ More precisely, $ m,n=m2+42+n2$,with $ m=i(2m+n-2)$, $ m=0,1,...,[ -n/2] $.Here, we focus on the discrete part of the spectrum, which is labeled bythe integer $ m$ and the corresponding eigenspace $ Am2,( Bn) $ defined in $ ( 1.1) $.",
"Taking intoaccount (3.6) and expressing the involved hypergeometric in terms of Jacobipolynomial, an orthonormal basis of $ Am2,( Bn) $ can be given explicitly by\\begin{equation}\\Phi _{p,q}^{\\nu ,m,j}\\left( z\\right) =\\kappa _{p,q}^{\\nu ,m,n}\\left(1-\\left| z\\right| ^{2}\\right) ^{\\nu -m}P_{m-q}^{( n+p+q-1,2(\\nu -m)-n) }\\left( 1-2\\left| z\\right| ^{2}\\right) h_{p,q}^{j}\\left( z,\\overline{z}\\right)\\end{equation}with\\begin{equation}\\kappa _{p,q}^{\\nu ,m,n}=\\left(\\frac{n\\Gamma (2\\nu -m-n-q+1)\\Gamma (p+n+m)}{(m-q)!",
"(2(\\nu -m)-n)\\Gamma (2\\nu -m+p)}\\right)^{-\\frac{1}{2}}.\\end{equation}for varying $ p=0,1,2,...$, $ q=0,1,....,m$ and $ j=1,....,d(n;p,q)$.Furthermore, the space $ Am2,(Bn) $ is a reproducing kernel Hilbert space.",
"That is, there exists a unique complex valued function $ K,m$ on $ BnBn$ such that, denoting $ K,mz(w)=K,m(w,z)$, $ K,mz$ belongs to $ Am2,(Bn)$ for any $ zBn$ and$$f(z)=<f,K^{\\nu ,m}_{z}>,$$for all functions $ f$ in $ Am2,(Bn)$ and all $ zBn$.Its expression can be given explicitly as function of the Bergman geodesic distance as\\begin{eqnarray}K^{\\nu ,m}(z,w) &=&\\frac{\\left( 2\\left( \\nu -m\\right) -n\\right) \\Gamma \\left( 2\\nu -m\\right) }{n!",
"\\Gamma \\left( 2\\nu -m-n+1\\right) }\\left( \\frac{(1-\\overline{< z,w>})}{1-<z,w>}\\right) ^{\\nu } \\\\&&\\times (\\cosh d(z,w))^{-2(\\nu -m)}) P_{m}^{\\left( n-1,2(\\nu -m)-n\\right) }( 1-2\\tanh ^{2}d(z,w)) \\end{eqnarray}$ Remark 3.1.",
"For $m=0,$ the space $\\mathcal {A}_{0}^{2,\\nu }\\left(\\mathbb {B}^{n}\\right) $ reduces further to be isomorphic to the weighted Bergman space of holomorphic function $\\psi $ on $\\mathbb {B}^{n}$ satisfying the growth condition $\\int _{\\mathbb {B}^{n}}\\left| \\psi \\left( z\\right) \\right|^{2}((1-\\left\\langle z,z\\right\\rangle )^{2\\nu -n-1}d\\mu \\left( z\\right)<+\\infty .$ This fact justify why the eigenspace $\\mathcal {A}_{m}^{2,\\nu }\\left( \\mathbb {B}^{n}\\right) $ have been also called a generalized Bergman spaces of index $m$ ."
],
[
"Coherent states quantization",
"Now, to adapt the defintion (2.1) of coherent states for the context of the generalized Bergman spaces in $\\left( 1.1\\right) $ we first list the following notations.",
"$\\left( X,d\\eta \\right) :=\\left( \\mathbb {B}^{n},\\left( 1-\\left|z\\right| ^{2}\\right) ^{-\\left( n+1\\right) }d\\mu \\right) ,d\\eta \\equiv d\\mu _{n}$ is the volume element on $\\mathbb {B}^{n}.$ $x\\equiv z\\in \\mathbb {B}^{n}.$ $\\mathcal {A}^{2}:=\\mathcal {A}_{m}^{2,\\nu }\\left( \\mathbb {B}^{n}\\right) \\subset L^{2}(\\mathbb {B}^{n},\\left( 1-\\left| z\\right|^{2}\\right) ^{-n-1}d\\mu ).$ $\\left\\lbrace \\Phi _{k}\\left( x\\right) \\right\\rbrace \\equiv \\left\\lbrace \\Phi _{p,q,j}^{\\nu ,m}\\left( z\\right) \\right\\rbrace $ is the orthonormal basis of $\\mathcal {A}_{m}^{2,\\nu }\\left( \\mathbb {B}^{n}\\right) $ in $\\left( 3.8\\right)$ $\\mathcal {N}\\left( x\\right) \\equiv \\mathcal {N}\\left( z\\right) $ is a normalization factor.",
"$\\left\\lbrace \\varphi _{k}\\right\\rbrace \\equiv \\left\\lbrace \\varphi _{p,q,j}\\right\\rbrace $ is an orthonormal basis of another (functional) Hilbert space $\\mathcal {H}$ .",
"Definition 4.1.",
"For each fixed integer $m=0,1,...,\\left[n-\\nu /2\\right] ,$ a class of generalized coherent states associated with the space $A_{m}^{2,\\nu }\\left( \\mathbb {B}^{n}\\right) $ is defined according to $\\left( 2.1\\right) $ by the form $\\phi _{z}^{\\nu ,m}\\equiv \\mid z,\\nu ,m>:=\\left( \\mathcal {N}\\left( z\\right)\\right) ^{-\\frac{1}{2}}\\sum \\limits _{\\begin{array}{c} 0\\le q\\le m,0\\le p<+\\infty \\\\ 1\\le j\\le d\\left( n,p,q\\right) \\end{array}}\\Phi _{p,q,j}^{\\nu ,m}\\left( z\\right)\\varphi _{p,q,j}$ where $\\mathcal {N}\\left( z\\right) $ is a normalization factor.",
"Proposition 4.1.",
"The factor in $\\left( 4.1\\right) $ is given by $\\mathcal {N}\\left( z\\right) =\\frac{\\left( 2(\\nu -m)-n\\right) \\Gamma \\left(2\\nu -m\\right) }{n!\\Gamma \\left( 2\\nu -m-n+1\\right) }\\frac{\\Gamma \\left( m+n\\right) }{m!\\Gamma \\left( n\\right) }$ for every $z\\in \\mathbb {B}^{n}.$ Proof.",
"To calculate this factor, we start by writing the condition $\\left\\langle \\phi _{z}^{\\nu ,m},\\phi _{z}^{\\nu ,m}\\right\\rangle _{\\mathcal {H}}=1.$ Equation $\\left( 4.3\\right) $ is equivalent to $\\left( \\mathcal {N}\\left( z\\right) \\right) ^{-1}\\sum \\limits _{p=0}^{+\\infty }\\sum \\limits _{q=0}^{m}\\sum \\limits _{j=1}^{d(n,p,q)}\\Phi _{p,q,j}^{\\nu ,m}\\left( z\\right) \\overline{\\Phi _{p,q,j}^{\\nu ,m}\\left( z\\right) }=1$ Making use of (3.9) and (3.11) for the particular case $z=w,$ we get that $\\mathcal {N}\\left( z\\right) =\\frac{\\left( 2(\\nu -m)-n\\right) \\Gamma \\left(2\\nu -m\\right) }{n!\\Gamma \\left( 2\\nu -m-n+1\\right) }P_{m}^{\\left(n-1,2(\\nu -m)-n\\right) }\\left( 1\\right)$ Next, by the following fact on Jacobi polynomial [14]: $P_{m}^{\\left( \\alpha ,\\beta \\right) }\\left( 1\\right) =\\frac{\\Gamma \\left(m+\\alpha +1\\right) }{m!\\Gamma \\left( \\alpha +1\\right) }$ for $\\alpha =n-1$ to arrive at the announced result.The states $\\phi _{z}^{\\nu ,m}\\equiv \\mid z,\\nu ,m>$ satisfy the resolution of the identity $1_{\\mathcal {H}}=\\int \\limits _{\\mathbb {B}^{n}}\\mid z,\\nu ,m><z,\\nu ,m\\mid \\mathcal {N}\\left( z\\right) d\\nu $ and with the help of them we can achieve the coherent states quantization scheme described in Sec.2 to rederive the Berezin transform ${B}_{m}^{\\nu ,n}$ in $\\left( 1.4\\right) $ which was defined by Toeplitz operators formalism in [13].",
"For this let us associate to any arbitrary function $\\varphi \\in L^{2}(\\mathbb {B}^{n},(1-\\left| \\xi \\right|^{2})^{-n-1}d\\mu )$ the operator-valued integral $A_{\\varphi }:=\\int \\limits _{\\mathbb {B}^{n}}\\mid z,\\nu ,m><z,\\nu ,m\\mid \\varphi \\left( z\\right) \\mathcal {N}\\left( z\\right) (1-\\left| z\\right|^{2})^{-n-1}d\\mu $ The function $\\varphi \\left( z\\right) $ is a upper symbol of the operator $A_{\\varphi }.$ On the other hand, we need to calculate the expectation value $\\mathbb {E}_{\\left\\lbrace \\mid z,\\nu ,m>\\right\\rbrace }\\left( A_{\\varphi }\\right):=<z,\\nu ,m\\mid A_{\\varphi }\\mid z,\\nu ,m>$ of $A_{\\varphi }$ with respect to the set of coherent states $\\left\\lbrace \\mid z,\\nu ,m>\\right\\rbrace _{z\\in \\mathbb {B}^{n}}$ defined in $\\left( 4.1\\right)$ .",
"This will constitute a lower symbol of the operator $A_{\\varphi }.$ Proposition 4.2.",
"Let $\\varphi \\in L^{2}(\\mathbb {B}^{n},(1-\\left| \\xi \\right| ^{2})^{-n-1}d\\mu ).$ Then, the expectation value in $\\left( 4.9\\right) $ has the following expression $\\mathbb {E}_{\\left\\lbrace \\mid z,\\nu ,m>\\right\\rbrace }\\left( A_{\\varphi }\\right) &=&\\frac{\\Gamma \\left( n\\right) m!\\left( 2\\left( \\nu -m\\right) -n\\right) \\Gamma \\left( 2\\nu -m\\right) }{n!\\Gamma \\left( n+m\\right) \\Gamma \\left( 2\\nu -m-n+1\\right) }\\int \\limits _{\\mathbb {B}}\\left( \\frac{(1-\\left| z\\right|^{2}(1-\\left| \\xi \\right| ^{2})}{\\left| 1-\\left\\langle z,\\xi \\right\\rangle \\right| ^{2}}\\right) ^{2\\left( \\nu -m\\right) } \\\\&&\\times \\left( P_{m}^{\\left( n-1,2(\\nu -m)-n\\right) }\\left( 1-2\\left| \\xi \\right| ^{2}\\right) \\right) ^{2}\\frac{\\varphi \\left( \\xi \\right) }{\\left(1-\\left| \\xi \\right| ^{2}\\right) ^{n+1}}d\\mu \\left( \\xi \\right) $ for every $z\\in \\mathbb {B}^{n}.$ Proof.",
"We first write the action of the operator $A_{\\varphi }$ in $\\left( 4.8\\right) $ on an arbitrary coherent state $\\mid z,\\nu ,m>$ in terms of Dirac's bra-ket notation as $A_{\\varphi }\\mid z,\\nu ,m>=\\int \\limits _{\\mathbb {B}^{n}}\\mid w,\\nu ,m><w,\\nu ,m\\mid z,\\nu ,m>\\frac{\\mathcal {N}\\left( w\\right) }{(1-\\left| w\\right|^{2})^{n+1}}d\\mu \\left( w\\right)$ Therefore, the expectation value reads $<z,\\nu ,m\\mid A_{\\varphi }\\mid z,\\nu ,m>& \\qquad =\\int \\limits _{\\mathbb {B}^{n}}<z,\\nu ,m\\mid w,\\nu ,m>\\overline{<z,\\nu ,m\\mid w,\\nu ,m>}\\frac{\\mathcal {N}\\left( w\\right) }{(1-\\left| w\\right| ^{2})^{n+1}}d\\mu \\left( w\\right)\\\\& \\qquad =\\int \\limits _{\\mathbb {B}^{n}}\\left| <z,\\nu ,m\\mid w,\\nu ,m>\\right|^{2}\\varphi \\left( w\\right) \\frac{\\mathcal {N}\\left( w\\right) }{(1-\\left|w\\right| ^{2})^{n+1}}d\\mu \\left( w\\right).$ Now, we need to evaluate the quantity $\\left| <z,\\nu ,m\\mid w,\\nu ,m>\\right|^{2}$ in $\\left( 4.13\\right)$ .",
"For this, we write the scalar product as $<z,\\nu ,m\\mid w,\\nu ,m>=\\sum _{p=0}^{+\\infty }\\sum _{q=0}^{m}\\sum \\limits _{j=1}^{d\\left( n;p,q\\right) }\\sum _{r=0}^{+\\infty }\\sum _{s=0}^{m}\\sum \\limits _{l=1}^{d\\left( n;p,q\\right) }\\frac{\\Phi _{p,q,j}^{\\nu ,m}\\left( z\\right) \\overline{\\Phi _{p,q,j}^{\\nu ,m}\\left(w\\right) }}{\\sqrt{\\mathcal {N}\\left( z\\right) \\mathcal {N}\\left( w\\right) }}\\left\\langle \\varphi _{p,q,j},\\varphi _{p,q,l}\\right\\rangle _{\\mathcal {H}}$ Recalling that $\\left\\langle \\varphi _{p,q,j},\\varphi _{p,q,l}\\right\\rangle _{\\mathcal {H}}=\\delta _{j,l}\\delta _{p,r}\\delta _{q,s}$ since $\\left\\lbrace \\varphi _{p,q,j}\\right\\rbrace $ is an orthonormal basis of $\\mathcal {H}$ , the above sum in $\\left( 4.14\\right) $ reduces to $<z,\\nu ,m\\mid w,\\nu ,m>=\\left( \\mathcal {N}\\left( z\\right) \\mathcal {N}\\left(w\\right) \\right) ^{-\\frac{1}{2}}\\sum \\limits _{\\begin{array}{c} 0\\le q\\le m,0\\le p<+\\infty \\\\ 1\\le j\\le d\\left( n,p,q\\right) \\end{array}}\\Phi _{p,q,j}^{\\nu ,m}\\left( z\\right) \\overline{\\Phi _{p,q,j}^{\\nu ,m}\\left( w\\right) }.$ Now, taking account (3.9) and (3.11), Equation ( 4.16) takes the form $<z,\\nu ,m\\mid w,\\nu ,m> &=\\frac{\\left( 2\\left( \\nu -m\\right) -n\\right) \\Gamma \\left( 2\\nu -m\\right) }{n!\\Gamma \\left( 2\\nu -m-n+1\\right) }\\left(\\mathcal {N}\\left( z\\right) \\mathcal {N}\\left( w\\right) \\right) ^{-\\frac{1}{2}}\\left( \\frac{1-\\overline{\\left\\langle z,w\\right\\rangle }}{1-\\left\\langle z,w\\right\\rangle }\\right) ^{\\nu }\\\\&\\times \\left( \\cosh \\left( d\\left( z,w\\right) \\right) \\right) ^{-2\\left( \\nu -m\\right) }P_{m}^{\\left( n-1,2\\left( \\nu -m\\right) -n\\right) }\\left(1-2\\tanh ^{2}\\left( d\\left( z,w\\right) \\right) \\right).",
"\\nonumber $ So that the square modulus of the scalar product in $\\left( 4.17\\right) $ reads $\\left| <z,\\nu ,m\\mid w,\\nu ,m>\\right| ^{2}&=\\left( \\frac{\\left( 2\\left( \\nu -m\\right) -n\\right) \\Gamma \\left( 2\\nu -m\\right) }{n!\\Gamma \\left( 2\\nu -m-n+1\\right) }\\right) ^{2}\\left( \\mathcal {N}\\left( z\\right) \\mathcal {N}\\left( w\\right) \\right) ^{-1}\\\\&\\times \\left( \\cosh \\left( d\\left( z,w\\right) \\right) \\right) ^{-4\\left( \\nu -m\\right) }\\left( P_{m}^{\\left( n-1,2\\left(\\nu -m\\right) -n\\right) }\\left(1-2\\tanh ^{2}\\left( d\\left( z,w\\right) \\right) \\right) \\right) ^{2}.",
"\\nonumber $ Returning back to $\\left( 4.12\\right),$ we get $\\mathbb {E}_{\\left\\lbrace \\mid z,\\nu ,m>\\right\\rbrace }\\left( A_{\\varphi }\\right)&=\\int \\limits _{\\mathbb {B}^{n}}\\varphi \\left( w\\right) \\left( \\frac{\\left( 2\\left[ \\nu -m\\right] -n\\right) \\Gamma \\left( 2\\nu -m\\right) }{n!\\Gamma \\left( 2\\nu -m-n+1\\right) }\\right) ^{2}\\left( \\mathcal {N}\\left( z\\right)\\mathcal {N}\\left( w\\right) \\right) ^{-1}\\frac{\\mathcal {N}\\left( w\\right) }{(1-\\left| w\\right| ^{2})^{n+1}}\\\\&\\times \\left( \\cosh \\left( d\\left( z,w\\right) \\right) \\right) ^{-4\\left( \\nu -m\\right) }\\left( P_{m}^{\\left( n-1,2\\left( \\nu -m\\right) -n\\right) }\\left(1-2\\tanh ^{2}\\left( d\\left( z,w\\right) \\right) \\right) \\right) ^{2}d\\mu \\left( w\\right), \\nonumber $ which can be also written as $\\mathbb {E}_{\\left\\lbrace \\mid z,\\nu ,m>\\right\\rbrace }\\left( A_{\\varphi }\\right)&=\\int \\limits _{\\mathbb {B}^{n}}\\varphi \\left( w\\right) \\left( \\frac{\\left( 2\\left( \\nu -m\\right) -n\\right) \\Gamma \\left( 2\\nu -m\\right) }{n!\\Gamma \\left( 2\\nu -m-n+1\\right) }\\right) ^{2}\\frac{\\left( \\mathcal {N}\\left(z\\right) \\right) ^{-1}}{(1-\\left| w\\right| ^{2})^{n+1}}\\\\&\\qquad \\times \\left( \\cosh \\left( d\\left( z,w\\right) \\right) \\right) ^{-4\\left( \\nu -m\\right) }\\left( P_{m}^{\\left( n-1,2\\left( \\nu -m\\right) -n\\right) }\\left(1-2\\tanh ^{2}\\left( d\\left( z,w\\right) \\right) \\right) \\right) ^{2}d\\mu \\left( w\\right)\\nonumber \\\\& =\\frac{\\left( 2(\\nu -m)-n\\right) \\Gamma \\left( 2\\nu -m\\right) m!\\Gamma \\left(n\\right) }{n!\\Gamma \\left( 2\\nu -m-n+1\\right) \\Gamma \\left( m+n\\right)}\\int \\limits _{\\mathbb {B}^{n}}\\frac{\\varphi \\left( w\\right) }{(1-\\left| w\\right| ^{2})^{n+1}}\\\\& \\qquad \\times \\left( \\cosh \\left( d\\left( z,w\\right) \\right) \\right) ^{-4\\left( \\nu -m\\right) }\\left( P_{m}^{\\left( n-1,2\\left( \\nu -m\\right) -n\\right) }\\left(1-2\\tanh ^{2}\\left( d\\left( z,w\\right) \\right) \\right) \\right) ^{2}d\\mu \\left( w\\right).",
"\\nonumber $ Finally, we summarize the above discussion by considering the following definition.",
"Definition 4.3.",
"The Berezin transform of the classical observable $\\varphi \\in L^{2}(\\mathbb {B}^{n},(1-\\left| \\xi \\right|^{2})^{-n-1}d\\mu )$ constructed according to the quantization by the coherent states $\\left\\lbrace \\mid z,\\nu ,m>\\right\\rbrace $ in $\\left(4.1\\right) $ is obtained by associating to $\\varphi $ the obtained mean value in $\\left( 4.10\\right) .$ That is, ${B}_{m}^{\\nu ,n}\\left[ \\varphi \\right] \\left( z\\right) =\\mathbb {E}_{\\left\\lbrace \\mid z,\\nu ,m>\\right\\rbrace }\\left( A_{\\varphi }\\right)$ for every $z\\in \\mathbb {B}^{n}.$ Remark 4.4.",
"For $m=0,$ the transform $\\left( 4.10\\right) $ reduces to the well known Berezin transform attached to the weighted Bergman space $\\mathcal {A}_{0}^{2,\\nu }\\left( \\mathbb {B}^{n}\\right) $ of holomorphic function $\\psi $ on $\\mathbb {B}^{n}$ satisfying the growth condition $\\left(3.12\\right) $ and given by ${B}_{0}^{\\nu ,n}\\left[ \\varphi \\right] \\left( z\\right) =\\frac{\\left(2\\nu -n\\right) \\Gamma \\left( 2\\nu \\right) }{n!\\Gamma \\left( 2\\nu -n+1\\right) }\\int \\limits _{\\mathbb {B}^{n}}\\left( \\cosh d\\left( z,\\xi \\right)\\right) ^{-4\\nu }\\frac{\\varphi \\left( \\xi \\right) }{\\left( 1-\\left| \\xi \\right| ^{2}\\right) ^{n+1}}d\\mu \\left( \\xi \\right)$ The latter one have also been written as a function of the Bergman Laplacian $\\Delta _{\\mathbb {B}^{n}}$ as ${B}_{0}^{\\nu ,n}=\\frac{1}{\\Gamma \\left( \\alpha +1\\right) \\Gamma \\left(\\alpha +n+1\\right) }\\left| \\Gamma \\left( \\alpha +1+\\frac{n}{2}+\\frac{i}{2}\\sqrt{-\\Delta _{\\mathbb {B}^{n}}-n^{2}}\\right) \\right| ^{2}$ firstly by Berezin.",
"The above form, involving gamma factors, was derived by Peetre in [18], so that $\\alpha $ there occurring in the weight of the Bergman space, corresponds to $2\\nu -n-1.$"
],
[
"An expression of ${B}_{m}^{\\protect \\nu ,n}$ as function of {{formula:d988f79b-05ba-4780-99de-bd6d09b9f852}}",
"Then Berezin transform ${B}_{m}^{\\nu ,n}$ associated the generalized Bergman space $\\mathcal {A}_{m}^{2,\\nu }\\left( \\mathbb {B}^{n}\\right)$ is given by ${B}_{m}^{\\nu ,n}\\left[ \\varphi \\right] \\left( z\\right) =c_{m}^{\\nu ,n}\\int \\limits _{\\mathbb {B}^{n}}\\frac{\\left( P_{m}^{\\left( n-1,2(\\nu -m)-n\\right) }\\left( 1-2\\tanh ^{2}d\\left( z,\\xi \\right) \\right) \\right)^{2}}{\\left( \\cosh d\\left( z,\\xi \\right) \\right) ^{4\\left( \\nu -m\\right) }}\\varphi \\left( \\xi \\right) \\frac{d\\mu \\left( \\xi \\right) }{\\left( 1-\\left|\\xi \\right| ^{2}\\right) ^{n+1}},$ with $c_{m}^{\\nu ,n}=\\frac{\\Gamma \\left( n\\right) m!\\left( 2\\left( \\nu -m\\right)-n\\right) \\Gamma \\left( 2\\nu -m\\right) }{n!\\Gamma \\left( n+m\\right)\\Gamma \\left( 2\\nu -m-n+1\\right) }$ Let $B^{\\nu ,n}_{m}(z,w)$ be the kernel function of the above integral operator and set $h^{\\nu ,n}_{m}(g)=B^{\\nu ,n}_{m}(z,0)$ , $z=g.0$ .",
"Then the integral operator (5.1) can be written as a convolution product over $G$ : ${B}_{m}^{\\nu ,n}\\left[ \\varphi \\right] \\left( z\\right)= c_{m}^{\\nu ,n}(\\varphi \\ast h^{\\nu ,n}_{m})(g),\\quad z=g.0,$ from which it follows easily that the Berezin operator is an $L^{2}$ -bounded operator.",
"Since $B^{\\nu ,n}_{m}(z,w)$ is a $G$ bi-invariant function it follows that ${B}_{m}^{\\nu ,n}$ is a $G$ -invariant operator.",
"That is $U(g)\\circ {B}_{m}^{\\nu ,n}={B}_{m}^{\\nu ,n}\\circ U(g)$ , for every $g\\in G$ .",
"Therefore ${B}_{m}^{\\nu ,n}$ is, in the spectral theoretic sense, a function of the $G$ -invariant Laplacian $\\Delta _{\\mathbb {B}^{n}}$ of the unit ball.",
"Below we give it explicitly.",
"Theorem 5.1.The Berezin transform ${B}_{m}^{\\nu ,n}$ can be expressed as a function of the Laplace-Beltrami operator $\\Delta _{\\mathbb {B}^{n}\\text{ }}$ as $\\begin{split}{B}_{m}^{\\nu ,n}&=\\left| \\Gamma \\left( 2\\left( \\nu -m\\right) -\\frac{n}{2}+\\frac{i}{2}\\sqrt{-\\Delta _{\\mathbb {B}^{n}}-n^{2}}\\right) \\right| ^{2}\\\\&\\quad \\sum \\limits _{k=0}^{2m}\\gamma _{k}^{\\nu ,m,n}W_{k}(-\\frac{1}{4}\\Delta _{\\mathbb {B}^{n}}-\\frac{n^2}{4};2(\\nu -m)-\\frac{n}{2},1+\\frac{n}{2},\\frac{n}{2},\\frac{n}{2})\\end{split}$ where $W_{k}(.",
")$ are Wilson polynomials, $\\gamma _{k}^{\\nu ,n,m}=\\frac{2m!\\Gamma \\left( n\\right) \\left( 2\\left( \\nu -m\\right) -n\\right) \\Gamma \\left( 2\\nu -m\\right) \\left( -1\\right) ^{k}}{\\Gamma \\left( n+m\\right) \\Gamma \\left( 2\\nu -m-n+1\\right) k!\\Gamma ^{2}\\left( 2\\left( \\nu -m\\right) +k\\right) }\\times A_{k}^{\\nu ,n,m},$ and the coefficients $A_{k}^{\\nu ,n,m}$ are given by (5.10) below.",
"Proof.",
"Recall that ${B}_{m}^{\\nu ,n}\\left[ \\varphi \\right] \\left( z\\right)= c_{m}^{\\nu ,n}(\\varphi \\ast h^{\\nu ,n}_{m})(g),\\quad z=g.0,$ where $h_{m}^{\\nu ,n}\\left( \\xi \\right) :=\\left( 1-\\left| \\xi \\right| ^{2}\\right)^{2\\left( \\nu -m\\right) }\\left( P_{m}^{\\left( n-1,2(\\nu -m)-n\\right) }\\left(1-2\\left| \\xi \\right| ^{2}\\right) \\right) ^{2},\\xi \\in \\mathbb {B}^{n},$ By this way, we have to compute the spherical transform $\\mathcal {F}[h_{m}^{\\nu ,n}]$ of $h_{m}^{\\nu ,n}$ .",
"Namely $\\mathcal {F}[h_{m}^{\\nu ,n}](\\lambda ):=\\int \\limits _{\\mathbb {B}^{n}}h_{m}^{\\nu ,n}(z)\\phi _{-\\lambda }(z)d\\mu _{n}(z),\\lambda \\in {\\mathbb {R}}$ where $\\phi _{\\lambda }$ is the spherical function associated to $\\Delta _{\\mathbb {B}^{n}}$ , given by $\\phi _{\\lambda }(z)=(1-\\mid z\\mid )^{\\frac{i\\lambda +n}{2}}\\quad _{2}F_{1}(\\frac{i\\lambda +n}{2},\\frac{i\\lambda +n}{2},n;\\mid z\\mid ^{2}).$ Using Pfaff's transformation [14] $_{2}\\digamma _{1}\\left( a,b,c;x\\right) =\\left( 1-x\\right) ^{-b}\\, \\, _{2}\\digamma _{1}\\left( b,c-a,c;\\frac{x}{x-1}\\right)$ we rewrite $\\phi _{-\\lambda }$ as $\\phi _{-\\lambda }(z)=\\quad _{2}F_{1}\\left(\\frac{-i\\lambda +n}{2},\\frac{i\\lambda +n}{2},n;\\frac{\\mid z\\mid ^{2}}{\\mid z\\mid ^{2}-1}\\right)$ So that returning back to (5.5) we get $\\mathcal {F}[h_{m}^{\\nu ,n}](\\lambda )=2n\\int \\limits _{0}^{1}\\frac{\\rho ^{2n-1}}{\\left( 1-\\rho ^{2}\\right)^{n+1-2\\left( \\nu -m\\right) }}\\left( P_{m}^{\\left( n-1,2(\\nu -m)-n\\right)}\\left( 1-2\\rho ^{2}\\right) \\right) ^{2}$ $\\times _{2}\\digamma _{1}\\left( \\frac{n+i\\lambda }{2},\\frac{n-i\\lambda }{2},n;\\frac{\\rho ^{2}}{\\rho ^{2}-1}\\right) d\\rho $ To calculate this last integral, we first use a linearisation of the square of Jacobi polynomial in (5.8) by making appeal to the following Clebsh-Gordon type formula see [8], $P_{s}^{\\left( \\kappa ,\\epsilon \\right) }\\left( u\\right) P_{l}^{\\left( \\tau ,\\eta \\right) }\\left( u\\right) =\\sum \\limits _{k=0}^{s+l}A_{s,l}\\left(k\\right) P_{k}^{\\left( \\alpha ,\\delta \\right) }\\left( u\\right)$ for the particular case of parameters $s=l=m$ , $\\kappa =\\tau =\\alpha =n-1$ , $\\epsilon =\\eta =2\\left( \\nu -m\\right) -n$ and $\\delta = 2\\left( \\nu -m\\right) .$ In our setting, the linearisation coefficients $A_{s,l}\\left(k\\right) $ are of the form $A_{k}^{\\nu ,n,m}=\\frac{\\left( 2\\left( \\nu -m\\right) +n\\right) _{k}\\left(n\\right) _{2m}\\left( 2k+2\\left( \\nu -m\\right) +n\\right) \\left( -1\\right)^{k}\\left( 2m\\right) !\\left( \\left( 2\\left( \\nu -m\\right) \\right)_{2m}\\right) ^{2}}{\\left( n\\right) _{k}\\left( 2\\left( \\nu -m\\right)+n\\right) _{2m+k+1}\\left( m!\\right) ^{2}\\left( 2m-k\\right) !\\left( \\left(2\\left( \\nu -m\\right) \\right) _{m}\\right) ^{2}}$ $\\times \\digamma _{2:1}^{2:2}\\left(\\begin{array}{c}-2m+k,-2\\nu -k-n:-m,-n-m+1;-m,-m-n+1 \\\\-2m,-2m-n+1:1-2\\nu ,1-2\\nu \\end{array}\\mid 1,1\\right)$ Here $\\digamma _{l:l^{\\prime }}^{p:p^{\\prime }}\\left( .\\right) $ denotes the Kampé de Fériet double hypergeometric function defined by [20] $\\digamma _{l:l^{\\prime }}^{p:p^{\\prime }}\\left(\\begin{array}{c}\\left( a_{p}\\right) :\\left( b_{p^{\\prime }}\\right) ,\\left( c_{p^{\\prime }}\\right) \\\\\\left( d_{l}\\right) :\\left( \\kappa _{l^{\\prime }}\\right) ,\\left( \\varrho _{l^{\\prime }}\\right)\\end{array}\\mid x,y\\right) =\\sum \\limits _{q,s=0}^{+\\infty }\\frac{\\left[ a_{p}\\right]_{q+s}\\left[ b_{p^{\\prime }}\\right] _{q}\\left[ c_{p^{\\prime }}\\right] _{s}}{\\left[ d_{l}\\right] _{q+s}\\left[ \\kappa _{l^{\\prime }}\\right] _{q}\\left[\\varrho _{l^{\\prime }}\\right] _{s}}\\frac{x^{q}}{q!}\\frac{y^{s}}{s!",
"}$ where $\\left[ a_{p}\\right] _{s}=\\prod _{j=1}^{p}\\left( a_{j}\\right) _{s}$ in which $\\left( x\\right) _{s}=x\\left( x+1\\right) ...\\left( x+s-1\\right) $ is the Pochhammer symbol.",
"Therefore, inserting $\\left( P_{m}^{\\left( n-1,2(\\nu -m)-n\\right) }\\left( 1-2\\rho ^{2}\\right)\\right) ^{2}=\\sum \\limits _{k=0}^{2m}A_{k}^{\\nu ,n,m}P_{k}^{\\left( n-1,2(\\nu -m)\\right) }\\left( 1-2\\rho ^{2}\\right)$ into (5.8) the Fourier-Helgason transform of $h_{m}^{\\nu ,n}$ takes the form $\\mathcal {F}[h_{m}^{\\nu ,n}](\\lambda )=\\sum \\limits _{k=0}^{2m}A_{k}^{\\nu ,n,m}{I}_{k}^{\\nu ,n,m}\\left( \\lambda \\right),$ where the last term in this sum is defined by the integral ${I}_{k}^{\\nu ,m}\\left( \\lambda \\right) &=&\\int \\limits _{0}^{1}\\frac{2n\\rho ^{2n-1}}{\\left( 1-\\rho ^{2}\\right) ^{n+1-2\\left( \\nu -m\\right) }}P_{k}^{\\left( n-1,2(\\nu -m)\\right) }\\left( 1-2\\rho ^{2}\\right)\\\\&&\\times _{2}F_{1}\\left( \\frac{1}{2}\\left( n+i\\lambda \\right) ,\\frac{1}{2}\\left( n-i\\lambda \\right) ,n;\\frac{\\rho ^{2}}{\\rho ^{2}-1}\\right) d\\rho $ To calculate this last integral we make the change of variable $\\rho =\\tanh t.$ Therefore (5.14) reads ${I}_{k}^{\\nu ,m}\\left( \\lambda \\right) &=&\\int \\limits _{0}^{+\\infty }2n\\left( \\sinh t\\right) ^{2n-1}P_{k}^{\\left( n-1,2(\\nu -m)\\right) }\\left(1-2\\tanh ^{2}t\\right)\\\\&&\\times \\left( \\cosh t\\right) ^{-4\\left( \\nu -m\\right) +1}._{2}F_{1}\\left(\\frac{n+i\\lambda }{2},\\frac{n-i\\lambda }{2},n;-\\sinh ^{2}t\\right) dt $ Now, we make use of the result established by Koornwinder [17] $& \\int \\limits _{0}^{+\\infty }(\\cosh t)^{-\\alpha +\\beta -\\delta -\\mu ^{\\prime }-1}\\left( \\sinh t\\right) ^{2\\alpha +1}P_{k}^{\\left( \\alpha ,\\delta \\right)}\\left( 1-2\\tanh ^{2}t\\right) \\nonumber \\\\& \\qquad \\times _{2}F_{1}\\left( \\frac{\\alpha +\\beta +1+i\\lambda }{2},\\frac{\\alpha +\\beta +1-i\\lambda }{2},\\alpha +1;-\\sinh ^{2}t\\right) dt \\\\&=\\frac{\\Gamma \\left( \\alpha +1\\right) \\left( -1\\right) ^{k}\\Gamma \\left(\\frac{1}{2}\\left( \\delta +\\mu ^{\\prime }+1+i\\lambda \\right) \\right) \\Gamma \\left( \\frac{1}{2}\\left( \\delta +\\mu ^{\\prime }+1-i\\lambda \\right) \\right) }{k!\\Gamma \\left( \\frac{1}{2}\\left( \\alpha +\\beta +\\delta +\\mu ^{\\prime }+2\\right) +k\\right) \\Gamma \\left( \\frac{1}{2}\\left( \\alpha -\\beta +\\delta +\\mu ^{\\prime }+2\\right) +k\\right) }\\\\& \\qquad \\times W_{k}\\left( \\frac{1}{4}\\lambda ^{2};\\frac{1}{2}\\left( \\delta +\\mu ^{\\prime }+1\\right) ,\\frac{1}{2}\\left( \\delta -\\mu ^{\\prime }+1\\right) ,\\frac{1}{2}\\left( \\alpha +\\beta +1\\right) ,\\frac{1}{2}\\left( \\alpha -\\beta +1\\right) \\right) \\nonumber $ where $\\beta ,\\delta ,\\lambda \\in \\mathbb {R}$ , $\\alpha ,\\delta >-1$ , $\\delta +\\Re (\\mu )^{\\prime }>-1$ and $W_{k}\\left( .\\right) $ is the Wilson polynomial given in terms of the $_{4}F_{3}$ -sum as ([1],p. 158): $W_{k}\\left( x^{2},a,b,c,d\\right) &:=\\left( a+b\\right) _{k}\\left( a+c\\right)_{k}\\left( a+d\\right) _{k} \\\\ & \\qquad \\times _{4}F_{3}\\left(\\begin{array}{c}-k,k+a+b+c+d-1,a+ix,a-ix \\\\a+b,a+c,a+d\\end{array}\\mid 1\\right)$ for the parameters $\\alpha =n-1,\\delta =2(\\nu -m)-n,\\beta =0$ and $\\mu ^{\\prime }=2(\\nu -m)-n-1.$ We find that ${I}_{k}^{\\nu ,m}\\left( \\lambda \\right) =\\frac{2n\\Gamma \\left( n\\right)\\left( -1\\right) ^{k}}{k!\\Gamma ^{2}\\left( 2\\left( \\nu -m\\right) +k\\right) }\\left| \\Gamma \\left( 2\\left( \\nu -m\\right) -\\frac{n}{2}+i\\frac{\\lambda }{2}\\right) \\right| ^{2}$ $\\times W_{k}\\left( \\frac{1}{4}\\lambda ^{2};2\\left( \\nu -m\\right) -\\frac{n}{2},1+\\frac{n}{2},\\frac{n}{2},\\frac{n}{2}\\right)$ Summarizing the above calculations $\\mathcal {F}\\left[ h_{m}^{\\nu ,n}\\right] \\left( \\lambda \\right) =\\left|\\Gamma \\left( 2\\left( \\nu -m\\right) -\\frac{n}{2}+i\\frac{\\lambda }{2}\\right)\\right| ^{2}$ $\\times \\sum \\limits _{k=0}^{2m}\\gamma _{k}^{\\nu ,n,m}W_{k}\\left( \\frac{\\lambda }{4}^{2};2\\left( \\nu -m\\right) -\\frac{n}{2},1+\\frac{n}{2},\\frac{n}{2},\\frac{n}{2}\\right)$ with the constants $\\gamma _{k}^{\\nu ,n,m}:=\\frac{2m!\\Gamma \\left( n\\right) \\left( 2\\left( \\nu -m\\right) -n\\right) \\Gamma \\left( 2\\nu -m\\right) \\left( -1\\right)^{k}A_{k}^{\\nu ,n,m}}{\\Gamma \\left( n+m\\right) \\Gamma \\left( 2\\nu -m-n+1\\right) k!\\Gamma ^{2}\\left( 2\\left( \\nu -m\\right) +k\\right) },$ where the constants $A_{k}^{\\nu ,n,m}$ is given by (5.10).",
"Finally, replacing $\\lambda $ by $\\sqrt{-\\Delta _{\\mathbb {B}^{n}}-n^{2}}$ , we arrive at the announced result.",
"Remark 5.1.",
"Setting $m=0$ in the formula (5.3) in Theorem 5.1, we recover the result of Peetre [18].",
"Remark 5.2.",
"We should note that the transform ${B}_{m}^{\\nu ,n}$ have been expressed in [13] as a function of the Laplace-Beltrami operator $\\Delta _{\\mathbb {B}^{n}}$ in terms of the $_{3}\\digamma _{2}$ -sum as ${B}_{m}^{\\nu ,n} &=\\sum \\limits _{j=0}^{2m}C_{j}^{\\nu ,n,m}\\frac{\\Gamma \\left( 2\\left( \\nu -m\\right) -\\frac{1}{2}\\left( n-i\\sqrt{-\\Delta _{\\mathbb {B}^{n}}-n^{2}}\\right) \\right) }{\\Gamma \\left( 2\\left( \\nu -m\\right) +j+\\frac{1}{2}\\left( n+i\\sqrt{-\\Delta _{\\mathbb {B}^{n}}-n^{2}}\\right) \\right) }\\\\& \\quad \\times _{3}\\digamma _{2}\\left[\\begin{array}{c}\\frac{1}{2}\\left( n+i\\sqrt{-\\Delta _{\\mathbb {B}^{n}}-n^{2}}\\right) ,n+j,\\frac{1}{2}\\left( n+i\\sqrt{-\\Delta _{\\mathbb {B}^{n}}-n^{2}}\\right) \\\\\\left( \\nu -m\\right) +j+\\frac{1}{2}\\left( n+i\\sqrt{-\\Delta _{\\mathbb {B}^{n}}-n^{2}}\\right) ,n\\end{array}\\mid 1\\right] \\nonumber $ where $C_{j}^{\\nu ,n,m}&=\\frac{\\left( 2\\left( \\nu -m\\right) -n\\right) \\Gamma \\left(n+m\\right) \\left( -1\\right) ^{j}\\Gamma \\left( n+j\\right) }{m!\\Gamma \\left(2\\nu -n-m+1\\right) \\Gamma \\left( 2\\nu -n\\right) }\\\\&\\times \\sum \\limits _{p=\\max \\left( 0,j-m\\right) }^{\\min \\left( m,j\\right) }\\frac{\\left( m!\\right) ^{2}\\Gamma \\left( 2\\nu -m\\right) \\Gamma \\left( 2\\nu -m+j-p\\right) }{\\left( j-p\\right) !\\left( m+p-j\\right) !p!\\left( m-p\\right)!\\Gamma \\left( n+j-p\\right) \\Gamma \\left( n+p\\right) }.",
"\\nonumber $"
]
] | 1204.0934 |
[
[
"Time-dependent electron transport through a strongly correlated quantum\n dot: multiple-probe open boundary conditions approach"
],
[
"Abstract We present a time-dependent study of electron transport through a strongly correlated quantum dot.",
"The time-dependent current is obtained with the multiple-probe battery method, while adiabatic lattice density functional theory in the Bethe ansatz local-density approximation to the Hubbard model describes the dot electronic structure.",
"We show that for a certain range of voltages the quantum dot can be driven into a dynamical state characterized by regular current oscillations.",
"This is a manifestation of a recently proposed dynamical picture of Coulomb blockade.",
"Furthermore, we investigate how the various approximations to the electron-electron interaction affect the line-shapes of the Coulomb peaks and the I-V characteristics.",
"We show that the presence of the derivative discontinuity in the approximate exchange-correlation potential leads to significantly different results compared to those obtained at the simpler Hartree level of description.",
"In particular, a negative differential conductance (NDC) in the I-V characteristics is observed at large bias voltages and large Coulomb interaction strengths.",
"We demonstrate that such NDC originates from the combined effect of electron-electron interaction in the dot and the finite bandwidth of the electrodes."
],
[
"Introduction",
"Electron transport through nanoscale devices is a diverse subject, which is currently the focus of extensive experimental and theoretical research.",
"The fuel of such interest is the expectation that nanoscale objects, such as quantum dots [1] and even single molecules, [2] are to become active components in novel electronic devices, which potentially offer unique advantages over existing technologies.",
"[3] At the fundamental level, the physics of such reduced-dimensional systems is dominated by quantum effects.",
"Among them are electron correlations, which strongly affect the electron transport at this level of confinement, giving rise to prototypical quantum phenomena, such as Coulomb blockade [4], [5] and the Kondo effect.",
"[6], [7], [8] While the Landauer formula is the solution to the non-interacting quantum transport problem, [9] the interacting case continues to be challenging to the theory.",
"The latter is typically approached with the non-equilibrium Green's function (NEGF) formalism, [10] which allows, in principle, the derivation of an interacting many-body Landauer-type formula for the steady-state current in the case where interaction is limited to a finite region in space.",
"[11] In practice, for the majority of the state-of-the-art ab initio transport calculations and numerical algorithms, [12], [13], [14], [15] the method of choice for the electronic structure description is the density functional theory (DFT).",
"However, typical steady-state DFT+NEGF transport schemes have a range of limitations, both conceptual and technical.",
"[16] At the fundamental level it has been recently demonstrated, at least for the case of a single Anderson impurity model, that the linear response conductance calculated from the Kohn-Sham levels for the exact exchange-correlation (XC) functional reproduces closely that computed with many-body approaches.",
"[17], [18] If the same holds true for ab initio DFT, then the DFT+NEGF scheme will provide a complete solution for the zero-bias limit.",
"Still, on the practical side, the commonly used approximations to the XC functional, lacking the so-important derivative discontinuity, [19] fail to capture essential physics for the transport in molecular junctions, qualitatively mispredicting the conduction regime.",
"[20], [21] Different is the situation at finite bias, where, let alone the implementation, conceptual concerns reflect on the very applicability of a ground-state electronic structure theory to an intrinsically non-equilibrium problem especially if electron correlations are significant.",
"[22], [23] One strategy to avoid some of the shortcomings of using equilibrium DFT has been sought in its natural extension, time-dependent (TD) DFT, [24] with practical schemes for TD transport having been developed.",
"[16] In general, real-time TD schemes for quantum transport can be roughly divided into two categories based on their assumption for the initial conditions.",
"In one case the electrodes are prepared in equilibrium with the poles of a battery, but not yet connected to the nanoscopic device.",
"The current then starts to flow when the connection is made.",
"In the other the system electrodes+device is initially at equilibrium and subsequently an electric field is applied to the electrodes.",
"The former assumption, where two initial electrochemical potentials are well defined, is more in the spirit of the Landauer transport picture.",
"The latter is instead more DFT-friendly, as the starting point is the ground state of the system.",
"[16] There has been evidence that these two TD transport variants agree in the non-interacting case, i.e.",
"they lead to the same history-independent steady-state current.",
"[25], [26] More recently, the latter variant combined with the TDDFT, further equipped with a novel XC functional carrying the physical derivative discontinuity, has been applied to study the transport through a quantum dot in the Coulomb blockade (CB) regime by Kurth et al.",
"in Ref. [Kurth].",
"In particular that work has put forward an important novel description of CB as a dynamical process with rapidly oscillating local currents, inaccessible by conventional steady-state transport models.",
"In this work we adopt another recently proposed TD transport scheme, the so-called, multiple-probe battery (MPB) method, [28], [29] to study electron transport through a strongly correlated quantum dot.",
"The MPB scheme was first proposed in the context of correlated electron-ion dynamics and was applied to a wide range of problems, such as current-induced heating in atomic wires.",
"[28], [30] This method belongs to the first of the fore-mentioned categories and enables the realization of an external battery within the finite system of electrodes+device.",
"The external bias is introduced through the difference in the electrochemical potentials of the set of reservoirs, or probes, attached individually to each atom in a pair of large but finite metallic electrodes (leads).",
"The scheme is very tractable computationally and has the control knobs to be an arbitrarily close approximation to the non-interacting Landauer transport in the long-time dc limit.",
"The MPB time-propagation scheme is based on the integration of the Liouville-von Neumann equation of motion for the reduced density matrix of the system, in which the open boundaries are described explicitly by a source and a drain term.",
"For the TD Hamiltonian of the quantum dot, entering the equation of motion, we adopt the description used by Kurth et al.. [27] This is based on the adiabatic Bethe ansatz local-density approximation [31] (adiabatic BALDA, or ABALDA) to the XC functional, which exhibits a derivative discontinuity at half-filling.",
"By investigating the real-time evolution of the current through the quantum dot, we find an agreement with Ref.",
"[Kurth], i.e.",
"for a certain set of parameters the system does not reach a steady state but rather remains in a dynamical state, characterized by oscillations in the current.",
"Furthermore, we try to interpret the TD results in terms of the more familiar steady-state picture of transport.",
"In particular, we construct the current-voltage, I-V, characteristics of the quantum dot from the long-time average of the current and the voltage obtained from the TD simulations.",
"This is done for a wide range of parameters, even in the cases when a steady state is not achieved.",
"Importantly, we observe a drop of the current as a function of the source-drain voltage and, as a consequence, a negative differential conductance (NDC) above a critical bias voltage.",
"We demonstrate that such an effect is not possible if the derivative-discontinuity is not included in the one-particle potential.",
"This is particularly interesting in view of some recent contrasting results.",
"On the one hand a number of studies, based on several distinct many-body approaches, [32], [33], [34] attribute the NDC mainly to electron-electron interaction.",
"On the other hand, it has been demonstrated by Bâldea and Köppel [35] that in the case of an exactly solvable model for a non-interacting dot within the steady-state formalism, the finite bandwidth of the electrodes can alone lead to pronounced NDC for a wide range of parameters.",
"Here we find a numerical proof that this result can be generalized to the interacting case and time-dependent transport.",
"Our calculations suggest, however, that for the system considered here, the NDC is due to a combination of two effects, namely electron-electron interaction on the dot and the finite bandwidth of the electrodes.",
"Our paper is organized as follows.",
"In the next section we introduce the model system and our theoretical framework, i.e.",
"the Hamiltonian and the computational scheme for MPB quantum transport.",
"In the first part of Section the I-V characteristics of a non-interacting quantum dot calculated by using the TD-MPB method is compared to analytic NEGF results.",
"We then discuss the finite electrode bandwidth as a source of NDC.",
"In the second part of Section , we present the TD results for a strongly correlated dot in the CB regime.",
"Finally, we propose an explanation for the observed NDC in the I-V characteristics."
],
[
"Methods",
"The model system considered in this work is presented in Fig.",
"REF .",
"This consists of a central region, which contains the quantum dot surrounded by two $N_{\\mathrm {d}}$ -site long atomic chains at both sides, and two one-dimensional finite leads, each counting $N_\\mathrm {L(R)}$ atoms.",
"The physics of the quantum dot connected to two leads is described by the Anderson impurity model.",
"[36], [11] The Hamiltonian of the total system thus reads $\\hat{H}_\\mathrm {S} = \\sum _{\\scriptsize \\alpha =\\mathrm {L,R}}\\hat{H}_{\\alpha }+\\hat{H}_\\mathrm {T}+\\hat{H}_\\mathrm {QD}\\:.$ Here the first term is the nearest-neighbors single-orbital tight-binding (TB) Hamiltonian describing respectively the left-hand side ($\\alpha $ =L) and right-hand side ($\\alpha $ =R) lead.",
"This is written as $\\hat{H}_{\\alpha }=\\sum _{\\scriptsize i,\\sigma } \\varepsilon _{i\\alpha }\\, \\hat{c}^{\\sigma \\dagger }_{i\\alpha } \\,\\hat{c}^{\\sigma }_{i\\alpha }+\\sum _{\\scriptsize i,\\sigma } \\gamma _0\\left(\\,\\hat{c}^{\\sigma \\dagger }_{i\\alpha } \\,\\hat{c}^{\\sigma }_{i+1\\alpha } + h.c.\\right)\\:,$ where $\\varepsilon _{i\\alpha }$ are the on-site energies and $\\gamma _0$ is the hopping integral; $\\hat{c}_{i\\alpha }^{\\sigma \\dagger }(\\hat{c}_{i\\alpha }^{\\sigma })$ is the creation (annihilation) operator for an electron with spin $\\sigma $ ($\\sigma $ =$\\uparrow ,\\downarrow $ ) at the atomic site $i$ of the lead $\\alpha $ (the index $i=1,..,N_{\\alpha }$ runs from left to right for $\\alpha $ =R and from right to left for $\\alpha $ =L).",
"Note that two atomic chains on each side of the quantum dot are also described by a TB model with the hopping integral $\\gamma _0$ and therefore they are included in the Hamiltonian of the leads.",
"Figure: (Color online) Schematic of the model system considered in this work: the central region consists of a quantum dot (QD) surroundedby two N d N_\\mathrm {d}-site long atomic chains, which in turns are attached to two one-dimensional leads comprising respectively N L N_\\mathrm {L}and N R N_\\mathrm {R} sites.",
"Here γ 0 \\gamma _0 is the hopping integral in the leads and in the two chains, and γ c \\gamma _\\mathrm {c} is thelead to dot hopping.",
"V g V_\\mathrm {g} denotes the gate voltage, acting locally on the dot, and V sd V_\\mathrm {sd} is the source-drain voltageapplied across the entire system.The second term in Eq.",
"(REF ) describes the tunneling between the quantum dot and the two adjacent sites and it is given by $\\hat{H}_\\mathrm {T}=\\sum _{\\scriptsize \\sigma } \\gamma _\\mathrm {c}\\left(\\,\\hat{c}^{\\sigma \\dagger }_{0} \\,\\hat{c}^{\\sigma }_{1L} +\\, \\hat{c}^{\\sigma \\dagger }_{0} \\,\\hat{c}^{\\sigma }_{1R} + h.c.\\right),$ where $\\hat{c}_{0}^{\\sigma \\dagger }(\\hat{c}_{0}^{\\sigma })$ is the creation (annihilation) operator for an electron with spin $\\sigma $ on the dot and $\\gamma _\\mathrm {c}$ is the hopping integral between the dot and site $i$ =1 in the lead $\\alpha $ .",
"Finally, the Hamiltonian of the quantum dot reads $\\hat{H}_\\mathrm {QD}=\\sum _{\\scriptsize \\sigma } V_\\mathrm {g}\\,\\hat{n}_{0}^{\\sigma }+U\\,\\hat{n}_{0}^{\\uparrow }\\hat{n}_{0}^{\\downarrow },$ where $V_\\mathrm {g}$ is the on-site energy of the dot, which acts as a local gate voltage; $U$ ($U\\ge 0$ ) is the charging energy, which expresses the strength of the Coulomb repulsion on the dot; $\\hat{n}_{0}^{\\sigma }$ =$\\hat{c}^{\\sigma \\dagger }_{0}\\,\\hat{c}^{\\sigma }_{0}$ is the site-occupation operator.",
"Within the lattice DFT framework [37] the many-body Hamiltonian in Eq.",
"(REF ) is mapped onto an effective single-particle Kohn-Sham Hamiltonian which, in the local density approximation, reads $\\hat{H}_\\mathrm {0}=\\sum _{\\scriptsize \\sigma } v_\\mathrm {KS}\\left[ n_{0}\\right] \\,\\hat{n}_{0}^{\\sigma }.$ Here $n_{0}$ is the charge density of the dot and $v_\\mathrm {KS}$ is the effective Kohn-Sham potential, which can be written as a sum of three terms $v_\\mathrm {KS}\\left[n_{0}\\right] =V_\\mathrm {g}+\\frac{n_{0}}{2}U+v_\\mathrm {XC}\\left[ n_{0}\\right] .$ The second and third terms are respectively the Hartree and the XC potential.",
"The latter is approximated by a modified BALDA potential, specifically tailored to a nonuniform configuration with a weakly coupled dot (we refer to Ref.",
"[Kurth] for the exact expression and the parametrization).",
"Notably, such $v_\\mathrm {XC}$ exhibits a derivative discontinuity at $n_0$ =1, i.e.",
"at the phase transition of the 1D Hubbard model.",
"In practice, however, we use a continuous approximation to the BALDA potential [27] where the true discontinuity, expressed through a Heaviside step function $\\theta (n_0)$ , is replaced by a function $f(n_0)=1/(e^{(n_0-1)/a}+1)$ with $a$ being a smoothing parameter.",
"We use $a=10^{-7}$ , which guarantees a very sharp slope at $n_0=1$ .",
"In our simulations we consider three levels of description: (i) $U=0$ , or non-interacting case, for which the effective potential of the dot is simply given by $v_\\mathrm {KS}$ =$V_\\mathrm {g}$ , (ii) $v_\\mathrm {XC}\\rightarrow 0$ , or the Hartree approximation, where the potential on the dot is $v_\\mathrm {H}$ =$V_\\mathrm {g}+U\\,n_{0}/2$ ; and (iii) the full discontinuous effective potential, given by Eq.",
"(REF ), which we refer to as $v_\\mathrm {KS}$ for clarity.",
"In order to introduce the time-dependence in the Hamiltonian of the quantum dot, we use the adiabatic approximation, where $v_\\mathrm {0}$ is assumed to depend on time only through the instantaneous charge density of the dot $v_\\mathrm {KS}(t)=v_\\mathrm {KS}[n_0(t)]\\:.$ The question of the applicability of such adiabatic local approximation to the description of non-equilibrium transport in strongly correlated systems has been addressed in recent two works respectively by Uimonen et al.",
"[38] and Khorsavi et al.",
"[39] In particular, a comparative study between the TDDFT approach with ABALDA (TDDFT+ABALDA) and the many-body perturbation theory, applied to out-of-equilibrium Anderson impurity model, has been carried out in Ref. [Uimonen].",
"The results obtained with both approaches have been tested against numerically exact results produced by time-dependent density matrix renormalization group theory.",
"It was found that, in general, the TDDFT+ABALDA approach is in good qualitative agreement with many-body perturbation theory over a wide range of parameters.",
"However, in many cases it overestimates the steady-state currents.",
"This problem was linked to the shortcomings of the local approximation to the XC functional and, in particular, to the absence of electron correlations inside the electrodes.",
"Moreover, it was demonstrated in Ref.",
"[Khorsavi] that the inclusion of dynamical correlations, or memory effects, might eliminate the multistability in the density and the current, which can be found within the TDDFT+ABALDA approach.",
"These are strong indications that more advanced non-local, both in space and time, approximations to the XC functional are required.",
"However, as was demonstrated in Ref.",
"[Kurth] and as it will be shown in this paper, the ABALDA already provides valuable insights into time-dependent transport in strongly correlated systems.",
"We now discuss, following the work of Todorov and co-workers, [28], [29] how the open boundary conditions are introduced in the MPB setup.",
"In the MPB method, each atom $i$ of the leads (with the exception of the $N_\\mathrm {d}$ atoms at both sides of the quantum dot) is connected to an external probe $P_i$ (see Fig.",
"REF ).",
"All the probes attached to the sites in the left (right) lead are kept at the electrochemical potential $\\mu _\\mathrm {L}$ ($\\mu _\\mathrm {R}$ ) and are occupied according to the Fermi-Dirac distribution $f_\\mathrm {L}$ ($f_\\mathrm {R})$ .",
"The source-drain voltage $V_\\mathrm {sd}$ is introduced as $V_\\mathrm {sd}=\\mu _\\mathrm {L}-\\mu _\\mathrm {R}$ (here $V_\\mathrm {sd}$ is in units of eV).",
"For symmetrically applied bias $\\mu _\\mathrm {L}=\\varepsilon _\\mathrm {F}+V_\\mathrm {sd}/2$ and $\\mu _\\mathrm {R}=\\varepsilon _\\mathrm {F}-V_\\mathrm {sd}/2$ , where $\\varepsilon _\\mathrm {F}$ is the Fermi level of the electrodes (assumed identical).",
"The time-dependent equation of motion for the density matrix of the system coupled to the probes reads $i\\hbar \\,\\dot{\\hat{\\rho }}_\\mathrm {S}(t) & = & \\left[\\hat{H}_\\mathrm {S}(t), \\hat{\\rho }_\\mathrm {S}(t)\\right]+\\hat{\\Sigma }^{+}\\,\\hat{\\rho }_\\mathrm {S}(t)-\\hat{\\rho }_\\mathrm {S}(t)\\,\\hat{\\Sigma }^{-}+\\\\& + & \\int _{-\\infty }^{\\infty } \\left[ \\hat{\\Sigma }^{<}(E)\\,\\hat{G}^{-}_\\mathrm {S}(E)-\\hat{G}^{+}_\\mathrm {S}(E)\\,\\hat{\\Sigma }^{<}(E) \\right] dE\\:.",
"\\nonumber $ The last two terms on the right-hand side are extraction (drain) and injection (source) terms, respectively; $\\hat{G}^{+}$ ($\\hat{G}^{-}$ ) is the retarded (advanced) Green's function of the system and it is given by $\\hat{G}^{\\pm }=\\left( E\\,\\hat{I}_\\mathrm {S} - \\hat{H}_\\mathrm {S_0} -\\hat{\\Sigma }^{\\pm } \\pm i\\,\\hat{I}_\\mathrm {S}\\,\\Delta \\right)^{-1}\\:,$ where $\\hat{H}_\\mathrm {S_0}= \\sum _{\\scriptsize \\alpha =\\mathrm {L,R}}\\hat{H}_{\\alpha }+\\hat{H}_\\mathrm {T}+\\sum _{\\scriptsize \\sigma } V_\\mathrm {g}\\,\\hat{n}_{0}^{\\sigma }$ is the time-independent part of $\\hat{H}_\\mathrm {S}(t)$ and $\\Delta $ is a dephasing factor (see later for an exact definition).",
"The self-energies due to the presence of the external probes and the in-scattering self-energy are written as $\\hat{\\Sigma }^{\\pm } & = & \\mp i\\, \\frac{\\Gamma }{2}\\,\\hat{I}_\\mathrm {L} \\mp i\\, \\frac{\\Gamma }{2}\\,\\hat{I}_\\mathrm {R}\\:,\\\\\\hat{\\Sigma }^{<} & = & \\frac{\\Gamma }{2\\pi }\\,f_\\mathrm {L}(E)\\,\\hat{I}_\\mathrm {L} + \\frac{\\Gamma }{2\\pi }\\,f_\\mathrm {R}(E)\\,\\hat{I}_\\mathrm {R}\\:,$ with the broadening $\\Gamma $ defined as $\\Gamma =2\\pi \\gamma _{P}^{2}d$ , where $\\gamma _P$ is the coupling to the probes, assumed to be identical for all sites in the leads, and $d$ is an energy-independent constant, which represents the surface density of states of the probes within the wide-band limit; $\\hat{I}_\\mathrm {M}$ is the identity operator in region M (M=L, R, S).",
"Equation (REF ) is derived from a general Liouville-von Neumann equation for the total density matrix of the system and the probes combined.",
"It incorporates two main approximations: (i) the wide-band limit in the probes and (ii) the decoherence in the injection process, introduced through the relaxation time $\\tau _{\\Delta }$ , with $\\Delta =\\hbar /\\tau _{\\Delta }$ [see Eq.",
"(REF )].",
"The second approximation essentially decouples, over the time interval $\\tau _\\Delta $ , the injection of electrons from the probes into the leads and their subsequent scattering from the time-dependent potential inside the central region, provided that the latter is long enough.",
"In other words the dephasing factor imposes a restriction on the size of the central region ($2 N_\\mathrm {d}+1$ sites).",
"Therefore the inclusion of $N_\\mathrm {d}$ buffer sites on both sides of the dot is essential within the time-dependent formalism.",
"The value of $\\Delta $ is determined in such way that the distance traveled by the electrons during the time interval $\\tau _\\Delta $ is smaller than the distance between the electrodes and the interior of the central region, i.e.",
"the quantum dot.",
"This condition can be written as $v_e\\tau _{\\Delta }<N_\\mathrm {d}a$ , where $v_e$ is the electron group velocity and $a$ the lattice constant ($a=1$ ).",
"In practical terms, the introduction of the dephasing factor allows one to write down the injection term, which is in general non-local in time, in a rather simple time-independent form [see Eq.",
"(REF )].",
"This, however, also introduces an additional broadening, proportional to $\\Delta $ , in the steady-state I-V characteristics, which is absent in the standard static NEGF formalism.",
"We note that in the steady-state MPB formalism, the NEGF result is recovered in the limit of infinitely long leads and weak lead-probe coupling.",
"[29] In order to investigate the open-boundary electron dynamics in the time domain, Eq.",
"(REF ) is numerically-integrated using the fourth-order Runge-Kutta (RK4) algorithm.",
"[40] As initial condition, we use the density matrix $\\hat{\\rho }_\\mathrm {S}(t_0)$ of an isolated system (not coupled to the probes), constructed from the eigenstates of the Hamiltonian $\\hat{H}_\\mathrm {S}$ .",
"The open boundary terms are switched on over a short time interval of 5 fs and maintained throughout the simulation.",
"The current through the dot is then calculated as a bond current between the dot and the adjacent site.",
"[41] The typical parameters of the MPB setup used in our simulations, unless specified otherwise, are $N_\\mathrm {L/R}=90$ and $N_\\mathrm {d}=20$ .",
"We have tested that further increasing the size of the system does not lead to significant difference in the I-V characteristics.",
"In order to have one free parameter instead of two, we use the condition $\\Delta $ =$\\Gamma /2$ , which has been discussed in detail in Ref.",
"[Todorov1], and $\\Gamma =0.35$ eV in our simulations."
],
[
"Non-interacting case",
"As a test of the applicability of the TD MPB method we first examine the non-interacting case ($U=0$ ).",
"For this situation, we directly compare the I-V characteristics obtained from the time-dependent simulations to the ones calculated by using the standard NEGF-based Landauer solution, which we refer to as exact NEGF.",
"[10] The comparison is presented in Fig.",
"REF , where the current is plotted as a function of the source-drain voltage for the non-interacting level aligned with the Fermi level in the leads ($V_\\mathrm {g}=0$ ).",
"In the case of the TD MPB approach, the value for the steady-state current is obtained from the time-dependent simulation for the corresponding value of $V_\\mathrm {sd}$ after the steady-state has been established, i.e.",
"when the variation of the current with time becomes negligible.",
"In the case of the exact NEGF method, we use the well-known analytical expression for the non-equilibrium current through a non-interacting resonant level coupled to two semi-infinite electrodes [10], [35] $I_\\mathrm {EN}=\\frac{2e}{h}\\int \\,&dE&\\,\\frac{\\Gamma _{\\mathrm {L}}(E)\\Gamma _{\\mathrm {R}}(E)}{\\left[E-V_\\mathrm {g}-\\Lambda (E)\\right]^2+\\left[\\Gamma (E)/2\\right]^2}\\times \\nonumber \\\\&\\times &\\left[f_\\mathrm {L}(E)-f_\\mathrm {R}(E)\\right].$ Here $\\Lambda (E)=\\Lambda _\\mathrm {L}(E)+\\Lambda _\\mathrm {R}(E)$ and $\\Gamma (E)=\\Gamma _\\mathrm {L}(E)+\\Gamma _\\mathrm {R}(E)$ represent, respectively, the real and imaginary part of the total self-energy due to the presence of electrodes, with $\\Lambda _\\mathrm {L(R)}$ and $\\Gamma _\\mathrm {L(R)}$ given by $\\Lambda _\\mathrm {L(R)}(E)&=&\\frac{\\gamma _\\mathrm {c}^2}{2\\gamma _0^2}E_\\mathrm {L(R)},\\\\\\Gamma _\\mathrm {L(R)}(E)&=&\\frac{\\gamma _\\mathrm {c}^2}{\\gamma _0^2}\\theta (2\\gamma _0-|E_\\mathrm {L(R)}|)\\sqrt{4\\gamma _0^2-E_\\mathrm {L(R)}^2}\\:,$ where $E_\\mathrm {\\alpha }\\equiv E-\\varepsilon _\\mathrm {\\alpha }$ , $\\varepsilon _\\mathrm {\\alpha }$ being the on-site energy in the lead ($\\alpha =$ L, R).",
"Figure: (Color online) Current through the quantum dot, I 0 I_0, as a function of the source-drain voltage, V sd V_\\mathrm {sd}, for zerogate voltage (V g V_\\mathrm {g}=0), calculated using the both exact NEGF and the TD MPB method, and for two configurations of the leads:ε L /R\\varepsilon _\\mathrm {L/R}=0 and ε L /R\\varepsilon _\\mathrm {L/R}=±V sd /2\\pm V_\\mathrm {sd}/2.",
"The inset shows the I-Vcharacteristics obtained with the TD MPB approach for ε L /R\\varepsilon _\\mathrm {L/R}=0 and three different gate voltages,V g V_\\mathrm {g}=1.01.0, 1.51.5 and 2.02.0 eV.",
"The following parameters are used: γ 0 \\gamma _0=-1.0-1.0 eV, γ c \\gamma _\\mathrm {c}=-0.1-0.1 eV andε F \\varepsilon _\\mathrm {F}=0.",
"The source-drain voltage is applied symmetrically, μ L /R\\mu _\\mathrm {L/R}=ε F ±V sd /2\\varepsilon _\\mathrm {F}\\pm V_\\mathrm {sd}/2.In order to achieve a better agreement with the exact NEGF results we use the improved MPB setup with N L(R) =250N_{\\mathrm {L(R)}}=250,N d =90N_{\\mathrm {d}}=90 and Γ=0.15\\Gamma =0.15.We consider two possible limits for the on-site energies in the electrodes: (i) the highly conducting regime with $\\varepsilon _\\mathrm {\\alpha }$ =0 for all atoms in $\\alpha =$ L, R and (ii) the weakly conducting regime for which the on-site energies in $\\mathrm {L(R)}$ are shifted in accordance with the respective electrochemical potential, $\\varepsilon _\\mathrm {L(R)}$ =$\\pm V_\\mathrm {sd}/2$ .",
"As expected, the difference between the I-V curves calculated in these two limits becomes significant at large bias, since the transmission in case ($ii$ ) rapidly drops to zero once the bias voltage exceeds the bandwidth of the leads ($4|\\gamma _0|$ ).",
"This high-bias NDC effect, stemming entirely from the finite electrode band-with, is a well-understood feature of steady-state transport in low-dimensional yet uncorrelated electron systems.",
"[35] We also note that the low-bias agreement between the two transport limits can, in principle, be extended to arbitrarily high biases $V_\\mathrm {sd}$ by increasing $\\gamma _0>V_\\mathrm {sd}/4$ .",
"We have established that if $|\\gamma _0|$ is increased from 1 eV to $3.88$ eV, the result for the current, obtained using two limits for the on-site energies of the leads, differ by at most $3\\%$ for $V_\\mathrm {sd}$ =4 eV and for $V_\\mathrm {g}$ between 0 and 1 eV.",
"An encouraging result is that for both the transport limits the TD MPB method reconstructs rather well the exact NEGF I-V.",
"The agreement is particularly good in the highly conducting limit.",
"The smearing of the abrupt I-V features at low bias and again the NDC drop at $V_\\mathrm {sd}\\lesssim 4\\gamma _0$ for the weakly conducting limit are inherent to the TD MPB method.",
"[28] These are due to the explicit dephasing factor, which simplifies the equation of motion for the density matrix by eliminating temporal non-localities of the injection.",
"In order to eliminate the drop in the current at large bias voltages and to focus on the electron interaction at the quantum dot, we will use the $\\varepsilon _\\mathrm {L(R)}$ =0 limit in all the further calculations presented.",
"In this case, the saturation current at high voltages is entirely determined by the position of the resonant level, set by the gate voltage $V_\\mathrm {g}$ (see the inset of Fig REF ), relatively to the electrodes band center.",
"As the resonant the level approaches the band-edge of the leads ($V_g \\lesssim 2\\gamma _0$ ), the saturation current decreases.",
"In Section REF we will recognize the contribution of the latter effect to the drop in the current as a function of the source-drain voltage.",
"While in the non-interacting case the TD current through the dot always reaches the steady-state, in the case when electron-electron interaction is considered this is not guaranteed.",
"In fact for certain values of the source-drain voltage, for which the charge density of the dot approaches unity, the system is driven into a dynamical state, where current, density and on-site potential oscillate [27] without ever reaching a steady-state.",
"Figure: (Color online) Real-time evolution of the quantum dot: (a) Charge density of the dot (n 0 n_0) for four different values of the source-drain voltage, V sd V_\\mathrm {sd}=1.31.3, 1.61.6, 1.71.7, and 1.91.9 eV.",
"The inset shows the fluctuation of the density around unity, δn 0 \\delta n_0, defined as δn 0 =(n 0 -1)×10 3 \\delta n_0=(n_0-1)\\times 10^{3}.",
"(b) Current through the dot, I 0 I_0, for two values of V sd V_\\mathrm {sd}: V sd V_\\mathrm {sd}=1.61.6 eV (black solid line), which corresponds to the oscillating regime, and V sd V_\\mathrm {sd}=1.31.3 eV (black dashed line) where no oscillations are observed.",
"Note that the corresponding Kohn-Sham potential (v KS v_\\mathrm {KS}) [red solid line] is also in the oscillating regime (V sd V_\\mathrm {sd}=1.61.6 eV).",
"The following parameters are used: γ 0 \\gamma _0=-1.5-1.5 eV, γ c \\gamma _\\mathrm {c}=-0.3-0.3 eV, ε F \\varepsilon _\\mathrm {F}=1.51.5 eV, UU=2.02.0 eV, ε L (R)\\varepsilon _\\mathrm {L(R)}=0 is taken as a reference of energy.",
"The source-drain voltage is applied asymmetrically (μ L \\mu _\\mathrm {L}=ε F +V sd \\varepsilon _\\mathrm {F}+V_\\mathrm {sd}, μ R \\mu _\\mathrm {R}=ε F \\varepsilon _\\mathrm {F}).The question we address here is whether such dynamical state can be captured by the MPB method.",
"The results of our calculations are shown in Fig.",
"REF .",
"For all values of the source-drain voltage below a critical value $V^\\mathrm {cr}_\\mathrm {sd}$ a steady-state is achieved.",
"However, for source-drain voltages above $V^\\mathrm {cr}_\\mathrm {sd}$ , oscillations indeed develop in all transport-related quantities.",
"As shown in Fig.",
"REF for this range of $V_\\mathrm {sd}$ the density quickly reaches a critical value of $n_0=1$ .",
"At the same time the first jump of the on-site potential occurs, followed by a series of almost rectangular pulses [see Fig.",
"REF (b)].",
"Due to the derivative discontinuity at $n_0=1$ , the on-site potential reaches an oscillating regime, abruptly alternating in time between two values, one just below and the other just above the discontinuity.",
"This translates into oscillations of the charge density around $n_0=1$ [see the inset in see Fig.",
"REF (a)] and also into oscillations in the current [Fig.",
"REF (b)].",
"Below, we elaborate on the dynamical features observed for different values of $V_\\mathrm {sd}$ .",
"The height of the pulses in $v_\\mathrm {KS}(t)$ is equal to the height of the jump of $v_\\mathrm {KS}[n_0]$ at the derivative discontinuity and it is mainly governed by the value of the charging energy $U$ .",
"The width of the pulses increases with increasing $V_\\mathrm {sd}$ .",
"This essentially means that for larger $V_\\mathrm {sd}$ the system tends to stay longer in the state with a larger on-site potential, corresponding to the density above 1.",
"Further increasing $V_\\mathrm {sd}$ will finally lead to a steady-state.",
"The exact value of the threshold voltage, $V^\\mathrm {cr}_\\mathrm {sd}$ , is difficult to determine since the on-site potential changes with time.",
"From simple considerations, however, we established that $V^\\mathrm {cr}_\\mathrm {sd}\\ge v_\\mathrm {KS}[\\bar{n}]$ , where $\\bar{n}$ is a value of the charge density just below 1.",
"For the set of parameters used here $V^\\mathrm {cr}_\\mathrm {sd}\\approx 1.5$ eV.",
"As discussed by Kurth et al., the dynamical state of the quantum dot described above is a manifestation of dynamical Coulomb blockade.",
"By applying a large enough source-drain voltage the dot can be charged.",
"However, when the charge reaches the critical value $n_0=1$ , the on-site potential immediately increases by an amount, determined by Coulomb repulsion $U$ , thus preventing further charging.",
"This essentially corresponds to the CB regime.",
"In addition, the time-dependent simulations reveal that in this regime the quantum dot is alternating between two states, separated by an energy barrier determined by $U$ .",
"These two states correspond to the fluctuation of the charge on the dot around $n_0=1$ , which originates from the fact that the ABALDA potential has a derivative discontinuity at $n_0=1$ but it is a smoothly varying function of $n_0$ away from this occupation.",
"It follows from the discussion that the dynamics of the quantum dot in the CB regime, calculated with the TD MPB method, is in a good agreement with the results reported in Ref.",
"[Kurth] both qualitatively and quantitatively.",
"We have established numerically that the two different methods reproduce practically identical dynamical trajectories for all the observables in the long-time limit in the case of an interacting system.",
"The remaining differences are limited to the early stage of the time-evolution.",
"A characteristic feature of the on-site potential of the dot, observed in Ref.",
"[Kurth], is a transition period just after the start of the oscillations, where the series of rectangular pulses in the time-dependent $v_\\mathrm {KS}$ is preceded by a larger pulse whose width increases with $V_\\mathrm {sd}$ .",
"This characteristic transient pulse is not present in our calculations (see Fig.",
"REF ).",
"In order to establish to what extent the transient pulse is determined by the initial conditions, we performed TD simulations for the same system as shown in Fig.",
"REF but without attaching the external probes, i.e.",
"for a closed-boundary finite system.",
"Instead, we applied the source-drain voltage as a rigid shift of the on-site energies in the left lead, i.e.",
"a term $V_\\mathrm {sd}\\sum _{\\scriptsize i,\\sigma }\\hat{c}^{\\sigma \\dagger }_{i\\alpha } \\,\\hat{c}^{\\sigma }_{i\\alpha }$ has been added to the Hamiltonian $\\hat{H}_\\mathrm {\\alpha }$ for $\\alpha =\\mathrm {L}$ [see Eq.",
"(REF )] at the start of the TD simulation.",
"We used longer leads ($N_\\mathrm {L/R}$ =220) and limited the time of the simulations to 100 fs, which is sufficient to observe the time propagation before the reflections from the finite boundaries start to affect the dynamics.",
"The time-dependence of the charge density, current and on-site potential, obtained from the closed-boundary simulation, is presented in Fig.",
"REF .",
"In contrast to our open-boundary simulations, we indeed observed qualitatively the same transient regime as in Ref. [Kurth].",
"This is mainly characterized by an earlier onset of the CB oscillations for larger source-drain voltages and by the increase of the width of the first pulse in the time-dependence of the Kohn-Sham potential with increasing $V_\\mathrm {sd}$ .",
"Figure: (Color online) Real-time evolution of the quantum dot in the closed-boundary setup:(a) Charge density of the dot (n 0 n_0) for three different values of thesource-drain voltage, V sd V_\\mathrm {sd}=1.21.2, 1.31.3, and 1.41.4 eV.",
"Current through the dot, I 0 I_0, [thick lines] and thecorresponding Kohn-Sham potential, v KS v_\\mathrm {KS}, [thin lines]for (b) V sd V_\\mathrm {sd}=1.21.2 eV, (c) V sd V_\\mathrm {sd}=1.31.3 eV, and (d) V sd V_\\mathrm {sd}=1.41.4 eV.The parameters are the same as in Fig.",
".The source-drain voltage is applied as a rigid shift of the on-site energies in the left lead."
],
[
"Steady-state transport",
"In the previous section we demonstrated that, within a certain range of parameters, the derivative discontinuity prevents the quantum dot to evolve towards the steady-state.",
"Outside this range, however, a steady-state is achievable.",
"Here we determine the steady-state current through the dot for various gate voltages and map out the corresponding I-V curves.",
"For situations, where the dot is trapped in oscillations, we take as steady-state current its time-average in the long-time limit.",
"The linear response conductance as a function of $V_\\mathrm {g}$ is depicted in Fig.",
"REF .",
"This is calculated as the finite-difference ratio $\\Delta I_0/\\Delta V_\\mathrm {sd}$ close to zero bias (for a very low but finite bias $\\Delta V_\\mathrm {sd}=0.01$ eV) and represents an approximation to the zero-bias differential conductance.",
"In the non-interacting case, the conductance is composed of a single peak centered around $V_\\mathrm {g}$ =$1.5$ eV, which corresponds to the Fermi level of the leads.",
"This is expected from the steady-state picture of transport through a non-interacting resonant level.",
"In principle the width of the resonance peak is given by the dot-lead hopping integral $\\gamma _\\mathrm {c}$ .",
"In our TD MPB calculations, however, there is an additional resonance broadening factor ($\\tau _\\Delta $ ) related to the dephasing condition in the equations of motion.",
"Its corresponding energy unit, $\\Delta =\\hbar /\\tau _\\Delta $ , can be associated to a fictitious temperature, smearing the electronic energy distributions in the leads.",
"[28] As a result, a suppression of the transmission resonance proportional to $1/\\Delta $ is also expected.",
"This is the reason of why the amplitude of non-interacting resonance conductance in Fig.",
"REF is below one quantum of conductance, $G_0=2e^2/h$ .",
"Figure: (Color online) Differential conductance of the dot as a function of the gate voltage (V g V_\\mathrm {g}) for the Kohn-Shampotential, v KS v_\\mathrm {KS}, [thick solid lines] and for the Hartree potential, v H v_\\mathrm {H}, [dashed lines] with UU=1, 2, and 3 eV,and for the non-interacting case (thin solid line).",
"The inset shows a comparison between the density-dependence of v KS v_\\mathrm {KS}(solid lines) and v H v_\\mathrm {H} (dashed lines) for the same values of UU.",
"Parameters are the same as those of Fig.",
"andV sd =0.01V_\\mathrm {sd}=0.01 eV.In the interacting case the Anderson impurity model predicts two distinct Coulomb peaks [42] in the conductance as a function of the gate voltage [43].",
"These are manifestation of charge quantization at the dot and correspond to each of the two integer electron number states, in which the dot is inhabited by one or two electrons, respectively.",
"Although the ABALDA potential succeeds in describing some important properties of strongly correlated systems, [44] due to the presence of the derivative discontinuity, it is a single-particle potential and, as such, cannot describe fully these charge states.",
"As a result, the gate-voltage dependence of the conductance, calculated using the full discontinuous effective potential ($v_\\mathrm {KS}$ ), does not show two distinct peaks.",
"However, it presents a structure, bearing the signature of two broadened and overlapping peaks (see Fig.",
"REF ).",
"The distance between these quasi-peaks increases with increasing $U$ and corresponds to the value of the jump of the on-site potential $v_\\mathrm {KS}[n_0]$ at the derivative discontinuity.",
"In the case of the Hartree approximation, the two-peak structure is less pronounced and the two resonances merge into an asymmetric plateau.",
"The width of this plateau is also proportional to $U$ .",
"It should be mentioned that for the TD calculations with $v_\\mathrm {KS}$ and for values of $V_\\mathrm {g}$ between the position of the $U=0$ resonance level $V_\\mathrm {res}\\equiv \\varepsilon _F$ and $V_\\mathrm {res}-U$ (roughly corresponding to the region between the two quasi-peaks) no steady-state is achieved.",
"Hence, the conductance curves in this region of $V_\\mathrm {g}$ carry some degree of arbitrariness, associated with the interpretation of the average TD current.",
"In fact, for those gate voltages driving a charge density at the dot close to unity, even the calculation of the ground-state is problematic from a numerical viewpoint, because of the derivative discontinuity.",
"In such cases we used the following iterative procedure.",
"Let $V_\\mathrm {g}^{0}$ be the value of the gate voltage, for which the ground-state (initial) density is calculated self-consistently, while $V_\\mathrm {g}^{0}+\\delta V_\\mathrm {g}$ is the value of the gate voltage for which the self-consistent calculation does not converge.",
"In this case, the final density, obtained at the end of the time-dependent simulation with $V_\\mathrm {g}$ =$V_\\mathrm {g}^{0}$ , is taken as initial density for the simulation with $V_\\mathrm {g}$ =$V_\\mathrm {g}^{0}+\\delta V_\\mathrm {g}$ .",
"In the same way, from the time-averages in the long time-limit, we map out the I-V characteristics of the interacting dot ($v_\\mathrm {KS}$ ) at a given $V_\\mathrm {g}$ (see Fig.",
"REF ).",
"A remarkable feature of the I-V curves is the drop of the current (NDC) at large source-drain voltages, which is almost negligible for small $U$ but increases with increasing $U$ .",
"Figure: (Color online) Current through the dot, I 0 I_0, as a function of the source-drain voltage, V sd V_\\mathrm {sd},for v KS v_\\mathrm {KS} and different values of UU.",
"The horizontal dashed lines represent the corresponding saturationcurrents I S I_\\mathrm {S} (see text for the exact definition).",
"The following parameters are used: γ 0 \\gamma _0=-3.88-3.88 eV,γ c \\gamma _\\mathrm {c}=-0.5-0.5 eV, ε F \\varepsilon _\\mathrm {F}=1.51.5 eV, V g V_\\mathrm {g}=2.02.0 eV.Figure: Current (a), density (b) and on-site potential (c) of the dot as a function of the source-drain voltage, V sd V_\\mathrm {sd},for v KS v_\\mathrm {KS} with U=5U=5 eV and for v H v_\\mathrm {H} with U=9U=9 eV.",
"The inset shows v KS v_\\mathrm {KS}and v H v_\\mathrm {H} as functions of the dot density for the corresponding values of UU.",
"The parameters are the same asthose in Fig.",
".For all values of $U$ the current initially increases with increasing $V_\\mathrm {sd}$ as the dot is charging.",
"It then reaches its maximum value as the charge density approaches $n_0=1$ .",
"This point corresponds to a threshold source-drain voltage $V^\\mathrm {cr}_\\mathrm {sd}$ , which is roughly the same for all values of $U$ .",
"Beyond $V^\\mathrm {cr}_\\mathrm {sd}$ , the system is driven into a dynamical state (where the steady-state current is calculated by averaging out the oscillations).",
"In the limit of very large $V_\\mathrm {sd}$ , the dot recovers its long-time tendency to a steady state and the average current saturates.",
"At saturation and beyond the dot occupation is above 1 and the on-site potential assumes a value above the discontinuity.",
"Hence, the on-site energy at the dot is proportional to the the jump of the $v_\\mathrm {KS}$ at $n_0=1$ , i.e.",
"it is proportional to $U$ .",
"As discussed in Section REF for the non-interacting case, the saturation current decreases with increasing the dot on-site potential, because of the finite bandwidth of the electrodes.",
"For the same reason here the drop of the current becomes larger when $U$ increases.",
"In fact a large $U$ corresponds to a large value of the steady-state on-site potential, which then approaches the electrodes' band-edge.",
"In order to confirm this conjecture, we compare the saturation current $I_\\mathrm {S}$ calculated at finite $U$ , with that for $U$ =0 and $V_\\mathrm {g}$ equal to the steady-state on-site potential corresponding to that obtained at the same $U$ .",
"Indeed $I_\\mathrm {S}$ matches quite well the value of the current obtained at large source-drain voltages in the I-V characteristics of the interacting dot (see horizontal dashed lines next to each curve in Fig.",
"REF ).",
"This argument can obviously be reversed, i.e.",
"the NDC cannot be observed, if the variation of the on-site potential at the derivative discontinuity, determined by $U$ , is much smaller than the electrodes' bandwidth.",
"For instance, for the same set of parameters used before for the dot+electrodes system, such NDC-free situation is found for $U=2$ eV ($U\\ll 4|\\gamma _0|$ for $\\gamma _0=3.88$ eV).",
"In this case the drop of the current above $V^\\mathrm {cr}_\\mathrm {sd}$ is practically negligible.",
"Importantly, the NDC displayed in Fig.",
"REF is not found in I-V's calculated within the Hartree approximation, even for large values of $U$ (see Fig.",
"REF ).",
"When comparing calculations at the Hartree level with those performed with the complete Kohn-Sham potential we intentionally use different $U$ .",
"These are selected in such a way that the value of the potential at $n_0=1$ is identical in the two calculations [see the inset in Fig.",
"REF (c)], i.e.",
"in such a way that the two calculations give the same saturation current.",
"At variance with the complete Kohn-Sham case, in the Hartree only problem the current, as well as the density and the on-site potential, monotonically increase with $V_\\mathrm {sd}$ until the saturation is reached.",
"Based on these numerical results we can argue that the self-interaction-free shape of the on-site potential $v_\\mathrm {KS}$ at the dot is a necessary condition for the occurrence of the NDC in the I-V.",
"The shallow increase of the on-site potential with the charging, produced by the opening of the bias window, keeps the resonant level away from the electrodes band edge and allows the current to rise.",
"Once the on-site charge exceeds $n_0=1$ and the resonant level energy shoots up towards the band-edge, the currents drops.",
"The averaged dynamical current monotonically approaches its saturation value corresponding to a steady-state solution."
],
[
"Conclusions",
"We have investigated the electronic transport through a strongly-correlated quantum dot by using a recently proposed multiple-probe battery method for time-dependent simulations of open systems.",
"Our aim was two-fold.",
"Firstly, we wanted to assess the outcomes of a TD transport scheme conceptually different from what used so far in literature, for a problem involving strong electron correlation as in Coulomb blockade.",
"Clearly our MPB-based simulations agree well with previous findings.",
"[27] In particular we have demonstrated self-sustained oscillations in the current, density and effective on-site potential, originating from the derivative discontinuity of the approximate exchange-correlation potential used.",
"As a further aspect we have addressed the question of whether the peculiar dynamics obtained from the time-dependent simulations can be related to the more accessible steady-state picture of transport.",
"In particular, we have shown the presence of Coulomb peaks in the linear response differential conductance and extracted the TD version of I-V characteristics, based on the time-averaged current through the dot in the long-time limit.",
"The resulting I-V curves, at a critical voltage, exhibit a drop in the average current through the dot.",
"This drop corresponds to the range of parameters where no steady state is found and the dot is in the oscillatory Coulomb blockade state.",
"Such an NDC is however present only when the calculation is performed at a DFT level in which the potential includes the derivative discontinuity at unitary occupation.",
"We are grateful to A. Hurley and I. Rungger for careful reading of the manuscript.",
"We thank T. N. Todorov for very helpful discussions.",
"This work is sponsored by Science Foundation of Ireland (Grant No.",
"07/IN.1/I945).",
"Computational resources have been provided by the Trinity Center for High Performance Computing."
]
] | 1204.0937 |
[
[
"Globular Cluster Systems of Early-type Galaxies in Low-density\n Environments"
],
[
"Abstract Deep images of 10 early-type galaxies in low-density environments have been obtained with the Advanced Camera for Surveys (ACS) on the Hubble Space Telescope.",
"The global properties of the globular cluster (GC) systems of the galaxies have been derived in order to investigate the role of the environment in galaxy formation and evolution.",
"Using the ACS Virgo Cluster Survey (ACSVCS) as a high-density counterpart, the similarities and differences between the GC properties in high- and low-density environments are presented.",
"We find a strong correlation of the GC mean colours and the degree of colour bimodality with the host galaxy luminosity in low-density environments, in good agreement with high-density environments.",
"In contrast, the GC mean colours at a given host luminosity are somewhat bluer (\\Delta(g-z) ~ 0.05) than those for cluster galaxies, indicating more metal-poor (\\Delta[Fe/H] ~ 0.10-0.15) and/or younger (\\Delta age > 2 Gyr) GC systems than those in dense environments.",
"Furthermore, with decreasing host luminosity, the colour bimodality disappears faster, when compared to galaxies in cluster environments.",
"Our results suggest that: (1) in both high- and low-density environments, the mass of the host galaxy has the dominant effect on GC system properties, (2) the local environment has only a secondary effect on the history of GC system formation, (3) GC formation must be governed by common physical processes across a range of environments."
],
[
"Introduction",
"The study of extragalactic globular cluster (GC) systems is now well-established as a powerful probe of the formation histories of elliptical galaxies (e.g.",
"[63]; [12]).",
"Because each cluster is essentially of a single age, a single metallicity, and a simple stellar population (SSP), the colours and line strengths of individual clusters are more easily interpreted using evolutionary synthesis stellar population models than is the integrated starlight from the spheroidal component.",
"The distribution of the aforementioned properties over the entire cluster population also allows for a unique examination of the relative importance of cluster formation episodes as a function of age and metallicity.",
"Furthermore, the well-documented correlations between the properties of GC systems and those of their host galaxies (e.g.",
"[3]; [42]; [12]; [48], hereafter PJC06) indicate that a close connection must exist between the formation of the diffuse stellar component of a galaxy and that of its GC system.",
"Table: The galaxy propertiesMost of the extant studies on globular systems in early-type galaxies have concentrated on high-density environments where these morphological types are most prevalent.",
"Such studies include the ACS Virgo Cluster Survey [18], the ACS Fornax Cluster Survey [34], and the ACS Coma Cluster Survey [14].",
"Semi-analytic models of galaxy formation (e.g.",
"[7]; [8]) predict that the largest spread in galaxy properties such as metallicity and age should occur for early-type galaxies (L $<$ L$_\\ast $ ) in low-density environments outside of rich clusters.",
"Low-luminosity ellipticals also provide a critical test for current theories on the origin of bimodal colour distributions in GC systems.",
"In some scenarios, clusters in the blue (low-metallicity) peak are primarily associated with the progenitor galaxy; red cluster populations form later via multi-phase collapse [22] or mergers [1].",
"It has also been proposed by [17] that red clusters represent the original population and blue clusters have been accreted from a lower metallicity dwarf galaxy population, in which case the low-luminosity ellipticals would be expected to contain any smaller population of red, metal-rich clusters [39].",
"In N-body simulations (e.g.",
"[46]), metallicity bimodality arises from the history of galaxy assembly.",
"For instance, early mergers are frequently involving relatively low mass protogalaxies, which preferentially produce low metallicity blue clusters.",
"Late mergers are more infrequent but involve more massive galaxies and just a few late massive mergers can produce a significant number of red clusters.",
"Alternatively, [64], [65], and [66] suggested that a broad, single-peaked metallicity distribution presumably as a result of continuous chemical enrichment can produce colour bimodality due to the non-linear metallicity-to-colour conversion, without invoking two distinct sub-populations.",
"In fact, the study of GC populations in low-luminosity ellipticals may be the only way of distinguishing between the above models since the predictions from all of them are very similar for more massive galaxies (e.g.",
"[38]).",
"It is crucial to test these predictions for galaxies in both rich clusters and in the field where, for example, accretion processes may be by far less efficient.",
"The current level of knowledge regarding GC populations in low-luminosity ellipticals is quite poor.",
"[37] concluded on the basis of limited ground-based data that the colours of their GC populations were consistent with a unimodal distribution, i.e., the colours were very different from those of luminous ellipticals.",
"However, ground-based imaging is severely compromised by the need for large statistical background corrections.",
"Compounding the problem is the proportionately smaller number of clusters associated with lower luminosity galaxies.",
"Using compilations of archival WFPC2 HST data, [24] and [39] both found evidence for possible bimodality in a small sample of lower luminosity ellipticals.",
"However, the limited resolution and depth of the WFPC2 data produced discrepancies in the classification of individual galaxies.",
"The properties of low-luminosity galaxies in high-density environments has been addressed by the ACS Virgo Cluster Survey ([48]; [50]).",
"However, such a sample is, by its nature, biased towards environments that are atypical of a general field population (Figure REF ).",
"A complementary field galaxy comparison sample selected from low-density environments is required to investigate the role of environmental effects on the physical processes that control the formation of GC systems and their host galaxies.",
"Figure: Distribution of the local density and the absolute magnitude in B for our sample and the ACSVCS.",
"left panel: The number densities are taken from the NBG .",
"A clear separation in the environment between our sample (solid histogram) and the ACSVCS (dashed histogram) is shown.",
"The distribution of allearly-type galaxies in the NBG is plotted with a dotted line on an arbitrary scale.",
"right panel: The solid histogram represents the distribution of the absolute B magnitude for our sample (Table ), while the dashed histogram indicates that of the ACSVCS galaxies.Figure: Growth curves of bright GCs in the different sample galaxies.",
"The dotted lines are growth curves forthe ACS PSF.",
"The GCs are clearly more extended than a point source.",
"The mostextended profile is from the closest galaxy (NGC 3377)."
],
[
"Sample Selection and Observation",
"In order to obtain a well-defined sample of early-type galaxies with low-luminosities in low-density environments, a complete sample of galaxies was first compiled from the Nearby Galaxy Catalogue (NBG) [61] with the following parameters: $-5$ $\\le $ $H_{code}$ $\\le $ $-2$This is a morphological type code.",
"Elliptical and lenticular galaxies are included in this range., $-18.0$ $ <$ $M_B$ $ <$ $-19.5$The absolute magnitudes of galaxies listed in the NBG scatter by $\\sim $ 0.5 mag compared to the values from the NASA Extragalactic Database (NED).",
"We adopt the values from the NED for the presents analysis., distance $D $ 30 Mpc, Galactic latitude $|b| 45 \\deg $ , and local density $\\rho _0$ $<$ 1.0 $Mpc^{-3}$ .",
"[61] determines the local density, $\\rho _0$ , on a three dimensional grid with a scale size of 0.5 Mpc after Gaussian smoothing, to estimate the contribution of each member of the galaxy population brighter than $M_B$ $<$ $-16$ mag.",
"After a visual inspection of the Digitized Sky Survey images, 10 of the brightest (in apparent magnitude) targets with no nearby bright stars within the ACS field of view were selected.",
"The properties of the final sample of early-type galaxies in low-density environments are listed in Table REF in descending order of absolute luminosity.",
"The local densities of 47 galaxies from the ACSVCS sample are also available in the NBG, including all ACSVCS galaxies within the luminosity range of our sample.",
"As shown in the left panel of Figure REF , there is not only a clear separation in the local density between our field galaxy sample and the ACSVCS sample, but the sample in low-density environments is also more representative of early-type galaxies in general.",
"The right panel of Figure REF shows the distribution in the absolute magnitude of our sample and the ACSVCS sample.",
"The observations (Program ID: 10554) were carried out with the ACS Wide Field Camera (WFC) on the Hubble Space Telescope during Cycle 14 (Oct 05 – Sep 06).",
"The WFC is composed of two 4K $\\times $ 2K chips with a pixel scale of $0.05/pix$ covering a field of view of 202$$ $\\times $ 202$$ .",
"For each galaxy, 2 orbits were used to obtain images in two bands, F475W and F850LP, which correspond to Sloan filters g and z, respectively.",
"These two filters were also used by the ACSVCS, which has the advantage of being able to compare results directly without the use of photometric transformations.",
"In this work, a four-point line dithering pattern in which the telescope pointing shifts 5 $\\times $ 60 pixels between sub-exposures was employed.",
"Such a pattern allows the gap between the two chips to be filled and eliminates hot pixels during data processing.",
"Because dithering also effectively removes cosmic rays, CR-SPLIT was set to NO.",
"The total exposure times were maximized by arranging 8 sub-exposures in the following way.",
"In the first orbit, four sub-exposures of F475W and one sub-exposure of F850LP were allocated.",
"In the second orbit, three sub-exposures of F850LP were obtained.",
"We aim to achieve a S/N $\\sim $ 30 at 50% completeness, corresponding to estimated exposure times of 23 minutes in F475W and 53 minutes in F850LP.",
"The actual total exposure times depend on the visibility of the target and were typically 1300 s and 3000 s for F475W and F850LP, respectively.",
"The observation logs are listed in Table REF .",
"For NGC 3156, the pointing and orientation were slightly adjusted so as to avoid adjacent bright stars.",
"Table: The observation logs"
],
[
"Basic image reduction",
"Each target has four sub-exposure images with different dithering positions in each filter.",
"These raw images were reduced using the default ACS pipeline, CALACS, to produce bias subtracted and flat-field corrected images.",
"The default processing was found to slightly overestimate the sky background because of the large contribution of galaxy light in the images.",
"This resulted in negative background levels on the outskirts of a galaxy in the final drizzled images.",
"An improved sky background was determined by adopting the minimum median value from four 200 pix $\\times $ 200 pix corners of each sub-exposure.",
"The four sub-exposure images were then combined and geometrically corrected with the MultiDrizzle package.",
"The final drizzled image consists of a 4096 $\\times $ 4096 pixel science image in units of electrons/s and an error map in the second extension that contains all error sources such as readout noise, dark current, and photon background noise."
],
[
"Galaxy light subtraction",
"The smooth galaxy light of each target was fitted with elliptical isophotes using the IRAF/ELLIPSE routine.",
"Fitted ellipse parameter tables were then employed in order to build a galaxy model image using the IRAF/BMODEL task.",
"The model galaxy was subtracted from the original image.",
"It was found that subtracting the galaxy continuum light improves the efficiency of GC detection in the central regions of a galaxy.",
"It is because that ellipse fitting does better job to model galaxy light in the central regions than SExtractor [9], which models background (galaxy light in our case) using interpolation of mean/median values in a specified mesh size, thus the galaxy light in the centre may be underestimated.",
"Figure: The colour-magnitude diagram of GC candidates; the parent galaxies are ordered from most luminous (NGC 3818) to least luminous (IC 2035).",
"The dotted boxindicates the applied colour and the magnitude cut applied.",
"The number of GCs inthis box is denoted along with the galaxy name.",
"The background grey scalerepresents the completeness at a given magnitude and colour.",
"Thebottom-center panel shows an example of contamination per field, after oneout of six objects is randomly selected from 6 compiled blank fields."
],
[
"Globular cluster detection and selection",
"SExtractor was run on the galaxy-light subtracted images so as to detect GC candidates with ${\\tt DETECT\\_THRESH} > 3\\sigma $ above the background.",
"In order to properly account for noise due to diffuse galaxy light, the error map from the drizzled images was used to create a weight image in SExtractor.",
"Among the detected objects, those with ${\\tt ELONGATION} > 2$ in either the F475W or F850LP band and very diffuse objects with ${\\tt CLASS\\_STAR} < 0.9$ in the F475W band were rejected.",
"Objects were matched within a 2 pixel radial separation across the two bands.",
"Some spurious detections that were found at the edge of the images were manually removed."
],
[
"Globular cluster photometry",
"Once the list of GC candidates had been generated, aperture photometry was performed on the galaxy-subtracted images using the IRAF/PHOT package with 1 to 10 pixel radius apertures in one pixel steps.",
"The background was estimated locally from a 10 to 20 pixel radius annulus.",
"This locally estimated background has the advantage that the photometry of GCs is hardly affected by the global sky background estimation and the modeling of underlying galaxy light.",
"Because GCs are marginally resolved by ACS/WFC at the distance of our sample galaxies, the aperture correction from stellar PSFs would not be applicable to our GCs.",
"Instead, a representative GC model profile was constructed from several moderately bright GCs for each galaxy using the IRAF/PSF task.",
"From this process, growth curves were extracted.",
"As clearly shown in Figure REF , the GCs are not point sources, which can be verified by the fact that the light profiles of the GCs in the closest galaxy, NGC 3377, are the most extended.",
"An aperture correction for the GCs in each galaxy was determined from the magnitude difference between the 3 pixel and 10 pixel radius apertures on each growth curve.",
"This aperture correction was then applied to the 3 pixel radius aperture photometric measurements for all of the GC candidates.",
"Such a process can be justified by the fact that the sizes of GCs are independent of their luminosity [32].",
"The final corrections from the 10 pixel radius to the total magnitudes are primarily a function of the ACS/WFC PSF and are 0.095 mag for F475W and 0.117 mag for F850LP; the zero-points of the ABmag scale are 26.068 and 24.862 for F475W and F850LP, respectively [54].",
"A correction for Galactic extinction was implemented based on the [52] extinction map values (listed in Table REF ) and the extinction ratios, $A_{F475W}=3.634E(B-V)$ and $A_{F850LP}=1.485E(B-V)$ , from [54].",
"Hereafter, g and z magnitudes refer to the F475W and F850LP extinction-corrected total ABmag.",
"Colour-magnitude diagrams for the GC candidates, in order of their host galaxy luminosity, are shown in Figure REF , with the most luminous galaxy in the top-left panel."
],
[
"Globular cluster completeness tests",
"To test the effectiveness of the proposed GC detection and selection method, a list of 1000 artificial GCs with a uniform luminosity function and a spatial distribution in both filters was independently generated.",
"These GCs were then added onto a galaxy subtracted image that contains all relevant noise sources such as the readout noise and poisson errors from sky and galaxy light.",
"The artificial GCs were simulated using IRAF/MKOBJECTS with the representative GC light profile detailed in §REF .",
"The GC detection and selection procedures described in § REF were then repeated and the initial input coordinates of the artificial GCs were matched with those of the recovered GCs.",
"For a given magnitude bin, the number fractions of recovered GCs were calculated.",
"A completeness curve can often be expressed by the analytic function, $ f=\\frac{1}{2}\\Big (1-\\frac{a(m-m_0)}{\\sqrt{1+a^2(m-m_0)^2}}\\Big )$ where $m_0$ is a magnitude when $f$ is 0.5 and $a$ controls how quickly $f$ declines (the larger the value of $a$ , the steeper the transformation from 1 to 0) [21].",
"An example of completeness in one of the galaxies is shown in Figure REF .",
"The fitted parameters of Equation REF are listed in Table REF .",
"The completeness in each galaxy at a given magnitude and colour is displayed as a grey scale in Figure REF .",
"It is calculated by multiplying the completeness in $g$ and $z$ over a grid of small boxes on the colour-magnitude plane.",
"As seen in Figure REF and Table REF , the completeness levels in $g$ decline more steeply than in z.",
"This is because the additional selection criterion in g, CLASS_STAR, rejects faint GCs more quickly due to the increasing uncertainty of CLASS_STAR in fainter GCs.",
"Figure: The completeness of GC detections in NGC 3377.",
"The filled and open circles denote the completeness of the gg and zz bands respectively.",
"The solid and dotted lines are the fitted curves from Equation .Table: The parameters of the completeness functions"
],
[
"Contamination of background galaxies",
"Although our selection criteria are quite strict, it is inevitable that some amount of contamination by background galaxies will be present (contamination by foreground stars is small at these magnitudes with our selection criteria).",
"To estimate this contamination statistically, the ACS archive was searched for high Galactic latitude blank-fields (proposal ID: 9488) that were observed with the same filters and were at least as deep as our observations.",
"In total 30 drizzled images from six blank fields were retrieved.",
"Table REF lists RA, DEC and the number of exposures of each band for the chosen blank fields.",
"One field usually consists of 3-6 sub-exposures that were combined into one image by taking median values via IRAF/IMCOMBINE.",
"The aforementioned GC detection and selection techniques were applied to these images.",
"Since the observation conditions for the blank fields are not identical to ours (e.g.",
"exposure time and dithering), this difference must be taken into account.",
"It was found that some hot pixels were misclassified as GCs in the blank fields.",
"These were removed by applying an additional classification, ${\\tt FWHM\\_IMAGE} >1.5$ .",
"In the bottom-center panel in Figure REF , one out of every six objects in the 6 blank fields that passed the selection criteria is randomly chosen and plotted.",
"Because the blank fields are deeper than ours, the completeness of each galaxy was applied so as to obtain the colour distribution and luminosity function of the contamination for each galaxy.",
"These contamination profiles were used in the subsequent analysis.",
"The typical contamination per field is $\\sim $ 14 objects.",
"Table: Control fields"
],
[
"Colour distributions",
"To construct the GC colour distributions, the sample of GC candidates was refined by applying the colour cut, $0.6 <(g-z)_0 < 1.7$ , and the magnitude cut, $M_g >-12.0$ , and $g < 25.5$ , where the completeness is $\\sim 80\\%$ .",
"These colour and magnitude selections are indicated by the dotted boxes in Figure REF .",
"In Figure REF , the raw distribution is drawn with a black histogram.",
"The red histogram represents the expected contamination level mentioned in the above section.",
"For each colour bin, the number of contaminating objects was removed from the raw distribution; the green histogram represents the contamination-corrected colour distribution.",
"The panels in Figure REF are placed in the same order as in Figure REF .",
"Figure: Colour distributions of GC candidates.",
"The black histograms arecolour distributions of the raw data.",
"The red histograms are contamination estimates.The green histograms are after correcting for contamination.",
"The KMMresults for sub-populations are drawn with dashed lines, and the solidlines are the sum of the two sub-populations.Table: The properties of globular cluster systems: colour distributionsIn order to statistically test the significance of any colour bimodality, the Kaye's mixture model (KMM) test [2] was employed.",
"The KMM test uses likelihood ratio test statistics to estimate the probability (P-value) that two distinct Gaussians with the same dispersion are a better fit to the observational data than a single Gaussian.",
"After running the KMM test for 100 bootstrap resamples of the contamination corrected colours, the peaks of the colour histograms, the $\\sigma $ of the sub-populations, and the red population fraction were estimated.",
"Also estimated were the P-value and the fraction of P-values that are less than 0.05 (Table REF (a)); the representative values are median values of the 100 KMM outputs and the uncertainties are half a width within which 68% of the data are contained relative to the median.",
"Since the KMM outputs (especially the P-values) have a somewhat skewed distribution, the median is a more robust estimation than the mean.",
"The statistical interpretation of the P-value is that a distribution with a P-value of 0.05 favors bimodality rather than unimodality with a 95% confidence.",
"In Figure REF , the results of the KMM tests for each subgroup are plotted with dashed lines, and the sum of the two groups is plotted with solid lines.",
"The significance of colour bimodality is decided on the basis of $\\langle P \\rangle $ and the fraction of $P<0.05$ .",
"If $\\langle P \\rangle < 0.05$ and $f_{(P<0.05)} > 0.90$ , a distribution is interpreted as a strong bimodal distribution and for $0.05 < \\langle P \\rangle < 0.10$ or $0.50 < f_{(P<0.05)} < 0.90$ , a distribution is likely bimodal; otherwise it is unimodal.",
"Despite the small P-value for IC 2035, this galaxy is regarded as unimodal because of the insignificant number of red clusters.",
"The significance of bimodality is listed as S(trong), L(ikely), or U(nimodal) in Table REF .",
"Figure: Luminosity functions of GCs in the gg band.",
"The black histograms are luminosity functions before correction.",
"The red histograms are contamination estimates, and the green ones are luminosity functions corrected for background contamination and completeness.",
"The solid curves are the best fit Gaussian luminosity functions, and the arrows indicate the expected turn-over position for each galaxy based on the GCLF parameters from .Figure: Luminosity functions of GCs in the zz band.",
"The histogram colours and fitted curves have the same meaning as for the gg band luminosity functions in Figure .The ability of the KMM test to detect bimodality depends on the sample number and the normalized separation between the two peaks (see [2]).",
"A larger sample size and a larger separation between two subcomponents will increase the chance of detecting bimodality.",
"Furthermore, the extended tails of the distribution can also affect the results.",
"Therefore, an alternative GC sample with a narrower colour range of $0.7<(g-z)_0<1.6$ was tested.",
"The corresponding results are listed in Table REF (b).",
"Clipping the tails of the distributions was found to decrease the P-value.",
"In other words, without the extended tails, the KMM test is more likely to detect bimodality.",
"This is not surprising because fitting extended tails results in a larger sigma for each subgroup, thus making it harder to separate two subgroups.",
"However, in this experiment, the values of most of the fitted parameters do not vary significantly and remain within the uncertainty that was initially estimated.",
"There is one case where a fitted parameter, the red peak colour, has been changed significantly after clipping the tails.",
"This scenario occurs in NGC 3156, where one isolated and very red GC (see middle-bottom panel in Figure REF ) makes the red peak significantly redder compared to the value when it is removed.",
"This is potentially a background galaxy contaminant because, unlike other red GCs, it lies at a large projected distance from the galaxy center.",
"For further analysis, the values from Table REF (a) were adopted in this study except for NGC 3156, where the value from Table REF (b) is used.",
"The KMM results are overplotted in Figure REF .",
"It should be noted that colour bimodality in GCs is more common in the more luminous galaxies.",
"The red population becomes weaker and moves to bluer colours with decreasing galaxy luminosity.",
"Eventually, only a blue peak appears in the faintest galaxies.",
"The richness of the GCs also decreases with the host galaxy luminosity.",
"None of our GC colour distributions have a single broad peak located between the normal blue and red peaks.",
"This is in contrast to the results of some studies on early-type galaxies in other environments (e.g.",
"[42]; [39]; [48]).",
"The behavior of the GC colour with host galaxy luminosity will be discussed in more detail below."
],
[
"Luminosity functions",
"Luminosity functions in the $g$ and $z$ bands were constructed using GCs in the colour range of $0.6 <(g-z)_0 < 1.7$ (black histograms in Figure REF and Figure REF ).",
"The contamination (red histograms) estimated previously was subtracted from the raw distribution and incompleteness corrections were applied.",
"The corrected luminosity functions (green histograms) were fitted with a Gaussian function up to the 50% completeness limit.",
"The fitted parameters (peak magnitude and standard deviation) and their uncertainties are listed in Table REF ; the missing data correspond to parameters with a large uncertainty caused by the small number of GC samples and an excess of faint GCs.",
"For the luminous galaxies, Gaussian functions aptly represent the luminosity functions of the GC systems.",
"The arrows in Figures REF and REF indicate the expected turn-over magnitudes based on a peak absolute magnitude of the GC luminosity function (GCLF) of $\\mu _g=-7.2$ and $\\mu _z=-8.4$ as given by [35].",
"As can be seen in Figure REF , there is an excess of faint GCs around $g\\sim 25.5$ mag in the GCLF of NGC 3377.",
"These faint GCs deviate significantly from the fitted Gaussian function.",
"This deviation is also found at $z\\sim 24.5$ mag in the $z$ -band GCLF shown in Figure REF .",
"The properties of these objects are discussed in more detail below.",
"In Figure REF , the spatial distribution, size, and colour of NGC 3377 GCs fainter than $g=25$ mag are plotted along with those brighter than $g=25$ mag.",
"In the top-left panel, the faint GCs (large closed circles) appear to be uniformly distributed, unlike the bright GCs (small open circles) which are concentrated at the galaxy center.",
"Shown in the top-right panel of Figure 8 are histograms of the FWHM returned by Sextractor for the faint and bright GCs.",
"The size distribution of the faint GCs is also quite different from that of the bright GCs in the sense that the sizes are either smaller or larger than the medium size for all clusters.",
"The colour distribution of the faint GCs appears to be rather flat, as shown in the bottom-left panel.",
"Any dependence of the colour on the size of the faint GCs was examined by separating the GCs at $FWHM=2.8$ pixels.",
"In the bottom-right panel of Figure 8, the thin histogram represents the colour distribution of faint GCs with $FWHM >2.8$ , while the thick histogram represents GCs with $FWHM<2.8$ .",
"No obvious correlation between size and colour was found in the plot.",
"We can speculate on the possible origin of the faint objects that deviate from the normal Gaussian GCLF.",
"[49] found a class of diffuse star clusters (DSCs) in the ACSVCS and investigated the nature of these objects, which are characterized by their low-luminosity, broad distribution of sizes, low surface brightness, and redder colour when compared to normal metal-rich GCs.",
"The spatial distribution of the objects was also closely associated with the host galaxy light.",
"However, not all of the aforementioned characteristics are found in our faint objects, which possess a wide distribution of sizes, but a significant number of blue GCs and a random spatial distribution.",
"Another possibility is that the faint objects may be field stars (bright giants) that belong to the parent galaxy.",
"By comparing isochrones of various ages and metallicities ([25]), the locus of giant stars ($M_V$ $<$ $-5$ ) of a very young age ($\\sim $ 10 Myr) was found to overlap with that of the faint objects on the colour-magnitude diagram (Figure REF ).",
"However, it is quite unlikely that such young stars exist in significant numbers in old elliptical galaxies like NGC 3377.",
"Furthermore, horizontal branch stars are not sufficiently luminous to account for the faint objects.",
"[30] found no evidence of young stellar populations ($<3$ Gyr) in the halo star CM diagram of NGC 3377 from deep HST/ACS photometry, although their field of view barely overlaps that used in this study.",
"The number density of foreground Milky Way stars in the direction of NGC 3377 is also very low ($N_{star}<10$ ) within the ACS field ([51]) and even lower in the magnitude range of the faint GCs.",
"Visual inspection of the faint objects reveals that some are more likely to be misclassified background galaxies.",
"Assuming the faint objects are not GCs belonging to NGC3377 shifts the mean colour of the GC system blueward by ($\\Delta (g-z)_{0}\\sim 0.015$ ) compared to the value including all the objects.",
"Note that the mean colours in Table REF have been estimated after a faint magnitude cut ($g=25.5$ mag) so that any colour change by faint objects is negligible."
],
[
"Host galaxy properties vs. globular cluster colours",
"It is well known that there is a strong correlation between mean colours/red peaks and host galaxy luminosities (e.g.",
"[42]; [39]).",
"However, few studies had been performed to support a correlation between the blue peak and the host galaxy luminosity until the recent ACSVCS data found such a relationship with their large dynamic range of galaxy luminosity [48].",
"In Figure REF , the peak colours of the sub-populations and the mean GC colours are plotted against the host galaxy luminosity for our sample.",
"Each relationship is fitted to a straight line using weighted chi-squre minimization.",
"It is clear that strong dependencies of the red peak and the mean GC colour on the host galaxy luminosity do exist in our field galaxies with almost identical slopes.",
"On the other hand, the blue peak is almost independent of the galaxy luminosity.",
"In fact, the fitted slope exhibits a weak anti-correlation, the statistical significance of which is within 1$\\sigma $ of a zero slope.",
"From the attained results alone, it is uncertain whether the absence of a blue peak gradient is due to the narrow dynamic range of the luminosity in our field galaxy sample or an intrinsic effect across a wide range of luminosity.",
"The obtained results clearly show that the observed trends in the GC colours with host galaxy luminosity are a consequence of the gradual change in the fraction and colour of the red GCs and not a systematic change in the colours of all GCs within a galaxy."
],
[
"Comparison with the ACS Virgo Cluster Survey",
"The ACS Virgo Cluster Survey (ACSVCS) was a large program used to image 100 early-type galaxies in the Virgo cluster with the ACS in two filters ($g$ and $z$ bands) [18].",
"The main scientific goals were to study the properties of GC systems in these galaxies (e.g., PJC06, [33]), analyze their central structures [20], and obtain accurate surface brightness fluctuation distances ([45]; see also [10]).",
"The observation conditions and data reduction procedures of the ACSVCS are somewhat different from those adopted in this study.",
"The ACSVCS used 100 orbits of the HST, allocating only one orbit for each galaxy.",
"Each galaxy of the ACSVCS had an exposure time of 750 s in F475W and 1210 s in the F850LP filter, while our exposure times are almost twice as long (typically 1300 s in F457W and 3000 s in F850LP).",
"The data reduction procedures of the ACSVCS are described in detail by [31].",
"In brief, the exposures were split into two or three sub-exposures, without repositioning, so as to remove cosmic rays.",
"These sub-exposures were then reduced and combined using the standard ACS pipeline.",
"Model galaxies were created using the ELLIPROF program [59] and subtracted from the original images.",
"The KINGPHOT program developed by Jordan et al.",
"(2005) was used for photometry of the GCs and to measure their sizes.",
"KINGPHOT fits King models convolved with a given point spread function in each filter to individual GC candidates and finds the best fit parameters of the King model.",
"Total magnitudes for the GCs were calculated by integrating these best-fit convolved King models.",
"All GCs were selected based on their magnitude ($g\\ge 19.1$ or $z\\ge 18.0$ ) and mean elongation ($\\langle e \\rangle \\le 2$ , where $\\langle e \\rangle \\equiv a/b$ ) in the two filters.",
"The main results regarding the colour distributions of the ACSVCS GC systems were published by PJC06 and are compared with the results of this study below.",
"Before directly comparing results from the ACSVCS with ours, it is necessary to check consistency of photometry, because the applied photometry methods are different; aperture photometry has been used in this work, while the ACSVCS used fitted King models.",
"We applied our photometry procedure to 24 galaxies of the ACSVCS within our galaxy luminosity range and directly compared magnitudes and colours of GCs we measured with those in the GC catalogue from [36].",
"Their catalogue lists two kinds of photometry values, PSF convolved King model and aperture photometry.",
"Since PJC06 used the former for their analysis, it was taken here for comparison.",
"Figure REF shows differences of $g$ and $z$ magnitudes ($\\Delta g=g_{ap}-g_{King}$ ,$\\Delta z=z_{ap}-z_{King}$ ) and colour ($\\Delta (g-z)_{0}=(g-z)_{0,ap}-(g-z)_{0,King}$ ) for all the GCs matched in 24 ACSVCS galaxies.",
"It is found that mean magnitude offsets in $g$ and $z$ band are $\\langle \\Delta g\\rangle \\sim 0.025$ and $\\langle \\Delta z\\rangle \\sim 0.024$ respectively, revealing that our photometry appears to slightly overestimate total flux of GCs.",
"However, no offsets in mean colour between the two methods are seen, $\\langle \\Delta (g-z)_{0}\\rangle <0.01$ .",
"This test verifies that there is no systematic colour offset between the two methods so that direct comparison of GC colours between PCJ06's results and our results can be justified.",
"In Figure REF , the mean colours for the 10 GC systems in our sample are compared with the results from PJC06.",
"At a given host galaxy luminosity, our mean colours are generally located at the bluer end of the mean colour distribution of the ACSVCS galaxies.",
"However, unlike the other GC systems in our sample, the mean colour of the NGC 3377 GC system lies in the middle of the range of ACSVCS mean colours.",
"Because the distribution of red GCs is more centrally concentrated and NGC 3377 is the nearest galaxy in our sample, it is possible that the GCs detected within our ACS field are biased toward red GCs.",
"To test whether this selection bias is large enough to shift the mean colour, the GCs in other galaxies were resampled with a smaller field of view and their mean colours were recalculated.",
"For this test NGC 4033 and NGC 1172, which have a similar absolute magnitude but are as twice as distant as NGC 3377, were chosen.",
"The mean colour of the GCs in these two galaxies within a 50$$ radius from their galaxy centers is $\\sim 0.03$ mag redder than our original estimations for the full spatial coverage.",
"Noting that the full field of view of the ACS is $202\\times 202$ so that a circle with a 50$$ radius covers almost half of the original aperture, the intrinsic mean colour of the GCs in NGC 3377 could be about $\\sim 0.03$ mag bluer than our original estimation.",
"However, this is a relatively small correction when compared to the wide range of mean colours observed in the ACSVCS.",
"Note that the error bar of NGC 3377 plotted in Figure REF is also $\\pm 0.02$ mag.",
"PJC06 analyzed their colour distributions with two different methods.",
"One method separates the two sub-populations in each host galaxy and tests for bimodality using the KMM routine.",
"The other method coadds the colour distributions of GCs in bins of the host galaxy luminosity and separates the two sub-populations by nonparametric decomposition.",
"The results from the first method of PJC06 were compared with our results because our colour distributions were also analyzed with the KMM method and the number of our sample galaxies is not large enough or wide enough to bin by galaxy luminosity.",
"A comparison of our results and those of PJC06 with respect to the colour peaks of the two sub-populations is shown in Figure REF .",
"A comparison of the fractions of red population GCs detected was also plotted; the results are shown in Figure REF .",
"In Figure REF , only galaxies with a bimodal colour distribution are plotted.",
"Figure: Comparison of the global properties of GC systems in our sample of low-luminosity field E/S0s with those of the ACSVCS.",
"The first, second, and third columns show plots versus the absolute magnitude, the local density of galaxies from Tully (1988), and the morphological type of the host galaxy, respectively.",
"The rows show the colour of the blue and red peaks, mean colour, and the fraction of red GCs.",
"The coloured symbols represent our data, while the grey symbols are from the ACSVCS.",
"For the second and third columns, only galaxies from the ACSVCS within the luminosity range of our sample are plotted.",
"For the first and third rows, closed circles show the GCs in which colour bimodality is strongly detected using the KMM test, while open symbols show GCs with weak colour bimodality.",
"In the top-left panel, black lines are the best linear fit to our data and the grey lines are the best-fit to the ACSVCS data.Overall trends for the colour distributions with respect to the host galaxy luminosity in both samples appear to be similar.",
"As evident in Figures REF and REF , the probability of bimodality decreases with decreasing host galaxy luminosity as the red population becomes weaker.",
"There is a strong correlation between the peak of the red sub-population and the host galaxy luminosity.",
"Moreover, the fraction of red GCs with respect to the entire population appears to increase with host galaxy luminosity.",
"Some differences do exist between the results of this study and those of PJC06.",
"In Figure REF , our slope for the red peak against galaxy absolute magnitude appears to be steeper than that of the ACSVCS.",
"In other words, the bimodal colour distributions in our sample disappear more quickly as galaxies become fainter.",
"Furthermore, we found no correlation between the blue peaks and the galaxy absolute magnitude, while PJC06 found a weak correlation (although it was not as steep as for the red peaks).",
"Our fraction of red GCs was also found to be slightly lower than that of the ACSVCS at a given galaxy luminosity.",
"The mean colours of the GCs in our sample are slightly bluer than those of the ACSVCS and the red and blue peak positions in our sample are almost identical to those in the ACSVCS.",
"This suggests that the difference in the mean colours is due to differences in the relative fraction of the red population and not an overall shifting of colours.",
"In the middle bottom panel of Figure REF it appears as if the ACSVCS galaxies have more red objects.",
"To test whether this is real or not, a K-S test was performed, but its results give a low significance level to any difference.",
"Figure: Gaussian fit parameters of luminosity functions against host galaxy luminosity.",
"Turn-over magnitudes (left) and dispersions (right) in the gg and zz bands against host galaxy luminosity.",
"The open circles represent the Gaussian fits to galaxy-binned GCLFs from the ACSVCS by , whereas the results from this work are plotted with filled circles.One might argue that because the majority of our sample galaxies are more distant than the Virgo cluster galaxies, and red GCs are more concentrated on the galaxy center, our detected GC samples would be less biased toward red GCs.",
"Consequently, our red GC fraction could be systematically lower than that in the ACSVCS.",
"Indeed NGC 3377, the nearest galaxy in our sample and closer than the Virgo cluster, has the highest red GC fraction.",
"To test this hypothesis, we resampled GCs in NGC 3818, which is as twice distant as Virgo, within a circle with 50radius that covers half of the ACS field of view.",
"Therefore, the actual physical overage of this circle is similar to that of the ACSVCS.",
"The new estimation of the red GC fraction within this circle is slightly higher by $\\sim $ 0.02 mag than our initial estimate.",
"In the case of NGC 7173, the original image contains 3 galaxies (NGC 7173, NGC 7174 and NGC 7176) and the actual coverage of NGC 7173 used for GC detection is only half of the full image.",
"Considering NGC 7173 is as twice distant as Virgo, both physical coverages are similar, but the red fraction of GCs in NGC 7173 is still lower than that of the ACSVCS at the same galaxy luminosity.",
"We therefore conclude that, although the ACSVCS samples are possibly biased to red GCs due to smaller physical coverage, the derived red GC fraction does not shift significantly and we still find a small systematic difference with the ACSVCS which may be related to the effects of local environment.",
"The shapes of the colour distributions in the ACSVCS are more varied than those in our sample.",
"For instance, PJC06 found one single broad peak for the colour distributions of VCC 1664 and VCC 1619.",
"They also found examples in which the red population completely dominates; red GC fractions of 0.84 in VCC1146 and NGC 4458 were measured.",
"These types of “abnormal” colour distributions are not found in our smaller field sample.",
"To summarize the comparison of the properties of the GC systems in our sample with those of the ACSVCS, plots of the GC properties against host galaxy absolute magnitude (identical to Figures REF , REF , and REF ) are shown in the first column of Figure REF .",
"The second and third columns in this figure are comparisons of the GC properties against the local galaxy number density (environment) and morphological type, respectively.",
"In these two columns, only results from the ACSVCS within the range of the galaxy luminosity of our sample ($-18.4\\ge M_{B}^{T}\\ge -20.4$ ) are plotted so as to minimize any galaxy luminosity dependence.",
"In the third column, the GC systems with the same host galaxy morphological type are randomly positioned within the morphological type bins.",
"As can be seen in the second column, our sample is clearly distinct from the ACSVCS in terms of the galaxy environment.",
"Because the GC system properties of both the ACSVCS and our sample are widely spread however, distinct differences with respect to the galaxy number density are not obvious in these plots.",
"In the plots of the GC properties against morphological type (third column of Figure REF ) no obvious trends are observed with morphological type for the ACSVCS.",
"However, for our sample, the GC systems in SB0/S0 type galaxies appear to have little/no bimodal colour distributions.",
"In other words, the SB0/S0 galaxies have smaller red sub-populations when compared to early-type galaxies.",
"Our GCLFs have also been compared with those of the ACSVCS and the results are shown in Figure REF .",
"[35] fitted the GCLFs of 89 galaxies with both a Gaussian function and a Schechter function.",
"They found that the GCLF dispersions were correlated with the galaxy luminosity, while the GCLF turn-over magnitudes were rather constant (in both bands).",
"Later [62] reanalysed [35]'s results for the Fornax cluster, finding that the turn-over magnitude of GCLF gets fainter with decreasing galaxy luminosity.",
"Using distance moduli from Table REF and apparent turn-over magnitudes from Table REF , we directly compared our results from §REF with the Gaussian fitting parameters of the galaxy-binned GCLFs reported by [35].",
"In Figure REF , our results are overplotted on the fitting parameters from [35] and appear to be more widely spread than those from the ACSVCS.",
"However, one must keep in mind that the ACSVCS results come from galaxy-binned GCLFs, which are less noisy than our individual GCLF fitting parameters.",
"Using only our GCLF dispersions, it is hard to say whether or not there is a trend with galaxy luminosity because of the narrower range of galaxy luminosity in our sample.",
"Our GCLF dispersions agree well with the ACSVCS, at least within the range of galaxy luminosities spanned by our sample."
],
[
"Effect of environment on galaxy formation",
"From the analysis presented above, it appears that the mean colours of GC systems in field environments are slightly bluer than those in cluster environments at a given host galaxy luminosity.",
"The simplest interpretation of this finding would be that these GC systems are either less metal-rich or younger than their counterparts in rich clusters (or both metal-poor and young).",
"Simple stellar population models from Giraradi (2006, hereafter GIR06) and the Yonsei Evolutionary Population Synthesis (YEPS, http://web.yonsei.ac.kr/cosmic/data/YEPS.htm; C. Chung et al., in prep.)",
"model were adopted in order to estimate the magnitude of these effects and compare them with our mean colours.",
"The prediction of $(g-z)_{0}$ colour in various age and metallicity ranges based on the GIR06 and YEPS models is shown in Figure REF .",
"The mean metallicity of the blue and red peaks shows good agreement with predictions based on the empirical transformation ($g-z$ ) to [Fe/H] ([48]; [53]).",
"Figure: Evolution of g-zg-z colour for various metallicities and ages.",
"The grey dashed tracks are taken from simple stellar population models by , while the black ones are from YEPS.",
"The metallicity noted at the end of each track is in units of [Fe/H].",
"The two stripes represent the colour ranges of red and blue peaks of GC systems in our sample galaxies.",
"For old stellar populations (>> 8 Gyr), (g-z) 0 (g-z)_{0} colour is mainly governed by metallicity rather than age.Figure: Metallcity and age differences of GC systems between high density (cluster) and low density (field) environments.",
"The black solid, dotted, dashed and dot-dashed lines represent 10, 11, 12 and 13Gyr of cluster GCs from the YEPS models respectively.",
"The grey solid, dotted, dashed and dot-dashed lines are 10, 11.2, 12.6 and 14.1 Gyr models from GIR06 respectively.",
"These model lines were derived from our results that the mean colours of GC systems in cluster galaxies are 0.05 mag redder than in field galaxies.With only one colour ($g-z$ ), it is almost impossible to break the well-known age-metallicity degeneracy.",
"Thus, an attempt was made to find any combination of metallicity and age differences that reproduce the mean colour offset between the high- and low-density GC systems.",
"We first assumed an age range from 10 Gyr to 14 Gyr for the cluster environment GC systems and a minimum age of 8 Gyr for the field environment GC systems.",
"We then took the mean colour of the cluster and field GC systems from Figure REF and derived the metallicity difference for a certain age combination from the YEPS and GIR06 models.",
"The metallicity and age differences for cluster GC systems of various ages is shown in Figure REF .",
"As an example, if the mean age of the cluster GC systems is 10 Gyr (the black solid line in Figure REF ) and that of the field GC systems is 8 Gyr, then there seems to be no metallicity difference between the two GC systems according to the YEPS model.",
"However, if the cluster and field GC systems are coeval, the cluster GC systems are, on average, more metal-rich by $\\sim $ $0.10-0.15$ dex than the field GC systems.",
"The YEPS model predicts a slightly lower metallicity difference than the GIR06 model when the two samples are coeval.",
"This metallicity offset between the two samples disappears if the GC systems in the field galaxies are younger than those in the clusters.",
"The age differences with no metallicity offset depend on the assumed ages of the GC systems in high-density environments and on the adopted simple stellar population model.",
"However, the colour offset can be successfully explained by at least a $\\sim $ 2 Gyr age offset when there is no metallicity difference.",
"Therefore, we tentatively conclude that GC systems in low-density environments are either less metal-rich, $\\Delta [Fe/H]\\sim 0.10-0.15$ , or younger, $\\Delta age>2$ Gyr, than those in high-density regions.",
"Of course, any combination of age and metallicity within the above ranges can be a possible solution, as seen in Figure REF .",
"Metallicity difference between the low- and high-density GC systems has been also derived from a fully empirical colour-metallicity relation by [11], in which a quartic polynomial is fit to compiled data from PJC06.",
"Based on the relation, the colour offset is found to correspond to $\\Delta [Fe/H]\\sim 0.12$ .",
"Although the empirical relation might contains age effects, the metallicity difference is well consistent with that from the SSP models.",
"Even though we managed to detect the colour offset of GC systems in low- and high-density environments, and quantify the age and metallicity difference using the simple stellar population models and the empirical colour-metallicity relation, the detected colour offset ($\\Delta (g-z)_{0}\\sim 0.05$ ) is not comparable with the colour range due to host galaxy luminosity.",
"For example, within our sample, the largest difference in mean colour is $\\Delta (g-z)_{0}\\sim 0.13$ (see Table REF ) and for the entire ACSVCS sample, the mean GC system colour has a range of $0.76\\le (g-z)_{0}\\le 1.25$ ([48]).",
"Therefore, stellar populations in GC systems are mainly governed by their host galaxy mass (luminosity) and environmental effects are less important in determining the star formation history of early-type galaxies and their GC systems.",
"There have been a few previous attempts to detect differences in GC properties depending on the environments to which their host galaxies belong.",
"[24] used 50 mostly early-type galaxies from the HST/WFPC2 archive and found no correlation between the colour peaks and the host galaxy properties in 15 field sample galaxies.",
"However, these researchers found a strong correlation of GC system peak colours versus galaxy velocity dispersion and Mg2 index in their cluster galaxy sample.",
"In our field galaxy results, a strong relationship between the mean colour and the galaxy luminosity was certainly observed (see Figure REF ).",
"One possible reason why [24] did not find such a relationship in their field galaxy sample is that their galaxies were too luminous ($M_{V}-20$ ) and only small differences are to be expected for luminous ($M_{B}-21$ ) early-type galaxies (e.g., [6]; [16]).",
"Another possible reason is that the photometric accuracy of the HST/WFPC2 data was not sufficient to detect the correlation.",
"In fact, the present work and the research of [24] have two field galaxies in common: NGC 1426 and NGC 3377.",
"When compared to [24], we detected twice as many GCs in these two galaxies.",
"[42] also briefly investigated the dependency of colour bimodality on the environment in an independent sample of 17 early-type galaxies obtained from HST/WFPC2, but did not find any correlation.",
"As can be seen from our results, the colour bimodality of GC systems mainly depends on the galaxy luminosity (mass), although this trend disappears at slightly higher galaxy luminosities in low-density regions.",
"[50] found a large spread in the specific frequency ($S_{N}$ ) of dwarf galaxies from the ACSVCS.",
"Almost all dwarfs with a high $S_{N}$ were located within a projected radius of 1 Mpc from M87 (Figure 4 in [50]), whereas no $S_{N}$ difference between the central region and the outskirts of the Virgo cluster was found for intermediate luminosity galaxies (i.e.",
"similar to those in our low-density sample).",
"While [50] found a dependency of $S_{N}$ for the dwarfs on the environment, the cumulative GC colour distributions of these two groups show no obvious difference.",
"It is possible that the effect of environment on the colour distribution is so subtle that it could not be detected within a cluster environment such as Virgo.",
"Regarding the general dependence of galaxy formation on environment, many observational and theoretical studies have been conducted without including constraints from GC system data.",
"It is now well-established that late-type galaxies are biased to low-density environments and giant elliptical galaxies are preferentially located in galaxy clusters ([19]).",
"Cluster galaxies also exhibit lower star formation rates than field galaxies at a given redshift, luminosity, and bulge-to-disk mass ratio ([4]; [43]; [26]).",
"Early-type galaxies in low-density environments also appear to be younger ($\\Delta age\\sim $ 2 Gyr) and more metal-rich ($\\Delta [Fe/H]\\sim $ 0.1 dex) when compared to their counterparts in dense environments ([41]; [58]), although [15] found no environmental influence on the metallicity (but still found younger ages for field galaxies).",
"These observational findings are contradictory to the semi-analytic model of hierarchical galaxy formation (e.g.",
"[6], [16]), which predicts a larger spread of age and metallicity and younger ages and lower metallicities (on average) in low-density environments, especially in low-luminosity early-type galaxies ([6], [16]).",
"Our results for the GC systems are somewhat inconsistent with those of [41] and [58], who found higher metallicities for galaxies in field environments (which is consistent with the semi-analytic models from [6] and [16]).",
"The question remains as to whether GC formation history is different from field star formation history or does the environment have different effects on GCs and field star formation.",
"[29] have suggested that GCs formed somewhat earlier than most field stars based on their discovery of a higher $S_{N}$ at a low metallicity in NGC 5128.",
"This argument is also supported by [50], who estimated GC formation rates from the Millennium Simulation ([55]) and showed that cluster formation peaks earlier than field star formation.",
"Such results would explain why GCs are more metal-poor than field stars in early-type dwarf galaxies.",
"The higher $S_{N}$ of dwarf ellipticals in the Virgo cluster center was successfully explained by the higher peak star formation rate (SFR) and star formation surface density ($\\Sigma _{SFR}$ ) in the cluster central region when compared to the outskirts and a model where GC formation rate $CFR\\propto SFR(\\Sigma _{SFR})^{0.8}$ .",
"As described in §REF , the colour of GC systems is mainly determined by the mass of the host galaxy in both cluster and field environments.",
"Subtle environmental differences also exist; the mean metallicities of GC systems in field galaxies are slightly lower than those in cluster galaxies at a given host galaxy luminosity (mass), while the fraction of red clusters is higher in dense environments.",
"From our findings, it is suggested that GC systems in field galaxies form later than those in cluster galaxies and/or have a lower metallicity.",
"Furthermore, we expect that in dense environments, cluster formation history is complicated by disturbing or interacting neighbor galaxies.",
"This is reflected by the larger degree of variation in the shapes of the colour distributions that are found in dense environments.",
"While [50] found no difference in the $S_{N}$ of low-luminosity ellipticals (intermediate luminosity according to their terminology) in their Virgo cluster sample, in the future it would be worth comparing the $S_{N}$ of our sample with that of the ACSVCS."
],
[
"Metal-poor GC colour–galaxy luminosity relation",
"The existence (or non-existence) of a correlation between the peak of the blue GC colours and the host galaxy luminosity can have implications for different galaxy formation scenarios.",
"Major merger [1] and accretion models [17] would have difficulty explaining this correlation because, in the merger model, metal-poor GCs in two lower equal mass spiral galaxies are still a primary resource of metal-poor GCs in the final ellipticals.",
"In the accretion model, metal-poor GCs come from dwarf galaxies, so the mean metallicities of GCs in dwarf galaxies and metal-poor GCs in ellipticals are more or less the same.",
"Many early studies found a strong correlation between the colours of metal-rich GCs and the host galaxy luminosity, but failed to detect any evidence of a similar correlation with the mean colours of metal-poor GCs (e.g., [22]; [39]; [23]).",
"However, [42] found a weak correlation between the mean colours of metal-poor GCs and the galaxy absolute magnitude in their HST/WFPC2 sample.",
"[13] also suggested that such a relationship may exist.",
"[44] found that the slope of the GC peak of their dwarf elliptical sample ($M_{B}\\ge -18$ ), when plotted against galaxy luminosity, is consistent with that of the GC blue peak in early-type galaxies found by [42].",
"Recent studies of the correlation between metal-poor GCs and host galaxy luminosity have been carried out by [57] and [48].",
"The former compiled blue GC peak data for early-type galaxies and local spirals from various sources ([42]; [39], [40]; [28]; [5]; [47]) and found a significant ($>5 \\sigma $ ) correlation between the blue colour peak and galaxy luminosity.",
"[48] also detected this blue peak trend in homogenous ACSVCS data, but it was not as strong as that for the red peak.",
"Using their own relationship between $g-z$ colour and [Fe/H], [48] found that the slope of the blue peak metallicity-galaxy luminosity relationship is even steeper than that found in the colour-galaxy luminosity plane due to the steeper colour-metallicity relationship in the metal-poor region (see Figure 12 in [48]).",
"However, they noted that this blue slope varies depending on the adopted colour-metallicity relationship (simple stellar population model or empirical transformation) by a factor of $\\sim 3$ .",
"In Figure 2 of [12], the peak metallicities of GC sub-populations are plotted against galaxy luminosity with the peak positions taken from [56] and [57].",
"In this plot, the peak metallicities of both metal-poor and metal-rich GCs appear to be correlated with host galaxy luminosity with similar slopes.",
"The metal-poor GC trend can be accounted for with an in-situ scenario [22] in which metal-poor GCs formed in the Universe first, and after a sudden truncation (possibly by reionization) metal-rich GCs then formed along with the bulk of the field stars in galaxies.",
"From our results on the blue peak colours alone, it is not clear whether or not the blue peak colours are correlated with the host galaxy luminosity because of the narrow galaxy luminosity range in our sample (only $\\sim 2$ mag range in $M_{B}$ compared to a range of $\\sim 6$ mag in the ACSVCS).",
"To observe any trend in the blue peak colour with galaxy luminosity, we would need to study GC systems in fainter galaxies ($M_{B}\\ge -18$ , i.e., dwarf ellipticals) in low-density environments (similar to those studied by [44]).",
"This would serve to strengthen the correlation between the GC blue peak and galaxy luminosity in the higher density Virgo and Fornax clusters.",
"It is thus unclear whether the slope of the blue (metal-poor) GC peak in field environments is different from that in clusters.",
"In fact, [56] predict that the slope of metal-poor GC metallicity-galaxy luminosity plots in dense environments should be steeper than that in low-density environments if cosmic reionization caused a quenching of blue GC formation.",
"The role of cosmic reionization in GC formation is, however, uncertain.",
"Whether cosmic reionization can really truncate GC formation and how efficient GC formation is prior to the reionization is still controversial (e.g.",
"[46]; [27])."
],
[
"Conclusions",
" High spatial resolution images of 10 early-type galaxies in low-density (field) environments have been obtained using the HST/ACS in the F457W and F850LP bands.",
"The properties of the GC systems associated with these galaxies and their connection with host galaxy properties have been investigated.",
"Our results have been compared with those of the ACS Virgo Cluster Survey (ACSVCS) in order to study the role of the environment in galaxy formation.",
"The GC system properties of our low-density sample exhibit trends with respect to the host galaxy luminosity that are similar to those in clustered environments.",
"There are more total GCs and more red GCs in luminous galaxies, while the mean colour and the colour of the red peak GCs are strongly correlated with host galaxy luminosity.",
"Colour bimodality becomes less clear with a decrease in galaxy luminosity.",
"The mean colours in low-density (field) regions appear to be slightly bluer than those around galaxies of equivalent luminosities in high-density regions.",
"The fraction of red GCs in field galaxies is found to be lower than that in clustered galaxies of similar luminosities.",
"When compared to the trend observed in the ACSVCS, the slope of the red peaks against host galaxy luminosity is steeper.",
"In other words, colour bimodality disappears more quickly as galaxies get fainter, in low density environments.",
"Luminosity functions of the GCs in most of our sample galaxies are well-fit by a Gaussian function whose fitting parameters (turn-over magnitude and dispersion) agree well with those of the ACSVCS.",
"We find no independent evidence for a trend of dispersion in the fitted Gaussian luminosity functions against galaxy luminosity.",
"We have investigated the possible origins of the unexpected excess of GCs in NGC 3377 near $g$ $\\sim 25.5$ , which deviate from a normal Gaussian luminosity function.",
"The origin of this population is still not fully understood, but some of clusters are likely to be background galaxies from visual inspection and some are possibly diffuse star clusters.",
"At a given luminosity, the mean colours of our GC systems in low-density environments are slightly bluer ($\\Delta (g-z)_{0}\\sim 0.05$ ) than their counterparts in high-density environments.",
"By assuming various combinations of age and employing simple stellar population models, this colour offset corresponds to a metallicity difference of $\\Delta [Fe/H]\\sim 0.10-0.15$ or an age difference of at least $\\Delta age\\sim 2$ Gyr on average.",
"Therefore, GCs in field galaxies appear to be either less metal-rich or younger than those in cluster galaxies of the same luminosity.",
"Whilst a correlation between the blue peak colour of GCs and host galaxy luminosity was found in the Virgo cluster (ACSVCS), no such correlation was detected in our low-density galaxy sample.",
"It is not clear whether the absence of such a trend is because our galaxy luminosity range is simply too narrow or because intrinsically the slope of the relationship depends on the environment.",
"For a more definitive determination, observations of GCs in dwarf ellipticals in low-density environments are needed.",
"The greater variation in the shapes of the colour distributions for GC systems in the Virgo cluster sample could imply that more complex galaxy formation processes (e.g., interactions/harassment with adjacent galaxies) are taking place in the galaxy clusters.",
"The higher fraction of red GCs in cluster galaxies also supports this possibility.",
"Although we found that the galaxy environment has a subtle effect on the formation and metal enrichment of GC systems, host galaxy mass is the primary factor that determines the stellar populations of both the GCs and the galaxy itself.",
"The processes which determine GC formation must therefore be largely common across a range of galaxy environments.",
"This remains a challenge for GC formation theories."
],
[
"Acknowledgments",
"JC acknowledges support by the Yonsei University Research Fund of 2009.",
"SJY acknowledges support from Basic Science Research Program (No.",
"2009-0086824) through the National Research Foundation (NRF) of Korea grant funded by the Ministry of Education, Science and Technology (MEST), and support by the NRF of Korea to the Center for Galaxy Evolution Research and the Korea Astronomy and Space Science Institute Research Fund 2011.",
"SEZ acknowledges support for this work from HST grant HST-GO-10554.01.",
"A. Kundu acknowledges support from HST archival program HST-AR-11264."
]
] | 1204.1054 |
[
[
"Efficient computational noise in GLSL"
],
[
"Abstract We present GLSL implementations of Perlin noise and Perlin simplex noise that run fast enough for practical consideration on current generation GPU hardware.",
"The key benefits are that the functions are purely computational, i.e.",
"they use neither textures nor lookup tables, and that they are implemented in GLSL version 1.20, which means they are compatible with all current GLSL-capable platforms, including OpenGL ES 2.0 and WebGL 1.0.",
"Their performance is on par with previously presented GPU implementations of noise, they are very convenient to use, and they scale well with increasing parallelism in present and upcoming GPU architectures."
],
[
"Introduction and background",
"Perlin noise [1], [3] is one of the most useful building blocks of procedural shading in software.",
"The natural world is largely built on or from stochastic processes, and manipulation of noise allows a variety of natural materials and environments to be procedurally created with high flexibility, at minimal labor and at very modest computational costs.",
"The introduction of Perlin Noise revolutionized the offline rendering of artificially-created worlds.",
"Hardware shading has not yet adopted procedural methods to any significant extent, because of limited GPU performance and strong real time constraints.",
"However, with the recent rapid increase in GPU parallelism and performance, texture memory bandwidth is often a bottleneck, and procedural patterns are becoming an attractive alternative and a complement to traditional image-based textures.",
"Simplex noise [2] is a variation on classic Perlin noise, with the same general look but with a different computational structure.",
"The benefits include a lower computational cost for high dimensional noise fields, a simple analytic derivative, and an absence of axis-aligned artifacts.",
"Simplex noise is a gradient lattice noise just like classic Perlin noise and uses the same fundamental building blocks.",
"Some examples of noise on a sphere are shown in Figure REF .",
"This presentation assumes the reader is familiar with classic Perlin noise and Perlin simplex noise.",
"A summary of both is presented in [6].",
"We will focus on how our approach differs from software implementations and from the previous GLSL implementations in [4], [5]."
],
[
"Platform constraints",
"GLSL 1.20 implementations usually do not allow dynamic access of arrays in fragment shaders, lack support for 3D textures and integer texture lookups, have no integer logic operations, and don't optimize conditional code well.",
"Previous noise implementations rely on many of these features, which limits their use on these platforms.",
"Integer table lookups implemented by awkward floating point texture lookups produces unnecessarily slow and complex code and consumes texture resources.",
"Supporting code outside of the fragment shader is needed to generate these tables or textures, preventing a concise, encapsulated, reusable GLSL implementation independent of the application environment.",
"Our solutions to these problems are: Replace permutation tables with computed permutation polynomials.",
"Use computed points on a cross polytope surface to select gradients.",
"Replace conditionals for simplex selection with rank ordering.",
"These concepts are explained below.",
"The resulting noise functions are completely self contained, with no references to external data and requiring only a few registers of temporary storage."
],
[
"Permutation polynomials",
"Previously published noise implementations have used permutation tables or bit-twiddling hashes to generate pseudo-random gradient indices.",
"Both of these approaches are unsuitable for our purposes, but there is another way.",
"A permutation polynomial is a function that uniquely permutes a sequence of integers under modulo arithmetic, in the same sense that a permutation lookup table is a function that uniquely permutes a sequence of indices.",
"A more thorough explanation of permutation polynomials can be found in the online supplementary material to this article.",
"Here, we will only point out that useful permutations can be constructed using polynomials of the simple form $(A x^2 + B x) \\bmod {M}$ .",
"For example, The integers modulo-9 admit the permutation polynomial $(6 x^2 + x) \\bmod {9}$ giving the permutation $( 0\\ 1\\ 2\\ 3\\ 4\\ 5\\ 6\\ 7\\ 8 ) \\mapsto ( 0\\ 7\\ 8\\ 3\\ 1\\ 2\\ 6\\ 4\\ 5 )$ .",
"The number of possible polynomial permutations is a small subset of all possible shufflings, but there are more than enough of them for our purposes.",
"We need only one that creates a good shuffling of a few hundred numbers, and the particular one we chose for our implementation is $(34 x^2 + x) \\bmod {289}$ .",
"What is more troublesome is the often inadequate integer support in GLSL 1.20 that effectively forces us to use single precision floats to represent integers.",
"There are only 24 bits of precision to play with (counting the implicit leading 1), and a floating point multiplication doesn't drop the highest bits on overflow.",
"Instead it loses precision by dropping the low bits that do not fit and adjusts the exponent.",
"This would be fatal to a permutation algorithm, where the least significant bit is essential and must not be truncated in any operation.",
"If the computation of our chosen polynomial is implemented in the straightforward manner, truncation occurs when $34 x^2 + x > 2^{24}$ , or $|x| > 702$ in the integer domain.",
"If we instead observe that modulo-$M$ arithmetic is congruent for modulo-$M$ operation on any operand at any time, we can start by mapping $x$ to $x \\bmod {289}$ and then compute the polynomial $34 x^2 + x$ without any risk for overflow.",
"By this modification, truncation does not occur for any $x$ that can be exactly represented as a single precision float, and the noise domain is instead limited by the remaining fractional part precision for the input coordinates.",
"Any single precision implementation of Perlin noise, in hardware or software, shares this limitation."
],
[
"Gradients on $N$ -cross-polytopes",
"Lattice gradient noise associates pseudo-random gradients with each lattice point.",
"Previous implementations have used pre-computed lookup tables or bit manipulations for this purpose.",
"We use a more floating-point friendly way and make use of geometric relationships between generalized octahedrons in different numbers of dimensions to map evenly distributed points from an ($N$ -1)-dimensional cube onto the boundary of the $N$ -dimensional equivalent of an octahedron, an $N$ -cross polytope.",
"For $N=2$ , points on a line segment are mapped to the perimeter of a rotated square, see Figure REF .",
"For $N=3$ , points in a square map to an octahedron, see Figure REF , and for $N=4$ , points in a cube are mapped to the boundary of a 4-D truncated cross polytope.",
"Equation (REF ) presents the mappings for the 2-D, 3-D and 4-D cases.",
"$\\textbf {2-D:\\quad } & x_0 \\in [ -2, 2 ], \\quad y = 1 - | x_0 | \\\\&\\textbf {if~} y > 0 \\textbf {~then~} x = x_0 \\textbf {~else~} x = x_0 - \\text{sign}(x_0)\\\\\\textbf {3-D:\\quad } & x_0, y_0 \\in [ -1, 1 ], \\quad z = 1 - | x_0 | - | y_0 | \\\\&\\textbf {if~} z > 0 \\textbf {~then~} x = x_0, ~ y = y_0\\\\&\\textbf {~else~} x = x_0 - \\text{sign}(x_0), ~ y = y_0 - \\text{sign}(y_0)\\\\\\textbf {4-D:\\quad } & x_0, y_0, z_0 \\in [ -1, 1 ], \\quad w = 1.5 - | x_0 | - | y_0 | - | z_0 | \\\\&\\textbf {if~} w > 0 \\textbf {~then~} x = x_0, ~ y = y_0, ~ z = z_0\\\\&\\textbf {~else~} x = x_0 - \\text{sign}(x_0), ~ y = y_0 - \\text{sign}(y_0), ~ z = z_0 - \\text{sign}(z_0)\\\\$ The mapping for the 4-D case doesn't cover the full polytope boundary – it truncates six of the eight corners slightly.",
"However, the mapping covers enough of the boundary to yield a visually isotropic noise field, and it is a simple mapping.",
"The 4-D mapping is difficult both to understand and to visualize, but it is explained in more detail in the supplementary material.",
"Figure: Mapping from a 1-D line segment to the boundary ofa 2-D diamond shape.Figure: Mapping from a 2-D square to the boundary of a 3-D octahedron.Blue points in the quadrant x>0,y>0x>0, y>0 where |x|+|y|<1|x|+|y| < 1 map to the facex,y,z>0x, y, z > 0, while red points where |x|+|y|>1|x|+|y| > 1 map to the opposite facex,y,z<0x, y, z < 0.Most implementations of Perlin noise use gradient vectors of equal length, but the longest and shortest vectors on the surface of an $N$ -dimensional cross polytope differ in length by a factor of $\\sqrt{N}$ .",
"This does not cause any strong artifacts, because the generated pattern is irregular anyway, but for higher dimensional noise the pattern becomes less isotropic if the vectors are not explicitly normalized.",
"Normalization needs only to be performed in an approximate fashion, so we use the linear part of a Taylor expansion for the inverse square root $1 / \\sqrt{r}$ in the neighborhood of $r = 0.7$ .",
"The built-in GLSL function inversesqrt() is likely to be just as fast on most platforms.",
"Normalization can even be skipped entirely for a slight performance gain."
],
[
"Rank ordering",
"Simplex noise uses a two step process to determine which simplex contains a point $\\mathbf {p}$ .",
"First, the N-simplex grid is transformed to an axis-aligned grid of $N$ -cubes, each containing $N!$ simplices.",
"The determination of which cube contains $\\mathbf {p}$ only requires computing the integer part of the transformed coordinates.",
"Then, the coordinates relative to the origin of the cube are computed by inverse transforming the fractional part of the transformed coordinates, and a rank ordering is used to determine which simplex contains $x$ .",
"Rank ordering is the first stage of the unusual but classic rank sorting algorithm, where the values are first ranked and then rearranged into their sorted order.",
"Rank ordering can be performed efficiently by pair-wise comparisons of components of $\\mathbf {p}$ .",
"Two components can be ranked by a single comparison, three components by three comparisons and four components can be ranked by six comparisons.",
"In GLSL, up to four comparisons can be performed in parallel using vector operations.",
"The ranking can be determined in a reasonably straightforward manner from the results of these comparisons.",
"The rank ordering approach was used in a roundabout way in the software 4D noise implementation of [6] and the GLSL implementation of [5], later improved and generalized by contributions from Bill Licea-Kane at AMD (then ATI).",
"The 3D noise of [6] and Perlin's original software implementation presented in [2] instead use a decision tree of conditionals.",
"For details on the rank ordering algorithm used for 3-D and 4-D simplex noise, which generalizes to $N$ -D, we refer to the supplementary material."
],
[
"Performance and source code",
"The performance of the presented algorithms is good, as presented in Table REF .",
"With reasonably recent GPU hardware, 2-D noise runs at a speed of several billion samples per second.",
"3-D noise attains about half that speed, and 4-D noise is somewhat slower still, with a clear speed advantage for 3-D and 4-D simplex noise compared to classic noise.",
"All variants are fast enough to be considered for practical use on current GPU hardware.",
"Procedural texturing scales better than traditional texturing with massive amounts of parallel execution units, because it is not dependent on texture bandwidth.",
"Looking at recent generations of GPUs, parallelism seems to increase more rapidly in GPUs than texture bandwidth.",
"Also, embedded GPU architectures designed for OpenGL ES 2.x have limited texture resources and may benefit from procedural noise despite their relatively low performance.",
"The full GLSL source code for 2D simplex noise is quite compact, as presented in Table REF .",
"For the gradient mapping, this particular implementation wraps the integer range $\\lbrace 0 \\ldots 288\\rbrace $ repeatedly to the range $\\lbrace 0 \\ldots 40\\rbrace $ by a modulo-41 operation.",
"41 has no common prime factors with 289, which improves the shuffling, and 41 is reasonably close to an even divisor of 289, which creates a good isotropic distribution for the gradients.",
"Counting vector operations as a single operation, this code amounts to just six dot operations, three mod, two floor, one each of step, max, fract and abs, seventeen multiplications and nineteen additions.",
"The supplementary material contains source code for 2-D, 3-D and 4-D simplex noise, classic Perlin noise and a periodic version of classic noise with an explicitly specified arbitrary integer period, to match the popular and useful pnoise() function in RenderMan SL.",
"The source code is licensed under the MIT license.",
"Attribution is required where substantial portions of the work is used, but there are no other limits on commercial use or modifications.",
"Managed and tracked code and a cross-platform benchmark and demo for Linux, MacOS X and Windows can be downloaded from the public git repository [email protected]:ashima/webgl-noise.git, reachable also by: http://www.github.com/ashima/webgl-noise Table: Performance benchmarks for selected GPUs, in Msamples per second"
],
[
"Old versus new",
"The described noise implementations are fundamentally different from previous work, in that they use no lookup tables at all.",
"The advantage is that they scale very well with massive parallelism and are not dependent on texture memory bandwidth.",
"The lack of lookup tables makes them suitable for a VLSI hardware implementation in silicon, and they can be used in vertex shader environments where texture lookup is not guaranteed to be available, as in the baseline OpenGL ES 2.0 and WebGL 1.0 profiles.",
"In terms of performance, this purely computational noise is not quite as fast on current GPUs as the previous implementation by Gustavson [5], which made heavy use of 2-D texture lookups both for permutations and gradient generation.",
"Most real time graphics of today is very texture intensive, and modern GPU architectures are designed to have a high texture bandwidth.",
"However, it should be noted that noise is mostly just one component of a shader, and a computational noise algorithm can make good use of unutilised ALU resources in an otherwise texture intensive shader.",
"Furthermore, we consider the simplicity that comes from independence of external data to be an advantage in itself.",
"A side by side comparison of the new implementation against the previous implementation is presented in Table REF .",
"The old implementation is roughly twice as fast as our purely computational version, although the gap appears to be closing with more recent GPU models with better computing power.",
"It is worth noting that 4D classic noise needs 16 pseudo-random gradients, which requires 64 simple quadratic polynomial evaluations and 16 gradient mappings in our new implementation, and a total of 48 2-D texture lookups in the previous implementation.",
"The fact that the old version is faster despite its very heavy use of texture lookups shows that current GPUs are very clearly designed for streamlining texture memory accesses."
],
[
"Supplementary material",
"http://www.itn.liu.se/~stegu/jgt2011/supplement.pdf Table: Performance of old vs. new implementation, in Msamples per second.Table: Complete, self-contained source code for 2D simplexnoise.",
"Code for 2D, 3D and 4D versions of classic and simplex noiseis in the supplementary material and in the online repository.Ian McEwan, David Sheets and Mark Richardson, Ashima Research, 600 S. Lake Ave., Suite 104, Pasadena CA 91106, USA ([email protected], [email protected], [email protected]) Stefan Gustavson, Media and Information Technology, ITN, Linköping University, 60174 Norrköping, Sweden ([email protected]) Received [DATE]; accepted [DATE]."
]
] | 1204.1461 |
[
[
"Gaseous Material Orbiting the Polluted, Dusty White Dwarf HE1349-2305"
],
[
"Abstract We present new spectroscopic observations of the polluted, dusty, helium-dominated atmosphere white dwarf star HE1349-2305.",
"Optical spectroscopy reveals weak CaII infrared triplet emission indicating that metallic gas debris orbits and is accreted by the white dwarf.",
"Atmospheric abundances are measured for magnesium and silicon while upper limits for iron and oxygen are derived from the available optical spectroscopy.",
"HE1349-2305 is the first gas disk-hosting white dwarf star identified amongst previously known polluted white dwarfs.",
"Further characterization of the parent body polluting this star will require ultraviolet spectroscopy."
],
[
"Introduction",
"White dwarf stars are now known to be polluted by remnant rocky bodies from planetary systems that otherwise stably orbited their host star while it was on the main sequence [42], [26], [17], [14], [16], [10], [29], [35], [43], [28].",
"Prior to being accreted, these rocky bodies are tidally shredded into disks of dusty material [7], [25].",
"A subset of dusty white dwarfs are also host to disks of gaseous metals which similarly have their origin in the disintegration of remnant rocky bodies from the white dwarf planetary system [21], [20], [19], [35], [15], [4].",
"[31] and [41] describe the atmospheric properties of the DBAZ (helium-dominated atmosphere with hydrogen and heavy element pollution) white dwarf star HE 1349$-$ 2305 (J2000 R.A. and Decl.",
"of 13 52 44.12 $-$ 23 20 05.3; [13]).",
"[23] detect excess infrared emission toward HE 1349$-$ 2305 indicating that it hosts and accretes from a dusty circumstellar disk.",
"As a result of parallels between this source and the potentially water-rich object GD 61 [18], [27], we obtained spectroscopic data for HE 1349$-$ 2305 to constrain its heavy element abundances and oxygen content.",
"An unexpected discovery in these spectroscopic data was the detection of emission lines indicating the presence of an orbiting gaseous disk.",
"Here we describe observations of HE 1349$-$ 2305 and place it in the context of other gas disk-hosting white dwarfs.",
"Optical imaging was performed on UT 24 March 2010 at Lick Observatory with the 40\" Nickel telescope.",
"These observations used the facility's Direct Imaging Camera (CCD-C2), a 2048 $\\times $ 2048 pixel detector with 15 $\\mu $ m pixels.",
"The 0.184$^{\\prime \\prime }$ pixel$^{-1}$ plate scale affords a field of view of roughly 6.3$^{\\prime }$ squared.",
"The detector was binned by two in rows and columns and was readout in fast mode.",
"HE 1349$-$ 2305 was observed in the $V$ -band [2].",
"A 4-step dither pattern with 10$^{\\prime \\prime }$ steps was repeated with 60 second integrations per step position.",
"Similar observations were performed for the flux calibrator source Gl 529 [1].",
"Images are reduced by median-combining all frames to obtain a sky frame and subtracting this sky frame from each image.",
"Sky-subtracted images are then divided by flat field frames obtained by imaging the twilight sky.",
"Each science frame is registered using bright stars in the field and then all science frames are median combined to yield the final reduced image.",
"Detector counts for HE 1349$-$ 2305 and Gl 529 are extracted with an aperture that yields $\\approx $ 85% encircled energy (with a negligible correction between the two sources).",
"This is achieved by extracting counts for both stars with a 4 binned-pixel (1.5$^{\\prime \\prime }$ ) radius circular aperture.",
"The sky is sampled with an annulus extending from 20-60 pixels.",
"Uncertainties are derived from the dispersion of measurements made from individually reduced frames.",
"The uncertainties for Gl 529 are propagated into the final quoted uncertainty for HE 1349$-$ 2305.",
"Nickel photometry is reported in Table which also includes the DENIS [13] $I$ -band magnitude and Galaxy Evolution Explorer [33] fluxes."
],
[
"Gemini Imaging at the Shane 3-m",
"Observations of HE 1349$-$ 2305 in the $J$ -, $H$ -, and $K^{\\prime }$ -bands were performed UT 28 March 2010 with the Gemini Twin-Arrays Infrared Camera [34] mounted on the 3 m Shane telescope at Lick Observatory.",
"We used a 4 position dither pattern with exposure times of 10, 5, and 7 seconds with coadds of 15, 30, and 21 per position for each of $JHK^{\\prime }$ , respectively.",
"Total on source integration times of 1800 seconds were accrued at each of $JH$ while 3528 seconds were obtained for $K^{\\prime }$ .",
"The $\\approx $ 3$^{\\prime }$ field-of-view of the Gemini instrument enabled simultaneous observations of two 2MASS [39] sources for use in flux calibration.",
"Data are reduced using in-house IDL software routines.",
"For each filter, science frames were median combined to generate a sky-median frame which is then subtracted from each science frame.",
"Sky-subtracted frames are then flat-fielded using exposures of the twilight sky for $JK^{\\prime }$ and the illuminated telescope dome for $H$ .",
"Reduced science frames are registered with bright point sources within the field.",
"Although data were recorded for the longer wavelength chip ($K^{\\prime }$ ), they are not usable for accurate photometric measurements due to instrumental difficulties.",
"The Gemini photometric results for HE 1349$-$ 2305 are reported in Table .",
"$JH$ -band results presented herein are consistent with those presented by [23]."
],
[
"MagE Optical Spectroscopy",
"Moderate resolution optical spectroscopy of HE 1349$-$ 2305 was obtained on UT 19 March 2011 with the Magellan Echellette (MagE; [32]) mounted on the 6.5 m Landon Clay Telescope at Las Campañas Observatory.",
"One 900 s exposure was obtained with the 0.5$^{\\prime \\prime }$ slit aligned with the parallactic angle; this setup provided 3200-10050 Å spectroscopy with a resolving power of $\\approx $ 11000.",
"Data are reduced using the MASE reduction pipeline [3] following standard procedures for order tracing, flat field correction, wavelength calibration (with ThAr lamp spectra), heliocentric wavelength correction, optimal source extraction, order stitching, and flux calibration via observations of Hiltner 600 [24]."
],
[
"SpeX Near-Infrared Spectroscopy",
"Near-infrared spectroscopy was obtained with SpeX [38] mounted on the 3 m NASA IRTF telescope on UT 22 April 2011.",
"Prism-mode observations covering 0.8-2.5 $\\mu $ m were performed with a 0.8$^{\\prime \\prime }$ slit aligned with the parallactic angle.",
"Six ABBA nod-patterns were obtained for HE 1349$-$ 2305 with 60 seconds of integration time and 2 coadds per nod position.",
"A single AB nod pair of 0.51 seconds integration time and 10 coadds per nod position was obtained for the telluric calibration source HD 119752 (A0 V).",
"Data are reduced with SpeXTool [6], [40].",
"Absolute flux calibration of the HE 1349$-$ 2305 prism data is accomplished by scaling its spectrum to the Gemini $JH$ -band measurements.",
"It is noted that the SpeX data smoothly connect all available near-infrared photometric data for HE 1349$-$ 2305 (Table and [23])."
],
[
"VLT X-shooter Spectroscopy",
"Moderate resolution spectroscopy of HE 1349$-$ 2305 was obtained in service mode with X-shooter [8] mounted on the 8.2 m VLT UT2 (Kueyen) telescope.",
"One observation was obtained on UT 26 May 2011 while two more were obtained on UT 28 May 2011.",
"UVB arm (3000-5600 Å) observations were performed with the 0.5$^{\\prime \\prime }$ slit resulting in a resolving power of 9900 and exposed for 1475 seconds per observation.",
"VIS arm (5500-10200 Å) observations were performed with the 0.4$^{\\prime \\prime }$ slit resulting in a resolving power of 18200 and exposed for 1420 seconds per observation.",
"Raw frames are reduced using the X-shooter pipeline version 1.3.7 within ESOREX The ESO Recipe Execution Tool $-$ http://www.eso.org/sci/software/cpl/esorex.html; version 3.9.0 is used.",
".",
"Standard X-shooter data reduction techniques are employed with default settings to extract and wavelength calibrate each spectrum.",
"Relative flux calibration on the science spectrum is performed with the use of the spectrophotometric standards LTT 3218 (May 26th) and EG 274 (May 28th) to derive the instrumental response function.",
"Although X-shooter coverage extends to the thermal-infrared, data beyond $\\approx $ 1 $\\mu $ m are unusable due to low recorded signal."
],
[
"Results and Modeling",
"Physical parameters for HE 1349$-$ 2305 are adopted from analysis performed on VLT UVES spectra in [31] and [41], namely T$_{\\rm eff}$ of 18173 K, log $g$ of 8.13 (cgs units), and a mass of 0.673 M$_{\\odot }$ .",
"By matching a model white dwarf atmosphere with these parameters to the observed spectra and photometry we derive a distance to the white dwarf of 120$\\pm $ 10 pc (the uncertainty here does not take into account uncertainties on the white dwarf parameters).",
"Spectral observations that extend to wavelengths of $\\approx $ 2.5 $\\mu $ m confirm the results of [23], but are not capable of further restricting the dusty disk parameters.",
"From absorption lines detected in the optical spectra we derive observed radial velocities (which include contributions from gravitational redshift and stellar motion) of 40$\\pm $ 30 and 40$\\pm $ 5 km s$^{-1}$ from the MagE and UVES data, respectively.",
"Difficulties in setting the wavelength zero-point in the X-shooter data prevent any meaningful radial velocity measurement $-$ these data are corrected to the white dwarf reference frame by assuming the radial velocity is the same as that measured for the MagE and UVES data.",
"The contribution from gravitational redshift is estimated to be 35 km s$^{-1}$ and hence the white dwarf systemic motion is $\\sim $ 5 km s$^{-1}$ ."
],
[
"Abundances",
"No absorption lines other than those from H I and He I are significantly detected in the MagE spectrum (Ca II H and K lines are marginally detected, but not useful for abundance modeling).",
"The X-shooter data enable the additional detection of Mg II, Si II (Figure REF ), and Ca II.",
"[31] and [41] report detections of H I and Ca II in their UVES spectra.",
"To try and explore the water content of the body polluting HE 1349$-$ 2305 we calculated upper limits for the abundances of oxygen and iron, the two major constituents of terrestrial rocky minerals not detected (Figure REF ).",
"We use a local thermodynamic equillibrium (LTE) model atmosphere code similar to that described in [9], [12].",
"Absorption line data are taken from the Vienna Atomic Line Database http://vald.astro.univie.ac.at/~vald/php/vald.php .",
"We calculate grids of synthetic spectra for each element of interest.",
"The grids cover a range of abundances typically from log[$n$ (Z)/$n$ (He)]= $-$ 3.0 to $-$ 7.0 in steps of 0.5 dex.",
"We determine abundances or limits by fitting the expected position of various lines in the spectra using a similar method to that described in [9].",
"Briefly, this is done by minimizing the value of $\\chi $$^2$ which is taken to be the sum of the difference between the normalized observed and model fluxes over the frequency range of interest with all frequency points being given an equal weight.",
"Upper limits are derived by comparing model lines of a given abundance with their expected position in the spectra and determining whether such a line would be detectable given the local signal-to-noise ratio of the spectrum.",
"All abundance measurements for HE 1349$-$ 2305 are reported in Table .",
"Abundance measurements for hydrogen and calcium agree within the errors with those reported in [31] and [41]."
],
[
"Gas Emission Lines",
"Broad emission lines from the Ca II infrared triplet (IRT) are detected in the MagE and X-shooter spectra (Figure REF ).",
"For each feature we measure the maximum gas velocity in the blue and red wings of the detected emission lines, full velocity width at zero power, and the line flux; these values are reported in Table .",
"It is not possible to place robust constraints on the gas disk inner and outer radii with the available spectra.",
"Modeling similar to that described in [21] suggests that the gas disk outer radius is similar to those of the other gas disk-hosting white dwarf stars, $\\sim $ 100 R$_{\\rm WD}$ .",
"From the maximum velocity gas seen in the disk, and assuming a disk inclination angle of 60$^{\\circ }$ (in accordance with inclination angle values used in modeling of white dwarf dust disks $-$ see e.g., [17]), we estimate a disk inner radius of $\\sim $ 15 R$_{\\rm WD}$ .",
"If we employ the inclination angle derived by [23] for the dust disk orbiting HE 1349$-$ 2305 $-$ $i$$\\approx $ 85$^{\\circ }$ $-$ then the inner radius of the gas disk is slightly higher at $\\sim $ 20 R$_{\\rm WD}$ .",
"More precise constraints on the gas disk structure will require higher signal-to-noise ratio observations.",
"These observations are likely better carried out using lower resolution optical spectrographs than those used herein."
],
[
"Discussion",
"HE 1349$-$ 2305 is found to host Ca II infrared triplet emission lines.",
"The lack of hydrogen or helium emission lines in the optical and infrared spectra suggest that this material is metal-rich.",
"Material orbiting the star would be expected to exhibit a double-peaked emission line morphology similar to that seen for the other gas disk-hosting white dwarfs [21], [20], [19], [35].",
"Although only a single peak of emission is significantly detected in the MagE data, a weaker second peak is evident in the X-shooter data securing the Keplerian origin of the gas emission.",
"No significant radial velocity shift is evident between the various epochs of optical spectroscopy, making the origin of the emission lines from a binary companion (or interactions therewith) unlikely.",
"As a result, we interpret these emission lines as emanating from a gaseous debris disk similar to that described by [21].",
"Support of such an interpretation is found through similarities observed for other such sources [5], [35] and the detection of thermal-infrared excess emission toward HE 1349$-$ 2305 indicating that a dusty debris disk also orbits the star [23].",
"The asymmetry of the emission line peak intensities is curious (Figure REF ).",
"Similar peak intensity contrast is evident at a slightly weaker level for SDSS J1228 [21], [35] and at a slightly stronger level in the 2004 epoch spectrum of Ton 345 [19].",
"In regards to the sharp cutoff in line intensity on one edge of the emission complex and the slower roll-off in intensity at the other edge, the emission structure of this source resembles those of the other gas disk white dwarfs studied at high spectral resolution [35].",
"The rough gas disk inner and outer radii are reminiscent of those measured for the other gas disk-hosting white dwarfs [21], [20], [19], [35].",
"The outer disk radius is consistent with the location of the Roche limit for HE 1349$-$ 2305, suggesting that its orbiting gas disk is generated by rocky objects that impinge on a pre-existing dusty debris disk (e.g., [26], [35]).",
"This is further supported by the fact that the gas disk inner and outer radii are roughly consistent with the dust disk inner and outer radii reported in [23].",
"Thus, despite the inexact characterization of the structure of its orbiting gas disk, it can be concluded that HE 1349$-$ 2305 is being polluted by rocky objects from its planetary system similar to other gas disk-hosting white dwarf stars [35].",
"An attempt to constrain the composition of the body polluting this star is made by examining the abundances of the major elemental constituents of rocky minerals: magnesium, silicon, iron, and oxygen [29].",
"These values are reported in Table ; from the available data we are unable to make any significant claims about the water content of the body polluting HE 1349$-$ 2305.",
"Comparing the measured heavy element abundances (calcium, silicon, and magnesium) to those of well-studied polluted white dwarf stars suggests that the body polluting HE 1349$-$ 2305 has experienced radiative weathering.",
"In particular, the accreted body exhibits the characteristic Si/Mg deficiency and Ca/Mg enhancement (relative to CI Chondrites) shown by GD 40 that is interpreted as evidence for silicate vaporization of the parent rocky body by the intense radiation field from its evolving host star [29], [36].",
"These results suggest that HE 1349$-$ 2305 could be accreting the remnants of a differentiated rocky body, although tighter constraints on or detections of oxygen and iron are necessary before solidifying any such claim.",
"Comparing the pollution of HE 1349$-$ 2305 to that of the other gas disk-hosting white dwarf stars (Table ) reveals that HE 1349$-$ 2305 is among the lowest accretors of the group.",
"Of the helium-dominated atmosphere white dwarf stars in Table , HE 1349$-$ 2305 has a time-averaged accretion rate that is lower than the other two (SDSS J0738 and Ton 345) by more than an order of magnitude.",
"The identification of a metallic gas disk orbiting HE 1349$-$ 2305 shows that there is possibly a significant population of gas disk-hosting white dwarf stars that remain to be discovered, perhaps even within the currently known population of polluted white dwarf stars."
],
[
"Conclusions",
"We have observed the heavy element and hydrogen polluted white dwarf HE 1349$-$ 2305 in the optical and near-infrared.",
"Our principal results are that the star hosts Ca II infrared triplet emission lines indicative of orbiting gaseous debris and that there is little evidence of heavy element pollution in its optical spectrum (compared to other disk-hosting helium dominated atmosphere white dwarfs; [42], [29], [28], [11]).",
"Observations in the ultraviolet will be necessary to further examine the elemental composition of the parent body polluting this white dwarf star.",
"C.M.",
"acknowledges support from the National Science Foundation under award No.",
"AST-1003318.",
"P.D.",
"is a CRAQ postdoctoral fellow.",
"This paper includes data gathered with the 6.5 meter Magellan Telescopes located at Las Campañas Observatory, Chile.",
"This work is based partly on observations made with ESO Telescopes at Paranal Observatory under the program 087.D-0858(A).",
"This publication makes use of data products from the Two Micron All Sky Survey, which is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center/California Institute of Technology, funded by the National Aeronautics and Space Administration and the National Science Foundation.",
"This research has made use of the SIMBAD database and VizieR service.",
"Based on observations made with the NASA Galaxy Evolution Explorer.",
"$GALEX$ is operated for NASA by the California Institute of Technology under NASA contract NAS5-98034.",
"This work is supported in part by the NSERC Canada and by the Fund FQRNT (Québec).",
"Facilities: Magellan:Clay (MagE), VLT:Kueyen (X-shooter), Nickel (Direct Imaging Camera), Shane (Gemini), IRTF (SpeX) cccc 4 0pt Broad-band Fluxes for HE 1349$-$ 2305 Band $\\lambda $ $m$ a F$_{\\rm obs}$ (nm) (mag) (mJy) Lick-Gemini cameraa $-$ NIR $H$ 1650 16.87$\\pm $ 0.08 0.19$\\pm $ 0.01 $J$ 1240 16.85$\\pm $ 0.06 0.30$\\pm $ 0.02 DENISa $-$ Infrared $I$ 798.2 16.70$\\pm $ 0.09 0.50$\\pm $ 0.04 Nickel 40$^{\\prime \\prime }$ a $-$ Optical $V$ 544.8 16.22$\\pm $ 0.11 1.17$\\pm $ 0.12 $GALEX$ a $-$ Ultraviolet $NUV$ 227.1 16.24$\\pm $ 0.03 1.16$\\pm $ 0.04 $FUV$ 152.8 16.54$\\pm $ 0.08 0.88$\\pm $ 0.06 aGemini, DENIS [13], and Nickel magnitudes are on the Vega system.",
"$GALEX$ measurements are in AB magnitudes.",
"The $GALEX$ $NUV$ uncertainty is as suggested in [37] while the $FUV$ uncertainty is representative of the scatter between the two separate $GALEX$ detections of HE 1349$-$ 2305. ccccccccc 9 0pt Atmospheric Pollution of Gas Disk-hosting White Dwarfs Star [H/He]a [O/H(e)] [Mg/H(e)] [Si/H(e)] [Ca/H(e)] [Fe/H(e)] $\\dot{M}_{\\rm acc,Mg}$ b Ref 6c(logarithmic abundances by number) (10$^{8}$ g s$^{-1}$ ) HE 1349$-$ 2305 $-$ 4.9$\\pm $ 0.2 $<$$-$ 5.6 $-$ 6.5$\\pm $ 0.2 $-$ 7.0$\\pm $ 0.2 $-$ 7.4$\\pm $ 0.2 $<$$-$ 5.9 1.3 1,2,3 SDSS J0738 $-$ 5.73$\\pm $ 0.17 $-$ 3.81$\\pm $ 0.19 $-$ 4.68$\\pm $ 0.07 $-$ 4.90$\\pm $ 0.16 $-$ 6.23$\\pm $ 0.15 $-$ 4.98$\\pm $ 0.09 146.4 4,5 SDSS J0959 $-$ $-$ $-$ 5.2 $-$ $-$ 7.0 $-$ 0.32 6 Ton 345 $<$$-$ 4.5 $-$ $-$ 5.2$\\pm $ 0.2 $-$ 5.1$\\pm $ 0.2 $-$ 6.9$\\pm $ 0.2 $-$ 18.3 7 SDSS J1043 $-$ $-$ $-$ 4.94$\\pm $ 0.24 $-$ $-$ $-$ 0.73 8 SDSS J1228 $-$ $-$ $-$ 4.58$\\pm $ 0.06 $-$ $-$ 5.76$\\pm $ 0.08 $-$ 2.2 9,10 (1) [31], (2) [41], (3) This work, (4) [10], (5) [11], (6) [15], (7) [19], (8) [20], (9) [21], (10) [22].",
"aHydrogen pollution for helium-dominated atmosphere (DB) white dwarfs.",
"A “$-$ ” in this column indicates that the star has a hydrogen-dominated atmosphere (DA) and that each elemental abundance listed is relative to hydrogen by number.",
"In other columns a “$-$ ” indicates that no measurement exists in the literature.",
"b$\\dot{M}$$_{\\rm acc,Mg}$ =$M_{\\rm env,Mg}$ /$\\tau $$_{\\rm diff,Mg}$ where $M_{\\rm env,Mg}$ is the mass of magnesium in each star's envelope and $\\tau $$_{\\rm diff,Mg}$ is the diffusion constant for magnesium (see [30]).",
"For helium-dominated atmosphere white dwarfs this quantity is averaged over the $\\sim $ 10$^{5}$ yr settling times.",
"lcccc 5 0pt HE 1349$-$ 2305 Emission Line Measurements Transition Equivalent Widtha $v_{max}$ sin$i$ b Full Widthb Total Line Fluxc (Å) (km s$^{-1}$ ) (km s$^{-1}$ ) (10$^{-15}$ ergs cm$^{-2}$ s$^{-1}$ ) 5lMagE $-$ 19 March 2011 Ca II $\\lambda $ 8498 1.4$\\pm $ 0.3 $-$ 190$\\pm $ 140/+410$\\pm $ 110 600$\\pm $ 180 0.30 Ca II $\\lambda $ 8542 2.1$\\pm $ 0.3 $-$ 160$\\pm $ 140/+510$\\pm $ 140 670$\\pm $ 200 0.45 Ca II $\\lambda $ 8662 1.7$\\pm $ 0.4 $-$ 220$\\pm $ 140/+400$\\pm $ 70 620$\\pm $ 160 0.37 5lX-shooter $-$ Average of 26 and 28 May 2011 Ca II $\\lambda $ 8498 1.9$\\pm $ 0.2 $-$ 780$\\pm $ 110/+380$\\pm $ 35 1160$\\pm $ 120 0.41 Ca II $\\lambda $ 8542 1.7$\\pm $ 0.2 $-$ 710$\\pm $ 70/+450$\\pm $ 70 1160$\\pm $ 100 0.37 Ca II $\\lambda $ 8662 1.6$\\pm $ 0.3 $-$ 740$\\pm $ 110/+400$\\pm $ 70 1140$\\pm $ 130 0.35 aEquivalent widths are not corrected for line absorption.",
"bThe two different values reported for $v_{max}$ sin$i$ correspond to the maximum velocity gas seen in the blue and red wings of the double-peaked emission features, respectively.",
"The blue wing is measured at the continuum blueward of the line while the red wing is measured at the continuum redward of the line.",
"Full velocity width of the emission feature is the velocity extent from the blue to the red wings.",
"cThese values are computed by multiplying the reported emission line equivalent width measurements by the stellar continuum flux (as deduced from the SpeX spectrum) at the emission line location."
]
] | 1204.1132 |
[
[
"Diphotons from Tetraphotons in the Decay of a 125 GeV Higgs at the LHC"
],
[
"Abstract Recently the ATLAS and CMS experiments have presented data hinting at the presence of a Higgs boson at $m_h\\simeq125$ GeV.",
"The best-fit $h\\rightarrow\\gamma\\gamma$ rate averaged over the two experiments is approximately $2.1\\pm0.5$ times the Standard Model prediction.",
"We study the possibility that the excess relative to the Standard Model is due to $h\\rightarrow aa$ decays, where $a$ is a light pseudoscalar that decays predominantly into $\\gamma\\gamma$.",
"Although this process yields $4\\gamma$ final states, if the pseudoscalar has a mass of the order tens of MeV, the two photons from each $a$ decay can be so highly collimated that they may be identified as a single photon.",
"Some fraction of the events then contribute to an effective $h\\rightarrow\\gamma\\gamma$ signal.",
"We study the constraints on the parameter space where the net $h\\rightarrow\\gamma\\gamma$ rate is enhanced over the Standard Model by this mechanism and describe some simple models that give rise to the pseudoscalar-photon interaction.",
"Further tests and prospects for searches in the near future are discussed."
],
[
"Introduction",
"Determining the nature of electroweak symmetry breaking (EWSB) has been a primary goal of particle physics for several decades.",
"In the Standard Model (SM), EWSB occurs when the neutral component of a single scalar weak isospin doublet possesses a vacuum expectation value (VEV).",
"The Higgs boson corresponds to the physical excitations of this field.",
"Recent experimental advances make it clear that we are entering a new phase in searches for the Higgs boson, with tantalizing excesses appearing in several SM-like Higgs search channels at both the CERN Large Hadron Collider (LHC) and the Fermilab Tevatron.",
"These hints point to a relatively light state with a mass in the range $122.5\\lesssim m_h\\lesssim 127.5$ GeV [1], [2].",
"In the SM, such a light Higgs boson has an extremely narrow width of the order $10^{-5}\\times m_h$ .",
"The tiny width of the SM Higgs makes it a sensitive probe of new physics (NP) beyond the SM, especially in the only loosely constrained scalar sector [3].",
"New states coupled to a SM-like Higgs can have appreciable effect on its decays without ruining the excellent agreement of the SM with data observed elsewhere thus far (see, e.g., [4]).",
"In fact, the excesses seen at the LHC and Tevatron may already be pointing toward interesting deviations from the SM predictions for the Higgs branching ratios.",
"The largest statistical power thus far comes from the searches for $h\\rightarrow \\gamma \\gamma $ at the LHC, where the ATLAS experiment finds a 2.8$\\sigma $ excess at $m_h\\simeq 126$ GeV with a best-fit signal strength relative to the SM of approximately $2.0\\pm 0.8$ [5], [1].",
"The CMS experiment finds a 3.1$\\sigma $ excess in $h\\rightarrow \\gamma \\gamma $ at $m_h\\simeq 124$ GeV with a best-fit signal strength relative to the SM of approximately $2.1\\pm 0.6$ [6], [2].",
"Under the assumption that these excesses are due to the same new particle, we can naively combine the diphoton rates from the two experiments and estimate Rate(h)/SM 2.10.5.",
"This estimate is not rigorous and the error bar is far from conclusive.",
"However, the uncertainties will continue to decrease as data accumulate, and the central value may very well remain high.",
"Therefore, it is of interest to classify models that can alter the $h\\rightarrow \\gamma \\gamma $ signal and study their other predictions.",
"A number of recent papers have contributed to this program, including discussions of the effects of superpartners on the diphoton rate [7], , , , , the effects of more general new fermion and scalar states [12], the effects of singlet-doublet mixing [13], , the predictions in the case of minimal universal extra dimensions [15], interference effects from charged Higgs contributions [16], , and the possibility that the signal is due to the Randall-Sundrum radion [18], to name a few.",
"In this paper we consider a different scenario in which a 125 GeV Higgs boson can appear to have a larger branching ratio into $\\gamma \\gamma $ .",
"We introduce a very light pseudoscalar of mass in the range $m_a\\in (\\sim 10{~\\rm MeV}$ , $\\sim m_\\pi )$ , and we allow the Higgs boson to decay in the channel $h\\rightarrow aa$ .",
"The pseudoscalars produced in these decays are extremely boosted in the lab frame, causing their decay products to be quite collimated.",
"Furthermore, for such light pseudoscalars, there are no kinematically available hadronic decay modes.",
"As first studied in [19], the decay $a\\rightarrow \\gamma \\gamma $ induced by the effective coupling $aF_{\\mu \\nu }\\tilde{F}^{\\mu \\nu }$ can easily dominate, leading to a $h\\rightarrow aa\\rightarrow 4\\gamma $ signature (in contrast to the “buried Higgs\" scenario where the dominant decays are hadronic [20], , ).",
"Because of the large boost for the pseudoscalars produced in $h$ decays, the photon pairs from each $a$ can be so highly collimated that a significant fraction may be identified as single photons, even in the ATLAS and CMS detectors which are very good at distinguishing two closely separated photons (e.g.",
"originating from $\\pi ^0$ or $\\eta $ decays).",
"A related scenario, although not in the context of Higgs boson decays, was considered in [23], which studied decays of a new heavy vector through pseudoscalars to extremely collimated “photon jets.\"",
"Since their vector mass is much larger than 125 GeV, they consider much heavier pseudoscalars.",
"Our analysis of the conditions necessary for photon jets to be identified as single photons is complementary to that of [23] and provides further motivation for the detailed study of photon jets.",
"Previously, the $h\\rightarrow aa\\rightarrow 4\\gamma $ scenario was considered in [19], [4], [24] and, in particular, the relevance of this channel to $h\\rightarrow \\gamma \\gamma $ searches at the Tevatron was studied in [19].",
"While Ref.",
"[19] focused mainly on pseudoscalars heavier than 200 MeV and heavier Higgs bosons, we will find that a significant $4\\gamma \\rightarrow 2\\gamma $ fake rate at the LHC with a 125 GeV Higgs boson requires a pseudoscalar lighter than about the mass of the $\\pi ^0$ .",
"In contrast to previous works, we also study the interplay between the net expected $h\\rightarrow \\gamma \\gamma $ signal and the signal in other Higgs search channels, provide an analysis of the compatibility of the light pseudoscalar scenario with the current Higgs data from the LHC and Tevatron, and study in detail the constraints from LEP and low-energy experiments.",
"For further discussion of Higgs decays to light pseudoscalars, see, e.g., [25].",
"This paper is organized as follows.",
"In Sec.",
"we discuss the basic features of the model.",
"In Sec.",
"we analyze the changes in the expected rates for SM Higgs search channels at the LHC.",
"We give a detailed estimate of the probability for a collimated photon pair to be identified as a single photon, and perform a basic statistical analysis of the constraints on the light pseudoscalar parameter space coming from current Higgs searches at the Tevatron and LHC.",
"In Sec.",
"we study the constraints following from the muon anomalous magnetic moment, direct searches at LEP for $e^+e^-\\rightarrow \\gamma +{\\rm inv.",
"}$ , meson decays, beam dump experiments, and nuclear physics.",
"In Sec.",
"we survey model building for the light $a$ and its coupling to photons, and consider the possibility that the latter is generated by $\\tau $ loops, or by $a-\\pi ^0$ mixing.",
"In Sec.",
"we discuss the prospects for directly producing pseudoscalars in the mass and coupling range of interest at Primakoff-type experiments.",
"In Sec.",
"we conclude, and an Appendix generalizes our main results to the case where $a$ has a substantial invisible branching fraction."
],
[
"Model",
"In addition to the SM, we consider a real pseudoscalar $a$ which is the pseudo Nambu-Goldstone boson of some spontaneously broken approximate global symmetry.A frequently-discussed model containing a light PNGB is the NMSSM in the approximate PQ- or R-symmetric limits.",
"In that model the pseudoscalar has sufficiently large fermionic couplings that in the mass range we consider, it is completely ruled out by low-energy experiments such as those considered in Sec. [26].",
"It has a small mass $m_a$ coming from small explicit breaking of the symmetry, and it interacts with the SM via Lint=12(a)2HH-e24Ma FF, where $H$ is the SM Higgs doublet, $F^{\\mu \\nu }$ is the photon field strength, and $e$ is the positron charge.",
"$\\Lambda $ and $M$ are scales describing the strength of the higher dimensional operators.",
"Expanding around the Higgs VEV $v\\simeq 246{~\\rm GeV}$ , $H^0=\\left(v+h\\right)/\\sqrt{2}$ , leads to a rate for the Higgs to decay to pseudoscalars of (haa)=v2mh3324 =1.18 MeV(mh125 GeV)3( TeV)-4, where we have assumed that $m_a\\ll m_h$ .",
"Using Eq.",
"() we can also calculate the rate for $a$ to decay to $\\gamma \\gamma $ , (a)=2ma34 M2 =2.6810-8 MeV(M10 GeV)-2(ma40 MeV)3.",
"Pseudoscalars produced by the decay of $125{~\\rm GeV}$ Higgs at rest then have a decay length of c1.15 cm (M10 GeV)2(ma40 MeV)-4 (mh125 GeV).",
"If we require $\\gtrsim 90\\%$ of the decays to occur inside the electromagnetic calorimeters of ATLAS and CMS ($\\simeq 1$ m radius), $\\gamma c\\tau $ should be less than about a half meter.",
"We can express the scale $M$ in terms of the decay length, $m_a$ , and $m_h$ , M=9.3 GeV (c1 cm)1/2(ma40 MeV)2 (mh125 GeV)-1/2.",
"In the Appendix, we expand this simple model to allow for $a$ to additionally decay invisibly.",
"We discuss the benefits and further constraints that this entails there."
],
[
"Higgs Signal",
"The interactions described in Sec.",
"do not appreciably affect the production cross section of the SM Higgs boson at the LHC.",
"In this section we study the effects of the decays $h\\rightarrow aa$ , $a\\rightarrow \\gamma \\gamma $ on the expected Higgs-to-diphotons rate.",
"We also estimate the constraints on such decays from existing SM Higgs searches at the Tevatron and LHC."
],
[
"Modified Rates in SM Higgs Search Channels",
"We define the ratios of the Higgs branchings to photons and to SM fermions, $f$ , or gauge bosons, $V=W,~Z$ , to their values in the SM as B(h)eff=RBSM(h), B(hff,VV)=RXXBSM(hff,VV).",
"The presence of the $h\\rightarrow aa$ decay suppresses uniformly the rates into all SM final states, yielding RXX=1-B(haa).",
"Note that to avoid cluttering notation, and because the rescaling is the same in both cases, we use $R_{XX}$ to represent both the fermionic and $W,Z$ rescalings relative to the SM.",
"The suppression in Eq.",
"REF is also present for the pure $\\gamma \\gamma $ final state; however, the subscript “${\\rm eff}$ \" in Eq.",
"(REF ) indicates that in pseudoscalar models there can be an additional, effective contribution to the measured $h\\rightarrow \\gamma \\gamma $ rate.",
"Since the LHC detectors have finite resolution, a fraction of $h\\rightarrow aa\\rightarrow 4\\gamma $ events may have sufficiently boosted photon pairs that each pair is identified as a single photon.",
"Assuming a SM production cross section for $h$ and a 100% branching of $a\\rightarrow \\gamma \\gamma $ , the measured diphoton branching ratio will be B(h)eff=B(h)+B(haa) or R=1+B(haa)(BSM(h)-1).",
"In these formulae, $\\epsilon $ is the probability that both $\\gamma \\gamma $ pairs are identified as single photons, so that four photons appear as two.",
"We see that to achieve an effective diphoton rate greater than or equal to the SM rate requires BSM(h)0.0023 for $m_h=125{~\\rm GeV}$ ."
],
[
"$4\\gamma \\rightarrow 2\\gamma $ Misidentification Rate",
"Estimating $\\epsilon $ is complicated by several factors.",
"We would like to be conservative in our estimate of the expected rate; on the other hand, underestimating the rate by too much might falsely indicate that some model points are allowed, when in fact the true rates are so large that the points are already ruled out by the LHC.",
"We base our estimate of $\\epsilon $ on the ATLAS selection criteria used to identify isolated photons in their cut-based analysis, and we comment on differences with CMS.",
"ATLAS uses a number of calorimeter variables to parametrize the shape of an electromagnetic shower, which can then be used to discriminate true isolated photons from backgrounds.",
"The background from isolated $\\pi ^0\\rightarrow \\gamma \\gamma $ decays bears strong similarity to our $a\\rightarrow \\gamma \\gamma $ process, and ATLAS efficiently vetoes isolated pions using information from the first layer of the calorimeter, which has finely-segmented strips in $\\eta $ [27].",
"The primary discrimination variables in the first layer are ${\\bf E_{\\rm ratio}}$ , which is the difference in energies between two strips containing energy maxima normalized to their sum; ${\\bf \\Delta E}$ , which measures the difference in energies between the strip with the second-largest energy maximum and the strip with the minimum energy between the first two maxima; ${\\bf F_{\\rm side}}$ , which is the fraction of the energy deposited in seven strips in $\\eta $ around the maximum that does not fall into the central three strips; ${\\bf w_{s \\rm 3}}$ , which measures the energy deposition in the two strips adjacent in $\\eta $ to a strip with an energy peak, relative to the total energy in the three strips; and ${\\bf w_{s \\rm tot}}$ , which generalizes $w_{s3}$ to approximately twenty strips in $\\eta $ and two strips in $\\phi $ [28].",
"To simplify our analysis, we begin by restricting our attention to photons that do not convert to $e^+e^-$ pairs in the tracker, and which are so highly collimated that a second energy maximum does not appear in the first-layer calorimeter strips.",
"In this case $\\Delta E$ is set to 0, $E_{\\rm ratio}$ is set to 1, and most of the energy will be deposited into just a few adjacent strips.",
"Therefore we expect that the most sensitive variable will be $w_{s3}$ , defined precisely as ws3i Ei(i-imax)2/iEi, where $i$ labels the strips.",
"On average the photon pair from a $\\pi ^0$ decay generates a larger $w_{s3}$ value than a true single-photon event and thus may be efficiently rejected.",
"On the other hand, the probability that a photon pair passes the $w_{s3}$ cut should increase with the collimation of the pair.",
"To easily estimate $\\epsilon $ , we would like to approximate the cut on $w_{s3}$ by a cut on the photon pair separation.",
"For unconverted events, ATLAS uses a weakly $\\eta $ -dependent cut on $w_{s3}$ that is typically between $0.6-0.7$ , and is about $0.66$ for the most central strips in the barrel.",
"The average value of $w_{s3}$ for true photons in these strips is about 0.58 [29].",
"We can reproduce this number with the following simple model.",
"First, we assume that a single photon lands in the center of one of these most central strips, and deposits its energy according to a Gaussian distribution in $\\eta $ with standard deviation $0.52$ times the smallest strip width.",
"This value is chosen so that the photon deposits about 70% of the energy into the central strip and 15% into each adjacent strip, giving a $w_{s3}$ value of 0.58.",
"Subsequently, we compute $w_{s3}$ for a pair of photons as a function of $\\eta $ separation, averaging over the impact point in the central strip.",
"We find that for $\\Delta \\eta _{\\gamma \\gamma }=0.0015$ , $w_{s3}\\simeq 0.66$ , equal to the ATLAS cut in these strips.",
"On the other hand, the central strips have a width in $\\eta $ of $\\Delta \\eta _{\\rm strip}=0.0031$ (the smallest in the calorimeter.)",
"Therefore, we estimate that the cut on $w_{s3}$ can be simulated by a cut on the photon separation, given by <1/2strip.",
"For larger $\\eta $ , where larger strips are present, we also require a separation less than $1/2$ of the relevant strip size, which is probably mildly conservative since on average the energy leakage into adjacent strips from a single-photon hit will be less.",
"There are two simple ways in which $w_{s3}$ may become insufficient.",
"First, photons that are less highly collimated (such as those that appear in the decays of more massive pseudoscalars) may deposit energy in strips that are sufficiently separated (more than three strips apart) that $w_{s3}$ becomes insensitive.",
"We assume that such events are very efficiently rejected by the other first-layer discriminators described above, and therefore our cut on $\\Delta \\eta _{\\gamma \\gamma }$ is still a good proxy for the cuts on those variables.",
"Secondly, photons may be closely spaced in $\\eta $ , but more broadly spaced in $\\phi $ .",
"Since the segmentation in $\\phi $ is much coarser than in $\\eta $ , such events must be very separated in $\\phi $ in order for cuts on the variable $R_{\\phi }$ (which measures the energy distribution across several second-layer $\\phi $ cells) to reject them.",
"A cut on $\\Delta \\phi _{\\gamma \\gamma }$ can simulate the cut on $R_{\\phi }$ , but since such events are geometrically rare, the net efficiency is quite insensitive to the precise value of this cut.",
"For definiteness we set the cut on $\\Delta \\phi _{\\gamma \\gamma }$ to be equal to the smallest second-layer cell size in $\\phi $ as a function of $\\eta $ ($\\Delta \\phi _{\\rm {cell}}=0.025$ for $\\eta <1.4$ ): <cell.",
"So far we have discussed only events which contain no converted photons.",
"Conversions happen with an $\\eta $ - and $E_T$ -dependent probability that ranges from about 10% at low $\\eta $ to more than 50% at larger $\\eta $ [28].",
"Since $h\\rightarrow aa$ events produce four photons in the final state instead of two, a larger fraction of our events will contain at least one converted photon than are contained in pure $h\\rightarrow \\gamma \\gamma $ events.",
"In our conversion events, one cluster in the calorimeter may contain one $e^+e^-$ pair and one $\\gamma $ , or two $e^+e^-$ pairs, so we might imagine that these events would be easy to distinguish from pure single-$\\gamma $ events and may even be vetoed.",
"In the case with $\\gamma e^+e^-$ , there is a mismatch between the momentum measured by the tracker (sensitive only to the charged particles) and the energy deposit in the calorimeter.",
"In the case with $2e^+e^-$ , two conversion vertices may be reconstructed in the tracker.",
"Both of these signatures have been considered for rejecting isolated $\\pi ^0$ backgrounds [30]; however, at present, the ATLAS photon identification analysis does not use an $E/p$ cut, and does not automatically veto events with multiple conversion vertices [28] (only the “best\" reconstructed conversion vertex is used, and is determined by the conversion radius and the number of tracks associated with the vertex.)",
"Furthermore, the cuts on the calorimeter variables are relaxed somewhat to accommodate the fact that the energy deposits for single-photon conversions tend to spread mildly in $\\eta $ and considerably in $\\phi $ (due to the magnetic field.)",
"Therefore, we will make the approximation that the value of $\\epsilon $ relevant for $4\\gamma $ events containing conversions is the same as the value of $\\epsilon $ for the unconverted sample; namely, it is determined by applying the collimation cuts in Eqs.",
"(REF ,REF ) to the sample of parent photons before they convert.",
"In summary, for all classes of photon events, we make the approximation coll, where $\\epsilon _{\\rm coll}$ is the rate at which the $4\\gamma $ events are expected to pass the $\\eta $ -dependent collimation cuts $\\Delta \\eta _{\\gamma \\gamma }<1/2\\times \\Delta \\eta _{\\rm {strip}}$ , $\\Delta \\phi _{\\gamma \\gamma }<\\Delta \\phi _{\\rm {cell}}$ .",
"The collimation of a photon pair produced in an $a$ decay is controlled by the ratio $2m_a/E_a$ , where $E_a$ is determined by the momentum of the Higgs boson.",
"We simulate 7 TeV $gg\\rightarrow h\\rightarrow aa\\rightarrow 4\\gamma $ events in Madgraph 5 [31], and on the subsample of events with all photons satisfying $|\\eta |\\in (0,1.37)\\cup (1.52,2.37)$ (the region in which ATLAS reconstructs photon candidates for the $h\\rightarrow \\gamma \\gamma $ analysis), and which pass the ATLAS photon $p_T$ cuts [5], we compute the efficiency $\\epsilon _{\\rm coll}$ .",
"In Fig.",
"REF we plot $\\epsilon _{\\rm coll}$ as a function of $m_a$ .Production mechanisms other than gluon fusion result in somewhat different kinematic distributions for $h$ .",
"However, gluon fusion dominates the production with O(10%) contribution from all other channels [5], so the effects on $\\epsilon _{\\rm coll}$ from these processes will be second-order.",
"We neglect them in this study.",
"In addition, we have checked that the effect of raising $\\sqrt{s}$ to 8 TeV is negligible.",
"Figure: Fraction of events where both photon pairs are sufficiently collimated to pass ATLAS isolation cuts, as a function of the pseudoscalar mass.To cross-check that our cuts effectively reproduce the rejection power of $w_{s3}$ and the other discrimination variables, we use the known isolated $\\pi ^0$ rejection power of ATLAS as a function of $(E_T,\\eta )$ (given in Fig.",
"11 of Sec.",
"5 of [27]) to estimate $\\epsilon $ from our simulation at the mass point $m_a=m_{\\pi ^0}$ .",
"We find good agreement between the $\\epsilon $ derived from this method and $\\epsilon _{\\rm coll}$ obtained from the collimation cuts.",
"Since the ATLAS $\\pi ^0$ rejection power was defined on events containing both converted and unconverted photons, we are reassured that Eq.",
"(REF ) is plausible.",
"Unlike the ATLAS analysis, the CMS analysis does place cuts on the ratio of the calorimeter energy measurement to the momentum measured in the tracker in order to isolate single photons from $\\pi ^0$ decays [32].",
"The cut may reduce $\\epsilon $ relative to $\\epsilon _{\\rm coll}$ .",
"However, the CMS ECAL has a barrel granularity $\\simeq 6\\times $ larger in $\\eta $ than that of ATLAS [33], increasing $\\epsilon _{\\rm coll}$ .",
"For this analysis we assume that the net efficiencies $\\epsilon $ will be comparable between the two detectors, but in principle any difference between them translates into an experiment-dependent and possibly conversion-dependent prediction for the $\\gamma \\gamma $ rate and could be used to probe the value of $m_a$ .",
"To summarize this section: (1) High collimation is sufficient for a significant number of $2\\gamma $ clusters to be misidentified as single isolated photons by the experimental analyses.",
"(2) In order to produce an interesting misidentification rate, $m_a\\lesssim m_{\\pi ^0}$ .",
"For larger $m_a$ there may still be misidentified events in the $h\\rightarrow \\gamma \\gamma $ signal, particularly if ${\\cal B}(h\\rightarrow aa)$ is not small, but the scenario should be more properly studied in the context of an $h\\rightarrow 4\\gamma $ signal.",
"(3) Contamination of $h\\rightarrow \\gamma \\gamma $ by misidentified $4\\gamma $ events will be visible as an excess of the total number of conversion events relative to the $\\gamma \\gamma $ expectation, as an excess of conversion events with multiple reconstructed vertices, and as an excess of conversion events with a mismatch between the track $p_T$ and the energy deposit in the ECAL."
],
[
"Current Constraints and Fits",
"In this study we are interested in the possibility that the current excesses in Higgs searches at ATLAS, CMS, and the Tevatron are due to a Higgs boson with mass near 125 GeV.",
"Under this hypothesis, some parts of the $(m_a,{\\cal B}(h\\rightarrow aa))$ parameter space are disfavored: they produce too few vector boson or fermion events, or too many or too few diphoton-like events.",
"We estimate the constraints with a matched-filter, taking each point in parameter space as a template and estimating a best-fit amplitude $\\hat{R}$ for the signal strength at the point.For other recent studies of the Higgs best-fit cross section data using similar $\\chi ^2$ analyses, see [34], , , .",
"In the Gaussian limit $\\hat{R}$ can be estimated as R=2 ti C-1ij dj , where $d_j$ is the “data,\" which we take to be the best-fit amplitudes relative to the SM for the channels $h\\rightarrow \\gamma \\gamma $ [5], $h\\rightarrow ZZ\\rightarrow 4l$ [38], $h\\rightarrow WW\\rightarrow l\\nu l\\nu $ [39], $(W/Z)h\\rightarrow (ll,l\\nu ,\\nu \\nu )b\\bar{b}$ [40], and $h\\rightarrow \\tau \\tau $ [41], presented by ATLAS at $m_h=126$ GeV [42] , the channels $gg\\rightarrow h\\rightarrow \\gamma \\gamma $ [6], [43], $qqh\\rightarrow qq\\gamma \\gamma $ [6], [43], $h\\rightarrow ZZ\\rightarrow 4l$ [44], $h\\rightarrow WW\\rightarrow l\\nu l\\nu $ [45], $(W/Z)h\\rightarrow (ll,l\\nu ,\\nu \\nu )b\\bar{b}$ [46], and $h\\rightarrow \\tau \\tau $ [47], presented by CMS at $m_h=124$ GeV [48], and the channels $h\\rightarrow b\\bar{b}$ and $h\\rightarrow WW$ presented by CDF and DZero at $m_h=125$ GeV [49].",
"For ATLAS and CMS we chose slightly different mass points based on where their respective $\\gamma \\gamma $ excesses are most significant; we assume that the current experimental resolution is large enough that both excesses can come from the same Higgs-like particle.",
"The vector $t_i$ gives the theoretical prediction for each data point (either $R_{\\gamma \\gamma }$ or $R_{XX}$ ), and $C^{-1}_{ij}$ is the inverse covariance matrix set by the squared symmetrized error bars (we include also correlations which we estimate from the luminosity uncertainty and the theoretical uncertainty on the gluon fusion cross section).",
"The error on the estimate $\\hat{R}$ is given by ( ti C-1ij tj )-1/2.",
"Figure: Regions of light pseudoscalar parameter space that are favored (blue/light) and disfavored (red/dark) by the current best-fit signal strengths in the h→γγ,ZZ,WW,bb,ττh\\rightarrow \\gamma \\gamma ,ZZ,WW,bb,\\tau \\tau channels.",
"Contours are overlaid for the net diphoton (solid green) and ZZ,WW,bb,ττZZ,WW,bb,\\tau \\tau rates (dashed yellow) expected at the LHC relative to the SM rates.Model points where $R=1$ is outside the 90% CL band around $\\hat{R}$ are then disfavored.",
"Of course, this statistical procedure is quite approximate at this stage, and serves only to get an estimated exclusion region if the data sets are large and $m_h$ actually is at 125 GeV.",
"A precise calculation would construct the full Poisson likelihood with all correlations.",
"With these caveats in mind, in Fig.",
"REF we present the regions of $(m_a,{\\cal B}(h\\rightarrow aa))$ parameter space disfavored by the current LHC and Tevatron results, including contours of $R_{\\gamma \\gamma }$ and $R_{XX}$ .",
"The contours make clear the fact that the disfavored regions are mainly controlled by the excesses in the diphoton channels: increasing it above a few times the SM rate is in tension with the current excesses.",
"We also maximize the likelihood over the plane, fixing the signal strength parameter to $R=1$ and constraining $m_a<m_\\pi $ .",
"The best-fit point lies near the $R_{\\gamma \\gamma }=2$ contour at the $m_\\pi $ boundary; relaxing the constraint, it would move beyond to near the intersection of the $R_{\\gamma \\gamma }=2$ and $R_{XX}=0.5$ contours.",
"However, the data are over-fit, $\\chi ^2/{\\rm d.o.f.",
"}=7.3/10$ , and the statistic is very shallow, particularly in the direction of constant $R_{\\gamma \\gamma }$ .",
"Therefore, the precise location of the best-fit point has little meaning, and we do not show it in Fig.",
"REF .",
"For comparison, we find that the SM $\\chi ^2/{\\rm d.o.f.",
"}=12.2/12$ ."
],
[
"Direct Constraints on $a$",
"We now discuss existing experimental constraints on the model in Eq. ().",
"Similar considerations have been undertaken with light pseudoscalars in [50], , [26].",
"Constraints from $e^+e^-\\rightarrow \\gamma a$ , quarkonia decays, and beam dump experiments are robust in the sense that they are controlled largely or entirely by the coupling of $a$ to photons.",
"Constraints from the muon anomalous magnetic moment and meson decays involving flavor-changing neutral currents are sensitive not only to the $aF\\tilde{F}$ coupling, but also any small coupling of $a$ to SM fermions that might be present in the Lagrangian.",
"These contributions may interfere constructively or destructively with the $aF\\tilde{F}$ terms– we can even envision situations where some constraints are mitigated by finely tuned tree level couplings of $a$ to SM fermions.",
"We assume for the purpose of this section that the fermionic couplings at the scale $M$ are small enough that the leading contributions to the constraints come from photon loops.",
"Since $aF\\tilde{F}$ is dimension-5, the amplitudes are typically logarithmically divergent.",
"To remove the divergence, we cut the integrals off at momenta $\\sim M$ , which is motivated by the view that the $aF\\tilde{F}$ interaction is an effective coupling resulting from some new physics at a scale $M$ —whether $a$ is a composite with constituents with masses of that order or is fundamental and coupled to photons via particles with such masses.",
"The phenomenological need for $M$ to be tens of GeV poses some model-building puzzles in this respect and we will sketch a few potential scenarios leading to such a scale in Sec. .",
"Taken together, these low energy constraints should be viewed mainly illustratively.",
"We attempt to be as conservative as possible, erring on the side of presenting stronger limits on the parameter space of the model and comment that these limits can likely be circumvented with relatively straightforward extensions of the model under consideration."
],
[
"Muon Anomalous Magnetic Moment",
"A light pseudoscalar that couples to two photons will contribute to the anomalous magnetic moment of the muon.",
"The leading correction, shown in Fig.",
"REF , occurs at order $\\alpha ^3$ and is analogous to the $\\pi ^0$ -pole portion of the hadronic light-by-light contribution to $(g-2)_\\mu $ .",
"Figure: A representative diagram of the leading contribution of the pseudoscalar, aa, to (g-2) μ (g-2)_\\mu .To estimate this effect as a function of $m_a$ and $M$ , we make use of the expression for the $\\pi ^0$ -pole contribution to $(g-2)_\\mu $ in [52], rescale it by the strength of the photon coupling relative to that of $\\pi ^0$ , $\\left(4\\pi ^2 F_\\pi /M\\right)^2$ , replace the pion mass with $m_a$ , and cut off the logarithmic divergence at the scale $M$ instead of at $\\sim 4\\pi ^2 F_\\pi $ .",
"To set limits we require that this contribution is less than the current deviation between experiment and theory on $\\Delta a_\\mu =(g-2)_\\mu /2$ of about $25\\times 10^{-10}$ [53], .",
"This sets a lower limit on $M$ around $4\\pi ^2 F_\\pi =3.64{~\\rm GeV}$ that is not very sensitive to $m_a$ for the range of pseudoscalar masses we consider.",
"Because of the lepton mass dependence of corrections to $g-2$ , the electron anomalous magnetic moment is less constraining and does not appear in our limits."
],
[
"$e^+e^-\\rightarrow \\gamma a$ Limits",
"The reaction $e^+e^-\\rightarrow \\gamma a$ can proceed through the $s$ -channel exchange of a virtual photon.",
"The cross section for this process is independent of the center-of-mass energy, dd=2233M2 3(1+2)8, where $\\theta $ is the angle between the photon and the beam axis in the center-of-mass.",
"If $a$ decays without being detected, then this process is subject to limits from $e^+e^-\\rightarrow \\gamma +{\\rm inv}$ .",
"We apply limits from the DELPHI collaboration [55], using $650~{\\rm pb}^{-1}$ of data at center-of-mass energies $\\sqrt{s}=180{~\\rm GeV}$ –$209{~\\rm GeV}$ .We note here that limit on $e^+e^-\\rightarrow \\gamma +{\\rm inv.",
"}$ from the ASP experiment [57] that has been used in previous studies to constrain light pseudoscalars coupled to photons do not actually apply.",
"This is because their analysis imposed a cut on the photon's energy of $E_\\gamma <\\sqrt{s}/2-4.5{~\\rm GeV}=10{~\\rm GeV}$ to eliminate backgrounds from $e^+e^-\\rightarrow \\gamma \\gamma $ , rendering it insensitive to invisibly decaying particles produced in association with a photon with a mass less than $16{~\\rm GeV}$ .",
"In the photon energy range of interest for $m_a\\ll \\sqrt{s}$ , $E_\\gamma \\simeq \\sqrt{s}/2$ , we require that there are fewer than 20 events and assume that the product of angular acceptance and efficiency is $\\sim 0.5$ .",
"Assuming that the $a$ decay length must be larger than 2 m to be unseen at DELPHI and a boost of $\\gamma \\sim 90{~\\rm GeV}/m_a$ excludes the following region of parameter space: 110 GeV(ma40 MeV)2$\\sim $ $<$ M$\\sim $ $<$ 125 GeV."
],
[
"Meson Decays",
"The interactions in Eq.",
"() lead to effective couplings of $a$ to SM fermions at the one-loop level.",
"This can have effects on rare meson decays involving photons or missing energy if $a$ decays too late to be detected.",
"Effective couplings to up-type quarks can then lead to flavor-changing transitions in the down sector such as $s\\rightarrow d+a$ as in Fig.",
"REF .",
"Figure: Diagram that gives the leading contribution to s→d+as\\rightarrow d+a from an effective interaction between aa and the top quark.This leads to the flavor-changing decay $K^\\pm \\rightarrow \\pi ^\\pm a$ which must be confronted with experimental data on $K^\\pm \\rightarrow \\pi ^\\pm \\gamma \\gamma $ .",
"The amplitude for $s\\rightarrow d+a$ can be very roughly estimated as M(sd+a)2M2(M2mt2)(GF mt2422) VtdVts (msdL sR-mddR sL), where we have again cut off the logarithmic divergence at the scale $M$ .",
"Using this we arrive at a rate for $K^\\pm \\rightarrow \\pi ^\\pm a$ of (Ka)4ms2M24(M2mt2)(GF2 mt42105) pmK2|VtdVts|2(mK2-m2ms-md)2|f0K(0)|2, where the form factor $f_0^{K\\pi }\\left(0\\right)\\simeq 1$ [58] and $p_\\pi $ is the pion momentum in the kaon rest frame.",
"We apply the limits on hypothetical particles with masses less than $100{~\\rm MeV}$ decaying to two photons from [59] (generally at the $10^{-7}$ level) to ${\\cal B}\\left(K^+\\rightarrow \\pi ^+a\\right)$ , resulting in $M\\mathrel {\\unknown.",
"{\\hspace{1.0pt}\\sim }}$ >$$ 1$--$ 2 GeV$.",
"In addition, if $ a$ is sufficiently long-lived so that its decay is not detected, it will give a contribution to the process $ K+++inv$.",
"The current experimental upper limit on the branching ratio for $ K+$ to decay to $ +$ and a light invisible particle is $ 7.310-11$~\\cite {Anisimovsky:2004hr}.",
"For this limit to apply, $ a$ must have a decay length larger than about 1~m when produced with a $ +$ in the decay of a $ K+$ at rest.",
"The {\\em excluded} region is then\\begin{align}1500{~\\rm GeV}\\left(\\frac{m_a}{40{~\\rm MeV}}\\right)^2\\mathrel {\\unknown.",
"{\\hspace{1.0pt}\\sim }}<\\end{align}$ M$\\sim $ $<$ 17 GeV.",
"Analogous limits from $B^\\pm \\rightarrow K^\\pm $ transitions are less stringent in the pseudoscalar mass range of interest.",
"$^3{\\rm S}_1$ quarkonium states can decay to $\\gamma a$ through an $s$ -channel virtual photon, just as in $e^+e^-$ annihilation.",
"In the $\\bar{c}c$ and $\\bar{b}b$ systems, assuming $m_a\\ll m_{c,b}$ , the branching ratios are B(J/a)2mc2M2B(J/e+e-) 4.610-6(10 GeVM)2, B((1S)a)2mb2M2B((1S)e+e-) 2.310-5(10 GeVM)2.",
"We require that the contribution to $J/\\psi \\rightarrow 3\\gamma $ from $J/\\psi \\rightarrow \\gamma a$ , $a\\rightarrow \\gamma \\gamma $ is less than the experimental result ${\\cal B}\\left(J/\\psi \\rightarrow 3\\gamma \\right)=\\left(1.2\\pm 0.4\\right)\\times 10^{-5}$ [61].",
"This translates into a limit $M\\mathrel {\\unknown.",
"{\\hspace{1.0pt}\\sim }}$ >$$ 6.2 GeV$.",
"Current limits on $ B((1S)a)$ at the $ 10-6$ to $ 10-5$ level assume $ a$ decays to leptons or hadrons~\\cite {Love:2008aa,*McKeen:2008gd} and therefore are not constraining on a pseudoscalar that decays promptly to $$.",
"In addition, if $ a$ is long-lived enough to decay after passing through the detector these decays are also subject to limits on $ J/, (1S)+inv$ as in the $ e+e-+inv.$ case in Sec.~\\ref {sec:ee}.",
"The current experimental limits are $ B(J/a+inv.",
")<4.610-6$ for $ ma<150 MeV$~\\cite {Insler:2010jw} and $ B((1S)a+inv.",
")<1.410-5$ for $ ma<5 GeV$~\\cite {Balest:1994ch}.",
"To estimate the region of parameter space that these limits are sensitive to, we assume that the $ a$ decay length needs to be greater than 1~m to go undetected in these searches.",
"This {\\em excludes} the regions\\begin{align}&590{~\\rm GeV}\\left(\\frac{m_a}{40{~\\rm MeV}}\\right)^2\\mathrel {\\unknown.",
"{\\hspace{1.0pt}\\sim }}<\\end{align}$ M$\\sim $ $<$ 10 GeV, from $J/\\psi \\rightarrow \\gamma +{\\rm inv.",
"}$ and 330 GeV(ma40 MeV)2$\\sim $ $<$ M$\\sim $ $<$ 13 GeV, from $\\Upsilon (1{\\rm S})\\rightarrow \\gamma +{\\rm inv}$ .",
"We note that the regions of $\\left(m_a,M\\right)$ ruled out by $K^+\\rightarrow \\pi ^++{\\rm inv}$ and $J/\\psi ,\\, \\Upsilon (1{\\rm S})\\rightarrow \\gamma +{\\rm inv.",
"}$ are all contained within the region excluded by $e^+e^-\\rightarrow \\gamma +{\\rm inv.",
"}$ in Sec.",
"REF ."
],
[
"Beam Dump Experiments",
"Beam dump experiments, where weakly coupled particles are searched for in the collision of proton or electron beams with fixed targets, also provide limits on light pseudoscalars (often called axion-like particles in this context).",
"To be successfully probed, the particles under consideration need to decay visibly and live long enough that they escape the target, but not so long that they do not decay before passing the detectors downstream, typically tens of meters away from the target.",
"This results in mass-dependent exclusion bands in the pseudoscalar's lifetime.",
"Light pseudoscalars are produced in beam dump experiments either through the Primakoff process via their coupling to two photons or by being radiated via direct couplings to SM fermions as the beam constituents are stopped, in a manner analogous to bremsstrahlung.",
"Each mode typically provides an ${\\cal O}(1)$ fraction of the total production cross section.",
"The final states that are considered in searches for light pseudoscalars include $\\gamma \\gamma $ , $e^+e^-$ , and $\\mu ^+\\mu ^-$ .",
"In the model we consider, the pseudoscalars are predominantly produced by the Primakoff process since they do not directly couple to SM fermions and the dominant decay mode is $\\gamma \\gamma $ .",
"In placing limits on our parameter space, however, we simply apply the experiments' reported exclusions [66], , , to generic axion-like particles, ignoring subtleties in the slightly different production cross sections and different branching fractions.",
"This approximation results in conservative exclusion regions since reducing the direct couplings to SM fermions should not make for more stringent limits.",
"In the region of parameter space of interest we find upper limits on the pseudoscalar lifetime, corresponding to prompt decays inside the targets, or, equivalently, upper limits on the inverse coupling $M$ in Eq. ().",
"The values of these upper limits tend to roughly coincide with those coming from the requirement that the pseudoscalar also decays promptly at the LHC."
],
[
"Further Limits",
"The coupling of $a$ to SM fermions becomes negligible in the non-relativistic limit.",
"Therefore, the strict constraints light scalars and vectors are subject to from diffractive low energy neutron scattering on nuclei do not apply in this case.",
"For the same reason, these particles are also not constrained by measurements of D–P transitions in muonic Si and Mg. Additionally, light pseudoscalars can also be produced in the decays of excited nuclear states in nuclear reactors.",
"However, this production mode is kinematically limited to $m_a\\mathrel {\\unknown.",
"{\\hspace{1.0pt}\\sim }}$ <$$ 10 MeV$, and does not extend the exclusions we have considered.$"
],
[
"Allowed Regions",
"In Fig.",
"REF , we show regions of $\\left(M,m_a\\right)$ excluded by the constraints outlined in Secs.",
"REF –REF .",
"We also display contours of $1~{\\rm cm}$ and $50~{\\rm cm}$ decay lengths, assuming $\\gamma =m_h/2m_a$ with $m_h=125{~\\rm GeV}$ .",
"Figure: Regions of pseudoscalar mass and inverse aγγa\\gamma \\gamma coupling MM excluded by the constraints in Secs. –.",
"The dashed curves are contours of 1 cm 1~{\\rm cm} and 50 cm 50~{\\rm cm} decay lengths for the pseudoscalar, assuming a boost γ=m h /2m a \\gamma =m_h/2m_a with m h =125 GeV m_h=125{~\\rm GeV}."
],
[
"Model Building",
"Having investigated the phenomenology associated with a light pseudoscalar coupled to the Higgs, we speculate on a few scenarios where the $aF\\tilde{F}$ interaction in Eq.",
"() could result.",
"As we have seen, given a 125 GeV Higgs, $m_a$ needs to be on the order of tens of MeV for the photons from its decay to be collimated enough to plausibly fake a single photon at the LHC.",
"Using this fact in Eqs.",
"() and (), we observe that $M$ cannot be at the TeV scale for $a$ to decay promptly as well to evade constraints from beam dump experiments.",
"Obtaining a relatively low scale for a dimension-5 operator such as this involving interactions beyond the SM poses a model-building challenge.",
"A simple scenario where an interaction $aF\\tilde{F}$ is generated is if $a$ interacts with particles of mass $m$ that have electric charge $q$ in units of the electron's charge.",
"One can imagine that $a$ is either a composite made up of these charged particles or is fundamental and has a renormalizable coupling to them.",
"After integrating out these charged particles, the dimension-5 interaction between $a$ and two photons is generated with a scale $M\\sim 4\\pi ^2 m/q^2$ .",
"Given the requirement that $a$ decay promptly seems to require $a$ to couple to charged particles with a mass not much larger than tens of GeV.We note that a scenario where $a\\rightarrow \\gamma \\gamma $ is mediated by a coupling of $a$ to a new, light vector boson, $V$ , that kinetically mixes with the photon (thereby lifting the requirement that $a$ couple to electrically charged particles) is not viable because the amplitude for such a process is proportional to $q_1^2 q_2^2\\left(q_1^2-m_V^2\\right)^{-1}\\left(q_2^2-m_V^2\\right)^{-1}$ where $q_{1,2}$ are the photons' momenta, which vanishes for on-shell photons.",
"If $m_V$ vanishes, then $a\\rightarrow VV$ decays will strongly dominate the branching ratios.",
"Candidates for such charged particles are likely limited to SM fermions other than the $t$ quark.",
"Significant direct couplings to electrons or muons can be ruled out by anomalous magnetic moment measurements.",
"Couplings to heavy quarks $c$ and $b$ likely cause issues with well-studied quarkonium transitions.",
"We are left with $\\tau $ and light quarks as SM mediators of the $aF\\tilde{F}$ interaction.",
"For a recent analysis of new light bosons decaying to $\\gamma \\gamma $ see Ref.",
"[70]."
],
[
"Coupling to $\\tau $",
"We can also imagine that the effective $aF\\tilde{F}$ interaction is due to a coupling of $a$ to $\\tau $ leptons, writing this coupling asWe write the interaction in terms of a pseudoscalar instead of an axial vector coupling here for simplicity.",
"La=-iga5.",
"This leads to an effective operator mediating $a\\rightarrow \\gamma \\gamma $ in Eq.",
"() with (assuming $m_a\\ll m_\\tau $ ) M=42mg.",
"The experimental information about the anomalous magnetic moment of $\\tau $ [71] -0.052<(g-2)2<0.013, limits $g_\\tau \\mathrel {\\unknown.",
"{\\hspace{1.0pt}\\sim }}$ <$$ 2.5$ for $ ma>10 MeV$.$ A similar limit of $g_\\tau \\mathrel {\\unknown.",
"{\\hspace{1.0pt}\\sim }}$ <$$ 2-3$ also comes from the OPAL Collaboration^{\\prime }s result~\\cite {Acton:1991dq}\\begin{align}{\\cal B}\\left(Z\\rightarrow \\tau ^+\\tau ^-\\gamma \\right)<7.3\\times 10^{-4},\\end{align}where $ Z+-a+-$ can mimic $ Z+-$ events~\\cite {McKeen:2011aa}.$ Without a dedicated search for these pseudoscalars in $\\tau $ decays, it is difficult to be certain about constraints on this scenario coming from $\\tau $ decay.",
"This is especially true if $a$ has a significant invisible width, a scenario which we discuss in the Appendix.",
"For example, the branching ratio of $\\tau $ to $a$ and $\\mu $ can be estimated to be [70] B(a )3.510-5g2.",
"Some of these decays could appear as $\\tau ^\\pm \\rightarrow \\mu ^\\pm \\bar{\\nu }\\nu \\gamma $ decays, given the decay $a\\rightarrow \\gamma \\gamma $ .",
"Even with the large Yukawa allowed by $\\left(g-2\\right)_\\tau $ and if all of the $\\tau ^\\pm \\rightarrow a \\mu ^\\pm \\bar{\\nu }\\nu $ were registered as $\\tau ^\\pm \\rightarrow \\mu ^\\pm \\bar{\\nu }\\nu \\gamma $ decays, the contribution is still below the uncertainty on the measurement of this branching of $\\left(3.61\\pm 0.38\\right)\\times 10^{-3}$ [73] which agrees well with expectations.",
"Limits from $\\tau ^\\pm \\rightarrow e^\\pm \\bar{\\nu }\\nu \\gamma $ are less stringent and hadronic $\\tau $ decays pose a bit more difficulty in prediction.",
"In this scenario, the scale $M$ is bounded by M=42mg$\\sim $ $>$ 35 GeV."
],
[
"Mixing with $\\pi ^0$",
"Coupling $a$ to light quarks so that $a$ mixes with $\\pi ^0$ could provide a solution.",
"Given the couplings (ignoring $s$ quarks for clarity) Laq=-igu au5u-igd ad5d, using the partially conserved axial current, one can estimate the mixing angle, (gu-gd)|q q|F(m02-ma2)(gu-gd)(400 MeV)2(m02-ma2), where $\\langle \\bar{q} q\\rangle $ is the value of the light quark condensate and $F_\\pi =92.2{~\\rm MeV}$ is the pion decay constant.",
"Here, the scale $M$ is given by its analogue in the $\\pi ^0$ case scaled by the mixing angle, M=42F.",
"If there are no symmetry conditions on $g_u$ and $g_d$ , then we will in general expect additional isospin violation due to interactions involving $a$ .",
"The relative size of these effects is of the order $g_{u,d}^2/\\left(m_{u,d}^2/F_\\pi ^2\\right)$ .",
"If we demand that these not exceed $\\sim 1\\%$ then we require $g_{u,d}\\mathrel {\\unknown.",
"{\\hspace{1.0pt}\\sim }}$ <$$ 10-3$.",
"Furthermore, one must then contend with corrections to pion beta decay $ 0e$ which is shifted by\\begin{align}\\Gamma \\left(\\pi ^\\pm \\rightarrow \\pi ^0e^\\pm \\nu \\right)=\\cos ^2\\theta ~\\Gamma _{\\rm SM}\\left(\\pi ^\\pm \\rightarrow \\pi ^0e^\\pm \\nu \\right).\\end{align}The $ 0.6%$ agreement of theory and experiment for this rate~\\cite {Pocanic:2003pf} implies that $$\\sim $ $<$ 0.08$.",
"Such a bound is compatible with the limits above on the Yukawa couplings from isospin violation.$ The current measurement of the $\\pi ^0$ lifetime is at the 3% level [75], and therefore does not pose any stricter constraints on the strength of this mixing.",
"Furthermore, given a mixing angle of this size, the contribution to $\\pi ^0\\rightarrow e^+e^-$ from $a$ –$\\pi ^0$ mixing is negligible.",
"There are potentially strict limits in this scenario coming from the $K^+\\rightarrow \\pi ^+\\gamma \\gamma $ limits described in Sec.",
"REF since the rate for $K^+\\rightarrow \\pi ^++a$ is enhanced.",
"The rate can be estimated as B(K++a)2 B(K++0) =20.21.",
"The limits on this branching at the $10^{-7}$ level [59] imply that $\\theta \\mathrel {\\unknown.",
"{\\hspace{1.0pt}\\sim }}$ <$$ 10-3$, or equivalently, $ M$\\sim $ $>$ 1 TeV$.",
"This does not allow for prompt decays at the LHC if $ a$ is produced in the decay of a $ 125 GeV$ Higgs and is problematic with respect to beam dump constraints.",
"However, the analysis in Ref.~\\cite {Kitching:1997zj} only selected events corresponding to $ ma<100 MeV$ because of the very large background from $ K++0$.",
"Thus, to avoid the strict limit on $ M$ that results from this analysis, $ a$ must have a mass greater than $ 100 MeV$ if its coupling to photons is mediated by a mixing with $ 0$.$ The limit from pion beta decay implies that $M\\mathrel {\\unknown.",
"{\\hspace{1.0pt}\\sim }}$ >$$ 47 GeV$ in this case.$"
],
[
"Other Scenarios",
"In addition to the two scenarios described above we briefly mention a few other potential models that could lead to the interactions in Eq.",
"() with a scale $M$ on the order of tens to a few hundred GeV.",
"As in [19], the pseudoscalar could be coupled to heavy vector-like matter that mediates its interaction with photons.",
"However, ensuring that the scale $M$ is small enough in this situation so that $a$ decays promptly is difficult.",
"This could be solved by giving the heavy matter a large electric charge or gauging it under a non-Abelian group with a large number of colors.",
"We also speculate that sterile neutrinos with very large transition magnetic moments coupled to $a$ could also offer a solution since their masses need not be at the weak scale."
],
[
"Outlook",
"As discussed in Sec.",
"REF , the scenario analyzed in this paper can be tested in $h\\rightarrow \\gamma \\gamma $ events, since the expected rate depends sensitively on the photon identification criteria used in the analysis.",
"Furthermore, the presence of more photons implies that the fraction of events with a photon conversion is higher than in a pure $\\gamma \\gamma $ sample.",
"Although CMS has already presented the best-fit rates in conversion and unconverted events separately, the statistics remain low, and we leave such analysis for the future.",
"We also note that in the case where the $a$ is long-lived, $\\gamma c\\tau \\sim 50~{\\rm cm}$ , it is imaginable that the decay length may be determined from the distribution of conversion radii, even though since the states are highly boosted they do not display displaced vertices in the traditional sense.",
"The light pseudoscalar hypothesis can also be tested by looking for a $3\\gamma $ signal with invariant mass matching that of the Higgs boson.",
"If the Higgs is produced with a sizable boost, some decays will yield one slow $a$ and one fast $a$ in the lab frame, and the former will produce widely separated photons.",
"Alternatively, if $m_a$ is much larger than the values studied here, the pseudoscalar can still contribute some events to $h\\rightarrow 2\\gamma $ if ${\\cal B}(h\\rightarrow aa)$ is large enough, but then the particle may be more easily seen in searches for $h\\rightarrow 4\\gamma $ [24].",
"A different way to search for a light pseudoscalar coupling to photons is to produce the $a$ directly in Primakoff-type experiments, $\\gamma +{\\rm Nuc.",
"}\\rightarrow a+{\\rm Nuc.",
"}$ , which would allow the mass of $a$ to be directly reconstructed.",
"A proposal for an upgrade to the PrimEx Experiment at Jefferson Lab to measure $\\Gamma \\left(\\eta \\rightarrow \\gamma \\gamma \\right)$ envisions collisions of photons with an energy $E_\\gamma \\simeq 11{~\\rm GeV}$ on a liquid hydrogen target with a luminosity of ${\\cal L}\\sim 10^{-2}~{\\rm nb}^{-1}{\\rm s}^{-1}$ with a run of 45 days [76].",
"Using the interaction in Eq.",
"(), the cross section for the Primakoff production of $a$ in the collision of a photon with a proton can be found to be ddQ2223M2|Fem(Q2)|2Q2, where $Q$ is the momentum transferred to the proton and $F_{\\rm em}$ is its electromagnetic form factor.",
"Using a simple dipole form factor and a proton charge radius of $0.87~{\\rm fm}$ (the results do not depend sensitively on the form factor and charge radius—using $0.84~{\\rm fm}$ as measured in muonic hydrogen does not change the estimate) and a detection efficiency of 60%, this cross section would lead to N(a)104(10 GeVM)2 pseudoscalars collected with $m_a=40{~\\rm MeV}$ .",
"For comparison, there would be about $10^4$ $\\eta $ 's produced via the Primakoff process and subsequently decaying to two photons under the same conditions.",
"It appears likely, at least statistically, that pseudoscalars with parameters in the range that is interesting in the context of the $h\\rightarrow \\gamma \\gamma $ signal can be probed at future Primakoff experiments."
],
[
"Conclusions",
"In this paper we have investigated a simple model that can give rise to an apparent excess relative to the SM in the $h\\rightarrow \\gamma \\gamma $ channel.",
"In our study the two photon excess comes from Higgs decays to two light, boosted pseudoscalars, $h\\rightarrow aa$ , and each pseudoscalar decays into two photons.",
"For very light pseudoscalars, each pair of photons is highly collimated and a non-negligible fraction of the pairs can appear as single photons, even at high-resolution detectors like those at the LHC.",
"This scenario serves as an example where photon jets [23] are produced and helps to motivate the future experimental study of these objects in more detail.",
"We have estimated the fraction of $4\\gamma $ events that appear as $2\\gamma $ , taking into account the fine granularity of the ATLAS detector and the tight photon selection criteria.",
"Additionally, we have investigated subtleties that may arise when one photon or more converts.",
"We have assumed that the different analyses used at CMS and ATLAS do not give rise to large deviations between the $h\\rightarrow \\gamma \\gamma $ measurements, but we note that the amount of $4\\gamma $ contamination can be analysis-dependent and thus the differing analyses may be used as a test of the hypothesis.",
"In addition to a diphoton excess, the model also predicts a deficit relative to the SM for Higgs decays into other final states.",
"Although the uncertainties are large, the data that have been collected so far at the LHC and Tevatron are consistent with both of these predictions, and with more data the model will be easily tested.",
"The relative size of a potential excess in the $\\gamma \\gamma $ channel and a decrement in the remaining channels will be a direct probe of the mass of the pseudoscalar, with a lighter pseudoscalar better able to accommodate a larger signal in the non-$\\gamma \\gamma $ channels for a fixed $\\gamma \\gamma $ rate.",
"Lower bounds around $m_a\\simeq 10{~\\rm MeV}$ from $\\gamma +{\\rm inv.",
"}$ searches and beam dump experiments offer a complimentary sensitivity to the model.",
"Finally, the mass of the light pseudoscalar can be well-measured at Primakoff experiments.",
"As we discussed in Sec.",
", future Primakoff experiments appear well-poised to test models involving new light bosons coupled to photons like the one we have considered, underscoring the complementarity of experiments at the intensity frontier with high-energy studies of the Higgs boson.",
"The authors would like to thank J. Albert, V. Bansal, K. Jensen, M. Lefebvre, J. Mitrevski, J. Nielsen, M. Pospelov, J. Redondo, and A. Ritz for helpful discussions.",
"The work of DM was supported by NSERC, Canada.",
"PD is supported by the DOE under Grant No.",
"DE-FG02-04ER41286.",
"*"
],
[
"Invisible Decays",
"In this Appendix, we describe a simple extension of the model in Sec.",
"to allow for $a$ to decay invisibly.",
"The Higgs phenomenology and direct constraints change somewhat from the case where $a$ decays purely to $\\gamma \\gamma $ .",
"As before, we present fits to the Higgs data and show allowed regions of parameter space."
],
[
"Framework",
"In addition to the two photon decay channel, the pseudoscalar $a$ could have an appreciable invisible decay width.",
"As a concrete example, we can add an interaction with a stable (on collider scales) SM singlet Dirac fermion $\\chi $ , La=aM 5, where $M^\\prime $ is another scale describing the strength of this interaction.",
"The rate for $a\\rightarrow \\chi \\bar{\\chi }$ is then (a)=m2ma8M2 When this channel is open, the branching ratio for $a\\rightarrow \\gamma \\gamma $ becomes B(a)=[1+2(1mMma M)2]-1, where $\\beta _\\chi ^2=1-4m_\\chi ^2/m_a^2$ .",
"The decay length (for a boost $\\gamma =m_h/2m_a$ ) is shortened by a factor of this branching ratio, c=1.15 mm (M10 GeV)2(ma40 MeV)-4 (mh125 GeV)(B(a)0.1), and the scale $M$ can then be expressed as M=29 GeV (c1 cm)1/2(ma40 MeV)2 (mh125 GeV)-1/2(B(a)0.1)-1/2.",
"For a fixed decay length, a sizable invisible branching fraction for $a$ allows the scale $M$ to be somewhat larger than it would be in the case where $a$ only decays to photons."
],
[
"Higgs Signal",
"The primary changes to Higgs phenomenology are in ${\\cal B}\\left(h\\rightarrow \\gamma \\gamma \\right)_{\\rm eff}$ and $R_{\\gamma \\gamma }$ , now given by B(h)eff=B(h) +B(haa)B(a)2 and R=1+B(haa)(B(a)2BSM(h)-1).",
"Figure: Regions of light pseudoscalar parameter space that are favored (blue/light) and disfavored (red/dark) by the current best-fit signal strengths in the h→γγ,ZZ,WW,bb,ττh\\rightarrow \\gamma \\gamma ,ZZ,WW,bb,\\tau \\tau channels.",
"Contours are overlaid for the net diphoton (solid green) and ZZ,WW,bb,ττZZ,WW,bb,\\tau \\tau rates (dashed yellow) expected at the LHC relative to the SM rates.",
"ℬa→γγ{\\cal B}\\left(a\\rightarrow \\gamma \\gamma \\right) is fixed to 0.10.1, indicating a 90% branching of a→ invisible a\\rightarrow \\rm {invisible}.Fixing ${\\cal B}\\left(a\\rightarrow \\gamma \\gamma \\right)=0.1$ , we regenerate the exclusion contours and contours of fixed $R_{\\gamma \\gamma ,XX}$ in Fig.",
"REF .",
"The best-fit $\\chi ^2/{\\rm d.o.f.",
"}=6.5/10$ , and the best-fit point is close to the intersection of the $R_{\\gamma \\gamma }=2$ and $R_{XX}=0.5$ contours.",
"Like the ${\\cal B}\\left(a\\rightarrow \\gamma \\gamma \\right)=1$ case, the $\\chi ^2$ is shallow and the best-fit point is not yet a meaningful quantity, so we do not show it in Fig.",
"REF .",
"In addition to an increased apparent branching to $\\gamma \\gamma $ , there are contributions to the invisible width of the Higgs, which can be related to the increased diphoton branching ratio, B(hinvisible)=B(haa)B(a)2 =B(haa)[1-B(a)]2.",
"There is also a “monophoton\" branching, B(h+invisible)=B(haa)B(a) B(a)2' =2' B(haa)B(a)[1-B(a)], where $\\epsilon ^{\\prime }$ is the efficiency for the one photon jet in the event to be reconstructed as a single photon.",
"Future LHC monophoton searches with low MET cuts may be able to probe this channel."
],
[
"Direct Constraints on $a$",
"We now analyze changes to the direct limits given in Secs.",
"REF –REF that arise if $a$ has an appreciable invisible branching fraction.",
"Figure: Same as Fig.",
"in the case that ℬa→γγ=0.1{\\cal B}\\left(a\\rightarrow \\gamma \\gamma \\right)=0.1.",
"The dashed curves are contours of 10 and 50 cm decay lengths for γ=m h /2m a \\gamma =m_h/2m_a and m h =125 GeV m_h=125{~\\rm GeV}.",
"To the left of the vertical dotted line, R γγ >1R_{\\gamma \\gamma }>1 [Eqs.",
"() and ()] while to its right, R γγ <1R_{\\gamma \\gamma }<1.",
"In this case, only the beam dump and e + e - →γ+ inv .e^+e^-\\rightarrow \\gamma +{\\rm inv.}",
"limits, which are the most constraining, are shown.Signals involving missing energy now apply regardless of $a$ 's decay length.",
"The $e^+e^-\\rightarrow \\gamma +{\\rm inv.",
"}$ limit then becomes the strongest lower limit on $M$ .",
"The limits on $M$ from beam dumps must be rescaled by a factor of ${\\cal B}\\left(a\\rightarrow \\gamma \\gamma \\right)^{-1/2}$ since they are limits on the $a$ lifetime.",
"We conservatively ignore any shrinking of the exclusion region due to the smaller visible branching ratio.",
"An invisibly decaying $a$ could contribute to star cooling by providing an additional channel for energy to leave a star.",
"However, such limits are not important in the mass range that we consider here.",
"If $a$ mixes with the $\\pi ^0$ and can decay invisibly, there are stringent limits on the invisible decay rate of the $\\pi ^0$ , with the collider upper limit measured to be $2.7\\times 10^{-7}$ [77] and a limit from Big Bang Nucleosynthesis that is several orders of magnitude stronger [78].",
"These are quite constraining and make the scenario where the $aF\\tilde{F}$ coupling is generated by a mixing with $\\pi ^0$ described in Sec.",
"REF unlikely if $a$ has an appreciable invisible branching fraction.",
"A large invisible branching fraction for $a$ renders the model where the coupling to photons is mediated by a coupling to $\\tau $ more plausible.",
"This is because $\\tau $ decays necessarily involve at least one neutrino in the final state so observing missing energy in a $\\tau $ decay does not signal new physics or a rare SM decay.",
"We reproduce the low-energy exclusions of Fig.",
"REF in Fig.",
"REF for ${\\cal B}\\left(a\\rightarrow \\gamma \\gamma \\right)=0.1$ , showing only the most stringent limits from beam dumps and $e^+e^-\\rightarrow \\gamma +{\\rm inv}$ ."
]
] | 1204.1061 |
[
[
"Explicit reduction modulo p of certain 2-dimensional crystalline\n representations, II"
],
[
"Abstract We complete the calculations begun in [BG09], using the p-adic local Langlands correspondence for GL2(Q_p) to give a complete description of the reduction modulo p of the 2-dimensional crystalline representations of G_{Q_p} of slope less than 1, when p > 2."
],
[
"Introduction",
"This paper is a sequel to [6], and we refer the reader to the introduction to that paper for a detailed discussion of (and the motivation for) the problem solved in this paper.",
"Another good reference is §5.2 of [4].",
"Let $p$ be a prime, choose an algebraic closure $\\overline{{\\mathbb {Q}}}_p$ of ${\\mathbb {Q}}_p$ , let $\\overline{{\\mathbb {Z}}}_p$ be the integers in $\\overline{{\\mathbb {Q}}}_p$ and let $\\overline{{\\mathbb {F}}}_p$ be the residue field of $\\overline{{\\mathbb {Z}}}_p$ .",
"We let $v$ be the $p$ -adic valuation on $\\overline{{\\mathbb {Q}}}_p^\\times $ , normalised so that $v(p)=1$ .",
"We set $v(0)=+\\infty $ .",
"We decree that the cyclotomic character has Hodge–Tate weight $+1$ .",
"We recall that given a positive integer $k\\ge 2$ and an element $a\\in \\overline{{\\mathbb {Q}}}_p$ with $v(a)>0$ there is a uniquely determined two-dimensional crystalline representation $V_{k,a}$ of ${\\operatorname{Gal}\\,}(\\overline{{\\mathbb {Q}}}_p/{\\mathbb {Q}}_p)$ with Hodge–Tate weights 0 and $k-1$ , determinant the cyclotomic character to the power of $k-1$ , and with the characteristic polynomial of crystalline Frobenius on the contravariant Dieudonne module being $X^2-aX+p^{k-1}$ (see for example §3.1 of [8] for a detailed construction of this representation).",
"Let $\\overline{V}_{k,a}$ denote the semisimplification of the reduction of $V_{k,a}$ modulo the maximal ideal of $\\overline{{\\mathbb {Z}}}_p$ .",
"Let $\\omega $ denote the mod $p$ cyclotomic character, and if $p+1\\nmid n$ let ${\\operatorname{ind}}(\\omega _2^n)$ denote the unique irreducible 2-dimensional representation of $G_{{\\mathbb {Q}}_p}$ with determinant $\\omega ^n$ and with restriction to inertia equal to $\\omega _2^n\\oplus \\omega _2^{pn}$ , with $\\omega _2$ the “niveau 2” character of inertia (see for example §1.1 of [4]).",
"Our main result is the following, which is an immediate consequence of Theorem 1.6 of [6] (the case $k\\lnot \\equiv 3$ mod $p-1$ ), Theorem 3.2.1 of [2] (the cases $k=3$ and $k=p+2$ ), and Corollary REF below.",
"Recall $k\\ge 2$ ; let $[k-2]$ denote denote the integer in the set $\\lbrace 0,1,\\ldots ,p-2\\rbrace $ congruent to $k-2$ mod $p-1$ , and set $t=[k-2]+1$ , so $1\\le t\\le p-1$ .",
"Theorem A Assume that $p>2$ and that $0<v(a)<1$ .",
"Then $\\overline{V}_{k,a}\\cong {\\operatorname{ind}}(\\omega _2^t)$ is irreducible, unless $k>3$ , $k\\equiv 3\\pmod {p-1}$ , and $v(k-3)+1+v(a)\\le v(a^2-(k-2)p)$ , in which case $\\omega ^{-1}\\otimes \\overline{V}_{k,a}$ is unramified, and the trace of a geometric Frobenius ${\\operatorname{Frob}}_p$ on $\\omega ^{-1}\\otimes \\overline{V}_{k,a}$ is $\\overline{\\tau }$ , where $\\tau =\\frac{(k-2)p-a^2}{ap(k-3)}$ .",
"Note that when $k\\equiv 3 \\pmod {p-1}$ and $v(k-3)+1+v(a)\\le v(a^2-(k-2)p)$ , the $\\tau $ in the theorem is in $\\overline{{\\mathbb {Z}}}_p$ , and its reduction is also the trace of an arithmetic Frobenius, because $\\omega ^{-1}\\otimes \\overline{V}_{k,a}$ has trivial determinant in this case.",
"For a fixed $k$ one can look at the behaviour of the representation $V_{k,a}$ as $a$ varies through the annulus $0<|a|<1$ , and we can give a more prosaic description of what our theorem says: it says that often $V_{k,a}={\\operatorname{ind}}(\\omega _2)$ is constant on this annulus, the only exception being when $k>3$ , $k\\equiv 3$ mod $p-1$ and $k\\lnot \\equiv 2$ mod $p$ , in which case the representation is ${\\operatorname{ind}}(\\omega _2)$ everywhere other than two small closed discs with centre $\\pm \\sqrt{(k-2)p}$ and radius $p^{-1-v(k-3)}$ .",
"Note that both these small discs are contained in the annulus $v(a)=1/2$ , and that as $k>3$ tends to 3 $p$ -adically the radius of the discs tends to zero.",
"Note also that the limit of $\\pm \\sqrt{(k-2)p}$ as $k>3$ tends to 3 $p$ -adically is $\\pm \\sqrt{p}$ , however $a=\\pm \\sqrt{p}$ is not in any of the discs; furthermore the intersection of all these discs as $k$ varies is empty, and in particular our result does not contradict the local constancy results of [5], contrary to one's initial reaction.",
"This theorem was proved in the case $k\\lnot \\equiv 3\\pmod {p-1}$ in [6] using the $p$ -adic local Langlands correspondence for $\\operatorname{GL}_2({\\mathbb {Q}}_p)$ .",
"In the present paper we build on the results and methods of [6] to handle the case $k\\equiv 3\\pmod {p-1}$ ; as one might expect from the statement of the theorem, the necessary calculations are more complicated in this case, because we have to control what is going on modulo an arbitrarily large power of $p$ in the auxiliary calculations.",
"We would like to thank Christophe Breuil for sharing with us the details of his unpublished calculations for $k=2p+1$ , which were the starting point for this article.",
"We would also like to thank Mathieu Vienney for pointing out a howler of a typo in the statement of the main theorem in an earlier version of this paper."
],
[
"Notation",
"Throughout the paper, $p$ denotes an odd prime, and $r$ and $n$ are integers.",
"If $\\lambda \\in {{\\mathbb {F}}_p}$ , we write $[\\lambda ]\\in {\\mathbb {Z}}_p$ for its Teichmueller lift."
],
[
"Combinatorial Lemmas",
"In this section we prove some elementary lemmas about congruences of binomial coefficients, that we will make repeated use of in the rest of the paper.",
"Lemma 2.1 Assume that $p>2$ and that $r\\in {\\mathbb {Z}}_{\\ge 2}$ , and write $t=v(r-1)\\ge 0$ .",
"Then for all integers $n\\ge 2$ we have $v\\left(\\binom{r}{n}\\right)+n\\ge t+2$ , and for all integers $n\\ge 1$ , $v\\left(\\binom{r-1}{n}\\right)+n\\ge t+1$ .",
"The left hand side is $v(r(r-1)(r-2)\\cdots (r-(n-1)))-v(n!",
")+n$ which is at least $v(r-1)-v(n!)+n=t+n-v(n!",
")$ .",
"If $n=2$ then we're OK as $p>2$ .",
"If $n\\ge 3$ then we need to check $n-v(n!",
")\\ge 2$ but this is clear because $v(n!",
")\\le n/(p-1)$ and hence $n-v(n!",
")\\ge n(p-2)/(p-1)\\ge 3(p-2)/(p-1)$ , and it is enough to prove that $\\lceil 3(p-2)/(p-1)\\rceil \\ge 2$ , which is true (by an explicit check for $p=3$ and true even without the $\\lceil \\cdot \\rceil $ for $p\\ge 5$ ).",
"The left hand side is $v((r-1)(r-2)\\cdots (r-n))-v(n!",
")+n\\ge v(r-1)-v(n!)+n=t+n-v(n!",
")$ .",
"Again, $v(n!",
")\\le n/(p-1)$ , and hence the left hand side is at least $t+n(p-2)/(p-1)$ .",
"Now the result is true for $n=1$ so it suffices to prove that $\\lceil 2(p-2)/(p-1)\\rceil \\ge 1$ , which follows as $p>2$ .",
"Lemma 2.2 Assume that $p>2$ and that $r>1$ , and write $t=v(r-1)\\ge 0$ .",
"Assume that $r\\equiv 1\\pmod {p-1}$ .",
"Then $r\\ge t+3$ .",
"$r-1\\ge (p-1)p^t\\ge 2\\cdot 3^t\\ge t+2$ by easy induction.",
"Lemma 2.3 Assume that $p>2$ and $r>1$ , and that $r\\equiv 1\\pmod {p-1}$ .",
"Write $t=v(r-1)\\ge 0$ .",
"If $\\mu \\in {{\\mathbb {F}}_p}$ , then $(-[\\mu ] x+py)^r-x^{r-1}(-[\\mu ] x+py)$ is congruent modulo $p^{t+2}\\overline{{\\mathbb {Z}}}_p[x,y]$ to $-px^{r-1}y$ if $\\mu =0$ , and to $(r-1)px^{r-1}y$ if $\\mu \\ne 0.$ If $\\mu =0$ then we just need to check that $r\\ge t+2$ , which follows from Lemma REF .",
"If however $\\mu \\ne 0$ then we expand via the binomial theorem and use part Lemma REF (1) (and the fact that $(p-1)|(r-1)$ , so $[\\mu ]^{r-1}=1$ ) to get that modulo $p^{t+2}$ we have $&\\phantom{\\equiv }(-[\\mu ] x+py)^r-x^{r-1}(-[\\mu ] x+py)\\\\&\\equiv -[\\mu ] x^r+rpx^{r-1}y+[\\mu ] x^r-x^{r-1}py\\\\&\\equiv (r-1)px^{r-1}y,$ as required.",
"Lemma 2.4 If $p>2$ and $r>1$ with $r\\equiv 1\\pmod {p-1}$ , and if $t=v(r-1)$ , then $\\sum _{\\mu \\in {{\\mathbb {F}}_p}}(1+[\\mu ])^r\\equiv rp\\pmod {p^{t+2}}$ , and $\\sum _{\\mu \\in {{\\mathbb {F}}_p}}(1+[\\mu ])^{r-1}\\equiv p-1\\pmod {p^{t+1}}$ .",
"(1) We rewrite $(1+[\\mu ])^r$ as $([1+\\mu ]+(1+[\\mu ]-[1+\\mu ]))^r$ and expand using the binomial theorem.",
"Since $(1+[\\mu ]-[1+\\mu ])$ is divisible by $p$ , by Lemma REF (1) we only need to look at the first two terms in the binomial expansion to compute it modulo $p^{t+2}$ , and we see that the sum is congruent modulo $p^{t+2}$ to $\\sum _{\\mu \\in {{\\mathbb {F}}_p}}\\left([1+\\mu ]^r+r[1+\\mu ]^{r-1}(1+[\\mu ]-[1+\\mu ])\\right).$ Since $r\\equiv 1\\pmod {p-1}$ , we have $\\sum _{\\mu \\in {{\\mathbb {F}}_p}}[1+\\mu ]^r=\\sum _{\\mu \\in {{\\mathbb {F}}_p}}[\\mu ]=0$ , and since $[1+\\mu ]^{r-1}=1$ unless $\\mu =-1$ , and if $\\mu =-1$ then $1+[\\mu ]-[1+\\mu ]=0$ , the sum is congruent modulo $p^{t+2}$ to $\\sum _\\mu r[1+\\mu ]^{r-1}(1+[\\mu ]-[1+\\mu ])&=r\\sum _{\\mu \\ne -1}(1+[\\mu ]-[1+\\mu ])\\\\&=r\\sum _\\mu (1+[\\mu ]-[1+\\mu ])\\\\&=r\\sum _\\mu 1=rp$ and we are done.",
"(2) We do the same trick using Lemma REF (2), which implies that we only have to look at the first term of the binomial expansion.",
"Modulo $p^{t+1}$ we have $\\sum _\\mu (1+[\\mu ])^{r-1}&=\\sum _\\mu ([1+\\mu ]+(1+[\\mu ]-[1+\\mu ]))^{r-1}\\\\&\\equiv \\sum _\\mu [1+\\mu ]^{r-1}\\\\&=p-1,$ as required.",
"Corollary 2.5 If $p>2$ and $r>1$ with $r\\equiv 1\\pmod {p-1}$ , and if $t=v(r-1)$ , then for all $\\lambda \\in {{\\mathbb {F}}_p}$ we have $\\sum _{\\mu \\in {{\\mathbb {F}}_p}}([\\mu ]-[\\lambda ])^r\\equiv -[\\lambda ]rp\\pmod {p^{t+2}}$ , and $\\sum _{\\mu \\in {{\\mathbb {F}}_p}}([\\mu ]-[\\lambda ])^{r-1}\\equiv p-1\\pmod {p^{t+1}}$ .",
"If $\\lambda =0$ then both statements are obvious.",
"If $\\lambda \\ne 0$ then we simply take out a factor of $(-[\\lambda ])^r$ (resp.",
"$(-[\\lambda ])^{r-1}$ ) and observe that as $[\\mu ]$ runs over the Teichmueller lifts, so does $-[\\mu ]/[\\lambda ]$ .",
"This reduces both claims to the case $\\lambda =-1$ , which is Lemma .",
"Corollary 2.6 If $p>2$ and $r>1$ with $r\\equiv 1\\pmod {p-1}$ , and if $t=v(r-1)$ , then for all $\\lambda \\in {{\\mathbb {F}}_p}$ we have $\\sum _{\\mu \\in {{\\mathbb {F}}_p}}([\\mu ] x-[\\lambda ] x+py)^r\\equiv -[\\lambda ] rpx^r+rp(p-1)x^{r-1}y\\pmod {p^{t+2}\\overline{{\\mathbb {Z}}}_p[x,y]}.$ Again by Lemma REF (1), in order to compute modulo $p^{t+2}$ we only need to expand out the first two terms of $(([\\mu ] x-[\\lambda ] x)+py)^r$ , giving that the sum is congruent to $\\sum _{\\mu \\in {{\\mathbb {F}}_p}}\\left(([\\mu ] x-[\\lambda ] x)^r+rp([\\mu ] x-[\\lambda ]x)^{r-1}y\\right).$ The result then follows from Corollary ."
],
[
"$p$ -adic local Langlands: definitions and lemmas",
"In this section we recall some of the basic definitions and properties of the $p$ -adic local Langlands correspondence.",
"For more details the reader could consult section 2 of [6] or any of the references therein.",
"Say $r\\in {\\mathbb {Z}}_{\\ge 0}$ .",
"Let $K$ be the group $\\operatorname{GL}_2({\\mathbb {Z}}_p)$ , and for $R$ a ${\\mathbb {Z}}_p$ -algebra let ${\\operatorname{Symm}}^r(R^2)$ denote the space $\\oplus _{i=0}^rRx^{r-i}y^i$ of homogeneous polynomials in two variables $x$ and $y$ , with the action of $K$ given by $\\begin{pmatrix}a&b\\\\c&d\\end{pmatrix}x^{r-i}y^i=(ax+cy)^{r-i}(bx+dy)^i.$ Set $G=\\operatorname{GL}_2({\\mathbb {Q}}_p)$ , and let $Z$ be its centre.",
"If $V$ is an $R$ -module with an action of $K$ , then extend the action of $K$ to the group $KZ$ by letting $\\bigl ({\\begin{matrix}{p}&{0}\\\\{0}&{p}\\end{matrix}}\\bigr )$ act trivially, and let $I(V)$ denote the representation ${\\operatorname{ind}}_{KZ}^G(V)$ (compact induction).",
"Explicitly, $I(V)$ is the space of functions $f:G\\rightarrow V$ which have compact support modulo $Z$ and which satisfy $f(\\kappa g)=\\kappa .",
"(f(g))$ for all $\\kappa \\in KZ$ .",
"This space has a natural action of $G$ , defined by $(gf)(\\gamma ):=f(\\gamma g)$ .",
"Note that §2.2 of [7] explains how an $R$ -linear $G$ -endomorphism of $I(V)$ can be interpreted as a certain function $G\\rightarrow {\\operatorname{End}\\,}_R(V)$ (by Frobenius reciprocity).",
"If $V={\\operatorname{Symm}}^r(R^2)$ for some integer $r\\ge 0$ and ${\\mathbb {Z}}_p$ -algebra $R$ , then there is a certain endomorphism $T$ of $I(V)$ which corresponds to the function $G\\rightarrow {\\operatorname{End}\\,}_R(V)$ which is supported on $KZ\\bigl ({\\begin{matrix}{p}&{0}\\\\{0}&{1}\\end{matrix}}\\bigr )KZ$ and sends $\\bigl ({\\begin{matrix}{p}&{0}\\\\{0}&{1}\\end{matrix}}\\bigr )$ to the endomorphism of ${\\operatorname{Symm}}^r(R^2)$ sending $F(x,y)$ to $F(px,y)$ .",
"We now establish some notation, following [8].",
"Recall that for $V$ a ${\\mathbb {Z}}_p[K]$ -module, the space $I(V)$ was defined previously to be a certain space of functions $G\\rightarrow V$ .",
"We let $[g,v]$ denote the (unique) element of $I(V)$ which is supported on $KZg^{-1}$ , and which satisfies $[g,v](g^{-1})=v$ .",
"Note that $g[h,v]=[gh,v]$ for $g,h\\in G$ , that $[g\\kappa ,v]=[g,\\kappa v]$ for $\\kappa \\in KZ$ , and that the $[g,v]$ span $I(V)$ as an abelian group, as $g$ and $v$ vary.",
"Now let $V={\\operatorname{Symm}}^r(R^2)$ for some ${\\mathbb {Z}}_p$ -algebra $R$ .",
"An easy consequence of the definition of $T$ (cf.",
"section 2 of [8]) is that $T[g,v]=\\sum _{\\lambda \\in {{\\mathbb {F}}_p}}\\left[g\\bigl ({\\begin{matrix}{p}&{[\\lambda ]}\\\\{0}&{1}\\end{matrix}}\\bigr ),v(x,-[\\lambda ]x+py)\\right]+\\left[g\\bigl ({\\begin{matrix}{1}&{0}\\\\{0}&{p}\\end{matrix}}\\bigr ),v(px,y)\\right].\\qquad \\mathrm {(\\ding {37})}$ Again we assume that $r\\ge p$ and $r\\equiv 1$ mod $p-1$ .",
"By Lemma 3.2 of [1], there is a $\\operatorname{GL}_2({{\\mathbb {F}}_p})$ -equivariant map $\\Psi :{\\operatorname{Symm}}^r\\overline{{\\mathbb {F}}}_p^2\\rightarrow \\det \\otimes {\\operatorname{Symm}}^{p-2}\\overline{{\\mathbb {F}}}_p^2$ , such that (using $X,Y$ for variables in ${\\operatorname{Symm}}^{p-2}$ ) $\\Psi (f)=\\sum _{s,t\\in {{\\mathbb {F}}_p}}f(s,t)(tX-sY)^{p-2}.$ We now move on to the $p$ -adic part of the story.",
"Say $k\\in {\\mathbb {Z}}_{\\ge 2}$ and $a\\in \\overline{{\\mathbb {Z}}}_p$ with $v(a)>0$ .",
"Definition 3.1 Let $\\Pi _{k,a}:={\\operatorname{ind}}_{KZ}^G{\\operatorname{Symm}}^{k-2}(\\overline{{\\mathbb {Q}}}_p^2)/(T-a)$ (compact induction, as before), and let $\\Theta _{k,a}$ be the image of ${\\operatorname{ind}}_{KZ}^G{\\operatorname{Symm}}^{k-2}(\\overline{{\\mathbb {Z}}}_p^2)$ in $\\Pi _{k,a}$ .",
"If $a\\ne \\pm p^{k-2}(1+p^{-1})$ then $\\Pi _{k,a}$ is irreducible and $\\Theta _{k,a}$ is a lattice in it.",
"Because of Theorem 3.2.1 of [3] (which deals with $k=3$ and $k=p+2$ ), and Theorem 1.6 of [6] and the comments following it (which deal with $k\\lnot \\equiv 3$ mod $p-1$ ), we are only really concerned in this paper in the case $k\\ge 2p+1$ , $k\\equiv 3$ mod $p-1$ and $0<v(a)<1$ , which implies $a\\ne \\pm (1+p^{-1})p^{k/2}$ anyway.",
"So let us assume $k\\ge 2p+1$ and $k\\equiv 3$ mod $p-1$ .",
"To simplify notation set $r=k-2$ , so $r\\equiv 1$ mod $p-1$ .",
"Now by Corollary 5.1 of [6], the natural surjection $I({\\operatorname{Symm}}^r\\overline{{\\mathbb {F}}}_p^2)\\rightarrow \\overline{\\Theta }_{k,a}$ factors through the map $I({\\operatorname{Symm}}^r\\overline{{\\mathbb {F}}}_p^2)\\rightarrow I(\\det \\otimes {\\operatorname{Symm}}^{p-2}\\overline{{\\mathbb {F}}}_p^2)$ induced by $\\Psi $ .",
"The key input we need from the $p$ -adic local Langlands correspondence is the following lemma.",
"Lemma 3.2 Assume $k\\ge 2p+1$ , $k\\equiv 3$ mod $p-1$ , and $0<v(a)<1$ .",
"If $\\overline{\\Theta }_{k,a}$ is a quotient of $I(\\det \\otimes {\\operatorname{Symm}}^{p-2}\\overline{{\\mathbb {F}}}_p^2)/T$ , then $\\overline{V}_{k,a}\\cong {\\operatorname{ind}}(\\omega _2)$ is irreducible.",
"If $\\overline{\\Theta }_{k,a}$ is a quotient of $I(\\det \\otimes {\\operatorname{Symm}}^{p-2}\\overline{{\\mathbb {F}}}_p^2)/(T^2-cT+1)$ for some $c\\in \\overline{{\\mathbb {F}}}_p$ , then $\\overline{V}_{k,a}$ is reducible, and $\\omega ^{-1}\\otimes \\overline{V}_{k,a}$ is an unramified reducible representation, and the trace of (both arithmetic and geometric) ${\\operatorname{Frob}}_p$ is $c$ .",
"This may be proved in exactly the same way as Proposition 3.3 of [6].",
"Note that both arithmetic and geometric Frobenius have the same trace in case (2), because $\\omega ^{-1}\\otimes \\overline{V}_{k,a}$ has trivial determinant (as $k\\equiv 3$ mod $p-1$ )."
],
[
"Computations with Hecke operators",
"We again assume throughout this section that $p>2$ is an odd prime and $r>p$ is an integer such that $r\\equiv 1\\pmod {p-1}$ .",
"We start with a couple of results about the map $\\Psi :{\\operatorname{Symm}}^r(\\overline{{\\mathbb {F}}}_p^2)\\rightarrow \\det \\otimes {\\operatorname{Symm}}^{p-2}(\\overline{{\\mathbb {F}}}_p^2)$ defined in the previous section.",
"Lemma 4.1 $\\Psi (y^r)=0$ .",
"$\\Psi (x^r)=0$ .",
"$\\Psi (x^{r-1}y)=X^{p-2}$ .",
"$\\Psi (y^r)=\\sum _{s,t\\in {{\\mathbb {F}}_p}}t^r(tX-sY)^{p-2}=\\sum _{s,t\\in {{\\mathbb {F}}_p}}t(tX-sY)^{p-2}$ .",
"Expanding out using the binomial theorem and using the fact that $\\sum _ss^n=0$ for $n=0,1,2,\\ldots ,p-2$ , this sum is zero.",
"Since $\\Psi $ is $\\operatorname{GL}_2({{\\mathbb {F}}_p})$ -equivariant, $\\Psi (x^r)$ is also zero.",
"Since $r\\equiv 1\\pmod {p-1}$ , we have $\\Psi (x^{r-1}y)&=\\sum _{s,t\\in {{\\mathbb {F}}_p}}s^{p-1}t(tX-sY)^{p-2}\\\\&=\\sum _{s\\ne 0,t\\in {{\\mathbb {F}}_p}}t(tX-sY)^{p-2}$ and $\\sum _{s\\ne 0}s^n$ is not zero if $n=0$ (although it is for $1\\le n\\le p-2$ as before) so expanding out we get $-\\sum _{t\\in {{\\mathbb {F}}_p}}t(tX)^{p-2}$ which is $-\\sum _{t\\ne 0}X^{p-2}=X^{p-2}$ .",
"Lemma 4.2 In $I({\\operatorname{Symm}}^{p-2}(\\overline{{\\mathbb {F}}}_p^2))$ we have $T[1,X^{p-2}]=\\sum _{\\mu \\in {{\\mathbb {F}}_p}}[\\bigl ({\\begin{matrix}{p}&{[\\mu ]}\\\\{0}&{1}\\end{matrix}}\\bigr ),X^{p-2}]$ , and $T^2[1,X^{p-2}]=\\sum _{\\lambda ,\\mu \\in {{\\mathbb {F}}_p}}[\\bigl ({\\begin{matrix}{p^2}&{p[\\mu ]+[\\lambda ]}\\\\{0}&{1}\\end{matrix}}\\bigr ),X^{p-2}]$ .",
"This is immediate from (REF ).",
"Lemma 4.3 Assume that $p>2$ and that $r>p$ with $r\\equiv 1\\pmod {p-1}$ .",
"Set $t=v(r-1)$ and say $a\\in \\overline{{\\mathbb {Q}}}_p$ with $v(a)>0$ .",
"Then $(T-a)[g,y^r-x^{r-1}y]&\\equiv [g\\bigl ({\\begin{matrix}{p}&{0}\\\\{0}&{1}\\end{matrix}}\\bigr ),-px^{r-1}y]\\\\&+\\sum _{\\lambda \\ne 0}[g\\bigl ({\\begin{matrix}{p}&{\\lambda }\\\\{0}&{1}\\end{matrix}}\\bigr ),(r-1)px^{r-1}y]+[g\\bigl ({\\begin{matrix}{1}&{0}\\\\{0}&{p}\\end{matrix}}\\bigr ),y^r]-[g,a(y^r-x^{r-1}y)]$ modulo $p^{t+2}$ .",
"This is immediate from (REF ) and Lemmas REF and REF .",
"We have $t=v(r-1)$ ; say $a\\in \\overline{{\\mathbb {Q}}}_p$ satisfies $0<v(a)<1$ , and set $t_0=\\min \\lbrace t+1+v(a),v(a^2-rp)\\rbrace $ .",
"Corollary 4.4 Assume that $p>2$ and that $r>p$ with $r\\equiv 1\\pmod {p-1}$ .",
"If $\\varphi _g=\\sum _{j=0}^{N}[g\\bigl ({\\begin{matrix}{p^j}&{0}\\\\{0}&{1}\\end{matrix}}\\bigr ),a^j(y^r-x^{r-1}y)]$ where $N>t_0/v(a)$ , then $(T-a)\\varphi _g\\equiv \\sum _{\\lambda \\in {{\\mathbb {F}}_p}}[g\\bigl ({\\begin{matrix}{p}&{[\\lambda ]}\\\\{0}&{1}\\end{matrix}}\\bigr ),(r-1)px^{r-1}y]+[g\\bigl ({\\begin{matrix}{1}&{0}\\\\{0}&{p}\\end{matrix}}\\bigr ),y^r]+[g,ax^{r-1}y]\\pmod {p^{t_0}}.$ Throughout the proof we will write $\\sum _{j\\ge 0}$ rather than keeping track of the upper index of our sums, as the implied terms will all be zero modulo $p^{t_0}$ .",
"Since $t+2>t+1+v(a)\\ge t_0$ , we can apply Lemma REF .",
"Noting that if $j\\ge 1$ , $v(a^j(r-1)p)\\ge t+1+v(a)\\ge t_0$ , we see that modulo $p^{t_0}$ , $(T-a)\\varphi _g$ is just $&[g\\bigl ({\\begin{matrix}{p}&{0}\\\\{0}&{1}\\end{matrix}}\\bigr ),-px^{r-1}y]+\\sum _{\\lambda \\ne 0}[g\\bigl ({\\begin{matrix}{p}&{[\\lambda ]}\\\\{0}&{1}\\end{matrix}}\\bigr ),(r-1)px^{r-1}y]+[g\\bigl ({\\begin{matrix}{1}&{0}\\\\{0}&{p}\\end{matrix}}\\bigr ),y^r]-[g,a(y^r-x^{r-1}y)]\\\\&+\\sum _{j\\ge 1}[g\\bigl ({\\begin{matrix}{p^{j+1}}&{0}\\\\{0}&{1}\\end{matrix}}\\bigr ),-pa^jx^{r-1}y]\\\\&+\\sum _{j\\ge 1}[g\\bigl ({\\begin{matrix}{p^{j-1}}&{0}\\\\{0}&{1}\\end{matrix}}\\bigr ),a^jy^r]\\\\&+\\sum _{j\\ge 1}[g\\bigl ({\\begin{matrix}{p^j}&{0}\\\\{0}&{1}\\end{matrix}}\\bigr ),-a^{j+1}y^r+a^{j+1}x^{r-1}y]$ which rearranges to $&\\sum _{\\lambda \\ne 0}[g\\bigl ({\\begin{matrix}{p}&{[\\lambda ]}\\\\{0}&{1}\\end{matrix}}\\bigr ),(r-1)px^{r-1}y]\\\\&+[g\\bigl ({\\begin{matrix}{1}&{0}\\\\{0}&{p}\\end{matrix}}\\bigr ),y^r]+[g,ax^{r-1}y]\\\\&+\\sum _{s\\ge 1}[g\\bigl ({\\begin{matrix}{p^s}&{0}\\\\{0}&{1}\\end{matrix}}\\bigr ),-pa^{s-1}x^{r-1}y]\\\\&+\\sum _{s\\ge 1}[g\\bigl ({\\begin{matrix}{p^s}&{0}\\\\{0}&{1}\\end{matrix}}\\bigr ),a^{s+1}y^r]\\\\&+\\sum _{s\\ge 1}[g\\bigl ({\\begin{matrix}{p^s}&{0}\\\\{0}&{1}\\end{matrix}}\\bigr ),-a^{s+1}y^r+a^{s+1}x^{r-1}y]$ where we have changed variables from $j$ to $s$ to make all the sums involve $g\\bigl ({\\begin{matrix}{p^s}&{0}\\\\{0}&{1}\\end{matrix}}\\bigr )$ , and put two of the “initial” terms into the sums.",
"Pressing on, we get two terms in the sums cancelling and we are left with $&\\sum _{\\lambda \\ne 0}[g\\bigl ({\\begin{matrix}{p}&{[\\lambda ]}\\\\{0}&{1}\\end{matrix}}\\bigr ),(r-1)px^{r-1}y]\\\\&+[g\\bigl ({\\begin{matrix}{1}&{0}\\\\{0}&{p}\\end{matrix}}\\bigr ),y^r]+[g,ax^{r-1}y]\\\\&+\\sum _{s\\ge 1}[g\\bigl ({\\begin{matrix}{p^s}&{0}\\\\{0}&{1}\\end{matrix}}\\bigr ),a^{s-1}(a^2-p)x^{r-1}y]\\\\$ By the definition of $t_0$ we have $ap(r-1)\\equiv 0\\pmod {p^{t_0}}$ and $a^2-rp\\equiv 0\\pmod {p^{t_0}}$ , so we see that if $s\\ge 2$ we have $a^{s-1}(a^2-p)\\equiv a^{s-2}(a(a^2-rp)+ap(r-1))\\equiv 0 \\pmod {p^{t_0}}$ .",
"Thus we can simplify further to $&\\sum _{\\lambda \\ne 0}[g\\bigl ({\\begin{matrix}{p}&{[\\lambda ]}\\\\{0}&{1}\\end{matrix}}\\bigr ),(r-1)px^{r-1}y]\\\\&+[g\\bigl ({\\begin{matrix}{1}&{0}\\\\{0}&{p}\\end{matrix}}\\bigr ),y^r]+[g,ax^{r-1}y]\\\\&+[g\\bigl ({\\begin{matrix}{p}&{0}\\\\{0}&{1}\\end{matrix}}\\bigr ),(a^2-p)x^{r-1}y].$ Finally, since $a^2\\equiv rp\\pmod {p^{t_0}}$ , the last term can be inserted into the sum by allowing $\\lambda =0$ .",
"Corollary 4.5 Assume that $p>2$ and that $r>p$ with $r\\equiv 1\\pmod {p-1}$ .",
"If $\\varphi =-p\\varphi _1+\\sum _{\\mu \\in {{\\mathbb {F}}_p}} a\\varphi _{\\bigl ({\\begin{matrix}{p}&{[\\mu ]}\\\\{0}&{1}\\end{matrix}}\\bigr )}+[1,\\sum _{\\mu \\in {{\\mathbb {F}}_p}} ([\\mu ] x+y)^r-rpx^{r-1}y]$ (where $\\varphi _g$ is as in the statement of Corollary REF ), and if $t_1=t_0+\\min \\lbrace v(a),1-v(a)\\rbrace >t_0$ , then $(T-a)\\varphi &\\equiv \\sum _{\\lambda ,\\mu \\in {{\\mathbb {F}}_p}}[\\bigl ({\\begin{matrix}{p^2}&{p[\\lambda ]+[\\mu ]}\\\\{0}&{1}\\end{matrix}}\\bigr ),a(r-1)px^{r-1}y]\\\\&+[1,ap(r-1)x^{r-1}y]\\\\&+\\sum _{\\lambda \\in {{\\mathbb {F}}_p}}[\\bigl ({\\begin{matrix}{p}&{[\\lambda ]}\\\\{0}&{1}\\end{matrix}}\\bigr ),(a^2-rp)x^{r-1}y]\\pmod {p^{t_1}}.$ First we note that $t+2\\ge t_1$ (because $t+2=t+1+v(a)+(1-v(a))\\ge t_0+(1-v(a))\\ge t_1$ ).",
"By Lemma REF we have $r\\ge t+3$ and hence $r\\ge t_1+1>t_1$ , so $p^r\\equiv 0\\pmod {p^{t_1}}$ .",
"We also see from Lemma REF and the inequality $r-1\\ge t_1$ that $\\sum _{\\mu \\in {{\\mathbb {F}}_p}}([\\mu ]px+y)^r\\equiv py^r\\pmod {p^{t_1}}.$ Using these facts and Corollary REF (for the first two terms in the definition of $\\varphi $ ) and (REF ) (for the final term), we see that modulo $p^{t_1}$ , we have $(T-a)\\varphi &\\equiv \\left[\\bigl ({\\begin{matrix}{1}&{0}\\\\{0}&{p}\\end{matrix}}\\bigr ),-py^r\\right]+\\left[1,-pax^{r-1}y\\right]\\\\&+[1,a\\sum _{\\mu \\in {{\\mathbb {F}}_p}} ([\\mu ] x+y)^r]+\\sum _{\\mu \\in {{\\mathbb {F}}_p}}\\left[\\bigl ({\\begin{matrix}{p}&{[\\mu ]}\\\\{0}&{1}\\end{matrix}}\\bigr ),a^2x^{r-1}y\\right]\\\\&+\\sum _{\\lambda ,\\mu \\in {{\\mathbb {F}}_p}}\\left[\\bigl ({\\begin{matrix}{p^2}&{p[\\lambda ]+[\\mu ]}\\\\{0}&{1}\\end{matrix}}\\bigr ),a(r-1)px^{r-1}y\\right]\\\\&+[1,-a\\sum _{\\lambda \\in {{\\mathbb {F}}_p}}([\\lambda ] x+y)^r+arpx^{r-1}y]\\\\&+\\sum _{\\lambda \\in {{\\mathbb {F}}_p}}[\\bigl ({\\begin{matrix}{p}&{[\\lambda ]}\\\\{0}&{1}\\end{matrix}}\\bigr ),(\\sum _{\\mu \\in {{\\mathbb {F}}_p}}([\\mu ] x-[\\lambda ] x+py)^r)-rpx^{r-1}(-[\\lambda ] x+py)]+[\\bigl ({\\begin{matrix}{1}&{0}\\\\{0}&{p}\\end{matrix}}\\bigr ),py^r].$ Now some terms cancel, and we get $&\\sum _{\\lambda ,\\mu \\in {{\\mathbb {F}}_p}}[\\bigl ({\\begin{matrix}{p^2}&{p[\\lambda ]+[\\mu ]}\\\\{0}&{1}\\end{matrix}}\\bigr ),a(r-1)px^{r-1}y]\\\\&+[1,ap(r-1)x^{r-1}y]\\\\&+\\sum _{\\lambda \\in {{\\mathbb {F}}_p}}[\\bigl ({\\begin{matrix}{p}&{[\\lambda ]}\\\\{0}&{1}\\end{matrix}}\\bigr ),(\\sum _{\\mu \\in {{\\mathbb {F}}_p}}([\\mu ] x-[\\lambda ] x+py)^r)+a^2x^{r-1}y-rpx^{r-1}(-[\\lambda ] x+py)].$ Again noting that $t+2\\ge t_1$ , by Corollary we have $\\sum _{\\mu \\in {{\\mathbb {F}}_p}}([\\mu ] x-[\\lambda ] x+py)^r\\equiv -[\\lambda ]rpx^r+rp(p-1)x^{r-1}y\\pmod {p^{t_1}},$ and the result follows.",
"Corollary 4.6 Assume that $p>2$ and that $r>p$ with $r\\equiv 1\\pmod {p-1}$ , and that $0<v(a)<1$ .",
"If $v(r-1)+1+v(a)>v(a^2-rp)$ , then $\\overline{\\Theta }_{k,a}$ is a quotient of $I(\\det \\otimes {\\operatorname{Symm}}^{p-2}\\overline{{\\mathbb {F}}}_p^2)/T$ .",
"If $v(r-1)+1+v(a)\\le v(a^2-rp)$ , then $\\overline{\\Theta }_{k,a}$ is a quotient of $I(\\det \\otimes {\\operatorname{Symm}}^{p-2}\\overline{{\\mathbb {F}}}_p^2)/(T^2-\\overline{\\tau }T+1)$ where $\\tau =\\frac{rp-a^2}{ap(r-1)}$ .",
"Set $\\psi =p^{-t_0}\\varphi $ , with $\\varphi $ as in Corollary REF .",
"By the definition of $t_0$ , we see that both $v(a(r-1)p)$ , and $v(a^2-rp)$ are at least $t_0$ , so that $(T-a)\\psi $ is integral, by Corollary REF .",
"Thus $\\overline{(T-a)\\psi }$ is in the kernel of the natural map $I({\\operatorname{Symm}}^r\\overline{{\\mathbb {F}}}_p^2)\\twoheadrightarrow \\overline{\\Theta }_{k,a}$ .",
"We will now compute $\\Psi (\\overline{(T-a)\\psi })$ in both cases, and hence deduce the claim.",
"If $v(r-1)+1+v(a)>v(a^2-rp)$ , then we see from Corollary REF that $\\overline{(T-a)\\psi }$ is a unit times $\\sum _{\\lambda \\in {{\\mathbb {F}}_p}}[\\bigl ({\\begin{matrix}{p}&{[\\lambda ]}\\\\{0}&{1}\\end{matrix}}\\bigr ),x^{r-1}y]$ .",
"By Lemma REF (3) and Lemma REF (1), we see that $\\Psi (\\overline{(T-a)\\psi })$ is a unit times $T[1,X^{p-2}]$ and the result follows.",
"(Note that $\\det \\otimes {\\operatorname{Symm}}^{p-2}\\overline{{\\mathbb {F}}}_p^2$ is irreducible, and in particular is generated by $X^{p-2}$ .)",
"If $v(r-1)+1+v(a)\\le v(a^2-rp)$ , then writing $\\tau =\\frac{rp-a^2}{ap(r-1)}$ , we see that $\\overline{(T-a)\\psi }$ is a unit times $&\\sum _{\\lambda ,\\mu \\in {{\\mathbb {F}}_p}}[\\bigl ({\\begin{matrix}{p^2}&{p[\\lambda ]+[\\mu ]}\\\\{0}&{1}\\end{matrix}}\\bigr ),x^{r-1}y]\\\\&+[1,x^{r-1}y]\\\\&-\\overline{\\tau }\\sum _{\\lambda \\in {{\\mathbb {F}}_p}}[\\bigl ({\\begin{matrix}{p}&{[\\lambda ]}\\\\{0}&{1}\\end{matrix}}\\bigr ),x^{r-1}y].$ By Lemma REF (3) and Lemma REF , we see that $\\Psi (\\overline{(T-a)\\psi })$ is a unit times $(T^2-\\overline{\\tau }T+1)[1,X^{p-2}]$ , as required.",
"Corollary 4.7 Assume that $p>2$ and that $r>p$ with $r\\equiv 1\\pmod {p-1}$ , and that $0<v(a)<1$ .",
"If $v(r-1)+1+v(a)>v(a^2-rp)$ , then $\\overline{V}_{k,a}\\cong {\\operatorname{ind}}(\\omega _2)$ is irreducible.",
"If $v(r-1)+1+v(a)\\le v(a^2-rp)$ , then $\\omega ^{-1}\\otimes \\overline{V}_{k,a}$ is unramified, and the trace of ${\\operatorname{Frob}}_p$ on this representation is $\\overline{t}$ , where $t=\\frac{a^2-rp}{ap(r-1)}$ .",
"This is immediate from Lemma and Corollary REF ."
]
] | 1204.1565 |
[
[
"High CO depletion in southern infrared-dark clouds"
],
[
"Abstract Infrared-dark high-mass clumps are among the most promising objects to study the initial conditions of the formation process of high-mass stars and rich stellar clusters.",
"In this work, we have observed the (3-2) rotational transition of C18O with the APEX telescope, and the (1,1) and (2,2) inversion transitions of NH3 with the Australia Telescope Compact Array in 21 infrared-dark clouds already mapped in the 1.2 mm continuum, with the aim of measuring basic chemical and physical parameters such as the CO depletion factor (fD), the gas kinetic temperature and the gas mass.",
"In particular, the C18O (3-2) line allows us to derive fD in gas at densities higher than that traced by the (1-0) and (2-1) lines, typically used in previous works.",
"We have detected NH3 and C18O in all targets.",
"The clumps possess mass, H2 column and surface densities consistent with being potentially the birthplace of high-mass stars.",
"We have measured fD in between 5 and 78, with a mean value of 32 and a median of 29.",
"These values are, to our knowledge, larger than the typical CO depletion factors measured towards infrared-dark clouds and high-mass dense cores, and are comparable to or larger than the values measured in low-mass pre-stellar cores close to the onset of the gravitational collapse.",
"This result suggests that the earliest phases of the high-mass star and stellar cluster formation process are characterised by fD larger than in low-mass pre-stellar cores.",
"Thirteen out of 21 clumps are undetected in the 24 {\\mu}m Spitzer images, and have slightly lower kinetic temperatures, masses and H2 column densities with respect to the eight Spitzer-bright sources.",
"This could indicate that the Spitzer-dark clumps are either less evolved or are going to form less massive objects."
],
[
"Introduction",
"An ever increasing number of observational evidences indicates that the earliest phases of massive star and stellar cluster formation occur within infrared dark clouds (IRDCs).",
"These are dense molecular clouds seen as extinction features against the bright mid-infrared Galactic background (e.g.",
"Simon et al [48], Ragan et al.",
"[44], [43], Rathborne et al.",
"[45], [47], Butler & Tan [8]).",
"IRDCs are characterised by very high gas column densities ($10^{23} - 10^{25}$ cm$^{-3}$ ) and low temperatures ($\\le 25~K$ ), so that they are believed to be the place where most of the stars in our Galaxy are being formed.",
"From an observational point of view, the spatial distribution of the IRDCs in the Galaxy follows that of the molecular galactic component (with a concentration in the so-called 5 kpc molecular ring), and they are strong emitters of both far-IR/millimetre continuum and rotational molecular transitions, especially those characterised by a high critical density.",
"In many of them, observations of the dense gas at sub-parsec linear scales revealed the presence of on-going star formation, both in isolated and clustered mode (e.g.",
"Beuther & Sridharan [5], Zhang et al.",
"[54], Fontani et al.",
"[18], Jiménez-Serra et al.",
"[29], Pillai et al.",
"[38]), including clear signs of high-mass star formation like hot cores (e.g.",
"Rathborne et al.",
"[46]) and/or Ultracompact Hii regions (Battersby et al. [2]).",
"This demonstrates that IRDCs are indeed the birthplace of stars and stellar clusters of all masses.",
"Therefore, the IRDCs in the earliest evolutionary stages are the best locations where to study the initial conditions of the star formation process and put constraints on current theories.",
"Despite the identification of thousands of IRDCs, the number of studies devoted to unveiling their physical and chemical properties remains still limited.",
"There are very few targeted studies, especially in the southern hemisphere (e.g.",
"Vasyunina et al.",
"[51], Miettinen et al.",
"[36]), where there are fewer groundbased facilities operating in the (sub-)millimetre and centimetre domain than there are in the north.",
"In particular, very little is known about the chemistry of IRDCs: studies suggest chemical compositions similar to those observed in low-mass pre–stellar cores (Vasyunina et al.",
"[51]), including large abundances of deuterated species (Pillai et al.",
"[39]; Pillai et al.",
"[38]; Fontani et al. [17]).",
"However, the amount of CO freeze-out, a key chemical parameter for pre–stellar cores (see e.g.",
"Bergin & Tafalla [4] for a review) remains controversial: some works indicate high levels (factor of 5, Hernández et al.",
"[25]) of CO freeze-out, while others do not reveal significant CO depletion (e.g.",
"Miettinen et al. [36]).",
"In this work we present observations of the rotational transition (3–2) of the dense gas tracer C$^{18}$ O, performed with the Atacama Pathfinder EXperiment (APEX) 12-m Telescope, and of the inversion transitions (1,1) and (2,2) of NH$_3$ carried out with the Australia Telescope Compact Array (ATCA), towards 21 IRDCs with declination lower than $-30^{\\circ }$ already mapped in the 1.2 mm continuum.",
"The C$^{18}$ O observations allow to compute the CO depletion factor, while the ammonia inversion transitions can be used to derive the temperature in dense and cold gas (see e.g.",
"Ho & Townes [28]).",
"Observations of N$_{2}$ H$^{+}$ and N$_{2}$ D$^{+}$ in some of the target sources, useful to derive the amount of deuterated fraction (by comparing the N$_{2}$ D$^{+}$ and N$_{2}$ H$^{+}$ column densities), are also presented.",
"In Sect.",
"we describe the criteria applied to select the targets.",
"Sect.",
"gives an overview of the observations.",
"The results are presented in Sect.",
"and discussed in Sect. .",
"Conclusions and a summary of the main findings are given in Sect.",
"."
],
[
"Target selection",
"We selected 21 IRDCs from the 95 massive millimetre clumps detected by Beltrán et al.",
"(2006) in the 1.2 mm continuum and non-MSX emitters (neither diffuse nor point-like).",
"The targets were chosen according to these criteria: (i) source declination $\\delta \\le -30^{\\circ }$ ; (ii) clumps isolated or having the emission peak separated by more than the SIMBA half power beam width to that of MSX-emitter objects, to limit confusion and select the most quiescent sources; (iii) clump masses above $\\sim 35 M_{\\odot }$ to deal with possible massive star formation.",
"In this work we have recomputed the masses utilising the gas temperature derived from ammonia for each clump, and our results confirm that all targets are high-mass clumps (see Sect REF ).",
"The list of IRDCs is given in Table REF .",
"The coordinates correspond to the peak of the 1.2 millimetre continuum emission mapped by Beltrán et al. ([3]).",
"In Table REF we also give some basic information like the distance to the Sun, the Galactocentric distance, and the (non-)detection in the Spitzer-MIPS 24 $\\mu $ m images.",
"A comparison between the 1.2 mm continuum maps and the Spitzer 24 $\\mu $ m images of each target (except for 13039–6108c6, for which the MIPS images are not available) is shown in Fig.",
"REF of Appendix ."
],
[
"APEX",
"Single-point spectra of the C$^{18}$ O (3–2), N$_{2}$ H$^{+}$ (3–2) and N$_{2}$ D$^{+}$ (4–3) lines towards the sources listed in Table REF were obtained with the APEX Telescope in service mode between the 20th and the 28th of June, 2008.",
"The observations were performed in the wobbler-switching mode with a 150 azimuthal throw and a chopping rate of 0.5 Hz.",
"The receiver used for all lines was SHFI/APEX-2.",
"The backend provided a total bandwidth of 1000 MHz.",
"Details about the observed lines and the observational parameters are given in Table REF .",
"The telescope pointing and focusing were checked regularly by continuum scans on planets and the corrections were applied on-line.",
"Calibration was done by the chopper-wheel technique.",
"Spectra were obtained in antenna temperature units (corrected for atmospheric attenuation), $T_A^*$ , and then converted to main beam temperature units through the relation $T_{\\rm MB}=\\frac{F_{\\rm eff}}{B_{\\rm eff}} T_A^*$ ($F_{\\rm eff}$ and $B_{\\rm eff}$ are given in Col. 5 of Table REF ).",
"The velocities adopted to centre the backends are given in Col. 6 of Table REF .",
"For most sources we knew the radial Local Standard of Rest velocities ($V_{\\rm LSR}$ ) from previous CS observations.",
"The spectra of 16435–4515c3 (for which we did not have the CS data) were centred at $V_{\\rm LSR} = 0$ ; in any case the total bandwidth of the backend ($\\sim 1000$ km s$^{-1}$) was larger than the velocity gradient across the Galaxy.",
"Table: Source list and detection summary.Table: Transitions observed with APEX and observational parameters."
],
[
"ATCA",
"We observed the inversion transitions $(J,K)=$ (1,1) and (2,2) of ammonia at 23694.5 MHz and 23722.6 MHz (K-band at $\\sim 1.2$ cm), respectively, with the ATCAThe Australia Telescope Compact Array is part of the Australia Telescope which is funded by the Commonwealth of Australia for operation as a National Facility managed by CSIRO.",
"towards all targets in Table REF .",
"The observations were performed between the 4th and the 8th of March, 2011, for a total telescope time of 48 hours.",
"We used the configuration 750D, which provides baselines between 31 and 4469 m. The primary beam was $\\sim 2.5$ at the line frequencies.",
"The flux density scale was established by observing the standard primary calibrator 1934–638, and the uncertainty is expected to be of the order of $\\sim 10\\%$ .",
"Gain calibration was ensured by frequent observations of nearby compact quasars.",
"The quasar 0537–441 was used for passband calibration.",
"Pointing corrections were derived from nearby quasars and applied online.",
"Atmospheric conditions were generally good (weather path noise $\\simeq 400\\;\\mu $ m or better).",
"The total time was broken up into series of 3-/5-minute snapshots in order to improve the coverage of the uv-plane for each target.",
"As a consequence of this observing strategy, the integration time on source is variable, and generally in between $\\sim 30$ mins and $\\sim 1$ hour.",
"The CABBhttp://www.narrabri.atnf.csiro.au/observing/CABB.html correlator provided two \"zoom\" bands of 64 MHz each, with a spectral resolution in each zoom of 32 kHz ($\\sim $ 0.4 km s$^{-1}$ at the frequencies of the lines).",
"The ammonia lines were observed in one zoom band.",
"The data were edited and calibrated following standard tasks and procedures of the MIRIAD software package.",
"After the editing and calibration in MIRIAD, the data were imported in AIPS.",
"Imaging and deconvolution were performed using the 'imagr' task, applying natural weighting to the visibilities.",
"The ammonia emission was detected only on the short baselines, thus we discarded all baselines $>30\\,\\mathrm {k}\\lambda $ .",
"In order to obtain images with the same angular resolution, and comparable to the APEX beam, we reconstructed the images with a circular beam of 20 of diameter for all sources, except for 17195-3811c2 and 17040-3959c1, that have a poorer UV-coverage, and for which we get a beam of roughly $20\\times 40$ .",
"Moreover, 16254-4844c1 and 16573-4214c2 were observed only once, making the clean impossible and thus forcing us to use the dirty image deconvolved with the dirty beam to determine the spectrum.",
"In this work the ATCA data will be used only to derive the gas temperature from the NH$_3$ (1,1) and (2,2) spectra at the dust emission peak (in Sect.",
"REF ).",
"A complete presentation and analysis of the ATCA observations will be given in a forthcoming paper (Giannetti et al., in prep)."
],
[
"Detection summary",
"The detection summary is given in Table REF .",
"C$^{18}$ O (3–2) was observed and detected in all sources.",
"Twelve targets were also observed and detected in N$_{2}$ H$^{+}$ (3–2).",
"Ten clumps were observed in N$_{2}$ D$^{+}$ (4–3), and only clump 15557–5215c2 was marginally detected.",
"This is, to our knowledge, the first detection of this line towards an IRDC.",
"All targets were observed and detected in the ammonia inversion transitions, with the exception of 14183–6050c3 which was undetected in the (2,2) line.",
"Table: Detection summary: Y = detected, N = not detected, – = not observed;"
],
[
"Line profiles and parameters",
"All spectra of the C$^{18}$ O and N$_{2}$ H$^{+}$ lines are shown in Appendix .",
"Fig.",
"REF shows the N$_{2}$ H$^{+}$ (3–2) and C$^{18}$ O (3–2) transitions in the twelve clumps observed and detected in both lines.",
"Fig.",
"REF shows the nine sources observed and detected in C$^{18}$ O (3–2) only.",
"Most of the C$^{18}$ O lines are well fitted by single Gaussians.",
"The results of these fits are reported in Table REF .",
"Slightly asymmetric profiles deviating from the Gaussian shape can be noted in 15470–5419c1, 16061–5048c1, 16482–4443c2, 16573–4214c2 and 17040–3959c1.",
"This could be due either to the superposition of multiple blended velocity components, or to high optical depth effects.",
"Non-Gaussian wings in one (or both) side(s) of the line indicating presence of outflows are noticeable towards two sources, 08477–4359c1 and 17195–3811c2 (see Fig.",
"REF ).",
"Four sources show multiple lines well-separated in velocity, 14183–6050c3, 16093–5128c2, 16093–5128c8, 16435–4515c3.",
"Because we had the previous CS observations (Fontani et al.",
"[20]) as reference, in Table REF we have identified the line associated with the star-forming region of our interest.",
"The other components are almost certainly arising from clouds along the line of sight but not associated with the star-forming region, given the large separation in velocity (from $\\sim 20$ to $\\sim 40$ km s$^{-1}$, see Figs.",
"REF and REF ).",
"Figure: C 18 ^{18}O (3–2) spectra of 08477–4359c1 and17195–3811c2.",
"The Gaussian fits are superimposed on thespectra and highlight significant emission in non-Gaussian wings.Table: C 18 ^{18}O line parameters derived from Gaussian fits.",
"For sources with more thanone velocity component, the line labelled as '–a' is the one associated with the IRDC (basedon previous CS observations, Fontani et al.",
").The N$_{2}$ H$^{+}$ (3–2) lines roughly peak at the same position as the C$^{18}$ O (3–2) ones and have comparable line widths.",
"Because of the hyperfine structure present in the transition, it is not straightforward to derive conclusions on blended multiple velocity components.",
"However, we note hints of possible secondary velocity components in all the lines that show similar features in the C$^{18}$ O (3–2) line (15470–5419c1, 16061–5048c1, 16573–4214c2) except for 16482–4443c2.",
"Table: N 2 _{2}H + ^{+} line parameters and column densities.",
"For optically thin transitionswithout well-constrained opacity (τ=0.1\\tau =0.1), to compute NN(N 2 _{2}H + ^{+})we adopted a T ex T_{\\rm ex}= 6 K (see text).",
"For sources with more thanone velocity component, the line labelled as '–a' is the one having thepeak velocity consistent with that of the clump of our interest,as in Table .In Table REF we give the N$_{2}$ H$^{+}$ (3–2) line parameters: in Cols.",
"3 – 7 we list integrated intensity ($\\int T_{\\rm MB}{\\rm d}v$ ), peak velocity ($V_{\\rm LSR}$ ), FWHM, opacity ($\\tau $ ), and excitation temperature ($T_{\\rm ex}$) of the N$_{2}$ H$^{+}$ (3–2) line, respectively.",
"Because the rotational transitions of N$_{2}$ H$^{+}$ possess hyperfine structure, we fitted the lines using the METHOD HFS of the CLASS packageThe CLASS program is part of the GILDAS software, developed at the IRAM and the Observatoire de Grenoble, and is available at http://www.iram.fr/IRAMFR/GILDAS.",
"This method fits all the hyperfine components simultaneously assuming that they have the same excitation temperature and width, that the opacity has a Gaussian dependence on frequency and fixing the separation of the components to the laboratory value.",
"The line parameters listed in Table REF have been derived from this method, except the integrated intensity that has been computed by simple integration over the velocity range given in Col. 2.",
"The line opacity is deduced from the intensity ratio of the different hyperfine components, most of which are blended due to the fact that the line widths are larger ($\\sim 1 - 3$ km s$^{-1}$) than the separation in velocity of the components.",
"However, the fit residuals are generally low indicating that the procedure provides good results despite the blending."
],
[
"Column densities of N$_{2}$ H{{formula:9f307b36-be0c-48bf-be26-efa942c5971b}}",
"The N$_{2}$ H$^{+}$ column densities were derived following the method outlined in the Appendix A of Caselli et al. ([11]).",
"Specifically, we adopted equations (A-1) and (A-4) for optically thick and optically thin lines, respectively.",
"The method assumes a constant excitation temperature, $T_{\\rm ex}$.",
"An estimate of $T_{\\rm ex}$ can be derived from the fitting procedure to the hyperfine structure See the CLASS user manual for the derivation of $T_{\\rm ex}$ from the output parameters: http://iram.fr/IRAMFR/GILDAS/doc/html/class-html/class.html/, but not for optically thin lines or lines with opacity not well-constrained.",
"For these, we have assumed the average $T_{\\rm ex}$ ($\\sim 6$ K; Table REF ) derived for the sources with well-constrained opacity.",
"For all sources but 16573–4214c2 and 16482–4443c2, the clump diameter derived from the 1.2 mm continuum is comparable to or larger than the beam size (see Table 2 in Beltrán et al.",
"[3], see also Table REF ), so that we assumed unity filling factor.",
"This factor was applied also to the unresolved source 16482–4443c2.",
"We stress that this is an approximation, because the sources could be clumpy.",
"However, due to the lack of observations at higher-angular resolution, neither in the lines observed in this work nor in other high-density gas tracers, the size of the effective emitting region cannot be determined.",
"For 16573–4214c2, for which $\\theta _{\\rm s}\\sim 7$ , we applied a correction factor $1/ \\eta _{\\nu }= (\\theta _{\\rm s}^2+HPBW^2)/ \\theta _{\\rm s}^2 \\sim 10$ .",
"The N$_{2}$ H$^{+}$ total column densities are given in Col. 8 of Table REF , and span a range from $10^{12}$ to $10^{14}$ cm$^{-2}$ .",
"These values are in good agreement with those derived from the same line in other IRDCs (Ragan et al.",
"[44], Miettinen et al.",
"[36]), as well as in other massive starless and star-forming cores (Fontani et al.",
"[17], Chen et al.",
"[14])."
],
[
"Upper limits on the column density of N$_{2}$ D{{formula:9e1cb757-b277-4f7a-bf68-1a254a724628}} and on N{{formula:67d4c09e-bd46-404e-ab90-b72d3060e650}} D{{formula:cc18fd8d-56da-4d8b-89b3-5f9c4dbf3ce3}} /N{{formula:8b066ba1-3fab-4b94-a96d-63393af5848f}} H{{formula:7686b9c8-93ee-4a0f-acf6-895695d89ace}} column density ratio",
"The N$_{2}$ D$^{+}$ (4–3) transition was marginally detected at $\\sim 3.2\\,\\sigma $ rms only towards 15557–5215c2.",
"This represents, to our knowledge, the first detection of this line in an IRDC.",
"The spectrum is shown in Fig.",
"REF .",
"For this source we have computed the N$_{2}$ D$^{+}$ column density fitting the line with a Gaussian (the results are given in Table REF ), and then following the approach of Appendix A in Caselli et al.",
"([11]) to derive the N$_{2}$ D$^{+}$ column density for the optically thin case.",
"This line is also a blend of hyperfine components, therefore the line width derived from the Gaussian fit is an upper limit to the intrinsic value.",
"However, because of the poor signal-to-noise ratio of the spectrum, METHOD HFS did not provide reliable results.",
"As $T_{\\rm ex}$, we used that derived from N$_{2}$ H$^{+}$ (3–2) for this source (see Table REF ).",
"We find $N$ (N$_{2}$ D$^{+}$ )$\\sim 8\\times 10^{11}$ cm$^{-2}$ and the corresponding deuterated fraction (column density ratio $N$ (N$_{2}$ D$^{+}$ )/$N$ (N$_{2}$ H$^{+}$ ) = $D_{\\rm frac}$ ) is $\\sim 0.003$ (Table REF ).",
"This value is consistent with studies performed in both IRDCs (e.g.",
"Miettinen et al.",
"[36], Chen et al.",
"[14]) and massive star-forming clumps containing more evolved objects (e.g.",
"Fontani et al.",
"[19], [17]).",
"Table: N 2 _{2}D + ^{+} (4–3) line parameters and N 2 _{2}D + ^{+} column density in 15557–5215c2Figure: N 2 _{2}D + ^{+} (4–3) spectrum of the only sourcedetected in this transition, 15557–5215c2.For all the other undetected sources, we have derived upper limits on the deuterated fraction in between 0.003 and 0.5.",
"These upper limits were computed from the integrated intensity upper limits assuming the lines to be Gaussian from the formula $\\int T_{\\rm MB}{\\rm d}v= \\frac{\\Delta V}{2\\sqrt{{\\rm ln}2/\\pi }}T_{\\rm MB}^{\\rm peak}$ .",
"We adopted the 3$\\sigma $ rms level in the spectrum as peak temperature $T_{\\rm MB}^{\\rm peak}$ , and the line width measured in the detected source 15557–5215c2 (Table REF ) as $\\Delta V$ .",
"The $D_{\\rm frac}$ upper limits are comparable to the values commonly measured in massive clumps (Fontani et al.",
"2006, Chen et al.",
"[14], Miettinen et al.",
"[36]), so that our sensitivity does not allow to conclude if our targets have deuterated fraction similar to or lower than the other clumps in the literature.",
"We believe that the very low detection rate in the N$_{2}$ D$^{+}$ (4–3) line is likely due to the high densities required to excite this transition: having a critical density of $\\sim 3 \\times 10^{7}$ cm$^{-3}$ , this line is expected to come from very compact regions, thus suffering enormous beam dilution effects."
],
[
"Rotation temperatures and NH$_3$ column densities at dust emission peak position",
"We have extracted spectra of the NH$_3$ (1,1) and (2,2) transitions from the ATCA channel maps towards the positions given in Table REF .",
"The spectra obtained this way were then imported in CLASS, and fitted using METHOD NH$_3$ to fit the (1,1) lines, which takes into account the line hyperfine structure, similarly to METHOD HFS (see Sect.",
"REF ).",
"This was also used for the (2,2) emission, even though the hyperfine components were not always visible, to obtain an upper limit for $\\tau $ .",
"In Table REF we list the rotation temperatures ($T_{\\rm rot}$), kinetic temperatures ($T_{\\rm k}$), and total column densities of ammonia ($N_{\\rm NH_3}$ ) derived from the NH$_3$ (2,2)/(1,1) line ratios, as well as the line peak velocities.",
"We have derived $T_{\\rm rot}$ and $N_{\\rm NH_3}$ from the output parameters given by the fitting method outlined above, and using the formulae given in the Appendix of Busquet et al. ([9]).",
"The formulae, which result from the discussion in Ho & Townes (1983, Eq.",
"(4)), have been derived assuming that the transitions between the metastable inversion doublets are approximated as a two-level system, and that the excitation temperature and line width are the same for both NH$_3$ (1, 1) and NH$_3$ (2, 2).",
"Note that the assumption of a two-level system is reasonable because transitions between the metastable inversion doublets are usually much faster than those of other rotational states (Ho & Townes [28]).",
"$T_{\\rm k}$ was extrapolated from $T_{\\rm rot}$ following the empirical approximation method described in Tafalla et al.",
"([50], see also Eq.",
"(A.5) in Busquet et al. [9]).",
"We find kinetic temperatures in between 13 and 25 K, with both mean and median $T_{\\rm k}$ of $\\sim 17$ K. These values are consistent, within the errors, with typical temperatures measured towards IRDCs (Pillai et al.",
"[40]) from ammonia.",
"Table: Rotation temperature, kinetic temperature, total NH 3 _3 columndensities and peak velocities derived from the NH 3 _3 inversion transitions observedwith the ATCA.",
"The errors on T rot T_{\\rm rot} and T k T_{\\rm k} are given in parentheses and are computedfrom the propagation of errors.",
"The uncertainty on the column densities includes the erroron the flux calibration, and is estimated to be of the order of 20 – 30%\\%."
],
[
"H$_2$ masses and other physical parameters from 1.2 mm continuum emission",
"Gas masses were calculated by Beltrán et al.",
"([3]) for all clumps from the 1.2 mm continuum integrated flux density, assuming optically thin emission and a reasonable dust temperature of 30 K for all clumps.",
"This latter assumption was due to the fact that a temperature estimate for each clump was lacking.",
"In this work we can profit from the temperatures derived from ammonia (Col. 3 of Table REF ) and recompute the H$_2$ masses assuming that the dust temperature equals the kinetic temperature.",
"This method implies coupling between gas and dust, which is a realistic assumption for gas densities as those of our clumps ($\\ge 10^4$ cm$^{-3}$ ).",
"The gas mass, $M$ , has been derived using Eq.",
"(1) in Beltrán et al.",
"([3]), and adopting the same assumptions, except the dust temperature for which we have utilised $T_{\\rm k}$ derived from ammonia.",
"From $M$ , for each source we have calculated the source averaged H$_2$ volume and column densities, $n_{\\rm H_2}$ and $N_{t}({\\rm H_2})$ , assuming the clumps to be spherical and homogeneous.",
"The H$_2$ column densities, $N_{p}({\\rm H_2})$ , have been estimated also from the 1.2 mm continuum peak flux using the equations in Beuther et al.",
"([6]) and adopting the same assumptions made to derive the mass (same $\\beta $ , dust mass opacity and gas-to-dust ratio).",
"These represent average values in the telescope beam of the 1.2 mm continuum observations ($\\sim 24$$^{\\prime \\prime }$ ), and we will use these estimates to derive the CO depletion in Sect.",
"REF , because the APEX C$^{18}$ O observations have a comparable angular resolution ($\\sim 19$$^{\\prime \\prime }$ ).",
"Also, we have computed the source averaged surface density, $\\Sigma ({\\rm H_2}) = 4 M/\\pi (\\theta _{\\rm s} d)^2$ , where $d$ and $\\theta _{\\rm s}$ are the source distance (see Table REF ) and angular diameter (see Table REF ), respectively.",
"For 16435–4515c3, for which we did not have a kinematic source distance, we computed it from the velocity of the C$^{18}$ O (3–2) line of the '–a' component (see Table REF ) following the method explained in Fontani et al. ([20]).",
"The method, which assumes the rotation curve of Brand & Blitz ([7]), is valid for distances from the Galactic centre between 2 and 25 kpc, and provides two kinematic distances, 3.1 and 12.6 kpc.",
"As for the other sources with distance ambiguity, we adopted the `near' value, i.e.",
"3.1 kpc.",
"The results are listed in Table REF , where we give $M$ , $n_{\\rm H_2}$ , $N({\\rm H_2})$ (both $N_{t}({\\rm H_2})$ and $N_{p}({\\rm H_2})$ ) and $\\Sigma ({\\rm H_2})$ .",
"For completeness, in Cols.",
"2 and 3 of Table REF we list the full width half maximum diameters, $\\theta _{\\rm s}$ (deconvolved with the telescope beam of 24$^{\\prime \\prime }$ ) adopted to derive $n_{\\rm H_2}$ , $N_{t}({\\rm H_2})$ and $\\Sigma ({\\rm H_2})$ , as well as the integrated flux densities used to calculate the masses, $S_{\\nu }$ .",
"Both parameters are taken from Beltrán et al. ([3]).",
"The mean mass turns out to be 439 M$_\\odot $, and the median is 244 M$_\\odot $.",
"As expected, these values are systematically higher than those derived by Beltrán et al.",
"([3]) given that the kinetic temperatures from ammonia (see Table REF ) are systematically lower than the representative value (30 K) assumed by Beltrán et al. ([3]).",
"The clump-averaged column densities are of the order of $10^{22}-10^{23}$ cm$^{-2}$ , with a mean value of $1.6 \\times 10^{23}$ cm$^{-2}$ and a median of $8.5 \\times 10^{22}$ cm$^{-2}$ .",
"The volumn densities are of the order of $10^{4}-10^{5}$ cm$^{-3}$ , with a mean value of $3.1 \\times 10^{5}$ cm$^{-3}$ and a median of $3.9 \\times 10^{4}$ cm$^{-3}$ .",
"Finally, the mean surface density is 0.37 g cm$^{-2}$ , and its median value is 0.19 g cm$^{-2}$ .",
"The difference between mean and median values for all parameters is due to the clump 16573–4214c2, which has an angular diameter ($\\sim 7$ ) more than twice smaller than any other source of the sample, despite the comparable mass.",
"The clump-averaged column densities $N_{t}({\\rm H_2})$ are generally smaller than the theoretical threshold given by Krumholz & McKee (2008) to avoid cloud fragmentation, which is $\\sim 1$ g cm$^{-2}$ , corresponding to $\\sim 3\\times 10^{23}$ cm$^{-2}$ .",
"Also, the distance independent parameter $\\Sigma ({\\rm H_2})$ is smaller, on average, than the theoretical values predicted for clumps that are going to form high-mass stars or super star clusters, which are expected to be larger than $\\sim 0.7$ g cm$^{-2}$ (Chakrabarti & McKee [13], Krumholz & McKee 2008).",
"Rather, the values measured in this work are consistent to those predicted for clumps that are going to form intermediate-mass stars and stellar clusters (Chakrabarti & McKee [13]).",
"We stress, however, that $\\Sigma ({\\rm H_2})$ and $N_{t}({\\rm H_2})$ represent average values across the whole clumps, which could in reality be fragmented in smaller and denser cores.",
"Therefore, our $\\Sigma ({\\rm H_2})$ and $N_{t}({\\rm H_2})$ are to be considered as lower limits for the individual embedded cores.",
"On the other hand, the column densities calculated from the 1.2 mm continuum peak flux, $N_{p}({\\rm H_2})$ , are closer to the theoretical threshold proposed by Krumholz & McKee (2008), indicating that the central regions of the clumps could more easily form massive stars.",
"Moreover, we have compared the clump masses to the threshold proposed by Kauffmann & Pillai ([30]) based on an empirical mass-radius relation which predicts that a cloud must exceed the relation $M > M_{\\rm thr} = 870 {\\rm M_{\\odot }} (R/{\\rm pc})^{1.33}$ to form massive stars.",
"The mass thresholds $M_{\\rm thr}$ are listed in Col. 9 of Table REF .",
"As $R$ , we have used half of the diameters listed in the same Table.",
"We can see that our clump masses are all above the predicted threshold, with four exceptions: 13039–6108c6, 15557–5215c3, 16164–4929c3 and 16482–4443c2, for which, in any case, the measured mass and the corresponding threshold value are similar, and for 16482–4443c2 the mass threshold is even an upper limit.",
"Based on these results, we claim that the targets have the potential to form massive stars.",
"Table: Parameters derived from the 1.2 mm continuum emission: 1.2 mm continuumflux density (S ν S_{\\nu }) integrated over the 3σ\\sigma rms contour level,clump angular diameter (θ s \\theta _{\\rm s}), gas mass (MM),H 2 _2 volume (n H 2 n_{\\rm H_2}), column (N(H 2 N({\\rm H_2}))and mass surface density (Σ(H 2 )\\Sigma ({\\rm H_2})).The last column lists the mass threshold, M thr M_{\\rm thr}, for a cloud that can form massive starsbased on the empirical mass-radius relation proposed by Kauffmann & Pillai (,see text)."
],
[
"The C$^{18}$ O column density has been derived from line intensities assuming optically thin lines and LTE conditions.",
"Under these assumptions one can demonstrate that the total column density of C$^{18}$ O, $N_{\\rm C^{18}O}$ , is related to the integrated intensity of a rotational transition $J \\rightarrow J-1$ according to (e.g.",
"Pillai et al.",
"[39]): ${N_{\\rm C^{18}O} = \\frac{N_J}{g_J}Q(T_{\\rm ex}){\\rm e}^{\\left(\\frac{E_J}{kT_{\\rm ex}}\\right)} = } \\nonumber \\\\{ = \\frac{3 h}{8 \\pi ^3}\\frac{1}{S \\mu ^2}\\frac{I({\\rm C^{18}O})}{J_{\\nu }(T_{\\rm ex})-J_{\\nu }(T_{\\rm BG})}\\frac{Q(T_{\\rm ex}){\\rm e}^{\\left(\\frac{E_J}{kT_{\\rm ex}}\\right)}}{{\\rm e}^{h \\nu / k T_{ex}}-1}\\frac{1}{\\eta _{\\nu }} }$ where: $I({\\rm C^{18}O})$ is the integrated intensity of the line; $N_J$ is the column density of the upper level; $g_J$ , $E_J$ and $S$ are statistical weight (=$2J+1$ ), energy of the upper level and line strength, respectively; $Q(T_{\\rm ex})$ is the partition function at the temperature $T_{\\rm ex}$ ; $\\nu $ the line rest frequency; $J_{\\nu }(T_{\\rm ex})$ and $J_{\\nu }(T_{\\rm BG})$ the equivalent Rayleigh-Jeans temperatures at frequency $\\nu $ computed for the excitation and background temperature ($T_{\\rm BG}\\sim 2.7~K$ ), respectively; $\\mu $ the molecule's dipole moment ($0.1102$ Debye for C$^{18}$ O); $\\eta _{\\nu }$ the beam filling factor.",
"The total integrated intensity $I({\\rm C^{18}O})$ was derived from the integral over all the channels with emission instead of the area given by the Gaussian fits to take into account also non-Gaussian features.",
"As for the derivation of the N$_{2}$ H$^{+}$ column density, we assumed a unity filling factor for all sources, so that $N_{\\rm C^{18}O}$ is a beam-averaged value.",
"As excitation temperature, we have used the kinetic temperatures listed in Col. 3 of Table REF (i.e.",
"we are assuming LTE conditions for C$^{18}$ O (3–2)).",
"The resulting column densities are listed in Table REF .",
"To check if the assumption of optically thin lines is reasonable, we have estimated the line optical depths from the solution of the line radiative transfer equation: $\\tau =-{\\rm ln}\\left(1-\\frac{T_{\\rm MB}}{J_{\\nu }(T_{\\rm ex})-J_{\\nu }(T_{\\rm BG})}\\right)\\;$ and found values smaller than $\\simeq 0.6$ , consistent with our assumption of optically thin lines.",
"The CO depletion factor, $f_{\\rm D}$ , is defined as the ratio between the 'expected' abundance of CO relative to H$_2$ ($X_{\\rm C^{18}O}^{E}$ ) and the 'observed' value: $f_{\\rm D}=\\frac{X_{\\rm C^{18}O}^{E}}{X_{\\rm C^{18}O}^{O}}\\;\\;.$ $X_{\\rm C^{18}O}^{O}$ is the ratio between the observed C$^{18}$ O column density and the observed H$_2$ column density.",
"For this latter, we have used the value derived from the 1.2 mm continuum peak flux, $N_{p}({\\rm H_2})$ (Col. 7 of Table REF ), which is averaged over a beam comparable to that of the C$^{18}$ O observations (24$^{\\prime \\prime }$ against 19$^{\\prime \\prime }$ ).",
"To compute $X_{\\rm C^{18}O}^{E}$ , we have taken into account the variation of atomic carbon and oxygen abundances with distance to the Galactic Center following the same procedure adopted in Fontani et al.",
"([19]; see also Miettinen et al. [36]).",
"Assuming the canonical abundance of $\\sim 8.5 \\times 10^{-5}$ for the abundance of the main CO isotopologue in the neighbourhood of the solar system (Frerking et al.",
"[23], see also Langer et al.",
"[33] and Pineda et al.",
"[41]), we have computed the expected CO abundance at the Galactocentric distance ($D_{\\rm GC}$ ) of each source according to the relationship: $X_{\\rm C^{16}O}^{E}=8.5\\times 10^{-5}{\\rm exp}\\,(1.105-0.13\\,D_{\\rm GC}({\\rm kpc}))\\;,$ which has been derived according to the abundance gradients in the Galactic Disk for $^{12}$ C/H and $^{16}$ O/H listed in Table 1 of Wilson & Matteucci ([52]), and assuming that the Sun has a distance of 8.5 kpc from the Galactic Centre.",
"Then, following Wilson & Rood ([53]), we have assumed that the Oxygen isotope ratio $^{16}$ O/$^{18}$ O depends on $D_{\\rm GC}$ according to the relationship $^{16}$ O/$^{18}$ O=$58.8\\times D_{\\rm GC}({\\rm kpc})+37.1$ , which gives: $X_{\\rm C^{18}O}^{E}=\\frac{X_{\\rm C^{16}O}^{E}}{(58.8\\,D_{\\rm GC}+37.1)}\\;\\;\\;.$ The results are listed in Table REF : Cols.",
"2, 3 and 4 list the C$^{18}$ O (3–2) integrated line intensity ($I({\\rm C^{18}O})$ ), the C$^{18}$ O column density ($N({\\rm C^{18}O})$ ) and the observed C$^{18}$ O abundance ($X_{\\rm C^{18}O}^{O}$ ); Cols.",
"5 and 6 give the expected C$^{18}$ O abundance ($X_{\\rm C^{18}O}^{E}$ ) calculated at the Galactocentric distance of the source (Table REF ) and the CO depletion factor ($f_{\\rm D}=X_{\\rm C^{18}O}^{E}/X_{\\rm C^{18}O}^{O}$ ), respectively.",
"The mean $f_{\\rm D}$ is $\\sim 32$ and the median value is 29.",
"These values are remarkably higher than those obtained in other IRDCs (Pillai et al.",
"[39], Miettinen et al.",
"[36], Hernández et al.",
"[25]), and are among the highest obtained both in low-mass starless cores (Crapsi et al.",
"[15], Tafalla et al.",
"[49]) and in massive clumps from observations with low-angular resolution (Fontani et al.",
"[19], Chen et al. [14]).",
"Such a difference could be explained by the fact that the transition used in this work to derive $f_{\\rm D}$ has a critical density of $\\sim 5 \\times 10^{4}$ cm$^{-3}$ , comparable to that of the central region where the CO freeze-out is expected to be important.",
"In fact, the CO freeze-out timescale, which depends on the H$_2$ volume density, becomes shorter than the free-fall timescale at densities larger than a few $10^{4}$ cm$^{-3}$ (see e.g.",
"Bergin & Tafalla [4], Caselli [10]).",
"On the contrary, in most of the studies performed so far $f_{\\rm D}$ was derived from the CO (1–0) or (2–1) lines, which trace lower-density gas where CO is significantly non-depleted, resulting in values of $f_{\\rm D}$ smaller than or comparable to $\\sim 10$ (e.g.",
"Crapsi et al.",
"[15], Tafalla et al.",
"[49], Pillai et al. [39]).",
"However, the discussion of this result requires three main comments.",
"First, the values of the 'canonical' CO abundance measured by other authors in different objects vary by a factor of 2 (see e.g.",
"Lacy et al.",
"1994; Alves et al.",
"1999), but because the value assumed by us was derived from the Taurus star forming regions (Frerking et al.",
"[23]) and used in previous estimates (Crapsi et al.",
"[15]; Emprechtinger et al.",
"[16]), it is likely the most appropriate to make comparison with low-mass dense cores.",
"Second, the angular resolution of our observations allows us to derive only average values of $f_{\\rm D}$ over angular regions much larger than the typical fragmentation scales (few arcseconds at the distance of our targets), which in reality may have complex structures, so that our estimates should be considered as lower limits.",
"Finally, high depletion of C$^{18}$ O could be due to mechanisms other than freeze-out in regions where young stellar objects are embedded, and can decrease the CO abundance even in warm environments (for a detailed explanation, see Fuente et al. [24]).",
"This comment especially concerns the clumps where the average kinetic temperature exceeds the CO sublimation temperature of $\\sim 20$ K. Table: Parameters used to determine the CO depletion factor: integratedintensity of the C 18 ^{18}O (3–2) line (I(C 18 O))I({\\rm C^{18}O}))), C 18 ^{18}O total column density (N(C 18 O)N({\\rm C^{18}O})),observed C 18 ^{18}O abundance (X C 18 O O X_{\\rm C^{18}O}^{O}), 'expected'C 18 ^{18}O abundance (X C 18 O E X_{\\rm C^{18}O}^{E}), and CO depletion factor(f D f_{\\rm D} = X C 18 O E X_{\\rm C^{18}O}^{E}/X C 18 O O X_{\\rm C^{18}O}^{O}).The errors are given between parentheses.",
"The uncertainty on f D f_{\\rm D}based on the calibration errors affecting the C 18 ^{18}O and H 2 _2 column densities is of theorder of the 40-50%\\%."
],
[
"Virial state of the clumps",
"In order to investigate if our targets are gravitationally stable, we have computed the virial masses of the clumps assuming virial equilibrium and negligible contributions of magnetic field and surface pressure.",
"Assuming also that the cores are spherical, one can demonstrate that the gas mass is given by: $M_{\\rm VIR}(M_{\\odot })\\simeq k\\, \\Delta v^{2} {\\rm (\\mbox{km~s$^{-1}$})}\\,R ({\\rm pc})\\;\\;,$ where $R$ is the clump radius, $\\Delta v^{2}$ the line width and $k$ is a multiplicative factor that depends on the gas density distribution as a function of the distance to the clump centre (see Eq.",
"(3) in MacLaren et al. [35]).",
"For a homogeneous clump (i.e.",
"$\\rho = const.$ ), $k\\simeq 210$ , while for $\\rho \\sim r^{-2}$ , $k\\simeq 126$ .",
"The millimetre continuum maps do not allow to derive the density profile of the clumps because of the low angular resolution and sensitivity to extended emission.",
"However, previous studies suggest that the density structure of massive clumps reasonably can be approximated by a constant density in the inner regions, and by a steep power-law in the outer layers (see e.g.",
"Beuther et al.",
"[6], Fontani et al.",
"[21], Hill et al. [27]).",
"Figure: Virial masses against gas masses computed from the 1.2 mmdust continuum emission.",
"M VIR M_{\\rm VIR} is calculated assuming a density distributionof the type ρ=const.\\rho = const.",
"(left panel) and ρ∼r -2 \\rho \\sim r^{-2}(right panel).",
"In both panels, the line indicates M VIR M_{\\rm VIR}= MM.Figure: Virial masses against gas masses computed from the dust continuumemission for the clumps studied in this work (triangles) and the massiveclumps observed by López-Sepulcre et al.",
"(, squares).Filled and open triangles represent sources detected and undetected at24 μ\\mu m, respectively, while the filled and open squares representthe infrared-bright and infrared-dark sources selected by López-Sepulcre et al. (),respectively.",
"The cross indicatesthe clump with no 24 μ\\mu m image available, 13039–6108c6.M VIR M_{\\rm VIR} is calculated assuming homogeneous clumps.",
"The line indicates M VIR M_{\\rm VIR}= MM.In Fig.",
"REF we compare the mass derived from the dust continuum emission to $M_{\\rm VIR}$ obtained assuming $\\rho \\sim const.$ (left panel) and $\\rho \\sim r^{-2}$ (right panel).",
"We note that $M_{\\rm VIR}$ is on average larger than $M$ for the case $\\rho \\sim const.$ , while it is on average smaller for the other case.",
"Because the overall density distribution of the clumps is likely in between these two extreme cases, we suggest that, on average, the clumps are close to virial equilibium.",
"Our findings are in good agreement with those derived by López-Sepulcre et al.",
"([34]) in a sample of high-mass clumps supposed to span a wide range of evolutionary stages.",
"The gas masses of the high-mass clumps studied by López-Sepulcre et al.",
"([34]) were derived from millimetre continuum measurements, and the dust temperatures assumed are in agreement with ours (see their Table 1).",
"Moreover, they estimated the virial masses from C$^{18}$ O (2–1) line widths and assuming $\\rho \\sim const.$ As shown in Fig.",
"REF , we do not highlight systematic differences between the two sub-samples, and between infrared-dark and infrared-bright sources either.",
"On average, the sources of the López-Sepulcre et al.",
"'s sample appear to have slightly larger virial masses, so that potentially they might be less gravitationally bound than the clumps studied in this work.",
"However, we stress that the parameters from which the mass estimates are derived are affected by large errors (especially the dust mass opacity, the distance and the gas-to-dust ratio) and difficult to quantify.",
"For example, the dust mass opacity coefficient can introduce a factor 2 in the uncertainty (Beuther et al. [6]).",
"Hence our conclusions on the virial stability of the targets must be taken with caution."
],
[
"24 $\\mu $ m dark versus 24 {{formula:9dd68de2-fe1b-464d-a58f-d1c3d2ac397b}} m bright clumps",
"The presence of mid-infrared emission in molecular clumps is very often the sign of embedded star formation activity.",
"Therefore, the IRDCs detected at 24 $\\mu $ m could harbour objects more evolved than those undetected.",
"If this is the case, the two groups should have physical properties indicative of a different evolutionary stage.",
"In Fig.",
"REF we show histograms which compare some physical and chemical properties of the IRDCs detected and undetected at 24 $\\mu $ m: line widths, kinetic temperature, column ($N_{t}$ (H$_2$ )), volume and surface density of H$_2$ , gas masses (both from continuum and the virial theorem), CO depletion factor and clump diameter.",
"For $M_{\\rm VIR}$, we have considered the values calculated assuming $\\rho \\sim const$ , bearing in mind that those obtained in the case $\\rho \\sim r^{-2}$ are just systematically lower by a factor $\\sim 1.7$ .",
"For the unresolved source 16482–4443c2, we have not included the $n$ (H$_2$ ) and $\\Sigma $ (H$_2$ ) lower limits, and replaced $N_{t}$ (H$_2$ ) with $N_{p}$ (H$_2$ ).",
"Although the statistics is poor because the two sub-samples contain eight 24 $\\mu $ m bright and twelve 24 $\\mu $ m dark objects, the comparative histograms are useful to check if one can notice clear systematic differences.",
"An inspection of the comparative histograms indicates that the 24 $\\mu $ m dark clumps have lower $T_{\\rm k}$ and higher $f_{\\rm D}$ .",
"In fact, the average $T_{\\rm k}$ and $f_{\\rm D}$ of the infrared-bright group turns out to be $\\sim 20$ K and 28, respectively, while that of the infrared-dark group is $\\sim 16$ K and 35, respectively.",
"Also, if we exclude from the statistical analysis the clear outliers 17040–3959c1 and 16573–4214c2, the 24 $\\mu $ m dark clumps tend to have lower mass (average $M$ of $\\sim 260$ M$_\\odot $ from the 1.2 mm continuum), smaller source-averaged H$_2$ column density (average $N_{t}$ (H$_2$ ) of $8.9\\times 10^{22}$ cm$^{-2}$ ), and smaller H$_2$ surface density (average $\\Sigma $ (H$_2$ ) of 0.19 gr cm$^{-2}$ ) than the infrared-bright clumps (average $M$ of $\\sim 370$ M$_\\odot $, average $N_{t}$ (H$_2$ ) of $1.3\\times 10^{23}$ cm$^{-2}$ , average $\\Sigma $ (H$_2$ ) of 0.27 g cm$^{-2}$ ).",
"The fact that on average the group of the 24 $\\mu $ m-dark sources has lower $T_{\\rm k}$ and higher CO depletion factor than the 24 $\\mu $ m-bright objects suggests that it likely contains objects less evolved, and this is in accordance with other comparative studies of massive clumps with or without indications of embedded star formation (Hill et al. [27]).",
"However, the dark sources also have lower $M$ and H$_2$ column and surface density, and this could indicate that the 24 $\\mu $ m dark clumps may be destined to form less massive stars/clusters, thus explaining the non-detection at mid-infrared wavelengths, although material could still be accreting.",
"Figure: Histograms comparing some of the physical andchemical properties of the clumps detected(solid line) and undetected (dashed line) in the Spitzer-MIPS 24 μ\\mu mimage (see Figs. ).",
"In the panelsshowing NN(H 2 _2), nn(H 2 _2) and Σ\\Sigma (H 2 _2) wehave not included the lower limits calculated for 16482–4443c2."
],
[
"Relation between ",
"Two indicators of active star formation are the line width and the gas temperature, which are both expected to become higher with increasing star formation activity.",
"In fact, the line widths of dense gas tracers are found to be higher in more evolved star-forming clumps than in quiescent regions of infrared-dark clouds (e.g.",
"Hill et al.",
"[26], Ragan et al.",
"[42]), while the cores associated with protostars are on average warmer than the starless cores, both in low- and high-mass star forming regions (e.g.",
"Foster et al.",
"[22], Emprechtinger et al.",
"[16], Ragan et al. [43]).",
"Fig.",
"REF shows that there is a slight correlation between $T_{\\rm k}$ and the C$^{18}$ O line width.",
"If we exclude from the statistical analysis 15038–5828c1, for which $\\Delta V$ is much higher than that of the other targets and has a spectrum with poor S/N (see Fig.",
"REF ), a linear least square fit to the data yields a slope of $\\sim 0.09$ .",
"Statistical correlation between $T_{\\rm k}$ and line width can be investigated also through non-parametric statistical methods, like the Kendall's $\\tau $ correlation coefficient.",
"This measures the rank correlation, namely how two quantities are ordered similarly when ranked by each of themfor details see, e.g., http://www.statsoft.com/textbook/nonparametric-statistics/).",
"$\\tau $ can range between 1 (perfect agreement between the two rankings) and $-1$ (one ranking is the reverse of the other).",
"We find $\\tau \\sim 0.23$ between $T_{\\rm k}$ and line width.",
"These results suggest a faint correlation between the two parameters, which indicates that the warmer clumps tend to also have larger line widths, i.e.",
"tend to be more turbulent.",
"However, the large dispersion cannot allow us to draw any firm conclusion."
],
[
"Relation between CO depletion and other physical parameters",
"One of the main results of this work is the high CO depletion factor measured in our targets, with values higher than those found in comparable high-mass clumps (e.g.",
"Miettinen et al.",
"[36], Chen et al.",
"[14]), and even in low-mass pre–stellar cores (see e.g.",
"Crapsi et al. [15]).",
"The fact that we have derived $f_{\\rm D}$ from the C$^{18}$ O (3–2) transition, which has a critical density of $5 \\times 10^{4}$ cm$^{-3}$ , certainly makes the difference with respect to previous works in which $f_{\\rm D}$ was calculated from the lower excitation transitions tracing material where freeze-out of CO is expected to be less important.",
"In fact, freeze-out of CO and other neutrals onto dust grains is favoured in gas characterised by low temperatures ($T\\le 20~K$ ) and high densities ($n \\ge 10^{5}$ cm$^{-3}$ ), in which the freeze-out timescale is generally shorter than the free-fall timescale (e.g.",
"Bergin & Tafalla [4]).",
"In fact, high CO depletion factors (of the order of 10 or more) were measured towards the dense and cold nuclei of low-mass pre–stellar cores (see e.g.",
"Crapsi et al. [15]).",
"After the formation of the protostellar object(s) at the core centre, evaporation of CO starts and the CO depletion factor drops (Caselli et al. [12]).",
"This theoretical prediction has been partly confirmed by observations of both low- and high-mass dense cores (e.g.",
"Emprechtinger et al.",
"[16], Fontani et al. [19]).",
"In Fig.",
"REF we show $f_{\\rm D}$ against $T_{\\rm k}$ (in the left panel) and $n{\\rm (H_2)}$ (in the right panel).",
"At a first glance, the two plots do not reveal clear (anti-)correlations between the parameters.",
"As made in Sect.",
"REF , we have performed a closer inspection of the data using the non-parametric ranking statistical test Kendall's $\\tau $ (see Sect.",
"REF ).",
"If we consider all points, we find a faint anti-correlation between $f_{\\rm D}$ and $T_{\\rm k}$ (Kendall's $\\tau $ = –0.34) and also between $f_{\\rm D}$ and $n{\\rm (H_2)}$ (Kendall's $\\tau $ = –0.12).",
"The anti-correlation between $T_{\\rm k}$ and $f_{\\rm D}$ is consistent with the increase of CO depletion with decreasing temperature, as already found in both low- and high-mass dense cores (Emprechtinger et al.",
"[16], Fontani et al.",
"[19]), while that between $f_{\\rm D}$ and $n{\\rm (H_2)}$ is the opposite of what expected from chemical models.",
"Even if we exclude from the statistical analysis 16573–4214c2, which has $n{\\rm (H_2)}$ markedly much higher than that of the other members of the sample, the correlation between $f_{\\rm D}$ and $n{\\rm (H_2)}$ is not observed (Kendall's $\\tau $ = –0.11).",
"We point out, however, that all the parameters obtained are average values over angular regions much larger than the size expected for the embedded condensations, while the correlations are predicted for single cores.",
"Moreover, clumps embedded in different clouds may be affected by different environmental conditions, so increasing the dispersion.",
"In fact, Emprechtinger et al.",
"([16]) found that dense cores in Perseus showed the best trends between the physical parameters, whereas when including cores from other star-forming regions the dispersion was significantly larger.",
"Finally, Fig.",
"REF suggests that $f_{\\rm D}$ is not clearly (anti-)correlated to either the clump mass $M$ and and the ratio $M_{\\rm VIR}$/$M$ which should show the tendency of a clump to be stable against gravitational collapse.",
"When excluding the clear outlier 17040–3959c1 (whose mass is statistically much larger than the rest of the sample), a slight correlation is found between $f_{\\rm D}$ and $M$ (Kendall's $\\tau $ = 0.4).",
"However, again the large dispersion of the data and the big errors on both parameters do not allow us to claim firm conclusions.",
"Figure: Gas kinetic temperature, T k T_{\\rm k}, derived from ammonia, againstthe C 18 ^{18}O line widths, ΔV\\Delta V. The straight line is a least square fit to the dataexcluding 15038–5828c1, the clump with ΔV>5\\Delta V > 5 km s -1 ^{-1}, i.e.",
"much higher thanany other clump and based on a very noisy spectrum (see Fig.",
").The slope of the linear fit is ∼0.09\\sim 0.09.",
"Filled and open symbols representclumps detected and undetected at 24 μ\\mu m, respectively.",
"The cross indicatesthe clump with no 24 μ\\mu m image available, 13039–6108c6.",
"Typical errorbarsare depicted in the top-right corner.Figure: CO depletion factor (f D f_{\\rm D}) against gas kinetictemperature (left panel) and H 2 _2 volume density (right panel).The symbols have the same meaning as in Fig.",
".Figure: CO depletion factor (f D f_{\\rm D}) against gasmass derived from the 1.2 mm continuum (top panel) and M VIR M_{\\rm VIR}/MM mass ratio(bottom panel).",
"The symbols have the same meaning as in Fig.",
"."
],
[
"Conclusions",
"We have undertaken a molecular line study, through observations of C$^{18}$ O (3–2), NH$_3$ (1,1) and (2,2), N$_{2}$ H$^{+}$ (3–2) and N$_{2}$ D$^{+}$ (4–3), of 21 IRDCs in the southern hemisphere with the aim of characterising the initial conditions of the high-mass star and cluster formation process.",
"The sample targeted was selected from the high-mass millimetre clumps not detected in the MSX images identified by Beltrán et al.",
"(2006), and includes sources with and without Spitzer 24 $\\mu $ m emission.",
"The NH$_3$ inversion transitions have been observed with the ATCA.",
"The rotational transitions of the other molecular species have been observed with the APEX Telescope.",
"We have detected C$^{18}$ O and ammonia emission in all clumps, and N$_{2}$ H$^{+}$ emission in all the 12 sources observed in this line.",
"Only one source has been marginally detected in N$_{2}$ D$^{+}$ (4–3), which is, to our knowledge, the first detection of this line in an IRDC.",
"The clumps have a median mass of $\\sim 244$ M$_\\odot $, appear to be gravitationally bound and possess mass, H$_2$ column and surface densities consistent with being potentially the birthplace of high-mass stars.",
"The most striking result of the work is the high average CO depletion factor (derived from the expected C$^{18}$ O-to-H$_2$ column density ratio compared to the observed value), which is in between 5 and 78, with a mean value of 32 and a median of 29.",
"These values, derived from the C$^{18}$ O (3–2) transition (which traces gas denser than the lower-excitation lines commonly used in previous works), are larger than the typical CO depletion factors measured towards other IRDCs, and are comparable to or larger than the values derived in the low-mass pre–stellar cores closest to the onset of gravitational collapse.",
"Also, a faint anti-correlation is found between $f_{\\rm D}$ and the gas kinetic temperature.",
"These results suggest that the earliest phases of the high-mass star and stellar cluster formation process are characterised by CO depletions larger than in low-mass pre–stellar cores.",
"The clumps have an average temperature around 17 K. We have found marginal statistical differences between the clumps detected and undetected in the 24 $\\mu $ m Spitzer images, with the Spitzer-dark clumps being on average colder, less massive and having lower H$_2$ column and surface densities, but higher CO depletion factors.",
"This indicates that the Spitzer-dark clumps are either less evolved or destined to form stars and stellar clusters less massive than the Spitzer-bright ones."
],
[
"Acknowledgments",
"This publication is based on data acquired with the Atacama Pathfinder Experiment (APEX).",
"The Atacama Pathfinder Experiment is a collaboration between the Max-Planck-Institut für Radioastronomie, the European Southern Observatory, and the Onsala Space Observatory.",
"We thank the staff at the APEX telescope for performing the service mode observations presented in this paper.",
"FF is deeply grateful to Ana López-Sepulcre for providing us with the parameters of the massive cores in López-Sepulcre et al.",
"([34])."
]
] | 1204.0907 |
[
[
"Encoding Universal Computation in the Ground States of Ising Lattices"
],
[
"Abstract We characterize the set of ground states that can be synthesized by classical 2-body Ising Hamiltonians.",
"We then construct simple Ising planar blocks that simulates efficiently a universal set of logic gates and connections, and hence any boolean function.",
"We therefore provide a new method of encoding universal computation in the ground states of Ising lattices, and a simpler alternative demonstration of the known fact that finding the ground state of a finite Ising spin glass model is NP complete.",
"We relate this with our previous result about emergence properties in infinite lattices."
],
[
"Introduction",
"The Physical Church-Turing thesis [1] provides a deep connection between the science of computation and the physical universe.",
"It posits that the dynamics of any known physical system can be simulated by a Turing machine [2], a theoretical device that consists of a finite state machine together with an infinite tape.",
"Upon reflection, this is a remarkable result, widely believed to be correct.",
"An arbitrary physical system is governed by a vast variety of different forces, from Coulomb interactions to gravity, and there is no reason, a priori, to suspect that all of these effects can be replicated on one particular machine.",
"This presents the idea of universality: a physical system is universal if its dynamics can be used to simulate any other physical system.",
"The prevalence of universality in commonly studied systems is not only a theoretical curiosity, but also has consequences of practical significance.",
"Recent results in computer science restrict our ability to predict the behavior of such systems.",
"Observations of universal systems led to the Strong Church-Turing thesis [3], which postulates that a Turing Machine together with a source of randomness is computationally as powerful as any other existing universal system.",
"Formally speaking, we say that a task lies in P, or is tractable, if the task can be performed efficiently by a Turing Machine, i.e., the time required to perform it scales as a polynomial of the size of the input [4].",
"This thesis then postulates that any problem which lies outside P cannot be solved with resources that scale polynomially with respect to the size of the problem, regardless of the method of computation used.",
"While the existence of Shor's algorithm in quantum mechanics may provide an exception to this thesis [5], it applies to all current classical models of computation.",
"This leads to deep insights into any universal system that simulates a Turing machine efficiently.",
"Suppose such a system simulates a Turing machine operating on an intractable problem as input.",
"If one could efficiently compute every physical property of this system, then one can use it to solve the encoded problem and therefore violate the Strong Church-Turing thesis.",
"Thus, such universal systems must necessarily exhibit properties which no classical algorithm can efficiently compute.",
"Many other universal systems have been proposed, for example, logic circuits [6], the Game of Life [7], Rule 110 [8], and measurement based quantum computation [9].",
"In addition to these abstract mathematical constructs, many surprisingly simple physical systems capable of universal computation have also been discovered.",
"These include billiard balls [10], simple dynamical systems [11] and the dynamics of 3-dimensional majority voting cellular automata [12].",
"This motivates an interesting question: how simple can a physical system be to still exhibit universality and thus complex behaviour?",
"In particular, we explore what limits can be placed on a class of Hamiltonians such that the evaluation of their ground states still requires the capacity to perform universal computation.",
"We relate this to the ground state decision problem: given a Hamiltonian $H$ and some number $E$ , does there exist a state with energy at most $E$ ?",
"Interestingly, the ground state decision problem is difficult to solve even for the simple Ising lattice, which is a widely used model to describe collective behaviour in diverse systems, as magnetism [13], lattice gases [14], neural activity [15] and even protein folding [16].",
"While an efficient solution is known in the case of one dimension, F. Barahona showed in 1982 that the computational task is generally NP-complete in higher dimensions [17], whenever some of the bonds are antiferromagnetic.",
"Here, NP denotes the class of non-deterministic polynomial time problems; an abstract class of problems whose solutions can be verified, but not necessarily found, in polynomial time.",
"Indeed, this connection has even allowed the engineering of spin lattice Hamiltonians whose ground states help model and study NP-complete problems [18].",
"The complexity of the ground state decision problem suggested that such ground states could also embed universal computation.",
"Indeed, this was first proven with the adiabatical model of quantum computation, where a simple Hamiltonian with known ground state is adiabatically evolved to the complex Hamiltonian whose ground state encodes the solution to the computational problem [19], [20].",
"To further simplify the models and make them more suitable to be recreated in real experiments, it has been proven that it is enough to consider just 2-body interactions in the Hamiltonian to obtain the capability of universal computation [21], [22], [23], [24].",
"In this paper we extend those studies in the classical case and derive a general result on what ground state sets can be synthesized by a $m$ -body Hamiltonian on a system of $n$ spins.",
"Using the circuit model of computation, we construct simple designer circuit blocks that can be combined to encode a universal computer in the ground state of 2-body Ising Hamiltonians, in such a way that there is a map between any given logic circuit to the ground states of some Hamiltonian.",
"This encoding, together with the strong Church-Turing thesis, provides immediate implications on the computational complexity of evaluating such ground states.",
"Furthermore, this allows us to provide a simple alternative proof of Barahona's result that the ground state decision problem is NP-complete [17].",
"We explore the connection of this result with the infinite lattice case we studied in a previous work [25].",
"We showed that there are undecidable properties in the infinite Ising model that give rise to emergent properties in the physical Ising lattice.",
"Besides, the circuit blocks presented here simplify the technical parts of that work.",
"This paper is organized as follows.",
"Section introduces the required background and notation.",
"Section introduced the ground synthesis problem, whilst Section gives an alternate proof of the universality of Ising ground states.",
"Section explores the consequences in complexity of the computational difficulty of the ground state problem and the the relation with the infinite case and emergence.",
"Section presents the main conclusions."
],
[
"Background and Notation",
"To explore how ground states can embed universal computation, we first address a related practical problem of ground state synthesis, i.e., given a set of desired states, is it possible to engineer a Hamiltonian whose set of ground states correspond to those in the desired set?",
"In particular, when reality dictates certain limits on the interactions available, what are the corresponding restrictions on the possible ground states that can be achieved?",
"For example, denote the state of each spin by either 0 or 1, is it possible to find a Hamiltonian with ground states given by $\\lbrace 000, 011, 101, 110\\rbrace $ ?",
"If so, is it possible to engineer this Hamiltonian from Ising interactions?",
"The solution to the above question gives us the tools to engineer a set of states that are capable of encoding a universal circuit.",
"Let us first define the nomenclature used in this paper.",
"Denote the state of each spin by either 0 or 1.",
"A system of $n$ spins is described by a binary number $\\mathbf {b} = b_1\\ldots b_n \\in \\mathbb {Z}_2^n$ , where $b_i \\in \\lbrace 0,1\\rbrace $ denotes the state of the $i^{th}$ spin.",
"Given a state $\\mathbf {b}$ , we make the following definitions: Weight: $|\\mathbf {b}|$ is the number of 1s in $\\mathbf {b}$ .",
"1-Sites Ones(b) is the set of indices whose corresponding spins are 1.",
"$\\mathrm {Ones}(\\mathbf {b}) := \\lbrace i: \\, b_i = 1\\rbrace $ .",
"Descendant $\\mathbf {a}$ is a descendent of $\\mathbf {b}$ iff $\\mathrm {Ones}(\\mathbf {a}) \\subseteq \\mathrm {Ones}(\\mathbf {b})$ , i.e., the 1-sites of $\\mathbf {a}$ are subsets of 1-sites of $\\mathbf {b}$ .",
"We write this as a partial order, $\\mathbf {a} \\preceq \\mathbf {b}$ .",
"$\\mathrm {Dsc}(\\mathbf {b})$ defines the set of all descendants of $\\mathbf {b}$ , and $\\mathrm {Dsc}(\\mathbf {b},k) := \\lbrace \\mathbf {a}: \\mathbf {a} \\preceq \\mathbf {b}, |\\mathbf {a}| = k\\rbrace $ are all descendants of $\\mathbf {b}$ with weight $k$ .",
"A Hamiltonian on this system is defined by a function $H: \\mathbb {Z}_2^n \\rightarrow \\mathbb {R}$ that maps each state of the system to a corresponding energy.",
"A general Hamiltonian is of the form: $H(b_1,\\ldots ,b_n) =\\sum _{\\mathbf {\\mathbf {a}} \\in \\mathbb {Z}_2^n}c_\\mathbf {\\mathbf {a}}b_1^{a_1}b_2^{a_2}\\ldots b_n^{a_n},$ where $\\mathbf {a} = a_1a_2\\ldots a_n \\in \\mathbb {Z}_2^n$ , $a_i \\in \\lbrace 0,1\\rbrace $ , $c_{\\mathbf {a}}$ are arbitrary constants, and the summation is taken over all binary strings of length $n$ .",
"Since we can always choose a labeling of the spin states such that one of the ground state corresponds to $\\mathbf {0}$ , we assert that $\\mathbf {0}$ is a ground state of $H$ (i.e.",
"$H(\\mathbf {0}) = 0$ ) without loss of generality.",
"A Hamiltonian $H$ is $m$ -body if it does not contain interactions involving $m+1$ spins or greater, i.e: $c_{\\mathbf {a}} = 0$ $\\forall \\mathbf {a}$ such that $|\\mathbf {a}| > m$ .",
"The general Ising model with an external magnetic field is a 2-body Hamiltonian of the form [14]: $H = \\sum c_{jk} b_jb_k + \\sum M_j b_j,$ where $c_{jk}$ are the interaction energies between spins $j$ and $k$ , and $M_j$ describes the external field at site $j$ .",
"Interaction graphs provide a convenient tool to visualize Ising Hamiltonians.",
"Given a system of $n$ spins, we associate with it a graph of $n$ vertices where each spin corresponds to a single vertex.",
"We draw an edge between two vertices $v_i$ and $v_j$ if the interaction energy between them, $c_{jk}$ is non-zero.",
"A square Ising model of size $N$ is described by an interaction graph with vertices $v_{j,k}$ where $j,k = 1,\\ldots ,N$ , with edge set $E = \\lbrace (v_{j,k},v_{j+1,k}), (v_{j,k},v_{j,k+1})\\rbrace $ with $j,k = 1,\\ldots ,N+1$ .",
"The main idea of our approach is as follows.",
"To embed a binary function on two bits $b_{out} = f(b_1,b_2)$ , we construct a Hamiltonian $H_f$ on $b_1$ ,$b_2$ ,$b_{out}$ with the ground state set $\\mathcal {G}_{f} = \\lbrace 00f(00),01f(01),10f(10),11f(11)\\rbrace .$ We see that each element of $\\mathcal {G}_{f}$ satisfy $b_{out} = f(b_1,b_2)$ .",
"We define the spins in state $b_1$ and $b_2$ as input spins, and the bit in state $b_{out}$ as the output spin.",
"We say that the ground state $\\mathcal {G}_{f}$ encodes $f$ .",
"We can then evaluate the action of $f$ on particular input, i.e., $f(x,y)$ , by introducing the external biases on the input spins that breaks the degeneracy of $H_f$ such that the state $x,yf(x,y)$ has lower energy than the other elements of $\\mathcal {G}$ .",
"For example, the Hamiltonian $H_{f(00)} = H_f + b_1 + b_2$ would have the unique ground state $\\lbrace 00f(00)\\rbrace $ .",
"Therefore, cooling such a system to ground state would allow us evaluate $f(0,0)$ , and the computational task of solving for a ground state of this system is at least as hard as evaluate $f(0,0)$ ."
],
[
"Ground State Synthesis",
"This motivates the problem of ground state synthesis, i.e., given a set of desired states, is it possible to engineer an $m$ -body Hamiltonian with a coinciding set of ground states, and if so, how?",
"The answer of this question can be directly applied to designer ground states, a set of ground states $G_f$ specifically designed to encode a desired binary function $f$ .",
"Should we be able to construct $m$ -body Hamiltonians for arbitrary $f$ , we can establish the universality of the Ising model.",
"We can represent $H(\\mathbf {b})$ and $c_{\\mathbf {b}}$ as vectors in $\\mathbb {R}^{2n}$ , where their components are indexed by all possible values of $\\mathbf {b} \\in \\lbrace 0,1\\rbrace ^{n}$ .",
"Eq.",
"(REF ) implies that $H(\\mathbf {b}) = L c_{\\mathbf {b}}$ , where $L$ is some invertible linear map.",
"Thus, the restriction of $H$ to $m$ -body interactions leads to a set of linear equations that constrain $H(\\mathbf {b})$ .",
"More precisely, $H$ is an $m$ -body Hamiltonian iff for each $H(\\mathbf {b})$ with $|\\mathbf {b}| = k > m$ , $H(\\mathbf {b}) &= \\sum _{p=1}^m a_p \\left[ \\sum _{\\mathbf {d} \\in \\mathrm {Dsc}(\\textbf {b},p)} H(\\mathbf {d}) \\right],$ where $a_p$ is given by the recurrence relation (see appendix): $a_p = \\left\\lbrace \\begin{array}{ll} 1 & p = m\\\\1 - \\sum _{j=1}^{m-p}a_{p+j}\\left(\\begin{array}{c}|\\mathbf {b}|-p \\\\j \\\\\\end{array} \\right) & 1 \\le p < m\\\\\\end{array} \\right.",
"$ This leads immediately to constraints on the ground state set $\\mathcal {G}$ if it can be $m$ -synthesized: Theorem 1 Suppose $H$ is an $m$ -body Hamiltonian on a system of $n$ spins.",
"For each $\\mathbf {b}$ with $|\\mathbf {b}| = k > m$ , define the sets $\\mathcal {A} = \\lbrace \\mathbf {b}\\rbrace \\cup \\mathrm {Dsc}(\\mathbf {b},m-1) \\cup \\mathrm {Dsc}(\\mathbf {b},m-3) \\cup \\ldots $ and $\\mathcal {B} = \\mathrm {Dsc}(\\mathbf {b},m) \\cup \\mathrm {Dsc}(\\mathbf {b},m-2) \\cup \\ldots $ Then the ground state set $\\mathcal {G}$ of $H$ must satisfy: $\\mathcal {A} \\subset \\mathcal {G} \\Leftrightarrow \\mathcal {B} \\subset \\mathcal {G}$ for every $\\mathbf {b}$ with $k > m$ .",
"Proof: Observe that $a_p$ alternates signs for each value of $p$ in Eq.",
"(REF ), thus we can write Eq.",
"$(\\ref {eqn:hrelmain})$ in the form $\\sum _{\\mathbf {b} \\in \\mathcal {A}} c_{\\mathbf {b}} H(\\mathbf {b}) = \\sum _{\\mathbf {b} \\in \\mathcal {B}} c_{\\mathbf {b}} H(\\mathbf {b})$ .",
"If $\\mathcal {A} \\subset \\mathcal {G}$ , then the left hand of this equation is 0.",
"Since $H(\\mathbf {b}) \\ge 0$ by assumption, it follows that the right hand side must also be 0, and vice versa.",
"$\\Box $ This theorem immediately implies that restrictions to $m$ -body Hamiltonians, for any $m$ , will also restrict the sets of ground states that we can synthesize.",
"In particular, an $m$ -body can only implement $m$ -wise correlations.",
"Consider for example the case of an $n$ -body system, then any ground state set $\\mathcal {G}$ that does not satisfy $\\lbrace \\mathbf {b}: \\mathrm {wt}(\\mathbf {b}) \\textrm { odd} \\rbrace \\subseteq \\mathcal {G} \\Leftrightarrow \\lbrace \\mathbf {b}: \\mathrm {wt}(\\mathrm {b})\\textrm { even}\\rbrace \\subseteq \\mathcal {G}$ can only be synthesized by a Hamiltonian with all $n$ bodies interacting together.",
"One observes that the ground state set corresponding to the parity function on a binary string (i.e: $f(\\mathbf {b}) = |\\mathbf {b}| \\mod {2}$ ) violates the above condition, and hence cannot be simulated by any 2-body Hamiltonian.",
"Thus, we cannot simulate all binary functions directly.",
"The above problem can be circumvented by introducing ancillae, additional bits within the Ising lattice that are not designated as either input or output bits.",
"For example, consider simulation of the NAND gate, defined by $\\mathrm {NAND}(b_1,b_2) = (b_1 \\otimes b_2) \\oplus 1$ , where all arithmetic is done modulo 2.",
"Directly, a Hamiltonian $H_{NAND}$ with ground state set $\\mathcal {G}_{\\mathrm {NAND}} = \\lbrace 001, 011, 101, 110\\rbrace $ simulates NAND.",
"However, NAND can also be simulated any Hamiltonian on $k + 3$ spins, with a ground state set of the form $\\mathcal {G} = \\lbrace 00\\mathbf {s}_{00}1, 01\\mathbf {s}_{01}1, 10\\mathbf {s}_{10}1, 11\\mathbf {s}_{11}0\\rbrace $ , where each $\\mathbf {s}_{ij}$ denote binary strings of length $k$ .",
"Now consider binary functions $f$ , $g$ , $h$ , simulated by Hamiltonians $H_f,H_g,H_h$ , with outputs $b_f$ , $b_g$ and $b_h$ .",
"The functional composition $f(g(b_1,b_2),h(b_3,b_4))$ on the four input bits $b_i$ where $i = 1,\\ldots 4$ , can be simulated by the Hamiltonian $H_g(b_1,b_2,b_g) + H_h(b_3,b_4,b_h) + H_f(b_g,b_h,b_{out})$ , where $b_g$ and $b_h$ are introduced as ancillae."
],
[
"Universality of Ising Ground States",
"An arbitrary Boolean circuit that takes $n$ input bits and maps them to $m$ output bits can be decomposed a basic logic circuit composed of the following components: Wires that takes a spin as input, and copies its state to a neighboring spin; and NAND Gate that can generate all Boolean functions.",
"These require the synthesis of the following ground state sets $\\mathcal {G}_{WIRE} = \\lbrace 00, 11\\rbrace $ and $\\mathcal {G}_{NAND} = \\lbrace 001, 011, 101, 110\\rbrace $ (In standard literature, the FANOUT gate that copies an input bit onto two outputs spins is also normally required.",
"However, this operation can be decomposed in spin systems into two wires that connect to the same input spin.)",
"We the convert this to a planar circuit, that is, one in which no wires may intersect.",
"This requires the replacement of each section where a wires intersects with a SWAP gate, $\\mathrm {SWAP}(b_1,b_2) = (b_2,b_1)$ .",
"We observe that this operation can be decomposed into a network of three XOR gates i.e: $\\mathrm {SWAP}(b_1,b_2) = \\mathrm {XOR}_1(\\mathrm {XOR}_2(\\mathrm {XOR}_1(b_1,b_2)))$ , where $\\mathrm {XOR}_1(b_1,b_2) = (b_1 \\oplus b_2, b_2)$ and $\\mathrm {XOR}_2(b_1,b_2) = (b_1 , b_1 \\otimes b_2)$ .",
"We call this the planar circuit representation of $f$ .",
"Therefore, we can construct a square Ising Hamiltonian that synthesizes $f$ provided there exists square Ising Hamiltonians that implement each of basic aforementioned components, i.e., (1) wires, (2) NAND gates and (3) XOR gates.",
"To see that each of these can be simulated by a 2-body Hamiltonian, we prove the following lemma: Lemma 2 Given a set of states $\\mathcal {G}$ on a system of three spins with $000 \\in \\mathcal {G}$ , there exists a 2-body Hamiltonian that synthesizes $\\mathcal {G}$ if and only if $\\lbrace 111\\rbrace \\cup \\mathrm {Desc}(111,1) \\subseteq \\mathcal {G} \\Leftrightarrow \\mathrm {Desc}(111,2) \\subseteq \\mathcal {G}.$ Proof: The forward direction is special case of Eq.",
"(REF ) for $n = 3$ .",
"To observe the converse, assume Eq.",
"(REF ) is true.",
"Eq.",
"(REF ) implies that $H$ is a 2-body Hamiltonian iff $H$ satisfies: $\\sum _{\\mathbf {b} \\in \\mathcal {A}}H(\\mathbf {b}) = \\sum _{\\mathbf {b} \\in \\mathcal {B}} H(\\mathbf {b}),$ where $\\mathcal {A} = \\lbrace 111\\rbrace \\cup \\mathrm {Desc}(111,1)$ and $\\mathcal {B} = \\mathrm {Desc}(111,2)$ .",
"To see that Eq.",
"($\\ref {eqn:H32rel}$ ) is true, observe that if $\\mathcal {A}, \\mathcal {B} \\subseteq \\mathcal {G}$ then Eq.",
"(REF ) is satisfied trivially.",
"Otherwise, construct the Ising Hamiltonian that has assignments $H(\\mathbf {b}) = \\frac{1}{|\\mathcal {A}/\\mathcal {G}|}, \\ \\ \\ \\ \\ \\ H(\\mathbf {d}) = \\frac{1}{|\\mathcal {B}/\\mathcal {G}|},$ for all $\\mathbf {b} \\in \\mathcal {A}/\\mathcal {G}$ , $\\mathbf {d} \\in \\mathcal {B}/\\mathcal {G}$ .",
"Here $|\\mathcal {A}/\\mathcal {G}|$ is the number of elements that lie in $\\mathcal {A}$ but outside $\\mathcal {G}$ .",
"$\\Box $ The above lemma gives us a method to construct all the elements of a universal circuit from 2-body nearest neighbor Hamiltonians.",
"Wires can be simulated through $H_{I} = b_1 + b_2 - 2b_1b_2$ .",
"Lemma REF implies that the NAND can be simulated directly (to see NAND can be simulated, relabel the third bit).",
"XOR cannot be implemented by the ground state of a 2-body Hamiltonian on three spins.",
"However, the Hamiltonian on four spins $\\nonumber H_{XOR}(b_1,b_2,b_A,b_o) &= (4 b_A - 3)(b_1 + b_2 + b_o) - 4 b_A\\\\& + 2(b_1b_2 + b_2b_o + b_1b_o) + 4$ with ground states $\\lbrace 0010,0101,1001,1100\\rbrace $ simulates XOR using $b_A$ as an ancilla.",
"Thus, all the above gates can be simulated by two-body Hamiltonians.",
"Since these Hamiltonians also involve at most four spins, their interaction graphs must also be planar with vertices of degree at most three.",
"Thus they can all be embedded in a square Ising Lattice with additional ancillae (See Fig.",
"REF ), and hence so can $f$ .",
"Finally, we observe that each gate can be simulated by a Hamiltonian on at most $k$ spins, where $k$ is a fixed number.",
"Thus, the number of spins used to simulate $f$ is at most some polynomial for the number of logic gates used to construct $f$ .",
"Therefore, the square Ising model can simulate an arbitrary circuit efficiently.",
"Figure: The Hamiltonian that synthesizes the XOR Gate, H XOR H_{XOR}, with its corresponding interaction graph (i) can be embedded into a 3×33\\times 3 square Ising Lattice.",
"(ii) Each original spin b i b_i is mapped to a set of spins B i B_i which are linked by H WIRE H_{WIRE} interactions.",
"At the ground state, all spins in each set B i B_i are of the same state, and hence behave as if they are a single bit.Theorem 3 Consider an arbitrary binary function $f$ .",
"There always exists a square Ising Hamiltonian $H$ whose ground states encode $f$ .",
"The above theorem allows us to encode any logic circuit, and thus computational task, into the ground state of an Ising Hamiltonian.",
"Not only is it remarkable that the ground state of such simple lattices are capable of simulating all physical processes, but this fact also allows us to apply the many results of computational complexity directly onto the task for computing ground states for an Ising Hamiltonian."
],
[
"Computational Complexity and Emergence",
"Any Boolean function $f$ can be encoded as the ground state of an Ising Hamiltonian $H_f$ .",
"Suppose now that $f$ is intractable, then the Strong Church-Turing thesis would necessarily imply that computing a ground state of $H_f$ would also be intractable.",
"In fact, the assertion is stronger.",
"Since we can potentially encode the output of $f$ in the state of any spin state, the process of determining the ground state of any particular spin would also be intractable.",
"In this final section, we will use the above intuition to provide lower bounds on the computational difficulty of the ground state problem, i.e., finding the ground state of some suitable two-dimensional, nearest neighbor Ising Hamiltonian.",
"In computational complexity [4], NP denotes the class of problems whose solutions can be verified, but not necessarily found, in polynomial time.",
"It encapsulates many computational tasks that we would like to be able to solve efficiently, such as prime factoring and the traveling salesman problem [26].",
"The hardest of such problems lie in the class NP-complete.",
"Should any NP-complete problem be solved efficiently, then it could be used as a subroutine to efficiently solve all problems in NP and imply that $\\mathrm {P} = \\mathrm {NP}$ .",
"While, this remains one of the biggest theoretical questions in computer science, popular opinion tends to favor that P is distinct from NP, and hence efficient solutions of NP-complete are unlikely.",
"One particular well known NP-complete problem is the circuit satisfiability (CSAT) problem [27]: given a circuit with $n$ input bits and a single output bit described by a binary function $f$ , is there a set of inputs such that the output is 1?",
"Consider a given CSAT problem with a circuit $f$ .",
"Theorem REF implies that we can construct a Hamiltonian $H_f$ together with a predefined output bit $b_o$ such that $b_o = f(x)$ for any ground state of $H_f$ .",
"Since we can modify the any Hamiltonian by a constant without affecting its set of ground states, we can always choose $H_f$ such that its ground state energy is 0.",
"Consider the ground state decision problem, does there exist a state with energy at most 0 under the Hamiltonian $H^{\\prime }_f = H_f + (1 - b_{out})$ ?",
"The perturbation $1 - b_{out}$ lifts the degeneracy in $H_f$ such that the resulting Hamiltonian $H^{\\prime }_f$ will have a zero energy state iff there is a set of inputs to $f$ such that it outputs 1.",
"Therefore, knowledge of the ground state of $H_f$ and hence $b_o$ clearly allows one to solve CSAT.",
"Therefore, the ground state decision problem is at least NP-hard.",
"Furthermore, since $H_f$ is a Hamiltonian on a square Ising lattice that grows at most polynomially with the size of the circuit, it is easy to check whether the energy of a given state is greater than 0.",
"Thus the ground state decision problem is NP-complete.",
"We see that the above result, originally derived by Barahona [17], flows as a natural consequence of applying Ising lattices to solve a particular NP-complete problem.",
"It is stimulating to speculate then, what other important results could be obtained by applying the Ising model to other non-trivial computational problems.",
"The Halting problem [2] is an exciting candidate; it and its generalizations [28] prove that there exist many properties of Turing machines that are undecidable.",
"Such properties would necessarily correspond to certain properties of the Ising model, and it would be interesting to see if these properties are physically relevant.",
"Another promising avenue of research is to consider what the limitations on the computation of ground states imply about the macroscopic properties of the resulting Ising lattice.",
"For example, it is easy to see how our results can be extended to show that computing the correlation length of such Ising lattices is also NP-complete.",
"This leads to the concept of emergence in the infinite case in [25], following the path established by P. Anderson in 1972 with his celebrated paper `More is Different' [29], where he postulated that the ground state of a spin glass may be non-computable.",
"Emergent properties of a physical system are properties which arise from the whole and are not deducible from the physical interactions of the component parts.",
"In `More Really Is Different' [25], a special case of this technique was applied to show that certain macroscopic properties of a properly chosen, 2-dimensional, infinite periodic Ising lattice are emergent.",
"That is, it is possible to embed universal circuits within infinite periodic Ising lattices, such that should certain macroscopic properties be computed, one would be able to decided whether a arbitrary computer program would halt.",
"The result naturally motivated the question: “What would happen should such lattices be finite?”.",
"In this paper we see that in such sceneries these emergent macroscopic properties are connected with the known NP-complete properties of finite lattice Ising spin glasses.",
"This relation (infinite $\\rightarrow $ undecidable, finite $\\rightarrow $ NP-complete) was previously proved as well in planar tiling problems [30], what suggests that it could be a common feature of complex universal systems."
],
[
"Conclusion",
"We have derived the general conditions for a desired set of states to be the ground state of a classical Hamiltonian constrained to interact with a finite number of spins—including 2-body interactions, i.e., the Ising Model.",
"We have presented a new and simple way of encoding universal circuit computation in the ground states of Ising lattices through the construction of Ising blocks that implement the necessary logical gates and connections.",
"This result can be immediately applied to derive a simple version of Barahona's original proof [17] that the problem of finding states on Ising Hamiltonians is, in general, NP-complete.",
"We thank Michael Nielsen for suggesting us the topic of this work and acknowledge fruitful discussions with Christian Weedbrook, Jacob Biamonte and Chip Neville."
],
[
"Derivation of equation (",
"We first define $g^\\mathbf {b}_c(a_1,a_2,\\ldots ,a_k)&= \\sum _{p=1}^k a_p \\left[ \\sum _{\\mathbf {d} \\in \\mathrm {Dsc}(b,p)} H(\\mathbf {d}) \\right]$ Thus for any $\\mathbf {b}$ such that $\\Vert \\mathbf {b}\\Vert > m$ , we have the relation: $H(\\mathbf {b}) &= g^{\\mathbf {b}}_c\\left(1,1,\\ldots ,\\alpha _m = 1, 0, \\ldots ,\\alpha _k = 0\\right) \\nonumber \\\\&= \\sum _{k=m+1}^{\\Vert \\mathbf {b}\\Vert -1} \\left( \\sum _{\\mathbf {d} \\in \\mathrm {Dsc}(\\mathbf {b},k)}c_{\\mathbf {d}} \\right) \\nonumber \\\\& + g^{\\mathbf {b}}_c\\left(1,1,\\ldots ,\\alpha _m = 1, 0, \\ldots ,\\alpha _k = 0\\right)$ Now, we note the fact $\\sum _{\\mathbf {d} \\in \\mathrm {Dsc}(b,m)} H(\\mathbf {d}) =g^{\\mathbf {b}}_c\\left(\\beta _1,\\beta _2,\\ldots ,\\beta _{m-1},0\\ldots ,0\\right)$ To compute $\\beta _j$ , Consider $H(\\mathbf {d})$ which has exactly $^mC_j$ terms of the form $c_\\mathbf {d^{\\prime }}$ with $\\Vert \\mathbf {d^{\\prime }}\\Vert = m$ .",
"Also there exists $^{\\Vert \\mathbf {b}\\Vert }C_m$ terms of the form $H(\\mathbf {d})$ .",
"Thus to total number $c_\\mathbf {d^{\\prime }}$ terms is $^{\\Vert \\mathbf {b}\\Vert }C_m ^mC_j$ .",
"Dividing this by the total number of $c_\\mathbf {d^{\\prime }}$ that are descendant from $\\mathbf {b}$ gives: $\\beta _j =\\frac{\\left(\\begin{array}{c}\\Vert \\mathbf {b}\\Vert \\\\m \\\\\\end{array} \\right)\\left(\\begin{array}{c}m \\\\j \\\\\\end{array} \\right)}{\\left(\\begin{array}{c}\\Vert \\mathbf {b}\\Vert \\\\j \\\\\\end{array} \\right)} = \\left(\\begin{array}{c}\\Vert \\mathbf {b}\\Vert -j \\\\\\Vert \\mathbf {b}\\Vert -m \\\\\\end{array} \\right) \\qquad 1\\le j \\le m$ So that $\\sum _{\\mathbf {d} \\in \\mathrm {Dsc}(b,m)} H(\\mathbf {d}) & = g^{\\mathbf {b}}_c \\bigg [ \\left(\\begin{array}{c}k-1 \\\\k-m \\\\\\end{array} \\right),\\left(\\begin{array}{c}k-2 \\\\k-m \\\\\\end{array} \\right),\\ldots \\nonumber \\\\& \\ldots ,k-m+1,1,0\\ldots ,0 \\bigg ]$ Substituting into Eq.",
"(REF ) $H(\\mathbf {b})= & \\sum _{\\mathbf {d} \\in \\mathrm {Dsc}(b,m)}H(\\mathbf {d}) + \\nonumber \\\\& g^{\\mathbf {b}}_c\\big [1-^{k-1}C_{k-m},1-^{k-2}C_{k-m},\\ldots \\nonumber \\\\& \\ldots ,1-^{k-(m-2)}C_{k-m},-(k-m),0,\\ldots ,0\\big ],$ we eliminate the $a_m$ term in the argument of $g^\\mathbf {b}_c$ .",
"By writing: $\\sum _{\\mathbf {d} \\in \\mathrm {Dsc}(b,m-1)} H(\\mathbf {d}) = g^{\\mathbf {b}}_c \\bigg [ \\left(\\begin{array}{c}k-1 \\\\k-(m-1) \\\\\\end{array} \\right), \\nonumber \\\\\\left(\\begin{array}{c}k-2 \\\\k-(m-1) \\\\\\end{array} \\right), \\ldots ,k-m+2,1,0\\ldots ,0 \\bigg ]$ etc, we can eliminate each of $a_j$ , $1\\le j \\le m$ recursively, and write out an equation for $H(d)$ entirely from the sum of its descendants: $H(\\mathbf {b}) = g^\\mathbf {b}_c(a_1,a_2,\\ldots ,a_j,\\ldots ,a_m = 1,0,\\ldots ,0)$ with $a_m &= 1\\\\a_{m-j} &= 1 - a_m\\left(\\begin{array}{c}k-(m-j) \\\\k-m \\\\\\end{array} \\right) - \\nonumber \\\\& a_{m-1}\\left(\\begin{array}{c}k-(m-j) \\\\k-1 \\\\\\end{array} \\right) - a_{m-2} \\left(\\begin{array}{c}k-(m-j) \\\\k-2 \\\\\\end{array} \\right) -\\nonumber \\\\& \\ldots - a_{m-(j-1)}(k-m-j)$ substituting indices $p = m-j$ , we get: $a_p =& 1 - a_{p+1}(k-p) - a_{p+2}\\left(\\begin{array}{c}k-p \\\\2 \\\\\\end{array} \\right) - \\ldots \\nonumber \\\\& \\ldots - a_{m}\\left(\\begin{array}{c}k-p \\\\m-p \\\\\\end{array} \\right) \\qquad 1 \\le p < m$ which is the recurrence relation featured.",
"Thus, if $H$ is $m$ -body, then the required equation is implied.",
"Conversely, if Eq.",
"(REF ) is satisfied, we have: $\\sum _{k=m+1}^{\\Vert \\mathbf {b}\\Vert -1} \\left( \\sum _{\\mathbf {d} \\in \\mathrm {Dsc}(\\mathbf {b},k)}c_{\\mathbf {d}} \\right) = 0 \\qquad \\forall \\mathbf {b}:\\, \\Vert \\mathbf {b}\\Vert > m$ which has no non-trivial solutions."
]
] | 1204.1084 |
[
[
"The relative and absolute timing accuracy of the EPIC-pn camera on\n XMM-Newton, from X-ray pulsations of the Crab and other pulsars"
],
[
"Abstract Reliable timing calibration is essential for the accurate comparison of XMM-Newton light curves with those from other observatories, to ultimately use them to derive precise physical quantities.",
"The XMM-Newton timing calibration is based on pulsar analysis.",
"However, as pulsars show both timing noise and glitches, it is essential to monitor these calibration sources regularly.",
"To this end, the XMM-Newton observatory performs observations twice a year of the Crab pulsar to monitor the absolute timing accuracy of the EPIC-pn camera in the fast Timing and Burst modes.",
"We present the results of this monitoring campaign, comparing XMM-Newton data from the Crab pulsar (PSR B0531+21) with radio measurements.",
"In addition, we use five pulsars (PSR J0537-69, PSR B0540-69, PSR B0833-45, PSR B1509-58 and PSR B1055-52) with periods ranging from 16 ms to 197 ms to verify the relative timing accuracy.",
"We analysed 38 XMM-Newton observations (0.2-12.0 keV) of the Crab taken over the first ten years of the mission and 13 observations from the five complementary pulsars.",
"All the data were processed with the SAS, the XMM-Newton Scientific Analysis Software, version 9.0.",
"Epoch folding techniques coupled with \\chi^{2} tests were used to derive relative timing accuracies.",
"The absolute timing accuracy was determined using the Crab data and comparing the time shift between the main X-ray and radio peaks in the phase folded light curves.",
"The relative timing accuracy of XMM-Newton is found to be better than 10^{-8}.",
"The strongest X-ray pulse peak precedes the corresponding radio peak by 306\\pm9 \\mus, which is in agreement with other high energy observatories such as Chandra, INTEGRAL and RXTE.",
"The derived absolute timing accuracy from our analysis is \\pm48 \\mus."
],
[
"Introduction",
"A reliable timing calibration is essential for all timing data analyses and the physics derived from those.",
"Irregularities in the spacecraft time correlation, the on-board instrument oscillators or data handling unit and the ground processing and data analysis software can lead to errors in relative and absolute information pertaining to the timing behaviour of astrophysical objects.",
"The timing of the XMM-Newton observatory is evaluated using XMM-Newton's EPIC-pn camera that has been extensively ground calibrated with respect to relative timing, but due to a limited calibration time budget, the end-to-end system for absolute timing was never checked on the ground.",
"The relative timing for fast sources like the Crab was expected to have an accuracy of $\\Delta \\,P/P\\,\\lesssim \\,10^{-8}$ before launch.",
"For the absolute timing a requirement of $\\Delta \\,T\\,\\lesssim \\,1 \\,ms$ was given.",
"XMM-Newton [18] was launched in December 1999 with an Ariane 5 rocket from French Guayana.",
"It operates six instruments in parallel on its 48 hour highly elliptical orbit: three Wolter type 1 telescopes, with 58 nested mirror shells each, focus X-ray photons onto the three X-ray instruments of the EPIC (European Photon Imaging Camera) [60], [62] and the two Reflecting Grating Spectrometers [17].",
"In addition, a 30 cm Ritchey Chrétien optical telescope, the Optical Monitor, is used for optical observations [34].",
"EPIC consists of three cameras: the two EPIC-MOS cameras use Metal-Oxide Semiconductor CCDs as X-ray detectors, while the EPIC-pn camera is equipped with a pn-CCD.",
"All three have been especially developed for XMM-Newton [48], [36], [62].",
"In this paper we determine the relative timing accuracy of XMM-Newton's EPIC-pn camera using all available observations of the Crab pulsar in combination with other isolated pulsars in order to extend our analysis to a broader variety of sources.",
"Preliminary results on the relative timing accuracy of XMM-Newton using the Crab pulsar and the other pulsars can be found in [4].",
"In this work we use only the Crab pulsar X-ray observations to determine the absolute timing accuracy.",
"However, as this is done in reference to radio timing, it is limited to the accuracy of the radio ephemerides.",
"We also see the paper as a summary of \"how to perform\" relative and absolute timing analysis with XMM-Newton and what timing accuracy the user can expect for different targets.",
"This paper is organised as follows.",
"In Sect.",
"we give a description of the targets used for the timing evaluation, followed by some technical comments on our data analysis in Sect. .",
"The relative and absolute timing results are presented in Sect.",
"and Sect. .",
"A short description of the XMM-Newton's EPIC-pn camera is given in Appendix ."
],
[
"Observations",
"All pulsars used in our analysis are isolated.",
"We concentrated primarily on the Crab pulsar (PSR B0531+21) as radio ephemerides are provided monthly by the Jodrell Bank Observatoryhttp://www.jb.man.ac.uk [28].",
"The other pulsars have been chosen to include a range of periods and pulse profiles, with which to check the relative timing.",
"Some of these pulsar observations were reported by [3] as a summary of first results from XMM-Newton.",
"Tables REF and REF summarise the data used and the results obtained from all the Crab observations studied and all the other pulsars respectively.",
"Column 1 gives the observation ID (OBSID) used for identifying XMM-Newton observations, followed by the satellite revolution (\"Rev.\")",
"in which the observation was done, the data mode, and the filter used.",
"Column 5 indicates whether the observation is affected by telemetry gaps (due to a full science buffer), and column 6 gives information on time jumps during the observation (see the footnote of the table for explanation).",
"Column 7 lists the start times (\"Epoch\") of the observations in MJD, followed by the exposure (\"Obs.",
"Time\") in ks.",
"Columns 9 and 10 list the pulse periods of the Crab pulsar in the radio at the time of the XMM-Newton observations (interpolated using the information provided by the Jodrell Bank Observatory) and the measured X-ray period, respectively.",
"Red.",
"$\\chi ^{2}$ (column 11) gives the reduced $\\chi ^{2}$ values found at the maximum of the respective $\\chi ^{2}$ distribution of the period search (the number of degrees of freedom, dof, was always 100 for the Crab pulsar), and \"FWHM\" is the full width at half maximum of the $\\chi ^{2}$ distribution.",
"$\\Delta $ P/P is the relative difference between the radio and the X-ray period (Eq.",
"REF in Sect.",
"REF ).",
"The \"Phase Shift\" (last column) shows the measured time shift of the main peak in the pulse profile between the X-ray and radio profiles, as explained in Sect.",
"REF .",
"All uncertainties given are at the 1 $\\sigma $ (68%) level.",
"The ephemerides of all the targets used in the analysis are shown in Table REF .",
"Fig.",
"REF shows the pulse profiles for all the pulsars analysed in this paper."
],
[
"The main XMM-Newton timing monitoring source: PSR B0531+21 (The Crab pulsar)",
"Since the discovery of the Crab Pulsar [57], the Crab has been one of the best studied objects in the sky and it remains one of the brightest X-ray sources regularly observed.",
"As a standard candle for instrument calibration, the 33 ms Crab pulsar has been repeatedly studied (monitored) by many astronomy missions in almost every energy band of the electromagnetic spectrum.",
"However, recent analysis presented by [66] showed that the flux of the Crab is not constant on long timescales at high energies.",
"These flux variations seem to be related to the nebula and correspond to a flux drop of $\\sim $ 7 % (70 mCrab) over two years (2008-2010).",
"This might affect the status of the Crab as a standard candle in the future.",
"In the X-ray regime its pulse profile exhibits a double peaked structure with a phase separation of 0.4 between the first (main) and the second peak.",
"X-ray emission at all phases, including the pulse minimum, was discovered by [61] using the Chandra observatory.",
"Measurements of X-ray to radio delays between the arrival times of the main pulse in each energy range of the Crab pulsar have been reported using all high-energy instruments aboard INTEGRAL [25] and RXTE [54].",
"The time delays were determined to be 280$\\pm $ 40 $\\mu $ s and 344$\\pm $ 40 $\\mu $ s respectively.",
"The Crab pulsar has been observed bianually to monitor the timing capabilities of XMM-Newton.",
"Over the years an observation strategy has been established that makes very efficient use of the limited calibration time budget.",
"XMM-Newton generally observed the Crab pulsar three times per orbit for 5 ks: at the beginning, in the middle, and the end of that orbit.",
"These campaigns were carried out in spring and autumn when XMM-Newton has a different location in its orbit with respect to the Sun-Earth system.",
"This guarantees the monitoring of the dependency of the timing with respect to XMM-Newton's orbital position.",
"Eventual irregularities in relative timing with respect to the orbital position could then be identified.",
"A total of 38 observations with exposure times between 2 ks and 40 ks have been analysed in this paper.",
"See Table REF for details of these observations.",
"PSR J0537-69 is a young pulsar, about 5000 years in age, located in the Large Magellanic Cloud.",
"It is embedded in the supernova remnant N157B and is considered to be the oldest known Crab-like pulsar.",
"It is a very fast-spinning pulsar with a period of 16 ms, discovered by [32] using RXTE.",
"No significant radio signal above a 5$\\sigma $ threshold has been detected from the pulsar [8].",
"In the X-ray energy range, RXTE has monitored PSR J0537-69 for seven years [33], [37], providing a complete study of the behaviour of the pulsar.",
"[37] reported 23 sudden increases in frequency, called glitches and present in most of young pulsars.",
"Due to this highly irregular activity (a glitch every $\\sim $ 4 months) a contemporaneous ephemeris is important.",
"Its pulse profile in the X-ray regime is characterized by a single narrow peak.",
"See Table REF for details of the observations.",
"Our 36 ks observation coincides with the RXTE monitoring campaign presented by [37] and a good ephemeris was therefore guaranteed."
],
[
"PSR B0540-69",
"This young pulsar ($\\sim $ 1500 years) was discovered in soft X-rays by [56] with a period of 51 ms, in the field of the Large Magellanic Cloud and it is considered to be a Crab-like pulsar.",
"Its pulse shape does not appear to change significantly from optical to hard X-rays [47].",
"The pulsed radio emission was discovered in late 1989 appearing as a faint source [31] and presenting a complex profile, very different from the simple sinusoidal one seen in X-rays (Fig.REF ).",
"A glitch was reported by [68] before the XMM-Newton observations and confirmed by [30] using a 7.6 year RXTE campaign.",
"The glitch activity of PSR B0540-69 is known to be less than that of the Crab pulsar [30] but the presence of considerable timing noise was reported by [7] using ASCA, BeppoSAX and RXTE observations made over a time interval of about 8 years.",
"Therefore, despite the low glitch activity, long extrapolations of the ephemeris might not be reliable.",
"See Table REF for details of the observations."
],
[
"PSR B0833-45 (Vela pulsar)",
"The Vela pulsar with a period of 89 ms was discovered by [27] and it is associated with a supernova remnant.",
"It is, with the Crab pulsar, one of the most active young/middle-age pulsars known, showing regular glitches.",
"These glitches have been intensively studied for the Vela pulsar, where a dozen events in different energy ranges have been recorded and analysed over the past three decades [16], [10].",
"Due to these important irregularities close radio ephemerides are needed.",
"No Vela timing mode observations have been performed with XMM-Newton, but since the period is 89 ms the data in Small Window mode (time resolution of 5.7 ms) can be used for our purposes.",
"Thus we have analysed the four observations listed in Table REF ."
],
[
"PSR B1509-58",
"This young pulsar ($\\sim $ 1700 years) is one of the most energetic pulsars known and has a pulse period of $\\sim $ 151 ms.",
"It is associated with the supernova remnant G320.4-1.2 and it has been well studied in all wavelengths since it was discovered in the soft X-ray band using Einstein [55].",
"The pulse profile in X-rays of appears to be much broader than in radio, changing from a narrow peak shape into a more sinusoidal shape at high energies.",
"Monitored by RXTE since its launch, and covering a 21 year time interval and in conjunction with radio data from the MOST and Parkes observatories, a detailed timing study has been carried out [29], but no glitch was found in the entire data sample.",
"This result makes PSR B1509-58 probably the only known young pulsar that does not present any glitches over long periods of time.",
"This property means that it is well adapted to extrapolation over long time intervals and useful for absolute timing analyses [53].",
"See Table REF for details of the observations."
],
[
"PSR B1055-52",
"PSR B1055-52, one of the Three Musketeers together with PSR B0656+14 and Geminga, is a middle-aged pulsar with a period of 197 ms.",
"It was discovered by [63] but it was only in 1983 that X-ray emission was first detected by [5] using the Einstein Observatory.",
"[44] detected sinusoidal pulsations in X-rays up to 2.4 keV.",
"More recently, [9] showed using XMM-Newton data that the pulsed emission is detectable up to 6 keV.",
"Most middle-aged pulsars like PSR B1055-52 show reduced timing noise and fewer glitches compared to younger ones.",
"See Table REF for details of the observations.",
"Results concerning these data have been originally published by [9].",
"Figure: XMM-Newton pulse profiles of the different pulsars analysed.",
"From top to bottom: PSR J0537-69 (obs.",
"ID: 0113020201),the Crab pulsar (obs.",
"ID: 0122330801), PSR B0540-69 (obs.",
"ID: 0125120201), the Vela pulsar (obs.",
"ID: 0111080201), PSR B1509-58 (obs.",
"ID: 0312590101) andPSR B1055-52 (obs.",
"ID: 0113050201) with periods of 16 ms, 33 ms, 51 ms, 89 ms, 151 ms and 197 ms, respectively.The energy band in all cases is 0.2-12 keV.Table: Individual observations from the Crab monitoring.",
"(A description of the columns is given in the text.",
")Table: Individual observations of the other objects."
],
[
"Data analysis",
"The data sets were processed using the XMM-Newton Scientific Analysis Software, SAS 9.0 [12].",
"Event times were corrected to the solar system barycentre using the SAS tool barycen."
],
[
"Relative timing data analysis",
"We define the relative timing accuracy as the difference between the period measured with XMM-Newton and the period measured at radio wavelengths evaluated at the epoch of the X-ray observations.",
"This difference is normalised to the pulse period measured in radio.",
"$\\centering \\texttt {Rel.",
"timing}:=\\dfrac{P_{\\texttt {X-ray}}(T_{\\texttt {X-ray}})-P_{\\texttt {radio}}(T_{\\texttt {X-ray}})}{P_{\\texttt {radio}}(T_{\\texttt {X-ray}})}=\\dfrac{\\Delta P}{P}$ where $P_{\\texttt {X-ray}}:$ period derived from XMM-Newton $P_{\\texttt {radio}}:$ period extrapolated from radio ephemeris $T_{\\texttt {X-ray}}:$ time of the first X-ray event of the XMM-Newton observation [MJD] We determined the period of the Crab pulsar in X-rays using the epoch folding software XRONOSXRONOS is part of the HEARSAC software (http://heasarc.gsfc.nasa.gov)..",
"The closest available radio ephemeris (supplied by the Jodrell Bank Crab Pulsar Monthly Ephemeris) before and after the X-ray observation were used to interpolate the radio period P for the time of the first X-ray event of the XMM-Newton observation in MJD.",
"The interpolated radio periods are then used as an initial trial value for the epoch folding.",
"The period derivative $\\dot{P}$ provided by Jodrell Bank is taken into account when doing the folding of the X-ray data.",
"All relevant initial and final values are listed in Tables REF , REF , REF and REF .",
"All X-ray pulse profiles shown in Fig.",
"REF have been produced using the best fit X-ray period.",
"The detailed steps of our data reduction are presented below in order to provide an example for proper XMM-Newton relative timing data analysis.",
"calibrate the XMM-Newton event list using the SAS routine epproccommand line set up of epproc: timing=YES burst=YES srcra=83.633216667 srcdec=22.014463889 withsrccoords=yes perform barycentre correction using precise coordinates with the SAS routine barycencommand line set up of barycen: withtable=yes table='bary.ds:EVENTS' timecolumn='TIME' withsrccoordinates=yes srcra='83.633216667' srcdec='22.014463889' processgtis=yes time=0 extract sourceDetector coordinates used in the extraction process a) timing mode: (RAWX,RAWY) IN box(35,101,12,100,0), b) burst mode: (RAWX,RAWY) IN box(35.,71.5,20,70,0) extrapolate the radio ephemeris period search using efsearch from XRONOS (see Table REF ) which gives the $\\chi ^{2}$ against the period period determined through a weighted mean of all values within 65$\\%$ of the efsearch $\\chi ^{2}$ maximum Table: Settings for the epoch folding using efsearch.Table: Radio and RXTE ephemerides used in the analysis.The number of phase bins per period (nphase) in each pulse profile was chosen such that the count rate uncertainties in each bin (determined using the Poisson error on the count rate per bin normalised by the bin size) are, on average, not bigger than approximately 10% of the total count rate variation in the pulse profile of the shortest observation for each pulsar.",
"This value, determined for each pulsar, is used for all the observations of that object.",
"In this way the signal to noise in each bin is sufficient to reliably determine any 'smearing out' of the pulse profile due to the use of an inaccurate period/ephemeris, essential for determining the relative timing precision, as described in Sect.",
"."
],
[
"Absolute timing data analysis",
"The XMM-Newton EPIC-pn absolute timing accuracy was determined using only observations of the Crab.",
"The ephemeris (epoch, P, $\\dot{P}$ , $\\ddot{P}$ ) of the nearest radio observation from the Jodrell Bank Observatory was used as a reference to obtain the phase shift between the time of arrival of the main peak in the X-ray profile and the time of arrival of the main peak in the radio profile, as described in Eq.",
"REF .",
"The phase shift was then multiplied by the corresponding X-ray period found during the relative timing analysis, as shown in Table REF .",
"$\\centering \\texttt {Phase Shift [\\mu \\,s]} := {T_{0}}_{\\texttt {X-ray}}-{T_{0}}_{\\texttt {radio}}$ where ${T_{0}}_\\texttt {{X-ray}}:$ Time of arrival of the main peak of the X-ray profile ${T_{0}}_{\\texttt {radio}}:$ Time of arrival of the main peak of the radio profile The phase of the main X-ray peak was determined using a pulse profile with 1000 phase bins which was then fitted with an asymmetrical Moffat function.",
"The explicit formula for the Moffat function is given in Appendix C. We also demonstrate how its shape varies when different parameters are modified.",
"Fig REF shows an example of how the phase of one Crab X-ray pulse profile (obs.",
"ID: 0122330801) is slightly shifted in phase with respect to the radio phase (shown as a red line).",
"Figure: Crab pulse profile (obs.",
"ID: 0122330801) with 1000 phase bins which was used to determine the phase of the main peak.",
"The phase of the main peak at radio wavelengths is shown as a solid red vertical line.The following steps describe an example of the data reduction carried out on the XMM-Newton data in order to assess the absolute timing precision: Steps 1-3 are the same as described in Sect.",
"REF fold the X-ray data on the radio period fit the X-ray pulse profile with a Moffat function determine the shift between the radio phase zero and the X-ray peak."
],
[
"Evaluating the efficiency of automatic corrections made to event time jumps by the SAS",
"In order to do proper timing analysis with the EPIC data, every event detected on board has to be assigned a correct photon arrival time.",
"The transformation from readout sequences by the EPIC camera to photon arrival times of each photon is performed by the EPEA (European Photon Event Analyser, [26]).",
"The absolute timing adjustment from On Board Time to UTC is done with the XMM-Newton time correlation [21].",
"A hardware problem in the EPEA can produce time jumps in some observations, which have to be corrected.",
"A time jump can affect the timing accuracy by broadening the $\\chi ^{2}$ -distribution during the epoch folding search or by producing 'ghost peaks' [21].",
"Figure: The time difference between consecutive events determined using the slope of a linear fit to the time differences ofconsecutive events modulo the frame time.",
"This is plotted against time.",
"Upper panel with old frame times,lower panel with the refined ones.",
"The different modes of EPIC-pn are represented by different colours: Full Frame (black,Extended Full Frame (dark blue), Large Window (green), Small Window (red), Timing (light blue), Burst (yellow).To find time jumps reliably one needs to have as accurate CCD frame times as possible.",
"The frame times for all EPIC-pn modes using all available archive data [11] were remeasured and cross-checked with an independent method, analysing the evolving differences between consecutive events.",
"If these differences are correct, the slope of the relation between the difference between the arrival times of consecutive events and the same quantity modulo the frame time is zero.",
"Fig.",
"REF shows the effect of the frame time recalibration, which brought the values of this slope very close to zero for all observational modes.",
"Constant 'timediff' values' indicate constant frame time.",
"Recalibration of the time jump detection algorithm of the XMM-Newton SAS has been done with the refined frame times.",
"A search for time jumps in all available XMM-Newton archive data up to revolution 1061 showed a significant number of time jumps in the data for each observational mode.",
"The application of the new algorithm reduced the number of remaining time jumps significantly.",
"The effects of the SAS_JUMP_TOLERANCE parameter in the new algorithm (Versions 2-7) are shown in Fig.",
"REF for each EPIC-pn mode.",
"The Version 0 shown in Fig.",
"REF represents the remaining time jumps with the old frame times and the old algorithm and it is included for comparison.",
"Version 1 was obtained using the new frame times but with the old algorithm.",
"While the rate of non-corrected time jumps (averaged over all pn-modes) was 2.8 per 100 ks before the implementation of the SASv8.0 time jump correction, just 0.3 time jumps per 100 ks remained uncorrected after its implementation.",
"A breakdown of time jumps for each EPIC-pn mode is given in Table REF .",
"This new improved algorithm has been implemented in the SAS as the default setting since version 8.0 (SAS_JUMP_TOLERANCE = 22.0).",
"Table REF indicates for which Crab observation a time jump has been corrected, and where data had to be excluded from the analysis.",
"In order to identify possible remaining time jumps, the data can be processed without the \"fine-time\"-correction, i.e., epframes set=\"infile_pn\" eventset=events.dat gtiset=tmp_g.dat withfinetime=N.",
"Then the time $\\Delta $ t between successive events is calculated and divided by the frame time, FT, of the relevant mode (FF Mode: 73.36496 ms, eFF Mode: 199.19408 ms, LW mode: 47.66384 ms, SW Mode: 5.67184 ms, Timing Mode: 5.96464 ms, Burst Mode: 4.34448 ms).",
"A time jump is shown to exist when $\\Delta $ t/FT is different from an integer by a quantity larger than a tolerance parameter.",
"Only those time jumps which happen to be an integer multiple of the relevant FT would not be found with this method.",
"It is important to notice that the tolerance acceptable between $\\Delta $ t and the full frame time should not be bigger than (20/48828.125$\\times $FT).",
"Table: Mean rate of residual uncorrected time jumps per 100 ks.Figure: Remaining time jumps in the EPIC-pn data after all SAS correction algorithms.",
"The triangles correspond to the remaining time jumps (in percent) for each Epic-pn mode: FF (black), eFF (blue), SW (red), LW green), Timing (light blue) and Burst yellow).",
"Filled circles represent the overall remaining time jumps for all XMM-Newton observations up to revolution 1061.The numbers stand for the different processing versions of the algorithm.Versions:0: old frame times (oft) and old algorithm,1: new frame times (nft) and old algorithm,2: nft and SAS_JUMP_TOLERANCE (SJT) 19.0,3: nft and SJT 20.0,4: nft and SJT 21.0,5: nft and SJT 22.0,6: nft and SJT 23.0 and7: nft and SJT 24.0"
],
[
"Relative Timing accuracy of ",
"The relative timing accuracy of the EPIC-pn camera has been studied using all six pulsars (see Fig.",
"REF ), presented in Sect. .",
"As described in Appendix B, the FWHM of the $\\chi ^{2}$ curve obtained during the period search analysis can be expressed in terms of the period and the exposure time of the observation (Eq.",
"REF ).",
"From this expression, and using the Independent Fourier Space [45], approach discussed in the appendix, an empirical formula for the error on the X-ray period was found (Eq.",
"REF ).",
"$\\centering \\delta P=\\dfrac{\\texttt {FWHM}}{dof}$ where dof is the number of degrees of freedom (number of phase bins used to construct the pulse profiles minus the number of variables).",
"The number of bins used in the pulse profiles are shown in Table REF .",
"The relative timing accuracy was defined by Eq.",
"REF and therefore its error will depend mostly on how accurately the radio and X-ray periods can be measured.",
"Other factors that could affect the relative timing accuracy are discussed in Appendix B.",
"Considering that the radio period measurements are more accurate than the X-ray periods (usually by 1-3 orders of magnitude; however, DM variations can sometimes cause problems and the time resolution of the radio telescopes has to be monitored), it was assumed in our analysis that their errors were negligible compared to the error on the X-ray period.",
"Thus, it can be found that the relative timing, $\\Delta $ P depends exclusively on the error of the X-ray period, $\\delta $ P as shown in Eq.",
"REF .",
"$\\centering \\Delta P \\approx \\delta P$ The relationship described in Eq.",
"REF allows a goodness of fit study of our measured $\\Delta $ P compared to the empirical $\\delta $ P described in Eq.",
"REF .",
"The empirical $\\delta $ P was considered as an upper limit on an accurate $\\Delta $ P measurement.",
"A comparison between the observed relative timing accuracy in absolute value and normalised by the corresponding period, $|\\Delta $ P/P$|$ (symbols) and its “expected” value obtained from Eq.",
"REF and Eq.",
"REF (lines) is presented in Fig.",
"REF as a function of the exposure time and in Fig.",
"REF as a function of date.",
"Figure: The relative timing accuracy of XMM-Newton's EPIC-pn camera.Comparison of the expected|Δ|\\Delta P/P|| using the assumption that the relative timing depends exclusively on the error of the X-ray period.",
"δP\\delta P can be empirically expressed as shown in Eq.",
"and (given as the lines in the Figure) and the measured one obtained by comparingour X-ray period with the extra-(inter-)polated radio period (symbols).For observations where we have very reliable radio/X-ray ephemerides, the observed data points are below the lines of the estimated accuracies.",
"The outliers above the respective theoretical lines for each individual pulsar are described below: Radio ephemerides extrapolated over long time intervals appear to be unreliable.",
"Therefore, we exclude the following observations in the final calculation regarding the relative timing accuracy: Vela pulsar: 015395140; PSR B1509-58: 0128120401; PSR B0540-69: 0413180201, 0413180301.",
"The $\\delta $ P approximation given in Eq.",
"REF was found to be unreliable in some cases (e.g.",
"for PSR B1509-58, observation: 0312590101 and the Vela pulsar, observations: 0111080101, 0111080201).",
"As the Vela pulsar is quite active, we may have under-estimated the error by simply extrapolating the ephemeris, however, the same can not be said about PSR B1509-58 which is one of the most stable young pulsars known.",
"As seen in Fig.",
"REF there is no obvious change in the relative timing accuracy of the EPIC-pn camera over its lifetime.",
"Figure: The relative timing accuracy of XMM-Newton's EPIC-pn camera:|Δ|\\Delta P/P|| for all pulsars as a function of date [MJD].The results for the Crab pulsar alone are shown in Fig.",
"REF .",
"As expected, there is a tendency towards smaller uncertainties for longer observations.",
"For a quantitative measure of the timing accuracy the standard deviation for the $\\Delta $ P/P distribution (shown in Fig.",
"REF ) was used.",
"Fitting the distributions with a Gaussian normal distribution, we found a standard deviation of $7\\times 10^{-9}$ for all the pulsars together (including the Crab pulsar) and $5\\times 10^{-9}$ for the Crab pulsar alone.",
"While the distribution for Crab pulsar is centred at zero (within uncertainties) the mean value of the distribution for all the pulsars combined is slightly offset, in the sense that the X-ray period is slightly shorter on average than the radio period.",
"Thus, the relative deviation of the observed pulse period with respect to the most accurate radio data available is $\\Delta $ P/P $\\lesssim 10^{-8}$ .",
"Figure: Relative timing using Crab monitoring data: upper panel as afunction of observing time, lower panel as a function of MJD.",
"These plots are regularly updated in the EPIC Calibration Status Document using the routine calibration observations of the Crab."
],
[
"Absolute timing accuracy of ",
"The Crab pulsar shows a shift of -306 $\\mu $ s (shown in Fig.",
"REF ) between the peak of the first X-ray pulse with respect to the radio peak.",
"We hereby confirm the similar results of other missions like Chandra [61], INTEGRAL [25], and RXTE [54].",
"The XMM-Newton values (\"stars\" in Fig.",
"REF ) show a considerable scatter with a standard deviation of 48 $\\mu $ s. The formal error on the mean value of -306 $\\mu $ s is $\\pm \\,9$ $\\mu $ s. The scatter found is consistent with the previously determined maximum integrated error for the time correlation of less than 100 $\\mu $ s [22].",
"The original requirement for an absolute timing accuracy of 1 ms for XMM-Newton, defined before launch, has clearly been reached and even improved on by at least a factor of 20.",
"This scatter is likely to be due to uncertainties in the time correlation process since the phase of the main peak can be measured with an accuracy of $\\mu $ s. Upper limits for these processes were reported by [21]http://xmm2.esac.esa.int/docs/documents/CAL-TN-0045-1-0.pdf who gave a detailed description of all kinds of instrumental delays considered while converting between observing time and UTC time and estimated the spacecraft clock error to be $\\sim $ 11 $\\mu $ s, the uncertainty in ground station delays to be $\\sim $ 5 $\\mu $ s, the interpolation errors to be $\\sim $ 10 $\\mu $ s, the error between latching observing time and the start of frame transmission as $\\sim $ 9 $\\mu $ s, and the uncertainties in the spacecraft orbit ephemeris to be $\\sim $ 30 $\\mu $ s. All these errors will be random for our data, and hence the fluctuations observed.",
"The 48 $\\mu $ s 1 $\\sigma $ scatter measured with respect to the mean may then be attributed uniquely to the above errors and no other systematic effect.",
"This value can then be considered to be the minimal significant time separation between two arrival times to be considered independent.",
"From the initial 38 Crab observations, 32 were considered for the absolute analysis.",
"Six of the Crab observations were excluded for the reasons given below: Observation 0122330801: was early in the mission (rev.",
"56) and appears to have problems in the time correlation that can no longer be recovered.",
"Observation 0160960201: too much data had to be excluded due to time jumps which caused a dramatic reduction in the number of counts (rev 698).",
"Observations 0160960401 and 0160960601: these correspond to rev.",
"874 which shows the X-ray pulse peak displaced from the expected radio position.",
"This is likely to have been caused by a glitch shortly before the XMM-Newton observations.",
"This offset is more dramatic in the second observation, which has poorer statistics due to a shorter exposure time.",
"None of these observations fall into the range shown in Fig.",
"REF .",
"Observations 0412590601 and 0412590701: these correspond to rev.",
"1325.",
"The reason for the offsets is unclear.",
"They may be due to a small, non-reported glitch or an anomalously large ground segment error, but because of the uncertainty, we excluded these observations when determining the absolute timing precision.",
"It was found that some observations presented pulse profiles with an excess of counts in the interpulse region of the Crab profile.",
"Numerical simulations have been used to study the effect that this excess could have caused in determining the peak of the X-ray profile and thus, in determining the difference in phase between the X-ray and the radio.",
"Using the typical 0.2-15.0 keV Crab profile as the input, 10000 light curves were created using Monte Carlo simulations.",
"The strength of both peaks in the pulse profile (keeping the ratio between them constant) as well as the strength of the interpulse were selected as input parameters.",
"As shown in Fig.",
"REF in Appendix C, the Moffat fit, used to determine the phase of the main peak, can be used to reliably fit different tails.",
"To fit the excess in the interpulse we added a Lorentzian function to the pulse profile in order to take into account the excess in the tail of the main peak.",
"The secondary peak was omitted in the fit.",
"No phase shift was found in any of the models tried, which implies that the strength of the interpulse region plays no role in determining the phase of the peak of the profile and thus all 32 of the retained Crab observations could be used to derive the absolute timing accuracy reliably.",
"Figure: Absolute timing as a function of MJD using Crab-pulsar monitoring data.Shown are the offsets of the main X-ray peak with respect to the main radio peak in the Crabpulse profile in time units (left scale) and phase units (right scale).",
"The colored lines give the mean values for XMM-Newton (solid black line,this work), Chandra (green dashed line), RXTE (red dot line) and INTEGRAL (blue dashed-dot line) respectively, all taken from the literature.",
"The yellow area indicatesthe standard deviation of the XMM-Newton data points.",
"The superscript numbers near to each XMM-Newtondata point give the XMM-Newton revolution in which the observation was carried out."
],
[
"Discussion and Conclusion",
"The Crab pulsar has been used by many missions as a calibration source for timing accuracy ([25], [54], [46], [1], [40]).",
"The XMM-Newton observatory began observing the Crab pulsar during its earliest orbits, monitoring its X-ray pulsation with high time resolution.",
"38 Crab observations spread over 10 years have been analysed in this paper (from revolution 56 until revolution 1687).",
"Measurements of the period were made with an accuracy of $\\sim $ 10$^{-11}$ s. A relative timing accuracy smaller than 10$^{-8}$ and stable with time was established for the EPIC-pn camera.",
"This result was achieved by comparing our X-ray measurements of the Crab pulsar with high precision radio measurements at each corresponding epoch.",
"Five isolated pulsars showing a wide range of periods and completely different pulse profiles (PSR J0537-69, PSR B0540-69, Vela pulsar, PSR B1509-58 and PSR B1055-52) were analysed to complement the study of the relative timing accuracy, confirming the results obtained with the Crab pulsar.",
"For the case of PSR B0540-69 a long term phase-coherent study of its period was reported by [30].",
"Due to its stability we considered it a good candidate to use for an extrapolation over a long time period.",
"As shown in Table REF the long extrapolation made in two of the three observations of PSR B0540-69 show poor results suggesting that a small glitch between the ephemeris and our observation may have occurred, rendering this pulsar less stable than anticipated.",
"An improved algorithm to detect and correct sporadic \"jumps” in the flow of the photon arrival times has been implemented with SASv8.0 [13]http://xmm2.esac.esa.int/docs/documents/CAL-TN-0018.pdf.",
"This method is based on a more accurate determination of the frame times for all pn modes and on a correction of frame time drifts due to temperature variation and aging of the on-board clock [11].",
"The total reduction of the rate per 100 ks of observation affected by residual uncorrected time jumps for all pn instrumental modes dropped from 2.8 before the improved algorithm to 0.3 once it was implemented.",
"For the absolute timing analysis, only Crab pulsar observations have been analysed since a high number of stable observations need to be considered to provide a reliable result.",
"We have considered the phase of the first (main) peak of the X-ray profile and measured the phase difference with respect to the corresponding peak of the radio profile.",
"Considering 32 of 38 Crab EPIC-pn observations (0.2-12 keV energy range) analysed in this paper, we confirmed previous results demonstrating that the first X-ray peak from the Crab pulsar leads the radio peak by 306 $\\pm $ 9 $\\mu $ s (statistical error) with $\\pm $ 48 $\\mu $ s (1 $\\sigma $ ) scatter.",
"This error is similar to the Ground Segment accuracy and defines the absolute timing accuracy of the instrument.",
"The observed shift is consistent within 1$\\sigma $ with those presented by [25] using INTEGRAL and by [54] using RXTE, as shown in Fig.",
"REF .",
"Figure: The peak pulse lead time (μ\\mu s) of various observations are plotted against energy (keV) in optical, X-rays and γ\\gamma -ray energy bands and a constant model is fitted to the data, which is found to be 284.4 μ\\mu s. The data points shown above cover over 7 orders of magnitude in energy and come from different observations and experiments mentioned on the graph (,, , , ).",
"Only the central energy corresponding to an individual observation was plotted along with corresponding lead time measured within the observed energy band.A systematic comparison of our measurements in the X-ray band with respect to other accurate measurements carried out in different energy bands from earlier observations in the optical, X-ray, and $\\gamma $ -ray parts of the spectrum are shown in Fig.",
"REF .",
"Differences in the shifts observed over 7 decades in energy are marginal with an average value of 284.4 $\\mu $ s. It is important to note that the large error bars quoted in the X-ray band for XMM-Newton and RXTE include systematic errors from the radio measurements, carried out at the Jodrell Bank Observatory.",
"The origin of the electromagnetic radiation emitted from pulsars is still unclear.",
"Several models have been proposed to explain the origin of the high energy radiation based on different regions of acceleration in the pulsar magnetosphere, such as the polar cap, the slot gap and the outer gap models [14], [2], [6], [67], [15].",
"The radio emission model is an empirical one and the radiation is usually assumed to come from a core beam centered on the magnetic axis and one or more hollow cones surrounding the core [51].",
"The estimated average delay between the emission from differing wavelengths is therefore very significant.",
"It implies that the site of radio production is distinctly different from that of the non-radio emission.",
"The difference in phase between the radio and the X-ray radiation is about 0.008, or three degrees in phase angle.",
"This time delay of about 300 $\\mu $ s most naturally implies that emission regions differ in position by about 90 km between radio and X-rays energy bands in a simplistic geometrical model neglecting any relativistic effects, with the radio being emitted from closer to the surface of the neutron star.",
"Such high time resolution, high precision absolute timing, multiwavelength observations are therefore essential for understanding the origin of the pulsar emission.",
"The XMM-Newton project is an ESA Science Mission with instruments and contributions directly funded by ESA Member States and the USA (NASA).",
"The German contribution of the XMM-Newton project is supported by the Bundesministerium für Bildung und Forschung/Deutsches Zentrum für Luft- und Raumfahrt.",
"The UK involvement is funded by the Particle Physics and Astronomy Research Council (PPARC).",
"The Parkes radio telescope is part of the Australia Telescope which is funded by the Commonwealth of Australia for operation as a National Facility managed by CSIRO.",
"We wish to thank Dr. L. Kuiper for all his help on the absolute timing analysis procedure and results and all the discussions and comments that he provided to the development of this paper.",
"We would also like to thank the anonymous referee for his/her useful comments and suggestions.",
"A. Martin-Carrillo also wish to thank the ESAC Faculty group for their financial support during the investigation and creation of this publication."
],
[
"EPIC is capable of providing moderate energy resolution spectroscopy in the energy band from 0.2 to 15 keV for as many as several hundred sources in its $30^{\\prime }$ field-of-view.",
"The EPIC cameras can be operated in different observational modes related to different readout procedures.",
"Detailed descriptions of the various readout modes of EPIC-pn and their limitations are given by [20], [26] and [43]http://xmm.esac.esa.int/external/xmm_user_support/ documentation/uhb/XMM_UHB.pdf .",
"The EPIC-pn camera provides the highest time resolution in its fast Timing and Burst modes (Timing mode: 29.52 $\\mu $ s, Burst mode: 7 $\\mu $ s) and moderate energy resolution ($E/\\text{d}E = 10$ –50) in the 0.2–15 keV energy band.",
"The pile-up limit (see Sec.",
"3.3.9 of [43]) for a point source is $800\\,\\text{counts}\\,\\text{s}^{-1}$ (85 mCrab) in Timing mode and $60000\\,\\text{counts}\\,\\text{s}^{-1}$ (6.3 Crab) in Burst mode.",
"Thus, the observations of the Crab suffer from pile-up only in Timing Mode, such that spectral analysis of the Crab can only be carried out accurately in Burst mode.",
"However for timing purposes the effect of pile up can be neglected in the Timing mode.",
"Figure: Upper: RAWX and RAWY image of a Crab observation in Burst mode.",
"The rows 180-200 are not transmitted.Bottom: RAWX and RAWY image of a Crab observation in Timing mode.As shown in Fig.",
"REF , in both timing and burst modes, EPIC-pn loses spatial resolution in the shift-direction.",
"In Timing mode, this is because 10 lines of events are fast-shifted towards the anodes and then the integrated signal is read out as one line.",
"In Burst Mode, it is because 200 lines are fast-shifted within 14.4 $\\mu $ s while still accumulating information from the source.",
"The stored information is then read out as normal, where the last 20 lines have to be deleted due to contamination by the source during the readout.",
"The CCD is then erased by a fast shift of 200 lines, and immediately after that the next Burst readout cycle starts.",
"Moreover, the lifetime in Burst mode is only 3% and therefore, the use of this mode has been limited to observations of very bright sources such as the Crab or X-ray transients.",
"For our analysis we use mainly Timing and Burst mode observations.",
"The images seen in Fig.",
"REF for the Timing and Burst mode are produced in CCD coordinates using RAWX and RAWY, which are simply the pixel co-ordinates, where each pn pixel is 4.1x4.1\" aside.",
"The source appears as a stripe in the CCD RAWY direction.",
"Source extraction regions in both modes will therefore always be boxes.",
"Operated in Timing mode EPIC-pn data show a bright line in the RAWX direction at $RAWY=19$ , that is related to a feature in the on-board clock sequence.",
"In the clocking scheme of the Timing mode 10 lines are shifted to the CAMEXs (CMOS Amplierand Multiplexer Chip) and then read out as one so called macro line, such that the integration time for a normal macro line is 29.52 $\\mu $ s. Within a frame time, 200 macro lines are read (corresponding to 2000 physical CCD lines).",
"During the first CCD readout the first macro line contains only one CCD line and is set to bad.",
"However, the integration time during the readout of the second macro-line is $29.52+23$ $\\mu $ s due to electronic implementation of the sequencer.",
"Therefore the integration time for a point source at $RAWY=189$ is a factor 1.8 higher than for all other macro line and macro line 19 receives a factor 1.8 higher flux from the point source.",
"The feature only shows up for bright point sources.",
"There is no effect on the scientific quality of the data as long as the integration time for spectra and light curves is higher than the frame time in Timing mode (5.96464 ms; [11]http://xmm2.esac.esa.int/docs/documents/CAL-TN-0081.pdf.",
"Caution should be used for pulse phase spectroscopy with bin sizes below the frame time (5.96464 ms), but only if the pulse period is a multiple of the frame time."
],
[
"Treatment of uncertainties and reliability of radio extrapolations",
"The $\\chi ^{2}$ distribution obtained from the period search can be approximated by a triangle where the maximum corresponds to the true period P$_{0}$ and the points P$_{1}$ and P$_{2}$ where the legs of the triangle meet the level of constant $\\chi ^{2}$ defining the total width of the $\\chi ^{2}$ distribution.",
"For a pulse profile with a small single peak, P$_{1}$ and P$_{2}$ can be calculated using Eq.",
"REF , where T$_{obs}$ is the elapsed observational time and N$_{per}$ is the number of pulse periods in this time.",
"$\\centering P_{1}=\\dfrac{T_{\\texttt {obs}}}{N_{\\texttt {per}}+1}; \\;\\; P_{2}=\\dfrac{T_{\\texttt {obs}}}{N_{\\texttt {per}}-1}\\;\\;\\; \\texttt {where} \\; N_{\\texttt {per}}=\\dfrac{T_{\\texttt {obs}}}{P}$ For a triangular function the Full Width Half Maximum (FWHM) is equal to $(P_{2} - P_{1})/2$ and can be expressed as in Eq.",
"REF as a function of the period and the elapsed observation time.",
"$\\centering \\texttt {FWHM}=\\dfrac{P_{2}-P_{1}}{2} \\Rightarrow \\texttt {FWHM}=\\dfrac{P^{2}}{T_{\\texttt {obs}}}$ Figure: Sample χ 2 \\chi ^{2} distributions for one observation of each of thestudied pulsars.An expected FWHM of the $\\chi ^{2}$ distribution can be estimated using Eq.",
"REF .",
"A comparison of such estimations (lines) and the measured FWHMs from all the observations is shown in Fig.",
"REF .",
"All values were normalised using the pulsar period to be able to present all the pulsars on the same diagram.",
"All measured values of FWHM/P are about a factor of 3 smaller than those predicted for the Crab and Vela pulsars.",
"In the other four pulsars the ratio between the measured and predicted values is $\\sim $ 1.3.",
"This would suggest that this approximation works better in single peaked pulse profiles.",
"Figure: Comparison of the predicted FWHM of the χ 2 \\chi ^{2}distributions (lines) and the observed ones (symbols).",
"All values were normalised using the pulsar period to be able to present all the pulsars on the same diagram.Empirically, two periods can be considered completely independent from each other when their difference is at least P$^{2}$ /T (one Independent Fourier Space, IFS [45]), which is identical to the FWHM definition in Eq.",
"REF .",
"One IFS can be seen, then, as the delta in pulse period which will smear a perfect pulse profile to a flat profile when folded over the complete observing time $T_{\\texttt {obs}}$ .",
"This approach is quite conservative and smaller changes than one IFS in period can be easily seen.",
"We have found that a rough estimate of the uncertainty in the measured period can be found by dividing the FWHM by the number of phase bins used to construct the pulse profile (degrees of freedom, see Table REF ).",
"Thus, two periods will be considered different when the pulse profile is smeared by one bin instead of one whole phase.",
"The error on the X-ray period can then be written as shown in Eq.",
"REF .",
"$\\centering \\delta P=\\dfrac{\\texttt {FWHM}}{dof}$ Besides providing a good estimate of the error on the period, the Independent Fourier Space approach can also provide a good indication of how reliable the extrapolation (or interpolation in the case of the Crab pulsar) of a period can be.",
"Since we defined the relative timing accuracy in Sect.",
"REF based on the reference period (normally obtained from radio observations) at the time of the XMM-Newton observation, it is critical to understand how reliable, and in some form, how accurate, this parameter really is.",
"For clarification, and due to the huge amount of data available we will focus on the Crab pulsar only.",
"However the same applies to all the pulsars studied in this paper.",
"Eq.",
"REF establishes that two periods are completely different if the pulse profile is smeared by one bin.",
"By studying the phase smear, we can determine whether the pulse profile has been affected by a glitch or whether extrapolating the ephemeris over (long) time periods leads to inaccuracies.",
"If a simple period evolution with time (including the second derivative) is assumed, the phase smear is then defined as in Eq.",
"REF .",
"$\\centering \\texttt {Phase Smear}=\\dfrac{(P_{\\texttt {extrap}}-P_{0})T_{\\texttt {obs}}}{P_{\\texttt {extrap}}\\times P_{0}}$ where P$_{\\texttt {extrap}}$ is the extrapolated period at the time T$_{0}$ , P$_{0}$ is the actual period at that time and T$_{\\texttt {obs}}$ is the exposure time of the observation.",
"Using radio data from the Jodrell Bank Observatory, the phase smear versus time of extrapolated periods is shown in Fig.",
"REF .",
"For the relative timing analysis 100 phase bins have been used and therefore a limit of 1% smearing is imposed by the criteria described above.",
"Extrapolating the period, the Crab pulsar would reach that limit in 4 months.",
"Considering that the Jodrell Bank Observatory provides an updated ephemeris every month, the actual smearing effect will be much lower ($\\sim $ 0.1%) and other properties such as timing noise will not affect our relative timing analysis.",
"For the Crab, we actually used interpolation rather than extrapolation, see Sect.",
"REF , so the phase smear was further minimised ($\\sim $ 0.09%).",
"Figure: Phase smear of the Crab pulsar versus time.",
"The timing noise seems to be a prominent feature for the Crab pulsar and therefore extrapolation over periods of 4 months will already reach our 1% limit (dashed line)."
],
[
"The Moffat function",
"The Moffat function is a modified Lorentzian with a variable power law index [39].",
"In Fig.",
"REF the behaviour of the function is shown as a function of its parameters.",
"The function presents different tails on each side of the maximum which fit the main pulse of the Crab profile better than a normal Lorentzian or Gaussian function.",
"The explicit formula of the Moffat function is the following: $\\centering y={\\dfrac{A_{0}}{(((x-A_{1})/A_{2})^{2}+1)^{A_{3}}}+A_{4}+A_{5}x}$ The different parameters represent: $A_{0}$ : normalization $A_{1}$ : Peak Centroid $A_{2}$ : HWHM $A_{3}$ : Moffat index $A_{4}$ : offset $A_{5}$ : slope The variation in the shape of the Moffat function for different values of the important parameters is shown in Fig.",
"REF .",
"Upper left: $A_{2}$ changes from 0.02 to 0.06; upper right: $A_{5}$ changes from 500 to 1400; lower left: $A_{1}$ changes from 0.4 to 1.12; and lower right: $A_{3}$ changes from 1.0 to 2.80.",
"Figure: Variations in the shape of the Moffat function when parameter values are changed."
]
] | 1204.0978 |
[
[
"Reheating after f(R) inflation"
],
[
"Abstract The reheating dynamics after the inflation induced by $R^2$-corrected $f(R)$ model is considered.",
"To avoid the complexity of solving the fourth order equations, we analyze the inflationary and reheating dynamics in the Einstein frame and its analytical solutions are derived.",
"We also perform numerical calculation including the backreaction from the particle creation and compare the results with the analytical solutions.",
"Based on the results, observational constraints on the model are discussed."
],
[
"Introduction",
"Seeking the physical origin of two accelerated expansion regimes of the Universe, namely, the primordial inflation and the present cosmic acceleration, is one of the most important theoretical challenges of cosmology today.",
"These unknown physical origins are referred by the primordial dark energy (DE) and the present DE.",
"Various theoretical models have been proposed to accelerate the cosmological expansion.",
"Among those theoretical models, $f(R)$ gravity is a simple and nontrivial generalization of General Relativity.",
"For a recent review, see Refs.",
"[1], [2].",
"It introduces a function $f(R)$ in the action, where $R$ is Ricci curvature.",
"This additional degree of freedom plays a role of a scalar field, which is called scalaron, and it can cause the accelerated expansion of the Universe.",
"The original idea recognized as $R^2$ inflation model was proposed in Ref.",
"[3], where de Sitter expansion was derived as a solution for the Einstein equation with quantum one-loop correction.",
"After the accelerating expansion, the particles are gravitationally created through the oscillation of the scalaron, and it leads radiation dominated Universe [4], [5], [6], [7], [8].",
"The $R^2$ model predicts an almost scale-invariant spectrum, whose scalar and tensor components are consistent with recent observational data (See e.g., [9]).",
"Later, the $R^2$ gravity was extended to a general function of $R$ , namely $f(R)$ , to describe the late time cosmic acceleration.",
"After some early challenges, the viable $f(R)$ models were proposed [10], [11], [12], which can realize stable matter-dominated regime and subsequent late time acceleration.",
"In these models, the expansion history of the Universe is close to that in the ${\\rm \\Lambda }$ -cold dark matter (${\\rm \\Lambda }$ CDM) model.",
"From the model selection point of view, the key to distinguish the models is focusing on small deviation from the ${\\rm \\Lambda }$ CDM model.",
"As for background quantity, the equation of state parameter $w_{\\rm DE}$ well characterizes the models.",
"While it remains constant $w_{\\rm DE}=-1$ in the ${\\rm \\Lambda }$ CDM model, it is time dependent in the $f(R)$ gravity and even crosses the phantom divide at redshift $z\\sim O(1)$ [10], [13], [14].",
"On the other hand, the growth of the matter density fluctuations is also useful to measure the deviation.",
"Since in $f(R)$ gravity the effective gravitational constant depends on time and scale, the matter power spectrum is enhanced [15], [16], [10], [12], [17], [18], [19], [20].",
"This enhancement not only measure the deviation but also provides another interesting consequence: $f(R)$ gravity admits larger neutrino mass.",
"This is because massive neutrinos suppress evolution of matter fluctuations by free streaming, which cancels the enhancement in $f(R)$ gravity.",
"As a result, $f(R)$ gravity allows larger neutrino mass up to $\\sim 0.5$ eV [21].",
"The constraint on sterile neutrino mass is also relaxed up to $\\sim 1$ eV, which is consistent with recent experiments [22].",
"Other distinguishable features of $f(R)$ gravity would be imprinted on cosmological gravitational waves [23], [24], [25].",
"Future pulsar timing experiments and gravitational-wave detectors will be able to probe them directly and test gravity theories [26], [27], [28], [29].",
"However, the above viable $f(R)$ models still have theoretical problems [12].",
"If we start from some initial condition and calculate back to the past, the scalaron mass diverges quickly and the scalaron oscillates rapidly [30].",
"Another problem is that the Ricci scalar also diverges at the past even if we include nonlinear effect [31].",
"This curvature singularity was also pointed out in Refs.",
"[32], [33].",
"To solve these problems, $R^2$ -corrected $f(R)$ model has been proposed [34].",
"This model is constructed from the late time acceleration part and $R^2$ term.",
"Consequently, the reheating followed by inflation in this model is different from that in $R^2$ model, i.e., the scalaron does not oscillate harmonically.",
"Of course, we can construct the other specific functional forms that avoid singularities and describe both the primordial DE and the present DE.",
"However, these functions belong to the same class and their behavior are similar [35].",
"In this sense, it is worth to study one specific model in detail as an example of such a class of extended $f(R)$ models.",
"The aim of this paper is to investigate the evolution during the inflation and reheating regimes of the $R^2$ -corrected $f(R)$ model in detail.",
"In the previous work [34], the reheating dynamics has been analyzed in the Jordan frame.",
"However, the field equation is fourth order differential equation, which makes the physical interpretation unclear.",
"In addition, in their analysis, the inflation and the reheating are separately solved with different initial conditions.",
"Therefore, we focus on the evolution of the scalaron rolling on a potential in the Einstein frame, which clarifies the physical picture and allows us to understand the dynamics intuitively.",
"We start from a certain initial condition imposed during inflationary regime and numerically solve the transition from the inflation to the reheating and the following reheating dynamics.",
"Thus, our analysis is more accurate than the previous work [34] and is complementary to the analysis in the Jordan frame.",
"This paper is organized as follows.",
"In Sec.",
", we review the basic equations in $f(R)$ gravity in both the Jordan frame and the Einstein frame.",
"We present the Einstein frame potential in the $R^2$ -corrected $f(R)$ model and consider its characteristics analytically.",
"In Sec.",
", we derive the analytic solutions of the inflation and reheating in the Einstein frame.",
"We shall adopt the slow roll and the fast roll approximations and solve the field equations in each era.",
"Sec.",
"contains the results of the numerical calculation.",
"We confirm that the analytic solutions are sufficiently in agreement with the numerical results.",
"We also consider the behavior at the end of the reheating.",
"In Sec.",
", we consider the connection between observables and model parameters and discuss its allowed ranges.",
"Sec.",
"is devoted to conclusions and discussion.",
"Throughout the paper, we adopt units $c=\\hbar =1$ ."
],
[
"Basic equations",
"We start to review the basic equations of $f(R)$ gravity in both the Jordan frame and the Einstein frame.",
"To avoid confusion of the frames, we fix the subscript $J$ and $E$ to physical quantities in the Jordan frame and the Einstein frame, respectively.",
"Otherwise we declare the frame in which the quantity is defined.",
"$f(R)$ gravity is defined by the action $S=\\int d^4x \\sqrt{-g_J} \\left[ {2} f(R_J) + {\\cal L}_M(g^J_{\\mu \\nu }) \\right] \\;, $ where ${\\cal L}_M$ is the Lagrangian density for the matter sector and $M_{\\rm {Pl}}$ is the reduced Planck mass.",
"By varying the action, we obtain the field equation in the Jordan frame $R^J_{\\mu \\nu } F(R_J) -\\frac{1}{2} g^J_{\\mu \\nu } f(R_J) + ( g^J_{\\mu \\nu } \\Box -\\nabla _{\\mu } \\nabla _{\\nu } ) F(R_J)= \\frac{T^J_{\\mu \\nu }}{M_{\\rm {Pl}}^2} \\;,$ with $F(R_J) \\equiv \\frac{d\\,f(R_J)}{dR_J} \\;, \\quad \\quad T_{\\mu \\nu }^J \\equiv -\\frac{2}{\\sqrt{-g_J}} \\frac{\\delta (\\sqrt{-g_J} {\\cal {L}}_{\\rm {M}})}{\\delta g_J^{\\mu \\nu }} \\;.",
"$ We regard the Jordan frame as the physical frame.",
"However, for our purpose analyzing the inflation and the reheating in $f(R)$ gravity, the formulation in the Einstein frame is useful because it contains the additional degree of freedom more explicitly as a scalar field and enables us to use the analogy of the single-field inflation.",
"We can recast the theory to the Einstein gravity with a scalar field by choosing the conformal transformation of the metric as $g^E_{\\mu \\nu }\\equiv F(R_J) g^J_{\\mu \\nu }$ .",
"The canonical scalar field $\\phi $ is defined by $F(R_J)\\equiv e^{\\sqrt{{3}}{M_{\\rm Pl}}}.",
"$ By the conformal transformation, the action is rewritten as $S=\\int d^4x \\sqrt{-g_E} \\left[ {2} R_E-{2}g_E^{\\mu \\nu } \\partial _\\mu \\phi \\partial _\\nu \\phi -V(\\phi ) +{\\cal L}_M \\left(e^{-\\sqrt{{3}}{M_{\\rm Pl}}} g^E_{\\mu \\nu } \\right) \\right], $ with the potential term $V(\\phi )=\\frac{M_{\\rm Pl}^2}{2} {F(R_J(\\phi ))^2}.",
"$ Then, the Einstein equation in the Einstein frame reduces to $H_E^2&=&{3M_{\\rm Pl}^2}\\left[ {2}\\left( {dt_E} \\right)^2 + V(\\phi ) + \\rho _E \\right], \\\\{dt_E}&=&-{2M_{\\rm Pl}^2}\\left[ \\left( {dt_E} \\right)^2 + \\rho _E+P_E \\right].",
"$ The equation of motion for the scalar field is ${dt_E^2}+3H_E{dt_E}+V_{,\\phi }(\\phi )={\\sqrt{6} M_{\\rm Pl}} (\\rho _E-3P_E).",
"$ From the conformal transformation, the time and scale factor in both frames are connected by $dt_J=e^{-{\\sqrt{6}}{M_{\\rm Pl}}}dt_E, \\quad a_J=e^{-{\\sqrt{6}}{M_{\\rm Pl}}}a_E.",
"$ The transformation of the Hubble parameter is derived from the above definitions, $H_J=e^{{\\sqrt{6}}{M_{\\rm Pl}}}\\left( H_E-{\\sqrt{6}M_{\\rm Pl}}{dt_E} \\right).",
"$ By definition in Eq.",
"(REF ), the energy momentum tensors of the matter sector in both frames are connected by $T^E_{\\mu \\nu }= e^{-\\sqrt{{3}}{M_{\\rm Pl}}} T^J_{\\mu \\nu }.",
"$ For perfect fluid $T^{\\mu }_{\\nu }={\\rm diag}(-\\rho , P,P,P)$ in each frame, the energy density and the pressure are related as $\\rho _E=e^{-2\\sqrt{{3}}{M_{\\rm Pl}}}\\rho _J,\\quad P_E=e^{-2\\sqrt{{3}}{M_{\\rm Pl}}}P_J.",
"$ Note that the energy density in the Einstein frame couples with the scalaron.",
"In the inflation and reheating in $f(R)$ gravity, there is no inflaton field from the point of view in the Jordan frame.",
"Consequently, particle creation occurs not through the decay of the inflaton but through the gravitational reheating [4], [5], [6], [7], [8].",
"Let us consider the gravitational particle creation in the Jordan frame.",
"We introduce a minimally or nonminimally coupled massless scalar field $\\chi $ , which describes the created particles, into the matter action, $S=\\int d^4x \\sqrt{-g_J} \\left[ {2}f(R_J) -{2}g_J^{\\mu \\nu } \\partial _{\\mu } \\chi \\partial _{\\nu } \\chi -{2} \\xi R\\chi ^2 \\right].",
"$ Since the scalar field $\\chi $ couples with the metric in the Jordan frame, the radiation (massless scalar particle) is created purely gravitationally.",
"Adopting the standard treatment of the quantum field theory in curved spacetime, we can expand $\\chi $ in Fourier modes with the annihilation and creation operators.",
"Then, computing the Bogolubov coefficients in the expanding Universe, we obtain the number density of the created scalar particles [4], [5], [6], [7], $n_J(t_J)={576\\pi a_J^3} \\int _{-\\infty }^{t_J} dt^{\\prime }_J a_J^3R_J^2.",
"$ The above equation holds regardless of the functional form of $f(R)$ .",
"Note that the particle creation is sourced by Ricci curvature.",
"Since the Ricci curvature is significantly suppressed during the reheating era as we shall see below, particle creation hardly occurs.",
"On the other hand, inflationary dynamics is the same as that of the $R^2$ inflation.",
"Therefore, we can use the approximated formula for the $R^2$ model and turn off the particle creation during the reheating era when we perform numerical calculation.",
"In the $R^2$ model with $f(R_J)=R_J+R_J^2/(6M^2)$ , the energy density of the created particles is $\\rho _J(t_J)={1152\\pi a_J^4} \\int _{-\\infty }^{t_J} dt^{\\prime }_J a_J^4 R_J^2 , $ where $g_*$ denotes the relativistic degree of freedom.",
"In the present paper, we consider minimally coupled massless scalar field and set $\\xi =0$ hereafter.",
"The evolution equation for the energy density of radiation is then ${dt_J}=-4H_J\\rho _J+{1152\\pi }\\;.",
"$ The pressure is obtained from the energy conservation equation, $P_J={3}-{3456\\pi H_J}.",
"$ Finally, we introduce $gR^2$ -AB model [34], which describes the accelerated expansions in both the early and the present Universes, $f (R_J)=(1-g)R_J+g M^2\\delta \\log \\left[ {\\cosh b} \\right]+{6M^2}, $ where $g,~b,~\\delta $ , and $M$ are positively-defined model parameters.",
"$\\delta $ describes the ratio between the energy scale of the present DE to the primordial DE and takes dramatically small value.",
"$M$ will be fixed to $M/M_{\\rm Pl}\\approx 1.2\\times 10^{-5}$ later in Sec.",
"by the temperature fluctuations of the cosmic microwave background (CMB) anisotropy.",
"$g$ is constrained in the range of $0<g<1/2$ by the stability conditions of $f(R)$ gravity: $F(R_J)>0$ and $dF(R_J)/dR_J>0$ .",
"Moreover, $g$ and $b$ are further constrained by the other stability condition as we shall show in Sec. .",
"The function $F(R_J)$ is given by $F(R_J)=1-g+{3M^2}+g \\tanh (R_J/M^2\\delta -b).",
"$ The above $f(R_J)$ function is equivalently written in the following form: $f(R_J) &=R_J-\\frac{R_{\\rm {vac}}}{2} + g M^2 \\delta \\log \\left[ 1+e^{-2(R_J/M^2\\delta -b)} \\right]+\\frac{R_J^2}{6M^2} \\;, \\\\R_{\\rm {vac}} &\\equiv 2 g M^2 \\delta \\left\\lbrace b+\\log (2\\cosh b) \\right\\rbrace .$ In Eq.",
"(REF ), the fourth term dominates at high curvature regime $R_J \\gg M^2$ and causes inflation [3].",
"The third term alters the reheating dynamics after the inflation, which is characteristic of the $gR^2$ -AB model.",
"The second term plays the same role as the current cosmological constant.",
"By substituting the action into Eq.",
"(REF ), the equation of motion in the Jordan frame is given by $&H_J^{^{\\prime \\prime }} H_J-\\frac{1}{2} (H_J^{^{\\prime }})^2+3 H_J^{^{\\prime }} H_J^2 +\\frac{1-g}{2} M^2 H_J^2 -\\frac{g}{2} M^2 (H_J^{^{\\prime }}+H_J^2) \\tanh \\left[ \\frac{R_J}{M^2\\delta }-b \\right] \\nonumber \\\\&+\\frac{g}{12} M^4\\delta \\ln \\left[ \\frac{\\cosh (R_J/M^2\\delta -b)}{\\cosh b} \\right] + \\frac{3g(H_J^{^{\\prime \\prime }} H_J + 4 H_J^{^{\\prime }} H_J^2)}{\\delta \\cosh ^2 (R_J/M^2\\delta -b)}= \\frac{M^2}{6M_{\\rm {Pl}}^2} \\rho _J \\;.$ The prime denote time derivative with respect to $t_J$ .",
"As expected from the specific form of $f(R_J)$ function, the equation of motion in the Jordan frame is nonlinear differential equation and quite complicated to solve.",
"The dynamics of inflation and reheating can be more intuitively understood from the potential in the Einstein frame, which is depicted in Fig.",
"REF (a).",
"From the definition of Eq.",
"(REF ), we can interpret the shape of the potential in the following way.",
"First, we notice that from Eq.",
"(REF ), $F(R_J)$ is almost a step function at $R_J/M^2\\simeq b\\delta $ with the change of the amplitude from $F=1-2g$ to 1.",
"In terms of the scalaron, the step corresponds from $\\phi /M_{\\rm Pl}=\\sqrt{6}\\log \\gamma $ to 0, where $\\gamma \\equiv \\sqrt{1-2g}$ .",
"In other words, during the scalaron moving in the range $\\sqrt{6}\\log \\gamma \\le \\phi /M_{\\rm Pl}\\le 0$ , $R_J$ remains almost constant value, $R_J/M^2\\simeq b\\delta $ .",
"Outside this interval, $F(R_J)$ is approximated by $F\\simeq 1+R/3M^2$ for $R_J/M^2 > b\\delta $ , i.e., $\\phi /M_{\\rm Pl}>0$ , and $F\\simeq 1-2g+R/3M^2$ for $R_J/M^2 < b\\delta $ , i.e., $\\phi /M_{\\rm Pl}<\\sqrt{6}\\log \\gamma $ , respectively.",
"By using these approximations, we can derive the potential analytically in terms of $\\phi $ : ${M_{\\rm Pl}^2M^2}\\simeq \\left\\lbrace \\begin{array}{ll}\\displaystyle {4}\\left( 1-e^{-\\sqrt{{3}}{M_{\\rm Pl}}} \\right)^2, &\\quad (\\phi /M_{\\rm Pl}> 0) \\\\\\displaystyle {4}\\left( 1-\\gamma ^2 e^{-\\sqrt{{3}}{M_{\\rm Pl}}} \\right)^2.",
"&\\quad (\\phi /M_{\\rm Pl}< \\sqrt{6}\\log \\gamma )\\end{array}\\right.",
"$ In these two regions, the potential is the same as that in the $R^2$ inflation.",
"On the other hand, the characteristic plateau shows up for $\\sqrt{6}\\log \\gamma <\\phi /M_{\\rm Pl}<0$ .",
"By using $R_J/M^2\\simeq b\\delta $ , we can approximate the potential as ${M_{\\rm Pl}^2M^2}\\simeq {2 e^{2\\sqrt{{3}}{M_{\\rm Pl}}}}.",
"\\quad (\\sqrt{6}\\log \\gamma <\\phi /M_{\\rm Pl}< 0) $ From this expression, we can estimate the height of the bump in the plateau.",
"We can obtain the position of the local maximum by solving $V^{\\prime }=0$ .",
"The solution is $\\phi /M_{\\rm Pl}\\simeq \\sqrt{3/2}\\log [2(1-2g)+b\\delta /3]\\equiv \\phi _m/M_{\\rm Pl}$ .",
"The potential has the local maximum when $\\phi _m$ satisfies $\\sqrt{6}\\log \\gamma < \\phi _m/M_{\\rm Pl}< 0$ , namely, $(3+b\\delta )/12 \\lesssim g \\lesssim (3+b\\delta )/6$ .",
"For $\\delta \\ll 1$ , this condition amounts to $1/4 \\lesssim g \\lesssim 1/2$ .",
"Therefore, in Fig.",
"REF (c), the potential for $g=0.35$ possesses the local maximum and the false vacuum.",
"However, as $g$ approaches $g=1/4$ , the local maximum is likely to disappear.",
"The potential heights at the right edge, the local maximum, and the left edge are given by ${M_{\\rm Pl}^2M^2}\\simeq bg\\delta ,\\quad {M_{\\rm Pl}^2M^2}\\simeq {8(1-2g)} , \\quad {M_{\\rm Pl}^2M^2}\\simeq 0, $ where we used $\\delta \\ll 1$ and $\\log (\\cosh b)\\simeq b$ .",
"These estimations well explain the parameter dependences of the potential shape in Fig.",
"REF (b) - (d).",
"Next, let us consider the evolution of the scalaron.",
"The scalaron starts slow rolling from $\\phi >0$ and plays a role of the inflaton.",
"For $\\phi >0$ , the potential is the same as that of a pure $R^2$ model and is almost independent of model parameters $g,~b$ and $\\delta $ .",
"Thus, the scale factor in the Einstein frame experiences quasi-de-Sitter expansion.",
"In this case, the scale factor in the Jordan frame also evolves exponentially, because the amplitude of $\\phi $ slowly varies and thus the scale factors in both frames are related by multiplying an almost constant factor.",
"As the scalaron approaches $\\phi =0$ , it rolls faster and enters the potential plateau with the kinetic energy larger than the potential energy.",
"Then the scalaron oscillates in the plateau, gradually loses its kinetic energy, and finally reaches the false vacuum at $\\phi =0$ because the chameleon effect lifts up the potential when the energy density of matter does not negligible [35].",
"During this oscillation, the scale factor in the Jordan frame undergoes the periodic evolution due to the exponential factor in Eq.",
"(REF ).",
"We shall see these situations in the next section.",
"Figure: Inflaton potential of the gR 2 gR^2-AB model in the Einstein frame.",
"(a): Potential for g=0.35,b=5,δ=5×10 -8 g=0.35, b=5, \\delta =5\\times 10^{-8}.",
"For φ>0\\phi >0 and φ<6M Pl logγ\\phi <\\sqrt{6}M_{\\rm Pl}\\log \\gamma , the potential is similar to that in pure R 2 R^2 inflation.",
"On the other hand, for 6M Pl logγ<φ<0\\sqrt{6}M_{\\rm Pl}\\log \\gamma <\\phi <0, there is the characteristic plateau.",
"The scalaron starts slow rolling from φ>0\\phi >0, and enters the plateau with large kinetic energy and oscillates inside it.",
"(b) - (d): How plateau changes its shape when one parameter is changed from the value in (a).",
"The typical height of the plateau is bgδbg\\delta from the analytic estimation."
],
[
"Analytic Solutions",
"To investigate the reheating dynamics in the Jordan frame, we work in the Einstein frame and derive the analytical expressions for the motion of the scalaron in the inflaton potential.",
"For $\\phi >0$ , the potential is almost reduced to that of a pure $R^2$ model, then we can use the slow-roll approximation in Sec.",
"REF and derive the analytic solutions.",
"After the end of the inflation, the kinetic energy of the scalaron is dominant.",
"Therefore, we can neglect the small structure of the potential during the oscillation in the plateau and use the approximation, $R_J\\sim b\\delta $ .",
"We shall explore this case in Sec.",
"REF .",
"Since in both the slow-roll and oscillation regimes the energy density of the created radiation is subdominant in comparison with the energy density of the inflaton, we neglect its backreaction to the background dynamics to derive the analytic solutions.",
"However, the energy density of radiation eventually becomes the same order as the total energy of the inflaton, and the reheating ends.",
"In the following, we define dimensionless variables $\\hat{t}=Mt,~\\hat{\\phi }=\\phi /M_{\\rm Pl},~\\hat{V}=V/M^2M_{\\rm Pl}^2,~\\hat{H}=H/M,~\\hat{R}=R/M^2,~\\hat{\\rho }_{\\rm {r}}=\\rho _{\\rm {r}}/M_{\\rm Pl}^2M^2$ and abbreviate the hat in this section to avoid the complexity of notation."
],
[
"Slow roll approximation",
"The inflaton starts to roll down the potential at $t_E=t_{E,{\\rm ini}}$ ($t_J=t_{J,{\\rm ini}}$ ) with small kinetic energy compared to the height of the potential.",
"The energy density of the radiation is also negligible.",
"The field equations (REF ) – (REF ) with slow roll approximations are $H_E&=&\\sqrt{{3}},\\\\\\dot{H}_E&=&-{6V},\\\\\\dot{\\phi }&=&-{\\sqrt{3V(\\phi )}},$ where the dot denotes the derivative with respect to the time in Einstein frame and the potential is approximated by Eq.",
"(REF ).",
"We set the initial condition as $\\phi =\\phi _{\\rm ini}$ .",
"Other initial quantities are fixed from the above equations with the approximated potential.",
"By substituting the potential into the field equations, we can derive the following solutions: $\\phi (t_E)&=&\\sqrt{{2}}\\log \\left[ d_{\\rm ini}^2-{3}(t_E-t_{E,{\\rm ini}}) \\right], \\\\\\dot{\\phi }(t_E)&=&-\\sqrt{{3}}\\left[ d_{\\rm ini}^2-{3}(t_E-t_{E,{\\rm ini}}) \\right]^{-1}, \\\\H_E(t_E)&=&{2}\\left[ 1-\\left( d_{\\rm ini}^2-{3}(t_E-t_{E,{\\rm ini}}) \\right)^{-1} \\right], \\\\a_E(t_E)&=& a_{E,{\\rm ini}} e^{(t_E-t_{E,{\\rm ini}})/2} \\left[ 1-{3}d_{\\rm ini}^{-2}(t_E-t_{E,{\\rm ini}}) \\right]^{3/4}, \\\\R_J(t_E)&=&3(e^{\\sqrt{{3}}\\phi (t_E)}-1), $ where we use $d_{\\rm ini}\\equiv e^{{\\sqrt{6}}}$ instead of $\\phi _{\\rm ini}$ itself.",
"The remaining task is to write down $t_E$ in terms of $t_J$ and to convert the above solutions.",
"Jordan frame time is derived by integrating Eq.",
"(REF ), $t_J(t_E)=t_{J,{\\rm ini}}-3\\sqrt{d_{\\rm ini}^2-{3}(t_E-t_{E,{\\rm ini}})}+3d_{\\rm ini}, $ and its inverse function is $t_E(t_J)=t_{E,{\\rm ini}}+(t_J-t_{J,{\\rm ini}})\\left[ d_{\\rm ini}-{6}(t_J-t_{J,{\\rm ini}}) \\right].",
"$ Thus, substituting Eq.",
"(REF ) into the solutions (REF ) – () and using Eqs.",
"(REF ) and (REF ), we obtain the analytic solutions in terms of the Jordan frame quantities.",
"Next, we derive an analytic solution for $\\rho _J$ by performing integration in Eq.",
"(REF ) with Eqs.",
"(REF ), (REF ), (), () and (REF ), $\\rho _r(t_J)&={1152\\pi }\\left( M̑{M_{\\rm Pl}} \\right)^2{16S^2} \\nonumber \\\\& \\times \\left[ -2\\sqrt{3}S(4S^4-14S^2+15)+3e^{S^2}\\left\\lbrace -6d_{\\rm ini}e^{-3d_{\\rm ini}^2}(12d_{\\rm ini}^4-14d_{\\rm ini}^2+5)+5\\sqrt{3\\pi }\\left( {\\rm erf}(\\sqrt{3}d_{\\rm ini})+{\\rm erf}(S) \\right) \\right\\rbrace \\right], $ where $S\\equiv (t_J-t_{J,{\\rm ini}}-3d_{\\rm ini})/\\sqrt{3}$ and ${\\rm erf}(x)$ is the error function.",
"Hereafter, we refer $\\rho _J$ as $\\rho _r$ , especially denoting radiation component.",
"Finally, we focus on boundary conditions.",
"By using the above analytic solutions, we define the time $t_E=t_{E0}$ or $t_J=t_{J0}$ when the inflaton reaches $\\phi =0$ for the first time.",
"Strictly speaking, when $\\phi \\simeq 0$ , the slow-roll approximation is not valid anymore and the boundary conditions are very sensitive due to the the sudden transition of the potential at $\\phi \\simeq 0$ .",
"Therefore, we should use the boundary conditions obtained from numerical computation.",
"We only rely on the analytical boundary values as the estimator.",
"We shall revisit this point in Sec. .",
"The time $t_{E0}$ is analytically estimated by using the analytic solution (REF ), $t_{E0}=t_{E,{\\rm ini}}+{2}(d_{\\rm ini}^2-1).",
"$ In terms of Jordan frame time, $t_{J0}=t_{J,{\\rm ini}}+3(d_{\\rm ini}-1).",
"$ The boundary conditions are $\\phi _0=0, \\quad \\dot{\\phi }_0=-\\sqrt{{3}}, \\quad a_{E0}=a_{E,{\\rm ini}}d_{\\rm ini}^{-2}e^{{4}(d_{\\rm ini}^2-1)}.$ From Eq.",
"(REF ), $H_{E0}={3},\\quad H_{J0}={3}.$ The energy density $\\rho _r$ at $t_J=t_{J0}$ is given by setting $S=-\\sqrt{3}$ in Eq.",
"(REF ) and taking large $\\phi _{\\rm {ini}}$ limit, i.e., $d_{\\rm ini}\\gg 1$ , $\\rho _{r0} = {1152\\pi }\\left( M̑{M_{\\rm Pl}} \\right)^2 \\;.",
"\\nonumber $ The coefficient $c_0$ is analytically given by $\\left[ 18+5e^3\\sqrt{3\\pi }\\left( 1-{\\rm erf}(\\sqrt{3}) \\right) \\right]/16 \\approx 1.4$ , but at this intermediate stage from the slow roll to the fast roll, the slow-roll approximation does not give the precise value.",
"So we will use the result of the numerical calculation for the precise determination of $c_0$ later.",
"We notice that since $\\rho _{r0}\\ll 3H_{J0}^2$ for typical values $g_*=106.75$ and $M \\ll M_{\\rm {Pl}}$ , we can neglect the energy density of the created radiation for the purpose of deriving analytic formulas of reheating dynamics in the next subsection."
],
[
"Fast roll approximation",
"When the inflaton falls down to the plateau on the bottom of the potential that lies $\\sqrt{6}\\log \\gamma <\\phi <0$ , its kinetic energy is much greater than the potential energy, which is typically of the order of $bg\\delta $ .",
"Moreover, it is also greater than the energy density of radiation.",
"Thus, we will use two approximations to derive the analytic solutions.",
"First, the energy density created at the slow roll regime is not dominant as we mentioned above.",
"Second, particle creation is negligible during the fast-roll regime because the source term $R_J^2$ keeps an almost constant value $b^2\\delta ^2 \\ll 1$ .",
"Thus, we can use the fast-roll approximation in Eqs.",
"(REF ) – (REF ): $&&H_E^2={6},\\\\&&\\dot{H}_E=-{2},\\\\&&\\ddot{\\phi }+3H_E\\dot{\\phi }=0.$ In the plateau, the inflaton repeatedly oscillates between the left and right walls of the potential.",
"We separately consider the time intervals dependent on the direction of the motion of the inflaton and derive the analytic solution in each regime.",
"The inflaton is reflected at the left side of the plateau at $\\phi =\\sqrt{6}\\log \\gamma $ .",
"We regard the reflection occurs instantly and define the first reflection at $t=t_{E1}$ .",
"After that, the inflaton reaches the right wall at $\\phi =0$ and is reflected at $t=t_{E2}$ .",
"Thus, we can periodically define $t_{En}$ until the inflaton stops somewhere.",
"First, let us consider the interval $t_{E0}<t_E<t_{E1}$ .",
"Eliminating $\\dot{\\phi }$ from Eq.",
"(REF ) and () and then solving it, we obtain the Hubble parameter and the scale factor $H_E(t_E)&=&{3H_{E0}(t_E-t_{E0})+1}, \\\\a_E(t_E)&=&a_{E0}\\left[ 3H_{E0}(t_E-t_{E0})+1 \\right]^{1/3}.",
"$ From Eq.",
"(REF ), we obtain $\\dot{\\phi }=\\pm \\sqrt{6}H_E$ .",
"For $t_{E0}<t_E<t_{E1}$ , since the inflaton moves in the left direction, we choose $\\dot{\\phi }=-\\sqrt{6} H_E$ .",
"Therefore, the solution is given by $\\phi (t_E)=-\\sqrt{{3}}\\log \\left[ 3H_{E0}(t_E-t_{E0})+1 \\right].",
"$ Next, we move to the interval $t_{E1}<t_E<t_{E2}$ .",
"By using $\\dot{\\phi }=+\\sqrt{6}H_E$ , we can derive $H_E(t_E)&=&{3H_{E1}(t_E-t_{E1})+1}, \\\\a_E(t_E)&=&a_{E1}\\left[ 3H_{E1}(t_E-t_{E1})+1 \\right]^{1/3}, \\\\\\phi (t_E)&=&\\sqrt{{3}}\\log \\left[ 3H_{E1}(t_E-t_{E1})+1 \\right]+\\sqrt{6}\\log \\gamma \\;.",
"$ The matching conditions between the two intervals are determined by setting $\\phi (t_{E1})=\\sqrt{6}\\log \\gamma $ in the solution for $t_{E0}<t_E<t_{E1}$ as $t_{E1}&=&t_{E0}+{3H_{E0}}, \\\\H_{E1}&=&H_{E0}\\gamma ^{3},\\\\a_{E1}&=&a_{E0}\\gamma ^{-1}.$ By substituting them into the solutions (REF ) – (), we obtain the same expressions as in Eqs.",
"(REF ) and () for $H_E(t_E)$ and $a_E(t_E)$ .",
"However, $\\phi (t_E)$ is different from Eq.",
"(REF ).",
"It is given by $\\phi (t_E)=\\sqrt{{3}}\\log \\left[ 3H_{E0}(t_E-t_{E0})+1 \\right]+2\\sqrt{6}\\log \\gamma .",
"$ Likewise, by solving the recurrence equations, we conclude that the boundary conditions for general $n$ are $t_{En}&=t_{E0}+{3H_{E0}}, \\\\H_{En}&=H_{E0}\\gamma ^{3n}, \\\\a_{En}&=a_{E0}\\gamma ^{-n},\\\\\\phi _{En}&=\\left\\lbrace \\begin{array}{ll}\\displaystyle 0 &\\quad (n:{\\rm even}) \\\\\\displaystyle \\sqrt{6}\\log \\gamma &\\quad (n:{\\rm odd})\\end{array}\\right.",
"\\;.$ Especially, it is noteworthy that the following relation holds regardless of the parity of $n$ : $3H_{En}(t_{E}-t_{En})+1=\\gamma ^{3(n-1)}[3H_{E0}(t_{E}-t_{E0})+1] \\;.",
"$ Thus, the solutions for $t_{En-1}<t_E<t_{En}$ are $H_E(t_E)&=&{3H_{E0}(t_E-t_{E0})+1}, \\\\a_E(t_E)&=&a_{E0}\\left[ 3H_{E0}(t_E-t_{E0})+1 \\right]^{1/3}, \\\\\\phi (t_E)&=&\\left\\lbrace \\begin{array}{ll}\\displaystyle -\\sqrt{{3}}\\log \\left[ 3H_{E0}(t_E-t_{E0})+1 \\right]-(n-1)\\sqrt{6}\\log \\gamma &\\quad (n:{\\rm odd}) \\\\\\displaystyle \\sqrt{{3}}\\log \\left[ 3H_{E0}(t_E-t_{E0})+1 \\right]+n\\sqrt{6}\\log \\gamma &\\quad (n:{\\rm even})\\end{array}\\right.",
"\\;, \\\\\\dot{\\phi }(t_E)&=&\\left\\lbrace \\begin{array}{ll}\\displaystyle -\\sqrt{6}H_E(t_E) &\\quad (n:{\\rm odd}) \\\\\\displaystyle \\sqrt{6}H_E(t_E) &\\quad (n:{\\rm even})\\end{array}\\right.",
"\\;.",
"$ Notice that the solutions for $\\phi $ and $\\dot{\\phi }$ have different expressions for the different parity of $n$ because of its direction of the motion.",
"Once the solutions in the Einstein frame are at hand, it is straightforward to convert them to those in the Jordan frame.",
"The Jordan frame time $t_J$ for $t_{En-1}<t_E<t_{En}$ is obtained by integrating $e^{-\\phi /\\sqrt{6}}$ , $t_J(t_E)=\\left\\lbrace \\begin{array}{ll}\\displaystyle t_{Jn-1}+{4H_{E0}} \\left[ \\left\\lbrace 3 H_{E0}(t_E-t_{E0})+1\\right\\rbrace ^{4/3}-\\gamma ^{-4(n-1)} \\right] &\\quad (n:{\\rm odd}) \\\\\\displaystyle t_{Jn-1}+{2H_{E0}} \\left[ \\left\\lbrace 3 H_{E0}(t_E-t_{E0})+1\\right\\rbrace ^{2/3}-\\gamma ^{-2(n-1)} \\right] &\\quad (n:{\\rm even})\\end{array}\\right.",
"\\;, $ where $t_{Jn}$ is given by $t_{Jn}=\\left\\lbrace \\begin{array}{ll}\\displaystyle t_{J0}+{4H_{E0}(\\gamma ^4+\\gamma ^2+1)}+{4H_{E0}\\gamma ^{3(n-1)}} &\\quad (n:{\\rm odd}) \\\\\\displaystyle t_{J0}+{4H_{E0}(\\gamma ^4+\\gamma ^2+1)} &\\quad (n:{\\rm even})\\end{array}\\right.",
"\\;.$ From Eqs.",
"(REF ), (REF ), (REF ), (), and (REF ), the Hubble parameter and the scale factor in the Jordan frame evolve as $H_J(t_J)&=&\\left\\lbrace \\begin{array}{ll}\\displaystyle {4\\gamma ^{3(n-1)}H_{E0}(t_J-t_{Jn-1})+1} &\\quad (n:{\\rm odd}) \\\\\\displaystyle 0 &\\quad (n:{\\rm even})\\end{array}\\right.",
"\\;, \\\\a_J(t_J)&=&\\left\\lbrace \\begin{array}{ll}\\displaystyle a_{J0}\\gamma ^{-(n-1)}\\left[ 4\\gamma ^{3(n-1)}H_{E0}(t_J-t_{Jn-1})+1 \\right]^{1/2} &\\quad (n:{\\rm odd}) \\\\\\displaystyle a_{J0}\\gamma ^{-n} &\\quad (n:{\\rm even})\\end{array}\\right.",
"\\;.",
"\\\\$ $H_J(t_J)$ periodically takes discrete behaviors at $t_J=t_{Jn}$ and vanishes for even $n$ .",
"However, to be precise, $H_J$ is not exactly zero because we here neglect the contribution from the energy density of created radiation and the potential.",
"We shall numerically confirm this point in the next section.",
"We define $H_{Jn}$ by $\\lim _{\\epsilon \\rightarrow \\pm 0} H(t_{Jn}+\\epsilon )$ where $\\pm $ for even and odd $n$ , respectively.",
"From the Hubble parameter (REF ), it is given by $H_{Jn}=2H_{En} = \\gamma ^{3n} H_{J0}.",
"$ Here we used Eqs.",
"(REF ) and ().",
"Let us remark the time averaged behavior during the oscillation.",
"Since Eq.",
"(REF ) implies $\\log (t_{En}-t_{E0}) \\propto n$ , the time intervals when inflaton goes to the left and the right are equal in terms of the logarithmic Einstein-frame time.",
"On the other hand, Eq.",
"(REF ) yields $\\log (t_J-t_{J0})\\simeq p\\log (t_E-t_{E0})$ with $p=4/3$ and $2/3$ for the left-directed regime and the right-directed regime, respectively.",
"Hence, the left-directed regime is twice as long as the right-directed regime in terms of the logarithmic Jordan-frame time.",
"Taking care of these facts, we can derive the average power of the Jordan frame quantities.",
"For instance, as for time duration, since the average power is $(4/3+2/3)/(1+1)=1$ , we can say $t_E$ is proportional to $t_J$ in average, i.e., $\\langle t_J-t_{J0}\\rangle \\propto t_E-t_{E0}$ .",
"Since $\\log a_J\\simeq q \\log t_J$ with $q=1/2$ and 0 for the left-directed regime and the right-directed regime respectively, the averaged power is $(1/2\\times 2+0\\times 1)/(2+1)=1/3$ , namely, $\\langle a_J(t_J)\\rangle \\propto (t_J-t_{J0})^{1/3}$ .",
"Hence, the averaged Hubble parameter is written as $\\langle H_J(t_J) \\rangle = \\frac{H_{J0}}{3 H_{J0} (t_J-t_{J0}) +1} \\;.$ Integrating this gives the averaged scale factor $\\langle a_J(t_J) \\rangle = a_{J0} \\left[ 3 H_{J0} (t_J-t_{J0}) + 1 \\right]^{1/3} \\;.$ In summary, neglecting the small structure of the plateau and the energy density of the radiation enables us to solve the differential equations analytically.",
"However, in the final phase of the reheating, they become important.",
"We shall consider this effect in the next section."
],
[
"Numerical Results",
"As we derived in Sec.",
", the typical height of the local maximum of the potential plateau is $V_{\\rm max}\\propto \\delta $ .",
"The oscillation regime ends when the kinetic energy of scalaron is of the same order of $V_{\\rm max}$ .",
"Thus, $\\delta $ determines the end time of the reheating.",
"However, since $\\delta $ is the ratio between the energy scales of DE and of inflation, it has to be dramatically tiny value $\\sim 10^{-120}$ .",
"Therefore, it is difficult to carry out numerical calculation for realistic model parameters.",
"In this section, using the modest value of $\\delta $ , we confirm the validity of the analytic solutions obtained in the previous section by comparing them with numerical results.",
"Then we extrapolate the analytic solutions to the end of the reheating and estimate the reheating temperature as a function of the model parameters.",
"Note that hereafter we explicitly fix the hat to normalized dimensionless physical variables, which was abbreviated in the previous section."
],
[
"Comparison with analytic solutions",
"We performed numerical calculation in the Einstein frame, i.e., we solved the coupled evolution equations: Eqs.",
"(), (REF ), and (REF ).",
"In particular, to avoid the complexity due to nonminimal couplings in radiation and matter sectors, we use Eq.",
"(REF ), instead of the continuous equation in the Einstein frame, by translating it to the Einstein frame with Eqs.",
"(REF ), (REF ), and (REF ).",
"After the numerical calculation, we obtain physical quantities in the Jordan frame by the inverse conformal transformations.",
"Figure REF illustrates the numerical results and the analytic solutions for a model parameter set: $g=0.35, b=5, \\delta =5\\times 10^{-8}, M/M_{\\rm Pl}=1.2\\times 10^{-5}$ .",
"Since observationally required value for $\\delta $ is too tiny to perform the numerical calculation, we chose this $\\delta $ just for demonstration and compared its results with the analytic solutions.",
"As mentioned before, the slow-roll approximation does not hold at the intermediate stage from the slow roll to the fast roll.",
"So we used the boundary conditions obtained by the numerical calculation as the initial conditions for the fast-roll approximated solutions: $c_0 = 0.72$ and $\\hat{H}_{J0} =0.26$ (These are different from the values obtained from slow-roll analytic solutions by about a factor of 2).",
"The analytic solutions well approximate the numerical results.",
"In the slow-roll regime, the scale factor undergoes quasi-de Sitter expansion.",
"In the fast-roll regime, the inflaton oscillates two times inside the potential plateau.",
"We see that the scale factor evolves as $a_J(t_J)\\propto (t_J-t_{J0})^{1/2}$ and $a_J(t_J)\\simeq \\text{const}$ in order.",
"Ricci scalar stays constant value $\\simeq b\\delta $ except for several spikes during the reheating.",
"In contrast to the analytical solution, $H_J$ does not vanish when the inflaton moves to the right direction in the potential plateau because of the contribution from the energy density $\\rho _r$ of the created radiation.",
"At late time, the fast-roll approximation becomes worse as the inflaton loses its kinetic energy.",
"Finally, the inflaton reaches a false vacuum at $\\phi =0$ ."
],
[
"End of the reheating",
"Since the above calculation have performed for $\\delta =5\\times 10^{-8}$ , the scalaron ends the oscillation quickly before radiation dominates the Universe.",
"However, the scenario is different for $\\delta \\sim 10^{-120}$ : the reheating appropriately ends by radiation domination.",
"This is because $V(0)$ and $V_{\\rm {max}}$ is too small to compete with the inflaton kinetic energy.",
"For the following, by using the analytic solutions, we estimate both the times when the scalaron stops the oscillation and when radiation dominates the Universe.",
"First, let us estimate when the oscillation stops.",
"It is estimated from the equality time of the kinetic energy of the scalaron and the local maximum of the plateau: $\\dot{\\phi }^2/2\\sim V_{\\rm max}$ .",
"We use the analytic solution for $\\dot{\\phi }$ in Eq.",
"() and the asymptotic behavior $\\langle \\hat{H}_E \\rangle \\sim 1/3(\\hat{t}_E-\\hat{t}_{E0})\\sim 1/3(\\hat{t}_J-\\hat{t}_{J0})$ .",
"Thus, the scalaron ceases the oscillation at $\\hat{t}_{Js}-\\hat{t}_{J0} =\\sqrt{{3b\\delta }} , $ where we used $V_{\\rm max}$ by Eq.",
"(REF ).",
"Substituting $\\delta \\sim 10^{-120}$ , $M=1.2\\times 10^{-5}M_{\\rm Pl}$ , and $g,b \\sim {\\cal {O}}(1)$ yields $t_{Js}\\sim 10^{46}~\\text{GeV}^{-1}$ , which is close to the Hubble time $H_0^{-1}$ .",
"To be precise, we should take the radiation and matter dominated epochs into account in the following evolution of the Universe, but it does not change the conclusion that the scalaron oscillation continues for the order of the Hubble time.",
"Next, let us estimate when the radiation dominates the Universe and the reheating ends.",
"We compare the energy density of radiation due to particle creation and that of gravity.",
"Since $R_J\\simeq b\\delta $ during the oscillation in the plateau of the potential, we can expand $F$ around $R_J= b\\delta $ and take its linear order only, $F\\simeq 1-g+{3}+\\left( {3}+g̑{\\delta } \\right)(\\hat{R}_J-b\\delta )\\equiv e^{\\sqrt{{3}}\\hat{\\phi }}.",
"$ Thus, we can explicitly connect the Ricci scalar in the Jordan frame with the inflaton as $\\hat{R}_J(t_E)={3g +\\delta }.",
"$ As we mentioned at the beginning of Sec.",
"REF , the particle creation during the plateau oscillation phase is negligible because $\\hat{R}_J\\simeq b\\delta \\ll 1$ .",
"Therefore, $\\rho _r$ is approximately given by $\\langle \\rho _r(t_J) \\rangle =\\rho _{r0}\\left( {a_{J0}} \\right)^{-4} \\approx \\frac{\\rho _{r0}}{\\left[ 3 H_{J0} (t_J-t_{J0})\\right]^{4/3}}, $ where we used Eq.",
"(REF ).",
"As for gravitational contribution, it is convenient to define the effective energy density of gravity by the equation of motion in the Jordan frame in Eq.",
"(REF ): $H_J^2 &= \\frac{1}{3M_{\\rm {Pl}}^2} (\\rho _r+\\rho _g) \\;, \\\\\\rho _g &\\equiv \\frac{3M_{\\rm {Pl}}^2}{M^2} \\left( g M^2 H_J^2 -2 H_J^{^{\\prime \\prime }} H_J +(H_J^{^{\\prime }})^2-6 H_J^{^{\\prime }} H_J^2 +g M^2 (H_J^{^{\\prime }}+H_J^2) \\tanh \\left[ \\frac{R_J}{M^2 \\delta }-b \\right] \\right.",
"\\nonumber \\\\&\\left.",
"-\\frac{g}{6} M^4\\delta \\log \\left[ \\frac{\\cosh (R_J/M^2\\delta -b)}{\\cosh b} \\right] - \\frac{6g (H_J^{^{\\prime \\prime }} H_J + 4 H_J^{^{\\prime }} H_J^2)}{\\delta \\cosh ^2 (R_J/M^2\\delta -b)} \\right) \\;.$ When $\\rho _r$ is negligible compared to $\\rho _g$ , from Eq.",
"(REF ), the energy density of gravity is reduced to $\\langle \\hat{\\rho }_g \\rangle \\approx 3 \\langle \\hat{H}_J \\rangle ^2 \\approx \\frac{1}{3(\\hat{t}_J-\\hat{t}_{J0})^2}\\;.$ Figure: Evolution of energy density of radiation ρ ^ r \\hat{\\rho }_r (blue) and effective energy density of gravity ρ ^ g ≈3H ^ J 2 \\hat{\\rho }_g\\approx 3\\hat{H}_J^2 (red) for model parameter g=0.35,b=5,δ=5×10 -8 ,M/M Pl =1.2×10 -5 g=0.35, b=5, \\delta =5\\times 10^{-8}, M/M_{\\rm Pl}=1.2\\times 10^{-5}.",
"For energy density of radiation, the analytic solutions for slow roll regime (magenta, dashed) and fast roll regime (green, dot-dashed) are also presented.Figure REF represents the evolution of $\\hat{\\rho }_r$ and $\\hat{\\rho }_g\\approx 3 \\hat{H}_J^2$ .",
"Throughout the inflation and reheating, the energy density of the radiation is subdominant for our choice of $\\delta =5\\times 10^{-8}$ .",
"The radiation is mainly created at the end of inflation and decays as $\\rho _r\\propto a^{-4}$ .",
"Since the scale factor evolves as $t_J^{1/2}$ and constant periodically, $\\rho _r$ correspondingly evolves as $t_J^{-2}$ and constant during the left-directed and right-directed regimes, respectively.",
"As we mentioned, $3 \\hat{H}_J^2$ does not vanish because of the contribution from the radiation.",
"At $Mt_J\\simeq 50$ and 1000, $3 \\hat{H}_J^2$ is smaller than $\\rho _r$ .",
"This is because the required step width at the abrupt transition is so tiny that $\\hat{H}_J$ over-decreases.",
"This tiny discrepancy is unimportant and does not affect the subsequent evolution.",
"The end time of the reheating and the reheating temperature $T_{\\rm {reh}}$ is defined by the condition $\\langle \\rho _r \\rangle =\\langle \\rho _g \\rangle $ .",
"Using Eqs.",
"(REF ) and (REF ), we obtain $\\hat{t}_{J,\\rm {reh}} - \\hat{t}_{J0} \\approx \\frac{\\sqrt{3} \\hat{H}_{J0}^2}{\\hat{\\rho }_{r0}^{3/2}}\\approx 3.8 \\times 10^5 (c_0 g_*)^{-3/2} \\hat{H}_{J0}^2 \\left( \\frac{M}{M_{\\rm {Pl}}}\\right)^{-3} \\;,$ and $\\frac{T_{\\rm {reh}}}{M} = \\hat{H}_{J0} \\left( \\frac{a_{J0}}{\\langle a_{J,\\rm {reh}} \\rangle } \\right) \\approx \\sqrt{\\frac{\\hat{\\rho }_{r0}}{3}} \\approx 9.6 \\times 10^{-3} (c_0 g_*)^{1/2} \\left( \\frac{M}{M_{\\rm {Pl}}} \\right) \\;.$ Since these equations include $H_{J0}$ and $\\rho _{r0}$ , the end of the reheating is sensitive to the boundary conditions at the transition from the slow-roll regime to the fast-roll regime.",
"For $g_*=106.75$ , $M/M_{\\rm Pl}=1.2\\times 10^{-5}$ , and numerically determined values, $c_0=0.72$ , $\\hat{H}_{J0}\\approx 0.26$ , the time and temperature of the reheating are $\\hat{t}_{J,\\rm {reh}} - \\hat{t}_{J0} \\approx 2.2\\times 10^{16}$ and $T_{\\rm {reh}} \\approx 3.0 \\times 10^7\\,{\\rm {GeV}}$ , respectively."
],
[
"Constraints on the model parameters",
"In this section, based on our analytic solutions and numerical results obtained in the previous sections, we discuss the allowed ranges of the model parameters.",
"First, we present the constraint on the energy scale $M$ in Sec.",
"REF .",
"It is fixed from the amplitude normalization of the CMB.",
"Second, we consider the other parameters, $g$ , $b$ , and $\\delta $ in Sec.",
"REF .",
"Once $M$ is determined, we can predict the other observable of the inflation and discuss its consistency with observations.",
"$g$ , $b$ , and $\\delta $ , cannot be constrained from observational data because the reheating temperature in Eq.",
"(REF ) is independent of those parameters.",
"However, since the $gR^2$ -AB model must realize current cosmic acceleration, its magnitude and stability constrain the allowed range of $g$ , $b$ , and $\\delta $ ."
],
[
"Constraint on $M$",
"When $R_J \\gg M^2$ , the $gR^2$ -AB model (REF ) can be approximated to $R^2$ inflation, in which the primordial spectrum of a scalar mode at the leading order in the slow-roll parameter $\\epsilon _1$ is given by [36] ${\\cal {P}}_S \\approx \\frac{1}{96 \\pi ^2 \\epsilon _1^2} \\left( \\frac{M}{M_{\\rm {Pl}}} \\right)^2 \\;,$ where $\\epsilon _1 \\equiv -H_J^{^{\\prime }}/H_J^2$ .",
"This slow-roll parameter is related to the $e$ -folding number between the end of inflation and the horizon crossing of the mode whose comoving wave number $k$ corresponds to the CMB scale today.",
"From the analytic solutions during the slow-roll regime, Eqs.",
"(REF ), (REF ) - (), the Hubble parameter in the Jordan frame is written as $\\hat{H}_J(t_J) = \\frac{3\\, \\tau (\\hat{t}_J)-1}{6\\,\\tau ^{1/2}(\\hat{t}_J)} \\;, \\quad \\quad \\tau (\\hat{t}_J) \\equiv d_{\\rm {ini}}^2 -\\frac{2}{3} (\\hat{t}_J-\\hat{t}_{J,{\\rm {ini}}}) \\left[ d_{\\rm {ini}} - \\frac{1}{6} (\\hat{t}_J-\\hat{t}_{J,{\\rm {ini}}}) \\right] \\;.$ For $d_{\\rm {ini}} \\gg 1$ , expanding Eq.",
"(REF ) in powers of $t_J$ around $t_{J,{\\rm {ini}}}$ gives $\\hat{H}_J(\\hat{t}_J) \\approx \\hat{H}_{J,{\\rm {ini}}} -\\frac{1}{6} (\\hat{t}_J-\\hat{t}_{J,{\\rm {ini}}}) \\;,$ where $\\hat{H}_{J,{\\rm {ini}}} = d_{\\rm {ini}}/2$ .",
"The $e$ -folding number between the end of inflation at $t_{J,{\\rm {end}}}$ and the horizon crossing of the CMB mode at $t_{Jk}$ is given by $N_k = \\int _{t_{Jk}}^{t_{J,\\rm {end}}} H_J dt_J \\approx -\\frac{H_{Jk}^2}{2H^{^{\\prime }}_{Jk}} = \\frac{1}{2 \\epsilon _1 (t_{Jk})} \\;,$ where $H_{Jk} \\equiv H_J(t_k)$ and we used the fact that $H^{^{\\prime }}_{Jk}$ is constant when we performed the integration.",
"Then Eq.",
"(REF ) is expressed as ${\\cal {P}}_S \\approx \\frac{N_k^2}{24\\pi ^2} \\left( \\frac{M}{M_{\\rm {Pl}}} \\right)^2 \\;.$ Using Eq.",
"(REF ), we obtain the $e$ -folding number when the comoving scale of CMB crosses the horizon during inflation: $N_k \\approx 66.2 -\\frac{1}{2} \\log \\left( \\frac{1-\\Omega _m}{0.7} \\right) -\\frac{1}{4} \\log \\left[ \\left( \\frac{c_0}{0.72} \\right) \\left( \\frac{g_{\\ast }}{106.75} \\right) \\right] \\;.$ Since the parameters $\\Omega _m$ and $g_{\\ast }$ hardly change $N_k$ , we set it to $N_k=66$ .",
"From the temperature fluctuation of CMB anisotropy [9], the amplitude of the power spectrum, ${\\cal {P}}_S =(2.445 \\pm 0.096) \\times 10^{-9}$ at $k=0.002\\,{\\rm {Mpc}}^{-1}$ , fixes the parameter $M$ to $\\frac{M}{M_{\\rm {Pl}}} \\approx 1.2 \\times 10^{-5} \\;.$ At the CMB scale, the spectral indices of the scalar and tensor modes and the tensor-to-scalar ratio are given by [36] $n_S-1 &\\equiv \\left.",
"\\frac{d \\log {\\cal {P}}_S(k)}{d \\log k} \\right|_{k=aH}\\approx -\\frac{2}{N_k} \\;, \\\\n_T &\\equiv \\left.",
"\\frac{d \\log {\\cal {P}}_T(k)}{d \\log k} \\right|_{k=aH} \\approx -\\frac{3}{2 N^2_k} \\;, \\\\r &\\equiv \\frac{{\\cal {P}}_T}{{\\cal {P}}_S} \\approx \\frac{12}{N_k^2} \\;,$ at the leading order in the slow-roll parameter.",
"For the above choice of $N_k$ , $n_S \\approx 0.97$ and $r \\approx 2.8 \\times 10^{-3}$ , which are consistent with observational bounds [9]."
],
[
"Constraints on $g$ , {{formula:e5addc5e-1f5b-43c0-abf7-2154eca00a06}} , and {{formula:8960baf8-459b-46ed-8249-d6a391439640}}",
"The parameters $g$ , $b$ and $\\delta $ considerably alter the dynamics of the reheating in the $gR^2$ -AB model.",
"However, as seen from Eqs.",
"(REF ) and (REF ), the reheating temperature and the radiation energy density at that time does not depends on these parameters.",
"So the constraint on $g$ , $b$ , and $\\delta $ comes not from the CMB observation but from a stability condition of a de-Sitter vacuum.",
"From Eq.",
"(REF ), $3 \\Box F(R_J) +R_J F(R_J) -2 f(R_J) = 8\\pi G T_J \\;,$ where $T_J \\equiv g_J^{\\mu \\nu } T^J_{\\mu \\nu }$ and $\\Box \\equiv (1/\\sqrt{-g_J})\\, \\partial _{\\mu } \\left[ \\sqrt{-g_J}\\, \\partial ^{\\mu } \\right] $ .",
"For the existence of a stable solution of a de-Sitter vacuum ($R_J={\\rm {const}}.$ , $T_J=0$ ), the following equation has to be satisfied: $R_J F(R_J) -2 f(R_J)=0 \\;.$ For $R_J \\ll M^2$ , substituting Eq.",
"(REF ) into Eq.",
"(REF ) and using $R_{\\rm {vac}} \\approx 4 g b M^2 \\delta $ lead to the equation $Q(y) \\equiv y-4gb +2 g \\left[ \\log \\left( 1+e^{-2(y-b)} \\right) +\\frac{y}{1+e^{2(y-b)}} \\right] =0 \\;.$ where $y \\equiv R_J/M^2\\delta $ .",
"This function $Q(y)$ typically has the shape shown in Fig.",
"REF .",
"Therefore, a stable de-Sitter vacuum exists if $Q^{^{\\prime }}(y_0)=0$ has the solution $y=y_0>1$ at which $Q^{^{\\prime \\prime }}(y_0)>0$ and $Q(y_0) \\le 0$ .",
"The boundary of the allowed parameter region of $b$ and $g$ can be obtained by solving $Q(y_0)=0$ and $Q^{^{\\prime }}(y_0)=0$ under the condition $Q^{^{\\prime \\prime }}(y_0)>0$ .",
"We cannot solve the above equations analytically.",
"Instead, we fit the numerical solution and obtain the allowed region for $g$ as $\\frac{1}{4} + \\frac{0.28}{(b-0.46)^{0.81}} \\le g \\le \\frac{1}{2} \\;.",
"$ This region is shown in Fig.",
"REF .",
"Figure: Function Q(y)Q(y) for b=10b=10 and g=0.1g=0.1 (blue), 0.20.2 (red), 0.30.3 (yellow), and 0.40.4 (green).Figure: Allowed parameter region of gg and bb.",
"Numerical solution (solid curve), fitting (dotted curve).",
"Above this curve, stable de-Sitter solutions exist.Once the parameters $M$ , $g$ , and $b$ are fixed, $\\delta $ should be determined so that the current observation of accelerating expansion is reproduced.",
"With the Ricci curvature of the present universe, $R_{\\rm {vac}} \\sim 10^{-120} M_{\\rm {Pl}}^2$ , the parameter $\\delta $ is given by $\\delta = \\frac{R_{\\rm {vac}}}{2g M^2 (b+\\log [ 2\\cosh b])} \\approx \\frac{1}{4g b} \\frac{R_{\\rm {vac}}}{M^2} \\;.$"
],
[
"Conclusions and discussion",
"We have studied the inflation and reheating dynamics in $f(R)$ gravity, especially in $gR^2$ -AB model.",
"This model is capable to describe both accelerated expansions in the early Universe and the present time.",
"In the Einstein frame, the inflaton potential of this model possesses a plateau and a false vacuum in the bottom of the potential.",
"These are different features from original $R^2$ inflation model and they significantly change the reheating process.",
"We have derived the analytic solutions in the slow-roll inflation regime and the fast-roll oscillation (reheating) regime.",
"We have also carried out the numerical computation including the backreaction from particle creation, and have confirmed that both results agree well.",
"According to the existence of the potential plateau, the particle creation via gravitational reheating mainly occurs in the slow-roll regime and is inefficient during the the fast-roll oscillation regime.",
"Consequently, in contrast to the $R^2$ inflationary scenario, the reheating era lasts longer.",
"Another interesting feature of this model is that the averaged time evolution of a scale factor is proportional to $t_J^{1/3}$ because of the periodic abrupt changes of the Hubble parameter.",
"Based on these results obtained from our analytic and numerical calculations, we have given the constraints on the model parameters.",
"The parameter $M$ is pinned down by the observational amplitude of CMB temperature fluctuations.",
"Also the value of $\\delta $ is selected to correctly reproduce the current accelerated expansion of the Universe.",
"On the other hand, the parameters $g$ and $b$ are poorly constrained because these parameters affect only the reheating dynamics after the inflation.",
"To more tightly constrain $g$ and $b$ , we need observations that can probe at much smaller scales than those of CMB and large-scale galaxy surveys.",
"In the future searches for primordial black holes and the direct detection experiments of gravitational waves would provide new observational windows for the reheating dynamics in modified gravity.",
"We would like to thank A. A.",
"Starobinsky, T. Suyama and J. Yokoyama for helpful discussions and valuable comments.",
"This work was supported in part by JSPS Research Fellowships for Young Scientists (H.M.) and Grant-in-Aid for JSPS Fellows (A.N.",
")."
]
] | 1204.1472 |
[
[
"Efficient method of designing optically-pumped vertical external cavity\n surface emitting lasers having equally excited quantum wells"
],
[
"Abstract Even distribution of carriers allows to maximize optical gain of the Optically-Pumped Vertical External Cavity Surface Emitting Laser.",
"In this paper we show how to distribute the quantum wells and blocking layers in order to compensate the exponential decay of the pumping beam intensity.",
"Our model says whether it is possible at all (for an assumed length of the device) and, if it is, allows to find positions of the blocking layers.",
"No iterations nor numerical calculations more sophisticated than a standard calculator can do are required to use the model."
],
[
"Introduction",
"The Optically Pumped Vertical External Cavity Surface Emitting Lasers (OP-VECSELs) are able to emit high quality beams of multi-watt powers [1], thus combining the most important advantages of Edge-Emitting and Vertical Cavity Surface Emitting Lasers (EELs and VCSELs).",
"Moreover their external cavity may contain additional elements like non-linear crystals or semiconductor saturable absorber mirrors (SESAMs), which may be used for instance for frequency doubling and short pulse generation.",
"Usually, optical gain is provided by several quantum-well (possibly double- or multi-quantum-well) active regions, located at the successive anti-nodes of the standing wave.",
"The cavity is formed by a Distributed Bragg Reflector (DBR) and an external mirror [2].",
"The pumping beam penetrates the device in the direction nearly perpendicular to the layers.",
"Absorption of the pumping beam generates carriers necessary to achieve optical gain, but also causes exponential decay of the intensity in the deeper regions.",
"If we simply placed, at each anti-node, the same number of wells, the gains provided by the active regions would differ significantly.",
"Since optical gain is a concave function of carrier concentration (roughly $\\sim \\log (\\mathcal {N}/\\mathcal {N}_0)$ , where $\\mathcal {N}$ is the carrier concentration and $\\mathcal {N}_0$ is the transparency concentration), the highest possible total gain (for a fixed, arbitrary number of the carriers) is highest when all the wells provide the same material gain.",
"If the temperatures of the wells can be assumed to be equal, we should try to have the same carrier concentration in each well.",
"The temperature rise in the VECSEL can be high, most of the temperature drop takes place in the substrate and DBR as these part are much thicker than the active part.",
"Generally, we want to direct the same number of carriers to all the wells.",
"In order to do so, we have to increase the volume from which the distant wells collect the carriers or increase the number of the wells in the stronger pumped regions.",
"To do the first thing one can use so called blocking layers—thin wide-gap layers which block carrier diffusion (in case of AlAs blocking layers in GaAs, the thickness of a few nanometers is sufficient to block the carrier diffusion).",
"They define the volume from which the well(s) between them collect the carriers. [3].",
"The question how to place them is not trivial, mainly because positions of the wells are restricted to the anti-nodes, which strongly restricts positions of the blocking layers.",
"If there are two or more anti-nodes between two subsequent blocking layers, one has to take into account the carrier diffusion in order to find the actual carrier concentration in the two (or more) active regions.",
"This makes the analysis more complicated [3].",
"Our goal is to build an analytical model which allows to construct the desired scheme without using complicated calculation."
],
[
"The model",
"As we mentioned, in order to avoid consideration of carrier diffusion, we restrict our interest to the case, where in each segment (area bounded by the neighbouring blocking layers) there is only one active region (we treat the DBR as the 0th blocking layer, see Fig.",
"REF ), and the active regions are placed at each anti-node.",
"The window layer acts as the last blocking layer and due to the optical reasons must be located at an anti-node.",
"This means that the last segment must be thicker than $d$ (see Fig.",
"REF ).",
"Therefore, in order to have the same number of generated carrier per a QW, we have to put more wells in this segment and treat it in a different manner in our calculations.",
"In order to obtain a handy result we base our model on the following simplifying assumptions: Widths of the wells and blocking layers are negligible.",
"More precisely, their presence (the blocking layers do not absorb the pump, on the other hand the wells have higher absorption than the adjacent bulk material) introduces a tolerable error.",
"The blocking layers block totally carrier diffusion We neglect reflection of the pumping light from the DBR and from the blocking layers Carrier losses in the absorbing barrier are negligible (relative to the losses in the QWs) In Fig.",
"REF a scheme of the VECSEL is presented.",
"Distance $d$ must be a multiple of a half of the emitted wavelength.",
"Usually $d\\approx \\lambda /(2n_r)$ , where $\\lambda $ is vacuum wavelength and $n_r$ is refractive index of the barrier.",
"Small deviations from the exact equality come from the presence of the wells and blocking layers.",
"Figure: Scheme of wells and blocking layers distribution in a VECSEL.",
"Wells are blue (downwards), blocking layers red (upwards).",
"Numbers 1,2,⋯,N+11,2,\\dots ,N+1 denote the segments.In our scheme the pumping light comes from the right, so under our assumptions the pumping wave intensity is described by the following formula: $I(z) = I_0\\exp (\\alpha z)$ where $\\alpha $ is the absorption in the barriers, $I_0$ is a normalisation constant, defining the pump power.",
"Number of the carriers generated in $n$ -th segment is simply $P_n=I_0 \\big (\\exp (\\alpha z_n) - \\exp (\\alpha z_{n-1})\\big )$ We assume that in segments $1,2,\\dots ,N$ there are the same number of wells (in this paper—just one well), and in the last segment, $N+1$ , we put $k$ wells.",
"we want to distribute the blocking layers such that $P_1=P_2=\\dots = P_N = \\frac{1}{k}P_{N+1}$ As we have $k$ times more wells in the last segment, we want to have $k$ times more carriers generated there.",
"Possible values of numbers $N$ and $k$ will be determined in our calculations.",
"Position $z_n$ can be written as (see Fig.",
"REF ) $z_n = nd + \\delta _n$ Then we have $P_1 &= I_0\\big [\\exp \\big (\\alpha (d +\\delta _1)\\big )-1\\big ] =\\\\&=I_0\\big [\\exp (\\alpha d)\\exp (\\alpha \\delta _1)-1\\big ]\\\\P_2 &= I_0\\big [\\exp \\big (\\alpha (2d +\\delta _2)\\big )-[\\exp \\big (\\alpha (d +\\delta _1)\\big )\\big ] =\\\\&=I_0\\big [\\exp (2\\alpha d)\\exp (\\alpha \\delta _2)-\\exp (\\alpha d)\\exp (\\alpha \\delta _1)\\big ]\\\\P_3 &= I_0\\big [\\exp \\big (\\alpha (3d +\\delta _3)\\big )-[\\exp \\big (\\alpha (2d +\\delta _2)\\big )\\big ] =\\\\&=I_0\\big [\\exp (3\\alpha d)\\exp (\\alpha \\delta _3)-\\exp (2\\alpha d)\\exp (\\alpha \\delta _2)\\big ]$ Let us introduce the following symbols: $a=\\exp (-\\alpha d)\\qquad x_n = \\exp (\\alpha \\delta _n)$ Note that $0<a<1$ , regardless of $\\alpha $ and $d$ .",
"Using these symbols we can write the system of equations (REF ) as: $0 &= x_2 - 2ax_1 + a^2 \\\\ 0 &=x_3 - 2ax_2 + a^2x_1 \\\\ 0 &=x_4 - 2ax_3 + a^2x_2 \\\\& \\vdots \\\\ 0 &=x_{N} -2ax_{N-1} + a^2x_{N-2} \\\\ 0 &=x_{N+1} -(k+1)ax_{N} + ka^2x_{N-1} \\\\ \\frac{1}{a} &=x_{N+1} \\qquad \\text{because }\\delta _{N+1} = d $ As one can see, thanks to the extraordinary properties of exponential function we got something as simple as a system of $N+1$ linear equations with $N+1$ unknowns.",
"Because we do not know what are the values of $N$ and $k$ , it is convenient to consider first only equations concerning first $N$ segments, i.e.",
"those defined by the blocking layers which position we can choose.",
"Thus we consider the system of $N-1$ equation with $N$ unknowns: $x_2 - 2ax_1 + a^2 &= 0 \\\\ x_3 - 2ax_2 + a^2x_1 &= 0 \\\\ x_4 - 2ax_3 + a^2x_2 &= 0 \\\\& \\vdots \\\\ x_{N} -2ax_{N-1} + a^2x_{N-2} &= 0 \\\\ $ If we treat one of the unknowns as a parameter we can solve the system.",
"As the parameter we choose $x_1$ as the first blocking layer must be always present.",
"This way we obtain: $x_2 &= a(2x_1-a)\\\\ x_3 &= a^2(3x_1-2a)\\\\&\\vdots \\\\ x_N &= a^{N-1}\\big (Nx_1-(N-1)a\\big )\\\\ $ Although the system (REF ) has always the solution in real numbers, we must check if the solution fulfils the additional conditions, i.e.",
": $1 < x_n < 1/a \\quad \\forall n = 1,2,\\dots ,N$ The above conditions say simply that $0 < \\delta _n < d$ .",
"It assures that in each segment there is one active region.",
"Substituting (REF ) to (REF ) we get: $1 &< x_1 < \\frac{1}{a} \\\\ \\frac{1}{2}\\left(a+\\frac{1}{a}\\right) &< x_1 < \\frac{1}{2}\\left(a+\\frac{1}{a^2}\\right) \\\\ \\frac{1}{3}\\left(2a+\\frac{1}{a^2}\\right) &< x_1 < \\frac{1}{3}\\left(2a+\\frac{1}{a^3}\\right) \\\\&\\vdots \\\\ \\frac{1}{N}\\left((N-1)a+\\frac{1}{a^{N-1}}\\right) &< x_1 < \\frac{1}{N}\\left((N-1)a+\\frac{1}{a^{N}}\\right) \\\\ $ If the above inequalities are inconsistent, it is impossible to build a system of $N$ active regions (of the same number of QWs), having equal carrier concentrations.",
"The above system can be significantly simplified.",
"Let us denote $L_n=\\frac{1}{n}\\left((n-1)a + \\frac{1}{a^{n-1}}\\right)\\\\R_n = \\frac{1}{n}\\left((n-1)a+\\frac{1}{a^n}\\right)$ being just the left and right hand side of the $n$ -th inequality.",
"Basic calculations show that: $L_{n+1} - L_{n} = \\frac{1}{n(n+1)a^n}\\left(a^{n+1} - (n+1)a + n\\right)$ The sign of the difference is determined by polynomial $l_n(a) = a^{n+1} - (n+1)a + n$ .",
"It is easy to see that $l_n(1) = 0$ , $l_n^{\\prime }(a) = (n+1)(a^n-1) \\le 0 \\ \\forall a\\in [0,1]$ .",
"It means that $l_n(a) \\le 0\\ \\forall a\\in [0,1]$ (in our case $0<a<1$ ), and hence $L_1<L_2<\\dots <L_N$ Thanks to that property we can replace all the inequalities $L_n<x_1, n=1,2,\\dots ,N$ by one: $L_N<x_1$ .",
"There is no similar properties of the right hand sides.",
"Now we can write the system of inequalities (REF ) in a more compact way: $\\frac{1}{N}\\left((N-1)a+\\frac{1}{a^{N-1}}\\right) < x_1 < \\\\ \\min _{n=1,\\dots ,N}\\left(\\frac{1}{n}\\left((n-1)a+\\frac{1}{a^n}\\right)\\right) = \\mathcal {R}(N)$ Which of the right hand sides is the actual minimum depends on $a$ (i.e.",
"on the material absorption and distance between the active regions).",
"If for a certain $N$ $L_N\\ge \\mathcal {R}(N)$ , it is impossible to build $N$ (and any greater number, since $L_N$ is increasing and $\\mathcal {R}(N)$ is non-increasing with $N$ ) equally pumped segments.",
"This way we can find the possible values of $N$ .",
"When we choose one of them, we can return to the full system of equations (REF ).",
"Solving it we can find values of all $x_1,x_2,\\dots ,x_N$ , but now we are interested only in $x_1$ : $x_1 = \\frac{1}{(N+k)a^{N+1}} + a\\frac{N+k-1}{N+k}$ If, for the chosen $N$ , we can find a natural number $k$ such that $x_1$ calculated with the above formula fulfils condition: $L_N < x_1 < \\mathcal {R}(N)$ we know that we can build $N$ one-well segments concluded by one $k$ -well segment, having equally pumped quantum wells.",
"If there is no such $k$ , or it is too high from the practical point of view, one has to decrease number $N$ and repeat the procedure.",
"Finally, when we have suitable $N$ , $k$ and hence $x_1$ , we can calculate all the other $x_2,x_3,\\dots ,x_N$ using (REF ) and then the actual positions $z_n, n=1,2,\\dots ,N+1$ of the blocking layers using formulae (REF ) and (REF )."
],
[
"Example",
"Let us consider an important example: a VECSEL with GaAs barirers, emitting at $980\\,$ nm, pumped by a $808\\,$ nm laser.",
"Assuming $\\alpha = 1.3\\cdot 10^4\\,1/\\mathrm {cm}$ [4], and $n_r=3.52$ we get the following parameters: $d \\approx 139.2\\,\\mathrm {nm}\\qquad a\\approx 0.8345$ Of course in a real calculations one should use more precise values, because the optical properties of the structure are more sensitive to the distances between the wells.",
"First we will check whether it is possible to have seven one-well segments.",
"We calculate, using (REF ): $L_7 = 1.13837 \\qquad \\mathcal {R}(7) = \\min _{n=1,\\dots ,7} R_n = R_3 = 1.12997$ We see that $L_7 > \\mathcal {R}(7)$ , so we cannot construct seven such segments.",
"But since $L_6 = 1.10730 < \\mathcal {R}(6) = R_3 = 1.12997$ we can find out if we can close such 6-segment sequence by the multi-well one.",
"Using (REF ) for $N=6$ we calculate: $x_1(k=3) = 1.13612 &> \\mathcal {R}(6)\\\\x_1(k=4) = 1.10595 &< L_6$ Since $x_1$ is a decreasing function of $k$ , there is no possibility to find a suitable $k$ .",
"It means that we cannot build the final segment with an integer number of wells such that it has the same carrier concentration as in all the others.",
"The non-integer values of $k$ could have physical meaning as $k$ is actually the ratio between number of wells in the last segment and number of wells in the other ones.",
"If we assumed the other segments to contain not one, but two wells each, we could consider $k=3.5$ .",
"But it would mean that we have to put as many as seven quantum wells in the last segment, which is too many, because the peripheral wells would be placed far from the anti-node of the standing wave.",
"If not 6, let us try $N=5$ .",
"Now we have $L_5 = 1.08005$ , $\\mathcal {R}(5) = 1.12997$ .",
"For the new $N$ : $x_1(k=2) = 1.13837 &> \\mathcal {R}(5)\\\\L_5 < x_1(k=3) = 1.10038 &< \\mathcal {R}(5)\\\\x_1(k=4) = 1.07083 &< L_5$ Number $x_1(k=3)$ fulfils all the conditions, so we can build 5 one-well segments and one 3-well segment on the top, with all the 8 wells equally pumped.",
"Now we calculate $x_2,\\dots ,x_N$ using (REF ), extract $\\delta _n$ given by: $\\delta _n=\\frac{\\log (x_n)}{\\alpha }$ Finally we get $z_1,\\dots ,z_{N+1}$ from formula (REF ).",
"Positions of the wells (which are independent on $x_1$ ) and blocking layers are presented in table REF .",
"One should remember that positions of the wells ($nd$ ) and hence numbers $z_n$ are only approximations.",
"The reliable values are $\\delta _n$ —positions of the blocking layers relative to the adjacent quantum well (see Fig.",
"REF ).",
"Table: Positions of the wells and blocking layers in the 6-segment scheme,rounded to full nanometersLooking at the values in the table one can see that the lowest distance between a well and a blocking layer is 37 nm.",
"This is a safe distance from the technological point of view, even taking into account non-zero thicknesses of the wells and the blocking layers.",
"The total thickness of the absorbing area is $7d$ , $\\exp (-7d\\alpha )\\approx 0.28$ , and hence over $70\\%$ of the pumping power is absorbed.",
"As the blocking layers are generally located near the nodes of the standing wave, and their thickness can be as low as a few nanometers, their presence modifies the optical properties of the resonator in a very limited degree."
],
[
"Summary",
"We have shown an efficient and simple way to design the VECSEL structure such that the carrier concentration, and hence optical gain (with the assumption that the temperature differences between the wells are not high), is equal in all the quantum wells.",
"This configuration gives the highest modal gain for given number of carriers, so is highly desirable.",
"We presented an example of a GaAs-based structures with 6 active regions in subsequent anti-nodes of the standing wave.",
"This is the longest possible design in which one use 1-well segments except the last one.",
"Our calculations can be easily modified to describe segments with different number of wells.",
"In this case longer absorbing areas can be achieved, if necessary.",
"However, the longer the area is, the higher temperature differences appear there, which spoils the desired gain uniformity."
],
[
"Acknowledgement",
"This work was partially supported by COST Action MP0805."
]
] | 1204.1494 |
[
[
"Direct minimization of electronic structure calculations with\n Householder reflections"
],
[
"Abstract We consider a minimization scheme based on the Householder transport operator for the Grassman manifold, where a point on the manifold is represented by a m x n matrix with orthonormal columns.",
"In particular, we consider the case where m >> n and present a method with asymptotic complexity mn^2.",
"To avoid explicit parametrization of the manifold we use Householder transforms to move on the manifold, and present a formulation for simultaneous Householder reflections for S-orthonormal columns.",
"We compare a quasi-Newton and nonlinear conjugate gradient implementation adapted to the manifold with a projected nonlinear conjugate gradient method, and demonstrate that the convergence rate is significantly improved if the manifold is taken into account when designing the optimization procedure."
],
[
"Introduction",
"We consider the optimization problem $\\min _{\\mathbf {X}^T\\mathbf {X}=\\mathbf {I}} f(\\mathbf {X}),$ that is, we attempt to minimize the real valued function $f$ of $\\mathbf {X}\\in \\mathbb {R}^{m\\times n}$ where $m \\gg n$ , subject to the constraint $\\mathbf {X}^T\\mathbf {X}= \\mathbf {I}$ , and with the computable derivative $df(\\mathbf {X})$ .",
"The constraint on $\\mathbf {X}$ ensures that $f$ has a minimum, but this minimum is not necessarily unique.",
"The method we present requires that $m \\ge 2n$ .",
"Extending the method to cover cases where just $m > n$ is, however, possible.",
"A special property we assume of $f$ is the homogeneity condition: $f(\\mathbf {X}) = f(\\mathbf {X}\\mathbf {Q})$ , where $\\mathbf {Q}$ is any $n\\times n$ orthonormal matrix.",
"This property means that $f$ only depends on the span of the columns of $\\mathbf {X}$ .",
"The set of subspaces that are spanned by the columns of orthonormal $m\\times n$ matrices is called the Grassman manifold, $\\mathcal {M}$ .",
"While the solution to (REF ) is an equivalence class, we choose an arbitrary representative of the class since we are interested in the value of $f$ .",
"The closely related Stiefel manifold consists of the same problem without the homogeneity condition.",
"While the Householder transformation is suitable for both the Grassman and Stiefel manifolds, the optimization method presented does not optimize with respect to the basis and is therefore suitable only for the Grassman manifold.",
"In the applications we have in mind the evaluation of $f$ and $df$ is expensive.",
"For this reason we cannot employ a high quality line search to decide the step length, and the optimization method must be robust.",
"Furthermore, we will measure the number of evaluations of $f$ and $df$ necessary to obtain a solution.",
"One iteration of the optimization procedure requires the two evaluations, once to evaluate the solution candidate and once to construct a quadratic approximation along the search direction.",
"For computational reasons we choose an $m \\times n$ matrix, $\\mathbf {X}$ , as a representative of a point on $\\mathcal {M}$ , and enforce the requirement $\\mathbf {X}^T\\mathbf {X}= \\mathbf {I}.$ While this representation includes more degrees of freedom than strictly necessary, the approach is suitable for use in practice [10].",
"We use the inner product for matrices $(\\mathbf {A}, \\mathbf {B}) = \\mathrm {trace} (\\mathbf {A}^T \\mathbf {B}),$ and the tangent spaces at $\\mathbf {X}$ satisfying $\\mathbf {X}^T\\mathbf {X}=\\mathbf {I}$ $\\lbrace \\mathbf {Z}=\\mathbf {X}\\mathbf {A}+\\mathbf {Y}\\, | \\,\\mathbf {Y}^T \\mathbf {X}= \\mathbf {0}\\;\\mathrm {and}\\; \\mathbf {A}^T=-\\mathbf {A}\\rbrace .$ On the Grassman manifold the value of $f$ depends only on the space spanned by the columns of $\\mathbf {X}$ .",
"We can therefore ignore the $\\mathbf {X}\\mathbf {A}$ component of the tangent space, and denote $\\mathcal {T}_{\\mathbf {X}}\\mathcal {M} = \\lbrace \\mathbf {Y}\\, | \\,\\mathbf {Y}^T \\mathbf {X}= \\mathbf {0}\\rbrace .$ On the Grassman manifold it is possible to substitute equivalence classes for the representatives we have chosen, but practical computations require us to always use a specific matrix.",
"We also assume that we are given a direction $\\mathbf {W}$ by the minimization method in which we want to move on the manifold.",
"We project the direction on to $\\mathcal {T}_{\\mathbf {X}}\\mathcal {M}$ by $\\mathbf {Y}= (\\mathbf {I}- \\mathbf {X}\\mathbf {X}^T)\\mathbf {W}.$ The motivation for the problem under consideration comes from density functional theory (DFT) electronic structure calculations [17], [21].",
"We construct a simultaneous Householder operator that can be used to ensure that the optimization method naturally enforces the orthogonality constraint.",
"This approach differs from several other approaches in that it does not solve the canonical electron orbitals [13], [18], [22], [4], [23], [21], instead we only solve the electron density that would be given by the orbitals.",
"To obtain the canonical orbitals from the electron density a linear eigenvalue problem must then be solved in the space spanned by the columns of $\\mathbf {X}$ .",
"A similar approach using polynomial filtering can be found in [24], [3].",
"It is also possible to solve a nonlinear eigenvalue problem instead of the minimization problem [20], [13], [16], [21].",
"In [10] a framework for optimization methods on the Stiefel and Grassmann manifolds is presented, while [7] discusses a Newton-like iteration scheme on a more general manifold.",
"Univariate optimization methods for the Stiefel manifold is presented in [5], where identity plus rank one Householder transforms are given as one possible choice for moving on the manifold.",
"The choice of coordinates can also be based on a QR factorization and polar decompositions [6], [9] or Lie groups [14].",
"An overview of geometric numerical integration techniques can be found in [15].",
"First, we present a simultaneous Householder transformation in Section that we can use to move on both the Stiefel and Grassman manifolds.",
"Then in Section we recall the method of steepest descent, the quasi-Newton (QN), and the nonlinear conjugate gradient (NLCG) methods adapted for use with the Householder operator.",
"In Section we numerically demonstrate the method on a model problem that includes nonlinearities similar to a DFT problem.",
"Finally, Section presents the conclusion."
],
[
"Householder operator",
"We ensure that $\\mathbf {X}\\in \\mathcal {M}$ during the solution process by using the Householder transformation to move from one solution candidate to the next.",
"Figure: Conceptual difference between reorthogonalization and Householder approach.",
"In Subfigure a) the step is taken without regard to the manifold, and after the step is taken the new solution candidate 𝐗 k+1 \\mathbf {X}_{k+1} is constructed by reorthogonalizing 𝐗 k +𝐘 k \\mathbf {X}_k+\\mathbf {Y}_k.",
"Subfigure b) illustrates the case where the Householder operator, 𝐇\\mathbf {H}, constructs an update 𝐗 k+1 \\mathbf {X}_{k+1} that immediately satisfies the orthogonality condition.To do this we need to find an operator $\\mathbf {H}(\\tau ) = \\mathbf {I}- 2 \\mathbf {Q}(\\tau )\\mathbf {Q}(\\tau )^T,$ where $\\mathbf {Q}(\\tau )^T\\mathbf {Q}(\\tau ) = \\mathbf {I}\\quad \\forall \\tau ,$ and $\\tau $ is a parametrization of $\\mathbf {H}$ such that $\\mathbf {H}(0)\\mathbf {X}= \\mathbf {X}.$ This requirement leads to $\\mathbf {Q}(0)^T\\mathbf {X}=\\mathbf {0}.$ $\\mathbf {H}$ is unitary, and if we let $\\mathbf {X}_{k+1} = \\mathbf {H}(\\tau )\\mathbf {X}_k,$ we obtain a sequence $\\lbrace \\mathbf {X}_k\\rbrace $ that satisfies $\\mathbf {X}_{k+1}^T\\mathbf {X}_{k+1} = \\mathbf {X}_k^T\\mathbf {X}_k \\quad \\forall \\; k, \\tau .$ The orthonormality requirement (REF ) on $\\mathbf {Q}(\\tau )$ leads to the constraint $\\frac{\\partial \\mathbf {Q}}{\\partial \\tau }^T\\mathbf {Q}+ \\mathbf {Q}^T\\frac{\\partial \\mathbf {Q}}{\\partial \\tau } = \\mathbf {0}.$ We set the initial condition for $\\tfrac{\\partial \\mathbf {Q}}{\\partial \\tau }$ by requiring that $\\frac{\\partial }{\\partial \\tau } \\left( \\mathbf {H}(\\tau )\\mathbf {X}\\right)\\Big |_{\\tau =0} = \\mathbf {Y},$ where $\\mathbf {Y}\\in \\mathcal {T}_{\\mathbf {X}}\\mathcal {M}$ is a projected direction given by the minimization method we choose to employ.",
"Differentiating with respect to $\\tau $ , and using property (REF ), we obtain from (REF ) $-2 \\mathbf {Q}(0)\\frac{\\partial \\mathbf {Q}}{\\partial \\tau }(0)^T \\mathbf {X}= \\mathbf {Y}.$ We set $\\mathbf {Q}(0) = $ where $$ is the compact QR decomposition of $\\mathbf {Y}$ , and choose $\\frac{\\partial \\mathbf {Q}}{\\partial \\tau }(0) = - \\tfrac{1}{2} \\mathbf {X}\\mathbf {R}^T.$ A solution satisfying Equation (REF ) and conditions (REF ) and (REF ) is $\\tilde{\\mathbf {Q}}(\\tau ) = \\tilde{\\mathbf {Q}}_0 \\exp \\Bigl (\\tau \\begin{bmatrix}\\mathbf {0}&\\frac{1}{2}\\mathbf {R}\\\\ -\\frac{1}{2}\\mathbf {R}^T&\\mathbf {0}\\end{bmatrix}\\Bigr ).$ Here $\\tilde{\\mathbf {Q}}_0=\\begin{bmatrix} \\mathbf {X}\\end{bmatrix}$ and $\\mathbf {Q}(\\tau )$ corresponds to the first $n$ columns of $\\tilde{\\mathbf {Q}}(\\tau )$ .",
"Given any $\\mathbf {Q}$ with orthonormal columns $\\mathbf {H}$ constructed by (REF ) ensures that $\\mathbf {X}_{k+1}^T\\mathbf {X}_{k+1} = \\mathbf {I}$ .",
"For this reason we also consider a Householder operator constructed from a second order expansion of the matrix exponential that has subsequently been orthonormalized by the QR method to ensure that the orthogonality constraint is satisfied.",
"This approach is similar to the orbital transformation but includes orthogonalization after every evaluation of the matrix exponential function [23].",
"To distinguish these from the basic algorithms we use AEQN and AENLCG to denote the approximate exponential versions.",
"Remark: If $m < 2 n$ the requirement $\\mathbf {Y}^T\\mathbf {X}= \\mathbf {0}$ restricts the number of columns in $\\mathbf {Y}$ to below $n$ .",
"In this case the size of the first block in Equation (REF ) should be reduced accordingly.",
"A similar modification must be made if $\\mathbf {Y}$ is not full column rank.",
"For the Householder transformation to work we must still have $m > n$ ."
],
[
"Descent methods with orthogonality constraints",
"In this section we consider the method of steepest descent, a quasi-Newton method, and a nonlinear conjugate gradient method for minimization with orthogonality constraints.",
"We also present the Householder operator for an $\\mathbf {S}$ -orthonormal basis, and combine this with the optimization methods."
],
[
"The method of steepest descent",
"The method of steepest descent for the Stiefel manifold is also known as the projected gradient method [8].",
"At each step we simply set $\\mathbf {Y}_k = -\\sigma (\\mathbf {I}- \\mathbf {X}_k\\mathbf {X}_k^T)\\nabla f(\\mathbf {X}_k).$ The parameter $\\sigma > 0$ is almost redundant for the method of steepest descent, but will become important for the quasi-Newton methods presented later.",
"To decide the step length we evaluate $f(\\mathbf {H}(\\tau _k^e)\\mathbf {X}_k)$ , where $\\tau _k^e$ is an estimate step length, and construct the quadratic approximation $p(\\tau )$ of $f(\\mathbf {H}(\\tau )\\mathbf {X}_k)$ .",
"We then solve $\\tau _\\mathrm {min} = \\mathrm {argmin}\\;p(\\tau )$ from the system $p(0) &= f(\\mathbf {H}(0)\\mathbf {X}),\\\\p(\\tau _k^e) &= f(\\mathbf {H}(\\tau _k^e)\\mathbf {X}),\\\\p^{\\prime }(0) &= (\\nabla f(\\mathbf {H}(0)\\mathbf {X}), \\mathbf {Y}).$ This permits us to compute $\\mathbf {H}(\\beta \\tau _\\mathrm {min})$ and evaluate $f(\\mathbf {H}(\\beta \\tau _\\mathrm {min})\\mathbf {X})$ as well as $\\nabla f(\\mathbf {H}(\\beta \\tau _\\mathrm {min})\\mathbf {X})$ , where $\\beta $ is an underrelaxation parameter.",
"To ensure that we obtain a non-increasing iteration we choose the step length $\\tau _k$ from the set $\\lbrace 0, \\beta \\tau _\\mathrm {min}, \\tau _k^e\\rbrace $ , such that we obtain the lowest value of $f$ evaluated so far.",
"If $\\tau _{k} = 0$ we set $\\tau _{k+1}^e = 0.25\\times \\tau _k^e$ and otherwise we set $\\tau _{k+1}^e = \\mathrm {min} (|\\tau _\\mathrm {min}|, 2\\,\\tau _k^e)$ .",
"Remark: It turns out that $\\tau _k^e$ often is an acceptable choice for step length, and we believe that it is possible to construct a completely line search free minimization method with adaptive step length [2].",
"However, it must be tuned for a real-world problem, and we will not explore this option here."
],
[
"Vector transport on the manifold",
"To reduce the number of iterations we use information from previous evaluations to improve the search direction.",
"To this end we construct a transport operator, $\\mathbf {T}: \\mathcal {T}_{\\mathbf {X}_k}\\mathcal {M} \\rightarrow \\mathcal {T}_{\\mathbf {X}_{k+1}}\\mathcal {M}$ , that moves tangent vectors at $\\mathbf {X}_k$ to the tangent space at $\\mathbf {X}_{k+1}$ .",
"Any given vector, $\\mathbf {Z}_k \\in \\mathbb {R}^{m\\times n}$ , associated with a candidate solution, $\\mathbf {X}_k$ , and search direction, $, can be decomposed into\\begin{equation}\\mathbf {Z}_k = \\mathbf {X}_k\\mathbf {A}+ + \\mathbf {U}\\end{equation}and we demand that $ XkTU= TU= 0$, and $ UTU= I$ in addition to $ XkT 0$ which is satisfied by construction.",
"For simplicity we assume that $ Zk$ is such that $ URmn$ and $ A, B, Rnn$.$ A$ and $ B$ can be computed by projecting $ Zk$ onto $ Xk$ and $ respectively, while $\\mathbf {U} is for example the QR-decomposition of the remainder.In practice the decomposition is not explicitly constructed.$ The different parts of the decompostion () must be transported separately when the position is updated to $\\mathbf {X}_{k+1} = \\mathbf {H}\\mathbf {X}_k$ .",
"The component spanned by $\\mathbf {X}_k$ is reflected correctly by $\\mathbf {H}$ , however a reflection gives the wrong sign to the $ component of $ Zk$.",
"The final component $ U$ is orthogonal to both $ Xk$ and $ , and should not change when $\\mathbf {H}$ is applied.",
"The transport operation is shown in Figure REF .",
"We construct the transport operator by separating the components of $\\mathbf {Z}_k$ by projection and subsequent application of $\\mathbf {H}$ .",
"The transport operator therefore becomes $\\mathbf {H}\\mathbf {X}_k\\mathbf {X}_k^T - \\mathbf {H}^T + (\\mathbf {I}- \\mathbf {X}_k\\mathbf {X}_k^T - ^T).$ In practice we apply this for vectors $\\mathbf {Z}_{k} \\in \\mathcal {T}_{\\mathbf {X}_k}\\mathcal {M}$ which satisfy $\\mathbf {Z}_{k}^T\\mathbf {X}_k = \\mathbf {0}$ , and we can use the simplified transport operator $\\mathbf {T}(\\tau ) = - \\mathbf {H}(\\tau )^T + (\\mathbf {I}- ^T),$ and the tangent vector corresponding to $\\mathbf {Z}_k$ at $\\mathbf {X}_{k+1}$ is $\\mathbf {Z}_{k+1} = \\mathbf {T}(\\tau _k)\\mathbf {Z}_{k} \\in \\mathcal {T}_{\\mathbf {X}_{k+1}}\\mathcal {M}.$"
],
[
"$\\mathbf {S}$ -orthonormal Householder and transport operator",
"In practice $\\mathbf {X}$ is often represented by a discretization that requires a generalization of the orthonormality constraint (REF ).",
"The generalized constraint is $\\mathbf {X}^T\\mathbf {S}\\mathbf {X}= \\mathbf {I},$ where $\\mathbf {S}$ is symmetric and positive definite.",
"This constraint arises for example as an overlap matrix in DFT or a mass matrix in finite element calculations.",
"If $\\mathbf {S}$ is sparse or has other structure that can be exploited it can be preferable not to change basis for the optimization procedure.",
"We therefore also present the Householder transform and optimization procedure for the $\\mathbf {S}$ -orthonormal case.",
"We can construct both the Householder transform and the transport operator using the same argument as for the regular orthonormal case, as long as we account for $\\mathbf {S}$ orthonormality (REF ).",
"However, if $\\mathbf {S}$ is full, and lacks exploitable structure, the operation $\\mathbf {S}\\mathbf {X}$ has asymptotic complexity $m^2n$ and overshadows the rest of the procedure.",
"The projection of the direction of steepest descent onto the $\\mathbf {S}$ -orthogonal manifold is $\\mathbf {Y}= -\\sigma (\\mathbf {I}- \\mathbf {X}\\mathbf {X}^T\\mathbf {S})\\nabla f(\\mathbf {X})$ instead of (REF ) and satisfies $\\mathbf {Y}^T\\mathbf {S}\\mathbf {X}= \\mathbf {0}$ .",
"We must also use the $\\mathbf {S}$ weighted compact QR decomposition to compute a factorization $\\mathbf {Y}=$ where $T\\mathbf {S} \\mathbf {I}$ , and $\\mathbf {R}$ is upper triangular.",
"With these modifications, the Householder operator in (REF ) becomes $\\mathbf {H}_\\mathbf {S}(\\tau ) = \\mathbf {I}- 2 \\mathbf {Q}(\\tau )\\mathbf {Q}(\\tau )^T\\mathbf {S},$ where $\\mathbf {Q}(\\tau )$ is as in Equation (REF ), and remains unchanged.",
"The $\\mathbf {S}$ -orthonormal transport operator corresponding to (REF ) is $\\mathbf {T}_\\mathbf {S}(\\tau ) = - \\mathbf {H}_\\mathbf {S}(\\tau )^T\\mathbf {S}+ (\\mathbf {I}- ^T\\mathbf {S}).$"
],
[
"The quasi-Newton method based on Householder transforms",
"The method of steepest descent generally performs poorly if the minimum of the target function is at the bottom of a narrow valley.",
"Newton's method solves this problem, but requires that the Hessian of the function is available to determine the search direction.",
"When the Hessian is not available we can replace it with an approximation of the true inverse Hessian of the system to obtain a quasi-Newton method.",
"We base our method on Broyden's second or bad update to construct the approximate inverse Hessian, $\\mathbf {G}_k$ , of $f$ at $\\mathbf {X}_k$ .",
"While Broyden's second update does not construct a symmetric approximation, or ensure that the approximation is positive definite it is a robust choice for electronic structure calculations [16], [1], [2].",
"However, we must take into account that our vectors are actually $\\mathbb {R}^{m\\times n}$ matrices, which will lead to a method identical to the generalized Broyden update.",
"The secant condition is then $\\mathbf {G}_{k+1} \\Delta \\mathbf {F}_k = \\Delta \\mathbf {X}_k,$ where we project the orbital differences $\\Delta \\mathbf {X}_k = (\\mathbf {I}- \\mathbf {X}_{k+1}\\mathbf {X}_{k+1}^T\\mathbf {S})(\\mathbf {X}_{k+1}-\\mathbf {X}_{k}),$ and gradient differences $\\Delta \\mathbf {F}_k = (\\mathbf {I}- \\mathbf {X}_{k+1}\\mathbf {X}_{k+1}^T\\mathbf {S})\\nabla f(\\mathbf {X}_{k+1})-\\mathbf {T}(\\tau _k)(\\mathbf {I}-\\mathbf {X}_k\\mathbf {X}_k^T)\\nabla f(\\mathbf {X}_{k})$ onto $\\mathcal {T}_{\\mathbf {X}_{k+1}}\\mathcal {M}$ .",
"The no change condition is now $\\mathbf {G}_k\\mathbf {Z}= \\mathbf {G}_{k+1}\\mathbf {Z}\\quad \\forall \\,\\mathbf {Z}\\;:\\,\\mathbf {Z}^T\\Delta \\mathbf {F}_k = \\mathbf {0}.$ These conditions corresponds to the generalized Broyden's second update for groups of size $n$ .",
"We can therefore use the generalized update formula [11] $\\mathbf {G}_{k+1} = \\mathbf {G}_{k} + (\\Delta \\mathbf {X}_k - \\mathbf {G}_k \\Delta \\mathbf {F}_k) (\\Delta \\mathbf {F}_k^T\\mathbf {S}\\Delta \\mathbf {F}_k)^{-1}\\Delta \\mathbf {F}_k^T\\mathbf {S}.$ As initial guess we use $\\mathbf {G}_0 = \\sigma \\mathbf {I}$ .",
"With this choice, the quasi-Newton method is identical with the method of steepest descent if we do not enforce any secant conditions (REF ).",
"When secant conditions are enforced we can use $\\sigma $ to control the influence of $\\mathbf {G}_0$ compared to the information gained from the secant conditions.",
"In general, the information gained from these is reliable, and therefore $\\sigma $ should be small [1], [2].",
"To construct the search direction we use $\\mathbf {Y}_k = -\\mathbf {G}_k(\\mathbf {I}- \\mathbf {X}_k\\mathbf {X}_k^T\\mathbf {S})\\nabla f(\\mathbf {X}_k).$ In practice, we do not store $\\mathbf {G}_k$ as a full matrix.",
"Instead we represent it as a low rank update.",
"Details on recursive or low rank implementation of $\\mathbf {G}_k$ can be found in [12], [19], [1], [2].",
"We also limit the number of secant conditions used to construct $\\mathbf {G}_k$ .",
"Each condition requires storage of two $m\\times n$ matrices, $\\Delta \\mathbf {X}$ and $\\Delta \\mathbf {F}$ , and these matrices must be transported to $\\mathcal {T}_{\\mathbf {X}_k}\\mathcal {M}$ after each step.",
"This is done with the transport operator, $\\mathbf {T}_\\mathbf {S}(\\tau )$ , defined in Equation (REF ).",
"As we demonstrate in Section the first few secant conditions offer dramatic improvement over the method of steepest descent, but further secant conditions do not give the same benefit.",
"For this reason we limit the secant conditions by a pre-determined history length, and simply discard older conditions.",
"We use the same line search as the one presented in Section REF .",
"If, however $(\\mathbf {Y}_k,\\mathbf {S}(\\mathbf {I}- \\mathbf {X}_k\\mathbf {X}_k^T\\mathbf {S})\\nabla f(\\mathbf {X}_k)) \\ge 0$ then the proposed direction is not a descent direction.",
"In this case we restart the optimization method and forget the secant history.",
"If the line search returns the current point $\\mathbf {X}_k$ , we update $\\mathbf {G}_k$ but stay at $\\mathbf {X}_k$ ."
],
[
"Nonlinear conjugate gradients",
"The linear conjugate gradient (CG) method can be viewed as a optimization method for a quadratic problem.",
"Several generalizations of the CG method have been presented to solve optimization problems that are not of quadratic form [19].",
"Below, we review a nonlinear CG method adapted to account for the curvature of the manifold [10].",
"Given $\\mathbf {X}_0$ which satisfies $\\mathbf {X}_0^T\\mathbf {X}_0 = \\mathbf {I}$ , the gradient projected onto $\\mathcal {T}_{\\mathbf {X}_0}\\mathcal {M}$ is $\\mathbf {Y}_0 = (\\mathbf {I}- \\mathbf {X}_0\\mathbf {X}_0^T\\mathbf {S})\\nabla f(\\mathbf {X}_0),$ and the initial search direction is the direction of steepest descent $\\mathbf {P}_0 = -\\mathbf {Y}_0.$ On the manifold the NLCG method then proceeds by minimizing $f$ along the path defined by the search direction $\\mathbf {P}_k$ .",
"In practice we evaluate $f$ once along the search direction and minimize the quadratic approximation as in Section REF .",
"The next candidate is chosen as the best evaluated step, $\\tau _k$ , $\\mathbf {X}_{k+1} = \\mathbf {H}(\\tau _k)\\mathbf {X}_k,$ and the gradient and conjugate directions are transported to $\\mathcal {T}_{\\mathbf {X}_{k+1}}\\mathcal {M}$ by $\\mathbf {T}(\\tau _k)$ in Equation (REF ).",
"The new gradient $\\mathbf {Y}_{k+1} = (\\mathbf {I}- \\mathbf {X}_{k+1}\\mathbf {X}^T_{k+1}\\mathbf {S})\\nabla f(\\mathbf {X}_{k+1}),$ and conjugate direction $\\mathbf {P}_{k+1} = -\\mathbf {Y}_{k+1} + \\gamma _k \\mathbf {T}_\\mathbf {S}(\\tau _k) \\mathbf {P}_k,$ are then computed where $\\gamma _k = \\frac{(\\mathbf {Y}_{k+1}-\\mathbf {T}_\\mathbf {S}(\\tau _k)\\mathbf {Y}_k,\\mathbf {Y}_{k+1})}{(\\mathbf {Y}_k,\\mathbf {Y}_k)}.$ For comparison we also implement a projected NLCG (PNLCG) method.",
"Instead of ensuring that orbital updates satisfy $\\mathbf {X}_{k+1}^T\\mathbf {X}_{k+1} = \\mathbf {I}$ we orthogonalize $\\mathbf {X}_{k+1}$ after every update with the QR method.",
"The conjugate directions and gradient are not transported to $\\mathcal {T}_{\\mathbf {X}_{k+1}}\\mathcal {M}$ , instead they are updated with the $\\mathbf {I}- \\mathbf {X}_{k+1}\\mathbf {X}_{k+1}^T\\mathbf {S}$ projector onto $\\mathcal {T}_{\\mathbf {X}_{k+1}}\\mathcal {M}$ ."
],
[
"Numerical experiments",
"We use a two dimensional model problem with the condition $\\mathbf {X}^T\\mathbf {S}\\mathbf {X}= \\mathbf {I}$ to compare the projected NLCG method with the NLCG and QN methods where satisfaction the orthonormality condition is ensured by the update operator.",
"The model problem is inspired by electronic structure theory, and corresponds to a three dimensional system constrained to two dimensions without exchange-correlation terms.",
"The target function is [17] $f(\\mathbf {X}) = -\\tfrac{1}{2}\\mathrm {tr}((\\mathbf {S}^{1/2}\\mathbf {X})^T\\mathbf {L}\\mathbf {S}^{1/2}\\mathbf {X}) + \\mathbf {v}^T\\mathbf {n}+\\tfrac{1}{2} \\mathbf {n}^T\\mathbf {P}\\mathbf {n},$ where $\\mathbf {L}\\in \\mathbb {R}^{m\\times m}$ is the discretized Laplace operator, $\\mathbf {v}\\in \\mathbb {R}^{m}$ the external potential, $\\mathbf {n}\\in \\mathbb {R}^m$ the electron density, and $\\mathbf {P}\\mathbf {n}$ the Hartree potential.",
"The electron density is $\\mathbf {n}_i = \\sum _{j=1}^n((\\mathbf {S}^{1/2}\\mathbf {X})\\circ (\\mathbf {S}^{1/2}\\mathbf {X}))_{ij},$ where $\\circ $ is the entrywise, or Hadamard, product.",
"We use the overlap matrix $\\mathbf {S}= \\frac{1}{9h^2}\\begin{bmatrix}\\mathbf {M}& \\tfrac{\\mathbf {M}}{4} & 0 & \\cdots \\\\\\tfrac{\\mathbf {M}}{4} & \\mathbf {M}& \\tfrac{\\mathbf {M}}{4} & \\ddots \\\\0 & \\tfrac{\\mathbf {M}}{4} & \\mathbf {M}& \\ddots \\\\\\vdots & \\ddots & \\ddots & \\ddots \\\\\\end{bmatrix},$ where $h$ is the one dimensional grid size and $\\mathbf {M}=\\begin{bmatrix}4 & 1 & 0 & \\cdots \\\\1 & 4 & 1 & \\ddots \\\\0 & 1 & 4 & \\ddots \\\\\\vdots & \\ddots & \\ddots & \\ddots \\\\\\end{bmatrix}.$ This corresponds to the mass matrix of a finite element discretization with bilinear quadratic element.",
"The calculations have also been performed with a symmetric and positive definite random matrix.",
"It turns out that as long as $\\mathbf {S}$ is well conditioned, it has only a small effect on the rate of convergence.",
"To calculate the potentials we use $\\mathbf {v}_i = -\\sum _{j=1}^N \\frac{Z_j}{||\\mathbf {r}_i - \\mathbf {R}_j||+\\alpha },$ where the sum is over the nuclei with charge $Z_j$ and position $\\mathbf {R}_j$ .",
"The position corresponding to the discretization point $i$ is $\\mathbf {r}_i$ , and the parameter $\\alpha $ is used to regularize the potential.",
"$\\mathbf {P}\\in \\mathbb {R}^{m\\times m}$ is similarly given by $\\mathbf {P}_{ij} = \\frac{1}{||\\mathbf {r}_i-\\mathbf {r}_j|| + \\alpha }.$ We solve the problem in the unit square with zero boundary conditions corresponding to an infinite potential well.",
"We use a uniform finite difference discretization with $m$ inner points to obtain a system where $\\mathbf {X}\\in \\mathbb {R}^{m\\times n}$ .",
"Here $n$ corresponds to the number of electrons.",
"As initial guess we use the solution of the quadratic problem using the first two terms of (REF ), and choose $\\tau _0^e = 1.0$ to initialize the minimization procedure, cf.",
"Equation (REF ).",
"As generators for the external potential we use two nuclei, where one is placed at the grid point closest to $(\\tfrac{1}{3},\\tfrac{1}{3})$ , and the other at the grid point closest to $(\\tfrac{2}{3},\\tfrac{13}{24})$ .",
"The off diagonal placement is chosen to break the symmetry of the system.",
"We use $\\epsilon _\\mathbf {X}= ||(\\mathbf {I}- \\mathbf {X}\\mathbf {X}^T)\\nabla f(\\mathbf {X})||/\\sqrt{mn},$ to measure convergence, and consider the system converged when $\\epsilon _\\mathbf {X}< 10^{-2}.$ At this point $|f(\\mathbf {X}_\\mathrm {Ref})-f(\\mathbf {X})| \\approx 10^{-4},$ where the reference solution has been calculated such that $\\epsilon _{\\mathbf {X}_\\mathrm {Ref}} < 10^{-5}.$ Figure: Iterations required for convergence as a function of the spatial degrees of freedom, mm.",
"For subfigure a) the external potential is generated by two nuclei, Z 1 =3Z_1 = 3, Z 2 =3Z_2 = 3, and the parameters of the calculation are n=6n=6, β=0.5\\beta = 0.5, σ=10 -4 \\sigma = 10^{-4}, α=2×10 -2 \\alpha = 2\\times 10^{-2}, and history length is 6.",
"For subfigure b) Z 1 =4Z_1 = 4, Z 2 =3Z_2 = 3, and n=7n = 7.Here NLCG corresponds to the nonlinear conjugate gradient method, PNLCG to the projected NLCG, QN to the quasi-Newton method and AEQN to the approximate exponent QN method.The number of iterations required for convergence is identical for the AENLCG and NLCG methods, where AENLCG is the approximate exponent NLCG method.Figure REF presents the iterations necessary for convergence for a six and seven electron system.",
"These iterations roughly grows as the square root of the degrees of freedom for the NLCG methods.",
"However, the projected NLCG method performs significantly worse than the NLCG and QN methods adapted for the manifold.",
"It turns out that the AENLCG method requires the same number of iterations to converge as the NLCG method, while a small difference is visible between QN and AEQN methods.",
"Figure: Iterations required for convergence of the QN method as a function of the spatial degrees of freedom, mm.",
"For the both figures the external potential is generated by two nuclei, Z 1 =3Z_1 = 3, Z 2 =3Z_2 = 3, and the parameters of the calculation are n=6n=6, α=2×10 -2 \\alpha = 2\\times 10^{-2}, and history length is 6.Unless otherwise indicated in the figure β=0.5\\beta = 0.5 and σ=10 -4 \\sigma = 10^{-4}.Figure: Iterations required for convergence as a function of history length for the QN method for a six and seven electron system for subfigure a) and b) respectively.",
"Spatial degrees of freedom is 2500 and history length varies, parameters are otherwise identical to Figure .In Figure REF the effect of weight, $\\sigma $ , of the initial approximation of the inverse Hessian and the underrelaxation, $\\beta $ are presented.",
"From Figure REF b) it is clear that $\\sigma $ is particularly important for fast convergence of the quasi-Newton method.",
"The effect of $\\beta $ is much smaller, and while $\\beta $ can in some cases improve convergence the effect of $\\sigma $ is significantly more important.",
"A low $\\sigma $ results in slower convergence as long as the more aggressive parameter choice converges well.",
"However, when the rate of convergence begins to suffer from the more aggressive parameter choice the rate of convergence can be improved by a more conservative choice.",
"These results agree with earlier work [1], which indicate that the secant conditions offer reliable information of the electronic structure problem, while the initial approximation of $\\mathbf {G}$ is less reliable.",
"The history length of the QN method must also be sufficient for the method to perform well.",
"This is illustrated in Figure REF ."
],
[
"Conclusion",
"We have presented a Householder update scheme which ensures that the columns remains orthogonal, and is suitable for both NLCG and QN methods.",
"Furthermore, the operator allows us to transport gradient information and construct secant conditions from previous evaluations of $f$ to the tangent space of the best candidate solution.",
"This approach eliminates the need to parametrize the manifold, and permits us to use standard linear algebra routines to update the solution candidate.",
"We have demonstrated the methods numerically on a model problem inspired by the electronic structure problem, and compared them to a projected NLCG method.",
"Taking the underlying manifold into account significantly improves convergence rate of the optimization methods, and using a second order orthonormal approximate matrix exponent does not decrease performance of the QN or NLCG methods.",
"The QN method is significantly improved by taking the first few secant conditions into account when constructing the approximation of the Hessian of $f$ .",
"However, for the secant condition history to improve convergence speed of the QN method the manifold must be taken into account.",
"The update of the secant conditions is also based on the Householder operator that is used to update the solution candidate.",
"While the performance of the QN method depends on the weight of the initial approximate Hessian the QN method performs well once the weight is correctly set.",
"While the QN method is sensitive to the correct choice of $\\sigma $ , the performance of the NLCG method does not depend on parameter choice.",
"Furthermore, the NLCG method is relatively simple to implement, and only requires a one step history.",
"For these reasons we believe that the NLCG method is a good general purpose optimization method for electronic structure problems if the method is adapted to the manifold."
],
[
"Acknowledgments",
"We are grateful towards Dr. Mika Juntunen for suggestions and comments on the manuscript."
]
] | 1204.1204 |
[
[
"Light-by-light scattering sum rules constraining meson transition form\n factors"
],
[
"Abstract Relating the forward light-by-light scattering to energy weighted integrals of the \\gamma* \\gamma -fusion cross sections, with one real photon (\\gamma) and one virtual photon (\\gamma*), we find two new exact super-convergence relations.",
"They complement the known super-convergence relation based on the extension of the GDH sum rule to the light-light system.",
"We also find a set of sum rules for the low-energy photon-photon interaction.",
"All of the new relations are verified here exactly at leading order in scalar and spinor QED.",
"The super-convergence relations, applied to the \\gamma* \\gamma -production of mesons, lead to intricate relations between the \\gamma \\gamma -decay widths or the \\gamma* \\gamma -transition form factors for (pseudo-) scalar, axial-vector and tensor mesons.",
"We discuss the phenomenological implications of these results for mesons in both the light-quark sector and the charm-quark sector."
],
[
"Introduction",
"Light-by-light (LbL) scattering is a prediction of the quantum theory [1], [2] which thus-far has not been directly observed, mainly due to smallness of the cross section.",
"On the other hand, the process of $\\gamma ^\\ast \\gamma ^\\ast $ fusion (by quasi-real photons $\\gamma $ or virtual photons $\\gamma ^\\ast $ ) into leptons and hadrons has been observed at nearly all high-energy colliders, see e.g.",
"[3], [4], [5] for reviews.",
"The two phenomena — LbL scattering and $\\gamma \\gamma $ fusion — must be related by causality, similar to how the refraction index of light is related to its absorption in the Kramers-Kronig relation.",
"The main goal of this work is to establish such relations and use them to investigate the structure of hadrons in the realm of quantum chromo-dynamics (QCD).",
"The electromagnetic interaction provides a clean probe and the two-photon state allows to produce hadrons with nearly all quantum numbers (with $C = +$ ), in contrast to the well studied single-photon scattering or production processes, which only accesses the vector states.",
"When producing exclusive final states such as in the $\\gamma ^\\ast \\gamma ^\\ast \\rightarrow {\\rm meson}$ process, one accesses meson transition form factors (FFs), which are some of the simplest observables where the approach to the asymptotic limit of QCD is studied along with the quark content of mesons described by distribution amplitudes (DAs).",
"The non-perturbative dynamics of QCD is also playing a profound role in these FFs at low momentum transfers.",
"For example, the transition FFs of the $\\eta $ and $\\eta ^\\prime $ mesons depend on the interplay of various symmetry breaking mechanisms in QCD, i.e.",
": $U_A(1)$ symmetry breaking [6], dynamical and explicit chiral symmetry breaking.",
"In addition, the $\\gamma ^\\ast \\gamma ^\\ast \\rightarrow {\\rm meson}$ transition FFs are important for providing and improving constraints on the light-by-light hadronic contribution to the anomalous magnetic moment of the muon, $(g - 2)_\\mu $ .",
"The hadronic contributions to $(g - 2)_\\mu $ are at present the major uncertainty in the search for new, beyond Standard Model, physics in this high-precision quantity [7].",
"In recent years, new experiments at high luminosity $e^+ e^-$ colliders such as BABAR and Belle have vastly expanded the field of $\\gamma \\gamma $ physics.",
"The result of a measurement of the $\\gamma ^\\ast \\gamma \\rightarrow \\pi ^0$ FF at large momentum transfers by the BABAR Collaboration [8] came as a surprise, as this form factor seems to rise much faster than the perturbative QCD predictions for momentum transfers up to 40 GeV$^2$ .",
"A $\\gamma \\gamma $ physics program is planned now by the BES-III Collaboration [9], which will allow to provide high-statistics results at intermediate momentum transfers for a multitude of $\\gamma ^\\ast \\gamma ^\\ast \\rightarrow {\\rm hadron}$ observables.",
"In this work we use the dispersion theory to relate the two phenomena of LbL scattering and $\\gamma ^\\ast \\gamma $ fusion, and express the low-energy LbL scattering as integrals over the $\\gamma ^\\ast \\gamma $ -fusion cross sections, where one photon is real while the second may have arbitrary (space-like) virtuality.",
"These integrals, or `sum rules', lead to interesting constraints on $\\gamma \\gamma $ decay widths or $\\gamma ^\\ast \\gamma $ transition FFs of $q\\bar{q}$ states, and more general meson states.",
"The first sum rule of this type involves the helicity-difference cross-section for real photons and reads as: $\\int \\limits _{s_0}^\\infty \\frac{ds}{s} \\, \\Big [ \\sigma _2(s) - \\sigma _0(s) \\Big ] =0,$ where $s$ is the total energy squared, $s_0$ is the first inelastic threshold for the $\\gamma \\gamma $ fusion process, and the subscripts 0 or 2 for the $\\gamma \\gamma $ cross sections indicate the total helicity of the state of two circularly polarized photons.",
"This sum rule was originallyAn earlier version of this sum rule had been proposed in Ref.",
"[12], where a contribution from $\\pi ^0$ production appears on the right-hand side (rhs) of Eq.",
"(REF ), while integration on the lhs starts at the 2$\\pi $ production threshold.",
"That version would be fully compatible with Eq.",
"(REF ), if it were not for the sign of the $\\pi ^0$ contribution obtained in [12].",
"inferred [10], [11] from the the Gerasimov–Drell–Hearn (GDH) sum rule, using the fact that the photon has no anomalous moments.",
"Parameterizing the lowest energy LbL interaction by means of an effective Lagrangian (which contains operators of dimension eight at lowest order) as $\\mathcal {L}^{(8)} = c_1 (F_{\\mu \\nu }F^{\\mu \\nu })^2 + c_2 (F_{\\mu \\nu }\\tilde{F}^{\\mu \\nu })^2,$ with $F$ and $\\tilde{F}$ being the electromagnetic field strength and its dual, one finds sum rules for the LbL low-energy constants (LECs) [13]: $c_1 = \\frac{1}{8 \\pi }\\int \\limits _{s_0}^{\\infty } {\\rm d} s\\, \\frac{ \\sigma _\\parallel (s)}{s^2}\\, ,\\quad \\quad \\quad c_2 = \\frac{1}{8 \\pi }\\int \\limits _{s_0}^{\\infty } {\\rm d} s\\, \\frac{\\sigma _\\perp (s)}{s^2} \\, ,$ where the subscripts $||$ or $\\perp $ indicate if the colliding photons are polarized parallel or perpendicular to each other.",
"While the GDH-type sum rule provides a stringent constraint on the polarized $\\gamma \\gamma $ fusion, the sum rules for the LECs allow one in principle to fully determine the low-energy LbL interaction through measuring the linearly polarized $\\gamma \\gamma $ fusion.",
"In this work we extend the GDH type sum rule to the case where one of the colliding photons is virtual, with arbitrary (space-like) virtuality.",
"Furthermore, we find two additional sum rules, involving the longitudinally polarized $\\gamma ^\\ast \\gamma $ cross sections.",
"All details of sum rule derivation are gathered in Sec. .",
"In Sec.",
", all of the newly derived sum rules are verified at leading order in scalar and spinor quantum electrodynamcis (QED).",
"Next we apply these results to the $\\gamma ^\\ast \\gamma ^\\ast $ fusion to mesons.",
"Using the available data, we quantitatively study the new sum rules derived in this paper for the case of production of light quark mesons as well as mesons containing charm quarks, both by real photons in Sec.",
"REF , and by virtual photons in Sec.",
"REF .",
"We demonstrate the intricate cancellations that must occur among the (pseudo-) scalar, tensor, and axial-vector mesons in order to satisfy these sum rules.",
"In the case of production of virtual photons, we use these relations to provide estimates of hitherto unmeasured $\\gamma ^\\ast \\gamma $ transition form factors of tensor mesons, such as $f_2(1285)$ and $a_2(1320)$ .",
"The conclusion and outlook is given in Sec. .",
"The Appendices contain () a review of the kinematical notations and $e^\\pm + e^- \\rightarrow e^\\pm + e^- + X$ cross section conventions; () expressions for the tree-level $\\gamma ^\\ast \\gamma ^\\ast $ cross sections for the case of scalar and spinor QED (Sec.",
"); () general formalism for the $\\gamma ^\\ast \\gamma \\rightarrow {\\rm meson}$ transitions with different quantum numbers ($J^{PC}$ ), i.e.",
": pseudo-scalars ($0^{-+}$ ), scalars ($0^{++}$ ), axial-vectors ($1^{++}$ ), and tensors ($2^{++}$ ).",
"In the most general case we consider the forward scattering of virtual photons on virtual photons: $\\gamma ^\\ast (\\lambda _1, q_1) + \\gamma ^\\ast (\\lambda _2, q_2) \\rightarrow \\gamma ^\\ast (\\lambda ^\\prime _1, q_1) + \\gamma ^\\ast (\\lambda ^\\prime _2, q_2),$ where $q_1$ , $q_2$ are photon four-momenta, and $\\lambda _1, \\lambda _2$ ($\\lambda ^\\prime _1, \\lambda ^\\prime _2$ ) are the helicities of the initial (final) virtual photons, which can take on the values $\\pm 1$ (transverse polarizations) and zero (longitudinal).",
"The total helicity in the $\\gamma ^\\ast \\gamma ^\\ast $ c.m.",
"system is given by $\\Lambda = \\lambda _1 - \\lambda _2 = \\lambda ^\\prime _1 - \\lambda ^\\prime _2$ .",
"To define the kinematics, we firstly introduce the photon virtualities $Q_1^2 = - q_1^2$ , $Q_2^2 = -q_2^2$ , the Mandelstam invariants: $s = (q_1 + q_2)^2$ , $u = (q_1 - q_2)^2$ , and the following crossing-symmetric variable: $\\nu \\equiv \\mbox{$\\frac{1}{4}$}(s - u) = q_1 \\cdot q_2,$ such that $s = 2 \\nu - Q_1^2 - Q_2^2$ , $u = - 2 \\nu - Q_1^2 - Q_2^2$ .",
"The $\\gamma ^\\ast \\gamma ^\\ast \\rightarrow \\gamma ^\\ast \\gamma ^\\ast $ forward scattering amplitudes, denoted as $M_{\\lambda ^\\prime _1 \\lambda ^\\prime _2, \\lambda _1 \\lambda _2}$ , are functions of $\\nu $ , $Q_1^2$ , $Q_2^2$ .",
"Parity invariance ($P$ ) and time-reversal invariance ($T$ ) imply the following relations among the matrix elements with different helicities : $P: \\quad M_{\\lambda ^\\prime _1 \\lambda ^\\prime _2, \\lambda _1 \\lambda _2} &=&M_{- \\lambda ^\\prime _1 - \\lambda ^\\prime _2, - \\lambda _1 - \\lambda _2},\\\\T: \\quad M_{\\lambda ^\\prime _1 \\lambda ^\\prime _2, \\lambda _1 \\lambda _2} &=&M_{\\lambda _1 \\lambda _2, \\lambda ^\\prime _1 \\lambda ^\\prime _2},$ which leaves out only eight independent amplitudes [14]: $M_{++,++}, \\, M_{+-,+-}, \\, M_{++,--}, \\,M_{00,00}, \\, M_{+0,+0}, \\, M_{0+,0+}, \\, M_{++,00}, \\, M_{0+,-0}.$ We next look at the constraint imposed by crossing symmetry, which requires that the amplitudes for the process (REF ) equal the amplitudes for the process where the photons with e.g.",
"label 2 are crossed: $\\gamma ^\\ast (\\lambda _1, q_1) + \\gamma ^\\ast (- \\lambda ^\\prime _2, - q_2) \\rightarrow \\gamma ^\\ast (\\lambda ^\\prime _1, q_1) + \\gamma ^\\ast (- \\lambda _2, - q_2).$ As under photon crossing $\\nu \\rightarrow -\\nu $ , one obtains $M_{\\lambda ^\\prime _1 \\lambda ^\\prime _2, \\lambda _1 \\lambda _2}(\\nu , Q_1^2, Q_2^2) &=&M_{\\lambda ^\\prime _1 - \\lambda _2, \\lambda _1 - \\lambda ^\\prime _2}(-\\nu , Q_1^2, Q_2^2),$ it becomes convenient to introduce amplitudes which are either even or odd in $\\nu $ (at fixed $Q_1^2$ and $Q_2^2$ ).",
"One easily verifies that the following six amplitudes are even in $\\nu $ : $\\left( M_{++,++} + M_{+-,+-} \\right), \\quad M_{++,--}, \\quad M_{00,00}, \\quad M_{+0,+0}, \\quad M_{0+,0+}, \\quad \\left( M_{++,00} + M_{0+,-0} \\right),$ whereas the following two amplitudes are odd in $\\nu $ : $\\left( M_{++,++} - M_{+-,+-} \\right), \\quad \\quad \\left( M_{++,00} - M_{0+,-0} \\right).$"
],
[
"Fusion of two virtual photons",
"The optical theorem allows one to relate the absorptive part of the $\\gamma ^\\ast \\gamma ^\\ast \\rightarrow \\gamma ^\\ast \\gamma ^\\ast $ forward scattering amplitudes to cross sections for the process $\\gamma ^\\ast \\gamma ^\\ast \\rightarrow \\mathrm {X}$ , where X stands for any possible final state.",
"Denoting the absorptive part as $W_{\\lambda ^\\prime _1 \\lambda ^\\prime _2, \\lambda _1 \\lambda _2} \\equiv \\mathrm {Abs}\\,M_{\\lambda ^\\prime _1 \\lambda ^\\prime _2, \\lambda _1 \\lambda _2},$ the optical theorem yields: $W_{\\lambda ^\\prime _1 \\lambda ^\\prime _2, \\lambda _1 \\lambda _2} =\\frac{1}{2} \\int d \\Gamma _\\mathrm {X} (2 \\pi )^4 \\delta ^4(q_1 + q_2 - p_\\mathrm {X}) \\,{\\cal M}_{\\lambda _1 \\lambda _2} (q_1, q_2; p_\\mathrm {X}) \\,{\\cal M}^\\ast _{\\lambda ^\\prime _1 \\lambda ^\\prime _2} (q_1, q_2; p_\\mathrm {X}),$ where ${\\cal M}_{\\lambda _1 \\lambda _2} (q_1, q_2; p_\\mathrm {X})$ denotes the invariant amplitude for the process $\\gamma ^\\ast (\\lambda _1, q_1) + \\gamma ^\\ast (\\lambda _2, q_2) \\rightarrow \\mathrm {X}(p_\\mathrm {X}).$ As a result, the absorptive parts are expressed in terms of eight independent $\\gamma ^\\ast \\gamma ^\\ast \\rightarrow \\mathrm {X}$ cross sections (see Ref.",
"[3] for details): $W_{++,++} + W_{+-,+-} &\\equiv & 2 \\sqrt{X} \\, \\left(\\sigma _0 + \\sigma _2 \\right) = 2 \\sqrt{X} \\, \\left(\\sigma _\\parallel + \\sigma _\\perp \\right)\\equiv 4 \\sqrt{X} \\, \\sigma _{TT}, \\\\W_{++,++} - W_{+-,+-} &\\equiv & 2 \\sqrt{X} \\, \\left(\\sigma _0 - \\sigma _2 \\right) \\equiv 4 \\sqrt{X} \\, \\tau ^a_{TT} , \\\\W_{++,--} &\\equiv & 2 \\sqrt{X} \\, \\left(\\sigma _\\parallel - \\sigma _\\perp \\right) \\equiv 2 \\sqrt{X} \\, \\tau _{TT} , \\\\W_{00,00} &\\equiv & 2 \\sqrt{X} \\, \\sigma _{LL}, \\\\W_{+0,+0} &\\equiv & 2 \\sqrt{X} \\, \\sigma _{TL}, \\\\W_{0+,0+} &\\equiv & 2 \\sqrt{X} \\, \\sigma _{LT}, \\\\W_{++,00} + W_{0+,-0} &\\equiv & 4 \\sqrt{X} \\, \\tau _{TL}, \\\\W_{++,00} - W_{0+,-0} &\\equiv & 4 \\sqrt{X} \\, \\tau ^a_{TL},$ where the virtual photon flux factor is defined through $X \\equiv (q_1 \\cdot q_2)^2 - q_1^2 q_2^2 = \\nu ^2 - Q_1^2 Q_2^2.$ In Eq.",
"(REF ), $\\sigma _0 (\\sigma _2)$ are the $\\gamma ^\\ast \\gamma ^\\ast \\rightarrow \\mathrm {X}$ cross sections for total helicity 0 (2) respectively, and $\\sigma _\\parallel (\\sigma _\\perp )$ are the cross sections for linear photon polarizations with both photon polarization directions parallel (perpendicular) to each other respectively.",
"The remaining cross sections (positive definite quantities $\\sigma $ ) involve either one transverse ($T$ ) and one longitudinal ($L$ ) photon polarization, or two longitudinal photon polarizations, with $\\sigma _{LT}$ and $\\sigma _{TL}$ related as : $\\sigma _{LT}(\\nu , Q_1^2, Q_2^2) = \\sigma _{TL}(\\nu , Q_2^2, Q_1^2).$ The quantities $\\tau _{TT}, \\tau ^a_{TT}, \\tau _{TL}, \\tau ^a_{TL}$ denote interference cross sections (which are not sign-definite) with either both photons transverse ($TT$ ), or for one transverse and one longitudinal photon ($TL$ ), where the superscript $a$ indicates the combinations which are odd in $\\nu $ ."
],
[
"Dispersion relations",
"The principle of (micro-)causality is known to translate into exact statements about analytic properties of the scattering amplitude in the complex energy plane.",
"In our case this principle translates into the statement of analyticity of the forward $\\gamma ^\\ast \\gamma ^\\ast $ scattering amplitude in the entire $\\nu $ plane, except for the real axis where the branch cuts associated with particle production are located.",
"Assuming that the threshold for particle production is $\\nu _0 > 0$ , one can write down the usual dispersion relations, in which the amplitude is given by integrals over the non-analyticities, which in this case are branch cuts extending from $\\pm \\nu _0$ to $\\pm \\infty $ .",
"Finally, for amplitudes that are even or odd in $\\nu $ we can write (for any fixed values of $Q_1^2, Q_2^2 > 0$ ): $f_{even}(\\nu ) & = & \\frac{2}{\\pi } \\int _{\\nu _0}^\\infty \\!",
"d \\nu ^\\prime \\frac{\\nu ^\\prime }{\\nu ^{\\prime \\, 2} - \\nu ^2-i0^+} \\mathrm {Abs} \\, f_{even}(\\nu ^\\prime ),\\\\f_{odd}(\\nu ) & = & \\frac{2 \\nu }{\\pi } \\int _{\\nu _0}^\\infty \\!",
"d \\nu ^\\prime \\frac{ 1}{\\nu ^{\\prime \\, 2} - \\nu ^2 - i0^+} \\mathrm {Abs}\\, f_{odd}(\\nu ^\\prime ),$ where $0^+$ is an infinitesimal positive number.",
"These dispersion relations are derived with the provision that the integrals converge.",
"If they do not, subtractions must be made; e.g., the once-subtracted dispersion relation for the even amplitudes reads: $f_{even}(\\nu ) & = & f_{even}(0) + \\frac{2\\nu ^2}{\\pi } \\int _{\\nu _0}^\\infty \\!",
"d \\nu ^\\prime \\frac{1}{\\nu ^\\prime (\\nu ^{\\prime \\, 2} - \\nu ^2-i0^+)} \\mathrm {Abs} \\, f_{even}(\\nu ^\\prime ).$ We are thus led to examine the high-energy behavior ($\\nu \\rightarrow \\infty $ at fixed $Q_1^2, Q_2^2$ ) of the absorptive parts given by Eq.",
"(REF ).",
"In Ref.",
"[14], a Regge pole model assumption for the high-energy asymptotics of the light-by-light forward amplitudes yielded: $\\left(W_{++,++} + W_{+-,+-}\\right), \\quad W_{+0,+0}, \\quad W_{0+,0+}, \\quad W_{00,00} \\quad &\\sim & \\nu ^{\\alpha _P(0)}, \\nonumber \\\\\\left(W_{++,++} - W_{+-,+-}\\right), \\quad W_{++,--} \\quad &\\sim & \\nu ^{\\alpha _\\pi (0)}, \\\\\\left(W_{++,00} + W_{0+,-0}\\right), \\quad \\left(W_{++,00} - W_{0+,-0}\\right) \\quad &\\sim & \\nu ^{\\alpha _\\pi (0) - 1}, \\nonumber $ where $\\alpha _P(0) \\simeq 1.08$ is the intercept of the Pomeron trajectory, and $\\alpha _\\pi (0) \\simeq -0.014 $ is the intercept of the pion trajectory.",
"This means that for all the even amplitudes, except $M_{++,00} + M_{0+,-0}$ , one can only use the subtracted dispersion relation Eq.",
"(REF ).",
"We therefore need the information about these amplitudes at zero energy $\\nu $ .",
"Anticipating the discussion of the low-energy expansion of the LbL scattering, we can state that at $\\nu =0$ these amplitudes vanish when one of the photons is real [cf.",
"Eq.",
"(REF )].",
"Using Eq.",
"(REF ) then to substitute the cross sections in place of the absorptive parts, we obtain the following sum rules for the case of one real and one virtual photon (when the virtual photon flux factor becomes $X = \\nu ^2$ ): $M_{++,++} (\\nu )+ M_{+-,+-}(\\nu ) &=&\\frac{4\\nu ^2}{\\pi } \\int _{\\nu _0}^\\infty \\!\\!",
"d \\nu ^\\prime \\, \\frac{ \\sigma _\\parallel (\\nu ^\\prime ) + \\sigma _\\perp (\\nu ^\\prime ) }{\\nu ^{\\prime \\, 2} - \\nu ^2-i0^+} , \\\\M_{++,--} (\\nu ) &=&\\frac{4\\nu ^2}{\\pi } \\int _{\\nu _0}^\\infty \\!\\!d \\nu ^\\prime \\, \\frac{ \\sigma _\\parallel (\\nu ^\\prime ) - \\sigma _\\perp (\\nu ^\\prime ) }{\\nu ^{\\prime \\, 2} - \\nu ^2-i0^+} , \\\\M_{0+,0+} (\\nu ) &=&\\frac{4\\nu ^2}{\\pi } \\int _{\\nu _0}^\\infty \\!\\!",
"d \\nu ^\\prime \\, \\frac{ \\sigma _{LT}(\\nu ^\\prime ) }{\\nu ^{\\prime \\, 2} - \\nu ^2-i0^+} , \\\\M_{+0,+0} (\\nu ) &=&\\frac{4\\nu ^2}{\\pi } \\int _{\\nu _0}^\\infty \\!\\!",
"d \\nu ^\\prime \\, \\frac{ \\sigma _{TL}(\\nu ^\\prime ) }{\\nu ^{\\prime \\, 2} - \\nu ^2-i0^+}.$ We cannot write such a subtracted sum rule for $M_{00,00}$ , since it trivially vanishes when one of the photons is real.",
"Instead, considering an unsubtracted dispersion relation, we find the following sum rule: $M_{00,00} (\\nu ) &=&\\frac{4}{\\pi } \\int _{\\nu _0}^\\infty \\!\\!",
"d \\nu ^\\prime \\, \\frac{\\nu ^\\prime \\sqrt{X^\\prime } \\, \\sigma _{LL}(\\nu ^\\prime ) }{\\nu ^{\\prime \\, 2} - \\nu ^2-i0^+},$ with $X^{\\prime }=\\nu ^{\\prime \\, 2} - Q_1^2 Q_2^2$ .",
"At least in perturbative QED calculations (cf.",
"Appendix ), the above integral converges which seems to validate this sum rule in a renormalizable, perturbative field theory.",
"We emphasize however that this observation is in contradiction with the expectation of non-convergence from the Regge pole model shown above.",
"A validation of this sum rule in non-perturbative field theory, particularly in QCD, is therefore an open issue.",
"For all the remaining amplitudes the asymptotic behavior of Eq.",
"(REF ) justifies the use of unsubtracted dispersion relations which, upon substituting Eq.",
"(REF ), lead to the following sum rules, valid for both photon virtual: $M_{++,++}(\\nu ) - M_{+-,+-} (\\nu ) &=&\\frac{4\\nu }{\\pi } \\int _{\\nu _0}^\\infty \\!\\!d \\nu ^\\prime \\,\\frac{\\sqrt{X^\\prime }\\,\\big [ \\sigma _0(\\nu ^\\prime ) - \\sigma _2(\\nu ^\\prime ) \\big ] }{\\nu ^{\\prime \\, 2} - \\nu ^2-i0^+} , \\\\M_{++,00}(\\nu ) - M_{0+,-0}(\\nu ) &=&\\frac{8\\nu }{\\pi } \\int _{\\nu _0}^\\infty \\!\\!d \\nu ^\\prime \\,\\frac{\\sqrt{X^\\prime } \\, \\tau ^a_{TL} (\\nu ^\\prime )}{\\nu ^{\\prime \\, 2} - \\nu ^2-i0^+} ,\\\\M_{++,00} (\\nu )+ M_{0+,-0} (\\nu ) &=&\\frac{8}{\\pi } \\int _{\\nu _0}^\\infty \\!\\!d \\nu ^\\prime \\, \\frac{\\nu ^\\prime \\sqrt{X^\\prime }\\, \\tau _{TL} (\\nu ^\\prime ) }{\\nu ^{\\prime \\, 2} - \\nu ^2-i0^+} ,$ where the dependence on virtualities $Q_1^2$ , $Q_2^2$ is tacitly assumed.",
"The above sum rules, relating all the forward $\\gamma ^\\ast \\gamma ^\\ast $ elastic scattering amplitudes to the energy integrals of the $\\gamma ^\\ast \\gamma ^\\ast $ fusion cross sections, should hold for any space-like photon virtualities in the unsubtracted cases, and for one of the virtualities equal to zero in the subtracted cases.",
"In the following we examine the low-energy expansion of these sum rules."
],
[
"Low-energy expansion via effective Lagrangian",
"To obtain more specific relations from the sum rules established in Eq.",
"(REF ), we parametrize the low-energy (small $\\nu $ ) behavior of the $\\gamma ^\\ast \\gamma ^\\ast \\rightarrow \\gamma ^\\ast \\gamma ^\\ast $ forward scattering amplitudes $M$ .",
"At lowest order in the energy, the self-interactions of the electromagnetic field are described by an effective Lagrangian (of fourth order in the photon energy and/or momentum, and fourth order in the electromagnetic field): $\\mathcal {L}^{(8)} = c_1 (F_{\\mu \\nu }F^{\\mu \\nu })^2 + c_2 (F_{\\mu \\nu }\\tilde{F}^{\\mu \\nu })^2,$ where $F_{\\mu \\nu } = \\partial _\\mu A_\\nu - \\partial _\\nu A_\\mu $ , $\\tilde{F}^{\\mu \\nu } =\\varepsilon ^{\\mu \\nu \\alpha \\beta } \\partial _\\alpha A_\\beta $ , and where $c_1, c_2$ are two low-energy constants (LECs) which contain the structure dependent information.",
"It is often referred to as Euler-Heisenberg Lagrangian due to the seminal work [1].",
"At the next order in energy, one considers the terms involving two derivatives on the field tensors, corresponding with the sixth order in the photon energy and/or momentum.",
"Writing down all such dimension-ten operators and reducing their number using the antisymmetry of the field tensors, the Bianchi identities, as well as adding or removing total derivative terms, we find that there are 6 independent terms at that order, which we choose as : $\\mathcal {L}^{(10)}&=&c_{3} (\\partial _{\\alpha } F_{\\mu \\nu }) (\\partial ^{\\alpha } F^{\\lambda \\nu }) F_{\\lambda \\rho }F^{\\mu \\rho }+c_4 (\\partial _{\\alpha } F_{\\mu \\nu }) (\\partial ^{\\alpha } F^{\\mu \\nu }) F_{\\lambda \\rho }F^{\\lambda \\rho }\\nonumber \\\\&+ & c_5 (\\partial ^{\\alpha } F_{{\\alpha }\\nu }) (\\partial _\\beta F^{\\beta \\nu }) F_{\\lambda \\rho }F^{\\lambda \\rho }+c_{6} (\\partial _{\\alpha }\\partial ^{\\alpha } F_{\\mu \\nu }) F^{\\lambda \\nu }F_{\\lambda \\rho }F^{\\mu \\rho }\\nonumber \\\\&+&c_7 ( \\partial _{\\alpha }\\partial ^{\\alpha } F_{\\mu \\nu }) F^{\\mu \\nu }F_{\\lambda \\rho }F^{\\lambda \\rho }+c_{8} (\\partial ^{\\alpha } F_{\\alpha \\mu }) (\\partial _{\\beta } F^{\\beta \\lambda }) F_{\\rho \\lambda }F^{\\rho \\mu },$ where $c_3, \\ldots , c_8$ are the new LECs arising at this order.",
"Only $c_3$ and $c_4$ appear in the case of real photons.",
"We can now specify the low-energy limit of the light-by-light scattering amplitudes in terms of the LECs describing the low-energy self-interactions of the electromagnetic field: $M_{++,++} + M_{+-,+-} &=& Q_1^2Q_2^2 \\left[ 64(c_1-c_2)+ 4(Q_1^2+Q_2^2)(-c_3-8c_4-4c_5+8c_7-c_8)+{\\cal O}(Q^4) \\right] \\nonumber \\\\& + & 8 \\nu ^2\\left[ 8 (c_1 + c_2) + \\left(Q_1^2 + Q_2^2 \\right) (-c_3+3c_{6}+4c_{7}) + {\\cal O}(Q^4) \\right] + {\\cal O}(\\nu ^4), \\\\M_{++,--} &=& Q_1^2Q_2^2 \\left[ 64c_2+4(Q_1^2+Q_2^2)(-c_3+2c_6-c_8)+{\\cal O}(Q^4) \\right] \\nonumber \\\\&+& 8\\nu ^2 \\left[ 8 (c_1 - c_2) + \\left(Q_1^2 + Q_2^2\\right) ( c_{6}+4c_7 )+{\\cal O}(Q^4) \\right] + {\\cal O}(\\nu ^4), \\\\M_{0+,0+} &=& Q_1^2Q_2^2\\left[-32 c_1+4Q_1^2 c_8+4(Q_1^2+Q_2^2)(c_3+4c_4+2c_5-2c_6-4c_7) +{\\cal O}(Q^4) \\right] \\nonumber \\\\&+& \\nu ^2 \\left[-4Q_1^2 c_{8} +{\\cal O}(Q^4) \\right] + {\\cal O}(\\nu ^4), \\\\M_{+0,+0} &=& Q_1^2Q_2^2\\left[-32 c_1+4Q_2^2 c_8+4(Q_1^2+Q_2^2)(c_3+4c_4+2c_5-2c_6-4c_7) +{\\cal O}(Q^4) \\right] \\nonumber \\\\&+& \\nu ^2 \\left[-4Q_2^2 c_{8} +{\\cal O}(Q^4) \\right] + {\\cal O}(\\nu ^4), \\\\M_{00,00} &=& Q_1^2 Q_2^2 \\left[ 96 c_1 +4(Q_1^2+Q_2^2)(-2c_3-4c_4-2c_5+6c_6+12c_7-c_8) + {\\cal O}(Q^4) \\right] \\nonumber \\\\&+& {\\cal O}(\\nu ^2), \\\\M_{++,++} - M_{+-,+-} &=& 8 \\nu Q_1^2 Q_2^2 \\left[ -c_3 - 4 c_{5} + c_{8} + {\\cal O}(Q^2) \\right] + \\nu ^3\\left[-64c_4 + {\\cal O}(Q^2) \\right]+ {\\cal O}(\\nu ^5), \\\\M_{++,00} - M_{0+,-0} &=& \\nu Q_1 Q_2 \\left[ - 64 c_1+ \\left(Q_1^2 + Q_2^2\\right) (4c_3-16c_{6}-32c_{7}+4c_{8}) + {\\cal O}(Q^4) \\right] \\nonumber \\\\&+& {\\cal O}(\\nu ^3), \\\\M_{++,00} + M_{0+,-0} &=& Q_1^3 Q_2^3 \\left[ 4c_5-12c_{8} + {\\cal O}(Q^2) \\right] +4\\nu ^2 Q_1Q_2 \\left[ 2 c_3+16c_4+4c_5+c_8+ {\\cal O}(Q^2) \\right]\\nonumber \\\\&+& {\\cal O}(\\nu ^4).",
"$ These expressions can be treated as a simultaneous expansion in $\\nu $ and the virtualities $Q_i^2$ of the lhs of the sum rules Eq.",
"(REF ).",
"Concerning the $Q$ dependence, it is important that the leading in $\\nu $ term, in any of the amplitudes, is proportional to $Q_1 Q_2$ and hence vanishes for at least one real photon.",
"The latter statement is valid for any values of virtualities, not just when they are small.",
"For example, let us show for the amplitude $( M_{++,++} - M_{+-,+-} )$ its leading term in $\\nu $ is proportional to the combination $Q_1^2Q_2^2$ , to all orders in $Q_1$ and $Q_2$ .",
"Since all photons are transversely polarized the only non-vanishing structures involving polarization vectors of photons $\\varepsilon ( \\lambda _i)$ are their mutual scalar products $\\varepsilon (\\lambda _i) \\cdot \\varepsilon (\\lambda _j)$ .",
"Due to gauge invariance, the electromagnetic fields enter the Lagrangian through the field tensor $F_{\\mu \\nu }$ , which contributes to the amplitude as $q_\\mu \\varepsilon _\\nu -q_\\nu \\varepsilon _\\mu $ .",
"Thus an arbitrary term in the effective Lagrangian contributes to $( M_{++,++} - M_{+-,+-} )$ as: $M_{++,++} - M_{+-,+-} \\sim q_1^\\mu q_2^\\nu q_1^\\lambda q_2^\\rho T_{\\mu \\nu \\lambda \\rho },$ where the tensor $T_{\\mu \\nu \\lambda \\rho }$ is constructed from four-vectors $q_{i}$ and the metric tensor.",
"Since this amplitude is odd with respect to $\\nu $ , it is required to be proportional to at least $\\nu ^1$ .",
"Assuming that one factor $\\nu $ comes from contraction of two of the $q$ 's in Eq.",
"(REF ), we are left with $q_1^\\mu q_2^\\nu $ .",
"Now, if we suppose that $q_1$ is contracted with $q_2$ we obtain an extra power of $\\nu $ , and such an amplitude vanishes when taking the limit $\\nu \\rightarrow 0$ .",
"Thus, both $q_1$ and $q_2$ must be contracted with another $q_1$ and $q_2$ respectively, giving a global factor $Q_1^2Q_2^2$ .",
"We are now in position to examine the sum rules in Eq.",
"(REF ) order by order in $\\nu $ .",
"For this we expand the rhs of Eq.",
"(REF ) using $1/(\\nu ^{\\prime \\, 2} - \\nu ^2) = 1/\\nu ^{\\prime \\, 2} + \\nu ^2/\\nu ^{\\prime \\, 4} + {\\cal O}(\\nu ^4)$ .",
"As the result we obtain from Eqs.",
"(REF ,,) the following set of super-convergence relations, valid for at least one real photon (e.g., $Q_1\\ge 0$ , $Q_2^2 = 0$ ): $0 &=& \\int \\limits _{s_0}^\\infty d s \\frac{ 1 }{(s + Q_1^2)} \\, \\tau _{TT}^a (s, Q_1^2, 0), \\\\0 &=& \\int \\limits _{s_0}^\\infty d s \\, \\frac{1}{(s + Q_1^2)^2}\\left[ \\sigma _\\parallel + \\sigma _{LT} + \\frac{(s + Q_1^2)}{Q_1 Q_2} \\tau ^a_{TL}\\right]_{Q_2^2 = 0}, \\\\0 &=& \\int \\limits _{s_0}^\\infty d s \\, \\left[ \\frac{\\tau _{TL} (s, Q_1^2, Q_2^2) }{Q_1 Q_2}\\right]_{Q_2^2 = 0}.$ and the following set of sum rules for the LECs of the dimension-8 (Euler-Heisenberg) Lagrangian, valid when both photons are quasi-real: $c_1 & = & \\frac{1}{8 \\pi }\\int \\limits _{s_0}^{\\infty } \\frac{d s}{s^2}\\,\\sigma _\\parallel (s,0,0), \\\\&=& - \\frac{1}{ 8 \\pi } \\int \\limits _{s_0}^\\infty \\frac{ds }{s} \\, \\left[ \\frac{\\tau ^a_{TL} (s, Q_1^2, Q_2^2) }{Q_1 Q_2}\\right]_{Q_1^2 = Q_2^2 = 0}, \\\\&=& \\frac{1}{ 8 \\pi } \\int \\limits _{s_0}^\\infty d s\\, \\left[ \\frac{\\sigma _{LL} (s, Q_1^2, Q_2^2) }{Q_1^2 Q_2^2}\\right]_{Q_1^2 = Q_2^2 = 0}, \\\\c_2 & = & \\frac{1}{8 \\pi }\\int \\limits _{s_0}^{\\infty } \\frac{d s}{s^2}\\, \\sigma _\\perp (s,0,0),$ where $s_0 = 2 \\nu _0 - Q_1^2 - Q_2^2$ .",
"We emphasize again that, unlike the other sum rules, the sum rule of Eq.",
"() is only shown to hold in perturbative field theory.",
"There are as well the sum rules for the LECs of the dimension-10 Lagrangian, most notably: $c_4 &=& - \\frac{1}{ 4 \\pi }\\int \\limits _{s_0}^\\infty \\frac{ ds }{s^3} \\, \\tau _{TT}^a (s, 0, 0),$ but presently they are of far lesser importance and we do not write them out here explicitly.",
"Let us remark again that the relation of Eq.",
"(REF ), obtained by combining Eqs.",
"(REF ) and (), is essentially a GDH sum rule for the photon target, see [10], [11], [12].",
"For large virtuality $Q_1^2$ , it leads to the sum rule for the photon structure function $g_1^\\gamma $ [15]: $\\int _0^1 d x g_1^\\gamma (x, Q^2) = 0$ .",
"The sum rules in Eqs.",
"(REF ) and (), first established in [13], are obtained by combining Eqs.",
"(REF ) with (REF ) and Eqs.",
"() with (), respectively.",
"All the other relations presented above are new.",
"In the following section we verify these sum rules in QED at leading order in the fine-structure constant $\\alpha $ ."
],
[
"Sum rules in perturbation theory",
"We will subsequently discuss a pair production in scalar QED (e.g., Born approximation to $\\gamma ^\\ast \\gamma ^\\ast \\rightarrow \\pi ^+ \\pi ^-$ ) and in spinor QED ($\\gamma ^\\ast \\gamma ^\\ast \\rightarrow q \\bar{q}$ where $q$ stands for a charged lepton or a quark)."
],
[
"Scalar QED",
"The response functions for the case of scalar QED at lowest order in the electromagnetic coupling can be found in Appendix REF .",
"We firstly study the three sum rules of Eqs.",
"(REF , , ) for the case of one real or quasi-real photon ($Q_2^2 \\rightarrow 0$ ) and for arbitrary space-like virtuality ($Q_1^2 \\ge 0$ ) of the other photon.",
"To better see the cancellation which must take place in these sum rules between contributions at low and higher energies, we show the integrands of the three sum rules in Figs.",
"REF , REF , REF multiplied by $s$ .",
"In this way, when plotted logarithmically, one can clearly see how the low and high energy contributions cancel each other.",
"For the sum rule of Eq.",
"(), we denote the integrand as : $I = \\frac{1}{(s + Q_1^2)^2}\\left[ \\sigma _\\parallel + \\sigma _{LT} + \\frac{(s + Q_1^2)}{Q_1 Q_2} \\tau ^a_{TL}\\right]_{Q_2^2 = 0}.$ All three sum rules of Eqs.",
"(REF , , ) are exactly verified in scalar QED for arbitrary space-like values of $Q_1^2$ .",
"One notices from Figs.",
"REF , REF , REF that for larger values of $Q_1^2$ the zero crossing of the integrands shifts to larger values of $s$ , requiring higher energy contributions for the cancellation to take place.",
"For the helicity difference sum rule of Eq.",
"(REF ), one notices that at low energies $\\sigma _0$ dominates while with increasing energies $\\sigma _2$ overtakes.",
"Figure: The γ * γ→π + π - \\gamma ^\\ast \\gamma \\rightarrow \\pi ^+ \\pi ^- tree level result (scalar QED) for theintegrand in the Δσ≡σ 2 -σ 0 \\Delta \\sigma \\equiv \\sigma _2 - \\sigma _0 sum rule of Eq.",
"(),multiplied by ss, where one of the photons is real.The different curves are for different virtualities for the other photon :Q 1 2 =0Q_1^2 = 0 (solid black curve), Q 1 2 =m 2 Q_1^2 = m^2 (short-dashed red curve), Q 1 2 =5m 2 Q_1^2 = 5 m^2 (long-dashed blue curve).Figure: The γ * γ→π + π - \\gamma ^\\ast \\gamma \\rightarrow \\pi ^+ \\pi ^- tree level result (scalar QED) for the integrandin the τ TL \\tau _{TL} sum rule of Eq.",
"() multiplied by ss,where one of the photons is quasi-real.",
"The different curves are for different virtualities for the other photon :Q 1 2 =0Q_1^2 = 0 (solid black curve), Q 1 2 =m 2 Q_1^2 = m^2 (short-dashed red curve), Q 1 2 =5m 2 Q_1^2 = 5 m^2 (long-dashed blue curve).Figure: The γ * γ→π + π - \\gamma ^\\ast \\gamma \\rightarrow \\pi ^+ \\pi ^- tree level result (scalar QED) for the integrandII in the sum rule of Eq.",
"() multiplied by ss, with II given by Eq.",
"(),where one of the photons is quasi-real.",
"The different curves are for different virtualities for the other photon :Q 1 2 =0Q_1^2 = 0 (solid black curve), Q 1 2 =m 2 Q_1^2 = m^2 (short-dashed red curve), Q 1 2 =5m 2 Q_1^2 = 5 m^2 (long-dashed blue curve).Besides exactly verifying the sum rules which integrate to zero, we can also use the above derived sum rules to study the low-energy coefficients for light-by-light scattering in scalar QED.",
"Using Eqs.",
"(REF , ), we obtain for the tree-level contributions to the lowest order coefficients $c_1$ and $c_2$ in scalar QED: $c_1 = \\frac{\\alpha ^2}{m^4} \\frac{7}{1440},\\quad \\quad \\quad c_2 = \\frac{\\alpha ^2}{m^4} \\frac{1}{1440}.$"
],
[
"Spinor QED",
"The response functions for the case of spinor QED at lowest order in the electromagnetic coupling can be found in Appendix REF .",
"We again study the three sum rules of Eqs.",
"(REF , , ) for the case of one real or quasi-real photon ($Q_2^2 \\rightarrow 0$ ) for different space-like virtualities of the other photon.",
"As the tree level contribution to $\\tau _{TL}$ in spinor QED vanishes for one quasi-real photon, one notices that the sum rule of Eq.",
"() is trivially satisfied.",
"For the sum rules involving the helicity difference of Eq.",
"(REF ), and involving the integrand $I$ of Eq.",
"(REF ), we show the corresponding integrands multiplied by $s$ in Figs.",
"REF , REF for the case of one real or quasi-real photon and for different virtualities of the other photon.",
"We again verify that the sum rules involve an exact cancellation between low and high energy contributions.",
"Figure: The γ * γ→e + e - \\gamma ^\\ast \\gamma \\rightarrow e^+ e^- tree level result (spinor QED) for theintegrand in the Δσ≡σ 2 -σ 0 \\Delta \\sigma \\equiv \\sigma _2 - \\sigma _0 sum rule of Eq.",
"(),multiplied by ss, where one of the photons is real.The different curves are for different virtualities for the other photon :Q 1 2 =0Q_1^2 = 0 (solid black curve), Q 1 2 =m 2 Q_1^2 = m^2 (short-dashed red curve), Q 1 2 =5m 2 Q_1^2 = 5 m^2 (long-dashed blue curve).Figure: The γ * γ→e + e - \\gamma ^\\ast \\gamma \\rightarrow e^+ e^- tree level result (spinor QED) for the integrandII in the sum rule of Eq.",
"() multiplied by ss, with II given by Eq.",
"(),where one of the photons is quasi-real.",
"The different curves are for different virtualities for the other photon :Q 1 2 =0Q_1^2 = 0 (solid black curve), Q 1 2 =m 2 Q_1^2 = m^2 (short-dashed red curve), Q 1 2 =5m 2 Q_1^2 = 5 m^2 (long-dashed blue curve).Using Eqs.",
"(REF , ), we obtain for the tree-level contributions to the lowest order coefficients $c_1$ and $c_2$ for light-by-light scattering in spinor QED : $c_1 = \\frac{\\alpha ^2}{m^4} \\frac{1}{90},\\quad \\quad \\quad c_2 = \\frac{\\alpha ^2}{m^4} \\frac{7}{360}.$ In these case we also were able to verify the sum rule in Eq.",
"(REF ), yielding $c_4 = - \\frac{\\alpha ^2}{m^6} \\frac{1}{315},$ in agreement with the result obtained in Ref.",
"[16] for the low-energy photon-photon scattering.",
"A more detailed study of the LbL sum rules in field theory, including loop effects, production of vector bosons, etc., is the subject of our forthcoming publication [17]."
],
[
"Meson production in $\\gamma \\gamma $ collision",
"In the previous section, the sum rules of Eqs.",
"(REF , , ) integrating to zero have been shown to hold exactly in perturbative calculations (e.g., in QED or QCD in the perturbative regime).",
"However as their derivation is general, their realization in QCD, in its non-perturbative regime, allows to gain insight in the $\\gamma ^\\ast \\gamma \\rightarrow {\\rm hadrons}$ cross-sections.",
"This was illustrated in Ref.",
"[13] for the sum rule of Eq.",
"(REF ).",
"In the remainder of this paper, we will elaborate on the discussion of Ref.",
"[13] and extend it to the other sum rules presented above.",
"The required non-perturbative input for the absorptive parts of the sum rules are the $\\gamma ^\\ast \\gamma \\rightarrow {\\rm hadrons}$ response functions.",
"In this paper, we will perform a first analysis by estimating the hadronic contributions to these response functions by the corresponding $\\gamma ^\\ast \\gamma ^\\ast \\rightarrow M$ (with $M$ a meson) production processes, which are described in terms of the $\\gamma ^\\ast \\gamma ^\\ast \\rightarrow M$ transition form factors.",
"In Appendix we detail the formalism and the available data for the $\\gamma ^\\ast \\gamma ^\\ast \\rightarrow M$ transition FFs, and successively discuss the $C$ -even pseudo-scalar ($J^{PC} = 0^{-+}$ ), scalar ($J^{PC} = 0^{++}$ ), axial-vector ($J^{PC} = 1^{++}$ ), and tensor ($J^{PC} = 2^{++}$ ) mesons."
],
[
"Real photons",
"We first consider the helicity sum rule of Eq.",
"(REF ) with two real photons producing a meson, as well as the sum rules of Eq.",
"() for the mesonic contributions to the low-energy constants $c_1$ and $c_2$ describing the forward light-by-light scattering amplitude.",
"When producing mesons, the sum rules will hold separately for states of given intrinsic quantum numbers.",
"Therefore, we will separately study the sum rule contributions for light quark isovector mesons (Table REF ), for light quark isoscalar mesons (Table REF ), as well as $c \\bar{c}$ mesons (Table REF ).",
"For the isoscalar mesons, one could in principle separate the contributions according to singlet or octet states (or alternatively according to $(u \\bar{u} + d \\bar{d})/\\sqrt{2}$ or $s \\bar{s}$ states).",
"The corresponding mesons involve mixings however which complicate such separation, as this mixing is not known well enough for some of the states.",
"We will postpone such a separation for a future work and add all isoscalar meson contributions in the present work.",
"The pseudo-scalar mesons contribute to the helicity-0 cross section only, given by Eq.",
"(REF ).",
"The corresponding contributions to the helicity sum rule of Eq.",
"(REF ) as well as the $c_1$ and $c_2$ sum rules are shown for the $\\pi ^0$ in Table REF , for the $\\eta , \\eta ^\\prime $ in Table REF , and for the $\\eta _c(1S)$ state in Table REF .",
"Besides the pseudo-scalar mesons, also scalar mesons can only contribute to $\\sigma _0$ .",
"We show the contributions of the $a_0(980)$ in Table REF , for the $f_0(980)$ and $f_0^\\prime (1370)$ in Table REF , and for the $\\chi _{c0}(1P)$ state in Table REF .",
"For the scalar mesons, only the $f_0^\\prime (1370)$ state gives a sizable contribution due to its large $2 \\gamma $ decay width.",
"For the helicity sum rule, one notices that in order to compensate the large negative contribution from the pseudo-scalar mesons, and to lesser extent from the scalar meson states, an equal strength is required in the helicity-2 cross section, $\\sigma _2$ .",
"For light quark mesons, the dominant feature of the helicity-2 cross section in the resonance region arises from the multiplet of tensor mesons $f_2(1270)$ , $a_2(1320)$ , and $f_2^\\prime (1525)$ .",
"For $c \\bar{c}$ tensor mesons, the dominant tensor contribution is given by the $\\chi _{c2}(1P)$ state.",
"Measurements at various $e^+ e^-$ colliders, notably the recent high statistics measurements by the BELLE Collaboration of the $\\gamma \\gamma $ cross sections to $\\pi ^+ \\pi ^-$ , $\\pi ^0 \\pi ^0$ , $\\eta \\pi ^0$ , and $K^+ K^-$ channels [19], [20], [21] have allowed to accurately establish their parameters.",
"For the light quark mesons, the experimental analyses of decay angular distributions have found [22] that the tensor mesons are produced predominantly (around 95% or more) in a state of helicity $\\Lambda = 2$ .",
"We will therefore assume in all of the following analyses that $\\Gamma _{\\gamma \\gamma }\\left({\\cal T}(\\Lambda = 0) \\right) \\approx 0$ , and that $\\Gamma _{\\gamma \\gamma }\\left({\\cal T}(\\Lambda = 2) \\right) \\approx \\Gamma _{\\gamma \\gamma }({\\cal T})$ in Tables REF , REF , REF .",
"We show all tensor meson contributions to the helicity difference sum rule as well as the $c_1, c_2$ sum rules for which the $2 \\gamma $ decay widths are known.",
"For the isovector meson contributions to the helicity sum rule, shown in Table REF , we conclude that the lowest isovector tensor meson composed of light quarks, $a_2(1320)$ , compensates to around 70% the contribution of the $\\pi ^0$ , which is entirely governed by the chiral anomaly.",
"For the isoscalar states composed of light quarks, the cancellation is even more remarkable: the sum of $f_2(1270)$ and $f_2^\\prime (1525)$ , within the experimental accuracy, entirely compensates the combined contribution of the $\\eta $ and $\\eta ^\\prime $ mesons.",
"Table: γγ\\gamma \\gamma sum rule contributions of the light quark isovector mesons based on the present PDG values of the meson masses (m M m_M) and their 2γ2 \\gamma decay widths Γ γγ \\Gamma _{\\gamma \\gamma }.",
"Fourth column: σ 2 -σ 0 \\sigma _2 - \\sigma _0 sum rule of Eq.",
"().Fifth, sixth columns: c 1 ,c 2 c_1, c_2 sum rules of Eqs.",
"(, ) respectively.Table: γγ\\gamma \\gamma sum rule contributions of the light quark isoscalar mesons based on the present PDG values of the meson masses (m M m_M) and their 2γ2 \\gamma decay widths Γ γγ \\Gamma _{\\gamma \\gamma }.",
"Fourth column: σ 2 -σ 0 \\sigma _2 - \\sigma _0 sum rule of Eq.",
"().Fifth, sixth columns: c 1 ,c 2 c_1, c_2 sum rules of Eqs.",
"(, ) respectively.Table: γγ\\gamma \\gamma sum rule contributions of the lowest cc ¯c \\bar{c} mesons based on the present PDG values of the meson masses (m M m_M) and their 2γ2 \\gamma decay widths Γ γγ \\Gamma _{\\gamma \\gamma }.Fourth column: the σ 2 -σ 0 \\sigma _2 - \\sigma _0 sum rule of Eq.",
"(), for which we also show the duality estimateof Eq.",
"()for the continuum contribution above DD ¯D \\bar{D} threshold, as well as the sum of resonances and continuumcontributions.Fifth, sixth columns: c 1 ,c 2 c_1, c_2 sum rules of Eqs.",
"(, ) respectively.For the $c \\bar{c}$ states, one notices that the known strength in the tensor channel from the $\\chi _{c2}(1P)$ state only compensates about 20% of the strength arising from the $\\eta _c(1S)$ and $\\chi _{c0}(1P)$ states.",
"We can however expect a sizable contribution to this sum rule from states above the nearby $D \\bar{D}$ threshold, which we denote by $s_D = 4 m_D^2 \\approx 14$ GeV$^2$ , using the $D$ -meson mass $m_D \\approx 1.87$ GeV.",
"So far, the helicity cross sections have not been measured above $D \\bar{D}$ threshold.",
"To estimate this continuum contribution to the helicity sum rule, which we denote by $I_{cont}$ , we use a quark-hadron duality argument [23] , which amounts to replacing the integral of the helicity difference cross section for the $\\gamma \\gamma \\rightarrow X$ process (with $X$ any hadronic final state containing charm quarks) by the corresponding integral of the helicity difference cross section for the perturbative $\\gamma \\gamma \\rightarrow c \\bar{c}$ process : $I_{cont} \\equiv \\int \\limits _{s_D}^{\\infty } \\, ds \\, \\frac{1}{s} \\left[ \\sigma _2 - \\sigma _0 \\right] (\\gamma \\gamma \\rightarrow X)\\approx \\int \\limits _{s_D}^{\\infty } \\, ds \\, \\frac{1}{s} \\left[ \\sigma _2 - \\sigma _0 \\right] (\\gamma \\gamma \\rightarrow c \\bar{c}),$ where the perturbative cross section is given in Appendix REF .",
"The duality expressed by the approximate equality in Eq.",
"(REF ) is meant to hold in a global sense, i.e.",
"after integration over the energy of the helicity difference cross section above the threshold $s_D$ .",
"As we have verified in Section that the perturbative cross section satisfies the helicity sum rule exactly, i.e.",
"$0 = \\int \\limits _{4 m_c^2}^{\\infty } \\, ds \\, \\frac{1}{s} \\left[ \\sigma _2 - \\sigma _0 \\right] (\\gamma \\gamma \\rightarrow c \\bar{c}),$ with $m_c$ the charm quark mass, we can re-express Eq.",
"(REF ) as : $I_{cont}\\approx - \\int \\limits _{4 m_c^2}^{s_D} \\, ds \\, \\frac{1}{s} \\left[ \\sigma _2 - \\sigma _0 \\right] (\\gamma \\gamma \\rightarrow c \\bar{c}).$ Using Eq.",
"(REF ) for the $\\gamma \\gamma \\rightarrow c \\bar{c}$ helicity difference cross section, we finally obtain: $I_{cont}\\approx - 8 \\pi \\, \\alpha ^2 \\, \\int \\limits _{4 m_c^2}^{s_D} \\, ds \\, \\frac{1}{s^2}\\left\\lbrace -3 \\, \\sqrt{1 - \\frac{4 m_c^2}{s}} \\,+ 2 \\, \\ln \\left( \\frac{\\sqrt{s}}{2 m_c} \\left[ 1 + \\sqrt{1 - \\frac{4 m_c^2}{s}} \\right] \\right)\\right\\rbrace .$ Using the PDG value $m_c \\approx 1.27$ GeV [18], we show the duality estimate for $-I_{cont}$ in Fig.",
"REF , as function of the integration limit $s_D$ (solid red curve).",
"Using the physical value of the $D \\bar{D}$ threshold, $s_D \\approx 14$ GeV$^2$ , we obtain: $I_{cont} \\approx 15.1$ nb.",
"We notice that within the experimental uncertainty, this fully cancels the sum of the $\\eta _c(1S), \\chi _{c0}(1P)$ , and $\\chi _{c2}(1P)$ resonance contributions to the $\\sigma _2 - \\sigma _0$ sum rule, as is shown in Table REF .",
"This cancellation quantitatively illustrates the interplay between resonances with hidden charm ($c \\bar{c}$ states) and production of charmed mesons in order to satisfy the sum rule.",
"It will be interesting to further test this experimentally by measuring the $\\gamma \\gamma $ production cross sections above $D \\bar{D}$ threshold, where a plethora of new states (so-called $XYZ$ states) have been found in recent years, see e.g.",
"Ref.",
"[24] for a review.",
"Figure: Solid (red) curve:duality estimate for the negative of the continuum contribution of Eq.",
"() to the helicity difference sum rule forcharm quarks as function of the integration limit s D s_D, which represents the threshold for charmed meson production(DD ¯D \\bar{D} threshold).",
"For reference, the dashed (blue) horizontal curve indicates the sum of theη c (1S),χ c0 (1P)\\eta _c(1S), \\chi _{c0}(1P), and χ c2 (1P)\\chi _{c2}(1P) resonance contributions to the σ 2 -σ 0 \\sigma _2 - \\sigma _0 sum rule,as listed in Table .The intersection between both curves near the physical DD ¯D \\bar{D} threshold, s D ≈14s_D \\approx 14 GeV 2 ^2 indicates a perfect cancellation between these resonance contributions and the duality estimate for the continuum contribution.We have also computed the meson contributions to the forward light-by-light scattering coefficients $c_1$ and $c_2$ (fifth and sixth columns respectively in Tables REF , REF , REF ).",
"The dimensionality of these coefficients requires them to scale with the meson mass $m_M$ as $1/m_M^4$ .",
"Therefore, the higher mass mesons contribute very insignificantly to these coefficients.",
"One notes that the coefficient $c_1$ , which involves the cross section $\\sigma _\\parallel $ , does not receive any contributions from pseudo-scalar mesons, and is dominated by the tensor mesons $a_2(1320)$ and $f_2(1270)$ , with smaller contributions from the scalar states around 1 GeV.",
"On the other hand, the coefficient $c_2$ , which involves the cross section $\\sigma _\\perp $ , is totally dominated by the contributions from pseudo-scalar mesons, especially the light $\\pi ^0$ , with contributions of $\\eta $ and $\\eta ^\\prime $ at the 10% level of the $\\pi ^0$ contribution."
],
[
"Virtual photons",
"We next discuss the sum rule of Eq.",
"() when both photons are quasi-real.",
"One immediately observes that pseudo-scalar mesons do not contribute to this sum rule.",
"However scalar, axial-vector and tensor mesons will contribute to this sum rule.",
"The sum rule will therefore require a cancellation mechanisms between scalar, axial-vector and tensor mesons, which we will study subsequently.",
"According to Eq.",
"(REF ), scalar mesons (with mass $m_S$ ) can only contribute to the $\\sigma _\\parallel $ term in the sum rule, and their contribution is given by: $\\int ds \\, \\frac{1}{s^2} \\left[ \\sigma _\\parallel \\right]_{Q_1^2 = Q_2^2 = 0}= 16 \\pi ^2 \\, \\frac{\\Gamma _{\\gamma \\gamma } ({\\cal S})}{m_S^5}.$ In contrast, Eq.",
"(REF ) shows that axial-vector mesons (with mass $m_A$ ) can only contribute to the $\\tau ^a_{TL}$ term in the sum rule as: $\\int ds \\, \\frac{1}{s} \\left[\\frac{\\tau ^a_{TL}}{Q_1 Q_2} \\right]_{Q_1^2 = Q_2^2 = 0} = - 8 \\pi ^2 \\,\\frac{3 \\, \\tilde{\\Gamma }_{\\gamma \\gamma } ({\\cal A})}{m_A^5},$ where we introduced the equivalent $2 \\gamma $ decay width $\\tilde{\\Gamma }_{\\gamma \\gamma } ({\\cal A})$ of Eq.",
"(REF ).",
"The tensor mesons in general contribute to both terms of the sum rule of Eq. ().",
"For the $\\sigma _\\parallel $ contribution, we will use the experimental observation that light tensor mesons are produced predominantly (around 95 % or more) in a state of helicity $\\Lambda = 2$ , as discussed above.",
"Neglecting therefore the much smaller $\\sigma _0$ term, we obtain from Eq.",
"(REF ): $\\int ds \\, \\frac{1}{s^2} \\left[ \\sigma _\\parallel \\right]_{Q_1^2 = Q_2^2 = 0}= \\int ds \\, \\frac{1}{s^2} \\frac{1}{2} \\left[ \\sigma _2 \\right]_{Q_1^2 = Q_2^2 = 0}= 8 \\pi ^2 \\, \\frac{5 \\, \\Gamma _{\\gamma \\gamma } ({\\cal T})}{m_T^5},$ with tensor meson mass $m_T$ .",
"For the $\\tau ^a_{TL}$ contribution to the sum rule of Eq.",
"(), one sees from Eq.",
"(REF ) that it involves a helicity-1 amplitude for tensor meson production by quasi-real photons, which unfortunately is not known experimentally for any tensor meson.",
"It is reasonable to assume that for quasi-real photons this amplitude is much smaller than the helicity-2 amplitude which is known to dominate in the real photon limit.",
"We will therefore neglect the helicity-1 contribution in the following analysis.",
"One notes from Eqs.",
"(REF , REF , REF ) that only axial-vector mesons give a negative contribution to the sum rule of Eq.",
"(), whereas scalar and tensor mesons contribute positively.",
"As the sum rule has to integrate to zero, one therefore obtains a cancellation mechanism between axial-vector mesons on one hand, and scalar and tensor mesons on the other.",
"In Table REF , we show the contributions of the lowest lying scalar, axial-vector and tensor mesons, for which the $2 \\gamma $ widths are known experimentally.",
"One sees from Table REF that the two lowest lying axial-vector mesons $f_1 (1285)$ and $f_1(1420)$ are entirely cancelled, within error bars, by the contribution of the dominant tensor meson $f_2 (1270)$ .",
"Using the experimentally known $2 \\gamma $ widths, the deviation of the (zero) sum rule value is at the $2 \\sigma $ level, which hints at a moderate contribution of either another higher mass axial-vector meson state or a non-resonant contribution with axial-vector quantum numbers.",
"Table: Light isoscalar meson contributions to the sum rule of Eq.",
"() based on the present PDG values of the meson masses (m M m_M) and their 2γ2 \\gamma decay widths Γ γγ \\Gamma _{\\gamma \\gamma }.For the axial-vector mesons, we quote the equivalent 2γ2 \\gamma decay width Γ ˜ γγ \\tilde{\\Gamma }_{\\gamma \\gamma } ofTable .Fourth column: σ ∥ \\sigma _\\parallel contribution,fifth column: τ TL a \\tau ^a_{TL} contribution,sixth column: total contribution to the sum rule of Eq.",
"().At finite $Q_1^2$ , for $Q_2^2 = 0$ , the three sum rules of Eqs.",
"(REF , , ) imply relations between the transition form factors for the contributing mesons.",
"To date, experimental results for the $\\gamma ^\\ast \\gamma \\rightarrow {\\rm meson}$ FFs only exist for the pseudo-scalar mesons $\\pi ^0, \\eta , \\eta ^\\prime $ , and $\\eta _c(1S)$ , as well as for the axial-vector mesons $f_1(1285)$ , and $f_1(1420)$ .",
"For other mesons, in particular the tensor mesons, the corresponding form factors still wait to be extracted.",
"We have seen from Table REF that for real photons the dominant contributions to the helicity sum rule of Eq.",
"(REF ) come from $\\eta , \\eta ^\\prime $ , and $f_2(1270)$ mesons, where the $f_2(1270)$ contribution cancels to 90% the contribution from the $\\eta $ and $\\eta ^\\prime $ mesons.",
"We will therefore use the corresponding sum rule of Eq.",
"(REF ) at finite $Q_1^2$ to estimate the $\\gamma ^\\ast \\gamma \\rightarrow f_2(1270)$ helicity-2 FF from the measured $\\eta $ and $\\eta ^\\prime $ FFs, given by Eq.",
"(REF ).",
"Assuming that the helicity sum rule of Eq.",
"(REF ) is saturated by the $\\eta $ , $\\eta ^\\prime $ , and $f_2(1270)$ mesons, we then obtain: $\\frac{5 \\, \\Gamma _{\\gamma \\gamma }(f_2)}{m^3_{f_2}} \\, \\left[ \\frac{T^{(2)}_{f_2}(Q_1^2, 0)}{T^{(2)}_{f_2}(0, 0)} \\right]^2\\simeq c_\\eta \\, \\frac{1}{\\left( 1 + Q_1^2 / \\Lambda ^2_\\eta \\right)^2}+ c_{\\eta ^\\prime } \\, \\frac{1}{\\left( 1 + Q_1^2 / \\Lambda ^2_{\\eta ^\\prime } \\right)^2} ,$ where we have introduced the shorthand notation: $c_P \\equiv \\frac{\\Gamma _{\\gamma \\gamma }({\\cal P})}{m_P^3}.$ For $Q_1^2 = 0$ , the $f_2(1270)$ meson contribution cancels to 90% the $\\eta + \\eta ^\\prime $ contributions to the helicity sum rule.",
"We can therefore use $\\frac{5 \\, \\Gamma _{\\gamma \\gamma }(f_2)}{m^3_{f_2}} \\simeq c_\\eta + c_\\eta ^\\prime ,$ which allows us to express Eq.",
"(REF ) as: $\\frac{T^{(2)}_{f_2}(Q_1^2,0)}{T^{(2)}_{f_2}(0,0)} \\simeq \\left[ \\frac{c_\\eta }{c_\\eta + c_{\\eta ^\\prime }} \\, \\frac{1}{\\left( 1 + Q_1^2 / \\Lambda ^2_\\eta \\right)^2}+ \\frac{c_{\\eta ^\\prime }}{c_\\eta + c_{\\eta ^\\prime }} \\, \\frac{1}{\\left( 1 + Q_1^2 / \\Lambda ^2_{\\eta ^\\prime } \\right)^2}\\right]^{1/2}.$ We can obtain a second estimate for the $T^{(2)}$ FF for the $f_2(1270)$ meson from the sum rule of Eq. ().",
"We have seen from Table REF that for quasi-real photons the dominant contributions to this sum rule come from $f_1(1285), f_1(1420)$ , and $f_2(1270)$ mesons, where the $f_2(1270)$ contribution cancels to 95 % the contribution from the $f_1(1285)$ and $f_1(1420)$ mesons.",
"We can then also use the corresponding sum rule of Eq.",
"() at finite $Q_1^2$ to estimate the $\\gamma ^\\ast \\gamma \\rightarrow f_2(1270)$ helicity-2 FF from the measured $f_1(1285)$ and $f_1(1420)$ FFs, using Eqs.",
"(REF , REF ).",
"Assuming that the helicity sum rule of Eq.",
"() is saturated by the $f_1(1285)$ , $f_1(1420)$ , and $f_2(1270)$ mesons, which we denote by $f_1, f_1^\\prime $ , and $f_2$ respectively, and retaining only the supposedly dominant $\\Lambda = 2$ FF for the tensor mesons, we obtain: $\\frac{5 \\, \\Gamma _{\\gamma \\gamma }(f_2)}{m^5_{f_2}} \\, \\frac{1}{\\left( 1 + \\frac{Q_1^2}{m^2_{f_2}} \\right)} \\,\\left[ \\frac{T^{(2)}_{f_2}(Q_1^2, 0)}{T^{(2)}_{f_2}(0, 0)} \\right]^2\\simeq c_{f_1} \\, \\frac{1}{\\left( 1 + Q_1^2 / \\Lambda ^2_{f_1} \\right)^4}+ c_{f_1^\\prime } \\, \\frac{1}{\\left( 1 + Q_1^2 / \\Lambda ^2_{f_1^\\prime } \\right)^4} ,$ where $c_A \\equiv \\frac{3 \\, \\tilde{\\Gamma }_{\\gamma \\gamma }({\\cal A})}{m_A^5}.$ For $Q_1^2 = 0$ , the $f_2(1270)$ meson contribution cancels to 95% the $f_1(1285) + f_1(1420)$ contributions to the sum rule of Eq.",
"(), which implies: $\\frac{5 \\, \\Gamma _{\\gamma \\gamma }(f_2)}{m^5_{f_2}} \\simeq c_{f_1} + c_{f_1^\\prime }.$ This allows to obtain a second estimate for the $T^{(2)}$ FF for the $f_2(1270)$ meson as: $\\frac{T^{(2)}_{f_2}(Q_1^2,0)}{T^{(2)}_{f_2}(0,0)} \\simeq \\left( 1 + \\frac{Q_1^2}{m^2_{f_2}} \\right)^{1/2} \\,\\left[ \\frac{c_{f_1}}{c_{f_1} + c_{f_1^\\prime }} \\, \\frac{1}{\\left( 1 + Q_1^2 / \\Lambda ^2_{f_1} \\right)^4}+ \\frac{c_{f_1^\\prime }}{c_{f_1} + c_{f_1^\\prime }} \\, \\frac{1}{\\left( 1 + Q_1^2 / \\Lambda ^2_{f^\\prime _1} \\right)^4}\\right]^{1/2}.$ In Fig.",
"REF we show the two sum rule estimates of Eqs.",
"(REF ) and (REF ) for the FF $T^{(2)}$ for the tensor meson $f_2(1270)$ using the known experimental information for either $\\eta , \\eta ^\\prime $ in Eq.",
"(REF ), or $f_1(1285), f_1(1420)$ in Eq.",
"(REF ).",
"When taking the ratio of both estimates, one sees that it is larger than 80% below 1 GeV$^2$ and around 65% around $Q^2 = 2$ GeV$^2$ .",
"It will be interesting to confront these estimates with a direct measurement of the $T^{(2)}$ FF for the $f_2(1270)$ tensor meson.",
"Figure: Sum rule estimates for the form factor T (2) (Q 2 ,0)/T (2) (0,0)T^{(2)}(Q^2,0) / T^{(2)}(0,0) with helicity Λ=2\\Lambda = 2 for the tensor meson f 2 (1270)f_2(1270).Red solid curve: sum rule estimate from Eq.",
"(), using the experimental input from the η\\eta and η ' \\eta ^\\prime FFs.Blue dashed curve: sum rule estimate from Eq.",
"(), using the experimental input from the f 1 (1285)f_1(1285) and f 1 (1420)f_1(1420) FFs.In an analogous way, we can provide an estimate for the $a_2(1320)$ FF from the $\\pi ^0$ FF.",
"We have seen from Table REF that $\\pi ^0$ and $a_2(1320)$ provide the dominant isovector contributions to the helicity sum rule of Eq.",
"(REF ), where the $a_2(1320)$ contribution cancels to 70% the contribution from the $\\pi ^0$ .",
"We can therefore use the sum rule of Eq.",
"(REF ) for one virtual photon to estimate the helicity-two FF $T^{(2)}$ for the $a_2(1320)$ meson in terms of the $\\pi ^0$ FF, given by Eq.",
"(REF ), as: $\\frac{T^{(2)}_{a_2}(Q_1^2,0)}{T^{(2)}_{a_2}(0,0)} \\simeq \\frac{1}{\\left( 1 + Q_1^2 / \\Lambda ^2_\\pi \\right)} .$ As empirically the $\\gamma ^\\ast \\gamma \\rightarrow \\pi ^0$ FF is the best known meson transition FF, it will be interesting to test the above prediction for the $a_2(1320)$ FF experimentally."
],
[
"Conclusions and outlook",
"We have studied the forward light-by-light scattering and derived three sum rules which involve energy weighted integrals of $\\gamma ^\\ast \\gamma $ fusion cross sections, measurable at $e^+ e^-$ colliders, which integrate to zero (super-convergence relations): $0 &=& \\int \\limits _{s_0}^\\infty d s \\frac{1}{(s + Q_1^2)} \\left[ \\sigma _0 - \\sigma _2 \\right]_{Q_2^2 = 0}, \\nonumber \\\\0 &=& \\int \\limits _{s_0}^\\infty d s \\, \\frac{1}{(s + Q_1^2)^2}\\left[ \\sigma _\\parallel + \\sigma _{LT} + \\frac{(s + Q_1^2)}{Q_1 Q_2} \\tau ^a_{TL}\\right]_{Q_2^2 = 0},\\nonumber \\\\0 &=& \\int \\limits _{s_0}^\\infty d s \\, \\left[ \\frac{\\tau _{TL} }{Q_1 Q_2}\\right]_{Q_2^2 = 0}.",
"\\nonumber $ In these sum rules the $\\gamma ^\\ast \\gamma $ fusion cross sections are for one (quasi-) real photon and a second virtual photon which can have arbitrary (space-like) virtuality.",
"The first of the sum rules generalizes the GDH sum rule for the helicity-difference $\\gamma \\gamma $ fusion cross section to the case of one real and one virtual photon.",
"The two further sum rules are for $\\gamma ^\\ast \\gamma $ fusion cross sections which involve longitudinal photon amplitudes.",
"We have shown that these sum rules are exactly verified for the tree level scalar and spinor QED cross sections.",
"Verifications beyond the tree-level in various field theories are underway [17].",
"We have performed a detailed quantitative study of the new sum rules for the case of the production of light quark mesons as well as for the production of mesons in the charm quark sector.",
"Using the empirical information in evaluating the sum rules, we have found that the helicity-difference sum rule requires cancellations between different mesons, implying non-perturbative relations.",
"For the light quark isovector mesons, the $\\pi ^0$ contribution was found to be compensated to around 70% by the contribution of the lowest lying isovector tensor meson $a_2(1320)$ .",
"For the isoscalar light quark mesons, the $\\eta $ and $\\eta ^\\prime $ contributions were found to be entirely compensated within the experimental accuracy by the two lowest-lying tensor mesons $f_2(1270)$ and $f_2^\\prime (1525)$ .",
"In the charm quark sector, the situation is different as it involves the narrow resonance contributions below $D \\bar{D}$ threshold, and the continuum contribution above $D \\bar{D}$ threshold.",
"For the narrow resonances, the $\\eta _c$ was found to give by far the dominant contribution.",
"When using a duality estimate for the continuum contribution, we found that it entirely cancels the narrow resonance contributions, verifying the sum rule, and pointing to large tensor strength (helicity 2) in the cross sections above $D \\bar{D}$ threshold.",
"It will be interesting to test this property experimentally.",
"The helicity difference sum rule has also been applied for the case of one real and one virtual photon.",
"In this case the $\\gamma ^\\ast \\gamma $ fusion cross sections depend on the meson transition form factors (FFs).",
"We have reviewed the general formalism and parameterization for the $\\gamma ^\\ast \\gamma \\rightarrow {\\rm meson}$ transition FFs for (pseudo-) scalar, axial-vector, and tensor mesons.",
"Because for scalar and tensor mesons the $\\gamma ^\\ast \\gamma $ transition FFs have not yet been measured, a direct test of the sum rules for finite virtuality is not possible at present.",
"However, we were able to show that the helicity-difference sum rule allows to provide an estimate for the $f_2(1270)$ tensor FF in terms of the $\\eta $ , and $\\eta ^\\prime $ FFs, and for the $a_2(1320)$ tensor FF in terms of the $\\pi ^0$ FF.",
"Since empirical information on pseudo-scalar meson FFs is available, these relations provide predictions for tensor meson FFs which will be interesting to confront with experiment.",
"The second new sum rule derived in this paper, involving the $\\sigma _\\parallel , \\sigma _{LT}$ , and $\\tau ^a_{TL}$ $\\gamma ^\\ast \\gamma $ response functions, has also been tested for the case of quasi-real photons.",
"As pseudo-scalar mesons cannot contribute to this sum rule, a cancellation between scalar and tensor mesons on one hand and axial-vector mesons on the other hand is at work.",
"Using the existing empirical information for quasi-real photons, the contribution of the two lowest lying axial-vector mesons $f_1 (1285)$ and $f_1(1420)$ was found to be entirely cancelled, within error bars, by the contribution of the dominant tensor meson $f_2 (1270)$ .",
"When applying this sum rule to the case of one virtual photon, it again allows one to relate the $f_2(1270)$ tensor FF, this time to the transition FFs for the $f_1 (1285)$ and $f_1(1420)$ mesons, which have both been measured.",
"The predictions from the two different sum rules for the $f_2(1270)$ FF were found to agree within 20% for a virtuality below 1 GeV$^2$ , and within 35% up to about 2 GeV$^2$ .",
"Besides the three super-convergence relations, we have also derived sum rules which express the coefficients in a low-energy expansion of the forward light-by-light scattering amplitude in terms of $\\gamma ^\\ast \\gamma \\rightarrow X$ cross sections.",
"These evaluations may be used as a cross-check for models of the non-forward light-by-light scattering which are applied to evaluate the hadronic LbL contribution to $(g - 2)_\\mu $ .",
"On the experimental side, the ongoing $\\gamma \\gamma $ physics programs by the BABAR and Belle Collaborations, as well as the upcoming $\\gamma \\gamma $ physics program by the BES-III Collaboration, will allow to further improve the data situation significantly.",
"In particular, the extraction of the $\\gamma ^\\ast \\gamma $ response functions through their different azimuthal angular dependencies, and the measurements of multi-meson final states ($\\pi \\pi $ , $\\pi \\eta , \\ldots $ ) promise to access besides the pseudo-scalar meson FFs also the scalar, axial-vector and tensor meson FFs, thus allowing direct tests of the sum rule predictions presented in this work."
],
[
"Acknowledgements",
"This work was supported by the Deutsche Forschungsgemeinschaft DFG through the Collaborative Research Center “The Low-Energy Frontier of the Standard Model\" (SFB 1044).",
"Furthermore, the work of V. Pauk is also supported by the graduate school Graduate School “Symmetry Breaking in Fundamental Interactions\" (DFG/GRK 1581)."
],
[
"Kinematics and cross sections of the $e^\\pm + e^- \\rightarrow e^\\pm + e^- + X$ process",
"The kinematics of the process $e (p_1) + e (p_2) \\rightarrow e (p^\\prime _1) + e (p^\\prime _2) + X$ , with $X$ the produced hadronic state, in the lepton c.m.",
"system, i.e.",
"the c.m.",
"system of the colliding beams (which we denote by c.m.",
"ee) is characterized by the four-vectors of the incoming leptons : $p_1 (E, \\vec{p}_1), \\quad \\quad \\quad p_2 (E, - \\vec{p}_1),$ with beam energy $E = \\sqrt{s} / 2$ , and $s = (p_1 + p_2)^2$ .",
"The kinematics of the outgoing leptons can be related to the virtual photon four-momenta as : $q_1 = p_1 - p^\\prime _1, \\quad \\quad \\quad q_2 = p_2 - p^\\prime _2 .$ The kinematics of the outgoing leptons then determines five kinematical quantities : the energies of both virtual photons : $\\omega _1 \\equiv q_1^0 = E - E^\\prime _1, \\quad \\quad \\quad \\omega _2 = q_2^0 \\equiv E - E^\\prime _2,$ with $E^\\prime _1$ and $E^\\prime _2$ the energies of both outgoing leptons; the virtualities of both virtual photons : $Q_1^2 \\equiv - q_1^2 = 4 E E^\\prime _1 \\sin ^2 \\theta _1 / 2 + Q_{1, \\, min}^2 \\; ,\\quad \\quad \\quad Q_2^2 \\equiv - q_2^2 = 4 E E^\\prime _2 \\sin ^2 \\theta _2 / 2 + Q_{2, \\, min}^2 \\; ,$ where $\\theta _1$ and $\\theta _2$ are the (polar) angles of the scattered electrons relative to the respective beam directions, and where the minimal values of the virtualities are given by (in the limit where $E^\\prime _1 > > m$ and $E^\\prime _2 > > m$ , with $m$ the lepton mass) : $Q_{1, \\, min}^2 \\simeq m^2 \\frac{\\omega _1^2}{E E^\\prime _1},\\quad \\quad \\quad Q_{2, \\, min}^2 \\simeq m^2 \\frac{\\omega _2^2}{E E^\\prime _2};$ the azimuthal angle $\\phi $ between both lepton planes, which in the lepton c.m.",
"frame can be obtained as : $\\bigl ( \\cos \\phi \\bigr )_{c.m.",
"e e} \\equiv - \\frac{p^\\prime _{1 \\perp } \\cdot p^\\prime _{2 \\perp }}{\\left[(p^\\prime _{1 \\perp })^2 \\; (p^\\prime _{2 \\perp })^2 \\right]^{1/2}},$ where $p^\\prime _{1 \\perp }$ and $p^\\prime _{2 \\perp }$ denote the components of the outgoing lepton four-vectors which are perpendicular to the respective beam directions, and are defined in the lepton c.m.",
"frame as : $\\left( p^\\prime _{1 \\perp } \\right)^\\mu = - R^{\\mu \\nu } (p_1, p_2) \\, \\left( p^\\prime _1 \\right)_\\nu ,\\quad \\quad \\quad \\left( p^\\prime _{2 \\perp } \\right)^\\mu = - R^{\\mu \\nu } (p_1, p_2) \\, \\left( p^\\prime _2 \\right)_\\nu ,$ with $R^{\\mu \\nu } (p_1, p_2) = - g^{\\mu \\nu } + \\frac{1}{\\left[ (p_1 \\cdot p_2)^2 - m^4 \\right]} \\;\\bigl \\lbrace (p_1 \\cdot p_2) \\left( p_1^\\mu \\, p_2^\\nu + p_2^\\mu \\, p_1^\\nu \\right)- m^2 \\left( p_1^\\mu \\, p_1^\\nu + p_2^\\mu \\, p_2^\\nu \\right)\\bigr \\rbrace .$ In the following it will also turn out to be useful to determine kinematical quantities in the c.m.",
"system of the virtual photons ( which we denote by c.m.",
"$\\gamma \\gamma $).",
"In particular, the azimuthal angle between both lepton planes, in the $\\gamma \\gamma $ c.m.",
"frame, which we denote by $\\tilde{\\phi }$ is given by : $\\cos \\tilde{\\phi }\\equiv - \\frac{\\tilde{p}_{1 \\perp } \\cdot \\tilde{p}_{2 \\perp }}{\\left[(\\tilde{p}_{1 \\perp })^2 \\; (\\tilde{p}_{2 \\perp })^2 \\right]^{1/2}},$ where $\\tilde{p}_{1 \\perp }$ and $\\tilde{p}_{2 \\perp }$ denote the transverse components of the incoming lepton four-vectors in the $\\gamma \\gamma $ c.m.",
"frame and are defined in a covariant way as : $\\left( \\tilde{p}_{1 \\perp } \\right)^\\mu = - R^{\\mu \\nu } (q_1, q_2) \\, \\left( p_1 \\right)_\\nu ,\\quad \\quad \\quad \\left(\\tilde{p}_{2 \\perp } \\right)^\\mu = - R^{\\mu \\nu } (q_1, q_2) \\, \\left( p_2 \\right)_\\nu ,$ with $R^{\\mu \\nu } (q_1, q_2) = - g^{\\mu \\nu } + \\frac{1}{\\left[ (q_1 \\cdot q_2)^2 - q_1^2 q_2^2 \\right]} \\;\\bigl \\lbrace (q_1 \\cdot q_2) \\left( q_1^\\mu \\, q_2^\\nu + q_2^\\mu \\, q_1^\\nu \\right)- q_1^2 \\, q_2^\\mu \\, q_2^\\nu - q_2^2 \\, q_1^\\mu \\, q_1^\\nu \\bigr \\rbrace .$ As the rhs of Eq.",
"(REF ) is expressed in a Lorentz invariant way, one can then evaluate all four-momenta in the lepton c.m.",
"frame, to obtain the expression of $\\cos \\tilde{\\phi }$ in terms of the lepton c.m.",
"kinematics.",
"The cross section for the process $e (p_1) + e (p_2) \\rightarrow e (p^\\prime _1) + e (p^\\prime _2) + X$ , with $X$ the produced hadronic state, can be expressed in terms of eight cross sections for the $\\gamma ^\\ast \\gamma ^\\ast \\rightarrow X$ process, which where defined in Eq.",
"(REF ), as : $d \\sigma &=& \\frac{\\alpha ^2}{16 \\pi ^4 \\, Q_1^2 \\, Q_2^2} \\, \\frac{2 \\sqrt{X}}{s (1 - 4 m^2 / s)}\\cdot \\frac{d^3 \\vec{p}_1^{\\, \\prime }}{E_1^{\\prime }}\\cdot \\frac{d^3 \\vec{p}_2^{\\, \\prime }}{E_2^\\prime } \\nonumber \\\\&\\times & \\left\\lbrace 4 \\, \\rho _1^{++} \\, \\rho _2^{++} \\, \\sigma _{TT}+ \\rho _1^{00} \\, \\rho _2^{00} \\, \\sigma _{LL}+ 2 \\, \\rho _1^{++} \\, \\rho _2^{00} \\, \\sigma _{TL}+ 2 \\, \\rho _1^{00} \\, \\rho _2^{++} \\, \\sigma _{LT}\\right.",
"\\nonumber \\\\&&+ 2 \\, \\left( \\rho _1^{++} - 1 \\right) \\, \\left( \\rho _2^{++} - 1 \\right) \\, \\left( \\cos 2 \\tilde{\\phi }\\right) \\, \\tau _{TT}+ 8 \\, \\left[ \\frac{\\left( \\rho _1^{00} + 1 \\right) \\, \\left( \\rho _2^{00} + 1 \\right)}{\\left( \\rho _1^{++} - 1 \\right) \\, \\left( \\rho _2^{++} - 1 \\right)}\\right]^{1/2} \\, \\left( \\cos \\tilde{\\phi }\\right) \\, \\tau _{TL}\\nonumber \\\\&&\\left.+ h_1 h_2 \\, 4 \\left[ \\left( \\rho _1^{00} + 1 \\right) \\, \\left( \\rho _2^{00} + 1 \\right) \\right]^{1/2} \\, \\tau ^a_{TT}+ h_1 h_2 \\, 8 \\left[ \\left( \\rho _1^{++} - 1 \\right) \\, \\left( \\rho _2^{++} - 1 \\right) \\right]^{1/2} \\, \\left( \\cos \\tilde{\\phi }\\right) \\, \\tau ^a_{TL}\\right\\rbrace ,$ where $h_1 = \\pm 1$ and $h_2 = \\pm 1$ are both lepton beam helicities, and where we have defined kinematical coefficients : $\\rho _1^{++} &=& \\frac{1}{2} \\left\\lbrace 1 - \\frac{4 m^2}{Q_1^2} + \\frac{1}{X} \\left( 2 \\, p_1 \\cdot q_2 - \\nu \\right)^2 \\right\\rbrace \\, , \\nonumber \\\\\\rho _2^{++} &=& \\frac{1}{2} \\left\\lbrace 1 - \\frac{4 m^2}{Q_2^2} + \\frac{1}{X} \\left( 2 \\, p_2 \\cdot q_1 - \\nu \\right)^2 \\right\\rbrace \\, , \\nonumber \\\\\\rho _1^{00} &=& \\frac{1}{X} \\left( 2 \\, p_1 \\cdot q_2 - \\nu \\right)^2 - 1 \\, , \\nonumber \\\\\\rho _2^{00} &=& \\frac{1}{X} \\left( 2 \\, p_2 \\cdot q_1 - \\nu \\right)^2 - 1\\, .$"
],
[
"Scalar QED",
"The $\\gamma ^\\ast \\gamma ^\\ast \\rightarrow {\\cal S} \\bar{\\cal S}$ cross sections (with ${\\cal S}$ an electrically charged structureless scalar particle) to lowest order in $\\alpha $ are given by : $\\sigma _0 + \\sigma _2 &=& \\sigma _\\parallel + \\sigma _\\perp \\nonumber \\\\&=& \\alpha ^2 \\frac{\\pi }{2} \\frac{s^2 \\nu ^3}{X^3} \\left\\lbrace \\sqrt{a} \\left[ 2 - a - \\left( 1 - \\frac{2 X}{s \\nu } \\right)^2 \\right]- \\left( 1 - a \\right) \\left( 3 - \\frac{4 X}{s \\nu } + a \\right) L\\right\\rbrace , \\\\\\sigma _\\parallel - \\sigma _\\perp &=& \\alpha ^2 \\frac{\\pi }{4} \\frac{s^2 \\nu ^3}{X^3} \\left\\lbrace \\sqrt{a} \\left[ 1 - a + 2 \\left( 1 - \\frac{2 X}{s \\nu } \\right)^2 \\right] - \\left( 1 - a \\right)\\left( 3 - \\frac{8 X}{s \\nu } + a \\right) L\\right\\rbrace , \\\\\\sigma _0 - \\sigma _2&=& \\alpha ^2 2 \\pi \\frac{s \\nu ^2}{X^2} \\left\\lbrace - \\sqrt{a} \\left( 1 - \\frac{X}{s \\nu } \\right) + \\left( 1 - a \\right)L \\right\\rbrace , \\\\\\sigma _{LL}&=& \\alpha ^2 \\pi Q_1^2 Q_2^2 \\frac{s^2 \\nu }{X^3} \\left\\lbrace \\sqrt{a} \\left[ 2 + \\frac{1}{1 - a} \\left( 1 - \\frac{X}{s \\nu }\\right)^2 \\right] - \\left( 3 + \\frac{X}{s \\nu } \\right) \\left( 1 - \\frac{X}{s \\nu } \\right)L \\right\\rbrace , \\\\\\sigma _{LT}&=& \\alpha ^2 \\frac{\\pi }{2} Q_1^2 \\frac{s \\nu (\\nu - Q_2^2)^2}{X^3} \\left\\lbrace - 3 \\sqrt{a} + \\left( 3 - a \\right)L \\right\\rbrace , \\\\\\tau _{TL}&=& \\alpha ^2 \\frac{\\pi }{2} Q_1 Q_2 \\frac{s \\nu }{X^2} \\left\\lbrace - \\sqrt{a} + \\left( 1 - \\frac{2 X}{s \\nu } + a \\right)L \\right\\rbrace , \\\\\\tau ^a_{TL}&=& \\alpha ^2 \\frac{\\pi }{2} Q_1 Q_2 \\frac{s^2 \\nu ^2}{X^3} \\left\\lbrace \\sqrt{a} \\left(3 - \\frac{4 X}{s \\nu } \\right) - \\left[ 1 - a + 2 \\left( 1 - \\frac{X}{s \\nu } \\right)^2 \\right]L \\right\\rbrace ,$ with $L \\equiv \\ln \\left( \\frac{1 + \\sqrt{a} }{\\sqrt{1 - a } } \\right),\\quad \\quad \\quad a \\equiv \\frac{X}{\\nu ^2} \\left( 1 - \\frac{4 m^2}{s} \\right).$ In the limit where one of the virtual photons becomes real ($Q_2^2 = 0$ ) in case of the response functions involving only transverse photons, or becomes quasi-real ($Q_2^2 \\approx 0$ ) in case of the response functions involving a longitudinal photon, the above expressions simplify to : $\\left[ \\sigma _\\parallel + \\sigma _\\perp \\right]_{Q_2^2 = 0} &=& \\alpha ^2 4\\pi \\frac{s^2}{(s + Q_1^2)^3} \\left\\lbrace \\sqrt{1 - \\frac{4 m^2}{s} } \\left( 1 + \\frac{4 m^2}{s} + \\frac{Q_1^4}{s^2} \\right)- \\frac{8 m^2}{s} \\left( 1 - \\frac{2 m^2}{s} - \\frac{Q_1^2}{s} \\right) L\\right\\rbrace , \\\\\\left[ \\sigma _\\parallel - \\sigma _\\perp \\right]_{Q_2^2 = 0}&=& \\alpha ^2 4 \\pi \\frac{s^2}{(s + Q_1^2)^3} \\left\\lbrace \\sqrt{1 - \\frac{4 m^2}{s} } \\left( \\frac{2 m^2}{s} + \\frac{Q_1^4}{s^2} \\right)+ \\frac{8 m^2}{s} \\left( \\frac{m^2}{s} + \\frac{Q_1^2}{s} \\right) L\\right\\rbrace , \\\\\\left[ \\sigma _0 - \\sigma _2 \\right]_{Q_2^2 = 0}&=& \\alpha ^2 4 \\pi \\frac{s}{(s + Q_1^2)^2} \\left\\lbrace - \\sqrt{1 - \\frac{4 m^2}{s} } \\left( 1 - \\frac{Q_1^2}{s} \\right) + \\frac{8 m^2}{s}L \\right\\rbrace , \\\\\\left[ \\frac{1}{Q_1^2 Q_2^2} \\sigma _{LL} \\right]_{Q_2^2 = 0}&=& \\alpha ^2 8 \\pi \\frac{s^2}{(s + Q_1^2)^5} \\left\\lbrace \\sqrt{1 - \\frac{4 m^2}{s} } \\left( 8 + \\frac{s}{4 m^2} \\left( 1 - \\frac{Q_1^2}{s} \\right)^2 \\right) - \\left(7 + \\frac{Q_1^2}{s} \\right)\\left(1 - \\frac{Q_1^2}{s} \\right)L \\right\\rbrace , \\\\\\left[ \\frac{1}{Q_1^2} \\sigma _{LT} \\right]_{Q_2^2 = 0}&=& \\alpha ^2 4 \\pi \\frac{s}{(s + Q_1^2)^3} \\left\\lbrace - 3 \\sqrt{1 - \\frac{4 m^2}{s} } + 2 \\left( 1 + \\frac{2 m^2}{s} \\right)L \\right\\rbrace , \\\\\\left[ \\frac{1}{Q_1 Q_2} \\tau _{TL} \\right]_{Q_2^2 = 0}&=& \\alpha ^2 4 \\pi \\frac{s}{(s + Q_1^2)^3} \\left\\lbrace - \\sqrt{1 - \\frac{4 m^2}{s} } + \\left( 1 - \\frac{Q_1^2}{s} - \\frac{4 m^2}{s} \\right)L \\right\\rbrace , \\\\\\left[\\frac{1}{Q_1 Q_2} \\tau ^a_{TL} \\right]_{Q_2^2 = 0}&=& \\alpha ^2 8 \\pi \\frac{s^2}{(s + Q_1^2)^4} \\left\\lbrace \\sqrt{1 - \\frac{4 m^2}{s} } \\left( 1 - \\frac{2 Q_1^2}{s} \\right) - \\left[ \\frac{1}{2} \\left(1 - \\frac{Q_1^2}{s}\\right)^2 + \\frac{4 m^2}{s} \\right]L \\right\\rbrace ,$ with $\\left[ L \\right]_{Q_2^2 = 0} = \\ln \\left( \\frac{\\sqrt{s}}{2 m} \\left[ 1 + \\sqrt{1 - \\frac{4 m^2}{s}} \\right] \\right),$"
],
[
"Spinor QED",
"The $\\gamma ^\\ast \\gamma ^\\ast \\rightarrow q \\bar{q}$ cross sections (with $q$ an electrically charged structureless spin-1/2 particle) to lowest order in $\\alpha $ are given by : $\\sigma _0 + \\sigma _2 &=& \\sigma _\\parallel + \\sigma _\\perp \\nonumber \\\\&=& \\alpha ^2 \\pi \\frac{s^2 \\nu ^3}{X^3} \\left\\lbrace \\sqrt{a} \\left[ - 4 \\left( 1 - \\frac{X}{s \\nu } \\right)^2 - (1 - a)+ \\frac{ Q_1^2 Q_2^2}{\\nu ^2} \\left( 2 - \\frac{1}{(1 - a)} \\, \\frac{4 X^2}{s^2 \\nu ^2} \\right)\\right] \\right.",
"\\nonumber \\\\&&\\left.",
"\\hspace{42.67912pt} + \\left[ 3 - a^2 + 2 \\left( 1 - \\frac{2 X}{s \\nu } \\right)^2- \\frac{2 Q_1^2 Q_2^2}{\\nu ^2} (1 + a) \\right] \\, L\\right\\rbrace , \\\\\\sigma _\\parallel - \\sigma _\\perp &=& \\alpha ^2 \\frac{\\pi }{2} \\frac{s^2 \\nu ^3}{X^3} \\left\\lbrace \\sqrt{a} \\left[ - (1 - a) - 2 \\left( 1 - \\frac{2 X}{s \\nu } \\right)^2 \\right] \\right.",
"\\nonumber \\\\&&\\left.",
"\\hspace{42.67912pt} + \\left[ - (1 - a)^2 + 4 (1 - a) \\left( 1 - \\frac{2 X}{s \\nu } \\right)+ \\frac{ Q_1^2 Q_2^2}{\\nu ^2} \\, \\frac{8 X^2}{s^2 \\nu ^2}\\right] \\, L\\right\\rbrace , \\\\\\sigma _0 - \\sigma _2&=& \\alpha ^2 4 \\pi \\frac{s \\nu ^2}{X^2} \\left\\lbrace \\sqrt{a} \\left[ 2 - \\frac{X}{s \\nu }- \\frac{ Q_1^2 Q_2^2}{\\nu ^2} \\frac{1}{(1 - a)} \\, \\frac{X}{s \\nu } \\right]\\, - 2 \\left( 1 - \\frac{X}{s \\nu } \\right) \\, L\\right\\rbrace , \\\\\\sigma _{LL}&=& \\alpha ^2 2 \\pi Q_1^2 Q_2^2 \\frac{s^2}{\\nu X^2} \\left\\lbrace \\sqrt{a} \\left[ -2 - \\frac{(3 - 2 a)}{(1 - a)} \\frac{Q_1^2 Q_2^2}{X}\\right]+ \\left( 2 + \\frac{3 Q_1^2 Q_2^2}{X}\\right) \\, L\\right\\rbrace , \\\\\\sigma _{LT}&=& \\alpha ^2 \\pi Q_1^2 \\frac{s}{\\nu X^2} \\left\\lbrace \\sqrt{a} \\left[ (\\nu - Q_2^2)^2 \\left( 2 + \\frac{3 Q_1^2 Q_2^2}{X}\\right)- 2 \\nu Q_2^2 + Q_2^4 \\frac{(3 - a)}{(1 - a)}\\right] \\right.",
"\\nonumber \\\\&&\\left.\\hspace{42.67912pt}+\\left[ (\\nu - Q_2^2)^2 \\left( -2 (1 - a) - (3 - a) \\frac{Q_1^2 Q_2^2}{X} \\right)+ 2 \\nu Q_2^2 (1 + a) - Q_2^4 (3 + a)\\right] \\, L\\right\\rbrace , \\\\\\tau _{TL}&=& \\alpha ^2 2 \\pi \\left( Q_1 Q_2 \\right)^{3} \\frac{s}{\\nu X^2} \\left\\lbrace \\frac{\\sqrt{a}}{1 - a} - L \\right\\rbrace , \\\\\\tau ^a_{TL}&=& \\alpha ^2 \\pi Q_1 Q_2 \\frac{s^2 \\nu ^2}{X^3} \\left\\lbrace - \\sqrt{a} \\left( 3 - \\frac{4 X}{s \\nu } \\right) + \\left(3 - \\frac{4 X}{s \\nu } - a \\right) L \\right\\rbrace ,$ with $L \\equiv \\ln \\left( \\frac{1 + \\sqrt{a} }{\\sqrt{1 - a } } \\right),\\quad \\quad \\quad a \\equiv \\frac{X}{\\nu ^2} \\left( 1 - \\frac{4 m^2}{s} \\right).$ In the limit where one of the virtual photons becomes real ($Q_2^2 = 0$ ) in case of the response functions involving only transverse photons, or becomes quasi-real ($Q_2^2 \\approx 0$ ) in case of the response functions involving a longitudinal photon, the above expressions simplify to : $\\left[ \\sigma _\\parallel + \\sigma _\\perp \\right]_{Q_2^2 = 0} &=& \\alpha ^2 8 \\pi \\frac{s^2}{(s + Q_1^2)^3}\\left\\lbrace \\sqrt{1 - \\frac{4 m^2}{s} } \\left[ - \\left( 1 - \\frac{Q_1^2}{s} \\right)^2 - \\frac{4 m^2}{s} \\right]+ 2 \\left( 1 + \\frac{4 m^2}{s} - \\frac{8 m^4}{s^2} + \\frac{Q_1^4}{s^2} \\right) L\\right\\rbrace , \\\\\\left[ \\sigma _\\parallel - \\sigma _\\perp \\right]_{Q_2^2 = 0}&=& - \\, \\alpha ^2 8 \\pi \\frac{s^2}{(s + Q_1^2)^3} \\left\\lbrace \\sqrt{1 - \\frac{4 m^2}{s} } \\left( \\frac{2 m^2}{s} + \\frac{Q_1^4}{s^2} \\right)+ \\frac{8 m^2}{s} \\left( \\frac{m^2}{s} + \\frac{Q_1^2}{s} \\right) L\\right\\rbrace , \\\\\\left[ \\sigma _0 - \\sigma _2 \\right]_{Q_2^2 = 0}&=& \\alpha ^2 8 \\pi \\frac{s}{(s + Q_1^2)^2} \\left\\lbrace \\sqrt{1 - \\frac{4 m^2}{s} } \\left( 3 - \\frac{Q_1^2}{s} \\right) - 2 \\left( 1 - \\frac{Q_1^2}{s} \\right)L \\right\\rbrace ,\\\\\\left[ \\frac{1}{Q_1^2 Q_2^2} \\sigma _{LL} \\right]_{Q_2^2 = 0}&=& \\alpha ^2 128 \\pi \\frac{s^2}{(s + Q_1^2)^5} \\left\\lbrace - \\sqrt{1 - \\frac{4 m^2}{s} } + L\\right\\rbrace , \\\\\\left[ \\frac{1}{Q_1^2} \\sigma _{LT} \\right]_{Q_2^2 = 0}&=& \\alpha ^2 16 \\pi \\frac{s}{(s + Q_1^2)^3} \\left\\lbrace \\sqrt{1 - \\frac{4 m^2}{s} } - \\frac{4 m^2}{s} \\, L\\right\\rbrace , \\\\\\left[ \\frac{1}{Q_1 Q_2} \\tau _{TL} \\right]_{Q_2^2 = 0}&=& 0, \\\\\\left[\\frac{1}{Q_1 Q_2} \\tau ^a_{TL} \\right]_{Q_2^2 = 0}&=& \\alpha ^2 16 \\pi \\frac{s^2}{(s + Q_1^2)^4} \\left\\lbrace - \\sqrt{1 - \\frac{4 m^2}{s} } \\left( 1 - \\frac{2 Q_1^2}{s} \\right) + \\left( - \\frac{2 Q_1^2}{s} + \\frac{4 m^2}{s} \\right)L \\right\\rbrace ,$ with $\\left[ L \\right]_{Q_2^2 = 0} = \\ln \\left( \\frac{\\sqrt{s}}{2 m} \\left[ 1 + \\sqrt{1 - \\frac{4 m^2}{s}} \\right] \\right).$"
],
[
"$\\gamma ^\\ast \\gamma ^\\ast \\rightarrow $ meson transition form factors",
"In this Appendix we detail the formalism and the available data for the $\\gamma ^\\ast \\gamma ^\\ast \\rightarrow $ meson transition form factors (FFs), and successively discuss the $C$ -even pseudo-scalar ($J^{PC} = 0^{-+}$ ), scalar ($J^{PC} = 0^{++}$ ), axial-vector ($J^{PC} = 1^{++}$ ), and tensor ($J^{PC} = 2^{++}$ ) mesons."
],
[
"Pseudo-scalar mesons",
"The process $\\gamma ^\\ast (q_1, \\lambda _1) + \\gamma ^\\ast (q_2, \\lambda _2) \\rightarrow {\\cal P}$ , describing the transition from an initial state of two virtual photons, with four-momenta $q_1, q_2$ and helicities $\\lambda _1, \\lambda _2 = 0, \\pm 1$ , to a pseudo-scalar meson ${\\cal P} = \\pi ^0, \\eta , \\eta ^\\prime , \\eta _c, ...$ ($J^{PC} = 0^{-+}$ ) with mass $m_P$ , is described by the matrix element : ${\\cal M}(\\lambda _1, \\lambda _2) = - i \\, e^2 \\, \\varepsilon _{\\mu \\nu \\alpha \\beta } \\,\\varepsilon ^\\mu (q_1, \\lambda _1) \\, \\varepsilon ^\\nu (q_2, \\lambda _2) \\,q_1^\\alpha \\, q_2^{\\beta } \\,F_{{\\cal P} \\gamma ^\\ast \\gamma ^\\ast } (Q_1^2, Q_2^2),$ where $ \\varepsilon ^\\alpha (q_1, \\lambda _1)$ and $\\varepsilon ^\\beta (q_2, \\lambda _2)$ are the polarization vectors of the virtual photons, and where the meson structure information is encoded in the form factor (FF) $F_{{\\cal P} \\gamma ^\\ast \\gamma ^\\ast }$ , which is a function of the virtualities of both photons, satisfying $F_{{\\cal P} \\gamma ^\\ast \\gamma ^\\ast } (Q_1^2, Q_2^2) = F_{{\\cal P} \\gamma ^\\ast \\gamma ^\\ast } (Q_2^2, Q_1^2)$ .",
"From Eq.",
"(REF ), one can easily deduce that the only non-zero $\\gamma ^\\ast \\gamma ^\\ast \\rightarrow {\\cal P}$ helicity amplitudes, which we define in the rest frame of the produced meson, are given by : ${\\cal M}(\\lambda _1 = +1, \\lambda _2 = +1) = - {\\cal M}(\\lambda _1 = -1, \\lambda _2 = -1) =- e^2 \\, \\sqrt{X} \\, F_{{\\cal P} \\gamma ^\\ast \\gamma ^\\ast }(Q_1^2, Q_2^2) \\, .$ The FF at $Q_1^2 = Q_2^2 = 0$ , $F_{{\\cal P} \\gamma ^\\ast \\gamma ^\\ast }(0, 0)$ , describes the two-photon decay width of the pseudo-scalar meson : $\\Gamma _{\\gamma \\gamma }({\\cal P}) = \\frac{\\pi \\alpha ^2}{4} \\, m_P^3 \\,| F_{{\\cal P} \\gamma ^\\ast \\gamma ^\\ast }(0, 0) | ^2,$ with $m_P$ the pseudo-scalar meson mass, and $\\alpha = e^2 / (4 \\pi ) \\simeq 1/137$ .",
"In this paper, we study the sum rules involving cross sections for one real photon and one virtual photon.",
"For one real photon ($Q_2^2 = 0$ ), the only non-vanishing cross sections in Eq.",
"(REF ) are given by : $\\left[ \\sigma _0 \\right]_{Q_2^2 = 0} = \\left[ \\sigma _\\perp \\right]_{Q_2^2 = 0}= 2 \\left[ \\sigma _{TT} \\right]_{Q_2^2 = 0} = - \\left[ \\tau _{TT} \\right]_{Q_2^2 = 0}= \\delta (s - m_P^2) \\, 16 \\, \\pi ^2 \\, \\frac{\\Gamma _{\\gamma \\gamma } ( {\\cal P})}{m_P} \\,\\left(1 + \\frac{Q_1^2}{m_P^2} \\right) \\, \\left[ \\frac{F_{{\\cal P} \\gamma ^\\ast \\gamma ^\\ast }(Q_1^2, 0)}{F_{{\\cal P} \\gamma ^\\ast \\gamma ^\\ast }(0, 0)} \\right]^2\\, .$ For massless quarks, the divergence of the isovector axial current, $A_3^\\mu \\equiv \\frac{1}{\\sqrt{2}}(\\bar{u} \\gamma ^\\mu \\gamma _5 u - \\bar{d} \\gamma ^\\mu \\gamma _5 d)$ , does not vanish but exhibits an anomaly due to the triangle graphs which allow the $\\pi ^0$ to couple to two vectors currents (Wess-Zumino-Witten anomaly).",
"For the $\\pi ^0$ , the chiral (isovector axial) anomaly, predicts that its transition FF at $Q_1^2 = Q_2^2 = 0$ is given by : $F_{\\pi ^0 \\gamma ^\\ast \\gamma ^\\ast }(0, 0) = \\frac{1}{4 \\pi ^2 f_\\pi },$ where the pion decay constant $f_\\pi $ is defined through the isovector axial current matrix element : $\\langle 0 | A_3^\\mu (0) | \\pi ^0 (p) \\rangle = i \\, ( \\sqrt{2} \\, f_\\pi ) \\, p^\\mu .$ When using the current empirical value of the pion decay constant $f_\\pi \\simeq 92.4$ MeV to evaluate the chiral anomaly prediction of Eq.",
"(REF ), one obtains the value $F_{M \\gamma ^\\ast \\gamma }(0) \\simeq 0.274$ GeV$^{-1}$ , which yields through Eq.",
"(REF ) a $2 \\gamma $ decay width in very good agreement with the experimental value (see Table REF ).",
"The form factors $F_{{\\cal P} \\gamma ^\\ast \\gamma ^\\ast }(Q_1^2, 0)$ for one virtual photon and one real photon have been measured for $\\pi ^0$ , $\\eta $ , $\\eta ^\\prime $ by the CELLO [25] , CLEO [26], and BABAR [8], [27] Collaborations, and for $\\eta _c(1S)$ by the BABAR Collaboration [28].",
"In the $Q_1^2$ range up to 10 GeV$^2$ , a good parameterization of the data is obtained by the monopole form : $\\frac{F_{{\\cal P} \\gamma ^\\ast \\gamma ^\\ast }(Q_1^2, 0)}{F_{{\\cal P} \\gamma ^\\ast \\gamma ^\\ast }(0, 0)} = \\frac{1}{1 + Q_1^2 / \\Lambda _P^2},$ where $\\Lambda _P$ is the monopole mass parameter.",
"In Table REF , we show the experimental extraction of $\\Lambda _P$ for the $\\pi ^0, \\eta , \\eta ^\\prime $ , and $\\eta _c(1S)$ mesons.",
"Table: Experimental extraction of the monopole mass parameter in the γ * γ→𝒫\\gamma ^\\ast \\gamma \\rightarrow {\\cal P} form factors,according to the fit of Eq.",
"().The measured value of Λ P \\Lambda _P for 𝒫=π 0 ,η,η ' {\\cal P} = \\pi ^0, \\eta , \\eta ^\\prime is from the CLEO Collaboration .For the η c (1S)\\eta _c(1S) state, the measured value is from the BABAR Collaboration ."
],
[
"Scalar mesons",
"We next consider the process $\\gamma ^\\ast (q_1, \\lambda _1) + \\gamma ^\\ast (q_2, \\lambda _2) \\rightarrow {\\cal S}$ , describing the transition from an initial state of two virtual photons, with four-momenta $q_1, q_2$ and helicities $\\lambda _1, \\lambda _2 = 0, \\pm 1$ , to a scalar meson ${\\cal S}$ ($J^{PC} = 0^{++}$ ) with mass $m_S$ .",
"Scalar mesons can be produced either by two transverse photons or by two longitudinal photons [4], [30].",
"Therefore, the $\\gamma ^\\ast \\gamma ^\\ast \\rightarrow {\\cal S}$ transition can be described by the matrix element : ${\\cal M}(\\lambda _1, \\lambda _2) &=& e^2 \\, \\varepsilon ^\\mu (q_1, \\lambda _1) \\, \\varepsilon ^\\nu (q_2, \\lambda _2) \\,\\, \\nonumber \\\\&\\times & \\left( \\frac{\\nu }{m_S} \\right) \\left\\lbrace - R^{\\mu \\nu } (q_1, q_2) F^T_{{\\cal S} \\gamma ^\\ast \\gamma ^\\ast } (Q_1^2, Q_2^2) \\,+\\,\\frac{\\nu }{X} \\left( q_1^\\mu + \\frac{Q_1^2}{\\nu } q_2^{\\mu } \\right) \\left( q_2^\\nu + \\frac{Q_2^2}{\\nu } q_1^{\\nu } \\right)F^L_{{\\cal S} \\gamma ^\\ast \\gamma ^\\ast } (Q_1^2, Q_2^2)\\right\\rbrace ,$ where we introduced the symmetric transverse tensor $R^{\\mu \\nu }$ : $R^{\\mu \\nu } (q_1, q_2) \\equiv - g^{\\mu \\nu } + \\frac{1}{X} \\,\\bigl \\lbrace \\nu \\left( q_1^\\mu \\, q_2^\\nu + q_2^\\mu \\, q_1^\\nu \\right)+ Q_1^2 \\, q_2^\\mu \\, q_2^\\nu + Q_2^2 \\, q_1^\\mu \\, q_1^\\nu \\bigr \\rbrace ,$ which projects onto both transverse photons, having the properties : $q_{1 \\mu } R^{\\mu \\nu } (q_1, q_2) = 0, \\quad q_{1 \\nu } R^{\\mu \\nu } (q_1, q_2) = 0,\\quad q_{2 \\mu } R^{\\mu \\nu } (q_1, q_2) = 0, \\quad q_{2 \\nu } R^{\\mu \\nu } (q_1, q_2) = 0.\\nonumber $ In Eq.",
"(REF ), the scalar meson structure information is encoded in the form factors $F^T_{{\\cal S} \\gamma ^\\ast \\gamma ^\\ast }$ and $F^L_{{\\cal S} \\gamma ^\\ast \\gamma ^\\ast }$ , which are a function of the virtualities of both photons, where the superscripts indicate the situation where either both photons are transverse ($T$ ) or longitudinal ($L$ ).",
"Note that the pre-factor $\\nu /m_S$ in Eq.",
"(REF ) is chosen such that the FFs are dimensionless.",
"Furthermore, both form factors are symmetric under interchange of both virtualities : $F^{T, L}_{{\\cal S} \\gamma ^\\ast \\gamma ^\\ast }(Q_1^2, Q_2^2) &=& F^{T, L}_{{\\cal S} \\gamma ^\\ast \\gamma ^\\ast }(Q_2^2, Q_1^2) .$ From Eq.",
"(REF ), one can easily deduce that the only non-zero $\\gamma ^\\ast \\gamma ^\\ast \\rightarrow {\\cal S}$ helicity amplitudes are given by : ${\\cal M}(\\lambda _1 = +1, \\lambda _2 = +1) &=& {\\cal M}(\\lambda _1 = -1, \\lambda _2 = -1) =e^2 \\, \\frac{\\nu }{m_S} \\, F^T_{{\\cal S} \\gamma ^\\ast \\gamma ^\\ast }(Q_1^2, Q_2^2) \\, , \\nonumber \\\\{\\cal M}(\\lambda _1 = 0, \\lambda _2 = 0) &=& - \\, e^2 \\, \\frac{Q_1 Q_2}{m_S} \\,F^L_{{\\cal S} \\gamma ^\\ast \\gamma ^\\ast }(Q_1^2, Q_2^2) \\, .$ The transverse FF at $Q_1^2 = Q_2^2 = 0$ , $F^T_{{\\cal S} \\gamma ^\\ast \\gamma ^\\ast }(0,0)$ , describes the two-photon decay width of the scalar meson : $\\Gamma _{\\gamma \\gamma }({\\cal S}) = \\frac{\\pi \\alpha ^2}{4} m_S \\,| F^T_{{\\cal S} \\gamma ^\\ast \\gamma ^\\ast }(0,0) | ^2.$ In this paper, we study the sum rules involving cross sections for one real photon and one virtual photon.",
"For one real photon ($Q_2^2 = 0$ ), the only non-vanishing cross sections in Eq.",
"(REF ) are given by : $\\left[ \\sigma _0 \\right]_{Q_2^2 = 0} = \\left[ \\sigma _\\parallel \\right]_{Q_2^2 = 0}= 2 \\left[ \\sigma _{TT} \\right]_{Q_2^2 = 0} = \\left[ \\tau _{TT} \\right]_{Q_2^2 = 0}= \\delta (s - m_S^2) \\, 16 \\, \\pi ^2 \\, \\frac{\\Gamma _{\\gamma \\gamma }({\\cal S})}{m_S} \\,\\left( 1 + \\frac{Q_1^2}{m_S^2} \\right)\\, \\left[ \\frac{F^T_{{\\cal S} \\gamma ^\\ast \\gamma ^\\ast }(Q_1^2, 0)}{F^T_{{\\cal S} \\gamma ^\\ast \\gamma ^\\ast }(0, 0)} \\right]^2 \\, .$"
],
[
"Axial-vector mesons",
"We next discuss the two-photon production of an axial vector meson.",
"Due to the symmetry under rotational invariance, spatial inversion as well as the Bose symmetry of a state of two real photons, the production of a spin-1 resonance by two real photons is forbidden, which is known as the Landau-Yang theorem [29].",
"However the production of an axial-vector meson by two photons is possible when one or both photons are virtual.",
"The matrix element for the process $\\gamma ^\\ast (q_1, \\lambda _1) + \\gamma ^\\ast (q_2, \\lambda _2) \\rightarrow {\\cal A}$ , describing the transition from an initial state of two virtual photons, with four-momenta $q_1, q_2$ and helicities $\\lambda _1, \\lambda _2 = 0, \\pm 1$ , to an axial-vector meson ${\\cal A}$ ($J^{PC} = 1^{++}$ ) with mass $m_A$ and helicity $\\Lambda = \\pm 1, 0$ (defined along the direction of $\\vec{q}_1$ ), is described by three structures [4], [30], and can be parameterized as : ${\\cal M}(\\lambda _1, \\lambda _2; \\Lambda ) &=& e^2 \\,\\varepsilon _\\mu (q_1, \\lambda _1) \\, \\varepsilon _\\nu (q_2, \\lambda _2) \\,\\varepsilon ^{\\alpha \\ast }(p_f, \\Lambda ) \\, \\nonumber \\\\&\\times & i \\, \\varepsilon _{\\rho \\sigma \\tau \\alpha } \\, \\left\\lbrace R^{\\mu \\rho } (q_1, q_2) R^{\\nu \\sigma } (q_1, q_2) \\,(q_1 - q_2)^\\tau \\, \\frac{\\nu }{m_A^2} \\, F^{(0)}_{{\\cal A} \\gamma ^\\ast \\gamma ^\\ast }(Q_1^2, Q_2^2)\\right.",
"\\nonumber \\\\&&\\hspace{28.45274pt} + \\, R^{\\nu \\rho }(q_1, q_2) \\left( q_1^\\mu + \\frac{Q_1^2}{\\nu } q_2^{\\mu } \\right)q_1^\\sigma \\, q_2^\\tau \\, \\frac{1}{m_A^2} \\, F_{{\\cal A} \\gamma ^\\ast \\gamma ^\\ast }^{(1)}(Q_1^2, Q_2^2) \\nonumber \\\\&&\\left.",
"\\hspace{28.45274pt} + \\, R^{\\mu \\rho }(q_1, q_2) \\left( q_2^\\nu + \\frac{Q_2^2}{\\nu } q_1^{\\nu } \\right)q_2^\\sigma \\, q_1^\\tau \\, \\frac{1}{m_A^2} \\, F^{(1)}_{{\\cal A} \\gamma ^\\ast \\gamma ^\\ast }(Q_2^2, Q_1^2)\\right\\rbrace .$ In Eq.",
"(REF ), the axial-vector meson structure information is encoded in the form factors $F^{(0)}_{{\\cal A} \\gamma ^\\ast \\gamma ^\\ast }$ and $F^{(1)}_{{\\cal A} \\gamma ^\\ast \\gamma ^\\ast }$ , where the superscript indicates the helicity state of the axial-vector meson.",
"Note that only transverse photons give a non-zero transition to a state of helicity zero.",
"The form factors are functions of the virtualities of both photons, and $F^{(0)}_{{\\cal A} \\gamma ^\\ast \\gamma ^\\ast }$ is symmetric under the interchange $Q_1^2 \\leftrightarrow Q_2^2$ .",
"In contrast, $F^{(1)}_{{\\cal A} \\gamma ^\\ast \\gamma ^\\ast }$ does not need to be symmetric under interchange of both virtualities, as can be seen from Eq.",
"(REF ).",
"From Eq.",
"(REF ), one can easily deduce that the only non-zero $\\gamma ^\\ast \\gamma ^\\ast \\rightarrow {\\cal A}$ helicity amplitudes are given by : ${\\cal M}(\\lambda _1 = +1, \\lambda _2 = +1; \\Lambda = 0) &=& - {\\cal M}(\\lambda _1 = -1, \\lambda _2 = -1; \\Lambda = 0) =e^2 \\, (Q_1^2 - Q_2^2) \\, \\frac{\\nu }{m_A^3} \\, F^{(0,T)}_{{\\cal A} \\gamma ^\\ast \\gamma ^\\ast }(Q_1^2, Q_2^2) \\, , \\nonumber \\\\{\\cal M}(\\lambda _1 = 0, \\lambda _2 = +1; \\Lambda = -1) &=& - \\, e^2 \\, Q_1 \\, \\left( \\frac{X}{\\nu m_A^2} \\right) \\,F^{(1)}_{{\\cal A} \\gamma ^\\ast \\gamma ^\\ast }(Q_1^2, Q_2^2) \\, , \\nonumber \\\\{\\cal M}(\\lambda _1 = -1, \\lambda _2 = 0; \\Lambda = -1) &=& - \\, e^2 \\, Q_2 \\, \\left( \\frac{X}{\\nu m_A^2} \\right) \\,F^{(1)}_{{\\cal A} \\gamma ^\\ast \\gamma ^\\ast }(Q_2^2, Q_1^2) \\, .$ Note that the helicity amplitude with two transverse photons vanishes when both photons are real, in accordance with the Landau-Yang theorem.",
"The matrix element $F^{(1)}_{{\\cal A} \\gamma ^\\ast \\gamma }(0, 0)$ allows to define an equivalent two-photon decay width for an axial-vector meson to decay in one quasi-real longitudinal photon and a (transverse) real photon as In defining the equivalent two-photon decay width for an axial-vector meson, we follow the convention of Ref.",
"[30], which is also followed in experimental analyses [31], [32].",
"Note however that the definition for $\\tilde{\\Gamma }_{\\gamma \\gamma }$ adopted here is one half of that used in Ref. [33].",
": $\\tilde{\\Gamma }_{ \\gamma \\gamma }({\\cal A}) \\equiv \\lim \\limits _{Q_1^2 \\rightarrow 0} \\, \\frac{m_A^2}{Q_1^2} \\, \\frac{1}{2} \\,\\Gamma \\left( {\\cal A} \\rightarrow \\gamma ^\\ast _L \\gamma _T \\right)= \\frac{\\pi \\alpha ^2}{4} \\, m_A \\, \\frac{1}{3} \\left[ F^{(1)}_{{\\cal A} \\gamma ^\\ast \\gamma ^\\ast }(0, 0) \\right]^2,$ where we have introduced the decay width $\\Gamma \\left( {\\cal A} \\rightarrow \\gamma ^\\ast _L \\gamma _T \\right)$ for an axial-vector meson to decay in a virtual longitudinal photon, with virtuality $Q_1^2$ , and a real transverse photon ($Q_2^2 = 0$ ), as : $\\Gamma \\left( {\\cal A} \\rightarrow \\gamma ^\\ast _L \\gamma _T \\right)= \\frac{\\pi \\alpha ^2}{2} \\, m_A \\, \\frac{1}{3} \\, \\frac{Q_1^2}{m_A^2} \\, \\, \\left(1 + \\frac{Q_1^2}{m_A^2} \\right)^3 \\,\\left[ F^{(1)}_{{\\cal A} \\gamma ^\\ast \\gamma ^\\ast }(Q_1^2, 0) \\right]^2.$ In this paper, we study the sum rules involving cross sections for one real photon and one virtual photon.",
"For one quasi-real photon ($Q_2^2 \\rightarrow 0$ ), we can obtain from the above helicity amplitudes and using Eq.",
"(REF ) the axial-vector meson contributions to the response functions of Eq.",
"(REF ) as : $\\left[ \\sigma _0 \\right]_{Q_2^2 = 0} = \\left[ \\sigma _\\perp \\right]_{Q_2^2 = 0}&=& 2 \\left[ \\sigma _{TT} \\right]_{Q_2^2 = 0} = - \\left[ \\tau _{TT} \\right]_{Q_2^2 = 0}= \\delta (s - m_A^2) \\, 4 \\, \\pi ^3 \\alpha ^2 \\, \\frac{Q_1^4}{m_A^4} \\left(1 + \\frac{Q_1^2}{m_A^2} \\right) \\,\\left[ F^{(0)}_{{\\cal A} \\gamma ^\\ast \\gamma ^\\ast }(Q_1^2, 0) \\right]^2\\, ,\\nonumber \\\\\\left[ \\sigma _{LT} \\right]_{Q_2^2 = 0}&=& \\delta (s - m_A^2) \\, 16 \\, \\pi ^2 \\, \\frac{3 \\, \\tilde{\\Gamma }_{\\gamma \\gamma }({\\cal A})}{m_A} \\, \\frac{Q_1^2}{m_A^2} \\,\\left(1 + \\frac{Q_1^2}{m_A^2} \\right) \\,\\left[ \\frac{F^{(1)}_{{\\cal A} \\gamma ^\\ast \\gamma ^\\ast }(Q_1^2, 0)}{F^{(1)}_{{\\cal A} \\gamma ^\\ast \\gamma ^\\ast }(0, 0) } \\right]^2\\, ,\\nonumber \\\\\\left[ \\tau _{TL} \\right]_{Q_2^2 = 0} = - \\left[ \\tau ^a_{TL} \\right]_{Q_2^2 = 0}&=& \\delta (s - m_A^2) \\, 8 \\, \\pi ^2\\, \\frac{3 \\, \\tilde{\\Gamma }_{\\gamma \\gamma }({\\cal A})}{m_A} \\, \\frac{Q_1 Q_2}{m_A^2} \\, \\left(1 + \\frac{Q_1^2}{m_A^2} \\right) \\,\\left[ \\frac{F^{(1)}_{{\\cal A} \\gamma ^\\ast \\gamma ^\\ast }(Q_1^2, 0)}{F^{(1)}_{{\\cal A} \\gamma ^\\ast \\gamma ^\\ast }(0, 0)}\\cdot \\frac{F^{(1)}_{{\\cal A} \\gamma ^\\ast \\gamma ^\\ast }(0, Q_1^2)}{F^{(1)}_{{\\cal A} \\gamma ^\\ast \\gamma ^\\ast }(0, 0)} \\right] \\, .$ Extracting the FFs $F^{(1)}$ , and $F^{(0)}$ separately from experiment requires the measurements of $\\sigma _{LT}$ and $\\sigma _{TT}$ respectively.",
"As experiments to date have not achieved this separation, one is so far only sensitive to the quantity $\\sigma _{TT} + \\varepsilon _1 \\, \\sigma _{LT}$ , where $\\varepsilon _1$ is a kinematical parameter (so-called virtual photon polarization parameter) defined as $\\varepsilon _1 \\equiv \\rho _1^{00} / 2 \\rho _1^{++}$ , see Appendix .",
"Note that in high-energy collider experiments, one typically has $\\varepsilon _1 \\approx 1$ .",
"From Eq.",
"(REF ) one then obtains for this experimentally accessible combination : $\\left[ \\sigma _{LT} \\left( 1 + \\frac{1}{\\varepsilon _1} \\frac{\\sigma _{TT}}{\\sigma _{LT}}\\right) \\right]_{Q_2^2 = 0} &=&\\delta (s - m_A^2) \\, 16 \\, \\pi ^2 \\, \\frac{3 \\, \\tilde{\\Gamma }_{\\gamma \\gamma }({\\cal A})}{m_A} \\, \\frac{Q_1^2}{m_A^2} \\,\\left(1 + \\frac{Q_1^2}{m_A^2} \\right) \\, \\nonumber \\\\&\\times & \\left(\\left[ \\frac{F^{(1)}_{{\\cal A} \\gamma ^\\ast \\gamma ^\\ast }(Q_1^2, 0)}{F^{(1)}_{{\\cal A} \\gamma ^\\ast \\gamma ^\\ast }(0, 0) } \\right]^2\\,+ \\frac{1}{\\varepsilon _1} \\, \\frac{Q_1^2}{2 \\, m_A^2} \\, \\left[ \\frac{F^{(0)}_{{\\cal A} \\gamma ^\\ast \\gamma ^\\ast }(Q_1^2, 0)}{F^{(1)}_{{\\cal A} \\gamma ^\\ast \\gamma ^\\ast }(0, 0) } \\right]^2\\,\\right),$ We can compare the above general formalism for the two-photon production of an axial-vector meson with the description of Ref.",
"[33], which is commonly used in the literature, and is based on a non-relativistic quark model calculation leading to only one independent amplitude for the $\\gamma ^\\ast \\gamma ^\\ast \\rightarrow {\\cal A}$ process as : ${\\cal M}(\\lambda _1, \\lambda _2; \\Lambda ) &=& e^2 \\,\\varepsilon ^\\mu (q_1, \\lambda _1) \\, \\varepsilon ^\\nu (q_2, \\lambda _2) \\,\\varepsilon ^{\\alpha \\ast }(p_f, \\Lambda ) \\,\\, i \\varepsilon _{\\mu \\nu \\tau \\alpha } \\,\\left( -Q_1^2 \\, q_2^\\tau + Q_2^2 \\, q_1^\\tau \\right) \\, A(Q_1^2, Q_2^2),$ where the independent form factor $A$ satisfies : $A(Q_1^2, Q_2^2) = A(Q_2^2, Q_1^2)$ .",
"In such a non-relativistic quark model limit, we can recover Eq.",
"(REF ) from Eq.",
"(REF ) through the identifications : $F^{(0)}(Q_1^2, Q_2^2) &=& m_A^2 \\, A(Q_1^2, Q_2^2), \\nonumber \\\\F^{(1)}(Q_1^2, Q_2^2) &=& - \\frac{\\nu }{X} (\\nu + Q_2^2) \\, m_A^2 \\, A(Q_1^2, Q_2^2), \\nonumber \\\\F^{(1)}(Q_2^2, Q_1^2) &=& - \\frac{\\nu }{X} (\\nu + Q_1^2) \\, m_A^2 \\, A(Q_1^2, Q_2^2),$ in which $2 \\nu = m_A^2 + Q_1^2 + Q_2^2$ .",
"In such model, the experimentally measured two-photon cross section combination of Eq.",
"(REF ), where $Q_2^2 = 0$ , is proportional to : $\\left[ \\sigma _{LT} \\left( 1 + \\frac{1}{\\varepsilon _1} \\frac{\\sigma _{TT}}{\\sigma _{LT}}\\right) \\right]_{Q_2^2 = 0} =\\delta (s - m_A^2) \\, 16 \\, \\pi ^2 \\, \\frac{3 \\, \\tilde{\\Gamma }_{\\gamma \\gamma }({\\cal A})}{m_A} \\, \\frac{Q_1^2}{m_A^2} \\,\\left(1 + \\frac{Q_1^2}{m_A^2} \\right) \\,\\left( 1 + \\frac{1}{\\varepsilon _1} \\, \\frac{Q_1^2}{2 \\, m_A^2} \\right) \\,\\left[ \\frac{A(Q_1^2, 0)}{A(0, 0) } \\right]^2.$ To apply this formula to experimental results where the axial-vector meson has a finite width, one commonly replaces the delta-function in Eq.",
"(REF ) by a Breit-Wigner form, yielding : $\\left[ \\sigma _{LT} \\left( 1 + \\frac{1}{\\varepsilon _1} \\frac{\\sigma _{TT}}{\\sigma _{LT}}\\right) \\right]_{Q_2^2 = 0} =48 \\, \\pi \\, \\frac{\\tilde{\\Gamma }_{\\gamma \\gamma }({\\cal A}) \\, \\Gamma _{total}}{(s - m_A^2)^2 + m_A^2 \\, \\Gamma ^2_{total}} \\,\\frac{Q_1^2}{m_A^2} \\,\\left(1 + \\frac{Q_1^2}{m_A^2} \\right) \\,\\left( 1 + \\frac{1}{\\varepsilon _1} \\, \\frac{Q_1^2}{2 \\, m_A^2} \\right) \\,\\left[ \\frac{A(Q_1^2, 0)}{A(0, 0) } \\right]^2,$ where $\\Gamma _{total}$ is the total decay width of the axial-vector meson.",
"Phenomenologically, the two-photon production cross sections have been measured for the two lowest lying axial-vector mesons : $f_1(1285)$ and $f_1(1420)$ .",
"The most recent measurements were performed by the L3 Collaboration [31], [32].",
"In those works, the non-relativistic quark model expression of Eq.",
"(REF ) in terms of a single FF $A$ has been assumed, and the resulting FF has been parameterized by a dipole : $\\frac{A(Q_1^2, 0)}{A(0, 0) }= \\frac{1}{\\left( 1 + Q_1^2/ \\Lambda _A^2 \\right)^2}\\, ,$ where $\\Lambda _A$ is a dipole mass.",
"By fitting the resulting expression of Eq.",
"(REF ) to experiment (for which $\\varepsilon _1 \\approx 1$ , and for a $Q_1^2$ range which extends up to 6 GeV$^2$ ), one can then extract the parameters $\\tilde{\\Gamma }_{\\gamma \\gamma }$ and $\\Lambda _1$ .",
"Table REF shows the present experimental status of the equivalent $2 \\gamma $ decay widths of the axial-vector mesons $f_1(1285)$ , and $f_1(1420)$ , which we use in this work.",
"Table: Present values of thef 1 (1285)f_1(1285) meson and f 1 (1420)f_1(1420) meson masses m A m_A, theirequivalent 2γ2 \\gamma decay widths Γ ˜ γγ \\tilde{\\Gamma }_{\\gamma \\gamma }, defined according to Eq.",
"(), as well as theirdipole masses Λ A \\Lambda _A entering the FF of Eq.",
"().For Γ ˜ γγ \\tilde{\\Gamma }_{\\gamma \\gamma }, we use the experimental results from the L3 Collaboration :f 1 (1285)f_1(1285) from Ref.",
",f 1 (1420)f_1(1420) from Ref.",
".Note that for the f 1 (1420)f_1(1420) state, only the branching ratio Γ ˜ γγ ×Γ KK ¯π /Γ total \\tilde{\\Gamma }_{\\gamma \\gamma } \\times \\Gamma _{K \\bar{K} \\pi } / \\Gamma _{total} is measured so far, which we use as a lower limit on Γ ˜ γγ \\tilde{\\Gamma }_{\\gamma \\gamma }."
],
[
"Tensor mesons",
"The process $\\gamma ^\\ast (q_1, \\lambda _1) + \\gamma ^\\ast (q_2, \\lambda _2) \\rightarrow {\\cal T}(\\Lambda )$ , describing the transition from an initial state of two virtual photons to a tensor meson ${\\cal T}$ ($J^{PC} = 2^{++}$ ) with mass $m_T$ and helicity $\\Lambda = \\pm 2, \\pm 1, 0$ (defined along the direction of $\\vec{q}_1$ ), is described by five independent structures [4], [30], and can be parameterized as : ${\\cal M}(\\lambda _1, \\lambda _2; \\Lambda ) &=& e^2 \\,\\varepsilon _\\mu (q_1, \\lambda _1) \\, \\varepsilon _\\nu (q_2, \\lambda _2) \\,\\varepsilon ^\\ast _{\\alpha \\beta }(p_f, \\Lambda ) \\, \\nonumber \\\\&\\times & \\left\\lbrace \\left[ R^{\\mu \\alpha } (q_1, q_2) R^{\\nu \\beta } (q_1, q_2)+ \\frac{s}{8 X} \\, R^{\\mu \\nu }(q_1, q_2)(q_1 - q_2)^\\alpha \\, (q_1 - q_2)^\\beta \\right] \\, \\frac{\\nu }{m_T} \\,T^{(2)}(Q_1^2, Q_2^2)\\right.",
"\\nonumber \\\\&&+ \\, R^{\\nu \\alpha }(q_1, q_2) (q_1 - q_2)^\\beta \\left( q_1^\\mu + \\frac{Q_1^2}{\\nu } q_2^{\\mu } \\right)\\, \\frac{1}{m_T} \\, T^{(1)}(Q_1^2, Q_2^2) \\nonumber \\\\&&+ R^{\\mu \\alpha }(q_1, q_2) (q_2 - q_1)^\\beta \\left( q_2^\\nu + \\frac{Q_2^2}{\\nu } q_1^{\\nu } \\right)\\, \\frac{1}{m_T} \\, T^{(1)}(Q_2^2, Q_1^2)\\nonumber \\\\&&+ \\, R^{\\mu \\nu }(q_1, q_2) (q_1 - q_2)^\\alpha \\, (q_1 - q_2)^\\beta \\, \\frac{1}{m_T} \\,T^{(0, T)}(Q_1^2, Q_2^2) \\, \\nonumber \\\\&&\\left.",
"+ \\, \\left( q_1^\\mu + \\frac{Q_1^2}{\\nu } q_2^{\\mu } \\right) \\left( q_2^\\nu + \\frac{Q_2^2}{\\nu } q_1^{\\nu } \\right)(q_1 - q_2)^\\alpha (q_1 - q_2)^\\beta \\, \\frac{1}{m_T^3} \\, T^{(0, L)}(Q_1^2, Q_2^2)\\right\\rbrace ,$ where $\\varepsilon _{\\alpha \\beta }(p_f, \\Lambda )$ is the polarization tensor for the tensor meson with four-momentum $p_f$ and helicity $\\Lambda $ .",
"Furthermore in Eq.",
"(REF ) $T^{(\\Lambda )}$ are the $\\gamma ^\\ast \\gamma ^\\ast \\rightarrow {\\cal T}$ transition form factors, for tensor meson helicity $\\Lambda $ .",
"For the case of helicity zero, there are two form factors depending on whether both photons are transverse (superscript $T$ ) or longitudinal (superscript $L$ ).",
"From Eq.",
"(REF ), we can easily calculate the different helicity amplitudes as : $&&{\\cal M}(\\lambda _1 = +1, \\lambda _2 = -1; \\Lambda = +2) ={\\cal M}(\\lambda _1 = -1, \\lambda _2 = +1; \\Lambda = -2) = e^2 \\, \\frac{\\nu }{m_T} \\, T^{(2)}(Q_1^2, Q_2^2) \\, , \\nonumber \\\\&&{\\cal M}(\\lambda _1 = 0, \\lambda _2 = +1; \\Lambda = -1) = - e^2 \\, Q_1 \\, \\frac{1}{\\sqrt{2}} \\, \\left( \\frac{2 X}{\\nu m_T^2} \\right)\\, T^{(1)}(Q_1^2, Q_2^2) \\, , \\nonumber \\\\&&{\\cal M}(\\lambda _1 = -1, \\lambda _2 = 0; \\Lambda = -1) = e^2 \\, Q_2 \\, \\frac{1}{\\sqrt{2}} \\, \\left( \\frac{2 X}{\\nu m_T^2} \\right)\\, T^{(1)}(Q_2^2, Q_1^2) \\, , \\nonumber \\\\&&{\\cal M}(\\lambda _1 = +1, \\lambda _2 = +1; \\Lambda = 0) ={\\cal M}(\\lambda _1 = -1, \\lambda _2 = -1; \\Lambda = 0) = - e^2 \\, \\sqrt{\\frac{2}{3}} \\, \\left( \\frac{4 X}{m_T^3} \\right) \\,T^{(0, T)}(Q_1^2, Q_2^2) \\, , \\nonumber \\\\&&{\\cal M}(\\lambda _1 = 0, \\lambda _2 = 0; \\Lambda = 0) = - e^2 \\, Q_1 Q_2 \\, \\sqrt{\\frac{2}{3}} \\,\\left( \\frac{4 X^2}{\\nu ^2 m_T^5} \\right) \\, T^{(0, L)}(Q_1^2, Q_2^2)\\, .$ The transverse FFs $T^{(2)}$ and $T^{(0, T)}$ at $Q_1^2 = Q_2^2 = 0$ describe the two-photon decay widths of the tensor meson with helicities $\\Lambda = 2$ and $\\Lambda = 0$ respectively : $\\Gamma _{\\gamma \\gamma } \\left({\\cal T}( \\Lambda = 2) \\right) &=& \\frac{\\pi \\alpha ^2}{4} \\, m_T \\, \\frac{1}{5} \\,| T^{(2)} (0,0) |^2 \\, , \\nonumber \\\\\\Gamma _{\\gamma \\gamma } \\left({\\cal T}(\\Lambda = 0) \\right) &=& \\frac{\\pi \\alpha ^2}{4} \\, m_T \\, \\frac{2}{15} \\,| T^{(0, T)} (0,0) |^2 \\, .$ In this work, we study the sum rules involving cross sections for one real photon and one virtual photon.",
"For one quasi-real photon ($Q_2^2 \\rightarrow 0$ ), we can obtain from the above helicity amplitudes and using Eq.",
"(REF ) the tensor meson contributions to the response functions of Eq.",
"(REF ) as : $\\left[ \\sigma _2 \\right]_{Q_2^2 = 0}&=& \\delta (s - m_T^2) \\, 16 \\, \\pi ^2 \\frac{5 \\, \\Gamma _{\\gamma \\gamma }({\\cal T}(\\Lambda = 2))}{m_T} \\,\\left( 1 + \\frac{Q_1^2}{m_T^2} \\right)\\, \\left[ \\frac{T^{(2)}(Q_1^2, 0)}{T^{(2)}(0, 0)} \\right]^2 \\, ,\\nonumber \\\\\\left[ \\sigma _0 \\right]_{Q_2^2 = 0}&=& \\delta (s - m_T^2) \\, 16 \\, \\pi ^2 \\, \\frac{5 \\, \\Gamma _{\\gamma \\gamma }({\\cal T}(\\Lambda = 0))}{m_T} \\,\\left( 1 + \\frac{Q_1^2}{m_T^2} \\right)^3\\, \\left[ \\frac{T^{(0, T)}(Q_1^2, 0)}{T^{(0, T)}(0, 0)} \\right]^2 \\, ,\\nonumber \\\\\\left[ \\sigma _\\parallel \\right]_{Q_2^2 = 0} &=& \\left[ \\frac{1}{2} \\sigma _2 + \\sigma _0 \\right]_{Q_2^2 = 0} \\, ,\\nonumber \\\\\\left[ \\sigma _\\perp \\right]_{Q_2^2 = 0} &=& \\left[ \\frac{1}{2} \\sigma _2 \\right]_{Q_2^2 = 0} \\, ,\\nonumber \\\\\\left[ \\sigma _{LT} \\right]_{Q_2^2 = 0} &=& \\delta (s - m_T^2) \\, 8 \\, \\pi ^3 \\alpha ^2 \\, \\frac{Q_1^2}{m_T^2} \\, \\left( 1 + \\frac{Q_1^2}{m_T^2} \\right) \\,\\left[ T^{(1)}(Q_1^2,0) \\right]^2\\, ,\\nonumber \\\\\\left[ \\frac{1}{Q_1 Q_2} \\tau _{TL} \\right]_{Q_2^2 = 0} &=& \\delta (s - m_T^2) \\,8 \\, \\pi ^3 \\alpha ^2 \\,\\frac{1}{m_T^2} \\, \\left( 1 + \\frac{Q_1^2}{m_T^2} \\right) \\nonumber \\\\&\\times & \\left\\lbrace \\frac{2}{3} \\left( 1 + \\frac{Q_1^2}{m_T^2} \\right)^2 T^{(0, T)}(Q_1^2, 0) \\, T^{(0, L)}(Q_1^2, 0) - \\frac{1}{2} T^{(1)}(Q_1^2,0) \\, T^{(1)}(0, Q_1^2) \\right\\rbrace \\, ,\\nonumber \\\\\\left[ \\frac{1}{Q_1 Q_2} \\tau ^a_{TL} \\right]_{Q_2^2 = 0} &=& \\delta (s - m_T^2) \\,8 \\, \\pi ^3 \\alpha ^2 \\,\\frac{1}{m_T^2} \\, \\left( 1 + \\frac{Q_1^2}{m_T^2} \\right) \\nonumber \\\\&\\times & \\left\\lbrace \\frac{2}{3} \\left( 1 + \\frac{Q_1^2}{m_T^2} \\right)^2 T^{(0, T)}(Q_1^2, 0) \\, T^{(0, L)}(Q_1^2, 0) + \\frac{1}{2} T^{(1)}(Q_1^2,0) \\, T^{(1)}(0, Q_1^2) \\right\\rbrace \\, ,\\nonumber \\\\\\left[ \\sigma _{LL} \\right]_{Q_2^2 = 0} &=& 0\\, .$"
]
] | 1204.0740 |
[
[
"Persistence and Uncertainty in the Academic Career"
],
[
"Abstract Understanding how institutional changes within academia may affect the overall potential of science requires a better quantitative representation of how careers evolve over time.",
"Since knowledge spillovers, cumulative advantage, competition, and collaboration are distinctive features of the academic profession, both the employment relationship and the procedures for assigning recognition and allocating funding should be designed to account for these factors.",
"We study the annual production n_{i}(t) of a given scientist i by analyzing longitudinal career data for 200 leading scientists and 100 assistant professors from the physics community.",
"We compare our results with 21,156 sports careers.",
"Our empirical analysis of individual productivity dynamics shows that (i) there are increasing returns for the top individuals within the competitive cohort, and that (ii) the distribution of production growth is a leptokurtic \"tent-shaped\" distribution that is remarkably symmetric.",
"Our methodology is general, and we speculate that similar features appear in other disciplines where academic publication is essential and collaboration is a key feature.",
"We introduce a model of proportional growth which reproduces these two observations, and additionally accounts for the significantly right-skewed distributions of career longevity and achievement in science.",
"Using this theoretical model, we show that short-term contracts can amplify the effects of competition and uncertainty making careers more vulnerable to early termination, not necessarily due to lack of individual talent and persistence, but because of random negative production shocks.",
"We show that fluctuations in scientific production are quantitatively related to a scientist's collaboration radius and team efficiency."
],
[
"Scientific production and the career trajectory",
" The academic career depends on many factors, such as cumulative advantage [16], [19], [22], [23], the “sacred spark,” [24], [25], and other complex aspects of knowledge transfer manifest in our techno-social world [26].",
"To exemplify this complexity, a recent case study on the impact trajectories of Nobel prize winners shows that “scientific career shocks” marked by the publication of an individual's “magnum opus” work(s) can trigger future recognition and reward, resembling the cascading dynamics of earthquakes [27].",
"We model the career trajectory as a sequence of scientific outputs which arrive at the variable rate $n_{i}(t)$ .",
"Since the reputation of a scientist is typically a cumulative representation of his/her contributions, we consider the cumulative production $N_{i}(t) \\equiv \\sum _{t^{\\prime }=1}^{t}n_{i}(t^{\\prime })$ as a proxy for career achievement.",
"Fig.",
"REF (A) shows the cumulative production $N_{i}(t)$ of six notable careers which display a temporal scaling relation $N_{i}(t) \\approx A_{i} t^{\\alpha _{i}}$ where $\\alpha _{i}$ is a scaling exponent that quantifies the career trajectory dynamics.",
"The average and standard deviation of the $\\alpha _{i}$ values calculated for each dataset are $\\langle \\alpha _{i}\\rangle = 1.42 \\pm 0.29$ [A], $1.44 \\pm 0.26$ [B], and $1.30 \\pm 0.31$ [C].",
"We justify this 2-parameter model in the SI Appendix text using scaling methods and data collapse.",
"There are also numerous cases of $N_{i}(t)$ which do not exhibit such regularity (see Fig.",
"S1), but instead display marked non-stationarity and non-linearity arising from significant exogenous career shocks.",
"Positive shocks, possibly corresponding to just a single discovery, can spur significant productivity and reputation growth [24], [27].",
"Negative shocks, such as in the case of scientific fraud, can end the career rather suddenly.",
"We also acknowledge that the end of the career is a difficult phase to analyze, since such an event can occur quite abruptly, and so our analysis is mainly concerned with the growth phase and not the termination phase.",
"Figure: Persistent accelerating career growth.",
"(A) The career trajectory N i (t)∼t α i N_{i}(t) \\sim t^{\\alpha _{i}} ofsix stellar careers from varying age cohorts.",
"The α i \\alpha _{i} value characterizes the career persistence, wherecareers with α>1\\alpha >1 are accelerating.",
"α i \\alpha _{i} values calculated using OLS regression in alphabetical order are:α=1.25±0.02\\alpha = 1.25\\pm 0.02,α=1.72±0.02\\alpha = 1.72\\pm 0.02,α=1.62±0.04\\alpha = 1.62\\pm 0.04,α=1.23±0.02\\alpha = 1.23\\pm 0.02,α=1.34±0.05\\alpha = 1.34\\pm 0.05,α=1.35±0.04\\alpha = 1.35\\pm 0.04.",
"(B) Defined in Eq.",
"[], the average career trajectory 〈N ' (t)〉\\langle N^{\\prime }(t) \\rangle calculated from 100 individual N i (t)N_{i}(t) in eachdataset demonstrates robust accelerating career growth within each cohort.",
"We use the normalized career trajectory N i ' (t)N^{\\prime }_{i}(t) in order to aggregate N i (t)N_{i}(t) with varying publication rates 〈n i 〉\\langle n_{i} \\rangle .",
"As a result, the aggregate scaling exponent α ¯\\overline{\\alpha } quantifies the acceleration ofthe typical career over time, independent of 〈n i 〉\\langle n_{i} \\rangle .",
"For the scientific careers, we calculateα ¯\\overline{\\alpha } values: 1.28±0.011.28\\pm 0.01 [A], 1.31±0.011.31\\pm 0.01 [B], and 1.15±0.021.15\\pm 0.02 [C].These values are all significantly greater than unity, α ¯>1\\overline{\\alpha }>1, indicating that cumulative advantage in science is closely related to knowledge and productionspillovers.",
"We calculate α ¯\\overline{\\alpha } using OLS regression andplot the corresponding best-fit lines (dashed) for each dataset.In order to analyze the average properties of $N_{i}(t)$ for all 300 scientists in our sample, we define the normalized trajectory $N^{\\prime }_{i}(t) \\equiv N_{i}(t) / \\langle n_{i} \\rangle $ .",
"The quantity $\\langle n_{i} \\rangle $ is the average annual production of author $i$ , with $N^{\\prime }_{i}(L_{i}) = L_{i}$ by construction ($L_{i}$ corresponds to the career length of individual $i$ ).",
"Fig.",
"REF (B) shows the characteristic production trajectory obtained by averaging together the 100 $N^{\\prime }_{i}(t)$ belonging to each dataset, $\\langle N^{\\prime }(t) \\rangle \\equiv \\Big \\langle \\frac{N_{i}(t)}{ \\langle n_{i} \\rangle } \\Big \\rangle \\equiv \\frac{1}{100}\\sum _{i=1}^{100} \\frac{N_{i}(t)}{ \\langle n_{i} \\rangle } \\ .$ The standard deviation $\\sigma (N^{\\prime }(t))$ shown in Fig.",
"S2(B) begins to decrease after roughly 20 years for dataset [A] and [B] scientists.",
"Over this horizon, the stochastic arrival of career shocks can significantly alter the career trajectory [20], [24], [27], [28].",
"Each $N^{\\prime }_{i}(t)$ exhibits robust scaling corresponding to the scaling law $\\langle N^{\\prime }(t) \\rangle \\sim t^{\\overline{\\alpha }}$ .",
"This regularity reflects the abundance of of careers with $\\alpha _{i} >1$ corresponding to accelerated career growth.",
"This acceleration is consistent with increasing returns arising from knowledge and production spillovers."
],
[
"Fluctuations in scientific output over the academic career",
" Individuals are constantly entering and exiting the professional market, with birth and death rates depending on complex economic and institutional factors.",
"Due to competition, decisions and performance at the early stages of the career can have long lasting consequences [16], [29].",
"To better understand career uncertainty portrayed by the common saying “publish or perish\" [30], we analyze the outcome fluctuation $r_{i}(t) \\equiv n_{i}(t)-n_{i}(t-\\Delta t) \\ $ of career $i$ in year $t$ over the time interval $\\Delta t=1$ year.",
"Fig.",
"REF (A) and (B) show the unconditional pdf of $r$ values which are leptokurtic but remarkably symmetric, illustrating the endogenous frequencies of positive and negative output growth.",
"Output fluctuations arise naturally from the lulls and bursts in both the mental and physical capabilities of humans [31], [32].",
"Moreover, the statistical regularities in the annual production change distribution indicate a striking resemblance to the growth rate distribution of countries, firms, and universities [33], [34].",
"Figure: Empirical evidence for the proportional growth model of career production.",
"(A) Probability density function (pdf) of the annual production change rr in the number of papers published overa Δt=1\\Delta t =1 year period.In the bulk of each P(r)P(r), the growth distribution is approximately double-exponential (Laplace).",
"(B) To test the stability of the distribution over career trajectory subintervals, we separate r i (t)r_{i}(t) valuesinto 5 non-overlapping 10-year periods and verify the stability of the Laplace P(r)P(r).",
"For each P(r)P(r), we also plot thecorrespondingLaplace distribution (solid line) with standard deviation σ\\sigma and mean μ≈0\\mu \\approx 0 calculated using the maximumlikelihood estimator method.",
"To improve graphical clarity, we vertically offset each P(r)P(r) by a constant factor.For visual comparison, we also plot a Normal distribution (dashed black curve) with σ≡1\\sigma \\equiv 1which instead decays parabolically on the log-linear axes.",
"(C) Accounting for individual production factors by using the normalized production change r ' r^{\\prime }, the resulting pdfs P(r ' )P(r^{\\prime })collapse onto a Gaussian distribution with unit variance.Deviations in the tails likely correspond to extreme “career shocks.” (D)The cumulative distribution CDF(X≥S i )CDF(X\\ge S_{i}) is exponential, indicating that the unconditional distributions P(r)P(r) in (A) and (B) follow from anexponential mixing of conditional Gaussian distributions P(r|S i )P(r|S_{i}).To better account for individual growth factors, we next define the normalized production change $r^{\\prime }_{i}(t) \\equiv [r_{i}(t)- \\langle r_{i} \\rangle ] / \\sigma _{i}(r)$ which is measured in units of the fluctuation scale $\\sigma _{i}(r)$ unique to each career.",
"We measure the average $\\langle r_{i} \\rangle $ and the standard deviation $\\sigma _{i}(r)$ of each career using the first $L_{i}$ available years for each scientist $i$ .",
"$r^{\\prime }_{i}(t)$ is a better measure for comparing career uncertainty, since individuals have production factors that depend on the type of research, the size of the collaboration team, and the position within the team.",
"Fig.",
"REF (C) show that $P(r^{\\prime })$ , the probability density function (pdf) of $r^{\\prime }$ measured in units of standard deviation, is well approximated by a Gaussian distribution with unit variance.",
"The data collapse of each $P(r^{\\prime })$ onto the predicted Gaussian distribution (solid green curve) indicates that individual output fluctuations are consistent with a proportional growth model.",
"We note that the remaining deviations in the tails for $\\vert r^{\\prime } \\vert \\ge 3$ are likely signatures of the exogenous career shocks that are not accounted for by an endogenous proportional growth model.",
"The ability to collaborate on large projects, both in close working teams and in extreme examples as remote agents (i.e.",
"Wikipedia [35]), is one of the foremost properties of human society.",
"In science, the ability to attract future opportunities is strongly related to production and knowledge spillovers [28], [36], [37] that are facilitated by the collaboration network [7], [12], [38], [39], [40], [41], [42].",
"Indeed, there is a tipping point in a scientific career that occurs when a scientist's knowledge investment reaches a critical mass that can sustain production over a long horizon, and when a scientist becomes an attractor (as opposed to a pursuer) of new collaboration/production opportunities.",
"To account for collaboration, we calculate for each author the number $k_{i}(t)$ of distinct coauthors per year and then define his/her collaboration radius $S_{i}$ as the median of the set of his/her $k_{i}(t)$ values, $S_{i} \\equiv Med[k_{i}(t)]$ .",
"We use the median instead of the average $\\langle k_{i}(t) \\rangle $ since extremely large $k_{i}(t)$ values can occur in specific fields such as high-energy physics and astronomy.",
"Figure: Quantitative relations between career growth, career risk, and collaboration efficiency.The fluctuations in production reflect theunpredictable horizon of “career shocks” which can affect the ability of a scientists to access newcreative opportunities.",
"(A) Relation between average annual production 〈n i 〉\\langle n_{i} \\rangle and collaboration radius S i ≡Med[k i ]S_{i}\\equiv Med[ k_{i}] shows a decreasing marginal output per collaborator as demonstrated by sublinear ψ<1\\psi <1.",
"Interestingly,dataset [A] scientists have on average a larger output-to-input efficiency.",
"(B) The production fluctuation scale σ i (r)\\sigma _{i}(r) is a quantitative measure for uncertainty in academiccareers, withscaling relation σ i (r)∼S i ψ/2 \\sigma _{i}(r) \\sim S_{i}^{\\psi / 2}.",
"(C) Management, coordination, and training inefficiencies can result in a γ<1\\gamma <1corresponding to a decreasing marginal return with each additional coauthor input.",
"The significantly larger γ\\gamma value fordataset [A] scientists seems to suggest that managerial abilities related to output efficiency is a common attribute of top scientists.Given the complex scientific coauthorship network, we ask the question: what is the typical number of unique coauthors per year?",
"Fig.",
"REF (D) shows the cumulative distribution function $CDF(S_{i})$ of $S_{i}$ values for each data set.",
"The approximately linear form on log-linear axes indicates that $S_{i}$ is exponentially distributed, $P(S_{i}) \\sim \\exp [-\\lambda S_{i}]$ .",
"We calculate $\\lambda = 0.15 \\pm 0.01$ [A], $\\lambda = 0.11\\pm 0.01$ [B], and $\\lambda = 0.11 \\pm 0.01$ [C].",
"The exponential size distribution has been shown to emerge in complex systems where linear preferential attachment governs the acquisition of new opportunities [43].",
"This result shows that the leptokurtic “tent-shaped” distribution $P(r)$ in Fig.",
"REF follows from the exponential mixing of heterogenous conditional Gaussian distributions [44].",
"The exponential mixture of Gaussians decomposes the unconditional distribution $P(r)$ into a mixture of conditional Gaussian distributions $P(r| S_{i}) = \\exp [ -r^{2}/2VS_{i}^{\\psi }] / \\sqrt{2\\pi VS_{i}^{\\psi }} \\ ,$ each with a fluctuation scale $\\sigma _{i}(r)$ depending on $S_{i}$ by the scaling relation $\\sigma ^{2}_{i}(r) \\approx V S_{i}^{\\psi } \\ .$ Hence, the mixture is parameterized by $\\psi $ $P_{\\psi }(r) = \\int _{0}^{\\infty } P(r|S) P(S) dS \\approx \\sum _{i=1}P_{i}(r|S_{i}) P(S_{i}) \\ .$ The independent case $\\psi =0$ results in a Gaussian $P_{\\psi }(r)$ and the linear case $\\psi =1$ results in a Laplace (double-exponential) $P_{\\psi }(r)$ .",
"See the SI Appendix text and ref.",
"[44] for further discussion of the $\\psi $ dependence of $P_{\\psi }(r)$ ."
],
[
"The size-variance relation and group efficiency",
"The values of $\\psi $ for scientific and athletic careers follow from the different combination of physical and intellectual inputs that enter the production function for the two distinct professions.",
"Academic knowledge is typically a non-rival good, and so knowledge-intensive professions are characterized by spillovers, both over time and across collaborations [36], [37], consistent with $\\alpha _{i}>1$ and $\\psi >0$ .",
"Interestingly, Azoulay et al.",
"show evidence for production spillovers in the 5–8% decrease in output by scientists who were close collaborators with a “superstar” scientists who died suddenly [28].",
"We now formalize the quantitative link between scientific collaboration [38], [39] and career growth given by the size-variance scaling relation in Eq.",
"[REF ] visualized in the scatter plot in Fig.",
"REF (B).",
"Using ordinary least squares (OLS) regression of the data on log-log scale, we calculate $\\psi /2 \\approx 0.40\\pm 0.03$ ($R =0.77$ ) for dataset [A], $\\psi /2 \\approx 0.22 \\pm 0.04$ ($R = 0.51$ ) [B], and $\\psi /2 \\approx 0.26 \\pm 0.05$ ($R =0.45$ ) [C].",
"Interdependent tasks characteristic of group collaborations typically involve partially overlapping efforts.",
"Hence, the empirical $\\psi $ values are significantly less than the value $\\psi = 1$ that one would expect from the sum of $S_{i}$ independent random variables with approximately equal variance $V$ .",
"Collectively, these empirical evidences serve as coherent motivations for the the preferential capture growth model that we propose in the following section.",
"Alternatively, it is also possible to estimate $\\psi $ using the relation between the average annual production $\\langle n_{i} \\rangle $ and the collaboration radius $S_{i}$ .",
"The input-output relation $\\langle n_{i} \\rangle \\sim S_{i}^{\\psi }$ quantifies the collaboration efficiency, with $\\psi = 0.74 \\pm 0.04$ ($R=0.87$ ) for dataset [A] and $\\psi = 0.25 \\pm 0.04$ ($R=0.37$ ) for dataset [B].",
"If the autocorrelation between sequential production values $n_{i}(t)$ and $n_{i}(t+1)$ is relatively small, then we expect the scaling exponents calculated for $\\langle n_{i} \\rangle $ and $\\sigma ^{2}_{i}(r)$ to be approximately equal.",
"This result follows from considering $r_{i}(t)$ as the convolution of an underlying production distribution $P_{i}(n)$ for each scientist that is approximately stable.",
"Interestingly, the larger $\\psi $ values calculated for dataset [A] scientists suggests that prestige is related to the increasing returns in the scientific production function [45].",
"Next we use an alternative method to estimate the annual collaboration efficiency by relating the number of publications $n_{i}(t)$ in a given year to the number of distinct coauthors $k_{i}(t)$ over the same year.",
"We use a single-factor production function, $n_{i}(t) \\approx q_{i}[k_{i}(t)]^{\\gamma _{i}} \\ ,$ to quantify the relation between output and labor inputs with a scaling exponent $\\gamma _{i}$ .",
"We estimate $q_{i}$ and $\\gamma _{i}$ for each author using OLS regression, and define the normalized output measure $Q_{i} \\propto n_{i}(t)/k_{i}(t)^{\\gamma _{i}}$ using the best-fit $q_{i}$ and $\\gamma _{i}$ values calculated for each scientist $i$ .",
"Fig.",
"REF (C) shows the efficiency parameter $\\gamma $ calculated by aggregating all careers in each dataset, and indicates that this aggregate $\\gamma $ is approximately equal to the average $ \\langle \\gamma _{i}\\rangle $ calculated from the $\\gamma _{i}$ values in each career dataset: $\\gamma = 0.68 \\pm 0.01$ [A], $\\gamma = 0.52 \\pm 0.01$ [B], and $\\gamma = 0.51 \\pm 0.02$ [C].",
"Furthermore, the $\\psi $ and $\\gamma $ values are approximately equal, which is not surprising, since both scaling exponents are efficiency measures that relate the scaling relation of output $n_{i}(t)$ per input $k_{i}(t)$ ."
],
[
"A Proportional growth model for scientific output",
" We develop a stochastic model as a heuristic tool to better understand the effects of long-term versus short-term contracts.",
"In this competition model, opportunities (i.e.",
"new scientific publications) are captured according to a general mechanism whereby the capture rate $\\mathcal {P}_{i}(t)$ depends on the appraisal $w_{i}(t)$ of an individual's record of achievement over a prescribed history.",
"We define the appraisal to be an exponentially weighted average over a given individual's history of production $w_{i}(t) \\equiv \\sum _{\\Delta t =1}^{t-1} n_{i}(t-\\Delta t) e^{-c\\Delta t} \\ ,$ which is characterized by the appraisal horizon $1/c$ .",
"We use the value $c=0$ to represent a long-term appraisal (tenure) system and a value $c \\gg 1$ to represent a short-term appraisal system.",
"Each agent $i=1...I$ simultaneously attracts new opportunities at a rate $\\mathcal {P}_{i}(t) = \\frac{w_{i}(t)^{\\pi }}{\\sum _{i=1}^{I} w_{i}(t)^{\\pi }} \\ .$ until all $P$ opportunities for a given period $t$ are captured.",
"We assume that each agent has the production potential of one unit per period, and so the total number of opportunities distributed per period $P$ is equal to the number of competing agents, $P\\equiv I$ .",
"Figure: Monte Carlo simulation of the linear preferential capture model (π=1\\pi = 1) for varying contract length parametrized by cc.We plot the probability distributions for (i) N i N_{i}, the total number of opportunities captured by the end period TT, (ii) thegrowth acceleration exponent α i \\alpha _{i}, (iii) the single period growth fluctuation r i (t)r_{i}(t) including for comparison the Laplace (solid green) and Gaussian (dashed red) best-fit distributions calculated using the respective MLE estimator, and (iv) the career longevity L i L_{i} defined as the time difference between an agent's first and last captured opportunity.",
"Results for c→0c \\rightarrow 0 systems shows that for a “long-term appraisal” scenario careers are less vulnerable to low-productionphases, and as a result, most agents sustain production throughout the career.",
"Conversely, results for c≥1c\\ge 1 systems show that for a “short-term appraisal” scenario the labor system is driven by fluctuations that can cause career“sudden death” for a large fraction of the population.",
"In this short-term appraisal model, there are typically a small number of agents who are able to capture the majority of the production opportunities with remarkably accelerating career growth reflected by significantly large α i ≥1\\alpha _{i} \\ge 1.",
"Thus, a few “lucky” agents are able to survive theinitial fluctuations and end up dominating the system.",
"In the SI text and Figs.",
"S12-S16, we further show that systems with increased levels of competition (π>1\\pi >1) mimic systems with short term contracts, resultingin productivity “death traps” whereby most careers stagnate and terminate early.We use Monte Carlo (MC) simulation to analyze this 2-parameter model over the course of $t=1...T$ sequential periods.",
"In each production period (i.e.",
"representing a characteristic time to publication), a fixed number of $P$ production units are captured by the competing agents.",
"At the end of each period, we update each $w_{i}(t)$ and then proceed to simulate the next preferential capture period $t+1$ .",
"Since $\\mathcal {P}_{i}(t)$ depends on the relative achievements of every agent, the relative competitive advantage of one individual over another is determined by the parameter $\\pi $ .",
"In the SI Appendix text we elaborate in more detail the results of our simulation of synthetic careers dynamics.",
"We vary $\\pi $ and $c$ for a labor force of size $I\\equiv 1000$ and maximum lifetime $T\\equiv 100$ periods as a representative size and duration of a real labor cohort.",
"Our results are general, and for sufficiently large system size, the qualitative features of the results do not depend significantly on the choice of $I$ or $T$ .",
"The case with $\\pi = 0$ corresponds to a random capture model that has (i) no appraisal and (ii) no preferential capture.",
"Hence, in this null model, opportunities are captured at a Poisson rate $\\lambda _{p} = 1$ per period.",
"The results of this model (see Fig.",
"S13) shows that almost all careers obtain the maximum career length $T$ with a typical career trajectory exponent $\\langle \\alpha _{i} \\rangle \\approx 1$ .",
"Comparing to simulations with $\\pi >0$ and $c\\ge 0$ , the null model is similar to a “long-term” appraisal system ($c\\rightarrow 0$ ) with sub-linear preferential capture ($\\pi <1$ ).",
"In such systems, the long-term appraisal timescale averages out fluctuations, and so careers are significantly less vulnerable to periods of low production and hence more sustainable since they are not determined primarily by early career fluctuations.",
"However, as $\\pi $ increases, the strength of competitive advantage in the system increases, and so some careers are “squeezed out” by the larger more dominant careers.",
"This effect is compounded by short-term appraisal corresponding to $c\\approx 1$ .",
"In such systems with super-linear capture rates and/or relatively large $c$ , most individuals experience “sudden death” termination relatively early in the career.",
"Meanwhile, a small number of “stars” survive the initial selection process, which is governed primarily by random chance, and dominate the system.",
"We found drastically different lifetime distributions when we varied the appraisal (contract) length (see Figs.",
"S12 – S16).",
"In the case of linear preferential capture with a long-term appraisal system $c=0$ , we find that 10% of the labor population terminates before reaching career age $0.94T$ (where $T$ is the maximum career length or “retirement age”), and only 25% of the labor population terminates before reaching career age $0.98T$ .",
"On the contrary, in a short-term appraisal system with $c=1$ , we find that 10% of the labor population terminates before reaching age $0.01T$ , and 25% of the labor population dies before reaching age $0.02T$ (see Table S1).",
"Hence, in model short contract systems, the longevity, output, and impact of careers are largely determined by fluctuations and not by persistence.",
"Fig.",
"REF shows the MC results for $\\pi =1$ .",
"For $c\\ge 1$ we observe a drastic shift in the career longevity distribution $P(L)$ , which becomes heavily right-skewed with most careers terminating extremely early.",
"This observation is consistent with the predictions of an analytically solvable Matthew effect model [16] which demonstrates that many careers have difficulty making forward progress due to the relative disadvantage associated with early career inexperience.",
"However, due to the nature of zero-sum competition, there are a few “big winners” who survive for the entire duration $T$ and who acquire a majority of the opportunities allocated during the evolution of the system.",
"Quantitatively, the distribution $P(N)$ becomes extremely heavy-tailed due to agents with $\\alpha >2$ corresponding to extreme accelerating career growth.",
"Despite the fact that all the agents are endowed initially with the same production potential, some agents emerge as superstars following stochastic fluctuations at relatively early stages of the career, thus reaping the full benefits of cumulative advantage."
],
[
"Discussion",
"An ongoing debate involving academics, university administration, and educational policy makers concerns the definition of professorship and the case for lifetime tenure, as changes in the economics of university growth have now placed tenure under the review process [3], [6].",
"Critics of tenure argue that tenure places too much financial risk burden on the modern competitive research university and diminishes the ability to adapt to shifting economic, employment, and scientific markets.",
"To address these changes, universities and other research institutes have shifted away from tenure at all levels of academia in the last thirty years towards meeting staff needs with short-term and non-tenure track positions [3].",
"For knowledge intensive domains, production is characterized by long-term spillovers both through time and through the knowledge network of associated ideas and agents.",
"A potential drawback of professions designed around short-term contracts is that there is an implicit expectation of sustained annual production that effectively discounts the cumulative achievements of the individual.",
"Consequently, there is a possibility that short-term contracts may reduce the incentives for a young scientist to invest in human and social capital accumulation.",
"Moreover, we highlight the importance of an employment relationship that is able to combine positive competitive pressure with adequate safeguards to protect against career hazards and endogenous production uncertainty an individual is likely to encounter in his/her career.",
"In an attempt to render a more objective review process for tenure and other lifetime achievement awards, quantitative measures for scientific publication impact are increasing in use and variety [20], [17], [18], [19], [24], [27], [46], [47].",
"However, many quantifiable benchmarks such as the $h$ -index [17] do not take into account collaboration size or discipline specific factors.",
"Measures for the comparison of scientific achievement should at least account for variable collaboration, publication, and citation factors [19], [46], [47].",
"Hence, such open problems call for further research into the quantitative aspects of scientific output using comprehensive longitudinal data for not just the extremely prolific scientists, but the entire labor force.",
"Current scientific trends indicate that there will be further increases in typical team sizes that will forward the emergent complexity arising from group dynamics [7], [12], [42].",
"There is an increasing need for individual/group production measures, such as the output measure $Q$ , following from Eq.",
"[REF ], which accounts for group efficiency factors.",
"Normalized production measures which account for coauthorship factors have been proposed in [19], [46], but the measures proposed therein do not account for the variations in team productivity.",
"The complexity of large collaborations raises open questions concerning scientific productivity and the organization of teams.",
"We measure a decreasing marginal returns $\\gamma <1$ with increasing group size which identifies the importance of team management.",
"A theory of labor productivity can help improve our understanding of institutional growth, for organizations ranging in size from scientific collaborations to universities, firms, and countries [33], [34], [44], [47], [48], [49], [50]."
],
[
"Acknowledgements",
"We thank D. Helbing, N. Dimitri, and O. Penner and an anonymous PNAS Board Member for insightful comments.",
"We gratefully acknowledge support from the IMT and Keck Foundations, the U.S. Defense Threat Reduction Agency (DTRA), Office of Naval Research (ONR), and the NSF Chemistry Division (grants CHE 0911389 and CHE 0908218).",
"Supporting Information Appendix Persistence and Uncertainty in the Academic Career Alexander M. Petersen,$^{1}$ Massimo Riccaboni,$^{2}$ , H. Eugene Stanley$^{3}$ , Fabio Pammolli $^{1,2,3}$ $^{1}$ Laboratory for the Analysis of Complex Economic Systems, IMT Lucca Institute for Advanced Studies, Lucca 55100, Italy $^{2}$ Laboratory of Innovation Management and Economics, IMT Lucca Institute for Advanced Studies, Lucca 55100, Italy $^{3}$ Center for Polymer Studies and Department of Physics, Boston University, Boston, Massachusetts 02215, USA (2012) [1] Corresponding author: Alexander M. Petersen E-mail: [email protected]"
],
[
"Data",
"To test the intriguing possibility that competition leads to common growth patterns in complex systems of arbitrary size $S$ , we analyze the production dynamics of two professions that are dissimilar in many regards, but share the common underlying driving force of competition for limited resources.",
"In order to establish empirical facts that we believe are independent of the details of a given competitive profession, we analyze a large dataset of production $n_{i}(t)$ values and corresponding growth fluctuation $r_{i}(t) \\equiv n_{i}(t)-n_{i}(t-1)$ values.",
"We define the appropriate measures for $n_{i}(t)$ to be (a) the annual number of papers published by scientist $i$ and (b) the seasonal performance metrics of professional athlete $i$ .",
"While these two professions both display a high level of competition, they differ in their employment term structure and salary scale.",
"In the case of academia, the tenure system rewards high performance levels with lifelong employment (tenure).",
"In contrast, professional sports are characterized by relatively short contracts that emphasize continued performance over a shorter time frame and thereby exploit the high levels of athletic prowess in a player's peak years.",
"The large number of careers in these two professions readily lend themselves to quantitative analysis because the data that quantify the career production trajectory are precisely defined and comprehensive throughout an individual's entire career.",
"Furthermore, because of the generic nature of competition, we use these two distinct professions to compare and contrast the distribution of career impact measures across a cohort of competitors.",
"The datasets we analyze are: Academia: We analyze the publication careers of 300 physicists which we categorize in 3 subsets each consisting of 100 individuals: (A) Dataset A corresponds to the 100 most-cited physicists according to the citation shares metric [19] (with average $h$ -index $\\langle h \\rangle = 61 \\pm 21$ ).",
"These 100 careers constitute 3,951 $r_{i}(t)$ values.",
"(B) Dataset B corresponds to the 100 other “control\" scientists, taken approximately randomly from the same physics database (with average $h$ -index $\\langle h \\rangle = 44 \\pm 15$ ).",
"In the selection process for dataset B, we only consider scientists who have published between 10 and 50 articles in PRL over the 50-year period 1958-2008.",
"These 100 careers constitute 3,534 $r_{i}(t)$ values.",
"(C) Dataset C corresponds to 100 Assistant Professors (with average $h$ -index $\\langle h \\rangle = 15 \\pm 7$ ), where we select two physicists from each of the top-50 U.S Physics & Astronomy Departments (according to the U.S. News rankings).",
"These Asst.",
"Profs.",
"are assumed to be early in their career and relatively accomplished given the difficulty in obtaining such a position in any given university.",
"These 100 careers constitute 1,050 $r_{i}(t)$ values.",
"In order to control for discipline-specific citation patterns, we select individuals in dataset A and B from set of all scientists who have published in Physical Review Letters (PRL) over the 50-year period 1958–2008.",
"As a measure of output, we define $n_{i}(t)$ as the number of papers published in year $t$ of the career of individual $i$ , where year $t=1$ corresponds to the year of the first publication on record for author $i$ .",
"We downloaded the complete publication records of the scientists in datasets A and B from ISI Web of Science (http://www.isiknowledge.com/) in Jan. 2010, and we downloaded the complete publication records of the scientists in dataset C from ISI Web of Science in Oct. 2010.",
"We used the “Distinct Author Sets” function provided by ISI in order to increase the likelihood that only papers published by each given author are analyzed.",
"Major League Baseball (MLB): We analyze 17,292 baseball players over the 90-year period 1920-2009 using comprehensive league data obtained from Sean Lahman's Baseball Archive accessed at http://baseball1.com/index.php.",
"We separate the career data into two distinct subsets: non-pitchers (players not on record as having pitched during a game) and pitchers.",
"(A) For non-pitchers, we analyze two batting metrics: an “opportunity metric” - at-bats (AB), and a “success” metric - hits (H).",
"Together, these 8,993 careers constitute 43,043 $r_{i}(t)$ values.",
"(B) For pitchers, we analyze two pitching metrics: an “opportunity metric” - innings-pitched measured in outs (IPO), and a “success” metric - strikeouts (K).",
"Together, these 8,299 careers constitute 33,965 $r_{i}(t)$ values.",
"National Basketball Association (NBA): We analyze 3,864 basketball careers, constituting 15,316 $r_{i}(t)$ values, over the 63-year period 1946–2008 using data obtained from Data Base Sports Basketball Archive accessed at http://www.databasebasketball.com/.",
"We analyze two player metrics: (A) an “opportunity metric” - minutes played (Min.",
"), and (B) a “success” metric - points scored (Pts.)",
"Since sports careers typically peak for athletes around age 30, we account for a time-dependent career trajectory which is dominant in most sports careers by “detrended\" the measures for career growth fluctuations.",
"In the case where we do not account for a individual fluctuation scale, $R_{i} \\equiv [r_{i}(t)-\\overline{r}(t)] / \\sigma (t) \\ .$ In this case we detrend with respect to the average production difference $\\overline{r}(t)$ and the standard deviation of production difference $\\sigma (t)$ which are calculated using all careers from a given sports league, conditional on the career year $t$ .",
"In the case where we do account for individual variations, we first define $z_{i}(t) \\equiv (r_{i}(t)-\\langle r_{i} \\rangle )/\\sigma _{i}$ to be normalized with respect to the individual career scales $\\langle r_{i} \\rangle $ and $\\sigma _{i}$ which are the average and standard deviation of the production change of athlete career $i$ .",
"Then we define the detrended growth rate as $R^{\\prime }_{i} \\equiv [z_{i}(t)- \\langle z(t) \\rangle ] / \\sigma _{z(t)} \\ ,$ where in this case we detrend with respect to the average $\\langle z(t) \\rangle $ and standard deviation $\\sigma _{z(t)}$ calculated by collecting all $z_{i}(t)$ values for a given career year $t$ .",
"This detrending better accounts for the relatively strong time-dependent growth patterns in sports.",
"In this section we analyze the annual production of scientists measured as the number of papers published $n_{i}(t)$ over the period of a year.",
"Using this measure does not account for the variability in the length of production, say in the number of pages, nor does it account for the impact of the paper, a quantity commonly approximated by a paper's citation number.",
"Instead, we consider a simple definition that a scientific product is a final output of a collection of inputs.",
"Furthermore, in science it is assumed that the peer review process establishes a quality threshold so that only manuscripts above a certain quality and novelty standard can be published and incorporated into the scientific body of knowledge.",
"Prior theories of scientific production have also used the number of publications as a proxy for scientific output.",
"In particular, the Shockely model [14] proposed a simple multiplicative factor model for the production $n_{i}(t)$ which predicts a log-normal distribution for $P(n)$ .",
"An alternative null model for $n_{i}(t)$ is the Poisson process, which assumes that each individual is endowed with a rate parameter $\\omega $ related to an individual's production factors.",
"This model predicts a Poisson distribution for $P(n)$ .",
"However, a shortfall of these models is that multiplicative parameters in the Shockley model and the rate parameter $\\omega $ are difficult to measure, especially if the set of individuals span a large range of production factors, and moreover, if the careers are non-stationary.",
"Fig.",
"REF shows the unconditional probability distribution $P(n)$ calculated by aggregating all $n_{i}(t)$ values for all scientists and all years into an aggregate dataset.",
"Naively, the distributions are well-fit by the Log-normal distribution, and so there is an apparent agreement with the multiplicative factor Shockley model.",
"However, the distribution $P(n) = \\sum _{i=1}^{100} P(n|S_{i})$ is the aggregate distribution constructed from 100 individual career trajectories $n_{i}(t)$ , each with varying size $S_{i}$ .",
"Indeed, we demonstrate in Figs.",
"REF and REF to be non-linear, with time-dependent residuals around the moving average.",
"Hence, it is not possible from the unconditional pdf $P(n)$ to determine if the process underlying scientific production corresponds to a simple multiplicative process or a Poisson process.",
"In order to better account for the variable size $S_{i}$ of each career which affects the rate at which an individual is able to capture publication opportunities, we plot in Fig.",
"REF the pdf of the normalized output $Q_{i}=\\frac{n_{i}(t)}{f_{i}(k)} \\ .$ We calculate the normalization factor $f_{i}(k) = q_{i}[k_{i}(t)]^{\\gamma _{i}}$ for each individual $i$ by estimating the parameters $q_{i}$ and $\\gamma _{i}$ for each scientist $i$ from the single-factor model $n_{i} = q_{i}k_{i}^{\\gamma _{i}} \\ .$ where $n_{i}(t)$ is the annual production in year $t$ and $k_{i}(t)$ is the total number of distinct coauthors in year $t$ .",
"Hence, $Q_{i}$ represents the production factor above $Q>1$ or below $Q<1$ what would be expected from the author $i$ given the fact that he/she had additional inputs from $k_{i}(t)-1$ individuals that year.",
"This model assumes that the major component contributing to production is the collaboration degree $k$ of the research output, and also assumes that the input of each coauthor contributes equally to the final output.",
"Clearly, these assumptions neglect some important idiosyncratic details affecting scientific publication, but given the incomplete information associated with every publication, it is a decent approximation.",
"We estimate $q_{i}$ and $\\gamma _{i}$ by performing a linear regression of $\\log n_{i}$ and $\\log k_{i}$ using the first $L_{i}$ years of each career, neglecting years with $n_{i}=0$ .",
"We use $L_{i}=35$ years for dataset [A] and [B] scientists, and $L_{i} = 10$ years for dataset [C] scientists.",
"In Fig.",
"REF (c) we approximate $\\gamma $ using all $n(t)$ within each dataset with $k\\le 50$ , and performing a regression of the model $\\ln n = \\ln q + \\gamma \\ln k + \\epsilon $ to estimate $\\gamma $ , where $\\epsilon $ is the residual due to other unaccounted production factors.",
"For each dataset we find that the aggregate efficiency parameter $\\gamma $ is approximately equal to the average $ \\langle \\gamma _{i}\\rangle $ calculated from the 100 $\\gamma _{i}$ values in each career dataset: $\\gamma = 0.68 \\pm 0.01$ [A], $\\gamma = 0.52 \\pm 0.01$ [B], and $\\gamma = 0.51 \\pm 0.02$ [C].",
"Furthermore, the $\\psi \\approx \\gamma $ since the size-variance scaling parameter $\\psi $ is also an efficiency measure that relates the scaling of output $n$ to input $k$ .",
"As a result of this analysis, we quantify the scaling exponent $\\gamma <1$ of the decreasing marginal returns in the scientific production function for projects with $k\\le 50$ .",
"This likely stems from the inefficient management costs associated with large group collaborations which typically manifest in a larger production timescale.",
"In fact, for years with $k\\ge 50$ coauthors, scientific output shows decreasing returns to scale.",
"Interestingly, the star scientists in dataset [A] display significantly larger efficiency, quantitatively showing the importance of management skills in scientific success.",
"The normalized production values are normalized to units of “expected production\" conditional on the $k_{i}$ inputs for author $i$ .",
"We aggregate all data from each dataset and show in Fig.",
"REF that the $Q$ values are well-described by the Gamma distribution $P(Q) = Q^{m-1} \\frac{\\exp [-Q/\\theta ]}{\\theta ^{m} \\Gamma (m)}$ where $m$ is the shape parameter and $\\theta $ is the scale parameter.",
"Surprisingly, we find that dataset [A] and [B] have approximately equal Gamma parameters, indicating that besides their production efficiency, top scientists are virtually indistinguishable with average normalized output $\\langle Q \\rangle = m \\theta >1$ .",
"For each dataset we calculate the Gamma parameters using the maximum likelihood estimator method: $m=5.45$ and $\\theta = 0.21$ [A], $m=5.60$ and $\\theta =0.20$ [B], and $m =7.00$ and $\\theta = 0.15$ [C].",
"We leave it as an open question to determine why the Gamma distribution describes so well the production statistics.",
"We ponder the intriguing possibility that the stochastic dynamics underlying individual production corresponds to an increasing Lévy process with variable jump length which is known to produce a Gamma distribution."
],
[
"Quantifying the Career Trajectory",
"The reputation of an individual is typically cumulative, based on the total sum of achievements, which we approximate by the cumulative output $N_{i}(t)$ (e.i.",
"number of papers published by year $t$ ).",
"In Figs.",
"REF and REF we plot $N_{i}(t)$ for several individuals.",
"The careers presented in Fig.",
"REF are more linear, indicating quantifiable career trajectory that has the approximate form $N_{i}(t) = \\sum _{t^{\\prime }=1}^{t} n_{i}(t^{\\prime }) \\approx A_{i} \\ t^{\\alpha _{i}} \\ , \\ \\ t < T_{i}$ where $n_{i}(t)$ are the number of papers in year $t$ of the scientist's career which begins with $t \\equiv 1$ in the year of his/her first publication, and begins to decline around time $T_{i}$ which is the time horizon over which the scaling regularity holds before termination and aging effects begin to dominate the career.",
"In our analysis of academic career trajectories $N_{i}(t)$ , we only analyze $N_{i}(t)$ for $t\\le 40$ years in order to account for such termination affects.",
"The smooth career trajectories which appear as a linear curve when plotted on log-log scale are characterized by an amplitude parameter $A_{i}$ and a scaling exponent $\\alpha _{i}$ .",
"However, as indicated by Fig.",
"REF , there are also non-stationary $N_{i}(t)$ which are dominated by “career shocks” that significantly alter the career trajectory.",
"Such career shocks have been demonstrated using publication impact measures (e.i.",
"citations, and h-index sequences) [24], [27], [20], and here we show that they even occur at the more fundamental level of individual production dynamics.",
"In order to analyze the characteristic properties of $N_{i}(t)$ for all 300 scientists analyzed, we define the normalized trajectory $N^{\\prime }_{i}(t) \\equiv N_{i}(t) / \\langle n_{i} \\rangle $ , where $\\langle n_{i}(t) \\rangle $ is the average annual production rate of author $i$ , and so by construction $N^{\\prime }_{i}(L_{i}) = L_{i}$ .",
"Fig.",
"REF (A) shows the characteristic production trajectory obtained by averaging the 100 individual $N^{\\prime }_{i}(t)$ for each dataset, $\\langle N^{\\prime }(t) \\rangle \\equiv \\Big \\langle \\frac{N_{i}(t)}{ \\langle n_{i} \\rangle } \\Big \\rangle \\equiv \\frac{1}{100}\\sum _{i=1}^{100} \\frac{N_{i}(t)}{ \\langle n_{i} \\rangle } \\ .$ The standard deviation $\\sigma ( N^{\\prime }(t))$ is shown in Fig.",
"REF (B), which has a broad peak that is a likely signature of career shocks that can significantly alter the career trajectory.",
"The characteristic trajectory for each dataset are well-approximated by the scaling relation $\\langle N^{\\prime }(t) \\rangle \\sim t^{\\overline{\\alpha }}$ with characteristic scaling exponents $ \\overline{\\alpha }>1$ that are significantly greater than unity: $ \\overline{\\alpha }= 1.28\\pm 0.01$ for Dataset A, $ \\overline{\\alpha }=1.31\\pm 0.01$ for Dataset B, and $ \\overline{\\alpha }=1.15\\pm 0.02$ for Dataset C. This fact implies that there is a significant cumulate advantage in scientific careers which allows for the career trajectory to be accelerating.",
"In Fig.",
"REF (C) and REF (D) we plot the analogous $\\langle N^{\\prime }(t) \\rangle $ curves for professional sports metrics, where for this profession, $\\overline{\\alpha }\\approx 1$ for all measures analyzed.",
"This quantitative feature is likely due to the fact that annual production in professional sports is capped by the limited number of opportunities provided by a season, whereas in academics, the number of publications a scientist can publish is in principle unlimited.",
"Also, in more labour-intensive activities are likely to experience smaller returns since physical labor is non-cumulative with less spillover through time.",
"In Fig.",
"REF we plot each individual career trajectory using the rescaled time $t^{\\prime }_{i} = t^{\\alpha _{i}}$ as an additional visual test of the scaling model given by Eq.",
"REF .",
"We show that on average, all curves $i=1..300$ approximately collapse onto the expected curve $N_{i}(t)/A_{i} = t^{\\prime }$ , where the residual difference $ \\epsilon _{i}(t^{\\prime }) \\equiv N_{i}(t)/A_{i}-t^{\\prime }$ are likely due to career shocks of various magnitudes.",
"We plot the average and standard deviation of each set of 100 $N_{i}(t)/A_{i}$ curves which show that most of the shocks $\\epsilon _{i}(t^{\\prime })$ , with some significant exceptions, lie within the 1$\\sigma $ standard deviation denoted by the error bars.",
"In Fig.",
"REF we plot the probability distributions $P(\\alpha _{i})$ for each academic dataset.",
"For each dataset, the average value $\\langle \\alpha _{i} \\rangle $ is in good agreement with $\\overline{\\alpha }$ , the scaling parameter calculated for the corresponding trajectory $\\langle N^{\\prime }(t) \\rangle $ ."
],
[
"Exponential Mixing of Gaussians",
"The idea that entities are independent and identically distributed is an unrealistic assumption commonly made in analyses of complex systems.",
"The unconditional pdf $P(r)$ is commonly analyzed in empirical studies where insufficient data are present to define normalized $r_{i}^{\\prime }$ measures for each sample constituent $i$ .",
"Nevertheless, when modeling the evolution of complex based on empirical data corresponding to distinct subunits (such as individual careers, companies, or nation regions), unconditional quantities that account for variations in underlying production factors should be used.",
"In the case of scientific output, there are many production factors that combine together and determine the amount of human efforts needed to produce a unit of production.",
"In general, consider the value $f_{i,j}$ of individual $i$ corresponding to his/her relative abilities in the production factor $j=1...J$ corresponding to a variety of attributes: knowledge, genius, persistence, reputation, mental and physical health, communication skills, organization skills, and access to technology, equipment and data, etc.",
"In this study, we compare scientists who publish in similar journals.",
"Still, the scientific input required for each scientific output can vary by a large amount, largely depending on the technology needed to perform the analysis, ranging from particle accelerators to just a pencil and paper.",
"In a very generalized representation, an unconditional distributions $P(r)$ , such as shown in Fig.",
"REF (a-d) for production change $r$ , may follow from a mixture of conditional Gaussian distributions $P(r|S_{i})$ $P_{\\psi }(r) = \\int _{0}^{\\infty } P(r|S) P(S) dS \\approx \\sum _{i=1}^{I}P_{i}(r|S_{i}) P(S_{i}) \\ .$ The underlying conditional distributions are characterized by the average $\\langle r \\rangle _{S_{i}}$ and variance $\\sigma _{i}^{2} \\equiv VS_{i}^{\\psi }$ $P(r| S_{i}) = \\exp [ -(r-\\langle r \\rangle )^{2}/2VS_{i}^{\\psi }] / \\sqrt{2\\pi VS_{i}^{\\psi }} \\ .$ which are each parameterized by the characteristic collaboration size $S_{i}$ .",
"In cases where the average change $\\langle r \\rangle \\approx 0$ , then the distribution $P(r| S_{i})$ is characterized by only the fluctuation scale $\\sigma _{i}(r)$ .",
"Fig.",
"REF demonstrates that the normalized production change $r^{\\prime }_{i}(t) = (r-\\langle r_{i} \\rangle )/\\sigma _{i}$ is distributed according to a Gaussian distribution.",
"Hence, using normalized variables, we have mapped the process to a universal scaling distribution $P(r|S_{i})$ .",
"When the distribution $P(S_{i})$ is exponential, $P(S_{i}) = \\lambda e^{-\\lambda S_{i}}$ then mixture is termed an “exponential mixture of Gaussians” [44], where the units have characteristic size $\\overline{S_{i}} = 1/\\lambda $ .",
"Fig.",
"REF shows that the distribution of collaboration radius $S_{i}$ is approximately exponential for each dataset, supporting the case for exponential mixing.",
"Using the cumulative distribution of $S$ for each data set we calculate $\\lambda = 0.15 \\pm 0.01$ [A], $\\lambda = 0.11\\pm 0.01$ [B], and $\\lambda = 0.11 \\pm 0.01$ [C].",
"While the tail behavior of $P(r)$ can be used to better discriminate the value of $\\psi $ , we do not have sufficient data in this analysis to perform a more rigorous test of the tail dependencies, or in general, to investigate the distribution of significantly large $r_{i}(t)$ values.",
"The scaling relation $\\sigma _{i}(r) \\sim S_{i}^{\\psi /2}$ determines the functional form of the aggregate $P_{\\psi }(r)$ .",
"Clearly, $\\sigma (r)$ increases for $\\psi >0$ values, whereas for values $\\psi <0$ , $\\sigma (r)$ decreases with size $S_{i}$ .",
"This latter case is empirically observed for countries and firms [49], whereby in general, large economic entities are able to decrease growth volatility by increasing and diversifying their portfolio of growth products.",
"In our analysis of scientific careers we define $S_{i} \\equiv Med[k_{i}(t)]$ , the median number of distinct coauthors per year, as a proxy for the ability of the career to attract new opportunities, and hence, as a proxy for the size $S_{i}$ of an academic career.",
"For professional athletes, we define the career size as the average number of points scored over the career $S_{i} \\equiv \\langle p_{i}(t) \\rangle $ .",
"In Fig.",
"REF we calculate $\\psi /2 \\approx 0.40\\pm 0.03$ (regression coefficient $R =0.77$ ) for dataset [A], $\\psi /2 \\approx 0.22 \\pm 0.04$ ($R = 0.51$ ) [B], and $\\psi /2 \\approx 0.26 \\pm 0.05$ ($R =0.45$ ) [C].",
"The role of mental, physical, and group spillovers is quite different in professional sports.",
"Athletes attract future opportunities largely through their historical track record, which is heavily weighted on performance in the near past, and less on the cumulative history.",
"Hence, for this performance-based labor force, we use a simple definition of “team value” to define the career size $S_{i}$ .",
"This quantity is easier to define for basketball, since there are smaller differences between players of different team position than in other sports.",
"For NBA player $i$ we define $S_{i}$ as the average number of points scored per year, $S_{i} \\equiv \\langle p_{i} \\rangle $ .",
"Fig.",
"S9 shows a crossover value $S_{c}$ which we interpret to reflect the fact that sports players typically fall into one of two categories: starters (everyday players) and replacement (game filler) players.",
"We calculate $\\psi /2 \\approx 0.38 \\pm 0.02$ for emerging and “second string” careers with $S_{i}< S_{c}$ , and a decreasing size variance relation ($\\psi <0$ ) for high-value careers with $S_{i}>S_{c}$ .",
"Similar values occur in the MLB.",
"These two $\\psi $ regimes reflect the crucial balance of risk and reward in short-term contract professions.",
"A variety of pdfs $P_{\\psi }(r)$ can result from the exponential mixture of Gaussians $P_{\\psi }(r) = \\int _{0}^{\\infty } \\lambda e^{-\\lambda S} \\frac{1}{\\sqrt{2\\pi \\sigma ^{2}(r)}} \\exp [ -r^{2}/2\\sigma ^{2}(r)] dS$ depending on the value of $\\psi $ which quantifies the size-variance relation.",
"The functional form of $P_{\\psi }(r)$ can vary in both the bulk and the tails of the distribution [44].",
"A simple result which follows from the case $\\psi =1$ is the Laplace (double-exponential) distribution $P_{\\psi = 1}(r) = \\sqrt{\\frac{\\lambda }{2V}} \\exp \\Big [-\\sqrt{\\frac{2 \\lambda }{ V}} \\vert r \\vert \\Big ] \\ .$ This distribution is a member of the family of Exponential power distributions which follow from the range of values $\\psi \\ge 0$ [44].",
"In general, if the scaling values are in the range $\\psi \\ge 0$ , then the exponential mixture leads to an Exponential power distribution $P(r) = \\frac{\\beta }{\\sqrt{2} \\sigma \\Gamma (1/ \\beta )} \\exp [ -\\sqrt{2}(\\vert r \\vert /\\sigma )^{\\beta }]$ with shape parameter $\\beta $ in the range $\\beta \\in (0,2]$ [44].",
"The pure exponential $P(r)$ with $\\beta =1$ corresponds to the case $\\psi = 1$ .",
"The pure Gaussian $P(r)$ with $\\beta =2$ corresponds to the case $\\psi = 0$ .",
"Furthermore, if the annual production is logarithmically related to an underlying production potential, $n_{i}(t) \\propto \\ln U_{i}(t)$ , then $r_{i}(t) \\propto \\ln U_{i}(t) - \\ln U_{i}(t-1)$ quantifies the logarithmic change (“growth rate”) of $U_{i}(t)$ .",
"This forms the analogy with growth dynamics of large institutions with size $S\\gg 1$ .",
"For example, in the case of financial securities such as the stock of a company $i$ , the growth rate $r_{i}(t)$ measure the logarithmic change in the market's expectations of the company's future earnings potential captured by the market capitalization and price [50].",
"As a result, distributions $P(r)$ of career growth fluctuation $r$ , which we plot in Figs.",
"REF (a-d), can be seen as a bridge between the micro level and the macro level of economic growth fluctuation.",
"A theory of micro growth processes can help improve the growth forecasts for economic organizations ranging in size from scientific collaborations to universities and firms [48], [34], [49], [33], [50], [44], [47]."
],
[
"Nonlinear preferential capture model",
"Here we describe a stochastic system in which a finite number of opportunities are distributed to a system of individual competing agents $i = 1...I$ .",
"The opportunities are distributed in batches of $P$ opportunities per arbitrary time interval.",
"This model has two parameters.",
"(i) $\\pi $ determines the preferential capture mechanism (the value $\\pi =1$ corresponds to the traditional “linear” preferential attachment model) and (ii) $c$ determines the performance timescale $1/c$ which is incorporated into the calculation of the capture rates of each individual.",
"The value $c=0$ corresponds to a long-term memory and $c \\gg 1$ corresponds to short-term memory.",
"We use this simple model to show that a system governed by a preferential capture can become dominated by fluctuations when $c$ is large.",
"The value $1/c$ quantifies the “performance appraisal timescale”: a small $c$ corresponds to a labor system with long contracts, or some alternative mechanism that provides employment insurance through periods of low production, so that the ability to attract future opportunities is largely based on the cumulative record of career achievement.",
"Conversely, a large $c$ corresponds to a labor system with short contracts in which the ability to attract future opportunities is largely based on the accomplishments in the near past, requiring an agent to maintain relatively high levels of production in order to survive.",
"In this latter case, we find that (natural) fluctuations in the annual production can cause a significant fraction of the careers to “fizzle out” leaving behind only a few “super careers” who attract almost all of the opportunities.",
"In other words, short contracts can tip the level of competition into dangerous territory whereby careers are largely determined by fluctuations and not persistence."
],
[
"System of competing agents",
"1) The system consists of $I \\equiv 1000$ agents competing for $P$ opportunities that are allocated in a single period.",
"There is no entry, hence the number $I$ is kept constant.",
"Also, $P$ is also kept constant, so there is no growth in the labor supply.",
"2) We run the Monte Carlo (MC) simulation for $T \\equiv 100$ time periods and all agents are by construction from the same age cohort (born at same time).",
"3) Each time period corresponds to the allocation of $P \\equiv \\sum _{i=1}^{I}n_{0,i}$ opportunities, sequentially one at a time, to randomly assigned agents $i$ , where $n_{0,i} \\equiv 1$ is the potential production capacity of a given individual.",
"4) The assignment of a given opportunity is proportional to the time-dependent weight (capture rate) $w_{i}(t)$ of each agent.",
"Hence, the assignment of 1 opportunity to agent $i$ at period $t$ results in the production (achievement) $n_{i}(t)$ to increase by one unit: $n_{i}(t) \\rightarrow n_{i}(t)+1$ .",
"In the next time period $t+1$ , we update the weight $w_{i}(t+1)$ to include the performance $n_{i}(t)$ in the current period."
],
[
"Initial Condition",
"The initial weight at the beginning of the simulation is $w_{i}(t=0) \\equiv n_{c}$ for each agent $i$ with $n_{c}\\equiv 1$ .",
"The value $n_{c} >0$ ensures that competitors begin with a non-zero production potential, and corresponds to a homogenous system where all agents begin with the same production capacity.",
"Hence, we do not analyze the more complicated model wherein external factors (i.e.",
"collaboration factors) can result in a heterogeneous production capacity across scientists.",
"By construction, each agent begins with one unit of achievement $n_{i}(t=1) \\equiv 1$ ."
],
[
"System Dynamics",
"1) In each Monte Carlo step we allocate one opportunity to a randomly chosen individual $i$ so that $n_{i}(t) \\rightarrow n_{i}(t)+1$ 2) The individual $i$ is chosen with probability $\\mathcal {P}_{i}(t)$ proportional to $[w_{i}(t)]^{\\pi }$ $\\mathcal {P}_{i}(t) = \\frac{w_{i}(t)^{\\pi }}{\\sum _{i=1}^{I} w_{i}(t)^{\\pi }}$ where the value $w_{i}(t)$ is given by an exponentially weighted sum over the entire achievement history $w_{i}(t) \\equiv \\sum _{\\Delta t =1}^{t-1} n_{i}(t-\\Delta t) e^{-c\\Delta t} \\ .$ The parameter $c \\ge 0$ is a memory parameter which determines how the record of accomplishments in the past affect the ability to obtain new opportunities in the current period, and therefore, the future.",
"The limit $c = 0$ rewards long-term accomplishment by equally weighting the entire history of accomplishments.",
"Conversely, when $c \\gg 1$ the value of $w_{i}(t)$ is largely dominated by the performance $n_{i}(t-1)$ in the previous period, corresponding to increased emphasis on short-term accomplishment in the immediate past.",
"Intermediate values $0 < c < 1$ weight more equally the immediate past and the entire history of accomplishment.",
"3) The exponent $\\pi $ determines how the relative ability to attract opportunities $\\mathcal {P}_{i}/\\mathcal {P}_{j} = [w_{i}(t)/w_{j}(t)]^{\\pi }$ depends on the weights $w_{i}(t)$ and $w_{j}(t)$ between two individuals $i$ and $j$ .",
"The linear capture case follows from $\\pi = 1$ , uniform capture $\\pi = 0$ , super linear capture $\\pi >1$ , and sub-linear capture $\\pi < 1$ .",
"4) At the end of each time period, the weight $w_{i}(t)$ is recalculated and used for the entirety of the next MC time period corresponding to the allocation of the next $I \\times n_{c}$ achievement opportunities."
],
[
"Model Results",
"We simulate this system for a realistic labor force size $I=1000$ with the assumption that in any given period, an individual has the capacity for one unit of production ($n_{c}\\equiv 1$ ).",
"We evolve the system for $T=100$ periods corresponding to $I \\times n_{c} \\times T$ Monte Carlo time steps.",
"The timescale $T$ represents the (production) lifetime of individuals with finite longevity.",
"In this model we do not include exogenous shocks (career hazards) that can result in career death [16].",
"Here we analyze four quantities: 1) The distribution $P(N)$ of the total number of opportunities $N_{i}(T)\\equiv \\sum _{t=1}^{T} n_{i}(t)$ captured by agent $i$ over the course of the $T-$ period simulation.",
"2) The distribution $P(\\alpha )$ of the career trajectory scaling exponent $\\alpha _{i}$ defined in Eq.",
"REF which quantifies the (de)acceleration of production over the course of the career.",
"3) The distribution $P(r)$ of production outcome change $r$ defined in Eq.",
"REF which quantifies the size of endogenous production shocks.",
"4) The distribution $P(L)$ of career length $L_{i}$ which measures the active production period of each career starting from $t=0$ .",
"We define activity as the largest period value $L_{i}$ for which $n_{i}(L_{i})=0$ , which in other words, corresponds to truncating all 0 production values from the end of the trajectory $n_{i}(t)$ and defining $L_{i}$ as the length of this time series.",
"We display these four distributions, from left to right, for varying $\\pi $ and $c$ values, in each panel of Figs.",
"REF – REF .",
"Empirical distributions calculated from MC simulations are plotted as blue dots, with benchmark distributions described below plotted as solid green curves.",
"For each $\\pi $ and $c$ value we simulate 10 MC systems, and combine the results into aggregate distributions which are shown.",
"For simulations with $\\pi >1$ the pdf data are aggregated over the results of 50 MC simulations.",
"We list below some of our main observations.",
"For $\\pi =1$ , independent of $c$ , we observe exponential $P(N)$ , consistent with the prediction of the linear preferential capture model in the case of no firm entry ($b=0$ ) in the model of Kazuko et al. [43].",
"However, the distribution $P(L)$ and the distribution $P(\\alpha )$ does depend strongly on $c$ , reflecting the possibility of career “sudden death” for large $c$ .",
"For the $P(\\alpha )$ distributions (middle-left panels), the solid green line is a best-fit Gaussian distribution (using the MLE method) for the set of $\\alpha _{i}$ values computed for careers that did not undergo “sudden death.” For the $P(r)$ distributions (middle-right panels), the solid green curve corresponds to a best-fit Laplace distribution (using the MLE method) and the dashed red curve corresponds to a best-fit Guassian distribution (using the MLE method) which we show only for benchmark comparison.",
"Typical empirical distributions (values shown as blue dots) range from being distributions that are Gaussian to distributions that are Laplacian in the bulk but with heavy tails.",
"For the $P(L)$ distributions (right most panels), we note that the most likely career length $L$ is typically either $L=1$ or $L=T$ for all systems analyzed.",
"However, there are likely $c$ and $\\pi $ parameter values corresponding to $P(L)$ that is uniform distributed over the entire range of $L$ values, which may be an interesting class of system to analyze in future analyses since such a system promotes diversity across the entire longevity spectrum.",
"The system we show for $\\pi = 1.2$ and $c=1$ appears to be close to this scenario.",
"Fig.",
"REF shows the null model with no preferential capture ($\\pi = 0$ ).",
"We confirm that the careers in this model are driven by a stochastic accumulation process that is equivalent to a Poisson process with rate $\\lambda _{p}\\equiv 1$ .",
"In this homogenous system, each career gains on average one opportunity each time period, so that at the end of the simulation, the distribution $P(N)$ is a Poisson distribution with $\\langle N\\rangle = \\lambda _{p} T$ (shown as the solid blue line) which fits the model data excellently.",
"For these careers, the typical $\\alpha = 1$ , the production changes are well-approximated by a Gaussian distribution, and most careers are sustained for the maximum possible lifetime corresponding to $T$ periods.",
"Fig.",
"REF shows the system with $c=0$ corresponding to comprehensive career appraisal corresponding to a long-term memory system.",
"We analyze this system for 4 values of $\\pi = {0.8, 1.0, 1.2, 1.4}$ .",
"This “long-term memory” scenario corresponds to a long-term contract profession whereby careers are less vulnerable to periods of low production.",
"As a result, most careers sustain production throughout the career.",
"Fig.",
"REF shows the system with $c=0.1$ corresponding to an effective memory timescale of $1/c =10$ periods.",
"We analyze this system for 4 values of $\\pi = {0.8, 1.0, 1.2, 1.4}$ .",
"This “medium-term memory” scenario yields a rich variety of careers for $\\pi =1$ , but for $\\pi =1.2$ the system becomes quickly dominated by “rich-get-richer” effects which results in careers being vulnerable to low production fluctuations.",
"Fig.",
"REF shows the system with $c=1$ corresponding to an effective memory timescale of $1/c=1$ period.",
"We analyze this system for 4 values of $\\pi = {0.8, 0.9, 1.0, 1.1}$ .",
"For all values of $\\pi $ analyzed, we observe a system that is dominated by careers that are cut short by the high levels of competition induced by the relatively high value placed on continued production.",
"Fig.",
"REF shows the extreme case of a “no memory” scenario in which $w_{i}(t) \\approx n_{i}(t-1)$ whereby most careers experience sudden death due to endogenous negative production shocks early in their career.",
"The lucky few careers who survive this period end up as rich-get-richer “superstars.” This behavior occurs for all systems analyzed using 4 values of $\\pi = {0.8, 0.9, 1.0, 1.05}$ ."
],
[
"Discussion of the model in relation to the Academic labor market",
"One serious drawback of short-term contracts are the tedious employment searches, which displace career momentum by taking focus energy away from the laboratory, diminishing the quality of administrative performance within the institution, and limiting the individual's time to serve the community through external outreach [3], [6].",
"These momentum displacements can directly transform into negative productivity shocks to scientific output.",
"As a result, there may be increased pressure for individuals in short-term contracts to produce quantity over quality, which encourages the presentation of incomplete analysis and diminishes the incentives to perform sound science.",
"These changing features may precipitate in a “tragedy of the scientific commons.” Aside from promoting circumspect research, job security in academia diminishes the incentives for scientists to “save and store” their knowledge for future liquidation in the case of employment emergency, and thus promotes the institution of “open science” [1].",
"However, a policy shift towards short-term contracts, along with the heightened value of intellectual property, may alter the course of publicly funded “open science.” This scientific commons emerged from the noble courts during the Renaissance as a hallmark of the scientific revolution and now faces pressure from what has been termed “intellectual capitalism,” with the vast privatization of knowledge and innovation (“closed science”) occurring in public universities and corporate R&D [1].",
"An academic system that is dominated by short term contracts, stymied by production incentives that favor quantity over quality, and jeopardized at the level of the “open knowledge” commons, presents a new institutional scenario revealing selection pressures that could alter the birth and death rates of high-impact careers.",
"The purpose of this stochastic model is to show how careers can become very susceptible to negative production shocks if the labor market is driven by a preferential capture mechanism with $\\gamma >1$ whereby early success of an individual can lead to future advantage.",
"However, this model also shows that the onset of a fluctuation-dominant (volatile) labor market can also be amplified when the labor market is governed by short-term contracts reinforced by a short-term appraisal system.",
"In such a system, career sustainability relies on continued recent short-term production, which can encourage rapid publication of low-quality science.",
"In professions where there is a high level of competition for employment, bottlenecks form whereby most careers stagnate and fail to rise above an initial achievement barrier.",
"Instead, these careers stagnate, and in a profession that shows no mercy for production lulls, these careers undergo a “sudden death” because they were “frozen out” by a labor market that did not provide insurance against endogenous fluctuations.",
"Such a system is an employment “death trap” whereby most careers stagnate and “flat-line” at zero production.",
"However, at the same time, a small fraction of the population overcomes the initial selection barrier and are championed as the “big winners”, possibly only due to random chance.",
"Table demonstrates how the life expectancy decreases with increasing $c$ even for the linear preferential capture model corresponding to $\\pi =1$ .",
"With increasing $c$ , the model simulates systems with shorter contracts (shorter appraisal “memory” timescales), and so larger percentages of the population die before characteristic ages $T_{c}(p)$ , values that decrease with increasing $c$ for a given $p$ .",
"Table: Decrease in career life expectancy as a result of short-term contract length in the π=1\\pi =1 linearpreferential capture model.",
"The fraction pp of the population that experienced career termination before thecrossover age T c (p)T_{c}(p): “pp percent of the population died before reaching the age L=T c (p)L=T_{c}(p).”As cc increases (recall the appraisal “memory” timescale is 1/c1/c) towards a short-term contract scenario, asignificant fraction of the population (increasing pp) dies before reaching a smaller and smaller T c (p)T_{c}(p).",
"Theempirical value of T c (p)T_{c}(p) is given as a percentage of the maximum career length TT corresponding to the stoppingtime of the Monte Carlo simulation.",
"The value T c (p)T_{c}(p) is calculated using the equality p=CDF(T<T c (p))p = CDF(T<T_{c}(p)), whereCDF(T<L)CDF(T<L) is the cumulative distribution function of career length LL.",
"To estimate CDF(T<L)CDF(T<L), we combine an ensembleof 10 MC simulations for each cc value.",
"In the model simulations we use T≡100T\\equiv 100 periods.Figure: Positive career shocks likely associated with reputation boosts.",
"Examples of career productiontrajectories N i (t)N_{i}(t) that have significant deviations from the scalinghypothesis in Eq.",
".",
"These significant deviations likely follow extraordinary scientific discoveries (andthe publicity and reputation that are typically rewarded) which can vault a career and result in lasting benefits to theindividual.Figure: Regularities in the career trajectory N i (t)N_{i}(t).",
"We analyze the normalized career trajectoryN i ' (t)≡N i (t)/〈n i 〉N^{\\prime }_{i}(t)\\equiv N_{i}(t) / \\langle n_{i} \\rangle which allows us to aggregate N i (t)N_{i}(t) with varying publication rates 〈n i 〉\\langle n_{i} \\rangle .",
"As a result, we can better quantify the scaling exponent α ¯\\overline{\\alpha } which quantifies theacceleration of the typical career over time.",
"We calculate α ¯\\overline{\\alpha } using OLS regression on log-log scale ofthe average normalized career trajectory 〈N ' (t)〉≡N i (t) 〈n i 〉 \\langle N^{\\prime }(t) \\rangle \\equiv \\Big \\langle \\frac{N_{i}(t)}{ \\langle n_{i}\\rangle } \\Big \\rangle .",
"For reference, each N i ' (t)N^{\\prime }_{i}(t) trajectory in panels A, B, and C has a corresponding best-fitcurve that is a dashed line.",
"(A) For the scientific careers, we calculate α ¯\\overline{\\alpha } values: 1.28±0.011.28\\pm 0.01 for Dataset A, 1.31±0.011.31\\pm 0.01for Dataset B, and 1.15±0.021.15\\pm 0.02 for Dataset C.These values are all significantly greater than unity, α ¯>1\\overline{\\alpha }>1, indicative of a systematic cumulativeadvantage effect in science.",
"(B)The standard deviation σN ' (t)\\sigma N^{\\prime }(t) has a broad peak, likely related to career shocks that can significantly alterthe career trajectory.",
"(C) The average normalized career trajectory for NBA careers has α ¯≈1\\overline{\\alpha }\\approx 1 (D) The averagenormalized career trajectory for MLB careers has α ¯≈1\\overline{\\alpha }\\approx 1.",
"For visual comparison, the solidstraight black line in panels A,B and C correspond to a linear function with α=1\\alpha =1.Figure: Using scaling methods to show approximate data collapse of each N i (t)N_{i}(t).Normalized trajectory N ˜ i (t)≡N i (t)/A i \\tilde{N}_{i}(t)\\equiv N_{i}(t)/A_{i} plotted using the scaled time t ' ≡t α i t^{\\prime }\\equiv t^{\\alpha _{i}} for each career over the time horizon t∈[1,40]t \\in [1,40] years.",
"We plot the 100 N ˜ i (t)\\tilde{N}_{i}(t) curvesbelonging to datasets [A], [B], and [C] in the corresponding panels.",
"There is approximate data collapse of all thenormalized trajectories N ˜ i (t)\\tilde{N}_{i}(t) along the dashed green line corresponding to the rescaled career trajectoryN ˜ i (t)=t ' \\tilde{N}_{i}(t) = t^{\\prime } with α ' ≡1\\alpha ^{\\prime } \\equiv 1 by construction.",
"We also plot in red the correspondingaverage value 〈N ˜ i (t)〉\\langle \\tilde{N}_{i}(t)\\rangle with 1σ\\sigma error bars for logarithmically spaced t ' t^{\\prime } intervals.Deviations from 〈N ˜ i (t)〉\\langle \\tilde{N}_{i}(t)\\rangle are indicative of career shocks which can significantly alter thecareer trajectory.Figure: Increasing returns to scale α>1\\alpha >1.",
"Probability distribution of the individual α i \\alpha _{i} valuescalculated for each career using the scalingmodel N i (t)∼t α i N_{i}(t) \\sim t^{\\alpha _{i}} over time horizon t∈[1,40]t \\in [1,40] years.",
"The average 〈α i 〉\\langle \\alpha _{i} \\rangle and standard deviation σ(α i )\\sigma (\\alpha _{i}) for each dataset are: 1.42±0.291.42 \\pm 0.29 [A], 1.44±0.261.44 \\pm 0.26 [B], 1.30±0.311.30 \\pm 0.31 [C].",
"The distribution of α i \\alpha _{i} values indicate that career trajectories are typically accelerating(α i >1)(\\alpha _{i}>1), most likely the result of a cumulative advantage effect.Figure: Universal patterns in underlying production fluctuations of scientists.",
"Accounting for variableindividual publication factors, such as academic subfield or group collaboration size,we find that the normalized annual production change r i ' (t)≡[r i (t)-〈r〉 i ]/σ i r^{\\prime }_{i}(t) \\equiv [r_{i}(t)- \\langle r \\rangle _{i} ] /\\sigma _{i} is distributedaccording to a Gaussian distribution, with 〈r ' 〉=0\\langle r^{\\prime } \\rangle =0 and σ(r ' )=1\\sigma (r^{\\prime })=1 by construction (solid linesshowbest-fit Guassian distributions using the maximum likelihood estimator method).",
"This results indicates that theLaplace distribution shown in Fig.",
"resultsfrom a mixture of Gaussian distributions P i (r=σ i r ' )P_{i}(r=\\sigma _{i}r^{\\prime }) that indicate that annual production is consistent with a proportional growth model..Figure: Universal patterns in the production fluctuations of athletes.For athlete careers in the NBA and MLB we define production change for (A,C) the change in the number of in-gameopportunities and (B,D) the change in the number of in-game successes.",
"(A,B) Since the detrended production change RR isdefined to have standard deviation σ≡1\\sigma \\equiv 1, the pdfs P(R)P(R) approximately collapse onto a universal “tent-shaped” Laplace pdf (solidgreen line).",
"(C,D) For sports careers, we alsodefine a measure R ' R^{\\prime } which account for variable individual production factors,such as propensity for injury, team position, etc.As a result normalized annual growth rate R i ' ≡[z i (t)-〈z(t)〉]/σ z(t) R^{\\prime }_{i} \\equiv [z_{i}(t)- \\langle z(t) \\rangle ] / \\sigma _{z(t)} is normalizedtwice, once to account for age factors and once to account for individual factors.",
"The quantity z i (t)≡(r i (t)-〈r i 〉)/σ i z_{i}(t) \\equiv (r_{i}(t)-\\langle r_{i} \\rangle )/\\sigma _{i} is normalized with respect toindividual factors, where 〈r i 〉\\langle r_{i} \\rangle and σ i \\sigma _{i} are the average and standard deviation of theproduction change of career ii.",
"Then, we aggregate all z i (t)z_{i}(t) values for a given career year tt in order tocalculate the average 〈z(t)〉\\langle z(t) \\rangle and standard deviation σ z(t) \\sigma _{z(t)} over all careers.",
"The final quantityR i ' R^{\\prime }_{i} represents a normalized annual production change which is distributedin the bulk according to a Gaussian distribution, with 〈R ' 〉≈0\\langle R^{\\prime } \\rangle \\approx 0 and σ(r ' )≈1\\sigma (r^{\\prime }) \\approx 1 byconstruction (solid lines showbest-fit Guassian distributions using the maximum likelihood estimator method).",
"This results indicates that thetent-shaped distributions in (A,B) resultsfrom a mixture of conditional Gaussian distributions P i (R=σ i R ' )P_{i}(R=\\sigma _{i}R^{\\prime }) that indicate that annual production is consistent with a proportional growth model.Figure: Universal micro-scale output distribution P(Q)P(Q) which accounts for coauthorship variability.",
"Thenormalizedoutput Q∝n i /k i γ i Q \\propto n_{i}/k_{i}^{\\gamma _{i}} is a residual output after we quantitatively account for the collaborationsize k i k_{i} correspondingto the number of distinct coauthors of author ii.",
"Each pdf is well-approximated by the Gamma distribution P(Q)∝Q m-1 exp[-Q/θ]P(Q)\\propto Q^{m-1}\\exp [-Q/\\theta ]which suggests that production at the micro scale is governed by a Gamma Lévy process.",
"We calculate the Gammadistribution parameters using themaximum likelihood estimator method (distributions shown by solid and dashed curves), and find an insignificantdifference between [A] and [B]scientists with Gamma shape parameter mm and scale parameter θ\\theta .",
"However, for dataset [C] scientists, theoutput distribution is moreskewed towards smaller QQ values, possibly reflecting the relative advantage that senior scientists gain due toreputation, experience, andknowledge spillover factors.Figure: Aggregate production distributions can be deceiving.",
"Unconditional distribution of annual publicationrate n(t)n(t) appears as log-normaldistributions because it is a mixture ofunderlying distributions that depend strongly on collaboration factors.",
"We define n i (t)n_{i}(t) as the number of paperspublished in (A) Δt=1\\Delta t = 1 and (B) Δt=2\\Delta t=2 year periods, which reduces the finite-size effects arising fromthe calendar year labeling of publication dates.",
"(A) We combine n i (t)n_{i}(t) values for all values of tt, and findexcellent agreement between the empiricalP(n(t))P(n(t)) data points and the log-normal model.We use the maximum likelihood estimator method to calculate the log-normal parameters σ L ≡σ(lnn)\\sigma _{L} \\equiv \\sigma (\\ln n ) and μ=〈lnn〉\\mu = \\langle \\ln n \\rangle .",
"(B) In order to analyze the time-dependence of P(n(t))P(n(t)), we separaten i (t)n_{i}(t) values from Dataset A into 5 subsets, depending on the range tt years into the career, as indicated in thefigure legend.",
"We offset each pdf by a constant factor in order to distinguish each pdf, which are alsowell-approximated by log-normal distributions (shown as solid curves).Figure: Quantifying the growth fluctuations of sports careers.",
"The size variance relation for sports careers is similarto academic careers for small S i S_{i}.However, for relatively large S i S_{i} the relation becomes decreasing corresponding to ψ<0\\psi <0, analogous to what isfound for firm growth , , , .",
"The decreasing relation for S i >S c S_{i}> S_{c} likelyfollows from the fact that in sports, there is a hard upper limit to the number of opportunities available to a playerin a given year.",
"Hence, individuals with large S i S_{i} are likely the starters on their teams, since it is neithereconomical nor in the strategy of winning to keep players above a threshold value S c S_{c} out of the game, and so theseplayers typically remainas positional starters except for episodic leaves of absence due to injury.",
"Hence, these players experience smallerσ i (r)\\sigma _{i}(r) due to limitations to their potential for further career growth.",
"However, players with S i <S c S_{i}< S_{c}are typically on the fringe of being released or provide alternative value to the team, and so these individualsexperience larger fluctuations in team play because they are easily dispensable, especially in a profession dominated byshort-contracts lasting sometimes less than a year.",
"For each dataset, we use careers with career length L i ≥3L_{i} \\ge 3seasons.",
"(A) NBA basketball players: Units of σ i (R)\\sigma _{i}(R) are normalized minutes played.",
"We define the scalingrelation σ i (R)∼〈p i 〉 ψ/2 \\sigma _{i}(R) \\sim \\langle p_{i} \\rangle ^{\\psi /2} between the average number of points scored per season 〈p i 〉=∑ t=1 L i p i (t)/L i \\langle p_{i} \\rangle = \\sum _{t=1}^{L_{i}}p_{i}(t)/L_{i} and the standard deviation σ i (R)\\sigma _{i}(R).",
"In this way, weutilize the average points per season as the proxy for the ability of a player to obtain future opportunities which arerealized as minutes played.",
"Using S c ≡720S_{c} \\equiv 720 points, we calculate ψ/2=0.38±0.02\\psi /2= 0.38 \\pm 0.02 (regressioncoefficient R=0.50R=0.50 and ANOVA F-test significance level p≈0p \\approx 0) for S i <S c S_{i}<S_{c} and ψ/2=-0.25±0.07\\psi /2= -0.25 \\pm 0.07(R=0.15R=0.15 and p≈10 -3 p \\approx 10^{-3}) for S i >S c S_{i}>S_{c}.",
"(B) MLB pitchers: Units of σ i (R)\\sigma _{i}(R) are normalized IPO (innings pitched in outs).",
"Interestingly,σ i (R)\\sigma _{i}(R) continues toincrease for S i >S c S_{i}>S_{c}, possibly due to the relatively high career risk attributed to throwing arm injury.",
"UsingS c ≡65S_{c} \\equiv 65 strikeouts, we calculate ψ/2=0.37±0.01\\psi /2= 0.37 \\pm 0.01 (R=0.48R=0.48 and p≈0p \\approx 0) for S i <S c S_{i}<S_{c} andψ/2=+0.15±0.07\\psi /2= +0.15 \\pm 0.07 (R=0.07R=0.07 and p≈0.02p \\approx 0.02) for S i >S c S_{i}>S_{c}.",
"(C) MLB batters: Units ofσ i (R)\\sigma _{i}(R) are normalized AB (at bats).",
"Using S c ≡68S_{c} \\equiv 68 hits, we calculate ψ/2=0.44±0.01\\psi /2= 0.44 \\pm 0.01(R=0.59R=0.59 and p≈0p \\approx 0) for S i <S c S_{i}<S_{c} and ψ/2=-0.37±0.03\\psi /2= -0.37 \\pm 0.03 (R=0.21R=0.21 and p≈0p \\approx 0) forS i >S c S_{i}>S_{c}.",
"The dashed black (blue) line in each panel is a least squares linear regression on log-log scale for alldata values with S i S_{i} less (greater) than S c S_{c}.",
"The data shown with error bars represent the average 〈σ i (R)〉\\langle \\sigma _{i}(R) \\rangle and corresponding 1 standard deviation values calculated using equally spaced S i S_{i} bins onthe logarithmic scale."
]
] | 1204.0752 |
[
[
"Fubini instantons in curved space"
],
[
"Abstract We study Fubini instantons of a self-gravitating scalar field.",
"The Fubini instanton describes the decay of a vacuum state under tunneling instead of rolling in the presence of a tachyonic potential.",
"The tunneling occurs from the maximum of the potential, which is a vacuum state, to any arbitrary state, belonging to the tunneling without any barrier.",
"We consider two different types of the tachyonic potential.",
"One has only a quartic term.",
"The other has both the quartic and quadratic terms.",
"We show that, there exist several kinds of new O(4)-symmetric Fubini instanton solution, which are possible only if gravity is taken into account.",
"One type of them has the structure with $Z_2$ symmetry.",
"This type of the solution is possible only in the de Sitter background.",
"We discuss on the interpretation of the solutions with $Z_2$ symmetry."
],
[
"Introduction ",
"The very first picture of an inflationary multiverse scenario was proposed in Ref.",
"[1], in which it would seem that the author wanted to suggest a universe without the cosmological singularity problem using an interesting feature of self-reproducting or regenerating exponential expansion of the universe.",
"A major development in this scenario was triggered by the discovery of the eternal inflationary scenario [2], [3], [4], [5] and a paradigm for string theory landscape [6], [7].",
"The eternal inflation is related to the expanding false vacuum solution with a positive cosmological constant, which in turn means that the inflation is eternal into the future.",
"If the theory has multiple minima then the false vacuum state decays into the true vacuum state, i.e.",
"the phase transition proceeded via the nucleation of a vacuum bubble.",
"In this scenario the universe is situated within some bubble called a pocket universe [4] having a certain value of the cosmological constant and the whole universes are referred to as multiverse.",
"The description of self-reproduction including tunneling process and random walk was combined into a scenario called recycling universe [8].",
"These scenarios seem to provide an escape from the question of the initial conditions of the universe, i.e.",
"it seems to be eternal into the past.",
"Unfortunately, inflationary spacetimes cannot be made complete in the past direction [9], even though the universe is eternal into the future.",
"There are still interesting arguments on the beginning of the universe [10].",
"The string theory landscape is a setting that involves a huge number of different metastable and stable vacua [11], [12], originated from different choices of Calabi-Yau manifolds and generalized magnetic fluxes.",
"The huge number of different vacua can be approximated by the potential of a scalar field.",
"The important thing is the fact that, once the de Sitter vacuum can exist, the inflationary expansion is eternal into the future and has the possibility of self-reproduction.",
"On the other hand, there are theories of gauged $d=4$ , $N=8$ supergravity having de Sitter(dS) solution, in which all SUSYs are spontaneously broken.",
"It is well known that the dS solution corresponds to a M/sting theory solution with a non-compact 7- or 6-dimensional internal space, in which a small value of the cosmological constant stems from the 4-form flux.",
"The simplest representative of these kind of theories has a tachyonic potential with the dS maxima [13], [14], [15].",
"The potential in the vicinity of the maximum reduces to a form having a quadratic term, that is not metastable but unstable.",
"However, according to some authors, the time for collapse giving rise to the tachyonic potential can be much greater than the age of the universe for anthropic reasoning.",
"If the curvature radius of the potential in the vicinity of the maximum is greater than that used in the above theory, then that will be all together different story.",
"The supergravity analogue of the tachyonic potential could be constructed also by using an exact supergravity solution representing the $D_p$ -$\\bar{D}_p$ system [16].",
"From the above scenarios, the study of the possibility of the tunneling process for the potential with stable and metastable vacua, or even tachyonic behavior has acquired renewed interest.",
"In the present paper, we will study the tunneling process under a simple tachyonic potential governed by a quartic term both without the quadratic term and with the term as a toy model.",
"To obtain the general solution including the effect of the backreaction, we solve the coupled equations for the gravity and the scalar field simultaneously.",
"Although the model has a tachyonic potential, it might still be an useful example to show how the tunneling process occurs in various shapes of the potential provided by the above scenarios.",
"A quantum particle can tunnel through a finite potential barrier via the so-called barrier penetration.",
"This process can be described by the Euclidean solution obeying appropriate boundary conditions.",
"There exist two kinds of Euclidean solutions describing quantum tunneling phenomena.",
"One corresponds to an instanton solution representing a stable pseudoparticle configuration characterized by the existence of a nontrivial topological charge.",
"It does not change even if we continuously deform the field, as long as the boundary conditions remain the same.",
"The instanton solution corresponds to the minimum of the Euclidean action to pass from the initial to final state [17].",
"The solution, in case of a double well potential, describes a general shift in the ground state energy of the classical vacuum due to the presence of an additional potential well, then lifting the so-called classical degeneracy.",
"The other is a bounce solution representing an unstable nontopological configuration that corresponds to a saddle point rather than a minimum of the Euclidean action.",
"The second derivative of the Euclidean action around the bounce has one negative eigenvalue which leads to the imaginary part of the energy.",
"The existence of the negative eigenvalue implies that the vacuum state is unstable, i.e.",
"the state decays into other states [18].",
"The Euclidean solutions can also mediate phase transitions.",
"The phase transition describes the sudden change of a physical system from one state to another.",
"The transition are of two different types transition accompanied by temperature or zero temperature.",
"The competition between the entropy and the energy terms in the thermodynamic potential cause thermal phase transitions in which dynamics is irrelevant.",
"In the modern classification scheme, thermal phase transitions are divided into two broad categories either with a discontinuous jump in the first-order derivatives of the free energy or without it.",
"A first-order phase transition is characterized by the discontinuity in the first derivative of the free energy and is associated with the existence of latent heat, whereas a $n$ th-order phase transition is characterized by the continuity in the first derivative while there is a discontinuity in the $n$ th-order derivative.",
"A quantum phase transition describes a transition between different phases by quantum fluctuation, which occurs at zero temperature, unlike the case of a thermal phase transition which is governed by a thermal fluctuation [19].",
"To simplify things, we consider an asymmetric double well potential to distinguish two different phase transitions at zero temperature.",
"If the initial state is the metastable vacuum state and the tunneling occurs from that state to the other vacuum state, then the transition corresponds to a tunneling process [20], [21], [22], [23], [24], [25].",
"On the other hand, if the initial state is the local maxima of the potential and the field is rolling down to one vacuum state continuously rather than any discontinuous jump, then the transition corresponds to the rolling.",
"However, one more channel exists as tunneling and that corresponds to the one without a barrier.",
"In this kind of transition, the initial state on the top of a potential can tunnel to the other state rather than rolling down the potential [26], [27], [28], [30].",
"There are two different kind of transitions in this case.",
"One is the tunneling without a barrier representing the tunneling from the local maximum of the potential to the vacuum state [28], [29], [30], [31].",
"Recently, an analytic study on this type of solution was performed in [31].",
"The other is a tunneling without a barrier representing the tunneling from the maximum of the potential to any arbitrary state.",
"This case corresponds to the Fubini instanton [26], [27], where the tachyonic potential is employed.",
"Can we describe the rolling corresponding to the transition between the initial metastable vacuum state and the other final vacuum state?",
"This may look similar to a superfluid motion by the liquid helium.",
"Although to establish the phase transition corresponding to the superfluid motion is itself a very challenging problem, we concentrate on the Fubini instantons in this work.",
"The Fubini instanton [26], [27] describes the decay of a vacuum state by the quantum phase transition instead of rolling down the tachyonic potential consisted of a quartic term only.",
"On the other hand, one can consider a tachyonic potential consisted of a quadratic term only, the point $\\Phi =0$ is unstable.",
"A small perturbation will cause it to roll down the hill of the potential.",
"Originally, it was Fubini who introduced a fundamental scale of hadron phenomena by means of the dilatation noninvariant vacuum state in the framework of a scale invariant Lagrangian field theory [26].",
"However, the solution is a one-parameter family of instanton solutions representing a tunneling without a barrier as an interpolating solution from the maximum of the potential to any arbitrary state.",
"The instanton solution was studied in a conformally invariant model, i.e.",
"a fixed background was used and the effect of the backreaction by instantons was neglected [32], [33], [34].",
"This is a good approximation, when the variation of the potential during the transition is much smaller than the maximum of the potential.",
"The instanton has gained much interest now-a-days in the context of anti-de Sitter(AdS)/conformal field theory correspondence [35], [36].",
"The paper is organized as follows: In Sec.",
"2, we review the Fubini instanton in the absence of gravity.",
"We present numerical solutions including the Euclidean energy density as an example and analyze the structure of the solution in the theory with a potential having only the quartic self-interaction term.",
"We stress the fact that there is no such solutions with the potential containing both the quartic and the quadratic terms.",
"In Sec.",
"3, we show that the instanton solutions exist in the curved space.",
"We perform a numerical study to solve the coupled equations for the gravity and the scalar field simultaneously.",
"We show that there exist numerical solutions without oscillation in the initial AdS space in the potential with only the quartic term.",
"We also show that there exist numerical solutions in the potential both with the quartic and the quadratic terms irrespective of the value of the cosmological constant, which is possible only when the gravity is switched on.",
"In order to estimate the decay rate of the background state, we compute the action difference between that of the solution and the background obtained by numerical means.",
"We present an oscillating numerical solutions in the potential with only the quartic term with various values of the cosmological constant.",
"One type of these solutions has the structure with $Z_2$ symmetry.",
"We will discuss on the interpretation of the solutions with $Z_2$ symmetry in the final Section.",
"We analyze the behavior of the solutions using the phase diagram method.",
"In Sec.",
"4, to observe the dynamics of the solutions, we briefly sketch the causal structures of the solutions in the Lorentzian spacetime.",
"Finally in Sec.",
"5, we summarize and discuss our results."
],
[
"Fubini instanton in the absence of gravity ",
"One can consider the following action in the absence of gravity $S= \\int _{\\mathcal {M}} \\sqrt{-g} d^4 x \\left[-\\frac{1}{2}{\\nabla ^\\alpha }\\Phi {\\nabla _\\alpha }\\Phi -U(\\Phi )\\right] , $ where $g=\\det \\eta _{\\mu \\nu }$ , $\\eta _{\\mu \\nu }=diag(-1,1,1,1)$ is the Minkowski metric, and the tachyonic potential has a quartic self-interaction term and also a quadratic term as follows: $U(\\Phi )=-\\frac{\\lambda }{4} \\Phi ^4 + \\frac{m^2}{2}\\Phi ^2 +U_o , $ where $m^2 > 0$ and $\\lambda > 0$ .",
"The plots of potentials (a) without the quadratic term and (b) with the quadratic term are shown in Fig.",
"REF .",
"The potential has a metastable vacuum state at $\\Phi =0$ and no other stationary state in Fig.",
"REF (a), while Fig.",
"REF (b) illustrates that the potential has a local minimum at $\\Phi =0$ and two maxima [27], [37], [38].",
"In both the cases, the potential is not bounded from below.",
"Before going to the tunneling problem in four dimensions, we briefly describe the problem in one dimension.",
"One can consider the simplest quantum tunneling problem in one dimension.",
"Quantum field theory in one dimension is nothing but ordinary quantum mechanics.",
"In case of $m^2 = 0$ , the amplitude for transmission obeys the WKB formula in the semiclassical approximation, in which $\\Phi _{\\pm }={\\pm }(\\frac{4U_o}{\\lambda })^{1/4}$ are the classical turning points.",
"On the other hand, the double-hump potential with $m^2 > 0$ and $U_o =0$ can be considered as an inverted double-well potential for a bounce solution representing the tunneling from $\\Phi =0$ to $\\Phi _{\\pm }= \\pm m\\sqrt{{2}/{\\lambda }}$ .",
"The solution is given by $\\Phi _s(\\tau )= \\pm m\\sqrt{{2}/{\\lambda }} \\mathrm {sech}[m(\\tau -\\tau _o)]$ , where $\\tau _o$ is an integration constant.",
"The bounce solutions can be easily understood in the Euclidean space.",
"The particle can only reach the point $\\Phi =0$ at $\\tau = \\pm \\infty $ and it bounces off $\\Phi _{\\pm }= \\pm m\\sqrt{{2}/{\\lambda }}$ at $\\tau =0$ with a vanishing velocity [39].",
"Figure: Potentials for the case of (a) Fubiniinstantons and (b) generalized Fubini instantons.We now turn to the tunneling problem in four dimensions.",
"It is an well-known fact that the massless theory has an instanton [26].",
"Actually, the instanton corresponds to the bounce solution representing the decay of the background vacuum state.",
"The equation of motion with $O(4)$ symmetry, obtained by varying the Euclidean action, is then; $\\frac{d^2\\Phi }{d\\eta ^2} + \\frac{3}{\\eta } \\frac{d\\Phi }{d\\eta }=-\\frac{d(-U)}{d\\Phi }\\,, $ where $\\eta (=\\sqrt{\\tau ^2 + x^2})$ plays the role of the evolution parameter in Euclidean space and the second term in the left-hand side plays the role of a damping term.",
"The boundary conditions are $\\frac{d\\Phi }{d\\eta }\\Big |_{\\eta =0}=0 \\,\\,\\,\\, {\\rm and}\\,\\,\\,\\,\\Phi |_{\\eta =\\infty }=0\\,.$ The particle in the classical mechanics problem starts at $\\Phi =\\Phi _o$ with zero velocity in the inverted potential, and stops at $\\Phi |_{\\eta =\\infty }=0$ without any oscillation.",
"For the potential with $m^2 = 0$ , the analytic solution of the Fubini instanton has the form $\\Phi (\\eta )= \\sqrt{\\frac{8}{\\lambda }}\\frac{b}{\\eta ^2+b^2}\\,,$ where $\\eta $ is the radial length in the Euclidean space, $b$ is any arbitrary length scale which characterizes the size of the instanton and is related to the initial value $\\Phi _o$ .",
"In addition, the value of the scalar field of the center of the solution depends on $b$ as $\\Phi (0)=\\sqrt{\\frac{8}{\\lambda }}\\frac{1}{b}$ .",
"This solution was used in the related perturbation theory [40].",
"Figure: The analytic solution of the Fubini instantonin absence of gravity.The characteristic behavior of the analytic solution is ploted in Fig.",
"REF in terms of the value of the parameter $b$ .",
"We take $\\lambda =1$ for all the cases.",
"The solid line denotes the solution with $b=1$ , the dashed line with $b=3$ , and the dotted line with $b=5$ .",
"The corresponding Euclidean action is given by $S_{\\mathrm {E}}= \\frac{32\\pi ^2 b^2}{\\lambda } \\int ^{\\infty }_{0}\\frac{\\eta ^5(1-\\frac{b^2}{\\eta ^2})}{(\\eta ^2+b^2)^4} d\\eta =\\frac{8\\pi ^2}{3\\lambda }\\,, $ where the action does not depend on the parameter $b$ due to the consequence of the conformal invariance of the potential and we take that value to be $U_o=0$ .",
"The action has the same value irrespective of the starting point $\\Phi _o$ .",
"In other words, the tunneling from the maximum of the potential to any arbitrary state always happens with same probability.",
"The numerical solutions for $\\Phi $ and $\\Phi ^{\\prime }$ and the Euclidean energy including the density variation with $\\eta $ are as shown in Fig.",
"REF .",
"Figure REF (a) illustrates the numerical solution for $\\Phi $ , in which the initial value set as $\\Phi _o =-1$ and the solution asymptotically approaches the value $\\Phi = 0$ .",
"Figure REF (b) illustrates $\\Phi ^{\\prime }$ with respect to $\\eta $ .",
"There is a peak of $\\Phi ^{\\prime }$ near $\\eta = 2.31$ .",
"Figure REF (c) depicts the volume energy density, when the density has got the form $\\xi =[\\frac{1}{2}\\Phi ^{\\prime 2}+U]$ .",
"The lower right box in the same figure shows the magnification of the small region clearly representing the existence of a smooth hill.",
"The smooth peak of the volume energy density exists at $\\eta =5.17$ .",
"There is a disagreement between the location of the peak for the energy density and that for $\\Phi ^{\\prime }$ .",
"It clearly reveals the fact that the position with the maximum value for $\\Phi ^{\\prime }$ is still not the same as the maximum of the energy density due to non-trivial contribution coming from the potential, $U=-\\frac{\\lambda }{4}\\Phi ^4$ .",
"Figure REF (d) shows the Euclidean energy $E_{\\xi }$ for each slice of constant $\\eta $ .",
"The Euclidean energy signifies the value of energy after the full integration of variables except for $\\eta $ in the present work, $E_{\\xi }=2\\pi ^2\\eta ^3\\xi $ .",
"There are one minimum and one maximum point for $E_{\\xi }$ .",
"The location of the minimum of $E_{\\xi }$ is around $\\eta =2.31$ , whereas that of the maximum is near $\\eta =6.93$ .",
"Ironically, the location of the minimum of $E_{\\xi }$ coincides with that of the maximum of $\\Phi ^{\\prime }$ .",
"These solutions can be considered as a ball consisting of only a thick wall except for one point at the center of the solution with a lower arbitrary state than the outer vacuum state unlike a vacuum bubble that consists of an inside part with a lower vacuum state and a wall.",
"Figure: (a) The numerical solution for Φ\\Phi in thecase of m 2 =0m^2=0, (b) the variation of Φ ' \\Phi ^{\\prime } with respect toη\\eta , (c) the energy density ξ\\xi and (d) the Euclidean energyE ξ E_{\\xi } evaluated at constant η\\eta .For a theory with $m^2 > 0$ , the conformal invariance is broken and any solution with a finite action is forbidden by scaling argument.",
"In other worlds, the particle can not have enough energy to reach the hill overcome the barrier near $\\Phi =0$ since the damping term has got a large value due to a large value of $\\Phi ^{\\prime }$ near the initial point [41]."
],
[
"Fubini instantons of a self-gravitating scalar field ",
"Let us consider the following action: $S= \\int _{\\mathcal {M}} \\sqrt{-g} d^4 x \\left[ \\frac{R}{2\\kappa }-\\frac{1}{2}{\\nabla ^\\alpha }\\Phi {\\nabla _\\alpha }\\Phi -U(\\Phi )\\right] +\\oint _{\\partial \\mathcal {M}} \\sqrt{-h} d^3 x \\frac{K-K_o}{\\kappa }\\,,$ where $g=\\det g_{\\mu \\nu }$ , $\\kappa \\equiv 8\\pi G$ , $R$ denotes the scalar curvature of the spacetime $\\mathcal {M}$ , $h$ is the induced boundary metric, $K$ and $K_o$ are the traces of the extrinsic curvatures of $\\partial \\mathcal {M}$ for the metric $g_{\\mu \\nu }$ and $\\eta _{\\mu \\nu }$ , respectively.",
"The second term on the right-hand side is the boundary term [42].",
"It is necessary to have a well-posed variational problem including the Einstein-Hilbert term.",
"Here we adopt the notations and sign conventions of Misner, Thorne and Wheeler [43].",
"We study the creation process of Fubini instantons in curved spacetime.",
"In the first place, we consider the massless case, and then we will also consider generalized Fubini instantons, the so-called massive case (see the form of the potential in Eq.",
"(REF )).",
"The cosmological constant is given by $\\Lambda =\\kappa U_{0}$ , such that background space will be dS, flat or AdS depending on the signs of $U_{0}$ .",
"In order to solve the coupled equations, we assume an $O(4)$ symmetry for the geometry and the scalar field similar to Ref.",
"[22] $ds^{2} = d\\eta ^{2} + \\rho ^{2}(\\eta ) \\left[ d\\chi ^{2} + \\sin ^{2}\\chi \\left(d\\theta ^{2} + \\sin ^{2}\\theta d\\phi ^{2} \\right) \\right] \\,.$ And then, $\\Phi $ and $\\rho $ depends only on $\\eta $ , and the Euclidean equation can be written respectively as follows: $\\Phi ^{\\prime \\prime } + \\frac{3\\rho ^{\\prime }}{\\rho }\\Phi ^{\\prime }=\\frac{dU}{d\\Phi } \\,\\,\\, {\\rm and} \\,\\,\\,\\rho ^{\\prime \\prime } = - \\frac{\\kappa }{3}\\rho (\\Phi ^{\\prime 2} +U)\\,, $ and the Hamiltonian constraint is then given by $\\rho ^{\\prime 2} - 1 - \\frac{\\kappa \\rho ^2}{3}\\left(\\frac{1}{2}\\Phi ^{\\prime 2}-U\\right) = 0 \\,.$ In order to yield a meaningful solution, the constraint requires a delicate balance among all the different terms.",
"Otherwise the solution can yield qualitatively incorrect behavior [44].",
"To solve the Eqs.",
"(REF ), we have to impose suitable boundary conditions.",
"When the gravity is switched off, boundary conditions for the Fubini instanton are $\\frac{d\\Phi }{d\\eta }\\Big |_{\\eta =0}=0$ and $\\Phi |_{\\eta =\\infty }=0$ as in Ref.",
"[26].",
"While gravity is taken into account, we can write boundary conditions as follows: $\\rho |_{\\eta =0}=0, \\,\\,\\,\\, \\frac{d\\rho }{d\\eta }\\Big |_{\\eta =0}=1, \\,\\,\\,\\,\\frac{d\\Phi }{d\\eta }\\Big |_{\\eta =0}=0, \\,\\,\\,\\, {\\rm and}\\,\\,\\,\\,\\Phi |_{\\eta =\\eta _{max}}=0 \\,,$ where $\\eta _{max}$ is the maximum value of $\\eta $ .",
"For the flat and AdS background $\\eta _{max}=\\infty $ , while $\\eta _{max}$ is finite for the dS background.",
"The first condition is to obtain a geodesically complete spacetime.",
"The second condition is nothing but Eq.",
"(REF ).",
"The third condition is the regularity condition as can be seen from the first equation in Eq.",
"(REF ).",
"One should find the undetermined initial value of $\\Phi $ , i.e.",
"$\\Phi |_{\\eta =0}=\\Phi _o$ , using the undershoot-overshoot procedure [21], [30], to satisfy the fourth condition $\\Phi |_{\\eta =\\eta _{max}}=0$ .",
"We employ these conditions for Fubini instantons in Sec.",
"III A, B.",
"If the background space is dS, we can impose conditions specified at $\\eta = 0$ and $\\eta =\\eta _{max}$ .",
"For this purpose, we choose the values of the field $\\rho $ and derivatives of the field $\\Phi $ as follows: $\\rho |_{\\eta =0}=0 ,\\,\\,\\,\\, \\rho |_{\\eta =\\eta _{max}} =0, \\,\\,\\,\\,\\frac{d\\Phi }{d\\eta }\\Big |_{\\eta =0}=0, \\,\\,\\,\\, {\\rm and}\\,\\,\\,\\,\\frac{d\\Phi }{d\\eta }\\Big |_{\\eta =\\eta _{max}} = 0.",
"$ The first two conditions are for the background space.",
"The last two conditions are for the scalar field.",
"In general, the solutions satisfying Eq.",
"(REF ) do not guaranty $\\Phi _{\\eta =\\eta _{max}}$ to be zero.",
"For the solution having $\\Phi _{\\eta =\\eta _{max}}=0$ , the conditions Eq.",
"(REF ) are equivalent to the conditions Eq.",
"(REF ).",
"If $\\Phi _{\\eta =\\eta _{max}} = \\pm \\Phi _o$ , they represent completely new type of solutions with $Z_2$ symmetry.",
"We will discuss this case more in detail in Sec.",
"III C. In order to solve the Euclidean field Eqs.",
"(REF ) and (REF ) numerically, we rewrite the equations in terms of dimensionless variables as in Ref.",
"[30].",
"In the present work, we employ the shooting method using the adaptive step size Runge-Kutta as the integrator similar to the treatment in Ref.",
"[45].",
"For this procedure we choose the initial values of $\\tilde{\\Phi }(\\tilde{\\eta }_{\\mathrm {initial}})$ , $\\tilde{\\Phi }^{\\prime }(\\tilde{\\eta }_{\\mathrm {initial}})$ , $\\tilde{\\rho }(\\tilde{\\eta }_{\\mathrm {initial}})$ , and $\\tilde{\\rho }^{\\prime }(\\tilde{\\eta }_{\\mathrm {initial}})$ at $\\tilde{\\eta }=\\tilde{\\eta }_{\\mathrm {initial}}$ as follows: $\\tilde{\\Phi }(\\tilde{\\eta }_{\\mathrm {initial}}) &\\sim & \\tilde{\\Phi }_{o} - \\frac{\\epsilon ^2}{8}\\tilde{\\Phi }_o(\\tilde{\\Phi }^2_o-1) + \\cdot \\cdot \\cdot \\,, \\nonumber \\\\\\tilde{\\Phi }^{\\prime }(\\tilde{\\eta }_{\\mathrm {initial}}) &\\sim & - \\frac{\\epsilon }{4}\\tilde{\\Phi }_o(\\tilde{\\Phi }^2_o-1) + \\cdot \\cdot \\cdot \\,, \\\\\\tilde{\\rho }(\\tilde{\\eta }_{\\mathrm {initial}}) &\\sim & \\epsilon + \\cdot \\cdot \\cdot \\,, \\nonumber \\\\\\tilde{\\rho }^{\\prime }(\\tilde{\\eta }_{\\mathrm {initial}}) &\\sim & 1 + \\cdot \\cdot \\cdot \\,, \\nonumber $ where $\\tilde{\\eta }_{\\mathrm {initial}} = 0+\\epsilon $ for $\\epsilon \\ll 1$ .",
"The minus sign in front of the second formula is due to the negative value of the $\\tilde{\\Phi }^{\\prime \\prime }$ determined by the sign of $dU/d\\Phi $ at $\\tilde{\\eta }=0$ .",
"However, the initial value of $\\tilde{\\Phi }^{\\prime }$ is taking to be positive in the present work.",
"Once we specify the initial value $\\tilde{\\Phi }_{0}$ , the remaining conditions can be exactly determined from Eqs.",
"(REF ).",
"Furthermore we impose additional conditions implicitly.",
"To avoid a singular solution at $\\tilde{\\eta } = \\tilde{\\eta }_{\\mathrm {max}}$ for the Euclidean field equations and to demand a $Z_2$ symmetry, the conditions $d\\tilde{\\Phi }/{d\\tilde{\\eta }} \\rightarrow 0$ and $\\tilde{\\rho }\\rightarrow 0$ as $\\tilde{\\eta }\\rightarrow \\tilde{\\eta }_{\\mathrm {max}}$ are needed in the next section.",
"In this work, we require that the value of $d\\tilde{\\Phi }/{d\\tilde{\\eta }}$ goes to a value smaller than $10^{-6}$ as $\\tilde{\\eta } \\rightarrow \\tilde{\\eta }_{\\mathrm {max}}$ , as the exact value of $\\tilde{\\eta }_{\\mathrm {max}}$ is not known [30].",
"The parameter $\\tilde{\\kappa }$ is the ratio between the gravitational constant or Planck mass and the mass scale in the theory, $\\tilde{\\kappa }=\\frac{m^2}{\\lambda }\\kappa =\\frac{8\\pi m^2}{M^{2}_{p}\\lambda }$ , and the parameter $\\tilde{\\kappa }\\tilde{U}_o$ is related to the rescaled cosmological constant $\\Lambda /m^2$ .",
"To find the probability of the instanton solution, we only consider the Euclidean action for the bulk part in Eq.",
"(REF ) to get, $S_E= \\int _{\\mathcal {M}} \\sqrt{g_{\\mathrm {E}}} d^4 x_{\\mathrm {E}}\\left[ -\\frac{R_\\mathrm {E}}{2\\kappa } +\\frac{1}{2}\\Phi ^{\\prime 2} +U \\right]= 2\\pi ^2 \\int \\rho ^3 d\\eta [-U]\\,, $ where $R_{\\mathrm {E}} =6[1/\\rho ^2 - \\rho ^{\\prime 2}/\\rho ^2 - \\rho ^{\\prime \\prime }/\\rho ]$ .",
"We used Eqs.",
"(REF ) and (REF ) to arrive at this.",
"The volume energy density has the form: $\\xi =-U$ , which has a different sign compared to the sign of the density used in Ref.",
"[30].",
"The Euclidean energy signifies the energy value after the full integration of variables except for $\\eta $ in the present case as $E_{\\xi }=2\\pi ^2\\rho ^3\\xi $ .",
"In the beginning, we obtain the numerical solution for $m^2=0$ .",
"And then we obtain the numerical solution for $m^2 > 0$ .",
"We call the space dS when the initial vacuum state has a positive cosmological constant, $U_o > 0$ , flat when $U_o = 0$ and AdS when $U_o < 0$ .",
"The rate of decay of a metastable state can be evaluated in terms of the classical configuration and represented as $Ae^{-B}$ in this approximation, in which the leading semiclassical exponent $B=S^{\\mathrm {cs}} - S^{\\mathrm {bg}}$ is the difference between the Euclidean action corresponding to the classical solution $S^{\\mathrm {cs}}$ and the background action $S^{\\mathrm {bg}}$ .",
"The prefactor $A$ is evaluated from the Gaussian integral over fluctuations around the background classical solution [46], [47].",
"Figure: (color online).",
"The numerical solutions ofFubini instantons with m 2 =0m^2=0 in the AdS space.Table: The dimensionless variables and color of plotused and the actions obtained in Fig.",
"."
],
[
"Solutions without oscillation",
"We perform the numerical work with $m^2=0$ and take $\\tilde{\\kappa } =0.1$ .",
"The solutions without oscillation are only possible in the initial AdS background as shown in Fig.",
"REF .",
"We guess that there is no solution without any oscillation for the initial flat and dS background.",
"In the given $\\tilde{\\kappa }$ , there may exist the phase space of solutions having the region of an arbitrary $\\Phi _o$ .",
"If $\\tilde{\\kappa }$ is increased, the oscillating behavior is appearing in the phase space of solutions [48].",
"Figure 4(a) illustrates the solution for $\\tilde{\\Phi }$ , in which the right box in the same figure shows the magnification of a small region representing the initial values of $\\tilde{\\Phi }$ and the behavior of the curves.",
"The curves move upwards with increasing value of $\\tilde{U}_o$ , then overlap near $\\tilde{\\eta }=1$ , and more downwards with increasing value of $\\tilde{U}_o$ .",
"Figure 4(b) shows the solutions of $\\tilde{\\rho }$ .",
"The curves move downwards with increasing $\\tilde{U}_o$ .",
"The shape of the numerical solution $\\tilde{\\rho }$ can be easily understood if one thinks of the shape of the solution in a fixed AdS space as $\\rho =\\sqrt{\\frac{3}{\\Lambda }}\\sinh \\sqrt{\\frac{\\Lambda }{3}} \\eta $ .",
"Table 1 shows the dimensionless variables and the color of plot used, and also the actions obtained from Fig.",
"REF .",
"From the numerical data, one can easily see that the magnitude of $\\tilde{\\Phi }_o$ approaches the vacuum state $\\tilde{\\Phi }=0$ as $\\tilde{U}_o$ approaches a vanishing value.",
"The vanishing of $\\tilde{U}_o$ means the background geometry which serves as the initial vacuum state is flat.",
"The action difference $\\tilde{B}$ between the action of the solution $\\tilde{S}^{\\mathrm {cs}}$ and that of the background $\\tilde{S}^{\\mathrm {bg}}$ has positive values.",
"We carry out the action integral in the range $0 \\leqq \\tilde{\\eta } \\leqq 25$ numerically as the action difference $\\tilde{B}$ diverges to infinity if we perform the integration for an infinite $\\tilde{\\eta }$ value.",
"This divergence is due to the fact that the size of the solution including the outside part in the evolution parameter space decrease compared to the size of the initial background similar to what happens for the case of the nucleation of a vacuum bubble.",
"In the analytic computation, the outside part and the background are simply canceled at the same radius.",
"In the present numerical work, it is difficult to decide the exact size of the solution.",
"Thus we straightforwardly compute the action difference and then the difference $\\tilde{B}$ has got an approximate behavior $\\delta (\\sinh ^3{\\tilde{\\eta }})= 3\\sinh ^2\\tilde{\\eta } \\cosh \\tilde{\\eta }$ which cause the divergence at infinity.",
"If this minor error is cured, the action difference has a finite value.",
"Figure: (color online).",
"The numerical solution ofΦ\\Phi , the derivative of Φ\\Phi with respect to η\\eta , ρ\\rho , andthe derivative of ρ\\rho with respect to η\\eta for the generalizedFubini instantons with m 2 >0m^2 > 0.Table: The dimensionless variables and color of plotused and the actions obtained in Fig.",
".Now we perform the numerical work with $m^2 > 0$ and take $\\tilde{\\kappa } =0.3$ .",
"This type of solutions belongs to usual tunneling with a barrier.",
"We obtained the numerical solutions with an arbitrary cosmological constant as shown in Fig.",
"REF .",
"The solutions are only possible for specific $\\tilde{\\Phi }_o$ s. The figures represent the vary fact that the solution is only possible in curved spacetime irrespective of the value of the cosmological constant.",
"Figure 5(a) illustrates the solution of $\\tilde{\\Phi }$ .",
"The upper right box in the same figure shows the magnification of the small region representing behavior of the curves which move to the left with an increase in $\\tilde{U}_o$ .",
"Figure REF (b) shows $\\tilde{\\Phi }^{\\prime }$ with respect to $\\tilde{\\eta }$ .",
"The upper right box in the same figure shows the magnification of the small region representing behavior of the curves moving below with increasing value of $\\tilde{U}_o$ .",
"Figure REF (c) illustrates the solutions of $\\tilde{\\rho }$ .",
"The curves move downwards with increasing value of $\\tilde{U}_o$ .",
"The shape of the numerical solutions of $\\tilde{\\rho }$ can be easily understood if one consider a fixed space.",
"In the fixed flat space, $\\rho = \\eta $ .",
"In the dS space, $\\rho = \\sqrt{\\frac{3}{\\Lambda }}\\sin \\sqrt{\\frac{\\Lambda }{3}}\\eta $ .",
"In the AdS space, $\\rho =\\sqrt{\\frac{3}{\\Lambda }}\\sinh \\sqrt{\\frac{\\Lambda }{3}} \\eta $ .",
"Figure REF (d) depicts the variation of $\\tilde{\\rho }$ with respect to $\\tilde{\\eta }$ .",
"The curves move below with increase in $\\tilde{U}_o$ .",
"The horizontal line with $\\tilde{U}_o=0$ indicate a flat space with $\\tilde{\\rho }^{\\prime }=1$ .",
"Table REF shows the dimensionless variables and color of the plot used among with the action obtained from Fig.",
"REF .",
"From the numerical data, one can infer that the magnitude of $\\tilde{\\Phi }_o$ decreases as $\\tilde{U}_o$ increases.",
"We carry out the action integral in the range $0 \\leqq \\tilde{\\eta } \\leqq 30.58$ numerically.",
"In the dS space, the solution and the background have their own periods for $\\tilde{\\eta }$ , which we take the period as the integration limit.",
"For the background dS space, we take $\\tilde{\\eta } = \\pi \\sqrt{\\frac{3}{\\tilde{\\kappa }\\tilde{U}_o}}$ .",
"The action for $\\tilde{S}^{\\mathrm {cs}}$ and $\\tilde{S}^{\\mathrm {bg}}$ are positive or zero as long as $\\tilde{U}_o \\leqq 0$ .",
"The background action is zero for $\\tilde{U}_o = 0$ .",
"In this work, we do not check for the special case $\\tilde{S}^{\\mathrm {cs}}=0$ .",
"Simply, the action has a negative value for the dS space.",
"It is related to the fact that the Euclidean action for Einstein gravity is not bounded from below, and this is known as the conformal factor problem in Euclidean quantum gravity [49].",
"It was argued in [50] that the conformal divergence due to the unboundedness of the action might get cancelled with a similar term of opposite sign caused by the measure of the path integral.",
"However, the difference between the action of the solution and that of the background remains positive-valued."
],
[
"Oscillating Fubini instantons ",
"The oscillating instanton and the bounce solutions with an $O(4)$ symmetry between the dS-dS vacuum states was first studied in Ref.",
"[51], in which the authors found the solutions in a fixed background geometry and showed how does the maximum allowed number $n_{\\mathrm {max}}$ depend on the parameters of the theory, where $n$ denotes the crossing number of the potential barrier by the oscillating solutions.",
"The oscillating bounce solutions in the presence of gravity was also studied in Ref.",
"[52], in which the authors analyzed the negative modes and the fluctuations around the oscillating solutions.",
"The instanton was interpreted as the thermal tunneling [53].",
"The oscillating instanton solutions under a symmetric double-well potential in the curved space with an arbitrary vacuum energy was also investigated in detail in [30], where a numerical solution is possible as long as the local maximum value of the potential remains positive.",
"The solutions have a thick wall and can be interpreted as a mechanism for the nucleation of the thick wall for topological inflation [54].",
"Similarly, the process for the tunneling without a barrier in curved space, was studied in Ref.",
"[28], [29].",
"The existence of numerical solutions was shown in Ref.",
"[30], in which the case representing the tunneling from flat to AdS space shows an oscillating behavior.",
"The solution oscillates around $\\Phi =0$ in the inverted potential and the oscillating behavior die away unlike the case under a harmonic potential.",
"In the present paper, the oscillation means that the field in the solutions oscillates around the minimum of the inverted potential and die away asymptotically to the minimum $\\Phi =0$ for the case with $m^2=0$ .",
"Thus the resulting geometry of the initial state has wrinkles due to the variation of the volume energy density and the instanton simultaneously.",
"The behavior of the solutions representing the resulting geometry with wrinkles is quite different from those in Ref.",
"[30].",
"Figure REF shows the numerical solutions representing an oscillatory behavior in (A) the initial flat background and (B) the initial AdS background.",
"We take $\\tilde{\\kappa } =0.30$ , $\\tilde{U}_o = 0$ (for the flat case), and $\\tilde{U}_o = -0.0001$ (for the AdS case), respectively.",
"Figures REF (a) and (c) illustrate the numerical solutions of $\\tilde{\\Phi }$ .",
"The lower right box in those figure shows the magnification of a small region representing the behavior of the solution around $\\tilde{\\Phi }=0$ .",
"The peak corresponds to the first turning point of the particle similar to a classical mechanics problem in the presence of an inverted potential.",
"For the case with $\\tilde{\\Phi }_o = -7$ the first turning point reaches furthermost point away from $\\tilde{\\Phi }=0$ among all the other cases, as one can easily see from the figure.",
"The curves oscillate around $\\tilde{\\Phi }=0$ and eventually stop at $\\tilde{\\Phi }=0$ in the flat and AdS space.",
"We take the initial point as an arbitrary $\\tilde{\\Phi }_o$ , which means that the number of oscillations for each solution can be different.",
"However, there is the tendency that the number of oscillations is decreased as the value of $\\tilde{\\Phi }_o$ is decreased in the given $\\tilde{\\kappa }$ .",
"Figures REF (b) and (d) illustrate the numerical solutions for $\\tilde{\\rho }$ .",
"The upper left box in those figure shows the magnification of an initial small region representing behavior of the curves which move below with the decrease in $\\tilde{\\Phi }_o$ .",
"Table: The dimensionless variables and color of plotused and the actions obtained in Fig.",
".Table REF shows the initial values of $\\tilde{\\Phi }$ , the actions for the AdS, and flat background which are obtained from Fig.",
"REF .",
"In the flat case, the background action is zero as $\\tilde{U}_o=0$ and therefore $\\tilde{S}^{\\mathrm {cs}}$ is equal to $\\tilde{B}$ .",
"In the present case, we cut all the data at a certain point which is $\\tilde{\\eta }=200$ .",
"We now analyze the behavior of the solutions using a phase diagram method.",
"After plugging the value of $\\frac{\\rho ^{\\prime }}{\\rho }$ from Eq.",
"(REF ) into Eq.",
"(REF ) and using $\\Phi ^{\\prime \\prime }=\\Phi ^{\\prime }\\frac{d\\Phi ^{\\prime }}{d\\Phi }$ , the equation becomes $\\frac{d\\Phi ^{\\prime }}{d\\Phi } = - \\frac{3\\sqrt{\\frac{1}{\\rho ^2} +\\frac{\\kappa }{3}(\\frac{1}{2}\\Phi ^{\\prime 2}+\\frac{\\lambda }{4}\\Phi ^4-U_o)}\\Phi ^{\\prime } +\\lambda \\Phi ^3}{\\Phi ^{\\prime }}\\,.$ First, we consider the situation where the kinetic energy is small compared to the potential energy such that $|U| \\gg \\Phi ^{\\prime 2}$ , $U_o\\ll 1$ and the term $1/\\rho ^2$ is smaller than other terms.",
"In other words, the last term is the most dominant among other terms in the numerator.",
"Then the equation reduces to the form $\\Phi ^{\\prime } \\simeq \\sqrt{\\frac{\\lambda }{2}(\\Phi ^4_o-\\Phi ^4)} \\,.",
"$ The above relation shows that the first stage of the curve has got such kind of form.",
"Second, we consider the situation where $d\\Phi ^{\\prime }/d\\Phi =0$ , i.e.",
"with vanishing acceleration and then we impose all the above mentioned conditions.",
"It will then describe the special points in the phase diagram.",
"Then the equation reduces to the form $\\Phi ^{\\prime } \\simeq - 2\\sqrt{\\frac{\\lambda }{3\\kappa }}\\Phi \\,.",
"$ Third, we consider the situation where $d\\Phi ^{\\prime }/d\\Phi = -c $ , i.e.",
"a negative constant.",
"We impose all the above mentioned conditions among with $\\Phi ^2 \\gg 2c/\\sqrt{3\\kappa \\lambda }$ .",
"Thus, we obtain the above equation again.",
"This relation implies that the special points with a vanishing acceleration and some of the region with a negative constant acceleration in the phase diagram have got a linear function type behavior in the phase diagram as shown in Figs.",
"REF (a) and (c).",
"Figure REF illustrates the behavior of the solutions in the $\\tilde{\\Phi }$ -$\\tilde{\\Phi }^{\\prime }$ plane.",
"Each trajectory represents the behavior of the solution in the phase diagram.",
"The trajectories begin with zero velocity as $\\tilde{\\Phi }^{\\prime }=0$ shown in Figs.",
"REF (a) and (c).",
"The velocity increases rapidly to the maximum and then decreases linearly up to the turning point.",
"Figures REF (b) and (d) show the magnification of the small region representing the behavior of the solution around $\\tilde{\\Phi }^{\\prime }=0$ and $\\tilde{\\Phi }=0$ .",
"Figure: The behavior of the solutionsrepresented in the phase diagram.Basically, the Fubini solution has an asymptotic condition to be satisfied.",
"We expect that there exist an oscillating solutions although the dS background has got a finite size in the Euclidean signature.",
"However, if we consider an analytic continuation not of the angle parameter $\\chi $ but of the Euclidean evolution parameter $\\eta =it$ , then the meaning becomes clearer.",
"When there is an `even' symmetry for the oscillating instantons, we can see the half-way point $\\eta _{0}$ as $\\dot{\\rho }(\\eta _{0})=\\dot{\\Phi }(\\eta _{0})=0$ .",
"Then, we can paste the Lorentzian manifold $t=0$ at the $\\eta =\\eta _{0}$ surface.",
"This is possible only for the case $\\dot{\\rho }(\\eta _{0})=\\dot{\\Phi }(\\eta _{0})=0$ , because of the Cauchy-Riemann theorem of complex analysis; otherwise, the Lorentzian manifold should be complex valued functions (for exceptional cases, we might be able to consider complex valued instantons, the so-called fuzzy instantons [55], [56]).",
"In this procedure, an event shows a spontaneous creation of the universe from `nothing' [2], in which nothing means a state without the concept of classical spacetime [57].",
"We already know that there is such a solution when the scalar field is exactly on top of the local maximum.",
"However, now we observe a creation from nothing with highly non-trivial field dynamics.",
"This is worthwhile to be highlighted and we postpone further analysis for the future work."
],
[
"Fubini instantons with $Z_2$ symmetry",
"We now shift our attention to the new type of solutions in the initial background as the dS space, i.e.",
"$U_o > 0$ .",
"The Euclidean dS space has a compact geometry.",
"Thus the solutions can have $Z_2$ symmetry.",
"We consider the boundary conditions in Eq.",
"(REF ).",
"To obtain the solutions with $Z_2$ symmetry, we need to impose additional conditions.",
"For the background geometry, $\\rho ^{\\prime }=0$ at $\\eta =\\frac{\\eta _{max}}{2}$ .",
"On the other hand, for the scalar field, we impose $\\Phi =0$ at $\\eta =\\frac{\\eta _{max}}{2}$ for the solutions with odd number of crossings of the potential well and $\\Phi ^{\\prime }=0$ at $\\eta =\\frac{\\eta _{max}}{2}$ for the solutions with even number of crossings.",
"The solutions with odd number of crossing have the opposite state of the value $\\Phi $ at $\\eta =0$ and $\\eta =\\eta _{max}$ , i.e.",
"$\\Phi |_{\\eta =\\eta _{max}}=-\\Phi _o$ .",
"The solutions with even number of crossing have the same state of the value $\\Phi $ at $\\eta =0$ and $\\eta =\\eta _{max}$ , i.e.",
"$\\Phi |_{\\eta =\\eta _{max}}=\\Phi _o$ .",
"We stress that the boundary conditions in Eq.",
"(REF ) gives rise to completely new type of solutions of Fubini instanton.",
"Figure: (color online).",
"The numerical solutionsof the Fubini instanton with Z 2 Z_2 symmetry.Figure REF shows the numerical solutions of the Fubini instanton with $Z_2$ symmetry.",
"We take $\\tilde{\\kappa } =0.50$ and $\\tilde{U}_o = 0.03$ .",
"Thus the dS region in the $\\tilde{\\Phi }$ -space spans the region $ -0.589 \\lesssim \\tilde{\\Phi } \\lesssim 0.589$ .",
"We consider four cases with different initial positions of $\\tilde{\\Phi }$ .",
"Figure REF (a) illustrates the numerical solution of $\\tilde{\\Phi }$ .",
"The trajectories with the blue and red color go back to the same position of $\\tilde{\\Phi }$ in the presence of the inverted potential, i.e.",
"they have even number of crossings.",
"The trajectories with the black and green color go back to the opposite position of $\\tilde{\\Phi }$ , i.e.",
"they have odd number of crossings.",
"Figure REF (b) depicts the numerical solution of $\\tilde{\\rho }$ .",
"Figures REF (c) and (d) illustrate the behavior of the solutions in the $\\tilde{\\Phi }$ -$\\tilde{\\Phi }^{\\prime }$ plane.",
"Each trajectory represents the behavior of the solution in the phase diagrams.",
"The blue and red lines indicate that the interior part of two instantons has got the same state as $\\tilde{\\Phi }$ , whereas the green and black lines indicate that the interior part has got the opposite state of $\\tilde{\\Phi }$ .",
"Figure REF (d) illustrates the magnification of a small region representing the behavior of the solution around $\\tilde{\\Phi }^{\\prime }=0$ and $\\tilde{\\Phi }=0$ .",
"Figure REF (e) illustrates the volume energy density, where the density has got a form $\\tilde{\\xi }=-\\tilde{U}$ .",
"The box shows the magnification of a small region representing behavior of curves.",
"The densities in each of the case have got positive values near the initial starting point $\\tilde{\\Phi }_o$ far away from the point $\\tilde{\\Phi }=0$ , because the densities have the form $\\tilde{\\xi }=-\\tilde{U}$ and $\\tilde{U}_o > 0$ .",
"The solutions oscillate in the dS region as found the present work.",
"The density always negative values for the case of the black line.",
"Figure REF (f) illustrates the Euclidean energy $\\tilde{E}_{\\tilde{\\xi }}=2\\pi ^2\\tilde{\\rho }^3\\tilde{\\xi }$ for each slice of constant $\\tilde{\\eta }$ values.",
"The negative energy parts in each of the case signifies a rolling state in the dS region.",
"Table REF shows the initial value of $\\tilde{\\Phi }$ , colors of the plot used, and the actions obtained from Fig.",
"REF .",
"Table: The dimensionless variables and the color ofplot used, and the actions obtained in Fig.",
"."
],
[
"Causal structures ",
"In this section, we briefly outline the causal structure of the solutions in the Lorentzian signature.",
"Due to the pressure difference, the nucleated AdS region will expand over the background and hence the boundary of the nucleated AdS region will be time-like.",
"Figure REF shows the schematic diagrams representing the causal structures of the Fubini instantons and the related solutions.",
"The $\\chi =\\pi /2$ surface can be analytically continued to the surface $t=0$ in the Lorentzian signature.",
"The lower vacuum region in the instanton (green colored region) will be unstable during the Lorentzian time evolution (orange colored region).",
"Due to the instability of the Fubini type potential, the whole causal structure may depend on the shape of the potential or the vacuum structure i.e.",
"whether the left or the right side of the potential has true vacua or not.",
"Therefore, the followings are meaningful only as reasonable estimations for general behavior and these may be different for special examples.",
"Figure REF (a) illustrates the instanton solution in an AdS background.",
"It will form time-like $r=0$ and $r=\\infty $ boundaries in the Lorentzian signature.",
"However, the AdS region may be unstable to form a kind of singularity.",
"Figure REF (b) illustrates the instanton solution in the dS background.",
"The dS region has a cosmological horizon and will this form a future infinity.",
"The AdS region (orange colored region) will expand over the dS region due to the pressure difference.",
"Figure REF (c) is the pair creation by the oscillating instanton solutions.",
"Therefore, in the instanton part, the dS region around the $\\rho =\\rho _{\\mathrm {max}}$ is surrounded by the AdS (green colored region) part.",
"In the Lorentzian signature, we can interpret these two AdS parts as being nucleated in a dS background.",
"In Fig.",
"REF (c), we infer that, there still remains a dS region and a future infinity.",
"Figure: (color online).",
"The schematic diagramsrepresenting the causal structure of the Fubini instantons and therelated solutions.",
"(a) Tunneling in an AdS background.",
"(b)oscillating instanton solution in a dS background.",
"(c) Pair creationby oscillating instantons in dS background.The pair creation of the instantons in this work is quite different from the ordinary quantum process of pair creation of particles.",
"We take the initial background as the dS space, i.e.",
"$U_o > 0$ .",
"The Euclidean dS space has a compact geometry.",
"Thus, the geometry has two poles.",
"If one object is created on the north pole and the other on the south pole, we can interpret that process as the pair creation of objects.",
"As an example of the process, the two-crossing solution between the sides of the potential barrier in the double-well potential was considered as a type of a double-bounce solution or an anti-double-bounce solution [58], in which the authors interpreted the double-bounce solution as the spontaneous pair-creation of true vacuum bubbles, one at each pole in the dS space.",
"We adopt a similar interpretation for our solutions with $Z_2$ symmetry."
],
[
"Summary and Discussions ",
"In this paper we have studied Fubini instantons of a self-gravitating scalar field representing the tunneling without a barrier.",
"There are two kinds of Euclidean solutions representing the tunneling without any barrier.",
"One of them is the tunneling from the local maximum of the potential to the vacuum state.",
"The other one is the tunneling from the maximum to any arbitrary state.",
"The latter corresponds to the Fubini instanton solution.",
"We have shown that there exist several new kinds of Fubini instanton solutions of a self-gravitating scalar field found as numerical solutions, which are possible only if gravity is taken into account.",
"We also computed the action difference $B$ , in each case, between the Euclidean action corresponding to a classical solution $S^{\\mathrm {cs}}$ and the background action $S^{\\mathrm {bg}}$ for the rate of decay.",
"In Sec.",
"2, we reviewed the Fubini instanton in the absence of gravity from the viewpoint of a tunneling problem.",
"We have presented a numerical solution including the Euclidean energy density for example.",
"We analyzed the structure of the solution in a theory with the potential having only a quartic self-interaction term.",
"These solutions can be considered as a ball consisting of only a thick wall except for the one point at the center of the solution with a lower arbitrary state than the outer vacuum state unlike a vacuum bubble which consists of an inner part with a lower vacuum state and a wall.",
"In Sec.",
"3, we have studied the instanton solutions in curved space.",
"We performed careful numerical study to solve the coupled equations for the gravity and the scalar field simultaneously.",
"We have shown that there exist numerical solutions without any oscillation in the initial AdS space for the potential with only the quartic term.",
"We have also shown that there exist numerical solutions for the potential with both a quartic and a quadratic term irrespective of the value of the cosmological constant.",
"For this particular case, there is no solution with an $O(4)$ symmetry when gravity is switched off.",
"In order to estimate the decay rate of the background state, we calculated the action difference between the action of the solution and that of the background obtained using numerical means.",
"We have obtained oscillating Fubini instantons as new types of solutions.",
"We have shown that there exist oscillating numerical solutions for the potential with only the quartic term in the flat and AdS space, except for the solution without oscillation in the initial AdS space with the specific value of a cosmological constant and the parameters.",
"We have analyzed the behavior of the solutions using the phase diagram method.",
"The oscillation dies away asymptotically in both the flat and the AdS space.",
"We have obtained numerical solutions representing the Fubini instanton with $Z_2$ symmetry.",
"We stress that they represent completely new type of solutions of Fubini instanton.",
"These solutions can be interpreted as the pair creation with each one having the same state and with each one having the opposite state, respectively.",
"The solutions can lead to more interesting interpretation as follows: any arbitrary state can tunnel into another arbitrary state with an $O(4)$ -symmetry in the curved spacetime, although no vacuum state exists as the instanton solution.",
"The solutions are possible as long as the maximum of the potential remains positive.",
"The subject on the pair creation of bubbles was first considered in Ref.",
"[59].",
"The numerical solution representing the pair of solutions is in Fig.",
"2 in Ref.",
"[58], which can be interpreted as the pair creation of the bubbles, one at each pole in the dS space.",
"However, there is a different interpretation on the solutions [53], [60], in which the authors studied a decay channel of de Sitter vacua.",
"The solutions with $O(3)$ symmetry can be understood as describing tunneling in a finite horizon volume at finite temperature.",
"The solutions maybe correspond to thermal production of a bubble in their interpretation.",
"In this stage, the comparative analysis between the O(4)-symmetric solution and O(3)-symmetric solution with respect to the pair creation is needed to be studied more.",
"We leave this for future work.",
"In Sec.",
"4, we have analyzed the schematic diagrams representing the causal structures of the Fubini instantons and the related solutions in the Lorentzian signature.",
"For the special case representing the solution with $Z_2$ symmetry, the dS region around the $\\rho =\\rho _{\\mathrm {max}}$ is surrounded by an AdS part.",
"We now mention on the negative mode problem.",
"It was known that the bounce solution has one negative mode in the spectrum of small perturbations about the solution [46], [61].",
"The bounce solution with one negative mode corresponds to the tunneling process in the lowest WKB approximation.",
"In Ref.",
"[61], Coleman argued that the Euclidean solution with only one negative mode is related to the tunneling process in the flat Minkowski spacetime.",
"However, there is no rigorous proof on extension of Coleman's argument to the curved space claiming the physical irrelevance of the solutions with additional negative modes.",
"For example, the time-translation invariance, or zero modes, is one of crucial elements to prove the uniqueness of the negative mode in his argument.",
"However, the existence of zero modes is not guaranteed in curved spacetime.",
"Another point is that the Euclidean time interval is at most of $O(H^{-1})$ in de Sitter space.",
"Hence, only a finite number of the bounces can be placed far apart from each other.",
"Therefore, the dilute gas approximation may become invalid easily, which leads to the breakdown of the WKB approximation [62].",
"There appears diverse situations on the negative modes when the gravity is taken into account [63], [64], [62], [65].",
"Although, the bounce solution with one negative mode in curved space dominates the tunneling process, the solutions with additional negative modes may also contribute to the tunneling process.",
"There exist some works including the physical interpretation on the oscillating solutions with more than one negative mode.",
"One can naturally interpret the system in de Sitter background as a thermal system.",
"The authors in Refs.",
"[51], [53] interpreted that the existence of additional negative modes represents the solutions as unstable intermediate thermal configuration.",
"They seem to observe the clue to support this idea on the other point of view.",
"It is known that the $N$ times oscillating solutions have $N$ negative modes [51], [53], [66].",
"The even numbers of negative modes of the form $4N$ and $4N+2$ do not have imaginary part of the energy, while the odd numbers of negative modes of the form $4N+3$ have the imaginary part of the energy with the wrong sign.",
"However, $4N+1$ negative modes may have a meaning for a tunneling process even if the solution may not be related to the lowest WKB approximation.",
"Recently the analysis on the negative modes of oscillating instantons has been investigated [66].",
"The oscillating instantons as homogeneous tunneling channels have been also studied [67].",
"In conclusion, we believe many Euclidean solutions in curved space with zero and negative modes may have physical significance and deserves further investigation.",
"In summary, we illustrate the following finding in our new contribution regarding this issue: In the absence of gravity, a $-\\phi ^{4}$ -type potential has infinitely many instanton solutions whereas a $-\\phi ^{4}+\\phi ^{2}$ -type potential has no instanton solution.",
"However, the inclusion of the gravity changes all the situation abruptly: for the former case, the solution space get reduced to a finite space and for the latter case, there exists solutions.",
"We also confirm that $-\\phi ^{4}$ -type potentials have oscillating instanton solutions as well as the solutions with $Z_2$ symmetry.",
"Therefore, the Fubini instanton is one of the few examples that shows the effect of gravity bringing drastic changes to the tunneling process.",
"There can be more applications of the oscillating instantons and we confirm that the Fubini-type potentials also contribute largely towards these processes.",
"We postpone any possible application of such oscillating solutions including the phase space of solutions for our future work [48]."
],
[
"Acknowledgements",
"We would like to thank Andrei Linde for kind historic comments on the inflationary multiverse scenario and Fubini instanton.",
"We would like to thank Erick J. Weinberg, George Lavrelashvili, Hongsu Kim, Yunseok Seo, and Dong-il Hwang for helpful discussions and comments, and thank Chaitali Roychowdhury for a careful English revision of the manuscript.",
"We would like to thank Manu B. Paranjape and Richard MacKenzie for their hospitality during our visit to Université de Montréal.",
"This work was supported by the Korea Science and Engineering Foundation (KOSEF) grant funded by the Korea government(MEST) through the Center for Quantum Spacetime(CQUeST) of Sogang University with grant number R11 - 2005 - 021.",
"WL was upported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education, Science and Technology(2012R1A1A2043908).",
"DY is supported by the JSPS Grant-in-Aid for Scientific Research (A) No.",
"21244033.",
"We appreciate APCTP for its hospitality during completion of this work."
]
] | 1204.1521 |
[
[
"Discrete Sampling and Interpolation: Universal Sampling Sets for\n Discrete Bandlimited Spaces"
],
[
"Abstract We study the problem of interpolating all values of a discrete signal f of length N when d<N values are known, especially in the case when the Fourier transform of the signal is zero outside some prescribed index set J; these comprise the (generalized) bandlimited spaces B^J.",
"The sampling pattern for f is specified by an index set I, and is said to be a universal sampling set if samples in the locations I can be used to interpolate signals from B^J for any J.",
"When N is a prime power we give several characterizations of universal sampling sets, some structure theorems for such sets, an algorithm for their construction, and a formula that counts them.",
"There are also natural applications to additive uncertainty principles."
],
[
"Introduction",
"In this paper and in a sequel [1] we consider the problem of interpolating all values of a discrete, periodic signal $f\\colon \\mathbb {Z}_N \\longrightarrow \\mathbb {C}$ , $N \\ge 2$ , when $d<N$ values of $f$ are known.",
"One solution is a discrete form of the classical Nyquist-Shannon theorem, where the spectrum of the signal is assumed to vanish outside a contiguous band of frequencies; see [2], for example.",
"At the other extreme is the new and important area of compressed sensing, where no assumptions on the spectrum are made.",
"For this, of the many papers we mention only [3], [4] and [5], since we will refer to this work later.",
"Our approach to the problem is in between, though we begin by formulating a very general definition.",
"Definition 1 Let $\\mathbb {Y}$ be a $d$ -dimensional subspace of $\\mathbb {C}^N$ , let $\\mathcal {I}\\subset [0: N-1]$ be an index set of size $d$ , and let $\\mathcal {U}_\\mathcal {I}=\\lbrace u_i \\colon i \\in \\mathcal {I}\\rbrace $ be a set of $d$ vectors in $\\mathbb {Y}$ .",
"We say that $(\\mathcal {I},\\mathcal {U}_\\mathcal {I})$ is an interpolating system if each $f\\in \\mathbb {Y}$ can be written as $ f=\\sum _{i\\in \\mathcal {I}} f(i)u_i.$ We call $\\mathcal {I}$ a sampling set and $\\mathcal {U}_\\mathcal {I}$ an interpolating basis.",
"When we refer simply to a sampling set we always mean that it is associated with an interpolating basis.",
"If the vectors $u_i$ are orthogonal we say that $(\\mathcal {I},\\mathcal {U}_\\mathcal {I})$ is an orthogonal interpolating system and that $\\mathcal {U}_\\mathcal {I}$ is an orthogonal interpolating basis.",
"The point of the definition is that the interpolation of all values of $f$ uses the sampled values $f(i)$ , $i\\in \\mathcal {I}$ , which might be thought of as measurements of $f$ with respect to the fixed, natural basis of the ambient space $\\mathbb {C}^N$ , while the basis $\\mathcal {U}_\\mathcal {I}$ is tailored to $\\mathbb {Y}$ and $\\mathcal {I}$ .We could make the definition even more general and allow $\\mathbb {Y}$ to be a subspace of any finite-dimensional vector space $\\mathbb {X}$ , and sample $f\\in \\mathbb {Y}$ with respect to any fixed basis of $\\mathbb {X}$ , but the present definition suffices.",
"Note that $\\mathcal {I}$ need not consist of uniformly spaced indices, so the sampling may be irregular.",
"Indeed, the results described here and in [1] were originally motivated by questions from colleagues in medical imaging who had observed that irregular sampling patterns could often give excellent results with less computation.",
"For us, to solve the interpolation problem for $\\mathbb {Y}$ is to find an interpolating system.",
"It is a linear theory in all aspects.",
"Every subspace has an interpolating system, though it may not be unique, but not every subspace has an orthogonal interpolating system.",
"For a given subspace it is also not true that any index set is a sampling set for some interpolating basis, so the intervals between samples are not arbitrary.",
"The only subspaces that have orthonormal interpolating systems are the coordinate subspaces.",
"All of this is discussed in Section .",
"Orthogonal interpolating systems are the subject of [1], and we find interesting connections with difference sets, perfect graphs, tiling, and we answer affirmatively a discrete version of a conjecture of Fuglede.",
"In Section we provide some basic results on interpolating systems in general.",
"We quickly move, in Section , to study bandlimited spaces, $\\mathbb {B}^\\mathcal {J}$ , defined as signals whose discrete Fourier transforms are supported on $\\mathcal {J}$ .",
"We do not require that $\\mathcal {J}$ be a set of contiguous indices, so this is more general than the situation in the discrete Nyquist-Shannon theorem (though we continue to use the term “bandlimited\" for short).",
"In Section we begin to concentrate on universal sampling sets, namely index sets $\\mathcal {I}$ that are sampling sets for any bandlimited space $\\mathbb {B}^\\mathcal {J}$ with $|\\mathcal {J}|=|\\mathcal {I}|$ .",
"That is, $\\mathcal {I}$ is universal if the sampling pattern specified by $\\mathcal {I}$ can be used for interpolation of signals from any $\\mathbb {B}^\\mathcal {J}$ .",
"Universal sampling sets were used in [4] for multicoset sampling and in [5] in connection with compressed sensing.",
"Here our central result gives several necessary and sufficient conditions for an index set to be universal when $N$ is a prime power.",
"A mathematical consequence of our result is a generalization of Chebotarev's theorem on the invertibility of submatrices of the Fourier matrix.",
"In Section we show that a universal sampling set has an interesting structure as a disjoint union of what we call elementary universal sets, and through this analysis we are able to count the number of universal sampling sets of a given size.",
"We also introduce maximal (and minimal) universal sampling sets which in turn enter naturally into the uncertainty principles that we discuss in Section .",
"As an application of uncertainty and universality we prove a “random” uncertainty principle, and deduce a generalization of the Cauchy-Davenport theorem from additive number theory.",
"Our debt to the work in [6] and [3] is clear.",
"Many of our results assume that $N$ is a prime power, and naturally we wonder whether this can be generalized.",
"The definitions we introduce and the methods we use are based primarily on properties of index sets when the elements are reduced modulo powers of a prime.",
"With a few exceptions (e.g., minimal and cyclotomic polynomials) these can be considered elementary, and it is surprising (to us) how far they lead.",
"The methods here also seem rather different from those of compressed sensing.",
"In compressed sensing, which is nonlinear in theory and practice, the recovery of a signal from samples does not require knowledge of the frequency spectrum, whereas linear theories like ours cannot do without knowledge of the spectrum.",
"Nevertheless, with universality the sampling patterns in our approach do not depend on the frequencies, the reconstruction of a signal from its samples is by linear operations, and the samples are “samples” in the classical sense instead of random projections of the signal onto a measurement basis as is done in compressed sensing.",
"Both approaches start with discrete signals, but one needs to sample an analog signal in the first place and this analog sampling generally needs some knowledge of the frequency spectrum.",
"Works such as [4] and [5] confront this issue through “spectrum blind\" sampling, and they end up needing the idea of universality in the process.",
"It is also interesting that the linear theory here can be used to prove a random uncertainty principle without the necessity of nonlinear techniques, though our result is not as strong as the result in [3].",
"We hope to pursue the connections and differences further.",
"We refer to [2] and [7] for additional results, discussion, and examples.",
"See also Appendix for references to papers on universality for continuous-time signals."
],
[
"General Properties, Existence of Interpolating Systems",
"This section is a summary of elementary properties of interpolating systems, including existence theorems in both an algebraic and geometric formulation.",
"The ideas are simple enough, but they fit together nicely and are an essential foundation for the less simple work to follow.",
"We fix some notation.",
"Without further comment we will identify a vector in $\\mathbb {C}^N$ with its $N$ -periodic extension and vice versa, and we typically index vectors from 0 to $N-1$ .",
"(We assume periodicity because the discrete Fourier transform will soon enter the picture.)",
"For $i\\in [0 : N-1]$ we let $\\delta _i\\colon \\mathbb {Z}_N\\longrightarrow \\mathbb {C}$ be the (periodized) discrete $\\delta $ -function shifted to $i$ , so that $\\lbrace \\delta _0,\\delta _1,\\dots , \\delta _{N-1}\\rbrace $ is the natural basis of $\\mathbb {C}^N$ .",
"The components of a vector in $\\mathbb {C}^N$ will always be in terms of the natural basis, but any fixed basis of $\\mathbb {C}^N$ would do for the following development.",
"If $\\mathcal {I}\\subset [0:N-1]$ we let $\\mathbb {C}^\\mathcal {I}= \\rm {span}\\lbrace \\delta _i\\colon i \\in \\mathcal {I}\\rbrace .$ Our first goal is to establish Theorem 1 Any subspace $\\mathbb {Y}$ of $\\mathbb {C}^N$ has an interpolating system.",
"We will give two proofs, one geometric and one algebraic, and both are straightforward.",
"In the following, $\\mathbb {Y}$ is always a subspace of dimension $d$ and $\\mathcal {I}$ is always an index set of size $d$ .",
"Let $\\mathcal {I}^{\\prime }=[0 :N-1]\\setminus \\mathcal {I}$ .",
"We record several facts.",
"An interpolating basis for a subspace $\\mathbb {Y}$ is trying to be the natural basis in the slots specified by the index set.",
"In fact this is a characterization of interpolating bases.",
"Proposition 1 (i) A basis $\\mathcal {U}= \\lbrace u_i\\colon i\\in \\mathcal {I}\\rbrace $ for $\\mathbb {Y}$ is an interpolating basis if and only if $u_j(i) =\\delta _j(i) \\quad i,j\\in \\mathcal {I}.$ (ii) Any natural basis vector $\\delta _k$ lying in $\\mathbb {Y}$ is an element of any interpolating basis of $\\mathbb {Y}$ .",
"(iii) An interpolating basis is determined by its index set, more precisely, if $\\lbrace u_i\\colon i\\in \\mathcal {I}\\rbrace $ and $\\lbrace v_i\\colon i\\in \\mathcal {I}\\rbrace $ are interpolating bases for $\\mathbb {Y}$ then $u_i=v_i$ .",
"for all $i \\in \\mathcal {I}$ Expanding on the first point in Proposition REF , the elements of an interpolating basis are perturbations of the natural basis vectors by vectors outside $\\mathbb {Y}$ : Proposition 2 (i) Any interpolating basis $\\lbrace u_i \\colon i \\in \\mathcal {I}\\rbrace $ of $\\mathbb {Y}$ is of the form $u_i = \\delta _i + v_i,$ where $v_i \\in \\mathbb {C}^{\\mathcal {I}^{\\prime }}$ .",
"If $v_i \\in \\mathbb {Y}$ then $v_i=0$ .",
"(ii) The subspaces of $\\mathbb {C}^N$ having an orthogonal interpolating system are of the form $\\mathbb {Y}= {\\rm span}\\lbrace \\delta _i+v_i\\colon i \\in \\mathcal {I}\\rbrace $ where the nonzero $v_i$ are orthogonal vectors in $\\mathbb {C}^{\\mathcal {I}^{\\prime }}$ .",
"We omit the proofs of Propositions REF and REF .",
"Part (ii) of Proposition REF can be applied in the negative to find examples of subspaces that do not have an orthogonal interpolating basis – this is a much larger topic – and it also follows from part (ii) that the only subspaces having an orthonormal interpolating basis are the coordinate subspaces.",
"Both of these points were raised in the introduction.",
"[Geometric Proof of Theorem REF ] It is easy to see that there is an index set $\\mathcal {J}$ of size $N-d$ such that $\\mathbb {C}^N = \\mathbb {Y} \\oplus \\mathbb {C}^\\mathcal {J}$ .",
"Let $P\\colon \\mathbb {Y}\\oplus \\mathbb {C}^\\mathcal {J}\\rightarrow \\mathbb {Y}$ be the projection of $\\mathbb {C}^N$ onto $\\mathbb {Y}$ along $\\mathbb {C}^\\mathcal {J}$ .",
"If $f\\in \\mathbb {Y}$ then, on the one hand, $f = \\sum _{i=1}^N f(i) \\delta _i.$ On the other hand, since $\\mathbb {C}^\\mathcal {J}= {\\rm ker} P$ and $Pf = f$ we have $f = Pf = \\sum _{i=1}^N f(i)P\\delta _i =\\sum _{i\\notin \\mathcal {J}} f(i) P\\delta _i.$ Thus the $u_i=P\\delta _i$ form an interpolating basis of $\\mathbb {Y}$ indexed by $\\mathcal {I}= [0 : N-1]\\setminus \\mathcal {J}$ .",
"We see from this why an interpolating basis need not be unique.",
"The ambiguity in choosing an interpolating basis arises from the ambiguity in choosing a complement; if there is not a unique choice of the complement $\\mathbb {C}^\\mathcal {J}$ of $\\mathbb {Y}$ , and generally there is not, then there is not a unique interpolating basis for $\\mathbb {Y}$ .",
"However, the existence of an interpolating basis produces a complement to $\\mathbb {Y}$ : Proposition 3 Let $\\mathcal {U}=\\lbrace u_i\\colon i \\in \\mathcal {I}\\rbrace $ be an interpolating basis of $\\mathbb {Y}$ .",
"Then $\\mathbb {C}^N = \\mathbb {Y} \\oplus \\mathbb {C}^{\\mathcal {I}^{\\prime }}$ .",
"If we show that $\\mathbb {Y}\\cap \\mathbb {C}^{\\mathcal {I}^{\\prime }}=\\lbrace {0}\\rbrace $ then $\\mathcal {U} \\cup \\lbrace \\delta _j\\colon j \\in \\mathcal {I}^{\\prime }\\rbrace $ forms a basis for $\\mathbb {C}^N$ .",
"For this, let $f \\in \\mathbb {Y}\\cap \\mathbb {C}^{\\mathcal {I}^{\\prime }}$ .",
"Then $ f = \\sum _{i\\in \\mathcal {I}} f(i)u_i$ because $\\mathcal {U}$ is an interpolating basis for $\\mathbb {Y}$ , and also $f = \\sum _{j\\in \\mathcal {I}^{\\prime }} f(j)\\delta _j.$ Thus $\\sum _{i\\in \\mathcal {I}} f(i)u_i = \\sum _{j\\in \\mathcal {I}^{\\prime }} f(j)\\delta _j.$ Let $k\\in \\mathcal {I}$ and evaluate both sides at $k$ : $\\begin{aligned}\\sum _{i\\in \\mathcal {I}} f(i)u_i(k) &= \\sum _{j\\in \\mathcal {I}^{\\prime }} f(j)\\delta _j(k),\\\\f(k) &= 0.\\end{aligned}$ By (REF ), $f = {0}$ and we are done.",
"The algebraic proof of Theorem REF is in terms of matrices.",
"Associate with an index set $\\mathcal {I}=\\lbrace i_1,i_2,\\dots , i_d\\rbrace $ the $N \\times d$ matrix $E_\\mathcal {I}$ whose $d$ columns are the basis vectors $\\delta _{i_1}$ , $\\delta _{i_2}$ , ..., $\\delta _{i_d}$ .",
"If $R$ is an a $N\\times M$ matrix then $E_\\mathcal {I}^\\textsf {T}R$ is $d \\times M$ submatrix of $R$ obtained by choosing the rows indexed by $\\mathcal {I}$ .",
"In particular, operating by $E_\\mathcal {I}^\\textsf {T}$ on an $N$ -vector $f$ produces the $d$ -vector with components $f(i_1)$ , $f(i_2)$ ,..., $f(i_d)$ .",
"If $R$ is an $M \\times N$ matrix then $RE_\\mathcal {I}$ is the $M\\times d$ submatrix of $R$ obtained by choosing the columns indexed by $\\mathcal {I}$ .",
"We note three general facts.",
"First, $E_\\mathcal {I}^\\textsf {T} E_\\mathcal {I}= I_d,$ where $I_d$ is the $d\\times d$ identity matrix.",
"Second, if $S$ is a $d\\times d$ matrix then $E_\\mathcal {I}^\\textsf {T}(RS)=(E_\\mathcal {I}^\\textsf {T} R)S\\,.$ Finally, if $\\mathcal {U}=\\lbrace u_{i_1},u_{i_2},\\dots , u_{i_d}\\rbrace $ is a basis for $\\mathbb {Y}$ and $U$ is the $N \\times d$ matrix whose columns are the $u_i$ then the condition (REF ) that $\\mathcal {U}$ be an interpolating basis can be written in matrix form as $ f = UE_\\mathcal {I}^\\textsf {T}f$ for all $f \\in \\mathbb {Y}$ .",
"Here $UE_\\mathcal {I}^\\textsf {T}$ is an $N \\times N$ matrix and we see that $\\mathcal {U}$ is an interpolating basis for $\\mathbb {Y}$ with sampling set $\\mathcal {I}$ if and only if $\\mathbb {Y} = {\\rm ker}(I_N-UE_\\mathcal {I}^\\textsf {T})$ .",
"Now we have [Algebraic Proof of Theorem REF ] Take any basis $\\mathcal {V}=\\lbrace v_1,v_2,\\dots ,v_d\\rbrace $ of $\\mathbb {Y}$ and let $R$ be the $N \\times d$ matrix whose columns are the basis vectors $v_k$ ; thus $R_{jk}=v_k(j)$ .",
"Since $R$ has rank $d$ it has a $d\\times d$ invertible submatrix, and possibly many such submatrices.",
"Let $\\mathcal {I}$ be the index set corresponding to the $d$ rows chosen from $R$ to form the invertible submatrix $E_\\mathcal {I}^\\textsf {T} R$ .",
"The columns of the $N\\times d$ matrix $R(E_\\mathcal {I}^\\textsf {T} R)^{-1}$ are again a basis of $\\mathbb {Y}$ .",
"We write them as $u_{i_1}$ , $u_{i_2}$ , ..., $u_{i_d}$ , indexed by $\\mathcal {I}$ .",
"Since $E_\\mathcal {I}^\\textsf {T}(R(E_\\mathcal {I}^\\textsf {T} R)^{-1}) = (E_\\mathcal {I}^\\textsf {T} R)(E_\\mathcal {I}^\\textsf {T} R)^{-1} = I_d\\,,$ the $u_{i_j}$ are as in Proposition REF , and hence comprise an interpolating basis of $\\mathbb {Y}$ .",
"This proof shows how to produce an interpolating basis provided one can find a $d\\times d$ invertible submatrix $E_\\mathcal {I}^\\textsf {T} R$ , indexed by $\\mathcal {I}$ .",
"The more such submatrices the more interpolating bases for $\\mathbb {Y}$ .",
"On the opposite side, in general not every index set $\\mathcal {I}$ is sampling set for an interpolating basis since, in general, not every choice of a $d \\times d$ submatrix is invertible.",
"A slightly different way of arranging the algebraic proof also gives an interpolation formula, making (REF ) more explicit.",
"As above, let $\\mathcal {V}=\\lbrace v_1,v_2,\\dots ,v_d\\rbrace $ be a basis of $\\mathbb {Y}$ and let $R$ be the corresponding $N \\times d$ matrix.",
"If $f \\in \\mathbb {Y}$ then $f = \\sum _{k=1}^N f(k)\\delta _k \\quad \\text{and also}\\quad f = \\sum _{k=1}^d \\alpha _k v_k,$ for some constants $\\alpha _k$ .",
"We want to solve for the $\\alpha _k$ in terms of $d$ of the values $f(k)$ .",
"Write the second equation for $f$ as $f = R{\\alpha }, \\quad \\alpha =(\\alpha _1,\\alpha _2, \\dots , \\alpha _d)^\\textsf {T}.$ Now $R$ has an invertible $d \\times d$ submatrix, say $E_I^\\textsf {T} R$ for an index set $I$ , and so $E_I^\\textsf {T} f = E_I^\\textsf {T}(R {\\alpha }) = (E_I^\\textsf {T} R){\\alpha }.$ We can then solve for ${\\alpha }$ via ${\\alpha } = (E_I^\\textsf {T} R)^{-1}(E_I^\\textsf {T} f),$ resulting in $ f = R(E_I^\\textsf {T} R)^{-1}(E_I^\\textsf {T}f).$ This equation writes $f$ in terms of the components $f(i)$ , $i \\in I$ .",
"Carrying the algebraic line of reasoning a little further, we also see how two interpolating bases for $\\mathbb {Y}$ are related to each other.",
"Theorem 2 Fix an interpolating basis of $\\mathbb {Y}$ , indexed by $J$ , and let $R$ be the corresponding $N \\times d$ matrix.",
"If $S$ is the matrix of another interpolating basis of $\\mathbb {Y}$ , indexed by $I$ , then $E_I^\\textsf {T} R$ is invertible and $S=R(E_I^\\textsf {T} R)^{-1}\\,.$ Let $\\lbrace v_i\\colon i\\in I\\rbrace $ be the interpolating basis of $\\mathbb {Y}$ that are the columns of $S$ and let $\\lbrace u_j\\colon j\\in J\\rbrace $ be the columns of $R$ .",
"Since the $u_j$ are an interpolating basis we can write, for each $i\\in I$ , $v_i= \\sum _{j\\in J} v_i(j) u_j.$ In matrix form this is $S = R(E_J^\\textsf {T} S).$ Now multiply on the left by $E_I^\\textsf {T}$ , resulting in $E_I^\\textsf {T} S= E_I^\\textsf {T}(R(E_J^\\textsf {T} S)) = (E_I^\\textsf {T} R)(E_J^\\textsf {T} S).$ But $E_I^\\textsf {T} S$ is the $d\\times d$ identity matrix, so this shows that $E_I^\\textsf {T}R$ is invertible, that $(E_I^\\textsf {T} R)^{-1} = E_J^\\textsf {T} S$ , and then that $S=R(E_I^\\textsf {T} R)^{-1}.$ Finally, we look a little more closely at the interpolating basis provided by $R(E_I^\\textsf {T} R)^{-1}$ in relation to the geometric construction.",
"From the $d\\times d$ matrix $(E_I^\\textsf {T} R)^{-1}$ form a $d \\times N$ matrix by adding $N-d$ columns of zeros in the slots $I^{\\prime }$ .",
"Call this matrix $T$ .",
"Then $RT$ is an $N \\times N$ matrix and one sees that $RT\\delta _i={\\left\\lbrace \\begin{array}{ll}u_i,&\\quad i \\in I\\\\{0},& \\quad i \\in I^{\\prime }\\end{array}\\right.",
"}$ Thus $RT$ is the projection of $\\mathbb {C}^N$ onto $\\mathbb {Y}$ along $\\mathbb {C}^{I^{\\prime }}$ and we are back to the idea of the geometric argument.",
"Observe that whereas the geometric argument started with a complement $\\mathbb {C}^{I^{\\prime }}$ to $\\mathbb {Y}$ and produced the interpolating basis via projection, here we started with an interpolating basis for $\\mathbb {Y}$ and produced the projection and the complement."
],
[
"Discrete Bandlimited Spaces",
"Bandlimited signals are defined by the vanishing of the discrete Fourier transform outside a set of specified indices.",
"They form a particularly interesting class of subspaces.",
"For notation, let $\\zeta _n= e^{-2\\pi i /n},$ simplified to just $\\zeta $ when $n=N$ , and let $\\omega \\colon \\mathbb {Z}_N \\longrightarrow \\mathbb {C}$ be the discrete complex exponential, $\\omega (m) = \\zeta ^m.$ The discrete Fourier transform is then ${{\\mathcal {F}}}f = \\sum _{n=0}^{N-1} f(n) \\omega ^{n}.$ As usual, we also regard ${{\\mathcal {F}}}$ as an $N \\times N$ matrix whose $mn$ -entry is ${{\\mathcal {F}}}_{mn}=\\omega ^{n}(m) = \\zeta ^{mn}$ .",
"We recall that ${{\\mathcal {F}}}^{-1} = (1/N){{\\mathcal {F}}}^*$ (the adjoint of ${{\\mathcal {F}}}$ ).",
"Definition 2 Let $\\mathcal {J}\\subseteq [0:N-1]$ .",
"The $|\\mathcal {J}|$ -dimensional space of bandlimited signals with frequency support $\\mathcal {J}$ is $\\mathbb {B}^\\mathcal {J}= {{\\mathcal {F}}}^{-1}(\\mathbb {C}^\\mathcal {J}).$ In words, $f \\in \\mathbb {B}^\\mathcal {J}$ if ${{\\mathcal {F}}}f$ has zeros in the slots $\\mathcal {J}^{\\prime }=[1:N]\\setminus \\mathcal {J}$ .",
"There might be more zeros of ${{\\mathcal {F}}}f$ for a given $f$ but there are at least these zeros.",
"We do not assume that the indices in $\\mathcal {J}$ are contiguous, so ${{\\mathcal {F}}}f$ is not necessarily supported on a “band” of frequencies, but we maintain the use of the term “bandlimited” in all cases.",
"Since ${{\\mathcal {F}}}^* \\delta _n(m) =\\zeta ^{-mn}$ , we get a basis for $\\mathbb {B}^\\mathcal {J}$ by pulling out of ${{\\mathcal {F}}}^*$ the columns indexed by $\\mathcal {J}$ .",
"Thus we get an interpolating basis with sampling set $\\mathcal {I}$ if and only if $E_\\mathcal {I}^\\textsf {T}{{\\mathcal {F}}}^*E_\\mathcal {J}$ is invertible, or equivalently if and only if $E_\\mathcal {I}^\\textsf {T}{{\\mathcal {F}}}E_\\mathcal {J}$ is invertible.",
"We prefer to use the latter, with ${{\\mathcal {F}}}$ instead of ${{\\mathcal {F}}}^*$ .",
"For the remainder of this paper, interpolating systems for bandlimited spaces will be our main concern.",
"Spaces of bandlimited functions having orthogonal interpolating bases are the subject of [1], but we do have one general observation here: such spaces cannot be too big.",
"Proposition 4 If $\\mathbb {B}^\\mathcal {J}$ has an orthogonal interpolating basis then $|\\mathcal {J}| \\le N/2$ .",
"Suppose $\\mathbb {B}^\\mathcal {J}$ has an orthogonal interpolating basis indexed by $\\mathcal {I}$ .",
"Then $|\\mathcal {I}|=|\\mathcal {J}|$ .",
"Let $\\mathcal {I}^{\\prime }=[0\\ : N-1]\\setminus \\mathcal {I}$ .",
"By Proposition REF we can write $\\mathbb {B}^{\\mathcal {J}}= {\\rm span}\\lbrace \\delta _i+v_i\\colon i \\in \\mathcal {I}\\rbrace ,$ where the $v_i$ are orthogonal vectors in $\\mathbb {C}^{\\mathcal {I}^{\\prime }}$ , or some possibly 0.",
"But none of the $v_i$ can be zero, for ${{\\mathcal {F}}}\\delta _k = \\omega ^{-k}$ which never vanishes.",
"There are $|\\mathcal {I}|$ of the $v$ 's, and if $|\\mathcal {J}|=|\\mathcal {I}| > N/2$ then $|\\mathcal {I}^{\\prime }| <N/2$ and we would have more than $N/2$ orthogonal vectors in a space of dimension less than $N/2$ ."
],
[
"Necklaces and Bracelets",
"Sampling sets for bandlimited spaces have more algebraic structure than it might appear.",
"Namely, the property of being a sampling set for a particular $\\mathbb {B}^\\mathcal {J}$ is preserved under the action of the dihedral group.",
"To explain, on $\\mathbb {Z}_N$ we denote the operations of translation (by 1) and reflection by $\\tau $ and $\\rho $ , respectively: $\\begin{aligned}&\\tau \\colon \\mathbb {Z}_N \\longrightarrow \\mathbb {Z}_N, \\quad \\tau (n) = n-1 \\mod {N},\\\\& \\rho \\colon \\mathbb {Z}_N \\longrightarrow \\mathbb {Z}_N, \\quad \\rho (n) = -n \\mod {N} .\\end{aligned}$ Then $\\tau ^N = \\rm {id}.\\quad \\rho ^2 = \\rm {id} \\quad \\text{and} \\quad \\rho \\tau \\rho = \\tau ^{-1} \\quad \\text{or} \\quad (\\rho \\tau )^2 = \\rm {id},$ so $\\tau $ and $\\rho $ generate the dihedral group ${\\rm Dih}_N$ .",
"Clearly ${\\rm Dih}_N$ can act on an index set $\\mathcal {I}$ via $\\tau \\mathcal {I}= \\lbrace \\tau (i) \\colon i \\in \\mathcal {I}\\rbrace , \\quad \\rho \\mathcal {I}= \\lbrace \\rho (i) \\colon i \\in \\mathcal {I}\\rbrace .$ We define the bracelet of $I$ to be the orbit of $I$ under the action of ${\\rm Dih}_N$ .",
"The necklace of $I$ is the orbit of $I$ under the action of the cyclic subgroup $\\langle \\tau \\rangle $ of ${\\rm Dih}_N$ .",
"Think of $\\mathcal {I}\\subset [0:N-1]$ as specifying a pattern of $N$ beads on a loop, with black beads in the locations in $\\mathcal {I}$ separated by white beads in the locations in the complement $\\mathcal {I}^{\\prime }$ , as in Figure REF .",
"A necklace is worn around the neck, and if the cyclic group acts then the spacing of the black and white beads is the same however the necklace is rotated.",
"But a bracelet can be worn on either wrist, introducing a reflection, and the symmetry group is ${\\rm Dih}_N$ .",
"See Appendix for a formula that counts distinct bracelets, and for references.",
"Figure: Two different bracelets with N=12N=12 and |ℐ|=4|\\mathcal {I}|=4.",
"On top the index set is ℐ={0,2,5,7}\\mathcal {I}=\\lbrace 0,2,5,7\\rbrace , on the bottom the index set is ℐ={0,3,5,6}\\mathcal {I}= \\lbrace 0,3,5,6\\rbrace With these definitions we now have Proposition 5 If $\\mathcal {I}$ is a sampling set for $\\mathbb {B}^J$ then any index set in the bracelet of $I$ is a sampling set for $\\mathbb {B}^J$ .",
"Let $\\mathcal {I}= \\lbrace m_1, m_2, \\ldots , m_d\\rbrace $ , $\\mathcal {J}= \\lbrace n_1, n_2, \\ldots , n_d\\rbrace $ and let $\\mathcal {K}=\\tau \\mathcal {I}$ .",
"Then the new submatrix $E_\\mathcal {K}^\\textsf {T}{{\\mathcal {F}}}E_\\mathcal {J}$ is given by $E_\\mathcal {K}^T{{\\mathcal {F}}}E_\\mathcal {J}&=\\begin{bmatrix}\\zeta ^{ (m_1-1)n_1} & \\zeta ^{(m_1-1)n_2} & \\cdots & \\zeta ^{(m_1-1)n_d} \\\\\\zeta ^{(m_2-1)n_1} & \\zeta ^{(m_2-1)n_2} & \\cdots & \\zeta ^{(m_2-1)n_d}\\\\\\vdots & \\vdots & \\ddots & \\vdots \\\\\\zeta ^{(m_d-1)n_1} & \\zeta ^{(m_d-1)n_2}& \\cdots & \\zeta ^{(m_d-1)n_d}\\end{bmatrix}\\\\\\vspace{7.22743pt}& = \\begin{bmatrix}\\zeta ^{m_1n_1} & \\zeta ^{m_1n_2} & \\cdots & \\zeta ^{m_1n_d} \\\\\\zeta ^{m_2n_1} & \\zeta ^{m_2n_2} & \\cdots & \\zeta ^{m_2n_d} \\\\\\vdots & \\vdots & \\ddots & \\vdots \\\\\\zeta ^{m_dn_1} & \\zeta ^{m_dn_2}& \\cdots & \\zeta ^{m_dn_d}\\end{bmatrix} \\times \\\\& \\hspace{25.29494pt} \\begin{bmatrix}\\zeta ^{-n_1}& 0 & 0 & \\cdots & 0 \\\\0 & \\zeta ^{-n_2} & 0 & \\cdots & 0 \\\\\\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\0 & 0 & 0 & \\cdots & \\zeta ^{-n_d}\\end{bmatrix}\\\\& = (E_\\mathcal {I}^\\textsf {T}{{\\mathcal {F}}}E_\\mathcal {J}) \\times \\text{an invertible diagonal matrix}.$ Hence $E_\\mathcal {K}^\\textsf {T}{{\\mathcal {F}}}E_\\mathcal {J}$ is invertible whenever $E_\\mathcal {I}^\\textsf {T}{{\\mathcal {F}}}E_\\mathcal {J}$ is, and the same is true for any translation of $\\mathcal {I}$ .",
"Next suppose $\\mathcal {K}$ is obtained by reversing $\\mathcal {I}$ , namely $\\mathcal {K}= \\lbrace N-m_1,N-m_2, \\ldots , N-m_d\\rbrace $ .",
"Then $E_\\mathcal {K}^\\textsf {T}{{\\mathcal {F}}}E_\\mathcal {J}$ is just the conjugate of $E_\\mathcal {I}^\\textsf {T}{{\\mathcal {F}}}E_\\mathcal {J}$ , so again, $E_\\mathcal {K}^\\textsf {T}{{\\mathcal {F}}}E_\\mathcal {J}$ is invertible whenever $E_\\mathcal {I}^\\textsf {T}{{\\mathcal {F}}}E_\\mathcal {J}$ is."
],
[
"Universal Sampling Sets",
"There is a kind of interchange duality for bandlimited spaces between sampling sets and frequency support sets.",
"On the one hand, the sampling problem is to start with $\\mathbb {B}^\\mathcal {J}$ and ask which index sets $\\mathcal {I}$ are sampling sets.",
"On the other hand, one could also start with an index set $\\mathcal {I}$ and ask which $\\mathbb {B}^\\mathcal {J}$ result from this sampling pattern.",
"These two questions are equivalent.",
"Proposition 6 $\\mathbb {B}^\\mathcal {J}$ has $\\mathcal {I}$ as a sampling set if and only if $\\mathbb {B}^\\mathcal {I}$ has $\\mathcal {J}$ as a sampling set.",
"The subspace $\\mathbb {B}^\\mathcal {J}$ has $\\mathcal {I}$ as a sampling set if and only if $E_\\mathcal {I}^\\textsf {T}{{\\mathcal {F}}}E_\\mathcal {J}$ is invertible, and this is true if and only if its transpose $E_\\mathcal {J}^\\textsf {T}{{\\mathcal {F}}}E_\\mathcal {I}$ is invertible.",
"Though the sampling problem may seem the more natural one, we will concentrate on the second, equivalent question and ask which frequency patterns, that is which $\\mathbb {B}^\\mathcal {J}$ , can arise from a given sampling set $\\mathcal {I}$ .",
"It may be that the space $\\mathbb {B}^\\mathcal {J}$ is not known exactly, or that we may have some erroneous estimate $\\tilde{\\mathcal {J}}$ of $\\mathcal {J}$ .",
"The question is whether we can pick sampling locations $\\mathcal {I}$ that are robust for these estimation errors.",
"We will find some interesting phenomena, and the results can easily be translated to apply to the sampling problem.",
"The extreme case is captured by the following definition.",
"Definition 3 An index set $\\mathcal {I}\\subset [0:N-1]$ is a universal sampling set if $\\mathcal {I}$ is a sampling set for each $\\mathbb {B}^{\\mathcal {J}}$ with $|\\mathcal {J}| = |\\mathcal {I}|$ .",
"See also [4] and [5].",
"If $\\mathcal {I}$ is a universal sampling set, then while an interpolating basis of a space $\\mathbb {B}^\\mathcal {J}$ still depends on $\\mathcal {J}$ , where the samples are taken does not depend on $\\mathcal {J}$ .",
"In Section we will show that there are universal sampling sets of any given size; in fact, we will count them.",
"Very concretely, to ask if $\\mathcal {I}$ is a universal sampling set is to ask if there are rows of ${{\\mathcal {F}}}$ indexed by $\\mathcal {I}$ , $|\\mathcal {I}|=d$ , such that any $d \\times d$ submatrix of ${{\\mathcal {F}}}$ formed with these rows is invertible.",
"Phrased this way, standard properties of Vandermonde determinants applied to ${{\\mathcal {F}}}$ allow us to conclude: Proposition 7 (i) If $\\mathcal {I}$ is a set of $d$ consecutive indices, reduced mod $N$ , $\\mathcal {I}=\\lbrace i_0,i_0+1,\\dots , i_0+(d-1)\\rbrace \\mod {N},$ then $\\mathcal {I}$ is a universal sampling set.",
"(ii) If $\\mathcal {I}$ is a set of $d$ indices in arithmetic progression, reduced mod $N$ , $\\mathcal {I}=\\lbrace i_0, i_0+s, i_0+2s,\\dots , i_0+(d-1)s\\rbrace \\mod {N},$ where $s$ is coprime to $N$ , then $\\mathcal {I}$ is a universal sampling set.",
"$\\Box $ Much deeper is the following theorem of Chebotarev.",
"Theorem 3 (Chebotarev) If $N$ is prime, then every square submatrix of ${{\\mathcal {F}}}$ is invertible.",
"And so, if $N$ is prime then any index set $\\mathcal {I}$ is a universal sampling set.",
"Chebotarev's theorem dates to 1948 (the original paper is in Russian) and there are now several published (and unpublished) proofs, see, e.g., [8], [9], but this is by no means a trivial result.",
"We will generalize Chebotarev's theorem when $N$ is a prime power, and we will offer several characteristic properties of universal sampling sets.",
"We are indebted to the works of Tao [6] and Delvaux and Van Barel [10].",
"The key is a quantitative, almost statistical comparison of $\\mathcal {I}$ to the simplest universal sampling set, $\\mathcal {I}^* = [0 : d-1],$ when the elements of both $\\mathcal {I}$ and $\\mathcal {I}^*$ are reduced modulo prime powers.",
"We need several additional definitions to state our main results."
],
[
"Multisets and the Size of Congruence Classes",
"We have found it conceptually helpful to use multisets in the description of one of the central ideas, and we briefly review this concept.",
"Informally, a multiset is a finite, unordered list $\\widetilde{A}$ whose elements are drawn from a finite set $A$ , and where, to distinguish a multiset from simply a set, elements of the list may be repeated.",
"More formally, a multiset is a pair $(A, \\widetilde{\\chi }_A)$ where $\\widetilde{\\chi }_A$ is the multiplicity function (generalizing the characteristic function): $\\begin{aligned}&\\widetilde{\\chi }_A \\colon A \\longrightarrow \\mathbb {N}, \\\\\\widetilde{\\chi }_A(a) &= \\text{the number of times $a\\in A$ is listed in $\\widetilde{A}$}.\\end{aligned}$ Two multisets $\\widetilde{A}$ and $\\widetilde{B}$ are equal if $\\widetilde{\\chi }_A = \\widetilde{\\chi }_B$ , so the individual elements are the same and so are their multiplicities.",
"The cardinality of $\\widetilde{A}$ is $|\\widetilde{A}| = \\sum _{a\\in A} \\widetilde{\\chi }_A(a).$ It is common practice to use the standard set notation in writing a multiset.",
"Thus, for example, drawing from $\\lbrace a,b,c,d\\rbrace $ we write a multiset as $\\lbrace a,a,c,d,d\\rbrace $ .",
"The tilde notation $\\widetilde{A}$ for a multiset drawn from $A$ is helpful in discussing general principles but, like all general notations, it has its limitations in particular cases.",
"It is a notation often used for covering spaces, as we comment on below.",
"Associated with a multiset $\\widetilde{A}$ is another multiset $\\mathcal {M}(\\widetilde{A}) = \\lbrace \\widetilde{\\chi }_A(a) \\colon a \\in A\\rbrace ,$ which we call the multiplicity multiset of $\\widetilde{A}$ .",
"Thus $\\mathcal {M}(\\widetilde{A})$ records as a multiset the counts of the elements of $\\widetilde{A}$ and also includes a zero for each element of $A$ that does not appear in $\\widetilde{A}$ .",
"One can think of $\\mathcal {M}(\\widetilde{A})$ as providing some statistics of $\\widetilde{A}$ , a kind of histogram of $\\widetilde{A}$ with bins from $A$ , except that the bins are not ordered.",
"Next, let $p$ be a prime, $k \\ge 0$ an integer, and for $x \\in \\mathbb {N}$ let $[x]_k$ be the residue of $x$ reduced mod $p^k$ .",
"For an index set $\\mathcal {I}$ let $\\mathcal {I}/{p^k}=\\lbrace [i]_k \\colon i \\in \\mathcal {I}\\rbrace $ be the set of residues mod $p^k$ of the elements of $\\mathcal {I}$ , and let $(\\mathcal {I}/p^k)^\\sim $ be the corresponding multiset, meaning that each residue is listed according to its multiplicity, i.e, the size of its congruence class.",
"We regard the elements of $(\\mathcal {I}/p^k)^\\sim $ to be drawn from $[0:p^k-1]$ , all possible residues, and we write $\\widetilde{\\chi }_k \\colon [0:p^k-1] \\longrightarrow \\mathbb {N}$ for the multiplicity function for the multiplicity multiset $\\mathcal {M}( (\\mathcal {I}/p^k)^\\sim )$ .",
"To be explicit, for $a \\in [0:p^k-1]$ $ \\begin{aligned}\\widetilde{\\chi }_k(a) = &\\,\\text{the number of elements of $\\mathcal {I}$}\\\\&\\,\\text{that leave a remainder of $a$ on dividing by $p^k$.",
"}\\end{aligned}$ In particular, $\\widetilde{\\chi }_k(a)=0$ means that no element of $\\mathcal {I}$ leaves a remainder of $a$ on dividing by $p^k$ .",
"In this case we speak of an empty congruence class in $\\mathcal {I}/p^k$ .",
"For $ a \\in [0:p^k-1]$ it will also be helpful to use the notation $\\mathcal {I}_{ka} = \\lbrace i\\in I\\colon i \\equiv a \\text{ mod $p^k$}\\rbrace $ for the elements of the congruence class of $a$ mod $p^k$ .",
"Then $\\widetilde{\\chi }_k(a) = |\\mathcal {I}_{ka}|$ .",
"When we need to emphasize the index set, especially in Section , we will write $\\widetilde{\\chi }_k(a\\,;\\mathcal {I})$ .",
"We note the obvious properties: If $\\mathcal {I}$ and $\\mathcal {J}$ are disjoint then $\\widetilde{\\chi }_k(a\\,;\\mathcal {I}\\cup \\mathcal {J}) = \\widetilde{\\chi }_k(a\\,;\\mathcal {I}) + \\widetilde{\\chi }_k(a\\,;\\mathcal {J})$ .",
"$\\mathcal {I}\\subseteq \\mathcal {J}\\Rightarrow \\widetilde{\\chi }_k(a\\,;\\mathcal {I}) \\le \\widetilde{\\chi }_k(a\\,;\\mathcal {J})$ .",
"Observe for $k=0$ that $(\\mathcal {I}/1)^\\sim $ just consists of $|\\mathcal {I}|$ zeros and $\\widetilde{\\chi }_0(0)=|\\mathcal {I}|$ .",
"More generally, $ | \\mathcal {I}| = \\sum _{a=0}^{p^k-1} \\widetilde{\\chi }_k(a).$ We also note that the multiplicity multiset $\\mathcal {M}((\\mathcal {I}/p^k)^\\sim )$ depends only on the bracelet of $\\mathcal {I}$ .",
"While the multisets $(\\mathcal {I}/p^k)^\\sim $ will generally change if $\\mathcal {I}$ is shifted or reversed, the counts of the residues on dividing by $p^k$ will be the same: $ \\begin{aligned}\\mathcal {M}((\\tau \\mathcal {I}/p^k)^\\sim ) & = \\mathcal {M}((\\mathcal {I}/p^k)^\\sim ) \\quad \\text{and}\\\\\\mathcal {M}((\\rho \\mathcal {I}/p^k)^\\sim ) & = \\mathcal {M}((\\mathcal {I}/p^k)^\\sim ).\\end{aligned}$ Remark 1 Introducing the multiset $(\\mathcal {I}/p^k)^\\sim $ is reminiscent of introducing covering spaces (for Riemann surfaces) to resolve the problem of multivalued functions.",
"Here we have the remainder map $r\\colon \\mathcal {I}\\longrightarrow \\mathcal {I}/p^k$ , $r(i)=[i]$ , which is generally not injective and so has a multivalued inverse.",
"Think of the residues (with multiplicity) in $(\\mathcal {I}/p^k)^\\sim $ as tagged by the number they come from, say as a pair $([i],i)$ , which serves to distinguish them much as we think of tagging points on different sheets of a covering space of a Riemann surface.",
"Then we have the commutative diagram ${& (\\mathcal {I}/p^k)^\\sim [d]^{\\text{pr}}\\\\\\mathcal {I}[ur]^{\\widetilde{r}} [r]_r &\\mathcal {I}/p^k }$ where $\\text{pr}$ is the projection map, $([i],i) \\mapsto [i]$ and the lift $\\widetilde{r}(i) =([i],i)$ , of $r$ is bijective.",
"The value of the multiplicity function $\\widetilde{\\chi }_k(i)$ is then the number of elements in the preimage $\\text{pr}^{-1}([i])$ , analogous to the number of sheets over $[i]$ .",
"It will generally vary with $[i]$ .",
"Returning to our primary considerations, we write $\\widetilde{\\chi }_k^*$ to distinguish the special case when $\\mathcal {I}=\\mathcal {I}^*$ .",
"We will need the following property of $\\widetilde{\\chi }_k^*$ : $ |\\widetilde{\\chi }_k^{*}(a) - \\widetilde{\\chi }_k^*(b)| \\le 1,$ for all $a,b \\in [0:p^k-1]$ and all $k$ .",
"In words, when reducing the elements of $\\mathcal {I}^*=[0:d-1]$ modulo $p^k$ for any $k$ , the conjugacy classes are all of about the same size.",
"Or, pursuing the analogy above, the preimages $\\text{pr}^{-1}([i])$ of the individual residues all have approximately the same number of elements and one might say that $\\mathcal {I}^*/p^k$ is uniformly covered for each $k$ .",
"The inequality in (REF ) is easy to see.",
"For some background calculations we have found it helpful to have a formula for $\\widetilde{\\chi }_k^*$ (from which (REF ) also follows).",
"If $\\ell \\in \\mathcal {I}^*$ with $[\\ell ]_k= a \\in [0:p^k-1]$ then $\\ell = a + \\alpha p^k$ for an integer $\\alpha \\ge 0$ , and since $\\ell \\le d-1$ we must have $0 \\le \\alpha \\le (d-1-a)/p^k$ .",
"The number of integers $\\alpha $ for which this inequality holds is the number of $\\ell $ whose residue is $a$ .",
"Thus $ \\widetilde{\\chi }_k^*(a) = \\left\\lfloor \\frac{d-1-a}{p^k} +1\\right\\rfloor .$"
],
[
"A Characterization of Universal Sampling Sets",
"Our main result is: Theorem 4 Let $\\mathcal {I}$ be an index set in $[0:p^M-1]$ .",
"The following are equivalent: $\\widetilde{\\chi }_k = \\widetilde{\\chi }_k^*$ for all $0 \\le k \\le M$ .",
"$|\\widetilde{\\chi }_k(a)-\\widetilde{\\chi }_k(b)| \\le 1$ for all $a,b \\in [0:p^k-1]$ and $0 \\le k \\le M$ .",
"$\\mathcal {I}$ is a universal sampling set.",
"According to Proposition REF and the relations (REF ), any index set in the bracelet of $\\mathcal {I}$ is also a universal sampling set.",
"Likewise, any index set in the bracelet of $\\mathcal {I}^*$ can serve as a model universal sampling set.",
"Only condition (i) directly compares $\\mathcal {I}$ to $\\mathcal {I}^*$ , and in terms of multisets it could be stated equivalently as $\\mathcal {M}(({\\mathcal {I}/p^k})^\\sim ) = \\mathcal {M}(({\\mathcal {I}^*/p^k})^\\sim ).$ Condition (i) for $k=0$ guarantees that $\\mathcal {I}$ and $\\mathcal {I}^*$ have the same size, from (REF ).",
"Computing $ \\mathcal {M}(({\\mathcal {I}/p^k})^\\sim )$ and $\\mathcal {M}(({\\mathcal {I}^*/p^k})^\\sim )$ for $k \\ge M$ is redundant; since all elements in $\\mathcal {I}$ and $\\mathcal {I}^*$ are in $[0:p^M-1]$ , $\\mathcal {M}(({\\mathcal {I}/p^k})^\\sim )$ for $k \\ge M$ is just indicative of the cardinality of $\\mathcal {I}$ and $\\mathcal {I}^*$ .",
"Namely, for $k \\ge M$ , each of $\\mathcal {M}(({\\mathcal {I}/p^k})^\\sim )$ and $\\mathcal {M}(({\\mathcal {I}^*/p^k})^\\sim )$ contains $|\\mathcal {I}|$ ones and $p^k-|\\mathcal {I}|$ zeros.",
"Condition (ii), a property only of $\\mathcal {I}$ , indirectly compares $\\mathcal {I}$ to $\\mathcal {I}^*$ via (REF ).",
"It says that $\\mathcal {I}/p^k$ , like $\\mathcal {I}^*/p^k$ , is uniformly covered for each $k$ .",
"Before we embark on the proof of the theorem, here is an example.",
"Let $N=2^3$ , and $\\mathcal {I}= \\lbrace 0, 1, 3, 4, 6 \\rbrace $ .",
"The following are the multisets for $k=1,2,3$ : $\\begin{aligned}&({\\mathcal {I}/2})^\\sim = \\lbrace 0, 1, 1, 0, 0\\rbrace , \\quad \\mathcal {M}(({\\mathcal {I}/2})^\\sim )= \\lbrace 3, 2\\rbrace ; \\nonumber \\\\& ({\\mathcal {I}/2^2})^\\sim = \\lbrace 0, 1, 3, 0, 2\\rbrace , \\quad \\mathcal {M}(({\\mathcal {I}/2^2})^\\sim ) = \\lbrace 2, 1, 1, 1\\rbrace ;\\nonumber \\\\&({\\mathcal {I}/2^3})^\\sim = \\lbrace 0, 1, 3, 4, 6\\rbrace , \\\\& \\hspace{21.68121pt} \\mathcal {M}(({\\mathcal {I}/2^3})^\\sim ) = \\lbrace 1, 1, 0, 1, 1, 0, 1, 0\\rbrace \\nonumber .\\end{aligned}$ The computations for $\\mathcal {I}^*=\\lbrace 0,1,2,3,4\\rbrace $ yield $\\begin{aligned}&({\\mathcal {I}^*/2})^\\sim = \\lbrace 0, 1, 0, 1, 0\\rbrace , \\quad \\mathcal {M}(({\\mathcal {I}^*/2})^\\sim ) = \\lbrace 3, 2\\rbrace ;\\nonumber \\\\&({\\mathcal {I}^*/2^2})^\\sim = \\lbrace 0, 1, 2, 3, 0\\rbrace , \\quad \\mathcal {M}(({\\mathcal {I}^*/2^2})^\\sim ) = \\lbrace 2, 1, 1, 1\\rbrace ; \\nonumber \\\\&({\\mathcal {I}^*/2^3})^\\sim = \\lbrace 0, 1, 2, 3, 4\\rbrace , \\\\& \\hspace{21.68121pt} \\mathcal {M}(({\\mathcal {I}^*/2^3})^\\sim ) = \\lbrace 1, 1, 1, 1,1 , 0, 0, 0\\rbrace .",
"\\nonumber \\end{aligned}$ We see that $ \\mathcal {M}(({\\mathcal {I}/2^k})^\\sim ) = \\mathcal {M}(({\\mathcal {I}^*/2^k})^\\sim )$ for $k=1,2,3$ , and hence $\\mathcal {I}$ is a universal sampling set.",
"So in case the reader has ever wondered, for the $8 \\times 8$ Fourier matrix any $5\\times 5$ submatrix built from the rows indexed by $\\mathcal {I}$ , or from the rows of an index set in the bracelet of $\\mathcal {I}$ , is invertible.",
"[Proof of Theorem REF , (i) $\\Longleftrightarrow $ (ii)] Note: This equivalence does not require that $N$ be a prime power.",
"The implication (i) $\\Rightarrow $ (ii) is immediate from (REF ).",
"Assume (ii) holds and let $\\chi = \\min _a\\widetilde{\\chi }_k(a).$ From (ii) it follows that any $\\widetilde{\\chi }_k(a)$ is either $\\chi $ or $\\chi +1$ .",
"Suppose $r$ of the $p^k$ numbers $\\widetilde{\\chi }_k(a) $ are equal to $\\chi + 1$ and the rest are equal to $\\chi $ .",
"The cardinality equation, (REF ), $\\sum _{a=0}^{p^k-1} \\widetilde{\\chi }_k(a) = |\\mathcal {I}|=d,$ then gives $p^k \\chi + r = d, \\quad \\text{ with } \\quad 0\\le r < p^k.$ This means that $\\chi $ is the quotient on dividing $d$ by $p^k$ and $r$ is the remainder.",
"In other words, (ii) and (REF ) together uniquely determine the multiset $\\mathcal {M}(({\\mathcal {I}/p^k})^\\sim ) = \\lbrace \\widetilde{\\chi }_k(a)\\colon a \\in [0,p^k-1] \\rbrace $ .",
"Since $\\mathcal {I}$ and $\\mathcal {I}^*$ both satisfy (ii) and (REF ), we must have $\\mathcal {M}(({\\mathcal {I}/p^k})^\\sim ) = \\mathcal {M}(({\\mathcal {I}^*/p^k})^\\sim )$ , or $\\widetilde{\\chi }_k = \\widetilde{\\chi }_k^*$ .",
"We need two lemmas to prove that condition (i) implies that $\\mathcal {I}$ is a universal sampling set.",
"The first is a very old theorem on Vandermonde determinants, [11], as updated in [10]: Lemma 1 (Delvaux and Van Barel) Let $V=\\begin{bmatrix}x_1^{m_1} & x_2^{m_1} & x_3^{m_1} & \\cdots & x_d^{m_1} \\\\x_1^{m_2} & x_2^{m_2} & x_3^{m_2} & \\cdots & x_d^{m_2} \\\\\\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\x_1^{m_d} & x_2^{m_d} & x_3^{m_d} & \\cdots & x_d^{m_d}\\end{bmatrix}$ be a $d \\times d$ generalized Vandermonde matrix.",
"Then the determinant of $V$ is given by $\\det V = \\left(\\prod _{i<j}(x_j - x_i)\\right)S(x_1, x_2, \\ldots , x_d),$ where $S(x_1, x_2, \\ldots , x_d)$ is a symmetric polynomial in $x_1, x_2, \\ldots x_d$ with integer coefficients such that $S(1,1,\\ldots ,1) = \\frac{\\prod _{0\\le i <j \\le d-1}(m_j-m_i)}{\\prod _{0\\le i <j \\le d-1}(j-i)}.$ The polynomial $S$ is called a Schur polynomial, see, for example, [12].",
"Based on this lemma we deduce a second result that is itself already a sufficient condition for an index set to be a universal sampling set.",
"Lemma 2 Let $\\mathcal {I}= \\lbrace m_0, m_1, m_2, \\ldots , m_{d-1}\\rbrace $ .",
"If $\\mu = \\frac{\\prod _{0\\le i <j \\le d-1}(m_j-m_i)}{\\prod _{0\\le i <j \\le d-1}(j-i)}$ is coprime to $p$ , then $\\mathcal {I}$ is a universal sampling set.",
"Note that without Lemma REF , it would not even be clear that $\\mu $ is an integer.",
"An intuitive idea for why this should be so is given below.",
"The proof of Lemma REF is along the lines of the proof of Chebotarev's theorem in [13], and also in [6].",
"[Proof of Lemma REF ] We make use of Lemma REF in the case when $V = E_{\\mathcal {I}}^T \\mathcal {F} E_{\\mathcal {J}} $ .",
"Each $x_\\ell $ in (REF ) is then a power of $\\zeta =e^{-2\\pi i /N}$ , $x_\\ell = \\zeta ^{j_\\ell }$ , where $\\mathcal {J}= \\lbrace j_1,j_2,\\ldots ,j_d\\rbrace $ .",
"Suppose $\\det V = 0$ .",
"From (REF ), this means that $S(x_1, x_2, \\ldots , x_d) = 0$ .",
"Substituting $x_\\ell = \\zeta ^{j_\\ell }$ in $S(x_1, x_2, \\ldots , x_d) = 0$ , we obtain an equation of the form $s(\\zeta ) = 0$ , where $s(x)$ is a polynomial in one variable with integer coefficients.",
"This means that $\\zeta $ is a root of $s(x)$ and since $s(x)$ has only integer coefficients, $s(x)$ must contain the minimal polynomial of $\\zeta $ over $\\mathbb {Z}$ as a factor.",
"For $N=p^M$ , the minimal polynomial of $\\zeta $ over $\\mathbb {Z}$ is $\\phi _N(x) = 1 + x^{p^{M-1}} + x^{2p^{M-1}} + x^{3p^{M-1}} +\\cdots +x^{(p-1)p^{M-1}}$ (the $N$ 'th cyclotomic polynomial).",
"So we have $\\phi _N(x) \\mid s(x)$ , where $\\phi _N(x) = 1 + x^{p^{M-1}} + x^{2p^{M-1}} + x^{3p^{M-1}} + \\cdots +x^{(p-1)p^{M-1}} .$ Now $\\phi _N$ and $s$ are both polynomials with integer coefficients, hence $\\phi _N(1) \\mid s(1)$ .",
"However, $\\phi _N(1) = p$ , and $s(1) = S(1,1,\\ldots ,,1) = \\mu $ .",
"Thus $p \\mid \\mu \\, \\text{ if $\\det V = 0$}.$ This proves the lemma.",
"Chebotarev's theorem follows from this result.",
"If $N$ is a prime $p$ then $\\mu $ is coprime to $p$ because every factor in the numerator and denominator of $\\mu $ is an integer strictly between $-p$ and $p$ .",
"We can now complete the proof of one direction of the implications in Theorem REF .",
"[Proof of Theorem REF : (i) $\\Rightarrow $ (iii)] Let $\\mathcal {I}= \\lbrace m_1,m_2,m_3,\\ldots ,m_d\\rbrace $ and consider the product of differences $A=\\prod _{1\\le i < j \\le d}(m_j - m_i).$ There are $\\widetilde{\\chi }_k(\\ell )$ elements of $\\mathcal {I}$ that leave a remainder of $\\ell $ when divided by $p^k$ .",
"Moreover, $m_i \\equiv m_j \\mod {p}^k$ if and only if $p^k \\mid (m_j - m_i)$ .",
"The number of differences that have a factor of $p^k$ (or higher: $p^r$ for $r>k$ ) is $\\sum _{l=0}^{p^k-1} \\binom{\\widetilde{\\chi }_k(l)}{2},$ and hence the number of differences that have a factor of exactly $p^k$ is given by $\\sum _{l=0}^{p^k-1} \\binom{\\widetilde{\\chi }_k(l)}{2} - \\sum _{l=0}^{p^{k+1}-1} \\binom{\\widetilde{\\chi }_{k+1}(l)}{2}.$ The largest power of $p$ that divides $A$ is then $p$ raised to $ {\\sum _k k\\left( \\sum _{l=0}^{p^k-1} \\binom{\\widetilde{\\chi }_k(l)}{2} - \\sum _{l=0}^{p^{k+1}-1} \\binom{\\widetilde{\\chi }_{k+1}(l)}{2}\\right)} .$ The expression (REF ) depends only on the values of $\\widetilde{\\chi }_k$ , but the hypothesis is that $\\widetilde{\\chi }_k = \\widetilde{\\chi }_k^*$ for $0 \\le k \\le N$ , and therefore the products $A=\\prod (m_j - m_i)$ and $B = \\prod (j-i)$ have the same powers of $p$ as factors.",
"Hence $\\mu = A/B$ is coprime to $p$ and from Lemma REF we conclude that $\\mathcal {I}$ is a universal sampling set.",
"Remark 2 The argument above also gives an insight, if not a proof, as to why $\\mu =A/B$ in (REF ) is an integer.",
"Suppose $\\mathcal {M}((\\mathcal {I}/p^k)^\\sim )= \\lbrace r_1, r_2, r_3, \\ldots , r_d\\rbrace $ .",
"The power of $p^k$ in $A = \\prod (m_i - m_j)$ is given by $\\sum _{i=1}^d {r_i \\atopwithdelims ()2} = \\frac{1}{2}\\left(\\sum _{i=1}^d r_i^2 - \\sum _{i=1}^d r_i\\right).$ Now, $\\sum _{i=1}^d r_i$ is the cardinality of $\\mathcal {I}$ so $\\sum _{i=1}^d r_i = d.$ Hence for a set $\\mathcal {I}$ which has the minimum power of $p^k$ in $A$ it must be that $\\mathcal {M}((\\mathcal {I}/p^k)^\\sim ) = \\lbrace r_1, r_2,\\ldots r_d\\rbrace $ is a solution to $&\\text{minimize }r_1^2 + r_2^2 + \\cdots +r_d^2 \\\\&\\text{subject to }r_1 + r_2 + \\cdots r_d = d.$ On the reals the optimal solution satisfies $r_1 = r_2 = \\cdots = r_d$ .",
"This suggests that the set $\\mathcal {I}$ with the smallest power of $p^k$ in $A$ must have roughly an equal number of elements in each congruence class.",
"$\\mathcal {I}^* = \\lbrace 0,1,2,\\ldots , d-1\\rbrace $ is one such set.",
"Thus the power of $p^k$ is smaller in $B = \\prod (i-j)$ than in $A = \\prod (m_i - m_j)$ for each $p$ and $k$ , and, if the reasoning is to trusted, $\\mu = A/B$ is an integer.",
"To finish the proof of Theorem REF we will derive the following bounds on $\\widetilde{\\chi }_k$ .",
"Lemma 3 If $\\mathcal {I}\\subseteq [0:p^M-1]$ is a universal sampling set of size $d$ then $ \\left\\lfloor \\frac{d}{p^k}\\right\\rfloor \\le \\widetilde{\\chi }_k(s) \\le \\left\\lceil \\frac{d}{p^k}\\right\\rceil , \\quad s\\in [0:p^k-1]\\,, 0 \\le k \\le M.$ It follows immediately from (REF ) that if $\\mathcal {I}$ is a universal sampling set then $|\\widetilde{\\chi }_k(a) - \\widetilde{\\chi }_k(b)| \\le 1, \\quad a, b \\in [0:p^k-1].$ This is condition (ii), and with this result the proof of Theorem REF will be complete.",
"Incidentally, for the case $\\mathcal {I}=\\mathcal {I}^*$ , (REF ) is a simple consequence of (REF ) and (REF ).",
"The argument for Lemma REF is through constructing submatrices of the Fourier matrix of known rank to obtain upper and lower bounds for $\\widetilde{\\chi }_k$ .",
"The first step is to build a particular model submatrix, and this requires some bookkeeping.",
"Let $\\mathcal {I}\\subseteq [0:p^M-1]$ , at this point not assumed to be a universal sampling set.",
"Fix $k\\le M$ and $s\\in [0:p^k-1]$ , and recall that we let $\\mathcal {I}_{ks} = \\lbrace i \\in \\mathcal {I}\\colon i \\equiv s \\text{ mod $p^k$}\\rbrace .$ The set $\\mathcal {I}_{ks}$ has $\\widetilde{\\chi }_k(s)$ elements.",
"List them, in numerical order, as $i_0, i_1,i_2,\\dots , i_c$ , where we put $c=\\chi _k(s)-1$ to simplify notation.",
"Let $r$ be a positive integer and define the column vector of length $c$ by $\\mathfrak {z}^r = \\begin{bmatrix}\\zeta _{N}^{i_0r} & \\zeta _{N}^{i_1r} & \\zeta _{N}^{i_2r} & \\cdots & \\zeta _N^{i_cr}\\end{bmatrix}^\\textsf {T}.$ Now let $\\mathfrak {Z}^r$ be the $c \\times p^k$ matrix obtained by repeating $p^k$ copies of the column $\\mathfrak {z}^r$ : $\\mathfrak {Z}^r =\\underbrace{\\begin{bmatrix} \\mathfrak {z}^r & \\mathfrak {z}^r & \\mathfrak {z}^r & \\cdots \\mathfrak {z}^r \\end{bmatrix}}_\\text{$p^k$ times},$ and let $\\mathfrak {D}^s$ be the $p^k \\times p^k$ diagonal matrix $\\mathfrak {D}^s =\\begin{bmatrix}1 & 0 & 0 & \\dots & 0\\\\0 & \\zeta _{p^k}^{s} & 0 \\dots & 0\\\\0 & 0 & \\zeta _{p^k}^{2s} & \\dots & 0\\\\\\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\0 & 0 & 0 & \\cdots & \\zeta _{p^k}^{(p^k-1)s}\\end{bmatrix} .$ Finally, let $k^{\\prime }=M-k$ , and set $ \\mathcal {J}_{k^{\\prime }r} = \\lbrace 0\\cdot p^{k^{\\prime }}+r, 1\\cdot p^{k^{\\prime }} +r, 2\\cdot p^{k^{\\prime }}+r, \\dots , (p^k-1)p^{k^{\\prime }}+r\\rbrace .$ From the Fourier matrix ${{\\mathcal {F}}}$ we choose $c$ rows indexed by $\\mathcal {I}_{ks}$ and $p^k$ columns indexed by $\\mathcal {J}_{k^{\\prime }r}$ .",
"The result of these choices, we claim, results in $ E_{\\mathcal {I}_{ks}}^\\textsf {T} {{\\mathcal {F}}}E_{\\mathcal {J}_{k^{\\prime }r}} = \\mathfrak {Z}^r\\mathfrak {D}^s.$ After the preparations, the derivation of (REF ) is straightforward.",
"The $(a,b)$ -entry of $E_{\\mathcal {I}_{ks}}^\\textsf {T} {{\\mathcal {F}}}E_{\\mathcal {J}_{k^{\\prime }r}}$ is $\\begin{aligned}\\zeta _{N}^{i_a(bp^{k^{\\prime }}+r)}&=\\exp \\left(-\\frac{(2\\pi i)i_a(bp^{M-k}+r)}{p^M}\\right)\\\\&= \\exp \\left(-\\frac{(2\\pi i ) i_ab}{p^k}\\right)\\exp \\left(-\\frac{(2\\pi i )i_ar}{p^M}\\right) .\\end{aligned}$ But now recall that, by definition, when $i_a\\in \\mathcal {I}_{ks}$ is divided by $p^k$ it leaves a remainder of $s$ , and thus $\\begin{aligned}&\\exp \\left(-\\frac{(2\\pi i ) i_ab}{p^k}\\right)\\exp \\left(-\\frac{(2\\pi i )i_ar}{p^M}\\right)\\\\&= \\exp \\left(-\\frac{(2\\pi i ) sb}{p^k}\\right)\\exp \\left(-\\frac{(2\\pi i )i_ar}{p^M}\\right) \\\\&=\\zeta _{p^k}^{sb}\\,\\zeta _N^{i_a r}.\\end{aligned}$ This construction is the basis for the proof of Lemma REF , but applied in block form.",
"[Proof of Lemma REF ] To deduce the upper bound $\\widetilde{\\chi }(s) \\le \\lceil d/p^k\\rceil $ we begin by letting $\\mathcal {J}= \\mathcal {J}_{k^{\\prime }0}\\cup \\mathcal {J}_{k^{\\prime }1} \\cup \\mathcal {J}_{k^{\\prime }2} \\cup \\cdots \\cup \\mathcal {J}_{k^{\\prime }d^{\\prime }}, \\quad d^{\\prime }=\\left\\lceil \\frac{d}{p^k}\\right\\rceil -1,$ where $\\mathcal {J}_{k^{\\prime }r}$ is defined as in (REF ).",
"Note that $\\mathcal {J}$ is a union of $\\lceil d/p^k\\rceil $ disjoint sets.",
"Each $\\mathcal {J}_{k^{\\prime }r}^{\\prime }$ , $0\\le r \\le d^{\\prime }=\\lceil d/p^k\\rceil -1$ indexes the choice of $p^k$ columns from ${{\\mathcal {F}}}$ and applying (REF ) we have $\\begin{aligned}&E_{\\mathcal {I}_{ks}}^\\textsf {T}{{\\mathcal {F}}}E_\\mathcal {J}= \\\\&\\begin{bmatrix}E_{\\mathcal {I}_{ks}}^\\textsf {T}{{\\mathcal {F}}}E_{\\mathcal {J}_{k^{\\prime }0}} & E_{\\mathcal {I}_{ks}}^\\textsf {T}{{\\mathcal {F}}}E_{\\mathcal {J}_{k^{\\prime }1}} & \\cdots & E_{\\mathcal {I}_{ks}}^\\textsf {T}{{\\mathcal {F}}}E_{\\mathcal {J}_{k^{\\prime }d^{\\prime }}}\\end{bmatrix}\\\\&=\\begin{bmatrix}\\mathfrak {Z}^0\\mathfrak {D}^s & \\mathfrak {Z}^1\\mathfrak {D}^s &\\cdots & \\mathfrak {Z}^{d^{\\prime }}\\mathfrak {D}^s\\end{bmatrix}\\\\&=\\begin{bmatrix}\\mathfrak {Z}^0 & \\mathfrak {Z}^1 & \\cdots & \\mathfrak {Z}^{d^{\\prime }}\\end{bmatrix}\\begin{bmatrix}\\mathfrak {D}^s & 0 & \\cdots & 0 \\\\0 & \\mathfrak {D}^s & \\cdots &0 \\\\\\vdots & \\vdots & \\ddots & \\vdots \\\\0 & 0 & \\cdots & \\mathfrak {D}^s\\end{bmatrix} .\\end{aligned}$ The diagonal matrix in this product is invertible, hence $ \\begin{aligned}&\\text{Rank of $E_{\\mathcal {I}_{ks}}^\\textsf {T}{{\\mathcal {F}}}E_\\mathcal {J}$} = \\text{Rank of $\\begin{bmatrix}\\mathfrak {Z}^0 & \\mathfrak {Z}^1 & \\mathfrak {Z}^2 & \\cdots & \\mathfrak {Z}^{d^{\\prime }}\\end{bmatrix}$} \\\\&\\hspace{36.135pt} \\le \\text{Number of distinct columns} = \\left\\lceil \\frac{d}{p^k}\\right\\rceil .\\end{aligned}$ Now, the number of columns of $E_{\\mathcal {I}_{ks}}^\\textsf {T}\\mathcal {F}E_{\\mathcal {J}}$ is equal to $|\\mathcal {J}| &= |\\mathcal {J}_{k^{\\prime }0} \\cup \\mathcal {J}_{k^{\\prime }1} \\cup \\mathcal {J}_{k^{\\prime }2} \\cup \\ldots \\cup \\mathcal {J}_{k^{\\prime }d^{\\prime }}| \\\\& = \\sum _{r=0}^{ \\lceil d/p^k \\rceil -1} |\\mathcal {J}_{k^{\\prime }r}| = p^k\\lceil d/p^k \\rceil \\ge d,$ so there are at least $d$ columns.",
"Hence if $\\mathcal {I}$ is a universal sampling set of size $d$ then $E_{\\mathcal {I}}^\\textsf {T}\\mathcal {F}E_{\\mathcal {J}}$ must be of full row rank.",
"In particular, since $\\mathcal {I}_{ks} \\subseteq \\mathcal {I}$ , it must be that $E_{\\mathcal {I}_{ks}}^\\textsf {T}\\mathcal {F}E_{\\mathcal {J}}$ is also of full row rank, for each $s$ .",
"Next, the number of rows in $E_{\\mathcal {I}_{ks}}^\\textsf {T}\\mathcal {F}E_{\\mathcal {J}}$ is equal to $|\\mathcal {I}_{ks}| = \\widetilde{\\chi }_k(s)$ by definition.",
"From (REF ) we know that the rank of $E_{\\mathcal {I}}^\\textsf {T}\\mathcal {F}E_{\\mathcal {J}}$ is at most $\\lceil d /p^k \\rceil $ , and so we have $\\text{(Number of rows) $\\widetilde{\\chi }_k(s)$} \\le \\left\\lceil \\frac{d}{p^k} \\right\\rceil .$ The proof of the lower bound $\\widetilde{\\chi }_k(s) \\ge \\lfloor d/p^k \\rfloor $ is very similar.",
"This time we construct a set $\\mathcal {J}$ with $|\\mathcal {J}| \\le d$ , and observe that if $\\mathcal {I}$ is a universal sampling set of size $d$ , then $E_{\\mathcal {I}}^\\textsf {T}\\mathcal {F}E_{\\mathcal {J}}$ is of full column rank.",
"Let $\\mathcal {J}= \\mathcal {J}_{k^{\\prime }0} \\cup \\mathcal {J}_{k^{\\prime }1} \\cup \\mathcal {J}_{k^{\\prime }2} \\cup \\ldots \\cup \\mathcal {J}_{k^{\\prime } d^{\\prime \\prime }}, \\quad d^{\\prime \\prime }=\\lfloor d/p^k \\rfloor -1.$ Then just as above, $\\begin{aligned}\\text{Rank of $E_{\\mathcal {I}_{ks}}^\\textsf {T}{{\\mathcal {F}}}E_\\mathcal {J}$} &= \\text{Rank of $\\begin{bmatrix}\\mathfrak {Z}^0 & \\mathfrak {Z}^1 & \\mathfrak {Z}^2 & \\cdots & \\mathfrak {Z}^{d^{\\prime \\prime }}\\end{bmatrix}$} \\\\&\\le \\text{Number of distinct columns} = \\left\\lfloor \\frac{d}{p^k}\\right\\rfloor .\\end{aligned}$ The number of rows of $E_{\\mathcal {I}_{ks}}^\\textsf {T}\\mathcal {F}E_{\\mathcal {J}}$ is $|\\mathcal {I}_{ks}| = \\widetilde{\\chi }_k(s)$ , and so we must have $\\text{Rank of }E_{\\mathcal {I}_{ks}}^\\textsf {T}\\mathcal {F}E_{\\mathcal {J}} \\le \\min \\lbrace \\lfloor d/p^k \\rfloor ,\\widetilde{\\chi }_k(s)\\rbrace .",
"$ Furthermore, $E_{\\mathcal {I}}^\\textsf {T}\\mathcal {F}E_{\\mathcal {J}} \\nonumber \\\\= \\begin{bmatrix}E_{\\mathcal {I}_{k0}}^\\textsf {T}\\mathcal {F}E_{\\mathcal {J}} \\\\ E_{\\mathcal {I}_{k1}}^\\textsf {T}\\mathcal {F}E_{\\mathcal {J}} \\\\\\vdots \\\\E_{\\mathcal {I}_{k(p^k-1)}}^\\textsf {T}\\mathcal {F}E_{\\mathcal {J}}\\\\\\end{bmatrix} , \\nonumber $ whence $&\\text{Row rank of }E_{\\mathcal {I}}^\\textsf {T}\\mathcal {F}E_{\\mathcal {J}} \\nonumber \\\\&\\quad \\le \\text{Rank of }E_{\\mathcal {I}_{k0}}^\\textsf {T}\\mathcal {F}E_{\\mathcal {J}} + \\text{Rank of }E_{\\mathcal {I}_{k1}}^\\textsf {T}\\mathcal {F}E_{\\mathcal {J}} \\nonumber \\\\& \\hspace{36.135pt}+ \\ldots + \\text{Rank of }E_{\\mathcal {I}_{k(p^k-1)}}^\\textsf {T}\\mathcal {F}E_{\\mathcal {J}} \\nonumber \\\\&\\quad \\le \\sum _{s=0}^{p^k-1} \\min \\lbrace \\lfloor d/p^k \\rfloor ,\\widetilde{\\chi }_k(s)\\rbrace .",
"$ Now, the number of columns indexed by $\\mathcal {J}$ is $p^k \\lfloor d/p^k \\rfloor \\le d$ .",
"Hence if $\\mathcal {I}$ is a universal sampling set of size $d$ , we need $E_{\\mathcal {I}}^\\textsf {T}\\mathcal {F}E_{\\mathcal {J}} $ to be of full column rank.",
"From (REF ), this means we must have $\\text{(Number of columns) } p^k \\lfloor d/p^k \\rfloor \\le \\sum _{s=0}^{p^k-1} \\min \\lbrace \\lfloor d/p^k \\rfloor ,\\widetilde{\\chi }_k(s)\\rbrace .$ This inequality will not be satisfied unless $\\lfloor d/p^k \\rfloor \\le \\widetilde{\\chi }_k(s)$ for all $s$ .",
"This completes the proof.",
"Remark 3 For many values of $d$ , it is enough to prove one side of the inequality (REF ).",
"If we know that $\\widetilde{\\chi }_k(s) \\le \\lceil d /p^k \\rceil $ , then from $\\sum _s\\widetilde{\\chi }_k(s) =d$ and a recurrence relation (REF ), below, it is possible to prove that $\\lfloor d /p^k \\rfloor \\le \\widetilde{\\chi }_k(s)$ .",
"Such cases include $N=p^M$ , $d = c_0 p^k + c_1 p^{k-1}$ for $c_0, c_1 \\in \\lbrace 0,1,2,\\ldots ,p-1\\rbrace $ .",
"$N=2^M$ , $d = c_0 2^k + c_1 2^{k-1} + c_22^{k-2}$ for $c_0, c_1, c_2 \\in \\lbrace 0,1\\rbrace $ $N=2^M$ , $d = 2^k + 2^{k-1} + 2^{k-2} + \\ldots + 2^{k-r+1}$ for some $r$ ,"
],
[
"Digit Reversal and Universal Sampling Sets",
"There is another interesting characterization of universal sampling sets in terms of digit reversal.",
"Expanding in base $p$ , any integer $a \\in [0:p^m-1]$ , $m \\ge 1$ , can be written uniquely as $a=\\alpha _0+\\alpha _1p+\\alpha _2p^2+\\cdots \\alpha _{m-1}p^{m-1},$ where the $\\alpha $ 's are in $[0:p-1]$ .",
"We define a permutation $\\pi _m\\colon [0:p^m-1] \\longrightarrow [0:p^m-1]$ by $\\begin{aligned}&\\pi _m(\\alpha _0+\\alpha _1p+\\alpha _2p^2+\\cdots \\alpha _{m-1}p^{m-1}) = \\\\&\\hspace{36.135pt} \\alpha _{m-1}+\\alpha _{m-2}p+\\alpha _{m-3}p^2+\\cdots +\\alpha _0p^{m-1}.\\end{aligned}$ The $\\alpha $ 's are the digits in the base $p$ expansion of $a \\in [0:p^m-1]$ and applying $\\pi _m$ to $a$ produces the number in $[0:p^m-1]$ with the digits reversed.",
"For example (an example we will use again in Section ), take $[0:7]$ .",
"Then $\\pi _3([0:7])=\\lbrace 0,4,2,6,1,5,3,7\\rbrace $ in that order.",
"Such digit reversing permutations were used in [10] to find rank-one submatrices of the Fourier matrix.",
"The issue for universal sampling sets is how the numbers $\\pi _M(\\mathcal {I})$ are dispersed within the interval $[0:p^M-1]$ , where, as before, $N=p^M$ .",
"To make this precise, take $k\\ge 1$ and partition $[0:p^M-1]$ into $p^k$ equal parts: $[0:p^M-1] = \\bigcup _{a=0}^{p^k-1}[ap^{k^{\\prime }}:(a+1)p^{k^{\\prime }}-1], \\quad k^{\\prime }=M-k.$ For any $\\mathcal {J}\\subseteq [0:p^M-1]$ and $a \\in [0:p^k-1]$ , let $\\phi _k(a\\,;\\mathcal {J}) = |\\mathcal {J}\\cap [ap^{k^{\\prime }},(a+1)p^{k^{\\prime }}-1]|.$ We say that $\\mathcal {J}$ is uniformly dispersed in $[0,p^M-1]$ if $ |\\phi _k(a\\,;\\mathcal {J}) - \\phi _k(b\\,;\\mathcal {J})| \\le 1$ for all $a, b\\in [0:p^k-1]$ , and $1\\le k \\le M$ .",
"Thus $\\mathcal {J}$ is uniformly dispersed if roughly equal numbers of its elements are in each of the intervals $[ap^{k^{\\prime }}: (a+1)p^{k^{\\prime }}-1]$ for all $1 \\le k \\le M$ , $k^{\\prime }=M-k$ .",
"We will show $ \\phi _k(\\pi _k(a)\\,; \\pi _M(\\mathcal {I})) = \\widetilde{\\chi }_k(a), \\quad a \\in [0:p^k-1].$ Thus, to the three equivalent conditions in Theorem REF we can add a fourth: $\\pi _M(\\mathcal {I})$ is uniformly dispersed.",
"The derivation of (REF ) uses the following lemma.",
"Lemma 4 If $j \\in [0:p^M-1]$ is given by $j = b + ap^{k^{\\prime }}$ , $0\\le b \\le p^{k^{\\prime }}-1 $ , then $\\pi _M(j) = \\pi _k(a) + p^k\\pi _{k^{\\prime }}(b)$ .",
"The proof is straightforward, and the argument for (REF ) then goes very quickly.",
"As defined, for any index set $\\mathcal {J}$ , $\\phi _k(a\\,; \\mathcal {J})$ is the number of elements in $\\mathcal {J}$ that lie in $[ap^{k^{\\prime }} : (a+1)p^{k^{\\prime }}-1]$ , and these are precisely the $j \\in \\mathcal {J}$ of the form $ap^{k^{\\prime }}+b$ with $0\\le b \\le p^{k^{\\prime }}-1$ .",
"Thus for $i \\in [0:p^k-1]$ , $\\begin{aligned}&{\\phi }_k(\\pi _k(i)\\,;{\\pi _M(\\mathcal {I})}) = \\\\&\\hspace{18.06749pt}\\text{the number of }j \\in \\pi _M(\\mathcal {I}) \\\\&\\hspace{25.29494pt}\\text{ of the form }\\pi _k(i) p^{k^{\\prime }} + b, \\ 0\\le b \\le p^{k^{\\prime }}-1 \\nonumber \\\\&= \\text{ number of }j \\in \\mathcal {I}\\text{ of the form } \\\\&\\hspace{25.29494pt}p^{k}\\pi _{k^{\\prime }}(b) + i, \\ 0\\le b \\le p^{k^{\\prime }}-1 \\text{ (from Lemma \\ref {lemma:univ-alt-2})} \\\\& = \\text{ number of }j \\in \\mathcal {I}\\text{ that leave a remainder of } \\\\&\\hspace{25.29494pt} i \\text{ on dividing by }p^k\\nonumber \\\\& = \\widetilde{\\chi }_k(i).",
"\\nonumber \\end{aligned}$"
],
[
"Structure and Enumeration of Universal Sampling Sets",
"In this section we analyze in detail the structure of universal sampling sets.",
"Specifically we show that when $N=p^M$ is a prime power such a set $\\mathcal {I}$ is the disjoint union of smaller, elementary universal sets that depend on the base $p$ expansion of $|\\mathcal {I}|$ .",
"The method is algorithmic, allowing us to construct universal sets of a given size, and to find a formula that counts the number of universal sets as a function of $p^M$ and $|\\mathcal {I}|$ .",
"In particular the formula answers the question: How likely is it that a randomly chosen index set is universal?",
"Not very likely, but there are several subtle aspects to the answer.",
"For example, we exhibit plots of the counting function showing some striking phenomena depending on the prime $p$ .",
"Our approach is via maximal universal sampling sets which, in turn, enter naturally in studying the relationship between universal sampling sets and uncertainty principles.",
"We take up the latter topic in the next section."
],
[
"A Recurrence Relation and Tree for $\\widetilde{\\chi }$",
"When $N=p^M$ the condition that an index set be a universal sampling set depends on the values of $\\widetilde{\\chi }_k$ for different $k$ .",
"To study this we use a recurrence relation in $k$ for $\\widetilde{\\chi }_k(a)$ .",
"The formula holds even when $N$ is not a prime power.",
"Lemma 5 Let $\\mathcal {I}\\subseteq [0:N-1]$ .",
"Then $ \\widetilde{\\chi }_{k-1}(a) = \\sum _{j=0}^{p-1}\\widetilde{\\chi }_k(a + jp^{k-1}), \\quad $ for all $a\\in [0:p^{k-1}-1]$ .",
"An integer $x \\in \\mathcal {I}$ that leaves a remainder of $a$ when divided by $p^{k-1}$ is of the form $x=\\alpha p^{k-1} + a$ .",
"Let $\\alpha = \\beta p + \\gamma $ for $\\gamma \\in [0:p-1]$ .",
"Then $x = \\beta p^k + \\gamma p^{k-1} + a$ , that is, $x$ leaves a remainder of either $0\\cdot p^{k-1}+a, 1\\cdot p^{k-1}+a, 2\\cdot p^{k-1}+ a,\\dots $ or $(p-1)\\cdot p^{k-1}+a$ on dividing by $p^k$ .",
"The result follows.",
"When $N=p^M$ the recurrence formula and the relation it expresses between conjugacy classes has an appealing interpretation in terms of a $p$ -ary tree.",
"Several arguments in this section will be based on this configuration.",
"Let $\\mathcal {I}\\subseteq [0: p^M-1]$ .",
"We construct a tree with $M+1$ levels and $p^k$ nodes in level $k$ , $0 \\le k \\le M$ .",
"The nodes in level $k$ are identified by a pair $(k,a)$ , with $a \\in [0:p^{k-1}]$ .",
"Call the nodes at the level $M$ the leaves.",
"At the node $(k,a)$ we imagine placing the congruence class $\\mathcal {I}_{ka} =\\lbrace i\\in \\mathcal {I}\\colon i \\equiv a \\mod {p}^k\\rbrace $ .",
"The root is $\\mathcal {I}_{00}=\\mathcal {I}$ and the nodes at the leaves host the sets $\\mathcal {I}_{Ma}$ , $a \\in [0:p^M-1]$ , each of which is either a singleton or empty.",
"We assign a weight of $\\widetilde{\\chi }_k(a) = |I_{ka}|$ to the node $(k,a)$ .",
"Further, at each level we arrange the nodes according to the digit reversing permutation, i.e., nodes at level $k$ are arranged as $\\pi _k([0:p^k-1])$ , where $\\pi _k$ is the digit reversing permutation from Section REF .",
"(This is similar to the starting step of the FFT algorithm, where the indices are sorted according to the reversed digits.)",
"Figure REF shows the case $N=2^3$ , a binary tree with four levels, $k=0, 1, 2, 3$ .",
"In the third level of the tree the nodes are ordered $0, 4, 2, 6, 1, 5, 3, 7$ , which is $\\pi _3([0:7])$ .",
"Then: The set $\\mathcal {I}_{ka}$ at level $k$ is the disjoint union of the sets at its children nodes at level $k+1$ .",
"The value of $\\widetilde{\\chi }_k(a)$ at the node $(k,a)$ is the sum of the values of $\\widetilde{\\chi }_{k+1}$ at its children nodes at level $k+1$ .",
"In other words, the weight of a parent is the sum of the weights of its children; this is the recurrence relation.",
"Consequently, the value of $\\widetilde{\\chi }_k$ at any node is the sum of the values of $\\widetilde{\\chi }_M$ at the leaves at level $M$ descended from the node.",
"For example, in Figure REF we have $\\begin{aligned}\\widetilde{\\chi }_0(0) &= \\sum _{a=0}^7\\widetilde{\\chi }_3(a), \\\\\\widetilde{\\chi }_1(0) &=\\sum _{a=0}^3\\widetilde{\\chi }_3(2a),\\\\\\widetilde{\\chi }_1(1) &= \\sum _{a=0}^3\\widetilde{\\chi }_3(2a+1),\\end{aligned}$ and so on.",
"In fact, a more general conclusion is the following: Fix a level $k$ .",
"Then the value of $\\widetilde{\\chi }_r$ at any node $(r,a)$ , for $r \\le k$ is the sum of the values of $\\widetilde{\\chi }_k$ at the level-$k$ nodes descending from the tree node $(r,a)$ .",
"When the root is $[0:p^M-1]$ , the extreme case, the leaves are all singletons and the nodes at level $k$ are each of weight $p^{M-k}$ .",
"Figure: A tree representing the relations between the congruence classes, and the recurrence relation satisfied byχ ˜ k (a)\\widetilde{\\chi }_k(a).",
"The value ofχ ˜ k (a)\\widetilde{\\chi }_k(a) at any node is the sum ofthe values of χ ˜ k (a)\\widetilde{\\chi }_k(a) at its children nodes in level k+1k+1."
],
[
"Elementary and Maximal Sets",
"To study the structure of universal sampling sets we need a series of definitions.",
"When $N$ is a prime power the building blocks are the elementary sets: Definition 4 A set $\\mathcal {E}\\subseteq [0:p^M-1]$ is a $k$ -elementary set if $\\widetilde{\\chi }_k(a) = 1, \\quad \\text{~for all $a\\in [0:p^k-1]$}.$ Note that $|\\mathcal {E}|=p^k$ .",
"As a first application of the formula (REF ) we can add the adjective “universal” to the description of elementary sets.",
"Lemma 6 A $k$ -elementary set $\\mathcal {E}$ is a universal sampling set.",
"From $\\widetilde{\\chi }_k(a) = 1$ and (REF ) it follows that $\\mathcal {E}$ has an equal number of elements in each congruence class modulo $p^s$ , $s \\le k$ .",
"More precisely, $ \\widetilde{\\chi }_s(a) = p^{k-s},$ for all $s \\le k$ .",
"Also from (REF ), for $s > k$ all the congruence classes are of size 0 or 1, i.e.",
"$ \\widetilde{\\chi }_s(a) \\in \\lbrace 0,1\\rbrace .$ Therefore $|\\widetilde{\\chi }_s(a)-\\widetilde{\\chi }_s(b)| \\le 1,$ for all $a,b \\in [0: p^k-1]$ and all $s$ , and we conclude that $\\mathcal {E}$ is a universal sampling set.",
"Next, a fruitful approach to understanding the structure of universal sampling sets is to ask how well an arbitrary index set is approximated from within by universal sets.",
"Definition 5 Let $\\mathcal {I}\\subseteq [0:N-1]$ .",
"A maximal universal sampling set for $\\mathcal {I}$ is a universal sampling set of largest cardinality that is contained in $\\mathcal {I}$ .",
"Note that the definition does not require $N$ to be a prime power, though this will most often be the case.",
"There is an allied notion of a minimal universal set.",
"We define this in Subsection REF below, and show how they are related to maximal sets.",
"Maximal and minimal sets enter naturally and together in connection with uncertainty principles, discussed in Section .",
"Finding a maximal universal sampling set for a given $\\mathcal {I}$ is a finitary process, so existence is not an issue.",
"However, maximal universal sampling sets need not be unique.",
"For example, take $N=3^2$ and $\\mathcal {I}=\\lbrace 0, 1, 2, 3, 6 \\rbrace $ .",
"The set $\\mathcal {I}$ is not itself a universal sampling set, and both $\\lbrace 0, 1, 2, 3\\rbrace $ and $\\lbrace 0, 1,2,6 \\rbrace $ are maximal universal sampling sets contained in $\\mathcal {I}$ .",
"Despite the lack of uniqueness it will be convenient to have a notation, and we let $\\Omega (\\mathcal {I})$ denote a generic maximal universal sampling set in $\\mathcal {I}$ .",
"The cardinality $|\\Omega (\\mathcal {I})|$ is well-defined; by definition $|\\mathcal {J}| \\le |\\Omega (\\mathcal {I})|$ for any universal sampling set $\\mathcal {J}\\subseteq \\mathcal {I}$ .",
"Elementary sets and maximal sets are related through an important construction of an elementary set.",
"Definition 6 Let $\\mathcal {I}\\subseteq [0:p^M-1]$ and let $\\bar{k}$ be the largest integer such that no congruence class in $\\mathcal {I}/p^{\\bar{k}}$ is empty.",
"(It might be that $\\bar{k}=0$ .)",
"Let $\\mathcal {I}_{\\bar{k}}^\\dagger $ denote an elementary set obtained by choosing one element from each congruence class in $\\mathcal {I}/p^{\\bar{k}}$ .",
"By Lemma REF , $\\mathcal {I}_{\\bar{k}}^\\dagger $ is a universal sampling set, and is of order $p^{\\bar{k}}$ .",
"We now have Theorem 5 Let $\\mathcal {\\mathcal {I}} \\subseteq [0:p^M-1]$ , and $\\mathcal {I}_{\\bar{k}}^\\dagger $ as above.",
"Then $p^{\\bar{k}}\\le |\\Omega (\\mathcal {I}) | < p^{\\bar{k}+1}$ .",
"There exists a maximal universal sampling set contained in $\\mathcal {I}$ and containing $\\mathcal {I}_{\\bar{k}}^\\dagger $ .",
"The lower bound in (i) follows from the definition of a maximal set and the comments above, $p^{\\bar{k}} = |\\mathcal {I}_{\\bar{k}}^\\dagger | \\le |\\Omega (\\mathcal {I})|.$ To prove the upper bound, suppose $\\mathcal {J}\\subseteq \\mathcal {I}$ has $|\\mathcal {J}| \\ge p^{\\bar{k}+1}$ .",
"By the definition of $\\bar{k}$ at least one congruence class in $\\mathcal {J}/p^{\\bar{k}+1}$ is empty, so $\\widetilde{\\chi }_{\\bar{k}+1}(a\\,;\\mathcal {J}) = 0$ for some $a\\in [0:p^{\\bar{k}+1}-1]$ .",
"From the cardinality equation (REF ), $\\sum _{\\ell =0}^{p^{\\bar{k}+1}-1}\\widetilde{\\chi }_{\\bar{k}+1}(\\ell \\,; \\mathcal {J}) = |\\mathcal {J}| \\ge p^{\\bar{k}+1}.$ This implies that at least one congruence class in $\\mathcal {J}/p^{\\bar{k}+1}$ has at least two elements, or $\\widetilde{\\chi }_{\\bar{k}+1}(b\\,;\\mathcal {J}) \\ge 2$ for some $b$ .",
"We then have $|\\widetilde{\\chi }_{\\bar{k}+1}(b\\,;\\mathcal {J}) - \\widetilde{\\chi }_{\\bar{k}+1}(a\\,;\\mathcal {J})| = 2 > 1,$ and $\\mathcal {J}$ cannot be a universal sampling set.",
"For part (ii), we first show that any maximal universal sampling set $\\Omega (\\mathcal {I})$ set must contain at least one element from each congruence class in $\\mathcal {I}/p^{\\bar{k}}$ .",
"By way of contradiction, suppose that $\\widetilde{\\chi }_{\\bar{k}}(a\\,;\\Omega (\\mathcal {I}))=0$ for some $a$ .",
"Since $\\Omega (\\mathcal {I})$ is universal we must then have $\\widetilde{\\chi }_{\\bar{k}}(b\\,;\\Omega (\\mathcal {I}))\\le 1$ for all $b$ .",
"By (REF ), $|\\Omega (\\mathcal {I})|= \\sum _{b=0}^{p^{\\bar{k}-1}} \\widetilde{\\chi }_{\\bar{k}}(b\\,;\\Omega (\\mathcal {I})) <p^{\\bar{k}},$ contradicting the lower bound in (i).",
"Let $\\mathcal {K}\\subseteq \\Omega (\\mathcal {I})$ be an elementary set, of size $p^{\\bar{k}}$ , that contains one element from each congruence class in $\\mathcal {I}/p^{\\bar{k}}$ , guaranteed to exist from what we just showed.",
"Assuming $\\mathcal {K}\\ne \\mathcal {I}_{\\bar{k}}^\\dagger $ , since otherwise we are done, we will use $\\mathcal {K}$ and $\\Omega (\\mathcal {I})$ to construct a (new) maximal universal set that contains $\\mathcal {I}_{\\bar{k}}^\\dagger $ .",
"Set up a $p$ -ary tree, as above, with root $\\Omega _{00}=\\Omega (\\mathcal {I})$ and $(\\ell ,a)$ -node the congruence class $\\Omega _{\\ell a}= \\lbrace i \\in \\Omega (\\mathcal {I}) \\colon i \\equiv a \\text{ mod $p^\\ell $}\\rbrace , \\quad |\\Omega _{\\ell a}|=\\widetilde{\\chi }_\\ell (a),$ for $a \\in [0:p^\\ell -1]$ .",
"Recall that $\\Omega _{\\ell a}$ , at level $\\ell $ , is the disjoint union of the sets at its children nodes at level $\\ell +1$ .",
"Figure REF is an example for $p=3$ and $M\\ge 3$ , showing only three levels for reasons of space.",
"The shading has to do with the rest of the proof, as we now explain.",
"Figure: The congruence class tree for Ω(ℐ)\\Omega (\\mathcal {I}).",
"The node Ω ℓa {\\Omega }_{\\ell a} is the congruence class of aa modulo p ℓ p^\\ell in Ω(ℐ)\\Omega (\\mathcal {I}), so that Ω 00 =Ω(ℐ)\\Omega _{00}=\\Omega (\\mathcal {I}) and |Ω ℓa |=χ ˜ ℓ (a)|{\\Omega }_{\\ell a} | = \\widetilde{\\chi }_\\ell (a).Both $\\mathcal {I}_{\\bar{k}}^\\dagger $ and $\\mathcal {K}$ are assembled by choosing single elements from sets at the nodes in the $\\bar{k}$ -level (call these the assembly nodes) for a total of $p^{\\bar{k}}$ elements for $I_{\\bar{k}}^\\dagger $ and $\\mathcal {K}$ each.",
"Observe that the sets at the nodes in the $\\bar{k}+1$ level are either empty or singletons.",
"This is so because by definition of $\\bar{k}$ there must be some $a\\in [0:p^{\\bar{k}+1}-1]$ for which $\\widetilde{\\chi }_{\\bar{k}+1}(a) = 0$ , and hence by universality $\\widetilde{\\chi }_{\\bar{k}+1}(b) \\le 1$ for all $b\\in [0:p^{\\bar{k}+1}-1]$ .",
"And then, according to how the tree is structured, the sets at all nodes farther down in the tree must as well be either empty or singletons.",
"Let $\\mathcal {L}\\supseteq \\mathcal {I}_{\\bar{k}}^\\dagger $ be the set of elements in $\\mathcal {I}$ that leave the same remainders as do the elements in $\\mathcal {I}_{\\bar{k}}^\\dagger $ when divided by $p^{\\bar{k}+1}$ , more precisely, $\\mathcal {L}= \\lbrace j \\in \\mathcal {I}\\colon j \\equiv i \\text{ mod $p^{\\bar{k}+1}$ for some $i\\in \\mathcal {I}_{\\bar{k}}^\\dagger $}\\rbrace .$ Likewise let $\\mathcal {L}^{\\prime } \\supseteq \\mathcal {K}$ be $\\mathcal {L}^{\\prime }= \\lbrace j \\in \\mathcal {I}\\colon j \\equiv i \\text{ mod $p^{\\bar{k}+1}$ for some $i\\in \\mathcal {K}$}\\rbrace .$ $\\mathcal {L}$ is the union of the assembly nodes for $I_{\\bar{k}}^\\dagger $ and $\\mathcal {L}^{\\prime }$ is the union of the assembly nodes for $\\mathcal {K}$ .",
"The collections may overlap.",
"We color a node red if it contributes to $\\mathcal {L}$ and blue if it contributes to $\\mathcal {L}^{\\prime }$ , and both red and blue (otherwise known as purple) if it contributes to both $\\mathcal {L}$ and $\\mathcal {L}^{\\prime }$ .",
"In the figure we take $\\bar{k}=1$ , so $\\mathcal {I}_{\\bar{k}}^\\dagger $ and $\\mathcal {K}$ live at the middle level in the tree, as shown.",
"Focus on each red node in turn.",
"The red node contains an element in $\\mathcal {I}^\\dagger _{\\bar{k}}$ , say $i$ .",
"If $\\Omega (\\mathcal {I})$ contains an element from this red node, say $j$ (which may or may not be equal to $i$ ), we replace $j\\in \\Omega (\\mathcal {I})$ with $i$ .",
"This neither changes the size of $\\Omega (\\mathcal {I})$ nor the universality.",
"Now suppose $\\Omega (\\mathcal {I})$ does not contain an element from this red node.",
"We know that the sibling blue node (i.e.",
"the blue node that shares the parent with this red node) contains an element of $\\mathcal {K}$ (and hence of $\\Omega (\\mathcal {I})$ ), say $j$ .",
"Replace $j \\in \\Omega (\\mathcal {I})$ with $i$ .",
"This neither changes the size, nor the universality; we are just exchanging one element from a node with its sibling, so the value of $\\widetilde{\\chi }$ at the parent node does not change.",
"These operations preserve size and universality, and repeating them for each red node ensures that the resultant set contains $\\mathcal {I}^\\dagger _k$ .",
"A stronger version of the upper bound in (i) is the following.",
"Corollary 1 Let $\\mathcal {I}\\subseteq [0:p^M-1]$ and let $\\underline{k}$ be the smallest integer such that $\\widetilde{\\chi }_{\\underline{k}}(a\\,;\\mathcal {I})=0$ for some $a$ .",
"Then, $|\\Omega (\\mathcal {I})| \\le |\\lbrace a: \\widetilde{\\chi }_{\\underline{k}}(a; \\mathcal {I})\\ne 0 \\rbrace |.$ From the definition of $\\underline{k}$ , we have $\\widetilde{\\chi }_{\\underline{k}}(a_0\\,; \\mathcal {I}) = 0 $ for some $a_0$ .",
"Hence by universality, $\\Omega (\\mathcal {I})$ must satisfy $|\\widetilde{\\chi }_{\\underline{k}}(b\\,; \\Omega (\\mathcal {I}))| \\le 1$ for all $b$ , an observation we used above and will use again.",
"From the cardinality equation (REF ) $|\\Omega (\\mathcal {I})| = \\sum _b \\widetilde{\\chi }_{\\underline{k}}(b\\,; \\Omega (\\mathcal {I})) \\le |\\lbrace a: \\widetilde{\\chi }_{\\underline{k}}(a\\,; \\mathcal {I})\\ne 0 \\rbrace |.$ Ultimately we will show that when $N=p^M$ any maximal universal sampling set, and in particular any universal sampling set, is a disjoint union of elementary sets.",
"In general, however, the union of two disjoint, elementary sets need not be universal.",
"For example, take $N=2^3$ , $\\mathcal {E}=\\lbrace 0, 1\\rbrace $ , $\\mathcal {E}^{\\prime }=\\lbrace 4,5\\rbrace $ .",
"Then $\\mathcal {E}$ and $\\mathcal {E}^{\\prime }$ are elementary but their union $\\mathcal {E}\\cup \\mathcal {E}^{\\prime } =\\lbrace 0,1,4,5\\rbrace $ is not universal.",
"What is needed is a kind of independence condition on a collection of elementary sets.",
"The following lemma, whose converse we will also show, makes this latter point precise and introduces the main features of the structure of universal sets.",
"Lemma 7 Let $N=p^M$ .",
"Suppose there exists a finite sequence of nonincreasing integers $k_1 \\ge k_2 \\ge \\cdots \\ge 0$ and sets $\\mathcal {E}_{r}\\subseteq [0:N-1]$ , $r =1, 2, \\dots $ , such that $\\mathcal {E}_{r}$ is $k_r$ -elementary.",
"For each $r\\ge 1$ $\\mathcal {E}_{r} \\cap \\left(\\bigcup _{j=1}^{r-1} \\mathcal {L}_j\\right) = \\emptyset ,$ where $\\mathcal {L}_j =\\lbrace x \\in [0:N-1] \\colon x \\equiv e \\text{ mod $p^{k_j+1}$ for some $ e \\in \\mathcal {E}_{j}$}\\rbrace .$ Let $\\mathcal {I}= \\bigcup _r \\mathcal {E}_{r}.$ Then $\\mathcal {I}$ is a universal sampling set.",
"Obviously it is condition (ii) that requires further comment.",
"The set $\\mathcal {L}_r$ is defined much as in the proof of Theorem REF , and we will illustrate the point of (ii) again by means of a tree.",
"Observe first that the $\\mathcal {E}_{r}$ are disjoint.",
"This follows from (ii), since $\\mathcal {L}_j \\supseteq \\mathcal {E}_{j}$ .",
"We build a congruence tree with root the full interval $[0:N-1]$ .",
"Write this as $\\mathcal {N}_{00}$ and write $\\mathcal {N}_{ka}$ for the congruence class of $a$ modulo $p^k$ in $[0:N-1]$ , so that $|\\mathcal {N}_{ka} | = \\widetilde{\\chi }_k(a\\,;[0:N-1])$ .",
"All the nodes represent non-singletons, except the bottom-most level, $M$ .",
"As before, Figure REF has $p=3$ , $M\\ge 3$ and shows the tree only up to the third level.",
"Suppose $k_1 =1$ , so $\\mathcal {E}_{1}$ , as an elementary set, contains one element from each node at the middle level in the figure.",
"In turn, suppose $\\mathcal {E}_1$ comes from picking one element from each of the red nodes.",
"The set $\\mathcal {L}_1$ is the union of the red nodes.",
"Now, the set $\\mathcal {E}_2$ comes from choosing one element from each node at the $k_2$ -level, and the sequence $k_r$ is nonincreasing so $\\mathcal {E}_2$ is drawn from nodes in a level at or higher up in the tree than $\\mathcal {E}_1$ (in this example $k_2$ is either 1 or 0).",
"Condition (ii) requires that $\\mathcal {E}_2$ be disjoint from the red nodes, not just from $\\mathcal {E}_1$ which is a (small) subset of the red nodes.",
"In the general case, think of $k_1$ as large (eventually it will be chosen as in Theorem REF ), so $\\mathcal {E}_1$ comes from a level far down the tree from the root, and then $\\mathcal {E}_2, \\mathcal {E}_3, \\dots $ are, at least, no further down since $k_1 \\ge k_2 \\ge \\cdots $ .",
"Condition (ii) requires that $\\mathcal {E}_r$ be assembled from nodes that were not used in assembling any of the $\\mathcal {E}_s$ for $s<r$ .",
"It is this property that we exploit to show that $\\bigcup _r\\mathcal {E}_r$ is universal.",
"Figure: Similar to Figure but with root 𝒩 00 =[0:N-1]\\mathcal {N}_{00}=[0:N-1], this tree shows the relationship between 𝒩 ka \\mathcal {N}_{ka}, forp=3p=3, M≥3M\\ge 3.",
"[Proof of Lemma REF ] Fix $r$ and $s$ with $k_r< s$ , and note that $ \\widetilde{\\chi }_s(a\\,;\\mathcal {E}_{r}) \\in \\lbrace 0,1\\rbrace , \\quad a\\in [0:p^{s}-1],$ from (REF ).",
"Now suppose $\\widetilde{\\chi }_s(a\\,;\\mathcal {E}_{r}) =1$ , so one element in $\\mathcal {E}_{r}$ leaves a remainder of $a$ on dividing by $p^s$ .",
"Then $ \\widetilde{\\chi }_s(a\\,;\\mathcal {E}_{t})=0 \\quad \\text{for all $t>r$},$ i.e., none of the $\\mathcal {E}_{t}$ for $t>r$ will have an element from the congruence class of $a$ modulo $p^s$ .",
"This follows (just as described for the tree) from $\\mathcal {E}_{t} \\cap \\mathcal {L}_r=\\emptyset $ , and also $\\mathcal {L}_r &= \\lbrace x \\in [0:N-1] \\colon x \\equiv e \\text{ mod $p^{k_r+1}$ for some $e \\in \\mathcal {E}_{r}$}\\rbrace \\nonumber \\\\&\\supseteq \\lbrace x \\in [0:N-1]: x \\equiv e \\text{ mod $p^s$ for some }e \\in \\mathcal {E}_{r} \\rbrace \\nonumber .$ From (REF ) and (REF ) we conclude that $ \\sum _{r} \\widetilde{\\chi }_s(a\\,;\\mathcal {E}_r) \\in \\lbrace 0,1\\rbrace $ for all $a$ , where the sum is over all $r$ with $k_r <s$ .",
"With this we can show that $\\mathcal {I}= \\bigcup _r \\mathcal {E}_r$ is universal.",
"For any $s$ , and for any $a,b \\in [0:p^s-1]$ , $&\\widetilde{\\chi }_s(a\\,;\\mathcal {I}) - \\widetilde{\\chi }_s(b\\,;\\mathcal {I})= \\sum _r \\widetilde{\\chi }_s(a\\,;\\mathcal {E}_r) - \\sum _r \\widetilde{\\chi }_s(b\\,;\\mathcal {E}_r) \\nonumber \\\\& = \\left(\\sum _{r \\,(k_r\\ge s)} \\widetilde{\\chi }_s(a\\,;\\mathcal {E}_r) - \\sum _{r\\,(k_r\\ge s)} \\widetilde{\\chi }_s(b\\,;\\mathcal {E}_r)\\right) +\\\\& \\hspace{36.135pt} \\left(\\sum _{r\\,(k_r<s)} \\widetilde{\\chi }_s(a\\,;\\mathcal {E}_r) - \\sum _{r\\,( k_r<s)}\\widetilde{\\chi }_s(b\\,;\\mathcal {E}_r) \\right)\\nonumber \\\\& = \\sum _{r\\,(k_r<s)} \\widetilde{\\chi }_s(a\\,;\\mathcal {E}_r) - \\sum _{r\\, (k_r<s)}\\widetilde{\\chi }_s(b\\,;\\mathcal {E}_r)\\nonumber \\\\&\\hspace{36.135pt}\\text{ (the first two sums cancel, by (\\ref {eq:chi-constant}))}.\\nonumber $ From (REF ) we have that $|\\widetilde{\\chi }_s(a\\,;\\mathcal {I}) - \\widetilde{\\chi }_s(b\\,;\\mathcal {I})| \\le 1$ , so $\\mathcal {I}$ is universal."
],
[
"An Algorithm to Construct Maximal Universal Sets",
"Consider now the problem of finding a maximal universal sampling set contained in a given $\\mathcal {I}\\subseteq [0:p^M-1]$ .",
"Build the congruence class tree with root $\\mathcal {I}$ , as in Figure REF , up to level $M$ .",
"The leaves having weight 1 are singletons in $\\mathcal {I}$ , and $\\widetilde{\\chi }_k(a\\,;\\mathcal {I})$ , $a \\in [0:p^k-1]$ , is the total weight at node $(k,a)$ .",
"The problem of constructing $\\Omega (\\mathcal {I})$ is to pick a subset of the leaves so that the tree with root $\\Omega (\\mathcal {I})$ is well balanced at each level.",
"By `well balanced' we mean that at any given level, all the subtrees have roughly equal weight, corresponding to the condition $|\\widetilde{\\chi }_k(a\\,;\\Omega (\\mathcal {I})) -\\widetilde{\\chi }_k(b\\,;\\Omega (\\mathcal {I}))| \\le ~1$ .",
"The following algorithm realizes this and provides the value of $|\\Omega (\\mathcal {I})|$ .",
"It marries the construction of elementary sets in Theorem REF with an iterative version of the method used in the proof of Lemma REF .",
"Let $\\mathcal {I}\\subseteq [0:p^M-1]$ .",
"Initialize with $\\mathcal {I}_1 = \\mathcal {I}$ , and $r=1$ .",
"Let ${k}_r$ be the largest integer such that no congruence class in $\\mathcal {I}_r/p^{{k_r}}$ is empty.",
"Construct an elementary set $\\mathcal {I}_{r}^\\dagger \\subseteq \\mathcal {I}_r$ by choosing one element of $\\mathcal {I}_r$ from each congruence class modulo $p^{{k}_r}$ .",
"(There may not be a unique choice, and this is the reason why there may be many universal sets contained in $\\mathcal {I}$ .)",
"Define $\\mathcal {L}_r \\supseteq \\mathcal {I}_r^\\dagger $ by $\\mathcal {L}_r= \\lbrace j \\in \\mathcal {I}\\colon j \\equiv i \\text{ mod $p^{{k}_r+1}$ for some $i\\in \\mathcal {I}_r^\\dagger $}\\rbrace .$ Let $\\mathcal {I}_{r+1} = \\mathcal {I}_r \\setminus \\mathcal {L}_r$ .",
"Stop if $\\mathcal {I}_{r+1} = \\emptyset $ .",
"Else increment $r$ to ${r+1}$ and go to (1).",
"Note the following: At each step of the algorithm the size of $\\mathcal {I}_r$ is reduced by $|\\mathcal {L}_r|\\ge |\\mathcal {I}_r^\\dagger |=p^{k_r} \\ge 1$ : $|\\mathcal {I}_{r+1}| \\le |\\mathcal {I}_r| -p^{k_r}.$ Since $\\mathcal {I}= \\mathcal {I}_1$ is a finite set, the algorithm terminates at some point.",
"The $k_r$ are nonincreasing: $k_1 \\ge k_2 \\ge k_3 \\ge \\ldots .$ We can now state Theorem 6 With $k_r$ , $r \\ge 1$ , defined as above, we have $ |\\Omega (\\mathcal {I})| = \\sum _r p^{k_r}.$ One possible maximal universal sampling set is $ \\Omega (\\mathcal {I}) = \\bigcup _r \\mathcal {I}_r^\\dagger .$ By construction this is a disjoint union.",
"Here is an example of the algorithm in action.",
"Let $N=2^5$ and $\\mathcal {I}= \\lbrace 0,1,2,3,4,6,7,8,9,10,12,14, 15\\rbrace = \\mathcal {I}_1.$ $(r=1)$ Note that $\\widetilde{\\chi }_3(5\\,;\\mathcal {I}_1) = 0$ , and that no values $\\widetilde{\\chi }_2(i\\,;\\mathcal {I}_1)$ are zero.",
"Hence $k_1 = 2$ .",
"Form $\\mathcal {I}_1^\\dagger $ by taking one element from each congruence class in $\\mathcal {I}_1$ modulo $2^{k_1} = 4$ , e.g.",
"$\\mathcal {I}_1^\\dagger = \\lbrace 0,1,2,3\\rbrace $ .",
"Then $\\mathcal {L}_1=\\lbrace 0,1,2,3,8,9,10 \\rbrace $ is the set of all elements of $\\mathcal {I}_1$ that leave a remainder of $0,1,2$ or 3 on dividing by $2^{k_1+1} = 8$ .",
"Removing such numbers from $\\mathcal {I}_1$ , we have $\\mathcal {I}_2 = \\mathcal {I}_1 \\setminus \\mathcal {L}_1 = \\lbrace 4, 6, 7, 12, 14, 15 \\rbrace $ .",
"$(r=2)$ Now $\\widetilde{\\chi }_2(1\\,;\\mathcal {I}_2) = 0$ while $\\widetilde{\\chi }_1(0\\,;\\mathcal {I}_2)$ , $\\widetilde{\\chi }_1(1\\,;\\mathcal {I}_2) \\ne 0$ so $k_2 = 1$ .",
"Let $\\mathcal {I}_2^\\dagger = \\lbrace 4, 7\\rbrace $ .",
"Then $\\mathcal {L}_2=\\lbrace 4,7,12,15\\rbrace $ is the set of all elements in $\\mathcal {I}_2$ that leave a remainder of $4 \\text{ mod }4 = 0 $ or $7 \\text{ mod }4 = 3$ on dividing by $2^{k_2+1} = 4$ .",
"Removing such numbers from $\\mathcal {I}_2$ , we have $\\mathcal {I}_3 = \\mathcal {I}_2 \\setminus \\mathcal {L}_2 = \\lbrace 6, 14 \\rbrace $ .",
"$(r=3)$ Now clearly $k_3 = 0$ .",
"Let $\\mathcal {I}_3^\\dagger = \\lbrace 6\\rbrace $ .",
"Then $\\mathcal {L}_4 = \\lbrace 6,14\\rbrace $ , $\\mathcal {I}_4 = \\emptyset $ and the algorithm terminates.",
"According to the theorem, we have $|\\Omega (\\mathcal {I})| = 2^{k_1} + 2^{k_2} + 2^{k_3} = 7 $ , and an example $\\Omega (\\mathcal {I})$ is given by $\\mathcal {I}_1^\\dagger \\cup \\mathcal {I}_2^\\dagger \\cup \\mathcal {I}_3^\\dagger = \\lbrace 0,1,2,3,4, 6, 7\\rbrace $ .",
"We have several additional comments.",
"First, we can say more about the formula for $|\\Omega (\\mathcal {I})|$ .",
"Since the $k_r$ 's are nonincreasing, a typical sequence is, say, $\\underbrace{l_1, l_1,\\ldots }_{\\alpha _1 \\text{ times }} \\quad \\underbrace{l_2, l_2, \\ldots }_{ \\alpha _2 \\text{ times }} \\quad \\underbrace{l_3, l_3, \\ldots }_{ \\alpha _3 \\text{ times }}, \\quad \\dots $ with $l_1 > l_2 > l_3$ .",
"Given this, equation (REF ) appears as $ |\\Omega (\\mathcal {I})| = \\alpha _1p^{l_1} + \\alpha _2p^{l_2} + \\alpha _3p^{l_3} + \\ldots .$ In fact, effectively, Theorem REF constructs a base $p$ expansion of $|\\Omega (\\mathcal {I})|$ because each power of $p$ appears at most $p-1$ times.",
"Corollary 2 Let $\\mathcal {I}\\subseteq [0:p^M-1]$ .",
"The formula (REF ) is of the form, $|\\Omega (\\mathcal {I})| = \\sum _r p^{k_r} = \\sum _s \\alpha _s p^{l_s},$ with $l_1 > l_2 > l_3 > \\ldots $ and $\\alpha _s \\in [0:p-1]$ for all $s$ .",
"Begin with $\\Omega (\\mathcal {I}) = \\bigcup _r\\mathcal {I}_r^\\dagger $ .",
"Since the $\\mathcal {I}_r^\\dagger $ are disjoint, we have $\\begin{aligned}\\sum _{r=1}^{\\alpha _1} \\widetilde{\\chi }_{l_1+1}(a\\,;\\mathcal {I}_r^\\dagger ) &= \\widetilde{\\chi }_{l_1+1}(a\\,;\\bigcup _{r=1}^{\\alpha _1}\\mathcal {I}_r^\\dagger ) \\nonumber \\\\& \\le \\widetilde{\\chi }_{l_1+1}(a\\,; \\Omega (\\mathcal {I})), \\quad a\\in [0:p^{l_1+1}-1].\\end{aligned}$ Summing this over all $a \\in [0:p^{l_1 +1}-1]$ we have $\\begin{aligned}\\alpha _1p^{l_1} &= \\sum _{r=1}^{\\alpha _1} |\\mathcal {I}_r^\\dagger | = \\sum _{r=1}^{\\alpha _1} \\sum _a\\widetilde{\\chi }_{l_1+1}(a\\,;\\mathcal {I}_r^\\dagger )\\\\& \\le \\sum _i\\widetilde{\\chi }_{l_1+1}(a\\,; \\Omega (\\mathcal {I})) = |\\Omega (\\mathcal {I})| < p^{l_1+1},\\end{aligned}$ so $\\alpha _1 < p$ .",
"For the last inequality in (REF ) we have used the upper bound from part (ii) in Theorem REF .",
"We have also used that $|\\mathcal {I}_r| = p^{k_r}$ .",
"The proof for other $\\alpha _s$ is similar.",
"For example, to prove that $\\alpha _2<p$ we start with $\\sum _{r=\\alpha _1+1}^{\\alpha _2} \\widetilde{\\chi }_{l_2+1}(a\\,;\\mathcal {I}_r^\\dagger ) \\le \\widetilde{\\chi }_{l_2+1}\\left(a\\,; \\Omega (\\mathcal {I}_{a_1+1})\\right)$ instead of (REF ).",
"If the algorithm above were initialized with a universal set $\\mathcal {I}$ , then from Theorem REF we would obtain $\\mathcal {I}= \\Omega (\\mathcal {I}) =\\bigcup _ r \\mathcal {I}_r^\\dagger $ .",
"This allows us to conclude that any universal set $\\mathcal {I}$ is a union of elementary universal sets.",
"Moreover, the sets $\\mathcal {I}^\\dagger _r$ defined by the algorithm satisfy conditions in Lemma REF .",
"For condition (ii), note that in the algorithm the set $\\mathcal {I}_r$ is recursively defined as $ \\mathcal {I}_r = \\mathcal {I}_{r-1}\\setminus \\mathcal {L}_{r-1}$ , so that $\\mathcal {I}_r = \\left(\\left(\\left(\\mathcal {I}\\setminus \\mathcal {L}_1\\right)\\setminus \\mathcal {L}_2\\right) \\ldots \\setminus \\mathcal {L}_{r-1}\\right)= \\mathcal {I}\\setminus \\left(\\bigcup _{j=1}^{r-1}\\mathcal {L}_j \\right).$ Hence $\\mathcal {I}_r \\cap \\left(\\bigcup _{j=1}^{r-1} \\mathcal {L}_j\\right) = \\emptyset $ .",
"Then the sets $\\mathcal {I}^\\dagger _r$ , obtained by the algorithm, satisfy $\\mathcal {I}_r^\\dagger \\cap \\left(\\bigcup _{j=1}^{r-1} \\mathcal {L}_j\\right) = \\emptyset $ , since $\\mathcal {I}^\\dagger _r \\subseteq \\mathcal {I}_r$ .",
"Putting all these comments together we have the converse of Lemma REF , and then adding Theorem REF we can state Corollary 3 $\\mathcal {I}\\subseteq [0:p^M-1]$ is universal if and only if there exist A nonincreasing finite sequence $k_1 \\ge k_2 \\ge \\cdots \\ge 0$ , with each value of $k_r$ repeating at most $p-1$ times; Sets $\\mathcal {I}^\\dagger _r\\subseteq \\mathcal {I}$ with $\\mathcal {I}= \\bigcup _r \\mathcal {I}^\\dagger _r$ ; such that $\\mathcal {I}_r^\\dagger $ is a $k_r$ -elementary universal set; $\\mathcal {I}_r^\\dagger \\cap \\left(\\bigcup _{j=1}^{r-1} \\mathcal {L}_j\\right)= \\emptyset $ , where $\\mathcal {L}_j=\\lbrace x \\in [0:N-1]: x \\equiv i \\text{ mod $p^{k_j}+1$ for some $i \\in \\mathcal {I}_j^\\dagger $}\\rbrace .",
"\\nonumber $ Note that from (i), (ii) and (iii) we can also conclude that $|\\mathcal {I}| = \\sum _r |\\mathcal {I}_r^\\dagger | = \\sum _r p^{k_r},$ so the $k_r$ are the powers of $p$ appearing in the base-$p$ expansion of $|\\mathcal {I}|$ , taken with repetitions.",
"For example with $N=9$ , $|\\mathcal {I}| = 7 = 2\\cdot 3^1 + 1 \\cdot 3^0$ , we expect the universal set $\\mathcal {I}= \\mathcal {I}_1^\\dagger \\cup \\mathcal {I}_2^\\dagger \\cup \\mathcal {I}_3^\\dagger $ with $\\mathcal {I}_1^\\dagger $ and $\\mathcal {I}^\\dagger _2$ being 1-elementary, and $\\mathcal {I}_3^\\dagger $ being 0-elementary.",
"Corollary REF implies that the $k_r$ read off from the base-$p$ expansion of $|\\mathcal {I}|$ must be the same as the $k_r$ generated by the algorithm if $\\mathcal {I}$ is universal.",
"Remark 4 (Universal sets of prescribed order) As it stands, the algorithm finds a universal set of the largest size contained in $\\mathcal {I}$ .",
"With Corollary REF we can now modify the algorithm to solve the following problem: Given a set $\\mathcal {I}\\subseteq [0:p^M-1]$ , and $d \\le |\\Omega (\\mathcal {I})|$ , find a universal set $\\mathcal {J}\\subseteq \\mathcal {I}$ with $|\\mathcal {J}| = d$ .",
"We follow the algorithm as in steps 1-4, but we change the definition of $k_r$ in Step 1.",
"Write the base-p expansion of $d$ with repetitions, $d = \\sum _r p^{k_r}$ , read off the $k_r$ as the powers of $p$ that appear in the expansion, and arrange the $k_r$ in nonincreasing order.",
"This ensures that condition (i) in Corollary REF is satisfied.",
"The construction of the $\\mathcal {I}_r^\\dagger $ in Steps 2-4 of the algorithm will ensure that (iii) and (iv) are satisfied.",
"We conclude that with the $\\mathcal {I}_r^\\dagger $ so obtained by the algorithm the set $\\mathcal {J}= \\bigcup _r \\mathcal {I}_r^\\dagger \\subset \\mathcal {I}$ is universal, and it is of the right size by definition of the $k_r$ .",
"Finally, we have [Proof of Theorem REF ] As observed above, the $\\mathcal {I}_r^\\dagger $ generated by the algorithm satisfy the hypotheses of Lemma REF , so the set $\\bigcup _r \\mathcal {I}^\\dagger _r$ is universal.",
"If we show $|\\Omega (I)| \\le \\sum _rp^{k_r},$ then Theorem REF follows.",
"For this we prove $ |\\Omega (\\mathcal {I}_r)| \\le p^{k_r}+ |\\Omega (\\mathcal {I}_{r+1})|.$ We appeal to Theorem REF to find a maximal universal sampling set $\\mathcal {A}$ with $\\mathcal {I}_r^{\\dagger } \\subseteq \\mathcal {A}\\subseteq \\mathcal {I}_r$ , and we will show $\\mathcal {A}\\setminus \\mathcal {I}_r^\\dagger \\quad \\text{is universal},$ $\\mathcal {A}\\setminus \\mathcal {I}_r^\\dagger \\subseteq \\mathcal {I}_{r+1}.",
"$ Since $|\\mathcal {A}| = |\\Omega (\\mathcal {I}_r)|$ These imply $|\\Omega (\\mathcal {I}_r)| - p^{k_r} = |\\mathcal {A}\\setminus \\mathcal {I}_r^\\dagger | \\le |\\Omega (\\mathcal {I}_{r+1})|,$ which is (REF ).",
"First (REF ).",
"Now, $\\widetilde{\\chi }_s(a \\,;\\mathcal {A}\\setminus \\mathcal {I}_r^\\dagger ) = \\widetilde{\\chi }_s(a \\,;\\mathcal {A}) - \\widetilde{\\chi }_s(a \\,;\\mathcal {I}_r^\\dagger ), \\quad a \\in [0:p^s-1],$ and for $s \\le k_r$ the second term is constant, $\\widetilde{\\chi }_s(a\\,;\\mathcal {I}_r^\\dagger ) = p^{k_r-s}, $ from (REF ).",
"Since $\\mathcal {A}$ is universal, $|\\widetilde{\\chi }_{s}(a\\,;\\mathcal {A}) - \\widetilde{\\chi }_{s}(b\\,;\\mathcal {A})| \\le 1$ for all $a, b \\in [0:p^s-1]$ and for all $s$ , so we at least have $|\\widetilde{\\chi }_s(a\\,;\\mathcal {A}\\setminus \\mathcal {I}_r^\\dagger ) - \\widetilde{\\chi }_s(b\\,;\\mathcal {A}\\setminus \\mathcal {I}_r^\\dagger )| \\le 1,$ for all $s \\le k_r$ .",
"We need to check that this inequality continues to hold for $s \\ge k_r+1$ .",
"As we have argued before, by the definition of $k_r$ at least one congruence class in $\\mathcal {I}_r/p^{s}$ is empty when $s \\ge k_r+1$ , so $\\widetilde{\\chi }_s(a_0\\,;\\mathcal {I}_r)=0$ for some $a_0$ , and because $\\mathcal {A}\\subseteq \\mathcal {I}_r$ is universal we have $\\widetilde{\\chi }_s(a\\,;\\mathcal {A}) \\le 1$ for all $a$ .",
"Furthermore, $I_r^\\dagger \\subseteq \\mathcal {A}$ implies $0 \\le \\widetilde{\\chi }_s(a\\,;\\mathcal {A}) - \\widetilde{\\chi }_s(a\\,;\\mathcal {I}_r^\\dagger ) = \\widetilde{\\chi }_s(a\\,;\\mathcal {A}\\setminus \\mathcal {I}_r^\\dagger ).$ Hence the values of $\\widetilde{\\chi }_s(a\\,;\\mathcal {A}\\setminus \\mathcal {I}_r^\\dagger )$ are in $\\lbrace 0,1\\rbrace $ and consequently $|\\widetilde{\\chi }_s(a\\,;\\mathcal {A}\\setminus \\mathcal {I}_r^\\dagger )-\\widetilde{\\chi }_s(b\\,;\\mathcal {A}\\setminus \\mathcal {I}_r^\\dagger )| \\le 1,$ for all $s \\ge k_r+1$ .",
"This establishes that $\\mathcal {A}\\setminus \\mathcal {I}_r^\\dagger $ is universal.",
"We prove (REF ) by contradiction.",
"If it were not true that $\\mathcal {A}\\setminus \\mathcal {I}_r^\\dagger \\subseteq \\mathcal {I}_{r+1}$ then there would exist an $x \\in (\\mathcal {A}\\setminus \\mathcal {I}_r^\\dagger ) \\cap \\mathcal {L}_r$ .",
"Then $\\widetilde{\\chi }_{k_r+1}([x]_{k_r+1}\\,;\\mathcal {A}) = 2$ , for on dividing by $p^{k_r+1}$ , $x$ leaves a remainder of $[x]_{k_r+1}$ (by definition) and so does one other element in $\\mathcal {I}_r^\\dagger $ .",
"But this contradicts $\\widetilde{\\chi }_s(a\\,;\\mathcal {A}) \\le 1$ for $s \\ge k_r+1$ from the preceding paragraph.",
"This completes the proof of Theorem REF .",
"Remark 5 We can give an upper bound for the computational complexity of the algorithm for constructing a universal sampling set of size $d$ (including constructing a maximal universal sampling set).",
"Within an iteration, in the worst case the algorithm makes a complete pass over all the nodes of the tree once, and the the number of nodes is $O(N)$ .",
"Further, the number of iterations is $\\alpha _1+\\alpha _2+\\cdots +\\alpha _M$ where $d = \\alpha _1p^{M-1} + \\alpha _{2}p^{M-2} + \\ldots + \\alpha _{M-1}p + \\alpha _M.$ Hence the largest number of iterations is $(p-1)M$ , and the complexity of the algorithm is at most $O(N \\log N)$ ."
],
[
"Counting Universal Sets",
"The preceding structure theorems allow us to find the number of universal sampling sets $\\mathcal {I}\\subseteq [0:p^M-1]$ of size $d$ .",
"The formula uses the digits from the base-$p$ expansion of $d$ , and as above we let $\\nonumber d = \\alpha _1p^{M-1} + \\alpha _{2}p^{M-2} + \\ldots + \\alpha _{M-1}p + \\alpha _M,$ where $0 \\le \\alpha _i < p$ .",
"For $i=0,1,\\dots , M$ define $d_i= \\sum _{j=i+1}^M\\alpha _jp^{M-j}.$ Hence $d_0=d$ and $d_M=0$ .",
"Theorem 7 The number of universal sampling sets in $[0:p^M-1]$ of size $d$ is $\\mathcal {C}(d,p^M) = \\prod _{i=1}^M \\binom{p}{\\alpha _i+1}^{d_i}\\binom{p}{\\alpha _i}^{p^{M-i}-d_i} .$ The proof goes by establishing a recurrence relation for $\\mathcal {C}$ in the $d_i$ .We are grateful to a reviewer for suggesting a way to make greater use of the recursive aspect of our original argument, resulting in a much shorter and cleaner proof.",
"Let $\\mathcal {I}$ be a universal sampling set of size $d$ and construct the congruence tree as in Figure REF with root $\\mathcal {I}_{00}=\\mathcal {I}$ .",
"We first note that $d_1$ of the nodes at level $M-1$ have weight $\\alpha _1+1$ and the remaining $p^{M-1}-d_1$ nodes have weight $\\alpha _1$ , where $d_1 = \\sum _{i=2}^M \\alpha _i p^{M-i}.$ The proof for this is along the same lines as the argument in the proof of Theorem REF , $(i) \\Longleftrightarrow (ii)$ .",
"Figure REF illustrates this.",
"The singleton blue nodes at the bottom level are the elements of $\\mathcal {I}$ , and the other nodes (which would be the singletons $\\lbrace 6\\rbrace $ and $\\lbrace 7\\rbrace $ ) are empty.",
"The red nodes at the penultimate level represent the nodes that have weight $\\alpha _1 +1$ (and there are $d_1$ of them).",
"Figure: The congruence class tree for N=8N=8.",
"The universal sampling set {0,1,2,3,4,5}\\lbrace 0,1,2,3,4,5\\rbrace of sized=6d=6 is represented by the blue nodes at the bottom level.",
"Thered nodes at the penultimate level represent the nodes thathave weight 2, the rest of the nodes at the penultimate level haveweight 1.Now remove the bottom level of the tree, effectively making $N=p^{M-1}$ , and resulting in Figure REF .",
"If the starting set (the blue nodes in Figure REF ) is universal, then so must be the set formed by the red nodes in Figure REF .",
"Hence the number of ways of choosing the red nodes is the same as the number of universal sampling sets of size $d_1$ in $[0:p^{M-1}-1]$ , that is $\\mathcal {C}(d_1,p^{M-1})$ .. Once the red nodes are chosen, we need to choose the blue nodes by taking $\\alpha _1+1$ elements from the red nodes and $\\alpha _1$ elements from the remaining (non-red) nodes, which can be done in $\\binom{p}{\\alpha _1+1}^{d_1}\\binom{p}{\\alpha _1}^{p^{M-1}-d_1}$ ways.",
"Hence $ \\mathcal {C}(d, p^M) = \\binom{p}{\\alpha _1+1}^{d_1}\\binom{p}{\\alpha _1}^{p^{M-1}-d_1}\\mathcal {C}(d_1, p^{M-1}).$ This full formula follows.",
"Figure: Remove the bottom level of the tree in Figure .",
"The resulting red nodesare a universal sampling set in [0:3][0:3]One special case of the counting formula is easy to evaluate.",
"Corollary 4 Let $d = p^k$ where $k <M$ .",
"Then the number of universal sets of size $d$ in $[0:p^M-1]$ is $(p^M/d)^d$ .",
"In particular when $N=2^M$ , and $d = 2^{M-1}=N/2$ , the number of universal sets is $2^{N/2}$ .",
"On the other hand, the total number of sets of size $2^{M-1}$ in $[0:2^N-1]$ is $\\binom{N}{N/2} \\approx 2^N/\\sqrt{\\pi N}$ by Stirling's approximation.",
"Hence the fraction of sets that are universal is approximately $\\sqrt{\\pi N}/2^{N/2}$ , which decreases exponentially with $N$ .",
"The function $\\mathcal {C}(d,p^M)$ is certainly complicated, but it has some remarkable properties.",
"Though not clear from the formula, we have $\\mathcal {C}(d,p^M) = \\mathcal {C}(p^M-d,p^M).$ This follows from the following lemma, which is itself a simple but interesting property of universal sampling sets.",
"Lemma 8 If $\\mathcal {A}\\subseteq [0:p^M-1]$ is a universal sampling set then so is $\\mathcal {A}^{\\prime }=[0:p^M-1] \\setminus \\mathcal {A}$ .",
"This extends the bracelet property of universal sampling sets, though for bracelets we need not assume that $N$ is a prime power.",
"For any $0 \\le k \\le M$ and $a \\in [0:p^k-1]$ , $\\begin{aligned}\\widetilde{\\chi }_k(a\\,;\\mathcal {A}^{\\prime }) &= \\widetilde{\\chi }_k(a\\,;[0:p^M-1]) - \\widetilde{\\chi }_k(a\\,;\\mathcal {A})\\\\&= p^{M-k} - \\widetilde{\\chi }_k(a\\,;\\mathcal {A}).\\end{aligned}$ Next, since $|\\widetilde{\\chi }_k(a\\,;\\mathcal {A}) - \\widetilde{\\chi }_k(b\\,;\\mathcal {A})| \\le 1,$ for all $a,b \\in [0:p^k-1]$ , it follows that $|\\widetilde{\\chi }_k(a\\,;\\mathcal {A}^{\\prime }) - \\widetilde{\\chi }_k(b\\,;\\mathcal {A}^{\\prime })| \\le 1 .$ Figure REF displays $\\log \\mathcal {C}(d,5^M)$ as a function of $d$ as $M$ takes increasing values.",
"The plots show the symmetry, $\\mathcal {C}(d,p^M) = \\mathcal {C}(p^M-d,p^M)$ , but they show much more.",
"We can observe the following: There are a series of bumps on several (visible) scales.",
"One cannot fail to notice that at each scale the number of bumps in the graph is 5, which is the prime $p$ here.",
"Experiments with other primes have similar plots and in each case indicate that the number of bumps is equal to the prime.",
"Figure: Plots of log𝒞(d,p M )\\log \\mathcal {C}(d,p^M) vs dd for powers of p=5p=5.",
"Note the 5 bumps on different scales.",
"With increasing $M$ the plots of the count are somehow converging in shape – they all start to look similar.",
"The second point can indeed be quantified.",
"One can show that for each $\\alpha \\in [0,1]$ , $\\lim _{M \\rightarrow \\infty }\\frac{\\log \\mathcal {C}(\\lfloor \\alpha p^M\\rfloor ,p^M)}{p^M}$ exists.",
"See [7].",
"This compares nicely with the fact that a similar function with $\\mathcal {C}(d,N)$ replaced by $\\binom{N}{d}$ also converges, and to the entropy function: $\\begin{aligned}\\lim _{M \\rightarrow \\infty }\\left(\\frac{1}{p^M}\\log \\binom{p^M}{\\lfloor \\alpha p^M\\rfloor }\\right) &= \\alpha \\log \\frac{1}{\\alpha } + (1-\\alpha )\\log \\frac{1}{1-\\alpha }\\\\& =H(\\alpha ).\\end{aligned}$ This is the limiting case of counting all index sets.",
"Plots of $ \\mathcal {H}_p(\\alpha ) = \\lim _{M \\rightarrow \\infty }\\frac{\\log \\mathcal {C}(\\lfloor \\alpha p^M\\rfloor ,p^M)}{p^M}, \\quad 0\\le \\alpha \\le 1,$ are shown in Figures REF and REF for several values of $p$ , along with a plot of $H(\\alpha )$ .",
"Figure: Plots of the limit of the counting functions for p=2,5p=2,5 compared to the Entropy function.",
"Note the self-similarity as it depends on the prime.Figure: Similar to Figure with p=3,7p=3,7The plots of $H_p(\\alpha )$ seem to satisfy observation (i), that the curves have $p$ bumps at each scale.",
"Here is an explanation.",
"In the notation of Theorem REF , suppose $\\alpha _1=0$ (i.e., $d <p^{M-1}$ ).",
"Then $d_1=d$ and we have, as in (REF ), $\\begin{aligned}\\mathcal {C}(d,p^M) &= \\binom{p}{1}^{d_1}\\binom{p}{0}^{p^{M-1}-d_1}\\mathcal {C}(d_1,p^{M-1})\\\\& = p^d\\mathcal {C}(d,p^{M-1}).\\end{aligned}$ Let $M \\rightarrow \\infty $ , so $d/p^M \\rightarrow \\alpha $ (with $\\alpha < p$ ).",
"Then with reference to (REF ), $\\mathcal {H}_p(\\alpha ) = \\lim _{M\\rightarrow \\infty }\\left(\\frac{d}{p^M}\\log p + \\mathcal {H}_p(p\\alpha )\\right) = \\alpha \\log p +\\mathcal {H}_p(p\\alpha ),$ leading to the self-similar plots we observe."
],
[
"Maximal and Minimal Universal Sampling Sets",
"Along with maximal universal sets is the allied notion of minimal universal sets.",
"Definition 7 Let $\\mathcal {I}\\subseteq [0:N-1]$ .",
"A minimal universal sampling set for $\\mathcal {I}$ is a universal sampling set of smallest cardinality that contains $\\mathcal {I}$ .",
"Again we need a notation and we let $\\Phi (\\mathcal {I})$ denote a generic minimal universal sampling set containing $\\mathcal {I}$ .",
"Thus $|\\Phi (\\mathcal {I})| \\le |\\mathcal {J}|$ for any universal sampling set $\\mathcal {J}\\supseteq \\mathcal {I}$ .",
"Let us show one way that maximal and minimal universal sampling sets are related.",
"The proof relies on Lemma REF from the previous subsection.",
"Theorem 8 Let $\\mathcal {I}\\subset [0:p^M-1]$ , $\\mathcal {I}^{\\prime }=[0:p^M-1]\\setminus \\mathcal {I}$ .",
"Then $|\\Phi (\\mathcal {I})| = p^M-|\\Omega (\\mathcal {I}^{\\prime })|.$ Let $\\mathcal {A}^{\\prime }=[0:p^M-1]\\setminus \\Phi (I)$ .",
"Then $\\mathcal {A}^{\\prime }$ is universal by Lemma REF .",
"Since $\\Phi (\\mathcal {I}) \\supseteq \\mathcal {I}$ we have $\\mathcal {A}^{\\prime } \\subset [0:N-1] \\setminus \\mathcal {I}=\\mathcal {I}^{\\prime }$ and hence $p^M-|\\Phi (\\mathcal {I})| = |\\mathcal {A}^{\\prime }| \\le |\\Omega (\\mathcal {I}^{\\prime })|.$ Similarly, let $\\mathcal {B}^{\\prime }=[0:p^M-1]\\setminus \\Omega (\\mathcal {I}^{\\prime })$ .",
"Then $\\mathcal {B}^{\\prime }$ is universal, it contains $[0:p^M-1]\\setminus \\mathcal {I}^{\\prime }=\\mathcal {I}$ and so $p^M-|\\Omega (\\mathcal {I}^{\\prime })| = |\\mathcal {B}^{\\prime }| \\ge |\\Phi (\\mathcal {I})|.$ Taken together the two inequalities prove the theorem."
],
[
"An Uncertainty Principle, Random Signals, and Sumsets",
"Generally speaking, an “uncertainty principle” is an inequality relating the supports of a nonzero function and its Fourier transform, in the present setting $f\\colon \\mathbb {Z}_N \\longrightarrow \\mathbb {C}$ , and ${{\\mathcal {F}}}f\\colon \\mathbb {Z}_N \\longrightarrow \\mathbb {C}$ .",
"The notions of maximal and minimal universal sampling sets lead immediately to an additive uncertainty principle.",
"Without the language of universality, Tao [6] made this connection in the case when $N$ is a prime using Chebotarev's theorem, see Corollary REF , though, as he states, it was probably already known as a folk theorem.",
"Let $\\mathcal {Z}(f) = \\lbrace i \\colon f(i) = 0\\rbrace $ be the zero set of $f$ .",
"The support is the complement of the zero set, and we denote it by ${\\rm supp}(f)$ .",
"Our result is Theorem 9 If $f$ is not the zero function then $ \\begin{aligned}|{\\rm supp}({{\\mathcal {F}}}f)| &\\ge 1+|\\Omega (\\mathcal {Z}(f))|, \\; \\\\|{\\rm supp}(f)| &\\ge 1+|\\Omega (\\mathcal {Z}({{\\mathcal {F}}}f))|;\\end{aligned}$ and $ \\begin{aligned}|\\mathcal {Z}({{\\mathcal {F}}}f)| +1 &\\le |\\Phi ({\\rm supp}(f))|,\\; \\\\|\\mathcal {Z}( f)| +1 &\\le |\\Phi ({\\rm supp}({{\\mathcal {F}}}f))|.\\end{aligned}$ We are not assuming that $N$ is a prime power here.",
"However, we immediately deduce Corollary 5 (Tao) If $N$ is prime and $f$ is not the zero function then $|{\\rm supp}({{\\mathcal {F}}}{f})| + |{\\rm supp}(f)| \\ge N +1.$ If $N$ is prime, then by Chebotarev's theorem every index set is universal.",
"In particular the set $\\mathcal {Z}(f)$ is universal.",
"Hence $\\Omega (\\mathcal {Z}(f)) = \\mathcal {Z}(f)$ .",
"From Theorem REF , $\\begin{aligned}|{\\rm supp}({{\\mathcal {F}}}{f})| &\\ge 1 + |\\Omega (\\mathcal {Z}(f))| \\\\&= 1 + |\\mathcal {Z}(f)| = 1 + N - |{\\rm supp}(f)|.\\end{aligned}$ We also have Corollary 6 Suppose $f$ vanishes on a set of consecutive integers $\\mathcal {I}$ .",
"Then $|{\\rm supp}({{\\mathcal {F}}}{f})| \\ge |\\mathcal {I}|+1$ .",
"If $\\mathcal {J}$ is a set of integers such that ${{\\mathcal {F}}}f(\\mathcal {J}) = 0$ , then $|\\mathcal {I}| + |\\mathcal {J}| \\le N - 1$ .",
"We observed previously that any set of consecutive integers, $\\mathcal {I}$ in this case, is universal.",
"Since $\\mathcal {I}\\subseteq \\mathcal {Z}(f)$ , we have $|\\Omega (\\mathcal {Z}(f))| \\ge |\\mathcal {I}|$ .",
"From Theorem REF , this implies $|{\\rm supp}({{\\mathcal {F}}}{f})| \\ge |\\mathcal {I}| + 1$ .",
"Further, if ${{\\mathcal {F}}}{f}(\\mathcal {J}) = 0$ then $N - |\\mathcal {J}| \\ge |{\\rm supp}({{\\mathcal {F}}}f)|$ and so $N - |\\mathcal {J}| \\ge |\\mathcal {I}| + 1$ .",
"The proof of Theorem REF itself is very brief.",
"[Proof of Theorem REF ] Suppose $|{\\rm supp}({{\\mathcal {F}}}{f})| \\le |\\Omega (\\mathcal {Z}(f))|$ .",
"From $\\Omega (\\mathcal {Z}(f)) \\subseteq \\mathcal {Z}(f)$ it follows that $f$ vanishes on $\\Omega (\\mathcal {Z}(f))$ .",
"Since $\\Omega (Z)$ is a universal sampling set this implies that ${{\\mathcal {F}}}{f}\\equiv 0$ , contradicting the assumption that $f$ is not the zero function.",
"This proves the first statement in (REF ).",
"A similar argument establishes the second statement.",
"For the proof of (REF ), write $\\mathcal {Z} = \\mathcal {Z}(\\mathcal {F}f)$ and $\\mathcal {A}=\\Phi (\\text{supp}(f))$ .",
"Then $\\mathcal {F}f (\\mathcal {Z}) = 0 \\text{ and so } E_{\\mathcal {Z}}^\\textsf {T}\\mathcal {F}f = 0.$ However $f$ is supported within $\\mathcal {A}$ , and so we may write $f = E_{\\mathcal {A}}g$ , where $ g = f(\\mathcal {A}) \\ne 0$ .",
"This means we must have $E_{\\mathcal {Z}}^\\textsf {T}\\mathcal {F}E_{\\mathcal {A}} g = 0, \\text{ for some } g \\ne 0,$ i.e.",
"the columns of $E_{\\mathcal {Z}}^\\textsf {T}\\mathcal {F}E_{\\mathcal {A}}$ are dependent.",
"This is expected if $|\\mathcal {Z}|< |\\mathcal {A}|$ .",
"However, if $|\\mathcal {Z}| \\ge |\\mathcal {A}|$ , this contradicts the universality of $\\mathcal {A}$ .",
"Hence we must have $|\\mathcal {Z}| \\le |\\mathcal {A}|-1$ , which is the first inequality in (REF ).",
"A similar argument establishes the second statement.",
"It is interesting that when $N$ is a prime power the two statements (REF ) and (REF ) are equivalent.",
"To see this we first derive (REF ) from (REF ) when $N=p^M$ .",
"This appeals to Theorem REF on the relation between maximal and minimal sets, with ${\\rm supp}(f) = [0:N-1]\\setminus \\mathcal {Z}(f)$ .",
"Thus, from (REF ), $|{\\rm supp}({{\\mathcal {F}}}f)| \\ge 1+|\\Omega (\\mathcal {Z}(f))|$ , and substituting from Theorem REF , $|{\\rm supp}({{\\mathcal {F}}}f)| \\ge 1 + N - |\\Phi ({\\rm supp}(f))|.$ But $|{\\rm supp}({{\\mathcal {F}}}f)| = N-|\\mathcal {Z}({{\\mathcal {F}}}f)|$ , so $N-|\\mathcal {Z}({{\\mathcal {F}}}f)| \\ge 1+N-|\\Phi ({\\rm supp}(f))|,$ which is the same as the first statement in (REF ).",
"Again, the second statement in (REF ) follows in a similar manner.",
"We could have started instead with (REF ) and from this derived (REF ).",
"In cases where $\\mathcal {Z}(f)$ itself is a universal sampling set, the uncertainty principle in Theorem REF can be as strong as the uncertainty principle for the prime $N$ case.",
"Remark 6 Readers familiar with the seminal paper of Donoho and Stark [14] will wonder if the additive uncertainty principle in Theorem REF can be applied to the problem of reconstruction of a signal corrupted by sparse noise.",
"(See also [15] for more recent work.)",
"The answer is yes, and we refer to [16]."
],
[
"Random Index Sets and Random Signals",
"We will give several applications of these ideas.",
"First we combine Theorem REF with a probabilistic estimate on the size of a maximal universal sampling set for randomly chosen index sets.",
"We must revert to the assumption that $N$ is a prime power.",
"Theorem 10 Let $N=p^M$ .",
"Let $\\mathcal {R}_s$ be an index set of $s$ numbers chosen at random from $[0:N-1]$ .",
"Let $\\lambda = (N-s)/N$ .",
"If $d, \\delta >0$ satisfy $N\\log (1/\\lambda ) \\ge (1+ \\delta ) d \\log d,$ then $|\\Omega (\\mathcal {R}_s)| \\ge d$ with probability at least $1- d^{-\\delta }$ .",
"This means that if we can choose a large $d$ satisfying (REF ), which is possible, for example, if $N$ is large and $\\lambda $ is small, then $|\\Omega (\\mathcal {R}_s)| \\ge d$ with high probability.",
"Thus while it is unlikely that a randomly chosen index set will be universal, it is quite likely that such an index set will contain a large universal set as a subset.",
"We will apply Theorem REF to the case when $\\mathcal {R}_s$ is the zero set of $f\\colon \\mathbb {Z}_N \\longrightarrow \\mathbb {C}$ .",
"Then $\\lambda = |\\text{supp}(f)|/N$ , i.e., $\\lambda $ is the fraction of nonzero entries in $f$ .",
"The proof uses the bound in part (ii) of Theorem REF .",
"Let $k$ be the largest integer such that no congruence classes in $\\mathcal {R}_s/p^k$ are empty.",
"Note that $k$ is random since $\\mathcal {R}_s$ is random.",
"Then $|\\Omega (\\mathcal {R}_s)| \\le d-1$ implies $p^k \\le |\\Omega (\\mathcal {R}_s)| \\le d-1,$ by Theorem REF .",
"Therefore $&\\text{Prob}\\left(|\\Omega (\\mathcal {R}_s)| \\le d-1 \\right) \\le \\text{Prob}(p^k \\le d-1) \\nonumber \\\\&\\quad = \\text{Prob}(k \\le \\lfloor \\log _p(d-1) \\rfloor ) \\nonumber \\\\&\\quad = \\text{Prob}(\\text{at least one congruence class in } \\nonumber \\\\& \\hspace{36.135pt} \\mathcal {R}_s/p^{\\lfloor \\log _p(d-1) \\rfloor +1} \\text{ is empty}).$ We will compute the last probability.",
"Let $b = {\\lfloor \\log _p(d-1) \\rfloor +1}$ , and let $\\mathcal {N}_{ba}$ be the set of elements in $[0:N-1]$ that leave a remainder of $a\\in [0:p^b-1]$ when divided by $p^b$ .",
"Since $N=p^M$ all of the $\\mathcal {N}_{ba}$ have size $t = N/p^b = p^M / p^{\\lfloor \\log _p(d-1) \\rfloor +1}$ .",
"Fix a particular residue $a$ .",
"The probability that $\\mathcal {N}_{ba}\\cap \\mathcal {R}_s $ is empty (in words, the probability that a particular congruence class goes missing in $\\mathcal {R}_s$ ) is $\\binom{N-t}{s}/\\binom{N}{s}$ .",
"This is because the number of ways of picking $\\mathcal {R}_s$ is $\\binom{N}{s}$ while the number of ways of picking $\\mathcal {R}_s $ so that $\\mathcal {N}_{ba} \\cap \\mathcal {R}_s = \\emptyset $ is the number of ways of picking $s$ elements from $ |[0:N-1] \\setminus \\mathcal {N}_{ba}| = N- t$ elements.",
"Then $&\\text{Prob}\\left(\\mathcal {N}_{ba}\\cap \\mathcal {R}_s = \\emptyset \\right) \\nonumber \\\\&= \\binom{N-t}{s}\\Big {/} \\binom{N}{s} \\nonumber \\\\&= \\frac{(N-(t-1)-s)(N-(t-2)-s)\\ldots (N-s)}{(N-t+1)(N-t+2)\\ldots N} \\nonumber \\\\&\\quad = \\left(1- \\frac{s}{N-t+1} \\right) \\left(1- \\frac{s}{N-t+2}\\right)\\ldots \\left(1- \\frac{s}{N} \\right) \\nonumber \\\\&\\quad \\le \\left(1- \\frac{s}{N} \\right) \\left(1- \\frac{s}{N}\\right)\\ldots \\left(1- \\frac{s}{N} \\right) =\\left(1-\\frac{s}{N}\\right)^t.",
"$ From this, $&\\text{Prob}(\\text{at least one congruence class in} \\nonumber \\\\& \\hspace{50.58878pt} \\mathcal {R}_s/p^{\\lfloor \\log _p(d-1) \\rfloor +1} \\text{ is empty})\\nonumber \\\\&\\quad = \\text{Prob}\\left(\\bigcup _i \\left(\\mathcal {N}_{ba}\\cap \\mathcal {R}_s = \\emptyset \\right)\\right)\\nonumber \\\\&\\quad \\le \\sum _i \\text{Prob}\\left(\\mathcal {N}_{ba}\\cap \\mathcal {R}_s = \\emptyset \\right) \\nonumber \\\\&\\quad \\le \\frac{N}{t}\\left(1-\\frac{s}{N}\\right)^t = N\\lambda ^t/t.$ Hence we have from (REF ), $\\text{Prob}\\left(|\\Omega (\\mathcal {R}_s)|\\le d-1 \\right) \\le N\\lambda ^t/t.$ Now, $ t = N / p^{\\lfloor \\log _p(d-1) \\rfloor +1} \\ge N/d$ , since $\\lfloor x \\rfloor \\le x$ .",
"Using this in (REF ), $&\\text{Prob}\\left(|\\Omega (\\mathcal {R}_s)| \\le d-1 \\right) \\nonumber \\\\& \\le N\\lambda ^t/t \\le d\\lambda ^{N/d} \\nonumber \\\\& =\\exp \\left(\\log d - \\frac{N\\log (1/\\lambda )}{d} \\right) \\nonumber \\\\& = \\exp \\left(\\log d\\left(1- \\frac{N\\log (1/\\lambda )}{d\\log d} \\right)\\right) \\nonumber \\\\& \\le \\exp \\left(-\\delta \\log d\\right) \\text{ (from the hypothesis of the theorem) } \\nonumber \\\\& = d^{-\\delta }.$ We conclude that $\\text{Prob}\\left(|\\Omega (\\mathcal {R}_s)| \\ge d \\right) \\ge 1- d^{-\\delta }$ .",
"We can now state a probabilistic uncertainty principle.",
"Afterward we will comment on how this compares to the result of Candes, Romberg and Tao [3].",
"Theorem 11 Let $N=p^M$ .",
"Let $\\mathcal {G}_{N,r}$ be the set of all signals $g\\colon \\mathbb {Z}_N \\longrightarrow \\mathbb {C}$ with support of size $r$ .",
"Let $g \\in \\mathcal {G}_{N,r}$ be a signal whose support is drawn at random from the set of all index sets of size $r$ .",
"Let the values of $g$ on the support set be drawn according to some arbitrary distribution.",
"For $\\delta >0$ let $a_{N, \\delta } = \\frac{N}{(1+\\delta )\\log N}\\left(1+ \\log (1+ \\delta ) + \\log \\log N\\right).$ Then $|\\text{supp}(g)| + |\\text{supp}(\\mathcal {F}g)| \\ge 1 + a_{N,\\delta }$ with probability at least $1- (a_{N,\\delta }-r)^{-\\delta }$ .",
"If $r$ is small compared to $a_{N,\\delta }$ , Theorem REF states that almost all signals $g$ in $\\mathcal {G}_{N,r}$ satisfy the uncertainty principle above; roughly speaking $|\\text{supp}(g)| + |\\text{supp}(\\mathcal {F}g)| \\ge N(1+ \\log \\log N)/\\log N$ for most $g$ .",
"Picking the support of $g$ at random among sets of size $r$ is equivalent to picking the zero set of $g$ at random among all index sets of size $N-r$ .",
"The proof now makes use of Theorem REF to get a lower bound on $|\\Omega (\\mathcal {Z}(g))|$ .",
"For this we need to choose $d, \\delta $ so that $N \\log (1/\\lambda ) = N \\log N/r > (1+\\delta )d \\log d.$ Fix any $\\delta >0$ and let $d = N \\log (N/r)/(1+\\delta )\\log N$ .",
"We check that $d, \\delta $ satisfy $(\\ref {eq:ran-uncen-repeat})$ : $(1+\\delta )d\\log d &= \\frac{N \\log (N/r)}{\\log N} \\log \\left(\\frac{N \\log (N/r)}{(1+\\delta )\\log N} \\right)\\\\&<\\frac{N \\log (N/r)}{\\log N} \\log N = N \\log N/r,$ Then from Theorem REF , $|\\Omega (\\mathcal {Z}(g))| \\ge N\\log (N/r) / (1+\\delta )\\log N$ with probability $1-d^{-\\delta }$ .",
"From the uncertainty principle Theorem REF , we now have $\\begin{aligned}|\\text{supp}(\\mathcal {F}g)| &\\ge 1 + |\\Omega (\\mathcal {Z}(g))|\\\\& \\ge 1 + N\\log (N/r) / (1+\\delta )\\log N\\end{aligned}$ with probability $1-d^{-\\delta }$ .",
"The final step in the proof uses a lower bound on $d = N \\log (N/r)/(1+\\delta )\\log N$ .",
"We have set apart this technical result as Lemma REF , below.",
"This gives $|\\text{supp}(\\mathcal {F}g)| \\ge 1 + a_{N,\\delta } - r \\quad $ with probability $1-d^{-\\delta }$ .",
"Since $1-d^{-\\delta } \\ge 1- (a_{N,\\delta } - r)^{-\\delta }$ , we can say $|\\text{supp}(\\mathcal {F}g)| \\ge 1 + a_{N,\\delta } - r$ with probability $1-(a_{N,\\delta } - r)^{-\\delta }$ .",
"The result follows since $ r = |\\text{supp}(g)|$ .",
"Lemma 9 Let $d = \\frac{N\\log (N/r)}{(1+\\delta )\\log N}$ and $a_{N, \\delta } = \\frac{N}{(1+\\delta )\\log N}\\left(1+ \\log (1+ \\delta ) + \\log \\log N\\right),$ as in Theorem REF .",
"Then $d \\ge a_{N, \\delta } - r$ .",
"The convex function $\\log (N/r)$ is bounded below by its tangent at any point $r_0>0$ .",
"Thus $\\log (N/r) \\ge \\log (N/r_0) + \\left(-\\frac{1}{r_0}(r - r_0)\\right).$ For $r_0 = \\frac{N}{(1+\\delta )\\log N}\\,,$ this reads $\\begin{aligned}\\log (N/r) &\\ge \\log \\left((1+\\delta )\\log N\\right)\\\\& \\hspace{18.06749pt} + \\left(-\\frac{(1+\\delta )\\log N}{N}\\left(r - \\frac{N}{(1+\\delta )\\log N}\\right)\\right).\\end{aligned}$ Multiplying by $N/(1+\\delta )\\log N$ , we have $d &= \\frac{N\\log (N/r)}{(1+\\delta )\\log N} \\\\&\\ge \\frac{N\\log \\left((1+\\delta )\\log N\\right)}{(1+\\delta )\\log N} - \\left(r - \\frac{N}{(1+\\delta )\\log N}\\right) \\\\&= \\frac{N}{(1+\\delta )\\log N}\\left(\\log (1+\\delta ) + 1 + \\log \\log N \\right) - r \\\\&= a_{N, \\delta } - r.$ Remark 7 The robust uncertainty principle of Candes, Romberg and Tao in [3] is as follows: for $M >0$ there exists a constant $C_M$ such that $|\\text{supp}(g)| + |\\text{supp}(\\mathcal {F}g)| \\ge C_M N(\\log N)^{-1/2},$ with probability $1- O(N^{-M})$ .",
"This inequality is stronger than that of Theorem REF by about $(\\log N)^{-1/2}$ .",
"Also, Theorem REF holds for $N=p^M$ , whereas the inequality above holds for all $N$ .",
"In our proof of Theorem REF we have only used the bound $|\\Omega (\\mathcal {Z}(g))| \\ge p^k$ from Theorem REF .",
"By using the exact formula for $|\\Omega (\\mathcal {Z}(g))|$ in Theorem REF (or by a better lower bound) it might be possible to tighten the uncertainty principle of Theorem REF and remove the factor $(\\log N)^{-1/2}$ ."
],
[
"Sumsets and the Cauchy-Davenport Theorem",
"Our final application is a generalization of the Cauchy-Davenport theorem [17], from additive number theory, on the size of sumsets.",
"Again the inspiration comes from Tao's approach, [6], to the original Cauchy-Davenport theorem via Chebotarev's theorem.",
"Theorem 12 Let $\\mathcal {X}, \\mathcal {Y}\\subseteq [0:N-1]$ .",
"If either $\\mathcal {X}$ or $\\mathcal {Y}$ is a universal sampling set, then $|\\mathcal {X}+\\mathcal {Y}| \\ge |\\mathcal {X}| + |\\mathcal {Y}| -1,$ when $|\\mathcal {X}| + |\\mathcal {Y}| -1 \\le N$ .",
"Here $\\mathcal {X}+\\mathcal {Y}$ is the sumset defined as $\\mathcal {X}+ \\mathcal {Y}= \\lbrace x + y: x \\in \\mathcal {X}, y \\in \\mathcal {Y}\\rbrace ,$ where the addition is modulo $N$ .",
"We are not assuming that $N$ is a prime power, while the classical theorem has $N=p$ and there are no assumptions on $\\mathcal {X}$ or $\\mathcal {Y}$ .",
"That form of the result follows from Theorem REF , since all index sets in $[0:N-1]$ are universal when $N$ is prime.",
"As a corollary we get a statement on the size of $|\\mathcal {X}+\\mathcal {Y}|$ without making an assumption on $\\mathcal {X}$ or $\\mathcal {Y}$ .",
"Corollary 7 Let $\\mathcal {X}, \\mathcal {Y}\\subseteq [0:N-1]$ be index sets.",
"Then, $|\\mathcal {X}+\\mathcal {Y}| \\ge \\max \\lbrace |\\Omega (\\mathcal {X})| + |\\mathcal {Y}| -1, |\\mathcal {X}| + |\\Omega (\\mathcal {Y})| -1 \\rbrace .$ Since $\\Omega (\\mathcal {X}) \\subseteq \\mathcal {X}$ , it follows that $\\Omega (\\mathcal {X}) + \\mathcal {Y}\\subseteq \\mathcal {X}+ \\mathcal {Y}$ .",
"Now, $|\\mathcal {X}+ \\mathcal {Y}| \\ge |\\Omega (\\mathcal {X}) + \\mathcal {Y}| \\ge |\\Omega (\\mathcal {X})| + |\\mathcal {Y}| -1 \\text{ from Theorem \\ref {thm:sumset-univ}}.$ The inequality $|\\mathcal {X}+ \\mathcal {Y}| \\ge |\\mathcal {X}| + |\\Omega (\\mathcal {Y})| -1 $ follows similarly.",
"[Proof of Theorem REF ] First note that (REF ) follows trivially when either $X$ or $Y$ is a singleton.",
"(More precisely, if, say, $\\mathcal {X}$ is a singleton, then $\\mathcal {X}+\\mathcal {Y}$ is just a translate of $\\mathcal {Y}$ , and so (REF ) holds with equality).",
"For the rest of the proof, we assume that $|\\mathcal {X}|, |\\mathcal {Y}| \\ge 2$ .",
"Let $|\\mathcal {X}|=r$ , $|\\mathcal {Y}| =s$ .",
"Assume without loss of generality that $\\mathcal {X}$ is universal.",
"Let $f_1 \\in \\mathbb {B}^{\\mathcal {X}} \\text{ be such that }f_1(\\left[1:r\\right]) = (\\underbrace{0, 0, \\ldots , 0}_{r-1 \\text{ times }}, 1).$ Such an $f_1$ exists because the set $[1:r]$ , as an index set of $r$ consecutive integers, is a universal sampling set, so is in particular a sampling set for $\\mathbb {B}^\\mathcal {X}$ .",
"Similarly let $f_2 \\in \\mathbb {B}^\\mathcal {Y}\\text{ be such that }f_2(\\left[r:r+s-1\\right]) = (\\underbrace{0, 0, \\ldots , 0}_{s-1 \\text{ times }}, 1),$ again possible because $[r:r+s-1]$ is a set of $s$ consecutive integers, and hence a sampling set for $\\mathbb {B}^\\mathcal {Y}$ .",
"Note that $f_1f_2 \\in \\mathbb {B}^{\\mathcal {X}+\\mathcal {Y}}$ and so $|\\mathcal {X}+\\mathcal {Y}| \\ge {\\rm supp}(\\mathcal {F}(f_1f_2))$ .",
"Note also that the zero set $\\mathcal {Z}(f_1f_2)$ of $f_1f_2$ contains $[1:r+s-2]$ , and hence, since the latter is a universal sampling set, $ |\\Omega \\left(\\mathcal {Z}(f_1f_2)\\right)| \\ge r+s-2 = |\\mathcal {X}|+|\\mathcal {Y}|-2$ .",
"Now we apply the uncertainty principle of Theorem REF to $f_1f_2$ .",
"We have, so long as $f_1f_2 \\ne 0$ , $ |\\mathcal {X}+\\mathcal {Y}| &\\ge {\\rm supp}\\left(\\mathcal {F}(f_1f_2)\\right) \\nonumber \\\\&\\ge 1 + |\\Omega \\left(\\mathcal {Z}(f_1f_2)\\right)| \\nonumber \\\\& \\ge 1 + |\\mathcal {X}|+|\\mathcal {Y}|-2 = |\\mathcal {X}| + |\\mathcal {Y}| -1,$ So we have proved that $|\\mathcal {X}+\\mathcal {Y}| \\ge |\\mathcal {X}| + |\\mathcal {Y}| -1 $ if we know that $ f_1 f_2 \\ne 0$ .",
"For this, again from Theorem REF we have $|\\mathcal {Z}(f_1)| \\le |\\Phi ({\\rm supp}(\\mathcal {F}f_1))| - 1 \\le |\\Phi (\\mathcal {X})| - 1,$ since $f_1 \\in \\mathbb {B}^\\mathcal {X}$ .",
"But $\\mathcal {X}$ is universal, so $\\Phi (\\mathcal {X})=\\mathcal {X}$ and $|\\mathcal {Z}(f_1)| \\le |\\mathcal {X}| -1.$ By definition of $f_1$ , the set $[1:r-1]=[1:|\\mathcal {X}|-1]$ is already in $\\mathcal {Z}(f_1)$ .",
"Together with (REF ), this implies that $f_1$ cannot have any more zeros.",
"In particular, $f_1(r+s -1) \\ne 0$ .",
"Since $f_2(r+s -1) = 1$ , $f_1f_2$ cannot be identically zero and (REF ) applies.",
"An important generalization of the Cauchy-Davenport theorem to any finite abelian group, not necessarily of prime order, is due to Kneser, [18].",
"Theorem 13 (Kneser) Let $G$ be a finite abelian group.",
"Let $A, B \\subseteq G$ be non empty subsets of $G$ .",
"Let $H$ be the set of periods, defined by $H = \\lbrace h \\in G : h + (A+B) = A+B \\rbrace $ .",
"(Thus $A+B$ is periodic if $H \\ne \\lbrace 0\\rbrace $ .)",
"Then $|A+B| \\ge |A| + |B| - |H|.$ Hence unless $A + B$ is periodic, $|A+B| \\ge |A| + |B| - 1$ .",
"Though the form is similar, this result neither implies nor is implied by Theorem REF .",
"We give two examples.",
"Let $N=8$ , $\\mathcal {X}= \\lbrace 0,1\\rbrace $ , $\\mathcal {Y}= \\lbrace 0,4\\rbrace $ .",
"Then $\\mathcal {X}$ is universal and $\\mathcal {X}+ \\mathcal {Y}= \\lbrace 0, 1, 4, 5\\rbrace $ is periodic with period 4.",
"So Theorem REF applies, but Kneser's theorem does not.",
"Next let $N=16$ , $\\mathcal {X}= \\lbrace 0,2\\rbrace , \\mathcal {Y}= \\lbrace 0,2,4\\rbrace $ .",
"Then $\\mathcal {X}+\\mathcal {Y}= \\lbrace 0,2,4,6,8,10\\rbrace $ , which is not periodic, and neither $\\mathcal {X}$ nor $\\mathcal {Y}$ is universal.",
"So Kneser's theorem applies, but Theorem REF does not.",
"We hope to understand this more thoroughly."
],
[
"Condition Number Associated with the Universal Sampling Set $\\mathcal {I}^*$",
"An index set of consecutive integers is the simplest universal sampling set, but there is a catch in using it.",
"Let $\\mathcal {I}$ be a universal sampling set of size $d$ , $f\\in \\mathbb {C}^N$ , and $f_\\mathcal {I}$ the $d$ -vector obtained from $f$ by sampling at locations in $\\mathcal {I}$ .",
"If $f$ is in some bandlimited space $\\mathbb {B}^\\mathcal {J}$ , $|\\mathcal {J}|=d$ , then the interpolation formula (REF ) reads $f = \\mathcal {F}E_\\mathcal {J}(E_\\mathcal {I}^T \\mathcal {F}E_\\mathcal {J})^{-1} f_\\mathcal {I}.$ The practical difficulty is the computation of the inverse of $E_\\mathcal {I}^T \\mathcal {F}E_\\mathcal {J}$ .",
"Suppose we use $\\mathcal {I}= \\mathcal {I}^* =[0:d-1]$ as a universal sampling set.",
"We give a lower bound on the condition number of $E_\\mathcal {I}^T \\mathcal {F}E_\\mathcal {J}$ that can be quite large for some $\\mathcal {J}$ , even though the matrix $E_\\mathcal {I}^T \\mathcal {F}E_\\mathcal {J}$ is invertible for all $\\mathcal {J}$ .",
"For $\\mathcal {I}= [0:d-1]$ , note that $|\\det \\left(E_{\\mathcal {I}}^T \\mathcal {F} E_{\\mathcal {J}} \\right)| &= |\\det (\\zeta _N^{ij})_{i \\in \\mathcal {I}, j \\in \\mathcal {J}} |\\\\&= \\prod _{j_1, j_2 \\in \\mathcal {J}}|\\zeta _N^{j_1} - \\zeta _N^{j_2}| \\\\&= \\prod _{j_1, j_2 \\in \\mathcal {J}}\\left|2\\sin \\frac{2\\pi (j_1-j_2)}{N}\\right|.$ If $\\lbrace \\sigma _i\\rbrace $ are the singular values of $A = E_{\\mathcal {I}}^T \\mathcal {F} E_{\\mathcal {J}}$ , then $\\det (A) = \\sigma _1\\sigma _2\\sigma _3\\ldots \\sigma _d \\ge \\sigma _{\\min }^d.$ Also if $a_{rk} = \\exp (-2\\pi i rj_k/N)$ are the entries of $A$ , then $d^2 = \\sum _{r,k = 0}^{d-1}|a_{rk}|^2 = \\text{tr}(A^*A) = \\sum _{r=0}^{d-1} \\sigma _r^2 \\le d\\sigma _{\\text{max}}^2,$ and so $\\sigma _{\\text{max}}^2 \\ge d$ .",
"From (REF ) and (REF ), the condition number satisfies $\\frac{\\sigma _{\\max }}{\\sigma _{\\min }} \\ge \\sqrt{d}\\left(\\frac{1}{\\prod _{j_1, j_2 \\in \\mathcal {J}}|2\\sin \\frac{2\\pi (j_1-j_2)}{N}|}\\right)^{1/2d}.$ A possible scenario may be when $d$ is very small and $N$ is very large.",
"In this case, the condition number can be very large if the frequency slots $\\mathcal {J}$ are clustered."
],
[
"Counting Bracelets",
"Several of our results, Theorem REF for example, depend only on the bracelet of an index set rather than on the index set itself.",
"Thus it is useful to know how many bracelets there are and how to enumerate them.",
"Counting bracelets – actually, multicolored bracelets – is a standard application in combinatorics of the orbit stabilizer theorem, and the problem is treated in many places.",
"Our situation is slightly different because we want a count that specifies the number of black beads in a black-and-white bracelet, corresponding to the size of the index set that determines the locations of the black beads.",
"Nevertheless, the orbit stabilizer theorem can still be applied, and we have the following results.",
"Theorem 14 Let $\\phi $ denote Euler's totient function.",
"When $N$ is odd, the number of black-and-white bracelets of length $N$ with exactly $d$ black beads is $\\begin{array}{ll}\\frac{1}{2} { { {(N-1)}/{2} } \\atopwithdelims (){ d/2 } } + \\frac{1}{2N} \\sum _{k|N , k|d} \\frac{\\phi (k)}{N} { {N/k} \\atopwithdelims (){d/k}}& \\quad \\textrm {for even } d, \\\\ \\\\\\frac{1}{2} { { {(N-1)}/{2} } \\atopwithdelims (){ ({d-1})/{2} } } + \\frac{1}{2N} \\sum _{k|N , k|d} \\frac{\\phi (k)}{N} { {N/k} \\atopwithdelims (){d/k}}& \\quad \\textrm {for odd } d.\\end{array}$ When $N$ is even, the number of black-and-white bracelets of length $N$ with exactly $d$ black beads is $\\begin{array}{ll}\\frac{1}{2} {{ N/2 } \\atopwithdelims (){ d/2 } } + \\frac{1}{2N} \\sum _{k|N , k|d} \\frac{\\phi (k)}{N} { {N/k} \\atopwithdelims (){d/k}}& \\quad \\textrm {for even } d, \\\\ \\\\\\frac{1}{2} {{(N/2) - 1} \\atopwithdelims (){{(d-1)}/{2} }} + \\frac{1}{2N} \\sum _{k|N , k|d} \\frac{\\phi (k)}{N} { {N/k} \\atopwithdelims (){d/k}}& \\quad \\textrm {for odd } d.\\end{array}$ We omit the proof; see [2].",
"An efficient algorithm for enumerating bracelets has been devised only recently by Sawada [19].",
"An algorithm for determining when two index sets are in the same necklace is due to J.P. Duval [20].",
"It can also be used for bracelets.",
"See [2] for examples of both of these."
],
[
"Additional References",
"Though our work has concerned discrete-time signals exclusively, there is also a notion of universal sampling sets for continuous-time signals.",
"We will not give the definition; it is interesting and not clear what the relations between the two may be.",
"Here we cite only a few sources, starting with the paper of Landau [21] that featured the renowned necessary density condition on sampling sets.",
"More recently, many interesting results have been obtained by Olevskii and Ulanovskii [22], [23] on universal sampling and stable reconstruction, by Matei and Meyer [24], who work with lattices and make contact with compressed sensing, and by Bass and Gröchenig [25], who consider random sampling.",
"Of course, anyone writing on so fundamental a topic as sampling and interpolation will encounter an enormous literature, and most probably miss an equal or greater amount.",
"We apologize to the authors of works we have missed."
],
[
"Acknowledgments",
"There are many people to thank for their interest, insight, and encouragement over quite some time, in particular S. Boyd, M. Chudnovsky, A. El Gamal, J.T.",
"Gill, S. Gunturk, B. Hassibi, J. Sawada, J. Smith, and M. Tygert.",
"We also thank the reviewers for their thorough and thoughtful comments.",
"[Figure: NO_CAPTION [Figure: NO_CAPTION [Figure: NO_CAPTIONwuriddles.com, an archive of puzzles."
]
] | 1204.0992 |
[
[
"Reactive-infiltration instabilities in rocks. Fracture dissolution"
],
[
"Abstract A reactive fluid dissolving the surface of a uniform fracture will trigger an instability in the dissolution front, leading to spontaneous formation of pronounced well-spaced channels in the surrounding rock matrix.",
"Although the underlying mechanism is similar to the wormhole instability in porous rocks there are significant differences in the physics, due to the absence of a steadily propagating reaction front.",
"In previous work we have described the geophysical implications of this instability in regard to the formation of long conduits in soluble rocks.",
"Here we describe a more general linear stability analysis, including axial diffusion, transport limited dissolution, non-linear kinetics, and a finite length system."
],
[
"Introduction",
"Fracture dissolution is an important component of a number of geological processes, including the early stages of karstification [1], diagenesis [2], and the evolution of carbonate aquifers [3].",
"It also plays an important role in geoengineering applications such as dam stability [4], oil reservoir stimulation methods [5] and leakage of sequestered ${\\rm CO}_2$ [6].",
"The dynamics of evolving fractures is complex, due to the highly nonlinear couplings between morphology, flow and dissolution.",
"Theoretical [1], [7], [8] and experimental studies [10], [11], [9] have shown that the positive feedback between fluid transport and mineral dissolution leads to an instability in an initially uniform reaction front and the subsequent formation of pronounced dissolution channels, deeply etched into the rock surfaces.",
"These processes were shown to be important in the development of limestone caves [8], and also in the assessment of subsidence hazards, since they dramatically speed up the growth of long conduits.",
"Understanding spontaneous flow focusing during fracture dissolution is also important to the petroleum industry, for efficient acidization of natural fractures and for acid fracturing of porous rocks.",
"In the former process, acid is pumped into the fractured reservoir to dissolve material blocking the pathways between the wellbore and the reservoir.",
"Spontaneous channeling increases the effectiveness of the process by creating highly permeable pathways, minimizing the amount of acid needed.",
"In acid fracturing the fluid pressure is high enough to induce hydrofracturing; the newly created fractures are then etched with acid to increase the permeability of the system.",
"Nonuniform dissolution is crucial in this process, since a uniformly etched fracture will close tightly under the overburden once the fluid pressure is removed; significant permeability will only be created by inhomogeneous etching when the less dissolved regions act as supports to keep more dissolved regions open.",
"In this paper we investigate the initiation of the instability in a fracture dissolution front and assess the wavelength and growth rate of the most unstable mode as a function of physical parameters characterizing the rates of transport and reaction in the fracture.",
"In Sec.",
"we present the two-dimensional averaged equations for fracture dissolution; a detailed justification of the transport equation (REF ) is given in Appendix .",
"Next we consider a uniform fracture where an analytic solution is possible; this forms the base state for the subsequent stability analysis in Sec. .",
"Results are presented in Sec.",
", extending our previous analysis [8] in several directions.",
"We now consider axial diffusion of reactant as well as lateral diffusion and also the effect of cross-aperture diffusion on the effective reaction rate.",
"After that we lift the assumptions that the fracture is of infinite length and that the reaction kinetics are linear.",
"We finish with a summary of our results and conclusions.",
"In a subsequent paper we will describe an analysis of the instability in the dissolution of a porous matrix."
],
[
"Equations for fracture dissolution",
"Fractures are geometrically characterized by a short dimension ($z$ direction), the aperture, and two much longer dimensions, length ($x$ direction) and width ($y$ direction).",
"In natural fractures the aperture is typically less than $1 \\, \\rm mm$ , while the length ($L$ ) and width ($W$ ) are of the order of meters (see Fig.",
"REF ).",
"It is typical to exploit this difference in scales by introducing approximate two-dimensional equations for fluid flow, reactant transport, and erosion.",
"Fluid flow is described by the Reynolds equation for the local volume flux (per unit length across the fracture), ${\\mbox{${q}$}}(x,y,t) = \\int _0^h {\\mbox{${v}$}}(x,y,z,t) dz$ : ${\\mbox{${q}$}}= - \\frac{h^3}{12 \\mu } \\mbox{${\\nabla }$}p, ~~ \\mbox{${\\nabla }$}\\cdot {\\mbox{${q}$}}= 0,$ where $\\mu $ is the fluid viscosity.",
"The essence of the Reynolds approximation is to assume that the exact result for stationary flow between parallel plates can be applied locally to a varying aperture.",
"In this approximation the pressure is independent of height and reduces to the two-dimensional field $p(x,y)$ .",
"The validity of the Reynolds approximation for rough fractures has been examined in [12] and [13].",
"The key requirements are: (i) low Reynolds number flow, $Re \\ll 1$ (ii) slow variation in aperture $\\left| \\nabla h \\right| \\ll 1$ .",
"We will assume these conditions hold in what follows.",
"The incompressibility condition in Eq.",
"(REF ) ignores effects of the reactant (or product) concentration on the mass density of the fluid.",
"This assumption is valid for the majority of natural systems; for example, in limestone dissolution the density correction due to the dissolved species is of the order of 0.01%.",
"However, dissolution of halite (rock salt) is a notable exception; here the increase in mass density can be as large as 25%.",
"The transport of reactant can be described in terms of a two-dimensional concentration field that has been averaged over the aperture.",
"The most important average is the “cup-mixing” or velocity-averaged concentration [14], $c(x,y,t) = \\frac{1}{\\left|{\\mbox{${q}$}}(x,y,t)\\right|} \\int _0^{h(x,y,t)} \\left|{\\mbox{${v}$}}(x,y,z,t)\\right| c_{3d}(x,y,z,t) dz,$ where we use $c_{3d}$ to identify the three-dimensional concentration field.",
"Under certain conditions, discussed in Appendix , the three-dimensional convection-diffusion equation for reactant transport in the fracture can be reduced to a two-dimensional convection-diffusion-reaction equation for the cup-mixing concentration [1], [7], [8], ${\\mbox{${q}$}}\\cdot \\mbox{${\\nabla }$}c = D \\mbox{${\\nabla }$}h \\cdot \\mbox{${\\nabla }$}c - 2R(c),$ where $R(c)$ accounts for reactant transfer at each of the fracture surfaces.",
"The slow dissolution of the rock surfaces allows the time-dependence in Eq.",
"(REF ) to be neglected (Appendix REF ).",
"In this paper we will usually assume a first-order dissolution reaction at the fracture surfaces $R = kc_w$ , where $k$ is the rate constant and $c_w$ is the reactant concentration at the fracture surface.",
"The reactive flux $R$ must balance the diffusive flux at the surface $R_{diff} = -D (\\nabla c)_w,$ where the gradient is pointing towards the surface.",
"Alternatively, and more usefully, the diffusive flux can be expressed in terms of the difference between the surface concentration, $c_w$ , and the cup-mixing concentration, $c$ by using a mass-transfer coefficient or Sherwood number [14], $R_{diff} = \\frac{D {\\rm Sh}}{2 h}(c-c_w).$ The Sherwood number, ${\\rm Sh}$ , depends on reaction rate at the fracture surfaces ($k$ ) but the variation is relatively small [16], [15], bounded by two asymptotic limits: high reaction rates (transport limit), ${\\rm Sh}=7.54$ , and low reaction rates (reaction limit), ${\\rm Sh}=8.24$ .",
"In the numerical calculations we approximate the Sherwood number by a constant value ${\\rm Sh}=8$ .",
"By equating the reactive and diffusive fluxes $R = R_{diff}$ we obtain the standard relationship between $c_w$ and $c$ [15], $c_w = \\dfrac{c}{1+2kh/D{\\rm Sh}}.$ The reactive flux can then be expressed in terms of the cup-mixing concentration, $R(c) = k_{eff}c,$ where the effective reaction rate is given by $k_{eff}(h) = \\dfrac{k}{1+2kh/D{\\rm Sh}},$ In sufficiently narrow apertures the dissolution kinetics are reaction limited and the concentration field is almost uniform across the aperture so that $k_{eff} \\approx k$ .",
"However, as the fracture opens the reaction rate becomes hindered by diffusive transport of reactant across the aperture.",
"When $k h/D Sh \\gg 1$ , dissolution can become entirely diffusion limited with $k_{eff} \\approx D{\\rm Sh}/2h$ .",
"A derivation of Eq.",
"(REF ), with the kinetics described in Eqs.",
"(REF ) and (REF ), will be given in Appendix , starting from the full three-dimensional transport equations.",
"In particular, the diffusive term in Eq.",
"(REF ) is shown to be purely molecular for either convective ($q/D \\rightarrow \\infty $ ) or reaction-limited ($2kh/D \\rightarrow 0$ ) transport.",
"In taking the Sherwood number to be independent of the distance from the inlet, we are assuming that entrance effects are negligible.",
"For a flat plat geometry the entrance length scale $l_{in}$ is given by [17] $l_{in} = 0.016\\frac{q h}{D},$ taking $l_{in}$ as the distance over which the Sherwood number is within $5\\%$ of its asymptotic value.",
"This length is small compared to the reactant penetration length under the typical conditions of fracture dissolution (see Sec. ).",
"Equations (REF ) and (REF ) describe a dissolution reaction controlled by the concentration of reactant; a typical example is dissolution of fractures (or porous rocks) by a strong acid.",
"However, when calcite is dissolved by aqueous ${\\rm CO}_2$ at pH values similar to those of natural groundwater, the dissolution rate is limited by the calcium ion undersaturation $c_{sat} - c_{ca}$ [18], $R(c_{ca}) = -k_{eff}(c_{sat} - c_{ca}),$ where $c_{ca}$ is the flow-averaged concentration of dissolved calcium ions.",
"The sign of $R$ accounts for a dissolution flux into the fluid rather than a reactive flux into the surface and so the transport equation for the undersaturation takes the same form as (REF ).",
"In the rest of the paper we will use $c$ to represent either the concentration of reactant or the undersaturation of dissolved minerals.",
"A reactive fluid with an inlet ($x = 0$ ) concentration $c_{in}$ dissolves the surrounding rock, increasing the fracture aperture at a rate $\\partial _th = 2 k_{eff} \\gamma \\frac{c}{c_{in}},$ where $\\gamma = c_{in}/\\nu c_{sol}$ is the acid capacity number or volume of solid dissolved by a unit volume of reactant.",
"Here $c_{sol}$ is the molar concentration of soluble material and $\\nu $ accounts for the stoichiometry of the reaction.",
"Mineral concentrations in the solid phase, are typically much higher than reactant concentrations in the aqueous phase and the characteristic dissolution time, $t_d = h/2k_{eff}\\gamma ,$ is large for natural minerals in typical groundwater conditions; for limestone fractures it is approximately 2 months [8].",
"Thus there is a significant separation between the dissolution time scale and the relaxation of the concentration field ($t \\sim h^2/D$ ), which justifies dropping the time dependence in Eq.",
"(REF ); for further discussion see Appendix REF ."
],
[
"Concentration profile in a uniform fracture",
"Let us first consider a uniform aperture $h(x,y)=h_0$ and find the corresponding concentration profile; the solutions will form the base state for the stability analysis.",
"The flow rate $q_0$ is independent of space and the transport equation is $q_0 \\partial _x c - D h_0 \\partial _x^2 c = - \\frac{2 kc}{1 + G},$ where we have absorbed the transport correction into a single factor, $G=\\frac{2 k h_0}{D{\\rm Sh}}.$ For an inlet concentration $c_{in}$ , Eq.",
"(REF ) has an exponentially decaying solution, $c(x) = c_{in}e^{-\\kappa x},$ with a penetration length $l_p = \\kappa ^{-1}$ given by $\\kappa h_0 = \\frac{{\\rm Pe}}{2} \\left(\\sqrt{1+\\frac{4 {\\rm Da}_{eff}}{{\\rm Pe}}}-1\\right).$ The Péclet number, ${\\rm Pe}=\\frac{q_0}{D},$ measures the relative magnitude of convective and diffusive transport of solute, and the effective Damköhler number, ${\\rm Da}_{eff}=\\frac{2k_{eff} h_0}{q_0} = \\frac{2k h_0}{(1+G)q_0},$ relates the effective surface reaction rate, Eqs.",
"(REF ) and (REF ), to the rate of convective transport.",
"It will be convenient to frame our results in terms of the transport correction $G$ (REF ) and the convective parameter $H=\\frac{{\\rm Da}_{eff}}{{\\rm Pe}}.$ A discussion of the natural length scales of the problem and their relation to $H$ can be found in Appendix .",
"The inverse penetration length can be written in terms of $H$ , $\\kappa h_0 = \\frac{{\\rm Pe}}{2} \\left(\\sqrt{1+4 H}-1\\right),$ with the important limiting cases: convection dominated ($H \\rightarrow 0$ ) $\\kappa h_0 ={\\rm Da}_{eff},$ diffusion dominated ($H \\rightarrow \\infty $ ) $\\kappa {h_0} = \\sqrt{{\\rm Pe}{\\rm Da}_{eff}} = \\sqrt{\\frac{G {\\rm Sh}}{1 + G}},$ In Appendix we show that (REF ) is valid for all $G$ when $H = 0$ (Sec.",
"REF ) and for all $H$ when $G \\ll 1$ (Sec.",
"REF ).",
"For long fractures, the reactant penetration length is the natural length scale for dissolution.",
"On the scale of $\\kappa ^{-1}$ the entrance length (REF ) is $\\kappa l_{in} = 0.008\\,{\\rm Pe}^2(\\sqrt{1+4H}-1).$ In the convective ($H \\rightarrow 0$ ) limit, $\\kappa l_{in} = 0.016G {\\rm Sh}/(1+G) < 0.12$ over the whole range of reaction rates; it is vanishingly small in the reaction ($G \\rightarrow 0$ ) limit.",
"In the diffusive ($H \\rightarrow \\infty $ ) limit $\\kappa l_{in} = 0.016 {\\rm Pe}\\sqrt{G {\\rm Sh}/(1+G)} < 0.05 {\\rm Pe}$ , which is again small (since ${\\rm Pe}\\ll 1$ ).",
"In Sec.",
"REF we will examine the instability in finite-length fractures $\\kappa L < 1$ , but only in the reaction limit ($G \\rightarrow 0$ ), in which case $l_{in}/L \\rightarrow 0$ , even for finite $\\kappa L$ ."
],
[
"Linear stability analysis of a uniform profile",
"The discussion in Sec.",
", supported by the derivations in Appendix , leads to the following average equations for the concentration, aperture and flow fields in an evolving fracture: $& q_x \\partial _x c + q_y \\partial _y c - D\\left[\\partial _x (h \\partial _x c) + \\partial _y (h \\partial _y c)\\right] = - \\frac{c_{in}}{\\gamma } \\partial _t h & \\text{(transport)} \\\\& c_{in}\\partial _t h= \\frac{2 k \\gamma c}{1+ 2 kh/D{\\rm Sh}}\\ \\ & \\text{(erosion)} \\\\& \\partial _x q_x + \\partial _y q_y = 0 & \\text{(continuity)} \\\\& \\partial _y q_x - \\frac{3}{h} q_x \\partial _y h = \\partial _x q_y - \\frac{3}{h} q_y\\partial _x h \\ \\ & \\text{(compatibility)}$ Here the Reynolds equation (REF ) has been replaced by the more convenient equations for continuity () and compatibility () (see Appendix ).",
"When supplemented by appropriate boundary conditions: $c(x=0,y,t)=c_{in}, \\ \\ \\ \\ c(x\\rightarrow \\infty ,y,t)=0,$ $q_x(x\\rightarrow \\infty ,y,t)=q_0, \\ \\ \\ \\ q_y(x=0,y,t)=0,$ Eqs.",
"(REF )–() form a complete, albeit approximate, description of the erosion of a single fracture (in the domain $x>0$ ).",
"The constant pressure condition at the inlet has been replaced by the boundary condition $q_y(x=0) = 0.$ The above equations allow one-dimensional solutions in which the fields depend only on $x$ and $t$ .",
"This corresponds to uniform dissolution of the fracture, an assumption still commonly found in models of fracture dissolution [19], [20].",
"For example, in the reaction-limited, convection-dominated case ($G \\rightarrow 0$ , $H \\rightarrow 0$ ), the solution is $c(x,t) = c_{in} e^{- 2k x/q},$ $h(x,t) = h_0 + 2 k \\gamma t e^{- 2k x/q},$ ${{\\mbox{${q}$}}}(x,t) = q_0 {{\\mbox{${e}$}}}_x.$ In [8] we showed that the solution represented by Eqs.",
"(REF )–(REF ) is unstable to infinitesimal perturbations along the $y$ direction.",
"Here we will not limit ourselves to the reaction-limited, convection dominated regime, but consider more general kinetics and transport.",
"Thus $\\kappa $ will no longer be equal to $2k/q$ , as in (REF ) and (REF ), but instead it will be given by the general expression (REF ).",
"An important detail in the stability analysis is that the base state for the aperture (REF ) is itself time-dependent.",
"The stability of nonautonomous systems is in general a difficult problem [21] and in [8] we adopted an approximate approach [22] in which the base state is frozen at a specific time, $t_0$ , and the growth rate is then determined as if the base state were time-independent (the quasi-steady-state approximation).",
"The validity of this approach was tested by comparing the results of the quasi-steady-state approximation with a numerical solution of the complete system of equations (REF )–().",
"In particular, we were able to show that the most relevant instability is obtained by freezing the base state at $t_0=0$ and in the present paper we will focus on this case.",
"The solution at $t=0$ is $c_b(x) = c_{in} e^{- \\kappa x}, \\ \\ \\ \\ h_b(x)=h_0, \\ \\ \\ \\ {{\\mbox{${q}$}}}_b(x) = q_0 {{\\mbox{${e}$}}}_x,$ which simplifies the subsequent calculations.",
"The linear stability analysis proceeds by considering infinitesimal perturbations to the base profile (REF ): $h=h_b+\\delta h$ , $c=c_b + \\delta c$ and ${{\\mbox{${q}$}}} = {{\\mbox{${q}$}}}_b + \\delta {{\\mbox{${q}$}}}$ .",
"This gives the following linearized equations for the aperture, concentration and flow fields: $\\delta q_x \\partial _x c_b + q_b \\partial _x \\delta c - D \\left[h_b \\partial _x^2 \\delta c + h_b \\partial _y^2 \\delta c + \\delta h \\partial _x^2 c_b + (\\partial _x \\delta h) (\\partial _x c_b)\\right] = - \\frac{c_{in}}{\\gamma } \\partial _t \\delta h,$ $c_{in}\\left(1+ \\frac{2 k h_b}{D{\\rm Sh}} \\right) \\partial _t \\delta h + \\left(1+ \\frac{2 k h_b}{D{\\rm Sh}} \\right)^{-1} \\frac{(2 k)^2 \\gamma c_b}{D{\\rm Sh}} \\delta h = 2 k \\gamma \\delta c,$ $\\partial _x \\delta q_x + \\partial _y \\delta q_y = 0,$ $\\partial _y \\delta q_x - \\frac{3}{h_b} q_b \\partial _y \\delta h = \\partial _x \\delta q_y.$ Terms in $\\partial _x h_b$ have been omitted from Eqs.",
"(REF ) and (REF ), since the expansion is about an $x-$ independent aperture field.",
"In Eq.",
"(REF ) we have made use of the erosion equation for the base field, $c_{in} (1+2kh_b/DSh) \\partial _t h_b = 2 k \\gamma c_b$ .",
"The linearized equations for fracture dissolution can be simplified by transforming to dimensionless variables.",
"We take the penetration length $\\kappa ^{-1}$ as the unit of length, and the characteristic inlet dissolution time, $t_d$ (REF ), as the unit of time.",
"The dimensionless variables are then: $\\xi = \\kappa x, \\ \\ \\ \\ \\ \\eta = \\kappa y, \\ \\ \\ \\ \\ \\tau = \\frac{2k\\gamma t}{(1+G)h_0}.$ The concentration is scaled by the inlet concentration $c_{in}$ , while the aperture and flow rate are scaled by their (constant) values in the base state: ${\\hat{c}}=\\frac{c}{c_{in}}, \\ \\ \\ \\ {\\hat{h}}= \\frac{h}{h_0}, \\ \\ \\ \\ {\\hat{{\\mbox{${q}$}}}}=\\frac{{\\mbox{${q}$}}}{q_0}.$ The dimensionless base-state solution is: ${\\hat{c}}_b=e^{-\\xi }, \\ \\ \\ \\ {\\hat{h}}_b = 1, \\ \\ \\ \\ {\\hat{{\\mbox{${q}$}}}}_b = {\\mbox{${e}$}}_\\xi ,$ and the dimensionless perturbations can be found from the following equations: $\\frac{2k}{q_0\\kappa (1+G)} \\partial _\\tau \\delta {\\hat{h}}= e^{-\\xi } \\delta {\\hat{q}}_\\xi - \\partial _\\xi \\delta {\\hat{c}}+ \\frac{D\\kappa h_0}{q_0} \\left(\\partial _\\xi ^2 \\delta {\\hat{c}}+ \\partial _\\eta ^2 \\delta {\\hat{c}}+ e^{-\\xi } \\delta {\\hat{h}}- e^{-\\xi } \\partial _\\xi \\delta {\\hat{h}}\\right),$ $\\partial _\\tau \\delta {\\hat{h}}+ \\frac{G}{1+G} e^{-\\xi } \\delta {\\hat{h}}= \\delta {\\hat{c}},$ $\\partial _\\xi ^2 \\delta {\\hat{q}}_\\xi + \\partial _\\eta ^2 \\delta {\\hat{q}}_\\xi = 3\\partial _\\eta ^2 \\delta {\\hat{h}}.$ In deriving (REF ) we have combined the continuity equation (REF ) and the compatibility equation (REF ) to eliminate $\\delta {\\hat{q}}_\\eta $ .",
"The transport equation (REF ) involves two new dimensionless constants, each one based on the penetration length $\\kappa ^{-1}$ , ${\\rm Pe}_\\kappa &=& \\dfrac{q_0}{D \\kappa h_0} = \\frac{2}{\\sqrt{1+4H}-1}, \\\\{\\rm Da}_\\kappa &=& \\frac{2k_{eff}}{q_0 \\kappa } = \\frac{2H}{\\sqrt{1+4H}-1}.",
"$ ${\\rm Pe}_\\kappa $ is the ratio of convective to diffusive fluxes on the length scale $\\kappa ^{-1}$ , while ${\\rm Da}_\\kappa $ is the ratio of convective to reactive fluxes on the same scale.",
"The physical significance of these parameters is discussed in Appendix .",
"Rewriting the transport equation in terms of ${\\rm Pe}_\\kappa $ and ${\\rm Da}_\\kappa $ and rearranging to isolate the term in $\\delta {\\hat{q}}_\\xi $ , $\\delta {\\hat{q}}_\\xi = e^\\xi \\left[{\\rm Da}_\\kappa \\partial _\\tau + {\\rm Pe}_\\kappa ^{-1}\\partial _\\xi e^{-\\xi } \\right]\\delta {\\hat{h}}+ e^\\xi \\left[\\partial _\\xi - {\\rm Pe}_\\kappa ^{-1} (\\partial _\\xi ^2 + \\partial _\\eta ^2) \\right]\\delta {\\hat{c}}.$ Assuming that the perturbations are sinusoidal in $\\eta $ and exponential in $\\tau $ , $\\delta {\\hat{c}}&=& f_c(\\xi )\\cos ({\\hat{u}}\\eta )e^{{\\hat{\\omega }}\\tau }, \\\\\\delta {\\hat{h}}&=& f_h(\\xi )\\cos ({\\hat{u}}\\eta )e^{{\\hat{\\omega }}\\tau }, \\\\\\delta {\\hat{q}}_\\xi &=& f_q(\\xi )\\cos ({\\hat{u}}\\eta )e^{{\\hat{\\omega }}\\tau }.",
"$ Note that ${\\hat{\\omega }}$ and ${\\hat{u}}$ are dimensionless quantities related to the instability growth rate $\\omega $ and wavelength $\\lambda $ by the relations ${\\hat{\\omega }}= \\omega t_d, ~~ {\\hat{u}}= \\frac{2\\pi }{\\kappa \\lambda }.$ Substituting the expansions (REF )-() into Eqs.",
"(REF ), (REF ), and (REF ) leads to coupled equations for the one-dimensional fields $f_c(\\xi )$ , $f_h(\\xi )$ , and $f_q(\\xi )$ : $f_q = e^\\xi \\left[{\\rm Da}_\\kappa {\\hat{\\omega }}+ {\\rm Pe}_\\kappa ^{-1}\\partial _\\xi e^{-\\xi } \\right]f_h + e^\\xi \\left[\\partial _\\xi - {\\rm Pe}_\\kappa ^{-1} (\\partial _\\xi ^2 - {\\hat{u}}^2) \\right]f_c.$ $\\left({\\hat{\\omega }}+ \\frac{Ge^{-\\xi }}{1+G}\\right)f_h = f_c.$ $(\\partial _\\xi ^2 - {\\hat{u}}^2)f_q = -3{\\hat{u}}^2f_h.$ Eliminating $f_c$ , we express $f_q$ in terms of $f_h$ only $f_q = e^\\xi \\left\\lbrace \\left[{\\rm Da}_\\kappa {\\hat{\\omega }}+ {\\rm Pe}_\\kappa ^{-1}\\partial _\\xi e^{-\\xi } \\right] + \\left[\\partial _\\xi - {\\rm Pe}_\\kappa ^{-1} (\\partial _\\xi ^2 - {\\hat{u}}^2) \\right]\\left[{\\hat{\\omega }}+ \\frac{Ge^{-\\xi }}{1+G}\\right]\\right\\rbrace f_h,$ and, substituting into (REF ), obtain a fourth-order equation for the $\\xi $ dependence of the aperture field, $(\\partial _\\xi ^2 - {\\hat{u}}^2)e^\\xi \\left\\lbrace \\left[{\\rm Da}_\\kappa {\\hat{\\omega }}+ {\\rm Pe}_\\kappa ^{-1}\\partial _\\xi e^{-\\xi } \\right] + \\left[\\partial _\\xi - {\\rm Pe}_\\kappa ^{-1} (\\partial _\\xi ^2 - {\\hat{u}}^2) \\right]\\left[{\\hat{\\omega }}+ \\frac{Ge^{-\\xi }}{1+G}\\right]\\right\\rbrace f_h + 3{\\hat{u}}^2f_h = 0.$ The boundary conditions on the perturbations can be found from Eqs.",
"(REF ) and (REF ).",
"From the inlet and outlet conditions (REF ) it follows that dissolution at the inlet is uniform (because ${\\hat{c}}=1$ ), $f_h(\\xi =0) = 0,$ and that far downstream the aperture is unperturbed, $f_h(\\xi \\rightarrow \\infty ) = 0.$ The boundary conditions on the flow (REF ) also impose conditions on $f_h$ through Eq.",
"(REF ).",
"The uniform pressure at the inlet leads to a condition on $q_\\xi $ , $f_q(\\xi =0) = \\left[\\partial _\\xi f_q\\right]_{\\xi =0}= 0,$ which, by means of (REF ), imposes a third-order boundary condition on $f_h$ , $\\left[\\partial _\\xi e^\\xi \\left\\lbrace \\left[{\\rm Da}_\\kappa {\\hat{\\omega }}+ {\\rm Pe}_\\kappa ^{-1}\\partial _\\xi e^{-\\xi } \\right] + \\left[\\partial _\\xi - {\\rm Pe}_\\kappa ^{-1} (\\partial _\\xi ^2 - {\\hat{u}}^2) \\right]\\left[{\\hat{\\omega }}+ \\frac{Ge^{-\\xi }}{1+G}\\right]\\right\\rbrace f_h \\right]_{\\xi =0} = 0,$ The outlet condition $f_q(\\xi \\rightarrow \\infty ) = 0,$ imposes a further restriction on $f_h$ , through Eq.",
"(REF ), namely that it must decay at least as fast as $e^{-\\xi }$ , $e^{\\xi } f_h(\\xi \\rightarrow \\infty ) = A.$ In most cases the constant $A$ must be zero in order for (REF ) to be satisfied, but in the convective limit ($H = 0$ ), the solution $f_h = Ae^{-\\xi }$ is an eigensolution of (REF ) with zero eigenvalue, and therefore satisfies the far-field boundary condition on $f_q$ .",
"Since the initial amplitude of the instability is arbitrary, the four boundary conditions impose an additional constraint which can be used to solve for the eigenvalue ${\\hat{\\omega }}({\\hat{u}})$ .",
"We have used a spectral method, which we summarize in Sec.",
", to find the dispersion relation numerically.",
"In certain limiting cases further analysis is feasible; we describe these on a case by case basis in Sec."
],
[
"Spectral method",
"The solution of equation (REF ), together with the boundary conditions (REF ), (REF ), and (REF ), was obtained using the pseudospectral, boundary-bordering method [23], [24].",
"For a given linear operator, ${\\cal H}$ , the differential equation ${\\cal H} f(\\xi ) = g(\\xi ), \\ \\ \\ \\ \\ \\ \\ 0 \\le \\xi \\le \\infty ,$ is represented as a linear system ${{\\mbox{${H}$}}} {{\\mbox{${f}$}}} = {{\\mbox{${g}$}}}$ where the elements of the vector ${{\\mbox{${f}$}}}$ are the coefficients of the expansion of $f(\\xi )$ in the basis functions $\\Psi _j(\\xi )$ , $f(\\xi ) = \\sum _{j=1}^N f_j \\Psi _{j-1}(\\xi ).$ Matrix elements of $\\cal H$ are calculated at $N-2$ collocation points, $\\xi _i$ , $H_{i+2,j} = [{\\cal H} \\Psi _{j-1}(\\xi )]_{\\xi =\\xi _i}$ and the corresponding elements of the right-hand-side vector are $g_{i+2} = g(\\xi _i).$ The first two rows of ${\\mbox{${H}$}}$ are used impose the boundary conditions at $\\xi =0$ .",
"If the boundary conditions are expressed in terms of the linear operators ${\\cal B}_{i^\\prime }$ , ${\\cal {B}}_{i^\\prime }(f) = \\alpha _{i^\\prime }, \\ \\ \\ \\ \\ \\ \\ \\ i^\\prime =1,2,$ then in the matrix representation $H_{i^\\prime ,j} = \\left[{\\cal {B}}_{i^\\prime }\\Psi _{j-1}(\\xi )\\right]_{\\xi =0}, \\ \\ \\ \\ \\ \\ \\ \\ g_{i^\\prime }=\\alpha _{i^\\prime },$ where $i^\\prime = 1,2$ .",
"The basis functions are rational Chebyshev functions in ${\\cal R}^+ = [0,\\infty ]$ , defined as $\\Psi _n(\\xi ) = T_n\\left(\\frac{\\xi -L}{\\xi +L}\\right),$ where $T_n(t)$ , with $n = 0, 1, 2, \\ldots $ , is a Chebyshev polynomial of the first kind, defined in the range $-1 \\le t < 1$ .",
"The convergence of the solution depends on a suitable choice of the mapping parameter, $L$ , which varies somewhat with wavelength.",
"For small numbers of basis functions ($N < 20$ ), we took $L= 1$ at short wavelengths (${\\hat{u}}> 1$ ) and $L=10$ at long wavelengths (${\\hat{u}}< 1$ ).",
"However, for larger numbers of basis functions ($N > 50$ ), a constant $L=10$ was suitable for the whole range of wavelengths, $0.01 < {\\hat{u}}< 10$ .",
"For a given $L$ and $N$ , the $N-2$ collocation points are [23], $\\xi _i=L \\cot ^2 \\left(\\frac{\\pi }{4} \\frac{2i-1}{(N-2)}\\right), \\ \\ \\ \\ i=1,\\dots ,N-2.$ The dispersion relation can be found by solving the linear system of equations represented by (REF )–(REF ), with boundary conditions $f(\\xi =0) = 0$ (REF ) and $\\partial _\\xi f(\\xi =0) = 1$ , which fixes the amplitude of the perturbation.",
"Then, we iteratively seek the largest value of ${\\hat{\\omega }}$ for which the boundary condition in (REF ) is satisfied and hence find the dispersion relation ${\\hat{\\omega }}(u)$ .",
"There is no need to separately impose the far-field regularity conditions, Eqs.",
"(REF ) and (REF ), since this is automatically incorporated by the basis functions [23].",
"We have cross-checked the spectral code with analytic solutions in a number of special cases (see Sec.",
"), and a Maple version of the spectral code is included in the Supplementary Material."
],
[
"Results",
"In general, the dispersion relation (REF ) must be solved numerically; for example, using the spectral method described in Sec. .",
"However, in the important limiting case of convection-dominated ($H\\rightarrow 0$ ), reaction-limited ($G\\rightarrow 0$ ) dissolution, it is possible to obtain a tractable analytic dispersion relation, as shown in Sec.",
"REF .",
"We can also obtain analytic solutions in other limiting cases, but the solutions are too lengthy to be reproduced in print, although we include Maple workbooks as Supplementary Material.",
"Analytic calculations from Maple [25] and Mathematica [26] were crosschecked with each other and with the spectral code (Sec. )",
"in many cases."
],
[
"Convection-dominated dissolution: $H \\rightarrow 0$ .",
"In convection-dominated flows ($H\\rightarrow 0$ ), the Damköhler number on the scale of the penetration length ${\\rm Da}_\\kappa = 1$ , and the corresponding Péclet number ${\\rm Pe}_\\kappa \\rightarrow \\infty $ .",
"The dispersion relation (REF ) then simplifies to $(\\partial _\\xi ^2 - {\\hat{u}}^2)e^\\xi \\left\\lbrace {\\hat{\\omega }}+ \\partial _\\xi \\left[{\\hat{\\omega }}+ \\frac{Ge^{-\\xi }}{1+G}\\right]\\right\\rbrace f_h + 3{\\hat{u}}^2f_h = 0.$ There is an analytic solution of Eq.",
"(REF ) in terms of a linear combination of three generalized hypergeometric functions $z^\\alpha (z-1)_3F_2(\\lbrace a_1,a_2,a_3\\rbrace ,\\lbrace b_1,b_2\\rbrace ; z)$ , where $a_k$ and $b_k$ are complicated algebraic functions of $G$ and ${\\hat{u}}$ , $z = - G{\\hat{\\omega }}^{-1}\\exp (-\\xi )/(1+G)$ , and $\\alpha $ is a simple function of ${\\hat{u}}$ .",
"As the solution is lengthy and not very informative we do not include it here, but a Maple notebook is included as Supplementary Material.",
"Figure: Growth rates of the inlet instability in the purely convective case (H=0)(H = 0).",
"The solid line corresponds to the reaction-limited case (G=0G = 0), whereas the dash-dotted curve corresponds to the diffusive limit (G=∞G=\\infty ) and the dashed curve is for mixed kinetics (G=1G=1).",
"The dimensionless growth rate ω ^=ωt d {\\hat{\\omega }}= \\omega t_d, Eq.",
"(), is plotted against the dimensionless wavevector, u ^=2π/κλ{\\hat{u}}= 2\\pi /\\kappa \\lambda .A much simpler equation is obtained in the reaction limit ($G \\rightarrow 0$ ) of (REF ) [8], $(\\partial _\\xi ^2-{\\hat{u}}^2) {\\hat{\\omega }}e^\\xi (1 + \\partial _\\xi ) f_h + 3 {\\hat{u}}^2 f_h =0.$ The general solution of (REF ) is $f_h(\\xi ) = Ae^{-\\xi }\\, _0F_2\\left(1+{\\hat{u}}, 1-{\\hat{u}}; 3{\\hat{\\omega }}^{-1}{\\hat{u}}^2e^{-\\xi }\\right) \\\\ + Be^{({\\hat{u}}-1)\\xi }\\, _0F_2\\left(1+{\\hat{u}}, 1-2{\\hat{u}}; 3{\\hat{\\omega }}^{-1}{\\hat{u}}^2e^{-\\xi }\\right) \\\\ + Ce^{-({\\hat{u}}+1)\\xi }\\, _0F_2\\left(1+{\\hat{u}}, 1+2{\\hat{u}}; 3{\\hat{\\omega }}^{-1}{\\hat{u}}^2e^{-\\xi }\\right),$ where $A$ , $B$ , and $C$ are constants and $_0F_2(p, q; z)$ is a generalized hypergeometric function.",
"The far field boundary condition (REF ) requires that $B = 0$ , while the condition $f_h(0) = 0$ (REF ) is then sufficient to determine the function $f_h(\\xi )$ to within an arbitrary constant, which is the initial amplitude of the perturbation.",
"Imposing the final boundary condition (REF ) gives a dispersion relation for ${\\hat{\\omega }}({\\hat{u}})$ , $\\left[{\\hat{\\omega }}^2 \\,_0\\tilde{F}_2\\left(1+{\\hat{u}},1+2{\\hat{u}};3{\\hat{\\omega }}^{-1}{\\hat{u}}^2\\right) + \\right.",
"\\\\ 3 (1+2{\\hat{u}}) {\\hat{\\omega }}\\,_0\\tilde{F}_2\\left(2+{\\hat{u}},2+2{\\hat{u}};3{\\hat{\\omega }}^{-1}{\\hat{u}}^2\\right) \\\\ \\left.",
"+ 9 {\\hat{u}}^2 \\, _0\\tilde{F}_2\\left(3+{\\hat{u}},3+2{\\hat{u}};3{\\hat{\\omega }}^{-1}{\\hat{u}}^2\\right) \\right] {_0\\tilde{F}_2}\\left(1+{\\hat{u}},1-{\\hat{u}};3{\\hat{\\omega }}^{-1}{\\hat{u}}^2\\right) = \\\\ 3\\left[{\\hat{\\omega }}\\, _0\\tilde{F}_2\\left(2+{\\hat{u}},2-{\\hat{u}};3{\\hat{\\omega }}^{-1}{\\hat{u}}^2\\right) + \\right.",
"\\\\ \\left.",
"3{\\hat{u}}^2 \\, _0\\tilde{F}_2\\left(3+{\\hat{u}},3-{\\hat{u}};3{\\hat{\\omega }}^{-1}{\\hat{u}}^2\\right)\\right]\\, _0\\tilde{F}_2\\left(1+{\\hat{u}},1+2{\\hat{u}};3{\\hat{\\omega }}^{-1}{\\hat{u}}^2\\right),$ where $_0{\\tilde{F}}_2(p,q;z) = {_0F_2}(p,q;z)/\\Gamma (p)\\Gamma (q)$ is a regularized hypergeometric function [27].",
"The maximum growth rate (largest positive root) at each ${\\hat{u}}$ from (REF ) corresponds to the solid line ($G = 0$ ) in Fig.",
"REF .",
"The positive growth rates show that the front is unstable across the whole spectrum of wavelengths, with a well-defined maximal growth rate, ${\\hat{\\omega }}_{max} = 0.79 t_d^{-1}$ , at a wavelength $\\lambda _{max} = 4.74 \\kappa ^{-1}$ .",
"An individual fracture will therefore develop a strongly heterogeneous permeability during dissolution, with an inherent length scale that depends on the kinetics and flow rate (via $\\kappa $ ), but not the initial topography.",
"There is no lower limit to the reaction rate for unstable dissolution if the scale of the fracture is sufficiently large.",
"Figure REF also shows the impact of reaction kinetics (controlled by the parameter $G$ ) on the dispersion relation.",
"For wider apertures (i.e.",
"$G\\gg 1$ ), diffusional transport of reactant across the aperture has a stabilizing effect on the growth of the instability.",
"The fastest-growing wavelength, $\\lambda _{max}$ , is pushed towards longer wavelengths and at sufficiently short wavelengths perturbations in the front are stable."
],
[
"Reaction-limited dissolution: $G \\rightarrow 0$ .",
"The dispersion relation (REF ) can also be solved analytically in the reaction limit, $G = 0$ ; the solution of the dispersion equation, $(\\partial _\\xi ^2 - {\\hat{u}}^2)e^\\xi \\left\\lbrace \\left[{\\rm Da}_\\kappa {\\hat{\\omega }}+ {\\rm Pe}_\\kappa ^{-1}\\partial _\\xi e^{-\\xi } \\right] + {\\hat{\\omega }}\\left[\\partial _\\xi - {\\rm Pe}_\\kappa ^{-1} (\\partial _\\xi ^2 - {\\hat{u}}^2) \\right]\\right\\rbrace f_h + 3{\\hat{u}}^2f_h = 0,$ is again a combination of hypergeometric functions $_3F_3(\\lbrace a_1,a_2,a_3\\rbrace ,\\lbrace b_1,b_2,b_3\\rbrace ; z) \\exp (g\\xi )$ , where $a_k$ , $b_k$ and $g$ are functions of ${\\hat{u}}$ and $H$ , and $z = -{\\hat{\\omega }}^{-1}\\exp (-\\xi )$ .",
"Again we have included a Maple notebook in the Supplementary Material.",
"In the diffusive limit ($H \\rightarrow \\infty $ ), ${\\rm Da}_\\kappa \\rightarrow {\\rm Pe}_\\kappa ^{-1} \\rightarrow \\sqrt{H}$ and the dispersion relation contains only a single length scale ($\\kappa ^{-1}$ ); $(\\partial _\\xi ^2 - {\\hat{u}}^2)e^\\xi \\left\\lbrace \\partial _\\xi e^{-\\xi } - {\\hat{\\omega }}(\\partial _\\xi ^2 - {\\hat{u}}^2 -1)\\right\\rbrace f_h = 0.$ It is possible to show analytically that the only root of the dispersion relation is $\\hat{\\omega }=0$ , which means that dissolution is neutrally stable in the diffusive limit ($H \\rightarrow \\infty $ ).",
"On the other hand, the numerical results in Fig.",
"REF imply that the dissolution is unstable for even an infinitesimal convective flux.",
"Figure: Growth rates of the inlet instability in the reaction-limited case (G=0)(G=0).",
"The solid line corresponds to the purely convective case (H=0H = 0).",
"The other lines show results for increasing HH: H=0.1H = 0.1, H=1H = 1, and H=10H = 10.",
"The dimensionless growth rate ωt d \\omega t_d is plotted against the dimensionless wavevector 2π/κλ2\\pi /\\kappa \\lambda ."
],
[
"Geophysical implications",
"Figure REF summarizes the most important results of this study.",
"Here we plot the (dimensionless) wavevector and growth rate of the dominant (most unstable) mode of the fracture instability.",
"The convective limit extends up to $H \\approx 0.01$ ; in this range both the dimensionless wavelength and growth rate are nearly constant.",
"Thus for convection-dominated infiltration, the wavelength and timescale are simply related to the underlying geophysical parameters: $\\lambda _{max} \\approx \\frac{2.4 q_0}{k_{eff}}, \\ \\ \\ \\ \\omega _{max} \\approx \\frac{1.6 k_{eff} \\gamma }{h_0}.",
"$ In order to put these results in a geological context, we consider typical values of the physical parameters characterizing dissolving fractures.",
"Fracture apertures are between $0.005\\, \\rm cm$ and $0.1\\, \\rm cm$ [28], [29], [20], and hydraulic gradients are of the order of $10^{-3}$ to $10^{-1}$ [31], [30].",
"This gives a range of characteristic flow velocities in undissolved fractures from $10^{-4} \\, \\rm cm\\, \\rm s^{-1}$ to $10 \\, \\rm cm\\, \\rm s^{-1}$ .",
"The corresponding Péclet numbers are $10^{-1} < {\\rm Pe}< 10^5$ , taking the solute diffusion coefficient $D=10^{-5} \\, \\rm cm^2\\, \\rm s^{-1}$ .",
"The reaction rates vary widely, depending on the mineral.",
"For example, relatively fast dissolving gypsum has a reaction rate $k$ of the order of $0.01 \\, \\rm cm\\, \\rm s^{-1}$ [32], whereas siliceous minerals have surface reaction rates of the order of $10^{-9} \\, \\rm cm\\, \\rm s^{-1}$ [33], [30].",
"The typical reaction rates for calcite are in the range $10^{-5} \\, \\rm cm\\, \\rm s^{-1} - 10^{-4} \\, \\rm cm\\, \\rm s^{-1}$ [31], [20].",
"Thus the limitations imposed by the diffusion of reactant across the fracture aperture vary widely, resulting in a broad range of possible $G$ values: from $G \\sim 10^{-7}$ in quartz, through $G \\approx 0.1$ for a typical calcite fracture, up to $G \\sim 1-10$ in gypsum.",
"Nevertheless, Fig.",
"REF shows that both the maximal growth rate and the position of the maximally unstable wavelength depend only weakly on G; ${\\hat{\\omega }}_{max}$ changes by 25% in the range $0 < G < \\infty $ , with a similar change in the corresponding wavelength.",
"However, the data in Fig.",
"REF is given in terms of dimensionless quantities and the absolute growth rates vary dramatically across different minerals.",
"For quartz, with $\\gamma =6 \\cdot 10^{-5}$ [30], the time unit $t_d \\sim 5000$ years, whereas the relevant timescale for calcite is a few months [34].",
"The same holds for the instability wavelengths, $\\lambda $ , which vary from centimeters (gypsum) to kilometers (quartz).",
"It is important to realize that the initial instability wavelength will in general be different from the spacing between protrusions in a mature formation.",
"This is due to a coarsening of the pattern that is characteristic of this kind of dynamics [35]; the fingers compete with each other for the flow such that the longer ones grow more rapidly but the shorter ones become stagnant.",
"As a result, the characteristic length between active (growing) protrusions increases with time.",
"In geophysical systems, diffusion has only a small effect on the instability.",
"Although $H$ can vary from $\\sim 10^{-15}$ (for wide fractures in siliceous formations) up to about 1 for narrow fractures in gypsum, fracture dissolution is typically convection dominated ($H \\ll 1$ ).",
"The residual diffusion leads to a slight shift of the peak growth rate towards longer wavelength, as observed in Fig.",
"REF , but the wavelength and growth rate depend primarily on ${\\rm Da}_{eff}$ (REF ), via the penetration length $l_p$ and the dissolution time scale $t_d$ , with just small corrections from $H$ .",
"These considerations refer to fracture dissolution in a natural geological setting.",
"For carbonate acidization (e.g.",
"with hydrochloric acid) the corresponding reaction rates are significantly higher than for dissolution with aqueous ${\\rm CO}_2$ ; in acidization $k \\sim 10^{-1} \\, \\rm cm\\, \\rm s^{-1}$ [36], so that $G$ can be larger than 100 (for $h_0 \\approx 0.1 \\, \\rm cm$ ), which means that the dissolution rate is strongly limited by diffusion across the aperture.",
"In the transport limit ($G \\gg 1$ ), $H = {\\rm Sh}/{\\rm Pe}^2$ is small under the typical flow rates used in acidization."
],
[
"Reaction order",
"Experiments on the dissolution of limestone suggest that, near saturation, dissolution follows a nonlinear rate law, c.f.",
"Eq.",
"(REF ): $R(c_{ca}) = - k c_{sat} \\left(1-\\frac{c_{ca}}{c_{sat}}\\right)^n, \\ \\ \\ \\ \\ n>1,$ where $c_{sat}$ is the saturation concentration of calcium ions.",
"If we define a relative undersaturation ${\\hat{c}}= (c_{sat}-c_{ca})/(c_{sat}-c_{in})$ , where $c_{in}$ is the concentration of calcium ions at the inlet, then the transport equation, from (REF ), is $q_x \\partial _x {\\hat{c}}+ q_y \\partial _y {\\hat{c}}= - 2 k\\left(1-\\frac{c_{in}}{c_{sat}}\\right)^{n-1} {\\hat{c}}^n.$ For simplicity, we only consider reaction-limited, convection-dominated dissolution.",
"The equation describing aperture opening, analogous to (REF ), is $\\partial _t h = 2k \\gamma \\left(1-\\frac{c_{in}}{c_{sat}}\\right)^{n-1} {\\hat{c}}^n,$ where $\\gamma = (c_{sat}-c_{in})/\\nu c_{sol}$ .",
"The remaining equations, continuity and compatibility, are given by Eqs.",
"() and ().",
"Assuming the aperture in the base state is uniform, $h_b(x) = h_0$ , the base concentration profile is ${\\hat{c}}_b(x) = \\left(1+\\frac{2 k (n-1) (1-c_{in}/c_{sat})^{n-1} x}{q_0}\\right)^{\\tfrac{1}{1-n}} = \\left[1+ (1-n^{-1}) \\kappa x\\right]^{\\tfrac{1}{1-n}}$ where $\\kappa = 2kn(1-c_{in}/c_{sat})^{n-1}/q_0$ .",
"In the limit $n \\rightarrow 1$ , Eq.",
"(REF ) approaches the exponential base profile for linear reaction kinetics (REF ) and the expression for $\\kappa $ reduces to Eq.",
"(REF ).",
"A dispersion equation for the growth rate can be obtained for non-linear kinetics by following the procedure in Sec.",
", starting with the analogues of Eqs.",
"(REF )–(REF ): $\\delta q_x \\partial _x {\\hat{c}}_b + q_b \\partial _x \\delta {\\hat{c}}= - \\frac{1}{\\gamma } \\partial _t \\delta h,$ $\\partial _t \\delta h = 2 k \\gamma \\left(1-\\frac{c_{in}}{c_{sat}}\\right)^{n-1} n {\\hat{c}}_b^{n-1} \\delta {\\hat{c}}.$ The continuity and compatibility relations are the same as Eqs.",
"(REF )–(REF ).",
"Introducing dimensionless variables: $\\xi = \\kappa x, \\ \\ \\ \\ \\ \\eta = \\kappa y, \\ \\ \\ \\ \\ \\tau = \\frac{2k\\gamma t (1-c_{in}/c_{sat})^{n-1} }{h_0},$ and scaling $\\delta h$ and ${\\mbox{${q}$}}$ as in (REF ), we obtain the following equations for $f_c$ , $f_h$ , and $f_q$ , defined in Eqs.",
"(REF )–(): $& f_q \\partial _\\xi \\hat{c}_b + \\partial _\\xi f_c = - \\frac{{\\hat{\\omega }}}{n} f_h, \\\\& \\frac{{\\hat{\\omega }}}{n} f_h = \\hat{c}_b^{n-1} f_c, \\\\& (\\partial _\\xi ^2 - {\\hat{u}}^2)f_q = -3{\\hat{u}}^2f_h.$ The inlet saturation, $c_{in}$ , has been absorbed into the length and time scales (REF ).",
"The base concentration (${\\hat{c}}_b$ ) can be eliminated from the equations for transport (REF ) and erosion () by using (REF ): $& f_q = \\left[1+ (1-n^{-1}) \\xi \\right]^{\\tfrac{n}{n-1}} \\left( {\\hat{\\omega }}f_h + n \\partial _\\xi f_c \\right) \\\\& {\\hat{\\omega }}f_h = n\\left[1+(1-n^{-1}) \\xi \\right]^{-1} f_c.$ Combining these equations with () we get a dispersion equation for arbitrary kinetic order, ${\\hat{\\omega }}(\\partial _\\xi ^2-{\\hat{u}}^2) \\left[1+ (1-n^{-1}) \\xi \\right]^{\\tfrac{n}{n-1}} \\left[1 + \\partial _\\xi (1+(1-n^{-1})\\xi )\\right] f_h + 3 {\\hat{u}}^2 f_h =0,$ which is well behaved in the limits $n \\rightarrow 1$ and $n \\rightarrow \\infty $ .",
"Figure: The impact of kinetic order on the growth rates.",
"The growth rate of the instability is shown for various powers of nn, including the linear rate law, n=1n=1, and the limit of high reaction order, n→∞n \\rightarrow \\infty .The impact of kinetic order is illustrated in Fig.",
"REF , which shows that even strongly non-linear reaction kinetics ($n \\rightarrow \\infty $ ) do not suppress the instability.",
"The dimensionless growth rate depends only weakly on reaction order, reflecting our choice of scaling for the dimensionless length and time.",
"Thus, as a first approximation we can take the peak growth rate as ${\\hat{\\omega }}_{max} \\sim 1$ and the corresponding wavevector ${\\hat{u}}_{max} \\sim 1$ , independent of reaction order.",
"Then, in absolute terms, the wavelength corresponding to maximum growth is roughly proportional to $n^{-1}$ ; $\\lambda _{max}^{(n)} \\approx \\lambda _{max}/n (1-c_{in}/c_{sat})^{n-1}$ , where $\\lambda _{max} \\sim 2\\pi q_0/2k$ is the peak wavelength for linear kinetics.",
"This is slightly counterintuitive since increasing reaction order tends to increase the penetration of reactant into the fracture.",
"Nevertheless its effect on the instability is to shorten the wavelength of the most unstable mode.",
"However the wavelength is also strongly dependent on $c_{in}$ , and a partially saturated solution at the inlet increases the wavelength of the most unstable mode.",
"The inlet solution to the fracture must be nearly saturated ($c_{in} \\rightarrow c_{sat}$ ) for non-linear kinetics to apply [18], so the wavelength in such cases is almost entirely dependent on the extent of the (small) undersaturation.",
"The corresponding growth rate of the instability $\\omega _{max}^{(n)} = \\omega _{max} (1-c_{in}/c_{sat})^{n-1}$ is sharply limited by the degree of undersaturation."
],
[
"Finite length fractures",
"The previous analysis corresponds to a semi-infinite system, $x \\ge 0$ , which is the relevant limit for geophysical systems where the length of the system, $L$ , is usually many orders of magnitude larger than the penetration length $\\kappa ^{-1}$ .",
"However, in laboratory experiments as well as in petroleum reservoir stimulation, the relevant length scales are much smaller and finite-size effects may be important.",
"In this case, the far-field boundary condition $q_x(x\\rightarrow \\infty ,y,t)=q_0$ must be replaced by a constant pressure condition at the outlet; then $q_y(x=L,y,t)=0$ or, in terms of perturbations, $\\delta q_y (x=L) = 0.$ Figure: Growth rate in the long-wavelength (u ^→0{\\hat{u}}\\rightarrow 0) limit, ω 0 \\omega _0, in the convection-reaction (G=H=0)(G = H = 0) limit.Figure REF shows the effect of a finite length aperture in reaction limited, convection-dominated dissolution ($H=G=0$ ).",
"Now all three solutions from Eq.",
"(REF ) are needed; Eqs.",
"(REF ) and (REF ) fix the perturbation to within an arbitrary amplitude, while Eq.",
"(REF ) enforces the eigenvalue condition.",
"The additional length scale leads to a richer spectrum of possibilities; in particular, the longest wavelengths are now less stable than in unbounded ($L \\rightarrow \\infty $ ) fractures.",
"The shape of the dispersion curve changes considerably as the length of the system is reduced and for short fractures, ($\\kappa L < 2$ ), the growth rate is maximum at zero wavevector.",
"As the length of the fracture increases, the wavelength of the most unstable mode shifts to larger ${\\hat{u}}$ and the longest wavelengths are only weakly unstable; as $L \\rightarrow \\infty $ the growth rate at zero wavevector vanishes altogether.",
"In fact, the growth rate at ${\\hat{u}}=0$ has a particularly simple analytical form ${\\hat{\\omega }}(u=0) = \\frac{3 \\left[1-(1+\\kappa L)e^{-\\kappa L}\\right]}{\\kappa L},$ which is shown in Fig.",
"REF .",
"Both for very small and very large lengths the long-wavelength growth rate is relatively small, with a maximum at $\\kappa L \\approx 1.8$ .",
"An analysis of Fig.",
"REF , together with Fig.",
"REF , offers some insight into the typical dispersion curve for a fracture dissolution instability, which exhibits a strong wavelength selection with a well-defined maximum in the growth rate for $\\lambda _{max} \\approx \\kappa ^{-1}$ .",
"The results presented in this section show that stabilization of the growth of long wavelength instabilities is connected with the far-field boundary condition (REF ), which imposes a uniform flow at large distances from the inlet.",
"However, in a finite system, the constant pressure condition at $x=L$ does not require $q_x$ to be uniform, and hence does not lead to a stabilization of long-wavelength modes, as shown in Figs.",
"REF and REF .",
"On the other hand, Fig.",
"REF shows that the shape of the short-wavelength spectrum is controlled by reaction kinetics.",
"In particular, transport-limited kinetics decreases the short-wavelength growth rates, since in this regime dissolution slows down as the fracture opens (REF )."
],
[
"Conclusions",
"In this paper, we have analyzed the stability of a one-dimensional reaction front in dissolving fractures.",
"Strikingly, the dissolution front turns out to be unstable over a wide range of wavelengths, suggesting that fracture dissolution is an inherently two-dimensional process.",
"The maximal growth rate corresponds to wavelengths of the order of the penetration length $\\kappa ^{-1}$ and this result turns out to be remarkably insensitive to the details of the reaction and transport mechanisms in the fracture: the maximum is shifted towards longer wavelengths when strong diffusion is present or for strongly nonlinear reaction kinetics, but the shift is relatively small and $\\kappa \\lambda _{max}$ remains within the same order of magnitude.",
"The only case where there is a qualitative change in the dispersion curve is a finite-length system.",
"For relatively short fractures, $\\kappa L \\le 3$ , the maximum growth rate occurs at zero wavevector and long-wavelength modes remain unstable.",
"In summary, the reactive front instability has been shown to be a generic phenomenon in the dissolution of fractured rock.",
"Hence the predictions of fracture breakthrough times, crucial for speleogenesis and for the assessment of subsidence hazards, cannot be based on one-dimensional models.",
"Instead, a two-dimensional model is necessary to take into account the highly localized dissolution front.",
"Numerical [1], [7], [8] and theoretical [35] work has suggested that the dissolutional instability leads to a strong focusing of the fluid flow into a few active channels, which advance in the fracture while competing with each other for the available reactant.",
"However, a quantitative characterization of this non-linear process, which is essential for the prediction of fracture breakthrough times, remains elusive.",
"This work was supported by the US Department of Energy, Chemical Sciences, Geosciences and Biosciences Division, Office of Basic Energy Sciences (DE-FG02-98ER14853).",
"Computations in this paper were performed by using Maple${13}^\\mathrm {TM}$ and Mathematica${7.0}^\\mathrm {TM}$ ."
],
[
"Convection-diffusion equation for reactant transport",
"In this Appendix we will investigate the validity of the two-dimensional steady-state transport equation (REF ) in various parameter ranges.",
"In the spirit of the Reynolds approximation, we will then assume that the global solution for parallel plates can be applied locally, if the fracture aperture field varies sufficiently slowly, $\\left| \\nabla h \\right| \\sim {\\cal O}(1)$ .",
"Taking the flow to be along the $x$ -axis and the normal to the fracture surfaces along the $z$ -axis, the convection-diffusion equation for the three-dimensional concentration field $c_{3d}(x,z,t)$ can be written as $\\partial _tc_{3d} + v_x \\partial _xc_{3d} = D \\left( \\partial _x^2 c_{3d} + \\partial _z^2 c_{3d} \\right),$ where $v_x = 6 v_a (z/h - {z^2}/{h^2})$ and $v_a = q_x/h$ is the aperture-averaged fluid velocity.",
"In addition we have boundary conditions on the fracture surfaces $D \\partial _zc_{3d}|_{z=0} = kc_{3d}, ~ D \\partial _zc_{3d}|_{z=h} = -kc_{3d},$ and at the inlet, $c_{3d}|_{x=0} = c_{in},$ where $c_{in}$ is the inlet concentration.",
"A direct integration of (REF ) over the $z$ coordinate gives a two-dimensional averaged convection-diffusion-reaction equation involving three different concentrations, $\\partial _tc_a + v_a \\partial _xc = D \\partial _x^2 c_a -\\frac{2k}{h} c_w;$ $c_a(x,t) = h^{-1}\\int _0^h c_{3d}(x,z,t) dz$ , is the aperture-averaged concentration, $c$ is the cup-mixing concentration (REF ), and $c_w(x,t) = c_{3d}(x,0,t) = c_{3d}(x,h,t)$ is the reactant concentration at the fracture surfaces.",
"Following standard procedures for averaging the convection-diffusion equation, we will solve Eqs.",
"(REF )–(REF ) to find relations between these average concentration fields in different parameter ranges.",
"In particular we will show that Eq.",
"(REF ) is correct in the important limits of convection-dominated transport (Sec.",
"REF ) and reaction-limited (Sec.",
"REF ) kinetics."
],
[
"Scaling and steady state",
"The steady state approximation in (REF ) can be justified by the time-scale separation between the transport of reactants and the consequent change in fracture aperture.",
"The dissolution time scale is characterized by $\\tilde{t}_d = h/2k\\gamma = t_d/(1+G)$ (REF ), where the acid capacity number $\\gamma = c_{in}/\\nu c_{sol}$ is usually small, because of the high molar concentration of the solid phase.",
"For example calcite contains roughly 25 moles per liter, whereas even a strong acid is rarely used in more than 1 molar concentrations; in the natural dissolution of calcite by atmospheric $\\rm CO_2$ , $\\gamma \\sim 10^{-4}$ .",
"To see how a small $\\gamma $ leads to the steady-state limit we scale the time by $\\tilde{t}_d$ in addition to the usual scaling of lengths: $\\xi = \\frac{x}{l}, ~ \\zeta = \\frac{z}{h}, ~ \\tau = \\frac{t}{\\tilde{t}_d}.$ The axial distance is scaled by the characteristic length $l = v_a h/2k$ , and the transverse distance is scaled by $h$ .",
"In addition the fluid velocity is scaled by $v_a$ and the concentration by $c_{in}$ : ${\\hat{v}}_\\zeta = \\frac{v_x}{v_a} = 6 \\zeta - 6 \\zeta ^2, ~ {\\hat{c}}_{3d} = \\frac{c_{3d}}{c_{in}}.$ The scaled convection-diffusion equation, $\\gamma \\partial _\\tau {\\hat{c}}_{3d} + {\\hat{v}}_\\xi \\partial _\\xi {\\hat{c}}_{3d} = {\\tilde{H}}\\partial _\\xi ^2 {\\hat{c}}_{3d} + {\\tilde{G}}^{-1}\\partial _\\zeta ^2 {\\hat{c}}_{3d},$ is then be characterized by $\\gamma $ and two new dimensionless groups: ${\\tilde{G}}= 2kh/D$ and ${\\tilde{H}}= 2kD/v_a^2 h$ .",
"${\\tilde{G}}$ and ${\\tilde{H}}$ are related to the corresponding parameters defined in the main body of the paper by ${\\tilde{G}}= G{\\rm Sh}$ (REF ), and ${\\tilde{H}}= H(1+G)$ (REF ).",
"In this appendix we consider the transverse ($z$ ) direction explicitly and so the Sherwood number does not appear in the defining equations; the ratio of diffusive and reactive fluxes is then characterized by ${\\tilde{G}}$ rather than $G$ .",
"Since the reactive flux appears in the boundary conditions rather than the underlying equations, the ratio of diffusive and convective fluxes is more naturally defined by ${\\tilde{H}}$ rather than $H$ .",
"The steady-state convection-diffusion equation ${\\hat{v}}_\\xi \\partial _\\xi {\\hat{c}}_{3d} = {\\tilde{H}}\\partial _\\xi ^2 {\\hat{c}}_{3d} + {\\tilde{G}}^{-1}\\partial _\\zeta ^2 {\\hat{c}}_{3d}$ is reached in the limit $\\gamma \\rightarrow 0$ , and is valid under most circumstances arising in fracture dissolution.",
"The boundary conditions in the dimensionless variables are $\\partial _\\zeta {\\hat{c}}_{3d} = \\pm \\frac{{\\tilde{G}}}{2} {\\hat{c}}_{3d}, ~~\\zeta = 0,1;$ and the average equation for steady-state reactant transport is $\\partial _\\xi {\\hat{c}}= {\\tilde{H}}\\partial _\\xi ^2 {\\hat{c}}_a -{\\hat{c}}_w.$"
],
[
"Convective limit: ${\\tilde{H}}=0$ .",
"Fracture dissolution is usually characterized by small ${\\tilde{H}}$ , corresponding to the convective limit ${\\tilde{H}}\\rightarrow 0$ (Sec.",
"REF ).",
"The time-independent convection-diffusion equation is then ${\\hat{v}}_\\xi \\partial _\\xi {\\hat{c}}_{3d} = {\\tilde{G}}^{-1} \\partial _\\zeta ^2 {\\hat{c}}_{3d},$ which can be solved by separation of variables, ${\\hat{c}}_{3d} = f(\\xi )g(\\zeta )$ [15].",
"The decay in the axial direction is a sum of exponentials, $\\exp (-\\lambda _n \\xi )$ , where $\\lambda _n$ are related to the positive eigenvalues of the equation $\\partial _\\zeta ^2 g + 16 r^2(\\zeta - \\zeta ^2)g = 0,$ with $r_n = \\sqrt{3{\\tilde{G}}\\lambda _n/8}$ .",
"This equation has a single solution that satisfies the symmetry condition $g(0) = g(1)$ , $g(\\zeta ) = \\,_1F_1\\left(\\frac{1-r}{4},\\frac{1}{2};r(2\\zeta -1)^2\\right)e^{-2r\\zeta (\\zeta -1)}.$ Applying the boundary conditions from (REF ) leads to the eigenvalue equation for $r({\\tilde{G}})$ , $r\\left(r-1\\right)\\,_1F_1\\left(\\frac{5-r}{4},\\frac{3}{2};r\\right) + \\left(r-\\frac{{\\tilde{G}}}{4}\\right)\\,_1F_1\\left(\\frac{1-r}{4},\\frac{1}{2};r\\right) = 0.$ The average equation for the concentration, $\\partial _\\xi {\\hat{c}}= -{\\hat{c}}_w,$ implies that for a single mode $\\lambda {\\hat{c}}= {\\hat{c}}_w$ (the same result follows from integrating Eq.",
"(REF ) over $\\zeta $ ).",
"Using the Sherwood number to connect ${\\hat{c}}_w$ and ${\\hat{c}}$ (REF ), ${\\hat{c}}_w = \\dfrac{{\\hat{c}}}{1+{\\tilde{G}}/{\\rm Sh}},$ we can relate the eigenvalue $\\lambda = 8r^2/3{\\tilde{G}}$ to ${\\rm Sh}$ ${\\rm Sh}= \\dfrac{\\lambda {\\tilde{G}}}{1-\\lambda }.$ Thus the steady-state convection-reaction equation is simply $\\partial _\\xi {\\hat{c}}= - \\dfrac{{\\hat{c}}}{1+{\\tilde{G}}/{\\rm Sh}},$ where ${\\rm Sh}({\\tilde{G}})$ is determined from the smallest root of (REF ).",
"For reaction-limited kinetics $r \\rightarrow 0$ , and the hypergeometric functions in Eq.",
"(REF ) can be expanded around $r = 0$ ; solving for ${\\tilde{G}}$ we obtain a quadratic equation for $\\lambda $ , ${\\tilde{G}}= \\dfrac{8}{3}r^2 + \\dfrac{272}{315}r^4 + {\\cal O}(r^6) = \\lambda {\\tilde{G}}+ \\dfrac{17}{140}\\lambda ^2 {\\tilde{G}}^2,$ with a solution $\\lambda = 1 - 17{\\tilde{G}}/140 + {\\cal O}({\\tilde{G}}^2)$ .",
"The concentration is nearly uniform across the aperture and decays axially as a single exponential $e^{-\\lambda \\xi }$ .",
"From Eq.",
"(REF ) we find the Sherwood number for reaction-limited kinetics ${\\rm Sh}^0 = 140/17 \\approx 8.24$ .",
"In the transport limit the concentration at the walls vanishes (Graetz problem) and the eigenvalues $\\lambda _n = 8 r_n^2/3{\\tilde{G}}$ can be found from the roots of the equation $\\,_1F_1\\left(\\frac{1-r}{4},\\frac{1}{2};r\\right) = 0.$ The transport-limited Sherwood number, ${\\rm Sh}^\\infty \\approx 7.541$ , follows from the smallest eigenvalue $r_0 \\approx 1.6816$ .",
"In the numerical work we will ignore the weak dependence of Sherwood number on ${\\tilde{G}}$ and take ${\\rm Sh}= 8$ throughout."
],
[
"Reaction-limit: ${\\tilde{G}}\\rightarrow 0$ .",
"Away from the convective limit, the diffusive flux prevents a solution of the transport equation (REF ) by separation of variables.",
"However, when the reaction rate is small, such that ${\\tilde{G}}\\ll 1$ , the deviation in concentration from the average concentration, $c_{3d} - c_a$ , can be expanded in powers of ${\\tilde{G}}$ [37], [38], ${\\hat{c}}_{3d} - {\\hat{c}}_a = {\\tilde{G}}c^{(1)} + {\\tilde{G}}^2 c^{(2)} + \\ldots ;$ it follows that $\\int _0^1 c^{(i)} d\\zeta = 0.$ From Eq.",
"(REF ), the zeroth order convection-diffusion equation is $(6\\zeta - 6 \\zeta ^2) \\partial _\\xi {\\hat{c}}_a = {\\tilde{H}}\\partial _\\xi ^2 {\\hat{c}}_a + \\partial _\\zeta ^2 c^{(1)}.$ Integrating Eq.",
"(REF ) across the aperture and using the boundary condition (REF ) $\\partial _\\zeta {\\hat{c}}^{(1)} = \\pm c_a/2$ , we obtain the average equation $\\partial _\\xi {\\hat{c}}_a = {\\tilde{H}}\\partial _\\xi ^2 {\\hat{c}}_a - {\\hat{c}}_a,$ which is the reaction limit of (REF ).",
"In this limit the concentration profile is uniform across the aperture and all three concentrations, $c$ , $c_a$ , and $c_w$ are equal.",
"Equation (REF ) can be subtracted from (REF ) to eliminate the diffusion term, $(6\\zeta - 6 \\zeta ^2 - 1) \\partial _\\xi {\\hat{c}}_a - c_a = \\partial _\\zeta ^2 c^{(1)}.$ Solving for $c^{(1)}$ , $c^{(1)} = \\partial _\\xi {\\hat{c}}_a \\left(\\zeta ^3 - \\frac{1}{2} \\zeta ^4 - \\frac{1}{2} \\zeta ^2 + \\frac{1}{60} \\right) - {\\hat{c}}_a \\left(\\frac{\\zeta ^2}{2} - \\frac{\\zeta }{2} + \\frac{1}{12} \\right);$ the linear term in $\\zeta $ is introduced to satisfy the boundary conditions in (REF ) and the constant term is to enforce the condition in (REF ).",
"Finally, we use Eq.",
"(REF ) to relate ${\\hat{c}}$ and ${\\hat{c}}_w$ to ${\\hat{c}}_a$ : ${\\hat{c}}&=& \\int _0^1 (6 \\zeta - 6 \\zeta ^2) ({\\hat{c}}_a + {\\tilde{G}}c^{(1)}) d\\zeta = \\left(1 + \\frac{{\\tilde{G}}}{60}\\right){\\hat{c}}_a - \\frac{{\\tilde{G}}}{210}\\partial _\\xi {\\hat{c}}_a, \\\\{\\hat{c}}_w &=& {\\hat{c}}_a + {\\tilde{G}}c^{(1)}(\\zeta =0) = \\left(1 - \\frac{{\\tilde{G}}}{12}\\right){\\hat{c}}_a + \\frac{{\\tilde{G}}}{60}\\partial _\\xi {\\hat{c}}_a.",
"$ These are the equivalents of the results in [38] (67c & d), but for flat plates instead of tubes.",
"Using Eqs.",
"(REF ) and () to eliminate $c_a$ and $c_w$ from the average equation (REF ), the transport equation becomes $\\partial _\\xi {\\hat{c}}= {\\tilde{H}}\\left(1-\\frac{4{\\tilde{G}}}{105}\\right) \\partial _\\xi ^2 {\\hat{c}}+ \\frac{{\\tilde{H}}{\\tilde{G}}}{210} \\partial _\\xi ^3 {\\hat{c}}- \\left(1-\\frac{17 {\\tilde{G}}}{140}\\right){\\hat{c}}.$ The third-order term in Eq.",
"(REF ), $\\frac{{\\tilde{H}}{\\tilde{G}}}{210} \\partial _\\xi ^3 {\\hat{c}}= \\frac{v_a h}{2k}\\frac{h^2}{210} \\partial _x^3 {\\hat{c}},$ is small compared with the convective term, $\\partial _\\xi {\\hat{c}}= \\frac{v_a h}{2k}\\partial _x {\\hat{c}},$ on all scales larger than the aperture $h$ .",
"Since $h$ is small on scales of interest in fracture dissolution we can safely ignore this term.",
"Similarly, the diffusive term $(4{\\tilde{H}}{\\tilde{G}}/105) \\partial _\\xi ^2 {\\hat{c}}$ is small compared to ${\\hat{c}}$ .",
"Dropping these terms leaves the renomalization of the reaction term as the leading-order correction for finite ${\\tilde{G}}$ (in the steady-state limit), $\\partial _\\xi {\\hat{c}}= {\\tilde{H}}\\partial _\\xi ^2 {\\hat{c}}- \\frac{{\\hat{c}}}{1+{\\tilde{G}}/{\\rm Sh}^0}.$ The average equation for the cup-mixing concentration has no Taylor dispersion term, but only the contribution from molecular diffusion.",
"This is true both in the convective limit (arbitrary ${\\tilde{G}}$ ) and the reaction limit (arbitrary ${\\tilde{H}}$ )."
],
[
"Summary",
"In this appendix we have examined the structure of the depth-averaged convection-diffusion equation across a range of Damköhler and Péclet numbers.",
"The dimensionless parameter $H = {\\rm Da}_{eff}/{\\rm Pe}$ is usually small in fracture dissolution, which implies a convection-dominated process.",
"In such cases the steady-state convection-reaction equation (REF ) follows (see Sec.",
"REF ), with only a weak dependence of the Sherwood number on reaction rate and entrance length.",
"When diffusion plays a significant role, the structure of the average equations is more complex, and it is not possible to rigorously treat transport in the case of significant transverse and axial diffusion (${\\tilde{G}}\\gg 1$ , ${\\tilde{H}}\\gg 1$ ) without considering more than one average concentration [37], [38].",
"Nevertheless, in Sec.",
"REF we showed that in the reaction limit ($G \\ll 1$ ) the structure of Eq.",
"(REF ) is preserved (REF )."
],
[
"Scale-dependent Péclet and Damköhler numbers",
"The one-dimensional transport equation (REF ) can be non-dimensionalized by the penetration length $\\kappa ^{-1}$ , $q_0 \\kappa \\partial _\\xi c - D h_0 \\kappa ^2 \\partial _\\xi ^2 c = - \\frac{2 kc}{1 + G},$ where $\\xi = \\kappa x$ .",
"Dividing Eq.",
"(REF ) by $q_0 \\kappa $ suggests two new dimensionless constants: ${\\rm Pe}_\\kappa = \\frac{q_0}{D \\kappa h_0} = \\frac{{\\rm Pe}}{\\kappa h_0}, ~~ {\\rm Da}_\\kappa = \\frac{2k}{q_0 \\kappa (1+G)} = \\frac{{\\rm Da}_{eff}}{\\kappa h_0}.$ ${\\rm Pe}_\\kappa $ is the ratio of convective to diffusive fluxes on the length scale $\\kappa ^{-1}$ , while ${\\rm Da}_\\kappa $ is the ratio of convective to reactive fluxes on the same scale; ${\\rm Da}_\\kappa $ is based on the effective reaction rate $k_{eff}$ (REF ).",
"The transport equation on the scale of the penetration length $\\kappa ^{-1}$ is then $\\partial _\\xi c - {\\rm Pe}_\\kappa ^{-1} \\partial _\\xi ^2 c = - {\\rm Da}_\\kappa c.$ The parameter $H$ retains the same meaning with the new definitions of Péclet and Damköhler number, $H = \\frac{{\\rm Da}_{eff}}{{\\rm Pe}} = \\frac{{\\rm Da}_\\kappa }{{\\rm Pe}_\\kappa },$ and the two new parameters can be written solely in terms of $H$ : ${\\rm Pe}_\\kappa = \\frac{2}{\\sqrt{1+4H}-1}, \\ \\ \\ \\ {\\rm Da}_\\kappa = \\frac{2H}{\\sqrt{1+4H}-1}.$ Although ${\\rm Pe}_\\kappa $ and ${\\rm Da}_\\kappa $ are not independent, ${\\rm Da}_\\kappa = 1 + {\\rm Pe}_\\kappa ^{-1}$ , it is a notational convenience to treat them so; however the results are discussed in terms of the independent parameters $G$ and $H$ .",
"On the relevant length scale for fracture dissolution, $\\kappa ^{-1}$ , the ratio of convective and diffusive fluxes is characterized by ${\\rm Pe}_\\kappa $ .",
"Nevertheless we prefer to characterize the dissolution in terms of $G$ and $H$ rather than $G$ and ${\\rm Pe}_\\kappa $ , since both ${\\rm Pe}_\\kappa $ and ${\\rm Da}_\\kappa $ have simple expressions in terms of $H$ .",
"In the convective limit (the most important for fracture dissolution) $H \\rightarrow {\\rm Pe}_\\kappa ^{-1}$ , while in the diffusive limit $H \\rightarrow {\\rm Pe}_\\kappa ^{-2}$ .",
"Thus the convective limit implies ${\\rm Pe}_\\kappa \\rightarrow \\infty $ and $H \\rightarrow 0$ , while the diffusive limit is the opposite, but the mapping is not a simple inverse relation."
],
[
"Derivation of the compatibility relation",
"Throughout the paper we will frequently make use of the compatibility relation (), which can be derived by noting that, from (REF ), $\\partial _y q_x = - \\frac{1}{12 \\mu } h^3 \\partial _{xy} p - 3 \\frac{1}{12 \\mu } h^2 \\partial _y h \\partial _x p =- \\frac{1}{12 \\mu } h^3 \\partial _{xy} p + \\frac{3}{h} q_x \\partial _y h.$ Similarly $\\partial _x q_y = - \\frac{1}{12 \\mu } h^3 \\partial _{xy} p - 3 \\frac{1}{12 \\mu } h^2 \\partial _x h \\partial _y p =- \\frac{1}{12 \\mu } h^3 \\partial _{xy} p + \\frac{3}{h} q_y \\partial _x h.$ Subtracting (REF ) from (REF ) leads to the compatibility relation $\\partial _y q_x - \\frac{3}{h} q_x \\partial _y h = \\partial _x q_y - \\frac{3}{h} q_y\\partial _x h.$"
]
] | 1204.0917 |
[
[
"Thermodynamics in f(R,T) Theory of Gravity"
],
[
"Abstract A non-equilibrium picture of thermodynamics is discussed at the apparent horizon of FRW universe in $f(R,T)$ gravity, where $R$ is the Ricci scalar and $T$ is the trace of the energy-momentum tensor.",
"We take two forms of the energy-momentum tensor of dark components and demonstrate that equilibrium description of thermodynamics is not achievable in both cases.",
"We check the validity of the first and second law of thermodynamics in this scenario.",
"It is shown that the Friedmann equations can be expressed in the form of first law of thermodynamics $T_hdS'_h+T_hd_{\\jmath}S'=-dE'+W'dV$, where $d_{\\jmath}S'$ is the entropy production term.",
"Finally, we conclude that the second law of thermodynamics holds both in phantom and non-phantom phases."
],
[
"Introduction",
"Cosmic observations from anisotropy of the Cosmic Microwave Background (CMB) [1], supernova type Ia (SNeIa) [2], large scale structure [3], baryon acoustic oscillations [4] and weak lensing [5] indicate that expansion of the universe is speeding up rather than decelerating.",
"The present accelerated expansion is driven by gravitationally repulsive dominant energy component known as dark energy (DE).",
"There are two representative directions to address the issue of cosmic acceleration.",
"One is to introduce the \"exotic energy component\" in the context of General Relativity (GR).",
"Several candidates have been proposed [6]-[10] in this perspective to explore the nature of DE.",
"The other direction is to modify the Einstein Lagrangian i.e., modified gravity theory such as $f(R)$ gravity [11].",
"The discovery of black hole thermodynamics set up a significant connection between gravity and thermodynamics [12].",
"The Hawking temperature $T=\\frac{|\\kappa _{sg}|}{2\\pi }$ , where $\\kappa _{sg}$ is the surface gravity, and horizon entropy $S=\\frac{A}{4G}$ obey the first law of thermodynamics [12]-[14].",
"Jacobson [15] showed that it is indeed possible to derive the Einstein field equations in Rindler spacetime by using the Clausius relation $TdS={\\delta }Q$ and proportionality of entropy to the horizon area.",
"Here ${\\delta }Q$ and $T$ are the energy flux across the horizon and Unruh temperature respectively, viewed by an accelerated observer just inside the horizon.",
"Frolov and Kofman [16] employed this approach to quasi-de Sitter geometry of inflationary universe with the relation $-dE=TdS$ to calculate energy flux of slowly rolling background scalar field.",
"Cai and Kim [17] derived the Freidmann equations of the FRW universe with any spatial curvature from the first law of thermodynamics for the entropy of the apparent horizon.",
"Later, Akbar and Cai [18] showed that the Friedmann equations in GR can be written in the form $dE=TdS+WdV$ at the apparent horizon, where $E={\\rho }V$ is the total energy inside the apparent horizon and $W=\\frac{1}{2}(\\rho -p)$ is the work density.",
"The connection between gravity and thermodynamics has been revealed in modified theories of gravity including Gauss-Bonnet gravity [18], Lovelock gravity [19], [20], braneworld gravity [21], non-linear gravity [22]-[27] and scalar-tensor gravity [19], [28], [29].",
"In $f(R)$ gravity and scalar-tensor theory, non-equilibrium description of thermodynamics is required [22]-[29] so that the Clausius relation is modified in the form $TdS={\\delta }Q+d\\bar{S}$ .",
"Here, $d\\bar{S}$ is the additional entropy production term.",
"Recently, Harko et al.",
"[30] generalized $f(R)$ gravity by introducing an arbitrary function of the Ricci scalar $R$ and the trace of the energy-momentum tensor $T$ .",
"The dependence of $T$ may be introduced by exotic imperfect fluids or quantum effects (conformal anomaly).",
"As a result of coupling between matter and geometry motion of test particles is nongeodesic and an extra acceleration is always present.",
"In $f(R,T)$ gravity, cosmic acceleration may result not only due to geometrical contribution to the total cosmic energy density but it also depends on matter contents.",
"This theory can be applied to explore several issues of current interest and may lead to some major differences.",
"Houndjo [31] developed the cosmological reconstruction of $f(R,T)$ gravity for $f(R,T)=f_1(R)+f_2(T)$ and discussed transition of matter dominated phase to an acceleration phase.",
"In a recent paper [26], Bamba and Geng investigated laws of thermodynamics in $f(R)$ gravity.",
"It is argued that equilibrium description exists in $f(R)$ gravity.",
"The equilibrium description of thermodynamics in modified gravitational theories is still under debate as various alternative treatments [32] have been proposed to reinterpret the non-equilibrium picture.",
"These recent studies have motivated us to explore whether the equilibrium description can be obtained in the framework of $f(R,T)$ gravity.",
"The study of connection between gravity and thermodynamics in $f(R,T)$ may provide some specific results which would discriminate this theory from various theories of modified gravity.",
"In this paper, we examine whether an equilibrium description of thermodynamics is possible in such a modified theory of gravity.",
"The horizon entropy is constructed from the first law of thermodynamics corresponding to the Friedmann equations.",
"We explore the generalized second law of thermodynamics (GSLT) and find out the necessary condition for its validity.",
"The paper is organized as follows: In the next section, we review $f(R,T)$ gravity and formulate the field equations of FRW universe.",
"Section 3 investigates the first and second laws of thermodynamics.",
"In section 4, the Friedmann equations are reformulated by redefining the dark components to explore the possible change in thermodynamics.",
"Finally, section 5 is devoted to the concluding remarks."
],
[
"$f(R,T)$ Gravity",
"The action of $f(R,T)$ theory of gravity is given by [30] $\\mathcal {A}=\\int {dx^4\\sqrt{-g}\\left[\\frac{f(R,T)}{16{\\pi }G}+\\mathcal {L}_{(matter)}\\right]},$ where $\\mathcal {L}_{(matter)}$ determines matter contents of the universe.",
"The energy-momentum tensor of matter is defined as [33] $T_{{\\mu }{\\nu }}^{(matter)}=-\\frac{2}{\\sqrt{-g}}\\frac{\\delta (\\sqrt{-g}{\\mathcal {\\mathcal {L}}_{(matter)}})}{\\delta {g^{{\\mu }{\\nu }}}}.$ We assume that the matter Lagrangian density depends only on the metric tensor components $g_{{\\mu }{\\nu }}$ so that $T_{{\\mu }{\\nu }}^{(matter)}=g_{{\\mu }{\\nu }}\\mathcal {L}_{(matter)}-\\frac{2{\\partial }{\\mathcal {L}_{(matter)}}}{\\partial {g^{{\\mu }{\\nu }}}}.$ Variation of the action (REF ) with respect to the metric tensor yields the field equations of $f(R,T)$ gravity as $&&R_{{\\mu }{\\nu }}f_{R}(R,T)-\\frac{1}{2}g_{{\\mu }{\\nu }}f(R,T)+(g_{{\\mu }{\\nu }}{\\Box }-{\\nabla }_{\\mu }{\\nabla }_{\\nu })f_{R}(R,T)\\nonumber \\\\&=&8{\\pi }GT_{{\\mu }{\\nu }}^{(matter)}-f_{T}(R,T)T_{{\\mu }{\\nu }}^{(matter)}-f_{T}(R,T)\\Theta _{{\\mu }{\\nu }},$ where ${\\nabla }_{\\mu }$ is the covariant derivative associated with the Levi-Civita connection of the metric and ${\\Box }={\\nabla }_{\\mu }{\\nabla }^{\\mu }$ .",
"We denote $f_{R}(R,T)={\\partial }f(R,T)/{\\partial }R$ , $f_{T}(R,T)={\\partial }f(R,T)/{\\partial }T$ and $\\Theta _{{\\mu }{\\nu }}=\\frac{g^{\\alpha {\\beta }}{\\delta }T_{{\\alpha }{\\beta }}}{{\\delta }g^{\\mu {\\nu }}}.$ The choice of $f(R,T){\\equiv }f(R)$ results in the field equations of $f(R)$ gravity.",
"The energy-momentum tensor of matter is defined as $T_{{\\mu }{\\nu }}^{(matter)}=({\\rho }_m+p_m)u_{\\mu }u_{\\nu }+p_{m}g_{{\\mu }{\\nu }},$ where $u_{\\mu }$ is the four velocity of the fluid.",
"If we take $\\mathcal {L}_{(matter)}=-p_m$ , then $\\Theta _{{\\mu }{\\nu }}$ becomes $\\Theta _{{\\mu }{\\nu }}=-2T_{{\\mu }{\\nu }}^{(matter)}-p_{m}g_{{\\mu }{\\nu }}.$ Consequently, the field equations (REF ) lead to $&&R_{{\\mu }{\\nu }}f_{R}(R,T)-\\frac{1}{2}g_{{\\mu }{\\nu }}f(R,T)+(g_{{\\mu }{\\nu }}{\\Box }-{\\nabla }_{\\mu }{\\nabla }_{\\nu })f_{R}(R,T)\\nonumber \\\\&=&8{\\pi }GT_{{\\mu }{\\nu }}^{(matter)}+T_{{\\mu }{\\nu }}^{(matter)}f_{T}(R,T)+p_{m}g_{{\\mu }{\\nu }}f_{T}(R,T).$ The field equations in $f(R,T)$ gravity depend on a source term, representing the variation of the energy-momentum tensor of matter with respect to the metric.",
"We consider only the non-relativistic matter (cold dark matter and baryons) with $p_{m}=0$ , therefore the contribution of $T$ comes only from ordinary matters.",
"Thus, Eq.",
"(REF ) can be written as an effective Einstein field equation of the form $R_{{\\mu }{\\nu }}-\\frac{1}{2}Rg_{{\\mu }{\\nu }}=8{\\pi }G_{eff}T_{{\\mu }{\\nu }}^{(matter)}+{T^{\\prime }}_{{\\mu }{\\nu }}^{(d)},$ where $G_{eff}=\\frac{1}{f_{R}(R,T)}\\left(G+\\frac{f_{T}(R,T)}{8\\pi }\\right)$ is the effective gravitational matter dependent coupling in $f(R,T)$ gravity and ${T^{\\prime }}_{{\\mu }{\\nu }}^{(d)}=\\frac{1}{f_{R}(R,T)}\\left[\\frac{1}{2}g_{\\mu \\nu }(f(R,T)-Rf_{R}(R,T))+({\\nabla }_{\\mu }{\\nabla }_{\\nu }-g_{{\\mu }{\\nu }}{\\Box })f_{R}(R,T)\\right]$ is the energy-momentum tensor of dark components.",
"Here, prime means non-equilibrium description of the field equations.",
"The FRW universe is described by the metric $ds^{2}=h_{\\alpha \\beta }dx^{\\alpha }dx^{\\beta }+\\tilde{r}^{2}d{\\Omega }^2,$ where $\\tilde{r}=a(t)r$ and $x^{0}=t,~x^{1}=r$ with the 2-dimensional metric $h_{\\alpha {\\beta }}=diag(-1,a^2/(1-kr^2))$ .",
"Here $a(t)$ is the scale factor, $k$ is the cosmic curvature and $d{\\Omega }^2$ is the metric of 2-dimensional sphere with unit radius.",
"In FRW background, the gravitational field equations are given by $3\\left(H^2+\\frac{k}{a^2}\\right)&=&8{\\pi }G_{eff}{\\rho }_m+\\frac{1}{f_{R}}\\left[\\frac{1}{2}(Rf_{R}-f)-3H(\\dot{R}f_{RR}\\right.\\\\\\nonumber &+&\\left.\\dot{T}f_{RT})\\right],\\\\\\nonumber -\\left(2\\dot{H}+3H^2+\\frac{k}{a^2}\\right)&=&\\frac{1}{f_{R}}\\left[-\\frac{1}{2}(Rf_{R}-f)+2H(\\dot{R}f_{RR}+\\dot{T}f_{RT})+\\ddot{R}f_{RR}\\right.\\\\&+&\\left.\\dot{R}^2f_{RRR}+2\\dot{R}\\dot{T}f_{RRT}+\\ddot{T}f_{RT}+\\dot{T}^2f_{RTT}\\right].$ These can be rewritten as $3\\left(H^2+\\frac{k}{a^2}\\right)&=&8{\\pi }G_{eff}({\\rho }_m+{\\rho }^{\\prime }_d),\\\\-2\\left(\\dot{H}-\\frac{k}{a^2}\\right)&=&8{\\pi }G_{eff}({\\rho }_m+{\\rho }^{\\prime }_d+{p}^{\\prime }_d),$ where ${\\rho }^{\\prime }_d$ and ${p}^{\\prime }_d$ are the energy density and pressure of dark components ${\\rho }^{\\prime }_d&=&\\frac{1}{8{\\pi }G\\mathcal {F}}\\left[\\frac{1}{2}(Rf_{R}-f)-3H(\\dot{R}f_{RR}+\\dot{T}f_{RT})\\right],\\\\\\nonumber {p}^{\\prime }_d&=&\\frac{1}{8{\\pi }G\\mathcal {F}}\\left[-\\frac{1}{2}(Rf_{R}-f)+2H(\\dot{R}f_{RR}+\\dot{T}f_{RT})+\\ddot{R}f_{RR}+\\dot{R}^2f_{RRR}\\right.\\\\&+&\\left.2\\dot{R}\\dot{T}f_{RRT}+\\ddot{T}f_{RT}+\\dot{T}^2f_{RTT}\\right].$ Here $\\mathcal {F}=1+\\frac{f_{T}(R,T)}{8{\\pi }G}$ .",
"The equation of state (EoS) parameter of dark fluid ${\\omega }^{\\prime }_d$ is obtained as ($p^{\\prime }_d=\\omega ^{\\prime }_d\\rho ^{\\prime }_d$ ) ${\\omega }^{\\prime }_d=-1+\\frac{\\ddot{R}f_{RR}+\\dot{R}^2f_{RRR}+2\\dot{R}\\dot{T}f_{RRT}+\\ddot{T}f_{RT}+\\dot{T}^2f_{RTT}-H(\\dot{R}f_{RR}+\\dot{T}f_{RT})}{\\frac{1}{2}(Rf_{R}-f)-3H(\\dot{R}f_{RR}+\\dot{T}f_{RT})}.$ The semi-conservation equation of ordinary matter is given by $\\dot{\\rho }+3H\\rho =q.$ The energy-momentum tensor of dark components may satisfy the similar conservation laws $\\dot{\\rho }_d+3H(\\rho _d+p_d)&=&q_d,\\\\\\dot{\\rho }_{tot}+3H(\\rho _{tot}+p_{tot})&=&q_{tot},$ where $\\rho _{tot}=\\rho _m+\\rho _d,~p_{tot}=p_d$ and $q_{tot}=q+q_d$ is the total energy exchange term and $q_d$ is the energy exchange term of dark components.",
"Substituting Eqs.",
"(REF ) and () in the above equation, we obtain $q_{tot}=\\frac{3}{8{\\pi }G}(H^2+\\frac{k}{a^2})\\partial _t\\left(\\frac{f_{R}}{\\mathcal {F}}\\right).$ The relation of energy exchange term in $f(R)$ gravity can be recovered if $\\mathcal {F}=1$ .",
"In GR, $q_{tot}=0$ for the choice $f(R,T)=R$ ."
],
[
"Laws of Thermodynamics",
"In this section, we examine the validity of the first and second law of thermodynamics in $f(R,T)$ gravity for FRW universe."
],
[
"First Law of Thermodynamics",
"Here we investigate the validity of the first law of thermodynamics in $f(R,T)$ gravity at the apparent horizon of FRW universe.",
"The dynamical apparent horizon is determined by the relation $h^{\\alpha \\beta }\\partial _{\\alpha }\\tilde{r}\\partial _{\\beta }\\tilde{r}=0$ which leads to the radius of apparent horizon for FRW universe $\\tilde{r}_A=\\frac{1}{\\sqrt{H^2+\\frac{k}{a^2}}}.$ The associated temperature of the apparent horizon is defined through the surface gravity $\\kappa _{sg}$ as $T_h=\\frac{|\\kappa _{sg}|}{2\\pi },$ where $\\kappa _{sg}$ is given by [17] $\\kappa _{sg}&=&\\frac{1}{2\\sqrt{-h}}\\partial _{\\alpha }(\\sqrt{-h}h^{\\alpha \\beta }\\partial _{\\beta }\\tilde{r}_A)=-\\frac{1}{\\tilde{r}_A}(1-\\frac{\\dot{\\tilde{r}}_A}{2H\\tilde{r}_A})\\nonumber \\\\&=&-\\frac{\\tilde{r}_A}{2}(2H^2+\\dot{H}+\\frac{k}{a^2}).$ In GR, the horizon entropy is given by the Bekenstein-Hawking relation $S_h=A/4G$ , where $A=4{\\pi }\\tilde{r}^2_A$ is the area of the apparent horizon [12]-[14].",
"In the context of modified gravitational theories, Wald [34] proposed that entropy of black hole solutions with bifurcate Killing horizons is a Noether charge entropy.",
"It depends on the variation of Lagrangian density of modified gravitational theories with respect to Riemann tensor.",
"Wald entropy is equal to quarter of horizon area in units of effective gravitational coupling i.e, ${S}^{\\prime }_h=A/4G_{eff}$ [35].",
"In $f(R,T)$ gravity, the Wald entropy is expressed as ${S}^{\\prime }_h=\\frac{Af_{R}}{4G\\mathcal {F}}.$ Taking the time derivative of Eq.",
"(REF ) and using (), it follows that $f_Rd\\tilde{r}_A=4{\\pi }G\\tilde{r}^3_A({\\rho }^{\\prime }_{tot}+{p}^{\\prime }_{tot})H\\mathcal {F}dt,$ where ${\\rho }^{\\prime }_{tot}={\\rho }^{\\prime }_m+{\\rho }^{\\prime }_d$ and ${p}^{\\prime }_{tot}={p}^{\\prime }_d$ .",
"$d\\tilde{r}_A$ is the infinitesimal change in radius of the apparent horizon during a time interval $dt$ .",
"Using Eqs.",
"(REF ) and (REF ), we obtain $\\frac{1}{2{\\pi }\\tilde{r}_A}d{S}^{\\prime }_h=4{\\pi }\\tilde{r}^3_A({\\rho }^{\\prime }_{tot}+{p}^{\\prime }_{tot})Hdt+\\frac{\\tilde{r}_A}{2G\\mathcal {F}}df_R+\\frac{\\tilde{r}_Af_R}{2G}d\\left(\\frac{1}{\\mathcal {F}}\\right).$ If we multiply both sides of this equation with a factor $(1-\\dot{\\tilde{r}}_A/2H\\tilde{r}_A)$ , it follows that $T_hd{S}^{\\prime }_h&=&4{\\pi }\\tilde{r}^3_A({\\rho }^{\\prime }_{tot}+{p}^{\\prime }_{tot})Hdt-2{\\pi }\\tilde{r}^2_A({\\rho }^{\\prime }_{tot}+{p}^{\\prime }_{tot})d\\tilde{r}_A+\\frac{{\\pi }\\tilde{r}^2_AT_hdf_R}{G\\mathcal {F}}\\nonumber \\\\&+&\\frac{{\\pi }\\tilde{r}^2_AT_hf_R}{G}d\\left(\\frac{1}{\\mathcal {F}}\\right).$ Now, we define energy of the universe within the apparent horizon.",
"The Misner-Sharp energy [36] is defined as $E=\\frac{\\tilde{r}_A}{2G}$ which can be written in $f(R,T)$ gravity as [24], [28] $E^{\\prime }=\\frac{\\tilde{r}_A}{2G_{eff}}.$ In terms of volume $V=\\frac{4}{3}{\\pi }\\tilde{r}^3_A$ , we obtain $E^{\\prime }=\\frac{3V}{8{\\pi }G_{eff}}\\left(H^2+\\frac{k}{a^2}\\right)=V{\\rho }^{\\prime }_{tot}$ which represents the total energy inside the sphere of radius $\\tilde{r}_A$ .",
"It is obvious that $E^{\\prime }>0$ , if $G_{eff}=\\frac{G\\mathcal {F}}{f_R}>0$ so that the effective gravitational coupling constant in $f(R,T)$ gravity should be positive.",
"It follows from Eqs.",
"(REF ) and (REF ) that $d{E}^{\\prime }=-4{\\pi }\\tilde{r}^3_A({\\rho }^{\\prime }_{tot}+{p}^{\\prime }_{tot})Hdt+4{\\pi }\\tilde{r}^2_A{\\rho }^{\\prime }_{tot}d\\tilde{r}_A+\\frac{\\tilde{r}_Adf_R}{2G\\mathcal {F}}+\\frac{\\tilde{r}_Af_R}{2G}d\\left(\\frac{1}{\\mathcal {F}}\\right).$ Using Eq.",
"(REF ) in (REF ), it follows that $T_hd{S}^{\\prime }_h=-d{E}^{\\prime }+{W}^{\\prime }dV+\\frac{(1+2{\\pi }\\tilde{r}_AT_h)\\tilde{r}_Adf_R}{2G\\mathcal {F}}+\\frac{(1+2{\\pi }\\tilde{r}_AT_h)\\tilde{r}_Af_R}{2G}d\\left(\\frac{1}{\\mathcal {F}}\\right),$ which includes the work density ${W}^{\\prime }=\\frac{1}{2}({\\rho }^{\\prime }_{tot}-{p}^{\\prime }_{tot})$ [37].",
"This can be rewritten as $T_hd{S}^{\\prime }_h+T_hd_{\\jmath }{S}^{\\prime }_h=-d{E}^{\\prime }+{W}^{\\prime }dV,$ where $d_{\\jmath }{S}^{\\prime }_h=-\\frac{\\tilde{r}_A}{2GT_h}(1+2{\\pi }\\tilde{r}_AT_h)d\\left(\\frac{f_R}{\\mathcal {F}}\\right)=-\\frac{\\mathcal {F}(E^{\\prime }+S^{\\prime }_hT_h)}{T_hf_R} d\\left(\\frac{f_R}{\\mathcal {F}}\\right).$ When we compare the cosmological setup of $f(R,T)$ gravity with GR, Gauss-Bonnet gravity and Lovelock gravity [18]-[20], we obtain an auxiliary term in the first law of thermodynamics.",
"This additional term $d_{\\jmath }S^{\\prime }_h$ may be interpreted as entropy production term developed due to the non-equilibrium framework in $f(R,T)$ gravity.",
"This result corresponds to the first law of thermodynamics in non-equilibrium description of $f(R)$ gravity [26] for $f(R,T)=f(R)$ .",
"If we assume $f(R,T)=R$ , then the traditional first law of thermodynamics in GR can be achieved."
],
[
"Generalized Second Law of Thermodynamics",
"Recently, the GSLT has been studied in the context of modified gravitational theories [25]-[28].",
"It may be interesting to investigate its validity in $f(R,T)$ gravity.",
"For this purpose, we have to show that [28] $\\dot{S}^{\\prime }_h+d_\\jmath \\dot{S}^{\\prime }_h+\\dot{S}^{\\prime }_{tot}\\ge 0,$ where ${S}^{\\prime }_h$ is the horizon entropy in $f(R,T)$ gravity, $d_\\jmath \\dot{S}^{\\prime }_h=\\partial _t(d_\\jmath {S}^{\\prime }_h)$ and $S^{\\prime }_{tot}$ is the entropy due to all the matter and energy sources inside the horizon.",
"The Gibb's equation including all matter and energy fluid is given by [38] $T_{tot}dS^{\\prime }_{tot}=d({\\rho }^{\\prime }_{tot}V)+{p}^{\\prime }_{tot}dV,$ where $T_{tot}$ is the temperature of total energy inside the horizon.",
"We assume that $T_{tot}$ is proportional to the temperature of apparent horizon [25], [28], i.e., $T_{tot}=bT_h$ , where $0<b<1$ to ensure that temperature being positive and smaller than the horizon temperature.",
"Substituting Eqs.",
"(REF ) and (REF ) in Eq.",
"(REF ), we obtain $\\dot{S}^{\\prime }_h+d_\\jmath \\dot{S}^{\\prime }_h+\\dot{S}^{\\prime }_{tot}=\\frac{24\\pi {\\Xi }}{\\tilde{r}_AbR}\\ge 0,$ where $\\Xi =(1-b)\\dot{\\rho }^{\\prime }_{tot}V+(1-\\frac{b}{2})(\\rho ^{\\prime }_{tot}+p^{\\prime }_{tot})\\dot{V}$ is the universal condition to protect the GSLT in modified gravitational theories [28].",
"Using Eqs.",
"(REF ) and (), condition (REF ) is reduced to $\\frac{12\\pi \\mathcal {X}}{bRG\\mathcal {F}(H^2+\\frac{k}{a^2})^2}\\ge 0,$ where $\\mathcal {X}&=&2(1-b)H(\\dot{H}-\\frac{k}{a^2})(H^2+\\frac{k}{a^2})f_R+(2-b)H(\\dot{H}-\\frac{k}{a^2})^2f_R\\\\\\nonumber &+&(1-b)(H^2+\\frac{k}{a^2})^2\\mathcal {F}\\partial _t(\\frac{f_R}{\\mathcal {F}}).$ Thus the condition to satisfy the GSLT is equivalent to $\\mathcal {X}\\ge 0$ .",
"In flat FRW universe, the GSLT is valid with the constraints $\\partial _t(\\frac{f_R}{\\mathcal {F}})\\ge 0,~H>0$ and $\\dot{H}\\ge 0$ .",
"Also, $\\mathcal {F}$ and $f_R$ are positive in order to keep $E>0$ .",
"If $b=1$ , i.e., temperature between outside and inside the horizon remains the same then the GSLT is valid only if $\\mathcal {J}=\\left(\\dot{H}-\\frac{k}{a^2}\\right)^2\\frac{f_R}{\\mathcal {F}}\\ge 0.$ For Eq.",
"(REF ), the effective EoS is defined as $\\omega _{eff}=-1-{2(\\dot{H}-\\frac{k}{a^2})}/{3(H^2+\\frac{k}{a^2})}$ .",
"Here $\\dot{H}<\\frac{k}{a^2}$ corresponds to quintessence region while $\\dot{H}>\\frac{k}{a^2}$ represents the phantom phase of the universe.",
"It follows that GSLT in $f(R,T)$ gravity is satisfied in both phantom and non-phantom phases.",
"This result is compatible with [39] according to which entropy may be positive even at the phantom era.",
"Bamba and Geng [26], [27] also shown that second law of thermodynamics can be satisfied in $f(R)$ and $f(T)$ theories of gravity."
],
[
"Redefining the Dark Components",
"In previous section, we have seen that an additional entropy term $d_{\\jmath }S^{\\prime }_h$ is produced in laws of thermodynamics.",
"This can be considered as the result of non-equilibrium description of the field equations.",
"If we redefine the dark components so that the extra entropy production term is vanished, then such formulation is referred as an equilibrium description.",
"It has been seen so far that the equilibrium description does exist in modified theories of gravity [26], [27], [29] and extra entropy production term can be removed.",
"Here, we discuss whether the equilibrium description of $f(R,T)$ gravity can be anticipated.",
"In fact, we may reduce the entropy production term through this description but it cannot be wiped out entirely.",
"We redefine the energy density and pressure of dark components.",
"The (00) and (11) components of the field equations can be rewritten as $3\\left(H^2+\\frac{k}{a^2}\\right)&=&8{\\pi }G_{eff}({\\rho }_m+\\rho _d),\\\\-2\\left(\\dot{H}-\\frac{k}{a^2}\\right)&=&8{\\pi }G_{eff}(\\rho _m+\\rho _d+p_d),$ where $G_{eff}=\\left(G+\\frac{f_{T}(R,T)}{8\\pi }\\right)$ is the effective gravitational coupling, $\\rho _d$ and $p_d$ are the energy density and pressure of dark components given by $\\nonumber \\rho _d&=&\\frac{1}{8{\\pi }G\\mathcal {F}}\\left[\\frac{1}{2}(Rf_{R}-f)-3H(\\dot{R}f_{RR}+\\dot{T}f_{RT})+3(1-f_R)(H^2\\right.\\\\&+&\\left.\\frac{k}{a^2})\\right],\\\\\\nonumber p_d&=&\\frac{1}{8{\\pi }G\\mathcal {F}}\\left[-\\frac{1}{2}(Rf_{R}-f)+2H(\\dot{R}f_{RR}+\\dot{T}f_{RT})+\\ddot{R}f_{RR}+\\dot{R}^2f_{RRR}\\right.\\\\&+&\\left.2\\dot{R}\\dot{T}f_{RRT}+\\ddot{T}f_{RT}+\\dot{T}^2f_{RTT}-(1-f_R)(2\\dot{H}+3H^2+\\frac{k}{a^2})\\right].$ The EoS parameter $\\omega _d$ in this description turns out to be $\\nonumber \\omega _d&=&-1+\\lbrace \\ddot{R}f_{RR}+\\dot{R}^2f_{RRR}+2\\dot{R}\\dot{T}f_{RRT}+\\ddot{T}f_{RT}+\\dot{T}^2f_{RTT}-H(\\dot{R}f_{RR}\\\\\\nonumber &+&\\dot{T}f_{RT})-2(1-f_R)(\\dot{H}-\\frac{k}{a^2})\\rbrace /\\lbrace \\frac{1}{2}(Rf_{R}-f)-3H(\\dot{R}f_{RR}+\\dot{T}f_{RT})\\\\&+&3(1-f_R)(H^2+\\frac{k}{a^2})\\rbrace ,$ In this case the expression of total energy exchange is given by $q_{tot}=\\frac{3}{8{\\pi }G}(H^2+\\frac{k}{a^2})\\partial _t\\left(\\frac{1}{\\mathcal {F}}\\right).$ Since $\\partial _t(f_T(R,T))\\ne 0$ in $f(R,T)$ gravity, so that $q_{tot}$ does not vanish.",
"So, we may not establish the equilibrium picture of thermodynamics in this modified gravity.",
"Hence, again we need to consider the non-equilibrium treatment of thermodynamics.",
"This result differ from other modified gravitational theories due to the matter dependence of the Lagrangian density.",
"In $f(R)$ gravity the redefinition of dark components result in local conservation of energy momentum tensor of dark components [26].",
"It is clear from Eqs.",
"(REF ) and (REF ) that the EoS parameter of dark components is not unique in both cases.",
"Thus, one should consider both formulations of the field equations in cosmic discussions.",
"Now we check the validity of the first and second laws of thermodynamics in this scenario."
],
[
"First Law of Thermodynamics",
"In this representation of the field equations, the time derivative of radius $\\tilde{r}_A$ at the apparent horizon is given by $d\\tilde{r}_A=4{\\pi }\\tilde{r}^3_A{G\\mathcal {F}}(\\rho _{tot}+p_{tot})Hdt.$ Since in $f(R,T)$ gravity, the equilibrium description is not feasible as it can be seen in modified gravitational theories such that $f(R),~f(T)$ and scalar tensor gravity etc.",
"Thus, we use the Wald entropy relation $S_h=A/(4G_{eff})$ rather than introducing Bekenstein-Hawking entropy.",
"Using Eq.",
"(REF ), the horizon entropy becomes $\\frac{1}{2{\\pi }\\tilde{r}_A}dS_h=4{\\pi }\\tilde{r}^3_A(\\rho _{tot}+p_{tot})Hdt+\\frac{\\tilde{r}_A}{2G}d\\left(\\frac{1}{\\mathcal {F}}\\right).$ The associated temperature of the apparent horizon is $T_h=\\frac{1}{2\\pi \\tilde{r}_A}(1-\\frac{\\dot{\\tilde{r}}_A}{2H\\tilde{r}_A}).$ Equations (REF ) and (REF ) imply that $T_hdS_h=4{\\pi }\\tilde{r}^3_A(\\rho _{tot}+p_{tot})Hdt-2{\\pi }\\tilde{r}^2_A(\\rho _{tot}+p_{tot})d\\tilde{r}_A+\\frac{{\\pi }\\tilde{r}^2_AT_h}{G}d\\left(\\frac{1}{\\mathcal {F}}\\right).$ Introducing the Misner-Sharp energy $E=\\frac{\\tilde{r}_A}{2G\\mathcal {F}}=V\\rho _{tot},$ we obtain $dE=-4{\\pi }\\tilde{r}^3_A(\\rho _{tot}+p_{tot})Hdt+4{\\pi }\\tilde{r}^2_A\\rho _{tot}d\\tilde{r}_A+\\frac{\\tilde{r}_A}{2G}d\\left(\\frac{1}{\\mathcal {F}}\\right).$ The total work density is defined as [37] $W=-\\frac{1}{2}T^{(tot)\\alpha \\beta }h_{\\alpha \\beta }=\\frac{1}{2}(\\rho _{tot}-p_{tot}),$ By combining Eqs.",
"(REF ), (REF ) and (REF ), we obtain the following expression of first law of thermodynamics $T_hdS_h+T_hd_{\\jmath }S_h=-dE+WdV,$ where $\\nonumber d_{\\jmath }S_h&=&-\\frac{\\tilde{r}_A}{2T_hG}(1+2{\\pi }\\tilde{r}_AT_h)d\\left(\\frac{1}{\\mathcal {F}}\\right)=-\\mathcal {F}\\left(\\frac{E}{T_h}+S_h\\right)d\\left(\\frac{1}{\\mathcal {F}}\\right)\\\\&=&-\\frac{{\\pi }(4H^2+\\dot{H}+3k/a^2)}{G(H^2+k/a^2)(2H^2+\\dot{H}+k/a^2)}d\\left(\\frac{1}{\\mathcal {F}}\\right)$ is the additional term of entropy produced due to the matter contents of the universe.",
"It involves derivative of $f(R,T)$ with respect to the trace of the energy-momentum tensor.",
"Notice that the first law of thermodynamics $T_hdS_h=-dE+WdV$ holds at the apparent horizon of FRW universe in equilibrium description of modified theories of gravity [26], [27], [29].",
"However, in $f(R,T)$ gravity, this law does not hold due to the presence of an additional term $d_{\\jmath }S_h$ .",
"This term vanishes if we take $f(R,T)=f(R)$ which leads to the equilibrium description of thermodynamics in $f(R)$ gravity."
],
[
"Generalized Second Law of Thermodynamics",
"To establish the GSLT in this formulation of $f(R,T)$ gravity, we consider the Gibbs equation in terms of all matter field and energy contents $T_{tot}dS_{tot}=d(\\rho _{tot}V)+p_{tot}dV,$ where $T_{tot}$ denotes the temperature of total energy inside the horizon and $S_{tot}$ is the entropy of all the matter and energy sources inside the horizon.",
"In this case, the GSLT can be expressed as $\\dot{S}_h+d_\\jmath \\dot{S}_h+\\dot{S}_{tot}\\ge 0$ which implies that $\\frac{12\\pi \\mathcal {Y}}{bRG\\mathcal {F}(H^2+\\frac{k}{a^2})^2}\\ge 0,$ where $\\mathcal {Y}&=&2(1-b)H(\\dot{H}-\\frac{k}{a^2})(H^2+\\frac{k}{a^2})+(2-b)H(\\dot{H}-\\frac{k}{a^2})^2\\\\\\nonumber &+&(1-b)(H^2+\\frac{k}{a^2})^2\\mathcal {F}\\partial _t(\\frac{1}{\\mathcal {F}}).$ Thus the GSLT is satisfied only if $\\mathcal {Y}\\ge 0$ .",
"In case of flat FRW universe, the GSLT is met with the conditions $\\partial _t(\\frac{1}{\\mathcal {F}})\\ge 0$ , $H>0$ and $\\dot{H}\\ge 0$ .",
"In thermal equilibrium $b=1$ , the above condition is reduced to the following form $\\mathfrak {B}=\\frac{12{\\pi }H\\left(\\dot{H}-\\frac{k}{a^2}\\right)^2}{G\\left(H^2+\\frac{k}{a^2}\\right)^2R}\\frac{1}{\\mathcal {F}}\\ge 0,$ for $V=\\frac{4}{3}{\\pi }\\tilde{r}^3_A$ and $R=6(\\dot{H}+2H^2+k/a^2)$ .",
"$\\mathfrak {B}\\ge 0$ clearly holds when the Hubble parameter and scalar curvature have same signatures.",
"It can be seen that main difference of results of $f(R,T)$ gravity with $f(R)$ gravity is the term $\\mathcal {F}=1+\\frac{f_{T}(R,T)}{8{\\pi }G}$ .",
"We remark that in both definitions of dark components, the GSLT is valid both in phantom and non-phantom phases of the universe."
],
[
"Concluding Remarks",
"The fact, $f(R,T)$ gravity is the generalization of $f(R)$ gravity is based on coupling between matter and geometry [30].",
"This theory can be applied to explore several issues of current interest in cosmology and astrophysics.",
"We have discussed the laws of thermodynamic at the apparent horizon of FRW spacetime in this modified gravity.",
"Akbar and Cai [23] have shown that the Friedmann equations for $f(R)$ gravity can be written into a form of the first law of thermodynamics, $dE=TdS+WdV+Td\\bar{S}$ , where $d\\bar{S}$ is the additional entropy term due to non-equilibrium thermodynamics.",
"Bamba and Geng [26], [27] established the first and second laws of thermodynamics at the apparent horizon of FRW universe with both non-equilibrium and equilibrium descriptions.",
"We have found that the picture of equilibrium thermodynamics is not feasible in $f(R,T)$ gravity even if we specify the energy density and pressure of dark components (see Section 4).",
"Thus the non-equilibrium treatment is used to study the laws of thermodynamics in both forms of the energy-momentum tensor of dark components.",
"In $f(R,T)$ gravity, $q_{tot}$ does not vanish so there exists some energy exchange with the horizon.",
"The non-equilibrium description can be interpreted as due to some energy flow between inside and outside the apparent horizon.",
"The first law of thermodynamics is obtained at the apparent horizon in FRW background for $f(R,T)$ gravity.",
"We observe that the additional entropy term is produced as compared to GR, Gauss-Bonnet gravity [18], Lovelock gravity [19], [20] and braneworld gravity [21].",
"The equilibrium and non-equilibrium description of thermodynamics in $f(R)$ gravity can be obtained if the term $f_T(R,T)$ vanishes i.e., Lagrangian density depends only on geometric part.",
"We have established the GSLT with the assumption that the total temperature inside the horizon $T_{tot}$ is proportional to the temperature of the apparent horizon $T_h$ and evaluated its validity conditions.",
"The GSLT in $f(R)$ gravity follows from condition (REF ) if $\\mathcal {F}=1$ .",
"It is concluded that in thermal equilibrium, GSLT is satisfied in both phantom and non-phantom phases."
]
] | 1204.0848 |
[
[
"Simulation of radiation driven wind from disc galaxies"
],
[
"Abstract We present 2-D hydrodynamic simulation of rotating galactic winds driven by radiation.",
"We study the structure and dynamics of the cool and/or warm component($T \\simeq 10^4$ K) which is mixed with dust.",
"We have taken into account the total gravity of a galactic system that consists of a disc, a bulge and a dark matter halo.",
"We find that the combined effect of gravity and radiation pressure from a realistic disc drives the gas away to a distance of $\\sim 5$ kpc in $\\sim 37$ Myr for typical galactic parameters.",
"The outflow speed increases rapidly with the disc Eddington parameter $\\Gamma_0(=\\kappa I/(2 c G \\Sigma)$) for $\\Gamma_0 \\ge 1.5$.",
"We find that the rotation speed of the outflowing gas is $\\lesssim 100$ km s$^{-1}$.",
"The wind is confined in a cone which mostly consist of low angular momentum gas lifted from the central region."
],
[
"Introduction",
"Many galaxies are observed to have moving extraplanar gas, generally termed as galactic superwinds (see Veilleux et al.",
"2005 for a recent review).",
"Initial observations showed the H$\\alpha $ emitting gas above the plane of M82 (e.g.",
"Lynds & Sandage 1963).",
"The advent of X-ray astronomy established yet another phase of galactic outflows, namely the hot plasma, emitting X-rays in the temperature range $0.3\\hbox{--}2$ keV (Strickland et al.",
"2004).",
"Also recent observations have revealed the existence of molecular gas in these outflows (Veilleux et al.",
"2009, walter et al.",
"2002).",
"Earlier observations were limited to local dwarf starburst galaxies that showed these winds.",
"However, in recent years, the observations of outflows in Ultra Luminous Infra-red Galaxies (ULIGs) have extended the range of galaxies in which outflows are found (Martin 2005, Rupke et al.",
"2005, Rupke et al.",
"2002).",
"On the theoretical side, there have been speculations on winds from starburst galaxies (Burke 1968, Mathews & Baker 1971, Johnson & Axford 1971).",
"In these models the large scale winds are a consequence of energy injection by multiple supernovae (Larson 1974, Chevalier & Clegg 1985, Dekel & Silk 1986, Heckman 2002).",
"In the context of the multiphase structure of the outflows, the results of these theoretical models are more relevant for the X-ray emitting hot wind.",
"On the other hand, observations of the cold outflows are better explaind by the radiation driving (Murray et al.",
"2005, Martin 2005).",
"If only Thompson scattering is considered, then radiation from galaxies does not seem to be a reasonable wind driving candidate because opacities would be small; however one should consider that these winds are heavily enriched.",
"Murray et al.",
"2005 proposed a wind driving mechanism based on the scattering of dust-grains by the photons from the galaxy (see also Chiao & Wickramasinghe 1972; Davies et al.",
"1998).",
"This mechanism can be quite effective since the opacities in dust-photon scattering can be of the order of hundred cm$^2$ g$^{-1}$ and gas in turn, being coupled with the dust, is driven out of the galaxy if the galaxy posseses a certain critical luminosity.",
"Bianchi & Ferrara (2005) argued that dust grains ejected from galaxies by radiation pressure can enrich the intergalactic medium.",
"Nath & Silk (2009) then described a model of outflows with radiation and thermal pressure, in the context of outflows from Lyman break galaxies observed by Shapely et al.",
"(2005).",
"Murray et al.",
"(2010) have also described a similar model in which radiation pressure is important for the first few million years of the starburst phase, after which SN heated hot gas pushes the outflowing material.",
"Sharma & Nath (2011) have also shown that radiation pressure is important for outflows from high mass galaxies with a large SFR (with $v_c \\ge 200$ km s$^{-1}$ , SFR $\\ge 100$ M$_{\\odot }$ yr$^{-1}$ ), particularly in ULIGs.",
"In this paper, we study the effect of radiation pressure in driving cold and/or warm gas outflows from disc galaxies with numerical simulations.",
"Recently, Sharma et al.",
"(2011) calculated the terminal speed of such a flow along the pole of a disc galaxy, taking into account the gravity of disc, stellar bulge and dark matter halo.",
"They determined the minimum luminosity (or, equivalently, the maximum mass-to-light ratio of the disc) to drive a wind, and also showed that the terminal speed lies in the range of $2\\hbox{--}4 \\, V_c$ (where $V_c$ is the rotation speed of the disc galaxy), consistent with observations (Rupke et al.",
"2005, Martin 2005), and the ansatz used by numerical simulations in order to explain the metal enrichment of the IGM (Oppenheimer et al.",
"2006).",
"We investigate further the physical processes for a radiation driven wind.",
"Rotation is yet another aspect of the winds that we address in our simulation.",
"As the wind material is lifted from a rotating disc, it should be rotating inherently which is seen in observations as well (Greve 2004, Westmoquette et al.",
"2009, Sofue et al.",
"1992, Seaquist & Clark 2001, Walter et al.",
"2002).",
"Previous simulations of galactic outflows have considered the driving force of a hot ISM energized by the effects of supernovae (Kohji & Ikeuchi 1988; Tomisaka & Bregman 1993; Mac Low & Ferrara 1999; Suchkov et al.",
"1994, 1996 ; Strickland & Stevens 2000; Fragile et al.",
"2004; Cooper et al.",
"2008, Fujita et al.",
"2009).",
"However the detailed physics of a radiatively driven galactic outflow is yet to be studied with a simulation.",
"In this work, we study the dynamics of an irradiated gas above an axisymmetric disc galaxy by using hydrodynamical simulation.",
"Recently Hopkins et al.",
"(2011) have explored the relative roles of radiation and supernovae heating in galactic outflows, and studied the feedback on the star formation history of the galaxy.",
"Our goal here is different in the sense that we focus on the structure and dynamics, particularly the effect of rotation, of the wind.",
"In order to disentangle the effects of various processes involved, we intentionally keep the physical model simple.",
"For example, we begin with a constant density and surface brightness disk, then study the effect of a radial density and radiation profile, and finally introduce rotation of the disk, in order to understand the effect of each detail separately, instead of performing one single simulation with many details put together."
],
[
"Gravitational and radiation fields",
"The main driving force is radiation force and the containing force is due to gravity.",
"We take the system to be composed of three components disc, bulge & dark matter halo.",
"We describe the forces due to these three constituents below.",
"We take a thin galactic disc and a spherical bulge.",
"All these forces are given in cylindrical coordinates because we solve the fluid equations in cylindrical geometry."
],
[
"Gravitational field from the disc",
"Consider a thin axisymmetric disc in $r\\phi $ plane with surface mass density $\\Sigma (r)$ .",
"As derived in the Appendix, the vertical and radial components of gravity due to the disc material at a point $Q$ above the disc with coordinates $(r, 0, z)$ , are given by $f_{disc,z} &=& \\int _{\\phi \\prime } \\int _{r\\prime } d\\phi \\prime \\, dr\\prime \\, \\frac{z G \\Sigma (r\\prime )\\ r\\prime }{[r^2+z^2+r\\prime ^2-2rr\\prime cos\\phi \\prime ]^{3/2}} \\nonumber \\\\f_{disc,r} &=& \\int _{\\phi \\prime }\\int _{r\\prime } \\, d\\phi \\prime \\,dr\\prime \\frac{(r-r\\prime cos\\phi \\prime )\\ G \\Sigma (r\\prime )\\ r\\prime }{[r^2+z^2+r\\prime ^2-2rr\\prime cos\\phi \\prime ]^{3/2}} \\ $ The azimuthal coordinate of $Q$ is taken to be zero, because of axisymmetry.",
"The integration limit for $\\phi \\prime =0$ to $2\\pi $ .",
"We consider two types of disc in our simulations, one with uniform surface mass density and radius $r_d$ (UD), and another with an exponential distribution of surface mass density (ED) with a scale radius $r_s$ .",
"The surface mass density of uniform surface density disc (i. e.,UD) is $\\Sigma =\\Sigma _0=\\mbox{constant}$ and in the case of a disc with exponentially falling density distribution (ED) $\\Sigma ={\\tilde{\\Sigma }}_0\\mbox{exp}(-r\\prime /r_s), ~~ r_s \\equiv \\mbox{scale length}\\,.$ In case of UD (eqn REF ), the integration limit would be $r\\prime =0$ to $r_d$ , while for ED (eqn REF ), the limits of the integration run from $r\\prime =0$ to $\\infty $ .",
"Numerically this means, we integrate up to a large number, increasing which will not change the gravitational field by any significant amount.",
"We have chosen the $\\Sigma $ s in such a way that the total disc mass remains same for the UD or ED.",
"Therefore, $\\tilde{\\Sigma }_0=\\frac{\\Sigma _0}{2}\\left(\\frac{r_d}{r_s}\\right)^2 \\,.",
"$ In Figure 1, we plot the contours of gravitational field strength and its direction vectors due to a UD (left panel), and that for the ED (right panel).",
"Interestingly, discs with same mass but different surface density distributions, produces different gravitational fields.",
"For the UD the gravitational field is not spherical and the gravitational acceleration is maximum at the edge of the disc.",
"On the other hand, the field due to ED is closer to spherical configuration with the maximum being closer to the centre of the disc and falling off outwards.",
"Figure: Magnitude of gravitational force of the (a) uniform disc (UD)(b) exponential disc (ED) in colours withdirection in arrows.",
"Values are in the units of GΣ 0 (=4.5×10 -9 G\\Sigma _0 (= 4.5\\times 10^{-9}) dyne."
],
[
"Bulge and the dark matter halo",
"We consider a bulge with a spherical mass distribution and constant density, with mass $M_b$ and radius $r_b$ .",
"The radiation force due to the bulge is negligible as it mostly hosts the old stars.",
"The gravitational force of the bulge is given by $f_{bulge,r} = \\left\\lbrace \\begin{array}{rl}-\\frac{G M_b r}{r_b^3} &\\mbox{ if $R<r_b$} \\\\\\\\-\\frac{G M_b r}{R^3} &\\mbox{ otherwise}\\end{array} \\right.$ $f_{bulge,z} = \\left\\lbrace \\begin{array}{rl}-\\frac{G M_b z}{r_b^3}\\,, &\\mbox{ if $R<r_b$} \\\\\\\\-\\frac{G M_b z}{R^3} \\,, &\\mbox{ otherwise}\\end{array} \\right.$ where R = $\\sqrt{r^2+z^2}$ .",
"We consider a NFW halo with a scaling with disc mass as given by Mo, Mao and White (1998; hereafter referred to as MMW98) where the total halo mass is $\\sim 20$ times the total disc mass.",
"The mass of an NFW halo has the following functional dependence on R $M(R)= 4 \\pi \\rho _{crit} \\delta _{0} R_s^3 \\left[ \\ln {(1+cx)}-\\frac{cx}{1+cx}\\right] \\,$ where $x = \\frac{R}{R_{200}}\\,, c = \\frac{R_{200}}{R_s} \\,,\\delta _0 = \\frac{200}{3}\\frac{c^3}{ln(1+c)-c/(1+c)}$ .",
"Here $\\rho _{crit}$ is the critical density of the universe at present epoch, R$_s$ is scale radius of NFW halo and R$_{200}$ is the limiting radius of virialized halo within which the average density is 200$\\rho _{crit}$ .",
"This mass distribution corresponds to the following potential, $\\Phi _{NFW} = -4 \\pi \\rho _{crit} \\delta _{0} R_s^3 \\Bigl [ \\ln {(1+R/R_s)} / R\\Bigr ]$ The gravitational force due to the dark matter halo is therefore given by, $f_{halo,r} = -\\frac{\\partial \\Phi _{NFW}}{\\partial r} = -\\frac{r\\ G M(R)}{(r^2+z^2)^{3/2}} ; \\nonumber \\\\f_{halo,z} = -\\frac{\\partial \\Phi _{NFW}}{\\partial z} = -\\frac{z\\ G M(R)}{(r^2+z^2)^{3/2}}\\,.$ The net gravitational acceleration is therefore given by $& & F_{grav,r}=f_{disc,r}+f_{bulge,r}+f_{halo,r}=G\\Sigma _0 f_{g,r}(r,z) \\\\ \\nonumber & & F_{grav,z}=f_{disc,z}+f_{bulge,z}+f_{halo,z}=G\\Sigma _0 f_{g,z}(r,z)\\,.$ The gravitational field for both bulge and halo is spherical in nature, although, that due to the bulge maximises at $r_b$ .",
"However, the net gravitational field will depend on the relative strength of the three components.",
"In Figure REF (left panel), we plot the contours of total gravitational field strength due to the bulge, the halo and an UD.",
"The non-spherical nature of the gravitational field is evident.",
"A more interesting feature appears due to the bulge gravity.",
"The net gravitational intensity maximizes in a spherical shell of radius $r_b (=0.2 L_{ref}$ ; see section §3.1).",
"Therefore, there is a possibility of piling up of outflowing matter at around a height $z\\sim r_b$ near the axis.",
"In the right panel of Figure (REF ), we present the contours of net gravitational field due to an embedded exponential disc within a halo and a bulge.",
"Figure: Total gravitational force of the (a)uniform disc (b) exponential disc in colors withdirection in arrows.",
"The values are in the same units as in Figure ."
],
[
"Radiation from disc and the Eddington factor",
"We treat the force due to radiation pressure as it interacts with charged dust particles that are assumed to be strongly coupled to gas by Coulomb interactions and which drags the gas with it.",
"The strength of the interaction is parameterized by the dust opacity $\\kappa $ which has the units cm$^2$ gm$^{-1}$ .",
"Gravitational pull on the field point $Q(R,Z)$ due to the disc point $P(r\\prime ,\\phi \\prime ,0)$ is along the direction ${\\overrightarrow{QP}}$ (see appendix).",
"The difference in computing the radiation force arises due to the fact that one needs to account for the projection of the intensity at $Q$ (for radiation force from more complicated disc, see Chattopadhyay 2005).",
"For a disc with surface brightness $I(r)$ , we can find the radiation force by replacing $G\\Sigma (r \\prime )$ in eqn REF by $I(r\\prime )\\kappa /c$ , and take into account the projection factor $z / \\sqrt{r^2 +z^2 +r \\prime ^2 -2 r r\\prime \\cos \\phi \\prime }$ .",
"Similar to the disc gravity, the net radiation force ${\\overrightarrow{F}_{rad}}$ at any point will have the radial component ($F_{rad,r}$ ) and the axial component ($F_{rad,z}$ ) and are given by, $F_{rad,r}(r,z) &=& \\frac{\\kappa z}{c}\\int \\int \\frac{d\\phi \\prime dr\\prime I(r\\prime )(r-r\\prime cos\\phi \\prime )\\ r\\prime }{[r^2+z^2+r\\prime ^2-2rr\\prime cos\\phi \\prime ]^2} \\\\ \\nonumber &=& \\frac{\\kappa I_0}{c} f_{r,r}(r,z)$ $F_{rad,z}(r,z) &=& \\frac{\\kappa z^2}{c}\\int \\int \\frac{d\\phi \\prime dr\\prime I(r\\prime )r\\prime }{[r^2+z^2+r\\prime ^2-2rr\\prime cos\\phi \\prime ]^2} \\\\ \\nonumber &=& \\frac{\\kappa I_0}{c} f_{r,z}(r,z)$ Since we have two models for disc gravity, we also consider two forms of disc surface brightness.",
"$I=I_0=\\mbox{constant, for UD}$ and $I={\\tilde{I}}_0\\mbox{exp}(-r\\prime /r_s) \\,, \\mbox{for ED}$ If the two disc types are to be compared for identical luminosity, then one finds ${\\tilde{I}}_0=\\frac{I_0}{2}\\left(\\frac{r_d}{r_s}\\right)^2 \\,.$ Figure: Magnitude of force due to radiation from the (a) uniform disc, (b) exponential disc for Γ 0 =0.5\\Gamma _0 =0.5, with arrows for direction.The disc Eddington factor is defined as the ratio of the radiation force and the gravitational force (MQT05).",
"In spherical geometry this factor is generally constant at each point because both gravity and radiation has an inverse square dependence on distance.",
"Although in the case of a disc, the two forces have different behaviour, we can still define an Eddington parameter as $\\Gamma = \\frac{F_{rad}}{F_{grav}}$ .",
"In this case this parameter depends on the coordinates $r,\\phi , z$ of the position under consideration.",
"We can however define a parameter whose value is the Eddington factor at the centre of the disc, i.e., $\\Gamma _0 = \\frac{\\kappa I}{2 c G \\Sigma }.$ If $\\Gamma _0 = 1$ , then the radiation and gravity of the disc will cancel each other at the centre of the disc.",
"We will parameterize our results in terms of $\\Gamma _0$ .",
"Therefore, the components of the net external force due to gravity and radiation is given by $& & {\\cal R}_r=F_{grav,r}-F_{rad,r}=G\\Sigma _0\\left(f_{g,r}-2\\Gamma _0f_{r,r}\\right) \\\\ \\nonumber & & {\\cal R}_z=F_{grav,z}-F_{rad,z}=G\\Sigma _0\\left(f_{g,z}-2\\Gamma _0f_{r,z}\\right)$ In Figure REF , we plot the contours of radiative acceleration from an UD, and the same from an ED.",
"There is a significant difference between the radiation field above an ED and that above an UD.",
"While the radiation field from an UD is largely vertical for small radii, but starts to diverge at the disc edge, at $r\\sim r_d$ .",
"One can therefore expect that for high enough $I$ , the wind trajectory will diverge.",
"In case of ED, the radiation field above the inner portion of the disc is strong and decreases rapidly towards the outer disc.",
"The hydrodynamic equations have been solved in this paper by using the TVD (i. e.,Total Variation Diminishing) code, which has been quite exhaustively used in cosmological and accretion disc simulations (see, Ryu et al.",
"1993, Kang et al.",
"1994, Ryu et al.",
"1995, Molteni et al.",
"1996) and is based on a scheme originally developed by Harten (1983).",
"We have solved the equations in cylindrical geometry in view of the axial symmetry of the problem.",
"This code is based on an explicit, second order accurate scheme, and is obtained by first modifying the flux function and then applying a non-oscillatory first order accurate scheme to obtain a resulting second order accuracy (see, Harten 1983 and Ryu et al.",
"1993 for details).",
"The equations of motion which are being solved numerically in the non-dimensional form is given by $\\frac{\\partial {\\bf q}}{\\partial t}+\\frac{1}{r}\\frac{\\partial (r{\\bf F}_1)}{\\partial r}+\\frac{\\partial {\\bf F}_2}{\\partial r}+\\frac{\\partial {\\bf G}}{\\partial z}={\\bf S}$ where, the state vector is ${\\bf q}=\\left({\\rho \\cr \\rho ~v_r \\cr \\rho ~v_{\\phi } \\cr \\rho ~v_z \\cr E}\\right),$ and the fluxes are ${\\bf F}_1=\\left({\\rho ~v_r \\cr \\rho ~v^2_r \\cr \\rho v_r v_{\\phi } \\cr \\rho v_z v_r \\cr (E+p)v_r}\\right), ~~ {\\bf F}_2=\\left({0 \\cr p \\cr 0 \\cr 0 \\cr 0}\\right),~~ {\\bf G}= \\left({\\rho v_z \\cr \\rho v_r v_z \\cr \\rho v_{\\phi } v_z \\cr \\rho v^2_z+p \\cr (E+p)v_z} \\right)$ and the source function is given by ${\\bf S}=\\left[{0 \\cr \\frac{\\rho v^2_{\\phi }}{r}-\\rho {\\cal R}_r \\cr -\\frac{\\rho v_r v_{\\phi }}{r}\\cr -\\rho {\\cal R}_z \\cr -\\rho [v_r{\\cal R}_r+v_z{\\cal R}_z)}\\right]$"
],
[
"Initial and boundary conditions",
"We do not include the disc in our simulations and only consider the effect of disc radiation and total gravity on the gas being injected from the disc.",
"We choose the disc mass to be $M_d=10^{11}$ M$_{\\odot }$ and assume it to be the unit of mass (i. e.,$M_{ref}$ ).",
"The unit of length (i. e.,$L_{ref}$ ) and velocity (i. e.,$v_{ref}$ ) are $r_d=10$ kpc and $v_c=200$ km s$^{-1}$ , respectively.",
"Therefore, the unit of time is $t_{ref}=48.8$ Myr.",
"We introduce a normalization parameter $\\xi $ such that $GM_d/v_c^2=\\xi r_d$ , which turns out to be $\\xi =1.08$ .",
"Hence the unit of density is $\\rho _{ref}=6.77{\\times }10^{-24}$ g cm$^{-3}$ ($\\sim 4 m_p$ cm$^{-3}$ ).",
"All the flow variables have been made non-dimensional by the choice of unit system mentioned above.",
"It is important to choose an appropriate initial condition to study the relevant physical phenomenon.",
"We note that previous simulations of galactic outflows have considered a variety of gravitational potential and initial ISM configurations.",
"For example, Cooper et al.",
"(2008) considered the potential of a spherical stellar bulge and an analytical expression for disc potential, but no dark matter halo, and an ISM that is stratified in $z$ -direction with an effective sound speed that is $\\sim 5$ times the normal gas sound speed.",
"Suchkov et al.",
"(1994) considered the potential of a spherical bulge and a dark matter halo and an initial ISM that is spherically stratified.",
"Fragile et al.",
"(2004) considered a spherical halo and a $z$ -stratified ISM.",
"However, in a recent simulation of outflows driven by supernovae from disc galaxies, Dubois & Teyssier (2008) found that the outflowing gas has to contend with infalling material from halo, which inhibits the outflow for a few Gyr.",
"Fujita et al.",
"(2004) also studied outflows from pre-formed disc galaxies in the presence of a cosmological infall of matter.",
"We choose a $z$ -stratified gas to fill the simulation box, with a scale height of 100 pc.",
"For the M$_2$ and M$_3$ case (of exponential disc), we also assume a radial profile for the initial gas, with a scale length of 5 kpc.",
"For the M$_3$ case, we further assume this gas to rotate with $v_\\phi $ decreasing with a scale height of 5 kpc.",
"These values are consistent with the observations of Dickey & Lockman (1990) and Savage et al.",
"(1997) for the warm neutral gas ($T \\sim 10^4$ K) in Milky Way.",
"We note that although the scale height for the warm neutral gas in our Galaxy is $\\sim 400$ pc at the solar vicinity, this is expected to be smaller in the central region because of strong gravity due to bulge.",
"The density of the gas just above the disc is assumed to be $0.1$ particles /cc ($0.025$ in simulation units).",
"Furthermore, the adiabatic index of the gas is $5/3$ and the gas is assumed initially to be at the same temperature corresponding to an initial sound speed $c_s(ini)=0.1v_{ref}$ , a value which is consistent with the values in our Galaxy for the warm ionized gas with sound speed $\\sim 18$ km s$^{-1}$ .",
"Figure: Rotation curves corresponding to the gravitational fields of an exponentialdisc, bulge and haloare shown here in the units of v ref [=200v_{ref} [=200 km s -1 ]^{-1}], along with thetotal rotation curve.",
"The approximation used inour simulation is shown by thick red line.Our computation domain is $r_d~\\times ~r_d$ in the $r-z$ plane, with a resolution $512~\\times ~512$ cells.",
"The size of individual computational cell is $\\sim 20$ pc.",
"We have imposed reflective boundary condition around the axis and zero rotational velocity on the axis.",
"Continuous boundary conditions are imposed at $r=r_d$ and $z=r_d$ .",
"The lower boundary is slightly above the galactic disc with an offset $z_0=0.01$ .",
"We impose fixed boundary condition at lower $z$ boundary.",
"The velocity of the injected matter is $v_z(r,z_0)=v_0=10^{-5} v_{ref}$ , and its density is given by, $\\rho (r,z_0) &=& \\rho _{z_0}, ~~ \\mbox{for UD} \\\\ \\nonumber &=& \\rho _{z_0}\\mbox{exp}\\left(-\\frac{r}{r_s}\\right), ~~ \\mbox{for ED} \\,.$ The density of the injected matter at the base $\\rho _{z_0}=0.025$ (corresponding to $0.1$ protons per cc).",
"For the case of exponential disc with rotation (M$_3$ ), we assume for the injected matter to have an angular momentum corresponding to an equilibrium rotation profile.",
"We show in Figure REF the rotation curves at $z=0$ for all components (disc, bulge and halo) separately and the total rotation curve.",
"We use the following approximation (shown by thick red line in Figure REF ) which matches the total rotation curve, $v_{\\phi }(r,z_0) = 1.6 \\, v_c~[1-\\mbox{exp}(-r/0.15r_d)] \\,.",
"$ We assume a bulge of mass $M_b=0.1 M_{ref}$ and radius $r_b=0.2L_{ref}$ .",
"The scale radius for NFW halo (R$_s$ ) is determined for a halo mass $M_h=20 M_d$ , as prescribed by MMW98.",
"The corresponding disc scale radius is found to be $r_s\\sim 5.8$ kpc, again using MMW98 prescriptions.",
"Therefore we set the disc scale length for the ED case to be $r_s\\sim 0.58 L_{ref}$ .",
"The above initial conditions have been chosen to satisfy the following requirements in order to sustain a radiatively driven wind as simulated here.",
"The strong coupling between dust grains and gas particles require that there are of order $\\sim m_d/m_p$ number of collisions between protons and dust grains of mass $m_d \\sim 10^{-14}$ g, for size $a \\sim 0.1 \\, \\mu $ m with density $\\sim 3 $ g cm$^{-3}$ .",
"To ensure sufficient number of collisions, the number density of gas particles should be $n \\ge {m_d \\over m_p} {1 \\over \\pi a^2} {1 \\over L_{ref}} \\sim 10^{-3} $ cm$^{-3}$ , for $L_{ref}=10$ kpc.",
"The time scale for radiative cooling of the gas, assumed to be at $T\\sim 10^4$ K, is $t_{cool}\\sim {1.5 kT \\over n \\Lambda }$ , where $\\Lambda \\sim 10^{-23}$ erg cm$^3$ s$^{-1}$ (Sutherland & Dopita 1993; Table 6) for solar metallicity.",
"The typical density filling up the wind cone in the realistic case (M$_3$ ) is $\\sim 10^{-3}\\hbox{--}10^{-4}$ cm$^{-3}$ , which gives $t_{cool}\\sim 8\\hbox{--}80$ Myr and the dynamical time scale of the wind is $t_{ref}\\sim 50$ Myr.",
"Hence radiative cooling is marginally important and we will address the issue of radiative cooling in a future paper.",
"Radiative transfer effects are negligible since the total opacity along a vertical column of length $L_{ref}$ is $\\kappa (n m_p) L_{ref}\\sim 0.003$ , for $n\\sim 10^{-3}$ cm$^{-3}$ and $\\kappa \\sim 100$ cm$^2$ g$^{-1}$ .",
"The mediation of the radiation force by dust grains also implies that the gas cannot be too hot for the dust grains to be sputtered.",
"The sputtering radius of grains embedded in even in a hot gas of temperature T$\\sim 10^5$ K is $\\sim 0.05 (n/0.1 \\, / {\\rm cc}) \\, \\mu $ m in a time scale of 100 Myr (Tielens et al.",
"1994), and this effect is not important for the temperature and density considered here.",
"Table: Models."
],
[
"Simulation set up",
"We present 3 models with parameters listed in the Table REF .",
"The initial condition for all the models are described in §3.1.",
"The boundary condition is essentially same, except that the mass flux into the computational domain from the lower $z$ boundary depends on the type of disc.",
"As has been mentioned in section 3.1, we keep the velocity of injected matter very low, $v_z(r,z_0)=v_z(ini)=10^{-5} v_{ref}$ , so that it does not affect the dynamics.",
"The three models have been constructed by a combination of different values of three parameters $\\Gamma _0$ , $v_{\\phi }$ and the distribution of the density in the disc.",
"Model M$_3$ has been run for different values of $\\Gamma _0$ , to ascertain the effect of radiation.",
"Figure: M 1 :_1: Logarithmic density contours for radiation driven wind from UD for four snapshots runningup to t=98t=98 Myr, with velocity vectors shown with arrows.Densities are colour-coded according to the computational unit of density, 6.7×10 -24 6.7\\times 10^{-24} g cm -3 ∼4m p ^{-3} \\sim 4 m_p cm -3 ^{-3}."
],
[
"Results",
"In Figure REF , we present the model $M_1$ for a constant surface density disc (UD).",
"The density contour and the velocity vectors for the wind are shown in four snapshots in Figure (REF ) upto a time $t=98$ Myr (corresponding to $t=2$ in computational time units).",
"There are a few aspects of the gaseous flow that we should note here.",
"Firstly, the disc and the outflowing gas in this case has no rotation ($v_\\phi =0$ ).",
"In the absence of the centrifugal force due to rotation which might have reduced the radial gravitational force, there is a net radial force driving the gas inward.",
"At the same time, the radiation force, here characterized by $\\Gamma _0=2$ , propels the gas upward (the radial component of radiation being weak).",
"The net result after a few Myr is that the gas in the region near the pole moves in the positive $z$ direction, and there is a density enhancement inside a cone around the pole, away from which the density and velocities decrease.",
"Figure: M 2 :_2: Logarithmic density contours for radiation driven wind from ED for four snapshots runningup to t=98t=98 Myr, with velocity vectors shown with arrows.Also, because of the strong gravity of the bulge, the gas tends to get trapped inside the bulge region, and even the gas at larger $r$ tends to get dragged towards the axis.",
"This region puffs due to accumulation of matter.",
"Ultimately the radiative force drives matter outwards in the form of a plume.",
"Next, we change the disc mass distribution and simulate the case of wind driven out of an exponential disc (ED).",
"We show the results in Figure REF .",
"Since both gravity and radiation forces in this case of exponential disc are quasi-spherical in nature, therefore in the final snapshot the flow appears to follow almost radial streamlines.",
"Although in the vicinity of the disc, the injected matter still falls towards the axis, but this is not seen at large height as was seen in the previous case of M$_1$ .",
"This makes the wind cone of rising gas more diverging than in the case of UD (M$_1$ )."
],
[
"Rotating wind from exponential disc",
"The direction of the fluid flow in M$_1$ and M$_2$ is by and large towards the axis, and this flow is mitigated in the presence of rotation in the disc and injected gas.",
"In the next model M$_3$ , we consider rotating matter being injected into the computational domain and which follows a $v_{\\phi }$ distribution given by Eq.",
"(REF ).",
"This is reasonable to assume since the disc from which the wind is supposed to blow, is itself rotating.",
"In M$_3$ , we simulate rotating gas being injected above a ED and being driven by a radiation force of $\\Gamma _0=2$ .",
"We present nine snapshots of the M$_3$ case in Figure REF .",
"The first six snapshots of Figure REF show the essential dynamics of the outflowing gas.",
"The fast rotating matter from the outer disc is driven outward because the radial gravity component is balanced by rotation.",
"Near the central region, rotation is small and also the radial force components are small.",
"Therefore the gas is mostly driven vertically.",
"The injected gas reaches a vertical height of $\\sim 5$ kpc in a time scale of $\\sim 37$ Myr (t=0.75).",
"The flow reaches a steady state after $\\sim 60$ Myr (t=1.25).",
"In the steady state we find a rotating and mildly divergent wind.",
"We show the azimuthal velocity contours in Figure REF in colour for the fully developed wind (last snapshot in M$_3$ ), and superpose on it the contour lines of $\\rho $ .",
"The density contours clearly show a conical structure for outflowing gas.",
"The rotation speed of the gas peaks at the periphery of the cone, and is of order $\\sim 50\\hbox{--}100$ km s$^{-1}$ .",
"Compared to the disc rotation speed, the rotation speed of the wind region is somewhat smaller.",
"In other words, we find the wind mostly consisting of low-angular momentum gas lifted from the disc.",
"Figure: M 3 _3: Contours of log 10 (ρ)\\log _{10} (\\rho ) and 𝐯{\\bf v}-field of radiation driven wind with Γ 0 =2.0\\Gamma _0=2.0 from anED.",
"t = 2 corresponds to 98 Myr.Figure: The rotation velocity v φ v_\\phi for the case M 3 _3 at a time of 98 Myris shown in colours.Contour lines of log 10 _{10}(ρ)(\\rho ) are plotted over it.Figure: The axial velocity v z (0 + ,z)v_z(0^+,z) with zz at different time steps for the model M 3 _3.",
"t = 2.0 corresponds to a time of 98Myr.Figure: The axial velocity v z (0 + ,10𝑘𝑝𝑐)v_z(0^+,10 {\\it kpc}) in simulation units v ref =200v_{ref} = 200 km s -1 ^{-1} with Γ 0 \\Gamma _0,at a time t∼10 2 t\\sim 10^2 Myr.We plot the velocity of gas close to the axis in Figure REF for different times in this model (M$_3$ ), using ${\\bf v}(0,z)\\sim v_z(0^+,z)$ .",
"The velocity profile in the snapshots at earlier time fluctuates at different height, but becomes steady after t $\\ge 1.5$ , as does the density profile.",
"We have run this particular case of ED with rotation (model M$_3$ ) for different values of $\\Gamma _0$ .",
"In order to illustrate the results of these runs, we plot the $z$ -component of velocity ($v_z (0^+,10\\ kpc)$ ) at 10 kpc and at simulation time, $t=2$ as a function of $\\Gamma _0$ in Figure REF .",
"We find that significant wind velocities are obtained for $\\Gamma _0 \\gtrsim 1.5$ and wind velocities appear to rise linearly with $\\Gamma _0$ after this critical value is acheived.",
"Sharma et al.",
"(2011) found this critical value to be $\\Gamma _0 \\sim 2$ for a constant density disc and wind launched above the bulge.",
"For the realistic case of an exponential disc, we find in the present simulation the critical value to be somewhat smaller than but close to the analytical result.",
"The important point is that the critical $\\Gamma _0$ is not unity.",
"This is because the parameter $\\Gamma _0$ is not a true Eddington parameter since it is defined in terms of disc gravity and radiation, whereas halo and bulge also contribute to gravity."
],
[
"Discussions",
"Our simulation differs from earlier works (e.g.",
"Suchkov et.al.",
"1994) mainly in that we specifically target warm outflows and the driving force is radiation pressure.",
"Most of the previous simulations of galactic wind have used energy injected from supernovae blasts as a driving force.",
"However, with the ideas presented in Murray et al.",
"(2005), which worked out the case of radiation pressure in a spherical symmetric set-up, it beomes important to study the physics of this model in an axisymmetric set up, as has been done analytically by Sharma et al.",
"(2011) (see also, Zhang & Thompson 2010).",
"Also we have tried to capture all features of a typical disc galaxy like a bulge and a dark matter halo, and a rotating disc.",
"Recent analytical works (Sharma & Nath 2011) and simulations (Hopkins et al.",
"2011) have shown that outflows from massive galaxies ($M_{halo}\\ge 10^{12}$ M$_{\\odot }$ ) have different characteristics than those from low mass galaxies.",
"Outflows from massive galaxies are mostly driven by radiation pressure and the fraction of cold gas in the halos of massive galaxies is large (van de Voort & Schaye 2011).",
"Our simulations presented here addresses these outflows in particular.",
"We have parameterized our simulation runs with the disc Eddington factor $\\Gamma _0$ , and it is important to know the corresponding luminosity for a typical disc galaxy, or the equivalent star formation rate.",
"For a typical opacity of a dust and gas mixture ($\\kappa \\sim 200$ cm$^2$ g$^{-1}$ ) (Draine 2011), the correspondig mass-to-light ratio requirement for $\\Gamma _0 \\gtrsim 1.5$ is that $M/L \\le 0.03$ .",
"Sharma et al.",
"(2011) showed that for the case of an instantaneous star formation, $\\Gamma _0 \\gtrsim 2$ is possible for an initial period of $\\sim 10$ Myr after the starburst.",
"However for a continuous star formation, which is more realistic for disc galaxies, Sharma & Nath (2011) found that only ultra luminous infrared galaxies (ULIGs), with star formation rate larger than $\\sim 100$ M$_{\\odot }$ yr$^{-1}$ and which are also massive, are suitable candidates for such large values of $\\Gamma _0$ , and for radiatively driven winds.",
"The results presented in the previous sections show that the outflowing gas within the central region of a few kpc tends to stay close to the pole, and does not move outwards because of its low angular momentum.",
"This makes the outflow somewhat collimated.",
"Although outflows driven by SN heated hot wind also produces a conical structure (e.g., Fragile et al.",
"2004) emanating from a breakout point of the SN remnants, there is a qualitative difference between this case and that of radiatively driven winds as presented in our simulations.",
"While it is the pressure of the hot gas that expands gradually as it comes out of a stratified atmosphere, in the case of a radiation driven wind, it is the combination of mostly the lack of rotation and almost vertical radiation driving force in the central region that produce the collimation effect.",
"We also note that the conical structure of rotation in the outflowing gas is similar to the case of outflow in M82 (Greve 2004), where one observes a diverging and rotating periphery of conical outflow.",
"We have not considered radiative cooling in our simulations, since for typical density in the wind the radiative cooling time is shorter or comparable than the dynamical time.",
"However, there are regions of higher density close to the base and radiative cooling can be important there.",
"We will address this point in a future paper.",
"From our results of the exponential and rotating disc model, we find the wind comprising of low-angular momentum gas lifted from the disc.",
"It is interesting to note that recent simulations of supernovae driven winds have also claimed a similar result (Governato et al.",
"2010).",
"Such loss of low angular momentum gas from the disc may have important implication for the formation and evolution of the bulge, since the bulge population is deficient in stars with low specific angular momentum.",
"Binney, Gerhardt & Silk (2001) have speculated that outflows from disc that preferentially removes low angular momentum material may resolve some discrepancies between observed properties of disc and results of numerical simulations.",
"As a caveat, we should finally note that the scope and predictions of our simulation is limited by the simple model of disc radiation adoped here.",
"In reality, radiation from disks is likely to be confined in the vicinity of star clusters, and not spread throughout the disk as we have assumed here.",
"This is likely to increase the efficacy of radiation pressure, but which is not possible within the scope of an axisymmetric simulation."
],
[
"Summary",
"We have presented the results of hydrodynamical (Eulerian) simulations of radiation driven winds from disc galaxies.",
"After studying the cases of winds from a constant surface density disc and exponential disc without rotation, we have studied a rotating outflow originating from an exponential disc with rotation.",
"We find that the outflow speed increases rapidly with the disc Eddington parameter $\\Gamma _0=\\kappa I/(2 c G \\Sigma )$ for $\\Gamma _0 \\ge 1.5$ , consistent with theoretical expectations.",
"The density structure of the outflow has a conical appearance, and most of the ouflowing gas consists of low angular momentum gas.",
"We thank Yuri Shchekinov for constructive comments and critical reading of the manuscript.",
"IC acknowledges the hospitality of the Astronomy and Astrophysics Group of Raman Research Institute, where the present work was conceived.",
"DR was supported by National Research Foundation of Korea through grant 2007-0093860."
],
[
"Appendix",
"Consider a razor thin disc in r$\\phi $ plane as illustrated in the fig.",
"Now our task is to calculate the force components at any arbitrary point above the disc.",
"Let us consider an annulus of the disc between $r\\prime $ and $r\\prime +dr\\prime $ .",
"Area of the element at point P($r\\prime ,\\phi \\prime ,0$ ) is $r\\prime dr\\prime d\\phi \\prime $ .",
"Also take a field point Q(r,0,z) above the disc plane.",
"Azimuthal coordinate of Q is taken to be zero for simplicity as we know that azimuthal force components are zero due to symmetry.",
"Let QN and QM be perpendiculars from Q on the x and the z axis, respectively.",
"So we can write, $PN^2 &=& (r-r\\prime cos\\phi \\prime )^2 + (r\\prime sin\\phi \\prime )^2 \\nonumber \\\\PQ^2 &=& PN^2 + z^2 = r^2+z^2+r\\prime ^2-2rr\\prime cos\\phi \\prime \\nonumber \\\\sin\\angle PQN &=& \\frac{PN}{PQ}$ The gravitational force due to the small area element at P is given by $d{\\bf F_g} = \\frac{G\\ dm\\ PQ}{(PQ)^3} \\hat{n}\\ ;\\\\dm = r\\prime dr\\prime d\\phi \\prime \\Sigma (r\\prime )$ Here $\\Sigma (r\\prime )$ is the surface density of the disc.",
"Now the z component of this force is $dF_{g,z} = |d{\\bf F_g}| \\frac{z}{PQ} = \\frac{z G \\Sigma (r\\prime )\\ r\\prime dr\\prime d\\phi \\prime }{[r^2+z^2+r\\prime ^2-2rr\\prime cos\\phi \\prime ]^{3/2}}$ To calculate the radial component, let PS be the perpendicular from P on the x-axis.",
"Then, we have sin$\\angle $ SPN = SN/PN = (r-$r\\prime cos\\phi \\prime $ )/PN.",
"Component of the force along the direction of PN is $dF_{g,PN} = |d{\\bf F_g}| sin\\angle PQN = |d{\\bf F_g}|\\frac{SN}{PN}$ So the radial component is $dF_{g,r} &=& d F_{g,PN} sin\\angle SPN = |d{\\bf F_g}| \\frac{SN}{PN}\\frac{PN}{PQ} \\nonumber \\\\&=& \\frac{(r-r\\prime cos\\phi \\prime )\\ G \\Sigma (r\\prime )\\ r\\prime dr\\prime d\\phi \\prime }{[r^2+z^2+r\\prime ^2-2rr\\prime cos\\phi \\prime ]^{3/2}}$ Figure: Schematic diagram for the calculation of gravitational force due to discin the xyxy-plane.We consider an annulus in the disc andan element of area around the point P (r',φ,0)r \\prime , \\phi , 0) in thisannulusis considered here in order to compute the force at a point Q (r,0,zr,0,z)whose azimuthal coordinate φ=0\\phi =0.",
"The point S (r'cosφ,0,0r\\prime \\cos \\phi ,0,0) is the foot of the perpendicular drawn from P on the xx-axis.The point S'\\prime (r',0,zr \\prime ,0,z) is at the intersection of the vertical from S (along zz-axis)and the line parallel to xx-axis at height zz.",
"The angle∠SQS'=cos -1 S'Q PQ\\angle SQS\\prime = \\cos ^{-1} \\Bigl [ {S\\prime Q \\over PQ} \\Bigr ],and ∠PQN=cos -1 QN PQ\\angle PQN =\\cos ^{-1} \\Bigl [ {QN \\over PQ} \\Bigr ]."
]
] | 1204.1133 |
[
[
"State/Operator Correspondence in Higher-Spin dS/CFT"
],
[
"Abstract A recently conjectured microscopic realization of the dS$_4$/CFT$_3$ correspondence relating Vasiliev's higher-spin gravity on dS$_4$ to a Euclidean $Sp(N)$ CFT$_3$ is used to illuminate some previously inaccessible aspects of the dS/CFT dictionary.",
"In particular it is argued that states of the boundary CFT$_3$ on $S^2$ are holographically dual to bulk states on geodesically complete, spacelike $R^3$ slices which terminate on an $S^2$ at future infinity.",
"The dictionary is described in detail for the case of free scalar excitations.",
"The ground states of the free or critical $Sp(N)$ model are dual to dS-invariant plane-wave type vacua, while the bulk Euclidean vacuum is dual to a certain mixed state in the CFT$_3$.",
"CFT$_3$ states created by operator insertions are found to be dual to (anti) quasinormal modes in the bulk.",
"A norm is defined on the $R^3$ bulk Hilbert space and shown for the scalar case to be equivalent to both the Zamolodchikov and pseudounitary C-norm of the $Sp(N)$ CFT$_3$."
],
[
"1.3 State/Operator Correspondence in Higher-Spin DS/CFT Gim Seng Ng and Andrew Strominger Center for the Fundamental Laws of Nature, Harvard University, Cambridge, MA 02138, USA A recently conjectured microscopic realization of the dS$_4$ /CFT$_3$ correspondence relating Vasiliev's higher-spin gravity on dS$_4$ to a Euclidean $Sp(N)$ CFT$_3$ is used to illuminate some previously inaccessible aspects of the dS/CFT dictionary.",
"In particular it is argued that states of the boundary CFT$_3$ on $S^2$ are holographically dual to bulk states on geodesically complete, spacelike $R^3$ slices which terminate on an $S^2$ at future infinity.",
"The dictionary is described in detail for the case of free scalar excitations.",
"The ground states of the free or critical $Sp(N)$ model are dual to dS-invariant plane-wave type vacua, while the bulk Euclidean vacuum is dual to a certain mixed state in the CFT$_3$ .",
"CFT$_3$ states created by operator insertions are found to be dual to (anti) quasinormal modes in the bulk.",
"A norm is defined on the $R^3$ bulk Hilbert space and shown for the scalar case to be equivalent to both the Zamolodchikov and pseudounitary C-norm of the $Sp(N)$ CFT$_3$ .",
"section1-3.5ex plus-1ex minus-.2ex2.3ex plus.2exIntroduction The conjectured dS/CFT correspondence attempts to adapt the wonderful successes of the AdS/CFT correspondence to universes (possibly like our own) which exponentially expand in the far future.",
"The hope [1], [2], [3], [4], [5] is to define bulk de Sitter (dS) quantum gravity in terms of a holographically dual CFT living at ${\\cal I}^+$ of dS, which is the asymptotic conformal boundary at future null infinity.",
"A major obstacle to this program has been the absence of any explicit microscopic realization.",
"This has so far prevented the detailed development of the dS/CFT dictionary.",
"This situation has recently been improved by an explicit proposal [6] relating Vasiliev's higher-spin gravity in dS$_4$ [7], [8] to the dual $Sp(N)$ CFT$_3$ described in [9].",
"In this paper we will use this higher-spin context to write some new entries in the dS/CFT dictionary.",
"The recent proposal [6] for a microscopic realization of dS/CFT begins with the duality relating the free (critical) $O(N)$ CFT$_3$ to higher-spin gravity on AdS$_4$ with Neumann (Dirichlet) boundary conditions on the scalar field.",
"Higher-spin gravity - unlike string theory [2] - has a simple analytic continuation from negative to positive cosmological constant $\\Lambda $ .",
"Under this continuation, AdS$_4 \\rightarrow $ dS$_4$ and the (singlet) boundary CFT$_3$ correlators are simply transformed by the replacement of $N\\rightarrow -N$ .",
"These same transformed correlators arise from the $Sp(N)$ models constructed from anticommuting scalars.",
"It follows that the free (critical) $Sp(N)$ correlators equal those of higher-spin gravity on dS$_4$ with future Neumann (Dirichlet) scalar boundary conditions (of the type described in [10]) at ${\\cal I}^+$ .",
"This mathematical relation between the bulk dS and boundary $Sp(N)$ correlators may provide a good starting point for understanding quantum gravity on dS, but so far important physical questions remain unanswered.",
"For example we do not know how to relate these physically $un$ measurable correlators to a set of true physical observables or to the dS horizon entropy.",
"These crucial entries in the dS/CFT dictionary are yet to be written.",
"As a step in this direction, in this paper we investigate the relation between quantum states in the bulk higher-spin gravity and those in the boundary CFT$_3$ .",
"Bulk higher-spin gravity has fields of $\\Phi ^s$ with all even spins $s=0,2,....$ , which are dual to CFT$_3$ operators ${\\cal O}^s$ with the same spins.",
"In the CFT$_3$ , we can also associate a state to each operator by the state-operator correspondence.",
"One way to do this is to take the southern hemisphere of $S^3$ , insert the operator ${\\cal O}^s$ at the south pole, and then define a state $\\Psi ^s_{S^2}$ as a functional of the boundary conditions on the equatorial $S^2$ .",
"For every object in the CFT$_3$ , we expect a holographically dual object in the bulk dS$_4$ theory.",
"This raises the question: what is the bulk representation of the spin-$s$ state $\\Psi ^s_{S^2}$ ?",
"In Lorentzian AdS$_4$ holography, the state created by a primary operator ${\\cal O}$ in the CFT$_{3}$ on S$^{2}$ has, at weak coupling, a bulk representation as the single particle state of the field $\\Phi $ dual to ${\\cal O}$ with a smooth minimal-energy wavefunction localized near the center of AdS$_4$ .",
"The form of the wavefunction is dictated by the conformal symmetry.",
"In dS$_4$ holography, the situation is rather different.",
"States in dS$_4$ quantum gravity are usually thought of as wavefunctions on complete spacelike slices which are topologically $S^3$ .As explained in [4], [5], such states do play an important role in dS/CFT, but as generating functions for correlators rather than as duals to CFT$_3$ states on $S^2$ .",
"The relation between the $R^3$ and $S^3$ bulk states in our example is detailed below.",
"These do not seem to be good candidates for bulk duals to $\\Psi ^s_{S^2}$ because, among other reasons, they are not associated to any $S^2$ in ${\\cal I}^+$ .",
"However, dS$_4$ also has everywhere spacelike and geodesically complete $R^3$ slices which end at an $S^2$ in ${\\cal I}^+$ .",
"Here we propose a construction of the bulk version of $\\Psi ^s_{S^2}$ on these slices, again as single particle states whose form is dictated by the conformal symmetry.",
"Interestingly, the classical wavefunction for the particle turns out to be the (anti) quasinormal modes for the static patch of de Sitter, as constructed in [11], [12].We are grateful to D. Anninos for pointing this out [13].",
"This relation between bulk and boundary states has a potentially profound nonperturbative consequence briefly mentioned in section 4.1 [14].",
"The operator ${\\cal O}^0$ dual to a scalar $\\Phi $ is bilinear in boundary fermions and hence obeys $({\\cal O}^0)^{{N 2}+1}=0$ .",
"Under bulk-boundary duality this translates into an $N 2$ -adicity relation for $\\Phi $ : one cannot put more than $N 2$ bulk scalar quanta into the associated quasinormal mode.",
"Further investigation of this dS exclusion principle is deferred to later work.",
"We also construct a norm for these bulk states and show that it is the Zamolodchikov norm on $S^3$ of the CFT$_3$ operator ${\\cal O}^s$ .",
"Explicit formulae are exhibited only for the scalar $s=0$ case but we expect the construction to generalize to all $s$ .",
"This paper is organized as follows.",
"In section 2 we revisit the issue of the usual global dS-invariant vacua for a free massive scalar field, paying particular attention to the case of $m^2\\ell ^2=2$ (where $\\ell $ is the de Sitter radius) arising in higher spin gravity.",
"The invariant vacua include the familiar Bunch-Davies Euclidean vacuum $|0_E\\rangle $ , as well as a pair of $|0^\\pm \\rangle $ of in/out vacua with no particle production.",
"As the scalar field acting on $|0^-\\rangle $ ($|0^+\\rangle $ ) obeys Dirichlet (Neumann) boundary conditions on ${\\cal I}^+$ , these are related to the critical (free) $Sp(N)$ model.",
"Generically all dS-invariant vacua are Bogolyubov transformations of one another, but we find that at $m^2\\ell ^2=2$ the transformation is singular and the in/out vacua are non-normalizable plane-wave type states.",
"In section 3 we use the conformal symmetries to find the classical bulk wavefunctions associated to an operator insertion on ${\\cal I}^+$ , and note the relation to (anti) quasinormal modes.",
"The construction uses a rescaled bulk-to-boundary Green function defined with Neumann or Dirichlet ${\\cal I}^+$ boundary conditions.",
"We also show that the Klein-Gordon inner product of these wavefunctions agrees with the conformally-covariant CFT$_3$ operator two-point function.",
"In section 4 we consider the Hilbert space on $R^3$ slices ending on an $S^2$ on ${\\cal I}^+$ .",
"This Hilbert space was explicitly constructed in [15] for a free scalar on hyperbolic slices ending on ${\\cal I}^+$ .",
"There are two such Hilbert spaces, which we denote the northern and southern Hilbert space, which live on spatial $R^3$ slices extending to the north or south of the $S^2$ .",
"The northern and southern slices add up to a global $S^3$ .",
"Hence the tensor product of the northern and southern Hilbert spaces is the global Hilbert space on $S^3$ , much as the left and right Rindler Hilbert spaces tensor to the global Minkowski Hilbert space.",
"We show that the global $|0^\\pm \\rangle $ vacua are simple tensor products of the northern and southern Dirichlet and Neumann vacua.",
"We then use symmetries to uniquely identify the states of the southern Hilbert space with those of the free and critical $Sp(N)$ models on an $S^2$ .",
"This leads directly to the dS exclusion principle.",
"We further construct an inner product for the southern Hilbert space which agrees, for states dual to ${\\cal I}^+$ operator insertions, to the conformal two-point function on ${\\cal I}^+$ .",
"In section 5 we discuss the restriction of Euclidean vacuum to a southern state and recall from [15], that this is a mixed state which is thermal with respect to an $SO(3,1)$ Casimir.",
"It would be interesting to relate this result to dS entropy in the present context.",
"In section 6 we show that the standard CFT$_3$ state-operator correspondence maps the known pseudo-unitary C-norm of the $Sp(N)$ model to the Zamolodchikov two-point function.",
"This completes the demonstration that the bulk states on $R^3_S$ have the requisite properties to be dual to the boundary $Sp(N)$ CFT$_3$ states on $S^2$ .",
"Speculations are made on the possible relevance of pseudo-unitarity to the consistency of dS/CFT in general.",
"An appendix gives some explicit formulae for the $SO(4,1)$ Killing vectors of dS$_4$ .",
"section1-3.5ex plus-1ex minus-.2ex2.3ex plus.2exGlobal dS vacua at $m^2\\ell ^2=2$ In this section we describe the quantum theory of a free scalar field $\\Phi $ in dS$_4$ with wave equation $(\\nabla ^2-m^2)\\Phi =0, $ and mass $m^2\\ell ^2=2.",
"$ This is the case of interest for Vasiliev's higher-spin gravity.",
"While there have been many general discussions of this problem, peculiar singular behavior as well as simplifications appear at the critical value $m^2\\ell ^2=2$ which are highly relevant to the structure of dS/CFT.",
"A parallel discussion of de Sitter vacua and scalar Green functions in the context of dS/CFT was given in [16].",
"However that paper in many places specialized to the large mass regime $m^2\\ell ^2>{9 4}$ , excluding the region of current interest.",
"The behavior in the region $m^2\\ell ^2<{9 4}$ divides into three cases $m^2\\ell ^2>{2}$ , $m^2\\ell ^2=2$ and $m^2\\ell ^2<2$ .",
"Much of the structure we describe below pertains to the entire range $m^2\\ell ^2<{9 4}$ with an additional branch-cut prescription for the Green functions.",
"subsection2-3.25explus-1ex minus-.2ex1.5ex plus.2exModes We will work in the dS$_4$ global coordinates $ {ds^2 \\ell ^2}=-dt^2+\\cosh ^2 t d^2\\Omega _3=-dt^2+\\cosh ^2 t \\left[d\\psi ^2+\\sin ^2{\\psi } \\left( d\\theta ^2+\\sin ^2{\\theta } d\\phi ^2\\right) \\right], $ where ${\\Omega }^i \\sim (\\psi ,\\theta ,\\phi )$ are coordinates on the global $S^3$ slices.",
"Following the notation of [16] solutions of the wave equation can be expanded in modes $\\phi _{Lj}(x)=y_L(t)Y_{Lj}(\\Omega ) $ of total angular momentum $L$ and spin labeled by the multi-index $j$ .",
"The spherical harmonics $Y_{Lj}$ obey $Y^*_{Lj}(\\Omega )&=&(-)^LY_{Lj}(\\Omega )=Y_{Lj}(\\Omega _A),\\cr D^2Y_{Lj}(\\Omega )&=&-L(L+2)Y_{Lj}(\\Omega ),\\cr \\int _{S^3}\\sqrt{h}d^3\\Omega Y^*_{Lj}(\\Omega )Y_{L^{\\prime }j^{\\prime }}(\\Omega )&=&\\delta _{L,L^{\\prime }}\\delta _{j,j^{\\prime }},\\cr \\sum Y^*_{Lj}(\\Omega )Y_{Lj}(\\Omega ^{\\prime })&=&{1{\\sqrt{h}}}\\delta ^3(\\Omega -\\Omega ^{\\prime }),$ where $\\sqrt{h}$ and $D^2$ are the measure and Laplacian on the unit $S^3$ , $\\Omega _A$ is the antipodal point of $\\Omega $ , and here and hereafter $\\sum $ denotes summation over all allowed values of $L$ and $j$ .",
"The time dependence is then governed by the second order ODE $ \\partial _t^2y_L+3\\tanh t \\partial _t y_L+\\left(m^2\\ell ^2+{L(L+2) \\cosh ^2t}\\right)y_L=0.$ Eq.",
"(REF ) has the $real$ solutions $y^\\pm _L={2^{L+h_\\pm +{1 2}}}(L+1)^{\\pm {1 2}}\\cosh ^L te^{-(L+h_\\pm ) t}F(L+{32},L+h_\\pm , h_\\pm -\\frac{1}{2};-e^{-2t}) $ where $h_{\\pm } \\equiv \\frac{3}{2} \\pm \\sqrt{\\frac{9}{4}-m^2 \\ell ^2}.$ We are interested in $m^2\\ell ^2=2$ , which implies $h_-=1,~~~~h_+=2,$ and $y^\\pm _L = \\frac{ (-i)^{{1 2}\\pm {1 2}} 2^L}{\\sqrt{1+L}}\\cosh ^L t e^{-(L+1)t}\\left[\\frac{1}{\\left(1-i e^{-t}\\right)^{2L+2}}\\mp \\frac{1}{\\left(1+i e^{-t}\\right)^{2L+2}}\\right].$ The modes behave near ${\\cal I}^+$ as $e^{-h_\\pm t}$ $ t&\\rightarrow & \\infty ,~~~~~\\cr y_L^-&\\rightarrow &(2(L+1)^{- {1 2}} )e^{-t}+{\\cal O}(e^{-3t})~~~~~{\\rm Neumann},\\cr ~~~~~y_L^+&\\rightarrow & (4(L+1)^{ {1 2}} )e^{-2t}+{\\cal O}(e^{-4t})~~~~~{\\rm Dirichlet}.$ Accordingly we refer to the $+$ modes as Dirichlet and the $-$ modes as Neumann.",
"We have normalized so that the Klein-Gordon inner product is $ \\langle \\phi ^+_{Lj} |\\phi ^-_{L^{\\prime }j^{\\prime }}\\rangle _{S^3}\\equiv i \\int _{S^3}d^3\\Sigma ^\\mu \\phi ^{+*}_{Lj}\\overleftrightarrow{ \\partial _\\mu }\\phi ^-_{L^{\\prime }j^{\\prime }} = i \\delta _{LL^{\\prime }}\\delta _{jj^{\\prime }},$ with $d^3\\Sigma ^\\mu $ the induced measure times the normal to the $S^3$ slice.",
"Under time reversal $y_L^\\pm (t)=\\pm (-)^Ly_L^\\pm (-t),$ so that $\\phi ^\\pm _{Lj}(x)=\\pm \\phi ^\\pm _{Lj}(x_A)=(-)^L\\phi ^{\\pm *}_{Lj}(x),$ where the point $x_A$ is antipodal to the point $x$ .",
"This implies that an incoming Dirichlet (Neumann) mode propagates to an outgoing Dirichlet (Neumann) mode.",
"This is not the case for generic $m^2$ and, as will be seen below, allows for Dirichlet and Neumann vacua with no particle production.",
"Euclidean modes are defined by the condition that when dS$_4$ is analytically continued to $S^4$ they remain nonsingular on the southern hemisphere.",
"In other words $y^E_L(t=-{i\\pi 2})={\\rm nonsingular}.$ One finds that the combination $y^E_L=\\frac{y^-_L+iy^+_L}{\\sqrt{2}}=\\frac{2^{L+1}}{\\sqrt{2L+2}}\\frac{\\cosh ^L t e^{-(L+1)t}}{\\left(1-i e^{-t}\\right)^{2L+2}}$ is nonsingular at $t=-i\\pi /2$ .",
"Hence $y_L^\\mp $ are simply the real and imaginary parts of the $y^E_L$ .",
"(REF ) and (REF ) imply the relations $y^{E*}_L(t)=(-)^{L+1}y^E_L(-t)$ $\\phi ^{E}_{Lj}(x_A)=(-)^{L+1}\\phi ^{E*}_{Lj}(x).$ $\\langle \\phi ^E_{Lj}|\\phi ^E_{L^{\\prime }j^{\\prime }}\\rangle _{S^3}=\\delta _{LL^{\\prime }}\\delta _{jj^{\\prime }}$ subsection2-3.25explus-1ex minus-.2ex1.5ex plus.2exVacua In the quantum theory $\\Phi $ is promoted to an operator which we denote $\\hat{\\Phi }$ obeying the equal time commutation relation $[ \\hat{\\Phi }(\\Omega ,t),\\partial _t\\hat{\\Phi }(\\Omega ^{\\prime },t)]={i \\sqrt{h}\\cosh ^3 t}\\delta ^3(\\Omega -\\Omega ^{\\prime }).$ Defining annihilation and creation operators $a^E_{Lj}=\\langle \\phi ^E_{Lj}|\\hat{\\Phi }\\rangle _{S^3},~~~~ a^{E\\dagger }_{Lj}=-\\langle \\phi ^{E*}_{Lj}|\\hat{\\Phi }\\rangle _{S^3},$ the global Euclidean (or Bunch-Davies) vacuum is defined by $a^E_{Lj}|0_E\\rangle =0.",
"$ We normalize so that $\\langle 0_E |0_E\\rangle =1$ .",
"For any $m^2$ there is a family of dS-invariant vacua labeled by a complex parameter $\\alpha $ .",
"They are annihilated by the normalized Bogolyubov-transformed oscillators $a^\\alpha _{Lj}={1 \\sqrt{1-e^{\\alpha +\\alpha ^*}}}\\left( a^{E}_{Lj} -e^{\\alpha ^*} a^{E\\dagger }_{Lj}\\right).",
"$ We are interested in the vacua annihilated by the Dirichlet or Neumann modes for the case of $m^2 \\ell ^2=2$ , which correspond to $e^\\alpha =\\pm 1$ .",
"In that case the Bogolyubov transformation is singular.",
"Nevertheless we can still construct non-normalizable plane-wave type vacua as follows.",
"The field operator may be decomposed as $\\hat{\\Phi }=\\hat{\\Phi }^++\\hat{\\Phi }^-,$ where $\\hat{\\Phi }^\\pm \\sim e^{-h_\\pm t}$ near ${\\cal I}^+$ .",
"The squeezed states $|0^\\pm \\rangle = e^{\\pm {1 2}\\sum (-)^L (a^{E\\dagger }_{Lj})^2}|0_E\\rangle $ then obey $ \\hat{\\Phi }^-|0^-\\rangle =0~~~~~~{\\rm Dirichlet}, $ $ \\hat{\\Phi }^+|0^+\\rangle =0~~~~~~{\\rm Neumann}.",
"$ Since only Dirichlet (Neumann) modes act non-trivially on $|0^-\\rangle $ ($ |0^+ \\rangle $ ) we refer to it as the Dirichlet (Neumann) vacuum.",
"These vacua are dS invariant.",
"With the conventional norm, $\\hat{\\Phi }^\\pm $ are hermitian and their eigenstates are non-normalizable.",
"Generalized dS non-invariant plane-wave type Neumann states with nonzero eigenvalues for $\\hat{\\Phi }^+$ $\\hat{\\Phi }^+|\\Phi ^+\\rangle = \\Phi ^+ | \\Phi ^+\\rangle $ are constructed as $|\\Phi ^+\\rangle =e^{-\\langle \\Phi ^+|\\hat{\\Phi }^-\\rangle _{S^3}} |0^+\\rangle .$ $\\Phi ^+$ here is an arbitrary solution of the classical wave equation, which can be parameterized by an arbitrary function $\\Phi ^+(\\Omega )$ on ${\\cal I}^+$ $t\\rightarrow \\infty ,~~~\\Phi ^+(\\Omega ,t)\\rightarrow \\Phi ^+(\\Omega )e^{-h_+t}.$ The states are delta-functional normalizable with respect the usual inner product $\\langle \\Phi ^+|{\\Phi ^{+}}^{\\prime }\\rangle =\\delta \\left( \\Phi ^+-{\\Phi ^+}^{\\prime } \\right),$ where the delta function integrates to one with the measure ${\\cal D}\\Phi ^+ \\equiv \\prod _{L,j}\\frac{dc^+_{Lj}}{\\sqrt{\\pi }}, ~~~\\Phi ^+(x)=\\sum c^+_{Lj}\\phi ^+_{Lj}(x).$ The $c^+_{Lj}$ satisfies the reality condition $c^{+*}_{Lj}=c^+_{Lj}(-)^L$ .",
"One may similarly define generalized Dirichlet states obeying $\\hat{\\Phi }^-|\\Phi ^-\\rangle = \\Phi ^- | \\Phi ^-\\rangle .$ The Euclidean vacuum can be expressed in terms of $|0^\\pm \\rangle $ as $|0_E>=\\int {\\cal D}\\Phi ^\\pm e^{\\mp {116} \\int d^3\\Omega d^3\\Omega ^{\\prime }\\Phi ^\\pm (\\Omega )\\Delta _\\mp (\\Omega ,\\Omega ^{\\prime })\\Phi ^\\pm (\\Omega ^{\\prime })}|\\Phi ^\\pm \\rangle ,$ where $\\Delta _{\\pm } (\\Omega , \\Omega ^{\\prime })&=&\\mp \\sum Y^*_{Lj}(\\Omega )Y_{Lj} (\\Omega ^{\\prime }) (2L+2)^{\\pm 1}=\\frac{1}{2^{2\\mp 1}\\pi ^2}\\frac{1}{\\left(1-\\cos {\\Theta _3}\\right)^{h_\\pm }}, \\nonumber \\\\\\cos {\\Theta _3(\\Omega ,\\Omega ^{\\prime })} &\\equiv &\\cos {\\psi }\\cos {\\psi ^{\\prime }} + \\sin {\\psi }\\sin {\\psi ^{\\prime }}(\\cos {\\theta }\\cos {\\theta ^{\\prime }} + \\sin {\\theta }\\sin {\\theta ^{\\prime }}\\cos {(\\phi -\\phi ^{\\prime })}).\\nonumber \\\\$ $\\Delta _{\\pm } $ are the (everywhere positive) two-point functions for a CFT$_3$ operator with $h_+=2$ and $h_-=1$ .Here, we regulate the expressions of $\\Delta _{\\pm }$ as sums over the spherical harmonics by introducing $e^{-L \\varepsilon }$ in each term in the sum and take the limit of $\\varepsilon \\rightarrow 0$ at the end after the summation.",
"These satisfy $-\\int \\sqrt{h}d^3\\Omega ^{\\prime \\prime } \\Delta _+(\\Omega ,\\Omega ^{\\prime \\prime })\\Delta _-(\\Omega ^{\\prime \\prime },\\Omega ^{\\prime })=\\frac{1}{\\sqrt{h}}\\delta ^3(\\Omega -\\Omega ^{\\prime }).$ We also have the relations $|\\Phi ^+\\rangle = \\frac{1}{N_0}\\int {\\cal D}\\Phi ^-e^{\\langle \\Phi ^-|\\Phi ^+\\rangle _{S^3}} |\\Phi ^-\\rangle ,\\quad N_0 \\equiv \\prod _{L,j} \\sqrt{2},$ $\\langle \\Phi ^- |\\Phi ^+\\rangle _{S^3} = \\frac{1}{N_0}e^{\\langle \\Phi ^-|\\Phi ^+\\rangle _{S^3}} .$ In particular $\\langle 0^- |0^+\\rangle _{S^3} =\\frac{1}{N_0}.$ The Wightman function in the Euclidean vacuum is $ G_E(x;x^{\\prime })&=&\\sum \\phi ^{E}_{Lj}(x)\\phi ^{E*}_{Lj}(x^{\\prime })\\cr &=&\\sum (-)^{L+1}\\phi ^{E}_{Lj}(x)\\phi ^E_{Lj}(x^{\\prime }_A)\\cr &=&{1 2}\\sum (-)^{L} \\left( \\phi ^{-}_{Lj}(x)\\phi ^-_{Lj}(x^{\\prime })+\\phi ^{+}_{Lj}(x)\\phi ^+_{Lj}(x^{\\prime })+i\\phi ^{+}_{Lj}(x)\\phi ^-_{Lj}(x^{\\prime })-i\\phi ^{-}_{Lj}(x)\\phi ^+_{Lj}(x^{\\prime })\\right).\\cr && $ In terms of the dS-invariant distance function $P(t,\\Omega ;t^{\\prime },\\Omega ^{\\prime }) = \\cosh {t} \\cosh {t^{\\prime }} \\cos \\Theta _3(\\Omega ,\\Omega ^{\\prime })-\\sinh {t} \\sinh {t^{\\prime }},$ this becomes simply $G_E(x;x^{\\prime })= \\frac{1}{8\\pi ^2} \\frac{1}{1-P(x;x^{\\prime })},$ with the usual $i \\varepsilon $ prescription for the singularity.",
"section1-3.5ex plus-1ex minus-.2ex2.3ex plus.2exBoundary operators and quasinormal modes According to the dS$_4$ /CFT$_3$ dictionary, for every spin zero primary CFT$_3$ operator ${\\cal O}$ of conformal weight $h$ there is a bulk scalar field $\\Phi $ with mass $m^2\\ell ^2=h(3-h)$ .",
"Boundary correlators of ${\\cal O}$ are then related by a rescaling to bulk $\\Phi $ correlators whose arguments are pushed to the boundary at ${\\cal I}^+$ .",
"As in AdS/CFT, a particular classical bulk wavefunction of $\\Phi $ can be associated to a boundary insertion of ${\\cal O}$ (at the linearized level) by symmetries: it must scale with weight $h$ under the isometry corresponding to dilations, obey the lowest-weight condition, and be invariant under rotations around the point of the boundary insertion.",
"The resulting wavefunction is a type of bulk-to-boundary Green function.",
"Interestingly[13], the (lowest) highest-weight modes can also be identified as (anti) quasinormal modes for the static patch of de Sitter, as constructed in [11], [12].",
"In this section we determine this wavefunction explicitly, regulate the singularities, generalize it to multi-particle insertions and define a symplectic product.",
"In the following section we will then use these classical objects to construct the associated dual bulk quantum states and their inner products.",
"subsection2-3.25explus-1ex minus-.2ex1.5ex plus.2exHighest and lowest weight wavefunctions In this subsection we give expressions for the classical wavefunctions, associated to lowest (highest) weight primary operator insertions at the south (north) pole of ${\\cal I}^+$ in terms of rescaled Green functions in the limit that one argument is pushed to ${\\cal I}^+$ .",
"These wavefunctions each comes in a Neumann and a Dirichlet flavor, denoted $\\Phi ^\\pm _{lw}(x)$ ($\\Phi ^\\pm _{hw}(x)$ ) depending on whether the weight of the dual operator insertion is $h_+$ or $h_-$ .",
"The relevant Green functions areNote that $G_+ (G_-)$ is even (odd) under the antipodal map.",
"Combinations of the Euclidean Green function with such properties have been previously studied in the context of elliptic $Z_2$ -identification of de Sitter space [17], [18], [19] which may be related to our construction.",
"$G_\\pm (x;x^{\\prime })\\equiv G_E(x;x^{\\prime })\\pm G_E(x;x^{\\prime }_A) =\\frac{1}{8\\pi ^2}\\left( \\frac{1}{1-P(x;x^{\\prime })}\\pm \\frac{1}{1+P(x;x^{\\prime })}\\right) $ with $G_E$ the Wightman function for the Euclidean vacuum given in equation (REF ).",
"$G_-$ ($G_+$ ) obeys Neumann (Dirichlet) boundary conditions at ${\\cal I}^+$ away from $x=x^{\\prime }$ .",
"These are for $m^2\\ell ^2=2$ the Green functions with future boundary conditions as discussed in [10].",
"We have normalized them so that they have the Hadamard form for the short-distance singularity.",
"In the Neumann case we begin with $G_-$ , which is (using the mode decomposition (REF )) given by $G_-(x;x^{\\prime })=\\sum (-)^{L} \\left( \\phi ^{-}_{Lj}(x)\\phi ^-_{Lj}(x^{\\prime })+i\\phi ^{+}_{Lj}(x)\\phi ^-_{Lj}(x^{\\prime })\\right).$ From this we construct the rescaled Green function $\\Phi ^-_{lw}(x;t^{\\prime })=e^{h_-t^{\\prime }}G_-(x;t^{\\prime },\\Omega _{SP}),$ in which the second argument is placed at the south pole $\\Omega _{SP}$ where $\\psi ^{\\prime }=0$ .",
"One may then check that (ignoring singularity prescriptions) $\\Phi ^-_{lw}(x)\\equiv \\lim _{t^{\\prime }\\rightarrow \\infty } \\Phi ^-_{lw}(x;t^{\\prime })= \\frac{1}{2\\pi ^2(\\sinh t-\\cos \\psi \\cosh t)}.$ Using (REF ) and the asymptotics (REF ) one finds that near ${\\cal I}^+$ (not ignoring singularities) $\\Phi ^-_{lw}(x)&=&8\\sum (-)^{L} \\left( {e^{-t} 2L+2} Y_{Lj}(\\Omega )Y_{Lj}(\\Omega _{SP})+ie^{-2t}Y_{Lj}(\\Omega )Y_{Lj}(\\Omega _{SP})+{\\cal O}(e^{-3t})\\right)\\cr &=&8e^{-t}\\Delta _-(\\Omega ,\\Omega _{SP})+8i {e^{-2t}\\sqrt{h}}\\delta ^3(\\Omega -\\Omega _{SP})+{\\cal O}(e^{-3t})$ Let us now confirm that $\\Phi ^-_{lw}(x)$ has the same symmetries as an insertion of a primary operator ${\\cal O}(\\Omega _{SP})$ at the south pole of ${\\cal I}^+$ .",
"First we note that the choice of a point on ${\\cal I}^+$ breaks $SO(4,1)$ to $SO(3)\\times SO(1,1)$ .",
"Both $\\Phi ^-_{lw}(x)$ and ${\\cal O}(\\Omega _{SP})$ are manifestly invariant under the $SO(3)$ spatial rotations.",
"The generator of $SO(1,1)$ dilations, denoted $L_0$ , acts on ${\\cal O}(\\Omega _{SP})$ as $[L_0,{\\cal O}(\\Omega _{SP})]=h_-{\\cal O}(\\Omega _{SP}).$ In the bulk it is generated by the Killing vector field $L_0=\\cos \\psi \\partial _t-\\tanh t \\sin \\psi \\partial _\\psi ,$ where the south pole is $\\psi =0$ .",
"dS invariance implies $(L_0-L_0^{\\prime })G_-(x,x^{\\prime })=0.$ It follows from this together with the definition (REF ) that the wavefunction obeys $L_0\\Phi ^-_{lw}(x)=h_-\\Phi ^-_{lw}(x).",
"$ By construction it obeys the wave equation $(\\nabla ^2-m^2)\\Phi ^-_{lw}(x)=0.",
"$ Acting on $SO(3)$ invariant symmetric functions we have $\\ell ^2\\nabla ^2=-L_0(L_0-3) +M_{-k}M_{+k},$ where the 6 Killing vector fields $M_{\\pm k}$ (given in Appendix REF ) are the raising and lowering operators for $L_0$ and we sum over $k$ .",
"It then follows that $M_{-k}M_{+k}\\Phi ^-_{lw}(x)=\\left( m^2\\ell ^2-h_-(3-h_-)\\right)\\Phi ^-_{lw}(x)=0,$ and hence $M_{+k}\\Phi ^-_{lw}(x)=0.$ which corresponds to the lowest-weight condition for the ${\\cal O}$ $[M_{+k},{\\cal O}(\\Omega _{SP})]=0.$ It may be shown that these symmetries uniquely determine the solution.",
"Hence $\\Phi ^-_{lw}$ is identified as the classical wavefunction associated to the insertion of the primary ${\\cal O}$ at the south pole.",
"A parallel argument leads to the dual of a highest weight operator insertion at the north pole .",
"The wavefunction is $\\Phi ^-_{hw}(x;t^{\\prime })=\\lim _{t^{\\prime }\\rightarrow \\infty } e^{h_-t^{\\prime }}G_-(x;t^{\\prime },\\Omega _{NP}).$ This obeys the relations $M_{-k}\\Phi ^-_{hw}(x)=0,~~~~ L_0\\Phi ^-_{hw}(x)=-h_-\\Phi ^-_{hw}(x), $ and has the asymptotic behavior $\\Phi ^-_{hw}(x)=8e^{-t}\\Delta _-(\\Omega ,\\Omega _{NP})+8i {e^{-2t}\\sqrt{h}}\\delta ^3(\\Omega -\\Omega _{NP})+{\\cal O}(e^{-3t}).$ Similar formulae apply to the Dirichlet case by beginning with $G_+$ in the above construction and replacing $+\\leftrightarrow -$ .",
"For example $\\Phi ^+_{hw}(x)=-8e^{-2t}\\Delta _+(\\Omega ,\\Omega _{NP})-8i {e^{-t}\\sqrt{h}}\\delta ^3(\\Omega -\\Omega _{NP})+{\\cal O}(e^{-3t}).$ We see from the above that the highest-weight wavefunction is smooth on the future horizon of the southern static patch dS$_4$ , and hence related to the quasinormal modes found in [11], [12].",
"The lowest quasinormal mode which is invariant under the $SO(3)$ of the static dS$_4$ is exactly the $\\Phi _{hw}^-$ with $h_-=1$ while the second lowest $SO(3)$ -invariant quasinormal mode corresponds to $\\Phi _{hw}^+$ with $h_+=2$ .",
"Lowest weight states are smooth on the past horizon and hence related to anti-quasinormal modes.",
"subsection2-3.25explus-1ex minus-.2ex1.5ex plus.2exGeneral multi-operator insertions In the preceding subsections we found the bulk duals of primary operators inserted at the north/south pole in the coordinates (REF ).",
"This can be generalized to insertions at an arbitrary point on ${\\cal I}^+$ with a general time slicing near ${\\cal I}^+$ .",
"Let us introduce coordinates $x\\sim ({y}^i,t)$ such that near ${\\cal I}^+$ $ ds_4^2 \\rightarrow -dt^2+e^{2t}h_{ij}(y)d{y}^id{y}^j,~~i,j=1,2,3.$ The dual wavefunction is then the $t^{\\prime }\\rightarrow \\infty $ limit of the rescaled Green function, denoted by $\\Phi ^\\pm _{y_1}(x)=\\lim _{t^{\\prime }\\rightarrow \\infty } e^{h_\\pm t^{\\prime }}G_\\pm (x;t^{\\prime },y_1).$ For the special cases of operator insertions at the north or south pole in global coordinates these reduce to our previous expressions.",
"Note that coordinate transformations of the form $t\\rightarrow t+f(y)$ induce a conformal transformation on ${\\cal I}^+$ $ h_{ij} \\rightarrow e^{2f(y)} h_{ij}, ~~~~ \\Phi ^\\pm _{y_1}(x)\\rightarrow e^{h_\\pm f(y)} \\Phi _{y_1}(x), $ as appropriate for a conformal field of weight $h_\\pm $ .",
"Hence the relative normalization in (REF ) will depend on the conformal frame at ${\\cal I}^+$ .",
"One may also consider multi-operator insertions such as ${\\cal O}(y_1){\\cal O}(y_2)$ in the CFT$_3$ at ${\\cal I}^+$ .",
"At the level of free field theory considered here these are associated to a bilocal wavefunction in the product of two bulk scalar fields $\\Phi _{ y_1}(x_1)\\Phi _{y_2} (x_2).$ We will use $\\Phi _\\Omega $ to denote these wavefunctions when working in global coordinates (REF ).",
"We note that in such coordinates near ${\\cal I}^+$ for an insertion at a general point $ \\Phi ^\\pm _{\\Omega _1}(t,\\Omega )=\\mp 8e^{-h_\\pm t}\\Delta _\\pm (\\Omega ,\\Omega _{1})\\mp 8i {e^{-h_\\mp t}\\sqrt{h}}\\delta ^3(\\Omega -\\Omega _{1})+{\\cal O}(e^{-3t}).$ subsection2-3.25explus-1ex minus-.2ex1.5ex plus.2exKlein-Gordon inner product We wish to define an inner product between e.g.",
"two Neumann wavefunctions $\\Phi ^-_{{\\Omega }_1}$ and $\\Phi ^-_{{\\Omega }_2} $ .",
"Later on we will compare this to the inner product on the CFT$_3$ Hilbert space and the two-point function of ${\\cal O}$ on $S^3$ .",
"One choice is to take a global spacelike $S^3$ slice in the interior and define the Klein-Gordon inner product $\\left\\langle \\Phi ^-_{{\\Omega }_1}|\\Phi ^-_{{\\Omega }_2}\\right\\rangle _{S^3}\\equiv i \\int _{S^3}d^3\\Sigma ^\\mu \\Phi ^{-*}_{{\\Omega }_1}\\overleftrightarrow{ \\partial _\\mu }\\Phi ^-_{{\\Omega }_2} .$ This integral does not depend on the choice of $S^3$ which can be pushed up to ${\\cal I}^+$ .",
"One may then see immediately from (REF ) that there are two nonzero terms proportional to $\\Delta _-$ giving $\\left\\langle \\Phi ^-_{{\\Omega }_1}|\\Phi ^-_{{\\Omega }_2}\\right\\rangle _{S^3}= 16 \\Delta _-(\\Omega _1-\\Omega _2) $ One may also define an inner product not on global spacelike $S^3$ slices, but on a spacelike $R^3$ slice which ends on an $S^2$ on ${\\cal I}^+$ .",
"The result is invariant under any deformation of the $S^2$ which does not cross the insertion point.",
"To be definite, we take the $S^2$ to be the equator, ${\\Omega }_1$ to be in the northern hemisphere and ${\\Omega }_2$ to be in the southern hemisphere, and the slice to be $R^3_S$ which intersects the south pole.",
"One then finds, pushing $R_S^3$ up to the southern hemisphere of ${\\cal I}^+$ $\\left\\langle \\Phi ^-_{{\\Omega }_1}|\\Phi ^-_{{\\Omega }_2}\\right\\rangle _{R_S^3}\\equiv i \\int _{R_S^3}d^3\\Sigma ^\\mu \\Phi ^{-*}_{{\\Omega }_1}\\overleftrightarrow{ \\partial _\\mu }\\Phi ^-_{{\\Omega }_2}= 8 \\Delta _-(\\Omega _1-\\Omega _2) .$ Similarly, the inner product between two Dirichlet wavefunctions is given by $\\left\\langle \\Phi ^+_{{\\Omega }_1}|\\Phi ^+_{{\\Omega }_2}\\right\\rangle _{R_S^3}=- 8 \\Delta _+(\\Omega _1-\\Omega _2) .$ section1-3.5ex plus-1ex minus-.2ex2.3ex plus.2exThe southern Hilbert space We now turn to the issue of bulk quantum states.",
"Quantum states in dS are often discussed, as in section 2, in terms of a Hilbert space built on the global $S^3$ slices.",
"The structure of the vacua and Green functions for such states was described in section 2.",
"However dS has the unusual feature that there are geodesically complete topologically $R^3$ spacelike slices which end on an $S^2$ in ${\\cal I}^+$ , which we will typically take to be the equator.",
"Examples of these are the hyperbolic slices, the quantization on which was studied in detail in [15].",
"We will see that the quantum states built on these $R^3$ slices are natural objects in dS/CFT.",
"An $S^2$ in ${\\cal I}^+$ is in general the boundary of a “northern\" slice, denoted $R^3_N$ and a “southern\" slice denoted $R^3_S$ .",
"The topological sum obeys $R^3_S \\cup R^3_N=S^3$ .",
"Hence the relation of the southern and northern Hilbert spaces on $R^3_S$ and $R^3_N$ to that on $S^3$ is like that of the left and right Rindler wedges to that of global Minkowski space.",
"It is also like the relation of the Hilbert spaces of the northern and southern causal diamonds to that of global dS.",
"However the diamond Hilbert spaces in dS quantum gravity are problematic in quantum gravity with a fluctuating metric because it is hard to find sensible boundary conditions.",
"A strong motivation for considering the $R^3_{S,N}$ slices comes from the picture of a state in the boundary CFT$_3$ .",
"The state-operator correspondence in CFT$_3$ begins with an insertion of a (primary or descendant) operator ${\\cal O}$ at the south pole of $S^3$ , and then defines a quantum state as a functional of the boundary conditions on an $S^2$ surrounding the south pole.",
"For every object in the CFT$_3$ , we expect a holographically dual object in the bulk dS$_4$ theory.",
"The dual bulk quantum state must somehow depend on the choice of $S^2$ in ${\\cal I}^+$ .",
"Hence it is natural to define the bulk state on the $R^3$ slice which ends on this $S^2$ in ${\\cal I}^+$ .",
"This is how holography works in AdS/CFT: CFT states live on the boundaries of the spacelike slices used to define the bulk states.",
"subsection2-3.25explus-1ex minus-.2ex1.5ex plus.2exStates In order to define quantum states on $R^3_S$ , we first note that modes of the scalar field operator $\\hat{\\Phi }(\\Omega ,t)$ are labeled by operators $\\hat{\\Phi }^\\pm ( \\Omega )$ defined on ${\\cal I}^+$ via the relation $ \\lim _{t\\rightarrow \\infty }\\hat{\\Phi }(\\Omega ,t)=e^{-h_+t}\\hat{\\Phi }^+(\\Omega )+e^{-h_-t}\\hat{\\Phi }^-(\\Omega ).$ They satisfy the following commutation relation $\\left[\\hat{\\Phi }^+(\\Omega ),\\hat{\\Phi }^-(\\Omega ^{\\prime })\\right]={8i \\sqrt{h}}\\delta ^3(\\Omega -\\Omega ^{\\prime }).$ We may then decompose these ${\\cal I}^+$ operators as the sum of two terms $ \\hat{\\Phi }^\\pm (\\Omega )=\\hat{\\Phi }^\\pm _N(\\Omega )+\\hat{\\Phi }^\\pm _S(\\Omega ) $ where the first (second) acts only on $R^3_N$ ($R^3_S$ ).",
"Defining the northern and southern Dirichlet and Neumann vacua by $\\hat{\\Phi }^\\pm _N|0^\\pm _N\\rangle =0,~~~~\\hat{\\Phi }^\\pm _S|0^\\pm _S>=0, $ it follows from the decomposition (REF ) that the global vacua have a simple product decompositionOf course a general quantum state on $S^3$ is a sum of products of northern and southern states, and reduces to a southern density matrix, not a pure state, after tracing over the northern Hilbert space.",
"We shall see this explicitly below for the case of the Euclidean vacuum.",
"$|0^\\pm \\rangle =|0^\\pm _N\\rangle |0^\\pm _S\\rangle .$ Excited southern states may then be built by acting on one of these southern vacua with $\\hat{\\Phi }_S$ .",
"We wish to identify these states with those of the CFT$_3$ on $S^2$ .",
"In the higher-spin dS/CFT correspondence there are actually two CFT$_3$ 's living on ${\\cal I}^+$ : the free $Sp(N)$ model, associated to Neumann boundary conditions, and the critical $Sp(N)$ model, associated to Dirichlet boundary conditions.",
"Since the field operators $\\hat{\\Phi }_S $ acting on $|0_S^+\\rangle $ ($|0_S^-\\rangle $ ) obeys, according to equation (REF ), Neumann (Dirichlet) boundary conditions near the southern hemisphere of ${\\cal I}^+$ , it is natural to identify $|0_S^+\\rangle &\\sim & {\\rm free}~~Sp(N)~~{\\rm vacuum}\\cr |0_S^-\\rangle &\\sim & {\\rm critical}~~Sp(N)~~{\\rm vacuum}.$ Next we want to consider excited states and their duals.",
"To be specific we consider the Neumann theory built on $|0_S^+\\rangle $ .",
"Parallel formulae apply to the Dirichlet case.",
"Operator versions of the classical wavefunctions $\\Phi ^-_\\Omega (x)$ are constructed as $\\hat{\\Phi }^-_{\\Omega _S} \\equiv \\left\\langle \\Phi ^-_{\\Omega _S}|\\hat{\\Phi }\\right\\rangle _{R_S^3},$ where $\\Omega _S$ is presumed to lie on the southern hemisphere.",
"We can make a quantum state $|\\Omega ^-_S\\rangle \\equiv \\hat{\\Phi }^-_{\\Omega _S} |0^+_S \\rangle =\\hat{\\Phi }^-(\\Omega _S)|0^+_S \\rangle , $ where in the last line we used (REF ).",
"By construction this will be a lowest weight state, and we therefore identify it as the bulk dual to the CFT$_3$ state created by the primary operator ${\\cal O}$ dual to the field $\\Phi $ .",
"This connection leads to an interesting nonperturbative dS exclusion principle [14].",
"The operator ${\\cal O}$ has a representation in the $Sp(N)$ theory as ${\\cal O}=\\Omega _{AB}\\eta ^A\\eta ^B,~~~A,B=1,...N,$ where $\\eta ^A$ are $N$ anticommuting real scalars and $\\Omega _{AB}$ is the quadratic form on $Sp(N)$ .",
"It follows that ${\\cal O}^{{N2} +1}=0.$ Bulk-boundary duality and the state-operator relation described above then implies the nonperturbative relation $\\left[ \\hat{\\Phi }^\\pm (\\Omega )\\right]^{{N 2}+1}=0.$ Hence the quantum field operators $\\hat{\\Phi }^\\pm (\\Omega )$ are $N 2$ -adic.",
"One is not allowed to put more than $N 2$ quanta in any given quasinormal mode.",
"This is similar to the stringy exclusion principle for AdS [20] and may be related to the finiteness of dS entropy.",
"Nonperturbative phenomena due to related finite $N$ effects in the $O(N)$ case have been discussed in [21].",
"We hope to investigate further the consequences of this dS exclusion principle.",
"subsection2-3.25explus-1ex minus-.2ex1.5ex plus.2exNorm Having identified the bulk duals of the boundary CFT$_3$ states, we wish to describe the bulk dual of the CFT$_3$ norm.",
"The standard bulk norm is defined by $\\Phi (x)=\\Phi ^\\dagger (x)$ .",
"However this norm is not unique.",
"It has been argued for a variety of reasons beginning in [3] that it is appropriate to modify the norm in the context of dS – see also [16], [22].",
"Here we have the additional problem that this standard norm is divergent for states of the form (REF ).",
"We now construct the modified norm for states on $R_3^S$ by demanding that it is equivalent to the CFT$_3$ norm.",
"The construction here generalizes to dS$_4$ the one given in [16] for dS$_3$ .",
"The bulk action of dS Killing vectors $K_A^\\mu \\partial _\\mu $ on a scalar field is generated by the integral over any global $S^3$ slice $\\hat{\\cal L}_{A}= \\int _{S^3} d^3\\Sigma ^\\mu T_{\\mu \\nu }K^{\\nu }_A,$ where $T_{\\mu \\nu }$ is the bulk stress tensor constructed from the operator $\\hat{\\Phi }$ .",
"If we take $\\hat{\\Phi }^\\dagger (x)=\\hat{\\Phi }(x)$ , then $\\hat{\\cal L}_A=\\hat{\\cal L}_{A}^\\dagger $ which is not what we want.",
"The CFT$_3$ states are in representations of the $SO(3,2)$ conformal group.",
"These arise from analytic continuation of the 10 $SO(4,1)$ conformal Killing vectors on $S^3$ which are the boundary restrictions of the bulk dS$_4$ Killing vectors $K^{\\mu }_A \\partial _\\mu $ .",
"Usually, the standard CFT$_3$ norm has a self-adjoint dilation operator ${\\cal L}_0$ generating $-i \\sin \\psi \\partial _\\psi $ as well as 3 self-adjoint $SO(3)$ rotation operators ${\\cal J}_k$ .",
"The remaining 6 raising and lowering operators ${\\cal L}_{\\pm k}$ arising from the Killing vectors $i M_{\\pm k}$ (described in the appendix) then obey ${\\cal L}_{\\pm k}^\\dagger = {\\cal L}_{\\mp k}$ in the conventional CFT$_3$ norm.",
"To obtain an adjoint with the desired properties, we define the modified adjoint $ \\hat{\\Phi }^{ \\dagger }(x)= {\\cal R}\\hat{\\Phi }(x) {\\cal R}=\\hat{\\Phi }( R x) ,$ where here and hereafter $\\dagger $ denotes the bulk modified adjoint.",
"The reflection operator ${\\cal R}$ is the discrete isometry of $S^3$ which reflects through the $S^2$ equator $R(\\psi ,\\theta ,\\phi )=(\\pi -\\psi ,\\theta , \\phi )$ along with complex conjugation.",
"In particular, it maps the south pole to the north pole while keeping the equator invariant.",
"This implies that ${\\cal L}_0$ (generating $i L_0$ ) and ${\\cal J}_k$ are self adjoint while ${\\cal L}_{\\pm k}^\\dagger =-i\\int _{S^3} d\\Sigma ^\\mu (x) T_{\\mu \\nu }(R x) M^{\\nu }_{\\pm k}(x)=-i\\int _{S^3} d\\Sigma ^\\mu (x) T_{\\mu \\nu }(x) M^{\\nu }_{\\pm k}(R x) = {\\cal L}_{\\mp k}.$ Hence we have constructed an adjoint admitting the desired $SO(3,2)$ action.",
"We do not know whether or not it is unique.",
"The action of ${\\cal R}$ maps an operator defined on the southern hemisphere to one defined on the southern hemisphere of ${\\cal I}^+$ according to $\\hat{\\Phi }^{\\pm \\dagger }(\\Omega )= \\hat{\\Phi }^\\pm (\\Omega _R),$ Hence the action of ${\\cal R}$ exchanges the northern and southern hemispheres, and maps a southern ${\\cal I}^+$ state to a northern one.",
"Therefore it cannot on its own define an adjoint within the southern Hilbert space.",
"For this we must combine (REF ) with a map from the north to the south.",
"Such a map is provided by the Euclidean vacuum.",
"The global Euclidean bra state (constructed with the standard adjoint) can be decomposed in terms of a basis of northern and southern bra states $\\langle 0_E|=\\sum _{m,n}E_{mn}\\langle m_S|\\langle n_N|.$ We then define the modified adjoint of an arbitrary southern state $|\\Psi _S\\rangle $ by $|\\Psi _S\\rangle ^\\dagger \\equiv \\langle 0_E| {\\cal R}|\\Psi _S\\rangle .$ We will denote the corresponding inner product by an $S$ subscript $\\langle \\Psi ^{\\prime }_S|\\Psi _S\\rangle _S\\equiv (|\\Psi ^{\\prime }_S\\rangle ^\\dagger )|\\Psi _S\\rangle .$ For example choosing the basis so that ${\\cal R}|m_S\\rangle =|m_N\\rangle $ we have $\\langle m_S|n_S\\rangle _S= E_{nm}.$ In particular one finds $\\langle 0^+_S|0^+_S\\rangle _S= \\langle 0_E|(|0^+_S\\rangle {\\cal R}|0^+_S\\rangle )=1.$ Let us now compute the norm of the southern state $|\\Omega ^-_S\\rangle $ in (REF ).",
"The action of ${\\cal R}$ gives a northern state which we will denote $|R \\Omega ^-_S\\rangle $ .",
"The norm is then $\\langle \\Omega ^-_S|\\Omega ^-_S\\rangle _S= \\langle 0_E|(|\\Omega ^-_S\\rangle |R\\Omega ^-_S\\rangle )= \\langle 0_E|\\hat{\\Phi }^-(\\Omega _S)\\hat{\\Phi }^-(R\\Omega _S)|0^+\\rangle .",
"$ Using the relation $|0_E>=N_0e^{-{116} \\int d^3\\Omega d^3\\Omega ^{\\prime }\\hat{\\Phi }^+(\\Omega )\\Delta _-(\\Omega ,\\Omega ^{\\prime })\\hat{\\Phi }^+(\\Omega ^{\\prime })}|0^-\\rangle $ we find $ \\langle \\Omega ^-_S|\\Omega ^-_S\\rangle _S=8\\Delta _-(\\Omega _S,R \\Omega _S).$ This is proportional to the $S^3$ two-point function of a dimension $h_-$ primary at the points $\\Omega _S$ and $R \\Omega _S$ .",
"The analogous computation in the Dirichlet theory gives $ \\langle \\Omega ^+_S|\\Omega ^+_S\\rangle _S=-8\\Delta _+(\\Omega _S,R \\Omega _S).$ section1-3.5ex plus-1ex minus-.2ex2.3ex plus.2ex Boundary dual of the bulk Euclidean vacuum In the preceding section we have argued that dS/CFT maps CFT$_3$ states on an $S^2$ in ${\\cal I}^+$ to bulk states on the southern slice ending on the $S^2$ .",
"A generic state in a global dS slice does not restrict to a pure southern state.",
"However we can always define a density matrix by tracing over the northern Hilbert space.",
"In particular, such a southern density matrix $\\rho ^E_S$ can be associated to the global Euclidean vacuum $|0_E\\rangle $ .",
"The choice of an equatorial $S^2$ in ${\\cal I}^+$ breaks the $SO(4,1)$ symmetry group down to $SO(3,1)$ , which also preserves the hyperbolic slices ending on the $S^2$ .",
"$\\rho _E^S$ must be invariant under this $SO(3,1)$ .",
"In fact $\\rho _E^S$ follows from results in [15].",
"Writing the quadratic Casimir of $SO(3,1)$ as $C_2=-(1+p^2)$ , it was shown, in a basis which diagonalizes $p$ , that $\\rho _E^S=N_1 e^{-2\\pi p},$ where $N_1$ is determined by ${\\rm Tr}\\rho _E^S=1$ .",
"It would be interesting to investigate this further and compute the entropy $S=-{\\rm Tr}\\rho _E^S\\ln \\rho _E^S$ in the $Sp(N)$ model.",
"section1-3.5ex plus-1ex minus-.2ex2.3ex plus.2exPseudounitarity and the C-norm in the $Sp(N)$ CFT$_3$ In this section we consider the $Sp(N)$ model (where $N$ is even) and compare the norms to those computed above.",
"The action is $I_{Sp(N)}=\\frac{1}{8\\pi }\\int d^3 x\\left[\\delta ^{ij}\\delta _{ab}\\partial _i \\bar{\\chi }^a \\partial _j \\chi ^b+m^2 \\bar{\\chi } \\chi +\\lambda \\left( \\bar{\\chi } \\chi \\right)^2\\right],$ where $\\chi ^a (a=1,\\ldots ,{ N 2})$ is a complex anticommuting scalar and $\\bar{\\chi }\\chi \\equiv \\delta _{ab} \\bar{\\chi }^a\\chi ^b$ .",
"This has a global $Sp(N)$ symmetry and we restrict to $Sp(N)$ singlet operators.The $U({N 2})$ theory has the same action but is restricted only to $U({N 2})$ singlets.",
"For the free theory $m=\\lambda =0$ while the critical theory is obtained by flowing to a nontrivial fixed point $\\lambda _F$ .",
"The $Sp(N)$ theory is not unitary in the sense that in the standard norm following from (REF ) one has that [9] $H\\ne H^\\dagger $ and $\\langle \\Psi ^{\\prime }|\\Psi \\rangle $ is not preserved.",
"Nevertheless, as detailed in [9], there exists an operator $C$ with the properties $ C^\\dagger C=C^2=1,~~~C\\chi ^\\dagger C=\\overline{\\chi }, ~~~CH^\\dagger C=H,~~~C|0\\rangle =|0\\rangle .$ To write it in real fields, for e.g., in the case of $Sp(2)$ , writing the real and imaginary part of $\\chi $ as $\\eta _1$ and $\\eta _2$ , the action of $C$ becomes $\\eta _2 = C \\eta _1^\\dagger C$ .",
"One may then define a “pseudounitary\" $C$ -inner product $\\langle \\Psi ^{\\prime }|\\Psi \\rangle _C\\equiv \\langle \\Psi ^{\\prime }|C|\\Psi \\rangle $ which is preserved under hamiltonian time evolution.",
"Such hamiltonians are pseudohermitian and are similar to those studied in [23].",
"We note that the norm is not positive definite.",
"Inserting an operator $O_i$ constructed from $\\chi ^a$ at the south pole gives a functional of the boundary conditions on the equatorial $S^2$ which we define as the state $|O_i \\rangle $ .",
"This is the standard state-operator correspondence.",
"An inner product for such states associated to $O_i$ and $O_j$ can be defined by the two point function with $O_i$ at one pole and $O_j$ at the other.",
"It follows from (REF ) that this is the $C$ -inner product for the states $|O_i\\rangle $ and $|O_j\\rangle $ : $\\langle O_i|O_j\\rangle _C= \\langle O_i|C|O_j\\rangle = \\langle {O_i}^\\dagger C {O_j}\\rangle =\\langle O_i O_j\\rangle .$ In the last line, we used the fact that the (singlet) currents in the $Sp(N)$ models satisfy $C {O_i}^\\dagger C=O_i$ since $ C\\left(\\bar{\\chi }\\chi \\right)^{\\dag } C=\\bar{\\chi }\\chi $ .",
"For primary operators of weight $h_i$ we then have [6] $\\langle O_i|O_i\\rangle _C=-N\\Delta _{h_i}(\\Omega _{NP},\\Omega _{SP}).",
"$ Hence it is the C-norm which maps under the state-operator correspondence to the Zamolodchikov norm defined as the Euclidean two point function on $S^3$ .",
"As seen in [6] this C-norm then agrees with the bulk inner product (REF )-(REF ) of the dual state for the scalar case.A factor of $i^h$ explained in [6] relating operator insertions to bulk fields makes the two-point function (REF ) negative.",
"Moreover, as the bulk and CFT$_3$ norms assign the same hermiticity properties to the $SO(4,1)$ generators, this result will carry over to descendants of the primaries.",
"A generalization of this construction to all spins seems possible.",
"One of the puzzling features of dS/CFT is that the dual CFT cannot be unitary in the ordinary sense.",
"This is not a contradiction of any kind because unitarity of the Euclidean CFT is not directly connected to any spacetime conservation law.",
"At the same time quantum gravity in dS – and its holographic dual – should have some good property replacing unitarity in the AdS case.",
"It is not clear what that good property is.",
"The appearance of a pseudounitary structure in the case of dS/CFT analyzed here is perhaps relevant in this regard.",
"section1-3.5ex plus-1ex minus-.2ex2.3ex plus.2ex*Acknowledgements It has been a great pleasure discussing this work with Dionysios Anninos, Daniel Harlow, Tom Hartman, Daniel Jafferis, Matt Kleban and Steve Shenker.",
"This work was supported in part by DOE grant DE-FG02-91ER40654 and the Fundamental Laws Initiative at Harvard.",
"section1-3.5ex plus-1ex minus-.2ex2.3ex plus.2exAppendix: dS$_4$ Killing vectors The 10 Killing vectors of dS$_4$ are given by: $L_0&=&\\cos \\psi \\partial _t-\\tanh t \\sin \\psi \\partial _\\psi \\nonumber \\\\M_{\\mp 1}&=&\\pm \\sin {\\psi }\\sin {\\theta }\\sin {\\phi } \\partial _t+ \\left( 1 \\pm \\tanh {t} \\cos {\\psi }\\right)\\sin {\\theta }\\sin {\\phi }\\partial _\\psi \\nonumber \\\\& &+\\left( \\cot {\\psi } \\pm \\tanh {t} \\csc {\\psi } \\right)\\left(\\cos {\\theta }\\sin {\\phi }\\partial _\\theta +\\csc {\\theta }\\cos {\\phi } \\partial _\\phi \\right)\\nonumber \\\\M_{\\mp 2} &=&\\pm \\sin {\\psi } \\sin {\\theta } \\cos {\\phi }\\partial _t+ \\left( 1 \\pm \\tanh {t} \\cos {\\psi }\\right)\\sin {\\theta }\\cos {\\phi }\\partial _\\psi \\nonumber \\\\& & +\\left( \\cot {\\psi } \\pm \\tanh {t} \\csc {\\psi }\\right) \\left(\\cos {\\theta }\\cos {\\phi }\\partial _\\theta -\\csc {\\theta }\\sin {\\phi } \\partial _\\phi \\right)\\nonumber \\\\M_{\\mp 3}&=& \\pm \\sin {\\psi } \\cos {\\theta } \\partial _t+ \\left(1\\pm \\tanh {t}\\cos {\\psi } \\right) \\cos {\\theta }\\partial _\\psi -\\left(\\cot {\\psi } \\pm \\tanh {t} \\csc {\\psi }\\right)\\sin {\\theta } \\partial _\\theta \\nonumber \\\\J_1 &=& \\cos {\\phi }\\partial _\\theta -\\sin {\\phi }\\cot {\\theta }\\partial _\\phi \\nonumber \\\\J_2 &=& -\\sin {\\phi }\\partial _\\theta -\\cos {\\phi }\\cot {\\theta }\\partial _\\phi \\nonumber \\\\J_3 &=& \\partial _\\phi .$ For each $k$ , the $M_{\\pm k}$ and $L_0$ form a $SO(2,1)$ subalgebra satisfying $[M_{+k},M_{-k}]=2L_0,~~[L_0,M_{+k}]=-M_{+k},~~[L_0,M_{-k}]=M_{-k}.~~$ As mentioned in the text, acting on $SO(3)$ -invariant functions, we have $\\ell ^2\\nabla ^2=-L_0(L_0-3) +M_{-k}M_{+k},$ where $k$ is summed over $k=1,2,3$ .",
"The conformal Killing vectors of the $S^3$ are given by the restriction of dS$_4$ Killing vectors on ${\\cal I}^+$ : $L_0&=& -\\sin {\\psi } \\partial _\\psi \\nonumber \\\\M_{\\mp 1}&=&\\left( 1 \\pm \\cos {\\psi }\\right)\\sin {\\theta }\\sin {\\phi }\\partial _\\psi +\\left( \\cot {\\psi } \\pm \\csc {\\psi } \\right)\\left(\\cos {\\theta }\\sin {\\phi }\\partial _\\theta +\\csc {\\theta }\\cos {\\phi } \\partial _\\phi \\right)\\nonumber \\\\M_{\\mp 2} &=& \\left( 1 \\pm \\cos {\\psi }\\right)\\sin {\\theta }\\cos {\\phi }\\partial _\\psi + \\left( \\cot {\\psi } \\pm \\csc {\\psi }\\right) \\left(\\cos {\\theta }\\cos {\\phi }\\partial _\\theta -\\csc {\\theta }\\sin {\\phi } \\partial _\\phi \\right)\\nonumber \\\\M_{\\mp 3}&=& \\left(1\\pm \\cos {\\psi } \\right) \\cos {\\theta }\\partial _\\psi -\\left(\\cot {\\psi } \\pm \\csc {\\psi }\\right)\\sin {\\theta } \\partial _\\theta .\\nonumber \\\\J_1 &=& \\cos {\\phi }\\partial _\\theta -\\sin {\\phi }\\cot {\\theta }\\partial _\\phi \\nonumber \\\\J_2 &=& -\\sin {\\phi }\\partial _\\theta -\\cos {\\phi }\\cot {\\theta }\\partial _\\phi \\nonumber \\\\J_3 &=& \\partial _\\phi .$"
]
] | 1204.1057 |
[
[
"Modelling an observer's branch of extremal consciousness"
],
[
"Abstract Extreme-order statistics is applied to the branches of an observer in a many-worlds framework.",
"A unitary evolution operator for a step of time is constructed, generating pseudostochastic behaviour with a power-law distribution when applied repeatedly to a particular initial state.",
"The operator models the generation of records, their dating, the splitting of the wavefunction at quantum events, and the recalling of records by the observer.",
"Due to the huge ensemble near an observer's end, the branch with the largest number of records recalled contains almost all \"conscious dimension\"."
],
[
"Introduction",
"Extreme-order statistics, dealing with distributions of largest, second-largest values, etc., in a random sample [1], has not received much attention in quantum theory.",
"However, statistics of outliers can be striking in a many-worlds scenario, due to the huge number of branches, providing the statistics is of the power-law type.",
"If a random draw of some information-related quantity is made in each branch, the excess $l$ of the largest over the second-largest draw would be huge.",
"That excess exponentiates to $2^l$ if information is processed in qubits, each of which has two Hilbert-space dimensions.",
"Thus, using the dimension as a weight of a branch [2], the weight of the extremal branch may exceed by far the total weight of all other branches.",
"This might be a realisation, for an entire history of an observer rather than for a single measurement, of the idea that “massive redundancy can cause certain information to become objective, at the expense of other information” [3].",
"The many-worlds scenario [4] is implicit in decoherence theory [5] which has been able to explain why macroscopic superpositions evolve, rapidly, into states with the properties of classical statistical ensembles.",
"There remains, however, the “problem of outcomes” [6]: Why is it that an observer of a superposition always finds himself ending up in a pure state instead of a mixture?",
"The problem is not with objective facts, but with the consciousness of the observer.",
"For, if any statement is derived from observations alone, it can only involve observations from one branch of the world because of vanishing matrix elements between different branches.",
"Nowhere on the branching tree an objective contradiction arises.",
"As to a conscious observer, however, we cannot tell by present knowledge whether he needs to “observe” his various branches in order to become aware of them, or whether it suffices for him to “be” the branches, whatever that means in physical terms [7].",
"In the model constructed below, the “observer” could in principle be aware of all of his branches, but, due to statistics of extremes under a power law, this amounts to being aware of one extremal branch plus fringe effects.",
"We shall be modelling, rather abstractly, an observer “as a finite information-processing structure” [8], with an emphasis on “processing”, because it means nothing to the observer if a bit of information “has” a certain value unless the value is revealed by interaction.",
"We shall take a history-based approach mainly for reasons of extreme-value statistics under a power-law: outliers are most pronounced in large random samples, suggesting to use as a sample the entire branching tree.",
"In addition, “the history of a brain's functioning is an essential part of its nature as an object on which a mind supervenes” [9].",
"Ideally, a model of quantum measurement would be based on some Hamiltonian without explicit time dependence; in particular, without an external random process in time.",
"Such an approach via the Schrödinger equation, however, would be hindered by difficulties in solving the equation.",
"We shall greatly simplify our task by considering time evolution only in discrete steps, and by constructing a unitary operator directly for a time step.",
"It would be possible, though of little use, to identify a Hamiltonian generating the evolution operator.",
"A further simplification will be to consider evolution only during an observer's lifetime.",
"If we describe measurements in the aforementioned way, we must show how stochastic behaviour can emerge in the observation of quantum systems.",
"It will be sufficient to use random numbers in the construction of the time-step operator, in presumed approximation to real-world dynamical complexity.",
"Under repeated application of the (constant) operator, evolution is deterministic in principle.",
"When the application is to a particular initial state, however, the built-in randomness becomes effective.",
"The evolution operator is constructed as a product of unitaries.",
"This facilitates evaluations; in particular, it enables a straightforward definition of “conscious dimension”.",
"The scenario of the model is as follows.",
"A quantum system is composed of a record-generating part, like some kind of moving body; of records keeping track of the motion; and of subsystems associated with the records, allowing for demolition-free reading.",
"The observer appears through the subsystems and through part of the evolution operator.",
"At his “birth”, all records are in blank states, while the record-generating body is in some quasiclassical state which determines the subsequent evolution.",
"The evolution operator provides four kinds of event: Quasiclassical motion for most of the time, accompanied by the writing of records; dating of records by conservative ageing; splitting of the motion into a superposition of two equal-amplitude branches at certain points of the body's orbit; and the reading of records by a “scattering” interaction with the subsystems of the records.",
"Random elements in the evolution are: Duration of quasiclassical sections; the states of the body at which evolution continues after a split; and most crucially, the number of records being recalled within a timestep.",
"For the latter number, a power-law distribution is assumedThe standard mechanism for generating such a distribution is a supercritical chain reaction stopped by an event with a constant rate of incidence [10].",
"Phenomenologically, power-law distributions are not uncommon in neurophysics, but it seems they are always discussed in highly specialised contexts.. No attempt is made here to justify the distribution—it should be regarded as a working hypothesis for the purpose of demonstrating the potential relevance of power-law statistics for quantum measurement.",
"The state of superposition, emerging by repeated steps of evolution from an initial state of the chosen variety, can be made explicit to a sufficient degree.",
"It can be put in correspondence with a branching tree of the general statistical theory of branching processes.",
"Statistical independence as required by that theory is exactly satisfied by the model evolution, due to random draws employed in constructing the evolution operator.",
"Consciousness is assumed to reside in unitary rotations of the recalling subsystems of the records, triggered when a record of a special class is encountered.",
"The trigger is associated with a random draw determining the number of “redundant” records to be processed.",
"Somewhere on the branching tree (that is, in some factor of the tensor products superposed) that number takes the extremal value.",
"Because of the branching structure, the probability is large for that value to occur near the end of an observer's lifetime.",
"Hence, it singles out (almost) an entire history.",
"A study into the probability distribution for the difference between the largest and the second-largest draw finally shows that the dimension of the subspace affected by conscious rotations in all branches of the superposition is almost certainly exhausted by the dimension of conscious rotations in the extremal branch.",
"The “objective” factors of the evolution operator are constructed in sections REF to REF .",
"Their effect on the initial state is evaluated in section REF .",
"Conscious processing of records is modelled in section REF , while its statistics is analysed in section .",
"Section shows how the model would generalise to branching with non-equal amplitudes or into more than two branches.",
"Conclusions are given in section ."
],
[
"Record-generating system",
"A basic assumption of the model is that for all but a sparse subset of time steps, evolution is quasi-classical, like a moving body represented by a coherent state.",
"Evolution on such a section is determined by a small set of dynamical variables, while a large number of “redundant” records is written along the path.",
"In the model's approximation, instantaneous dynamical variables are represented by one number from the orbital set $ \\lbrace 1,\\ldots ,K \\rbrace = : {\\cal O}$ The ordering is such that quasi-classical evolution takes the record-generating system from index $k$ to index $k+1$ within a step of time.",
"The corresponding basis vectors of the record-generating system are denoted by $ \\psi _k \\qquad k \\in {\\cal O}$ They are assumed to be orthonormal.",
"Figure: States of a record represented by NN dots on a circle,ageing (without loss of information) under repeated writing operations."
],
[
"Structure of records and the writing operation",
"The focus of the model is on an observer's interaction with records generated during his personal history.",
"The term “record” will refer to observer's individual memories as well as to more objective forms of recording.",
"Moreover, it will be convenient to use the term “record” as an abbreviation for “recording unit”.",
"Thus, records can be in “blank” or “written” states.",
"Each recording unit $r_i$ decomposes into a subsystem holding the information, and another subsystem allowing the observer to interactively read the information without destroying it.",
"The information consists, firstly, of some quality represented by the index of the record, and secondly, in the time of recording, or rather the age of the record.",
"The age is encoded in canonical basis vectors as follows.",
"$ \\begin{array}{l} e_0 = \\mbox{blank state} \\\\e_j = \\mbox{written since $j$ steps of time} \\qquad j = 1,\\ldots ,N-1\\end{array}$ This is illustrated in figure REF .",
"The information about age will be crucial for composing an observer's conscious history by one extremal reading.",
"Only ages up to $N-1$ steps of time are possible, which is sufficient if we are dealing with a single observer.",
"Both the generation of a record and its ageing can be described by a writing operation $W$ .",
"Acting on an indicated record, it acts on the first factor of (REF ) according to $ \\begin{array}{l}W e_i = e_{i+1} \\qquad i=0,1,\\ldots ,N-1 \\\\W e_N = e_0\\end{array}$ The second of these equalities is unwarranted, expressing an erasure of the record and thus limiting the model to less than $N$ steps of time, but there does not seem to be any better choice consistent with unitarity.",
"Obviously, writing operations on different records commute, $ W_i W_j = W_j W_i \\qquad \\mbox{for all }i,j$ To allow for the observer's conscious interaction with a record, a two-dimensional factor space is provided, vaguely representing the firing and resting states of a neuron.",
"Reading is modelled as a scattering of some unit vector $s$ into some other unit vector $s^{\\prime }$ , in a way that could, in an extended model, depend on the index and the age of the record.",
"In the present model, only dimensions will be counted, so no further specification of $s$ and $s^{\\prime }$ is required.",
"The Hilbert space of a single record is thus spanned by product vectors of the form $ r_i = e_{n_i} \\otimes s_i \\qquad \\mbox{with }~\\left\\lbrace \\begin{array}{l} \\displaystyle e_{n_i}\\mbox{ an $N$-dimensional canonical unit vector}\\\\s_i \\mbox{ a $2$-dimensional unit vector} \\end{array}\\right.$ The Hilbert space of all records possible is spanned by product vectors $r_1 \\otimes r_2 \\otimes \\cdots \\otimes r_I$ where $I$ , in view of the redundancy required, is much larger than $K$ .",
"The set of all indices of records will be denoted by $\\cal I$ .",
"It will be convenient to use the following abbreviation.",
"$ V({\\cal A}) ~ = ~ \\parbox {60mm}{any tensor product of records in which all r_iwith i\\in {\\cal A} are blank}$"
],
[
"Initial state",
"We assume that when the observer is “born” the record-generating system is in some quasiclassical state $\\psi _{k_\\mathrm {in}}$ in which also the observer's identity is encoded.",
"All records of the personal history are initially blank.",
"Using abbreviation (REF ), the assumed initial state can be written as $ |\\mathrm {in}\\rangle = \\psi _{k_\\mathrm {in}} \\otimes V({\\cal I})$ This choice of an initial state will imply that, in the terms of [3], we are restricting to a “branching-state ensemble”.",
"Such a restriction is necessary for pseudorandom behaviour to emerge under evolution by repeated action of a unitary time-step operator.",
"By contrast, eigenstates of that operator would evolve without any randomness.",
"$k_\\mathrm {in}$ is located on some string of quasiclassical events, as defined in section REF .",
"It is this string, chosen out of many similar ones, that acts as a “seed” which determines the observer's pseudo-random history."
],
[
"Quasiclassical evolution and quantum events",
"The idea of quasiclassical evolution, assumed to prevail for most of the time, is $\\psi _k\\rightarrow \\psi _{k+1}$ in a timestep (section REF ).",
"This is to be accompanied by the writing of records.",
"When the record-generating system is in the state $\\psi _k$ we assume that writing operations $W_i$ act on all records $r_i$ whose indices, or addresses, are in a set ${\\cal A}_{\\mathrm {W}}(k)$ .",
"While these records are redundant, we assume that $k$ can be retrieved from each of them, which requires $ {\\cal A}_\\mathrm {W}(k) \\cap {\\cal A}_\\mathrm {W}(l) = \\emptyset ~~\\mbox{for}~~ k\\ne l$ When the record-generating system arrives at an index $k$ in a sparse subset ${\\cal Q}\\subset {\\cal O}$ , we assume that a superposition of two branches (“up” and “down”) is formed, with equal amplitudes in both branches.",
"Quasiclassical evolution is assumed to jump from $k$ to $u(k)$ or $d(k)$ , respectively, and continue there, as illustrated in figure REF .",
"Figure: (a) Sections of quasi-classical evolution (thick lines) which, when anindex in the set 𝒬\\cal Q is encountered, split into superpositions of continued quasi-classical evolution.Length of lines, and addresses after splitting, are random elements of the evolution operator.When an initial state is chosen, defining an observer's origin and the “seed” for pseudo-randomevolution, a branching tree results whose first and second branches are shown in (b).For simplicity of evaluation, we choose $u(k)$ and $d(k)$ to coincide with starting points of quasiclassical sections.",
"These are indices in ${\\cal O}$ subsequent to an index in ${\\cal Q}$ .",
"We include index 1 as a starting point and assume $K\\in {\\cal Q}$ to avoid “boundary” effects.",
"Also, we must avoid temporal loops to occur within an observer's lifetime.",
"That is, evolution operator $U_\\mathrm {O}$ should be constructed such that no loops arise within $T$ repeated applications.",
"Loops cannot be avoided entirely because for every $k\\in {\\cal Q}$ there are two jumping addresses, so every section of quasi-classical evolution will have to be targeted by two jumps on average.",
"To keep branches apart for a number $n$ of splittings, assume ${\\cal Q}$ to be decomposable into $2^n$ subsets ${\\cal J}[s]$ , mutually disjoint, and each big enough to serve as an ensemble for a random draw.",
"Let $s$ be a register of the form $ s = [s_1,s_2,\\ldots ,s_n] ~~ \\mbox{ where }~~ s_j \\in \\lbrace u,d \\rbrace $ Then, for $k\\in {\\cal J}[s_1,s_2,\\ldots ,s_n]$ , define the jumping address $u(k)$ as follows.",
"$ \\begin{array}{l}\\mbox{Draw $k^{\\prime }$ at random from } {\\cal J}[s_2,s_3,\\ldots ,s_n,u].",
"\\\\\\mbox{Put }u(k)\\mbox{ at first point of quasiclassical section leading to }k^{\\prime }.\\end{array}$ Likewise for $v(k)$ .",
"$ \\begin{array}{l}\\mbox{Draw $k^{\\prime }$ at random from } {\\cal J}[s_2,s_3,\\ldots ,s_n,v].",
"\\\\\\mbox{Put }v(k)\\mbox{ at first point of quasiclassical section leading to }k^{\\prime }.\\end{array}$ The new entry, $u$ or $v$ , will be in the register for $n$ subsequent splittings.",
"Later on, the $u/v$ information is lost, allowing for inevitable loops to close.",
"Given observer's lifetime $T$ , and a minimal length $d_\\mathrm {min}$ of quasiclassical sections, parameter $n$ should be chosen as $ n = T/d_\\mathrm {min}$ .",
"The splitting addresses are collected in a set, $ {\\cal A}_\\mathrm {S}(k) = \\lbrace u(k),d(k)\\rbrace $ Quasiclassical motion $k\\rightarrow k+1$ and the branching $k\\rightarrow (u,d)$ constitute the “orbital” factor of evolution which is to be described by a unitary operator $U_\\mathrm {O}$ .",
"Under its action, images of orthonormal vectors must be orthonormal.",
"Since loops are inevitable in the set of indices, orthogonality of images cannot be ensured by the states of the record-generating system alone, but can be accomplished by orthogonalities (blank vs. written) in the accompanying states of recordsThere is a rudiment of decoherence in this model.. To this effect, certain records must be blank before the action of $U_\\mathrm {O}$ .",
"Their addresses are $ {\\cal B}(k) = \\mbox{union of all }{\\cal A}_\\mathrm {W}(l)\\mbox{ with nonempty }{\\cal A}_\\mathrm {S}(l) \\cap {\\cal A}_\\mathrm {S}(k) \\qquad k\\in {\\cal Q}$ Using this, definitions (REF ), (REF ), and abbreviation (REF ), we define $ \\begin{array}{ll}\\displaystyle U_\\mathrm {O}~\\psi _k \\otimes V({\\cal B}(k)) =\\frac{ \\psi _{u(k)} + \\psi _{d(k)} }{\\sqrt{2}}\\otimes \\left(\\prod _{i\\in {\\cal A}_\\mathrm {W}(k)} W_i\\right) V({\\cal B}(k))& \\quad k\\in {\\cal Q} \\\\\\displaystyle U_\\mathrm {O}~\\psi _k \\otimes V({\\cal A}_{\\mathrm {W}}(k)) =\\psi _{k+1} \\otimes \\left(\\prod _{i\\in {\\cal A}_\\mathrm {W}(k)} W_i\\right) V({\\cal A}_{\\mathrm {W}}(k)) & \\quad k\\notin {\\cal Q}\\end{array}$ This is a mapping of orthonormal vectors onto orthonormal image vectors, so it can be extended to a definition of a unitary operator $U_\\mathrm {O}$ by choosing any unitary mapping between the orthogonal complements of the originals and images.",
"However, due to the loop-avoiding construction, only properties (REF ) are required for the evolution of the initial states of section REF ."
],
[
"Dating of records",
"The time at which a record was written can be retrieved from information about its age.",
"An ageing operator for the $i$ th record, $U_\\mathrm {A}(i)$ , is defined by its action on the first tensorial factor of (REF ) as follows.",
"$ \\begin{array}{l}U_\\mathrm {A}(i) e_0 = e_0 \\\\U_\\mathrm {A}(i) e_i = e_{i+1} \\qquad i=1,\\ldots ,N-2 \\\\U_\\mathrm {A}(i) e_{N-1} = e_1\\end{array}$ In particular, a record in blank state $e_0$ remains unchanged.",
"Also, tensorial factors with indices different from $i$ are unaffected by $U_\\mathrm {A}(i)$ .",
"At the limiting age, corresponding to $N$ steps of time, $U_\\mathrm {A}(i)$ becomes senseless.",
"The ageing operator for the entire system of recording units is $ U_\\mathrm {A} = \\prod _{i=1}^I U_\\mathrm {A}(i)$ The ageing of records is conservative, without loss of information."
],
[
"Explicit form of superposition ",
"The operators constructed in (REF ) and (REF ) define the “objective” part of time evolution.",
"They do not act on the second tensorial factor of a record (equation (REF )).",
"They are assumed to multiply in the order $ U_\\mathrm {O} U_\\mathrm {A} = U_\\mathrm {obj}$ Starting from our preferred initial product state $|\\mathrm {in}\\rangle $ (section REF ), let us formulate the sequences $\\lbrace b_n\\rbrace $ of indices to which evolution branches under the action of $U_\\mathrm {obj}$ .",
"If quasiclassical evolution has promoted the record-generating system to an index $k\\in {\\cal Q}$ , branching to addresses $j\\in {\\cal A}_\\mathrm {S}(k)$ occurs in the next step of time.",
"A choice of $j$ , here renamed $b_n$ , characterises the branch.",
"From $b_n$ , quasiclassical evolution proceeds through indices numerically increasing until the next index in ${\\cal Q}$ is reached, and branching to $b_{n+1}$ occurs.",
"This takes a number of steps, $d_n$ .",
"The possible sequences of branching addresses $b_n$ and intervals $d_n$ of quasiclassical evolution must satisfy the following recursion relations.",
"$ \\begin{array}{rcl}b_0 &=& k_\\mathrm {in} \\\\q_n & = & \\min \\big \\lbrace q \\in {\\cal Q} ~ | ~ q > b_n \\big \\rbrace \\quad \\mbox{(auxiliary)} \\\\b_{n+1} & \\in & {\\cal A}_\\mathrm {S}(q_n) \\\\d_n &=& q_n - b_n + 1\\end{array}$ The time $t_n(b)$ at which a branching index $b_n$ is reached, depending on the branch considered, is $ t_n(b) = \\sum _{m=0}^{n-1} d_m(b)$ For a convenient representation of the stages of evolution in various branches, let us use the following abbreviation.",
"$ [t] = \\left\\lbrace \\begin{array}{ll} 0 & \\quad t \\le 0 \\\\t & \\quad t > 0\\end{array} \\right\\rbrace = \\Theta (t-\\epsilon )$ Moreover, let $p(b,t)$ be the number of branching points passed by time $t$ .",
"Referring to definitions (REF ) and (REF ), the record-generating system then is in the state $ \\psi _{k(b,t)} ~~ \\mbox{with}~~ k(b,t) = b_{p(b,t)} + t - t_{p(b,t)}$ Denoting by $V({\\cal I})$ the state in which all records are blank, the evolved state after $t$ steps of time may be expressed as $ \\left( U_\\mathrm {obj}\\right)^t |\\mathrm {in}\\rangle = \\sum _b \\left(\\frac{1}{\\sqrt{2}}\\right)^{p(b,t)}\\psi _{k(b,t)} \\otimes \\left( \\prod _{n=0}^\\infty \\prod _{l=0}^{d_n-1}~ \\prod _{i\\in {\\cal A}_{\\mathrm {W}}(b_n+l)} W_i^{[t - t_n - l]} \\right) V({\\cal I})$ To see this, first note that once a record is written, its ageing is the same as repeated writing by (REF ) and (REF ).",
"Writing operations can be assembled to powers because they commute (equation (REF )).",
"Thus, the linear rise of the powers with $t$ is the result of the ageing operator $U_\\mathrm {A}$ .",
"It remains to consider the time of the first writing of a record.",
"At time $t_n+l$ , the record-generating system is in the state with index $b_n+l$ .",
"Corresponding records, with indices in ${\\cal A}_{\\mathrm {W}}(b_n+l)$ , are written at the next step of time, that is, when the exponent $[t - t_n - l]$ of the writing operator is nonzero for the first time.",
"For later reference we note that the product vectors constituting different branches are orthogonal.",
"This is because two branches differ by at least one record, so that there is at least one tensorial factor $r_i$ which is in the blank state in one branch and written, hence orthogonal, in the other."
],
[
"Consciousness modelled as triggered recall",
"A third factor of the evolution operator is supposed to model reflections in the observer's mind.",
"Neurophysical detail is beyond the scope of this paper, but consciousness “supervenes” on neural dynamics [8].",
"The other ingredient of the present model is power-law statistics, which appears to be common in neurophysics, but is usually discussed in highly specialised context.",
"It is essentially a working hypothesis here.",
"As this part of the evolution operator is going to make its dominant impact near the end of an observer's histories, it must be prevented from writing records of an objective sort, that is, from writing any records at all in the model's terms.",
"Otherwise, a scenario would result in which the facts constituting an observer's histories would be generated within a step of time.",
"Thus the factor $U_\\mathrm {C}$ , defined below, should be only reflective, like reading a record by elastic scattering.",
"Activities like writing this article would be regarded as “subconscious”, that is, rather a matter of $U_\\mathrm {O} U_\\mathrm {A}$ within the scope of the model."
],
[
"Triggering records",
"Conscious reflection is assumed to be triggered by the reading of a record $r_m$ in a sparse index set ${\\cal M} \\subset {\\cal I}$ .",
"Moreover, it is assumed for simplicity that $ \\mbox{for each }k\\in {\\cal O}\\backslash {\\cal Q}\\mbox{ there is exactly one such $m$ in } {\\cal A}_\\mathrm {W}(k)$ If $r_m$ is blank, no reflection occurs.",
"If $r_m$ is in a written state, a “scattering” operation $S_l$ will be triggered on all $r_l$ with indices in a set ${\\cal A}_{\\mathrm {R}}(m)$ specified in equation (REF ) below.",
"Let $P^0_m$ denote the projection on the blank state of $r_m$ , and $P^\\perp _m$ the projection on all written states of $r_m$ .",
"We define the reflection triggered by $r_m$ as $ U_\\mathrm {C}(m) = P^0_m + P^\\perp _m \\prod _{l\\in {\\cal A}_{\\mathrm {R}}(m)} S_l\\qquad m \\notin {\\cal A}_{\\mathrm {R}}(m)$ with implicit unities for all tensor factors whose indices do not appear.",
"A scattering operation $S$ , in the indicated space, is assumed to modify the second factor of (REF ) in a way dependent on the first factor, $ \\begin{array}{l}S_l ~ e_0 \\otimes s = e_0 \\otimes s \\\\S_l ~ e_i \\otimes s = e_{i} \\otimes u_{li} s \\qquad u_{li} \\ne 1 \\qquad i=1,\\ldots ,N-1\\end{array}$ For records in a written state, all we assume about the unitary $2\\times 2$ matrices $u_{li}$ is that they be different from $\\bf 1$ so as to make “something” go on in the observer's mind.",
"A crucial assumption is made on the statistics, in random draws for the construction, of the lengths $L({\\cal A}_{\\mathrm {R}})$ of the address sets ${\\cal A}_{\\mathrm {R}}$ .",
"Let $\\overline{F}(L)$ be the complementary cumulative distribution function, that is the fraction of sets whose length is greater than $L$ .",
"We assume a capped Pareto distribution $ \\overline{F}_1(L) = \\left\\lbrace \\begin{array}{cr} (L_0/L)^\\alpha & L_0 \\le L \\le I \\\\0 & L > I \\end{array} \\right\\rbrace \\qquad 1 < \\alpha < 2$ where the cap is assumed to be practically irrelevant due to the size of the index set $\\cal I$ .",
"To ensure the statistical independence required for the theorems of order statistics to apply, let us construct the index sets by explicit use of independent random draws.",
"In a first step, the lengths of sets are determined.",
"$ \\mbox{For all $m\\in {\\cal M}$,} ~ L(m) = \\mbox{random draw from distribution (\\ref {AkPowerLaw})}$ In a second step, $L(m)$ indices are selected by another random procedure, and collected into ${\\cal A}_{\\mathrm {R}}(m)$ .",
"The procedure is as follows."
],
[
"Searching for potential records ",
"Operator $U_\\mathrm {C}$ must select $L(m)$ records that may have been written during time evolution.",
"In fact, if recall operations were searching for records irrespective of causal relations, the scenario envisioned would not work statistically.",
"The search would be based on mere chance—on a probability proportional to $L(m)$ , which would either have to be very small, or could not be power-law distributed, since probabilities are bounded above.",
"Tracing back histories that may have lead to a memory index $m$ , there emerges a backward-branching structure because there are, on average, two indices of ${\\cal Q}$ from which quantum jumps are directed to a given section of quasiclassical evolution; see section REF .",
"Starting from the memory-triggering index $m$ , all sequences $\\lbrace c^m_n\\rbrace _{n=0,1,\\ldots }$ of branching points that may have lead to the writing at $m$ must satisfy the following relations.",
"$ \\begin{array}{rcl}c^m_0 &=& \\lbrace k\\in {\\cal O} ~|~ m \\in {\\cal A}_\\mathrm {W}(k)\\rbrace \\\\j_n & = & 1 + \\max \\big \\lbrace q \\in {\\cal Q} ~ | ~ q < c^m_n \\big \\rbrace \\quad \\mbox{(auxiliary)} \\\\d^m_n &=& c^m_n - j_n \\quad \\mbox{(length of quasiclassical section)} \\\\j_n & = & {\\cal A}_\\mathrm {S}(c^m_{n+1}) \\quad \\mbox{(preceding points of branching)}\\end{array}$ Tracing back quasiclassical evolution, which is index-increasing, $j_n$ is the first index encountered to which evolution may have jumped from somewhere.",
"Indices $c^m_n,c^m_n-1,\\ldots ,c^m_n-d^m_n$ constitute the $n$ th section on a branch of possible evolution.",
"The average length of such a section is $K/Q$ .",
"We wish to distribute $L(m)$ conscious recalls equally over a lifetime.",
"Hence there are $L(m)/T && \\mbox{ recalls per time} \\\\l(m) = K L(m)/ TQ && \\mbox{ recalls per section} $ Thus, for every sequence $c$ branching backward from $m$ and for every section number $n$ let us define $\\begin{array}{rl}{\\cal C}(m,c,n) ~ = & \\mbox{set of $l(m)$ randomly chosen elements unequal $m$} \\\\ & \\mbox{of }{\\cal A}_\\mathrm {W}(c^m_n) \\cup {\\cal A}_\\mathrm {W}(c^m_n-1) \\cup \\cdots \\cup {\\cal A}_\\mathrm {W}(c^m_n-d^m_n)\\end{array}$ In terms of ${\\cal C}(m,c,n)$ we can specify the index sets already used in (REF ).",
"$ {\\cal A}_\\mathrm {R}(m) = \\bigcup _{n=0}^{TQ/K} ~ \\bigcup _{\\lbrace c\\rbrace } ~ {\\cal C}(m,c,n)$ The full consciousness-generating part of the evolution operator is, referring to (REF ) again, $ U_\\mathrm {C} = \\prod \\limits _{m\\in {\\cal M}} U_\\mathrm {C}(m)$"
],
[
"Branch of extremal consciousness",
"By $T$ steps of evolution, a superposition of product states builds up, which in equation (REF ) was expressed as a sum over branches, each branch being generated by a product of writing operations.",
"One-to-one correspondence to a branching tree can be seen by factoring out $W$ operations of common parts of the branches.",
"The loop-avoiding construction of section REF is important here.",
"On the branching tree, certain memory-triggering records are in a written state.",
"One of those records will trigger the maximal number of recalls, whose excess we wish to quantify statistically.",
"It would be straightforward to estimate the excess on the basis of mean values alone, similar to the argument given in [11], but fluctuations in branching processes are as big as the mean values [12] so analysis in terms of probability distributions is required."
],
[
"Statistics of branching and recall-triggering",
"The general theory of Galton-Watson processes [12] deals with familiy trees whose members are grouped in generations $n=1,2,3,\\ldots $ .",
"Each member generates a number $j=0,1,\\ldots $ of members of the next generation with probability $p_j$ .",
"In our model, a new generation occurs at each step of time.",
"The number of members in a generation, $Z_t$ , is the number of product states superposed at time $t$ .",
"The probability $p_0$ , corresponding to an end of a branch of the observer's history, is zero within the lifetime $T$ considered.",
"The probability $p_1$ , corresponding to a product state continuing as a product state after a step of time, is close to one.",
"The probability $p_2$ , corresponding to the splitting of a branch into a superposition of two product states, is small but nonzero.",
"Probabilities $p_3,p_4,\\ldots $ are zero by the model assumptions.",
"In our model, splitting in two branches occurs at $Q$ randomly distributed points of $K$ , so the parameters for the branching process here are $ p_2 = \\frac{Q}{K} = : \\sigma ~~~~~~~~~ p_1 = 1 - p_2 ~~~~~~~~~~p_j = 0 \\mbox{ for }j = 0,3,4,5,\\ldots $ The mean number of offspring generated by a member thus is $ \\mu = 1 + \\sigma > 1$ Because of $p_0=0$ , we are dealing with zero “probability of extinction”.",
"For the statistics of the extremes, we need to know the total number of recall-triggering factors on the tree.",
"By assumption (REF ) that number equals the “total progeny” $Y_t = \\sum _{\\tau =1}^t Z_\\tau $ .",
"By Theorem 6 of [13], the probability distribution for the values of $Y_t$ has an asymptotic form which can be described as follows.",
"There exists a sequence of positive constants $C_t$ , $t=1,2,\\ldots $ , with $C_{t+1}/C_t \\rightarrow \\mu $ for $t\\rightarrow \\infty $ such that $ \\lim _{t\\rightarrow \\infty } P\\lbrace Y_t \\le x C_t\\rbrace = P\\lbrace W\\le x\\sigma /\\mu \\rbrace $ where $W$ is a non-degenerate random variable which has a continuous distribution on the set of positive real numbers.",
"Let $w$ be the probability density for $W$ .",
"We shall treat $Y$ as continuous, too, and assume that by an observer's lifetime $T$ the limiting form of (REF ) already applies.",
"Then the probability of $Y$ is, by differentiating (REF ) and using $\\mu \\approx 1$ , $ w\\left(\\frac{\\sigma Y}{C_T}\\right)\\,\\frac{\\sigma }{C_T} \\, \\mathrm {d}Y$ Each occurrence of a memory-triggering index $m$ is characterised by the location on the tree, in particular the time $t$ , and by the length $L(m)$ of the recalling sequence according to (REF ).",
"Since the location results from random draws in $U_\\mathrm {O}$ , and the length from a random draw in $U_\\mathrm {C}$ , they are statistically independent, so their joint probability is the product of the separate probabilities.",
"The time $t$ of occurrence, that is the generation number in the general theory, has a probability $Z_t/Y_t$ whose asymptotic form, under the same conditions as for (REF ), is given by Lemma 2.2 of [14].",
"If $j = 0,1,2,\\ldots $ denotes the distance form the latest time on the tree, the probability is $P_j = (1-\\mu ^{-1}) \\mu ^{-j}$ Taking the latest time on the branching tree to be $T$ , and treating $L$ as continuous, $P_j$ and the Pareto distribution (REF ) give the joint probability of $t$ and $L$ , $ P(t,L)\\, \\mathrm {d}L = P_{T-t} \\,\\alpha L_0^\\alpha L^{-\\alpha - 1} \\, \\mathrm {d}L\\qquad 0\\le t \\le T, ~ L \\ge L_0$ If the memory-triggering $m$ occurs at $t$ , then by (REF ) the number of records recalled is $ R = \\frac{t}{T}\\,L(m)$ It is the extreme-order statistics of this quantity that matters.",
"The density of $R$ is obtained by taking the expectation of $\\delta (R - tL/T)$ with the probability distribution (REF ).",
"In the range $R>L_0$ this gives another Pareto distribution with complementary cumulative distribution function $ \\overline{F}_2(R) = (R_0/R)^\\alpha ~ \\mbox{ where } ~ R_0^\\alpha =L_0^\\alpha \\sum _{t=0}^T P_{T-t} \\left(\\frac{t}{T}\\right)^\\alpha $ Thus, with probability given by (REF ), we have a number $Y$ of memory-triggering indices on the branching tree of a lifetime, each of which with a probability given by (REF ) induces $R$ recalls along its branch.",
"We now use a result of order statistics, conveniently formulated for our purposes in [1], table 3.4.2 and corollary 4.2.13, which relates the number of random draws, here $Y$ (different letters used in [1]), to the spacing $D$ between the largest and the second-largest draw of $R$ from an ensemble given by (REF ).",
"$ D = R_{\\mbox{\\small largest}} - R_{\\mbox{\\small second-largest}} = R_0 \\, Y^{1/\\alpha } \\, X$ where $X$ is a random variable independent of $Y$ .",
"The probability density $g(x)$ of $X$ , given in integral representation, can be seen to be uniformly bounded.",
"The cumulative distribution function for $D$ is, for a given value of $Y$ , of the form $G\\left(Y^{-1/\\alpha }D/R_0\\right)$ where $G^{\\prime }(x)=g(x)$ .",
"Hence, the joint probability of $Y$ and $D$ , expressed by density (REF ) for $Y$ and the cumulative distribution function for $D$ , is $ G\\left(Y^{-1/\\alpha }D/R_0\\right)w\\left(\\frac{\\sigma Y}{C_T}\\right)\\,\\frac{\\sigma }{C_T}\\,\\mathrm {d}Y$"
],
[
"Dimension of conscious subspace",
"Consciousness, in the model's approximation, is assumed to reside in unitary rotations $u\\ne 1$ of the right tensor factors of (REF ).",
"Transformations of the left factors, as generated by $U_\\mathrm {obj}$ , are assumed to be unconscious.",
"In the superposition generated by $U_\\mathrm {obj}$ , equation (REF ), the branches (product vectors) are mutually orthogonal by the “objective” left factors alone, as was noted at the end of section REF .",
"Hence, the unitary rotations of consciousness take place in subspaces which, for different branches, are orthogonal.",
"Thus, a “conscious dimension” $d_\\mathrm {C}$ can be assigned to each branch.",
"$ d_\\mathrm {C} ~ = ~ \\parbox [t]{100mm}{Hilbert-space dimension of the tensor factors rotating underU_\\mathrm {C} while the remainder of factors is constant.",
"}$ The number of tensor factors rotating in a branch is $R$ , as defined in equation (REF ), so the dimension of the conscious subspace in a branch is $2^R$ .",
"It should be noted that the subspace as such depends on the vectorial value taken by the nonrotating factors.",
"The proposition of the paper is that the conscious dimension in the branch with the largest $R$ exceeds, by a huge factor $E$ , the sum of conscious dimensions in all other branches.",
"The latter sum can be estimated, denoting by $Z_T$ the number of branches (terms of superposition) at time $T$ , as $< ~ 2^{R_{\\mbox{\\small second-largest}}} Z_T$ Evaluating this would require a joint distribution of $R$ , $Y$ , and $Z$ , so a more convenient estimate, using $Z_T < Y_T$ , is $< ~ 2^{R_{\\mbox{\\small second-largest}}} Y_T$ Taking binary logarithms, the customised proposition is that the last term in the equation $ R_{\\mbox{\\small largest}} = R_{\\mbox{\\small second-largest}} + \\log _2 Y + \\log _2 E$ almost certainly takes a large value.",
"By (REF ), $\\log _2 E = D - \\log _2 Y$ , so the relevant cumulative distribution function is obtained from (REF ) as $ F_3(x) = P\\lbrace D - \\log _2 Y < x \\rbrace = \\int _1^\\infty \\!\\!",
"G\\Big (Y^{-1/\\alpha }(x + \\log _2 Y)/R_0\\Big )w\\left(\\frac{\\sigma Y}{C_T}\\right)\\,\\frac{\\sigma }{C_T} \\,\\mathrm {d}Y$ Substituting $ \\frac{\\sigma Y}{C_T} = y ~~~~~~~~~~~~ \\left(\\frac{\\sigma }{C_T}\\right)^{1/\\alpha } \\frac{x}{R_0} = z$ and putting $\\sigma /C_T\\approx 0$ in the lower limit of integration, the integral becomes $\\int _0^\\infty G\\left(y^{-1/\\alpha } z + \\left(\\frac{\\sigma }{C_T}\\right)^{1/\\alpha } R_0^{-1}y^{-1/\\alpha }(\\log _2 y + \\log _2 C_T - \\log _2\\sigma )\\right)w(y) \\,\\mathrm {d}y$ Asymptotically for $C_T\\rightarrow \\infty $ , which represents an exponentially grown number of branches, the integral simplifies to $F_3(x) = \\int _0^\\infty G\\left(y^{-1/\\alpha } z )\\right) w(y) \\,\\mathrm {d}y$ because $G$ has the uniformly bounded derivative $g$ (see text following (REF )) while $\\int _0^\\infty y^{-1/\\alpha } \\log _2 y \\, w(y) \\,\\mathrm {d}y$ converges for $\\alpha $ in the range given by (REF ) and the coefficients $C_T^{-1}$ and $C_T^{-1}\\log _2 C_T$ become vanishingly small.",
"Inserting $z$ from (REF ), and $x=\\log _2 E$ from (REF ), the cumulative distribution function for the excess factor $E$ is given by $ F_3\\left(\\left(\\frac{\\sigma }{C_T}\\right)^{1/\\alpha }R_0^{-1} \\log _2 E\\right)$ Due to logarithmation, followed by a rescaling which broadens the distribution by a large factor, $E$ almost certainly takes a huge value, rather independently of the exact form of the distribution function $F_3$ ."
],
[
"Complying with Born's rule",
"In section REF the wavefunction is modelled to split in two branches with equal amplitudes.",
"Born's rule is trivially satisfied in this case.",
"Does the model generalise correctly to a splitting with unequal amplitudes?",
"Technically, this is accomplished by a unitary transformation devised in [15], [16], [17] which entangles a two-state superposition with a large number of auxiliary states so as to form another equal-amplitude superposition.",
"It will be argued that in this way the model scenario is consistent with Born's rule in general.",
"The crucial point here is that “if you believe in determinism, you have to believe it all the way” [18].",
"When an observer encounters a wavefunction for a measurement, like $ a |A\\rangle + b |B\\rangle $ that wavefunction is given to him by the total operator of evolution.",
"The operator is thus only required to handle wavefunctions that it provides itself.",
"Extending the model accordingly would be based on the following considerations.",
"In the course of measurement, a result $A$ or $B$ is obtained, but it always comes with many irrelevant properties of the constituents, like the number of photons scattered off the apparatus, the number of observer's neurons firing, etc.",
"Let $n$ be the number of irrelevant properties to be taken into account.",
"Then, after the measurement, we have a state vector of the form $ \\sum _{k=1}^m c_k |A,k\\rangle + \\sum _{k=m+1}^n c_k |B,k\\rangle $ The measuring evolution should commute with the projections on the spaces defined by $A$ and $B$ , so we have constraints on the absolute values, $ \\sum _{k=1}^m |c_k|^2 = |a|^2 \\qquad \\qquad \\sum _{k=m+1}^n |c_k|^2 = |b|^2$ On the other hand, state vectors differing only in the phases of the $c_k$ can be regarded as equivalent for the measurement process, as has been shown by different arguments in [15] and [16], and elaborately in [17].",
"Since the $k$ -properties in (REF ) are “irrelevant”, we expect the evolution to produce a state belonging to the equivalence class at the peak number of representatives.",
"A measure of this number is given by the surface element in the space of $n$ -dimensional normalised states $\\delta \\left(1-\\sum _{k=1}^n |c_k|^2\\right) \\prod _{k=1}^n \\mathrm {d}^2 c_k$ which is defined uniquely, up to a constant, by its invariance under unitary changes of basis for the span of the vectors.",
"The number of representatives is obtained by integrating over the phases, which gives $ \\delta \\left(1-\\sum _{k=1}^n |c_k|^2\\right) \\prod _{k=1}^n 2\\pi |c_k|\\mathrm {d}|c_k|$ At the maximum, all moduli must be nonzero because of the $|c_k|$ factors.",
"It follows by the permutation symmetry of constraints (REF ) that $|c_k| = \\frac{|a|}{\\sqrt{m}} \\quad k=1,\\ldots ,m \\qquad |c_k| = \\frac{|b|}{\\sqrt{n-m}}\\quad k=m+1,\\ldots ,n$ Finally extremising in the parameter $m$ , by extremising the product of moduli in (REF ) we find $ |c_k| = \\frac{1}{\\sqrt{n}} ~~ \\mbox{ for all }k$ So the number $m$ of branches with property $A$ equals $|a|^2 n$ .",
"A similar argument, with a discussion of fluctuations about the maximum, was given for a different scenario in [19].",
"Equations (REF ) and (REF ), in conjunction with an arbitrary choice of phases, like $c_k=\\sqrt{1/n}$ , now specify the state vector that an evolution operator for a measurement should generate.",
"Instead of splitting into “up” and “down” branches we now have splitting into $n$ branches, of which $m$ correspond to result $A$ of the measurement, and $n-m$ to result $B$ .",
"Information about $A$ or $B$ can be regarded as implicit in the indices of the records, so parameter $m$ need not even appear.",
"In the equations of section the following replacements have to be made.",
"In (REF ), $\\psi _u+\\psi _d$ extends to a sum over $n$ equal parts.",
"In (REF ) the entries $u,d$ of the register change to $1,\\ldots ,n$ , and the same holds for the address sets (REF ).",
"Normalisation factors change from $\\sqrt{1/2}$ to $\\sqrt{1/n}$ in equations (REF ) and (REF ).",
"In section the nonzero branching probabilities become $p_n \\ll 1$ and $p_1 = 1-p_n$ , and all powers of 2 change to powers of $n$ ."
],
[
"Conclusion",
"Observer's consciousness playing a role in the projection postulate has been pondered since the beginnings of quantum theory [20].",
"The model presented here is a proposal of how the idea might be realised, albeit with modifications, in a framework of unitary quantum-mechanical evolution.",
"In the model scenario, the projection involved is on the conscious subspace at the time of the extremal draw as defined in section REF .",
"But this projection assigns an outcome, “up” or “down”, not only to a single measurement (or quantum event, as it was called here) but to all measurements of an observer's lifetime.",
"Moreover, an assumption on statistics in the dynamics of consciousness was crucial to the functioning of the model.",
"It might be objected then that we have only replaced the projection postulate with a statistical postulate based on speculation about consciousness.",
"But there are well-known mechanisms for generating power-law statistics, some of which may be adaptable so as replace the postulate by the workings of an extended model.",
"For the present model, it was important also to demonstrate that the preconditions of extreme-order theorems can be realised exactly in a framework of unitary evolution.",
"This was made obvious by employing random draws in the construction of the time-step operator.",
"A rather complicated part in that construction, section REF , was to identify by backward branching the records that had a chance to be written before a given record.",
"This might suggest to alternatively accumulate the extremal value in the course of evolution.",
"But “when the sum of [...] independent heavy-tail random variables is large, then it is very likely that only one summand is large” [21], so the alternative approach is very likely to reduce to the one already taken.",
"A scenario of consciousness, all generated by one extremal event in a short interval of time, may also contribute to improving the physical notion of the metaphysical present.",
"In the physical approach, time is parameterised by the reading of a clock, and it is possible to quantify the time intervals of cognitive processing.",
"But outside the science community, such an approach is often felt to inadequately represent the existential quality of a moment.",
"The model scenario would suggest a more complex relation to physical time.",
"Physicswise, it is a moment indeed, but since it covers all individual experience, the moment appears to be lasting.",
"The undifferentiated usage of “consciousness” in this paper will be unsatisfactory from a biological or psychological point of view, although consciousness as a processing of memories was discussed first in those fields [22].",
"For the present purpose, the meaning of the term was defined in section REF .",
"As a consequence, activities of an observer which are conscious in the usual sense had to be regarded as “unconscious”.",
"Apparently, various levels of consciousness should be taken into account by an extended model.",
"References"
]
] | 1204.0760 |
[
[
"A game interpretation of the Neumann problem for fully nonlinear\n parabolic and elliptic equations"
],
[
"Abstract We provide a deterministic-control-based interpretation for a broad class of fully nonlinear parabolic and elliptic PDEs with continuous Neumann boundary conditions in a smooth domain.",
"We construct families of two-person games depending on a small parameter which extend those proposed by Kohn and Serfaty (2010).",
"These new games treat a Neumann boundary condition by introducing some specific rules near the boundary.",
"We show that the value function converges, in the viscosity sense, to the solution of the PDE as the parameter tends to zero.",
"Moreover, our construction allows us to treat both the oblique and the mixed type Dirichlet-Neumann boundary conditions."
],
[
"Introduction",
"In this paper, we propose a deterministic control interpretation, via “two persons repeated games”, for a broad class of fully nonlinear equations of elliptic or parabolic type with a continuous Neumann boundary condition in a smooth (not necessarily bounded) domain.",
"In their seminal paper [21], Kohn and Serfaty focused on the one hand on the whole space case in the parabolic setting and on the other hand on the Dirichlet problem in the elliptic framework.",
"They construct a monotone and consistent difference approximation of the operator from the dynamic programming principle associated to the game.",
"Our motivation here is to adapt their approach to the Neumann problem in both settings.",
"Furthermore, once this issue is solved, we will see how the oblique or the mixed type Dirichlet-Neumann boundary problem can also be treated by this analysis.",
"We consider equations in a domain $\\Omega \\subset \\mathbb {R}^N$ having the form $-u_t+f(t,x,u,Du,D^2u)=0 $ or $f(x,u,Du,D^2u)+\\lambda u=0 , $ where $f$ is elliptic in the sense that $f$ is monotone in its last variable, subject to the Neumann boundary condition $\\frac{\\partial u}{\\partial n}=h.$ As in [21], the class of functions $f$ considered is large, including those that are non-monotone in the $u$ argument and degenerate in the $D^2u$ argument.",
"We make the same hypotheses on the continuity, growth, and $u$ -dependence of $f$ imposed in [21].",
"They are recalled at the end of the section.",
"In the stationary setting (REF ), we focus on the Neumann problem, solving the equation in a domain $\\Omega $ with (REF ) at $\\partial \\Omega $ .",
"In the time-dependent setting (REF ), we address the Cauchy problem, solving the equation with (REF ) at $\\partial \\Omega $ for $t<T$ and $u=g$ at terminal time $t=T$ .",
"The PDEs and boundary conditions are always interpreted in the “viscosity sense” (Section presents a review of this notion).",
"Our games have two opposite players, Helen and Mark, who always make decisions rationally and deterministically.",
"The rules depend on the form of the equation, but there is always a small parameter $\\varepsilon $ , which governs the spatial step size and (in time-dependent problems) the time step.",
"Helen's goal is to optimize her worst-case outcome.",
"When $f$ is independent of $u$ , we shall characterize her value function $u^\\varepsilon $ by the dynamic programming principle.",
"If $f$ depends also on $u$ , the technicality of ours arguments requires to introduce a level-set formulation since the uniqueness of the viscosity solution is no longer guaranteed.",
"The score $U^\\varepsilon $ of Helen now depends on a new parameter $z \\in \\mathbb {R}$ .",
"In the parabolic setting, it is defined by an induction backward in time given by $\\forall z\\in \\mathbb {R}, \\quad U^\\varepsilon (x,z,t)=\\max _{p,\\Gamma } \\min _{\\Delta \\hat{x}} U^\\varepsilon (x+\\Delta x, z+\\Delta z, t+\\Delta t),$ endowed with the final-time condition $ U^\\varepsilon (x,z,t)=g(x)-z$ .",
"The max on $p$ , $\\Gamma $ and the min on $\\Delta \\hat{x}$ are given by some constrains depending on the rules of the game and some powers of $\\varepsilon $ .",
"This dynamic programming principle is similar to the one given in [21].",
"In that case, our value functions $u^\\varepsilon $ of interest are defined through the 0-level set of $U^\\varepsilon $ with respect to $z$ as the maximal and the minimal solutions of $U^\\varepsilon (x,z,t)=0$ .",
"They satisfy two dynamic programming inequalities (for the details of our games and the definition of Helen's value function, see Section ).",
"Roughly speaking, our main result states that $& \\limsup _{\\varepsilon \\rightarrow 0} u^\\varepsilon \\text{ is a viscosity subsolution of the PDE, and } \\\\& \\liminf _{\\varepsilon \\rightarrow 0} u^\\varepsilon \\text{ is a viscosity supersolution of the PDE.",
"}$ For the general theory of viscosity solutions to fully nonlinear equations with Neumann (or oblique) boundary condition the reader is referred to [12], [3], [19].",
"As for the Neumann boundary condition, its relaxation in the viscosity sense was first proposed by Lions [22].",
"Our result is most interesting when the PDE has a comparison principle, i.e.",
"when every subsolution must lie below any supersolution.",
"For such equations, we conclude that $\\lim u^\\varepsilon $ exists and is the unique viscosity solution of the PDE.",
"In the case when $f$ is continuous in all its variable, there are already a lot of comparison and existence results for viscosity solutions of second order parabolic PDEs with general Neumann type boundary conditions.",
"We refer for this to [3], [5], [22], [19] and references therein.",
"For homogeneous Neumann conditions, Sato [27] has obtained such a comparison principle for certain parabolic PDEs.",
"We are interested here in giving a game interpretation for fully nonlinear parabolic and elliptic equations with a Neumann condition.",
"Applications of the Neumann condition to deterministic optimal control and differential games theory in [22] rely much on a reflection process, the solution of the deterministic Skorokhod problem.",
"Its properties in differents situations are studied in many articles such as [28], [24], [13].",
"The case of the Neumann problem for the motion by mean curvature was studied by Giga and Liu [17].",
"There, a billiard game was introduced to extend the interpretation made by Kohn and Serfaty [20] via the game of Paul and Carol.",
"It was based on the natural idea that a homogeneous Neumann condition will be well-modeled by a reflection on the boundary.",
"Liu also applies this billiard dynamics to study some first order Hamilton-Jacobi equations with Neumann or oblique boundary conditions [25].",
"Nevertheless, in our case, if we want to give a billiard interpretation with a bouncing rule which can send the particle far from the boundary, we can only manage to solve the homogeneous case.",
"This is not too surprising because the reflection across $\\partial \\Omega $ is precisely associated to a homogeneous Neumann condition.",
"Another approach linked to the Neumann condition is to proceed by penalization on the dynamics.",
"For a bounded convex domain, Lions, Menaldi and Sznitman [23] construct a sequence of stochastic differential equations with a term in the drift coefficients that strongly penalizes the process from leaving the domain.",
"Its solution converges towards a diffusion process which reflects across the boundary with respect to the normal vector.",
"Barles and Lions [7] also treat the oblique case by precisely establishing the links between some approximated processes and the elliptic operators associated to the original oblique stochastic dynamics.",
"Instead of a billiard, our approach here proceeds by a suitable penalization on the dynamics depending on the Neumann boundary condition.",
"It will be favorable to one player or the other according to its sign.",
"We modify the rules of the game only in a small neighborhood of the boundary.",
"The particle driven by the players can leave the domain but then it is projected within.",
"This particular move, combined with a proper weight associated to the Neumann boundary condition, gives the required penalization.",
"Outside this region, the usual rules are conserved.",
"Therefore the previous analysis within $\\Omega $ done by Kohn and Serfaty can be preserved.",
"We focus all along this article on the changes near the boundary and their consequences on the global convergence theorem.",
"In this context, the modification of the rules of the original game introduces many additional difficulties intervening at the different steps of the proof.",
"Most of all, they are due to the geometry of the domain or the distance to the boundary.",
"As a result, our games seem like a natural adaptation of the games proposed by Kohn and Serfaty by permitting to solve an inhomogeneous Neumann condition $h$ depending on $x$ on the boundary.",
"We only require $h$ to be continuous and uniformly bounded, the domain to be $C^2$ and to satisfy some natural geometric conditions in order to ensure the well-posedness of our games.",
"Moreover our approach can easily be extended both to the oblique and the mixed Neumann-Dirichlet boundary conditions in both parabolic and elliptic settings.",
"Our games can be compared to those proposed in [21] for the elliptic Dirichlet problem: if the particle crosses the boundary, the game is immediately stopped and Helen receives a bonus $b(x_F)$ where $b$ corresponds to the Dirichlet boundary condition and $x_F$ is the final position.",
"Meanwhile, our games cannot stop unexpectedly, no matter the boundary is crossed or not.",
"Our games, like the ones proposed by Kohn and Serfaty, are deterministic but closely related to a recently developed stochastic representation due to Cheridito, Soner, Touzi and Victoir [11] (their work uses a backward stochastic differential equation, BSDE, whose structure depends on the form of the equation).",
"Another interpretation is to look our games as a numerical scheme whose solution is an approximation of a solution of a certain PDE.",
"This aspect is classical and has already been exploited in several contexts.",
"We mention the work of Peres, Schramm, Sheffield and Wilson [26] who showed that the infinity Laplace equation describes the continuum limit of the value function of a two-player, random-turn game called $\\varepsilon $-step tug-of-war.",
"In related work, Armstrong, Smart and Sommersille [2] obtained existence, uniqueness and stability results for an infinity Laplace equation with mixed Dirichlet-Neumann boundary terms by comparing solutions of the PDE to subsolutions and supersolutions of a certain finite difference scheme, by following a previous work of Armstrong and Smart for the Dirichlet case [1].",
"This paper is organized as follows: Section presents the two-person games that we associate with the PDEs (REF ) and (REF ), motivating and stating our main results.",
"The section starts with a simple case before adressing the general one.",
"Understanding our games is still easy, though the technicality of our proofs is increased.",
"Since $f$ depends on $u$ , the game determines a pair of value functions $u^\\varepsilon $ and $v^\\varepsilon $ .",
"Section REF gives a formal argument linking the principle of dynamic programming to the PDE in the limit $\\varepsilon \\rightarrow 0$ and giving the optimal strategies for Helen that will be essential to obtain consistency at Section .",
"Section addresses the link between our game and the PDE with full rigor.",
"The proofs of convergence follow the background method of Barles and Souganidis [10], i.e.",
"they use the stability, monotonicity and consistency of the schemes provided by our games.",
"Their theorem states that if a numerical scheme is monotone, stable, and consistent, then the associated “lower semi-relaxed limit” is a viscosity supersolution and the associated “upper semi-relaxed limit” is a viscosity subsolution.",
"The main result in Section is a specialization of their theorem in our framework: if $v^\\varepsilon $ and $u^\\varepsilon $ remain bounded as $\\varepsilon \\rightarrow 0$ then the lower relaxed semi-limit of $v^\\varepsilon $ is a viscosity supersolution and the upper relaxed semi-limit of $u^\\varepsilon $ is a viscosity subsolution.",
"We also have $v^\\varepsilon \\le u^\\varepsilon $ with no extra hypothesis in the parabolic setting, or if $f$ is monotone in $u$ in the elliptic setting.",
"If the PDE has a comparison principle (see [10]) then it follows that $\\lim u^\\varepsilon =\\lim v^\\varepsilon $ exists and is the unique viscosity solution of the PDE.",
"The analysis in Section shows that consistency and stability imply convergence.",
"Sections and provide the required consistency and stability results.",
"The new difficulties due to the penalization corresponding to the Neumann condition arise here.",
"The main difficulty is to control the degeneration of the consistency estimate obtained in [21] with respect to the penalization.",
"Therefore we will mainly focus on the consistency estimates whereas the needed changes for stability will be simply indicated.",
"Section describes the games associated on the one hand to the oblique problem in the parabolic setting and on the other hand to the mixed type Dirichlet-Neumann boundary conditions in the elliptic framework.",
"By combining the results associated to the game associated to the Neumann problem in Section with the ideas already presented in [21], we can obtain the results of convergence.",
"Notation: The term domain will be reserved for a nonempty, connected, and open subset of $\\mathbb {R}^N$ .",
"If $x,y \\in \\mathbb {R}^N$ , $\\displaystyle \\left\\langle x,y \\right\\rangle $ denotes the usual Euclidean inner product and $\\Vert x \\Vert $ the Euclidean length of $x$ .",
"If $A$ is a $N \\times N$ matrix, $\\Vert A \\Vert $ denotes the operator norm $\\displaystyle \\Vert A \\Vert = \\sup _{\\Vert x \\Vert \\le 1} \\Vert Ax \\Vert $ .",
"$\\mathcal {S}^N$ denotes the set of symmetric $N \\times N$ matrices and $E_{ij}$ the $(i,j)$ -th matrix unit, the matrix whose only nonzero element is equal to 1 and occupies the $(i,j)$ -th position.",
"Let $\\mathcal {O}$ be a domain in $\\mathbb {R}^N$ and $C^k_b(\\mathcal {O})$ be the vector space of $k$ -times continuously differentiable functions $u$ : $\\mathcal {O} \\rightarrow \\mathbb {R}$ , such that all the partial derivatives of $u$ up to order $k$ are bounded on $\\mathcal {O}$ .",
"For a domain $\\Omega $ , we define $C^k_b(\\overline{\\Omega })= \\left\\lbrace u \\in L^\\infty (\\overline{\\Omega }) : \\exists \\mathcal {O} \\supset \\overline{\\Omega }, \\mathcal {O} \\text{ domain},\\exists v \\in C^k_b(\\mathcal {O})\\text{ s.t. }",
"u=v_{|\\overline{\\Omega }} \\right\\rbrace .$ It is equipped with the norm $\\Vert \\cdot \\Vert _{C^k_b(\\overline{\\Omega })}$ given by $\\displaystyle \\Vert \\phi \\Vert _{C^k_b(\\overline{\\Omega })} =\\sum _{i=0}^k\\Vert D^i\\phi \\Vert _{L^\\infty (\\overline{\\Omega })}$ .",
"If $\\Omega $ is a smooth domain, say $C^2$ , the distance function to $\\partial \\Omega $ is denoted by $d= d(\\cdot , \\partial \\Omega )$ , and we recall that, for all $x \\in \\partial \\Omega $ , the outward normal $n(x)$ to $\\partial \\Omega $ at $x$ is given by $n(x) = - Dd(x)$ .",
"Observe that, if $\\partial \\Omega $ is assumed to be bounded and at least of class $C^{2}$ , any $x\\in \\mathbb {R}^N$ lying in a sufficiently small neighborhood of the boundary admits a unique projection onto $\\partial \\Omega $ , denoted by $\\bar{x} = \\operatorname{proj}_{\\partial \\Omega }(x).$ In particular, the vector $x- \\bar{x}$ is parallel to $n(\\bar{x})$ .",
"The projection onto $\\overline{\\Omega }$ will be denoted by $\\operatorname{proj}_{\\overline{\\Omega }}$ .",
"When it is well-defined, it can be decomposed as $\\operatorname{proj}_{\\overline{\\Omega }} (x) ={\\left\\lbrace \\begin{array}{ll}\\operatorname{proj}_{\\partial \\Omega }(x), & \\text{ if } x \\notin \\Omega , \\\\x, & \\text{ if } x\\in \\Omega .\\end{array}\\right.",
"}$ For each $a>0$ , we define $ \\Omega (a)=\\lbrace x\\in \\overline{\\Omega }, d(x) < a\\rbrace $ .",
"We recall the following classical geometric condition (see e.g.",
"[14]).",
"Definition 1.1 (Interior ball condition) The domain $\\Omega $ satisfies the interior ball condition at $x_0 \\in \\partial \\Omega $ if there exists an open ball $B\\subset \\Omega $ with $x_0\\in \\partial B$ .",
"We close this introduction by listing our main hypotheses on the form of the PDE.",
"First of all we precise some hypotheses on the domain $\\Omega $ .",
"Throughout this article, $\\Omega $ will denote a $C^2$ -domain.",
"In the unbounded case, we impose the following slightly stronger condition than the interior ball condition.",
"Definition 1.2 (Uniform interior/exterior ball condition) The domain $\\Omega $ satisfies the uniform interior ball condition if there exists $r>0$ such that for all $x \\in \\partial \\Omega $ there exists an open ball $B\\subset \\Omega $ with $x\\in \\partial B$ and radius $r$ .",
"Moreover, the domain $\\Omega $ satisfies the uniform exterior ball condition if $\\mathbb {R}^N \\backslash \\overline{\\Omega }$ satisfies the uniform interior ball condition.",
"We observe that the uniform interior ball condition implies the interior ball condition and that both the uniform interior and exterior ball conditions hold automatically for a $C^2$ -bounded domain.",
"The Neumann boundary condition $h$ is assumed to be continuous and uniformly bounded on $\\partial \\Omega $ .",
"Similarly, in the parabolic framework, the final-time data $g$ is supposed to be continuous and uniformly bounded on $\\overline{\\Omega }$ .",
"The real-valued function $f$ in (REF ) is defined on $\\mathbb {R}\\times \\overline{\\Omega }\\times \\mathbb {R}\\times \\mathbb {R}^N \\times \\mathcal {S}^N$ .",
"It is assumed throughout to be a continuous function of all its variables, and also that $f$ is monotone in $\\Gamma $ in the sense that $ f(t,x,z,p,\\Gamma _1+\\Gamma _2) \\le f(t,x,z,p,\\Gamma _1) \\quad \\text{ for } \\Gamma _2 \\ge 0.$ In the time-dependent setting (REF ) we permit $f$ to grow linearly in $|z|$ (so solutions can grow exponentially, but cannot blow up).",
"However we require uniform control in $x$ (so solutions remain bounded as $\\Vert x \\Vert \\rightarrow \\infty $ with $t$ fixed).",
"In fact we assume that $f$ has at most linear growth in $z$ near $p=0$ , $\\Gamma =0$ , in the sense that for any $K$ we have $ |f(t,x,z,p,\\Gamma )|\\le C_K(1+|z|) ,$ for some constant $C_K \\ge 0$ , for all $x\\in \\overline{\\Omega }$ and $t,z \\in \\mathbb {R}$ , when $\\Vert (p,\\Gamma ) \\Vert \\le K$ .",
"$f$ is locally Lipschitz in $p$ and $\\Gamma $ in the sense that for any $K$ we have $ |f(t,x,z,p,\\Gamma )-f(t,x,z,p^{\\prime },\\Gamma ^{\\prime })| \\le C_K(1+|z|) \\Vert (p,\\Gamma )-(p^{\\prime },\\Gamma ^{\\prime }) \\Vert ,$ for some constant $C_K \\ge 0$ , for all $x \\in \\overline{\\Omega }$ and $t,z\\in \\mathbb {R}$ , when $\\Vert (p,\\Gamma ) \\Vert +\\Vert (p^{\\prime },\\Gamma ^{\\prime }) \\Vert \\le K$ .",
"$f$ has controlled growth with respect to $p$ and $\\Gamma $ , in the sense that for some constants $q, r\\ge 1$ , $C>0$ , we have $ |f(t,x,z,p,\\Gamma )|\\le C(1+|z|+\\Vert p \\Vert ^q+ \\Vert \\Gamma \\Vert ^r),$ for all $t,x,z,p$ and $\\Gamma $ .",
"In the stationary setting (REF ) our solutions will be uniformly bounded.",
"To prove the existence of such solutions we need the discounting to be sufficiently large.",
"We also need analogues of (REF )–(REF ) but they can be local in $z$ since $z$ will ultimately be restricted to a compact set.",
"In fact, we assume that There exists $\\eta >0$ such that for all $K\\ge 0$ , there exists $C_K^\\ast >0$ satisfying $ |f(x,z,p, \\Gamma )| \\le (\\lambda -\\eta ) |z|+ C_{K}^\\ast ,$ for all $x\\in \\overline{\\Omega }$ , $z\\in \\mathbb {R}$ , when $\\Vert (p,\\Gamma ) \\Vert \\le K$ ; here $\\lambda $ is the coefficient of $u$ in the equation (REF ).",
"$f$ is locally Lipschitz in $p$ and $\\Gamma $ in the sense that for any $K$ and $L$ we have $ |f(x,z,p,\\Gamma )-f(x,z,p^{\\prime },\\Gamma ^{\\prime })| \\le C_{K,L} \\Vert (p,\\Gamma )-(p^{\\prime },\\Gamma ^{\\prime }) \\Vert ,$ for some constant $C_{K,L} \\ge 0$ , for all $x \\in \\overline{\\Omega }$ , when $\\Vert (p,\\Gamma ) \\Vert +\\Vert (p^{\\prime },\\Gamma ^{\\prime }) \\Vert \\le K$ and $|z|\\le L$ .",
"$f$ has controlled growth with respect to $p$ and $\\Gamma $ , in the sense that for some constants $q, r\\ge 1$ and for any $L$ we have $ |f(x,z,p,\\Gamma )|\\le C_L(1+ \\Vert p \\Vert ^q+ \\Vert \\Gamma \\Vert ^r),$ for some constant $C_L \\ge 0$ , for all $x$ , $p$ and $\\Gamma $ , and any $|z|\\le L$ ."
],
[
"The games",
"This section present our games.",
"We begin by dealing with the linear heat equation.",
"Section REF adresses the time-dependent problem depending non linearly on $u$ ; our main rigorous result for the time-dependent setting is stated here (Theorem REF ).",
"Section REF discusses the stationary setting and states our main rigorous result for that case (Theorem REF )."
],
[
"The linear heat equation",
"This section offers a deterministic two-persons game approach to the linear heat equation in one space dimension.",
"More precisely, let $a<c$ and $\\Omega =]a,c[$ .",
"We consider the linear heat equation on $\\Omega $ with continuous final time data $g$ and Neumann boundary condition $h$ given by ${\\left\\lbrace \\begin{array}{ll}u_t+u_{xx}=0 , & \\text{for }x \\in \\Omega \\text{ and } t<T, \\\\\\dfrac{\\partial u}{ \\partial n}(x,t)=h(x), & \\text{for }x \\in \\partial \\Omega =\\lbrace a,c\\rbrace \\text{ and } t<T, \\\\u(x,T)=g(x) , & \\text{for } x \\in \\overline{\\Omega }\\text{ and } t=T.\\end{array}\\right.",
"}$ Our goal is to capture, in the simplest possible setting, how a homogeneous Neumann condition can be retrieved through a repeated deterministic game.",
"The game discussed here shares many features with the ones we will introduce in Sections REF –REF , though it is not a special case.",
"In particular, it allows to understand the way we need to modify the rules of the pioneering games proposed by Kohn and Serfaty in [21] in order to model the Neumann boundary condition.",
"There are two players, we call them Mark and Helen.",
"A small parameter $\\varepsilon >0$ is fixed as are the final time $T$ , “Helen's payoff” (a continuous function $g$ : $[a,c] \\rightarrow \\mathbb {R}$ ) and a “coupon profile” close to the boundary (a function $h$ : $\\lbrace a,c\\rbrace \\rightarrow \\mathbb {R})$ .",
"The state of the game is described by its “spatial position” $x\\in \\overline{\\Omega }$ and “Helen's score” $y\\in \\mathbb {R}$ .",
"We suppose the game begins at time $t_0$ .",
"Since time steps are increments of $\\varepsilon ^2$ , it is convenient to assume that $T-t_0=K\\varepsilon ^2$ , for some $K$ .",
"When the game begins, the position can have any value $x_0 \\in \\overline{\\Omega }$ ; Helen's initial score is $y_0=0$ .",
"The rules are as follows: if, at time $t_j=t_0+j\\varepsilon ^2$ , the position is $x_j$ and Helen's score is $y_j$ , then Helen chooses a real number $p_j$ .",
"After seeing Helen's choice, Mark chooses $b_j=\\pm 1$ which gives an intermediate position $\\hat{x}_{j+1}=x_j+\\Delta \\hat{x}_j$ where $\\Delta \\hat{x}_j=\\sqrt{2} \\varepsilon b_j \\in \\mathbb {R}.$ This position $\\hat{x}_{j+1}$ determines the next position $x_{j+1}=x_j + \\Delta x_j$ at time $t_{j+1}$ by the rule $x_{j+1}=\\operatorname{proj}_{\\overline{\\Omega }}(\\hat{x}_{j+1}) \\in \\overline{\\Omega },$ and Helen's score changes to $y_{j+1}=y_j+ p_j \\Delta \\hat{x}_j- \\Vert x_{j+1} - \\hat{x}_{j+1} \\Vert h(x_j+\\Delta x_j) .$ The clock moves forward to $t_{j+1}=t_j+\\varepsilon ^2$ and the process repeats, stopping when $t_K=T$ .",
"At the final time $t_K=T$ a bonus $g(x_K)$ is added to Helen's score, where $x_K$ is the final-time position.",
"Remark 2.1 To give a sense to (REF ) for all $\\Delta x_j$ , the function $h$ , which is defined only on $\\lbrace a,c\\rbrace $ , can be extended on $]a,c[$ by any function $\\Omega \\rightarrow \\mathbb {R}$ since $\\Vert x_{j+1} - \\hat{x}_{j+1} \\Vert $ is different from zero if and only if $\\hat{x}_{j+1}\\notin \\overline{\\Omega }$ .",
"Moreover, by comparing the two moves $\\Delta \\hat{x}_j$ and $\\Delta x_j$ , it is clear that $\\Vert x_{j+1} - \\hat{x}_{j+1} \\Vert = \\Vert \\Delta x_j - \\Delta \\hat{x}_j \\Vert $ .",
"Helen's goal is to maximize her final score, while Mark's goal is to obstruct her.",
"We are interested in Helen's “value function” $u^\\varepsilon (x_0,t_0)$ , defined formally as her maximum worst-case final score starting from $x_0$ at time $t_0$ .",
"It is determined by the dynamic programming principle $u^{\\varepsilon }(x,t_j)=\\max _{p\\in \\mathbb {R}} \\min _{b=\\pm 1} \\left[ u^{\\varepsilon }(x+\\Delta x,t_{j+1}) - p \\Delta \\hat{x} + \\Vert \\Delta \\hat{x} - \\Delta x \\Vert h(x+\\Delta x) \\right],$ where $\\Delta \\hat{x} = \\sqrt{2} \\varepsilon b$ and $\\Delta x= \\operatorname{proj}_{\\overline{\\Omega }}(x+\\Delta \\hat{x}) -x$ , associated with the final-time condition $u^\\varepsilon (x,T)=g(x).$ Evidently, if $t_0=T - K\\varepsilon ^2$ then $u^\\varepsilon (x_0,T_0) = \\max _{p_{0} \\in \\mathbb {R}} \\min _{b_{0}=\\pm 1} \\cdots \\max _{p_{K-1} \\in \\mathbb {R}} \\min _{b_{K-1}=\\pm 1}\\left\\lbrace g(x_K) + \\sum _{j=0}^{K-1} - \\sqrt{2} \\varepsilon b_j p_j + \\Vert \\Delta \\hat{x}_j - \\Delta x_j \\Vert h(x_j+\\Delta x_j) \\right\\rbrace ,$ where $\\Delta \\hat{x}_j = \\sqrt{2} \\varepsilon b_j$ and $\\Delta x_j= \\operatorname{proj}_{\\overline{\\Omega }}(x_j+\\Delta \\hat{x}_j) -x_j$ .",
"In calling this Helen's value function, we are using an established convention from the theory of discrete-time, two person games (see e.g. [15]).",
"By introducing the operator $L_\\varepsilon $ defined by $L_{\\varepsilon } [x,\\phi ] = \\max _{p\\in \\mathbb {R}} \\min _{b= \\pm 1}\\left[\\phi \\left( x +\\Delta x \\right) - p \\Delta \\hat{x} + \\Vert \\Delta \\hat{x} - \\Delta x \\Vert h (x+\\Delta x )\\right],$ where $\\Delta \\hat{x} = \\sqrt{2} \\varepsilon b$ and $\\Delta x= \\operatorname{proj}_{\\overline{\\Omega }}(x+\\Delta \\hat{x}) -x$ , the dynamic programming principle (REF ) can be written in the form $u^{\\varepsilon }(x,t)=L_{\\varepsilon } [x, u^{\\varepsilon }(\\cdot ,t_{}+\\varepsilon ^2)].$ We now formally argue that $u^\\varepsilon $ should converge as $\\varepsilon \\rightarrow 0$ to the solution of the linear heat equation (REF ).",
"The procedure for formal passage from the dynamic programming principle to the associated PDE is familiar: we suppress the dependence of $u^\\varepsilon $ on $\\varepsilon $ and we assume $u$ is smooth enough to use the Taylor expansion.",
"The first step leads to $u^{}(x,t)\\approx L_{\\varepsilon } [x, u^{}(\\cdot ,t_{}+\\varepsilon ^2)].$ For the second step we need to compute $L^\\varepsilon $ for a $C^2$ -function $\\phi $ .",
"By the Taylor expansion $\\phi (x+\\Delta x) & =\\phi (x)+ \\phi _x(x) \\Delta x +\\frac{1}{2} \\phi _{xx}(x) (\\Delta x)^2 +O(\\varepsilon ^{3}) \\\\& =\\phi (x)+ \\phi _x(x) \\Delta \\hat{x} + \\Vert \\Delta \\hat{x}-\\Delta x \\Vert \\phi _x(x) n(\\overline{x})+\\frac{1}{2} \\phi _{xx}(x) (\\Delta x)^2 +O(\\varepsilon ^{3}),$ where $\\overline{x}= \\operatorname{proj}_{\\partial \\Omega }(x)$ , $\\Delta \\hat{x} - \\Delta x=\\Vert \\Delta \\hat{x} - \\Delta x \\Vert n(\\overline{x})$ with $n$ defined on $\\partial \\Omega $ by $n(x)=1$ if $x=c$ and $n(x)= - 1$ if $x=a$ .",
"Substituting this expression in (REF ), we deduce that for all $C^2$ -function $\\phi $ , $ L_{\\varepsilon } [x,\\phi ]=\\phi (x) +\\max _{p\\in \\mathbb {R}} \\min _{b= \\pm 1}\\left[(\\phi _x - p) \\Delta \\hat{x} +\\frac{1}{2} \\phi _{xx} (\\Delta x)^2 + \\Vert \\Delta \\hat{x} - \\Delta x \\Vert \\big \\lbrace h(x+\\Delta x ) - n(\\overline{x})\\phi _x \\big \\rbrace \\right]+o(\\varepsilon ^2).$ It remains to compute the max min.",
"If $d(x)>\\sqrt{2} \\varepsilon $ , we always have $\\Delta x=\\Delta \\hat{x}=\\sqrt{2}\\varepsilon b$ , so that the boundary is never crossed and we retrieve the usual situation detailed in [21]: Helen's optimal choice is $p=\\phi _x$ and $L_{\\varepsilon }[x,\\phi ] = \\phi (x) +\\varepsilon ^2 \\phi _{xx}(x)+o(\\varepsilon ^2)$ .",
"If $d(x)<\\sqrt{2} \\varepsilon $ , we still have $\\Delta \\hat{x}=\\sqrt{2} b \\varepsilon $ but there is a change: if the boundary is crossed, $\\Delta x= d(x)$ and $\\Vert \\Delta \\hat{x} - \\Delta x \\Vert =\\sqrt{2} \\varepsilon -d(x)$ .",
"Suppose that Helen has chosen $p\\in \\mathbb {R}$ .",
"Considering the min in (REF ), Mark only has two possibilities $b\\in \\lbrace \\pm 1\\rbrace $ .",
"More precisely, suppose that $x$ is close to $c$ so that $\\overline{x}=c$ and $n(\\overline{x})=1$ ; the case when $x$ is close to $a$ is strictly parallel.",
"If Mark chooses $b=1$ , the associated value is $V_{p,+}= \\sqrt{2}(\\phi _x - p) \\varepsilon +\\frac{1}{2} \\phi _{xx} d^2(x) + (\\sqrt{2} \\varepsilon - d(x)) ( h(c) - \\phi _x ),$ while if Mark chooses $b=-1$ , the associated value is $V_{p,-} = - \\sqrt{2}(\\phi _x - p) \\varepsilon + \\phi _{xx} \\varepsilon ^2.$ To determine his strategy, Mark compares $V_{p,-}$ to $ V_{p,+}$ .",
"He chooses $b=-1$ if $V_{p,-}< V_{p,+}$ , i.e.",
"if $\\sqrt{2}(\\phi _x - p) \\varepsilon +\\frac{1}{2} \\phi _{xx} d^2(x) + (\\sqrt{2} \\varepsilon - d(x)) ( h(c)-\\phi _x)> -\\sqrt{2}(\\phi _x - p) \\varepsilon + \\phi _{xx} \\varepsilon ^2,$ that we can rearrange into $2 \\sqrt{2}(\\phi _x - p) \\varepsilon > \\phi _{xx} \\left(\\varepsilon ^2- \\frac{d^2(x)}{2}\\right) - (\\sqrt{2} \\varepsilon - d(x)) \\left[ h(c) - \\phi _x \\right].$ This last inequality yields an explicit condition on the choice of $p$ previously made by Helen $p < p_\\text{opt}:= \\phi _x + \\frac{1}{2} \\left(1- \\frac{d(x)}{\\sqrt{2}\\varepsilon } \\right) \\left[ h(c)-\\phi _x \\right]+ \\frac{1}{2\\sqrt{2}} \\phi _{xx}\\left(1- \\frac{d^2(x)}{2\\varepsilon ^2}\\right)\\varepsilon .$ Meanwhile Mark chooses $b=1$ if $V_{p,+}< V_{p,-}$ , which leads to the reverse inequality $p > p_\\text{opt}$ .",
"The situation when $V_{p,+}= V_{p,-}$ obviously corresponds to $p = p_\\text{opt} $ .",
"We deduce that $L_\\varepsilon [x,\\phi ] = \\max \\left[ \\max _{p\\le p_\\text{opt}} V_{p,-}, V_{p_\\text{opt},-}, \\max _{p\\ge p_\\text{opt}} V_{p,+} \\right].$ Helen wants to optimize her choice of $p$ .",
"The functions $V_{p,+}$ and $V_{p,-}$ are both affine on $\\phi _x-p$ .",
"The first one is decreasing while the second is increasing with respect to $p$ .",
"As a result, we deduce that Helen's optimal choice is $p=p_\\text{opt}$ as defined in (REF ) and $L_\\varepsilon [x,\\phi ] = V_{p_\\text{opt},+} = V_{p_\\text{opt},-} $ .",
"We notice that Helen behaves optimally by becoming indifferent to Mark's choice; our games will not always conserve this feature, which was observed in [21].",
"Finally, for all $C^2$ -function $\\phi $ , we have $ L_\\varepsilon [x,\\phi ]= \\phi (x) \\\\+ {\\left\\lbrace \\begin{array}{ll}\\displaystyle \\dfrac{\\varepsilon }{\\sqrt{2}} \\left(1- \\frac{d(x)}{\\sqrt{2}\\varepsilon }\\right) \\left[ h(\\overline{x}) - n (\\overline{x}) \\phi _x(x) \\right]+ \\dfrac{\\varepsilon ^2}{2} \\phi _{xx}(x) \\left(1+ \\frac{d^2(x)}{2\\varepsilon ^2} \\right) +o(\\varepsilon ^2), & \\text{if } d(x)\\le \\sqrt{2} \\varepsilon , \\\\\\displaystyle \\varepsilon ^2 \\phi _{xx}(x) +o(\\varepsilon ^2), & \\text{if } d(x)\\ge \\sqrt{2} \\varepsilon .\\end{array}\\right.",
"}$ Since $u$ is supposed to be smooth, the Taylor expansion on $t$ yields that $u(\\cdot , t+\\varepsilon ^2)=u(\\cdot ,t)+ u_t(\\cdot ,t) \\varepsilon ^2+o(\\varepsilon ^2)$ and we formally derive the PDE by plugging (REF ) in (REF ).",
"This gives $0\\approx \\varepsilon ^2 u_t+{\\left\\lbrace \\begin{array}{ll}\\displaystyle \\dfrac{\\varepsilon }{\\sqrt{2}} \\left(1- \\frac{d(x)}{\\sqrt{2}\\varepsilon }\\right) \\left[ h(\\overline{x}) - n (\\overline{x}) u_x \\right]+ \\dfrac{\\varepsilon ^2}{2} u_{xx} \\left(1+ \\frac{d^2(x)}{2\\varepsilon ^2} \\right) +o(\\varepsilon ^2) , & \\text{if } d(x)\\le \\sqrt{2} \\varepsilon , \\\\\\displaystyle \\varepsilon ^2 u_{xx} +o(\\varepsilon ^2) , & \\text{if } d(x)\\ge \\sqrt{2} \\varepsilon .\\end{array}\\right.",
"}$ If $x \\in \\Omega $ , for $\\varepsilon $ small enough, the second alternative in (REF ) is always valid so that we deduce from the $\\varepsilon ^2$ -order terms in (REF ) that $u_t+u_{xx}=0$ .",
"If $x$ is on the boundary $\\partial \\Omega $ , then $d(x)=0$ , $\\overline{x}=x$ and the first possibility in (REF ) is always satisfied.",
"We observe that the $\\varepsilon $ -order term is predominant since $\\varepsilon \\gg \\varepsilon ^2$ .",
"By dividing by $\\varepsilon $ and letting $\\varepsilon \\rightarrow 0$ , we obtain $h(x)-u_x(x) \\cdot n(x)=0$ .",
"Now we present a financial interpretation of this game.",
"Helen plays the role of a hedger or an investor, while Mark represents the market.",
"The position $x$ is a stock price which evolves in $\\overline{\\Omega }$ as a function of time $t$ , starting at $x_0$ at time $t_0$ and the boundary $\\partial \\Omega $ plays the role of barriers which additionally determine a coupon when the stock price crosses $\\partial \\Omega $ .",
"The small parameter $\\varepsilon $ determines both the stock price increments $\\Delta \\hat{x} \\le \\sqrt{2} \\varepsilon $ and the time step $\\varepsilon ^2$ .",
"Helen's score keeps track of the profits and losses generated by her hedging activity.",
"Helen's situation is as follows: she holds an option that will pay her $g(x(T))$ at time $T$ ($g$ could be negative).",
"Her goal is to hedge this position by buying or selling the stock at each time increment.",
"She can borrow and lend money without paying or collecting any interest, and can take any (long or short) stock position she desires.",
"At each step, Helen chooses a real number $p_j$ (depending on $x_j$ and $t_j$ ), then adjusts her portfolio so it contains $-p_j$ units of stock (borrowing or lending to finance the transaction, so there is no change in her overall wealth).",
"Mark sees Helen's choice.",
"Taking it into account, he makes the stock go up or down (i.e.",
"he chooses $b_j= \\pm 1$ ), trying to degrade her outcome.",
"The stock price changes from $x_j$ to $x_{j+1}=\\operatorname{proj}_{\\overline{\\Omega }}(x_j+\\Delta \\hat{x}_j)$ , and Helen's wealth changes by $-\\sqrt{2} \\varepsilon b_j p_j + \\Vert \\Delta \\hat{x}_j - \\Delta x_j \\Vert h(x_j+\\Delta x_j)$ (she has a profit if it is positive, a loss if it is negative).",
"The term $\\Vert \\Delta \\hat{x}_j - \\Delta x_j \\Vert h(x_j+\\Delta x_j)$ is a coupon that will be produced only if the special event $\\Delta \\hat{x}_j \\notin \\Omega $ happens.",
"The hedger must take into account the possibility of this new event.",
"The hedging parameter $p_j$ is modified close to the boundary but the hedger's value function is still independent from the variations of the market.",
"At the final time Helen collects her option payoff $g(x_K)$ .",
"If Helen and Mark both behave optimally at each stage, then we deduce by (REF ) that $u^\\varepsilon (x_0,t_0)+ \\sum _{j=0}^{K-1} \\sqrt{2} \\varepsilon b_j p_j - \\Vert \\Delta \\hat{x}_j - \\Delta x_j \\Vert h(x_j+\\Delta x_j)=g(x_K).$ Helen's decisions are in fact identical to those of an investor hedging an option with payoff $g(x)$ and coupon $h(x)$ if the underlying asset crosses the barrier $\\partial \\Omega $ in a binomial-tree market with $\\Delta \\hat{x} =\\sqrt{2} \\varepsilon $ at each timestep."
],
[
"General parabolic equations",
"This section explains what to do when $f$ depends on $Du$ , $D^2u$ and also on $u$ .",
"We also permit dependence on $x$ and $t$ , so we are now discussing a fully-nonlinear (degenerate) parabolic equation of the form $ {\\left\\lbrace \\begin{array}{ll}\\partial _t u - f(t,x,u,Du,D^2u)=0, & \\text{ for } x\\in \\Omega \\text{ and }t<T , \\\\\\left\\langle D u(x,t), n(x) \\right\\rangle =h(x), & \\text{ for } x\\in \\partial \\Omega \\text{ and }t<T , \\\\u(x,T) =g(x), &\\text{ for } x \\in \\overline{\\Omega },\\end{array}\\right.",
"}$ where $\\Omega $ is a $C^2$ -domain satisfying both the uniform interior and exterior ball conditions and the boundary condition $h$ and the final-time data $g$ are uniformly bounded, continuous, depending only on $x$ .",
"There are two players, Helen and Mark; a small parameter $\\varepsilon $ is fixed.",
"Since the PDE is to be solved in $\\Omega $ , Helen's final-time bonus $g$ is now a function of $x\\in \\overline{\\Omega }$ and Helen's coupon profile $h$ is a function of $x\\in \\partial \\Omega $ .",
"The state of the game is described by its spatial position $x\\in \\overline{\\Omega }$ and Helen's debt $z\\in \\mathbb {R}$ .",
"Helen's goal is to minimize her final debt, while Mark's is to obstruct her.",
"The rules of the game depend on three new parameters, $\\alpha , \\beta , \\gamma >0$ whose presence represents no loss of generality.",
"Their role will be clear in a moment.",
"The requirements $ \\alpha <1/3,$ and $\\alpha +\\beta <1,\\qquad 2\\alpha +\\gamma <2, \\qquad \\max (\\beta q, \\beta r)<2 ,$ will be clear in the explanation of the game.",
"However, the proof of convergence in Section and consistency in Section needs more: there we will require $\\gamma <1-\\alpha , \\quad \\beta (q-1)<\\alpha +1, \\quad \\gamma (r-1)<2 \\alpha ,\\quad \\gamma r<1+\\alpha .$ These conditions do not restrict the class of PDEs we consider, since for any $q$ and $r$ there exist $\\alpha $ , $\\beta $ and $\\gamma $ with the desired properties.",
"Using the language of our financial interpretation: First we consider $U^\\varepsilon (x,z,t)$ , Helen's optimal wealth at time $T$ , if initially at time $t$ the stock price is $x$ and her wealth is $-z$ .",
"Then we define $u^\\varepsilon (x,t)$ or $v^\\varepsilon (x,t)$ as, roughly speaking, the initial debt Helen should have at time $t$ to break even at time $T$ .",
"The proper definition of $U^\\varepsilon (x,z,t)$ involves a game similar to that of Section REF .",
"The rules are as follows: if at time $t_j=t_0+j \\varepsilon ^2$ , the position is $x_j$ and Helen's debt is $z_j$ , then Helen chooses a vector $p_j \\in \\mathbb {R}^N$ and a matrix $\\Gamma _j \\in \\mathcal {S}^N$ , restricted by $\\Vert p_j \\Vert \\le \\varepsilon ^{-\\beta }, \\Vert \\Gamma _j \\Vert \\le \\varepsilon ^{-\\gamma } .$ Taking Helen's choice into account, Mark chooses the stock price $x_{j+1}$ so as to degrade Helen's outcome.",
"Mark chooses an intermediate point $\\hat{x}_{j+1}=x_j +\\Delta \\hat{x}_j \\in \\mathbb {R}^N$ such that $\\left\\Vert \\Delta \\hat{x}_j\\right\\Vert \\le \\varepsilon ^{1-\\alpha }.$ This position $\\hat{x}_{j+1}$ determines the new position $x_{j+1}=x_j+\\Delta x_j \\in \\overline{\\Omega }$ at time $t_{j+1}$ by the rule $x_{j+1}= \\operatorname{proj}_{\\overline{\\Omega }} (\\hat{x}_{j+1}).$ Helen's debt changes to $z_{j+1}=z_j + p_j\\cdot \\Delta \\hat{x}_j +\\frac{1}{2} \\left\\langle \\Gamma _j \\Delta \\hat{x}_j,\\Delta \\hat{x}_j \\right\\rangle +\\varepsilon ^2 f(t_j,x_j,z_j,p_j,\\Gamma _j) - \\Vert \\Delta \\hat{x}_j - \\Delta x_j \\Vert h(x_j+\\Delta x_j ) .$ The clock steps forward to $t_{j+1}=t_j+\\varepsilon ^2$ and the process repeats, stopping when $t_K=T$ .",
"At the final time Helen receives $g(x_K)$ from the option.",
"This game is well-posed for all $\\varepsilon >0$ small enough.",
"As mentioned in the introduction, the uniform exterior ball condition holds automatically for a $C^2$ -bounded domain.",
"In this case, by compactness of $\\partial \\Omega $ , there exists $\\varepsilon _\\ast >0$ such that $\\operatorname{proj}_{\\overline{\\Omega }} $ is well-defined for all $x\\in \\Omega $ such that $d(x) \\le \\varepsilon _\\ast $ .",
"It can be noticed that an unbounded $C^2$ -domain, even with bounded curvature, does not generally satisfy this condition.",
"Since the domain $\\Omega $ satisfy the uniform exterior ball condition given by Definition REF for a certain $r$ , the projection is well-defined on the tubular neighborhood $\\lbrace x\\in \\mathbb {R}^N \\backslash \\Omega , d(x)< r/2\\rbrace $ of the boundary.",
"Figure: Rules of the game, near the boundary and inside the domain.Remark 2.2 To give a sense to (REF ) for all $\\Delta x_j$ , the function $h$ which is defined only on the boundary can be extended on $\\overline{\\Omega }$ by any function $\\Omega \\rightarrow \\mathbb {R}$ since $\\Vert x_{j+1} - \\hat{x}_{j+1} \\Vert $ is different from zero if and only if $\\hat{x}_{j+1} \\notin \\overline{\\Omega }$ .",
"Moreover, by comparing $\\Delta \\hat{x}_j$ and $\\Delta x_j$ , one gets the relation $x_{j+1}=\\hat{x}_{j+1} + \\Delta x_j - \\Delta \\hat{x}_j.$ If $\\hat{x}_{j+1}\\in \\Omega $ , then $x_{j+1}= \\hat{x}_{j+1}$ and the rules of the usual game [21] are retrieved.",
"Figure REF presents the two geometric situations for the choice for Mark: $B(x, \\varepsilon ^{1-\\alpha }) \\subset \\Omega $ or not.",
"Helen's goal is to maximize her worst-case score at time $T$ , and Mark's is to work against her.",
"Her value function is $ U^\\varepsilon (x_0,z_0,t_0) =\\max _{\\text{Helen's choices}} \\left[ g(x_K) - z_K \\right].$ It is characterized by the dynamic programming principle $ U^\\varepsilon (x,z,t_j) = \\max _{p,\\Gamma } \\min _{\\Delta \\hat{x}} U^{\\varepsilon } (x+\\Delta x, z+\\Delta z, t_{j+1})$ together with the final-time condition $U^\\varepsilon (x,z,T)= g(x) - z$ .",
"Here $\\Delta \\hat{x}$ is $\\hat{x}_{j+1} - x_j$ , $\\Delta x$ is determined by $\\Delta x= x_{j+1} - x_j= \\operatorname{proj}_{\\overline{\\Omega }} (x_j+\\Delta \\hat{x}_j) - x_j,$ and $\\Delta z=z_{j+1} - z_j$ is given by (REF ), and the optimizations are constrained by (REF ) and (REF ).",
"It is easy to see that the max/min is achieved and is a continuous function of $x$ and $z$ at each discrete time (the proof is by induction backward in time, like the argument sketched in [21]).",
"When $f$ depends on $z$ , the function $z\\mapsto U^\\varepsilon (x,z,t)$ can be nonmonotone, so we must distinguish between the minimal and maximal debt with which Helen breaks even at time $T$ .",
"Thus, following [11], we define $u^\\varepsilon (x_0,t_0)=\\sup \\lbrace z_0 : U^\\varepsilon (x_0,z_0,t_0)\\ge 0 \\rbrace $ and $v^\\varepsilon (x_0,t_0)=\\inf \\lbrace z_0 : U^\\varepsilon (x_0,z_0,t_0)\\le 0 \\rbrace ,$ with the convention that the empty set has $\\sup = - \\infty $ and $\\inf = \\infty $ .",
"Clearly $v^\\varepsilon \\le u^\\varepsilon $ , and $u^\\varepsilon (x,T) = v^\\varepsilon (x,T)=g(x)$ .",
"Since the definitions of $u^\\varepsilon $ and $v^\\varepsilon $ are implicit, these functions can not be characterized by a dynamic programming principle.",
"However we still have two “dynamic programming inequalities”.",
"Proposition 2.3 If $u^\\varepsilon (x,t)$ is finite then $u^\\varepsilon (x,t) \\le \\sup _{p,\\Gamma } \\inf _{\\Delta \\hat{x}} \\left[u^\\varepsilon (x+\\Delta x ,t+\\varepsilon ^2) \\right.",
"\\\\- \\left.\\left(p\\cdot \\Delta \\hat{x} +\\frac{1}{2} \\left\\langle \\Gamma \\Delta \\hat{x} ,\\Delta \\hat{x} \\right\\rangle +\\varepsilon ^2 f(t,x, u^\\varepsilon (x,t),p ,\\Gamma ) - \\Vert \\Delta \\hat{x} - \\Delta x \\Vert h(x+\\Delta x)\\right) \\right].$ Similarly, if $v^\\varepsilon (x,t)$ is finite then $v^\\varepsilon (x,t) \\ge \\sup _{p,\\Gamma } \\inf _{\\Delta \\hat{x}} \\left[v^\\varepsilon (x+\\Delta x ,t+\\varepsilon ^2)\\right.",
"\\\\- \\left.",
"\\left(p\\cdot \\Delta \\hat{x}+\\frac{1}{2} \\left\\langle \\Gamma \\Delta \\hat{x},\\Delta \\hat{x} \\right\\rangle +\\varepsilon ^2 f(t,x, v^\\varepsilon (x,t),p ,\\Gamma ) - \\Vert \\Delta \\hat{x} - \\Delta x \\Vert h(x+\\Delta x) \\right) \\right] .$ The sup and inf are constrained by (REF ) and (REF ) and $\\Delta x$ is determined by (REF ).",
"The argument follows the same lines as the proof of the dynamic programming inequalities given in [21].",
"For sake of completeness we give here the details.",
"To prove (REF ), consider $z=u^\\varepsilon (x,t)$ .",
"By the definition of $u^\\varepsilon $ (and remembering that $U^\\varepsilon $ is continuous) we have $U^\\varepsilon (x,z,t)=0$ .",
"Hence writing (REF ), we have $0=\\max _{p,\\Gamma } \\min _{\\Delta \\hat{x}} U^{\\varepsilon } \\left(x+\\Delta x, z+ p\\cdot \\Delta \\hat{x} +\\frac{1}{2} \\langle \\Gamma \\Delta \\hat{x}, \\Delta \\hat{x} \\rangle +\\varepsilon ^2 f(t,x,z,p, \\Gamma ) - \\Vert \\Delta \\hat{x} - \\Delta x \\Vert h(x+\\Delta x), t+\\varepsilon ^2 \\right).$ We conclude that there exist $p, \\Gamma $ (constrained by (REF )) such that for all $\\Delta \\hat{x}$ constrained by (REF ), determining $\\Delta x$ by (REF ), we have $U^{\\varepsilon }\\left(x+\\Delta x, z+ p\\cdot \\Delta \\hat{x} +\\frac{1}{2} \\langle \\Gamma \\Delta \\hat{x}, \\Delta \\hat{x} \\rangle +\\varepsilon ^2 f(t,x,z,p, \\Gamma ) - \\Vert \\Delta \\hat{x} - \\Delta x \\Vert h(x+\\Delta x) , t+\\varepsilon ^2\\right) \\ge 0.$ By the definition of $u^\\varepsilon $ given by (REF ), this implies that $z+ p\\cdot \\Delta \\hat{x} +\\frac{1}{2} \\langle \\Gamma \\Delta \\hat{x}, \\Delta \\hat{x} \\rangle +\\varepsilon ^2 f(t,x,z,p, \\Gamma ) - \\Vert \\Delta \\hat{x} - \\Delta x \\Vert h(x+\\Delta x)\\le u^{\\varepsilon }(x+\\Delta x, t+\\varepsilon ^2).$ In other words, there exist $p, \\Gamma $ such that for every $\\Delta \\hat{x}$ , determining $\\Delta x$ by (REF ), $z \\le u^{\\varepsilon }(x+\\Delta x, t+\\varepsilon ^2) - \\left( p\\cdot \\Delta \\hat{x} +\\frac{1}{2} \\langle \\Gamma \\Delta \\hat{x}, \\Delta \\hat{x} \\rangle +\\varepsilon ^2 f(t,x,z,p, \\Gamma ) - \\Vert \\Delta \\hat{x} - \\Delta x \\Vert h(x+\\Delta x) \\right).$ Recalling that $z=u^\\varepsilon (x,t)$ and passing to the inf and sup, we get (REF ).",
"The proof of (REF ) follows exactly the same lines.",
"To define viscosity subsolutions and supersolutions, we shall follow the Barles and Perthame procedure [8], let us recall the upper and lower relaxed semi-limits defined for $(t,x)\\in [0,T]\\times \\overline{\\Omega }$ as $\\bar{u}(x,t) =\\limsup _{\\begin{array}{c} y \\rightarrow x, y \\in \\overline{\\Omega }\\\\ t_j \\rightarrow t \\\\ \\varepsilon \\rightarrow 0\\end{array}} u^\\varepsilon (y,t_j) \\quad \\text{ and } \\quad \\underline{v}(x,t) =\\liminf _{\\begin{array}{c} y \\rightarrow x, y \\in \\overline{\\Omega }\\\\ t_j \\rightarrow t \\\\ \\varepsilon \\rightarrow 0\\end{array}} v^\\varepsilon (y,t_j) ,$ where the discrete times are $t_j=T-j\\varepsilon ^2$ .",
"We shall show, under suitable hypotheses, that $\\underline{v}$ and $\\overline{u}$ are respectively viscosity super and subsolutions of (REF ).",
"Before stating our rigorous result in Section REF , the next section presents the heuristic derivation of the PDE (REF ) through the optimal strategies of Helen and Mark."
],
[
"Heuristic derivation of the optimal player strategies",
"We now formally show that $u^\\varepsilon $ should converge as $\\varepsilon \\rightarrow 0$ to the solution of (REF ).",
"Roughly speaking, the PDE (REF ) is the formal Hamilton Jacobi Bellman equation associated to the two-persons game presented at the beginning of the present section.",
"The procedure for formal derivation from the dynamic programming principle to a corresponding PDE is classical: we assume $u^\\varepsilon $ and $v^\\varepsilon $ coincide and are smooth to use Taylor expansion, suppress the dependence of $u^\\varepsilon $ and $v^\\varepsilon $ on $\\varepsilon $ and finally make $\\varepsilon \\rightarrow 0$ .",
"That has already been done for $x$ far from the boundary in [21] for $f$ depending only on $(Du,D^2u)$ .",
"We now suppose that $x$ is close enough of the boundary so that $\\hat{x}$ can be nontrivial.",
"By assuming $u^\\varepsilon = v^\\varepsilon $ as announced and suppressing the dependence of $u^\\varepsilon $ on $\\varepsilon $ , the two dynamic programming inequalities (REF ) and (REF ) give the programming equality $u(x,t) \\approx \\sup _{p,\\Gamma } \\inf _{\\Delta \\hat{x}} \\left[u(x+\\Delta x ,t+\\varepsilon ^2) \\right.",
"\\\\- \\left.",
"\\left(p\\cdot \\Delta \\hat{x}+\\frac{1}{2} \\left\\langle \\Gamma \\Delta \\hat{x} ,\\Delta \\hat{x} \\right\\rangle +\\varepsilon ^2 f(t,x, u(x,t),p ,\\Gamma ) - \\Vert \\Delta \\hat{x} - \\Delta x \\Vert h(x+\\Delta x) \\right) \\right].$ Remembering that $\\Delta \\hat{x}$ is small, if $u$ is assumed to be smooth, we obtain $u ( x +\\Delta x,t+\\varepsilon ^2)& + \\Vert \\Delta \\hat{x} - \\Delta x \\Vert h(x+\\Delta x ) \\\\& \\approx u(x,t)+\\varepsilon ^2 u_t + Du\\cdot \\Delta x +\\frac{1}{2} \\left\\langle D^2u \\Delta x, \\Delta x \\right\\rangle + \\Vert \\Delta \\hat{x} - \\Delta x \\Vert h(x+\\Delta x ) \\\\& \\approx u(x,t)+\\varepsilon ^2 u_t + Du \\cdot \\Delta \\hat{x} + \\Vert \\Delta \\hat{x} - \\Delta x \\Vert \\left\\lbrace h(x+\\Delta x )- Du \\cdot n(x+\\Delta x) \\right\\rbrace +\\frac{1}{2} \\left\\langle D^2u \\Delta x, \\Delta x \\right\\rangle ,$ since the outer normal can be expressed by $\\displaystyle n(x+\\Delta x)= - \\frac{\\Delta x - \\Delta \\hat{x}}{\\Vert \\Delta \\hat{x} - \\Delta x \\Vert }$ if $\\hat{x} \\notin \\Omega $ .",
"Substituting this computation in (REF ), and rearranging the terms, we get $ 0 \\approx \\varepsilon ^2 u_t+ \\max _{p,\\Gamma } \\min _{\\Delta \\hat{x}}\\left[ (Du - p)\\cdot \\Delta \\hat{x} + \\Vert \\Delta \\hat{x} - \\Delta x \\Vert \\lbrace h(x+\\Delta x ) - Du \\cdot n(x+\\Delta x) \\rbrace \\right.",
"\\\\\\left.",
"+\\frac{1}{2} \\left\\langle D^2u \\Delta x, \\Delta x \\right\\rangle - \\frac{1}{2} \\left\\langle \\Gamma \\Delta \\hat{x}, \\Delta \\hat{x} \\right\\rangle - \\varepsilon ^2 f \\left(t, x, u, p,\\Gamma \\right) \\right],$ where $u$ , $Du$ , $D^2u$ are evaluated at $(x,t)$ .",
"We have ignored the upper bounds in (REF ) since they allow $p$ , $\\Gamma $ to be arbitrarily large in the limit $\\varepsilon \\rightarrow 0$ (we shall of course be more careful in Section ).",
"If the domain $\\Omega $ does not satisfy the uniform interior ball condition, $\\Omega $ can present an infinity number of “neck pitchings” of neck size arbitrarily small.",
"To avoid this situation, the uniform interior ball condition is used to impose a strictly positive lower bound on these necks.",
"If $x$ is supposed to be extremely close to the $C^2$ -boundary and $\\Vert \\Delta \\hat{x} \\Vert \\le \\varepsilon ^{1-\\alpha }$, the boundary looks like a hyperplane orthogonal to the outer normal vector $n(\\bar{x})$ , where $\\bar{x}$ is the projection of $x$ on the boundary $\\partial \\Omega $ (see Figure REF ).",
"By Gram-Schmidt process, we can find some vectors $e_2, \\cdots , e_N$ such that $(e_1=n(\\bar{x}), e_2, \\cdots , e_N)$ form an orthonormal basis of $\\mathbb {R}^N$ .",
"In this basis, denote $ p=p_1 n(\\bar{x}) + \\widetilde{p} \\quad \\text{ and } \\quad \\Gamma = \\left(\\left\\langle \\Gamma e_i,e_j \\right\\rangle \\right)_{1\\le i,j\\le N} =\\left(\\begin{array}{c|ccc}\\Gamma _{11} & \\cdots & (\\Gamma _{1i})_{2 \\le i\\le N} & \\cdots \\\\\\hline \\vdots & & & \\\\(\\Gamma _{i1})_{2 \\le i\\le N} & & \\widetilde{\\Gamma }& \\\\\\vdots & &&\\end{array}\\right),$ where $ p_1\\in \\mathbb {R}$ , $\\widetilde{p}\\in V^\\perp =\\text{span}(e_2, \\cdots , e_N)$ and $\\widetilde{\\Gamma }=\\left(\\left\\langle \\Gamma e_i,e_j \\right\\rangle \\right)_{2 \\le i,j\\le N}\\in \\mathcal {S}^{N-1}$ .",
"Figure: Formal derivation for xx near the boundary ∂Ω\\partial \\Omega , notation: x ¯=proj ∂Ω (x)\\bar{x} =\\operatorname{proj}_{\\partial \\Omega }(x).Let us focus on the Neumann penalization term in (REF ) denoted by $P(x)= \\Vert \\Delta \\hat{x} - \\Delta x \\Vert m(\\Delta x) \\quad \\text{ with } \\quad m(\\Delta x)={\\left\\lbrace \\begin{array}{ll}h(x+\\Delta x) - Du(x)\\cdot n(x+\\Delta x), & \\text{ if } \\hat{x} \\notin \\overline{\\Omega }, \\\\\\tilde{m}(\\Delta x) , & \\text{ if } \\hat{x} \\in \\overline{\\Omega },\\end{array}\\right.",
"}$ where $m(\\Delta x)$ is extended for $\\hat{x} \\in \\overline{\\Omega }$ by any function $\\tilde{m}(\\Delta x)$ (see Remark REF ).",
"This contribution is favorable to Helen, $P(x)>0$ , if $m(x)>0$ , or to Mark, $P(x)<0$ , if $m(x)<0$ , and its size depends on the magnitude of the vector $\\Delta \\hat{x} - \\Delta x$ .",
"Our formal derivation is local and essentially geometric, in the sense that our target is to determine the optimal choices for Helen by considering all the moves $\\Delta \\hat{x}$ that Mark can choose.",
"By continuity of $h$ and smoothness of $u$ , the function $m(\\Delta x)$ is close to $m=h(\\bar{x}) - Du(x)\\cdot n(\\bar{x})$ if $\\hat{x} \\notin \\overline{\\Omega }$ .",
"We shall assume here that $m(\\Delta x)$ , which serves to model the Neumann boundary condition, is locally constant on the boundary and equal to $m$ .",
"This hypothesis corresponds in the game to assume that in a small neighborhood, crossing the boundary is always favorable to one player.",
"In order to focus only on the geometric aspects, this approach seems formally appropriate since it freezes the dependence of $p(x)$ on $m(x)$ by eliminating the difficulties linked to the local variations of $m(x)$ like the change of sign.",
"Hence, it is sufficient to examine $\\max _{p,\\Gamma } \\min _{\\Delta \\hat{x}}\\left[ (Du - p)\\cdot \\Delta \\hat{x} + \\Vert \\Delta \\hat{x} - \\Delta x \\Vert m + \\frac{1}{2} \\left\\langle D^2u \\Delta x, \\Delta x \\right\\rangle - \\frac{1}{2} \\left\\langle \\Gamma \\Delta \\hat{x}, \\Delta \\hat{x} \\right\\rangle - \\varepsilon ^2 f \\left(t, x, u , p,\\Gamma \\right) \\right].$ The formal proof will be performed in three steps.",
"Step 1: To determine the optimal choice for Helen of $p$ , we consider the $\\varepsilon $ -order optimization problem $\\mathcal {M}$ obtained from (REF ) by neglecting the second $\\varepsilon $ -order terms $\\mathcal {M}= \\max _{p} \\min _{\\Delta \\hat{x}} \\left[ (Du - p) \\cdot \\Delta \\hat{x} +\\Vert \\Delta \\hat{x} - \\Delta x \\Vert m \\right].$ By writing $\\Delta \\hat{x}= (\\Delta \\hat{x})_1 n(\\overline{x}) + \\widetilde{\\Delta \\hat{x}}$ with $\\widetilde{\\Delta \\hat{x}}\\in V^\\perp $ and observing that $\\Vert \\Delta \\hat{x} - \\Delta x \\Vert $ depends only on $(\\Delta \\hat{x})_1$ , we decompose the max min (REF ) into $\\mathcal {M} & = \\max _{p_1,\\widetilde{p} } \\min _{\\Delta \\hat{x}} \\left[ (\\widetilde{Du} - \\widetilde{p})\\cdot \\widetilde{\\Delta \\hat{x}} +(Du_1-p_1)(\\Delta \\hat{x})_1+\\Vert \\Delta \\hat{x} - \\Delta x \\Vert \\right] \\\\& = \\max _{p_1 } \\min _{|( \\Delta \\hat{x})_1 | \\le \\varepsilon ^{1-\\alpha }} \\left[ (Du_1-p_1)(\\Delta \\hat{x})_1+ \\Vert \\Delta \\hat{x} - \\Delta x \\Vert m+ \\max _{\\widetilde{p} } \\min _{\\Vert \\widetilde{\\Delta \\hat{x}} \\Vert \\le \\sqrt{ \\varepsilon ^{2-2\\alpha } - |(\\Delta \\hat{x})_1|^2} }(\\widetilde{Du} - \\widetilde{p})\\cdot \\widetilde{\\Delta \\hat{x}} \\right].$ Noticing that the choices of $\\widetilde{p}$ and $p_1$ are independent from each other, we can successively solve the optimization problems.",
"First of all, in order to choose $\\widetilde{p}$ , let us determine $\\widetilde{\\mathcal {M}}= \\max _{\\widetilde{p} } \\min _{\\Vert \\widetilde{\\Delta \\hat{x}} \\Vert \\le \\sqrt{ \\varepsilon ^{2-2\\alpha } - |(\\Delta \\hat{x})_1|^2} }(\\widetilde{Du} - \\widetilde{p})\\cdot \\widetilde{\\Delta \\hat{x}}.$ If $\\Delta \\hat{x} = \\pm \\varepsilon ^{1-\\alpha } n(\\overline{x})$ , $\\widetilde{\\Delta \\hat{x}}=0$ and the min is always zero: Helen's choice is irrelevant.",
"Otherwise, Helen should take $ \\widetilde{p}= \\operatorname{proj}_{V^\\perp } Du = \\widetilde{Du} $ , since otherwise Mark can make this max min strictly negative and minimal by choosing $\\widetilde{\\Delta \\hat{x}} = -\\sqrt{ \\varepsilon ^{2-2\\alpha } - |(\\Delta \\hat{x})_1|^2} \\frac{(Du - p)_{V^{\\perp }}}{\\Vert Du- p \\Vert }$ with $\\Delta \\hat{x} \\ne \\pm \\varepsilon ^{1-\\alpha } n(\\overline{x})$ .",
"Thus Helen chooses $\\widetilde{p}=\\widetilde{Du}$ , $\\widetilde{\\mathcal {M}}=0$ and $\\mathcal {M}$ reduces to $\\mathcal {M}= \\max _{p_1} \\min _{\\Delta \\hat{x}} \\left[ ((Du)_1 - p_1) (\\Delta \\hat{x})_1 +\\Vert \\Delta \\hat{x} - \\Delta x \\Vert m \\right].$ To determine the remaining coordinate $p_1=p\\cdot n(\\overline{x})$ of $p$ , we now consider the optimization problem (REF ) by restricting the possible choices made by Mark to the moves $\\Delta \\hat{x}$ which belong to the subspace $V=\\mathbb {R}n(\\bar{x})$ .",
"Since $\\Vert \\Delta \\hat{x} \\Vert \\le \\varepsilon ^{1-\\alpha }$ and $\\Delta \\hat{x} \\in V$ , we use the parametrization $\\Delta \\hat{x}=\\lambda \\varepsilon ^{1-\\alpha } n(\\bar{x})$ , $\\lambda \\in [-1,1]$ .",
"If $\\hat{x} \\in \\Omega $ , the boundary is not crossed and $\\Vert \\Delta x-\\Delta \\hat{x} \\Vert =0$ , while if $\\hat{x} \\notin \\Omega $ the boundary is crossed and $\\Vert \\Delta x - \\Delta \\hat{x} \\Vert =\\lambda \\varepsilon ^{1-\\alpha } - d(x)$ .",
"The intermediate point $\\hat{x} = \\bar{x} \\in \\partial \\Omega $ separating the two regions corresponds to $\\lambda _0= \\frac{d(x)}{\\varepsilon ^{1-\\alpha }}$ and $\\Vert \\Delta x - \\Delta \\hat{x} \\Vert =0$ .",
"As a result, to compute the min in (REF ), we shall distinguish these two regions by decomposing the global minimization problem into two minimization problems respectively on each region $\\mathcal {M}= \\max _{s_p} \\kappa (s_p) \\quad \\text{ with } \\quad \\kappa (s_p) =\\min (\\mathcal {M}_1(s_p), \\mathcal {M}_2(s_p)),$ where $s_p= (Du - p) \\cdot n(\\bar{x})$ and $& \\mathcal {M}_1(s_p)= \\min _{\\lambda _0 \\le \\lambda _1 \\le 1 }M_1 (\\lambda _1) \\quad \\text{ with } \\quad M_1(\\lambda _1)=(s_p+ m)\\varepsilon ^{1-\\alpha } \\lambda _1 - d(x) m, \\\\& \\mathcal {M}_2(s_p) =\\min _{-1 \\le \\lambda _2 \\le \\lambda _0}M_2(\\lambda _2) \\quad \\text{ with } \\quad M_2(\\lambda _2)= s_p \\varepsilon ^{1-\\alpha } \\lambda _2.$ For fixed $p$ , the functions defining $M_1$ and $M_2$ are affine and can easily be minimized separately: If $s_p+m\\ge 0$ , $\\mathcal {M}_1(s_p)$ is attained for $\\lambda _1=\\lambda _0$ and $\\mathcal {M}_1(s_p)=d(x)s_p$ .",
"If $s_p+m <0$ , $\\mathcal {M}_1(s_p)$ is attained for $\\lambda _1=1$ and $\\mathcal {M}_1(s_p)=\\varepsilon ^{1-\\alpha } s_p +(\\varepsilon ^{1-\\alpha } - d(x))m$ .",
"If $s_p \\ge 0$ , $ \\mathcal {M}_2(s_p)$ is attained for $\\lambda _2=- 1$ and $ \\mathcal {M}_2(s_p) = - \\varepsilon ^{1-\\alpha }s_p $ .",
"If $s_p< 0$ , $ \\mathcal {M}_2(s_p)$ is attained for $\\lambda _2= \\lambda _0$ and $ \\mathcal {M}_2(s_p)=d(x)s_p$ .",
"Geometrically, $\\lambda \\in \\lbrace -1,1,\\lambda _0\\rbrace $ corresponds to three particular moves: $\\Delta \\hat{x}=\\pm \\varepsilon ^{1-\\alpha } n(\\bar{x})$ and $\\Delta \\hat{x}= d(x) n(\\bar{x})$ .",
"We are going to distinguish several cases to compute the max min according to the sign of $s_p$ and $m$ .",
"First of all, let us assume that $m$ is positive.",
"If $s_p \\ge 0$ then $s_p + m \\ge 0$ and the optimal choices are $(\\lambda _1,\\lambda _2)=(\\lambda _0,-1) $ .",
"It remains to minimize between (REF ) and ().",
"Taking into account that $d(x) \\le \\varepsilon ^{1-\\alpha }$ and $s_p \\ge 0$ , we get by the definition of $\\kappa (s_p)$ given by (REF ) that $\\kappa (s_p) = \\min \\lbrace d(x) s_p, - \\varepsilon ^{1-\\alpha } s_p \\rbrace = -\\varepsilon ^{1-\\alpha } s_p$ .",
"If $-m\\le s_p < 0$ then $(\\lambda _1,\\lambda _2)=(\\lambda _0,\\lambda _0)$ and $\\kappa (s_p) =\\mathcal {M}_1(s_p)=\\mathcal {M}_2(s_p)=d(x) s_p$ .",
"If $s_p<-m<0$ then $(\\lambda _1,\\lambda _2)=(1,\\lambda _0)$ and $\\mathcal {M}_1(s_p)= \\varepsilon ^{1-\\alpha } s_p +(\\varepsilon ^{1-\\alpha } - d(x)) m$ and $\\mathcal {M}_2(s_p)=d(x) s_p$ .",
"By multiplying the inequality $s_p<-m<0$ by $(\\varepsilon ^{1-\\alpha } - d(x))$ , we get $\\kappa (s_p) = \\min \\lbrace \\varepsilon ^{1-\\alpha } s_p +(\\varepsilon ^{1-\\alpha } - d(x)) m , d(x) s_p \\rbrace = d(x) s_p.$ By combining cases REF –REF , we conclude that if $m>0$ , $\\kappa (s_p)={\\left\\lbrace \\begin{array}{ll}\\varepsilon ^{1-\\alpha } s_p +(\\varepsilon ^{1-\\alpha }-d(x)) m, & \\text{if }s_p \\le -m , \\\\d(x) s_p, & \\text{if } -m \\le s_p \\le 0, \\\\- \\varepsilon ^{1-\\alpha } s_p, & \\text{if }s_p \\ge 0 .\\end{array}\\right.",
"}$ The max of $\\kappa $ is zero and reached at the unique value $s_p=Du\\cdot n (\\overline{x}) -p_1=0$ .",
"Since $\\tilde{p}=\\widetilde{Du}$ by the previous analysis, we conclude in (REF ) that if $m> 0$ , Helen's optimal choice is $p=Du$ .",
"Let us now suppose that $m$ is negative.",
"If $s_p < 0$ then $s_p+m < 0$ and the optimal choices are $(\\lambda _1,\\lambda _2)=(1,\\lambda _0)$ .",
"By the definition of $\\kappa (s_p)$ given by (REF ), we obtain $ \\kappa (s_p)=\\min \\lbrace \\varepsilon ^{1-\\alpha } s_p+(\\varepsilon ^{1-\\alpha }-d(x))m, d(x)s_p \\rbrace = \\varepsilon ^{1-\\alpha } s_p +(\\varepsilon ^{1-\\alpha }-d(x))m.$ If $s_p \\ge -m > 0$ then $(\\lambda _1, \\lambda _2)=(\\lambda _0,-1)$ and $\\mathcal {M}_1(s_p)=d(x) s_p$ and $\\mathcal {M}_2(s_p)=-\\varepsilon ^{1-\\alpha } s_p$ .",
"By the definition of $\\kappa (s_p)$ given by (REF ), we obtain $\\kappa (s_p) = \\min \\left\\lbrace d(x) s_p , - \\varepsilon ^{1-\\alpha } s_p \\right\\rbrace = - \\varepsilon ^{1-\\alpha } s_p$.",
"If $0< s_p < -m$ , then $(\\lambda _1, \\lambda _2)=(1,-1)$ and $\\mathcal {M}_1(s_p)=\\varepsilon ^{1-\\alpha }s_p +(\\varepsilon ^{1-\\alpha }-d(x)) m$ and $\\mathcal {M}_2(s_p)=-\\varepsilon ^{1-\\alpha } s_p$ .",
"By the definition of $\\kappa (s_p)$ given by (REF ), we obtain $\\kappa (s_p)= \\min \\left\\lbrace \\varepsilon ^{1-\\alpha }s_p +(\\varepsilon ^{1-\\alpha }-d(x)) m , -\\varepsilon ^{1-\\alpha }s_p \\right\\rbrace .$ The target for Helen is to maximize this minimum with respect to $s_p$ .",
"Both functions intervening in the minimum are affine: the first one is affine, strictly increasing and is equal to $(\\varepsilon ^{1-\\alpha }-d(x)) m<0$ for $s_p=0$ and to $d(x)m>0$ for $s_p=-m$ whereas the second function is linear and strictly decreasing and is equal to $m\\varepsilon ^{1-\\alpha }<0$ for $s_p=-m$ .",
"As a result, there is a unique $s^\\ast $ such that these two functions are equal and this value precisely realizes the max of $\\kappa $ on $[0, -m]$ .",
"Thus, the best that Helen can hope corresponds to $ \\varepsilon ^{1-\\alpha } s^\\ast +(\\varepsilon ^{1-\\alpha }-d(x)) m = - \\varepsilon ^{1-\\alpha } s^\\ast $ .",
"This gives $s^\\ast = (Du - p) \\cdot n(\\bar{x} )= - \\frac{1}{2}\\Big (1 - \\frac{d(x)}{\\varepsilon ^{1-\\alpha }} \\Big ) m.$ We immediately check that $s^\\ast \\in \\left[0,-\\frac{m}{2}\\right]$ , which implies the condition $s^\\ast +m \\le \\frac{1}{2}m< 0$ .",
"Thus, $\\displaystyle \\max _{s_p \\in [0,-m]} \\kappa (s_p)= \\frac{1}{2} (\\varepsilon ^{1-\\alpha }-d(x)) m$ is greater than the minimum obtained in (REF ).",
"By combining cases REF –REF , we conclude that if $m \\le 0$ , $\\kappa (s_p)={\\left\\lbrace \\begin{array}{ll}\\varepsilon ^{1-\\alpha } s_p +(\\varepsilon ^{1-\\alpha } - d(x)) m, & \\text{if }s_p < s^\\ast , \\\\- \\varepsilon ^{1-\\alpha } s_p, & \\text{if }s_p \\ge s^\\ast .\\end{array}\\right.",
"}$ The max of $\\kappa $ is equal to $\\kappa (s^\\ast )$ and reached for $s_p=Du\\cdot n (\\overline{x}) -p_1=s^\\ast $ .",
"Let us give an intermediate conclusion: if $m>0$ , Helen chooses $p=Du$ whereas if $m \\le 0$ , she chooses $ p = Du + \\frac{m}{2} \\left( 1-\\frac{d(x)}{\\varepsilon ^{1-\\alpha }} \\right) n(\\bar{x} ).$ Step 2: We are now going to take into account the second order terms in $\\varepsilon $ in the optimization problem.",
"If $m \\ge 0$ , once Helen has chosen $p=Du$ , the optimization problem (REF ) reduces to computing $ \\max _{\\Gamma } \\min _{\\Delta \\hat{x}}\\left[ \\Vert \\Delta \\hat{x} - \\Delta x \\Vert m + \\frac{1}{2} \\left\\langle D^2u \\Delta x, \\Delta x \\right\\rangle - \\frac{1}{2} \\left\\langle \\Gamma \\Delta \\hat{x}, \\Delta \\hat{x} \\right\\rangle - \\varepsilon ^2 f \\left(t, x, u , Du,\\Gamma \\right) \\right] .$ Mark is going to choose $\\Delta \\hat{x}\\cdot n(\\overline{x}) \\le 0$ , because otherwise the first $\\varepsilon $ -order quantity $\\Vert \\Delta \\hat{x} - \\Delta x \\Vert m $ will be favorable to Helen.",
"Then considering $\\Delta \\hat{x}\\cdot n(\\overline{x}) \\le 0$ , we have $\\Delta \\hat{x}=\\Delta x$ and by symmetry of the quadratic form associated to $ (D^2u - \\Gamma )$ , the optimization problem (REF ) reduces to $\\max _{\\Gamma } \\min _{\\Delta \\hat{x}\\cdot n(\\overline{x}) \\le 0}\\left[ \\frac{1}{2} \\left\\langle (D^2u - \\Gamma )\\Delta \\hat{x}, \\Delta \\hat{x} \\right\\rangle - \\varepsilon ^2 f \\left(t, x, u , Du,\\Gamma \\right) \\right]\\\\= \\varepsilon ^2 \\max _{\\Gamma } \\min _{\\Delta \\hat{x}}\\left[ \\frac{1}{2} \\varepsilon ^{-2} \\left\\langle (D^2u - \\Gamma )\\Delta \\hat{x}, \\Delta \\hat{x} \\right\\rangle -f \\left(t, x, u, Du,\\Gamma \\right) \\right].$ Helen should choose $\\Gamma \\le D^2u$ , since otherwise Mark can drive $\\varepsilon ^{-2}\\langle (D^2u -\\Gamma ) \\Delta \\hat{x}, \\Delta \\hat{x}\\rangle $ to $-\\infty $ by a suitable choice of $\\Delta \\hat{x}$ .",
"Thus, the min attainable by Mark is zero and is at least realized for the choice $\\Delta \\hat{x}=0$ .",
"Helen's maximization reduces to $\\max _{\\Gamma \\le D^2u}[u_t-f(t,x,u,Du,\\Gamma )].$ Since the PDE is parabolic, i.e.",
"since $f$ satisfies (REF ), Helen's optimal choice is $\\Gamma =D^2u$ and (REF ) reduces formally to $u_t- f(t,x,u,Du, D^2u)=0$ .",
"If $m<0$ , Helen must now choose $\\Gamma $ .",
"In fact, we are going to see that the choice of $p_1=p\\cdot n(\\overline{x})$ obtained at (REF ) can be slightly improved by taking into account the additional terms containing $D^2u$ and $\\Gamma $ .",
"Suppose Helen chooses $p$ such that $(p-Du)_{|V^\\perp }=0$ (notice that our first order computation (REF ) fulfills this condition) and Mark chooses a move $\\Delta \\hat{x}^\\ast $ realizing the minimum on $\\Delta \\hat{x}$ in (REF ).",
"We consider two cases depending on $\\Delta \\hat{x}^\\ast $ .",
"Case a: if $\\Delta \\hat{x}^\\ast \\in V^\\perp $ , we can restrain the minimization problem to the moves $\\Delta \\hat{x}$ which belong to $V^\\perp $ , $\\Delta \\hat{x}=\\Delta x $ .",
"Thus, the optimization problem (REF ) reduces to computing $\\mathcal {M}_{V^\\perp }= \\varepsilon ^2 \\max _{\\Gamma } \\min _{\\begin{array}{c}\\Delta \\hat{x} \\in V^\\perp \\\\ \\Vert \\Delta \\hat{x} \\Vert \\le \\varepsilon ^{1-\\alpha }\\end{array}} \\left[\\frac{1}{2} \\varepsilon ^{-2} \\langle (D^2u - \\widetilde{\\Gamma }) \\Delta x, \\Delta x \\rangle - f \\left(t, x, u, p,\\Gamma \\right) \\right],$ where $\\widetilde{\\Gamma }= \\Gamma _{|V^\\perp }$ .",
"Helen should choose $\\widetilde{\\Gamma }\\le \\widetilde{D^2u}$ , since otherwise Mark can drive $\\varepsilon ^{-2}\\langle (D^2u - \\widetilde{\\Gamma }) \\Delta x, \\Delta x \\rangle $ to $-\\infty $ by a suitable choice of $\\Delta \\hat{x}$ .",
"By repeating the same argument of ellipticity of $f$ already used for $m> 0$ , Helen's optimal choice is $\\widetilde{\\Gamma }= \\widetilde{D^2u}$ .",
"Case b: if $\\Delta \\hat{x}^\\ast \\notin V^\\perp $ , there exists an unit vector $v$ orthogonal to $n(\\bar{x})$ such that $\\Delta \\hat{x}^\\ast \\in \\text{span} (n (\\bar{x}), v)$ .",
"Thus, we restrain the minimization problem on $\\Delta \\hat{x}$ given by (REF ) to the moves $\\Delta \\hat{x}$ which belong to the disk $D=\\text{span} (n (\\bar{x}), v) \\cap B(\\varepsilon ^{1-\\alpha })$ .",
"This gives the optimization problem $\\mathcal {M}_D$ given by $\\mathcal {M}_D= \\max _{p_1, \\Gamma } \\min _{\\Delta \\hat{x} \\in D} \\Big [(Du-p)_1 (\\Delta \\hat{x})_1 +\\Vert \\Delta \\hat{x} - \\Delta x \\Vert m + \\frac{1}{2} \\left\\langle D^2u \\Delta x, \\Delta x \\right\\rangle - \\frac{1}{2} \\left\\langle \\Gamma \\Delta \\hat{x}, \\Delta \\hat{x} \\right\\rangle - \\varepsilon ^2 f \\left(t, x, u, p ,\\Gamma \\right) \\Big ]$ by taking into account that $\\widetilde{p} = \\widetilde{Du}$ and $\\widetilde{\\Gamma }= \\widetilde{D^2u}$ .",
"Neglecting $-\\varepsilon ^2 f(t,x,z,p, \\Gamma )$ , we want to compute $\\displaystyle \\max _{s_p,\\Gamma } \\min _{\\Delta \\hat{x}} \\mathcal {N}(s_p, \\Gamma , \\Delta \\hat{x})$ with $\\mathcal {N}(s_p, \\Gamma , \\Delta \\hat{x}) = s_p ( \\Delta \\hat{x})_1+ \\Vert \\Delta \\hat{x} - \\Delta x \\Vert m + \\frac{1}{2} \\left\\langle D^2u \\Delta x, \\Delta x \\right\\rangle - \\frac{1}{2} \\left\\langle \\Gamma \\Delta \\hat{x}, \\Delta \\hat{x} \\right\\rangle .$ Since $\\Vert \\Delta \\hat{x} \\Vert \\le \\varepsilon ^{1-\\alpha }$ , we parametrize the disk $D$ by $\\Delta \\hat{x}= \\lambda \\varepsilon ^{1-\\alpha } n(\\bar{x})+ \\mu \\varepsilon ^{1-\\alpha } v$ with $\\lambda ^2+\\mu ^2 \\le 1 $ .",
"Notice that the calculation of $\\Gamma _{1v}:= \\langle \\Gamma n(\\bar{x}), v\\rangle $ for $v$ orthogonal to $n(\\bar{x})$ implies the computation of $(\\Gamma n(\\bar{x}))_{|V^\\perp }$ .",
"If Mark chooses $\\Delta \\hat{x}$ such that $\\lambda \\ge \\lambda _0$ for which the boundary is crossed, $\\mathcal {N}(s_p, \\Gamma , \\Delta \\hat{x}) = (s_p +m)\\varepsilon ^{1-\\alpha } \\lambda - d(x) m+\\frac{1}{2} d^2(x) (D^2u)_{11} - \\frac{1}{2} \\lambda ^2 \\varepsilon ^{2-2\\alpha } \\Gamma _{11} \\\\- \\mu \\left(\\frac{d(x)}{\\varepsilon ^{1-\\alpha }} (D^2u)_{1v} - \\lambda \\Gamma _{1v} \\right) \\varepsilon ^{2-2\\alpha },$ whereas for $\\Delta \\hat{x}$ such that $\\lambda \\le \\lambda _0$ for which the boundary is not crossed, $\\mathcal {N}(s_p, \\Gamma , \\Delta \\hat{x}) = s_p \\varepsilon ^{1-\\alpha } \\lambda + \\frac{1}{2} \\lambda ^2 ((D^2u)_{11}- \\Gamma _{11}) \\varepsilon ^{2-2\\alpha } - \\lambda \\mu ((D^2u)_{1v} - \\Gamma _{1v} ) \\varepsilon ^{2-2\\alpha }.$ For fixed $\\lambda $ , Mark will always choose $\\mu $ so that the last term is negative and maximal which leads to $\\mu ={\\left\\lbrace \\begin{array}{ll}\\text{sgn}(\\frac{d(x)}{\\varepsilon ^{1-\\alpha }} (D^2u)_{1v}- \\lambda \\Gamma _{1v}) \\sqrt{1-\\lambda ^2}, & \\text{if } \\lambda _0 \\le \\lambda \\le 1, \\\\\\text{sgn}(\\lambda ((D^2u)_{1v} - \\Gamma _{1v} )) \\sqrt{1-\\lambda ^2}, & \\text{if } - 1 \\le \\lambda < \\lambda _0.\\end{array}\\right.",
"}$ The min of $\\mathcal {N}(s_p, \\Gamma , \\Delta \\hat{x})$ on $\\mu $ depends only on $\\lambda = \\Delta \\hat{x} \\cdot n(\\bar{x})$ and will be denoted below by $\\mathcal {N}(s_p, \\Gamma , \\lambda )$ .",
"By virtue of (REF ) and (REF ) it corresponds, for $\\lambda \\ge \\lambda _0$ , to $\\mathcal {N}(s_p, \\Gamma , \\lambda ) = (s_p +m)\\varepsilon ^{1-\\alpha } \\lambda - d(x) m +\\frac{1}{2} d^2(x) (D^2u)_{11}- \\frac{1}{2} \\lambda ^2 \\varepsilon ^{2-2\\alpha } \\Gamma _{11} - \\sqrt{1-\\lambda ^2} \\Big | \\frac{d(x)}{\\varepsilon ^{1-\\alpha }} (D^2u)_{1v} - \\lambda \\Gamma _{1v} \\Big | \\varepsilon ^{2-2\\alpha } $ and, for $\\lambda \\le \\lambda _0$ , to $\\mathcal {N}(s_p, \\Gamma , \\lambda ) =s_p \\varepsilon ^{1-\\alpha } \\lambda + \\frac{1}{2} \\lambda ^2 ((D^2u)_{11} - \\Gamma _{11}) \\varepsilon ^{2-2\\alpha }- |\\lambda | \\sqrt{1-\\lambda ^2} \\Big |(D^2u)_{1v} - \\Gamma _{1v} \\Big | \\varepsilon ^{2-2\\alpha } .$ The second order terms containing $D^2u$ and $\\Gamma $ being a little perturbation for $\\varepsilon >0$ small enough compared to $(\\varepsilon ^{1-\\alpha } - d(x))m$ for $d(x)\\ll \\varepsilon ^{1-\\alpha }$ , it is sufficient to consider the case REF which led to (REF ) and $(\\lambda _1, \\lambda _2)=(1,-1)$ corresponding to $\\Delta \\hat{x}= \\pm \\varepsilon ^{1-\\alpha } n(\\overline{x})$ .",
"Therefore, we are going to compare the moves close to the optimal choices $\\Delta \\hat{x}= \\pm \\varepsilon ^{1-\\alpha } n(\\overline{x})$ previously obtained by considering only the first terms in the Taylor expansion.",
"More precisely, we may assume $\\lambda \\approx \\pm 1$ , which leads to making the change of variables $\\lambda _1= 1-\\rho _1$ , $\\lambda _2= -1+ \\rho _2$ and take $\\displaystyle \\rho _i \\operatornamewithlimits{\\longrightarrow }_{\\varepsilon \\rightarrow 0} 0$ for $i=1,2$ .",
"After some computations, we get a Taylor expansion in $\\rho _i$ , $i=1,2$ , in the form $ \\mathcal {N} (s_p, \\Gamma , 1- \\rho _1) =(s_p +m)\\varepsilon ^{1-\\alpha } - d(x) m +\\frac{1}{2} d^2(x) (D^2u)_{11} - \\frac{1}{2} \\varepsilon ^{2-2\\alpha } \\Gamma _{11} \\\\- \\sqrt{2\\rho _1} \\Big | \\frac{d(x)}{\\varepsilon ^{1-\\alpha }} (D^2u)_{1v} - \\Gamma _{1v} \\Big | \\varepsilon ^{2-2\\alpha }+\\rho _1 \\varepsilon ^{1-\\alpha } \\Big [- (s_p +m) + \\varepsilon ^{1-\\alpha } \\Gamma _{11} \\Big ] +O(\\varepsilon ^{2-2\\alpha } \\rho _1^{3/2}),$ and $\\mathcal {N} (s_p, \\Gamma , -1+ \\rho _2) = -s_p \\varepsilon ^{1-\\alpha } - \\frac{1}{2} ((D^2u)_{11}- \\Gamma _{11}) \\varepsilon ^{2-2\\alpha } \\\\- \\sqrt{2\\rho _2} \\Big |(D^2u)_{1v} - \\Gamma _{1v} \\Big | \\varepsilon ^{2-2\\alpha }+ \\rho _2 \\varepsilon ^{1-\\alpha } ( s_p - ((D^2u)_{11} - \\Gamma _{11}) \\varepsilon ^{1-\\alpha } ) +O(\\varepsilon ^{2-2\\alpha } \\rho _2^{3/2}).$ First of all, we are now going to focus on the 0-order terms on the $\\rho _1,\\rho _2$ variables.",
"Dropping the next terms corresponds to the two particular moves $\\Delta \\hat{x}= \\pm \\varepsilon ^{1-\\alpha } n(\\bar{x})$ ($\\rho _1=\\rho _2=0 $ ).",
"For fixed $\\Gamma $ , since these terms containing $\\Gamma $ have the same contributions, we can omit the dependence of $\\mathcal {N}(\\cdot , \\Gamma ,1)$ and $\\mathcal {N}(\\cdot , \\Gamma , - 1)$ on $\\Gamma $ .",
"Then, by repeating the same arguments already used, there exists a unique $s_2^\\ast $ realizing the max of $\\displaystyle \\min _{}(\\mathcal {N}(\\cdot , \\Gamma ,1),\\mathcal {N}(\\cdot , \\Gamma ,-1) )$ for which both functions are equal.",
"After some calculations, we find $s_2^\\ast = -\\frac{1}{2}\\left(1-\\frac{ d(x)}{\\varepsilon ^{1-\\alpha }}\\right) m+ \\frac{1}{4} \\left( \\varepsilon ^{1-\\alpha }- \\frac{d^2(x)}{\\varepsilon ^{1-\\alpha }} \\right) (D^2u)_{11}.$ If $m< 0$ , Helen will finally choose $p_\\text{opt}(x) = Du + \\left[ \\frac{1}{2}\\left(1-\\frac{ d(x)}{\\varepsilon ^{1-\\alpha }}\\right) m- \\frac{1}{4} \\left( \\varepsilon ^{1-\\alpha }- \\frac{d^2(x)}{\\varepsilon ^{1-\\alpha }} \\right) (D^2u)_{11}\\right] n(\\bar{x}).$ To complete the analysis on the 0-order terms on the $\\rho _1,\\rho _2$ variables, we are now going to see how Helen must choose $\\Gamma _{11}$ .",
"By conserving only the 0-order terms, we obtain $\\mathcal {M}_D \\approx \\frac{1}{2}(\\varepsilon ^{1-\\alpha } - d(x) ) m+ \\max _{\\Gamma _{11}}\\left[ \\frac{1}{4} (\\varepsilon ^{2-2\\alpha }+ d^2(x)) (D^2u)_{11} -\\frac{1}{2} \\varepsilon ^{2-2\\alpha } \\Gamma _{11}- \\varepsilon ^2 f \\left(t, x, u, p_{\\text{opt}},\\Gamma \\right) \\right].$ Since $\\Gamma _{11}$ cannot counterbalance the first order term, the $\\Gamma _{11}$ -term and the second order terms are gathered.",
"Helen wants to make the best choice, so she is going to choose $ \\Gamma _{11} $ such that $\\frac{1}{4} (\\varepsilon ^{2-2\\alpha }+ d^2(x)) (D^2u)_{11} -\\frac{1}{2} \\varepsilon ^{2-2\\alpha } \\Gamma _{11} \\ge 0 .$ By ellipticity of $f$ , Helen will choose $\\Gamma _{11} $ such that this upper bound on $\\Gamma _{11}$ is attained.",
"She takes $\\Gamma _{11} = \\frac{1}{2} \\left(1+ \\frac{d^2(x)}{\\varepsilon ^{2-2\\alpha }}\\right) (D^2u)_{11}.$ It remains to determine $\\Gamma _{1v}$ .",
"By plugging the optimal choices $s_2^\\ast $ , corresponding to $p_\\text{opt}$ , and $\\Gamma _{11}$ , respectively given by (REF ) and (REF ) in (REF )–(REF ), we have $\\mathcal {N} (s_2^\\ast , \\Gamma , 1- \\rho _1)&= \\frac{1}{2} (\\varepsilon ^{1-\\alpha } - d(x) )m - \\sqrt{2\\rho _1} \\Big | \\frac{d(x)}{\\varepsilon ^{1-\\alpha }} (D^2u)_{1v} - \\Gamma _{1v} \\Big | \\varepsilon ^{2-2\\alpha }-(s_2^\\ast +m) \\rho _1 \\varepsilon ^{1-\\alpha } +O(\\varepsilon ^{2-2\\alpha } \\rho _1), \\\\\\mathcal {N} (s_2^\\ast , \\Gamma , -1 + \\rho _2)&= \\frac{1}{2} (\\varepsilon ^{1-\\alpha } - d(x) )m - \\sqrt{2\\rho _2} \\left|(D^2u)_{1v} -\\Gamma _{1v} \\right| \\varepsilon ^{2-2\\alpha }+ s_2^\\ast \\rho _2 \\varepsilon ^{1-\\alpha } +O(\\varepsilon ^{2-2\\alpha } \\rho _2).$ Dropping the $O(\\varepsilon ^{2-2\\alpha } \\rho _i)$ terms and noticing that $s_2^\\ast >0$ and $ - (s_2^\\ast +m)>0 $ for $\\varepsilon $ small enough, the two minimization problems $\\displaystyle \\min _{\\rho _i} \\mathcal {N} (s_2^\\ast , \\Gamma , 1- \\rho _i)$ , $i\\in \\lbrace 1,2\\rbrace $ for Mark reduce to find $\\min _{0< \\rho \\le 1} f(\\rho ), \\quad \\text{ where } f(\\rho ) = a \\sqrt{\\rho }+b\\rho ,$ with $a<0<b$ .",
"Differentiating $f$ , the minimum of $f$ is attained at $\\displaystyle \\sqrt{\\rho ^\\ast } = - \\frac{a}{2b}$ .",
"We can notice that this computation is equivalent to formally differentiating the Taylor expansion of (REF )–(REF ).",
"Conserving the predominant terms and dropping the next terms, the minimum of $\\mathcal {N} (s_2^\\ast , \\Gamma , 1- \\rho _1)$ and $\\mathcal {N} (s_2^\\ast , \\Gamma , -1+ \\rho _2)$ are respectively attained at $\\sqrt{\\rho _{1}^\\ast } \\simeq \\frac{1}{\\sqrt{2}} \\frac{ |\\frac{d}{\\varepsilon ^{1-\\alpha }}(D^2u)_{1v}-\\Gamma _{1v} | }{|s_2^\\ast +m|} \\varepsilon ^{1-\\alpha }\\quad \\text{ and } \\quad \\sqrt{\\rho _2^\\ast } \\simeq \\frac{1}{\\sqrt{2}}\\frac{|(D^2u)_{1v}-\\Gamma _{1v}|}{|s_2^\\ast |} \\varepsilon ^{1-\\alpha } .$ Assuming formally that these approximations are in fact equalities, we obtain $\\mathcal {N} (s_2^\\ast , \\Gamma , 1- \\rho _{1}^\\ast ) & = \\frac{1}{2} (\\varepsilon ^{1-\\alpha } - d(x) )m-\\frac{1}{2} \\left|\\frac{d(x)}{\\varepsilon ^{1-\\alpha }}(D^2u)_{1v} - \\Gamma _{1v} \\right|^2 \\frac{\\varepsilon ^{3-3\\alpha }}{|s_2^\\ast +m|} +O(\\varepsilon ^{4-4\\alpha }), \\\\\\mathcal {N} (s_2^\\ast , \\Gamma , -1+\\rho _2^\\ast ) & = \\frac{1}{2} (\\varepsilon ^{1-\\alpha } - d(x) )m-\\frac{1}{2} \\left|(D^2u)_{1v} - \\Gamma _{1v} \\right|^2 \\frac{\\varepsilon ^{3-3\\alpha }}{|s_2^\\ast |} +O(\\varepsilon ^{4-4\\alpha }).$ Helen now has to choose $\\Gamma _{1v}$ such that $\\min \\lbrace \\mathcal {N} (s_2^\\ast , \\Gamma , 1 - \\rho _{1}^\\ast ),\\mathcal {N}(s_2^\\ast ,\\Gamma , -1+\\rho _{2}^\\ast ) \\rbrace $ is maximal.",
"We could compute the optimal value of $\\Gamma _{1v}$ on the $\\varepsilon ^{3-3\\alpha }$ -terms.",
"However, it is not very useful.",
"Since $m$ is a constant and $\\varepsilon ^{3-3\\alpha } \\ll \\varepsilon ^2$ by (REF ), the $\\varepsilon ^{3-3\\alpha }$ -terms are negligible compared to $- \\varepsilon ^2 f(t,x,u, p_\\text{opt}, \\Gamma )$ that we have omitted until now.",
"For instance Helen can fix $\\Gamma _{1v}$ such that one of the two terms depending on $\\Gamma _{1v}$ in (REF ) and () is equal to zero: $\\Gamma _{1v} = (D^2u)_{1v}$ or $ \\Gamma _{1v} = \\frac{d(x)}{\\varepsilon ^{1-\\alpha }}(D^2u)_{1v}$ .",
"The two choices are equivalent because Mark can reverse his move $\\Delta \\hat{x}$ .",
"For sake of simplicity, we assume Helen chooses $\\Gamma _{1v} = (D^2u)_{1v}$ .",
"It is worth noticing that this expansion holds if $m$ is far from zero and we shall modify our arguments very carefully in Section when $m$ is negative but small with respect to a certain power of $\\varepsilon $ .",
"Thus, if $m<0$ , Helen will choose $ \\Gamma _\\text{opt}(x) = D^2 u + \\left[ \\frac{1}{2} \\left(-1+\\frac{d^2 (x) }{\\varepsilon ^{2-2\\alpha }} \\right) (D^2 u)_{11} \\right] E_{11}.$ Unlike the usual game [21], when Helen chooses $p$ and $\\Gamma $ optimally, she does not become indifferent to Mark's choice of $\\Delta \\hat{x}$ .",
"More precisely, it depends on the projection of $\\Delta \\hat{x} $ with respect to $n(\\bar{x})$ .",
"Our games always have this feature.",
"Step 3: Now let us go back to the original optimization problem (REF ).",
"If $m=0$ , by letting $\\varepsilon \\rightarrow 0$ , we get $h(x) - Du(x)\\cdot n(x)=0$ .",
"Otherwise, (REF ) formally reduces to $ 0\\approx \\varepsilon ^2 u_t+{\\left\\lbrace \\begin{array}{ll}\\displaystyle \\frac{1}{2}(\\varepsilon ^{1-\\alpha } - d(x) ) m - \\varepsilon ^2 f (t, x, u, p_{\\text{opt}}(x),\\Gamma _\\text{opt}(x))+o(\\varepsilon ^2),& \\text{if } d(x)\\le \\varepsilon ^{1-\\alpha } \\text{ and } m < 0, \\\\\\displaystyle - \\varepsilon ^2 f (t, x,u,Du,D^2u), & \\text{if } d(x)\\ge \\varepsilon ^{1-\\alpha } \\text{ or } m > 0,\\end{array}\\right.",
"}$ with $p_\\text{opt}$ and $\\Gamma _\\text{opt}$ respectively defined by (REF ) and (REF ).",
"If $x\\in \\Omega $ , for $\\varepsilon $ small enough, the second relation in (REF ) is always valid so that we deduce from the $\\varepsilon ^2$ -order terms in (REF ) that $u_t- f(t,x,u,Du,D^2u)=0$ .",
"If $x\\in \\partial \\Omega $ , $d(x)=0$ and we distinguish the cases $m>0 $ and $m< 0$ .",
"If $m>0$ , one more time the second relation in (REF ) is always valid so that $u_t- f(t,x,u,Du,D^2u)=0$ .",
"Otherwise, if $m< 0$ , the first relation in (REF ) is always satisfied.",
"We observe that the $\\varepsilon $ -order term is predominant since $\\varepsilon ^{1-\\alpha } \\gg \\varepsilon ^2$ .",
"By dividing by $\\varepsilon ^{1-\\alpha }$ and letting $\\varepsilon \\rightarrow 0$ , we obtain $ m = 0$ that leads to a contradiction since we assumed $m< 0$ .",
"Therefore, we have formally shown that on the boundary $h(x) - Du(x)\\cdot n(x)=0$ or $u_t- f(t,x,u,Du,D^2u)=0$ ."
],
[
"Main parabolic result",
"We shall show, under suitable hypotheses, that $\\overline{u}$ and $\\underline{v}$ are respectively viscosity sub and supersolutions.",
"A natural question is to compare $\\overline{u}$ and $\\underline{v}$ .",
"This is a global question, which we can answer only if the PDE has a comparison principle.",
"Such a principle asserts that if $u$ is a subsolution and $v$ is a supersolution then $u\\le v$ .",
"If the PDE has such a principle then it follows that $\\overline{u} \\le \\underline{v}$ .",
"The opposite inequality is immediate from the definitions, so it follows that $\\overline{u}= \\underline{v}$ , and we get a viscosity solution of the PDE.",
"It is in fact the unique viscosity solution, since the comparison principle implies uniqueness.",
"Theorem 2.4 Consider the final-value problem (REF ) where $f$ satisfies (REF )–(REF ), $g$ and $h$ are continuous, uniformly bounded, and $\\Omega $ is a $C^2$ -domain satisfying both the uniform interior and exterior ball conditions.",
"Assume the parameters $\\alpha $ , $\\beta $ , $\\gamma $ satisfy (REF )-(REF ).",
"Then $\\overline{u}$ and $\\underline{v}$ are uniformly bounded on $\\overline{\\Omega }\\times [t_\\ast , T]$ for any $t_\\ast <T$ , and they are respectively a viscosity subsolution and a viscosity supersolution of (REF ).",
"If the PDE has a comparison principle (for uniformly bounded solutions), then it follows that $u^\\varepsilon $ and $v^\\varepsilon $ converge locally uniformly to the unique viscosity solution of (REF ).",
"This theorem is an immediate consequence of Propositions REF and REF .",
"In this theorem, we require the domain $\\Omega $ to be $C^2$ .",
"This assumption is crucial for the proof of Proposition REF case REF corresponding to the convergence at the final-time in the viscosity sense (see Remark REF ).",
"It can also be noticed that it is this part of Proposition REF which allows to use a comparison principle for the parabolic PDE.",
"On the other hand, since the game already requires the uniform interior and exterior ball conditions, the domain $\\Omega $ is in fact at least $C^{1,1}$ .",
"It remains an open question to overcome the analysis in this case.",
"As mentioned in [21], some sufficient conditions for the PDE to have a comparison result can be found in Section 4.3 of [11].",
"In our framework, we can emphasise on the comparison principle obtained by Sato [27] for a fully nonlinear parabolic equation with a homogeneous condition.",
"The reader is also referred to the introduction for other references about comparison and existence results.",
"Note that most comparison results require $f(t,x,z,p, \\Gamma )$ to be nondecreasing in $z$ .",
"We close this section with the observation that if $U^\\varepsilon (x,z,t)$ is a strictly decreasing function of $z$ then $v^\\varepsilon (x,t)=u^\\varepsilon (x,t)$ .",
"A sufficient condition for this to hold is that $f$ be nondecreasing in $z$ : Lemma 2.5 Suppose $f$ is non-decreasing in $z$ in the sense that $f(t,x,z_1,p, \\Gamma )\\ge f(t,x,z_0,p, \\Gamma ) \\quad \\text{ whenever }z_1>z_0.$ Then $U^\\varepsilon $ satisfies $U^\\varepsilon (x,z_1,t_j)\\le U^\\varepsilon (x,z_0,t_j) - (z_1-z_0) \\quad \\text{ whenever }z_1>z_0,$ at each discrete time $t_j=T - j\\varepsilon ^2$ .",
"In particular, $U^\\varepsilon $ is strictly decreasing in $z$ and $v^\\varepsilon = u^\\varepsilon $ .",
"The whole space case is provided in [21].",
"For our game, it suffices to add $- \\Vert \\Delta \\hat{x} - \\Delta x \\Vert h(x+\\Delta x)$ in the expressions of $\\Delta z_0$ and $\\Delta z_1$ defined in the proof of [21].",
"The rest of the proof remains unchanged."
],
[
"Nonlinear elliptic equations",
"This section explains how our game can be used to solve stationary problems with Neumann boundary conditions.",
"The framework is similar to the parabolic case, but one new issue arises: we must introduce discounting as in [21], to be sure Helen's value function is finite.",
"Therefore we focus on ${\\left\\lbrace \\begin{array}{ll}f(x, u,Du,D^2u)+ \\lambda u =0, & \\text{ in } \\Omega , \\\\\\left\\langle D u, n \\right\\rangle =h, & \\text{ on } \\partial \\Omega ,\\end{array}\\right.",
"}$ where $\\Omega $ is a domain with $C^2$ -boundary and satisfies both the uniform interior and exterior ball condition presented in the introduction.",
"The constant $\\lambda $ (which plays the role of an interest rate) must be positive, and large enough so that (REF ) holds.",
"Notice that if $f$ is independent of $z$ then any $\\lambda $ will do.",
"We now present the game.",
"The main difference with Section REF is the presence of discounting.",
"The boundary condition $h$ is assumed to be a bounded continuous function on $\\partial \\Omega $ .",
"Besides the parameters $\\alpha $ , $\\beta $ , $\\gamma $ introduced previously, in the stationary case we need two new parameters, $m$ and $M$ , and a $C_b^2(\\overline{\\Omega })$ -function $\\psi $ such that $\\frac{\\partial \\psi }{\\partial n}=\\Vert h \\Vert _\\infty +1 \\quad \\text{ on } \\partial \\Omega .$ It suffices to construct $\\psi _1$ such that it is $C_b^2(\\overline{\\Omega })$ and satisfies $\\frac{\\partial \\psi _1}{\\partial n}=1$ on the boundary.",
"Then we can define $\\psi $ by $\\psi =( \\Vert h \\Vert _\\infty +1)\\psi _1$ .",
"The existence and construction of such a function $\\psi _1$ for a $C^2$ -domain $\\Omega $ satisfying the uniform interior ball condition is discussed at the end of this section.",
"From $m$ and $\\psi $ we construct a function $\\chi $ defined by $\\chi (x)=m+ \\Vert \\psi \\Vert _{L^\\infty (\\overline{\\Omega })} +\\psi (x).$ Both $m$ and $M$ are positive constants, which also yield that $\\chi $ is positive.",
"$M$ serves to cap the score, and the function $\\chi $ determines what happens when the cap is reached.",
"We shall in due course choose $m$ such that $m + 2 \\Vert \\psi \\Vert _{L^\\infty } =M-1$ and require that $M$ is sufficiently large.",
"Like the choices of $\\alpha $ , $\\beta $ , $\\gamma $ , the parameters $M$ , $m$ and the function $\\psi $ are used to define the game but they do not influence the resulting PDE.",
"As in Section REF , we proceed in two steps: First we introduce $U^\\varepsilon (x,z)$ , the optimal worst-case present value of Helen's wealth if the initial stock price is $x$ and her initial wealth is $-z$ .",
"Then we define $u^\\varepsilon (x)$ and $v^\\varepsilon (x)$ as the maximal and minimal initial debt Helen should have at time $t$ to break even upon exit.",
"The definition of $U^\\varepsilon (x,z)$ for $x\\in \\overline{\\Omega }$ involves a game similar to that of the last section: Initially, at time $t_0=0$ , the stock price is $x_0=x$ and Helen's debt is $z_0=z$ .",
"Suppose, at time $t_j=j\\varepsilon ^2$ , the stock price is $x_j$ and Helen's debt is $z_j$ with $|z_j|<M$ .",
"Then Helen chooses a vector $p_j \\in \\mathbb {R}^N$ and a matrix $\\Gamma _j \\in \\mathcal {S}^N$ , restricted in magnitude by (REF ).",
"Knowing these choices, Mark determines the next stock price $ x_{j+1}=x_j+\\Delta x$ so as to degrade Helen's outcome.",
"The increment $\\Delta x$ allows to model the reflection exactly as in the previous subsections.",
"Mark chooses an intermediate point $\\hat{x}_{j+1}=x_j +\\Delta \\hat{x}_j \\in \\mathbb {R}^N$ such that $\\left\\Vert \\Delta \\hat{x}_j\\right\\Vert \\le \\varepsilon ^{1-\\alpha }.$ This position $\\hat{x}_{j+1}$ determines the new position $x_{j+1}=x_j +\\Delta x_j$ at time $t_{j+1}$ by $x_{j+1}= \\operatorname{proj}_{\\overline{\\Omega }} (\\hat{x}_{j+1}).$ Helen experiences a loss at time $t_j$ of $ \\delta _j = p_j\\cdot \\Delta \\hat{x}_j +\\frac{1}{2} \\left\\langle \\Gamma _j \\Delta \\hat{x}_j,\\Delta \\hat{x}_j \\right\\rangle +\\varepsilon ^2 f(x_j,z_j,p_j,\\Gamma _j) - \\Vert \\Delta \\hat{x}_j - \\Delta x_j \\Vert h(x_j+\\Delta x_j ) .$ As a consequence, her time $t_{j+1}=t_j+\\varepsilon ^2$ debt becomes $z_{j+1}=e^{\\lambda \\varepsilon ^2} (z_j+\\delta _j),$ where the factor $e^{\\lambda \\varepsilon ^2}$ takes into account her interest payments.",
"If $z_{j+1} \\ge M$ , then the game terminates, and Helen pays a “termination-by-large-debt penalty” worth $e^{\\lambda \\varepsilon ^2}( \\chi (x_j) -\\delta _j)$ at time $t_{j+1}$ .",
"Similarly, if $z_{j+1} \\le - M$ , the the game terminates, and Helen receives a “termination-by-large-wealth bonus” worth $e^{\\lambda \\varepsilon ^2}(\\chi (x_j) +\\delta _j)$ at time $t_{j+1}$ .",
"If the game stops this way we call $t_{j+1}$ the “ending index” $t_K$ .",
"If the game has not terminated then Helen and Mark repeat this procedure at time $t_{j+1}=t_j+\\varepsilon ^2$ .",
"If the game never stops the “ending index” $t_K$ is $+\\infty $ .",
"Helen's goal is a bit different from before, due to the presence of discounting: she seeks to maximize the minimum present value of her future income, using the discount factor of $e^{-j\\lambda \\varepsilon ^2}$ for income received at time $t_j$ .",
"If the game ends by capping at time $t_K$ with $z_K \\ge M$ , then the present value of her income is $U^\\varepsilon (x_0,z_0) & = - z_0 -\\delta _0 - e^{-\\lambda \\varepsilon ^2} \\delta _1 - \\cdots - e^{-(K-1) \\lambda \\varepsilon ^2} \\delta _{K-1}- e^{-(K-1) \\lambda \\varepsilon ^2} ( \\chi (x_{K-1}) -\\delta _{K-1}) \\\\& = e^{-(K-1) \\lambda \\varepsilon ^2} (-z_{K-1}- \\chi (x_{K-1})).$ Similarly, if the game ends by capping at time $t_K$ with $z_K \\le - M$ , then the present value of her income is $U^\\varepsilon (x_0,z_0) & = - z_0 -\\delta _0 - e^{-\\lambda \\varepsilon ^2} \\delta _1 - \\cdots - e^{-(K-1) \\lambda \\varepsilon ^2} \\delta _{K-1}+ e^{-(K-1) \\lambda \\varepsilon ^2} (\\chi (x_{K-1}) + \\delta _{K-1}) \\\\& = e^{-(K-1) \\lambda \\varepsilon ^2} (-z_{K-1}+ \\chi (x_{K-1}) ).$ If the game never ends (since $z_j$ and $\\chi (x_j)$ are uniformly bounded), we can take $K=\\infty $ in the preceding formula to see that the present value of her income is 0.",
"To get a dynamic programming characterization of $U^\\varepsilon $ , we observe that if $|z_0|<M$ then $U^{\\varepsilon }(x_0,z_0)=\\sup _{p, \\Gamma } \\inf _{\\Delta \\hat{x} }{\\left\\lbrace \\begin{array}{ll}e^{-\\lambda \\varepsilon ^2} U^{\\varepsilon }(x_1,z_1), & \\text{if } |z_1|< M , \\\\-z_0 - \\chi (x_0) , & \\text{if } z_1 \\ge M, \\\\-z_0 + \\chi (x_0) , & \\text{if } z_1 \\le -M.\\end{array}\\right.",
"}$ Since the game is stationary (nothing distinguishes time 0), the associated dynamic programming principle is that for $|z| <M$ , $U^{\\varepsilon }(x,z)=\\sup _{p, \\Gamma } \\inf _{\\Delta \\hat{x} }{\\left\\lbrace \\begin{array}{ll}e^{-\\lambda \\varepsilon ^2} U^{\\varepsilon }(x^{\\prime },z^{\\prime }), & \\text{if } |z^{\\prime }|< M , \\\\-z - \\chi (x), & \\text{if } z^{\\prime } \\ge M, \\\\-z + \\chi (x), & \\text{if } z^{\\prime } \\le -M,\\end{array}\\right.",
"}$ where $x^{\\prime }=\\operatorname{proj}_{\\overline{\\Omega }}(x+\\Delta \\hat{x})$ and $z^{\\prime }=e^{\\lambda \\varepsilon ^2}(z+\\delta )$ , with $\\delta $ defined as in (REF ).",
"Here $p$ , $\\Gamma $ and $\\Delta \\hat{x}$ are constrained as usual by (REF )–(REF ), and we write $\\sup /\\inf $ rather than $\\max /\\min $ since it is no longer clear that the optima are achieved (since the right-hand side is now a discontinuous function of $p$ , $\\Gamma $ and $\\Delta \\hat{x}$ ).",
"The preceding discussion defines $U^\\varepsilon $ only for $|z|<M$ ; it is natural to extend the definition to all $z$ by $U^{\\varepsilon }(x,z) ={\\left\\lbrace \\begin{array}{ll}-z - \\chi (x), & \\text{ for } z\\ge M,\\\\-z + \\chi (x), & \\text{ for } z\\le - M,\\end{array}\\right.",
"}$ which corresponds to play being “capped” immediately.",
"Notice that when extended this way, $U^\\varepsilon $ is strictly negative for $z\\ge M$ and strictly positive for $z\\le - M$ .",
"The definitions of $u^\\varepsilon $ and $v^\\varepsilon $ are slightly different from those in Section REF : $u^\\varepsilon (x_0) & =\\sup \\lbrace z_0\\, :\\, U^\\varepsilon (x_0,z_0)> 0 \\rbrace , \\\\v^\\varepsilon (x_0) & =\\inf \\lbrace z_0\\, :\\, U^\\varepsilon (x_0,z_0)< 0 \\rbrace .$ The change from Section REF is that the inequalities in (REF )–(REF ) are strict.",
"Proposition 2.6 Let $m_1$ , $M$ be two constants such that $0<m_1<M$ .",
"Then whenever $x\\in \\overline{\\Omega }$ and $-m_1 \\le u^{\\varepsilon }(x) <M$ we have $ u^\\varepsilon (x) \\le \\sup _{p,\\Gamma } \\inf _{\\Delta \\hat{x}} \\left[e^{ - \\lambda \\varepsilon ^2} u^\\varepsilon (x+\\Delta x) \\right.",
"\\\\\\left.",
"- \\left(p\\cdot \\Delta \\hat{x} +\\frac{1}{2} \\left\\langle \\Gamma \\Delta \\hat{x} ,\\Delta \\hat{x} \\right\\rangle +\\varepsilon ^2 f(x, u^\\varepsilon (x),p ,\\Gamma ) - \\Vert \\Delta \\hat{x} - \\Delta x \\Vert h(x+\\Delta x) \\right) \\right] ,$ for $\\varepsilon $ small enough (depending on $m_1$ and the parameters of the game but not on $x$ ).",
"Similarly, if $x\\in \\overline{\\Omega }$ and $-M<v^\\varepsilon (x)<m_1$ then for $\\varepsilon $ small enough $ v^\\varepsilon (x) \\ge \\sup _{p,\\Gamma } \\inf _{\\Delta \\hat{x}} \\left[e^{ - \\lambda \\varepsilon ^2} v^\\varepsilon (x+\\Delta x) \\right.",
"\\\\\\left.",
"- \\left(p\\cdot \\Delta \\hat{x} +\\frac{1}{2} \\left\\langle \\Gamma \\Delta \\hat{x},\\Delta \\hat{x} \\right\\rangle +\\varepsilon ^2 f(x, v^\\varepsilon (x),p ,\\Gamma ) - \\Vert \\Delta \\hat{x} - \\Delta x \\Vert h(x+\\Delta x) \\right) \\right] .$ As usual, the sup and inf are constrained by (REF ) and (REF ) and $\\Delta x$ is determined by (REF ).",
"We shall focus on (REF ); the proof for (REF ) follows exactly the same lines.",
"Since $-m_1 \\le u^{\\varepsilon }(x)<M$ , there is a sequence $z^k \\uparrow u^{\\varepsilon }(x) $ such that $U^\\varepsilon (x,z^k)>0$ .",
"Since $u^{\\varepsilon }(x)$ is bounded away from $-M$ , we may suppose that $z^k$ also remains bounded away from $-M$ .",
"Dropping the index $k$ for simplicity of notation, consider any such $z=z^k$ .",
"The fact that $U^\\varepsilon (x,z)>0$ tells us that the right-hand side of the dynamic programming principle (REF ) is positive.",
"So there exist $p$ , $\\Gamma $ constrained by (REF ) such that for any $\\Delta \\hat{x}$ satisfying (REF ), $ 0 < \\sup _{p, \\Gamma }\\inf _{\\Delta \\hat{x} }{\\left\\lbrace \\begin{array}{ll}e^{-\\lambda \\varepsilon ^2} U^{\\varepsilon }(x^{\\prime },z^{\\prime }), & \\text{if } |z^{\\prime }|< M , \\\\-z- \\chi (x), & \\text{if } z^{\\prime } \\ge M, \\\\-z+ \\chi (x), & \\text{if } z^{\\prime } \\le -M,\\end{array}\\right.",
"}$ where $x^{\\prime }=\\operatorname{proj}_{\\overline{\\Omega }}(x+\\Delta \\hat{x})$ and $z^{\\prime }=e^{\\lambda \\varepsilon ^2}(z+\\delta )$ .",
"Capping above, the alternative $z^{\\prime }\\ge M$ , cannot happen, since otherwise we compute $-z- \\chi (x) = - e^{-\\lambda \\varepsilon ^2}z^{\\prime } - \\delta - \\chi (x)\\le - M e^{-\\lambda \\varepsilon ^2} - \\delta - m \\le - \\delta - m< 0,$ for $\\varepsilon $ small enough because $\\delta $ is bounded by a positive power of $\\varepsilon $ .",
"This sign is a contradiction to our assumption (REF ).",
"If $\\varepsilon $ is sufficiently small, capping below (the alternative $z^{\\prime }\\le - M$ ) cannot occur either, because $z$ is bounded away from $-M$ and $\\delta $ is bounded by a positive power of $\\varepsilon $ .",
"Therefore only the first case can take place $0<U^{\\varepsilon }(x+\\Delta x, e^{\\lambda \\varepsilon ^2}(z+\\delta )),$ whence by the definition of $u^\\varepsilon $ given by (REF ), we deduce that $u^\\varepsilon (x+\\Delta x) \\ge e^{\\lambda \\varepsilon ^2} (z+\\delta ).$ Thus, we have shown the existence of $p$ , $\\Gamma $ such that for every $\\Delta \\hat{x}$ , $ z\\le e^{ - \\lambda \\varepsilon ^2} u^\\varepsilon (x+\\Delta x)- \\left( p \\cdot \\Delta \\hat{x} +\\frac{1}{2} \\langle \\Gamma \\Delta \\hat{x}, \\Delta \\hat{x} \\rangle +\\varepsilon ^2 f(x,z,p, \\Gamma ) - \\Vert \\Delta \\hat{x} - \\Delta x \\Vert h(x+\\Delta x) \\right).$ Recalling that $z=z^k \\uparrow u^\\varepsilon (x)$ , we pass to the limit on both sides of (REF ), with $p$ , $\\Gamma $ held fixed, to see that $u^\\varepsilon (x) \\le e^{-\\lambda \\varepsilon ^2} u^\\varepsilon (x+\\Delta x) - \\left( p\\cdot \\Delta \\hat{x} +\\frac{1}{2} \\langle \\Gamma \\Delta \\hat{x}, \\Delta \\hat{x} \\rangle +\\varepsilon ^2 f(x,u^{\\varepsilon }(x),p, \\Gamma ) - \\Vert \\Delta \\hat{x} - \\Delta x \\Vert h(x+\\Delta x) \\right).$ Since this is true for some $p$ , $\\Gamma $ and for every $\\Delta \\hat{x}$ , we have established (REF ).",
"The PDE (REF ) is the formal Hamilton-Jacobi-Bellman equation associated with the dynamic programming inequalities (REF )–(REF ), by the usual Taylor expansion, if one accepts $-M<v^\\varepsilon \\approx u^\\varepsilon <M$ .",
"Rather than giving that heuristic argument which is quite similar to the one proposed in the parabolic setting, we now state our main result in the stationary setting, which follows from the results in Sections and .",
"It concerns the upper and lower relaxed semi-limits, defined for any $x \\in \\overline{\\Omega }$ , by $ \\overline{u}(x)= \\limsup _{\\begin{array}{c}y\\rightarrow x \\\\ \\varepsilon \\rightarrow 0\\end{array}} u^{\\varepsilon }(y) \\quad \\text{ and } \\quad \\underline{v}(x)= \\liminf _{\\begin{array}{c}y\\rightarrow x \\\\ \\varepsilon \\rightarrow 0\\end{array}} v^{\\varepsilon }(y),$ with the convention that $y$ approaches $x$ from $\\overline{\\Omega }$ (since $u^\\varepsilon $ and $v^\\varepsilon $ are defined on $\\overline{\\Omega }$ ).",
"Theorem 2.7 Consider the stationary boundary value problem (REF ) where $f$ satisfies (REF ) and (REF )–(REF ), $g$ and $h$ are continuous, uniformly bounded, and $\\Omega $ is a $C^2$ -domain satisfying both the uniform interior and exterior ball conditions.",
"Assume the parameters of the game $\\alpha $ , $\\beta $ , $\\gamma $ fulfill (REF )–(REF ), $\\psi \\in C^2(\\overline{\\Omega })$ satisfies (REF ), $\\chi \\in C^2(\\overline{\\Omega })$ is defined by (REF ), $M$ is sufficiently large and $m=M-1 -2 \\Vert \\psi \\Vert _{L^\\infty (\\overline{\\Omega })}$ .",
"Then $u^\\varepsilon $ and $v^\\varepsilon $ are well-defined when $\\varepsilon $ is sufficiently small, and they satisfy $ |u^\\varepsilon | \\le \\chi $ and $ |v^\\varepsilon | \\le \\chi $ .",
"Their relaxed semi-limits $\\overline{u}$ and $\\underline{v}$ are respectively a viscosity subsolution and a viscosity supersolution of (REF ).",
"If in addition we have $\\underline{v} \\le \\overline{u}$ and the PDE has a comparison principle, then it follows that $u^\\varepsilon $ and $v^\\varepsilon $ converge locally uniformly in $\\overline{\\Omega }$ to the unique viscosity solution of (REF ).",
"This is an immediate consequence of Propositions REF and REF .",
"A sufficient condition for $\\underline{v} \\le \\bar{u}$ is that $f$ is nondecreasing in $z$ .",
"As mentioned in [21], sufficient conditions for the PDE to have a comparison principle can be found for example in Section 5 of [12], and (for more results) in [6]–[9].",
"Let us now go back to the existence and the construction of $\\psi _1\\in C_b^2(\\overline{\\Omega })$ such that $\\frac{\\partial \\psi _1}{\\partial n}= 1$ on $\\partial \\Omega $ , that we will need at various points of the paper.",
"If $\\Omega $ is of class $C^2$ and satisfies the uniform interior ball condition of Definition REF for a certain $r$ , $d$ is $C^2$ on $\\Omega (3r/4)$ and an explicit suitable function is $\\psi _1(x) ={\\left\\lbrace \\begin{array}{ll} \\exp \\left[ - \\frac{d(x)}{1 - \\frac{d(x)}{r/2} }\\right], & \\text{ if } d(x)<r/2, \\\\0 , & \\text{ if } d(x)\\ge r/2.\\end{array}\\right.",
"}$ It is clear that $\\text{supp } \\psi _1 \\subset \\Omega (r/2)$ , $\\psi _1(\\overline{\\Omega }) \\subset [0,1] $ and $\\psi _1$ is $C^2$ on $\\Omega (r/2)$ .",
"Then, for all $x$ such that $d(x)=\\frac{r}{2}$ , $D\\psi _1$ and $D^2\\psi _1$ are continuous at $x$ .",
"Thus $\\psi _1$ is $C^2$ on $\\overline{\\Omega }$ .",
"It is easy to check that the two first derivatives of $\\psi _1$ are also bounded and that $\\frac{\\partial \\psi _1}{\\partial n} = 1$ on the boundary.",
"Hence, the function $\\psi _1$ defined by (REF ) has all the desired properties.",
"Remark 2.8 If $\\Omega $ is a domain with $C^{2,\\alpha }$ -boundary where $\\alpha >0$ , the Schauder theory [18] ensures the solution $\\psi $ of the elliptic problem ${\\left\\lbrace \\begin{array}{ll}\\Delta \\psi - \\psi =0 , & \\text{ in } {\\Omega } , \\\\\\dfrac{\\partial \\psi }{\\partial n}= \\Vert h \\Vert _{L^\\infty } +1, & \\text{ on } \\partial \\Omega , \\\\\\end{array}\\right.",
"}$ is $C^{2,\\alpha }(\\overline{\\Omega })$ .",
"In addition, the estimate $\\Vert \\psi \\Vert _{C^{2,\\alpha }(\\overline{\\Omega })} \\le C_\\Omega (1 + \\Vert h \\Vert _{L^\\infty })$ holds for a certain constant $ C_\\Omega $ depending only on the domain."
],
[
"Convergence",
"This section presents our main convergence results, linking the limiting behavior of $v^\\varepsilon $ and $u^\\varepsilon $ as $\\varepsilon \\rightarrow 0$ to the PDE.",
"The argument uses the framework of [10] and is basically a special case of the theorem proved there.",
"Convergence is a local issue: in the time-dependent setting, Proposition REF shows that in any region where the lower and upper semi-relaxed limits $\\underline{v}$ and $\\bar{u}$ are finite they are in fact viscosity super and subsolutions respectively.",
"The analogous statement for the stationary case is more subtle.",
"In fact, we will need global hypotheses on $f$ at Section REF to ensure that $u^{\\varepsilon }$ and $v^{\\varepsilon }$ are well-defined and satisfy the dynamic programming inequalities (REF )–(REF ).",
"Thus, we cannot discuss about $\\underline{v}$ or $\\bar{u}$ without global assumptions on $f$ ."
],
[
"Viscosity solutions with Neumann condition",
"Our PDEs can be degenerate parabolic, degenerate elliptic, or even first order equations.",
"Therefore, we cannot expect a classical solution, and we cannot always impose boundary data in the classical sense.",
"The theory of viscosity solutions provides the proper framework for handling these issues.",
"We review the basic definitions in our setting for the reader's convenience.",
"We refer to [4], [12] and [16] for further details about the general theory.",
"Consider first the final-value problem (REF ) in $\\Omega $ , ${\\left\\lbrace \\begin{array}{ll}- u_t + f(t,x,u, Du, D^2 u ) =0, & \\text{ for } x \\in \\Omega \\text{ and }t<T, \\\\\\langle Du(x,t),n(x) \\rangle = h(x), & \\text{ for } x \\in \\partial \\Omega \\text{ and }t<T, \\\\u(x,T)=g(x), & \\text{ for } x\\in \\overline{\\Omega }.",
"\\end{array}\\right.",
"}$ where $f(t,x,z,p, \\Gamma )$ is continuous in all its variables and satisfies the monotonicity condition (REF ) in its last variable.",
"We must be careful to impose the boundary condition in the viscosity sense.",
"Definition 3.1 A real-valued lower-semicontinuous function $u(x,t)$ defined for $x\\in \\overline{\\Omega }$ and $t_\\ast \\le t \\le T$ is a viscosity supersolution of the final-value problem (REF ) if for any $(x_0,t_0)$ with $x_0\\in \\Omega $ and $t_\\ast \\le t_0 <T$ and any smooth $\\phi (x,t)$ such that $u-\\phi $ has a local minimum at $(x_0,t_0)$ , we have $\\partial _t \\phi (x_0,t_0) - f(t_0,x_0,u(x_0,t_0),D\\phi (x_0,t_0),D^2\\phi (x_0,t_0))\\le 0 ,$ for any $(x_0,t_0)$ with $x_0\\in \\partial \\Omega $ and $t_\\ast \\le t_0 <T$ and any smooth $\\phi (x,t)$ such that $u-\\phi $ has a local minimum on $\\overline{\\Omega }$ at $(x_0,t_0)$ , we have $\\max \\lbrace -( \\partial _t \\phi (x_0,t_0) - & f(t_0,x_0,u(x_0,t_0),D\\phi (x_0,t_0),D^2\\phi (x_0,t_0)) ) ,\\left\\langle D\\phi (x_0,t_0), n(x_0)\\right\\rangle - h(x_0)\\rbrace \\ge 0 ,$ $u\\ge g$ at the final time $t=T$ .",
"Similarly, a real-valued upper-semicontinuous function $u(x,t)$ defined for $x\\in \\overline{\\Omega }$ and $t_\\ast \\le t \\le T$ is a viscosity subsolution of the final-value problem (REF ) if for any $(x_0,t_0)$ with $x_0\\in \\Omega $ and $t_\\ast \\le t_0 <T$ and any smooth $\\phi (x,t)$ such that $u-\\phi $ has a local maximum at $(x_0,t_0)$ , we have $\\partial _t \\phi (x_0,t_0) - f(t_0,x_0,u(x_0,t_0),D\\phi (x_0,t_0),D^2\\phi (x_0,t_0))\\ge 0 ,$ for any $(x_0,t_0)$ with $x_0\\in \\partial \\Omega $ and $t_\\ast \\le t_0 <T$ and any smooth $\\phi (x,t)$ such that $u-\\phi $ has a local maximum on $\\overline{\\Omega }$ at $(x_0,t_0)$ , we have $\\min \\lbrace -( \\partial _t \\phi (x_0,t_0) -& f(t_0,x_0,u(x_0,t_0),D\\phi (x_0,t_0),D^2\\phi (x_0,t_0)) ),\\left\\langle D\\phi (x_0,t_0), n(x_0) \\right\\rangle - h(x_0) \\rbrace \\le 0 ,$ $u\\le g$ at the final time $t=T$ .",
"A viscosity solution of (REF ) is a continuous function $u$ that is both a viscosity subsolution and a viscosity supersolution of (REF ).",
"In the stationary problem (REF ), the definitions are similar to the time-dependent setting.",
"Definition 3.2 A real-valued lower-semicontinuous function $u(x)$ defined on $\\overline{\\Omega }$ is a viscosity supersolution of the stationary problem (REF ) if for any $x_0\\in \\Omega $ and any smooth $\\phi (x)$ such that $u-\\phi $ has a local minimum at $x_0$ , we have $f(x_0,u(x_0),D\\phi (x_0),D^2\\phi (x_0)) +\\lambda u(x_0) \\ge 0,$ for any $x_0 \\in \\partial \\Omega $ and any smooth $\\phi (x)$ such that $u-\\phi $ has a local minimum on $\\overline{\\Omega }$ at $x_0$ , we have $\\max \\lbrace f(x_0,u(x_0),D\\phi (x_0),D^2\\phi (x_0)) +\\lambda u(x_0), \\left\\langle D\\phi (x_0), n(x_0)\\right\\rangle - h(x_0) \\rbrace \\ge 0 .$ Similarly, a real-valued upper-semicontinuous function $u(x)$ defined on $\\overline{\\Omega }$ is a viscosity subsolution of the stationary problem (REF ) if for any $x_0\\in \\Omega $ and any smooth $\\phi (x)$ such that $u-\\phi $ has a local maximum at $x_0$ , we have $f(x_0,u(x_0),D\\phi (x_0),D^2\\phi (x_0)) +\\lambda u(x_0) \\le 0,$ for any $x_0 \\in \\partial \\Omega $ and any smooth $\\phi (x)$ such that $u-\\phi $ has a local maximum on $\\overline{\\Omega }$ at $x_0$ , we have $\\min \\lbrace f(x_0,u(x_0),D\\phi (x_0),D^2\\phi (x_0)) +\\lambda u(x_0), \\left\\langle D\\phi (x_0), n(x_0) \\right\\rangle - h(x_0) \\rbrace \\le 0 .$ A viscosity solution of (REF ) is a continuous function $u$ that is both a viscosity subsolution and a viscosity supersolution of (REF ).",
"In stating these definitions, we have assumed that the final-time data $g$ and the boundary Neumann condition $h$ are continuous.",
"In Definition REF , the pointwise inequality in part REF can be replaced by an apparently weaker condition analogous to part REF .",
"The equivalence of such a definition with the one stated above is standard, the argument uses barriers of the form $ \\phi (x,t)=\\Vert x-x_0 \\Vert ^2/\\delta + (T-t)/\\mu + K d(x)$ with $\\delta $ and $\\mu $ sufficiently small, and is contained in our proof of Proposition REF REF .",
"We shall be focusing on the lower and upper semi-relaxed limits of $v^{\\varepsilon }$ and $u^{\\varepsilon }$ , defined by (REF ) in the time-dependent setting and (REF ) in the stationary case.",
"We now provide a key definition to deal with the Neumann boundary condition within viscosity solutions framework which will be essential all along the article.",
"We introduce some applications which give bounds on the Neumann penalization term for a smooth function and $x$ close to the boundary.",
"This approach is well-suited to the viscosity solutions framework.",
"More precisely, we define the applications $ m_\\varepsilon $ and $M_\\varepsilon $ , for all $x\\in \\Omega (\\varepsilon ^{1- \\alpha })$ and $\\phi \\in C^1 (\\overline{\\Omega })$ , by $m_\\varepsilon ^{x}[\\phi ] &:=\\inf _{\\begin{array}{c}x+\\Delta \\hat{x} \\notin \\Omega \\\\ \\Delta \\hat{x}\\end{array}}\\left\\lbrace h(x+\\Delta x) - D \\phi (x)\\cdot n(x+\\Delta x) \\right\\rbrace , \\\\M_\\varepsilon ^{x}[\\phi ] &:=\\sup _{\\begin{array}{c}x+\\Delta \\hat{x} \\notin \\Omega \\\\ \\Delta \\hat{x} \\end{array}}\\left\\lbrace h(x+\\Delta x) - D \\phi (x)\\cdot n(x+\\Delta x) \\right\\rbrace , $ where $\\Delta \\hat{x}$ is constrained by (REF ) and determines $\\Delta x$ by (REF ).",
"Notice that the functions $ m_\\varepsilon ^{\\cdot }[\\phi ]$ and $M_\\varepsilon ^{\\cdot }[\\phi ]$ are bounded by $\\Vert h \\Vert _{L^\\infty }+ \\Vert D\\phi \\Vert _{L^\\infty (\\overline{\\Omega })}$ .",
"Since $h$ is supposed to be continuous, the following property clearly holds.",
"Lemma 3.3 Let $x\\in \\partial \\Omega $ and $\\phi \\in C^1(\\overline{\\Omega })$ .",
"Suppose there exists a sequence $(\\varepsilon _k, x_k)_{k\\in \\mathbb {N}}$ in $\\mathbb {R}_+^\\ast \\times \\overline{\\Omega }$ convergent to $(0,x)$ such that for all $k$ large enough, $x_k\\in \\Omega (\\varepsilon _k^{1-\\alpha })$ .",
"Then $\\lim _{k\\rightarrow +\\infty } m_{\\varepsilon _k}^{x_k}[\\phi ] =\\lim _{k\\rightarrow +\\infty } M_{\\varepsilon _k}^{x_k}[\\phi ] =h(x) - D\\phi (x)\\cdot n(x).$ Similarly, let $\\phi \\in C^1(\\overline{\\Omega }\\times [0,T])$ .",
"Suppose there exists a sequence $(\\varepsilon _k, x_k,t_k)_{k\\in \\mathbb {N}}$ in $\\mathbb {R}_+^\\ast \\times \\overline{\\Omega }\\times [0,T]$ convergent to $(0,x,t)$ such that for all $k$ large enough, $x_k\\in \\Omega (\\varepsilon _k^{1-\\alpha })$ .",
"Then $\\lim _{k\\rightarrow +\\infty } m_{\\varepsilon _k}^{x_k}[\\phi (\\cdot , t_k)] =\\lim _{k\\rightarrow +\\infty } M_{\\varepsilon _k}^{x_k}[\\phi (\\cdot , t_k)]=h(x) - D\\phi (x,t)\\cdot n(x) .$"
],
[
"The parabolic case",
"We are ready to state our main convergence result in the time-dependent setting.",
"At first sight, the proof seems to use the monotonicity condition (REF ).",
"The proof relies on the consistency of the numerical scheme, Propositions REF , REF and REF , which are proved in Section .",
"Proposition REF is necessary to deal with the degeneration of the consistency estimates due to the Neumann boundary condition.",
"So we also require that $f(t,x,z,p, \\Gamma )$ satisfy (REF )–(REF ), and that the parameters $\\alpha $ , $\\beta $ , $\\gamma $ satisfy (REF )–(REF ).",
"Proposition 3.4 Suppose $f$ and $\\alpha $ , $\\beta $ , $\\gamma $ satisfy the hypotheses just listed.",
"Assume furthermore that $\\bar{u}$ and $\\underline{v}$ are finite for all $x$ near $x_0$ and all $t\\le T$ near $t_0$ .",
"Then if $t_0<T$ and $x_0\\in \\Omega $ , then $\\bar{u}$ is a viscosity subsolution at $x_0$ and $\\underline{v}$ is a supersolution at $x_0$ (i.e.",
"each one satisfies part REF of the relevant half of Definition REF at $x_0$ ).",
"if $t_0<T$ and $x_0\\in \\partial \\Omega $ , then $\\bar{u}$ is a viscosity subsolution at $x_0$ and $\\underline{v}$ is a supersolution at $x_0$ (i.e.",
"each one satisfies part REF of the relevant half of Definition REF at $x_0$ ).",
"if $t_0=T$ , then $\\bar{u}(x_0,T)=g(x_0)$ and $\\underline{v}(x_0,T)=g(x_0)$ (in particular, each one satisfies part REF of the relevant half of Definition REF at $x_0$ ).",
"In particular, if $\\bar{u}$ and $\\underline{v}$ are finite for all $x\\in \\overline{\\Omega }$ and $t_\\ast <t\\le T$ , then they are respectively a viscosity subsolution and a viscosity supersolution of (REF ) on this time interval.",
"When $x_0\\in \\Omega $ , since we can find in $\\Omega $ a $\\delta $ -neighborhood of $x_0$ , the proof follows from [21].",
"Therefore we shall focus on the case when $x_0\\in \\partial \\Omega $ .",
"We give the proof for $\\overline{u}$ .",
"The argument for $\\underline{v}$ is entirely parallel, relying on Proposition REF .",
"Consider a smooth function $\\phi $ such that $\\bar{u}- \\phi $ has a local maximum at $(x_0,t_0)$ .",
"Adding a constant, we can assume $\\overline{u}(x_0,t_0)=\\phi (x_0,t_0)$ .",
"Replacing $\\phi $ by $\\phi + \\Vert x-x_0 \\Vert ^4 + |t-t_0|^2$ if necessary, we can assume that the local maximum is strict, i.e.",
"that $\\bar{u}(x,t)<\\phi (x,t) \\quad \\text{ for } 0< \\Vert (x,t)-(x_0,t_0) \\Vert \\le r,$ for some $r>0$ .",
"By the definition of $\\bar{u}$ , there exist sequences $\\varepsilon _k$ , $\\tilde{y}_k$ , $\\tilde{t}_k=T-\\tilde{N}_k \\varepsilon ^2_k$ such that $\\tilde{y}_k\\rightarrow x_0, \\quad \\tilde{t}_k\\rightarrow t_0, \\quad u^{\\varepsilon _k}(\\tilde{y}_k,\\tilde{t}_k) \\rightarrow \\bar{u}(x_0,t_0).$ Let $y_k$ and $t_k=T-N_k \\varepsilon ^2_k$ satisfying $(u^{\\varepsilon _k}-\\phi )(y_k,t_k) \\ge \\sup _{\\Vert (x,t)-(x_0,t_0) \\Vert \\le r} (u^{\\varepsilon _k}-\\phi )(x,t) - \\varepsilon _k^3.$ Notice that since $u^{\\varepsilon _k}$ is defined only at discrete times, the sup is taken only over such times.",
"Evidently, $(u^{\\varepsilon _k} - \\phi )(y_k,t_k)\\ge (u^{\\varepsilon _k}-\\phi )(\\tilde{y}_k,\\tilde{t}_k) -\\varepsilon _k^3$ and the right-hand side tends to 0 as $k\\rightarrow +\\infty $ .",
"It follows using (REF ) that $(y_k,t_k) \\rightarrow (y_0,t_0) \\quad \\text{ and } \\quad u^{\\varepsilon _k}(y_k,t_k) \\rightarrow \\bar{u}(x_0,t_0) ,$ as $k\\rightarrow +\\infty $ .",
"Setting $\\xi _k=u^{\\varepsilon _k}(y_k,t_k) -\\phi (y_k,t_k)$ , we also have by construction that $\\xi _k \\rightarrow 0 \\text{ and } u^{\\varepsilon _k}(x,t) \\le \\phi (x,t)+\\xi _k +\\varepsilon _k^3 \\quad \\text{ whenever }t=T-n_k\\varepsilon _k^2 \\text{ and }\\Vert (x,t)-(x_0,t_0) \\Vert \\le r.$ Now we use the dynamic programming inequality (REF ) at $(y_k,t_k)$ , which can be written concisely as $u^{\\varepsilon _k}(y_k,t_k)\\le \\sup _{p,\\Gamma } \\inf _{\\Delta \\hat{x}} \\left\\lbrace u^{\\varepsilon _k}(y_k +\\Delta x, t_k +\\varepsilon _k^2)-\\Delta z \\right\\rbrace ,$ with the convention $\\Delta z= p\\cdot \\Delta \\hat{x} +\\frac{1}{2} \\langle \\Gamma \\Delta \\hat{x}, \\Delta \\hat{x} \\rangle +\\varepsilon _k^2 f(t_k,y_k, u^{\\varepsilon _k}(y_k,t_k), p, \\Gamma ) - \\Vert \\Delta \\hat{x} - \\Delta x \\Vert h(y_k+\\Delta x) .$ Using the definition of $\\xi _k$ , (REF ), and the fact that $\\Delta x$ is bounded by a positive power of $\\varepsilon $ , we conclude that $ \\phi (y_k,t_k) +\\xi _k \\le \\sup _{p,\\Gamma } \\inf _{\\Delta \\hat{x}} \\left\\lbrace \\phi (y_k+\\Delta x, t_k+\\varepsilon ^2_k) +\\xi _k+\\varepsilon ^3_k - \\Delta z \\right\\rbrace ,$ when $k$ is sufficiently large.",
"Dropping $\\xi _k$ from both sides of (REF ), we deduce, by introducing the operator $S_\\varepsilon $ defined by (REF ), that $\\phi (y_k ,t_k) \\le S_\\varepsilon [y_k,t_k,u^{\\varepsilon _k}(y_k,t_k), \\phi (\\cdot , t_k+\\varepsilon ^2_k)] +o(\\varepsilon _k^2).$ According to the consistency estimates provided by Proposition REF , we shall introduce four sets $(A_i)_{1\\le i \\le 4}$ respectively defined by $A_1 & :=\\left\\lbrace k\\in \\mathbb {N}: d(y_k)\\le \\varepsilon _k^{1-\\alpha } \\text{ and } M_{\\varepsilon _k}^{y_k}[\\phi (\\cdot , t_k+\\varepsilon _k^2)]\\ge \\frac{4}{3} \\Vert D^2\\phi (y_k,t_k+\\varepsilon _k^2) \\Vert \\varepsilon _k^{1-\\alpha } \\right\\rbrace , \\\\A_2 & :=\\left\\lbrace k\\in \\mathbb {N}: \\varepsilon _k^{1-\\alpha }- \\varepsilon _k^{\\rho } \\le d(y_k) \\le \\varepsilon _k^{1-\\alpha }\\text{ and } M_{\\varepsilon _k}^{y_k}[\\phi (\\cdot , t_k+\\varepsilon _k^2)]\\le \\frac{4}{3} \\Vert D^2\\phi (y_k,t_k+\\varepsilon _k^2) \\Vert \\varepsilon _k^{1-\\alpha } \\right\\rbrace \\\\& \\phantom{=============}\\bigcup \\bigg \\lbrace k\\in \\mathbb {N}: d(y_k)\\ge \\varepsilon _k^{1-\\alpha } \\bigg \\rbrace , \\\\A_3 & :=\\left\\lbrace k\\in \\mathbb {N}: d(y_k) \\le \\varepsilon _k^{1-\\alpha } - \\varepsilon _k^{ \\rho } \\text{ and }-\\varepsilon _k^{1-\\alpha - \\kappa } \\le M_{\\varepsilon _k}^{y_k}[\\phi (\\cdot , t_k+\\varepsilon _k^2)] \\le \\frac{4}{3} \\Vert D^2\\phi (y_k,t_k+\\varepsilon _k^2) \\Vert \\varepsilon _k^{1-\\alpha } \\right\\rbrace , \\\\A_4 & :=\\left\\lbrace k\\in \\mathbb {N}: d(y_k) \\le \\varepsilon _k^{1-\\alpha } - \\varepsilon _k^{ \\rho } \\text{ and }M_{\\varepsilon _k}^{y_k}[\\phi (\\cdot , t_k+\\varepsilon _k^2)] \\le -\\varepsilon _k^{1-\\alpha - \\kappa } \\right\\rbrace ,$ where $\\rho $ and $\\kappa $ are defined in Section REF by (REF ) and (REF ) and satisfy $0 < \\kappa <1-\\alpha < \\rho <1 $ .",
"Since $\\cup _{1\\le i \\le 4} A_i=\\mathbb {N}$ , at least one set among $A_{1}$ , $A_{2}$ , $A_{3}$ and $A_4$ is necessarily unbounded.",
"We shall consider these four cases.",
"If $A_1$ is unbounded, up to a subsequence, we may assume that $A_1=\\mathbb {N}$ .",
"Taking the limit $k\\rightarrow +\\infty $ , we deduce that $\\displaystyle \\liminf _{k\\rightarrow +\\infty } M_{\\varepsilon _k}^{y_k}[\\phi (\\cdot , t_k+\\varepsilon _k^2)] \\ge 0$ .",
"Since $M_{\\varepsilon _k}^{y_k}[\\phi (\\cdot , t_k+\\varepsilon _k^2)]\\rightarrow h(x_0) - D \\phi (x_0, t_0)\\cdot n(x_0) $ by Lemma REF , it follows in the limit $k\\rightarrow \\infty $ that $D \\phi (x_0, t_0)\\cdot n(x_0) - h(x_0) \\le 0.$ We can notice this result also holds through (REF ).",
"We can apply the second alternative given by (REF ) in Proposition REF to evaluate the right-hand side of (REF ).",
"This gives $\\phi (y_k,t_k) - \\phi (y_k, t_k+\\varepsilon _k^2)\\le 3 \\varepsilon _k^{1-\\alpha } M_{\\varepsilon _k}^{y_k}[\\phi (\\cdot , t_k+\\varepsilon _k^2)] + C \\varepsilon _k^2 (1+|u^{\\varepsilon _k}(y_k,t_k)|)+o(\\varepsilon _k^2),$ where $C$ depends only on $\\Vert h \\Vert _{L^\\infty }$ and $\\Vert D\\phi (\\cdot ,t_k+\\varepsilon _k^2) \\Vert _{C^1_b(\\overline{\\Omega }\\cap B(y_k,\\varepsilon _k^{1-\\alpha }))}$ .",
"Since for $k$ large enough, $\\Vert D\\phi (\\cdot ,t_k+\\varepsilon _k^2) \\Vert _{C^1_b(\\overline{\\Omega }\\cap B(y_k,\\varepsilon _k^{1-\\alpha }))}\\le \\sup _{|t-t_0|\\le r}\\Vert D\\phi (\\cdot ,t) \\Vert _{C^1_b(\\overline{\\Omega }\\cap B(x_0,r))},$ we can suppose that $C$ depends only on $\\Vert h \\Vert _{L^\\infty }$ and this sup, which is finite (since $\\phi $ is smooth) and independent of $k$ .",
"By smoothness of $\\phi $ we have $-\\varepsilon _k^2 \\partial _t \\phi (y_k,t_k)+o(\\varepsilon ^2_k)-C(1+|u^{\\varepsilon _k}(y_k,t_k)|)\\varepsilon _k^2 \\le 3 \\varepsilon _k^{1-\\alpha }M_{\\varepsilon _k}^{y_k}[\\phi (\\cdot , t_k+\\varepsilon _k^2)].$ By dividing by $\\varepsilon _k^{1-\\alpha }$ we obtain $-\\varepsilon _k^{1+\\alpha } \\Big (\\partial _t \\phi (y_k,t_k)-C(1+|u^{\\varepsilon _k}(y_k,t_k)|)\\Big ) + o(\\varepsilon ^{1+\\alpha }_k)\\le 3 M_{\\varepsilon _k}^{y_k}[\\phi (\\cdot ,t_k+\\varepsilon _k^2)].$ The sequences $(u^{\\varepsilon _k}(y_k,t_k))_{k\\in \\mathbb {N}}$ and $(\\partial _t\\phi (y_k,t_k))_{k\\in \\mathbb {N}}$ are respectively bounded by definition of $\\overline{u}(x_0,t_0)$ and smoothness of $\\phi $ .",
"By passing to the limit on $k$ , $\\displaystyle \\liminf _{k\\rightarrow +\\infty } M_{\\varepsilon _k}^{y_k}[\\phi (\\cdot ,t_k+\\varepsilon _k^2)] \\ge 0$ .",
"By Lemma REF , we know that $M_{\\varepsilon _k}^{y_k} [\\phi (\\cdot ,t_k+\\varepsilon _k^2)]\\rightarrow h(x_0) - D \\phi (x_0, t_0)\\cdot n(x_0) $ and (REF ) is retrieved.",
"If $A_{2}$ is unbounded, up to a subsequence, we may assume that $A_2=\\mathbb {N}$ .",
"We can apply Proposition REF case REF to evaluate the right-hand side of (REF ).",
"This gives $\\phi (y_k ,t_k)& \\le \\phi (y_k, t_k+\\varepsilon _k^2) - \\varepsilon _k^2 f(t_k,y_k,u^{\\varepsilon _k}(y_k,t_k), D\\phi (y_k, t_k+\\varepsilon _k^2), D^2\\phi (y_k, t_k+\\varepsilon _k^2)) +o(\\varepsilon ^2_k).$ By smoothness of $\\phi $ and Lipschitz continuity of $f$ with respect to $p$ and $\\Gamma $ , we obtain $\\phi (y_k,t_k) - \\phi (y_k, t_k+\\varepsilon _k^2) \\le - \\varepsilon _k^2 f(t_k,y_k,u^{\\varepsilon _k}(y_k,t_k), D\\phi (y_k, t_k), D^2\\phi (y_k, t_k)) +o(\\varepsilon _k^2).$ It follows in the limit $k\\rightarrow \\infty $ that $\\partial _t\\phi (x_0,t_0) - f(t_0,x_0, \\bar{u} (x_0,t_0), D\\phi (x_0,t_0), D^2\\phi (x_0,t_0)) \\ge 0.$ If $A_3$ is unbounded, up to a subsequence, we may assume that $A_3=\\mathbb {N}$ .",
"By passing to the limit on $k$ , we have that $M_{\\varepsilon _k}^{y_k,t_k+\\varepsilon _k^2}[\\phi ]$ tends to zero when $\\varepsilon _k$ tends to zero.",
"Since $M_{\\varepsilon _k}^{y_k}[\\phi (\\cdot ,t_k+\\varepsilon _k^2)]\\rightarrow h(x_0) - D \\phi (x_0, t_0)\\cdot n(x_0)$ by Lemma REF , it follows in the limit $k\\rightarrow \\infty $ that $D \\phi (x_0, t_0)\\cdot n(x_0) - h(x_0) = 0$ .",
"If $A_{4}$ is unbounded, up to a subsequence, we may assume that $A_4=\\mathbb {N}$ .",
"Hence, taking the limit $k \\rightarrow +\\infty $ , we have $ \\displaystyle \\limsup _{k\\rightarrow +\\infty } M_{\\varepsilon _k}^{y_k}[\\phi (\\cdot ,t_k+\\varepsilon _k^2)] \\le 0.$ Moreover, by applying the fourth alternative given by (REF ) in Proposition REF to evaluate the right-hand side of (REF ), we obtain $\\phi (y_k ,t_k)\\le \\phi (y_k, t_k+\\varepsilon _k^2) + \\dfrac{1}{4} \\varepsilon _k^{ \\rho } M_{\\varepsilon _k}^{y_k} [\\phi (\\cdot ,t_k+\\varepsilon _k^2)] +C \\varepsilon _k^2 (1+|u^{\\varepsilon _k}(y_k,t_k)|)+o(\\varepsilon _k^2),$ where $C$ depends only on $\\Vert h \\Vert _{L^\\infty }$ and $\\displaystyle \\sup _{|t-t_0|\\le r}\\Vert D\\phi (\\cdot ,t) \\Vert _{C^1_b(\\overline{\\Omega }\\cap B(x_0,r))}$ by the same arguments used above for $A_1$ .",
"By smoothness of $\\phi $ we have $-\\varepsilon _k^2 \\partial _t \\phi (y_k,t_k)+o(\\varepsilon ^2_k) - C (1+|u^{\\varepsilon _k}(y_k,t_k)|)\\varepsilon _k^2\\le \\frac{1}{4} \\varepsilon _k^{ \\rho } M_{\\varepsilon _k}^{y_k}[\\phi (\\cdot ,t_k+\\varepsilon _k^2)].$ By dividing by $\\varepsilon _k^{\\rho }$ we get $-\\varepsilon _k^{2-\\rho } \\Big (\\partial _t \\phi (y_k,t_k) - C(1+|u^{\\varepsilon _k}(y_k,t_k)|) \\Big ) + o(\\varepsilon _k^{2- \\rho }) &\\le \\frac{1}{4} M_{\\varepsilon _k}^{y_k}[ \\phi (\\cdot ,t_k+\\varepsilon _k^2)] .$ The sequences $(u^{\\varepsilon _k}(y_k,t_k))_{k \\in \\mathbb {N}}$ and $(\\partial _t\\phi (y_k,t_k))_{k \\in \\mathbb {N}}$ are respectively bounded by definition of $\\overline{u}(x_0,t_0)$ and smoothness of $\\phi $ .",
"By passing to the limit as $k \\rightarrow +\\infty $ , we have $\\liminf _{k\\rightarrow +\\infty } M_{\\varepsilon _k}^{y_k}[\\phi (\\cdot ,t_k+\\varepsilon _k^2)] \\ge 0.$ By comparing this inequality with (REF ) and using Lemma REF , we deduce that $D \\phi (x_0, t_0)\\cdot n(x_0) - h(x_0) = 0.$ Moreover, we can also apply Lemma REF since $\\varepsilon ^{1-\\alpha }_k \\ll \\varepsilon _k^{1-\\alpha -\\kappa }$ .",
"By the same manipulations as those done for the set $A_2$ , the inequality (REF ) holds also true.",
"Thus $\\bar{u}$ is a viscosity subsolution at $(x_0,t_0)$ .",
"We turn now to case REF , i.e.",
"the case $t_0=T$ .",
"If $x_0\\in \\Omega $ , the analysis led in [21] gives the result.",
"It remains to study $\\overline{u}$ on the boundary.",
"We want to show that $\\overline{u}(\\cdot , T)=g$ is also satisfied on $\\partial \\Omega $ .",
"By the definition of $\\overline{u}$ given by (REF ) and considering a particular sequence $(\\varepsilon _k, x_k, t_k=T)_{k \\in \\mathbb {N}}$ which converges to $(0, x_0, T)$ , it is clear that $\\overline{u}(\\cdot , T)\\ge g$ on $\\partial \\Omega $ (using the continuity of $g$ and the fact that each $u^\\varepsilon $ has final value $g$ ).",
"If this sequence realizes the sup, we have in fact the equality.",
"The preceding argument can still be used provided $t_k<T$ for all sufficiently large $k$ .",
"Thus, considering the different possibilities according to $t_k<T$ or $t_k=T$ and also on $x_k\\in \\Omega $ or $x_k\\in \\partial \\Omega $ , we know that for any smooth $\\phi $ such that $\\bar{u}-\\phi $ has a local maximum at $(x_0,T)$ , $ \\text{either } \\bar{u}(x_0,T) = g(x_0) \\text{ or else } \\\\\\max \\left( \\partial _t\\phi (x_0,T) - f(t_0,x_0, \\bar{u}(x_0,T), D\\phi (x_0,T), D^2\\phi (x_0,T)),h(x_0) - D\\phi (x_0, T)\\cdot n(x_0) \\right)\\ge 0.$ Moreover this statement applies not only at the given point $x_0$ , but also at any point nearby.",
"Now consider the functions $\\psi (x,t) =\\bar{u}(x,t) - \\frac{\\Vert x-x_0 \\Vert ^2}{\\eta }- \\frac{T-t}{\\mu } + K d(x)$ and $ \\phi (x,t) = \\frac{\\Vert x-x_0 \\Vert ^2}{\\eta }+ \\frac{T-t}{\\mu } - K d(x),$ where the parameters $\\eta $ , $\\mu $ are small and positive and $K=\\Vert h \\Vert _{L^\\infty }+1$ .",
"Suppose $\\overline{u}$ is uniformly bounded on the closed half-ball $\\lbrace \\Vert (x,t)-(x_0,T) \\Vert \\le r, t\\le T \\rbrace $ and let $\\psi $ attain its maximum on this half-ball at $(x_{\\eta , \\mu }, t_{\\eta ,\\mu })$ .",
"We assume $r$ is small enough such that $d$ is $C^2$ on this half-ball so that $\\phi $ can be taken as a test function.",
"We clearly have $ \\bar{u}(x_{\\eta ,\\mu }, t_{\\eta , \\mu }) + K d(x_{\\eta ,\\mu }) \\ge \\psi (x_{\\eta ,\\mu }, t_{\\eta , \\mu }) \\ge \\psi (x_0,T)=\\overline{u} (x_0,T) .$ By plugging the expression of $\\psi (x_{\\eta ,\\mu }, t_{\\eta , \\mu })$ in the right-hand side of inequality (REF ), we obtain $0\\le \\frac{\\Vert x_{\\eta , \\mu } - x_0 \\Vert ^2}{\\eta }+\\frac{T-t_{\\eta , \\mu }}{\\mu } \\le \\bar{u}(x_{\\eta ,\\mu }, t_{\\eta , \\mu }) - \\overline{u} (x_0,T)+K d(x_{\\eta , \\mu }).$ Since $\\overline{u}$ is bounded on the half-ball and $x_{\\eta ,\\mu }$ belongs to the half ball for all $\\eta $ and $\\mu $ , the right-hand side of (REF ) is bounded independently of $\\eta $ , $\\mu $ , which yields $ (x_{\\eta , \\mu }, t_{\\eta ,\\mu } ) \\rightarrow (x_0,T) \\quad \\text{ as } \\eta , \\mu \\rightarrow 0.$ By using the upper semicontinuity of $\\overline{u}$ and taking the limit (REF ) in (REF ), we get $\\overline{u} (x_{\\eta ,\\mu }, t_{\\eta , \\mu }) \\rightarrow \\overline{u} (x_0,T) \\quad \\text{ as } \\eta , \\mu \\rightarrow 0.$ By combining (REF ) and (REF ), we finally obtain by (REF ) that $\\frac{\\Vert x_{\\eta , \\mu } - x_0 \\Vert ^2}{\\eta }+\\frac{T-t_{\\eta , \\mu }}{\\mu }\\rightarrow 0 \\quad \\text{ as } \\eta , \\mu \\rightarrow 0.$ If $t_{\\eta , \\mu }<T$ and $x_{\\eta , \\mu }\\in \\Omega $ then part REF of Proposition REF applied to $\\phi $ defined by (REF ) assures us that $-\\frac{1}{\\mu } - f(t_{\\eta , \\mu }, x_{\\eta , \\mu }, \\overline{u} (x_{\\eta ,\\mu }, t_{\\eta , \\mu }) ,2 \\frac{x_{\\eta ,\\mu }-x_0}{\\eta }-KDd(x_{\\eta ,\\mu }) , \\frac{2}{\\eta }I -KD^2d(x_{\\eta ,\\mu })) \\ge 0.$ Since $f$ is continuous, for any $\\eta >0$ there exists $\\mu >0$ such that (REF ) cannot happen.",
"Restricting our attention to such choices of $\\eta $ and $\\mu $ , it remains to examine two situations: on the one hand $t_{\\eta , \\mu }<T$ and $x_{\\eta , \\mu }\\in \\partial \\Omega $ and on the other hand $t_{\\eta , \\mu }=T$ .",
"Arguing by contradiction, let us assume that $t_{\\eta , \\mu }<T$ and $x_{\\eta , \\mu }\\in \\partial \\Omega $ .",
"By the Taylor expansion on the distance function close to $x_0$ , we have $d(x)=d(x_0)+Dd(x_0)\\cdot (x-x_0) +O(\\Vert x-x_0 \\Vert ^2).$ By using that $x_0$ and $x_{\\eta , \\mu }$ are on the boundary $ \\partial \\Omega $ , $d(x_0)= d(x_{\\eta , \\mu })=0$ and $Dd(x_0)=-n(x_0)$ , this relation reduces to $n(x_0)\\cdot (x_{\\eta , \\mu }-x_0)= O(\\Vert x_{\\eta , \\mu }-x_0 \\Vert ^2).$ By combining (REF ) and (REF ), we compute $D \\phi (x_{\\eta , \\mu }, t_{\\eta , \\mu })\\cdot n(x_0) & = \\frac{2}{ \\eta } (x_{\\eta , \\mu }-x_0)\\cdot n(x_0)- K Dd(x_{\\eta , \\mu }) \\cdot n(x_0) \\\\& =O\\left(\\dfrac{\\Vert x_{\\eta , \\mu } - x_0 \\Vert ^2}{\\eta }\\right) +K n(x_{\\eta , \\mu }) \\cdot n(x_0) \\rightarrow K, \\text{ as } \\eta , \\mu \\rightarrow 0.$ By smoothness of $\\phi $ and continuity of $n$ on $\\partial \\Omega $ , we deduce that $D \\phi (x_{\\eta , \\mu }, T)\\cdot n(x_{\\eta , \\mu })\\rightarrow \\Vert h \\Vert _{L^\\infty }+1>h(x_{\\eta , \\mu })$ which denies the second alternative proposed at (REF ).",
"As a result, the only remaining possibility for (REF ) is $\\overline{u} (x_{\\eta , \\mu },T)=g(x_{\\eta , \\mu })$ .",
"By continuity of $g$ , it follows in the limit $\\eta , \\mu \\rightarrow 0$ that $\\overline{u} (x_0,T)=g(x_0)$ , as asserted.",
"Remark 3.5 In the proof of the convergence at the final-time in Theorem REF , we needed in a essential way that the domain was assumed to be at least $C^2$ .",
"More precisely, in this case, since the distance function $d$ is $C^2$ in a neighborhood of the boundary, it allows us to take $\\phi $ given by (REF ) as a test function."
],
[
"The elliptic case",
"We turn now to the stationary setting discussed in Section REF .",
"As in the time-dependent setting, our convergence result depends on the fundamental consistency result Proposition REF .",
"So we require that the parameters $\\alpha $ , $\\beta $ , $\\gamma $ satisfy (REF )–(REF ), and that $f(x,z,p,\\Gamma )$ satisfy not only the monotonicity condition (REF ) but also the Lipschitz continuity and growth conditions (REF )–(REF ).",
"To prove that $U^\\varepsilon $ is well defined, we require that the interest rate $\\lambda $ be large enough, condition (REF ), and that $h$ be uniformly bounded.",
"Finally, concerning the parameters $m$ and $M$ and the function $\\psi $ associated to the termination of the game, we assume that $\\psi \\in C^2 (\\overline{\\Omega })$ fulfills $\\frac{\\partial \\psi }{\\partial n}=\\Vert h \\Vert _{L^\\infty }+1$ on $\\partial \\Omega $ , $m=M-1-2\\Vert \\psi \\Vert _{L^\\infty (\\overline{\\Omega })}$ , $\\chi =m+\\Vert \\psi \\Vert _{L^\\infty (\\overline{\\Omega })}+\\psi $ and $M$ is sufficiently large.",
"These hypotheses ensure us the availability of the dynamic programming inequalities stated in Proposition REF .",
"Proposition 3.6 Suppose $f$ , $g$ , $\\lambda $ and $\\alpha $ , $\\beta $ , $\\gamma $ , $m$ , $M$ , $\\psi $ satisfy the hypotheses just listed (from which it follows that $\\underline{v}$ and $\\overline{u}$ are bounded away from $\\pm M$ for all $x\\in \\overline{\\Omega }$ ).",
"Then $\\overline{u}$ is a viscosity subsolution and $\\underline{v}$ is a viscosity supersolution of (REF ) in $ \\overline{\\Omega }$ .",
"More specifically: if $x_0\\in \\Omega $ then each of $\\overline{u}$ and $\\underline{v}$ satisfies part REF of relevant half of Definition REF at $x_0$ , and if $x_0\\in \\partial \\Omega $ then each of $\\overline{u}$ and $\\underline{v}$ satisfies part REF of relevant half of Definition REF at $x_0$ .",
"When $x_0 \\in \\Omega $ , the proof is similar to that of Theorem REF .",
"Therefore we shall focus on the case when $x_0\\in \\partial \\Omega $ .",
"We give the proof for $\\bar{u}$ , the arguments for $\\underline{v}$ being similar and even easier due to fewer cases to distinguish.",
"Consider a smooth function $\\phi $ such that $\\overline{u} - \\phi $ has local maximum on $\\overline{\\Omega }$ at $x_0 \\in \\partial \\Omega $ .",
"We may assume that $\\langle D\\phi (x_0), n(x_0) \\rangle > h(x_0)$ since otherwise the assertion is trivial.",
"Adjusting $\\phi $ if necessary, we can assume that $\\overline{u} (x_0)=\\phi (x_0)$ and that the local maximum is strict, i.e.",
"$ \\overline{u} (x) <\\phi (x) \\quad \\text{ for } x\\in \\overline{\\Omega }\\cap \\lbrace 0<\\Vert x-x_0 \\Vert \\le r \\rbrace ,$ for some $r>0$ .",
"By the definition of $\\overline{u}$ given by (REF ), there exist sequences $\\varepsilon _k>0$ and $\\tilde{y}_k \\in \\overline{\\Omega }$ such that $\\tilde{y}_k \\rightarrow x_0, \\quad u^{\\varepsilon _k}(\\tilde{y}_k) \\rightarrow \\overline{u} (x_0).$ We may choose $y_k \\in \\overline{\\Omega }$ such that $\\displaystyle (u^{\\varepsilon _k} - \\phi ) (y_k) \\ge \\sup _{\\overline{\\Omega }\\cap \\lbrace \\Vert x-x_0 \\Vert \\le r \\rbrace } (u^{\\varepsilon _k} - \\phi ) (x) -\\varepsilon _k^3$ .",
"Evidently $(u^{\\varepsilon _k} - \\phi ) (y_k) \\ge (u^{\\varepsilon _k} - \\phi ) (\\tilde{y}_k) - \\varepsilon _k^3$ and the right-hand side tends to 0 as $k \\rightarrow \\infty $ .",
"It follows using (REF ) that $y_k \\rightarrow x_0 \\quad \\text{ and } \\quad u^{\\varepsilon _k}(y_k) \\rightarrow \\bar{u}(x_0),$ as $k \\rightarrow \\infty $ .",
"Setting $\\xi _k = (u^{\\varepsilon _k} - \\phi )(y_k)$ , we also have by construction that $ \\xi _k \\rightarrow 0 \\quad \\text{ and } \\quad u^{\\varepsilon _k}(x)\\le \\phi (x) +\\xi _k - \\varepsilon ^3_k \\quad \\text{ whenever } x\\in \\overline{\\Omega }\\text{ with } \\Vert x-x_0 \\Vert \\le r.$ We now use the dynamic programming inequality (REF ) at $y_k$ , which can be written concisely as $u^{\\varepsilon _k}(y_k) \\le \\sup _{p, \\Gamma } \\inf _{\\Delta \\hat{x}} \\left\\lbrace e^{-\\lambda \\varepsilon _k^2} u^{\\varepsilon _k}(y_k+\\Delta x) - \\delta _k \\right\\rbrace ,$ with the convention $\\delta _k= p\\cdot \\Delta \\hat{x} +\\frac{1}{2} \\langle \\Gamma \\Delta \\hat{x}, \\Delta \\hat{x}\\rangle +\\varepsilon ^2_k f(x, u^{\\varepsilon _k}(x), p, \\Gamma ) - \\Vert \\Delta \\hat{x} - \\Delta x \\Vert h(x+\\Delta x).$ By the rule (REF ) of the game, for every move $\\Delta \\hat{x}$ decided by Mark, the point $y_k+\\Delta x$ belongs to $\\overline{\\Omega }$ .",
"Combining this observation with (REF ) and the definition of $\\xi _k$ we conclude that $\\phi (y_k)+\\xi _k \\le \\sup _{p, \\Gamma } \\inf _{\\Delta \\hat{x}} \\left\\lbrace e^{-\\lambda \\varepsilon ^2_k} \\left[ \\phi (y_k+\\Delta x) +\\xi _k - \\varepsilon ^3_k \\right] - \\delta _k \\right\\rbrace .$ We may replace $e^{-\\lambda \\varepsilon _k^2}$ by $1-\\lambda \\varepsilon _k^2$ and $e^{-\\lambda \\varepsilon _k^2} \\xi _k$ by $\\xi _k$ while making an error which is only $o(\\varepsilon ^2)$ using the fact that $\\xi _k \\rightarrow 0$ .",
"Dropping $\\xi _k$ from both sides, we conclude that $\\phi (y_k)\\le \\sup _{p, \\Gamma } \\inf _{\\Delta \\hat{x}} \\left( e^{-\\lambda \\varepsilon _k^2} \\phi (y_k+\\Delta x) - \\delta _k \\right) +o(\\varepsilon _k^2).$ We can evaluate the right-hand side using Proposition REF case REF for $k$ large enough.",
"We introduce $\\rho $ and $\\kappa $ defined in Section REF by (REF ) and (REF ) and satisfying $0 < \\kappa <1-\\alpha < \\rho <1 $ .",
"If we may assume, up to a subsequence, that for all $k$ large enough, on the one hand $d(y_k)\\ge \\varepsilon _k^{1-\\alpha }$ or on the one hand $\\varepsilon _k^{1-\\alpha } - \\varepsilon _k^{\\rho } \\le d(y_k)\\le \\varepsilon _k^{1-\\alpha }$ and $ M_{\\varepsilon _k}^{y_k}[\\phi ] \\le \\frac{4}{3} \\Vert D^2\\phi (y_k) \\Vert \\varepsilon _k^{1-\\alpha }$ , we can apply Proposition REF case REF to evaluate the right-hand side $0\\le - \\varepsilon _k^2 f(y_k, u^{\\varepsilon _k}(y_k), D\\phi (y_k), D^2\\phi (y_k)) - \\varepsilon _k^2\\lambda u^{\\varepsilon _k}(y_k) +o(\\varepsilon _k^2).$ By passing to the limit $k\\rightarrow +\\infty $ , we get the required inequality in the viscosity sense.",
"Otherwise, recall that $\\langle D\\phi (x_0), n(x_0) \\rangle > h(x_0)$ .",
"By Lemma REF , we have $M_{\\varepsilon _k}^{y_k}[\\phi ] \\rightarrow h(x_0) - \\langle D\\phi (x_0), n(x_0) \\rangle < 0 ,$ and the condition $ M_{\\varepsilon _k}^{y_k}[\\phi ] \\le - \\varepsilon _k^{1-\\alpha -\\kappa }$ is satisfied for all $k$ sufficiently large.",
"Therefore, up to a subsequence, it remains to consider a sequence $(y_k, \\varepsilon _k)_{k\\in \\mathbb {N}}$ satisfying both $d(y_k) \\le \\varepsilon _k^{1-\\alpha } - \\varepsilon _k^{\\rho }$ and (REF ).",
"The last part of Proposition REF can be applied and we get by (REF ) that there exists a constant $C$ depending only on $M$ , $\\Vert D\\phi \\Vert _{C_b^1(\\overline{\\Omega })\\cap B(y_k,\\varepsilon _{k}^{1-\\alpha })}$ and $\\Vert h \\Vert _{L^\\infty }$ such that $0 \\le \\frac{1}{4} \\varepsilon _k^{\\rho } M_{\\varepsilon _k}^{y_k}[\\phi ] +C \\varepsilon _k^2 - \\lambda \\varepsilon _k^2 \\phi (y_k) +o(\\varepsilon _k^2),$ recalling that $\\left( \\varepsilon ^{1-\\alpha } - d(y_k) \\right)\\ge \\varepsilon _k^{\\rho }$ and $M_{\\varepsilon _k}^{y_k}[\\phi ] <0$ .",
"By dividing by $\\varepsilon _k^\\rho $ , it follows that $- \\varepsilon _k^{2-\\rho } \\left(C - \\lambda \\phi (y_k)\\right) +o(\\varepsilon _k^{2-\\rho }) \\le \\frac{1}{4} M_{\\varepsilon _k}^{y_k}[\\phi ].$ The sequence $(\\phi (y_k))_{k\\in \\mathbb {N}}$ is bounded by smoothness of $\\phi $ .",
"Since $\\Vert D\\phi \\Vert _{C_b^1(\\overline{\\Omega })\\cap B(y_k,\\varepsilon _{k}^{1-\\alpha })} \\le \\Vert D\\phi \\Vert _{C_b^1(\\overline{\\Omega })\\cap B(x_0,r)}$ holds for $k$ large enough, we can assume that $C$ is independent of $k$ depending only on $\\Vert D\\phi \\Vert _{C_b^1(\\overline{\\Omega })\\cap B(x_0,r)}$ , $M$ and $\\Vert h \\Vert _{L^\\infty }$ .",
"Taking the limit as $k \\rightarrow +\\infty $ , we deduce that $\\liminf _{k \\rightarrow \\infty }M_{\\varepsilon _k}^{y_k}[\\phi ] \\ge 0,$ which is a contradiction with (REF ).",
"Thus $\\overline{u} $ is a viscosity subsolution at $x_0$ ."
],
[
"Consistency",
"A numerical scheme is said to be consistent if every smooth solution of the PDE satisfies it modulo an error that tends to zero with the step size.",
"It is the idea of the argument used in [21].",
"In our case, we must understand how the Neumann condition interferes with the estimates proposed in [21].",
"The essence of our formal argument in Section REF was that the Neumann condition term is predominant compared to the PDE term at the boundary and produces a degeneracy in the consistency estimate.",
"The present section clarifies the connection between our formal argument and the consistency of the game, by discussing consistency in more conventional terms.",
"The main point is presented in Propositions REF and REF .",
"In order to explain very precisely how the consistency estimate of [21] degenerates, we establish the consistency of our game as a numerical scheme by focusing on different cases according to the values of the quantities $m^{x}_\\varepsilon [\\phi ]$ and $M^{x}_\\varepsilon [\\phi ]$ defined by (REF )–() and the distance $d(x)$ to the boundary $\\partial \\Omega $ ."
],
[
"The parabolic case",
"Consider the game discussed in Section REF for solving $-u_t +f(t,x,u,Du,D^2u)=0$ in $ \\Omega $ with final-time data $u(x,T)=g(x)$ for $x\\in \\overline{\\Omega }$ and boundary condition $\\frac{ \\partial u}{\\partial n}(x,t)=h(x) $ for $x\\in \\partial \\Omega , t\\in (0,T)$ .",
"The dynamic programming principles (REF )–(REF ) can be written as $u^{\\varepsilon }(x,t) & \\le S_{\\varepsilon } \\left[x,t,u^{\\varepsilon }(x,t), u^{\\varepsilon }(\\cdot ,t+\\varepsilon ^2) \\right], \\\\v^{\\varepsilon }(x,t) & \\ge S_{\\varepsilon } \\left[x,t,v^{\\varepsilon }(x,t), v^{\\varepsilon }(\\cdot , t+\\varepsilon ^2) \\right],$ where $S_{\\varepsilon } \\left[x,t,z, \\phi \\right]$ is defined for any $x\\in \\overline{\\Omega }$ , $z\\in \\mathbb {R}$ and $t\\le T$ and any continuous function $\\phi $ : $\\overline{\\Omega }\\rightarrow \\mathbb {R}$ by $S_{\\varepsilon } \\left[x,t,z, \\phi \\right] = \\max _{p,\\Gamma } \\min _{\\Delta \\hat{x}} \\left[\\phi \\left( x +\\Delta x \\right) \\right.",
"\\\\\\left.",
"- \\left( p \\cdot \\Delta \\hat{x} +\\frac{1}{2} \\left\\langle \\Gamma \\Delta \\hat{x}, \\Delta \\hat{x} \\right\\rangle +\\varepsilon ^2 f \\left(t,x,z,p,\\Gamma \\right) - \\Vert \\Delta \\hat{x} -\\Delta x \\Vert h(x+\\Delta x) \\right) \\right],$ subject to the usual constraints $\\Vert p \\Vert \\le \\varepsilon ^{-\\beta }$ , $\\Vert \\Gamma \\Vert \\le \\varepsilon ^{-\\gamma }$ , $\\Vert \\Delta \\hat{x} \\Vert \\le \\varepsilon ^{1-\\alpha }$ and $\\Delta x=\\operatorname{proj}_{\\overline{\\Omega }} (x+\\Delta \\hat{x}) - x$ .",
"The operator $S_\\varepsilon $ clearly satisfies the three following properties: For all $\\phi \\in C(\\overline{\\Omega })$ , $S_{0} \\left[x,t,z, \\phi \\right]=\\phi (x)$ .",
"$S_{\\varepsilon }$ is monotone, i.e.",
"if $\\phi _1 \\le \\phi _2$ , then $S_{\\varepsilon } \\left[x,t,z, \\phi _1 \\right]\\le S_{\\varepsilon } \\left[x,t,z, \\phi _2 \\right]$ .",
"For all $\\phi \\in C(\\overline{\\Omega })$ and $c\\in \\mathbb {R}$ , $S_{\\varepsilon } \\left[x,t,z, c + \\phi \\right]=c + S_{\\varepsilon }\\left[x,t,z, \\phi \\right] .$ Fixing $x,t,z$ and a smooth function $\\phi $ , a Taylor expansion shows that for any $\\Vert \\Delta x \\Vert \\le \\varepsilon ^{1-\\alpha }$ , $\\phi (x+\\Delta x)=\\phi (x)+ D\\phi (x)\\cdot \\Delta x +\\frac{1}{2} \\langle D^2\\phi (x) \\Delta x, \\Delta x \\rangle +O(\\varepsilon ^{3-3 \\alpha }).$ Since $\\alpha <1/3$ by hypothesis, $\\varepsilon ^{3-3\\alpha }=o(\\varepsilon ^{2})$ .",
"By rearranging the terms, we compute $\\phi \\left( x +\\Delta x \\right) - \\left( p \\cdot \\Delta \\hat{x}+\\frac{1}{2} \\left\\langle \\Gamma \\Delta \\hat{x}, \\Delta \\hat{x} \\right\\rangle +\\varepsilon ^2 f \\left(t,x, z, p,\\Gamma \\right)- \\Vert \\Delta \\hat{x} - \\Delta x \\Vert h(x+\\Delta x ) \\right) \\\\= \\phi (x) + (D\\phi (x) - p)\\cdot \\Delta \\hat{x} + \\Vert \\Delta \\hat{x} - \\Delta x \\Vert \\left[ h(x+\\Delta x ) - D\\phi (x) \\cdot n(x+\\Delta x) \\right] \\\\+\\frac{1}{2} \\left\\langle D^2\\phi (x) \\Delta x, \\Delta x \\right\\rangle - \\frac{1}{2} \\left\\langle \\Gamma \\Delta \\hat{x}, \\Delta \\hat{x} \\right\\rangle - \\varepsilon ^2 f(t,x, z, p ,\\Gamma ) +o(\\varepsilon ^2),$ since the outward normal can be expressed by $\\displaystyle n(x+\\Delta x)= - \\frac{\\Delta x - \\Delta \\hat{x}}{\\Vert \\Delta \\hat{x} - \\Delta x \\Vert }$ if $x+\\Delta \\hat{x} \\notin \\Omega $ and the move $\\Delta x$ can be decomposed as $\\Delta x=\\Delta \\hat{x} + ( \\Delta x - \\Delta \\hat{x} )$ .",
"Thus, we shall examine $S_{\\varepsilon } \\left[x,t,z, \\phi \\right] - \\phi (x)=\\max _{p,\\Gamma } \\min _{\\Delta \\hat{x}}\\left[ (D\\phi (x) - p)\\cdot \\Delta \\hat{x} +\\frac{1}{2} \\left\\langle D^2\\phi (x) \\Delta x, \\Delta x \\right\\rangle \\right.",
"\\\\\\left.",
"+ \\Vert \\Delta \\hat{x} - \\Delta x \\Vert \\lbrace h(x+\\Delta x ) - D\\phi (x) \\cdot n(x+\\Delta x) \\rbrace - \\frac{1}{2} \\left\\langle \\Gamma \\Delta \\hat{x}, \\Delta \\hat{x} \\right\\rangle - \\varepsilon ^2 f( t,x, z, p,\\Gamma )\\right] +o(\\varepsilon ^2).$"
],
[
"Preliminary geometric lemmas",
"This subsection is devoted to some geometric properties of the game that will be useful to show consistency in Section REF .",
"We start by some estimates, involving the geometric conditions on the domain, about the moves $\\Delta \\hat{x}$ decided by Mark.",
"Lemma 4.1 Suppose that $\\Omega $ is a $C^2$ -domain satisfying the uniform exterior ball condition for a certain $r>0$ .",
"Then, for all $0<\\varepsilon <r^{\\frac{1}{1-\\alpha }}$ and for all $\\Delta \\hat{x}$ constrained by (REF ), determining $\\Delta x$ by (REF ), we have $\\Vert \\Delta \\hat{x} - \\Delta x \\Vert \\le \\varepsilon ^{1-\\alpha } - d(x) \\quad \\text{ and } \\quad \\Vert \\Delta x \\Vert \\le 2 \\varepsilon ^{1-\\alpha } - d(x).$ Let us prove the first inequality, the second following immediately by the triangle inequality.",
"If the point $\\hat{x}= x+\\Delta \\hat{x}$ belongs to $ \\overline{\\Omega }$ , $\\Delta x= \\Delta \\hat{x}$ and the result is obvious.",
"Supposing now $\\hat{x}$ does not belong to $ \\overline{\\Omega }$ , the set $S=[x,\\hat{x}] \\cap \\partial \\Omega $ is not empty and we can consider a point $x_{\\partial } \\in S$ .",
"By the rule of the game, we have $\\Vert x-\\hat{x} \\Vert = \\Vert \\Delta \\hat{x} \\Vert \\le \\varepsilon ^{1-\\alpha }$ .",
"Since $x_{\\partial }\\in \\partial \\Omega $ by construction, it is clear that $\\Vert x - x_\\partial \\Vert \\ge d(x)$ .",
"We deduce that $\\Vert x_\\partial - \\hat{x} \\Vert = \\Vert x - \\hat{x} \\Vert - \\Vert x_\\partial - x \\Vert \\le \\varepsilon ^{1-\\alpha } - d(x) .$ By the uniform exterior ball condition, the orthogonal projection on $\\overline{\\Omega }$ is well-defined on $\\Omega (\\varepsilon ^{1-\\alpha }) \\subset \\Omega (r)$ .",
"By property of the orthogonal projection and since $\\hat{x}\\notin \\overline{\\Omega }$ , we can write $\\Vert \\Delta \\hat{x} - \\Delta x \\Vert =\\inf _{y\\in \\overline{\\Omega }} \\Vert y-\\hat{x} \\Vert =\\inf _{y\\in \\partial \\Omega } \\Vert y-\\hat{x} \\Vert \\le \\Vert x_\\partial - \\hat{x} \\Vert ,$ which gives directly the first estimate in (REF ).",
"The following lemma uses the crucial geometric fact that $\\Omega $ satisfies the interior ball condition introduced in Definition REF for which there is no neck pitching for $\\varepsilon $ sufficiently small.",
"Lemma 4.2 Let $\\sigma > 1-\\alpha $ and $B>0$ .",
"Suppose that $\\Omega $ is a domain with $C^2$ -boundary $\\partial \\Omega $ and satisfies the uniform interior ball condition.",
"Then, for all possible moves $\\Vert \\Delta \\hat{x} \\Vert \\le \\varepsilon ^{1-\\alpha }$ such that $\\Vert \\Delta \\hat{x} + \\varepsilon ^{1-\\alpha } n(\\bar{x}) \\Vert \\le B\\varepsilon ^\\sigma $ the intermediate point $\\hat{x}$ belongs to $\\Omega $ for all $\\varepsilon $ sufficiently small.",
"Moreover, for all possible moves $\\Vert \\Delta \\hat{x} \\Vert \\le \\varepsilon ^{1-\\alpha }$ such that $\\Vert \\Delta \\hat{x} - \\varepsilon ^{1-\\alpha } n(\\bar{x}) \\Vert \\le B\\varepsilon ^\\sigma $ and $\\Delta x$ determined by (REF ), we have $\\Vert \\Delta \\hat{x} - \\Delta x \\Vert \\ge \\varepsilon ^{1-\\alpha } - d(x)- B\\varepsilon ^\\sigma + O(\\varepsilon ^{2-2\\alpha }) .$ Furthermore, if in addition we assume $d(x)\\ge \\varepsilon ^{1-\\alpha } - \\varepsilon ^\\eta $ with $1-\\alpha < \\eta <\\sigma $ , the intermediate point $\\hat{x}$ is outside $\\Omega $ for all $\\varepsilon $ sufficiently small.",
"For the first assertion, since $\\Omega $ satisfies the uniform interior ball condition (there is no neck pitching for $\\varepsilon $ sufficiently small), we observe that the set $\\partial \\Omega \\cap B(x,2\\varepsilon ^{1-\\alpha })$ is below a paraboloid $P_1$ of opening $A$ and above a paraboloid $P_2$ of opening $ - A$ touching $\\partial \\Omega $ at $\\bar{x}$ .",
"By the Taylor expansion, if $T_{\\bar{x}} \\partial \\Omega $ denotes the tangent space to $\\partial \\Omega $ at $\\bar{x}$ , we get that for all $y\\in \\partial \\Omega \\cap B(x, 2\\varepsilon ^{1-\\alpha })$ , $|(y-\\bar{x}) \\cdot n(\\bar{x})|=d(y, T_{\\bar{x}} \\partial \\Omega ) \\le \\frac{1}{2}A (2\\varepsilon ^{1-\\alpha })^2,$ Since $(x+\\Delta \\hat{x}-\\bar{x}) \\cdot n(\\bar{x}) \\le - \\varepsilon ^{1-\\alpha } - d(x) + B\\varepsilon ^{\\sigma }$ , we deduce that for all $\\varepsilon $ sufficiently small, $(x+\\Delta \\hat{x}-\\bar{x}) \\cdot n(\\bar{x})< \\inf _{y\\in \\partial \\Omega \\cap B(x, 2\\varepsilon ^{1-\\alpha })} (y-\\bar{x}) \\cdot n(\\bar{x}),$ which yields that $x+\\Delta \\hat{x}$ belongs to $\\Omega $ .",
"For the second claim, we denote by $(\\kappa _i(x))_{1\\le i\\le N-1}$ the principal curvatures at $x$ on $\\partial \\Omega $ and by $(e_1, \\cdots , e_N)$ an orthonormal frame centered in $\\bar{x}$ with first vector $e_1=n(\\bar{x})$ .",
"Since $\\Omega $ is a $C^2$ -domain, $(e_2, \\cdots , e_N)$ form a basis of the tangent space $T_{\\bar{x}} \\partial \\Omega $ .",
"We compute $\\varepsilon ^{1-\\alpha } - B\\varepsilon ^{\\sigma } \\le \\Delta \\hat{x} \\cdot n(\\bar{x}) = (\\Delta \\hat{x} -\\varepsilon ^{1-\\alpha }n(\\bar{x})) \\cdot n(\\bar{x}) + \\varepsilon ^{1-\\alpha }.$ Thus $\\hat{x}$ is contained in the half-space $H_1$ determined by $(y-\\bar{x})\\cdot e_1 \\ge \\varepsilon ^{1-\\alpha } - d(x) - B\\varepsilon ^{\\sigma }$ and $d(\\hat{x} , T_{\\overline{x}} \\partial \\Omega ) \\ge \\varepsilon ^{1-\\alpha } - d(x) - B\\varepsilon ^{\\sigma }$ .",
"Moreover, we deduce from (REF ) and the triangle inequality that for each move $\\Delta \\hat{x}$ we have $x+\\Delta x \\in B(\\bar{x}, 2\\varepsilon ^{1-\\alpha })$ .",
"Assume $x_1= p(x_2,\\cdots , x_N)$ is a local $C^2$ -parametrization of $\\partial \\Omega $ around $x$ .",
"By a Taylor argument and by continuity of the principal curvatures on $\\partial \\Omega $ , it follows that, for $\\varepsilon >0$ small enough, $d(x+\\Delta x, T_{\\bar{x}} \\partial \\Omega ) \\le \\frac{1}{2}C_1 (2\\varepsilon ^{1-\\alpha })^2 =2 C_1 \\varepsilon ^{2-2\\alpha } ,$ where $\\displaystyle C_1:=2 \\max \\left\\lbrace | \\kappa _i(\\overline{x})| : 1\\le i \\le N-1 \\right\\rbrace $ .",
"By the triangle inequality, we deduce that $\\Vert x+\\Delta x-\\hat{x} \\Vert & \\ge \\Vert \\operatorname{proj}_{T_{\\bar{x}} \\partial \\Omega }( x+\\Delta x) -\\hat{x} \\Vert - \\Vert x+\\Delta x - \\operatorname{proj}_{T_{\\bar{x}} \\partial \\Omega }( x+\\Delta x) \\Vert \\\\& \\ge d(\\hat{x} , T_{\\overline{x}} \\partial \\Omega ) - d(x+\\Delta x, T_{\\overline{x}} \\partial \\Omega ) \\\\& \\ge \\varepsilon ^{1-\\alpha } - d(x) - B\\varepsilon ^{\\sigma } - 2 C_1 \\varepsilon ^{2-2\\alpha }.$ In particular, if $d(x)\\ge \\varepsilon ^{1-\\alpha } - \\varepsilon ^\\eta $ with $1-\\alpha < \\eta <\\sigma $ the right-hand side is strictly positive for $\\varepsilon $ sufficiently small and $\\hat{x}\\notin \\Omega $ .",
"The next lemmas gather some estimates which will be useful to establish our consistency estimates.",
"Lemma 4.3 Under the hypothesis of Lemma REF , for all moves $\\Delta \\hat{x}$ constrained by (REF ), determining $\\Delta x$ by (REF ), we have $-\\frac{1}{2} (\\varepsilon ^{1-\\alpha } - d(x ))\\le - \\frac{1}{2} \\left( 1-\\frac{ d(x)}{\\varepsilon ^{1-\\alpha }}\\right) (\\Delta \\hat{x})\\cdot n(\\bar{x})+ \\Vert \\Delta \\hat{x} - \\Delta x \\Vert \\le \\frac{3}{2}(\\varepsilon ^{1-\\alpha } - d(x)).$ The left-hand side of (REF ) can be written in the form $- \\frac{1}{2} \\left( 1-\\frac{ d(x)}{\\varepsilon ^{1-\\alpha }}\\right) (\\Delta \\hat{x})\\cdot n(\\bar{x})+ \\Vert \\Delta \\hat{x} - \\Delta x \\Vert =(\\varepsilon ^{1-\\alpha } - d(x)) \\left[-\\frac{1}{2}\\frac{(\\Delta \\hat{x})\\cdot n(\\bar{x})}{\\varepsilon ^{1-\\alpha }}+ \\frac{\\Vert \\Delta \\hat{x} - \\Delta x \\Vert }{\\varepsilon ^{1-\\alpha } - d(x)}\\right],$ which directly gives the desired estimates by using (REF ) and the first inequality given by (REF ).",
"Lemma 4.4 Let $A\\in \\mathcal {M}^N(\\mathbb {R})$ , $k\\in C_b(\\partial \\Omega )$ extended by some function $k: \\overline{\\Omega }\\rightarrow \\mathbb {R}$ , and $x\\in \\overline{\\Omega }$ .",
"Suppose in addition that $(3\\varepsilon ^{1-\\alpha }-d(x)) \\Vert A \\Vert \\le \\inf _{\\begin{array}{c}x+\\Delta \\hat{x} \\notin \\Omega \\\\ \\Delta \\hat{x}\\end{array}} k(x+\\Delta x),$ with $\\Delta \\hat{x}$ constrained by (REF ) and $\\Delta x$ determined by (REF ).",
"Then $ \\min _{\\Delta \\hat{x}} \\left\\lbrace \\left\\langle A \\Delta x, \\Delta x \\right\\rangle - \\left\\langle A \\Delta \\hat{x}, \\Delta \\hat{x} \\right\\rangle + \\Vert \\Delta \\hat{x} - \\Delta x \\Vert k(x+\\Delta x) \\right\\rbrace = 0,$ where $\\Delta \\hat{x}$ is constrained by (REF ) and determines $\\Delta x$ by (REF ).",
"If $\\hat{x} =x+\\Delta \\hat{x} \\in \\overline{\\Omega }$ , the function is equal to zero.",
"We now consider the moves for which $\\hat{x} \\notin \\overline{\\Omega }$ .",
"Then $ \\left\\langle A \\Delta x, \\Delta x \\right\\rangle - \\left\\langle A \\Delta \\hat{x}, \\Delta \\hat{x} \\right\\rangle = \\left\\langle A (\\Delta \\hat{x} - \\Delta x), \\Delta \\hat{x} - \\Delta x \\right\\rangle + 2 \\left\\langle A \\Delta \\hat{x}, \\Delta \\hat{x} - \\Delta x \\right\\rangle .$ By the Cauchy-Schwarz inequality, we obtain $| \\left\\langle A \\Delta x, \\Delta x \\right\\rangle - \\left\\langle A \\Delta \\hat{x} , \\Delta \\hat{x} \\right\\rangle |\\le \\Vert A \\Vert \\Vert \\Delta \\hat{x} - \\Delta x \\Vert \\left( \\Vert \\Delta \\hat{x} - \\Delta x \\Vert + 2 \\Vert \\Delta \\hat{x} \\Vert \\right).",
"$ By using (REF ) and $\\Vert \\Delta \\hat{x} \\Vert \\le \\varepsilon ^{1-\\alpha }$ , we get $ | \\left\\langle A \\Delta x, \\Delta x \\right\\rangle - \\left\\langle A \\Delta \\hat{x} , \\Delta \\hat{x} \\right\\rangle |\\le \\Vert A \\Vert \\Vert \\Delta \\hat{x} - \\Delta x \\Vert \\left( 3 \\varepsilon ^{1-\\alpha } - d(x) \\right).$ Thus $\\left\\langle A \\Delta x, \\Delta x \\right\\rangle - \\left\\langle A \\Delta \\hat{x}, \\Delta \\hat{x} \\right\\rangle + \\Vert \\Delta \\hat{x} - \\Delta x \\Vert k(x+\\Delta x)\\ge \\Vert \\Delta \\hat{x} - \\Delta x \\Vert \\left\\lbrace \\inf _{\\begin{array}{c}x+\\Delta \\hat{x} \\notin \\Omega \\\\ \\Delta \\hat{x}\\end{array}} k(x+\\Delta x) -\\Vert A \\Vert (3\\varepsilon ^{1-\\alpha } - d(x)) \\right\\rbrace .$ The right-hand side of this last inequality is strictly positive by the assumption (REF )."
],
[
"Consistency estimates",
"In this subsection we state our consistency estimates.",
"They explain precisely the conditions under which the usual estimate proposed in [21] holds for $x$ near the boundary and $\\phi \\in C^2 (\\overline{\\Omega })$ .",
"If it does not hold, there is a degeneration of the estimates respecting the final discussion of formal derivation of the PDE at Section REF .",
"For fixed $x\\in \\Omega (\\varepsilon ^{1-\\alpha })$ , these estimates take into account the size and the sign of the boundary condition in the small ball $B(x, \\varepsilon ^{1-\\alpha })$ and the distance $d(x)$ to the boundary.",
"In the heuristic derivation presented in Section REF , we assumed that $\\Delta \\hat{x}\\mapsto h(x+\\Delta x) - D\\phi (x) \\cdot n(x+\\Delta x)$ , with $\\Delta x$ determined by (REF ), was locally constant in a $\\delta $ -neighborhood of the boundary near $x$ .",
"In the general case, this hypothesis must be relaxed.",
"To do this, we observe that, for all $\\Delta \\hat{x}$ constrained by (REF ) satisfying $x+\\Delta \\hat{x} \\notin \\Omega $ and determining $\\Delta x$ by (REF ), $m_\\varepsilon ^x[\\phi ] \\le h(x+\\Delta x) - D\\phi (x) \\cdot n(x+\\Delta x) \\le M_\\varepsilon ^x[\\phi ],$ where $m_\\varepsilon ^{x}[\\phi ]$ and $M_\\varepsilon ^{x}[\\phi ]$ are defined by (REF )–().",
"Therefore we are going to specify some strategies for Helen which are associated to the two extreme situations $m_\\varepsilon ^{x}[\\phi ]$ and $M_\\varepsilon ^{x}[\\phi ]$ by following the optimal choices (REF ) and (REF ) obtained in the formal derivation at Section REF .",
"More precisely, for all $x\\in \\Omega (\\varepsilon ^{1-\\alpha })$ , we define the strategies $p_\\text{opt}^m(x)$ , $p_\\text{opt}^M(x)$ and $\\Gamma _\\text{opt}(x)$ in an orthonormal basis $\\mathcal {B}=(e_1=n(\\bar{x}), e_2,\\cdots , e_N)$ respectively by $p_\\text{opt}^m(x) & = D\\phi (x) +\\left[\\frac{1}{2}\\left(1-\\frac{ d(x)}{\\varepsilon ^{1-\\alpha }}\\right) m_\\varepsilon ^{x}[\\phi ]- \\frac{\\varepsilon ^{1-\\alpha }}{4} \\left(1 - \\frac{d^2(x)}{\\varepsilon ^{2-2\\alpha }} \\right) (D^2\\phi (x))_{11} \\right] n(\\bar{x}), \\\\p_\\text{opt}^M (x) & = D\\phi (x) +\\left[\\frac{1}{2}\\left(1-\\frac{ d(x)}{\\varepsilon ^{1-\\alpha }}\\right) M_\\varepsilon ^{x}[\\phi ]- \\frac{\\varepsilon ^{1-\\alpha }}{4} \\left( 1 - \\frac{d^2(x)}{\\varepsilon ^{2- 2\\alpha }} \\right) (D^2\\phi (x))_{11} \\right] n(\\bar{x}), $ and $\\Gamma _\\text{opt}(x) = D^2\\phi (x) +\\left[ \\frac{1}{2}\\left( - 1+\\frac{d^2(x)}{\\varepsilon ^{2-2\\alpha }} \\right) (D^2\\phi (x))_{11} \\right] E_{11},$ where $E_{11}$ denotes the unit-matrix $(1,1)$ in the basis $\\mathcal {B}$ .",
"These strategies depend on the local behavior of $\\phi $ (size and amplitude) around the boundary and on the geometry of the boundary itself.",
"Since there is a degeneration of the usual estimates, there is no hope for one simple estimate.",
"We are going to separate the study in two steps: Proposition REF provides the estimates for the lower bound and Proposition REF deals with the upper bound.",
"Moreover, Section REF is devoted to the technical proof of the upper bound distinguishing several cases according to the size of $M_\\varepsilon ^{x}[\\phi ]$ and $d(x)$ .",
"Proposition 4.5 Let $f$ satisfy (REF ) and (REF )–(REF ) and assume $\\alpha $ , $\\beta $ , $\\gamma $ satisfy (REF )–(REF ).",
"Let $p_\\text{opt}^m$ and $\\Gamma _\\text{opt}$ be respectively defined in the orthonormal basis $(e_1=n(\\bar{x}), e_2,\\cdots , e_N)$ by (REF ) and (REF ).",
"For any $x$ , $t$ , $z$ and any smooth function $\\phi $ defined near $x$ , $S_\\varepsilon [x,t,z,\\phi ]$ being defined by (REF ), we distinguish two cases: Big bonus: if $ d(x)\\ge \\varepsilon ^{1-\\alpha }$ or $m_\\varepsilon ^{x}[\\phi ] >\\frac{1}{2} (3\\varepsilon ^{1-\\alpha }-d(x)) \\Vert D^2\\phi (x) \\Vert $ , then $S_\\varepsilon [x,t,z, \\phi ] - \\phi (x) \\ge - \\varepsilon ^2 f(t,x,z, D\\phi (x), D^2\\phi (x)).$ Penalty or small bonus: if $d(x)\\le \\varepsilon ^{1-\\alpha }$ and $ m_\\varepsilon ^{x}[\\phi ] \\le \\frac{1}{2} (3\\varepsilon ^{1-\\alpha }-d(x)) \\Vert D^2\\phi (x) \\Vert $ , then $S_\\varepsilon [x,t,z,\\phi ] - \\phi (x) \\ge \\frac{1}{2} (\\varepsilon ^{1-\\alpha } - d(x)) \\left(s m_\\varepsilon ^{x}[\\phi ]- 4 \\Vert D^2 \\phi (x) \\Vert \\varepsilon ^{1-\\alpha } \\right)- \\varepsilon ^2 f(t,x,z, p_{\\text{opt}}^m(x), \\Gamma _\\text{opt}(x)) ,$ where $s=-1$ if $m_\\varepsilon ^x[\\phi ]\\ge 0$ and $s=3$ if $m_\\varepsilon ^x[\\phi ]<0$ .",
"If $d(x) \\ge \\varepsilon ^{1-\\alpha }$ , the usual estimate [21] holds.",
"We now focus on the case $d(x) \\le \\varepsilon ^{1-\\alpha }$ .",
"By the definition of $m_\\varepsilon ^{x}[\\phi ]$ given by (REF ) and the positivity of $\\Vert \\Delta \\hat{x} - \\Delta x \\Vert $ , for all $\\Vert \\Delta \\hat{x} \\Vert \\le \\varepsilon ^{1-\\alpha }$ , we have $\\Vert \\Delta \\hat{x} - \\Delta x \\Vert \\left\\lbrace h(x+\\Delta x) - D\\phi (x) \\cdot n(x+\\Delta x) \\right\\rbrace \\ge \\Vert \\Delta \\hat{x} - \\Delta x \\Vert m_\\varepsilon ^{x}[\\phi ].$ Therefore it is sufficient to find a lower bound for $\\max _{p, \\Gamma } \\min _{\\Delta \\hat{x}} \\left[(D\\phi (x) - p)\\cdot \\Delta \\hat{x} +m_\\varepsilon ^{x}[\\phi ] \\Vert \\Delta \\hat{x} - \\Delta x \\Vert + \\frac{1}{2} \\langle D^2\\phi (x) \\Delta x ,\\Delta x\\rangle - \\frac{1}{2} \\langle \\Gamma \\Delta \\hat{x},\\Delta \\hat{x}\\rangle \\\\- \\varepsilon ^2 f(t,x,z,p,\\Gamma ) \\right].$ where $p$ , $\\Gamma $ and $\\Delta \\hat{x}$ are constrained by (REF )–(REF ) and $\\Delta x$ determined by (REF ).",
"In other words, by taking advantage of the monotonicity of the operator $S_\\varepsilon $ with (REF ), we shall look for a lower bound for an approximated operator bounding $S_\\varepsilon $ from below and very close to it when $\\varepsilon \\rightarrow 0$ .",
"Then, we also observe that for every choice $p$ and $\\Gamma $ , $S_\\varepsilon [x,t, z, \\phi ] - \\phi (x) \\ge - \\varepsilon ^2 f \\left( t,x, z, p,\\Gamma \\right)\\\\+ \\min _{\\Delta \\hat{x}} \\left[(D\\phi (x) - p)\\cdot \\Delta \\hat{x} +\\frac{1}{2} \\left\\langle D^2\\phi (x) \\Delta x, \\Delta x \\right\\rangle - \\frac{1}{2} \\left\\langle \\Gamma \\Delta \\hat{x}, \\Delta \\hat{x} \\right\\rangle + \\Vert \\Delta \\hat{x} - \\Delta x \\Vert m_\\varepsilon ^{x}[\\phi ] \\right].$ We now distinguish two particular strategies for Helen.",
"For part REF , we consider the particular choice $p=D\\phi (x)$ , $\\Gamma =D^2\\phi (x)$ and obtain $S_\\varepsilon [x,t, z, \\phi ] - \\phi (x) & \\ge - \\varepsilon ^2 f \\left(t,x, z, D\\phi (x),D^2\\phi (x) \\right) \\\\& \\phantom{\\ge } + \\min _{\\Delta \\hat{x}}\\left[ \\frac{1}{2} ( \\left\\langle D^2\\phi (x)\\Delta x, \\Delta x \\right\\rangle - \\frac{1}{2}\\left\\langle D^2\\phi (x)\\Delta \\hat{x}, \\Delta \\hat{x} \\right\\rangle + \\Vert \\Delta \\hat{x} - \\Delta x \\Vert m_\\varepsilon ^{x}[\\phi ]\\right] \\\\& \\ge - \\varepsilon ^2 f \\left(t,x, z, D\\phi (x),D^2\\phi (x) \\right),$ by applying Lemma REF with $A=\\frac{1}{2}D^2\\phi (x)$ .",
"For part REF , we consider the choice $p=p_{\\text{opt}}^m(x)$ , $\\Gamma = \\Gamma _{\\text{opt}}(x)$ and find $S_\\varepsilon [x,t,z, \\phi ] &- \\phi (x) \\ge - \\varepsilon ^2 f(t,x, z, p_{\\text{opt}}^m(x), \\Gamma _{\\text{opt}}(x)) +l^{x}[\\phi ],$ with $l^{x}[\\phi ]$ defined by $l^{x} [\\phi ] = \\min _{\\Delta \\hat{x}} \\left[(D\\phi (x) - p_{\\text{opt}}^m) \\cdot (\\Delta \\hat{x})+ \\frac{1}{2} \\left\\langle D^2\\phi (x) \\Delta x, \\Delta x \\right\\rangle - \\frac{1}{2} \\left\\langle \\Gamma _{\\text{opt}}(x) \\Delta \\hat{x}, \\Delta \\hat{x} \\right\\rangle + \\Vert \\Delta \\hat{x} - \\Delta x \\Vert m_\\varepsilon ^{x}[\\phi ] \\right].$ It now remains to give a lower bound for $l^{x} [\\phi ]$ .",
"By plugging the expression (REF ) of $p_{\\text{opt}}^m(x)$ in (REF ), we have $l^{x} [\\phi ]= \\min _{\\Delta \\hat{x}} \\left[ \\left( - \\frac{\\varepsilon ^{1-\\alpha } - d(x) }{2\\varepsilon ^{1-\\alpha }}(\\Delta \\hat{x})_1 + \\Vert \\Delta \\hat{x} - \\Delta x \\Vert \\right) m_\\varepsilon ^{x}[\\phi ] \\right.",
"\\\\\\left.+ \\frac{1}{2} \\left\\langle D^2\\phi (x) \\Delta x, \\Delta x \\right\\rangle - \\frac{1}{2} \\left\\langle \\Gamma _\\text{opt}(x) \\Delta \\hat{x}, \\Delta \\hat{x} \\right\\rangle + \\frac{1}{4} \\left( \\varepsilon ^{1-\\alpha }- \\frac{d^2(x)}{\\varepsilon ^{1-\\alpha }} \\right) (D^2\\phi (x))_{11} (\\Delta \\hat{x})_1 \\right].$ It is clear that $ l^{x}[\\phi ] \\ge l_1^{x}[\\phi ]+ l_2^{x}[\\phi ]$ with $l_1^{x}[\\phi ]$ and $l_2^{x}[\\phi ]$ respectively defined by $ l_1^{x}[\\phi ] := \\min _{\\Delta \\hat{x}} \\left[\\left( - \\frac{\\varepsilon ^{1-\\alpha } - d(x) }{2\\varepsilon ^{1-\\alpha }} (\\Delta \\hat{x})_1 + \\Vert \\Delta \\hat{x} - \\Delta x \\Vert \\right) m_\\varepsilon ^{x}[\\phi ]\\right],$ and $l_2^{x}[\\phi ] : = \\frac{1}{2}\\min _{\\Delta \\hat{x}} \\left[ \\left\\langle D^2\\phi (x) \\Delta x, \\Delta x \\right\\rangle - \\left\\langle \\Gamma _\\text{opt}(x) \\Delta \\hat{x}, \\Delta \\hat{x} \\right\\rangle + \\frac{\\varepsilon ^{1-\\alpha }}{2} \\left( 1- \\frac{d^2(x)}{\\varepsilon ^{2- 2 \\alpha }} \\right) (D^2\\phi (x))_{11} (\\Delta \\hat{x})_1 \\right].$ By using Lemmas REF and REF stated below, giving lower bounds respectively for $l_1^{x}[\\phi ]$ and $l_2^{x}[\\phi ]$ , one obtains $l^{x}[\\phi ]& \\ge \\frac{s}{2}(\\varepsilon ^{1-\\alpha } - d(x)) m_\\varepsilon ^{x}[\\phi ]- 2 \\Vert D^2\\phi (x) \\Vert \\varepsilon ^{1-\\alpha } \\left( \\varepsilon ^{1-\\alpha }-d(x) \\right)= \\frac{1}{2}(\\varepsilon ^{1-\\alpha } - d(x)) \\left( s m_\\varepsilon ^{x}[\\phi ] - 4 \\Vert D^2\\phi (x) \\Vert \\varepsilon ^{1-\\alpha } \\right),$ which gives the desired estimate.",
"The three following lemmas provide the required estimates for $l_1^{x}[\\phi ]$ and $l_2^{x}[\\phi ]$ .",
"Lemma 4.6 For any $x \\in \\Omega (\\varepsilon ^{1-\\alpha })$ and any function $\\phi $ defined at $x$ , $l_1^{x}[\\phi ]$ being defined by (REF ), we have $\\frac{s}{2}(\\varepsilon ^{1-\\alpha } - d(x)) m_\\varepsilon ^{x}[\\phi ] \\le l_1^{x}[\\phi ] \\le 0,$ with $s= -1$ if $m_\\varepsilon ^{x}[\\phi ]$ is positive and $s= 3$ if $m_\\varepsilon ^{x}[\\phi ]$ is nonpositive.",
"By considering $\\Delta \\hat{x} = 0$ , $ l_1^{x}[\\phi ]$ is negative.",
"To find a lower bound on $ l_1^{x}[\\phi ]$ , if $m_\\varepsilon ^{x}[\\phi ]$ is negative, we may write $\\left[ - \\frac{\\varepsilon ^{1-\\alpha } - d(x) }{2 \\varepsilon ^{1-\\alpha }} (\\Delta \\hat{x})_1+ \\Vert \\Delta \\hat{x} - \\Delta x \\Vert \\right] m_\\varepsilon ^{x}[\\phi ] \\ge \\frac{3}{2}(\\varepsilon ^{1-\\alpha } - d(x)) m_\\varepsilon ^{x}[\\phi ],$ the last inequality being provided by the right-hand side inequality given in Lemma REF since by hypothesis $m_\\varepsilon ^{x}[\\phi ]$ is negative.",
"If $m_\\varepsilon ^{x}[\\phi ]$ is nonnegative, the result follows from applying the left-hand side inequality given in Lemma REF .",
"Lemma 4.7 Let $x\\in \\Omega (\\varepsilon ^{1-\\alpha })$ and $\\phi \\in C^2(\\overline{\\Omega })$ .",
"For all $\\Delta \\hat{x}$ constrained by (REF ), we have $ \\left| \\frac{1}{2} \\langle D^2\\phi (x) \\Delta x, \\Delta x \\rangle - \\frac{1}{2} \\langle D^2\\phi (x) \\Delta \\hat{x} ,\\Delta \\hat{x}\\rangle \\right|\\le \\frac{1}{2}\\Vert D^2\\phi (x) \\Vert \\left(3 \\varepsilon ^{1-\\alpha }-d(x)\\right) \\Vert \\Delta \\hat{x} -\\Delta x \\Vert ,$ and $\\left| \\frac{1}{2} \\langle D^2\\phi (x) \\Delta x,\\Delta x \\rangle - \\frac{1}{2} \\langle \\Gamma _\\text{opt}(x) \\Delta \\hat{x} ,\\Delta \\hat{x}\\rangle \\right|\\le \\frac{1}{4}\\Vert D^2\\phi (x) \\Vert \\left(\\varepsilon ^{1-\\alpha } - d(x)\\right) \\left(7 \\varepsilon ^{1-\\alpha } - d(x) \\right),$ where $\\Gamma _\\text{opt}(x)$ is the optimal choice defined by (REF ) in an orthonormal basis $\\mathcal {B}=(e_1=n(\\bar{x}), \\cdots , e_N)$ .",
"The first inequality is an immediate consequence of (REF ).",
"For the second inequality, all the coordinates $ \\langle (D^2\\phi (x)-\\Gamma _\\text{opt}(x)) e_i,e_j \\rangle $ in the basis $\\mathcal {B}$ are equal to zero, except for $i=j=1$ .",
"By using the vector decomposition given by (REF ), we have $\\frac{1}{2} \\langle D^2\\phi (x) \\Delta x , \\Delta x \\rangle - \\frac{1}{2} \\langle \\Gamma _\\text{opt}(x) \\Delta \\hat{x} ,\\Delta \\hat{x}\\rangle =\\frac{1}{2} (D^2\\phi (x)-\\Gamma _\\text{opt}(x))_{11} |(\\Delta \\hat{x})_1|^2\\\\+\\frac{1}{2} \\Vert \\Delta \\hat{x} - \\Delta x \\Vert ^2\\langle (D^2\\phi (x) n(x+\\Delta x), n(x+\\Delta x) \\rangle - \\Vert \\Delta \\hat{x} - \\Delta x \\Vert \\langle D^2\\phi (x) n(x+\\Delta x), \\Delta \\hat{x} \\rangle .$ Since $\\displaystyle (D^2\\phi (x)-\\Gamma _\\text{opt}(x))_{11}=\\frac{1}{2}\\Big (1-\\frac{d^2(x)}{\\varepsilon ^{2-2\\alpha }} \\Big ) (D^2\\phi (x))_{11}$ by (REF ), one obtains $\\left| \\frac{1}{2} \\langle D^2\\phi (x) \\Delta x , \\Delta x \\rangle -\\frac{1}{2} \\langle \\Gamma _\\text{opt}(x)\\Delta \\hat{x} ,\\Delta \\hat{x}\\rangle \\right| \\\\\\le \\Vert D^2\\phi (x) \\Vert \\left\\lbrace \\frac{1}{4} \\left(1-\\frac{d^2(x)}{\\varepsilon ^{2-2\\alpha }} \\right) |(\\Delta \\hat{x})_1|^2+\\frac{1}{2} \\Vert \\Delta \\hat{x} - \\Delta x \\Vert ^2 + \\Vert \\Delta \\hat{x} - \\Delta x \\Vert \\Vert \\Delta \\hat{x} \\Vert \\right\\rbrace .$ The estimate (REF ) now follows from (REF ) and (REF ).",
"Lemma 4.8 For any $x \\in \\Omega (\\varepsilon ^{1-\\alpha })$ and any function $\\phi $ defined at $x$ , $l_2^{x}[\\phi ]$ being defined by (REF ), we have $- 2 \\Vert D^2\\phi (x) \\Vert \\varepsilon ^{1-\\alpha } \\left( \\varepsilon ^{1-\\alpha }-d(x) \\right) \\le l_2^{x}[\\phi ] \\le 0.$ By considering $\\Delta \\hat{x} = 0$ , $l_2$ is negative.",
"We seek now to find a lower bound on $l_2$ .",
"By combining the triangle inequality with Lemma REF , the explicit expression of $\\Gamma _{\\text{opt}}(x)$ given by (REF ) and $\\Vert \\Delta \\hat{x} \\Vert \\le \\varepsilon ^{1-\\alpha }$ , we deduce that $\\frac{1}{2} \\Big | \\langle D^2\\phi (x) \\Delta x, \\Delta x \\rangle - & \\left\\langle \\Gamma _{\\text{opt}}(x) \\Delta \\hat{x}, \\Delta \\hat{x} \\right\\rangle + \\frac{1}{2} \\left( \\varepsilon ^{1-\\alpha }- \\frac{d^2(x))}{\\varepsilon ^{1-\\alpha }} \\right) (D^2\\phi (x))_{11} (\\Delta \\hat{x})_1 \\Big | \\\\& \\le \\frac{1}{4}\\Vert D^2\\phi (x) \\Vert (\\varepsilon ^{1-\\alpha } - d(x)) \\left(7 \\varepsilon ^{1-\\alpha } - d(x) \\right)+ \\frac{1}{4} \\Vert D^2\\phi (x) \\Vert \\left(\\varepsilon ^{1-\\alpha }-\\frac{d^2(x)}{\\varepsilon ^{1-\\alpha }} \\right) \\varepsilon ^{1-\\alpha } \\\\& \\le 2\\Vert D^2\\phi (x) \\Vert (\\varepsilon ^{1-\\alpha } - d(x)) \\varepsilon ^{1-\\alpha }, $ which is precisely the proposed estimate.",
"We shall now provide the consistency estimates about the upper bound of (REF ).",
"Before stating our main estimate in Proposition REF , we can give a simple case for which the usual estimate holds.",
"Lemma 4.9 Let $f$ satisfy (REF ) and (REF )–(REF ) and assume $\\alpha $ , $\\beta $ , $\\gamma $ satisfy (REF )–(REF ).",
"For any $x$ , $t$ , $z$ and any smooth function $\\phi $ defined near $x$ , $ S_\\varepsilon [x,t,z,\\phi ]$ being defined by (REF ), if $d(x)\\le \\varepsilon ^{1-\\alpha }$ and $M_\\varepsilon ^{x}[\\phi ] \\le - \\frac{1}{2} \\Vert D^2\\phi (x) \\Vert \\left(3\\varepsilon ^{1-\\alpha } - d(x) \\right)$ , then we have $S_\\varepsilon [x,t,z, \\phi ] - \\phi (x) \\le -\\varepsilon ^2 f(t,x,z, D\\phi (x), D^2\\phi (x)) +o(\\varepsilon ^2).$ Moreover, the implicit constant in the error term is uniform as $x$ , $t$ and $z$ range over a compact subset of $\\overline{\\Omega }\\times \\mathbb {R}\\times \\mathbb {R}$ .",
"In the rest of the section, we now accurately focus on the case $d(x)\\le \\varepsilon ^{1-\\alpha }$ .",
"The goal is to obtain precise estimates on (REF ) in the following three cases: $M_\\varepsilon ^{x}[\\phi ]$ very negative, $M_\\varepsilon ^{x}[\\phi ]$ very positive and $M_\\varepsilon ^{x}[\\phi ]$ close to zero, the bounds between the cases depending on some powers of $\\varepsilon $ .",
"We have formally shown in Section REF that the first case is favorable to Mark since Helen can undergo a big penalty if Mark chooses to cross the boundary.",
"On the contrary, the second case is preferable to Helen because she can receive a big coupon if the boundary is crossed.",
"In the last case, the boundary is transparent (think of $M_\\varepsilon ^{x}[\\phi ]=0$ ) and the penalization due to the boundary is to be considered only through second order terms.",
"In order to establish the precise estimates, we successively introduce two additional parameters $\\rho , \\kappa >0$ such that $ 1-\\alpha <\\rho < \\min \\left(1 -\\frac{\\gamma (r-1)}{2} , 2- 2 \\alpha - \\gamma \\right),$ and $ \\gamma +\\rho - (1-\\alpha ) < \\kappa < 1-\\alpha .$ These coefficients are well-defined by virtue of (REF ) and (REF ).",
"Proposition 4.10 Let $f$ satisfy (REF ) and (REF )–(REF ) and assume $\\alpha $ , $\\beta $ , $\\gamma $ , $\\rho $ , $\\kappa $ satisfy (REF )–(REF ) and (REF )–(REF ).",
"Let $p_\\text{opt}^M$ and $\\Gamma _\\text{opt}$ be respectively defined in the orthonormal basis $(e_1=n(\\bar{x}), e_2,\\cdots , e_N)$ by () and (REF ).",
"For any $x$ , $t$ , $z$ and any smooth function $\\phi $ defined near $x$ , $S_\\varepsilon [x,t,z,\\phi ]$ being defined by (REF ), we distinguish four cases: Big bonus: If $d(x) \\le \\varepsilon ^{1-\\alpha }$ and $M_\\varepsilon ^{x}[\\phi ] >\\frac{4}{3}\\Vert D^2\\phi (x) \\Vert \\varepsilon ^{1-\\alpha } $ , then $S_\\varepsilon [x,t,z, \\phi ] - \\phi (x) \\le 3 (\\varepsilon ^{1-\\alpha }-d(x)) M_\\varepsilon ^{x}[\\phi ] - \\varepsilon ^2 f(t,x,z,p_\\text{opt}^M(x), \\Gamma _\\text{opt}(x)) +o(\\varepsilon ^2).$ Far from the boundary with a small bonus: if $\\varepsilon ^{1-\\alpha } - \\varepsilon ^{\\rho } \\le d(x) \\le \\varepsilon ^{1-\\alpha }$ and $M_\\varepsilon ^{x}[\\phi ] \\le \\frac{4}{3} \\Vert D^2\\phi (x) \\Vert \\varepsilon ^{1-\\alpha }$ , or if $d(x)\\ge \\varepsilon ^{1-\\alpha }$ , then $S_\\varepsilon [x,t,z, \\phi ] - \\phi (x) \\le -\\varepsilon ^2 f(t,x,z,D\\phi (x), D^2\\phi (x)) +o(\\varepsilon ^2) .$ Close to the boundary with a small bonus/penalty: if $d(x) \\le \\varepsilon ^{1-\\alpha } - \\varepsilon ^{\\rho }$ and $-\\varepsilon ^{1-\\alpha -\\kappa }\\le M_\\varepsilon ^{x}[\\phi ]\\le \\frac{4}{3}\\Vert D^2\\phi (x) \\Vert \\varepsilon ^{1-\\alpha }$ , then $S_\\varepsilon [x,t,z, \\phi ] - \\phi (x) \\le -\\varepsilon ^2 f(t,x,z,D\\phi (x), D^2\\phi (x)+C_1 I) +o(\\varepsilon ^2),$ with $C_1=\\frac{20}{3} \\Vert D^2\\phi (x) \\Vert \\left(1 - \\frac{d(x)}{\\varepsilon ^{1-\\alpha }}\\right)$ .",
"Close to the boundary with a big penalty: if $d(x) \\le \\varepsilon ^{1-\\alpha } - \\varepsilon ^{\\rho }$ and $M_\\varepsilon ^{x}[\\phi ]\\le - \\varepsilon ^{1-\\alpha - \\kappa }$ , then $ S_\\varepsilon [x,t,z, \\phi ] - \\phi (x)\\le \\frac{1}{4} (\\varepsilon ^{1-\\alpha } - d(x)) M_\\varepsilon ^{x}[\\phi ]- \\varepsilon ^2 \\min _{p\\in B(p_\\text{opt}^M(x), r)} f(t,x,z,p, \\Gamma _\\text{opt}(x)) +o(\\varepsilon ^2),$ with $r$ defined by $r= 3\\Big (1-\\frac{d(x)}{\\varepsilon ^{1-\\alpha }}\\Big ) |M_\\varepsilon ^{x}[\\phi ]|$ .",
"Moreover, the implicit constants in the error term is uniform as $x$ , $t$ and $z$ range over a compact subset of $\\overline{\\Omega }\\times \\mathbb {R}\\times \\mathbb {R}$ .",
"Before proving these estimates, it is worth drawing a parallel with the formal derivation done at Section REF .",
"The lower bound proposed by Proposition REF case REF corresponds to the formal analysis when $m>0$ .",
"The upper bound proposed by Proposition REF case REF is associated to the formal analysis when $m<0$ .",
"Furthermore, we can observe in the proof that the factor $1/4$ in (REF ) could be replaced by any number in $[1/4,1/2)$ , the bound $1/2$ corresponding to the heuristic derivation given by (REF )."
],
[
"Proof of Lemma ",
"For sake of notational simplicity, we write $\\lambda _{\\text{min}}(A)$ for the smallest eigenvalue of the symmetric matrix $A$ and we omit the $x$ -dependence of $p_\\text{opt}^M(x)$ and $\\Gamma _\\text{opt}(x)$ .",
"Moreover, by the definition of $M_\\varepsilon ^{x}[\\phi ]$ given by () and the positivity of $\\Vert \\Delta \\hat{x} - \\Delta x \\Vert $ , for all $\\Vert \\Delta \\hat{x} \\Vert \\le \\varepsilon ^{1-\\alpha }$ , we have $\\Vert \\Delta \\hat{x} - \\Delta x \\Vert \\left\\lbrace h(x+\\Delta x) - D\\phi (x) \\cdot n(x+\\Delta x) \\right\\rbrace \\le \\Vert \\Delta \\hat{x} - \\Delta x \\Vert M_\\varepsilon ^{x}[\\phi ].$ Therefore it is sufficient to find an upper bound for $\\max _{p, \\Gamma } \\min _{\\Delta \\hat{x}} \\left[(D\\phi (x) - p)\\cdot \\Delta \\hat{x} +M_\\varepsilon ^{x}[\\phi ] \\Vert \\Delta \\hat{x} - \\Delta x \\Vert + \\frac{1}{2} \\langle D^2\\phi (x) \\Delta x ,\\Delta x\\rangle - \\frac{1}{2} \\langle \\Gamma \\Delta \\hat{x},\\Delta \\hat{x}\\rangle \\\\- \\varepsilon ^2 f(t,x,z,p,\\Gamma ) \\right].$ In other words, by taking advantage of the monotonicity of the operator $S_\\varepsilon $ with (REF ), we shall look for an upper bound for an approximated operator bounding $S_\\varepsilon $ above and very close to it as $\\varepsilon \\rightarrow 0$ ."
],
[
"Proof of Lemma ",
"We introduce $\\mathcal {A}^ x(p,\\Gamma ,\\Delta \\hat{x}) : = (D\\phi ( x) - p)\\cdot \\Delta \\hat{x} + \\Vert \\Delta \\hat{x} - \\Delta x \\Vert M_\\varepsilon ^ x[\\phi ]+\\frac{1}{2} \\langle D^2\\phi ( x) \\Delta x , \\Delta x \\rangle -\\frac{1}{2} \\langle \\Gamma \\Delta \\hat{x} ,\\Delta \\hat{x}\\rangle - \\varepsilon ^2 f(t,x,z,p,\\Gamma ),$ where $\\Delta x=\\operatorname{proj}_{\\overline{\\Omega }}(x+\\Delta \\hat{x}) - x$ .",
"We give the following useful decomposition: $\\frac{1}{2} \\langle D^2\\phi ( x) \\Delta x , \\Delta x\\rangle - \\frac{1}{2} \\langle \\Gamma \\Delta \\hat{x} , \\Delta \\hat{x} \\rangle = \\frac{1}{2} \\langle D^2\\phi ( x) \\Delta x , \\Delta x\\rangle - \\frac{1}{2} \\langle D^2\\phi (x) \\Delta \\hat{x} , \\Delta \\hat{x} \\rangle + \\frac{1}{2} \\langle (D^2\\phi (x) - \\Gamma ) \\Delta \\hat{x} , \\Delta \\hat{x} \\rangle ,$ which will be used repeatedly in this section.",
"We clearly have by (REF ) that $\\Vert \\Delta \\hat{x} - \\Delta x \\Vert M_\\varepsilon ^x[\\phi ] +\\frac{1}{2} \\langle D^2\\phi (x) \\Delta x , \\Delta x \\rangle - \\frac{1}{2} \\langle D^2\\phi (x) \\Delta \\hat{x} , \\Delta \\hat{x} \\rangle \\\\\\le \\Vert \\Delta \\hat{x} - \\Delta x \\Vert \\left( M_\\varepsilon ^{x}[\\phi ] +\\frac{1}{2} \\Vert D^2\\phi (x) \\Vert \\Big (3\\varepsilon ^{1-\\alpha } - d(x) \\Big ) \\right)\\le 0.$ From the previous inequality and (REF ) we deduce that for all $p,\\Gamma ,\\Delta \\hat{x}$ constrained by (REF )–(REF ), $\\mathcal {A}^x(p,\\Gamma ,\\Delta \\hat{x}) & \\le (D\\phi (x) - p)\\cdot \\Delta \\hat{x} +\\frac{1}{2} \\langle (D^2\\phi (x) - \\Gamma ) \\Delta \\hat{x} , \\Delta \\hat{x} \\rangle - \\varepsilon ^2 f(t,x,z,p, \\Gamma ).$ By monotonicity of the operator $ S_\\varepsilon $ and by using [21] to estimate the max min, we conclude that $S_\\varepsilon [x,t,z,\\phi ] - \\phi (x) & \\le \\max _{p, \\Gamma } \\min _{\\Delta \\hat{x}} \\Big [(D\\phi (x) - p)\\cdot \\Delta \\hat{x} +\\frac{1}{2} \\langle (D^2\\phi (x) - \\Gamma ) \\Delta \\hat{x} , \\Delta \\hat{x} \\rangle - \\varepsilon ^2 f(t,x,z,p, \\Gamma )\\Big ] \\\\& \\le - \\varepsilon ^2 f(t,x,z, D\\phi (x), D^2\\phi (x)) +o(\\varepsilon ^2),$ which gives the desired result."
],
[
"Proof of Proposition ",
"We define the function $\\mathcal {A}_b^x$ of $\\Delta \\hat{x}$ associated to the particular choice $p = p_\\text{opt}^M$ and $ \\Gamma = \\Gamma _\\text{opt}$ by $\\mathcal {A}^x_b(\\Delta \\hat{x}) = (D\\phi (x) - p_\\text{opt}^M)\\cdot \\Delta \\hat{x} + \\Vert \\Delta \\hat{x} - \\Delta x \\Vert M_\\varepsilon ^{x}[\\phi ]+\\frac{1}{2} \\langle D^2\\phi (x) \\Delta x , \\Delta x \\rangle - \\frac{1}{2}\\langle \\Gamma _\\text{opt} \\Delta \\hat{x} ,\\Delta \\hat{x}\\rangle ,$ where $\\Delta x=\\operatorname{proj}_{\\overline{\\Omega }}(x+\\Delta \\hat{x}) - x$ .",
"Thus, the operator $S_\\varepsilon $ can be written in the form $ S_{\\varepsilon } [x,t,z,\\phi ] - \\phi (x)=\\max _{p,\\Gamma } \\min _{\\Delta \\hat{x}} \\left[ \\mathcal {A}^x_b(\\Delta \\hat{x}) + (p_\\text{opt}^M - p)\\cdot \\Delta \\hat{x}+\\frac{1}{2} \\langle (\\Gamma _\\text{opt}- \\Gamma ) \\Delta \\hat{x} ,\\Delta \\hat{x}\\rangle - \\varepsilon ^2 f(t,x,z,p,\\Gamma )\\right].$ To compute an upper bound of (REF ), we now introduce two preliminary lemmas.",
"Lemma 4.11 Assume that $M_\\varepsilon ^{x}[\\phi ]\\ge 0$ .",
"Then $\\mathcal {A}^{x}_b$ defined by (REF ) is $\\Delta \\hat{x}$ -bounded by $ 0\\le \\sup _{\\Delta \\hat{x}}\\mathcal {A}^{x}_b(\\Delta \\hat{x}) \\le \\frac{1}{2}(\\varepsilon ^{1-\\alpha }- d(x)) \\left( 3 M_\\varepsilon ^{x}[\\phi ]+ 4 \\Vert D^2\\phi (x) \\Vert \\varepsilon ^{1-\\alpha } \\right),$ where $\\Delta \\hat{x}$ is constrained by (REF ).",
"This estimate follows exactly the same lines as for Lemmas REF –REF .",
"The sup is clearly positive by considering $\\Delta \\hat{x} =0$ .",
"Then, by plugging the expression of $p^M_\\text{opt}$ in $\\mathcal {A}_b ( \\Delta \\hat{x})$ , we have $\\mathcal {A}^{x}_b ( \\Delta \\hat{x})= \\Big \\lbrace -\\frac{\\varepsilon ^{1-\\alpha }- d(x)}{2 \\varepsilon ^{1-\\alpha } }(\\Delta \\hat{x})_1 + \\Vert \\Delta \\hat{x} - \\Delta x \\Vert \\Big \\rbrace M_\\varepsilon ^{x}[\\phi ] \\\\+\\frac{1}{4} \\Big (\\varepsilon ^{1-\\alpha } - \\frac{d^2(x)}{\\varepsilon ^{1-\\alpha }}\\Big ) (D^2\\phi (x))_{11} (\\Delta \\hat{x})_1+\\frac{1}{2} \\langle D^2\\phi (x) \\Delta x , \\Delta x \\rangle -\\frac{1}{2} \\langle \\Gamma _\\text{opt} \\Delta \\hat{x} ,\\Delta \\hat{x}\\rangle .$ Since $M_\\varepsilon ^{x}[\\phi ]\\ge 0$ , using the estimates (REF ) and (REF ), we obtained the desired estimate.",
"Lemma 4.12 Let $f$ satisfy (REF ) and (REF )–(REF ) and assume $\\alpha $ , $\\beta $ , $\\gamma $ satisfy (REF )- (REF ).",
"Let $(p_\\varepsilon )_{0<\\varepsilon \\le 1}$ and $(\\Gamma _\\varepsilon )_{0<\\varepsilon \\le 1}$ be two sequences bounded respectively in $\\mathbb {R}^N$ and $\\mathcal {S}^N$ .",
"Then for any $x$ , $t$ and $z$ , we have $\\max _{\\begin{array}{c}\\Vert p \\Vert \\le \\varepsilon ^{-\\beta }\\\\ \\Vert \\Gamma \\Vert \\le \\varepsilon ^{-\\gamma }\\end{array}} \\min _{\\Vert \\Delta \\hat{x} \\Vert \\le \\varepsilon ^{1-\\alpha }}\\left[ (p_\\varepsilon - p)\\cdot \\Delta \\hat{x} +\\frac{1}{2} \\langle (\\Gamma _\\varepsilon - \\Gamma ) \\Delta \\hat{x} ,\\Delta \\hat{x}\\rangle - \\varepsilon ^2 f \\left( t,x, z, p,\\Gamma \\right)\\right] \\\\= -\\varepsilon ^2 f(t,x,z,p_\\varepsilon , \\Gamma _\\varepsilon ) +o(\\varepsilon ^2) .$ Moreover, the implicit constant in the error term is uniform as $x$ , $t$ , and $z$ range over a compact subset of $\\overline{\\Omega }\\times \\mathbb {R}\\times \\mathbb {R}$ .",
"It is a direct adaptation of [21] by distinguishing three cases according to the size of $\\Vert p_\\varepsilon - p \\Vert $ and $\\lambda _\\text{min}(\\Gamma _\\varepsilon - \\Gamma )$ .",
"We can now provide an upper bound on (REF ).",
"By Lemma REF , $\\mathcal {A}_b$ is upper bounded independently of all possible moves $\\Delta \\hat{x}$ .",
"It follows from (REF ) that $S_{\\varepsilon } [x,t,z,\\phi ] - \\phi (x) \\le \\sup _{\\Delta \\hat{x}}\\mathcal {A}^x_b(\\Delta \\hat{x})+ \\max _{p,\\Gamma } \\min _{\\Delta \\hat{x}}\\left[ (p_\\text{opt}^M - p)\\cdot \\Delta \\hat{x} -\\frac{1}{2} \\langle \\Gamma _\\text{opt} \\Delta \\hat{x} ,\\Delta \\hat{x}\\rangle - \\varepsilon ^2 f \\left( t, x, z, p,\\Gamma \\right) \\right] .$ The consistency Lemma REF provides an estimate of the max min and one obtains $S_{\\varepsilon }[x,t,z,\\phi ]& -\\phi (x)\\le \\sup _{\\Delta \\hat{x}}\\mathcal {A}^x_b(\\Delta \\hat{x}) -\\varepsilon ^2 f(t,x,z,p_\\text{opt}^M,\\Gamma _\\text{opt})+o(\\varepsilon ^2).$ By plugging the upper bound in (REF ) of $\\mathcal {A}^x_b$ in the previous inequality, we obtained the desired result."
],
[
"Proof of Proposition ",
"It is sufficient to show that for any $\\Vert p \\Vert \\le \\varepsilon ^{-\\beta }$ and $\\Vert \\Gamma \\Vert \\le \\varepsilon ^{-\\gamma }$ , there exists $\\Vert \\Delta \\hat{x} \\Vert \\le \\varepsilon ^{1-\\alpha }$ , determining $\\Delta x$ by (REF ), such that $(D\\phi (x) - p)\\cdot \\Delta \\hat{x} +M_\\varepsilon ^{x}[\\phi ] \\Vert \\Delta \\hat{x} - \\Delta x \\Vert + \\frac{1}{2} \\langle D^2\\phi (x) \\Delta x ,\\Delta x\\rangle - \\frac{1}{2} \\langle \\Gamma \\Delta \\hat{x},\\Delta \\hat{x}\\rangle \\\\- \\varepsilon ^2 f(t,x,z,p,\\Gamma ) \\le -\\varepsilon ^2 f(t,x,z,D\\phi (x), D^2\\phi (x)) +o(\\varepsilon ^2) ,$ with an error estimate $o(\\varepsilon ^2)$ that is independent of $p$ and $\\Gamma $ and locally uniform in $x$ , $t$ , $z$ .",
"In view of the conditions (REF ) and (REF ), we can pick $\\mu >0$ and $\\delta >0$ such that $\\mu +\\gamma & <1-\\alpha \\text{ and } \\mu +\\gamma r <1+\\alpha , \\\\\\delta & <\\min (2\\alpha , \\rho - (1-\\alpha )).",
"$ Now we consider separately the following three cases: $\\Vert D\\phi (x) - p \\Vert \\le \\varepsilon ^\\mu $ and $\\lambda _\\text{min}(D^2\\phi (x)- \\Gamma )\\ge - \\varepsilon ^{\\delta }$ , $\\Vert D\\phi (x) - p \\Vert \\le \\varepsilon ^\\mu $ and $\\lambda _\\text{min}(D^2\\phi (x)- \\Gamma )\\le - \\varepsilon ^{\\delta }$ , $\\Vert D\\phi (x) - p \\Vert \\ge \\varepsilon ^\\mu $ .",
"For case REF , we choose $\\Delta \\hat{x}=0$ .",
"By a reasoning similar to Case 1 in the proof of [21], we obtained the inequality given by (REF ).",
"For cases REF and REF , in order to use the decomposition (REF ), we now give a preliminary inequality.",
"By the inequality (REF ) in Lemma REF , we have $\\left| \\frac{1}{2} \\langle D^2\\phi (x) \\Delta x ,\\Delta x\\rangle - \\frac{1}{2} \\langle D^2\\phi (x) \\Delta \\hat{x} ,\\Delta \\hat{x}\\rangle \\right|\\le \\frac{3}{2} \\Vert D^2\\phi (x) \\Vert \\Vert \\Delta \\hat{x} - \\Delta x \\Vert \\varepsilon ^{1- \\alpha } ,$ which yields with the assumption $M_\\varepsilon ^{x}[\\phi ] \\le \\frac{4}{3} \\Vert D^2 \\phi (x) \\Vert \\varepsilon ^{1-\\alpha }$ that $M_\\varepsilon ^{x} [\\phi ] \\Vert \\Delta \\hat{x} - \\Delta x \\Vert +\\frac{1}{2} \\langle D^2\\phi (x) \\Delta x ,\\Delta x\\rangle - \\frac{1}{2} \\langle D^2\\phi (x) \\Delta \\hat{x} ,\\Delta \\hat{x}\\rangle \\le \\frac{17}{6}\\Vert D^2\\phi (x) \\Vert \\varepsilon ^{1- \\alpha } \\Vert \\Delta \\hat{x} - \\Delta x \\Vert .$ By combining the geometric estimate (REF ) with the assumption $d(x)\\ge \\varepsilon ^{1-\\alpha } - \\varepsilon ^{\\rho }$ , we get that the left-hand side of (REF ) is upper bounded by $\\frac{17}{6}\\Vert D^2\\phi (x) \\Vert \\varepsilon ^{1- \\alpha +\\rho }$ .",
"By using the decomposition (REF ) we deduce that it is sufficient to show that there exists $\\Vert \\Delta \\hat{x} \\Vert \\le \\varepsilon ^{1-\\alpha }$ such that $(D\\phi (x) - p) \\cdot \\Delta \\hat{x} + \\frac{1}{2} \\langle (D^2\\phi (x)- \\Gamma ) \\Delta \\hat{x} , \\Delta \\hat{x} \\rangle + \\frac{17}{6} \\Vert D^2\\phi (x) \\Vert \\varepsilon ^{1-\\alpha +\\rho } - \\varepsilon ^2 f(t,x,z,p,\\Gamma ) \\\\\\le - \\varepsilon ^2 f(t,x,z,D\\phi (x), D^2\\phi (x)).$ For case REF , we choose $\\Delta \\hat{x}$ to be an eigenvector for the minimum eigenvalue $\\lambda = \\lambda _{\\text{min}}(D^2\\phi (x)- \\Gamma )$ of norm $\\varepsilon ^{1-\\alpha }$ .",
"Notice that since $f$ is monotone in its last input, we have $f(t,x,z,p,\\Gamma ) \\ge f(t,x,z, D^2\\phi (x) - \\lambda I).$ Choosing $\\Delta \\hat{x}$ as announced, and changing the sign if necessary to make $(D\\phi (x) - p) \\cdot \\Delta \\hat{x}\\le 0$ , we deduce that $(D\\phi (x) - p) \\cdot \\Delta \\hat{x} + \\frac{1}{2} \\langle (D^2\\phi (x)- \\Gamma ) \\Delta \\hat{x} , \\Delta \\hat{x} \\rangle + \\frac{17}{6} \\Vert D^2\\phi (x) \\Vert \\varepsilon ^{1-\\alpha +\\rho } - \\varepsilon ^2 f(t,x,z,p,\\Gamma ) \\\\\\le \\frac{1}{2} \\varepsilon ^{2-2\\alpha } \\lambda + \\frac{17}{6} \\Vert D^2\\phi (x) \\Vert \\varepsilon ^{1-\\alpha +\\rho }- \\varepsilon ^2 f(t,x,z,p, D^2\\phi (x) - \\lambda I).$ If $-1\\le \\lambda \\le -\\varepsilon ^\\delta $ then $\\varepsilon ^{2-2\\alpha } \\lambda \\le - \\varepsilon ^{2-2\\alpha +\\delta }$ and $f(t,x,z,p, D^2\\phi (x)- \\lambda I)$ is bounded.",
"Since $\\varepsilon ^{1-\\alpha +\\rho }\\ll \\varepsilon ^{2-2\\alpha +\\delta } $ by (), for such $\\lambda $ we have $\\frac{1}{2} \\varepsilon ^{2-2\\alpha } \\lambda + \\frac{17}{6} \\Vert D^2\\phi (x) \\Vert \\varepsilon ^{1-\\alpha +\\rho }- \\varepsilon ^2 f(t,x,z, p, D^2\\phi (x) - \\lambda I) \\le - \\frac{1}{4} \\varepsilon ^{2-2\\alpha +\\delta } +O(\\varepsilon ^2).$ In this case, we are done by (), since the right-hand side is $\\le \\varepsilon ^2 f(t,x,z,D\\phi (x), D^2\\phi (x)) $ when $\\varepsilon $ is small enough.",
"To complete case REF , suppose $\\lambda \\le - 1$ .",
"Then using the growth hypothesis (REF ) and recalling that $p$ is near $D\\phi (x)$ we have $\\frac{1}{2} \\varepsilon ^{2-2 \\alpha } \\lambda - \\varepsilon ^2 f(t,x,z,p, D^2\\phi (x) -\\lambda I)\\le -\\frac{1}{2} \\varepsilon ^{2-2 \\alpha } |\\lambda | +C\\varepsilon ^2 (1+|\\lambda |^r).$ Now notice that $|\\lambda | \\le C(1+\\Vert \\Gamma \\Vert ) \\le C\\varepsilon ^{-\\gamma }$ .",
"Since $\\gamma (r-1)<2\\alpha $ we have $\\varepsilon ^{2-2\\alpha } |\\lambda | \\gg \\varepsilon ^2 |\\lambda |^r$ .",
"Therefore we deduce by () that $-\\frac{1}{2} \\varepsilon ^{2-2 \\alpha } |\\lambda | +C\\varepsilon ^2|\\lambda |^r + \\frac{17}{6} \\Vert D^2\\phi (x) \\Vert \\varepsilon ^{1-\\alpha +\\rho }\\le - \\frac{1}{4} \\varepsilon ^{2-2 \\alpha } \\le - \\varepsilon ^2 f(t,x,z,D\\phi (x),D^2\\phi (x)),$ when $\\varepsilon $ is sufficiently small.",
"Case REF is now complete.",
"Finally, to treat case REF , we take $\\Delta \\hat{x}$ parallel to $D\\phi (x)-p$ with norm $\\varepsilon ^{1-\\alpha }$ , and with the sign chosen such that $(D\\phi (x)-p)\\cdot \\Delta \\hat{x} = -\\varepsilon ^{1-\\alpha } \\Vert D\\phi (x)-p \\Vert \\le - \\varepsilon ^{1-\\alpha +\\mu }.$ By observing that $\\displaystyle \\frac{17}{6} \\Vert D^2\\phi (x) \\Vert \\varepsilon ^{1-\\alpha +\\rho } \\ll \\varepsilon ^{1-\\alpha } \\Vert D\\phi (x)-p \\Vert $ , this case follows exactly the sames lines as [21]."
],
[
"Proof of Proposition ",
"This proof is quite similar to case REF .",
"Since this estimate will not be needed in the rest of the paper, we just indicate that we need to distinguish three cases according to the respective sizes of $\\Vert D\\phi (x) - p \\Vert $ and $\\lambda _\\text{min}(D^2\\phi (x) - \\Gamma )$ with respect to $\\varepsilon ^\\mu $ and $- C_1 - \\varepsilon ^{\\alpha }$ , where $\\mu $ is defined by (REF )."
],
[
"Proof of Proposition ",
"This case corresponds to the heuristic derivation presented at Section REF when $m<0$ .",
"Recalling that $p_\\text{opt}^M$ and $\\Gamma _\\text{opt}$ are defined by ()–(REF ), our task is to show that for any $\\Vert p \\Vert \\le \\varepsilon ^{-\\beta }$ and $\\Vert \\Gamma \\Vert \\le \\varepsilon ^{-\\gamma }$ , there exists $\\Vert \\Delta \\hat{x} \\Vert \\le \\varepsilon ^{1-\\alpha }$ , determining $\\Delta x$ by (REF ), such that $ (D\\phi (x) - p) \\cdot \\Delta \\hat{x} + \\Vert \\Delta \\hat{x} - \\Delta x \\Vert M_\\varepsilon ^{x}[\\phi ]+ \\frac{1}{2} \\langle D^2\\phi (x) \\Delta x ,\\Delta x\\rangle - \\frac{1}{2} \\langle \\Gamma \\Delta \\hat{x} ,\\Delta \\hat{x}\\rangle \\\\- \\varepsilon ^2 f \\left(t,x,z,p,\\Gamma \\right)\\le \\frac{1}{4} (\\varepsilon ^{1-\\alpha } - d(x)) M_\\varepsilon ^{x}[\\phi ]-\\varepsilon ^2 \\min _{p\\in B (p_\\text{opt}^M,r)} f(t,x,z,p, \\Gamma _\\text{opt}) +o(\\varepsilon ^2),$ with an error estimate $o(\\varepsilon ^2)$ that is independent of $p$ and $\\Gamma $ and locally uniform in $x$ , $t$ , $z$ .",
"We can notice in (REF ) that the function $M_\\varepsilon ^{x}[\\phi ]$ is $\\varepsilon , x$ -bounded by $\\Vert h \\Vert _{L^\\infty }+\\Vert D\\phi \\Vert _{L^\\infty }$ .",
"Moreover, by Lemma REF we have $\\frac{1}{2} \\langle D^2\\phi (x) \\Delta x , \\Delta x \\rangle -\\frac{1}{2} \\langle \\Gamma _\\text{opt} \\Delta \\hat{x} ,\\Delta \\hat{x}\\rangle \\le \\frac{7}{4}\\Vert D^2\\phi (x) \\Vert (\\varepsilon ^{1-\\alpha } - d(x)) \\varepsilon ^{1-\\alpha }.",
"$ Thus, it is sufficient to examine, for any $\\Vert p \\Vert \\le \\varepsilon ^{-\\beta }$ and $\\Vert \\Gamma \\Vert \\le \\varepsilon ^{-\\gamma }$ , $ \\min _{\\Delta \\hat{x}} \\left[(D\\phi (x) - p)\\cdot \\Delta \\hat{x} + \\Vert \\Delta \\hat{x} - \\Delta x \\Vert M_\\varepsilon ^{x}[ \\phi ]+\\frac{1}{2}\\langle (\\Gamma _\\text{opt} - \\Gamma ) \\Delta \\hat{x}, \\Delta \\hat{x} \\rangle - \\varepsilon ^2 f(t,x,z,p, \\Gamma )\\right].$ We consider separately the following three cases: $\\Vert p_\\text{opt}^M - p \\Vert \\le 3 \\Big (1 - \\frac{d(x)}{\\varepsilon ^{1-\\alpha }}\\Big ) |M_\\varepsilon ^{x}[\\phi ]|$ , and $\\lambda _\\text{min}(\\Gamma _\\text{opt} - \\Gamma )\\ge - \\varepsilon ^{\\alpha }$ , $\\Vert p_\\text{opt}^M - p \\Vert \\le 3 \\Big (1 - \\frac{d(x)}{\\varepsilon ^{1-\\alpha }}\\Big ) |M_\\varepsilon ^{x}[\\phi ]|$ , and $\\lambda _\\text{min}(\\Gamma _\\text{opt} - \\Gamma )\\le - \\varepsilon ^{\\alpha }$ , $\\Vert p_\\text{opt}^M - p \\Vert \\ge 3 \\Big (1 - \\frac{d(x)}{\\varepsilon ^{1-\\alpha }}\\Big ) |M_\\varepsilon ^{x}[\\phi ]|$ .",
"For case REF , we choose $\\Delta \\hat{x}=\\pm \\varepsilon ^{1-\\alpha } n(\\bar{x})$ with the sign chosen such that $(p-p_\\text{opt}^M)\\cdot \\Delta \\hat{x} \\le 0.$ Since $\\lambda _\\text{min}(\\Gamma _\\text{opt}- \\Gamma )\\ge - \\varepsilon ^{\\alpha }$ we have $\\Gamma _\\text{opt}- \\Gamma + \\varepsilon ^{\\alpha } I \\ge 0$ and thus $\\Gamma \\le \\Gamma _\\text{opt}+ \\varepsilon ^{\\alpha } I$ .",
"Using the monotonicity of $f$ with respect to its last entry, this gives $f(t,x,z,p, \\Gamma )\\ge f(t,x,z,p, \\Gamma _\\text{opt} +\\varepsilon ^\\alpha I)$ .",
"Since $f$ is locally Lipschitz, we conclude that $ f(t,x,z,p,\\Gamma )\\ge f(t,x,z,p, \\Gamma _\\text{opt})+O(\\varepsilon ^\\alpha ) \\ge \\min _{p\\in B (p_\\text{opt}^M,r)} f(t,x,z,p, \\Gamma _\\text{opt}) +O(\\varepsilon ^\\alpha ).$ The constant in the error term is independent of $p$ and $\\Gamma $ , since we are assuming in case REF that $\\Vert p- p_\\text{opt}^M \\Vert \\le 3 (\\Vert h \\Vert _{L^\\infty }+\\Vert D\\phi \\Vert _{L^\\infty })$ .",
"Moreover we directly compute $ (D\\phi (x) - p_\\text{opt}^M)\\cdot \\Delta \\hat{x} + \\Vert \\Delta \\hat{x} - \\Delta x \\Vert M_\\varepsilon ^{x}[\\phi ]= \\frac{1}{2} (\\varepsilon ^{1-\\alpha } - d(x)) M_\\varepsilon ^{x}[ \\phi ] .$ Since $\\varepsilon ^{1-\\alpha } - d(x) \\ge \\varepsilon ^{\\rho }$ and $M_\\varepsilon ^{x}[\\phi ]< 0$ , we have $\\frac{1}{2} (\\varepsilon ^{1-\\alpha }-d(x)) M_\\varepsilon ^{x}[\\phi ]\\le \\frac{1}{2}\\varepsilon ^{\\rho } M_\\varepsilon ^{x}[\\phi ]\\le -\\frac{1}{2} \\varepsilon ^{1 -\\alpha -\\kappa + \\rho }.$ By noticing that $\\varepsilon ^{2-2\\alpha - \\gamma }\\ll \\varepsilon ^{1 - \\alpha - \\kappa + \\rho }$ using (REF ), we deduce from (REF ) that $\\Big | \\frac{1}{2} \\langle (\\Gamma _\\text{opt} - \\Gamma ) \\Delta \\hat{x}, \\Delta \\hat{x} \\rangle \\Big |\\le \\frac{1}{2}(\\Vert D^2 \\phi (x) \\Vert +\\varepsilon ^{-\\gamma }) \\varepsilon ^{2-2\\alpha }\\le \\frac{3}{4}\\varepsilon ^{2-2\\alpha - \\gamma }\\ll (\\varepsilon ^{1-\\alpha } - d(x)) M_\\varepsilon ^{x}[\\phi ].$ Therefore, by combining (REF ), (REF ) and (REF ), the choice $\\Delta \\hat{x}= \\pm \\varepsilon ^{1-\\alpha } n(\\bar{x})$ in the left-hand side of (REF ) yields $(D& \\phi (x) - p) \\cdot \\Delta \\hat{x} + \\Vert \\Delta \\hat{x} - \\Delta x \\Vert M_\\varepsilon ^{x}[\\phi ] + \\frac{1}{2} \\langle D^2\\phi (x) \\Delta x ,\\Delta x\\rangle - \\frac{1}{2} \\langle \\Gamma \\Delta \\hat{x} ,\\Delta \\hat{x}\\rangle - \\varepsilon ^2 f \\left(t,x, z, p,\\Gamma \\right) \\\\& \\le \\frac{1}{2} ( \\varepsilon ^{1-\\alpha } - d(x)) \\big ( M_\\varepsilon ^{x}[ \\phi ] + \\frac{7}{2}\\Vert D^2\\phi (x) \\Vert \\varepsilon ^{1-\\alpha } \\big )+ \\frac{3}{4}\\varepsilon ^{2-2\\alpha - \\gamma }- \\varepsilon ^2 \\min _{p\\in B (p_\\text{opt}^M,r)} f(t,x,z,p, \\Gamma _\\text{opt}) +o(\\varepsilon ^2) \\\\& \\le \\frac{1}{4} (\\varepsilon ^{1-\\alpha }-d(x)) M_\\varepsilon ^{x}[\\phi ] - \\varepsilon ^2 \\min _{p\\in B (p_\\text{opt}^M,r)} f(t,x,z,p, \\Gamma _\\text{opt}) +o(\\varepsilon ^2),$ as desired.",
"For case REF , in view of the condition (REF ), we can pick $\\sigma >1-\\alpha $ such that $\\rho <\\sigma <1 - \\frac{\\gamma (r-1)}{2}.$ Let $v^\\lambda $ be a unit eigenvector for the minimum eigenvalue $\\lambda = \\lambda _{\\text{min}}(\\Gamma _\\text{opt}- \\Gamma )$ .",
"We choose $\\Delta \\hat{x}$ of the form $\\Delta \\hat{x}=\\pm \\left[\\left(\\varepsilon ^{1-\\alpha }-\\varepsilon ^{\\sigma }\\right)n(\\bar{x})+\\text{sgn}(\\langle n(\\bar{x}),v^\\lambda \\rangle ) \\varepsilon ^{\\sigma } v^\\lambda \\right]= \\pm \\left[ a_1 n(\\bar{x}) +b v^\\lambda \\right],$ where $a_1= \\left(\\varepsilon ^{1-\\alpha } - \\varepsilon ^{\\sigma }\\right)$ , $ b =\\text{sgn} (\\langle n(\\bar{x}), v^\\lambda \\rangle ) \\varepsilon ^{\\sigma }$ and sgn denotes the sign function with the convention that $\\text{sgn}(0)=1$ .",
"The sign $\\pm $ will be chosen later.",
"This move fulfills the following estimate.",
"Lemma 4.13 The move $\\Delta \\hat{x}$ defined by (REF ) is authorized by the game and satisfies $(D\\phi (x) - p_\\text{opt}^M) \\cdot \\Delta \\hat{x} + \\Vert \\Delta \\hat{x} - \\Delta x \\Vert M_\\varepsilon ^{x}[\\phi ]\\le \\frac{\\varepsilon ^{1-\\alpha } -d(x)}{2} ( M_\\varepsilon ^{x}[\\phi ] + \\Vert D^2\\phi (x) \\Vert \\varepsilon ^{1-\\alpha } )- 4 \\varepsilon ^{\\sigma } M_\\varepsilon ^{x}[\\phi ],$ independently of the choice on $\\pm $ in (REF ).",
"To authorize this move, it suffices to check that $\\Vert \\Delta \\hat{x} \\Vert \\le \\varepsilon ^{1-\\alpha }$ .",
"After some calculations and by rearranging the terms, we compute $\\Vert \\Delta \\hat{x} \\Vert ^2 & =\\varepsilon ^{2-2\\alpha } +2 \\varepsilon ^{2 \\sigma }- 2 \\varepsilon ^{1-\\alpha +\\sigma } + \\varepsilon ^{\\sigma } \\Big ( \\varepsilon ^{1-\\alpha } - \\varepsilon ^{\\sigma } \\Big ) |\\langle n(\\bar{x}), v^\\lambda \\rangle | \\\\& = \\varepsilon ^{2-2\\alpha } - 2 \\varepsilon ^{1-\\alpha +\\sigma } ( 1 - \\varepsilon ^{ \\sigma -1+\\alpha } ) \\Big (1 - \\frac{1}{2} |\\langle n(\\bar{x}), v^\\lambda \\rangle | \\Big )\\le \\varepsilon ^{2-2\\alpha }.$ For the second part, we distinguish successively the two cases $\\pm $ .",
"By (REF ), we directly compute $\\Delta \\hat{x} \\cdot n(\\bar{x})= \\pm \\left[ \\left(\\varepsilon ^{1-\\alpha } - \\varepsilon ^{\\sigma }\\right) +|\\langle n(\\bar{x}), v^\\lambda \\rangle | \\varepsilon ^{\\sigma } \\right]= \\pm \\left[ \\varepsilon ^{1-\\alpha } - \\left(1- |\\langle n(\\bar{x}), v^\\lambda \\rangle | \\right) \\varepsilon ^{\\sigma } \\right].$ If $\\Delta \\hat{x} \\cdot n(\\bar{x}) \\le 0$ , this move corresponds to the sign $-$ in (REF ) by (REF ) and we observe that $\\hat{x} \\in \\Omega $ by Lemma REF .",
"As a result, by introducing the explicit expressions of $p_\\text{opt}^M$ and $(\\Delta \\hat{x})_1$ respectively given by () and (REF ), we get $(D\\phi (x) - p_\\text{opt}^M) \\cdot \\Delta \\hat{x} + \\Vert \\Delta \\hat{x} - \\Delta x \\Vert M_\\varepsilon ^{x}[ \\phi ]= (D\\phi (x) - p_\\text{opt}^M)_1 (\\Delta \\hat{x})_1 \\\\= - \\left( -\\frac{1}{2} (1 -\\frac{d(x)}{\\varepsilon ^{1-\\alpha }}) M_\\varepsilon ^{x}[ \\phi ]+\\frac{1}{4} (\\varepsilon ^{1-\\alpha } - \\frac{d^2(x)}{\\varepsilon ^{1-\\alpha }}) (D^2\\phi (x))_{11} \\right)\\left(\\varepsilon ^{1-\\alpha } - (1-|\\langle n(\\bar{x}), v^\\lambda \\rangle |) \\varepsilon ^{\\sigma }\\right) .$ Since $0 \\le \\varepsilon ^{1-\\alpha } - (1- |\\langle n(\\bar{x}), v^\\lambda \\rangle |) \\varepsilon ^{\\sigma }\\le \\varepsilon ^{1-\\alpha }$ , we observe that $\\left| \\frac{1}{4} (\\varepsilon ^{1-\\alpha } - \\frac{d^2(x)}{\\varepsilon ^{1-\\alpha }}) (D^2\\phi (x))_{11}\\left(\\varepsilon ^{1-\\alpha } - (1- |\\langle n(\\bar{x}), v^\\lambda \\rangle |) \\varepsilon ^{\\sigma }\\right) \\right|& \\le \\frac{1}{4} \\Vert D^2\\phi (x) \\Vert (\\varepsilon ^{2-2 \\alpha } - d^2(x)) \\\\& \\le \\frac{1}{2} \\Vert D^2\\phi (x) \\Vert (\\varepsilon ^{1- \\alpha } - d(x))\\varepsilon ^{1- \\alpha }.$ By plugging this inequality in (REF ) and rearranging the terms, we obtain $(D\\phi (x) - p_\\text{opt}^M) \\cdot \\Delta \\hat{x} + & \\Vert \\Delta \\hat{x} - \\Delta x \\Vert M_\\varepsilon ^{x}[ \\phi ] \\\\& \\le (\\varepsilon ^{1-\\alpha } -d(x) ) \\left\\lbrace \\frac{1}{2} \\left(1 - (1- |\\langle n(\\bar{x}), v^\\lambda \\rangle |) \\varepsilon ^{\\sigma -1+\\alpha }\\right) M_\\varepsilon ^{x}[\\phi ]+ \\frac{1}{2} \\Vert D^2\\phi (x) \\Vert \\varepsilon ^{1-\\alpha } \\right\\rbrace \\\\& \\le \\frac{1}{2} (\\varepsilon ^{1-\\alpha } -d(x)) ( M_\\varepsilon ^{x}[\\phi ] + \\Vert D^2\\phi (x) \\Vert \\varepsilon ^{1-\\alpha } )- \\frac{1}{2} \\varepsilon ^{\\sigma } M_\\varepsilon ^{x}[\\phi ] .$ Otherwise, if $\\Delta \\hat{x} \\cdot n(\\bar{x})\\ge 0$ , this move corresponds to the sign $+$ in (REF ) by (REF ).",
"We have $\\Vert \\Delta \\hat{x}-\\varepsilon ^{1-\\alpha } n(\\bar{x}) \\Vert =\\Vert -\\varepsilon ^{\\sigma } n(\\bar{x}) +\\text{sgn}(\\langle n(\\bar{x}), v^\\lambda \\rangle )\\varepsilon ^{\\sigma }v^\\lambda \\Vert = \\sqrt{2} \\varepsilon ^{ \\sigma } \\sqrt{ 1 - |\\langle n(\\bar{x}), v^\\lambda \\rangle |}\\le \\sqrt{2} \\varepsilon ^{ \\sigma }.$ By using Lemma REF , we deduce from the previous inequality that, for $\\varepsilon $ small enough, the intermediate point $\\hat{x}=x +\\Delta \\hat{x}$ is outside $\\Omega $ and $ \\varepsilon ^{1-\\alpha } - d(x) - \\sqrt{2}\\varepsilon ^\\sigma -2 C_1 \\varepsilon ^{2-2\\alpha } \\le \\Vert \\Delta \\hat{x} - \\Delta x \\Vert ,$ where $C_1$ is a certain constant depending on the principal curvatures of $\\partial \\Omega $ in a neighborhood of $x$ .",
"By repeating the computations above, we find $(D\\phi (x) - p_\\text{opt}^M)_1 (\\Delta \\hat{x})_1& \\le \\frac{1}{2} (\\varepsilon ^{1-\\alpha } -d(x) ) \\left\\lbrace - (1 - ( 1- |\\langle n(\\bar{x}), v^\\lambda \\rangle |)\\varepsilon ^{\\sigma -1 +\\alpha }) M_\\varepsilon ^{x}[\\phi ]+ \\Vert D^2\\phi (x) \\Vert \\varepsilon ^{1-\\alpha } \\right\\rbrace \\\\& \\le \\frac{1}{2} (\\varepsilon ^{1-\\alpha } -d(x) ) (- M_\\varepsilon ^{x}[\\phi ] + \\Vert D^2\\phi (x) \\Vert \\varepsilon ^{1-\\alpha } ).$ Recalling that $M_\\varepsilon ^{x}[\\phi ]<0$ , by combining (REF ) with the previous estimate, we are led to $(D\\phi (x) & - p_\\text{opt}^M) \\cdot \\Delta \\hat{x} + \\Vert \\Delta \\hat{x} - \\Delta x \\Vert M_\\varepsilon ^{x}[\\phi ] \\\\& \\le \\frac{1}{2} (\\varepsilon ^{1-\\alpha } -d(x) ) (- M_\\varepsilon ^{x}[\\phi ] + \\Vert D^2\\phi (x) \\Vert \\varepsilon ^{1-\\alpha } )+\\left(\\varepsilon ^{1-\\alpha } - d(x) - \\sqrt{2} \\varepsilon ^{\\sigma }- 2 C_1\\varepsilon ^{2-2\\alpha } \\right) M_\\varepsilon ^{x}[\\phi ] \\\\& \\le \\frac{1}{2} (\\varepsilon ^{1-\\alpha } -d(x) ) ( M_\\varepsilon ^{x}[\\phi ] + \\Vert D^2\\phi (x) \\Vert \\varepsilon ^{1-\\alpha } )- \\varepsilon ^{\\sigma } M_\\varepsilon ^{x}[\\phi ] ( \\sqrt{2} + 2 C_1\\varepsilon ^{2-2\\alpha - \\sigma } ).$ Putting together the two cases, the proof of the inequality given by (REF ) is complete.",
"Now we turn back to the analysis of case REF .",
"Note that since $f$ is monotone in its last input $f(t,x,z,p,\\Gamma ) \\ge f(t,x,z,p, \\Gamma _\\text{opt} - \\lambda I).$ The direct evaluation of the second order terms in $\\Delta \\hat{x}$ of (REF ) gives $\\langle ( \\Gamma _{\\text{opt}}- \\Gamma ) \\Delta \\hat{x} , \\Delta \\hat{x} \\rangle & = a_1^2 \\langle (\\Gamma _{\\text{opt}}- \\Gamma ) n(\\bar{x}) , n(\\bar{x}) \\rangle + 2 a_1 b \\langle (\\Gamma _{\\text{opt}}- \\Gamma ) v^\\lambda , n(\\bar{x}) \\rangle + b^2 \\langle (\\Gamma _{\\text{opt}}- \\Gamma ) v^\\lambda , v^\\lambda \\rangle \\\\& \\le a_1^2 (\\Vert \\Gamma _{\\text{opt}} \\Vert + \\Vert \\Gamma \\Vert )+ 2 a_1 b \\lambda \\langle v^\\lambda , n(\\bar{x}) \\rangle + b^2 \\lambda .$ With our choice for $\\Delta \\hat{x}$ , we have $a_1 b \\langle v^\\lambda , n(\\bar{x}) \\rangle \\ge 0$ .",
"Hence, since $\\lambda \\le 0$ in case REF , it follows that $\\langle ( \\Gamma _{\\text{opt}}- \\Gamma ) \\Delta \\hat{x} , \\Delta \\hat{x} \\rangle & \\le a_1^2 (\\Vert \\Gamma _{\\text{opt}} \\Vert + \\Vert \\Gamma \\Vert ) + b^2 \\lambda \\le \\varepsilon ^{2- 2\\alpha } (\\Vert D^2\\phi (x) \\Vert + \\varepsilon ^{-\\gamma } ) + \\varepsilon ^{2 \\sigma } \\lambda .$ Choosing $\\Delta \\hat{x}$ as announced, using (REF ) and (REF ) and changing the sign $\\pm $ in (REF ) if necessary to make $(p_{\\text{opt}} - p) \\cdot \\Delta \\hat{x}\\le 0$ , $(D\\phi (x)& - p) \\cdot \\Delta \\hat{x} + \\Vert \\Delta \\hat{x} - \\Delta x \\Vert M_\\varepsilon ^{x}[\\phi ] + \\frac{1}{2} \\langle D^2\\phi (x) \\Delta x ,\\Delta x\\rangle - \\frac{1}{2} \\langle \\Gamma \\Delta \\hat{x} ,\\Delta \\hat{x}\\rangle - \\varepsilon ^2 f \\left(t,x, z, p,\\Gamma \\right) \\\\& \\le \\frac{1}{2} (\\varepsilon ^{1-\\alpha } -d(x) ) \\left(M_\\varepsilon ^{x}[\\phi ] +\\frac{9}{2}\\Vert D^2\\phi (x) \\Vert \\varepsilon ^{1-\\alpha } \\right)+ \\frac{1}{2} \\varepsilon ^{2-2\\alpha } (\\Vert D^2\\phi (x) \\Vert + \\varepsilon ^{-\\gamma } ) - 4 \\varepsilon ^{\\sigma } M_\\varepsilon ^{x}[\\phi ] \\\\& \\phantom{\\le \\frac{1}{2} (\\varepsilon ^{1-\\alpha } -d(x))====================}+ \\frac{1}{2} \\varepsilon ^{\\sigma } \\lambda - \\varepsilon ^2 f(t,x,z,p, \\Gamma _\\text{opt}- \\lambda I).",
"$ Since $d(x) \\le \\varepsilon ^{1-\\alpha }-\\varepsilon ^\\rho $ in case REF , we deduce from the assumption (REF ) that $\\varepsilon ^{1-\\alpha } -d(x) \\ge \\varepsilon ^{\\rho } \\gg \\varepsilon ^{\\sigma } .$ Since $M_\\varepsilon ^{x}[\\phi ] \\le - \\varepsilon ^{1-\\alpha - \\kappa }$ and $\\varepsilon ^{2-2\\alpha -\\gamma }\\ll \\varepsilon ^{1-\\alpha -\\kappa + \\rho }$ using (REF ), we conclude by (REF ) that $\\frac{1}{2} (\\varepsilon ^{1-\\alpha } -d(x) ) \\Big (M_\\varepsilon ^{x}[\\phi ] +\\frac{9}{2}\\Vert D^2\\phi (x) \\Vert \\varepsilon ^{1-\\alpha } \\Big )+ \\frac{1}{2} \\varepsilon ^{2-2\\alpha } (\\Vert D^2\\phi (x) \\Vert + \\varepsilon ^{-\\gamma } )- 4 \\varepsilon ^{\\sigma } M_\\varepsilon ^{x}[\\phi ] \\\\\\le \\frac{1}{4} (\\varepsilon ^{1-\\alpha } - d(x)) M_\\varepsilon ^{x}[\\phi ] .$ It remains to control the terms in (REF ) depending on $\\lambda $ .",
"If $-1\\le \\lambda \\le -\\varepsilon ^\\alpha $ , then $\\varepsilon ^{2 \\sigma } \\lambda \\le - \\varepsilon ^{2 \\sigma + \\alpha }$ and $f(t,x,z,p, \\Gamma _\\text{opt} - \\lambda I)$ is bounded.",
"So for such $\\lambda $ we have $\\frac{1}{2} \\varepsilon ^{2 \\sigma } \\lambda - \\varepsilon ^2 f(t,x,z, p, \\Gamma _\\text{opt} - \\lambda I) \\le - \\frac{1}{2} \\varepsilon ^{2 \\sigma + \\alpha } +O(\\varepsilon ^2).$ In this case, the right-hand side is $\\displaystyle \\le - \\varepsilon ^2 \\min _{p\\in B (p_\\text{opt}^M,r)} f(t,x,z,p, \\Gamma _\\text{opt})$ when $\\varepsilon $ is sufficiently small since $\\varepsilon ^{2 \\sigma + \\alpha }\\gg \\varepsilon ^2$ by (REF ).",
"To complete case REF , suppose $\\lambda \\le - 1$ .",
"Then using the growth hypothesis (REF ) and recalling that $p$ is near $p_\\text{opt}$ we have $\\frac{1}{2} \\varepsilon ^{2 \\sigma } \\lambda - \\varepsilon ^2 f(t,x,z, p, D^2\\phi (x) - \\lambda I)\\le -\\frac{1}{2} \\varepsilon ^{2 \\sigma } |\\lambda | +C\\varepsilon ^2 (1+|\\lambda |^r).$ Now notice that $|\\lambda | \\le C(1+\\Vert \\Gamma \\Vert ) \\le C\\varepsilon ^{-\\gamma }$ .",
"Since $\\gamma (r-1)<2 - 2 \\sigma $ by (REF ), we have $\\varepsilon ^{2 \\sigma } |\\lambda | \\gg \\varepsilon ^2 |\\lambda |^r$ .",
"Therefore $-\\frac{1}{2} \\varepsilon ^{2 \\sigma } |\\lambda | +C\\varepsilon ^2|\\lambda |^r \\le - \\frac{1}{4} \\varepsilon ^{2 \\sigma }\\le - \\varepsilon ^2 \\min _{p\\in B (p_\\text{opt}^M,r)} f(t,x,z,p, \\Gamma _\\text{opt}),$ for $\\varepsilon $ small enough.",
"Case REF is now complete.",
"Finally in case REF , we take $\\Delta \\hat{x}$ to be parallel to $p_\\text{opt}^M - p$ with norm $\\varepsilon ^{1-\\alpha }$ , and with the sign chosen such that $(p_\\text{opt}^M-p)\\cdot \\Delta \\hat{x} = -\\varepsilon ^{1-\\alpha } \\Vert p_\\text{opt}^M -p \\Vert \\le - 3 (\\varepsilon ^{1-\\alpha } - d(x)) |M_\\varepsilon ^{x}[\\phi ]|\\le - 3 \\varepsilon ^{1-\\alpha - \\kappa + \\rho }.$ Estimating the other terms on the left-hand side of (REF ), some manipulations analogous to those made in Lemma REF led us to $\\left|(D\\phi (x) - p_\\text{opt}^M) \\cdot \\Delta \\hat{x} + \\Vert \\Delta \\hat{x} - \\Delta x \\Vert M_\\varepsilon ^{x}[ \\phi ] \\right|& \\le \\frac{1}{2}(\\varepsilon ^{1-\\alpha }- d(x)) \\left( 3 |M_\\varepsilon ^{x}[\\phi ]| + 4 \\Vert D^2\\phi (x) \\Vert \\varepsilon ^{1-\\alpha } \\right) .$ From (REF ), we deduce that $(D\\phi (x) - p) \\cdot \\Delta \\hat{x} + \\Vert \\Delta \\hat{x} - \\Delta x \\Vert M_\\varepsilon ^{x}[ \\phi ]\\le - \\frac{1}{2}\\varepsilon ^{1-\\alpha } \\Vert p_\\text{opt}^M -p \\Vert +2 \\Vert D^2\\phi (x) \\Vert \\varepsilon ^{2-2\\alpha } .$ Estimating the other terms $ | \\langle (\\Gamma _\\text{opt}(x)-\\Gamma )\\Delta \\hat{x}, \\Delta \\hat{x} \\rangle |\\le (C+\\Vert \\Gamma \\Vert )\\Vert \\Delta \\hat{x} \\Vert ^2 \\le C\\varepsilon ^{-\\gamma +2-2\\alpha },$ and $ \\varepsilon ^2 |f(t,x,z,p, \\Gamma )|\\le C\\varepsilon ^2 (1+\\Vert p \\Vert ^q+\\Vert \\Gamma \\Vert ^r) \\le C ( \\varepsilon ^2+\\varepsilon ^2 \\Vert p \\Vert ^q + \\varepsilon ^{2-\\gamma r} ) .$ Thus $(D\\phi (x) - p) \\cdot \\Delta \\hat{x} + \\Vert \\Delta \\hat{x} - \\Delta x \\Vert M_\\varepsilon ^{x}[\\phi ] + \\frac{1}{2} \\langle D^2\\phi (x) \\Delta x ,\\Delta x\\rangle - \\frac{1}{2} \\langle \\Gamma \\Delta \\hat{x} ,\\Delta \\hat{x}\\rangle - \\varepsilon ^2 f \\left(t,x, z, p,\\Gamma \\right) \\\\\\le -\\frac{1}{2} \\varepsilon ^{1-\\alpha } \\Vert p_\\text{opt}^M - p \\Vert +C\\varepsilon ^2 \\Vert p \\Vert ^q +O(\\varepsilon ^{2-2\\alpha } + \\varepsilon ^{-\\gamma +2-2\\alpha }+\\varepsilon ^{2-\\gamma r} ).$ Since $ \\varepsilon ^{1-\\alpha } \\Vert p_\\text{opt}^M - p \\Vert \\ge 2 \\varepsilon ^{1-\\alpha -\\kappa + \\rho } $ by using (REF ), we obtain that $\\varepsilon ^{-\\gamma +2-2\\alpha } +\\varepsilon ^{2-\\gamma r} \\ll \\varepsilon ^{1-\\alpha } \\Vert p_\\text{opt}^M-p \\Vert ,$ noticing that $\\min (- \\gamma +2 - 2 \\alpha , 2-\\gamma r) >1-\\alpha - \\kappa + \\rho $ by using (REF ) and (REF ).",
"Thus, by combining (REF )–(REF ), we conclude that $(D\\phi (x) - p) \\cdot \\Delta \\hat{x} + \\Vert \\Delta \\hat{x} - \\Delta x \\Vert M_\\varepsilon ^{x}[\\phi ] + \\frac{1}{2} \\langle D^2\\phi (x) \\Delta x ,\\Delta x\\rangle - \\frac{1}{2} \\langle \\Gamma \\Delta \\hat{x} ,\\Delta \\hat{x}\\rangle - \\varepsilon ^2 f \\left(t,x, z, p,\\Gamma \\right) \\\\\\le -\\frac{1}{2\\sqrt{2}} \\varepsilon ^{1-\\alpha } \\Vert p_\\text{opt}^M - p \\Vert +C\\varepsilon ^2 \\Vert p \\Vert ^q .",
"$ If $\\Vert p \\Vert \\le 2 \\Vert p_\\text{opt}^M \\Vert $ , then $\\varepsilon ^2 \\Vert p \\Vert ^q \\ll \\varepsilon ^{1-\\alpha -\\kappa + \\rho }$ .",
"If $\\Vert p \\Vert \\ge 2 \\Vert p_\\text{opt}^M \\Vert $ , we infer from the condition on $\\beta $ in (REF ) that $\\varepsilon ^{1-\\alpha } \\Vert p_\\text{opt}^M -p \\Vert \\sim \\varepsilon ^{1-\\alpha } \\Vert p \\Vert \\gg \\varepsilon ^2 \\Vert p \\Vert ^q$ .",
"In either case the term $\\varepsilon ^{1-\\alpha } \\Vert p_\\text{opt}^M - p \\Vert $ dominates and we get $( p_\\text{opt}^M - p)\\cdot \\Delta \\hat{x} +\\frac{1}{2} \\langle (\\Gamma _\\text{opt}^M -\\Gamma ) \\Delta \\hat{x}, \\Delta \\hat{x} \\rangle - \\varepsilon ^2 f(t,x,z,p, \\Gamma )\\le -\\frac{1}{4} \\varepsilon ^{1-\\alpha } \\Vert p_\\text{opt}^M - p \\Vert \\le \\frac{3}{4} (\\varepsilon ^{1-\\alpha } - d(x)) M_\\varepsilon ^{x}[\\phi ] .$ The right-hand side of this inequality is certainly $\\displaystyle \\le \\frac{1}{4} (\\varepsilon ^{1-\\alpha } - d(x)) M_\\varepsilon ^{x}[\\phi ]- \\varepsilon ^2 \\min _{p\\in B (p_\\text{opt}^M,r)} f(t,x,z,p, \\Gamma _\\text{opt})$ when $\\varepsilon $ is small.",
"Case REF is now complete which finishes the proof of Proposition REF ."
],
[
"Application to stability",
"To prove stability in Section , we will need some global variants of Propositions REF and REF .",
"It is at this point that the uniformity of the constants in (REF )–(REF ) in $x$ and $t$ , and the growth condition (REF ) intervene.",
"We must also take care of the Neumann boundary condition.",
"Unlike the Dirichlet problem solved in [21], it is no longer appropriate to consider constant functions as test functions.",
"For this reason, we are going to consider a $C_b^2(\\overline{\\Omega })$ -function $\\psi $ such that $ \\frac{\\partial \\psi }{\\partial n}=\\Vert h \\Vert _{L^\\infty }+1 \\quad \\text{ on } \\partial \\Omega .$ It is worth noticing that $\\psi $ has exactly the same properties as the function introduced in Section REF for the game associated to the elliptic PDE with Neumann boundary condition.",
"If we take $\\psi =(\\Vert h \\Vert _{L^\\infty }+1)\\psi _1$ where $\\psi _1 \\in C_b^2(\\overline{\\Omega })$ such that $\\dfrac{\\partial \\psi _1}{\\partial n}= 1$ on $\\partial \\Omega $ , it is clear that $ \\Vert \\psi \\Vert _{C_b^{2}(\\overline{\\Omega })} =\\Vert \\psi _1 \\Vert _{C_b^{2}(\\overline{\\Omega })} (1 + \\Vert h \\Vert _{L^\\infty })$.",
"The next lemma is the crucial point to obtain stability in both parabolic and elliptic settings.",
"Lemma 4.14 If $\\psi \\in C_b^2(\\overline{\\Omega })$ satisfies (REF ), then there exists $\\varepsilon _0>0$ such that for all $\\varepsilon <\\varepsilon _0$ and for all $x\\in \\Omega (\\varepsilon ^{1-\\alpha })$ , $ - \\Vert h \\Vert _{L^\\infty } - \\Vert D\\psi \\Vert _{L^\\infty (\\overline{\\Omega })} \\le M_\\varepsilon ^x[ \\psi ] \\le - \\frac{1}{2}\\quad \\text{ and } \\quad \\frac{1}{2} \\le m_\\varepsilon ^x[ - \\psi ] \\le \\Vert h \\Vert _{L^\\infty } +\\Vert D\\psi \\Vert _{L^\\infty (\\overline{\\Omega })} .$ We shall demonstrate the bounds on $M_\\varepsilon ^x[ \\psi ]$ in (REF ); the proof for $m_\\varepsilon ^x[- \\psi ]$ is entirely parallel.",
"The left-hand side inequality is clear by the Cauchy-Schwarz inequality.",
"Let us consider $0<\\varepsilon <\\varepsilon _0$ , where $\\displaystyle \\varepsilon _0= \\left(4 \\Vert D^2\\psi \\Vert _{L^\\infty (\\overline{\\Omega })}+2 \\right)^{-\\frac{1}{1-\\alpha }} $.",
"By the geometric relation (REF ), we observe that every move $\\Delta x$ associated to the move $\\Delta \\hat{x}$ decided by Mark satisfies $\\Vert \\Delta x \\Vert \\le 2\\varepsilon ^{1-\\alpha } \\le \\frac{1}{2 \\Vert D^2\\psi \\Vert _{L^\\infty (\\overline{\\Omega })} +1}.$ By the Cauchy-Schwarz inequality and using that $\\psi \\in C_b^{2}(\\overline{\\Omega })$ , we have $h(x+\\Delta x) - D\\psi (x) \\cdot n(x+\\Delta x)& \\le \\Vert h \\Vert _{L^\\infty } - D\\psi (x+\\Delta x)\\cdot n(x+\\Delta x)+ (D\\psi (x+\\Delta x)-D\\psi (x))\\cdot n(x+\\Delta x) \\\\& \\le -1+ \\Vert D^2\\psi \\Vert _{L^\\infty (\\overline{\\Omega })} \\Vert \\Delta x \\Vert \\le - \\frac{1}{2}.$ Then, by passing to the sup, we get the desired result.",
"Lemma 4.15 Let $\\phi \\in C_b^{2}(\\overline{\\Omega })$ .",
"Assume that $p_\\text{opt}^m$ $p_\\text{opt}^M$ and $\\Gamma _\\text{opt}$ are the strategies, associated to $\\phi $ , respectively defined by (REF ), () and (REF ).",
"Then, for all $x\\in \\Omega (\\varepsilon ^{1-\\alpha })$ , we have $\\max \\left(\\Vert p_\\text{opt}^m(x) \\Vert , \\Vert p_\\text{opt}^M(x) \\Vert \\right) \\le \\frac{1}{2} \\left( \\Vert h \\Vert _{L^\\infty }+3 \\Vert D\\phi \\Vert _{C_b^1(\\overline{\\Omega })} \\right)\\quad \\text{ and } \\quad \\Vert \\Gamma _\\text{opt}(x) \\Vert \\le \\frac{3}{2} \\Vert D^2\\phi \\Vert _{L^\\infty (\\overline{\\Omega })} .$ The proof being exactly the same for $p_\\text{opt}^m$ , it is sufficient to show the result for $p_\\text{opt}^M$ .",
"By the triangle inequality and (), we have $\\Vert p_\\text{opt}^M(x) - D\\phi (x) \\Vert & \\le \\frac{1}{2}\\left(1-\\frac{ d(x)}{\\varepsilon ^{1-\\alpha }}\\right) \\left( | M_\\varepsilon ^{x}[\\phi ] |+ \\frac{1}{2}\\varepsilon ^{1-\\alpha } \\left( 1+ \\frac{d(x)}{\\varepsilon ^{1-\\alpha }} \\right) \\Vert D^2\\phi (x) \\Vert \\right)\\\\ &\\le \\frac{1}{2} \\left( | M_\\varepsilon ^{x}[\\phi ] | + \\varepsilon ^{1-\\alpha } \\Vert D^2\\phi (x) \\Vert \\right).$ Since $ M_\\varepsilon ^{x}[\\phi ]$ is $\\varepsilon ,x$ -bounded by $\\Vert h \\Vert _{L^\\infty } + \\Vert D\\phi \\Vert _{L^\\infty (\\overline{\\Omega })} $ , we deduce the desired inequality on $\\Vert p_\\text{opt}^M (x) \\Vert $ .",
"Similarly, the estimate on $ \\Vert \\Gamma _\\text{opt}(x) \\Vert $ stems directly from (REF ) and the triangle inequality.",
"In preparation for stability, we need to compute the action of $S_\\varepsilon $ on $\\psi $ .",
"According to Lemma REF , only some cases proposed in Proposition REF must be considered.",
"The next proposition gives the required estimates for $S_\\varepsilon $ concerning these cases.",
"Proposition 4.16 Let $f$ satisfy (REF ) and (REF )–(REF ) and assume $\\alpha $ , $\\beta $ , $\\gamma $ , $\\rho $ fulfill (REF )–(REF ) and (REF ).",
"Then for any $x$ , $t$ , $z$ and any $C_b^{2}(\\overline{\\Omega })$ -function $\\phi $ defined near $x$ , $S_\\varepsilon [x,t,z,\\phi ]$ being defined by (REF ), we have $S_\\varepsilon [x,t,z, \\phi ] - \\phi (x) \\\\\\le {\\left\\lbrace \\begin{array}{ll}C \\varepsilon ^2 (1+|z|) , & \\text{ if } d(x) \\ge \\varepsilon ^{1-\\alpha }, \\\\3 \\varepsilon ^{1-\\alpha } M_\\varepsilon ^{x}[\\phi ] +C \\varepsilon ^2 (1+|z|), &\\text{ if } d(x)\\le \\varepsilon ^{1-\\alpha } \\text{ and } M_\\varepsilon ^{x}[\\phi ] \\ge \\frac{4}{3} \\Vert D^2\\phi (x) \\Vert \\varepsilon ^{1-\\alpha } , \\\\C \\varepsilon ^2 (1+|z|) , & \\text{ if } \\varepsilon ^{1-\\alpha } - \\varepsilon ^{\\rho } \\le d(x) \\le \\varepsilon ^{1-\\alpha }\\text{ and } M_\\varepsilon ^{x}[\\phi ] \\le \\frac{4}{3} \\Vert D^2\\phi (x) \\Vert \\varepsilon ^{1-\\alpha } , \\\\\\frac{1}{4} \\varepsilon ^{\\rho } M_\\varepsilon ^{x}[\\phi ] +C \\varepsilon ^2 (1+|z|), &\\text{ if }d(x) \\le \\varepsilon ^{1-\\alpha } - \\varepsilon ^{\\rho } \\text{ and }M_\\varepsilon ^{x}[\\phi ] \\le - \\varepsilon ^{1-\\alpha - \\kappa } , \\\\\\end{array}\\right.",
"}$ with a constant $C$ that depends on $\\Vert D\\phi \\Vert _{C_b^{1}(\\overline{\\Omega })}+ \\Vert h \\Vert _{L^\\infty }$ but is independent of $x$ , $t$ and $z$ .",
"Moreover, if $d(x)\\ge \\varepsilon ^{1-\\alpha }$ , or if $ d(x)\\le \\varepsilon ^{1-\\alpha }$ and $m_\\varepsilon ^{x}[\\phi ] >\\frac{1}{2} (3\\varepsilon ^{1-\\alpha }-d(x)) \\Vert D^2\\phi (x) \\Vert $ , then $- C\\varepsilon ^2 (1+|z|) \\le S_\\varepsilon [x,t,z, \\phi ] - \\phi (x), $ with a constant $C$ that depends on $\\Vert D\\phi \\Vert _{C_b^{1} (\\overline{\\Omega })}$ but is independent of $x$ , $t$ and $z$ .",
"The arguments in the different cases are the same as those given in the proof of Proposition REF but we must pay attention to the uniformity of the constant.",
"For the second part, since $f$ grows linearly by (REF ) and $\\Vert (D\\phi (x), D^2\\phi (x)) \\Vert \\le \\Vert D\\phi \\Vert _{C_b^{1} (\\overline{\\Omega })} $ , we have $|f(t,x,z,D\\phi (x), D^2\\phi (x))| \\le C(1+|z|),$ with a constant $C$ that depends on $\\Vert D\\phi \\Vert _{C_b^{1} (\\overline{\\Omega })}$ but is independent of $x$ , $t$ and $z$ .",
"The lower bound $S_\\varepsilon [x,t,z,\\phi ] - \\phi (x) \\ge - \\varepsilon ^2 f(x,t,z,D\\phi (x), D^2\\phi (x)) \\ge - C \\varepsilon ^2 (1+|z|)$ is a consequence of Proposition REF and (REF ).",
"Similarly, since we know by Lemma REF that $\\max (\\Vert p_\\text{opt}^m(x) \\Vert , \\Vert p_\\text{opt}^M(x) \\Vert )+ \\Vert \\Gamma _\\text{opt}(x) \\Vert $ is uniformly bounded by $ \\frac{1}{2} \\Vert h \\Vert _{L^\\infty }+3 \\Vert D\\phi \\Vert _{C_b^{1}(\\overline{\\Omega })}$ , we get that $\\max (|f(t,x,z,p_\\text{opt}^m(x), \\Gamma _\\text{opt}(x))|, | f(t,x,z,p_\\text{opt}^M(x), \\Gamma _\\text{opt}(x))|) \\le C(1+|z|),$ with a constant $C$ that depends on $\\Vert D\\phi \\Vert _{C_b^{1}(\\overline{\\Omega })}$ and $\\Vert h \\Vert _{L^\\infty }$ but is independent of $x$ , $t$ and $z$ .",
"We shall prove the estimate for the fourth alternative of (REF ) by examining the proof of Proposition REF case REF , the proofs for the other alternatives being quite similar.",
"Since $f$ is locally Lipschitz by (REF ), $\\min _{p \\in B(p_\\text{opt}^M,r)} f(t,x,z,p, \\Gamma _\\text{opt}) \\ge f(t,x,z,p_\\text{opt}^M(x), \\Gamma _\\text{opt}(x))- C(1+|z|) \\left( 1-\\frac{d(x)}{\\varepsilon ^{1-\\alpha }}\\right) \\left( \\Vert h \\Vert _{L^\\infty }+\\Vert D\\phi \\Vert _{L^\\infty (\\overline{\\Omega })} \\right),$ where $C$ depends only on $\\Vert D\\phi \\Vert _{C_b^{1}(\\overline{\\Omega })}$ and $\\Vert h \\Vert _{L^\\infty }$ by the estimates on $p_\\text{opt}^M$ and $\\Gamma _\\text{opt}$ given by Lemma REF .",
"By using (REF ), we deduce that there exists a constant $C$ depending only on $\\Vert D\\phi \\Vert _{C_b^{1}(\\overline{\\Omega })}$ and $\\Vert h \\Vert _{L^\\infty }$ such that $\\min _{p \\in B(p_\\text{opt}^M,r)} f(t,x,z,p, \\Gamma _\\text{opt}) \\ge - C(1+|z|).$ In case REF , by combining (REF ) and the locally Lipschitz character (REF ) of $f$ on $\\Gamma $ , the estimate (REF ) gets replaced by $f(t,x,z,p,\\Gamma ) \\ge - C(1+|z|) (1+\\varepsilon ^\\alpha ),$ whence by (REF ) there exists a constant $C$ depending on $\\Vert D\\phi \\Vert _{C_b^{1} (\\overline{\\Omega })}+\\Vert h \\Vert _{L^\\infty }$ such that $-\\varepsilon ^2 f(t,x,z,p,\\Gamma )\\le C (1+|z|)\\varepsilon ^2.$ In case REF , since the domain satisfies both the uniform interior and exterior ball conditions, we notice that the constant $C_1$ corresponding to the curvature of the boundary (see Lemma REF ) is $x$ -bounded.",
"This implies that the first order estimate (REF ) is valid independently of $x$ for $\\varepsilon $ sufficiently small.",
"Thus, the estimate (REF ) is valid uniformly in $x$ .",
"Besides, the estimate (REF ) gets replaced by $\\frac{1}{2} \\varepsilon ^{2 \\sigma } \\lambda - \\varepsilon ^2 f(t,x,z, p, \\Gamma _\\text{opt}(x) - \\lambda I)\\le - \\frac{1}{2} \\varepsilon ^{2 \\sigma + \\alpha } +C\\varepsilon ^2(1+|z|) \\Vert p \\Vert \\Vert \\Gamma _\\text{opt} (x)- \\lambda I \\Vert ,$ where $C$ depends on $\\Vert D\\phi \\Vert _{C_b^{1} (\\overline{\\Omega })}+\\Vert h \\Vert _{L^\\infty }$ .",
"We obtain an estimate of the desired form by dropping the first term and observing that $\\lambda $ is bounded.",
"In second half of case REF and in case REF we used the growth estimate (REF ); since $z$ enters linearly on the right-hand side of (REF ), the previous calculation still applies but we get an additional term of the form $C|z|\\varepsilon ^2$ in (REF )–(REF ).",
"The following corollary provides the key estimate for stability in the parabolic setting.",
"Corollary 4.1 Let $f$ satisfy (REF ) and (REF )–(REF ) and assume $\\alpha $ , $\\beta $ , $\\gamma $ fulfill (REF )–(REF ).",
"Then, for any $x$ , $t$ , $z$ and $\\psi \\in C_b^2(\\overline{\\Omega })$ satisfying (REF ), we have $S_\\varepsilon [x,t,z,\\psi ] -\\psi (x) \\le C(1+|z|) \\varepsilon ^2 \\quad \\text{ and } \\quad S_\\varepsilon [x,t,z, - \\psi ] -(-\\psi )(x) \\ge - C(1+|z|) \\varepsilon ^2,$ with a constant $C$ that is independent of $x$ , $t$ , $z$ but depends on $\\Vert D\\psi \\Vert _{C_b^{1}(\\overline{\\Omega })}$ and $\\Vert h \\Vert _{L^\\infty }$ .",
"We shall prove the first estimate, the second follows exactly the same lines.",
"By applying Lemma REF , we have that $M_\\varepsilon ^{x}[ \\psi ] \\le - \\frac{1}{2}$ for all $x \\in \\Omega (\\varepsilon ^{1-\\alpha })$ .",
"We introduce $\\rho $ fulfilling (REF ).",
"By putting together the estimates obtained from (REF ) and the third alternative in (REF ), we get that there exists a constant $C$ depending only on $\\Vert D\\psi \\Vert _{C_b^{1}(\\overline{\\Omega })}$ and $\\Vert h \\Vert _{L^\\infty }$ such that $S_\\varepsilon [x,t,z, \\psi ] - \\psi (x) \\le {\\left\\lbrace \\begin{array}{ll}C \\varepsilon ^2 (1+|z|), &\\text{ if } d(x)\\ge \\varepsilon ^{1-\\alpha } - \\varepsilon ^{\\rho }, \\\\\\frac{1}{4} \\varepsilon ^{\\rho } M_\\varepsilon ^{x}[\\psi ] +C \\varepsilon ^2 (1+|z|) , &\\text{ if } d(x)\\le \\varepsilon ^{1-\\alpha } - \\varepsilon ^{\\rho }.\\end{array}\\right.",
"}$ Noticing that $M_\\varepsilon ^{x}[\\psi ]$ is negative, we get the proposed result."
],
[
"The elliptic case",
"For the game corresponding to the stationary equation, we consider the operator $Q_{\\varepsilon }$ defined for any $x\\in \\overline{\\Omega }$ , $z\\in \\mathbb {R}$ , and any continuous function $\\phi $ : $\\overline{\\Omega }\\rightarrow \\mathbb {R}$ , by $Q_{\\varepsilon }[x,z,\\phi ]=\\sup _{p,\\Gamma } \\inf _{\\Delta \\hat{x}} \\left[ e^{ - \\lambda \\varepsilon ^2} \\phi (x+\\Delta x) \\right.",
"\\\\\\left.",
"- \\left( p\\cdot \\Delta \\hat{x} +\\frac{1}{2} \\left\\langle \\Gamma \\Delta \\hat{x},\\Delta \\hat{x} \\right\\rangle +\\varepsilon ^2 f(x, z,p ,\\Gamma ) - \\Vert \\Delta \\hat{x} - \\Delta x \\Vert h(x+\\Delta x) \\right) \\right] ,$ with the usual conventions that $p$ , $\\Gamma $ and $\\Delta \\hat{x}$ are constrained by (REF ) and (REF ) and that $\\Delta x$ is determined by (REF ).",
"We can easily check that the operator $Q_\\varepsilon $ is still monotone but its action on shifted functions by a constant is described by the following way: for all function $\\phi \\in C(\\overline{\\Omega })$ and $c\\in \\mathbb {R}$ , $Q_{\\varepsilon } \\left[x,z, c + \\phi \\right]=e^{-\\lambda \\varepsilon ^2}c + Q_{\\varepsilon } \\left[x,z, \\phi \\right] .$ The dynamic programming inequalities (REF )–(REF ) can be concisely written as $u^\\varepsilon (x)\\le Q_\\varepsilon [x, u^\\varepsilon (x), u^\\varepsilon ] \\quad \\text{ and } \\quad v^\\varepsilon (x)\\ge Q_\\varepsilon [x, v^\\varepsilon (x), v^\\varepsilon ].$ In the elliptic setting, we can formally derive the PDE by following the same lines as for the parabolic framework.",
"We keep the optimal strategies $p_\\text{opt}^m$ , $p_\\text{opt}^M$ and $\\Gamma _\\text{opt}$ for Helen, defined by (REF ), () and (REF ) in an orthonormal basis $\\mathcal {B}=(e_1=n(\\bar{x}), e_2,\\cdots , e_N)$ .",
"The next proposition is the elliptic analogue of Propositions REF and REF .",
"It establishes the consistency estimates for $Q_\\varepsilon $ defined by (REF ).",
"Proposition 4.17 Let $f$ satisfy (REF ) and (REF )–(REF ) and assume $\\alpha $ , $\\beta $ , $\\gamma $ and $\\rho $ fulfill (REF )–(REF ) and (REF ).",
"Let $p_\\text{opt}^m$ , $p_\\text{opt}^M$ and $\\Gamma _\\text{opt}$ be respectively defined in the orthonormal basis $\\mathcal {B}=(e_1=~n(\\bar{x}), e_2,\\cdots , e_N)$ by (REF )–(REF ).",
"For any $x$ , $z$ and any smooth function $\\phi $ defined near $x$ , we distinguish two cases for the lower bound estimate: Big bonus: if $ d(x)\\ge \\varepsilon ^{1-\\alpha }$ or $ m_\\varepsilon ^x[\\phi ] >\\frac{1}{2} (3\\varepsilon ^{1-\\alpha }-d(x)) \\Vert D^2\\phi (x) \\Vert $ , then $- \\varepsilon ^2 (f(x,z, D\\phi (x), D^2\\phi (x))+\\lambda \\phi (x)) \\le Q_\\varepsilon [x,z,\\phi ] - \\phi (x) .",
"$ Penalty or small bonus: if $d(x)\\le \\varepsilon ^{1-\\alpha }$ and $ m_\\varepsilon ^x[\\phi ] \\le \\frac{1}{2} (3\\varepsilon ^{1-\\alpha }-d(x)) \\Vert D^2\\phi (x) \\Vert $ , then $\\frac{1}{2} (\\varepsilon ^{1-\\alpha } - d(x)) \\left( s m_\\varepsilon ^x[\\phi ] - 4 \\Vert D^2\\phi (x) \\Vert \\varepsilon ^{1-\\alpha } \\right)- \\varepsilon ^2 (f(x,z, p_{\\text{opt}}^m(x), \\Gamma _\\text{opt}(x)) +\\lambda \\phi (x)) \\le Q_\\varepsilon [x,z,\\phi ] - \\phi (x), $ where $s=-1$ if $m_\\varepsilon ^x[\\phi ]\\ge 0$ and $s=3$ if $m_\\varepsilon ^x[\\phi ]< 0$ .",
"For the upper bound estimate, we distinguish four cases: Big bonus: if $d(x) \\le \\varepsilon ^{1-\\alpha }$ and $M_\\varepsilon ^x[\\phi ] >\\frac{4}{3} \\Vert D^2\\phi (x) \\Vert \\varepsilon ^{1-\\alpha }$ , then $Q_\\varepsilon [x,z,\\phi ]-\\phi (x) \\le 3(\\varepsilon ^{1-\\alpha }-d(x))M_\\varepsilon ^x[\\phi ] -\\varepsilon ^2 \\left(f(x,z,p_\\text{opt}^M(x),\\Gamma _\\text{opt}(x))+\\lambda \\phi (x)\\right) +o(\\varepsilon ^2).$ Far from the boundary with a small bonus: if $\\varepsilon ^{1-\\alpha }-\\varepsilon ^{\\rho } \\le d(x)\\le \\varepsilon ^{1-\\alpha }$ and $M_\\varepsilon ^x[\\phi ] \\le \\frac{4}{3} \\Vert D^2\\phi (x) \\Vert \\varepsilon ^{1-\\alpha }$ , or if $d(x)\\ge \\varepsilon ^{1-\\alpha }$ , then $Q_\\varepsilon [x,z,\\phi ] - \\phi (x) \\le -\\varepsilon ^2 \\left(f(x,z, D\\phi (x), D^2\\phi (x))+\\lambda \\phi (x) \\right) +o(\\varepsilon ^2).$ Close to the boundary with a small bonus/penalty: if $d(x)\\le \\varepsilon ^{1-\\alpha } - \\varepsilon ^{\\rho }$ and $- \\varepsilon ^{1-\\alpha -\\kappa } \\le M_\\varepsilon ^x[\\phi ] \\le \\frac{4}{3} \\Vert D^2\\phi (x) \\Vert \\varepsilon ^{1-\\alpha } $ , then $Q_\\varepsilon [x,z,\\phi ] - \\phi (x) \\le -\\varepsilon ^2 \\left(f(x,z, D\\phi (x), D^2\\phi (x)+C_1I)+\\lambda \\phi (x) \\right) +o(\\varepsilon ^2),$ with $C_1=\\frac{20}{3} \\Vert D^2\\phi (x) \\Vert \\left(1-\\frac{d(x)}{\\varepsilon ^{1-\\alpha }}\\right)$ .",
"Close to the boundary with a big bonus: if $d(x)\\le \\varepsilon ^{1-\\alpha }-\\varepsilon ^{\\rho }$ and $M_\\varepsilon ^x[\\phi ] \\le - \\varepsilon ^{1-\\alpha -\\kappa }$ , then $Q_\\varepsilon [x,z,\\phi ] - \\phi (x) \\le \\frac{1}{4} (\\varepsilon ^{1-\\alpha } - d(x)) M_\\varepsilon ^x[\\phi ]-\\varepsilon ^2 \\left( \\min _{p\\in B(p_\\text{opt}^M(x),r)} f(x,z,p, \\Gamma _\\text{opt}(x))+\\lambda \\phi (x)\\right) +o(\\varepsilon ^2),$ with $r$ defined by $r=3 \\left(1-\\frac{d(x)}{\\varepsilon ^{1-\\alpha }} \\right) |M_\\varepsilon ^x[\\phi ]|$ .",
"Moreover the implicit constants in the error term are uniform as $x$ and $z$ range over a compact subset of $\\overline{\\Omega }\\times \\mathbb {R}$ .",
"The arguments are entirely parallel to the proofs of Propositions REF and REF .",
"For stability we will need a variant of the preceding lemma.",
"This is where we use the hypothesis (REF ) on the $z$ -dependence of $f$ .",
"Lemma 4.18 Let $f$ satisfy (REF ) and (REF )–(REF ) and assume as always that $\\alpha $ , $\\beta $ , $\\gamma $ satisfy (REF )–(REF ).",
"Let $\\psi \\in C_b^2 (\\overline{\\Omega })$ satisfy (REF ).",
"Fix $M$ and $m$ two positive constants such that $ m + 2 \\Vert \\psi \\Vert _{L^\\infty (\\overline{\\Omega })} \\le M$ .",
"Then, there exists $C_\\ast =C_\\ast (\\Vert D\\psi \\Vert _{C_b^1(\\overline{\\Omega })}, \\Vert h \\Vert _{L^\\infty })$ such that for any $|z|\\le M$ and any $x\\in \\overline{\\Omega }$ , we have $Q_\\varepsilon \\left[x,z, m +\\psi \\right] - \\left(m + \\psi (x) \\right) \\le \\varepsilon ^2 \\left(1+ (\\lambda -\\eta )|z|+C_\\ast \\right) - \\lambda \\varepsilon ^2 \\left(m+ \\psi (x)\\right) ,$ and $Q_\\varepsilon \\left[x,z,- m- \\psi \\right] - \\left(- m -\\psi (x)\\right) \\ge -\\varepsilon ^2 \\left(1+ (\\lambda -\\eta )|z|+C_\\ast \\right) - \\lambda \\varepsilon ^2 \\left(-m-\\psi (x)\\right),$ for all sufficiently small $\\varepsilon $ (the smallness condition on $\\varepsilon $ depends on $M$ , but not on $x$ ).",
"Moreover, if $\\phi \\in C_b^2 (\\overline{\\Omega })$ , then there exists $C=C(M, \\Vert D\\phi \\Vert _{C_b^1(\\overline{\\Omega })}, \\Vert h \\Vert _{L^\\infty })$ such that for any $|z|\\le M$ and any $x\\in \\overline{\\Omega }$ such that $d(x)\\le \\varepsilon ^{1-\\alpha }-\\varepsilon ^{\\rho }$ and $M_\\varepsilon ^x[\\phi ] \\le - \\varepsilon ^{1-\\alpha -\\kappa }$ , $Q_\\varepsilon [x,z, \\phi ] - \\phi (x) \\le \\frac{1}{4} \\left( \\varepsilon ^{1-\\alpha } - d(x) \\right) M_\\varepsilon ^x[\\phi ] +C\\varepsilon ^2 - \\lambda \\varepsilon ^2 \\phi (x) ,$ for all sufficiently small $\\varepsilon $ (the smallness condition on $\\varepsilon $ depends on $M$ , but not on $x$ ).",
"We shall prove the first inequality, the proof of the second being entirely parallel.",
"The assumption $|z| \\le M$ ensures that the constants in (REF ) and (REF ) are uniform.",
"Then the implicit constants in the error terms of (REF ) and (REF ) are $x$ ,$z$ -uniform for $\\varepsilon $ small enough, and the smallness condition depends only on $M$ .",
"Since $ m + 2 \\Vert \\psi \\Vert _{L^\\infty (\\overline{\\Omega })}\\le M$ we can use the dynamic programming inequalities (REF )–(REF ).",
"First of all, by the action of $Q_\\varepsilon $ on constant functions provided by (REF ), we have $Q_\\varepsilon [x,z, m +\\psi ] - (m + \\psi (x))=(e^{-\\lambda \\varepsilon ^2} - 1)m + Q_\\varepsilon [x,z, \\psi ] - \\psi (x),$ and noticing that $e^{-\\lambda \\varepsilon ^2}m=(1-\\lambda \\varepsilon ^2)m +O(\\varepsilon ^4m)$ , it is sufficient to get the estimate corresponding to $m=0$ .",
"By Lemma REF , we observe that every $x\\in \\Omega (\\varepsilon ^{1-\\alpha })$ satisfies $M_\\varepsilon ^{x}[ \\psi ] \\le - \\frac{1}{2}$ .",
"We now need to distinguish two cases according to the distance to the boundary by introducing $\\rho $ fulfilling (REF ).",
"If $x\\in \\overline{\\Omega }$ such that $d(x) \\ge \\varepsilon ^{1-\\alpha } - \\varepsilon ^{\\rho }$ , since $\\Vert (D\\psi (x), D^2 \\psi (x)) \\Vert \\le K_1= \\Vert D\\psi \\Vert _{C_b^1(\\overline{\\Omega })}$ , we deduce by assumption (REF ) on $f$ that there exists $C_{K_1}^\\ast $ such that for all $x$ we have $|f(x,z, D\\psi (x), D^2 \\psi (x))| \\le (\\lambda -\\eta )|z|+C_{K_1}^\\ast , $ which gives by (REF ) that for all $x\\in \\overline{\\Omega }$ such that $d(x)\\ge \\varepsilon ^{1-\\alpha }$ , $ Q_\\varepsilon [x,z,\\psi ] - \\psi (x) \\le \\varepsilon ^2 \\left((\\lambda -\\eta )|z|+C_{K_1}^\\ast \\right) - \\lambda \\varepsilon ^2 \\psi (x)+o(\\varepsilon ^2).$ If $x\\in \\overline{\\Omega }$ such that $d(x) \\le \\varepsilon ^{1-\\alpha } - \\varepsilon ^{\\rho }$ , combining the triangle inequality with the inequalities given by Lemma REF gives that, for all $ p\\in B \\left(p_\\text{opt}^M(x),r \\right)$ with $ r=3\\left(1-\\frac{d(x)}{\\varepsilon ^{1-\\alpha }}\\right) |M_\\varepsilon ^x[\\psi ]|$ , $\\Vert (p, \\Gamma _\\text{opt}(x) ) \\Vert \\le \\Vert p_\\text{opt}^M(x) \\Vert _{L^\\infty }+r+\\Vert \\Gamma _\\text{opt}(x) \\Vert _{L^\\infty }\\le K_2= \\frac{7}{2} \\Vert h \\Vert _{L^\\infty }+6 \\Vert D\\psi \\Vert _{C_b^1(\\overline{\\Omega })},$ since $ M_\\varepsilon ^{x}[\\psi ]$ is $\\varepsilon ,x$ -bounded by $\\Vert h \\Vert _{L^\\infty } + \\Vert D\\psi \\Vert _{L^\\infty } $ .",
"The assumption (REF ) on $f$ yields that there exists $C_{K_2}^\\ast $ such that, $ \\left|\\min _{p\\in B(p_\\text{opt}^M(x),r)} f(x,z,p, \\Gamma _\\text{opt}(x) ) \\right|\\le (\\lambda -\\eta )|z|+C_{K_2}^\\ast ,$ By using this inequality in (REF ) and recalling that $M_\\varepsilon ^x[\\psi ]\\le -\\frac{1}{2}$ , we conclude that, for all $x\\in \\overline{\\Omega }$ such that $d(x) \\le \\varepsilon ^{1-\\alpha } - \\varepsilon ^{\\rho }$ , $ Q_\\varepsilon [x,z,\\psi ] - \\psi (x) \\le \\varepsilon ^2 \\left( (\\lambda -\\eta )|z|+C_{K_2}^\\ast \\right) - \\lambda \\varepsilon ^2 \\psi (x) +o(\\varepsilon ^2).$ By comparing (REF ) and (REF ) we get the desired result by taking $C_\\ast =\\max (C^\\ast _{K_1},C^\\ast _{K_2})$ .",
"To prove the third inequality, it is sufficient to replace the assumption (REF ) by (REF ) in the previous estimates.",
"For instance, instead of (REF ), there exists a constant $C$ depending only on $M$ , $\\Vert h \\Vert _{L^\\infty }$ , and $\\Vert D\\phi \\Vert _{C_b^1(\\overline{\\Omega })}$ such that $\\displaystyle \\left|\\min _{p\\in B(p_\\text{opt}^M(x),r)} f(x,z,p, \\Gamma _\\text{opt}(x) ) \\right| \\le C$ .",
"The rest of the proof remains unchanged."
],
[
"Stability",
"In the time-dependent setting, we showed in Section REF that if $v^\\varepsilon $ and $u^\\varepsilon $ remain bounded as $\\varepsilon \\rightarrow 0$ then $\\underline{v}$ is a supersolution and $\\bar{u}$ is a subsolution.",
"The argument was local, using mainly the consistency of the game as a numerical scheme.",
"It remains to prove that $v^\\varepsilon $ and $u^\\varepsilon $ are indeed bounded; this is achieved in Section REF .",
"For the stationary setting, we must do more.",
"Even the existence of $U^\\varepsilon (x,z)$ remains to be proved.",
"We also need to show that the associated functions $u^\\varepsilon $ and $v^\\varepsilon $ are bounded, away from $M$ , so that we can apply the dynamic programming inequalities at each $x \\in \\overline{\\Omega }$ .",
"These goals will be achieved in Section REF , provided the parameters $M$ and $m$ satisfy (i) $m=M-1 - 2 \\Vert \\psi \\Vert _{L^\\infty }$ and (ii) $M$ is sufficiently large.",
"We also show in Section REF that if $f$ is a nondecreasing function on $z$ then $U^\\varepsilon $ is strictly decreasing on $z$ .",
"As a consequence, this result implies that $\\underline{v} \\le \\bar{u}$ , allowing us to conclude that $\\underline{v} = \\bar{u}$ is the unique viscosity solution if the boundary value problem has a comparison principle."
],
[
"The parabolic case",
"To obtain stability, we are going to consider one more time a $C^2_b(\\overline{\\Omega })$ -function $\\psi $ such that $\\frac{\\partial \\psi }{\\partial n }=\\Vert h \\Vert _{L^\\infty }+1$ in order to take care of the Neumann boundary condition.",
"Proposition 5.1 Assume the hypotheses of Propositions REF and REF hold, and suppose furthermore that the final-time data are uniformly bounded: $|g(x)| \\le B \\quad \\text{ for all } x\\in \\overline{\\Omega }.$ Then there exists a constant $s=s(\\Vert \\psi \\Vert _{C_b^2(\\overline{\\Omega })})$ , independent of $\\varepsilon $ , such that $u^{\\varepsilon }(x,t) \\le (B+\\Vert \\psi \\Vert _{L^\\infty (\\overline{\\Omega })}) s^{T-t} +\\psi (x) \\quad \\text{ for all } x\\in \\overline{\\Omega },$ and $v^{\\varepsilon }(x,t) \\ge - (B+\\Vert \\psi \\Vert _{L^\\infty (\\overline{\\Omega })}) s^{T-t} - \\psi (x) \\quad \\text{ for all } x\\in \\overline{\\Omega },$ for every $t<T$ .",
"We shall demonstrate the lower bound on $v^{\\varepsilon }$ ; the proof of the upper bound on $u^\\varepsilon $ is entirely parallel.",
"The argument proceeds backward in time $t_k=T-k \\varepsilon ^2$ .",
"At $k=0$ , we have a uniform bound $v^{\\varepsilon }(x,T)=g(x)\\ge - B$ by hypothesis, and we may assume without loss of generality that $B\\ge 1$ .",
"Since $\\psi $ is bounded on $\\overline{\\Omega }$ , we can suppose that $v^{\\varepsilon }(x,T)=g(x)\\ge - B_0 - \\psi (x),$ where $B_0= B + \\Vert \\psi \\Vert _{L^\\infty (\\overline{\\Omega })}$ .",
"Now suppose that for fixed $k\\ge 0$ we already know a bound $v^\\varepsilon (\\cdot ,t_k)\\ge - B_k -\\psi $ .",
"By the dynamic programming inequality (REF ), we have $v^{\\varepsilon }(x,t_k-\\varepsilon ^2) \\ge S_{\\varepsilon } \\left[x,t,v^{\\varepsilon }(x,t_k-\\varepsilon ^2), v^{\\varepsilon }(.,t_k) \\right] .$ Since $S_{\\varepsilon }$ is monotone in its last argument, we have $v^{\\varepsilon }(x,t_k-\\varepsilon ^2) \\ge S_{\\varepsilon } \\left[x,t,v^{\\varepsilon }(x,t_k-\\varepsilon ^2), - B_k - \\psi \\right] .$ By applying successively (REF ) and Corollary REF , we deduce that $S_{\\varepsilon } \\left[x,t,v^{\\varepsilon }(x,t_k-\\varepsilon ^2), - B_k - \\psi \\right] & = - B_k + S_{\\varepsilon } \\left[x,t,v^{\\varepsilon }(x,t_k-\\varepsilon ^2), - \\psi \\right] \\\\& \\ge - B_k - \\psi (x) - C(1+|v^{\\varepsilon }(x,t_k-\\varepsilon ^2)|)\\varepsilon ^2 ,$ where $C$ depends only on $\\Vert D\\psi \\Vert _{C_b^1(\\overline{\\Omega })}$ .",
"If $v^{\\varepsilon }(x,t_k-\\varepsilon ^2) \\ge 0$ , then it is over (recall we are looking for a lower bound $ - B_{k+1}\\le -1$ ).",
"Otherwise, we have $(1-C\\varepsilon ^2)v^{\\varepsilon }(x,t_k-\\varepsilon ^2) \\ge - B_k - C\\varepsilon ^2 - \\psi (x).$ By dividing by $1-C\\varepsilon ^2$ , we get $v^{\\varepsilon }(x,t_k-\\varepsilon ^2) \\ge - \\frac{B_k + C\\varepsilon ^2}{1-C\\varepsilon ^2} - \\frac{1}{1-C\\varepsilon ^2}\\psi (x)=- \\frac{B_k + C\\varepsilon ^2 (1+ \\psi (x)) }{1 - C\\varepsilon ^2} - \\psi (x).$ Then, by setting $\\displaystyle B_{k+1}= \\frac{B_k + C(1+ \\Vert \\psi \\Vert _{L^\\infty (\\overline{\\Omega })}) \\varepsilon ^2 }{1 - C\\varepsilon ^2},$ we obtain $v^{\\varepsilon }(x,t_k-\\varepsilon ^2) \\ge -B_{k+1} - \\psi (x).$ As it is clear that $\\displaystyle B_{k+1}\\le B_k \\frac{1 + C(1+ \\Vert \\psi \\Vert _{L^\\infty (\\overline{\\Omega })}) \\varepsilon ^2}{1 - C\\varepsilon ^2}$ , we deduce that $v^{\\varepsilon }(x,T-k\\varepsilon ^2)\\ge \\tilde{B}_k - \\psi (x)$ for all $k$ with $\\tilde{B}_k=B_0 \\left( \\frac{1 +C(1+ \\Vert \\psi \\Vert _{L^\\infty (\\overline{\\Omega })})\\varepsilon ^2}{1 - C\\varepsilon ^2} \\right)^k.$ Since $k=(T-t)/\\varepsilon ^2$ and recalling that $B_0= B + \\Vert \\psi \\Vert _{L^\\infty (\\overline{\\Omega })}$ , we have shown that $v_{\\varepsilon }(x,t)\\ge - (B + \\Vert \\psi \\Vert _{L^\\infty (\\overline{\\Omega })}) s_\\varepsilon ^{T-t} - \\psi (x)$ with $s_{\\varepsilon }=\\left( \\frac{1 + C(1+ \\Vert \\psi \\Vert _{L^\\infty (\\overline{\\Omega })}) \\varepsilon ^2}{1 - C\\varepsilon ^2} \\right)^{1/\\varepsilon ^2}.$ Since $s_{\\varepsilon }$ has a finite limit as $\\varepsilon \\rightarrow 0$ we obtain a bound on $v^\\varepsilon $ of the desired form.",
"Remark 5.2 By following the construction of the elliptic game we can take $\\psi =(\\Vert h \\Vert _{L^\\infty }+1)\\psi _1$ where $\\psi _1$ is defined by (REF ).",
"In that case, $\\Vert D\\psi \\Vert _{C_b^1(\\overline{\\Omega })}= \\Vert D\\psi _1 \\Vert _{C_b^1(\\overline{\\Omega })} (1+\\Vert h \\Vert _{L^\\infty })$ .",
"This expression can be compared for a $C^{2,\\alpha }$ -domain to the estimate given by Remark REF provided by the Schauder theory for which $\\Vert D\\psi _1 \\Vert _{C_b^1(\\overline{\\Omega })}$ plays the role of the constant $C_\\Omega $ depending only on the domain."
],
[
"The elliptic case",
"We shall assume throughout this section that the parameters $M$ and $m$ controlling the termination of the game are related by $m=M-1- 2 \\Vert \\psi \\Vert _{L^\\infty (\\overline{\\Omega })}$ ; in addition, we need to assume $M$ is sufficiently large.",
"Our plan is to show, using a fixed point argument, the existence of a function $U^\\varepsilon (x,z)$ (defined for all $x\\in \\overline{\\Omega }$ and $|z|<M$ ) satisfying (REF ) and also $- z - \\chi (x) \\le U^\\varepsilon (x,z) \\le - z+ \\chi (x).$ This implies that $U^\\varepsilon (x,z)<0$ when $z>\\chi (x)$ , and $U^\\varepsilon (x,z)>0$ when $z<- \\chi (x)$ .",
"Recalling the definitions of $u^\\varepsilon $ and $v^\\varepsilon $ , it follows from (REF )–() that $ |v^\\varepsilon (x)| \\le \\chi (x) ,\\quad |u^\\varepsilon (x) | \\le \\chi (x),$ for all $x \\in \\overline{\\Omega }$ .",
"It is convenient to work with $V^\\varepsilon (x,z) =U^\\varepsilon (x,z) + z$ rather than $U^\\varepsilon $ , since this turns (REF ) into $|V^\\varepsilon (x,z)| \\le \\chi (x),$ whose right-hand side is not constant.",
"The dynamic programming principle (REF ) for $U^\\varepsilon $ is equivalent (after a bit of manipulation) to the statement that $V^\\varepsilon $ is a fixed point of the mapping $\\phi (\\cdot , \\cdot ) \\mapsto R_\\varepsilon [\\cdot ,\\cdot , \\phi ]$ where the operator $R_\\varepsilon $ is defined for any $L^\\infty $ -function $\\phi $ defined on $\\overline{\\Omega }\\times (-M,M)$ by $R^\\varepsilon [x,z,\\phi ]= \\sup _{p, \\Gamma }\\inf _{\\Delta \\hat{x} }{\\left\\lbrace \\begin{array}{ll}e^{-\\lambda \\varepsilon ^2} \\phi (x^{\\prime },z^{\\prime }) - \\delta , & \\text{if } |z^{\\prime }|< M, \\\\- \\chi (x), & \\text{if } z^{\\prime }\\ge M, \\\\\\chi (x), & \\text{if } z^{\\prime }\\le -M.\\end{array}\\right.",
"}$ where $x^{\\prime }=x+\\Delta x$ and $z^{\\prime }=e^{\\lambda \\varepsilon ^2}(z+\\delta )$ , with $\\delta $ defined as in (REF ).",
"Here $p$ , $\\Gamma $ and $\\Delta \\hat{x}$ are constrained as usual by (REF )–(REF ).",
"We shall identify $V^\\varepsilon $ as the unique fixed point of the mapping $\\phi (\\cdot , \\cdot ) \\mapsto R_\\varepsilon [\\cdot ,\\cdot , \\phi ]$ in $F_\\chi $ defined by $ F_\\chi = \\left\\lbrace \\phi \\in L^\\infty \\left(\\overline{\\Omega }\\times \\left(-M,M \\right)\\right) : \\forall (x,z)\\in \\overline{\\Omega }\\times (-M,M), |\\phi (x,z)| \\le \\chi (x) \\right\\rbrace .$ Lemma 5.3 Let $f$ satisfy (REF ) and (REF )–(REF ) and assume as always that $\\alpha $ , $\\beta $ , $\\gamma $ fulfill (REF )–(REF ) and that $\\Omega $ is a $C^2$ -domain satisfying both the uniform interior and exterior ball conditions.",
"Then, there exists $M_0>0$ such that for all two positive constants $m$ and $M> M_0$ satisfying $m + 2\\Vert \\psi \\Vert _{L^\\infty (\\overline{\\Omega })} = M-1$ , for any $|z|\\le M$ and any $x\\in \\overline{\\Omega }$ , we have $Q_\\varepsilon [x,z,\\chi ] \\le \\chi (x) \\quad \\text{ and } \\quad Q_\\varepsilon [x,z, - \\chi ] \\ge -\\chi (x).$ We are going to establish the upper estimate for $\\chi $ .",
"By Lemma REF , we deduce that $Q_\\varepsilon [x,z,\\chi ] - \\chi (x) \\le \\varepsilon ^2 \\Big (1+ (\\lambda -\\eta )|z|+C_\\ast \\Big ) - \\lambda \\varepsilon ^2 (m+ \\Vert \\psi \\Vert _{L^\\infty (\\overline{\\Omega })} + \\psi (x)) .$ Since $m + 2\\Vert \\psi \\Vert _{L^\\infty (\\overline{\\Omega })} = M-1$ and $|z|\\le M$ , we compute $Q_\\varepsilon [x,z,\\chi ] - \\chi (x)\\le \\varepsilon ^2 \\Big (1+ (\\lambda -\\eta )M +C_\\ast \\Big ) - \\lambda \\varepsilon ^2 \\left(M-1 - \\Vert \\psi \\Vert _{L^\\infty (\\overline{\\Omega })} + \\psi (x) \\right).$ By rearranging the terms, we obtain that $Q_\\varepsilon [x,z,\\chi ] - \\chi (x) \\le \\varepsilon ^2 \\left(1+ \\lambda (1+2\\Vert \\psi \\Vert _{L^\\infty (\\overline{\\Omega })}) + C_\\ast -\\eta M \\right) .$ We can choose $M$ large enough such that the right-hand side is negative.",
"It suffices to take $M> M_0:=\\frac{1}{\\eta } \\left( 1 + \\lambda (1+2\\Vert \\psi \\Vert _{L^\\infty (\\overline{\\Omega })}) +C_\\ast \\right).$ The case for $Q_\\varepsilon [x,z, - \\chi ] \\ge - \\chi (x)$ is analogous.",
"Proposition 5.4 Assume the hypotheses of Lemma REF hold.",
"Suppose further that $m=M - 1-2 \\Vert \\psi \\Vert _{L^\\infty (\\overline{\\Omega })}$ .",
"Then for all sufficiently small $\\varepsilon $ , the map $\\phi (\\cdot ,\\cdot ) \\mapsto R_\\varepsilon [\\cdot , \\cdot , \\phi ]$ is a contraction in the $L^\\infty $ -norm, which preserves $F_\\chi $ .",
"In particular, it has a unique fixed point, which has $L^\\infty $ -norm at most $m+2 \\Vert \\psi \\Vert _{L^\\infty (\\overline{\\Omega })}$ .",
"By the arguments already used in [21], the map is a contraction for any $\\varepsilon $ (this part of the proof works for any $M$ ).",
"More precisely, if $\\phi _i$ , $i=1,2$ are two $L^\\infty $ -functions defined on $\\overline{\\Omega }\\times (-M,M)$ to $\\mathbb {R}$ , then $\\displaystyle \\Vert R_\\varepsilon [\\cdot ,\\cdot , \\phi _1] - R_\\varepsilon [\\cdot , \\cdot , \\phi _2] \\Vert _{L^\\infty } \\le e^{-\\lambda \\varepsilon ^2} \\Vert \\phi _1 - \\phi _2 \\Vert _{L^\\infty }$ .",
"Now we prove that if $M$ is large enough and $m + 2\\Vert \\psi \\Vert _{L^\\infty (\\overline{\\Omega })} = M-1$ , the map preserves the ball $F_\\chi $ defined by (REF ).",
"Since $R_\\varepsilon [x,z, \\phi ]$ is monotone in its last argument, it suffices to show that $ R_\\varepsilon [x,z, \\chi ] \\le \\chi (x) \\quad \\text{ and } \\quad R_\\varepsilon [x,z, -\\chi ] \\ge - \\chi (x).$ For the first inequality of (REF ), let $p$ and $\\Gamma $ be fixed, and consider $\\inf _{\\Delta \\hat{x} }{\\left\\lbrace \\begin{array}{ll}e^{-\\lambda \\varepsilon ^2} \\chi (x^{\\prime }) - \\delta , & \\text{if } |z^{\\prime }|< M , \\\\- \\chi (x) , & \\text{if } z^{\\prime } \\ge M, \\\\\\chi (x) , & \\text{if } z^{\\prime } \\le -M.\\end{array}\\right.",
"}$ If a minimizing sequence uses the second or third alternative then the inf is less than $\\chi (x)$ .",
"In the remaining case, when all minimizing sequences use the first alternative, we apply Lemma REF to see that (REF ) is bounded above by $ \\chi (x) $ .",
"It follows that for all $x\\in \\overline{\\Omega }$ , $R_\\varepsilon [x,z, \\chi ] \\le \\chi (x)$ , as asserted.",
"For the second inequality of (REF ), the argument is strictly parallel by considering the function $-\\chi $ .",
"We have shown that the map $\\phi (\\cdot ,\\cdot ) \\mapsto R_\\varepsilon [\\cdot ,\\cdot , \\phi ]$ preserves the ball $F_{\\chi }$ .",
"Since it is also a contraction, the map has a unique fixed point.",
"This result justify the discussion of the stationary case given in Section , by showing that the value functions $u^\\varepsilon $ and $v^\\varepsilon $ are well-defined, and bounded independently of $\\varepsilon $ , and they satisfy the dynamic programming inequalities: Proposition 5.5 Suppose $f$ satisfies (REF ) and (REF )–(REF ), the $C^2$ -domain $\\Omega $ fulfills both the uniform interior and exterior ball conditions, and the boundary condition $h$ is continuous, uniformly bounded.",
"Assume the parameters of the game $\\alpha , \\beta , \\gamma $ fulfill (REF )–(REF ), $\\psi \\in C_b^2(\\overline{\\Omega })$ satisfy (REF ), $M$ large enough, $m=M-1 - 2 \\Vert \\psi \\Vert _{L^\\infty (\\overline{\\Omega })}$ , and $\\chi \\in C_b^2(\\overline{\\Omega })$ is defined by (REF ).",
"Let $V^\\varepsilon $ be the solution of (REF ) obtained by Proposition REF and let $U^\\varepsilon (x,z)=V^\\varepsilon (x,z)-z$ .",
"Then the associated functions $u^\\varepsilon $ , $v^\\varepsilon $ defined by (REF )–() satisfy $ |u^\\varepsilon | \\le \\chi $ and $ |v^\\varepsilon | \\le \\chi $ for all sufficiently small $\\varepsilon $ , and they satisfy the dynamic programming inequalities (REF ) and (REF ) at all points $x\\in \\overline{\\Omega }$ .",
"The bounds on $u^\\varepsilon $ and $v^\\varepsilon $ were demonstrated in (REF ).",
"The bounds assure that the dynamic programming inequalities hold for all $x\\in \\overline{\\Omega }$ , as a consequence of Proposition REF .",
"We close this section with the stationary analogue of Lemma REF .",
"Lemma 5.6 Under the hypotheses of Proposition REF , suppose in addition that $f(x,z_1,p, \\Gamma )\\ge f(x,z_0,p, \\Gamma ) \\quad \\text{ whenever }z_1>z_0.$ Then $U^\\varepsilon $ satisfies $U^\\varepsilon (x,z_1)\\le U^\\varepsilon (x,z_0) - (z_1-z_0) \\quad \\text{ whenever }z_1>z_0.$ In particular, $U^\\varepsilon $ is strictly decreasing in $z$ and $v^\\varepsilon = u^\\varepsilon $ .",
"The Dirichlet case is provided in [21].",
"For our game, it suffices to add $ - \\Vert \\Delta \\hat{x} - \\Delta x \\Vert h(x+\\Delta x)$ in the expression of $\\delta _0$ and $\\delta _1$ defined in the proof of [21].",
"Then the arguments can be repeated on the operator $R_\\varepsilon $ defined by (REF ), noticing that the function $\\chi $ is independent of $z$ ."
],
[
"Some natural generalizations",
"In the precedent sections, we solved the Neumann boundary problem in both parabolic and elliptic settings.",
"In the present section, we are going to explain without full proof how the previous work can be used to solve on the one hand the mixed Dirichlet-Neumann boundary conditions in the elliptic framework and on the other hand the oblique problem in the parabolic setting.",
"For the definitions of the viscosity solutions on these frameworks which are the natural extensions of those presented in Section REF , the interested reader is referred to [12] or [4]."
],
[
"Elliptic PDE with mixed Dirichlet-Neumann boundary conditions",
"We extend the games of Section REF devoted to the single Neumann problem to the mixed Dirichlet-Neumann boundary-value problem ${\\left\\lbrace \\begin{array}{ll}f(x,u,Du,D^2u)+\\lambda u=0, & \\text{ in } \\Omega , \\\\u =g, & \\text{ on } \\Upsilon _D, \\\\\\dfrac{\\partial u}{\\partial n}=h, & \\text{ on } \\Upsilon _N, \\\\\\end{array}\\right.",
"}$ where $\\Omega \\subsetneq \\mathbb {R}^N$ is a domain, $\\Upsilon _D \\cup \\Upsilon _N= \\partial \\Omega $ is a partition of $\\partial \\Omega $ with $\\Upsilon _D$ nonempty and closed and $\\Upsilon _N$ is assumed to be $C^2$ .",
"Then, $\\Omega $ is assumed to satisfy the uniform exterior ball condition and, in a neighborhood of $\\Upsilon _N$ , the uniform interior ball condition explained in Definition REF .",
"We will need a $C_b^2(\\overline{\\Omega })$ -function $\\psi $ such that $ \\frac{\\partial \\psi }{\\partial n} = \\Vert h \\Vert _{L^\\infty }+1 \\quad \\text{on } \\Upsilon _N.$ From $m$ and $\\psi $ , we construct a function $\\chi $ defined by $\\chi (x) = m+\\Vert \\psi \\Vert _{L^\\infty }+\\psi (x).$ As in Section REF , we introduce $U^\\varepsilon (x,z)$ , the optimal worst-case present value of Helen's wealth if the initial stock is $x$ and her initial wealth is $-z$ .",
"The definition of $U^\\varepsilon (x,z)$ for $x\\in \\Omega \\cup \\Upsilon _N$ involves here a game similar to that of Section REF .",
"The rules are as follows: Initially, at time $t_0=0$ , the stock price is $x_0=x$ and Helen's debt is $z_0=z$ .",
"Suppose, at time $t_j=j\\varepsilon ^2$ , the stock price is $x_j$ and Helen's debt is $z_j$ with $|z_j|<M$ .",
"Then Helen chooses $p_j \\in \\mathbb {R}^N$ and $\\Gamma _j\\in \\mathcal {S}^N$ , restricted in magnitude by (REF ).",
"Knowing these choices, Mark determines the next stock price $ x_{j+1}=x_j+\\Delta x$ so as to degrade Helen's outcome.",
"Mark chooses an intermediate point $\\hat{x}_{j+1}=x_j +\\Delta \\hat{x}_j \\in \\mathbb {R}^N$ such that $\\left\\Vert \\Delta \\hat{x}_j\\right\\Vert \\le \\varepsilon ^{1-\\alpha }$ .",
"This position $\\hat{x}_{j+1}$ determines the new position $x_{j+1}=x_j +\\Delta x_j$ by $x_{j+1}= \\operatorname{proj}_{\\overline{\\Omega }} (\\hat{x}_{j+1}) \\in \\overline{\\Omega }.$ Helen experiences a loss at time $t_j$ of $\\delta _j = p_j\\cdot \\Delta \\hat{x}_j +\\frac{1}{2} \\left\\langle \\Gamma _j \\Delta \\hat{x}_j,\\Delta \\hat{x}_j \\right\\rangle +\\varepsilon ^2 f(x_j,z_j,p_j,\\Gamma _j) - \\Vert \\Delta \\hat{x}_j - \\Delta x_j \\Vert h(x_j+\\Delta x_j ) .$ As a consequence, her time $t_{j+1}=t_j+\\varepsilon ^2$ debt becomes $z_{j+1}=e^{\\lambda \\varepsilon ^2} (z_j+\\delta _j) $ .",
"If $z_{j+1} \\ge M$ , the the game terminates, and Helen pays a “termination-by-large-debt penalty” worth $e^{\\lambda \\varepsilon ^2}( \\chi (x_j) -\\delta _j)$ at time $t_{j+1}$ .",
"Similarly, if $z_{j+1} \\le - M$ , the the game terminates, and Helen receives a “termination-by-large-wealth bonus” worth $e^{\\lambda \\varepsilon ^2}(\\chi (x_j) + \\delta _j)$ at time $t_{j+1}$ .",
"If the game ends this way, we call $t_{j+1}$ the “ending index” $t_K$ .",
"If $|z_{j+1}|<M$ and $ x_{j+1}\\in \\Upsilon _D$ , then the game terminates, and Helen gets an “exit payoff” worth $g( x_{j+1})$ at time $t_{j+1}$ .",
"If the game ends this way, we call $t_{j+1}$ the “exit index” $t_E$ .",
"If the game has not terminated then Helen and Mark repeat this procedure at time $t_{j+1}=t_j+\\varepsilon ^2$ .",
"If the game never stops, the “ending index” $t_K$ is $+\\infty $ .",
"All the possibilities, apart the end by exit, had already been investigated at Section REF .",
"If the game ends by exit at time $t_E$ , then the present value of her income is $U^\\varepsilon (x_0,z_0) & = -z_0 - \\delta _0 - e^{-\\lambda \\varepsilon ^2} \\delta _1 - \\cdots - e^{-\\lambda (E-1)\\varepsilon ^2} \\delta _{E-1} + e^{-\\lambda E \\varepsilon ^2} g(x_E) \\\\& = e^{-\\lambda E \\varepsilon ^2} (g(x_E) - z_E).$ Since the game is stationary, the associated dynamic programming principle is that for $|z|<M$ , $U^{\\varepsilon }(x,z)=\\sup _{p, \\Gamma }\\min _{\\Delta \\hat{x} }{\\left\\lbrace \\begin{array}{ll}e^{-\\lambda \\varepsilon ^2} U^{\\varepsilon }(x^{\\prime },z^{\\prime }), & \\text{if } x^{\\prime }\\in \\Omega \\cup \\Gamma _N \\text{ and } |z^{\\prime }|< M, \\\\e^{-\\lambda \\varepsilon ^2} (g(x^{\\prime }) -z^{\\prime }) , & \\text{if } x^{\\prime }\\in \\Gamma _D \\text{ and }|z^{\\prime }|< M, \\\\-z - \\chi (x) , & \\text{if } z^{\\prime } \\ge M, \\\\-z + \\chi (x) , & \\text{if } z^{\\prime } \\le -M,\\end{array}\\right.",
"}$ where $x^{\\prime }=\\operatorname{proj}_{\\overline{\\Omega }} (x+\\Delta \\hat{x})$ and $z^{\\prime }=e^{\\lambda \\varepsilon ^2} (z+\\delta )$ , with $\\delta $ defined by (REF ).",
"Here $p$ , $\\Gamma $ and $\\Delta \\hat{x}$ are constrained as usual by (REF )–(REF ).",
"The definitions (REF )–() of $u^{\\varepsilon }$ and $v^{\\varepsilon }$ on $\\Omega \\cup \\Gamma _N$ are conserved.",
"The corresponding semi-relaxed limits are defined for any $x\\in \\overline{\\Omega }$ by $\\overline{u}(x)= \\limsup _{\\begin{array}{c}y\\rightarrow x \\\\ \\varepsilon \\rightarrow 0\\end{array}} u^{\\varepsilon }(y) \\quad \\text{ and } \\quad \\underline{v}(x)= \\liminf _{\\begin{array}{c}y\\rightarrow x \\\\ \\varepsilon \\rightarrow 0\\end{array}} v^{\\varepsilon }(y),$ with the convention that $y$ approaches $x$ from $\\Omega \\cup \\Gamma _N$ (since $u^{\\varepsilon }$ and $v^{\\varepsilon }$ are only defined on $\\Omega \\cup \\Gamma _N$ ).",
"Proposition REF still holds without any modification for mixed-type Dirichlet-Neumann boundary conditions.",
"Moreover, the definition of viscosity subsolutions and supersolutions is clear by relaxing the PDE condition on $\\Upsilon _D$ with the Dirichlet condition in the same way that has been done in [21].",
"Following the same steps as our proof for the Neumann problem (the main modification consists in the proof of convergence on $\\Upsilon _D$ but has already been done in [21]), the following theorem is now immediate.",
"Theorem 6.1 Consider the stationary boundary value problem (REF ) where $f$ satisfies (REF ) and (REF )–(REF ), $g$ and $h$ are continuous, uniformly bounded and $\\Omega $ is a $C^2$ -domain satisfying the uniform exterior ball condition and the uniform interior ball condition in a neighborhood of $\\Upsilon _N$ .",
"Assume the parameters of the game $\\alpha $ , $\\beta $ , $\\gamma $ fulfill (REF )–(REF ), $\\psi \\in C_b^2 (\\overline{\\Omega })$ satisfies (REF ), $\\chi \\in C^2(\\overline{\\Omega })$ is defined by (REF ), $M$ is sufficiently large, and $m=M-1 -2 \\Vert \\psi \\Vert _{L^\\infty (\\overline{\\Omega })} $ .",
"Then $u^\\varepsilon $ and $v^\\varepsilon $ are well-defined when $\\varepsilon $ is sufficiently small, and they satisfy $|u^\\varepsilon | \\le \\chi $ and $|v^\\varepsilon |\\le \\chi $ .",
"Their relaxed semi-limits $\\overline{u}$ and $\\underline{v}$ are respectively a viscosity subsolution and a viscosity supersolution of (REF ).",
"If in addition we have $\\underline{v} \\le \\overline{u}$ and the PDE has a comparison principle, then it follows that $u^\\varepsilon $ and $v^\\varepsilon $ converge locally uniformly in $\\overline{\\Omega }$ to the unique viscosity solution of (REF )."
],
[
"Parabolic PDE with an oblique boundary condition",
"The target of this section is to construct a game which could interpret the PDE with an oblique condition $h$ and final-time data $g$ given by $\\left\\lbrace \\begin{array}{ll}\\partial _t u - f(t,x,u,Du, D^2u)=0, & \\text{for }x \\in \\Omega \\text{ and } t<T, \\\\\\dfrac{\\partial u}{\\partial \\varsigma }(x,t)=h(x), & \\text{for }x \\in \\partial \\Omega \\text{ and } t<T, \\\\u(x,T)=g(x), & \\text{for } x \\in \\overline{\\Omega },\\end{array}\\right.$ where $\\varsigma $ defines a smooth vector field, say $C^2$ , on $\\partial \\Omega $ pointing outward such that $\\langle \\varsigma (x) , n(x)\\rangle \\ge \\theta >0 \\quad \\text{ for all }x\\in \\partial \\Omega .$ As usual, the domain $\\Omega $ is supposed to be at least of boundary $C^2$ and to satisfy both the uniform and the exterior ball conditions.",
"First of all, following P.L.",
"Lions [22], P.L.",
"Lions and A.S. Sznitman [24], we introduce some smooth functions $a_{ij}(x)=a_{ji}(x)$ , say $C_b^2(\\mathbb {R}^N)$ , such that $&\\exists \\theta >0, \\forall x\\in \\mathbb {R}^N, (a_{ij}(x)) \\ge \\theta I_N , \\\\&\\forall x\\in \\partial \\Omega , \\sum _{j=1}^N a_{ij}(x) \\varsigma _j(x)=n_i(x) \\quad \\text{ for } 1\\le i \\le N. $ Clearly if we had $ \\varsigma =n$ , we would just take $a_{ij}(x)=\\delta _{ij}$ .",
"Next, the matrices induce a metric $d_\\varsigma $ on $\\mathbb {R}^N$ defined by $d_\\varsigma (x,y) = \\inf \\left\\lbrace \\int _0^1 \\left[ \\sum _{1 \\le i,j \\le N} a_{ij}(\\xi (t))\\dot{\\xi }_i(t)\\dot{\\xi }_j(t)\\right]^{1/2} dt :\\xi \\in C^1([0,1], \\mathbb {R}^N), \\xi (0)=y,\\xi (1)=x \\right\\rbrace .$ Then it is well known that for $\\Vert x-y \\Vert $ small, there exists a unique minimizer in (REF ).",
"The interested reader is referred to [22] for additional properties about $d_\\varsigma $ .",
"For this specific metric, we can now define for any $x$ lying on a small $\\delta $ -neighborhood of the boundary a unique projection according the vector field $\\gamma $ along the boundary by $\\bar{x}^\\gamma = \\operatorname{proj}_{\\overline{\\Omega }}^\\varsigma (x) \\in \\partial \\Omega ,$ which corresponds to the unique minimum of $d_\\varsigma (x,y)$ for $y$ lying on the boundary.",
"Finally, $B_\\varsigma (x,r)$ denotes the ball of center $x$ and radius $r$ induced by the metric $d_\\varsigma $ .",
"We can now explain the rules of the game corresponding to the oblique problem (REF ).",
"Let the parameters $\\alpha $ , $\\beta $ , $\\gamma $ satisfy (REF )–(REF ).",
"When the game begins, the position can have any value $x_0\\in \\overline{\\Omega }$ ; Helen's initial score is $y_0=0$ .",
"The rules are as follows: if at time $t_j=t_0+j\\varepsilon ^2$ Helen's debt is $z_j$ and the stock price is $x_j$ , then Helen chooses a vector $p_j \\in \\mathbb {R}^N$ and a matrix $\\Gamma _j\\in \\mathcal {S}^N$ , restricted in magnitude by (REF ).",
"Taking Helen's choice into account, Mark chooses the stock price $x_{j+1}=x_{j}+\\Delta x_j$ so as to degrade Helen's outcome.",
"Mark is going to choose an intermediate point $\\hat{x}_{j+1}=x_j +\\Delta \\hat{x}_j \\in \\mathbb {R}^N$ such that $\\hat{x}_{j+1} \\in B_\\varsigma (x_j, \\varepsilon ^{1-\\alpha }),$ which determines the new position $x_{j+1}=x_j+\\Delta x_j \\in \\overline{\\Omega }$ by the rule $x_{j+1} = \\operatorname{proj}_{\\overline{\\Omega }}^\\varsigma (\\hat{x}_{j+1}),$ where $\\operatorname{proj}_{\\bar{\\Omega }}^\\varsigma $ is the projection defined by (REF ).",
"Helen's debt is changed to $z_{j+1}=z_j + p_j\\cdot \\Delta \\hat{x}_j +\\frac{1}{2} \\left\\langle \\Gamma _j \\Delta \\hat{x}_j,\\Delta \\hat{x}_j \\right\\rangle +\\varepsilon ^2 f(t_j,x_j,z_j,p_j,\\Gamma _j) - d_\\varsigma (\\hat{x}_{j+1}, x_{j+1}) h(x_j+\\Delta x_j ) .$ The clock steps forward to $t_{j+1}=t_j+\\varepsilon ^2$ and the process repeats, stopping when $t_K=T$ .",
"At the final time Helen receives $g(x_K)$ from the option.",
"Rather than repeating the arguments already used, we are going to explain the modifications to carry out the analysis.",
"First of all, by the boundedness of the $a_{ij}$ and (REF ), the distance $d_\\varsigma $ defined by (REF ) is equivalent to the euclidean distance.",
"Since $\\Omega $ satisfies the uniform exterior ball condition, there exists, for a certain $r_\\varsigma >0$ , a tubular neighborhood $\\lbrace x\\in \\mathbb {R}^N\\backslash \\Omega , d(x) < r_\\varsigma \\rbrace $ of the boundary on which $\\operatorname{proj}_{\\bar{\\Omega }}^\\varsigma $ is well-defined.",
"This guarantees the well-posedness of this game for all $\\varepsilon >0$ small enough.",
"Then, if $d_\\varsigma $ or the euclidean distance is used to compute $D\\phi $ and $D^2\\phi $ for a smooth function $\\phi $ , we will get the same results.",
"Therefore, we can introduce the oblique analogues $m_{\\varsigma , \\varepsilon }^{x}[\\phi ]$ and $M_{\\varsigma , \\varepsilon }^{x}[\\phi ]$ of (REF )–() by $m_{\\varsigma , \\varepsilon }^{x}[\\phi ] & :=\\inf _{\\begin{array}{c}x+\\Delta \\hat{x} \\notin \\Omega \\\\ \\Delta \\hat{x}\\end{array}}\\left\\lbrace h(x+\\Delta x) - D \\phi (x)\\cdot \\varsigma (x+\\Delta x) \\right\\rbrace , \\\\M_{\\varsigma , \\varepsilon }^{x}[\\phi ] & :=\\sup _{\\begin{array}{c}x+\\Delta \\hat{x} \\notin \\Omega \\\\ \\Delta \\hat{x}\\end{array}}\\left\\lbrace h(x+\\Delta x) - D \\phi (x)\\cdot \\varsigma (x+\\Delta x) \\right\\rbrace , $ where $\\Delta \\hat{x}$ is constrained by (REF ) and $\\Delta x$ is determined by $\\Delta x = \\operatorname{proj}_{\\bar{\\Omega }}^\\varsigma (x+\\Delta \\hat{x}) - x$ .",
"Thus, the particular choices $p_\\text{opt}^{m_ \\varsigma }$ , $p_\\text{opt}^{M_\\varsigma }$ and $\\Gamma _\\text{opt}^\\varsigma $ will be now respectively defined in the orthonormal basis $\\mathcal {B}_\\varsigma =(e_1=\\varsigma (\\bar{x}^\\gamma ),e_2, \\cdots , e_N)$ by $p_\\text{opt}^{m_\\varsigma }(x) & = D\\phi (x) +\\left[\\frac{1}{2}\\left(1-\\frac{ d_\\varsigma (x)}{\\varepsilon ^{1-\\alpha }}\\right) m_{\\varsigma , \\varepsilon }^{x}[\\phi ]- \\frac{\\varepsilon ^{1-\\alpha }}{4} \\left(1 - \\frac{d^2_\\varsigma (x)}{\\varepsilon ^{2-2\\alpha }} \\right) (D^2\\phi (x))_{11} \\right] \\varsigma (\\bar{x}^\\gamma ), \\\\p_\\text{opt}^{M_\\varsigma } (x) & = D\\phi (x) +\\left[\\frac{1}{2}\\left(1-\\frac{ d_\\varsigma (x)}{\\varepsilon ^{1-\\alpha }}\\right) M_{\\varsigma , \\varepsilon }^{x}[\\phi ]- \\frac{\\varepsilon ^{1-\\alpha }}{4} \\left( 1 - \\frac{d^2_\\varsigma (x)}{\\varepsilon ^{2- 2\\alpha }} \\right) (D^2\\phi (x))_{11} \\right] \\varsigma (\\bar{x}^\\gamma ),$ and $\\Gamma _\\text{opt}^\\varsigma (x) = D^2\\phi (x) +\\left[ \\frac{1}{2}\\left( - 1+\\frac{d_\\varsigma ^2(x)}{\\varepsilon ^{2-2\\alpha }} \\right) (D^2\\phi (x))_{11} \\right] E_{11},$ where $m_{\\varsigma , \\varepsilon }^{x}[\\phi ]$ and $M_{\\varsigma , \\varepsilon }^{x}[\\phi ]$ are defined by (REF )–(), and $E_{11}$ denotes the unit-matrix $(1,1)$ in the basis $\\mathcal {B}_\\varsigma $ .",
"The definitions of $u^\\varepsilon $ , $v^\\varepsilon $ and their relaxed semi-limits $\\overline{u}$ and $\\underline{v}$ , given by (REF )–(REF ) and (REF ), are conserved.",
"The only change on the dynamic programming inequalities (REF )–(REF ) concerning $u^\\varepsilon $ and $v^\\varepsilon $ is to replace $\\Vert \\Delta \\hat{x} - \\Delta x \\Vert $ by $d_\\varsigma (x+\\Delta \\hat{x}, x+\\Delta x)$ , and to constrain $\\Delta \\hat{x} $ by (REF ).",
"For stability, we need to consider a $C_b^2(\\overline{\\Omega })$ -function $\\psi $ such that $\\frac{\\partial \\psi }{\\partial \\varsigma }(x)= \\Vert h \\Vert _{L^\\infty }+1 \\quad \\text{ on } \\partial \\Omega .$ It is still allowed by the uniform interior ball condition applied to the $C^2$ -domain $\\Omega $ .",
"By using exactly the same ingredients already used for the Neumann problem and adapting the geometric estimates given by Section REF in the oblique framework, we obtain the following theorem.",
"Theorem 6.2 Consider the final-value problem (REF ) where $f$ satisfies (REF )–(REF ), $g$ and $h$ are continuous, uniformly bounded, $\\Omega $ is a $C^2$ -domain satisfying both the uniform interior and exterior ball conditions, and $ \\varsigma $ is a continuous vector field on $\\partial \\Omega $ and satisfy (REF ).",
"Assume the parameters $\\alpha $ , $\\beta $ , $\\gamma $ fulfill (REF )-(REF ).",
"Then $\\overline{u}$ and $\\underline{v}$ are uniformly bounded on $\\overline{\\Omega }\\times [t_\\ast , T]$ for any $t_\\ast <T$ , and they are respectively a viscosity subsolution and a viscosity supersolution of (REF ).",
"If the PDE has a comparison principle (for uniformly bounded solutions), then it follows that $u^\\varepsilon $ and $v^\\varepsilon $ converge locally uniformly to the unique viscosity solution of (REF ).",
"Acknowledgements: I thank Sylvia Serfaty for bringing the problem to my attention and numerous helpful discussions.",
"I thank Scott N. Armstrong for fruitful and encouraging talks and Guy Barles for helpful comments about viscosity solutions.",
"Finally, I gratefully acknowledge support from the European Science Foundation through a EURYI award of Sylvia Serfaty.",
"Jean-Paul Daniel UPMC Univ Paris 06, UMR 7598 Laboratoire Jacques-Louis Lions, Paris, F-75005 France ; CNRS, UMR 7598 LJLL, Paris, F-75005 France [email protected]"
]
] | 1204.1459 |
[
[
"Quantum widening of CDT universe"
],
[
"Abstract The physical phase of Causal Dynamical Triangulations (CDT) is known to be described by an effective, one-dimensional action in which three-volumes of the underlying foliation of the full CDT play a role of the sole degrees of freedom.",
"Here we map this effective description onto a statistical-physics model of particles distributed on 1d lattice, with site occupation numbers corresponding to the three-volumes.",
"We identify the emergence of the quantum de-Sitter universe observed in CDT with the condensation transition known from similar statistical models.",
"Our model correctly reproduces the shape of the quantum universe and allows us to analytically determine quantum corrections to the size of the universe.",
"We also investigate the phase structure of the model and show that it reproduces all three phases observed in computer simulations of CDT.",
"In addition, we predict that two other phases may exists, depending on the exact form of the discretised effective action and boundary conditions.",
"We calculate various quantities such as the distribution of three-volumes in our model and discuss how they can be compared with CDT."
],
[
"Introduction",
"Causal dynamical triangulations (CDT) [1], [2], [3] is an attempt to construct a non-perturbative theory of quantum gravity.",
"Rather than postulating the existence of new degrees of freedom or new physical principles at the Planck scale, CDT uses a standard quantum field theory method — path integrals — to sum over space-time geometries weighted by the Einstein-Hilbert action.",
"The path integrals are regularised by discretisation of space-time geometry into piece-wise flat manifolds with temporal foliation.",
"Usually, space-time is divided into discrete spatial slices, each having the topology of the three-sphere, which ensures global, proper-time foliation consistent with the Lorentzian signature of the metric.",
"Each spatial slice is represented as a triangulation of the three-sphere, made of equilateral tetrahedra.",
"The tetrahedra from neighbouring spatial slices are then glued together, thus forming a complicated 4d manifold, with periodic boundary conditions in time direction.",
"This lattice regularisation provides a suitable ultraviolet cut-off and simultaneously reproduces classical general relativity in the infrared limit.",
"Although analytic calculations do not seem to be feasible in the full 3+1 dimensional CDT, the model can be studied by means of computer simulations.",
"After the Wick rotation to the Euclidean signature, the sum over geometries can be performed by standard Monte Carlo methods developed earlier for Euclidean quantum gravity [4], [5], [6], [7].",
"In recent years, it has been shown that this computational approach has a potential to bring many interesting results.",
"In particular, the existence of three phases has been observed [8].",
"These phases have different profiles of the three-volume $N_3(t)$ as a function of time (slice index) $t$ .",
"Depending on the values of parameters in the Einstein-Hilbert action, the system is either in phase “A”, in which $N_3(t)$ fluctuates randomly from slice to slice, phase “B” in which $N_3(t)$ is localised in a single spatial slice, or in phase “C” in which a macroscopic “quantum universe” is formed [9], [10], [11], [12].",
"In this last phase, the average value of $N_3(t)$ at each time slice $t$ is well described by the following formula: $\\langle N_3(t)\\rangle = \\left\\lbrace \\begin{array}{ll} \\frac{N_4^{3/4}}{2s} \\cos ^3\\left(\\frac{t}{s N_4^{1/4}} \\right) & \\mbox{for}\\; |t|<\\frac{\\pi s N_4^{1/4}}{2}, \\\\0 & \\mbox{for}\\; |t|\\ge \\frac{\\pi s N_4^{1/4}}{2},\\end{array} \\right.",
"$ where $N_4=\\sum _t N_3(t)$ is the total (fixed) four-volume of the universe; $s$ is obtained by fitting to the results of simulations; the centre of mass is assumed to be at $t=0$ .",
"The last formula means that the universe produces a “droplet” of $\\cos ^3(x)$ shape, and that this droplet extends as $\\pi s N_4^{1/4}$ in time direction.",
"This shape is equivalent to the classical de-Sitter solution.",
"By making a connection with the mini-superspace model [13] it has been concluded in Refs.",
"[14], [8], [10], [11], [12] that, when only the three-volume is concerned, the full CDT model effectively reduces to a 1d model with three-volumes $\\lbrace N_3(t)\\rbrace $ as the sole degrees of freedom, and with the following discrete action: $S = c_1 \\sum _t \\frac{(N_3(t+1)-N_3(t))^2}{N_3(t)} + c_2 \\sum _t N_3^{1/3}(t), $ Here $c_1,c_2$ are new coupling constants related to those in the full Einstein-Hilbert action.",
"An important fact is that although the action (REF ) completely neglects the internal structure of each spatial slice $t$ , it gives an excellent agreement with simulations of the full model.",
"In this paper we introduce a statistical-physics model which reproduces the de-Sitter phase of the CDT.",
"Our model consist of a certain number of particles which occupy sites of a 1d lattice, and microstates (configurations of particles) are weighted with the factor $e^{-S}$ .",
"We identify the emergence of the de-Sitter universe with a condensation-like transition known from similar statistical models [15], [16].",
"We show (both analytically and via computer simulations) that a symmetrised version of the action (REF ) reproduces the shape of the macroscopic universe observed in CDT.",
"We calculate the width (temporal extension) of this universe and show that quantum corrections make it wider as compared to the classical solution.",
"Moreover, we show that the effective action (REF ) describes not only phase C of CDT but also phases A and B, in the space of the coupling constants $c_1,c_2$ .",
"In addition, we suggest that two further phases may exist: “antiferromagnetic” phase D in which thin spatial slices of extended three-volume are separated by slices of minimal size, and “correlated fluid” phase E which emerges from phase C for large four-volume $N_4$ as a result of merging boundaries of the $\\cos ^3(x)$ -shaped universe.",
"In all these phases we calculate quantities such as the probability distribution of the three-volume or the correlation function for different three-volumes.",
"Lastly, we suggest that by determining analogous quantities in CDT it should be possible to test whether the effective action (REF ) is valid in all phases."
],
[
"Model",
"In our model, we consider a one-dimensional closed ring of $N$ sites, each of them carrying a positive number of particles $m_1\\ge 1,\\dots ,m_N\\ge 1$ .",
"The total number of particles is equal to $M$ .",
"We denote the density of particles by $\\rho =M/N$ .",
"The numbers of sites $N$ and particles $M$ correspond to the numbers of spatial slices and four-volume $N_4$ , respectively, while the occupation numbers $\\lbrace m_i\\rbrace $ correspond to three-volumes $\\lbrace N_3(t)\\rbrace $ of spatial slices in CDT.",
"We assume that the probability of a microstate $P(m_1,\\dots ,m_N)$ factorizes into the product of two-point kernels for pairs of neighbouring sites, $P(m_1,\\dots ,m_N) = g(m_1,m_2) g(m_2,m_3)...g(m_{N-1},m_N)g(m_N,m_1), $ where $g(m,n) = \\exp \\left(-c_1 \\frac{2(m-n)^2}{m+n} -c_2 \\frac{m^{1/3}+n^{1/3}}{2}\\right) \\ .$ The kernel $g(m,n)$ plays the role of a reduced transfer matrix between neighbouring slices of CDT.",
"The above choice guarantees that the partition function $Z(N,M) &=& \\sum _{m_1=1}^M ... \\sum _{m_N=1}^M g(m_1,m_2) g(m_2,m_3)...g(m_{N-1},m_N)g(m_N,m_1) \\delta \\left(\\sum _i m_i-M\\right)\\nonumber \\\\&= & \\sum _{m_1=1}^M ... \\sum _{m_N=1}^M \\exp \\left[ - \\sum _i \\left(c_1\\frac{(m_{i+1}-m_i)^2}{(m_i+m_{i+1})/2} + c_2 m_i^{1/3}\\right)\\right] \\delta \\left(\\sum _i m_i-M\\right) \\nonumber \\\\ &= & \\sum _{m_1=1}^M ... \\sum _{m_N=1}^M \\exp \\big [ -S\\left[\\lbrace m_i\\rbrace \\right]\\big ] \\delta \\left(\\sum _i m_i-M\\right) $ corresponds to that of CDT with the effective action (REF ) in the limit of large systems.",
"Our choice (REF ) is however symmetric in $n,m$ as opposed to (REF ).",
"We shall see later that this symmetry is necessary to reproduce full-CDT simulation results.",
"Equation (REF ) has the same form as the steady-state probability of a recently introduced non-equilibrium statistical physics model of particles hopping between sites of a 1d lattice [15], [16].",
"A key feature of this model is the condensation phenomenon in which a finite fraction of particles becomes localised in a small region of the lattice if the density of particles $\\rho =M/N$ exceeds some critical value $\\rho _c$ .",
"In particular, in Ref.",
"[16] the following two-point function $g(m,n)$ has been analysed: $g(m,n)=K(|m-n|)\\sqrt{f(m)f(n)}, $ with two functions $K(x),f(m)$ playing the role of surface stiffness and on-site potential, respectively.",
"This model has a rich phase diagram which depends on the choice of $K(x)$ and $f(m)$ .",
"We will briefly discuss some results of Ref.",
"[16] because they are important for the model discussed in this paper.",
"Let us begin with defining the grand-canonical partition function $Z_N(z) = \\sum _M Z(N,M) z^M = \\sum _{\\lbrace m_i\\rbrace } z^{\\sum _i m_i} \\prod _i g(m_i,m_{i+1}),$ in which the fugacity $z$ is determined from $\\rho =\\frac{1}{N}\\left<\\sum _i m_i\\right> = \\frac{z}{N}\\frac{\\partial \\ln Z_N(z)}{\\partial z} .",
"$ We note that the left-hand side of Eq.",
"(REF ) grows monotonously with $z$ .",
"Since phase transitions are related to singularities of $Z(z)= \\lim _{N\\rightarrow \\infty } Z_N(z)$ and its derivatives, we are interested in the behaviour of this function as $z$ approaches the radius of convergence $z_c$ of $Z(z)$ .",
"If $z_c$ is infinite, there is always some $z>0$ which obeys Eq.",
"(REF ) for any $\\rho $ .",
"This means that $Z(z)$ does not have a singularity for $0<z<\\infty $ which is the physically relevant range of $z$ .",
"Also, both ensembles, the canonical and the grand-canonical one, are equivalent in the thermodynamic limit in this case.",
"The partition function $Z_N(z)$ can be expressed as $Z_N(z) = \\sum _{m_1,\\dots ,m_N} T_{m_1 m_2} T_{m_2 m_3} \\cdots T_{m_N m_1} = \\mbox{Tr}\\, T(z)^N, $ where $T(z)$ is a square $M\\times M$ matrix defined as $T_{mn}(z)=z^{(m+n)/2}g(m,n).",
"$ If we now define $\\phi _m(z)$ to be the normalised eigenvector of $T_{mn}(z)$ to the largest eigenvalue $\\lambda _{\\rm max}(z)$ , $\\sum _n T_{mn}(z) \\phi _n(z) = \\lambda _{\\rm max}(z) \\phi _m(z) ,$ we obtain for large $N$ that $Z_N(z) \\cong \\lambda _{\\rm max}(z)^N$ .",
"We can also calculate the probability $p(m)$ that a randomly chosen site has $m$ particles: $p(m) = \\lim _{N\\rightarrow \\infty } \\frac{1}{Z_N(z)} \\sum _{m_2,\\dots ,m_N} T_{m m_2} T_{m_2 m_3} \\cdots T_{m_N m} = \\phi _m^2(z), \\\\$ The eigenvector $\\phi _m(z)$ decays with $m$ , and so does $p(m)$ .",
"This is guaranteed by the fact that $\\rho $ from Eq.",
"(REF ) is finite.",
"In this case, the system has a finite number of particles at every site – we say that the system is in the “fluid” phase.",
"One can also show that there are only local correlations between different $m_i$ 's in this phase.",
"We shall therefore call this phase a weakly-correlated fluid.",
"On the other hand, if $Z(z) = \\lim _{N\\rightarrow \\infty } Z_N(z)$ has some finite radius of convergence $z_c<\\infty $ , the derivative in (REF ) can either grow to infinity for $z\\rightarrow z_c$ , or tend to a finite constant.",
"In the first case, we again have no phase transition, because for any $\\rho $ there exists some real $z<z_c$ which obeys Eq.",
"(REF ).",
"However, if $z_c<\\infty $ and $dZ(z)/dz|_{z\\rightarrow z_c}\\rightarrow \\rm const$ , there is a critical density $\\rho _c = \\sum _m m \\phi _m(z_c)^2, $ above which the grand-canonical ensemble does not exist.",
"This indicates a phase transition from the fluid to the condensed state.",
"The nature of the condensate depends on $K(x)$ and $f(m)$ which define $g(m,n)$ .",
"If $K(x)=\\rm const$ and $f(m)$ falls off sufficiently fast, the condensate spontaneously forms on one randomly chosen site, breaking translational invariance.",
"This is precisely the balls-in-boxes (B-in-B) [17] or zero-range process (ZRP) [18] condensation.",
"We shall note here that the B-in-B model has already been successfully applied to the transition between crumpled and elongated phase in Euclidean quantum gravity models [19], [20].",
"If $K(x)$ decays with $x$ , the condensate extends to more than one site.",
"The width $W$ of the condensate grows as some power $\\alpha $ of its volume, $W\\sim M^\\alpha $ .",
"The condensate can be either bell-shaped, or rectangular, depending on exact forms of $K(x)$ and $f(m)$ .",
"We see that this closely resembles the features of the macroscopic universe from phase C in CDT.",
"This type of phase, which we shall call a “droplet” phase, will be discussed extensively in Section .",
"However, if the extension $W$ of the condensate becomes comparable to the linear extension of the system $N$ , both ends of the condensate merge and the particles spread uniformly in the system.",
"This phase differs from the weakly-correlated fluid phase which exists for $\\rho <\\rho _c$ in that the occupation numbers $\\lbrace m_i\\rbrace $ are correlated.",
"We shall call this phase, rather obviously, a “correlated fluid” phase.",
"It is important to note that the existence of these phases does not depend on the details of $K(x)$ and $f(m)$ (REF ), which often only affect the shape of the condensate and the value of the critical density.",
"In what follows we shall use the analogy between this model and the effective 1d model of CDT to study the emergence of the bell-shaped quantum universe.",
"We shall also investigate the phase diagram of the model, assuming that the effective action is valid in all phases.",
"A small difference between our model and the model from Refs.",
"[15], [16] is that the two-point kernel $g(m,n)$ of the 1d effective CDT model (as given in Eq.",
"(REF )) has a slightly different form that Eq.",
"(REF ), because $K(x)$ depends not only on the difference between two consecutive occupation numbers $m,n$ but also on their absolute magnitudes $m,n$ : $g(m,n)=K(|m-n|/\\sqrt{m+n})\\sqrt{f(m)f(n)}$ However, as we shall see, the only new result is the existence of the antiferromagnetic phase which is not observed in the model with kernels of the form (REF )."
],
[
"Phase diagram",
"We begin with presenting the results of Monte Carlo simulations of our model (see Appendix for details), which reveal its rich phase structure.",
"Anticipating the existence of condensed/fluid phase, and also extended/localised condensates, we define the following quantities which allow us to detect phase transitions: $\\sigma &=& \\left\\langle \\frac{\\sum _i^N m_i^2}{M^2} \\right\\rangle , \\\\\\gamma &=& 1 - \\left\\langle \\frac{1}{\\min m_i} \\right\\rangle , \\\\\\delta &=& 1 - \\left\\langle \\frac{\\sum _i^N |m_i - m_{i+1}|}{2M} \\right\\rangle .$ These quantities assume values between 0 and 1 and play the role of order parameters in the limit $N,M\\rightarrow \\infty $ .",
"The parameter $\\sigma $ is the inverse participation ratio for site occupation numbers and it measures the degree of localisation: for a delocalised microstate in which all $m_i$ 's are roughly the same, $\\sigma \\approx 1/N \\rightarrow 0$ for $N\\rightarrow \\infty $ .",
"However, if one $m_i$ is much larger than others, $\\sigma \\rightarrow 1$ .",
"The value of the parameter $\\gamma $ indicates whether there are any sites with minimal number of particles $m_i=1$ in typical configurations (slices with the smallest possible three-volume in CDT): $\\gamma = 0$ if there are such sites, whereas $\\gamma >0$ if all sites are occupied by larger numbers of particles.",
"The parameter $\\delta $ is related to the surface roughness or stiffness of typical configurations and is close to zero for configurations in which $\\lbrace m_i\\rbrace $ dramatically change from site to site, and close to one for relatively smooth configurations.",
"We have used the parameters $\\sigma ,\\gamma $ and $\\delta $ to determine the phase diagram shown in Fig.",
"REF (see also Fig.",
"REF for examples of plots of $\\sigma ,\\gamma ,\\delta $ ) in the phase plane of the parameters $c_1$ and $c_2$ , by simulating our model for fixed $N=80, M=18100$ , and different pairs of $c_1,c_2$ .",
"Snapshots of typical configurations in each phase are shown in Fig.",
"REF .",
"Our phase diagram includes both positive and negative $c_1,c_2$ .",
"One might be worried that negative coupling constants should not have any physical meaning in the CDT, because the effective action $S_{\\rm eff}$ would be unbounded from below for negative $c_1,c_2$ and hence the partition function was ill-defined.",
"However, as we consider here the system with a finite number of sites $N$ and particles $M$ , the action is bounded and the partition function is well defined.",
"Figure: Phase diagram determined from Monte Carlo simulations for N=80N=80 and M=18100M=18100.Figure: Order parameters: (a) σ(c 2 )\\sigma (c_2) for c 1 =-0.8c_1=-0.8, (b) γ(c 2 )\\gamma (c_2) for c 1 =0.5c_1=0.5 and (c) δ(c 1 )\\delta (c_1) for c 2 =-0.5c_2=-0.5.Figure: Typical configurations for all phases from Fig.",
": (a)-droplet, (b)-correlated fluid, (c)-antiferromagnetic, (d)-localised, (e)-uncorrelated fluid.Looking at Fig.",
"REF , we can distinguish five different phases in the $(c_1,c_2)$ plane for fixed $N,M$ : Droplet phase: a finite fraction of particles (typically almost all particles) form a bell-shaped condensate extended over $W\\gg 1$ sites of the lattice.",
"The shape of the condensate can be approximated by Eq.",
"(REF ).",
"The droplet phase is observed for $c_1>0$ and $c_2>c_{2,\\rm crit}(c_1)$ , where the shape of the critical curve $c_{2,\\rm crit}$ depends also on $N$ and $M$ .",
"This phase corresponds to the macroscopic universe phase “C” in CDT.",
"The width $W$ and other properties of the condensate will be discussed in Section .",
"The values of the order parameters are as follows: $\\sigma $ is of order $1/W$ , $\\gamma = 0$ and $\\delta > 0$ .",
"Correlated fluid: particles are distributed approximately uniformly over all sites of the lattice.",
"The occupation numbers fluctuate around the average value $\\left<m_i\\right>=\\rho $ , but the typical size of fluctuations is small as compared to the average.",
"This phase is observed for $c_1>0$ and $c_2 < c_{2,\\rm crit}(c_1)$ .",
"In the thermodynamic limit, we expect the order parameters to be $\\sigma =0$ (of order $1/N$ for finite system), $\\gamma >0$ and $\\delta >0$ .",
"Antiferromagnetic fluid: typical configurations contain alternated occupied/empty (i.e., containing only one particle) sites.",
"This phase is observed when both $c_1$ and $c_2$ are negative.",
"The number of empty sites increases when $c_1$ or $c_2$ grow.",
"In the thermodynamic limit, the order parameters in this phase are $\\sigma =0$ (of order $2/N$ for finite $N$ ), $\\gamma =0$ and $\\delta =0$ .",
"Localised phase: in a typical configuration, almost all particles occupy a single site, while the remaining sites have only small numbers of particles of order $O(1)$ .",
"This phase is observed for $c_1<0$ and $c_2>0$ .",
"The order parameters are $\\sigma =1$ , $\\gamma =0$ and $\\delta >0$ .",
"This phase may correspond to phase “B” in CDT.",
"Uncorrelated fluid: Particle occupation numbers are uncorrelated and there is no condensation regardless of the density of particles $\\rho $ .",
"This phase is observed in a small region close to the origin of the $(c_1,c_2)$ plane: $c_1 \\approx 0, c_2\\approx 0$ and it may correspond to “A” of the CDT model.",
"Interestingly, as we have already mentioned, there are two new phases: the correlated-fluid phase and the antiferromagnetic-fluid phase, which have not been observed in computer simulations of CDT.",
"In next sections we shall present some arguments supporting the existence of these new phases in the full CDT quantum gravity model.",
"We shall now give a crude mean-field argument supporting our phase diagram, based on estimating the value of the action $S =\\sum _i \\left(c_1\\frac{(m_{i+1}-m_i)^2}{(m_i+m_{i+1})/2} + c_2 m_i^{1/3}\\right), $ for typical configurations in different phases, and assuming that, for given $c_1$ and $c_2$ , the phase with the least value of the action is selected.",
"Although we neglect quantum fluctuations of $m_i$ 's in this section, we shall see that our approach reproduces the phase diagram quite well.",
"Quantum fluctuations will be discussed in the next section.",
"The mean-field action for the droplet of width $W$ shown in Fig.",
"REF a can be approximated as $S_{\\rm droplet} \\approx 2c_1 \\frac{M}{W} \\int _0^W \\frac{(h((i+1)/W)-h(i/W))^2}{h((i+1)/W)+h(i/W)} di + c_2 \\left(\\frac{M}{W}\\right)^{1/3} \\int _0^W h(i)^{1/3} di, $ where we have assumed that the average shape of the droplet is $m_i = (M/W)h(i/W)$ and that fluctuations can be neglected in the limit of large $M$ .",
"We assume that $h(x)$ is fixed and the only degree of freedom is the width $W$ of the droplet.",
"Equation (REF ) can be further simplified if $h((i+1)/W) \\cong h(i/W) + h^{\\prime }(i/W)/W$ , $S_{\\rm droplet} \\approx c_1 \\frac{M}{W^2} \\int _0^1 dx \\frac{h^{\\prime }(x)^2}{h(x)} + c_2 W^{2/3} M^{1/3} \\int _0^1 dx h(x)^{1/3} \\ .",
"$ The integrals over $dx$ will be explicitly calculated later, now we just treat them as two unknown constants.",
"Searching for $W$ which minimizes the action we obtain $W\\sim (c_1/c_2)^{3/8} M^{1/4}$ and, finally, $S_{\\rm droplet} \\propto c_1^{1/4} c_2^{3/4} M^{1/2} \\ .$ We see that the above calculation predicts the extension $W$ of the condensate to grow as $\\sim M^{1/4}$ .",
"We shall come back to that later.",
"Now, let us consider the energy of the correlated fluid phase (see Fig.",
"REF b): $S_{\\rm corr.",
"fluid} \\approx N c_1 \\frac{\\left<(m_{i+1}-m_i)^2\\right>}{\\rho } + c_2 N \\rho ^{1/3} \\ .$ Assuming that $m_i = \\rho + \\Delta m_i$ where $\\Delta m_i$ is of order $\\sqrt{\\rho }$ due to stochastic fluctuations, we obtain $S_{\\rm corr.",
"fluid} \\propto N c_1 + c_2 N^{2/3} M^{1/3} \\ .$ The value of the action for a typical configuration in the antiferromagnetic phase (see Fig.",
"REF c) is $S_{\\rm antiferr.}",
"\\approx 2c_1 M + c_2 n^{2/3} M^{1/3}$ where we have assumed that there are $n$ peaks of height $\\approx M/n$ , separated by empty sites.",
"We can use the last formula also to estimate the action in the localised phase (see Fig.",
"REF d) by setting $n=1$ : $S_{\\rm localised} \\approx 2c_1 M + c_2 M^{1/3} \\ .$ Comparing the values of the action for different $c_1,c_2$ and taking the least one, we obtain for large $N,M$ the phase diagram shown in Fig.",
"REF .",
"The diagram agrees qualitatively with the experimentally obtained one in Fig.",
"REF .",
"The lines separating different phases are at $c_1=0$ and $c_2=0$ , except for a line between the droplet and the correlated fluid phase, which has a more complicated shape and will be discussed in Sec. .",
"Figure: Phase diagram obtained by comparing the action of typical configurations in different phases.The reader may wonder why we did not estimate the action in the uncorrelated fluid phase.",
"The reason is that this phase is dominated by fluctuations (entropy) rather than by the action (energy) (REF ) which vanishes for $c_1=c_2=0$ .",
"Although this phase exists only at a single point $(c_1,c_2)=(0,0)$ in the phase space in the thermodynamic limit, we expect that for finite systems we discuss here, the uncorrelated-fluid phase extends to a small region around $(c_1,c_2)=(0,0)$ .",
"We conclude this section with a technical remark.",
"Because our model is motivated by the CDT model of quantum gravity, we prefer to use the language of quantum physics rather than that of statistical physics in the paper.",
"If one used statistical physics language instead, one would replace the action $S$ by $\\beta E$ , where $\\beta =c_1$ would be the inverse temperature, $E$ would be the energy of configurations, and $c_2/c_1$ would be the second parameter (besides $\\beta $ ) of our model.",
"The partition function could then be written as $Z = e^{-\\beta F} = \\sum _{\\lbrace m\\rbrace } e^{-\\beta E}$ , where $F$ would correspond to the free energy of the system, including the entropic contribution coming from the sum over all microstates.",
"In quantum physics, $F$ is rather referred to as the effective action and the entropic contribution as to the contribution from quantum fluctuations.",
"In the next section we shall estimate the contribution from quantum fluctuations to the droplet phase and show that these fluctuations lead to the widening of the effective universe as compared to the classical de-Sitter solution."
],
[
"Droplet phase - the macroscopic universe",
"In the droplet phase, which exists for positive coupling constants $c_1,c_2$ , the condensate takes the form of an extended “droplet”.",
"Figure: Shape of the universe in the droplet phaseaveraged over n=1000000n=1000000 configurations for c 1 =1c_1=1 and c 2 =5c_2=5 andN=256N=256 and M=50000M=50000 (blue points), compared to the cos 3 \\cos ^3 shape (Eqs.",
"() and ()) with the width parameter given by the classical solution Eq.",
"() W=W 0 (M)=55.5W=W_0(M)=55.5 (red curve), and a more accurate resultincluding quantum and finite size corrections Eq.",
"()W=W 2 (M)=59.64W=W_2(M)=59.64 (black curve).In Fig.",
"REF we show the average shape of this droplet obtained in numerical simulations (see the appendix for details).",
"The envelope of the droplet has a $\\cos ^3$ form and its extension scales as $\\sim M^{1/4}$ (see Fig.",
"REF ) as determined already in the previous section.",
"We will now find the function $h(x)$ and calculate the integrals from Eq.",
"(REF ) to find the coefficient in the power law $W\\sim M^{1/4}$ .",
"Let us first assume that in the limit of large system sizes $N,M\\rightarrow \\infty $ and $\\rho =\\rm const$ , fluctuations of $\\lbrace m_i\\rbrace $ can be neglected, so that $m_i \\equiv \\bar{m}_i,$ where $\\bar{m}_i$ denotes the average occupation number at site $i$ .",
"The shape of the condensate can be obtained by minimising the action $S(\\lbrace \\bar{m}_i \\rbrace ) = \\sum _{i=1}^N \\left[c_1\\frac{(\\bar{m}_{i+1}-\\bar{m}_i)^2}{(\\bar{m}_i+\\bar{m}_{i+1})/2} + c_2 \\bar{m}_i^{1/3}\\right].$ Going into the continuous limit: $\\bar{m}_i \\rightarrow m(t)$ and $\\bar{m}_{i+1}-\\bar{m}_i \\rightarrow m^{\\prime }(t)$ , with $m(t)$ defined on the interval $0\\le t\\le N$ , we see that the following functional has to be minimised with respect to $m(t)$ : $S[m(t)] = \\int _{0}^{N} \\left[ c_1 \\frac{m^{\\prime }(t)^2}{m(t)}+c_2 m(t)^\\frac{1}{3} \\right] dt,$ with an additional constraint that $\\int _0^N m(t) dt = M$ .",
"Using the method of Lagrange multipliers we obtain the following Euler-Lagrange differential equation for $m(t)$ : $\\frac{c_2}{3}m(t)^{-\\frac{2}{3}}+c_1 \\left(\\frac{m^{\\prime }(t)}{m(t)}\\right)^2-2 c_1 \\frac{m^{\\prime \\prime }(t)}{m(t)} - a = 0.$ where $a$ is the Lagrange multiplier used to fix the total number of particles $M$ .",
"This equation is exactly soluble: $m(t) = \\frac{M}{W} h(t/W), $ where $h(x)$ is the “$\\cos ^3$ ” shape of the droplet, $h(x) = \\left\\lbrace \\begin{array}{ll} \\frac{3\\pi }{4} \\cos ^3 (\\pi (x-1/2)) = \\frac{3\\pi }{4} \\sin ^3 (\\pi x), & 0 <x< 1 \\\\ 0, & x<0\\;\\mbox{or}\\; x>1 \\end{array} \\right.",
"\\\\$ and $W$ is the width of the droplet, $W = W_0(M) = M^{1/4} \\frac{3 \\pi }{\\sqrt{2}} \\left(\\frac{c_1}{c_2}\\right)^{3/8} \\approx 6.66432 \\times M^{1/4} \\left(\\frac{c_1}{c_2}\\right)^{3/8} .$ Equations (REF ) and (REF ) are equivalent to Eq.",
"(REF ) up the position of the centre of mass which is shifted from $x=0$ to $x=1/2$ (we have used the freedom of shifting the droplet to $x=1/2$ for the future convenience).",
"The width $W$ is uniquely determined by $M,c_1,c_2$ and it grows as expected as $\\sim M^{1/4}$ for large systems.",
"Equation (REF ) shows that the average height of the droplet $M/W$ scales as $\\sim M^{3/4}$ .",
"Remembering that $M$ plays the role of four-volume of the corresponding CDT model, we see that the height is proportional to the three-volume of spatial slices.",
"This is one of the reasons why the droplet is considered to be a manifestation of a macroscopic universe in Refs.",
"[9], [10], [11], [12].",
"The shape observed numerically closely follows the classical solution (REF ), see the red curve in Fig.",
"REF .",
"However, the width of the droplet $W$ observed in numerical simulations is larger than the one calculated from Eq.",
"(REF ), as shown in Fig.",
"REF (red curves).",
"The reason is that calculation that lead to Eq.",
"(REF ) neglect quantum fluctuations.",
"Figure: The width WW versus the number of particles MM for different pairs (c 1 ,c 2 )=(1,2),(2,2),(1,5),(2,5)(c_1,c_2)=(1,2),(2,2),(1,5),(2,5) (from left to right).",
"Black symbols correspond to numerical data, red line shows the classical solution W=W 0 (M)W=W_0(M) (), black dashed line corresponds to the solution W=W 1 (M)W=W_1(M) of Eq.",
"() taking into account quantum corrections and black solid line shows the quantum solution including interface effects W=W 2 (M)W=W_2(M) from Eq.",
"().We will now calculate quantum corrections to $W$ assuming that they leave the shape of the droplet intact.",
"This assumption, as we have mentioned, is corroborated by simulations.",
"Our reasoning follows in part the lines of Ref.",
"[16] in which the spatial extension of the condensate has been calculated analytically by splitting the system into two parts: the condensate and the fluid background.",
"Proceeding in a similar way, we assume that the total free energy $F(W)$ of the system having a condensate of width $W$ can be approximated by $F(W) \\approx F_{\\rm background}(N-W) + F_{\\rm droplet}(W) = \\ln Z_{\\rm crit}(N-W,\\rho _c(N-W)) + \\ln Z_{\\rm droplet}(W,\\tilde{M}) \\nonumber \\\\= -W \\ln \\lambda _{\\rm max} + \\ln Z_{\\rm droplet}(W,\\tilde{M}) + O(N), $ where $Z_{\\rm crit}$ is the canonical partition function for the system with $N-W$ sites being at the critical density, and $Z_{\\rm droplet}$ is the partition function for the condensate extended over $W$ sites and containing $\\tilde{M} = \\left(1 - [1\\!-\\!W/N] \\rho _{\\rm crit}/\\rho \\right) M$ particles.",
"If the density is high (the case relevant for CDT), $\\rho _{\\rm crit}/\\rho \\ll 1$ and we can assume $\\tilde{M} \\approx M$ .",
"Equation (REF ) states that the free energy of the system is the sum of free energies of the fluid and the droplet, and neglects contributions from the boundaries between these two coexisting states.",
"The partition function for the bulk reads $Z_{\\rm crit}(N-W)\\sim \\lambda _{\\rm max}^{N-W}$ , where $\\lambda _{\\rm max}$ is the maximal eigenvalue of the matrix $T_{mn}(z_c)$ defined in Eq.",
"(REF ).",
"The partition function of the condensate $Z_{\\rm droplet}(W)$ reads: $Z_{\\rm droplet}(W,M) = \\sum _{m_1=1}^\\infty \\cdots \\sum _{m_W=1}^\\infty \\exp \\left[-S_W(\\lbrace m_i\\rbrace )\\right] \\delta \\left(M-\\sum _i m_i\\right),$ where $m_0=m_{W+1}=1$ and $S_W[\\lbrace m_i\\rbrace ] = \\sum _{i=0}^{W} \\left(2c_1\\frac{(m_{i+1}-m_i)^2}{m_i+m_{i+1}} + c_2 m_i^{1/3}\\right), $ is the action for the droplet of size $W$ .",
"The standard way of estimating the contribution of fluctuations is to expand each $m_i$ around its average value $\\bar{m}_i$ , $m_i = \\bar{m}_i + \\Delta m_i$ , and to assume that the fluctuations $\\Delta m_i$ are Gaussian.",
"In this approximation, $Z_{\\rm droplet}(W) \\cong e^{-S_W[\\lbrace \\bar{m}_i\\rbrace ]}\\int \\exp \\left[ -\\frac{1}{2}\\sum _{ij} \\Delta m_i \\bar{A}_{ij} \\Delta m_j \\right]\\delta \\left(\\sum _i \\Delta m_i\\right) \\prod _i d \\Delta m_i ,$ where $\\Delta m_i$ are now continuous variables and the matrix $\\bar{A}$ is the matrix of second derivatives (the Hessian), $\\bar{A}_{ij} = \\frac{\\partial ^2 S_W}{\\partial m_i \\partial m_j}\\bigg |_{m_i=\\bar{m}_i} ,$ calculated for $\\lbrace m_i\\rbrace $ which correspond to the classical solution $\\bar{m}_i=M/W h(i/W)$ (see Eqs.",
"(REF ) and (REF )).",
"Using the integral representation of the Dirac delta $\\delta (k) = \\int ^{\\infty }_{-\\infty } \\frac{dq}{2\\pi } e^{iqk} ,$ we obtain $Z_{\\rm droplet}(W) \\cong e^{-S_W[\\lbrace \\bar{m}_i\\rbrace ]}\\int _{-\\infty }^{+\\infty } \\frac{dq}{2\\pi }\\int \\exp \\left[ -\\frac{1}{2}\\sum _{ij} \\Delta m_i \\bar{A}_{ij} \\Delta m_j + i q \\sum _i m_i \\right] \\prod _i d \\Delta m_i .$ We can now calculate the Gaussian integral over $\\Delta m_i$ 's using the standard result: $\\int d^W n \\exp \\left[-\\frac{1}{2} \\sum _{i,j} \\bar{A}_{ij} n_i n_j + \\sum _j n_j b_j\\right] = \\sqrt{\\frac{(2\\pi )^W}{\\det \\bar{A}}}\\exp \\left[\\frac{1}{2} \\sum _{i,j} b_i b_j (\\bar{A}^{-1})_{ij} \\right],$ where $\\bar{A}^{-1}$ denotes the inverse of $\\bar{A}$ .",
"Taking $b_j = i q$ for all $j$ we have $Z_{\\rm droplet}(W) \\cong e^{-S_W[\\lbrace \\bar{m}_i\\rbrace ]} \\sqrt{\\frac{(2\\pi )^W}{\\det \\bar{A}}} \\int _{-\\infty }^{+\\infty } \\frac{dq}{2\\pi } \\exp \\left[-\\frac{1}{2}q^2 \\sum _{i,j} (\\bar{A}^{-1})_{ij} \\right],$ and, performing the last Gaussian integral over $q$ , we obtain that $F_{\\rm droplet}(W) = \\ln Z_{\\rm droplet}(W) \\cong - S_W[\\lbrace \\bar{m}_i\\rbrace ] + Q(W), $ where $Q(W)$ correspond to a quantum correction to the free energy: $Q(W) = \\frac{W}{2} \\ln (2\\pi ) - \\frac{1}{2}\\ln \\det \\bar{A} - \\frac{1}{2} \\ln \\left( \\sum _{i,j} (\\bar{A}^{-1})_{ij} \\right).",
"$ The first term in Eq.",
"(REF ) is just the action (REF ) calculated along the classical trajectory and it can be easily evaluated in the continuous approximation: $S_W[\\lbrace \\bar{m}_i\\rbrace ] \\cong \\int _0^W \\left(c_1\\frac{(m^{\\prime }(t))^2}{m(t)} + c_2 m(t)^{1/3}\\right) = \\frac{9\\pi ^2 c_1 M}{2W^2} + \\frac{6^{1/3} c_2 M^{1/3} W^{2/3}}{\\pi ^{2/3}}, $ where we have inserted $m(t)$ from Eqs.",
"(REF ) and (REF ).",
"The quantum contribution $Q(W)$ to the effective action from Eq.",
"(REF ) consists of three terms.",
"The first term $ \\frac{W}{2} \\ln (2\\pi )$ is trivial.",
"The second term $- \\frac{1}{2}\\ln \\det \\bar{A}$ is more complicated because it contains the determinant of $\\bar{A}$ .",
"To evaluate this determinant, we first observe that the matrix $\\bar{A}$ is tridiagonal, with only non-zero elements being $\\bar{A}_{ii} &=& -\\frac{2 c_2}{9 \\bar{m}_i^{5/3}}+\\frac{4 c_1 (-\\bar{m}_{i-1}+\\bar{m}_i)^2}{(\\bar{m}_{i-1}+\\bar{m}_i)^3}-\\frac{8 c_1 (-\\bar{m}_{i-1}+\\bar{m}_i)}{(\\bar{m}_{i-1}+\\bar{m}_i)^2}+\\frac{4 c_1}{\\bar{m}_{i-1}+\\bar{m}_i} \\nonumber \\\\& &+ \\frac{4 c_1 (-\\bar{m}_i+\\bar{m}_{i+1})^2}{(\\bar{m}_i+\\bar{m}_{i+1})^3}+\\frac{8 c_1 (-\\bar{m}_i+\\bar{m}_{i+1})}{(\\bar{m}_i+\\bar{m}_{i+1})^2}+\\frac{4 c_1}{\\bar{m}_i+\\bar{m}_{i+1}} \\cong \\frac{4c_1 W}{M h(i/W)}, \\\\\\bar{A}_{i,i\\pm 1} &=& \\frac{4 c_1 (-\\bar{m}_i+\\bar{m}_{i\\pm 1})^2}{(\\bar{m}_i+\\bar{m}_{i\\pm 1})^3}-\\frac{4 c_1}{\\bar{m}_i+\\bar{m}_{i\\pm 1}} \\cong -\\frac{2c_1 W}{M h(i/W)}.$ We see that $\\bar{A}_{i,i\\pm 1} \\approx -\\frac{1}{2} \\bar{A}_{ii}$ so the determinant $\\det \\bar{A}$ can be approximated by $\\det \\bar{A} \\approx (\\det \\bar{a}) \\prod _{i=1}^W \\bar{A}_{ii}$ , where the matrix $\\bar{a}$ is a tridiagonal matrix with diagonal elements $\\bar{a}_{ii}=1$ and off-diagonal ones $\\bar{a}_{i,i\\pm 1}=-1/2$ .",
"One should note that due to the periodic boundary conditions also the corner elements $\\bar{a}_{1N}$ and $\\bar{a}_{N1}$ of this matrix should be in principle equal to $-1/2$ .",
"In this case the matrix $\\bar{a}$ would have a zero mode.",
"The zero mode has been however removed by fixing the position of the centre of mass to be at $N/2$ .",
"With this choice one can safely set $\\bar{a}_{1N}=\\bar{a}_{N1}=0$ leaving only the tridiagonal structure of the matrix $\\bar{a}$ .",
"The determinant of this matrix $\\det \\bar{a} = (N+1) 2^{-N}$ is independent of $W$ , hence the whole dependence of quantum corrections on $W$ is in the factor $\\prod _{i=1}^W \\bar{A}_{ii}$ .",
"We can now estimate that $\\ln \\det \\bar{A}\\cong \\sum _{i=1}^W \\ln \\frac{4c_1 W}{M h(i/W)} +O(N) \\cong W \\int _0^1 \\ln \\frac{4c_1 W}{M h(x)} dx +O(N) = W \\ln \\frac{128c_1 W}{3\\pi M} +O(N).$ This is the leading term in $Q(W)$ .",
"We shall now argue that the last term $\\sum _{i,j} (\\bar{A}^{-1})_{ij}$ in the quantum correction $Q(W)$ can be neglected.",
"The reason is that because $\\bar{A}_{ij} \\propto W/(M h(i/W))$ , elements of the inverse matrix $\\bar{A}^{-1}$ have to be proportional to a product of different powers of $W,M$ .",
"Therefore, the sum $\\sum _{i,j}^W (\\bar{A}^{-1})_{ij}$ will also be proportional to a certain power of $M$ times a certain power of $W$ (one can show using the fact that $\\bar{A}_{ij}$ is a Laplacian matrix times a diagonal matrix with elements $\\sim 1/h(i/W)$ that $\\sum _{i,j}^W (\\bar{A}^{-1})_{ij} = M W^2 O(1)$ ), and its logarithm will give only a sub-leading correction $\\sim \\ln W$ to $Q(W)$ , whose leading behaviour is $\\sim W\\ln W$ .",
"In summary, the quantum correction approximately reads $Q(W) \\cong \\frac{W}{2}\\left( \\ln \\frac{3\\pi ^2}{64c_1} - \\ln W + \\ln M\\right) +O(N), $ and, inserting Eqs.",
"(REF ) and (REF ) into Eq.",
"(REF ), and then Eq.",
"(REF ) into Eq.",
"(REF ) we obtain the final expression for the free energy of the system: $F(W) \\cong -W\\ln \\lambda _{\\rm max} - \\frac{9\\pi ^2 c_1 M}{2W^2} - \\frac{6^{1/3} c_2 M^{1/3} W^{2/3}}{\\pi ^{2/3}} + \\frac{W}{2}\\left( \\ln \\frac{3\\pi ^2}{64c_1} - \\ln W + \\ln M\\right) + O(N).$ The width $W$ of the droplet is determined by the maximum of $F(W)$ .",
"Taking a derivative with respect to $W$ we finally arrive at an equation for the spatial extension $W$ : $-\\ln \\lambda _{\\rm max}+c_1 9\\pi ^2 \\frac{M}{W^3}- c_2 \\frac{2\\cdot 6^{1/3} M^{1/3}}{3\\pi ^{2/3} W^{1/3}} + \\frac{1}{2} \\left(\\ln \\frac{3 \\pi ^2}{64 c_1}+\\ln M-\\ln W -1\\right) =0.",
"$ In the limit of large $M$ , this equation leads to the same expression as Eq.",
"(REF ).",
"For finite $M$ we solve it numerically for $W$ .",
"The solution gives a function $W=W_1(M)$ which includes quantum corrections.",
"The maximal eigenvalue $\\lambda _{\\rm max}$ of the matrix $T_{mn}$ from Eq.",
"(REF ) which is necessary to solve Eq.",
"(REF ) can be determined by numerical diagonalisation of $T_{mn}$ truncated at $m,n\\approx 50$ .",
"In Fig.",
"REF we compare $W=W_1(M)$ calculated as a root of Eq.",
"(REF ) and $W=W_0(M)$ obtained from the classical formula (REF ).",
"In the same plot we also show values of $W$ measured in simulations of the model for different $c_1,c_2$ .",
"We see that the solution $W=W_1(M)$ which takes into account quantum corrections reproduces the data much better than the classical solution $W=W_0(M)$ from Eq.",
"(REF ).",
"The agreement could be further improved by taking into account interactions on the interface between the droplet and the fluid, where the fluctuations $\\Delta m_i$ become non-Gaussian.",
"We will not do this here but instead we observe that subtracting a small correction from $W_1(M)$ , $W_2(M) = W_1(M) - 2, $ is enough to almost perfectly reproduce the data as shown in Figs.",
"REF and REF .",
"A physical meaning of this correction could be that interactions at the interface droplet-fluid exert a pressure on the droplet that shifts its boundaries towards the centre of mass by one lattice unit on each side of the droplet.",
"Figure: Plots of a normalised deviation between theoretical W th W_{th} and experimental W exp W_{exp} results versus MM.",
"Experimental results were obtained by MC simulations of the model.",
"Blue symbols show the ratio (W th -W exp )/W exp (W_{th}-W_{exp})/W_{exp} for classical prediction W th =W 0 (M)W_{th}=W_0(M) from Eq. ().",
"Green and red symbols show the ratio for quantum predictions W th =W 1 (M)W_{th}=W_1(M) (calculated from Eq.",
"()) and W th =W 2 (M)W_{th}=W_2(M) (Eq.",
"()), correspondingly without and with interface corrections.",
"Circles correspond to (c 1 ,c 2 )=(1,5)(c_1,c_2)=(1,5), squares to (c 1 ,c 2 )=(2,5)(c_1,c_2)=(2,5).",
"One can see that the expression W=W 2 (M)W=W_2(M) almost ideally reproduces results of simulations."
],
[
"Correlated-fluid phase",
"In models such as B-in-B [17] or ZRP [18] one usually fixes the density $\\rho =M/N$ of particles and takes the thermodynamic limit $M,N\\rightarrow \\infty $ .",
"The condensate emerges in this limit above the critical density $\\rho _c$ .",
"The same remains true in our model.",
"However, there is another important limit here, namely $M^{1/4}/N \\equiv w = \\rm const$ and $M,N\\rightarrow \\infty $ .",
"In this limit, the width $W\\sim M^{1/4}$ of the condensate becomes a finite fraction of the system size $N$ .",
"It turns out that there is a new phase transition as a function of the parameter $w$ : when the width of the condensate becomes equal to $N$ , both borders of the condensate merge together.",
"The envelope of the condensate loses its $\\cos ^3$ shape and becomes flat: mean occupation numbers $\\bar{m}_i = M/N \\propto N^3$ are much larger than 1, and fluctuations which are of order $\\sqrt{M/N}$ are not powerful enough to cause $\\lbrace m_i\\rbrace $ to drop to $m_i \\approx 1$ .",
"Therefore, the condensate no longer separates from the background.",
"We shall stress that the existence of this phase is possible only due to periodic boundary conditions.",
"If boundary conditions were fixed, i.e., $m_1=m_N=\\rm const$ , the droplet would not disappear but only changed its shape for $W> N$ .",
"The correlated-fluid phase is not the same as the weakly-correlated fluid phase below $\\rho _c$ .",
"In particular, correlations between different $m_i$ 's are very strong in this phase.",
"To calculate correlations ${\\rm cov}(m_j,m_k) = \\overline{m_j m_k}-\\bar{m}_j\\bar{m}_k$ , let us first observe that the partition function (REF ) can be approximated in this phase as $Z_{\\rm corr.",
"fluid}(N,M) \\approx \\sum _{m_1=-\\infty }^\\infty ... \\sum _{m_N=-\\infty }^\\infty \\exp \\left[ - \\sum _i \\left(c_1\\frac{(m_{i+1}-m_i)^2}{\\rho } + c_2 \\rho ^{1/3}\\right)\\right] \\delta \\left(\\sum _i m_i-M\\right), $ because the average occupation numbers $\\bar{m}_i \\approx \\rho $ and, since we anticipate that $\\sqrt{{\\rm var}(m_i)} \\ll \\bar{m}_i$ , we can focus on small deviations only.",
"If we now replace the sum by an $N$ -dimensional integral over $m_1,\\dots ,m_N$ , Eq.",
"(REF ) reduces to a Gaussian integral with the constraint on the total number of particles.",
"We can subsequently get rid of the Dirac delta by replacing it by $\\delta \\left(x\\right) = \\lim _{\\sigma \\rightarrow 0} \\frac{1}{\\sqrt{2\\pi }\\sigma } e^{-\\frac{x^2}{2\\sigma ^2}} \\ .$ We now define an auxiliary function $G(M,N,\\vec{u})$ with auxiliary variables $\\vec{u}$ : $G(M,N,\\vec{u}) = \\lim _{\\sigma \\rightarrow 0} \\int {\\rm d}^N m \\frac{1}{\\sqrt{2\\pi }\\sigma } \\exp \\left[-\\frac{M^2}{2\\sigma ^2} -\\frac{1}{2} \\vec{m}^T A\\vec{m} + (\\vec{b} + \\vec{u})^T \\vec{m} \\right], $ in which $b_i = M/\\sigma ^2$ and $A_{ij} = -\\frac{2c_1}{\\rho } \\Delta _{ij} + \\frac{1}{\\sigma ^2} \\delta _{ij}$ , where $\\Delta _{ij}$ denotes a 1d discrete Laplacian with periodic boundary conditions, and $\\delta _{ij}$ is the Kronecker delta.",
"We have: ${\\rm cov}(m_j,m_k) = \\overline{m_j m_k} - \\bar{m}_j \\bar{m}_k = \\left[\\frac{{\\rm d}}{{\\rm d}u_j} \\frac{{\\rm d}}{{\\rm d}u_k} \\ln G(M,N,\\vec{u}) \\right]_{\\vec{u}=0}.",
"$ The Gaussian integral in Eq.",
"(REF ) can be performed exactly: $G(M,N,\\vec{u}) = \\lim _{\\sigma \\rightarrow 0} \\sqrt{\\frac{(2\\pi )^{N-1}}{\\sigma ^2 \\det A}} \\exp \\left[ \\vec{b}^T A^{-1} \\vec{u} + \\frac{1}{2} \\vec{u}^T A^{-1} \\vec{u}\\right],$ and we obtain that ${\\rm var}(m_j) &=& {\\rm cov}(m_j,m_j) = \\lim _{\\sigma \\rightarrow 0} \\left(A^{-1}\\right)_{jj}, \\\\{\\rm cov}(m_j,m_k) &=& \\lim _{\\sigma \\rightarrow 0} \\left(A^{-1}\\right)_{jk}.$ The inverse matrix $A^{-1}$ which appears in these formulas can be calculated using spectral decomposition of the matrix $A$ : $A_{jk} &=& \\sum _i \\lambda _i \\psi _{i,j} \\psi _{i,k}, \\\\A^{-1}_{jk} &=& \\sum _i \\lambda _i^{-1} \\psi _{i,j} \\psi _{i,k}, $ in which $\\lbrace \\lambda _i\\rbrace $ and $\\lbrace \\vec{\\psi }_{i}\\rbrace $ are the eigenvalues and the corresponding normalised eigenvectors of $A$ , respectively, $\\lambda _k = \\left\\lbrace \\begin{array}{ll} N/\\sigma ^2 & k=1\\\\ 8c_1/\\rho & k=2\\\\ \\frac{8c_1}{\\rho } \\sin ^2(\\pi (k-1)/2N) & k=3,5,7 \\dots \\\\\\frac{8c_1}{\\rho } \\sin ^2(\\pi (k-2)/2N) & k=4,6,8 \\dots \\\\\\end{array}\\right.\\quad (\\vec{\\psi _{k}})_j = \\left\\lbrace \\begin{array}{ll} 1/\\sqrt{N} & k=1\\\\ (-1)^j/\\sqrt{N} & k=2\\\\ \\frac{\\cos (\\pi j(k-1)/N)}{\\sqrt{N/2}} & k=3,5,7,\\dots \\\\ \\frac{\\sin (\\pi j(k-2)/N)}{\\sqrt{N/2}} & k=4,6,8,\\dots \\end{array}\\right.$ Using the expansion (REF ) and taking the limit $\\sigma \\rightarrow 0$ we obtain for large $N$ : ${\\rm var}(m_j) &=& M/(24c_1),\\\\{\\rm cov}(m_j,m_k) &=& \\frac{M}{4c_1\\pi ^2} \\sum _{n=1}^\\infty \\frac{1}{n^2} \\cos \\left(\\frac{2\\pi n (k-j)}{N}\\right) \\ .$ This means that the correlation function $A(k)={\\rm cov}(m_1,m_{k+1})/{\\rm var}(m_1)$ behaves as $A(k) = \\frac{6}{\\pi ^2} \\sum _{n=1}^\\infty \\frac{1}{n^2} \\cos \\left(\\frac{2\\pi n k}{N}\\right), $ and does not depend either on $c_1,c_2$ , or the number of particles.",
"In Fig.",
"REF we show that $A(k)$ calculated from the above equation agrees very well with the result of numerical simulations.",
"Figure: Plot of A(k)A(k) in the correlated-fluid phase (solid red line) calculated from Eq.",
"() and compared to Monte-Carlo simulations for M=72400,N=80,c 1 =1.0,c 2 =-0.5M=72400, N=80, c_1=1.0, c_2=-0.5 (dashed green line).We shall now discuss the phase transition between the droplet and the correlated fluid phase.",
"For any fixed $w=M^{1/4}/N$ , there is a critical line in the $(c_1,c_2)$ phase plane which separates these phases.",
"In the limit of large $N,M$ and fixed $w$ , the line can be determined from the condition that $W(M)=N$ (Eq.",
"(REF )): $c_{2,\\rm crit}(c_1) \\approx \\frac{M^{2/3}}{N^{8/3}} r^{8/3} c_1, $ where $r$ is the proportionality coefficient $r\\approx 6.66432$ from Eq.",
"(REF ).",
"This means that if we plot the transition lines determined in computer simulations for different $M,N$ , and rescale $c_1\\rightarrow \\frac{M^{2/3}}{N^{8/3}} c_1$ , all of them should collapse onto a single line.",
"We show in Fig.",
"REF that such a collapse indeed seems to take place for large system sizes.",
"However, for the largest $M=289600$ for which we were able to obtain the phase diagram numerically, the data points are still quite far from the theoretical line.",
"We believe that this is caused by a very slow convergence towards the asymptotic result (REF ) due to finite-size corrections which are very strong in the region between the droplet and the fluid.",
"Figure: Rescaled phase line between the droplet phase and the correlated-fluid phase for systems with different number of particles M=4525,9050,18100,⋯,289600M=4525,9050,18100,\\dots ,289600, and N=80N=80.",
"Solid black line is our theoretical prediction ().The phase transition is of the first order.",
"One can see this by observing that the two phases coexist at the transition point with a characteristic binomial structure of the distribution of the order parameter.",
"The double maximum seen in Fig.",
"REF indicates that the system jumps from one phase to another.",
"This is a typical feature of the 1st-order transition.",
"Figure: Probability of order parameter γ\\gamma for droplet-correlated fluid phase transition, N=80N=80, M=18100M=18100, c1=0.5c1=0.5, c2=1.29c2=1.29."
],
[
"Other phases",
"We shall now briefly discuss three other phases which appear in our model: localised (condensed) phase, antiferromagnetic fluid, and uncorrelated fluid.",
"For positive $c_1,c_2$ , the width $W$ of the droplet decreases with decreasing $c_1$ as seen from Eq.",
"(REF ).",
"Finally, at the point of $c_1=0$ , the width formally reaches zero.",
"This means that the condensate becomes localised at a single site.",
"For $c_1=0$ , the partition function reads $Z(N,M) = \\sum _{m_1=1}^M ... \\sum _{m_N=1}^M \\exp \\left[ - c_2 \\sum _i m_i^{1/3}\\right] \\delta \\left(\\sum _i m_i-M\\right),$ and the probability of microstates factorizes over sites, $P(\\lbrace m_i\\rbrace )=f(m_1)\\cdots f(m_N)$ , with $f(m)=\\exp (-c_2 m^{1/3})$ .",
"In this limit, our model corresponds to the B-in-B/ZRP model with a stretched-exponential weight function $f(m)$ [18].",
"In particular, following [17], [18], the critical density is given by $\\rho _c = \\frac{F^{\\prime }(1)}{F(1)}, $ with $F(z) = \\sum _{m=1}^\\infty z^m \\exp (-c_2 m^{1/3}) .$ The above series does not admit a closed form, but it can be evaluated numerically for any $z$ and hence the critical density (REF ) can be computed for any $c_2>0$ .",
"An important observable in this phase is the distribution of particles $p(m)$ - the probability that a randomly chosen node has $m$ particles.",
"This corresponds to the distribution of three-volume in CDT.",
"This distribution can be approximated as follows for $\\rho \\gg \\rho _c$ : $p(m) \\approx \\exp (-c_2 m^{1/3})/F(1) + (1/N) p_{\\rm cond}(m),$ in which the first term corresponds to the critical distribution in the liquid bulk, and $p_{\\rm cond}(m)$ denotes the probability of finding $m$ particles in the condensate.",
"We can use the method of Ref.",
"[17], [21] to express this probability as follows: $p_{\\rm cond}(m) = N f(m) \\frac{I(N-1,m,M-m)}{Z(N,M)}.",
"$ Here $I(N-1,m,M-m)/Z(N,M)$ is the probability that the condensate has $m$ or less particles, $I(N,m,M) = \\sum _{m_1=1}^m \\dots \\sum _{m_N=1}^m \\delta \\left[M-\\sum _i m_i\\right] f(m_1) \\dots f(m_N).$ Following Ref.",
"[21], we replace the Delta function by its integral representation, and perform the sum over $\\lbrace m_i\\rbrace $ .",
"This gives $I(N-1,m,M-m) \\approx \\int _{-i\\infty }^{i\\infty } \\frac{ds}{2\\pi i} \\exp \\left[ N\\left( \\rho s - ms/N + \\ln F(e^{-s})\\right)\\right].$ The integral over $ds$ is dominated by its small-$s$ behaviour.",
"We therefore expand $\\ln F(e^{-s})$ at $s=0$ , $\\ln F(e^{-s}) \\cong \\ln F(1) -s \\frac{F^{\\prime }(1)}{F(1)} + \\frac{s^2}{2} \\left(\\frac{F^{\\prime }(1)}{F(1)} - \\frac{F^{\\prime 2}(1)}{F^2(1)} + \\frac{F^{\\prime \\prime }(1)}{F(1)}\\right) = \\ln F(1) -s\\rho _c + \\frac{s^2}{2} \\left(\\rho _c - \\rho _c^2 + \\frac{F^{\\prime \\prime }(1)}{F(1)}\\right),$ and evaluate the resulting Gaussian integral.",
"We obtain that the distribution of mass in the condensate (REF ) is approximately Gaussian for $m$ close to $N(\\rho -\\rho _c)$ : $p_{\\rm cond}(m) \\propto \\exp \\left[-c_2 m^{1/3} - \\frac{(m/N-(\\rho -\\rho _c))^2}{2(\\rho _c - \\rho _c^2 + \\frac{F^{\\prime \\prime }(1)}{F(1)})}\\right].",
"$ This result agrees qualitatively with the simulations, see Fig.",
"REF .",
"Figure: Probability p cond (m)p_{\\rm cond}(m) that the localised condensate has mm particles.",
"Blue symbols: computer simulations for N=50,M=2000,c 1 =0,c 2 =5N=50,M=2000,c_1=0,c_2=5.",
"Black line: exact probability distribution obtained from Z(N,M)Z(N,M) calculated recursively as in Ref. .",
"Red line: approximate formula () normalised so that ∑ m p cond =1/N\\sum _m p_{\\rm cond} = 1/N.When $c_1<0$ and $c_2>0$ , the condensate is still localised but the critical density is now zero, i.e., all particles go into the condensed phase.",
"This so-called complete (or strong) condensation has its origin in the fact that the radius of convergence $z_c$ of $Z_N(z)$ from Eq.",
"(REF ) is becomes zero.",
"This is because $T_{mn}(z)$ is unbound as either $m$ or $n$ approach infinity.",
"The number of particles in the condensate is $\\approx M$ and virtually does not fluctuate.",
"The transition between the droplet phase and the localised phase is of second order, because the order parameters are continuous at $c_1=0$ .",
"We shall now briefly discuss the antiferromagnetic phase.",
"This phase exists in the region of both coupling constants being negative: $c_1<0,c_2<0$ .",
"The two-point weight $g(m,n)$ from Eq.",
"(REF ) has now two positive terms: $(m-n)^2/(m+n)$ which prefers large differences in occupation numbers on neighbouring sites, and $m^{1/3}$ which prefers large occupations but itself does not lead to condensation.",
"In Fig.",
"REF we show the correlation function $A(k)$ for this phase.",
"Its oscillatory behaviour reflects altered arrangement of occupied/empty sites.",
"Interestingly, the correlation length is quite long, which may indicate a possible coupling between two neighbouring occupied sites via a not-completely-empty site between them.",
"Finally, let us consider the uncorrelated fluid phase which exists for $c_1=c_2=0$ .",
"The action $S[\\lbrace m_i\\rbrace ]$ equals zero and the partition function can be calculated exactly: $Z(N,M) = \\sum _{m_1=1}^M ... \\sum _{m_N=1}^M \\delta \\left(\\sum _i m_i-M\\right) = \\binom{M-1}{M-N} \\ .$ We can now calculate the distribution of particles as follows (cf.",
"Ref.",
"[22]): $p(m) = \\frac{Z(N-1,M-m)}{Z(N,M)} = \\frac{(N-1)(M-N)!(M-m-1)!}{(M-1)!(M-m-N+1)!}",
"\\cong \\frac{1}{\\rho }\\exp (-m/\\rho ),$ where the last formula holds for $M\\ll N$ , i.e.",
"for large density $\\rho =M/N$ we typically deal with in this work.",
"The distribution of particles (which corresponds to the distribution of three-volume) falls off exponentially with $m$ .",
"Figure: Correlations in the antiferromagnetic phase obtained in MC simulations N=80N=80, M=18100M=18100, c 1 =-0.5c_1=-0.5, c 2 =-0.5c_2=-0.5."
],
[
"Uniqueness of $g(m,n)$",
"The choice of the transfer matrix $g(m,n)$ made in Eq.",
"(REF ) to reproduce the bell-shaped quantum universe is not unique.",
"In fact, there is a whole family of functions $g(m,n)$ which lead to the following continuous limit: $P(m_1,\\dots ,m_N) \\rightarrow P(m(t)) = \\exp \\left[-\\int dt \\left(c_1\\frac{m(t)^{\\prime 2}}{m(t)} +c_2 m(t)^{1/3}\\right) \\right],$ and reproduce the shape given by Eq.",
"(REF ).",
"In particular, two other forms of $g(m,n)$ , the asymmetric one $g(m,n)=\\exp {\\left(-c_1 \\frac{(m-n)^2}{m}-c_2 m^{1/3}\\right)}, $ and the symmetric one with the geometric mean $\\sqrt{mn}$ rather than the arithmetic mean $(m+n)/2$ in the denominator, $g(m,n)=\\exp {\\left(-c_1 \\frac{(m-n)^2}{\\sqrt{mn}} -c_2 \\frac{m^{1/3} + n^{1/3}}{2}\\right)} \\ , $ have the same asymptotic behaviour as Eq.",
"(REF ).",
"Our simulations show (see Fig.",
"REF ) that the shape of the droplet is reproduced well by all three forms of $g(m,n)$ in the large-$N,M$ limit.",
"However, the shape is slightly asymmetric in the case of Eq.",
"(REF ), whereas it is perfectly symmetric for symmetric forms of $g(m,n)$ as those given in Eqs.",
"(REF ) or (REF ).",
"However, the data from the full CDT model are perfectly symmetric (excluding small statistical fluctuations).",
"We thus conclude that the asymmetry is of finite-size origin and that the effective transfer matrix in CDT has to be symmetric as in Eqs.",
"(REF ) or (REF ).",
"Interestingly, although Eq.",
"(REF ) leads to exactly the same envelope (REF ) in the droplet phase as Eq.",
"(REF ), it does not permit the existence of the antiferromagnetic phase.",
"Indeed, the corresponding action in the antiferromagnetic phase, $S_{\\rm antiferr.}",
"\\approx 2c_1 K^{-1/2} M^{3/2} + c_2 K^{2/3} M^{1/3},$ is bigger than the corresponding action in the localised phase, $S_{\\rm localised} \\approx 2c_1 M^{3/2} + c_2 M^{1/3},$ for $c_1<0$ and for any $c_2$ , and therefore antiferromagnetic states are disfavoured in this case.",
"In other words, the localised phase extends to all $c_2$ (positive and negative) in the phase plane (compare with Fig.",
"REF ) for the model with the transfer matrix given by Eq.",
"(REF ).",
"We see that the existence of the antiferromagnetic phase depends on the behaviour of the kernel $g(m,n)$ for small values of the arguments.",
"Figure: Comparison between the average droplet shape m ¯ i \\bar{m}_i (top) and quantum fluctuations var (m i )=m i 2 ¯-m ¯ i 2 \\sqrt{{\\rm var} (m_i)}=\\sqrt{\\overline{m_i^2}-\\overline{m}_i^2} (bottom) for asymmetric and symmetric g(m,n)g(m,n), for N=80,M=367200,c 1 =0.01,c 2 =0.59N=80, M=367200, c_1=0.01, c_2=0.59.",
"Black curves: MC simulations of the full CDT model (courtesy of A. Görlich) for T=80T=80 time slices and the total volume V 4 =367200V_4=367200 equivalent to the number of sites and particles in our simulations.",
"Left: symmetric g(m,n)g(m,n) from Eq. ().",
"The same result is obtained for the symmetric Eq.",
"() studied in previous sections.",
"Small asymmetry in var (m i )\\sqrt{{\\rm var} (m_i)} is caused by statistical fluctuations.",
"Right: asymmetric g(m,n)g(m,n) from Eq.",
"()."
],
[
"Conclusions",
"In this paper we have analysed a simple model of particles residing on sites of a 1d lattice, in which the probability of microstate (REF ) equals $e^{-S}$ , where $S$ corresponds to the effective action (REF ) of the CDT model.",
"We have shown that our model reproduces not only the average shape of the droplet – the macroscopic universe of CDT – but also quantum fluctuations around it.",
"We have calculated the extension of this droplet and shown that the quantum universe is bigger than classical de-Sitter solution.",
"The droplet phase is one of five different phases which exists in our model.",
"Two of these phases, localised condensate and uncorrelated fluid, can be identified as phases “B” and “A” of CDT.",
"In each of these phases, we have calculated the distribution of particles $p(m)$ which corresponds to the distribution of three-volume in CDT.",
"By measuring this distribution in the original CDT model and comparing it to our predictions one could validate our hypothesis that all phases can be described by the same effective action.",
"Furthermore, we have predicted the existence of at least one more phase – the correlated fluid phase.",
"This phase, although yet unobserved, must surely exists in CDT as a simple consequence of periodic boundary conditions ensured by the global topology of CDT.",
"We have calculated two observables: $p(m)$ and the correlation function $A(k)$ , which can be easily measured in CDT.",
"The agreement with our predictions would provide further evidence for the effective action (REF ).",
"Lastly, we have suggested that, depending on the behaviour of the action for small three-volumes, the fifth, antiferromagnetic phase can exist.",
"Our predictions can be tested in the CDT model, even without the knowledge of the mapping between the effective coupling constants $c_1,c_2$ and the parameters in the Einstein-Hilbert action of CDT.",
"In particular, the values of $c_1,c_2$ can be determined by fitting Eq.",
"(REF ) to the data from computer simulations in the macroscopic-universe phase, calculating $c_1/c_2$ from $W$ , and resolving for $c_1,c_2$ using the equation for the background density $\\rho _c$ .",
"Then, the correlated-fluid phase can be reached by increasing $M$ .",
"In other phases, equations derived in this paper for some quantities can be used to determine $c_1,c_2$ .",
"These values in turn can be applied to calculate other quantities and compare them to those estimated in full CDT simulations."
],
[
"Acknowledgments",
"We thank A. Görlich and J. Jurkiewicz for discussions and A. Görlich for providing us with data from CDT simulations.",
"BW was supported by the EPSRC under grant EP/E030173 and ZB by the Polish Ministry of Science Grant No.",
"N N202 229137 (2009-2012).",
"Our model can be simulated using standard Monte Carlo techniques.",
"We start each simulation from some initial, random configuration of particles and construct a Markov chain in the space of configurations by moving particles between sites with probability depending on the current configuration.",
"More specifically, we construct a new configuration $B=\\lbrace m_1,\\dots ,m_i-1,\\dots ,m_j+1,\\dots ,m_N\\rbrace $ from the old one $A=\\lbrace m_1,\\dots ,m_i,\\dots ,m_j,\\dots ,m_N\\rbrace $ by picking two random sites $i$ and $j$ with $m_i>1$ , and moving one particle from site $i$ to site $j$ with probability given by the Metropolis formula $P(A \\rightarrow B)= \\min {\\left\\lbrace 1,\\frac{P(B)}{P(A)} \\right\\rbrace } = \\min { \\left\\lbrace 1,\\frac{g(m_{i-1},m_i-1)g(m_i-1,m_{i+1})g(m_{j-1},m_j+1)g(m_j+1,m_{j+1})}{g(m_{i-1},m_i)g(m_i,m_{i+1})g(m_{j-1},m_j)g(m_j,m_{j+1})} \\right\\rbrace }$ if $i,j$ are not nearest neighbours, and with probability $P(A \\rightarrow B) = \\min { \\left\\lbrace 1,\\frac{g(m_{i-1},m_i-1)g(m_i-1,m_{i+1}+1)g(m_{i+1}+1,m_{i+2})}{g(m_{i-1},m_i)g(m_i,m_{i+1})g(m_{i+1},m_{i+2})} \\right\\rbrace } \\quad \\mbox{for} j=i+1, \\\\P(A \\rightarrow B) = \\min { \\left\\lbrace 1,\\frac{g(m_{i-2},m_{i-1}+1)g(m_{i-1}+1,m_i-1)g(m_i-1,m_{i+1})}{g(m_{i-2},m_{i-1})g(m_{i-1},m_i)g(m_i,m_{i+1})} \\right\\rbrace } \\quad \\mbox{for} j=i-1, $ if they are neighbours, i.e., if $|i-j|=1$ .",
"Such form of the acceptance probability guarantees that the probability of microstate $P(\\lbrace m_i\\rbrace )$ will be given by Eq.",
"(REF ).",
"It is convenient to introduce the following notation: $\\alpha (m,n)=\\frac{g(m,n-1)}{g(m,n)}, \\,\\,\\, \\beta (m,n)=\\frac{g(m-1,n)}{g(m,n)}, \\,\\,\\, \\gamma (m,n)=\\frac{g(m-1,n+1)}{g(m,n)}, \\,\\,\\, \\delta (m,n)=\\frac{g(m+1,n-1)}{g(m,n)}.$ Then, the acceptance probabilities can be rewritten as: $P(A \\rightarrow B)=\\min \\left\\lbrace 1, \\frac{\\alpha (m_{i-1},m_i) \\beta (m_i, m_i+1)}{\\alpha (m_{j-1},m_j+1) \\beta (m_j+1,m_{j+1})} \\right\\rbrace , \\quad \\mbox{for} \\; |i-j|>1, \\\\P(A \\rightarrow B)=\\min \\left\\lbrace 1, \\frac{\\alpha (m_{i-1},m_i) \\gamma (m_i, m_{i+1})}{\\beta (m_{i+1}+1,m_{i+2})} \\right\\rbrace , \\quad \\mbox{for} \\; j=i+1, \\\\P(A \\rightarrow B)=\\min \\left\\lbrace 1, \\frac{\\beta (m_{i},m_{i+1}) \\delta (m_{i-1}, m_{i})}{\\alpha (m_{i-2},m_{i-1}+1)} \\right\\rbrace , \\quad \\mbox{for} \\; j=i-1.",
"\\\\$ In our simulations, we calculate and store the values of $\\alpha (m,n)$ , $\\beta (m,n)$ , $\\gamma (m,n)$ , $\\delta (m,n)$ for $m,n = 1,...,m_{\\rm max}$ , with some $m_{\\rm max}<M$ .",
"This allows us to use Eqs.",
"(REF )-() and to avoid time-consuming computations of the ratios of $g(m,n)$ in Eqs.",
"(REF )-(), if only the number of particles at sites $i,j$ does not exceed $m_{\\rm max}$ .",
"Otherwise, we calculate the acceptance probability directly from Eqs.",
"(REF )-().",
"The value of $m_{\\rm max}$ - typically a few thousands - is chosen as big as possible given available computer memory.",
"To reduce the autocorrelation time, measurements are made every $M$ moves.",
"All measurements of the average shape of the condensate and fluctuations around it are performed by shifting the condensate for each sample to a common centre of mass at site $i=N/2$ .",
"In order to account for periodic boundary conditions, the centre of mass is found in a 2d plane, assuming that the sites reside on a circle in this plane, and then the coordinates $(x,y)$ of that point are mapped to the index $i$ of a site closest to the centre of mass.",
"We have checked that other procedures of finding the centre lead to very similar results."
]
] | 1204.1356 |
[
[
"Formation of nonequilibrium modulated phases under local energy input"
],
[
"Abstract We study numerically an inhomogeneous Ising lattice gas with short-range interactions where different sectors are in contact with thermal baths at different temperatures.",
"Inside the different sectors particles jump to empty sites following the familiar Kawasaki dynamics.",
"In addition, particles can freely hop from one sector to the other.",
"This crossing between the sectors breaks detailed balance and yields a local energy influx that drives the system to a nonequilibrium steady state.",
"When the low-temperature sector is cooled below the equilibrium critical temperature, a complicated nonequilibrium phase diagram emerges, dominated by unusual modulated nonequilibrium stationary states.",
"These steady states result from the interplay of phase separation and convection."
],
[
"Formation of nonequilibrium modulated phases under local energy input Formation of nonequilibrium modulated phases under local energy input Linjun Li Michel Pleimling Linjun Li and Michel Pleimling 1 Department of Physics, Virginia Tech, Blacksburg, VA 24061-0435 USA 05.70.LnNonequilibrium and irreversible thermodynamics 47.55.pbThermal convection 64.60.CnOrder-disorder transformations, statistical mechanics of model systems We study numerically an inhomogeneous Ising lattice gas with short-range interactions where different sectors are in contact with thermal baths at different temperatures.",
"Inside the different sectors particles jump to empty sites following the familiar Kawasaki dynamics.",
"In addition, particles can freely hop from one sector to the other.",
"This crossing between the sectors breaks detailed balance and yields a local energy influx that drives the system to a nonequilibrium steady state.",
"When the low-temperature sector is cooled below the equilibrium critical temperature, a complicated nonequilibrium phase diagram emerges, dominated by unusual modulated nonequilibrium stationary states.",
"These steady states result from the interplay of phase separation and convection.",
"Systems far from equilibrium display a large variety of novel and unexpected features that often defy our common sense.",
"Enhancing our understanding of generic properties of systems far from equilibrium therefore remains one of the main challenges faced by contemporary physics.",
"Recent years witnessed some major progress in the theoretical studies of various typical nonequilibrium situations, encompassing steady-state properties of paradigmatic transport models [1] and driven diffusive systems [2], aging phenomena during relaxation processes [3] as well as fluctuation relations and theorems both for steady-state systems and for systems driven out of a steady state (see, e.g., [4], [5], [6], [7], [8], [9]).",
"Still, a common theoretical framework for nonequilibrium systems remains elusive.",
"The role of surfaces and interfaces during nonequilibrium processes remains poorly understood as the overwhelming majority of studies focuses on bulk systems (see [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21] for some examples where the effect of surfaces was discussed).",
"However, as long-range correlations are usually present far from equilibrium, and this even in systems with only short-range interactions, the influence of surfaces and interfaces is expected to be non negligible in many instances, thereby changing the physical properties even far away from the interface.",
"In this Letter we discuss the intriguing phase diagram that emerges in an interacting many-body system with conserved dynamics when energy is pumped into the system locally at an interface.",
"This nonequilibrium phase diagram is found to be dominated by modulated phases with a modulation vector parallel to the interface.",
"The model we consider in the following is the standard ferromagnetic two-dimensional Ising model with the Hamiltonian $H=-J \\sum _{x,y}{(S_{x,y}S_{x+1,y}+S_{x,y}S_{x,y+1})}$ where the spin $S_{x,y} = \\pm 1$ characterizes the state of the site $(x,y)$ , whereas $J > 0$ is the coupling constant between nearest neighbor spins.",
"We supplement this model with conserved dynamics that only allows spin exchanges.",
"Alternatively, we can cast our model in the language of the Ising lattice gas where the spin value $+1$ corresponds to a particle, whereas the value $-1$ indicates an empty site (hole).",
"The total magnetization $M$ of our system (or, equivalently, the particle density) is therefore kept fixed.",
"In this communication we restrict ourselves to the case $M = 0$ .",
"This system is brought into contact with two different thermal baths, see Fig.",
"REF : whereas the lower sector is immersed into a heat bath that is at some high temperature $T^{\\prime }$ (the data presented below have been obtained for $T^{\\prime } = \\infty $ , for the sake of simplicity), the upper sector is in contact with a much colder heat bath, with temperature $T_0$ .",
"Using periodic boundary conditions in both directions, our systems consist of $2 W \\times 2 L$ sites, with two interfaces separating the hot and cold sectors.",
"Inside the two sectors we use Kawasaki dynamics, so that a particle can jump to an empty neighboring site (spins of different signs on neighboring sites can be exchanged) with the Metropolis rates at the given temperature.",
"In addition, particles freely jump from one sector to the other.",
"This breaks detailed balance at the interface separating the sectors and results in an energy flow from the hot sector to the cold one.",
"As a result the system is driven to a nonequilibrium stationary state [22].",
"Figure: (Color online)Schematic sketch of our system.",
"The lower (upper) sector is in contact with a hot (cold) thermal bath.Disorder prevails in the hot sector, whereas phase separation is possiblein the cold sector when cooled below the critical temperature of the two-dimensional Ising system.The arrows indicate possible moves of some particles.",
"Note that the particles can freely jumpfrom one sector to the other, thereby breaking detailed balance.We carefully checked that the results of our Monte Carlo simulations reported in the following were indeed obtained in the steady state of our systems.",
"Starting from different initial conditions, the system settles in the same stationary state after a long transient.",
"For the half width $W = 60$ discussed in detail below we typically discard the first 7 million time steps (one time step corresponds to $2W \\times 2L$ proposed updates) before computing time averages.",
"Longer relaxation times are needed for larger system sizes.",
"Our data acquisition runs were started from an initial state with half of the particles randomly distributed in the lower sector (corresponding to an equilibrium state at infinite temperature), whereas in the upper sector we considered a fully phase separated configuration (corresponding to the $T=0$ equilibrium state), see Fig.",
"REF .",
"In order to get a first impression of the complexity of the stationary states encountered in our system, we show in Fig.",
"REF some steady state configurations for various temperatures $T_0$ of the upper sector as well as for various aspect ratios $L/W$ , with $W = 60$ .",
"We readily identify different nonequilibrium phases.",
"Figure: (Color online)Typical steady state configurations, showing (a) the disordered phase, (b) and (c) two modulated phases with different wave numbers, and(d) the fully ordered phase, all encountered when changing the temperature T 0 T_0 of the cold sector and/or the aspect ratio of the system.The width of the system is always 2W=1202W = 120.",
"The system parameters are:(a) T 0 =2.4T_0=2.4 and L=32L = 32, (b) T 0 =1.0T_0=1.0 and L=60L=60, (c) T 0 =0.4T_0=0.4 andL=60L=60, and (d) T 0 =1.0T_0=1.0 and L=12L=12.",
"The gray (cyan online) lines indicate the boundaries between the two sectors.For the purpose of showing both boundaries we have shifted the configurations by L/2L/2 in the yy-direction.Fig.",
"REF a shows a typical configuration for the disordered phase that prevails when $T_0$ is larger than the critical temperature $T_c = 2.269 \\ldots $ of the two-dimensional Ising model (we set $J/k_B = 1$ , with $k_B$ being the Boltzmann constant).",
"Due to the contact with the hot sector, correlated fluctuations inside the cold sector are taking place close to the interface on much shorter length scales than for the corresponding equilibrium system.",
"When moving further away from the interface, the correlation length increases and approaches the correlation length of an equilibrium Ising system at temperature $T_0$ .",
"When $T_0$ is below $T_c$ , phase separation sets in in the low temperature half.",
"It is well known that the equilibrium Ising lattice gas at half filling separates into high and low density regions.",
"In our system, however, a very complicated ordered state emerges where the particles form stripes that span the cold sector in the direction perpendicular to the interface, the different stripes being separated by empty regions, see Fig.",
"REF b and REF c. As shown in the figures for two cases with 3 and 5 stripes, and as discussed in more detail below, the number of stripes changes when the temperature $T_0$ and/or the aspect ratio changes, yielding a complex nonequilibrium phase diagram.",
"Finally, for very asymmetric shapes, i.e.",
"for small values of $L/W$ , another phase emerges where the particles and holes are surprisingly well separated.",
"Fig.",
"REF d shows a configuration where the particles are almost completely sucked out of the low-temperature sector (due to the symmetry of the Ising Hamitonian, one can of course also end up with the situation where particles are accumulating in this sector).",
"All these particles end up in the infinite temperature region which therefore has an extremely high particle density.",
"Let us pause here for a second to consider an equilibrium system that at first look might seem very similar to the nonequilibrium system under investigation.",
"In this system we set the coupling constants to be one in the upper half of the system but zero in the lower half, with the coupling constants along bonds connecting the two sub-systems being equal to one.",
"The system itself is at a fixed temperature $T < T_c$ and the updates are done using Kawasaki dynamics.",
"As a result of the zero coupling constants in the lower sector, every proposed exchange will be accepted in that sector.",
"The main difference to our model can be found at the interface separating the sectors: for the equilibrium model the exchanges across the interface fulfill detailed balance, whereas detailed balance is broken in our case.",
"As a result the upper sector of the equlibrium system simply phase separates, whereas the lower sector remains disordered, and none of the intriguing non-equilibrium features discussed in this paper show up.",
"Especially, if we start for the equilibrium system with a phase separated upper sector and a disordered lower sector, phase separation will simply persist, as there is no mechanism in the equilibrium case that would allow to break up this well ordered large domains.",
"Consequently, modulated phases do not show up in this equilibrium system.",
"In order to fully characterize the periodic structures observed in our nonequilibrium system we analyze the vertically resolved structure factor, i.e.",
"the norm squared of the Fourier transform of $S_{x,y}$ in the $x$ -direction for fixed values of $y$ A more complete information is contained in the two-dimensional structure factor, but as we are interested here in the study of the modulated structures that form in $x$ -direction, we restrict ourselves to the Fourier transform in that direction.",
": $F_y(k) = \\left| \\frac{1}{2W}\\sum _{x=1}^{2W} \\, S_{x,y} \\, e^{i k x} \\right|^2~.$ For the sake of obtaining good statistics, we perform both a time and ensemble average.",
"After reaching the steady state we sample the quantity for another million time steps, repeating this procedure typically ten times.",
"In order to reduce the noise in the data we found it useful to average our quantity over the middle third of the upper sector.",
"It is this averaged quantity $F_{ave}(k) = \\frac{3}{L} \\sum \\limits _{y=4L/3}^{5L/3} \\langle F_y(k) \\rangle ~,$ where $\\langle \\cdots \\rangle $ indicates both the time and ensemble averages, that we use in order to construct the nonequilibrium phase diagram discussed below.",
"The temperature dependence of the averaged structure factor $F_{ave}(k)$ for the system with $W = L =60$ is discussed in Fig.",
"REF .",
"In most of the cases shown in that figure $F_{ave}(k)$ exhibits a single pronounced maximum at some value $k = k_{max}$ , as for example for the temperature $T_0 = 1.4$ , corresponding to a stable phase characterized by stripes of width $\\pi /k_{max}$ .",
"Thus, for $T_0 = 1.4$ we have 2 vertical stripes composed of particles with an average horizontal size of 30 lattice sites, separated by stripes of the same size that are formed by holes.",
"We will in the following characterize the different phases by the number of particle stripes.",
"Figure: (Color online) Averaged structure factor F ave (k)F_{ave}(k) for a system with W=L=60W = L =60,at various temperatures T 0 T_0 of the cold sector.",
"The peaks show up for well defined wave numbers kk that correspondto the number of stripes in the system.For $T_0=0.4$ the average structure factor has two pronounced maxima, thus revealing the coexistence of two nonequilibrium phases with different numbers of stripes (4 and 5 in our example).",
"The presence of coexistence regions indicates that the transitions between different phases are discontinuous.",
"We use the information from the structure factor in order to construct the phase diagram shown in Fig.",
"REF .",
"We can distinguish three different regions, corresponding to the different types of steady states illustrated in Fig.",
"REF : the disordered phase without long range order, the almost perfectly phase separated state where an overwhelmingly large fraction of particles is sucked into one part of the system, and the modulated region with a multitude of striped phases.",
"As inside the modulated region the transitions between the phases are discontinuous, the lines shown in that part of the diagram stand for the larger coexistence regions that separate different modulated phases.",
"It is worth noting that in our finite systems phases with strip lengths that are incommensurable with the lattice size are suppressed.",
"In the infinite volume limit $L, W \\longrightarrow \\infty $ , with $L/W$ kept constant, the stripe width is expected to change continuously.",
"Figure: (Color online) Phase diagram for a system of width 2W=1202W=120 as a function of T 0 /T c T_0/T_c and L/WL/W.Three different regions are readilyidentified: the disordered phase at large values of T 0 /T c T_0/T_c and small values of L/WL/W without long range order,the phase separated region where almost all particles can be found in one sector of the system, and themodulated region where different periodic arrangements show up in the cold temperaturesector of the system.",
"The crosses indicate the temperatures and system sizes at which the numericalsimulations were done.",
"The lines separating the different phases result from cubic splines throughthe midpoints between crosses that yield different phases.Studying systems with sizes ranging from $W = 30$ and $W=120$ , we observe the same three regions in the phase diagram.",
"The areas occupied by the disordered phase and by the phase separated region in the $T/T_c-L/W$ phase diagram decrease when the lateral extension of the system increases, indicating that they will vanish in the infinite volume limit.",
"Inside the modulated region the number of stable phases increases with $W$ .",
"Thus for $W=30$ we only observe phases that contain up to 5 stripes, whereas for $W=120$ we can identify phases with up to 20 stripes.",
"All this indicates that the modulated regions are not due to finite size effects and should therefore also persist in larger systems.",
"The apparent persistence of the modulated region for temperatures slightly above $T_c$ , the critical temperature of the infinite system, is due to the well known shift of the (pseudo-)critical temperature in finite systems.",
"In order to understand the appearance of stripes in this system with conserved dynamics, we point out that a recent study [22] showed the emergence of convection cells, driven by spontaneous symmetry breaking, when the cold sector temperature $T_0$ is set below the equilibrium critical temperature.",
"In that work pinned boundary conditions at the $y$ -boundaries were used in order to facilitate the measurement of the convection cells through the vorticity $\\omega $ and the stream function.",
"For the periodic boundary conditions used in our work the time and ensemble averaged vorticity $\\langle \\omega \\rangle $ is trivially zero.",
"Non-trivial insights into the emergence and persistence of convection cells, as well as of their relationship to the modulated phases, can be gained by studying (we here go back to the language of magnetic spin systems) the spin-vorticity correlation function $\\langle S_{x^{\\prime },y^{\\prime }} \\omega (x,y)\\rangle $ where $\\omega (x,y)$ is the vorticity related to the plaquette centered at $(x+\\frac{1}{2},y+\\frac{1}{2})$ .",
"The vorticity is thereby measured through $\\omega (x,y) = j_y(x,y) + j_x(x,y+1) - j_y(x+1,y) - j_x(x,y)$ where the $j$ 's are the net currents across each bond around the plaquette [22], i.e.",
"$\\omega (x,y) > 0$ for a counter-clockwise rotation around the plaquette.",
"Figure: (Color online) Temperature dependence of thespin-vorticity correlation 〈S W/2,3L/2 ω(W,L)〉\\langle S_{W/2,3L/2} \\, \\omega (W,L) \\rangle that relates the vorticity around a plaquette in the middle of theinterface separating the hot and cold sectors to the spin S W/2,3L/2 S_{W/2,3L/2} located insidethe cold sector.",
"The system contains 2W×2L2W\\times 2 L sites, with W=L=60W = L = 60.The inset shows the positions of the plaquette (square) and of the spin S W/2,3L/2 S_{W/2,3L/2} (full circle)in the sample.",
"The correlation 〈S 3W/2,3L/2 ω(W,L)〉\\langle S_{3W/2,3L/2} \\, \\omega (W,L) \\rangle with the spin S 3W/2,3L/2 S_{3W/2,3L/2}, shown by the open circle in the inset,only differs from 〈S W/2,3L/2 ω(W,L)〉\\langle S_{W/2,3L/2} \\, \\omega (W,L) \\rangle by the sign.These data result both from a time average and an ensemble average over 30different realizations of the noise.Fig.",
"REF shows the temperature dependence of $\\langle S_{W/2,3L/2} \\, \\omega (W,L) \\rangle $ that relates the vorticity around a plaquette in the middle of the interface between the hot and cold sectors to a spin located in the middle of the cold sector.",
"We immediately note that the spin-vorticity correlation is non-vanishing, thereby revealing the presence of convection cells when $T_0$ is brought below the critical temperature.",
"Due to the symmetry between the locations of the two spins $S_{W/2,3L/2}$ and $S_{3W/2,3L/2}$ the two correlation functions $\\langle S_{W/2,3L/2} \\, \\omega (W,L) \\rangle $ and $\\langle S_{3W/2,3L/2} \\, \\omega (W,L) \\rangle $ have the same magnitude but different signs.",
"The spin-vorticity correlation also shows features that are due to the transitions between modulated phases.",
"We can restrict the following discussion to configurations where the plaquette is located at a stripe boundary, as only these configurations yield a non-vanishing vorticity.",
"Thus the change in sign around $T_0 \\approx 0.26~T_c$ is readily understood by the phase sequence 3 stripes $-$ 4 stripes $-$ 5 stripes when lowering $T_0$ .",
"Writing the 3 stripes phase as $+-+|-+-$ , where the vertical line indicates the position of the plaquette along the $x$ axis whereas a $+$ ($-$ ) sign corresponds to a $+$ ($-$ ) stripe, we expect $\\omega > 0$ around our plaquette, due to the sequence $+|-$ [22].",
"The spin $S_{W/2,3L/2}$ being located in a $-$ region, this then yields a negative sign for $\\langle S_{W/2,3L/2} \\, \\omega (W,L) \\rangle $ .",
"This is different for the case of 5 stripes, $+-+-+|-+-+-$ , that yields again a counter-clockwise rotation with $\\omega > 0$ , but now with the spin $S_{W/2,3L/2}$ in a $+$ region.",
"These two phases are separated by the phase with four stripes, $+-+-|+-+-$ , where our spin is located at the boundary between $+$ and $-$ stripes, yielding a strong suppression of the corresponding spin-vorticity correlations.",
"These results indicate that it is the presence of convection cells that ultimately leads to the formation of the modulated phases in our nonequilibrium system.",
"Whereas the corresponding equilibrium system separates below the critical temperature into two domains with high and low densities, these large domains are broken up in our system due to the subtle interplay of phase separation and convection facilitated by the presence of the high temperature region.",
"An analytical description of this complicated situation remains a challenge and is left for future studies.",
"We expect this type of behavior to be generic for systems undergoing phase separation where different sectors are coupled to different thermal baths and where detailed balance is broken locally.",
"In this situation spontaneous symmetry breaking leads to the emergence of convection cells that favor the formation of modulated nonequilibrium phases.",
"We have restricted ourselves in this communication to the situation where the hot sector is at infinite temperatures.",
"A detailed discussion of the situation where that sector is at a finite temperature $T^{\\prime } \\ge T_c$ will be published elsewhere.",
"In summary, our study reveals a complex nonequilibrium phase diagram for an Ising lattice gas where different sectors of the system are in contact with thermal baths with markedly different temperatures.",
"In case that one sector is at a temperature above $T_c$ whereas the other is at a temperature below $T_c$ , two mechanisms set in, namely spontaneous symmetry breaking and convection, that together lead to the formation of remarkable nonequilibrium modulated stationary states.",
"We thank Beate Schmittmann and Royce Zia for many enlightening discussions and Uwe Täuber for a critical reading of the manuscript.",
"This work was supported by the US National Science Foundation through DMR-0904999."
]
] | 1204.1219 |
[
[
"Theory of Spin Torque Assisted Thermal Switching of Single Free Layer"
],
[
"Abstract The spin torque assisted thermal switching of the single free layer was studied theoretically.",
"Based on the rate equation, we derived the theoretical formulas of the most likely and mean switching currents of the sweep current assisted magnetization switching, and found that the value of the exponent $b$ in the switching rate formula significantly affects the estimation of the retention time of magnetic random access memory.",
"Based on the Fokker-Planck approach, we also showed that the value of $b$ should be two, not unity as argued in the previous works."
],
[
"Introduction",
"Magnetic random access memory (MRAM) using tunneling magnetoresistance (TMR) effect [1],[2] and spin torque switching [3],[4] has attracted much attention for spintronics device applications due to its non-volatility and fast writing time with a low switching current.",
"A high thermal stability ($\\Delta _{0}$ ) (more than 60) of magnetic tunnel junctions (MTJs) is also important to keep the information in MRAM more than ten years.",
"Recently, Hayakawa et al.",
"[5] and Yakata et al.",
"[6],[7] respectively reported that the anti-ferromagnetically (AF) and ferromagnetically (F) coupled synthetic free layers show high thermal stabilities ($\\Delta _{0}>80$ for AF coupled layer and $\\Delta _{0}=146$ for F coupled layer) compared to a single free layer.",
"The thermal stability has been determined by measuring the spin torque assisted thermal switching of the free layer and analyzing the time evolution of the switching probability by Brown's formula [8] with the spin torque term.",
"The theoretical formula of the switching probability is generally given by $P=1-\\exp [-\\int _{0}^{t} t̥^{\\prime }\\nu (t^{\\prime })]$ , where $\\nu (t)=f_{0}\\exp [-\\Delta _{0}(1-I/I_{\\rm c})^{b}]$ .",
"Here, $f_{0}$ , $I$ , and $I_{\\rm c}$ are the attempt frequency, current magnitude, and critical current of the spin torque switching at zero temperature, respectively.",
"$b$ is the exponent of the current term in the switching rate $\\nu $ , and was argued to be unity by Koch et al.",
"in 2004 [9].",
"On the other hand, recently, Suzuki et al.",
"[10] and we [11],[12] independently studied the spin torque assisted thermal switching theoretically, and showed that the exponent $b$ should be two.",
"Since the estimation of the thermal stability strongly depends on the value of $b$ , as discussed in this paper, the determination of $b$ is important for the spintronics applications.",
"In this paper, we study the spin torque assisted thermal switching of the single free layer theoretically.",
"In Sec.",
", we derive the theoretical formulas of the most likely and mean switching currents of the sweep current assisted magnetization switching, and study the effect of the value of the exponent $b$ on the estimation of the retention time of the MRAM.",
"In Sec.",
", the differences of the theories in Refs.",
"[9],[10],[11] are discussed by analyzing the solution of the Fokker-Planck equation.",
"Section is devoted to the conclusions."
],
[
"Theory of Magnetization Switching due to Sweep Current",
"In this section, we consider the spin torque assisted thermal switching of the uniaxially anisotropic free layer, which has two minima of its magnetic energy.",
"At the initial time $t=0$ , the system stays one minimum.",
"From $t=0$ , the electric current $I(t)=\\varkappa t$ is applied to the free layer which exerts the spin torque on the magnetization and assists its switching.",
"In this section, the current is assumed to increase linearly in time with the sweep rate $\\varkappa $ , as done in the experiments [7],[13],[14].",
"The magnitude of the current $I(t)=\\varkappa t$ should be less than $I_{\\rm c}$ because we are interested in the thermally activated region.",
"The time evolution of the survival probability of the initial state, $R(t)$ , is described by the rate equation, $\\frac{R̥(t)}{t̥}=-\\nu (t)R(t),$ where the switching rate $\\nu (t)$ is given by $\\nu (t)=f_{0}\\exp \\left[-\\Delta _{0}\\left(1-\\frac{I(t)}{I_{\\rm c}}\\right)^{b}\\right].$ We assume that the attempt frequency is constant.",
"$b$ is the exponent of the current term, $(1-I/I_{\\rm c})$ .",
"The switching probability is given by $P(t)=1-R(t)$ .",
"Also, we define the probability density $p(t)$ by $p(t)=-R̥/t̥=P̥/t̥$ .",
"Equation (REF ) describes the escape from one equilibrium to the others in many physical systems, and the value of $b$ reflects their energy landscape: $b=1$ for the Bell's approximation [15], $b=3/2$ for the linear-cubic potential [16], and $b=2$ for the parabolic potential [17],[18].",
"The determination of the value of $b$ has been discussed not only in spintronics but also the other fields of physics [19].",
"The form of Eq.",
"(REF ) is the special case of the model of Garg ($a$ in Ref.",
"[20] corresponds to $1-b$ ).",
"Figure: The time evolutions of(a) the switching probability P(t)P(t) and (b) its density p(t)p(t)for b=1b=1 (solid) and b=2b=2 (dotted).The solution of Eq.",
"(REF ) with the initial condition $R(0)=1$ is given by $R(t)\\!=\\!\\exp \\left\\lbrace \\!-\\frac{f_{0}I_{\\rm c}}{b \\varkappa \\Delta _{0}^{1/b}}\\!\\!\\left[\\gamma \\!\\left(\\!\\frac{1}{b},\\Delta _{0}\\!\\right)\\!-\\!\\gamma \\!\\left(\\!\\frac{1}{b}, \\Delta _{0}\\!",
"\\left(\\!",
"1 \\!-\\!",
"\\frac{I}{I_{\\rm c}} \\!\\right)^{b}\\right)\\!\\right]\\!\\right\\rbrace ,$ where $\\gamma (\\beta ,z)=\\int _{0}^{z} t̥ t^{\\beta -1}{\\rm e}^{-t}$ is the lower incomplete $\\Gamma $ function.",
"Figure REF the time evolutions of (a) the switching probability $P(t)$ and (b) its density $p(t)$ .",
"The values of the parameters are taken to be $f_{0}=1.0$ GHz, $I_{\\rm c}=1.0$ mA, $\\varkappa =1.0$ mA/s, and $\\Delta _{0}=60$ , respectively, which are typical values found in the experiments [6],[7],[13],[14].",
"As shown, $P(t)$ suddenly changes from 0 to 1 at a certain time $t=\\tilde{t}$ at which $p(t)$ takes its maximum.",
"We call $\\tilde{t}$ the switching time.",
"The switching time $\\tilde{t}$ is determined by the condition $(p̥(t)/t̥)_{t=\\tilde{t}}=0$ , i.e., $/t̥=\\nu ^{2}$ , and is given by $\\frac{\\varkappa \\tilde{t}}{I_{\\rm c}}=1-\\frac{1}{\\Delta _{0}}\\log \\left(\\frac{f_{0}I_{\\rm c}}{\\varkappa \\Delta _{0}}\\right),$ for $b=1$ , and $\\frac{\\varkappa \\tilde{t}}{I_{\\rm c}}=1-\\left\\lbrace \\frac{b-1}{b\\Delta _{0}}{\\rm plog}\\left[\\frac{b}{b-1}\\left(\\frac{f_{0}I_{\\rm c}}{b \\varkappa \\Delta _{0}^{1/b}}\\right)^{b/(b-1)}\\right]\\right\\rbrace ^{1/b},$ for $b>1$ .",
"Here ${\\rm plog}(z)$ is the product logarithm which satisfies ${\\rm plog}(z)\\exp [{\\rm plog}(z)]=z$ .",
"For a large $z \\gg 1$ , ${\\rm plog}(z) \\simeq \\log z$ , and $\\tilde{t}$ ($b>1$ ) can be approximated to $\\frac{\\varkappa \\tilde{t}}{I_{\\rm c}}\\simeq 1-\\left\\lbrace \\frac{1}{\\Delta _{0}}\\log \\left[\\left(\\frac{b}{b-1}\\right)^{1-1/b}\\frac{f_{0}I_{\\rm c}}{b\\varkappa \\Delta _{0}^{1/b}}\\right]\\right\\rbrace ^{1/b}.$ The current at $t=\\tilde{t}$ , $I(\\tilde{t})=\\varkappa \\tilde{t}$ , is the most likely switching current for the thermal switching.",
"Since we are interested in the switching after the injection of the current at $t=0$ , $\\tilde{t}$ should be larger than zero.",
"Thus, the above formula is valid in the sweep rate range $\\varkappa > \\varkappa _{\\rm c}$ , where the critical sweep rate $\\varkappa _{\\rm c}$ is given by $\\varkappa _{\\rm c}=\\frac{f_{0}I_{\\rm c}}{b\\Delta _{0}}{\\rm e}^{-\\Delta _{0}}.$ The value of $\\varkappa _{\\rm c}$ estimated by using the above parameter values is on the order of $10^{-19}$ mA/s, which is much smaller than the experimental values ($0.01-1.0$ mA/s in Ref.",
"[14]).",
"Thus, the above analysis is applicable to the conventional experiments.",
"We also define the mean switching current $\\langle I \\rangle $ by $\\langle I \\rangle =\\int _{0}^{1} R̥I=-\\int _{0}^{\\infty } t̥\\frac{R̥}{t̥}\\varkappa t=\\varkappa \\int _{0}^{\\infty } t̥R.$ Since $p(t)$ takes its maximum at $t=\\tilde{t}$ , we approximate that $\\begin{split}\\nu (t)&\\simeq \\tilde{\\nu }+\\frac{}{t̥}\\bigg |_{t=\\tilde{t}}\\left(t-\\tilde{t}\\right)=\\tilde{\\nu }\\left[1+\\tilde{\\nu }\\left(t-\\tilde{t}\\right)\\right]\\simeq \\tilde{\\nu }{\\rm e}^{\\tilde{\\nu }(t-\\tilde{t})},\\end{split}$ where $\\tilde{\\nu }=\\nu (\\tilde{t})$ .",
"Then, $R(t)=\\exp [-\\int _{0}^{t} t̥^{\\prime } \\nu (t^{\\prime })]$ can be approximated to $R(t)\\simeq \\exp \\left\\lbrace -\\Lambda \\left[\\exp \\left(\\tilde{\\nu }t\\right)-1\\right]\\right\\rbrace ,$ where $\\Lambda ={\\rm e}^{-\\tilde{\\nu }\\tilde{t}}$ .",
"Thus, $\\langle I \\rangle $ is given by $\\begin{split}\\langle I \\rangle &\\simeq \\varkappa {\\rm e}^{\\Lambda }\\int _{0}^{\\infty } t̥\\exp \\left(-\\Lambda {\\rm e}^{\\tilde{\\nu }t}\\right)=\\frac{\\varkappa {\\rm e}^{\\Lambda }}{\\tilde{\\nu }}E_{1}(\\Lambda ),\\end{split}$ where $E_{\\beta }(z)=\\int _{1}^{\\infty } t̥ {\\rm e}^{-zt}/t^{\\beta }$ is the exponential integral.",
"It should be noted that $E_{1}(\\Lambda )$ is expanded as [21] $E_{1}(\\Lambda )=-\\gamma -\\log \\Lambda -\\sum _{k=1}^{\\infty }\\frac{(-\\Lambda )^{k}}{kk!",
"},$ where $\\gamma =0.57721...$ is the Euler constant.",
"In general, the moment $\\langle I^{n} \\rangle = \\int _{0}^{1} R̥ I^{n}=n \\varkappa ^{n}\\int _{0}^{\\infty }t̥ Rt^{n-1}$ is given by $\\begin{split}\\langle I^{n} \\rangle &=n \\varkappa ^{n}{\\rm e}^{\\Lambda }\\int _{0}^{\\infty } t̥\\ t^{n-1}\\exp \\left(-\\Lambda {\\rm e}^{\\tilde{\\nu }t}\\right)\\\\&=n\\left(\\frac{\\varkappa }{\\tilde{\\nu }}\\right)^{n}{\\rm e}^{\\Lambda }\\int _{1}^{\\infty } x̥\\frac{(\\log x)^{n-1}}{x}{\\rm e}^{-\\Lambda x}.\\end{split}$ Then, the standard deviation of the current, $\\sigma _{I} \\!=\\!",
"\\sqrt{\\langle I^{2} \\rangle \\!-\\!",
"\\langle I \\rangle ^{2}}$ , is given by $\\begin{split}\\sigma _{I}^{2}\\!=\\!\\left(\\frac{\\varkappa }{\\tilde{\\nu }}\\right)^{2}\\!&\\left\\lbrace \\frac{\\pi ^{2}}{6}\\!+\\!\\Lambda \\!\\left[\\frac{\\pi ^{2}}{6}\\!-\\!2\\!+\\!\\gamma \\left(2\\!-\\!\\gamma \\right)\\!+\\!\\log \\!",
"\\Lambda \\!\\left(2\\!-\\!2 \\gamma \\!-\\!\\log \\!",
"\\Lambda \\right)\\right]\\right.\\\\&+\\!\\frac{\\Lambda ^{2}}{2}\\!\\left[\\frac{\\pi ^{2}}{6}\\!-\\!\\frac{11}{2}\\!+\\!\\gamma \\left(7 \\!-\\!",
"3\\gamma \\right)\\!+\\!\\log \\!",
"\\Lambda \\!\\left(7 \\!-\\!",
"6 \\gamma \\!-\\!",
"3 \\log \\!",
"\\Lambda \\right)\\right]\\\\&+\\!\\frac{\\Lambda ^{3}}{3!",
"}\\!\\left[\\frac{\\pi ^{2}}{6}\\!-\\!\\frac{247}{18}\\!+\\!\\frac{7 \\gamma (8 \\!-\\!",
"3 \\gamma ) \\!+\\!",
"7 \\log \\!",
"\\Lambda \\!",
"(8 \\!-\\!",
"6 \\gamma \\!- \\!",
"3 \\log \\!",
"\\Lambda )}{3}\\right]\\\\&+\\left.",
"{O}(\\Lambda ^{4})\\right\\rbrace .\\end{split}$ Since the thermal stability can be estimated by evaluating the parameter $\\tilde{\\nu }$ , as shown below, let us derive the relations between $\\tilde{\\nu }$ and experimentally measurable variables.",
"The difference between the most likely switching current $I(\\tilde{t})=\\varkappa \\tilde{t}$ and mean switching current $\\langle I \\rangle $ is given by $\\langle I \\rangle -I(\\tilde{t})=-\\frac{\\varkappa {\\rm e}^{\\Lambda }}{\\tilde{\\nu }}\\left[\\gamma +\\sum _{k=1}^{\\infty }\\frac{(-\\Lambda )^{k}}{kk!",
"}\\right]+\\left({\\rm e}^{\\Lambda }-1\\right)\\varkappa \\tilde{t}.$ For $b=1$ , $\\tilde{\\nu }$ and $\\tilde{\\nu }\\tilde{t}$ are, respectively, given by $\\tilde{\\nu }=\\frac{\\varkappa \\Delta _{0}}{I_{\\rm c}},$ $\\tilde{\\nu }\\tilde{t}=\\Delta _{0}\\left[1-\\frac{1}{\\Delta _{0}}\\log \\left(\\frac{f_{0}I_{\\rm c}}{\\varkappa \\Delta _{0}}\\right)\\right].$ As shown in Refs.",
"[22],[23] $I(\\tilde{t})/I_{\\rm c}=\\tilde{\\nu }\\tilde{t}/\\Delta _{0}$ is around $0.4 \\sim 1.0$ in the experimentally reasonable temperature and sweep rate regions (so called fast pulling regime or Garg's limit [24],[25]).",
"Thus, we can approximate that $\\Lambda ={\\rm e}^{-\\tilde{\\nu }\\tilde{t}} \\simeq 0$ and ${\\rm e}^{\\Lambda }={\\rm e}^{{\\rm e}^{-\\tilde{\\nu }\\tilde{t}}}\\simeq {\\rm e}^{{\\rm e}^{-\\Delta _{0}}}\\simeq 1$ for $\\Delta _{0}\\gg 1$ .",
"Then, $\\langle I \\rangle - I(\\tilde{t})$ for $b=1$ is given by $\\langle I \\rangle -I(\\tilde{t})=-\\gamma \\frac{I_{\\rm c}}{\\Delta _{0}}.$ Similarly, for $b>1$ , by using the approximation ${\\rm plog}(z) \\simeq \\log z$ , $\\tilde{\\nu }$ and $\\tilde{\\nu }\\tilde{t}$ are, respectively, given by $\\tilde{\\nu }\\simeq \\left(\\frac{b-1}{b}\\right)^{1-1/b}\\frac{b \\varkappa \\Delta _{0}^{1/b}}{I_{\\rm c}},$ $\\begin{split}\\tilde{\\nu }\\tilde{t}&\\simeq \\left(\\frac{b-1}{b}\\right)^{1-1/b}\\Delta _{0}^{1/b}\\\\&\\ \\ \\ \\ \\ \\ \\times \\left(1-\\left\\lbrace \\frac{1}{\\Delta _{0}}\\log \\left[\\left(\\frac{b}{b-1}\\right)^{1-1/b}\\frac{f_{0}I_{\\rm c}}{b \\varkappa \\Delta _{0}^{1/b}}\\right]\\right\\rbrace ^{1/b}\\right),\\\\&\\simeq \\left(\\frac{b-1}{b}\\right)^{1-1/b}\\Delta _{0}^{1/b}\\end{split}$ Then, $\\langle I \\rangle - I(\\tilde{t})$ for $b>1$ is given by $\\langle I \\rangle -I(\\tilde{t})\\simeq -\\gamma \\left(\\frac{b}{b-1}\\right)^{1-1/b}\\frac{I_{\\rm c}}{b \\Delta _{0}^{1/b}}.$ $[\\langle I \\rangle - I(\\tilde{t})]/I_{\\rm c}$ is approximately zero for a sufficiently high thermal stability ($\\Delta _{0} \\gg 1$ ) which means a narrow width of the probability density.",
"We also find $\\frac{\\tilde{\\nu } \\left[ \\langle I \\rangle - I(\\tilde{t}) \\right]}{\\varkappa }\\simeq -\\gamma =-0.57721...$ $\\frac{\\tilde{\\nu } \\sqrt{\\langle I^{2} \\rangle -\\langle I \\rangle ^{2}}}{\\varkappa }\\simeq \\sqrt{\\frac{\\pi ^{2}}{6}}=1.28254...$ for arbitrary $b$ and $\\Delta _{0} \\gg 1$ .",
"We numerically verify Eqs.",
"(REF ) and (REF ) among the temperature region $0< T \\le 500$ K, where the values of the parameters are same with those in Fig.",
"REF ($\\Delta _{0}\\propto 1/T$ is taken to be 60 for $T=300$ K).",
"Equation (REF ) or (REF ) can be used to determine the value of $\\tilde{\\nu }$ experimentally.",
"Otherwise, $\\tilde{\\nu }$ can be estimated by using the relation $\\tilde{\\nu }=-\\frac{1}{R}\\frac{R̥}{t̥}\\bigg |_{t=\\tilde{t}}=-\\frac{{t̥}\\log R\\bigg |_{t=\\tilde{t}}.",
"}{}$ Let us discuss the effect of the value of $b$ on the estimation of the retention time of MRAM.",
"We assume that the value of $I_{\\rm c}$ is experimentally determined by some other experiments [5].",
"Then, the unknown parameter in Eq.",
"(REF ) or (REF ) is only the thermal stability.",
"As mentioned above, $\\tilde{\\nu }$ can be experimentally determined by using Eq.",
"(REF ), (REF ), or (REF ).",
"By setting $\\tilde{\\nu }(b=1)=\\tilde{\\nu }(b=2)$ , we found that the estimated values of the thermal stability with $b=1$ ($\\Delta _{1}$ ) and $b=2$ ($\\Delta _{2}$ ) satisfy the relation $\\Delta _{1}=\\sqrt{2\\Delta _{2}}$ .",
"Let us define the retention time of MRAM by $t^{*}={\\rm e}^{\\Delta _{0}}/f_{0}$ .",
"Then, the ratio of the estimated values of the retention time by $b=1$ ($t_{1}^{*}$ ) and $b=2$ ($t_{2}^{*}$ ) is given by $t_{2}^{*}/t_{1}^{*}={\\rm e}^{\\Delta _{2}-\\sqrt{2\\Delta _{2}}}$ , which is on the order of $10^{21}$ for $\\Delta _{2}=60$ and increases with increasing $\\Delta _{2}$ .",
"Thus, the determination of the value of $b$ is important for the accurate estimation of the retention time of MRAM."
],
[
"Comparison with Theory of Koch ",
"In this section, we investigate the difference of the value of $b$ between Koch et al.",
"[9] and Refs.",
"[10],[11],[12] by comparing the solutions of the Fokker-Planck equation, and show that $b$ should be two.",
"For simplicity, in this section, the current magnitude is assumed to be constant in time [9],[10],[11],[12].",
"First of all, it should be mentioned that the analytical solution of the switching probability can be obtained only for the two special cases.",
"The first one is the uniaxially anisotropic system [10].",
"The second one is the in-plane magnetized thin film in which the switching path in the thermally activated region is completely limited to the film plane, and thus, the effect of the demagnetization field normal to film plane is neglected [11].",
"In these systems, the magnetization dynamics can be described by one variable (the angle from the easy axis, $\\theta $ ), although, in general, the magnetization dynamics is described by two angles (the zenith angle $\\theta $ and azimuth angle $\\varphi $ ).",
"Then, the thermal switching of the magnetization can be regarded as the one dimensional Brownian motion of a point particle.",
"Although the effect of the demagnetization field of an in-plane magnetized system is taken into account in the definition of the critical current of Ref.",
"[9], the model of Ref.",
"[9] should be regarded as the identical with the models in Refs.",
"[10],[11] because the assumption $\\mathbf {H}\\parallel \\mathbf {p}$ in Ref.",
"[9] is valid for the two special cases mentioned above, where $\\mathbf {H}$ and $\\mathbf {p}$ are the total magnetic field acting on the free layer and magnetization direction of the pinned layer, respectively.",
"The difficulty to calculate the spin torque assisted thermal switching probability arises from the fact that the spin torque cannot be expressed as the torque due to the conserved energy.",
"Mathematically, it means that we cannot find any function $\\tilde{F}(\\theta ,\\varphi )$ whose two gradients, $\\partial \\tilde{F}/\\partial \\varphi $ and $\\partial \\tilde{F}/\\partial \\theta $ , simultaneously give the spin torque terms of the Landau-Lifshitz-Gilbert equation in $(\\theta ,\\varphi )$ coordinate.",
"Then, the steady state solution of the Fokker-Planck equation deviates from the Boltzmann distribution.",
"However, in the two special cases mentioned above, since the magnetization dynamics depends on only $\\theta $ , $\\tilde{F}$ can be obtained by integrating the spin torque term with respect to $\\theta $ .",
"Then, the Fokker-Planck equation, $\\begin{split}\\frac{\\partial W}{\\partial t}\\!=\\!\\frac{\\alpha \\gamma ^{\\prime }}{\\sin \\theta }\\frac{\\partial }{\\partial \\theta }&\\!\\left\\lbrace \\sin \\theta \\!\\left[\\!\\left(\\!\\!H_{\\rm appl}\\!+\\!\\frac{H_{\\rm s}}{\\alpha }\\!+\\!H_{\\rm K}\\!\\cos \\theta \\!\\!\\right)\\!\\sin \\theta W\\right.\\right.\\\\&\\ \\ \\ \\ \\ +\\!\\left.\\left.\\frac{k_{\\rm B}T}{MV}\\frac{\\partial W}{\\partial \\theta }\\right]\\!\\right\\rbrace ,\\end{split}$ has a steady state solution of the Boltzmann distribution form, $W \\propto \\exp [-{F}/(k_{\\rm B}T)]$ .",
"Here $M$ , $V$ , $H_{\\rm appl}$ , $H_{\\rm K}$ , $H_{\\rm s}(\\propto I)$ , $\\gamma _{0}=(1+\\alpha ^{2})\\gamma ^{\\prime }$ , and $\\alpha $ are the magnetization, volume of the free layer, applied field, uniaxial anisotropy field, strength of the spin torque in the unit of the magnetic field, gyromagnetic ratio, and the Gilbert damping constant, respectively.",
"$F=-MH_{\\rm appl}V \\cos \\theta -(MH_{\\rm K}V/2) \\cos ^{2}\\theta $ is the magnetic energy, and ${F}$ is the effective magnetic energy given by $\\frac{{F}}{MV}=-H_{\\rm appl}\\cos \\theta -\\frac{H_{\\rm s}}{\\alpha }\\cos \\theta -\\frac{1}{2}H_{\\rm K}\\cos ^{2}\\theta .$ The term $-(MH_{\\rm s}V/\\alpha )\\cos \\theta $ in Eq.",
"(REF ) corresponds to $\\tilde{F}$ mentioned above.",
"By using the steady state solution of the Fokker-Planck equation, we can calculate the switching probability, according to Refs.",
"[8],[10],[11].",
"Koch et al.",
"argued that Brown's formula with the magnetic energy $F$ is applicable to the spin torque switching problem by replacing $\\alpha $ and $T$ with $\\tilde{\\alpha } \\!=\\!",
"\\alpha [1 \\!+\\!",
"H_{\\rm s}/(\\alpha H)]$ and $\\tilde{T} \\!=\\!",
"T/[1 \\!+\\!",
"H_{\\rm s}/(\\alpha H)]$ , where $H=|\\mathbf {H}|=|H_{\\rm appl}+H_{\\rm K}\\cos \\theta |$ .",
"These replacements arise from the assumption that the directions of the spin torque ($\\propto \\!",
"\\mathbf {M} \\!\\times \\!",
"(\\mathbf {M} \\!\\times \\!",
"\\mathbf {p})$ ) and the Landau-Lifshitz damping ($\\propto \\!",
"\\mathbf {M} \\!\\times \\!",
"(\\mathbf {M} \\!\\times \\!",
"\\mathbf {H})$ ) are parallel, i.e., $\\mathbf {H} \\parallel \\mathbf {p}$ .",
"At the minimum of the magnetic energy $F$ , $\\tilde{T} \\!=\\!",
"T/(1-I/I_{\\rm c})$ , and thus, Ref.",
"[9] argued that the exponent of the current term of the potential barrier height ($\\propto \\!",
"MH_{\\rm K}V/(2k_{\\rm B}\\tilde{T})$ ) is unity.",
"However, it should be noted that the definition of the the potential barrier height requires not only the minimum of the magnetic energy $F_{\\rm min} \\!=\\!",
"F(0)$ but also its maximum $F_{\\rm max} \\!=\\!",
"F(\\theta _{\\rm m})$ divided by the temperature, where $\\theta _{\\rm m}=\\cos ^{-1}(-H_{\\rm appl}/H_{\\rm K})$ .",
"We can easily verify that $H$ , and also $\\tilde{T}$ , are zero at $\\theta =\\theta _{\\rm m}$ .",
"Thus, $F_{\\rm max}/[k_{\\rm B}\\tilde{T}(\\theta _{\\rm m})]$ is not well defined, and the relation argued in Ref.",
"[9] is not satisfied, as shown below: $\\frac{F_{\\rm max}}{k_{\\rm B}\\tilde{T}(\\theta _{\\rm m})}-\\frac{F_{\\rm min}}{k_{\\rm B}\\tilde{T}(0)}\\ne \\frac{(F_{\\rm max}-F_{\\rm min})(1-I/I_{\\rm c})}{k_{\\rm B}T}.$ The origin of the problem in Ref.",
"[9] is that $\\exp [-F/(k_{\\rm B}\\tilde{T})]$ is not a steady state solution of the Fokker-Planck equation (REF ): the steady state solution is $\\exp [-{F}/(k_{\\rm B}T)]$ .",
"Since the effect of the spin torque can be regarded as an additional term to the applied field, as shown in Eq.",
"(REF ), the potential barrier height of the spin torque assisted thermal switching is, similar to Brown's formula [8], given by $\\frac{{F}_{\\rm max}-{F}_{\\rm min}}{k_{\\rm B}T}=\\Delta _{0}\\left(1+\\frac{H_{\\rm appl}+H_{\\rm s}/\\alpha }{H_{\\rm K}}\\right)^{2},$ where the thermal stability is defined by $\\Delta _{0}=MH_{\\rm K}V/(2k_{\\rm B}T)$ .",
"By using the relation $\\left(1\\!+\\!\\frac{H_{\\rm appl} \\!+\\!",
"H_{\\rm s}/\\alpha }{H_{\\rm K}}\\right)\\!=\\!\\left(1\\!+\\!\\frac{H_{\\rm appl}}{H_{\\rm K}}\\right)\\!\\!\\left[1\\!+\\!\\frac{H_{\\rm s}}{\\alpha (H_{\\rm K} \\!+\\!",
"H_{\\rm appl})}\\right],$ and defining the critical current $I_{\\rm c}$ by $H_{\\rm s}/[\\alpha (H_{\\rm K}+H_{\\rm appl})]=-I/I_{\\rm c}$ , we find that [11] $\\frac{{F}_{\\rm max}-{F}_{\\rm min}}{k_{\\rm B}T}=\\Delta _{0}\\left(1+\\frac{H_{\\rm appl}}{H_{\\rm K}}\\right)^{2}\\left(1-\\frac{I}{I_{\\rm c}}\\right)^{2},$ Thus, the exponent of the current term should be two."
],
[
"Conclusion",
"In conclusion, we studied the spin torque assisted thermal switching of the single free layer theoretically.",
"We derived the theoretical formulas of the most likely and averaged switching currents of the sweep current assisted magnetization reversal, and showed that the value of the exponent $b$ in the switching rate significantly affects the estimation of the retention time of MRAM.",
"We also discussed the difference between the theories in Ref.",
"[9] and Refs.",
"[10],[11] from the Fokker-Planck approach, and showed that the exponent should be two."
],
[
"Acknowledgment",
"The authors would like to acknowledge H. Kubota and S. Yuasa for the valuable discussions they had with us."
]
] | 1204.1190 |
[
[
"Post-outburst X-ray flux and timing evolution of Swift J1822.3-1606"
],
[
"Abstract Swift J1822.3-1606 was discovered on 2011 July 14 by the Swift Burst Alert Telescope following the detection of several bursts.",
"The source was found to have a period of 8.4377 s and was identified as a magnetar.",
"Here we present a phase-connected timing analysis and the evolution of the flux and spectral properties using RXTE, Swift, and Chandra observations.",
"We measure a spin frequency of 0.1185154343(8) s$^{-1}$ and a frequency derivative of $-4.3\\pm0.3\\times10^{-15}$ at MJD 55761.0, in a timing analysis that include significant non-zero second and third frequency derivatives that we attribute to timing noise.",
"This corresponds to an estimated spin-down inferred dipole magnetic field of $B\\sim5\\times10^{13}$ G, consistent with previous estimates though still possibly affected by unmodelled noise.",
"We find that the post-outburst 1--10 keV flux evolution can be characterized by a double-exponential decay with decay timescales of $15.5\\pm0.5$ and $177\\pm14$ days.",
"We also fit the light curve with a crustal cooling model which suggests that the cooling results from heat injection into the outer crust.",
"We find that the hardness-flux correlation observed in magnetar outbursts also characterizes the outburst of Swift J1822.3-1606.",
"We compare the properties of Swift J1822.3-1606 with those of other magnetars and their outbursts."
],
[
"Introduction",
"Over the past two decades, several new classes of neutron stars have been discovered [28].",
"Perhaps the most exotic is that of the magnetars, which exhibit some highly unusual properties, often including violent outbursts and high persistent X-ray luminosities that exceed their spin-down powers [56], [35].",
"These objects, while previously classified as anomalous X-ray pulsars (AXPs) and soft gamma repeaters (SGRs), are now generally accepted as a unified class of neutron stars powered by the decay of ultra-strong magnetic fields [51].",
"To date, there are roughly two dozen magnetars and candidates observed,See the magnetar catalog at http://www.physics.mcgill.ca/$\\sim $ pulsar/magnetar/main.html.",
"with spin periods between 2 and 12 s, and high spin-down rates that generally suggest dipole $B$ -fields of order $10^{14}$ to $10^{15}$ G [45].",
"Thanks to the Swift satellite, several new magnetars have been discovered in recent years via their outbursts [44], [22], [27].",
"Once a new source has been identified, long-term monitoring is crucial to measure its timing properties, and hence to constrain the dipole magnetic field strength.",
"Also, the flux evolution following an outburst could provide insights into many physical properties, such as the location of energy deposition during an outburst, the crust thickness and heat capacity [40], or the physics of a highly active magnetosphere [5], [38].",
"One of the latest additions to the list of magnetars is Swift J1822.3$-$ 1606.",
"This source was first detected by Swift Burst Alert Telescope (BAT) on 2011 July 14 (MJD 55756) via its bursting activities [10].",
"It was soon identified as a new magnetar upon the detection of a pulse period $P$ =8.4377 s [21].",
"No optical counterpart was found, with 3$\\sigma $ limit down to a z-band magnitude of 22.2 [43].",
"In [34], we reported initial timing and spectroscopic results using follow-up X-ray observations from Swift, Rossi X-ray Timing Explorer (RXTE), and Chandra X-ray Observatory.",
"We found a spin-down rate of $\\dot{P}=2.54\\times 10^{-13}$ which implies a surface dipole magnetic fieldThe surface dipolar component of the $B$ -field can be estimated by $B=3.2\\times 10^{19}(P\\dot{P})^{1/2}$ G. $B=4.7\\times 10^{13}$ G, the second lowest $B$ -field among magnetars.",
"Using an additional 6 months of Swift and XMM-Newton data, [46] present a timing solution and spectral analysis.",
"They find a spin-down rate of $\\dot{P}=8.3\\times 10^{-14}$ which implies a magnetic field of $B=2.7\\times 10^{13}$ , slightly lower than that found in Paper I.",
"In this paper, we present an updated timing solution, and the latest flux evolution using new observations from the same X-ray instruments as in Paper I.",
"The additional two Chandra and 18 Swift observations provide a timing baseline that is over four times longer and allows a detailed study of the flux decay.",
"We also report on an archival ROSAT observation to constrain the pre-outburst flux.",
"We discuss the effects of timing noise on our timing solution and the properties and implications of this outburst within the magnetar model."
],
[
"Observations",
"lccccc Summary of observations of Swift J1822.3$-$ 1606.",
"0pt ObsID Mode Obs Date MJD Exposure Days since trigger (TDB) (ks) 6cChandra 12612 ASIS-S CC 2011-07-27 55769.2 15.1 12.6 13511 HRC-I 2011-07-28 55770.8 1.2 14.2 12613 ASIS-S CC 2011-08-04 55777.1 13.5 20.5 12614 ASIS-S CC 2011-09-18 55822.7 10.1 66.1 12615 ASIS-S CC 2011-11-02 55867.1 16.3 110.5 14330 ASIS-S CC 2012-04-19 56036.9 20.0 280.4 6cROSAT rp500311n00 1993-09-12 49242 6.7 – 6cSwift 00032033001 PC 2011-07-15 55757.7 1.6 1.2 00032033002 WT 2011-07-16 55758.7 2.0 2.1 00032033003 WT 2011-07-17 55759.7 2.0 3.1 00032033005 WT 2011-07-19 55761.1 0.5 4.6 00032033006 WT 2011-07-20 55762.0 1.8 5.5 00032033007 WT 2011-07-21 55763.2 1.6 6.7 00032033008 WT 2011-07-23 55765.8 2.2 9.2 00032033009 WT 2011-07-24 55766.2 1.7 9.7 00032033010 WT 2011-07-27 55769.5 2.1 12.9 00032033011 WT 2011-07-28 55770.3 2.1 13.8 00032033012 WT 2011-07-29 55771.2 2.1 14.7 00032033013 WT 2011-07-30 55772.3 2.1 15.7 00032051001 WT 2011-08-05 55778.0 1.7 21.5 00032051002 WT 2011-08-06 55779.0 1.7 22.5 00032051003 WT 2011-08-07 55780.4 2.3 23.9 00032051004 WT 2011-08-08 55781.4 2.3 24.8 00032051005 WT 2011-08-13 55786.4 2.2 29.8 00032051006 WT 2011-08-14 55787.6 2.2 31.0 00032051007 WT 2011-08-15 55788.1 2.3 31.6 00032051008 WT 2011-08-16 55789.5 2.2 32.9 00032051009 WT 2011-08-17 55790.3 2.2 33.8 00032033015 WT 2011-08-27 55800.8 2.9 44.2 00032033016 WT 2011-09-03 55807.2 2.4 50.6 00032033017 PC 2011-09-18 55822.7 4.9 66.2 00032033018 WT 2011-09-20 55824.5 1.5 68.0 00032033019 WT 2011-09-25 55829.1 2.3 72.6 00032033020 WT 2011-10-01 55835.1 2.6 78.5 00032033021 WT 2011-10-07 55841.7 4.2 85.2 00032033022 WT 2011-10-15 55849.2 3.4 92.7 00032033023 WT 2011-10-22 55856.2 2.2 99.7 00032033024 PC 2011-10-28 55862.2 10.2 105.6 00032033025 PC 2012-02-19 55976.4 6.2 219.8 00032033026 WT 2012-02-20 55977.0 10.2 220.5 00032033027 PC 2012-02-21 55978.1 11.0 221.6 00032033028 WT 2012-02-24 55981.9 6.7 225.4 00032033029 WT 2012-02-25 55982.8 7.0 226.3 00032033030 WT 2012-02-28 55985.0 7.0 228.5 00032033031 WT 2012-03-05 55991.1 6.8 234.5 00032033032 WT 2012-04-14 56031.1 4.3 274.6 00032033033 WT 2012-05-05 56052.6 5.1 296.0 00032033034 WT 2012-05-26 56073.0 4.9 316.5 00032033035 WT 2012-06-17 56095.5 5.6 338.9 00032033036 WT 2012-06-26 56104.1 6.2 347.6 00032033037 WT 2012-07-06 56114.2 6.8 357.6 00032033039 WT 2012-08-17 56156.1 4.9 399.6 00032033040 WT 2012-08-22 56161.5 5.0 405.0 lcccc Summary of RXTE observations of Swift J1822.3$-$ 1606.",
"0pt ObsID Obs Date MJD Exposure Days since (TDB) (ks) trigger D96048-02-01-00 2011-07-16 55758.49 6.5 1.96 D96048-02-01-05 2011-07-18 55760.81 1.7 4.28 D96048-02-01-01 2011-07-19 55761.57 5.1 5.04 D96048-02-01-02 2011-07-20 55762.48 4.9 5.95 D96048-02-01-04 2011-07-21 55763.42 3.3 6.89 D96048-02-01-03 2011-07-21 55763.64 6.0 7.11 D96048-02-02-00 2011-07-22 55764.62 6.1 8.09 D96048-02-02-01 2011-07-23 55765.47 6.8 9.94 D96048-02-02-02 2011-07-25 55767.60 3.0 11.07 D96048-02-03-00 2011-07-29 55771.35 6.8 14.82 D96048-02-03-01 2011-08-01 55774.35 6.9 17.82 D96048-02-03-02 2011-08-04 55777.85 1.9 21.32 D96048-02-03-04 2011-08-04 55777.92 1.8 21.39 D96048-02-04-00 2011-08-07 55780.49 6.9 23.96 D96048-02-04-01 2011-08-09 55782.58 6.5 26.05 D96048-02-04-02 2011-08-11 55784.97 3.7 28.44 D96048-02-05-02 2011-08-12 55785.03 3.3 28.50 D96048-02-05-00 2011-08-15 55788.05 5.9 31.52 D96048-02-05-01 2011-08-16 55789.96 6.0 33.43 D96048-02-06-00 2011-08-21 55794.46 6.6 37.93 D96048-02-07-00 2011-08-26 55799.61 6.8 43.1 D96048-02-08-00 2011-09-06 55810.38 6.0 53.8 D96048-02-10-00 2011-09-16 55820.24 6.7 63.7 D96048-02-10-01 2011-09-22 55826.18 5.6 69.6 D96048-02-09-00 2011-09-25 55829.38 6.2 72.8 D96048-02-11-00 2011-10-01 55835.90 7.1 79.4 D96048-02-12-00 2011-10-08 55842.23 5.9 85.7 D96048-02-13-00 2011-10-15 55849.67 5.6 93.1 D96048-02-14-00 2011-10-29 55863.11 6.7 106.6 D96048-02-16-00 2011-11-13 55878.90 5.9 122.4 D96048-02-17-00 2011-11-20 55885.21 6.0 128.7 D96048-02-15-00 2011-11-28 55893.18 6.7 136.6"
],
[
"The Swift X-Ray Telescope (XRT) consists of a Wolter-I telescope and an XMM-Newton EPIC-MOS CCD detector [9].",
"Swift is optimized to provide rapid follow-up of gamma-ray bursts and other X-ray transients.",
"Following the 2011 July 14 outburst of Swift J1822.3$-$ 1606, the XRT was used to obtain 46 observations for a total exposure time of 175 ks.",
"Data were collected in two different modes, Photon Counting (PC) and Windowed Timing (WT).",
"While the former gives full imaging capability with a time resolution of 2.5 s, the latter forgoes imaging to provide 1.76-ms time resolution by reading out events in a collapsed one-dimensional strip.",
"For each observation, the unfiltered Level 1 data were downloaded from the Swift quicklook archivehttp://swift.gsfc.nasa.gov/cgi-bin/sdc/ql.",
"For a summary of observations used, see Table .",
"The standard XRT data reduction script, xrtpipeline, was then run using HEASOFT 6.11 and the Swift 20110725 CALDB.",
"We reduced the events to the barycenter using the position of RA$=18^{\\rm {h}}$ $22^{\\rm {m}}$ $18^{\\rm {s}}$ , Dec$=-16^{\\circ }$ 04 268 [37].",
"Source and background events were extracted using the following regions: for WT mode, a 40-pixel long strip centered on the source was used to extract the source events and a strip of the same size positioned away from the source was used to extract the background events.",
"For PC mode, a circular region with radius 20 pixels was used for the source region and an annulus with inner radius 40 pixels and outer radius 60 pixels was used as the background region.",
"For the first PC mode observation (00032033001), a circular region with radius 6 pixels was excluded to avoid pileup.",
"For the subsequent PC mode observation (00032033017), a region with radius 2 pixels was excluded.",
"We estimate the maximum pileup fraction of the remaining PC observations be less than 5%.",
"For the spectral analysis, Swift ancillary response files, which provide the effective area as a function of energy, were created using the FTOOL xrtmkarf and the spectral redistribution matrices from the Swift CALDB were used."
],
[
"The RXTE Proportional Counter Array (PCA) comprised five proportional counting units and provided a large collecting area and high timing precision [25].",
"We downloaded 32 observations from the HEASARC archive spanning an MJD range from 55758 to 55893, for a total of 174 ks of integration time.",
"The data were collected in GoodXenon mode which records each event with 1-$\\mu $ s time resolution.",
"The observations are summarized in Table .",
"We selected events in the 2–10 keV energy range (PCA channels 6–14) from the top xenon layer of each PCU for our analysis, to maximize signal-to-noise ratio.",
"The data from all the active PCUs were then merged.",
"If more than one observation occurred in a 24-hr period, the observations were combined into a single data set.",
"Photon arrival times were adjusted to the solar system barycenter using the same position as the for Swift data.",
"Events were then binned into time series with resolution 1/32 s for use in the following analysis."
],
[
"Following the outburst, we triggered our ToO program with the Chandra X-ray Observatory.",
"The telescope onboard Chandra has an effective area $\\sim $ 3 times larger than that of Swift XRT, when used with the ACIS detector in continuous clocking (CC) mode.",
"This mode has a time resolution of 2.85 ms and sensitivity between 0.3 and 10 keVhttp://cxc.harvard.edu/proposer/POG/html/.",
"Five ACIS CC-mode observations were obtained between days 13 and 281 after the outburst, with exposures ranging from 10 to 20 ks.",
"The observation parameters are summarized in Table .",
"For imaging purposes, we also processed a short (1.2 ks) archival Chandra HRC-I observation taken 14 days after the outburst.",
"All Chandra data were processed using CIAO 4.3 with CALDB 4.4.6.",
"We extracted the source events with a 6-long strip region, and the remainder of the collapsed strip ($\\sim $ 7 long), excluding the region within 1 of the source in order to minimize any contamination from the wings of the PSF, was used for the background.",
"We restricted the timing analysis to events between 0.3 and 8 keV.",
"Photon arrival times were corrected to the solar system barycenter.",
"The source spectrum was extracted using the tool specextract."
],
[
"The only existing X-ray image that covers the field prior to the outburst is a 6.5-ks ROSAT PSPC [3] observation of the nearby Hii region M17 (Omega Nebula, G15.1$-$ 0.8).",
"The observation has a time resolution of 130 ms. We downloaded the filtered event list from the HEASARC data archivehttp://heasarc.gsfc.nasa.gov/W3Browse/ and carried out the analysis using FTOOLS."
],
[
"Imaging",
"Figure REF shows the ROSAT and Chandra images.",
"Swift J1822.3$-$ 1606 is the only source detected in the Chandra HRC image and its radial profile is fully consistent with that of a model PSF.",
"Hence, there is no evidence for any surrounding nebula or dust scattering halo.",
"We find a source position of RA$=18^{\\rm {h}}$ $22^{\\rm {m}}$ $18.06^{\\rm {s}}$ , Dec$=-16^{\\circ }$ 04 2555 from the HRC image which is consistent with the XRT position from [37] used above.",
"We assume an error radius of $0.6$ which is the uncertainty in the absolute astrometry of Chandra for a 90% confidence interval According to http://cxc.harvard.edu/cal/ASPECT/celmon/.",
"In the ROSAT image, an unresolved source is clearly detected at the position of the magnetar, as first reported by [14].",
"Since the Chandra image shows no other bright X-ray sources in the field, we take this source to be Swift J1822.3$-$ 1606 in quiescence.",
"Using a $4\\times 2$ elliptical aperture, we obtain $113\\pm 11$ total counts in 0.1–2.4 keV range, of which $48\\pm 7$ counts are due to background.",
"Finally, we note that the diffuse X-ray emission $\\sim $ 20 southwest of the magnetar is from M17, which contains the young stellar cluster NGC 6618 with over 100 OB stars [30].",
"Barycentered events were used to derive a pulse time-of-arrival (TOA) for each Swift WT mode, RXTE, and Chandra observation.",
"For the RXTE observations, events were binned into time series with 31.25-ms resolution.",
"The time series were then folded with 128 phase bins using the ephemeris from Paper I.",
"A TOA was then measured from each profile by cross-correlation with a template profile.",
"We verified that the RXTE pulse profiles were consistent with each other except in one isolated observation which was handled accordingly (see Section REF ).",
"For Swift and Chandra observations, TOAs were extracted using a Maximum Likelihood (ML) method, as it yields more accurate TOAs than the traditional cross-correlation technique [33].",
"This method was not used for the RXTE observations as their high number of counts (due to the large collecting area and background count rates of the PCA) make the ML method computationally expensive.",
"The ML method for measuring TOAs requires a continuous model of the template pulse profile for which we used a Fourier model.",
"The discrete Fourier Transform of the binned template profile was first calculated.",
"The template was then fitted by $f(\\phi )=\\sum _{j=0}^n\\alpha _je^{i 2\\pi j\\phi }$ where $\\alpha _j$ is the Fourier coefficient for the $j^{th}$ harmonic, and $\\phi $ the phase between 0 and 1.",
"The number of harmonics used was optimized to account for the features of the light curve while ignoring small fluctuations caused by the finite number of counts.",
"For Swift J1822.3$-$ 1606, we used five harmonics to derive the TOAs.",
"For each observation, a probability or likelihood for a grid of trial offsets, $\\phi _\\mathrm {off}$ , can be calculated using $P(\\phi _\\mathrm {off}) = \\prod ^N_{i=0} f(\\phi _i - \\phi _\\mathrm {off})$ , where $\\phi _i$ is the phase of each photon folded at the best ephemeris of the pulsar.",
"The likelihood distribution that results then describes the probability density for the average pulse arrival time.",
"A TOA can be calculated from the optimal phase offset.",
"We estimated TOA uncertainties by simulating one-hundred sets of events drawn from the pulse profile of the observation and measured an offset for each set using the ML method.",
"The standard deviation of the simulated offset distribution was then taken as the TOA uncertainty.",
"The ML derived TOAs were consistent with those derived for Paper I using the cross-correlation method.",
"Timing solutions were then fit to the TOAs using TEMPOhttp://www.atnf.csiro.au/people/pulsar/tempo/.",
"Three solutions, one with a single freqency derivative, one with two frequency derivatives and one with three frequency derivatives, are given in Table REF .",
"The top panel of Figure REF shows the timing residuals with just $\\nu $ and $\\dot{\\nu }$ fitted (Solution 1), the middle panel shows the residuals with $\\ddot{\\nu }$ also fitted (Solution 2), and the bottom panel shows the residuals with $\\ddot{\\nu }$ and $\\dddot{\\nu }$ also fitted (Solution 3).",
"Solution 1 is a poor fit with a $\\chi ^2_\\nu /\\nu $ of 5.02/72.",
"This is likely due to timing noise, a common phenomenon in young neutron stars including magnetars [12], [32].",
"The best-fit $\\nu $ and $\\dot{\\nu }$ values for Solution 1 imply a surface dipolar magnetic field of $B = 2.43\\pm 0.03 \\times 10^{13}$ G. Solution 2, with a significant non-zero $\\ddot{\\nu }$ , gives a better fit with a $\\chi ^2_\\nu /\\nu $ of 1.94/71.",
"An $F$ -test gives a probability of $2 \\times 10^{-16}$ that the addition of a second derivative does not significantly improve the fit.",
"The surface dipolar magnetic field implied by Solution 2 is $B = 3.84\\pm 0.08 \\times 10^{13}$ G. Solution 3, with a significant non-zero $\\dddot{\\nu }$ , provides still a better fit than Solution 2 with a $\\chi ^2_\\nu /\\nu $ of 1.44/70.",
"An $F$ -test gives a probability of $3 \\times 10^{-6}$ that the addition of a third derivative does not significantly improve the fit.",
"The best-fit parameters from Solution 3 imply a surface dipolar magnetic field of $B = 5.1\\pm 0.2 \\times 10^{13}$ G. Note that the fit is heavily influenced by the very high quality Chandra TOAs.",
"However, omitting them and including only TOAs from Swift and RXTE still yields significant second and third derivatives and an implied $B$ -field of $B = 4.8\\pm 0.2 \\times 10^{13}$ G which is consistent with that of Solution 3.",
"The above-quoted uncertainties in $B$ and other derived quantities in Table REF reflect only the statistical uncertainties in $\\nu $ and its derivatives and do not include any contributions from the simplified assumptions in the standard formulae used to determine such quantities.",
"Note that even with the addition of highly significant second and third derivatives, Solution 3 still does not provide an adequate fit.",
"Adding additional derivatives reduces the $\\chi ^2$ with marginal significance and results in larger values of the spin-down rate and hence $B$ .",
"For example, including a fourth frequency derivative does not result in significant improvement in $\\chi ^2$ ($\\chi ^2_\\nu /\\nu = 1.31/69$ ) and yields $B=6.0\\times 10^{13}$ G. To search for pulsations in the ROSAT observation, we applied a barycenter correction to the event arrival times, then used the $Z_m^2$ test [8] to search for pulsations.",
"We searched in the frequency range from zero to 3.8 kHz in steps of 1.3 $\\mu $ Hz, oversampling the independent Fourier spacing by a factor of 10; however, we found no significant signal.",
"By simulating a pulsar with a background subtracted count rate of that of the ROSAT observation, we find that the pulsar would be undetectable even with a pulsed fraction of 100%, therefore we cannot constrain the pulsed fraction.",
"Figure: NO_CAPTIONlc Spin Parameters for Swift J1822.3$-$ 1606.",
"0pt Parameter Value Dates (Modified Julian Day) 55759 – 56161 Epoch (Modified Julian Day) 55761.0 Number of TOAs - RXTE 31 Number of TOAs - Swift 40 Number of TOAs - Chandra 5 Solution 1 - one frequency derivative $\\nu $ (s$^{-1}$ ) 0.1185154253(3) $\\dot{\\nu }$ (s$^{-2}$ ) $-9.6(3)\\times 10^{-16}$ RMS residuals (ms) 52.2 $\\chi ^2_\\nu /\\nu $ 5.02/72 $B$ (G) $2.43(3)\\times 10^{13}$ $\\dot{E}$ (erg s$^{-1}$ ) $4.5(1)\\times 10^{30}$ $\\tau _c$ (kyr) 1963(51) Solution 2 - two frequency derivatives $\\nu $ (s$^{-1}$ ) 0.1185154306(5) $\\dot{\\nu }$ (s$^{-2}$ ) $-2.4(1)\\times 10^{-15}$ $\\ddot{\\nu }$ (s$^{-3}$ ) $1.12(8)\\times 10^{-22}$ RMS residuals (ms) 32.2 $\\chi ^2_\\nu /\\nu $ 1.94/71 $B$ (G) $3.84(8)\\times 10^{13}$ $\\dot{E}$ (erg s$^{-1}$ ) $1.12(5)\\times 10^{31}$ $\\tau _c$ (kyr) 784(33) Solution 3 - three frequency derivatives $\\nu $ (s$^{-1}$ ) 0.1185154343(8) $\\dot{\\nu }$ (s$^{-2}$ ) $-4.3(3)\\times 10^{-15}$ $\\ddot{\\nu }$ (s$^{-3}$ ) $4.4(6)\\times 10^{-22}$ $\\dddot{\\nu }$ (s$^{-4}$ ) $-2.2(4)\\times 10^{-29}$ RMS residuals (ms) 27.5 $\\chi ^2_\\nu /\\nu $ 1.44/70 $B$ (G) $5.1(2)\\times 10^{13}$ $\\dot{E}$ (erg s$^{-1}$ ) $2.0(2)\\times 10^{31}$ $\\tau _c$ (kyr) 442(33) Errors are formal 1$\\sigma $ TEMPO uncertainties."
],
[
"Pulse Profile Analysis",
"Here we search for time and energy variability in the pulse profile of Swift J1822.3$-$ 1606 using the RXTE, Swift, and Chandra observations.",
"We created pulse profiles for each RXTE observation for energy ranges of 2 – 6 keV, 6 – 10 keV (with photons selected from only the top xenon layer), 10 – 15 keV, 15 – 20 keV, 20 – 40 keV, and 20 – 60 keV (with photons selected from all three xenon layers) using the Solution 2 ephemeris.",
"For the Chandra data we produced pulse profiles with the energy ranges of 0.5 – 6 keV, 0.5 – 2 keV and 2 – 6 keV.",
"For the Swift data we created 0.5 – 10 keV profiles, using only WT mode observations as PC mode does not have sufficient time resolution.",
"As in Paper I, we searched for time variability in pulse profiles but found that all the RXTE profiles are consistent with the template in each case except for the one profile from the very first observation after the outburst (Obsid D96048-02-01-00).",
"The difference is due primarily to the off-pulse feature which had slightly different structure between the template profile and the first RXTE observation.",
"We therefore did not use D96048-02-01-00 in the timing analysis.",
"The Chandra profiles, however, do show evidence for low-level variability.",
"Figure REF shows the 0.5 – 6 keV pulse profile for each Chandra observation of Swift J1822.3$-$ 1606.",
"We produced residuals between each pair of Chandra profiles by normalizing each profile and taking the difference between each normalized pair.",
"A comparison of the profile residuals between each set of profiles shows that there is significant low-level evolution of the small `pulse' that precedes the main pulse.",
"The main pulse does not exhibit any significant variation.",
"The most significant variability is that between the first (MJD 55769) and last (MJD 56036) Chandra observation.",
"The residuals between those two profiles have a ${\\chi }_\\nu ^2$ of 16.8 for 63 degrees of freedom.",
"We note however that these low-level variations in the smaller component likely do not have a significant impact on the timing analysis, since the TOA extraction is heavily weighted toward the unchanging primary component.",
"Indeed our simulations of the effects of such low-level profile variations on the TOAs (see below) strongly support this conclusion.",
"The Swift profiles also show evidence for low-level variability.",
"As above, we produced residuals between each pair of profiles and calculated a ${\\chi }_\\nu ^2$ for the null hypothesis.",
"The measured values are not consistent with a ${\\chi }^2$ distribution, so there is significant variation between profile pairs.",
"A closer look at the residuals shows that the variation is due primarily to the small interpulse, as in the Chandra data.",
"To investigate the dependence of pulse morphology on energy, we created a single high significance profile by aligning and summing individual profiles for each energy range.",
"Figure REF shows a summary of the results, with the summed profiles for 0.5 – 2 keV (top panel, with 64 phase bins, Chandra data), 2 – 6 keV (middle panel, with 64 phase bins, Chandra data), 6 – 10 keV and 10 – 15 keV (bottom two panels, RXTE data, with 64 and 16 phase bins, respectively).",
"No pulsations were detected above 15 keV with the PCA.",
"We then calculated residuals between pairs of profiles, and calculated ${\\chi }_\\nu ^2$ values of the resulting residuals in order to identify energy dependence of the pulse morphology.",
"The most significant variability is between the 0.5 – 2 keV Chandra profile and the 6 – 10 keV RXTE profile.",
"This can been seen in Figure REF as a change in the phase of the interpulse, arriving later for higher energies.",
"For this profile pair, the ${\\chi }_\\nu ^2$ of the residuals is 46.2 (for 28 degrees of freedom), excluding the null hypothesis.",
"The interpulse variability causes significant differences between each pair of profiles, except the 10 – 15 keV profile, likely because of the lower statistics of the latter."
],
[
"Comparison to Previously Reported Results",
"[46] present a timing solution with a spin-down implied magnetic field of $B=2.7\\times 10^{13}$ G. Their data set is similar to ours, although they use proprietary XMM-Newton data whereas we use proprietary Chandra data and our data set includes seven additional Swift observations.",
"Their timing solution is similar to our Solution 1.",
"They, however, do not find a significant second frequency derivative.",
"A possible cause of this discrepancy could be the difference in TOA extraction methods.",
"Instead of using a pulse profile template, [46] fit the folded profile for each observation with two sine functions with periods equal to the fundamental and the first harmonic of the pulse period.",
"They then assign the ascending node of the fundamental sine function as the time-of-arrival of the pulse.",
"This method was used to attempt to account for pulse-profile changes.",
"We implemented this method and derived an additional set of TOAs to compare to our ML derived TOAs.",
"We found that the sine-model derived TOAs provided similar timing solutions as our ML TOAs and the addition of a second frequency derivative did significantly improve the fit, reducing the $\\chi ^2_\\nu /\\nu $ from 7.91/72 to 2.72/71.",
"The addition of a third derivative in this case, results in only marginal improvement with a $\\chi ^2_\\nu /\\nu $ of 2.47/70.",
"If we limit our dataset to the Swift and RXTE data used in [46], we find that the addition of a second frequency derivative is not necessary, which is consistent with their findings.",
"In order to investigate the effects of pulse profile changes on both TOA extraction methods, we simulated pulse profiles with an unchanging (other than noise) primary component and a varying secondary component, as is observed in the pulse profile evolution of Swift J1822.3$-$ 1606.",
"We modelled the profile using two gaussians and modified the amplitude of the smaller gaussian in order to vary the secondary component.",
"We found that for both the sine-model and the ML methods, as the amplitude of the secondary component was varied, the measured phase offsets varied by less than their uncertainties.",
"Hence we conclude that the observed pulse profile variations do not have an appreciable affect on the TOA determination, independent of which TOA extraction method was used.",
"lcccc Models of the Flux Evolution of Swift J1822.3$-$ 1606.",
"0pt Model $\\tau _1$ (days) $\\tau _2$ (days) $F_q$ (erg cm$^{-2}$ s$^{-1}$ ) $\\chi ^2_\\nu /\\nu $ Single Exponential $23.8\\pm 0.5$ – $3\\times 10^{-14}$ (fixed) 20.1/43 Single Exponential $19.5\\pm 0.4$ – $4.7\\pm 0.2\\times 10^{-12}$ 4.48/42 Double Exponential $15.5\\pm 0.5$ $177\\pm 14$ $3\\times 10^{-14}$ (fixed) 2.17/41 Double Exponential $9\\pm 1$ $39\\pm 3$ $4.0\\pm 0.2\\times 10^{-12}$ 1.06/40 Figure: NO_CAPTIONFigure: NO_CAPTIONSpectral models were fit to the Swift, Chandra and ROSAT data using XSPEChttp://xspec.gfsc.nasa.gov v12.7.",
"The quiescent flux of Swift J1822.3$-$ 1606 was determined by first extracting the source spectrum from the ROSAT data, then fitting it with an absorbed blackbody model.",
"The absorption column density $N_{\\rm H}$ was fixed during the fit at the best-fit value ($4.53\\times 10^{21}\\, \\textrm {cm}^{-2}$ ) determined from the Swift and Chandra spectra (see below).",
"We obtained a quiescent blackbody temperature of $kT=0.12\\pm 0.02$ keV and a radius of $5_{-2}^{+7}d_{1.6}$ km, where $d_{1.6}$ is the distance to the source in units of 1.6 kpc, the estimated distance as discussed in Section REF .",
"We found an absorbed flux of $9_{-9}^{+20}\\times 10^{-14}$ erg cm$^{-2}$ s$^{-1}$ in the 0.1–2.4 keV range.",
"The Chandra and Swift spectra were grouped with a minimum of 100 and 20 counts per bin, respectively.",
"The spectra were fitted jointly to a photoelectrically absorbed blackbody model with an added power-law component.",
"The model was fit with a single $N_{\\rm H}$ using the XSPEC phabs model assuming abundances from [1] and photoelectric cross-sections from [4].",
"All the other parameters were allowed to vary from observation to observation.",
"The blackbody plus power-law model provided an acceptable fit to the Swift and Chandra data.",
"The model had $\\chi ^2_\\nu /\\nu $ of 1.07/5451 and a best-fit $N_{\\rm H}$ of $4.53\\pm 0.08\\times 10^{21}\\, \\textrm {cm}^{-2} $ .",
"Figure REF shows the evolution of the post-outburst spectral parameters.",
"The spectrum is softening following the outburst as the 1–10 keV absorbed flux (top panel of Figure REF ) decays: the photon index of the power-law component increases and the temperature of the blackbody decreases.",
"This is clear from the high-quality Chandra data alone but is also apparent in the Swift data, and is consistent with the behavior of other magnetars post-outburst (see Section REF ) Figure REF shows the flux decay and pulsed fraction evolution following the Swift J1822.3$-$ 1606 outburst.",
"In Paper I, we showed that both a double exponential and an exponential model provided acceptable fits to both the total and pulsed flux decays, whereas a power-law decay model was excluded.",
"Here we fitted a double-exponential decay and an exponential decay to the total and pulsed flux evolutions.",
"The exponential decay model is described by $F(t) = F_p \\exp ^{-(t-t_0)/\\tau } + F_q$ where $t$ is in MJD, $F_p$ is the peak absorbed flux, $F_q$ is the quiescent flux, $t_0$ is the time of the BAT trigger in MJD, and $\\tau $ is decay timescale in days.",
"The double-exponential decay is described by $F(t) = F_1 \\exp ^{-(t-t_0)/\\tau _1} + F_2 \\exp ^{-(t-t_0)/\\tau _2}+F_q$ where $t_0$ is the time of the BAT trigger in MJD, $F_1$ and $F_2$ are the absorbed fluxes at $t_0$ of each exponential component and $\\tau _1$ and $\\tau _2$ are the decay timescales in days of each component.",
"For both models we fit the data both by using the quiescent flux as a free parameter, and by using a fixed quiescent flux $F_q = 3\\times 10^{-14}$ erg cm$^{-2}$ s$^{-1}$ .",
"This is the approximate 1–10 keV flux assuming the 0.1–2.4 keV flux and spectral model from the ROSAT observation.",
"Table REF shows the results of these fits.",
"With $\\chi _\\nu ^2/\\nu $ of 1.06/40, the double-exponential decay with a free quiescent flux provides the best fit to the data.",
"However, the best-fit value for the quiescent flux, $4.0\\pm 0.2\\times 10^{-12}$ erg cm$^{-2}$ s$^{-1}$ , is more than two orders of magnitude higher than the quiescent flux implied by the ROSAT observation.",
"We therefore take the double-exponential decay with the fixed quiescent flux as the best model of the total absorbed flux decay with timescales of $\\tau _1 = 15.5\\pm 0.5$ days and $\\tau _2 = 177\\pm 14$ days.",
"In both cases, the single-exponential fit is much worse than the double exponential.",
"In Figure REF , the absorbed flux measured with Chandra is always larger than that with Swift at a similar epoch by about 10–15%.",
"The discrepancy could be attributed to insufficient cross-calibration between instruments.",
"[53] found that flux measurements from different X-ray telescopes could differ by as much as 20%, and that Chandra appears to give a higher flux than others, as well as a harder photon-index, which is consistent with our findings.",
"To determine the 2–10 keV pulsed count rate from Swift, RXTE, and Chandra observations, the barycentered events were folded using the Solution 2 ephemeris in Table REF with 16 phase bins.",
"For the RXTE observations, only data from the first xenon layer of PCU2 were used.",
"Both PCU0 and PCU1 lost their propane layers and there is minimal PCU3 and PCU4 data for this source.",
"The inclusion of only a single detector in the analysis reduces instrumental biases.",
"For observations from all three telescopes, the pulsed count rate was then measured from each folded profile using a RMS method as described in [12], where the pulsed count rate, $F$ , is given by: $F = \\sqrt{2 \\sum ^n_{k=1} [(a_k^2 + b_k^2) - (\\sigma ^2_{a_k} + \\sigma ^2_{b_k})]}\\,\\textrm {,}$ where $a_k$ is the even Fourier component and is equal to $(1/N)\\sum ^N_{i=1}p_i\\cos (2\\pi ki/N)$ , $\\sigma _{a_k}$ is the uncertainty in $a_k$ , $b_k$ is the odd Fourier component and is equal to $(1/N)\\sum ^N_{i=1}p_i\\sin (2\\pi ki/N)$ , $\\sigma _{b_k}$ is the uncertainty in $b_k$ , $i$ is an index over phase bins, $N$ is the total number of phase bins, $p_i$ is the count rate in the $i$ th phase bin, and $n$ is the maximum number of Fourier harmonics used.",
"In this case, $n=5$ .",
"This technique is equivalent to the simple RMS formula, $F=(1/\\sqrt{N})[\\sum ^N_{i=1}(p_i-\\bar{p})^2]^{1/2}$ , except only statistically significant Fourier components are used and the upward statistical bias is removed by subtracting the variances [12].",
"For Swift and Chandra observations, pulsed fractions were determined by dividing the pulsed count rate by the total count rate.",
"The middle panel of Figure REF shows the 2–10 keV pulsed-flux evolution of Swift J1822.3$-$ 1606.",
"The pulsed count rates measured by each instrument depend on the different instrumental responses.",
"The RXTE PCA and Chandra pulsed count rates were therefore scaled to the Swift WT mode values by including factors between each data set as free parameters in the double and single-exponential fits.",
"For the pulsed-flux evolution, the double-exponential fit also provided the best fit with $\\chi _\\nu ^2/\\nu $ of 5.15/74 and decay timescales of $\\tau _1 = 15.3\\pm 0.2$ days and $\\tau _2 = 182\\pm 6$ days.",
"The exponential model had a $\\chi _\\nu ^2/\\nu $ of 60.5/76 with a best-fit decay timescale of $25.1\\pm 0.2$ days.",
"This is the opposite of what we found in Paper I where the exponential model was a better fit to the data available at the time."
],
[
"X-ray Bursts",
"To search for X-ray bursts in RXTE data of Swift J1822.3$-$ 1606, we created a time series for each active PCU from GoodXenon data for each observation, selecting events in the 2 – 20 keV range (PCA channels 6–24) and from all three detection layers [15], [17].",
"For the Swift observations, binned time series were made for each Good Timing Interval (GTI) in an observation.",
"For both Swift and RXTE, time series were made at 15.625-ms, 31.25-ms, 62.5-ms and 125-ms time resolutions to provide sensitivity to bursts on a hierarchy of time scales.",
"Bursts were identified by comparing the count rate in the ith bin to the average count rate as described in [17].",
"Because the background rate of the PCA typically varies over a single observation, we calculated a local mean around the ith bin for RXTE.",
"For Swift data, a mean was calculated for each GTI.",
"We then compared the count rate in the ith bin to the mean.",
"If the count rate in a single bin was larger than the local/GTI average, the probability of such a count rate occuring by chance was calculated.",
"For RXTE data, the probability of the count rate in the corresponding bin in the other active PCUs was also calculated (whether or not the count rate in that bin was greater than the local average).",
"If a PCU was off during the bin of interest, its probability was set to 1.",
"We then found the total probability that a burst was observed, by multipying the probabilities for each PCU together.",
"If the total probability of an event was $P_{i,{\\rm {tot}}} \\le 0.01/N$ (where $N$ is the total number of time bins searched), it was flagged as a burst.",
"We found six bursts in RXTE data of Swift J1822.3$-$ 1606.",
"The burst properties are summarized in Table REF .",
"In the Table are the MJDs of each burst, the number of counts in a 31.25-ms bin, and the probability that the burst would occur by chance given the local mean count rate.",
"An insufficient number of bursts was detected to perform a detailed statistical analysis of the burst properties for Swift J1822.3$-$ 1606.",
"The bursts found were very narrow, typically only one or two 31.25-ms bins wide, and not very fluent compared to typical magnetar bursts [15], [17], [47], [31].",
"No significant changes in the long-term flux decay were observed at the times of these bursts.",
"Although in certain Swift observations we detected several bursts, these had much softer spectra than typical magnetar bursts and were also seen in the background region.",
"Therefore, we do not believe they originate from Swift J1822.3$-$ 1606.",
"No other bursts were detected in any of the Swift data.",
"lccc X-ray Bursts from Swift J1822.3$-$ 1606.",
"0pt RXTE ObsidMJDTotal counts Chance Prob$^a$ RXTE bursts D96048-02-01-01 55761.53224 $15\\pm 4$ $7.8\\times 10^{-7}$ D96048-02-01-01 55761.57082 $36\\pm 6$ $8.6\\times 10^{-33}$ D96048-02-01-02 55762.49919 $21\\pm 5$ $1.1\\times 10^{-13}$ D96048-02-03-04 55777.91627 $12\\pm 3$ $4.5\\times 10^{-5}$ D96048-02-04-01 55782.53122 $13\\pm 4$ $2.4\\times 10^{-5}$ D96048-02-05-01 55789.96209 $11\\pm 3$ $2.2\\times 10^{-4}$ $^a$ The probability of the detected signal being due to noise.",
"We have presented Swift, RXTE, Chandra observations following the discovery of Swift J1822.3$-$ 1606 during its outburst in 2011 July.",
"We presented a phase-connected timing solution which suggests a spin-down inferred $B\\sim 5\\times 10^{13}$ G, the second lowest measured for a magnetar thus far, although we note that timing noise may significantly contaminate this estimate.",
"The flux of the magnetar was found to be decaying, both in total and pulsed flux, according to a double-exponential model.",
"The spectrum softened following the outburst.",
"Swift J1822.3$-$ 1606 also emitted several short bursts during its period of outburst.",
"We also analysed an archival ROSAT observation from which [14] previously reported that Swift J1822.3$-$ 1606 is detected in quiescence.",
"We note that the source had a similar absorption column density to the nearby Galactic Hii region M17.",
"In the following we discuss the above results."
],
[
"Timing Behavior",
"In Section REF we presented a timing solution for Swift J1822.3$-$ 1606 with just $\\nu $ and $\\dot{\\nu }$ fitted (Solution 1).",
"However, this solution appears significantly contaminated by timing noise, a common phenomenon in pulsars.",
"Most pulsars seem to display some unexplained `wandering' in their spin evolution [23].",
"A measure of the amount of timing noise displayed by a pulsar is $\\Delta _8$ and is defined as $\\Delta _8=\\log [(1/6\\nu )|\\ddot{\\nu }|(10^8{\\mathrm {s}})^3]$ [2].",
"[23] measured a correlation between $\\dot{P}$ and $\\Delta _8$ using timing solutions for 366 rotation-powered pulsars.",
"Magnetars are very noisy timers, generally having more timing noise, as measured by $\\Delta _8$ , than those rotation-powered pulsars of similar properties [16], [55].",
"Here, for Swift J1822.3$-$ 1606, we measure $\\Delta _8=2.8$ (using $\\ddot{\\nu }$ from Solution 3) which is much higher than the value predicted from the correlation in [23] of $\\sim -2$ .",
"However, we caution that in general the $\\ddot{\\nu }$ used to calculate $\\Delta _8$ is measured for a data span of $10^8$ s, whereas our data span in much shorter.",
"The large value of $\\Delta _8$ we measured may be biased by the short span, or by unmodelled relaxation following a hypothetical glitch that could have occured at the BAT trigger.",
"Glitches are commonly seen to accompany radiative outbursts from magnetars [29], [13].",
"Due to the presence of timing noise, we take the timing and derived parameters of Solution 3 not as the `true' spin-inferred values, but as a `best guess' given the data thus far.",
"As such, the uncertainties in the parameters presented, which do not take into account the effect of contamination by timing noise, likely underestimate the true uncertainties.",
"Further timing observations of Swift J1822.3$-$ 1606 will help to average over the effects of timing noise and thus provide improved estimates of the spin-inferred magnetic field of the pulsar.",
"The $B$ -field measured by Solution 1 would be the second lowest measured for a magnetar to date, higher than only SGR 0418+5729 [45].",
"Solution 3, although still the second lowest yet measured, gives a higher value of $B$ that is close to that of magnetar 1E 2259+586 and the magnetically active rotation-powered pulsar PSR J1846$-$ 0258.",
"It is also similar to the quantum critical field of $B_{\\mathrm {Q}ED}= 4.4\\times 10^{13}$ G [50] which has been viewed in the past of being a lower limit on the magnetic field of magnetars, although SGR 0418+5729 has shown that it is not a necessary condition for magnetar-like activity."
],
[
"Flux and Spectral Evolution",
"In the twisted-magnetosphere model of magnetars, the thermal emission is thought to originate from heating within the star, caused by the decay of strong internal magnetic fields [51].",
"Currents in the magnetosphere, which are due to twists in the magnetic field [51], [5], scatter the thermal surface photons to higher energies.",
"In addition to scattering, the currents provide a source of surface heating in the form of a return current.",
"The flux increase that accompanies a magnetar outburst is theorized to be due a rapid heating which could originate from magnetospheric, internal, or crustal reconfiguration of the neutron star.",
"This release may result in a significant increase in the surface temperature, in the return-current heating, and in the twisting of the magnetic field.",
"Thus, an increase in flux due to an internal heat release should result in an increase of the hardness of the emission.",
"This hardness-flux correlation is in agreement with observations of several magnetars [19], [48], [60], [47].",
"[47] explored the hardness-flux correlation for magnetar outbursts by comparing the relation between fractional increase in 4–10/2–4 keV hardness ratio and fractional increase in 2–10 keV unabsorbed flux for six different outbursts in four different magnetars.",
"We present a similar plot here in Figure REF for Swift J1822.3$-$ 1606.",
"Here, however, the hardnesses and fluxes are absolute quantities and not fractional increases over quiescent values as in [47], as there is no quiescent observation of Swift J1822.3$-$ 1606 with the appropriate spectral coverage.",
"Figure REF shows that Swift J1822.3$-$ 1606 softens as the flux decreases following the outburst and so is in broad agreement with the hardness-flux correlation observed in other magnetar outbursts.",
"This spectral softening with flux decline is clear also in Figure REF , where $kT$ declines and the power-law index $\\Gamma $ increases as the flux drops."
],
[
"Magnetars in quiescence",
"The quiescent flux of Swift J1822.3$-$ 1606 measured by ROSAT in 1993 is about 3 orders of magnitude lower than the peak flux measured following the outburst.",
"Such large flux variations have been observed in several other magnetars (e.g.",
"1E 1547$-$ 5408, XTE J1810$-$ 197, AX J1845$-$ 0258; [24], [20], [49], [47], [6]).",
"Other magnetars, such as 1E 1841$-$ 045 [59], [31], 4U 0142+61 [18], and 1RXS J170849.0$-$ 400910 [11] have not exhibited large flux variations, but are much brighter in quiescence than are the magnetars with large outbursts.",
"The cause of this difference is unclear.",
"[40] suggest that there is a maximum luminosity that can be reached by a magnetar during an outburst due to neutrino cooling dominating at high crust temperatures.",
"This helps to explain the differences in outburst magnitudes, but does not address the wide range of quiescent luminosities.",
"Case in point, the magnetar 1E 2259+586 has spin properties that are likely quite similar to those of Swift J1822.3$-$ 1606 but has a much higher quiescent luminosity.",
"The magnetic field measured from spin-down for 1E 2259+586 is $5.9 \\times 10^{13}$ G [16], close to $B = 5.1\\times 10^{13}$ G for Swift J1822.3$-$ 1606 as estimated by our Solution 3.",
"1E 2259+586 also went into a period of outburst on 2002 June 18 where the flux increased by a factor of $\\stackrel{>}{_{\\sim }}$ 20 [57].",
"However, in quiescence, 1E 2259+586 is much brighter than Swift J1822.3$-$ 1606 with a quiescent 2–10 keV luminosity of $~2 \\times 10^{34}$ erg s$^{-1}$ [60] compared to $\\stackrel{<}{_{\\sim }}$ $10^{31}$ erg s$^{-1}$ for Swift J1822.3$-$ 1606.",
"One possibility is that the `true' magnetic fields of the more luminous magnetars are higher than those of the fainter magnetars.",
"The spin-down of the neutron star is only sensitive to the dipole component of the magnetic field.",
"If the magnetic field had significant components in higher multipoles or a toroidal component [50], [39], the true magnetic field could be higher.",
"Another possibility is that neutrino cooling in the core is setting a long-term luminosity limit, and that the neutrino cooling properties of the stars are different, e.g.",
"due to different masses.",
"For example, consider first the case where the neutrino emission in the core is due to the modified URCA process, with an emissivity $\\epsilon _\\nu \\sim 10^{20}\\ {\\rm erg\\ cm^{-3}\\ s^{-1}}\\ T_9^8$ [58].",
"If we take the magnetic-field decay time to be $\\tau =10^4\\ {\\rm yrs}$ , then the luminosity from magnetic field decay is roughly $L_B=(B^2/8\\pi )(4\\pi R^3/3)(1/\\tau )=10^{34}\\ {\\rm erg\\ s^{-1}}$ for $B=10^{14}\\ {\\rm G}$ .",
"Balancing this with the neutrino losses $L_\\nu =(4\\pi R^3/3)\\epsilon _\\nu $ , we find a core temperature $T_c=2.5\\times 10^8\\ {\\rm K}$ or, using the core temperature-luminosity relation from [42], a luminosity $L\\approx 4\\times 10^{33}\\ {\\rm erg\\ s^{-1}}$ .",
"On the other hand, if the neutrino emission is by the direct URCA process, with $\\epsilon _\\nu \\sim 10^{26}\\ {\\rm erg\\ cm^{-3}\\ s^{-1}}\\ T_9^6$ [58], we find a core temperature $T_c=1.5\\times 10^7\\ {\\rm K}$ , corresponding to a surface luminosity of $\\approx 2\\times 10^{31}\\ {\\rm erg\\ s^{-1}}$ .",
"This shows that we might reasonably expect a factor of $\\gtrsim 200$ in luminosity between different stars if one has slow neutrino emission in the core, and the other fast, for example if the mass of one of the stars is large enough for direct URCA reactions to occur in the core.",
"Even in the case where external currents dominate the quiescent luminosity, thermal emission from the neutron star provides a baseline luminosity, so that the low quiescent luminosity of Swift J1822.3$-$ 1606 suggests a low core temperature which implies either a low heating rate or efficient neutrino emission."
],
[
"The observed luminosity decay of Swift J1822.3$-$ 1606",
"We find that the observed luminosity decay is well reproduced by models of thermal relaxation of the neutron-star crust following the outburst.",
"An example is shown in Figure REF , which shows the cooling of the crust after an injection of $\\approx 3\\times 10^{42}\\ {\\rm ergs}$ of energy at low density $\\approx 10^{10}\\ {\\rm g\\ cm^{-3}}$ in the outer crust at the start of the outburst.",
"We follow the evolution of the crust temperature profile by integrating the thermal diffusion equation.",
"The calculation and microphysics follow [7] who studied transiently accreting neutron stars, but with the effects of strong magnetic fields on the thermal conductivity included [41] and for the outer boundary condition using the $T_{\\rm eff}$ –$T_{\\rm int}$ relation appropriate for a magnetized envelope following [42].",
"The calculation follows the radial structure only; we assume that the magnetic field geometry is a dipole and take appropriate spherical averages to account for the variation in thermal conductivity across the star [41].",
"We assume $B=6\\times 10^{13}\\ {\\rm G}$ , similar to the value inferred from the spin down, a 1.6 $M_\\odot $ , $R=11.2\\ {\\rm km}$ neutron star, and take an impurity parameter for the inner crust of $Q_{\\rm imp}=10$ [26].",
"We set the neutron-star core temperature to $2\\times 10^7\\ {\\rm K}$ , which is needed to obtain a quiescent luminosity $<10^{32}\\ {\\rm erg\\ s^{-1}}$ .",
"With the neutron-star parameters fixed, we then vary the location and strength of the heating and find that we obtain good agreement with the observed light curve for times $<100$ days, if the initial heating event is located at low densities $\\lesssim 3\\times 10^{10}\\ {\\rm g\\ cm^{-3}}$ .",
"This conclusion comes from matching the observed timescale of the decay, and is not very sensitive to the choice of neutron-star parameters.",
"For example, changing the neutron-star gravity changes the crust thickness and therefore cooling time, giving an inferred maximum density $\\rho _{\\rm max}\\propto g^{-2}$ .",
"This means that the inferred density can change by a factor of a few but cannot be moved into the neutron drip region, for instance.",
"That only a shallow part of the outer crust is heated is an interesting constraint on models of crust heating in a magnetar outburst.",
"We find that it is difficult to match the observed light curve at times $\\gtrsim 200$ days, but the late time behaviour of the light curve is sensitive to a number of physics inputs associated with the inner crust, including the thermal conductivity and superfluid parameters, as well as modification due to the angular distribution of the heating over the surface of the star.",
"We will investigate the late-time behaviour in more detail in future work.",
"Figure REF suggests that the source could undergo significant further cooling in the coming years."
],
[
"Distance Estimate and Possible Association",
"As shown in the ROSAT image (Figure REF ), the Galactic Hii region M17 is located $\\sim $ 20 southwest of Swift J1822.3$-$ 1606.",
"It has a distance of $1.6\\pm 0.3$ kpc [36] and an absorption column density $N_{\\rm H}=4\\pm 1\\times 10^{21}\\,$ cm $^{-2}$ [52] which is consistent with our best-fit value of $4.53\\pm 0.08\\times 10^{21}$ cm $^{-2}$ .",
"This suggests that Swift J1822.3$-$ 1606 could have a comparable distance to that of M17While there are two molecular clouds surrounding M17 [54], they are confined to the north and west, such that they should not contribute to the $N_{\\mathrm {H}}$ of either M17 or Swift J1822.3$-$ 1606..",
"If so, then Swift J1822.3$-$ 1606 would be one of the closest magnetars detected thus far.",
"The above argument does not necessitate a direct association between M17 and Swift J1822.3$-$ 1606.",
"However, if Swift J1822.3$-$ 1606 is associated with M17, then its angular separation of 26 from the cluster center, where the X-ray emission peaks in the ROSAT image, implies a physical distance of 12 pc.",
"For a pulsar age of $10^5$ yr, this requires a space velocity of only $\\sim $ 100 km s$^{-1}$ (corresponding to a proper motion of 0016 yr$^{-1}$ ).",
"This would make a direct proper motion measurement difficult.",
"From timing, the characteristic age appears to be larger than $10^5$ yr which would further reduce the implied proper motion.",
"On the other hand, characteristic ages can be large overestimates of the true age.",
"However, even if the true age were as low as $10^4$ yr, the proper motion would be difficult to measure even with Chandra.",
"Additionally, if the magnetar was born near an edge of the cluster, the angular separation from its birthplace could be larger or smaller by up to $\\sim 10$ ."
],
[
"Conclusions",
"We have presented an analysis of the post-outburst radiative evolution and timing behavior of Swift J1822.3$-$ 1606, following its discovery on 2011 July 14.",
"Following a timing analysis for the source post-outburst, we estimate the surface dipolar component of the $B$ -field to be $\\sim 5 \\times 10^{13}$ G, slightly higher than that inferred in Paper I.",
"However, as this measurement is contaminated by timing noise, the true value of the magnetic field could be well outside of the uncertainties quoted in Table REF .",
"Futher monitoring of Swift J1822.3$-$ 1606 as it fades following the outburst will allow us to better account for the timing noise and measure more robust timing parameters.",
"The quiescent flux of Swift J1822.3$-$ 1606 measured using a 1993 ROSAT observation of M17 was found to be roughly three orders of magnitude lower than the peak flux during the outburst.",
"The flux evolution following the outburst was well characterized by a double-exponential decay.",
"By applying a crustal cooling model to the flux decay, we found that the energy deposition likely occured in the outer crust at a density of $\\sim 10^{10}$ g cm$^{-3}$ .",
"The spectral properties of Swift J1822.3$-$ 1606 were observed to soften following the outburst, with the power-law index increasing and the temperature of the blackbody decreasing.",
"Indeed, a hardness-flux correlation, similar to what is observed in other magnetars, was clearly observed.",
"Based on the similarity in $N_H$ to that of the Hii region M17, we argue for a source distance of $1.6\\pm 0.3$ kpc, one of the closest distances yet inferred for a magnetar.",
"We are grateful to the Swift, Chandra, and RXTE teams for their flexibility in scheduling TOO observations.",
"We thank the anonymous referee for helpful comments and suggestions.",
"V.M.K.",
"holds the Lorne Trottier Chair in Astrophysics and Cosmology and a Canadian Research Chair in Observational Astrophysics.",
"This work is supported by NSERC via a Discovery Grant, by FQRNT via the Centre de Recherche Astrophysique du Québec, by CIFAR, and a Killam Research Fellowship."
]
] | 1204.1034 |
[
[
"Lag length identification for VAR models with non-constant variance"
],
[
"Abstract The identification of the lag length for vector autoregressive models by mean of Akaike Information Criterion (AIC), Partial Autoregressive and Correlation Matrices (PAM and PCM hereafter) is studied in the framework of processes with time varying variance.",
"It is highlighted that the use of the standard tools are not justified in such a case.",
"As a consequence we propose an adaptive AIC which is robust to the presence of unconditional heteroscedasticity.",
"Corrected confidence bounds are proposed for the usual PAM and PCM obtained from the Ordinary Least Squares (OLS) estimation.",
"The volatility structure of the innovations is used to develop adaptive PAM and PCM.",
"We underline that the adaptive PAM and PCM are more accurate than the OLS PAM and PCM for identifying the lag length of the autoregressive models.",
"Monte Carlo experiments show that the adaptive $AIC$ have a greater ability to select the correct autoregressive order than the standard AIC.",
"An illustrative application using US international finance data is presented."
],
[
"Introduction",
"The analysis of time series using linear models is usually carried out following three steps.",
"First the model is identified, then estimated and finally we proceed to the checking of the goodness-of-fit of the model (see Brockwell and Davis (1991, chapters 8 and 9)).",
"Tools for the three phases in the specification-estimation-verification modeling cycle of time series with constant unconditional innovations variance are available in any of the specialized softwares as for instance R, SAS or JMulTi.",
"The identification stage is important for the choice of a suitable model for the data.",
"In this step the partial autoregressive and correlation matrices (PAM and PCM hereafter) are often used to identify VAR models with stationary innovations (see Tiao and Box (1981)).",
"Information criteria are also extensively used.",
"In the framework of stationary processes numerous information criteria have been studied (see e.g.",
"Hannan and Quinn (1979), Cavanaugh (1997) or Boubacar Mainassara (2012)).",
"One of the most commonly used information criterion is the Akaike Information Criterion ($AIC$ ) proposed by Akaike (1973).",
"Nevertheless it is widely documented in the literature that the constant variance assumption is unrealistic for many economic data.",
"Reference can be made to Mc-Connell, Mosser and Perez-Quiros (1999), Kim and Nelson (1999), Stock and Watson (2002), Ahmed, Levin and Wilson (2002), Herrera and Pesavento (2005) or Davis and Kahn (2008).",
"In this paper we investigate the lag length identification problem of autoregressive processes in the important case where the unconditional innovations variance is time varying.",
"The statistical inference of processes with non constant variance has recently attracted much attention.",
"Horváth and Steinebach (2000), Sanso, Arago and Carrion (2004) or Galeano and Pena (2007) among other contributions proposed tests to detect variance and/or covariance breaks in the residuals.",
"Francq and Gautier (2004) studied the estimation of ARMA models with time varying parameters, allowing a finite number of regimes for the variance.",
"Mikosch and Stărică (2004) give some theoretical evidence that financial data may exihibit non constant variance.",
"In the context of GARCH models reference can be made to the works of Kokoszka and Leipus (2000), Engle and Rangel (2008), Dahlhaus and Rao (2006) or Horvath, Kokoszka and Zhang (2006) who investigated the inference for processes with possibly unconditional time varying variance.",
"In the multivariate framework Bai (2000), Qu and Perron (2007) or Kim and Park (2010) among others studied autoregressive models with unconditionally non constant variance.",
"Aue, Hörmann, Horvàth and Reimherr (2009) studied the break detection in the covariance structure of multivariate processes.",
"Xu and Phillips (2008) studied the estimation of univariate autoregressive models whose innovations have a non constant variance.",
"Patilea and Raïssi (2012) generalized their findings in the case of Vector AutoRegressive (VAR) models with time-varying variance and found that the asymptotic covariance matrix obtained if one take into account of the non constant variance can be quite different from the standard covariance matrix expression.",
"As a consequence they also provided Adaptive Least Squares (ALS) estimators which achieve a more efficient estimation of the autoregressive parameters.",
"Patilea and Raïssi (2011) proposed tools for checking the adequacy of the autoregressive order of VAR models when the unconditional variance is non constant.",
"In this paper modified tools for lag length identification in the case of multivariate autoregressive processes with time-varying variance are introduced.",
"The unreliability of the use of the standard $AIC$ for the identification step in VAR modeling in presence of non constant variance is first highlighted.",
"Consequently a modified $AIC$ based on the adaptive estimation of the non constant variance structure is proposed.",
"We establish the suitability of the adaptive $AIC$ to identify the autoregressive order of non stationary but stable VAR processes through theoretical results and numerical illustrations.",
"On the other hand it is also shown that the standard results on the OLS estimators of the PAM and PCM can be quite misleading.",
"Consequently corrected confidence bounds are proposed.",
"Using the adaptive approach more efficient estimators of the PAM and PCM are proposed.",
"Therefore the identification tools proposed in this paper may be viewed as a complement of the above mentioned results on the estimation and diagnostic testing in the important framework of autoregressive models with non constant variance.",
"The structure of the paper is as follow.",
"In Section we define the model and introduce assumptions which give the general framework of our study.",
"The asymptotic behavior of different estimators of the autoregressive parameters is given.",
"We also describe the adaptive estimation of the variance.",
"In Section it is shown that the standard $AIC$ is irrelevant for model selection when the innovations variance is not constant.",
"The adaptive $AIC$ is derived taking into account the time-varying variance in the Kullback-Leibler discrepancy.",
"In Section some Monte Carlo experiments results are given to examine the performances of the studied information criteria for VAR model identification in our non standard framework.",
"We also investigate the lag length selection of a bivariate system of US international finance variables."
],
[
"Estimation of the model",
"In this paper we restrict our attention to VAR models since they are extensively used for the analysis of multivariate time series (see e.g.",
"Lütkepohl (2005)).",
"Let us consider the $d$ -dimensional autoregressive process $(X_t)$ satisfying $&&X_t=A_{01}X_{t-1}+\\dots +A_{0p}X_{t-p_0}+u_t\\\\&&u_t=H_t\\epsilon _t,\\nonumber $ where the $A_{0i}$ 's, $i\\in \\lbrace 1,\\dots ,p_0\\rbrace $ , are such that $\\det (A(z))\\ne 0$ for all $|z|\\le 1$ , with $A(z)=1-\\sum _{i=1}^{p_0}A_{0i} z^i$ and $\\det (.",
")$ denotes the determinant of a square matrix.",
"We suppose that $X_{-p+1},\\dots ,X_0,\\dots ,X_n$ are observed with $p>p_0$ .",
"Now let us denote by $[.",
"]$ the integer part.",
"For ease of exposition we shall assume that the process $(\\epsilon _t)$ is iid multivariate standard Gaussian.",
"Throughout the paper we assume that the following conditions on the unconditional variance structure of the innovations process $(u_t)$ hold.",
"Assumption A1: The $d\\times d$ matrices $H_{t}$ are positive definite and satisfy $H_{[Tr]}=G(r)$ , where the components of the matrix $G(r):=\\lbrace g_{kl}(r)\\rbrace $ are measurable deterministic functions on the interval $(0,1]$ , such that $\\sup _{r\\in (0,1]}|g_{kl}(r)|<\\infty $ , and each $g_{kl}$ satisfies a Lipschitz condition piecewise on a finite number of some sub-intervals that partition $(0,1]$ .",
"The matrix $\\Sigma (r)=G(r)G(r)^{\\prime }$ is assumed positive definite for all $r$ .",
"The rescaling method of Dahlhaus (1997) is considered to specify the unconditional variance structure in Assumption A1.",
"Note that one should formally use the notation $X_{t,n}$ with $0<t\\le n$ and $n\\in \\mathbb {N}$ .",
"Nevertheless we do not use the subscript $n$ to lighten the notations.",
"This specification allows to consider kinds of time-varying variance which are commonly considered in the literature as for instance abrupt shifts, smooth transitions or periodic heteroscedasticity.",
"Note that Sensier and Van Dijk (2004) found that approximately 80% among 214 US macro-economic data they investigated exhibit a variance break.",
"Stărică (2003) hypothesized that the returns of the Standard and Poors 500 stock market index have a non constant unconditional variance.",
"Then considering the framework given by A1 is important given the strong empirical evidence of non-constant unconditional variance in many macro-economic and financial data.",
"Our assumption is similar to that of recent papers in the literature.",
"For instance similar structure for the variance was considered by Xu and Phillips (2008) or Kim and Park (2010) among others.",
"Our framework encompass the important case of piecewise constant variance as considered in Pesaran and Timmerman (2004) or Bai (2000).",
"Finally it is important to underline that the framework induced by A1 is different from the case of autoregressive processes with conditionally heteroscedastic but (strictly) stationary errors.",
"For instance models like the GARCH or the All-Pass models cannot take into account for non constant unconditional variance in the innovations.",
"The model identification problem for stationary processes which may display nonlinearities has been recently investigated by Boubacar Mainassara (2012) in a quite general framework.",
"In this part we introduce estimators of the autoregressive parameters.",
"Let us rewrite (REF ) as follow $&& X_t=(\\tilde{X}_{t-1}^{\\prime }\\otimes I_d)\\theta _0+u_t\\\\&&u_t=H_t\\epsilon _t,\\nonumber $ where $\\theta _0=(\\mbox{vec}\\:(A_{01})^{\\prime },\\dots ,\\mbox{vec}\\:(A_{0p_0})^{\\prime })^{\\prime }\\in \\mathbb {R}^{p_0d^2}$ is the vector of the true autoregressive parameters and $\\tilde{X}_{t-1}=(X_{t-1}^{\\prime },\\dots ,X_{t-p_0}^{\\prime })^{\\prime }$ .",
"For a fitted autoregressive order $p\\ge p_0$ , the OLS estimator is given by $\\hat{\\theta }_{OLS}=\\hat{\\Sigma }_{\\tilde{X}}^{-1}\\mbox{vec}\\:\\left(\\hat{\\Sigma }_{X}\\right),$ where $\\hat{\\Sigma }_{\\tilde{X}}=n^{-1}\\sum _{t=1}^n\\tilde{X}_{t-1}^p(\\tilde{X}_{t-1}^{p})^{\\prime }\\otimes I_d\\quad \\mbox{and}\\quad \\hat{\\Sigma }_X=n^{-1}\\sum _{t=1}^nX_t(\\tilde{X}_{t-1}^{p})^{\\prime },$ and $\\tilde{X}_{t-1}^p=(X_{t-1}^{\\prime },\\dots ,X_{t-p}^{\\prime })^{\\prime }$ .",
"If we suppose that the true unconditional covariance matrices $\\Sigma _t:=H_tH_t^{\\prime }$ are known, we can define the following Generalized Least Squares (GLS) estimator $\\hat{\\theta }_{GLS}=\\hat{\\Sigma }_{\\tilde{\\underline{X}}}^{-1}\\mbox{vec}\\:\\left(\\hat{\\Sigma }_{\\underline{X}}\\right),$ with $\\hat{\\Sigma }_{\\tilde{\\underline{X}}}=n^{-1}\\sum _{t=1}^n\\tilde{X}_{t-1}^{p}(\\tilde{X}_{t-1}^{p})^{\\prime }\\otimes \\Sigma _t^{-1}\\quad \\mbox{and}\\quad \\hat{\\Sigma }_{\\underline{X}}=n^{-1}\\sum _{t=1}^n\\Sigma _t^{-1}X_t(\\tilde{X}_{t-1}^{p})^{\\prime }.$ Let us define $u_t(\\theta )=X_t-\\lbrace (\\tilde{X}_{t-1}^{p})^{\\prime }\\otimes I_d\\rbrace \\theta $ with $\\theta \\in \\mathbb {R}^{pd^2}$ .",
"Note that $\\hat{\\theta }_{GLS}$ maximizes the conditional log-likelihood function (up to a constant and divided by $n$ ) $\\mathcal {L}_{GLS}(\\theta )=-\\frac{1}{2n}\\sum _{t=1}^n\\ln \\left\\lbrace \\det (\\Sigma _t)\\right\\rbrace -u_t(\\theta )^{\\prime }\\Sigma _t^{-1}u_t(\\theta ),$ (see Lütkepohl (2005, p 589)).",
"If we assume that the innovations process variance is constant ($\\Sigma _t=\\Sigma _{u}$ for all $t$ ) and unknown, the standard conditional log-likelihood function $\\mathcal {L}_{OLS}(\\theta ,\\Sigma )=-\\frac{1}{2}\\ln (\\det (\\Sigma ))-\\frac{1}{2n}\\sum _{t=1}^nu_t(\\theta )^{\\prime }\\Sigma ^{-1}u_t(\\theta )$ where $\\Sigma $ is a $d\\times d$ invertible matrix, is usually used for the estimation of the parameters.",
"The estimator obtained by maximizing $\\mathcal {L}_{OLS}$ with respect to $\\theta $ corresponds to $\\hat{\\theta }_{OLS}$ .",
"In this case the estimator of the constant variance $\\Sigma _{u}$ is given by $\\hat{\\Sigma }_u:=n^{-1}\\sum _{t=1}^n\\hat{u}_t\\hat{u}_t^{\\prime }$ where $\\hat{u}_t:=u_t(\\hat{\\theta })$ are the residuals of the OLS estimation of (REF ).",
"In practice the assumption of known variance is unrealistic.",
"Therefore we consider an adaptive estimator of the autoregressive parameters.",
"We may first define adaptive estimators of the true unconditional variances $\\Sigma _t:=H_tH_t^{\\prime }$ as in Patilea and Raïssi (2012) $\\check{\\Sigma }_t=\\sum _{i=1}^n w_{ti} (b)\\hat{u}_i\\hat{u}_i^{\\prime },$ where the weights $w_{ti}$ are given by $w_{ti}(b)= \\left(\\sum _{i=1}^nK_{ti}(b)\\right)^{-1} K_{ti}(b),$ with $b$ the bandwidth and $K_{ti} (b) =\\left\\lbrace \\begin{array}{c}K(\\frac{t-i}{nb})\\quad \\mbox{if}\\quad t\\ne i\\\\0 \\quad \\mbox{if}\\quad t=i,\\\\\\end{array}\\right.$ where $K(.",
")$ is the kernel function which is such that $\\int _{-\\infty }^{+\\infty } K(z) dz=1$ .",
"The bandwidth $b$ is taken in a range $\\mathcal {B}_n = [c_{min} b_n, c_{max} b_n]$ with $c_{max}>c_{min}>0$ some constants and $b_n \\downarrow 0$ at a suitable rate.",
"Alternatively one can use different bandwidths cells for the $\\check{\\Sigma }_t$ 's (see Patilea and Raïssi (2012) for more details).",
"The results in this paper are given uniformly with respect to $b\\in \\mathcal {B}_n$ .",
"This justifies the approach which consists in selecting the bandwidth on a grid defined in a range using for example the cross validation criterion.",
"Note also that the $\\check{\\Sigma }_t$ 's are positive definite.",
"Of course our results do not rely on a particular bandwidth choice procedure and are valid provided estimators of the $\\Sigma _t$ 's with similar asymptotic properties of the $\\check{\\Sigma }_t$ 's are available.",
"The non parametric estimator of the covariance matrices employed in this paper is similar to the variance estimators used in Xu and Phillips (2008) among others.",
"Considering the $\\check{\\Sigma }_t$ 's, we are in position to introduce the ALS estimators $\\hat{\\theta }_{ALS}=\\check{\\Sigma }_{\\tilde{X}}^{-1}\\mbox{vec}\\:\\left(\\check{\\Sigma }_{X}\\right),$ with $\\check{\\Sigma }_{\\tilde{X}}=n^{-1}\\sum _{t=1}^n\\tilde{X}_{t-1}^p(\\tilde{X}_{t-1}^{p})^{\\prime }\\otimes \\check{\\Sigma }_t^{-1}\\quad \\mbox{and}\\quad \\check{\\Sigma }_{X}=n^{-1}\\sum _{t=1}^n\\check{\\Sigma }_t^{-1}X_t(\\tilde{X}_{t-1}^{p})^{\\prime }.$ Now we have to state the asymptotic behavior of the estimators and introduce some notations.",
"Define $\\Delta =\\left(\\begin{array}{cccc}A_1 & \\dots & A_{p-1} & A_p \\\\I_d & 0 & \\dots & 0 \\\\& \\ddots & \\ddots & \\vdots \\\\0 & & I_d & 0 \\\\\\end{array}\\right)$ and $e_p(1)$ the vector of dimension $p$ such that the first component is equal to one and zero elsewhere.",
"Under A1 it is shown in Patilea and Raïssi (2012) that $\\sqrt{n}(\\hat{\\theta }_{OLS}-\\theta _0)\\Rightarrow \\mathcal {N}(0,\\Lambda _3^{-1}\\Lambda _2\\Lambda _3^{-1}),$ with $\\Lambda _2=\\int _0^1\\sum _{i=0}^{\\infty }\\left\\lbrace \\Delta ^i(e_p(1)e_p(1)^{\\prime }\\otimes \\Sigma (r))\\Delta ^{i^{\\prime }}\\right\\rbrace \\otimes \\Sigma (r)\\:dr,$ $\\Lambda _3=\\int _0^1\\sum _{i=0}^{\\infty }\\left\\lbrace \\Delta ^i(e_p(1)e_p(1)^{\\prime }\\otimes \\Sigma (r))\\Delta ^{i^{\\prime }}\\right\\rbrace \\otimes I_d\\:dr,$ and $\\sqrt{n}(\\hat{\\theta }_{GLS}-\\theta _0)\\Rightarrow \\mathcal {N}(0,\\Lambda _1^{-1}),$ where $\\Lambda _1=\\int _0^1\\sum _{i=0}^{\\infty }\\left\\lbrace \\Delta ^i(e_p(1)e_p(1)^{\\prime }\\otimes \\Sigma (r))\\Delta ^{i^{\\prime }}\\right\\rbrace \\otimes \\Sigma (r)^{-1}\\:dr.$ In addition we may use the following consistent estimators for the covariance matrices: $\\check{\\Sigma }_{\\tilde{X}}=\\Lambda _1+o_p(1),\\:\\hat{\\Sigma }_{\\tilde{X}}=\\Lambda _3+o_p(1)\\quad \\mbox{and}\\quad \\hat{\\Lambda }_2:=n^{-1}\\sum _{t=1}^n\\tilde{X}_{t-1}^p\\tilde{X}_{t-1}^{p^{\\prime }}\\otimes \\hat{u}_t\\hat{u}_t^{\\prime }=\\Lambda _2+o_p(1).$ We make the following assumptions to state the asymptotic equivalence between the ALS and GLS estimators.",
"Assumption A1': Suppose that all the conditions in Assumption A1 hold true.",
"In addition $\\inf _{r\\in (0,1]} \\lambda _{min}(\\Sigma (r)) >0$ where for any symmetric matrix $M$ the real value $ \\lambda _{min}(M)$ denotes its smallest eigenvalue.",
"Assumption A2: (i) The kernel $K(\\cdot )$ is a bounded density function defined on the real line such that $K(\\cdot )$ is nondecreasing on $(-\\infty , 0]$ and decreasing on $[0,\\infty )$ and $\\int _\\mathbb {R} v^2K(v)dv < \\infty $ .",
"The function $K(\\cdot )$ is differentiable except a finite number of points and the derivative $K^\\prime (\\cdot )$ is an integrable function.",
"Moreover, the Fourier Transform $\\mathcal {F}[K](\\cdot )$ of $K(\\cdot )$ satisfies $\\int _{\\mathbb {R}} \\left| s \\mathcal {F}[K](s) \\right|ds <\\infty $ .",
"(ii) The bandwidth $b$ is taken in the range $\\mathcal {B}_n = [c_{min} b_n, c_{max} b_n]$ with $0<c_{min}< c_{max}< \\infty $ and $b_n + 1/Tb_n^{2+\\gamma } \\rightarrow 0$ as $n\\rightarrow \\infty $ , for some $\\gamma >0$ .",
"(iii) The sequence $\\nu _n$ is such that $n\\nu _n^2 \\rightarrow 0.$ Under these additional assumptions Patilea and Raïssi (2012) also showed that $\\sqrt{n}(\\hat{\\theta }_{ALS}-\\hat{\\theta }_{GLS})=o_p(1),$ and $\\check{\\Sigma }_{[Tr]} \\stackrel{P}{\\longrightarrow }\\Sigma (r-)\\int _{-\\infty }^0 K(z) dz + \\Sigma (r+)\\int _0^\\infty K(z)dz,$ where $\\Sigma (r-):=\\lim _{\\tilde{r}\\uparrow r}\\Sigma (\\tilde{r})$ and $\\Sigma (r+):=\\lim _{\\tilde{r}\\downarrow r}\\Sigma (\\tilde{r})$ .",
"As a consequence $\\hat{\\theta }_{ALS}$ and $\\hat{\\theta }_{GLS}$ have the same asymptotic behavior and we can also write $\\check{\\Sigma }_{t}=\\Sigma _t+o_p(1)$ , unless at the break dates where we have $\\check{\\Sigma }_{t}=\\Sigma _t+O_p(1)$ .",
"Using these asymptotic results we underline the unreliability of the standard $AIC$ and develop a criterion which is adapted to the case of non stationary but stable processes.",
"Corrected confidence bounds for the PAM and PCM in our non standard framework are also proposed."
],
[
"Derivation of the adaptive AIC",
"In the standard case (the variance of the innovations is constant with true variance $\\Sigma _u$ ) the Kullback-Leibler discrepancy between the true model and the approximating model with parameter vector $\\hat{\\theta }_{OLS}$ is given by $d_n(\\hat{\\theta }_{OLS},\\theta _0)=E_{\\theta _0,\\Sigma _u}\\left\\lbrace -2\\mathcal {L}_{OLS}(\\theta ,\\Sigma _u)\\right\\rbrace \\mid _{\\theta =\\hat{\\theta }_{OLS}},$ see Brockwell and Davis (1991, p 302).",
"Akaike (1973) proposed the following approximately unbiased estimator of (REF ) to compare the discrepancies between competing VAR($p$ ) models $AIC(p)=-2\\mathcal {L}_{OLS}(\\hat{\\theta }_{OLS},\\hat{\\Sigma }_u)+\\frac{2pd^2}{n},$ where the term $2pd^2$ penalizes the more complicated models fitted to the data (see Lütkepohl (2005), p 147).",
"The terms corresponding to the nuisance parameters are neglected in the previous expressions since they do not interfere in the model selection when the $AIC$ is used.",
"The identified model corresponds to the model which minimizes the $AIC$ .",
"However in our non standard framework it is clear that the $\\mathcal {L}_{OLS}$ cannot take into account the non constant variance in the observations.",
"In addition if we assume that the variance of the innovations is constant $\\Sigma _t=\\Sigma _u$ , we obtain $\\sqrt{n}(\\hat{\\theta }_{OLS}-\\theta _0)\\Rightarrow \\mathcal {N}(0,\\Lambda _4^{-1}),$ with $\\Lambda _4=E(\\tilde{X}_{t-1}\\tilde{X}_{t-1}^{\\prime })\\otimes \\Sigma _u^{-1},$ so that the following result is used for the derivation of the standard $AIC$ $E_{\\theta _0,\\Sigma _u}\\left\\lbrace n(\\hat{\\theta }_{OLS}-\\theta _0)^{\\prime }\\hat{\\Lambda }_4(\\hat{\\theta }_{OLS}-\\theta _0)\\right\\rbrace \\approx pd^2,$ for large $n$ , where $\\hat{\\Lambda }_4=n^{-1}\\sum _{t=1}^n\\tilde{X}_{t-1}^p(\\tilde{X}_{t-1}^{p})^{\\prime }\\otimes \\hat{\\Sigma }_u^{-1}$ is a consistent estimator of $\\Lambda _4$ .",
"In view of (REF ) this property is obviously not verified in our case.",
"Indeed Patilea and Raïssi (2012) pointed out that $\\Lambda _4$ and $\\Lambda _3^{-1}\\Lambda _2\\Lambda _3^{-1}$ can be quite different.",
"Therefore the standard $AIC$ have no theoretical basis in our non standard framework and we can expect that the use of the standard $AIC$ can be misleading in such a situation.",
"To remedy to this problem we shall use the more appropriate expression (REF ) in our framework for the Kullback-Leibler discrepancy between the fitted model and the true model $\\Delta _n(\\hat{\\theta }_{GLS},\\theta _0)=E_{\\theta _0}\\left\\lbrace -2\\mathcal {L}_{GLS}(\\theta )\\right\\rbrace \\mid _{\\theta =\\hat{\\theta }_{GLS}}.$ Using a second order Taylor expansion of $\\mathcal {L}_{GLS}$ about $\\hat{\\theta }_{GLS}$ and since $\\frac{\\mathcal {L}_{GLS}(\\hat{\\theta }_{GLS})}{\\partial \\theta }=0$ , we obtain $E_{\\theta _0}\\left\\lbrace -2\\mathcal {L}_{GLS}(\\theta _0)\\right\\rbrace &=&\\frac{1}{n}E_{\\theta _0}\\left\\lbrace n(\\hat{\\theta }_{GLS}-\\theta _0)^{\\prime }\\hat{\\Sigma }_{\\underline{\\tilde{X}}}(\\hat{\\theta }_{GLS}-\\theta _0)\\right\\rbrace \\nonumber \\\\&&+E_{\\theta _0}\\left\\lbrace -2\\mathcal {L}_{GLS}(\\hat{\\theta }_{GLS})\\right\\rbrace +o(1).$ Using again the second order Taylor expansion and taking the expectation we also write $E_{\\theta _0}\\left[E_{\\theta _0}\\left\\lbrace -2\\mathcal {L}_{GLS}(\\theta )\\right\\rbrace \\mid _{\\theta =\\hat{\\theta }_{GLS}}\\right]&=&\\frac{1}{n}E_{\\theta _0}\\left[n(\\hat{\\theta }_{GLS}-\\theta _0)^{\\prime }E_{\\theta _0}\\left\\lbrace \\hat{\\Sigma }_{\\underline{\\tilde{X}}}\\right\\rbrace (\\hat{\\theta }_{GLS}-\\theta _0)\\right]\\nonumber \\\\&&+E_{\\theta _0}\\left\\lbrace -2\\mathcal {L}_{GLS}(\\theta _0)\\right\\rbrace +o(1).$ From (REF ) we have for large $n$ $E_{\\theta _0}\\left[n(\\hat{\\theta }_{GLS}-\\theta _0)^{\\prime }\\hat{\\Sigma }_{\\underline{\\tilde{X}}}(\\hat{\\theta }_{GLS}-\\theta _0)\\right]\\approx pd^2$ and $E_{\\theta _0}\\left[n(\\hat{\\theta }_{GLS}-\\theta _0)^{\\prime }E_{\\theta _0}\\left\\lbrace \\hat{\\Sigma }_{\\underline{\\tilde{X}}}\\right\\rbrace (\\hat{\\theta }_{GLS}-\\theta _0)\\right]\\approx pd^2.$ Noting that $&&E_{\\theta _0}\\left[E_{\\theta _0}\\left\\lbrace -2\\mathcal {L}_{GLS}(\\theta )\\right\\rbrace \\mid _{\\theta =\\hat{\\theta }_{GLS}}\\right]=E_{\\theta _0}\\left\\lbrace -2\\mathcal {L}_{GLS}(\\hat{\\theta }_{GLS})\\right\\rbrace \\\\&&+E_{\\theta _0}\\left[E_{\\theta _0}\\left\\lbrace -2\\mathcal {L}_{GLS}(\\theta )\\right\\rbrace \\mid _{\\theta =\\hat{\\theta }_{GLS}}\\right]-E_{\\theta _0}\\left\\lbrace -2\\mathcal {L}_{GLS}(\\theta _0)\\right\\rbrace \\\\&&+E_{\\theta _0}\\left\\lbrace -2\\mathcal {L}_{GLS}(\\theta _0)\\right\\rbrace -E_{\\theta _0}\\left\\lbrace -2\\mathcal {L}_{GLS}(\\hat{\\theta }_{GLS})\\right\\rbrace $ and using (REF ) and (REF ), we see that the following criterion based on the GLS estimator $AIC_{GLS}=-2\\mathcal {L}_{GLS}(\\hat{\\theta }_{GLS})+\\frac{2pd^2}{n}$ is an approximately unbiased estimator of $\\Delta _n(\\hat{\\theta }_{GLS},\\theta _0)$ .",
"Nevertheless the $AIC_{GLS}$ is infeasible since it depends on the unknown variance of the errors.",
"Thereby we will use the adaptive estimation of the variance structure to propose a feasible selection criterion.",
"Recall that $-2\\mathcal {L}_{GLS}(\\hat{\\theta }_{GLS})=\\frac{1}{n}\\sum _{t=1}^n\\ln \\left\\lbrace \\det (\\Sigma _t)\\right\\rbrace +u_t(\\hat{\\theta }_{GLS})^{\\prime }\\Sigma _t^{-1}u_t(\\hat{\\theta }_{GLS}),$ and define $-2\\mathcal {L}_{ALS}(\\hat{\\theta }_{ALS})=\\frac{1}{n}\\sum _{t=1}^n\\ln \\left\\lbrace \\det (\\check{\\Sigma }_t)\\right\\rbrace +u_t(\\hat{\\theta }_{ALS})^{\\prime }\\check{\\Sigma }_t^{-1}u_t(\\hat{\\theta }_{ALS}).$ In view of (REF ) and (REF ) we have $-2\\mathcal {L}_{GLS}(\\hat{\\theta }_{GLS})=-2\\mathcal {L}_{ALS}(\\hat{\\theta }_{ALS})+o_p(1),$ since we allowed for a finite number of variance breaks for the innovations.",
"Therefore we can introduce the adaptive criterion $AIC_{ALS}=-2\\mathcal {L}_{ALS}(\\hat{\\theta }_{ALS})+\\frac{2pd^2}{n},$ which gives an approximately unbiased estimation of (REF ) for large $n$ .",
"Finally note that if we suppose that $(X_t)$ is cointegrated, Kim and Park (2010) showed that the long run relationships estimated by reduced rank are $n$ -consistent.",
"Therefore our approach for building information criteria can be straightforwardly extended to the cointegrated case since it is clear that the estimated long run relationships can be replaced by the true relationships in the preceding computations."
],
[
"Identifying the lag length using partial autoregressive and partial correlation matrices",
"In this part we assume $p>p_0$ , so that the cut-off property of the presented tools can be observed.",
"Following the approach described in Reinsel (1993) chapter 3, one can use the estimators of the autoregressive parameters to identify the lag length of (REF ).",
"Consider the regression of $X_t$ on its past values ${X}_t={A}_{01}{X}_{t-1}+\\dots +{A}_{0p}{X}_{t-p}+u_t.$ We can remark that the partial autoregressive matrices $A_{0p_0+1},\\dots ,A_{0p}$ are equal to zero.",
"The PAM are estimated using OLS or ALS estimation.",
"Confidence bounds for the PAM can be proposed as follow.",
"Let us introduce the $d^2(p-p_0)\\times d^2p$ dimensional matrix $R=(0,I_{d^2(p-p_0)})$ , so that from (REF ), (REF ) and (REF ) we write $\\sqrt{n}R\\hat{\\theta }_{OLS}\\Rightarrow \\mathcal {N}(0,R\\Lambda _{3}^{-1}\\Lambda _{2}\\Lambda _{3}^{-1}R^{\\prime })\\quad \\mbox{and}\\quad \\sqrt{n}R\\hat{\\theta }_{ALS}\\Rightarrow \\mathcal {N}(0,R\\Lambda _{1}^{-1}R^{\\prime }),$ where $R\\hat{\\theta }_{OLS}$ and $R\\hat{\\theta }_{ALS}$ correspond to the OLS and ALS estimators of the null matrices $A_{0p_0+1},\\dots ,A_{0p}$ .",
"Denote by $\\upsilon _{i}^{OLS}$ (resp.",
"$\\upsilon _{i}^{ALS}$ ) the asymptotic standard deviation of the $i$ th component of $\\hat{\\theta }_{OLS}$ (resp.",
"$\\hat{\\theta }_{ALS}$ ) for $i\\in \\lbrace d^2p_0+1,\\dots ,d^2p\\rbrace $ with obvious notations.",
"From (REF ) the $i$ -th component of $\\hat{\\theta }_{OLS}$ (resp.",
"$\\hat{\\theta }_{ALS}$ ) are usually compared with the 95% approximate asymptotic confidence bounds $\\pm 1.96\\hat{\\upsilon }_{i}^{OLS}$ (resp.",
"$\\pm 1.96\\hat{\\upsilon }_{i}^{ALS}$ ) as suggested in Tiao and Box (1981), and where the $\\hat{\\upsilon }_{i}^{OLS}$ 's and $\\hat{\\upsilon }_{i}^{ALS}$ 's can be obtained using the consistent estimators in (REF ).",
"Therefore the identified lag length for model (REF ) correspond to the higher order of the matrix $A_{0i}$ which have an estimator of a component which is clearly beyond its confidence bounds about zero.",
"The identification of the lag length of standard VAR processes is usually performed using also the partial cross-correlation matrices which are the extension of the partial correlations of the univariate case.",
"Consider the regressions $X_{t-p}=\\phi _1X_{t-p+1}+\\dots +\\phi _{p-1}X_{t-1}+w_t,$ ${X}_t={A}_{01}{X}_{t-1}+\\dots +{A}_{0p-1}{X}_{t-p+1}+u_t,$ with $p>1$ .",
"In our framework it is clear that the error process $(w_t)$ is unconditionally heteroscedastic, and then we define $\\underline{\\Sigma }_w=\\lim _{n\\rightarrow \\infty }n^{-1}\\sum _{t=1}^nE(w_tw_t^{\\prime })$ which converge under A1, and the consistent estimator of $\\underline{\\Sigma }_w$ : $\\hat{\\underline{\\Sigma }}_w&=&n^{-1}\\sum _{t=p}^nX_{t-p}X_{t-p}^{\\prime }\\\\&-&\\left\\lbrace n^{-1}\\sum _{t=p}^nX_{t-p}(\\tilde{X}_{t-p}^{p-1})^{\\prime }\\right\\rbrace \\left\\lbrace n^{-1}\\sum _{t=p}^n\\tilde{X}_{t-p}^{p-1}(\\tilde{X}_{t-p}^{p-1})^{\\prime }\\right\\rbrace ^{-1}\\left\\lbrace n^{-1}\\sum _{t=p}^n\\tilde{X}_{t-p}^{p-1}X_{t-p}^{\\prime }\\right\\rbrace $ with obvious notations.",
"The consistency of this estimator can be proved from standard computations and using lemmas 7.1-7.4 of Patilea and Raïssi (2012).",
"We also define the 'long-run' innovations variance $\\underline{\\Sigma }_u=\\lim _{n\\rightarrow \\infty }n^{-1}\\sum _{t=1}^nE(u_tu_t^{\\prime })$ where the $E(u_tu_t^{\\prime })$ 's are non constant and the consistent estimator $\\hat{\\underline{\\Sigma }}_u=n^{-1}\\sum _{t=1}^n\\hat{u}_t\\hat{u}_t^{\\prime }$ of $\\underline{\\Sigma }_u$ where we recall that the $\\hat{u}_t$ 's are the OLS residuals.",
"Several definitions for the partial cross-correlations are available in the literature.",
"In the sequel we concentrate on the definition given in Ansley and Newbold (1979) which is used in the VARMAX procedure of the software SAS.",
"We propose to extend the partial cross-correlation matrices in our framework as follow $P(p)=\\left(\\underline{\\Sigma }_u^{-\\frac{1}{2}}\\otimes \\underline{\\Sigma }_w^{-\\frac{1}{2}}\\right)\\mbox{vec}\\left\\lbrace n^{-1}\\sum _{t=1}^nE(w_tu_t^{\\prime })\\right\\rbrace =\\left(\\underline{\\Sigma }_u^{-\\frac{1}{2}}\\otimes \\underline{\\Sigma }_w^{\\frac{1}{2}}\\right)\\mbox{vec}(A_p)$ and it is clear that for $p>p_0$ we have $P(p)=0$ .",
"The expression (REF ) may be viewed as the 'long-run' relation between the $X_t$ 's and the $X_{t-p}$ 's corrected for the intermediate values for each date $t$ .",
"Consider the OLS and ALS consistent estimators $\\hat{P}_{OLS}(p)=\\left(\\hat{\\underline{\\Sigma }}_u^{-\\frac{1}{2}}\\otimes \\hat{\\underline{\\Sigma }}_w^{\\frac{1}{2}}\\right)\\mbox{vec}(\\tilde{R}\\hat{\\theta }_{OLS})$ $\\hat{P}_{ALS}(p)=\\left(\\hat{\\underline{\\Sigma }}_u^{-\\frac{1}{2}}\\otimes \\hat{\\underline{\\Sigma }}_w^{\\frac{1}{2}}\\right)\\mbox{vec}(\\tilde{R}\\hat{\\theta }_{ALS}),$ where $\\tilde{R}=(0,I_{d^2})$ is of dimension $d^2\\times d^2p$ , so that $\\tilde{R}\\hat{\\theta }_{OLS}$ and $\\tilde{R}\\hat{\\theta }_{ALS}$ correspond to the ALS and OLS estimators of $A_p$ in (REF ).",
"Using again (REF ), (REF ), (REF ) and from the consistency of $\\hat{\\underline{\\Sigma }}_u$ and $\\hat{\\underline{\\Sigma }}_w$ we obtain $n^{\\frac{1}{2}}\\hat{P}_{OLS}(p)\\Rightarrow \\mathcal {N}\\left(0,\\left(\\underline{\\Sigma }_u^{-\\frac{1}{2}}\\otimes \\underline{\\Sigma }_w^{\\frac{1}{2}}\\right)\\left(\\tilde{R}\\Lambda _{3}^{-1}\\Lambda _{2}\\Lambda _{3}^{-1}\\tilde{R}^{\\prime }\\right)\\left(\\underline{\\Sigma }_u^{-\\frac{1}{2}}\\otimes \\underline{\\Sigma }_w^{\\frac{1}{2}}\\right)\\right)$ $n^{\\frac{1}{2}}\\hat{P}_{ALS}(p)\\Rightarrow \\mathcal {N}\\left(0,\\left(\\underline{\\Sigma }_u^{-\\frac{1}{2}}\\otimes \\underline{\\Sigma }_w^{\\frac{1}{2}}\\right)\\left(\\tilde{R}\\Lambda _{1}^{-1}\\tilde{R}^{\\prime }\\right)\\left(\\underline{\\Sigma }_u^{-\\frac{1}{2}}\\otimes \\underline{\\Sigma }_w^{\\frac{1}{2}}\\right)\\right).$ Hence approximate confidence bounds can be built using (REF ) and (REF ).",
"Similarly to the partial autoregressive matrices the highest order $p$ for which a cut-off is observed for an element of $\\hat{P}_{OLS}(p)$ ( resp.",
"$\\hat{P}_{ALS}(p)$ ) correspond to the identified lag length for the VAR model.",
"Note that for $p=1$ ($p_0=0$ , so that the observed process is uncorrelated and $w_t=X_{t-1}$ , $u_t=X_t$ ), we have $\\lim _{n\\rightarrow \\infty }n^{-1}\\sum _{t=1}^nE(X_tX_t^{\\prime })=\\underline{\\Sigma }_w=\\underline{\\Sigma }_u$ .",
"In this case similar results to (REF ) and (REF ) can be used.",
"Let us end this section with some remarks on the OLS and ALS estimation approaches of the PAM and PCM.",
"If we assume that the variance of the error process is constant, the result (REF ) is used to identify the autoregressive order using the partial autoregressive and correlation matrices obtained from the OLS estimation.",
"However as pointed out in the previous section this standard result can be misleading in our framework.",
"The simulations carried out in the next section show that the standard bounds are not reliable in our framework.",
"From the real example below it appears that the OLS PAM and PCM with the standard confidence bounds seem to select a too large lag length.",
"Note that since the tools presented in this section are based on the results (REF ), (REF ) and on the adaptive estimation of the autoregressive parameters, they are able to take into account changes in the variance.",
"In the univariate case the partial autocorrelation function is used for identifying the autoregressive order.",
"In such a case the asymptotic behavior of $\\hat{\\theta }_{ALS}$ does not depend on the variance structure (see equation (REF ) below).",
"Hence the ALS estimators of the partial autocorrelations do not depend on the variance function on the contrary to its OLS counterparts.",
"In the general VAR case Patilea and Raïssi (2012) also showed that that $\\Lambda _3^{-1}\\Lambda _2\\Lambda _3^{-1} - \\Lambda _1^{-1} $ is positive semi-definite (the same result is available in the univariate case).",
"Therefore the tools based on the ALS estimator are more accurate than the tools based on the OLS estimator for identifying the autoregressive order.",
"We illustrate the above remarks by considering the following simple case where we assume that $\\Sigma (r)=\\sigma (r)^2I_d$ with $\\sigma (\\cdot )^2$ a real-valued function.",
"The univariate stable autoregressive processes are a particular case of this specification of the variance.",
"In this case we obtain $ \\Lambda _1=\\Lambda _4=\\sum _{i=0}^{\\infty }\\left\\lbrace \\Delta ^i(e_p(1)e_p(1)^{\\prime }\\otimes I_d)\\Delta ^{i^{\\prime }}\\right\\rbrace \\otimes I_d$ and $\\Lambda _2=\\int _0^1\\sigma (r)^4dr\\Lambda _1,\\qquad \\Lambda _3=\\int _0^1\\sigma (r)^2dr \\Lambda _1,$ so that we have $\\Lambda _3^{-1}\\Lambda _2\\Lambda _3^{-1}=\\frac{\\int _0^1\\sigma (r)^4dr}{\\left(\\int _0^1\\sigma (r)^2dr\\right)^2}\\Lambda _1^{-1}$ with $\\int _0^1\\sigma (r)^4dr\\ge \\left(\\int _0^1\\sigma (r)^2dr\\right)^2$ .",
"Hence from (REF ) it is clear that the adaptive PAM and PCM are more reliable than the PAM and PCM obtained using the OLS approach.",
"In addition from (REF ) the asymptotic behavior of the adaptive partial autoregressive and partial correlation matrices does not depend on the variance function.",
"On the other hand we also see from (REF ) that the matrices $\\Lambda _4$ and $\\Lambda _3^{-1}\\Lambda _2\\Lambda _3^{-1}$ can be quite different."
],
[
"Empirical results",
"For our empirical study the $AIC_{ALS}$ is computed using an adaptive estimation of the variance as described in Section .",
"The ALS estimators of the PAM and PCM are obtained similarly.",
"In particular the bandwidth is selected using the cross-validation method.",
"The OLS partial autoregressive and partial correlation matrices used with the standard confidence bounds are denoted by $PAM_S$ and $PCM_S$ .",
"Similarly we also introduce the $PAM_{OLS}$ , $PAM_{ALS}$ and the $PCM_{OLS}$ , $PCM_{ALS}$ with obvious notations.",
"In the simulation study part the infeasible and $GLS$ tools are used only for comparison with the feasible $ALS$ tools.",
"It is important to note that when the $PAM$ and $PCM$ are used, the practitioners base their decision on the visual inspection of these tools.",
"Results concerning automatically selected lag lengths over the iterations using several $PAM$ and $PCM$ do not really reflect their ability to identify the lag length in practice.",
"For instance it is well known that there are cases where some $PAM$ or $PCM$ are beyond the confidence bounds but not taken into account for the lag length identification.",
"Therefore we provide instead some simulation results which assess the ability of the studied methods to provide reliable confidence bounds for the choice of the lag length.",
"The use of the modified $PAM$ and $PCM$ is also illustrated in the real data study below.",
"For a given tool we assume in our experiments that when the selected autoregressive order is such that $p>5$ , the model identification is suspected to be not reliable.",
"For instance the more complicated models may appear not enough penalized by the information criterion, or the number of estimated parameters becomes too large when compared to the number of observations.",
"In such situations the practitioner is likely to stop the procedure."
],
[
"Monte Carlo experiments",
"In this part $N=1000$ independent trajectories of bivariate VAR(2) ($p_0=2$ ) processes of length $n=50$ , $n=100$ and $n=200$ are simulated with autoregressive parameters given by $A_{01}=\\left(\\begin{array}{cc}-0.4 & 0.1 \\\\0 & -0.7 \\\\\\end{array}\\right)\\quad \\mbox{and}\\quad A_{02}=\\left(\\begin{array}{cc}-0.6 & 0 \\\\0 & -0.3 \\\\\\end{array}\\right).$ Recall that the process $(\\epsilon _t)$ is assumed iid standard Gaussian.",
"Two kinds of non constant volatilities are used.",
"When the variance smoothly change in time we consider the following specification $\\Sigma (r)=\\left(\\begin{array}{cc}(1+\\gamma _1 r)(1+\\rho ^2) & \\rho (1+\\gamma _1 r)^{\\frac{1}{2}}(1+\\gamma _2 r)^{\\frac{1}{2}} \\\\\\rho (1+\\gamma _1 r)^{\\frac{1}{2}}(1+\\gamma _2 r)^{\\frac{1}{2}} & (1+\\gamma _2 r) \\\\\\end{array}\\right).$ In case of abrupt change the following variance specification is used $\\Sigma (r)=\\left(\\begin{array}{cc}(1+f_1(r))(1+\\rho ^2) & \\rho (1+f_1(r))^{\\frac{1}{2}}\\rho (1+f_2(r))^{\\frac{1}{2}} \\\\\\rho (1+f_2(r))^{\\frac{1}{2}}(1+f_1(r))^{\\frac{1}{2}} & (1+f_2(r))(1+\\rho ^2) \\\\\\end{array}\\right),$ with $f_i(r)=(\\gamma _i-1)\\mathbf {1}_{(r\\ge 1/2)}(r)$ .",
"In this case we have a common variance break at the date $t=n/2$ .",
"In all the experiments we take $\\gamma _1=20$ , $\\gamma _2=\\gamma _1/3$ and $\\rho =0.2$ .",
"Note that the autoregressive parameters as well as the variance structure are inspired by the real data study below.",
"More precisely the autoregressive parameters in (REF ) are taken close to the two first adaptive PAM obtained for the government securities and foreign direct investment system.",
"In addition the ratio between the first and last adaptive estimation of the residual variance of the studied real data are of the same order of the ratio for the residual variance of the simulated series.",
"The results are given in Tables REF -REF and REF -REF for the variance specification (REF ) and in Tables REF -REF and REF -REF when specification (REF ) is used.",
"To facilitate the comparison of the studied identification tools, the most frequently selected lag length for the studied information criteria and the correct order ($p_0=2$ ) are in bold type.",
"The small sample properties of the different information criteria for selecting the autoregressive order is first analyzed.",
"According to Tables REF -REF we can remark that the $AIC_{ALS}$ and $AIC_{GLS}$ are selecting most frequently the true autoregressive order.",
"However we note that the modified $AIC$ have a slight tendency to select $p>p_0$ .",
"On the other hand we can see that the classical $AIC$ selects too large lags lengths in our framework.",
"This is in accordance with the fact that $AIC$ is not consistent (see e.g.",
"Paulsen (1984) or Hurvich and Tsai (1989)).",
"We also note that the frequency of selected true lag length $p=p_0=2$ increase with $n$ for the $AIC_{GLS}$ and $AIC_{ALS}$ .",
"The infeasible $AIC_{GLS}$ provide slightly better results than the $AIC_{ALS}$ .",
"As expected it can be seen that the difference between the $AIC_{GLS}$ and $AIC_{ALS}$ seems more marked when the processes display an abrupt variance change.",
"Indeed note that from (REF ) the variance is not consistently estimated at the break dates.",
"Nevertheless such bias is divided by $n$ , and we note that the behavior of the $AIC_{GLS}$ and $AIC_{ALS}$ become similar as the samples increase in all the studied cases.",
"According to our simulation results it appears that the adaptive $AIC$ is more able to select the appropriate autoregressive order than the standard $AIC$ when the underlying process is indeed a VAR process.",
"Now we turn to the analysis of the results for the PAM and PCM in Tables REF -REF .",
"Note that we used the 5% (asymptotic) confidence bounds in our study.",
"From our results it emerges that the standard bounds do not provide reliable tools for the identification of the lag length when the variance is non constant.",
"It can be seen that the frequencies of PAM and PCM with lag greater than $p_0$ beyond the standard bounds do not converge to 5%.",
"On the other hand it is found that the PAM and PCM based on the OLS and adaptive approaches give satisfactory results.",
"Indeed the frequencies of PAM and PCM with lag greater than $p_0$ beyond the standard and adaptive bounds converge to 5%.",
"As above we can remark that the results when the variance is smooth are better than the case where the variance exihibits an abrupt break.",
"When the PAM and PCM are equal to zero it seems that the adaptive and OLS method give similar results.",
"In accordance with the theoretical the more accurate adaptive method is more able than the OLS method to detect the significant PAM and PCM with lag smaller or equal to $p_0$ .",
"We can draw the conclusion that the standard bounds have to be avoided in our non standard framework.",
"The modified adaptive and OLS methods give reliable approaches for identifying the lag length of a VAR model with non constant variance.",
"It emerges that the more elaborated adaptive approach is preferable."
],
[
"Real data study",
"In this part we try to identify the VAR order of a bivariate system of variables taken from US international finance data.",
"The first differences of the quarterly US Government Securities (GS hereafter) hold by foreigners and the Foreign Direct Investment (FDI hereafter) in the US in billions dollars are studied from January 1, 1973 to October 1, 2009.",
"The length of the series is $n=147$ .",
"The studied series are plotted in Figure REF and can be downloaded from website of the research division of the federal reserve bank of Saint Louis: www.research.stlouisfed.org.",
"We first highlight some features of the studied series.",
"The OLS residuals and the variances of the errors estimated by kernel smoothing are plotted in Figure REF .",
"From Figure REF it appears that the data do not have a random walk behavior, while from Figure REF the estimated volatilities seem not constant.",
"The residuals plotted in Figure REF show that the variance of the first component of the residuals seems constant from January 1973 to October 1995 and then we may suspect an abrupt variance change.",
"Similarly the variance of the second component of the residuals seems constant from January 1973 to July 1998 and then we remark an abrupt variance change.",
"Therefore it clearly appears that the standard homoscedasticity assumption turns out to be not realistic for the studied series.",
"We fitted VAR($p$ ) models with $p\\in \\lbrace 1,\\dots ,5\\rbrace $ to the data and computed the $AIC$ and $AIC_{ALS}$ for each $p$ .",
"In our VAR system the first component corresponds to the GS and the second corresponds to the FDI.",
"From Table REF the $AIC$ is decreasing as $p$ is increased so that the higher autoregressive order $p=5$ is selected, while the minimum value for the $AIC_{ALS}$ is attained for $p=2$ .",
"If it is assumed that the studied processes follow a VAR model and since we noted that the variance of the studied processes seems non constant, it is likely that the $AIC$ is not reliable and selects a too large autoregressive order.",
"In view of our above results the model identification with the more parsimonious $AIC_{ALS}$ seems more reliable.",
"We also considered the $PCM$ obtained from the standard, OLS and ALS estimation methods.",
"The $PCM$ are plotted in Figures REF and REF and it appear that we can identify $p=2$ using the modified tools while $p=3$ could be identified using the standard $PCM$ .",
"We also see that the standard and OLS confidence bounds can be quite different.",
"The PAM are given below with the 95% confidence bounds into brackets.",
"We base our lag length choice on the $PAM$ which are clearly greater than its 95% confidence bounds (in bold type).",
"The $PAM_S$ give for the studied data: $\\hat{A}_1^S=\\left(\\begin{array}{cc}{\\bf -0.35}_{[\\pm 0.16]} & 0.12_{[\\pm 0.19]} \\\\0.06_{[\\pm 0.14]} & {\\bf -0.72}_{[\\pm 0.16]} \\\\\\end{array}\\right)\\quad \\hat{A}_2^S=\\left(\\begin{array}{cc}{\\bf -0.55}_{[\\pm 0.17]} & 0.08_{[\\pm 0.23]} \\\\0.08_{[\\pm 0.14]} & {\\bf -0.27}_{[\\pm 0.20]} \\\\\\end{array}\\right)$ $\\hat{A}_3^S=\\left(\\begin{array}{cc}{\\bf -0.32}_{[\\pm 0.18]} & 0.07_{[\\pm 0.23]} \\\\0.15_{[\\pm 0.16]} & -0.20_{[\\pm 0.20]} \\\\\\end{array}\\right)\\quad \\hat{A}_4^S=\\left(\\begin{array}{cc}{\\bf -0.25}_{[\\pm 0.17]} & 0.03_{[\\pm 0.24]} \\\\-0.04_{[\\pm 0.15]} & -0.03_{[\\pm 0.21]} \\\\\\end{array}\\right)$ $\\hat{A}_5^S=\\left(\\begin{array}{cc}-0.06_{[\\pm 0.17]} & -0.04_{[\\pm 0.19]} \\\\-0.02_{[\\pm 0.15]} & -0.05_{[\\pm 0.17]} \\\\\\end{array}\\right).$ We obtain the following $PAM_{OLS}$ $\\hat{A}_1^{OLS}=\\left(\\begin{array}{cc}{\\bf -0.35}_{[\\pm 0.23]} & 0.12_{[\\pm 0.23]} \\\\0.06_{[\\pm 0.19]} & {\\bf -0.72}_{[\\pm 0.19]} \\\\\\end{array}\\right)\\quad \\hat{A}_2^{OLS}=\\left(\\begin{array}{cc}{\\bf -0.55}_{[\\pm 0.27]} & 0.08_{[\\pm 0.35]} \\\\0.08_{[\\pm 0.24]} & -0.27_{[\\pm 0.36]} \\\\\\end{array}\\right)$ $\\hat{A}_3^{OLS}=\\left(\\begin{array}{cc}-0.32_{[\\pm 0.31]} & 0.07_{[\\pm 0.35]} \\\\0.15_{[\\pm 0.18]} & -0.20_{[\\pm 0.31]} \\\\\\end{array}\\right)\\quad \\hat{A}_4^{OLS}=\\left(\\begin{array}{cc}-0.25_{[\\pm 0.24]} & 0.03_{[\\pm 0.26]} \\\\-0.04_{[\\pm 0.26]} & -0.03_{[\\pm 0.27]} \\\\\\end{array}\\right)$ $\\hat{A}_5^{OLS}=\\left(\\begin{array}{cc}-0.06_{[\\pm 0.25]} & -0.04_{[\\pm 0.17]} \\\\-0.02_{[\\pm 0.19]} & -0.05_{[\\pm 0.24]} \\\\\\end{array}\\right).$ and the following $PAM_{ALS}$ $\\hat{A}_1^{ALS}=\\left(\\begin{array}{cc}{\\bf -0.42}_{[\\pm 0.18]} & 0.06_{[\\pm 0.24]} \\\\0.02_{[\\pm 0.11]} & {\\bf -0.70}_{[\\pm 0.21]} \\\\\\end{array}\\right)\\quad \\hat{A}_2^{ALS}=\\left(\\begin{array}{cc}{\\bf -0.58}_{[\\pm 0.19]} & 0.07_{[\\pm 0.30]} \\\\0.03_{[\\pm 0.12]} & -0.26_{[\\pm 0.26]} \\\\\\end{array}\\right)$ $\\hat{A}_3^{ALS}=\\left(\\begin{array}{cc}-0.21_{[\\pm 0.21]} & 0.07_{[\\pm 0.30]} \\\\0.08_{[\\pm 0.13]} & -0.21_{[\\pm 0.26]} \\\\\\end{array}\\right)\\quad \\hat{A}_4^{ALS}=\\left(\\begin{array}{cc}-0.20_{[\\pm 0.20]} & 0.06_{[\\pm 0.31]} \\\\0.02_{[\\pm 0.13]} & -0.01_{[\\pm 0.27]} \\\\\\end{array}\\right)$ $\\hat{A}_5^{ALS}=\\left(\\begin{array}{cc}-0.13_{[\\pm 0.19]} & 0.06_{[\\pm 0.26]} \\\\-0.01_{[\\pm 0.12]} & -0.07_{[\\pm 0.22]} \\\\\\end{array}\\right).$ It can be seen that the $PAM_{OLS}$ and $PAM_{ALS}$ in the $\\hat{A}_i^{OLS}$ and $\\hat{A}_i^{ALS}$ for $i=3,4$ and 5 seem not significant, so that one can identify $p=2$ using our modified tools.",
"The cut-off at $p=2$ is clearly marked for the $PAM_{OLS}$ and $PAM_{ALS}$ .",
"If the $PAM_{S}$ are used we note that one could select again $p=3$ or even $p=4$ , and we note that the cut-off is not so clearly marked in this case.",
"We also see that the 95% standard and OLS confidence bounds can be quite different.",
"If the length was automatically selected using the $PAM$ and $PCM$ , larger lag lengths would have been chosen.",
"Indeed we note that some of the $PAM$ and $PCM$ are only slightly beyond the 95% confidence bounds (see for instance the $\\hat{P}_{OLS}(3)$ , $\\hat{P}_{OLS}(4)$ or $\\hat{P}_{ALS}(3)$ in Figures REF and REF ).",
"In general it emerges from our empirical study part that the standard identification tools lead to select large lag lengths for the VAR models with non constant variance.",
"This may be viewed as a consequence to the fact that the standard tools are not adapted to our non standard framework.",
"Note that the identification of the model is the first step of the VAR modeling of time series.",
"In such situation the practitioner is likely to adjust a VAR model with a too large number of parameters which can affect the analysis of the series.",
"The identification tools developed in this paper take into account for unconditional heteroscedasticity.",
"From our real data study we found that the modified tools are more parsimonious."
],
[
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]
] | 1204.0757 |
[
[
"Towards a Security Engineering Process Model for Electronic Business\n Processes"
],
[
"Abstract Business process management (BPM) and accompanying systems aim at enabling enterprises to become adaptive.",
"In spite of the dependency of enterprises on secure business processes, BPM languages and techniques provide only little support for security.",
"Several complementary approaches have been proposed for security in the domain of BPM.",
"Nevertheless, support for a systematic procedure for the development of secure electronic business processes is still missing.",
"In this paper, we pinpoint the need for a security engineering process model in the domain of BPM and identify key requirements for such process model."
],
[
"Introduction",
"Business processes are the way organizations do their work – a set of activities carried out to accomplish a defined objective.",
"Therefore, the design, administration, enactment, and analysis of business processes – subsumed under the term business process management (BPM) – are vital challenges to organizations.",
"Business process management systems (BPMS) are seen as important facilitators for the necessary alignment of people and organizational resources.",
"BPMSs enable organizations to become adaptive enterprises: They allow for a faster reaction to environmental and market changes and support proactive innovation of products and services.",
"[1], [2] Consequently, BPM supported by information systems has seen an ongoing development in the last decades.",
"With respect to modeling languages and techniques, a multitude of approaches has been introduced.",
"At the same time, BPMS developed from simple information systems to capture and administrate process models to feature-rich BPM suites that support also simulation, execution, and controlling of business process instances.",
"Hence, the ability for organizations to manage business processes is well supported by today's software industry.",
"[3] Business processes are closely connected with assets of the respective organization.",
"Observation, manipulation, and disruption of business processes might threaten these assets or even the existence of the organization itself.",
"Thus, security of business processes ought to be of high importance for every organization.",
"But in spite of this dependence on secure business processes, BPM languages and techniques provide only little support to express security needs or controls applied."
],
[
"Security and Electronic Business Processes",
"In reaction to this need, several approaches have been developed to support security in the context of BPM.",
"Most approaches address one of two issues: the analysis of security (and safety) properties of business process models or the specification of security requirements and controls for electronic business processes [4].",
"By contrast, the support for a systematic procedure to develop secure electronic business processes is weak.",
"The few existing approaches do not address actual runtime environments and enforceable controls [5], [6], support only specific activities like security requirements engineering [7], [8], or do not provide any guidance for their application [9].",
"Also general approaches applied to electronic business processes display similar issues [10].",
"This situation might be attributed to security engineering in general: As a discipline – commonly defined as “building systems to remain dependable in the face of malice, error, or mischance” [11] – it is considered to be still in its infancy.",
"At present, mostly top-down approaches from the software engineering domain are adopted and enhanced with security-specific technologies and methods.",
"With regard to BPM, lack of support for security engineering is endangering one of its main objectives: allowing enterprises to react faster and to continuously innovate products and services.",
"Currently, enterprises either have to choose to focus on the protection of their assets or to develop and deploy their electronic business processes with little security expertise and support by applicable methods.",
"The first option requires security professionals to secure the electronic business processes individually and manually which implies investments in terms of time and money and threatens the adaptability gained with the application of BPMS.",
"The second option exposes the enterprises' assets to malice and mischance.",
"Industrial experiences from our Security Test Lab as well as academic studies analyzing industrial security engineering practices indicate that enterprises tend to choose the latter option [12]."
],
[
"Requirements for a Security Engineering Process Model Addressing BPM",
"As a first step to bridge the gap between (executable) business process models and secure electronic business processes, we provide a set of requirements for a security engineering process model in the domain of BPM.",
"These requirements stem from fundamental ideas of BPM: separation of technical and domain aspects (allowing domain experts to work largely independently from developers), independence from development methodologies, and applicability notwithstanding environmental heterogeneity.",
"Key issues for a general security engineering process model are separation of requirements and controls, traceability, correctness, completeness, and iterative applicability on different levels of abstraction.",
"Core activities encompass security requirements elicitation, threat modeling and evaluation, control design, and validation.",
"[13] To align a security engineering process model with BPM we identify the following requirements: Separation of (initial) activities for security professionals and (recurring) activities for security nonprofessionals Consistent coverage of all activities with detailed guidance Utilization of models and adequate tooling to separate the security analysis from design and implementation Possibility to integrate the security engineering process model with different development approaches We envision a security engineering process model that aids security professionals to prepare an environment for domain experts, providing common threats, evaluation criteria, and their countermeasures supported by business process engines.",
"The security engineering process model supports domain experts to identify and evaluate security requirements utilizing business process models as primary input, to select from a restricted set of applicable controls, and to configure the business process engines correspondingly."
],
[
"Conclusion",
"BPMS enable enterprises to become adaptive and are well supported by today's software industry.",
"Although an active research community proposed several approaches to address the need for security in the domain of BPM, support for a systematic procedure for the development of secure electronic business processes is still missing.",
"We identify four key requirements for a security engineering process model that is able to bridge the gap between (executable) business process models and secure electronic business processes.",
"Currently we are working on such a process model as well as supporting modeling languages and tooling."
],
[
"Acknowledgment",
"The work presented was developed in the context of the project Innovative Services for the Internet of the Future (InDiNet, ID 01IC10S04F) which is funded by the German Federal Ministry of Education and Research.",
"8"
]
] | 1204.1296 |
[
[
"Approximation Limits of Linear Programs (Beyond Hierarchies)"
],
[
"Abstract We develop a framework for approximation limits of polynomial-size linear programs from lower bounds on the nonnegative ranks of suitably defined matrices.",
"This framework yields unconditional impossibility results that are applicable to any linear program as opposed to only programs generated by hierarchies.",
"Using our framework, we prove that O(n^{1/2-eps})-approximations for CLIQUE require linear programs of size 2^{n^\\Omega(eps)}.",
"(This lower bound applies to linear programs using a certain encoding of CLIQUE as a linear optimization problem.)",
"Moreover, we establish a similar result for approximations of semidefinite programs by linear programs.",
"Our main ingredient is a quantitative improvement of Razborov's rectangle corruption lemma for the high error regime, which gives strong lower bounds on the nonnegative rank of certain perturbations of the unique disjointness matrix."
],
[
"Introduction",
"Linear programs (LPs) play a central role in the design of approximation algorithms, see, e.g., [44], [46], [32].",
"Therefore, understanding the limitations of LPs as tools for designing approximation algorithms is an important question.",
"The first generation of results studied the limitations of specific LPs by seeking to determine their integrality gaps.",
"The second generation of results, pioneered by [1], studied the limitations of structured LPs such as those generated by lift-and-project procedures or hierarchies (e.g., [42] and [35]).",
"In this work, we start a third generation of results that apply to any LP for a given problem.",
"For example, our lower bounds address the following question: Is there a polynomial-size linear programming relaxation $\\mathsf {LP}_n$ for CLIQUE that achieves a $n^{\\Theta (1)}$ -approximation for all graphs with at most $n$ vertices?",
"We develop a framework for reducing questions of this kind to lower bounds on the nonnegative rankThe nonnegative rank of a matrix $M$ , denoted $\\operatorname{rank}_{+}(M)$ , is the minimum $r$ such that $M = TU$ where $T$ and $U$ are nonnegative matrices with $r$ columns and $r$ rows, respectively.",
"of certain matrices associated to the problem, and then prove lower bounds for the matrices corresponding to CLIQUE.",
"The matrices studied here are related to the unique disjointness problem, a variant of the famous disjointness problem from communication complexity (see, e.g., [16] for a survey).",
"In the disjointness problem (DISJ), both Alice and Bob receive a subset of $[n] := \\lbrace 1,\\ldots ,n\\rbrace $ .",
"They have to determine whether the two subsets are disjoint.",
"The unique disjointness problem (UDISJ) is the promise version of the disjointness problem where the two subsets are guaranteed to have at most one element in common.",
"Denoting the binary encoding of the sets of Alice and Bob by $a, b \\in \\lbrace 0,1\\rbrace ^n$ , respectively, this amounts to computing the Boolean function $\\text{UDISJ}(a,b) 1-a^{\\intercal }b$ on the set of pairs $(a,b) \\in \\lbrace 0,1\\rbrace ^n \\times \\lbrace 0,1\\rbrace ^n$ with $a^{\\intercal } b \\in \\lbrace 0,1\\rbrace $ .",
"Viewing it as a partial $2^n \\times 2^n$ matrix, we call $\\text{UDISJ}$ the unique disjointness matrix.",
"It is known that the communication complexity of UDISJ is $\\Omega (n)$ bits for deterministic, nondeterministic and even randomized communication protocols [26], [38], [4].",
"One consequence of this is that the nonnegative rank of any matrix obtained from UDISJ by filling arbitrarily the blank entries (for pairs $(a,b)$ with $a^{\\intercal } b > 1$ ) and perhaps adding rows and/or columns is still $2^{\\Omega (n)}$ .",
"Indeed: (i) the support of the resulting matrix has $\\Omega (n)$ nondeterministic communication complexity because it contains UDISJ, (ii) for every matrix $M$ , $\\log \\operatorname{rank}_{+}(M)$ is lower bounded by the nondeterministic communication complexity of (the support matrix of) $M$ [48].",
"In a recent paper [18] proved strong lower bounds on the size of LPs expressing the traveling salesman problem (TSP), or more precisely on the size of extended formulations of the TSP polytope (see Section for definitions of concepts related to polyhedra, extended formulations and slack matrices).",
"Their proof works by embedding UDISJ in a slack matrix of the TSP polytope of the complete graph on $\\Theta (n^2)$ vertices.",
"This solved a question left open in [48].",
"We use a similar approach for approximate extended formulations.",
"In case of CLIQUE, our approach requires lower bounds on the nonnegative rank of partial matrices obtained from the UDISJ matrix by adding a positive offset to all the entries."
],
[
"Related Work",
"Our results are closely related to previous work in communication complexity for the (unique) disjointness problem and related problems.",
"Lower bounds of $\\Omega (n)$ on the randomized, bounded error communication complexity of disjointness were established in [26].",
"In [38] the distributional complexity of unique disjointness problem was analyzed, which in particular implies the result of [26].",
"In that famous paper, Razborov proved the following rectangle corruption lemma: for every large rectangle within UDISJ, the number of 0-entries is proportional to the number of 1-entries.",
"The most recent proof that the randomized, bounded error communication complexity of DISJ is $\\Omega (n)$ is due to [4] and is based on information theoretic arguments.",
"This leads to a lower bound for randomized communication within a high-error regime, that is, when the error probability is close to $1/2$ .",
"Here we derive a strong generalization dealing with shifts for approximate EFs and we recover the high-error regime bound.",
"There has been extensive work on LP and SDP hierarchies/relaxations and their limitations; we will be only able to list a few here.",
"In [14], strong lower bounds (of $2-\\epsilon $ ) on the integrality gap for $n^{\\epsilon }$ rounds of the Sherali-Adams hierarchy when applied to (natural relaxations of) VERTEX COVER, Max CUT, SPARSEST CUT have been been established via embeddings into $\\ell _2$ ; see also [15] for limits and tradeoffs in metric embeddings.",
"For integrality gaps of relaxations for the KNAPSACK problem see [27].",
"A nice overview of the differences and similarities of the Sherali-Adams, the Lovász-Schrijver and the Lasserre hierarchies/relaxations can be found in [33].",
"Similar to the level of a hierarchy, we have the notion of rank for the Lovász-Schrijver relaxation and rank correspond to a similar complexity measure as the level.",
"The rank is the minimum number of application of the Lovász-Schrijver operator $N$ until we obtain the integral hull of the polytope under consideration.",
"Rank lower bounds of $n$ for Lovász-Schrijver relaxations of CLIQUE have been obtained in [17]; a similar result for Sherali-Adams hierarchy can be found in [33].",
"In [43] integrality gaps, after adding few rounds of Chvátal-Gomory cuts, have been studied for problems including $k$ -CSP, Max CUT, VERTEX COVER, and UNIQUE LABEL COVER showing that in some cases (e.g., $k$ -CSP) the gap can be significantly reduced whereas in most other cases the gap remains high.",
"In the context of SDP relaxations, in particular formulations derived from the Lovász-Schrijver $N_+$ hierarchies (see [35]) and the Lasserre hierarchies (see [31]) there has been significant work in recent years.",
"For example, [2] obtained a $O(\\sqrt{\\log n})$ upper bound on a suitable SDP relaxation of SPARSEST CUT.",
"For lower bounds in terms of rank, see e.g., [40] for the $k$ -CSP in the Lasserre hierarchy or [41] for VERTEX COVER in the semidefinite Lovász-Schrijver hierarchy.",
"Motivated by the Unique Games Conjecture, several works studied upper and lower bounds for SDP hierarchy relaxations of Unique Games (see for example, [24], [5], [7], [6]).",
"Approximate extended formulations have been studied before, for specific problems, e.g., KNAPSACK in [9], or as a general tool, see [45].",
"For recent results on computing the nonnegative rank see, e.g., [3]."
],
[
"Contribution",
"The contribution of the present paper is threefold.",
"We develop a framework for proving lower bounds on the sizes of approximate EFs.",
"Through a generalization of Yannakakis's factorization theorem, we characterize the minimum size of a $\\rho $ -approximate extended formulations as the nonnegative rank of any slack matrix of a pair of nested polyhedra.",
"Thus we reduce the task of proving approximation limits for LPs to the task of obtaining lower bounds on the nonnegative ranks of associated matrices.",
"Typically, these matrices have no zeros, which renders it impossible to use nondeterministic communication complexity.",
"We emphasize the fact that the results obtained within our framework are unconditional.",
"In particular, they do not rely on P $\\ne $ NP.",
"We extend Razborov's rectangle corruption lemma to deal with shifts of the UDISJ matrix.",
"As a consequence, we prove that the nonnegative rank of any matrix obtained from the UDISJ matrix by adding a constant offset to every entry is still $2^{\\Omega (n)}$ .",
"Moreover, we show that the nonnegative rank is still $2^{\\Omega (n^{2\\epsilon })}$ when the offset is at most $n^{1/2-\\epsilon }$ .",
"To our knowledge, these are the first strong lower bounds on the nonnegative rank of matrices that contain no zeros.",
"Our extension of Razborov's lemma allow us to recover known lower bounds for DISJ in the high-error regime of [4].",
"We obtain a strong hardness result for CLIQUE w.r.t.",
"a natural linear encoding of the problem.",
"From the results described above, we prove that the size of every $O(n^{1/2-\\epsilon })$ -approximate EF for CLIQUE is $2^{\\Omega (n^{2\\epsilon })}$ .",
"Finally, we observe that the same bounds hold for approximations of SDPs by LPs.",
"This suggests that SDP-based approximation algorithms can be significantly stronger than LP-based approximation algorithms.",
"The inapproximability of SDPs by LPs has some interesting consequences.",
"In particular we cannot expect to convert SDP-based approximation algorithms into LP-based ones by approximating the PSD-cone via linear programming.",
"We point out that our framework readily generalizes to SDPs by replacing nonnegative rank with PSD rank (see [22] for a definition of the PSD rank).",
"However, no strong bound on PSD rank seems to be currently in sight.",
"Finally, we report that the results of this paper have inspired further research.",
"[11] improved our lower bound on the nonnegative rank of shifted UDISJ matrices and obtain super-polynomial lower bounds for shifts up to $O(n^{1-\\epsilon })$ , hence matching the algorithmic hardness of approximation for CLIQUE.",
"This was achieved by pioneering information-theoretic methods for proving lower bounds on the nonnegative rank.",
"An alternative information theoretic approach for lower bounding the nonnegative rank which simplifies and slightly improves the results in [11] has been presented in [10].",
"This last paper also establishes that matrices obtained from shifts of UDISJ by removing rows and columns, or flipping entries, still have high nonnegative rank.",
"[13] obtain lower bounds on the size of LPs approximating Max CSP.",
"In particular, they prove that approximating Max CUT (with nonnegative weights) with a constant factor less than 2 requires $n^{\\Omega (\\log n / \\log \\log n)}$ .",
"This solves a conjecture we stated in an earlier version of this text.",
"[39] proved a $2^{\\Omega (n)}$ lower bound on the nonnegative rank of the slack matrix of the perfect matching polytope by a significant modification of Razborov's lemma.",
"This exciting result essentially proves that there are is no small LP that can solve all weighted instance of the matching problem on a $n$ -vertex complete graph."
],
[
"Outline",
"We begin in Section by setting up our framework for studying approximate extended formulations of combinatorial optimization problems.",
"Then we extend Razborov's rectangle corruption lemma in Section and use this to prove strong lower bounds on the nonnegative rank of shifts of the UDISJ matrix.",
"Finally, we draw consequences for CLIQUE and approximations of SDPs by LPs in Section ."
],
[
"Framework for Approximation Limits of LPs",
"In this section we establish our framework for studying approximation limits of LPs.",
"First, we define in details the concepts of linear encodings and approximate extended formulations.",
"Second, we prove a factorization theorem for pairs of nested polyhedra reducing existential questions on approximate extended formulations to the computation of nonnegative ranks of corresponding slack matrices."
],
[
"Preliminaries",
"A (convex) polyhedron is a set $P \\subseteq \\mathbb {R}^d$ that is the intersection of a finite collection of closed halfspaces.",
"In other words, $P$ is a polyhedron if and only if $P$ is the set of solutions of a finite system of linear inequalities and possibly equalities.",
"(Note that every equality can be represented by a pair of inequalities.)",
"Equivalently, a set $P \\subseteq \\mathbb {R}^d$ is a polyhedron if and only if $P$ is the Minkowski sum of the convex hull $\\operatorname{conv}\\left(V\\right)$ of a finite set $V$ of points and the conical hull $\\operatorname{cone}\\left(R\\right)$ of a finite set $R$ of vectors, that is, $P = \\operatorname{conv}\\left(V\\right) + \\operatorname{cone}\\left(R\\right)$ .",
"Let $P \\subseteq \\mathbb {R}^d$ be a polyhedron.",
"The dimension of $P$ is the dimension of its affine hull $\\operatorname{aff}(P)$ .",
"A face of $P$ is a subset $F \\lbrace x \\in P \\mid w^{\\intercal } x = \\delta \\rbrace $ such that $P$ satisfies the inequality $w^{\\intercal } x \\leqslant \\delta $ .",
"Note that face $F$ is again a polyhedron.",
"A vertex is a face of dimension 0, i.e., a point.",
"A facet is a face of dimension one less than $P$ .",
"The inequality $w^{\\intercal } x \\leqslant \\delta $ is called facet-defining if the face $F$ it defines is a facet.",
"The recession cone $\\operatorname{rec}\\left(P\\right)$ of $P$ is the set of directions $v \\in \\mathbb {R}^d$ such that, for a point $p$ in $P$ , all points $p + \\lambda v$ where $\\lambda \\geqslant 0$ belong to $P$ .",
"The recession cone of $P$ does not depend on the base point $p$ , and is again a polyhedron (even more, it is a polyhedral cone).",
"The elements of the recession cone are sometimes called rays.",
"A (convex) polytope $P \\subseteq \\mathbb {R}^d$ is a bounded polyhedron.",
"Equivalently, $P$ is a polytope if and only if $P$ is the convex hull $\\operatorname{conv}\\left(V\\right)$ of a finite set $V$ of points.",
"Let $P \\subseteq \\mathbb {R}^d $ be a polytope.",
"Every (finite or infinite) set $V$ such that $P = \\operatorname{conv}\\left(V\\right)$ contains all the vertices of $P$ .",
"Letting $\\operatorname{vert}(P)$ denote the vertex set of $P$ , then we have $P = \\operatorname{conv}\\left(\\operatorname{vert}(P)\\right)$ .",
"Every (finite) system describing $P$ contains all the facet-defining inequalities of $P$ , up to scaling by positive numbers and adding equalities satisfied by all points of $P$ .",
"Conversely, a linear description of $P$ can be obtained by picking one defining inequality per facet and adding a system of equalities describing $\\operatorname{aff}(P)$ .",
"A $0/1$ -polytope in $\\mathbb {R}^d$ is simply the convex hull of a subset of $\\lbrace 0,1\\rbrace ^d$ .",
"For more about convex polytopes and polyhedra, see the standard reference [49]."
],
[
"Linear Encodings of Problems and Approximate EFs",
"A linear encoding of a (combinatorial optimization) problem is a pair $(\\mathcal {L},\\mathcal {O})$ where $\\mathcal {L} \\subseteq \\lbrace 0,1\\rbrace ^*$ is the set of feasible solutions to the problem and $\\mathcal {O} \\subseteq \\mathbb {R}^*$ is the set of admissible objective functions.",
"An instance of the linear encoding is a pair $(d,w)$ where $d$ is a positive integer and $w \\in \\mathcal {O}\\cap \\mathbb {R}^d$ .",
"Solving the instance $(d,w)$ means finding $x \\in \\mathcal {L} \\cap \\lbrace 0,1\\rbrace ^d$ such that $w^{\\intercal } x$ is either maximum or minimum, according to the type of problem under consideration.",
"Example 1 (Linear encoding of metric TSP) In the natural linear encoding of the metric traveling salesman problem (metric TSP), the feasible solutions $x \\in \\mathcal {L}$ are the characteristic vectors (or incidence vectors) of tours of the complete graph over $[n]$ for some $n \\geqslant 3$ , and the admissible objective functions $w \\in \\mathcal {O}$ are all nonnegative vectors $w = (w_{ij})$ such that $w_{ik} \\leqslant w_{ij} + w_{jk}$ for all distinct $i$ , $j$ and $k$ in $[n]$ .",
"All vectors are encoded in $\\mathbb {R}^d$ , where $d = \\binom{n}{2}$ .",
"By considering all possible $n \\geqslant 3$ , we obtain the pair $(\\mathcal {L},\\mathcal {O})$ corresponding to metric TSP.",
"(Recall that metric TSP is a minimization problem.)",
"For every fixed dimension $d$ , a linear encoding $(\\mathcal {L},\\mathcal {O})$ naturally defines a pair of nested convex sets $P \\subseteq Q$ where $P &\\operatorname{conv}\\left(\\lbrace x \\in \\lbrace 0,1\\rbrace ^d \\mid x \\in \\mathcal {L}\\rbrace \\right), \\quad \\text{and}\\\\Q &\\lbrace x \\in \\mathbb {R}^d \\mid \\forall w \\in \\mathcal {O} \\cap \\mathbb {R}^d :w^{\\intercal } x \\leqslant \\max \\lbrace w^{\\intercal } z \\mid z \\in P\\rbrace \\rbrace $ if the goal is to maximize and $Q \\lbrace x \\in \\mathbb {R}^d \\mid \\forall w \\in \\mathcal {O} \\cap \\mathbb {R}^d :w^{\\intercal } x \\geqslant \\min \\lbrace w^{\\intercal } z \\mid z \\in P\\rbrace \\rbrace $ if the goal is to minimize.",
"Intuitively, the vertices of $P$ encode the feasible solutions of the problem under consideration and the defining inequalities of $Q$ encode the admissible objective functions.",
"Notice that $P$ is always a 0/1-polytope but $Q$ might be unbounded and, in some pathological cases, nonpolyhedral.",
"Below, we will mostly consider the case where $Q$ is polyhedral, that is, defined by a finite number of “interesting” inequalities.",
"Given a linear encoding $(\\mathcal {L},\\mathcal {O})$ of a maximization problem, and $\\rho \\geqslant 1$ , a $\\rho $ -approximate extended formulation (EF) is an extended formulation $Ex + Fy = g$ , $y \\geqslant \\mathbf {0}$ with $(x,y) \\in \\mathbb {R}^{d+r}$ such that $\\max \\lbrace w^{\\intercal } x \\mid Ex + Fy = g,\\ y \\geqslant \\mathbf {0}\\rbrace &\\geqslant \\max \\lbrace w^{\\intercal } x \\mid x \\in P\\rbrace \\quad \\text{for all} \\quad w \\in \\mathbb {R}^d \\quad \\text{and}\\\\\\max \\lbrace w^{\\intercal } x \\mid Ex + Fy = g,\\ y \\geqslant \\mathbf {0}\\rbrace &\\leqslant \\rho \\max \\lbrace w^{\\intercal } x \\mid x \\in P\\rbrace \\quad \\text{for all} \\quad w \\in \\mathcal {O} \\cap \\mathbb {R}^d.$ Letting $K \\lbrace x \\in \\mathbb {R}^d \\mid \\exists y \\in \\mathbb {R}^r:Ex + Fy = g,\\ y \\geqslant \\mathbf {0}\\rbrace $ , we see that this is equivalent to $P \\subseteq K \\subseteq \\rho Q$ .",
"For a minimization problem, we require $\\min \\lbrace w^{\\intercal } x \\mid Ex + Fy = g,\\ y \\geqslant \\mathbf {0}\\rbrace &\\leqslant \\min \\lbrace w^{\\intercal } x \\mid x \\in P\\rbrace \\quad \\text{for all} \\quad w \\in \\mathbb {R}^d \\quad \\text{and}\\\\\\min \\lbrace w^{\\intercal } x \\mid Ex + Fy = g,\\ y \\geqslant \\mathbf {0}\\rbrace &\\geqslant \\rho ^{-1} \\min \\lbrace w^{\\intercal } x \\mid x \\in P\\rbrace \\quad \\text{for all} \\quad w \\in \\mathcal {O} \\cap \\mathbb {R}^d.$ This is equivalent to $P \\subseteq K \\subseteq \\rho ^{-1} Q$ .",
"Example 2 (Approximate extended formulation of metric TSP) We return to Example REF .",
"It is known that the Held-Karp relaxation $K$ of the metric TSP has integrality gap at most $3/2$ (see [25], [47]).",
"In geometric terms, this means that $P \\subseteq K \\subseteq 2/3 \\cdot Q$ .",
"Although $K$ is defined by an exponential number of inequalities, it is known that it can be reformulated with a polynomial number of constraints by adding a polynomial number of variables, see, e.g., [12].",
"That is, the Held-Karp relaxation $K$ has a polynomial-size extended formulation.",
"Thus, the pair $(\\mathcal {L},\\mathcal {O})$ for the metric TSP has a polynomial-size $3/2$ -approximate EF.",
"We require the following faithfulness condition: every instance of the problem can be mapped to an instance of the linear encoding in such a way that feasible solutions to an instance of the problem can be converted in polynomial time to feasible solutions to the corresponding instance of the linear encoding without deteriorating their objective function values, and vice-versa.",
"Roughly speaking, we ask that each instance of the problem can be encoded as an instance of the linear encoding.",
"For linear encoding of graph problems, such as the maximum clique problem (CLIQUE), the set of feasible solutions is not allowed to depend on the input graph, which therefore must be encoded solely in the objective function.",
"The set of feasible solutions is only allowed to depend on the size $n$ of the ground set.",
"Example 3 (Max $k$ -SAT) Consider the maximum $k$ -SAT problem (Max $k$ -SAT), where $k$ is constant.",
"Letting $u_1$ , ..., $u_n$ denote the variables of a Max $k$ -SAT instance, we encode the problem in dimension $d = \\Theta (n^k)$ .",
"For each nonempty clause $C$ of size at most $k$ , we introduce a variable $x_C$ .",
"Collectively, these variables define a point $x \\in \\mathbb {R}^d$ .",
"Given a truth assignment, we set $x_C$ to 1 if $C$ is satisfied and otherwise we set $x_C$ to 0.",
"Letting $n$ vary, this defines a language $\\mathcal {L} \\subseteq \\lbrace 0,1\\rbrace ^*$ .",
"We let $\\mathcal {O} \\lbrace 0,1\\rbrace ^*$ .",
"The pair $(\\mathcal {L},\\mathcal {O})$ defines a linear encoding of Max $k$ -SAT because each instance of Max $k$ -SAT can be encoded as an instance of $(\\mathcal {L},\\mathcal {O})$ .",
"More precisely, to any given set of clauses over $n$ variables, we can associate a dimension $d = \\Theta (n^k)$ and weight vector $w \\in \\lbrace 0,1\\rbrace ^d$ such that maximizing $\\sum w_C x_C$ for $x \\in \\mathcal {L} \\cap \\lbrace 0,1\\rbrace ^d$ corresponds to finding a truth assignment that maximizes the number of satisfied clauses.",
"Finally, we remark that the EF defined by the inequalities $0 \\leqslant x_C \\leqslant 1$ and $x_C \\leqslant \\sum _{u_i \\in C}x_{\\lbrace u_i\\rbrace } + \\sum _{\\bar{u}_i \\in C} (1-x_{\\lbrace u_i\\rbrace })$ for all clauses $C$ is a polynomial-size $4/3$ -approximate EF for Max $k$ -SAT, as follows from [20]."
],
[
"Factorization Theorem for Pairs of Nested Polyhedra",
"Let $P$ and $Q$ be polyhedra with $P \\subseteq Q\\subseteq \\mathbb {R}^d$ .",
"An extended formulation (EF) of the pair $P,Q$ is a system $Ex + Fy = g$ , $y \\geqslant \\mathbf {0}$ defining a polyhedron $K \\lbrace x \\in \\mathbb {R}^d \\mid Ex + Fy = g,\\ y \\geqslant \\mathbf {0}\\rbrace $ such that $P \\subseteq K \\subseteq Q$ .",
"We denote by $\\operatorname{xc}(P,Q)$ the minimum size of an EF of the pair $P,Q$ .",
"Now consider an inner description $P \\operatorname{conv}\\left(\\lbrace v_{1}, \\cdots , v_{n}\\rbrace \\right)+ \\operatorname{cone}\\left(\\lbrace r_{1}, \\dots , r_{k}\\rbrace \\right)$ of $P$ and an outer description $Q \\lbrace x \\in \\mathbb {R}^d \\mid A x \\leqslant b\\rbrace $ of $Q$ , where the system $Ax \\leqslant b$ consists of $m$ inequalities: $A_{1} x \\leqslant b_{1}, \\cdots , A_{m} x \\leqslant b_{m}$ .",
"The slack matrix of the pair $P,Q$ w.r.t.",
"these inner and outer descriptions is the $m \\times (n + k)$ matrix $S^{P,Q} =\\left[{\\begin{matrix}S^{P,Q}_{\\mathrm {vertex}} & S^{P,Q}_{\\mathrm {ray}}\\end{matrix}}\\right]$ given by block decomposition into a vertex and ray part: $S^{P,Q}_{\\mathrm {vertex}}(i,j) &b_{i} - A_{i} v_{j},& i \\in [m],\\ j \\in [n], \\\\S^{P,Q}_{\\mathrm {ray}}(i,j) &- A_{i} r_{j},& i \\in [m],\\ j \\in [k].$ A rank-$r$ nonnegative factorization of an $m \\times n$ matrix $M$ is a decomposition of $M$ as a product $M = TU$ of nonnegative matrices $T$ and $U$ of sizes $m \\times r$ and $r \\times n$ , respectively.",
"The nonnegative rank $\\operatorname{rank}_{+}(M)$ of $M$ is the minimum rank $r$ of nonnegative factorizations of $M$ .",
"In case $M$ is zero, we let $\\operatorname{rank}_{+}(M) = 0$ .",
"It is quite useful to notice that the nonnegative rank of $M$ is also the minimum number of nonnegative rank-1 matrices whose sum is $M$ .",
"From this, we see immediately that the nonnegative rank of $M$ is at least the nonnegative rank of any of its submatrices.",
"Our first result gives an essentially exact characterization of $\\operatorname{xc}(P,Q)$ in terms of the nonnegative rank of the slack matrix of the pair $P,Q$ .",
"It states that the minimum extension complexity $\\operatorname{xc}(P,Q)$ of a polyhedron sandwiched between $P$ and $Q$ equals the nonnegative rank of $S^{P,Q}$ (minus 1, in some cases).",
"The result readily generalizes Yannakakis's factorization theorem [48], which concerns the case $P = Q$ .",
"The idea of considering a pair $P,Q$ as we do here first appeared in [37] and similar ideas appeared earlier in [19].",
"Theorem 1 With the above notations, we have $\\operatorname{rank}_{+}(S^{P,Q}) - 1 \\leqslant \\operatorname{xc}(P, Q) \\leqslant \\operatorname{rank}_{+}(S^{P,Q})$ for every slack matrix of the pair $P, Q$ .",
"If the affine hull of $P$ is not contained in $Q$ and $\\operatorname{rec}\\left(Q\\right)$ is not full-dimensional, we have $\\operatorname{xc}(P, Q) = \\operatorname{rank}_{+}(S^{P,Q})$ .",
"In particular, this holds when $P$ and $Q$ are polytopes of dimension at least 1.",
"First, we deal with degenerate cases.",
"Observe that $\\operatorname{xc}(P,Q) = 0$ if and only if there exists an affine subspace containing $P$ and contained in $Q$ , that is, if and only if the affine hull of $P$ is contained in $Q$ .",
"In this case, we have $\\operatorname{rank}_{+}(S^{P,Q})\\in \\lbrace 0,1\\rbrace $ , so the theorem holds.",
"Now assume that the affine hull of $P$ is not contained in $Q$ .",
"Then, $\\operatorname{rank}_{+}(S^{P,Q})\\geqslant 1$ because having $\\operatorname{rank}_{+}(S^{P,Q}) = 0$ means either that $S^{P,Q}$ is empty, that is, $m = 0$ or $n+k = 0$ , or that $S^{P,Q}$ is the zero matrix.",
"In all cases, this contradicts our assumption that the affine hull of $P$ is not contained in $Q$ .",
"Next, let $S^{P,Q} = TU$ be any rank-$r$ nonnegative factorization of $S^{P,Q}$ with $r = \\operatorname{rank}_{+}(S^{P,Q}) \\geqslant 1$ .",
"This factorization decomposes into blocks: $S^{P,Q}_{\\mathrm {vertex}} = T U_{\\mathrm {vertex}}$ and $S^{P,Q}_{\\mathrm {ray}} = T U_{\\mathrm {ray}}$ .",
"Consider the system $Ax + Ty = b,\\ y \\geqslant \\mathbf {0}$ and the corresponding polyhedron $K \\lbrace x \\in \\mathbb {R}^d \\mid Ax + Ty = b,\\ y \\geqslant \\mathbf {0}\\rbrace $ .",
"We verify now that $P \\subseteq K \\subseteq Q$ .",
"The inclusion $K \\subseteq Q$ simply follows from $Ty \\geqslant \\mathbf {0}$ .",
"For the inclusion $P \\subseteq K$ , pick a vertex $v_j$ of $P$ and observe that $(x,y) =(v_j,U_{\\mathrm {vertex}}^j)$ satisfies (REF ), where $U_{\\mathrm {vertex}}^j$ denotes the $j$ th column of $U_{\\mathrm {vertex}}$ , because $Av_j + TU_{\\mathrm {vertex}}^{j} =Av_j + b - Av_j = b$ and $U^j \\geqslant \\mathbf {0}$ .",
"Similarly, for every ray $r_{j}$ we obtain a ray $(r_{j}, U_{\\mathrm {ray}}^{j})$ of $K$ as $A r_{j} + T U_{\\mathrm {ray}}^{j} = 0$ and $U_{\\mathrm {ray}}^{j} \\geqslant \\mathbf {0}$ .",
"Thus we obtain that (REF ) is a size-$r$ EF of the pair $P,Q$ .",
"Therefore, $\\operatorname{xc}(P,Q) \\leqslant \\operatorname{rank}_{+}(S^{P,Q})$ .",
"Finally, suppose that the system $Ex + Fy = g,\\ y \\geqslant \\mathbf {0}$ defines a size-$r$ EF of the pair $P, Q$ .",
"Let $L \\subseteq \\mathbb {R}^{d+r}$ denote the polyhedron defined by (REF ), and let $K \\subseteq \\mathbb {R}^d$ denote the orthogonal projection of $L$ into $x$ -space.",
"Since $P \\subseteq K$ , for each point $v_j$ , there exists $w_{j} \\in \\mathbb {R}_{+}^{r}$ such that $(v_{j}, w_{j}) \\in L$ .",
"Similarly, for each ray $r_{j}$ there exists a $z_{j} \\in \\mathbb {R}_{+}^{r}$ with $(r_{j}, z_{j})$ a ray of $L$ .",
"Let $W$ be the matrix with columns $w_{j}$ , and $Z$ be the matrix with columns $z_{j}$ .",
"Since $K \\subseteq Q$ , by Farkas's lemma, $Ax \\leqslant b$ can be derived from (REF ), i.e., there exists a matrix $T$ and a vector $c \\geqslant \\mathbf {0}$ with $A = T E$ , $b = T g + c$ and $T F \\geqslant 0$ .",
"This gives the factorizations $S^{P,Q}_{\\mathrm {vertex}} = (T F) W + c \\mathbf {1}^\\intercal $ and $S^{P, Q}_{\\mathrm {ray}} = (T F) Z$ , resulting in the rank-$(r+1)$ nonnegative factorization $S^{P, Q} =\\left[{\\begin{matrix}T F & c\\end{matrix}}\\right]\\cdot \\left[{\\begin{matrix}W & Z \\\\\\mathbf {1}^\\intercal & \\mathbf {0}^\\intercal \\end{matrix}}\\right].$ Taking $r = \\operatorname{xc}(P,Q)$ , we find $\\operatorname{rank}_{+}(S^{P,Q}) \\leqslant \\operatorname{xc}(P,Q) + 1$ .",
"Finally, when $\\operatorname{rec}\\left(Q\\right)$ is not full-dimensional, then $c$ above can be chosen to be $\\mathbf {0}$ .",
"This simplifies the factorization, and yields the sharper inequality $\\operatorname{rank}_{+}(S^{P,Q}) \\leqslant \\operatorname{xc}(P,Q)$ .",
"Let $P,Q$ be as above and $\\rho \\geqslant 1$ .",
"Then $\\rho Q =\\lbrace x \\in \\mathbb {R}^d \\mid Ax \\leqslant \\rho b\\rbrace $ and the slack matrix of the pair $P,\\rho Q$ is related to the slack matrix of the pair $P,Q$ in the following way: $S^{P,\\rho Q}_{\\mathrm {vertex}}(i,j) = \\rho b_{i} - A_{i} v_{j}= (\\rho - 1) b_{i} + b_{i} - A_{i} v_{j}= S^{P,Q}_{ij} + (\\rho -1) b_{i}, \\\\S^{P,\\rho Q}_{\\mathrm {ray}}(i,j)= S^{P,Q}_{ij}.$ Theorem REF directly yields the following result.",
"Theorem 2 Consider a maximization problem with a linear encoding.",
"Let $P, Q \\subseteq \\mathbb {R}^d$ be the pair of polyhedra associated with the linear encoding, and let $\\rho \\geqslant 1$ .",
"Consider any slack matrix $S^{P,Q}$ for the pair $P, Q$ and the corresponding slack matrix $S^{P,\\rho Q}$ for the pair $P, \\rho Q$ .",
"Then the minimum size of a $\\rho $ -approximate EF of the problem, w.r.t.",
"the considered linear encoding, is $\\operatorname{rank}_{+}(S^{P,\\rho Q}) +\\Theta (1)$ , where the constant is 0 or 1.",
"For a minimization problem, the minimum size of a $\\rho $ -approximate EF is $\\operatorname{rank}_{+}(S^{P,\\rho ^{-1}Q}) + \\Theta (1)$ .",
"Fixing $\\rho \\geqslant 1$ , Theorem REF characterizes the minimum number of inequalities in any LP providing a $\\rho $ -approximation for the problem under consideration.",
"We point out that the theorem directly generalizes to SDPs, by replacing nonnegative rank by PSD rank [22].",
"Here, we focus on LPs and nonnegative rank.",
"As a matter of fact, strong lower bounds on the PSD rank seem to be currently lacking."
],
[
"A Problem with no Polynomial-Size Approximate EF",
"We conclude this section with an example showing the necessity to restrict the set of admissible objective functions rather than allowing every $w \\in \\mathbb {R}^*$ (that is $P = Q$ ).",
"Let $K_n = (V_n,E_n)$ denote the $n$ -vertex complete graph.",
"For a set $X$ of vertices of $K_n$ , we let $\\delta (X)$ denote the set of edges of $K_n$ with one endpoint in $X$ and the other in its complement $\\bar{X}$ .",
"This set $\\delta (X)$ is known as the cut defined by $X$ .",
"For a subset $F$ of edges of $K_n$ , we let $\\chi ^{F} \\in \\mathbb {R}^{E_n}$ denote the characteristic vector (or incidence vector) of $F$ , with $\\chi ^F_e = 1$ if $e \\in F$ and $\\chi ^F_e = 0$ otherwise.",
"The cut polytope $\\operatorname{CUT}(n)$ is defined as the convex hull of the characteristic vectors of all cuts in the complete graph $K_n = (V_n,E_n)$ .",
"That is, $\\operatorname{CUT}(n) \\operatorname{conv}\\left(\\lbrace \\chi ^{\\delta (X)} \\in \\mathbb {R}^{E_n} \\mid X \\subseteq V_n\\rbrace \\right).$ A related object is the cut cone, defined as the cone generated by the cut-vectors $\\chi ^{\\delta (X)}$ : $\\operatorname{CUT-CONE}(n) \\operatorname{cone}\\left(\\lbrace \\chi ^{\\delta (X)} \\in \\mathbb {R}^{E_n}\\mid X \\subseteq V_n\\rbrace \\right).$ Consider the maximum cut problem (Max CUT) with arbitrary weights, and its usual linear encoding.",
"With this encoding we have $P = Q = \\operatorname{CUT}(n)$ .",
"Our next result states that this problem has no $\\rho $ -approximate EF, whatever $\\rho \\geqslant 1$ is.",
"Intuitively, this phenomenon stems from the fact that, because $\\mathbf {0}$ is a vertex of the cut polytope, every approximate EF necessarily `captures' all facets of the cut polytope incident to $\\mathbf {0}$ (see Figure REF ).",
"These facets define the cut cone, which turns out to have high extension complexity.",
"Although this follows rather easily from ideas of [18], we include a proof here for completeness.",
"Figure: CUT(3)\\operatorname{CUT}(3) and a dilate ρCUT(3)\\rho \\operatorname{CUT}(3) for ρ=1.5\\rho = 1.5.Proposition 3 For every $\\rho \\geqslant 1$ , every $\\rho $ -approximate EF of the Max CUT problem with arbitrary weights has size $2^{\\Omega (n)}$ .",
"More precisely, disregarding the value of $\\rho \\geqslant 1$ , we have $\\operatorname{xc}(\\operatorname{CUT}(n),\\rho \\operatorname{CUT}(n)) = 2^{\\Omega (n)}$ .",
"Let $Ex + Fy = g$ , $y \\geqslant \\mathbf {0}$ denote a minimum size $\\rho $ -approximate EF of $\\operatorname{CUT}(n)$ .",
"We claim that $Ex + Fy - \\lambda g = \\mathbf {0},\\ y \\geqslant \\mathbf {0},\\ \\lambda \\geqslant 0$ is an EF of the cut cone.",
"Let $K$ be the polyhedron obtained by projecting the set of solutions of (REF ) into $x$ -space.",
"Clearly, $K$ is a cone containing all the cut-vectors $\\chi ^{\\delta (X)}$ , from which we get that $\\operatorname{CUT-CONE}(n) \\subseteq K$ .",
"Now take any point $(x,y,\\lambda )$ satisfying (REF ).",
"If $\\lambda = 0$ then necessarily $x = \\mathbf {0}$ because $Ex + Fy = \\mathbf {0}$ , $y \\geqslant \\mathbf {0}$ defines the recession cone of a polyhedron that projects into $\\rho \\operatorname{CUT}(n)$ , which is bounded.",
"In this case we have $x = \\mathbf {0} \\in \\operatorname{CUT-CONE}(n)$ .",
"Assume that $\\lambda > 0$ .",
"Then $E\\lambda ^{-1}x + F\\lambda ^{-1}y = g$ and $\\lambda ^{-1}y \\geqslant \\mathbf {0}$ which implies that $\\lambda ^{-1}x$ is in $\\rho \\operatorname{CUT}(n)$ .",
"Thus $\\rho ^{-1}\\lambda ^{-1}x$ is in $\\operatorname{CUT}(n)$ and $x$ is thus a positive combination of cut-vectors, hence $x \\in \\operatorname{CUT-CONE}(n)$ .",
"This yields $K \\subseteq \\operatorname{CUT-CONE}(n)$ .",
"In conclusion, $K = \\operatorname{CUT-CONE}(n)$ and (REF ) is an EF of the cut cone.",
"The size of this EF is at most $r + 1$ , where $r$ denotes the size of the given $\\rho $ -approximate EF of $\\operatorname{CUT}(n)$ .",
"Thus $\\operatorname{xc}(\\operatorname{CUT-CONE}(n)) \\leqslant r +1$ .",
"By using the correlation mapping (see [34]), the cut cone has the same extension complexity as its corresponding correlation cone, defined as $\\operatorname{COR-CONE}(n-1) \\operatorname{cone}\\left(\\left\\lbrace \\binom{b_0}{b}\\binom{b_0}{b}^\\intercal \\,|\\,b_0 \\in \\lbrace 0,1\\rbrace , b \\in \\lbrace 0,1\\rbrace ^{n-2}\\right\\rbrace \\right).$ We claim that the unique disjointness matrix on $[n-2]$ can be embedded in a slack matrix of $\\operatorname{COR-CONE}(n-1)$ .",
"To prove this, consider the $(n-1) \\times (n-1)$ rank-1 positive semidefinite matrices $T_a \\binom{-1}{a}\\binom{-1}{a}^\\intercal \\qquad \\text{and}\\qquad U^b \\binom{1}{b}\\binom{1}{b}^\\intercal $ where $a, b \\in \\lbrace 0,1\\rbrace ^{n-2}$ .",
"The Frobenius inner product $\\langle T_a,z \\rangle \\geqslant 0$ of $T_a$ with any correlation matrix $z=\\binom{b_0}{b}\\binom{b_0}{b}^\\intercal $ is nonnegative because both matrices are positive semidefinite.",
"Thus $\\langle T_a,z \\rangle \\geqslant 0$ is valid for all points $z \\in \\operatorname{COR-CONE}(n-1)$ , for all $a \\in \\lbrace 0,1\\rbrace ^{n-2}$ .",
"Moreover, $\\langle T_a,U^b \\rangle = (1 - a^\\intercal b)^2$ for all $a, b \\in \\lbrace 0,1\\rbrace ^{n-2}$ and thus $\\langle T_a,U^b \\rangle = \\text{UDISJ}(a,b)$ provided $a^\\intercal b \\in \\lbrace 0,1\\rbrace $ .",
"From what precedes, the slack of correlation matrix $U^b$ with respect to the valid inequality $\\langle T_a,z \\rangle \\geqslant 0$ is $\\text{UDISJ}(a,b)$ provided $a^\\intercal b \\in \\lbrace 0,1\\rbrace $ .",
"Therefore, $\\operatorname{COR-CONE}(n-1)$ has a slack matrix that contains UDISJ on $[n-2]$ .",
"Because the nonnegative rank of any matrix containing UDISJ is $2^{\\Omega (n)}$ (this follows from [38], see [18]), we conclude that the nonnegative rank of some slack matrix of $\\operatorname{COR-CONE}(n-1)$ is $2^{\\Omega (n)}$ .",
"From Theorem REF applied to $P = Q = \\operatorname{COR-CONE}(n-1)$ , it follows that $\\operatorname{xc}(\\operatorname{COR-CONE}(n-1))= 2^{\\Omega (n)}$ .",
"Thus we get $r + 1 \\geqslant \\operatorname{xc}(\\operatorname{CUT-CONE}(n)) = \\operatorname{xc}(\\operatorname{COR-CONE}(n-1)) = 2^{\\Omega (n)},$ from which we obtain $r = 2^{\\Omega (n)}$ .",
"The result then follows immediately."
],
[
"Extension of Razborov's Lemma and Shifts of Unique Disjointness",
"In the first subsection we generalize Razborov's famous lemma on the disjointness problem (see [38] or [30] for the original version).",
"In the next subsection we apply it to shift the UDISJ matrix without significantly decreasing its nonnegative rank, which will be used in later sections to obtain lower bounds on approximate extended formulations.",
"The main improvements to Razborov's lemma are threefold: the dependence on the error parameter $\\epsilon $ is made explicit; better analytical estimations are employed to improve overall strength of the statement; probabilities are generalized to expected values to homogenize the proof and yield a stronger lemma.",
"Extension of Razborov's Rectangle Corruption Lemma Suppose that $n \\equiv 3 \\pmod {4}$ and let $\\ell &\\frac{n+1}{4},\\\\A &\\lbrace (a,b) \\in 2^{[n]} \\times 2^{[n]} \\mid |a| = |b| = \\ell ,\\ \\left|a \\cap b\\right| = 0\\rbrace ,\\\\B &\\lbrace (a,b) \\in 2^{[n]} \\times 2^{[n]} \\mid |a| = |b| = \\ell ,\\ \\left|a \\cap b\\right| = 1\\rbrace .$ Thus $A$ is the set of disjoint pairs of $\\ell $ -subsets and $B$ is the set of barely intersecting pairs of $\\ell $ -subsets.",
"Furthermore, let $\\mu $ be any distribution on pairs $(a,b)$ of subsets of $[n]$ that is supported on $A \\cup B$ and uniform when conditioned to either $A$ or $B$ .",
"Lemma 4 Let $n$ , $\\ell $ , $A$ , $B$ and $\\mu $ be as above.",
"For every nonnegative functions $f$ and $g$ defined on $2^{[n]} \\times 2^{[n]}$ we introduce a random variable $X f(a)g(b)$ .",
"Then for every $0 < \\epsilon < 1$ : $(1 - \\epsilon ) \\operatorname{\\mathbb {E}}\\left[X\\,|\\,A\\right] - \\operatorname{\\mathbb {E}}\\left[X\\,|\\,B\\right] \\leqslant \\left\\Vert X \\upharpoonright (A \\cup B)\\right\\Vert _{\\infty } 2^{-\\frac{\\epsilon ^{2}}{16 \\ln 2} \\ell + O(\\log \\ell )},$ where the constant in the $O(\\log \\ell )$ is absolute, and $X \\upharpoonright (A \\cup B)$ denotes the restriction of $X$ to $A \\cup B$ .",
"Let us write $I_{C}$ for the indicator of an event $C$ .",
"In case $f$ and $g$ are both binary, $X$ is the indicator of a rectangle $R$ , that is $X = I_R$ , and (REF ) becomes $(1 - \\epsilon ) \\operatorname{\\mathbb {P}}\\left[R\\,|\\,A\\right] - \\operatorname{\\mathbb {P}}\\left[R\\,|\\,B\\right]\\leqslant 2^{-\\frac{\\epsilon ^{2}}{16 \\ln 2} \\ell + O(\\log \\ell )},$ which is a strengthened version of Razborov's original lemma.",
"For concreteness, the reader might find it helpful to imagine that $X$ is the indicator of a rectangle in the proof below.",
"Our proof is inspired by the version in [30] and we adopt similar notations.",
"[Proof of Lemma REF ] The proof is in four main steps.",
"Step 1: Expressing $\\operatorname{\\mathbb {E}}\\left[X\\,|\\,A\\right]$ and $\\operatorname{\\mathbb {E}}\\left[X\\,|\\,B\\right]$ in an alternative framework.",
"The statement of the lemma does not depend on the actual probabilities of $A$ and $B$ , hence for convenience, we fix them as $\\operatorname{\\mathbb {P}}\\left[A\\right] = \\frac{3}{4} \\qquad \\text{and} \\qquad \\operatorname{\\mathbb {P}}\\left[B\\right] = \\frac{1}{4}.$ This brings the advantage of the following alternative description of $\\mu $ .",
"Let $T = (T_1,T_2,\\lbrace i\\rbrace )$ be a uniformly chosen partition of $[n]$ into two subsets $T_{1}$ , $T_{2}$ with $2 \\ell - 1$ elements each and one singleton $\\lbrace i\\rbrace $ .",
"Given $T$ we choose $a$ as a uniform $\\ell $ -subset of $T_1 \\cup \\lbrace i\\rbrace = [n] \\setminus T_2$ and $b$ as a uniform $\\ell $ -subset of $T_2 \\cup \\lbrace i\\rbrace = [n]\\setminus T_1$ , independently.",
"This defines a distribution $\\mu $ that is supported on $A \\cup B$ , uniform when conditioned to either $A$ or $B$ and satisfies $\\operatorname{\\mathbb {P}}\\left[[\\right]T]{B} = \\operatorname{\\mathbb {P}}\\left[[\\right]T]{i \\in a, i \\in b}= \\operatorname{\\mathbb {P}}\\left[[\\right]T]{i \\in a} \\operatorname{\\mathbb {P}}\\left[[\\right]T]{i \\in b} = (1/2)^2 = 1/4$ and thus $\\operatorname{\\mathbb {P}}\\left[[\\right]T]{A} = 1 - 1/4 = 3/4$ .",
"In particular, $\\operatorname{\\mathbb {P}}\\left[A\\right] = 3/4$ and $\\operatorname{\\mathbb {P}}\\left[B\\right] = 1/4$ , as required.",
"We begin by rewriting $\\operatorname{\\mathbb {E}}\\left[X\\,|\\,B\\right]$ and then $\\operatorname{\\mathbb {E}}\\left[X\\,|\\,A\\right]$ in terms of the following functions of $T$ : $\\operatorname{Row}_{0}(T) &\\operatorname{\\mathbb {E}}\\left[f(a)\\,|\\,T, i \\notin a\\right], &\\operatorname{Row}_{1}(T) &\\operatorname{\\mathbb {E}}\\left[f(a)\\,|\\,T, i \\in a\\right], \\\\\\operatorname{Col}_{0}(T) &\\operatorname{\\mathbb {E}}\\left[g(b)\\,|\\,T, i \\notin b\\right], &\\operatorname{Col}_{1}(T) &\\operatorname{\\mathbb {E}}\\left[g(b)\\,|\\,T, i \\in b\\right].$ We note the following nice interpretation of $\\operatorname{Row}_{0}(T) + \\operatorname{Row}_{1}(T)$ and $\\operatorname{Col}_{0}(T) + \\operatorname{Col}_{1}(T)$ , that we will use at the end of the proof: $\\operatorname{\\mathbb {E}}\\left[f(a)\\,|\\,T\\right] &\\begin{aligned}[t]&=\\underbrace{\\operatorname{\\mathbb {E}}\\left[f(a)\\,|\\,T, i \\in a\\right]}_{\\operatorname{Row}_{1}(T)}\\cdot \\underbrace{\\operatorname{\\mathbb {P}}\\left[i \\in a\\,|\\,T\\right]}_{1/2}+\\underbrace{\\operatorname{\\mathbb {E}}\\left[f(a)\\,|\\,T, i \\notin a\\right]}_{\\operatorname{Row}_{0}(T)}\\cdot \\underbrace{\\operatorname{\\mathbb {P}}\\left[i \\notin a\\,|\\,T\\right]}_{1/2}\\\\&= \\frac{\\operatorname{Row}_{0}(T) + \\operatorname{Row}_{1}(T)}{2},\\end{aligned}\\\\\\operatorname{\\mathbb {E}}\\left[g(b)\\,|\\,T\\right] &= \\frac{\\operatorname{Col}_{0}(T) + \\operatorname{Col}_{1}(T)}{2}.$ Note that: the distribution of $(a,b)$ conditioned on a given $T$ is a product distribution (this local independence property is the main reason why we reinterpret the distribution $\\mu $ ); the marginal distributions of $a$ conditioned on $(T, i \\in a, i \\in b)$ and $(T, i \\in a)$ are the same (and similarly for $b$ , we can remove the condition $i \\in a$ ).",
"From these facts, we get $\\begin{aligned}\\operatorname{\\mathbb {E}}\\left[X\\,|\\,B\\right]&= \\operatorname{\\mathbb {E}}\\left[f(a)g(b)\\,|\\,i \\in a, i \\in b\\right]\\\\&= \\operatorname{\\mathbb {E}}\\left[\\operatorname{\\mathbb {E}}\\left[f(a)g(b)\\,|\\,T, i \\in a, i \\in b\\right]\\right]\\\\&= \\operatorname{\\mathbb {E}}\\left[\\operatorname{\\mathbb {E}}\\left[f(a)\\,|\\,T, i \\in a, i \\in b\\right] \\operatorname{\\mathbb {E}}\\left[g(b)\\,|\\,T, i \\in a, i \\in b\\right]\\right]\\\\&= \\operatorname{\\mathbb {E}}\\left[\\operatorname{\\mathbb {E}}\\left[f(a)\\,|\\,T, i \\in a\\right] \\operatorname{\\mathbb {E}}\\left[g(b)\\,|\\,T, i \\in b\\right]\\right]\\\\&= \\operatorname{\\mathbb {E}}\\left[\\operatorname{Row}_1(T) \\operatorname{Col}_1(T)\\right].\\end{aligned}$ By similar arguments, we find $\\operatorname{\\mathbb {E}}\\left[X\\,|\\,A\\right]&= \\frac{1}{3} \\operatorname{\\mathbb {E}}\\left[f(a)g(b)\\,|\\,i \\notin a, i \\notin b\\right] +\\frac{1}{3} \\operatorname{\\mathbb {E}}\\left[f(a)g(b)\\,|\\,i \\in a, i \\notin b\\right] +\\frac{1}{3} \\operatorname{\\mathbb {E}}\\left[f(a)g(b)\\,|\\,i \\notin a, i \\in b\\right]\\\\&= \\frac{1}{3} \\operatorname{\\mathbb {E}}\\left[\\operatorname{Row}_0(T) \\operatorname{Col}_0(T)\\right] +\\frac{1}{3} \\operatorname{\\mathbb {E}}\\left[\\operatorname{Row}_1(T) \\operatorname{Col}_0(T)\\right] +\\frac{1}{3} \\operatorname{\\mathbb {E}}\\left[\\operatorname{Row}_0(T) \\operatorname{Col}_1(T)\\right].$ Pick a $(2\\ell -1)$ -subset $T_2$ of $[n]$ , that we consider fixed for the time being.",
"The marginal distributions of $a$ conditioned on the events $T_2$ , $(T_2, i \\in a)$ and $(T_2, i \\notin a)$ are the same, namely, the uniform distribution on the $\\ell $ -subsets of $[n] \\setminus T_2$ .",
"(Note that fixing $T_2$ does not fix $i$ , which could be any element of $[n] \\setminus T_2$ .)",
"Thus, we get $\\operatorname{\\mathbb {E}}\\left[f(a)\\,|\\,T_2, i \\notin a\\right] = \\operatorname{\\mathbb {E}}\\left[f(a)\\,|\\,T_2, i \\in a\\right] = \\operatorname{\\mathbb {E}}\\left[f(a)\\,|\\,T_2\\right].$ On the other hand, we have $\\begin{aligned}\\operatorname{\\mathbb {E}}\\left[\\operatorname{Row}_0(T)\\,|\\,T_2\\right]& = \\operatorname{\\mathbb {E}}\\left[\\operatorname{\\mathbb {E}}\\left[f(a)\\,|\\,T, i \\notin a\\right]\\,|\\,T_2\\right]\\\\& = \\operatorname{\\mathbb {E}}\\left[\\frac{\\operatorname{\\mathbb {E}}\\left[f(a) I_{i \\notin a}\\,|\\,T\\right]}{\\operatorname{\\mathbb {P}}\\left[i \\notin a\\,|\\,T\\right]}\\,|\\,T_2\\right]\\\\& = 2 \\operatorname{\\mathbb {E}}\\left[f(a) I_{i \\notin a}\\,|\\,T_2\\right]\\\\& = \\operatorname{\\mathbb {E}}\\left[f(a)\\,|\\,T_2, i \\notin a\\right]\\end{aligned}$ and similarly $\\operatorname{\\mathbb {E}}\\left[\\operatorname{Row}_1(T)\\,|\\,T_2\\right] = \\operatorname{\\mathbb {E}}\\left[f(a)\\,|\\,T_2, i \\in a\\right].$ From (REF ), we conclude $\\operatorname{\\mathbb {E}}\\left[ \\operatorname{Row}_{0}(T) \\,|\\,T_{2}\\right] = \\operatorname{\\mathbb {E}}\\left[ \\operatorname{Row}_{1}(T) \\,|\\,T_{2}\\right].$ Therefore (letting $T_2$ vary), $\\operatorname{\\mathbb {E}}\\left[\\operatorname{Row}_1(T) \\operatorname{Col}_0(T)\\right]& = \\operatorname{\\mathbb {E}}\\left[\\operatorname{\\mathbb {E}}\\left[\\operatorname{Row}_1(T) \\operatorname{Col}_0(T)\\,|\\,T_2\\right]\\right]\\\\& = \\operatorname{\\mathbb {E}}\\left[\\operatorname{\\mathbb {E}}\\left[\\operatorname{Row}_1(T)\\,|\\,T_2\\right] \\operatorname{Col}_0(T)\\right]\\\\& = \\operatorname{\\mathbb {E}}\\left[\\operatorname{\\mathbb {E}}\\left[\\operatorname{Row}_0(T)\\,|\\,T_2\\right] \\operatorname{Col}_0(T)\\right]\\\\& = \\operatorname{\\mathbb {E}}\\left[\\operatorname{\\mathbb {E}}\\left[\\operatorname{Row}_0(T)\\operatorname{Col}_0(T)\\,|\\,T_2\\right]\\right]\\\\& = \\operatorname{\\mathbb {E}}\\left[\\operatorname{Row}_0(T) \\operatorname{Col}_0(T)\\right].$ The second and fourth equalities above are due to the fact that $\\operatorname{Col}_0(T)$ is constant when $T_2$ is fixed.",
"This is because $\\operatorname{Col}_0(T) = \\operatorname{\\mathbb {E}}\\left[g(b)\\,|\\,T, i \\notin b\\right]$ depends only on $T_2$ , as the marginal distribution of $b$ given $(T, i \\notin b)$ is uniform on the $\\ell $ -subsets of $T_2$ .",
"Exchanging the roles of rows and columns, we have $\\operatorname{\\mathbb {E}}\\left[\\operatorname{Row}_1(T) \\operatorname{Col}_0(T)\\right] = \\operatorname{\\mathbb {E}}\\left[\\operatorname{Row}_0(T) \\operatorname{Col}_0(T)\\right].$ In conclusion, we find the following simple expression for $\\operatorname{\\mathbb {E}}\\left[X\\,|\\,A\\right]$ : $\\operatorname{\\mathbb {E}}\\left[X\\,|\\,A\\right] = \\operatorname{\\mathbb {E}}\\left[\\operatorname{Row}_0(T) \\operatorname{Col}_0(T)\\right].$ Step 2: Estimation of $\\operatorname{\\mathbb {E}}\\left[X\\,|\\,A\\right] - \\operatorname{\\mathbb {E}}\\left[X\\,|\\,B\\right]$ .",
"Via obvious estimates: $\\begin{aligned}\\operatorname{Row}_0(T) \\operatorname{Col}_0(T) - \\operatorname{Row}_1(T) \\operatorname{Col}_1(T)\\\\&\\leqslant \\operatorname{Row}_0(T) \\operatorname{Col}_0(T) - \\min \\lbrace \\operatorname{Row}_{0}(T), \\operatorname{Row}_{1}(T)\\rbrace \\cdot \\min \\lbrace \\operatorname{Col}_{0}(T), \\operatorname{Col}_{1}(T)\\rbrace \\\\&=\\begin{aligned}[t]&\\operatorname{Row}_{0}(T) (\\operatorname{Col}_{0}(T) - \\min \\lbrace \\operatorname{Col}_{0}(t), \\operatorname{Col}_{1}(T)\\rbrace )\\\\&+ (\\operatorname{Row}_{0}(T) - \\min \\lbrace \\operatorname{Row}_{0}(t), \\operatorname{Row}_1(T)\\rbrace )\\min \\lbrace \\operatorname{Col}_{0}(t), \\operatorname{Col}_1(T)\\end{aligned}\\\\&\\leqslant \\operatorname{Row}_0(T) |\\operatorname{Col}_0(T) - \\operatorname{Col}_1(T)| + |\\operatorname{Row}_0(T) - \\operatorname{Row}_1(T)| \\operatorname{Col}_0(T).\\end{aligned}$ This argument is depicted on Figure REF .",
"Figure: The estimation of Row 0 (T)Col 0 (T)-Row 1 (T)Col 1 (T)\\operatorname{Row}_0(T) \\operatorname{Col}_0(T) - \\operatorname{Row}_1(T) \\operatorname{Col}_1(T).In Step 3 below, we will define two events, $\\operatorname{row-big}(T)$ and $\\operatorname{column-big}(T)$ .",
"The event $\\operatorname{small}(T)$ holds if and only if not both of $\\operatorname{row-big}(T)$ and $\\operatorname{column-big}(T)$ hold.",
"Thus $1 = I_{\\operatorname{row-big}(T) \\cap \\operatorname{column-big}(T)} + I_{\\operatorname{small}(T)}.$ From (REF ), $&(\\operatorname{Row}_0(T) \\operatorname{Col}_0(T) - \\operatorname{Row}_1(T) \\operatorname{Col}_1(T)) \\cdot I_{\\operatorname{row-big}(T) \\cap \\operatorname{column-big}(T)}\\\\&\\leqslant (\\operatorname{Row}_0(T) |\\operatorname{Col}_0(T) - \\operatorname{Col}_1(T)| + |\\operatorname{Row}_0(T) - \\operatorname{Row}_1(T)| \\operatorname{Col}_0(T)) \\cdot I_{\\operatorname{row-big}(T) \\cap \\operatorname{column-big}(T)}\\\\&\\leqslant \\operatorname{Row}_0(T) |\\operatorname{Col}_0(T) - \\operatorname{Col}_1(T)| \\cdot I_{\\operatorname{column-big}(T)} + |\\operatorname{Row}_0(T) - \\operatorname{Row}_1(T)| \\operatorname{Col}_0(T) \\cdot I_{\\operatorname{row-big}(T)}.$ Moreover, we obviously have $(\\operatorname{Row}_0(T) \\operatorname{Col}_0(T) - \\operatorname{Row}_1(T) \\operatorname{Col}_1(T)) \\cdot I_{\\operatorname{small}(T)} \\leqslant \\operatorname{Row}_0(T) \\operatorname{Col}_0(T) \\cdot I_{\\operatorname{small}(T)}.$ Below, we will prove $\\operatorname{\\mathbb {E}}\\left[\\operatorname{Row}_0(T) |\\operatorname{Col}_0(T) - \\operatorname{Col}_1(T)| \\cdot I_{\\operatorname{column-big}(T)}\\right]&\\leqslant \\frac{\\epsilon }{2} \\operatorname{\\mathbb {E}}\\left[\\operatorname{Row}_0(T) \\operatorname{Col}_0(T)\\right],\\\\\\operatorname{\\mathbb {E}}\\left[|\\operatorname{Row}_0(T) - \\operatorname{Row}_1(T)| \\operatorname{Col}_0(T) \\cdot I_{\\operatorname{row-big}(T)}\\right]&\\leqslant \\frac{\\epsilon }{2} \\operatorname{\\mathbb {E}}\\left[\\operatorname{Row}_0(T) \\operatorname{Col}_0(T)\\right], \\quad \\text{and}\\\\\\operatorname{\\mathbb {E}}\\left[\\operatorname{Row}_0(T) \\operatorname{Col}_0(T) \\cdot I_{\\operatorname{small}(T)}\\right]&\\leqslant \\left\\Vert X \\upharpoonright (A \\cup B)\\right\\Vert _{\\infty } 2^{-\\frac{\\epsilon ^2}{16 \\ln 2} - O(\\log \\ell )}$ By (REF ), (REF ) and (REF ), these upper bounds imply $&\\operatorname{\\mathbb {E}}\\left[X\\,|\\,A\\right] - \\operatorname{\\mathbb {E}}\\left[X\\,|\\,B\\right]\\\\&= \\operatorname{\\mathbb {E}}\\left[\\operatorname{Row}_0(T) \\operatorname{Col}_0(T) - \\operatorname{Row}_1(T) \\operatorname{Col}_1(T)\\right]\\\\&= \\operatorname{\\mathbb {E}}\\left[(\\operatorname{Row}_0(T) \\operatorname{Col}_0(T) - \\operatorname{Row}_1(T) \\operatorname{Col}_1(T))\\cdot (I_{\\operatorname{row-big}(T) \\cap \\operatorname{column-big}(T)} + I_{\\operatorname{small}(T)})\\right]\\\\&\\leqslant 2 \\frac{\\epsilon }{2} \\operatorname{\\mathbb {E}}\\left[\\operatorname{Row}_0(T) \\operatorname{Col}_0(T)\\right] +\\left\\Vert X \\upharpoonright (A \\cup B)\\right\\Vert _{\\infty } 2^{-\\frac{\\epsilon ^2}{16 \\ln 2} \\ell - O(\\log \\ell )}\\\\&= \\epsilon \\operatorname{\\mathbb {E}}\\left[X\\,|\\,A\\right] +\\left\\Vert X \\upharpoonright (A \\cup B)\\right\\Vert _{\\infty } 2^{-\\frac{\\epsilon ^2}{16 \\ln 2} \\ell - O(\\log \\ell )}$ from which the result clearly follows, by rearranging.",
"Step 3.",
"One-sided error estimation via entropy argument in the “big” case.",
"Let $\\delta > 0$ be a constant to be chosen later.",
"Essentially, $\\delta $ will be the coefficient of $\\ell $ in the exponent.",
"Let $\\operatorname{row-big}(T)$ denote the event $\\operatorname{\\mathbb {E}}\\left[f(a)\\,|\\,T_2\\right] > 2^{- \\delta \\ell -1} \\left\\Vert f \\upharpoonright \\binom{[n] \\setminus T_{2}}{\\ell }\\right\\Vert _{\\infty }$ where $f \\upharpoonright \\binom{[n] \\setminus T_{2}}{\\ell }$ denotes the restriction of $f$ to $\\ell $ -subsets of $[n] \\setminus T_{2}$ .",
"The event $\\operatorname{column-big}(T)$ is defined in a similar way.",
"These events depend only on $T_2$ and $T_1$ , respectively.",
"Let $T_{2}$ be fixed and assume that $\\operatorname{row-big}(T)$ holds.",
"In particular $\\operatorname{\\mathbb {E}}\\left[f(a)\\,|\\,T_2\\right]$ is positive.",
"Because $\\binom{2\\ell - 1}{\\ell -1} = \\binom{2\\ell - 1}{\\ell }$ , the distribution of $a$ given $T_2$ is the same as the distribution of $a$ given $T$ , for every fixed choice of $i$ .",
"Thus, we have $\\operatorname{\\mathbb {E}}\\left[f(a)\\,|\\,T\\right] =\\operatorname{\\mathbb {E}}\\left[f(a)\\,|\\,T_{2}\\right] =\\sum _{\\begin{array}{c}x \\subseteq [n] \\setminus T_{2} \\\\ \\left|x\\right| = \\ell \\end{array}}\\frac{1}{\\binom{2\\ell }{\\ell }} f(x).$ (This holds when $f(a)$ is replaced by any function of $a$ .)",
"We define $s$ as a random $\\ell $ -subset of $[n] \\setminus T_{2}$ with distribution $\\operatorname{\\mathbb {P}}\\left[s = x\\,|\\,T_{2}\\right]= \\frac{f(x)}{\\binom{2\\ell }{\\ell } \\operatorname{\\mathbb {E}}\\left[f(a)\\,|\\,T_{2}\\right]}= \\frac{f(x)}{\\sum _{\\begin{array}{c}y \\subseteq [n] \\setminus T_{2} \\\\ \\left|y\\right| = \\ell \\end{array}} f(y)}\\leqslant \\frac{2^{\\delta \\ell + 1}}{\\binom{2\\ell }{\\ell }}.$ Let us introduce the shorthand notation $\\lambda \\operatorname{\\mathbb {P}}\\left[i \\in s\\,|\\,T_2\\right]$ .",
"Then $\\lambda = \\frac{\\sum _{\\begin{array}{c}x \\subseteq [n] \\setminus T_{2}\\\\ \\left|x\\right| = \\ell ,\\ x \\ni i\\end{array}} f(x)}{\\sum _{\\begin{array}{c}y \\subseteq [n] \\setminus T_{2} \\\\ \\left|y\\right| = \\ell \\end{array}} f(y)}= \\frac{\\frac{1}{\\binom{2\\ell }{\\ell }}\\sum _{\\begin{array}{c}x \\subseteq [n] \\setminus T_{2}\\\\ \\left|x\\right| = \\ell ,\\ x \\ni i\\end{array}} f(x)}{\\frac{1}{\\binom{2\\ell }{\\ell }}\\sum _{\\begin{array}{c}y \\subseteq [n] \\setminus T_{2} \\\\ \\left|y\\right| = \\ell \\end{array}} f(y)}= \\frac{\\operatorname{\\mathbb {E}}\\left[f(a) I_{i \\in a}\\,|\\,T\\right]}{\\operatorname{\\mathbb {E}}\\left[f(a)\\,|\\,T_2\\right]}.$ Hence, $&\\operatorname{Row}_{1} (T)= 2 \\operatorname{\\mathbb {E}}\\left[f(a) I_{i \\in a}\\,|\\,T\\right]= 2 \\operatorname{\\mathbb {E}}\\left[f(a)\\,|\\,T_{2}\\right] \\cdot \\operatorname{\\mathbb {P}}\\left[i \\in s\\,|\\,T_2\\right]= 2 \\lambda \\operatorname{\\mathbb {E}}\\left[f(a)\\,|\\,T_{2}\\right],\\\\[1ex]&\\operatorname{Row}_{0} (T)= 2 \\operatorname{\\mathbb {E}}\\left[f(a) I_{i \\notin a}\\,|\\,T\\right]= 2 \\operatorname{\\mathbb {E}}\\left[f(a)\\,|\\,T_{2}\\right] \\cdot \\operatorname{\\mathbb {P}}\\left[i \\notin s\\,|\\,T_2\\right]= 2 (1 - \\lambda ) \\operatorname{\\mathbb {E}}\\left[f(a)\\,|\\,T_{2}\\right].$ We now estimate the entropy of $s$ .",
"On the one hand, by subadditivity of the entropy, we get the following upperbound on $H\\left(s\\,|\\,T_{2}\\right)$ : $H\\left(s\\,|\\,T_{2}\\right)\\leqslant \\sum _{j \\in [n] \\setminus T_{2}} H\\left(I_{j \\in s}\\,|\\,T_{2}\\right)= 2 \\ell \\operatorname{\\mathbb {E}}\\left[H\\left(\\lambda \\right)\\,|\\,T_{2}\\right].$ In this last equation, $H\\left(\\lambda \\right)$ denotes the binary entropy of $\\lambda $ .",
"On the other hand, we get a lower bound on $H\\left(s\\,|\\,T_2\\right)$ from our upper bound on the distribution of $s$ (which induces “flatness” of the distribution): $\\begin{split}H\\left(s\\,|\\,T_{2}\\right) &= \\sum _{x}\\operatorname{\\mathbb {P}}\\left[s = x\\,|\\,T_{2}\\right] \\log \\frac{1}{\\operatorname{\\mathbb {P}}\\left[s = x\\,|\\,T_{2}\\right]}\\\\&\\geqslant \\sum _{x}\\operatorname{\\mathbb {P}}\\left[s = x\\,|\\,T_{2}\\right] \\log \\frac{\\binom{2\\ell }{\\ell }}{2^{\\delta \\ell + 1}}= \\log \\frac{\\binom{2\\ell }{\\ell }}{2^{\\delta \\ell + 1}}= 2 \\ell \\left( 1 - \\frac{\\delta }{2}- O \\genfrac(){}{}{\\log \\ell }{\\ell } \\right).\\end{split}$ This implies $\\frac{\\delta }{2} + O \\genfrac(){}{}{\\log \\ell }{\\ell }\\geqslant \\operatorname{\\mathbb {E}}\\left[1 - H\\left(\\lambda \\right)\\,|\\,T_{2}\\right].$ To estimate this expression, we use the Taylor expansion of the binary entropy function at $1/2$ : $1 - H\\left(x\\right)\\geqslant \\frac{{\\left( 1 - 2 x \\right)}^{2}}{2 \\ln 2}.$ Hence (REF ) yields $\\frac{\\delta }{2} + O \\genfrac(){}{}{\\log \\ell }{\\ell }\\geqslant \\frac{\\operatorname{\\mathbb {E}}\\left[{\\left(1 - 2 \\lambda \\right)}^{2}\\,|\\,T_{2}\\right]}{2 \\ln 2}\\geqslant \\frac{{\\left(\\operatorname{\\mathbb {E}}\\left[ \\left|1 - 2 \\lambda \\right| \\,|\\,T_{2}\\right]\\right)}^{2}}{2 \\ln 2}.$ From (REF ), (REF ) we have $\\operatorname{\\mathbb {E}}\\left[f(a)\\,|\\,T_2\\right] = \\operatorname{\\mathbb {E}}\\left[\\operatorname{Row}_0(T)\\,|\\,T_2\\right]$ .",
"Using () and (REF ), we derive $\\begin{aligned}\\operatorname{\\mathbb {E}}\\left[\\left|\\operatorname{Row}_0(T) - \\operatorname{Row}_{1}(T)\\right|\\,|\\,T_{2}\\right]&= \\operatorname{\\mathbb {E}}\\left[|2(1-\\lambda )\\operatorname{\\mathbb {E}}\\left[f(a)\\,|\\,T_2\\right] - 2 \\lambda \\operatorname{\\mathbb {E}}\\left[f(a)\\,|\\,T_2\\right]|\\,|\\,T_2\\right]\\\\&= 2 \\operatorname{\\mathbb {E}}\\left[|1 - 2\\lambda |\\,|\\,T_2\\right] \\operatorname{\\mathbb {E}}\\left[f(a)\\,|\\,T_2\\right]\\\\& \\leqslant 2 \\sqrt{\\delta ^{\\prime }} \\operatorname{\\mathbb {E}}\\left[\\operatorname{Row}_0(T)\\,|\\,T_2\\right].\\end{aligned}$ with $\\delta ^{\\prime } \\left(\\delta + O \\genfrac(){}{}{\\log \\ell }{\\ell }\\right) \\ln 2.$ We now globalize to prove (): $&\\operatorname{\\mathbb {E}}\\left[|\\operatorname{Row}_0(T) - \\operatorname{Row}_1(T)| \\operatorname{Col}_0(T) I_{\\operatorname{row-big}(T)}\\right]\\\\&= \\operatorname{\\mathbb {E}}\\left[\\operatorname{\\mathbb {E}}\\left[|\\operatorname{Row}_0(T) - \\operatorname{Row}_1(T)| \\operatorname{Col}_0(T) I_{\\operatorname{row-big}(T)}\\,|\\,T_2\\right]\\right]\\\\&= \\operatorname{\\mathbb {E}}\\left[\\operatorname{\\mathbb {E}}\\left[|\\operatorname{Row}_0(T) - \\operatorname{Row}_1(T)| I_{\\operatorname{row-big}(T)}\\,|\\,T_2\\right]\\operatorname{Col}_0(T)\\right]\\\\&\\leqslant \\operatorname{\\mathbb {E}}\\left[2 \\sqrt{\\delta ^{\\prime }} \\operatorname{\\mathbb {E}}\\left[\\operatorname{Row}_0(T)\\,|\\,T_2\\right] \\operatorname{Col}_0(T)\\right]\\\\&= 2 \\sqrt{\\delta ^{\\prime }} \\operatorname{\\mathbb {E}}\\left[\\operatorname{Row}_0(T) \\operatorname{Col}_0(T)\\right]$ We require $\\frac{\\epsilon }{2} = 2 \\sqrt{\\delta ^{\\prime }}$ , from which we express $\\delta $ in terms of $\\epsilon $ using (REF ): $\\delta =\\frac{\\delta ^{\\prime }}{\\ln 2} - O \\genfrac(){}{}{\\log \\ell }{\\ell }=\\frac{\\epsilon ^{2}}{16 \\ln 2}- O \\genfrac(){}{}{\\log \\ell }{\\ell }$ This concludes the proof of ().",
"Equation (REF ) follows by exchanging rows and columns.",
"Step 4: Error estimation in the “small” case.",
"Suppose that for some given $T$ , $\\operatorname{small}(T)$ holds because $\\operatorname{row-big}(T)$ does not hold (the argument is similar in case $\\operatorname{column-big}(T)$ does not hold).",
"Then, using (REF ), $\\operatorname{Row}_0(T)\\leqslant \\operatorname{Row}_0(T) + \\operatorname{Row}_1(T)=2 \\operatorname{\\mathbb {E}}\\left[f(a)\\,|\\,T\\right].$ Thus $\\operatorname{Row}_0(T) \\operatorname{Col}_0(T)&\\leqslant 2 \\operatorname{\\mathbb {E}}\\left[f(a)\\,|\\,T\\right] \\cdot \\operatorname{\\mathbb {E}}\\left[g(b)\\,|\\,T, i \\notin b\\right]\\\\&\\leqslant 2^{- \\delta \\ell } \\left\\Vert f(a) \\upharpoonright \\binom{[n] \\setminus T_2}{\\ell }\\right\\Vert _{\\infty } \\cdot \\left\\Vert g(b) \\upharpoonright \\binom{T_2}{\\ell }\\right\\Vert _{\\infty }\\\\&\\leqslant 2^{-\\delta \\ell } \\left\\Vert f(a)g(b) \\upharpoonright (A \\cup B)\\right\\Vert _{\\infty }$ This is easily seen to imply ().",
"Lower Bounds for Shifts of Unique Disjointness Now we apply Lemma REF to show that the nonnegative rank (and hence the communication complexity in expectation) of any shifted version of the unique disjointness matrix remains high.",
"More precisely, let $M \\in \\mathbb {R}_+^{2^n \\times 2^n}$ ; for convenience we index the rows and columns with elements in $\\lbrace 0,1\\rbrace ^n$ .",
"We say that $M$ is a $\\rho $ -extension of UDISJ, if $M_{ab} = \\rho $ whenever $\\left|a \\cap b\\right| = 0$ and $M_{ab} = \\rho - 1$ whenever $\\left|a \\cap b\\right| = 1$ with $a,b \\in \\lbrace 0,1\\rbrace ^n$ .",
"Note that for these pairs $M$ has exclusively positive entries whenever $\\rho >1$ .",
"For $\\rho = 1$ a nonnegative rank of $2^{\\Omega (n)}$ was already shown in [18] via nondeterministic communication complexity.",
"We now extend this result for a wide range of $\\rho $ using Lemma REF .",
"Theorem 5 (Nonnegative rank of UDISJ shifts) Let $M \\in \\mathbb {R}_+^{2^n \\times 2^n}$ be a $\\rho $ -extension of UDISJ as above.",
"If $\\rho $ is a fixed constant, then $\\operatorname{rank}_{+}(M) = 2^{\\Omega (n)}$ .",
"If $\\rho = O(n^\\beta )$ for some constant $\\beta < 1/2$ then $\\operatorname{rank}_{+}(M) = 2^{\\Omega (n^{1 - 2 \\beta })}$ .",
"Without loss of generality, assume $n \\equiv 3 \\pmod {4}$ .",
"Let $r = \\operatorname{rank}_{+}(M)$ .",
"Regarding the $2^n \\times 2^n$ matrix $M$ as a function from $2^{[n]} \\times 2^{[n]}$ to $\\mathbb {R}$ , we can write $M =\\sum _{i=1}^r X_i$ where $X_i(a,b) = f_i(a)g_i(b)$ for some nonnegative functions $f_i$ and $g_i$ defined over $2^{[n]}$ .",
"Then $\\operatorname{\\mathbb {E}}\\left[M\\,|\\,A\\right] = \\rho \\quad \\text{and}\\quad \\operatorname{\\mathbb {E}}\\left[M\\,|\\,B\\right] = \\rho - 1.$ On the other hand, by applying Lemma REF to each $i \\in [r]$ and summing up all equations we find $(1-\\epsilon ) \\operatorname{\\mathbb {E}}\\left[M\\,|\\,A\\right] - \\operatorname{\\mathbb {E}}\\left[M\\,|\\,B\\right]&\\leqslant \\sum _{i=1}^r \\left\\Vert X_i \\upharpoonright (A \\cup B)\\right\\Vert _{\\infty } 2^{-\\frac{\\epsilon ^{2}}{16 \\ln 2} \\ell + O(\\log \\ell )}\\\\&\\leqslant r \\left\\Vert M \\upharpoonright (A \\cup B)\\right\\Vert _{\\infty } 2^{-\\frac{\\epsilon ^{2}}{16 \\ln 2} \\ell + O(\\log \\ell )}$ where $\\ell = \\frac{n+1}{4}$ as before.",
"By plugging in the values of $\\operatorname{\\mathbb {E}}\\left[M\\,|\\,A\\right]$ , $\\operatorname{\\mathbb {E}}\\left[M\\,|\\,B\\right]$ and $\\left\\Vert M \\upharpoonright (A \\cup B)\\right\\Vert _{\\infty }$ , we get $(1-\\epsilon ) \\rho - \\rho + 1&\\leqslant r \\cdot \\rho \\cdot 2^{-\\frac{\\epsilon ^{2}}{16 \\ln 2} \\ell + O(\\log \\ell )},\\multicolumn{2}{l}{\\text{which provides the lower bound}}\\\\r &\\geqslant \\left(\\frac{1}{\\rho } - \\epsilon \\right)2^{\\frac{\\epsilon ^{2}}{16 \\ln 2} \\ell - O(\\log \\ell )}.$ If $\\rho $ is constant, this last expression is $2^{\\Omega (n)}$ provided $\\epsilon $ is chosen sufficiently close to 0.",
"This proves part REF of the theorem.",
"If $\\rho \\leqslant C n^{\\beta }$ for some positive constant $C$ , then we can take $\\epsilon = \\frac{1}{2C n^{\\beta }}$ .",
"Thus $\\frac{1}{\\rho } - \\epsilon \\geqslant \\frac{1}{2C n^{\\beta }}= \\Omega (n^{-\\beta })$ .",
"This leads to the lower bound $r \\geqslant 2^{\\Omega (n^{1 - 2 \\beta })}$ as claimed in part REF .",
"Polyhedral Inapproximability of CLIQUE and SDPs We will now use Theorem REF in combination with Theorem REF to lower bound the sizes of approximate EFs for CLIQUE and some SDPs.",
"First, we pinpoint a pair $P,Q$ of nested polyhedra that will be the source of our polyhedral inapproximability results.",
"Second, we give a faithful linear encoding of CLIQUE and prove strong lower bounds on the sizes of approximate EFs for CLIQUE w.r.t.",
"this encoding.",
"Third, we focus on approximations of SDPs by LPs.",
"A Hard Pair Let $n$ be a positive integer.",
"The correlation polytope $\\operatorname{COR}(n)$ is defined as the convex hull of all the $n \\times n$ rank-1 binary matrices of the form $bb^\\intercal $ where $b \\in \\lbrace 0,1\\rbrace ^n$ .",
"In other words, $\\operatorname{COR}(n) = \\operatorname{conv}\\left(\\left\\lbrace bb^\\intercal \\,|\\,b \\in \\lbrace 0,1\\rbrace ^n\\right\\rbrace \\right).$ This will be our inner polytope $P$ .",
"Next, let $Q = Q(n) \\lbrace x \\in \\mathbb {R}^{n \\times n} \\mid \\langle 2\\operatorname{diag}(a)-aa^\\intercal ,x \\rangle \\leqslant 1,\\ a \\in \\lbrace 0,1\\rbrace ^n\\rbrace ,$ where $\\langle \\cdot ,\\cdot \\rangle $ denotes the Frobenius inner product.",
"This will be our outer polyhedron $Q$ .",
"Then the following is known, see [18].",
"First, $P \\subseteq Q$ .",
"Second, denoting by $S^{P,Q}$ the slack matrix of the pair $P,Q$ , we have $S^{P,Q}_{ab} = (1-a^\\intercal b)^2$ .",
"Thus, for $\\rho \\geqslant 1$ , we have $S^{P,\\rho Q}_{ab}= (1-a^\\intercal b)^2 + \\rho - 1$ .",
"Observe that the matrix $S^{P,\\rho Q}$ is a $\\rho $ -extension of UDISJ and therefore has high nonnegative rank via Theorem REF ; moreover it has positive entries everywhere for $\\rho > 1$ .",
"Together with Theorem REF this implies that every polyhedron sandwiched between $P = \\operatorname{COR}(n)$ and $\\rho Q$ has large extension complexity.",
"We obtain the following theorem.",
"Theorem 6 (Lower bounds for approximate EFs of the hard pair) Let $\\rho \\geqslant 1$ , let $n$ be a positive integer and let $P = \\operatorname{COR}(n)$ , $Q = Q(n)$ be as above.",
"Then the following hold: If $\\rho $ is a fixed constant, then $\\operatorname{xc}(P,\\rho Q) = 2^{\\Omega (n)}$ .",
"If $\\rho = O(n^\\beta )$ for some constant $\\beta < 1/2$ , then $\\operatorname{xc}(P,\\rho Q) = 2^{\\Omega (n^{1 - 2 \\beta })}$ .",
"Polyhedral Inapproximability of CLIQUE We define a natural linear encoding for the maximum clique problem (CLIQUE) as follows.",
"Let $n$ denote the number of vertices of the input graph.",
"We define a $d = n^2$ dimensional encoding.",
"The variables are denoted by $x_{ij}$ for $i, j \\in [n]$ .",
"Thus $x \\in \\mathbb {R}^{n \\times n}$ .",
"The interpretation is that a set of vertices $X$ is encoded by $x_{ij} = 1$ if $i, j \\in X$ and $x_{ij} = 0$ otherwise.",
"Note that $X = \\lbrace i : x_{ii} = 1\\rbrace $ can be recovered from only the diagonal variables.",
"This defines the set $\\mathcal {L} \\subseteq \\lbrace 0,1\\rbrace ^*$ of feasible solutions.",
"Notice that $x \\in \\lbrace 0,1\\rbrace ^{n \\times n}$ is feasible if and only if it is of the form $x = bb^\\intercal $ for some $b \\in \\lbrace 0,1\\rbrace ^n$ , the characteristic vector of $X$ .",
"Thus we have $P = \\operatorname{COR}(n)$ for the inner polytope.",
"The admissible objective functions are chosen as follows to encode the CLIQUE problem for graphs $G$ supported on $[n]$ .",
"Given a graph $G$ such that $V(G) \\subseteq [n]$ , we let $w_{ii} 1$ for $i \\in V(G)$ , $w_{ii} 0$ for $i \\in [n] \\setminus V(G)$ , $w_{ij} = w_{ji} -1$ when $ij$ is a non-edge of $G$ (that is, $i, j \\in V(G)$ , $i \\ne j$ and $ij \\notin E(G)$ ), and $w_{ij} = w_{ji} 0$ otherwise.",
"We denote the resulting weight vector by $w^G$ .",
"Notice that for a graph $G$ with $V(G) = [n]$ , we have $w^G =I - A(\\overline{G})$ where $I$ is the $n \\times n$ identity matrix, $A(\\overline{G})$ is the adjacency matrix of the complement of $G$ .",
"A feasible solution $x = bb^\\intercal \\in \\lbrace 0,1\\rbrace ^{n \\times n}$ maximizes $\\langle w^G,x \\rangle $ only if $b$ is the characteristic vector (or incidence vector) of a clique of $G$ .",
"Indeed, if $b = \\chi ^X$ and $ij$ is a non-edge of $G$ with $i, j \\in X$ then removing $i$ or $j$ from $X$ increases $\\langle w^G,x \\rangle $ .",
"Moreover, the maximum of $\\langle w^G,x \\rangle $ over $x \\in \\lbrace 0,1\\rbrace ^{n \\times n}$ feasible is the clique number $\\omega (G)$ .",
"The admissible objective functions are the ones of the form $w^{G}$ , i.e., $\\mathcal {O} = \\lbrace w^{G} : V(G) \\subseteq [n]\\rbrace $ is the set of admissible functions.",
"Therefore, $(\\mathcal {L},\\mathcal {O})$ defines a valid linear encoding of CLIQUE.",
"We denote the outer convex set of this linear encoding by $Q^{\\mathrm {all}}$ .",
"It is actually the polyhedron defined as $Q^{\\mathrm {all}} = \\lbrace x \\in \\mathbb {R}^{n \\times n} \\mid \\forall $ graphs $G$ s.t.",
"$V(G) \\subseteq [n] : \\langle w^G,x \\rangle \\leqslant \\omega (G),\\ \\forall i \\ne j \\in [n] : x_{ij} \\geqslant 0\\rbrace $ .",
"We will now show that $Q^{\\mathrm {all}} \\subseteq Q$ .",
"Lemma 7 Let $Q^{\\mathrm {all}},Q$ be as above, then $Q^{\\mathrm {all}}\\subseteq Q$ .",
"Let $x \\in Q^{\\mathrm {all}}$ .",
"We want to prove that $x$ satisfies all the constraints defining $Q$ .",
"We show this by restricting to graphs $G$ with $\\omega (G) = 1$ .",
"For a given $a \\in \\lbrace 0,1\\rbrace ^n$ , let $G$ be the graph with $\\chi ^{V(G)} = a$ and $E(G) = \\emptyset $ .",
"Then, $\\langle 2\\operatorname{diag}(a) - aa^T,x \\rangle = \\langle w^G,x \\rangle \\leqslant \\omega (G) = 1.$ The lemma follows.",
"Because $Q^{\\mathrm {all}}$ is contained in the polyhedron $Q$ defined above, every $K$ satisfying $P \\subseteq K \\subseteq \\rho Q^{\\mathrm {all}}$ also satisfies $P \\subseteq K \\subseteq \\rho Q$ .",
"Hence, Theorem REF yields the following result.",
"Theorem 8 (Polyhedral inapproximability of CLIQUE) W.r.t.",
"the linear encoding defined above, CLIQUE has an $O(n^2)$ -size $n$ -approximate EF.",
"Moreover, every $n^{1/2-\\epsilon }$ -approximate EF of CLIQUE has size $2^{\\Omega (n^{2\\epsilon })}$ , for all $0 < \\epsilon < 1/2$ .",
"The $n$ -approximate EF of CLIQUE is trivial: it is defined by the system $\\mathbf {0} \\leqslant x \\leqslant \\mathbf {1}$ , or in slack form $x - y = \\mathbf {0}$ , $x + z = \\mathbf {1}$ , $y \\geqslant \\mathbf {0}$ , $z \\geqslant \\mathbf {0}$ .",
"We claim that this defines a $n$ -approximate EF of CLIQUE of size $2n^2$ .",
"Indeed, letting $K = [0,1]^{n \\times n}$ denote the polytope defined by this EF, we have $P \\subseteq K$ .",
"Moreover, $\\max \\lbrace \\langle w,x \\rangle \\mid x \\in K\\rbrace \\leqslant n \\leqslant n \\cdot \\max \\lbrace \\langle w,x \\rangle \\mid x \\in P\\rbrace $ for all admissible objective functions $w$ of dimension $n \\times n$ with a nonzero diagonal.",
"In case an admissible $w$ has $w_{ii} = 0$ for all $i \\in [n]$ , we have $\\max \\lbrace \\langle w,x \\rangle \\mid x \\in K\\rbrace = 0 = \\max \\lbrace \\langle w,x \\rangle \\mid x \\in P\\rbrace $ .",
"Our claim and the first part of the theorem follows.",
"The second part of the theorem follows directly from Theorem REF and the fact that $Q^{\\mathrm {all}}\\subseteq Q$ .",
"Polyhedral Inapproximability of SDPs In this section we show that there exists a spectrahedron with small semidefinite extension complexity but high approximate extension complexity; i.e., any sufficiently fine polyhedral approximation is large.",
"This indicates that in general it is not possible to approximate SDPs arbitrarily well using small LPs, so that SDPs are indeed a much stronger class of optimization problems.",
"(The situation looks quite different for SOCPs, see [8].)",
"The result follows from Theorem REF and [18].",
"We denote the vector space of all $r \\times r$ symmetric matrices by $\\mathbb {S}^r$ and the cone of all $r \\times r$ symmetric positive semidefinite matrices (shortly, the PSD cone) by $\\mathbb {S}_+^r$ .",
"A semidefinite EF of a convex set $S \\subseteq \\mathbb {R}^d$ is a linear system $\\langle E_i,x \\rangle + \\langle F_i,Y \\rangle = g_i$ ($i \\in [k]$ ), $Y \\in \\mathbb {S}_+^r$ where $E_i \\in \\mathbb {R}^d$ and $F_i \\in \\mathbb {S}^r$ , such that $x \\in S$ if and only if $\\exists Y \\in \\mathbb {S}_+^r$ with $\\langle E_i,x \\rangle + \\langle F_i,Y \\rangle = g_i$ for all $i \\in [m]$ .",
"Thus a convex set admits a semidefinite EF if and only if it is a spectrahedron.",
"The size of the semidefinite EF $\\langle E_i,x \\rangle + \\langle F_i,Y \\rangle = g_i$ ($i \\in [k]$ ), $Y \\in \\mathbb {S}_+^r$ is simply $r$ .",
"The semidefinite extension complexity of a spectrahedron $S \\subseteq \\mathbb {R}^d$ is the minimum size of a semidefinite EF of $S$ .",
"This is denoted by $\\operatorname{xc}_{SDP}(S)$ .",
"A rank-$r$ PSD-factorization of a nonnegative matrix $M \\in \\mathbb {R}^{m \\times n}$ is given by matrices $T_1, \\ldots , T_m \\in \\mathbb {S}_+^r$ and $U^1, \\ldots , U^n \\in \\mathbb {S}_+^r$ , so that $M_{ij} =\\langle T_i,U^j \\rangle $ ; the PSD-rank of $M$ is the smallest $r$ such that there exists such a factorization.",
"Yannakakis's factorization theorem can be generalized to the SDP-case (see [22]), i.e., the semidefinite extension complexity of a pair of polyhedra $P, Q$ equals the PSD-rank of any associated slack matrix, in most cases (e.g., [23] prove the equality under the hypothesis that $Q$ does not contain any line).",
"Let $P = \\operatorname{COR}(n)$ be the correlation polytope and $Q = Q(n) \\subseteq \\mathbb {R}^{n \\times n}$ be the polyhedron defined above in Section REF .",
"Although every polyhedron $K$ sandwiched between $P$ and $Q$ has super-polynomial extension complexity, and by Theorem REF this even applies to polyhedra sandwiched between $P$ and $\\rho Q$ for $\\rho = O(n^{1/2-\\epsilon })$ , there exists a spectrahedron $S$ sandwiched between $P$ and $Q$ with small semidefinite extension complexity.",
"Lemma 9 (Existence of spectrahedron) Let $n$ be a positive integer and let $P = \\operatorname{COR}(n)$ , $Q = Q(n)$ be as above.",
"Then there exists a spectrahedron $S$ in $\\mathbb {R}^{n \\times n}$ with $P \\subseteq S \\subseteq Q$ and $\\operatorname{xc}_{SDP}(S) \\leqslant n+1$ .",
"For $a,b \\in \\lbrace 0,1\\rbrace ^{n}$ , the matrices $T^{a},U_b \\in \\mathbb {S}_+^{n+1}$ defined in (REF ) satisfy $\\langle T^{a},U_{b} \\rangle = (1 - a^\\intercal b)^2$ .",
"Let $M = M(n) \\in \\mathbb {R}^{2^n \\times 2^n}$ be the matrix defined as $M_{ab} = (1 - a^\\intercal b)^2$ .",
"The matrix $M$ is an $O(n^2)$ -rank nonnegative matrix extending the UDISJ matrix, and also the slack matrix of the pair $P$ , $Q$ .",
"Then $M_{ab} = \\langle T_a,U^b \\rangle $ is a rank-$(n+1)$ PSD-factorization of $M$ .",
"Consider the system $\\langle 2\\operatorname{diag}(a)-aa^\\intercal ,x \\rangle + \\langle T_a,Y \\rangle = 1\\quad (a \\in \\lbrace 0,1\\rbrace ^n), \\qquad Y \\in \\mathbb {S}_+^{n+1}$ and $S$ be the projection to $\\mathbb {R}^{n \\times n}$ of the pairs $(x,Y) \\in \\mathbb {R}^{n \\times n} \\times \\mathbb {S}^{n+1}$ satisfying (REF ).",
"First observe that $S \\subseteq Q$ : since $T_{a} \\in \\mathbb {S}_+^{n+1}$ for all $a \\in \\lbrace 0,1\\rbrace ^n$ and $Y \\in \\mathbb {S}_+^{n+1}$ we have $\\langle T_a,Y \\rangle \\geqslant 0$ and thus $\\langle 2\\operatorname{diag}(a)-aa^\\intercal ,x \\rangle \\leqslant 1$ holds for all $x \\in S$ .",
"In order to show that $P \\subseteq S$ recall that $M$ is the slack matrix of the pair $P$ , $Q$ .",
"Therefore, for each vertex $x bb^\\intercal $ of $P$ , we can pick $Y U^{b}$ from the factorization such that $\\langle 2\\operatorname{diag}(a)-aa^\\intercal ,x \\rangle +\\langle T_a,Y \\rangle = 1$ and $Y \\in \\mathbb {S}_+^{n+1}$ .",
"It follows that $P \\subseteq S$ .",
"Our final result is the following inapproximability theorem for spectrahedra.",
"Let us denote the closed $\\varepsilon $ -neighbourhood of $S$ in the $\\ell _{1}$ -norm by $S^{\\varepsilon } \\left\\lbrace x \\in \\mathbb {R}^{n \\times n}\\,|\\,\\exists x_{0} \\in S \\colon \\Vert x - x_{0}\\Vert _{1} \\leqslant \\varepsilon \\right\\rbrace $ .",
"Theorem 10 (Polyhedral inapproximability of SDPs) Let $\\rho \\geqslant 1$ , and let $n$ be a positive integer.",
"Then there exists a spectrahedron $S \\subseteq \\mathbb {R}^{n \\times n}$ with $\\operatorname{xc}_{SDP}(S) \\leqslant n+1$ such that for every polyhedron $K$ with $S \\subseteq K \\subseteq S^{\\rho - 1}$ the following hold: If $\\rho $ is a fixed constant, then $\\operatorname{xc}(K) = 2^{\\Omega (n)}$ .",
"If $\\rho = O(n^\\beta )$ for some constant $\\beta < 1/2$ , then $\\operatorname{xc}(K) = 2^{\\Omega (n^{1 - 2 \\beta })}$ .",
"By Lemma REF , there is a spectrahedron $S$ with $P \\subseteq S \\subseteq Q$ and $\\operatorname{xc}_{SDP}(S) \\leqslant n+1$ .",
"We now show $S^{\\rho - 1} \\subseteq \\rho Q$ .",
"Let $x \\in S^{\\rho - 1}$ , and let $x_{0} \\in S$ with $\\Vert x - x_{0}\\Vert _{1} \\le \\rho - 1$ .",
"As $S \\subseteq Q$ , we also have $x_{0} \\in Q$ , hence for every $a \\in \\lbrace 0,1\\rbrace ^{n}$ we obtain $\\begin{split}\\langle 2\\operatorname{diag}(a)-aa^\\intercal ,x \\rangle &=\\langle 2\\operatorname{diag}(a)-aa^\\intercal ,x - x_{0} \\rangle +\\langle 2\\operatorname{diag}(a)-aa^\\intercal ,x_{0} \\rangle \\\\&\\le \\underbrace{\\left\\Vert 2\\operatorname{diag}(a)-aa^\\intercal \\right\\Vert _{\\infty }}_{\\le 1}\\cdot \\underbrace{\\Vert x - x_{0}\\Vert _{1}}_{\\le \\rho - 1} + 1\\le \\rho .\\end{split}$ Therefore $x \\in \\rho Q$ .",
"Therefore, $S^{\\rho - 1} \\subseteq \\rho Q$ for $\\rho \\geqslant 1$ .",
"If now $K$ is a polyhedron such that $S \\subseteq K \\subseteq S^{\\rho - 1}$ then also $P \\subseteq K \\subseteq \\rho Q$ .",
"The result thus follows from Theorem REF .",
"Concluding Remarks We have introduced a general framework to study approximation limits of small LP relaxations.",
"Given a polyhedron $Q$ encoding admissible objective functions and a polytope $P$ encoding feasible solutions, we have proved that any LP relaxation sandwiched between $P$ and a dilate $\\rho Q$ has extension complexity at least the nonnegative rank of the slack matrix of the pair $P$ , $\\rho Q$ .",
"This yields a lower bound depending only on the linear encoding of the problem at hand, and applies independently of the structure of the actual relaxation.",
"By doing so, we obtain unconditional lower bounds on integrality gaps for small LP relaxations, which hold even in the unlikely event that $P = NP$ .",
"We have proved that every polynomial-size LP relaxation for (a natural linear encoding of) CLIQUE has essentially an $\\Omega (\\sqrt{n})$ integrality gap.",
"As mentioned above, this was recently improved by [11] to a tight $\\Omega (n^{1-\\epsilon })$ integrality gap, see [10] for a short proof and many generalizations.",
"Finally, our work sheds more light on the inherent limitations of LPs in the context of combinatorial optimization and approximation algorithms, in particular, in comparison to SDPs.",
"We provide strong evidence that certain approximation guarantees can only be achieved via non-LP-based techniques (e.g., SDP-based or combinatorial).",
"Actually, our work has inspired [13] to prove lower bounds on the size of LPs for approximating Max CUT, Max $k$ -SAT and in fact any Max CSP.",
"Among other results, they obtain a $n^{\\Omega (\\log n / \\log \\log n)}$ lower bound on the size of any $(2-\\epsilon )$ -approximate EF for Max CUT (of course, with nonnegative weights).",
"[13] thus proving the following conjecture on Max CUT that we stated in an earlier version of this text: Theorem 11 [13] It is not possible to approximate Max CUT with LPs of poly-size within a factor better than 2.",
"This is in stark contrast with the ratio achieved by the SDP-based algorithm of [21] which is known to be optimal, assuming the Unique Games Conjecture [28], [29], [36].",
"Next, about CLIQUE itself, here is an interesting question that this paper leaves open, as pointed out by one of the referees: find an $n$ -vertex graph $G$ for which the clique polytope $\\operatorname{CLIQUE}(G) \\operatorname{conv}\\left(\\lbrace \\chi ^K \\in \\mathbb {R}^{V(G)} \\mid K \\subseteq V(G) \\text{ is a clique}\\rbrace \\right)$ has no polynomial-size $n^{1-\\epsilon }$ -approximate EF.",
"Note that encoding CLIQUE through the clique polytope does not satisfy our faithfulness condition.",
"Finally, so far no strong lower bounding technique for semidefinite EFs are known.",
"It is plausible that in the near future we will see lower bounding techniques on the PSD rank that would be suited for studying approximation limits of SDPs.",
"(We remark however that such bounds should not only argue on the zero/nonzero pattern of a slack matrix.)",
"Acknowledgements We would like to thank the two referees for their time and comments which contributed to improve the text."
],
[
"Polyhedral Inapproximability of CLIQUE and SDPs",
"We will now use Theorem REF in combination with Theorem REF to lower bound the sizes of approximate EFs for CLIQUE and some SDPs.",
"First, we pinpoint a pair $P,Q$ of nested polyhedra that will be the source of our polyhedral inapproximability results.",
"Second, we give a faithful linear encoding of CLIQUE and prove strong lower bounds on the sizes of approximate EFs for CLIQUE w.r.t.",
"this encoding.",
"Third, we focus on approximations of SDPs by LPs."
],
[
"A Hard Pair",
"Let $n$ be a positive integer.",
"The correlation polytope $\\operatorname{COR}(n)$ is defined as the convex hull of all the $n \\times n$ rank-1 binary matrices of the form $bb^\\intercal $ where $b \\in \\lbrace 0,1\\rbrace ^n$ .",
"In other words, $\\operatorname{COR}(n) = \\operatorname{conv}\\left(\\left\\lbrace bb^\\intercal \\,|\\,b \\in \\lbrace 0,1\\rbrace ^n\\right\\rbrace \\right).$ This will be our inner polytope $P$ .",
"Next, let $Q = Q(n) \\lbrace x \\in \\mathbb {R}^{n \\times n} \\mid \\langle 2\\operatorname{diag}(a)-aa^\\intercal ,x \\rangle \\leqslant 1,\\ a \\in \\lbrace 0,1\\rbrace ^n\\rbrace ,$ where $\\langle \\cdot ,\\cdot \\rangle $ denotes the Frobenius inner product.",
"This will be our outer polyhedron $Q$ .",
"Then the following is known, see [18].",
"First, $P \\subseteq Q$ .",
"Second, denoting by $S^{P,Q}$ the slack matrix of the pair $P,Q$ , we have $S^{P,Q}_{ab} = (1-a^\\intercal b)^2$ .",
"Thus, for $\\rho \\geqslant 1$ , we have $S^{P,\\rho Q}_{ab}= (1-a^\\intercal b)^2 + \\rho - 1$ .",
"Observe that the matrix $S^{P,\\rho Q}$ is a $\\rho $ -extension of UDISJ and therefore has high nonnegative rank via Theorem REF ; moreover it has positive entries everywhere for $\\rho > 1$ .",
"Together with Theorem REF this implies that every polyhedron sandwiched between $P = \\operatorname{COR}(n)$ and $\\rho Q$ has large extension complexity.",
"We obtain the following theorem.",
"Theorem 6 (Lower bounds for approximate EFs of the hard pair) Let $\\rho \\geqslant 1$ , let $n$ be a positive integer and let $P = \\operatorname{COR}(n)$ , $Q = Q(n)$ be as above.",
"Then the following hold: If $\\rho $ is a fixed constant, then $\\operatorname{xc}(P,\\rho Q) = 2^{\\Omega (n)}$ .",
"If $\\rho = O(n^\\beta )$ for some constant $\\beta < 1/2$ , then $\\operatorname{xc}(P,\\rho Q) = 2^{\\Omega (n^{1 - 2 \\beta })}$ ."
],
[
"Polyhedral Inapproximability of CLIQUE",
"We define a natural linear encoding for the maximum clique problem (CLIQUE) as follows.",
"Let $n$ denote the number of vertices of the input graph.",
"We define a $d = n^2$ dimensional encoding.",
"The variables are denoted by $x_{ij}$ for $i, j \\in [n]$ .",
"Thus $x \\in \\mathbb {R}^{n \\times n}$ .",
"The interpretation is that a set of vertices $X$ is encoded by $x_{ij} = 1$ if $i, j \\in X$ and $x_{ij} = 0$ otherwise.",
"Note that $X = \\lbrace i : x_{ii} = 1\\rbrace $ can be recovered from only the diagonal variables.",
"This defines the set $\\mathcal {L} \\subseteq \\lbrace 0,1\\rbrace ^*$ of feasible solutions.",
"Notice that $x \\in \\lbrace 0,1\\rbrace ^{n \\times n}$ is feasible if and only if it is of the form $x = bb^\\intercal $ for some $b \\in \\lbrace 0,1\\rbrace ^n$ , the characteristic vector of $X$ .",
"Thus we have $P = \\operatorname{COR}(n)$ for the inner polytope.",
"The admissible objective functions are chosen as follows to encode the CLIQUE problem for graphs $G$ supported on $[n]$ .",
"Given a graph $G$ such that $V(G) \\subseteq [n]$ , we let $w_{ii} 1$ for $i \\in V(G)$ , $w_{ii} 0$ for $i \\in [n] \\setminus V(G)$ , $w_{ij} = w_{ji} -1$ when $ij$ is a non-edge of $G$ (that is, $i, j \\in V(G)$ , $i \\ne j$ and $ij \\notin E(G)$ ), and $w_{ij} = w_{ji} 0$ otherwise.",
"We denote the resulting weight vector by $w^G$ .",
"Notice that for a graph $G$ with $V(G) = [n]$ , we have $w^G =I - A(\\overline{G})$ where $I$ is the $n \\times n$ identity matrix, $A(\\overline{G})$ is the adjacency matrix of the complement of $G$ .",
"A feasible solution $x = bb^\\intercal \\in \\lbrace 0,1\\rbrace ^{n \\times n}$ maximizes $\\langle w^G,x \\rangle $ only if $b$ is the characteristic vector (or incidence vector) of a clique of $G$ .",
"Indeed, if $b = \\chi ^X$ and $ij$ is a non-edge of $G$ with $i, j \\in X$ then removing $i$ or $j$ from $X$ increases $\\langle w^G,x \\rangle $ .",
"Moreover, the maximum of $\\langle w^G,x \\rangle $ over $x \\in \\lbrace 0,1\\rbrace ^{n \\times n}$ feasible is the clique number $\\omega (G)$ .",
"The admissible objective functions are the ones of the form $w^{G}$ , i.e., $\\mathcal {O} = \\lbrace w^{G} : V(G) \\subseteq [n]\\rbrace $ is the set of admissible functions.",
"Therefore, $(\\mathcal {L},\\mathcal {O})$ defines a valid linear encoding of CLIQUE.",
"We denote the outer convex set of this linear encoding by $Q^{\\mathrm {all}}$ .",
"It is actually the polyhedron defined as $Q^{\\mathrm {all}} = \\lbrace x \\in \\mathbb {R}^{n \\times n} \\mid \\forall $ graphs $G$ s.t.",
"$V(G) \\subseteq [n] : \\langle w^G,x \\rangle \\leqslant \\omega (G),\\ \\forall i \\ne j \\in [n] : x_{ij} \\geqslant 0\\rbrace $ .",
"We will now show that $Q^{\\mathrm {all}} \\subseteq Q$ .",
"Lemma 7 Let $Q^{\\mathrm {all}},Q$ be as above, then $Q^{\\mathrm {all}}\\subseteq Q$ .",
"Let $x \\in Q^{\\mathrm {all}}$ .",
"We want to prove that $x$ satisfies all the constraints defining $Q$ .",
"We show this by restricting to graphs $G$ with $\\omega (G) = 1$ .",
"For a given $a \\in \\lbrace 0,1\\rbrace ^n$ , let $G$ be the graph with $\\chi ^{V(G)} = a$ and $E(G) = \\emptyset $ .",
"Then, $\\langle 2\\operatorname{diag}(a) - aa^T,x \\rangle = \\langle w^G,x \\rangle \\leqslant \\omega (G) = 1.$ The lemma follows.",
"Because $Q^{\\mathrm {all}}$ is contained in the polyhedron $Q$ defined above, every $K$ satisfying $P \\subseteq K \\subseteq \\rho Q^{\\mathrm {all}}$ also satisfies $P \\subseteq K \\subseteq \\rho Q$ .",
"Hence, Theorem REF yields the following result.",
"Theorem 8 (Polyhedral inapproximability of CLIQUE) W.r.t.",
"the linear encoding defined above, CLIQUE has an $O(n^2)$ -size $n$ -approximate EF.",
"Moreover, every $n^{1/2-\\epsilon }$ -approximate EF of CLIQUE has size $2^{\\Omega (n^{2\\epsilon })}$ , for all $0 < \\epsilon < 1/2$ .",
"The $n$ -approximate EF of CLIQUE is trivial: it is defined by the system $\\mathbf {0} \\leqslant x \\leqslant \\mathbf {1}$ , or in slack form $x - y = \\mathbf {0}$ , $x + z = \\mathbf {1}$ , $y \\geqslant \\mathbf {0}$ , $z \\geqslant \\mathbf {0}$ .",
"We claim that this defines a $n$ -approximate EF of CLIQUE of size $2n^2$ .",
"Indeed, letting $K = [0,1]^{n \\times n}$ denote the polytope defined by this EF, we have $P \\subseteq K$ .",
"Moreover, $\\max \\lbrace \\langle w,x \\rangle \\mid x \\in K\\rbrace \\leqslant n \\leqslant n \\cdot \\max \\lbrace \\langle w,x \\rangle \\mid x \\in P\\rbrace $ for all admissible objective functions $w$ of dimension $n \\times n$ with a nonzero diagonal.",
"In case an admissible $w$ has $w_{ii} = 0$ for all $i \\in [n]$ , we have $\\max \\lbrace \\langle w,x \\rangle \\mid x \\in K\\rbrace = 0 = \\max \\lbrace \\langle w,x \\rangle \\mid x \\in P\\rbrace $ .",
"Our claim and the first part of the theorem follows.",
"The second part of the theorem follows directly from Theorem REF and the fact that $Q^{\\mathrm {all}}\\subseteq Q$ ."
],
[
"Polyhedral Inapproximability of SDPs",
"In this section we show that there exists a spectrahedron with small semidefinite extension complexity but high approximate extension complexity; i.e., any sufficiently fine polyhedral approximation is large.",
"This indicates that in general it is not possible to approximate SDPs arbitrarily well using small LPs, so that SDPs are indeed a much stronger class of optimization problems.",
"(The situation looks quite different for SOCPs, see [8].)",
"The result follows from Theorem REF and [18].",
"We denote the vector space of all $r \\times r$ symmetric matrices by $\\mathbb {S}^r$ and the cone of all $r \\times r$ symmetric positive semidefinite matrices (shortly, the PSD cone) by $\\mathbb {S}_+^r$ .",
"A semidefinite EF of a convex set $S \\subseteq \\mathbb {R}^d$ is a linear system $\\langle E_i,x \\rangle + \\langle F_i,Y \\rangle = g_i$ ($i \\in [k]$ ), $Y \\in \\mathbb {S}_+^r$ where $E_i \\in \\mathbb {R}^d$ and $F_i \\in \\mathbb {S}^r$ , such that $x \\in S$ if and only if $\\exists Y \\in \\mathbb {S}_+^r$ with $\\langle E_i,x \\rangle + \\langle F_i,Y \\rangle = g_i$ for all $i \\in [m]$ .",
"Thus a convex set admits a semidefinite EF if and only if it is a spectrahedron.",
"The size of the semidefinite EF $\\langle E_i,x \\rangle + \\langle F_i,Y \\rangle = g_i$ ($i \\in [k]$ ), $Y \\in \\mathbb {S}_+^r$ is simply $r$ .",
"The semidefinite extension complexity of a spectrahedron $S \\subseteq \\mathbb {R}^d$ is the minimum size of a semidefinite EF of $S$ .",
"This is denoted by $\\operatorname{xc}_{SDP}(S)$ .",
"A rank-$r$ PSD-factorization of a nonnegative matrix $M \\in \\mathbb {R}^{m \\times n}$ is given by matrices $T_1, \\ldots , T_m \\in \\mathbb {S}_+^r$ and $U^1, \\ldots , U^n \\in \\mathbb {S}_+^r$ , so that $M_{ij} =\\langle T_i,U^j \\rangle $ ; the PSD-rank of $M$ is the smallest $r$ such that there exists such a factorization.",
"Yannakakis's factorization theorem can be generalized to the SDP-case (see [22]), i.e., the semidefinite extension complexity of a pair of polyhedra $P, Q$ equals the PSD-rank of any associated slack matrix, in most cases (e.g., [23] prove the equality under the hypothesis that $Q$ does not contain any line).",
"Let $P = \\operatorname{COR}(n)$ be the correlation polytope and $Q = Q(n) \\subseteq \\mathbb {R}^{n \\times n}$ be the polyhedron defined above in Section REF .",
"Although every polyhedron $K$ sandwiched between $P$ and $Q$ has super-polynomial extension complexity, and by Theorem REF this even applies to polyhedra sandwiched between $P$ and $\\rho Q$ for $\\rho = O(n^{1/2-\\epsilon })$ , there exists a spectrahedron $S$ sandwiched between $P$ and $Q$ with small semidefinite extension complexity.",
"Lemma 9 (Existence of spectrahedron) Let $n$ be a positive integer and let $P = \\operatorname{COR}(n)$ , $Q = Q(n)$ be as above.",
"Then there exists a spectrahedron $S$ in $\\mathbb {R}^{n \\times n}$ with $P \\subseteq S \\subseteq Q$ and $\\operatorname{xc}_{SDP}(S) \\leqslant n+1$ .",
"For $a,b \\in \\lbrace 0,1\\rbrace ^{n}$ , the matrices $T^{a},U_b \\in \\mathbb {S}_+^{n+1}$ defined in (REF ) satisfy $\\langle T^{a},U_{b} \\rangle = (1 - a^\\intercal b)^2$ .",
"Let $M = M(n) \\in \\mathbb {R}^{2^n \\times 2^n}$ be the matrix defined as $M_{ab} = (1 - a^\\intercal b)^2$ .",
"The matrix $M$ is an $O(n^2)$ -rank nonnegative matrix extending the UDISJ matrix, and also the slack matrix of the pair $P$ , $Q$ .",
"Then $M_{ab} = \\langle T_a,U^b \\rangle $ is a rank-$(n+1)$ PSD-factorization of $M$ .",
"Consider the system $\\langle 2\\operatorname{diag}(a)-aa^\\intercal ,x \\rangle + \\langle T_a,Y \\rangle = 1\\quad (a \\in \\lbrace 0,1\\rbrace ^n), \\qquad Y \\in \\mathbb {S}_+^{n+1}$ and $S$ be the projection to $\\mathbb {R}^{n \\times n}$ of the pairs $(x,Y) \\in \\mathbb {R}^{n \\times n} \\times \\mathbb {S}^{n+1}$ satisfying (REF ).",
"First observe that $S \\subseteq Q$ : since $T_{a} \\in \\mathbb {S}_+^{n+1}$ for all $a \\in \\lbrace 0,1\\rbrace ^n$ and $Y \\in \\mathbb {S}_+^{n+1}$ we have $\\langle T_a,Y \\rangle \\geqslant 0$ and thus $\\langle 2\\operatorname{diag}(a)-aa^\\intercal ,x \\rangle \\leqslant 1$ holds for all $x \\in S$ .",
"In order to show that $P \\subseteq S$ recall that $M$ is the slack matrix of the pair $P$ , $Q$ .",
"Therefore, for each vertex $x bb^\\intercal $ of $P$ , we can pick $Y U^{b}$ from the factorization such that $\\langle 2\\operatorname{diag}(a)-aa^\\intercal ,x \\rangle +\\langle T_a,Y \\rangle = 1$ and $Y \\in \\mathbb {S}_+^{n+1}$ .",
"It follows that $P \\subseteq S$ .",
"Our final result is the following inapproximability theorem for spectrahedra.",
"Let us denote the closed $\\varepsilon $ -neighbourhood of $S$ in the $\\ell _{1}$ -norm by $S^{\\varepsilon } \\left\\lbrace x \\in \\mathbb {R}^{n \\times n}\\,|\\,\\exists x_{0} \\in S \\colon \\Vert x - x_{0}\\Vert _{1} \\leqslant \\varepsilon \\right\\rbrace $ .",
"Theorem 10 (Polyhedral inapproximability of SDPs) Let $\\rho \\geqslant 1$ , and let $n$ be a positive integer.",
"Then there exists a spectrahedron $S \\subseteq \\mathbb {R}^{n \\times n}$ with $\\operatorname{xc}_{SDP}(S) \\leqslant n+1$ such that for every polyhedron $K$ with $S \\subseteq K \\subseteq S^{\\rho - 1}$ the following hold: If $\\rho $ is a fixed constant, then $\\operatorname{xc}(K) = 2^{\\Omega (n)}$ .",
"If $\\rho = O(n^\\beta )$ for some constant $\\beta < 1/2$ , then $\\operatorname{xc}(K) = 2^{\\Omega (n^{1 - 2 \\beta })}$ .",
"By Lemma REF , there is a spectrahedron $S$ with $P \\subseteq S \\subseteq Q$ and $\\operatorname{xc}_{SDP}(S) \\leqslant n+1$ .",
"We now show $S^{\\rho - 1} \\subseteq \\rho Q$ .",
"Let $x \\in S^{\\rho - 1}$ , and let $x_{0} \\in S$ with $\\Vert x - x_{0}\\Vert _{1} \\le \\rho - 1$ .",
"As $S \\subseteq Q$ , we also have $x_{0} \\in Q$ , hence for every $a \\in \\lbrace 0,1\\rbrace ^{n}$ we obtain $\\begin{split}\\langle 2\\operatorname{diag}(a)-aa^\\intercal ,x \\rangle &=\\langle 2\\operatorname{diag}(a)-aa^\\intercal ,x - x_{0} \\rangle +\\langle 2\\operatorname{diag}(a)-aa^\\intercal ,x_{0} \\rangle \\\\&\\le \\underbrace{\\left\\Vert 2\\operatorname{diag}(a)-aa^\\intercal \\right\\Vert _{\\infty }}_{\\le 1}\\cdot \\underbrace{\\Vert x - x_{0}\\Vert _{1}}_{\\le \\rho - 1} + 1\\le \\rho .\\end{split}$ Therefore $x \\in \\rho Q$ .",
"Therefore, $S^{\\rho - 1} \\subseteq \\rho Q$ for $\\rho \\geqslant 1$ .",
"If now $K$ is a polyhedron such that $S \\subseteq K \\subseteq S^{\\rho - 1}$ then also $P \\subseteq K \\subseteq \\rho Q$ .",
"The result thus follows from Theorem REF ."
],
[
"Concluding Remarks",
"We have introduced a general framework to study approximation limits of small LP relaxations.",
"Given a polyhedron $Q$ encoding admissible objective functions and a polytope $P$ encoding feasible solutions, we have proved that any LP relaxation sandwiched between $P$ and a dilate $\\rho Q$ has extension complexity at least the nonnegative rank of the slack matrix of the pair $P$ , $\\rho Q$ .",
"This yields a lower bound depending only on the linear encoding of the problem at hand, and applies independently of the structure of the actual relaxation.",
"By doing so, we obtain unconditional lower bounds on integrality gaps for small LP relaxations, which hold even in the unlikely event that $P = NP$ .",
"We have proved that every polynomial-size LP relaxation for (a natural linear encoding of) CLIQUE has essentially an $\\Omega (\\sqrt{n})$ integrality gap.",
"As mentioned above, this was recently improved by [11] to a tight $\\Omega (n^{1-\\epsilon })$ integrality gap, see [10] for a short proof and many generalizations.",
"Finally, our work sheds more light on the inherent limitations of LPs in the context of combinatorial optimization and approximation algorithms, in particular, in comparison to SDPs.",
"We provide strong evidence that certain approximation guarantees can only be achieved via non-LP-based techniques (e.g., SDP-based or combinatorial).",
"Actually, our work has inspired [13] to prove lower bounds on the size of LPs for approximating Max CUT, Max $k$ -SAT and in fact any Max CSP.",
"Among other results, they obtain a $n^{\\Omega (\\log n / \\log \\log n)}$ lower bound on the size of any $(2-\\epsilon )$ -approximate EF for Max CUT (of course, with nonnegative weights).",
"[13] thus proving the following conjecture on Max CUT that we stated in an earlier version of this text: Theorem 11 [13] It is not possible to approximate Max CUT with LPs of poly-size within a factor better than 2.",
"This is in stark contrast with the ratio achieved by the SDP-based algorithm of [21] which is known to be optimal, assuming the Unique Games Conjecture [28], [29], [36].",
"Next, about CLIQUE itself, here is an interesting question that this paper leaves open, as pointed out by one of the referees: find an $n$ -vertex graph $G$ for which the clique polytope $\\operatorname{CLIQUE}(G) \\operatorname{conv}\\left(\\lbrace \\chi ^K \\in \\mathbb {R}^{V(G)} \\mid K \\subseteq V(G) \\text{ is a clique}\\rbrace \\right)$ has no polynomial-size $n^{1-\\epsilon }$ -approximate EF.",
"Note that encoding CLIQUE through the clique polytope does not satisfy our faithfulness condition.",
"Finally, so far no strong lower bounding technique for semidefinite EFs are known.",
"It is plausible that in the near future we will see lower bounding techniques on the PSD rank that would be suited for studying approximation limits of SDPs.",
"(We remark however that such bounds should not only argue on the zero/nonzero pattern of a slack matrix.)"
],
[
"Acknowledgements",
"We would like to thank the two referees for their time and comments which contributed to improve the text."
]
] | 1204.0957 |
[
[
"The scaling limit of polymer pinning dynamics and a one dimensional\n Stefan freezing problem"
],
[
"Abstract We consider the stochastic evolution of a 1+1-dimensional interface (or polymer) in presence of a substrate.",
"This stochastic process is a dynamical version of the homogeneous pinning model.",
"We start from a configuration far from equilibrium: a polymer with a non-trivial macroscopic height profile, and look at the evolution of a space-time rescaled interface.",
"In two cases, we prove that this rescaled interface has a scaling limit on the diffusive scale (space rescaled by $L$ in both dimensions and time rescaled by $L^2$ where $L$ denotes the length of the interface) which we describe: when the interaction with the substrate is such that the system is unpinned at equilibrium, then the scaling limit of the height profile is given by the solution of the heat equation with Dirichlet boundary condition ; when the attraction to the substrate is infinite, the scaling limit is given a free-boundary problem which belongs to the class of Stefan problems with contracting boundary, also referred to as Stefan freezing problems.",
"In addition, we prove the existence and regularity of the solution to this problem until a maximal time, where the boundaries collide."
],
[
"The dynamical pinning model",
"Random polymer models have been used for a long time by physicists to describe a large variety of physical phenomena.",
"Among the numerous models that have been introduced by theoretical physicists and rigorously studied by mathematicians (see e.g.",
"[13] for a survey of the most studied polymer models), the polymer pinning model, that involves a simple random walk interacting with a defect line, has focused a lot of interest both of the mathematics and physics community.",
"The phase transition phenomenon between a pinned phase and a depinned one is now well understood, even in presence of disorder (see [9] for a seminal paper concerning the homogeneous case, and [10], [11] for recent reviews).",
"On the other hand, dynamical pinning (which has some importance in biophysical application see [1], [2] and references therein) has attracted attention of mathematicians only more recently and a lot of questions concerning relaxation to equilibrium and its connection properties are still unsolved.",
"The object of most of the mathematical studies on the dynamical pinning model up to now (see [3], [4]) has been the mixing time and the relaxation time for the dynamics.",
"It has been shown there that the mixing property of the system depends in a crucial way of the pinning parameter $\\lambda $ , or more precisely, on whether the polymer at equilibrium is pinned or unpinned.",
"In the present paper, we choose to study a different aspect, that is, the dynamical scaling limit of the height-profile of the polymer.",
"We start from an initial condition that is very far from equilibrium and approximates a macroscopic deterministic profile and we want to describe the evolution of the profile under diffusive scaling.",
"Our aim is to show that the nature of the limit of the rescaled process depends only on whether one lies in the pinned or depinned phase, and to describe explicitly the scaling limit in each case.",
"The scaling limit is easier to guess in the unpinned phase.",
"As in this case, there is no contact point with the substrate at equilibrium, one can infer that the scaling limit is the same that for a system with no substrate, for which it is known that the height profile converges to the solution of the heat-equation on the segment with zero Dirichlet boundary condition.",
"In the pinned phase, a more interesting phenomenon takes place.",
"In this case the dynamical picture should be the following: there are macroscopic region where the polymer is pinned to the substrate and other regions where the polymer stays at a macroscopic distance from it; the boundary between the pinned and the unpinned region is moving in time and in the unpinned region, the polymer profile evolves according to the heat-equation.",
"The system reaches equilibrium when the unpinned region has totally vanished.",
"In the present paper, we prove that this picture holds when the pinning parameter is infinite (or tending to infinity sufficiently fast with the size of the system).",
"An important step to establish this result is to prove existence and regularity of the free-boundary problem that appears in the scaling limit."
],
[
"A one dimensional Stefan freezing problem",
"The free-boundary problem of unknown $(f,l,r)$ ($f$ is the function and $l$ and $r$ are the boundary of the unpinned region) that appears as the scaling limit of the pinning model in the pinned phase is the following ${\\left\\lbrace \\begin{array}{ll}\\partial _t f- f_{xx}=0 \\quad \\text{on } (l(t),r(t)),\\\\f(\\cdot ,t)\\equiv 0 \\quad \\text{on } [-1,1]\\setminus (l(t),r(t)),\\\\f_x(l(t),t)=- f_x(r(t),t)=1,\\\\l^{\\prime }(t)=- f_{xx}(l(t),t),\\quad r^{\\prime }(t)=f_{xx}(r(t),t),\\\\f(\\cdot ,0)=f_0,\\ l(0)=l_0,\\ r_0.\\end{array}\\right.",
"}$ We are exclusively interested in the case of a 1-Lipshitz initial condition that vanishes outside of the interval $(r_0,l_0)\\subset [-1,1]$ and is positive inside.",
"This problem belongs to the class of Stefan problem, which have been introduced in mathematics to describe the evolution of a multiphase medium.",
"The boundary condition for $f_x$ and the fact that the heat equation preserves Lipshitzianity imply that $f$ cannot be convex in the neighborhood of the moving boundaries, and thus the boundary points $l(t)$ and $r(t)$ are moving towards each other ($l^{\\prime }\\;\\geqslant \\;0$ and $r^{\\prime }\\;\\geqslant \\;0$ ).",
"These problems with contracting boundary are referred to as freezing problems whereas those with expanding boundary are called melting problems.",
"Most of the work in the literature on Stefan problems concerns melting problems.",
"One of the reason for this is that these problem can be rewritten as a diffusion equation for an enthalpy function with a diffusion coefficient that is monotonous increasing function of the enthalpy.",
"This monotonicity allows to derive uniqueness of the solution with some generality (we refer to the Introduction of [7] for more precision).",
"The freezing problems like (REF ) on the contrary are more challenging even in the one dimensional setup.",
"Even though one dimensional freezing problems have attracted some attention in a recent past [5], [6], some amount of work is required to establish the existence and the unicity of a solution to (REF ) up to a maximal time."
],
[
"A simple model for interface motion with no constraint",
"To introduce our reader to polymer dynamics, we first introduce the simplest version of the model where no substrate is present: this is the so-called corner-flip dynamics.",
"Let $\\Omega =\\Omega _{L}$ denote the set of all lattice paths (polymers) starting at 0 and ending at 0 after $2L$ steps $\\Omega ^{0}_L :=\\lbrace \\eta \\in {\\mathbb {Z}} ^{2L+1} \\ | \\ \\eta _{-L}=\\eta _L=0\\,,\\;|\\eta _{x+1}-\\eta _x|= 1, \\;x=-L,\\dots ,L-1\\rbrace \\,.$ The stochastic dynamics is defined by the natural spin-flip continuous time Markov chain with state space $\\Omega ^0_L$ .",
"Namely, sites $x=-L,\\dots ,L$ are equipped with independent Poisson clocks which ring with rate one: when a clock rings at $x$ the path $\\eta $ is replaced by $\\eta ^{(x)}$ , defined by $\\eta _y^{(x)}=\\eta _x^{(x)}$ for all $y\\ne x$ and $\\eta _x^{(x)}:={\\left\\lbrace \\begin{array}{ll}\\eta _x+2 \\text{ if } \\eta _{x\\pm 1}=\\eta _x+1,\\\\\\eta _x-2 \\text{ if } \\eta _{x\\pm 1}=\\eta _x-1,\\\\\\eta _x \\text{ if } |\\eta _{x+1}-\\eta _{x-1}|=2.\\end{array}\\right.",
"}$ One denotes by ($\\widetilde{\\eta }(\\cdot ,t)$ , $t\\;\\geqslant \\;0$ ), the trajectory of the Markov chain.",
"By doing linear interpolation between the integer values of $x$ , one can consider $\\widetilde{\\eta }(\\cdot ,t)$ as a function of the real variable $x\\in [-L,L]$ .",
"The unique invariant measure for this dynamics is the uniform measure on $\\Omega ^{0}_L$ , and thus, from standard properties of the random walk, when the system is at equilibrium, the rescaled interface $(\\eta (Lx)/L)_{x\\in [-1,1]}$ is macroscopically flat ($\\eta (x)$ has fluctuation of order $\\sqrt{L}$ ).",
"For this model, relaxation to equilibrium is well understood both in terms of mixing-time (see Wilson [20]) or scaling limits (see [15], weaker versions of these result had been known before, using connection with the one dimensional simple symmetric exclusion process).",
"We cite in full the result concerning the scaling limit for two reasons: it gives some point of reference to better understand results in presence of a substrate; and we use it as a fundamental building brick for the proof of our new results.",
"Given $f_0$ a Lipshitz function in $[-1,1]$ , with $f_0(-1)=f_0(1)=0$ , set $\\widetilde{f}$ defined on $[-1,1]\\times [0,\\infty )$ to be the solution of the heat equation with Dirichlet boundary condition ${\\left\\lbrace \\begin{array}{ll}\\partial _t \\widetilde{f}- \\widetilde{f}_{xx}&=0,\\\\\\widetilde{f}(\\cdot , 0)&=f_0,\\\\\\widetilde{f}(-1,t)&=\\widetilde{f}(1,t)=0, \\quad \\forall t>0.\\end{array}\\right.",
"}$ Theorem 2.1 ([15] Theorem 3.2) Let $\\widetilde{\\eta }^L$ be the dynamic described above, starting from a sequence of initial condition $\\eta ^L_0$ that satisfies, $ \\eta ^L_0(x)=L(f_0(x/L)+o(1)) \\text{ uniformly in $x$ when $L\\rightarrow \\infty $ }.$ Then, under diffusive scaling, $\\widetilde{\\eta }^L$ converges to $\\widetilde{f}$ in law for the uniform topology: for any $T>0$ , in probability, $\\lim _{L\\rightarrow \\infty } \\sup _{x\\in [-1,1],t\\in [0,T]} \\left|\\frac{1}{L}\\widetilde{\\eta }^L(Lx,L^2t)-\\widetilde{f}(x,t)\\right|=0.$ The notation in Equation (REF ) means that there exists a function $\\varepsilon _L$ tending to zero such that for all $x$ and $L$ $\\left|\\frac{\\eta ^L_0(x)-L(f_0(x/L))}{L}\\right|\\;\\leqslant \\;\\varepsilon _L.$ We keep this notation for the rest of the paper."
],
[
"Dynamical polymers interacting with a substrate",
"Let us now define precisely the model that is the object of study of this paper.",
"Our aim is to understand how the pattern of relaxation to equilibrium given by Theorem REF is modified (or not modified) when the dynamics has additional constraints.",
"We focus on the case of an interface interacting with a solid substrate.",
"This brings us to consider a dynamics with the following modifications: We consider that a solid wall fills the entire bottom half-plane so that our trajectories have to stay in the positive half-plane ($\\eta _x\\;\\geqslant \\;0$ , $\\forall x \\in \\lbrace -L,\\dots , L\\rbrace $ ).",
"The wall interacts with the interface $\\eta $ so that the rates of the transitions that modifies the number of contacts with the wall are changed.",
"More precisely one starts from a trajectory that lies entirely above the wall, i.e.",
"which belongs to the following subset of $\\Omega _L^0$ : $\\Omega _L :=\\lbrace \\eta \\in {\\mathbb {Z}} ^{2L+1}\\ | \\;\\eta _{-L}=\\eta _L=0\\,; \\ \\forall x\\in \\lbrace -L,\\dots ,L-1\\rbrace ,\\ |\\eta _{x+1}-\\eta _x|= 1, \\eta _x\\;\\geqslant \\;0 \\rbrace \\,.$ The rates of the transitions from $\\eta $ to $\\eta ^{(x)}$ are not uniformly equal to 1 as in the previous section but they are given by $c(\\eta ,\\eta ^{(x)})={\\left\\lbrace \\begin{array}{ll}0 \\text{ if } \\eta _x^{(x)}=-1 \\quad \\text{(interdiction to go through the wall)},\\\\\\frac{2\\lambda }{1+\\lambda } \\text{ if } \\eta _x=2 \\text{ and } \\eta _x^{(x)}=0,\\\\\\frac{2}{1+\\lambda } \\text{ if } \\eta _x=0 \\text{ and } \\eta _x^{(x)}=2,\\\\1 \\text{ in every other cases }.\\end{array}\\right.",
"}$ The generator $\\mathcal {L}=\\mathcal {L}^\\lambda _L$ of the Markov process is given by $\\mathcal {L}(f)=\\sum _{x=-L+1}^{L-1} c(\\eta ,\\eta ^{(x)})(f(\\eta ^{(x)})-f(\\eta ).$ The value of the parameter $\\lambda \\in [0,\\infty ]$ determines the nature of the interaction with the wall.",
"If $\\lambda >1$ , the transitions adding a contact are favored, which means that the wall is attractive, whereas if $\\lambda <1$ the wall is repulsive.",
"The process defined above is the heat-bath dynamics for the polymer pinning model, with equilibrium measure $\\pi =\\pi _{2L}^\\lambda $ on $\\Omega _L$ defined by $\\pi ^{\\lambda }_{L}(\\eta )= \\frac{\\lambda ^{N(\\eta )}}{Z_{2L}^\\lambda }\\,,$ where $N(\\eta ) = \\#\\lbrace x\\in [-L+1,L-1]\\,:\\;\\eta _x=0\\rbrace $ denotes the number of zeros in the path $\\eta \\in \\Omega $ and $Z_{2L}^\\lambda := \\sum _{\\eta ^{\\prime }\\in \\Omega _L}\\lambda ^{N(\\eta ^{\\prime })}$ is the partition function, which is the renormalization factor that makes $\\pi _{L}^\\lambda $ a probability.",
"For every $\\lambda >0$ , $L\\in {\\mathbb {N}} $ , $\\pi _{L}^\\lambda $ is the unique reversible invariant measure for the Markov chain.",
"For any value of $\\lambda $ , the rescaled version of $\\eta $ at equilibrium (that is, under the measure $\\pi _{L}^\\lambda $ ) is flat, but the microscopic properties of $\\eta $ vary with the value of $\\lambda $ : (i) When $\\lambda \\in [0,2)$ , the interface is repelled by the wall (i.e.",
"when $\\lambda \\in (1,2)$ , the entropic repulsion wins against the energetic attraction of the wall) and typical paths have a number of contacts with the wall which stays bounded when $L$ tends to infinity (the sequence of the laws of $N(\\eta )$ is tight).",
"(ii) When $\\lambda \\in (2,\\infty ]$ , the interface is pinned to the wall, and typical paths have a number of contacts with the wall which is of order $L$ .",
"(iii) When $\\lambda =2$ , $\\eta $ has a lot of contact with the wall (order $\\sqrt{L}$ ) but the longest excursion away from the wall has length of order $L$ .",
"For more precise statements and proofs, we refer to Chapter 2 in [10].",
"These three cases are respectively referred to as the depinned or unpinned phase, the pinned phase, and the critical point (or phase transition point).",
"Remark 2.2 The case $\\lambda =\\infty $ , that will be also considered in this paper is a bit particular.",
"Indeed, as seen in (REF ), when $\\eta _x=0$ (when $\\eta $ touches the wall at $x$ ), it sticks to it forever, so that $\\eta (\\cdot ,t)$ stops moving once it has reached the minimal configuration $ \\eta ^{\\min } $ defined by $\\eta ^{\\min }_x:={\\left\\lbrace \\begin{array}{ll}0 \\text{ if } x+L \\text{ is even },\\\\1 \\text{ if } x+L \\text{ is odd },\\end{array}\\right.",
"}$ which is the configuration with the maximal number of contact point with the wall.",
"In that case, the unique invariance probability measure is the Dirac mass on $\\eta ^{\\min }$ .",
"A question of interest is then to compute the time at which $\\eta (\\cdot ,t)$ stops to move: the hitting time of $\\eta ^{\\min }$ .",
"Our aim is to get a result similar to Theorem REF , describing how, starting from a non-flat profile, the system relaxes to equilibrium.",
"We are able to deduce results in two cases: in the depinned phase, when $\\lambda \\in [0,2)$ : in that case the scaling limit is the same one as for the model without wall.",
"The result can be obtained with rather soft comparison arguments when $\\lambda \\;\\leqslant \\;1$ but requires some additional work when $\\lambda \\in (1,2)$ .",
"when the wall is sticky, $\\lambda =\\infty $ : in that case, the attraction of the wall can be seen ot the macroscopic level, and the scaling limit is given by the solution of a partial differential equation with moving boundary: the free bounary problem (REF ).",
"We can understand what happens when $\\lambda =2$ (the critical point) and when $\\lambda \\in (2,\\infty )$ (the pinned phase) at a heuristic level, and formulate this as conjectures (see Section REF ).",
"There are a lot of technical reasons why bringing these conjectures to rigorous ground cannot be done only we the ideas exposed in this paper.",
"We might address this issue in future work."
],
[
"Scaling limit in the repulsive case",
"Our first result is an analog of Theorem REF for the dynamic in the depinned phase.",
"Theorem 2.3 Let $\\eta =\\eta ^{L,\\lambda }$ be the dynamic on $\\Omega _L$ with generator $\\mathcal {L}$ described above, with the parameter $\\lambda \\in [0,2)$ and starting from a sequence of initial condition $\\eta ^L_0$ satisfying $ \\eta ^L_0(x)=Lf_0(x/L)(1+o(1)) \\text{ uniformly in $x$ when $L\\rightarrow \\infty $ },$ where $f_0$ is a 1-Lipshitz non-negative function.",
"Then $\\eta ^{L,\\lambda }$ converges to $\\widetilde{f}$ defined by (REF ) in law for the uniform topology in the sense that for any $T>0$ , $\\lim _{L\\rightarrow \\infty } \\sup _{x\\in [-1,1],t\\in [0,T]} \\left|\\frac{1}{L}\\eta ^L(Lx,L^2t)-\\widetilde{f}(x,t)\\right|=0,$ in probability.",
"This result is not much of a surprise.",
"For this range of $\\lambda $ , the wall is pushing the trajectory $\\eta $ away, so that for most of the time $\\eta (t)$ lies in the wall-free zone.",
"This is the reason why the effect of the wall does not appear in the scaling limit.",
"We believe that this also to be the case for $\\lambda =2$ , but the fact that $\\sqrt{L}$ contact with the wall can appear at equilibrium instead of $O(1)$ brings an additional technical challenge."
],
[
"Toward the scaling limit for pinning on a sticky substrate",
"We move now to the case $\\lambda =\\infty $ .",
"In that case (recall (REF )), the corners on the interface flip with rate 1 if it does not change the number of contact with the wall, with rate 2 if it adds one contact, and the contacts with the wall cannot be removed and stay forever.",
"In that case, it is known that with large probability after a time $L^2$ the dynamics ends up with a path completely stuck to the substrate (with probability tending to 1), (see [3] or Lemma REF below).",
"This implies in particular that the scaling limit in this case cannot be given by (REF ).",
"Let us try to give some heuristic justification for the PDE problem that rules the evolution of scaling limit $f$ .",
"We suppose that the polymer path consists of a pinned region where it sticks to the wall and $f \\equiv 0$ and an unpinned region which corresponds to an interval $[Ll(t),Lr(t)]$ (i.e.",
"$(l(t),r(t))$ for the rescaled picture) so that $f(t,l(t))=f(t,r(t))=0$ .",
"In the unpinned region the wall has no influence and thus Theorem REF indicates that one should have $\\partial _t f-f_{xx}=0$ .",
"What is left to be determined is the speed at which the boundary of the pinned region moves (value of of the time derivative $l^{\\prime }(t)$ and $r^{\\prime }(t)$ ) and/or the boundary condition that $f$ has to satisfy at the boundary of $(l(t),r(t))$ .",
"Remark 2.4 With the boundary condition that one considers, $f$ is not space derivable at the extremities of $[l(t),r(t)]$ .",
"In what follows, when one talk about the space derivatives of $f$ at point $l(t)$ , resp.",
"$r(t)$ , we refer to right resp.",
"left derivatives.",
"What we want to justify here is that the slope of the scaling limit at the left (resp.",
"right) boundary of the pinned region, given by $f_x(l(t),t)$ (resp.",
"$f_x(r(t),t)$ ) has to be equal to $+1$ (resp.",
"$-1$ ).",
"The reason for this is that, as the scaling is diffusive, the mean-speed of the left-boundary of the pinned zone for the non-rescaled dynamics has to be of order $1/L$ .",
"This can be achieved only if the density of down-steps near the left boundary is vanishing, and hence if $f_x(l(t),t)=1$ .",
"Combining this boundary condition with $\\partial _t f-f_{xx}$ , and doing some trigonometry it implies (at least at the heuristic level) that one must also have $l^{\\prime }(t)=-f_{xx}(t,l(t))$ and $r^{\\prime }(t)= f_{xx}(t,l(t))$ .",
"Thus, $f$ should be the solution of (REF )."
],
[
"Solving the Stefan problem",
"The problem (REF ) is slightly overdetermined but this obstacle vanishes if one considers the derivative problem, ${\\left\\lbrace \\begin{array}{ll}\\partial _t \\rho - \\rho _{xx} \\text{ in } \\quad \\left(l(t);r(t)\\right),\\\\\\rho (l(t),t)\\equiv 1 \\text{ on } [-1,l(t)], \\quad \\rho (r(t),t)=-1 \\text{ on } [l(t),1]\\\\l^{\\prime }(t)=- \\rho _x(l(t),t), \\quad r^{\\prime }(t)=\\rho _x(r(t),t),\\\\\\rho (\\cdot ,0)=\\rho _0 \\text{ on } (r_0,l_0).\\end{array}\\right.",
"}$ A problem very similar to (REF ) has been considered by Chayes and Kim in [5] but with the third line replaced by $l^{\\prime }(t)=- \\rho _x(l(t),t)/2 \\text{ and } r^{\\prime }(t)=\\rho _x(r(t),t)/2.$ Note that the formulation in [5] is slightly differs but this is what one finds after appropriate rescaling).",
"This small change has big consequences on the behavior of the solution.",
"Whereas for the problem considered by [5], the solution exists until a maximal time where $r(t)=l(t)$ for all reasonable initial condition $\\rho _0$ , our problem might show some degeneracy when $l$ and $r$ are still well apart (see Section REF ).",
"What we show in this paper is that this kind of complication does not occur for the initial conditions we are interested in.",
"Furthermore, we establish regularity and further additional properties of interest.",
"Theorem 2.5 Suppose that $f_0$ is a 1-Lipshitz function positive and smooth on $(l_0,r_0)$ and satisfies the boundary condition $f^{\\prime }(l_0)=-f^{\\prime }(r_0)=1$ .",
"Then the free boundary problem (REF ) has a unique classical solution up to time $T^*=\\frac{1}{2}\\int ^{r_0}_{l_0} f(x)\\,\\text{\\rm d}x,$ at which the area below the curve vanishes.",
"Furthermore: (i) $r(t)$ and $l(t)$ are $C^\\infty $ on on the interval $(0,T^*)$ , (ii) $f$ becomes concave on $(l(t),r(t))$ before time $T^*$ , (iii) $\\lim _{t\\rightarrow T^*}( r(t)-l(t))=0$ .",
"Equivalently if $\\rho _0$ is the derivative of a function $f_0$ that has the above properties, then (REF ) has a solution until time $T^*$ where the two boundary meets.",
"The proof of short-time existence and regularity of the boundary motion strongly relies on the work and Kim and Chayes [6], and does not require $\\rho _0$ to be the derivative of positive function, but just reasonable regularity assumption (we suppose it Lipshitz but this could be relaxed).",
"The proof of existence until a maximal time uses ideas and is inspired by the work of Grayson concerning the shrinking of curves by curvature flow [12]."
],
[
"The scaling limit for $\\lambda =\\infty $",
"We are now ready to state our result concerning the dynamics with a wall and $\\lambda =\\infty $ .",
"Theorem 2.6 Let $\\eta ^{L,\\infty }$ denote the dynamic with wall and $\\lambda =\\infty $ , and starting from a sequence of initial condition $\\eta ^L_0$ which satisfies $\\eta ^L_0(x)=Lf_0(x/L)(1+o(1)) \\text{ uniformly in $x$ when $L\\rightarrow \\infty $ },$ where $f_0$ satisfies the assumption of Theorem REF .",
"Set $f$ to be the solution of (REF ).",
"Then $\\eta ^{L,\\infty }$ converges to $f$ in law for the uniform topology in the sense that $\\lim _{L\\rightarrow \\infty } \\sup _{x\\in [-1,1],t>0} \\left|\\frac{1}{L}\\eta ^{L,\\infty }(Lx,L^2t)-f(x,t)\\right|=0,$ in probability.",
"Moreover, one can precisely estimate the time at which the dynamics terminates $ \\mathcal {T}:=\\inf \\lbrace t\\;\\geqslant \\;0 \\ | \\ \\eta (\\cdot ,t)=\\eta ^{\\min }\\rbrace .$ We have that in probability $\\lim _{L\\rightarrow \\infty } \\frac{\\mathcal {T}}{L^2}=\\int _0^1 f_0(x)\\,\\text{\\rm d}x.$ The second part of the result (REF ) can be compared to the result obtained by Caputo et al.",
"[3], where is proved that the mixing time for this dynamics is of order $L^2$ .",
"Remark 2.7 The assumption that $f$ is strictly positive in $(l_0,r_0)$ is necessary and if $f$ cancels in the middle of the interval, the scaling limit depends on the microscopic details of the initial condition and the scaling limit might be a random object even for a deterministic initial condition."
],
[
"Discussion on the scaling limit in the attractive case for $\\lambda \\in (2,\\infty )$",
"Although we are quite far from being able to prove it, we do believe that Theorem REF extends in some way to the whole localized phase $\\lambda \\in (2,\\infty )$ .",
"We summarize here the full conjecture and give an idea of the technical difficulties that arises when trying to prove it.",
"What we believe is that the polymer consists of one (or several) unpinned region situated a macroscopic distance of the interface, and pinned regions where the polymers looks locally at equilibrium (recall that when $\\lambda >2$ the polymer has a density of contacts for $\\pi ^\\lambda _L$ .",
"As before it is natural to say that in the unpinned region, the rescaled polymer must satisfy $\\partial f-f_{xx}=0$ .",
"However, the argument giving the slope of $f$ at the boundary of the unpinned region in the $\\lambda =\\infty $ case is not valid when $\\lambda $ is finite, because the boundary can now microscopically move in both direction as unpinning is allowed.",
"To guess the value of the slope at the boundary $\\partial _x f(t,l(t))$ , $\\partial _x f(t,r(t))$ , we assume that the system close to the phase separation must be in a state of local-equilibrium.",
"From the equilibrium results on polymer pinning with elevated boundary condition proved in [16], one can infer that the equilibrium slope of a polymer at the boundary of a pinned and an unpinned phase is $d_\\lambda :=1-\\frac{2}{\\lambda }.$ and that one must have $ f_x(t,l(t))=-f_x (t,r(t))=d_{\\lambda }.$ Hence, the scaling limit $f$ of $\\eta ^{L,\\lambda }(x,t)$ with $\\lambda >2$ must satisfy the following free boundary problem ${\\left\\lbrace \\begin{array}{ll}\\partial _t f- f_{xx}=0 \\quad \\text{on } (l(t),r(t)),\\\\f(\\cdot ,t)\\equiv 0 \\quad \\text{on } [-1,1]\\setminus (l(t),r(t)),\\\\f_x(l(t),t)=- f_x(r(t),t)=d_{\\lambda },\\\\l^{\\prime }(t)=- f_{xx}(l(t),t)/d_{\\lambda },\\quad r^{\\prime }(t)=f_{xx}(r(t),t)/d_{\\lambda },\\\\f(\\cdot ,0)=f_0,\\ l(0)=l_0,\\ r_0.\\end{array}\\right.",
"}$ The problem (REF ) presents additional technical difficulties when compared to (REF ), even though the difference between them is just a scaling factor.",
"The reason for this is comes from the kind of initial condition that one wants to consider.",
"If one starts from an initial condition that is positive in $(l_0,r_0)$ and Lipshitz with Lipshitz constant $d_\\lambda $ , then Theorem REF ensures that the solution to (REF ) exists and is well behaved until the time when the pinned region has vanished.",
"However, if one consider a 1-Lipshitz boundary condition, $f_{xx}(l(t),t)$ can be positive for some times, and thus the boundary are not necessarily contracting... which makes the problem more difficult to solve.",
"When the wall is sticky, there is no loss of generality in considering that there is only one unpinned region, because two regions separated by a contact with the wall stay separated and behave independently.",
"When $\\lambda <\\infty $ the situation is different: if one starts with two distinct unpinned regions $(l_1,r_1)$ and $(l_2,r_2)$ with $l_2>r_1$ , this is possible that the two region merge at a positive time.",
"Besides these obstacles on the analytical side, there are many reasons why the proof of Theorem REF cannot easily be adapted to the case $\\lambda \\in (2,\\infty )$ .",
"Indeed a part of the strategy relies on the control of the area below $\\eta $ (see for instance Lemma REF ) and this kind of argument seems very difficult to adapt when the polymer is allowed to detach itself from the wall."
],
[
"Stefan problem and statistical mechanics",
"Free boundary problems similar to (REF ) appear naturally in thermodynamics to describe the motion of phase boundary in a multiphase medium (e.g.",
"water in solid and liquid state), and for this reason have been the object of extensive studies (see the seminal paper of Stefan [19] and [18] for a survey the subject).",
"There has been then some recent efforts in the area of statistical mechanics in order to prove to show that Stefan problem can be obtained as the limiting equation of evolution for particle systems whose microscopic behavior is random.",
"Among these work we can cite [7], where Chayes and Swindle have exhibited a particle system whose hydrodynamic limit is given by a Stefan problem, and [17] where Landim and Valle proposed a microscopic modeling of Stefan freezing/melting problem and proved the weak convergence of the particle density to the solution of a free boundary equation (for a more complete bibliography we refer to the monograph [14]).",
"The main difference of the problem we study here when compared to the one considered e.g.",
"in [7] and [17] is that, microscopically, the motion of the phase boundary does not depend only on the state of the system close to the boundary, but on the whole configuration (a contact to the wall can be added anywhere and not only near the boundary).",
"This makes control of the boundary motion more difficult, and for this reason we did not use an approach based on weak convergence like in most of the literature, but something based on a classical interpretation of the partial differential equations.",
"This is the reason why we have to prove the existence of a solution of (REF ) first: we cannot prove a convergence result a priori without knowing about the existence of a classical (and sufficiently regular) solution.",
"After a preliminary version of this paper was published, De Masi et al.",
"[8] proved a convergence result for a special version of the simple exclusion process with moving sources and sinks at the boundary.",
"They conjecture that the scaling limit they obtain is a solution of a problem similar to (REF ), but for which there exists a stationary solution."
],
[
"Organization of the paper",
"In Section we prove Theorem REF .",
"In Section , we introduce a few result to that will be of use both for the proof of Theorem REF and REF .",
"In Section we prove Theorem REF .",
"Finally Section contains the proof of Theorem REF"
],
[
"Decomposition of the proof",
"To prove the result we proceed in three steps.",
"Firstly, we use results and ideas from [6] to prove the existence and the unicity of a $C^\\infty $ solution for a short-time, and to extend this solution until a time where the second derivative of $f$ becomes unbounded.",
"Proposition 3.1 If $f_0$ is smooth and Lipshitz on $[l_0,r_0]$ and satisfies the boundary condition $f^{\\prime }(l_0)=-f^{\\prime }(r_0)=1$ .",
"Then there exists $t_1$ dor which the problem (REF ) has a classical solution on $[0,t_1)$ which satisfies $\\lim _{t\\rightarrow t_1} \\sup _{x\\in (l(t),r(t))} |f_{xx}(x,t)|=\\infty .$ Furthermore $l(t)$ and r$(t)$ are $C_{\\infty }$ on $(0,t_1)$ .",
"Then we show that if $f_0$ restricted to $(l_0,r_0)$ is a concave function, the solution exists up to a maximal time where the boundaries $l$ and $r$ meet.",
"Proposition 3.2 If $f_0$ is concave on $(l_0,r_0)$ , then $t_1=T^*=\\int ^{r_0}_{l_0} f_0(x)\\,\\text{\\rm d}x/2$ and $\\lim _{t\\rightarrow T^*}(r(t)-l(t))=0.$ Finally, we show that if $f_0$ is positive, then it becomes concave before $t_1$ .",
"Proposition 3.3 If $f_0$ is positive, then there exists a time $t_2<t_1$ such that $f(\\cdot ,t)$ is concave for $t\\in [t_2,t_1)$ .",
"Theorem REF is obtained by combining the three statements together."
],
[
"Discussion on the case of general $f_0$",
"Before going into the proofs, let us first discuss about why the positivity of $f$ is required for the existence of a solution until $T^*$ .",
"We show in this section with simple examples that if this condition is violated, the solution might degenerate and the boundary condition can stop being satisfied before $l(t)$ and $r(t)$ meet.",
"First note that the fact that the area below the curve vanishes at $T^*$ is a simple consequence of $\\partial _t \\left(\\int _{l(t)}^{r(t)} f(x,t) \\,\\text{\\rm d}x\\right)=\\int _{l(t)}^{r(t)} f_{xx}(x,t) \\,\\text{\\rm d}x=f_x(l(t),t)-f_x(r(t),t)=-2.$ so that for all $t\\in (0,t_1)$ for which the solution is defined $\\int _{l(t)}^{r(t)} f(x,t) \\,\\text{\\rm d}x=\\int _{l(t)}^{r(t)} f_0(x) \\,\\text{\\rm d}x-2t.$ Thus if $f_0$ is such that $\\int _{l(t)}^{r(t)} f_0(x)<0$ and satisfies the assumption of Proposition REF the signed area $\\int _{l(t)}^{r(t)} f(x,t) \\,\\text{\\rm d}x$ is bounded away from zero uniformly in time, and it implies that the solution must degenerate before $l(t)$ collides with $r(t)$ .",
"If it were not the case, the fact that $f$ is Lipshitz would imply $\\lim _{t\\rightarrow t_1} \\int _{l(t)}^{r(t)} f(x,t) \\,\\text{\\rm d}x=0$ which is impossible.",
"In fact, even if $\\int _{l(t)}^{r(t)} f(x,t) \\,\\text{\\rm d}x$ is initially positive, there is no guarantee that the solution exists until $l$ meets $r$ .",
"We give a simple counter-example here: set $l_0=-3\\pi /2$ , $r_0=3\\pi /2$ (we choose $[l_0,r_0]$ to be the interval of definition instead of $[-1,1]$ ), and set $f_0(x):=-\\cos (x).$ The function $f_0$ satisfies the assumption of Proposition REF which implies that a solution to (REF ) exists until a positive time $t_1$ for which the second derivative explodes.",
"A quick analysis of the problem shows that as long as $f$ is negative somewhere, the graph of $f$ consists of two positive bumps that frame a negative one (see Figure REF ).",
"We call the unique interval where $f$ is negative$(z_1(t),z_2(t))$ .",
"The geometric area of the negative bump satisfies $\\partial _t\\left(\\int _{z_1(t)}^{z_2(t)}|f(x,t)|\\,\\text{\\rm d}x\\right)=f_x(z_1(t),t)-f_x(z_2(t),t)> -2,$ where the last inequality comes from the fact that $|f_x|<1$ in $(r(t),l(t))$ for all positive time.",
"Hence, if the solution continues to exists, $f$ stays negative somewhere at least until a time $t^{\\prime }>\\left(\\int _{-\\pi /2}^{\\pi /2}|f_0(x)|\\,\\text{\\rm d}x\\right)/2=1$ .",
"However, from Equation (REF ), for $t=t^{\\prime }$ $\\int _{l(t^*)}^{r(t^*)} f(x,t^*)\\,\\text{\\rm d}x =2-2t^*<0,$ meaning that $l$ and $r$ cannot meet for $t\\;\\geqslant \\;t^*$ .",
"Figure: A simple example of an initial condition for which the solution stops to exits before the boundary meets.In fact a closer inspection to the proof of Proposition REF and REF reveals that at time $t_1$ , when the second derivative $\\Vert f_{xx}\\Vert _{\\infty }$ explodes, the two external positive bumps disappear and $f$ becomes completely negative.",
"As the boundary condition for $f_x$ is not satisfied anymore, it is impossible to define any reasonable notion of solution for $t\\;\\geqslant \\;t_1$ .",
"We refer the reader to [6] for some additional discussion on what can occur after $t_1$ for the derivative problem (REF ) with different boundary conditions."
],
[
"Proof of the short-time existence: Proposition ",
"Instead of solving (REF ) directly, we solve the corresponding derivative problem (REF ).",
"Given a solution $(\\rho ,l,r)$ to (REF ) with initial condition $\\rho _0=f^{\\prime }_0$ , the triplet $(f,l,r)$ with $f$ defined by $f(x,t)=\\int _{l(t)}^{r(t)}\\rho (x,t)\\,\\text{\\rm d}x,$ is a solution of (REF ).",
"Indeed, the initial condition is satisfied and we have $\\begin{split}\\partial _t f(r(t),t)&=-r^{\\prime }(t)+l^{\\prime }(t)+\\int _{l(t)}^{r(t)}\\rho _{xx}(x,t)\\,\\text{\\rm d}x=0,\\\\\\partial _t f(x,t)&=-r^{\\prime }(t)+\\int _{l(t)}^{x}\\rho _{xx}(x,t)\\,\\text{\\rm d}x=\\rho _x(x,t)=f_{xx}(x,t) \\ \\text{for} \\ x\\in (l(t),r(t)).\\end{split}$ To solve (REF ), we adapt a method developed in [6] for a similar contracting one dimensional Stefan problem.",
"In what follows we denote by $\\rho _0$ the initial condition $f_0$ .",
"Let us consider two (fixed) continuous functions $\\widetilde{l}(t)$ and $\\widetilde{r}(t)$ , $\\widetilde{l}$ increasing, $\\widetilde{r}$ decreasing, that satisfy $\\widetilde{l}(0)=l_0$ and $\\widetilde{r}(0)=r_0$ .",
"We consider the heat equation on the contracting domain $(\\widetilde{l},\\widetilde{r}):=\\cup _{t\\;\\leqslant \\;T} (\\widetilde{l}(t),\\widetilde{r}(t))\\times \\lbrace t \\rbrace $ with fixed boundary condition $+1$ on the left and $-1$ on the right, ${\\left\\lbrace \\begin{array}{ll}\\partial _t \\rho -\\rho _{xx}=0 \\text{ on } \\quad \\left(\\widetilde{l}(t); \\widetilde{r}(t)\\right),\\\\\\rho (\\widetilde{l}(t),t)=-\\rho (\\widetilde{r}(t),t)=1,\\ \\rho (0,\\cdot )=\\rho _0.\\end{array}\\right.",
"}$ We need the following technical result.",
"Lemma 3.4 Given $\\rho _0$ derivable that satisfies the boundary condition, and setting $\\bar{l}=(4\\Vert \\rho ^{\\prime }_0\\Vert )^{-1}$ .",
"If $\\tau $ is such that $\\forall t<\\tau ,\\ \\widetilde{l}(t)\\;\\leqslant \\;l_0+2\\bar{l}, \\quad \\text{and} \\quad \\widetilde{r}(t)\\;\\geqslant \\;r_0-2\\bar{l},$ then the solution $\\rho $ of (REF ) satisfies $\\begin{split}\\rho (t,x)&\\;\\geqslant \\;0, \\quad \\forall t\\;\\leqslant \\;\\tau ,\\ \\forall x\\in [\\widetilde{l}(t),l_0+2\\bar{l}], \\\\\\rho (t,x)&\\;\\leqslant \\;0, \\quad forall t\\;\\leqslant \\;\\tau ,\\ \\forall x\\in [r_0-2\\bar{l},\\widetilde{r}(t)] .\\end{split}$ Of course $\\bar{l}$ depends on the length the interval $[l_0,r_0]$ , but, as $\\rho _0$ satisfies the prescribed boundary condition, one has $r_0-l_0\\;\\geqslant \\;2(\\Vert \\rho ^{\\prime }_0\\Vert _{\\infty })^{-1}=8\\bar{l}.$ By symmetry we can restrict ourselves to the first statement.",
"Because of the boundary condition, the solution that we have to consider is larger than the solution of the heat-equation on $[l_0,l_0+4\\bar{l}]$ with initial condition $\\rho _0$ and Dirichlet boundary condition $+1$ at $l_0$ and $-1$ on $l_0+4\\bar{l}$ .",
"Then we notice that $\\rho _0(x)\\;\\geqslant \\;1-2(x-l_0)\\Vert \\rho ^{\\prime }_0\\Vert _{\\infty }=\\rho _{\\min }(x), \\quad \\forall x\\in [l_0,l_0+4\\bar{l}].$ Finally, we remark that $\\rho _{\\min }(x)\\;\\geqslant \\;0$ on $[l_0,l_0+2\\bar{l}]$ and that as $\\rho _{\\min }$ is a stationary solution of the heat-equation on $[l_0,l_0+4\\bar{l}]$ mentioned above, the inequality remains valid for all further time.",
"Given $\\tau $ that satisfies the assumption of Lemma REF and let us consider $\\mathcal {J}:=\\left\\lbrace (\\phi _1,\\phi _2) \\in \\left(L^\\infty ([0,t_0] \\right)^2 \\ | \\ \\forall t\\in [0,t_0],\\ \\phi _1(t)\\in [0,1], \\ \\phi _2(t)\\in [-1,0]\\right\\rbrace .$ We are going to construct an application $\\Phi : \\mathcal {J}\\rightarrow \\mathcal {J},$ in two steps.",
"First, given $(\\phi _1,\\phi _2)\\in \\mathcal {J}$ , we define two one-sided contracting Stefan problems, with respective initial domains $[l_0,l_0+2\\bar{l}]$ and $[r_0-2\\bar{l},r_0]$ , and an imposed boundary condition on the non moving side: $l_0+2\\bar{l}$ and $r_0-2\\bar{l}$ respectively, given by $\\phi _1$ and $\\phi _2$ .",
"$\\begin{split}{\\left\\lbrace \\begin{array}{ll}\\partial _t \\rho ^{(1)}- \\rho ^{(1)}_{xx}=0 \\quad \\text{ on } \\left(l(t); l_0+2\\bar{l}\\right),\\\\\\rho ^{(1)}(l(t),t)=1, \\quad l^{\\prime }(t)=-\\rho ^{(1)}_x(l(t),t),\\\\\\rho ^{(1)}(l_0+2\\bar{l},t)=\\phi _1(t), \\\\\\rho ^{(1)}(x,0)=\\rho _0(x) \\text{ on } [l_0,l_0+2\\bar{l}]\\end{array}\\right.",
"}\\\\{\\left\\lbrace \\begin{array}{ll}\\partial _t \\rho ^{(2)}-\\rho ^{(2)}_{xx}=0 \\quad \\text{ on } \\left(r_0-2\\bar{l},r(t)\\right),\\\\\\rho ^{(2)}(r(t),t)=-1, \\quad r^{\\prime }(t)=\\rho ^{(2)}_x(r(t),t),\\\\\\rho ^{(2)}(r_0-2\\bar{l}, t)=\\phi _2(t), \\\\\\rho ^{(2)}(x,0)=\\rho _0(x) \\text{ on } [r_0-2\\bar{l},r_0].\\end{array}\\right.",
"}\\end{split}$ According to [5], these two problem have a solution until the time when $l(t)$ meets $l_0+2\\bar{l}$ or $r(t)$ meets $r_0-2\\bar{l}$ respectively (more precisely, to fit exactly the setup of [5] where $\\rho \\equiv 0$ on the moving boundary, one must consider the problems solved by $1-\\rho ^{(1)}$ and $1+\\rho ^{(2)}$ ), and the boundary $r$ and $l$ are $C^\\infty $ on $(0,T)$ where $T$ is the time where the solution ceases to exist.",
"For technical purpose we want to guarantee that for any choice of $(\\phi _1,\\phi _2)$ , the boundaries $l$ and $r$ need some time to come half-way towards the fixed boundary.",
"Lemma 3.5 Let $(\\rho ^{(1)},l)$ and $(\\rho ^{(2)},r)$ be solutions of (REF ) with boundary condition $(\\phi _1,\\phi _2)\\in \\mathcal {J}$ , and initial condition $\\rho _0$ .",
"There exists a universal constant $c$ such that for all $t\\;\\leqslant \\;c \\Vert \\rho ^{\\prime }_0\\Vert ^2$ , $l(t)\\;\\leqslant \\;l_0+\\bar{l} \\quad \\text{and} \\quad r(t)\\;\\geqslant \\;r_0-\\bar{l}.$ By symmetry it is sufficient to perform the proof only for $l(t)$ .",
"We suppose also that $l_0=0$ .",
"Set for $x\\in (l(t),2\\bar{l}]$ , $f^{(1)}(x,t)=\\int _{l(t)}^{x}\\rho ^{(1)}(x,t)\\,\\text{\\rm d}x.$ The reader can check that $(f^{(1)},l)$ is a solution of ${\\left\\lbrace \\begin{array}{ll}\\partial _t f^{(1)}- f^{(1)}_{xx}=0 \\quad \\text{ on } \\left(l(t); l_0+2\\bar{l}\\right),\\\\f^{(1)}(l(t),t)=0, \\quad l^{\\prime }(t)=-f^{(1)}_{xx}(l(t),t),\\\\f_x^{(1)}(l_0+2\\bar{l},t)=\\phi _1(t), \\quad f_x^{(1)}(l(t),t)=1,\\\\f^{(1)}(x,0)=f_0(x) \\text{ on } [l_0,l_0+2\\bar{l}].\\end{array}\\right.",
"}$ The solution $f^{(1)}$ is monotone in $f_0$ and $\\phi _1$ : it decreases if $f_0$ and/or $\\phi _1$ are decreased.",
"Thus $l(t)$ is smaller than $l_{\\min }(t)$ which is obtained by taking $\\phi _1\\equiv 0$ and $f^{(1)}(\\cdot ,0)$ to be the integral of $\\rho _{\\min }$ $f^{(1)}(\\cdot ,0)=x-\\Vert \\rho ^{\\prime }_0\\Vert x^2.$ By diffusive scaling $\\inf \\lbrace t\\ | \\ l_{\\min }=\\bar{l} \\rbrace =c\\Vert \\rho ^{\\prime }_0\\Vert ^{-2}_{\\infty }.$ Given $\\phi _1$ and $\\phi _2$ , we generate $r(t)$ and $l(t)$ that are given by the solutions of problems (REF ) with initial condition $\\rho _0$ .",
"Then we define $\\bar{\\rho }$ to be the solution of the heat equation on the contracting domain $(l,r):=\\cup _{t\\;\\leqslant \\;T} (l(t),r(t))\\times \\lbrace t \\rbrace $ with $+1$ $-1$ boundary condition and set $\\Phi (\\phi _1,\\phi _2)(t):=\\left(\\bar{\\rho }(l_0+2\\bar{l},t), \\bar{\\rho }(r_0-2\\bar{l},t)\\right).$ The fundamental building brick of our proof is the following adaptation of [6].",
"Proposition 3.6 There exists $t_0(\\Vert \\rho _x\\Vert _{\\infty })$ such that the function $\\Phi : \\mathcal {J} \\rightarrow \\mathcal {J}$ is a contraction map for the $l_\\infty $ norm.",
"In other words, there exists $m(t_0)<1$ , such that for every $(\\phi _1,\\phi _2), (\\bar{\\phi }_1,\\bar{\\phi }_2)\\in \\mathcal {J}$ , $\\sup _{t \\in [0,t_0]} | \\Phi (\\phi _1,\\phi _2)(t)-\\Phi (\\bar{\\phi }_1,\\bar{\\phi }_2)(t)|\\;\\leqslant \\;m(t_0)|(\\phi _1,\\phi _2)(t)-(\\bar{\\phi }_1,\\bar{\\phi }_2)(t)|$ where $|\\cdot |$ stands for the $l_\\infty $ norm in ${\\mathbb {R}} ^2$ .",
"See [6].",
"Lemma REF is needed as one need that for small time $l(t)$ , and $r(t)$ stay at a positive distance from the fixed boundary uniformly in $(\\bar{\\phi }_1,\\bar{\\phi }_2)\\in \\mathcal {J}$ Note that in Proposition 3.1, the time $t_0$ is said to depend of four different quantities, but the reader can check that they can all be expressed in term of $\\Vert \\rho ^{\\prime }_0\\Vert _{\\infty }$ .",
"The solution of (REF ) until time $t_0$ choose $l(t)$ and $r(t)$ to be moving boundary condition generated by the problems (REF ) with initial condition $\\rho _0$ and boundary condition $(\\phi _1,\\phi _2)$ given by the unique fixed point of $\\Phi $ .",
"Then from the definition of $\\Phi $ , the solution of the heat equation in the contracting domain $(l,r)$ satisfies the boundary condition of (REF ).",
"The solution is unique because of unicity of the fixed point of the contraction and is smooth for positive times because as stated in [5] the one sided problems (REF ) generate boundaries that are $C_\\infty $ for positive time.",
"Let us now show that the solution can be extended until the derivative explodes.",
"Suppose that the solution exists and is smooth until a time $t_2$ and that $\\sup _{t< t_2} \\max _{x\\in (l(t),r(t))} |\\rho _{x}(x,t)|\\;\\leqslant \\;K<\\infty $ Then taking the $t_0$ corresponding to $K$ in Lemma REF , we can take the solution at time $t_2-(t_0/2)$ and extend it until time $t_2+(t_0/2)$ .",
"Smoothness of the motion of the boundary at the time $t_2-t_0/2$ is guaranteed by unicity of short time solutions.",
"Iterating this procedure, we see that the maximal time for which a smooth solution exists must satisfies (REF ).",
"$\\Box $"
],
[
"The convex case: proof of Proposition ",
"We introduce the notation $k(x,t):=-f_{xx}(t).$ When $f$ is concave $k$ is positive but as in this section we prove some technical results that are also valid also when $k$ is allowed to be negative, we will mention to the reader when we suppose that $k$ is positive.",
"The line of the proof is to show first that around time $t_1$ , we must have $\\lim _{t \\rightarrow T_1} \\int _{l(t)}^{r(t)} k\\log k=\\infty ,$ and then to show that $ \\int _{l(t)}^{r(t)} k\\log k \\,\\text{\\rm d}x$ can be large only if the area below the graph of $f$ is small.",
"Recalling Equation (REF ), the area is small only if $t$ is close to $T^*$ .",
"The combination of these two statements implies that one must have $t_1=T^*$ .",
"The inspiration for many ingredients of this proof comes from [12].",
"The first point can be stated as follows Lemma 3.7 At any positive time $t<t_1$ , we have $\\Vert k(\\cdot ,t)\\Vert _{\\infty }\\;\\leqslant \\;\\max \\left( \\Vert k(\\cdot ,0)\\Vert _{\\infty }, k(t,r(t)), k(t,l(t))\\right).$ and also $\\max _{s\\;\\leqslant \\;t} \\Vert k_x(\\cdot ,t)\\Vert _{\\infty }\\;\\leqslant \\;\\max _{s\\;\\leqslant \\;t}\\left( \\Vert k_x(\\cdot ,0)\\Vert _{\\infty }, k(s,r(s))^2, k(s,l(s))^2\\right).$ As a consequence $\\limsup _{t\\rightarrow t_1} \\int ^{r(t)}_{l(t)} (k\\log k)\\mathbf {1}_{k\\;\\geqslant \\;0} \\,\\text{\\rm d}x=\\infty .$ To prove the result, we notice that the boundary constraint for our problem yields a simple relation between $k$ and its derivative at the boundary.",
"Lemma 3.8 We have for all $t\\in (0,t_1)$ .",
"$k_x(l(t),t)=-k^2(l(t),t) \\text{ and } k_x(r(t),t)=k^2(r(t),t).$ The result is obtained by derivating the equality $f_x(t,l(t))=1$ which gives $\\partial t \\left(f_x(l(t),t)\\right)= -l^{\\prime }(t) k(l(t),t)- k_x(l(t),t)=-(k_x+k^2)(l(t),t)=0.$ The proof for $r(t)$ is similar.",
"Inside the interval $(l,r)$ , $k$ evolves according to the heat equation.",
"This implies that its local maxima decrease and its local minima increase.",
"Furthermore, equation (REF ) guarantees that $k_x$ never vanishes on the boundary at positive times so that local extrema cannot be created from the boundary.",
"Hence at all times, local minima and maxima of $k$ in $(l,r)$ are smaller than $ \\Vert k(\\cdot ,0)\\Vert _{\\infty }$ in absolute value.",
"Thus, if the overall maximum is larger than $\\Vert k(\\cdot ,0)\\Vert _{\\infty }$ , it must be reached on the boundary.",
"For the derivative $k_x$ there is no easy argument that prevents the creation of new local maxima $k_x$ from the boundary (although we do not believe it can occur).",
"However if $t$ is such that $\\Vert k_x(\\cdot ,t)\\Vert _{\\infty }=\\max _{s\\;\\leqslant \\;t} \\Vert k_x(\\cdot ,s)\\Vert _{\\infty },$ the fact that local maxima of $k_x$ in $(l,r)$ decrease and local minima increase implies that the overall maximum of $k_x$ is realized on the boundary.",
"This yields the second result.",
"Finally, consider $K\\;\\geqslant \\;\\max ( \\Vert k(\\cdot ,0)\\Vert _{\\infty }, \\sqrt{\\Vert k_x(\\cdot ,0)\\Vert _{\\infty }})$ , and $t_K$ the first time where $\\Vert k(\\cdot ,t)\\Vert _{\\infty }=K$ .",
"From the two first points, the maximum of $k$ is reached on the boundary of $[l,r]$ , and thus $k_x$ is bounded by $K^2$ .",
"It yields the following inequality $\\int ^{r(t_k)}_{l(t_k)} (k\\log k)\\mathbf {1}_{k\\;\\geqslant \\;0} \\,\\text{\\rm d}x\\;\\geqslant \\;e^{-1}(r_0-l_0)\\\\+\\int _{l(t_K)}^{l(t_K)+\\frac{1}{K}}(K-K^2(x-l(t_K)))\\log (K-K^2(x-l(t_K)))\\,\\text{\\rm d}x.$ After a change of variable the reader can check that second term is equal to $\\frac{1}{2}\\left(\\log K-1/2\\right)$ , which allows us to conclude by letting $K$ go to infinity.",
"Now are ready to prove that $t_1=T^*$ when $f_0$ is concave.",
"To do so, we suppose that $t_1\\;\\leqslant \\;T^*-\\varepsilon $ for some positive $\\varepsilon $ and show that this implies $\\int ^{r(t)}_{l(t)} (k\\log k) \\,\\text{\\rm d}x \\text{ is uniformly bounded on } [0,t_1),$ which contradicts Lemma REF .",
"Our strategy for the proof is to show that until time $t_1$ the time derivative of $\\int ^{r(t)}_{l(t)} (k\\log k)\\,\\text{\\rm d}x$ is uniformly bounded.",
"$\\partial _t\\left(\\int ^{r(t)}_{l(t)} k \\log k \\,\\text{\\rm d}x \\right)\\\\= -l^{\\prime }(t)(k\\log k)(l(t),t)+r^{\\prime }(t)(k\\log k)(r(t),t)+ \\int ^{r(t)}_{l(t)}\\partial _t(k \\log k)\\,\\text{\\rm d}x\\\\=-(k^2\\log k)(l(t),t)-(k^2\\log k)(r(t),t)+\\int ^{r(t)}_{l(t)} k_{xx} (\\log k+1)\\,\\text{\\rm d}x \\\\= -(k^2\\log k)(l(t),t)-(k^2\\log k)(r(t))+[k_x(\\log k+1)]_{l(t)}^{r(t)}-\\int ^{r(t)}_{l(t)} \\frac{k^2_{x}}{k}\\,\\text{\\rm d}x ,$ where the third inequality is obtained by using integration by parts.",
"The relation REF between $k$ and its derivative makes some of the terms cancel each other and we end up with $\\partial _t\\left(\\int ^{r(t)}_{l(t)} k \\log k \\,\\text{\\rm d}x\\right)=k^2(l(t),t)+k^2(r(t),t)-\\int ^{r(t)}_{l(t)} \\frac{k^2_{x}}{k}\\,\\text{\\rm d}x.$ In order to show that the r.h.s.",
"is uniformly finite, we first show that it is not possible to have, on the graph of $f$ , $k$ large on an arc whose total curvature is close to $\\pi /4$ near one of the boundaries.",
"If it were not the case, the concavity of $f$ would imply that the area below the graph of $f$ is small.",
"We need to introduce some definitions.",
"Set $\\begin{split}a(K,t)&:=\\inf \\lbrace x\\;\\geqslant \\;l(t) \\ | \\ k(x,t) \\;\\leqslant \\;K^2\\rbrace , \\\\b(K,t)&:=\\sup \\lbrace x\\;\\leqslant \\;r(t) \\ | \\ k(x,t) \\;\\leqslant \\;K^2\\rbrace .\\end{split}$ Lemma 3.9 We have for all $t$ , $f(a(K,t),t)\\;\\leqslant \\;K^{-2}.$ In addition, if $\\int _{l(t)}^{a(K,t)} k \\,\\text{\\rm d}x\\;\\geqslant \\;1-\\delta $ and $f$ is concave, $\\int _{l(t)}^{r(t)} f \\,\\text{\\rm d}x\\;\\leqslant \\;\\left(K^{-2}+\\delta (r_0-l_0)\\right)(r_0-l_0).$ If $\\int _{b(K,t)}^{r(t)} k \\,\\text{\\rm d}x \\;\\leqslant \\;1-\\delta $ then $f(B(K,t),t)\\;\\leqslant \\;K^{-2}$ and the inequality (REF ) also holds.",
"As a consequence, we get that if $K$ and $\\delta $ are sufficiently large resp.",
"small so that $ \\left(K^{-2}+\\delta (r_0-l_0)\\right)(r_0-l_0)\\;\\leqslant \\;2\\varepsilon ,$ then for all $t< t_1\\;\\leqslant \\;T^*-\\varepsilon $ , $\\int _{l(t)}^{a(K,t)} k \\,\\text{\\rm d}x< 1-\\delta \\quad \\text{ and } \\quad \\int _{b(K,t)}^{r(t)} k \\,\\text{\\rm d}x < 1-\\delta ,$ because if not, the conclusion of Lemma REF would contradict (REF ).",
"We are going to use this information to bound the r.h.s.",
"of (REF ) from above, by using the following functional inequality sometimes referred to as Agmon's inequality.",
"We include its proof at the end of the section for the sake of completeness.",
"Lemma 3.10 Let $\\gamma $ be a function in $L_2({\\mathbb {R}} _+)$ whose derivative is in $L_2({\\mathbb {R}} _+)$ .",
"Then $\\Vert \\gamma \\Vert ^4_{\\infty }\\;\\leqslant \\;4 \\left(\\int _{{\\mathbb {R}} _+} \\gamma ^2 \\,\\text{\\rm d}x\\right) \\left(\\int _{{\\mathbb {R}} _+}\\gamma _x^2 \\,\\text{\\rm d}x\\right).$ We apply the inequality to $\\gamma $ defined as $\\gamma (x)={\\left\\lbrace \\begin{array}{ll}\\sqrt{k(x+l(t),t)}-K &\\text{ if } x\\;\\leqslant \\;a(K,t)-l(t),\\\\0 &\\text{ if } x\\;\\geqslant \\;a(K,t)-l(t).\\end{array}\\right.",
"}$ With this definition, (REF ) reads $\\int _{{\\mathbb {R}} _+} \\gamma ^2 \\,\\text{\\rm d}x\\;\\leqslant \\;\\int _{l(t)}^{a(K,t)} k\\,\\text{\\rm d}x \\;\\leqslant \\;1-\\delta .$ Then we obtain that $\\int ^{a(K,t)}_{l(t)} \\frac{k^2_{x}}{k}\\,\\text{\\rm d}x=4\\int _{l(t)}^{a(K,t)} \\gamma ^2_{x}\\,\\text{\\rm d}x\\;\\geqslant \\;\\frac{\\Vert \\gamma \\Vert ^4_{\\infty }}{\\int _{{\\mathbb {R}} _+} \\gamma ^2 \\,\\text{\\rm d}x}\\;\\geqslant \\;\\frac{\\left(k(l(t))^{1/2}-K\\right)^4}{1-\\delta }.$ If $k(l(t))\\;\\geqslant \\;16 K^2/\\delta ^2$ this is larger than $k^2(l(t))$ , and in any case it is non-negative.",
"Hence $k^2(l(t))-\\int ^{a(K,t)}_{l(t)} \\frac{k^2_{x}}{k}\\,\\text{\\rm d}x\\;\\leqslant \\;256 K^4/\\delta ^4.$ Symmetrically $k^2(r(t))-\\int ^{r(t)}_{b(K,t)} \\frac{k^2_{x}}{k}\\,\\text{\\rm d}x\\;\\leqslant \\;256 K^4/\\delta ^4,$ and hence, combining these inequalities with (REF ), we have $\\partial _t\\left(\\int ^{r(t)}_{l(t)} k \\log k\\right)\\;\\leqslant \\;512 K^4/\\delta ^4.$ This implies that $\\int ^{r(t)}_{l(t)} k \\log k$ remains bounded, and gives a contradiction to Lemma REF .",
"To finish the proof of Proposition REF , we show now that, when $f_0$ is concave, $\\lim _{t\\rightarrow T^*} r(t)-l(t)=0.$ Because of (REF ) and the fact that $f$ is a Lipshitz function, we have $\\max _{x\\in [l,r]} f(x,t)\\;\\leqslant \\;\\sqrt{2(T^*-t)}.$ Since $\\min _{x\\in [l,r]} k(\\cdot ,t)$ is an increasing function of time, for $t\\;\\geqslant \\;T^*/2$ , $k$ is uniformly bounded away from zero, say $k\\;\\geqslant \\;\\eta >0$ .",
"This combined with with (REF ) implies that $(r-l)(t)\\;\\leqslant \\;2 \\left(\\frac{8(T^*-t)}{\\eta }\\right)^{1/4}.$ $\\Box $ We end this section with the proof of Lemma REF and Lemma REF As $f$ is a Lipshitz function $\\int ^{a(K,t)}_{l(t)} k \\,\\text{\\rm d}x =1-f_x(a(K,t),t)\\;\\leqslant \\;2$ .",
"Hence, using again Lipshitzianity and the definition of $a(K,t)$ , $f(a(K,t),t)\\;\\leqslant \\;a(K,t)-l(t)\\;\\leqslant \\;K^{-2}\\int ^{a(K,t)}_{l(t)} k \\,\\text{\\rm d}x \\;\\leqslant \\;2K^{-2}.$ If $f$ is concave and $\\int ^{a(K,t)}_{l(t)} k \\,\\text{\\rm d}x\\;\\geqslant \\;1-\\delta $ , then $\\forall x\\;\\geqslant \\;a(K,t),\\ f_x(x,t) \\;\\leqslant \\;f_x(a(K,t),t)=\\delta ,$ and hence for all $x\\;\\geqslant \\;(a(K,t))$ $f(x,t)\\;\\leqslant \\;f_x(a(K,t),t)+\\delta (x-a(K,t))\\;\\leqslant \\;2K^{-2}+\\delta (r_0-l_0).$ As the bound also holds for $x\\;\\leqslant \\;a(K,t)$ , we can integrate the inequality over $[r(t),l(t)]$ to conclude.",
"It is sufficient to prove the result when the maximum of $\\gamma $ is attained at 0 (if it is attained at a positive value $x_0$ , we can then consider $\\gamma ( \\cdot -x_0)$ restricted to ${\\mathbb {R}} _+$ which has the effect of making the r.h.s.",
"of (REF ) smaller).",
"As the inequality is invariant by the scalings $\\gamma \\rightarrow \\lambda \\gamma $ and $\\gamma \\rightarrow \\gamma ( \\lambda \\ \\cdot )$ with $\\lambda \\in (0,\\infty )$ , we can also assume that $\\int \\gamma ^2=\\int \\gamma _x^2=1$ .",
"Then we have $0\\;\\leqslant \\;\\int (\\gamma -\\gamma _x)^2\\,\\text{\\rm d}x=\\int \\gamma ^2\\,\\text{\\rm d}x+ \\int \\gamma _x^2\\,\\text{\\rm d}x -\\gamma (0)^2,$ and hence $\\gamma (0)^4\\;\\leqslant \\;4$ ."
],
[
"The concavification of positive initial condition: proof of Proposition ",
"Let us now move to the non-convex case.",
"We consider the inflection points on the graph of $f$ , that is, the points around which $k$ changes sign.",
"When $t>0$ there are only finitely many of them, and furthermore, their number is decreasing in time and they move continuously.",
"We want to show that the last inflection point disappears before $t_1$ .",
"We suppose that this is not the case, and then we show that $\\int ^{r(t)}_{l(t)} k \\log k\\mathbf {1}_{k\\;\\geqslant \\;0}$ remains bounded when $t$ approaches $t_1$ .",
"The first thing we do is to place ourselves in a neighborhood of $t_1$ where the number of inflection points is constant (it has to be even because they are the extremities of arcs where $k$ is negative).",
"Let $i_1,\\dots ,i_{2p}$ denote the abscissa of these inflection points.",
"Then $\\int ^{r(t)}_{l(t)} k \\log k\\mathbf {1}_{k\\;\\geqslant \\;0}\\,\\text{\\rm d}x= \\int ^{r(t)}_{i_1(t)} k \\log k\\,\\text{\\rm d}x +\\sum _{j=1}^{p-1} \\int _{i_{2j}(t)}^{i_{2j+1}(t)} k \\log k \\,\\text{\\rm d}x+\\int _{i_{2p}}^{r(t)} k \\log k \\,\\text{\\rm d}x.$ First notice that using integration by part, we have $\\partial _t\\left( \\int ^{i_{2j}(t)}_{i_{2j+1}(t)} k \\log k \\,\\text{\\rm d}x\\right)= -\\int ^{i_{2j}(t)}_{i_{2j+1}(t)} \\frac{k^2_x}{k} \\,\\text{\\rm d}x \\;\\leqslant \\;0.$ Hence to prove that $\\int ^{r(t)}_{l(t)} k \\log k\\mathbf {1}_{k\\;\\geqslant \\;0} \\,\\text{\\rm d}x$ remains bounded we just have to check that the extremal terms in equation (REF ) do not explode.",
"By symmetry we can concentrate on $\\int _{l(t)}^{i_1(t)} k \\log k \\,\\text{\\rm d}x$ .",
"Similarly to (REF ), we have $\\partial _t\\left(\\int ^{i_1(t)}_{l(t)} k \\log k \\,\\text{\\rm d}x \\right)=k^2(l(t),t)-\\int ^{i_1(t)}_{l(t)} \\frac{k^2_{x}}{k}\\,\\text{\\rm d}x.$ Note that $\\int _{l(t)}^{i_1(t)} k(l(t),t)\\,\\text{\\rm d}x=1-f_x(i_1(t))$ is decreasing, because $f_x(i_1(t))$ is a local minimum (thus increases).",
"Suppose that $\\lim _{t\\rightarrow t_1} \\int _{l(t)}^{i_1(t)} k \\,\\text{\\rm d}x< 1.$ Then for $t$ close to $t_1$ can use Lemma REF with $\\gamma (x)={\\left\\lbrace \\begin{array}{ll}\\sqrt{k(x+l(t),t)} &\\text{ if } x\\;\\leqslant \\;i_1(t)-l(t),\\\\0 &\\text{ if } x\\;\\geqslant \\;i_1(t)-l(t).\\end{array}\\right.",
"}$ and obtain that $\\int ^{i_1(t)}_{l(t)} \\frac{k^2_{x}}{k}\\,\\text{\\rm d}x< k^2(l(t),t).$ and hence that $\\int ^{i_1(t)}_{l(t)} k \\log k$ is decreasing in a neighborhood of $t_1$ .",
"Hence $\\int ^{i_1(t)}_{l(t)} k \\log k$ can only explode if $\\lim _{t\\rightarrow t_1} \\int _{l(t)}^{i_1(t)} k\\,\\text{\\rm d}x\\;\\geqslant \\;1.$ In (REF ) holds, let us consider $i_0(t)< i_1(t)$ such that $\\int _{l(t)}^{i_0(t)} k(l(t),t)=1.$ The point $i_0(t)$ is the point at which the leftest local maximum of $f$ is attained.",
"Let us first show that if $\\int ^{i_1(t)}_{l(t)} k \\log k \\,\\text{\\rm d}x$ is unbounded, then in a neighborhood of $t_1$ $f$ must reach its maximum at $i_0(t)$ and $f(i_0,t)$ is small.",
"Let us define $a(K,t)$ like in (REF ).",
"Then if $\\int _{l(t)}^{a(K,t)} k \\,\\text{\\rm d}x\\;\\leqslant \\;1-\\delta $ then the computation of the previous section (REF ) to (REF ) are still valid and thus $\\partial _t\\left(\\int ^{i_1(t)}_{l(t)} k \\log k\\right)\\;\\leqslant \\;\\frac{256 K^4}{\\delta ^4}.$ This implies that if $\\int ^{i_1(t)}_{l(t)} k \\log k \\,\\text{\\rm d}x$ is unbounded then for every $K$ and $\\delta $ , there is some $t\\;\\leqslant \\;t_1$ such that $\\int _{l(t)}^{a(K,t)} k(x,t) \\,\\text{\\rm d}x\\;\\geqslant \\;1-\\delta .$ If (REF ) holds, then Lemma REF and the concavity of $f$ restricted to $[0,i_1(t)]$ imply that $f(i_0(t),t)\\;\\leqslant \\;\\delta (i_0(t)-a(K,t))+f(a(K,t),t)\\;\\leqslant \\;K^{-2}+\\delta (r_0-l_0).$ If $f(i_0(t),t)$ is not the overall maximum of $f$ then it means that $f$ admits a local minimum which is smaller than $K^{-2}+\\delta (r_0-l_0)$ .",
"As local minima of $f$ are increasing, they must all be higher than local minima of $f_0$ at all time, and thus cannot be smaller than $K^{-2}+\\delta (r_0-l_0)$ , if $K$ is large enough and $\\delta $ small enough.",
"Let us write the conclusion of this reasoning as a lemma.",
"Lemma 3.11 If $\\limsup _{t\\rightarrow t_1} \\int ^{i_1(t)}_{l(t)} k \\log k\\,\\text{\\rm d}x= \\infty ,$ then in a neighborhood of $t_1$ , $f(\\cdot ,t)$ has its maximum at $i_0\\in (l(t),i_1(t))$ , and $f$ has no other local extremum.",
"Furthermore $\\lim _{t\\rightarrow t_1} f(i_0(t),t)=0.$ Our last task is to show that this is impossible, and thus that $f$ should become convex before $t_1$ .",
"As for $t$ sufficiently large $i_0(t)$ is the only local maximum of $f$ $\\lim _{t\\rightarrow t_1} \\int _{i_{2p}(t)}^{r(t)} k \\,\\text{\\rm d}x= \\lim _{t\\rightarrow t_1} (1+f_x(i_{2p}(t),t))<1.$ Indeed for sufficiently large $t$ , $f_x(i_{2p}(t),t)\\;\\leqslant \\;0$ because there is no local maximum of $f$ in $[i_{2p}(t),r(t)]$ ; and $f_x(i_{2p}(t),t)$ is a local maximum of $f_x$ and thus decreases strictly in time.",
"Note also that $\\int _{i_{2p}(t)}^{r(t)}k \\,\\text{\\rm d}x > \\int _{i_{2p-1}(t)}^{r(t)}k \\,\\text{\\rm d}x=1+f_x(i_{2p-1}(t),t).$ The r.h.s.",
"in the above equation is positive and strictly increasing in $t$ , because $f_x(i_{2p-1}(t),t)$ is a local maximum of $f_x$ $\\lim _{t\\rightarrow t_1} \\int _{i_{2p}(t)}^{r(t)} k \\,\\text{\\rm d}x= \\alpha \\in (0,1).$ For $x\\in (i_{2p}(t),r(t))$ , $f_x\\in [-1,-1+\\alpha ]$ , and thus $f(i_{2p}(t),t)\\;\\geqslant \\;(1-\\alpha )(r(t)-i_{2p}(t)).$ As $f(i_{2p}(t),t)\\;\\leqslant \\;f(i_{0}(t),t)$ , equation (REF ) implies that $\\lim _{t \\rightarrow t_1} r(t)-i_{2p}(t)=0.$ Thus the mean value of $k$ on the interval $[i_{2p}(t),r(t)]$ explodes: $\\lim _{t\\rightarrow t_1} \\bar{k}(t):= \\lim _{t\\rightarrow t_1} \\frac{\\int _{i_{2p}(t)}^{r(t)} k\\,\\text{\\rm d}x }{(r(t)-i_{2p}(t))}=\\infty .$ By Jensen's inequality, we have $\\int _{i_{2p}(t)}^{r(t)} k\\log k \\,\\text{\\rm d}x \\;\\geqslant \\;(r(t)-i_{2p}(t))(\\bar{k}\\log \\bar{k})(t)\\;\\geqslant \\;\\alpha \\log \\bar{k} (t),$ and thus $\\lim _{t\\rightarrow t_1}\\int _{i_{2p}(t)}^{r(t)} k\\log k \\,\\text{\\rm d}x=\\infty .$ However, with the same argument used to obtain Equation (REF ), the inequality (REF ) implies that $\\int _{i_{2p}(t)}^{r(t)} k\\log k \\,\\text{\\rm d}x \\quad \\text{ is uniformly bounded,}$ yielding a contradiction.",
"$\\Box $"
],
[
"Stochastic domination and monotonicity in $\\lambda $ /boundary condition",
"Our dynamics has quite enjoyable monotonicity properties that can be proved by standard coupling argument using the so-called graphical construction.",
"First introduce a natural order on $\\Omega _0^L$ .",
"Given for two elements $\\xi $ and $\\xi ^{\\prime }$ we say that $\\xi \\;\\geqslant \\;\\xi ^{\\prime }$ if $\\xi _x\\;\\geqslant \\;\\xi ^{\\prime }_x$ for every $x\\in [-L,L]$ .",
"We say that a dynamic $\\eta $ dominates stochastically $\\eta ^{\\prime }$ if one can couple the two dynamic on the same probability space and have with probability one $\\eta (\\cdot ,t)\\;\\geqslant \\;\\eta ^{\\prime }(\\cdot ,t), \\quad \\forall t>0$ We give some examples of monotonicity that we may use in what follows: The dynamic with a wall and $\\lambda =1$ dominates the one without wall.",
"If $\\lambda <\\lambda ^{\\prime }$ the dynamic with parameter $\\lambda $ dominates the one with parameter $\\lambda ^{\\prime }$ .",
"For the construction of the coupling, we refer to [3] where these things are very well explained."
],
[
"A general upper-bound",
"Using monotonicity, we prove here that the solution of (REF ) is a general upper-bound for the scaling limit.",
"This provides half of Theorem REF , and will be of use for the proof of Theorem REF .",
"Here and in what follows we say that an event $A_L$ (or more properly, a sequence of event) occurs with high probability (we may also write w.h.p.)",
"if the probability of $A_L$ tends to one when $L$ tends to infinity.",
"Proposition 4.1 For all choices of $\\lambda \\in [0,\\infty ]$ , the dynamic starting with initial condition satisfying $ \\eta ^L_0(x)=Lf_0(x/L)(1+o(1)) \\text{ uniformly in $x$ when $L\\rightarrow \\infty $ },$ is such that for any given $\\varepsilon >0$ and $T>0$ , w.h.p $\\frac{1}{L}\\eta ^L(Lx,L^2t)\\;\\leqslant \\;\\widetilde{f}(x,t)+\\varepsilon , \\forall x\\in [-1,1],t\\in [0,T].$ We construct an alternative dynamics $\\widehat{\\eta }$ that constitutes an upper bound for $\\eta $ .",
"The dynamics $\\widehat{\\eta }$ has the same transition rates as $\\eta $ except that transition $\\eta \\rightarrow \\eta ^{(x)}$ for $x$ at a distance smaller than $L^{3/4}$ from the boundary are rejected.",
"We also modify the initial condition slightly so that $\\widehat{\\eta }^L_0(x)=x+L$ , for $x\\in [-L,-L+2L^{3/4}]$ , $\\widehat{\\eta }^L_0(x)=L-x$ , for $x\\in [L-2L^{3/4},L]$ , $\\widehat{\\eta }^L_0\\;\\geqslant \\;\\eta ^L_0$ and $\\widehat{\\eta }^L_0(x)\\;\\geqslant \\;2L^{3/4}$ , $\\forall x\\in [-L+2L^{3/4},L-2L^{3/4}]$ , $\\widehat{\\eta }^L_0$ satisfies (REF ).",
"From its initial condition and constraint it follows that $\\widehat{\\eta }$ is an upper bound for $\\eta $ .",
"Moreover, up to the first time of contact with the wall, in $[-L+L^{3/4},L-L^{3/4}]$ , $\\widehat{\\eta }$ coincides with a corner-flip dynamics, and as seem in Lemma REF in the appendix, with large probability $\\widehat{\\eta }$ does not touch the wall before time $L^2T$ .",
"Thus we can apply Theorem REF to the corner-flip dynamics on the segment $[-L+L^{3/4},L-L^{3/4}]$ to get the result."
],
[
"The case $\\lambda \\in [0,1]$",
"The Proposition REF already provides the upper-bound part of the convergence, so what remains to do is to prove that for every $\\varepsilon $ with high probability $ \\forall x\\in [-1,1], \\forall t\\in [0,T], \\quad \\frac{1}{L}\\eta ^L(Lx,L^2t)\\;\\geqslant \\;\\widetilde{f}(x,t)-\\varepsilon .$ From Theorem REF the above inequality is satisfied, when $\\eta $ is replaced by the dynamics without wall $\\widetilde{\\eta }$ .",
"Moreover from Section REF , one can couple the two dynamics such that $\\widetilde{\\eta }(t) \\;\\leqslant \\;\\eta (t)$ for all $t$ , and hence the result follows."
],
[
"The case $\\lambda \\in (1,2)$",
"When $\\lambda >1$ , there is no simple stochastic comparison available and one must work harder to obtain the result.",
"The idea we use (which is also present in [3], but used to prove bounds on the mixing time), is that when $\\lambda \\;\\leqslant \\;2$ the function $\\Phi $ defined on the set of paths $\\Omega $ as $\\Phi (\\eta ):=\\sum _{x=-L}^L g(x)\\eta (x),$ where $g(x):= \\cos \\left(\\frac{x\\pi }{2L}\\right),$ is close to be an eigenfunction of the generator of our Markov chain.",
"In the remainder of the paper, for notional convenience we write $\\eta (t)$ for $\\eta (\\cdot , t)$ .",
"Proposition 5.1 When $L$ is large enough, for any $\\eta _0\\in \\Omega _L$ for all $t\\;\\leqslant \\;L^{2+\\varepsilon }$ $|{\\mathbb {E}} [\\Phi (\\eta (t))]-\\exp \\left(-t\\pi ^2/(2L)^2\\right)\\Phi (\\eta _0)|\\;\\leqslant \\;L^{7/4}.$ Furthermore, if $\\eta _0$ satisfies (REF ) for a given $f_0$ , then $\\lim _{L\\rightarrow \\infty }\\max _{t\\in [0,T]}\\left|\\frac{1}{L^2}\\Phi (\\eta (L^2t))-\\exp \\left(-t\\pi ^2/4\\right)\\int _{-1}^1 f_0(x)\\cos (\\pi x/2)\\,\\text{\\rm d}x\\right|=0.$ A way to reformulate (REF ) is that the Fourier coefficient of the rescaled interface converges to the one of $\\widetilde{f}$ .",
"We define $\\bar{\\eta }$ to be the rescaled version (defined on $[-1,1]$ ) $\\bar{\\eta }(x,t)=\\frac{1}{L}\\eta (Lx,L^2t).$ Then (REF ) can be read as $\\lim _{L\\rightarrow \\infty } \\int _{-1}^1 \\bar{\\eta }(x,t)\\cos (\\pi x/2)\\,\\text{\\rm d}x=\\int _{-1}^1 \\widetilde{f}(x,t) \\cos (\\pi x/2)\\,\\text{\\rm d}x,$ where convergence holds uniformly in $[0,T]$ , in probability.",
"As Proposition REF already provides one bound, this estimate turns out to be sufficient to prove convergence of $\\eta $ .",
"Using the fact that $|y|= 2y_+ -y$ (where $y_+=\\max (y,0)$ ) we have $\\int _{-1}^1|\\bar{\\eta }(x,t)-f(x,t)|\\cos (\\pi x/2)\\,\\text{\\rm d}x=\\\\2 {\\mathbb {E}} \\left[\\int _{-1}^1(\\bar{\\eta }(x,t)-f(x,t))_+\\cos (\\pi x/2)\\,\\text{\\rm d}x\\right]+ \\int _{-1}^1(f(x,t)- \\bar{\\eta }(x,t))\\cos (\\pi x/2)\\,\\text{\\rm d}x.$ Using Proposition REF , we know that the first term tends to zero uniformly on $[0,T]$ in probability.",
"This is also the case of for the second term thanks to (REF ) or (REF ).",
"Hence for any $\\varepsilon >0$ , w.h.p.",
"for all $t\\;\\leqslant \\;T$ $\\int _{-1}^1|\\bar{\\eta }(x,t)-f(x,t)|\\cos (\\pi x/2)\\,\\text{\\rm d}x\\;\\leqslant \\;\\varepsilon .$ To conclude, we use the fact $|\\bar{\\eta }(x,t)-f(x,t)|$ is a 2-Lipshitz function to show that (REF ) implies uniform convergence.",
"Note that $\\forall x\\in [-1,-1-\\delta ]\\cup [1-\\delta ,1],\\quad |\\bar{\\eta }(x,t)-f(x,t)|\\;\\leqslant \\;\\delta $ because both $f$ and $\\bar{\\eta }$ are in $[0,\\delta ]$ .",
"To control $|\\bar{\\eta }(x,t)-f(x,t)|$ on $[-1+\\delta ]\\cup [1-\\delta ]$ we notice that (REF ) implies that $\\int _{-1+\\delta }^{1-\\delta } |\\bar{\\eta }(x,t)-f(x,t)|\\,\\text{\\rm d}x\\;\\leqslant \\;\\frac{\\varepsilon }{\\sin (\\pi \\delta /2)}\\;\\leqslant \\;\\varepsilon /\\delta .$ As $\\bar{\\eta }-f$ is a 2-Lipshitz function in $x$ , this implies that whenever (REF ) holds, $|\\bar{\\eta }(x,t)-f(x,t)|\\;\\leqslant \\;\\sqrt{\\varepsilon /\\delta },\\quad \\forall x \\in [-1+\\delta ,1-\\delta ].$ Taking choosing $\\varepsilon =\\delta ^3$ we conclude that w.h.p.",
"for all $t\\in [0,T]$ $ |\\bar{\\eta }(x,t)-f(x,t)|\\;\\leqslant \\;\\delta .$ Using [3] (be careful that the definition for the discrete Laplacian differs by a factor 2 and the same apply to our transition rates), and the linearity of $\\mathcal {L}$ of we have $\\partial _t {\\mathbb {E}} \\left[\\Phi (\\eta (t))\\right]= {\\mathbb {E}} \\left[(\\mathcal {L}\\Phi )(\\eta (t))\\right]=\\\\{\\mathbb {E}} \\left[\\sum _{x=-L}^L g(x)(\\Delta \\eta )(x,t)\\right]+2{\\mathbb {E}} \\left[\\sum _{x=-L}^L g(x) \\mathbf {1}_{\\eta (x\\pm 1,t)=0}\\right]\\\\-\\frac{2(\\lambda -1)}{\\lambda +1}{\\mathbb {E}} \\left[\\sum _{x=-L}^L g(x) \\mathbf {1}_{\\eta (x\\pm 1,t)=1}\\right].$ Doing summation by part and using the fact that $\\Delta g(x)=-\\kappa _L g(x)$ where $\\kappa _L:=2\\left(1-\\cos \\left(\\pi /2L\\right)\\right))$ the first term is equal to $-\\kappa _L{\\mathbb {E}} [\\Phi (\\eta (t)]$ .",
"We can control the value of the two other terms with the following estimates: Lemma 5.2 For any $\\delta >0$ , there exists a constant $C_1(\\lambda ,\\delta )$ such that for any choice of initial configuration $\\eta _0\\in \\Omega ^0_L$ , one has for every $t\\;\\geqslant \\;0$ , and every $x$ $\\begin{split}{\\mathbb {P}} [\\eta (x\\pm 1,t)=1]&\\;\\leqslant \\;C_1 \\min (d_L(x), t^{1/2-\\delta })^{-3/2} \\text{ if $x+L$ is even},\\\\{\\mathbb {P}} [\\eta (x\\pm 1,t)=0]&\\;\\leqslant \\;C_1 \\min (d_L(x), t^{1/2-\\delta })^{-3/2} \\text{ if $x+L$ is odd},\\end{split}$ where $d_L(x)=\\min (|x+L|,|x-L|)$ denotes the distance between $x$ and the boundary of $[-L,L]$ .",
"We postpone the proof to the end of the Section.",
"Note that Lemma REF implies that for every $t\\;\\leqslant \\;L^{\\frac{2}{1-2\\delta }}$ $\\begin{split}{\\mathbb {E}} \\left[\\sum _{x=-L}^L g(x) \\mathbf {1}_{\\eta (x\\pm 1,t)=1}\\right]\\;\\leqslant \\;C_2 L t^{-3/4+3\\delta /2},\\\\{\\mathbb {E}} \\left[\\sum _{x=-L}^L g(x) \\mathbf {1}_{\\eta (x\\pm 1,t)=0}\\right]\\;\\leqslant \\;C_2 L t^{-3/4+3\\delta /2}.\\end{split}$ and thus $|\\partial _t {\\mathbb {E}} \\left[\\Phi (\\eta (t))\\right]+\\kappa _L{\\mathbb {E}} [\\Phi (\\eta (t)]|\\;\\leqslant \\;C_3 L t^{-3/4+3\\delta /2}.$ Integrating the above inequality, we obtain for all $t\\;\\leqslant \\;L^{\\frac{2}{1-2\\delta }}$ ${\\mathbb {E}} \\left[\\Phi (\\eta (t))\\right]\\;\\geqslant \\;\\exp (-t\\kappa _L)\\Phi (\\eta _0)-C_3\\int _0^t e^{\\kappa _L(s-t)} L t^{-3/4+3\\delta /2}\\,\\text{\\rm d}s\\\\\\;\\geqslant \\;\\exp (-t\\kappa _L)\\Phi (\\eta _0)-C_3L t^{1/4+3\\delta /2}\\;\\geqslant \\;\\exp (-t\\kappa _L)\\Phi (\\eta _0)-L^{7/4},$ if $\\delta $ is small enough.",
"A lower-bound can be obtained in the same manner accordingly.",
"And hence (REF ) holds.",
"Let us now use the convergence of ${\\mathbb {E}} \\left[\\Phi (\\eta (t))\\right]$ to prove (REF ).",
"As we already have the upper-bound which is just a consequence of Proposition REF , we only need to prove that for any $\\delta $ w.h.p.",
"$\\forall t\\in [0,T],\\quad \\frac{1}{L^2}\\Phi (\\eta (L^2t))-\\exp \\left(-t\\pi ^2/4\\right)\\int _{-1}^1 f_0(x)\\cos (\\pi x/2)\\,\\text{\\rm d}x\\;\\geqslant \\;-\\delta .$ Let us fix $T$ and $\\delta >0$ arbitrary $\\varepsilon =\\delta ^2/4$ .",
"Fom Proposition REF , we have for $L$ large enough, for any event $A$ ${\\mathbb {E}} \\left[\\Phi (\\eta (L^2T))\\mathbf {1}_A \\right]\\;\\leqslant \\;({\\mathbb {P}} [A]+\\varepsilon )\\exp (-\\pi ^2 T/4)\\Phi (\\eta _0).$ Let us define $\\tau := \\min \\lbrace t\\ | \\ \\Phi (\\eta (L^2t))\\;\\leqslant \\;(1-\\delta )\\exp (-\\pi ^2t/4)\\Phi (\\eta _0)\\rbrace .$ As $\\tau $ is a stopping time, we can apply the Markov property and (REF ) for the initial condition $\\eta _{\\tau }$ to obtain that for every $t\\;\\leqslant \\;T$ .",
"${\\mathbb {E}} \\left[ \\Phi (\\eta (L^2(\\tau +t)))\\ | \\ (\\tau ,\\eta _{\\tau }) \\right]\\;\\leqslant \\;\\exp (-\\pi ^2 t/4)\\Phi (\\eta _{\\tau })+L^{7/4}\\\\\\;\\leqslant \\;(1-\\delta )\\exp (-\\pi ^2 (\\tau +t)/4)\\Phi (\\eta _0)+L^{7/4}.$ This inequality remains also valid if $t$ is a function of $\\tau $ .",
"Taking $t=T-\\tau $ we have, on the event $\\lbrace \\tau \\;\\leqslant \\;T\\rbrace $ , ${\\mathbb {E}} \\left[ \\Phi (\\eta (L^2T)) | \\ \\tau \\right]\\;\\leqslant \\;(1-\\delta )\\exp (-\\pi ^2 T/4)\\Phi (\\eta _0)+L^{7/4}\\\\\\;\\leqslant \\;(1-\\delta /2)\\exp (-\\pi ^2 T/4)\\Phi (\\eta _0).$ Combining it with (REF ) for the event $\\lbrace \\tau >T\\rbrace $ we have ${\\mathbb {E}} \\left[ \\Phi (\\eta (L^2T))\\right]\\;\\leqslant \\;(1-\\delta /2)\\exp (-\\pi ^2 T/4)\\Phi (\\eta _0){\\mathbb {P}} [\\tau \\;\\leqslant \\;T]+ {\\mathbb {E}} \\left[ \\Phi (\\eta (L^2T))\\mathbf {1}_{\\tau >T}\\right]\\\\\\;\\leqslant \\;(1-\\delta /2{\\mathbb {P}} [\\tau \\;\\leqslant \\;T]+\\varepsilon )\\exp (-\\pi ^2 T/4)\\Phi (\\eta _0).$ On the other hand (REF ) implies that ${\\mathbb {E}} \\left[ \\Phi (\\eta (L^2T))\\right]\\;\\geqslant \\;(1-\\varepsilon )\\exp (-\\pi ^2 T/4)\\Phi (\\eta _0),$ and hence as $\\varepsilon =\\delta ^2/4$ , ${\\mathbb {P}} [\\tau \\;\\leqslant \\;T]\\;\\leqslant \\;4\\varepsilon /\\delta \\;\\leqslant \\;\\delta .$ Hence with probability larger than $1-\\delta $ $\\Phi (\\eta (L^2t))\\;\\geqslant \\;\\exp (-\\pi ^2 t/4)\\Phi (\\eta _0)(1-\\delta ) \\quad \\forall t\\in [0,T].$ Dividing by $L^2$ leads to (REF ).",
"We can assume $t\\;\\geqslant \\;C^{4/(3+2\\delta }_1)$ , as if not the result for holds trivially.",
"By monotonicity, it is sufficient to prove the result with the smallest possible initial condition, where $\\eta _0=\\eta ^{\\min }$ (recall (REF )) Let us treat the case $x+L$ even only as the second line of (REF ) can be proved in a similar manner.",
"Consider first $x$ which is at a distance larger than $t^{1/2-\\delta }$ from the boundary.",
"To avoid complicated notation we assume that $t^{1/2-\\delta }$ is an even integer (or else, we replace it by twice the integer part of its half).",
"By mononicity again, the dynamics $(\\eta (\\cdot ,s))_{s\\;\\geqslant \\;0}$ can be coupled with a dynamics $(\\eta ^{\\prime }(\\cdot ,s)_{s\\;\\geqslant \\;0}$ with a wall and pinning force $\\lambda $ on state space $\\Omega _{x,t}:=\\lbrace \\eta \\in {\\mathbb {Z}} ^{2t^{1/2-\\delta }+1}\\ | \\;\\eta _{x\\pm t^{1/2-\\delta }}=0\\,; \\\\\\forall y\\in \\lbrace x-t^{1/2-\\delta },\\dots ,x+t^{1/2-\\delta }-1\\rbrace , |\\eta _{x+1}-\\eta _x|= 1, \\eta _x\\;\\geqslant \\;0 \\rbrace \\, .$ We can perform the coupling in such a manner that $ \\forall y\\in \\lbrace x-t^{1/2-\\delta },\\dots ,x+t^{1/2-\\delta }\\rbrace ,\\ \\forall s\\;\\geqslant \\;0, \\eta (y,t)\\;\\geqslant \\;\\eta ^{\\prime } (y,t) $ Hence it is sufficient to prove the result for $\\eta ^{\\prime }$ .",
"From [3], we know that the mixing time $T_{\\rm mix}(1/2e)$ of such a dynamics is smaller than $t^{1-\\delta }$ .",
"Hence at time $s=t$ , the total variation distance between the distribution of $\\eta ^{\\prime }(\\cdot ,t)$ and the equilibrium distribution $\\pi ^{\\prime }:=\\pi ^{\\prime }_{x,t}$ satisfies $\\Vert {\\mathbb {P}} [\\eta ^{\\prime }(\\cdot ,t) \\in \\cdot ] -\\pi ^{\\prime }\\Vert _{TV}\\;\\leqslant \\;\\exp (\\lfloor t/ T_{\\rm mix}(1/2e)\\rfloor )\\;\\leqslant \\;\\exp (-t^{\\delta }/2).$ At equilibrium, the probability that the midpoint of a polymer of length $2t^{1/2-\\delta }$ is pinned satisfies $\\pi ^{\\prime }( \\eta ^{\\prime }(x\\pm 1)=1)=\\frac{1+\\lambda }{\\lambda } \\pi ^{\\prime }(\\bar{\\eta }(x)=0)=\\frac{1+\\lambda }{\\lambda }\\frac{(Z^{\\lambda }_{t^{1/2-\\delta }})^2}{(Z^{\\lambda }_{2t^{1/2-\\delta }})^2}\\;\\leqslant \\;C_\\lambda t^{-3/2+3\\delta }.$ In the last inequality, we used the asymptotic equivalence $Z^{\\lambda }_{2l}\\approx c_\\lambda l^{-3/2}$ which holds for an explicit constant $c_\\lambda $ , see [10].",
"Hence from (REF ) one has ${\\mathbb {P}} [\\eta ^{\\prime }(x,t)=0]\\;\\leqslant \\;C_\\lambda t^{-3/2+3\\delta }+\\exp (-t^{\\delta })\\;\\leqslant \\;C_1 t^{-3/2+3\\delta }.$ if $C_1$ is chosen in an appropriate manner.",
"When $x$ is at a distance smaller than $t^{1/2-\\delta }$ from the boundary (say from $-L$ ), we use the same idea and compare the dynamics to one on the restricted path space $\\Omega _{x}:=\\lbrace \\eta \\in {\\mathbb {Z}} ^{2(L+x)+1}\\ | \\;\\eta _{-L}=0, \\eta _{2x+L}=0\\,; \\\\\\forall y\\in \\lbrace -L,\\dots ,2x+L\\rbrace , |\\eta _{x+1}-\\eta _x|= 1, \\eta _x\\;\\geqslant \\;0 \\rbrace \\, .$ From [3], this restricted $\\eta ^{\\prime }$ on an interval of length $2(L+x)$ , for which the mixing time is smaller than $t^{1-\\delta }$ and thus similarly ${\\mathbb {P}} [\\eta ^{\\prime }(x,t)=0]\\;\\leqslant \\;\\pi ^{\\prime }( \\eta ^{\\prime }(x\\pm 1)=1) +\\exp (-t^\\delta )\\;\\leqslant \\;2C(L+x)^{-3/2}.$ where $\\pi ^{\\prime }$ denote the equilibrium measure for the restricted dynamics."
],
[
"Proof of Theorem ",
"Let us describe a bit the strategy behind the proof of Theorem REF .",
"The first step of the proof, Lemma REF , is to show that the area below $\\eta $ decays at least with a constant rate equal to $-2$ on the rescaled picture.",
"This is performed by computing the expected drift of the area, and using a martingale technique to bound the possible fluctuation.",
"Using this bound on the area, we show that the problem can be reduced to proving that the solution of (REF ) is asymptotically a lower bound for the evolution of the polymer (see Proposition REF and below).",
"The most difficult part then, when proving Proposition REF is to control the motion of the boundary of the unpinned zone for $\\eta (t)$ ."
],
[
"An upper-bound for the decay of the area below the graph of $\\bar{\\eta }$",
"Consider $a(\\bar{\\eta }(t))=\\int _{-1}^1 \\bar{\\eta }(x,t)\\,\\text{\\rm d}x$ the area below the rescaled polymer (recall (REF )) (recalle the notation $\\eta (t)=\\eta (\\cdot ,t)$ ).",
"Similarly let $a(f(t))=\\int _{-1}^1 f(x,t)\\,\\text{\\rm d}x$ denote the area below the graph of $f(\\cdot ,t)$ .",
"We have seen (recall (REF )) that $a(f(t))=a(f(0))-2t \\text{ when } t\\;\\leqslant \\;T^*.$ We want to prove a similar statement for $a(\\bar{\\eta }(t))$ .",
"To state our result, it is easier to consider the area below the non-rescaled curve: $A(\\eta (t)):=\\int _{-L}^L \\max (\\eta (x,t),1)\\,\\text{\\rm d}x.$ Note that $A$ is not exactly the area because of the $\\max $ present in the integral, but this detail is of no importance once rescaling is performed.",
"The quantity $\\max (\\eta (x,t),1)$ is present instead of $\\eta (x,t)$ in order to have a nice expression for the expected drift of $A$ (see below).",
"It can be checked by the reader (see also Section 5.4 in [3]) that the expected drift of $A(\\eta (t))$ is equal to (recall (REF ) $(\\mathcal {L} A)(\\eta )= 2|\\lbrace x\\in \\lbrace -L\\dots L\\rbrace \\ | \\ \\eta _x=\\eta _{x\\pm 1}-1 \\text{ and } \\eta _{x\\pm 1} \\ne 0\\rbrace |\\\\-2|\\lbrace x\\in \\lbrace -L\\dots L\\rbrace \\ | \\ \\eta _x=\\eta _{x\\pm 1}+1 \\text{ and }\\eta _x\\ne 1 \\rbrace |.$ This is because the transition $\\eta \\rightarrow \\eta ^{(x)}$ increases/decreases $A$ by $\\pm 2$ , except if it adds one contact with the wall, in which case $A$ is decreased by $-1$ (but this happens with a rate twice as big).",
"The right-hand side of (REF ) is equal to minus 2 times the number of excursions of length 4 or more away from the wall, and thus $(\\mathcal {L} A)(\\eta )\\;\\geqslant \\;-2, \\forall \\eta \\in \\Omega _L\\setminus \\lbrace \\eta _{\\min }\\rbrace .$ Lemma 6.1 One has w.h.p., $A(\\eta (t))\\;\\leqslant \\;A(\\eta (0))+\\int _0^{t} \\mathcal {L} A(\\eta (s))\\,\\text{\\rm d}s+ L^{7/4}, \\quad \\forall t\\in [0,L^2],$ so that in particular for all $t\\in [0,L^2]$ $A(\\eta (t))\\;\\leqslant \\;\\max (A(\\eta (t))-2t+L^{7/4}, 2L).$ As a consequence, w.h.p.",
"$\\mathcal {T}\\;\\leqslant \\;A(\\eta (0))/2+L^{7/4},$ and w.h.p.",
"uniformly for all time, $a(\\bar{\\eta }(t))\\;\\leqslant \\;(a(f_0)-2t)_+ + 2L^{-1/4}.$ It is a standard property of Markov chains that $M_t:=A(\\eta (t))- A(\\eta (0))-\\int _0^t (\\mathcal {L} A (\\eta (s)))\\,\\text{\\rm d}s$ is a martingale, for the filtration associated with the process $(\\eta (t),\\ t\\;\\geqslant \\;0)$ .",
"To prove (REF ) we have to show that $M_t$ cannot be too large.",
"We use Doob's maximal inequality $ {\\mathbb {P}} [\\max _{t\\in [0,L^2]} M_t\\;\\geqslant \\;L^{7/4}]\\;\\leqslant \\;L^{-7/2} {\\mathbb {E}} [M^2_{L^2}].$ As $M_0=0$ the expected value of $M^2_{L^2}$ is the one of the martingale bracket $\\langle M^2\\rangle _{L^2}$ , for which we have an explicit expression $\\langle M^2\\rangle _{t}=\\int _0^t F(\\eta (s))\\,\\text{\\rm d}s$ where $F(\\eta (s)):= 4 |\\lbrace x\\in \\lbrace -L\\dots L\\rbrace \\ | \\ \\eta _{x+1}=\\eta _{x-1}\\notin \\lbrace 0,1\\rbrace \\rbrace |\\\\+ 2 |\\lbrace x\\in \\lbrace -L\\dots L\\rbrace \\ | \\ \\eta _x=2,\\ \\eta _{x\\pm 1}= 1\\rbrace | \\;\\leqslant \\;8L.$ Hence from (REF ) ${\\mathbb {P}} [\\max _{t\\in [0,L^2]} M_t\\;\\geqslant \\;L^{7/4}]\\;\\leqslant \\;8 L^{-1/2}.$ For the second statement, we note that either the chain has already reached $\\eta _{\\min }$ at time $t$ and thus $A(t)=2L$ or it has not and from (REF ) $\\int ^t_0 \\mathcal {L} A(\\eta (s))\\,\\text{\\rm d}s \\;\\geqslant \\;-2t,$ so that (REF ) is a consequence (REF ).",
"Finally, to obtain (REF ) we notice that if $\\mathcal {T}\\;\\geqslant \\;A(\\eta (0))/2+L^{7/4}$ , then $\\int _0^{A(\\eta (0))+L^{7/4}} \\mathcal {L} A(\\eta (s))\\,\\text{\\rm d}s\\;\\leqslant \\;-2A(\\eta (0))-2L^{7/4}.$ Thus this cannot occur with non-vanishing probability, as it would bring a contradiction to (REF ).",
"Equation (REF ) is obtained by rescaling time and space in (REF ) (we a have to replace $A$ by the area below the curve, but the difference between the two is at most $L$ )."
],
[
"Reducing the problem to the proof of an asymptotic lower-bound for $\\eta (t)$",
"A consequence of equation (REF ) is that in order to prove Theorem REF , we only need a lower-bound result.",
"Indeed the upper-bound on the area plus the constraint that $\\bar{\\eta }(\\cdot , t)$ is a Lipshitz function are sufficient to deduce the upper-bound from the lower-bound.",
"For this reason, in the remainder of the paper we focus on proving Proposition 6.2 Let $\\eta ^{L,\\infty }$ be the dynamic with wall and $\\lambda =\\infty $ , and starting from a sequence of initial condition $\\eta ^L_0$ satisfying $ \\eta ^L_0(x)= Lf_0(x/L)(1+o(1)) \\text{ uniformly in $x$ when $L\\rightarrow \\infty $ }.$ Then for every choice of $\\varepsilon >0$ , the rescaled dynamics $\\bar{\\eta }$ satisfies w.h.p.",
"$\\bar{\\eta }^{L,\\infty }(x.t)\\;\\geqslant \\;f(x,t)-\\varepsilon , \\quad \\forall x\\in [-1,1], \\forall t>0.$ It is sufficient to prove that for any $\\delta >0$ , w.h.p $\\forall x\\in [-1,1], \\forall t>0 \\quad \\bar{\\eta }(x.t)\\;\\leqslant \\;f(x,t)-\\delta .$ Combining Equations (REF ) and (REF ) we have w.h.p $\\forall x\\in [-1,1], \\forall t>0,\\quad a(\\bar{\\eta }(t))\\;\\leqslant \\;a(f(\\cdot ,t))+\\delta ^2/32.$ Moreover, from Proposition REF we have w.h.p.",
"for all $x$ and $t$ $\\bar{\\eta }(x,t)\\;\\geqslant \\;f(x,t)-\\delta ^2/64.$ Hence w.h.p.",
"for all $t>0$ $x\\mapsto \\bar{\\eta }(x,t)- f(x,t)+\\delta ^2/32$ is a 2-Lifshitz positive function whose integral is smaller than $\\delta ^2/16$ .",
"This implies that $\\bar{\\eta }(x,t)- f(x,t)-\\delta ^2/32 \\;\\leqslant \\;\\delta /2.$ Concerning (REF ), the upper-bound on $\\mathcal {T}$ is proved in Lemma REF , and the lower-bound is a consequence of (REF )."
],
[
"Overall strategy for the proof of Proposition ",
"The main difficuly when trying to prove Proposition REF is to control the motion of the boundary between the pinned and the unpinned zone.",
"Indeed, for part of $\\eta $ that are far from the wall, Theorem REF can be used to control the drift of the rescaled polymer.",
"The first and most novel idea in the proof is to add a a small perturbation of amplitude $\\delta $ to the function $f$ and to the initial condition $\\eta _0^L$ (see the caption of Figure REF ).",
"The reason for adding this perturbation is that when adding flat parts of slope $+1/-1$ on the sides of $\\eta $ , the motion of the phase boundary i.e.",
"the boundary between the pinned and unpinned phases, is slowed down.",
"Of course a consequence of this modification is that the initial condition does not satisfy (REF ), and we have to find an equivalent formulation Proposition REF that deals with this problem: this is Proposition REF .",
"We show that it is enough to control where the dynamics goes during a time period $\\varepsilon $ and iterate the process.",
"This is the role of Lemmata REF and REF ."
],
[
"Modification of the initial function",
"Given $f_0$ with $l_0>-1$ , $r_0<-1$ , set (recall (REF )) $\\bar{\\delta }=a(f_0)\\delta /2 \\quad \\text{and} \\quad \\bar{\\delta }(t):=\\delta a(f(t))/2=\\delta (a(f_0)/2-t).$ We define $f^{\\delta }:[-1,1]\\times (0,\\infty )\\rightarrow {\\mathbb {R}} _+$ (for $\\delta $ small enough) by $f^{\\delta }(x,t):={\\left\\lbrace \\begin{array}{ll}f(x,t)+\\bar{\\delta }(t), \\text{ for } x\\in (l(t),r(t)),\\\\x-(l(t)-\\bar{\\delta }(t)), \\text{ for } x\\in (l(t)-\\bar{\\delta }(t),l(t)),\\\\-\\big (x-(r(t)+\\bar{\\delta }(t))\\big ), \\text{ for } x\\in (r(t),r(t)+\\bar{\\delta }(t)),\\\\0 \\quad \\text{elsewhere}.\\end{array}\\right.",
"}$ See Figure REF for a graphical vision of $f^{\\delta }$ .",
"We also set $f^\\delta _0:=f^{\\delta }(\\cdot ,0).$ Figure: The construction of the graph of f δ (t)f^{\\delta }(t) from the graph of f(t)f(t) is done by adding some kind of pedestal of heightδ ¯(t)\\bar{\\delta }(t) to support the original graph.From the fact that $f$ is the solution of (REF ), we can deduce that $f^\\delta $ satisfies ${\\left\\lbrace \\begin{array}{ll}\\partial _t f^\\delta =f^{\\delta }_{xx}-\\delta \\text{ for } x\\in (r(t),l(t)),\\\\\\partial _t f^\\delta (x,t)=f_{xx}(l(t),t)-\\delta \\text{ for } x\\in (l(t)-\\bar{\\delta }(t),l(t)),\\\\\\partial _t f^\\delta (x,t)=f_{xx}(r(t),t)-\\delta \\text{ for } x\\in (r(t),r(t)+\\bar{\\delta }(t)).\\end{array}\\right.",
"}$ Instead of proving Proposition REF , we prove a similar statement where $f$ is replaced by $f^\\delta $ .",
"Somewhere in the proof we will need some continuity assumption on the solution that are uniform for all time.",
"For this reason we will require our initial condition to be in the set $\\mathcal {E}:=\\lbrace (f_0,r_0,l_0) \\ |\\ f_0 \\text{ is positive, Lipshitz,} C^\\infty \\text{ on } (l_0,r_0), \\\\\\text{ and such that }l(t) \\text{ and } r(t) \\text{ are } C^\\infty \\text{ on } [0,T^*) \\rbrace .$ With this assumptions we are certain that there exists $C(f_0,c)$ that is such that uniformly on $t\\in [0,T^*-c/4]$ and $x\\in l(t),r(t)$ $\\begin{split}|f_xx(x,t)|& \\;\\leqslant \\;C,\\quad |(\\partial _x)^3 f(x,t)|\\;\\leqslant \\;C,\\\\|\\partial _t f(x,t)|=|(\\partial _x)^4 f(x,t)|\\&le C,\\\\\\max (|r^{\\prime }(t)|,|r^{\\prime \\prime }(t)|,|l^{\\prime }(t)|,|l^{\\prime \\prime }(t)|)&\\;\\leqslant \\;C,\\\\\\end{split}$ The set $\\mathcal {E}$ is dense for the uniform norm in the set of Lipshitz function that are positive on an interval.",
"This is because from Theorem REF , the solution of (REF ) $f(\\cdot ,t)$ belongs to $\\mathcal {E}$ for all positive times.",
"Proposition 6.3 Given a $f_0$ regular enough, and $\\delta >0$ starting from a (sequence of) initial condition $\\eta _0^L$ satisfying $\\forall x\\in [-1,1], \\quad \\bar{\\eta }^L_0(x)\\;\\geqslant \\;f_0^{\\delta }(x),$ we have, for every $\\varepsilon >0$ , w.h.p $\\forall x\\in [-1,1], \\ \\forall t>0,\\quad \\bar{\\eta }(x,t)\\;\\geqslant \\;f^{\\delta }(x,t)-\\varepsilon .$ The reason why Proposition REF is easier to prove than Proposition REF is that as can be seen from (REF ), $f^{\\delta }$ has a stronger “push-down” than $f$ .",
"Also even if $f^\\delta _{xx}\\equiv 0$ in a neighborhood $l(t)-\\bar{\\delta }(t)$ and $r(t)+\\bar{\\delta }(t)$ , there is a contration of the boundary of the unpinned region (meaning that the lateral push is also stronger).",
"However, it is not too difficult to prove that the two statements are in fact equivalent.",
"Given $f_0$ , and $\\varepsilon $ , we consider $\\delta >0$ small enough (depending on $f_0$ and $\\varepsilon $ ) and we define $\\widehat{f}_0$ to be a 1-Lipshitz function which is positive and smooth on $(\\widehat{l}_0, \\widehat{r}_0)\\subset (l_0,r_0)$ .",
"and that satisfies $\\widehat{f}_0^{2\\delta }\\;\\leqslant \\;f_0, \\quad \\text{ and } \\quad \\int _{-1}^1 (f_0-\\widehat{f}_0)(x)\\,\\text{\\rm d}x\\;\\leqslant \\;\\varepsilon ^2/4.$ Let $\\widehat{f}$ denote the solution of (REF ) with initial condition $\\widehat{f}_0$ and $\\widehat{f}^{\\delta }$ resp.",
"$\\widehat{f}^\\delta _0$ , is defined as in (REF ), replacing $f$ by $\\widehat{f}$ .",
"We remark that if $\\eta ^L_0$ satisfies (REF ) then for $L$ large enough $ \\forall x\\in [-1,1], \\quad \\bar{\\eta }^L_0\\;\\geqslant \\;\\widehat{f}^\\delta _0.$ Indeed it has to be checked only on the interval where $\\widehat{f}_0^{\\delta }$ is positive, and on this interval $f_0- \\widehat{f}_0^\\delta $ is uniformly bounded away from zero (from (REF )).",
"Applying Proposition REF to $\\widehat{f}_0$ we obtain that w.h.p.",
"$\\forall t\\;\\geqslant \\;0,\\ \\forall x\\in [-1,1], \\quad \\bar{\\eta }(x,t)\\;\\geqslant \\;\\widehat{f}^{\\delta }(x,t)- \\varepsilon \\;\\geqslant \\;\\widehat{f}(x,t)-\\varepsilon .$ Applying Equation (REF ) to $f$ and $\\widehat{f}$ and combining it with the assumption (REF ), we obtain that for all $t>0$ $\\int ^1_{-1} (f(x,t)-\\widehat{f}(x,t))\\,\\text{\\rm d}x\\;\\leqslant \\;\\varepsilon ^2/4.$ As $(f-\\widehat{f})(\\cdot ,t)$ is $2-$ Lipshitz and positive, this implies $\\forall t>0,\\ \\forall x\\in [-1,1], \\quad f \\;\\leqslant \\;\\widehat{f}+\\varepsilon .$ Hence (REF ) implies that $\\bar{\\eta }(x,t)\\;\\geqslant \\;f(x,t)-2\\varepsilon .$"
],
[
"Reduction to a statement for infinitesimal time",
"To prove Proposition (REF ), we slice time into short period of length $\\varepsilon $ during which it is easier to control the dynamics.",
"The proof can be decomposed in two steps: in Lemma REF we check that when $t$ is a multiple of $\\varepsilon $ , the polymer stays above $f^\\delta (\\cdot ,t)$ .",
"Then Lemma REF is used to fill the gap; it shows that during a period of time $\\varepsilon $ the polymer cannot go down too much.",
"Lemma 6.4 Given $f_0$ and $\\delta >0$ , and $c>0$ , there exists $\\varepsilon _0=\\varepsilon _0(f_0,\\delta ,c)>0$ such that for all $\\varepsilon \\;\\leqslant \\;\\varepsilon _0$ , $k$ satifying $k\\varepsilon \\;\\leqslant \\;(a(f_0)-c)/2$ , if w.h.p.",
"$, \\forall x\\in [-1,1], \\quad \\eta (x,k\\varepsilon )\\;\\geqslant \\;f^{\\delta }(x,k\\varepsilon ),$ then w.h.p.",
"$, \\forall x\\in [-1,1], \\quad \\bar{\\eta }(x,(k+1)\\varepsilon )> f^{\\delta }(x,(k+1)\\varepsilon ).$ Lemma 6.5 Given $f_0$ , $\\delta >0$ , $c>0$ , and $\\eta >0$ there exists $\\varepsilon _1=\\varepsilon _1(f_0,\\delta ,c,\\alpha )>0$ such that: for all $\\varepsilon \\;\\leqslant \\;\\varepsilon _1$ and $k$ satifying $k\\varepsilon \\;\\leqslant \\;(a(f_0)-c)/2$ , if w.h.p.",
"(REF ) holds then $\\forall t\\in (\\varepsilon k,\\varepsilon (k+1)), \\quad \\bar{\\eta }(x,t)> f^{\\delta }(x,k\\varepsilon )-\\alpha .$ Given $f_0,\\delta ,c,\\eta $ , let us choose $\\varepsilon =\\min (\\varepsilon _0,\\varepsilon _1)$ .",
"If $\\eta ^L_0$ satisfies (REF ) then reasoning by induction and using Lemma REF one obtains setting $k_{\\max }=\\lceil (a(f_0)-c)/2\\varepsilon \\rceil $ that w.h.p.",
"$\\forall k \\in [0,k_{\\max }], \\quad \\bar{\\eta }(x,k\\varepsilon )> f^{\\delta }(x,k\\varepsilon ).$ Then we can use Lemma REF to get a lower bound on $\\bar{\\eta }(x,t)$ for intermediate times and we obtain that w.h.p.",
"$\\forall t\\in [0,(a(f_0)-c)/2], \\quad \\bar{\\eta }(x,t)> f^{\\delta }(x,t)-\\alpha .$ Finally, for $t\\;\\geqslant \\;(a(f_0)-c)/2$ , Equation (REF ) ensures that $\\int _{-1}^1 f(x,t)\\,\\text{\\rm d}x \\;\\leqslant \\;c.$ As $f$ is Lipshitz and vanishes on the boundary of $[-1,1]$ this implies that $f(x,t)\\;\\leqslant \\;\\sqrt{c}$ uniformly and thus that $f^\\delta (x,t)\\;\\leqslant \\;\\sqrt{c}+\\delta c/2.$ This is enough to conclude, because $c$ can be chosen arbitrarily small."
],
[
"Proof of Lemmata ",
"The final step in the proof of Theorem REF is the proof of Lemma REF and Lemma REF .",
"For commodity reason, we chose to shift time from $\\varepsilon k$ to zero.",
"What we need to show is that if one start with an initial condition $\\eta ^L_0$ that satisfies $\\forall x\\in [-1,1], \\quad \\bar{\\eta }^L_0(x)\\;\\geqslant \\;f^{\\delta }(x,k\\varepsilon ),$ then w.h.p.",
"$\\forall x\\in [-1,1], \\quad \\bar{\\eta }(x,\\varepsilon )\\;\\geqslant \\;f^{\\delta }(x,(k+1)\\varepsilon ),$ and $\\quad \\forall t\\in (0,\\varepsilon ) \\bar{\\eta }(x,t)\\;\\geqslant \\;f^{\\delta }(x,k\\varepsilon )-\\alpha .$ Note that the assumption that we suppose is not exactly the same as in the Lemmata REF and REF as (REF ) holds only w.h.p.",
"but this is not a problem, as anything happening on a set of vanishing probability does not change the conclusion.",
"In the remainder of the proof, we shall consider only the case $k=0$ , to lighten the notation.",
"The reader can check then that the bound we use throughout the proof are in fact valid uniformly for all $k$ .",
"For instance for Proposition REF we can note $\\bar{\\delta }(\\varepsilon k)$ is bounded from below by $\\bar{\\delta }c$ , and for the rest (REF ) provides uniform bounds.",
"By monotonicity, it is sufficient to prove (REF ) and (REF ) starting from the smallest initial condition satisfying (REF ).",
"In in this case we habe $\\bar{\\eta }^L_0(x)= f_0^\\delta +\\sigma _L(x),$ where $0\\;\\leqslant \\;\\sigma _L(x)\\;\\leqslant \\;2/L$ .",
"The strategy is then quite simple.",
"Set $\\bar{l}=l_0-\\bar{\\delta }$ and $\\bar{r}=r_0-\\bar{\\delta }$ to be the left and right boundaries of the region where $f^{\\delta }(\\cdot ,\\varepsilon k)$ is positive.",
"First, we show that for a small time $\\varepsilon $ , the boundary of the unpinned region do not move more further than $\\varepsilon ^2$ from their original location, i.e.",
"that w.h.p.",
"$\\eta $ does not add contact points with the wall in the interval $L[\\bar{l}+\\varepsilon ^2,\\bar{r}-\\varepsilon ^2]$ .",
"Proposition 6.6 For $\\varepsilon $ small enough (depending only on $\\delta $ ) the dynamic started from an initial condition satisfying (REF ).",
"we have w.h.p.",
"$\\bar{\\eta }(x,t)>0, \\quad \\forall x\\in [\\bar{l}+\\varepsilon ^2,\\bar{r}-\\varepsilon ^2 ], \\forall t\\in [0,\\varepsilon ].$ The second step, at the end of the section, is then to say that if no contact is added, then the dynamics restricted to the interval $L[\\bar{l}+\\varepsilon ^2, \\bar{r}-\\varepsilon ^2]$ stochastically dominates a corner-flip dynamics like the one of Section REF for which the scaling limit is given by the solution of the heat equation with an initial condition that is close to $f_0$ .",
"What remains to do at last is to compare the solution of the heat-equation after time $\\varepsilon $ to $f(\\cdot ,\\varepsilon )$ to establish (REF ) and to $f_0-\\alpha $ to establish (REF ).",
"The trick is to show that if one touches the wall to soon, the area below the curve $a(\\bar{\\eta })$ shrinks too fast.",
"Set $\\mathcal {T}^{\\prime }$ to be the first time at which $\\eta $ touches the wall in the interval $L[\\bar{l}+\\varepsilon ^2,\\bar{r}-\\varepsilon ^2]$ and $\\tau ^{\\prime }=\\mathcal {T}^{\\prime }/L^2$ .",
"We decompose the proof in two lemmata which proofs are postponed.",
"The first step of our reasoning is to prove that Lemma REF is almost sharp up to time $\\mathcal {T}^{\\prime }$ .",
"Let $ \\widetilde{A}(\\eta )$ be defined as $\\widetilde{A}(\\eta (t)):=\\int _{x_l}^{x_r} \\max (\\eta (x,t),1)\\,\\text{\\rm d}x.$ where $x_l:=\\lceil L (\\bar{l}+\\varepsilon ^2) \\rceil \\text{ and } x_r:=\\lfloor L (\\bar{r}-\\varepsilon ^2)\\rfloor .$ Figure: A trajectory η\\eta , with the volume A d (η) A^d(\\eta ) darkened.The drift of the volume when there are no contact with the wall is either equal to 2, 1, 0, -1-1 or -2-2.As in Lemma REF , one can compute explicitly $\\mathcal {L} \\widetilde{A}$ and use the expression we obtain to prove, Lemma 6.7 One has w.h.p.",
"$\\widetilde{A}(\\eta ({\\mathcal {T}} ^{\\prime }))\\;\\geqslant \\;\\widetilde{A}(\\eta _0)-2{\\mathcal {T}} ^{\\prime }-L^{7/4},$ and as a consequence, w.h.p.",
"$a(\\bar{\\eta }(\\tau ^{\\prime }))\\;\\geqslant \\;a(f^{\\delta }_0)-2\\tau ^{\\prime }-\\varepsilon ^4-L^{-1/4}.$ The second step is to show that if $\\tau ^{\\prime }$ is to small, then we have a lower bound on $a(\\bar{\\eta }(\\tau ^{\\prime }))$ which brings a contradiction to Lemma REF .",
"This bound is obtained by combining Proposition REF with a simple geometric argument.",
"Set $X_{\\tau ^{\\prime }}$ to be the point where the first contact with the wall in $[x_l,x_r]$ is occurs.",
"It is the only $x$ that satisfies $L X_{\\tau ^{\\prime }}\\in [x_l,x_r] \\text{ and } \\bar{\\eta }(X_{\\tau ^{\\prime }},\\tau ^{\\prime })=0.$ Set also $\\widetilde{f}^\\delta $ to be the solution heat equation with Dirichlet boundary condition on $[\\bar{l},\\bar{r}]$ , and initial condition $f^\\delta _0$ .",
"Lemma 6.8 For the dynamics $\\eta $ starting from an initial condition that satisfies (REF ).",
"For every positive $\\alpha $ , w.h.p.",
"we have $\\forall x\\in [\\bar{l},\\bar{r}], \\quad \\bar{\\eta }(x,\\tau ^{\\prime })\\;\\leqslant \\;\\min (\\widetilde{f}^\\delta (x,\\tau ^{\\prime }),|x-X_{\\tau ^{\\prime }}|)+\\alpha .$ In addition if $\\tau ^{\\prime }\\;\\leqslant \\;\\varepsilon $ , and $\\varepsilon $ is small enough (depending on $\\delta $ and $c$ ) one has $\\int _{-1}^1\\min ( \\widetilde{f}^\\delta (x,\\tau ^{\\prime }),|x-X_{\\tau ^{\\prime }}|))\\,\\text{\\rm d}x \\;\\leqslant \\;a (f_0^{\\delta })-2\\tau ^{\\prime }-\\varepsilon ^2\\bar{\\delta }/16$ Suppose now that with a non vanishing probability $\\tau ^{\\prime }\\;\\leqslant \\;\\varepsilon $ .",
"Then Lemma REF implies (using (REF ) with $\\alpha =\\varepsilon ^2\\bar{\\delta }/64$ that with a non vanishing probability, $a(\\bar{\\eta }(\\tau ^{\\prime }))\\;\\leqslant \\;a (f^{\\delta }(\\varepsilon k))-2\\tau ^{\\prime }-\\varepsilon ^2\\bar{\\delta }/32.$ If $\\varepsilon $ is chosen such that $\\varepsilon ^2\\;\\leqslant \\;\\delta c/64$ then this brings a contradiction to (REF ) for $L$ sufficiently large.",
"Evaluating the effect of each transition on $\\widetilde{A}$ , and noticing in particular that corner flips involving either $x_l$ or $x_r$ modifies $\\widetilde{A}$ only by $\\pm 1$ , one obtains $\\mathcal {L} \\widetilde{A}(\\eta ):=2 \\big (|\\lbrace x\\in (-x_l,x_r) \\ | \\ \\eta _x\\;\\geqslant \\;1, \\eta _{x\\pm 1}=\\eta _x+1\\rbrace |\\\\-|\\lbrace x\\in (-x_l,x_r) \\ | \\ \\eta _x\\;\\geqslant \\;2, \\eta _{x\\pm 1}=\\eta _x-1\\rbrace |\\big )\\\\-\\mathbf {1}_{\\lbrace \\eta _{x_l}\\;\\geqslant \\;2, \\ \\eta _{x_l}=\\eta _{x_l\\pm 1}-1\\rbrace }+\\mathbf {1}_{\\lbrace \\eta _{x_l}\\;\\geqslant \\;1,\\ \\eta _{x_l}=\\eta _{x_l\\pm 1}+1\\rbrace }\\\\-\\mathbf {1}_{\\lbrace \\eta _{-x_r}\\;\\geqslant \\;2,\\ \\eta _{x_r}=\\eta _{x_r\\pm 1}-1\\rbrace }+\\mathbf {1}_{\\lbrace \\eta _{x_r}\\;\\geqslant \\;1,\\ \\eta _{x_r}=\\eta _{x_r\\pm 1}+1\\rbrace }.$ It is then easy to check that $t<\\mathcal {T}^{\\prime }$ $\\mathcal {L} \\widetilde{A}(\\eta (t))\\in \\lbrace 2,1,0,-1,-2\\rbrace .$ It is just a consequence of the fact that the difference between the number of local maximum and local minimum in $[x_l,x_r]$ is at most 1.",
"Next, as in Lemma REF , we use Doobs maximal inequality for the martingale $\\widetilde{M}_t:=\\widetilde{A}(\\eta (t))- \\widetilde{A}(\\eta _0)-\\int _0^t \\mathcal {L} \\widetilde{A}(\\eta (s))\\,\\text{\\rm d}s,$ and obtain that ${\\mathbb {E}} [\\min _{t\\in [0,L^2]} \\widetilde{M}_t\\;\\leqslant \\;-L^{7/4}]\\;\\leqslant \\;L^{-7/2}{\\mathbb {E}} \\left[\\widetilde{M}^2_{L^2}\\right].$ As $\\widetilde{M}_0=0$ , the expectation of $\\widetilde{M}^2_{L^2}$ is equal that of the martingale bracket $\\langle \\widetilde{M}^2\\rangle _{L^2}$ , which can be shown to be almost surely bounded by $8L^3$ (recall REF ).",
"It implies that the r.h.s.",
"of (REF ) vanishes when $L$ tends to infinity.",
"Next, we notice that ${\\mathbb {P}} [\\widetilde{M}_{\\mathcal {T}^{\\prime }}<-L^{7/4}]\\;\\leqslant \\;{\\mathbb {P}} [ \\mathcal {T}^{\\prime }<L^2]+ {\\mathbb {E}} [\\min _{t\\in [0,L^2]} \\widetilde{M}_t\\;\\leqslant \\;-L^{7/4}],$ and we note that by (REF ) (and the fact that ${\\mathcal {T}} ^{\\prime }\\;\\leqslant \\;{\\mathcal {T}} $ ), the first term in the right-hand side also vanishes.",
"Hence, we have w.h.p.",
"$\\widetilde{A}(\\eta ({\\mathcal {T}} ^{\\prime }))\\;\\geqslant \\;\\widetilde{A}(\\eta _0)-\\int _0^{\\mathcal {T}} \\mathcal {L} \\widetilde{A}(\\eta (s))\\,\\text{\\rm d}s-L^{7/4},$ and (REF ) allows us to obtain (REF ).",
"For the second-point we first notice that from the various definitions we have $a(\\bar{\\eta }(\\tau ^{\\prime }))\\;\\geqslant \\;\\frac{1}{L^2} \\widetilde{A}( \\eta (\\mathcal {T}^{\\prime })),$ Then, noticing that the area below the curve in $[-L,x_l]\\cup [x_r,L]$ cannot be larger $\\varepsilon ^4$ (the curve is 1-Lipshitz), we have $\\widetilde{A}(\\eta _0)\\;\\geqslant \\;L^2 (a(f^{\\delta }_0)-\\varepsilon ^4),$ and hence (REF ) holds by a combination of (REF ), (REF ) and (REF ).",
"The fact that $\\bar{\\eta }(x,\\tau ^{\\prime })\\;\\leqslant \\;|x-X_\\tau ^{\\prime }|$ is just derived from the fact that $\\bar{\\eta }$ is a Lipshitz function which equals zero at $X_\\tau ^{\\prime }$ .",
"The inequality $\\bar{\\eta }(x,\\tau ^{\\prime })\\;\\leqslant \\;\\widetilde{f}^\\delta (x,\\tau ^{\\prime })+\\alpha , \\quad \\forall x\\in (-1,1)$ is derived from Proposition REF , but one has to be careful since $\\tau ^{\\prime }$ is a random time.",
"One has ${\\mathbb {P}} [\\max _{x\\in [\\bar{l}, \\bar{r}]} [\\bar{\\eta }(x,\\tau ^{\\prime })- \\widetilde{f}^\\delta (x,\\tau ^{\\prime })]\\;\\geqslant \\;\\alpha ]\\;\\leqslant \\;{\\mathbb {P}} [\\tau ^{\\prime }\\;\\geqslant \\;1]+{\\mathbb {P}} \\left[\\max _{\\begin{array}{c}x\\in [\\bar{l}, \\bar{r}] \\\\ t\\in [0,1]\\end{array}} [\\bar{\\eta }(x,t)- \\widetilde{f}^\\delta (x,t)]\\;\\geqslant \\;\\alpha \\right].$ The first term has vanishing probability from (REF ) and the second one from Proposition REF .",
"The second point (REF ) is a bit more technical.",
"The first task is to show is that if $\\tau ^{\\prime }\\;\\leqslant \\;\\varepsilon $ $\\int _{-1}^1\\min (\\widetilde{f}^{\\delta }(x,\\tau ^{\\prime }),|x-X_{\\tau ^{\\prime }}|))\\,\\text{\\rm d}x \\;\\leqslant \\;\\int _{-1}^1\\widetilde{f}^{\\delta }(x,\\tau ^{\\prime })-\\frac{\\varepsilon ^2\\bar{\\delta }}{8}.$ We consider separately two cases, either $X_{\\tau ^{\\prime }}$ is far from the boundary say $\\min (|X_{\\tau ^{\\prime }}-\\bar{l} |,X_{\\tau ^{\\prime }}-\\bar{r}|)\\;\\geqslant \\;\\bar{\\delta }/4$ .",
"By symmetry we can suppose that $X_{\\tau ^{\\prime }}$ is closer to the left boundary.",
"A consequence of (REF ) in Lemma REF is that $x-X_{\\tau ^{\\prime }} \\;\\leqslant \\;\\widetilde{f}^{\\delta }(x,\\tau ^{\\prime })-\\bar{\\delta }/16, \\quad \\forall x\\in (X_{\\tau ^{\\prime }},X_{\\tau ^{\\prime }}+\\bar{\\delta }/16),$ which implies that $ \\int _{-1}^1\\min (\\widetilde{f}^{\\delta }(x,\\tau ^{\\prime }),|x-X_{\\tau ^{\\prime }}|))\\,\\text{\\rm d}x \\;\\leqslant \\;\\int _{-1}^1\\widetilde{f}^{\\delta }(x,\\tau ^{\\prime })-\\frac{\\bar{\\delta }^2}{256}.$ We can say that (REF ) holds for all value of $X_{\\tau ^{\\prime }}$ provided that $\\varepsilon ^2\\;\\leqslant \\;\\bar{\\delta }/32$ .",
"If $X_{\\tau ^{\\prime }}$ is close to one of the boundary say $X_{\\tau ^{\\prime }}\\in [\\bar{l}+\\varepsilon ^2,\\bar{l}+\\bar{\\delta }/4]$ , then (REF ) implies that for all $x\\;\\leqslant \\;\\in [X_\\tau , X_\\tau + \\bar{\\delta }/4]$ $x-X_{\\tau ^{\\prime }}\\;\\leqslant \\;\\widetilde{f}^{\\delta }(x,\\tau ^{\\prime })-(x-\\bar{l})(1-e^{-\\frac{\\bar{\\delta }^2}{16t}})+(x-X_{\\tau ^{\\prime }}).\\\\ \\;\\leqslant \\;\\widetilde{f}^{\\delta }(x,\\tau ^{\\prime })-(X_{\\tau ^{\\prime }}-\\bar{l})+(x-\\bar{l})e^{-\\frac{\\bar{\\delta }^2}{16t}}\\\\ \\;\\leqslant \\;\\widetilde{f}^{\\delta }(x,\\tau ^{\\prime })-\\varepsilon ^2+(\\bar{\\delta }/2)e^{-\\frac{\\bar{\\delta }^2}{16\\varepsilon }}\\;\\leqslant \\;\\widetilde{f}^{\\delta }(x,\\tau ^{\\prime })-\\varepsilon ^2/2.$ The last inequality is valid if $\\varepsilon $ is small enough (how small depending on $\\bar{\\delta }$ ).",
"Integrating the above inequality over $[X_\\tau , X_\\tau + \\bar{\\delta }/4]$ we obtain (REF ) also in that case To conclude, it is then sufficient to use (REF ) in the r.h.s of (REF ) with $\\varepsilon $ sufficiently small.",
"Figure: Figure representing the two function f ˜ δ (·,τ ' )\\widetilde{f}^{\\delta }(\\cdot ,\\tau ^{\\prime }), and |·-X τ ' ||\\cdot -X_\\tau ^{\\prime }|.",
"The difference of volume below the graph off ˜ δ (·,τ ' )\\widetilde{f}^{\\delta }(\\cdot ,\\tau ^{\\prime }) and min(f ˜ δ (·,τ ' ),|·-X τ ' |)\\min (\\widetilde{f}^{\\delta }(\\cdot ,\\tau ^{\\prime }),|\\cdot -X_\\tau ^{\\prime }|) is the dark area on the figure.",
"We prove a lower-bound on it using thatf ˜ δ \\widetilde{f}^{\\delta } is not too small if X τ ' X_{\\tau ^{\\prime }} lies in the middle of the interval, or that the slope of ff on the boundary is close to one if X τ ' X_{\\tau ^{\\prime }} is closerto the boundary.We just proved that the dynamic do not touch the wall in the interval $[\\bar{l}+\\varepsilon ^2,\\bar{r}-\\varepsilon ^2]$ .",
"This allows us to compare it with the dynamic with no-wall for which we know the scaling limit by Theorem (REF ).",
"If one runs the dynamic up to a time $\\varepsilon $ , according to Proposition REF the dynamic $\\eta $ coincide w.h.p.",
"with a modified one $\\eta ^{(\\varepsilon )}$ where there is no wall-constraint in the interval $[x_l,x_r]$ .",
"Using monotonicity of the dynamics, this second dynamic can be bounded from below by a dynamic with the domain reduced to $[x_l,x_r]$ and with a modified initial condition which satisfies $\\forall x\\in [x_l,x_r], \\quad \\eta ^{(\\varepsilon )}_0(x)\\;\\leqslant \\;\\eta _0(x).$ As $\\eta _0$ satisfies (REF ) we can choose a sequence of initial condition $\\eta ^{(\\varepsilon )}_0$ which satisfies $\\bar{\\eta }^{(\\varepsilon )}_0(x):= \\frac{1}{L}\\eta ^{(\\varepsilon )}_0(Lx)=f^{(\\varepsilon ,\\delta )}_0+o(1),$ where $f^{(\\varepsilon ,\\delta )}_0$ is defined on $[\\bar{l}+\\varepsilon ^2,\\bar{r}-\\varepsilon ^2]$ by $f^{(\\varepsilon ,\\delta )}_0:=f_0^{\\delta }(x)-\\varepsilon ^2.$ One calls $\\bar{\\eta }^{(\\varepsilon )}$ the resulting space-time rescaled dynamics.",
"By Theorem REF w.h.p.",
"$\\forall t>0 \\forall x\\in [-1,1],\\quad \\bar{\\eta }^{(\\varepsilon )}(x,t)=\\widetilde{f}^{(\\varepsilon ,\\delta )}(x,t)+o(1),$ where $\\widetilde{f}^{(\\varepsilon ,\\delta )}(x,t)$ denotes the solution of the heat equation on $[\\bar{l}+\\varepsilon ^2,\\bar{r}-\\varepsilon ^2]$ with Dirichlet boundary condition and initial condition $f^{(\\varepsilon ,\\delta )}_0$ .",
"Thus, to prove that Equation (REF ) and (REF ) hold for $k=0$ , it is sufficient to show that $\\forall x\\in [l(\\varepsilon )-\\bar{\\delta }(\\varepsilon ),r(\\varepsilon )+\\bar{\\delta }(\\varepsilon )],\\quad \\widetilde{f}^{(\\varepsilon ,\\delta )}(x,\\varepsilon )>f^{\\delta }(x,\\varepsilon ),$ and that $\\forall x\\in [\\bar{l}+\\varepsilon ^2,\\bar{r}-\\varepsilon ^2], \\forall t\\in [0,\\varepsilon ], \\quad \\widetilde{f}^{(\\varepsilon ,\\delta )}(x,t)>f^{\\delta }_0(x)-\\alpha .$ The second inequality is easier, it is a consequence of the fact that $\\partial _t \\widetilde{f}^{(\\varepsilon ,\\delta )}= \\widetilde{f}^{(\\varepsilon ,\\delta )}_{xx}$ is bounded from below by $-\\Vert f^{\\prime \\prime }_0\\Vert _\\infty $ because the minimum of the solution of the heat-equation with Dirichlet boundary condition is increasing.",
"Thus (REF ) holds provided $\\varepsilon \\Vert f^{\\prime \\prime }_0\\Vert _\\infty <\\alpha .$ To prove REF for $k>0$ , we use (REF ) and replace $\\Vert f^{\\prime \\prime }_0\\Vert _\\infty $ by $C$ .",
"Now we turn to the proof of (REF ).",
"To control the value of $f^{\\delta }(x,\\varepsilon )$ , we use (REF ), (REF ) and the Taylor-Young formula to obtain $f^{\\delta }(x,\\varepsilon )\\;\\leqslant \\;{\\left\\lbrace \\begin{array}{ll}f^{\\delta }_0(x)+\\varepsilon (f^{\\prime \\prime }_0(x)-\\delta )+C\\varepsilon ^2, \\quad \\forall x\\in (l_0,r_0),\\\\f^{\\delta }_0(x)+\\varepsilon ( f_0 ^{\\prime \\prime }(l(0))-\\delta )+C \\varepsilon ^2,\\quad \\forall x\\in [l(\\varepsilon )-\\bar{\\delta }(\\varepsilon ),l_0]\\\\f^{\\delta }_0(x)+\\varepsilon (f_0^{\\prime \\prime }(r(0))-\\delta )+C \\varepsilon ^2, \\quad \\forall x\\in [r_0,r(\\varepsilon )+\\bar{\\delta }(\\varepsilon )],\\end{array}\\right.",
"}$ and $\\begin{split}l(\\varepsilon )-l_0&\\;\\geqslant \\;\\varepsilon (-f_0^{\\prime \\prime }(l_0)+\\bar{\\delta })-C\\varepsilon ^2,\\\\r(\\varepsilon )-r_0&\\;\\leqslant \\;\\varepsilon (f_0^{\\prime \\prime }(r_0)-\\bar{\\delta }))+C\\varepsilon ^2.\\end{split}$ Controlling $ \\widetilde{f}^{(\\varepsilon ,\\delta )}(x,\\varepsilon )$ is more tedious.",
"As the initial condition is not $C^2$ , there is no continuity of the time derivatives.",
"Let $K$ denote the heat kernel on $I:=[\\bar{l}+\\varepsilon ^2, \\bar{r}-\\varepsilon ^2]$ with Dirichlet boundary condition.",
"We have $\\widetilde{f}^{(\\varepsilon ,\\delta )}(x,\\varepsilon )=f^{(\\delta )}_0(x)-\\varepsilon ^2+\\int ^\\varepsilon _0 \\widetilde{f}^{(\\varepsilon ,\\delta )}_{xx}(x,t) \\,\\text{\\rm d}t$ and $\\widetilde{f}^{(\\varepsilon ,\\delta )}_{xx}(x,t)=\\int _I f^{\\prime \\prime }_0(y)\\mathbf {1}_{\\lbrace y\\in [l_0,r_0]\\rbrace }K(x,y,t)\\,\\text{\\rm d}y.$ Now for $t\\;\\leqslant \\;\\varepsilon $ , and if $\\varepsilon $ is sufficiently small (how small depending on the value of $\\bar{\\delta }$ ) we have uniformly in $x\\in (l_0,r_0)$ $\\int _{|y-x|\\;\\leqslant \\;\\varepsilon ^{1/3}} K(x,y,t) \\,\\text{\\rm d}x \\;\\geqslant \\;1-\\varepsilon ,$ and from (REF ) for $x\\in (l_0,r_0)$ $f^{\\prime \\prime }_0(y)\\mathbf {1}_{\\lbrace y\\in [l_0,r_0]\\rbrace }\\;\\leqslant \\;f^{\\prime \\prime }_0(x)+C|x-y|.$ This implies that for $x\\in (l_0,r_0)$ $\\widetilde{f}^{(\\varepsilon ,\\delta )}_{xx}(x,t)\\;\\geqslant \\;f^{\\prime \\prime }_0(x)-C(\\varepsilon +\\varepsilon ^{1/3}).$ For $x\\notin (l_0,r_0)$ , using similar heat kernel estimates, we obtain that for all $t\\in [0,\\varepsilon ]$ $\\widetilde{f}^{(\\varepsilon ,\\delta )}_{xx}(x,t)\\;\\geqslant \\;{\\left\\lbrace \\begin{array}{ll} f^{\\prime \\prime }_0(l_0)-C\\varepsilon ^{1/3} \\text{ for } x\\in (l_0-\\bar{\\delta }+\\varepsilon ^2, l_0),\\\\f^{\\prime \\prime }_0(r_0)-C\\varepsilon ^{1/3} \\text{ for } x\\in (r_0, r_0+\\bar{\\delta }-\\varepsilon ^2).\\end{array}\\right.",
"})]$ Note that neither (REF ) nor (REF ) are optimal estimates but they are sufficient for our purpose.",
"Using these estimates in (REF ) we obtain that $\\widetilde{f}^{(\\varepsilon ,\\delta )}(x,\\varepsilon )\\;\\geqslant \\;{\\left\\lbrace \\begin{array}{ll}f^{(\\delta )}_0(x)-\\varepsilon ^2-\\varepsilon f^{\\prime \\prime }_0(x)-2C\\varepsilon ^{4/3} \\text{ for } x\\in (l_0,r_0),\\\\f^{(\\delta )}_0(x)-\\varepsilon ^2-\\varepsilon f^{\\prime \\prime }_0(l_0)-C\\varepsilon ^{4/3} \\text{ for } x\\in (l_0-\\bar{\\delta }+\\varepsilon ^2,l_0),\\\\f^{(\\delta )}_0(x)-\\varepsilon ^2-\\varepsilon f^{\\prime \\prime }_0(r_0)-C\\varepsilon ^{4/3} \\text{ for } x\\in (r_0, r_0+ \\bar{\\delta }-\\varepsilon ^2).\\end{array}\\right.",
"}$ Comparing the above inequalities with (REF ), we can conclude that (REF ) holds.",
"This ends the proof.",
"$\\Box $ Acknowledgement: The author would like to thank Inwon Kim for her usefull advice for the proof of the short time existence of a solution to the Stefan Problem, Jean Dolbeault for pointing out a simple proof for Lemma REF , Jimmy Lamboley for various discussions and Julien Sohier for his comments on the manuscript.",
"This research work has been initiated during the authors stay in Instituto Nacional de Matématica Pura e Aplicada, he acknowledges gratefully hospitality and support."
],
[
"In this section, we prove several technical results.",
"The first one concerns the time needed for a corner-flip dynamics to reach an atypically low position.",
"Lemma 1.1 Consider a corner-flip dynamics on $\\Omega ^0_L$ , started from an initial condition that satisfies $\\eta _0(x)\\;\\geqslant \\;\\min (x+L,-x+L, L^{3/4}).$ Then with large probability, for all $t\\;\\leqslant \\;\\exp (L^{1/4})$ $\\widetilde{\\eta }(x,t)\\;\\geqslant \\;-L^{3/4}, \\forall x\\in [-L,L]$ One couples $\\widetilde{\\eta }$ with a dynamic $\\widetilde{\\eta }^{2}$ starting from the uniform measure on $\\Omega ^{0}_{L}$ (we denote it by $\\pi $ ).",
"Because of our choice for the initial condition of $\\widetilde{\\eta }$ is above $\\widetilde{\\eta }^{2}$ at time zero with large probability and thus we can couple the two dynamics so that $\\widetilde{\\eta }\\;\\geqslant \\;\\widetilde{\\eta }^{2}$ for all time with large probability.",
"Hence it is sufficient to prove the result for $\\widetilde{\\eta }^{2}$ .",
"Consider the discrete-time dynamics $\\widehat{\\eta }^2(n)$ starting from the uniform measure and that at each step choses $x$ at random and flip $\\eta $ to $\\eta ^{(x)}$ .",
"As $\\pi $ is left invariant by this dynamics, the probability that $\\widehat{\\eta }(x,n)< -L^{3/4}$ for some $x$ after $\\exp (2L^{1/4})$ step is at most (by union bound) $\\exp (2L^{1/4})\\pi (\\exists x\\in [-L,L], \\eta (x)< -L^{3/4})=O(\\exp (-L^{1/2})).$ The dynamics in continuous time can then be construction from $\\widehat{\\eta }$ by considering $(\\tau _n)_{n\\;\\geqslant \\;0}$ a Poisson clock process of intensity $L$ and setting $\\widetilde{\\eta }^2(t)=\\widehat{\\eta }(n)$ if $t\\in [\\tau _n,\\tau _{n+1})$ .",
"Then we conclude by remarking that with high probability $\\tau _{\\exp (2L^{1/4})}\\;\\geqslant \\;\\exp (L^{1/4}).$ The second result concerns estimate on $\\widetilde{f}^{\\delta }$ , the solution of the heat-equation with initial condition $f^{\\delta }_0$ and Dirichlet boundary condition on $[\\bar{l},\\bar{r}]$ .",
"It says that in the time interval $[0,\\varepsilon ]$ the slope of $\\widetilde{f}^{\\delta }$ near the boundary stays close to one.",
"Lemma 1.2 For $\\varepsilon $ small enough, for all $t\\;\\leqslant \\;\\varepsilon $ the following three statements hold (i) For all $s\\in [0,\\bar{\\delta }/2]$ , $\\max \\left( \\widetilde{f}^{\\delta }(\\bar{l}+s,t) ,\\widetilde{f}^{\\delta }(\\bar{r}-s,t)\\right)\\;\\geqslant \\;s(1-e^{-\\frac{\\bar{\\delta }^2}{16t}}).$ (ii) For all $x\\in [\\bar{l}+\\bar{\\delta }/4, \\bar{r} -\\bar{\\delta }/4]$ $\\widetilde{f}^{\\delta }(x,t)\\;\\geqslant \\;\\bar{\\delta }/8.$ (iii) We also have $\\int _{\\bar{l}}^{\\bar{r}} \\widetilde{f}^{\\delta }(x,t)\\,\\text{\\rm d}x \\;\\geqslant \\;\\int _{\\bar{l}}^{\\bar{r}} f^\\delta _0(x)\\,\\text{\\rm d}x-2t+e^{-\\frac{\\bar{\\delta }^2}{16t}}.$ The important point is to control the value of $f_x$ near $\\bar{l}$ and $\\bar{r}$ and the rest follows.",
"We have $\\widetilde{f}^{\\delta }_x(x,t)={\\mathbf {E}} _x[(f^{\\delta }_0)^{\\prime }(B_t)],$ where ${\\mathbf {E}} _x$ is the expectation with respect to standard Brownian Motion reflected on the boundaries of $[\\bar{l}, \\bar{r}]$ .",
"As $f^{\\delta }_0)^{\\prime } \\in [-1,1]$ in the whole interval and is equal to 1 on $[\\bar{l}, \\bar{l}+\\bar{\\delta }]$ one has ${\\mathbf {E}} _x[(f^{\\delta }_0)^{\\prime }(B_t)]\\;\\geqslant \\;1-{\\mathbf {P}} _x[B_t \\;\\geqslant \\;\\bar{l}+\\delta ].$ Finally if $t$ is much smaller than $\\bar{\\delta }^2$ and $x\\in [\\bar{l}, \\bar{l}+\\bar{\\delta }/2]$ , ${\\mathbf {P}} _x[B_t \\;\\geqslant \\;\\bar{l}+\\bar{\\delta }]\\;\\leqslant \\;e^{-\\frac{\\bar{\\delta }^2}{16t}},$ which implies that for all $x\\in [\\bar{l}, \\bar{l}+\\bar{\\delta }/2]$ , $\\widetilde{f}^{\\delta }_x(x,t)\\;\\geqslant \\;1-e^{-\\frac{\\bar{\\delta }^2}{16t}}.$ Similarly for all $x\\in [\\bar{r}-\\bar{\\delta }/2,\\bar{r}]$ $\\widetilde{f}^{\\delta }_x(x,t)\\;\\leqslant \\;-1+e^{-\\frac{\\bar{\\delta }^2}{16t}}.$ Integrating these inequalities we obtain (REF ).",
"For $(ii)$ , we notice that (REF ) is a consequence of the first point for $x\\in \\lbrace \\bar{l}+\\bar{\\delta }/4, \\bar{r} -\\bar{\\delta }/4\\rbrace $ .",
"For points inside the interval, it is sufficient to notice that local minima of the solution of the heat equation increase, and that they are initially larger than $\\bar{\\delta }$ .",
"For $(iii)$ we notice that $\\partial _s\\left(\\int _{\\bar{l}}^{\\bar{r}} \\widetilde{f}^{\\delta }(x,s)\\,\\text{\\rm d}x\\right)= \\widetilde{f}^{\\delta }_x(\\bar{l},t)- \\widetilde{f}^{\\delta }_x(\\bar{r},t)\\;\\geqslant \\;-2+2e^{-\\frac{\\bar{\\delta }^2}{16s}},$ if $s$ is much smaller than $\\bar{\\delta }^2$ .",
"Then (REF ) follows by integrating over $s\\in [0,t]$ ."
]
] | 1204.1253 |
[
[
"Non-neutral global solutions for the electron Euler-Poisson system in 3D"
],
[
"Abstract We prove that small smooth irrotational but charged perturbations of a constant background are global and go back to equilibrium in the 3D electron Euler-Poissson equation."
],
[
"Presentation of the equation",
"Consider a three-dimensional plasma composed of fixed ions with uniform density, and a gas of moving electrons.",
"This situation can be described by the Euler-Poisson equation, which couples a compressible gas to an electrostatic field.",
"Letting $u$ be the velocity of the electron gas and $\\rho $ its density, it reads after a simple rescaling $(EP) \\quad \\left\\lbrace \\begin{array}{l} \\partial _t \\rho + \\nabla \\cdot (\\rho u) = 0 \\\\\\rho (\\partial _t u + u \\cdot \\nabla u) = -\\rho \\nabla \\rho + \\rho \\nabla \\Phi \\\\\\Delta \\Phi = \\rho - 1.",
"\\end{array}\\right.$ We took for simplicity the pressure law $p(\\rho ) = \\frac{1}{2} \\rho ^2$ ; the analysis is similar for other pressure laws.",
"We make the standing assumption that the fluid is irrotational $\\nabla \\times u = 0;$ this condition is of course conserved by the flow of (REF ).",
"Our aim is to understand the stability of the obvious stationary state $u = 0 \\quad ,\\quad \\rho = 1$ under perturbations in $\\rho $ which do not have mean zero (i.e.",
"are not electrically neutral).",
"It has been proposed that the non-neutral assumption could have important consequences for the asymptotic dynamics of the perturbation.",
"This is made plausible by the following remark: consider for each time a ball centered at the center of mass and containing one half of the total electric charge.",
"Far away from the ball, the action of the electric field generated by the electrons inside this ball is similar to the action of a single point charge at the center of mass of the same total electric charge.",
"Recall now that particles moving in a Coulombian electric field experience asymptotically a logarithmic correction to their free trajectory [9].",
"Based on this analogy, one could expect a correction to the linear scattering of the perturbation, which had been shown for electrically neutral perturbations (for which the electric field at infinity would be that of a dipole and hence integrable along the trajectories).",
"It turns out that this picture is not accurate, at least for irrotational perturbations.",
"To the best of our understanding, this is due to a combination of the following three facts The analogy with point particle is not completely accurate.",
"The generated perturbation of the electric field oscillates in time along with the particles.",
"However, the nonlinearities come from inertial terms (convection and pressure) and therefore oscillate “out of phase” with the density and velocity fields (in other words, the nonlinear terms oscillate in a non resonant way with respect to the linear terms).",
"This allows to use a normal form transformation and partially cancel their long-time influence (at least cancel the influence of the “short-range” part of the nonlinearity involving $\\beta $ ).",
"The electric field away from the origin has constant amplitude and oscillates (when there is a motion).",
"Its main effect is to periodically repel and attract the particles to the origin.",
"However, since its strength is not uniform, the net effect is to create a repulsive force.",
"To understand the effect of this force, assume that the center of mass remains at the origin.",
"When moving away from the origin, the particles experience a weaker force which takes more time to counteract their inertia and invert their velocity, so they move a long distance away from the origin; in contrast, when the particles move closer to the origin, they encounter a stronger force which sends them back faster and they move a shorter distance towards the origin.",
"However, this dispersion effect depends not on the amplitude of the electric field (which decays like $1/\\vert x\\vert $ ), but on the gradient of this field (which decays like $1/\\vert x\\vert ^2$ and is thus integrable).",
"Therefore, it has no long term effect either and we recover linear scattering."
],
[
"Main result",
"Our main result is that a constant equilibrium of charged electrons is stable, even under non neutral (but still irrotational) perturbations.",
"We denote $\\quad \\tilde{Q}=\\int _{\\mathbb {R}^3}\\left[\\rho _0(x)-1\\right]dx$ (which can thus be taken non zero) for the charge of the perturbation.",
"This extends the work of Guo [4] who assumed that $\\tilde{Q}=0$ .",
"Theorem 1.1 There exists $\\delta _0$ such that if $(\\rho _0,v_0)\\in C^\\infty _c$ satisfy $\\hbox{curl}[v_0]=0,\\quad \\Vert (\\rho _0-1,v_0)\\Vert _{H^{10}}+\\Vert (\\rho _0-1,v_0)\\Vert _{W^{5,1}}\\le \\delta _0,$ then there exists a unique global solution of (REF ) which converges to equilibrium in the sense that $\\Vert (\\rho (t)-1,v(t))\\Vert _{L^\\infty _x}\\rightarrow 0,\\quad \\text{as }t\\rightarrow +\\infty .$ Furthermore, it scatters in a sense which will be made precise in Corollary REF .",
"Again, as in Guo [4], note the contrast with the result in the absence of electric field in [11].",
"This theorem suggests that the neutral assumption made in [1], [8] might be removed.",
"Recall that the neutral assumption is not necessary either to get small-data/global existence for the Euler-Poisson equation for the ions [5], which corresponds to the large-time behavior of the system, but the ion case is more transparent since the non-neutral assumption has no implication on the decay of free solutions.",
"Our analysis relies on works on quasilinear dispersive equations, especially on normal form transform methods, starting from the work of Shatah [10] and following recent developments of the space-time resonance method in [2], [3], [6], [7], [8].",
"The main consequence of the non neutral assumption $\\tilde{Q}\\ne 0$ is that the solution to the linearized equation is no longer integrable in time.",
"However, we remark that its derivative still is.",
"Since the quadratic nonlinearities always involve at least one derivative, the main point is then to systematically track down the extra decay provided by this derivative term, thus giving a fairly simple proof of Theorem REF ."
],
[
"Notations",
"We adopt the following notations $A \\lesssim B$ if $A \\le C B$ for some implicit constant $C$ .",
"The value of $C$ may change from line to line.",
"We note $A\\simeq B$ if $A\\lesssim B\\lesssim A$ .",
"If $f$ is a function over $\\mathbb {R}^d$ then its Fourier transform, denoted $\\widehat{f}$ , or $\\mathcal {F}(f)$ , is given by $\\widehat{f}(\\xi ) = \\mathcal {F}f (\\xi ) = \\frac{1}{(2\\pi )^{d/2}} \\int e^{-ix\\xi } f(x) \\,dx \\;\\;\\;\\;\\mbox{thus} \\;\\;\\;\\;f(x) = \\frac{1}{(2\\pi )^{d/2}} \\int e^{ix\\xi }\\widehat{f}(\\xi ) \\,d\\xi .$ (in the text, we systematically drop the constants such as $\\frac{1}{(2 \\pi )^{d/2}}$ since they are not relevant).",
"The Fourier multiplier with symbol $m(\\xi )$ is defined by $m(i\\nabla )f = \\mathcal {F}^{-1} \\left[m \\mathcal {F} f \\right].$ The Littlewood-Paley projector $P_{\\le N}$ , $P_{\\ge N/2}$ and $P_N$ are defined as the Fourier multipliers of symbols $\\chi (\\xi /N),\\quad (1-\\chi (\\xi /N))\\hbox{ and }\\chi (\\xi /(2N))-\\chi (\\xi /N)$ for $\\chi \\in C^\\infty _c(\\mathbb {R}^3)$ such that $\\chi (x)=1$ when $\\vert x\\vert \\le 1$ and $\\chi (x)=0$ when $\\vert x\\vert \\ge 2$ .",
"In what follows, sums over capital letters are understood to be over dyadic numbers.",
"The bilinear Fourier multiplier with symbol $m$ is given by $T_m[f,g](x) \\overset{def}{=} \\int e^{ix(\\xi +\\eta )} \\widehat{f}(\\xi ) \\widehat{g}(\\eta ) m(\\xi ,\\eta )\\, d\\xi d\\eta = \\mathcal {F}^{-1} \\int m(\\xi -\\eta ,\\eta ) \\widehat{f}(\\xi -\\eta ) \\widehat{g}(\\eta )\\,d\\eta .$ We also define $\\tilde{T}_m$ to denote an operator “of the form” $T_m$ in the sense that $\\tilde{T}_m[f,g]\\in \\lbrace T_m[f,g],T_m[\\overline{f},g],T_m[f,\\overline{g}],T_m[\\overline{f},\\overline{g}]\\rbrace .$ The japanese bracket $\\langle \\cdot \\rangle $ stands for $\\langle x \\rangle = \\sqrt{1 + x^2}$ .",
"The Riesz transform is defined as the real operator $R_j=\\vert \\nabla \\vert ^{-1}\\partial _j$ .",
"The Besov spaces are defined by their norms as follows $\\Vert f\\Vert _{B^\\sigma _{p,q}}^q=\\sum _{N\\in 2^\\mathbb {Z}}\\langle N\\rangle ^{q\\sigma }\\Vert P_Nf\\Vert _{L^p}^q.$"
],
[
"Preliminary steps",
"In order to investigate the stability of $u=0$ , $\\rho =1$ , using (REF ), we introduce the new unknown function $\\alpha = \\langle \\nabla \\rangle \\vert \\nabla \\vert ^{-1} (\\rho -1) + i \\vert \\nabla \\vert ^{-1}\\hbox{div}[u].$ The original unknowns can be recovered by the formulas $\\rho -1=\\vert \\nabla \\vert \\langle \\nabla \\rangle ^{-1}{Re}[\\alpha ] \\quad \\mbox{and} \\quad u_j = - R_j{Im}[\\alpha ].$ The system (REF ) becomes $\\begin{split}(\\partial _t-i\\langle \\nabla \\rangle )\\alpha &= - \\frac{i}{4}\\sum _{j=1}^3R_j\\langle \\nabla \\rangle \\big [\\frac{|\\nabla |}{\\langle \\nabla \\rangle }(\\alpha +\\overline{\\alpha })\\cdot R_j(\\alpha -\\overline{\\alpha })\\big ]-\\frac{i}{8}\\sum _{j=1}^3|\\nabla |\\big [R_j(\\alpha -\\overline{\\alpha })\\cdot R_j(\\alpha -\\overline{\\alpha })\\big ].\\end{split}$ The above right-hand side is a sum of quadratic terms in $\\alpha $ and $\\bar{\\alpha }$ : $RHS = F(\\alpha ,\\alpha ) \\quad \\mbox{where} \\quad F(f,g) = Q_1 (f,g) + Q_2 (f,\\bar{g}) + Q_3 (\\bar{f},\\bar{g}).$ The bilinear operators $Q_1$ , $Q_2$ , and $Q_3$ are pseudo-products as in (REF ) whose symbols are linear combinations of the following multipliers $\\begin{split}m_p(\\xi _1,\\xi _2)&=\\vert \\xi _1\\vert \\frac{\\langle \\xi _1+\\xi _2\\rangle }{\\langle \\xi _1\\rangle }\\frac{\\left[ \\xi _2\\cdot (\\xi _1+\\xi _2)\\right]}{\\vert \\xi _2\\vert \\vert \\xi _1+\\xi _2\\vert }=\\vert \\xi _1\\vert \\langle \\xi _2\\rangle \\frac{\\langle \\xi _1+\\xi _2\\rangle }{\\langle \\xi _1\\rangle \\langle \\xi _2\\rangle }\\tilde{m}_p(\\xi _1,\\xi _2)\\\\m_t(\\xi _1,\\xi _2)&=\\vert \\xi _1\\vert \\left[\\frac{\\xi _1\\cdot (\\xi _1+\\xi _2)}{\\vert \\xi _1\\vert \\vert \\xi _1+\\xi _2\\vert }\\frac{\\xi _1\\cdot \\xi _2}{\\vert \\xi _1\\vert \\vert \\xi _2\\vert }\\right]=\\vert \\xi _1\\vert \\tilde{m}_t(\\xi _1,\\xi _2)\\end{split}$ or their symmetric $m_p(\\xi _2,\\xi _1)$ and $m_t(\\xi _2,\\xi _1)$ .",
"In (REF ), $m_t$ corresponds to the second term in (REF ) after using that $\\vert \\nabla \\vert =-\\sum _jR_j\\partial _j$ .",
"We now isolate the effect of the electric field at infinity.",
"Actually, for simplicity in our analysis, we need to replace $\\tilde{Q}$ by $Q$ in (REF ) and introduce $\\chi ^Q$ as followsbut one should essentially think of $\\chi ^Q$ as $Q\\vert \\nabla \\vert ^{-1}\\chi $ for $\\chi $ a nice bump function such that $\\int \\chi dx=1$ .",
"$\\beta (t) \\overset{def}{=} \\alpha (t) - e^{it\\langle \\nabla \\rangle } \\chi ^Q \\quad \\mbox{where} \\quad \\chi ^Q \\overset{def}{=} P_{\\le 1}{Re}\\left[\\alpha (0)\\right].$ It solves the system $\\begin{split}\\left(\\partial _t - i \\langle \\nabla \\rangle \\right) \\beta &= F(e^{it\\langle \\nabla \\rangle } \\chi ^Q,e^{it\\langle \\nabla \\rangle } \\chi ^Q) + \\left[F(e^{it\\langle \\nabla \\rangle } \\chi ^Q,\\beta )+F(\\beta ,e^{it\\langle \\nabla \\rangle } \\chi ^Q)\\right] + F(\\beta ,\\beta )\\\\&=I+II+III.\\end{split}$ which is forced by $e^{it\\langle \\nabla \\rangle } \\chi ^Q$ satisfying (due to (REF )): $\\begin{split}\\Vert e^{it\\langle \\nabla \\rangle } \\chi ^Q\\Vert _{B^0_{p,2}}&\\lesssim Q(1+t)^{-\\frac{3}{2}(1-\\frac{2}{p})},\\quad 2\\le p<3\\\\\\Vert \\nabla e^{it\\langle \\nabla \\rangle } \\chi ^Q\\Vert _{B^0_{p,2}}&\\lesssim Q(1+t)^{-\\frac{3}{2}(1-\\frac{2}{p})},\\quad 2\\le p<+\\infty \\\\\\Vert e^{it\\langle \\nabla \\rangle } \\chi ^Q\\Vert _{H^N}&\\lesssim Q\\\\\\end{split}$ uniformly in $t\\ge 0$ , where $Q:=\\Vert P_{\\le 1}\\rho (0)\\Vert _{L^1}\\\\$ is a substitute for $\\tilde{Q}$ in (REF ).",
"Fix $\\sigma \\ge 2$ and $N\\ge \\sigma +7$ .",
"We define our main norm for the global control: $\\begin{split}\\Vert f\\Vert _Y&:=\\Vert f\\Vert _{W^{\\sigma +2,10/9}_x}+\\Vert f\\Vert _{H^N_x}\\\\\\Vert \\beta \\Vert _{X_T}&:=\\sup _{0\\le t\\le T}\\left[(1+t)^{6/5}\\Vert \\beta (t)\\Vert _{B^{\\sigma }_{10,2}}+\\Vert \\beta (t)\\Vert _{H^N_x}\\right].\\end{split}$ Using (REF ), we see that, for all $T$ $\\Vert e^{it\\langle \\nabla \\rangle }f\\Vert _{X_T}\\lesssim \\Vert f\\Vert _Y$ uniformly in $T$ .",
"Local existence in $X_T$ follows from energy estimates, therefore we see that Theorem REF will be a consequence of the following Proposition.",
"Proposition 3.1 There exists $\\delta >0$ such that if $\\beta \\in C([0,T]:H^N)$ satisfies (REF ) on $[0,T]$ and $Q+\\sup _{0\\le t\\le T}\\Vert \\beta (t)\\Vert _{H^5}\\le \\delta ,$ then $\\Vert \\beta \\Vert _{X_T}\\lesssim \\Vert \\beta (0)\\Vert _Y+(Q+\\Vert \\beta \\Vert _{X_T})^2$ uniformly in $T$ .",
"In the proof, we have decided to use Besov spaces instead of more classical spaces in order to have a simple access to the estimates in Lemma REF .",
"More elaborate harmonic analysis techniques probably allow to replace Besov spaces with Lebesgue spaces.",
"In any case, the difference between the two should be thought of as inessential to the proof.",
"Also, we have made some effort to quantify the number of derivatives needed and to keep it reasonably low (around 10), notably through a “tame” estimatetame in the sense that most of the loss of derivative is on the low frequency term on bilinear operators in Lemma REF .",
"A slightly more efficient analysis could somewhat reduce this number, but to make it close to the number of derivatives in the physical energy would require significantly stronger results."
],
[
"Proof of Proposition ",
"We introduce the linear profile $b(t)=e^{-it\\langle \\nabla \\rangle }\\beta (t).$ This natural unknown is only affected by the nonlinearity $\\partial _tb=e^{-it\\langle \\nabla \\rangle }\\left[I+II+III\\right].$ Using Duhamel formula, we see that $\\beta (t)=e^{it\\langle \\nabla \\rangle }\\left[\\beta (0)+\\mathcal {N}\\right]$ where $\\mathcal {N}$ is a finite sum of operators of the form $\\begin{split}\\mathcal {I}_{\\varepsilon ,m}[c_1,c_2]&=\\int _0^te^{-is\\langle \\nabla \\rangle }T_m[\\mathcal {C}^{\\varepsilon _1}e^{is\\langle \\nabla \\rangle }c_1(s),\\mathcal {C}^{\\varepsilon _2}e^{is\\langle \\nabla \\rangle }c_2(s)]ds\\\\&=\\mathcal {F}^{-1}\\int _0^t\\int _{\\mathbb {R}^3}e^{is\\phi _\\varepsilon (\\xi -\\eta ,\\eta )}m(\\xi -\\eta ,\\eta )\\widehat{\\mathcal {C}^{\\varepsilon _1}c_1}(s,\\xi -\\eta )\\widehat{\\mathcal {C}^{\\varepsilon _2}c_2}(s,\\eta )d\\eta ds\\end{split}$ for some $m(\\xi _1,\\xi _2)$ as in (REF ), some $\\varepsilon =(\\varepsilon _1,\\varepsilon _2)\\in \\lbrace \\pm \\rbrace ^2$ , $c_1(t,x),c_2(t,x)\\in \\lbrace b(t,x),\\chi ^Q(x)\\rbrace $ , where $\\mathcal {C}^+=Id$ and $\\mathcal {C}^-$ denotes the complex conjugation and $\\phi _\\varepsilon (\\xi _1,\\xi _2)=-\\langle \\xi _1+\\xi _2\\rangle +\\varepsilon _1\\langle \\xi _1\\rangle +\\varepsilon _2\\langle \\xi _2\\rangle .$ We will obtain bounds which are uniform in $\\varepsilon $ and $m$ .",
"In all that follows, the worst term to keep in mind is a variation of $e^{it\\langle \\nabla \\rangle }\\chi ^Q\\cdot \\vert \\nabla \\vert P_{\\ge 1}e^{it\\langle \\nabla \\rangle }b(t)$ where $b$ has to provide both decay and regularity.",
"We start with a simple estimate.",
"Lemma 4.1 For any choice of $c\\in \\lbrace b,\\chi ^Q\\rbrace $ , we have $\\Vert e^{it\\langle \\nabla \\rangle }\\partial _tc(t)\\Vert _{H^{\\sigma +3}_x}\\lesssim (1+t)^{-9/10-\\varepsilon }\\left[Q+\\Vert \\beta \\Vert _{X_T}\\right]^2$ for $0<\\varepsilon <1/100$ and $\\Vert e^{it\\langle \\nabla \\rangle }\\partial _tc(t)\\Vert _{H^{\\sigma -1}_x}\\lesssim (1+t)^{-6/5}\\left[Q+\\Vert \\beta \\Vert _{X_T}\\right]^2.$ The case $c=\\chi ^Q$ is trivial.",
"Hence it suffices to treat the case $c=b$ .",
"From (REF ), (REF ), it suffices to show that $\\begin{split}\\Vert \\tilde{T}_{\\mu }[\\vert \\nabla \\vert e^{it\\langle \\nabla \\rangle }c_1,\\langle \\nabla \\rangle e^{it\\langle \\nabla \\rangle }c_2]\\Vert _{H^{\\sigma +3}_x}&\\lesssim (1+t)^{-9/10-\\varepsilon }(Q+\\Vert \\beta \\Vert _{X_T})^2,\\\\\\Vert \\tilde{T}_{\\mu }[\\vert \\nabla \\vert e^{it\\langle \\nabla \\rangle }c_1,\\langle \\nabla \\rangle e^{it\\langle \\nabla \\rangle }c_2]\\Vert _{H^{\\sigma -1}_x}&\\lesssim (1+t)^{-6/5}(Q+\\Vert \\beta \\Vert _{X_T})^2\\end{split}$ for $c_1,c_2\\in \\lbrace Rb,R\\chi ^Q\\rbrace $ , where $R$ denotes a Fourier multiplier coming from a symbol of order 0, (which we will omit for simplicity of notation) and for $\\mu \\in \\lbrace 1,\\frac{\\langle \\xi _1 + \\xi _2 \\rangle }{\\langle \\xi _2 \\rangle \\langle \\xi _1 \\rangle } \\rbrace $ , for which it is easily seen (for instance by Coifman Meyer theory) that Hölder bounds apply.",
"We first treat the worst case using (REF ) and (REF ).",
"$\\begin{split}\\Vert \\tilde{T}_\\mu [\\vert \\nabla \\vert e^{it\\langle \\nabla \\rangle }c_1,\\langle \\nabla \\rangle e^{it\\langle \\nabla \\rangle }\\chi ^Q]\\Vert _{H^{\\sigma -1}_x}&\\lesssim \\Vert \\vert \\nabla \\vert e^{it\\langle \\nabla \\rangle }c_1\\Vert _{W^{\\sigma -1,10}_x}\\Vert \\langle \\nabla \\rangle e^{it\\langle \\nabla \\rangle }\\chi ^Q\\Vert _{W^{\\sigma -1,5/2}_x}\\\\&\\lesssim (1+t)^{-3/2}(Q+\\Vert \\beta \\Vert _{X_T})^2.\\end{split}$ Independently, $\\begin{split}\\Vert \\tilde{T}_\\mu [\\vert \\nabla \\vert e^{it\\langle \\nabla \\rangle }c_1,\\langle \\nabla \\rangle e^{it\\langle \\nabla \\rangle }\\chi ^Q]\\Vert _{H^{\\sigma +6}_x}&\\lesssim \\Vert \\langle \\nabla \\rangle e^{it\\langle \\nabla \\rangle }\\chi ^Q\\Vert _{L^\\infty _x}\\Vert c_1\\Vert _{H^{\\sigma +7}_x}\\\\&\\lesssim \\left[\\sum _N\\min (N^\\frac{1}{2},N^{-1}(1+t)^{-3/2})\\right](Q+\\Vert \\beta \\Vert _{X_T})^2\\\\&\\lesssim (1+t)^{-1/2}(Q+\\Vert \\beta \\Vert _{X_T})^2.\\end{split}$ Interpolating at order $(4/10+\\varepsilon ,6/10-\\varepsilon )$ , we see that this term is acceptable.",
"It only remains to consider, using (REF ) $\\begin{split}\\Vert \\tilde{T}_\\mu [\\vert \\nabla \\vert \\beta ,\\langle \\nabla \\rangle \\beta ]\\Vert _{H^{\\sigma +4}_x}&\\lesssim \\Vert \\beta \\Vert _{W^{2,10}_x}\\Vert \\beta \\Vert _{H^{\\sigma +5}_x}\\lesssim (1+t)^{-6/5}\\Vert \\beta \\Vert _{X_T}^2.\\end{split}$ This ends the proof.",
"Corollary 4.2 Under the assumptions of the theorem, the solution scatters in $H^{\\sigma }$ ; namely there exists $\\alpha _\\infty $ in $H^{\\sigma }_x$ such that $\\left\\Vert \\alpha (t) - e^{it \\langle \\nabla \\rangle } \\alpha _\\infty \\right\\Vert _{H^{\\sigma }_x} \\longrightarrow 0,\\quad \\mbox{as $t \\rightarrow +\\infty $} .$ This follows immediately from the integrability in time of $\\Vert \\partial _t b(t)\\Vert _{H^\\sigma _x}\\lesssim (1+t)^{-19/14}$ which in turn follows by interpolation from the two bounds in Lemma REF .",
"Now, we are ready to prove Proposition REF .",
"We start with the dispersive estimate.",
"We claim that under the hypothesis of Proposition REF , there holds that $\\sup _{0\\le t\\le T}(1+t)^\\frac{6}{5}\\Vert \\beta (t)\\Vert _{W^{\\sigma ,10}_x}\\lesssim \\Vert \\beta (0)\\Vert _Y+(Q+\\Vert \\beta \\Vert _{X_T})^2.$ Using (REF ), and (REF ), it suffices to treat the nonlinear term $\\mathcal {N}$ .",
"The effect of the nonlinear terms $\\mathcal {I}_{\\varepsilon ,m}$ can be conveniently reformulated through a normal form transformation, which follows from a simple integration by parts.",
"Indeed, $\\begin{split}\\mathcal {I}_{\\varepsilon ,m}[c_1,c_2](t)&=-ie^{-it\\langle \\nabla \\rangle }T_{m/\\phi _\\varepsilon }[\\mathcal {C}^\\varepsilon _1e^{it\\langle \\nabla \\rangle }c_1(t),\\mathcal {C}^\\varepsilon _2e^{it\\langle \\nabla \\rangle }c_2(t)]+iT_{m/\\phi _\\varepsilon }[c_1(0),c_2(0)]\\\\&+i \\mathcal {I}_{\\varepsilon ,m/\\phi _\\varepsilon }[\\partial _sc_1,c_2](t)+ i \\mathcal {I}_{\\varepsilon ,m/\\phi _\\varepsilon }[c_1,\\partial _sc_2](t).\\end{split}$ The first term in (REF ) can be treated as follows.",
"First, from (REF ), Bernstein and Sobolev inequalities, we observe that $\\begin{split}&\\sum _{M\\ge 1}\\Vert P_{\\ge M/8}\\tilde{T}_{m/\\phi _\\varepsilon }[P_{M}e^{it\\langle \\nabla \\rangle }c_1(t),P_{\\ge M/8}e^{it\\langle \\nabla \\rangle }c_2(t)]\\Vert _{B^{\\sigma }_{10,2}}\\\\&\\lesssim \\sum _{ M\\ge 1}\\Vert e^{it\\langle \\nabla \\rangle }P_{M}c_1(t)\\Vert _{W^{1,\\infty }_x}\\Vert e^{it\\langle \\nabla \\rangle }c_2(t)\\Vert _{B^{\\sigma +5}_{10,2}}\\\\&\\lesssim (1+t)^{-6/5}(Q+\\Vert \\beta \\Vert _{X_T}) \\sum _{M\\ge 1}M^\\frac{3}{10}M^{1-\\sigma }\\Vert c_2\\Vert _{H^{\\sigma +31/5}_x}\\\\&\\lesssim (1+t)^{-6/5}(Q+\\Vert \\beta \\Vert _{X_T})^2\\end{split}$ Independently, using (REF ) with $p=\\infty $ , we compute that $\\begin{split}&\\sum _{M\\le 8}\\Vert P_{\\ge M/8}\\tilde{T}_{m/\\phi _\\varepsilon }[P_Me^{it\\langle \\nabla \\rangle }c_1(t),P_{\\ge M/8}e^{it\\langle \\nabla \\rangle }c_2(t)]\\Vert _{B^{\\sigma }_{10,2}}\\\\&\\lesssim \\sum _{M\\le 8}\\Vert P_Me^{it\\langle \\nabla \\rangle }c_1(t)\\Vert _{L^\\infty _x}\\Vert e^{it\\langle \\nabla \\rangle }\\vert \\nabla \\vert c_2(t)\\Vert _{B^{\\sigma }_{10,2}}\\\\&\\lesssim \\left[\\sum _{M\\le 8}\\min (M^{-1/2}(1+t)^{-1},M^{1/2})\\right](Q+\\Vert \\beta \\Vert _{X_T})\\Vert e^{it\\langle \\nabla \\rangle }\\vert \\nabla \\vert c_2(t)\\Vert _{B^{\\sigma }_{10,2}}\\\\&\\lesssim (1+t)^{-1/2}(Q+\\Vert \\beta \\Vert _{X_T})\\Vert e^{it\\langle \\nabla \\rangle }\\vert \\nabla \\vert c_2(t)\\Vert _{B^{\\sigma }_{10,2}}.\\end{split}$ If $c_2=\\chi ^Q$ , this is acceptable; otherwise, using complex interpolation, $\\begin{split}\\Vert e^{it\\langle \\nabla \\rangle }b(t)\\Vert _{B^{\\sigma +1}_{10,2}}&\\lesssim \\Vert e^{it\\langle \\nabla \\rangle }b(t)\\Vert _{B^{\\sigma }_{10,2}}^\\frac{7}{12}\\Vert e^{it\\langle \\nabla \\rangle }b(t)\\Vert _{B^{\\sigma +12/5}_{10,2}}^\\frac{5}{12}\\\\&\\lesssim (1+t)^{-7/10}(Q+\\Vert \\beta \\Vert _{X_T})^\\frac{7}{12}\\Vert b(t)\\Vert _{H^{\\sigma +18/5}}^\\frac{5}{12}\\lesssim (1+t)^{-7/10}(Q+\\Vert \\beta \\Vert _{X_T}),\\end{split}$ and we see that this term is acceptable.",
"Finally, using (REF ), $\\begin{split}&\\sum _{M}\\Vert P_{\\le M/8}\\tilde{T}_{m/\\phi _\\varepsilon }[P_Me^{it\\langle \\nabla \\rangle }c_1(t),P_{\\ge M/8}e^{it\\langle \\nabla \\rangle }c_2(t)]\\Vert _{B^{\\sigma }_{10,2}}\\\\&\\lesssim \\sum _{N\\le M\\sim O} N^\\frac{3}{5}\\Vert P_N\\tilde{T}_{m/\\phi _\\varepsilon }[P_Me^{it\\langle \\nabla \\rangle }c_1(t),P_Oe^{it\\langle \\nabla \\rangle }c_2(t)]\\Vert _{W^{\\sigma ,5}_x}\\\\&\\lesssim \\sum _{N\\le M\\sim O}N^{\\frac{3}{5}}\\langle N\\rangle ^{\\sigma +5} \\sum _{\\lbrace a,b\\rbrace =\\lbrace 1,2\\rbrace }\\Vert P_Me^{it\\langle \\nabla \\rangle }c_a(t)\\Vert _{L^{10}_x}\\Vert P_Oe^{it\\langle \\nabla \\rangle }\\vert \\nabla \\vert c_b(t)\\Vert _{L^{10}_x}\\\\&\\lesssim (1+t)^{-6/5}(Q+\\Vert \\beta \\Vert _{X_T})^2\\end{split}$ For the second term, using (REF ), we see that, for $\\kappa =1/1000$ , $\\begin{split}\\Vert e^{it\\langle \\nabla \\rangle }\\tilde{T}_{m/\\phi ,\\varepsilon }[c_1(0),c_2(0)]\\Vert _{B^{\\sigma }_{10,2}}&\\lesssim (1+t)^{-6/5}\\Vert \\tilde{T}_{m/\\phi ,\\varepsilon }[c_1(0),c_2(0)]\\Vert _{W^{\\sigma +2,10/9}_x}\\\\&\\lesssim (1+t)^{-6/5}\\Vert \\vert \\nabla \\vert ^{-\\kappa }c_1(0)\\Vert _{W^{\\sigma +5+2\\kappa ,20/9}}\\Vert \\vert \\nabla \\vert ^{-\\kappa }c_2(0)\\Vert _{W^{\\sigma +5+2\\kappa ,20/9}}\\\\&\\lesssim (1+t)^{-6/5}(Q+\\Vert b\\Vert _X)^2.\\end{split}$ We treat the third term in (REF ) in a similar way, using (REF ), Lemma REF and Lemma REF .",
"First, we have for the worst term $\\begin{split}&\\sum _{M}\\Vert P_{\\ge M/8}e^{it\\langle \\nabla \\rangle }\\mathcal {I}_{\\varepsilon ,m/\\phi _\\varepsilon }[P_{\\ge M/8}\\partial _sc_1,P_Mc_2]\\Vert _{B^{\\sigma }_{10,2}}\\\\&\\lesssim \\sum _{M}\\int _0^t(1+t-s)^{-6/5}\\Vert P_{\\ge M/8}\\tilde{T}_{m/\\phi _\\varepsilon }[P_{\\ge M/8}e^{is\\langle \\nabla \\rangle }\\partial _sc_1,P_Me^{is\\langle \\nabla \\rangle }c_2]\\Vert _{B^{\\sigma +2}_{10/9,2}}ds\\\\&\\lesssim \\sum _{M}\\int _0^t(1+t-s)^{-6/5}\\Vert e^{is\\langle \\nabla \\rangle }\\partial _sc_1\\Vert _{H^{\\sigma +3}_x}\\Vert P_Me^{is\\langle \\nabla \\rangle }c_2\\Vert _{W^{5,\\frac{5}{2}}_x}ds\\\\&\\lesssim \\int _0^t(1+t-s)^{-6/5}(1+s)^{-9/10-\\varepsilon }\\sum _{M}\\min (M^\\frac{3}{10}(1+M)^{5-N},(1+s)^{-3/10})ds(Q+\\Vert \\beta \\Vert _{X_T})^2\\\\&\\lesssim (1+t)^{-6/5}(Q+\\Vert \\beta \\Vert _{X_T})^2\\end{split}$ while the other terms are easier; we now turn to them.",
"Let $\\varepsilon $ be as in Lemma REF and $2_\\varepsilon $ and $(5/2)_\\varepsilon $ be such that $\\frac{1}{2_\\varepsilon }=\\frac{1}{2}-\\frac{\\varepsilon }{3},\\quad \\frac{1}{(5/2)_\\varepsilon }=\\frac{2}{5}+\\frac{\\varepsilon }{3},$ then $\\begin{split}&\\sum _{M}\\Vert P_{\\ge M/8}e^{it\\langle \\nabla \\rangle }\\mathcal {I}_{\\varepsilon ,m/\\phi _\\varepsilon }[P_{M}\\partial _sc_1,P_{\\ge M/8}c_2]\\Vert _{B^{\\sigma }_{10,2}}\\\\&\\lesssim \\sum _{M}\\int _0^t(1+t-s)^{-6/5}\\Vert P_{\\ge M/8}\\tilde{T}_{m/\\phi _\\varepsilon }[P_{M}e^{is\\langle \\nabla \\rangle }\\partial _sc_1,P_{\\ge M/8}e^{is\\langle \\nabla \\rangle }c_2]\\Vert _{B^{\\sigma +2}_{10/9,2}}ds\\\\&\\lesssim \\sum _{M}\\int _0^t(1+t-s)^{-6/5}\\Vert P_Me^{is\\langle \\nabla \\rangle }\\partial _sc_1\\Vert _{W^{4,2_\\varepsilon }_x}\\Vert \\vert \\nabla \\vert e^{is\\langle \\nabla \\rangle }c_2\\Vert _{W^{\\sigma +3,(5/2)_\\varepsilon }_x}ds\\\\&\\lesssim \\int _0^t(1+t-s)^{-6/5}\\sum _{M}M^\\varepsilon \\Vert P_M\\partial _sc_1\\Vert _{H^5_x}(1+s)^{-3/10+\\varepsilon }ds(Q+\\Vert \\beta \\Vert _{X_T})\\\\&\\lesssim (1+t)^{-6/5}(Q+\\Vert \\beta \\Vert _{X_T})^2\\end{split}$ and finally, $\\begin{split}&\\sum _{M\\ge 1}\\Vert P_{M}e^{it\\langle \\nabla \\rangle }\\mathcal {I}_{\\varepsilon ,m/\\phi _\\varepsilon }[P_{\\ge 4M}\\partial _sc_1,P_{\\ge 4M}c_2]\\Vert _{B^{\\sigma }_{10,2}}\\\\&\\lesssim \\sum _{M\\ge 1}\\int _0^t(1+t-s)^{-6/5}\\Vert P_{M}\\tilde{T}_{m/\\phi _\\varepsilon }[P_{\\ge 4M}e^{is\\langle \\nabla \\rangle }\\partial _sc_1,P_{\\ge 4M}e^{is\\langle \\nabla \\rangle }c_2]\\Vert _{B^{\\sigma +2}_{10/9,2}}ds\\\\&\\lesssim \\sum _{1\\le M\\le N\\sim O}M^{\\sigma +7}N\\int _0^t(1+t-s)^{-6/5}\\Vert P_N\\partial _sc_1\\Vert _{L^{2}_x}\\Vert P_Oe^{is\\langle \\nabla \\rangle }c_2\\Vert _{L^{\\frac{5}{2}}_x}ds\\\\&\\lesssim (1+t)^{-6/5}(Q+\\Vert \\beta \\Vert _{X_T})^2\\end{split}$ while $\\begin{split}&\\sum _{M\\le 1}\\Vert P_{M}e^{it\\langle \\nabla \\rangle }\\mathcal {I}_{\\varepsilon ,m/\\phi _\\varepsilon }[P_{\\ge 4M}\\partial _sc_1,P_{\\ge 4M}c_2]\\Vert _{B^{\\sigma }_{10,2}}\\\\&\\lesssim \\sum _{M\\le 1}\\int _0^t(1+t-s)^{-6/5}\\Vert P_{M}\\tilde{T}_{m/\\phi _\\varepsilon }[P_{\\ge 4M}e^{is\\langle \\nabla \\rangle }\\partial _sc_1,P_{\\ge 4M}e^{is\\langle \\nabla \\rangle }c_2]\\Vert _{L^{10/9}_x}ds\\\\&\\lesssim \\sum _{M\\le 1}M^{3/10}\\int _0^t(1+t-s)^{-6/5}\\left[\\sum _N\\Vert P_N\\partial _sc_1\\Vert _{L^{2}_x}\\Vert P_Nc_2\\Vert _{L^{2}_x}\\right]ds\\\\&\\lesssim (1+t)^{-6/5}(Q+\\Vert \\beta \\Vert _{X_T})^2.\\end{split}$ Now, to finish the proof, we appeal to the following result, which follows from a straightforward adaptation of the computations in [8].",
"There exists an energy $E_N:=\\sum _{\\vert \\gamma \\vert \\le N}\\int _{\\mathbb {R}^3}\\left\\lbrace \\vert \\partial ^\\gamma (\\rho -1)\\vert ^2+\\rho \\vert \\partial ^\\gamma v\\vert ^2+\\vert \\vert \\nabla \\vert ^{-1}\\partial ^\\alpha (\\rho -1)\\vert ^2\\right\\rbrace dx$ such that, if $\\sup _{0\\le t\\le T}\\Vert \\alpha (t)\\Vert _{H^5}\\lesssim 1$ then $E_N(t)\\simeq \\Vert \\alpha (t)\\Vert _{H^N}^2$ and for any $0\\le t\\le T$ , $E_N(t)\\lesssim E_N(0)+\\int _0^t\\Vert \\alpha (s)\\Vert _{Z^\\prime }E_N(s)ds,\\quad \\Vert f\\Vert _{Z^\\prime }\\lesssim \\sup _{M}(M^\\frac{3}{4}+M^\\frac{4}{3})\\Vert P_M\\alpha \\Vert _{L^\\infty }.$ Using that $\\begin{split}\\Vert P_M\\alpha \\Vert _{L^\\infty _x}&\\lesssim \\Vert P_Me^{it\\langle \\nabla \\rangle }\\chi ^Q\\Vert _{L^\\infty _x}+\\Vert P_M\\beta \\Vert _{L^\\infty _x}\\\\&\\lesssim \\min (M^2,1,M^{-1}\\langle M\\rangle ^{\\frac{5}{2}} t^{-\\frac{3}{2}})Q+\\min (M^{-N+3},1,M^\\frac{3}{10}\\langle M\\rangle ^{-\\sigma }t^{-\\frac{6}{5}})\\Vert \\beta \\Vert _X\\end{split}$ is integrable, we obtain an a priori bound on $E_N$ .",
"This ends the proof."
],
[
"Linear decay",
"The standard dispersive estimates for the linear Klein-Gordon equation follow from a straightforward application of the stationary phase estimates $\\left\\Vert e^{it\\langle \\nabla \\rangle } f \\right\\Vert _{B^0_{p,2}} \\lesssim t^{\\frac{3}{p}-\\frac{3}{2}}\\Vert f\\Vert _{B^{ 5 \\left( \\frac{1}{2} - \\frac{1}{p} \\right)}_{ p^{\\prime },2}}\\lesssim t^{\\frac{3}{p}-\\frac{3}{2}} \\Vert f\\Vert _{W^{5(\\frac{1}{2}-\\frac{1}{p}),p^\\prime }_x}\\;\\;\\;\\;\\;\\;\\mbox{if $2\\le p<\\infty $.",
"}$ We also use the simple product formula, valid for all $\\gamma \\in \\mathbb {N}$ and smooth functions $a$ , $b$ $\\Vert ab\\Vert _{H^\\gamma _x}\\lesssim _\\gamma \\Vert a\\Vert _{H^\\gamma _x}\\Vert b\\Vert _{W^{1,10}_x}+\\Vert a\\Vert _{W^{1,10}_x}\\Vert b\\Vert _{H^\\gamma _x}.$"
],
[
"Boundedness of pseudo-products",
"We need the following Lemma about boundedness of some pseudo-products as defined in (REF ).",
"The main feature here is that the extra loss of derivative coming from the fact that the phase $\\phi _\\varepsilon $ can be small only gives extra powers of the low frequency $L$ .",
"Lemma 5.1 Let $m$ be a symbol as in (REF ) and $\\varepsilon \\in \\lbrace (\\pm ,\\pm )\\rbrace $ .",
"Then for any $1\\le r\\le \\infty $ and $1<p,q\\le +\\infty $ with $\\frac{1}{r}=\\frac{1}{p}+\\frac{1}{q},$ we have Hölder's inequality $\\Vert P_{M}\\tilde{T}_{m/\\phi _\\varepsilon }[P_Na,P_Ob]\\Vert _{L^r_x}\\lesssim H(1+L)^5\\Vert a\\Vert _{L^p_x}\\Vert b\\Vert _{L^q_x},$ where $H=\\max (M,N,O)$ and $L=\\min (M,N,O)$ .",
"In particular, we find that for $\\kappa >0$ , $\\Vert T_{m/\\phi _\\varepsilon }[a,b]\\Vert _{W^{\\sigma ,r}_x}\\lesssim _\\kappa \\left[\\Vert a\\Vert _{W^{\\sigma +3+\\kappa ,p}_x}+\\Vert \\vert \\nabla \\vert ^{-\\kappa }a\\Vert _{L^p_x}\\right]\\left[\\Vert b\\Vert _{W^{\\sigma +3+\\kappa ,q}_x}+\\Vert \\vert \\nabla \\vert ^{-\\kappa }b\\Vert _{L^q_x}\\right].$ Furthermore, for $\\sigma \\ge 0$ , $2 \\le p \\le \\infty $ , $2<q<\\infty $ , $1<r<\\infty $ , we get $\\Vert P_{\\ge M/8}T_{m/\\phi _\\varepsilon }[P_Ma,P_{\\ge M/8}b]\\Vert _{B^{\\sigma }_{r,2}}\\lesssim \\Vert a\\Vert _{W^{\\gamma -\\theta ,p}_x}\\Vert \\vert \\nabla \\vert b\\Vert _{B^{\\sigma +\\theta }_{q,2}}.$ where $0\\le \\theta \\le \\gamma $ , $\\gamma =5$ if $M\\ge 1$ and $\\gamma =0$ if $M\\le 1$ .",
"Multiplying by a test function, we define $\\begin{split}I_{M,N,O}=\\langle P_Mc,T_{m/\\phi _\\varepsilon }[P_Na,P_Ob]\\rangle &=\\iiint _{\\mathbb {R}^3}K_{M,N,O}(y_1,y_2,y_3)\\overline{a}(y_2)\\overline{b}(y_3)c(y_1)dy_1dy_2dy_3\\\\K_{M,N,O}(y_1,y_2,y_3)&=\\iint _{\\mathbb {R}^3}e^{-i\\xi \\cdot \\left[y_1-y_2\\right]}e^{-i\\eta \\cdot \\left[y_2-y_3\\right]}\\varphi (\\frac{\\xi }{M})\\varphi (\\frac{\\xi -\\eta }{N})\\varphi (\\frac{\\eta }{O})\\frac{m(\\xi -\\eta ,\\eta )}{\\phi _\\varepsilon (\\xi -\\eta ,\\eta )}d\\xi d\\eta \\\\&=\\iint _{\\mathbb {R}^3}e^{-i\\xi \\cdot \\left[y_1-y_2\\right]}e^{-i\\eta \\cdot \\left[y_2-y_3\\right]}\\frac{\\mathfrak {m}_{M,N,O}(\\xi -\\eta ,\\eta )}{\\phi _\\varepsilon (\\xi -\\eta ,\\eta )}d\\xi d\\eta \\\\\\end{split}$ Changing variables $(\\xi ,\\eta )\\rightarrow (\\xi ,\\xi -\\eta )$ , we may assume that $\\min (M,O)/2\\le N\\le 2\\max (M,O)$ .",
"We compute that $\\begin{split}\\left|I_{M,N,O}\\right|&\\lesssim \\left|\\sum _{A,B}\\iiint _{\\vert y_1-y_2\\vert \\sim A,\\,\\vert y_2-y_3\\vert \\sim B}K_{M,N,O}(y_1,y_2,y_3)\\overline{a}(y_2)\\overline{b}(y_3)c(y_1)dy_1dy_2dy_3\\right|\\\\&\\lesssim \\sum _{A,B}\\iint _{\\vert y_1-y_2\\vert \\sim A}\\vert a(y_2)\\vert \\vert c(y_1)\\vert \\mathcal {M}b(y_2)\\left[\\sup _{\\vert y_2-y_3\\vert \\sim B}B^3\\vert K_{M,N,O}(y_1,y_2,y_3)\\vert \\right]dy_1dy_2\\\\&\\lesssim \\left[\\sum _{A,B}(AB)^3\\sup _{\\vert y_1-y_2\\vert \\sim A,\\,\\vert y_2-y_3\\vert \\sim B}\\vert K_{M,N,O}(y_1,y_2,y_3)\\vert \\right]\\int _{\\mathbb {R}^3}\\vert c(y_1)\\vert \\cdot \\mathcal {M}\\left[a\\cdot \\mathcal {M}b\\right](y_1)dy_1,\\end{split}$ where $\\mathcal {M}$ denotes the Maximal functionIf $r=1$ , we have to put the maximal function on $c$ instead of on $a\\cdot \\mathcal {M}b$ at the last line above, but this makes absolutely no change..",
"Using Hölder's inequality and the boundedness of the Maximal function, in order to prove (REF ), it suffices to show that $\\begin{split}\\sum _{A,B}(AB)^3H^{-1}(1+L)^{-5}c_{A,B}&\\lesssim 1,\\quad c_{A,B}=\\sup _{\\vert y_1-y_2\\vert \\sim A,\\,\\vert y_2-y_3\\vert \\sim B}\\vert K_{M,N,O}(y_1,y_2,y_3)\\vert .\\end{split}$ We observe that $\\left|\\partial ^\\alpha _\\xi \\partial ^\\beta _\\eta \\frac{\\mathfrak {m}_{M,N,O}(\\xi -\\eta ,\\eta )}{\\phi _\\varepsilon (\\xi -\\eta ,\\eta )}\\right|\\lesssim {\\left\\lbrace \\begin{array}{ll}H\\vert \\xi \\vert ^{-\\vert \\alpha \\vert }\\vert \\eta \\vert ^{-\\vert \\beta \\vert }&\\hbox{ if }L\\le 1\\\\HL^{-1}\\left[\\theta +L^{-1}\\right]^{-(\\vert \\alpha \\vert +\\vert \\beta \\vert +2)}\\vert \\xi \\vert ^{-\\vert \\alpha \\vert }\\vert \\eta \\vert ^{-\\vert \\beta \\vert }&\\hbox{ if }L\\ge 1\\end{array}\\right.",
"}$ where $\\theta =\\vert \\angle (\\xi ,\\eta )\\vert $ .",
"This follows from the fact that the left-hand side above can be written as a linear combinations of terms like $\\frac{\\partial _\\xi ^{\\gamma _1}\\partial _\\eta ^{\\delta _1}\\mathfrak {m}_{M,N,O}}{\\phi _\\varepsilon }\\frac{\\partial _\\xi ^{\\gamma _2}\\partial _\\eta ^{\\delta _2}\\phi _\\varepsilon }{\\phi _\\varepsilon }\\dots \\frac{\\partial _\\xi ^{\\gamma _k}\\partial _\\eta ^{\\delta _k}\\phi _\\varepsilon }{\\phi _\\varepsilon },\\qquad \\gamma _i,\\delta _i\\ge 0,\\,\\,\\gamma _1+\\dots +\\gamma _k=\\alpha ,\\,\\,\\delta _1+\\dots +\\delta _k=\\beta $ which is easily seen by induction on $\\vert \\alpha \\vert +\\vert \\beta \\vert $ and from the bounds $\\begin{split}&\\vert \\phi _\\varepsilon (\\xi -\\eta ,\\eta )\\vert \\gtrsim \\langle L\\rangle \\left[\\theta ^2+\\langle L\\rangle ^{-2}\\right],\\qquad \\vert \\partial _\\xi ^\\alpha \\partial ^\\beta _\\eta \\mathfrak {m}_{M,N,O}\\vert \\lesssim H\\vert \\xi \\vert ^{-\\vert \\alpha \\vert }\\vert \\eta \\vert ^{-\\vert \\beta \\vert }\\\\&\\left|\\frac{\\partial ^\\alpha _\\xi \\partial ^\\beta _\\eta \\phi _\\varepsilon }{\\phi _\\varepsilon }\\right|\\lesssim {\\left\\lbrace \\begin{array}{ll}\\vert \\xi \\vert ^{-\\vert \\alpha \\vert }\\vert \\eta \\vert ^{-\\vert \\beta \\vert }&\\text{if }\\,L\\le 1\\\\\\vert \\xi \\vert ^{-\\vert \\alpha \\vert }\\vert \\eta \\vert ^{-\\vert \\beta \\vert }\\left[\\theta +\\vert L\\vert ^{-1}\\right]^{-\\vert \\alpha \\vert -\\vert \\beta \\vert }&\\text{if }\\,L\\ge 1.\\end{array}\\right.",
"}\\end{split}$ These bounds follow from elementary but lengthy computations which we omitRemark however that one can easily obtain worst bounds which would allow for a similar Hölder estimate on the pseudo-product but with a worst loss in derivative (but still loosing less than 20 derivatives).",
"This would be sufficient to obtain the main result assuming more derivatives on the initial data.. From (REF ), we compute that $\\begin{split}A^aB^b\\vert c_{AB}\\vert &\\lesssim \\sup _{\\vert \\alpha \\vert =a;\\,\\vert \\beta \\vert =b}\\left|\\iint _{\\mathbb {R}^3}e^{-i\\xi \\cdot \\left[y_1-y_2\\right]}e^{-i\\eta \\cdot \\left[y_2-y_3\\right]}\\partial ^\\alpha _\\xi \\partial ^\\beta _\\eta \\frac{\\mathfrak {m}_{M,N,O}(\\xi -\\eta ,\\eta )}{\\phi _\\varepsilon (\\xi ,\\eta )} d\\xi d\\eta \\right|\\\\&\\lesssim {\\left\\lbrace \\begin{array}{ll}HM^{3-a}O^{3-b}&\\hbox{ if }L\\le 1,\\\\HL^{a+b-1}M^{3-a}O^{3-b}&\\hbox{ if }L\\ge 1,\\end{array}\\right.",
"}\\end{split}$ from which (REF ) follows easily, choosing $a,b=2$ or 4, $\\begin{split}\\sum _{A,B}(AB)^3H^{-1}\\langle L\\rangle ^{-5}c_{A,B}&\\lesssim \\sum _{A,B}(AM/\\langle L\\rangle )^{3-a}(BO/\\langle L\\rangle )^{3-b}\\lesssim 1.\\\\\\end{split}$ This finishes the proof of (REF ).",
"Estimate (REF ) then follows directly by summing (REF ) when $M\\sim O\\ge N$ ."
]
] | 1204.1536 |
[
[
"TeV Gamma-ray Astronomy: A Summary"
],
[
"Abstract The field of TeV gamma-ray astronomy has produced many exciting results over the last decade.",
"Both the source catalogue, and the range of astrophysical questions which can be addressed, continue to expand.",
"This article presents a topical review of the field, with a focus on the observational results of the imaging atmospheric Cherenkov telescope arrays.",
"The results encompass pulsars and their nebulae, supernova remnants, gamma-ray binary systems, star forming regions and starburst and active galaxies."
],
[
"Introduction",
"Teraelectronvolt (TeV) astronomy concerns the study of astrophysical sources of gamma-ray photons, with energies in the range between $\\sim 30{GeV}$ and $\\sim 30{TeV}$ .",
"The TeV range is one of the most recent windows of the electromagnetic spectrum to be opened for study, beginning with the identification of the first source, the Crab Nebula, in 1989 [1].",
"The results have been impressive, with significant advances in instrumentation leading to the detection of well over 100 sources over the last decade.",
"The goals of TeV astronomy are wide-ranging, but can broadly be described as the study of sites of relativistic particle acceleration in the Universe, both hadronic and leptonic.",
"This encompasses a huge range of size scales and energetics, from the interactions of galaxy clusters, to the magnetospheres of individual pulsars.",
"Numerous recent reviews of TeV gamma-ray astronomy have been written (e.g.",
"[2], [3], [4]), but the field is rapidly evolving.",
"For example, since the extensive review by Hinton & Hofmann in 2009 [2], the number of sources has grown from around 80 to more than 120, and a number of new source classes have been identified.",
"This paper therefore aims to provide an update, and to supplement the existing reviews with a summary of the current observational status.",
"The primary focus of the review is on the results from imaging atmospheric Cherenkov telescopes, with a relatively brief discussion of the air shower particle detector experiments.",
"We use the definitions outlined by Aharonian [5] as follows: “high energy”, or “GeV”, astronomy refers to the energy range from $30{MeV}$ to $30{GeV}$ , while “very high energy”, or “TeV”, astronomy, refers to the range from $30{GeV}$ to $30{TeV}$ ."
],
[
"A Brief History",
"The development of ground-based gamma-ray astronomy is closely linked to the study of cosmic rays and cosmic ray air showers.",
"The idea of searching for astrophysical gamma-ray sources at $\\sim 100{MeV}$ energies was first proposed by Morrison in 1958 [6].",
"A prediction of a very high TeV gamma-ray flux from various sources, including the Crab, which might be detectable with an air shower particle array at high altitude, was made by Cocconi in 1959 [7].",
"Cherenkov radiation associated with large cosmic ray air showers was first detected by Galbraith & Jelley in 1953 [8], and the possibility of using this phenomenon to study gamma-ray initiated showers led to the development of a number of dedicated facilities in the 1960s.",
"This effort was boosted by the apparent detection of a gamma-ray signal from the black-hole binary Cygnus X-3 by both particle air shower arrays and atmospheric Cherenkov detectors.",
"With hindsight, this detection was likely spurious, as were numerous unsubstantiated claims throughout the 1970s, 1980s and into the 1990s.",
"The field reached a firm experimental footing with the development of the imaging technique, which provides a method of effectively discriminating between gamma-ray initiated showers and the background of cosmic ray showers, based on the morphology of their Cherenkov images, and guided by the results of Monte Carlo simulations [9], [10].",
"This technique was applied by the Whipple collaboration to detect steady gamma-ray emission from the Crab Nebula using a 10m reflector with a 37-element photomultiplier tube camera in 1989 [1].",
"A number of imaging atmospheric Cherenkov telescopes (IACTs) were subsequently developed around the world (including Durham, CANGAROO, Telescope Array, Crimean Astrophysical Observatory, SHALON, TACTIC), with the northern hemisphere instruments (Whipple, HEGRA and CAT) leading the field.",
"The 1990s saw two particularly important developments: the detection of the first extragalactic sources by the Whipple Collaboration, starting with the nearby blazars Markarian 421 [11] and Markarian 501 [12], and the application of the stereo imaging technique by the HEGRA array [13].",
"HEGRA consisted of 5 telescopes of modest aperture ($<10{m}{2}$ ), and demonstrated that the combination of Cherenkov image information from multiple telescopes located within the same Cherenkov light pool could dramatically improve the sensitivity of the technique.",
"Despite this progress, the relative scarcity of bright TeV gamma-ray sources ($<10$ were identified by 2000) highlighted the necessity for improved instrumentation.",
"Cherenkov wavefront samplers such as CELESTE and STACEE attempted to probe to lower energies, and hence higher gamma-ray fluxes and larger distances, using converted solar farms; however, the difficulty of discriminating gamma-rays from the cosmic ray background using this technique limited its effectiveness.",
"Successful gamma-ray observations using a particle detector were made by the Milagro experiment, which ran from 2000 to 2008, providing a survey of the northern sky with modest sensitivity.",
"Starting with the commissioning of H.E.S.S.",
"in 2003, the new generation of IACTs - H.E.S.S., MAGIC and VERITAS - have provided the required order of magnitude improvement in sensitivity, and firmly established gamma-ray studies as an important astronomical discipline."
],
[
"Imaging Atmospheric Cherenkov Telescopes",
"The Cherenkov emission from an air shower forms a column of blue light in the sky, with the maximum emission occuring around $10{km}$ above sea level at TeV energies.",
"Cherenkov telescopes used to record these images are essentially rather simple devices, consisting of a large, segmented optical flux collector used to focus the Cherenkov light onto an array of fast photo-detectors.",
"The optical specifications are not terribly strict; an optical point spread function width of typically $\\sim 0.05^{\\circ }$ is adequate.",
"The photo-detector array (usually $<1000$ photomultiplier tubes) comprises a crude camera, covering a few degrees on the sky.",
"A Cherenkov flash triggers read-out of the photo-detectors, with a read-out window defined by the timescale of the arrival of the Cherenkov photons ($\\sim 10{ns}$ ).",
"There are currently three major imaging atmospheric Cherenkov telescope systems in operation.",
"H.E.S.S., located in the Khomas Highland of Namibia ($-23^{\\circ }$ N, $-16^{\\circ }$ W, altitude $1800{m}$ ), consists of four telescopes arranged on a square with 120m side length.",
"Each telescope has a mirror area of $107{m}{2}$ and is equipped with a 960 pixel camera covering a $5^{\\circ }$ field of view.",
"VERITAS, at the Fred Lawrence Whipple Observatory in southern Arizona ($32^{\\circ }$ N, $111^{\\circ }$ W, altitude $1275{m}$ ) has similar characteristics, with 4 telescopes of $107{m}{2}$ area and 499-pixel cameras, covering $3.5^\\circ $ .",
"MAGIC ($28^{\\circ }$ N, $17^{\\circ }$ W, altitude $2225{m}$ ) originally consisted of a single, very large reflector ($236{m}{2}$ ) on the Canary island of La Palma, with a $3.5^{\\circ }$ camera.",
"In 2009, a second telescope with the same mirror area was installed at a distance of $85{m}$ from the first.",
"Each of these facilities work in a similar fashion.",
"Cherenkov images of air showers are recorded at a rate of a few hundred Hz, and analyzed offline.",
"The overwhelming majority of these images are due to cosmic ray initiated air showers.",
"Gamma-ray showers can be discriminated from this background based on the image shape and orientation.",
"Gamma-ray images result from purely electromagnetic cascades and appear as narrow, elongated ellipses in the camera plane.",
"The long axis of the ellipse corresponds to the vertical extension of the air shower, and points back towards the source position in the field of view.",
"If multiple telescopes are used to view the same shower, the source position is simply the intersection point of the various image axes (illustrated schematically in Figure REF ).",
"Cosmic-ray primaries produce secondaries with large transverse momenta, which initiate sub-showers.",
"The Cherenkov images of cosmic-ray initiated air showers are consequently wider then those with $\\gamma $ -ray primaries, and form an irregular shape, as opposed to a smooth ellipse.",
"In addition, since the original charged particle has been deflected by galactic magnetic fields before reaching the Earth, cosmic-ray images have no preferred orientation in the camera.",
"Figure: A schematic illustration of an atmospheric Cherenkovtelescope array.",
"The primary particle initiates an air shower,resulting in a cone of Cherenkov radiation.",
"Telescopes within theCherenkov light pool record elliptical images; the intersection of thelong axes of these images indicates the arrival direction of theprimary, and hence the location of a γ\\gamma -ray source in the sky.Cherenkov light reaching the ground from air showers peaks at optical/UV wavelengths, and so IACTs operate only under clear, dark skies.",
"Both MAGIC and VERITAS have demonstrated that useful observations can be made when the moon is visible above the horizon, but the typical duty cycle of these instruments is still limited to $\\sim 1200{hours}$ per year ($<15\\%$ ).",
"Given the small field of view of IACTs, regions of the sky containing one or more source candidates are usually targeted for observations.",
"Surveys can only be accomplished slowly, by tiling regions of the sky with overlapping fields-of-view.",
"The sensitivity of the current generation of IACTs is sufficient to detect the Crab Nebula in under a minute, and a source with 1% of the Crab flux ($\\sim 2\\times 10^{-13}{m}{-2}{s}{-1}$ above $1{TeV}$ ) in $\\sim 25{hours}$ .",
"The angular and energy resolution of the technique are energy-dependent, with typical values of $<0.1^{\\circ }$ and $<15\\%$ per photon, respectively, at $1{TeV}$ .",
"The catalog of TeV sources grew rapidly with the commissioning of H.E.S.S.",
"in the southern hemisphere, which provided the first high sensitivity observations of the densely populated inner Galaxy.",
"It has continued to expand in recent years as MAGIC and VERITAS have come online, and now numbers 130 sources, as listed in the online catalog TeVCat [14] We note that a small fraction of these 130 sources may be duplicate detections of the same object, while other detections remain contentious or unconfirmed.",
"Full details are available in the individual source annotations in TeVCat, available at http://tevcat.uchicago.edu.",
"Figure REF shows the locations of these sources in Galactic coordinates.",
"Figure: A map of the catalog of localized sources of TeV gamma-rayemission in Galactic coordinates as of November 2011, provided bythe online catalog TeVCat"
],
[
"Particle Detectors",
"The direct detection of air shower particles using arrays of particle detectors at ground level offers some important advantages over the atmospheric Cherenkov technique.",
"In particular, observations can be made continuously, both day and night, and over the entire viewable sky (a field-of-view of $>1{steradian}$ ).",
"These advantages are offset, however, by the rather low sensitivity to point sources, which is primarily due to the difficulty of rejecting the substantial background of cosmic ray initiated air showers.",
"The angular and energy resolution of these detectors are also significantly worse than IACTs.",
"Early claims for emission from binary systems using sparse particle detector arrays were not confirmed by later, more sensitive instruments (e.g.",
"[15], [16]), indicating the need for a new approach to this problem.",
"Milagro, which operated between 2000 and 2008 in northern New Mexico ($36^{\\circ }$ N, $107^{\\circ }$ W), was the first successful attempt at this.",
"The Milagro detector consisted of a large water reservoir ($60\\times 80{m}$ ) at an altitude of $2630{m}$ , covered with a light-tight barrier, and instrumented with PMTs.",
"The central reservoir provided high-resolution sampling of air shower particles over a relatively small area (compared to the air shower footprint).",
"In 2004 an array of 175 small outrigger tanks were added, irregularly spread over an area of $200\\times 200{m}$ around the central reservoir.",
"This configuration, coupled with the development of analysis techniques for cosmic ray background discrimination, provided sufficient sensitivity for the first comprehensive survey of the northern TeV sky.",
"The results showed strong detections of the bright, known TeV sources Markarian 421 and the Crab Nebula, along with the detection of three extended sources in the Galactic plane, each with integrated fluxes comparable to the Crab Nebula at $20{TeV}$ .",
"A few less significant source candidates were also identified in the plane, and a reanalysis following the launch of Fermi-LAT also showed fourteen $3\\sigma $ excesses co-located with bright Galactic LAT sources [17].",
"The Milagro results for the region around the Galactic plane are shown in Figure REF .",
"Figure: The Milagro survey of the Galactic plane.",
"Thez-axis is the pre-trials statistical significance, with afixed maximum of 7σ7\\sigma .",
"Figure from Abdo etal.",
".Milagro ceased operation in 2008; however, two large particle detector arrays remain in operation at very high altitude in Tibet.",
"ARGO-YBJ is located at $4300{m}$ in Yangbajing, and consists of a single layer of resistive plate chambers completely covering an area of $110\\times 100{m}$ .",
"The results of 1265 days of observations were recently presented [19], showing $>5\\sigma $ detections of the Crab Nebula, Markarian 421 and two Milagro sources.",
"The fact that one of the brightest Milagro sources, MGRO J2019+37, is not detected in these observations presents something of a mystery.",
"The Tibet AS$\\gamma $ air shower array, also at Yangbajing, consists of $\\sim 750$ closely-spaced scintillation detectors covering an area of $36900{m}{2}$ , and has demonstrated that this technique is also practical for the detection of bright TeV sources [20]."
],
[
"Blazars",
"Approximately 1% of all galaxies host an active nucleus; a central compact region with much higher than normal luminosity.",
"Around 10% of these Active Galactic Nuclei (AGN) exhibit relativistic jets, powered by accretion onto a supermassive black hole.",
"Many of the observational characteristics of AGN can be attributed to the geometry of the system; in particular, the orientation of the jets with respect to the observer.",
"Blazars, which host a jet oriented at an acute angle to the line of sight, are of particular interest for gamma-ray astronomy, as the emission from these objects is dominated by relativistic beaming effects, which dramatically boost the observed photon energies and luminosity.",
"Figure: The location of the 49 AGN (BL Lac, FSRQs and radiogalaxies) with detected TeV emission (from TeVCat).",
"On the left is the spatial distribution,in Galactic coordinates; The right plot shows the redshiftdistribution, for 41 of the objects with redshifts listed in theliterature.The first extragalactic source discovered at TeV energies was Markarian 421 [11], a blazar of the BL Lacertae sub-class.",
"The extragalactic TeV catalog now comprises $\\sim 50$ objects, and continues to increase steadily (Figure REF ).",
"Blazar SEDs show a double-peaked structure in a $\\nu F_{\\nu }$ representation of their spectral energy distribution (SED), with the lower frequency peak usually attributed to synchrotron emission of energetic electrons, and the higher frequency peak to inverse Compton.",
"BL Lac objects are further classified as low-, intermediate- or high-frequency peaked, according to the location of the peak of their synchrotron emission.",
"The majority ($\\sim 80\\%$ ) of the known TeV blazars are high-frequency peaked objects, in part because of inherent biases in the target selection: initially, objects were chosen based primarily upon their radio and X-ray spectral properties (e.g.",
"[21]).",
"More recently, the TeV observatories have expanded their selection criteria, using additional guidance in the form of Fermi-LAT results, and multi-wavelength observation triggers.",
"This has broadened the catalog to include examples of intermediate- and low-frequency peaked objects.",
"The overall data quality has also improved markedly since the launch of Fermi: Figure REF shows a recent compilation of spectral measurements for the bright TeV blazar, Markarian 501 [22], taken during an extensive multiwavelength campaign in 2009.",
"The mechanisms which drive the high energy emission from blazars remain poorly understood, and a full discussion is beyond the scope of this review.",
"Briefly; in leptonic scenarios, a population of electrons is accelerated to TeV energies, typically through Fermi acceleration by shocks in the AGN jet.",
"These electrons then cool by radiating X-ray synchrotron photons.",
"TeV emission results from inverse Compton interactions of the electrons with either their self-generated synchrotron photons, or an external photon field.",
"The strong correlation between X-ray and TeV emission which is often observed provides evidence for a common origin such as this, although counter-examples do exist [23].",
"Another class of models has hadrons as the primary particle population, which can then produce TeV gamma-rays through subsequent interactions with target matter or photon fields.",
"Hadronic models are less favoured, typically, in part because the cooling times for the relevant processes are long, making rapid variability difficult to explain.",
"One exception to this is the case of proton synchrotron emission, which may provide a plausible alternative, in which the emission results from extremely high energy protons in highly magnetized ($B\\sim 100{G}$ ), compact regions of the jet [24].",
"Figure: Extensive multifrequency measurements showing the spectralenergy distribution of Markarian 501 for observations in2009.",
"Emission from the host galaxy is clearly visible atinfrared/optical frequencies.",
"The VERITAS data are divided to showboth the average spectrum (red circles), and the spectrum during a3-day flare (green triangles).",
"Figure from Abdo etal.",
"Many of the AGN detected at TeV energies exhibit extreme variability.",
"The timescales can range from years to minutes, and the observed flux can change by more than an order of magnitude.",
"Figure REF shows the H.E.S.S.",
"lightcurve from July 2006 for one of the most extreme examples, the BL Lac object PKS 2155-304 [18], [25].",
"Such rapid variability can be used to place constraints on the size of the emission region, which depend upon the Doppler factor, $\\delta $ .",
"$\\delta $ itself is constrained by the requirement that the emission region should be transparent to gamma-rays (e.g.",
"[26]).",
"Extremely rapid TeV gamma-ray variability of distant blazars can also be used to place limits on the energy-dependent violation of Lorentz invariance [27], [28], as predicted in some models of quantum gravity.",
"Figure: Integrated flux (>200GeV>200{GeV}) versus time forH.E.S.S.",
"observations of PKS 2155-304 on MJD 53944 (28 July,2006).",
"The data are binned in 1-minute intervals, and the horizontaldashed line shows the steady flux from the Crab Nebula forcomparison.",
"Figure from .Related to the BL Lacertae objects are Flat Spectrum Radio Quasars (FSRQs).",
"These are characterized primarily by their intense UV emission, associated with an accretion disk, strong broad emission lines in the optical spectrum, and infra-red emission associated with a dusty torus.",
"FSRQs are similar to low-frequency peaked BL Lacs, in that the X-ray emission is dominated by the inverse Compton peak of the SED.",
"Despite this, inverse Compton emission can extend up to TeV energies, particularly during intense flaring episodes, and three FSRQs have recently been detected by IACTs (PKS 1222+21 [30] , PKS 1510-089 [31] and 3C279 [32]).",
"A unique additional case is the TeV detection of IC 310, a “head-tail” radio galaxy in the Perseus cluster, possibly hosting a low-luminosity BL Lac nucleus [33].",
"Head-tail radio galaxies display a distinctive radio morphology, consisting of a bright “head” and a fainter “tail”, which is believed to be the result of their rapid motion with respect to the intracluster medium.",
"The source was originally identified as a VHE emitter in an analysis of the highest energy ($>30{GeV}$ ) Fermi photons by Neronov et al.",
"[34], and then subsequently detected from the ground by MAGIC [35].",
"Neronov et al.",
"considered the intriguing possibility that the emission might originate with particles accelerated at the bow shock formed by the interaction between the relativistic outflow from the galaxy and the intracluster medium.",
"This scenario is ruled out by the detection of variability in the TeV flux by MAGIC, and the more familiar BL Lac mechanisms are now favoured.",
"Figure REF also shows the distribution of measured redshifts for 41 TeV AGN.",
"Some of these measurements are rather uncertain, since BL Lac optical spectra, by definition, do not contain strong emission lines.",
"The most distant object detected is 3C279 [32], with a relatively modest redshift of $z=0.5362$ .",
"The population is truncated at large distances due to the absorption of TeV gamma-rays by electron-positron pair production with the low energy photons of the extragalactic background light (EBL).",
"This effect is energy dependent, and can strongly modify the observed VHE spectra of extragalactic sources.",
"While this limits the observation of more distant TeV sources, it also provides a mechanism by which to infer the intensity of the EBL, using reasonable assumptions about the intrinsic TeV spectra at the source [36].",
"The EBL provides a calorimetric measure of the complete history of star and galaxy formation in the Universe, but is difficult to measure directly, due to the presence of bright local foreground sources of emission, such as zodiacal light.",
"Presently, all of the TeV blazar measurements are consistent with a relatively low level of EBL, with the constraints derived from VHE measurements now approaching the lower limits derived from galaxy counts [37], [38], [39].",
"TeV blazar observations have also been suggested as probes of other physical phenomena, such as the acceleration and propagation of ultra-high energy cosmic rays [40], [41], or, more speculatively, the production of axion-like particles [42], [43].",
"Various authors have also discussed the possibility that TeV observations may be used to measure or constrain the strength of the intergalactic magnetic field (IGMF) (e.g.",
"[40], [44]).",
"Temporal, spectral and spatial signatures of the IGMF are all possible; however, the fact that blazars are intrinsically variable gamma-ray sources limits the power of this technique.",
"Accounting for this, Dermer et al.",
"derive a lower limit of $B_{IGMF}\\ge 10^{-18}{G}$ [45]."
],
[
"Radio Galaxies",
"As described above, the TeV fluxes from blazars are dramatically enhanced by the effects of Doppler boosting.",
"Nearby radio galaxies, in which the jet is not directly oriented towards the line-of-sight, provide an alternative method by which to investigate the particle acceleration and gamma-ray emission from relativistic outflows in AGN.",
"The advantage of studying such objects lies in the fact that the jets can be resolved from radio to X-ray wavelengths, allowing the possibility of correlating the gamma-ray emission state with observed changes in the jet structure.",
"Three radio galaxies have been identified as TeV emitters: M 87, Centaurus A and NGC 1275.",
"M 87 is the most well studied of these, and was first reported as a gamma-ray source by the HEGRA collaboration [46], with subsequent confirmation by H.E.S.S.",
"[47], VERITAS [48] and MAGIC [49].",
"M 87 is a giant radio galaxy at a distance of $16.7 \\pm 0.2 {Mpc}$ , displaying a prominent misaligned jet, with an orientation angle of $\\le 20^{\\circ }$ to the line-of-sight.",
"The mass of the central black hole is estimated to be $\\sim 3\\times 10^9$ M$_\\odot $ .",
"The TeV source is strongly variable, and has undergone three episodes of enhanced emission in 2005, 2008 and 2010 (Figure REF ).",
"The results are summarized by Abramoswski et al [50].",
"Causality arguments use the shortest variability timescale of around $1{day}$ to place strong constraints on the size of the TeV emission region, corresponding to only a few Schwarzschild radii.",
"The TeV emission region cannot be directly resolved with IACTs, but correlations with contemporaneous X-ray and radio observations provide some clues to its location.",
"Two structures are of particular interest: the inner region close to the central black hole (the “core”), and HST-1, a bright jet feature first resolved in the optical band by the Hubble Space Telescope.",
"HST-1 underwent a multi-year flare in radio, optical and X-rays, peaking around the time when the first short-term variability was detected at VHE energies [47].",
"In contrast to this, the 2008 VHE flare was accompanied by enhanced radio and X-ray fluxes from the core region [51].",
"The 2010 VHE flare showed no enhanced radio emission from the core, although an enhanced X-ray flux was observed 3 days after the VHE peak.",
"The results, therefore, remain somewhat ambiguous, and the possibility remains that the observed VHE flares may have different origins, or that the total VHE emission may be the sum of multiple components.",
"Given the burdensome observing requirements for instruments with a limited duty cycle, M 87 represents the best example of the importance of data-sharing and coordinated observing planning between the various IACTs.",
"Figure: VHE light curves of the three flares from M87 observed in2005, 2008 and 2010, showing integral fluxes above an energy of350GeV350{GeV}.",
"Figure from .The closest active galaxy, Centaurus A, at a distance of $3.8{Mpc}$ , was identified as a VHE source in a deep, 120 hour exposure by H.E.S.S.",
"[52].",
"Cen A is among the faintest VHE sources detected, with a flux of 0.8% of the Crab Nebula above $250{GeV}$ .",
"The emission is steady, although the variability is not strongly constrained as a result of the low flux level.",
"As with M 87, numerous sites for the production of the TeV emission have been suggested, from the immediate vicinity of the central black hole (with a mass of $\\sim 5\\times 10^7$ M$_\\odot $ ), to the AGN jet, or even beyond [53].",
"The final, and most recent, addition to the known TeV radio galaxies is NGC 1275, identified by MAGIC [54].",
"NGC 1275 is the central galaxy of the Perseus cluster, at a distance of $72.2{Mpc}$ .",
"The VHE emission has a flux of $\\sim 3\\%$ of the Crab Nebula above $100{GeV}$ and exhibits a very soft spectrum, with a power-law index of $3.96\\pm 0.37$ [55], indicating a sharp turnover from the measured Fermi spectrum at lower energies, where the index is $2.09\\pm 0.02$ [56]."
],
[
"Starburst Galaxies",
"Starburst galaxies are those which exhibit an extremely high rate of star formation, sometimes triggered by interaction with another galaxy.",
"High cosmic-ray and gas densities in the starburst region make these objects promising targets for gamma-ray observations, with emission predicted to result from the interactions of hadronic cosmic rays in the dense gas.",
"TeV emission has now been identified from two starburst galaxies: M 82 [57] and NGC 253 [58].",
"M 82 is a bright galaxy located at a distance of approximately $3.9{Mpc}$ , with an active starburst region at its centre.",
"The star formation rate in this region is approximately 10 times that of the Milky Way, with an estimated supernova rate of 0.1 to 0.3 per year.",
"A deep VERITAS exposure ($137{hours}$ ) in 2008-2009 resulted in a detection of gamma-ray emission from M 82 with a flux of $(3.7 \\pm 0.8_{stat} \\pm 0.7_{syst}) \\times 10^{-13} {cm}{-2} {s}{-1}$ above the 700GeV energy threshold of the analysis.",
"NGC 253 lies at a distance of $2.9-3.6{Mpc}$ , and also has a central, compact ($\\sim 100{pc}$ ) starburst region.",
"The supernova rate in this region is estimated at $\\sim 0.03$ per year.",
"TeV emission was detected by H.E.S.S.",
"with an integrated flux above $220{GeV}$ of $(5.5 \\pm 1.0_{stat} \\pm 2.8_{syst}) \\times 10^{-13} {cm}{-2} {s}{-1}$ [58].",
"The emission from both M 82 and NGC 253 is consistent with the predictions of models based on the acceleration and propagation of cosmic rays in the starburst core (e.g.",
"[59]), assuming that they act as efficient “proton calorimeters” (i.e.",
"cosmic rays lose the majority of their energy to collisions).",
"In this case, the estimated cosmic ray density in the starburst region is 2-3 orders of magnitude larger than that of the Milky Way.",
"An alternative explanation is proposed by Mannheim et al.",
"[60], who suggest that the TeV luminosity is consistent with the combined emission from a large population of pulsar wind nebulae, which result from the elevated supernova rate.",
"More accurate TeV spectra, and observation of other starburst classes, such as ultra-luminous infrared galaxies, should provide more insight in the future."
],
[
"The Large Magellenic Cloud",
"Galaxies of the Local Group are also of interest to TeV observatories, although the predicted fluxes due to cosmic ray acceleration and propagation lie below the current instrumental sensitivity in the TeV range.",
"Given their proximity, the possibility arises of detectable emission from individual objects, or from localized regions, particularly in the Milky Way's satellite galaxies.",
"H.E.S.S.",
"has recently identified the first such object in the Large Magellenic Cloud (LMC), at a distance of $48{kpc}$ [61].",
"An unresolved source with a flux of $\\sim 2\\%$ of the Crab Nebula ($1.5\\times 10^{-12} {erg} {cm}{-2} {s}{-1}$ between 1 and 10 TeV) was detected, consistent with the location of PSR J0357-6910.",
"This object is the most powerful pulsar known, with a spin-down energy of Ė$=4.8\\times 10^{38}{erg}{s}{-1}$ .",
"Given the positional coincidence, and based on comparisons with similar objects within our Galaxy, it seems likely that the TeV emission is due to inverse Compton emission from electrons in the pulsar's wind nebula interacting with a strong infrared target photon field.",
"This is in contrast to the extended GeV emission which has been observed from the LMC by Fermi-LAT, which is attributed to cosmic ray acceleration and interactions in the massive star forming region of 30 Doradus [62]."
],
[
"Other Extragalactic TeV targets",
"TeV observations have also been used to place important constraints on the gamma-ray emission from numerous undetected extragalactic source classes, including galaxy clusters and potential sources of ultra-high energy cosmic rays.",
"Here we summarize two of the most important: gamma-ray bursts, and the predicted sites of dense regions of dark matter particles.",
"Gamma-Ray Bursts: GRBs are the signatures of brief, extremely energetic explosions which occur at cosmological distances.",
"They are observationally divided into short and long classes, which are presumably the result of different progenitor systems.",
"Long duration GRBs are generally ascribed to the collapse of massive, rapidly rotating stars into black holes.",
"The origin of the short bursts is less certain, although neutron star - neutron star merger events are among the favoured candidates.",
"The search for $>100{GeV}$ emission has been a long-running goal of both the IACTs and particle detectors (see [63] for a review).",
"Given the brief duration of GRB emission, the wide field-of-view of particle detectors is particularly important in this regard, although observations so far have been hampered by limited sensitivity.",
"IACTs, conversely, must be re-pointed rapidly on receipt of an alert.",
"The detection of delayed high energy emission by Fermi-LAT , lasting hundreds to thousands of seconds longer than the sub-MeV emission, has provided additional impetus to the search [64], [65], and IACTs now regularly target burst locations within $<100{s}$ of the burst alert.",
"The task is difficult, since the burst must also be at a small enough redshift such that the high energy emission is not completely suppressed by photon-photon pair production ($z\\lesssim 0.5$ ).",
"No convincing signals have been detected as yet (e.g.",
"[66], [67], [68], [69]), although predictions based on the brightest bursts observed by the LAT indicate that the potential for detecting TeV emission associated with a GRB is promising, assuming no intrinsic spectral cut-off of the high energy emission [70].",
"Extragalactic Dark Matter: The search for the self-annihilation signature of dark matter particles in astrophysical objects is wide-ranging, and complementary to direct detection techniques on the Earth (see [71] for an excellent review).",
"As discussed later in this review, the centre of our own Galaxy is a natural target for dark matter searches, and provides the most stringent limits to date [72].",
"Objects outside of our own Galaxy are also worthy of investigation, however, and are potentially much less affected by contamination from unknown astrophysical background sources (supernova remnants, pulsar wind nebulae, etc.).",
"Dwarf spheroidal galaxies of the local group are among the most promising of these, due to their proximity and their presumed large dark matter content.",
"The Sloane Digital Sky Survey has more than doubled the known population of dwarf spheroidals, providing additional targets for the TeV searches.",
"No signals have been detected as yet, despite deep exposures on a number of objects [73], [74], [75], [76], [77].",
"Figure REF shows upper limits on the annihilation cross-section derived for various annihilation channels using VERITAS observations of the Segue I dwarf spheroidal.",
"Figure: Upper limits at the 95% confidence level on thevelocity-weighted annihilation neutralino cross-section fordifferent annihilation channels, based on VERITAS observations ofthe Segue I dwarf spheroidal galaxy.",
"The dark band represents thetypical range of predictions.",
"Figure from There are presently $\\sim 80$ known TeV sources within our Galaxy, as indicated either by their association with known Galactic sources at other wavelengths, or by their location in the Galactic plane - a particularly compelling argument when coupled with a resolvable angular extent.",
"The majority of these sources were identified as TeV emitters during the H.E.S.S.",
"survey of the inner Galaxy [78].",
"H.E.S.S.",
"is the only IACT currently operating in the southern hemisphere, which allows it to view the inner Galaxy at high elevation, and hence with a low energy threshold and good sensitivity.",
"H.E.S.S.",
"was the first instrument with sufficient sensitivity to observe sources with $\\lesssim 10\\%$ of the Crab Nebula flux in this region, and the results have been revelatory - the Galactic plane is densely populated with TeV sources, primarily clustered within the inner $\\pm 60^{\\circ }$ in Galactic longitude (Figure REF ).",
"The most recent survey results consist of 2300 hours of observations, allowing the detection of over 50 sources within the range $l=280^{\\circ }$ to $60^{\\circ }$ and $b=-3.5^{\\circ }$ to $+3.5^{\\circ }$ [79].",
"Observations of the outer Galactic regions by VERITAS, MAGIC, Milagro and ARGO-YBJ have revealed a less densely populated sky, but containing some unique objects of particular interest for TeV studies.",
"Many Galactic TeV sources are extended, allowing detailed studies of source morphology and spatially resolved spectra, while others are time variable and/or periodic.",
"The various source classes are discussed in some detail below.",
"Figure: Significance map of the Galactic Plane from the originalH.E.S.S.",
"survey in 2004, based on 230hours230{hours} ofobservations.",
"The complete survey now extends over the range froml=280 ∘ l=280^{\\circ } to 60 ∘ 60^{\\circ } and b=-3.5 ∘ b=-3.5^{\\circ } to+3.5 ∘ +3.5^{\\circ }, and comprises 2300hours2300{hours} of observations .Figure from"
],
[
"The Galactic Centre and Ridge",
"A TeV source at the location of the Galactic Centre has been reported by various IACTs [80], [81], [82], [83].",
"Determining the nature of this source is a difficult task, due to the complexity of the region, which includes multiple different potential counterparts.",
"The most detailed studies have been performed by H.E.S.S., which reveal that the emission is dominated by a bright central source, HESS J1745-290, lying close to the central supermassive black hole, Sgr A$^*$ .",
"An additional, fainter, component is also seen, which extends in both directions along the Galactic plane [84].",
"The extended component is spatially correlated with a complex of giant molecular clouds in the central $200{pc}$ of the Milky Way, and the TeV emission can be attributed to the decay of neutral pions produced in the interactions of hadronic cosmic rays with material in the clouds.",
"The central source is point-like, steady and exhibits a curved power-law spectrum [85].",
"Its location with respect to three of the most likely counterparts is shown in Figure REF .",
"This study reveals that the source centroid is displaced from the radio centroid of the supernova remnant Sgr A East, excluding this object with high probability as the dominant source of the VHE gamma-ray emission, and leaving Sgr A$^*$ and the pulsar wind nebula G359.95-0.04 as the most likely counterparts [86].",
"The Galactic Centre is also a prime candidate region in which to search for gamma-ray emission due to dark matter particle self-annihilation.",
"The analysis is complicated, however, because of the high background due to astrophysical sources.",
"An analysis by H.E.S.S.",
"using an optimized background subtraction technique shows no hint of a residual dark matter gamma-ray flux at a projected distance of $r\\sim 45-150{pc}$ from the Galactic Centre [72].",
"Figure: 90 cm VLA radio flux density map of the innermost20pc20{pc} of the Galactic Centre, showing emission from the SNRSgr A East.",
"Black contours denote radio flux levels; the centre ofthe SNR is marked by the white square, and the positions ofSgr A * ^* and G359.95-0.04 are given by the cross hairs and the blacktriangle, respectively.",
"The 68% CL total error contour of thebest-fit centroid position of HESS J1745-290 is given by the whitecircle.",
"Figure from : see that paper forfull details."
],
[
"The Crab Nebula and Pulsar",
"The Crab is the nearby ($2.0\\pm 0.2{kpc}$ ) remnant of a historical supernova explosion, observed in 1054 A.D.",
"There is no detected shell, and the broadband emission below $\\sim 100{MeV}$ is dominated by a bright synchrotron nebula, powered by a central pulsar (PSR B0531+21).",
"PSR B0531+21 is the most energetic pulsar in our Galaxy, with a pulse period of $33{ms}$ , and a spin-down power of $4.6\\times 10^{38}{erg}{s}{-1}$ .",
"The Crab Nebula and Pulsar hold a unique place in the development of TeV astronomy: the birth of the field as an astronomical discipline can be traced to the detection of the Crab Nebula TeV source by Weekes et al.",
"using the Whipple $10{m}$ telescope, in the first application of the imaging atmospheric Cherenkov technique [1].",
"Subsequently, the Crab has acted as a bright, standard candle for TeV observatories.",
"The SED of the non-thermal nebula emission displays two components (Figure REF ).",
"The dominant, low frequency component is explained by synchrotron radiation of high energy electrons spiraling in the magnetic field of the nebula [87], [88].",
"The higher frequency component is attributed to inverse Compton scattering of lower energy photons by these electrons, including microwave background photons, far infrared and the electron-synchrotron photons themselves.",
"The electron population reaches energies of at least $10^{15}{eV}$ , through acceleration occuring both in a relativistic particle outflow driven by the spin-down energy of the pulsar, and in shocks where this outflow encounters the surrounding nebula.",
"The highest energy particles likely require an alternative explanation for their origin, such as direct acceleration in intense electric fields associated with the pulsar itself [89].",
"Observations of the synchrotron nebula from radio to X-ray wavelengths provide high resolution imaging of the emission region; however, the synchrotron data alone only contain information concerning the product of the magnetic field strength and the relativistic electron density.",
"Since the inverse Compton component is independent of the magnetic field strength, the combined SED allows an estimate of the Nebula magnetic field, which is now constrained to be between 100 and $200{\\mu G}$ [90].",
"The search for a VHE component to the pulsed emission from the Crab has been long and, until recently, fruitless.",
"Despite discouraging model predictions, and the detection of spectral cut-offs below $10{GeV}$ in other pulsars, the fact that no super-exponential cut-off was observed by EGRET in the Crab Pulsar spectrum initially provided some encouragement for a continued search by IACTs [91].",
"Fermi-LAT subsequently extended the GeV spectrum and measured a sharp spectral cut-off at $6{GeV}$ [90].",
"A campaign by MAGIC, using a specially designed “analog sum” hardware trigger, provided the first ground-based measurement of gamma-ray emission from the Crab pulsar [92].",
"The initial MAGIC flux measurement above $25{GeV}$ was, like the Fermi-LAT result, consistent with an exponential cut-off.",
"The existence of an exponential cut-off is a natural consequence of emission due to curvature radiation, as favored by various models (e.g.",
"[93]).",
"Both VERITAS and MAGIC recently presented new results, which challenge this paradigm [94], [95], [96].",
"Pulsed emission is observed to extend up to well beyond $100{GeV}$ , and the combined LAT-IACT spectrum can best be fit with a broken power law.",
"The explanation for this high energy component is an open question, at present.",
"Gamma-ray opacity arguments require that the emission zone of the highest energy photons must be at least 10 stellar radii from the surface of the neutron star - much further than previously assumed.",
"The results require either a substantial revision of existing models of high energy pulsar emission, or the addition of a new component, not directly related to the MeV-GeV emission.",
"Figure: Left: The spectral energy distribution of the CrabNebula from Abdo et al.",
".",
"Their fit tothe synchrotron component is shown (blue dashed line), as well asinverse Compton spectra from Atoyan and Aharonian for assumed magnetic field strengths of100μG100{\\mu G} (solid red line), 200μG200{\\mu G} (dashed greenline) and 300μG300{\\mu G} (dotted blue line).",
"Right: HighEnergy spectrum of the Crab Pulsar.",
"The black dashed line shows afit of a power law with exponential cut-off to theFermi-LAT data alone; the solid line shows a broken powerlaw fit to the LAT and IACT data.",
"The red dashed line and bowtieshows a power law fit to the VERITAS points alone.",
"Figure courtesyof N. Otte (priv.",
"comm.",
").As mentioned above, the Crab has been used as a standard candle in TeV astronomy, on the assumption that its emission was steady.",
"This is now demonstrably false, at least at energies below $\\sim 1{GeV}$ , with the detection of multiple day-scale flaring events [98], [89], and long term variation in the hard X-ray/ soft gamma-ray regime [99].",
"The GeV spectrum during flares indicates that the emission is confined to the synchrotron component of the SED, a conclusion which is supported by the rapid timescale of the events (since the inverse Compton or Bremsstrahlung cooling time of the emitting electrons is much greater than the observed flare duration).",
"At higher energies, some evidence for an enhanced flux during HE flare states has been presented by ARGO-YBJ [100].",
"IACT measurements do not support these results, but are not necessarily in conflict, given the differing duty cycles.",
"Detailed measurements with IACTs during future flare states are required to resolve this question."
],
[
"Pulsar Wind Nebulae",
"Pulsar wind nebulae are the most abundant class of known VHE emitters in the Galaxy, with $\\sim 30$ firm examples, and numerous other sources where the PWN association is more tentative (for reviews see e.g.",
"[101], [102], [103]).",
"The essential emission mechanisms - shock accelerated leptons producing synchrotron and inverse Compton radiation - have already been described for the case of the Crab PWN, but the Crab is far from the typical object.",
"Understanding of the structure and evolution of PWN has advanced significantly over the past few years, in particular thanks to the high resolution X-ray imaging provided by Chandra (see Gaensler and Slane [104] for a detailed review).",
"Initially, the PWN expands uniformly from the central pulsar, while at later stages the nebula may be confined and distorted by the reverse shock from the expanding supernova remnant (SNR).",
"At TeV energies, young PWN are usually still embedded within their parent SNR and are point-like, within the angular resolution of IACTs.",
"They are positionally coincident with a bright X-ray synchrotron nebula powered by a pulsar with very high spin-down luminosity (e.g.",
"G0.9+0.1 [105], HESS J1813-178 [106], G54.1+0.3 [107]).",
"More evolved PWN, with ages $>10,000$ years, are usually much larger, and their TeV emission can be spatially resolved and mapped.",
"The pulsar powering the nebula is often offset from the center of the TeV emission, probably for reasons related to density gradients in the medium surrounding the SNR [108].",
"Remarkably, the TeV nebulae are often two or three orders of magnitude larger than the corresponding X-ray PWN, and the TeV PWN sizes tend to increase with age, while the X-ray PWN sizes show the opposite trend.",
"This can be understood as a result of the fact that the electron population which is responsible for the TeV inverse Compton flux has lower energies than the electrons which produce the X-ray synchrotron emission.",
"They therefore cool more slowly, and survive for longer so, while the X-ray nebula is dominated by freshly accelerated particles, the TeV nebula can record the entire history of particle propagation away from the pulsar.",
"A natural result of this is that the observed TeV spectrum should vary with distance from the pulsar.",
"One of the best examples of this is HESS J1825-137, associated with the PWN of the pulsar PSRJ1826-1334 [109].",
"Figure REF shows the spatially dependent spectra for this source, which soften with increasing distance from the pulsar.",
"This is interpreted as the natural effect of both inverse Compton and synchrotron cooling of the electron population during propagation.",
"A counter-example is the case of Vela-X [110], [111], in which no spectral variability is seen over the extended nebula, suggesting that cooling does not play an important role.",
"Figure: Inset: H.E.S.S.",
"gamma-ray excess map for HESS J1825-137.",
"Thewedges show the radial regions with radii in steps of0.1 ∘ 0.1^{\\circ } in which the energy spectra were determined.",
"Themain figure shows the differential energy spectra for the regionsillustrated in the inset, scaled by powers of 10 for the purposeof viewing.",
"The spectrum for the analysis at the pulsar positionis shown as a reference along with the other spectra as dashedline.",
"Figure from : see that paper forfull details."
],
[
"Supernova Remnants",
"The search for the origin of the cosmic rays triggered the development of gamma-ray astronomy, and continues to motivate many gamma-ray observations.",
"Chief among these is the study of supernova remnants, which are believed to efficiently accelerate particles at the shock front where the expanding SNR encounters the surrounding medium (e.g.",
"[112]).",
"This likely occurs through diffusive shock acceleration (first order Fermi acceleration), in which charged particles are reflected from magnetic inhomogeneities and repeatedly cross the shock front, gaining energy with each crossing (see e.g.",
"[113], [114]).",
"As well as plausibly providing enough energy to explain the observed Galactic cosmic ray population, this process naturally produces a power law distribution of particle energies with an index of $\\sim 2$ , which matches the cosmic ray spectrum (after accounting for diffusion and escape).",
"In recent years, the importance of magnetic field amplification by the accelerated particles themselves has been increasingly recognised, and plays a particular role in explaining the existence of the highest energy Galactic cosmic rays, around the cosmic ray knee region (at $\\sim 3\\times 10^{15}{eV}$ ).",
"The evidence for efficient leptonic acceleration in SNRs is now clearly established (e.g.",
"[115]); however, the question of whether SNR are efficient hadron accelerators is more difficult to answer.",
"A definitive measurement would be the detection of high energy neutrinos from an SNR, but the expected fluxes are likely below the sensitivity thresholds of current neutrino observatories.",
"Gamma-ray observations may provide the key, since the interactions of high energy nuclei with target material produce neutral pions, which decay immediately into gamma-rays.",
"Disentangling the spectral signature of this process from other sources of gamma-ray emission (i.e.",
"leptonic inverse Compton and bremsstrahlung processes) is difficult, but not impossible.",
"Two classes of gamma-ray source are of interest for these studies: those which can be clearly associated with SNR shells, based on the gamma-ray morphology, and sources which are coincident with a massive volume of target material, such as molecular clouds.",
"The definitive association of gamma-ray emission with an SNR shell is often difficult to make, due to the presence of other potential counterparts, particularly PWN.",
"A handful of shell-type SNRs have been unequivocally identified as gamma-ray sources by IACTs.",
"This identification can be made on the basis of the observed shell morphology, (RXJ 1713.7-3946 [116], RXJ 0852.0-4622 (Vela Jr) [117], HESS J1731-3467 [118], SN1006 [119] and, possibly, RCW 86 [120]), or, in the case of Tycho's SNR [121], on the positional coincidence, coupled with the fact that the progenitor was a known Type Ia exposion, and so no compact object is present.",
"Figure REF shows the gamma-ray map for the first SNR shell to be resolved, RXJ 1713.7-3946.",
"The recent addition of Fermi-LAT observations to the broadband spectrum of RXJ 1713.7-3946 [122] are consistent with a leptonic origin as the dominant mechanism for the gamma-ray emission.",
"A counter-example is illustrated by the spectrum in Figure REF , which corresponds to Tycho's SNR.",
"In this case, the fact that the broadband gamma-ray spectrum can be fit with a single hard power law from $500{MeV}$ to $10{TeV}$ favours a hadronic origin [123], [124].",
"This interpretation is not completely compelling, however, given the large statistical errors in the measurements, and the impact of various unknown parameters such as the SNR distance, and possible enhancements of the gamma-ray flux due to a nearby molecular cloud.",
"Additionally, Atoyan and Dermer [125] describe a two-zone leptonic model which provides an acceptable spectral fit.",
"Future measurements of the spectrum below $500{MeV}$ , and deeper exposure at TeV energies, will further test the differing interpretations.",
"Figure: Left: H.E.S.S.",
"map of gamma-ray excess events for RXJ 1713.7-3946 -the first SNR shell to be resolved at TeV energies.",
"Figure from.",
"Right: The broadband SED ofTycho's SNR from , together with modelsfor the various emission components (dominated by hadronicprocesses in the gamma-ray band).",
"See paper for details.The intensity of gamma-ray emission due to hadronic interactions depends upon the flux of high energy nuclei, and also upon the density of target material.",
"Regions of high matter density (e.g.",
"molecular clouds with densities $>100{cm}{-3}$ ), situated close to sites of particle acceleration (such as SNRs), can therefore be expected to produce a large gamma-ray flux due to hadronic interactions.",
"At TeV energies, $\\sim 10$ likely candidates for this process have been identified.",
"The task of identification is complicated, both for the usual reasons of source confusion, and also because the evidence for a molecular cloud / SNR interaction is only definitive in those cases where the cloud morphology is visibly deformed by the expanding SNR, and/or where sites of hydroxyl (OH) maser emission indicate the presence of shocked molecular material.",
"One of the best examples of this source class is the old remnant W28 [126].",
"H.E.S.S.",
"observations of this region show four distinct sites of emission, with three of the four showing a resolvable angular extent ($\\sim 10^{\\prime }$ ).",
"Each of the TeV sources is positionally coincident with a molecular cloud.",
"Assuming that the TeV emission is due to hadronic cosmic rays interacting with the cloud material, the cosmic ray density is inferred to be a factor of 10 to 30 times greater than in the solar neighbourhood."
],
[
"Star forming regions",
"The process of diffusive shock acceleration is not limited to supernova remnant shells.",
"An alternative scenario invokes particle acceleration at the shock formed by the collision between the supersonic stellar winds of massive stars in close binary systems.",
"Stellar winds may also become collectively important in large assemblies of massive stars.",
"The combined effect of the stellar winds, coupled with the effect of multiple SNRs, results in an overall wind from the cluster which forms a giant superbubble in the interstellar medium.",
"Particle acceleration can occur where the cluster wind interacts with the surrounding medium (e.g.",
"[127]).",
"Massive star associations are naturally likely to host other potential source counterparts for TeV emission, such as compact object binary systems, individual supernova remnants and pulsar wind nebulae.",
"A case in point is the young, open cluster Westerlund 2, containing the Wolf-Rayet binary system WR 20a.",
"This was originally suggested as a plausible counterpart to the unidentified source HESS J1023-575, with the emission presumed to be connected to either the Wolf-Rayet binary, or the combined cluster wind [128].",
"A re-assesment of this region, informed by a deeper H.E.S.S.",
"exposure and results from Fermi-LAT, alters the picture [129].",
"The LAT detects an energetic pulsar, PSR J1022-5746, which drives a PWN which is bright in GeV gamma-rays [130].",
"Given the ubiquity of bright TeV PWN, this now seems the most likely explanation for the TeV source.",
"A similar conclusion may arise for the first TeV source to be linked with a massive star association, TeV 2032+4130 (coincident with the Cyg OB2 association).",
"In this case, the LAT pulsar (PSR J2032+4127) is sufficiently energetic to explain the TeV emission, although no PWN has been detected as yet.",
"Other unidentified sources which have been linked with massive star clusters and associations include HESS J1646-458 (Westerlund 1) [131], HESS J1614-518 (Pismis 22) [132], HESS J1848-018 (W43, which hosts Wolf-Rayet star WR121a) [133] and W49A [134], a massive star forming region.",
"For all of these, however, the evidence that particle acceleration in stellar winds is the driving force behind the gamma-ray emission is not definitive (e.g.",
"[135])."
],
[
"Compact Object Binary Systems",
"Despite many early unconfirmed claims, the first definitive detection of a TeV gamma-ray binary system was not published until 2005.",
"The population has grown slowly, and now consists of four clearly identified systems, plus marginal evidence for transient emission associated with Cyg X-1 [136].",
"The gamma-ray emission from binaries is believed to be powered either by accretion (most likely onto a black hole), or by a pulsar wind.",
"In the case of accretion, particle acceleration takes place in relativistic jets (e.g.",
"[137]).",
"In the pulsar wind scenario, the acceleration occurs either in shocks where the pulsar wind encounters the circumstellar environment (e.g.",
"[138]), or possibly within the pulsar wind zone itself [139].",
"The detection of Cyg X-1, if confirmed, would be extremely important, since there is no doubt that this system hosts a black hole.",
"This is in contrast to all of the other TeV binaries, in which the compact object is either known to be, or may be, a neutron star.",
"Here we briefly summarize the results for each of the four well-studied objects.",
"PSR B1259-63/LS 2883: This was the first gamma-ray binary system to be firmly detected at TeV energies, and the first known variable VHE source in our Galaxy [140].",
"The system comprises a $48{ms}$ pulsar orbiting a massive B2Ve companion.",
"The orbit is highly eccentric ($e=0.87$ ), with a period of 3.4 years.",
"The TeV emission exhibits two peaks, approximately 15 days before and after periastron.",
"Various authors have attempted to explain the double bumped VHE lightcurve within a 'hadronic disk scenario', in which a circumstellar disk provides target material for accelerated hadrons, leading to $\\pi ^0$ production and subsequent TeV gamma-ray emission [141], [142].",
"The 2007 H.E.S.S.",
"observations disfavour this, since the onset of TeV emission occurs $\\sim 50{days}$ prior to periastron, well before interactions with the disk could be expected to play a significant role [143].",
"Leptonic scenarios have also been discussed in e.g.",
"Kangulyan et al.",
"[144].",
"The recent discovery of an extended and variable radio structure in PSR B1259-63/LS 2883 at phases far from periastron provides definitive evidence that non-accreting pulsars orbiting massive stars can produce variable and extended radio emission at AU scales [145].",
"This is important, since similar structures in LS 5039 and LS I +61$^{\\circ }$ 303, where the nature of the compact object is not certain, have been used to argue for the existence of jets driven by accretion onto a black hole.",
"LS 5039: LS 5039 consists of a compact object, either neutron star or black hole, orbiting a massive O6.5V ($\\sim $ 23 M$_\\odot $ ) star in a 3.9 day orbit.",
"Observations by H.E.S.S.",
"in 2004 revealed that LS 5039 is a bright source of VHE gamma-rays [146].",
"Unlike PSR B1259-63 (and, to a lesser extent, LS I +61${^\\circ }$ 303) LS 5039 is almost perfectly suited to TeV observations, with a short orbital period and a convenient declination angle, allowing sensitive observations at all phases over numerous orbits.",
"The VHE emission measured by H.E.S.S.",
"is modulated at the orbital period, peaking around inferior conjunction, when the compact object is closest to us and co-aligned with our line-of-sight (Figure REF ).",
"The spectrum is also orbitally modulated, appearing significantly harder around inferior conjuction ($\\Gamma =1.85\\pm 0.06_{\\mathrm {stat}}\\pm 0.1_{\\mathrm {syst}}$ ), but with an exponential cut-off at $E_o=8.7\\pm 2.0{TeV}$ .",
"At GeV energies, the source is detected by Fermi-LAT at all orbital phases, with the emission peaking close to superior conjuction, in apparent anti-phase with the VHE results [147].",
"A sharp spectral cut-off at $E_o=1.9\\pm 0.5{GeV}$ is observed in the LAT data near superior conjunction, indicating that the VHE spectra cannot be simply a smooth extrapolation of the lower energy emission.",
"Figure: The H.E.S.S.",
"flux (bottom) and photon index (top) for LS 5039 as a functionof orbital phase.",
"Figure from Aharonian et al.",
".LS I +61${^\\circ }$ 303: Similar to LS 5039, LS I +61${^\\circ }$ 303 consists of a compact object, either neutron star or black hole, in this case orbiting a B0Ve star with a circumstellar disk ($\\sim $ 12.5 M$_\\odot $ ) in a 26.5 day orbit.",
"The detection of a variable VHE source at the location of LS I +61${^\\circ }$ 303 with MAGIC [148], later confirmed by VERITAS [149], established this source as a gamma-ray binary.",
"The object is now one of the most heavily observed locations in the VHE sky, with deep exposures by the two observatories spread over half a decade.",
"Despite this, the VHE source is much less well-characterized than LS 5039, owing to its relatively weak VHE flux, and an inconvenient orbital period which closely matches the lunar cycle, making observations over all orbital phases almost impossible within a single observing season.",
"VHE emission was originally detected close to apastron, between phases $\\phi =0.5-0.8$ .",
"In contrast to this, VHE observations between 2008 and 2010 showed that, at least during the orbits when the source was observed, the apastron flux was much lower than during the previous detections [150], [151].",
"The detection of TeV emission by VERITAS during a single episode close to superior conjunction complicates the picture even further.",
"As with LS 5039, the Fermi-LAT GeV emission peaks closer to periastron, and the spectrum displays sharp cut-off, at $E_o=6.3\\pm 1.1{GeV}$ .",
"Long term variability in the GeV band has also been observed [152].",
"HESS J0632+057: This TeV source was serendipitously detected during H.E.S.S.",
"observations of the Monoceros Loop SNR region, and noted as a potential binary primarily because of its small angular extent [153].",
"Subsequent observations revealed it to be a variable TeV source [154], and co-located with a variable radio and X-ray source, at the position of a massive Be star, MWC 148 [155].",
"X-ray observations with Swift recently provided definitive proof of its binary nature, with the measurement of a $321\\pm 5{day}$ periodicity in the lightcurve [156].",
"The TeV light curve displays a broad gamma-ray flare close to the X-ray maximum, with a duration of $\\sim 40{days}$ [157].",
"No GeV source has been detected.",
"A number of competing processes likely contribute to the variability observed in the gamma-ray binaries.",
"In particular, the efficiency of inverse Compton gamma-ray production, as well that of VHE gamma-ray absorption (through pair production), changes as a function of orbital phase.",
"This can go some way towards explaining the apparent phase shift between the GeV and TeV lightcurves, for example in the case of LS 5039.",
"There are clearly other effects which contribute, however, as demonstrated by the long term instability of LS I +61${^\\circ }$ 303.",
"The sharp GeV cut-offs in these systems are also difficult to explain, and may indicate that the GeV and TeV emission components do not have the same origin.",
"A final comment should be reserved for the Fermi-LAT source 1 FGL J1018.6-5856, which was recently identified as a GeV binary system, with a period of $16.58\\pm {0.04}{days}$ [158].",
"This source resides in a complex region, coincident with the center of the SNR G284.3-1.8, and close to a LAT pulsar (at a distance of 35').",
"The extended, unidentified HESS source, HESS J1018-589, overlaps with the GeV binary location, and appears to consist of a point-like source overlaid on a diffuse structure [159].",
"While this is indicative of a new TeV binary there is, as yet, no evidence for variability in the point-like emission, making the identification still uncertain."
],
[
"Globular Clusters",
"A single globular cluster, Terzan 5, has been suggested as the probable counterpart of a TeV source (HESS J1747-248) [160].",
"If the association is correct, the emission is likely related to the large population of millisecond pulsars in this cluster, which provide an injection source of relativistic leptons [161].",
"Inverse Compton gamma-ray photons result when these electrons upscatter low energy photons of the intense stellar radiation field.",
"The H.E.S.S.",
"source is extended, and slightly offset from the cluster core (although there is significant overlap).",
"The probability of this being simply a chance positional coincidence is $\\sim 10^{-4}$ .",
"Globular clusters have also been favored targets in searches for dark matter particle annihilation signatures, since they may have been generated in dark matter mini-haloes before the formation of galaxies took place, and thus retain a significant dark matter component [162], [163].",
"Limits have been placed on NGC 6388, M15, Omega Centauri, 47 Tuc, M13, and M5 ([164], and references therein)."
],
[
"Unidentified objects",
"IACTs are able to locate point sources with reasonably good accuracy (typically $\\lesssim 1^{\\prime }$ for a moderately strong source).",
"Extragalactic TeV sources can therefore usually be firmly identified with a single counterpart at other wavelengths.",
"For Galactic sources, identification poses more of a problem, and around one third of the Galactic sources lack a firm identification.",
"While the diffuse Galactic background emission, which dominates at GeV energies, is not significant, the TeV sources themselves are mostly extended, and can often be plausibly associated with multiple counterparts.",
"PWN, in particular, often have their brightest TeV emission offset from the location of the parent pulsar or X-ray PWN, which may not yet have been detected (a number of new energetic pulsars have been located in follow-up observations of unidentified TeV sources).",
"In other cases, despite deep X-ray and radio follow-up observations, no reasonable counterpart has yet been found.",
"The unidentified Milagro sources also pose some interesting questions.",
"While IACTs have identified sources associated with these objects, the TeV sources are typically much smaller in angular extent, and cannot account for the entire Milagro flux.",
"One possible explanation for this is that there is a diffuse high energy component to the emission, which is difficult to resolve with IACTs, given their relatively small fields of view."
],
[
"The Future",
"The field of ground-based gamma-ray astronomy has expanded dramatically over the past 10 years, but it is worth noting that the observatories currently operating are far from reaching the physical limits of the detection techniques.",
"We therefore conclude this review with a brief discussion of some of the instrumental developments which can be expected over the next decade.",
"The IACTs currently operating have all made significant efforts to maintain, and improve, sensitivity since they were first commissioned.",
"H.E.S.S.",
"recoated the telescope mirrors, and successfully developed sophisticated analysis tools with which to exploit the data.",
"VERITAS relocated their original prototype telescope to provide a more favorable array layout, halving the time required to detect a weak source.",
"Most significantly, MAGIC added a second telescope of similar design to the first, providing a stereo pair with a baseline of $85{m}$ .",
"All of these observatories have further upgrade plans.",
"Both MAGIC and VERITAS are implementing camera upgrades - in the case of VERITAS this involves the replacement of all of the camera PMTs with more sensitive, super-bialkali devices in summer 2012.",
"H.E.S.S.",
"are constructing H.E.S.S.",
"II - the addition of a single large telescope, with $600{m}{2}$ mirror area, to the centre of the array.",
"Figure REF shows the telescope structure in November 2011.",
"The mirrors and camera will be installed in the first half of 2012.",
"Figure: The steel structure for the 600m2600{m}{2} H.E.S.S.",
"IItelescope.",
"Mirrors and camera will be installed in the first half of2012.",
"(Note that the H.E.S.S.",
"II telescope is located in between thetwo H.E.S.S.",
"I telescopes shown in the image, not in theforeground).",
"Figure courtesy of the H.E.S.S.",
"collaborationhttp://www.mpi-hd.mpg.de/hfm/HESS/Novel approaches are also being pursued by many smaller projects.",
"A particularly nice example of this is FACT (the First G-APD Cherenkov Telescope), which has recently demonstrated the application of Geiger-mode Avalanche Photodiodes for Cherenkov astronomy.",
"The FACT telescope consists of a 1440-pixel G-APD camera at the focus of one of the original HEGRA telescopes.",
"Geiger-APDs hold great promise as a potential replacement for PMTs, since they are robust, and offer much better photon conversion efficiency.",
"Figure REF shows some “first light” images from FACT.",
"Other projects in development include GAW (Gamma Air Watch), MACE (Major Atmospheric Cerenkov Telescope Experiment) and LHASSO (Large High Altitude Air Shower Observatory).",
"LHASSO is an ambitious project which will be located near the site of the ARGO-YBJ experiment in Tibet.",
"It is planned to consist of four water Cherenkov detectors, two IACTs, three fluorescence telescopes and a large scintillator array.",
"Figure: First light cosmic ray images from FACT (First G-APDCherenkov Telescope).",
"Four different events are shown; the upperright event shows the characteristic ring image produced by a localmuon.",
"The camera contains 1440 Geiger-mode avalanche photodiodes,installed on one of the original HEGRA telescopes at the Roque delos Muchachos on La Palma.",
"Figure courtesy of the FACTcollaboration http://fact.ethz.ch/firstFor the particle detectors, the next stage in instrumentation is HAWC (the High Altitude Water Cherenkov Observatory).",
"HAWC will consist of 300 individual water Cherenkov tanks, at an altitude of $4100{m}$ in Mexico.",
"The final array is expected to be 15 times more sensitive than Milagro, and will be a powerful tool for surveying, and for observations of transient phenomena.",
"A prototype system is already in operation, and science operations will start in spring 2012, with completion of the full array expected in 2014.",
"The most ambitious future project is CTA (the Cherenkov Telescope Array).",
"This is described in detail in [165].",
"Briefly, it comprises an array of imaging atmospheric Cherenkov telescopes covering $\\sim 1{km}{2}$ , providing a factor of 5-10 improvement in sensitivity in the $100{GeV}$ -$10{TeV}$ range, and extending the energy range both above and below these values.",
"Both a northern and a southern site are envisaged, and the array will be operated as an open observatory.",
"Multiple telescope designs are planned, including small (few${m}$ ), medium ($10-15{m}$ ) and large ($20-30{m}$ ) diameter reflectors, as well as two-mirror telescope designs, such as the Schwarzchild-Couder [166].",
"In conclusion, TeV gamma-ray astronomy now describes a broad astronomical discipline which addresses a wide, and expanding, range of astrophysical topics.",
"With planned instrumental developments, it is not unreasonable to expect the source catalogue to exceed 1000 objects within the next decade.",
"Much of the TeV sky remains relatively unexplored.",
"Less than $\\sim 10\\%$ of the sky has been observed with $\\sim 10{milliCrab}$ sensitivity at $1{TeV}$ , and the sensitive exposure to transient events and widely extended sources at these energies is much lower.",
"The likelihood of continued exciting results is certain, both for the known sources and source classes, and for new discoveries."
]
] | 1204.1267 |
[
[
"Phononic gaps in the charged incommensurate planes of Sr14Cu24O41"
],
[
"Abstract The terahertz (THz) excitations in the quantum spin-ladder system Sr14Cu24O41 have been determined along the c-axis using THz time-domain, Raman and infrared spectroscopy.",
"Low-frequency infrared and Raman active modes are observed above and below the charge-ordering temperature T_{co} ~ 200 K over a narrow interval of 1 - 2 meV .",
"A new infrared mode around 1 meV develops below ~ 100 K. The temperature dependence of these modes shows that they are coupled to the charge- and spin-density-wave correlations in this system.",
"These low-energy features are conjectured to originate in the gapped sliding-motion of the chain and ladder sub-systems, which are both incommensurate and charged."
],
[
"=1 Phononic gaps in the charged incommensurate planes of Sr$_{14}$ Cu$_{24}$ O$_{41}$ V. K. Thorsmølle [email protected] Laboratory for Photonics and Interfaces, École Polytechnique Fédérale de Lausanne (EPFL), CH-1015, Switzerland Département de Physique de la Matière Condensée, Université de Genève, CH-1211 Genève 4, Switzerland C. C. Homes [email protected] A. Gozar [email protected] Condensed Matter Physics and Materials Science Department, Brookhaven National Laboratory, Upton, New York 11973, USA G. Blumberg Department of Physics and Astronomy, Rutgers, The State University of New Jersey, Piscataway, New Jersey 08854, USA J. L. M. van Mechelen A.",
"B. Kuzmenko Département de Physique de la Matière Condensée, Université de Genève, CH-1211 Genève 4, Switzerland S. Vanishri C. Marin CEA Grenoble, INAC, SPSMS, IMAPEC, 17 rue des Martyrs, 38054 Grenoble, France H. M. Rønnow Laboratory for Quantum Magnetism, École Polytechnique Fédérale de Lausanne (EPFL), CH-1015, Switzerland The terahertz (THz) excitations in the quantum spin-ladder system Sr$_{14}$ Cu$_{24}$ O$_{41}$ have been determined along the c-axis using THz time-domain, Raman and infrared spectroscopy.",
"Low-frequency infrared and Raman active modes are observed above and below the charge-ordering temperature $T_{\\rm co} \\simeq 200$ K over a narrow interval $\\simeq 1 - 2$ meV ($\\simeq 8 - 16$ cm$^{-1}$ ).",
"A new infrared mode at $\\simeq 1$ meV develops below $\\simeq 100$ K. The temperature dependence of these modes shows that they are coupled to the charge- and spin-density-wave correlations in this system.",
"These low-energy features are conjectured to originate in the gapped sliding-motion of the chain and ladder sub-systems, which are both incommensurate and charged.",
"71.45.Lr, 78.30.-j, 78.70.Gq More than three decades ago, new normal modes were predicted to occur in ionic materials with incommensurate (IC) layers which can slide past each other [1], [2], [3], [4].",
"These new degrees of freedom allow separate phonons in each subsystem at high frequencies with a crossover to slow oscillations due to relative sliding motions of the two almost rigid subsystems at ultra-low frequencies, leading effectively to an extra acoustic mode.",
"If the IC layers are charged these sliding modes become gapped due to the restoring Coulomb forces.",
"These modes are the ionic complements of the electronic plasmons in metals and their dynamics also resemble the sliding motion in density wave (DW) systems [5].",
"Thus far,unambiguous experimental evidence for sliding gapped acoustic mode resonances has remained, to our knowledge, elusive.",
"A promising avenue of investigation is the low-dimensional quantum spin-ladder system Sr$_{14}$ Cu$_{24}$ O$_{41}$ containing such substructures in the form of Cu$_2$ O$_3$ ladders and one-dimensional (1D) CuO$_2$ chains [6], .",
"The chains and ladders run parallel along the c-axis with the rungs of the ladders along the a-axis [6], , shown in Fig.",
"REF .",
"The two subsystems are structurally IC, resulting in a buckling along the c-axis with a period $c=27.5\\,{\\rm Å}\\,\\simeq 10\\,c_{\\rm ch} \\simeq 7\\,c_{\\rm ld}$ , where $c_{\\rm ch}$ and $c_{\\rm ld}$ represent the lattice constants for the chain and ladder subcells, respectively.",
"This intrinsically hole-doped material exhibits a variety of unusual charge, magnetic and vibrational phenomena [8] that have been probed by several techniques, among them magnetic resonance [9], neutron scattering [10] and resonant x-ray scattering [11], [12], [13].",
"The unusual DW order is attributed to cooperative phenomena driven and stabilized by charge and spin correlations, in conjunction with the IC lattice degrees of freedom.",
"Low-energy features spanning frequencies from the kHz to the THz range and associated with the DW dynamics have also been observed.",
"However, while the microwave data have been consistently interpreted in terms of screened DW relaxational dynamics [14], [15], [16], [17], the nature of the excitations in the THz regime, which are seen to be strongly coupled to the charge/spin ordering, remains controversial [14], [15], [16], [17], [11], [18], [19], [12], [20], [13].",
"An important and open question is, what are the salient spectroscopic features in the $\\simeq 1$ meV energy region?",
"In this Letter we demonstrate using three experimental techniques that low-energy collective modes in Sr$_{14}$ Cu$_{24}$ O$_{41}$ consist of one Raman and one infrared (IR) active excitation present in the $5-300$ K temperature region, as well as a new mode appearing well below the charge-ordering temperature (T$_{\\rm co}\\simeq 200$ K) whose temperature-dependent behavior is tracked by terahertz time-domain spectroscopy (THz-TDS) [21].",
"An intuitive interpretation of these modes in relation to “phononic gaps” opened by Coulomb interactions in IC lattices and coupled to the DW ordering is able to consistently explain the range and relative energies of these excitations which are illustrated in Fig.",
"REF .",
"Figure: Perspective view of the ideal unit cell of Sr 14 _{14}Cu 24 _{24}O 41 _{41} (theSr atoms are omitted) in the a-b plane viewed along the c axis showing thestacking of the CuO 2 _2 chains and the Cu 2 _2O 3 _{3} ladders.",
"Beneath the unitcell are the symbolic representations of the low-frequency infrared and Raman modesin the ladders and chains (see text for details); the chains are depicted by the circles(q ch ≃-1.4eq_{\\rm ch}\\simeq -1.4\\,e), and the ladders by the squares (q ld ≃2eq_{\\rm ld}\\simeq 2\\,e).The dots and crosses refer to the oscillatory motion in and out of the a-bplane, respectively.Single crystals of Sr$_{14}$ Cu$_{24}$ O$_{41}$ were grown using the traveling-solvent floating-zone method.",
"The sample was oriented and cut to $4\\,{\\rm mm} \\times 6\\,{\\rm mm}$ , with a thickness of 440 $\\mu $ m and the a- and c-axes in the plane.",
"The THz-TDS experiments (TPI spectra 1000, TeraView Ltd.) were performed in transmission geometry with the sample mounted inside an optical cryostat capable of reaching 5 K. To obtain the complex conductivity and the transmittance window of the time-domain signals lasting beyond 60 ps (containing several Fabry-Perot internal sample reflections), a complete Drude-Lorentz time-domain analysis study is presented, in contrast to simple frequency-inversion [22].",
"The polarized reflectance was measured over a wide frequency range using an in-situ evaporation technique [23].",
"The complex conductivity is determined from a Kramers-Kronig analysis of the reflectance [24] requiring extrapolations in the $\\omega \\rightarrow 0$ limit; above 250 K a Hagen-Rubens form is employed $R(\\omega ) \\propto 1-\\sqrt{\\omega }$ , while below this temperature the reflectance is assumed to be constant, $R(\\omega \\rightarrow 0) \\simeq 0.68 - 0.76$ .",
"The Raman measurements were performed in (cc) and (aa) polarizations as described in Ref. Gozar2003.",
"The direction of propagation of the light was perpendicular to the a-c plane for all measurements.",
"Figure: Electric field E(t)E(t) of THz pulse in the time-domain transmitted througha 440-μ\\mu m thick Sr 14 _{14}Cu 24 _{24}O 41 _{41} crystal.",
"(a) E(t)E(t) polarized alongthe a- and c-axis at 300 and 5 K. The reference is recorded without thecrystal.",
"The inset shows the FFT amplitude spectrum of E(t)E(t) for the reference.",
"(b) E(t)E(t) (black lines) along the c-axis shown with the Drude-Lorentztime-domain fits (red lines) at three different temperatures.",
"(c) Temperaturedependence of the largest peak of the amplitude for the fast oscillation (blacksquares) and slow oscillation (red circles) of the THz signal.",
"The slow oscillationis only shown to 150 K.Figure REF (a) shows the electric field of the THz pulse transmitted through the Sr$_{14}$ Cu$_{24}$ O$_{41}$ sample comparing the response for polarizations along the a- and c-axis at high and low temperatures, as well as a reference signal without a sample.",
"The Fourier transformed (FFT) amplitude spectrum for the reference is shown in the inset.",
"For the electric field polarized along the a-axis, the shape of the THz pulse is essentially the same at 300 and 5 K, characteristic of insulating behavior along this direction.",
"However, for the electric field polarized along the c-axis, the THz signal is barely present at 300 K, indicative of metallic response at higher temperatures.",
"At low temperatures a distinct long-lived ringing is observed with a period of $\\simeq 1$ ps.",
"Figure REF (b) shows the transmitted THz pulse polarized along the c-axis for three different temperatures with time-domain fits revealing an additional $\\simeq 4$ ps oscillation.",
"The $\\simeq 4$ ps period oscillation is observed to appear below $\\simeq 200$ K, while the the $\\simeq 1$ ps oscillation below $\\simeq 170$ K, as shown in Fig.",
"REF (c).",
"Figure: (a) Infrared reflectance of Sr 14 _{14}Cu 24 _{24}O 41 _{41} for lightpolarized along the c-axis above and below T co T_{\\rm co} where thedashed lines indicate the ω→0\\omega \\rightarrow 0 extrapolations.",
"(b) Transmittance for light polarized along the c axis (T≲T co T \\lesssim T_{\\rm co}).",
"(c) Raman response in relative units for the aa polarization, and (d)cc polarization at various temperatures.",
"The Raman data wereadapted from Ref.",
"gozar2005a,*gozar2005b.The far-infrared reflectance for light polarized along the c-axis is shown in Fig.",
"REF (a), while (b) shows the transmittance for the electric field polarized along the c-axis of the THz oscillations presented in Fig.",
"REF ; Figs.",
"REF (c) and (d) show the aa and cc Raman responses, respectively [25], .",
"A metallic Drude-like behavior is observed in the reflectance above $T_{\\rm co}$ .",
"For $T \\lesssim T_{\\rm co}$ the reflectance changes from a metallic to an insulating character, allowing a strong vibrational feature to emerge.",
"For the transmittance shown in Fig.",
"REF (b) there is a sharply defined window where the transmittance is effectively blocked by at least 5 orders of magnitude between $\\simeq 13 - 33$ cm$^{-1}$ (1 meV $\\simeq 8.1$ cm$^{-1}$ , 1 THz $\\simeq 33.3$ cm$^{-1}$ ), corresponding to the pronounced slow and fast oscillations observed in the time-domain (Fig.",
"REF ).",
"As the temperature is raised, the upper limit of this window becomes less effective until $T \\gtrsim T_{\\rm co}$ , at which point the entire region becomes increasingly opaque.",
"In addition, one also notices an absorption at $\\simeq 8$ cm$^{-1}$ which also decreases with increasing temperature.",
"The loss of the electronic background for $T \\lesssim T_{\\rm co}$ signals the transition to a DW ground state and an insulating phase.",
"Figure: (a) THz-TDS conductivity along the c-axis ofSr 14 _{14}Cu 24 _{24}O 41 _{41} above and below T co T_{\\rm co}.",
"Above 180 Kthe conductivity was calculated from the reflectance (see text); thedashed lines indicate where the conductivity has been determinedfrom the extrapolations supplied for the reflectance [Fig. 2(a)].",
"(b) Contour plot of the conductivity along the c-axis (a) as a function offrequency and temperature.",
"(b) Temperature dependence of the two phonon modes observedin (a) shown together with the Raman-active mode [Figs.",
"(c),(d)].The strengths and linewidths are shown in (d) and (e), respectively, together with that of thea-axis phonon mode at ≃57\\simeq 57 cm -1 ^{-1}.",
"The amplitudes of the Raman data in (d) isin arbitrary units.",
"The open symbols indicate data obtained from reflectance measurements.Figure REF (a) shows the temperature-dependence of the real part of the optical conductivity along the c-axis.",
"A metallic response is observed for $T \\gtrsim T_{\\rm co}$ , while for $T \\lesssim T_{\\rm co}$ an insulating response develops.",
"The mode at 14.9 cm$^{-1}$ narrows and softens to 12.6 cm$^{-1}$ at low temperature with a slight kink around $T_{\\rm co}$ (we refer to this mode as IR1).",
"The Raman mode seen in aa and cc polarizations has the opposite behavior; its position increases from 9.5 to 12 cm$^{-1}$ upon cooling from $\\simeq 300$ to 5 K. Interestingly, below $T_{\\rm co}$ , a new infrared mode branches off below 10 cm$^{-1}$ towards lower frequencies, reaching 8.4 cm$^{-1}$ at $\\simeq 5$ K; the temperature dependence of this new mode (which we denote IR2) closely resembles the behavior reported for the weakly-dispersive magnetic chain excitations [10].",
"Along the a-axis the optical conductivity is an order of magnitude smaller than it is along the c axis, confirming the insulating behavior perpendicular to the chains and ladders (not shown); in addition to a number of weak features, a sharp infrared phonon is observed at $\\simeq 57$ cm$^{-1}$ .",
"The remaining panels in Figure REF show the details of the c-axis features in the optical conductivity with (b) a contour plot of optical conductivity as a function of frequency and temperature; (c) the frequency of IR1 and IR2, as well as the frequency of the aa and cc Raman excitations; their respective (d) strengths and (e) linewidths.",
"We note that the total oscillator strength of IR1 and IR2 appears to be conserved [Fig.",
"REF (d)].",
"It is tempting to relate these excitations to folded phonon modes due to the $10/7$ superstructure.",
"However, using $c \\simeq 27.5$ Å and the sound velocity $v_s \\simeq 13$ km/s [27], one would expect these excitations at energies $\\simeq 100$ cm$^{-1}$ .",
"This is a factor of ten higher than what we observe suggesting that a different mechanism is at work.",
"We then turn our attention to the scenario of the sliding motions of the chain and ladder subsystems presented in Fig.",
"REF .",
"The energy scale and the relative frequencies of the Raman and IR1 modes can be understood qualitatively by taking into account the c-axis incommensurability between the chain and ladder unit cells, $c_{\\rm ch} / c_{\\rm ld} \\simeq 0.699$ , in accordance with recent x-ray studies [28], [29] which demonstrated that consideration of the super space group is mandatory.",
"We assign $n$ , $m$ and $q$ to the corresponding atomic mass density, unit-cell mass and charge respectively.",
"Using the available crystallographic data we obtain without any fitting parameters $\\omega _{\\rm IR1} = \\sqrt{\\frac{n_{\\rm ch} q_{\\rm ch}^2}{\\varepsilon _0 \\varepsilon _{\\infty } m_{\\rm ch}}\\left( 1 + \\frac{c_{\\rm ld}}{c_{\\rm ch}} \\frac{m_{\\rm ch}}{m_{\\rm ld}} \\right)}\\simeq 29.5 \\, {\\rm cm}^{-1}$ for IR1 [2].",
"Here the chains and ladders are considered uniformly charged and $\\varepsilon _{\\infty } (75\\,{\\rm cm}^{-1}) \\simeq 15$ is the experimentally measured contribution of all other higher-energy phonons to the dielectric permittivity.",
"All holes were assumed to be located in the chain system, i.e.",
"$q_{\\rm ch} = -1.4\\,e$ and $q_{\\rm ld} = 2\\,e$ .",
"This is the ionic complement of the electronic zone-center plasmons in metals.",
"In general this mode is acoustic with a diffusive character at long wavelengths but it becomes gapped due to restoring Coulomb forces if the IC systems are oppositely charged [1], [2], [3], [4].",
"The Raman mode corresponds to the out-of-phase oscillation of the chain layers, phase shifted by $\\pi $ along the b-axis, with the ladders at rest (Fig.",
"REF ).",
"Its frequency can be estimated by Eq.",
"(REF ) in the $m_{\\rm ld} \\rightarrow \\infty $ limit.",
"Hence, $\\omega _{\\rm R} \\simeq 0.85\\,\\omega _{\\rm IR1}$ which is in good agreement with the experimental observations.",
"Removal of free carriers with decreasing temperature due to the activated nature of the conductivity would reduce screening effects, leading to the Raman mode hardening at low temperatures, also in agreement with our observations.",
"We suggest that the origin and energy of the IR2 mode can be understood by considering the effects of quasi-2D charge ordering in the chains [10].",
"Once the long-range hole ordering in the chains sets in below $\\simeq 100$ K restoring Coulomb forces will oppose the out-of-phase oscillation of adjacent chains.",
"Above 100 K this excitation is expected to have a vanishingly small energy because of the short range charge correlations along the a-axis resulting in the absence of net restoring electrostatic forces in the disordered state.",
"For a long-range sinusoidal charge modulation, the energy of this mode is $\\omega _{\\rm IR2}^2 \\propto \\sqrt{\\frac{\\lambda }{\\pi d}} \\,\\exp \\left( {-\\frac{\\pi d}{\\lambda }} \\right)\\frac{\\delta q^2}{m_{\\rm ch} c_{\\rm ch} \\lambda ^2},$ with a proportionality factor of the order of unity.",
"Here $d$ , $\\lambda $ and $\\delta q$ are the distance between two chains, the wavelength and depth of the harmonic charge modulation, respectively.",
"In Sr$_{14}$ Cu$_{24}$ O$_{41}$ this charge modulation is not a simple sinusoid; however, to first order a harmonic approximation may be used for two coupled chains.",
"Taking $\\lambda \\simeq 5\\, c_{\\rm ch}$ , $d \\simeq 5.7$ Å and an average charge modulation depth $\\langle \\delta q \\rangle \\simeq 0.37\\,{e}$ , we find $\\omega _{\\rm IR2} \\approx 3$ cm$^{-1}$ , again consistent with the experimental data.",
"From Eq.",
"(REF ) it is seen that IR2 is a direct probe of the charge order.",
"The situation is quite similar to the undoped La$_{6}$ Ca$_{8}$ Cu$_{24}$ O$_{41}$ where the staggered chain arrangement along the a-axis should generate restoring Coulomb forces $\\pi $ -shifted oscillations of adjacent chains and the analog of the IR2 mode in Sr$_{14}$ Cu$_{24}$ O$_{41}$ is expected to be present at all temperatures.",
"This is in agreement with our experimental observations.",
"A quantitative analysis of the IR2 mode and comparison to crystals of the “14-24-41” family will be the topic of a future study.",
"In conclusion, we propose that the low-energy Raman and IR1 excitations in Sr$_{14}$ Cu$_{24}$ O$_{41}$ originate from sliding motions of the IC chains and ladders which are gapped by Coulomb interactions due to the net charge carried by these sub-systems.",
"Long-range charge ordering in the chains will further generate low-energy infrared activity and we suggest this to be at the origin of the new IR2 mode observed in the time-domain THz data below $\\simeq 100$ K. The energy of this mode is as such a direct probe of the charge modulation depth as well as of the quasi-2D hole ordering pattern in the chain subsystem.",
"We gratefully acknowledge useful discussions with T. Maurice Rice and Jason Hancock.",
"We would like to thank H. Eisaki for providing us with samples.",
"Work at Brookhaven was supported by the U.S. Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering under Contract No.",
"DE-AC02-98CH10886.",
"Work at Rutgers was supported by NSF DMR-1104884.",
"Thanks to Prof. H. L. Bhat of IISc, Bangalore for the Indo-French collaborative project, CEFIPRA under project No.3408-4, which supported the crystal growth work at CEA Grenoble."
]
] | 1204.1579 |
[
[
"An Invertible Linearization Map for the Quartic Oscillator"
],
[
"Abstract The set of world lines for the non-relativistic quartic oscillator satisfying Newton's equation of motion for all space and time in 1-1 dimensions with no constraints other than the \"spring\" restoring force is shown to be equivalent (1-1-onto) to the corresponding set for the harmonic oscillator.",
"This is established via an energy preserving invertible linearization map which consists of an explicit nonlinear algebraic deformation of coordinates and a nonlinear deformation of time coordinates involving a quadrature.",
"In the context stated, the map also explicitly solves Newton's equation for the quartic oscillator for arbitrary initial data on the real line.",
"This map is extended to all attractive potentials given by even powers of the space coordinate.",
"It thus provides classes of new solutions to the initial value problem for all these potentials."
],
[
"INTRODUCTION",
"The set of world lines for the non-relativistic quartic oscillator satisfying Newton’s equation of motion for all space and time in 1-1 dimensions with no constraints other than the ”spring” restoring force is shown to be equivalent (1-1-onto) to the corresponding set for the harmonic oscillator.",
"This is established via an energy preserving invertible linearization map which consists of an explicit nonlinear algebraic deformation of coordinates and a nonlinear deformation of time coordinates involving a quadrature.",
"In the context stated, this result also explicitly solves Newton’s equation for the quartic oscillator for arbitrary initial data on the real line.",
"No approximations are involved!",
"Specifically, each world line for the non-relativistic harmonic oscillator satisfying Newton’s equation of motion for all space and time in 1-1 dimensions with no constraints other than the spring restoring force satisfies $m{\\frac{d^2}{d{\\hat{t}^2}}}{\\hspace{4.0pt}}x{\\hspace{4.0pt}}({\\hat{t}})=-k_2{\\hspace{4.0pt}}x{\\hspace{4.0pt}}({\\hat{t}})^2, $ where as usual $m$ =mass, $k_2$ = spring constant, and $k_2$ = $m{\\omega }^2$ and is given by $x{\\hspace{4.0pt}}({\\hat{t}})={\\sqrt{\\frac{2E}{k_{2}}}}{\\hspace{2.0pt}}cos{\\hspace{2.0pt}}{\\omega }{\\hspace{4.0pt}}({\\hat{t}}-{\\hat{t}_{max}}),$ for -${\\infty }$ $<{\\hat{t}}$ $<{\\infty }$ and $E$ equals the total energy.",
"With the invertible map given in Part III, we shall pair each member of the set of all world lines for the harmonic oscillator with one for the quartic oscillator with the same mass and total energy obeying Newton's equation of motion: $m{\\hspace{4.0pt}}{\\frac{d^{2}}{{dt}^{2}}}{\\hspace{4.0pt}}y{\\hspace{4.0pt}}({t})=-{k_{4}}{\\hspace{4.0pt}}y^{3}{\\hspace{4.0pt}}(t).$ The invertible map matches potential energies and momenta, hence energies.",
"It is important to comment that an extremal, i.e., each solution of the associated variational problem connecting the spacetime events $(x_{a}, {\\hat{t}}_{a})$ and $(x_{b}, {\\hat{t}}_{b})$ for the harmonic oscillator ( $(y_{a}, t_{a})$ and $(y_{b}, t_{b})$ for the quartic oscillator) is a segment of the world line with the same energy.",
"Hence, a fundamental role of our energy preserving linearization map is to map the set of extremals in spacetime connecting the initial point $(x_{a}, {\\hat{t}}_{a})$ and the final point $(x_{b}, {\\hat{t}}_{b})$ for the linear (harmonic) oscillator (ho), 1-1 onto the set of extremals in spacetime connecting the initial point $(y_{a}, t_{a})$ and the final point $(y_{b}, t_{b})$ for the quartic oscillator (qo).",
"R. C. Santos, J. Santos and J.A.S.",
"Lima first demonstrated the possibility of linearization of the nonrelativisitic quartic oscillator to that of the nonrelativistic harmonic oscillator [1].",
"In [1], they approach the problem of dealing with nonlinear \"oscillators\" via the classical Hamilton-Jacobi equation in contrast to our approach via Newton's equations of motion.",
"The most important difference is we have to add a fundamental bookkeeping system to their work; namely, the signs of what we call $x$ and $y$ over a cycle have to be matched as well as requiring the harmonic oscillator time $\\hat{t}$ and the quartic oscillator time $t$ to progress together in a positive manner.",
"Further, we match momenta which coupled with matching the potential energies allows us to match total energies.",
"(In [1], they match potential energies.)",
"This is critical since we not only map solutions of the quartic oscillator onto a solution of the harmonic oscillator, but this allows us to specify a 1-1 map of extremals in spacetime onto extremals in spacetime for the two systems.",
"In Part II it is shown that the world lines of any two harmonic oscillators can be mapped 1-1 onto each other.",
"So we can select any harmonic oscillator as representative in this context.",
"The essentials of the map we shall use in Part III to linearize the quartic oscillator are illustrated in the development in Part II where for this case both the space and time parts are algebraic.",
"In Part III, the set of world lines hence the spacetime extremals connecting the spacetime events ($y_{a}$ , $t_{a}$ ) and ($y_{b}$ , $t_{b}$ ) for the quartic oscillator particle system is shown to be equivalent to the set for the linear (harmonic) oscillator connecting the spacetime events ($x_{a}$ , ${\\hat{t}}_{a}$ ) and ($x_{b}$ , ${\\hat{t}}_{b}$ ) under an energy preserving map which is a nonlinear algebraic deformation of harmonic oscillator space coordinates and a nonlinear deformation of the harmonic oscillator time coordinates.",
"The latter is given by quadrature and we shall deal with this point in Part III.",
"Part III ends with the ratio of the period of the quartic oscillator (${\\tau }_{qo}$ ) to that of the harmonic oscillator period (${\\tau }_{ho}$ ) which is shown to be inversely proportional to the fourth power of energy.",
"In Part IV, a summary of the extension of these results to the hierarchy of attractive potentials given by even powers of the space coordinate is given.",
"With respect to [1], the remarks made in Part I generalize to potentials considered in Part IV.",
"In Part V, notable work in 1 + 1-Dim is briefly described and higher dimensional applications are referenced.",
"Finally in Part VI Concluding Remarks, three other possible applications are briefly described."
],
[
"EXTREMAL MAPPING $ho{\\leftrightarrow }ho$",
"The map to a second harmonic oscillator with mass $m$ and space coordinate ${\\tilde{x}}$ is stated in two parts:"
],
[
"(A)",
"This implements the physical requirement that ${\\frac{1}{2}}k_{2}x^{2}$ (${\\hat{t}}$ ) = ${\\frac{1}{2}}{\\tilde{k}}_{2}{\\tilde{x}}^{2}(\\tilde{t})$ matching the potential energies at the two different times, coupled with matching of the signs of the space coordinates."
],
[
"(B)",
"This follows by requiring $dx({\\hat{t}})/d{\\hat{t}}=d{\\tilde{x}}({\\tilde{t}})/d{\\tilde{t}}.$ Given the matching of the potential energies, the matching of the velocities and the masses of the oscillators for all values of $k_{2}$ and ${\\tilde{k}_{2}}$ implies physically matching the momentum and the kinetic energies at the two different times, i.e.",
"$p_{ho}$ (${\\hat{t}}$ ) = ${\\tilde{p}_{ho}}$ (${\\tilde{t}}$ ), $E_{ho}$ = ${\\tilde{E}_{ho}}$ = $E$ .",
"Equation (2.2) integrates to ${\\tilde{t}}-{\\tilde{t}_{max}}={\\sqrt{\\frac{k_{2}}{\\tilde{k_{2}}}}}{\\hspace{4.0pt}}({\\hat{t}}-{\\hat{t}_{max}})={\\frac{\\omega }{\\tilde{\\omega }}}({\\hat{t}}-{\\hat{t}_{max}})$ Thus $x({\\hat{t}})={x_{max}}cos{\\omega }({\\hat{t}}-{\\hat{t}_{max}}){\\hspace{6.0pt}}{\\Leftrightarrow }{\\hspace{6.0pt}}{\\tilde{x}}({\\tilde{t}})={\\tilde{x}_{max}}cos{\\tilde{\\omega }}({\\tilde{t}}-{\\tilde{t}_{max}})$ or this map implements the correspondence between Newton's equations of motion $m{\\frac{d^{2}}{d^{2}{\\hat{t}}}}x({\\hat{t}})=-k_{2}x({\\hat{t}}){\\hspace{6.0pt}}{\\Leftrightarrow }{\\hspace{6.0pt}}m{\\frac{d^{2}}{d^{2}{\\tilde{t}}}}{\\tilde{x}}({\\tilde{t}})=-{\\tilde{k}}_{2}{\\tilde{x}}({\\tilde{t}}).$ Finally, all the world lines of the hat system correspond to all the world lines of the tilde system, so we are free to pick any harmonic oscillator as a representative.",
"We designate the hat system as our choice."
],
[
"EXTREMAL MAPPING $ho{\\leftrightarrow }qo$",
"The map to the quartic oscillator with mass $m$ and space coordinate $y$ is stated in two parts:"
],
[
"(A)",
"where $y$ is the space coordinate of the quartic oscillator and we have used the representation sgn($x$ ) = $x$ /($x^{2}$ )$^{1/2}$ and similarly for sgn($y$ ).",
"This implements the physical requirement that ${\\frac{1}{2}}k_{2}x^{2}$ (${\\hat{t}}$ ) = ${\\frac{1}{4}}k_{4}y^{4}$ ($t$ ) i.e.",
"matching the potential energies at the two different times, coupled with matching of the signs of the space coordinates."
],
[
"(B)",
"which results by requiring $dx({\\hat{t}})/d{\\hat{t}}=dy(t)/dt.$ Given the matching of the potential energies, the matching of the velocities and the masses of the oscillators for all values of $k_{2}$ and $k_{4}$ implies physically matching the momentum and the kinetic energies at the two different times, i.e.",
"$p_{ho}$ (${\\hat{t}}$ ) = $p_{qo}$ ($t$ ), $E_{ho}$ = $E_{qo}$ = $E$ .",
"It is interesting to note that (2.2) is also equivalent to requiring $dt$ /$d{\\hat{t}}$ = [($-k_{4}y^{3}$ ($t$ ))/($-k_{2}x$ (${\\hat{t}}$ ))]$^{-1}$ = the inverse of the ratio of the forces.",
"This map implements the correspondence between Newton's equations of motion $m{\\frac{d^{2}}{d^{2}{\\hat{t}}}}x({\\hat{t}})=-k_{2}x({\\hat{t}}){\\hspace{6.0pt}}{\\Leftrightarrow }{\\hspace{6.0pt}}m{\\frac{d^{2}}{d^{2}t}}y(t)=-k_{4}y^{3}(t)$ in the following straight-forward way.",
"Differentiation of (2.1) shows that (2.1) and (2.2) are self-consistent.",
"Differentiating (2.1) a second time, invoking (2.2) and assuming the validity of one side of (2.6) then yields the validity of the other side of the equivalence in (2.6).",
"If one plots the $ho$ -potential and the $qo$ -potentials vs space coordinates, then the horizontal lines in such a plot allow one to graphically read off the corresponding coordinates.",
"Because of reflection symmetry in the vertical axis, one need only plot the potential for the positive space coordinate.",
"Further, for some $y$ -coordinate near enough to zero but not zero, the $qo$ -potential is less than the $ho$ -potential for all nonzero $y$ -coordinates less than that value.",
"This feature is illustrated in Figure 1.",
"Figure: Comparison of PotentialsIn [1], they approach the problem of dealing with nonlinear “oscillators\" via the classical Hamilton-Jacobi equation in contrast to our approach via Newton's equations of motion.",
"Another way that our work differs from theirs is that we do not identify the solutions of the HJ equation, rather we only assume their existence.",
"Another difference is we have to add a fundamental bookkeeping addition to their work; namely, the signs of what we call $x$ and $y$ over a cycle have to be matched as well as requiring the harmonic oscillator time ${\\hat{t}}$ and the quartic oscillator time $t$ to progress together in a positive manner.",
"Further, we match momenta which, coupled with matching the potential energies, allows us to match total energies.",
"(In [1], they match potential energies.)",
"This is critical since we not only map solutions of the quartic oscillator onto a solution of the harmonic oscillator, this allows us to specify a 1-1 map of world lines in spacetime onto world lines in spacetime for the two systems.",
"Figure: Phase SpaceThe corresponding phase space diagrams are interesting, but very case dependent in terms of $k_{4}$ /$k_{2}$ and $E_{ho}$ .",
"The horizontal coordinate scales are set by $x_{\\max }$ = ${\\sqrt{2E/k_{2}}}$ and $y_{\\max }$ = $\\@root 4 \\of {4E/k_{4}}$ .",
"As a result $y_{\\max }$ can be less than, greater than or equal to $x_{\\max }$ .",
"The vertical scales are set by the equality of the linear momentum at the different times, i.e.",
"$m{\\frac{dy}{dt}}$ ($t$ ) = $m{\\frac{d}{d{\\hat{t}}}}x$ (${\\hat{t}}$ ) which also implies that $m{\\frac{dy}{dt}}{\\biggl {\\vert }}_{\\max }$ = $m{\\frac{d}{d{\\hat{t}}}}x{\\biggl {\\vert }}_{\\max }$ .",
"If one were to draw a phase diagram, horizontal lines yield the corresponding phase space coordinates ($x$ (${\\hat{t}}$ ), ${\\frac{dx}{d{\\hat{t}}}}$ (${\\hat{t}}$ )) $\\Leftrightarrow $ ($y$ ($t$ ), ${\\frac{dy}{dt}}$ ($t$ )).",
"Thus, while both make one complete cycle together in terms of phase coordinates, they do it in different times.",
"This point is important to emphasize; one complete cycle for the $ho$ in the phase space diagram corresponds to one complete cycle for the $qo$ in the phase space diagram, but this is done in general in different periods.",
"These features are illustrated in Figure 2 for $y_{\\max }$ $<$ $x_{\\max }$ .",
"Because of symmetry, one need only graph the first quadrant of the phase diagram.",
"Now, the nonlinear deformation of time given by (3.2) is in terms of a definite integral.",
"$(1/2){\\hspace{6.0pt}}(2k_{2}/k_{4})^{1/4}{\\hspace{6.0pt}}{\\int ^{{\\hat{t}}_{b}}_{\\hat{t}_{a}}}{\\hspace{6.0pt}}{\\frac{1}{{\\biggl {(}}(x^{2}({\\hat{t}}))^{1/4}{\\biggr {)}}}}{\\hspace{6.0pt}}d{\\hat{t}}=t_{b}-t_{a}$ Equation (3.5) is the operative quadrature that establishes the connection between the two “times\" $t$ and ${\\hat{t}}$ .",
"Mathematica [2] provides a complete program for evaluating this integral.",
"Now we turn to evaluating (3.5) for the period of the quartic oscillator ${\\tau _{qo}}$ in terms of the period ${\\tau _{ho}}$ of the harmonic oscillator.",
"It is sufficient and convenient to set ${\\hat{t}_{max}}$ = ${\\pi }/2$ in (1.2) or take the world line corresponding to $x({\\hat{t}})=x_{\\max }{\\sin }{\\omega }{\\hat{t}},$ where as before $x_{\\max }$ = ${\\sqrt{(2E/k_{2})}}$ .",
"Then (3.5) becomes for one cycle of the quartic harmonic oscillator (or linear oscillator) $1/2{\\hspace{6.0pt}}(k^{2}_{2}/k_{4}E)^{1/4}{\\hspace{6.0pt}}{\\int ^{{\\tau }_{ho}}_{0}}{\\hspace{6.0pt}}{\\frac{1}{{\\biggl {(}}({\\sin }^{2}{\\omega }{\\hat{t}})^{1/4}{\\biggr {)}}}}{\\hspace{6.0pt}}d{\\hat{t}}={\\tau }_{qo}.$ Equation (3.7) is an elliptic function of the first kind [2].",
"A simplification is possible, namely the change of variable $\\theta $ = ${\\omega }{\\hat{t}}$ .",
"Equation (3.7) then becomes $1/2{\\hspace{6.0pt}}{\\biggl {(}}k^{2}_{2}/k_{4}E{\\biggr {)}}^{1/4}{\\hspace{6.0pt}}{\\omega }^{-1}{\\hspace{6.0pt}}{\\int ^{2{\\pi }}_{0}}{\\hspace{6.0pt}}{\\frac{1}{{\\biggl {(}}({\\sin }^{2}{\\theta })^{1/4}{\\biggr {)}}}}{\\hspace{6.0pt}}d{\\theta }={\\tau }_{qo},$ where, again, $\\omega $ = 2$\\pi $ /${\\tau }_{ho}$ .",
"Equation (3.8), using symmetry, yields the period ${\\tau }_{qo}$ of the oscillator in terms of the period ${\\tau }_{ho}$ as follows ${\\tau }_{qo}/{\\tau }_{ho}=(k^{2}_{2}/k_{4}E)^{1/4}{\\hspace{6.0pt}}[{\\pi }^{-1}{\\hspace{6.0pt}}{\\int ^{{\\pi }/2}_{0}}{\\hspace{6.0pt}}({\\sin }{\\theta })^{-1/2}{\\hspace{6.0pt}}d{\\theta }],$ where using Mathematica [2] the term in square brackets equals .83.",
"It follows from (3.8) that for fixed $k_{2}$ and $k_{4}$ we have ${\\tau }_{qo}$ /${\\tau }_{ho}$ $\\approx $ $E_{^{\\mbox{.",
"}}}^{-1/4}$"
],
[
"EXTREMAL MAPPING FOR ${\\frac{1}{2n}}$ {{formula:e04520f5-38b1-412c-bcc5-4149b57377ef}} ({{formula:ec5f3e46-36e6-441b-850e-2cf67c9ff221}} ) {{formula:c61c5b6d-4411-4d9e-82c5-b6123e1041e3}} HIERARCHY",
"In this section we present a summary of the extension of these results to the hierarchy of attractive potentials given by even powers of the space coordinate paralleling that given in Part II - Part III.",
"First, we shall outline the mapping of the harmonic oscillator extremals onto the extremals of a each member of an hierarchy of attractive oscillators with coordinates $y_{2n}$ ($t$ ); $n$ = 2, 3, 4, ... characterized by even positive power law potentials.",
"(The case, $y$ $\\equiv $ $y_{4}$ which is included in the hierarchy, has been the subject of the preceding paragraph.)",
"In a straight forward manner the mappings in Part III, generalize and yield the following relationships:"
],
[
"(A)",
"which is the generalization of (3.1).",
"The generalization of (3.2) is given by:"
],
[
"(B)",
"These mappings take the space-time extremals of the linear oscillator with coordinates ($x$ , ${\\hat{t}}$ ) and map them onto the space-time extremals of the 2$n$th oscillator with coordinates ($y_{2n}$ , $t$ ).",
"A straightforward calculation yields $m{\\frac{d^{2}}{d^{2}{\\hat{t}}}}x({\\hat{t}})=-k_{2}x({\\hat{t}}{\\hspace{4.0pt}}{\\Leftrightarrow }{\\hspace{4.0pt}}m{\\frac{d^{2}}{d^{2}t}}y_{2n}(t)=-k_{2n}y_{2n}^{2n-1}(t).$ Further, as a consequence of the above, we have the following equality for the conserved total energies $E_{2}={\\frac{1}{2}}m({\\frac{dx_{2}}{d{\\tau }}})^{2}{\\hspace{4.0pt}}+{\\hspace{4.0pt}}{\\frac{1}{2}}k_{2}x^{2}_{2}=E_{2n}={\\frac{1}{2}}m({\\frac{dx_{2n}}{dt}})^{2}{\\hspace{4.0pt}}+{\\hspace{4.0pt}}{\\frac{k_{2n}}{2n}}q_{2n}^{2n}=E,$ The operative deformation of time given by (4.2) becomes in integral form $t_{b}-t_{a}={\\int ^{{\\hat{t}}_{b}}_{{\\hat{t}}_{a}}}n^{-(2n-1)/2n}(k_{2}/k_{2n})^{1/2n}(x^{2}({\\hat{t}}))^{-(n-1)/2n}d{\\hat{t}}$ This is implemented using Mathematica [2].",
"All of the analyses presented in Part III can then be extended to the members of the hierarchy."
],
[
"NOTABLE CONSERVATIVE 1 + 1-DIM WORK",
"Whittaker [3, p. 64] solved all conservative 1 + 1-Dim mechanical systems up to a quadrature by solving the energy relationship for the velocity and integrating: $t_{b}-t_{a}={\\int ^{y_{b}}_{y_{a}}}{\\pm }{\\sqrt{2m}}{\\sqrt{E-V(y^{\\prime })}}dy^{\\prime }.$ For the quartic oscillator, we can cast (5.1) into the following form $t_{b}-t_{a}={\\int ^{{\\theta }_{b}}_{{\\theta }_{a}}}{\\frac{1}{2{\\sqrt{2mE}}(sin^2{\\theta })^{1/2}}}d{\\theta },$ where ${\\sqrt{\\frac{k_4}{4E}}}(y^2)^{1/2}y=sin{\\theta }.$ We have accounted for the sign of the velocity with $cos{\\theta }$ .",
"Similar forms can be derived for the rest of the attractive hierarchy studied in this paper.",
"One of the most interesting formulations is the one found by a \"regularization\" scheme for the differential equations [4, p. 14-17] describing the negative energy 1 + 1-Dim Kepler problem.",
"Interestingly, this leads to a cycloid solution and related harmonic oscillator solution.",
"This problem has an old and distinguished past and the author has not done justice to it here.",
"This solution is not obtainable by matching of potential and kinetic energies as employed in this paper.",
"The utility of the 1 + 2-Dim and the 1 + 3-Dim versions of their regularization in the Feynman Path Integral Method is prominent in the work of [5], [6].",
"The radical solutions of 1 + 2-Dim central force problems with angular momentum as one constant of the motion and energy as the other are addressed by Whittaker [3, p. 80-81].",
"He solves them in the spirit of the 1 + 1-Dim problem cited above, but in two dimensions it requires two quadratures."
],
[
"CONCLUDING REMARKS",
"Physically what we have done with our linearization map is to view the tape of the space-time evolution of these nonlinear systems using the right optical lens to accomplish the nonlinear deformation of space and playing each frame according to a nonlinear deformation of time.",
"When we do this we see that the quartic oscillator evolves like an harmonic oscillator.",
"The mappings given in Parts III-IV provide new classes of exact solutions for nonlinear spring systems.",
"The essence of the maps presented here is that they involved two conservative mechanical systems and if it makes sense they match the potential energies and momenta of the two systems.",
"Within this context, its utility arises if solutions (a solution) of one of the systems are (is) known.",
"This means e.g., that one could linearize the hierarchy $-{\\frac{K_{2n}}{2n}}{z^{2n}}{\\vert _{n>1}}{\\hspace{4.0pt}}to-{\\frac{K}{2}}{z^{2}},$ or one could treat each motion of a quartic oscillator bouncing off an infinite barrier as a sequence of maps to a harmonic oscillator capturing the discontinuous velocity at the barrier, or one could map the radial equation in the procession of Mercury onto that of the inverse law force (4), [7, p. 194].",
"The latter involves extracting the zeros of polynomials cubic in ${\\mu }={\\frac{1}{{r}}}$ for the algebraic part of the map.",
"This is a nonlinear problem to nonlinear problem in the context of boundstates.",
"Special thanks go to my Department of Mathematics colleague Robert Varley.",
"He spent enumerable hours over a four year period of time discussing this work with me.",
"His comments, questions and posing of challenging related problems helped to clarify for me many aspects of this work.",
"The author wishes to aknowledge the referee's role in particular, in directing him to the work in [5] and [6].",
"The author wishes to thank Professor Howard Lee for insightful discussions and his constant encouragement.",
"The idea to emphasize the quartic oscillator was his."
],
[
"References",
"[1] R. C. Santos, J. Santos and J.",
"A. S. Lima, “Hamilton-Jacobi Approach for Power-Law Potentials\", Braz.",
"J. Phys.",
"36 04.A, (2006) pp.",
"1257-1261.",
"[2] Stephen Wolfram, Mathematica Version 7.0.",
"[3] E. T. Whittaker, A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, 3rd ed.",
"(University of Cambridge Press, 1927).",
"[4] E. L. Stiefel and G. Scheifele, Linear and Regular Celestial Mechanics, (Berlin/Heidelberg/New York 1971.",
"Springer-Verlag 1971) Chapter 1.",
"[5] P. Kustaanheimo and E. Stiefel, “Perturbation Theory of Kepler Motion Based on Spinor Regularization\", (J. Reine Angew.",
"Math 218, 204-219, 1965).",
"[6] I.H.",
"Duru and H. Kleinert, “Quantum Mechanics of H-Atom from Path Integrals\", (Fortschr.",
"Physik 30, 401-435, 1982).",
"[7] James B. Hartle, Gravity/An Introduction to Einstein's General Relativity, (Addison Wesley 2003)."
]
] | 1204.0765 |
[
[
"A proof of the parabolic Schauder estimates using Trudinger's method and\n the mean value property of the heat equation"
],
[
"Abstract One method available to prove the Schauder estimates is Neil Trudinger's method of mollification.",
"In the case of second order elliptic equations, the method requires little more than mollification and the solid mean value inequality for subharmonic functions.",
"Our goal in this article is show how the mean value property of subsolutions of the heat equation can be used in a similar fashion as the solid mean value inequality for subharmonic functions in Trudinger's original elliptic treatment, providing a relatively simple derivation of the interior Schauder estimate for second order parabolic equations."
],
[
"Introduction",
"One method available to prove the Schauder estimates is Neil Trudinger's method of mollification ().",
"In the case of second order elliptic equations, the method requires little more than mollification and the solid mean value inequality for subharmonic functions.",
"The method was lated adapted to the parabolic setting by Xu-Jia Wang in , however in that presentation Wang uses an auxiliary estimate coming from the fundamental solution of the heat equation and the mean value of property of subsolutions of the heat equations is not used.",
"Our goal in this article is show how the mean value property of subsolutions of the heat equation can be used in a similar fashion as the solid mean value inequality for subharmonic functions in Trudinger's original elliptic treatment, providing a relatively simple derivation of the interior Schauder estimate for second order parabolic equations."
],
[
"Preliminaries",
"For an open subset $U \\in \\mathbb {R}^d$ , we denote the corresponding open parabolic domain $U \\times (0, T) \\subset \\mathbb {R}^{d+1}$ by $U_T$ .",
"We denote a (backwards) parabolic cylinder by $Q_R = B_R \\times (t - R^2, t)$ .",
"We often notate a point $(x,t) \\in U_T$ by $X$ .",
"A (second order parabolic) mollifier is a fixed smooth function $\\rho \\in C_c^{ \\infty }( \\mathbb {R}^{d+1})$ with $\\iint _{ \\mathbb {R}^{d+1} } \\rho \\, dx \\, dt = 1$ .",
"For $\\tau > 0$ we define the scaled mollifier $\\rho _{\\tau }(x,t) := \\frac{1}{\\tau ^{d+2}} \\rho \\left(\\frac{x}{\\tau },\\frac{t}{\\tau ^{2}}\\right).$ Let $U \\in \\mathbb {R}^{d+1}$ and $u \\in L^1_{\\text{loc}}(U)$ .",
"For $0 < \\tau < d(X, \\partial P)$ , the mollification of $u$ is given by $u_{\\tau }(x,t) := \\frac{1}{\\tau ^{d+2}} \\iint \\rho \\left(\\frac{x-y}{\\tau },\\frac{t-s}{\\tau ^{2}} \\right) u(y,s) \\, dy \\, ds$ and satisifes spt $u_{\\tau } \\subset U_{\\tau }$ , where $U_{\\tau } = \\lbrace X \\in U : d(X, \\partial U) > \\tau \\rbrace $ .",
"The parabolic distance between two points $X = (x,t)$ and $Y = (y,s)$ is defined to be $ d(X, Y) := \\max \\lbrace \\vert x-y\\vert , \\vert t-s\\vert ^{1/2} \\rbrace .",
"$ We use both $\\sup $ and $\\vert \\cdot \\vert _0$ to denote the supremum of a function.",
"The Hölder seminorm is defined by $ [ u ]_{\\alpha ; U_T} : = \\sup _{X \\ne Y \\in U_T} \\frac{ \\vert u(X) - u(Y) \\vert }{ d(X,Y)^{\\alpha } } $ and the Hölder norms by $&\\vert u \\vert _{2, 1; \\, U_T } := \\sum _{ k=0 }^{2} \\vert \\partial _x^k u \\vert _{ 0; \\, U_T } + \\vert \\partial _t u \\vert _{ 0; \\, U_T } \\\\&\\vert u \\vert _{2, 1, \\alpha ; \\, U_T } := \\vert u \\vert _{2, 1; \\, U_T } + [ \\partial _x^{2} u ]_{ \\alpha ; \\, U_T } + [ \\partial _t u ]_{ \\alpha ; \\, U_T } \\\\$ The set of functions $\\lbrace u \\in C^{2, 1}(U_T) : [u]_{ 2, 1, \\alpha ; \\, U_T } < \\infty \\rbrace $ endowed with the norm $\\vert u \\vert _{ 2m, 1, \\alpha ; \\, U_T }$ is called a Hölder space.",
"Written out in full the norm is $\\vert u \\vert _{ 2, 1, \\alpha ; \\, U_T } := \\sum _{ k=0 }^{2} \\vert \\partial _x^k u \\vert _{ 0; \\, U_T } + \\vert \\partial _t u \\vert _{ 0; \\, U_T }+ [ \\partial _x^{2} u ]_{ \\alpha ; \\, U_T } + [ \\partial _t u ]_{ \\alpha ; \\, U_T }.$ We sometimes use $ \\partial ^{2,1} := \\partial _x^2 + \\partial _t, $ e.g, $ [ \\partial ^{2,1}u ]_{\\alpha ; U_T} = [ \\partial _x^2 u ]_{\\alpha ; U_T} + [ \\partial _t u ]_{\\alpha ; U_T}, $ to compress notation a little.",
"We begin by establishing some very basic estimates that will be use throughout.",
"Proposition 2.1 We have $u_{\\tau } \\in C_c^{\\infty }$ .",
"Proposition 2.2 Let $u \\in L^1_{\\text{loc}}(U)$ .",
"The following estimates hold: $&\\vert u_{ \\tau } \\vert _{0; \\, U_{\\tau } } \\le \\vert u\\vert _{ 0; \\, U_{\\tau } } \\\\&\\vert \\partial _x^i \\partial _t^j u_{\\tau }(x,t) \\vert _{ 0; \\, U_{\\tau } } \\le C \\tau ^{-i-2j} \\vert u\\vert _{0; \\, U_{\\tau } }.",
"$ To prove (REF ), we have $u_{ \\tau }(x,t) &= \\frac{1}{ \\tau ^{d+2} } \\iint \\rho \\left( \\frac{x-y}{\\tau },\\frac{t-s}{ \\tau ^{2}} \\right) u(y,s) \\, dy \\, ds \\\\&\\le \\vert u \\vert _{0; \\, U_{\\tau } } \\cdot \\frac{1}{ \\tau ^{d+2} } \\iint \\rho \\left( \\frac{x-y}{\\tau },\\frac{t-s}{ \\tau ^{2}} \\right) \\, dy \\, ds \\\\&= \\vert u \\vert _{0; \\, U_{\\tau } }.$ And for (): $\\partial _x^i \\partial _t^j u_{\\tau }(x,t) &= \\frac{1}{ \\tau ^{d+2} } \\iint _{U_{\\tau }} \\partial _x^i \\partial _t^j \\rho \\left( \\frac{x-y}{\\tau },\\frac{t-s}{ \\tau ^{2}} \\right) u(y,s) \\, dy \\, ds \\\\&\\le C \\tau ^{-i-2j}\\vert u \\vert _{ 0; \\, U_{\\tau } }.$ Proposition 2.3 Let $u \\in C^{\\alpha }_{\\text{loc}}(U)$ .",
"The following estimates hold: $&\\vert u_{\\tau }(x,t) - u(x,t) \\vert _{ 0; \\, U_{\\tau } }\\le \\tau ^{\\alpha } [u]_{\\alpha ; \\, U_{\\tau } } \\\\&\\vert \\partial _x^i \\partial _t^j u_{\\tau }(x,t) \\vert _{ 0; \\, U_{\\tau } } \\le C \\tau ^{ \\alpha -i-2j } [u]_{\\alpha ; \\, U_{\\tau } }.",
"$ For estimate (REF ) we have $u_{\\tau }(x,t) - u(x,t) &= \\frac{1}{ \\tau ^{d+2} } \\iint \\rho \\left( \\frac{x-y}{\\tau },\\frac{t-s}{ \\tau ^{2}} \\right) (u(y,s) - u(x,t)) \\, dy \\, ds \\\\&\\le \\operatorname{osc}_{U_{\\tau } } u \\\\&\\le \\tau ^{\\alpha } [u]_{\\alpha ; U_{\\tau } }.$ To prove the second estimate we have $\\partial _x^i \\partial _t^j u_{\\tau }(x,t) &= \\frac{1}{ \\tau ^{d+2} } \\iint \\partial _x^i \\partial _t^j \\rho \\left( \\frac{x-y}{\\tau },\\frac{t-s}{ \\tau ^{2}} \\right) u(y,s) \\, dy \\, ds \\\\&= \\frac{1}{ \\tau ^{d+2} } \\iint \\partial _x^i \\partial _t^j \\rho \\left( \\frac{x-y}{\\tau },\\frac{t-s}{ \\tau ^{2}} \\right) (u(y,s) - u(x,t)) \\, dy \\, ds \\\\&\\quad + \\frac{u(x,t)}{ \\tau ^{d+2} } \\iint \\partial _x^i \\partial _t^j \\rho \\left( \\frac{x-y}{\\tau },\\frac{t-s}{ \\tau ^{2}} \\right) \\, dy \\, ds.$ The mollifier $\\rho $ is has compact support on $U_{\\tau }$ and so the last term vanishes by the Divergence Theorem.",
"Continuing, we have $\\partial _x^i \\partial _t^j u_{\\tau }(x,t) &= \\frac{1}{ \\tau ^{d+2} } \\iint _{U_{\\tau }} \\partial _x^i \\partial _t^j \\rho \\left( \\frac{x-y}{\\tau },\\frac{t-s}{ \\tau ^{2}} \\right) (u(y,s) - u(x,t)) \\, dy \\, ds \\\\&\\le C \\tau ^{-i-2j}\\operatorname{osc}_{Q_\\tau } u \\\\&\\le C \\tau ^{ \\alpha -i-2j } [u]_{ \\alpha ; \\, U_{\\tau } }.$"
],
[
"The interior elliptic Schauder estimate for functions of compact support",
"To motivate things a little for the parabolic setting, we first briefly show how Trudinger's method works in the elliptic realm by treating the Poisson equation.",
"The crucial ingredient in Trudinger's method is the following norm equivalence: Lemma 3.1 Let $U \\subset \\mathbb {R}^d$ be an open bounded domain and $u \\in C^{ \\alpha }(U)$ , where $\\alpha \\in (0,1)$ .",
"Let $R > 0$ and $\\tau _0$ be small constants, both of which will be fixed in the course of the proof.",
"There exists a constant $C = C(d, \\alpha )$ such that the norm equivalence $\\frac{1}{C}[u]_{ \\alpha ; \\, B_{R} } \\le \\sup _{ 0 < \\tau < \\tau _0 } \\tau ^{1-\\alpha } \\vert \\partial _x u_{\\tau }\\vert _{ 0; \\, B_{ R } } \\le C[u]_{ \\alpha ; \\, B_{R} }.$ is valid.",
"The inequality on the right follows directly from equation () (the elliptic version) by choosing the appropriate values for the indices $i$ : choosing $i = 1$ (there is no $j$ in the elliptic mollifier) gives $\\vert \\partial _x u_{\\tau }\\vert _{ 0; \\, B_{ R } } \\le C \\tau ^{ \\alpha - 1 } [u]_{\\alpha ; \\, B_{ R } }.$ The first inequality requires a little more work.",
"Let $x, y \\in U$ satisfy $\\vert x-y\\vert < d_x/2$ and let $\\tau $ satisfy $0 < \\tau <\\tau _0 < d_x/2$ , where $\\tau _0$ will be fixed shortly.",
"For $ \\vert x - y \\vert < R = d_x/2 $ , by the triangle inequality $\\vert u(x) - u(y)\\vert &\\le \\vert u(x) - u_{\\tau }(x)\\vert + \\vert u_{\\tau }(x) - u_{\\tau }(y) \\vert + \\vert u_{\\tau }(y) - u(y) \\vert \\\\&\\le 2\\tau ^{\\alpha }[u]_{ \\alpha ; \\, B_{ R }} + \\vert \\partial _x u_{\\tau } \\vert _{ 0; \\, B_{ R } }\\vert x-y\\vert .$ Set $\\tau = \\epsilon \\vert x-y\\vert $ , where $\\epsilon < 1/2$ .",
"Factoring out and dividing by $\\vert x-y\\vert ^{\\alpha }$ we find $\\vert u(x) - u(y) \\vert \\le \\vert x-y\\vert ^{\\alpha } \\left( 2\\epsilon ^{\\alpha }[u]_{\\alpha ; B_R} + \\epsilon ^{\\alpha - 1}\\tau ^{1-\\alpha } \\vert \\partial _x u_{\\tau } \\vert _{0; \\, B_R} \\right).$ Choose $\\epsilon = 4^{-1/\\alpha }$ then set $\\tau _0 = 4^{-1/\\alpha } .",
"\\, d_x/2$ .",
"Taking the supremum over $\\tau \\in (0, \\tau _0)$ completes the proof.",
"We now derive the interior Schauder estimate for Poisson's equation.",
"For simplicity, we only consider solutions with compact support in a ball $B_R$ .",
"The method extends in the usual way to more general equations and domains by using cutoff functions and Simon's absorption lemma (see, for example, ).",
"Suppose that $u \\in C_c^{2, \\alpha }(B_R)$ solves $-a^{ij}(x)\\partial _{ij}u(x) = f(x),$ where we assume $a^{ij}, f \\in C^{\\alpha }(B_R)$ , and there are constants $\\lambda >0$ and $\\Lambda < \\infty $ such that $\\lambda \\vert \\xi \\vert ^2 \\le a^{ij}\\xi _i\\xi _j \\le \\Lambda \\vert \\xi \\vert ^2$ .",
"We proceed by the method of freezing coefficients, and accordingly fix a point $x_0 \\in \\mathbb {R}^d$ a rewrite the above equation equation as $-a^{ij}(x_0)\\partial _{ij}u(x) - f (x_0)&= (a^{ij}(x) - a^{ij}(x_0) )\\partial _{ij}u(x) + f(x) - f(x_0)\\\\&:= g(x) $ After a coordinate transformation, we can assume without loss of generality that $a^{ij}(x_0) = \\delta ^{ij}$ , that is we can assume (REF ) is the Poisson equation.",
"Mollify equation (REF ) to get $-\\Delta u_{\\tau } - f(x_0) = g_{\\tau }$ and then differentiate thrice with respect to $x$ to obtain $-\\Delta \\partial _x^3 u_{\\tau } = \\partial _x^3 g_{\\tau }.$ Using inequality () we can estimate $\\vert \\partial _x^3 g_{\\tau } \\vert _{0; \\, B_R } &\\le C(d) \\tau ^{-3} \\vert g \\vert _{0; \\, B_R } \\\\&\\le C(d) \\tau ^{-3} R^{\\alpha } \\big ( [ a ]_{\\alpha ; \\, B_R } \\vert \\partial _x^2 u \\vert _{0; \\, B_R } + [ f ]_{\\alpha ; \\, B_R } \\big ).$ Recall the solid mean value inequality for subharmonic functions: If $v$ solves $-\\Delta v(x) \\le 0$ on a ball $B_R(x) \\subset \\mathbb {R}^d$ , then $v$ satisfies $ v(x) \\le \\frac{ C(d) }{ R^n } \\int _{B_R} v(y) \\, dy.",
"$ To apply this inequality to our situation, noting $\\Delta \\vert x \\vert ^2 = 2d$ , we have $ -\\Delta \\left( \\partial _x^3 u_{\\tau } + \\frac{ \\vert \\partial _x^3 g_{\\tau } \\vert _{0; \\, B_R } \\vert x \\vert ^2 }{ 2n } \\right) = -\\Delta \\partial _x^3 u_{\\tau } - \\vert \\partial _x^3 g_{\\tau } \\vert _{0; \\, B_R} \\le 0.",
"$ Thus the function $\\partial _x^3 u_{\\tau } + \\vert \\partial _x^3 g_{\\tau } \\vert _{0; \\, B_R } \\vert x \\vert ^2 / (2d)$ is subharmonic and applying the mean value inequality and estimating we obtain $\\vert \\partial _x^3 u_{\\tau }(x_0) \\vert &\\le C(d) \\left( \\frac{1}{R^d} \\left|\\int _{B_R} \\partial _y^3 u_{\\tau }(y) \\, dy \\right|+ R^2 \\vert \\partial _x^3 g_{\\tau } \\vert _{0; \\, B_R } \\right) \\\\&\\le C(d) \\left( \\frac{1}{R} \\operatorname{osc}_{ B_R } \\partial _x^2 u_{\\tau }(x) + \\tau ^{-3} R^{2+\\alpha } \\big ( [ a ]_{\\alpha ; B_{R} } \\vert \\partial _x^2 u \\vert _{0; \\, B_{R} } + [ f ]_{\\alpha ; \\, B_{R} } \\big ) \\right) \\\\&\\le C(d) \\Big ( R^{ \\alpha - 1 } [ \\partial _x^2 u ]_{ \\alpha ; \\, B_R } + \\tau ^{-3} R^{2+\\alpha } \\big ( [ a ]_{\\alpha ; B_{R} } \\vert \\partial _x^2 u \\vert _{0; \\, B_{R} } + [ f ]_{\\alpha ; \\, B_{R} } \\big ) \\Big ).$ We set $R = N\\tau $ and return to the original coordinates to find $ \\tau ^{1-\\alpha }\\vert \\partial _x^3 u_{\\tau }(x_0) \\vert \\le C(d, \\lambda , \\Lambda ) \\left( N^{ \\alpha - 1 } [ \\partial _x^2 u ]_{ \\alpha ; \\, B_R } + N^{2+\\alpha } \\big ( [ a ]_{\\alpha ; B_{R} } \\vert \\partial _x^2 u \\vert _{0; \\, B_{R} } + [ f ]_{\\alpha ; \\, B_{R} } \\big ) \\right), $ then taking the supremum over $\\tau $ and using the norm equivalence we obtain $ [ \\partial _x^2 u ]_{\\alpha ; \\, B_R } \\le C(d, \\lambda , \\Lambda , \\alpha ) \\left( N^{ \\alpha - 1 } [ \\partial _x^2 u ]_{ \\alpha ; \\, B_R } + N^{2+\\alpha } \\big ( [ a ]_{\\alpha ; B_R } \\vert \\partial _x^2 u \\vert _{0; \\, B_R } + [ f ]_{\\alpha ; \\, B_R } \\big ) \\right).",
"$ Choosing $N$ sufficiently large and using the Hölder space interpolation inequality on the right gives the desired estimate, namely $ [ \\partial _x^2 u ]_{\\alpha ; \\, B_R} \\le C \\big ([ f ]_{\\alpha ; \\, B_R } + \\vert u \\vert _{0; \\, B_R } \\big ), $ where $C$ depends on $d, \\lambda , \\Lambda $ , and $\\alpha $ ."
],
[
"The interior parabolic Schauder estimate for functions of compact support",
"Having given a feel for Trudinger's method, we move on to use this method to derive the Schauder estimates for second order parabolic equations.",
"The crucial equivalence of norms lemma in the parabolic setting is the following: Lemma 4.1 Let $U \\subset \\mathbb {R}^{d+1}$ be an open bounded domain and $u \\in C^{ \\alpha }(U)$ , where $\\alpha \\in (0,1)$ , and $R > 0$ and $\\tau _0$ be small constants, both of which will be fixed in the course of the proof.",
"There exists a constant $C = C(d, \\alpha )$ such that the norm equivalence $\\frac{1}{C}[u]_{ \\alpha ; \\, Q_{R} } \\le \\sup _{0 < \\tau < \\tau _0 } \\left\\lbrace \\tau ^{1-\\alpha } \\vert \\partial _x u_{\\tau }\\vert _{ 0; \\, Q_{ R } } + \\tau ^{2-\\alpha }\\vert \\partial _t u_{\\tau }\\vert _{0; Q_{R} } \\right\\rbrace \\le C[u]_{ \\alpha ; \\, Q_{R} }.$ is valid.",
"The second inequality follows directly from equation () by choosing the appropriate values for the indices $i$ and $j$ .",
"To prove the spatial part of the second inequality, choosing $i = 1$ and $j = 0$ in estimate () gives $\\vert \\partial _x u_{\\tau }(x,t) \\vert _{ 0; \\, Q_{ R } } \\le C \\tau ^{ \\alpha - 1 } [u]_{\\alpha ; \\, Q_{ R } }.$ The temporal estimate follows similarly.",
"Moving on to the proof of the first inequality, let $X,Y \\in U_T $ be points such that $d(X,Y) < d_X/2$ .",
"To simplify notation a little, set $R = d_X/2$ ; thus $d(X,Y) \\in Q_R \\subset U_T$ .",
"Let $\\tau $ satisfy $0 < \\tau <\\tau _0 < R$ , where $\\tau _0$ will be fixed shortly.",
"For $d(X,Y) < R$ , $\\vert u(X) - u(Y)\\vert &\\le \\vert u(X) - u_{\\tau }(X)\\vert + \\vert u_{\\tau }(Y) - u(Y) \\vert + \\vert u_{\\tau }(x,t) - u_{\\tau }(y,t) \\vert + \\vert u_{\\tau }(y,t) - u_{\\tau }(y,s) \\vert \\\\&\\le 2\\tau ^{\\alpha }[u]_{ \\alpha ; \\, Q_{ R }} + \\vert x-y\\vert \\vert \\partial _x u_{\\tau } \\vert _{ 0; \\, Q_{ R } } + \\vert t-s\\vert \\vert \\partial _t u_{\\tau } \\vert _{ 0; \\, Q_{ R } }.$ Set $\\tau = \\epsilon d(X,Y)$ , where $\\epsilon < 1/2$ .",
"Factoring out $d(X,Y)^{\\alpha }$ we have $\\vert u(X) - u(Y)\\vert \\le d(X,Y)^{\\alpha } \\left( 2\\epsilon ^{\\alpha }[u]_{\\alpha ; \\, Q_{ R }} + \\epsilon ^{\\alpha -1}\\tau ^{1-\\alpha }\\vert \\partial _x u_{\\tau } \\vert _{ 0; \\, Q_{ R } } + \\epsilon ^{\\alpha -2}\\tau ^{2-\\alpha }\\vert \\partial _t u_{\\tau } \\vert _{ 0; \\, Q_{ R } } \\right).$ Choose $\\epsilon = 4^{-1/\\alpha }$ then set $\\tau _0 = 4^{-1/\\alpha } \\cdot \\, d_X/2$ .",
"Taking the supremum over $\\tau \\in (0, \\tau _0)$ completes the proof.",
"We now proceed similarly to Poisson's equation to derive the Schauder estimate for the nonhomongeneous heat equation.",
"Again for simplicity, we only condisider solutions with compact support in a parabolic cylinder $Q_R$ as the more general estimates follow from this case using cutoff functions and Simon's adsorption lemma.",
"Suppose that $u \\in C_c^{2, \\alpha }(Q_R)$ solves $ \\partial _t u(x,t) - a^{ij}(x,t)\\partial _{ij}u(x,t) = f(x,t), $ where we assume $a^{ij}, f \\in C^{\\alpha }(Q_R)$ , and there are constants $\\lambda >0$ and $\\Lambda < \\infty $ such that $\\lambda \\vert \\xi \\vert ^2 \\le a^{ij}\\xi _i\\xi _j \\le \\Lambda \\vert \\xi \\vert ^2$ .",
"Again we freeze coefficients at a point $(x_0, t_0) \\in Q_R$ and perform a coordinate transformation if necessary to get $\\partial _t u(x,t) - \\Delta u(x,t) - f(x_0, t_0) &= ((a^{ij}(x,t) - a^{ij}(x_0,t_0) ) \\partial _{ij}u(x,t) + f(x,t) - f(x_0,t_0) \\\\&:= g(x,t),$ and then mollify the equation to obtain $\\partial _t u_{\\tau }(x,t) - \\Delta u_{\\tau }(x,t) = g_{\\tau }(x,t).",
"$ Given the form of the norm equivalence, the desired Schauder estimate will follow if we can establish the estimates (for the spatial component of the Schauder estimate) $\\vert \\partial _x^3 u_\\tau (x, t) \\vert &\\le C \\left( \\frac{1}{R} \\operatorname{osc}_{ Q_R } \\partial _x^2 u + R^2\\vert \\partial _x^3 g_{\\tau } \\vert _{0; \\, Q_R} \\right) \\\\\\vert \\partial _t\\partial _x^2 u_\\tau (x, t) \\vert &\\le C \\left( \\frac{1}{R^2} \\operatorname{osc}_{ Q_R } \\partial _x^2 u + R^2\\vert \\partial _t\\partial _x^2 g_{\\tau } \\vert _{0; \\, Q_R} \\right), $ and for the temporal part $\\vert \\partial _x\\partial _t u_\\tau (x, t) \\vert &\\le C \\left( \\frac{1}{R} \\operatorname{osc}_{ Q_R } \\partial _t u + R^2\\vert \\partial _x\\partial _t g_{\\tau } \\vert _{0; \\, Q_R} \\right) \\\\\\vert \\partial _t^2 u_\\tau (x, t) \\vert &\\le C \\left( \\frac{1}{R^2} \\operatorname{osc}_{ Q_R } \\partial _t u + R^2\\vert \\partial _t^2 g_{\\tau } \\vert _{0; \\, Q_R} \\right).",
"$ We show how to obtain estimate (REF ) as the other three estimates follow in the same way.",
"Recall the mean value property for subsolutions of the heat equation: If $v$ is a subsolution to the heat equation on $\\mathbb {R}^{d+1}$ , that is if $v$ solves $\\partial _v - \\Delta v \\le 0$ , then $v$ satisfies $v(x, t) \\le \\frac{1}{4r^n}\\iint _{E(x,t;\\,r)} v(y,s) \\frac{ \\vert x - y \\vert ^2 }{ (t-s)^2 } \\, dy \\, ds$ for each $E(x,t; r) \\subset \\mathbb {R}^{d+1}$ , where the heat ball $E(x,t;r)$ is the set given by $E(x,t;r) = \\lbrace (y,s) \\in \\mathbb {R}^{d+1} : \\vert x-y\\vert ^2 \\le \\sqrt{ -2\\pi s \\log [r^2/(-4\\pi s)] }, s \\in (t - r^2/(4\\pi s), t) \\rbrace $ .",
"The radius of the heat ball $\\sqrt{ -2\\pi s \\log [r^2/(-4\\pi s)] }$ is often denoted by $R_r(s)$ .",
"We refer the interested reader to and for more information on the mean value property and heat balls.",
"Let us now show (REF ): Differentiate (REF ) thrice in space.",
"Since $\\vert \\partial _x^3 g_{\\tau } \\vert _{0; \\, E}\\vert x\\vert ^2/(2n)$ is independent of time we see $\\partial _t \\left( \\partial _x^3 u_{\\tau } + \\vert \\partial _x^3 g_{\\tau } \\vert _{0; \\, E}\\frac{ \\vert x\\vert ^2 }{2n} \\right) - \\Delta \\left( \\partial _x^3 u_{\\tau } + \\vert \\partial _x^3 g_{\\tau } \\vert _{0; \\, E}\\frac{ \\vert x\\vert ^2 }{2n} \\right) &= \\partial _t(\\partial _x^3 u_{\\tau }) - \\Delta (\\partial ^3_x u_{\\tau }) - \\vert \\partial _x^3 g_{\\tau } \\vert _{0; \\, E} \\\\&= \\partial _x^3 g_{\\tau } - \\vert \\partial _x^3 g_{\\tau } \\vert _{0; \\, E} \\le 0,$ and hence the function $\\partial _x^3 u_{\\tau } + \\vert \\partial _x^3 g_{\\tau } \\vert _{0; \\, E(x_0, t_0; r) } \\vert x \\vert ^2 / (2n)$ is subsolution of the heat equation.",
"From the mean value property of subsolutions we have $\\partial _x^3 u_{\\tau }(x,t) \\le \\frac{1}{4r^n} \\iint _{E(x,t;r)} \\left( \\partial _y^3 u_{\\tau }(y,s) + \\vert \\partial _x^3 g_{\\tau }\\vert _{0} \\frac{ \\vert y \\vert ^2 }{ 2n } \\right) \\frac{ \\vert x - y \\vert ^2 }{ \\vert t - s \\vert ^2 } \\, dy ds.",
"$ By translating coordinates we can assume without loss of generality that $(x,t) = (0,0)$ .",
"Establishing the required estimates involves evaluation of an integral of the form $ \\frac{1}{r^n}\\int _{ \\frac{-r^2}{4\\pi } } \\frac{ R_r(s)^{\\alpha } }{ s^{\\beta } } \\, ds, $ where $\\alpha $ and $\\beta $ are given integers.",
"The constants can be computed explicitly, however we are only interested in the scaling behaviour with respect to the radius $r$ (and that the integral is finite).",
"We compute $\\frac{1}{r^n}\\int _{ \\frac{-r^2}{4\\pi } }^0 \\frac{ R_r(s)^{\\alpha } }{ s^{\\beta } } \\, ds &= \\frac{1}{r^n}\\int _{ \\frac{-r^2}{4\\pi } }^0 \\frac{ \\big ( -2ns \\log [ r^2/ (-4\\pi s ) ] \\big )^{ \\alpha /2} }{ -s^{\\beta } } \\\\&= C(n, \\alpha , \\beta ) r^{-n+\\alpha -2\\beta +2} \\int _{ \\frac{1}{4\\pi } }^0 t^{\\alpha /2 - \\beta } \\big ( \\log ( 4 \\pi t) \\big )^{\\alpha /2} \\,dt \\\\&= C(n, \\alpha , \\beta ) r^{-n+\\alpha -2\\beta +2} \\int _0^{\\infty } s^{\\alpha /2} e^{-\\alpha /2-\\beta +1} \\, ds \\\\&= C(n, \\alpha , \\beta ) r^{-n+\\alpha -2\\beta +2},$ where the last lines follows from properties of the Gamma function.",
"We point that the integral is independent of $\\alpha $ and $\\beta $ when $\\alpha =\\beta =2$ .",
"Returning to (REF ), we have $\\partial _x^3 u_{\\tau }(x, t) \\le 4r^{-n} \\iint _E \\partial _y^3 u_{\\tau }(y,s) \\frac{ \\vert y\\vert ^2 }{ s^2 } \\, dy ds + \\frac{4r^{-n}}{2n}\\vert \\partial _x^3 g_{ \\tau } \\vert _{0; \\, E} \\iint _E \\frac{ \\vert y\\vert ^4 }{ s^2 } \\, dy ds.",
"$ We estimate the first term on the right by $4r^{-n} \\iint _E \\partial _y^3 u_{\\tau }(y,s) \\frac{ \\vert y\\vert ^2 }{ s^2 } \\, dy ds &\\le Cr^{-n} \\int _{ \\frac{ -r^2 }{4\\pi } }^0 \\frac{ R_r(s)^2 }{s^2} \\left( \\int _{B_{R_r(s)}} \\partial _y^3 u_{\\tau } \\, dy \\right) ds \\\\&\\le Cr^{-n} \\int _{ \\frac{ -r^2 }{4\\pi } }^0 \\frac{ R_r(s)^2 }{s^2} \\left( \\int _{\\partial B_{R_r(s)}} \\operatorname{osc}\\partial _y^2 u \\, dy \\right) ds \\\\&\\le Cr^{-n} \\operatorname{osc}_E \\partial _x^2 u \\int _{ \\frac{ -r^2 }{4\\pi } }^0 \\frac{ R_r(s)^{n+1} }{s^2} ds\\\\&\\le \\frac{ C(d) }{ r } \\operatorname{osc}_E \\partial _x^2 u.$ The second term on the right of (REF ) can be estimated more simply to give $ 4\\vert \\partial _x^3 g_{ \\tau } \\vert _{0; \\, E} r^{-n} \\iint _E \\frac{ \\vert y\\vert ^4 }{ s^2 } \\, dy ds \\le C(d) r^2 \\vert \\partial _x^3 g_{ \\tau } \\vert _{0; \\, E}.",
"$ The other three estimates () - () follow in a similar way.",
"For example, to derive estimate (), we differentiate (REF ) twice in space and once in time.",
"Using the mean value property of subsolutions of the heat equation we obtain $\\partial _t \\partial _x^2 u_{\\tau }(x, t) \\le 4r^{-n} \\iint _E \\partial _s \\partial _y^2 u_{\\tau }(y,s) \\frac{ \\vert y\\vert ^2 }{ s^2 } \\, dy ds + \\frac{4r^{-n}}{2n}\\vert \\partial _t \\partial _x^2 g_{ \\tau } \\vert _{0; \\, E} \\iint _E \\frac{ \\vert y\\vert ^4 }{ s^2 } \\, dy ds.",
"$ The first part of the estimate follows by integrating by parts in time: $4r^{-n} \\iint _E \\partial _t\\partial _y^2 u_{\\tau }(y,s) \\frac{ \\vert y\\vert ^2 }{ s^2 } \\, dy ds &\\le Cr^{-n} \\iint _E \\operatorname{osc}\\partial _y^2 u \\frac{ \\vert y\\vert ^2 }{ s^3 } \\\\&\\le \\frac{ C(d) }{ r^2 } \\operatorname{osc}_E \\partial _x^2 u,$ and the second term on right can be estimate in the same way as before to give ().",
"The derivation of the parabolic Schauder estimate now continues in the the same way as for the Poisson equation, using the estimates (REF ) - (), the equivalence of norms lemma and the Hölder space interpolation inequality.",
"After completing these calculations we obtain obtain the desired parabolic Schauder estimate: $[\\partial ^{2,1} u ]_{\\alpha ; \\, Q_R } \\le C \\big ( [ f ]_{\\alpha ; \\, Q_R }+ \\vert u \\vert _{0; \\, Q_R } \\big ), $ where $C$ depends on $d, \\lambda , \\Lambda $ , and $\\alpha $ .",
"Eckbook author=Ecker, Klaus, label=Eck title=Regularity theory for mean curvature flow, series=Progress in Nonlinear Differential Equations and their Applications, 57, publisher=Birkhäuser Boston Inc., place=Boston, MA, date=2004, pages=xiv+165, Evabook author=Evans, Lawrence C., label=Eva, title=Partial differential equations, series=Graduate Studies in Mathematics, volume=19, publisher=American Mathematical Society, place=Providence, RI, date=1998, pages=xviii+662, Simarticle author=Simon, Leon, title=Schauder estimates by scaling, label=Sim journal=Calc.",
"Var.",
"Partial Differential Equations, volume=5, date=1997, number=5, pages=391–407, Truarticle author=Trudinger, Neil S., label=Tru title=A new approach to the Schauder estimates for linear elliptic equations, conference= title=, address=North Ryde, date=1986, , book= series=Proc.",
"Centre Math.",
"Anal.",
"Austral.",
"Nat.",
"Univ., volume=14, publisher=Austral.",
"Nat.",
"Univ., place=Canberra, , date=1986, pages=52–59, Wanarticle author=Wang, Xu Jia, label=Wan title=Schauder estimates for solutions to second-order linear parabolic equations, language=Chinese, journal=J.",
"Partial Differential Equations Ser.",
"B, volume=1, date=1988, number=2, pages=17–34,"
]
] | 1204.0882 |
[
[
"Distributed convergence to Nash equilibria in two-network zero-sum games"
],
[
"Abstract This paper considers a class of strategic scenarios in which two networks of agents have opposing objectives with regards to the optimization of a common objective function.",
"In the resulting zero-sum game, individual agents collaborate with neighbors in their respective network and have only partial knowledge of the state of the agents in the other network.",
"For the case when the interaction topology of each network is undirected, we synthesize a distributed saddle-point strategy and establish its convergence to the Nash equilibrium for the class of strictly concave-convex and locally Lipschitz objective functions.",
"We also show that this dynamics does not converge in general if the topologies are directed.",
"This justifies the introduction, in the directed case, of a generalization of this distributed dynamics which we show converges to the Nash equilibrium for the class of strictly concave-convex differentiable functions with locally Lipschitz gradients.",
"The technical approach combines tools from algebraic graph theory, nonsmooth analysis, set-valued dynamical systems, and game theory."
],
[
"Introduction",
"Recent years have seen an increasing interest on networked strategic scenarios where agents may cooperate or compete with each other towards the achievement of some objective, interact across different layers, have access to limited information, and are subject to evolving interaction topologies.",
"This paper is a contribution to this body of work.",
"Specifically, we consider a class of strategic scenarios in which two networks of agents are involved in a zero-sum game.",
"We assume that the objective function can be decomposed as a sum of concave-convex functions and that the networks have opposing objectives regarding its optimization.",
"Agents collaborate with the neighbors in their own network and have partial information about the state of the agents in the other network.",
"Such scenarios are challenging because information is spread across the agents and possibly multiple layers, and networks, by themselves, are not the decision makers.",
"Our aim is to design a distributed coordination algorithm that can be used by the agents to converge to the Nash equilibrium.",
"Note that, for a 2-player zero-sum game of the type considered here, a pure Nash equilibrium corresponds to a saddle point of the objective function.",
"Literature review.",
"Multiple scenarios involving networked systems and intelligent adversaries in sensor networks, filtering, finance, and wireless communications [18], [28] can be cast into the strategic framework described above.",
"In such scenarios, the network objective arises as a result of the aggregation of agent-to-agent adversarial interactions regarding a common goal, and information is naturally distributed among the agents.",
"The present work has connections with the literature on distributed optimization and zero-sum games.",
"The distributed optimization of a sum of convex functions has been intensively studied in recent years, see e.g.",
"[22], [28], [17], [31].",
"These works build on consensus-based dynamics [24], [25], [6], [21] to find the solutions of the optimization problem in a variety of scenarios and are designed in discrete time.",
"Exceptions include [29], [30] on continuous-time distributed optimization on undirected networks and [13] on directed networks.",
"Regarding zero-sum games, the works [2], [20], [23] study the convergence of discrete-time subgradient dynamics to a saddle point.",
"Continuous-time best-response dynamics for zero-sum games converges to the set of Nash equilibria for both convex-concave [15] and quasiconvex-quasiconcave [3] functions.",
"Under strict convexity-concavity assumptions, continuous-time subgradient flow dynamics converges to a saddle point [1], [2].",
"Asymptotic convergence is also guaranteed when the Hessian of the objective function is positive definite in one argument and the function is linear in the other [2], [9].",
"The distributed computation of Nash equilibria in noncooperative games, where all players are adversarial, has been investigated under different assumptions.",
"The algorithm in [19] relies on all-to-all communication and does not require players to know each other's payoff functions (which must be strongly convex).",
"In [10], [27], players are unaware of their own payoff functions but have access to the payoff value of an action once it has been executed.",
"These works design distributed strategies based on extremum seeking techniques to seek the set of Nash equilibria.",
"Statement of contributions.",
"We introduce the problem of distributed convergence to Nash equilibria for two networks engaged in a strategic scenario.",
"The networks aim to either maximize or minimize a common objective function which can be written as a sum of concave-convex functions.",
"Individual agents collaborate with neighbors in their respective network and have partial knowledge of the state of the agents in the other one.",
"Our first contribution is the introduction of an aggregate objective function for each network which depends on the interaction topology through its Laplacian and the characterization of a family of points with a saddle property for the pair of functions.",
"We show the correspondence between these points and the Nash equilibria of the overall game.",
"When the graphs describing the interaction topologies within each network are undirected, the gradients of these aggregate objective functions are distributed.",
"Building on this observation, our second contribution is the synthesis of a consensus-based saddle-point strategy for adversarial networks with undirected topologies.",
"We show that the proposed dynamics is guaranteed to asymptotically converge to the Nash equilibrium for the class of strictly concave-convex and locally Lipschitz objective functions.",
"Our third contribution focuses on the directed case.",
"We show that the transcription of the saddle-point dynamics to directed topologies fails to converge in general.",
"This leads us to propose a generalization of the dynamics, for strongly connected weight-balanced topologies, that incorporates a design parameter.",
"We show that, by appropriately choosing this parameter, the new dynamics asymptotically converges to the Nash equilibrium for the class of strictly concave-convex differentiable objective functions with globally Lipschitz gradients.",
"The technical approach employs notions and results from algebraic graph theory, nonsmooth and convex analysis, set-valued dynamical systems, and game theory.",
"As an intermediate result in our proof strategy for the directed case, we provide a generalization of the known characterization of cocoercivity of concave functions to concave-convex functions.",
"The results of this paper can be understood as a generalization to competing networks of the results we obtained in [13] for distributed optimization.",
"This generalization is nontrivial because the payoff functions associated to individual agents now also depend on information obtained from the opposing network.",
"This feature gives rise to a hierarchy of saddle-point dynamics whose analysis is technically challenging and requires, among other things, a reformulation of the problem as a constrained zero-sum game, a careful understanding of the coupling between the dynamics of both networks, and the generalization of the notion of cocoercivity to concave-convex functions.",
"Organization.",
"Section contains preliminaries on nonsmooth analysis, set-valued dynamical systems, graph theory, and game theory.",
"In Section , we introduce the zero-sum game for two adversarial networks involved in a strategic scenario and introduce two novel aggregate objective functions.",
"Section presents our algorithm design and analysis for distributed convergence to Nash equilibrium when the network topologies are undirected.",
"Section presents our treatment for the directed case.",
"Section gathers our conclusions and ideas for future work.",
"Appendix contains the generalization to concave-convex functions of the characterization of cocoercivity of concave functions."
],
[
"Preliminaries",
"We start with some notational conventions.",
"Let ${\\mathbb {R}}$ , ${\\mathbb {R}}_{\\ge 0}$ , $\\mathbb {Z}$ , $\\mathbb {Z}_{\\ge 1}$ denote the set of real, nonnegative real, integer, and positive integer numbers, respectively.",
"We denote by $ ||\\cdot || $ the Euclidean norm on ${\\mathbb {R}}^d$ , $ d \\in \\mathbb {Z}_{\\ge 1}$ and also use the short-hand notation $ \\mathbf {1}_d=(1,\\ldots ,1)^T $ and $\\mathbf {0}_d=(0,\\ldots ,0)^T \\in \\mathbb {R}^d$ .",
"We let $ \\mathsf {I}_{d} $ denote the identity matrix in $ \\mathbb {R}^{d\\times d}$ .",
"For matrices $ A\\in {\\mathbb {R}}^{d_1\\times d_2}$ and $ B \\in {\\mathbb {R}}^{e_1\\times e_2} $ , $ d_1,d_2,e_1,e_2 \\in \\mathbb {Z}_{\\ge 1}$ , we let $ A\\otimes B $ denote their Kronecker product.",
"The function $f:\\mathsf {X}_1\\times \\mathsf {X}_2 \\rightarrow \\mathbb {R} $ , with $ \\mathsf {X}_1 \\subset {\\mathbb {R}}^{d_1} $ , $ \\mathsf {X}_2\\subset {\\mathbb {R}}^{d_2} $ closed and convex, is concave-convex if it is concave in its first argument and convex in the second one [26].",
"A point $(x_1^*,x_2^*) \\in \\mathsf {X}_1\\times \\mathsf {X}_2 $ is a saddle point of $ f $ if $ f(x_1,x_2^*)\\le f(x^*_1,x^*_2)\\le f(x_1^*,x_2)$ for all $x_1 \\in \\mathsf {X}_1 $ and $ x_2 \\in \\mathsf {X}_2$ .",
"Finally, a set-valued map $f:{\\mathbb {R}}^d \\rightrightarrows {\\mathbb {R}}^d$ takes elements of ${\\mathbb {R}}^d$ to subsets of ${\\mathbb {R}}^d$ ."
],
[
"Nonsmooth analysis",
"We recall some notions from nonsmooth analysis [7].",
"A function $ f:{\\mathbb {R}}^d \\rightarrow {\\mathbb {R}} $ is locally Lipschitz at $x \\in {\\mathbb {R}}^d $ if there exists a neighborhood $ \\mathcal {U} $ of $ x$ and $ C_x \\in {\\mathbb {R}}_{\\ge 0}$ such that $ |f(y)-f(z)|\\le C_x||y-z|| $ , for $ y,z \\in \\mathcal {U} $ .",
"$f$ is locally Lipschitz on ${\\mathbb {R}}^d$ if it is locally Lipschitz at $x$ for all $x \\in {\\mathbb {R}}^d$ and globally Lipschitz on $ {\\mathbb {R}}^d $ if for all $ y,z \\in {\\mathbb {R}}^d $ there exists $ C \\in {\\mathbb {R}}_{\\ge 0}$ such that $|f(y)-f(z)|\\le C ||y-z|| $ .",
"Locally Lipschitz functions are differentiable almost everywhere.",
"The generalized gradient of $f $ is $\\partial f(x) = \\mathrm {co}\\Big \\lbrace \\lim _{k \\rightarrow \\infty } \\nabla f(x_k)\\ | \\ x_k \\rightarrow x, x_k \\notin \\Omega _f \\cup S\\Big \\rbrace ,$ where $ \\Omega _f $ is the set of points where $ f $ fails to be differentiable and $ S $ is any set of measure zero.",
"Lemma 2.1 (Continuity of the generalized gradient map): Let $ f:{\\mathbb {R}}^d \\rightarrow {\\mathbb {R}} $ be a locally Lipschitz function at $x \\in {\\mathbb {R}}^d $ .",
"Then the set-valued map $ \\partial f:{\\mathbb {R}}^d \\rightrightarrows {\\mathbb {R}}^d $ is upper semicontinuous and locally bounded at $ x \\in {\\mathbb {R}}^d $ and moreover, $ \\partial f(x) $ is nonempty, compact, and convex.",
"For $f:{\\mathbb {R}}^d \\times {\\mathbb {R}}^d \\rightarrow {\\mathbb {R}}$ and $z \\in {\\mathbb {R}}^d$ , we let $ \\partial _x f(x,z) $ denote the generalized gradient of $x\\mapsto f(x,z)$ .",
"Similarly, for $x \\in {\\mathbb {R}}^d$ , we let $ \\partial _zf(x,z) $ denote the generalized gradient of $z \\mapsto f(x,z)$ .",
"A point $ x \\in {\\mathbb {R}}^d $ with $ \\mathbf {0}\\in \\partial f(x) $ is a critical point of $ f $ .",
"A function $ f:{\\mathbb {R}}^d \\rightarrow {\\mathbb {R}}$ is regular at $ x \\in {\\mathbb {R}}$ if for all $ v \\in {\\mathbb {R}}^d $ the right directional derivative of $ f $ , in the direction of $ v $ , exists at $ x $ and coincides with the generalized directional derivative of $ f $ at $ x$ in the direction of $v$ .",
"We refer the reader to [7] for definitions of these notions.",
"A convex and locally Lipschitz function at $ x $ is regular [7].",
"The notion of regularity plays an important role when considering sums of Lipschitz functions.",
"Lemma 2.2 (Finite sum of locally Lipschitz functions): Let $ \\lbrace f^i\\rbrace _{i=1}^n$ be locally Lipschitz at $ x \\in {\\mathbb {R}}^d$ .",
"Then $\\partial (\\sum _{i=1}^nf^i)(x)\\subseteq \\sum _{i=1}^n\\partial f^i(x)$ , and equality holds if $ f^i $ is regular for $ i \\in \\lbrace 1,\\ldots , n\\rbrace $ .",
"A locally Lipschitz and convex function $f$ satisfies, for all $x,x^{\\prime }\\in {\\mathbb {R}}^d$ and $ \\xi \\in \\partial f(x) $ , the first-order condition of convexity, $f(x^{\\prime })-f(x) \\ge \\xi \\cdot (x^{\\prime }-x).$"
],
[
"Set-valued dynamical systems",
"Here, we recall some background on set-valued dynamical systems following [8].",
"A continuous-time set-valued dynamical system on $\\mathsf {X}\\subset \\mathbb {R}^d$ is a differential inclusion $\\dot{x}(t) \\in \\Psi (x(t))$ where $ t \\in {\\mathbb {R}}_{\\ge 0}$ and $\\Psi :\\mathsf {X}\\subset {\\mathbb {R}}^d \\rightrightarrows {\\mathbb {R}}^d$ is a set-valued map.",
"A solution to this dynamical system is an absolutely continuous curve $ x:[0,T]\\rightarrow \\mathsf {X}$ which satisfies (REF ) almost everywhere.",
"The set of equilibria of (REF ) is denoted by $\\operatorname{Eq}(\\Psi ) = \\lbrace x \\in \\mathsf {X} \\; | \\; 0 \\in \\Psi (x)\\rbrace $ .",
"Lemma 2.3 (Existence of solutions): For $\\Psi :{\\mathbb {R}}^d \\rightrightarrows {\\mathbb {R}}^d$ upper semicontinuous with nonempty, compact, and convex values, there exists a solution to (REF ) from any initial condition.",
"The LaSalle Invariance Principle for set-valued continuous-time systems is helpful to establish the asymptotic stability properties of systems of the form (REF ).",
"A set $ W \\subset \\mathsf {X}$ is weakly positively invariant with respect to $ \\Psi $ if for any $ x \\in W $ , there exists $ \\tilde{x} \\in \\mathsf {X}$ such that $\\tilde{x} \\in \\Psi (x) $ .",
"The set $ W $ is strongly positively invariant with respect to $ \\Psi $ if $ \\Psi (x) \\subset W $ , for all $ x \\in W $ .",
"Finally, the set-valued Lie derivative of a differentiable function $ V:{\\mathbb {R}}^d \\rightarrow {\\mathbb {R}} $ with respect to $ \\Psi $ at $ x\\in {\\mathbb {R}}^d $ is defined by $ \\widetilde{\\mathcal {L}}_{\\Psi }{V(x)}=\\lbrace v \\cdot \\nabla V(x) \\ | \\ v \\in \\Psi (x) \\rbrace $ .",
"Theorem 2.4 (Set-valued LaSalle Invariance Principle): Let $ W \\subset \\mathsf {X}$ be a strongly positively invariant under (REF ) and $ V: \\mathsf {X}\\rightarrow \\mathbb {R}$ a continuously differentiable function.",
"Suppose the evolutions of (REF ) are bounded and $ \\max \\widetilde{\\mathcal {L}}_{\\Psi }{V(x)} \\le 0 $ or $\\widetilde{\\mathcal {L}}_{\\Psi }{V(x)}=\\emptyset $ , for all $ x \\in W $ .",
"If $ S_{\\Psi ,V} = \\lbrace x\\in \\mathsf {X}\\ | \\ 0 \\in \\widetilde{\\mathcal {L}}_{\\Psi }{V(x)}\\rbrace , $ then any solution $ x(t) $ , $ t\\in \\mathbb {R}_{\\ge 0} $ , starting in $W$ converges to the largest weakly positively invariant set $ M$ contained in $ \\bar{S}_{\\Psi ,V}\\cap W$ .",
"When $ M $ is a finite collection of points, then the limit of each solution equals one of them."
],
[
"Graph theory",
"We present some basic notions from algebraic graph theory following the exposition in [6].",
"A directed graph, or simply digraph, is a pair $\\mathcal {G}=(\\mathcal {V},\\mathcal {E})$ , where $\\mathcal {V}$ is a finite set called the vertex set and $ \\mathcal {E}\\subseteq \\mathcal {V}\\times \\mathcal {V}$ is the edge set.",
"A digraph is undirected if $(v,u) \\in \\mathcal {E}$ anytime $(u,v) \\in \\mathcal {E}$ .",
"We refer to an undirected digraph as a graph.",
"A path is an ordered sequence of vertices such that any ordered pair of vertices appearing consecutively is an edge of the digraph.",
"A digraph is strongly connected if there is a path between any pair of distinct vertices.",
"For a graph, we refer to this notion simply as connected.",
"A weighted digraph is a triplet $\\mathcal {G}=(\\mathcal {V},\\mathcal {E},\\mathsf {A}) $ , where $ (\\mathcal {V},\\mathcal {E}) $ is a digraph and $ \\mathsf {A}\\in \\mathbb {R}^{n\\times n}_{\\ge 0} $ is the adjacency matrix of $\\mathcal {G}$ , with the property that $a_{ij}>0 $ if $ (v_i,v_j)\\in \\mathcal {E}$ and $ a_{ij}=0 $ , otherwise.",
"The weighted out-degree and in-degree of $v_i$ , $i \\in \\lbrace 1,\\dots ,n\\rbrace $ , are respectively, $ d_{\\textup {out}}^{\\textup {w}}(v_i) =\\sum _{j=1}^{n}a_{ij} $ and $d_{\\textup {in}}^{\\textup {w}}(v_i)=\\sum _{j=1}^n a_{ji} $ .",
"The weighted out-degree matrix $ \\mathsf {D}_{\\textup {out}}$ is the diagonal matrix defined by $(\\mathsf {D}_{\\textup {out}})_{ii}=d_{\\textup {out}}^{\\textup {w}}(i) $ , for all $ i \\in \\lbrace 1,\\ldots ,n\\rbrace $ .",
"The Laplacian matrix is $ \\mathsf {L}= \\mathsf {D}_{\\textup {out}}-\\mathsf {A}$ .",
"Note that $\\mathsf {L}\\mathbf {1}_n=0 $ .",
"If $ \\mathcal {G}$ is strongly connected, then zero is a simple eigenvalue of $\\mathsf {L}$ .",
"$\\mathcal {G}$ is undirected if $ \\mathsf {L}=\\mathsf {L}^T$ and weight-balanced if $ d_{\\textup {out}}^{\\textup {w}}(v) =d_{\\textup {in}}^{\\textup {w}}(v) $ , for all $ v\\in \\mathcal {V}$ .",
"Equivalently, $\\mathcal {G}$ is weight-balanced if and only if $ \\mathbf {1}_n^T \\mathsf {L}=0 $ if and only if $ \\mathsf {L}+\\mathsf {L}^T $ is positive semidefinite.",
"Furthermore, if $\\mathcal {G}$ is weight-balanced and strongly connected, then zero is a simple eigenvalue of $\\mathsf {L}+\\mathsf {L}^T $ .",
"Note that any undirected graph is weight-balanced."
],
[
"Zero-sum games",
"We recall basic game-theoretic notions following [4].",
"An $ n $ -player game is a triplet $ \\mathbf {G}=(P, \\mathsf {X},U) $ , where $ P$ is the set of players with $ |P| = n\\in \\mathbb {Z}_{\\ge 2} $ , $\\mathsf {X}=\\mathsf {X}_1\\times \\ldots \\times \\mathsf {X}_n $ , $ \\mathsf {X}_i \\subset {\\mathbb {R}}^{d_i} $ is the set of (pure) strategies of player $ v_i \\in P $ , $ d_i \\in \\mathbb {Z}_{\\ge 1}$ , and $ U=(u_1,\\ldots , u_n) $ , where $u_i:\\mathsf {X} \\rightarrow {\\mathbb {R}} $ is the payoff function of player $ v_i $ , $i\\in \\lbrace 1,\\ldots , n\\rbrace $ .",
"The game $ \\mathbf {G} $ is called a zero-sum game if $ \\sum _{i=1}^nu_i = 0 $ .",
"An outcome $ x^* \\in \\mathsf {X}$ is a (pure) Nash equilibrium of $ \\mathbf {G} $ if for all $ i \\in \\lbrace 1,\\ldots , n\\rbrace $ and all $ x_i \\in \\mathsf {X}_i $ , $u_i(x_i^*, x_{-i}^*)\\ge u_i(x_i, x_{-i}^*),$ where $ x_{-i} $ denotes the actions of all players other than $v_i$ .",
"In this paper, we focus on a class of two-player zero-sum games which have at least one pure Nash equilibrium as the next result states.",
"Theorem 2.5 (Minmax theorem): Let $ \\mathsf {X}_1\\subset {\\mathbb {R}}^{d_1} $ and $ \\mathsf {X}_2 \\subset {\\mathbb {R}}^{d_2}$ , $ d_1,d_2\\in \\mathbb {Z}_{\\ge 1}$ , be nonempty, compact, and convex.",
"If $ u:\\mathsf {X}_1\\times \\mathsf {X}_2 \\rightarrow \\mathbb {R} $ is continuous and the sets $ \\lbrace x^{\\prime }\\in \\mathsf {X}_1 \\ | \\ u(x^{\\prime },y) \\ge \\alpha \\rbrace $ and $ \\lbrace y^{\\prime }\\in \\mathsf {X}_2 \\ | \\ u(x,y^{\\prime }) \\le \\alpha \\rbrace $ are convex for all $ x \\in \\mathsf {X}_1 $ , $ y \\in \\mathsf {X}_2 $ , and $ \\alpha \\in {\\mathbb {R}}$ , then $\\max _x\\min _y u(x,y)=\\min _y\\max _x u(x,y).$ Theorem REF implies that the game $\\mathbf {G}=(\\lbrace v_1,v_2\\rbrace , \\mathsf {X}_1\\times \\mathsf {X}_2, (u,-u)) $ has a pure Nash equilibrium."
],
[
"Problem statement",
"Consider two networks $\\Sigma _1$ and $\\Sigma _2$ composed of agents $\\lbrace v_1,\\dots ,v_{n_1}\\rbrace $ and agents $\\lbrace w_1,\\dots ,w_{n_2}\\rbrace $ , respectively.",
"Throughout this paper, $\\Sigma _1 $ and $ \\Sigma _2 $ are either connected undirected graphs, c.f.",
"Section , or strongly connected weight-balanced digraphs, c.f.",
"Section .",
"Since the latter case includes the first one, throughout this section, we assume the latter.",
"The state of $\\Sigma _1$ , denoted $x_1$ , belongs to $ \\mathsf {X}_1 \\subset {\\mathbb {R}}^{d_1}$ , $ d_1 \\in \\mathbb {Z}_{\\ge 1}$ .",
"Likewise, the state of $\\Sigma _2$ , denoted $x_2$ , belongs to $\\mathsf {X}_2 \\subset {\\mathbb {R}}^{d_2}$ , $ d_2 \\in \\mathbb {Z}_{\\ge 1}$ .",
"In this paper, we do not get into the details of what these states represent (as a particular case, the network state could correspond to the collection of the states of agents in it).",
"In addition, each agent $v_i $ in $\\Sigma _1$ has an estimate $x_1^i \\in {\\mathbb {R}}^{d_1}$ of what the network state is, which may differ from the actual value $x_1$ .",
"Similarly, each agent $w_j $ in $\\Sigma _2$ has an estimate $x_2^j\\in {\\mathbb {R}}^{d_2}$ of what the network state is.",
"Within each network, neighboring agents can share their estimates.",
"Networks can also obtain information about each other.",
"This is modeled by means of a bipartite directed graph $\\Sigma _{\\textup {eng}}$ , called engagement graph, with disjoint vertex sets $\\lbrace v_1,\\ldots , v_{n_1}\\rbrace $ and $\\lbrace w_1,\\ldots , w_{n_2}\\rbrace $ , where every agent has at least one out-neighbor.",
"According to this model, an agent in $\\Sigma _1$ obtains information from its out-neighbors in $\\Sigma _{\\textup {eng}}$ about their estimates of the state of $\\Sigma _2$ , and vice versa.",
"Figure REF illustrates this concept.",
"Figure: Networks Σ 1 \\Sigma _1 and Σ 2 \\Sigma _2 engaged in astrategic scenario.",
"Both networks are strongly connected andweight-balanced, with weights of 1 on each edge.",
"Edges whichcorrespond to Σ eng \\Sigma _{\\textup {eng}} are dashed.For each $i \\in \\lbrace 1,\\ldots , n_1\\rbrace $ , let $f_1^i:\\mathsf {X}_1\\times \\mathsf {X}_2 \\rightarrow {\\mathbb {R}} $ be a locally Lipschitz concave-convex function only available to agent $ v_i \\in \\Sigma _1 $ .",
"Similarly, let $f_2^j:\\mathsf {X}_1\\times \\mathsf {X}_2 \\rightarrow {\\mathbb {R}} $ be a locally Lipschitz concave-convex function only available to agent $ w_j \\in \\Sigma _2$ , $ j \\in \\lbrace 1,\\ldots ,n_2\\rbrace $ .",
"The networks $\\Sigma _1$ and $\\Sigma _2$ are engaged in a zero-sum game with payoff function $U:\\mathsf {X}_1\\times \\mathsf {X}_2 \\rightarrow {\\mathbb {R}} $ $U(x_1,x_2)=\\sum _{i=1}^{n_1}f_1^{i}(x_1,x_2) =\\sum _{j=1}^{n_2}f_2^{j}(x_1,x_2),$ where $\\Sigma _1$ wishes to maximize $U$ , while $\\Sigma _2$ wishes to minimize it.",
"The objective of the networks is therefore to settle upon a Nash equilibrium, i.e., to solve the following maxmin problem $\\max _{x_1\\in \\mathsf {X}_1} \\min _{x_2 \\in \\mathsf {X}_2} U(x_1,x_2) .$ We refer to the this zero-sum game as the 2-network zero-sum game and denote it by $\\mathbf {G}_{\\textup {adv-net}}=(\\Sigma _1,\\Sigma _2,\\Sigma _{\\textup {eng}},U)$ .",
"We assume that $ \\mathsf {X}_1\\subset {\\mathbb {R}}^{d_1}$ and $ \\mathsf {X}_2 \\subset {\\mathbb {R}}^{d_2} $ are compact convex.",
"For convenience, let $ \\mathbf {x}_1=(x_1^1, \\ldots , x_1^{n_1})^T $ and $ \\mathbf {x}_2=(x_2^1, \\ldots , x_2^{n_2})^T $ denote vector of agent estimates about the state of the respective networks.",
"Remark 3.1 (Power allocation in communication channels in the presence of adversaries): Here we present an example from communications inspired by [5].",
"Consider $n$ Gaussian communication channels, each with signal power $p_i \\in {\\mathbb {R}}_{\\ge 0}$ and noise power $\\eta _i \\in {\\mathbb {R}}_{\\ge 0}$ , for $i \\in \\lbrace 1,\\dots ,n\\rbrace $ .",
"The capacity of each channel is proportional to $ \\log (1+\\beta p_i/(\\sigma _i+\\eta _i))$ , where $\\beta \\in {\\mathbb {R}}_{>0}$ and $ \\sigma _i >0 $ is the receiver noise.",
"Note that capacity is concave in $p_i$ and convex in $\\eta _i$ .",
"Both signal and noise powers must satisfy a budget constraint, i.e., $ \\sum _{i=1}^{n} p_i=P $ and $\\sum _{i=1}^{n}\\eta _i = C$ , for some given $ P, C \\in {\\mathbb {R}}_{>0}$ .",
"Two networks of $n$ agents are involved in this scenario, one, $\\Sigma _1$ , selecting signal powers to maximize capacity, the other one, $\\Sigma _2$ , selecting noise powers to minimize it.",
"The network $\\Sigma _1$ has decided that $m_1$ channels will have signal power $x_1$ , while $n-1-m_1$ will have signal power $x_2$ .",
"The remaining $n$ th channel has its power determined to satisfy the budget constraint, i.e., $P-m_1 x_1 -(n-1-m_1)x_2$ .",
"Likewise, the network $\\Sigma _2$ does something similar with $m_2$ channels with noise power $y_1$ , $n-1-m_2$ channels with noise power $y_2$ , and one last channel with noise power $C-m_2 y_1 - (n-1-m_2) y_2$ .",
"Each network is aware of the partition made by the other one.",
"The individual objective function of the two agents (one from $\\Sigma _1$ , the other from $\\Sigma _2$ ) making decisions on the power levels of the $i$ th channel is the channel capacity itself.",
"For $i \\in \\lbrace 1,\\dots ,n-1\\rbrace $ , this takes the form $f^i (x,y)= \\log \\Big (1+ \\frac{\\beta x_a}{\\sigma _i+y_b} \\Big ),$ for some $a,b \\in \\lbrace 1,2\\rbrace $ .",
"Here $x=(x_1,x_2)$ and $y=(y_1,y_2)$ .",
"For $i=n$ , it takes instead the form $f^{n}(x,y) = \\log \\Big (1+ \\frac{\\beta (P-m_1 x_1 -(n-1-m_1)x_2)}{\\sigma _{n}+C-m_2 y_1 - (n-1-m_2) y_2} \\Big ).$ Note that $ \\sum _{i=1}^{n}f^i(x,y) $ is the total capacity of the $n$ communication channels.",
"$\\bullet $"
],
[
"Reformulation of the 2-network zero-sum game",
"In this section, we describe how agents in each network use the information obtained from their neighbors to compute the value of their own objective functions.",
"Based on these estimates, we introduce a reformulation of the $ \\mathbf {G}_{\\textup {adv-net}}=(\\Sigma _1,\\Sigma _2,\\Sigma _{\\textup {eng}},U)$ which is instrumental for establishing some of our results.",
"Each agent in $\\Sigma _1$ has a locally Lipschitz, concave-convex function $ \\tilde{f}_1^i:{\\mathbb {R}}^{d_1}\\times {\\mathbb {R}}^{d_2n_2} \\rightarrow {\\mathbb {R}} $ with the properties: (Extension of own payoff function): for any $ x_1 \\in {\\mathbb {R}}^{d_1}$ , $x_2 \\in {\\mathbb {R}}^{d_2}$ , $\\tilde{f}_1^i(x_1,\\mathbf {1}_{n_2}\\otimes x_2)=f_1^i(x_1,x_2).$ (Distributed over $\\Sigma _{\\textup {eng}}$ ): there exists $\\mathfrak {f}_1^i:{\\mathbb {R}}^{d_1} \\times {\\mathbb {R}}^{d_2|\\mathcal {N}^{\\textup {in}}_{\\Sigma _{\\textup {eng}}}(v_i)|} \\rightarrow {\\mathbb {R}}$ such that, for any $ x_1 \\in {\\mathbb {R}}^{d_1}$ $ \\mathbf {x}_2 \\in {\\mathbb {R}}^{d_2n_2} $ , $\\tilde{f}_1^i(x_1,\\mathbf {x}_2) = \\mathfrak {f}_1^i(x_1,\\pi _1^i(\\mathbf {x}_2)) ,$ with $\\pi _1^i:{\\mathbb {R}}^{d_2n_2} \\rightarrow {\\mathbb {R}}^{d_2|\\mathcal {N}^{\\textup {out}}_{\\Sigma _{\\textup {eng}}}(v_i)|}$ the projection of $ \\mathbf {x}_2 $ to the values received by $ v_i $ from its out-neighbors in $ \\Sigma _{\\textup {eng}}$ .",
"Equation (REF ) states the fact that, when the estimates of all neighbors of an agent in the opponent's network agree, its evaluation should coincide with this estimate.",
"Equation (REF ) states the fact that agents can only use the information received from their neighbors in the interaction topology to compute their new estimates.",
"Each agent in $\\Sigma _2$ has a function $\\tilde{f}_2^j:{\\mathbb {R}}^{d_1n_1}\\times {\\mathbb {R}}^{d_2} \\rightarrow {\\mathbb {R}} $ with similar properties.",
"The collective payoff functions of the two networks are $\\tilde{U}_1(\\mathbf {x}_1,\\mathbf {x}_2) &=\\sum _{i=1}^{n_1}\\tilde{f}_1^i(x_1^i,\\mathbf {x}_2),\\\\\\tilde{U}_2(\\mathbf {x}_1,\\mathbf {x}_2) &=\\sum _{j=1}^{n_2}\\tilde{f}_2^j(\\mathbf {x}_1,x_2^j).$ In general, the functions $ \\tilde{U}_1 $ and $ \\tilde{U}_2 $ need not be the same.",
"However, $ \\tilde{U}_1 (\\mathbf {1}_{n_1} \\otimes x_1,\\mathbf {1}_{n_1}\\otimes x_2) = \\tilde{U}_2 (\\mathbf {1}_{n_1} \\otimes x_1,\\mathbf {1}_{n_1} \\otimes x_2)$ , for any $x_1 \\in {\\mathbb {R}}^{d_1}$ , $x_2 \\in {\\mathbb {R}}^{d_2}$ .",
"When both functions coincide, the next result shows that the original game can be lifted to a (constrained) zero-sum game.",
"Lemma 3.2 (Reformulation of the 2-network zero-sum game): Assume that the individual payoff functions $\\lbrace \\tilde{f}^i_1\\rbrace _{i=1}^{n_1}$ , $\\lbrace \\tilde{f}^j_2\\rbrace _{j=1}^{n_2}$ satisfying (REF ) are such that the network payoff functions defined in (REF ) satisfy $ \\tilde{U}_1 =\\tilde{U}_2 $ , and let $ \\tilde{U}$ denote this common function.",
"Then, the problem (REF ) on ${\\mathbb {R}}^{d_1}\\times {\\mathbb {R}}^{d_2} $ is equivalent to the following problem on ${\\mathbb {R}}^{n_1d_1}\\times {\\mathbb {R}}^{n_2d_2}$ , $& \\max _{\\mathbf {x}_1\\in \\mathsf {X}_1^{n_1}} \\min _{\\mathbf {x}_2 \\in \\mathsf {X}_2^{n_2}}\\tilde{U}(\\mathbf {x}_1,\\mathbf {x}_2), \\nonumber \\\\& \\qquad \\text{subject to} \\quad \\mathbf {L}_1 \\mathbf {x}_1 =\\mathbf {0}_{n_1d_1}, \\quad \\mathbf {L}_2 \\mathbf {x}_2 = \\mathbf {0}_{n_2d_2} ,$ with $ \\mathbf {L}_\\ell = \\mathsf {L}_\\ell \\otimes \\mathsf {I}_{d_\\ell }$ and $\\mathsf {L}_\\ell $ the Laplacian of $\\Sigma _\\ell $ , $ \\ell \\in \\lbrace 1, 2\\rbrace $ .",
"Proof.",
"The proof follows by noting that (i) $\\tilde{U}(\\mathbf {1}_{n_1}\\otimes x_1, \\mathbf {1}_{n_2}\\otimes x_2) = U(x_1,x_2)$ for all $ x_1 \\in {\\mathbb {R}}^{d_1} $ and $ x_2 \\in {\\mathbb {R}}^{d_2} $ and (ii) since $\\mathcal {G}_1$ and $\\mathcal {G}_2$ are strongly connected, $\\mathbf {L}_1 \\mathbf {x}_1 =\\mathbf {0}_{n_1d_1} $ and $\\mathbf {L}_2\\mathbf {x}_2 = \\mathbf {0}_{n_2d_2} $ iff $\\mathbf {x}_1 = \\mathbf {1}_{n_1} \\otimes x_1 $ and $\\mathbf {x}_2 = \\mathbf {1}_{n_2}\\otimes x_2 $ for some $x_1 \\in {\\mathbb {R}}^{d_1}$ and $x_2 \\in {\\mathbb {R}}^{d_2}$ .",
"$\\Box $ Remark 3.3 (Restrictions on extensions): The assumption of Lemma REF does not hold in general for all sets of extensions satisfying (REF ) and (REF ).",
"If the interaction topology is one-to-one (i.e., both networks have the same number of agents, the interaction topology is undirected, and each agent in the first network obtains information only from one agent in the opposing network), the natural extensions satisfy the assumption.",
"Example REF later provides yet another instance of a different nature.",
"In general, determining if it is always possible to choose the extensions in such a way that the assumption holds is an open problem.",
"$\\bullet $ We denote by $\\mathbf {\\tilde{G}}_{\\textup {adv-net}}=(\\Sigma _1,\\Sigma _2,\\Sigma _{\\textup {eng}},\\tilde{U})$ the constrained zero-sum game defined by (REF ) and refer to this situation by saying that $\\mathbf {G}_{\\textup {adv-net}}$ can be lifted to $\\mathbf {\\tilde{G}}_{\\textup {adv-net}}$ .",
"Our objective is to design a coordination algorithm that is implementable with the information that agents in $\\Sigma _1$ and $\\Sigma _2$ possess and leads them to find a Nash equilibrium of $ \\mathbf {\\tilde{G}}_{\\textup {adv-net}}$ , which corresponds to a Nash equilibrium of $ \\mathbf {G}_{\\textup {adv-net}}$ by Lemma REF .",
"Achieving this goal, however, is nontrivial because individual agents, not networks themselves, are the decision makers.",
"From the point of view of agents in each network, the objective is to agree on the states of both their own network and the other network, and that the resulting states correspond to a Nash equilibrium of $ \\mathbf {G}_{\\textup {adv-net}}$ .",
"The function $\\tilde{U}$ is locally Lipschitz and concave-convex.",
"Moreover, from Lemma REF , the elements of $\\partial _{\\mathbf {x}_1} \\tilde{U}(\\mathbf {x}_1,\\mathbf {x}_2)$ are of the form $\\tilde{g}_{(\\mathbf {x}_1,\\mathbf {x}_2)} = (\\tilde{g}^1_{(x^1_1, \\mathbf {x}_2)}, \\ldots ,\\tilde{g}^n_{(x^n_1, \\mathbf {x}_2)}) \\in \\partial _{\\mathbf {x}_1}\\tilde{U}(\\mathbf {x}_1,\\mathbf {x}_2),$ where $ \\tilde{g}^i_{(x^i_1, \\mathbf {x}_2)} \\in \\partial _{x_1}\\tilde{f}^i_1(x^i_1, \\mathbf {x}_2) $ , for $ i \\in \\lbrace 1,\\ldots , n_1\\rbrace $ .",
"Note that, because of (REF ), we have $\\partial _{x_1}\\tilde{f}^i_1(x^i_1, \\mathbf {1}_{n_2}\\otimes x_2 ) = \\partial _{x_1}f^i_1(x^i_1, x_2) $ .",
"A similar reasoning can be followed to describe the elements of $ \\partial _{\\mathbf {x}_2} \\tilde{U}(\\mathbf {x}_1,\\mathbf {x}_2) $ .",
"Next, we present a characterization of the Nash equilibria of $\\mathbf {\\tilde{G}}_{\\textup {adv-net}}$ , instrumental for proving some of our upcoming results.",
"Proposition 3.4 (Characterization of the Nash equilibria of $\\mathbf {\\tilde{G}}_{\\textup {adv-net}}$ ): For $\\Sigma _1$ , $\\Sigma _2$ strongly connected and weight-balanced, define $ F_1 $ and $ F_2 $ by $F_1(\\mathbf {x}_1,\\mathbf {z}_1,\\mathbf {x}_2)&=-\\tilde{U}(\\mathbf {x}_1,\\mathbf {x}_2) +\\mathbf {x}_1^T\\mathbf {L}_1 \\mathbf {z}_1 + \\frac{1}{2}\\mathbf {x}_1^T\\mathbf {L}_1\\mathbf {x}_1,\\\\F_2(\\mathbf {x}_2,\\mathbf {z}_2,\\mathbf {x}_1)& =\\tilde{U}(\\mathbf {x}_1,\\mathbf {x}_2) +\\mathbf {x}_2^T\\mathbf {L}_2 \\mathbf {z}_2 + \\frac{1}{2}\\mathbf {x}_2^T\\mathbf {L}_2\\mathbf {x}_2.$ Then, $F_1$ and $F_2$ are convex in their first argument, linear in their second one, and concave in their third one.",
"Moreover, assume $(\\mathbf {x}^*_1,\\mathbf {z}^*_1,\\mathbf {x}^*_2,\\mathbf {z}^*_2) $ satisfies the following saddle property for $(F_1,F_2)$ : $(\\mathbf {x}^*_1,\\mathbf {z}^*_1) $ is a saddle point of $(\\mathbf {x}_1,\\mathbf {z}_1) \\mapsto F_1(\\mathbf {x}_1,\\mathbf {z}_1,\\mathbf {x}^*_2) $ and $(\\mathbf {x}^*_2,\\mathbf {z}^*_2) $ is a saddle point of $(\\mathbf {x}_2,\\mathbf {z}_2) \\mapsto F_2(\\mathbf {x}_2,\\mathbf {z}_2, \\mathbf {x}^*_1) $ .",
"Then, $(\\mathbf {x}^*_1,\\mathbf {z}^*_1+\\mathbf {1}_{n_1}\\otimes a_1,\\mathbf {x}^*_2,\\mathbf {z}^*_2+\\mathbf {1}_{n_2}\\otimes a_2) $ satisfies the saddle property for $(F_1,F_2) $ for any $ a_1\\in {\\mathbb {R}}^{d_1} $ , $a_2 \\in {\\mathbb {R}}^{d_2} $ , and $(\\mathbf {x}^*_1,\\mathbf {x}^*_2) $ is a Nash equilibrium of $ \\mathbf {\\tilde{G}}_{\\textup {adv-net}}$ .",
"Furthermore, if $ (\\mathbf {x}^*_1,\\mathbf {x}^*_2) $ is a Nash equilibrium of $\\mathbf {\\tilde{G}}_{\\textup {adv-net}}$ then there exists $ \\mathbf {z}^*_1,\\mathbf {z}^*_2 $ such that $(\\mathbf {x}^*_1,\\mathbf {z}^*_1,\\mathbf {x}^*_2,\\mathbf {z}^*_2) $ satisfies the saddle property for $(F_1,F_2) $ .",
"Proof.",
"The statement (i) is immediate.",
"For (ii), since $(\\mathbf {x}^*_1,\\mathbf {z}^*_1,\\mathbf {x}^*_2,\\mathbf {z}^*_2) $ satisfies the saddle property, and the networks are strongly connected and weight-balanced, we have $\\mathbf {x}^*_1 = \\mathbf {1}_{n_1} \\otimes x^*_1 $ , $x^*_1 \\in {\\mathbb {R}}^{d_1} $ , $\\mathbf {x}^*_2 = \\mathbf {1}_{n_2}\\otimes x^*_2 $ , $x^*_2 \\in {\\mathbb {R}}^{d_2} $ , $\\mathbf {L}_1 \\mathbf {z}^*_1 \\in -\\partial _{\\mathbf {x}_1} \\tilde{U}(\\mathbf {x}^*_1,\\mathbf {x}^*_2) $ , and $\\mathbf {L}_2 \\mathbf {z}^*_2 \\in \\partial _{\\mathbf {x}_2} \\tilde{U}(\\mathbf {x}^*_1,\\mathbf {x}^*_2) $ .",
"Thus there exist $ g^i_{1,(x^*_1,x^*_2)} \\in \\partial _{x_1} f_1^i(x^*_1,x^*_2)$ , $ i \\in \\lbrace 1,\\ldots , n_1\\rbrace $ , and $ g^j_{2,(x^*_1,x^*_2)} \\in -\\partial _{x_2} f_2^j(x^*_1,x^*_2)$ , $ j \\in \\lbrace 1,\\ldots , n_2\\rbrace $ , such that $\\mathbf {L}_1\\mathbf {z}_1^*& =(g^1_{1,(x^*_1,x^*_2)}, , \\ldots ,g^n_{1,(x^*_1,x^*_2)}))^T, \\ \\mathrm {and} \\\\\\mathbf {L}_2 \\mathbf {z}_2^*&=(g^1_{2,(x^*_1,x^*_2)}, , \\ldots ,g^n_{2,(x^*_1,x^*_2)}))^T.$ Noting that, for $ \\ell \\in \\lbrace 1,2\\rbrace $ , $ (\\mathbf {1}_{n_\\ell }^T\\otimes \\mathsf {I}_{d_\\ell }) \\mathbf {L}_\\ell = (\\mathbf {1}_{n_\\ell }^T\\otimes \\mathsf {I}_{d_\\ell })(\\mathsf {L}_\\ell \\otimes \\mathsf {I}_{d_\\ell }) = \\mathbf {1}_{n_\\ell }^T \\mathsf {L}\\otimes \\mathsf {I}_{d_\\ell } = \\mathbf {0}_{d_\\ell \\times d_\\ell n_\\ell }$ , we deduce that $\\sum _{i=1}^{n_1} g^i_{1,(x^*_1,x^*_2)}=\\mathbf {0}_{d_1} $ and $\\sum _{j=1}^{n_2} g^j_{2,(x^*_1,x^*_2)}= \\mathbf {0}_{d_2} $ , i.e., $(\\mathbf {x}^*_1,\\mathbf {x}^*_2) $ is a Nash equilibrium.",
"Finally for proving (iii), note that $\\mathbf {x}_1^* = \\mathbf {1}_{n_1}\\otimes x_1^*$ and $\\mathbf {x}_2^* = \\mathbf {1}_{n_2}\\otimes x_2^*$ .",
"The result follows then from the fact that $0 \\in \\partial _{x_1}U(x_1^*,x_2^*)$ and $0 \\in \\partial _{x_2}U(x_1^*,x_2^*)$ implies that there exists $\\mathbf {z}_1^*\\in {\\mathbb {R}}^{n_1d_1} $ and $\\mathbf {z}_2^*\\in {\\mathbb {R}}^{n_2d_2} $ with $\\mathbf {L}_1 \\mathbf {z}_1^*\\in \\partial _{\\mathbf {x}_1}\\tilde{U}(\\mathbf {x}_1^*,\\mathbf {x}_2^*)$ and $\\mathbf {L}_2\\mathbf {z}_2^* \\in -\\partial _{\\mathbf {x}_2}\\tilde{U}(\\mathbf {x}_1^*,\\mathbf {x}_2^*)$ .",
"$\\Box $"
],
[
"Distributed convergence to Nash equilibria for undirected\ntopologies",
"In this section, we introduce a distributed dynamics which solves (REF ) when $ \\Sigma _1 $ and $ \\Sigma _2$ are undirected.",
"In particular, we design gradient dynamics to find points with the saddle property for $(F_1,F_2) $ prescribed by Proposition REF .",
"Consider the set-valued dynamics $\\Psi _{\\textup {Nash-undir}}:({\\mathbb {R}}^{d_1n_1})^2 \\times ({\\mathbb {R}}^{d_2n_2})^2 \\rightrightarrows ({\\mathbb {R}}^{d_1n_1})^2\\times ({\\mathbb {R}}^{d_2n_2})^2 $ , $\\dot{\\mathbf {x}}_1 + \\mathbf {L}_1 \\mathbf {x}_1 + \\mathbf {L}_1 \\mathbf {z}_1&\\in \\partial _{\\mathbf {x}_1} \\tilde{U}(\\mathbf {x}_1,\\mathbf {x}_2), \\\\\\dot{\\mathbf {z}}_1 &= \\mathbf {L}_1\\mathbf {x}_1, \\\\\\dot{\\mathbf {x}}_2 + \\mathbf {L}_2\\mathbf {x}_2+\\mathbf {L}_2\\mathbf {z}_2 & \\in - \\partial _{\\mathbf {x}_2}\\tilde{U}(\\mathbf {x}_1,\\mathbf {x}_2), \\\\\\dot{\\mathbf {z}}_2 &= \\mathbf {L}_2\\mathbf {x}_2, $ where $ \\mathbf {x}_\\ell ,\\mathbf {z}_\\ell \\in {\\mathbb {R}}^{n_\\ell d_\\ell } $ , $ \\ell \\in \\lbrace 1,2\\rbrace $ .",
"Note that (REF )-() and ()-() correspond to saddle-point dynamics of $F_1$ in $(\\mathbf {x}_1,\\mathbf {z}_1)$ and $F_2$ in $(\\mathbf {x}_2,\\mathbf {z}_2)$ , respectively.",
"Local solutions to this dynamics exist by virtue of Lemmas REF and REF .",
"We characterize next its asymptotic convergence properties.",
"Theorem 4.1 (Distributed convergence to Nash equilibria for undirected networks): Consider the zero-sum game $ \\mathbf {G}_{\\textup {adv-net}}=(\\Sigma _1,\\Sigma _2,\\Sigma _{\\textup {eng}},U)$ , with $ \\Sigma _1 $ and $ \\Sigma _2 $ connected undirected graphs, $ \\mathsf {X}_1 \\subset {\\mathbb {R}}^{d_1} $ , $ \\mathsf {X}_2 \\subset {\\mathbb {R}}^{d_2} $ compact and convex, and $U:\\mathsf {X}_1 \\times \\mathsf {X}_2 \\rightarrow {\\mathbb {R}} $ strictly concave-convex and locally Lipschitz.",
"Assume $\\mathbf {G}_{\\textup {adv-net}}$ can be lifted to $\\mathbf {\\tilde{G}}_{\\textup {adv-net}}$ .",
"Then, the projection onto the first and third components of the solutions of () asymptotically converge to agreement on the Nash equilibrium of $\\mathbf {G}_{\\textup {adv-net}}$ .",
"Proof.",
"Throughout this proof, since property (REF ) holds, without loss of generality and for simplicity of notation, we assume that agents in $ \\Sigma _1 $ have access to $ \\mathbf {x}_2 $ and, similarly, agents in $ \\Sigma _2 $ have access to $ \\mathbf {x}_1 $ .",
"By Theorem REF , a solution to (REF ) exists.",
"By the strict concavity-convexity properties, this solution is, in fact, unique.",
"Let us denote this solution by $ \\mathbf {x}_1^*=\\mathbf {1}_{n_1} \\otimes x^*_1$ and $ \\mathbf {x}_2^*=\\mathbf {1}_{n_2} \\otimes x^*_2 $ .",
"By Proposition REF (iii), there exists $\\mathbf {z}_1^*$ and $\\mathbf {z}_2^* $ such that $ (\\mathbf {x}_1^*,\\mathbf {z}_1^*, \\mathbf {x}_2^*,\\mathbf {z}_2^*)\\in \\operatorname{Eq}(\\Psi _{\\textup {Nash-undir}}) $ .",
"First, note that given any initial condition $ (\\mathbf {x}^0_1,\\mathbf {z}^0_1, \\mathbf {x}^0_2,\\mathbf {z}^0_2) \\in ({\\mathbb {R}}^{n_1d_1})^2 \\times ({\\mathbb {R}}^{n_2d_2})^2 $ , the set $W_{\\mathbf {z}^0_1,\\mathbf {z}^0_2} & = \\lbrace (\\mathbf {x}_1,\\mathbf {z}_1,\\mathbf {x}_2,\\mathbf {z}_2) \\ | \\nonumber \\\\& \\ (\\mathbf {1}_{n_\\ell }^T \\otimes \\mathsf {I}_{d_\\ell }) \\mathbf {z}_\\ell =(\\mathbf {1}_{n_\\ell }^T \\otimes \\mathsf {I}_{d_\\ell }) \\mathbf {z}^0_\\ell , \\; \\ell \\in \\lbrace 1,2\\rbrace \\rbrace $ is strongly positively invariant under ().",
"Consider the function $V:({\\mathbb {R}}^{d_1n_1})^2 \\times ({\\mathbb {R}}^{d_2n_2})^2 \\rightarrow {\\mathbb {R}}_{\\ge 0}$ defined by $V(&\\mathbf {x}_1,\\mathbf {z}_1,\\mathbf {x}_2,\\mathbf {z}_2) \\\\&= \\frac{1}{2}(\\mathbf {x}_1-\\mathbf {x}_1^*)^T(\\mathbf {x}_1-\\mathbf {x}_1^*) +\\frac{1}{2}(\\mathbf {z}_1-\\mathbf {z}_1^*)^T(\\mathbf {z}_1-\\mathbf {z}_1^*)\\\\& \\quad + \\frac{1}{2}(\\mathbf {x}_2-\\mathbf {x}_2^*)^T(\\mathbf {x}_2-\\mathbf {x}_2^*) +\\frac{1}{2}(\\mathbf {z}_2-\\mathbf {z}_2^*)^T(\\mathbf {z}_2-\\mathbf {z}_2^*).$ The function $ V $ is smooth.",
"Next, we examine its set-valued Lie derivative along $ \\Psi _{\\textup {Nash-undir}}$ .",
"Let $ \\xi \\in \\widetilde{\\mathcal {L}}_{\\Psi _{\\textup {Nash-undir}}}V(\\mathbf {x}_1,\\mathbf {z}_1,\\mathbf {x}_2,\\mathbf {z}_2) $ .",
"By definition, there exists $ v \\in \\Psi _{\\textup {Nash-undir}}(\\mathbf {x}_1,\\mathbf {z}_1,\\mathbf {x}_2,\\mathbf {z}_2) $ , given by $v=(&-\\mathbf {L}_1 \\mathbf {x}_1 -\\mathbf {L}_1 \\mathbf {z}_1 +g_{1,(\\mathbf {x}_1,\\mathbf {x}_2)}, \\\\& -\\mathbf {L}_2 \\mathbf {x}_2 -\\mathbf {L}_2 \\mathbf {z}_2 -g_{2,(\\mathbf {x}_1,\\mathbf {x}_2)} ,\\mathbf {L}_1\\mathbf {x}_1, \\mathbf {L}_2\\mathbf {x}_2),$ where $ g_{1,(\\mathbf {x}_1,\\mathbf {x}_2)} \\in \\partial _{\\mathbf {x}_1}U(\\mathbf {x}_1,\\mathbf {x}_2) $ and $ g_{2,(\\mathbf {x}_1,\\mathbf {x}_2)}\\in \\partial _{\\mathbf {x}_2}U (\\mathbf {x}_1,\\mathbf {x}_2) $ , such that $\\xi & =v\\cdot \\nabla V(\\mathbf {x}_1,\\mathbf {z}_1,\\mathbf {x}_2,\\mathbf {z}_2)\\\\&= (\\mathbf {x}_1-\\mathbf {x}_1^*)^T(-\\mathbf {L}_1 \\mathbf {x}_1 -\\mathbf {L}_1 \\mathbf {z}_1+g_{1,(\\mathbf {x}_1,\\mathbf {x}_2)}) \\\\&+ (\\mathbf {x}_2-\\mathbf {x}_2^*)^T(-\\mathbf {L}_2 \\mathbf {x}_2-\\mathbf {L}_2 \\mathbf {z}_2 -g_{2,(\\mathbf {x}_1,\\mathbf {x}_2)} )\\\\& \\quad + (\\mathbf {z}_1-\\mathbf {z}_1^*)^T \\mathbf {L}_1\\mathbf {x}_1 +(\\mathbf {z}_2-\\mathbf {z}_2^*)^T\\mathbf {L}_2\\mathbf {x}_2.$ Note that $ -\\mathbf {L}_1 \\mathbf {x}_1 -\\mathbf {L}_1 \\mathbf {z}_1+g_{1,(\\mathbf {x}_1,\\mathbf {x}_2)}\\in -\\partial _{\\mathbf {x}_1}F_1(\\mathbf {x}_1,\\mathbf {z}_1,\\mathbf {x}_2)$ , $\\mathbf {L}_1\\mathbf {x}_1\\in \\partial _{\\mathbf {z}_1}F_1(\\mathbf {x}_1,\\mathbf {z}_1,\\mathbf {x}_2) $ , $ -\\mathbf {L}_2 \\mathbf {x}_2 -\\mathbf {L}_2 \\mathbf {z}_2-g_{2,(\\mathbf {x}_1,\\mathbf {x}_2)} \\in -\\partial _{\\mathbf {x}_2}F_2(\\mathbf {x}_1,\\mathbf {z}_2,\\mathbf {x}_2) $ , and $\\mathbf {L}_2\\mathbf {x}_2\\in \\partial _{\\mathbf {z}_2}F_2(\\mathbf {x}_2,\\mathbf {z}_2,\\mathbf {x}_1)$ .",
"Using the first-order convexity property of $ F_1 $ and $ F_2 $ in their first two arguments, one gets $\\xi & \\le F_1(\\mathbf {x}_1^*,\\mathbf {z}_1,\\mathbf {x}_2)-F_1(\\mathbf {x}_1,\\mathbf {z}_1,\\mathbf {x}_2)+F_2(\\mathbf {x}_2^*,\\mathbf {z}_2,\\mathbf {x}_1)\\\\& -F_2(\\mathbf {x}_2,\\mathbf {z}_2,\\mathbf {x}_1)+F_1(\\mathbf {x}_1,\\mathbf {z}_1,\\mathbf {x}_2)-F_1(\\mathbf {x}_1,\\mathbf {z}^*_1,\\mathbf {x}_2)\\\\&+F_2(\\mathbf {x}_2,\\mathbf {z}_2,\\mathbf {x}_1)-F_2(\\mathbf {x}_2,\\mathbf {z}_2^*,\\mathbf {x}_1).$ Expanding each term and using the fact that $(\\mathbf {x}_1^*,\\mathbf {z}_1^*,\\mathbf {x}_2^*,\\mathbf {z}_2^*) $ $ \\in \\operatorname{Eq}(\\Psi _{\\textup {Nash-undir}})$ , we simplify this inequality as $\\xi & \\le -\\tilde{U}(\\mathbf {x}_1^*,\\mathbf {x}_2) +\\tilde{U}(\\mathbf {x}_1,\\mathbf {x}^*_2)-\\mathbf {z}^*_1\\mathbf {L}_1\\mathbf {x}_1\\\\& \\quad -\\frac{1}{2}\\mathbf {x}_1\\mathbf {L}_1\\mathbf {x}_1-\\mathbf {z}^*_2\\mathbf {L}_2\\mathbf {x}_2-\\frac{1}{2}\\mathbf {x}_2\\mathbf {L}_2\\mathbf {x}_2 .$ By rearranging, we thus have $\\xi \\le -F_2(\\mathbf {x}_2,\\mathbf {z}_2^*,\\mathbf {x}_1^*)-F_1(\\mathbf {x}_1,\\mathbf {z}_1^*,\\mathbf {x}_2^*) .$ Next, since $F_2(\\mathbf {x}_1^*,\\mathbf {z}_2^*,\\mathbf {x}_2^*)+F_1(\\mathbf {x}_2^*,\\mathbf {z}_2^*,\\mathbf {x}_1^*)=0$ , we have $\\xi & \\le F_1(\\mathbf {x}_1^*,\\mathbf {z}_1^*,\\mathbf {x}_2^*)-F_1(\\mathbf {x}_1,\\mathbf {z}_1^*,\\mathbf {x}_2^*)\\\\& \\quad +F_2(\\mathbf {x}_2^*,\\mathbf {z}_2^*,\\mathbf {x}_1^*)-F_2(\\mathbf {x}_2,\\mathbf {z}_2^*,\\mathbf {x}^*_1),$ yielding that $ \\xi \\le 0 $ .",
"As a result, $ \\max \\widetilde{\\mathcal {L}}_{\\Psi _{\\textup {Nash-undir}}}V (\\mathbf {x}_1,\\mathbf {z}_1, \\mathbf {x}_2,\\mathbf {z}_2) \\le 0.$ As a by-product, we conclude that the trajectories of () are bounded.",
"By virtue of the set-valued version of the LaSalle Invariance Principle, cf.",
"Theorem REF , any trajectory of () starting from an initial condition $ (\\mathbf {x}_1^0,\\mathbf {z}_1^0,\\mathbf {x}_2^0,\\mathbf {z}_2^0) $ converges to the largest positively invariant set $ M $ in $S_{\\Psi _{\\textup {Nash-undir}},V} \\cap V^{-1}(\\le V(\\mathbf {x}_1^0,\\mathbf {z}_1^0,\\mathbf {x}_2^0,\\mathbf {z}_2^0))$ .",
"Let $(\\mathbf {x}_1,\\mathbf {z}_1,\\mathbf {x}_2,\\mathbf {z}_2) \\in M$ .",
"Because $M \\subset S_{\\Psi _{\\textup {Nash-undir}},V} $ , then $F_1(\\mathbf {x}_1^*,\\mathbf {z}_1^*,\\mathbf {x}_2^*)-F_1(\\mathbf {x}_1,\\mathbf {z}_1^*,\\mathbf {x}_2^*)=0, $ i.e., $-\\tilde{U}(\\mathbf {x}_1^*,\\mathbf {x}_2^*) +\\tilde{U}(\\mathbf {x}_1,\\mathbf {x}_2^*)-\\mathbf {x}_1^T\\mathbf {L}_1\\mathbf {z}_1^* -\\frac{1}{2}\\mathbf {x}_1^T\\mathbf {L}_1\\mathbf {x}_1 = 0.$ Define now $ G_1:{\\mathbb {R}}^{n_1d_1} \\times {\\mathbb {R}}^{n_1d_1} \\times {\\mathbb {R}}^{n_2d_2} \\rightarrow {\\mathbb {R}} $ by $G_1(\\mathbf {x}_1,\\mathbf {z}_1,\\mathbf {x}_2) =F_1(\\mathbf {x}_1,\\mathbf {z}_1,\\mathbf {x}_2)-\\frac{1}{2}\\mathbf {x}_1^T\\mathbf {L}_1\\mathbf {x}_1$ .",
"$G_1$ is convex in its first argument and linear in its second.",
"Furthermore, for fixed $\\mathbf {x}_2$ , the map $(\\mathbf {x}_1,\\mathbf {z}_1)\\mapsto G_1(\\mathbf {x}_1,\\mathbf {z}_1,\\mathbf {x}_2)$ has the same saddle points as $(\\mathbf {x}_1,\\mathbf {z}_1) \\mapsto F_1(\\mathbf {x}_1,\\mathbf {z}_1,\\mathbf {x}_2)$ .",
"As a result, $ G_1(\\mathbf {x}_1^*,\\mathbf {z}_1^*,\\mathbf {x}_2^*) -G_1(\\mathbf {x}_1,\\mathbf {z}^*_1,\\mathbf {x}_2^*) \\le 0 $ , or equivalently, $-\\tilde{U}(\\mathbf {x}_1^*,\\mathbf {x}_2^*) + \\tilde{U}(\\mathbf {x}_1,\\mathbf {x}_2^*) - \\mathbf {x}_1^T\\mathbf {L}_1 \\mathbf {z}_1^* \\le 0 $ .",
"Combining this with (REF ), we have that $ \\mathbf {L}_1\\mathbf {x}_1=0 $ and $-\\tilde{U}(\\mathbf {x}_1^*,\\mathbf {x}_2^*)+\\tilde{U}(\\mathbf {x}_1,\\mathbf {x}_2^*)= 0 $ .",
"Since $ \\tilde{U}$ is strictly concave in its first argument $\\mathbf {x}_1=\\mathbf {x}_1^* $ .",
"A similar argument establishes that $\\mathbf {x}_2=\\mathbf {x}_2^* $ .",
"Using now the fact that $ M $ is weakly positively invariant, one can deduce that $ \\mathbf {L}_\\ell \\mathbf {z}_\\ell \\in -\\partial _{\\mathbf {x}_\\ell } \\tilde{U}(\\mathbf {x}_1,\\mathbf {x}_2) $ , for $ \\ell \\in \\lbrace 1,2\\rbrace $ , and thus $ (\\mathbf {x}_1,\\mathbf {z}_1, \\mathbf {x}_2,\\mathbf {z}_2)\\in \\operatorname{Eq}(\\Psi _{\\textup {Nash-undir}}) $ .",
"$\\Box $"
],
[
"Distributed convergence to Nash equilibria for directed\ntopologies",
"Interestingly, the saddle-point dynamics () fails to converge when transcribed to the directed network setting.",
"This observation is a consequence of the following result, which studies the stability of the linearization of the dynamics (), when the payoff functions have no contribution to the linear part.",
"Lemma 5.1 (Necessary condition for the convergence of () on digraphs): Let $\\Sigma _\\ell $ be strongly connected and $ f^i_\\ell =0$ , $ i\\in \\lbrace 1,\\ldots , n_\\ell \\rbrace $ , for $ \\ell \\in \\lbrace 1,2\\rbrace $ .",
"Then, the set of network agreement configurations $\\mathcal {S}_{\\text{agree}}= \\lbrace (\\mathbf {1}_{n_1} \\otimes x_1, \\mathbf {1}_{n_1} \\otimes z_1,\\mathbf {1}_{n_2} \\otimes x_2, \\mathbf {1}_{n_2}\\otimes z_2) \\in ({\\mathbb {R}}^{n_1 d_1})^2 \\times ({\\mathbb {R}}^{n_2d_2})^2 \\; | \\; x_\\ell ,z_\\ell \\in {\\mathbb {R}}^{d_\\ell }, \\ell \\in \\lbrace 1,2\\rbrace \\rbrace $ , is stable under () iff, for any nonzero eigenvalue $\\lambda $ of the Laplacian $ \\mathsf {L}_\\ell $ , $ \\ell \\in \\lbrace 1,2\\rbrace $ , one has $\\sqrt{3}|\\mathrm {Im}(\\lambda )| \\le \\mathrm {Re}(\\lambda ) $ .",
"Proof.",
"In this case, () is linear with matrix $\\begin{pmatrix}\\left({\\begin{matrix} -1 & -1\\\\1 & 0\\end{matrix}}\\right)\\otimes \\mathbf {L}_1& 0 \\\\0 & \\left({\\begin{matrix} -1 & -1\\\\1 & 0\\end{matrix}}\\right)\\otimes \\mathbf {L}_2\\end{pmatrix}$ and has $\\mathcal {S}_{\\text{agree}}$ as equilibria.",
"The eigenvalues of (REF ) are of the form $\\lambda _\\ell \\, \\big (\\frac{-1}{2}\\pm \\frac{\\sqrt{3}}{2}i\\big )$ , with $\\lambda _\\ell $ an eigenvalue of $\\mathbf {L}_\\ell $ , for $\\ell \\in \\lbrace 1,2\\rbrace $ (since the eigenvalues of a Kronecker product are the product of the eigenvalues of the corresponding matrices).",
"Since $\\mathbf {L}_\\ell = \\mathsf {L}_\\ell \\otimes \\mathsf {I}_{d_\\ell }$ , each eigenvalue of $\\mathbf {L}_\\ell $ is an eigenvalue of $ \\mathsf {L}_\\ell $ .",
"The result follows by noting that $ \\mathrm {Re}\\big (\\lambda _\\ell \\big (\\frac{-1}{2}\\pm \\frac{\\sqrt{3}}{2}i \\big )\\big ) = \\frac{1}{2} (\\mp \\sqrt{3}\\mathrm {Im}(\\lambda _\\ell )-\\mathrm {Re}(\\lambda _\\ell ))$ .$\\Box $ It is not difficult to construct examples of strictly concave-convex functions that have zero contribution to the linearization of () around the solution.",
"Therefore, such systems cannot be convergent if they fail the necessary condition identified in Lemma REF .",
"The counterexample provided in our recent paper [13] of strongly connected, weight-balanced digraphs that do not meet the stability criterium of Lemma REF is therefore valid in this context too.",
"From here on, we assume that the payoff functions are differentiable.",
"We elaborate on the reasons for this assumption in Remark REF later.",
"Motivated by the observation made in Lemma REF , we introduce a parameter $ \\alpha \\in {\\mathbb {R}}_{>0}$ in the dynamics of () as $\\dot{\\mathbf {x}}_1 + \\alpha \\mathbf {L}_1\\mathbf {x}_1 + \\mathbf {L}_1\\mathbf {z}_1 & = \\nabla \\tilde{U}(\\mathbf {x}_1,\\mathbf {x}_2),\\\\\\dot{\\mathbf {z}}_1 & = \\mathbf {L}_1 \\mathbf {x}_1,\\\\\\dot{\\mathbf {x}}_2 + \\alpha \\mathbf {L}_2\\mathbf {x}_2 + \\mathbf {L}_2s\\mathbf {z}_2 & = -\\nabla \\tilde{U}(\\mathbf {x}_1,\\mathbf {x}_2),\\\\\\dot{\\mathbf {z}}_2 & = \\mathbf {L}_2 \\mathbf {x}_2.$ We next show that a suitable choice of $\\alpha $ makes the dynamics convergent to the Nash equilibrium.",
"Theorem 5.2 (Distributed convergence to Nash equilibria for directed networks): Consider the zero-sum game $ \\mathbf {G}_{\\textup {adv-net}}=(\\Sigma _1,\\Sigma _2,\\Sigma _{\\textup {eng}},U)$ , with $ \\Sigma _1 $ and $ \\Sigma _2 $ strongly connected and weight-balanced digraphs, $ \\mathsf {X}_1 \\subset {\\mathbb {R}}^{d_1} $ , $ \\mathsf {X}_2 \\subset {\\mathbb {R}}^{d_2} $ compact and convex, and $U:\\mathsf {X}_1 \\times \\mathsf {X}_2 \\rightarrow {\\mathbb {R}} $ strictly concave-convex and differentiable with globally Lipschitz gradient.",
"Assume $\\mathbf {G}_{\\textup {adv-net}}$ can be lifted to $\\mathbf {\\tilde{G}}_{\\textup {adv-net}}$ such that $\\tilde{U}$ is differentiable and has a globally Lipschitz gradient.",
"Define $ h:{\\mathbb {R}}_{>0} \\rightarrow {\\mathbb {R}} $ by $h(r) =& \\frac{1}{2}\\Lambda _{*}^{\\min } \\Big (\\sqrt{\\Big (\\frac{r^4+3r^2+2}{r}\\Big )^2-4} -\\frac{r^4+3r^2+2}{r} \\Big )\\nonumber \\\\&+\\frac{K r^2}{(1+r^2)},$ where $ \\Lambda _{*}^{\\min } =\\min _{\\ell =1,2}\\lbrace \\Lambda _{*}(\\mathsf {L}_\\ell +\\mathsf {L}_\\ell ^T)\\rbrace $ , $\\Lambda _{*}(\\cdot ) $ denotes the smallest non-zero eigenvalue and $ K \\in {\\mathbb {R}}_{>0}$ is the Lipschitz constant of the gradient of $ \\tilde{U}$ .",
"Then there exists $ \\beta ^* \\in {\\mathbb {R}}_{>0}$ with $ h(\\beta ^*)=0 $ such that for all $ 0 < \\beta < \\beta ^* $ , the projection onto the first and third components of the solutions of () with $\\alpha =\\frac{\\beta ^2+2}{\\beta } $ asymptotically converge to agreement on the Nash equilibrium of $\\mathbf {G}_{\\textup {adv-net}}$ .",
"Proof.",
"Similarly to the proof of Theorem REF , we assume, without loss of generality, that agents in $ \\Sigma _1 $ have access to $ \\mathbf {x}_2 $ and agents in $ \\Sigma _2 $ to $ \\mathbf {x}_1 $ .",
"For convenience, we denote the dynamics described in () by $\\Psi _{\\textup {Nash-dir}}:({\\mathbb {R}}^{d_1n_1})^2 \\times ({\\mathbb {R}}^{d_2n_2})^2 \\rightarrow ({\\mathbb {R}}^{d_1n_1})^2 \\times ({\\mathbb {R}}^{d_2n_2})^2 $ .",
"Let $(\\mathbf {x}_1^0,\\mathbf {z}_1^0,\\mathbf {x}_2^0,\\mathbf {z}_2^0) $ be an arbitrary initial condition.",
"Note that the set $ W_{\\mathbf {z}^0_1,\\mathbf {z}^0_2} $ defined by (REF ) is invariant under the evolutions of ().",
"By an argument similar to the one in the proof of Theorem REF , there exists a unique solution to (REF ), which we denote by $ \\mathbf {x}_1^*=\\mathbf {1}_{n_1} \\otimes x^*_1$ and $\\mathbf {x}_2^*=\\mathbf {1}_{n_2} \\otimes x^*_2 $ .",
"By Proposition REF (i), there exists $(\\mathbf {x}^*_1,\\mathbf {z}^*_1,\\mathbf {x}^*_2,\\mathbf {z}^*_2) \\in \\operatorname{Eq}(\\Psi _{\\textup {Nash-dir}})\\cap W_{\\mathbf {z}^0_1,\\mathbf {z}^0_2} $ .",
"Consider the function $ V:({\\mathbb {R}}^{d_1n_1})^2\\times ({\\mathbb {R}}^{d_2n_2})^2 \\rightarrow {\\mathbb {R}}_{\\ge 0} $ , $V(\\mathbf {x}_1&,\\mathbf {z}_1,\\mathbf {x}_2,\\mathbf {z}_2)\\\\&=\\frac{1}{2}(\\mathbf {x}_1-\\mathbf {x}^*_1)^T(\\mathbf {x}_1-\\mathbf {x}^*_1)+\\frac{1}{2}(\\mathbf {x}_2-\\mathbf {x}^*_2)^T(\\mathbf {x}_2-\\mathbf {x}^*_2)\\\\& \\quad +\\frac{1}{2}(\\mathbf {y}_{(\\mathbf {x}_1,\\mathbf {z}_1)}-\\mathbf {y}_{(\\mathbf {x}^*_1,\\mathbf {z}^*_1)})^T(\\mathbf {y}_{(\\mathbf {x}_1,\\mathbf {z}_1)}-\\mathbf {y}_{(\\mathbf {x}^*_1,\\mathbf {z}^*_1)}),\\\\& \\quad +\\frac{1}{2}(\\mathbf {y}_{(\\mathbf {x}_2,\\mathbf {z}_2)}-\\mathbf {y}_{(\\mathbf {x}^*_2,\\mathbf {z}^*_2)})^T(\\mathbf {y}_{(\\mathbf {x}_2,\\mathbf {z}_2)}-\\mathbf {y}_{(\\mathbf {x}^*_2,\\mathbf {z}^*_2)}),$ where $ \\mathbf {y}_{(\\mathbf {x}_\\ell ,\\mathbf {z}_\\ell )}=\\beta \\mathbf {x}_\\ell +\\mathbf {z}_\\ell $ , $ \\ell \\in \\lbrace 1,2\\rbrace $ , and $ \\beta \\in {\\mathbb {R}}_{>0}$ satisfies $\\beta ^2-\\alpha \\beta +2=0 $ .",
"This function is quadratic, hence smooth.",
"Next, we consider $\\xi =\\mathcal {L}_{\\Psi _{\\textup {Nash-dir}}}V(\\mathbf {x}_1,\\mathbf {z}_1,\\mathbf {x}_2,\\mathbf {z}_2)$ given by $\\xi & =(-\\alpha \\mathbf {L}_1\\mathbf {x}_1-\\mathbf {L}_1 \\mathbf {z}_1 +\\nabla \\tilde{U}(\\mathbf {x}_1,\\mathbf {x}_2),\\mathbf {L}_1\\mathbf {x}_1,-\\alpha \\mathbf {L}_2\\mathbf {x}_2\\\\& \\quad -\\mathbf {L}_2 \\mathbf {z}_2 -\\nabla \\tilde{U}(\\mathbf {x}_1,\\mathbf {x}_2),\\mathbf {L}_2\\mathbf {x}_2) \\cdot \\nabla V(\\mathbf {x}_1,\\mathbf {z}_1,\\mathbf {x}_2,\\mathbf {z}_2).$ After some manipulation, one can show that $\\xi &= \\sum _{\\ell =1}^2\\frac{1}{2} (\\mathbf {x}_\\ell -\\mathbf {x}^*_\\ell ,\\mathbf {y}_{(\\mathbf {x}_\\ell ,\\mathbf {z}_\\ell )}-\\mathbf {y}_{(\\mathbf {x}^*_\\ell ,\\mathbf {z}^*_\\ell )})^TA_\\ell ( \\mathbf {x}_l, \\mathbf {y}_{(\\mathbf {x}_\\ell ,\\mathbf {z}_\\ell )} )\\\\&+\\sum _{\\ell =1}^2\\frac{1}{2}( \\mathbf {x}_\\ell ^T,\\mathbf {y}_{(\\mathbf {x}_\\ell ,\\mathbf {z}_\\ell )}^T ) A_\\ell ^T (\\mathbf {x}_\\ell -\\mathbf {x}^*_\\ell ,\\mathbf {y}_{(\\mathbf {x}_\\ell ,\\mathbf {z}_\\ell )}-\\mathbf {y}_{(\\mathbf {x}^*_\\ell ,\\mathbf {z}^*_\\ell )})\\\\&+\\sum _{\\ell =1}^2(-1)^{j-1}(\\mathbf {x}_\\ell -\\mathbf {x}^*_\\ell )^T\\nabla _{\\mathbf {x}_\\ell }\\tilde{U}(\\mathbf {x}_1,\\mathbf {x}_2)\\\\&+\\sum _{\\ell =1}^2(-1)^{j-1}\\beta (\\mathbf {y}_{(\\mathbf {x}_\\ell ,\\mathbf {z}_\\ell )} -\\mathbf {y}_{(\\mathbf {x}^*_\\ell ,\\mathbf {z}^*_\\ell )})^T\\nabla _{\\mathbf {x}_\\ell }\\tilde{U}(\\mathbf {x}_1,\\mathbf {x}_2),$ where $ A_\\ell $ , $ \\ell \\in \\lbrace 1,2\\rbrace $ , is $A_\\ell =\\begin{pmatrix}-(\\alpha -\\beta ) \\mathbf {L}_\\ell & - \\mathbf {L}_\\ell \\\\(-\\beta (\\alpha -\\beta )+1)\\mathbf {L}_\\ell & -\\beta \\mathbf {L}_\\ell \\end{pmatrix}.$ This equation can be written as $\\xi &=\\sum _{\\ell =1}^2\\frac{1}{2} ( \\mathbf {x}_\\ell -\\mathbf {x}^*_\\ell ,\\mathbf {y}_{(\\mathbf {x}_\\ell ,\\mathbf {z}_\\ell )}-\\mathbf {y}_{(\\mathbf {x}^*_\\ell ,\\mathbf {z}^*_\\ell )})^T\\\\&\\qquad \\qquad Q_\\ell ( \\mathbf {x}_\\ell -\\mathbf {x}^*_\\ell ,\\mathbf {y}_{(\\mathbf {x}_\\ell ,\\mathbf {z}_\\ell )}-\\mathbf {y}_{(\\mathbf {x}^*_\\ell ,\\mathbf {z}^*_\\ell )})\\\\& + \\sum _{\\ell =1}^2 (\\mathbf {x}_\\ell -\\mathbf {x}^*_\\ell ,\\mathbf {y}_{(\\mathbf {x}_\\ell ,\\mathbf {z}_\\ell )}-\\mathbf {y}_{(\\mathbf {x}^*_\\ell ,\\mathbf {z}^*_\\ell )})^T A_\\ell ( \\mathbf {x}^*_\\ell , \\mathbf {y}_{(\\mathbf {x}^*_\\ell ,\\mathbf {z}^*_\\ell )} )\\\\&+\\sum _{\\ell =1}^2(-1)^{j-1}(\\mathbf {x}_\\ell -\\mathbf {x}^*_\\ell )^T\\nabla _{\\mathbf {x}_\\ell }\\tilde{U}(\\mathbf {x}_1,\\mathbf {x}_2)\\\\&+\\sum _{\\ell =1}^2(-1)^{j-1}\\beta (\\mathbf {y}_{(\\mathbf {x}_\\ell ,\\mathbf {z}_\\ell )}-\\mathbf {y}_{(\\mathbf {x}^*_\\ell ,\\mathbf {z}^*_\\ell )})^T\\nabla _{\\mathbf {x}_\\ell } \\tilde{U}(\\mathbf {x}_1,\\mathbf {x}_2),$ where $ Q_\\ell $ , $ \\ell \\in \\lbrace 1,2\\rbrace $ , is given by $Q_\\ell =(\\mathbf {L}_\\ell +\\mathbf {L}_\\ell ^T)\\otimes \\begin{pmatrix}-(\\frac{\\beta ^2+2}{\\beta }-\\beta ) & -1\\\\-1 & -\\beta \\end{pmatrix}.$ Note that, we have $A_1( \\mathbf {x}^*_1, \\mathbf {y}_{(\\mathbf {x}^*_1,\\mathbf {z}^*_1)} )&=-(\\mathbf {L}_1\\mathbf {y}_{(\\mathbf {x}^*_1,\\mathbf {z}^*_1)},\\beta \\mathbf {L}_1\\mathbf {y}_{(\\mathbf {x}^*_1,\\mathbf {z}^*_1)} )\\\\&= -(\\nabla _{\\mathbf {x}_1} \\tilde{U}(\\mathbf {x}^*_1,\\mathbf {x}_2),\\beta \\nabla _{\\mathbf {x}_1} \\tilde{U}(\\mathbf {x}^*_1,\\mathbf {x}_2)),\\\\A_2( \\mathbf {x}^*_2, \\mathbf {y}_{(\\mathbf {x}^*_2,\\mathbf {z}^*_2)} )&=-(\\mathbf {L}_2\\mathbf {y}_{(\\mathbf {x}^*_2,\\mathbf {z}^*_2)},\\beta \\mathbf {L}_2\\mathbf {y}_{(\\mathbf {x}^*_2,\\mathbf {z}^*_2)} )\\\\&=(\\nabla _{\\mathbf {x}_2} \\tilde{U}(\\mathbf {x}_1,\\mathbf {x}^*_2),\\beta \\nabla _{\\mathbf {x}_2} \\tilde{U}(\\mathbf {x}_1,\\mathbf {x}^*_2)).$ Thus, after substituting for $ \\mathbf {y}_{(\\mathbf {x}_\\ell ,\\mathbf {z}_\\ell )} $ , we have $& \\xi =\\sum _{\\ell =1}^2\\frac{1}{2} (\\mathbf {x}_\\ell -\\mathbf {x}^*_\\ell ,\\mathbf {z}_\\ell -\\mathbf {z}^*_\\ell )^T \\tilde{Q}_\\ell (\\mathbf {x}_\\ell -\\mathbf {x}^*_\\ell , \\mathbf {z}_\\ell -\\mathbf {z}^*_\\ell ) \\nonumber \\\\&\\; +(1+\\beta ^2)(\\mathbf {x}_1-\\mathbf {x}^*_1)^T(\\nabla _{\\mathbf {x}_1}\\tilde{U}(\\mathbf {x}_1, \\mathbf {x}_2)-\\nabla _{\\mathbf {x}_1}\\tilde{U}(\\mathbf {x}^*_1,\\mathbf {x}_2)) \\nonumber \\\\&\\; -(1+\\beta ^2)(\\mathbf {x}_2-\\mathbf {x}^*_2)^T(\\nabla _{\\mathbf {x}_2}\\tilde{U}(\\mathbf {x}_1, \\mathbf {x}_2)-\\nabla _{\\mathbf {x}_2}\\tilde{U}(\\mathbf {x}_1,\\mathbf {x}^*_2)) \\nonumber \\\\&\\; +\\beta (\\mathbf {z}_1-\\mathbf {z}^*_1)^T(\\nabla _{\\mathbf {x}_1}\\tilde{U}(\\mathbf {x}_1,\\mathbf {x}_2) - \\nabla _{\\mathbf {x}_1}\\tilde{U}(\\mathbf {x}^*_1,\\mathbf {x}_2)) \\nonumber \\\\&\\; -\\beta (\\mathbf {z}_2-\\mathbf {z}^*_2)^T(\\nabla _{\\mathbf {x}_2}\\tilde{U}(\\mathbf {x}_1,\\mathbf {x}_2) - \\nabla _{\\mathbf {x}_2}\\tilde{U}(\\mathbf {x}_1,\\mathbf {x}^*_2)),$ where $\\tilde{Q}_\\ell =\\begin{pmatrix}-\\beta ^3-(\\frac{\\beta ^2+2}{\\beta })-\\beta & -(1+\\beta ^2)\\\\-(1+\\beta ^2) & -\\beta \\\\\\end{pmatrix}\\otimes (\\mathbf {L}_\\ell +\\mathbf {L}_\\ell ^T),$ for $ \\ell \\in \\lbrace 1,2\\rbrace $ .",
"Each eigenvalue of $ \\tilde{Q}_\\ell $ is of the form $\\tilde{\\eta }_\\ell = \\lambda _\\ell \\frac{-(\\beta ^4+3\\beta ^2+2) \\pm \\sqrt{(\\beta ^4+3\\beta ^2+2)^2-4\\beta ^2}}{2\\beta },$ where $ \\lambda _\\ell $ is an eigenvalue of $ \\mathsf {L}_\\ell +\\mathsf {L}_\\ell ^T$ , $ \\ell \\in \\lbrace 1,2\\rbrace $ .",
"Using now Theorem REF twice, one for $ (\\mathbf {x}_1,\\mathbf {x}_2) $ , $ (\\mathbf {x}_1^*,\\mathbf {x}_2)$ , and another one for $ (\\mathbf {x}_1,\\mathbf {x}_2) $ , $ (\\mathbf {x}_1,\\mathbf {x}_2^*) $ , we have $&(\\mathbf {x}_1-\\mathbf {x}^*_1)^T(\\nabla _{\\mathbf {x}_1} \\tilde{U}(\\mathbf {x}_1,\\mathbf {x}_2)-\\nabla _{\\mathbf {x}_1} \\tilde{U}(\\mathbf {x}^*_1,\\mathbf {x}_2))\\le \\\\&\\quad -\\frac{1}{K} \\left(||\\nabla _{\\mathbf {x}_1} \\tilde{U}(\\mathbf {x}_1,\\mathbf {x}_2)-\\nabla _{\\mathbf {x}_1}\\tilde{U}(\\mathbf {x}^*_1,\\mathbf {x}_2)||^2\\right),\\\\-&(\\mathbf {x}_2-\\mathbf {x}^*_2)^T(\\nabla _{\\mathbf {x}_2} \\tilde{U}(\\mathbf {x}_1,\\mathbf {x}_2)-\\nabla _{\\mathbf {x}_2} \\tilde{U}(\\mathbf {x}_1,\\mathbf {x}_2^*))\\le \\\\&\\quad -\\frac{1}{K} \\left(||\\nabla _{\\mathbf {x}_2} \\tilde{U}(\\mathbf {x}_1,\\mathbf {x}_2)-\\nabla _{\\mathbf {x}_2}\\tilde{U}(\\mathbf {x}_1,\\mathbf {x}^*_2)||^2\\right),$ where $ K \\in {\\mathbb {R}}_{>0}$ is the Lipschitz constant of $\\nabla \\tilde{U}$ .",
"We thus conclude that $&\\xi \\le \\sum _{\\ell =1}^2\\frac{1}{2} ( \\mathbf {x}_\\ell -\\mathbf {x}^*_\\ell ,\\mathbf {z}_\\ell -\\mathbf {z}^*_\\ell )^T \\tilde{Q}_\\ell (\\mathbf {x}_\\ell -\\mathbf {x}^*_\\ell , \\mathbf {z}_\\ell -\\mathbf {z}^*_\\ell )\\\\&-\\frac{(1+\\beta ^2)}{K} \\big ( ||\\nabla _{\\mathbf {x}_1}\\tilde{U}(\\mathbf {x}_1, \\mathbf {x}_2)-\\nabla _{\\mathbf {x}_1}\\tilde{U}(\\mathbf {x}^*_1,\\mathbf {x}_2)||^2 \\\\& +||\\nabla _{\\mathbf {x}_2} \\tilde{U}(\\mathbf {x}_1,\\mathbf {x}_2)-\\nabla _{\\mathbf {x}_2} \\tilde{U}(\\mathbf {x}_1,\\mathbf {x}^*_2)||^2 \\big )\\nonumber \\\\&+\\beta (\\mathbf {z}_1-\\mathbf {z}^*_1)^T(\\nabla _{\\mathbf {x}_1}\\tilde{U}(\\mathbf {x}_1,\\mathbf {x}_2) - \\nabla _{\\mathbf {x}_1}\\tilde{U}(\\mathbf {x}^*_1,\\mathbf {x}_2))\\\\&-\\beta (\\mathbf {z}_2-\\mathbf {z}^*_2)^T(\\nabla _{\\mathbf {x}_2}\\tilde{U}(\\mathbf {x}_1,\\mathbf {x}_2) - \\nabla _{\\mathbf {x}_2}\\tilde{U}(\\mathbf {x}_1,\\mathbf {x}^*_2)).\\nonumber $ One can write this inequality as displayed in (REF ), Figure: NO_CAPTIONwhere $X=(&\\mathbf {x}_1-\\mathbf {x}^*_1,\\mathbf {z}_1-\\mathbf {z}^*_1,\\mathbf {x}_2-\\mathbf {x}^*_2,\\mathbf {z}_2-\\mathbf {z}^*_2,\\\\&\\nabla _{\\mathbf {x}_1} \\tilde{U}(\\mathbf {x}_1, \\mathbf {x}_2) -\\nabla _{\\mathbf {x}_1}\\tilde{U}(\\mathbf {x}^*_1,\\mathbf {x}_2),\\\\&\\nabla _{\\mathbf {x}_2} \\tilde{U}(\\mathbf {x}_1, \\mathbf {x}_2) -\\nabla _{\\mathbf {x}_2} \\tilde{U}(\\mathbf {x}_1,\\mathbf {x}^*_2)) .$ Since $ (\\mathbf {x}_1, \\mathbf {z}_1,\\mathbf {x}_2,\\mathbf {z}_2) \\in W_{\\mathbf {z}^0_1,\\mathbf {z}^0_2} $ , we have $(\\mathbf {1}_{n_\\ell }^T \\otimes \\mathsf {I}_{d_\\ell }) (\\mathbf {z}_\\ell - \\mathbf {z}_\\ell ^*) =\\mathbf {0}_{d_\\ell }$ , $\\ell \\in \\lbrace 1,2\\rbrace $ , and hence it is enough to establish that $\\mathbf {Q}$ is negative semidefinite on the subspace $\\mathcal {W}=\\lbrace (v_1,v_2,v_3,v_4,v_5,v_6) \\in ({\\mathbb {R}}^{n_1d_1})^2\\times ({\\mathbb {R}}^{n_2d_2})^2\\times {\\mathbb {R}}^{n_1d_1}\\times {\\mathbb {R}}^{n_2d_2} \\; | \\; (\\mathbf {1}_{n_1}^T \\otimes \\mathsf {I}_{d_1}) v_2 = \\mathbf {0}_{d_1}, (\\mathbf {1}_{n_2}^T \\otimes \\mathsf {I}_{d_2}) v_4 =\\mathbf {0}_{n_2}\\rbrace $ .",
"Using the fact that $-\\tfrac{1}{K} (1+\\beta ^2) \\mathsf {I}_{n_\\ell d_\\ell } $ is invertible, for $ \\ell \\in \\lbrace 1,2\\rbrace $ , we can express $\\mathbf {Q}$ as $&\\mathbf {Q}=\\\\&\\ N\\underbrace{\\begin{pmatrix}\\bar{Q}_1 & 0 & 0 &0\\\\0 & \\bar{Q}_2 & 0 & 0\\\\0 & 0 & -\\tfrac{1}{K} (1+\\beta ^2) \\mathsf {I}_{n_1d_1} & 0\\\\0 & 0 & 0 & -\\tfrac{1}{K} (1+\\beta ^2) \\mathsf {I}_{n_2d_2}\\\\\\end{pmatrix}}_{\\mathbf {D}} N^T ,$ where $ \\bar{Q}_\\ell =\\tilde{Q}_\\ell +\\frac{K\\beta ^2}{(1+\\beta ^2)}\\left(\\begin{matrix}0 & 0\\\\0 & \\mathsf {I}_{n_\\ell d_\\ell }\\end{matrix}\\right)$ , $ \\ell \\in \\lbrace 1,2\\rbrace $ , and $N =\\begin{pmatrix}\\mathsf {I}_{n_1d_1} & 0 & 0 &0 &0 &0\\\\0 & \\mathsf {I}_{n_1d_1} & 0 & 0 &-\\frac{\\beta K}{1+\\beta ^2}\\mathsf {I}_{n_1d_1} &0 \\\\0 & 0 & \\mathsf {I}_{n_2d_2} & 0 & 0 &0 \\\\0 & 0 & 0 & \\mathsf {I}_{n_2d_2} & 0 & \\frac{\\beta K}{1+\\beta ^2}\\mathsf {I}_{n_2d_2}\\\\0 & 0 & 0 & 0 & \\mathsf {I}_{n_1d_1} & 0 \\\\0 & 0 & 0 & 0 & 0 & \\mathsf {I}_{n_2d_2}\\\\\\end{pmatrix}.$ Noting that $\\mathcal {W}$ is invariant under $N^T$ (i.e., $N^T \\mathcal {W}= \\mathcal {W}$ ), all we need to check is that the matrix $ \\mathbf {D} $ is negative semidefinite on $\\mathcal {W}$ .",
"Clearly, $\\left({\\begin{matrix}-\\tfrac{1}{K} (1+\\beta ^2) \\mathsf {I}_{n_1d_1} & 0\\\\0 & -\\tfrac{1}{K} (1+\\beta ^2) \\mathsf {I}_{n_2d_2}\\end{matrix}} \\right)$ is negative definite.",
"On the other hand, for $ \\ell \\in \\lbrace 1,2\\rbrace $ , on $({\\mathbb {R}}^{n_\\ell d_\\ell })^2$ , 0 is an eigenvalue of $\\tilde{Q}_\\ell $ with multiplicity $2d_\\ell $ and eigenspace generated by vectors of the form $(\\mathbf {1}_{n_\\ell } \\otimes a,0)$ and $(0,\\mathbf {1}_{n_\\ell } \\otimes b)$ , with $a,b \\in {\\mathbb {R}}^{d_\\ell }$ .",
"However, on $ \\lbrace (v_1,v_2) \\in ({\\mathbb {R}}^{n_\\ell d_\\ell })^2 \\; | \\; (\\mathbf {1}_{n_\\ell }^T \\otimes \\mathsf {I}_{d_\\ell }) v_2 =\\mathbf {0}_{d_\\ell }\\rbrace $ , 0 is an eigenvalue of $\\tilde{Q}_\\ell $ with multiplicity $d_\\ell $ and eigenspace generated by vectors of the form $(\\mathbf {1}_{n_\\ell } \\otimes a,0)$ .",
"Moreover, on $ \\lbrace (v_1,v_2) \\in ({\\mathbb {R}}^{n_\\ell d_\\ell })^2 \\; | \\; (\\mathbf {1}_{n_\\ell }^T \\otimes \\mathsf {I}_{d_\\ell }) v_2 =\\mathbf {0}_{d_\\ell }\\rbrace $ , the eigenvalues of $\\frac{K\\beta ^2}{(1+\\beta ^2)}\\left({\\begin{matrix}0 & 0\\\\0 & \\mathsf {I}_{n_\\ell d_\\ell }\\end{matrix}}\\right)$ are $\\frac{K\\beta ^2}{(1+\\beta ^2)}$ with multiplicity $n_\\ell d_\\ell -d_\\ell $ and 0 with multiplicity $n_\\ell d_\\ell $ .",
"Therefore, using Weyl's theorem [16], we deduce that the nonzero eigenvalues of the sum $ \\bar{Q}_\\ell $ are upper bounded by $\\Lambda _{*}(\\tilde{Q}_\\ell )+\\frac{K\\beta ^2}{(1+\\beta ^2)} $ .",
"Thus, the eigenvalues of $\\bar{Q}=\\left({\\begin{matrix}\\bar{Q}_1 & 0\\\\0 & \\bar{Q}_2\\end{matrix}} \\right) $ are upper bounded by $\\min _{\\ell =1,2}\\lbrace \\Lambda _{*}(\\bar{Q}_\\ell )\\rbrace +\\frac{K\\beta ^2}{(1+\\beta ^2)}$ .",
"From (REF ) and the definition of $h$ in (REF ), we conclude that the nonzero eigenvalues of $\\bar{Q} $ are upper bounded by $ h(\\beta ) $ .",
"It remains to show that there exists $ \\beta ^* \\in {\\mathbb {R}}_{>0}$ with $ h(\\beta ^*)=0$ such that for all $ 0 < \\beta < \\beta ^* $ we have $ h(\\beta ) < 0$ .",
"For $ r>0 $ small enough, $h(r)<0$ , since $ h(r) =-\\frac{1}{2}\\Lambda _{*}^{\\min }r+O(r^2)$ .",
"Furthermore, $\\lim _{r\\rightarrow \\infty } h(r) =K > 0 $ .",
"Hence, using the Mean Value Theorem, we deduce the existence of $ \\beta ^* $ .",
"Therefore we conclude that $ \\mathcal {L}_{\\Psi _{\\textup {Nash-dir}}}V(\\mathbf {x}_1,\\mathbf {z}_1,\\mathbf {x}_2,\\mathbf {z}_2) \\le 0$ .",
"As a by-product, the trajectories of () are bounded.",
"Consequently, all assumptions of the LaSalle Invariance Principle, cf.",
"Theorem REF , are satisfied.",
"This result then implies that any trajectory of () starting from an initial condition $(\\mathbf {x}_1^0,\\mathbf {z}_1^0,\\mathbf {x}_2^0,\\mathbf {z}_2^0) $ converges to the largest invariant set $ M $ in $ S_{\\Psi _{\\textup {Nash-dir}},V} \\cap \\mathcal {W}_{\\mathbf {z}^0_1, \\mathbf {z}^0_2}$ .",
"Note that if $(\\mathbf {x}_1,\\mathbf {z}_1,\\mathbf {x}_2,\\mathbf {z}_2) \\in S_{\\Psi _{\\textup {Nash-dir}},V} \\cap W_{\\mathbf {z}^0_1,\\mathbf {z}^0_2} $ , then $ N^T X \\in \\ker (\\bar{Q}) \\times \\lbrace 0\\rbrace $ .",
"From the discussion above, we know $\\ker (\\bar{Q})$ is generated by vectors of the form $(\\mathbf {1}_{n_1} \\otimes a_1,0,0,0)$ , $(0,0,\\mathbf {1}_{n_2}\\otimes a_2,0)$ , $ a_\\ell \\in {\\mathbb {R}}^{d_\\ell } $ , $ j\\in \\lbrace 1,2\\rbrace $ , and hence $\\mathbf {x}_\\ell = \\mathbf {x}^*_\\ell +\\mathbf {1}_{n_\\ell }\\otimes a_\\ell $ , $\\mathbf {z}_\\ell = \\mathbf {z}_\\ell ^*$ .",
"Using the strict concavity-convexity, this then implies that $\\mathbf {x}_\\ell = \\mathbf {x}^*_\\ell $ .",
"Finally, for $(\\mathbf {x}^*_1,\\mathbf {z}_1,\\mathbf {x}^*_2,\\mathbf {z}_2) \\in M$ , using the positive invariance of $M$ , one deduces that $ (\\mathbf {x}^*_1,\\mathbf {z}_1,\\mathbf {x}^*_2,\\mathbf {z}_2) \\in \\operatorname{Eq}(\\Psi _{\\textup {Nash-dir}}) $ .",
"$\\Box $ Remark 5.3 (Assumptions on payoff function): Two observations are in order regarding the assumptions in Theorem REF on the payoff function.",
"First, the assumption that the payoff function has a globally Lipschitz gradient is not too restrictive given that, since the state spaces are compact, standard boundedness conditions on the gradient imply the globally Lipschitz condition.",
"Second, we restrict our attention to differentiable payoff functions because locally Lipschitz functions with globally Lipschitz generalized gradients are in fact differentiable, see [13].$\\bullet $ Remark 5.4 (Comparison with best-response dynamics): Using the gradient flow has the advantage of avoiding the cumbersome computation of the best-response map.",
"This, however, does not come for free.",
"There are concave-convex functions for which the (distributed) gradient flow dynamics, unlike the best-response dynamics, fails to converge to the saddle point, see [9] for an example.$\\bullet $ We finish this section with an example.",
"Example 5.5 (Distributed adversarial selection of signal and noise power via ()): Recall the communication scenario described in Remark REF .",
"Consider 5 channels, $\\lbrace \\texttt {ch}_1,\\texttt {ch}_2,\\texttt {ch}_3,\\texttt {ch}_4,\\texttt {ch}_5\\rbrace $ , for which the network $ \\Sigma _1 $ has decided that $\\lbrace \\texttt {ch}_1, \\texttt {ch}_3\\rbrace $ have signal power $ x_1 $ and $\\lbrace \\texttt {ch}_2,\\texttt {ch}_4\\rbrace $ have signal power $ x_2 $ .",
"Channel $ \\texttt {ch}_5 $ has its signal power determined to satisfy the budget constraint $ P\\in {\\mathbb {R}}_{>0}$ , i.e., $P-2 x_1 -2x_2$ .",
"Similarly, the network $\\Sigma _2 $ has decided that $ \\texttt {ch}_1 $ has noise power $y_1$ , $\\lbrace \\texttt {ch}_2,\\texttt {ch}_3,\\texttt {ch}_4\\rbrace $ have noise power $y_2$ , and $ \\texttt {ch}_5 $ has noise power $C-y_1 -3 y_2$ to meet the budget constraint $ C\\in {\\mathbb {R}}_{>0}$ .",
"We let $ \\mathbf {x}=(x^1,x^2,x^3,x^4,x^5) $ and $\\mathbf {y}=(y^1,y^2,y^3,y^4,y^5) $ , where $ x^i=(x^i_1,x^i_2)\\in [0,P]^2 $ and $ y^i = (y^i_1,y^i_2)\\in [0,C]^2 $ , for each $ i\\in \\lbrace 1,\\ldots ,5\\rbrace $ .",
"The networks $ \\Sigma _1 $ and $ \\Sigma _2 $ , which are weight-balanced and strongly connected, and the engagement topology $\\Sigma _{\\textup {eng}}$ are shown in Figure REF .",
"Figure: Networks Σ 1 \\Sigma _1 , Σ 2 \\Sigma _2 and Σ eng \\Sigma _{\\textup {eng}} for the case study of Example .Edges which correspond to Σ eng \\Sigma _{\\textup {eng}} are dashed.",
"Fori∈{1,...,5} i\\in \\lbrace 1,\\ldots , 5\\rbrace , agents v i v_i and w i w_i are placed inchannel 𝚌𝚑 i \\texttt {ch}_i .Note that, according to this topology, each agent can observe the power employed by its adversary in its channel and, additionally, the agents in channel 2 can obtain information about the estimates of the opponent in channel 4 and vice versa.",
"The payoff functions of the agents are given in Remark REF , where for simplicity we take $\\sigma _i=\\sigma _1 $ , for $ i\\in \\lbrace 1,3, 5\\rbrace $ , and $\\sigma _i=\\sigma _2 $ , for $ i\\in \\lbrace 2,4\\rbrace $ , with $\\sigma _1,\\sigma _2 \\in {\\mathbb {R}}_{>0}$ .",
"This example fits into the approach described in Section REF by considering the following extended payoff functions: $\\tilde{f}^1_1(x^1,\\mathbf {y}) =&\\log (1+\\frac{\\beta x^1_1}{\\sigma _1+y^1_1}),\\\\\\tilde{f}^2_1(x^2,\\mathbf {y}) =&\\frac{1}{3}\\log (1+\\frac{\\beta x^2_2}{\\sigma _2+y^4_2}) +\\frac{2}{3}\\log (1+\\frac{\\beta x^2_2}{\\sigma _2+y^2_2}),\\\\\\tilde{f}^3_1(x^3,\\mathbf {y}) =&\\log (1+\\frac{\\beta x^3_1}{\\sigma _1+y^3_2}),\\\\\\tilde{f}^4_1(x^4,\\mathbf {y}) =&\\frac{1}{3}\\log (1+\\frac{\\beta x^4_2}{\\sigma _2+y^2_2})+\\frac{2}{3}\\log (1+\\frac{\\beta x^4_2}{\\sigma _2+y^4_2}),\\\\\\tilde{f}^5_1(x^5,\\mathbf {y}) =& \\log \\Big (1+ \\frac{\\beta (P-2x^5_1 -2x^5_2)}{\\sigma _1+C-y^5_1 - 3 y^5_2} \\Big ),\\\\\\tilde{f}^1_2(\\mathbf {x},y^1)=&\\tilde{f}^1_1(x^1,\\mathbf {y}), \\quad \\tilde{f}^3_2(\\mathbf {x},y^3)=\\tilde{f}^3_1(x^3,\\mathbf {y}),\\\\\\tilde{f}^2_2(\\mathbf {x},y^2) =&\\frac{2}{3}\\log (1+\\frac{\\beta x^2_2}{\\sigma _2+y^2_2})+\\frac{1}{3}\\log (1+\\frac{\\beta x^4_2}{\\sigma _2+y^2_2}),\\\\\\tilde{f}^4_2(\\mathbf {x},y^4) =&\\frac{1}{3}\\log (1+\\frac{\\beta x^2_2}{\\sigma _2+y^4_2})+\\frac{2}{3}\\log (1+\\frac{\\beta x^4_2}{\\sigma _2+y^4_2}),\\\\\\tilde{f}^5_2(\\mathbf {x},y^5)=&\\tilde{f}^5_1(x^5,\\mathbf {y}).$ Note that these functions are strictly concave and thus the zero-sum game defined has a unique saddle point on the set $[0,P]^2\\times [0,C]^2 $ .",
"These functions satisfy (REF ) and $ \\tilde{U}_1 = \\tilde{U}_2 $ .",
"Figure: Execution of () over thenetworked strategic scenario described inExample , with β=8 \\beta =8 , σ 1 =1 \\sigma _1=1 , σ 2 =4\\sigma _2=4 , P=6 P=6 , and C=4 C=4 .",
"(a) and (b) show theevolution of the agent's estimates of the state of networks Σ 1 \\Sigma _1 and Σ 2 \\Sigma _2 , respectively, and (c) showsthe value of the Lyapunov function.",
"Here, α=3 \\alpha =3 in () and initially, 𝐱 0 =((1,0.5),(0.5,1),(0.5,0.5),(0.5,1),(0.5,1)) T \\mathbf {x}^0=((1,0.5),(0.5,1),(0.5,0.5),(0.5,1),(0.5,1))^T , 𝐳 1 0 =0 10 \\mathbf {z}^0_1=\\mathbf {0}_{10} , 𝐲 0 =((1,0.5),(0.5,1),(0.5,1),(0.5,0.5),(1,0.5)) T \\mathbf {y}^0=((1,0.5),(0.5,1),(0.5,1),(0.5,0.5),(1,0.5))^T and 𝐳 2 0 =0 10 \\mathbf {z}^0_2=\\mathbf {0}_{10} .",
"The equilibrium (𝐱 * ,𝐳 1 * ,𝐲 * ,𝐳 2 * )(\\mathbf {x}^*,\\mathbf {z}^*_1,\\mathbf {y}^*,\\mathbf {z}^*_2) is 𝐱 * =(1.3371,1.0315) T ⊗1 5 \\mathbf {x}^*=(1.3371,1.0315)^T\\otimes \\mathbf {1}_5, 𝐲 * =(1.5027,0.3366) T ⊗1 5 \\mathbf {y}^* =(1.5027,0.3366)^T\\otimes \\mathbf {1}_5 , 𝐳 1 * =(0.7508,0.5084,0.1447,0.5084,0.1447,-0.1271,-0.5201,-0.1271,-0.5201,-0.7626) T \\mathbf {z}^*_1 =(0.7508,0.5084,0.1447,0.5084,0.1447,-0.1271,-0.5201,-0.1271,-0.5201,-0.7626)^T and 𝐳 2 * =(0.1079,-0.0987,-0.0002,0.2237,0.0358,0.2875,-0.0360,0.0087,-0.1076,-0.4213)\\mathbf {z}^*_2 =(0.1079,-0.0987,-0.0002,0.2237,0.0358,0.2875,-0.0360,0.0087,-0.1076,-0.4213).Figure REF shows the convergence of the dynamics () to the Nash equilibrium of the resulting 2-network zero-sum game.$\\bullet $"
],
[
"Conclusions and future work",
"We have considered a class of strategic scenarios in which two networks of agents are involved in a zero-sum game.",
"The networks aim to either maximize or minimize a common objective function.",
"Individual agents collaborate with neighbors in their respective network and have partial knowledge of the state of the agents in the other one.",
"We have introduced two aggregate objective functions, one per network, identified a family of points with a special saddle property for this pair of functions, and established their correspondence between the Nash equilibria of the overall game.",
"When the individual networks are undirected, we have proposed a distributed saddle-point dynamics that is implementable by each network via local interactions.",
"We have shown that, for a class of strictly concave-convex and locally Lipschitz objective functions, the proposed dynamics is guaranteed to converge to the Nash equilibrium.",
"We have also shown that this saddle-point dynamics fails to converge for directed networks, even when they are strongly connected and weight-balanced.",
"Motivated by this fact, we have introduced a generalization that incorporates a design parameter.",
"We have shown that this dynamics converges to the Nash equilibrium for strictly concave-convex and differentiable objective functions with globally Lipschitz gradients for appropriate parameter choices.",
"An interesting venue of research is determining whether it is always possible to choose the extensions of the individual payoff functions in such a way that the lifted objective functions coincide.",
"Future work will also include relaxing the assumptions of strict concavity-convexity and differentiability of the payoff functions, and the globally Lipschitz condition on their gradients, extending our results to dynamic interaction topologies and non-zero sum games, and exploring the application to various areas, including collaborative resource allocation in the presence of adversaries, strategic social networks, collective bargaining, and collaborative pursuit-evasion."
],
[
"Appendix",
"The following result can be understood as a generalization of the characterization of cocoercivity of concave functions [14].",
"Theorem 7.1 (Concave-convex differentiable functions with globally Lipschitz gradients): Let $f:{\\mathbb {R}}^{d_1}\\times {\\mathbb {R}}^{d_2} \\rightarrow {\\mathbb {R}} $ be a concave-convex differentiable function with globally Lipschitz gradient (with Lipschitz constant $ K \\in {\\mathbb {R}}_{>0}$ ).",
"For $(x,y), (x^{\\prime },y^{\\prime })\\in {\\mathbb {R}}^{d_1}\\times {\\mathbb {R}}^{d_2} $ , $(x& -x^{\\prime })^T (\\nabla _xf(x,y)-\\nabla _xf(x^{\\prime },y^{\\prime }))\\\\&\\quad +(y-y^{\\prime })^T(\\nabla _yf(x^{\\prime },y^{\\prime })-\\nabla _yf(x,y))\\\\\\le & - \\frac{1}{2K}\\Big (\\Vert \\nabla _xf(x,y^{\\prime })-\\nabla _xf(x^{\\prime },y^{\\prime })\\Vert ^2\\\\& +\\Vert \\nabla _yf(x^{\\prime },y)-\\nabla _yf(x^{\\prime },y^{\\prime })\\Vert ^2\\\\&+\\Vert \\nabla _xf(x^{\\prime },y)-\\nabla _xf(x,y)\\Vert ^2\\\\& + \\Vert \\nabla _yf(x,y^{\\prime })-\\nabla _yf(x,y)\\Vert ^2 \\Big ) .$ Proof.",
"We start by noting that, for a concave function $j:{\\mathbb {R}}^d \\rightarrow {\\mathbb {R}}$ with globally Lipschitz gradient, the following inequality holds, see [14], $j(x) \\le j^*-\\frac{1}{2M}||\\nabla j(x)||^2,$ where $ j^*=\\sup _{x\\in {\\mathbb {R}}^{d}} j(x) $ and $M$ is the Lipschitz constant of $\\nabla j$ .",
"Given $(x^{\\prime },y^{\\prime }) \\in {\\mathbb {R}}^{d_1}\\times {\\mathbb {R}}^{d_2}$ , define the map $\\tilde{f}:{\\mathbb {R}}^{d_1}\\times {\\mathbb {R}}^{d_2} \\rightarrow {\\mathbb {R}} $ by $\\tilde{f}(x,y)&=f(x,y)-f(x^{\\prime },y)-(x-x^{\\prime })^T\\nabla _x f(x^{\\prime },y)\\\\& \\quad +f(x,y)-f(x,y^{\\prime })+(y-y^{\\prime })^T\\nabla _y f(x,y^{\\prime }) .$ Since the gradient of $f$ is Lipschitz, the function $ \\tilde{f} $ is differentiable almost everywhere.",
"Thus, almost everywhere, we have $\\nabla _x\\tilde{f}(x,y) &= \\nabla _xf(x,y)-\\nabla _xf(x^{\\prime },y)+ \\nabla _xf(x,y)\\\\& \\quad -\\nabla _xf(x,y^{\\prime })-(y-y^{\\prime })^T\\nabla _x\\nabla _yf(x,y^{\\prime }),\\\\\\nabla _y\\tilde{f}(x,y) &= \\nabla _yf(x,y) -\\nabla _yf(x^{\\prime },y) + \\nabla _yf(x,y)\\\\& \\quad - \\nabla _yf(x,y^{\\prime })-(x-x^{\\prime })^T\\nabla _y\\nabla _xf(x^{\\prime },y).$ In particular, note that $\\nabla _x\\tilde{f}(x^{\\prime },y^{\\prime }) =\\nabla _y\\tilde{f}(x^{\\prime },y^{\\prime })=0$ .",
"Since $x \\mapsto \\tilde{f}(x,y^{\\prime })$ and $y \\mapsto \\tilde{f}(x^{\\prime },y)$ are concave and convex functions, respectively, we can use (REF ) to deduce $\\tilde{f}(x,y^{\\prime }) &\\le -\\frac{1}{2K}||\\nabla _xf(x,y^{\\prime })-\\nabla _xf(x^{\\prime },y^{\\prime })||^2,\\\\-\\tilde{f}(x^{\\prime },y) & \\le -\\frac{1}{2K}||\\nabla _yf(x^{\\prime },y)-\\nabla _yf(x^{\\prime },y^{\\prime })||^2,$ where we have used the fact that $\\sup _{x\\in {\\mathbb {R}}^{d_1}}\\tilde{f}(x,y^{\\prime }) = \\inf _{y \\in {\\mathbb {R}}^{d_2}}\\tilde{f}(x^{\\prime },y)=\\tilde{f}(x^{\\prime },y^{\\prime })=0 $ .",
"Next, by definition of $\\tilde{f} $ , $\\tilde{f}(x,y^{\\prime })&=f(x,y^{\\prime })-f(x^{\\prime },y^{\\prime })-(x-x^{\\prime })^T\\nabla _xf(x^{\\prime },y^{\\prime }),\\\\\\tilde{f}(x^{\\prime },y)&=f(x^{\\prime },y)-f(x^{\\prime },y^{\\prime })-(y-y^{\\prime })^T\\nabla _yf(x^{\\prime },y^{\\prime }).$ Using (), we deduce that $& f(x,y^{\\prime }) -f(x^{\\prime },y) \\nonumber \\\\& -(x-x^{\\prime })^T\\nabla _xf(x^{\\prime },y^{\\prime }) + (y-y^{\\prime })^T\\nabla _yf(x^{\\prime },y^{\\prime })\\nonumber \\\\& \\le -\\frac{1}{2K} \\big (\\Vert \\nabla _xf(x,y^{\\prime }) -\\nabla _xf(x^{\\prime },y^{\\prime })\\Vert ^2 \\nonumber \\\\& \\hspace*{56.9055pt} +\\Vert \\nabla _yf(x^{\\prime },y)-\\nabla _yf(x^{\\prime },y^{\\prime })\\Vert ^2\\big ) .$ The claim now follows by adding together (REF ) and the inequality that results by interchanging $ (x,y) $ and $(x^{\\prime },y^{\\prime })$ in (REF ).",
"$\\Box $"
]
] | 1204.0852 |
[
[
"The Gravity Dual of a Density Matrix"
],
[
"Abstract For a state in a quantum field theory on some spacetime, we can associate a density matrix to any subset of a given spacelike slice by tracing out the remaining degrees of freedom.",
"In the context of the AdS/CFT correspondence, if the original state has a dual bulk spacetime with a good classical description, it is natural to ask how much information about the bulk spacetime is carried by the density matrix for such a subset of field theory degrees of freedom.",
"In this note, we provide several constraints on the largest region that can be fully reconstructed, and discuss specific proposals for the geometric construction of this dual region."
],
[
"Introduction",
"The AdS/CFT correspondence [1], [2] relates states of a field theory on some fixed spacetime B to states of a quantum gravity theory for which the spacetime metric is asymptotically locally AdS with boundary geometry B.",
"The field theory provides a nonperturbative description of the quantum gravity theory that is manifestly local on the boundary spacetime: for a given spacelike slice of the boundary spacetime B, the degrees of freedom in one subset are independent from the degrees of freedom in another subset.",
"On the gravity side, identifying independent degrees of freedom is much more difficult; for example, the idea of black hole complementarity [3] suggests that local excitations inside the horizon of a black hole cannot be independent of the physics outside the horizon.",
"It is therefore interesting to ask whether we can use our knowledge of independent field theory degrees of freedom to learn anything about which degrees of freedom on the gravity side may be considered to be independent.",
"In this paper, we consider the following question: Given a CFT on B in a state $|\\Psi \\rangle $ dual to a spacetime $M$ with a geometrical description, and given a subset $A$ of a spatial slice of B, what part of the spacetime $M$ can be fully reconstructed from the density matrix $\\rho _A$ describing the state of the subset of the field theory degrees of freedom in $A$ ?",
"An immediate question is why we expect there to be any region that can be reconstructed if we know only about the degrees of freedom on a subset of the boundary.",
"If the map between boundary degrees of freedom and the bulk spacetime is sufficiently non-local, it could be that information from every region of the boundary spacetime is needed to reconstruct any particular subset of $M$ .",
"However, there are various reasons to be more optimistic.",
"It is well known that the asymptotic behavior of the fields in the bulk spacetime is given directly in terms of expectation values of local operators in the field theory (together with the field theory action).",
"Equipped with this boundary behavior of the bulk fields in some region of the boundaryAs we recall below, knowledge of the field theory density matrix for a spatial region $A$ allows us to compute any field theory quantities localized to a particular codimension-zero region of the boundary, the domain of dependence of $A$ .",
"and the bulk field equations, we should be able to integrate these field equations to find the fields in some bulk neighborhood of this boundary region.",
"We can also compute various other field theory quantities (e.g.",
"correlation functions, Wilson loops, entanglement entropies) restricted to the region $A$ or its domain of dependence.",
"According to the AdS/CFT dictionary, these give us direct information about nearby regions of the bulk geometry.",
"The notion that particular density matrices can be associated with certain patches of spacetime was advocated in [4].For an earlier discussion of mixed states in the context of AdS/CFT, see [5].",
"There, it was pointed out that a given density matrix may arise from many different states of the full system, or from a variety of different quantum systems that contain this set of degrees of freedom as a subset.",
"Different pure states that give rise to the same density matrix for the subset correspond to different spacetimes with a region in common; this common region can be considered to be the dual of the density matrix.As a particular example, it was pointed out in [4] that a CFT on $S^d$ in a thermal density matrix, commonly understood to be dual to an AdS/Schwarzchild black hole, cannot possibly know whether the whole spacetime is the maximally extended black hole; only the region outside the horizon is common to all states of larger systems for which the CFT on $S^d$ forms a subset of degrees of freedom described by a thermal density matrix.",
"In the bulk of this paper, we seek to understand in general the region of a bulk spacetime $M$ that can be directly associated with the density matrix describing a particular subset of the field theory degrees of freedom.",
"We begin in Section 2 by reviewing some relevant facts from field theory and arguing that the density matrix associated with a region $A$ may be more naturally associated with the domain of dependence $D_A$ (defined below).",
"In Section 3, we outline in more detail the basic question considered in the paper.",
"In Section 4, we propose several basic constraints on the region $R(A)$ dual to a density matrix $\\rho _A$ .",
"In Section 5, we consider two regions that are plausibly contained in $R(A)$ .",
"First, we argue that $z(D_A)$ , the intersection of the causal past and causal future of $D_A$ , satisfies our constraints and should be contained in $R(A)$ , as should its domain of dependence, $\\hat{z}(D_A)$ .We denote domains of dependence in the boundary with $D_\\cdot $ (for example, $D_A$ ), while domains of dependence in the bulk are marked with a hat $\\hat{\\phantom{.",
"}}$ .",
"We note that in some special cases, $R(A)$ cannot be larger than $\\hat{z}(D_A)$ .",
"However, in generic spacetimes, we argue that entanglement observables that can be calculated from the density matrix $\\rho _A$ certainly allow us to probe regions of spacetime beyond $\\hat{z}(D_A)$ .It is an open question whether these observables are enough to reconstruct the spacetime beyond $\\hat{z}(D_A)$ , so we cannot say with certainty that $R(A)$ is larger than $\\hat{z}(D_A)$ .",
"This motivates us to consider another region, $w(D_A)$ , defined as the union of surfaces used to calculate these entanglement observables (defined more precisely below) according to the holographic entanglement entropy proposal [6], [7].",
"We show that $w(D_A)$ (or more precisely, its domain of dependence $\\hat{w}(D_A)$ ) also satisfies our constraints, and that for a rather general class of spacetimes, there is a sense in which $R(A)$ cannot be larger than $\\hat{w}(D_A)$ .",
"On the other hand, we show that in some examples, $R(A)$ must be larger than $\\hat{w}(D_A)$ .",
"We conclude in Section 6 with a summary and discussion.",
"Note added: While this paper was in preparation, [8] appeared, which has some overlap with our discussion.",
"We also became aware of [9], which considers related questions."
],
[
"Field theory considerations",
"To begin, consider a field theory on some globally hyperbolic spacetime B, and consider a spacelike slice $\\Sigma $ that forms a Cauchy surface.",
"Then, classically, the fields on this hypersurface and their derivatives with respect to some timelike future-directed unit vector orthogonal to the hypersurface determine the complete future evolution of the field.",
"Quantum mechanically, the fields on this hypersurface can be taken as the basic set of variables for quantization and conjugate momenta defined with respect to the timelike normal vector.",
"Now consider some region $A$ of the hypersurface $\\Sigma $ .",
"Since the field theory is local, the degrees of freedom in $A$ are independent from the degrees of freedom in the complement $\\bar{A}$ of $A$ on $\\Sigma $ .",
"Thus, the Hilbert space can be decomposed as a tensor product ${\\cal H} = {\\cal H}_A \\otimes {\\cal H}_{\\bar{A}}$ , and we can associate a density matrix $\\rho _A = \\mathrm {Tr}_{\\bar{A}}(|\\Psi \\rangle \\langle \\Psi \\rangle )$ to the degrees of freedom in $A$ .",
"This density matrix captures all information about the state of the degrees of freedom in $A$ and can be used to compute any observables localized to $A$ .",
"In fact, the density matrix $\\rho _A$ allows us to compute field theory observables localized to a larger region $D_A$ known as the domain of dependence of $A$ .",
"The domain of dependence $D_A$ is the set of points $p$ in B for which every (inextensible) causal curve through $p$ intersects $A$ (see Figure REF ).",
"Classically, the region $D_A$ is the subspace of B in which the field values are completely determined in terms of the initial data on $A$ .",
"Quantum mechanically, any operator in $D_A$ can be expressed in terms of the fields in $A$ alone and therefore computed using the density matrix $\\rho _A$ .",
"Figure: A spacelike slice Σ\\Sigma of a boundary manifold B (=S 1 × = S^1 \\times time) with a region AA and its domain of dependence D A D_A.",
"The same domain of dependence arises from any spacelike boundary region A ˜\\tilde{A} homologous to AA with ∂A=∂A ˜\\partial A = \\partial \\tilde{A}.As can be seen from Figure REF , any other spacelike surface $\\tilde{A}$ homologous to $A$ with boundary $\\partial \\tilde{A} = \\partial A$ shares its domain of dependence.To see this, we note that since $A$ and $\\tilde{A}$ are homologous, we can deform $A$ into $\\tilde{A}$ and define $\\cal B$ to be the volume bound by $A$ and $\\tilde{A}$ .",
"Then for any point $p$ in ${\\cal B}$ , consider an inextensible causal curve through $p$ .",
"Such a curve must necessarily pass through $A$ .",
"But it cannot pass through $A$ twice, since $A$ is spacelike.",
"On the other hand, the curve must intersect the boundary of the region $\\cal B$ twice (on the past boundary and on the future boundary), so it must have an intersection with $\\tilde{A}$ .",
"Thus, in some other quantization of the theory based on a hypersurface $\\tilde{\\Sigma }$ with $\\tilde{A} \\subset \\tilde{\\Sigma }$ , we expect that the density matrix $\\rho _{\\tilde{A}}$ contains the same information as the density matrix ${\\rho _A}$ .",
"It is then perhaps more natural to associate density matrices directly with domain of dependence regions.",
"This observation is important for our considerations below: in constructing the bulk region dual to a density matrix $\\rho _A$ , it is more natural to use the boundary region $D_A$ as a starting point, rather than the surface $A$ .",
"It is useful to note that a quantum field theory on a particular domain of dependence can be thought of as a complete quantum system, independent of the remaining degrees of freedom of the field theory.",
"The observables of this field theory are the set of all operators built from the fields on $A$ .",
"The state of the theory is specified by a density matrix $\\rho _A$ , which allows us to compute any such observable.",
"The spectrum of this density matrix, and associated observables such as the von Neumann entropy, give additional information about the system.",
"We can interpret this in a thermodynamic way as giving information about the ensemble of pure states described by the density matrix.",
"Alternatively, viewing this system as a subset of a larger system that we assume is in a pure state, we can interpret this additional information as telling us about the entanglement between the degrees of freedom in our causal development region with other parts of the system."
],
[
"The gravity dual of $\\rho _A$",
"In this section, we consider the question of how much information the density matrix $\\rho _A$ carries about the dual spacetime.",
"We restrict the discussion to states of the full system that are dual to some spacetime M with a good classical description.",
"Specifically, we ask the question Question: Suppose that a field theory on a spacetime B in a state $|\\Psi \\rangle $ has a dual spacetime $M$ with a good geometrical description (e.g.",
"a solution to some low-energy supergravity equations).",
"How much of $M$ can be reconstructed given only the density matrix $\\rho _A$ for the degrees of freedom in a subset $A$ of some spacelike slice of the boundary?",
"Alternatively, we can ask: Consider all states $|\\Psi _\\alpha \\rangle $ with dual spacetimes $M_\\alpha $ that give rise to a particular density matrix $\\rho _A$ for region $A$ of the boundary spacetime.",
"What is the largest region common to all the $M_\\alpha $ s?",
"We recall that knowledge of the density matrix $\\rho _A$ allows us to calculate any field theory observable involving operators localized in the domain of dependence $D_A$ , plus additional quantities such as the entanglement entropy associated with the degrees of freedom on any subset of $A$ .",
"According to the AdS/CFT dictionary, these observables give us a large amount of information about the bulk spacetime, particularly near the boundary region $D_A$ , so it is plausible that at least some region of the bulk spacetime can be fully reconstructed from this data.",
"We will refer to this region as $R(A)$ .",
"We expect that in general the density matrix $\\rho _A$ carries additional information about some larger region $G(A)$ , but this additional information does not represent the complete information about $G(A) - R(A)$ .",
"In this paper, we do not attempt to come up with a procedure to reconstruct the region $R(A)$ ; rather we will attempt to use general arguments to constrain how large $R(A)$ can be."
],
[
"Constraints on the region dual to $\\rho _A$",
"Before considering specific proposals for $R(A)$ , it will be useful to point out various constraints that $R(A)$ should satisfy.",
"First, since the density matrices for any two subsets $A$ and $\\tilde{A}$ with the same domain of dependence $D$ correspond to the same information in the field theory, we expect that the region of spacetime that can be reconstructed from $\\rho _A$ is the same as the region that can be reconstructed from $\\rho _{\\tilde{A}}$ .",
"Thus we have: Constraint 1: If $A$ and $\\tilde{A}$ have the same domain of dependence $D$ , then $R(A)= R(\\tilde{A})$ .",
"For a particular boundary field theory, the bulk spacetime will be governed by some specific low-energy field equations.",
"We assume that we are working with a known example of AdS/CFT so that these equations are known.",
"If we know all the fields in some region $R$ of the bulk spacetime $M$ , we can use these field equations to find the fields everywhere in the bulk domain of dependence of $R$ (which we denote by $\\hat{R}$ ).",
"Since $R(A)$ is defined to be the largest region of the bulk spacetime that we can reconstruct from $\\rho _A$ , we must have: Constraint 2: $\\hat{R}(A) = R(A)$ .",
"Now, suppose we consider two non-intersecting regions $A$ and $B$ on some spacelike slice of the boundary spacetime.",
"The degrees of freedom in $A$ and $B$ are completely independent, so it is possible to change the state $|\\Psi \\rangle $ such that $\\rho _B$ changes but $\\rho _A$ does not.Further, we expect that for some subset of these variations, the dual spacetime continues to have a classical geometric description.",
"Changes in $\\rho _B$ will generally affect the region $R(B)$ in the bulk spacetime, but as a consequence can also affect any region in the causal future $J^+(R(B))$ or causal past $J^-(R(B))$ of $R(B)$ .",
"But these changes can have no effect on the region $R(A)$ since this region can be reconstructed from $\\rho _A$ , which does not change.",
"Thus, we have: Constraint 3: If $A$ and $B$ are non-intersecting regions of a spacelike slice of the boundary spacetime, then $R(A)$ cannot intersect $J(R(B))$ .",
"Here we have defined $J(R) = J^-(R) \\cup J^+(R)$ .",
"Note that whatever $R(B)$ is, it certainly includes $D_B$ so as a corollary, we can say that $R(A)$ cannot intersect $J(D_B)$ .",
"Taking $B = \\bar{A}$ (i.e.",
"as large as possible without intersecting $A$ ), we get a definite upper bound on the size of $R(A)$ : it cannot be larger than the complement of $J(D_{\\bar{A}})$ ."
],
[
"Possibilities for $R(A)$",
"Let us now consider some physically motivated possibilities for the region $R(A)$ .",
"An optimistic expectation is that we could reconstruct the entire region $G(A)$ of the bulk spacetime $M$ used in calculating any field theory observable localized in $D_A$ (for example, all points touched by any geodesic with boundary points in $D_A$ ).",
"However, this cannot be a candidate for $R(A)$ , since it is easy to find examples of non-intersecting $A$ and $B$ on some spacelike slice of a boundary spacetime such that geodesics with endpoints in $B$ intersect with geodesics with endpoints in $A$ .For example, suppose we consider the vacuum state of a CFT on a cylinder and take $A$ and $B$ to be the regions $\\theta \\in (0,\\pi /2)\\cup (\\pi ,3 \\pi /2)$ and $\\theta \\in (\\pi /2,\\pi ) \\cup (3\\pi /2 , 2\\pi )$ on the $\\tau =0$ slice.",
"Then the lines of constant $\\theta $ are spatial geodesics in the bulk, and the region covered by such geodesics anchored in $A$ clearly intersects the region of such geodesics anchored in $B$ .",
"Thus, $G(A) \\cap G(B) \\ne \\emptyset $ (which implies $G(A) \\cap J(G(B)) \\ne \\emptyset $ ) and so Constraint 3 is violated.",
"A lesson here is that even if field theory observables calculated from a boundary region $D_A$ probe a certain region of the bulk, they cannot necessarily be used to reconstruct that region.",
"Generally, we will have $R(A) \\subset G(A) \\subset M$ , where $\\rho _A$ contains complete information about $R(A)$ , some information about $G(A)$ and no information about $\\bar{G}(A)$ ."
],
[
"The causal wedge $z(D_A)$",
"A simple region that is quite plausibly included in $R(A)$ is the set of points $z(D_A)$ in the bulk that a boundary observer restricted to $D_A$ can communicate with (i.e.",
"send a light signal to and receive a signal back).",
"For example, such an observer could easily detect the presence or absence of an arbitrarily small mirror placed at any point in $z(D_A)$ .",
"Formally, this region in the bulk is defined as the intersection of the causal past of $D_A$ with the causal future of $D_A$ in the bulk, $z(D_A) \\equiv J^+(D_A) \\cap J^-(D_A)$ , as shown in Figure REF .Recall that the causal future $J^+(D_A)$ of $D_A$ in the bulk is the set of points reachable by causal curves starting in $D_A$ while the causal past $J^-(D_A)$ of $D_A$ is the set of points, from which $D_A$ can be reached along a causal curve.",
"These observations correspond to perturbing the spacetime at one point in the asymptotic region and observing the asymptotic fields at another point at a later time.",
"In the field theory language, such observations correspond to calculating response functions, in which the fields are perturbed at one point in $D_A$ and observed at another point in $D_A$ .",
"Such calculations can be done using only the density matrix $\\rho _A$ , thus we expect that $z(D_A)$ is included in the region $R(A)$ .",
"By condition 2, we can extend this expectation to the proposal that $\\hat{z}(D_A) \\subset R(A)$ .",
"It is straightforward to check that $\\hat{z}(D_A)$ also satisfies condition 3.Suppose subsets $A$ and $B$ of a boundary slice do not intersect and suppose $p \\in J(\\hat{z}(D_B))$ .",
"Then there exists a causal curve through $p$ that intersects $\\hat{z}(D_B)$ and therefore intersects some $q$ in $z(D_B)$ .",
"If $p$ is also in $\\hat{z}(D_A)$ , this same causal curve through $p$ must intersect a point $r$ in $z(D_A)$ .",
"Thus, there is a causal curve from $q$ in $z(D_B)$ to $r$ in $z(D_A)$ .",
"By definition of $z$ , we must be able to extend this curve to a causal curve connecting $D_A$ to $D_B$ .",
"But in this situation, perturbations to the fields in $D_A$ could affect the fields in $D_B$ (or vice versa), and this would violate field theory causality.",
"Thus, the suggestion that $\\hat{z}(D_A) \\subset R(A)$ is consistent with our Constraints 1, 2 and 3.",
"The boundary of the region $z(D_A)$ in the interior of the spacetime is a horizon with respect to the boundary region $D_A$ .",
"Thus, the statement that we can reconstruct the region $z(D_A)$ is equivalent to saying that the information in $D_A$ is enough to reconstruct the spacetime outside this horizon.",
"This horizon can be an event horizon for a black hole, but in general is simply a horizon for observers restricted to the boundary region $D_A$ .",
"In certain simple examples, it is straightforward to argue that $R(A)$ cannot be larger than $z(D_A)$ or $\\hat{z}(D_A)$ .",
"For example, if $M$ is pure global AdS spacetime and $A$ is a hemisphere of the $\\tau =0$ slice of the boundary cylinder, then $z(D_A)$ is the region bounded by the lightcones from the past and future tips of $D_A$ and the spacetime boundary, as shown in Figure REF .",
"Any point outside this region is in the causal future or causal past of the boundary region $D_{\\bar{A}}$ ,This relies on the fact that for pure global AdS, the forward lightcone from the past tip of $D_A$ (point b in Figure REF ) is the same as the backward lightcone from the future tip of $D_{\\bar{A}}$ (point c) and the backward lightcone from the future tip of $D_A$ (point a in Figure REF ) is the same as the forward lightcone from the past tip of $D_{\\bar{A}}$ (point d).",
"so by Constraint 3 (and the consequences discussed afterwards) such points cannot be in $R(A)$ .",
"Figure: In pure global AdS, causal wedges of complementary hemispherical regions of the τ=0\\tau =0 slice intersect along a codimension-two surface.",
"In generic asymptotically AdS spacetimes, they intersect only at the boundary."
],
[
"Information beyond the causal wedge $z(D_A)$",
"We might be tempted to guess that $R(A)=\\hat{z}(D_A)$ in general, but we will now see that $\\rho _A$ typically contains significant information about the spacetime outside the region $\\hat{z}(D_A)$ .",
"Consider the same example of a CFT on the cylinder with the same regions $A$ and $\\bar{A}$ , but now consider some other state for which the dual spacetime is not pure AdS.",
"A key observationWe are grateful to Veronika Hubeny and Mukund Rangamani for pointing this out.",
"is that, generically, the wedges $z(D_A)$ and $z(D_{\\bar{A}})$ do not intersect, except at the boundary of $A$ .",
"This follows from a result of Gao and Wald [10] that light rays through the bulk of a generic asymptotically AdS spacetime generally take longer to reach the antipodal point of the sphere than light rays along the boundary.",
"Thus, the backward lightcone from the point $a$ in the right panel of Figure REF is different from the forward lightcone from point $d$ .",
"We can still argue that $R(A)$ cannot overlap with the region $J^+(D_{\\bar{A}}) \\cup J^-(D_{\\bar{A} })$ , but the complement of this region no longer coincides with $\\hat{z}(D_A)$ .",
"Thus, it is possible that $R(A)$ is larger than $\\hat{z}(D_A)$ in these general cases.As an explicit example of a spacetime with this property, we can take a static spacetime with a spherically symmetric configuration of ordinary matter in the interior, e.g.",
"the boson stars studied in [11].",
"To see that the density matrix $\\rho _A$ typically does contain information about the spacetime outside the region $z(D_A)$ , we can take inspiration from Hubeny [12], who argued that in many examples, the field theory observables that probe deepest into the bulk of the spacetime are those computed by extremal codimension-one surfaces in the bulk.",
"According to the proposal of Ryu and Takayanagi [6] and the covariant generalization by Hubeny, Rangamani, and Takayanagi [7], the von Neumann entropy of a density matrix $\\rho _C$ corresponding to any spatial region $C$ of the boundary gives the area of a surface $W(C)$ in the bulk defined either as the extremal codimension-two surface $W$ in the bulk whose boundary is the boundary of $C$ .",
"In cases where more than one such extremal surface exists, we take the one with minimal area, or the surface of minimal area such that the light sheets from this surface intersect the boundary exactly at $\\partial D_C.$ In each case, it is assumed that the surface $W$ is homologous to $C$ .",
"In [7], it is argued that these two definitions are equivalent.",
"Now, consider the surface $W(A)$ that computes the entanglement entropy of the full density matrix $\\rho _A$ .",
"From the second definition, it is clear that the surface $W$ cannot have any part in the interior of $z(D_A)$ .",
"Otherwise, the light rays from any such point would reach the boundary in the interior of region $D_A$ , and it would not be true that the light sheet from $W$ intersects the boundary at $\\partial D_A$ .",
"By the same argument, the surface $W(\\bar{A})$ that computes the entanglement entropy of $\\rho _{\\bar{A}}$ cannot have any part in the interior of $z(D_{\\bar{A}})$ .",
"But by the first definition, the surface $W(\\bar{A})$ is the same as the surface $W(A)$ , since $\\partial \\bar{A} = \\partial A$ .The equivalence of these surfaces and hence their areas is consistent with the fact that for a pure state in a Hilbert space ${\\cal H} = {\\cal H}_A \\otimes {\\cal H}_{\\bar{A}}$ , the spectrum of eigenvalues of $\\rho _A$ must equal the spectrum of eigenvalues of $\\rho _{\\bar{A}}$ .",
"Thus, the entanglement entropies $S(\\rho _A)$ and $S(\\rho _{\\bar{A}})$ must agree.",
"We do not consider here the case where the entire theory is in a mixed state.",
"Since $z(D_A)$ and $z(D_{\\bar{A}})$ generally have no overlap in the bulk of the spacetime, it is now clear that the surface $W$ lies outside at least one of $z(D_A)$ and $z(D_{\\bar{A}})$ .",
"To summarize, the area of surface $W$ may be computed as the von Neumann entropy of either the density matrix $\\rho _A$ or the density matrix $\\rho _{\\bar{A}}$ .",
"In the generic case where $z(D_A)$ and $z(D_{\\bar{A}})$ do not intersect in the bulk, the surface $W$ must lie outside at least one of $z(D_A)$ and $z(D_{\\bar{A}})$ .",
"Thus, we can say that either the density matrix $\\rho _A$ carries some information about the spacetime outside $z(D_A)$ or the density matrix $\\rho _{\\bar{A}}$ carries information about the spacetime outside $z(D_{\\bar{A}})$ .Again, it is easy to check this in specific examples.",
"For explicit examples of spherically symmetric static star geometries asymptotic to global AdS with $A$ equal to a hemisphere of the $\\tau =0$ slice, the surface $W(A)$ lies at $\\tau =0$ and passes through the center of the spacetime, while the regions $z(D_A)$ and $z(D_{\\bar{A}})$ do not reach the center."
],
[
"The wedge of minimal-area extremal surfaces $w(D_A)$ .",
"Based on these observations, and the observation of Hubeny that the extremal surfaces probe deepest into the bulk in various examples, it is natural to define a second candidate for the region $R(A)$ based on extremal surfaces.",
"The surface $W(A)$ calculates the entanglement entropy associated with the entire domain of dependence $D_A$ (equivalently, the largest spacelike surface in $D_A$ ).",
"We can also consider the entanglement entropy associated with any smaller causal development region within $D_A$ .",
"For any such region $C$ , there will be an associated surface $W(C)$ (as defined above) whose area computes the entanglement entropy (according to the proposal).",
"Define a bulk region $w(D_A)$ as the set of all points contained on some minimal-areaHere, we mean minimal area among the set of extremal surfaces with the same boundary.",
"extremal codimension-two surface whose boundary coincides with the boundary of a spacelike codimension-one region in $D_A$ .",
"The area of each such codimension-two surface is (according to [7]) equal to the entanglement entropy of the corresponding boundary region.",
"Thus, the region $w(D_A)$ directly corresponds to the region of the bulk whose geometry is probed by entanglement observables.",
"As we have seen, the region $w(D_A)$ generally extends beyond the region $z(D_A)$ .",
"From the region $w(D_A)$ , we can define a larger region $\\hat{w}(D_A)$ as the domain of dependence of the region $w(D_A)$ .",
"As discussed above, knowing the geometry (and other fields) in $w(D_A)$ and the bulk gravitational equations should allow us to reconstruct the geometry in $\\hat{w}(D_A)$ .",
"We would now like to understand whether the region $\\hat{w}(D_A)$ obeys the constraints outlined above.",
"Constraints 1 and 2 are satisfied by definition.",
"It is straightforward to show that Constraint 3 is satisfied assuming that the following conjecture holds: Conjecture C1: If $D_A$ and $D_B$ are domains of dependence for non-intersecting regions $A$ and $B$ of a spacelike slice of the boundary spacetime, then $w(D_A)$ and $w(D_B)$ are spacelike separated.",
"Supposing that this holds, if $p$ is in $J(\\hat{w}(D_B))$ , then there exists a causal curve through $p$ intersecting $\\hat{w}(D_B)$ , and by definition of $\\hat{w}$ , this causal curve also intersects $w(D_B)$ .",
"If $p$ is also in $\\hat{w}(D_A)$ , then every causal curve through $p$ intersects $w(D_A)$ .",
"Thus, there exists a causal curve that intersects both $w(D_B)$ and $w(D_A)$ , which violates ${\\bf C1}$ .",
"We conclude that $\\hat{w}(D_A)$ satisfies Constraints 1, 2 and 3 assuming that Conjecture ${\\bf C1}$ holds."
],
[
"Aside: proving Conjecture C1",
"While a proof (or refutation) of Conjecture ${\\bf C1}$ is left to future work, we make a few additional comments here.",
"For the case of static spacetimes, it is straightforward to prove a result similar to ${\\bf C1}$ .",
"Let $A_1$ and $A_2$ be two non-intersecting regions of the $t=0$ boundary slice of a static spacetime, with $B_1$ and $B_2$ spacelike regions in $A_1$ and $A_2$ , respectively.",
"Let $W(B_1)$ and $W(B_2)$ be the minimal surfaces in the $t=0$ slice of the bulk spacetime with $\\partial W(B_1) = \\partial B_1$ and $\\partial W(B_2) = \\partial B_2$ .",
"Then $W(B_1)$ and $W(B_2)$ cannot intersect.",
"To show this, consider the part of $W(B_1)$ contained in the region of the $t=0$ slice bounded by $W(B_2)$ and $B_2$ , and the part of $W(B_2)$ contained in the region of the $t=0$ slice bounded by $W(B_1)$ and $B_1$ .",
"If these two pieces have different areas, then by swapping the two pieces, either the new surface $W(B_1)$ or the new surface $W(B_2)$ will have a smaller area than before, contradicting the assumption that these were minimal-area surfaces.",
"If the two pieces have the same area, the modified surfaces will have the same area as before, but the new surfaces will be cuspyThe surfaces $W(B_1)$ and $W(B_2)$ cannot be tangent at their intersection because there should be a unique extremal surface passing through a given point with a specified tangent plane to the surface at this point., such that we can decrease the area by smoothing the cusps.",
"In attempting a more general proof, it may be useful to note that Conjecture C1 is equivalent to the following statement (with some mild assumptions): Conjecture C2: For any spacelike boundary region $C$ , the surface $W(C)$ is spacelike separated from the rest of $w(D_C)$ .",
"To see the equivalence, assume first that C1 holds and let $A=C$ and $B=\\bar{C}$ .",
"If we assume the generic case that $W(C)$ is the same as $W(\\bar{C})$ , then $W(C) =W(B) \\subset w(D_B)$ must be spacelike separated from $w(D_A)=w(D_C)$ .",
"Conversely, for two disjoint regions $A$ and $B$ , let $C$ be any region such that $A \\subset C$ and $B \\subset \\bar{C}$ .",
"By definition, we have that $w(D_A)\\subset w(D_C)$ and $w(D_B)\\subset w(D_{\\bar{C}})$ .",
"Assuming again that $W(C) = W(\\bar{C})$ , Conjecture C2 implies that there is a spacelike path connecting any point in $w(D_A)\\subset w(D_C)$ with any point $p$ in $W(C)$ , and that there also exists a spacelike path connecting any point in $w(D_B)\\subset w(D_{\\bar{C}})$ with the same point $p$ .",
"Therefore, there is a spacelike path (through $p$ ) connecting any point in $w(D_A)$ with any point in $w(D_B)$ , as required for C1.",
"While C1 is immediately more useful, C2 might be easier to prove.",
"Consider any boundary region $C$ and any point $p$ in $w(D_C)$ .",
"Then there exists a spacelike codimension-one region $I_p$ in the domain of dependence $D_C$ such that $p \\in W(I_p)$ .",
"$I_p$ can be extended to a spacelike surface $A_I$ homologous with $C$ , with the same boundary as $C$ , $\\delta _{A_I} = \\delta _C$ .",
"The surface which calculates entanglement entropy is the same for $A_I$ and $C$ : $W(A_I)=W(C)$ .",
"Consider now a one-parameter family of surfaces $S(\\lambda )$ , which continuously interpolate between $A_I=S(0)$ and $I_p=S(1)$ , and the corresponding family of bulk minimal surfaces $W(S(\\lambda ))$ interpolating between $W(C)$ and $W(I_p)$ .",
"It is plausible that these bulk minimal surfaces change smoothly and that their deformations are spacelike; following the flow, we can find a spacelike path from $p$ to $W(C)$ , which would complete the proof of the Conjecture C2.",
"We leave further investigation of the general validity of ${\\bf C1}$ as a question for future work.We note here that the restriction to minimal extremal surfaces (rather than all extremal surfaces) is essential for the validity of this conjecture.",
"In static spacetimes with metric of the form $ds^2 = -f(r)dt^2 + dr^2/g(r) + r^2 d \\Omega ^2$ where $g(0)=1$ and $g(r) \\rightarrow r^2$ , it is possible that extremal surfaces bounded on one hemisphere intersect extremal surfaces bounded on the other hemisphere in cases where $g(r)$ is not monotonically increasing.",
"For these examples, ${\\bf C1}$ would fail if the definition of $w$ did not restrict to minimal surfaces.",
"To summarize the discussion so far, the region $\\hat{w}(D_A)$ satisfies conditions 1, 2 and 3 assuming that Conjecture ${\\bf C1}$ is correct.",
"Thus, $\\hat{w}(D_A)$ is a possible candidate for the region $R(A)$ .",
"A rather nice feature of this possibility is that $\\hat{w}(D_A)$ intersects $\\hat{w}(D_{\\bar{A}})$ along the codimension-two surface $W(A)=W(\\bar{A})$ defined above.",
"Thus, the surface $W$ represents the information in the bulk common to $\\hat{w}(D_A)$ and $\\hat{w}(D_{\\bar{A}})$ .",
"The area of this surface corresponds to the von Neumann entropy of $\\rho _A$ , which is the simplest information shared by $\\rho _A$ and $\\rho _{\\bar{A}}$ .",
"We might then conjecture that the full spectrum of $\\rho _A$ (which is the same as the spectrum of $\\rho _{\\bar{A}}$ and represents the largest set of information common to $\\rho _A$ and $\\rho _{\\bar{A}}$ ) encodes the full geometry of the surface $W$ (i.e.",
"the largest set of information common to $\\hat{w}(D_A)$ and $\\hat{w}(D_{\\bar{A}})$ ).",
"Before proceeding, let us ask whether it is even possible that the areas of extremal surfaces with boundary in some region $D_A$ carry enough information to reconstruct the geometry in $w(D_A)$ .",
"Consider the simple case of a 1+1 dimensional CFT on a cylinder with $D_A$ a diamond-shaped region on the boundary.",
"Given any state for the CFT, we could in principle compute the entanglement entropy associated with any smaller diamond-shaped region bounded by the past lightcone of some point in $D_A$ and the forward lightcone of some other point.",
"This would give us one function of four variables, since each of the two points defining the smaller diamond-shaped region is labeled by two coordinates.",
"Assuming the state has a geometrical bulk dual description, the bulk geometry will be described by a metric which consists of several functions of three variables.We are ignoring the possible extra compact dimensions in the bulk.",
"These functions allow us to determine the entanglement entropy from the geometry in the wedge $w(D_A)$ via the Takayanagi et.",
"al.",
"proposal, so we have a map from the space of metrics to the space of entropy functions.",
"Small changes in the geometry of the wedge $w(D_A)$ will generally affect the areas of some of the minimal surfaces, while small changes in the geometry outside the wedge will generally not affect these areas.",
"It is at least plausible that the entanglement information could be used to fully reconstruct the geometry in the wedge in some cases, since the map from wedge geometries into the entanglement information is a map from finitely many functions of three variables to a function of four variables, and it is possible for such a map to be an injection.",
"A proven result of this form in the mathematics literature [13] is that for two-dimensional simpleSee [13] for the definition of a simple manifold.",
"compact Riemannian manifolds with boundary, the bulk geometry is completely fixed by the distance function $d(x,y)$ between points on the boundary (the lengths of the shortest geodesics connecting various points).",
"This implies that for static three-dimensional spacetimes, the spatial metric of the bulk constant time slices can be reconstructed in principle if the entanglement entropy is known for arbitrary subsets of the boundary.",
"However, we are not aware of any results about the portion of a space that can be reconstructed if the distance function is known only on a subset of the boundary, or of any results that apply to Lorentzian spacetimes.",
"We saw above that in special cases, $z(D_A)$ together with $J(z(D_{\\bar{A}}))$ cover the entire spacetime, so Constraint 3 is just barely satisfied for $z$ (or $\\hat{z}$ ).",
"For these examples, if $z(D_{\\bar{A}})$ is in $R(\\bar{A})$ then $R(A)$ cannot possibly be larger than $z(D_A)$ .",
"On the other hand, for generic spacetimes, we argued that only a portion of the spacetime is covered by $z(D_A)$ and $J(z(D_{\\bar{A}}))$ , leaving the possibility that $R(A)$ could be larger than $z(D_A)$ .",
"In these examples, extremal surfaces from $A$ typically extend into the region not covered by $z(D_A)$ or $z(D_{\\bar{A}})$ (or the causal past/future of these), and this motivated us to consider $\\hat{w}(D_A)$ as a larger possibility for $R(A)$ .",
"We will now see that in a much wider class of examples, $\\hat{w}(D_A)$ together with $J(w(D_{\\bar{A}}))$ do cover the entire spacetime.",
"To see this, recall that the surfaces $W(A)$ and $W(\\bar{A})$ computing the entanglement entropy of the entire regions $A$ and $\\bar{A}$ are the same by definition, as long as $A$ and $\\bar{A}$ are homologous in the bulk.The only possible exception would be the case where there are two extremal surfaces with equal area having boundary $\\partial A$ .",
"In this case, we might call one $W(A)$ and the other $W(\\bar{A})$ .",
"Now, suppose that for a one-parameter family of boundary regions $B(\\lambda ) \\subset A$ interpolating between $A$ and a point (assuming $A$ is contractible), the surfaces $W(B(\\lambda ))$ change smoothly.",
"Similarly, suppose that for a one-parameter family of boundary regions $B^{\\prime }(\\lambda ) \\subset \\bar{A}$ interpolating between $\\bar{A}$ and a point (assuming $\\bar{A}$ is contractible), the surfaces $W(B^{\\prime }(\\lambda ))$ change smoothly.",
"Then the union of all surfaces $W(B(\\lambda ))$ and $W(B^{\\prime }(\\lambda ))$ covers an entire slice of the bulk spacetime.",
"In this case, for any point $p$ in the bulk spacetime, either there is a causal curve through $p$ that intersects $\\cup _\\lambda W(B(\\lambda )) \\subset w(D_A)$ or else every causal curve through $p$ intersects $\\cup _{\\lambda } W(B^{\\prime }(\\lambda )) \\subset w(D_{\\bar{A}})$ .",
"This shows that $\\hat{w}(D_A)$ together with $J(w(D_{\\bar{A}}))$ cover the entire spacetime.",
"To summarize, in cases where $W(B)$ varies smoothly with $B$ as described above, we have that $\\hat{w}(D_A)$ together with $J(w(D_{\\bar{A}}))$ cover the entire spacetime.",
"Thus, by Constraint 3, with this smoothness condition, if $\\hat{w}(\\bar{A}) \\subset R(\\bar{A})$ then $R(A)$ cannot be larger than $\\hat{w}(D_A)$ .An alternative condition that leads to the same conclusion is that $w(D_A) \\cup w(D_{\\bar{A}})$ includes a Cauchy surface.",
"While there are many examples of spacetimes for which this smooth variation does not occur (e.g.",
"as described in the next section), spacetimes satisfying the condition are not particularly special.",
"We have seen that $\\hat{w}(D_A)$ is in some sense a maximally optimistic proposal for $R(A)$ in cases where a particular smoothness condition is satisfied or when $w(D_A) \\cup w(D_{\\bar{A}})$ includes a Cauchy surface.",
"We will now see that these conditions can fail to be true in some cases, and that in these cases, $R(A)$ must be larger than $\\hat{w}(D_A)$ for some choice of $A$ .",
"Figure: Different possible behaviors of extremal surfaces in spherically symmetric static spacetimes.",
"Shaded region indicates w(D A )w(D_A) where AA is the right hemisphere.",
"The boundary of the shaded region on the interior of the spacetime is the minimal area extremal surface bounded by the equatorial S d-1 S^{d-1}.Consider the simple example of static spherically symmetric spacetimes with metric of the form $ds^2 = -f(r)dt^2 + dr^2/g(r) + r^2 d \\theta ^2$ where $g(0)=1$ and $g(r) \\rightarrow r^2$ for large $r$ .",
"For any spacetime of this form, the extremal codimension-two surfaces bounded by spherical regions on the boundary will be constant-time surfaces in the bulk that can easily be computed.",
"By symmetry, there always exists an extremal surface through the center of the spacetime whose boundary is an equatorial $S^{d-1}$ of the boundary $S^d$ .",
"Now, moving out towards the boundary along some radial geodesic, there will be a unique extremal surface passing through each point and normal to the radial line.",
"In some cases (e.g.",
"pure AdS), the boundary spheres for these extremal surfaces shrink monotonically as we approach the boundary, as shown in the left half of Figure REF .",
"However, there are other cases for which $g(r)$ is not monotonic where the extremal surfaces shrink in the opposite direction, then grow, then shrink again, as shown in the right half of Figure REF .As an explicit example, we have considered the case of a charged massive scalar field coupled to gravity, with scalar field of the form $\\phi (r) = e^{i \\omega t} f(r)$ .",
"Spherically-symmetric configurations of this type with non-zero charge are known as “boson-stars” [11].",
"We find that for fixed $\\psi (0)$ , the metric function $g(r)$ is monotonically increasing for sufficiently small values of the scalar field mass, while for sufficiently large values we can have the behavior shown on the right in Figure REF .",
"In these cases, boundary spheres with angular radius in a neighborhood of $\\pi /2$ will bound multiple extremal surfaces in the bulk.",
"The extremal surface of minimum area in these cases is always one that is contained within one half of the bulk space (otherwise we could construct intersecting minimal surfaces bounding disjoint regions of the boundary).",
"Considering only the minimal surfaces, we find that there exists a spherical region in the middle of the spacetime penetrated by no such surface.",
"Thus, even if we choose $D_A$ to be the entire spacetime boundary, the region $w(D_A)$ excludes the region $r<r_0$ for some $r_0$ .",
"In this case, we have all information about the field theory (assumed to be a pure state), so $R(A)$ should be the entire spacetime.",
"More generally, the region $w(D_A)$ in these cases will have a “hole” if $A$ is chosen to be any boundary sphere with angular radius between $\\pi /2$ and $\\pi $ , as shown in Figure REF .",
"Note, however, that the central region is included in $z(D_A)$ for sufficiently large $A$ , so $z(D_A) \\lnot \\subset w(D_A)$ in these cases.",
"Figure: Region w(D A )w(D_A) (shaded) where AA is a boundary sphere of angular size greater than π\\pi .",
"No minimal surface with boundary in AA penetrates the unshaded middle region."
],
[
"Discussion",
"In this note, we have presented various consistency constraints on the region $R(A)$ of spacetime which can in principle be reconstructed from the density matrix $\\rho _A$ for a spatial region $A$ of the boundary with domain of dependence $D_A$ .",
"We have argued that the $z(D_A) \\equiv J^+(D_A) \\cap J^-(D_A)$ and its domain of dependence $\\hat{z}(D_A)$ should be contained in $R(A)$ and that $\\hat{z}(D_A)$ satisfies our consistency constraints.",
"Since entanglement observables calculated from $\\rho _A$ correspond to extremal surfaces that typically probe a region of spacetime beyond $\\hat{z}(D_A)$ , we have also considered the union of these surfaces $w(D_A)$ and its domain of dependence $\\hat{w}(D_A)$ as a possibility for $R(A)$ that is often larger than $\\hat{z}(D_A)$ .",
"We have seen that $\\hat{w}(D_A)$ also satisfies our constraints (assuming Conjecture ${\\bf C1}$ ), and that if $\\hat{w}(D_A) \\subset R(A)$ generally, then $R(A) = \\hat{w}(D_A)$ for a broad class of spacetimes."
],
[
"A false constraint",
"The constraints discussed in this note are essentially consistency requirements that do not make use of details of the AdS/CFT correspondence.",
"It is interesting to ask whether there exist any more detailed conditions that could constrain the region $R(A)$ further.",
"It may be instructive to point out a somewhat plausible constraint that turns out to be false.",
"For two non-intersecting regions $A$ and $B$ of the boundary spacetime, it may seem that the region $G(B)$ of the spacetime used to construct field theory observables in $B$ should not intersect the region $R(A)$ dual to the density matrix $\\rho _A$ .",
"The argument might be that if the physics in $R(A)$ is the bulk manifestation of information in $\\rho _A$ , we cannot expect to learn anything about this region knowing only $\\rho _B$ .",
"It would seem that this would be telling us directly about $\\rho _A$ knowing only $\\rho _B$ .",
"Perhaps surprisingly, it is easy to find an example where neither $w(D_A)$ nor $z(D_A)$ satisfies this constraint, see Figure REF .",
"Figure: Spatial t=0t=0 slice of w(D A )w(D_A) (light shaded plus dark shaded) and z(D A )z(D_A) (dark shaded) for a planar AdS black hole.",
"The dashed curve is a spatial geodesic with endpoints in A ¯\\bar{A}.",
"Knowledge of observables obtained from ρ A ¯ \\rho _{\\bar{A}} alone allow us to compute the length of this geodesic.In the planar AdS black hole geometry, take the region $A$ to be a ball-shaped region on the boundary.",
"In this case, it is straightforward to check that spatial geodesics with endpoints in $\\bar{A}$ intersect both $w(D_A)$ and $z(D_A)$ .",
"Thus, the constraint $R(A) \\cap G(\\bar{A}) = \\emptyset $ can't be correct if $z(D_A) \\subset R(A)$ .",
"In hindsight, it is not difficult to understand the reason.",
"Knowledge of the density matrix $\\rho _{\\bar{A}}$ allows us to reconstruct $R(\\bar{A})$ .",
"There could be many states of the full theory that give rise to the same density matrix $\\rho _{\\bar{A}}$ .",
"For any such state with a classical gravity dual description, the dual spacetime geometry must be such that spatial geodesics anchored in $D_{\\bar{A}}$ have the same lengths as in the original spacetime we were considering.",
"But there can be many such spacetimes.",
"So using the information in $\\rho _{\\bar{A}}$ , we are not learning directly about $\\rho _A$ , only about the family of density matrices $\\rho _A^\\alpha $ such that the pair $(\\rho _{\\bar{A}}, \\rho _A^\\alpha )$ can arise from a pure state $|\\Psi \\rangle $ that has a geometrical gravity dual.Don Marolf has pointed out to us that the connection between two-point functions and the lengths of spatial geodesics has been argued to fail for spacetimes that do not satisfy certain analyticity properties [15].",
"It is likely that demanding such properties imposes even stronger constraints connecting $\\rho _A$ and $\\rho _{\\bar{A}}$ .",
"The observations in this note highlight the importance of entanglement in the emergence of the dual spacetime.",
"Consider a collection $\\lbrace A_i\\rbrace $ of subsets on the boundary such that $\\cup A_i$ covers an entire boundary Cauchy surface.",
"In a classical system, knowing the configuration and time derivatives of the fields in each of these regions would give us complete information about the physical system.",
"Quantum mechanically, however, complete information about the system consists of two ingredients: (i) the density matrices $\\rho _{A_i}$ , and (ii) the entanglement between the various regions.",
"If we subdivide a set $A \\rightarrow \\lbrace B,C\\rbrace $ and pass from $\\rho _A \\rightarrow \\lbrace \\rho _B,\\rho _C\\rbrace $ , we lose information about the entanglement between $B$ and $C$ .",
"In the bulk picture, the region of spacetime that we can reconstruct (for any $R$ satisfying our constraints) is significantly smaller than before, as we see in Figure REF .",
"The region of spacetime that we can no longer reconstruct corresponds to the information about the entanglement between the degrees of freedom in $B$ and $C$ that we lost when subdividing.",
"As we divide the boundary into smaller and smaller sets $A_i$ , we retain information about entanglement only at successively smaller scales, while the bulk space $\\cup R(A_i)$ that can be reconstructed retreats ever closer to the boundary (Figure REF ).",
"Conversely, knowledge of the bulk geometry at successively greater distance from the boundary requires knowledge of entanglement at successively longer scales.A very similar picture was advocated in [16].",
"In the limit where $A_i$ become arbitrarily small, we know nothing about the bulk spacetime even if we know the precise state for each of the individual degrees of freedom via the matrices $\\rho _{A_i}$ .",
"In this sense, the bulk spacetime is entirely encoded in the entanglement of the boundary degrees of freedom.",
"We are especially grateful to Veronika Hubeny and Mukund Rangamani for important comments and helpful discussions.",
"This work is supported in part by the Natural Sciences and Engineering Research Council of Canada and by the Canada Research Chairs Programme."
]
] | 1204.1330 |
[
[
"Nucleosynthesis and the Inhomogeneous Chemical Evolution of the Carina\n Dwarf Galaxy"
],
[
"Abstract The detailed abundances of 23 elements in nine bright RGB stars in the Carina dSph are presented based on high resolution spectra gathered at the VLT and Magellan telescopes.",
"A spherical model atmospheres analysis is applied using standard methods to spectra ranging from 380 to 680 nm.",
"The stars in this analysis range from -2.9 < [Fe/H] < -1.3, and adopting the ages determined by Lemasle et al.",
"(2012), we are able to examine the chemical evolution of Carina's old and intermediate-aged populations.",
"One of the main results from this work is the evidence for inhomogeneous mixing in Carina; a large dispersion in [Mg/Fe] indicates poor mixing in the old population, an offset in the [alpha/Fe] ratios between the old and intermediate-aged populations (when examined with previously published results) suggests that the second star formation event occurred in alpha-enriched gas, and one star, Car-612, seems to have formed in a pocket enhanced in SN Ia/II products.",
"This latter star provides the first direct link between the formation of stars with enhanced SN Ia/II ratios in dwarf galaxies to those found in the outer Galactic halo (Ivans et al.",
"2003).",
"Another important result is the potential evidence for SN II driven winds.",
"We show that the very metal-poor stars in Carina have not been enhanced in AGB or SN Ia products, and therefore their very low ratios of [Sr/Ba] suggests the loss of contributions from the early SNe II.",
"Low ratios of [Na/Fe], [Mn/Fe], and [Cr/Fe] in two of these stars support this scenario, with additional evidence from the low [Zn/Fe] upper limit for one star.",
"It is interesting that the chemistry of the metal-poor stars in Carina is not similar to those in the Galaxy, most of the other dSphs, or the UFDs, and suggests that Carina may be at the critical mass where some chemical enrichment events are lost through SN II driven winds."
],
[
"Introduction",
"Chemical analyses of stars in nearby classical dwarf galaxies have swelled in the past decade, with the advent of large aperture telescopes, high efficiency spectrographs, and multiplexing capabilities.",
"Detailed abundances are now available for dozens to hundreds of stars in classical dwarf spheroidal galaxies (dSph), ultra faint dwarf (UFD) systems, and even a few massive stars in some low mass dwarf irregular galaxies (dIrr); see the review by Tolstoy et al. (2009).",
"These chemical studies have shown that low mass dwarf galaxies have had a slower chemical evolution than the stellar populations in the Milky Way.",
"For example, the majority of stars in dSphs have mean metallicities and [$\\alpha $ /Fe][X/Fe] = log(X/Fe)$_*$ - log(X/Fe)$_\\odot $ .",
"ratios that are lower than those of the Sun.",
"Stars in dwarf galaxies also tend to have different heavy element ratios (e.g., higher [Ba/Y] or [La/Eu]) than similar metallicity stars in the Galaxy (e.g., Venn et al.",
"2004, [5], Letarte et al. 2010).",
"On the other hand, the most metal-poor stars in dwarf galaxies, with [Fe/H] $\\le -2.5$ , tend to have similar abundance ratios to Galactic halo stars, even for the $\\alpha $ and heavy element ratios (e.g., Sculptor, Fornax, and Sextans, Tafelmeyer et al. 2010).",
"This is also true of the very metal-poor stars in the UFDs (e.g., Com Ber, Boötes I, and Leo IV, Norris et al.",
"2008, Feltzing et al.",
"2009, Frebel et al.",
"2010a, Simon et al. 2010).",
"One exception are the $\\alpha $ and heavy element abundances for metal-poor stars in Sextans, which show both offsets and larger dispersions than the Galaxy and other dwarf galaxies [5].",
"The Carina dwarf galaxy provides a new opportunity for chemical evolution and nucleosynthetic studies.",
"It has a low mass that is similar to that of Sextans (Walker et al.",
"2009), and it has had an unusual, episodic star formation history.",
"Its colour-magnitude diagram (CMD) shows at least three main sequence turn-offs (Monelli et al.",
"2003, Bono et al.",
"2010) and is best described by a star formation history with a well-defined old population ($\\sim $ 10-12 Gyr), a dominant intermediate-aged population ($\\sim $ 5-7 Gyr), and a trace young population ($\\sim $ 2 Gyr; although the specific timescales are uncertain).",
"These star forming episodes are separated by long quiescent periods, seen as gaps between the main sequence turn-off points.",
"It is estimated that 70 to 80% of the stars in Carina have intermediate-ages, while most of the remaining stars are old and associated with the first star forming episode (also see Dolphin 2002, Hernandez et al.",
"2000, Hurley-Keller et al.",
"1998, and Mighell 1997).",
"One wonders how this unique star formation history may have affected the chemical evolution of Carina.",
"In spite of its punctuated star formation history, Carina has an extremely narrow red giant branch.",
"One possibly is that the RGB stars in Carina have a fortuitous alignment in the age-metallicity degeneracy, such that the older metal-poor stars overlay the metal enhanced intermediate-aged stars.",
"Alternatively, the narrow red giant branch could be dominated by intermediate-aged stars only, if that population formed on a relatively short timescale and with only a modest metallicity spread (Rizzi et al.",
"2003, Bono et al. 2010).",
"Low resolution spectra of 437 red giant branch (RGB) stars in Carina analysed by Koch et al.",
"(2006) showed the mean metallicity in Carina is [Fe/H] $\\sim -1.7 \\pm 0.9$ , in agreement with similar analyses by Helmi et al.",
"(2006) and Starkenburg et al. (2010).",
"This metallicity spread is larger than predicted by [19] from their CMD analysis ($\\Delta $ [Fe/H] $\\le \\pm 0.5$ dex).",
"Koch et al.",
"(2006) also found no significant gradients in metallicity or stellar population with position in Carina, unlike the results for Sculptor (Tolstoy et al.",
"2004) and Fornax (Battaglia et al. 2006).",
"High resolution analyses of five RGB stars (Shetrone et al.",
"2003) and ten RGB stars (Koch et al.",
"2008) supported the larger range in metallicity found by the low resolution spectral analyses.",
"These analyses also showed that Carina is like the other dSph galaxies in that the [$\\alpha $ /Fe] ratios are lower than in Galactic halo stars with similar metallicity.",
"Lanfranchi et al.",
"(2003, 2004, 2006) used these datasets to develop chemical evolution models for Carina, tuning the high wind efficiency and low star formation efficiency to reproduce Carina's metallicity distribution function, [$\\alpha $ /Fe] ratios, and low gas content.",
"In this paper, we present a new detailed abundance analysis of up to 23 elements in nine stars in the Carina dwarf galaxy.",
"This work increases the number of elements, the number of stars, and the metallicity range previously explored.",
"We also report the abundances for two newly discovered very metal-poor stars (with [Fe/H]$\\sim -2.85$ ), which allow us to examine the earliest epoch of star formation.",
"These results, with previously published abundances, are used to explore the unique star formation history and chemical evolution in Carina.",
"Figure: Colour-magnitude diagram, V vs (V-I), for Carinafrom our ESO WFI data.",
"Only stars with radial velocities>> 200 km s -1 ^{-1} (from Koch et al.",
"2006) are shown.Stars with high resolution spectral analyses are identifiedby red symbols, including nine stars from this paper, fivefrom Shetrone et al.",
"(2003), and ten from Koch et al.",
"(2008;four are in common with this analysis).One carbon-star is noted by the red cross,and two foreground objects are shown as black crosses.The faintest star in our sample is Car-7002."
],
[
"Observations and Data Reductions",
"The data presented in this paper were acquired at two observatories during separate time allocations in January 2005.",
"The FLAMES (Fibre Large Array Multi Element Spectrograph) multiobject spectrograph at the 8.2 meter UT2 (Kueyen) at the Very Large Telescope (VLT) of the European Southern Observatory (ESO) was used to collect high resolution spectra using both the UVES and GIRAFFE fiber modes (Pasquini et al. 2002).",
"The analysis of the FLAMES/GIRAFFE spectra is presented by Lemasle et al. (2012).",
"Also, the the MIKE (Magellan Inamori Kyocera Echelle) spectrograph at the Magellan Landon 6.5m Clay Telescope at the Las Campanas Observatory was used to collect high resolution spectra of individual stars outside of the central field of Carina.",
"Our targets and those previously analysed with high resolution spectral analyses by Koch et al.",
"(2008) and Shetrone et al.",
"(2003) are shown in Fig.",
"REF on the $V$ vs $(V-I)$ CMD.",
"These stars are also shown in Fig.",
"REF as metallicity (from Koch et al.",
"2006) versus their location within Carina.",
"Figure: Spatial location and metallicities of the nine Carina starsin this paper, as well as five from Shetrone et al.",
"(2003), and tenstars from Koch et al.",
"(2008; four are in common with this analysis).Symbols are the same as in Figure .These are compared to the distribution of 437 RGBstars from Koch et al.",
"(2006) which have a high (3 σ\\sigma )probability of Carina membership.The metallicitieson this plot are those determined by Koch et al.",
"(2006).Elliptical radii are calculated using the structuralparameters for Carina from Irwin & Hatzidimitrou (1995;central coordinates 6 h 40.6 ' ^h 40.6^{\\prime } and -50 o 56 ' -50^o 56^{\\prime } for epochB1950, position angle 65o, ellipticity ϵ\\epsilon =0.33),and listed in Table ,and core and tidal radii are indicated."
],
[
"Magellan MIKE Spectra ",
"Target selection for the Magellan run was from low resolution spectra of the Ca2 8600 Å feature (=CaT) for hundreds of stars in the Carina dSph in the ESO archive (Koch et al.",
"2006, Helmi et al. 2006).",
"The feature was used to ascertain membership from radial velocities, as well as estimate an initial metallicity for each target based on the CaT-metallicity calibration available at that time [13].",
"Exposure times, radial velocities (km s$^{-1}$ ), the signal-to-noise ratio (SNR) at three wavelengths, the elliptical radius from the center of Carina (in arcminutes), and alternative names for the targets are summarized in Table REF .",
"Spectra for three objects in the metal-poor globular cluster M68 were also taken as standard stars.",
"M68 was chosen as a standard since it has low metallicity (Harris 1996) and low reddening.",
"BVI magnitudes are from Stetson (2000, with corrections onto the Johnson Kron-Cousins scale of the Landolt standard stars as in Stetson 2005), which places these stars on the upper RGB of M68 and suggests they have intrinsic luminosities similar to our program stars.",
"On the red side, the double echelle design covers 4850 - 9400 Å, and on the blue side, 3800 -5050 Å.",
"The quality of the blue spectra is significantly higher in the overlapping wavelength region ($\\sim $ 4850 to 5050 Å).",
"Using a 1.0” slit, a spectral resolution R = 28,000 (blue) to R = 22,000 (red) was obtained.",
"The red chip had a gain of 1.0 electrons ADU$^{-1}$ and read noise of 4 electrons; the blue chip had a gain of 0.47 electrons ADU$^{-1}$ and a read noise of 2 electrons.",
"We binned on-chip with 2x2 pixels.",
"Table REF lists the SNR per pixel achieved in the final combined spectra near 4200, 5200, and 6200 Å.",
"The data were reduced using standard IRAFIRAF is distributed by the National Optical Astronomy Observatory, which is operated by the Association of Universities for Research in Astronomy, Inc., under cooperative agreement with the National Science Foundation.",
"routines.",
"Sky subtraction was done with a smooth fit perpendicular to the dispersion axis.",
"Heliocentric corrections were applied to each spectrum before determining their radial velocities.",
"Multiple spectra were median combined, which helped to remove cosmic ray strikes.",
"Spectra taken of the Th-Ar lamp provided the wavelength calibration.",
"Both quartz flats and screen flats were taken at varying exposure levels; no significant offsets were found between the well-illuminated quartz flats and the science exposures, thus the quartz flats were adopted for the data reductions.",
"A small dark current was noticed on one corner of the blue chip in a region that did not receive starlight and therefore did not affect this analysis.",
"The final spectra were normalized using k-sigma clipping with a non-linear filter (a combination of a median and a boxcar).",
"The effective scale length of the filter was set from 8 to 15 Å, dependent on the crowding of the spectral lines, and we found that this was sufficient to follow the continuum without affecting the presence of the lines when used in conjunction with iterative clipping.",
"This method was also used by [13] to normalize CaT spectra.",
"llrrcrrrccc Observing Information 0pt Star Other RA DEC R$_{\\rm ell}$ Exp RV RV SNR SNR SNR Name (2000) (2000) (arcmin) Time CaT HR 4200 5200 6200 (s) (km s$^{-1}$ ) (km s$^{-1}$ ) M68-6022 S195 12 39 17.0 -26 45 39 2x 600 -95.1 $\\pm $ 1.5 60 73 75 M68-6023 S239 12 39 37.6 -26 45 15 1x 600 -94.8 $\\pm $ 1.3 55 66 65 M68-6024 S225 12 39 30.1 -26 42 47 1x 600 -96.2 $\\pm $ 1.6 30 44 46 Car-1087 S12924 6 41 15.4 -51 01 16 5.0 4x 3600 229.3 220.1 $\\pm $ 4.1 20 30 32 LGO4c-006621 Car-5070 S24846 6 41 53.8 -50 58 11 3.5 3x 3600 213.5 211.5 $\\pm $ 3.5 20 30 32 car1-t213 Car-7002 S06496 6 40 49.1 -51 00 33 8.0 3x 3600 226.6 224.2 $\\pm $ 3.7 18 30 27 LGO4c-000826 car0619-a2 Car-484 car1-t057 (K) 6 41 39.6 -50 49 59 11.2 11x 3600 232.6 229.4 $\\pm $ 0.7 20 35 LGO4a-002181 Car-524 car1-t083 6 41 14.6 -50 51 10 11.4 11x 3600 219.4 218.6 $\\pm $ 0.7 20 40 LGO4a-002065 Car-612 car1-t076 (K) 6 40 58.6 -50 53 35 10.0 11x 3600 223.1 222.9 $\\pm $ 0.7 19 29 LGO4a-001826 Car-705 car1-t048 (K) 6 42 17.3 -50 55 55 6.8 11x 3600 221.3 220.9 $\\pm $ 0.9 18 28 LGO4a-001556 Car-769 car1-t069 6 41 19.7 -50 57 26 3.3 11x 3600 219.4 218.9 $\\pm $ 0.6 17 27 LGO4a-001364 Car-1013 car1-t152 (K) 6 41 22.0 -51 03 43 7.8 11x 3600 218.5 218.4 $\\pm $ 1.5 11 15 LGO4c-006477 Car-837 car1-t191 6 41 46.3 -50 58 56 11x 3600 232.6 222.4 $\\pm $ 0.7 20 30 (C-star) Car-489 (non-member) 6 41 37.0 -50 50 07 11x 3600 236.5 16 25 Car-X (non-member) 6 41 26.9 -51 00 34 11x 3600 4.9 $\\pm $ 0.8 15 25 Data for M68 and the first three Carina stars are from the Magellan MIKE spectrograph, with the remaining nine Carina stars observed with the FLAMES spectrograph at the VLT.",
"Stars partially analysed by Koch et al.",
"(2008) are noted with “(K)” beside their alternative names.",
"The S# target names are from P.B.",
"Stetson's online database of homogeneous photometry (at http://cadcwww.hia.nrc.ca/stetson).",
"Car-X is a bright target right next to Car-909, which was the actual target.",
"SNR are the signal-to-noise ratios per pixel."
],
[
"VLT FLAMES/UVES Data",
"Observations with the multi-object FLAMES spectrograph at the UT2 Kueyen ESO-VLT (Pasquini et al.",
"2002) were carried out for nine targets in the central region of Carina.",
"In two separate configurations, the FLAMES/UVES fibers were placed on bright RGB stars resulting in nine stellar spectra and two sky observations.",
"Two of the nine targets proved to be foreground RGB stars, while one is a carbon star that is unsuitable for our analysis.",
"Target coordinates, exposure times, radial velocities (km s$^{-1}$ ), elliptical radius from the center of Carina (in arcmins), and the SNR at two wavelengths for the final combined spectra are listed in Table REF .",
"Other names for each target are also listed.",
"FLAMES/GIRAFFE spectra were simultaneously obtained at high resolution (R=20,000) for 36 more stars in Carina, but over much shorter wavelength regions (3 wavelength settings that yielded $\\sim $ 250 Å each) - these are presented in a separate paper by Lemasle et al.",
"(2012).",
"Due to variable weather conditions, only eleven of the one-hour exposures had sufficient signal to be used further in this analysis.",
"On the red side, the double echelle design covers 5840$-$ 6815 Å, and on the blue, 4800$-$ 5760Å.",
"A 1.0” slit yields a spectral resolution R = 47,000.",
"Table REF lists the exposure times and SNR per pixel in the final combined spectra near 5200 and 6200 Å.",
"Unfortunately, six of the one-hour exposures did not have enough signal in the FLAMES/UVES fibers to be useful in this analysis.",
"The data were reduced using the ESO FLAMES pipeline.The GIRAFFE Base-Line Data Reduction, girBLDRS, was written at the Observatory of Geneva by A. Blecha and G. Simond, and is available through SourceForge at http://girbldrs.sourceforge.net/.",
"The sky spectrum was fit with a smoothly varying function, and this was subtracted from the stellar exposures per wavelength set up (to reduce adding more noise in already low SNR spectra).",
"Each spectrum was then heliocentric velocity corrected, radial velocities were determined, and the spectra were coadded (weighted by the SNR).",
"The final spectra were normalized using an iterative asymmetric k-sigma clipping routine with a non-linear filter (like for the Magellan/MIKE spectra)."
],
[
"Line List",
"A range of elements are detectable in our spectra, which enables a comprehensive abundance analysis.",
"Atomic lines for this analysis were selected from the literature, including the line lists from Shetrone et al.",
"(2003), Cayrel et al.",
"(2004), [6], Cohen et al.",
"(2008), Letarte et al.",
"(2009), Tafelmeyer et al.",
"(2010), and Frebel et al.",
"(2010a); see Table REF .",
"Atomic data for the Fe1 lines were updated from O'Brian et al.",
"(1991) when available, or the atomic data was updated to the latest values in the National Institute of Standards and Technology (NIST) database.http://physics.nist.gov/PhysRefData/ASD/index.html lrrrrrrrr 10 0pt Line List Wave $\\chi $ log gf CAR CAR CAR CAR CAR CAR (Å) (eV) 484 524 612 705 769 1013 Fe I 5001.864 3.88 0.010 124.9 116.7 124.0 134.0 104.3 165.1 5006.120 2.83 -0.615 194.7 166.1 185.2 193.8 158.2 5012.070 0.86 -2.642 195.0 179.3 5014.940 3.94 -0.300 113.4 103.4 117.0 114.3 65.4 97.2 Equivalent widths are in mÅ, and upper limits are noted.",
"When spectrum syntheses have been used in the abundance analysis, then an “S” is noted.",
"A second table includes the stars observed with the MIKE spectrograph at Magellan.",
"Both tables are published in their entirety in the electronic edition.",
"A portion is shown here for guidance regarding its form and content."
],
[
"Equivalent Width Measurements ",
"Most of the elemental abundances in this analysis are determined from equivalent width (EW) measurements.",
"Spectrum syntheses were used only for lines affected by hyperfine splitting or for elements with line measurements from low SNR spectra.",
"Equivalent widths were measured using the Gaussian fitting routine DAOSPEC (Stetson & Pancino 2008).",
"The placement of the continuum is a critical step, as it influences the measurements of the equivalent widths; thus some care was taken in assigning and testing the DAOSPEC fitting parameters.",
"For example, a low order polynomial (order 13) was adopted to allow DAOSPEC to measure the effective continuum, and tests of other low order values (order =5 and =1) resulted in nearly identical equivalent widths, with $\\Delta $ EW $<$ 1 %.",
"As a final exercise, the continuum was set to one so that DAOSPEC would not relocate the global continuum for the measurement (order = -1).",
"Results from this test showed a slight offset $\\le +12\\%$ in the equivalent width measurements, as expected.",
"We note this offset is similar to or less than our adopted equivalent width measurement errors from the Cayrel formula (described below).",
"For testing purposes, equivalent widths were also measured with splot in IRAF to determine the area under the continuum.",
"A comparison of these measurements with those from DAOSPEC for two of the stars observed with the MIKE spectrograph are shown in Fig.",
"REF .",
"Figure: Equivalent width measurement comparisonsfor M68-6022 and Car-1087, both observed with theMIKE spectrograph.",
"The three lines represent equalEW values and EW ±\\pm Δ\\Delta EW, whereΔ\\Delta EW = EW rms _{\\rm rms} + 10% EW, andEW rms _{\\rm rms} = 3 mÅ and 6 mÅ, respectively.The expected minimum measurement uncertainties on the equivalent widths (EW$_{\\rm rms}$ ) have been estimated using a revised Cayrel formula, i.e., a new derivation of the measurement errors on the equivalent widths was presented by [13], where it was found that the factor of 1.5 in the Cayrel (1988) formula is actually within the square root; thus, EW$_{\\rm rms}$ = SNR$^{-1}$ x $\\@root \\of {1.5 \\times FWHM \\times \\delta {\\rm x}}$ where SNR is the signal-to-noise ratio per pixel, FWHM is the line full width at half maximum, and $\\delta {\\rm x}$ is the pixel size.",
"An extra 10% of the EW was added to this for a more conservative measurement error, such that $\\Delta $ EW = EW$_{\\rm rms}$ + 0.10 $\\times $ EW.",
"For the M68 stellar spectra taken at Magellan the revised Cayrel formula gives EW$_{\\rm rms}$ = 3 mÅ, but EW$_{\\rm rms}$ = 6 mÅ for the lower SNR spectra of the Carina stars observed with MIKE.",
"For the spectra taken at the VLT with FLAMES/UVES, the higher resolution yields a minimum error of EW$_{\\rm rms}$ = 4 mÅ, with an exception for Car-1013 where EW$_{\\rm rms}$ = 7 mÅ due to its lower SNR spectrum.",
"The reported errors from the DAOSPEC program were lower than those from the revised Cayrel formula, by 1/3 to 1/2.",
"This is due to correlations in the noise estimates in DAOSPEC when the pixel data are rebinned/interpolated during the spectral extractions and wavelength calibrations (P. Stetson, priv.",
"communications).",
"When individual line measurements were significantly different between the DAOSPEC and splot measurements, those lines were checked by eye.",
"These lines tended to be unresolved blends and/or suffered from difficult continuum placement.",
"Since DAOSPEC uses a fixed FWHM and a consistent prescription for continuum placement, those EW measurements were adopted for this analysis, although the $\\Delta $ EW errors are from the revised Cayrel formula.",
"lrrr|rr|rr|rrrrr Target Information 0pt Star B V I V I J K$_{s}$ B V I J K PBS PBS PBS WFI WFI 2MASS 2MASS Gul Gul Gul Gul Gul M68-6022 15.127 14.205 13.106 $-$ $-$ 12.324 11.659 $-$ $-$ $-$ $-$ $-$ M68-6023 15.192 14.284 13.191 $-$ $-$ 12.363 11.712 $-$ $-$ $-$ $-$ $-$ M68-6024 15.230 14.354 13.289 $-$ $-$ 12.479 11.858 $-$ $-$ $-$ $-$ $-$ Car-1087 19.146 18.031 16.735 18.00 16.73 15.802 15.301 $-$ $-$ $-$ $-$ $-$ Car-5070 19.125 17.904 16.520 17.86 16.49 15.641 14.758 $-$ $-$ $-$ $-$ $-$ Car-7002 19.284 18.344 17.146 18.40 17.19 16.244 15.562 $-$ $-$ $-$ $-$ $-$ Car-484 19.009 17.603 16.179 17.56 16.16 15.322 14.627 19.055 17.607 16.178 15.290 14.406 Car-524 18.934 17.645 16.308 17.63 16.32 15.388 14.473 18.921 17.598 16.278 15.398 14.574 Car-612 19.077 17.811 16.416 17.82 16.42 15.456 14.763 19.103 17.779 16.403 15.609 14.828 Car-705 19.101 17.828 16.430 17.78 16.43 15.687 14.597 19.104 17.788 16.424 15.509 14.663 Car-769 19.202 18.002 16.693 18.01 16.71 15.698 14.921 19.224 18.025 16.725 15.758 15.040 Car-1013 19.048 17.783 16.424 17.77 16.42 15.515 14.888 19.056 17.767 16.421 15.528 14.726 Car-489 $-$ $-$ $-$ 17.85 16.53 15.544 14.933 $-$ $-$ $-$ $-$ $-$ Car-837 $-$ $-$ $-$ 17.71 16.47 15.456 14.509 19.111 17.705 16.509 15.427 14.621 Car-X $-$ $-$ $-$ 17.51 16.42 15.748 15.223 $-$ $-$ $-$ $-$ $-$ BVI from P.B.",
"Stetson's website of homogeneous photometry, VI from our WFI data, JK$_s$ from the 2MASS database, and BVIJK from M. Gulleuszik (private communications).",
"A maximum EW of 200 mÅ was adopted for all stars since deviations from the Gaussian profiles used in DAOSPEC become more significant for stronger lines.",
"In fact, for lines $\\ge $ 200 mÅ, we found that DAOSPEC would occassionally divide these lines into 2-3 lines, each with lower EWs.",
"The final line lists and measurements were carefully reviewed during this analysis."
],
[
"Radial Velocities",
"Radial velocities were initially measured from 5-7 strong lines and refined using DAOSPEC (which cross correlates the measured positions of the detected spectral lines with reference wavelengths from an input linelist).",
"Heliocentric corrections were then applied.",
"The radial velocities for the Carina stars are listed in Table REF ; the results for the Magellan/MIKE blue and red spectra were averaged together since each arm was reduced independently.",
"The three M68 stars have a mean radial velocity of $-95.4 \\pm 1.0$ km s$^{-1}$ , with an average measurement error of 1.5 km s$^{-1}$ .",
"This is in agreement with results in the literature, e.g., Lane et al.",
"(2009) report a radial velocity of $-94.93 \\pm 0.26$ km s$^{-1}$ from 123 RGB stars in M68 taken at the 3.9m Anglo Australian Telescope with the AAOmega spectrograph.",
"The mean radial velocity of our nine Carina stars is $220.5 \\pm 4.9$ km s$^{-1}$ ; the average measurement error of the FLAMES/UVES data are 0.9 km s$^{-1}$ , whereas it is 3.8 km s$^{-1}$ for the metal-poor stars with lower quality Magellan/MIKE spectra.",
"These results are in agreement with the mean radial velocity of 223.9 km s$^{-1}$ found by Koch et al.",
"(2006) for 437 RGB members of Carina, with a dispersion of 7.5 km s$^{-1}$ .",
"The radial velocities from both the high resolution (this paper) and lower resolution spectra (from Koch et al. )",
"are shown in Table REF ; again, these are in good agreement, with a mean difference in $\\langle $ RV$_{\\rm CaT}$ $-$ RV$_{\\rm HR}$ $\\rangle $ = 2.1 $\\pm $ 3.4 km s$^{-1}$ ."
],
[
"Photometric Parameters (physical gravity) ",
"BVIJK$_{s}$ photometric values for the Carina targets and M68 standard stars are listed in Table REF .",
"BVI magnitudes for the M68 stars are from Stetson (2000, on the Johnson Kron-Cousins scale, Stetson 2005).",
"J and K$_{s}$ magnitudes are from the 2MASS databaseThis publication makes use of data products from the Two Micron All Sky Survey which is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center/California Institute of Technology, funded by the National Aeronautics and Space Administration and the National Science Foundation.",
"The database can be found at http://www.ipac.caltech.edu/2mass/releases/allsky.. For the Carina members, VI magnitudes are from the ESO 2.2m WFI photometry; initial calibrations were on the default ESO zero-point values, but have been updated to the Johnson Kron-Cousins scale with Stetson's database and Gullieuszik's photometry (private communications).",
"Initial metallicities for the Carina stars are from the CaT measurements (Koch et al. 2006).",
"For the three M68 stars, the average metallicity from Lee et al.",
"(2005) is adopted, [Fe/H]=$-2.16$ .",
"Lee et al.",
"(2005) found a mean [Fe2/H]=$-2.16$ from a high resolution spectral analysis of eight stars in M68, using photometric gravities.",
"They also find a mean [Fe1/H]=$-2.56$ using spectroscopic gravities.. Color temperatures were found using the Ramirez & Melendez (2005) calibration, and adopted as the initial effective temperatures; see Table REF .",
"Deviations between the different color temperatures are quite small for the three M68 globular cluster stars, i.e., $\\sigma _T \\le $ 20 K from T$(B-V)$ , T$(V-I)$ , T$(V-J)$ , and T$(V-K)$ .",
"Deviations are larger for the Carina stars, with $\\sigma _T \\le $ 134 K, and $\\langle \\sigma _T \\rangle $ = 74 K. Physical (photometric) gravities are determined using the standard relation, log g = log g$_$ + log $\\left({\\rm M}_*\\over {{\\rm M}_}\\right)$ + 4$\\times $ log $\\left({T_{\\rm eff*}}\\over {T_{\\rm eff}}\\right)$ + 0.4$\\times $ (M$_{\\rm Bol*}$ - M$_{\\rm Bol}$ ) with log g$_$ = 4.44, T$_{\\rm eff}$ = 5777 K, M$_{\\rm Bol_}$ = 4.75, and adopting a reddening law of A(V)/E(B-V) = 3.24.",
"The values of M$_{\\rm Bol*}$ are based on the bolometric correction to the V magnitude from [7], with an assumed distance and stellar mass.",
"lrrrrrrrrr Photometric Parameters 0pt Target Tbv Tvi Tvj Tvk $\\langle $ T$\\rangle $ $\\sigma _T$ log g Mv Mbol (K) (K) (K) (K) (K) (K) M68-6022 4663 4671 4677 4694 4676 13 1.53 -0.98 -1.40 M68-6023 4686 4682 4654 4647 4667 20 1.55 -0.91 -1.33 M68-6024 4741 4732 4722 4701 4724 17 1.61 -0.84 -1.23 Car-1087 4391 4457 4572 4416 4459 80 0.95 -2.05 -2.59 Car-5070 4303 4314 4367 4364 4337 33 0.83 -2.19 -2.79 Car-7002 4677 4552 4492 4473 4548 92 1.16 -1.65 -2.15 Car-484 4073 4245 4383 4320 4255 134 0.65 -2.49 -3.14 Car-524 4228 4346 4235 4311 4280 58 0.70 -2.42 -3.05 Car-612 4229 4253 4301 4224 4252 35 0.75 -2.23 -2.88 Car-705 4194 4305 4221 4460 4295 120 0.77 -2.27 -2.89 Car-769 4308 4361 4278 4254 4300 46 0.86 -2.04 -2.65 Car-1013 4208 4280 4372 4250 4278 70 0.75 -2.28 -2.91 Photometric temperatures are from the Ramirez & Melendez (2005) calibration.",
"Each temperature is determined from a photometric colour, e.g., Tbv represents the temperature determined from the (B$-$ V) colour.",
"The average of the four colour temperatures ($\\langle $ T$\\rangle $ ) and standard deviation ($\\sigma _T$ ) are listed along with other parameters derived from the magnitudes (see Section REF ).",
"For M68, we adopt the distance and reddening in the Harris catalogue (1996), (m-M)v = 15.19 $\\pm $ 0.10 and E(B-V)=0.05 from isochrone fitting by McClure et al. (1987).",
"This value is in good agreement with [21], but the proper motion distance modulus, (m-M)v = 14.91 (Dinescu et al.",
"1999) is quite short if this reddening is adopted.",
"A turn off mass of 0.83 $\\pm $ 0.03 M$_\\odot $ is found for M68 using isochrones by Vandenberg & Bell (1985).",
"For Carina, Mighell (1997) determined the distance modulus and reddening from WFPC2 V and I band imaging, but also compares his results in an appendix to the many estimates available in the literature.",
"We adopt Mighell's determination of (m-M)v=20.05 $\\pm $ 0.11 and E(B-V)=0.06 $\\pm $ 0.02.",
"This reddening value is the same as from the Schlegel et al.",
"(1998) maps.",
"One of the highest values for the distance modulus was found by Monelli et al.",
"(2003; (m-M)v = 20.24, with E(B-V)=0.03), but as discussed below (also see Section 5); these differences have negligible effects on our spectral analysis.",
"An estimate of the turn off mass is more difficult in Carina due to its separate episodes of star formation, each of which has a different turn off mass; these estimates can have a larger effect on the physical gravity than the distance modulus.",
"Since Carina is dominated by an intermediate aged population (5-7 Gyr old), we examined isochrones (Fagotto et al.",
"1994, with Y=0.23 and Z=0.0004) to find that the age range of 5 to 12 Gyr corresponds to turn off masses of 1.0 to 0.8 M$_\\odot $ , respectively.",
"We adopted 0.8 M$_\\odot $ in this analysis, but note that if 1.0 M$_\\odot $ were used, this would change the physical gravities by $+0.10$ dex with no effect on temperature; this change has only a small effect on the chemical abundances."
],
[
"Spectroscopic Parameters ",
"The CaT metallicity and the photometric parameters for temperature and gravity were adopted as the initial stellar parameters for a model atmospheres analysis.",
"Spectroscopic indicators were used to refine the effective temperature and metallicity, as well as to determine the microturbulence values.",
"These values are listed in Table REF .",
"Only the physical gravity is unchanged from the photometric parameters analysis."
],
[
"Model Atmospheres",
"The new MARCS spherical models have been adopted in this analysis (Gustafsson et al.",
"2003, 2008, further expanded by B. Plez, private communications).",
"The grid covers the range of parameters (temperature, luminosity, microturbulence, and metallicity) of RGB stars.",
"The models used in this analysis adopt the Galactic abundance pattern, i.e., [$\\alpha $ /Fe] = +0.4 for [Fe/H] $\\le -1.0$ , which is the metallicity range of the stars in Carina.",
"We adopt the standard version of the MOOGThe 2009 version of MOOG is available from http://www.as.utexas.edu/ chris/moog.html.",
"(Sneden 1973) spectral synthesis code to determine the individual line abundances.",
"Corrections were applied when necessary for continuum scattering effects that can affect the results for lines below 5000 Å, as described in Section REF .",
"The photometric gravities were retained throughout the analysis (see Section REF ).",
"lrccclc Spectroscopic Parameters (photometric gravities) 0pt Target T$_{\\rm eff}$ log g$_*$ $\\xi $ 12+log(Fe1) [Fe1/H] $\\pm \\sigma $ (#) CaT (K) (cgs) (km/s) $\\pm \\delta $ (Fe)a (S10)b M68-6022 4550 $\\pm $ 44 1.53 $\\pm $ 0.04 1.50 $\\pm $ 0.12 4.99 $\\pm $ 0.02 $-$ 2.51 $\\pm $ 0.23 (92) $-$ 2.16 M68-6023 4667 $\\pm $ 55 1.55 $\\pm $ 0.04 1.60 $\\pm $ 0.12 5.09 $\\pm $ 0.03 $-$ 2.41 $\\pm $ 0.24 (92) $-$ 2.16 M68-6024 4650 $\\pm $ 56 1.61 $\\pm $ 0.04 1.80 $\\pm $ 0.12 4.86 $\\pm $ 0.03 $-$ 2.64 $\\pm $ 0.24 (81) $-$ 2.16 Car-1087 4600 $\\pm $ 98 0.95 $\\pm $ 0.10 2.45 $\\pm $ 0.25 4.69 $\\pm $ 0.06 $-$ 2.81 $\\pm $ 0.34 (38) $-$ 3.10 Car-5070 4550 $\\pm $ 81 0.83 $\\pm $ 0.10 2.30 $\\pm $ 0.20 5.35 $\\pm $ 0.05 $-$ 2.15 $\\pm $ 0.38 (55) $-$ 2.57 Car-7002 4548 $\\pm $ 96 1.16 $\\pm $ 0.10 2.00 $\\pm $ 0.24 4.64 $\\pm $ 0.05 $-$ 2.86 $\\pm $ 0.33 (38) $-$ 3.29 Car-484 4400 $\\pm $ 66 0.65 $\\pm $ 0.10 2.30 $\\pm $ 0.07 5.97 $\\pm $ 0.02 $-$ 1.53 $\\pm $ 0.21 (114) $-$ 1.62 Car-524 4500 $\\pm $ 64 0.70 $\\pm $ 0.10 2.40 $\\pm $ 0.07 5.75 $\\pm $ 0.02 $-$ 1.75 $\\pm $ 0.21 (121) $-$ 1.61 Car-612 4500 $\\pm $ 61 0.75 $\\pm $ 0.10 2.10 $\\pm $ 0.08 6.20 $\\pm $ 0.02 $-$ 1.30 $\\pm $ 0.23 (126) $-$ 1.84 Car-705 4500 $\\pm $ 61 0.77 $\\pm $ 0.10 2.10 $\\pm $ 0.07 6.15 $\\pm $ 0.02 $-$ 1.35 $\\pm $ 0.25 (127) $-$ 1.63 Car-769 4600 $\\pm $ 76 0.86 $\\pm $ 0.10 2.30 $\\pm $ 0.08 5.82 $\\pm $ 0.02 $-$ 1.68 $\\pm $ 0.25 (125) $-$ 1.63 Car-1013 4600 $\\pm $ 105 0.75 $\\pm $ 0.10 2.20 $\\pm $ 0.11 6.20 $\\pm $ 0.03 $-$ 1.30 $\\pm $ 0.37 (125) $-$ 1.58 a$\\delta $ (Fe) is the error in the mean of the Fe1 lines (see Section REF ).",
"bCaT metallicities from the calibration by Starkenburg et al. (2010).",
"For the three M68 stars, CaT metallicities are from Lee et al.",
"(2005)."
],
[
"Effective Temperatures, Microturbulence, & Metallicity ",
"The effective temperatures were refined from the photometric values through examination of the Fe1 lines.",
"Only lines redwards 5000 Å were considered to avoid uncertainties in the standard MOOG treatment of scattering in the continuous opacity (see Section REF ).",
"Microturbulence values were found by minimizing the slope in the Fe1 line abundances with equivalent width.",
"Similarly, the excitation temperatures were found by minimizing the slope in the same Fe1 line abundances with excitation potential ($\\chi $ , in eV).",
"While the M68 stars had lower excitation than colour temperatures (by an average of 67 K), the Carina stars had higher excitation temperatures (by an average of 200 K).",
"The final metallicity (for which we adopt the [Fe I/H] values) is also found at the end of these minimization iterations.",
"No significant relationship was found in the Fe1 abundances versus wavelength ($>$ 5000 Å) for either the globular cluster or the Carina targets; this indicates that the sky subtraction and overall data reduction methods were successful.",
"Eliminating lines with $\\chi < 1.4$ eV, which can be particularly sensitive to departures from local thermodynamic equilibrium (LTE) effects, also had no significant effect on the temperature or microturbulence values.",
"Samples of these relationships are shown in Fig.",
"REF for two stars, one observed with the FLAMES/UVES spectrograph and the other observed with the MIKE spectrograph.",
"Adjusting the metallicities from the initial CaT values had no significant effect on the excitation temperatures ($\\Delta $ T $<$ 10 K).",
"Adopting the spectroscopic temperatures had only a small effect on the photometric gravities ($\\le +0.08$ in the Carina stars, and $\\le -0.02$ for the M68 stars).",
"Four stars in Carina have been observed and analysed independently by Koch et al. (2008).",
"Their analysis with ATLAS9 models atmospheres results in excitation temperatures that are in excellent agreement with ours, with differences ranging from $\\Delta $ T = +116 to -70 K, and with an average offset of +32 K. The differences in microturbulence are more significant, with Koch et al.",
"'s values being $\\sim 0.50$ km s$^{-1}$ lower.",
"This is linked to differences in our gravity determinations discussed in Section REF ."
],
[
"Stellar Parameter Error Estimates",
"Stellar parameters and their uncertainties are listed in Table REF .",
"The one sigma uncertainty in the effective temperature is determined from the slope in the Fe1 line abundances vs excitation potential ($\\chi $ ) allowing the change in the iron abundance to equal the standard deviation $\\sigma $ (Fe1.",
"Similarly, the one sigma uncertainty in microturbulence is determined by setting the slope of the Fe1 line abundances vs equivalent width equal to $\\sigma $ (Fe1).",
"For the physical gravities, the uncertainties are determined from errors in the distance moduli, reddening, and stellar mass (see Section REF ).",
"For M68, the uncertainty in the turn off mass is tiny ($\\pm $ 0.03 M$_\\odot $ ), which has a negligible effect on the gravity ($\\Delta $ log g= $\\pm $ 0.04).",
"The uncertainties in gravity are also very small when the short distance modulus reported by Dinescu et al.",
"(1999) is adopted for M68 ($\\Delta $ log g= $\\pm $ 0.07).",
"For Carina, the dominant uncertainty is from the stellar mass.",
"The range in turn off masses for Carina is 0.8 to 1.0 M$_\\odot $ due to its episodic star formation history; the resulting uncertainties in log g can be as large as 0.10 dex.",
"Uncertainties in metallicity are adopted from the error in the mean of the Fe1 abundances (see Section REF ).",
"Figure: MOOG results for Car-484 (left panel), a moderate SNR highresolution FLAMES/UVES spectrum, and Car-7002, a moderate SNR andmoderate resolution Magellan/MIKE spectrum.",
"These plots show thatthe Fe1 line abundances are minimizedwith excitation potential (to determine surface temperature),equivalent width (to define microturbulence), and wavelength(a check on the sky subtraction)."
],
[
"Spectrum Syntheses and Hyperfine Structure Corrections ",
"Spectrum syntheses were carried out for Na1 (5688, 5889, 5895), Mg1 (3829, 3832), Sr2 (4077, 4205), Ba2 (4554, 5853, 6141, 6496), and Eu2 (4129, 4205, 6645).",
"In each case, the instrumental resolution of the spectrum was adopted as the main source of broadening, and checked with the shapes and EW abundance results from nearby spectral lines (Fe, Ca, Ti).",
"When an EW could be measured, the synthetic and EW abundance results were compared; when excellent agreement was found then the EW result was taken.",
"Line abundances taken from the synthetic results are noted in Table REF .",
"Hyperfine structure (HFS) corrections were determined from spectrum syntheses for the odd-Z elements (Sc2, V1, Mn1, Co1), Cu1, and the neutron capture elements Ba2, La2, and Eu2.",
"HFS components were taken from Lawler et al.",
"(2001a; La2) , Booth et al.",
"(1983; Mn1), Prochaska et al.",
"(2001; Sc2, V1, Mn1, and Co1), and the Kurucz databasehttp://kurucz.harvard.edu/LINELISTS/GFHYPERALL for the remaining lines.",
"HFS components and isotopic ratios for Cu1 are from [18], for Eu2 from Lawler et al.",
"(2001b), and for Ba2 from McWilliam et al.",
"(1998; r-process isotopic ratios were adopted for the most metal-poor stars, whereas the solar ratios were used for stars with [Fe/H]$>-2$ )."
],
[
"Spherical vs Plane Parallel ",
"Spherical MARCS models represent a significant improvement in the modeling of stellar atmospheres.",
"e.g., Heiter & Eriksson (2006) recommend the use of spherical model atmospheres in abundance analyses for stars with log g$< 2$ and 4000 $\\le $ T$_{\\rm eff}$$\\le $ 6500 K, which encompasses the range in stellar parameters of our target stars.",
"Tests performed by Tafelmeyer et al.",
"(2010) showed systematic offsets between the spherical and plane-parallel models are below 0.15 dex (this includes iron lines and other elements, e.g., [C/Fe]).",
"We performed similar tests, comparing the metallicities we determined from spherical MARCS models to those from plane parallel MARCS (pp-MARCS) and Kurucz models.",
"The plane parallel MARCS models had the effect of reducing the excitation temperature results by $\\sim $ 100 K, but the Kurucz models had a different effect, to increase the microturbulence values by $\\sim $ 0.1 dex.",
"In both cases, all other parameters were unchanged.",
"These offsets, when applied to the abundance analysis, changed the Fe1 abundances by only $\\sim $ 0.05 dex (+0.05 with pp-MARCS models, and -0.05 with Kurucz models).",
"This is very similar to the offsets found by Tafelmeyer et al.",
"(2010) and Heiter & Eriksson (2006) for iron."
],
[
"Continuum Scattering Effects ",
"The standard version of MOOG treats continuum scattering ($\\sigma _\\nu $ ) as if it were absorption ($\\kappa _\\nu $ ) in the source function, i.e., S$_\\nu $ = B$_\\nu $ (the Planck function), an approximation that is only valid at long wavelengths in RGB stars.",
"At shorter wavelengths ($\\le $ 5000 Å), Cayrel et al.",
"(2004) have shown that the scattering term must be taken into account such that the source function becomes S$_\\nu $ = ($\\kappa _\\nu \\, B_\\nu + \\sigma _\\nu \\, J_\\nu $ )/($\\kappa _\\nu + \\sigma _\\nu $ ).",
"A new version of MOOG (MOOG-SCAT, Sobeck et al.",
"2010) has been used to test our results and calculate corrections to our abundances line by line.",
"As shown in Fig.",
"REF , the scattering corrections are negligible for red lines, but can approach 0.4 dex in the blue.",
"Lines with the largest corrections are the resonance lines of Mn1.",
"Notice also that the corrections are sensitive to the atmospheric parameters, in particular metallicity; the three stars in Carina that are shown were choosen because they have the largest corrections and they are the most metal-poor.",
"We compared the elemental abundances between the programs MOOG and CALRAI (Spite 1967, Cayrel et al.",
"1991, Hill et al.",
"2012), and also the scattering corrections calculated with MOOG-SCAT to those determined with the program Turbospectrum (Alvarez & Plez 1998).",
"The standard abundance results and scattering corrections were nearly identical for all lines between these codes, in particular, the scattering corrections were in agreement to within $\\sim $ 0.01 dex.",
"Only the resonance lines of Mn1 showed slightly larger differences in the scattering corrections ($\\sim $ 0.04 dex); since resonance lines form over more atmospheric layers, we expect these lines are more sensitive to the details in the models and the line formation calculations between the codes.",
"Thus, we consider the consistency in the abundance results from the line formation codes MOOG and CALRAI, and the scattering corrections between MOOG-SCAT and Turbospectrum to be in excellent agreement.",
"As a final note, the scattering corrections had no effect on our spectroscopic temperature or microturbulence determinations, nor the metallicities for [Fe1/H], since only Fe1 lines with $\\lambda > 5000$ Å were used in those steps.",
"Figure: A comparson of the line abundance corrections fromMOOG-SCAT for one of the M68 standard star and the threemost metal-poor stars in Carina.The y-label “Corr” = MOOG-SCAT -- MOOG."
],
[
"Gravity is not from Fe1/Fe2 ",
"The analysis of ten stars in Carina by Koch et al.",
"(2008) and five more stars by Shetrone et al.",
"(2003) used the Fe2 lines to determine spectroscopic gravities from ionization equilibrium (where [Fe1/H] = [Fe2/H]).",
"For four stars in common with Koch et al.",
", our gravities are smaller than theirs by log g$\\sim 0.5$ dex.",
"Since turbulent velocity and ionization equilibrium are correlated in RGBs, then the higher microturbulent values found by Koch et al.",
"(see Section REF ) are correlated with their higher spectroscopic gravities.",
"There are sufficient Fe2 lines in many of the stars in our analysis to examine the ionization equilibrium of iron and determine spectroscopic gravities, however we do not use this method in this analysis.",
"For consistency with the FLAMES/GIRAFFE analysis, where a shorter wavelength interval meant fewer Fe2 lines, then we adopt photometric gravities using the same methods as in Lemasle et al. (2012).",
"On the other hand, we notice that the Fe1 and Fe2 abundances are in good agreement for our Carina targets, with a mean [Fe II/Fe I] = +0.02, and range of -0.24 to +0.20, which is $\\sim $ 1$\\sigma $ (Fe1).",
"The stars in M68 have higher offsets, with a mean [Fe II/Fe I] = +0.35, which may reflect uncertainties in its distance or reddening.",
"When the abundance of Fe2 is found to be larger than Fe1 in red giants, it can be due to the overionization of Fe1 by the radiation field which is neglected under the assumption of LTE.",
"Corrections for this effect can be as large as 0.3 dex (Mashonkina et al.",
"2010), which is similar to the offset in the M68 stars.",
"Thus, we consider our Fe1 and Fe2 results to be in good agreement for all of our targets when physical gravities are adopted.",
"Note, Fe2 lines at all wavelengths were examined in this analysis, i.e., they were not restricted to $\\lambda <$ 5000 Å like Fe1."
],
[
"Evaluating the CaT Metallicity ",
"The initial metallicities in our analysis are taken from a calibration of the near-IR Ca2 triplet near 8500 Å(= CaT).",
"A direct comparison between the low resolution CaT metallicity index and high resolution iron abundances for large samples of RGB stars in the dwarf galaxies Fornax and Sculptor was shown by [13] to provide consistent abundances in the range $-2.5 <$ [Fe/H] $< -0.5$ .",
"More recently, Starkenburg et al.",
"(2010) found that the metallicities deviate strongly from the linear relationship for RGB stars with [Fe/H] $< -2.0$ and have developed a new calibration for metal-poor stars that also considers the [Ca/Fe] ratio.",
"Figure: Iron abundances ([FeI/H]) from high resolution spectralanalyses versus those derived from the CaT calibration byStarkenburg et al. (2010).",
"A line of equal metallicitiesis shown (dashed), ±\\pm 0.2 dex (dotted).The CaT metallicities for our targets are listed in Table REF , and plotted against the [Fe1/H] values in Fig.",
"REF .",
"The high resolution [Fe1/H] abundances tend to be slightly larger than the metallicities from the CaT index, however, as Starkenburg et al.",
"(2010) show, the ratio of [Ca/Fe] becomes more important in the calibration of low metallicity stars.",
"The two most metal-poor stars in this analysis are found to have lower [Ca/Fe] ratios than the Galactic trend which was assumed in the Starkenburg et al.",
"calibration."
],
[
"Solar Abundances and Galactic Comparisons",
"For comparison purposes, solar abundances from [10] are adopted.",
"Galactic comparisons are from the compilation by Venn et al.",
"(2004), supplemented with Si, V, Cr, and Nd from Fulbright (2000).",
"The compilation was updated with thick disk stars from Reddy et al.",
"(2006), La from the compilation by Roederer et al.",
"(2010), and Cu and Zn from Mishenina et al. (2002).",
"All of these were rescaled to [10] solar abundances from the Grevesse & Sauval (1998) values.",
"The compilation of abundances of metal-poor stars in the Galactic halo and Local Group dwarf galaxies by Frebel (2010b, after eliminating the upper limit values) was also used here for comparison purposes.",
"Previously published abundances for Carina and M68 were re-scaled to the [10] solar abundances, i.e., Shetrone et al.",
"(2003) and Lee et al.",
"(2005) from Grevesse & Sauval (1989), and Koch et al.",
"(2008) from [11] values."
],
[
"Abundance Error Estimates ",
"lrrrrr 5 0pt Abundance Uncertainties for M68-6023 Elem $\\Delta $ T$_{\\rm eff}$ $\\Delta $ log g $\\Delta \\xi $ $\\Delta $ [Fe/H] Added in +55 $-$ 0.04 +0.12 $-$ 0.02 Quadrature Fe I +0.09 0.00 $-$ 0.05 0.01 0.10 Fe II $-$ 0.02 $-$ 0.02 $-$ 0.05 $-$ 0.01 0.06 CH +0.05 $-$ 0.04 0.00 0.00 0.06 Na I +0.08 +0.01 $-$ 0.03 0.01 0.09 Mg I +0.08 +0.02 $-$ 0.03 0.01 0.09 Ca I +0.08 +0.01 $-$ 0.04 0.01 0.09 Sc II +0.02 $-$ 0.01 $-$ 0.05 $-$ 0.01 0.06 Ti I +0.09 +0.01 $-$ 0.04 0.01 0.10 Ti II +0.02 $-$ 0.01 $-$ 0.05 $-$ 0.01 0.06 V I +0.06 0.00 $-$ 0.03 0.01 0.07 Cr I +0.10 +0.01 $-$ 0.05 0.01 0.11 Mn I +0.10 0.00 $-$ 0.07 0.01 0.12 Co I +0.10 +0.01 $-$ 0.08 0.01 0.13 Ni I +0.09 +0.01 $-$ 0.05 0.01 0.11 Zn I +0.01 0.00 $-$ 0.02 0.00 0.02 Sr II +0.04 $-$ 0.01 $-$ 0.08 0.01 0.09 Y II +0.02 $-$ 0.01 $-$ 0.03 $-$ 0.01 0.04 Zr II +0.02 $-$ 0.02 $-$ 0.02 $-$ 0.01 0.03 Ba II +0.04 $-$ 0.02 $-$ 0.09 $-$ 0.01 0.10 Eu II +0.03 $-$ 0.02 $-$ 0.09 $-$ 0.01 0.10 lrrrrr 5 0pt Abundance Uncertainties for CAR-1087 Elem $\\Delta $ T$_{\\rm eff}$ $\\Delta $ log g $\\Delta \\xi $ $\\Delta $ [Fe/H] Added in +98 $-$ 0.10 $-$ 0.25 $-$ 0.06 Quadrature Fe I +0.10 0.00 +0.08 +0.01 0.13 Fe II $-$ 0.03 $-$ 0.02 +0.08 $-$ 0.01 0.09 Na I +0.13 0.00 +0.06 +0.01 0.15 Mg I +0.05 +0.02 +0.04 +0.01 0.07 Ca I +0.08 +0.01 +0.02 +0.01 0.09 Sc II +0.03 $-$ 0.02 +0.06 $-$ 0.01 0.07 Ti I +0.17 +0.01 +0.08 +0.01 0.19 Ti II 0.00 $-$ 0.02 +0.08 $-$ 0.01 0.08 V I +0.13 +0.02 +0.04 +0.01 0.14 Cr I +0.18 +0.01 +0.09 +0.02 0.20 Mn I +0.18 +0.02 +0.10 +0.02 0.21 Ni I +0.07 +0.01 +0.03 +0.01 0.08 Ba II +0.08 $-$ 0.04 +0.02 $-$ 0.01 0.09 Eu II +0.04 $-$ 0.03 +0.02 $-$ 0.01 0.06 lrrrrr 5 0pt Abundance Uncertainties for Car-484 Elem $\\Delta $ T$_{\\rm eff}$ $\\Delta $ log g $\\Delta \\xi $ $\\Delta $ [Fe/H] Added in +66 $-$ 0.1 +0.07 $-$ 0.02 Quadrature Fe I +0.06 $-$ 0.01 $-$ 0.04 0.00 0.07 Fe II $-$ 0.07 $-$ 0.05 $-$ 0.03 0.00 0.09 O I 0.00 $-$ 0.05 $-$ 0.01 $-$ 0.01 0.05 Na I +0.06 +0.01 $-$ 0.01 0.00 0.06 Mg I +0.05 +0.01 $-$ 0.03 0.00 0.06 Si I* 0.00 $-$ 0.01 0.00 $-$ 0.01 0.01 Ca I +0.09 +0.01 $-$ 0.03 0.00 0.10 Sc II $-$ 0.02 $-$ 0.03 $-$ 0.02 0.00 0.04 Ti I +0.13 $-$ 0.01 $-$ 0.04 0.00 0.14 Ti II $-$ 0.03 $-$ 0.05 $-$ 0.04 0.00 0.07 V I +0.14 $-$ 0.01 $-$ 0.01 0.00 0.14 Cr I +0.14 $-$ 0.01 $-$ 0.05 0.00 0.15 Mn I +0.10 0.00 $-$ 0.03 0.00 0.11 Co I +0.08 $-$ 0.02 $-$ 0.03 0.00 0.08 Ni I +0.04 $-$ 0.01 $-$ 0.03 0.00 0.05 Cu I +0.08 $-$ 0.02 $-$ 0.04 0.00 0.08 Zn I $-$ 0.04 $-$ 0.02 $-$ 0.03 0.00 0.05 Y II 0.00 $-$ 0.04 $-$ 0.03 0.00 0.05 Ba II +0.01 $-$ 0.06 $-$ 0.06 0.00 0.08 La II +0.01 $-$ 0.06 $-$ 0.01 $-$ 0.01 0.06 Nd II 0.00 $-$ 0.05 $-$ 0.03 0.00 0.06 Eu II $-$ 0.01 $-$ 0.05 $-$ 0.01 $-$ 0.01 0.05 *Si1 errors determined from the lines and EWs in Car-524.",
"Errors in the abundances have been determined in two ways, with the maximum of these errors adopted for the analysis (Tables REF to REF ).",
"Firstly, the error in the equivalent width measurements are conservatively estimated as $\\Delta $ EW = EW$_{\\rm rms}$ + 10% x EW, where EW$_{\\rm rms}$ is from the Cayrel formula (Section REF ).",
"This $\\Delta $ EW was propagated through the abundance analysis to provide a $\\sigma (EW)$ in each line abundance.",
"This error is not necessarily symmetric, and we adopt the average.",
"For synthetic spectra, $\\sigma (EW)$ was estimated from the range of abundances for which a good fit of the observed line profile could be achieved.",
"The second method for estimating the error in the abundances is simply to calculate the standard deviation when more than five lines of an element are available, $\\sigma (X)$ ; when fewer lines are available, we set $\\sigma (X)$ = $\\sigma $ (Fe1).",
"The final error in [X/H] is adopted as the maximum of $\\sigma (X)$ or $\\sigma (EW)$ .",
"The corresponding error on the mean, $\\delta (X)$ , is calculated as $\\delta (X)$ = $\\sigma (X)$ /$\\sqrt{N_X}$ where N$_X$ is the number of lines available of element X.",
"To get the final error in [X/Fe], i.e., the ratio with iron, then the error on the mean ($\\delta $ (FeI)) was added in quadrature.",
"These line measurement errors do not take into account the errors due to the uncertainties in the stellar parameters.",
"These errors are difficult to ascertain and combine properly since they are not independent and their relationship(s) to one another are not well defined.",
"We report these errors for three representative stars, Car-6023, Car-1087, and Car-484, in Tables REF , REF , and REF .",
"These three stars represent the range in stellar parameters, resolution, and SNR of all of our spectra.",
"It can be seen that the stellar parameter errors are similar to or smaller than the line measurement errors (exceptions include Ca1, Ti1, Cr1, and Co1 where the stellar parameter errors can be slightly larger, but still $\\le $ 0.15 dex).",
"The stellar parameter errors reported for Car-6023 can be applied to Car-6022 and Car-6024, while those for Car-1087 can be applied to Car-5070 and Car-7002.",
"Those for Car-484 can be applied to the rest of the FLAMES/UVES sample, although the lower SNR for Car-1013 means the errors for that target are larger by $\\sim $ 2x (as seen in Table REF )."
],
[
"M68 Comparison Stars",
"llll 4 0pt M68 Chemical Abundances Elem 6022 6023 6024 [X/Fe] $\\pm \\sigma $ (#) [X/Fe] $\\pm \\sigma $ (#) [X/Fe] $\\pm \\sigma $ (#) Fe I 4.99 $\\pm $ 0.02 (92) 5.09 $\\pm $ 0.03 (92) 4.86 $\\pm $ 0.03 (81) Fe II 0.40 $\\pm $ 0.05 (19) 0.26 $\\pm $ 0.05 (19) 0.39 $\\pm $ 0.06 (17) CH $-$ 0.82 $\\pm $ 0.20 (S) $-$ 0.62 $\\pm $ 0.20 (S) $-$ 0.59 $\\pm $ 0.20 (S) O I $<$ 1.36 $<$ 1.33 $<$ 1.54 Na Ia 0.07 $\\pm $ 0.10 (S) 0.27 $\\pm $ 0.10 (S) $-$ 0.03 $\\pm $ 0.10 (S) Mg I 0.01 $\\pm $ 0.20 (S) $-$ 0.09 $\\pm $ 0.20 (S) 0.04 $\\pm $ 0.20 (S) Si I $<$ 1.22 $<$ 1.15 $<$ 1.36 Ca I 0.48 $\\pm $ 0.04 (17) 0.39 $\\pm $ 0.04 (16) 0.48 $\\pm $ 0.05 (15) Sc II 0.57 $\\pm $ 0.05 (10) 0.28 $\\pm $ 0.10 (10) 0.44 $\\pm $ 0.06 (9) Ti I 0.11 $\\pm $ 0.05 (23) 0.12 $\\pm $ 0.04 (21) 0.30 $\\pm $ 0.07 (20) Ti II 0.77 $\\pm $ 0.05 (52) 0.54 $\\pm $ 0.05 (51) 0.65 $\\pm $ 0.05 (46) V I $-$ 0.06 $\\pm $ 0.16 (2) 0.02 $\\pm $ 0.17 (2) 0.35 $\\pm $ 0.17 (2) Cr I $-$ 0.17 $\\pm $ 0.06 (12) $-$ 0.25 $\\pm $ 0.09 (12) $-$ 0.17 $\\pm $ 0.08 (9) Mn Ia $-$ 0.48 $\\pm $ 0.07 (7) $-$ 0.61 $\\pm $ 0.12 (7) $-$ 0.31 $\\pm $ 0.10 (4) Co I $-$ 0.27 $\\pm $ 0.09 (6) $-$ 0.26 $\\pm $ 0.06 (7) 0.10 $\\pm $ 0.10 (6) Ni I 0.06 $\\pm $ 0.03 (8) 0.20 $\\pm $ 0.10 (8) 0.04 $\\pm $ 0.14 (7) Zn I 0.47 $\\pm $ 0.16 (2) 0.33 $\\pm $ 0.17 (2) 0.43 $\\pm $ 0.24 (1) Sr II $-$ 0.08 $\\pm $ 0.16 (2) $-$ 0.26 $\\pm $ 0.17 (2) $-$ 0.23 $\\pm $ 0.17 (2) Y II $-$ 0.07 $\\pm $ 0.23 (1) $-$ 0.43 $\\pm $ 0.17 (2) $-$ 0.21 $\\pm $ 0.17 (2) Zr II 0.20 $\\pm $ 0.23 (1) $-$ 0.05 $\\pm $ 0.24 (1) 0.00 $\\pm $ 0.24 (1) Ba II 0.33 $\\pm $ 0.12 (4) 0.14 $\\pm $ 0.12 (4) 0.36 $\\pm $ 0.12 (4) La II $<$ 1.37 $<$ 1.36 $<$ 1.56 Nd II 1.27 $\\pm $ 0.24 (1) Eu II 0.24 $\\pm $ 0.17 (2) 0.47 $\\pm $ 0.17 (2) We calculate [X/Fe] = [X/H] $-$ [Fe1/H].",
"For Fe1, the abundance listed is 12$+$ log(Fe/H).",
"aA correction of $\\Delta $ [Na/Fe] = $-0.5$ has been applied to account for NLTE effects on the Na D lines [1].",
"Similarly, we apply a correction of $\\Delta $ log (Mn/H) = $+0.5$ to the Mn1 resonance lines to account for NLTE effects (Bergemann & Gehren 2008).",
"The stellar parameters and chemical abundances for our three stars in M68 are compared to the one star analysed by Shetrone et al.",
"(2003) and eight stars by Lee et al. (2005).",
"The targets in this analysis are fainter by $\\sim $ 1.4 magnitudes, thus they are located further down the RGB and have slightly higher temperatures and gravities.",
"Our microturbulence values are comparable or lower.",
"The metallicities from Fe1 for these three stars are in excellent agreement with one another, as expected for a single stellar population.",
"The mean Fe1 abundance is log(Fe1/H) = 5.04 $\\pm $ 0.10, where the error is the line to line abundance scatter in each star, added in quadrature, and divided by $\\sqrt{3}$ .",
"This is similar to the mean [Fe1/H] abundance determined by Lee et al.",
"(2005), where [Fe1/H] = 4.96 $\\pm $ 0.06 when they adopt the photometric gravities.",
"Higher iron abundances are found with lower spectroscopic gravities; this is also seen by Shetrone et al.",
"(2003) for one star.",
"Most of the elemental abundances determined here are in excellent agreement with those from Lee et al.",
"(2005) and Shetrone et al.",
"(2003), and our Na1 values are within the ranges of the others.",
"To compare the [X/Fe] $ratios$ for these elements with those from Lee et al.",
"requires knowing that their [X/Fe] results use Fe1 for ratios with neutral species and Fe2 for ratios with ionized species and oxygen In our analysis, we use only the Fe1 lines to indicate the metallicity, thus we adjust their [X/Fe2] abundances by adding their [Fe2/Fe1] ratio, = $+0.40$ .",
"The most significant differences are that our Mg and Mn abundances are lower than Lee et al.",
"'s (2005), and the Shetrone et al.",
"'s (2003) values for Zn, Y, Ba, and Eu are lower (see Figures in the next Sections).",
"For the heavy elements examined by Shetrone et al.",
", we track the differences to the model atmosphere parameters, e.g., ionized species that are more sensitive to gravity, and Zn abundances that are $\\sim $ 2x more sensitive to temperature than the other elements.",
"For Mn, we note that Lee et al.",
"examined only a single, very weak line (Mn I 6021.8), whereas ours are from several lines that are in good agreement with one another, thus we consider our result more reliable.",
"For Mg, our abundances have been determined from four spectral lines $-$ three lines from spectrum synthesis due to severe blending and difficulties in the continuum placements (two near 3830 Å, and one at $\\lambda $ 5172), and a fourth line at $\\lambda $ 5528 which is sufficiently unblended to be analysed from EW measurements (note that a fifth line is available in the FLAMES/UVES spectra at $\\lambda $ 5711 but is too weak to be measured in any of the Magellan/MIKE spectra).",
"The Mg abundances are consistent between these four lines and from star to star in M68, however we assign a larger uncertainty to account for the difficulties in setting the continuum in the three line syntheses.",
"The average [Mg/Fe] is slightly lower than found in the other analyses (Lee et al.",
"2005, Shetrone et al.",
"2003), than the Galactic trends at that metallicity, and than found in the other $\\alpha $ -elements.",
"While the offset is within the estimated uncertainties, we note that our [Na/Fe] are slightly higher than the Galactic trends $-$ it is possible these stars exhibit some deep mixing (e.g., Gratton et al. 2010).",
"llll 4 0pt Carina Chemical Abundances Elem 484 524 612 [X/Fe] $\\pm \\sigma $ (#) [X/Fe] $\\pm \\sigma $ (#) [X/Fe] $\\pm \\sigma $ (#) Fe I 5.97 $\\pm $ 0.02 (114) 5.75 $\\pm $ 0.02 (121) 6.20 $\\pm $ 0.02 (126) Fe II 0.07 $\\pm $ 0.04 (14) 0.09 $\\pm $ 0.03 (15) 0.12 $\\pm $ 0.05 (18) O I 0.40 $\\pm $ 0.21 (1) 0.41 $\\pm $ 0.21 (1) $<$ 0.09 Na I $-$ 0.25 $\\pm $ 0.21 (1) $<-$ 0.26 $<-$ 0.72 Mg I 0.19 $\\pm $ 0.15 (2) 0.27 $\\pm $ 0.15 (2) $-$ 0.50 $\\pm $ 0.16 (2) Si I 0.60 $\\pm $ 0.15 (2) 0.13 $\\pm $ 0.23 (1) Ca I 0.16 $\\pm $ 0.04 (20) 0.10 $\\pm $ 0.04 (19) $-$ 0.17 $\\pm $ 0.04 (19) Sc II 0.06 $\\pm $ 0.04 (12) $-$ 0.08 $\\pm $ 0.04 (10) $-$ 0.74 $\\pm $ 0.07 (8) Ti I 0.23 $\\pm $ 0.04 (25) $-$ 0.02 $\\pm $ 0.03 (22) $-$ 0.42 $\\pm $ 0.04 (18) Ti II 0.18 $\\pm $ 0.05 (13) 0.03 $\\pm $ 0.06 (13) $-$ 0.29 $\\pm $ 0.05 (12) V I 0.01 $\\pm $ 0.03 (10) 0.21 $\\pm $ 0.15 (2) $-$ 0.67 $\\pm $ 0.23 (1) Cr I 0.00 $\\pm $ 0.12 (4) $-$ 0.09 $\\pm $ 0.10 (4) $-$ 0.20 $\\pm $ 0.12 (4) Mn I $-$ 0.27 $\\pm $ 0.05 (7) $-$ 0.34 $\\pm $ 0.10 (4) $-$ 0.51 $\\pm $ 0.10 (6) Co I $-$ 0.13 $\\pm $ 0.21 (1) $-$ 0.27 $\\pm $ 0.21 (1) $<-$ 0.86 Ni I $-$ 0.07 $\\pm $ 0.04 (15) $-$ 0.08 $\\pm $ 0.04 (14) $-$ 0.46 $\\pm $ 0.04 (14) Cu I $-$ 0.47 $\\pm $ 0.21 (1) $<-$ 0.85 $<-$ 1.60 Zn I 0.16 $\\pm $ 0.21 (1) $-$ 0.17 $\\pm $ 0.21 (1) $-$ 0.83 $\\pm $ 0.23 (1) Y II $-$ 0.32 $\\pm $ 0.12 (3) $-$ 0.46 $\\pm $ 0.12 (3) $-$ 1.37 $\\pm $ 0.16 (2) Ba II 0.16 $\\pm $ 0.26 (1) 0.09 $\\pm $ 0.21 (1) $-$ 0.64 $\\pm $ 0.16 (S) La II 0.48 $\\pm $ 0.12 (3) 0.55 $\\pm $ 0.15 (2) $<$ 0.11 Nd II 0.63 $\\pm $ 0.15 (2) 0.52 $\\pm $ 0.15 (2) $-$ 0.56 $\\pm $ 0.23 (1) Eu II 0.47 $\\pm $ 0.21 (1) $<$ 0.63 $<$ 0.26 We calculate [X/Fe] = [X/H] $-$ [Fe1/H].",
"For Fe1, the abundance listed is 12$+$ log(Fe/H).",
"llll 4 0pt Carina Chemical Abundances continued Elem 705 769 1013 [X/Fe] $\\pm \\sigma $ (#) [X/Fe] $\\pm \\sigma $ (#) [X/Fe] $\\pm \\sigma $ (#) Fe I 6.15 $\\pm $ 0.02 (127) 5.82 $\\pm $ 0.02 (125) 6.20 $\\pm $ 0.03 (125) Fe II 0.02 $\\pm $ 0.06 (16) 0.06 $\\pm $ 0.05 (15) $-$ 0.24 $\\pm $ 0.06 (17) O I $<$ 0.12 $<$ 0.08 Na I $-$ 0.22 $\\pm $ 0.25 (1) 0.04 $\\pm $ 0.25 (1) $-$ 0.25 $\\pm $ 0.37 (1) Mg I 0.13 $\\pm $ 0.18 (2) 0.42 $\\pm $ 0.18 (2) 0.09 $\\pm $ 0.26 (2) Si I 0.23 $\\pm $ 0.25 (1) 0.45 $\\pm $ 0.18 (2) 0.06 $\\pm $ 0.26 (2) Ca I 0.11 $\\pm $ 0.04 (19) 0.24 $\\pm $ 0.06 (21) $-$ 0.06 $\\pm $ 0.05 (20) Sc II $-$ 0.03 $\\pm $ 0.05 (11) 0.15 $\\pm $ 0.06 (11) $-$ 0.24 $\\pm $ 0.07 (13) Ti I 0.02 $\\pm $ 0.05 (25) 0.27 $\\pm $ 0.05 (23) 0.27 $\\pm $ 0.06 (24) Ti II $-$ 0.01 $\\pm $ 0.06 (11) 0.13 $\\pm $ 0.06 (11) 0.11 $\\pm $ 0.09 (13) V I 0.02 $\\pm $ 0.06 (10) 0.41 $\\pm $ 0.11 (5) 0.09 $\\pm $ 0.08 (11) Cr I $-$ 0.23 $\\pm $ 0.12 (4) 0.08 $\\pm $ 0.12 (4) 0.25 $\\pm $ 0.21 (3) Mn I $-$ 0.30 $\\pm $ 0.07 (7) 0.02 $\\pm $ 0.11 (5) $-$ 0.51 $\\pm $ 0.17 (5) Co I $-$ 0.26 $\\pm $ 0.25 (1) 0.02 $\\pm $ 0.25 (1) $-$ 0.01 $\\pm $ 0.26 (2) Ni I $-$ 0.04 $\\pm $ 0.04 (17) 0.03 $\\pm $ 0.04 (13) $-$ 0.10 $\\pm $ 0.06 (17) Cu I $-$ 0.80 $\\pm $ 0.25 (1) $-$ 0.55 $\\pm $ 0.25 (1) Zn I $-$ 0.05 $\\pm $ 0.25 (1) $-$ 0.46 $\\pm $ 0.37 (1) Y II $-$ 0.63 $\\pm $ 0.18 (2) $-$ 0.27 $\\pm $ 0.14 (3) $-$ 0.32 $\\pm $ 0.21 (3) Ba II $-$ 0.61 $\\pm $ 0.14 (3) $-$ 0.17 $\\pm $ 0.14 (3) $-$ 0.15 $\\pm $ 0.26 (S) La II $<$ 0.15 $<$ 0.14 Nd II 0.26 $\\pm $ 0.18 (2) 0.27 $\\pm $ 0.18 (2) 0.11 $\\pm $ 0.26 (2) Eu II $<$ 0.31 $<$ 0.63 0.33 $\\pm $ 0.37 (1) We calculate [X/Fe] = [X/H] $-$ [Fe1/H].",
"For Fe1, the abundance listed is 12$+$ log(Fe/H).",
"llll 4 0pt Carina Chemical Abundances Elem 1087 5070 7002 [X/Fe] $\\pm \\sigma $ (#) [X/Fe] $\\pm \\sigma $ (#) [X/Fe] $\\pm \\sigma $ (#) Fe I 4.69 $\\pm $ 0.06 (38) 5.35 $\\pm $ 0.05 (55) 4.64 $\\pm $ 0.05 (38) Fe II 0.01 $\\pm $ 0.07 (7) $-$ 0.14 $\\pm $ 0.10 (11) 0.20 $\\pm $ 0.14 (6) CH $<-$ 1.02 (S) $<-$ 0.98 (S) $<-$ 1.07 (S) O I $<$ 1.55 $<$ 0.90 $<$ 1.61 Na Ia $-$ 0.82 $\\pm $ 0.24 (S) $-$ 1.18 $\\pm $ 0.38 (S) $-$ 0.45 $\\pm $ 0.23 (S) Mg I 0.48 $\\pm $ 0.34 (1) $-$ 0.36 $\\pm $ 0.38 (1) 0.22 $\\pm $ 0.23 (2) Si I $<$ 1.70 $<$ 1.01 $<$ 1.76 Ca I $-$ 0.03 $\\pm $ 0.20 (3) $-$ 0.01 $\\pm $ 0.10 (9) 0.18 $\\pm $ 0.10 (6) Sc II $-$ 0.01 $\\pm $ 0.15 (5) $-$ 0.20 $\\pm $ 0.24 (8) 0.10 $\\pm $ 0.15 (4) Ti I 0.65 $\\pm $ 0.17 (4) $-$ 0.08 $\\pm $ 0.15 (8) 0.11 $\\pm $ 0.18 (3) Ti II 0.29 $\\pm $ 0.07 (24) 0.04 $\\pm $ 0.07 (30) 0.39 $\\pm $ 0.08 (20) V I 0.72 $\\pm $ 0.34 (1) $-$ 0.26 $\\pm $ 0.38 (1) Cr I $-$ 0.82 $\\pm $ 0.17 (4) $-$ 0.47 $\\pm $ 0.08 (7) $-$ 0.34 $\\pm $ 0.16 (4) Mn I $-$ 0.72 $\\pm $ 0.24 (2) $-$ 0.81 $\\pm $ 0.18 (3) $-$ 0.33 $\\pm $ 0.23 (2) Co I $<$ 1.11 $<$ 0.31 $-$ 0.26 $\\pm $ 0.23 (2) Ni I 0.36 $\\pm $ 0.34 (1) 0.18 $\\pm $ 0.13 (7) 0.35 $\\pm $ 0.18 (3) Cu I $<$ 0.37 $<-$ 0.43 $<$ 0.39 Zn I $<$ 0.66 $<$ 0.01 $<$ 0.77 Sr II $-$ 1.56 $\\pm $ 0.24 (S) $-$ 1.21 $\\pm $ 0.32 (S) Y II $<-$ 0.05 $<-$ 0.69 $<-$ 0.12 Zr II $<$ 0.24 $<-$ 0.41 $<$ 0.36 Ba II $-$ 1.07 $\\pm $ 0.34 (S) $-$ 1.12 $\\pm $ 0.27 (S) $-$ 0.98 $\\pm $ 0.18 (S) La II $<$ 1.77 $<$ 1.15 $<$ 1.75 Nd II $<$ 0.99 $<$ 0.34 $<$ 1.09 Eu II $<$ 1.85 0.13 0.38 (1) $<$ 1.90 We calculate [X/Fe] = [X/H] $-$ [Fe1/H].",
"For Fe1, the abundance listed is 12$+$ log(Fe/H).",
"aA correction of $\\Delta $ [Na/Fe] = $-0.6$ has been applied to account for NLTE effects on the Na D lines [1].",
"Similarly, we apply a correction of $\\Delta $ log (Mn/H) = $+0.5$ to the Mn1 resonance lines to account for NLTE effects (Bergemann & Gehren 2008).",
"New elemental abundances not previously determined in M68 (Sr2 and Zr2) are in good agreement with the Galactic distributions.",
"Finally, our upper limits for Si1, O1, and La2 are in agreement with the abundances determined by Lee et al.",
"(2005) and Shetrone et al. (2003).",
"Thus, our M68 results appear to be in good agreement with other abundance determinations for this cluster, and Galactic cluster and field stars in general.",
"We conclude that our stellar parameter determinations and model atmospheres analyses can therefore be used reliably for elemental abundances in the Carina targets."
],
[
"Carina Stars",
"The chemical abundances of the stars in Carina are presented in Tables REF to REF , and discussed element by element in the following Sections."
],
[
"Carbon",
"Carbon forms during the helium burning phases, whether the hydrostatic helium core burning phases in massive stars, or helium shell burning phases in AGB stars (Woosley & Weaver 1995).",
"The chemical evolution of carbon is further complicated by CNO cycling, where carbon is reduced when exposed to hot protons and the CN-cycle runs to equilibrium values.",
"This evolution in carbon can be seen in RGB stars in globular clusters, where the [C/H] abundance and $^{12}$ C/$^{13}$ C ratio are both reduced as stars ascend the RGB due to internal (self) mixing (Gratton et al. 2004).",
"Carbon abundances in this analysis were determined from spectrum syntheses of portions of the $A^2\\Pi $ - $X^2\\Delta $ CH G-band near $\\lambda $ 4320 in the Magellan spectra.",
"The best fit was taken as the carbon abundance from the CH line list by Brown et al.",
"(1987) and Carbon et al.",
"(1982), with acceptable fits as the abundance uncertainty.",
"A C$^{\\rm 12}$ /C$^{\\rm 13}$ ratio of six was adopted (the typical value for bright RGB stars that have undergone the first dredge up); this ratio made only a small difference such that increasing the value to 50 caused $\\Delta $ log(C/H) = 0.06.",
"Similarly, reducing the oxygen abundance had only a small effect (since C can be locked into the CO molecule); we adopted [O/Fe] = +0.4, but reducing this by 3x lowered the carbon abundance by only $\\le 0.05$ dex.",
"The spectrum of M68-6023 around the CH 4320 Å feature is shown in Fig.",
"REF , with spectrum syntheses for three carbon abundances.",
"Figure: Spectrum synthesis of the CH 4320 Å feature inM68-6023.",
"The chemical composition determined in this paperwas used for the syntheses, along with three carbon abundances,[C/Fe] = -0.4,-0.6,-0.8-0.4, -0.6, -0.8.",
"Broadening parameters wereinitially taken from the resolution of the Magellan spectra(Gaussian FWHM ∼\\sim 0.15)and checked against the best fit well defined iron-features.The carbon abundances for the M68 stars are in good agreement with typical red giants in globular clusters above the RGB bump (e.g., Smith et al.",
"2005, Gratton et al.",
"2000), where [C/Fe] $\\sim -0.4$ at the RGB bump and slowly decreases to $\\sim -1.0$ dex at the RGB tip due to mixing of CNO-cycled gas.",
"This trend can be seen in Fig.",
"REF (lower panel, data from Gratton et al. 2000).",
"The carbon abundances compiled by Frebel (2010b) suggests a much larger scatter in carbon in metal-poor stars than seen in clusters (Fig.",
"REF , upper panel), however Frebel's compilation includes field stars of all evolutionary stages, including carbon enhanced metal-poor (CEMP) stars.",
"CEMPs have [C/Fe] $> 1.0$ and are thought to be both stars that have undergone mass transfer in a binary system with an AGB companion (CEMP-s, e.g., Beers & Christlieb 2005) and stars with high primordial carbon (e.g., from high mass rotating stars, Meynet et al. 2006).",
"[6] suggested that mixing on the RGB and extra mixing at the tip of the RGB will lower the surface carbon abundance, such that the definition of a carbon enhanced metal-poor (CEMP) star should depend on luminosity.",
"This luminosity dependent range is shown in Fig.",
"REF (lower panel), and clearly shows that stars in M68 and the Carina dSph are not CEMP stars by any definition.",
"The mean carbon abundance in M68 is $<$ [C/Fe]$>$ = $-0.7 \\pm 0.1$ , in agreement with typical upper RGB stars.",
"Upper limits on the carbon abundances in the metal-poor Carina stars are even lower ([C/Fe] $\\le -1.0$ ); this value is typical of upper RGB stars in globular clusters, but on the low side of the values found for the bright, metal-poor field stars in the Galaxy, as seen in Fig.",
"REF .",
"Figure: Carbon abundances in Galactic stars (small greydots; Frebel 2010b, Gratton et al.",
"2000), the three stars inM68 (empty black squares), and upper limits on the three metalpoor Carina stars (upper panel).CEMP stars are usually defined as having [C/Fe] >1.0> 1.0, althoughextra mixing at the RGB tip can lead to a luminosity dependenceas noted in the lower panel.",
"The low carbon abundances inthe Carina stars are consistent with those of luminous RGB starsin globular clusters (data from Gratton et al.",
"2000), which are notsimilar to CEMP stars."
],
[
"Alpha Elements",
"The alpha-elements (O, Mg, Si, Ca, Ti) are built from multiple captures of He nuclei ($\\alpha $ particle) during various stages in the evolution of massive stars ($>$ 8 M$_\\odot $ , e.g., carbon burning, neon burning, complete and incomplete Si burning), and dispersed during SN II events.",
"Thus, the [$\\alpha $ /Fe] ratio in a star is a way to trace the relative contributions from SN II to SN Ia products in the ISM when it formed (the nucleosynthesis of iron is discussed in the next Section).",
"Although Ti is not a true alpha-element, the dominant isotope $^{48}$ Ti forms through explosive Si-burning and the $\\alpha $ -rich freeze outThe $\\alpha $ -rich freeze out occurs when a strong shock travels through infalling matter at the inner most shell of a core-collapse supernova.",
"After being shocked, material is heated to $>5$ billion degrees, so that nuclei decompose into neutrons and tightly bound $\\alpha $ particles., thus it behaves like an alpha-element (Woosley & Weaver 1995).",
"Our [$\\alpha $ /Fe] results are plotted in Figs.",
"REF and REF .",
"Koch et al.",
"(2008) determined $\\alpha $ element abundances for 4 stars in common with ours, and a solid line connects those results in the Figures.",
"In general, the $\\alpha $ -element abundances in Carina tend to be lower than the Galactic distributions at all intermediate metallicities, i.e., $-2.2 <$ [Fe/H] $< -1.3$ .",
"In the most metal-poor stars at [Fe/H] = $-2.9$ , most of the $\\alpha $ elements are in good agreement with the Galactic distributions.",
"Two stars stand out in their [$\\alpha $ /Fe] distributions, Car-612 and Car-5070, which have very low ratios of [Mg,Ca,Ti/Fe] for their metallicities.",
"These two stars will be discussed further in the following Sections.",
"Oxygen: Oxygen abundances in this analysis are solely from the [O1] 6300 line.",
"Oxygen could only be measured in the higher metallicity stars in the sample where the line is stronger.",
"Excellent agreement was found for one star in common with Koch et al.",
"(2008; Car-484, where they found [O/Fe]=+0.39 and we find =+0.48).",
"Silicon: Silicon abundances are only available for the higher metallicity stars.",
"Line measurements of the Si1 lines at 5684 Å and 6155 Å are similar to those in Koch et al.",
"(2008), however our oscillator strength for one line is lower by 0.4 dex.",
"Only upper limits were available from the Magellan spectra taken of the metal-poor stars.",
"Magnesium: Mg1 abundances were calculated from the $\\lambda $ 5528 line in all of the Carina stars, combined with $\\lambda $ 5172 and $\\lambda $ 5183 when $<$ 200 mÅ, and $\\lambda $ 5711 in the FLAMES/UVES spectra (it was too weak to measure in the Magellan spectra).",
"The abundances between these lines are in good agreement.",
"We note that the Carina Mg abundances are in good agreement with those from Koch et al.",
"(2008) and Shetrone et al.",
"(2003), see Fig.",
"REF .",
"Figure: The LTE oxygen and silicon abundances of starsin Carina and M68, compared to the Galactic distribution.Results for Carina are from this paper (filled circles),Koch et al.",
"(2008, empty circles), andShetrone et al.",
"(2003, filled triangles);for four stars in common between this analysisand Koch et al.",
", the abundance results are connected by agrey line.M68 results are shown for one star fromShetrone et al.",
"(2003, grey triangle),the mean of 7 RGBs from Lee et al.",
"(2005, grey square).Usually our M68 results are shown by empty black squares,but for O and Si only upper limits are available, shownas grey arrows.",
"A representative error baris shown (Δ\\Delta [Fe/H] = ±0.1\\pm 0.1, Δ\\Delta [X/Fe] = ±\\pm 0.15),although the actual errors per star fromthis analysis are also plotted per point.Small grey points show the Galactic distributions summarizedby Venn et al.",
"(2004) and Frebel (2010b).Calcium: Many lines of Ca1 were available in all of the program stars, over a wide range of wavelengths, and of appropriate line strengths.",
"Results are in good agreement with Koch et al.",
"(2008), see Fig REF .",
"Titanium: Titanium abundances were determined from a range of Ti1 and Ti2 lines across the entire spectral region.",
"The abundances between the two species are in good agreement, typically within 1 $\\sigma $ of each other, despite the possibility that titanium can be overionized by the radiation field resulting in lower Ti1 abundances when LTE is assumed.",
"Letarte et al.",
"(2010) suggested that better agreement is found when [Ti1/Fe1] is compared to [Ti2/Fe2] in their sample of Fornax RGB stars, due to similarities in the NLTE effects and possibly temperature scale variations.",
"Examination of our M68 comparison stars are in agreement with their finding, such that the agreement improves when Ti2 is compared with Fe2, however this had no effect on our Carina abundances.",
"Thus, in this paper, the abundances from both species are averaged together.",
"We note that our Carina Ti abundances are in good agreement with those from Koch et al.",
"(2008) and Shetrone et al.",
"(2003); see Fig.",
"REF .",
"Figure: The LTE magnesium, calcium and titanium abundancesof stars in Carina and M68, compared to the Galactic distribution.Symbols are the same as in Fig.",
"."
],
[
"Sodium ",
"Sodium is produced primarily during the carbon burning stages, but as metallicity increases then a sufficient amount of neutron rich material also allows sodium to be produced through the Ne-Na cycle during H-burning (Woosley & Weaver 1995).",
"Thus, the chemical evolution of Na initially follows the $\\alpha $ elements, but then deviates from this (rising) when AGB stars contribute.",
"Furthermore, Herwig et al.",
"(2004) have shown that metal-poor AGB stars from 2 to 5 M$_\\odot $ will overproduce sodium by factors of 10 to 100.",
"This Na overproduction has multiple sites dependent on the AGB mass.",
"In the FLAMES/UVES spectra, sodium abundances were determined from a single Na1 line at 5688 Å; this was because the resonance Na D lines at 5895 and 5889 Å were too strong ($\\sim $ 300 mÅ), and the line at 5682 Å (used by Koch et al.",
"2008) was too weak.",
"Only the most metal-poor stars in Carina have Na D lines that are weak enough to be used in this analysis, but since those stars were observed with Magellan/MIKE spectra, their lower SNR spectra required spectrum syntheses for the line abundance determinations.",
"The Fe1 5615 and 5634 lines were examined to check the broadening parameters (Gaussian broadening with FWHM = 0.25 Å was adopted, which is the effective resolution of the Magellan MIKE spectra).",
"When the Na D lines could not be fit simultaneously, then an average of the best fit abundances for each line was adopted, and the range used to estimate the uncertainty.",
"[1] have examined the NLTE corrections for the Na D resonance lines in metal-poor RGB stars.",
"Using their Table 2, combined with their Fig.",
"6 showing the effect of the correction on the line equivalent widths, then we estimated a NLTE correction for our abundances.",
"For the M68 stars, the higher gravities and strong line strengths imply corrections of $\\sim -0.5$ dex.",
"For the three metal-poor Carina stars (Car-1087, Car-5070, and Car-7002) then the atmospheric parameters and line strengths suggest corrections of $\\sim -0.6$ dex.",
"These corrections are included in Fig.",
"REF .",
"Of course, no corrections are applied to the other Carina objects since we did not use the Na D lines for those.",
"For Car-612, we only determined an upper limit, EW $<$ 30 mÅ for Na1 $\\lambda $ 5688, which corresponds to a very low upper limit of [Na/Fe] = $-0.8$ .",
"This result is comparable to that from Koch et al.",
"(2008) who measured an EW (= 24 mÅ) for the same weak line and determined [Na/Fe] = $-0.5$ .",
"Our lower upper limit is due to small differences in the atmospheric parameters and adopted solar Na abundance.",
"Figure: Sodium abundance ratios of stars in Carina and M68,compared to the Galactic distribution.",
"For the metal-poorstars in Carina and M68, NLTE corrections have been appliedto the Na D line abundances; the symbols are the same as inFig. .",
"For the Galactic data, only a limitedamount of available data are shown; the results from and Gehren et al.",
"(2006) are includedsince these two studies considered NLTE effects on the NaDlines, and results from Reddy et al.",
"(2003, 2006) are shownsince they only used the subordinate Na1 lines."
],
[
"Iron peak Elements ",
"In the early Universe, the iron-peak elements (Sc to Zn: 23 $\\le $ Z $\\le $ 30) are exclusively synthesized during Type II supernovae explosions by explosive oxygen and neon burning, and complete and incomplete explosive Si burning.",
"Abundances of these elements show a strong odd-even effect (odd nuclei have lower abundances than the even nuclei).",
"Their yields depend on the mass of the progenitors (e.g., Woosley & Weaver 1995), the mass cut adopted (mass expelled relative to the mass that falls back onto the remnant, e.g., Nakamura et al.",
"1999) and SNe II explosive energies (e.g., Umeda & Nomoto 2005).",
"Only at later times ($\\sim $ 1 Gyr, e.g., Maoz et al.",
"2010), when lower mass stars reach the end of their life time, do SNe Ia become a significant, possibly the dominant, contributor to the total iron-group inventory.",
"The onset of SNe Ia in the chemical evolution of the Galaxy is observed as a knee in the [$\\alpha $ /Fe] vs [Fe/H] near [Fe/H] $\\sim -1.0$ (e.g., McWilliam et al. 1995).",
"At lower metallicities, SNe Ia [X/Fe] yields could vary (e.g., Kobayashi & Nomoto 2009).",
"We have measured several of the Fe-peak elements in the Carina stars to compare with the Galactic trends.",
"Scandium: Several (4-13) lines of Sc2 were measured and HFS corrections applied.",
"The distribution in the Sc abundances at intermediate metallicities is larger than most of the other iron peak elements (see Fig.",
"REF and REF ), although this includes Car-612 which has several chemical peculiarities and stars analysed by Shetrone et al. (2003).",
"Other than those objects, the Sc abundances are similar to the Galactic distribution.",
"Vanadium: For most of the intermediate metallicity stars, the V abundance is well determined from $\\sim $ 10 V1 lines and HFS corrections were applied.",
"However, in the metal-poor stars and Car-612, it is determined from only 1-2 lines.",
"Manganese: The Mn1 abundances were determined from several lines (4-7) in the intermediate metallicity stars, and these line abundances showed good agreement with one another.",
"Generally, [Mn/Fe] in Carina follows the Galactic distribution above [Fe/H] = $-2$ .",
"In the metal-poor Carina stars and M68 stars, a different set of spectral lines was used (because of the wavelength coverage and resolution of the Magellan/MIKE spectra).",
"The blue resonance lines at $\\lambda $ 4030, 4033, and 4034, as well as several additional subordinate lines were measured, with only $\\lambda $ 4823 in common between the MIKE/Magellan and FLAMES/UVES datasets.",
"The Mn results from the resonance lines were lower than from $\\lambda $ 4823 and the other subordinate lines in the M68 standard stars, most likely due to non-LTE effects.",
"Bergemann & Gehren (2008) have calculated non-LTE corrections for Mn1 lines and do find large corrections for the blue resonance lines that are metallicity dependent.",
"They do not give corrections for red giants, but for red main sequence stars the corrections can be as large as +0.5 dex at [Fe/H] = $-3$ .",
"These corrections are similar to the offsets that we find between resonance and subordinate lines, therefore we apply a correction of $\\Delta $ log(Mn) = +0.5 dex to the Mn1 resonance line abundances $-$ this correction should be checked with proper modelling of these metal-poor red giant atmospheres.",
"Correcting the resonance line abundances improves the mean [Mn/Fe] results, but the mean values in Car-5070 and Car-1087 remain lower than for similar stars in the Galactic halo.",
"Figure: The odd-Z LTE abundance ratios for scandium, vanadium,and manganese in Carina.",
"HFS corrections have been applied,as well as a NLTE correction to the resonance lines ofMn1.",
"Symbols are the same as in Fig.",
"with the exception of the Galactic standards for [Mn/Fe]which are taken from Sobeck et al.",
"(2006) and Cayrel et al.",
"(2004).Figure: LTE abundance ratios for chromium, cobolt, and nickel in Carina.The low abundances for Car-612, and low Cr in Car-5070, areexceptional, like in Fig.",
".Symbols are the same as in Fig.",
".Chromium: Cr abundances were determined from 4-7 lines of Cr1, which showed good agreement from line to line.",
"A recent calculation of the NLTE effects on the Cr1 lines by Bergemann & Cescutti (2010) showed that the corrections are small ($\\le 0.1$ dex), but that Cr1/Cr2 ionization equilibrium and the solar [Cr/Fe] ratio is regained for metal poor stars, rather than the downward trend seen in Fig.",
"REF .",
"Our Carina data generally follow the Galactic trend, with the exception of very low abundances in Car-5070, Car-1087, Car-705, and Car-612.",
"Cobalt: Co was determined from only 1-2 lines in this analysis.",
"The [Co/Fe] ratios lie slightly below the Galactic abundances, although they are not well constrained from so few lines.",
"The only exception is the very low upper limit to [Co/Fe] for Car-612 (see Fig.",
"REF ).",
"[17] calculated NLTE corrections for Co1 and Co2; they suggest the NLTE corrections depend on metallicity and can become as large as +0.6 to +0.8 dex at [Fe/H] $\\sim -3.0$ .",
"This would not affect the very low Co upper limit found for Car-612.",
"Nickel: Several Ni1 lines (13-17) were measured in the intermediate metallicity stars over a range of wavelengths, showing that [Ni/Fe] is in agreement with the Galactic distribution (Fig.",
"REF ).",
"Only 1-3 Ni1 lines were available in the most metal-poor stars though, thus the uncertainties are large because $\\sigma $ (Fe1) was adopted for the error estimates.",
"Only the Ni result for Car-612 stands out, but this result is robust, determined from 14 Ni1 lines with a small error in the mean.",
"Combined with the low Na upper limit, then Car-612 fits the Na-Ni relationship observed for some $\\alpha $ -poor stars in the Galaxy by Nissen & Schuster (1997, 2010)."
],
[
"Cu & Zn",
"While contiguous on the Periodic Table, the main nucleosynthetic sites of copper and zinc are difficult to ascertain.",
"In massive stars, Cu and Zn form during complete Si-burning, the $\\alpha $ -rich freeze out, and even the weak s-process (e.g., Timmes et al. 1995).",
"In SN Ia models, the yields of Cu and Zn are sensitive to the neutron excess and thus metallicity (Matteucci et al.",
"1993, Travaglio et al.",
"2005, Kobayashi & Nomoto 2009).",
"The contributions to the production of Cu and Zn in AGB stars are uncertain, though recent calculations suggest small amounts of Cu and even smaller amounts of Zn can be produced in more massive (5 M$_\\odot $ ) AGB stars (Karakas et al.",
"2008, Karakas 2010).",
"Precise estimates of the AGB yields can also depend on uncertain parameters such as the mass loss law and number of dredge up episodes (Travaglio et al. 2004).",
"Figure: The LTE copper and zinc abundances in the Carina stars.The Cu and Zn abundances in Car-612 are remarkably low.Symbols are the same as in Fig.",
".Copper: The Cu1 line at 5105 Å was observed in some of our target stars.",
"Upper limits were calculated adopting an EW of 30 mÅ for the FLAMES/UVES spectra and 50 mÅ for the most metal-poor Carina stars.",
"Hyperfine structure corrections were applied.",
"Generally, copper follows the Galactic trend, and though we have very few actual measurements, the upper limit for Car-612 is very low and provides a very strong constraint.",
"Zinc: The Zn1 line at $\\lambda $ 4810 was observed in most of our sample, or used to determine the upper limits to the Zn abundance (adopting $<$ 40 mÅ); see Fig REF .",
"The zinc abundances at intermediate metallicities were slightly lower than the Galactic distribution, especially Car-612."
],
[
"Neutron Capture Elements ",
"Neutron-capture elements (Sr to U, 38 $\\le $ Z $\\le $ 92) originate in the rapid neutron capture process (r-process) that occurs during explosive nucleosynthesis in SNe II.",
"The r-process has a main source in 8-10 M$_$ stars that form elements with Z$>$ 50, and a weak source in more massive progenitors ($>$ 20 M$_$ ) that contribute primarily to the lighter neutron-capture elements (e.g., Sr, Y, Zr, Travaglio et al. 2004).",
"A more detailed examination of the r-process shows that a more accurate representation is in terms of a high entropy wind model (Farouqi et al.",
"2009, Roederer et al.",
"2010); nucleosynthesis in low entropy winds proceeds primarily through a chared-particle ($\\alpha $ -) process, whereas a neutron-capture component (the classical weak and main r-processes) occur in the high entropy winds.",
"Thermal pulsing in AGB stars also contributes to these elements through the slow neutron capture process (s-process).",
"With intermediate masses (2-4 M$_\\odot $ ), AGB stars have longer lifetimes than the sites of the r-process, and therefore in simple chemical evolution models AGB stars do not produce any of the heavy elements in the most metal-poor stars, but dominate the formation of some of these elements at later times and higher metallicities.",
"In the Galaxy, this is seen as the rise in the s-process which begins near [Fe/H] = $-2.5$ (McWilliam et al.",
"1998, [24], Francois et al.",
"2007); this is in contrast to the knee in [$\\alpha $ /Fe] due to contributions from SNe Ia near [Fe/H] $= -1.0$ .",
"In the Sun, the r-process contributes (11%, 15%, 25%, 28%, 53%, and 97%) of (Sr, Ba, La, Y, Nd, and Eu) respectively [24].",
"Thus Eu is critically important as a nearly pure r-process indicator.",
"Yttrium: [Y/Fe] in the intermediate metallicity stars was determined from Y2 4883, 5087, and 5200, which gave very consistent abundances from line to line and star to star.",
"The Y2 4900 line could not be used because it is severely blended in all of the spectra.",
"The [Y/Fe] ratios are similar to those found by Shetrone et al.",
"(2003), i.e., slightly below the Galactic values at intermediate metallicities, see Fig.",
"REF .",
"One exception is Car-612 which has an extremely low [Y/Fe] result, and another is Car-5070 which has a low upper limit value.",
"The low [Y/Fe] in Car-612 can be seen directly in comparison with Car-705, where Fig.",
"REF which shows that the Fe1 line strengths are similar but the Y2 5087 line is much weaker in Car-612.",
"Upper limits were determined in the most metal-poor stars using 40 mÅ as the EW upper limit for the Y2 $\\lambda $ 4883 line.",
"Figure: The LTE abundances for the heavyelements Y, Ba, and La in Carina and M68, compared to theGalactic distribution.The Y abundance in Car-612 is remarkably low, butverified by a comparison of the spectra of Car-612 andCar-705, e.g., in Fig.",
".The Galactic star abundances for La are from the critically examinedcompilation by Roederer et al.",
"(2010).These ratios are again quite low for Car-612, and Ba in Car-5070.Symbols are the same as in Fig.",
".Figure: Spectra around the Y2 5087 Å lines in Car-612 and Car-705.Since the atmospheric parameters and metallicities of these twostars are very similar, then this figure shows yttrium is truly weakerin Car-612.",
"Spectrum synthesis in this region confirms the differentY2 abundances derived from the EW analyses.",
"The lower [Ni/Fe]abundances in Car-612 can also be seen directly bycomparing the 5084 Å lines.Barium: Five lines of Ba2 were analysed (at 4554, 4934, 5853, 6141, and 6496 Å), which yielded consistent abundances from line to line and star to star, although they were not all observed in one star.",
"Spectrum syntheses were used to confirm the abundances when the SNR was low.",
"Hyperfine structure corrections for three lines (5853, 6141, and 6496 Å) are negligible ($\\le 0.02$ ).",
"NLTE corrections are also negligible ($\\le 0.03$ , Short & Hauschlidt 2006), with the exception of the $\\lambda $ 4554 resonance line, however we have chosen to not correct that line since the estimated correction is not large ($\\le 0.15$ dex, Short & Hauschlidt 2006, [2]) and the LTE abundance from that line is consistent with the other Ba2 line results.",
"Most stars in Carina have the same [Ba/Fe] distribution as stars in the Galaxy, with the exception of three stars, Car-612, Car-5070, and Car-705; see Fig.",
"REF .",
"Lanthanum: La was determined from three lines of La2 at $\\lambda $ 6320, 6390, 6774 (additional lines at $\\lambda $ 4333, 5301, 5303 could not be used due to the SNR of the spectra).",
"Negligible HFS ($\\le 0.02$ ) were calculated for two La lines ($\\lambda $ 6320 and 6390).",
"[La/Fe] follows the Galactic distribution for the few stars where we could measure it (see Fig.",
"REF ).",
"Figure: LTE abundances for the heavy elementsSr, Nd, and Eu in Carina and M68, compared to the Galacticdistribution.Symbols the same as in Fig.",
".Strontium: [Sr/Fe] was determined from two Sr2 lines at very blue wavelengths (4077 and 4215 Å) reached only by our Magellan/MIKE spectra.",
"The SNR is poor in that region, thus we adopted the results from spectrum syntheses for those lines in the three Carina stellar spectra, see Fig.",
"REF .",
"Nevertheless, the interpretation of the synthetic results remains difficult; the $\\lambda $ 4077 line is the stronger of the Sr2 resonance lines, and yet the detection of the $\\lambda $ 4215 line is more clear in Car-7002, and possibly Car-1087.",
"The best estimates for the Sr abundances in these two stars are listed in Table REF .",
"A Sr upper limit for Car-5070 could not be determined due to noise near both the $\\lambda $ 4215 and 4077 lines.",
"NLTE corrections were considered, but not applied to our results.",
"Short & Hauschildt (2006) estimate abundance corrections of $-0.07$ dex, whereas [3] estimate corrections of $\\sim 0.0$ and +0.1 dex for $\\lambda $ 4077 and 4215, respectively.",
"We chose to not apply either correction since these estimates are small and in opposite directions.",
"Figure: Spectrum syntheses of the Sr2 4077 and 4215 lines in thethree most metal-poor Carina stars,for [Sr/Fe] = -0.5,-1.0,-1.5,and-2.0-0.5, -1.0, -1.5, and -2.0.The low SNR makes accurate Sr abundances very difficult.Zirconium: Upper limits for Zr were determined from the Zr2 $\\lambda $ 4208 line in the Magellan spectra.",
"These upper limits are quite high, and do not add the discussion on the heavy element abundances.",
"Neodymium: Nd was determined from two Nd2 lines ($\\lambda $ 5319 and 5249) for all stars with FLAMES/UVES spectra, and these lines were used to calculate the upper limits from the Magellan spectra.",
"Nd is consistent with the Galactic distribution, though it is very low in Car-612 (Fig.",
"REF ).",
"Europium: The europium abundance in the Sun is nearly entirely due to the r-process, therefore Eu is identified as an important indicator in any stellar analysis to establish the ratio of r-process to s-process contributions of the neutron capture elements.",
"In this analysis, three Eu2 lines were examined ($\\lambda $ 4129, 4205, & 6645).",
"Unfortunately, all three lines were not observed in any one star; the Magellan spectra have too low resolution to detect the weak 6645 Å line, and the FLAMES/UVES data do not cover the bluer wavelengths.",
"Nevertheless, the Eu results are consistent with the Galactic distribution (see Fig REF ).",
"Hyperfine structure and isotopic splitting corrections are negligible ($<$ 0.05 dex).",
"Figure: Neutron capture ratios with Eu of stars in Carina and M68,compared to the Galactic distribution.",
"The ratio with Eu permitsan assessment of the s-process contributions relative to the r-processcontents in the Sun (Burris et al.",
"2000, dashed line).The rise in the s-process can be seen in the Carina stars and theGalactic distributions.Upper and lower limits are shown for some stars.Symbols the same as in Fig.",
"."
],
[
"s-process to r-process Ratios: [X/Eu] ",
"To assess the relative amounts of s-process to r-process contributions amongst the neutron capture elements, then abundance ratios relative to Eu are examined.",
"In Fig.",
"REF , [Y/Eu] and [Ba/Eu] of stars in Carina and the Galaxy are shown and compared to the solar r-process fractions from Burris et al.",
"(2000; also see [9].",
"In both [Y/Eu] and [Ba/Eu] (as well as [La/Eu] and [Nd/Eu], not shown), the gradual rise in these ratios can be seen with increasing metallicity.",
"McWilliam et al.",
"(1998) showed that the rise in the s-process starts at [Fe/H] = $-2.5$ in the Galaxy.",
"In Carina, this rise appears to begin at a higher metallicity, [Fe/H] $\\ge -2.0$ .",
"This is most likely due to the metallicity dependence of the AGB yields, as modelled by Travaglio et al.",
"(2004, also Pignatari et al.",
"2010), where fewer iron seed nuclei in a high neutron density wind can collect more neutrons, thus underproducing the first s-process peak elementsThe s-process peaks are defined by the neutron magic numbers for filling nuclear shells (e.g., N=50, 82, 126 are full nuclear shells).",
"A full nuclear shell lowers the cross section for further neutron captures, thus elements collect at these neutron numbers defining the first, second, and third s-process peaks., and overproducing the second and/or third s-process peak elements.",
"The slightly lower [Y/Eu] and slightly higher [Ba/Eu] ratios in Carina are consistent with metal-poor AGB stars contributions.",
"As a final note, the high Ba abundance in M68 is somewhat surprising.",
"This was also found by Lee et al. (2005).",
"Since [Ba/Eu] is high, but [Eu/Fe] is normal, this suggests M68 has had an unusual chemical evolution, possibly having been enriched (or self-enriched) by metal-poor AGB stars."
],
[
"Discussion",
"Detailed chemical abundances for up to 23 elements in nine stars in the Carina dSph have been presented in this paper.",
"Previously, only five stars in this dSph had detailed abundance determinations for so many elements (Shetrone et al.",
"2003), while another ten had [$\\alpha $ /Fe] determinations from high resolution spectroscopy (Koch et al. 2008).",
"In a companion paper by Lemasle et al.",
"(2012), the VLT FLAMES/GIRAFFE spectra for 36 additional RGB stars are presented, but due to the limited wavelength coverage of those spectra, only four elements are analysed in detail (Mg, Ca, Ba, and Fe).",
"For the first time, Lemasle et al.",
"(2012) calculated the ages of all stars with detailed chemical abundances in the Carina dSph, including those in this paper, from isochrone fitting.",
"While the uncertainties in the ages can be quite large for any one star, the differential ages should be better, and any discussion regarding ages is therefore limited to two age bins, an old population and an intermediate-aged population.",
"These ages (two age bins) are adopted in this discussion.",
"In the previous Section, our results were compared to stars in the Galaxy, where some chemical peculiarities were noted.",
"In this Section, we will compare our abundance results to those of stars in other dwarf galaxies.",
"Three dSphs have been chosen for comparison: Sculptor, Fornax, and Sextans.",
"The datasets for the three dSphs are taken from Shetrone et al.",
"(2001, 2003), Geisler et al.",
"(2005), [5], Letarte et al.",
"(2006, 2010), Tafelmeyer et al.",
"(2010), and Frebel et al.",
"(2010c), all scaled to the [10] solar abundances.",
"These three dSphs were chosen since large datasets of detailed abundances from high resolution spectroscopy are available, and the analyses of those spectra used similar methods and model atmospheres to our analysis.",
"We do not include results from Kirby et al.",
"(2009, 2010, 2011) in these comparisons to avoid systematic differences due to the lower resolution of their spectra and differences in the analysis methods.",
"Also, we do not include higher mass dwarfs (e.g., the LMC, SMC, or Sgr) because the metallicity range of the stars in those galaxies do not overlap well with the stars in the Carina dSph.",
"We also compare several elemental abundances of the most metal-poor stars in Carina to stars in the ultra faint dwarf galaxies (UFDs) and the metal-poor stars in the Draco dSph.",
"Datasets for the UFDs are from the compilation by Frebel (2010b), and for Draco are from Cohen et al.",
"(2009), Fulbright (2004), and Shetrone et al. (2001).",
"These comparisons are valuable because Carina has a tiny dynamical mass within its half light radius (M$_{1/2}$ ), e.g., M$_{1/2}$ (Carina) = 6.1 $\\pm $ 2.3 $\\times $ 10$^6$ M$_\\odot $ (Walker et al.",
"2009), which is a factor of 2, 4, and 9 times smaller than Sculptor, Sextans, and Fornax, respectively, but is comparable to those of the more massive UFDs.",
"As an example, Böotes I has M$_{1/2}$ = 5.9 $\\pm $ 3.7 $\\times $ 10$^6$ M$_\\odot $ , which is essentially the same as Carina, although the masses of the UFDs continue to be revised downwards, e.g., the mass of Boötes I has been revised downwards by a factor of 14 due to the lower velocity dispersions found by Koposov et al. (2011).",
"Of course, Carina is not considered an UFD because of its luminosity - Carina is 8x brighter than Boötes I (Walker et al.",
"2009)."
],
[
"The Metallicity Distribution of the High Resolution Spectroscopic Sample",
"As shown in Fig.",
"4 of Lemasle et al.",
"(2012), the stars that have been analysed with high resolution spectroscopy are biased towards metallicities near [Fe/H] = $-1.5 \\pm 0.3$ .",
"This is the same as the mean metallicity and metallicity range predicted for the dominant intermediate-aged population by [19] from their CMD analysis; however this is not scientifically significant for two reasons.",
"Firstly, more than half of the high resolution sample comes from the FLAMES/GIRAFFE spectral analysis presented by Lemasle et al.",
"(2012), and as discussed in that paper, the low SNR of those spectra meant that the weaker lines of the more metal-poor stars were harder to detect.",
"This effectively removed those stars from the analysis.",
"Secondly, although the V magnitude at the tip of the RGB may be slightly brighter with higher metallicities, this is only a small effect and maximizing the fiber placement was a more significant concern.",
"Also, while it is true that the three stars targeted for Magellan/MIKE observations are located in the outer fields of Carina and are the most metal-poor stars sampled, this does not imply a population gradient in Carina as those stars were purposely selected for their low metallicities.",
"Koch et al.",
"(2006) found no significant difference in the mean metallicities of RGB stars in the inner and outer fields of Carina.",
"Thus, the results presented in this paper are not used to constrain the distributions in location or metallicity of the stars in Carina."
],
[
" Dispersions in [$\\alpha $ /Fe] ",
"In Fig.",
"REF , we show [Mg/Fe] and [Ca/Fe] for Carina, compared to the three other dSphs and the Galaxy, over the full metallicity range examined.",
"Looking at [Ca/Fe] alone suggests that the chemical evolution of Carina has been similar to Sculptor.",
"In Sculptor, a noticeable downward trend in these abundances begins at [Fe/H] $\\sim -1.8$ , whereas in Galactic stars this occurs at a higher metallicity.",
"This knee in the [$\\alpha $ /Fe] ratios is usually interpreted as the onset of contributions to [Fe/H] from SN Ia, with the shift in the knee to lower metallicities in dwarf galaxies attributed to their slower chemical evolution (e.g., Lanfranchi et al.",
"2008, Kirby et al. 2011).",
"In Carina, it is not clear where or if there is a knee.",
"The more remarkable result seen in Fig.",
"REF is the dispersion in [Mg/Fe] ($\\Delta $ [Mg/Fe]) in Carina, which is observed in both the FLAMES/UVES and FLAMES/GIRAFFE data, and is much larger than the dispersion in [Ca/Fe] ($\\Delta $ [Ca/Fe]).",
"A calculation of the intrinsic spreadsWe have estimated the intrinsic spread (N$_i$ ) from the formula N$_i^2$ = N$^2$ - $\\langle \\sigma \\rangle ^2$ , where N is the range in [X/Fe] and $\\langle \\sigma \\rangle $ the average error in [X/Fe].",
"suggests that this difference is real and signficant when considering the whole dataset, the data from Lemasle et al.",
"(2012) only, our nine UVES targets alone, or the data at a specific intermediate metallicity (such as [Fe/H]$=-1.2\\pm 0.1$ ), i.e., 0.4 $\\le $ [N$_i$ ([Mg/Fe])$-$ N$_i$ ([Ca/Fe])] $\\le $ 0.7.",
"Only for [Fe/H] $< -2.0$ are the intrinsic spreads between [Mg/Fe] and [Ca/Fe] similar.",
"Differences in the dispersions of [Mg/Fe] and [Ca/Fe] may be partially due to differences in their nucleosynthetic sites, but also may be the result of inhomogeneous mixing of the interstellar gas and therefore poor statistical sampling of the SN contributions when forming stars.",
"In terms of nucleosynthesis, SN yields of Ca and especially Mg depend on the progenitor mass (e.g., Woosley & Weaver 1995, Iwamoto et al.",
"1999); for example, Mg forms in hydrostatic core C and O burning, whereas Ca has contributions from the $\\alpha $ -rich freeze out and explosive Si burning during the SN II explosion Differences in the SN Ia models can also lead to differences in the [Si-Ca/Fe] ratios, e.g., the central density, metallicity, ignition source, flame speed, and even type of SN Ia model can play a role (e.g., Maeda et al.",
"2010, Röpke et al.",
"2006, Iwamoto et al. 1999).",
"In terms of inhomogeneous mixing, models by Revaz & Jablonka (2011, also Revaz et al.",
"2009) predict a large spread in the [Mg/Fe] ratios in low mass dwarf galaxies due to longer gas cooling times and subsequently longer mixing timescales for the interstellar medium.",
"As an example, the hot gas from SNe II can be subject to buoyant forces requiring up to 2 Gyr to cool and mix through a low mass galaxy.",
"This hot gas can also quench star formation.",
"Therefore, it is possible to find models of low mass dwarf galaxies that predict both a high dispersion in element ratios and an episodic star formation history, like Carina.",
"Surprisingly, the [Mg/Ca] ratios in Carina do not show a larger dispersion then the other dSphs in Fig.",
"REF .",
"This may imply that inhomogeneous mixing plays the dominant role (over differences in nucleosynthetic sites).",
"The one star with the extremely low [Mg/Ca] ratio is Car-743, analysed by Lemasle et al.",
"(2012) from their lowest SNR spectrum; the formal uncertainty in that one result is $\\sim 2$ x larger than the representative errorbar shown, and should be considered with caution.",
"The one star with the highest [Mg/Ca] ratio is Car-1087, analysed in this paper; with a metallicity of [Fe/H] $=-2.9$ and high [Mg/Ca] ratio then this star is very similar to the unusual star Draco-119 (discussed below).",
"Figure: Mg and Ca abundances of stars in Carina (red),Sculptor (green), Fornax (cyan), and Sextans (magenta).Symbols for Carina:solid circles are from this paper,squares are from Lemasle et al.",
"(2012),triangles are from Shetrone et al.",
"(2003),and asterisks are from Koch et al.",
"(2008).Symbols for Fornax:circles are from Tafelmeyer et al.",
"(2010),triangles are from Shetrone et al.",
"(2003),squares are from Letarte et al.",
"(2010).Symbols for Sculptor:large/small squares are UVES/GIRAFFE FLAMES data from Hill et al.",
"(2012),solid circles are from Tafelmeyer et al.",
"(2010),asterisk are from Giesler et al.",
"(2005),triangles are from Shetrone et al.",
"(2003),and the star is from Frebel et al.",
"(2010c).Symbols for Sextans:squares are from ,triangles are from Shetrone et al.",
"(2001),and circles are from Tafelmeyer et al.",
"(2010).Tafelmeyer et al.",
"(2010) note that the very low [Ca/Fe]ratio reported for the most metal-poor dSph star(at [Fe/H] ∼-4\\sim -4) is likely due to NLTE effectsin the formation of the strong resonance Ca1 4227Å line, a line that was not used throughout the rest oftheir analysis, and therefore should be regarded withcaution.Representative error bars of Δ\\Delta [Fe/H] = ±\\pm 0.1 andΔ\\Delta [X/Fe] = ±\\pm 0.15 are shown.Figure: [Mg/Fe] and [Ca/Fe] in the most metal-poor stars inthe Galaxy, Carina, Draco, and UFD galaxies.The data for Carina are the same as in Fig.",
",while for Draco, the small/large squares are fromShetrone et al.",
"(2001)/Cohen et al.",
"(2009).The large empty triangle is Draco-119,the metal-poor star found by Shetrone et al.",
"(2001)and reanalysed with higher SNR spectra by Fulbright (2004).The UFDs are presented as follows:solid/empty circles are for Böotes I fromFeltzing et al.",
"(2009)/Norris et al.",
"(2010),solid triangles are for Ursa Major II from Frebel et al.",
"(2010a),solid diamonds are for Com Ber from Frebel et al.",
"(2010a),plus signs are for Hercules from Koch et al.",
"(2008b),and the asterisk for one star in Leo IV by Simon et al.",
"(2010).The dashed line in the [Mg/Ca] panel represents stars with themore extreme enhancements (3 in UFDs, 2 in dSphs)."
],
[
" Metal-poor Stars and the UFDs ",
"In Fig.",
"REF , we show only the metal-poor tail in the Carina abundances and compare the [Mg/Fe] and [Ca/Fe] ratios to those of metal-poor stars in five UFD galaxies and the Draco dSph.",
"Most of the UFD stars have high values of [Mg/Fe] and [Ca/Fe], like the Galactic halo, whereas Carina and Draco show a range extending to very low values.",
"Some stars in all of these systems show very high [Mg/Ca] values ($>0.5$ ) e.g., two stars in the Hercules UFD (Koch et al.",
"2008), one in Böotes I (Feltzing et al.",
"2009), one in Draco (Fulbright 2004, Shetrone et al.",
"2001), and one in Carina (this paper).",
"In the UFDs, it has been proposed that high [Mg/Ca] may be due to the chemical enrichment of its interstellar gas by as few as one SN II explosion from a massive progenitor (e.g., a $>$ 35 M$_$ star).",
"In this scenario, a unique chemical signature can be imprinted onto the gas that is used to form stars, while the rest of the gas is expelled to quench further star formation (e.g., Koch et al 2008b, Frebel et al.",
"2010a, Simon et al. 2010).",
"This scenario is aided by peculiar neutron capture ratios, e.g., in the Hercules stars the Ba2 lines are not detected (Koch et al. 2008).",
"Similarly, the Ba2 (and Sr2) lines in the high [Mg/Ca] star in Draco (Draco-119) are also not detected (Fulbright et al.",
"2004 suggested that this star was enriched by a SN II in the mass range of $20 - 25$ M$_\\odot $ , and is lacking in contributions from higher mass progenitors).",
"However, Ba2 (and Sr2) lines are detected in the high [Mg/Ca] star in Carina.",
"Those lines are weak, leading to very low abundances of [Ba/Fe] and [Sr/Fe], but if the source of these elements is the same as in the other dSphs, then Carina appears to have been able to retain a bit more of those early (high mass progenitor) SN II enrichments."
],
[
" Measurement Errors? ",
"One question worth considering is whether the large dispersion in [Mg/Fe] is simply due to measurement errors.",
"For example, there are only 1-2 Mg1 lines available in each star, whereas there are 3-21 lines of Ca1 (depending on the metallicity) in this analysis.",
"Similarly, due to the limited wavelength coverage of the FLAMES/GIRAFFE spectra, there are $\\le $ 13 Ca1 lines measured by Lemasle et al.",
"(2012), and only one Mg1 line at $\\lambda $ 5528.",
"A larger number of spectral lines reduces the random measurement errors in an elemental abundance (by $\\sqrt{N}$ ).",
"This is reflected in the smaller average error in [Ca/Fe] than [Mg/Fe] in Table 10 of Lemasle et al. (2012).",
"In this paper, even if the measurement error from fitting a line is small, as in a high SNR spectrum, the analysis of a single line means that the adopted error is the standard deviation in the Fe1 line abundance ($\\sigma $ (Fe); see Section REF ).",
"These (conservative) errors are about as large as the [Mg/Ca] dispersion, but smaller than the [Mg/Fe] dispersion, therefore the large dispersion in [Mg/Fe] does not appear to be due to measurement errors alone.",
"Another consideration is that variations in the [Mg/Fe] and [Mg/Ca] ratios have been found in a number of metal-poor stars in the ultra faint dwarf galaxies, and attributed to incomplete mixing and poor sampling of the full mass function, as shown in Fig.",
"REF .",
"Only the largest outliers are examined in terms of pollution by a single SN II, and similarly large outliers are found in our [Mg/Ca] and [Mg/Fe] data as well.",
"The consistency between these analyses suggests that the signatures in the largest outliers are not due only to measurement errors."
],
[
"Age Considerations",
"The Carina dSph is an interesting galaxy partially because of its low mass and partially because of its episodic star formation history.",
"Lemasle et al.",
"(2012) were the first to attempt to interpret the chemical evolution of this dwarf galaxy after separating the stars into two populations, old ($>$ 10 Gyr) and intermediate-aged ($<$ 6 Gyr); stars with ages between 6 and 10 Gyr were examined by Lemasle et al.",
"2012, but we do not include those in this discussion.",
"The old population shows a large range in [Fe/H] and [Mg/Fe], whereas the intermediate-aged (IA) population appears to have a very small range in [Fe/H] and [Mg/Fe].",
"Lemasle et al.",
"(2012) were also struck by the significant overlap in [Fe/H] between these populations, which would imply that the IA stars formed from gas that was more metal-poor than the end-point metallicity of the old population.",
"Combined with the higher mean [Mg/Fe] abundance of the IA population, Lemasle et al.",
"(2012) suggested that the second epoch of star formation in Carina may have occurred after the late accretion of metal-poor, $\\alpha $ -element rich gas.",
"The overlap in the metallicities of the old and IA population can be seen in Figs.",
"REF to REF .",
"We note that the overlap is heavily weighted by the age assigned to Car-612, which is the chemically peculiar star discussed in Section 4.",
"Differences in the specific chemistry of stars are known to affect their isochrone ages (Dotter et al.",
"2007), and thus the age assignment to this star should be considered with caution.",
"The overlap is also heavily weighted by Car-524, which has a large uncertainty in its age assignment.",
"If the ages for these two stars are neglected, then there is a sharp transition at $-1.4 <$ [Fe/H] $< -1.6$ between the age groups, i.e., equivalent to the measurement errors in metallicity.",
"Thus, the overlap in metallicity between the two age groups is not sufficiently clear to indicate an infall of metal-poor gas to form the second generation of stars.",
"Figure: [α\\alpha /Fe] ratios of stars in the Carina dSphseparated into age groups.Symbols the same as in Fig.",
",though now red is used for the old populationand blue for the IA population;ages are from Lemasle et al.",
"(2012).Galactic comparison stars are shown as small grey dots (includingdata from Venn et al.",
"2004, Frebel et al.",
"2010b, andReddy et al.",
"2003, 2006).",
"Representative errorbars are shownbased on the mean error in the Carina dataset."
],
[
"[$\\alpha $ /Fe] Ratios Between the Age Groups",
"An offset in the mean [Mg/Fe] abundance in the IA population relative to the old population can clearly be seen in Fig.",
"REF .",
"A Kolmogorov-Smirnov (K-S) test shows that the distribution of the [Mg/Fe] values for the old population is broader and with lower mean values, i.e., <[Mg/Fe]>$_{\\rm old}$ = 0.05 $\\pm $ 0.32 and <[Mg/Fe]>$_{\\rm IA}$ = 0.22 $\\pm $ 0.15, both with moderately high probabilities of having “normal” distributions once the outlier in the IA dataset (at [Mg/Fe] = -0.95) is removed.",
"This was also seen by Lemasle et al.",
"(2012) for both [Mg/Fe] and [Ca/Fe], although the offset in [Ca/Fe] is not as large, which we confirm with a K-S test.",
"We also note that these offsets in the [Mg/Fe] and [Ca/Fe] ratios with age are still present when only the high resolution data are examined (this paper, Koch et al.",
"2008, and Shetrone et al.",
"2003), although they are not as clear.",
"The same signature is also hinted at in the [Ti/Fe] ratios (from the high resolution data), but there is insufficient data in the IA population for a meaningful K-S test.",
"Since the large disperions in the [$\\alpha $ /Fe] ratios discussed above (larger than seen in most dSph galaxies) are an indication of inhomogeneous mixing, the simplest explanation for the offset in [$\\alpha $ /Fe] between the old and IA population is that the second epoch of star formation occurred in $\\alpha $ -enriched gas.",
"The small range in the [$\\alpha $ /Fe] and [Fe/H] ratios suggests this gas was well mixed."
],
[
"The Rise in the s-Process",
"Clearly the old and IA stellar populations in Carina show different chemical signatures, and it is interesting to investigate how the two populations might be connected.",
"While we have discussed the $\\alpha $ -elements, possibly the most valuable abundance ratios for studying the chemical evolution of a dwarf galaxy are of the neutron capture elements.",
"In Fig.",
"REF , it is possible to see that AGB stars have contributed to both age groups in Carina, but not until [Fe/H] $> -2$ .",
"This can be seen by the [Ba/Eu] ratios, where the lower dashed line represents the r-process contributions (in the Sun), such that stars with this ratio have been enriched by SNe II products only.",
"Above [Fe/H] $= -2$ , the [Ba/Eu] ratios slowly increase from the low r-process value through contributions to Ba from the s-process (the rise in the s-process).",
"This is seen in the Galaxy, Fornax, and Sculptor as well, although the rise begins at higher metallicities in those systems.",
"From the [Ba/Fe] ratios, it is also possible to see how the AGB contributions begin to dominate the nucleosynthesis of Ba, e.g., the dispersion in [Ba/Fe] at low metallicities in the Galaxy is interpreted as the stochastic sampling of the (small amounts of) Ba from SNe II products early on, but this scatter lessens when the dominant contributions from AGB stars come at later times.",
"The flatness of the relationship in [Ba/Fe] at higher metallicities in the Galaxy implies that the timescale and yields of Ba from the AGB stars and Fe from SNe Ia are similar, and the small scatter implies that the gas is well mixed.",
"In Carina, [Ba/Fe] in the IA population may have a large scatter.",
"If the large scatter is real, then this would suggest that the AGB contributions were not well mixed in the ISM at any age, which would be an interesting contrast to the uniformity in the [$\\alpha $ /Fe] ratios in the IA population.",
"However, several of these data points are from Lemasle etal (2012) where the measurements of a single Ba2 line were very difficult due to the low SNR in that spectral region.",
"The data from our high resolution analysis is far more uniform above [Fe/H] = $-2$ , as shown in Fig.",
"REF .",
"Below [Fe/H] = $-2$ it is difficult to ascertain if there are any stars with AGB contributions.",
"There is one star (Car-5070) with [Ba/Eu] below the r-process ratio, implying that this star shows no signs of AGB products; however Carina was poorly mixed at early times, and metal-poor AGB stars could produce very little Ba.",
"The high [Ba/Y] ratios in stars over [Fe/H] = $-2$ show that most of the s-process elements came from metal-poor AGB stars, where the first s-process peak (Y) was bypassed in favour of the second s-process peak (Ba; as described further in Section REF ) $-$ at even lower metallicities, both can be bypassed for the third s-process peak.",
"Therefore we cannot ascertain the precise metallicity when AGB stars began to contribute in Carina, but certainly there are contributions from metal-poor AGB stars in the stars with [Fe/H] $> -2$ , and a hint of the rise in the s-process between -2.0 $<$ [Fe/H] $< -1.6$ .",
"Further evidence for the rise in the s-process can be seen in the evolution in the [Na/Fe]; see Fig.",
"REF .",
"Cayrel et al.",
"(2004) and [1] showed that the [Na/Mg] and [Na/Fe] ratios are flat and low in metal-poor stars in the Galaxy, suggesting that Na is produced with the $\\alpha $ -elements in SNe II at a (low) fixed ratio.",
"The Na abundance rises when AGB stars begin to contribute (see Section REF ).",
"We see a very similar trend in the [Na/Fe] ratios in Carina; in fact the rise in [Na/Fe] is very similar to the rise in the s-process elements.",
"One significant difference though is that the initial [Na/Fe] value appears to be much lower in Carina than in the Galaxy.",
"The final [Na/Fe] ratios may be similar in Carina as in the Galaxy (and Sculptor), and we see no offset in the Na abundances between the old and IA populations.",
"A dispersion in Na is not clear in this small sample.",
"It is interesting that the evolution of [Mn/Fe] and [Cr/Fe] are also similar to Na in Carina.",
"The abundances of these elements are much lower than in the Galaxy at low metallicities, but then rise above [Fe/H] = $-2.0$ to abundances that are similar to stars in the Galaxy with the same metallicity.",
"There is also no difference in these elements between the old and IA populations.",
"Like Na, once AGB stars and SNe Ia begin to produce iron-group elements, those contributions will dominate over the initial (low) SNe II yields.",
"The timescale for AGB contributions is thought to be equal to or shorter than that of SNe Ia ($\\sim $ 1 Gyr, Maoz et al.",
"2010), thus it is likely that both AGB stars and SNe Ia begin to contribute to the ISM in Carina near [Fe/H] $= -2.0$ .",
"The SNe Ia contributions appear to be well mixed in the ISM, and there is no offset between the old and IA stars."
],
[
"Early Chemical Evolution by SNe II",
"The earliest stages of chemical evolution in Carina can be examined from the element ratios of the stars below [Fe/H] = $-2$ .",
"As discussed above, these stars appear to have been enriched only in SNe II products.",
"While there is a large range in the [$\\alpha $ /Fe] and [Mg/Ca] ratios (when the whole dataset is examined, including previously published results) , the ratios of Mn, Cr, and Na all start out very low but then rise to the Galactic values at intermediate metallicities.",
"In addition, the [Sr/Fe] values are amongst the lowest of all stars analysed in the Galaxy, other dSphs, and the UFDs (see Fig.",
"REF ).",
"The [Sr/Fe] values for Car-1087 and Car-7002 are similar to the lowest values found for metal-poor stars in Draco, Boötes I, and Com Ber.",
"Tafelmeyer et al.",
"(2010) suggested that dwarf galaxies may have a lower floor in the [Sr/Fe] ratio ($\\sim -1.2$ ) when their mass is equal to or lower than Draco.",
"The low [Sr/Fe] ratios in Carina lead to very low [Sr/Ba] ratios as well.",
"[Sr/Ba] ratios are quite interesting because they tend to be high in metal-poor stars in the Galactic halo, which has been interpreted as evidence for an extra nucleosynthetic source in the formation of these elements in massive stars (Travaglio et al.",
"2004, Ishimaru et al.",
"2005, Qian & Wasserburg 2008, Farouqi et al.",
"2009, Pignatari et al. 2010).",
"The absence of a [Sr/Ba] enhancement at low metallicities in Carina suggests a lack of the excesses seen in the Galactic stars.",
"It is not clear from which stellar mass range these excesses arise: Travaglio et al.",
"(2004) suggest the main r-process occurs in 8-10 M$_\\odot $ stars, and that the excess must arise from the more massive SNe II, whereas Farouqi et al.",
"(2009) suggest the excess occurs in SN II with lower entropy winds where a neutron-rich, $\\alpha $ -freezeout can occur.",
"The specific mass or energy range in Farouqi's models is not clear.",
"If we follow Travaglio's suggestions, then the excess Sr may form in more massive, or higher energy SNe II (i.e., hypernovae), and therefore those SNe II seem to be missing in Carina $-$ either those stars did not form or their ejecta was driven out of CarinaLow upper limits on the [Sr/Fe] and [Ba/Fe] are also reported for Draco-119, however these cannot be used together to constrain [Sr/Ba].",
"As an exercise, if the upper limits are taken as the actual values, then [Ba/Fe] $=-2.6$ and [Sr/Ba] $=+0.1$ , which places this star squarely amongst the Galactic distribution.",
"Thus, even the lack of a detection of the Ba2 and Sr2 spectra lines in Draco-119 does not provide a strong constraint on the one shot hypernova model.",
"The low values could be consistent with inhomogeneous mixing and/or gas driven out by supernova winds..",
"Figure: [Sr/Fe] and [Sr/Ba] in low metallicity stars in the Galaxy,classical dSphs, and UFDs.",
"The [Sr/Ba] ratio is enhanced in metal-poorstars in the Galaxy, which is not always seen in the dwarf galaxies.The UFD symbols are the same as in Fig.",
", whilethe dSph data is from this paper and Tafelmeyer et al.",
"(2010).Mn and Cr also form in hypernovae as the decay products of complete and incomplete Si-burning (Umeda & Nomoto 2002, Nomoto et al.",
"2011), thus a lack of hypernovae might also explain the very low initial values of these elements.",
"A neutron-rich, $\\alpha $ -freezeout in hypernovae would also contribute to Na production, and thus a lack of hypernovae (i.e., if this is the source of the Sr excess, as in Farouqi et al.",
"'s 2009 models) could explain the very low [Na/Fe] ratios in the most metal-poor stars.",
"Evolution models (e.g., Salvadori et al.",
"2008, Brooks et al.",
"2009, 2007, Ferrara & Tolstoy 2000) suggest that very low mass galaxies can lose gas after the first star forming epoch through SN II driven winds (possibly reaccreting cold gas for later star formation events).",
"Koch et al.",
"(2006) also suggested that the loss of metal rich winds would help to explain the MDF of Carina, which suffers from the well known G dwarf problem.",
"Detailed models of the star formation and chemical evolution of Carina by Lanfranchi & Matteucci (2004) invoked two major epochs of outflows through winds.",
"Their best model included a wind efficiency that was 7x the star formation rate and associated with the initial and IA star formation episodes.",
"Thus, the detailed chemical composition of the most metal-poor stars in Carina suggest a lack of hypernovae contributions, possibly implying that these stars did not form, but more likely indicating that their gas was removed from Carina by SN driven winds.",
"One alternative to this scenario is that the metal-poor stars are displaying the imprint of the hypernovae, rather than the lack of hypernovae, which would provide a very strong constraint on the nucleosynthetic models.",
"An excellent test of this alternative would be the [Zn/Fe] abundances, which are predicted to be enriched by hypernovae because of an increase in the mass ratio between the complete and incomplete Si-burning regions (Nomoto et al. 2011).",
"We are only able to determine upper limits to the [Zn/Fe] ratios in three stars in Carina below [Fe/H] $= -2$ from our moderate SNR Magellan/MIKE spectra; however these limits are tantalizingly close to providing an interesting constraint (see Fig.",
"REF ).",
"In particular, Car-5070 has a low [Zn/Fe] upper limit, and suggests that Car-5070 lacks hypernova enrichments."
],
[
"Summary",
"Carina and Draco (and possibly Boötes I) appear to be quite different in their early chemical evolution from the other dSphs and UFDs.",
"These two (three) galaxies may be at the critical mass where SN driven winds remove the gas from the most massive or energetic SNe II progenitors, but the products of the remaining SNe II are retained, and contribute to the (inhomogeneous) chemical evolution of the host.",
"This is unlike the dSphs, which appear to retain the gas from the earliest epochs and undergo a smooth chemical evolution that is not too different from stars in the Galactic halo (other than contributions occurring at lower metallicities, e.g., the AGB yields are from metal-poor stars, and the SNe Ia contribute at lower metallicities).",
"Both of these galactic systems are unlike the UFDs, where a single massive SN II may remove all of the gas, quenching the star formation event, and imprinting their unique chemical signatures on the few stars that will complete the star formation process - in this case, any abundance variations within an UFD are due to stochastic sampling of that hypernova and not chemical evolution.",
"As discussed in Section 4, Car-612 is underabundant in nearly every element when examined relative to iron, i.e., [X/Fe].",
"Therefore, we propose that this star is iron enhanced, most likely due to an excess of SN Ia contributions in the gas cloud from which it formed.",
"In Fig.",
"REF , we show the abundance distribution relative to the Galactic averages at [Fe/H] = -1.3 (estimated from the Figures in Section 4).",
"Notice that very few element ratios lay in the grey band that would describe the [X/Fe] ratios of a typical Galactic star at that metallicity.",
"The only exceptions are Cr, Mn, and Fe, and possibly Si (though the error is large for that element).",
"If this star has an excess of iron by 0.7 dex (a factor of 5), then removing this in the [X/Fe] ratios would produce the values in the blue band (other than Fe itself which we do not adjust).",
"Now Cr, Mn, and Fe (itself) appear enhanced, however nearly all other [X/Fe] ratios are in agreement with the Galactic stars.",
"A direct comparison of spectral line strengths shows that Car-612 has similar Fe1 line strengths to stars with similar atmospheric parameters (e.g., Car-705 in Fig.",
"REF ; also to Car-1013 not shown), while the other lines are weaker.",
"This peculiar chemistry makes Car-612 similar to three iron-rich stars in the outer halo, studied by Ivans et al. (2003).",
"These stars are all near [Fe/H] $= -2$ and have low [$\\alpha $ /Fe] ratios, low ratios of Y, Sr, and Ba, and two show enhancements in Cr, Mn, Ni, and Zn.",
"Of those two stars, one is enhanced in Si and Eu.",
"Ivans et al.",
"concluded that these stars have larger SN Ia/II contributions, by factors of 3 to 7 relative to the average halo star.",
"This is similar to Car-612 where we suggest the enhancement is by a factor of 5.",
"A deep and thorough examination of the predicted yields from existing SN models showed that no combination could reproduce the detailed abundance patterns of the three outer halo stars, i.e., problems remained in the abundances of Ti, Cr, Mn, Ni, and Zn.",
"Since the SN yields are able to reproduce the chemistry of the majority of stars in the Galaxy, Ivans et al.",
"suggested that perhaps the SN yields should not be integrated over a Salpeter IMF for these three stars and/or the degree of mixing in the ISM may vary from region to region.",
"If so, they considered that it is also possible for these stars to have been deposited in the outer halo during a dwarf galaxy merger.",
"Variations in their detailed abundances are therefore related to differences in the chemical evolution of their hosts.",
"Car-612 fits with this hypothesis $-$ it appears to be iron-enhanced, and yet its detailed chemistry is again different from that of the three outer halo stars, e.g., Ni, Zn, and Eu are not enhanced.",
"Identifying this star within the Carina dwarf galaxy provides additional evidence for inhomogeneous mixing in this low mass galaxy, and provides a clear connection between the formation of these chemically peculiar stars in dwarf galaxies and their existence in the outer Galactic halo."
],
[
"Conclusions",
"Carina is an interesting galaxy for chemical evolution studies because of its low mass and its episodic star formation history.",
"In this paper, we have determined the abundances of 23 elements in the spectra of nine RGB stars in the Carina dSph galaxy taken with both the VLT/FLAMES-UVES and Magellan/MIKE spectrographs.",
"This is a significant increase from the previous number of stars with detailed analyses, e.g., by Shetrone et al.",
"(2003, where all stars had intermediate metallicities) and Koch et al.",
"(2008, where only the iron and $\\alpha $ -element abundances were determined).",
"Our analysis uses both photometric and spectroscopic techniques to determine the stellar parameters, and we use new spherical models, an expanded line list, continuum scattering corrections, and hyperfine structure and NLTE corrections (when available) to improve the precision in our abundances.",
"Adopting the ages determined by Lemasle et al.",
"(2012), we are able to examine the chemical evolution of Carina, separating chemical signatures in the old and intermediate-aged populations.",
"A summary of the most important results in this paper are as follows: Inhomogeneous mixing: A large dispersion in [Mg/Fe] indicates poor mixing in the old population.",
"An offset in the [$\\alpha $ /Fe] ratios between the old and intermediate-aged populations (when previously published data are included) also suggests that the second star formation event occurred in $\\alpha $ -enriched gas.",
"In addition, one star, Car-612, seems to have formed in a pocket enhanced in SN Ia/II products.",
"SN driven winds: Stars with [Fe/H] $< -2$ do not appear to have been enriched in AGB or SNe Ia products.",
"Their peculiar chemistry includes very low ratios of [Sr/Ba], an element ratio that usually shows an excess in metal-poor stars in the Galaxy and dSphs.",
"Adopting a scenario where the excess Sr forms in the more massive or energetic SN II, then the lack of this excess in Carina (also Draco and Boötes I) is consistent with the loss of those products by SN II driven winds.",
"Low ratios of [Na/Fe], [Mn/Fe], and [Cr/Fe] support this scenario, with additional evidence from the low [Zn/Fe] upper limit for one star, Car-5070.",
"The $\\alpha $ -elements ratios in Car-5070 are also lower than the Galactic distribution.",
"It is interesting that the chemistry of the metal-poor stars in Carina are not similar to those in the Galaxy, most of the other dSphs, or the UFDs.",
"The [Sr/Fe] and [Sr/Ba] ratios are clear indicators of the differences in the early chemical evolution of these systems: the Galaxy and dSphs appear to retain all of their SN II products, and show excesses in the [Sr/Ba] ratios, whereas Carina (with Draco and Boötes I) may be at the critical mass where some gas is lost through SN II driven winds, by showing very low [Sr/Fe] and [Sr/Ba] ratios.",
"In the UFDs, all of the gas may be lost with the first SN II, quenching star formation, and imprinting the unique chemical signature of that SN II on the remaining stars as they complete the formation process, thus [Sr/Fe] is low, but [Sr/Ba] can vary.",
"It is also interesting to find a star with an enhancement in the SN Ia/II products in Carina, similar to the three outer halo stars examined by Ivans et al. (2003).",
"This provides the first direct link between the formation of these stars in low mass galaxies and their presence in the outer Galactic halo We are grateful to the ESO VLT and Magellan Observatory support staff for outstanding help and hospitality during our visitor runs associated with this work.",
"Special thanks to the Stars Group at UVic for our frequent and lively discussions, and help with Magellan/MIKE spectroscopy from Mario Mateo and Andy McWilliam.",
"KAV and MD thank NSERC for the Discovery Grant that funded the majority this work.",
"KAV and CB thank the NSF for early support through award AST 99-84073.",
"Facilities: VLT:Kueyen(FLAMES), Magellan:I(MIKE), ESO:2.2m(WFI)"
]
] | 1204.0787 |
[
[
"On the Averaging Principle"
],
[
"Abstract Typically, models with a heterogeneous property are considerably harder to analyze than the corresponding homogeneous models, in which the heterogeneous property is replaced with its average value.",
"In this study we show that any outcome of a heterogeneous model that satisfies the two properties of differentiability and interchangibility is O(\\epsilon^2) equivalent to the outcome of the corresponding homogeneous model, where \\epsilon is the level of heterogeneity.",
"We then use this averaging principle to obtain new results in queueing theory, game theory (auctions), and social networks (marketing)."
],
[
"The Averaging Principle",
"Let $F(\\mu _1,\\dots , \\mu _k)$ be an outcome of a model with a heterogeneous property, captured by the $k$ parameters $\\lbrace \\mu _1,\\dots , \\mu _k\\rbrace $ , that satisfies the following two properties: Differentiability: $F$ is twice-differentiable at and near the diagonal $\\mu _1 = \\dots = \\mu _k$ .",
"Interchangeability: For every $(\\mu _1, \\dots , \\mu _k) \\in \\mathbb {R}^k$ and every $i \\ne j$ , $F(\\dots , \\mu _i,\\dots ,\\mu _j,\\dots )=F(\\dots , \\mu _j,\\dots ,\\mu _i,\\dots )$ .",
"Thus, the outcome $F$ is independent of the identities/indices of the heterogeneous parameters.For example, in the queueing-system example, switching the identities/locations of two servers does not affect the expected number of customers in the system.",
"Then, we have the following result:This and all other proofs are given in the Appendix.",
"[The Averaging Principle] Let $F$ satisfy the differentiability and interchangeability properties.",
"Let ${\\mbox{$\\mu $}} =(\\mu _1,\\dots , \\mu _k)$ be “sufficiently close to the diagonal”, i.e., $||{\\mbox{$\\mu $}}-\\bar{\\mbox{$\\mu $}}_A||<C_{{\\bar{\\mu }}_A},$ where ${\\bar{\\mbox{$\\mu $}}}_A = (\\underbrace{\\bar{\\mu }_A, \\dots , \\bar{\\mu }_A}_{\\times k})$ , $\\bar{\\mu }_A = \\frac{1}{k}\\sum _{j=1}^k \\mu _j$ is the arithmetic average, $||\\cdot ||$ is a vector norm on $\\mathbb {R}^k$ , and $C_{{\\bar{\\mu }}_A}$ is a positive constant that only depends on $\\bar{\\mu }_A$ (and of course on $F$ ).",
"Then, $F(\\mu _1,\\dots , \\mu _k) = F_{\\mathrm {homog.",
"}}(\\bar{\\mu }_A)+ O(||\\mbox{$\\mu $}-\\bar{\\mbox{$\\mu $}}_A||^2),$ where $ F_{\\mathrm {homog.",
"}}({\\mu }):= F(\\underbrace{{\\mu },\\dots , {\\mu }}_{\\times k}) $ .",
"Theorem remains valid if $\\lbrace \\mu _1, \\dots , \\mu _k\\rbrace $ are functions and not scalars, see the Game Theory (auctions) example below.",
"In Theorem we averaged the $\\lbrace \\mu _i\\rbrace $ s using the arithmetic mean.",
"It is well-known in homogenization theory that in some cases the correct homogenization is provided by the geometric or the harmonic mean.",
"To address the question of the “correct” averaging, we recall the following result: Let $\\mu >0$ , and let $\\lbrace h_1, \\dots ,h_k\\rbrace \\in \\mathbb {R}$ .",
"Then, as $\\epsilon \\rightarrow 0$ , the arithmetic, geometric and harmonic means of $\\lbrace {\\mu }+ \\epsilon h_1, \\dots ,{\\mu }+ \\epsilon h_k\\rbrace $ are $O(\\epsilon ^2)$ asymptotically equivalent.",
"Proof.",
"We can prove this result using the averaging principle.",
"Let $\\bar{\\mu }_{\\mathrm {A}}$ denote the arithmetic mean of $\\lbrace {\\mu }+ \\epsilon h_1, \\dots ,{\\mu }+\\epsilon h_k\\rbrace $ .",
"The geometric mean $\\mu _G({\\mu }+ \\epsilon h_1, \\dots ,{\\mu }+ \\epsilon h_k) = \\left(\\prod _{i=1}^k ({\\mu }+ \\epsilon h_i) \\right)^{1/k}$ satisfies the interchangeability and differentiability properties.",
"Therefore, application of Theorem gives $\\mu _G({\\mu }+ \\epsilon h_1, \\dots ,{\\mu }+ \\epsilon h_k) = \\mu _G(\\bar{\\mu }_{\\mathrm {A}}, \\dots ,\\bar{\\mu }_{\\mathrm {A}}) + O(\\epsilon ^2) = \\bar{\\mu }_{\\mathrm {A}} + O(\\epsilon ^2).$ The proof for the harmonic mean $\\bar{\\mu }_{\\mathrm {H}} =k/(\\frac{1}{\\mu _1}+\\cdots +\\frac{1}{\\mu _k})$ is similar.",
"$\\Box $ From Lemma REF and the differentiability of $F_{\\mathrm {homog.",
"}}(\\mu )$ it follows that $F_{\\mathrm {homog.",
"}}(\\bar{\\mu }_{\\mathrm {A}}) &=& F_{\\mathrm {homog.",
"}}(\\bar{\\mu }_{\\mathrm {G}}) + O(\\epsilon ^2)\\\\ &=& F_{\\mathrm {homog.",
"}}(\\bar{\\mu }_{\\mathrm {H}}) + O(\\epsilon ^2).$ Let $\\mu >0$ .",
"Then, the averaging principle (Theorem ) remains valid if we replace the arithmetic mean with the geometric or harmonic means.",
"A natural question is which of the three averages is “optimal”, in the sense that it minimizes the constant in the $O(||\\mbox{$\\mu $}-\\bar{\\mbox{$\\mu $}}||^2)$ error term.",
"The answer to this question is model specific.",
"It can be pursued by calculating explicitly the $O(\\epsilon ^2)$ term, as we will do later on."
],
[
"Weak Interchangeability",
"To extend the scope of the averaging principle, we define a weaker interchangeability property.See the social-networks application below for an example of a weakly-interchangeable outcome which is not interchangeable.",
"2A.",
"Weak interchangeability: For every $\\mu ,\\tilde{\\mu }$ and every $1 \\le j_0 \\le k$ , if $\\mu _{j_0} = \\tilde{\\mu }$ and $\\mu _j = \\mu $ for all $j \\ne j_0$ , then $F(\\mu _1,\\ldots ,\\mu _k)$ is independent of the value of $j_0$ .",
"Thus, $F(\\mu _1,\\cdots ,\\mu _k)$ is weakly interchangeable if, whenever all but one of the parameters are identical, the outcome $F$ is independent of the identity (coordinate) of the heterogeneous parameter.",
"Every interchangeable function $F$ is also weakly interchangeable, but not vice versa.",
"Nevertheless, the proof of Theorem implies that: The averaging principle (Theorem ) remains valid if we replace the assumption of interchangeability with the assumption of weak interchangeability.",
"Consider a system with $k$ servers.",
"Server $i$ has a random service time that is distributed according to an exponential distribution with rate $\\mu _{i}$ .",
"Customers arrive randomly according to a Poisson distribution with arrival rate $\\lambda $ .",
"An arriving customer is randomly allocated to one of the non-busy servers, if such a server exists.",
"Otherwise, the customer joins a waiting queue, which is unbounded in length.",
"Once a customer is allocated to a server, he gets the service he needs and then leaves the system.",
"This setup is known in the Queuing literature as M/M/k model.For an introduction to queueing theory, see e.g., [6].",
"Examples for such multi-server queuing systems are call centers, queues in banks, parallel computing, and communications in ISDN protocols.",
"Let $F(\\mu _{1}, \\dots ,\\mu _{k})$ denote the expected number of customers in the system (i.e., waiting in the queue or receiving service) in steady state.",
"In the case of two heterogeneous servers, $F(\\mu _{1},\\mu _{2})$ can be explicitly calculated (see Appendix): Consider an M/M/2 queue with heterogeneous servers.",
"The expected number of customers in the system is given by $F(\\mu _{1},\\mu _{2})=\\frac{1}{(1-\\rho )^{2}}\\frac{1}{\\frac{1}{\\rho }\\frac{2\\mu _1 \\mu _2}{(\\mu _1+\\mu _2)^2} +\\frac{1}{1-\\rho }} , \\qquad \\rho := \\frac{\\lambda }{\\mu _1+\\mu _2}.$ Finding an explicit solution for $F(\\mu _{1},\\dots ,\\mu _{k})$ when $k \\ge 3$ is computationally challenging, because it involves solving a system of $2^{k}-1$ linear equations.",
"In the homogeneous case $\\mu _{1}=\\cdots =\\mu _{k}=\\mu $ , however, it is well-known that $F(\\underbrace{\\mu ,\\dots ,\\mu }_{\\times k})=\\frac{\\frac{\\left(\\lambda /\\mu \\right) ^{k}}{k!",
"}\\frac{\\frac{\\lambda }{k\\mu }}{1-\\frac{\\lambda }{k\\mu }}}{\\sum _{n=0}^{k-1}\\frac{\\left( \\lambda /\\mu \\right) ^{n}}{n!",
"}+\\frac{\\left( \\lambda /\\mu \\right) ^{k}}{k!",
"}\\frac{1}{1-\\frac{\\lambda }{k\\mu }}}\\frac{1}{1-\\frac{\\lambda }{k\\mu }}+\\frac{\\lambda }{\\mu }.",
"$ The function $F(\\mu _{1}, \\dots ,\\mu _{k})$ can be written as a sum of solutions of a system of linear equations with coefficients that depend smoothly on $\\mu _{1}, \\dots ,\\mu _{k}$ (see the appendix).",
"Therefore, $F$ is differentiable.",
"Since customers are randomly allocated to the free servers, renaming the servers does not affect the expected number of customers in the system.",
"Hence, $F$ is also interchangeable.",
"Therefore, we can use the averaging principle to obtain an explicit $O(\\epsilon ^{2})$ approximation for $F(\\mu _{1},\\dots ,\\mu _{k})$ : Consider an M/M/k queue with heterogeneous servers whose service rates are $\\lbrace \\mu _{1}, \\dots ,\\mu _{k}\\rbrace $ .",
"The expected number of customers in the system is given by $F(\\mu _{1},\\dots ,\\mu _{k})=F_{\\mathrm {homog.",
"}}(\\bar{\\mu })+O(\\varepsilon ^{2}),$ where $F_{\\mathrm {homog.",
"}}(\\bar{\\mu }):=F(\\underbrace{\\bar{\\mu } ,\\dots ,\\bar{\\mu }}_{\\times k} )$ is given by (REF ), $\\bar{\\mu } := \\frac{1}{k} \\sum _{i=1}^k \\mu _i$ , and $\\epsilon $ is give by (REF ).",
"For example, by Theorem , the expected number of customers with 2 heterogeneous servers is $F(\\mu _1 ,\\mu _2 )=F(\\bar{\\mu } ,\\bar{\\mu })+O\\left(\\varepsilon ^{2}\\right)=\\frac{4\\lambda \\bar{\\mu } }{4\\ \\bar{\\mu }^{2}-\\lambda ^{2}}+O\\left( \\varepsilon ^{2}\\right),$ where $\\bar{\\mu } =\\frac{\\mu _{1}+\\mu _{2}}{2},\\qquad \\varepsilon =\\frac{\\mu _{2}-\\mu _{1}}{2}.$ Indeed, substituting $\\mu _{1,2}=\\bar{\\mu } \\pm \\epsilon $ in the exact expression (REF ) and expanding in $\\varepsilon $ gives (REF ).",
"In the case of $k=8$ heterogeneous servers, even writing the system of $2^{8}-1=255$ equations for the 255 unknowns is a formidable task, not to mention solving it explicitly.",
"By the averaging principle, however, $F(\\mu _{1},\\dots ,\\mu _{8}) = F_{\\mathrm {homog.",
"}}(\\bar{\\mu })+O\\left( \\varepsilon ^{2}\\right),$ where $F_{\\mathrm {homog.",
"}}(\\bar{\\mu }):=F(\\underbrace{\\bar{\\mu },\\dots ,\\bar{\\mu }}_{\\times 8 })$ is given by (REF ) with $k=8$ .",
"We ran stochastic simulations of an M/M/8 queuing system with 8 heterogeneous servers using the ARENA simulation software, and used it to calculate the expected number of customers in the system.",
"The simulation parameters were $\\lambda = \\frac{28}{\\mbox{hour}}, \\qquad \\mu = \\frac{5}{\\mbox{hour}}, \\quad \\mu _{i}=\\mu +\\varepsilon h_{i},\\quad i=1,\\ldots 8,$ $(h_{1},\\ldots ,h_{8}) =(1,1.5,2,3,3.5,-2.5,-4,-4.5) \\frac{1}{\\mbox{hour}},$ and $\\varepsilon $ varies between 0 and 1 in increments of $0.05$ .",
"Because $\\sum _{i=1}^{k}h_{i}=0$ , the average service rate is $\\bar{\\mu } =\\mu = 5$ .",
"Therefore, by Theorem , $F(\\mu +\\varepsilon h_{1}, \\dots , \\mu +\\varepsilon h_{8}) = F_{\\mathrm {homog.",
"}}(5)+O\\left(\\varepsilon ^{2}\\right).$ In addition, by Equation (REF ), $F_{\\mathrm {homog.",
"}}(5)= 6.2314$ .",
"To illustrate the accuracy of this approximation, we plot in Fig.",
"REF the relative error of the averaging-principle approximation $\\frac{F(\\mu +\\varepsilon h_{1}, \\dots , \\mu +\\varepsilon h_{8})-F_{\\mathrm {homog.",
"}}(5)}{F(\\mu +\\varepsilon h_{1}, \\dots , \\mu +\\varepsilon h_{8})}$ .",
"As expected, this error scales as $\\varepsilon ^{2}$ .",
"Note that even when the heterogeneity is not small, the averaging-principle approximation is quite accurate.",
"This is because the coefficient (0.594) of the $O(\\epsilon ^2)$ term is small.This coefficient will be computed analytically later on from eq.",
"(REF ).",
"For example, when $\\epsilon =0.5$ the relative error is $\\approx 2\\%$ , and for $\\epsilon =1$ it is below $10\\%$ .",
"Figure: The relative error of the averaging-principle approximation for the steady-state number of customersin a system with 8 heterogeneous servers, as a function of the heterogeneityparameter ϵ\\epsilon .",
"The solid line iserror=0.594ϵ 2 \\mbox{\\it error} = 0.594 \\epsilon ^2.",
"The crosses denote the relative errorof the improved approximation ().The dotted line is error=0.074ϵ 3 \\mbox{\\it error} = 0.074 \\epsilon ^3.",
"Remark.",
"We can also use the averaging principle to obtain $O(\\epsilon ^2)$ approximations of other quantities of interest that satisfy the interchangeability property, such as the average waiting time in the queue, or the probability that there are exactly $m$ customers in the queue."
],
[
"Game theory application: Asymmetric Auctions",
"Consider a sealed-bid first-price auction with $k$ bidders, in which the bidder who places the highest bid wins the object and pays his bid, and all other bidders pay nothing.See [7] for an introduction to auction theory.",
"A common assumption in auction theory is that of independent private-value auctions, which says that each bidder knows his own valuation for the object, does not know the valuation of the other bidders, but does know the cumulative distribution functions (CDF) of the valuations of the other bidders.",
"Bidders are also characterized by their attitude towards risk: The literature usually assumes that bidders are risk neutral, since this simplifies the analysis.",
"More often than not, however, bidders are risk averse.",
"A strategy of bidder $i$ is a function $b_i(\\cdot )$ that assigns a bid $b_i(v_i)$ to each possible valuation $v_i$ of that bidder.",
"The bid that a bidder places depends on his valuation $v_i$ , and on his beliefs about the distributions of the valuations of the other bidders and about their bidding behavior.",
"An equilibrium in this setup is a vector of $k$ strategies $\\lbrace b_i(\\cdot ) \\rbrace _{i=1}^k$ , such that no single bidder can profit by deviating from his bidding strategy, whatever his valuation might be, so long that all other bidders follow their equilibrium bidding strategies.",
"Most of the auction literature focuses on the symmetric (homogeneous) case, in which the beliefs of any bidder about any other bidder (e.g., about his distribution of valuations, his attitude towards risk, etc.)",
"are the same.",
"In this case, one can look for a symmetric equilibrium, in which all bidders adopt the same strategy.",
"In practice, however, bidders are usually asymmetric (heterogeneous), both in their attitude towards risk and in the distribution of their valuations.",
"Each bidder then faces a different competition.",
"As a result, the equilibrium strategies of the bidders are not the same.",
"The addition of asymmetry usually leads to a huge complication in the analysis.",
"For example, in the case of a first-price auction for a single object with risk-neutral bidders that have private values that are independently distributed in the unit interval $[0,1]$ according to a common function $F(v)$ , the symmetric Nash equilibrium inverse bidding strategy $v(b) = b^{-1}(v)$ satisfies the ordinary differential equation (ODE)Since we consider the case where all bidders use the same strategy, we omit the subscript $i$ from $b$ and $v$ .",
"$v^{\\prime }(b) = \\frac{1}{k-1}\\frac{F(v(b))}{F^{\\prime }(v(b))}\\frac{1}{v(b)-b},\\qquad v(0)=0.$ This equation can be solved explicitly, yielding $ b(v)=v-\\frac{\\int _0^v F^{k-1}(s)\\,ds}{F^{k-1}(v)}.$ Therefore, this case is “completely understood\".",
"From the seller's point of view, a key property of an auction is his expected revenue.",
"In the symmetric case, the expression (REF ) can be used to calculate the seller's expected revenue $R_{\\mathrm {homog.",
"}}[F]$ , yielding $ R_{\\mathrm {homog.",
"}}[F] = 1+(k-1) \\int _0^1 F^k(v) \\, dv - k \\int _0^1 F^{k-1}(v)\\, dv.$ In the asymmetric case, where the value of bidder $i$ is independently distributed in $[0,1]$ according to $F_i(v)$ , the inverse equilibrium strategies $\\lbrace v_{i}(\\cdot ) \\rbrace _{i=1}^k$ are the solutions of the system of ODE's $ v_{i}^{\\prime }(b)=\\frac{F_{i}(v_{i}(b))}{F_{i}^\\prime (v_{i}(b))}\\left[\\left( \\frac{1}{k-1}\\sum \\limits _{j=1}^{k}\\frac{1}{\\left( v_{j}(b)-b\\right) }\\right) -\\frac{1}{\\left( v_{i}(b)-b\\right) }\\right], $ for $ i=1,\\cdots ,k$ , subject to the initial conditions $ v_i(b=0)=0,\\qquad \\quad i=1,\\cdots ,k,$ and the “end condition” at some unknown $\\bar{b}$ $ v_i(\\bar{b})=1,\\quad \\qquad i=1,\\cdots ,k.$ Thus, the addition of asymmetry leads to a huge complication of the mathematical model: instead of a single ODE that can be explicitly integrated, the mathematical model consists of a system of coupled nonlinear ODE's with a non-standard boundary condition.",
"As a result, the system () cannot be explicitly solved, and it is poorly understood, compared with the symmetric case.",
"In [2], Fibich and Gavious considered the system () in the weakly-asymmetric case $F_i = F +\\epsilon H_i$ , $i=1, \\dots , k$ .",
"After several pages of perturbation-analysis calculations, they obtained $O(\\epsilon ^{2})$ asymptotic approximations of the inverse equilibrium strategies $\\lbrace v_{i}(b; \\epsilon ) \\rbrace _{i=1}^k$ .",
"Substituting these approximations in the expression for the seller's expected revenue, showed that it is given by $ &&R[F_1=F + \\epsilon H_1, \\dots ,F_k=F + \\epsilon H_k] = R_{\\mathrm {homog.",
"}}[F] \\\\&& -\\epsilon (k-1)\\int _{0}^{1}(1-F(v))F^{k-2}(v)\\sum \\limits _{i=1}^{k}H_{i}(v)\\,dv+O(\\epsilon ^{2}).",
"\\qquad $ Subsequently, Lebrun [8] proved that the function on the left-hand-side of (REF ) is differentiable in $\\epsilon $ , and used that to show that Eq.",
"(REF ) holds.",
"This is, in fact, a special case of the averaging principle.",
"Indeed, interchangeability holds since changing the indices of the bidders does not affect the revenue, and, as mentioned above, differentiability in $\\epsilon $ was proved in [8].",
"Therefore, by the averaging principle for functions (see the appendix), $R[F_1=F + \\epsilon H_1, \\dots ,F_k=F + \\epsilon H_k] = R_{\\mathrm {homog.",
"}}[\\bar{F}]+O(\\epsilon ^{2})$ where $\\bar{F} = F + \\frac{\\epsilon }{k} \\sum _{i=1}^k H_i$ .",
"Substituting $\\bar{F}$ in (REF ) and expanding in powers of $\\epsilon $ gives $&& R_{\\mathrm {homog.",
"}}[\\bar{F}] = R_{\\mathrm {homog.",
"}}[F]\\\\ && \\quad - \\epsilon (k-1)\\int _{0}^{1}(1-F(v))F^{k-2}(v)\\sum \\limits _{i=1}^{k}H_{i}(v)\\,dv+O(\\epsilon ^{2}).$ Hence, relation (REF ) follows.",
"Numerical calculations ([2] and [3]) show that the error of the averaging-principle approximation (REF ) is small (typically below 1%), even when the asymmetry level is mild (e.g., $\\epsilon =0.4$ ).",
"This provides another illustration that the averaging-principle approximation can be useful even when $\\epsilon $ is not very small.",
"The averaging principle does not only lead to a simpler derivation of relation (REF ), but also enables us to derive a more general novel result: Consider an anonymous auctioni.e., an auction in which the winner and the amount that each bidder pays depend solely on their bids, and not on the identity of the bidders.",
"in which all $k$ bidders have the same attitude towards risk, and all bidders follow the same “rules” when they determine their bidding strategies.For example, bidders may use bounded rationality [11] when determining their bidding strategies.",
"Thus, bidders may restrict themselves to a class of simple strategies, such as low-order polynomial functions of the valuation $v$ .",
"They may even not be aware of the concept of equilibrium.",
"Nevertheless, as long as all bidders have the “same” bounded rationality, the interchangeability requirement holds.",
"Let $F_1,\\cdots ,F_k$ be the cumulative distribution functions of the valuations of the bidders, and let $R[F_1,\\cdots ,F_k]$ be the expected revenue of the seller.",
"If $R$ is twice differentiable at and near the diagonal, then $&& R[F_1,\\cdots ,F_k] = R_{\\mathrm {homog.",
"}}[\\bar{F}] + O(\\epsilon ^2),$ where $R_{\\mathrm {homog.",
"}}[\\bar{F}] = R[\\bar{F},\\cdots ,\\bar{F}]$ , $\\bar{F}$ is the average of $F_1,\\cdots ,F_k$ , and $\\epsilon $ is the level of heterogeneity.",
"Indeed, the assumptions of the theorem imply that $F$ is interchangeable.",
"Therefore, if $F$ is also differentiable, the theorem follows from the averaging principle."
],
[
"Social-networks application: Diffusion of new products",
"Diffusion of new products is a fundamental problem in Marketing, which has been studied in diverse areas such as retail service, industrial technology, agriculture, and educational, pharmaceutical and consumer-durables markets [9].",
"Typically, the diffusion process begins when the product is first introduced into the market, and progresses through a series of adoption events.",
"An individual can adopt the product due to external influences such as mass-media or commercials, and/or due to internal influences by other individuals who have already adopted the product (word of mouth).",
"The internal influences depend on the underlying social-network structure, since adopters can only influence people that they “know”.",
"The social network is usually modeled by an undirected graph, where each vertex is an individual, and two vertices are connected by an edge if they can influence each other.",
"The first quantitative analysis of diffusion of new products was the Bass model [1], which inspired a huge body of theoretical and empirical research.",
"In this model and in many of the subsequent product-diffusion models: A new product is introduced at time $t=0$ .",
"Once a consumer adopts the product, he remains an adopter at all later times.",
"If consumer $j$ has not adopted before time $t$ , the probability that he adopts the product in the time interval $[t,t+s)$ , given that the product was already adopted by $n_j(t)$ people that are connected to $j$ , and that no other consumer adopts the product in the time interval $[t,t+s)$ , is $&& \\hspace{-28.45274pt} \\text{Prob}\\left( j\\text{ adopts in }[t,t+s)~\\Big |~n_j(t),\\begin{array}{c} \\text{ no other consumer} \\\\ \\text{adopts in }[t,t+s)\\end{array}\\right) \\\\&&\\qquad =\\left( p_j+{\\frac{n_j(t)}{m_j}}\\cdot q_j \\right) s+O(s^{2}),$ as $s\\rightarrow 0$ , where $m_j$ is the total number of individuals connected to consumer $j$ and the parameters $p_j$ and $q_j$ describe the likelihood of individual $j$ to adopt the product due to external and internal influences, respectively.",
"We say that a social network is translation invariant, if any individual sees exactly the same network structure.",
"Therefore, in particular, $m_j$ is independent of $j$ .",
"Examples of translation-invariant social networks are (see Fig.",
"REF ): A) A complete graph, in which any two individuals are connected.",
"B) A one-dimensional circle, in which each individual is connected to his two nearest neighbors.",
"C) A one-dimensional circle, in which each individual is connected to his four nearest neighbors.",
"D) A 2-dimensional torus, in which each individual is connected to his four nearest neighbors.",
"Figure: Examples of translation-invariant networks.We say that all individuals are homogeneous when all individuals share the same parameters, i.e., $p_j = p$ and $q_j=q$ for every individual $j$ .",
"Let $N(t)$ denote the number of adopters at time $t$ .",
"The expected aggregate adoption curve $E_{\\rm homog.",
"}[N(t;p,q)]$ in several translation-invariant social networks with homogeneous individuals were analytically calculated in [10], [4].",
"In these studies, the assumption that all individuals are homogeneous was essential for the analysis.",
"One of the fundamentals of marketing theory is that consumers are anything but homogeneous.",
"An explicit calculation of the expected aggregate adoption curve $E[N(t;\\lbrace p_j \\rbrace ,\\lbrace q_j \\rbrace )]$ in the heterogeneous case, however, is much harder than in the homogeneous case.",
"As a result, the effect of heterogeneity is not well understood.",
"The averaging principle allows us to approximate the heterogeneous model with the corresponding homogeneous model.",
"Consider a translation-invariant network.",
"Then, for $t \\ge 0$ the function $F(\\lbrace p_j\\rbrace ,\\lbrace q_j\\rbrace ):= E[N(t; \\lbrace p_j \\rbrace ,\\lbrace q_j \\rbrace )]$ is differentiable and weakly-interchangeable (see Appendix).",
"Therefore, by the averaging principle, The expected aggregate adoption curve in a translation invariant social network with heterogeneous individuals, can be approximated with $E[N(t;\\lbrace p_j \\rbrace ,\\lbrace q_j \\rbrace )] = E_{\\rm homogeneous}[N(t;\\bar{p},\\bar{q})]+O(\\epsilon ^2),$ where $\\bar{p}$ and $\\bar{q}$ are the averages of $\\lbrace p_j\\rbrace $ and $\\lbrace q_j\\rbrace $ , respectively, and $\\epsilon $ is the level of heterogeneity of $\\lbrace p_j \\rbrace $ and $\\lbrace q_j \\rbrace $ .",
"Theorem is consistent with previous numerical findings: In [5], simulations of an agent-based model with a complete graph showed that heterogeneity in $p$ and $q$ had a minor effect on the expected aggregate adoption curve.",
"Simulations of agent-based models with 1D and 2D translation-invariant networks [4] showed that when the values of $\\lbrace p_j\\rbrace $ and $\\lbrace q_j\\rbrace $ are uniformly distributed within $\\pm 20\\%$ of the corresponding values $\\bar{p}$ and $\\bar{q}$ of the homogeneous individuals, the heterogeneous and homogeneous adoption curves are nearly indistinguishable.",
"Even when the heterogeneity level was increased to $\\pm 50\\%$ , the two adoption curves were still very close."
],
[
"Calculating the $O(\\protect \\epsilon ^2)$ term",
"The averaging principle is based on a two-term Taylor expansion of $F$ .",
"Therefore, the error of this approximation is given, to leading order, by the quadratic term in this expansion.",
"When $F$ satisfies the differentiability and interchangeability propertiesHere we cannot assume that $F$ is only weakly interchangeable, since we require that $\\frac{\\partial ^2 F}{\\partial \\mu _i \\partial \\mu _j} =\\frac{\\partial ^2 F}{\\partial \\mu _1 \\partial \\mu _2}$ for all $i,j$ .",
"and $\\mu $ is the arithmetic mean, this error is given by (see the appendix): $F(\\mu _1, \\dots , \\mu _k) - F(\\bar{\\mu }_A, \\dots , \\bar{\\mu }_A) \\sim \\alpha \\sum _{i=1}^k (\\mu _i-\\bar{\\mu }_A)^2,$ where $\\alpha : = \\frac{1}{2}\\left(\\frac{\\partial ^2 F}{\\partial \\mu _1 \\partial \\mu _1}\\bigg |_{\\bar{\\mbox{$\\mu $}}_A}-\\frac{\\partial ^2 F}{\\partial \\mu _1 \\partial \\mu _2}\\bigg |_{\\bar{\\mbox{$\\mu $}}_A} \\right).$ Therefore, The magnitude of this error is $\\sim |\\alpha | \\,||\\mbox{$\\mu $}-\\bar{\\mbox{$\\mu $}}_A||^2 $ .",
"The sign of this error is the same as the sign of $ \\alpha $ .",
"A Taylor expansion in $h$ gives $&& \\hspace{-14.22636pt} F(\\bar{\\mu }_A+2h,\\bar{\\mu }_A,\\underbrace{\\bar{\\mu }_A, \\dots ,\\bar{\\mu }_A}_{\\times k-2 })-F(\\bar{\\mu }_A+h,\\bar{\\mu }_A+h,\\underbrace{\\bar{\\mu }_A \\dots ,\\bar{\\mu }_A}_{\\times k-2 })\\\\ && \\quad \\sim 2\\alpha h^2, \\qquad h \\ll 1.$ This shows that in order to determine the sign of $\\alpha $ , one can compare the effect of adding $h$ units to two parameters with the corresponding effect of adding $2h$ units to a single parameter.",
"The value of $\\alpha $ can be calculated as follows: Assume that $F(\\mu _1, \\dots .",
"\\mu _k)$ satisfies the differentiability and interchangeability properties.",
"Then, $\\alpha = \\frac{k}{2(k-1)} \\left( \\frac{\\partial ^2 F}{\\partial \\mu _1 \\mu _1}\\bigg |_{ \\bar{\\mbox{$\\mu $}}_A}-\\frac{1}{k^2} F_{\\mathrm {homog.",
"}}^{\\prime \\prime }(\\bar{\\mu }_A) \\right).$ Therefore, this calculation only requires the explicit calculations of $F$ in the homogeneous case $\\mu _1 = \\dots = \\mu _k$ , and in the case that the heterogeneity is limited to a single coordinate (i.e., when $\\mu _2 = \\dots = \\mu _k$ ).",
"In many cases, this is a considerably easier task than the explicit calculation of $F$ in the fully-heterogeneous case.",
"To illustrate this, consider again the M/M/k example of Fig.",
"REF with $k=8$ .",
"While the fully-heterogeneous case requires solving $2^{k}-1=255$ equations, the single-coordinate heterogeneous case requires solving only $2\\cdot k =16$ equations.",
"Solving these 16 equations symbolically and using Lemma yields, see the appendix, $\\alpha (k=8)=\\frac{1}{2\\lambda \\bar{\\mu }} \\frac{\\sum _{i=0}^{12} c_i \\left(\\frac{\\bar{\\mu }}{\\lambda }\\right)^i}{\\left( \\sum _{i=0}^{7} b_i \\left(\\frac{\\bar{\\mu }}{\\lambda }\\right)^i\\right)^2 },$ where the values of $\\lbrace c_i, b_i\\rbrace $ are listed in the following table: Table: Values of {c i ,b i }\\lbrace c_i, b_i\\rbrace .In particular, substituting $\\bar{\\mu }= 5$ and $\\lambda = 28$ yields $\\alpha \\approx 0.00837$ .",
"This leads to the improved approximation $\\nonumber F(\\mu _1, \\dots , \\mu _8) &\\approx & F_{\\mathrm {homog.",
"}}(\\bar{\\mu })+ \\alpha \\sum _{i=1}^{8} (\\mu _{i}-\\bar{\\mu })^2\\\\ &\\approx & F_{\\mathrm {homog.",
"}}(5)+ 0.594 \\epsilon ^2.$ The error of this improved approximation scales as $0.074\\epsilon ^3$ , see Fig.",
"REF , which is the next term in the Taylor expansion.",
"In particular, the relative error of (REF ) is below 1.5% for $0 \\le \\epsilon \\le 1$ ."
],
[
"Final remarks",
"The averaging principle is based on a simple observation: the leading-order effects of heterogeneity cancel out when the outcome is interchangeable.",
"Nevertheless, it can lead to a significant simplification of mathematical models in all branches of science.",
"The averaging principle is unrelated to averaging that originates from laws of large numbers in large populations, and it holds, e.g., when there are few servers in a queuing system, or a few bidders in an auction.",
"The interchangeability and the weak interchangeability properties are usually easy to check.",
"The differentiability of $F$ is easy to check in some cases, but can be quite a challenge in others.",
"We note, however, that more often than not, functions that arise in mathematical models are differentiable, unless there is a “very good reason” why they are not.",
"While this is a very informal statement, we make it in order to point out that the “generic” case is that the outcome $F$ is differentiable, rather then the other way around.",
"An important issue is the “level of heterogeneity” that is covered by the averaging principle.",
"Strictly speaking, the level of heterogeneity should be “sufficiently small”.",
"In practice, however, in many cases the averaging principle provides good approximations even when $\\epsilon = 0.5$ .",
"In other words, the coefficient of the $O(\\epsilon ^2)$ term is O(1).",
"While this is also an informal statement, we make it in order to point out that one should not be “surprised” that the averaging principle holds even when $\\epsilon $ is not very small.",
"We thank Uri Yechiali for suggesting the problem of M/M/k queues with heterogeneous servers and the solution of the M/M/2 problem.",
"The research of Solan was partially supported by the ISF grant #212/09 and by the Google Inter-university center for Electronic Markets and Auctions."
],
[
"Proof of Theorem ",
"Because of the differentiability of $F$ , there exists a positive constant $C_{\\bar{\\mu }_A}$ , such that for all $||\\mbox{$\\mu $}- \\bar{\\mbox{$\\mu $}}_A||<C_{\\bar{\\mu }}$ , we can expand $F(\\mbox{$\\mu $})$ as $F(\\mbox{$\\mu $})=F(\\bar{\\mbox{$\\mu $}}_A)+\\sum _{j=1}^{k}(\\mu _{j}-\\bar{\\mu }_A)\\frac{\\partial F}{\\partial \\mu _{j}}\\bigg |_{\\bar{\\mbox{$\\mu $}}_A}+O(||\\mbox{$\\mu $}- \\bar{\\mbox{$\\mu $}}_A||^{2}).$ Because $F$ is interchangeable, $\\frac{\\partial F}{\\partial \\mu _{i}}\\bigg |_{\\bar{\\mbox{$\\mu $}}}=\\frac{\\partial F}{\\partial \\mu _{1}}\\bigg |_{\\bar{\\mbox{$\\mu $}}_A},\\qquad j=1,\\dots ,k.$ Therefore, $F(\\mbox{$\\mu $})=F(\\bar{\\mbox{$\\mu $}}_A)+\\frac{\\partial F}{\\partial \\mu _{1}}\\bigg |_{\\bar{\\mbox{$\\mu $}}_A} \\sum _{j=1}^{k}(\\mu _{j}-\\bar{\\mu }_A)+O(||\\mbox{$\\mu $}- \\bar{\\mbox{$\\mu $}}_A||^{2}).$ Since $\\bar{\\mu }_A$ is the arithmetic average, $\\sum _{j=1}^{n}(\\mu _{j}-\\bar{\\mu }_A)=0$ .",
"Hence, the result follows."
],
[
"Proof of Lemma ",
"We calculate $F(\\mu _{1},\\mu _{2})$ explicitly using the steady-state transition diagram that is shown in Fig.",
"REF .",
"We denote by $p_{i}$ the steady-state probability for the system to be with $i$ customers, and by $p_{1}^{(1,0)}$ and $p_{1}^{(0,1)}$ the steady-state probability for the system to be with 1 customer in server 1 and 2, respectively.",
"In particular, $p_{1}=p_{1}^{(1,0)}+p_{1}^{(0,1)}$ .",
"Since in steady state the amount of inflow is equal to the amount of outflow, the following equalities hold: $&& \\lambda p_{0} = \\mu _{1}p_{1}^{(1,0)}+\\mu _{2}p_{1}^{(0,1)}, \\\\&&\\frac{\\lambda }{2}p_{0}+\\mu _{2}p_{2}=(\\lambda +\\mu _{1})p_{1}^{(1,0)}, \\\\&&\\frac{\\lambda }{2}p_{0}+\\mu _{1}p_{2} =(\\lambda +\\mu _{2})p_{1}^{(0,1)}, \\\\&&\\lambda p_{1}^{(1,0)}+\\lambda p_{1}^{(0,1)}+(\\mu _{1}+\\mu _{2})p_{3}=(\\lambda +\\mu _{1}+\\mu _{2})p_{2}, \\qquad \\qquad \\\\&&\\lambda p_{n}+(\\mu _{1}+\\mu _{2})p_{n+2} \\nonumber \\\\&& \\qquad =(\\lambda +\\mu _{1}+\\mu _{2})p_{n+1},\\quad n=2,3,\\dots $ We can view (REF )–() as a linear system for the three unknowns $\\lbrace p_{0},p_{1}^{(1,0)},p_{1}^{(0,1)}\\rbrace $ .",
"Solving this system for $p_{0}$ yields $p_{0}=\\frac{2\\mu _{1}\\mu _{2}}{\\lambda ^{2}}p_{2}.$ In addition, the solution of ()–() is $p_{n}=\\left( \\frac{\\lambda }{\\mu _{1}+\\mu _{2}}\\right) ^{n-2}p_{2}=\\rho ^{n-2}p_{2}$ for $n\\ge 1$ .",
"Substituting the above in $1=\\sum _{n=0}^{\\infty }p_{n}=p_{0}+\\sum _{n=1}^{\\infty }\\rho ^{n-2}p_{2}=\\left( \\frac{2\\mu _{1}\\mu _{2}}{\\lambda ^{2}}+\\frac{1}{\\rho }\\frac{1}{1-\\rho }\\right) p_{2},$ gives $p_{2}=\\left( \\frac{2\\mu _{1}\\mu _{2}}{\\lambda ^{2}}+\\frac{1}{\\rho }\\frac{1}{1-\\rho }\\right) ^{-1}.$ Therefore, $F(\\mu _{1},\\mu _{2}) &=&\\sum _{n=0}^{\\infty }np_{n}=\\sum _{n=0}^{\\infty }n\\rho ^{n-2}p_{2}=\\frac{p_{2}}{\\rho }\\sum _{n=0}^{\\infty }n\\rho ^{n-1}\\\\&=&\\frac{p_{2}}{\\rho }\\left( \\sum _{n=0}^{\\infty }\\rho ^{n}\\right) ^{\\prime } = \\frac{p_{2}}{\\rho }\\left( \\frac{1}{1-\\rho }\\right) ^{\\prime }=\\frac{p_{2}}{\\rho }\\frac{1}{(1-\\rho )^{2}},$ and the result follows."
],
[
"M/M/3 queue",
"Consider the case of three heterogeneous servers with average service times $\\mu _{1}$ , $\\mu _{2}$ and $\\mu _{3}$ .",
"Denote by $p_{0}$ , $p_{1}^{(1,0,0)}$ , $p_{1}^{(0,1,0)}$ , $p_{1}^{(0,0,1)}$ , $p_{2}^{(1,1,0)}$ , $p_{2}^{(1,0,1)}$ , $p_{2}^{(0,1,1)}$ , $p_{3}$ , $p_{4}$ , ..., the steady-state probabilities.",
"Thus, for example, $p_{2}^{(1,0,1)}$ is the steady-state probability that servers 1 and 3 are busy, server 2 is free, and there are no waiting customers in the queue (we denote by $p_n,~n \\ge 2$ the probability having $n$ customers in the system).",
"The transition diagram for $k=3$ servers is given in Fig.",
"REF .",
"The steady-state equations are $&&\\lambda p_{0}=\\mu _{1}p_{1}^{(1,0,0)}+\\mu _{2}p_{1}^{(0,1,0)}+\\mu _{3}p_{1}^{(0,0,1)}, \\\\&&\\frac{\\lambda }{3}p_{0}+\\mu _{2}p_{2}^{(1,1,0)}+\\mu _{3}p_{2}^{(1,0,1)}=(\\mu _{1}+\\lambda )p_{1}^{(1,0,0)}, \\\\&&\\frac{\\lambda }{3}p_{0}+\\mu _{1}p_{2}^{(1,1,0)}+\\mu _{3}p_{2}^{(0,1,1)}=(\\mu _{2}+\\lambda )p_{1}^{(0,1,0)}, \\\\&&\\frac{\\lambda }{3}p_{0}+\\mu _{1}p_{2}^{(1,0,1)}+\\mu _{2}p_{2}^{(0,1,1)}=(\\mu _{3}+\\lambda )p_{1}^{(0,0,1)}, \\\\&&\\frac{\\lambda }{2}p_{1}^{(1,0,0)}+\\frac{\\lambda }{2}p_{1}^{(0,1,0)}+\\mu _{3}p_{3}=(\\lambda +\\mu _{1}+\\mu _{2})p_{2}^{(1,1,0)}, \\\\&&\\frac{\\lambda }{2}p_{1}^{(1,0,0)}+\\frac{\\lambda }{2}p_{1}^{(0,0,1)}+\\mu _{2}p_{3}=(\\lambda +\\mu _{1}+\\mu _{3})p_{2}^{(1,0,1)}, \\\\&&\\frac{\\lambda }{2}p_{1}^{(0,1,0)}+\\frac{\\lambda }{2}p_{1}^{(0,0,1)}+\\mu _{1}p_{3}=(\\lambda +\\mu _{2}+\\mu _{3})p_{2}^{(0,1,1)}, \\\\&& \\qquad =(\\lambda +\\mu _{1}+\\mu _{2}+\\mu _{3})p_{3}, \\\\&&\\lambda p_{n}+\\left( \\mu _{1}+\\mu _{2}+\\mu _{3}\\right) p_{n+2} \\\\&& \\qquad =(\\lambda +\\mu _{1}+\\mu _{2}+\\mu _{3})p_{n+1},\\qquad n\\ge 3,\\\\&& \\sum _{n=0}^\\infty p_n =1.$ The solution of the last two equations is $p_{n}=\\left( \\frac{\\lambda }{\\mu _{1}+\\mu _{2}+\\mu _{3}}\\right)^{n-3} \\!\\!\\!\\!\\!\\!",
"p_{3}$ for $n\\ge 2$ .",
"The values of $\\lbrace p_{0},p_{1},p_{2}\\rbrace $ as a function of $p_{3}$ can be evaluated explicitly with MAPLE, by solving the first $2^{3}-1=7$ linear equations for $\\lbrace p_{0}$ , $p_{1}^{(1,0,0)}$ , $p_{1}^{(0,1,0)}$ , $p_{1}^{(0,0,1)}$ , $p_{2}^{(1,1,0)}$ , $p_{2}^{(1,0,1)}$ , $p_{2}^{(0,1,1)}\\rbrace $ .",
"The resulting expression for $F(\\mu _{1},\\mu _{2},\\mu _{3})$ , however, is extremely cumbersome and not informative."
],
[
"Proof of Theorem ",
"Since customers are randomly assigned to the available servers, $F(\\mu _{1},\\dots ,\\mu _{k})$ is interchangeable.",
"To see that $F$ is differentiable in $(\\mu _{1},\\dots ,\\mu _{k})$ , we note that $F=\\sum _{n=0}^{\\infty }np_{n}$ where $p_{n}$ is the steady-state probability that there are $n$ customers in the system.",
"In addition, $\\lbrace p_{n}\\rbrace _{n=1}^{k}$ are the solutions of a linear system with coefficients that depend smoothly on $(\\mu _{1},\\dots ,\\mu _{k})$ , and $p_{n}=(\\frac{\\lambda }{\\mu _{1}+\\dots +\\mu _{k}})^{n-k}p_{k}$ for $n\\ge k-1$ .",
"This was shown explicitly for the cases $k=2$ and $k=3$ ; the proof for $k>3$ is similar."
],
[
"Averaging principle for functions (proof of eq. (",
"Let $(F_j({\\bf x})))_{j=1}^k$ be functions in the same function space $\\mathcal {F}$ , and let $\\epsilon \\in {R}$ .",
"Let $R : (F_1, \\dots , F_k) \\mapsto R[F_1, \\dots , F_k] \\in {R}$ be a functional.",
"We say that the functional $R$ is interchangeable, if $R(\\dots , F_i,\\dots ,F_j,\\dots )=R(\\dots , F_j,\\dots ,F_i,\\dots )$ for all $i \\ne j$ .",
"We say that the functional $R$ is differentiable if the scalar function $\\tilde{R}(\\epsilon ):=R[F_1 = F+\\epsilon H_1, \\dots , F_k= F+\\epsilon H_k]$ is twice differentiable at and near $\\epsilon =0$ , for every $F \\in \\mathcal {F}$ and every $(H_j({\\bf x}))_{j=1}^k \\in \\mathcal {F}^k$ .",
"Given functions $(F_j({\\bf x})))_{j=1}^k$ in $\\mathcal {F}$ , denote $\\bar{F} = \\frac{1}{k}\\sum _{j=1}^k F_j$ and $H_j = F_j - \\bar{F}$ .",
"By Taylor expansion, $\\tilde{R}(\\epsilon ) = \\tilde{R}(0) + \\epsilon \\sum _{j=1}^k \\frac{\\delta R}{\\delta F_j} H_j+ O(\\epsilon ^2),$ where $\\frac{\\delta R}{\\delta F_j}$ is the variational derivative.",
"Because $R$ is interchangeable, $\\tilde{R}(\\epsilon ) = \\tilde{R}(0) + \\epsilon \\frac{\\delta R}{\\delta F_1}\\sum _{j=1}^k H_j+ O(\\epsilon ^2).$ In particular, if $F = \\bar{F}$ , then $\\sum _{j=1}^k H_j = 0$ .",
"Hence, $\\tilde{R}(\\epsilon ) = \\tilde{R}(0) + O(\\epsilon ^2),$ which is (REF )."
],
[
"Proof of Theorem ",
"We first prove that $F$ is differentiable.",
"Denote $\\delta _{i,i^{\\prime }}=1$ if individuals $i$ and $i^{\\prime }$ influence each other, and $\\delta _{i,i^{\\prime }}=0$ otherwise.",
"For every $k$ , every set of $k$ consumers $\\lbrace i_{1},i_{2},\\ldots ,i_{k}\\rbrace $ , and every increasing sequence of times $0\\le t_{1}\\le \\cdots \\le t_{k}$ , denote by $P(i_{1},t_{1},i_{2},t_{2},\\ldots ,i_{k},t_{k})$ the probability that consumer $i_{1}$ adopts the product before time $t_{1}$ , consumer $i_{2}$ adopts the product between times $t_{1}$ and $t_{2}$ , etc., and all consumers who are not in $\\lbrace i_{1},\\ldots ,i_{k}\\rbrace $ do not adopt the process by time $t_{k}$ .",
"Then, $P(i_{1},t_{1})=\\big (1-\\exp (-p_{i_{1}}t_{1})\\big )\\prod _{j\\ne i_{1}}\\exp (-p_{j}t_{1}).$ Similarly, $&&P(i_{1},t_{1},i_{2},t_{2},\\ldots ,i_{k},t_{k})= \\\\&&\\quad P(i_{1},t_{1},i_{2},t_{2},\\ldots ,i_{k-1},t_{k-1}) \\\\&&\\times \\left( 1-\\exp \\left( -\\Big (p_{i_{k}}+\\sum _{m=1}^{k-1}\\delta _{i_k,i_m}q_{i_{m}}\\Big )(t_{k}-t_{k-1})\\right) \\right)\\\\&&\\times \\prod _{j\\notin \\lbrace i_1,\\ldots ,i_k\\rbrace }\\exp \\left( -\\left(p_{j}+\\sum _{m=1}^{k-1}\\delta _{j,i_m}q_{i_{m}}\\right) (t_{k}-t_{k-1})\\right) .$ Hence, the function $P(i_{1},t_{1},i_{2},t_{2},\\ldots ,i_{k},t_{k})$ is differentiable in $\\lbrace p_{i},q_{i}\\rbrace $ .",
"Finally, $& &E[N(t;\\lbrace p_{j}\\rbrace ,\\lbrace q_{j}\\rbrace )] = \\frac{1}{M}\\sum _{\\pi }\\sum _{k=1}^{M}\\frac{k}{(M-k!}",
"\\hfil \\mbox{}\\\\&&\\hspace{-14.22636pt} \\times \\int _{t_{1}=0}^{t}\\int _{t_{2}=t_{1}}^{t} \\!\\!\\!\\!\\!\\!\\!\\!",
"\\cdots \\int _{t_{k-1}=t_{k-2}}^{t} \\hspace{-28.45274pt} P(i_{1},t_{1},\\ldots ,i_{k-1},t_{k-1},i_{k},t)\\,dt_{k-1}\\ldots dt_{1}, \\qquad \\mbox{}$ where $\\pi $ ranges over all permutations on the set of $M$ individuals.",
"Therefore, the differentiability of $E[N(t;\\lbrace p_{j}\\rbrace ,\\lbrace q_{j}\\rbrace )]$ follows.",
"Because the network is translation invariant, $F$ is weakly-interchangeable in $\\lbrace p_j \\rbrace $ and in $\\lbrace q_j\\rbrace $ .",
"By this we mean that If $p_m = \\tilde{p}$ , $p_j = p$ for all $j \\ne m$ , and $q_j = q$ for all $j$ , then $F$ is independent of the value of $m$ .",
"If $q_n = \\tilde{q}$ , $q_j = q$ for all $j \\ne n$ , and $p_j = p$ for all $j$ , then $F$ is independent of the value of $n$ .",
"Therefore, the result follows from a slight modification of the proof of Theorem ."
],
[
"Proof of equation (",
"Since $F$ is interchangeable, the quadratic term in the Taylor expansion of $F(\\mu _1, \\dots .",
"\\mu _k)$ around the arithmetic mean is equal to $&& \\sum _{i,j=1}^k (\\mu _i-\\bar{\\mu }_A) (\\mu _j-\\bar{\\mu }_A) \\frac{\\partial ^2 F}{\\partial \\mu _i \\partial \\mu _j}\\bigg |_{\\bar{\\mbox{$\\mu $}}_A} = \\\\\\nonumber && \\frac{\\partial ^2 F}{\\partial \\mu _1 \\partial \\mu _2}\\bigg |_{\\bar{\\mbox{$\\mu $}}}\\sum _{i,j=1, i\\ne j}^k (\\mu _i-\\bar{\\mu }_A) (\\mu _j-\\bar{\\mu }_A)+ \\frac{\\partial ^2 F}{\\partial \\mu _1 \\partial \\mu _1}\\bigg |_{\\bar{\\mbox{$\\mu $}}}\\sum _{i=1}^k (\\mu _i-\\bar{\\mu }_A)^2.$ Since $\\bar{\\mu }_A$ is the arithmetic mean, $\\sum _{i,j=1}^k (\\mu _i-\\bar{\\mu }_A) (\\mu _j-\\bar{\\mu }_A) =\\sum _{i=1}^k (\\mu _i-\\bar{\\mu }_A) \\sum _{j=1}^k(\\mu _j-\\bar{\\mu }_A)= 0.$ Therefore, the result follows."
],
[
"Proof of Lemma ",
"Consider the case where $\\mu _i = \\bar{\\mu }+ h$ for $i=1, \\dots , k$ .",
"By equation (), $&&\\frac{1}{2} \\sum _{i,j=1}^k (\\mu _i-\\bar{\\mu }) (\\mu _j-\\bar{\\mu }) \\frac{\\partial ^2 F}{\\partial \\mu _i \\partial \\mu _j}\\bigg |_{\\bar{\\mbox{$\\mu $}}} \\\\\\nonumber && \\quad =\\frac{1}{2} \\frac{\\partial ^2 F}{\\partial \\mu _1 \\partial \\mu _2}\\bigg |_{\\bar{\\mbox{$\\mu $}}} k(k-1)h^2+ \\frac{1}{2}\\frac{\\partial ^2 F}{\\partial \\mu _1 \\partial \\mu _1}\\bigg |_{\\bar{\\mbox{$\\mu $}}} k h^2 .$ On the other hand, since $F(\\bar{\\mu }+ h, \\dots ,\\bar{\\mu }+ h)= F_{\\mathrm {homog.",
"}}(\\bar{\\mu }+ h),$ we have $\\frac{1}{2} \\sum _{i,j=1}^k (\\mu _i-\\bar{\\mu }) (\\mu _j-\\bar{\\mu }) \\frac{\\partial ^2 F}{\\partial \\mu _i \\partial \\mu _j}\\bigg |_{\\bar{\\mbox{$\\mu $}}} = \\frac{h^2}{2} F_{\\mathrm {homog.",
"}}^{\\prime \\prime }(\\bar{\\mu }).$ Therefore, $\\frac{1}{2} \\frac{\\partial ^2 F}{\\partial \\mu _1 \\partial \\mu _2}\\bigg |_{\\bar{\\mbox{$\\mu $}}} k(k-1)h^2+ \\frac{1}{2}\\frac{\\partial ^2 F}{\\partial \\mu _1 \\partial \\mu _1}\\bigg |_{\\bar{\\mbox{$\\mu $}}} k h^2 = \\frac{h^2}{2} F_{\\mathrm {homog.",
"}}^{\\prime \\prime }(\\bar{\\mu }).$ Hence, $\\frac{\\partial ^2 F}{\\partial \\mu _1 \\partial \\mu _2}\\bigg |_{ \\bar{\\mbox{$\\mu $}}} = \\frac{1}{k-1} \\left( \\frac{1}{k} F_{\\mathrm {homog.",
"}}^{\\prime \\prime }(\\bar{\\mu }) - \\frac{\\partial ^2 F}{\\partial \\mu _1 \\partial \\mu _1}\\bigg |_{ \\bar{\\mbox{$\\mu $}}} \\right).$"
],
[
"Calculation of $\\alpha $",
"We illustrate the computation of the coefficient $\\alpha $ for a queue with 8 servers.",
"Consider then the case of a single server with service time $\\mu _1$ , and seven servers with service time $\\mu $ .",
"Denote by $p_{0,n}$ and $p_{1,n}$ , $n=1, \\dots , 6$ , the steady-state probabilities that $n$ out of the homogeneous servers are busy and that the single heterogeneous servers is free or busy, respectively.",
"The equations for the $2\\cdot 8-1 = 15$ variables $\\lbrace p_0, p_{0,1}, p_{1,0}, \\dots ,p_{1,6}, p_{0,7}\\rbrace $ are $&& \\lambda p_{0,0} = \\mu p_{0,1}+\\mu _{1}p_{1,0}\\text{\\ ,} \\\\&& -p_{0,0}\\frac{\\lambda }{8}+p_{1,0}(\\lambda +\\mu _{1})-p_{1,1}\\mu = 0, \\\\&& p_{0,n}\\left( \\lambda +n\\mu \\right) = p_{0,n-1}\\frac{8-n}{9-n}\\lambda +p_{1,n}\\mu _{1}+p_{0,n+1}(n+1)\\mu ,\\\\&& \\quad n=1,\\ldots ,6, \\\\&& p_{1,n}\\left( \\mu _{1}+\\lambda +n\\mu \\right) = p_{1,n-1}\\lambda +p_{1,n+1}(n+1)\\mu +p_{0,n}\\frac{\\lambda }{8-n},\\\\ && \\quad n=1,\\ldots ,5, \\\\&& p_{0,7}(\\lambda +7)\\mu )=p_{0,6}\\frac{\\lambda }{2}+p_{7}\\mu _{1}\\rho ,$ where $\\rho =\\frac{\\lambda }{7\\mu +\\mu _{1}}$ , $p_{n}=\\rho ^{n-7}p_{7}$ for $n \\ge 8$ , and $\\sum _{n=0}^{\\infty }p_{n} = 1$ .",
"These equations can be solved with Maple The Maple code is available at www.bgu.ac.il/~ariehg/averagingprinciple.html., and the solution can be used to calculate $F(\\mu _1,\\underbrace{\\mu , \\dots ,\\mu }_{\\times 7})$ explicitly.",
"Differentiating this expression twice with respect to $\\mu _1$ , differentiating $F_{\\mathrm {homog.",
"}}$ , see eq.",
"(REF ), twice with respect to $\\mu $ , and using Lemma , yields equation (REF ).",
"Substituting $\\bar{\\mu }=5$ and $\\lambda = 28$ gives $\\alpha \\approx 0.00837$ .",
"In addition, $\\sum _{i=1}^{8} (\\mu _{i}-\\bar{\\mu })^2 = \\epsilon ^2 \\sum _{i=1}^{8} h_{i}^2 = 71 \\epsilon ^2 .$ Therefore, $\\alpha \\sum _{i=1}^{8} (\\mu _{i}-\\bar{\\mu })^2 \\approx 0.594 \\epsilon ^2.$"
]
] | 1204.0813 |
[
[
"A Renormalization Program for Systems with Non-Perturbative Conditions"
],
[
"Abstract In this paper we introduce an alternative renormalization program for systems with non-perturbative conditions.",
"The non-perturbative conditions that we concentrate on in this paper are confined to be either the presence of non-trivial boundary conditions or non-perturbative background fields.",
"We show that these non-perturbative conditions have profound effects on all physical properties of the system and our renormalization program is consistent with these conditions.",
"We formulate the general renormalization program in the configuration space.",
"The differences between the free space renormalization program and ours manifest themselves in the counter-terms as well, which we shall elucidate.",
"The general expressions that we obtain for the counter-terms reduce to the standard results in the free space cases.",
"We show that the differences between these divergent counter-terms are extremely small.",
"Moreover we argue that the position dependences induced on the parameters of the renormalized Lagrangian via the loop corrections, however small, are direct and natural consequences of the non-perturbative position dependent conditions imposed on the system."
],
[
"Introduction",
"The presently successful theory of particle physics is the Standard Model which is a local Quantum Field Theory (QFT) including electromagnetic, weak, and strong interactions.",
"Most problems of physical interest are not exactly solvable and when the problem is amenable to perturbation theory and describable by a renormalizable local QFT, the calculations are usually done perturbatively.",
"The renormalization program for these theories are defined to remove the divergences order by order in perturbation theory, by redefining the physical parameters of the system.",
"Most of the renormalization programs presented up to now, as far as we know, have been for systems which are not subject to any non-perturbative conditions, such as the presence of non-trivial boundary conditions or non-perturbative background fields.",
"The appropriate renormalization program for these systems is the usual free space renormalization program.",
"The renormalization program started in Quantum Electrodynamics (QED).",
"The initial successes of QED were based upon the renormalization program initiated by Tomonaga [1], [2], [3], [4], Schwinger [5], [6], [7], [8] and Feynman [9], [10], [11] and developed to a general form by Dyson [12].",
"The amazing success of this free-space renormalization program to predict the radiative corrections for various quantities in QED, such as the electron $g$ -factor, has left little doubt on the validity of this program.",
"Analogous free space renormalization programs have been an essential and integral part of the development of the Standard Model.",
"It is extremely important to note that all of these QFT calculations including their renormalization programs are perturbative calculations based on the free space.",
"Before discussing our renormalization program we like to emphasize that the case of ($1+1$ ) dimensions is very special in the sense that the renormalization program is equivalent to normal ordering [13].",
"For a comprehensive review of free space renormalization program and its history see for example [14].",
"The main issue that we discuss in this paper is that the presence of non-perturbative conditions in the system, such as non-trivial boundary conditions, non-perturbative background fields such as solitons, and non-trivial background metrics have profound effects on all of the physical aspects of the system which cannot be taken into account perturbatively.",
"The boundary conditions that we consider include the phenomenological ones, such as the the ones appropriate for the bag model of nucleons which effectively provide the confinement mechanism of the low energy QCD [15], [16], [17], [18], [19].",
"The appropriate renormalization program should be self-contained, and automatically take into account the aforementioned conditions in a self-consistent manner.",
"Moreover, the renormalization program should not only be consistent with these non-perturbative conditions, but also should emerge naturally from the standard procedures.",
"The presence of these non-perturbative conditions breaks the translational symmetry of the system, which obviously could have many profound manifestations.",
"In particular the linear momentum will no longer be a conserved quantity.",
"For these systems, the use of free space renormalization program in which the momentum appears to all orders in perturbation theory, is certainly not appropriate in momentum space and might not be appropriate in general.",
"We should mention that recently much work has been done on the renormalization program for the system within the context of non-commutative geometry, and these fall with the category of non-trivial renormalization program [20], [21], [22].",
"Most importantly, the information about the non-perturbative conditions is carried by the full set of the $n$ -point functions.",
"We expect and shall show that the breaking of the translational symmetry could force all of the $n$ -point functions of the theory to have, in general, non-trivial position dependence in the coordinate representation.",
"This occurs with certainty for the case of non-trivial boundary conditions.",
"For the case of non-trivial background fields this occurs only in non-perturbative cases where the Green's functions are altered.",
"In this paper we concentrate on these two cases.",
"The case of non-trivial background metrics in finite volume has already been investigated [23], [24], [25], and position dependent counter-terms have been obtained.",
"The procedure to deduce the counter-terms from the $n$ -point functions in a renormalizable perturbation theory is standard and has been available for over half a century.",
"In our renormalization program, this will lead to a set of uniquely defined position dependent counter-terms.",
"However, as we shall explicitly show, the differences between the two divergent counter-terms, i.e.",
"the position dependent and the free ones, are generically not only finite but also extremely small.",
"In the absence of any non-perturbative conditions, the momentum space and configuration space renormalization programs are equivalent and can be used interchangeably [26].",
"In this paper we set up a general formalism for the configuration space renormalization programs which can be used even when the non-perturbative conditions are present.",
"In the process of removing the divergences, the counter-terms are determined self-consistently and unambiguously by the standard procedure itself.",
"These counter-terms turn out to be position dependent, as a direct consequence of position dependence of the $n$ -point functions.",
"In section we set up a renormalization program for the problems with non-perturbative conditions, and present four sets of reasonings in favor of our renormalization program.",
"In section we summarize our results.",
"In the appendix we illustrate our results our results using a very simple model."
],
[
"Renormalization program for problems with Non-perturbative conditions",
"Our starting point is the standard expression for the $n$ -point functions $\\langle \\Omega |T\\lbrace \\phi (x_1)\\ldots \\phi (x_n)\\rbrace |\\Omega \\rangle =\\lim \\limits _{T\\rightarrow \\infty (1-i\\epsilon )}\\frac{\\langle 0|T\\lbrace \\phi _{I}(x_1)\\ldots \\phi _{I}(x_n)~\\textrm {e}^{-i\\int ^{T}_{-T}\\textrm {d}^4x~\\mathcal {H}_{I}}\\rbrace |0\\rangle }{\\langle 0|T\\lbrace \\textrm {e}^{-i\\int ^{T}_{-T}\\textrm {d}^4x~\\mathcal {H}_{I}}\\rbrace |0\\rangle }.$ Since most problems of physical interest are not exactly solvable, when applicable, one usually resorts to perturbation theory by expanding the exponential, and uses the interaction picture for convenience.",
"This expansion in principle contains infinite number of terms all of which are propagators defined by $G(x,y)=\\langle \\Omega |T\\lbrace \\phi (x)\\phi (y)\\rbrace |\\Omega \\rangle ={}{\\phi }{(x_1)}{\\phi }\\phi {(x)}\\phi {(y)}.$ The modes appearing in the expansion of $\\phi (x)$ must be chosen to be the eigen-modes of the system.",
"These modes and the resulting Green's functions must satisfy the following differential equations, in addition to the boundary conditions imposed on the system, $D_x\\phi (x)=\\omega \\phi (x),\\quad D_xG(x,y)=-i\\delta (x-y),$ where the differential operator $D_x$ directly emerges from the Euler-Lagrange equation for the system.",
"In order to show the procedure, we need to be more concrete and therefore use the $\\lambda \\phi ^4$ theory as a generic example.",
"The Lagrangian density for a real scalar field with $\\lambda \\phi ^4$ self-interaction suitable for the trivial case or the case with non-trivial (NT) boundary condition is $\\mathcal {L}=\\frac{1}{2} [\\partial _\\mu \\phi (x)]^2-\\frac{\\lambda _{0}}{4!",
"}\\big [\\phi (x)^2+\\frac{6m_{0}^2}{\\lambda _{0}}\\big ]^2,$ where $m_0$ and $\\lambda _0$ are the bare mass and bare coupling constant, respectively.",
"For the case with a solitonic background one should replace $m_0^2\\rightarrow -m_0^2$ .",
"After rescaling the field by $\\phi =Z^\\frac{1}{2}\\phi _r$ and utilizing the standard procedure for setting up the renormalized perturbation theory, the Lagrangian becomes $\\mathcal {L}=\\frac{1}{2} [\\partial _\\mu \\phi _r]^2-\\frac{1}{2} m^2\\phi _r^2-\\frac{\\lambda }{4!",
"}\\phi _r^4-\\frac{3}{2}\\frac{m_0^4}{\\lambda _0}+\\frac{1}{2}\\delta _z [\\partial _\\mu \\phi _r]^2-\\frac{1}{2}\\delta _m\\phi _r^2-\\frac{\\delta _\\lambda }{4!",
"}\\phi _r^4,$ where $\\delta _m$ , $\\delta _\\lambda $ and $\\delta _z$ are the counter-terms, and $m$ and $\\lambda $ are the physical mass and physical coupling constant, respectively.",
"The relationship between the bare and physical quantities are $\\delta _z=Z-1,~~~\\delta _\\lambda =\\lambda _0 Z^2-\\lambda \\qquad \\textrm {and}\\qquad \\delta _m=m_0^2 Z-m^2.$ Notice that we can draw an important condition from Eq.",
"(REF ) that any position dependence induced in the counter-terms in the renormalization program, will necessarily induce an opposite position dependence in the parameters of the theory.",
"For example for the mass we conclude $m^{2}_{}=\\frac{\\delta _m^{\\textrm {NT}}+m^{2}_{\\textrm {NT}}}{Z_{\\textrm {NT}}}=\\frac{\\delta _m^{\\textrm {free}}+m^{2}_{\\textrm {free}}}{Z_{\\textrm {free}}}$ since the free case has no position dependence, neither should the combination of non-trivial quantities shown.",
"It is extremely important at this point to distinguish between three separate cases.",
"First, the $\\lambda \\phi ^4$ theory in the topologically trivial sector in free space, which we shall refer to as the “free” case.",
"Second, the $\\lambda \\phi ^4$ theory in the topologically trivial sector with non-trivial boundary conditions imposed, which we shall refer to as the “non-trivial boundary” case.",
"Third , the $\\lambda \\phi ^4$ theory in the topologically non-trivial sector, which we shall refer to as the “soliton” case.",
"Most of the material presented from this point on is common between these three cases.",
"However, there are delicate differences which we shall highlight at appropriate points.",
"In the context of renormalized perturbation theory, as indicated in Eq.",
"(REF ), we can symbolically represent the first few terms of the perturbation expansion of Eq.",
"(REF ).",
"The results for the two-point and four-point functions up to 2-loops in $\\lambda \\phi ^4$ theory are shown in Eq.",
"(REF ) and Eq. ().",
"In Eq.",
"() we have only shown the diagrams of the $s$ -channel for simplicity.",
"In these diagrams the counter-terms should eliminate the corresponding divergences.",
"The first argument in favor of our program is as follows.",
"$\\raisebox {-6mm}{\\includegraphics [width=12.7cm]{2p}}.\\\\\\raisebox {-6mm}{\\includegraphics [width=12.7cm]{4p}}.$ The propagators appearing in these diagrammatic expansions are the results of the contractions of the fields in Eq.",
"(REF ) after the perturbation expansion of the exponential.",
"Therefore, all of the propagators appearing in these perturbation expansions should be of the same kind, i.e.",
"the free ones or the non-trivial ones.",
"The form of the propagator appropriate for each case is dictated by the physical conditions which determine the problem.",
"The propagators appropriate for the problem directly appear in the renormalization procedure and thus both are fixed by the nature of the problem.",
"As we shall show, use of the free space renormalization program for the non-trivial cases requires using free propagators for all of the internal lines, and this might lead to inconsistencies.",
"We should note that the diagrammatic expansion illustrated in Eq.",
"(REF ) and Eq.",
"() could be representations of either one of the three cases mentioned above.",
"The difference lies only in the propagators.",
"For implementing the renormalization program for the $n$ -point functions, one must first study the Green's functions in the free and non-trivial systems.",
"To calculate the counter-terms we need the divergent parts of the $n$ -point functions.",
"In the free $\\lambda \\phi ^4$ case, obviously the position and momentum space renormalization programs are equivalent.",
"In this case these divergences in the momentum space renormalization program are due to integration over large momenta in the loops of the Feynman diagrams.",
"The integration over large momenta corresponds to integration over infinitesimal distances in the coordinate space, i.e.",
"when any of two internal points are close to each other.",
"Our second argument in favor of our program has two parts: first we show that a necessary requirement for self-consistency of the renormalization program is that all of the internal propagators have to be of the same form.",
"Second we show that in the process of connecting the internal and external propagators through the internal points immediately adjacent to the external ones, the requirement of consistency mandates that all of the propagators have to be of the same form.",
"Since the external propagators by definition have to be consistent with the non-perturbative conditions on the system, we conclude that all of the propagators and the resulting counter-terms have to be of such form.",
"we start by studying the two-point function.",
"Obviously the Green's function in the non-perturbative cases will have non-trivial position dependence in the coordinate representation.",
"These position dependences do not disappear when the end-points approach each other.",
"We start by considering the one-loop correction to the two-point function.",
"Consider the first, second and the last diagram on the r.h.s.",
"of Eq.",
"(REF ).",
"Our renormalization condition is identical to the usual ones, which states that the exact propagator close to its pole should be equal to the propagator represented by the first term.",
"This implies that the second term and the counter-term should cancel each other.",
"That is $\\int \\textrm {d}^4 x~G(x_1,x)\\left\\lbrace \\frac{\\lambda }{2}G(x,x)+\\delta ^{(1)}_m\\right\\rbrace G(x,x_2)=0.$ Since $x_1$ and $x_2$ are arbitrary, from this integral expression one concludes that the expression for the counter-term to one-loop is [28], $\\delta ^{(1)}_m(x)=\\frac{-i}{2}\\raisebox {-2.7mm}{\\includegraphics [width=1.3cm]{MC}}=\\frac{-\\lambda }{2}G(x,x).$ This is the general desired result for the trivial or non-trivial cases, i.e.",
"$G(x,y)$ could be the propagator of the real scalar field for any of those cases.",
"From this expression we conclude that the counter-terms in the non-trivial cases have non-trivial position dependences.",
"In the trivial case our expression for $\\delta _{m}^{(1)}$ reduces to the standard result, $\\delta _{m,\\textrm {free}}^{(1)}=-\\frac{\\lambda }{2}\\lim _{y\\rightarrow x}\\int \\frac{\\textrm {d}^{d}k}{(2\\pi )^{d}}\\frac{ie^{i(x-y)}}{k^2-m^2+i\\epsilon }=-\\frac{\\lambda }{2}\\int \\frac{\\textrm {d}^{d}k}{(2\\pi )^{d}}\\frac{i}{k^2-m^2+i\\epsilon }=-\\frac{\\lambda }{2(4\\pi )^{\\frac{d}{2}}}\\frac{\\Gamma (1-\\frac{d}{2})}{(m^2)^{1-\\frac{d}{2}}}.$ Next we consider the one loop contribution to the four-point function and the calculation of the first order correction to the vertex counter-term.",
"We show the calculations only for the $s$ -channel, since the $t$ and $u$ channels are calculated similarly.",
"For this purpose consider the first, second and the last diagrams on the r.h.s.",
"of Eq. ().",
"As usual the renormalization condition is that the divergent part of the second term and the counter-term should cancel each other.",
"That is, $\\lim _{z\\rightarrow y}\\int \\textrm {d}^{4}y\\int \\textrm {d}^{4}z~G(x_1,y)G(x_2,y)\\left\\lbrace \\frac{(-i\\lambda )^2}{2}G^2(y,z)-i\\delta _{\\lambda }^{(2)s}(y)\\delta ^{4}(y-z)\\right\\rbrace G(z,x_3)G(z,x_4)=0,$ where the limit $z \\rightarrow y$ has been implemented for the last two propagators and the superscript $s$ stands for the $s$ -channel.",
"Since $x_1,\\dots ,x_4$ are arbitrary, one concludes from this integral expression that the expression for the counter-term to one-loop should be $\\delta ^{(2)}_{\\lambda }(y)=\\lim _{z\\rightarrow y}\\frac{-i}{2}\\left(~~\\raisebox {-5.0mm}{\\includegraphics [width=4.5cm]{VC}}\\right)=\\lim _{z\\rightarrow y}\\frac{3i\\lambda ^2}{2}\\int \\textrm {d}^{4}z~G^2(y,z),$ which in the trivial case reduces to the standard result expression, $\\delta _{\\lambda ,\\textrm {free}}^{(2)}=\\frac{3i\\lambda ^2}{2}\\lim _{z\\rightarrow y}\\int \\textrm {d}^{d}z\\int \\frac{\\textrm {d}^{d}k_1}{(2\\pi )^{d}}\\frac{\\textrm {d}^{d}k_2}{(2\\pi )^{d}}\\frac{ie^{ik_1.",
"(y-z)}}{k_{1}^{2}-m^2+i\\epsilon }~\\frac{ie^{ik_2.",
"(y-z)}}{k_{2}^{2}-m^2+i\\epsilon }=\\frac{3\\lambda ^2}{2(4\\pi )^{\\frac{d}{2}}}\\frac{\\Gamma (2-\\frac{d}{2})}{(m^2)^{2-\\frac{d}{2}}}.$ To prove the first piece of our second argument we need to concentrate on higher order corrections.",
"First consider the second order correction to the two-point function.",
"It is easy to show that the divergent parts of the third and fifth diagram on the r.h.s.",
"of Eq.",
"(REF ) cancel each other exactly, provided that the counter-term in the fifth diagram is chosen in accordance with Eq.",
"(REF ), and the propagator in upper loop of the third diagram is of the same form as in the first order diagram shown in Eq.",
"(REF ).",
"This can be easily seen by combining the divergent parts of those two diagrams as follows, $\\frac{(-i\\lambda )}{2}\\int \\textrm {d}^4 x~\\textrm {d}^4 y~G(x_1,x)~G^{2}(x,y)\\left\\lbrace \\frac{(-i\\lambda )}{2}~\\textrm {pole}\\lbrace G(y,y)\\rbrace -i\\delta _m (y)\\right\\rbrace G(x,x_2)=0.$ The fourth and sixth diagram in Eq.",
"(REF ) both have divergences of order $1/\\epsilon $ and $1/\\epsilon ^2$ where $\\epsilon =4-d$ .",
"The $1/\\epsilon $ terms of these two diagrams should cancel each other, in order for the first order result Eq.",
"(REF ) to hold true.",
"For this to happen, the internal propagators in the loop diagram in Eq.",
"(REF ), which determine the first order vertex counter-term, should be of the same form as the propagators appearing in the fourth diagram of the two-point function.",
"Then the requirement of the cancellation of $1/\\epsilon ^2$ terms automatically determines the second order part of the mass counter-term, which we shall derive and present below.",
"Analogous arguments can be presented for the four-point function given in Eq. ().",
"Consider the third, sixth and seventh diagrams all in the $s$ -channel.",
"The $1/\\epsilon $ divergences of these diagrams should cancel each other, in order for the $O(\\lambda ^2)$ divergence in the vertex counter-terms given by Eq.",
"(REF ) to hold true.",
"Therefore, the internal propagators shown in that equation should be of the same form as those of the second diagram on the r.h.s.",
"in Eq. ().",
"Moreover, the divergences resulting from the collapse of the lower loop in the fourth and the upper loop in the fifth diagram should be canceled by the sixth and seventh diagrams (when the $t+u$ parts of the vertex counter-terms are used), respectively.",
"This would happen if and only if the $O(\\lambda ^2)$ divergence of these counter-terms are in accord with Eq.",
"(REF ).",
"This in turn implies that the mentioned propagators in the loops in the fourth and fifth diagrams should be of the same form as those displayed in Eq.",
"(REF ).",
"In that case the upper two propagators in the fourth diagram and lower two of the fifth diagram should also be of the same form as the propagators shown in the sixth and seventh diagrams in the corresponding channels.",
"Obviously if the internal propagators of the fourth and fifth diagram are not of the same kind one would encounter inconsistencies.",
"Next we can consider diagrams of order $\\lambda ^3$ and above.",
"In particular, consider diagrams with either an extra closed propagator or a mass counter-term inserted on any of its internal propagators.",
"Obviously the divergent parts of these terms should cancel each other, and this occurs only if the mass counter-term is chosen according to Eq.",
"(REF ).",
"The above arguments clearly show that all of internal propagators should be of the same kind.",
"This proves the first part of our second argument.",
"Now we are in a good position to determine the second order mass and field strength counter-terms (to first order $\\delta _z=0$ ).",
"Now consider the fourth, sixth and seventh diagrams on the r.h.s.",
"of Eq.",
"( REF ).",
"Using Eq.",
"( REF ), we obtain $&\\int \\textrm {d}^4 x~G(x_1,x)\\nonumber \\\\&\\times \\lim _{y\\rightarrow x}\\bigg \\lbrace \\lambda ^2\\int \\textrm {d}^4 y\\bigg [\\bigg (\\frac{3}{4}G(x,x)G^2(x,y)-\\frac{1}{6}G^3(x,y)\\bigg )G(x,x_2)-\\frac{1}{12} (y-x)^\\mu (y-x)^\\nu G^{3}(x,y){\\frac{\\partial ^2 G(y,x_2)}{\\partial y^\\mu \\partial y^\\nu }}\\mid _{y=x}\\bigg ]\\nonumber \\\\&+\\bigg ({\\overleftarrow{\\frac{\\partial }{\\partial x^\\mu }}} i{\\delta }^{(2)}_z(x){\\overrightarrow{\\frac{\\partial }{\\partial x_\\mu }}}-i\\delta ^{(2)}_m(x)\\bigg )G(x,x_2)\\bigg \\rbrace =0.$ From this integral expression we find the second order mass counter-term, $\\delta ^{(2)}_m(x)=\\lim _{y\\rightarrow x}\\frac{i\\lambda ^2}{6}\\int \\textrm {d}^4 y\\left\\lbrace G^{3}(x,y)-\\frac{9}{2}G(x,x)G^2(x,y)\\right\\rbrace .$ However, for the field strength counter-term the integral of the internal and external propagators are entangled in such a way that it is in general difficult to extract and isolate the infinite part of the integral which would yield the $\\delta ^{(2)}_z(x)$ counter-term.",
"We can simplify the expression involving $\\delta ^{(2)}_z(x)$ by using the coordinate transformations $x=x_0-\\frac{u}{2}$ and $y=x_0+\\frac{u}{2}$ to symmetrize the second order derivative term in Eq.",
"(REF ) to obtain, $\\int \\textrm {d}^4 x_0 ~\\frac{\\partial G(x_1,x_0)}{\\partial {x_0}^\\mu }\\bigg \\lbrace i\\delta ^{(2)}_z(x_0)\\frac{\\partial G(x_0,x_2)}{\\partial {x_0}_\\mu }+\\lim _{u\\rightarrow 0}\\frac{\\lambda ^2}{12}\\int \\textrm {d}^4 u~u^\\mu u^\\nu G^{3}(x_0-\\frac{u}{2},x_0+\\frac{u}{2})\\frac{\\partial G(x_0,x_2)}{\\partial {x_0}^\\nu }\\bigg \\rbrace =0.$ We obtain following expression involving $\\delta ^{(2)}_z(x)$ , $\\eta ^{\\mu \\nu }\\delta ^{(2)}_z(x)=i\\lim _{u\\rightarrow 0}\\frac{\\lambda ^2}{12}\\int \\textrm {d}^4 u~u^\\mu u^\\nu G^{3}(x-\\frac{u}{2},x+\\frac{u}{2}),$ which in the trivial and massless case reduces to the standard expression, $\\delta _{z,\\textrm {free}}^{(2)}=\\frac{i\\lambda ^2}{48}\\int \\textrm {d}^{4}u~u^2\\left[\\int \\frac{\\textrm {d}^{4}u}{(2\\pi )^4}\\frac{ie^{ik.u}}{k^2-m^2+i\\epsilon }\\right]^{3}~\\stackrel{{\\tiny {d=4-\\epsilon }}}{\\longrightarrow }~\\delta _{z,\\textrm {free}}^{(2)}=\\frac{\\lambda ^2}{12(4\\pi )^4}(-\\frac{1}{\\epsilon }).$ We expect the higher order contributions to $\\delta _z(x)$ to be even harder to isolate, if not impossible.",
"The expressions that we have obtained for $\\delta ^{(2)}_m(x)$ and $\\delta ^{(2)}_z(x)$ in Eq.",
"( REF ) and Eq.",
"( REF ), also indicate that the consistency of the renormalization program mandates that all of the internal propagators to be of the same kind.",
"Up to now we have only shown that a necessary condition for the consistency of renormalization program is that all of the internal propagators to be of the same kind, either trivial or non-trivial.",
"Now, we show that this condition is insufficient.",
"This is accomplished by showing that in order to connect the internal and external propagators, all of the propagators should be of the same form.",
"The argument is as follows.",
"The external propagators connect the external points to their adjacent internal points.",
"Obviously these propagators should satisfy the same set of conditions on all of these points.",
"If we insist that the propagators satisfy the non-trivial conditions imposed on the system at the external points, they should satisfy the same conditions on the internal ones.",
"This starts a cascade process forcing all of the propagators to be of the same form.",
"Even if one is willing to consider exotic propagators resulting from the contraction of ${}{\\phi }{\\,_{\\textrm {trivial}}}{\\phi }\\phi _\\textrm {trivial}\\phi _\\textrm {non-trivial}$ , one would run into mathematical difficulty of contracting fields of different nature and possibly with different number of degrees of freedom (e.g.",
"$\\aleph _0$ and $\\aleph _1$ ).",
"This proves our second set of reasoning.",
"The expression for $\\delta _z$ provides us with the third set of reasonings in favor of our renormalization program.",
"Since the expressions for the counter-terms in higher orders are extremely convoluted and entangled, e.g.",
"the expression for $\\delta ^{(2)}_z$ , one does not have the freedom to choose the counter-terms arbitrarily.",
"The forms of the counter-terms have to be consistent with the Green's functions appropriate to the problem, or else the renormalization program's self consistency might be compromised.",
"The fourth reason in favor of our program is that the loss of translational symmetry in problems with non-trivial conditions in general requires the cancellation of all of the divergences to occur locally.",
"That is, the divergences will not have in general a constant value throughout the space.",
"This implies that the counter-terms should be in general position dependent."
],
[
"Conclusion",
"In this paper we have shown that a consistent renormalization program for the problems with non-perturbative conditions can be formulated in the configuration space.",
"We have presented four sets of reasonings in favor of our renormalization program.",
"In our program all of the propagators appearing in the perturbative expansion of the $n$ -point functions should be consistent with the non-perturbative conditions imposed on the system.",
"This in turn implies that the radiative corrections to all of the input parameters of the theory, including the mass, will be position dependent.",
"As we shall show in the appendix, the induced position dependences on the counter-terms are extremely small.",
"We believe that the position dependences induced in the parameters of the Lagrangian, including the mass and coupling constants, via the loop corrections, however small, are of fundamental importance.",
"They are a direct and natural manifestation of the non-perturbative position dependent conditions imposed on the system.",
"It is important to note that usually the free space renormalization program also works for low orders in perturbation theory, in the sense that it eliminate divergences and render the processes within the problem computable.",
"However, the difference between the two programs leads to small finite differences in the final results.",
"We have explored some of the consequences of this renormalization program in connection with the NLO Casimir energy [28], [29], and radiative correction to the mass of the soliton [30]."
],
[
"Acknowledgment",
"We would like to thank the research office of the Shahid Beheshti University for financial support."
],
[
"A simple illustrative example for the comparison between trivial and non-trivial counter-terms",
"In this appendix we compute the mass counter-term for a simple $\\lambda \\phi ^4$ model in $1+1$ dimensions with Dirichlet boundary conditions (BC).",
"We then compare this with the analogous counter-term without any boundary conditions.",
"The Green's function for the case with the boundary conditions imposed is, Figure: The difference between the mass counter-terms of a scalar field confined with Dirichlet boundarycondition and the free case in 1+11+1 dimensions, when m=1m=1, λ=0.1(m 2 )\\lambda =0.1~(m^2) and a=10(m -1 )a=10~(m^{-1}).Note that the difference between these two infinite quantities is significant only very close to theboundaries.",
"As a→∞a\\rightarrow \\infty the difference goes to zero for -a 2<x<a 2-\\frac{a}{2}<x<\\frac{a}{2}.$G_{\\textrm {BC}}(x,x^\\prime )=\\frac{2}{a}\\int \\frac{\\textrm {d}\\omega }{2\\pi }~e^{\\omega (t-t^\\prime )}\\sum \\limits _n\\frac{\\textrm {sin}[k_n(x+\\frac{a}{2})]\\textrm {sin}[k_n(x^\\prime +\\frac{a}{2})]}{\\omega ^2+k_n^2+m^2},$ where $a$ denotes the distance between the plates.",
"The Green's function in the free or no boundary case is, $G_{\\textrm {free}}(x,x^\\prime )=\\int \\frac{\\textrm {d}^{2}k}{(2\\pi )^2}\\frac{ie^{-ik.",
"(x-x^\\prime )}}{k^2-m^2+i\\epsilon }.$ These two Green's functions diverge as $x\\rightarrow x^\\prime $ .",
"Using Eq.",
"(REF ), the difference between the corresponding counter-terms is, $\\Delta \\big [(\\delta m(x)\\big ]=\\delta m_{\\textrm {BC}}-\\delta m_{\\textrm {free}}=-\\frac{\\lambda }{2}\\left[G_{{\\textrm {BC}}}(x,x)-G_{\\textrm {free}}(x,x)\\right].$ In Fig.",
"(REF ) we plot $\\Delta \\big [\\delta m(x)\\big ]$ .",
"Note that this quantity is extremely small, as compared to each of the divergent counter-terms, except at the boundaries.",
"In general the value of $\\Delta \\big [\\delta m(x)\\big ]$ is only significant for distances within a Compton wave-length from the end points."
]
] | 1204.1117 |
[
[
"Progenitors of type Ia supernovae"
],
[
"Abstract Type Ia supernovae (SNe Ia) play an important role in astrophysics and are crucial for the studies of stellar evolution, galaxy evolution and cosmology.",
"They are generally thought to be thermonuclear explosions of accreting carbon-oxygen white dwarfs (CO WDs) in close binaries, however, the nature of the mass donor star is still unclear.",
"In this article, we review various progenitor models proposed in the past years and summarize many observational results that can be used to put constraints on the nature of their progenitors.",
"We also discuss the origin of SN Ia diversity and the impacts of SN Ia progenitors on some fields.",
"The currently favourable progenitor model is the single-degenerate (SD) model, in which the WD accretes material from a non-degenerate companion star.",
"This model may explain the similarities of most SNe Ia.",
"It has long been argued that the double-degenerate (DD) model, which involves the merger of two CO WDs, may lead to an accretion-induced collapse rather than a thermonuclear explosion.",
"However, recent observations of a few SNe Ia seem to support the DD model, and this model can produce normal SN Ia explosion under certain conditions.",
"Additionally, the sub-luminous SNe Ia may be explained by the sub-Chandrasekhar mass model.",
"At present, it seems likely that more than one progenitor model, including some variants of the SD and DD models, may be required to explain the observed diversity of SNe Ia."
],
[
"Introduction",
"Type Ia supernova (SN Ia) explosions are among the most energetic events observed in the Universe.",
"They are defined as those without hydrogen or helium lines in their spectra, but with strong SiII absorption lines around the maximum light (Filippenko, 1997).",
"They appear to be good cosmological distance indicators due to their high luminosities and remarkable uniformity, and thus are used for determining the cosmological parameters (e.g.",
"$\\Omega _{M}$ and $\\Omega _{\\Lambda }$; Riess et al., 1998; Perlmutter et al., 1999).",
"This leads to the discovery of the accelerating expansion of the Universe that is driven by the mysterious dark energy.",
"SNe Ia are also a key part of our understanding of galactic chemical evolution owing to the main contribution of iron to their host galaxies (e.g.",
"Greggio and Renzini, 1983; Matteucci and Greggio, 1986).",
"In addition, they are accelerators of cosmic rays and as sources of kinetic energy in galaxy evolution processes (e.g.",
"Helder et al., 2009; Powell et al., 2011).",
"The use of SNe Ia as standard candles is based on the assumption that all SNe Ia have similar progenitors and are highly homogeneous.",
"However, several key issues related to the nature of their progenitors and explosion mechanism are still not well understood (Branch et al., 1995; Hillebrandt and Niemeyer, 2000).",
"This may directly affects the reliability of the results of the current cosmological model and galactic chemical evolution model.",
"When SNe Ia are used as distance indicators, the Phillips relation is adopted, which is a phenomenological linear relation between the absolute magnitude of SNe Ia and the magnitude difference from its $B$ -band maximum to 15 days after that (Phillips, 1993).",
"The Phillips relation is based on the SN Ia sample of low redshift Universe ($z<0.05$ ) and assumed to be valid at high redshift.",
"This assumption is precarious since there is still no agreement on the nature of their progenitors.",
"If the properties of SNe Ia evolve with the redshift, the results for cosmology might be different.",
"In addition, more observational evidence indicates that not all SNe Ia obey the Phillips relation (e.g.",
"Wang et al., 2006).",
"Aside from the Phillips relation, many updated versions of this method are given to establish the relation between SN Ia intrinsic luminosities and the shape of their light curves.",
"The stretch factor $s$ method was proposed to measure the light curve shape by adjusting the scale on the time axis by a multiplicative factor (Perlmutter et al., 1997; Goldhaber et al., 2001).",
"In addition, an empirical method based on multicolor light curve shapes has been developed to estimate the luminosity, distance, and total line-of-sight extinction of SNe Ia (Riess et al., 1998).",
"Wang et al.",
"(2005) presented a single post-maximum color parameter $\\Delta C_{12}$ ($B - V$ color $\\sim $ 12 days after the $B$ -band light maximum), which empirically describes almost the full range of the observed SN Ia luminosities and gives tighter correlations with their luminosities, but the underpinning physics is still not understood.",
"Recently, Guy et al.",
"(2005) used an innovative approach to constrain the spectral energy distribution of SNe Ia, parameterized continuously as a function of color and stretch factor $s$ , and allow for the generation of light curve templates in arbitrary pass-bands.",
"This method was known as the spectral adaptive light curve template method, which offers several practical advantages that make it easily applicable to high redshift SNe Ia.",
"The k-corrections are built into the model but not applied to the data, which allows one to propagate all the uncertainties directly from the measurement errors.",
"It is widely accepted that SNe Ia arise from thermonuclear explosions of carbon–oxygen white dwarfs (CO WDs) in close binaries (Hoyle and Fowler, 1960; Nomoto et al., 1997).",
"This hypothesis is supported by the fact that the amount of energy observed in SN explosions is equal to the amount that would be produced in the conversion of carbon and oxygen into iron ($\\sim $$10^{51}$ erg; Thielemann et al., 2004).",
"The energy released from the nuclear burning completely destroys the CO WD and produces a large amount of $^{56}$ Ni.",
"The optical/infrared light curves are powered by the radioactive decay of $^{56}$ Ni $\\rightarrow $ $^{56}$ Co $\\rightarrow $ $^{56}$ Fe.",
"In order to trigger the carbon ignition, the mass of the CO WD must grow close to the Chandrasekhar (Ch) mass.",
"When the WD increases its mass close to the Ch mass, it is thought to ignite near the center; at first the flame propagates subsonically as a deflagration, and in a second phase a detonation triggers, which propagates supersonically and completely destroys the CO WD (Hillebrandt and Niemeyer, 2000).",
"The realistically conceivable way to make the WD grow to the Ch mass is via mass-transfer from a mass donor star in a close binary.",
"However, the nature of the mass donor star in the close binary is still uncertain, and no progenitor system before SN explosion has been conclusively identified.",
"Additionally, there is some observational evidence that a subset of SNe Ia have progenitors with a mass exceeding or below the standard Ch mass limit (e.g.",
"Howell et al., 2006; Foley et al., 2009; Wang et al., 2008a).",
"Many progenitor models of SNe Ia have been proposed in the past years.",
"The most popular progenitor models are single-degenerate (SD) and double-degenerate (DD) models.",
"In Sect.",
"2, we review various progenitor models, including some variants of the SD and DD models proposed in the literature.",
"We summarize some observational ways to test the current progenitor models in Sect.",
"3, and introduce some objects that may be related to the progenitors and the surviving companion stars of SNe Ia in Sect.",
"4.",
"We discuss the origin of SN Ia diversity and the impacts of SN Ia progenitors on some research fields in Sects.",
"5 and 6, respectively.",
"Finally, a summary is given in Sect.",
"7.",
"For more discussions on these subjects, see previous reviews on SN Ia progenitors (e.g.",
"Branch et al., 1995; Hillebrandt and Niemeyer, 2000; Livio, 2000; Nomoto et al., 2003; Podsiadlowski, 2010; Maoz and Mannucci, 2012).",
"In this model, a CO WD accretes hydrogen-rich or helium-rich material from a non-degenerate companion star, increases its mass to the Ch mass, and then explodes as a SN Ia (Whelan and Iben, 1973; Nomoto, 1982a).",
"The SD model may explain the similarities of most SNe Ia, since SN Ia explosions in this model occur when the CO WD increases its mass to the maximum stable mass (i.e.",
"the Ch mass).",
"In addition, the observed light curves and early time spectra of the majority of SNe Ia are in excellent agreement with the synthetic spectra of the SD Ch mass model (Nomoto et al., 1984; Höflich et al., 1996; Nugent et al., 1997).",
"The companion star in this model could be a main-sequence (MS) star or a subgiant star (WD + MS channel), or a red-giant star (WD + RG channel), or a helium star (WD + He star channel) (Hachisu et al., 1996, 1999a,b; Li and van den Heuvel, 1997; Langer et al., 2000; Han and Podsiadlowski, 2004; Fedorova et al., 2004; Meng et al, 2009; Wang et al., 2009a, 2010a).",
"The main problem for this class of models is that it is generally difficult to increase the mass of the WD by accretion.",
"Whether the WD can grow in mass depends crucially on the mass-transfer rate and the evolution of the mass-transfer rate with time.",
"(1) If the rate is too high, the system may enter into a common envelope (CE) phase; (2) if the rate is too low, the nuclear burning is unstable that leads to nova explosions in which all the accreted matter is ejected.",
"There is only a very narrow parameter range in which the WD can accrete H-rich or He-rich material and burn in a stable manner.",
"This parameter range may be increased if the rotation affects the WD mass-accretion process (Yoon and Langer, 2004).",
"An essential element in this model is the optically thick wind assumption, which enlarges the parameter space for producing SNe Ia (Hachisu et al., 1996, 1999a,b; Li and van den Heuvel, 1997; Han and Podsiadlowski, 2004; Wang et al., 2009a, 2010a).",
"In this assumption, taking a MS donor star for an example, if the mass-transfer rate from the MS star exceeds a critical value, $\\dot{M}_{\\rm cr}$ , it is assumed that the accreted H burns steadily on the surface of the WD and that the H-rich material is converted into He at the rate of $\\dot{M}_{\\rm cr}$ .",
"The unprocessed matter is assumed to be lost from the binary system as an optically thick wind.",
"However, this assumption is very sensitive to the Fe abundance, and it is likely that the wind does not work when the metallicity is lower than a certain value.At low enough metallicities (e.g.",
"$Z<0.002$ ), the optical depth of the wind would become small, and thus the wind-regulation mechanism would become ineffective (e.g.",
"Kobayashi et al., 1998; Kobayashi and Nomoto, 2009).",
"In this case, the binary system will pass through a CE phase before reaching the Ch mass.",
"Thus, if this is true then there would be an obvious low-metallicity threshold for SNe Ia in comparison with SNe II.",
"However, the metallicity threshold has not been found in observations (Prieto et al., 2008; Badenes et al., 2009a).",
"In the WD + MS channel (usually called the supersoft channel), a CO WD in a binary system accretes H-rich material from a MS or a slightly evolved subgiant star.",
"The accreted H-rich material is burned into He, and then the He is converted to carbon and oxygen.",
"When the CO WD increases its mass close to the Ch mass, it explodes as a SN Ia.",
"Based on the evolutionary phase of the primordial primary (i.e.",
"the massive star) at the beginning of the first Roche lobe overflow (RLOF), there are three evolutionary scenarios to form WD + MS systems and then produce SNe Ia (Fig.",
"1; for details see Wang et al., 2010a; also see Postnov and Yungelson, 2006; Meng et al, 2009).",
"Scenario A: The primordial primary first fills its Roche lobe when it is in the Hertzsprung gap (HG) or first giant branch (FGB) stage (i.e.",
"Case B mass-transfer defined by Kippenhahn and Weigert, 1967).",
"In this case, due to a large mass-ratio or a convective envelope of the mass donor star, a CE may be formed (Paczyński, 1976).",
"After the CE ejection, the primary becomes a He star and continues to evolve.",
"After the exhaustion of central He, the He star evolves to the RG stage.",
"The He RG star that now contains a CO-core may fill its Roche lobe again due to the expansion itself, and transfer its remaining He-rich envelope onto the surface of the MS companion star, eventually leading to the formation of a CO WD + MS system.",
"For this scenario, SN Ia explosions occur for the ranges $M_{\\rm 1,i}\\sim 4.0$$-$$7.0\\,M_\\odot $ , $M_{\\rm 2,i}\\sim 1.0$$-$$2.0\\,M_\\odot $ , and $P^{\\rm i} \\sim 5$$-$ 30 days, where $M_{\\rm 1,i}$ , $M_{\\rm 2,i}$ and $P^{\\rm i}$ are the initial masses of the primary and the secondary at zero age main-sequence (ZAMS), and the initial orbital period of the binary system.",
"Scenario B: If the primordial primary is on the early asymptotic giant branch (EAGB, i.e.",
"He is exhausted in the center of the star while this star has a thick He-burning layer and the thermal pulses have not yet started), a CE may be formed due to the dynamically unstable mass-transfer.",
"After the CE is ejected, a close He RG + MS binary may be produced; the binary orbit decays in the process of the CE ejection and the primordial primary may evolve to a He RG that contains a CO-core.",
"The He RG may fill its Roche lobe and start mass-transfer, which is likely stable and results in a CO WD + MS system.",
"For this scenario, SN Ia explosions occur for the ranges $M_{\\rm 1,i}\\sim 2.5-6.5\\,M_\\odot $ , $M_{\\rm 2,i}\\sim 1.5-3.0\\,M_\\odot $ and $P^{\\rm i} \\sim 200-900$ days.",
"Scenario C: The primordial primary fills its Roche lobe at the thermal pulsing asymptotic giant branch (TPAGB) stage.",
"A CE is easily formed owing to the dynamically unstable mass-transfer during the RLOF.",
"After the CE ejection, the primordial primary becomes a CO WD, then a CO WD + MS system is produced.",
"For this scenario, SN Ia explosions occur for the ranges $M_{\\rm 1,i}\\sim 4.5$$-$$6.5\\,M_\\odot $ , $M_{\\rm 2,i}\\sim 1.5$$-$$3.5\\,M_\\odot $ , and $P^{\\rm i}>1000$ days.",
"Among the three evolutionary scenarios above, models predict that scenario A is the more significant route for producing SNe Ia (e.g.",
"Wang et al., 2010a).",
"The WD + MS channel has been identified in recent years as supersoft X-ray sources and recurrent novae (van den Heuvel et al., 1992; Rappaport et al., 1994; Meng and Yang, 2010a).",
"Many works have been concentrated on this channel.",
"Some authors studied the WD + MS channel with a simple analytical method to treat binary interactions (e.g.",
"Hachisu et al., 1996, 1999a, 2008).",
"Such analytic prescriptions may not describe some mass-transfer phases well enough, especially those occurring on a thermal time-scale.",
"Li and van den Heuvel (1997) studied this channel from detailed binary evolution calculation with two WD masses (e.g.",
"1.0 and 1.2 $M_{\\odot }$ ).",
"Langer et al.",
"(2000) investigated this channel for metallicities $Z=0.001$ and 0.02, but they only studied Case A evolution (mass-transfer during the central H-burning phase).",
"Han and Podsiadlowski (2004) carried out a detailed study of this channel including Case A and early Case B for $Z=0.02$ .",
"The Galactic SN Ia birthrate from this study is $0.6-1.1\\times 10^{-3}\\,{\\rm yr}^{-1}$ .",
"Following the studies of Han and Podsiadlowski (2004), Meng et al.",
"(2009) studied the WD + MS channel comprehensively and systematically at various metallicities.",
"King et al.",
"(2003) inferred that the mass-accretion rate on to the WD during dwarf nova outbursts can be sufficiently high to allow steady nuclear burning of the accreted matter and growth of the WD mass.",
"Recently, Xu and Li (2009) also emphasized that, during the mass-transfer through the RLOF in the evolution of WD binaries, the accreted material can form an accretion disc surrounding the WD, which may become thermally unstable (at least during part of the mass-transfer lifetime), i.e.",
"the mass-transfer rate is not equivalent to the mass-accretion rate onto the WD.",
"By considering the effect of the thermal-viscous instability of accretion disk on the evolution of WD binaries, Wang et al.",
"(2010a) recently enlarged the regions of the WD + MS channel for producing SNe Ia, and confirmed that WDs in this channel with an initial mass as low as $0.6\\,M_\\odot $ can accrete efficiently and reach the Ch mass limit.",
"Based on a detailed binary population synthesis (BPS) approach,BPS is a useful tool to simulate a large population of stars or binaries and can help understand processes that are difficult to observe directly or to model in detail (e.g.",
"Han et al., 1995; Yungelson and Livio, 2000; Nelemans et al., 2001).",
"they found that this channel is effective for producing SNe Ia (up to $1.8\\times 10^{-3}\\,{\\rm yr}^{-1}$ in the Galaxy), which can account for about $2/3$ of the observations (see also Meng and Yang, 2010a).",
"However, the parameter regions for producing SNe Ia in this model depend on many uncertain input parameters, in particular the duty cycle during the nova outbursts that is still poorly known.",
"Additionally, whether dwarf nova outbursts can increase the mass of a WD is still a problem (e.g.",
"Hachisu et al., 2010)."
],
[
"WD + RG channel",
"The mass donor star in this channel is a RG star, which is also called the symbiotic channel.",
"There is one evolutionary scenario that can form WD + RG binaries and then produce SNe Ia (Fig.",
"2; for details see Wang et al, 2010a).",
"Compared with the WD + MS channel, SNe Ia in the WD + RG channel are from wider primordial binaries.",
"The primordial primary fills its Roche lobe at the TPAGB stage.",
"A CE is easily formed due to the dynamically unstable mass-transfer during the RLOF.",
"After the CE ejection, the primordial primary becomes a CO WD.",
"The MS companion star continues to evolve until the RG stage, i.e.",
"a CO WD + RG binary is formed.",
"For the WD + RG systems, SN Ia explosions occur for the ranges $M_{\\rm 1,i}\\sim 5.0$$-$$6.5\\,M_\\odot $ , $M_{\\rm 2,i}\\sim 1.0$$-$$1.5\\,M_\\odot $ , and $P^{\\rm i}>1500$ days.",
"Unfortunately, the WD + RG binary usually undergoes a CE phase when the RG star overflows its Roche lobe.",
"More importantly, the appropriate initial parameter space for producing SNe Ia in this channel is too small.",
"Thus, WD + RG binaries seem to unlikely become a major way to form SNe Ia.",
"Many authors claimed that the SN Ia birthrate via the WD + RG channel is much lower than that from the WD + MS channel (Yungelson and Livio, 1998; Han and Podsiadlowski, 2004; Lü et al., 2006; Wang et al., 2010a).",
"The lowest initial WD mass in this channel for producing SNe Ia is about $1.0\\,M_\\odot $ (e.g.",
"Wang and Han, 2010a).",
"In order to stabilize the mass-transfer process and avoid the formation of the CE, Hachisu et al.",
"(1999b) assumed a mass-stripping model in which a stellar wind from the WD collides with the RG surface and strips some of the mass from the RG.",
"They obtained a high SN Ia birthrate ($\\sim $$0.002\\,{\\rm yr}^{-1}$ ) for this channel.",
"Here, Hachisu et al.",
"(1999b) used equation (1) of Iben and Tutukov (1984) to estimate the birthrate, i.e.",
"$\\nu = 0.2\\,\\Delta q \\int _{M_{\\rm A}}^{M_{\\rm B}} {{d M} \\over M^{2.5}} \\Delta \\log A \\, \\mbox{yr}^{-1},$ where $\\Delta q$ , $\\Delta \\log A$ , $M_{\\rm A}$ and $M_{\\rm B}$ are the appropriate ranges of the initial mass ratio, the initial separation, and the lower and upper limits of the primary mass for producing SNe Ia in units of solar masses, respectively.",
"However, the birthrate is probably overestimated, since some parameter spaces considered to produce SNe Ia in equation (1) may not contribute to SNe Ia.",
"In symbiotic systems, WDs can accrete a fraction of the stellar wind from cool giants.",
"It is generally believed that the stellar wind from a normal RG is expected to be largely spherical owing to the spherical stellar surface and isotropic radiation.",
"However, the majority ($>$$80\\%$ ) of the observed planetary nebulae are found to have aspherical morphologies (Zuckerman and Aller, 1986).",
"Additionally, the stellar winds from cool giants in symbiotic systems flow out in two ways: an equatorial disc and a spherical wind.",
"In this context, by assuming an aspherical stellar wind with an equatorial disk from a RG, Lü et al.",
"(2009) investigated the production of SNe Ia via the symbiotic channel.",
"They estimated that the Galactic SN Ia birthrate via this channel is between $2.27\\times 10^{-5}$ yr$^{-1}$ and $1.03\\times 10^{-3}$ , and the theoretical SN Ia delay time (between the star formation and SN explosion) has a wide range from 0.07 to 5 Gyr.",
"However, these results are greatly affected by the outflow velocity and the mass-loss rate of the equatorial disk.",
"The stellar wind from RG stars might be enhanced by tidal or other interactions with a companion.",
"Tout and Eggleton (1988) brought the tidally enhanced stellar wind assumption to explain the mass inversion in RS CVn binaries.",
"This assumption has been widely used to explain many phenomena related to giant star evolution in binaries (e.g.",
"Han, 1998; van Winckel, 2003).",
"The tidally enhanced stellar wind assumption has two advantages in the studies of symbiotic systems: (1) The WD may grow in mass substantially by accretion from stellar wind before RLOF; (2) the mass-transfer may be stabilized because the mass ratio ($M_{\\rm giant}$ /$M_{\\rm WD}$ ) can be much reduced at the onset of RLOF.",
"By adopting the tidally enhanced stellar wind assumption, Chen et al.",
"(2011) recently argued that the parameter space of SN Ia progenitors can be extended to longer orbital periods for the WD + RG channel (compared to the mass-stripping model of Hachisu et al., 1999b), and thus increase the birthrate up to $6.9\\times 10^{-3}$ yr$^{-1}$ , which is also probably overestimated due to the use of equation (1).",
"Additionally, the parameter space of SN Ia progenitors strongly depends on the tidal wind enhancement parameter $B_{\\rm w}$ that is still poorly known.",
"In a variant of the symbiotic channel, the mass-transfer from carbon-rich AGB stars with WD components can occur via stellar winds or RLOF (Iben and Tutukov, 1985).",
"It has been suggested that an AGB donor star is in the progenitor system of SN 2002ic, which is an atypical SN Ia with evidence for substantial amounts of hydrogen associated with the system (Hamuy et al., 2003).",
"Recently, Chiotellis et al.",
"(2012) presented a WD with an AGB donor star for the SN remnant (SNR) of SN 1604, also known as Kepler's SNR.",
"They argued that its main features can be explained by the model of a symbiotic binary consisting of a WD and an AGB donor star with an initial mass of 4$-$ 5 $M_{\\odot }$ .",
"Detailed calculations of binary evolutionary model are needed to understand whether these WD components in WD + AGB binaries can result in SN Ia explosions."
],
[
"WD + He star channel",
"A CO WD may also accrete helium-rich material from a He star or a He subgiant to increase its mass to the Ch mass, which is also known as the He star donor channel.",
"There are three evolutionary scenarios to form WD + He star systems and then produce SNe Ia (see Fig.",
"3; for details see Wang et al., 2009b).",
"Scenario A: The primordial primary first fills its Roche lobe when it is in the HG or FGB stage.",
"At the end of the RLOF, the primary becomes a He star and continues to evolve.",
"After the exhaustion of central He, the He star evolves to the RG stage.",
"The He RG star that now contains a CO-core may fill its Roche lobe again due to the expansion of itself, and transfer its remaining He-rich envelope to the MS companion star, eventually leading to the formation of a CO WD + MS system.",
"After that, the MS companion star continues to evolve and fills its Roche lobe in the HG or FGB stage.",
"A CE is possibly formed due to the dynamically unstable mass-transfer.",
"If the CE can be ejected, a close CO WD + He star system is then produced.",
"The CO WD + He star system continues to evolve, and the He star may fill its Roche lobe again (due to the orbit decay induced by the gravitational wave radiation or the expansion of the He star itself), and transfer some material onto the surface of the CO WD.",
"The accreted He may be converted into carbon and oxygen via the He-shell burning, and the CO WD increases in mass and explodes as a SN Ia when its mass reaches the Ch mass.",
"For this scenario, SN Ia explosions occur for the ranges $M_{\\rm 1,i}\\sim 5.0-8.0\\,M_\\odot $ , $M_{\\rm 2,i}\\sim 2.0-6.5\\,M_\\odot $ and $P^{\\rm i} \\sim 10-40$ days.",
"Scenario B: If the primordial primary is on the EAGB stage at the onset of the RLOF, a CE may be formed due to the dynamically unstable mass-transfer.",
"After the CE is ejected, a close He RG + MS binary may be produced; the binary orbit decays in the procedure of the CE ejection and the primordial primary becomes a He RG.",
"The He RG may fill its Roche lobe and start the mass-transfer, which is likely stable and results in a CO WD + MS system.",
"The subsequent evolution of this system is similar to scenario A above, and may form a CO WD + He star system and finally produce a SN Ia.",
"For this scenario, SN Ia explosions occur for the ranges $M_{\\rm 1,i}\\sim 6.0-6.5\\,M_\\odot $ , $M_{\\rm 2,i}\\sim 5.5-6.0\\,M_\\odot $ and $P^{\\rm i} \\sim 300-1000$ days.",
"Scenario C: The primordial primary fills its Roche lobe at the TPAGB stage, and the companion star evolves to the He-core burning stage.",
"A double-core CE may be formed owing to the dynamically unstable mass-transfer during the RLOF.",
"After the CE ejection, the primordial primary becomes a CO WD, and the companion star is a He star at the He-core burning stage, i.e.",
"a CO WD + He star system is formed.",
"The subsequent evolution of this system is similar to that in the above two scenarios, i.e.",
"a SN Ia may be produced.",
"For this scenario, SN Ia explosions occur for the ranges $M_{\\rm 1,i}\\sim 5.5-6.5\\,M_\\odot $ , $M_{\\rm 2,i}\\sim 5.0-6.0\\,M_\\odot $ and $P^{\\rm i}>1000$ days.",
"SNe Ia from the He star donor channel can neatly avoid H lines, consistent with the defining spectral characteristic of most SNe Ia.",
"Yoon and Langer (2003) followed the evolution of a WD + He star binary with a $1.0\\,M_{\\odot }$ WD and a $1.6\\,M_{\\odot }$ He star in a 0.124 d orbit.",
"In this binary, the WD accretes He from the He star and grows in mass to the Ch mass.",
"Based on the optically thick wind assumption, Wang et al.",
"(2009a) systematically studied the He star donor channel.",
"In the study, they carried out binary evolution calculations of this channel for about 2600 close WD + He star binaries.",
"The study showed the initial parameter spaces for the progenitors of SNe Ia, and found that the minimum mass of CO WD for producing SNe Ia in this channel may be as low as $0.865\\,M_{\\odot }$ .",
"By using a detailed BPS approach, Wang et al.",
"(2009b) found that the Galactic SN Ia birthrate from this channel is $\\sim $$0.3\\times 10^{-3}\\,{\\rm yr}^{-1}$ and this channel can produce SNe Ia with short delay times ($\\sim $ 45$-$ 140 Myr).",
"Wang and Han (2010b) also studied the He star donor channel with different metallicities.",
"For a constant star-formation galaxy (like our own galaxy), they found that SN Ia birthrates increase with metallicity.",
"If a single starburst is assumed (like in an elliptical galaxy), SNe Ia occur systematically earlier and the peak value of the birthrate is larger for a higher metallicity."
],
[
"Double-degenerate model",
"In the DD model, SNe Ia arise from the merging of two close CO WDs that have a combined mass larger than or equal to the Ch mass (Tutukov and Yungelson, 1981; Iben and Tutukov, 1984; Webbink, 1984).",
"Both CO WDs are brought together by gravitational wave radiation on a timescale $t_{\\rm GW}$ (Landau and Lifshitz, 1971), $t_{\\rm GW}(\\rm yr \\it )=\\rm 8\\times 10^{7}\\it (\\rm yr \\it )\\times \\frac{(M_{\\rm 1}+M_{\\rm 2})^{\\rm 1/3}}{M_{\\rm 1}M_{\\rm 2}}P^{\\rm 8/3}(\\rm h),$ where $P$ is the orbital period in hours, $t_{\\rm GW}$ in years and $M_{\\rm 1}$ , $M_{\\rm 2}$ in $M_{\\odot }$ .",
"The delay time from the star formation to the occurrence of a SN Ia is equal to the sum of the timescale that the secondary star becomes a WD and the orbital decay time $t_{\\rm GW}$ .",
"For the DD model, there are three binary evolutionary scenarios to form double CO WD systems and then produce SNe Ia, i.e.",
"stable RLOF plus CE ejection scenario, CE ejection plus CE ejection scenario and exposed core plus CE ejection scenario (for details see Han, 1998).",
"The DD model has the advantage that the theoretically predicted merger rate is quite high, consistent with the observed SN Ia birthrate (e.g.",
"Yungelson et al., 1994; Han, 1998; Nelemans et al., 2001; Ruiter et al., 2009; Wang et al., 2010b).Badenes and Maoz (2012) recently calculated the merger rate of binary WDs in the Galactic disk based on the observational data in the Sloan Digital Sky Survey.",
"They claimed that there are not enough double WD systems with the super-Ch masses to reproduce the observed SN Ia birthrate in the context of the DD model.",
"Importantly, this model can naturally explain the lack of H or He emission in the spectra of SNe Ia.",
"As an additional argument in favor of the DD model, one may consider this model to explain some observed super-luminous SNe Ia (for more discussions see Sect.",
"2.3.2).",
"Furthermore, there are some double WD progenitor candidates that have been found in observations, and recent observations of a few SNe Ia seem to support the DD model (for more discussions see Sect.",
"3).",
"However, the DD model has difficulties in explaining the similarities of most SNe Ia, since the merger mass in this model varies for different binaries and has a relatively wide range ($\\sim $$1.4-2.0\\,M_{\\odot }$ ; Wang et al., 2010b).",
"Most importantly, the merger of two WDs may result in an accretion-induced collapse to form a neutron star rather than a thermonuclear explosion (Nomoto and Iben, 1985; Saio and Nomoto, 1985; Timmes et al., 1994).",
"In the process of a double-WD merger, once the less massive WD fills its Roche lobe, it is likely to be disrupted and rapidly accreted by the more massive one.",
"Meanwhile, the less massive WD is transformed into a disk-like structure around the more massive companion.",
"It is usually assumed that in this configuration the temperature maximum is located at the “disk-dwarf” interface and that carbon burning starts there.",
"In this process, the carbon burning front propagates inward and then the CO WD is transformed into an O-Ne-Mg WD, which collapses to form a neutron star by electron capture on $^{24}$ Mg.",
"There may be some parameter ranges where the accretion-induced collapse can be avoided (e.g.",
"Piersanti et al., 2003; Yoon et al., 2007).",
"Piersanti et al.",
"(2003) suggested that the double WD merger process could be quite violent, and might lead to a SN Ia explosion under the right conditions.",
"Pakmor et al.",
"(2010) argued that the violent mergers of two equal-mass CO WDs ($\\sim $$0.9\\,M_{\\odot }$ , critical conditions for the successful initiation of a detonation) can be obtained, and may explain the formation of sub-luminous 1991bg-like objects.",
"Although the light curve from the merger model is broader than that of SN 1991bg-like objects, the synthesized spectra, red color and low expansion velocities are all close to those observed for SN 1991bg-like objects (Pakmor et al., 2010).",
"In a further study, Pakmor et al.",
"(2011) claimed that a high mass-ratio is required for this model to work; for a primary mass of $0.9\\,M_{\\odot }$ a mass-ratio of at least about 0.8.",
"This result will affect the potential SN Ia birthrate of the DD model.",
"We note that van Kerkwijk et al.",
"(2010) came to a similar conclusion before Pakmor et al.",
"(2011), but that was in turn partially based on Pakmor et al.",
"(2010) and Lorín-Aguilar et al.",
"(2009).",
"Adopting the results of Pakmor et al.",
"(2011) with a detailed BPS approach, Meng et al.",
"(2011) estimated that the sub-luminous events from this model may only account for not more than $1\\%$ of all SNe Ia.",
"Recently, by assuming that the moment at which the detonation forms is an artificial parameter, Pakmor et al.",
"(2012) presented a fully three-dimensional simulation of a violent merger of two CO WDs with masses of $0.9\\,M_{\\odot }$ and $1.1\\,M_{\\odot }$ , by combining very high resolution and the exact initial conditions.",
"They estimated that the simulation produces about $0.62\\,M_{\\odot }$ of $^{56}$ Ni, and the synthetic multi-color light curves show good agreement with those observed for normal SNe Ia.",
"Due to the small number of such massive systems available, this model may only contribute a small fraction to the observed population of normal SNe Ia.",
"Future studies are needed to explore the parameter space of different WD masses and mass ratios in this scenario for normal SNe Ia, which is important in BPS studies."
],
[
"Potential progenitor models",
"Besides the SD and DD models above, some variants of SD and DD models have been proposed to explain the observed diversity of SNe Ia, such as the sub-Ch mass model, the super-Ch mass model, the single star model, the delayed dynamical instability model, the spin-up/spin-down model, the core-degenerate model, the model of the collisions between two WDs, and the model of WDs near black holes, etc."
],
[
"Sub-Chandrasekhar mass model",
"In this model, a CO WD accumulates a $\\sim $$0.15\\,M_{\\odot }$ He layer with a total mass below the Ch mass (Nomoto 1982b; Woosley et al., 1986).",
"In order to achieve the central densities necessary to produce iron-peak elements, the WD in this model needs a narrow mass range of $\\sim $$0.9-1.1\\,M_{\\odot }$ .",
"The He may ignite off-center at the bottom of the He layer, resulting in an event known as Edge Lit Detonation (or Indirect Double Detonation).",
"In this process, one detonation propagates outward through the He layer, while an inward propagating pressure wave compresses the CO core that ignites off-center, followed by an outward detonation (e.g.",
"Livne, 1990; Höflich and Khokhlov, 1996).",
"It is possible that sub-luminous 1991bg-like objects may be explained by this model (Branch et al., 1995).",
"Unfortunately, the sub-Ch mass model has difficulties in matching the observed light curves and spectroscopy of SNe Ia (Höflich and Khokhlov, 1996; Nugent et al., 1997), likely owing to the thickness of the He layer.",
"Recently, Shen and Bildsten (2009) argued that, under some suitable conditions, a detonation in the WD might be achieved for even lower He layer masses than that in previous studies.",
"By assuming that a detonation is successfully triggered in the He layer, Fink et al.",
"(2010) claimed that the double detonations in sub-Ch mass WDs with low-mass He layers can be a robust explosion, leading to normal SN Ia brightness.",
"Recent studies involving the sub-Ch mass WDs with subsequent nucleosynthesis and radiative transfer calculations also indicate that the sub-Ch mass model could account for the range of the observed SN Ia brightness (Sim et al., 2010; Kromer et al., 2010).",
"Additionally, BPS studies by Ruiter et al.",
"(2009) predicted that there are a sufficient number of binaries with sub-Ch primary WDs to explain the observed birthrate of SNe Ia.",
"However, it must be noted that it is difficult for the sub-Ch mass model to explain the similarities observed in most SNe Ia (e.g.",
"Branch et al., 1995)."
],
[
"Super-Chandrasekhar mass model",
"The $^{\\rm 56}$ Ni mass deduced from some SN Ia explosions strongly suggests the existence of super-Ch mass progenitors.",
"SN 2003fg was observed to be 2.2 times over-luminous than a normal SN Ia, and the amount of $^{\\rm 56}$ Ni was inferred to be $1.3\\,M_{\\odot }$ , which requires a super-Ch mass WD explosion ($\\sim $$2.1\\,M_{\\odot }$ ; Howell et al., 2006).",
"Following the discovery of SN 2003fg, three 2003fg-like events were also discovered, i.e.",
"SN 2006gz (Hicken et al., 2007), SN 2007if (Scalzo et al., 2010; Yuan et al., 2010), and SN 2009dc (Yamanaka et al., 2009; Tanaka et al., 2010; Silverman et al., 2011).",
"These super-luminous SNe Ia may raise the possibility that more than one progenitor model may lead to SNe Ia.",
"It is usually assumed that these super-luminous SNe Ia are from the mergers of double WD systems, where the total mass of the DD systems is over the Ch mass.",
"Meanwhile, a super-Ch WD may be also produced by a SD system, where the massive WD is supported by its rapid rotation, e.g.",
"Maeda and Iwamoto (2009) claimed that the properties of SN 2003fg may be consistent with the aspherical explosion of a super-Ch WD, which is supported by its rapid rotation.",
"Yoon and Langer (2004) argued that WDs can rotate differentially for high mass-accretion rates of $3.0\\times 10^{-7}\\,M_{\\odot }\\,\\rm yr^{-1}$ .",
"By adopting the results of Yoon and Langer (2004), Chen and Li (2009) calculated the evolution of close binaries consisting of a CO WD and a MS star, and obtained the initial parameter space for super-Ch mass SN Ia progenitors.",
"Within this parameter space, Meng et al.",
"(2011) estimated that the upper limit of the contribution rate of these super-luminous SNe Ia to all SN Ia is less than 0.3%.",
"Hachisu et al.",
"(2012) recently made a comprehensive study of these super-luminous SNe Ia via the WD + MS channel, and suggested that these SNe Ia are born in a low metallicity environment as more massive initial CO WDs are required in this model.",
"Meanwhile, Liu et al.",
"(2010) also studied the He star donor channel to the formation of super-luminous SNe Ia by considering the effects of rapid differential rotation on the accreting WD.",
"Aside from the differential rotation, a super-Ch WD may also be supported by the WDs with strong magnetic fields due to the lifting effect.",
"It has been found that $\\sim $ 10% of WDs have magnetic fields stronger than 1 MG (Liebert et al., 2003, 2005; Wickramasinghe and Ferrario, 2005).",
"The mean mass of these magnetic WDs is $\\sim $$0.93\\,M_{\\odot }$ , compared with the mean mass of all WDs that is $\\sim $$0.56\\,M_{\\odot }$ (e.g.",
"Parthasarathy et al., 2007).",
"Thus, the magnetic WDs are more easily to reach the Ch mass limit by accretion.",
"The magnetic field may also affect some properties of SD progenitor systems, e.g.",
"the mass-transfer rate, the critical mass-accretion rate and the thermonuclear reaction rate, etc.",
"However, these effects are still unclear.",
"Further studies are thus needed."
],
[
"Single star model",
"Single star progenitor models have been considered by Iben and Renzini (1983) and Tout et al.",
"(2008).",
"In the absence of mass-loss, single massive star less than about 7 $M_{\\odot }$ will develop a degenerate CO-core when the star evolves to the AGB stage.",
"The mass growing rate of the CO-core is controlled by the rate of the double shell burning.",
"If the CO-core can grow to the Ch mass, it will produce a SN Ia.",
"Under certain conditions, Tout et al.",
"(2008) claimed that carbon can ignite at the center of the CO-core and the subsequent explosion would appear as a SN Ia.",
"These single star progenitors are likely to be over 2 $M_{\\odot }$ , so this kind of SNe Ia should be associated with younger galaxies with recent star formation.",
"The single star model was also proposed to explain the strongly circumstellar-interacting SN 2002ic (Hamuy et al., 2003).",
"An important theoretical argument for this model is that the H-rich envelope in AGB star may be lost in a superwind before the CO-core grows to the Ch mass, based on the envelope ejection criteria by Han et al.",
"(1994) and Meng et al.",
"(2008).",
"Another problem for this model is that there should be far more SNe Ia than observed if a single star can naturally experience thermonuclear explosion."
],
[
"Delayed dynamical instability model",
"This model is a variant of the WD + MS channel, which requires that the donor star is initially a relatively massive MS star ($\\sim $ 3 $M_{\\odot }$ ) and that the system has experienced a delayed dynamical instability, resulting in a large amount of mass-loss from the system in the last a few $10^{4}$ yr before SN explosion (Han and Podsiadlowski, 2006).",
"The delayed dynamical instability model can reproduce the inferred H-rich circumstellar environment, most likely with a disc-like geometry.",
"Han and Podsiadlowski (2006) claimed that the unusual properties of SN 2002ic can be understood by the delayed dynamical instability model.",
"Observationally, this model seems to be consistent with SN 2005gj (another 2002ic-like object) found by Nearby Supernova Factory observations (Aldering et al., 2006).",
"However, in order for this model to be feasible, it requires a larger mass-accretion efficiency onto the WD than is assumed in present parametrizations.",
"Based on a detailed BPS simulation, Han and Podsiadlowski (2006) estimated that not more than 1% SNe Ia should belong to this subclass of SNe Ia.",
"Since this model requires an intermediate-mass secondary star, these SNe Ia should only be found in stellar populations with relatively recent star formation (e.g.",
"with the last $\\sim $$3\\times 10^{8}\\,{\\rm yr}$ )."
],
[
"Spin-up/spin-down model",
"In the SD model, since the continued accretion of angular momentum can prevent the explosion of a WD, Justham (2011) recently argued that it may be natural for the mass donor stars in the SD model to exhaust their envelopes and shrink rapidly before SN explosion, which may explain the lack of H or He in the spectra of SNe Ia, often seen as troublesome for the SD progenitor model.",
"Di Stefano et al.",
"(2011) also suggested that the CO WD is likely to achieve fast spin periods as the accreted mass carries angular momentum, which can increase the critical mass, $M_{\\rm cr}$ , needed for SN explosion.",
"When the $M_{\\rm cr}$ is higher than the maximum mass obtained by the WD, the WD must spin down before it explodes.",
"This leads to a delay between the time at which the WD has completed its epoch of mass gain and the time of the SN explosion.",
"However, the spin-down time is still unclear, which may have a large range from $<$ 1 Myr to $>$ 1 Gyr (Lindblom, 1999; Yoon and Langer, 2005).",
"The spin-down time may be important for the formation of the SNe Ia with long delay times.",
"The spin-up/spin-down model may provide a route to explain the similarities and the diversity observed in SNe Ia.",
"However, the birthrates, the delay times and the distributions of SN Ia explosion masses are still uncertain in this model.",
"A detailed BPS studies are needed for this."
],
[
"Core-degenerate model",
"Kashi and Soker (2011) recently investigated some possible outcomes of double WD mergers, in which these two components are made of CO.",
"Most simulations and calculations of double WD mergers assume that a merger occurs a long time after the CE ejection, when these two WDs are already cold.",
"In this model, Kashi and Soker (2011) proposed that, a merger occurs within the final stages of the CE, whereas the CO-core is still hot.",
"The merged hot core is supported by rotation until it slows down through the magnetic dipole radiation, and finally explodes.",
"Kashi and Soker (2011) named this as the core-degenerate model, and claimed that this is another scenario to form a massive WD with super-Ch mass that might explode as a super-luminous SN Ia (see also Ilkov and Soker, 2012).",
"A BPS study is required to determine the birthrate and delay time of this model, which are then compared with observations."
],
[
"Collisions of two WDs",
"The WD number densities in globular clusters allow $\\sim $ 10$-$ 100 times collisions between two WDs per year, and the observations of globular clusters in the nearby S0 galaxy NGC 7457 have detected a likely remnant of SNe Ia (Chomiuk et al., 2008).",
"In this context, Raskin et al.",
"(2009) explored collisions between two WDs as a way for producing SNe Ia.",
"They carried out simulations of the collisions between two WDs ($\\sim $$0.6\\,M_{\\odot }$ ) at various impact parameters (the vertical separation of the centers of the WDs).",
"By taking impact parameters less than half of the WD radius before collision, they claimed that the SN explosions induced by such collisions can produce $\\sim $$0.4\\,M_{\\odot }$ of $^{56}$ Ni, making such objects potential candidates for sub-luminous SN Ia events.",
"In a further study, Raskin et al.",
"(2010) argued that two WD collisions could also realize super-Ch mass WD explosions (see also Rosswog et al., 2009a).",
"However, this model predicts a very aspherical explosion, inconsistent with the small continuum polarization level in one of the observed super-luminous SNe Ia (i.e.",
"SN 2009dc; see Tanaka et al., 2010).",
"We note that collisions between two WDs are likely to happen in the dense environments of globular clusters, however the expected of which is still less frequent than that of the double WD mergers."
],
[
"WDs near black holes",
"Wilson and Mathews (2004) proposed a new mechanism for producing SNe Ia, in which relativistic terms enhance the self-gravity of a CO WD when it passes near a black hole.",
"They suggested that this relativistic compression can cause the central density of the WD to exceed the threshold for pycnonuclear reactions so that a thermonuclear runaway occurs.",
"Dearborn et al.",
"(2005) speculated that this mechanism might explain the observed `mixed-morphology' of the Sgr A East SN remnant in the Galactic center.",
"For more studies of this mechanism see Rosswog et al.",
"(2008, 2009b).",
"Due to the expected low rate of a WD passing near a black hole, the expected SN Ia birthrate from this mechanism should be significantly lower than that from normal SNe Ia."
],
[
"Observational constraints ",
"Many observational results can be used to constrain the SN Ia progenitor models, e.g.",
"the properties of SN Ia host galaxies, the birthrates and delay times of SNe Ia, the candidate progenitors of SNe Ia, the surviving companion stars of SNe Ia, the stripped mass of companions due to SN explosion, the signatures of gas outflows from some SN Ia progenitor systems, the wind-blown cavity in SN remnant, the early optical and UV emission of SNe Ia, the early radio and X-ray emission of SNe Ia, and the pre-explosion images and spectropolarimetry of SNe Ia, etc."
],
[
"SN Ia host galaxies",
"There are some observational clues from the galaxies that host SNe Ia.",
"SNe Ia have been known to occur both in young and old stellar populations (e.g.",
"Branch and van den Bergh, 1993), which implies that there is a time delay between the star formation and the SN explosion, ranging from much less than 1 Gyr to at least several Gyr.",
"In addition, SNe Ia in old population tend to be less luminous, and the most luminous SNe Ia appear to prefer young populations with recent star formation (Hamuy et al., 1996; Wang et al.",
"2008a).",
"This indicates that the age of SNe Ia is an important parameter controlling at least part of SN Ia diversity.",
"It was also established that super-luminous SNe Ia preferably occur in relatively metal poor environments with low-mass host galaxies, whereas sub-luminous SNe Ia occur in non star-forming host galaxies with large stellar masses, such as elliptical galaxies (Neill et al., 2009; Taubenberger et al., 2011).",
"The observational homogeneity of SNe Ia implies that a single progenitor system may produce most or all SNe Ia.",
"However, evidence for some observational diversity among SNe Ia, as well as evidence that SNe Ia can be produced by stellar populations that have a wide range of ages, raises the possibility that a variety of progenitor systems may be contributing."
],
[
"Birthrates of SNe Ia",
"The observed SN Ia birthrate in our Galaxy is $\\sim $ 3$\\times 10^{-3}\\,{\\rm yr}^{-1}$ (Cappellaro and Turatto, 1997), which can be used to constrain the progenitor models of SNe Ia.",
"Based on a detailed BPS study, Wang et al.",
"(2010b) systematically investigated Galactic SN Ia birthrates for the SD and DD models, where the SD model includes the WD + MS, WD + RG and WD + He star channels (see Fig.",
"4).",
"They found that the Galactic SN Ia birthrate from the DD model is up to $2.9\\times 10^{-3}\\,{\\rm yr}^{-1}$ by assuming that SNe Ia arise from the merging of two CO WDs that have a combined mass larger than or equal to the Ch mass, which is consistent with the birthrate inferred from observations, whereas the total birthrates from the SD models can only account for about 2/3 of the observations, in which the birthrate from the WD + MS channel is up to $1.8\\times 10^{-3}\\,{\\rm yr}^{-1}$ , the WD + RG channel is up to $3\\times 10^{-5}\\,{\\rm yr}^{-1}$ and the WD + He star channel is up to $0.3\\times 10^{-3}\\,{\\rm yr}^{-1}$ .",
"The Galactic SN Ia birthrate from the WD + RG channel is too low to be compared with that of observations, i.e.",
"SNe Ia from this channel may be rare.",
"However, further studies on this channel are necessary, since this channel may explain some SNe Ia with long delay times.",
"In addition, it has been suggested that both recurrent novae, i.e.",
"RS Oph and T CrB, are probable SN Ia progenitors and belong to the WD + RG channel (e.g.",
"Belczy$\\acute{\\rm n}$ ski and Mikolajewska, 1998; Hachisu et al., 1999b; Sokoloski et al., 2006; Hachisu et al., 2007; Patat et al., 2007a, 2011).",
"For other arguments in favour of the WD + RG channel see Sects.",
"4.2 and 4.3.",
"The SN Ia birthrate in galaxies is the convolution of the delay time distributions (DTDs) with the star formation history (SFH): $\\nu (t)=\\int ^t_0 SFR(t-t^{\\prime })DTDs(t^{\\prime })dt^{\\prime },$ where $SFR$ is the star formation rate, and $t^{\\prime }$ is the delay time of a SN Ia.",
"Due to a constant $SFR$ adopted here, the SN Ia birthrate $\\nu (t)$ is only related to the $DTDs$ , which can be expressed by $DTDs(t)=\\left\\lbrace \\begin{array}{lc}0, & t<{t_1},\\\\DTDs^{\\prime }(t) , & {t_1} \\le t \\le {t_2},\\\\0, & t>{t_2},\\\\\\end{array}\\right.$ where ${t_1}$ and ${t_2}$ are the minimum and maximum delay times of SNe Ia, respectively, and the $DTDs^{\\prime }$ is the distribution of the delay times between ${t_1}$ and ${t_2}$ .",
"If $t$ is larger than the ${t_2}$ , equation (3) can be written as $\\nu (t)={\\rm SFR}\\int ^{t_2}_{t_1}DTDs^{\\prime }(t^{\\prime })dt^{\\prime }={\\rm constant}.$ Therefore, the SN Ia birthrates shown in Fig.",
"4 seem to be completely flat after the first rise."
],
[
"Delay time distributions",
"The delay times of SNe Ia are defined as the time interval between the star formation and SN explosion.",
"The various progenitor models of SNe Ia can be examined by comparing the delay time distributions (DTDs) expected from a progenitor model with that of observations.",
"Many works involve the observational DTDs (e.g.",
"Scannapieco and Bildsten, 2005; Mannucci et al., 2006, 2008; Förster et al., 2006; Aubourg et al., 2008; Botticella et al., 2008; Totani et al., 2008; Schawinski, 2009; Maoz et al., 2011).",
"In recent years, three important observational results for SNe Ia have been proposed, i.e.",
"the strong enhancement of the SN Ia birthrate in radio-loud early-type galaxies, the strong dependence of the SN Ia birthrate on the colors of the host galaxies, and the evolution of the SN Ia birthrate with redshift.",
"Mannucci et al.",
"(2006) claimed that these observational results can be best matched by a bimodal DTD, in which about half of SNe Ia explode soon after starburst with a delay time less than 100 Myr, whereas others have a much wider distribution with a delay time $\\sim $ 3 Gyr.",
"In a further study, Mannucci et al.",
"(2008) suggested that 10% (weak bimodality) to 50% (strong bimodality) of all SNe Ia belong to the young SNe Ia.",
"The existence of the young SN Ia population has also been confirmed by many other observations (e.g.",
"Aubourg et al., 2008; Cooper et al., 2009; Thomson and Chary, 2011), although with a wide range in defining the delay times of the young population.",
"Maoz et al.",
"(2011) presented a new method to recover the DTD, which can avoid some loss of information.",
"In this method, the star formation history of every individual galaxy, or even every galaxy subunit, is convolved with a trial universal DTD, and the resulted current SN Ia birthrate is compared to the number of SNe Ia the galaxy hosted in their survey.",
"They reported that a significant detection of both a prompt SN Ia component, that explodes within 420 Myr of star formation, and a delayed SNe Ia with population that explodes after 2.4 Gyr.",
"Recently, a number of DTD measurements show that the DTD of SNe Ia follows the power-law form of $t^{-1}$ (Maoz and Mannucci, 2012).",
"The power-law form is even different from the strong bimodal DTD suggested by Mannucci et al.",
"(2006), which might indicate that the two-component model is an insufficient description for the observational data.",
"We also note that there are many uncertainties in the observed DTDs, which are dominated by the uncertainties in galactic stellar populations and star formation histories (Maoz and Mannucci, 2012).",
"Many BPS groups work on the theoretical DTDs of SNe Ia (e.g.",
"Yungelson and Livio, 2000; Nelemans et al., 2001; Han and Podsiadlowski, 2004; Wang et al., 2009b, 2010a,b; Ruiter et al., 2009, 2011; Meng and Yang, 2010a; Mennekens et al., 2010; Yu and Jeffery, 2011; Claeys and Pols, 2011).",
"Other theoretical DTDs of SNe Ia have been based on physically motivated mathematical parameterizations (e.g.",
"Greggio and Renzini, 1983; Madau et al., 1998; Greggio, 2005, 2010).",
"Recently, Nelemans et al.",
"(2011) collected data from different BPS groups and made a comparison.",
"They found that the DTDs of different research groups for the DD model agree reasonably well, whereas the SD model have rather different results (see Fig.",
"5).",
"One of the main differences in the results of the SD model is the mass-accretion efficiency with which the accreted H is added onto the surface of the WD (Nelemans et al., 2011).",
"However, the treatment of the mass-accretion efficiency cannot explain all the differences.",
"Nelemans et al.",
"are planning to do that in a forthcoming paper.",
"For the SD model, Nelemans et al.",
"(2011) only considered systems with H-rich donor stars, not including the He-rich donor stars (Wang et al., 2009a).",
"It is worth noting that the He star donor channel can produce SNe Ia effectively with short delay times (accounting for 14% of all SNe Ia in SD model; Wang et al., 2010b), which constitutes the weak bimodality as suggested by Mannucci et al.",
"(2008).",
"Hachisu et al.",
"(2008) recently investigated new binary evolutionary models for SN Ia progenitors, with introducing the mass-stripping effect on a massive MS companion star by winds from a mass-accreting WD.",
"This model can also provide a possible way for producing young SNe Ia, but the model significantly depends on the efficiency of the artificial mass-stripping effect.",
"Additionally, Chen and Li (2007) studied the WD + MS channel by considering a circumbinary disk which extracts the orbital angular momentum from the binary through tidal torques.",
"This study also provides a possible way to produce SNe Ia with long delay times ($\\sim $ 1$-$ 3 Gyr)."
],
[
"Single-degenerate progenitors",
"A number of WD binaries are known to be excellent candidates for SD progenitors of SNe Ia, e.g.",
"U Sco, RS Oph and TCrB (Parthasarathy et al., 2007).",
"All of these binaries contain WDs which are already close to the Ch mass, where the latter two systems are symbiotic binaries containing a giant companion star (see Hachisu et al., 1999b).",
"However, it is unclear whether these massive WD is a CO or an O-Ne-Mg WD; the latter is thought to collapse by forming a neutron star through electron capture on $^{24}$ Mg rather than experience a thermonuclear explosion (for more discussion see Sect.",
"4.2).",
"Meanwhile, there are also two massive WD + He star systems (HD 49798 with its WD companion and V445 Pup), which are good candidates of SN Ia progenitors.",
"HD 49798 is a H depleted subdwarf O6 star and also a single-component spectroscopic binary with an orbital period of 1.548 d (Thackeray, 1970), which contains an X-ray pulsating companion star (RX J0648.0-4418; Israel et al., 1997).",
"The X-ray pulsating companion star is suggested to be a massive WD (Bisscheroux et al., 1997).",
"Based on the pulse time delays and the inclination of the binary, constrained by the duration of the X-ray eclipse, Mereghetti et al.",
"(2009) recently derived the masses of these two components.",
"The corresponding masses are 1.50$\\pm $ 0.05$\\,M_{\\odot }$ for HD 49798 and 1.28$\\pm $ 0.05$\\,M_{\\odot }$ for the WD.",
"According to a detailed binary evolution model with the optically thick wind assumption, Wang and Han (2010c) found that the massive WD can increase its mass to the Ch mass after only a few $10^{4}$ years.",
"Thus, HD 49798 with its WD companion is a likely candidate of a SN Ia progenitor.",
"V445 Pup is the first, and so far only, helium nova detected (Ashok and Banerjee, 2003; Kato and Hachisu, 2003).",
"The outburst of V445 Pup was discovered on 30 December 2000 by Kanatsu (Kato et al., 2000).",
"After that time, a dense dust shell was formed in the ejecta of the outburst, and the star became a strong infrared source, resulting in the star's fading below 20 magnitudes in the $V$ -band (Goranskij et al., 2010).",
"From 2003 to 2009, $BVR$ observations by Goranskij et al.",
"(2010) suggest that the dust absorption minimum finished in 2004, and the remnant reappeared at the level of 18.5 magnitudes in the $V$ -band.",
"Goranskij et al.",
"(2010) reported that the most probable orbital period of the binary system is $\\sim $ 0.65 day.",
"Based on the optically thick wind theory, Kato et al.",
"(2008) presented a free-free emission dominated light curve model of V445 Pup.",
"The light curve fitting in their study shows that the mass of the WD is more than $1.35\\,M_{\\odot }$ , and half of the accreted matter remains on the WD, leading to the mass increase of the WD.",
"In addition, the massive WD is a CO WD instead of an O-Ne-Mg WD, since no indication of neon was observed in the nebula-phase spectrum (Woudt and Steeghs, 2005).",
"Therefore, V445 Pup is a strong candidate of a SN Ia progenitor (e.g.",
"Kato et al., 2008; Woudt et al., 2009)."
],
[
"Double-degenerate progenitors",
"Several systematic searches for double WD systems have been made.",
"The largest survey for this is SPY (ESO SN Ia Progenitor Survey; Napiwotzki et al., 2004; Nelemans et al., 2005; Geier et al., 2007), which aims at finding double WD systems as candidates of SN Ia progenitors.",
"The only likely SN Ia progenitor in this sample is not a double WD system, but the WD + sdB binary KPD 1930+2752 (Maxted et al., 2000).",
"The orbital period of this binary is 2.283 h, the mass of the sdB star is $\\sim $$0.55\\,M_{\\odot }$ , and the mass of the WD is $\\sim $$0.97\\,M_{\\odot }$ .",
"The total mass ($\\sim $$1.52\\,M_{\\odot }$ ) and the merging time ($<$ 0.2 Gyr) of the binary indicate that it is a good candidate of a SN Ia progenitor (Geier et al., 2007).",
"Recently, some other double WD systems have also been found, which may have the total mass close to the Ch mass, and possibly merge in the Hubble-time.",
"These include a binary WD 2020-425 with $P_{\\rm orb}\\sim 0.3$ day, $M_{\\rm 1}+M_{\\rm 2}=1.348\\pm 0.045\\,M_{\\odot }$ (Napiwotzki et al., 2007), V458 Vulpeculae with $P_{\\rm orb}\\sim 0.068$ day, $M_{\\rm 1}\\sim 0.6\\,M_{\\odot }$ , $M_{\\rm 2}>1.0\\,M_{\\odot }$ (Rodríguez-Gil et al., 2010), a close binary star SBS 1150+599A (double-degenerate nucleus of the planetary nebula TS 01) with $P_{\\rm orb}\\sim 0.163$ day, $M_{\\rm 1}=0.54\\pm 0.02\\,M_{\\odot }$ , $M_{\\rm 2}\\sim 0.86\\,M_{\\odot }$ (Tovmassian et al., 2010), and GD687 that will evolve into a double WD system and merge to form a rare supermassive WD with the total mass at least $1\\,M_{\\odot }$ (Geier et al., 2010).",
"There are also some ongoing projects searching for double WD systems, e.g.",
"the SWARMS survey by Badenes et al.",
"(2009b) which is searching for compact WD binaries based on the spectroscopic catalog of the Sloan Digital Sky Survey."
],
[
"Surviving companion stars",
"A SN Ia explosion following the merger of two WDs will leave no compact remnant behind, whereas the companion star in the SD model will survive after a SN explosion and potentially be identifiable by virtue of its anomalous properties.",
"Thus, one way to distinguish between the SD and DD models is to look at the center of a known SN Ia remnant to see whether any surviving companion star is present.",
"A surviving companion star in the SD model would evolve to a WD finally, and Hansen (2003) suggested that the SD model could potentially explain the properties of halo WDs (e.g.",
"their space density and ages).",
"Note that, there has been no conclusive proof yet that any individual object is the surviving companion star of a SN Ia.",
"It will be a promising method to test SN Ia progenitor models by identifying their surviving companions.",
"Han (2008) obtained many properties of the surviving companion stars of SNe Ia with intermediate delay times (100 Myr$-$ 1 Gyr) from the WD + MS channel, which are runaway stars moving away from the center of SN remnants.",
"Wang and Han (2009) studied the properties of the companion stars of the SNe Ia with short delay times ($<$ 100 Myr) from the He star donor channel, which are related to hypervelocity He stars (for more discussion see Sect.",
"4.5; also see Justham et al., 2009).",
"Moreover, Wang and Han (2010d) recently obtained the properties of the surviving companions of the SNe Ia with long delay times ($>$ 1 Gyr) from the WD + MS and WD + RG channels, providing a possible way to explain the formation of the population of single low-mass He WDs ($<$ 0.45$\\,M_{\\odot }$ ; for more discussion see Sect.",
"4.4; also see Justham et al., 2009).",
"The properties of the surviving companion stars (e.g.",
"the masses, the spatial velocities, the effective temperatures, the luminosities and the surface gravities, etc) can be verified by future observations.",
"Tycho G was taken as the surviving companion of Tycho's SN by Ruiz-Lapuente et al.",
"(2004).",
"It has a space velocity of $136\\,{\\rm km/s}$ , more than three times the mean velocity of the stars in the vicinity.",
"Its surface gravity is $\\log \\, (g/{\\rm cm}\\, {\\rm s}^{-2})=3.5\\pm 0.5$ , whereas the effective temperature is $T_{\\rm eff}=5750\\pm 250 {\\rm K}$ (Ruiz-Lapuente et al., 2004).",
"These parameters are compatible with the properties of SN Ia surviving companions from the SD model (e.g.",
"Han, 2008; Wang and Han, 2010d).",
"However, Fuhrmann (2005) argued that Tycho G might be a Milky way thick-disk star that is coincidentally passing the vicinity of the remnant of Tycho's SN.",
"Ihara et al.",
"(2007) also argued that Tycho G may not be the companion star of Tycho's SN, since this star does not show any special properties in its spectrum; the surviving companions of SNe Ia would be contaminated by SN ejecta and show some special characteristics.Pan et al.",
"(2012) studied the impact of SN Ia ejecta on MS, RG and He star companions with the FLASH code.",
"They quantified the amount of contamination on the companion star by the SN ejecta in their simulations, which might help to identify a companion star even a long time after the SN explosion.",
"Recently, Gonz$\\acute{\\rm a}$ lez-Hern$\\acute{\\rm a}$ ndez et al.",
"(2009) presented some evidence that Tycho G may be enriched in $^{56}$ Ni, which could be the result of pollution of the atmosphere with the SN ejecta.",
"By assuming that the companion star in the SD model is co-rotating with the binary orbit at the moment of the SN explosion, the predicted rotational velocity of Tycho G is $\\sim $$100\\,\\rm km/s$ (e.g.",
"Wang and Han, 2010d).",
"However, the rapid rotation predicted by the SD model is not observed in Tycho G ($7.5\\pm 2\\,\\rm km/s$ ; Kerzendorf et al., 2009).",
"This does not yet rule out that this star is the surviving companion.",
"The inferred slow rotation of Tycho G may be related to the angular momentum loss induced by the rapid expansion of its outer shell.",
"Recently, Pan et al.",
"(2012) claimed that the post-impact companion star loses about half of its initial angular momentum for Tycho G, with the rotational velocity decreasing to a quarter of its initial rotational velocity, $\\sim $$37\\,\\rm km/s$ , which is closer to the observed rotational velocity ($7.5\\pm 2\\,\\rm km/s$ ).",
"Therefore, whether Tycho G is the surviving companion of Tycho's SN is still quite debatable.",
"The confliction might be conquered by studying the interaction between the SN ejecta and the rotating companion star.",
"We also note that Lu et al.",
"(2011) recently claimed that the angle between the direction of the non-thermal X-ray arc in Tycho's SNR to the explosive center and the proper motion velocity of Tycho G is well consistent with the theoretical predictions and simulations.",
"This supports Tycho G as the surviving companion of Tycho's SN.",
"Lu et al.",
"(2011) also estimated the parameters of the binary system before the SN explosion, which is useful for constraining progenitor models of SNe Ia.",
"By investigating archival Hubble Space Telescope deep images, Schaefer and Pagnotta (2012) recently reported that the central region of SNR 0509-67.5 (the site of a 1991T-like SN Ia explosion that occurred $\\sim $ 400 years ago) in the Large Magellanic Cloud contains no surviving companion star.",
"Thus, they argued that the progenitor of this particular SN Ia is a double WD system.",
"In a subsequent work, Edwards et al.",
"(2012) used the same method as in Schaefer and Pagnotta (2012) on SNR 0519-69.0, which is a normal SN Ia remnant in the Large Magellanic Cloud with an age of 600$\\pm $ 200 years, and found that the 99.73% error circle contains no post-MS stars for SNR 0519-69.0.",
"Thus, Edwards et al.",
"(2012) claimed to rule out the symbiotic, recurrent nova, He star and spin-up/spin-down models for this particular SN.",
"They argued that SNR 0519-69.0 might be formed from either a supersoft channel or a double WD merge.",
"We note that, based on very short maximum spin-down times, Edwards et al.",
"(2012) excluded the spin-up/spin-down model.",
"However, if the spin-down time is much longer, the results in Edwards et al.",
"(2012) might be different."
],
[
"Stripped mass of companions",
"In the SD model, SN explosion will strip some mass of its non-degenerate companion star.",
"By using two-dimensional Eulerian hydrodynamics simulations, Marietta et al.",
"(2000) examined the interaction of SN ejecta with a MS star, a subgiant star and a RG star.",
"They claimed that the MS and subgiant companions lose $\\sim $ 10$-$ 20% of their mass after the SN explosion, and the RG companion loses about 96%$-$ 98% of its envelope.",
"In this process, these stripped material is mixed with the SN ejecta.",
"Since these stripped material is likely to be dominated by H, this should then lead to easily detectable H emission lines in the SN nebular phase.",
"Unfortunately, no H has ever been detected in a normal SN Ia.",
"The most recently observational upper limits on the amount of H detected are $\\sim $$0.01\\,M_{\\odot }$ (Leonard, 2007),Leonard (2007) obtained deep spectroscopy in the late nebular phase of two well observed SNe Ia (SN 2005am and SN 2005cf), in search of the trace amounts of H and He that would be expected from the SD model.",
"which may provide a strong constraint on the progenitor model of SNe Ia.",
"Additionally, based on the properties of the X-ray arc inside the Tycho's SNR, Lu et al.",
"(2011) also obtained a low stripped mass ($\\le $$0.0083\\,M_{\\odot }$ ), consistent with that from Leonard (2007).",
"These observational limits are inconsistent with Marietta's predictions.",
"Meng et al.",
"(2007) used a simple analytical method to calculate the amount of the stripped masses.",
"They obtained a lower limit of $0.035\\,M_{\\odot }$ for the stripped mass, but their analytic method used oversimplified physics of the interaction between SN Ia ejecta and a companion star.",
"Recently, many updated studies involve the effects of SN explosion on the companion star.",
"However, more realistic stellar models for the companion star do not show stripped mass as small as that close to the Leonard's observational limits, i.e.",
"they do not resolve the conflict between the theory and the observations (Pakmor et al., 2008; Pan et al., 2010, 2012; Liu et al., 2012b).",
"Thus, the high stripped mass from simulations may bring some problems for the SD model.",
"The spin-up/spin-down model may explain the lack of H or He in SNe Ia (Justham, 2011; Di Stefano et al., 2011; Hachisu et al., 2012).",
"In addition, the mixture degree between the SN ejecta and the stripped material may also influence the detection of H or He lines in the nebular spectra of SNe Ia."
],
[
"Circumstellar material after SN explosion",
"In the SD model, non-accreted material blown away from the binary system before SN explosion should remain as circumstellar matter (CSM).",
"Thus, the detection of CSM in SN Ia early spectra would support the SD model.",
"Patat et al.",
"(2007a) found some direct evidence on CSM in a normal SN Ia, i.e.",
"SN 2006X, which was also exceptional in its high ejecta velocity and high reddening (Wang et al., 2008b).",
"Patat et al.",
"(2007a) have observed a variation of Na I doublet lines immediately after the SN explosion, which is interpreted as arising from the ionization and subsequent recombination of Na in CSM.",
"This strongly favours a SD progenitor for this SN.",
"Patat et al.",
"(2007a) suggested that the narrow lines may be explained by a recurrent nova.",
"The time-variable Na I doublet absorption features are also found in SN 1999cl (Blondin et al., 2009) and SN 2007le (Simon et al., 2009).",
"Patat et al.",
"(2007a) argued that the CSM may be common in all SNe Ia, although there exists variation in its detect ability because of viewing angle effects.",
"However, in a subsequent work, Patat et al.",
"(2007b) did not find the same spectral features in SN 2000cx as they did with SN 2006X, which indicates that there might be multiple SD progenitor models.",
"Meanwhile, the derivation of smaller absorption ratio $R_{\\rm V}$ (the ratio of the total to selective absorption by dust) perhaps also suggests the presence of CSM dust around a subclass of SNe Ia (Wang et al., 2009c).",
"More encouragingly, Sternberg et al.",
"(2011) studied the velocity structure of absorbing material along the line of sight to 35 SNe Ia in nearby spiral galaxies via Na I doublet absorption features.",
"They found a strong statistical preference for blue shifted structures, which are likely signatures of gas outflows from the SN Ia progenitor systems.",
"They concluded that many SNe Ia in nearby spiral galaxies may originate in SD systems, and estimated that at least 20% of SNe Ia that occur in spiral galaxies are from the SD progenitors.",
"Recently, Foley et al.",
"(2012) reported that SNe Ia with blue shifted structures have higher ejecta velocities and redder colors at maximum brightness relative to the rest of the SN Ia population, which provides the link between the progenitor systems and properties of SN explosion.",
"This result adds additional confirmation that some SNe Ia are produced from the SD model.",
"However, Shen et al.",
"(2012) argued that such gas outflow signatures could also be induced by winds and/or the mass ejected during the coalescence in the double WDs."
],
[
"SN remnants",
"SN remnants (SNRs) are beautiful astronomical objects that are also of high scientific interest, since they provide direct insights into SN progenitor models and explosion mechanisms.",
"Recent studies by Lu et al.",
"(2011) suggested that the non-thermal X-ray arc in Tycho's SNR is a result of interaction between the SN ejecta and the stripped mass of the companion, strengthening the motivation of studying the progenitor of a SN by studying its SNR.",
"In addition, SNRs may reveal the metallicity of SN progenitors (Badenes et al., 2008).",
"Circumstellar matter (CSM) is predicted by the SD model, which was responsible for creating a low-density bubble (i.e.",
"wind-blown cavity; Badenes et al., 2007).",
"Its modification on larger scales will become apparent during the SNR phase.",
"One of the obstacles the SD model faces is to search for this signatures from SNR observations.",
"Badenes et al.",
"(2007) searched 7 young SN Ia remnants for the wind-blown cavities that would be expected in the SD model.",
"Unfortunately, in every case it appears that the remnant is expanding into a constant density interstellar matter (i.e.",
"there is no wind-blown cavity in these SN remnants).",
"However, Williams et al.",
"(2011) recently reported results from a multi-wavelength analysis of the Galactic SN remnant RCW 86 (remnant of SN 185 A.D.).",
"From hydrodynamic simulations, the observed characteristics of RCW 86 are successfully reproduced by an off-center SN explosion in a low-density cavity carved by the progenitor system (Williams et al., 2011).",
"This makes RCW 86 the first known case of a SN Ia in a wind-blown cavity."
],
[
"Early optical and UV emission of SNe Ia",
"The presence of a non-degenerate companion in the SD model could leave an observable trace in the form of the optical and ultraviolet (UV) emission.",
"Kasen (2010) showed that the collision of the SN ejecta with its companion should produce detectable optical and UV emission in the hours and days following the SN explosion, which can be used to infer the radius of the companion.",
"Thus, the early optical and UV observations of SN ejecta can directly test progenitor models.",
"The optical and UV emission at early times forms mainly in the outer shells of the SN ejecta, in which the unburned outer layers of the WD play an important role in shaping the appearance of the spectrum.",
"Kasen (2010) claimed that these emission would be observable only under favorable viewing angles, and its intensity depends on the nature of the companion star.",
"Hayden et al.",
"(2010) looked for this signal in the rising portion of the $B$ -band light curves of 108 SNe Ia from Sloan Digital Sky Survey, finding no strong evidence of a shock signature in the data.",
"They constrained the companion in the SD model to be less than a $6\\,M_{\\rm \\odot }$ MS star, strongly disfavouring a RG star undergoing RLOF.",
"Recently, Bianco et al.",
"(2011) searched for the signature of a non-degenerate companion star in three years of SN Legacy Survey data by generating synthetic light curves accounting for the shock effects and comparing true and synthetic time series with Kolmogorov-Smirnov tests.",
"Based on the constraining result that the shock effect is more prominent in rest-frame $B$ than $V$ band (for details see Fig.",
"3 of Kasen, 2010), Bianco et al.",
"(2011) excluded a contribution of WD + RG binaries to SN Ia explosions.",
"However, a rather contradictory result for the shock effects was obtained by Ganeshalingam et al.",
"(2011).",
"These shock signatures predicted in Kasen (2010) are based on the assumption that the companion star fills its Roche lobe at the moment of a SN explosion.",
"However, if the binary separation is much larger than the radius of the companion star, the solid angle subtended by the companion would be much smaller.",
"Thus, the shock effect would be lower.",
"Justham (2011) and Di Stefano et al.",
"(2011) argued that the donor star in the SD model may shrink rapidly before the SN explosion, since it would exhaust its H-rich envelope during a long spin-down time of the rapidly rotating WD until the SN explosion.",
"In this condition, the companion star would be a smaller target for the SN ejecta and produce a much smaller shock luminosity than the Roche lobe model considered in Kasen (2010) (see also Hachisu et al., 2012).",
"Therefore, the early optical and UV emission of SN ejecta may be compatible with the SD model.",
"In recent optical and UV observations, Wang et al.",
"(2012) presented UV and optical photometry and early time spectra of four SNe Ia (SNe 2004dt, 2004ef, 2005M, and 2005cf) by using Hubble Space Telescope.",
"One SN Ia in their sample, SN 2004dt, displays a UV excess (the spectra reveal an excess in the 2900$-$ 3500 Å wavelength range, compared with spectra of the other SN Ia events).",
"In their study, the comparison object SN 2006X may also exhibit strong UV emission.",
"The early UV emission may indicate the presence of a non-degenerate companion star in SN Ia progenitor systems."
],
[
"Early radio and X-ray emission of SNe Ia",
"Circumstellar matter (CSM) provides a medium with which the SN ejecta can interact and produce radio synchrotron emission.",
"Many authors have searched for early radio emission from SNe Ia, but no detection has been made (Weiler et al., 1989; Eck et al., 1995, 2002).",
"Hancock et al.",
"(2011) recently have used a stacking analysis of 46 archival Very Large Array observations by Panagia et al.",
"(2006) to set upper limits on the radio emission from SNe Ia in nearby galaxies.",
"They gave an upper limit on the SN Ia peak radio luminosity of $1.2\\times 10^{25}\\,{\\rm erg\\,s^{-1}\\,Hz^{-1}}$ at 5 GHz, which implies an upper limit on the average companion stellar wind mass-loss rate of $1.3\\times 10^{-7}\\,M_{\\odot }\\,\\rm yr^{-1}$ before a SN explosion.",
"Hancock et al.",
"(2011) argued that these limits challenge expectations if the SN ejecta were encountering a CSM from the SD model.",
"Aside from radio emission, the interaction of SN ejecta with the CSM can also produce X-ray emission.",
"SN shock would run into CSM and heat it to high enough temperatures ($\\sim $$10^{6}-10^{9}$ K), resulting in thermal X-rays (Chevalier, 1990).",
"Compared with radio emission, X-rays from SNe Ia result from a different process and from different regions in the shocked CSM.",
"Thus, it is a completely independent method to constrain progenitor model via detecting early X-ray emission of SNe Ia.",
"Russel and Immler (2012) recently considered 53 SNe Ia observed by the Swift X-Ray Telescope.",
"They gave an upper limit on the X-ray luminosity ($0.2-10$ keV) of $1.7\\times 10^{38}\\,{\\rm erg\\,s^{-1}}$ , which implies an upper limit on mass-loss rate of $1.1\\times 10^{-6}\\,M_{\\odot }\\,\\rm yr^{-1}\\times (\\nu _{w})/(10\\,km\\,s^{-1})$ , where $\\rm \\nu _{w}$ is the wind speed for red supergiants that ranges from 5 to 25 $\\rm km\\,s^{-1}$ .",
"Russel and Immler (2012) claimed that these limits exclude massive or evolved stars as the companions in progenitor systems of SNe Ia, but allow the possibility of MS and WD as the companion.",
"According to the spin-up/spin-down model of SNe Ia suggested by Justham (2011) and Di Stefano et al.",
"(2011), there is a delay between the time at which the WD has completed its mass-accretion and the time of the SN explosion.",
"Since the matter ejected from the binary system during the mass-transfer has a chance to become diffuse, the SN explosion will occur in a medium with a density similar to that of typical regions of the interstellar medium.",
"Therefore, the SD model may be compatible with the upper limits from SN Ia radio and X-ray detection."
],
[
"Pre-explosion images",
"One of the methods to clarify SN Ia progenitor models is to directly detect the progenitor of a SN Ia in pre-explosion images of the position where the SN occurred.",
"Voss and Nelemans (2008) first studied the pre-explosion archival X-ray images at the position of the recent SN 2007on, and considered that its progenitor may be a WD + RG system.",
"However, Roelofs et al.",
"(2008) did not detect any X-ray source in images taken six weeks after SN 2007on's optical maximum and found an offset between the SN and the measured X-ray source position.",
"Nelemans et al.",
"(2008) also obtained an ambiguous answer.",
"Nielsen et al.",
"(2011) recently derived the upper limits of the X-ray luminosities from the locations of ten SNe Ia in nearby galaxies ($<$ 25 Mpc) before the explosions, most above a few $10^{38}\\,{\\rm erg\\,s^{-1}}$ (for details see Fig.",
"1 of Nielsen et al., 2011), which indicates that the progenitors of these SNe Ia were not bright supersoft X-ray sources shortly before they exploded as SNe Ia.",
"However, the upper limits are not constraining enough to rule out less bright supersoft X-ray progenitors (Nielsen et al., 2011).",
"Future observations may shed light on the connection between SN Ia progenitors and X-ray emission.",
"SN 2011fe occurred in M101 at a distance of 6.4 Mpc is the second closest SN Ia in the digital imaging era,The closest SN Ia in the digital imaging era is SN 1986G that exploded in NGC 5128 at a distance of $\\sim $ 4 Mpc (Frogel et al., 1987).",
"which was discovered by the Palomar Transient Factory survey less than a day after its explosion (Nugent et al., 2011a), and quickly followed up in many wavebands (Li et al., 2011; Nugent et al., 2011b; Smith et al., 2011; Tammann and Reindl, 2011; Patat et al., 2011b; Liu et al., 2012; Horesh et al., 2012; Chomiuk et al., 2012; Bloom et al., 2012; Brown et al., 2012a; Margutti et al., 2012).",
"Li et al.",
"(2011) used extensive historical imaging obtained at the location of SN 2011fe to constrain the visible-light luminosity of the progenitor to be 10$-$ 100 times fainter than previous limits on other SN Ia progenitors.",
"This result rules out luminous RG stars and most He stars as the mass donor star of this SN progenitor.",
"These observations favour a scenario where the progenitor of SN 2011fe accreted material either from WD, or via RLOF from a MS or subgiant companion.",
"In a subsequent work, Liu et al.",
"(2012) also excluded its progenitor system with the most hottest photospheres by constraining X-ray properties prior to the SN explosion.",
"Very recently, Horesh et al.",
"(2012) set upper limits on both radio and X-ray emission from SN 2011fe, excluding the presence of a circumstellar matter from a giant donor star.",
"Based on deep radio observations, Chomiuk et al.",
"(2012) also excluded the presence of circumstellar matter.",
"By using early optical and UV observations of SN 2011fe, Nugent et al.",
"(2011b) excluded the presence of shock effects from SN ejecta hitting a companion, and put a strict upper limit to the exploding star radius ($\\le $$0.1\\,R_{\\odot }$ ), thus providing a direct evidence that the progenitor is a compact star.",
"A recent study by Bloom et al.",
"(2012) also ruled out a MS star as the mass donor star and seem to favor a DD progenitor for SN 2011fe (also see Brown et al., 2012a).",
"We note that the spin-up/spin-down model potentially affects the conclusions above."
],
[
"Polarization of SNe Ia",
"Spectropolarimetry provides a direct probe of early time SN geometry, which is an important diagnostic tool for discriminating among SN Ia progenitor systems and theories of SN explosion physics (see Livio and Pringle, 2011).",
"A hot young SN atmosphere is dominated by the electron scattering that is highly polarizing.",
"For an unresolved source with a spherical distribution of scattering electrons, the directional components of the electric vectors of the scattered photons counteract exactly, resulting in zero net linear polarization.",
"However, an incomplete cancelation will be derived from any asymmetry in the distribution of the scattering electrons, or of absorbing material overlying the electron-scattering atmosphere, which produces a net polarization (Leonard and Filippenko, 2005).",
"SN asymmetry can therefore be measured via spectropolarimetry, since asymmetric electron scattering leads to polarization vectors that do not cancel.",
"Most normal SNe Ia are found to be spherically symmetric (a rather low polarization, $\\lesssim $ 0.3%; Wang et al., 1996; Wang and Wheeler, 2008), but asymmetry has been detected at significant levels for a range of SN Ia subclasses, e.g.",
"sub-luminous SNe Ia with a continuum polarization about 0.3%$-$ 0.8% (Howell et al., 2001), and high-velocity (HV) SNe Ia with a high polarization about 2%, the spectra of which around maximum light are characterized by unusually broad and highly blueshifted absorption troughs in many line features (Leonard et al., 2005).",
"Leonard et al.",
"(2005) claimed that the following order emerges in terms of increasing strength of line-polarization features: normal/over-luminous SNe Ia $<$ sub-luminous SNe Ia $<$ HV SNe Ia.",
"They argued that the most convincing explanation for the linear polarization of all objects is partial obscuration of the photosphere by clumps of intermediate-mass elements forged in the SN explosion.",
"For a review of SN Ia polarimetric studies see Wang and Wheeler (2008).",
"The explosion mechanism itself may produce asymmetry due to off-center explosion, and thus a polarization spectrum is expected (Plewa et al., 2004; Kasen and Plewa, 2005).",
"Thus, it is possible to obtain insight into the SN explosion physics with spectropolarimetry.",
"Meanwhile, the progenitor systems may also cause the asymmetry.",
"The SD model provides a natural way to produce the asymmetry.",
"The existence of a companion in the SD model may change the configuration of the SN ejecta (e.g.",
"a cone-shaped hole shadowed by the companion), and thus a polarization spectrum is expected (Marietta et al., 2000; Kasen et al., 2004; Meng and Yang, 2010b).By using smoothed particle hydrodynamics simulations, García-Senz et al.",
"(2012) studied the interaction of the hole, SN material and ambient medium.",
"They concluded that the hole could remain open in the SNR for hundreds of years, suggesting the hole could affect its structure and evolution.",
"In addition, the DD model may also naturally result in an asymmetry of the distribution of SN ejecta.",
"One relevant mechanism is the rapid rotation of a WD before a SN explosion, which leads to a change in the stellar shape.",
"Another is that there may be a thick accretion disc around the CO WD, which may be an origin of asymmetry in the configuration of the SN ejecta (e.g.",
"Hillebrandt and Niemeyer, 2000).",
"Livio and Pringle (2011) argued that the nature of the correlation between the polarization and the observed SN Ia properties can be used to distinguish between the SD and DD models.",
"As a specific example, they considered possible correlations between the polarization and the velocity gradient; a SN explosion is viewed from one pole it is seen as a high velocity gradient event at early phases with redshifts in late-time emission lines, while if it is viewed from the other pole it is seen as a low-velocity gradient event with blueshifts at late phases (Maeda et al, 2010).",
"In the SD model, it is expected that the velocity gradient is a two-valued function of polarization, with the largest and smallest values corresponding to essentially zero polarization.",
"In the DD model, it is expected that the observed SN properties (i.e.",
"velocity gradient) is a single-valued and monotonic function of polarization.",
"For details see Fig.",
"1 of Livio and Pringle (2011)."
],
[
"Related objects",
"There are some objects that may be related to the progenitors and surviving companions of SNe Ia in observations, e.g.",
"supersoft X-ray sources, cataclysmic variables, symbiotic systems, single low-mass He WDs and hypervelocity He stars, etc."
],
[
"Supersoft X-ray sources",
"Supersoft X-ray sources (SSSs) are one of the most promising progenitor candidates of SNe Ia.",
"Binaries in which steady nuclear burning takes place on the surface of the WDs have been identified with bright SSSs, discovered by the ROSAT satellite (van den Heuvel et al., 1992; Rappaport et al., 1994; Kahabka and van den Heuvel, 1997).",
"Most of the known SSSs are located in the Large Magellanic Cloud, Small Magellanic Cloud and M31.",
"They typically emit $10^{36}-10^{38}\\,{\\rm erg\\,s^{-1}}$ in the form of very soft X-rays, peaking in the energy range 20$-$ 100 eV.",
"van den Heuvel et al.",
"(1992) proposed a model that the relatively massive WD sustains steady H-burning from a MS or subgiant donor star.",
"They suggested that the mass-accretion occurs at an appropriate rate, in the range of $1.0-4.0\\times 10^{-7}\\,M_{\\rm \\odot }{\\rm yr}^{-1}$ .",
"Meanwhile, a WD + He star system has luminosity around $10^{37}-10^{38}\\,{\\rm erg\\,s^{-1}}$ when the He-burning is stable on the surface of the WD, which is consistent with that of observed from SSSs.",
"Thus, WD + He star systems may also appear as SSSs before SN explosions (Iben and Tutukov, 1994; Yoon and Langer, 2003; Wang et al., 2009a).",
"In addition, in the context of SSSs, the time that elapses between the double WD merger and the SN explosion is about $10^{5}$ yr, and during this phase the merged object would look like as a SSS (with $T\\sim 0.5-1\\times 10^{6}$ K and $L_{\\rm X-ray}\\sim 10^{37}\\,{\\rm erg\\,s^{-1}}$ ), which could provide a potential test for the DD model (Yoon et al., 2007; Voss and Nelemans, 2008).",
"Note that the Galactic interstellar absorption and circumstellar matter may play an important role in the obscuration of X-rays.",
"Recently, Di Stefano (2010a,b) called attention to the fact that in the galaxies of different morphological types there exists a significant (up to 2 orders of magnitude) deficit of SSSs as compared with expectations based on SN Ia birthrates from the SD model.",
"Gilfanov and Bogdán (2010) also obtained the same conclusion, based on the study of the luminosity of elliptical galaxies in the supersoft X-ray range.",
"However, these authors did not consider the binary evolution.",
"A typical binary in the SD model undergoes three evolutionary stages in order of time before SN explosion, i.e.",
"the wind phase, the supersoft X-ray source phase and the recurrent nova phase, since the mass-accretion rate decreases with time as the mass of the donor star decreases.",
"The supersoft X-ray source phase is only a short time (e.g.",
"a few hundred thousand years), since the SD progenitor system spends a large part of lifetime in the wind phase or recurrent nova phase on its way to SN explosion (e.g.",
"Han and Podsiadlowski, 2004; Meng et al., 2009; Wang et al., 2009a, 2010a; Hachisu et al., 2010; Meng and Yang, 2011a).",
"Lipunov et al.",
"(2011) also considered that the theoretical SSS lifetimes and X-ray luminosities have been overestimated."
],
[
"Cataclysmic variables",
"Cataclysmic variable stars (CVs) are stars that irregularly increase in brightness by a large factor, then drop back down to a quiescent phase (Warner, 1995).",
"They consist of two component stars: a WD primary and a mass donor star.",
"CVs are usually divided into several types, such as classical novae, recurrent novae, nova-like variables, dwarf novae, magnetic CVs and AM CVns, etc (Warner, 1995).",
"Among these subclasses of CVs, recurrent novae and dwarf novae are the most probable candidates of SN Ia progenitors.",
"Recurrent novae have outbursts of about 4$-$ 9 magnitudes, and exhibit multiple outbursts at intervals of 10$-$ 80 years (Warner, 1995).",
"They contain a massive WD and a relatively high mass-accretion rate (but below steady burning rate).",
"The evolution of the outburst is very fast.",
"Since the heavy element enhancement is not detected in recurrent novae, their WD mass is supposed to increase after each outburst.",
"Additionally, nova outbursts require a relatively high mass-accretion rate onto a massive WD to explain the recurring nova outbursts.",
"Thus, these objects become some of the most likely candidates of SN Ia progenitors (Starrfield et al., 1985; Hachisu and Kato, 2001).",
"However, this class of objects are rare, with ten Galactic recurrent novae, two in the Large Magellanic Cloud and a few in M31.",
"Recurrent novae and SSSs differ in the mass-accretion rate from a mass donor star onto the WD; SSSs have steady nuclear burning on the surface of the WD, while recurrent novae happen at rates that allow shell flashes.",
"By modeling the decline of the outburst light curves of some recurrent novae (T CrB, RS Oph, V745 Sco and V3890 Sgr), Hachisu and Kato (2001) suggested that these WDs are approaching the Ch mass and will produce SNe Ia.",
"Recurrent nova systems like RS Oph have been proposed as possible SN Ia progenitors, based on the high mass of the accreting WD.",
"Patat et al.",
"(2011a) investigated the circumstellar environment of RS Oph and its structure, suggesting that the recurrent eruptions might create complex structures within the material lost by the donor star.",
"This may establish a strong link between RS Oph and the progenitor system of SN 2006X, for which similar features have been detected.",
"Recurrent nova U Sco contains a WD of $M_{\\rm WD}=1.55\\pm 0.24\\,M_{\\odot }$ and a secondary star with $M_{\\rm 2}=0.88\\pm 0.17\\,M_{\\odot }$ orbiting with a period $P_{\\rm orb}\\sim 0.163$ day (Thoroughgood et al., 2001).",
"The high mass of the WD implies that U Sco is a strong progenitor candidate of a SN Ia (Thoroughgood et al., 2001; also see Hachisu et al., 2000).",
"However, the nebular spectra of U Sco displays that the relative abundance of [Ne/O] is 1.69, which is higher than that of the typical [Ne/O] abundance found in classical novae from CO WDs and suggests that U Sco has a O-Ne-Mg WD (Mason, 2011).",
"Thus, U Sco may not explode as a SN Ia but rather collapse to a neutron star by electron capture on $^{24}$ Mg.",
"Dwarf novae have multiple outbursts ranging in brightness from 2 to 5 magnitudes, and exhibit intervals from days to decades.",
"The lifetime of an outburst is typically from 2 to 20 days and is related to the outburst interval.",
"Dwarf nova outbursts are usually attributed to the release of gravitational energy resulted from an instability in the accretion disk or by sudden mass-transfer via the disk (Warner, 1995).",
"Observationally, there are a number of dwarf novae in which the WD is about $1\\,M_{\\odot }$ (e.g.",
"GK Per, SS Aur, HL CMa, U Gem, Z Cam, SY Cnc, OY Car, TW Vir, AM Her, SS Cyg, RU Peg, GD 552 and IP Peg, etc).",
"The secondaries of these WD binaries are K or M stars ($<$$1\\,M_{\\odot }$ ).",
"A few of these systems with early K type secondaries may have the WD mass close to the Ch mass.",
"It has been suggested that the mass-accretion rate onto a WD during a dwarf nova outbursts can be sufficiently high to allow steady nuclear burning of the accreted matter and growth of the WD mass (King et al., 2003; Xu and Li, 2009; Wang et al., 2010a; Meng and Yang, 2010a).",
"However, whether dwarf nova outbursts can increase the mass of a WD close to Ch mass is still a problem (e.g.",
"Hachisu et al., 2010)."
],
[
"Symbiotic systems",
"Symbiotic systems are long-period binaries, consisting of a RG and a hot object that is usually a WD (Truran and Cameron, 1971).",
"The hot object accretes and burns material from the RG star via stellar wind in most cases, but could also be RLOF in some cases.",
"They usually show strong emission lines from surrounding circumstellar material ionized by the hot component, and low temperature absorption features from the RG.",
"Symbiotic systems are essential to understand the evolution and interaction of detached and semi-detached binaries.",
"There are two distinct subclasses of symbiotic stars, i.e.",
"the S-type (stellar) with normal RG stars and orbital periods of about 1$-$ 15 years, and the D-type (dusty) with Mira primaries usually surrounded by a warm dust shell and orbital periods longer than 10 years.",
"Symbiotic stars are thus interacting binaries with the longest orbital periods.",
"Tang et al.",
"(2012) recently found a peculiar symbiotic system J0757 that consists of an accreting WD and a RG.",
"In quiescent phase, however, it doesn't show any signature of “symbiotic”.",
"Thus, it is a missing population among symbiotic systems, which may contribute to a significant fraction of SN Ia.",
"Moreover, this object showed a 10 year flare in the 1940s, possibly from H-shell burning on the surface of the WD and without significant mass-loss.",
"Therefore, the WD could grow effectively.",
"The presence of both the accreting WD and the RG star makes symbiotic binaries a promising nursery for the production of SNe Ia.",
"However, due to the low efficiency of matter accumulation by a WD accreting material from the stellar wind, SN Ia birthrate from these symbiotic systems is relative low (e.g.",
"Yungelson and Livio, 1998)."
],
[
"Single low-mass He WDs",
"The existence of a population of single low-mass He WDs (LMWDs; $<$$0.45\\,M_{\\odot }$ ) is supported by some recent observations (e.g.",
"Marsh et al., 1995; Kilic et al., 2007).",
"However, it is still unclear how to form single LMWDs.",
"It has been suggested that single LMWDs could be produced by single old metal-rich stars that experience significant mass-loss before the central He flash (Kalirai et al., 2007; Kilic et al., 2007).",
"However, the study of the initial-final mass relation for stars by Han et al.",
"(1994) implied that only LMWDs with masses larger than $0.4\\,M_{\\odot }$ might be produced from such a single star scenario, even at high metallicity environment (Meng et al., 2008).",
"Thus, it would be difficult to conclude that single stars can produce LMWDs of $\\sim $$0.2\\,M_{\\odot }$ .",
"Justham et al.",
"(2009) inferred an attractive formation scenario for single LMWDs, which could be formed in binaries where their companions have exploded as SNe Ia.",
"Wang and Han (2010d) recently found that the surviving companions of the old SNe Ia from the WD + MS and WD + RG channels have low masses, providing a possible way to explain the formation of the population of single LMWDs (see also Meng and yang, 2010c).",
"Conversely, the observed single LMWDs may provide evidence that at least some SN Ia explosions have occurred with non-degenerate donors (such as MS or RG donors).",
"We note that Nelemans and Tauris (1998) also proposed an alternative scenario to form single LMWDs from a solar-like star accompanied by a massive planet, or a brown dwarf, in a relatively close binary orbit."
],
[
"Hypervelocity stars",
"In recent years, hypervelocity stars (HVSs) have been observed in the halo of the Galaxy.",
"HVSs are stars with velocities so high that they are able to escape the gravitational pull of the Galaxy.",
"However, it is still not clear how to form HVSs (for a review see Tutukov and Fedorova, 2009).",
"It has been suggested that such HVSs can be formed by the tidal disruption of a binary through interaction with the super-massive black hole (SMBH) at the Galactic center (GC) (Hills, 1988; Yu and Tremaine, 2003; Zhang et al., 2010).",
"The first three HVSs have only recently been discovered serendipitously (e.g.",
"Brown et al., 2005; Hirsch et al., 2005; Edelmann et al., 2005).",
"Up to now, about 17 confirmed HVSs have been discovered in the Galaxy (Brown et al., 2009; Tillich et al., 2009), most of which are B-type stars, probably with masses ranging from 3 to 5 $M_\\odot $ (Brown et al., 2005, 2009; Edelmann et al., 2005).",
"The HVS B-type stars are demonstrated short-lived B-type stars at 50$-$ 100 kpc distances that are significantly unbound based on radial velocity alone.",
"Their observed properties (ages, flight times, latitude distribution) are consistent with the Galactic center ejection scenario (Brown et al., 2012b).",
"One HVS, HE 0437-5439, is known to be an apparently normal early B-type star.",
"Edelmann et al.",
"(2005) suggested that the star could have originated in the Large Magellanic Cloud, since it is much closer to this galaxy ($\\sim $ 18 kpc) than to the GC (see also Przybilla et al., 2008).",
"Li et al.",
"(2012) recently reported 13 metal-poor F-type HVS candidates which are selected from 370,000 stars of the data release 7 of the Sloan Digital Sky Survey.",
"With a detailed analysis of the kinematics of these stars, they claimed that seven of them were likely ejected from the GC or the Galactic disk, four neither originated from the GC nor the Galactic disk, and the other two were possibly ejected from either the Galactic disk or other regions.",
"At present, only one HVS, US 708, is an extremely He-rich sdO star in the Galactic halo, with a heliocentric radial velocity of +$708\\pm 15$ km/s.",
"Hirsch et al.",
"(2005) speculated that US 708 was formed by the merger of two He WDs in a close binary induced by the interaction with the SMBH in the GC and then escaped.",
"Recently, Perets (2009) suggested that US 708 may have been ejected as a binary from a triple disruption by the SMBH, which later on evolved and merged to form a sdO star.",
"However, the evolutionary lifetime of US 708 is not enough if it originated from the GC.",
"Wang and Han (2009) found that the surviving companions from the He star donor channel have a high spatial velocity ($>$ 400 km/s) after a SN explosion, which could be an alternative origin for HVSs, especially for HVSs such as US 708 (see also Justham et al., 2009).",
"Considering the local velocity nearby the Sun ($\\sim $ 220 km/s), Wang and Han (2009) found that about 30$\\%$ of the surviving companions may be observed to have velocity above 700 km/s.",
"In addition, a SN asymmetric explosion may also enhance the velocity of the surviving companion.",
"Thus, a surviving companion star in the He star donor channel may have a high velocity like US 708.",
"SNe Ia have been successfully used as cosmological distance candles, but there exists spectroscopic diversity among SNe Ia that is presently not well understood, nor how this diversity is linked to the properties of their progenitors (e.g.",
"Branch et al., 1995; Livio, 2000).",
"When SNe Ia are applied as distance indicators, the Phillips relation is adopted (i.e.",
"the luminosity-width relation; brighter SNe Ia have wider light curves), which implies that SN Ia luminosity is mainly determined by one parameter.",
"In an attempt to quantify the rate of spectroscopically peculiar SNe Ia in the existing observed sample, Branch et al.",
"(1993) compiled a set of 84 SNe Ia and found that about $83\\% -89\\%$ of the sample are normal.",
"According to the study of Li et al.",
"(2001), however, only $64\\%\\pm 12\\%$ of the observed SNe Ia are normal in a volume-limited search.There is increasing evidence showing that even the normal SNe Ia exhibit diversity in their spectral features (e.g.",
"Branch et al., 2009; Wang et al., 2009c; Blondin et al., 2012).",
"Wang et al.",
"(2009c) investigated 158 relatively normal SNe Ia by dividing them into two groups in terms of the expansion velocity inferred from the absorption minimum of the Si${\\rm II}$ ${\\rm \\lambda }6355$ line around maximum light.",
"They claimed that, one group “Normal” consists of SNe Ia with an average expansion velocity $10,600\\pm 400\\rm km/s$ , but another group “HV” consists of objects with higher velocities $\\sim $$11,800\\rm km/s$ .",
"The HV SNe Ia are found to prefer a smaller extinction ratio $R_{V}$ (relative to the Normal ones), which might suggest the presence of circumstellar material (see Sect.",
"3.7).",
"The total rate of peculiar SNe Ia could be as high as $36\\%\\pm 9\\%$ ; the rates are $16\\%\\pm 7\\%$ and $20\\%\\pm 7\\%$ for SN 1991bg-like objects and SN 1991T-like objects, respectively.",
"SN 1991bg-like objects both rise to their maximum and decline more quickly, and are sub-luminous relative to normal SNe Ia, whereas SN 1991T-like objects both rise to their maximum and decline more slowly, and are more luminous relative to normal SNe Ia.",
"These two types of peculiar events obey the luminosity-width relation.",
"However, a subset of SNe Ia apparently deviate from the luminosity-width relation, e.g.",
"some were observed with exceptionally high luminosity or extremely low luminosity, which may have progenitors with masses exceeding or below the standard Ch mass limit (e.g.",
"Howell et al., 2006; Foley et al., 2009).",
"This implies that at least some SNe Ia can be produced by a variety of different progenitor systems, and probably suggests that SN Ia luminosity is not the single parameter of the light curve shape.",
"It has been suggested that the amount of $^{\\rm 56}$ Ni formed during a SN Ia explosion dominates its maximum luminosity (Arnett, 1982), but the origin of the variation of the amount of $^{\\rm 56}$ Ni for different SNe Ia is still unclear (the derived $^{\\rm 56}$ Ni masses for different SNe Ia could vary by a factor of ten; Wang et al., 2008a).",
"Many efforts have been paid to solve this problem.",
"Umeda et al.",
"(1999) suggested that the average ratio of carbon to oxygen (C/O) of a WD at the moment of a SN explosion is the dominant parameter for the Phillips relation, i.e.",
"the higher the C/O ratio, the larger the amount of $^{56}$ Ni, and then the higher the maximum luminosity (see also Meng and Yang, 2011b).",
"However, 3D simulations by Röpke and Hillebrandt (2004) suggest that different C/O ratios have a negligible effect on the amount of $^{56}$ Ni produced.",
"At present, the studies from the explosion models of SNe Ia indicate that the number of ignition points at the center of WDs or the transition density from deflagration to detonation dominates the production of $^{56}$ Ni, and consequently the maximum luminosity (e.g.",
"Hillebrandt and Niemeyer, 2000; Höflich et al., 2010; Kasen et al., 2010).",
"It was claimed that the ignition intensity (the number of ignition points) in the center of WDs is a useful parameter in interpreting the Phillips relation (Hillebrandt and Niemeyer, 2000).",
"Based on the SD model, Lesaffre et al.",
"(2006) carried out a systematic study of the sensitivity of carbon ignition conditions for the Ch mass WDs on various properties, and claimed that the central density of a WD at the carbon ignition may be the origin of the scatter of the maximum luminosity.",
"This suggestion was further supported by detailed multi-dimensional numerical simulations of SN explosions (Krueger et al., 2010).",
"We note that the WD cooling time before mass-accretion is less than 1 Gyr in the simulations of Lesaffre et al.",
"(2006) and Krueger et al.",
"(2010).",
"However, there are SNe Ia with the delay times $\\sim $ 10 Gyr in observations.",
"The WDs with such a long cooling time may become more degenerate before the onset of the mass-accretion phase.",
"Some other processes, such as carbon and oxygen separation or crystallization, may occur and dominate the properties of the CO WD (Fontaine et al., 2001).",
"How the extremely degenerate conditions affect the properties of SNe Ia still remains unclear.",
"The suggestion of Lesaffre et al.",
"(2006) should be checked carefully under extremely degenerate conditions.",
"Adopting the WD mass-accretion process in Lesaffre et al.",
"(2006), Chen et al.",
"(2012) recently studied the evolution of various CO WDs from the onset of mass-accretion to carbon ignition at Ch mass limit.",
"The study shows that the carbon ignition generally occurs at the center for hot low-mass CO WDs but off-center for cool massive ones, which may provide more information for the explosion models of SNe Ia.",
"Some numerical and synthetical results showed that the metallicity may have an effect on the final amount of $^{56}$ Ni, and thus the maximum luminosity of SNe Ia (Timmes et al., 2003; Podsiadlowski et al., 2006; Bravo et al., 2010).",
"There is also some other evidence of the correlation between the properties of SNe Ia and metallicity from observations (e.g.",
"Branch and Bergh, 1993; Hamuy et al., 1996; Wang et al, 1997; Gallagher et al., 2008; Sullivan, 2006; Howell et al., 2009a; Sullivan et al., 2010).",
"Podsiadlowski et al.",
"(2006) introduced metallicity as a second parameter that affects the light curve shape.",
"For a reasonable range of metallicity, this may account for the observed spread in the Phillips relation.",
"Since metallicity in the Universe has evolved with time, this introduces an undesirable evolutionary effect in the SN Ia distance method, which could mimic the effect of an accelerating Universe.",
"We also note that Maeda et al.",
"(2010) argued that the origin of spectral evolution diversity in SNe Ia can be understood by an asymmetry in the SN explosion combined with the observer's viewing angle.",
"Moreover, Parrent et al.",
"(2011) investigated the presence of C${\\rm II}$ ${\\rm \\lambda }6580$ in the optical spectra of 19 SNe Ia.",
"Most of the objects in their sample that exhibit C${\\rm II}$ ${\\rm \\lambda }6580$ absorption features are of the low-velocity gradient subtype.",
"This study indicates that the morphology of carbon-rich regions is consistent with either a spherical distribution or a hemispheric asymmetry, supporting the idea that SN Ia diversity may be a result of off-center ignition coupled with observer's viewing angle."
],
[
"Impacts of SN Ia progenitors on some fields",
"The identification of SN Ia progenitors also has important impacts on some other astrophysical fields, e.g.",
"cosmology, the evolution of galaxies, SN explosion models and binary evolution theories, etc (e.g.",
"Branch et al., 1995; Livio, 2000).",
"Cosmology.",
"It is feasible to improve SNe Ia as mature cosmological probes, since the dominant systematic errors are clear, which include photometric calibration, selection effects, reddening and population-dependent differences, etc.",
"In the next decade, SNe Ia are proposed to be cosmological probes for testing the evolution of the dark energy equation of state with time (Howell et al., 2009b).",
"The use of SNe Ia as one of the main ways to determine the Hubble constant ($H_0$ ) and cosmological parameters (e.g.",
"$\\Omega _{M}$ and $\\Omega _{\\Lambda }$; Riess et al., 1998; Perlmutter et al., 1999), requires our understanding of the evolution of the luminosities and birthrates of SNe Ia with cosmic epoch.",
"Both of these depend on the nature of their progenitors.",
"Meanwhile, the evolution of the progenitor systems or a changing mix of different progenitors may bias cosmological inferences.",
"For a recent review of this field see Howell (2011).",
"Galaxy evolution.",
"Aside from cosmology, the evolution of galaxies depends on the radiative, kinetic energy, nucleosynthetic outputs (e.g.",
"Kauffmann et al., 1993; Liu et al., 2012a) and the birthrates of SNe Ia with time, which all depend on the nature of the progenitor systems.",
"SNe Ia are also laboratories for some extreme physics, e.g.",
"they are accelerators of cosmic rays and as sources of kinetic energy in galaxy evolution processes (e.g.",
"Helder et al., 2009; Powell et al., 2011).",
"Especially, SNe Ia regulate galactic and cluster's chemical evolution.",
"Due to the main contribution of iron to their host galaxies, SNe Ia are a key part of our understanding of galactic chemical evolution (e.g.",
"Greggio and Renzini, 1983; Matteucci and Greggio, 1986).",
"The existence of young and old populations of SNe Ia suggested by recent observations may have an important effect on models of galactic chemical evolution, since they would return large amounts of iron to the interstellar medium either much earlier or much later than previously thought.",
"Explosion models.",
"SNe Ia provide natural laboratories for studying the physics of hydrodynamic and nuclear processes with extreme conditions.",
"The link between the progenitor models and the explosion models is presently one of the weakest points in our understanding of SNe Ia (Hillebrandt and Niemeyer, 2000).",
"Due to some uncertainties that still exist in the SN explosion mechanism itself, a knowledge of the initial conditions and the distribution of matter in the environment of the exploding star is essential for our understanding of SN explosion, e.g.",
"the ignition density may depend on the initial WD mass, the age of the progenitor, the metallicity and the treatment of rotation in the progenitor.",
"Moreover, different progenitor models may lead to different WD structures before SN explosion.",
"Lu et al.",
"(2011) recently studied the properties of the Tycho's SNR.",
"They estimated the parameters of the binary system before the SN explosion, which may shed lights on the possible explosion models.",
"Binary evolution theories.",
"The identification of SN Ia progenitors, coupled with observationally determined SN Ia birthrates and delay times will help to place meaningful constraints on some theories of binary evolution, e.g.",
"the mass-transfer between two stars, the mass-accretion efficiency of WDs, etc (e.g.",
"Hachisu et al., 1996; Han and Podsiadlowski, 2004; Wang et al., 2009a).",
"Especially, it is possible that the CE efficiency parameter may be constrained (e.g.",
"Meng et al, 2011), which is important in binary evolution and BPS studies."
],
[
"Summary",
"In this article, various progenitor models proposed in the literatures are reviewed, including some variants of SD and DD models.",
"We addressed some observational ways to test the current progenitor models and introduced some observed objects that may be related to the progenitors and the surviving companion stars of SNe Ia.",
"We also discussed the impacts of SN Ia progenitors on some fields.",
"The origin of the observed SN Ia diversity is still unclear.",
"It seems likely that SNe Ia can be produced by a variety of different progenitor systems, perhaps explaining part of the observed diversity.",
"SN asymmetric explosion coupled with observer's viewing angle may also produce the diversity.",
"Additionally, the metallicity of progenitors may be a second parameter that affects the light curve shape of SNe Ia.",
"At present, the SD model is the most widely accepted SN Ia progenitor model.",
"The advantages of this model can be summarized as follows: The SD model is in excellent agreement with the observed light curves and spectroscopy of SNe Ia, and this model may explain the similarities of most SNe Ia.",
"Observationally, there is increasing evidence indicating that some SNe Ia may come from the SD model (e.g.",
"the signatures of gas outflows from some SN Ia progenitor systems, the wind-blown cavity in SN remnant, and the early optical and UV emission of SNe Ia, etc).",
"In addition, the SD model may be compatible with some recent observations (e.g.",
"the lack of H or He seen in nebular spectra of SNe Ia, and the upper limits from SN Ia radio and X-ray detection, etc) by considering the spin-down time.",
"There are some SD progenitor candidates in observations, e.g.",
"supersoft X-ray sources, recurrent novae, dwarf novae and symbiotic systems, etc.",
"Meanwhile, a number of high mass WDs that have been accreting from a non-degenerate companion star have been found.",
"The observed single low-mass He WDs and hypervelocity He stars may be explained by the surviving companion stars predicted in the SD model.",
"SNe Ia with long delay times can be understood by the WD + MS and WD + RG channels.",
"In contrast, SNe Ia with short delay times may consist of systems with a He donor star in the WD + He channel, or even a massive MS donor star in the WD + MS channel.",
"Besides the DD model, these observed super-luminous SNe Ia can also be produced by the SD model by considering the effects of rapid differential rotation on the accreting WD.",
"However, the SD model is still suffering some problems from both theoretically and observationally that need to be resolved: The optically thick wind assumption, widely adopted in the studies of the SD model, is in doubt for very low metallicity; the low-metallicity threshold for SNe Ia predicted by theories has not been found in observations.",
"It is still difficult to reproduce the observed birth rates and delay times of SNe Ia.",
"This suggests that we need a better understanding of mass-accretion onto WDs.",
"There is still no conclusive proof that any individual object is the surviving companion star of a SN Ia, which is predicted by the SD model.",
"A likely surviving companion star for the progenitor of Tycho's SN has been identified, but the claim is still controversial.",
"Although a DD merger is thought to experience an accretion-induced collapse rather than a thermonuclear explosion, any definitive conclusion about the DD model is currently premature: There are some parameter ranges in which the accretion-induced collapse can be avoided.",
"Recent simulations indicate that the violent mergers of two massive WDs can closely resemble normal SN Ia explosion with the assumption of the detonation formation as an artificial parameter, although these mergers may only contribute a small fraction to the observed population of normal SNe Ia.",
"This model can naturally reproduce the observed birthrates and delay times of SNe Ia and may explain the formation of some observed super-luminous SNe Ia.",
"This model can explain the lack of H or He seen in the nebular spectra of SNe Ia.",
"Recent observational studies of SN 2011fe seem to favor a DD progenitor.",
"In addition, there is no signal of a surviving companion star from the central region of SNR 0509-67.5 (the site of a SN Ia explosion whose light swept Earth about 400 years ago), which may indicate that the progenitor for this particular SN Ia is a DD system.",
"Some observed double WD systems may have the total mass larger than the Ch mass, and possibly merge within the Hubble-time, although there are not enough double WD systems to reproduce the observed SN Ia birthrates in the context of the DD model.",
"Some variants of the SD and DD models have been proposed to explain the observed diversity of SNe Ia: The sub-luminous 1991bg-like objects may be explained by the sub-Ch mass model.",
"The unusual properties of 2002ic-like objects can be understood by the delayed dynamical instability model.",
"The spin-up/spin-down model may provide a route to explain the similarities and the diversity observed in SNe Ia.",
"The core-degenerate model could form a massive WD with super-Ch mass that might explode as a super-luminous SN Ia.",
"The collisions between two WDs in dense environments could also potentially lead to sub-luminous SN Ia explosions.",
"The mechanism of WDs exploding near black holes is also a potential progenitor model for thermonuclear runaway, despite of the expected low rate when a WD passes near a black hole.",
"To set further constraints on SN Ia progenitor models, large samples of SNe Ia with well-observed light curves and spectroscopy in nearby galaxies are required to establish the connection of SN Ia properties with the stellar environments of their host galaxies.",
"Many new surveys from ground and space have been proposed to make strides in SN Ia studies , e.g.",
"Palomar Transient Factory, Skymapper, La Silla QUEST, Pan-STARRS, the Dark Energy Survey, Large Synoptic Survey Telescope, the Joint Dark Energy Mission and the Gaia Astrometric Mission, etc (Howell et al., 2009; Altavilla et al., 2012).",
"These surveys will allow comparisons via large SN Ia subsamples, and start to connect SN Ia progenitors with the observed features of SN explosions themselves, and thus to unveil the nature of SN Ia progenitors."
],
[
"Acknowledgments",
"We acknowledge useful comments and suggestions from Shuangnan Zhang and Stephen Justham.",
"We also thank Simon Jeffery, Xiaofeng Wang, Xiangcun Meng, Xuefei Chen and Zhengwei Liu for their helpful discussions.",
"This work is supported by the National Natural Science Foundation of China (Grant Nos.",
"11033008 and 11103072), the National Basic Research Program of China (Grant No.",
"2009CB824800), the Chinese Academy of Sciences (Grant No.",
"KJCX2-YW-T24), the Western Light Youth Project and Youth Innovation Promotion Association of the Chinese Academy of Sciences."
]
] | 1204.1155 |
[
[
"Regularization of Linear Ill-posed Problems by the Augmented Lagrangian\n Method and Variational Inequalities"
],
[
"Abstract We study the application of the Augmented Lagrangian Method to the solution of linear ill-posed problems.",
"Previously, linear convergence rates with respect to the Bregman distance have been derived under the classical assumption of a standard source condition.",
"Using the method of variational inequalities, we extend these results in this paper to convergence rates of lower order, both for the case of an a priori parameter choice and an a posteriori choice based on Morozov's discrepancy principle.",
"In addition, our approach allows the derivation of convergence rates with respect to distance measures different from the Bregman distance.",
"As a particular application, we consider sparsity promoting regularization, where we derive a range of convergence rates with respect to the norm under the assumption of restricted injectivity in conjunction with generalized source conditions of H\\\"older type."
],
[
"Introduction",
"We aim for the solution of the problem $\\inf _{u\\in X} J(u)\\quad \\textnormal { s.t.",
"}\\quad Ku = g,$ where $K\\colon X\\rightarrow H$ is a linear and bounded mapping between a Banach space $X$ and a Hilbert space $H$ and where $J\\colon X\\rightarrow \\overline{\\mathbb {R}}$ is convex and lower semi-continuous.",
"We are particularly interested in the case when the right hand side in the linear constraint is not at hand but only an approximation $g^\\delta $ such that $\\left\\Vert g-g^\\delta \\right\\Vert \\le \\delta $ for some $\\delta >0$ .",
"A possible method for computing a stable approximation of solutions of intro:primal is the augmented Lagrangian method (ALM), an iterative method that, for a given initial value $p_0^\\delta \\in H$ and for $k=1,2,\\ldots $ , computes $u_k^\\delta & \\in & \\operatornamewithlimits{\\textnormal {argmin}}_{u\\in X}\\biggl [ \\frac{\\tau _k}{2}\\left\\Vert Ku - g^\\delta \\right\\Vert ^2 +J(u) - \\left\\langle p_{k-1}^\\delta , Ku - g^\\delta \\right\\rangle \\biggr ] \\\\p_k^\\delta & = & p_{k-1}^\\delta + \\tau _k(g^\\delta - Ku_k^\\delta ).$ Here, $\\left\\lbrace \\tau _k \\right\\rbrace _{k\\in \\mathbb {N}}$ denotes a pre-defined sequence of positive parameters such that $t_n := \\sum _{k=1}^n \\tau _k\\rightarrow \\infty \\quad \\textnormal { as }\\quad n\\rightarrow \\infty .$ The ALM was originally introduced in , (under the name method of multipliers) as a solution method for problems of type intro:primal with exact right hand side $g$ .",
"Since then, the ALM was developed further in various directions; see e.g.",
", and the references therein.",
"In the context of inverse problems, the ALM was first considered for the special case when $X$ is a Hilbert space and $J$ is a quadratic functional, i.e., $J(u) = {1\\over 2}\\left\\Vert Lu \\right\\Vert ^2$ for a densely defined and closed linear operator $L\\colon D(L)\\subset X\\rightarrow \\tilde{H}$ , where $\\tilde{H}$ is some further Hilbert space (here we set $J(u) = +\\infty $ if $u\\notin D(L)$ ).",
"For this special case, it is readily seen that the ALM can be rewritten into $u_k^\\delta = \\operatornamewithlimits{\\textnormal {argmin}}_{u\\in X}\\biggl [ \\tau _k \\left\\Vert Ku - g^\\delta \\right\\Vert ^2 +\\left\\Vert L(u-u_{k-1}^\\delta ) \\right\\Vert ^2_{\\tilde{H}}\\biggr ].$ The analysis of iteration intro:itertik dates back to the papers , .",
"The case when $L\\equiv \\textnormal {Id}$ is referred to as the iterated Tikhonov method and has been studied in , , , .",
"The regularization scheme that results for $K\\equiv \\textnormal {Id}$ is termed iterated Tikhonov–Morozov method and amounts to stably evaluate the (possibly unbounded) operator $L$ at $g$ given only an approximation $g^\\delta $ that satisfies intro:error.",
"For detailed analysis see e.g.",
", .",
"A generalization of the iteration in intro:itertik for total-variation based image reconstruction has been established in under the name Bregman iteration and convergence properties were studied in .",
"In it was pointed out that the Bregman iteration and the iterated Tikhonov(–Morozov) method are special instances of the ALM as it is stated in intro:alm, and an improved convergence analysis was developed.",
"In , Morozov's discrepancy principle was studied for the ALM.",
"The application of the ALM for the regularization of nonlinear operators has been considered in , .",
"Up to now, convergence rates for the ALM (in the context of inverse problems) have only been derived under the assumption that the solutions $u^\\dagger $ of intro:primal satisfy the standard source condition $K^*p^\\dagger \\in \\partial J(u^\\dagger )\\quad \\textnormal { for some }p^\\dagger \\in H.$ Here $K^*\\colon H \\rightarrow X^*$ denotes the adjoint operator of $K$ and $\\partial J(u^\\dagger )$ is the subdifferential of $J$ at $u^\\dagger $ .",
"This typically results in a convergence rate of $\\delta $ with respect to the Bregman distance (for a definition of the subdifferential and the Bregman distance, see Section ).",
"In this paper we will extend these results to convergence rates of lower order by replacing intro:sc by variational inequalities.",
"The analysis will apply for both a priori and a posteriori parameter selection rules, where the latter will be realized by Morozov's discrepancy principle.",
"In addition, our approach allows the derivation of convergence rates with respect to distance measures different from the Bregman distance.",
"The paper is organized as follows: In Section we state basic assumptions and review tools from convex analysis that are essential for our analysis.",
"In Section we establish variational inequalities and prove that these are sufficent for lower order convergence rates for the ALM with suitable a priori stopping rules.",
"In Section we reprove the same convergence rates when Morozov's discrepancy principle is employed as an a posteriori stopping rule.",
"In Section we finally consider some examples that clarify the connection of the variational inequalities in Section and more classic notions of source conditions, such as the standard source condition intro:sc or Hölder-type conditions.",
"Moreover, we show for the particular scenario of sparsity promoting regularization how our approach can be used to derive convergence rates with respect to the norm."
],
[
"Assumptions and Mathematical Prerequisites",
"In this section we fix some basic assumptions as well as review basic notions and facts from convex analysis.",
"We start by delimiting minimal functional analytic requirements.",
"Assumption 2.1 $X$ is a separable Banach space with topological dual $X^*$ .",
"We denote the duality pairing of $X$ and $X^*$ by $\\left\\langle \\xi , x \\right\\rangle _{X^*, X} = \\xi (x)$ .",
"The operator $K\\colon X\\rightarrow H$ is linear and continuous.",
"The functional $J\\colon X\\rightarrow \\overline{\\mathbb {R}}:=\\mathbb {R}\\cup \\lbrace +\\infty \\rbrace $ is convex, lower semicontinuous and proper with nonempty domain $D(J) = \\left\\lbrace u\\in X~:~ J(u)<\\infty \\right\\rbrace $ .",
"For each $g\\in H$ and $c>0$ the set $\\Lambda (g, c) = \\left\\lbrace u\\in X~:~ \\left\\Vert Ku - g \\right\\Vert ^2 + J(u)\\le c \\right\\rbrace $ is sequentially weakly pre-compact in $X$ .",
"For our analysis we will make extensive use of tools from convex analysis (here, we refer to as a standard reference).",
"We will henceforth denote by $\\partial J(u_0)$ the subdifferential of $J$ at $u_0\\in X$ , i.e., the set of all $\\xi \\in X^*$ such that $J(u)\\ge J(u_0) + \\left\\langle \\xi , u-u_0 \\right\\rangle _{X^*,X},\\quad \\textnormal { for all }u\\in X.$ In this case, we call $\\xi $ a subgradient of $J$ at $u_0$ .",
"We denote by $K^*\\colon H\\rightarrow X^*$ the adjoint operator of $K$ , where we identify the Hilbert space $H$ with its dual $H^*$ by means of Riesz' representation theorem.",
"Under Assumption REF it is guaranteed that solutions of intro:primal exist for all $g\\in K(D(J))$ and that the iteration intro:alm is well defined.",
"The proof is analogous to .",
"Recall that the Legendre-Fenchel conjugate $J^*\\colon X^*\\rightarrow \\overline{\\mathbb {R}}$ of $J$ is defined by $J^*(x^*) = \\sup _{x\\in X} \\left\\langle x^*, x \\right\\rangle _{X^*,X} - J(x)$ .",
"The dual problem to intro:primal is then defined by $\\inf _{p\\in H} \\Bigl [J^*(K^*p) - \\left\\langle p, g \\right\\rangle \\Bigr ].$ Sufficient and necessary conditions for guaranteeing the existence of a solution $u^\\dagger \\in X$ of intro:primal and a solution $p^\\dagger \\in H$ of dual:dual are the Karush-Kuhn-Tucker conditions, which read as $K^*p^\\dagger \\in \\partial J(u^\\dagger )\\quad \\textnormal { and }\\quad Ku^\\dagger = g.$ From an inverse problems perspective, these conditions are understood as source conditions that delimit a class of particular regular solutions $u^\\dagger $ of intro:primal that can be reconstructed from noisy data at a certain rate depending on the noise level $\\delta $ .",
"If the source condition dual:kkt does not hold, then solutions of intro:primal may still exist (e.g.",
"if Assumption REF holds) whereas dual:dual has no solutions.",
"The value of dual:dual, though, will still be finite: Lemma 2.2 Assume that Assumption REF holds and let $u^\\dagger \\in X$ be a solution of intro:primal.",
"Then $\\inf _{p\\in H} \\Bigl [J^*(K^*p) - \\left\\langle p, g \\right\\rangle \\Bigr ] = -J(u^\\dagger ).$ Define a function $\\Gamma \\colon X\\times H \\rightarrow \\overline{\\mathbb {R}}$ by setting $\\Gamma (u,p) = J(u)$ if $Ku = g+p$ and $G(u,p) = +\\infty $ else.",
"According to the assertion holds, if the function $p\\mapsto h(p) = \\inf _{u\\in X} \\Gamma (u,p)$ is finite and lower semicontinuous at $p = 0$ .",
"Since $p(0) = J(u^\\dagger )<\\infty $ it remains to prove lower semicontinuity.",
"Let therefore $\\left\\lbrace p_k \\right\\rbrace _{k\\in \\mathbb {N}}$ be a sequence in $H$ such that $p_k\\rightarrow 0$ .",
"Without loss of generality, we may, after possibly passing to a subsequence, assume that $h(p_k) < \\infty $ for every $k$ , which amounts to saying that the equation $Ku = g+p_k$ has a solution $u_k \\in X$ satisfying $J(u_k) < \\infty $ .",
"In addition, because of Assumption REF , we can choose $u_k$ such that the infimum in the definition of $h$ is realized at $u_k$ , that is, $h(p_k) = \\Gamma (u_k,p_k)$ .",
"Now, if $J(u_k)\\rightarrow \\infty $ as $k\\rightarrow \\infty $ , nothing remains to be proven.",
"Thus we can assume that there exists a subsequence of $\\left\\lbrace u_{k^{\\prime }} \\right\\rbrace $ such that $\\sup _{k^{\\prime }\\in \\mathbb {N}} J(u_{k^{\\prime }}) < \\infty $ .",
"It is not restrictive to assume that $\\lim _{{k^{\\prime }}\\rightarrow \\infty } J(u_{k^{\\prime }}) = \\liminf _{k\\rightarrow \\infty } J(u_k)$ .",
"Moreover, we observe that $\\left\\Vert Ku_k -g \\right\\Vert ^2 = \\left\\Vert p_k \\right\\Vert ^2$ is bounded, since $p_k\\rightarrow 0$ .",
"Thus it follows from Assumption REF that there exists a further subsequence $\\left\\lbrace u_{k^{\\prime \\prime }} \\right\\rbrace $ such that $u_{k^{\\prime \\prime }}\\rightharpoonup \\hat{u}$ for some $\\hat{u}\\in X$ .",
"This implies that $Ku_{k^{\\prime \\prime }}\\rightharpoonup K\\hat{u} = g$ , and the lower semicontinuity and convexity of $J$ finally proves that $\\liminf _{k\\rightarrow \\infty } h(p_{k}) = \\lim _{k^{\\prime }\\rightarrow \\infty } J(u_{k^{\\prime }}) =\\liminf _{k^{\\prime \\prime }\\rightarrow \\infty } J(u_{k^{\\prime \\prime }})\\ge J(\\hat{u})\\ge J(u^\\dagger ) = h(0).$ Similar to the duality relation between the optimization problems intro:primal and dual:dual such a relation can be established for the ALM: As it was first observed in , the dual sequence $\\left\\lbrace p_0^\\delta ,p_1^\\delta ,\\ldots \\right\\rbrace $ generated by the ALM can be characterized by the proximal point method (PPM).",
"To be more precise, for all $k\\ge 1$ , $p_k^\\delta = \\operatornamewithlimits{\\textnormal {argmin}}_{p\\in H}\\biggl [\\frac{1}{2}\\left\\Vert p - p_{k-1}^\\delta \\right\\Vert ^2 +\\tau _k\\left(J^*(K^*p) - \\left\\langle p, g^\\delta \\right\\rangle \\right)\\biggr ].$ The PPM was introduced by Martinet in for minimizing a convex functional, which in the present situation is the dual functional dual:dual.",
"The sequence $\\left\\lbrace p_k^\\delta \\right\\rbrace $ generated by the PPM is known to converge weakly to a solution of dual:dual if it exists, i.e., when dual:kkt holds.",
"If this is not the case, then still $J^*(K^*p_k^\\delta )-\\left\\langle p_k^\\delta , g^\\delta \\right\\rangle $ converges to the value of the program dual:dual which, in the general case, may be $-\\infty $ , of course."
],
[
"Convergence Rates",
"It is well known that linear convergence rates (with respect to the Bregman distance) for iterates of the ALM can be proven if the source condition dual:kkt holds (cf.",
", ).",
"In this section we prove lower order rates of convergence in the case, when the source condition dual:kkt does not hold.",
"Instead, we impose weaker regularity conditions on solutions $u^\\dagger $ of intro:primal in terms of variational inequalities.",
"We formulate this in the following Assumption 3.1 We are given an index function $\\Phi \\colon [0,\\infty )\\rightarrow [0,\\infty )$ , i.e., a non-negative continuous function that is strictly increasing and concave with $\\Phi (0) = 0$ .",
"Moreover, $D\\colon X\\times X\\rightarrow [0,\\infty ]$ satisfies $D(u,u) = 0$ whenever $u \\in X$ , and $u^\\dagger $ is a solution of intro:primal is such that $D(u,u^\\dagger )\\le J(u) - J(u^\\dagger ) + \\Phi (\\left\\Vert Ku - g \\right\\Vert ^2)\\quad \\textit { forall }u\\in X.$ We denote by $\\Psi $ the Legendre-Fenchel conjugate of $\\Phi ^{-1}$ .",
"A typical choice is $D(u,v) = \\beta D_J^\\xi (u,v)$ , where $\\beta \\in (0,1]$ and $D^{\\xi }_J(v,u) = J(v) - J(u) - \\left\\langle \\xi , v-u \\right\\rangle _{X^*,X}$ is the Bregman-distance of $u$ and $v$ w.r.t.",
"$\\xi \\in \\partial J(v)$ .",
"With this, rates:varineqcond is equivalent to the condition $\\left\\langle \\xi ^\\dagger , u^\\dagger -u \\right\\rangle _{X^*,X}\\le (1-\\beta ) D_J^{\\xi ^\\dagger }(u,u^\\dagger ) + \\Phi (\\left\\Vert Ku-g \\right\\Vert ^2)$ for all $u\\in X$ .",
"In this form, variational inequalities have been introduced in , with $\\Phi (s) = \\sqrt{s}$ , and for general index functions in , .",
"The following theorem asserts that the condition rates:varineqcond in Assumption REF imposes sufficient smoothness on the true solution $u^\\dagger $ that the iterates of the ALM approach $u^\\dagger $ with a certain rate (that depends on $\\Phi $ ).",
"Theorem 3.2 Let Assumptions REF and REF hold.",
"Then, there exists a constant $C>0$ such that $D(u_n^\\delta , u^\\dagger )\\le C t_n \\left(\\Psi \\left(\\frac{16}{t_n} \\right)+\\delta ^2\\right)$ and $\\left\\Vert Ku_n^\\delta - g^\\delta \\right\\Vert ^2 \\le C\\left(\\Psi \\left(\\frac{16}{t_n}\\right) + \\delta ^2\\right).$ In particular, if $t_n \\asymp {1\\over \\Psi ^{-1}(\\delta ^2)}$ , then $D(u_n^\\delta , u^\\dagger ) = \\mathcal {O}\\left(\\frac{\\delta ^2}{\\Psi ^{-1}(\\delta ^2)}\\right)\\quad \\textnormal { and }\\quad \\left\\Vert Ku_n^\\delta - g \\right\\Vert ^2 = \\mathcal {O}(\\delta ^2).$ Theorem REF is a consequence of the following two Lemmas.",
"Lemma 3.3 Let Assumptions REF and REF hold and define for $p\\in H$ , $t>0$ and $\\delta \\ge 0$ $\\psi (p,t,\\delta ) = \\left(t \\Psi (16 \\slash t) + t\\delta ^2 + J^*(K^*p) + J(u^\\dagger ) - \\left\\langle p, g \\right\\rangle + \\frac{\\left\\Vert p \\right\\Vert ^2}{2t}\\right).$ Then, there exists a constant $C>0$ such that $D(u_n^\\delta , u^\\dagger )\\le C \\psi (p,t_n,\\delta )\\quad \\textnormal { and }\\quad \\left\\Vert Ku_n^\\delta - g^\\delta \\right\\Vert ^2 \\le C \\frac{\\psi (p,t_n,\\delta )}{t_n}$ for all $p\\in H$ .",
"Without loss of generality we assume that $p_0^\\delta = 0$ and we shall agree upon $G(p,g) = J^*(K^*p) - \\left\\langle p, g \\right\\rangle $ .",
"In it was proved that for all $p\\in V$ $\\frac{t_n \\left\\Vert p_n^\\delta - p_{n-1}^\\delta \\right\\Vert ^2}{2\\tau _n^2} \\le G(p,g^\\delta )- G(p_n^\\delta ,g^\\delta ) - \\frac{\\left\\Vert p-p_n^\\delta \\right\\Vert ^2}{2t_n} +\\frac{\\left\\Vert p \\right\\Vert ^2}{2t_n}.$ Since $G(p,g^\\delta ) - G(p_n^\\delta ,g^\\delta ) = G(p,g) - G(p_n^\\delta ,g) +\\left\\langle p-p_n^\\delta , g-g^\\delta \\right\\rangle $ and $p_n^\\delta - p_{n-1}^\\delta =\\tau _n(g^\\delta - Ku_n^\\delta )$ , this implies that $\\frac{t_n}{2} \\left\\Vert Ku_n^\\delta - g^\\delta \\right\\Vert ^2 & \\le G(p,g)- G(p_n^\\delta ,g) - \\frac{\\left\\Vert p-p_n^\\delta \\right\\Vert ^2}{2t_n} +\\frac{\\left\\Vert p \\right\\Vert ^2}{2t_n} + \\left\\langle p-p_n^\\delta , g-g^\\delta \\right\\rangle \\nonumber \\\\& \\le G(p,g) + J(u^\\dagger ) - \\frac{\\left\\Vert p-p_n^\\delta \\right\\Vert ^2}{2t_n} +\\frac{\\left\\Vert p \\right\\Vert ^2}{2t_n} + \\left\\langle p-p_n^\\delta , g-g^\\delta \\right\\rangle ,$ where the second inequality follows from Lemma REF .",
"Setting $p =p_n^\\delta $ , this proves that $\\frac{t_n}{2}\\left\\Vert Ku_n^\\delta - g^\\delta \\right\\Vert ^2 \\le J^*(K^*p_n^\\delta ) -\\left\\langle p_n^\\delta , g \\right\\rangle + J(u^\\dagger )+ \\frac{\\left\\Vert p_n^\\delta \\right\\Vert ^2}{2t_n}.$ Since $K^*p_n^\\delta \\in \\partial J(u_n^\\delta )$ , we observe that $J^*(K^*p_n^\\delta ) + J(u_n^\\delta ) = \\left\\langle K^*p_n^\\delta , u_n^\\delta \\right\\rangle $ and conclude that $\\frac{t_n}{2}\\left\\Vert Ku_n^\\delta - g^\\delta \\right\\Vert ^2 & \\le J(u^\\dagger ) -J(u_n^\\delta ) + \\left\\langle p_n^\\delta , Ku_n^\\delta - g \\right\\rangle +\\frac{\\left\\Vert p_n^\\delta \\right\\Vert ^2}{2t_n} \\\\& = J(u^\\dagger ) -J(u_n^\\delta ) + \\left\\langle p_n^\\delta , Ku_n^\\delta - g^\\delta \\right\\rangle +\\left\\langle p_n^\\delta , g^\\delta - g \\right\\rangle + \\frac{\\left\\Vert p_n^\\delta \\right\\Vert ^2}{2t_n}.$ Applying Young's inequality $\\left\\langle a, b \\right\\rangle \\le \\left\\Vert a \\right\\Vert ^2\\slash 2 + \\left\\Vert b \\right\\Vert ^2\\slash 2$ first with $a = \\sqrt{2\\slash t_n}p_n^\\delta $ and $b = (Ku_n^\\delta -g^\\delta )\\sqrt{t_n\\slash 2}$ , and then with $a = p_n^\\delta \\slash \\sqrt{t_n}$ and $b = \\sqrt{t_n}(g^\\delta -g)$ , we obtain $\\frac{t_n}{4}\\left\\Vert Ku_n^\\delta - g^\\delta \\right\\Vert ^2 & \\le J(u^\\dagger ) -J(u_n^\\delta )+\\left\\langle p_n^\\delta , g^\\delta - g \\right\\rangle +\\frac{3\\left\\Vert p_n^\\delta \\right\\Vert ^2}{2t_n} \\\\& \\le J(u^\\dagger ) - J(u_n^\\delta )+\\frac{\\delta ^2 t_n}{2} +\\frac{2\\left\\Vert p_n^\\delta \\right\\Vert ^2}{t_n}$ Summarizing, we find that $\\left\\Vert Ku_n^\\delta - g^\\delta \\right\\Vert ^2 \\le \\frac{4}{t_n}\\left(J(u^\\dagger ) -J(u_n^\\delta )\\right) + 2\\delta ^2 + \\frac{8\\left\\Vert p_n^\\delta \\right\\Vert ^2}{t_n^2}.$ Now, we observe from rates:varineqcond that $J(u^\\dagger ) -J(u_n^\\delta ) \\le - D(u_n^\\delta , u^\\dagger ) + \\Phi (\\left\\Vert Ku_n^\\delta -g \\right\\Vert ^2)$ .",
"Plugging this inequality into the above estimate yields $\\left\\Vert Ku_n^\\delta - g^\\delta \\right\\Vert ^2 + \\frac{4}{t_n}D(u_n^\\delta ,u^\\dagger ) \\le \\frac{4}{t_n} \\Phi (\\left\\Vert K u_n^\\delta - g \\right\\Vert ^2) + 2\\delta ^2+ \\frac{8\\left\\Vert p_n^\\delta \\right\\Vert ^2}{t_n^2}.$ Since $\\Psi $ is the Legendre-Fenchel conjugate of $t\\mapsto \\Phi ^{-1}(t)$ , i.e., $\\Psi (s) = \\sup _{t\\ge 0} st - \\Phi ^{-1}(t)$ , it follows that $st\\le \\Psi (s) + \\Phi ^{-1}(t)$ for all $s,t\\ge 0$ , and in particular, for $t =\\Phi (r)$ , that $s\\Phi (r)\\le \\Psi (s) + r$ for all $s,r\\ge 0$ .",
"Setting $s =16\\slash t_n$ and $r = \\left\\Vert Ku_n^\\delta -g^\\delta \\right\\Vert ^2$ gives $\\frac{4}{t_n} \\Phi (\\left\\Vert Ku_n^\\delta - g \\right\\Vert ^2) & =\\frac{1}{4}\\frac{16}{t_n} \\Phi (\\left\\Vert Ku_n^\\delta - g \\right\\Vert ^2) \\\\& \\le \\frac{1}{4} \\Psi \\left(\\frac{16}{t_n}\\right) + \\frac{1}{4}\\left\\Vert Ku_n^\\delta - g \\right\\Vert ^2 \\\\& \\le \\frac{1}{4} \\Psi \\left(\\frac{16}{t_n}\\right) + \\frac{1}{2}\\left\\Vert Ku_n^\\delta - g^\\delta \\right\\Vert ^2 + \\frac{\\delta ^2}{2}.$ Combining this with rates:aux1 yields $\\frac{1}{2}\\left\\Vert Ku_n^\\delta - g^\\delta \\right\\Vert ^2+ \\frac{4}{t_n}D(u_n^\\delta , u^\\dagger ) \\le \\frac{1}{4}\\Psi \\left(\\frac{16}{t_n}\\right) + \\frac{5\\delta ^2}{2} + \\frac{8\\left\\Vert p_n^\\delta \\right\\Vert ^2}{t_n^2}.$ Finally, we observe again from gueler that for all $p\\in H$ $\\frac{\\left\\Vert p-p_n^\\delta \\right\\Vert ^2}{2t_n^2} & \\le \\frac{G(p,g^\\delta ) - G(p_n^\\delta ,g^\\delta )}{t_n} + \\frac{\\left\\Vert p \\right\\Vert ^2}{2t_n^2} \\\\&\\le \\frac{G(p,g) - G(p_n^\\delta , g)}{t_n} + \\frac{1}{t_n}\\left\\langle p -p_n^\\delta , g-g^\\delta \\right\\rangle + \\frac{\\left\\Vert p \\right\\Vert ^2}{2t_n^2} \\\\& \\le \\frac{G(p,g) - \\inf _{q\\in V} G(q, g)}{t_n} +\\frac{\\left\\Vert p-p_n^\\delta \\right\\Vert ^2}{4t_n^2} + \\delta ^2+ \\frac{\\left\\Vert p \\right\\Vert ^2}{2t_n^2}.$ This shows that $\\frac{\\left\\Vert p_n^\\delta \\right\\Vert ^2}{8t_n^2} & \\le \\frac{\\left\\Vert p-p_n^\\delta \\right\\Vert ^2}{4t_n^2} +\\frac{\\left\\Vert p \\right\\Vert ^2}{4t_n^2} \\\\& \\le \\frac{G(p,g) - \\inf _{q\\in V} G(q, g)}{t_n} + \\delta ^2+\\frac{3\\left\\Vert p \\right\\Vert ^2}{4t_n^2}.$ Combining rates:aux3 with rates:aux2 and applying Lemma REF finally gives $\\frac{1}{2}\\left\\Vert Ku_n^\\delta - g^\\delta \\right\\Vert ^2& + \\frac{4}{t_n}D(u_n^\\delta , u^\\dagger )\\le \\frac{1}{4}\\Psi \\left(\\frac{16}{t_n}\\right) + \\frac{5\\delta ^2}{2} + \\frac{8\\left\\Vert p_n^\\delta \\right\\Vert ^2}{t_n^2}\\\\&\\le \\frac{1}{4}\\Psi \\left(\\frac{16}{t_n}\\right) + 64\\frac{G(p,g) - \\inf _{q\\in V} G(q, g)}{t_n}+ \\frac{133\\delta ^2}{2} + \\frac{48\\left\\Vert p \\right\\Vert ^2}{t_n^2}.$ Lemma 3.4 Let Assumptions REF and REF hold.",
"Then, $\\inf _{p\\in H}\\biggl [ J^*(K^*p) + J(u^\\dagger ) - \\left\\langle p, g \\right\\rangle +\\frac{\\left\\Vert p \\right\\Vert ^2}{2t}\\biggr ] \\le \\frac{t}{2}\\Psi \\left(\\frac{2}{t}\\right).$ Classical duality theory (see ) implies that $\\mu :=\\inf _{p\\in H} \\biggl [J^*(K^*p)+J(u^\\dagger )-\\left\\langle p, g \\right\\rangle + \\frac{\\left\\Vert p \\right\\Vert ^2}{2t}\\biggr ]= -\\inf _{u \\in X} \\biggl [\\frac{t}{2}\\left\\Vert Ku-g \\right\\Vert ^2 + J(u)-J(u^\\dagger )\\biggr ],$ as the right hand side of this equation is the dual of the left hand side.",
"Using the variational inequality rates:varineqcond and the non-negativity of $D$ , we therefore find that $\\mu &\\le \\sup _{u\\in X}\\biggl [\\Phi \\bigl (\\left\\Vert Ku-g \\right\\Vert ^2\\bigr ) - D(u,u^\\dagger ) - \\frac{t}{2}\\left\\Vert Ku-g \\right\\Vert ^2\\biggr ]\\\\&\\le \\sup _{u\\in X}\\biggl [\\Phi \\bigl (\\left\\Vert Ku-g \\right\\Vert ^2\\bigr ) - \\frac{t}{2}\\left\\Vert Ku-g \\right\\Vert ^2\\biggr ].$ Replacing $\\left\\Vert Ku-g \\right\\Vert ^2$ by $s \\ge 0$ in the last term and using the definition of $\\Psi $ , we obtain $\\mu \\le \\sup _{s \\ge 0} \\biggl [\\Phi (s)-\\frac{ts}{2}\\biggr ]= \\frac{t}{2}\\sup _{s \\ge 0} \\biggl [\\frac{2\\Phi (s)}{t}-s\\biggr ]= \\frac{t}{2}\\sup _{s \\ge 0} \\biggl [\\frac{2s}{t}-\\Phi ^{-1}(s)\\biggr ]= \\frac{t}{2}\\Psi \\biggl (\\frac{2}{t}\\biggr ),$ which proves the assertion.",
"We close this section by a statement concerning the dual variables $\\left\\lbrace p_1^\\delta , p_2^\\delta , \\ldots \\right\\rbrace $ generated by the ALM.",
"It is well known (in the case when $\\delta = 0$ ) that these stay bounded if and only if the source condition dual:kkt holds.",
"Assumption REF , however, allows to control their growth, as the following result shows.",
"Corollary 3.5 Let Assumptions REF and REF hold.",
"Then, there exists a constant $C>0$ such that $\\left\\Vert p_n^\\delta \\right\\Vert ^2\\le C t_n^2\\left(\\Psi \\left(\\frac{2}{t_n}\\right) + \\delta ^2\\right)$ It follows from rates:aux3 that there exists a constant $C>0$ such that $\\left\\Vert p_n^\\delta \\right\\Vert ^2 \\le C t_n \\left(J^*(K^*p) + J(u^\\dagger ) - \\left\\langle p, g \\right\\rangle +\\frac{\\left\\Vert p \\right\\Vert ^2}{2t_n} + t_n \\delta ^2 \\right)$ for all $p\\in H$ .",
"Applying Lemma REF yields the desired estimate."
],
[
"Morozov's Discrepancy Principle",
"In this section we study Morozov's discrepancy principle as an a posteriori stopping rule for the ALM.",
"To be more precise, if $\\left\\lbrace u_1^\\delta ,u_2^\\delta ,\\ldots \\right\\rbrace $ is generated by the ALM, Morozov's rule suggests to stop the iteration at the index $n^*(\\delta ) = \\min \\left\\lbrace n\\in \\mathbb {N}~:~\\left\\Vert Ku_n^\\delta - g^\\delta \\right\\Vert \\le \\rho \\delta \\right\\rbrace ,$ where $\\rho >1$ .",
"In this section we prove convergence rates for the iterates $u_{n^*(\\delta )}^\\delta $ given that Assumption REF holds.",
"Morozov's principle for the case when the source condition dual:kkt holds was studied in .",
"Theorem REF below extends this result to regularity classes that are delimited by the variational inequality in Assumption REF .",
"Additionally to these, we will assume Assumption 4.1 Let Assumption REF hold.",
"The mapping $s\\mapsto \\Phi (s)^2/s$ is non-increasing.",
"The sequence of stepsizes $\\left\\lbrace \\tau _1,\\tau _2,\\ldots \\right\\rbrace $ in the ALM is bounded.",
"Theorem 4.2 Let Assumptions REF and REF hold and assume that $n^*(\\delta )$ is chosen according to Morozov's discrepancy principle morozov:rule for some $\\rho > 1$ .",
"Then there exists a constant $C >0$ independent of $\\rho $ such that $D(u_{n^*(\\delta )}^\\delta , u^\\dagger )\\le \\frac{C(\\rho +1)^2\\delta ^2}{\\Psi ^{-1}\\bigl ((\\rho ^2-1)\\delta ^2\\bigr )}+ C(\\rho +1)^2\\delta ^2\\sup _{k\\in \\mathbb {N}}\\tau _k.$ Remark 4.3 Assume that the variational inequality rates:varineqcond is satisfied with $\\Phi (s) = Cs^p$ for some $C > 0$ and $p > 0$ .",
"Then, setting $u = u^\\dagger + tz$ for some $z \\in X$ and $t > 0$ , the non-negativity of $D$ implies in particular the inequality $J(u^\\dagger ) - J(u^\\dagger +tz) \\le C t^{2p} \\left\\Vert Kz \\right\\Vert ^{2p}.$ Now assume that $p > 1/2$ .",
"Then we obtain, after dividing by $t$ and considering the limit $t \\rightarrow 0^+$ , that the directional derivative of $J$ satisfies $-J^{\\prime }(u^\\dagger )(z) \\le 0$ .",
"Because $z$ was arbitrary, this implies that $u^\\dagger $ minimizes the regularization term $J$ .",
"Thus the variational inequality can hold in non-trivial situations, if and only if $p \\le 1/2$ .",
"Now note that the same condition is required for the function $\\Phi (s)^2/s = C^2s^{2p-1}$ to be non-increasing.",
"Therefore, in the case of a variational inequality of Hölder type, Assumption REF imposes no relevant further restrictions on the index function.",
"Before we give the proof of Theorem REF , we state the following Lemma, which is interesting in its own right.",
"Lemma 4.4 Let Assumptions REF and REF hold and assume that $n^*(\\delta )$ is chosen according to Morozov's discrepancy principle morozov:rule.",
"Then, $t_{n^*(\\delta )} \\le \\frac{2}{\\Psi ^{-1}((\\rho ^2-1)\\delta ^2)} +\\tau _{n^*(\\delta )}.$ Without loss of generality we may assume that $n^*(\\delta ) > 1$ ; else the assertion is trivial.",
"Denote for the sake of simplicity $\\bar{n} := n^*(\\delta )-1$ .",
"Then it follows from morozov:rule that $\\left\\Vert Ku_{\\bar{n}}^\\delta - g^\\delta \\right\\Vert ^2 > \\rho ^2\\delta ^2$ .",
"Plugging in this relation into rates:aux0 yields $\\frac{\\rho ^2t_{\\bar{n}}\\delta ^2}{2} + \\frac{\\left\\Vert p-p_{\\bar{n}}^\\delta \\right\\Vert ^2}{2t_{\\bar{n}}}&< \\frac{t_{\\bar{n}}}{2}\\left\\Vert Ku_{\\bar{n}}^\\delta -g^\\delta \\right\\Vert ^2 + \\frac{\\left\\Vert p-p_{\\bar{n}}^\\delta \\right\\Vert ^2}{2t_{\\bar{n}}}\\\\&\\le J^*(K^*p)-\\left\\langle p, g \\right\\rangle + J(u^\\dagger ) + \\frac{\\left\\Vert p \\right\\Vert ^2}{2t_{\\bar{n}}} + \\left\\langle p-p_{\\bar{n}}^\\delta , g-g^\\delta \\right\\rangle $ for every $p \\in H$ .",
"Applying Young's inequality $\\left\\langle p-p_{\\bar{n}}^\\delta , g-g^\\delta \\right\\rangle \\le \\frac{\\left\\Vert p-p_{\\bar{n}}^\\delta \\right\\Vert ^2}{2t_{\\bar{n}}} + \\frac{t_{\\bar{n}}\\delta ^2}{2},$ we obtain with Lemma REF the estimate $\\frac{(\\rho ^2-1)t_{\\bar{n}}\\delta ^2}{2}\\le \\inf _{p\\in H}\\biggl [J^*(K^*p)-\\left\\langle p, g \\right\\rangle + J(u^\\dagger ) + \\frac{\\left\\Vert p \\right\\Vert ^2}{2t_{\\bar{n}}}\\biggr ]\\le \\frac{t_{\\bar{n}}}{2}\\Psi \\biggl (\\frac{2}{t_{\\bar{n}}}\\biggr ).$ This proves that $(\\rho ^2-1)\\delta ^2 \\le \\Psi (2\\slash t_{\\bar{n}})$ .",
"Now the assertion follows by applying the monotoneously increasing function $\\Psi ^{-1}$ to both sides of this inequality and adding the last step size $\\tau _{n^*(\\delta )}$ .",
"Next we need another lemma, which relates the condition on $\\Phi $ in Assumption REF to an equivalent condition on the function $\\Psi = (\\Phi ^{-1})^*$ .",
"Lemma 4.5 Let $\\Phi $ be an index function and $\\Psi $ the Fenchel conjugate of $\\Phi ^{-1}$ .",
"Then the mapping $s \\mapsto \\Phi (s)^2/s$ is non-increasing, if and only if the mapping $t \\mapsto t^2 \\Psi (2/t)$ is non-decreasing.",
"First note that, by means of the change of variables $t \\mapsto 2/t$ and ignoring the constant factor, the mapping $t \\mapsto t^2 \\Psi (2/t)$ is non-decreasing, if and only if the mapping $t \\mapsto H(t) := \\Psi (t)/t^2$ is non-increasing.",
"Because $\\Psi $ is convex and continuous, this condition is satisfied, if and only if $H^{\\prime }(t) \\le 0$ for every $t > 0$ for which $\\Psi ^{\\prime }(t)$ exists.",
"Now, $H^{\\prime }(t) = \\frac{\\Psi ^{\\prime }(t)}{t^2} - \\frac{2\\Psi (t)}{t^3} = \\frac{1}{t^3}\\bigl (t\\Psi ^{\\prime }(t)-2\\Psi (t)\\bigr ),$ and therefore $H^{\\prime }(t) \\le 0$ if and only if $t\\Psi ^{\\prime }(t)-2\\Psi (t) \\le 0$ .",
"Now recall that $\\Psi $ is the Fenchel conjugate of $\\Phi ^{-1}$ and therefore $t\\Psi ^{\\prime }(t) = \\Psi (t)+\\Phi ^{-1}\\bigl (\\Psi ^{\\prime }(t)\\bigr )$ .",
"Thus $H^{\\prime }(t) \\le 0$ , if and only if $\\Phi ^{-1}\\bigl (\\Psi ^{\\prime }(t)\\bigr )-\\Psi (t) \\le 0$ .",
"Similarly, the mapping $s \\mapsto \\Phi (s)^2/s$ is non-increasing, if and only if the mapping $s \\mapsto \\tilde{H}(s) := s^2/\\Phi ^{-1}(s)$ is non-increasing, which in turn is equivalent to the condition $\\tilde{H}^{\\prime }(s) = \\frac{2s}{\\Phi ^{-1}(s)} - \\frac{s^2{\\Phi ^{-1}}^{\\prime }(s)}{\\Phi ^{-1}(s)^2}= \\frac{s\\bigl (2\\Phi ^{-1}(s)-s{\\Phi ^{-1}}^{\\prime }(s)\\bigr )}{\\Phi ^{-1}(s)^2} \\le 0.$ Because of the equality $s{\\Phi ^{-1}}^{\\prime }(s) = \\Phi ^{-1}(s)+\\Psi \\bigl ({\\Phi ^{-1}}^{\\prime }(s)\\bigr )$ , this is the case, if and only if $\\Phi ^{-1}(s)-\\Psi \\bigl ({\\Phi ^{-1}}^{\\prime }(s)\\bigr ) \\le 0$ .",
"The assertion now follows from the fact that $s = \\Psi ^{\\prime }(t)$ if and only if $t = {\\Phi ^{-1}}^{\\prime }(s)$ , which, again, is a consequence of the fact that $\\Phi ^{-1}$ and $\\Psi $ are conjugate.",
"[Proof of Theorem REF ] Throughout the proof we use the abbreviation $n = n^*(\\delta )$ .",
"First observe that $K^* p_n^\\delta \\in \\partial J(u_n^\\delta )$ and thus $J(u_n^\\delta ) - J(u^\\dagger )\\le \\left\\langle p_n^\\delta , Ku_n^\\delta - g \\right\\rangle $ .",
"From the discrepancy rule morozov:rule it follows that $\\left\\Vert Ku_n^\\delta - g \\right\\Vert \\le \\left\\Vert Ku_n^\\delta - g^\\delta \\right\\Vert + \\delta \\le (\\rho + 1)\\delta ,$ and hence the variational inequality rates:varineqcond implies $D(u_n^\\delta , u^\\dagger )\\le \\left\\Vert p_n^\\delta \\right\\Vert (\\rho +1)\\delta +\\Phi ((\\rho +1)^2\\delta ^2).$ As in the proof of Lemma REF we observe that for all $s$ , $r\\ge 0$ one has $s\\Phi (r)\\le \\Psi (s) + r$ .",
"Setting $r = (\\rho +1)^2\\delta ^2$ and $s = \\Psi ^{-1}((\\rho ^2-1)\\delta ^2)$ , one finds, after dividing both sides of the inequality by $s$ , that $\\Phi \\bigl ((\\rho +1)^2\\delta ^2\\bigr ) \\le \\frac{(\\rho ^2-1)\\delta ^2}{\\Psi ^{-1}((\\rho ^2-1)\\delta ^2)} +\\frac{(\\rho +1)^2\\delta ^2}{\\Psi ^{-1}((\\rho ^2-1)\\delta ^2)}= \\frac{2\\rho (\\rho +1)\\delta ^2}{\\Psi ^{-1}((\\rho ^2-1)\\delta ^2)},$ which yields an estimate for the second term in morozov:aux1.",
"For estimating the first term, we note that Corollary REF implies the estimate $\\left\\Vert p_n^\\delta \\right\\Vert \\le \\tilde{C} t_n \\biggl (\\Psi \\biggl (\\frac{2}{t_n}\\biggr ) + \\delta ^2\\biggr )^{1/2}\\le \\tilde{C} t_n\\Psi \\biggl (\\frac{2}{t_n}\\biggr )^{1/2} + \\tilde{C}t_n\\delta $ for some constant $\\tilde{C} > 0$ .",
"By assumption, the mapping $x \\mapsto \\Phi (x)^2\\slash x$ is non-increasing, and therefore, using Lemma REF , the mapping $s \\mapsto s^2\\Psi (2\\slash s)$ is non-decreasing.",
"Thus we obtain, after using the estimate for $t_n$ of Lemma REF and the monotonicity of $\\Psi $ , $\\left\\Vert p_n^\\delta \\right\\Vert &\\le \\tilde{C} \\biggl (\\frac{2}{\\Psi ^{-1}((\\rho ^2-1)\\delta ^2)} + \\tau _n\\biggr )\\Psi \\biggl (\\frac{2\\Psi ^{-1}\\bigl ((\\rho ^2-1)\\delta ^2\\bigr )}{2+\\tau _n\\Psi ^{-1}\\bigl ((\\rho ^2-1)\\delta ^2\\bigr )}\\biggr )^{1/2}\\\\&\\qquad \\qquad + \\frac{2\\tilde{C}\\delta }{\\Psi ^{-1}\\bigl ((\\rho ^2-1)\\delta ^2\\bigr )} + \\tilde{C}\\tau _n\\delta \\\\&\\le \\frac{2\\tilde{C}(\\rho +1)\\delta }{\\Psi ^{-1}\\bigl ((\\rho ^2-1)\\delta ^2\\bigr )}+ \\tilde{C}\\tau _n(\\rho +1)\\delta .$ Consequently we have $D(u_n^\\delta ,u^\\dagger )&\\le \\frac{2\\tilde{C}(\\rho +1)^2\\delta ^2}{\\Psi ^{-1}\\bigl ((\\rho ^2-1)\\delta ^2\\bigr )}+ \\tilde{C}(\\rho +1)^2\\tau _n\\delta ^2 + \\frac{2\\rho (\\rho +1)\\delta ^2}{\\Psi ^{-1}\\bigl ((\\rho ^2-1)\\delta ^2\\bigr )}\\\\&\\le \\frac{2(\\tilde{C}+1)(\\rho +1)^2\\delta ^2}{\\Psi ^{-1}\\bigl ((\\rho ^2-1)\\delta ^2\\bigr )}+ \\tilde{C}(\\rho +1)^2\\delta ^2\\sup _k \\tau _k,$ which proves the assertion with $C := 2(\\tilde{C}+1)$ ."
],
[
"Examples",
"In this section we discuss particular instances of the variational inequality rates:varineqcond and the implications of the general results in Sections and for these special scenarios.",
"The first two examples shed some light on the relation of variational inequalities and more standard notions of source conditions: the KKT condition dual:kkt and Hölder-type conditions.",
"The third example shows an example from sparsity promoting regularization, where standard notions of source conditions together with an additional restricted injectivity assumption allow the derivation of convergence rates with respect to norm instead of the Bregman distance."
],
[
"Standard Source Condition",
"It is quite easy to see that the standard source condition dual:kkt implies the variational inequality rates:varineqbreg.",
"Indeed, assume that $u^\\dagger $ is a solution of intro:primal and that $K^*p^\\dagger \\in \\partial J(u^\\dagger )$ for some $p^\\dagger \\in H$ .",
"By defining $\\xi ^\\dagger =K^*p^\\dagger $ one observes $\\left\\langle \\xi ^\\dagger , u^\\dagger - u \\right\\rangle _{X^*,X} = \\left\\langle p^\\dagger , g - Ku \\right\\rangle \\le \\left\\Vert p^\\dagger \\right\\Vert \\left\\Vert Ku-g \\right\\Vert .$ Setting $\\beta = 1$ and $\\Phi (t) = \\left\\Vert p^\\dagger \\right\\Vert t^{1\\slash 2}$ gives rates:varineqbreg.",
"The converse is in general not true, i.e., rates:varineqbreg with $\\Phi (t) = \\gamma t^{1\\slash 2}$ ($\\gamma > 0$ ) does not imply the existence of a $p^\\dagger \\in V$ such that $K^*p^\\dagger \\in \\partial J(u^\\dagger )$ .",
"However, if rates:varineqbreg is replaced by the stronger condition $\\left\\langle \\xi ^\\dagger , u^\\dagger -u \\right\\rangle _{X^*,X}\\le (1-\\beta ) D_J(u,u^\\dagger ) +\\gamma \\left\\Vert Ku - g \\right\\Vert ,$ for all $u\\in X$ , then the two notions are equivalent.",
"Here, $D_J(u,v) = J(u) - J(v) - J^{\\prime }(v)(u-v)$ and $J^{\\prime }(v)(w)$ is the directional derivative of $J$ at $v$ in direction $w$ : $J^{\\prime }(v)(w) = \\lim _{h\\rightarrow 0^+} \\frac{1}{h}(J(v+hw) - J(v)).$ Note that for convex $J$ , the directional derivative is well-defined for every $v$ and $w$ (though it takes values in $[-\\infty , \\infty ]$ ) and is positively one-homogeneous, i.e.",
"$J^{\\prime }(v)(tw) = t J^{\\prime }(v)(w)$ for all $t > 0$ .",
"In order to see the aforementioned equivalence, let $v\\in X$ and set $u = u^\\dagger - t v$ in rates:eqnmod for some $t > 0$ .",
"Then, $\\left\\langle \\xi ^\\dagger , tv \\right\\rangle _{X^*,X} \\le (1-\\beta ) D_J(u^\\dagger - tv, u^\\dagger ) +\\gamma \\left\\Vert tKv \\right\\Vert .$ Since the mapping $w\\mapsto J^{\\prime }(u^\\dagger )(w)$ is positively one-homogeneous, this implies that $\\left\\langle \\xi ^\\dagger , v \\right\\rangle _{X^*,X} \\le (1-\\beta )\\left(\\frac{J(u^\\dagger - tv) -J(u^\\dagger )}{t} - J^{\\prime }(u^\\dagger )(-v) \\right) + \\gamma \\left\\Vert Kv \\right\\Vert ,$ for all $v\\in X$ and $t>0$ .",
"Letting $t\\rightarrow 0^+$ this shows that $\\left\\langle \\xi ^\\dagger , v \\right\\rangle _{X^*,X}\\le \\gamma \\left\\Vert Kv \\right\\Vert $ for all $v\\in X$ and hence $K^*p^\\dagger = \\xi ^\\dagger $ for some $p^\\dagger \\in H$ according to .",
"In the particular case where the mapping $J$ is Gâteaux differentiable at $u^\\dagger $ , the subdifferential $\\partial J(u^\\dagger )$ contains a single element $\\xi ^\\dagger $ , which coincides with the directional derivative, that is, $\\left\\langle \\xi ^\\dagger , v \\right\\rangle = J^{\\prime }(u^\\dagger )(v)$ for every $v \\in X$ .",
"Thus, in this case, the source condition is equivalent with the variational inequality.",
"If $\\Phi (t) = \\gamma t^{1\\slash 2}$ then the Fenchel conjugate $\\Psi $ of $\\Phi ^{-1}$ reads as $\\Psi (t) = \\gamma \\slash (2\\sqrt{2}) t^2$ .",
"Hence it follows from Theorem REF that there exists a constant $C>0$ such that $D_J^{K^*p^\\dagger }(u_n^\\delta , u^\\dagger ) \\le C \\delta $ given the a priori stopping rule $t_n \\asymp \\delta ^{-1}$ .",
"This is the well known convergence rate result for the standard source condition (see , ).",
"We note that the results in are slightly stronger, as they give $\\delta $ -rates for the symmetric Bregman distance (see also ).",
"If Morozov's discrepancy principle morozov:rule is applied as an a posteriori stopping rule, we obtain from Theorem REF that $D_J^{K^*p^\\dagger }(u_{n^*(\\delta )}^\\delta , u^\\dagger ) \\le C\\sqrt{\\frac{(\\rho +1)^3 }{\\rho -1}}\\delta + C (\\rho +1)^2\\delta ^2\\sup _{k\\in \\mathbb {N}}\\tau _k.$ This coincides with the results in , where Morozov's discrepancy rule for the standard source condition was studied."
],
[
"Hölder-type Conditions",
"In this section we study the relationship between the variational inequality rates:varineqbreg and Hölder-type source conditions for the iteration intro:itertik.",
"We first consider the case of the iterated Tikhonov method, i.e., $ L =\\textnormal {Id}$ and thus $J(u) = {1\\over 2}\\left\\Vert u \\right\\Vert ^2$ .",
"Then, a solution $u^\\dagger $ of intro:primal is said to satisfy a Hölder condition with exponent $0\\le \\nu < {1\\over 2}$ , if $(K^*K)^\\nu p^\\dagger = u^\\dagger = \\partial J(u^\\dagger )$ .",
"If $u^\\dagger $ satisfies a Hölder condition with exponent $\\nu $ , then rates:varineqbreg holds with $D_J^{u^\\dagger }(u,u^\\dagger ) ={1\\over 2}\\left\\Vert u-u^\\dagger \\right\\Vert ^2$ and $\\Phi (s) \\asymp s^\\frac{2\\nu }{1+2\\nu }$ .",
"To see this, observe that the interpolation inequality (cf. )",
"implies $\\left\\langle u^\\dagger , u^\\dagger - u \\right\\rangle & \\le \\left\\Vert p^\\dagger \\right\\Vert \\left\\Vert (K^*K)^\\nu (u^\\dagger - u) \\right\\Vert \\\\& \\le \\left\\Vert p^\\dagger \\right\\Vert \\left\\Vert (K^*K)^\\frac{1}{2}(u^\\dagger -u) \\right\\Vert ^{2\\nu } \\left\\Vert u^\\dagger - u \\right\\Vert ^{1-2\\nu } \\\\& = 2^{{1\\over 2} -\\nu } \\left\\Vert p^\\dagger \\right\\Vert \\bigl (\\left\\Vert Ku - g \\right\\Vert ^2\\bigr )^\\nu D_J^{u^\\dagger }(u,u^\\dagger )^{1-2\\nu \\over 2}.$ Using Young's inequality $ab\\le a^p\\slash p + b^q\\slash q$ with $q = 2\\slash (1-2\\nu )$ and $p = 2\\slash (1+2\\nu )$ shows for all $\\eta > 0$ $\\bigl (\\left\\Vert Ku - g \\right\\Vert ^2\\bigr )^\\nu D_J^{u^\\dagger }(u,u^\\dagger )^{1-2\\nu \\over 2}& = \\frac{1}{\\eta }\\bigl (\\left\\Vert Ku - g \\right\\Vert ^2\\bigr )^\\nu \\eta D_J^{u^\\dagger }(u,u^\\dagger )^{1-2\\nu \\over 2} \\\\& = \\frac{1+2\\nu }{2\\eta ^{2\\over (1+2\\nu )}}(\\left\\Vert Ku - g \\right\\Vert ^2)^{\\frac{2\\nu }{1+2\\nu }} +\\frac{\\eta ^{2\\over (1-2\\nu )}(1-2\\nu )}{2}D_J^{u^\\dagger }(u,u^\\dagger ).$ Choosing $\\eta $ such that $1-\\beta = \\eta ^{2\\over 1-2\\nu }\\left\\Vert p^\\dagger \\right\\Vert ({1-2\\nu \\over 2})2^{1-2\\nu \\over 2} < 1$ results in rates:varineqbreg after setting $\\Phi (s) = c s^\\frac{2\\nu }{1+2\\nu }$ with $c= {1+2\\nu \\over 2\\eta ^{2\\slash (1+2\\nu )}}\\left\\Vert p^\\dagger \\right\\Vert 2^{1-2\\nu \\over 2}$ .",
"In case of the iterated Tikhonov-Morozov method, we consider intro:itertik with $K=\\textnormal {Id}$ and $L\\colon D(L)\\subset X\\rightarrow \\tilde{H}$ being a densely defined, closed linear operator.",
"Recall that in this case $\\hat{L} =(\\textnormal {Id}+ LL^*)^{-1}$ and $\\tilde{L} = (\\textnormal {Id}+ L^*L)^{-1}$ are self-adjoint and bounded linear operators (cf.",
").",
"A solution $u^\\dagger $ of intro:primal is said to satisfy a Hölder condition with exponent $0\\le \\nu \\le {1\\over 2}$ if $Lu^\\dagger = \\hat{L}^\\nu \\omega ^\\dagger $ for some $\\omega ^\\dagger \\in \\tilde{H}$ .",
"We show that this condition implies rates:varineqcond when $D(u,u^\\dagger )$ equals ${\\gamma \\over 2}\\left\\Vert Lu - Lu^\\dagger \\right\\Vert ^2$ (for some $\\gamma \\in (0,1)$ ) whenever $u\\in D(L)$ and $+\\infty $ else.",
"To see this, recall that $J(u) = \\infty $ if $u\\notin D(L)$ .",
"Thus rates:varineqcond is equivalent to $\\left\\langle Lu^\\dagger , Lu^\\dagger - Lu \\right\\rangle \\le (1-\\gamma )\\left\\Vert Lu - Lu^\\dagger \\right\\Vert ^2 +\\Phi (\\left\\Vert u - u^\\dagger \\right\\Vert ^2)$ for all $u\\in D(L)$ .",
"Setting $Lu^\\dagger =\\hat{L}^\\nu \\omega ^\\dagger $ shows together with the interpolation inequality and that for all $u\\in D(L)$ $\\left\\langle Lu^\\dagger , Lu^\\dagger - Lu \\right\\rangle & = \\left\\langle \\omega ^\\dagger , \\hat{L}^{\\nu }(Lu^\\dagger - Lu) \\right\\rangle \\\\& \\le \\left\\Vert \\omega ^\\dagger \\right\\Vert \\left\\Vert \\hat{L}^{1\\over 2}(Lu^\\dagger -Lu) \\right\\Vert ^{2\\nu }\\left\\Vert Lu^\\dagger - Lu \\right\\Vert ^{1-2\\nu } \\\\& \\le \\left\\Vert \\omega ^\\dagger \\right\\Vert \\bigl \\Vert L \\tilde{L}^{1\\over 2}\\bigr \\Vert ^{2\\nu }\\left\\Vert u^\\dagger - u \\right\\Vert ^{2\\nu }\\left\\Vert Lu^\\dagger - Lu \\right\\Vert ^{1-2\\nu }.$ With the same arguments as in the case of the iterated Tikhonov method above, we conclude that ex:hoelaux1 holds with $\\Phi (s) = \\tilde{c} s^{2\\nu \\over 2\\nu +1}$ for some constant $\\tilde{c} > 0$ .",
"Now let again be $X$ a general Banach space and $J\\colon X\\rightarrow \\overline{\\mathbb {R}}$ be convex such that Assumptions REF are satisfied.",
"As revealed by the calculations above, the variational inequality rates:varineqcond with $\\Phi (s) \\asymp s^\\frac{2\\nu }{1+2\\nu }$ can be seen as a generalized Hölder condition.",
"Note, that in this case the Legendre conjugate $\\Psi $ of $\\Phi ^{-1}$ comes as $\\Psi (t)\\asymp t^{1+2\\nu }$ and thus Theorem REF amounts to say that there exists a constant $C>0$ such that $D(u_n^\\delta , u^\\dagger ) \\le C \\delta ^{4\\nu \\over 1+2\\nu }$ if $t_n \\asymp \\delta ^{-2\\over 1+2\\nu }$ and Morozov's discrepancy principle morozov:rule shows that $D(u_{n^*(\\delta )}^\\delta , u^\\dagger ) \\le C\\left(\\frac{(\\rho +1)^{1+4\\nu }}{\\rho -1}\\right)^{1\\over 1+2\\nu } \\delta ^{4\\nu \\over 1+2\\nu } + C(\\rho +1)^2\\delta ^2 \\sup _{k\\in \\mathbb {N}}\\tau _k.$ These results coincide with the lower order rates for the iterated Tikhonov method and iterated Tikhonov-Morozov method ."
],
[
"Sparsity Promoting Regularization",
"We now discuss the application of the results derived in this paper to sparsity promoting regularization.",
"To that end, we assume that $X$ is a Hilbert space with orthonormal basis $\\left\\lbrace \\phi _i : i \\in \\mathbb {N} \\right\\rbrace $ , and we consider the regularization term $J(u) := \\sum _i \\left| \\left\\langle \\phi _i, u \\right\\rangle \\right|^q$ for some $1 \\le q < 2$ (see ).",
"In , it has been shown that, for Tikhonov regularization, this setting allows the derivation of convergence rates of order $\\mathcal {O}(\\delta ^q)$ with respect to the norm, if $u^\\dagger $ satisfies the standard source condition $K^*p^\\dagger \\in \\partial J(u^\\dagger )$ for some $p^\\dagger \\in H$ , and, additionally, a restricted injectivity condition holds.",
"In the following, we will generalize these results to the Augmented Lagrangian Method and source conditions of Hölder type.",
"Assume that there exists $0 < \\nu \\le 1/2$ such that $(K^*K)^\\nu p^\\dagger = \\xi ^\\dagger \\in \\partial J(u^\\dagger )$ and that $\\textnormal {supp}(u^\\dagger ) := \\left\\lbrace i\\in \\mathbb {N}: \\left\\langle \\phi _i, u \\right\\rangle \\ne 0 \\right\\rbrace $ is finite.",
"In case $q > 1$ assume in addition that the restriction of $K$ to $\\textnormal {span}\\left\\lbrace \\phi _i : i \\in \\textnormal {supp}(x^\\dagger ) \\right\\rbrace $ , and in case $q = 1$ assume that the restriction of $K$ to $\\textnormal {span}\\left\\lbrace \\phi _i : \\left| \\left\\langle \\phi _i, \\xi ^\\dagger \\right\\rangle \\right| < 1 \\right\\rbrace $ is injective.",
"We will show in the following that, under these assumptions, there exists a constant $C > 0$ such that rates:varineqcond holds with $D(u,u^\\dagger ) = C\\left\\Vert u^\\dagger -u \\right\\Vert ^q$ and $\\Phi (s)\\asymp s^{\\frac{q\\nu }{q-1+2\\nu }}$ in case $q > 1$ , and with $D(u,u^\\dagger ) = C\\left\\Vert u^\\dagger -u \\right\\Vert $ and $\\Phi (s) \\asymp s^{\\frac{1}{2}}$ for $q = 1$ .",
"It has been shown in that the given assumptions imply the existence of constants $C_1$ , $C_2 > 0$ such that $C_1\\left\\Vert u^\\dagger -u \\right\\Vert ^q \\le C_2\\left\\Vert Ku-g \\right\\Vert ^q + J(u)-J(u^\\dagger )-\\left\\langle \\xi ^\\dagger , u-u^\\dagger \\right\\rangle $ for all $u \\in X$ .",
"Applying the interpolation inequality to $\\left\\langle \\xi ^\\dagger , u-u^\\dagger \\right\\rangle $ , we obtain, similarly as in Section REF , the estimate $C_1\\left\\Vert u^\\dagger -u \\right\\Vert ^q &\\le C_2\\left\\Vert Ku-g \\right\\Vert ^q + J(u)-J(u^\\dagger )+ \\left\\Vert p^\\dagger \\right\\Vert \\left\\Vert Ku-g \\right\\Vert ^{2\\nu }\\left\\Vert u^\\dagger -u \\right\\Vert ^{1-2\\nu }.$ Now Young's inequality with $p = q/(1-2\\nu )$ and $p_*=q/(q-1+2\\nu )$ shows that $\\Bigl [C_1 -\\left\\Vert p^\\dagger \\right\\Vert \\frac{1-2\\nu }{q}\\eta ^{\\frac{q}{1-2\\nu }}\\Bigr ]\\left\\Vert u^\\dagger -u \\right\\Vert ^q&\\le C_2 \\left\\Vert Ku-g \\right\\Vert ^q + J(u)-J(u^\\dagger )\\\\&\\quad + \\left\\Vert p^\\dagger \\right\\Vert \\frac{q-1+2\\nu }{q}\\eta ^{\\frac{q}{q-1+2\\nu }} \\left\\Vert Ku-g \\right\\Vert ^{\\frac{2\\nu q}{q-1+2\\nu }}.$ Choosing $\\eta > 0$ such that $C = C_1 - \\left\\Vert p^\\dagger \\right\\Vert \\frac{1-2\\nu }{q}\\eta ^{\\frac{q}{1-2\\nu }} > 0$ and setting $\\Phi (s) = C_2 s^{\\frac{q}{2}} + \\left\\Vert p^\\dagger \\right\\Vert \\frac{q-1+2\\nu }{q}\\eta ^{\\frac{q}{q-1+2\\nu }}s^{\\frac{2\\nu q}{q-1+2\\nu }},$ we obtain the variational inequality rates:varineqcond.",
"Because $\\frac{2\\nu q}{q-1-2\\nu } \\le q$ , the asymptotic behaviour of $\\Phi $ for $s \\rightarrow 0$ is governed by its second term, which shows that $\\Phi (s) \\asymp s^{\\frac{q\\nu }{q-1+2\\nu }}$ .",
"Moreover, in the special case $q = 1$ , the term $s^{\\frac{q\\nu }{q-1+2\\nu }}$ reduces to $s^{\\frac{1}{2}}$ independent of the type of the source condition.",
"For the function $\\Psi $ , we obtain the asymptotic behaviour $\\Psi (s) \\asymp s^{\\frac{q-1+2\\nu }{q-1+(2-q)\\nu }}$ .",
"Thus, Theorem REF shows that for $t_n \\asymp \\delta ^{-2\\frac{q-1+(2-q)\\nu }{q-1+2\\nu }}$ we have the estimate $\\left\\Vert u_n^\\delta - u^\\dagger \\right\\Vert \\le C\\delta ^{\\frac{2\\nu }{q-1+2\\nu }}$ for $\\delta > 0$ sufficiently small, and a similar estimate for Morozov's discrepancy principle.",
"Remark 5.1 In , it has been shown for Tikhonov regularization with $J(u) = \\sum _i \\left| \\left\\langle \\phi _i, u \\right\\rangle \\right|$ , which is the special case of the ALM with a single iteration step, that a linear convergence rate with respect to the norm is equivalent to the usual source condition.",
"Thus the results above imply that, in the case $q = 1$ , the Hölder type source condition $(K^*K)^\\nu p^\\dagger \\in \\partial J(u^\\dagger )$ in fact already implies the standard source condition $K^* \\tilde{p}^\\dagger \\in \\partial J(u^\\dagger )$ for some different source element $\\tilde{p}^\\dagger $ .",
"The second author would like to thank Axel Munk and the staff of the Institute for Mathematical Stochastics at the University of Göttingen for their hospitality during his stay in Göttingen.",
"This work was partially funded by the DFG-SNF Research Group FOR916 Statistical Regularization and Qualitative Constraints (Z-Project)."
]
] | 1204.0771 |
[
[
"White Light Interferometry for Quantitative Surface Characterization in\n Ion Sputtering Experiments"
],
[
"Abstract White light interferometry (WLI) can be used to obtain surface morphology information on dimensional scale of millimeters with lateral resolution as good as ~1 {\\mu}m and depth resolution down to 1 nm.",
"By performing true three-dimensional imaging of sample surfaces, the WLI technique enables accurate quantitative characterization of the geometry of surface features and compares favorably to scanning electron and atomic force microscopies by avoiding some of their drawbacks.",
"In this paper, results of using the WLI imaging technique to characterize the products of ion sputtering experiments are reported.",
"With a few figures, several example applications of the WLI method are illustrated when used for (i) sputtering yield measurements and time-to-depth conversion, (ii) optimizing ion beam current density profiles, the shapes of sputtered craters, and multiple ion beam superposition and (iii) quantitative characterization of surfaces processed with ions.",
"In particular, for sputter depth profiling experiments of 25Mg, 44Ca and 53Cr ion implants in Si (implantation energy of 1 keV per nucleon), the depth calibration of the measured depth profile curves determined by the WLI method appeared to be self-consistent with TRIM simulations for such projectile-matrix systems.",
"In addition, high depth resolution of the WLI method is demonstrated for a case of a Genesis solar wind Si collector surface processed by gas cluster ion beam: a 12.5 nm layer was removed from the processed surface, while the transition length between the processed and untreated areas was 150 {\\mu}m."
],
[
"Introduction",
"In many experiments designed to determine sputtering yields (SY) of various materials under specific ion bombardment conditions, uncertainties in ion beam parameters can propagate and result in uncertain sputtering yield values [1].",
"For example, it can be challenging to determine shapes of ion beam profiles and the corresponding operational current densities, especially when the projectile energy goes below 1 keV and then further approaches the sputtering threshold.",
"Moreover, under such conditions, the focusing of the ion beam is in question, and the relative spread $\\Delta \\varepsilon /\\varepsilon $ in the initial kinetic energy distribution of ions [2] can have strong influence on experimental results [3], [4].",
"The other aspect that has a great impact on the final results is the method used for quantitative analysis of the surface, being commonly scanning electron and atomic force microscopy (SEM and AFM, respectively).",
"Both techniques are valuable, but each has its own limitations, when used for surface morphology characterization.",
"The AFM can obtain three-dimensional (3D) imaging and thus the cross section profiles for sputtering craters, but AFM has rather narrow ranges in the maximum lateral and especially depth scanning.",
"The SEM has much greater flexibility in the size of field-of-view with large depth of focus, but obtaining 3D imaging is cumbersome [5].",
"Another technique widely used in secondary ion mass spectrometry is the Stylus Profilometry.",
"This technique is popular because of its simplicity, but it is a coarser contact tool able to scan along a single line at a time, which would make 3D surface imaging extremely time consuming.",
"The qualifier \"coarser\" means the Stylus has difficulty measuring surface features of high aspect ratio or of size comparable with its characteristic tip size that implies a tip radius along with a tip angle [6].",
"It should be mentioned that in the case of the trace analysis mass spectrometry (our case), it is undesirable to have a sample to be analyzed in physical contact with a Stylus tip, which may contaminate or even scratch the surface.",
"All these facts make researchers to look for alternative methods for surface topography measurements.",
"In this regard, the optical interference methods seem to be natural.",
"It is known that the main drawback of an optical technique (utilizing geometrical optics) is the limited lateral resolution against SEM and AFM.",
"This limitation is of fundamental nature in that a surface feature of characteristic size less than $\\sim \\lambda /2$ (where $\\lambda $ is a light wavelength) cannot be resolved correctly.",
"On the other hand, the interference approach gives a fascinating depth resolution of less than 1 nm.",
"This work reports on application of the white light profilometry based on a Mirau interferometer (which is common for most of the commercial instruments) to characterize solid surfaces eroded in ion sputtering experiments.",
"A few examples of applying this method are provided when used for (i) characterization of ion beam profiles and crater shapes yielding accurate SY estimates, (ii) overlap alignment of a multiple ion beams system, (iii) time-to-depth calibration in sputter depth profiling, and (iv) characterization of surface processing of materials by ion beams.",
"For sputtering yield and rate estimates, the presented results demonstrate an alternative experimental approach to generate reference data for many materials and technological applications [4], [7], [8], [9], [10], [11], [12], [13] under bombardment with both commonly used atomic ions and relatively new molecular and cluster ions and help to resolve the problem of time or primary ion fluence to depth conversion.",
"In regard to mass spectrometry experiments, investigation of WLI benefits is practically important for us, since WLI as non-contact optical technique is attractive for implementing as an in-situ characterization tool."
],
[
"Material and methods",
"Mirau interferometry is an optical technique that measures the phase shift between the reference light signal and the light reflected from the sample surface.",
"It provides an optical micrograph onto which constructive and destructive interference fringes (light/dark) are superimposed.",
"The fringes are used to reconstruct the three-dimensional surface profile.",
"A white light source supplies a broad spectrum light.",
"This eliminates the problem associated with certain specimen features where the correct interference order cannot be determined.",
"The lateral resolution of the WLI probe is determined mostly by the chosen numerical aperture of the objective (limited to $\\sim \\lambda /2$ at numerical aperture $\\sim $ 1).",
"Once the best focus is found by mechanical positioning of the sample stage and the objective (corresponding to the brightest and strongest interference fringes, see Fig.1), a piezo transducer inside the objective performs vertical scanning of heights over a specified range.",
"Then an array of phase shifts between the reference signal, with constant optical path, and the signal, with an optical path which depends on the depth, is used to reconstruct true 3D surface topography and morphology.",
"At first glance, it seems that optically transparent films on a reflective substrate pose a serious problem for WLI.",
"If a material is transparent for given wavelength $\\lambda $ , there is always a phase shift (optical path length change) due to multiple passes of the light inside a film of refractive index $n>1$ , which may yield artifacts in a 3D topographic image.",
"At the same time, the phase shift allows one to distinguish between the transparent film response and the signal originated from the reflective base.",
"By separating these two responses (either directly [14] or by a special post-processing algorithm [15]) and paying attention to an absorption characteristic (which can be obtained independently) [16], one can measure a transparent/semitransparent film thickness starting at an order of 10 nm or higher (up to several $\\mu $ m), so that the drawback may turn out to be an additional advantage.",
"Information on Mirau WLI can be found in Refs.",
"[17], [18] in great details.",
"In the experiments presented here a MicroXAM-1200 profilometer controlled via MapVue AE software was employed.",
"The images were visualized using the SPIP software.",
"Before every measurement, the profilometer was calibrated laterally by a precise sub-mm ruler and vertically by 500 nm step AFM standard from Ted Pella, Inc.",
"In the examples of application of the WLI to ion sputtering experiments that follow, small ($\\sim $ 10$\\times $ 10 mm$^2$ ) pieces of Si(001) (MEMC Electronics and Unisil), and Cu(110) and Cu(111) (MTI Corporation) monocrystals were utilized.",
"In this context, these Si and Cu samples (which are uniform and nontransparent materials) do not have the \"transparent sample\" problem described above.",
"In addition, it seems that in many sputtering experiments, including the present study, shapes of removed craters, spots, etc.",
"can be classified as low gradient or step-like, which favors WLI applicability [19].",
"Figure: Optical micrograph showing example of theoptimally-aligned interference fringes from a white lightprofilometer.",
"The sample is a sputtering semispherical craterformed by direct current ion irradiation of a small Si waferchip."
],
[
"Results and discussion",
"1.",
"Craters and ion beam profiles measurements to estimate Sputtering Yield As an alternative to the known and widely employed method for estimating sputtering yields using mass-loss method, based on direct weighing or quartz microcrystal balance [20], we propose to use the WLI method for direct visualization of the sputtered ion beam spots or craters obtained by static sputtering or by raster scanning of an ion beam, respectively.",
"For low energy ion beam irradiation, WLI can verify whether or not the entire beam was confined to the sample of interest.",
"By combining the WLI visualization with precise measurements of the total ion current by a Faraday cup, the SY and the operating current density can be obtained simultaneously.",
"Besides, this approach appears to be very helpful in estimating the extent of undesirable \"wings\" of the ion beam profile so that, as a feedback, it guides the alignment of an ion beam source.",
"The sputtering yield $Y$ is then estimated using the following expression $Y=\\frac{\\rho \\cdot V\\cdot e}{I\\cdot \\tau \\cdot M_{atom}}, $ where $I$ , direct current (dc) current of an ion beam; $\\tau $ , time of sputtering; $M_{atom}$ , mass of a matrix atom in grams; $\\rho $ , density; $e$ , the elementary charge.",
"$V$ is the volume of the removed sample material obtained by means of the WLI measurement.",
"Volume calculations can be performed either by using a histogram of heights typically available through an interferometer post-processing software called SPIP by Image Metrology that works with files type generated by MapVue AE or by three dimensional integration based on cross sections in two orthogonal directions centered on the eroded surface area (black lines in Fig.2a).",
"Figure 2 compares longitudinal cross sections of a spot (red dotted line) of a normally incident static 5 keV Ar$^+$ ion beam against a crater (green open squares) obtained by 100$\\times $ 100 pixels digital raster scanning of the same ion beam over the surface of a Cu(110) monocrystal.",
"The curve corresponding to the static beam overlaps one edge of the crater to demonstrate how raster scan of the ion beam generates the crater during sputter depth profiling.",
"Good alignment of the ion beam column manifests itself in a symmetric beam profile and FWHM of 120 $\\mu $ m at a total current of 2 $\\mu $ A.",
"The WLI approach allows one to characterize the ion sputtering with the same normally incident ion beam decelerated to 150 eV by the target potential.",
"In this case, the cross section of the static beam spot is shown by an orange solid line, and the crater cross section is shown by cyan open circles.",
"The ion column allowed delivery of the same 2 $\\mu $ A of Ar$^+$ current on the target because the deceleration of the beam from the nominal 5 keV energy to 150 eV occurred in the immediate vicinity of the target, and in such a way that its optimal focusing was maintained by an electrostatic lens (FWHM of 150 $\\mu $ m in Fig.2b proves that) [21].",
"The sputtered crater has in this case a larger lateral size because the deflection voltages of the raster-generating octupole were kept unchanged for the two primary ion impact energies, resulting in additional beam swinging due to the target potential.",
"Based on the WLI data, sputtering yields of Cu(110) at 5 keV and 150 eV ion impact energies were determined.",
"An obtained SY value of 1.8 at/ion for the former case was in good agreement with literature data [22].",
"For the latter one, the sputtering yield was 0.2 at/ion.",
"The SY values for Cu(111) at 50, 100, and 150 eV were also determined as 0.13, 0.27, and 0.42 at/ion, respectively.",
"The measured energy spread $\\Delta \\varepsilon $ of the low energy system [21] is 23 eV.",
"Figure: (a) Pseudocolor 2D top view of produced crater.",
"Blacklines are directions along which cross sections plotted in (b)were measured.",
"(b) Beam spot and crater cross sectionssuperimposed.",
"Measurements were made on Cu(110) sputtered bynormally incident Ar + ^+ ion beams with 5 keV (green squares andred dotted line) and 150 eV (cyan circles and orange solid line)energies.2.",
"Multiple beam system alignment and time-to-depth conversion In our previous work, we have introduced and demonstrated a new variant of dual-beam (DB) sputter depth profiling for time-of-flight secondary ion mass spectrometry (TOF SIMS), where we aimed at improving the depth resolution by using a normally incident low-energy direct current ion beam for sputtering, in combination with obliquely incident fine focused pulsed ion beam for TOF SIMS analysis.",
"The benefit of such an arrangement of the sputtering ion beam is two-fold: its low (a few hundred eV) energy reduces ion beam mixing, and its normal impact angle reduces surface roughening.",
"To make this concept work, it is needed to precisely overlap the crater created by raster scanning the low energy dc ion beam with the area probed by the pulsed analysis ion beam.",
"Moreover, (i) most of the bottom of the low energy crater must be flat (Fig.2b), and (ii) the analysis area must be confined within that flat part, in order to avoid distortions in the depth profile due to probing sloped areas or crater walls.",
"This can only be achieved by thorough optimization of both ion beams (current density profiles and focusing) as well as precise control of their steering.",
"The WLI technique helps to make this multi-step alignment much easier.",
"Results of the WLI characterization presented in Fig.3 give straightforward answers regarding mutual positioning of sputtering and analysis ion beams by showing two craters produced by raster scanning of these beams in dc mode.",
"The deep and narrow crater seen in Fig.3 was made by the analysis beam (5 keV Ar$^+$ ions with 60$^\\circ $ incident angle).",
"The wide and shallow crater was made by a normally incident 500 eV Ar$^+$ ion beam.",
"Fig.3 demonstrates that the 5 keV Ar$^+$ probing in the DB mode was conducted on the flat bottom part of the crater created by the low energy sputtering ion beam.",
"Figure: (a) Pseudocolor 3D topographic view of two superimposedcraters made by two separate raster scanned ion beams.",
"The largeone (1.5×\\times 1.5 mm 2 ^2) is sputtered by a 500 eV normallyincident Ar + ^+ ion beam.",
"The smaller one (500×\\times 500μ\\mu m 2 ^2) is produced by 5 keV Ar + ^+ ions with 60 ∘ ^\\circ incidence angle.",
"(b) Cross section of 3D image along one of theblack lines shown in (a)Another important application of the WLI method to sputter depth profiling is exemplified by the sputtering time to sputtered depth calibration procedure applied to this particular experiment.",
"The samples analyzed here were pieces of Si(001) wafer implanted with $^{25}$ Mg$^+$ , $^{44}$ Ca$^+$ and $^{53}$ Cr$^+$ ions at energy of 1 keV per atomic mass unit (25 keV for $^{25}$ Mg, 44 keV for $^{44}$ Ca and 53 keV for $^{53}$ Cr, all at 3$\\times $ 10$^{13}$ ions/cm$^2$ fluence) fabricated by Leonard Kroko Inc. A TOF MS analysis of these samples was performed by laser post-ionization of sputtered neutrals (secondary neutral mass spectrometry, SNMS) using resonantly enhanced multi-photon ionization to simultaneously detect all isotopes of Mg, Ca and Cr [23].",
"This was an experiment on sputter depth profiling which started in the DB mode as described above but, after the concentration peaks of the implants were passed (that is, after 170 nm on the depth scale in Fig.3, see also Fig.4), the experiment continued in the single beam (SB) mode by switching off the low energy sputtering beam, while the analysis beam performed both the ion milling (in dc mode) and the analysis (in pulsed mode).",
"The higher energy (5 keV) and 60$^\\circ $ incidence of the analysis beam allowed us to reduce the time needed for measuring the trailing edge of the implant depth profiles where high depth resolution was not needed.",
"The calibration procedure involved: (i) the WLI measurements of the depths of craters created by both ion beams, as shown in Fig.3, (ii) ion current measurements of both these beams with the Faraday cup, and (iii) calculating depth scale based on the total sputtering time with either of the two beams and the corresponding WLI measurements of crater depths.",
"To compare this depth calibration with a model estimate, TRIM simulations for 1 keV/amu ions of the same Mg, Ca and Cr isotopes implanted in a Si matrix with SiO$_2$ of 2 nm on top were performed.",
"After that, the experimental and simulated data were compared on the same plot, as shown in Fig.4.",
"This comparison revealed very good agreement between the depths of Ca and Mg implant peak concentrations determined by the WLI-based depth calibration and simulated by TRIM.",
"In the case of Cr, the shift between simulated and experimentally measured peak was $\\sim $ 5 nm.",
"Thus, the sputtering time to depth calibration using the WLI measurements proved to be satisfactorily accurate.",
"It proved also that, if a depth profile is made purely in SB manner, an elemental peak distribution appears to lie deeper (under the same time-to-depth conversion procedure by WLI) as compared to DB results shown in Fig.4.",
"This peak depth overestimation leads to an error in the fluence value obtained by integration of the depth profile curve.",
"This issue is not discussed here, since this fact is obvious and lies beyond the scope of this paper.",
"Figure: Symbols represent measured secondary neutralmass-spectrometry depth profiles of isotopes 25 ^{25}Mg (winecircles), 44 ^{44}Ca (green squares), 53 ^{53}Cr (blue diamonds)implanted in Si host matrix at 1 keV per nucleon (1 keV/amu).Lines (red, light green and gray) are independent TRIM simulationsof depth distributions for the same isotopes of 1keV/amu energiesin SiO 2 _2/Si sandwich (SiO 2 _2 thickness is of 2 nm)3.",
"Quantitative characterization of ultra-shallow surface processing with cluster ion beam In this example, the depth resolution of the WLI technique applied to characterization of Si surfaces irradiated with gas cluster ion beams (GCIB) is demonstrated.",
"The GCIB in these sputtering experiments was an argon cluster beam Ar$^+_N$ with $N$ =2000, where $N$ corresponds to the number of atoms in the peak distribution and, in general, can lie between 200 and 10000 [24], [25].",
"Irradiating materials surfaces with such cluster ions causes two unique effects.",
"First, because the impact energy of such projectiles equals to their kinetic energy divided by the number of constituent atoms, for a 20 keV Ar$^+_{2000}$ , for example, it will be only 10 eV/Ar, which significantly reduces the penetration of individual Ar atoms into the target and the sputtering process starts to strongly depend on the collective effects of many such impacts.",
"In essence, for GCIB irradiation, the sample damage is confined to a narrow near-surface layer.",
"Another effect is the surface \"polishing\" (or planarization), which manifests itself in a reduced roughness of the irradiated surfaces.",
"To summarize, at normal incidence the GCIB irradiation can literally \"shave off\" topmost layers from a target with minimal alteration of underlying regions.",
"These two effects are very beneficial for our efforts on quantitative analyses of the Genesis mission [26] solar wind (SW) collectors by resonance ionization mass spectrometry [23].",
"The Genesis mission samples present a serious analytical challenge because of abundant contamination which blanketed the collectors surface after the crash landing of the Genesis sample return capsule.",
"In addition to the crash-derived contamination, such as terrestrial dust particles, a highly refractory organic/silicon film, known as the \"brown stain\" [27], covers the top of Genesis samples.",
"While conventional methods such as megasonic cleaning with ultrapure water removes particulates $\\ge $ 1 $\\mu $ m loosely connected to surface [28], the remaining contamination must be dealt with differently.",
"The GCIB processing of surfaces of Genesis collectors has the potential to \"shave off\" this contamination blanket with minimal losses of the implanted SW species [29].",
"To our knowledge, this is possibly the most advanced cleaning method proposed so far for uniform removal of surface contamination.",
"In this WLI example the GCIB processed surface of the Genesis 60428 Si coupon is characterized in order to measure the exact depth removed.",
"Currently, by measuring $^{24}$ Mg, $^{40}$ Ca and $^{52}$ Cr solar wind distributions by DB SNMS, we know that the surface contamination covers the first $\\sim $ 10 nm of the depth profile [30].",
"By using the GCIB process to reduce the surface contamination, the contribution of contamination to the depth profile is significantly decreased, resolving the SW profile from it and permitting a more accurate integration of the SW depth profile curve to obtain elemental abundance fluences.",
"Thus, the precise thickness of the layer removed by GCIB is critical.",
"GCIB processing conditions on the Genesis 60428 Si coupon were as follows: operating current of 68 $\\mu $ A, GCIB raster area of 6.4$\\times $ 10$^{-3}$ m$^2$ , GCIB exposed Si surface area of 2.9$\\times $ 10$^{-5}$ m$^2$ , and the sample processing time under GCIB $T=\\frac{2.9\\times 10^{-5}}{6.4\\times 10^{-3}}\\times 153$ s (where 153 s is the total time during which GCIB source was switched on and raster scanned).",
"The measurement depicted in Fig.5 shows that the surface layer removed by GCIB irradiation was as low as 12.5 nm.",
"At the same time, the length of the transition region between irradiated and non-irradiated areas of the sample is as long as $\\sim $ 150 $\\mu $ m. This length is on the order of the full lateral scan of an AFM, and makes it essentially impossible to find such a step by means of AFM, while the depth is at the resolution limit of the best Stylus Profilometer, emphasizing the high value of the WLI method.",
"If we assume that, originally, the sample consists of only Si and use the literature data for sputtering yield of Si under a 20 keV Ar$^+_{2000}$ cluster ion beam ($Y$ =41.5 atoms per cluster ion [31]), the thickness that should have been removed would be 8.5 nm.",
"This estimate proves indirectly the presence of an extra layer that may contain submicron particulates, the \"brown stain\", and the native silicon oxide layer before the GCIB processing.",
"Figure: (a) Pseudocolor surface of the Genesis Si solar windcollector coupon 60428 \"cleaned\" by GCIB.",
"Black line is thedirection along which cross section plotted in (b) of the figurewas measured.",
"Vertical black arrows indicate the separationbetween processed/cleaned and original surfaces.",
"(b) WLI crosssectional profile gives the precise thickness of the removed layerover the irradiated surface area"
],
[
"Conclusions",
"The benefits of the white light optical profilometry based on a Mirau interferometer were demonstrated when it is applied to problems of quantitative characterization of ion sputtered surfaces.",
"The key advantages of this technique are high depth resolution in combination with flexible lateral field-of-view and the capability of true three-dimensional surface topography reconstruction.",
"Examples to prove the power of this method were provided here.",
"In particular, it was demonstrated how to use the WLI approach to determine sputtering yields of copper and silicon irradiated by ultralow energy argon ions over confined eroded area of controlled geometry.",
"Such measurements can be done both on focused static beam spots ($\\sim $ 10 $\\mu $ m dia.)",
"and on mm-scale raster scan craters with high extent of averaging the sputtering characteristic.",
"In addition, the WLI technique can significantly help with alignment of ion columns with multiple overlapping or superimposed ion (or ion and laser) beams, as demonstrated in presented example with the dual-beam sputter depth profiling.",
"Thus, the WLI technique facilitates better fundamental understanding of sputtering processes at ultra-low energies by helping to accurately determine sputtering yields (and by addressing problems of preferential sputtering), and by helping with precise conversion of ion fluence or sputtering time into depth.",
"Moreover, it lends scientists an ability to precisely characterize and, ultimately, to control materials' surface topography formed by ion sputtering under a wide variation of conditions (eV to tens of keV impact energy or atomic/cluster/molecular projectile species), which is a great benefit for ion sputtering based materials synthesis or characterization."
],
[
"Acknowledgments",
"The authors would like to thank Dr. James Norem (Argonne National Laboratory, USA) for providing the Cu monocrystals, and Profs.",
"Isao Yamada and Noriaki Toyoda (University of Hyogo, Japan) for GCIB processing of the Genesis sample surface.",
"This work was supported under Contract No.",
"DE-AC02-06CH11357 between UChicago Argonne, LLC and the U.S. Department of Energy and by NASA through grants NNH08AH761 and NNH08ZDA001N."
]
] | 1204.0808 |
[
[
"Impulsive gravitational waves of massless particles in extended theories\n of gravity"
],
[
"Abstract We investigate the vacuum pp-wave and Aichelburg-Sexl-type solutions in f(R) and the modified Gauss-Bonnet theories of gravity with both minimal and nonminimal couplings between matter and geometry.",
"In each case, we obtain the necessary condition for the theory to admit the solution and examine it for several specific models.",
"We show that the wave profiles are the same or proportional to the general relativistic one."
],
[
"Introduction",
"Extensions of the general theory of relativity in which functions of some geometric quantity are coupled either minimally or nonminimally to the matter part of the usual action for gravity have been widely studied recently, mainly as part of current efforts to explain the Universe with its observed late-time accelerated expansion [1], [2], [3].",
"The so-called $f(R)$ gravity comprises the largest subset of models constructed in this way, see Refs.",
"[4], [5], [6] for reviews.",
"In these models, a priori arbitrary functions of the scalar curvature of spacetime are included in the gravitational action in its various metric, Palatini, or metric-affine formulations.",
"The $f(R)$ function can also be coupled nonminimally to the matter Lagrangian, as suggested in Refs [7], [8].",
"An interesting consequence of such nonminimal coupling is the emergence of extra forces making the trajectories of otherwise-free particles nongeodesic [9].",
"Another well-known class consists of the so-called ${\\mathcal {F}}({\\mathcal {G}})$ gravity models, in which a function of the Gauss-Bonnet invariant, ${\\mathcal {G}}$ , is added to the Einstein-Hilbert action [10].",
"More general modified Gauss-Bonnet theories of gravity with nonminimal coupling have also been suggested [11], [12], see also Ref.",
"[13].",
"Various aspects of the above-mentioned models have been studied in recent years, including black hole solutions and their thermodynamics and cosmological solutions with accelerated expansion.",
"There is now a huge literature on this, and a selection is listed in Refs.",
"[4], [5], [6] and [14].",
"Massless and massive gravitational wave solutions have been presented, too [15].",
"The issue of linearized $f(R)$ gravity has been studied in Refs.",
"[16], [17], [18].",
"Compared with general relativity, several new features arise in this context, namely, the appearance of extra polarization modes and a nonlinear dispersion law, see, e.g., Ref.",
"[19].",
"Different models introduced in this context have been examined against several theoretical or observational criteria.",
"As an example, the study by Dolgov and Kawasaki [20] has shown that certain $f(R)$ models accommodate ghosts and hence are ruled out by the consequent instability.",
"One can also mention constraints coming from solar system effects such as planetary orbits or bending of light [21], and the bounds coming from imposing the energy conditions [22], in this regard.",
"The aim of the present work is to find classes of these extended theories of gravity which admit plane-fronted-parallel rays gravitational wave solutions the same or similar to the general relativistic vacuum or Aichelburg-Sexl solutions [23].",
"One motivation behind this work is the recent interest in gravitational pp-waves in general, and in gravitational shock waves of various sources, in particular.",
"In fact, in the context of general relativity, the problem has been studied by boosting the Kerr metric in Refs.",
"[24] and [25], by boosting the Kerr-Newman metric in Ref.",
"[26], in the presence of a nonvanishing cosmological constant [27], for motion in Schwarzschild-Nordström and Schwarzschild-de Sitter spacetimes [28], for particles with arbitrary multipoles in Ref.",
"[29], for motion in Nariai universe [30], and for motion in the presence of electromagnetic fields in Ref.",
"[31].",
"A reason for interest in such solutions lies in their role in the scattering of particles off of each other at ultrahigh energies [32], [33].",
"Extensions of the Aichelburg-Sexl solution outside the context of general relativity have also been of interest, namely, in the framework of the brane-induced gravity [34] and in the context of the ghost-free model of massive gravity [35].",
"Second, and at the same time, this study aims to put forward a new criterion for testing extended gravity models.",
"The basic idea is that the gravitational radiation by moving particles, massless ones in the present case, is detectable, at least in principle, and whether a given model admits the plane wave solution distinguishes it from other models.",
"For models admitting the solution, the explicit form of the wave profile provides further information.",
"In the next sections, after a brief review of the Aichelburg-Sexl solution of general relativity, we apply the above ideas to several well-known extended theories of gravity including minimal and nonminimal $f(R)$ gravities and ${\\mathcal {F}}({\\mathcal {G}})$ gravity."
],
[
"The Aichelburg-Sexl solution",
"The Aichelburg-Sexl (AS) solution, first introduced in Ref.",
"[23], represents the gravitational field of a massless particle moving in an otherwise empty space.",
"It belongs to the general class of plane-fronted gravitational waves with parallel rays, or pp-wave, solutions.",
"The general form of a pp-wave line element may be written as $ds^2=-dudv-K(u,x,y)du^2+dx^2+dy^2,$ in which $u=t-z$ and $v=t+z$ are the null coordinates, and $K(u,x,y)$ is obtained from the field equations.",
"For vacuum, the field equation results in the two-dimensional Laplace equation to be satisfied by $K(u,x,y)$ .",
"The AS solution may be obtained by inserting the energy-momentum tensor of a massless particle in the Einstein equation.",
"Starting with the action for the coupled gravity-massless particle system, $S=\\frac{1}{16\\pi G}\\int {\\sqrt{-g}}\\,R\\,d^4x+\\int {\\sqrt{-g}}\\,{\\mathcal {L}}_{p}\\,d^4x,$ one obtains $G_{\\mu \\nu }=8\\pi GT_{\\mu \\nu },$ where $R=\\delta ^\\alpha _\\mu g^{\\beta \\nu }(\\partial _\\alpha \\Gamma ^\\mu _{\\beta \\nu }-\\partial _\\beta \\Gamma ^\\mu _{\\alpha \\nu }+\\Gamma ^\\mu _{\\alpha \\gamma }\\Gamma ^\\gamma _{\\nu \\beta }-\\Gamma ^\\mu _{\\beta \\gamma }\\Gamma ^\\gamma _{\\nu \\alpha })$ is the scalar curvature, $G_{\\mu \\nu }$ is the Einstein tensor, and $T_{\\mu \\nu }=-\\frac{2}{\\sqrt{-g}} \\frac{\\delta ({\\mathcal {L}}_{p}{\\sqrt{-g}})}{\\delta g^{\\mu \\nu }}$ is the energy-momentum tensor.",
"The above action is expressed in a system of units in which $c=1$ .",
"For convenience, we also set $8\\pi G=1$ .",
"For a massless particle of momentum $p$ with ${\\mathcal {L}}_{p}=\\frac{p}{2}\\int {g_{\\mu \\nu }{\\dot{x}}^\\mu {\\dot{x}}^\\nu }\\delta ^4(x-x(\\tau ))d\\tau $ in which ${\\dot{x}}^\\mu =\\frac{dx^\\mu }{d\\tau }$ , we have $T^{\\mu \\nu }=p\\int \\delta ^4(x-x(\\tau )){\\dot{x}}^\\mu {\\dot{x}}^\\nu d\\tau ,$ where $x^\\mu (\\tau )$ corresponds to the trajectory of the particle which has to be a null geodesic of the spacetime under consideration.",
"Thus, we can take $x^\\mu (\\tau )$ to be of the form ($\\tau ,0,0,z(\\tau )=\\tau )$ which is null and satisfies the geodesic equation in the spacetime described by Eq.",
"(REF ).",
"Inserting these together with the ansatz given by Eq.",
"(REF ) into Eq.",
"(REF ) and making use of the following relation $\\left(\\frac{\\partial ^2}{\\partial x^2}+\\frac{\\partial ^2}{\\partial y^2}\\right)\\ln {\\sqrt{x^2+y^2}}=2\\pi \\delta (x)\\delta (y),$ we obtain $K_{AS}(u,x,y)=-\\frac{p}{\\pi }\\delta (u)\\ln {\\sqrt{x^2+y^2}}$ which upon substitution in Eq.",
"(REF ), describes a gravitational shock-wave propagating with the speed of light along the $z$ -direction."
],
[
"Gravitational waves in nonminimal $f(R)$ gravity",
"We start with the following action [9] $S=\\int {\\sqrt{-g}}\\left(\\frac{1}{2}f(R)+(1+\\lambda F(R)){\\mathcal {L}}_m\\right)d^4 x$ in which $f(R)$ and $F(R)$ are arbitrary functions of $R$ .",
"This represents an extension of the general theory of relativity in which the matter field Lagrangian density ${\\mathcal {L}}_m$ is coupled nonminimally with the geometric structure $F(R)$ , and the coupling constant $\\lambda $ controls the strength of the coupling.",
"This reduces to the usual $f(R)$ theories of gravity for $\\lambda =0$ , which in turn reduces to general relativity by choosing $f(R)=R$ .",
"The field equation associated with the above action reads ${\\mathcal {E}}_{\\mu \\nu }&=&\\nabla _\\mu \\nabla _\\nu f^\\prime (R)-g_{\\mu \\nu }\\square f^\\prime (R)\\nonumber \\\\&&+2\\lambda (\\nabla _\\mu \\nabla _\\nu -g_{\\mu \\nu }\\square ){\\mathcal {L}}_m F^\\prime (R)\\nonumber \\\\&&-2\\lambda F^\\prime (R){\\mathcal {L}}_m R_{\\mu \\nu }+(1+\\lambda F(R))T_{\\mu \\nu },$ where $\\nabla $ means covariant differentiation, $\\square =\\nabla _\\mu \\nabla ^\\mu $ , $R_{\\mu \\nu }$ is the Ricci tensor, and ${\\mathcal {E}}_{\\mu \\nu }=f^\\prime (R)R_{\\mu \\nu }-\\frac{1}{2}f(R)g_{\\mu \\nu }.$ Now, let us examine the above field equation to see if the vacuum pp-wave solution is admitted in this model.",
"For vacuum, the last three terms in the right-hand side of Eq.",
"(REF ) vanish, and if we insert the ansatz (REF ) into the resulting equation, we will reach the following relations $f^\\prime (0)\\left(\\frac{\\partial ^2}{\\partial x^2}+\\frac{\\partial ^2}{\\partial y^2}\\right)K(u,x,y)-f(0)K(u,x,y)=0,$ and $f(0)=0.$ Thus, the vacuum pp-wave solution is admitted only if the above condition holds.",
"This condition is in fact among the requirements needed to constrain the models from cosmological or solar system tests, which is also consistent with the limiting case of the $\\Lambda $ CDM phenomenology [36].",
"The other requirement is that $f(R)$ should tend to a constant when the scalar curvature tends to infinity.",
"When Eq.",
"(REF ) holds, Eq.",
"(REF ) reduces to the two-dimensional Laplace equation in the transverse plane, provided $f^\\prime (0)\\ne 0$ .",
"This means that the waveform is the same as the vacuum general relativistic one.",
"For the special case where $f^\\prime (0)=0$ , Eq.",
"(REF ) is trivially satisfied with arbitrary $K(u,x,y)$ .",
"But this corresponds to the absence of the linear term $R$ in the Lagrangian and hence is ruled out.",
"Now, we consider a massless particle moving along the $z$ -direction and seek solutions of the form given in Eq.",
"(REF ) (renaming $K(u,x,y)$ to $ K_{nR}(u,x,y)$ to avoid confusion).",
"Inserting Eqs.",
"(REF ) and (REF ) into the above field equation (with ${\\mathcal {L}}_m$ replaced by ${\\mathcal {L}}_{p}$ ) and noting that for massless particles $g_{\\mu \\nu }{\\dot{x}}^\\mu {\\dot{x}}^\\nu $ vanishes, we reach $\\nabla ^2_TK_{nR}(u,x,y)=-2p\\frac{1+\\lambda F(0)}{f^\\prime (0)}\\delta (u)\\delta (x)\\delta (y),$ provided Eq.",
"(REF ) holds and $f^\\prime (0)\\ne 0$ .",
"Here, $\\nabla ^2_T\\equiv \\frac{\\partial ^2}{\\partial x^2}+\\frac{\\partial ^2}{\\partial y^2}$ .",
"Thus, if Eq.",
"(REF ) is satisfied, the pp-wave solution (REF ) is admitted, and the waveform is given by $K_{nR}(u,x,y)=\\frac{1+\\lambda F(0)}{f^\\prime (0)}K_{AS}(u,x,y).$ If we also choose $F(0)=0$ , then the effect of the nonminimal coupling disappears totally.",
"For, $F(0)\\ne 0$ the nonminimal coupling has a contribution equal to $\\frac{F(0)}{f^\\prime (0)}\\lambda $ .",
"Because of the very small expected value of the coupling constant $\\lambda $ , this would be a small contribution; see Ref.",
"[37] for a discussion of bounds on the values of $\\lambda $ .",
"It should be noted here that for the pp-wave spacetime (REF ), the nonlinear field equation (REF ) reduces to a linear equation discussed above, and this allows the use of distributional expressions as in general relativity."
],
[
"Gravitational waves in $f(R)$ gravity",
"The well-studied $f(R)$ theories of gravity are in fact a subclass of the nonminimal theory considered above, with $\\lambda =0$ .",
"Thus, both the vacuum pp-wave and the AS solutions are admitted if Eq.",
"(REF ) holds.",
"To determine the waveform of the AS solution, we insert $\\lambda =0$ into Eq.",
"(REF ) (this time with $K_R(u,x,y)$ in place of $K_{nR}(u,x,y)$ ).",
"This yields $K_R(u,x,y)=\\frac{1}{f^\\prime (0)}K_{AS}(u,x,y).$ Since in general $f^\\prime (R) > 0$ , otherwise ghosts are allowed; the signs of $K(u,x,y)$ and $K_{AS}(u,x,y)$ are the same.",
"An example of the models satisfying the requirement given by Eq.",
"(REF ) is the broken power-law model $f(R)=R-m^2\\frac{c_1\\left(\\frac{R}{m^2}\\right)^n}{1+c_2\\left(\\frac{R}{m^2}\\right)^n}$ suggested in Ref.",
"[36], in which $c_1, c_2, m$ , and $n$ are constants, and $n > 0$ .",
"Note that the linear term above is included to reproduce the Einstein-Hilbert action in Eq.",
"(REF ).",
"For this model, we have $\\lim _{R\\rightarrow 0}\\frac{1}{f^\\prime (R)}=\\left\\lbrace \\begin{array}{cll}1 & \\mbox{if} & n>1\\\\ \\frac{1}{1-c_1} & \\mbox{if} & n=1 .\\\\ 0 & \\mbox{if} & n<1\\end{array}\\right.$ By inserting this into Eq.",
"(REF ), we conclude that for $n\\ge 1$ , the model admits the plane wave solution, with a wave profile the same as the general relativistic one for $n>1$ , and $\\frac{1}{1-c_1}$ times the general relativistic waveform for $n=1$ .",
"For $n < 1$ , in which the above limit equals zero, the solution reduces to the Minkowski spacetime.",
"In other words, for $n<1$ , the plane wave solution is not admitted, and this is in agreement with what we expect from the general requirement $f^\\prime (R)>0$ mentioned earlier.",
"Another example is the Starobinsky model [38] given by $f(R)=R+\\frac{R^2}{M^2}.$ Here, we have $\\frac{1}{f^\\prime (0)}=1,$ and hence the solution is exactly the same as the general relativistic one.",
"Also, the following cosmologically viable model proposed in Ref.",
"[39] (see also Ref.",
"[40]) $f(R)=R-\\lambda _0 R_0\\left(1-\\frac{1}{\\left(1+\\frac{R^2}{R^2_0}\\right)^n}\\right),$ in which $\\lambda _0, R_0, n$ are positive constants, satisfies the condition (REF ).",
"For this model, we have $f^\\prime (0)=1$ , i.e.",
"coincidence with general relativity, again.",
"However, because in this model $f^{\\prime \\prime }(0) < 0$ , it is unstable [41].",
"The model described by [42] $f(R)=R+\\frac{1}{a}\\ln (\\cosh (aR)+b\\sinh (aR)),$ where $a, b$ are constants, admits the solution, too.",
"Here, we have $\\frac{1}{f^\\prime (0)}=\\frac{1}{1+b}.$ An example of the models incompatible with the condition (REF ) is the so-called IR-modified gravity model of Refs.",
"[43], [44] described by $f(R)=R-\\frac{\\mu ^4}{R},$ where $\\mu \\sim H_0$ , $H_0$ being the Hubble constant.",
"However, such models are ruled out by the Dolgov-Kawasaki instability [20].",
"It is interesting to note that in fact the above model and the Starobinsky model (the second example model discussed above) are special cases of a more general model given by $f(R)=R-(1-n)\\mu ^2\\left(\\frac{R}{\\mu ^2}\\right)^n,$ which have been investigated in Ref.",
"[45].",
"Another model which does not admit the solution is given by [46] $f(R)=R+\\alpha \\ln \\left(\\frac{R}{\\mu ^2}\\right)+\\beta R^m, $ $\\alpha ,\\beta , m$ being constants."
],
[
"Gravitational waves in nonminimal ${\\mathcal {F}}({\\mathcal {G}})$ gravity",
"In this section, we consider a nonminimal ${\\mathcal {F}}({\\mathcal {G}})$ gravity model described by the following action [13] $S=\\int {\\sqrt{-g}}\\left(\\frac{1}{2}R+{\\mathcal {F}}({\\mathcal {G}})+(1+\\kappa {\\mathcal {H}}({\\mathcal {G}})){\\mathcal {L}}_m\\right)d^4 x,$ in which an arbitrary function of the Gauss-Bonnet invariant ${\\mathcal {G}}\\equiv R^2-4R_{\\mu \\nu }R^{\\mu \\nu }+R_{\\mu \\nu \\lambda \\sigma }R^{\\mu \\nu \\lambda \\sigma }$ is coupled to the matter field Lagrangian ${\\mathcal {L}}_m$ with a coupling constant $\\kappa $ .",
"This represents a generalized version of the actions introduced in Refs.",
"[11], [12].",
"The equation of motion resulting from the above action is given by $(1+\\kappa {\\mathcal {H}(\\mathcal {G})})T^{\\mu \\nu }&=&G^{\\mu \\nu }-g^{\\mu \\nu }{\\mathcal {F}}({\\mathcal {G}})\\nonumber \\\\&&+4H^{\\mu \\nu }({\\mathcal {F}}^\\prime ({\\mathcal {G}})+\\kappa {\\mathcal {L}}_m{\\mathcal {H}}^\\prime ({\\mathcal {G}})),$ where ${\\mathcal {F}}^\\prime ({\\mathcal {G}})=\\frac{d{\\mathcal {F}}({\\mathcal {G}})}{d{\\mathcal {G}}},$ and $H^{\\mu \\nu }&=&RR^{\\mu \\nu }+{R^\\mu }_{\\alpha \\beta \\gamma }R^{\\nu \\alpha \\beta \\gamma }-2R^\\mu _\\alpha R^{\\alpha \\nu }\\\\&&+2R^{\\mu \\alpha \\beta \\nu }R_{\\alpha \\beta }-2G^{\\mu \\nu }\\nabla ^2-R\\nabla ^\\mu \\nabla ^\\nu \\\\&&-2g^{\\mu \\nu }R^{\\alpha \\beta }\\nabla _\\alpha \\nabla _\\beta +2R^{\\alpha \\nu }\\nabla _\\alpha \\nabla ^\\mu \\\\&&+2R^{\\mu \\alpha }\\nabla _\\alpha \\nabla ^\\nu -2R^{\\mu \\alpha \\beta \\nu }\\nabla _\\alpha \\nabla _\\beta .$ Now, we consider the pp-wave anstaz, Eq.",
"(REF ).",
"First, we note that the Gauss-Bonnet invariant, Eq.",
"(REF ) above, vanishes identically for the spacetime described by Eq.",
"(REF ).",
"For vacuum, the above field equation reduces to $\\left(\\frac{\\partial ^2}{\\partial x^2}+\\frac{\\partial ^2}{\\partial y^2}\\right)K(u,x,y)-2{\\mathcal {F}}(0)K(u,x,y)=0,$ and ${\\mathcal {F}}(0)=0.$ Thus, the vacuum solution is admitted only if the above condition holds.",
"Then, Eq.",
"(REF ) reduces to the same equation governing the waveform in general relativity.",
"Hence, the waveform is the same as the general relativistic counterpart.",
"Back to the nonminimal coupling to a massless particle, by taking ${\\mathcal {L}}_{p}$ as the matter Lagrangian and inserting the associated energy-momentum tensor into the field equation (REF ), we obtain the plane wave solution (REF ) whenever Eq.",
"(REF ) is satisfied.",
"The relevant waveform is given by $K_{\\mathcal {G}}(u,x,y)=(1+\\kappa {\\mathcal {H}}(0))K_{AS}(u,x,y).$ If, in addition, the function ${\\mathcal {H}}({\\mathcal {G}})$ is chosen so that it satisfies ${\\mathcal {H}}(0)=0$ , then the waveform is not distinguishable from the general relativistic one."
],
[
"Gravitational waves in ${\\mathcal {F}}({\\mathcal {G}})$ gravity",
"The action for ${\\mathcal {F}}(\\mathcal {G})$ gravity is given by $S=\\int d^{4}x\\sqrt{-g}\\left[\\frac{1}{2}R+{\\mathcal {F}}({\\mathcal {G}})+{\\mathcal {L}}_m\\right]\\,,$ which is a particular case of the nonminimal model discussed in the previous section with $\\kappa =0$ .",
"Thus, all such models satisfying Eq.",
"(REF ) admit both the vacuum pp-wave solution and the AS solution.",
"Now, from Eq.",
"(REF ), with vanishing $\\kappa $ , it is obvious that the waveform $K_{\\mathcal {G}}(u,x,y)$ is the same as the one in general relativity.",
"Examples of the ${\\mathcal {F}}({\\mathcal {G}})$ models satisfying the requirement Eq.",
"(REF ) include ${\\mathcal {F}}({\\mathcal {G}})={\\mathcal {G}}^n,$ with $n > 0$ which is shown in Ref.",
"[47] that it could also pass solar system tests for $n\\lesssim 0.074$ .",
"Also, the following cosmologically viable models ${\\mathcal {F}}({\\mathcal {G}})&=&\\mu \\frac{\\mathcal {G}}{{\\mathcal {G}}_\\star }\\tan ^{-1}\\left(\\frac{\\mathcal {G}}{{\\mathcal {G}}_\\star }\\right)-\\frac{\\mu }{2}{\\sqrt{{\\mathcal {G}}_\\star }}\\ln \\left(1+\\frac{{\\mathcal {G}}^2}{{{\\mathcal {G}}_\\star }^2}\\right)\\nonumber \\\\&&-\\alpha \\mu {\\sqrt{\\mathcal {G}}_\\star },\\\\{\\mathcal {F}}({\\mathcal {G}})&=&\\mu \\frac{\\mathcal {G}}{{\\mathcal {G}}_\\star }\\tan ^{-1}\\left(\\frac{\\mathcal {G}}{{\\mathcal {G}}_\\star }\\right)-\\alpha \\mu {\\sqrt{\\mathcal {G}}_\\star },\\\\{\\mathcal {F}}({\\mathcal {G}})&=&\\mu {\\sqrt{{\\mathcal {G}}_\\star }}\\ln \\cosh \\left(\\frac{{\\mathcal {G}}^2}{{{\\mathcal {G}}_\\star }^2}\\right)-\\alpha \\mu {\\sqrt{\\mathcal {G}}_\\star },$ proposed in Ref.",
"[48] admit the plane wave solution for $\\alpha =0$ .",
"Here, $\\mu $ and ${\\mathcal {G}}_\\star $ are positive constants.",
"The model presented in Ref.",
"[49] provides another example which admits the wave solution.",
"It is described by ${\\mathcal {F}}({\\mathcal {G}})=\\frac{({\\mathcal {G}}-G_0)^{2n+1}+G^{2n+1}_0}{F_0+F_1\\lbrace ({\\mathcal {G}}-G_0)^{2n+1}+G^{2n+1}_0\\rbrace },$ where $F_0, F_1, G_0$ are constants.",
"It is interesting to note that both $f(R)$ and ${\\mathcal {F}}({\\mathcal {G}})$ theories considered above can be obtained from a more general theory described by the following action [50] $S=\\int {\\sqrt{-g}}(F(R,{\\mathcal {G}})+{\\mathcal {L}}_m)d^4x,$ which can be seen from the associated field equation which admits the above plane wave solution when similar conditions to those introduced above are satisfied."
],
[
"Discussion and conclusions",
"In this work, we studied plane-fronted gravitational waves with parallel rays in the context of some extended theories of gravity.",
"We considered $f(R)$ and modified Gauss-Bonnet gravity with minimal and nonminimal couplings to matter and showed that they admit the vacuum pp-wave solution and also an Aichelburg-Sexl-type solution if certain conditions are satisfied.",
"For $f(R)$ gravity, the required condition is that $f(R)$ vanishes for vanishing scalar curvature.",
"This condition is compatible with the requirements for such theories to pass local gravity tests.",
"It is also the same condition for a given model to admit the Schwarzschild solution.",
"Thus, for those models admitting the Schwarzschild solution, it should be possible to obtain the AS plane wave solution by boosting the black hole one as in general relativity.",
"The explicit form of the wave profile depends on $f^\\prime (0)$ and coincides with the general relativistic wave profile for some specific models including the well-known Starobinsky model.",
"A similar condition holds for the modified Gauss-Bonnet gravity.",
"In the latter case, the solution is the same as the general relativistic one.",
"This was examined for several specific cosmologically viable models.",
"For models with nonminimal coupling between the matter and geometry, more interesting options are available, including the possibility of (dis)appearance of (the)a contribution from the nonminimal coupling by choosing appropriate coupled function.",
"The gravitational wave solution presented here might be used as an experimentally testable, at least in principle, criterion to distinguish between various extended gravity models.",
"This can be achieved by looking at the behavior of two pointlike objects in the gravitational field of the massless source.",
"For models with minimal coupling, this can be seen by measuring the relative acceleration of two nearby test particles separated by $n^\\mu $ which is obtained from the geodesic deviation equation $\\frac{D^2n^\\mu }{D\\tau ^2}=-{R^\\mu }_{\\alpha \\nu \\beta }{\\dot{x}}^\\alpha n^\\nu {\\dot{x}}^\\beta $ and noting that for the spacetime under consideration, the components of the Riemann curvature tensor are proportional to the second derivatives of the wave profile $K(u,x,y)$ with respect to the transverse coordinates.",
"For models with nonminimal couplings where the particles do not move along geodesics as a result of extra forces coming from the coupling, the above equation should be modified by adding the relevant terms.",
"However, since the extra force is proportional to the gradient of the scalar curvature or the Gauss-Bonnet invariant, both of which vanish in the above considered spacetime, the same argument still holds.",
"The fact that the waveform obtained by application of the extended theories of gravity is proportional to the general relativistic waveform would then reflects in the observed relative accelerations.",
"The results obtained here might also be used to study the scattering of particles at high energies in the framework of extended theories of gravity.",
"Possible interesting extensions of the present work include a study of shock waves due to massless particles moving in curved backgrounds, and particles with arbitrary multipoles moving in curved spacetimes in the presence of matter fields and-/or a cosmological constant."
],
[
"Acknowledgments",
"The author gratefully acknowledges several comments by two anonymous referees of Physical Review D."
]
] | 1204.0744 |
[
[
"Study of Bc-> D_s^* \\ell^+ \\ell^- in Single Universal Extra Dimension"
],
[
"Abstract The rare semileptonic Bc-> D_s^* \\ell^+ \\ell^- decay is studied in the scenario of the universal extra dimension model with a single extra dimension in which inverse of the compactification radius R is the only new parameter.",
"The sensitivity of differential branching ratio, total branching ratio, polarization and forward-backward asymmetries of final state leptons, both for muon and tau, to the compactification parameter is presented.",
"For some physical observables uncertainty on the form factors and resonance contributions have been considered in the calculations.",
"Obtained results, compared with the available data, show that there appear new contributions due to the extra dimension."
],
[
"Introduction",
"Flavor-changing neutral current (FCNC) $b \\rightarrow s,d$ transitions which occur at loop level in the standard model (SM) provide us a powerful tool to test the SM and also a frame to study physics beyond the SM.",
"After the observation of $b \\rightarrow s \\, \\gamma $ [1], these transitions became more attractive and since then rare radiative, leptonic and semileptonic decays of $B_{u,d,s}$ mesons have been intensively studied [2].",
"Among these decays, semileptonic decay channels are significant because of having relatively larger branching ratio.",
"The experimental data for exclusive $B\\rightarrow K^{(*)} \\ell ^+ \\ell ^-$ also increased the interest in these decays.",
"These studies will be even more complete if similar studies for $B_c$ , discovered by CDF Collaboration [3], are also included.",
"The $B_c$ meson is the lowest bound state of two heavy quarks, bottom $b$ and charm $c$ , with explicit flavor that can be compared with the $c\\bar{c}$ and $b\\bar{b}$ - bound state which have implicit flavor.",
"The implicit-flavor states decay strongly and electromagnetically whereas the $B_c$ meson decays weakly.",
"$B_{u,d,s}$ are described very well in the framework of the heavy quark limit, which gives some relations between the form factors of the physical process.",
"In case of $B_c$ meson, the heavy flavor and spin symmetries must be reconsidered because of heavy $b$ and $c$ .",
"On the experimental side of the decay, for example, at LHC, $10^{10} B_c $ events per year is estimated [4]-[5].",
"This reasonable number is stimulating the work on the $B_c$ phenomenology and this possibility will provide information on rare $B_c$ decays as well as CP violation and polarization asymmetries.",
"In rare $B$ meson decays, effects of the new physics may appear in two different manners, either through the new contributions to the Wilson coefficients existing in the SM or through the new structures in the effective Hamiltonian which are absent in the SM.",
"Considering different models beyond the SM, extra dimensions are specially attractive because of including gravity and other interactions, giving hints on the hierarchy problem and a connection with string theory.",
"Those with universal extra dimensions (UED) have special interest because all the SM particles propagate in extra dimensions, the compactification of which allows Kaluza-Klein (KK) partners of the SM fields in the four-dimensional theory and also KK modes without corresponding the SM partners [6], [7], [8], [9].",
"Throughout the UED, a simpler scenario with a single universal extra dimension is the Appelquist-Cheng-Dobrescu (ACD) model [10].",
"The only additional free parameter with respect to the SM is the inverse of the compactification radius, $1/R$ .",
"In particle spectrum of the ACD model, there are infinite towers of KK modes and the ordinary SM particles are presented in the zero mode.",
"This only parameter have been attempted to put a theoretical or experimental restriction on it.",
"Tevatron experiments put the bound $1/R \\ge 300 GeV$ .",
"Analysis of the anomalous magnetic moment and $B\\rightarrow X_s \\gamma $ [11] also lead to the bound $1/R \\ge 300 GeV$ .",
"In the study of $B\\rightarrow K^* \\gamma $ decay [12], the results restrict R to be $1/R\\ge 250 GeV$ .",
"Also, in [13] this bound is $1/R\\ge 330 \\,GeV$ .",
"In two recent works, the theoretical study of $B \\rightarrow K \\eta \\gamma $ matches with experimental data if $1/R\\ge 250 \\,GeV$ [14] and using the experimental result [15] and theoretical prediction on the branching ratio of $\\Lambda _b \\rightarrow \\Lambda \\mu ^+ \\mu ^-$ , the lower bound was obtained to be approximately $1/R\\sim 250\\, GeV$ [16].",
"In this work, we will consider $1/R$ from 200 GeV up to 1000 GeV, however, under above consideration $1/R=250-350 \\,GeV$ region will be taken more common bound region.",
"In literature, effective Hamiltonian of several FCNC processes [17], [18], semileptonic and radiative decays have been investigated in the ACD model [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29].",
"Concentrating on $B_c \\rightarrow D_s^{*} \\ell ^+ \\ell ^-$ decay, it has been studied by using model independent effective Hamiltonian [30], in Supersymmetric models [31] and with fourth generation effects [32].",
"Also in [33], the UED effects on branching ratio and helicity fractions of the final state $D^*$ meson were calculated using the form factors obtained through the Ward identities for this process.",
"The weak annihilation contribution in addition to the FCNC transitions was taken into account.",
"We will, however, only consider the FCNC transitions and calculate the lepton asymmetries adding the resonance contributions.",
"The main aim of this paper is to find the effects of the ACD model on some physical observables related to the $B_c \\rightarrow D_s^{*} \\ell ^+ \\ell ^-$ decay, while doing this we also give the behavior of these observables by a couple of figures in the SM.",
"Measurement of final state lepton polarizations is an useful way in searching new physics beyond the SM.",
"Another tool is the study of forward-backward asymmetry $(A_{FB})$ , especially the position of zero value of $A_{FB}$ is very sensitive to the new physics.",
"In addition to differential decay rate and branching ratio, we study forward-backward asymmetry and polarization of final state leptons, including resonance contributions and uncertainty on form factors in as many as possible cases.",
"We analyze these observables in terms of the compactification factor and the form factors.",
"The form factors for $B_c \\rightarrow D_s^{*} \\ell ^+ \\ell ^-$ have been calculated using the light front, constitute quark models [34], the relativistic constituent quark model [35], relativistic quark model [36] and light-cone quark model [37].",
"In this work, we will use the form factors calculated in three-point QCD sum rules [38].",
"The paper is organized as follows.",
"In Sec.",
"II, we give the effective Hamiltonian for the quark level process $b \\rightarrow s \\ell ^+ \\ell ^-$ and mention briefly the Wilson coefficients in the ACD model; a detailed discussion is given in Appendix A.",
"We drive matrix element using the form factors and calculate the decay rate in Sec.",
"III.",
"In Sec.",
"IV, we present the forward-backward asymmetry and Sec.",
"V is devoted to lepton polarizations.",
"In the last section, we introduce our conclusions."
],
[
"Effective Hamiltonian and Wilson Coefficients",
"The quark-level transition of $B_c \\rightarrow D_s^{*} \\ell ^+ \\ell ^-$ decay is governed by $b\\rightarrow s \\ell ^+ \\ell ^- $ and given by the following effective Hamiltonian in the SM [39] $ {\\cal H}_{eff} &=&\\frac{G_F \\alpha }{\\sqrt{2} \\pi } V_{tb}V_{ts}^\\ast \\Bigg [ C_{9}^{eff} (\\bar{s} \\gamma _{\\mu } L\\, b)\\, \\bar{\\ell }\\gamma ^\\mu + C_{10} (\\bar{s} \\gamma _{\\mu } L\\, b)\\, \\bar{\\ell }\\gamma ^\\mu \\gamma _5 \\ell \\nonumber \\\\& & -2C_{7}^{eff} m_b (\\bar{s} i \\sigma _{\\mu \\nu } \\frac{q^{\\nu }}{q^2} R\\, b)\\,\\bar{\\ell }\\gamma ^\\mu \\ell \\Bigg ] \\,,$ where q is the momentum transfer, $L, R = (1 \\pm \\gamma _5)/2$ and ${C_i}$ s are the Wilson coefficients evaluated at the b quark mass scale.",
"The coefficient $C_9^{eff}$ has perturbative and resonance contributions.",
"So, $C_9^{eff}$ can be written as $C_9^{eff}(\\mu ) = C_9(\\mu ) \\Big ( 1+\\frac{\\alpha _{s}(\\mu )}{\\pi }\\omega (s^{\\prime }) \\Big )+ Y(\\mu , s^{\\prime })+C_9^{res}(\\mu ,s^{\\prime })$ where $s^{\\prime }=q^2/m_{b}^2$ .",
"The perturbative part, coming from one-loop matrix elements of the four-quark operators, is $ Y(\\mu ,s^{\\prime })&=& h(y,s^{\\prime }) [ 3 C_1(\\mu ) + C_2(\\mu ) + 3C_3(\\mu ) + C_4(\\mu ) + 3 C_5(\\mu ) + C_6(\\mu )] \\nonumber \\\\&-& \\frac{1}{2} h(1,s^{\\prime }) \\left( 4 C_3(\\mu ) + 4 C_4(\\mu )+ 3 C_5(\\mu ) + C_6(\\mu ) \\right)\\nonumber \\\\&- & \\frac{1}{2} h(0,s^{\\prime }) \\left[ C_3(\\mu ) + 3 C_4(\\mu ) \\right] \\nonumber \\\\&+& \\frac{2}{9} \\left( 3 C_3(\\mu ) + C_4(\\mu ) + 3 C_5(\\mu ) +C_6(\\mu ) \\right),$ with $y=m_c/m_b$ .",
"The explicit forms of the functions $\\omega (s^{\\prime })$ and $h(y,s^{\\prime })$ are given in [40]-[41].",
"The resonance contribution due to the conversion of the real $c \\bar{c}$ into lepton pair can be done by using a Breit-Wigner shape as [42], $C^{res}_{9}(\\mu ,s^{\\prime })&=&-\\frac{3}{\\alpha ^2_{em}}\\kappa \\sum _{V_i=\\psi _i}\\frac{\\pi \\Gamma (V_i\\rightarrow \\ell ^+ \\ell ^-)m_{V_i}}{s m_{b}^2 -m^{2}_{V_i}+i m_{V_i}\\Gamma _{V_i}} \\nonumber \\\\&\\times & [ 3 C_1(\\mu ) + C_2(\\mu ) + 3 C_3(\\mu ) + C_4(\\mu ) + 3C_5(\\mu ) + C_6(\\mu )]\\, .$ The normalization is fixed by the data in [43] and the phenomenological parameter $\\kappa $ is taken 2.3 to produce the correct branching ratio $BR (B \\rightarrow J/\\psi K^* \\rightarrow K^* \\ell ^+ \\ell ^-)=BR (B \\rightarrow J/\\psi K^* )B(J/\\psi \\rightarrow \\ell ^+ \\ell ^-)$ .",
"In the ACD model, there are not any new operators, therefore, new physics contributions appear by modifying the Wilson coefficients available in the SM.",
"In this model, the Wilson coefficients can be written in terms of some periodic functions, as a function of compactification factor $1/R$ .",
"The function $F(x_t, 1/R)$ which generalize the $F_0(x_t)$ SM functions according to $ F(x_t, 1/R) = F_0(x_t) + \\sum _{n=1} ^{\\infty } F_n(x_t, x_n)$ where $x_t=m_t^2/m_W^2$ , $x_n=m_n^2/m_W^2$ with the mass of KK particles $m_n=n/R$ .",
"n=0 corresponding the ordinary SM particles.",
"The modified Wilson coefficients in the ACD model, taken place in many works in literature, are discussed in Appendix A.",
"Briefly, for $C_9$ , in the ACD model and in the NDR scheme we have $ C_9(\\mu ,1/R)= P_0^{NDR} + \\frac{Y(x_t, 1/R)}{sin^2{\\theta _W}} - 4 Z(x_t, 1/R) + P_E E(x_t, 1/R).$ Instead of $C_7$ , a normalization scheme independent effective coefficient $C_7^{eff}$ can be written as $ C_7^{eff}(\\mu ,1/R) = &&\\eta ^{16/23} C_7(\\mu _W, 1/R) \\nonumber \\\\&&+ \\frac{8}{3} (\\eta ^{14/23} - \\eta ^{16/23}) C_8(\\mu _W, 1/R) +C_2(\\mu _W, 1/R) \\sum _{i=1} ^{8} h_i \\eta ^{a_i}.$ The Wilson coefficient $C_{10}$ is independent of scale $\\mu $ and given by $ C_{10}(1/R) = - \\frac{Y(x_t, 1/R)}{sin^2{\\theta _W}}.$ Figure: The variation of Wilson coefficients with respect to 1/R1/R at q 2 =14GeV 2 q^2=14 \\, GeV^2for the normalization scale μ=4.8GeV\\mu =4.8 \\,GeV.",
"(C 9 eff C_9^{eff} does not include resonance contributions.",
")The Wilson coefficients differ considerably from the SM values for small R. The variation of modified Wilson coefficients with respect to $1/R$ at $q^2=14\\,GeV^2$ , in which the normalization scale is fixed to $\\mu =\\mu _b\\simeq 4.8 \\,GeV$ , is given in Fig.",
"REF .",
"The suppression of $\\left|{C^{eff}_7}\\right|$ for $1/R=250-350 \\, GeV$ amount to $75\\%-86\\%$ relative to the SM value.",
"$\\left|{C_{10}}\\right|$ is enhanced by $23\\%-13\\%$ .",
"The impact of the ACD on $\\left|{C^{eff}_9}\\right|$ is very small.",
"For $1/R\\stackrel{>}{_\\sim }600\\, GeV$ the difference is less than $5\\%$ ."
],
[
"Matrix Elements and Decay Rate",
"The hadronic matrix elements in the exclusive $B_c \\rightarrow D_s^{*} \\ell ^+ \\ell ^-$ decay can be obtained by sandwiching the quark level operators in the effective Hamiltonian between the initial and the final state mesons.",
"The nonvanishing matrix elements are parameterized in terms of form factors as follows [44], [45] $ \\left<D_s^\\ast (p_{D_s^\\ast },\\varepsilon ) \\left|\\bar{s} \\gamma _\\mu (1 - \\gamma _5) b \\right|B_c(p_{B_c}) \\right>=- \\epsilon _{\\mu \\nu \\alpha \\beta } \\varepsilon ^{\\ast \\nu }p_{D_s^\\ast }^\\alpha q^\\beta \\frac{2 V(q^2)}{m_{B_c}+m_{D_s^\\ast }}- i \\varepsilon _\\mu ^\\ast (m_{B_c}+m_{D_s^\\ast })A_1(q^2) \\nonumber \\\\+ i (p_{B_c} + p_{D_s^\\ast })_\\mu (\\varepsilon ^\\ast q)\\frac{A_2(q^2)}{m_{B_c}+m_{D_s^\\ast }}+ i q_\\mu (\\varepsilon ^\\ast q)\\frac{2 m_{D_s^\\ast }}{q^2} [A_3(q^2) -A_0(q^2)] ,$ and $ \\left<D_s^\\ast (p_{D_s^\\ast },\\varepsilon ) \\left|\\bar{s} i \\sigma _{\\mu \\nu } q^\\nu (1 + \\gamma _5) b \\right|B_c(p_{B_c}) \\right>=2 \\epsilon _{\\mu \\nu \\alpha \\beta } \\varepsilon ^{\\ast \\nu }p_{D_s^\\ast }^\\alpha q^\\beta T_1(q^2) \\nonumber \\\\+ i \\Big [\\varepsilon _\\mu ^\\ast (m_{B_c}^2-m_{D_s^\\ast }^2) -(p_{B_c} + p_{D_s^\\ast })_\\mu (\\varepsilon ^\\ast q) \\Big ] T_2(q^2)+ i (\\varepsilon ^\\ast q) \\Big [ q_\\mu - (p_{B_c} +p_{D_s^\\ast })_\\mu \\frac{q^2}{m_{B_c}^2-m_{D_s^\\ast }^2} \\Big ] T_3 (q^2),$ where $q = p_{B_c}-p_{D_{s}^\\ast }$ is the momentum transfer and $\\varepsilon $ is the polarization vector of $D_{s}^\\ast $ meson.",
"The relation between the form factors $A_1(q^2)$ , $A_2(q^2)$ and $A_3(q^2)$ can be stated as $ A_3(q^2) = \\frac{m_{B_s} + m_{\\phi }}{2 m_{\\phi }} A_1(q^2)- \\frac{m_{B_s} - m_{\\phi }}{2 m_{\\phi }} A_2(q^2) \\nonumber $ and in order to avoid kinematical singularity in the matrix element at $q^2=0$ , it is assumed that $A_0(0) = A_3(0)$ and $T_1(0)=T_2(0)$ [45].",
"Using the effective Hamiltonian and matrix elements in Eqs.",
"(REF )–(REF ), the transition amplitude for $B_c \\rightarrow D_s^{*} \\ell ^+ \\ell ^-$ is written as ${ {\\cal M}(B_c \\rightarrow D_s^{*} \\ell ^+ \\ell ^-) =\\frac{G \\alpha }{2 \\sqrt{2} \\pi } V_{tb} V_{ts}^\\ast }\\nonumber \\\\&&\\times \\Bigg \\lbrace \\bar{\\ell }\\gamma ^\\mu \\ell \\, \\Big [-2 A \\epsilon _{\\mu \\nu \\alpha \\beta } \\varepsilon ^{\\ast \\nu }p_{D_s^\\ast }^\\alpha q^\\beta -i B \\varepsilon _\\mu ^\\ast + i C (\\varepsilon ^\\ast q) (p_{B_c}+p_{D_s^\\ast })_\\mu + i D (\\varepsilon ^\\ast q) q_\\mu \\Big ] \\nonumber \\\\&&+ \\bar{\\ell }\\gamma ^\\mu \\gamma _5 \\ell \\, \\Big [-2 E \\epsilon _{\\mu \\nu \\alpha \\beta } \\varepsilon ^{\\ast \\nu }p_{D_s^\\ast }^\\alpha q^\\beta -i F \\varepsilon _\\mu ^\\ast + i G (\\varepsilon ^\\ast q) (p_{B_c}+p_{D_s^\\ast })_\\mu + i H (\\varepsilon ^\\ast q) q_\\mu \\Big ]\\Bigg \\rbrace ,$ with the auxiliary functions $ A &=& C_9^{eff}\\frac{V(q^2)}{m_{B_c}+m_{D_s^\\ast }} +\\frac{2m_{b}}{q^2}C_7^{eff} T_1(q^2), \\nonumber \\\\B &=& C_9^{eff}(m_{B_c}+ m_{D_s^\\ast }) A_1(q^2)+\\frac{2 m_{b}}{q^2}C_7^{eff} (m_{B_c}^2-m_{D_s^\\ast }^2) T_2(q^2), \\nonumber \\\\C &=& C_9^{eff} \\frac{A_2(q^2)}{m_{B_c}+m_{D_s^\\ast }} +\\frac{2m_{b}}{q^2}C_7^{eff} \\Big (T_2(q^2) + \\frac{q^2}{m_{B_c}^2 - m_{D_s^\\ast }^2} T_3(q^2) \\Big ),\\nonumber \\\\D &=& 2 C_9^{eff} \\frac{m_{D_s^\\ast }}{q^2} (A_3(q^2) -A_0(q^2))-2 \\frac{m_{b}}{q^2} C_7^{eff} T_3(q^2), \\nonumber \\\\E &=& C_{10} \\frac{V(q^2)}{m_{B_c}+m_{D_s^\\ast }}, \\nonumber \\\\F &=& C_{10}(m_{B_c}+m_{D_s^\\ast }) A_1(q^2), \\nonumber \\\\G &=& C_{10} \\frac{A_2(q^2)}{m_{B_c}+m_{D_s^\\ast }} , \\nonumber \\\\H &=& 2 C_{10} \\frac{m_{D_s^\\ast }}{q^2} (A_3(q^2) -A_0(q^2)) .$ Integrating over the angular dependence of the double differential decay rate, following dilepton mass spectrum is obtained $\\frac{d\\Gamma }{ds} = \\frac{G^2 \\alpha ^2m_{B_c}}{2^{12} \\pi ^5 }\\left|V_{tb} V_{ts}^\\ast \\right|^2 \\sqrt{\\lambda } v \\Delta _{D_{s}^\\ast }$ where $s=q^2/m_{B_c}^2$ , $\\lambda = 1 + r^2 + s^2 -2r-2s-2rs$ , $r=m_{D_s^\\ast }^2/m_{B_c}^2$ , $v=\\sqrt{1-{4m_\\ell ^2}/{s m_{B_c}^2}}$ and $ \\Delta _{D_{s}^\\ast } =& & \\frac{8}{3} \\lambda m_{B_c}^6 s\\Big [(3-v^2) \\left|A \\right|^2 + 2v^2 \\left|E \\right|^2 \\Big ] + \\frac{1}{r}\\lambda m_{B_c}^4\\Bigg [\\frac{1}{3} \\lambda m_{B_c}^2 (3-v^2) \\left|C \\right|^2+m_{B_c}^2 s^2 (1-v^2) \\left|H \\right|^2 \\nonumber \\\\&& +\\frac{2}{3} \\Big [(3-v^2)(r+s-1)-3s(1-v^2) \\Big ]Re[FG^\\ast ]+2m_{B_c}^2 s (1-r) (1-v^2) Re[GH^\\ast ]\\nonumber \\\\&& -2s(1-v^2) Re[FH^\\ast ]+\\frac{2}{3}(3-v^2)(r+s-1)Re[BC^\\ast ] \\Bigg ]+\\frac{1}{3r}(3-v^2) m_{B_c}^2 \\Bigg [(\\lambda +12r s) \\left|B \\right|^2 \\nonumber \\\\&&+\\lambda m_{B_c}^4 \\Big [ \\lambda -3s(s-2r-2) (1-v^2) \\Big ] \\left|G \\right|^2+\\Big [ \\lambda + 24r s v^2 \\Big ]\\left|F \\right|^2 \\Bigg ].$ Figure: (color online) The dependence of differential branching ratio on s, including the uncertainities on form factors in non-resonance case.",
"(In the legend 1/R=200,350,500GeV1/R=200, 350, 500 \\,GeV.",
")Figure: (color online) The dependence of differential branching ratio on s with the central values of form factors including resonance contributions.Figure: (color online) The dependence of differential branching ratio on 1/R, with and without resonance contributions, including uncertainty on form factors at s=0.18s=0.18 for μ{\\mu }, and s=0.4s=0.4 for τ{\\tau }.",
"(The subscript R in the legend represents resonance contributions.",
")In the numerical analysis, we have used $m_{B_c}=6.28 \\, GeV$ , $m_{D_{s}^\\ast }=2.112 \\,GeV$ , $m_b =4.8 \\, GeV$ , $m_{\\mu } =0.105 \\, GeV$ , $m_{\\tau } =1.77 \\, GeV$ , $|V_{tb} V^*_{ts}|=0.041$ , $G_F=1.17 \\times 10^{-5}\\, GeV^{-2}$ , $\\tau _{B_{c}}=0.46 \\times 10^{-12} \\, s$ , and the values that are not given here are taken from [43].",
"In our work, we have used the numerical values of the form factors calculated in three point QCD sum rules [38], in which $q^2$ dependencies of the form factors are given as $F(q^2) = \\frac{F(0)}{1+a (q^2/m^{2}_{B_c}) + b (q^2/m^{2}_{B_c})^2}~, \\nonumber $ and the values of parameters $F(0)$ , $a$ and $b$ for the $B_c\\rightarrow D^\\ast $ decay are listed in Table REF .",
"Table: B c B_c meson decay form factors in the three point QCD sum rules.The differential branching ratio is calculated without resonance contributions, including uncertainty on form factors, and with resonance contributions, and s dependence for $1/R=200, 350, 500 \\,GeV$ is presented in Figs.",
"REF and REF , respectively.",
"The change in differential decay rate and difference between the SM results and new effects can be noticed in the figures.",
"The maximum effect is around $s=0.25 \\pm 0.05 \\,(0.37\\pm 0.02)$ for $\\mu \\,(\\tau )$ in Fig.",
"REF .",
"In spite of the hadronic uncertainty, for $1/R=200\\,GeV$ and $350 \\,GeV$ , studying differential decay rate can be a suitable tool for studying the effect of extra dimension.",
"Supplementary of these, $1/R$ dependence of differential branching ratio at s= 0.18 (0.4) for $\\mu \\,(\\tau )$ is plotted in Fig.",
"REF .",
"Considering any given bounds on the compactification factor the effect of universal extra dimension can be seen clearly for low values of R, with and without resonance contributions.",
"On the other hand, $1/R \\stackrel{>}{_\\sim }600\\,GeV$ the contribution varies between $\\sim 5-8 \\%$ more than the SM results.",
"To obtain the branching ratio, we integrate Eq.",
"(REF ) in the allowed physical region.",
"While taking the long-distance contributions into account we introduce some cuts around ${J/ \\psi }$ and $\\psi (2s)$ resonances to minimize the hadronic uncertainties.",
"The integration region for $q^2$ is divided into three parts for $\\mu $ as $4m_{\\mu }^2 \\le q^2 \\le (m_{J/{\\psi }}-0.02)^2$ , $(m_{J/{\\psi }}+0.02)^2 \\le q^2 \\le (m_{\\psi (2s)}-0.02)^2$ and $(m_{\\psi (2s)}+0.02)^2 \\le q^2 \\le (m_{B_c}-m_{D_{s}^\\ast })^2$ and for $\\tau $ we have $4m_{\\tau }^2 \\le q^2 \\le (m_{\\psi (2s)}-0.02)^2$ $(m_{\\psi (2s)}+0.02)^2 \\le q^2 \\le (m_{B_c}-m_{D_{s}^\\ast })^2$ , the same as in [48].",
"The results of branching ratio in the SM with resonance contributions and uncertainty on form factors, we obtain $&&Br(B_c \\rightarrow D_s^{*} \\mu ^+ \\mu ^-)=2.13^{+0.27}_{-0.25}\\times 10^{-7}\\nonumber \\\\&&Br(B_c \\rightarrow D_s^{*} \\tau ^+ \\tau ^-)=1.45^{+0.15}_{-0.14}\\times 10^{-8}.$ Figure: (color online) The dependence of branching ratio on 1/R1/R, including resonance contributions and uncertainty on form factors.Observing the contribution of the ACD, the $1/R$ dependent branching ratios, including resonance contributions and uncertainty on form factors, are given in Fig.",
"REF .",
"Comparing the SM results and our theoretical predictions on the branching ratio for both decay channels, the lower bound for $1/R$ is found approximately $250\\,GeV$ which is consistent with the previously mentioned results.",
"As $1/R$ increases, the branching ratios approach to their SM values.",
"For $1/R \\ge 550\\, GeV$ in both channels, they become less than $5\\%$ greater than that of the SM values.",
"Between $1/R=250-350\\, GeV$ the ratio is $(2.66-2.40)^{+0.30}_{-0.28}\\times 10^{-7}$ for ${\\mu }$ , $(1.75-1.61)^{+0.16}_{-0.15}\\times 10^{-8}$ for $\\tau $ decay.",
"Comparing these with the SM results, the differences worth to study and can be considered as a signal of new physics and an evidence of existence of extra dimension."
],
[
"Forward-Backward Asymmetry",
"Another efficient tool for establishing new physics is the study of forward-backward asymmetry.",
"The position of zero value of $A_{FB}$ is very sensitive to the new physics.",
"The normalized differential form is defined for final state leptons as $ A_{FB}(s) &=&\\frac{\\int _{0}^{1}\\frac{d^2\\Gamma }{dsdz} dz-\\int _{-1}^{0}\\frac{d^2\\Gamma }{dsdz}dz}{\\int _{0}^{1}\\frac{d^2\\Gamma }{dsdz}dz+\\int _{-1}^{0}\\frac{d^2\\Gamma }{dsdz}dz}\\nonumber \\\\$ where $z=\\cos \\theta $ and $\\theta $ is the angle between the directions of $\\ell ^{-}$ and $B_c$ in the rest frame of the lepton pair.",
"In the case of $B_c \\rightarrow D_s^{*} \\ell ^+ \\ell ^-$ , we get $ A_{FB}&=& \\frac{G^2 \\alpha ^2m_{B_c}}{2^{12} \\pi ^5 }\\left|V_{tb} V_{ts}^\\ast \\right|^2\\frac{8m_{B_c}^4\\sqrt{\\lambda }v s(Re[BE^\\ast ]+Re[AF^\\ast ])}{d\\Gamma /ds} \\nonumber \\\\&=& \\frac{8m_{B_c}^4 \\sqrt{\\lambda }v s (Re[BE^\\ast ]+Re[AF^\\ast ])}{\\Delta _{D_{s}^\\ast }}.$ Using above equation, we present the variation of lepton forward-backward asymmetry with s including uncertainty on form factors in Fig.",
"REF .",
"As $1/R$ gets smaller, there appears considerable difference between the SM and the ACD results for $s\\stackrel{<}{_\\sim }0.16$ in $\\mu $ and $0.33\\stackrel{<}{_\\sim }s\\stackrel{<}{_\\sim }0.43$ in $\\tau $ decays.",
"Considering the resonance contributions, the results are given in Fig.",
"REF , one can recognize a similar situation for $s\\stackrel{<}{_\\sim }0.23$ and $0.32\\stackrel{<}{_\\sim }s\\stackrel{<}{_\\sim }0.44$ , respectively.",
"Figure: (color online) The lepton forward-backward asymmetry including uncertainty on form factors.Figure: (color online) The lepton forward-backward asymmetry including resonance contributionsTo understand the dependence of $A_{FB}$ on $1/R$ for both lepton channels better, we perform calculation at $s=0.05 \\,(0.4)$ for $\\mu (\\tau )$ and present the results in Fig.",
"REF .",
"In the $\\mu $ channel, UED contribution on $A_{FB}$ gets important between $1/R=200-600\\,GeV$ , while in $\\tau $ decay the contribution is insignificant for $1/R\\stackrel{>}{_\\sim }400\\,GeV$ .",
"Figure: (color online) The dependence of lepton forward-backward asymmetryon 1/R1/R at s=0.05s=0.05 for μ{\\mu } and s=0.4s=0.4 for τ{\\tau }.The position of the zero of forward-backward asymmetry, $s_0$ , is determined numerically and the results are presented in Fig.",
"REF .",
"Both plots for $B_c \\rightarrow D_s^{*} \\mu ^+ \\mu ^-$ is for the zero point in the $s<0.1$ region; the lower (upper) one is for the resonance (non resonance) case, while the zero point for $B_c \\rightarrow D_s^{*} \\tau ^+ \\tau ^-$ is because of resonance contributions.",
"In the SM, resonance shifts the zero point of the asymmetry, $s_0=0.079$ , to a lower value, $s_0=0.068$ , in $B_c \\rightarrow D_s^{*} \\mu ^+ \\mu ^-$ , i.e., further corrections could shift $s_0$ to smaller values [12].",
"As $1/R\\rightarrow 200\\, GeV$ the $s_0$ approaches low values for both decay channels.",
"In the $1/R=250-350\\,GeV$ region, $s_0$ varies between ($0.058-0.068$ ) without resonance contributions and ($0.051-0.058$ ) with resonance contributions.",
"The $s_0$ shift is $\\sim 5\\%$ of the SM value for $1/R\\stackrel{>}{_\\sim }600 \\,GeV$ .",
"The variation of $s_0$ for $B_c \\rightarrow D_s^{*} \\tau ^+ \\tau ^-$ is negligible.",
"Figure: (color online) The variation of zero position of lepton forward-backward asymmetry with 1/R1/R."
],
[
"Lepton Polarization Asymmetries",
"We will discuss the possible effects of the ACD model in lepton polarization, a way of searching new physics.",
"Using the convention followed by previous works [46]-[47], in the rest frame of $\\ell ^-$ we define the orthogonal unit vectors $S_i^-$ , for the polarization of the lepton along the longitudinal, transverse and normal directions as $S_L^{-} &\\equiv & (0,\\vec{e}_L) =\\left(0,\\frac{\\vec{p}_{\\ell }}{\\left|\\vec{p}_{\\ell } \\right|} \\right), \\nonumber \\\\S_N^{-} &\\equiv & (0,\\vec{e}_N) =\\left(0,\\frac{\\vec{p}_{D_{s}^\\ast } \\times \\vec{p}_{\\ell }}{\\left|\\vec{p}_{D_{s}^\\ast } \\times \\vec{p}_{\\ell } \\right|} \\right), \\nonumber \\\\S_T^{-} &\\equiv & (0,\\vec{e}_T) =\\left(0, \\vec{e}_N \\times \\vec{e}_L \\right),$ where $\\vec{p}_{\\ell }$ and $\\vec{p}_{D_{s}^\\ast }$ are the three momenta of $\\ell ^-$ and $D_{s}^\\ast $ meson in the center of mass (CM) frame of $\\ell ^+ \\ell ^-$ system, respectively.",
"The longitudinal unit vector $S_L^-$ is boosted by Lorentz transformation, $S^{-\\mu }_{L,\\, CM} = \\left(\\frac{\\left|\\vec{p}_{\\ell } \\right|}{m_\\ell },\\frac{E_\\ell \\,\\vec{p}_{\\ell }}{m_\\ell \\left|\\vec{p}_{\\ell } \\right|} \\right),$ while vectors of perpendicular directions remain unchanged under the Lorentz boost.",
"The differential decay rate of $B_c \\rightarrow D_s^{\\ast } \\ell ^+ \\ell ^-$ for any spin direction $\\vec{n}^{-}$ of the $\\ell ^{-}$ can be written in the following form $\\frac{d\\Gamma (\\vec{n}^{-})}{ds} = \\frac{1}{2}\\left(\\frac{d\\Gamma }{ds}\\right)_0\\Bigg [ 1 + \\Bigg ( P_L^{} \\vec{e}_L^{\\,-} + P_N^{-}\\vec{e}_N^{\\,-} + P_T^{-} \\vec{e}_T^{\\,-} \\Bigg ) \\cdot \\vec{n}^{-} \\Bigg ]\\,.$ Here, $(d \\Gamma / ds)_0$ corresponds to the unpolarized decay rate, whose explicit form is given in Eqn.",
"(REF ).",
"The polarizations $P^{-}_L$ , $P^{-}_T$ and $P^{-}_N$ in Eq.",
"(REF ) are defined by the equation $P_i^{-}(s) = \\frac{\\displaystyle {\\frac{d \\Gamma }{ds}({\\bf {n}}^{-}={\\bf {e}}_i^{\\,-}) -\\frac{d \\Gamma }{ds}({\\bf {n}}^{-}=-{\\bf {e}}_i^{\\,-})}}{\\displaystyle {\\frac{d \\Gamma }{ds}({\\bf {n}}^{-}={\\bf {e}}_i^{\\,-}) +\\frac{d \\Gamma }{ds}({\\bf {n}}^{-}=-{\\bf {e}}_i^{\\,-})}}~, \\nonumber $ for $i=L,~N,~T$ .",
"Here, $P^{-}_L$ and $P^{-}_T$ represent the longitudinal and transversal asymmetries, respectively, of the charged lepton $\\ell ^{-}$ in the decay plane, and $P^{-}_N$ is the normal component to both of them.",
"The explicit form of longitudinal polarization for $\\ell ^-$ is $P^-_L&=&\\frac{1}{3\\Delta _{D^\\ast _s}} 4 m^2_{B_c} v \\Big [8 m^4_{B_c} s \\lambda Re[AE^\\ast ]+\\frac{1}{r}(12rs+\\lambda )Re[BF^\\ast ] \\nonumber \\\\& & -\\frac{1}{r}\\lambda m^2_{B_c}(1-r-s)\\Big [Re[BG^\\ast ]+ Re[CF^\\ast ] \\Big ]+ \\frac{1}{r} {\\lambda ^2} m^4_{B_c} Re[CG^\\ast ]\\Big ].$ Figure: (color online) The dependence of longitudinal polarization on s without resonance contributions using central values of form factors.Figure: (color online) The dependence of longitudinal polarization on s with resonance contributions using central values of form factors.Figure: (color online) The dependence of longitudinal polarization on 1/R1/R including uncertainty on form factors and resonance contributions.Similarly, the transversal polarization is given by $P^-_T&=&\\frac{1}{\\Delta _{D^\\ast _s}} m_{B_c} m_\\ell \\pi \\sqrt{s \\lambda } \\Big [-8 m^2_{B_c} Re[AB^\\ast ] + \\frac{(1-r-s)}{rs} Re[BF^\\ast ]-\\frac{m^2_{B_c}\\lambda }{rs} Re[CF^\\ast ]\\nonumber \\\\& &-\\frac{m^2_{B_c}}{rs}(1-r)(1-r-s) Re[BG^\\ast ]+\\frac{m^4_{B_c}}{rs}\\lambda (1-r) Re[CG^\\ast ] \\nonumber \\\\& &-\\frac{m^2_{B_c}}{r}(1-r-s)Re[BH^\\ast ]+\\frac{m^4_{B_c}\\lambda }{r} Re[CH^\\ast ]\\Big ]$ Figure: (color online) The dependence of transversal polarization on s without resonance contributions using central values of form factors.Figure: (color online) The dependence of transversal polarization on s with resonance contributions using central values of form factors.Figure: (color online) The dependence of transversal polarization on 1/R1/R, including uncertainty on form factors.and the normal polarization by $P^-_N&=&\\frac{1}{\\Delta _{D^\\ast _s}} m^3_{B_c} m_{\\ell }\\pi v \\sqrt{s \\lambda } \\Big [-4 Im[B E^\\ast ]-4 Im[AF^\\ast ]+\\frac{1}{r}(1-r-s) Im[F H^\\ast ] \\nonumber \\\\&&+\\frac{1}{r} (1+3r-s) Im[F G^\\ast ]-\\frac{1}{r}m^2_{B_c} \\lambda Im[G H^\\ast ] \\Big ].$ Figure: (color online) The dependence of normal polarization on s with resonance contributions using central values of form factors.Figure: (color online) The dependence of normal polarization on s without resonance contributions using central values of form factors.Figure: (color online) The dependence of normal polarization on 1/R1/R including uncertainty on form factors and resonance contributions.We eliminate the dependence of the lepton polarizations on $s$ in order to clarify dependence on $1/R$ , by considering the averaged forms over the allowed kinematical region.",
"The averaged lepton polarizations are defined by $ \\left<P_i \\right>=\\frac{\\displaystyle \\int _{(2 m_\\ell /m_{B_c})^2}^{(1-m_{D_{s}^\\ast }/m_{B_c})^2}P_i \\frac{d{\\cal B}}{ds} ds}{\\displaystyle \\int _{(2m_\\ell /m_{B_c})^2}^{(1-m_{D_{s}^\\ast }/m_{B_c})^2}\\frac{d{\\cal B}}{ds} ds}.$ The dependence of longitudinal polarizations on s with and without resonance contributions are given in Figs.",
"REF and REF , respectively.",
"For high values of s as $1/R$ approaches 200 GeV the deviation from the SM results get greater for $\\tau $ in both resonance and non-resonance cases, while for $\\mu $ channel this effect can be seen clearly for all s values when resonance contributions are not added; including resonance contributions, around the peaks this effect seems to be suppressed and only for low values of s we can mention a deviation.",
"Eliminating the dependence of polarization on $s$ , we get variation of longitudinal polarization with respect to $1/R$ , given by Fig.",
"REF .",
"For $1/R\\ge 500\\,GeV$ , the difference becomes less important for both channels.",
"The SM longitudinal polarization, $P_{L}=-0.599$ , develops into $-0.670$ ($-0.646$ ) for $1/R \\ge 250 (350)\\,GeV$ for $\\mu $ .",
"A similar aspect can also be noticed for $\\tau $ .",
"That is, $P_{L}=-0.321$ SM value vary to $-0.366$ ($-0.347$ ) for $1/R\\ge 250 (350)$ .",
"The dependence of transversal polarization on s with and without resonance contributions are given in Figs.",
"REF and REF , respectively.",
"The UED effect is unimportant in both decay channels.",
"In view of $1/R$ dependency, given by Fig.",
"REF , no difference is observed for $\\tau $ decay.",
"Up to $1/R=600\\,GeV$ the change is sizeable for $\\mu $ channel.",
"In particular, between $1/R=250-350 \\,GeV$ the difference might be checked for a signal of new physics.",
"We have plotted the variation of normal polarizations on s with and without resonance contributions in Figs.",
"REF and REF , respectively and on $1/R$ in Fig.",
"REF .",
"The SM value itself for $\\mu $ is tiny and as can be seen from the figures the effect of UED on normal polarization in this channel is irrelevant.",
"Additionally, the relatively greater value of normal polarization in the SM for $\\tau $ differs slightly."
],
[
"Conclusion",
"In this work, we discussed the $B_c \\rightarrow D_s^{*} \\ell ^+ \\ell ^-$ decay for $\\mu $ and $\\tau $ as final state leptons in the SM and the ACD model.",
"We used form factors calculated in QCD sum rules and throughout the work, we reflected the errors on form factors on calculations and demonstrate the results in possible plotting.",
"Comparing the SM results and our theoretical predictions on the branching ratio for both decay channels, we obtain the lower bound as $1/R\\sim 250\\,GeV$ .",
"Although this is consistent with the previously mentioned results, a detailed analysis, particularly with the data supplied by experiments, is necessary to put a precise bound on the compactification scale.",
"As an overall result, we can conclude that, as stated previous works in literature, as $1/R\\rightarrow 200\\, GeV$ the physical values differ from the SM results.",
"Up to a few hundreds GeV above the considered bounds, $1/R\\ge 250\\,GeV$ or $1/R\\ge 350)\\,GeV$ , it is possible to see the effects of UED.",
"Taking the differential branching ratio into consideration, for small values of $1/R$ there comes out essential difference comparing with the SM results.",
"Difference between the SM and the ACD results in the forward-backward asymmetry of final state leptons, particularly in the specified region, the obtained result is essential.",
"In addition, the position of the zero of forward-backward asymmetry, which is sensitive in searching new physics, can be a useful tool to check the UED contributions.",
"Polarization of the leptons have been studied comprehensively and we found that transversal and normal polarizations are not sensitive to the extra dimension, only dependence of transversal (normal) polarization on $1/R$ for $\\mu $ ($\\tau $ ) decay channel for low values of $1/R$ might be useful.",
"However, studying longitudinal polarization for both leptons up to $1/R=600\\,GeV$ will be a powerful tool establishing new physics effects.",
"Under the discussion throughout this work, the sizable discrepancies between the ACD model and the SM predictions at lower values of the compactification scale can be considered the indications of new physics and should be searched in the experiments.",
"The author would like to thank K. Azizi for valuable discussion and U. Kanbur for contributions on computer base works."
],
[
"Wilson Coefficients in the ACD Model",
"In the ACD model, the new physics contributions appear by modifying available Wilson coefficients in the SM.",
"The modified Wilson coefficients are calculated in [17]-[18] and can be expressed in terms of $F(x_t, 1/R)$ which generalize the corresponding SM functions $F_0(x_t)$ according to $ F(x_t, 1/R) = F_0(x_t) + \\sum _{n=1} ^{\\infty } F_n(x_t, x_n)$ where $x_t=m_t^2/m_W^2$ , $x_n=m_n^2/m_W^2$ and $m_n=n/R$ .",
"Instead of $C_7$ , an effective, normalization scheme independent, coefficient $C^{eff}_7 $ in the leading logarithmic approximation is defined as $ C_7^{eff}(\\mu _b, 1/R) = && \\eta ^{16/23} C_7(\\mu _W, 1/R) \\nonumber \\\\&& + \\frac{8}{3} (\\eta ^{14/23} - \\eta ^{16/23}) C_8(\\mu _W,1/R) +C_2(\\mu _W, 1/R) \\sum _{i=1} ^{8} h_i \\eta ^{a_i}$ with $\\eta =\\frac{\\alpha _s(\\mu _W)}{\\alpha _s(\\mu _b)}$ and $ \\alpha _s(x)=\\frac{\\alpha _s(m_Z)}{1-\\beta _0 \\frac{\\alpha _s(m_Z)}{2 \\pi } ln(\\frac{m_Z}{x})}$ where in fifth dimension $\\alpha _s(m_Z)=0.118$ and $\\beta _0=23/3$ .",
"The coefficients $a_i$ and $h_i$ are $a_i & = &\\Big (\\frac{14}{23},\\frac{16}{23},\\frac{6}{23},-\\frac{12}{23},0.4086,-0.4230,-0.8994,0.1456 \\Big ) \\nonumber \\\\h_{i} & = & \\Big (2.2996,-1.088,-\\frac{3}{7},-\\frac{1}{14},-0.6494,-0.0380,-0.0186,-0.0057 \\Big ).$ The functions in (REF ) are $ C_2 (\\mu _W) = 1,~~~C_7(\\mu _W, 1/R) = - \\frac{1}{2} D^\\prime (x_t, 1/R),~~~C_8(\\mu _W, 1/R) = - \\frac{1}{2} E^\\prime (x_t, 1/R).$ Here, $D^\\prime (x_t, 1/R)$ and $E^\\prime (x_t, 1/R)$ are defined by using (REF ) with the following functions $ D_0^{\\prime }(x_t) = - \\frac{(8x_t^3 + 5x_t^2 - 7x_t)}{ 12(1-x_t)^3} +\\frac{x_t^2 (2-3x_t)}{2(1-x_t)^4} lnx_t$ $ E_0^{\\prime } (x_t) = - \\frac{x_t (x_t^2 - 5x_t - 2)}{4(1-x_t)^3} +\\frac{3 x_t^2}{2(1-x_t)^4} lnx_t$ $ D_n^{\\prime } (x_t, x_n) = \\frac{ x_t (-37 + 44x_t + 17x_t^2 + 6x_n^2(10-9x_t+3x_t^2) - 3x_n(21-54x_t+17x_t^2))}{36(x_t - 1)^3} \\nonumber \\\\-\\, \\frac{(-2+x_n+3x_t)(x_t+3x_t^2 + x_n^2(3+x_t) - x_n(1+ (-10+x_t)x_t))}{6(x_t-1)^2} \\ln \\frac{x_n + x_t}{1+x_n}\\nonumber \\\\+\\, \\frac{x_n (2-7x_n+3x_n^2}{6} \\ln \\frac{x_n}{1+x_n}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$ $ E_n^{\\prime } (x_t, x_n)= \\frac{x_t(-17-8x_t+x_t^2-3x_n(21-6x_t+x_t^2) - 6x_n^2(10-9x_t+3x_t^2))}{12(x_t-1)^3} ~~~~~~~~~\\nonumber \\\\+\\, \\frac{(1+x_n)(x_t+3x_t^2+x_n^2(3+x_t)-x_n(1+(-10+x_t)x_t))}{2(x_t-1)^4} \\ln \\frac{x_n+x_t}{1+x_n} \\nonumber \\\\-\\, \\frac{1}{2}x_n(1+x_n)(-1+3x_n) \\ln \\frac{x_n}{1+x_n}.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$ Following [17] or directly from [12] one gets the expressions for the sum over n as $ \\sum _{n=1} ^{\\infty } D_n^{\\prime }(x_t, x_n)&&= -\\frac{x_t(-37+x_t(44+17x_t))}{72(x_t-1)^3} \\nonumber \\\\&&+\\frac{\\pi M_W R}{2} \\Bigg [ \\int _0^1 dy \\,\\frac{(2 y^{1/2} + 7 y^{3/2} + 3 y^{5/2})}{6} \\coth (\\pi M_W R \\sqrt{y})\\nonumber \\\\&&+\\, \\frac{(-2+3x_t)x_t(1+3x_t)}{6(x_t-1)^4} J(R, -1/2)\\nonumber \\\\&& -\\, \\frac{1}{6(x_t-1)^4} [x_t (1+3x_t)-(-2+3x_t)(1+(-10+x_t)x_t)]J(R, 1/2) \\nonumber \\\\&& +\\, \\frac{1}{6(x_t-1)^4}[(-2+3x_t)(3+x_t) - (1+(-10+x_t)x_t)]J(R, 3/2) \\nonumber \\\\&& -\\, \\frac{(3+x_t)}{6(x_t-1)^4} J(R, 5/2)\\Bigg ]$ and $ \\sum _{n=1} ^{\\infty } E_n^{\\prime }(x_t, x_n) &&= -\\frac{x_t(-17+(-8+x_t)x_t)}{24(x_t-1)^3} \\nonumber \\\\&&+\\frac{\\pi M_W R}{4} \\Bigg [ \\int _0^1 dy \\,( y^{1/2} + 2 y^{3/2} - 3 y^{5/2}) \\coth (\\pi M_W R \\sqrt{y})\\nonumber \\\\&&-\\, \\frac{x_t(1+3x_t)}{(x_t-1)^4} J(R, -1/2)\\nonumber \\\\&& +\\, \\frac{1}{(x_t-1)^4} [x_t (1+3x_t)-(1+(-10+x_t)x_t)]J(R, 1/2) \\nonumber \\\\&& -\\, \\frac{1}{(x_t-1)^4}[(3+x_t) - (1+(-10+x_t)x_t)]J(R, 3/2) \\nonumber \\\\&& +\\, \\frac{(3+x_t)}{(x_t-1)^4} J(R, 5/2)\\Bigg ]$ where $ J(R, \\alpha ) = \\int _{0}^{1} dy \\,y^{\\alpha } \\,[\\coth (\\pi M_W R \\sqrt{y}) - x_t^{1+\\alpha } \\coth (\\pi m_t R \\sqrt{y})].$ The Wilson coefficient $C_9$ in the ACD model and the NDR scheme is $ C_9(\\mu , 1/R)= P_0^{NDR} + \\frac{Y(x_t, 1/R)}{sin^2{\\theta _W}} - 4 Z(x_t, 1/R) + P_E E(x_t, 1/R)$ where $P_0^{NDR}=2.6 \\pm 0.25$ and $P_E$ is numerically negligible.",
"The functions $Y(x_t, 1/R)$ and $Z(x_t, 1/R)$ are defined as $ Y(x_t, 1/R) = Y_0(x_t) + \\sum _{n=1} ^{\\infty } C_n(x_t, x_n)$ $ Z(x_t, 1/R) = Z_0(x_t) + \\sum _{n=1} ^{\\infty } C_n(x_t, x_n)$ with $ Y_0(x_t) = \\frac{x_t}{8} \\Bigg [ \\frac{x_t - 4}{x_t - 1} +\\frac{3 x_t}{(x_t - 1)^2} lnx_t \\Bigg ]$ $ Z_0 (x_t) = \\frac{18x_t^4 - 163x_t^3 + 259x_t^2 - 108x_t}{144(x_t - 1)^3} +\\Bigg [ \\frac{32x_t^4 - 38x_t^3 - 15x_t^2 + 18x_t}{72(x_t - 1)^4} - \\frac{1}{9} \\Bigg ]lnx_t$ $ C_n(x_t, x_n) = \\frac{x_t}{8(x_t-1)^2} \\Bigg [x_t^2 - 8x_t +7 + (3 + 3x_t + 7x_n - x_t x_n)ln\\frac{x_t + x_n}{1+x_n}\\Bigg ]$ and $\\sum _{n=1} ^{\\infty } C_n(x_t, x_n) = \\frac{x_t (7-x_t)}{16(x_t-1)}- \\frac{\\pi M_W R x_t}{16(x_t-1)^2} [3(1+x_t) J(R, -1/2) + (x_t-7) J(R, 1/2)].$ The $\\mu $ independent $C_{10}$ is given by $ C_{10}(1/R) = - \\frac{Y(x_t, 1/R)}{sin^2{\\theta _W}}$ where $Y(x_t, 1/R)$ is defined in (REF )."
]
] | 1204.1261 |
[
[
"Nonlocal growth processes and conformal invariance"
],
[
"Abstract Up to now the raise and peel model was the single known example of a one-dimensional stochastic process where one can observe conformal invariance.",
"The model has one-parameter.",
"Depending on its value one has a gapped phase, a critical point where one has conformal invariance and a gapless phase with changing values of the dynamical critical exponent $z$.",
"In this model, adsorption is local but desorption is not.",
"The raise and strip model presented here in which desorption is also nonlocal, has the same phase diagram.",
"The critical exponents are different as are some physical properties of the model.",
"Our study suggest the possible existence of a whole class of stochastic models in which one can observe conformal invariance."
],
[
"39"
]
] | 1204.1001 |
[
[
"Multiplicity in pp and AA collisions: the same power law from\n energy-momentum constraints in string production"
],
[
"Abstract We show that the dependence of the charged particle multiplicity on the centre-of-mass energy of the collision is, in the String Percolation Model, driven by the same power law behavior in both proton-proton and nucleus- nucleus collisions.",
"The observed different growths are a result of energy- momentum constraints that limit the number of formed strings at low en- ergy.",
"Based on the very good description of the existing data, we provide predictions for future high energy LHC runs."
],
[
"Introduction",
"Measurements of particle multiplicities constrain the early time properties of colliding systems.",
"In the nucleus-nucleus case, these measurements are an essential ingredient for the estimation of the initial energy and entropy densities and thus the initial conditions from which the system will eventually thermalize and the quark gluon plasma be formed.",
"Data collected at RHIC and the LHC for proton-proton (pp) and nucleus-nucleus (AA) collisions establish unambiguously that nucleus-nucleus collisions are not a simple incoherent superposition of collisions of the participating nucleons, $\\left.",
"{dn_{AA}}/{d\\eta }\\right|_{\\eta =0}>N_{part} \\cdot \\left.",
"{dn_{pp}}/{d\\eta }\\right|_{\\eta =0}$ , and thus that multiple scattering plays an important role.",
"Further, the possible scaling with the number of nucleon-nucleon also does not hold $\\left.",
"{dn_{AA}}/{d\\eta }\\right|_{\\eta =0}\\ll N_{coll} \\cdot \\left.",
"{dn_{pp}}/{d\\eta }\\right|_{\\eta =0}$ , indicating that coherence effects among the relevant degrees of freedom at the nucleon level are at play during the collision process.",
"The collision centre-of-mass energy dependence of the charged particle multiplicity in both pp and AA collisions is well reproduced by a power law as suggested by models, e.g.",
"the Colour Glass Condensate or the String Percolation Model, where coherence effects play an important role.",
"The logarithmic growth consistent with pre-LHC data is ruled out.",
"Notably, this dependence is stronger [1] in the AA ($\\left.",
"{dn_{AA}}/{d\\eta }\\right|_{\\eta =0}\\sim {s}^{0.15}$ ) case than for pp ($\\left.",
"{dn_{pp}}/{d\\eta }\\right|_{\\eta =0}\\sim {s}^{0.11}$ ).",
"Several possible explanations for this difference in energy dependence, naively at odds with theoretical expectations, have been put forward recently [2], [3], [4], [5], [6], [7].",
"The String Percolation Model (SPM) [8], the Dual Parton model [9] including parton saturation effects, describes consistently properties of bulk multiparticle production [10], [11], [12], [13], [14], [15], [16].",
"This framework is closely related to descriptions based on the Colour Glass Condensate and Glasma flux tubes [17] for which the same issues have been discussed extensively [18], [19], [20].",
"In this short note, we detail how the SPM can provide for a joint description of the energy dependence of multiplicity in pp and AA collisions once energy-momentum conservation constraints are taken into account.",
"Further, we show the resulting dependences on the nuclear species $A$ and number of participating nucleon $N_{\\rm part}$ to be fully compatible with available data.",
"Finally, we give predictions for future higher energy LHC runs and discuss generic expectations at high energy."
],
[
"Mid-rapidity multiplicity in the SPM",
"The Glauber model and its generalizations (see [21] for a review) relate nucleus-nucleus collisions to collisions of their constituent nucleons.",
"In the single scattering limit the average number of participating nucleons (per nucleon) $N_{A}=N_{\\rm part}/2$ behave incoherently and the mid-rapidity multiplicities in nucleus-nucleus and proton-proton collisions are simply related by: ${\\frac{dn^{AA}}{dy}}\\bigg |_{y=0} \\sim N_{A} \\cdot \\frac{dn^{pp}}{dy}\\bigg |_{y=0} \\, .$ This result, which corresponds to the wounded nucleon model [22], is expected to dominate at sufficiently low centre-of-mass energies.",
"At higher energies, multiple scattering becomes important, and $\\frac{dn^{AA}}{dy}\\bigg |_{y=0} \\sim \\big (N_{A}^{4/3} -N_A\\big ) \\cdot \\frac{dn^{pp}}{dy}\\bigg |_{y=0} \\, .$ Here, $N_{A}^{4/3}$ is the total number of nucleon-nucleon collisions and single scattering has been subtracted [23], [24].",
"Energy-momentum conservation constrains the combinatorial factors of the Glauber model at low energy.",
"In the framework of SPM, these constraints translate into the sharing of energy-momentum of $N_A$ valence strings amongst $N_A^{4/3}$ (mostly sea) strings.",
"A possible solution to this problem, the reduction of the height of the plateau for sea strings, was pursued in [23], [24].",
"Here, we proceed differently and account for energy-momentum conservation by reducing the effective number of sea strings via $N_{A}^{4/3}\\rightarrow N_{A}^{1+\\alpha (\\sqrt{s})}\\, ,$ with $\\alpha (\\sqrt{s})=\\frac{1}{3}\\Bigg (1-\\frac{1}{1+\\ln (\\sqrt{s/s_{0}}+1)}\\Bigg )\\, .$ We thus can write $\\frac{dn^{AA}}{dy} \\bigg |_{y=0} \\sim N_A \\big (N_{A}^{\\alpha (\\sqrt{s})} - 1\\big ) \\cdot \\frac{dn^{pp}}{dy}\\bigg |_{y=0} \\, ,$ such that the wounded nucleon model eq.",
"(REF ) is recovered for $\\sqrt{s}\\ll \\sqrt{s_{0}}$ , and Glauber result eq.",
"(REF ) follows for $\\sqrt{s} \\gg \\sqrt{s_{0}}$ , $\\alpha (\\sqrt{s}) \\rightarrow \\frac{1}{3}$ .",
"In the SPM one considers Schwinger strings, which can fuse and percolate [25], [26], [27], [28], as the fundamental degrees of freedom.",
"Multiparticle production is described in terms of these colour strings which are formed in the collision and stretch between partons in the parting nuclei and are thus longitudinally extended in rapidity.",
"The multiplicity of produced particles $dn/dy$ is proportional to the average number of such strings (twice the number of elementary collisions) $N^s$ .",
"Thus, for a generic $N_A N_A$ collision, be it $pp$ or $AA$ , one has $\\frac{dn}{dy} \\bigg |_{y=0} \\sim N^s_{N_A}\\, .$ In the impact parameter plane, the colour content of the strings is confined within a small transverse area $S_1 = \\pi r^2_0$ , with $r_0\\sim 0.2\\div 0.3$ fm.",
"The strings decay via $q\\bar{q}$ and $qq-\\bar{q}\\bar{q}$ pair production and subsequently hadronize to the observed hadrons.",
"In the impact parameter plane, the strings appear as disks and with increasing energy-density these disks overlap, fuse and percolate leading to a reduction of the overall charge [29].",
"A cluster of $n$ strings behaves as a single string with energy-momentum corresponding to the sum of the individual strings and with a higher colour field corresponding to the vectorial sum in colour space of the colour fields of the strings.",
"In this way, the mean multiplicity $\\langle \\mu _{n}\\rangle $ and the mean transverse momentum squared $\\langle p^{2}_{T}\\rangle $ of the particles produced by a cluster in the limit of random distribution of strings are given by $\\langle \\mu _{n}\\rangle =N^{s}F(\\eta ^t)\\langle \\mu _{1}\\rangle $ and $\\langle p^{2}_{T}\\rangle =\\langle p^{2}_{T,1}\\rangle /F(\\eta ^t)$ where $\\langle \\mu _{1}\\rangle $ and $\\langle p^{2}_{T,1}\\rangle $ are the corresponding quantities for a single string.",
"For a random distribution of disks, the colour reduction factor $F(\\eta ^t)$ is $F(\\eta ^{t})=\\sqrt{\\frac{1-e^{-\\eta ^{t}}}{\\eta ^{t}}}\\, ,$ where $\\eta ^t$ is the impact parameter transverse density of strings (disks) $\\eta ^{t} \\equiv \\frac{S_1}{S_{N_A}}\\, {N}^{s} = \\frac{\\pi r^2_0}{S_{N_A}}{N}^{s}\\, ,$ with $S_{N_A}$ the area of the impact parameter projected overlap of the interaction.",
"For a $N_A N_A$ collision, eq.",
"(REF ) can be recast as $\\frac{dn}{dy} \\bigg |_{y=0} \\sim F(\\eta ^t)\\, N^s_{N_A}\\, , \\qquad N^s_{N_A} = N^s_{p} N_A^{1+\\alpha (\\sqrt{s})}\\, .$ The colour reduction factor $F(\\eta ^t)$ results in a slowdown of the increase of $dn/dy$ with energy and number of participating nucleons.",
"Note that if nucleons were to act incoherently, as in the limiting case eq.",
"(REF ), $S_{N_A}$ would be given by the area of a single nucleon $S_p$ , while once coherence is accounted for, eq.",
"(REF ), $S_{N_A}$ is the overall area of interaction.",
"In general, $S_{N_A}$ depends on the impact parameter $b$ of the collision ($0\\le b\\le 2 R_A$ ) with $N_A\\rightarrow 0, S_{N_A}\\rightarrow 0$ as $b\\rightarrow R_A$ and $N_A\\rightarrow A, S_{N_A}\\rightarrow S_A\\equiv \\pi R_A^2 \\equiv \\pi R^2_p A^{2/3}$ as $b\\rightarrow 0$ .",
"These constraints are satisfied by $\\frac{S_{N_{A}}}{S_{A}}=\\bigg (\\frac{N_{A}}{A}\\bigg )^{\\beta }\\, ,$ with $\\beta >0$ .",
"It follows that $S_{N_{A}}=\\pi R^{2}_{p}\\,A^{2/3}\\bigg (\\frac{N_{A}}{A}\\bigg )^{\\beta }\\, ,$ and $\\eta ^{t}_{N_{A}}=\\eta ^{t}_{p}N_{A}^{\\alpha (\\sqrt{s})}A^{1/3}\\bigg (\\frac{A}{N_{A}}\\bigg )^{\\beta -1}\\, .$ Here, motivated by the scaling limit of the number of vertices in a loop-erased random walk [30], we set $\\beta =5/3$ such that $\\eta ^{t}_{N_{A}}=\\eta ^{t}_{p}N_{A}^{\\alpha (\\sqrt{s})}\\bigg (\\frac{A}{N_{A}^{2/3}}\\bigg )\\, .$ From eqs.",
"(REF ), (REF ), (REF ), (REF ) we can write $\\frac{1}{N_{A}}\\, \\frac{dn}{dy}\\bigg |_{y=0}= \\frac{dn^{pp}}{dy}\\bigg |_{y=0} \\bigg (1+\\frac{F(\\eta ^{t}_{N_{A}})}{F(\\eta ^{t}_{p})}(N^{\\alpha (\\sqrt{s})}_{A}-1)\\bigg )\\, ,$ The dependence of the multiplicity on the centre-of-mass collision energy $\\sqrt{s}$ is fully specified once the average number of strings in a pp collision $N_p^s$ is known.",
"At low energy $N_p^s$ , is approximately equal to 2, growing with energy as $(\\sqrt{s}/m_p)^{2\\lambda }$ such that $N_p^s = 2 + 4\\bigg (\\frac{r_0}{R_p}\\bigg )^2 \\bigg (\\frac{\\sqrt{s}}{m_p}\\bigg )^{2\\lambda }\\, ,$ with $R_p$ the proton radius."
],
[
"Data description",
"The energy and number of participants dependences implied by eq.",
"(REF ) can be tested against the available experimental data.",
"Before doing so, two remarks on the multiplicity given by eq.",
"(REF ) are in order: (i) it is for all particles, while experimental data accounts only for charged particles; (ii) it is for mid-rapidity $y\\sim 0$ , while data is collected at mid pseudo-rapidity $\\eta \\sim 0$ and although these coincide, the multiplicity value should be rescaled by Jacobean of the $y\\rightarrow \\eta $ transformation.",
"Neglecting the dependence of $p_T$ , on which the Jacobean depends, on centre-of-mass energy and the (small) difference between the Jacobean in pp and AA cases, both rescaling factors above can be absorbed into an overall normalization constant $\\kappa $ .",
"We thus write the charged particle multiplicity at mid-rapidity in the pp case as $\\frac{dn^{pp}_{\\rm ch}}{d\\eta }\\bigg |_{\\eta =0} = \\kappa \\, F(\\eta ^{t}_{p})\\, N_p^s\\, ,$ such that, for the general $N_A N_A$ case, one has $\\frac{1}{N_A} \\frac{dn^{N_AN_A}_{\\rm ch}}{d\\eta }\\bigg |_{\\eta =0} = \\kappa \\, F(\\eta ^{t}_{p})\\, N_p^s \\bigg (1+\\frac{F(\\eta ^{t}_{N_{A}})}{F(\\eta ^{t}_{p})}(N^{\\alpha (\\sqrt{s})}_{A}-1)\\bigg )\\, .$ Eq.",
"(REF ) depends on three parameters: (i) the normalization $\\kappa $ ; (ii) the threshold scale $\\sqrt{s_0}$ in $\\alpha ({\\sqrt{s}})$ in eq.",
"(REF ); (iii) the power $\\lambda $ in eq.",
"(REF ).",
"We performed a global fit to pp data [31], [32], [33], [34], [35], [36], [37] in the range $ 53\\le \\sqrt{s}\\le 7000$ GeV, and to AA (AuAu, CuCu and PbPb) at different centralities [38], [39] for $19.6\\le \\sqrt{s}\\le 2760$ GeV.",
"The fitted data sample consists of 116 points (19 for pp and 97 for AA).",
"The best fit yields the parameter values: $\\kappa = 0.63\\pm 0.01$ , $\\lambda =0.201\\pm 0.003$ , and $\\sqrt{s_0}=245\\pm 29$ GeV.",
"One notices that while $\\kappa $ and $\\lambda $ are tightly constrained, the threshold scale $\\sqrt{s_0}$ is determined with a sizable associated error of $\\sim $ 10%.",
"This results from the mild dependence of eq.",
"(REF ), through eq.",
"(REF ), on $\\sqrt{s_0}$ compounded with the sizable errors in the existing measurements (see Fig.",
"REF below).",
"Fig.",
"REF shows a comparison of the evolution of the mid-rapidity multiplicity with energy given by eq.",
"(REF ) with data for pp and AA collisions.",
"Figure: Multiplicity dependence on s\\sqrt{s}.",
"pp data from , , , , , , (circles), CuCu (triangles) and AuAu (stars) from , PbPb (star) from .",
"Curves obtained from eq.",
"(): (N A =1N_A=1, A=1A=1) for pp (grey line); (N A =50N_A=50, A=63A=63) for CuCu (blue line); and (N A =175N_A=175, A=200A=200) for AuAu/PbPb (red line).",
"Color online.For reference we quote, in Table REF , the predicted values relevant for future LHC runs.",
"Table: Predicted multiplicities for pp and PbPb at future LHC energies.The dependence of the multiplicity on the number of participating nucleons implied by eq.",
"(REF ) is compared with data in Fig.",
"REF .",
"Figure: Multiplicity dependence on centrality (the number of participating nucleons N part =2N A N_{\\rm part}=2 N_A where N A N_A is the number of participants per nucleus).",
"CuCu (triangles) and AuAu (stars) data from , and PbPb (circles) from .",
"Curves obtained from eq.",
"(): (s=22.4,62.4,200\\sqrt{s}=22.4, 62.4, 200 GeV) for CuCu (blue); (s=19.6,62.4,130,200\\sqrt{s}=19.6, 62.4, 130, 200 GeV) for AuAu (green); and (s=2.76,3.2,3.9,5.5\\sqrt{s}=2.76, 3.2, 3.9, 5.5 TeV) for PbPb (red).",
"Color online."
],
[
"Conclusions",
"We have shown that, in the SPM, the power law dependence of the multiplicity on the collision energy is the same in pp and AA collisions.",
"The slower growth in the AA case is due to finite energy-momentum constraints which tamper string creation, and thus the multiplicity, at low energy.",
"In the high energy limit, $F(\\eta ^{t})\\rightarrow (\\eta ^t)^{-1/2}$ and $\\alpha (\\sqrt{s})\\rightarrow 1/3$ , eq.",
"(REF ) can be written $\\frac{1}{N_{A}}\\,\\frac{dn^{N_AN_A}_{\\rm ch}}{d\\eta }\\bigg |_{\\eta =0} = \\frac{dn^{pp}_{\\rm ch}}{d\\eta }\\bigg |_{\\eta =0} \\Bigg (1+\\bigg (\\frac{N_{A}}{A} \\bigg )^{1/2} \\bigg (1 - \\frac{1}{N_A^{1/3}}\\bigg )\\Bigg )\\, .$ With the obtained fit parameters, this asymptotic result becomes a good approximation (within 5% of that given by eq.",
"(REF )) for $\\sqrt{s}\\gtrsim 500$ GeV.",
"This indicates that finite energy corrections persist to fairly high energies.",
"In this high energy limit, the shape of $\\frac{dn^{N_AN_A}}{d\\eta }$ as a function of the number of participants is energy independent, that is to say that the $N_{\\rm part}$ and $\\sqrt{s}$ dependences factorize.",
"In the SPM, energy-momentum conservation results in violations of this factorization and they are the origin of the observed discrepancy in multiplicity growth with energy in pp and AA.",
"The arguments put forward in this short note can be readily adapted to the case of asymmetric (proton-nucleus) collisions.",
"Also in this case, energy-momentum constraints are expected to play an important role."
],
[
"Acknowledgments",
"We thank N. Armesto for useful discussions.",
"IB is supported by the grant SFRH/BD/51370/2011 from Fundação para a Ciência e a Tecnologia (Portugal).",
"JDD and JGM acknowledge the support of Fundação para a Ciência e a Tecnologia (Portugal) under project CERN/FP/116379/2010.",
"IB and CP were partly supported by the project FPA2008-01177 and FPA2011-22776 of MICINN, the Spanish Consolider Ingenio 2010 program CPAN and Conselleria de Educacion Xunta de Galicia."
]
] | 1204.1457 |
[
[
"A Weiszfeld-like algorithm for a Weber location problem constrained to a\n closed and convex set"
],
[
"Abstract The Weber problem consists of finding a point in $\\mathbbm{R}^n$ that minimizes the weighted sum of distances from $m$ points in $\\mathbbm{R}^n$ that are not collinear.",
"An application that motivated this problem is the optimal location of facilities in the 2-dimensional case.",
"A classical method to solve the Weber problem, proposed by Weiszfeld in 1937, is based on a fixed point iteration.",
"In this work a Weber problem constrained to a closed and convex set is considered.",
"A Weiszfeld-like algorithm, well defined even when an iterate is a vertex, is presented.",
"The iteration function $Q$ that defines the proposed algorithm, is based mainly on an orthogonal projection over the feasible set, combined with the iteration function of a modified Weiszfeld algorithm presented by Vardi and Zhang in 2001.",
"It can be seen that the proposed algorithm generates a sequence of feasible iterates that have descent properties.",
"Under certain hypotheses, the limit of this sequence satisfies the KKT optimality conditions, is a fixed point of the iteration function that defines the algorithm, and is the solution of the constrained minimization problem.",
"Numerical experiments confirmed the theoretical results."
],
[
"Introduction",
"Let $a^1, \\ldots , a^m$ be $m$ distinct points in the space $\\mathbb {R}^n$ , called vertices, and positive numbers $w_1, \\ldots , w_m$ , called weights.",
"The function $f : \\mathbb {R}^n \\rightarrow \\mathbb {R}$ defined by $f( x ) = \\sum _{j=1}^m w_j \\left\\Vert x - a^j \\right\\Vert ,$ is called the Weber function, where $\\left\\Vert \\cdot \\right\\Vert $ denotes the Euclidean norm.",
"It is well-known that this function is not differentiable at the vertices, and strictly convex if the vertices are not collinear (we will assume this hypothesis from now on).",
"The Weber problem (also known as the Fermat-Weber problem) is to find a point in $\\mathbb {R}^n$ that minimizes the weighted sum of Euclidean distances from the $m$ given points, that is, we have to find the solution of the following unconstrained optimization problem: $\\begin{array}{rl}\\displaystyle \\mathop { \\mathrm {argmin} }_x & \\displaystyle f( x ) \\\\ \\mathrm {subject \\, to} & x \\in \\mathbb {R}^n.\\end{array}$ This problem has a unique solution $x^u$ in $\\mathbb {R}^n$ .",
"The problem was also stated as a pure mathematical problem by Fermat [44], [27], Cavalieri [37], Steiner [14], Fasbender [20] and many others.",
"Several solutions, based on geometrical arguments, were proposed by Torricelli and Simpson.",
"In [30] historical details and geometric aspects were presented by Kupitz and Martini.",
"In [41] Weber formulated the problem (REF ) from an economical point of view.",
"The vertices represent customers or demands, the solution to the problem denotes the location of a new facility, and the weights are costs associated with the interactions between the new facility and the customers.",
"Among several schemes to solve the Weber location problem (see [12], [19], [28], [34]), one of the most popular methods was presented by Weiszfeld in [42], [43].",
"The Weiszfeld algorithm is an iterative method based on the first-order necessary conditions for a stationary point of the objective function.",
"If we define $T_0 : \\mathbb {R}^n \\rightarrow \\mathbb {R}^n$ by: $T_0(x) = \\left\\lbrace \\begin{array}{ll} \\frac{ \\displaystyle \\sum _{j=1}^m \\frac{ w_j a^j }{ \\left\\Vert x - a^j \\right\\Vert } }{ \\displaystyle \\sum _{j=1}^m \\frac{ w_j }{ \\left\\Vert x - a^j \\right\\Vert } }, & \\quad \\text{if $x \\ne a^1, \\ldots , a^m$}, \\\\ & \\\\ a^k, & \\quad \\text{if $x = a^k$, $k = 1, \\ldots , m$}, \\end{array} \\right.$ the Weiszfeld algorithm is: $x^{(l)} = T_0 \\left( x^{(l-1)} \\right), \\quad l \\in \\mathbb {N},$ where $x^{(0)} \\in \\mathbb {R}^n$ is a starting point.",
"The Weiszfeld algorithm (REF ), despite of its simplicity, has a serious problem if some $x^{(l)}$ lands accidentally in a vertex $a^k$ , because the algorithm gets stuck at $a^k$ , even when $a^k$ is not the solution of (REF ).",
"Many authors studied the set of initial points for which the sequence generated by the Weiszfeld algorithm yields in a vertex (see [29], [11], [6], [9], [7], [3]).",
"Vardi and Zhang [40] derived a simple but nontrivial modification of the Weiszfeld algorithm in which they solved the problem of landing in a vertex.",
"Generalizations and new techniques for the Fermat-Weber location problem have been developed in recent years.",
"In [18] Eckhardt applied the Weiszfeld algorithm to generalized Weber problems in Banach spaces.",
"An exact algorithm for a Weber problem with attraction and repulsion was presented by Chen et al.",
"in [13].",
"Kaplan and Yang [24] proved a duality theorem which includes as special cases a great variety of choices of norms in the terms of the Fermat-Weber sum.",
"In [10] Carrizosa et al.",
"studied the so called Regional Weber Problem, which allows the demand not to be concentrated onto a finite set of points, but follows an arbitrary probability measure.",
"In [17] Drezner and Wesolowsky studied the case where different $l_p$ norms are used for each demand point.",
"In [23] the so called Complementary Problem (the Weber problem with one negative weight) was studied by Jalal and Krarup, and geometrical solutions were given.",
"In [15] Drezner presented a Weiszfeld-like iterative procedure and convergence is proved if appropriate conditions hold.",
"In some practical problems it is necessary to consider barriers (forbidden regions).",
"Barriers were first introduced to location modeling by Katz and Cooper [25].",
"There exist several heuristic and iterative algorithms for single-facility location problems for distance computations in the presence of barriers (see [2], [8], [5], [4]).",
"In [35] Pfeiffer and Klamroth presented a unified formulation for problems with barriers and network location problems.",
"A complete reference to barriers in location problems can be found in [26].",
"Barriers can be applied to model real life problems where regions like lakes and mountains are forbidden.",
"On the other hand, there are location problems whose solution needs to lie within a closed set.",
"For example, see [39] for a discussion of the case when the solution is constrained to be within a maximum distance of each demand point.",
"Drezner and Wesolowsky [16] studied the problem of locating an obnoxious facility with rectangular distances ($l_1$ norm), where the facility must lie within some prespecified region (linear constraints).",
"A primal-dual algorithm to deal with the constrained Fermat-Weber problem using mixed norms was developed in [33] by Idrissi et al..",
"In [21] Hansen et al.",
"presented an algorithm for solving the Weber problem when the set of feasible locations is the union of a finite number of convex polygons.",
"In [36] Pilotta and Torres considered a Weber location problem with box constraints.",
"Constrained Weber problems arise when we require that the solution is in an area (feasible region) determined by, for example, environmental and/or political reasons.",
"It could be the case for a facility producing dangerous materials that must be installed in a restricted (constrained) area.",
"Another example could be the location of a plant in an industrial zone or of a hospital in a non-polluted area.",
"In this paper a constrained location problem is considered.",
"An algorithm is proposed to solve the following problem: $\\begin{array}{rl}\\displaystyle \\mathop { \\mathrm {argmin} }_x & \\displaystyle f( x ) \\\\ \\mathrm {subject \\, to} & x \\in \\Omega ,\\end{array}$ where $\\Omega $ is a closed and convex set, generalizing the problem formulated in [36].",
"Problem (REF ) could be seen as a nonlinear programming problem and solved by standard solvers, but they may fail since the Weber function is not differentiable at the vertices.",
"It can be proved that problem (REF ) has a unique solution $x^*$ , since the function $f$ is strictly convex and $\\Omega $ is a closed and convex set.",
"On the other hand, it is well-known that the convex hull of the given vertices $a^1, \\ldots , a^m$ contains the solution $x^u$ of the unconstrained Weber problem (see for instance [29]).",
"If $\\Omega $ contains the convex hull, both solutions $x^*$ and $x^u$ agree.",
"In other cases, the solution $x^*$ is not necessarily a projection of $x^u$ over $\\Omega $ (see [36]).",
"The algorithm is based basically on a slight variation of an orthogonal projection of the Weiszfeld algorithm presented in [40], that is well defined even when an iterate coincides with a vertex.",
"Properties of the sequence generated by the proposed algorithm related with the minimization problem REF will be proved in the following sections.",
"The paper is structured as follows: Section describes the results in [40] in which a modified Weiszfeld algorithm is presented and some notation is introduced.",
"In Section the proposed algorithm is defined.",
"Section is dedicated to definitions and technical lemmas.",
"In Section the main results about convergence to optimality are presented.",
"Numerical experiments are considered in Section .",
"Finally, conclusions are given in Section .",
"Some words about notation.",
"As it was mentioned, we will call $x^u$ the solution of problem (REF ) and $x^*$ the solution of problem (REF ).",
"The symbols $\\Vert \\cdot \\Vert $ and $\\langle \\cdot , \\cdot \\rangle $ will refer to the standard Euclidean norm and standard inner product in $\\mathbb {R}^n$ , respectively.",
"For a function $f : \\mathbb {R} \\rightarrow \\mathbb {R}$ we will denote by $f^{\\prime }(a-)$ the left-hand side derivative at $a$ , and by $f^{\\prime }(a+)$ the right-hand side derivative at $a$ ."
],
[
"The modified Weiszfeld algorithm",
"This section reviews the main results presented in [40] in which the authors generalize the Weiszfeld algorithm for the case that an iterate lands on a vertex.",
"From now on, this algorithm will be referred to as the modified Weiszfeld algorithm.",
"In order to make notation easier, we define the function $A : \\mathbb {R}^n \\rightarrow \\mathbb {R}$ by: $A(x) = \\left\\lbrace \\begin{array}{ll} \\displaystyle \\sum _{j=1}^m \\frac{ w_j }{ 2 \\left\\Vert x - a^j \\right\\Vert }, & \\quad \\text{if $x \\ne a^1, \\ldots , a^m$}, \\\\ & \\\\ \\displaystyle \\sum _{\\begin{array}{c}j=1 \\\\ j \\ne k\\end{array}}^m \\frac{ w_j }{ 2 \\left\\Vert a^k - a^j \\right\\Vert }, & \\quad \\text{if $x = a^k$, $k = 1, \\ldots , m$}.",
"\\end{array} \\right.$ Notice that $A(x) > 0$ for all $x \\in \\mathbb {R}^n$ .",
"In [40], the number $A(a^k)$ was called $A_k$ .",
"A generalization for the iteration function $T_0$ , defined in (REF ), is given by $\\widetilde{T} : \\mathbb {R}^n \\rightarrow \\mathbb {R}^n$ defined as follows: $\\widetilde{T}(x) = \\left\\lbrace \\begin{array}{ll} \\displaystyle \\frac{ \\displaystyle \\sum _{j=1}^m \\frac{ w_j a^j }{ \\left\\Vert x - a^j \\right\\Vert } }{ 2 A(x) }, & \\quad \\text{if $x \\ne a^1, \\ldots , a^m$}, \\\\ & \\\\ \\displaystyle \\frac{ \\displaystyle \\sum _{\\begin{array}{c}j=1 \\\\ j \\ne k\\end{array}}^m \\frac{ w_j a^j }{ \\left\\Vert a^k - a^j \\right\\Vert } }{ 2 A\\left(a^k\\right) }, & \\quad \\text{if $x = a^k$, $k = 1, \\ldots , m$}.",
"\\end{array} \\right.$ Notice that $\\widetilde{T}$ coincides with $T_o$ in $\\mathbb {R}^n - \\left\\lbrace a^1, \\ldots , a^m \\right\\rbrace $ .",
"Let $\\widetilde{R} : \\mathbb {R}^n \\rightarrow \\mathbb {R}^n$ and $r : \\mathbb {R}^n \\rightarrow \\mathbb {R}$ be: $\\widetilde{R}(x) & = & \\left\\lbrace \\begin{array}{ll} \\displaystyle \\sum _{j=1}^m \\frac{ w_j \\left( a^j - x \\right) }{ \\left\\Vert x - a^j \\right\\Vert }, & \\quad \\text{if $x \\ne a^1, \\ldots , a^m$}, \\\\ & \\\\ \\displaystyle \\sum _{\\begin{array}{c}j=1 \\\\ j \\ne k\\end{array}}^m \\frac{ w_j \\left( a^j - a^k \\right) }{ \\left\\Vert a^k - a^j \\right\\Vert }, & \\quad \\text{if $x = a^k$, $k = 1, \\ldots , m$}, \\end{array} \\right.\\\\& \\nonumber \\\\r(x) & = & \\Vert \\widetilde{R}(x) \\Vert , \\qquad \\forall \\, x \\in \\mathbb {R}^n.",
"\\nonumber $ The function $\\widetilde{R}$ generalizes the negative gradient of the Weber function since, for all $x \\ne a^1, \\ldots , a^m$ , $\\nabla f(x) = - \\widetilde{R}(x).$ The following lemma is very easy to prove (see [40]), and it relates the functionals $\\widetilde{T}$ and $\\widetilde{R}$ .",
"Lemma 1 For all $x \\in \\mathbb {R}^n$ we have $\\widetilde{R}(x) = 2 A(x) \\left[ \\widetilde{T}(x) - x \\right]$ .",
"If we define $\\gamma : \\mathbb {R}^n \\rightarrow \\mathbb {R}$ by: $\\gamma (x) = \\left\\lbrace \\begin{array}{ll} 0, & \\quad \\text{if $x \\ne a^1, \\ldots , a^m$}, \\\\ 0, & \\quad \\text{if $x = a^k$ and $r\\left(a^k\\right) = 0$ for some $k = 1, \\ldots , m$}, \\\\ \\displaystyle w_k / r\\left(a^k\\right), & \\quad \\text{if $x = a^k$ and $r\\left(a^k\\right) \\ne 0$ for some $k = 1, \\ldots , m$}, \\end{array} \\right.$ we can see that $\\gamma (x) \\ge 0$ for all $x \\in \\mathbb {R}^n$ .",
"The modified Weiszfeld algorithm presented in [40] is defined by: $x^{(l)} = T\\left( x^{(l-1)} \\right), \\quad l \\in \\mathbb {N},$ where $x^{(0)} \\in \\mathbb {R}^n$ and $T : \\mathbb {R}^n \\rightarrow \\mathbb {R}^n$ is given by: $T(x) = \\left( 1 - \\beta (x) \\right) \\widetilde{T}(x) + \\beta (x) x,$ where $\\beta : \\mathbb {R}^n \\rightarrow \\mathbb {R}$ is defined by $\\beta (x) = \\min \\left\\lbrace 1, \\gamma (x) \\right\\rbrace $ .",
"Remark 2 If $x \\ne a^1, \\ldots , a^m$ , then $\\beta (x) = 0$ because $\\gamma (x) = 0$ .",
"So, we can deduce that $T(x) = \\widetilde{T}(x)$ .",
"Notice that this fact implies that the functional $T$ is continuous in $\\mathbb {R}^n - \\left\\lbrace a^1, \\ldots , a^m \\right\\rbrace $ .",
"It can be seen that if $a^k \\ne x^u$ , then $0 < \\beta ( a^k ) < 1$ (see [40]).",
"From equation (REF ) we obtain that $T(x) - x = \\left( 1 - \\beta (x) \\right) \\left( \\widetilde{T}(x) - x \\right)$ for $x \\in \\mathbb {R}^n$ .",
"The main result in [40] is: Theorem 3 The following propositions are equivalent: $x = x^u$ .",
"$T(x) = x$ .",
"$r(x) \\le \\eta (x)$ .",
"where $\\eta (x) = \\left\\lbrace \\begin{array}{ll} 0, & \\quad \\text{if $x \\ne a^1, \\ldots , a^m$}, \\\\ w_k, & \\quad \\text{if $x = a^k$, $k = 1, \\ldots , m$}.",
"\\end{array} \\right.$"
],
[
"The proposed algorithm",
"This section is dedicated to describe the proposed algorithm, introducing some definitions and remarks.",
"First of all, we can notice that problem (REF ) has a unique solution, due to the fact that $f$ is a non-negative, strictly convex, and continuous function, $\\lim _{\\Vert x \\Vert \\rightarrow \\infty } f(x) = \\infty $ and $\\Omega $ is closed and convex.",
"In order to define the proposed algorithm at the vertices, we will need to determine which points of the segment that joins $a^k$ and $T(a^k)$ are in the feasible set $\\Omega $ .",
"If $k = 1, \\ldots , m$ , let the set $\\mathcal {S}_k$ be defined by: $\\mathcal {S}_k = \\left\\lbrace \\lambda \\in [0,1] : ( 1 - \\lambda ) T(a^k) + \\lambda a^k \\in \\Omega \\right\\rbrace .$ Notice that $\\mathcal {S}_k$ could be equal to the empty set in case that $a^k$ and $T(a^k)$ do not belong to $\\Omega $ .",
"On the other hand, if $a^k \\in \\Omega $ , then $1 \\in \\mathcal {S}_k$ , which means that $\\mathcal {S}_k \\ne \\emptyset $ .",
"Thus, we can define: $\\lambda ( a^k ) = \\inf \\mathcal {S}_k, \\quad a^k \\in \\Omega .$ In case a vertex $a^k$ is not in $\\Omega $ , there is no need to define the number $\\lambda ( a^k )$ .",
"In the following lemma, a set of basic properties of $\\lambda ( a^k )$ are listed: Lemma 4 If $k = 1, \\ldots , m$ and $a^k \\in \\Omega $ then: $\\lambda ( a^k ) \\in [0,1]$ .",
"If $T(a^k) \\in \\Omega $ then $\\lambda ( a^k ) = 0$ .",
"If $T(a^k) \\notin \\Omega $ then $\\lambda ( a^k ) \\in (0,1]$ .",
"The proof of (a) follows from the definition of $\\mathcal {S}_k$ .",
"If $T(a^k) \\in \\Omega $ , then $0 \\in \\mathcal {S}_k$ , so $\\lambda (a^k) = 0$ , and this proves (b).",
"Finally, for item (c), let us consider that $T(a^k) \\notin \\Omega $ .",
"Since $\\Omega $ is a closed set, there is an entire ball centered at $T(a^k)$ that does not intersect $\\Omega $ , which implies that there exists $\\epsilon $ such that $( 1 - \\lambda ) T(a^k) + \\lambda a^k \\notin \\Omega $ for all $\\lambda \\in [0,\\epsilon ]$ .",
"Thus, $\\lambda (a^k) \\in (0,1]$ and this concludes the proof.",
"$\\Box $ Let us call $P_{\\Omega } : \\mathbb {R}^n \\rightarrow \\Omega $ the orthogonal projection over $\\Omega $ .",
"Since $\\Omega $ is a nonempty, closed and convex set, the operator $P_{\\Omega }$ is a continuous function [1].",
"We define the iteration function $Q : \\Omega \\rightarrow \\Omega $ by: $Q(x) = \\left\\lbrace \\begin{array}{ll} P_{\\Omega } \\circ T(x), & \\quad \\text{if $x \\ne a^1, \\ldots , a^m$}, \\\\ \\left( 1 - \\lambda ( a^k ) \\right) T(a^k) + \\lambda ( a^k ) a^k, & \\quad \\text{if $x = a^k \\in \\Omega $, $k = 1, \\ldots , m$}.",
"\\end{array} \\right.$ There will be no need to define $Q$ outside $\\Omega $ since the proposed algorithm generates a sequence of feasible points.",
"The iteration function $Q$ at $x \\in \\Omega $ coincides with the orthogonal projection of $T(x)$ over the feasible set when $x$ is different from the vertices.",
"Only when $x$ is a vertex $a^k$ belonging to $\\Omega $ , $Q(x)$ is defined as the farthest possible feasible point of the segment that joins $x$ with $T(x)$ .",
"The following remark states some basic properties of the iteration function of the proposed algorithm.",
"Remark 5 (a) If $a^k \\in \\Omega $ and $T(a^k) \\in \\Omega $ , then $Q(a^k) = T(a^k) = P_{\\Omega } \\circ T(a^k)$ .",
"(b) If $a^k \\in \\Omega $ , it can be seen that: $Q(a^k) - a^k & = & \\left( 1 - \\lambda ( a^k ) \\right) \\left( T(a^k) - a^k \\right),\\\\Q(a^k) - T(a^k) & = & - \\lambda ( a^k ) \\left( T(a^k) - a^k \\right).$ (c) The functional $Q$ is continuous in $\\mathbb {R}^n - \\left\\lbrace a^1, \\ldots , a^m \\right\\rbrace $ .",
"The proofs of (a) and (b) are straightforward.",
"For (c), since $P_{\\Omega }$ is continuous in $\\mathbb {R}^n$ (see [1]) and $T$ is continuous in $\\mathbb {R}^n - \\left\\lbrace a^1, \\ldots , a^m \\right\\rbrace $ (see Remark REF ), we have that $Q$ is continuous in $\\mathbb {R}^n - \\left\\lbrace a^1, \\ldots , a^m \\right\\rbrace $ .",
"$\\Box $ The proposed algorithm is described below.",
"Algorithm 6 Let $\\Omega \\subset \\mathbb {R}^n$ be a closed and convex set.",
"Assume that $x^{(0)} \\in \\Omega $ is an initial approximation such that $f(x^{(0)}) \\le f(a^j)$ for all $j \\in \\left\\lbrace 1, \\ldots , m \\right\\rbrace $ and $a^j \\in \\Omega $ .",
"Given $\\varepsilon > 0$ a tolerance and $x^{(l-1)} \\in \\Omega $ , do the following steps to compute $x^{(l)}$ : Step 1: Compute: $x^{(l)} = Q \\left( x^{(l-1)} \\right).$ Step 2: Stop the execution if $\\left\\Vert x^{(l)} - x^{(l-1)} \\right\\Vert < \\varepsilon ,$ and declare $x^{(l)}$ as solution to the problem (REF ).",
"Otherwise return to Step 1.",
"From the definition of $Q$ it follows that Algorithm REF generates a sequence of feasible iterates.",
"Also notice that if there are vertices in the feasible set, $x^{(0)}$ can be one of them, for example, a vertex $a_s$ such that $f(a_s) \\le f(a^j)$ for all $a^j \\in \\Omega $ .",
"On the other hand, if there are no vertices in the feasible set, $x^{(0)}$ can be chosen as the projection over $\\Omega $ of the null vector.",
"Some definitions and technical results The purpose of this section is to define some entities and prove technical lemmas that will be important in the proof of the main results.",
"First of all, we will define some useful operators for making notation easier.",
"If $\\mathcal {A} \\subset \\left\\lbrace 1, \\ldots , n \\right\\rbrace $ , then we define $\\Vert \\cdot \\Vert _{\\mathcal {A}} : \\mathbb {R}^n \\rightarrow \\mathbb {R}$ and $\\langle \\cdot , \\cdot \\rangle _{\\mathcal {A}} : \\mathbb {R}^n \\times \\mathbb {R}^n \\rightarrow \\mathbb {R}$ by: $\\Vert x \\Vert _{\\mathcal {A}} = \\sqrt{ \\sum _{j \\in \\mathcal {A}} x_j^2 }, \\qquad \\langle x , y \\rangle _{\\mathcal {A}} = \\sum _{j \\in \\mathcal {A}} x_j y_j.$ Notice that, when $\\mathcal {A} \\subsetneq \\left\\lbrace 1, \\ldots , n \\right\\rbrace $ , $\\Vert \\cdot \\Vert _{\\mathcal {A}}$ is not necessarily a norm and $\\langle \\cdot , \\cdot \\rangle _{\\mathcal {A}}$ is not necessarily an inner product.",
"According to this definition, if $\\mathcal {A}$ and $\\mathcal {B}$ are sets such that $\\mathcal {A} \\cap \\mathcal {B} = \\emptyset $ and $\\mathcal {A} \\cup \\mathcal {B} = \\left\\lbrace 1, \\ldots , n \\right\\rbrace $ , it can be seen that: $\\Vert x \\Vert ^2 & = & \\Vert x \\Vert _{\\mathcal {A}}^2 + \\Vert x \\Vert _{\\mathcal {B}}^2,\\\\\\langle x , y \\rangle & = & \\langle x , y \\rangle _{\\mathcal {A}} + \\langle x , y \\rangle _{\\mathcal {B}},\\\\c \\langle x , y \\rangle _{\\mathcal {A}} & = & \\langle cx , y \\rangle _{\\mathcal {A}} = \\langle x , cy \\rangle _{\\mathcal {A}}.$ For $x \\in \\Omega $ , let us define the following sets of indices: $\\mathcal {N}(x) & = & \\left\\lbrace k \\in \\mathbb {N} : \\quad 1 \\le k \\le n, \\quad (T(x))_k \\ne (Q(x))_k \\right\\rbrace ,\\\\\\mathcal {E}(x) & = & \\left\\lbrace k \\in \\mathbb {N} : \\quad 1 \\le k \\le n, \\quad (T(x))_k = (Q(x))_k \\right\\rbrace ,$ Notice that for all $x \\in \\mathbb {R}^n$ we have that $\\mathcal {N}(x) \\cap \\mathcal {E}(x) = \\emptyset $ and $\\mathcal {N}(x) \\cup \\mathcal {E}(x) = \\left\\lbrace 1, \\ldots , n \\right\\rbrace $ .",
"Let $\\alpha : \\Omega \\rightarrow \\mathbb {R}^n$ be the following function: If $x \\ne a^1, \\ldots , a^m$ : $\\alpha (x) = \\displaystyle \\sum _{j=1}^m \\frac{ w_j }{ \\Vert x - a^j \\Vert } \\left[ Q(x) - a^j \\right].$ If $x = a^k \\in \\Omega $ for some $k = 1, \\ldots , m$ : $\\alpha (x) = \\displaystyle \\sum _{\\begin{array}{c}j=1 \\\\ j \\ne k\\end{array}}^m \\frac{ w_j }{ \\Vert a^k - a^j \\Vert } \\left[ Q(a^k) - (1 - \\beta (a^k)) a^j - \\beta (a^k) a^k \\right].$ It can be seen that the function $\\alpha $ is related to the iteration function $Q$ of the proposed algorithm, and the iteration function $T$ of the modified algorithm.",
"Lemma 7 If $x \\in \\Omega $ , then $\\alpha (x) = 2 A(x) \\left[ Q(x) - T(x) \\right].$ If $x \\ne a^1, \\ldots , a^m$ , then: $\\alpha (x) & = & \\sum _{j=1}^m \\frac{ w_j }{ \\Vert x - a^j \\Vert } \\left[ Q(x) - a^j \\right] = \\sum _{j=1}^m \\frac{ w_j Q(x) }{ \\Vert x - a^j \\Vert } - \\sum _{j=1}^m \\frac{ w_j a^j }{ \\Vert x - a^j \\Vert }\\\\&&\\\\& = & \\left( \\sum _{j=1}^m \\frac{ w_j }{ \\Vert x - a^j \\Vert } \\right) \\left[ Q(x) - \\frac{ \\sum _{j=1}^m \\frac{ w_j a^j }{ \\Vert x - a^j \\Vert } }{ \\sum _{j=1}^m \\frac{ w_j }{ \\Vert x - a^j \\Vert } } \\right] = 2 A(x) \\left[ Q(x) - \\widetilde{T}(x) \\right]\\\\&&\\\\& = & 2 A(x) \\left[ Q(x) - T(x) \\right].$ where in the last equalities we have used the definition of $\\widetilde{T}$ as in (REF ), and the fact that $\\widetilde{T}(x) = T(x)$ due to Remark REF .",
"If $x = a^k$ for some $k = 1, \\ldots , m$ , we follow a similar procedure than in the previous case.",
"$\\Box $ Now, we will define auxiliary functions that take into account the projection $P_{\\Omega }$ in order to prove a descent property of $f$ (see next sections).",
"If $x \\in \\Omega $ , we define: $E_x : \\mathbb {R}^n \\rightarrow \\mathbb {R}^n$ , where: $\\left( E_x(y) \\right)_k = \\left\\lbrace \\begin{array}{ll} (Q(x))_k, & \\quad \\text{if $k \\in \\mathcal {N}(x)$}, \\\\ y_k, & \\quad \\text{if $k \\in \\mathcal {E}(x)$}.",
"\\end{array} \\right.$ If $\\mathcal {E}(x) = \\left\\lbrace i_1, \\ldots , i_r \\right\\rbrace \\ne \\emptyset $ define $P_x : \\mathbb {R}^n \\rightarrow \\mathbb {R}^r$ where: $\\left( P_x(y) \\right)_k = y_{i_k}, \\quad k = 1, \\ldots , r.$ A useful property of $E_x$ , that follows from the definition, is pointed out in the following remark.",
"Remark 8 If $x \\in \\Omega $ then $E_x \\circ Q(x) = Q(x)$ .",
"The iteration function $Q$ inherits an important property from the orthogonal projection $P_{\\Omega }$ .",
"Lemma 9 If $x \\in \\Omega $ we have that $\\left\\langle Q(x) - x , Q(x) - T(x) \\right\\rangle \\le 0$ .",
"If $x \\ne a^1, \\ldots , a^m$ , then $Q(x) = P_{\\Omega } \\circ T(x)$ .",
"By a property of the orthogonal projection [1] we have that $\\langle Q(x) - x , Q(x) - T(x) \\rangle \\le 0$ .",
"If $x = a^k$ for some $k = 1, \\ldots , m$ , Remark REF and Lemma REF imply: $\\left\\langle Q(a^k) - a^k , Q(a^k) - T(a^k) \\right\\rangle = - \\lambda ( a^k ) \\left( 1 - \\lambda ( a^k ) \\right) \\left\\Vert T(a^k) - a^k \\right\\Vert ^2 \\le 0,$ and this concludes the proof.",
"$\\Box $ The next technical lemma will help us to save computations in other lemmas.",
"Lemma 10 If $x \\in \\Omega $ , $\\mathcal {A}$ is a subset of $\\left\\lbrace 1, \\ldots , n \\right\\rbrace $ and $j \\in \\left\\lbrace 1, \\ldots , m \\right\\rbrace $ , then: $\\left\\Vert Q(x) - a^j \\right\\Vert _{\\mathcal {A}}^2 = \\left\\Vert x - a^j \\right\\Vert _{\\mathcal {A}}^2 - \\left\\Vert Q(x) - x \\right\\Vert _{\\mathcal {A}}^2 + 2 \\left\\langle Q(x) - x, Q(x) - a^j \\right\\rangle _{\\mathcal {A}}.$ If $x \\in \\Omega $ , we have: $\\Vert Q(x) - a^j \\Vert _{\\mathcal {A}}^2 & = & \\langle Q(x) - a^j , Q(x) - a^j \\rangle _{\\mathcal {A}}\\\\& = & \\langle Q(x) - x + x - a^j , Q(x) - x + x - a^j \\rangle _{\\mathcal {A}}\\\\& = & \\Vert Q(x) - x \\Vert _{\\mathcal {A}}^2 + \\Vert x - a^j \\Vert _{\\mathcal {A}}^2 + 2 \\langle Q(x) - x , x - a^j \\rangle _{\\mathcal {A}}\\\\& = & \\Vert Q(x) - x \\Vert _{\\mathcal {A}}^2 + \\Vert x - a^j \\Vert _{\\mathcal {A}}^2 + 2 \\langle Q(x) - x , x - Q(x) \\rangle _{\\mathcal {A}}\\\\& + & 2 \\langle Q(x) - x , Q(x) - a^j \\rangle _{\\mathcal {A}}\\\\& = & \\Vert x - a^j \\Vert _{\\mathcal {A}}^2 - \\Vert Q(x) - x \\Vert _{\\mathcal {A}}^2 + 2 \\langle Q(x) - x , Q(x) - a^j \\rangle _{\\mathcal {A}}.$ $\\Box $ If $x \\in \\Omega $ , let us define $g_x : \\mathbb {R}^n \\rightarrow \\mathbb {R}$ by: $g_x(y) = \\left\\lbrace \\begin{array}{ll} \\displaystyle {\\sum _{j=1}^m} \\frac{ w_j }{ 2 \\Vert x - a^j \\Vert } \\Vert E_x(y) - a^j \\Vert ^2, & \\text{if $x \\ne a^1, \\ldots , a^m$}, \\\\ & \\\\ \\displaystyle { \\sum _{\\begin{array}{c}j=1 \\\\j \\ne k\\end{array}}^m} \\frac{ w_j }{ 2 \\Vert a^k - a^j \\Vert } \\Vert y - a^j \\Vert ^2 + w_k \\Vert y - a^k \\Vert , & \\text{if $x = a^k$}, \\\\ & \\text{$k = 1, \\ldots , m$}.",
"\\end{array} \\right.$ The values that $g_x$ assumes at $x$ and $Q(x)$ will play an important role in the proof of a property of the objective function $f$ .",
"Lemma 11 Let $x \\in \\Omega $ be.",
"If $x \\ne a^1, \\ldots , a^m$ then: $g_x(x) & = & \\frac{1}{2} f(x) + 2 A(x) \\left\\langle Q(x) - x , Q(x) - T(x) \\right\\rangle \\\\& - & A(x) \\left\\Vert Q(x) - x \\right\\Vert _{\\mathcal {N}(x)}^2.$ If $x = a^k$ for some $k = 1, \\ldots , m$ , then $g_{a^k}(a^k) = \\frac{1}{2} f(a^k)$ .",
"Let us suppose that $x \\ne a^1, \\ldots , a^m$ .",
"By property (REF ) and (REF ), we have for $j = 1, \\ldots , m$ : $\\left\\Vert E_x(x) - a^j \\right\\Vert ^2 = \\left\\Vert x - a^j \\right\\Vert _{\\mathcal {E}(x)}^2 + \\left\\Vert Q(x) - a^j \\right\\Vert _{\\mathcal {N}(x)}^2.$ Using Lemma REF , we can see that: $g_x(x) & = & \\sum _{j=1}^m \\frac{ w_j }{ 2 \\Vert x - a^j \\Vert } \\left[ \\left\\Vert x - a^j \\right\\Vert _{\\mathcal {E}(x)}^2 + \\left\\Vert x - a^j \\right\\Vert _{\\mathcal {N}(x)}^2 \\right.\\\\& - & \\left.",
"\\left\\Vert Q(x) - x \\right\\Vert _{\\mathcal {N}(x)}^2 + 2 \\left\\langle Q(x) - x , Q(x) - a^j \\right\\rangle _{\\mathcal {N}(x)} \\right].$ Due to (REF ), the definition of the Weber function $f$ , the definition of $A$ as in (REF ), the property () and the definition of $\\alpha $ as in (REF ), we obtain: $g_x(x) = \\frac{1}{2} f(x) - A(x) \\Vert Q(x) - x \\Vert _{\\mathcal {N}(x)}^2 + \\langle Q(x) - x , \\alpha (x) \\rangle _{\\mathcal {N}(x)}.$ By Lemma REF , the fact that $\\left( Q(x) \\right)_i = \\left( T(x) \\right)_i$ for all $i \\in \\mathcal {E}(x)$ and (), we get: $g_x(x) & = & \\frac{1}{2} f(x) - A(x) \\Vert Q(x) - x \\Vert _{\\mathcal {N}(x)}^2\\\\& + & 2 A(x) \\langle Q(x) - x , Q(x) - T(x) \\rangle ,$ which concludes the proof of (a).",
"Now, let us assume that $x = a^k$ for some $k = 1, \\ldots , m$ .",
"Then: $g_{a^k}(a^k) = \\sum _{\\begin{array}{c}j=1 \\\\ j \\ne k\\end{array}}^m \\frac{ w_j }{ 2 \\Vert a^k - a^j \\Vert } \\Vert a^k - a^j \\Vert ^2 = \\frac{1}{2} \\sum _{\\begin{array}{c}j=1 \\\\ j \\ne k\\end{array}}^m w_j \\Vert a^k - a^j \\Vert = \\frac{1}{2} f(a^k).$ This concludes the proof of (b).",
"$\\Box $ The number $g_x(Q(x))$ can be computed in the next lemma.",
"Lemma 12 Let $x \\in \\Omega $ be.",
"If $x \\ne a^1, \\ldots , a^m$ then: $g_x(Q(x)) & = & \\frac{1}{2} f(x) + 2 A(x) \\left\\langle Q(x) - x , Q(x) - T(x) \\right\\rangle \\\\& - & A(x) \\left\\Vert Q(x) - x \\right\\Vert ^2.$ If $x = a^k$ for some $k = 1, \\ldots , m$ , then $g_{a^k}(Q(a^k)) & = & \\frac{1}{2} f(a^k) - A(a^k) \\left\\Vert Q(a^k) - a^k \\right\\Vert ^2\\\\& + & 2 A(a^k) \\left\\langle Q(a^k) - a^k , Q(a^k) - T(a^k) \\right\\rangle \\\\& - & 2 \\beta (a^k) A(a^k) \\left\\langle Q(a^k) - a^k , \\widetilde{T}(a^k) - a^k \\right\\rangle \\\\& + & w_k \\left\\Vert Q(a^k) - a^k \\right\\Vert .$ First, let us consider $x \\ne a^1, \\ldots , a^m$ .",
"Due to Remark REF we have: $g_x(Q(x)) = \\sum _{j=1}^m \\frac{ w_j }{ 2 \\Vert x - a^j \\Vert } \\Vert Q(x) - a^j \\Vert ^2.$ By Lemma REF we obtain: $g_x(Q(x))& = & \\sum _{j=1}^m \\frac{ w_j }{ 2 \\Vert x - a^j \\Vert } \\left[ \\Vert x - a^j \\Vert ^2 - \\Vert Q(x) - x \\Vert ^2 \\right.\\\\& + & \\left.",
"2 \\langle Q(x) - x , Q(x) - a^j \\rangle \\right].$ Due to the definition of the Weber function $f$ , the definition of $A$ as in (REF ) and the definition of $\\alpha $ as in (REF ), we deduce that: $g_x(Q(x)) = \\frac{1}{2} f(x) - A(x) \\Vert Q(x) - x \\Vert ^2 + \\langle Q(x) - x , \\alpha (x) \\rangle .$ By Lemma REF we get: $g_x(Q(x)) = \\frac{1}{2} f(x) - A(x) \\Vert Q(x) - x \\Vert ^2 + 2 A(x) \\langle Q(x) - x , Q(x) - T(x) \\rangle ,$ concluding the proof of (a).",
"Now, consider $x = a^k$ for some $k = 1, \\ldots , m$ .",
"Due to (REF ) we have: $g_{a^k}(Q(a^k)) = \\sum _{\\begin{array}{c}j=1 \\\\ j \\ne k\\end{array}}^m \\frac{ w_j }{ 2 \\Vert a^k - a^j \\Vert } \\Vert Q(a^k) - a^j \\Vert ^2 + w_k \\Vert Q(a^k) - a^k \\Vert .$ By Lemma REF , the definition of the Weber function $f$ and the definition of $A$ as in (REF ) we obtain: $g_{a^k}\\left( Q(a^k) \\right) & = & \\frac{1}{2} f(a^k) - A(a^k) \\Vert Q(a^k) - a^k \\Vert ^2\\\\& + & \\left\\langle Q(a^k) - a^k , \\sum _{\\begin{array}{c}j=1 \\\\ j \\ne k\\end{array}}^m \\frac{ w_j }{ \\Vert a^k - a^j \\Vert } \\left[ Q(a^k) - a^j \\right] \\right\\rangle \\\\& + & w_k \\Vert Q(a^k) - a^k \\Vert .$ Manipulating algebraically, $Q(a^k) - a^j = Q(a^k) - ( 1 - \\beta (a^k) ) a^j - \\beta (a^k) a^k + \\beta (a^k) ( a^k - a^j ).$ Due to the definition of $\\alpha $ (see (REF )) and the definition of $\\widetilde{R}$ (see (REF )) we get: $g_{a^k} \\left( Q(a^k) \\right)& = & \\frac{1}{2} f(a^k) - A(a^k) \\left\\Vert Q(a^k) - a^k \\right\\Vert ^2\\\\& + & \\left\\langle Q(a^k) - a^k , \\alpha (a^k) \\right\\rangle - \\beta (a^k) \\left\\langle Q(a^k) - a^k , \\widetilde{R}(a^k) \\right\\rangle \\\\& + & w_k \\Vert Q(a^k) - a^k \\Vert .$ By Lemma REF and Lemma REF we have: $g_{a^k} \\left( Q(a^k) \\right) & = & \\frac{1}{2} f(a^k) - A(a^k) \\left\\Vert Q(a^k) - a^k \\right\\Vert ^2\\\\& + & 2 A(a^k) \\left\\langle Q(a^k) - a^k , Q(a^k) - T(a^k) \\right\\rangle \\\\& - & 2 A(a^k) \\beta (a^k) \\left\\langle Q(a^k) - a^k , \\widetilde{T}(a^k) - a^k \\right\\rangle + w_k \\Vert Q(a^k) - a^k \\Vert .$ which concludes the proof.",
"$\\Box $ The next lemma deals with the last two terms of $g_{a^k}(Q(a^k))$ .",
"Lemma 13 If $a^k \\in \\Omega $ for some $k = 1, \\ldots , m$ , the number $z = w_k \\left\\Vert Q(a^k) - a^k \\right\\Vert - 2 A(a^k) \\beta (a^k) \\left\\langle Q(a^k) - a^k , \\widetilde{T}(a^k) - a^k \\right\\rangle ,$ is equal to zero.",
"If $Q(a^k) = a^k$ the result is true.",
"So, from now on, let us consider that $Q(a^k) \\ne a^k$ .",
"First, let us check that $a^k \\ne x^u$ .",
"In case that $a^k = x^u$ , then $T(a^k) = a^k$ by Theorem REF .",
"Since $a^k \\in \\Omega $ , then $T(a^k) \\in \\Omega $ .",
"By Remark REF we have that $Q(a^k) = T(a^k) = a^k$ which is a contradiction.",
"By Remark REF , we have that $\\beta (a^k) \\in (0,1)$ (since $a^k \\ne x^u$ ) and: $z = w_k \\left\\Vert Q(a^k) - a^k \\right\\Vert - \\frac{ 2 A(a^k) \\beta (a^k) }{ 1 - \\beta (a^k) } \\left\\langle Q(a^k) - a^k , T(a^k) - a^k \\right\\rangle .$ Extracting common factors, using Remarks REF and REF , the fact that $T(a^k) \\ne a^k$ (if $T(a^k) = a^k$ then $a^k = x^u$ by Theorem REF ), and the fact that $\\widetilde{T}(a^k) \\ne a^k$ (if $\\widetilde{T}(a^k) = a^k$ then $T(a^k) = a^k$ by definition (REF )) we get that: $z & = & 2 A(a^k) \\left\\Vert Q(a^k) - a^k \\right\\Vert \\left\\Vert \\widetilde{T}(a^k) - a^k \\right\\Vert \\left[ \\frac{ w_k }{ 2 A(a^k) \\left\\Vert \\widetilde{T}(a^k) - a^k \\right\\Vert } \\right.\\\\& - & \\left.",
"\\beta (a^k) \\left\\langle \\frac{ ( 1 - \\lambda ( a^k ) ) ( T(a^k) - a^k ) }{ \\left\\Vert ( 1 - \\lambda ( a^k ) ) ( T(a^k) - a^k ) \\right\\Vert } , \\frac{ T(a^k) - a^k }{ \\left\\Vert T(a^k) - a^k \\right\\Vert } \\right\\rangle \\right].$ Simplifying and using the definition of $\\beta (a^k)$ we have that: $z = 2 A(a^k) \\left\\Vert Q(a^k) - a^k \\right\\Vert \\left\\Vert \\widetilde{T}(a^k) - a^k \\right\\Vert \\left[ \\beta (a^k) - \\beta (a^k) \\right] = 0,$ which concludes the proof.",
"$\\Box $ The purpose of the next two lemmas is to determine a strict inequality between the functions $g_x$ and $f$ at suitable points.",
"First of all, we have to prove the following result.",
"Lemma 14 Let $x \\in \\Omega $ be such that $x \\ne Q(x)$ .",
"If $x \\ne a^1, \\ldots , a^m$ , then $g_x(Q(x)) \\le g_x(x)$ .",
"Besides that, if $\\mathcal {E}(x) \\ne \\emptyset $ and $P_x \\circ Q(x) \\ne P_x(x)$ , then $g_x(Q(x)) < g_x(x)$ .",
"If $x = a^k$ for some $k = 1, \\ldots , m$ , then $g_{a^k}(Q(a^k)) < g_{a^k}(a^k)$ .",
"If $x \\ne a^1, \\ldots , a^m$ , then $g_x(Q(x)) - g_x(x) = - A(x) \\Vert Q(x) - x \\Vert _{\\mathcal {E}(x)}^2 \\le 0$ , by Lemma REF and Lemma REF .",
"Besides that, if $\\mathcal {E}(x) \\ne \\emptyset $ and $P_x \\circ Q(x) \\ne P_x(x)$ we deduce that $\\Vert Q(x) - x \\Vert _{\\mathcal {E}(x)} \\ne 0$ .",
"Thus, $g_x(Q(x)) < g_x(x)$ .",
"If $x = a^k$ for some $k = 1, \\ldots , m$ , by Lemmas REF , REF and REF we have: $g_{a^k}(Q(a^k)) - g_{a^k}(a^k) & = & - A(a^k) \\left\\Vert Q(a^k) - a^k \\right\\Vert ^2\\\\& + & 2 A(a^k) \\left\\langle Q(a^k) - a^k , Q(a^k) - T(a^k) \\right\\rangle .$ Due to Lemma REF and the fact that $A > 0$ we obtain: $g_{a^k}(Q(a^k)) - g_{a^k}(a^k) \\le - A(a^k) \\Vert Q(a^k) - a^k \\Vert ^2 < 0,$ and the proof is finished.",
"$\\Box $ Lemma 15 Let $x \\in \\Omega $ be such that $x \\ne Q(x)$ .",
"Then $g_x(Q(x)) < \\frac{1}{2} f(x)$ .",
"Let us consider the case when $x \\ne a^1, \\ldots , a^m$ .",
"By Lemmas REF , REF and REF we have that: $g_x(Q(x)) & \\le & g_x(x) = \\frac{1}{2} f(x) + 2 A(x) \\left\\langle Q(x) - x , Q(x) - T(x) \\right\\rangle \\\\& - & A(x) \\left\\Vert Q(x) - x \\right\\Vert _{\\mathcal {N}(x)}^2\\\\& \\le & \\frac{1}{2} f(x) - A(x) \\left\\Vert Q(x) - x \\right\\Vert _{\\mathcal {N}(x)}^2.$ If $\\mathcal {E}(x) = \\emptyset $ , then $\\Vert \\cdot \\Vert _{\\mathcal {N}(x)} = \\Vert \\cdot \\Vert $ .",
"Therefore: $g_x(Q(x)) \\le \\frac{1}{2} f(x) - A(x) \\Vert Q(x) - x \\Vert ^2 < \\frac{1}{2} f(x).$ If $\\mathcal {E}(x) \\ne \\emptyset $ and $P_x \\circ Q(x) = P_x(x)$ , then there exists an index $i \\in \\mathcal {N}(x)$ such that $x_i \\ne (Q(x))_i$ since $x \\ne Q(x)$ .",
"Thus, $\\Vert Q(x) - x \\Vert _{\\mathcal {N}(x)} \\ne 0$ , which implies: $g_x(Q(x)) \\le \\frac{1}{2} f(x) - A(x) \\left\\Vert Q(x) - x \\right\\Vert _{\\mathcal {N}(x)}^2 < \\frac{1}{2} f(x).$ If $\\mathcal {E}(x) \\ne \\emptyset $ and $P_x \\circ Q(x) \\ne P_x(x)$ , due to Lemmas REF , REF and REF , we have that: $g_x(Q(x)) < g_x(x) \\le \\frac{1}{2} f(x) - A(x) \\left\\Vert Q(x) - x \\right\\Vert _{\\mathcal {N}(x)}^2 \\le \\frac{1}{2} f(x).$ Now, when $x = a^k$ for some $k = 1, \\ldots , m$ , $g_{a^k}(Q(a^k)) < g_{a^k}(a^k) = \\frac{1}{2} f(a^k)$ due to Lemma REF and Lemma REF .",
"$\\Box $ The next lemma states an equality that relates the Weber function and $g_x$ at appropriate points when $x \\ne a^1, \\ldots , a^m$ .",
"Besides that, this result will be crucial in the next section.",
"Lemma 16 Let $x \\ne a^1, \\ldots , a^m$ be such that $x \\in \\Omega $ and $x \\ne Q(x)$ .",
"Then: $g_x \\circ Q(x) = \\frac{1}{2} f(x) + \\left( f(Q(x)) - f(x) \\right) + \\delta , \\quad \\delta \\ge 0.$ Due to the definition of $g_x$ as in (REF ) and Remark REF we get that: $g_x \\circ Q(x) = \\sum _{j=1}^m \\frac{ w_j }{ 2 \\Vert x - a^j \\Vert } \\Vert Q(x) - a^j \\Vert ^2.$ Adding and subtracting $\\Vert x - a^j \\Vert $ we have: $g_x \\circ Q(x) & = & \\sum _{j=1}^m \\frac{ w_j }{ 2 \\Vert x - a^j \\Vert } \\left[ \\left\\Vert x - a^j \\right\\Vert + \\left( \\left\\Vert Q(x) - a^j \\right\\Vert - \\left\\Vert x - a^j \\right\\Vert \\right) \\right]^2\\\\& = & \\frac{1}{2} \\sum _{j=1}^m w_j \\left\\Vert x - a^j \\right\\Vert + \\sum _{j=1}^m w_j \\left( \\left\\Vert Q(x) - a^j \\right\\Vert - \\left\\Vert x - a^j \\right\\Vert \\right)\\\\& + & \\sum _{j=1}^m \\frac{ w_j }{ 2 \\Vert x - a^j \\Vert } \\left( \\left\\Vert Q(x) - a^j \\right\\Vert - \\left\\Vert x - a^j \\right\\Vert \\right)^2.$ Notice that the first term of the last equality is the Weber function (divided by two), and the last term is a non-negative number, so we will define it as $\\delta $ .",
"So, using the definition of the Weber function in the middle term we obtain: $g_x \\circ Q(x) = \\frac{1}{2} f(x) + \\left( f(Q(x)) - f(x) \\right) + \\delta .$ $\\Box $ Convergence to optimality results This section states the main results about convergence of the sequence $\\left\\lbrace x^{(l)} \\right\\rbrace $ generated by Algorithm REF .",
"The next theorem establishes that if a point $x \\in \\Omega $ is not a fixed point of the iteration function, then the function $f$ strictly decreases at the next iterate.",
"Theorem 17 Let $x \\in \\Omega $ be such that $x \\ne Q(x)$ .",
"Then $f(Q(x)) < f(x)$ .",
"Let us consider that $x \\ne a^1, \\ldots , a^m$ .",
"By Lemma REF , we have that: $g_x \\circ Q(x) < \\frac{1}{2} f(x).$ By Lemma REF we get that: $\\frac{1}{2} f(x) + f(Q(x)) - f(x) + \\delta < \\frac{1}{2} f(x).$ Simplifying the last expression we obtain: $f(Q(x)) - f(x) + \\delta < 0.$ Finally, $f(Q(x)) - f(x) \\le f(Q(x)) - f(x) + \\delta < 0.$ Therefore, $f(Q(x)) < f(x)$ .",
"Now, consider that $x = a^k$ for some $k = 1, \\ldots , m$ .",
"Following a reasoning similar than in [40], using Lemma REF we have that: $g_{a^k} \\circ Q(a^k) - g_{a^k}(a^k) < 0.$ By definition of $g_{a^k}$ we know that: $g_{a^k} \\circ Q(a^k) - g_{a^k}(a^k) & = & w_k \\left\\Vert Q(a^k) - a^k \\right\\Vert \\\\& + & \\sum _{\\begin{array}{c}j=1 \\\\ j \\ne k\\end{array}}^m \\frac{ w_j }{ 2 \\left\\Vert a^k - a^j \\right\\Vert } \\left( \\left\\Vert Q(a^k) - a^j \\right\\Vert ^2 - \\left\\Vert a^k - a^j \\right\\Vert ^2 \\right).$ Using the fact that $\\left( a^2 - b^2 \\right)/ (2b) \\ge a - b$ for $a = \\left\\Vert Q(a^k) - a^j \\right\\Vert ^2 \\ge 0$ and $b = \\left\\Vert a^k - a^j \\right\\Vert ^2 > 0$ we obtain that: $g_{a^k} \\circ Q(a^k) - g_{a^k}(a^k) & \\ge & w_k \\left\\Vert Q(a^k) - a^k \\right\\Vert \\\\& - & \\sum _{\\begin{array}{c}j=1 \\\\ j \\ne k\\end{array}}^m w_j \\left\\Vert a^k - a^j \\right\\Vert + \\sum _{\\begin{array}{c}j=1 \\\\ j \\ne k\\end{array}}^m w_j \\left\\Vert Q(a^k) - a^j \\right\\Vert .$ Rearranging terms we deduce that: $0 & > & g_{a^k} \\circ Q(a^k) - g_{a^k}(a^k)\\\\& = & \\sum _{j=1}^m w_j \\left\\Vert Q(a^k) - a^j \\right\\Vert - \\sum _{j=1}^m w_j \\left\\Vert a^k - a^j \\right\\Vert = f(Q(a^k)) - f(a^k),$ and the proof is complete.",
"$\\Box $ Corollary 18 Let $\\left\\lbrace x^{(l)} \\right\\rbrace $ be the sequence generated by Algorithm REF .",
"Then the sequence $\\left\\lbrace f \\left( x^{(l)} \\right) \\right\\rbrace $ is not increasing.",
"Even more, each time $x^{(l)} \\ne Q \\left( x^{(l)} \\right)$ the sequence strictly decreases at the next iterate.",
"If the sequence $\\left\\lbrace x^{(l)} \\right\\rbrace $ generated by Algorithm REF were not bounded, then we could choose a subsequence $\\left\\lbrace y^{(l)} \\right\\rbrace $ such that $y^{(l)} \\rightarrow \\infty $ .",
"But this implies that $f(y^{(l)}) \\rightarrow \\infty $ , which is a contradiction since the sequence $\\left\\lbrace f \\left( x^{(l)} \\right) \\right\\rbrace $ is not increasing.",
"Remark 19 The sequence $\\left\\lbrace x^{(l)} \\right\\rbrace $ generated by Algorithm REF is bounded.",
"So, there exists a subsequence convergent to a point $x^* \\in \\Omega $ .",
"Hence, $x^*$ is a feasible point.",
"Due to the nondifferentiability of $f$ at the vertices $a^1, \\ldots , a^m$ , we can not use the KKT optimality conditions at $a^k$ .",
"Therefore, if $a^k$ and $z$ are in $\\Omega $ , let us define $G_{a^k}^z : [0,1] \\rightarrow \\mathbb {R}$ by: $G_{a^k}^z(t) = f( a^k + t(z-a^k) ).$ If $a^k \\in \\Omega $ , $z \\in \\Omega $ , $t \\in [0,1]$ and $\\Omega $ convex, we have that $a^k + t(z-a^k) \\in \\Omega $ .",
"Notice that the right-hand side derivative $G_{a^k}^z(0+)$ (or the directional derivative of $f$ in the direction of $z$ ) exists (see [22]).",
"Besides that, $G_{a^k}^{z \\ \\prime }(0+) = w_k \\left\\Vert z - a^k \\right\\Vert - \\left\\langle \\widetilde{R}(a^k) , z - a^k \\right\\rangle .$ The next lemma shows that if we are in a vertex $a^k$ , the directional derivative of $f$ at $a^k$ in the direction of $Q(a^k)$ is a descent direction.",
"Lemma 20 Let $a^k \\in \\Omega $ be such that $T(a^k) \\notin \\Omega $ .",
"Then: $G_{a^k}^{z \\ \\prime }(0+) \\ge G_{a^k}^{Q(a^k) \\ \\prime }(0+), \\quad \\forall \\, z \\in \\left[ a^k, T(a^k) \\right],$ where: $G_{a^k}^{Q(a^k) \\ \\prime }(0+) = - 2 \\left[ 1 - \\beta (a^k) \\right] A(a^k) \\left\\Vert \\widetilde{T}(a^k) - a^k \\right\\Vert \\left\\Vert Q(a^k) - a^k \\right\\Vert .$ If $T(a^k) = a^k$ then $T(a^k) \\in \\Omega $ , which is a contradiction.",
"Besides that, if $\\widetilde{T}(a^k) = a^k$ , we would have that $T(a^k) = a^k$ because of (REF ), and again it would be a contradiction.",
"So, we will consider $\\widetilde{T}(a^k) \\ne a^k$ and $T(a^k) \\ne a^k$ for the rest of the proof.",
"Since $T(a^k) \\ne a^k$ , then $\\beta (a^k) \\in (0,1)$ (see Remark REF and Theorem REF ).",
"Let us prove equation (REF ) first.",
"Now, by (REF ) we can see that: $G_{a^k}^{Q(a^k) \\ \\prime }(0+) = w_k \\left\\Vert Q(a^k) - a^k \\right\\Vert - \\left\\langle \\widetilde{R}(a^k) , Q(a^k) - a^k \\right\\rangle .$ Notice that if $Q(a^k) = a^k$ , equation (REF ) holds.",
"So, let us consider from now on that $Q(a^k) \\ne a^k$ .",
"By using Lemma REF we replace $\\widetilde{R}(a^k)$ and get: $G_{a^k}^{Q(a^k) \\ \\prime }(0+) = w_k \\left\\Vert Q(a^k) - a^k \\right\\Vert - 2 A(a^k) \\left\\langle \\widetilde{T}(a^k) - a^k , Q(a^k) - a^k \\right\\rangle .$ Extracting common factors and using the definition of $\\beta $ when it belongs to $(0,1)$ we obtain: $G_{a^k}^{Q(a^k) \\ \\prime }(0+) & = & 2 A(a^k) \\left\\Vert Q(a^k) - a^k \\right\\Vert \\left\\Vert \\widetilde{T}(a^k) - a^k \\right\\Vert \\Bigg [ \\beta (a^k)\\\\& - & \\left.",
"\\left\\langle \\frac{ \\widetilde{T}(a^k) - a^k }{ \\left\\Vert \\widetilde{T}(a^k) - a^k \\right\\Vert } , \\frac{ Q(a^k) - a^k }{ \\left\\Vert Q(a^k) - a^k \\right\\Vert } \\right\\rangle \\right].$ By Remarks REF and REF the vectors $Q(a^k) - a^k$ and $\\widetilde{T}(a^k) - a^k$ are parallel, so: $G_{a^k}^{Q(a^k) \\ \\prime }(0+) = 2 A(a^k) \\left\\Vert Q(a^k) - a^k \\right\\Vert \\left\\Vert \\widetilde{T}(a^k) - a^k \\right\\Vert \\left[ \\beta (a^k) - 1 \\right].$ which is equivalent to (REF ).",
"Now, let us prove (REF ).",
"If $z = a^k$ then $G_{a^k}^z(t) = f(a^k)$ for all $t \\in [0,1]$ , thus $G_{a^k+}^{a^k \\ \\prime }(0) = 0$ , and therefore the inequality (REF ) holds.",
"So, let us assume that $z \\ne a^k$ for the rest of the proof.",
"Using (REF ) and due to Lemma REF to replace $\\widetilde{R}(a^k)$ : $G_{a^k}^{z \\ \\prime }(0+) = w_k \\left\\Vert z - a^k \\right\\Vert - 2 A(a^k) \\left\\langle \\widetilde{T}(a^k) - a^k , z - a^k \\right\\rangle .$ Extracting common factors: $G_{a^k}^{z \\ \\prime }(0+) & = & 2 A(a^k) \\left\\Vert z - a^k \\right\\Vert \\left\\Vert \\widetilde{T}(a^k) - a^k \\right\\Vert \\left[ \\frac{ w_k }{ 2 A(a^k) \\left\\Vert \\widetilde{T}(a^k) - a^k \\right\\Vert } \\right.\\\\& - & \\left.",
"\\left\\langle \\frac{ \\widetilde{T}(a^k) - a^k }{ \\left\\Vert \\widetilde{T}(a^k) - a^k \\right\\Vert } , \\frac{ z - a^k }{ \\left\\Vert z - a^k \\right\\Vert } \\right\\rangle \\right].$ Using the expression for $\\beta (a^k) \\in (0,1)$ we obtain: $G_{a^k}^{z \\ \\prime }(0+) & = & 2 A(a^k) \\left\\Vert z - a^k \\right\\Vert \\left\\Vert \\widetilde{T}(a^k) - a^k \\right\\Vert \\Bigg [ \\beta (a^k)\\\\& - & \\left.",
"\\left\\langle \\frac{ \\widetilde{T}(a^k) - a^k }{ \\left\\Vert \\widetilde{T}(a^k) - a^k \\right\\Vert } , \\frac{ z - a^k }{ \\left\\Vert z - a^k \\right\\Vert } \\right\\rangle \\right].$ If $z$ belongs to the segment that joins $a^k$ and $T(a^k)$ we have that $z-a^k$ and $\\widetilde{T}(a^k) - a^k$ are parallel vectors, then: $G_{a^k}^{z \\ \\prime }(0+) \\ge - 2 \\left[ 1 - \\beta (a^k) \\right] A(a^k) \\left\\Vert z - a^k \\right\\Vert \\left\\Vert \\widetilde{T}(a^k) - a^k \\right\\Vert .$ We can write $z = (1-\\lambda )T(a^k) + \\lambda a^k$ where $\\lambda \\in [0,1]$ .",
"Therefore: $G_{a^k}^{z \\ \\prime }(0+) \\ge - 2 \\left[ 1 - \\beta (a^k) \\right] A(a^k) \\left( 1 - \\lambda \\right) \\left\\Vert T(a^k) - a^k \\right\\Vert \\left\\Vert \\widetilde{T}(a^k) - a^k \\right\\Vert .$ for all $\\lambda \\in [0,1]$ .",
"The minimum value of the right-hand side of the last expression happens when $\\lambda = \\lambda ( a^k )$ , so: $G_{a^k}^{z \\ \\prime }(0+) \\ge - 2 \\left[ 1 - \\beta (a^k) \\right] A(a^k) \\left( 1 - \\lambda ( a^k ) \\right) \\left\\Vert T(a^k) - a^k \\right\\Vert \\left\\Vert \\widetilde{T}(a^k) - a^k \\right\\Vert .$ Using Remark REF we conclude that: $G_{a^k}^{z \\ \\prime }(0+) \\ge - 2 \\left[ 1 - \\beta (a^k) \\right] A(a^k) \\left\\Vert Q(a^k) - a^k \\right\\Vert \\left\\Vert \\widetilde{T}(a^k) - a^k \\right\\Vert .$ $\\Box $ Now we will prove an equivalence that characterizes the solution of (REF ) in terms of the iteration function $Q$ .",
"Moreover, if $x^*$ is a regular point that is not a vertex, then $x^*$ is a KKT point.",
"From now on, let us consider that $\\Omega = \\left\\lbrace y \\in \\mathbb {R}^n : g(y) \\le 0, h(y) = 0 \\right\\rbrace ,$ where $g : \\mathbb {R}^n \\rightarrow \\mathbb {R}^s$ is a convex function and $h : \\mathbb {R}^n \\rightarrow \\mathbb {R}^p$ is an affine function.",
"Theorem 21 Let $\\Omega $ be defined as in (REF ) and $x \\in \\Omega $ .",
"Consider the following propositions: $x$ is a KKT point.",
"$x$ is the minimizer of the problem (REF ).",
"$Q(x) = x$ .",
"If $x \\ne a^1, \\ldots , a^m$ , $g$ and $h$ are continuously differentiable, and $x$ is a regular point, then (a), (b) and (c) are equivalent.",
"If $x = a^k$ for some $k = 1, \\ldots , m$ , then (b) implies (c).",
"Let $x \\ne a^1, \\ldots , a^m$ be.",
"Since $f$ is strictly convex and $\\Omega $ is convex, the KKT optimality conditions are necessary and sufficient.",
"Therefore, it holds that (a) is equivalent to (b).",
"Now we will prove that (b) implies (c).",
"Let us suppose that $x$ is the minimizer of the problem (REF ).",
"If $x$ were not a fixed point of the iteration function $Q$ , we would have that $x \\ne Q(x)$ , which means that $f(Q(x)) < f(x)$ by Theorem REF .",
"This contradicts the hypothesis.",
"To demonstrate that (c) implies (a), we will assume that $x$ is a fixed point of $Q$ , that is, $x = Q(x)$ .",
"Since $Q(x) = P_{\\Omega } \\circ T(x)$ , $x$ is the solution of: $\\begin{array}{rl}\\displaystyle \\mathop { \\mathrm {argmin} }_z & \\displaystyle F(z) = \\frac{1}{2} \\left\\Vert z - T(x) \\right\\Vert ^2 \\\\ \\mathrm {subject \\, to} & g(z) \\le 0, \\\\ & h(z) = 0.\\end{array}$ Since $F$ and $g$ are convex, $h$ is affine, and $x$ is a regular point, the KKT optimality conditions hold at $x$ .",
"That is, there exist multipliers $\\left\\lbrace \\mu _j \\right\\rbrace _{j=1}^s$ and $\\left\\lbrace \\lambda _j \\right\\rbrace _{j=1}^p$ such that (see [31], [38]): $x - T(x) + \\sum _{j=1}^s \\mu _j \\nabla g_j(x) + \\sum _{j=1}^p \\lambda _j \\nabla h_j(x) & = & 0,\\\\\\mu _j g_j( x ) & = & 0, \\quad j = 1, \\ldots , s,\\\\\\mu _j & \\ge & 0, \\quad j = 1, \\ldots , s,\\\\g(x) & \\le & 0,\\\\h(x) & = & 0.$ Multiplying these equations by $2 A(x)$ , using equation (REF ), Lemma REF and Remark REF , we obtain: $\\nabla f(x) + \\sum _{j=1}^s \\left( 2 A(x) \\mu _j \\right) \\nabla g_j(x) + \\sum _{j=1}^p \\left( 2 A(x) \\lambda _j \\right) \\nabla h_j(x) & = & 0,\\\\\\left( 2 A(x) \\mu _j \\right) g_j( x ) & = & 0, \\quad j = 1, \\ldots , s,\\\\\\left( 2 A(x) \\mu _j \\right) & \\ge & 0, \\quad j = 1, \\ldots , s,\\\\g(x) & \\le & 0,\\\\h(x) & = & 0.$ where $\\left\\lbrace 2 A(x) \\mu _j \\right\\rbrace _{j=1}^s$ and $\\left\\lbrace 2 A(x) \\lambda _j \\right\\rbrace _{j=1}^p$ are multipliers.",
"Therefore, $x$ is a KKT point of the problem (REF ) (see [31], [38]).",
"Now, let us suppose that $x = a^k$ for some $k = 1, \\ldots , m$ .",
"As before, if $x$ is a minimizer of the problem (REF ), then $Q(a^k) = a^k$ , otherwise $f(Q(a^k)) < f(a^k)$ , which would be a contradiction.",
"$\\Box $ Numerical experiments.",
"The purpose of this section is to discuss the efficiency and robustness of the proposed algorithm versus a solver for nonlinear programming problems.",
"A prototype code of Algorithm REF was programmed in MATLAB (version R2011a) and executed in a PC running Linux OS, Intel(R) Core(TM) i7 CPU Q720, 1.60GHz.",
"We have considered a closed and convex set $\\Omega \\subset \\mathbb {R}^2$ defined by the set $\\Omega = \\left\\lbrace y \\in \\mathbb {R}^n : g(y) \\le 0 \\right\\rbrace $ , where $g$ is given by: $g( x ) = \\left[ \\begin{array}{c} \\displaystyle - 4 - \\frac{ 1 }{ 8 } x + \\frac{ 7 }{ 72 } x^2 + \\frac{ 1 }{ 216 } x^2 ( x - 3 ) + y \\\\[3mm] \\displaystyle \\frac{ 4 }{ 5 } x + y - \\frac{ 59 }{ 10 } \\\\[3mm] \\displaystyle x - \\frac{ 11 }{ 2 } \\\\[3mm] \\displaystyle \\frac{ 3 }{ 2 } x - y - \\frac{ 35 }{ 4 } \\\\[3mm] \\displaystyle x - y - \\frac{ 13 }{ 2 } \\\\[3mm] \\displaystyle - 4 + \\frac{ 1 }{ 8 } ( x - 1 ) + \\frac{ 1 }{ 16 } ( x - 1 )^2 + \\frac{ 1 }{ 32 } ( x - 1 )^2 ( x - 3 ) - y \\\\[3mm] \\displaystyle - \\frac{ 1 }{ 3 } x - y - \\frac{ 11 }{ 3 } \\\\[3mm] \\displaystyle - \\frac{ 2 }{ 3 } x - y - \\frac{ 13 }{ 3 } \\\\[3mm] \\displaystyle - 4 x + y - 19 \\end{array} \\right].$ The feasible set is defined by linear and nonlinear constraints, as it can be seen in Figure REF .",
"Figure: Feasible set Ω\\Omega We have built 1000 different experiments where for each one: The number of vertices was $m = 50$ .",
"The vertices were normally distributed random vectors, with mean equal to 0 and standard deviation equal to 10.",
"The weights were uniformly distributed random positive numbers between 0 and 10.",
"Tolerance was set to $\\varepsilon = 0.00001$ .",
"On one hand, each experiment was solved using Algorithm REF and, on the other hand, it was considered as a nonlinear programming problem and solved using function $fmincon$ (see [32] and references therein).",
"Since the Weber function (REF ) is not differentiable at the vertices, nonlinear programming solvers may fail.",
"Let $x_m(i)$ be the solution of (REF ) obtained by $fmincon$ in experiment $i$ , and $f_{m}(i) = f(x_m(i))$ .",
"Analogously, let $x_p(i)$ be the solution of (REF ) obtained by Algorithm REF in experiment $i$ , and $f_{p}(i) = f(x_p(i))$ .",
"Figure REF shows the difference between the arrays $f_{m}$ and $f_{p}$ .",
"Both methods finished succesfully in all cases, however, Algorithm REF found equal or better results for all experiments.",
"For example, the difference $f_{m} - f_{p}$ was greater than $0.01$ in 35 experiments (the maximum difference ocurred in experiment 506).",
"Figure: Difference between minimum values found by Algorithm and fminconfmincon.Feasibility of the solutions $x_p(i)$ can be checked computing $\\max (g(x_m(i)))$ .",
"Results can be seen in Figure REF Figure: Feasibility of the solution x p (i)x_p(i) obtained by Algorithm .",
"Conclusions This paper proposes a Weiszfeld-like algorithm for solving the Weber problem constrained to a closed and convex set, and it is well defined even when an iterate is a vertex.",
"The algorithm consists of two stages: first, iterate using the fixed point modified Weiszfeld iteration (REF ), and second, either project onto the set $\\Omega $ when the iterate is different from the vertices, or, if the iterate is a vertex $a^k$ , take the point belonging to the line that joins $T(a^k)$ with $a^k$ as defined in (REF ).",
"It is proved that the constrained problem (REF ) has a unique solution.",
"Besides that, the definition of the iteration function $Q$ allows us to demonstrate that the proposed algorithm produces a sequence $\\left\\lbrace x^{(l)} \\right\\rbrace $ of feasible iterates.",
"Moreover, the sequence $\\left\\lbrace f \\left( x^{(l)} \\right) \\right\\rbrace $ is not increasing, and when $x^{(l)} \\ne Q\\left( x^{(l)} \\right)$ , the sequence decreases at the next iterate.",
"It can be seen that if a point $x^*$ is the solution of the problem (REF ) then $x^*$ is a fixed point of the iteration function $Q$ .",
"Even more, if $x^*$ is different from the vertices, the fact of being $x^*$ a fixed point of $Q$ is equivalent to the fact that $x^*$ satisfies the KKT optimality conditions, and equivalent to the fact that $x^*$ is the solution of the problem (REF ).",
"These properties allows us to connect the proposed algorithm with the minimization problem.",
"Numerical experiments showed that the proposed algorithm found equal or better solutions than a well-known standard solver, in a practical example with 1000 random choices of vertices and weights.",
"That is due to the fact that the proposed algorithm does not use of the existence of derivatives at the vertices, because the Weber function is not differentiable at the vertices."
],
[
"Some definitions and technical results",
"The purpose of this section is to define some entities and prove technical lemmas that will be important in the proof of the main results.",
"First of all, we will define some useful operators for making notation easier.",
"If $\\mathcal {A} \\subset \\left\\lbrace 1, \\ldots , n \\right\\rbrace $ , then we define $\\Vert \\cdot \\Vert _{\\mathcal {A}} : \\mathbb {R}^n \\rightarrow \\mathbb {R}$ and $\\langle \\cdot , \\cdot \\rangle _{\\mathcal {A}} : \\mathbb {R}^n \\times \\mathbb {R}^n \\rightarrow \\mathbb {R}$ by: $\\Vert x \\Vert _{\\mathcal {A}} = \\sqrt{ \\sum _{j \\in \\mathcal {A}} x_j^2 }, \\qquad \\langle x , y \\rangle _{\\mathcal {A}} = \\sum _{j \\in \\mathcal {A}} x_j y_j.$ Notice that, when $\\mathcal {A} \\subsetneq \\left\\lbrace 1, \\ldots , n \\right\\rbrace $ , $\\Vert \\cdot \\Vert _{\\mathcal {A}}$ is not necessarily a norm and $\\langle \\cdot , \\cdot \\rangle _{\\mathcal {A}}$ is not necessarily an inner product.",
"According to this definition, if $\\mathcal {A}$ and $\\mathcal {B}$ are sets such that $\\mathcal {A} \\cap \\mathcal {B} = \\emptyset $ and $\\mathcal {A} \\cup \\mathcal {B} = \\left\\lbrace 1, \\ldots , n \\right\\rbrace $ , it can be seen that: $\\Vert x \\Vert ^2 & = & \\Vert x \\Vert _{\\mathcal {A}}^2 + \\Vert x \\Vert _{\\mathcal {B}}^2,\\\\\\langle x , y \\rangle & = & \\langle x , y \\rangle _{\\mathcal {A}} + \\langle x , y \\rangle _{\\mathcal {B}},\\\\c \\langle x , y \\rangle _{\\mathcal {A}} & = & \\langle cx , y \\rangle _{\\mathcal {A}} = \\langle x , cy \\rangle _{\\mathcal {A}}.$ For $x \\in \\Omega $ , let us define the following sets of indices: $\\mathcal {N}(x) & = & \\left\\lbrace k \\in \\mathbb {N} : \\quad 1 \\le k \\le n, \\quad (T(x))_k \\ne (Q(x))_k \\right\\rbrace ,\\\\\\mathcal {E}(x) & = & \\left\\lbrace k \\in \\mathbb {N} : \\quad 1 \\le k \\le n, \\quad (T(x))_k = (Q(x))_k \\right\\rbrace ,$ Notice that for all $x \\in \\mathbb {R}^n$ we have that $\\mathcal {N}(x) \\cap \\mathcal {E}(x) = \\emptyset $ and $\\mathcal {N}(x) \\cup \\mathcal {E}(x) = \\left\\lbrace 1, \\ldots , n \\right\\rbrace $ .",
"Let $\\alpha : \\Omega \\rightarrow \\mathbb {R}^n$ be the following function: If $x \\ne a^1, \\ldots , a^m$ : $\\alpha (x) = \\displaystyle \\sum _{j=1}^m \\frac{ w_j }{ \\Vert x - a^j \\Vert } \\left[ Q(x) - a^j \\right].$ If $x = a^k \\in \\Omega $ for some $k = 1, \\ldots , m$ : $\\alpha (x) = \\displaystyle \\sum _{\\begin{array}{c}j=1 \\\\ j \\ne k\\end{array}}^m \\frac{ w_j }{ \\Vert a^k - a^j \\Vert } \\left[ Q(a^k) - (1 - \\beta (a^k)) a^j - \\beta (a^k) a^k \\right].$ It can be seen that the function $\\alpha $ is related to the iteration function $Q$ of the proposed algorithm, and the iteration function $T$ of the modified algorithm.",
"Lemma 7 If $x \\in \\Omega $ , then $\\alpha (x) = 2 A(x) \\left[ Q(x) - T(x) \\right].$ If $x \\ne a^1, \\ldots , a^m$ , then: $\\alpha (x) & = & \\sum _{j=1}^m \\frac{ w_j }{ \\Vert x - a^j \\Vert } \\left[ Q(x) - a^j \\right] = \\sum _{j=1}^m \\frac{ w_j Q(x) }{ \\Vert x - a^j \\Vert } - \\sum _{j=1}^m \\frac{ w_j a^j }{ \\Vert x - a^j \\Vert }\\\\&&\\\\& = & \\left( \\sum _{j=1}^m \\frac{ w_j }{ \\Vert x - a^j \\Vert } \\right) \\left[ Q(x) - \\frac{ \\sum _{j=1}^m \\frac{ w_j a^j }{ \\Vert x - a^j \\Vert } }{ \\sum _{j=1}^m \\frac{ w_j }{ \\Vert x - a^j \\Vert } } \\right] = 2 A(x) \\left[ Q(x) - \\widetilde{T}(x) \\right]\\\\&&\\\\& = & 2 A(x) \\left[ Q(x) - T(x) \\right].$ where in the last equalities we have used the definition of $\\widetilde{T}$ as in (REF ), and the fact that $\\widetilde{T}(x) = T(x)$ due to Remark REF .",
"If $x = a^k$ for some $k = 1, \\ldots , m$ , we follow a similar procedure than in the previous case.",
"$\\Box $ Now, we will define auxiliary functions that take into account the projection $P_{\\Omega }$ in order to prove a descent property of $f$ (see next sections).",
"If $x \\in \\Omega $ , we define: $E_x : \\mathbb {R}^n \\rightarrow \\mathbb {R}^n$ , where: $\\left( E_x(y) \\right)_k = \\left\\lbrace \\begin{array}{ll} (Q(x))_k, & \\quad \\text{if $k \\in \\mathcal {N}(x)$}, \\\\ y_k, & \\quad \\text{if $k \\in \\mathcal {E}(x)$}.",
"\\end{array} \\right.$ If $\\mathcal {E}(x) = \\left\\lbrace i_1, \\ldots , i_r \\right\\rbrace \\ne \\emptyset $ define $P_x : \\mathbb {R}^n \\rightarrow \\mathbb {R}^r$ where: $\\left( P_x(y) \\right)_k = y_{i_k}, \\quad k = 1, \\ldots , r.$ A useful property of $E_x$ , that follows from the definition, is pointed out in the following remark.",
"Remark 8 If $x \\in \\Omega $ then $E_x \\circ Q(x) = Q(x)$ .",
"The iteration function $Q$ inherits an important property from the orthogonal projection $P_{\\Omega }$ .",
"Lemma 9 If $x \\in \\Omega $ we have that $\\left\\langle Q(x) - x , Q(x) - T(x) \\right\\rangle \\le 0$ .",
"If $x \\ne a^1, \\ldots , a^m$ , then $Q(x) = P_{\\Omega } \\circ T(x)$ .",
"By a property of the orthogonal projection [1] we have that $\\langle Q(x) - x , Q(x) - T(x) \\rangle \\le 0$ .",
"If $x = a^k$ for some $k = 1, \\ldots , m$ , Remark REF and Lemma REF imply: $\\left\\langle Q(a^k) - a^k , Q(a^k) - T(a^k) \\right\\rangle = - \\lambda ( a^k ) \\left( 1 - \\lambda ( a^k ) \\right) \\left\\Vert T(a^k) - a^k \\right\\Vert ^2 \\le 0,$ and this concludes the proof.",
"$\\Box $ The next technical lemma will help us to save computations in other lemmas.",
"Lemma 10 If $x \\in \\Omega $ , $\\mathcal {A}$ is a subset of $\\left\\lbrace 1, \\ldots , n \\right\\rbrace $ and $j \\in \\left\\lbrace 1, \\ldots , m \\right\\rbrace $ , then: $\\left\\Vert Q(x) - a^j \\right\\Vert _{\\mathcal {A}}^2 = \\left\\Vert x - a^j \\right\\Vert _{\\mathcal {A}}^2 - \\left\\Vert Q(x) - x \\right\\Vert _{\\mathcal {A}}^2 + 2 \\left\\langle Q(x) - x, Q(x) - a^j \\right\\rangle _{\\mathcal {A}}.$ If $x \\in \\Omega $ , we have: $\\Vert Q(x) - a^j \\Vert _{\\mathcal {A}}^2 & = & \\langle Q(x) - a^j , Q(x) - a^j \\rangle _{\\mathcal {A}}\\\\& = & \\langle Q(x) - x + x - a^j , Q(x) - x + x - a^j \\rangle _{\\mathcal {A}}\\\\& = & \\Vert Q(x) - x \\Vert _{\\mathcal {A}}^2 + \\Vert x - a^j \\Vert _{\\mathcal {A}}^2 + 2 \\langle Q(x) - x , x - a^j \\rangle _{\\mathcal {A}}\\\\& = & \\Vert Q(x) - x \\Vert _{\\mathcal {A}}^2 + \\Vert x - a^j \\Vert _{\\mathcal {A}}^2 + 2 \\langle Q(x) - x , x - Q(x) \\rangle _{\\mathcal {A}}\\\\& + & 2 \\langle Q(x) - x , Q(x) - a^j \\rangle _{\\mathcal {A}}\\\\& = & \\Vert x - a^j \\Vert _{\\mathcal {A}}^2 - \\Vert Q(x) - x \\Vert _{\\mathcal {A}}^2 + 2 \\langle Q(x) - x , Q(x) - a^j \\rangle _{\\mathcal {A}}.$ $\\Box $ If $x \\in \\Omega $ , let us define $g_x : \\mathbb {R}^n \\rightarrow \\mathbb {R}$ by: $g_x(y) = \\left\\lbrace \\begin{array}{ll} \\displaystyle {\\sum _{j=1}^m} \\frac{ w_j }{ 2 \\Vert x - a^j \\Vert } \\Vert E_x(y) - a^j \\Vert ^2, & \\text{if $x \\ne a^1, \\ldots , a^m$}, \\\\ & \\\\ \\displaystyle { \\sum _{\\begin{array}{c}j=1 \\\\j \\ne k\\end{array}}^m} \\frac{ w_j }{ 2 \\Vert a^k - a^j \\Vert } \\Vert y - a^j \\Vert ^2 + w_k \\Vert y - a^k \\Vert , & \\text{if $x = a^k$}, \\\\ & \\text{$k = 1, \\ldots , m$}.",
"\\end{array} \\right.$ The values that $g_x$ assumes at $x$ and $Q(x)$ will play an important role in the proof of a property of the objective function $f$ .",
"Lemma 11 Let $x \\in \\Omega $ be.",
"If $x \\ne a^1, \\ldots , a^m$ then: $g_x(x) & = & \\frac{1}{2} f(x) + 2 A(x) \\left\\langle Q(x) - x , Q(x) - T(x) \\right\\rangle \\\\& - & A(x) \\left\\Vert Q(x) - x \\right\\Vert _{\\mathcal {N}(x)}^2.$ If $x = a^k$ for some $k = 1, \\ldots , m$ , then $g_{a^k}(a^k) = \\frac{1}{2} f(a^k)$ .",
"Let us suppose that $x \\ne a^1, \\ldots , a^m$ .",
"By property (REF ) and (REF ), we have for $j = 1, \\ldots , m$ : $\\left\\Vert E_x(x) - a^j \\right\\Vert ^2 = \\left\\Vert x - a^j \\right\\Vert _{\\mathcal {E}(x)}^2 + \\left\\Vert Q(x) - a^j \\right\\Vert _{\\mathcal {N}(x)}^2.$ Using Lemma REF , we can see that: $g_x(x) & = & \\sum _{j=1}^m \\frac{ w_j }{ 2 \\Vert x - a^j \\Vert } \\left[ \\left\\Vert x - a^j \\right\\Vert _{\\mathcal {E}(x)}^2 + \\left\\Vert x - a^j \\right\\Vert _{\\mathcal {N}(x)}^2 \\right.\\\\& - & \\left.",
"\\left\\Vert Q(x) - x \\right\\Vert _{\\mathcal {N}(x)}^2 + 2 \\left\\langle Q(x) - x , Q(x) - a^j \\right\\rangle _{\\mathcal {N}(x)} \\right].$ Due to (REF ), the definition of the Weber function $f$ , the definition of $A$ as in (REF ), the property () and the definition of $\\alpha $ as in (REF ), we obtain: $g_x(x) = \\frac{1}{2} f(x) - A(x) \\Vert Q(x) - x \\Vert _{\\mathcal {N}(x)}^2 + \\langle Q(x) - x , \\alpha (x) \\rangle _{\\mathcal {N}(x)}.$ By Lemma REF , the fact that $\\left( Q(x) \\right)_i = \\left( T(x) \\right)_i$ for all $i \\in \\mathcal {E}(x)$ and (), we get: $g_x(x) & = & \\frac{1}{2} f(x) - A(x) \\Vert Q(x) - x \\Vert _{\\mathcal {N}(x)}^2\\\\& + & 2 A(x) \\langle Q(x) - x , Q(x) - T(x) \\rangle ,$ which concludes the proof of (a).",
"Now, let us assume that $x = a^k$ for some $k = 1, \\ldots , m$ .",
"Then: $g_{a^k}(a^k) = \\sum _{\\begin{array}{c}j=1 \\\\ j \\ne k\\end{array}}^m \\frac{ w_j }{ 2 \\Vert a^k - a^j \\Vert } \\Vert a^k - a^j \\Vert ^2 = \\frac{1}{2} \\sum _{\\begin{array}{c}j=1 \\\\ j \\ne k\\end{array}}^m w_j \\Vert a^k - a^j \\Vert = \\frac{1}{2} f(a^k).$ This concludes the proof of (b).",
"$\\Box $ The number $g_x(Q(x))$ can be computed in the next lemma.",
"Lemma 12 Let $x \\in \\Omega $ be.",
"If $x \\ne a^1, \\ldots , a^m$ then: $g_x(Q(x)) & = & \\frac{1}{2} f(x) + 2 A(x) \\left\\langle Q(x) - x , Q(x) - T(x) \\right\\rangle \\\\& - & A(x) \\left\\Vert Q(x) - x \\right\\Vert ^2.$ If $x = a^k$ for some $k = 1, \\ldots , m$ , then $g_{a^k}(Q(a^k)) & = & \\frac{1}{2} f(a^k) - A(a^k) \\left\\Vert Q(a^k) - a^k \\right\\Vert ^2\\\\& + & 2 A(a^k) \\left\\langle Q(a^k) - a^k , Q(a^k) - T(a^k) \\right\\rangle \\\\& - & 2 \\beta (a^k) A(a^k) \\left\\langle Q(a^k) - a^k , \\widetilde{T}(a^k) - a^k \\right\\rangle \\\\& + & w_k \\left\\Vert Q(a^k) - a^k \\right\\Vert .$ First, let us consider $x \\ne a^1, \\ldots , a^m$ .",
"Due to Remark REF we have: $g_x(Q(x)) = \\sum _{j=1}^m \\frac{ w_j }{ 2 \\Vert x - a^j \\Vert } \\Vert Q(x) - a^j \\Vert ^2.$ By Lemma REF we obtain: $g_x(Q(x))& = & \\sum _{j=1}^m \\frac{ w_j }{ 2 \\Vert x - a^j \\Vert } \\left[ \\Vert x - a^j \\Vert ^2 - \\Vert Q(x) - x \\Vert ^2 \\right.\\\\& + & \\left.",
"2 \\langle Q(x) - x , Q(x) - a^j \\rangle \\right].$ Due to the definition of the Weber function $f$ , the definition of $A$ as in (REF ) and the definition of $\\alpha $ as in (REF ), we deduce that: $g_x(Q(x)) = \\frac{1}{2} f(x) - A(x) \\Vert Q(x) - x \\Vert ^2 + \\langle Q(x) - x , \\alpha (x) \\rangle .$ By Lemma REF we get: $g_x(Q(x)) = \\frac{1}{2} f(x) - A(x) \\Vert Q(x) - x \\Vert ^2 + 2 A(x) \\langle Q(x) - x , Q(x) - T(x) \\rangle ,$ concluding the proof of (a).",
"Now, consider $x = a^k$ for some $k = 1, \\ldots , m$ .",
"Due to (REF ) we have: $g_{a^k}(Q(a^k)) = \\sum _{\\begin{array}{c}j=1 \\\\ j \\ne k\\end{array}}^m \\frac{ w_j }{ 2 \\Vert a^k - a^j \\Vert } \\Vert Q(a^k) - a^j \\Vert ^2 + w_k \\Vert Q(a^k) - a^k \\Vert .$ By Lemma REF , the definition of the Weber function $f$ and the definition of $A$ as in (REF ) we obtain: $g_{a^k}\\left( Q(a^k) \\right) & = & \\frac{1}{2} f(a^k) - A(a^k) \\Vert Q(a^k) - a^k \\Vert ^2\\\\& + & \\left\\langle Q(a^k) - a^k , \\sum _{\\begin{array}{c}j=1 \\\\ j \\ne k\\end{array}}^m \\frac{ w_j }{ \\Vert a^k - a^j \\Vert } \\left[ Q(a^k) - a^j \\right] \\right\\rangle \\\\& + & w_k \\Vert Q(a^k) - a^k \\Vert .$ Manipulating algebraically, $Q(a^k) - a^j = Q(a^k) - ( 1 - \\beta (a^k) ) a^j - \\beta (a^k) a^k + \\beta (a^k) ( a^k - a^j ).$ Due to the definition of $\\alpha $ (see (REF )) and the definition of $\\widetilde{R}$ (see (REF )) we get: $g_{a^k} \\left( Q(a^k) \\right)& = & \\frac{1}{2} f(a^k) - A(a^k) \\left\\Vert Q(a^k) - a^k \\right\\Vert ^2\\\\& + & \\left\\langle Q(a^k) - a^k , \\alpha (a^k) \\right\\rangle - \\beta (a^k) \\left\\langle Q(a^k) - a^k , \\widetilde{R}(a^k) \\right\\rangle \\\\& + & w_k \\Vert Q(a^k) - a^k \\Vert .$ By Lemma REF and Lemma REF we have: $g_{a^k} \\left( Q(a^k) \\right) & = & \\frac{1}{2} f(a^k) - A(a^k) \\left\\Vert Q(a^k) - a^k \\right\\Vert ^2\\\\& + & 2 A(a^k) \\left\\langle Q(a^k) - a^k , Q(a^k) - T(a^k) \\right\\rangle \\\\& - & 2 A(a^k) \\beta (a^k) \\left\\langle Q(a^k) - a^k , \\widetilde{T}(a^k) - a^k \\right\\rangle + w_k \\Vert Q(a^k) - a^k \\Vert .$ which concludes the proof.",
"$\\Box $ The next lemma deals with the last two terms of $g_{a^k}(Q(a^k))$ .",
"Lemma 13 If $a^k \\in \\Omega $ for some $k = 1, \\ldots , m$ , the number $z = w_k \\left\\Vert Q(a^k) - a^k \\right\\Vert - 2 A(a^k) \\beta (a^k) \\left\\langle Q(a^k) - a^k , \\widetilde{T}(a^k) - a^k \\right\\rangle ,$ is equal to zero.",
"If $Q(a^k) = a^k$ the result is true.",
"So, from now on, let us consider that $Q(a^k) \\ne a^k$ .",
"First, let us check that $a^k \\ne x^u$ .",
"In case that $a^k = x^u$ , then $T(a^k) = a^k$ by Theorem REF .",
"Since $a^k \\in \\Omega $ , then $T(a^k) \\in \\Omega $ .",
"By Remark REF we have that $Q(a^k) = T(a^k) = a^k$ which is a contradiction.",
"By Remark REF , we have that $\\beta (a^k) \\in (0,1)$ (since $a^k \\ne x^u$ ) and: $z = w_k \\left\\Vert Q(a^k) - a^k \\right\\Vert - \\frac{ 2 A(a^k) \\beta (a^k) }{ 1 - \\beta (a^k) } \\left\\langle Q(a^k) - a^k , T(a^k) - a^k \\right\\rangle .$ Extracting common factors, using Remarks REF and REF , the fact that $T(a^k) \\ne a^k$ (if $T(a^k) = a^k$ then $a^k = x^u$ by Theorem REF ), and the fact that $\\widetilde{T}(a^k) \\ne a^k$ (if $\\widetilde{T}(a^k) = a^k$ then $T(a^k) = a^k$ by definition (REF )) we get that: $z & = & 2 A(a^k) \\left\\Vert Q(a^k) - a^k \\right\\Vert \\left\\Vert \\widetilde{T}(a^k) - a^k \\right\\Vert \\left[ \\frac{ w_k }{ 2 A(a^k) \\left\\Vert \\widetilde{T}(a^k) - a^k \\right\\Vert } \\right.\\\\& - & \\left.",
"\\beta (a^k) \\left\\langle \\frac{ ( 1 - \\lambda ( a^k ) ) ( T(a^k) - a^k ) }{ \\left\\Vert ( 1 - \\lambda ( a^k ) ) ( T(a^k) - a^k ) \\right\\Vert } , \\frac{ T(a^k) - a^k }{ \\left\\Vert T(a^k) - a^k \\right\\Vert } \\right\\rangle \\right].$ Simplifying and using the definition of $\\beta (a^k)$ we have that: $z = 2 A(a^k) \\left\\Vert Q(a^k) - a^k \\right\\Vert \\left\\Vert \\widetilde{T}(a^k) - a^k \\right\\Vert \\left[ \\beta (a^k) - \\beta (a^k) \\right] = 0,$ which concludes the proof.",
"$\\Box $ The purpose of the next two lemmas is to determine a strict inequality between the functions $g_x$ and $f$ at suitable points.",
"First of all, we have to prove the following result.",
"Lemma 14 Let $x \\in \\Omega $ be such that $x \\ne Q(x)$ .",
"If $x \\ne a^1, \\ldots , a^m$ , then $g_x(Q(x)) \\le g_x(x)$ .",
"Besides that, if $\\mathcal {E}(x) \\ne \\emptyset $ and $P_x \\circ Q(x) \\ne P_x(x)$ , then $g_x(Q(x)) < g_x(x)$ .",
"If $x = a^k$ for some $k = 1, \\ldots , m$ , then $g_{a^k}(Q(a^k)) < g_{a^k}(a^k)$ .",
"If $x \\ne a^1, \\ldots , a^m$ , then $g_x(Q(x)) - g_x(x) = - A(x) \\Vert Q(x) - x \\Vert _{\\mathcal {E}(x)}^2 \\le 0$ , by Lemma REF and Lemma REF .",
"Besides that, if $\\mathcal {E}(x) \\ne \\emptyset $ and $P_x \\circ Q(x) \\ne P_x(x)$ we deduce that $\\Vert Q(x) - x \\Vert _{\\mathcal {E}(x)} \\ne 0$ .",
"Thus, $g_x(Q(x)) < g_x(x)$ .",
"If $x = a^k$ for some $k = 1, \\ldots , m$ , by Lemmas REF , REF and REF we have: $g_{a^k}(Q(a^k)) - g_{a^k}(a^k) & = & - A(a^k) \\left\\Vert Q(a^k) - a^k \\right\\Vert ^2\\\\& + & 2 A(a^k) \\left\\langle Q(a^k) - a^k , Q(a^k) - T(a^k) \\right\\rangle .$ Due to Lemma REF and the fact that $A > 0$ we obtain: $g_{a^k}(Q(a^k)) - g_{a^k}(a^k) \\le - A(a^k) \\Vert Q(a^k) - a^k \\Vert ^2 < 0,$ and the proof is finished.",
"$\\Box $ Lemma 15 Let $x \\in \\Omega $ be such that $x \\ne Q(x)$ .",
"Then $g_x(Q(x)) < \\frac{1}{2} f(x)$ .",
"Let us consider the case when $x \\ne a^1, \\ldots , a^m$ .",
"By Lemmas REF , REF and REF we have that: $g_x(Q(x)) & \\le & g_x(x) = \\frac{1}{2} f(x) + 2 A(x) \\left\\langle Q(x) - x , Q(x) - T(x) \\right\\rangle \\\\& - & A(x) \\left\\Vert Q(x) - x \\right\\Vert _{\\mathcal {N}(x)}^2\\\\& \\le & \\frac{1}{2} f(x) - A(x) \\left\\Vert Q(x) - x \\right\\Vert _{\\mathcal {N}(x)}^2.$ If $\\mathcal {E}(x) = \\emptyset $ , then $\\Vert \\cdot \\Vert _{\\mathcal {N}(x)} = \\Vert \\cdot \\Vert $ .",
"Therefore: $g_x(Q(x)) \\le \\frac{1}{2} f(x) - A(x) \\Vert Q(x) - x \\Vert ^2 < \\frac{1}{2} f(x).$ If $\\mathcal {E}(x) \\ne \\emptyset $ and $P_x \\circ Q(x) = P_x(x)$ , then there exists an index $i \\in \\mathcal {N}(x)$ such that $x_i \\ne (Q(x))_i$ since $x \\ne Q(x)$ .",
"Thus, $\\Vert Q(x) - x \\Vert _{\\mathcal {N}(x)} \\ne 0$ , which implies: $g_x(Q(x)) \\le \\frac{1}{2} f(x) - A(x) \\left\\Vert Q(x) - x \\right\\Vert _{\\mathcal {N}(x)}^2 < \\frac{1}{2} f(x).$ If $\\mathcal {E}(x) \\ne \\emptyset $ and $P_x \\circ Q(x) \\ne P_x(x)$ , due to Lemmas REF , REF and REF , we have that: $g_x(Q(x)) < g_x(x) \\le \\frac{1}{2} f(x) - A(x) \\left\\Vert Q(x) - x \\right\\Vert _{\\mathcal {N}(x)}^2 \\le \\frac{1}{2} f(x).$ Now, when $x = a^k$ for some $k = 1, \\ldots , m$ , $g_{a^k}(Q(a^k)) < g_{a^k}(a^k) = \\frac{1}{2} f(a^k)$ due to Lemma REF and Lemma REF .",
"$\\Box $ The next lemma states an equality that relates the Weber function and $g_x$ at appropriate points when $x \\ne a^1, \\ldots , a^m$ .",
"Besides that, this result will be crucial in the next section.",
"Lemma 16 Let $x \\ne a^1, \\ldots , a^m$ be such that $x \\in \\Omega $ and $x \\ne Q(x)$ .",
"Then: $g_x \\circ Q(x) = \\frac{1}{2} f(x) + \\left( f(Q(x)) - f(x) \\right) + \\delta , \\quad \\delta \\ge 0.$ Due to the definition of $g_x$ as in (REF ) and Remark REF we get that: $g_x \\circ Q(x) = \\sum _{j=1}^m \\frac{ w_j }{ 2 \\Vert x - a^j \\Vert } \\Vert Q(x) - a^j \\Vert ^2.$ Adding and subtracting $\\Vert x - a^j \\Vert $ we have: $g_x \\circ Q(x) & = & \\sum _{j=1}^m \\frac{ w_j }{ 2 \\Vert x - a^j \\Vert } \\left[ \\left\\Vert x - a^j \\right\\Vert + \\left( \\left\\Vert Q(x) - a^j \\right\\Vert - \\left\\Vert x - a^j \\right\\Vert \\right) \\right]^2\\\\& = & \\frac{1}{2} \\sum _{j=1}^m w_j \\left\\Vert x - a^j \\right\\Vert + \\sum _{j=1}^m w_j \\left( \\left\\Vert Q(x) - a^j \\right\\Vert - \\left\\Vert x - a^j \\right\\Vert \\right)\\\\& + & \\sum _{j=1}^m \\frac{ w_j }{ 2 \\Vert x - a^j \\Vert } \\left( \\left\\Vert Q(x) - a^j \\right\\Vert - \\left\\Vert x - a^j \\right\\Vert \\right)^2.$ Notice that the first term of the last equality is the Weber function (divided by two), and the last term is a non-negative number, so we will define it as $\\delta $ .",
"So, using the definition of the Weber function in the middle term we obtain: $g_x \\circ Q(x) = \\frac{1}{2} f(x) + \\left( f(Q(x)) - f(x) \\right) + \\delta .$ $\\Box $"
],
[
"Convergence to optimality results",
"This section states the main results about convergence of the sequence $\\left\\lbrace x^{(l)} \\right\\rbrace $ generated by Algorithm REF .",
"The next theorem establishes that if a point $x \\in \\Omega $ is not a fixed point of the iteration function, then the function $f$ strictly decreases at the next iterate.",
"Theorem 17 Let $x \\in \\Omega $ be such that $x \\ne Q(x)$ .",
"Then $f(Q(x)) < f(x)$ .",
"Let us consider that $x \\ne a^1, \\ldots , a^m$ .",
"By Lemma REF , we have that: $g_x \\circ Q(x) < \\frac{1}{2} f(x).$ By Lemma REF we get that: $\\frac{1}{2} f(x) + f(Q(x)) - f(x) + \\delta < \\frac{1}{2} f(x).$ Simplifying the last expression we obtain: $f(Q(x)) - f(x) + \\delta < 0.$ Finally, $f(Q(x)) - f(x) \\le f(Q(x)) - f(x) + \\delta < 0.$ Therefore, $f(Q(x)) < f(x)$ .",
"Now, consider that $x = a^k$ for some $k = 1, \\ldots , m$ .",
"Following a reasoning similar than in [40], using Lemma REF we have that: $g_{a^k} \\circ Q(a^k) - g_{a^k}(a^k) < 0.$ By definition of $g_{a^k}$ we know that: $g_{a^k} \\circ Q(a^k) - g_{a^k}(a^k) & = & w_k \\left\\Vert Q(a^k) - a^k \\right\\Vert \\\\& + & \\sum _{\\begin{array}{c}j=1 \\\\ j \\ne k\\end{array}}^m \\frac{ w_j }{ 2 \\left\\Vert a^k - a^j \\right\\Vert } \\left( \\left\\Vert Q(a^k) - a^j \\right\\Vert ^2 - \\left\\Vert a^k - a^j \\right\\Vert ^2 \\right).$ Using the fact that $\\left( a^2 - b^2 \\right)/ (2b) \\ge a - b$ for $a = \\left\\Vert Q(a^k) - a^j \\right\\Vert ^2 \\ge 0$ and $b = \\left\\Vert a^k - a^j \\right\\Vert ^2 > 0$ we obtain that: $g_{a^k} \\circ Q(a^k) - g_{a^k}(a^k) & \\ge & w_k \\left\\Vert Q(a^k) - a^k \\right\\Vert \\\\& - & \\sum _{\\begin{array}{c}j=1 \\\\ j \\ne k\\end{array}}^m w_j \\left\\Vert a^k - a^j \\right\\Vert + \\sum _{\\begin{array}{c}j=1 \\\\ j \\ne k\\end{array}}^m w_j \\left\\Vert Q(a^k) - a^j \\right\\Vert .$ Rearranging terms we deduce that: $0 & > & g_{a^k} \\circ Q(a^k) - g_{a^k}(a^k)\\\\& = & \\sum _{j=1}^m w_j \\left\\Vert Q(a^k) - a^j \\right\\Vert - \\sum _{j=1}^m w_j \\left\\Vert a^k - a^j \\right\\Vert = f(Q(a^k)) - f(a^k),$ and the proof is complete.",
"$\\Box $ Corollary 18 Let $\\left\\lbrace x^{(l)} \\right\\rbrace $ be the sequence generated by Algorithm REF .",
"Then the sequence $\\left\\lbrace f \\left( x^{(l)} \\right) \\right\\rbrace $ is not increasing.",
"Even more, each time $x^{(l)} \\ne Q \\left( x^{(l)} \\right)$ the sequence strictly decreases at the next iterate.",
"If the sequence $\\left\\lbrace x^{(l)} \\right\\rbrace $ generated by Algorithm REF were not bounded, then we could choose a subsequence $\\left\\lbrace y^{(l)} \\right\\rbrace $ such that $y^{(l)} \\rightarrow \\infty $ .",
"But this implies that $f(y^{(l)}) \\rightarrow \\infty $ , which is a contradiction since the sequence $\\left\\lbrace f \\left( x^{(l)} \\right) \\right\\rbrace $ is not increasing.",
"Remark 19 The sequence $\\left\\lbrace x^{(l)} \\right\\rbrace $ generated by Algorithm REF is bounded.",
"So, there exists a subsequence convergent to a point $x^* \\in \\Omega $ .",
"Hence, $x^*$ is a feasible point.",
"Due to the nondifferentiability of $f$ at the vertices $a^1, \\ldots , a^m$ , we can not use the KKT optimality conditions at $a^k$ .",
"Therefore, if $a^k$ and $z$ are in $\\Omega $ , let us define $G_{a^k}^z : [0,1] \\rightarrow \\mathbb {R}$ by: $G_{a^k}^z(t) = f( a^k + t(z-a^k) ).$ If $a^k \\in \\Omega $ , $z \\in \\Omega $ , $t \\in [0,1]$ and $\\Omega $ convex, we have that $a^k + t(z-a^k) \\in \\Omega $ .",
"Notice that the right-hand side derivative $G_{a^k}^z(0+)$ (or the directional derivative of $f$ in the direction of $z$ ) exists (see [22]).",
"Besides that, $G_{a^k}^{z \\ \\prime }(0+) = w_k \\left\\Vert z - a^k \\right\\Vert - \\left\\langle \\widetilde{R}(a^k) , z - a^k \\right\\rangle .$ The next lemma shows that if we are in a vertex $a^k$ , the directional derivative of $f$ at $a^k$ in the direction of $Q(a^k)$ is a descent direction.",
"Lemma 20 Let $a^k \\in \\Omega $ be such that $T(a^k) \\notin \\Omega $ .",
"Then: $G_{a^k}^{z \\ \\prime }(0+) \\ge G_{a^k}^{Q(a^k) \\ \\prime }(0+), \\quad \\forall \\, z \\in \\left[ a^k, T(a^k) \\right],$ where: $G_{a^k}^{Q(a^k) \\ \\prime }(0+) = - 2 \\left[ 1 - \\beta (a^k) \\right] A(a^k) \\left\\Vert \\widetilde{T}(a^k) - a^k \\right\\Vert \\left\\Vert Q(a^k) - a^k \\right\\Vert .$ If $T(a^k) = a^k$ then $T(a^k) \\in \\Omega $ , which is a contradiction.",
"Besides that, if $\\widetilde{T}(a^k) = a^k$ , we would have that $T(a^k) = a^k$ because of (REF ), and again it would be a contradiction.",
"So, we will consider $\\widetilde{T}(a^k) \\ne a^k$ and $T(a^k) \\ne a^k$ for the rest of the proof.",
"Since $T(a^k) \\ne a^k$ , then $\\beta (a^k) \\in (0,1)$ (see Remark REF and Theorem REF ).",
"Let us prove equation (REF ) first.",
"Now, by (REF ) we can see that: $G_{a^k}^{Q(a^k) \\ \\prime }(0+) = w_k \\left\\Vert Q(a^k) - a^k \\right\\Vert - \\left\\langle \\widetilde{R}(a^k) , Q(a^k) - a^k \\right\\rangle .$ Notice that if $Q(a^k) = a^k$ , equation (REF ) holds.",
"So, let us consider from now on that $Q(a^k) \\ne a^k$ .",
"By using Lemma REF we replace $\\widetilde{R}(a^k)$ and get: $G_{a^k}^{Q(a^k) \\ \\prime }(0+) = w_k \\left\\Vert Q(a^k) - a^k \\right\\Vert - 2 A(a^k) \\left\\langle \\widetilde{T}(a^k) - a^k , Q(a^k) - a^k \\right\\rangle .$ Extracting common factors and using the definition of $\\beta $ when it belongs to $(0,1)$ we obtain: $G_{a^k}^{Q(a^k) \\ \\prime }(0+) & = & 2 A(a^k) \\left\\Vert Q(a^k) - a^k \\right\\Vert \\left\\Vert \\widetilde{T}(a^k) - a^k \\right\\Vert \\Bigg [ \\beta (a^k)\\\\& - & \\left.",
"\\left\\langle \\frac{ \\widetilde{T}(a^k) - a^k }{ \\left\\Vert \\widetilde{T}(a^k) - a^k \\right\\Vert } , \\frac{ Q(a^k) - a^k }{ \\left\\Vert Q(a^k) - a^k \\right\\Vert } \\right\\rangle \\right].$ By Remarks REF and REF the vectors $Q(a^k) - a^k$ and $\\widetilde{T}(a^k) - a^k$ are parallel, so: $G_{a^k}^{Q(a^k) \\ \\prime }(0+) = 2 A(a^k) \\left\\Vert Q(a^k) - a^k \\right\\Vert \\left\\Vert \\widetilde{T}(a^k) - a^k \\right\\Vert \\left[ \\beta (a^k) - 1 \\right].$ which is equivalent to (REF ).",
"Now, let us prove (REF ).",
"If $z = a^k$ then $G_{a^k}^z(t) = f(a^k)$ for all $t \\in [0,1]$ , thus $G_{a^k+}^{a^k \\ \\prime }(0) = 0$ , and therefore the inequality (REF ) holds.",
"So, let us assume that $z \\ne a^k$ for the rest of the proof.",
"Using (REF ) and due to Lemma REF to replace $\\widetilde{R}(a^k)$ : $G_{a^k}^{z \\ \\prime }(0+) = w_k \\left\\Vert z - a^k \\right\\Vert - 2 A(a^k) \\left\\langle \\widetilde{T}(a^k) - a^k , z - a^k \\right\\rangle .$ Extracting common factors: $G_{a^k}^{z \\ \\prime }(0+) & = & 2 A(a^k) \\left\\Vert z - a^k \\right\\Vert \\left\\Vert \\widetilde{T}(a^k) - a^k \\right\\Vert \\left[ \\frac{ w_k }{ 2 A(a^k) \\left\\Vert \\widetilde{T}(a^k) - a^k \\right\\Vert } \\right.\\\\& - & \\left.",
"\\left\\langle \\frac{ \\widetilde{T}(a^k) - a^k }{ \\left\\Vert \\widetilde{T}(a^k) - a^k \\right\\Vert } , \\frac{ z - a^k }{ \\left\\Vert z - a^k \\right\\Vert } \\right\\rangle \\right].$ Using the expression for $\\beta (a^k) \\in (0,1)$ we obtain: $G_{a^k}^{z \\ \\prime }(0+) & = & 2 A(a^k) \\left\\Vert z - a^k \\right\\Vert \\left\\Vert \\widetilde{T}(a^k) - a^k \\right\\Vert \\Bigg [ \\beta (a^k)\\\\& - & \\left.",
"\\left\\langle \\frac{ \\widetilde{T}(a^k) - a^k }{ \\left\\Vert \\widetilde{T}(a^k) - a^k \\right\\Vert } , \\frac{ z - a^k }{ \\left\\Vert z - a^k \\right\\Vert } \\right\\rangle \\right].$ If $z$ belongs to the segment that joins $a^k$ and $T(a^k)$ we have that $z-a^k$ and $\\widetilde{T}(a^k) - a^k$ are parallel vectors, then: $G_{a^k}^{z \\ \\prime }(0+) \\ge - 2 \\left[ 1 - \\beta (a^k) \\right] A(a^k) \\left\\Vert z - a^k \\right\\Vert \\left\\Vert \\widetilde{T}(a^k) - a^k \\right\\Vert .$ We can write $z = (1-\\lambda )T(a^k) + \\lambda a^k$ where $\\lambda \\in [0,1]$ .",
"Therefore: $G_{a^k}^{z \\ \\prime }(0+) \\ge - 2 \\left[ 1 - \\beta (a^k) \\right] A(a^k) \\left( 1 - \\lambda \\right) \\left\\Vert T(a^k) - a^k \\right\\Vert \\left\\Vert \\widetilde{T}(a^k) - a^k \\right\\Vert .$ for all $\\lambda \\in [0,1]$ .",
"The minimum value of the right-hand side of the last expression happens when $\\lambda = \\lambda ( a^k )$ , so: $G_{a^k}^{z \\ \\prime }(0+) \\ge - 2 \\left[ 1 - \\beta (a^k) \\right] A(a^k) \\left( 1 - \\lambda ( a^k ) \\right) \\left\\Vert T(a^k) - a^k \\right\\Vert \\left\\Vert \\widetilde{T}(a^k) - a^k \\right\\Vert .$ Using Remark REF we conclude that: $G_{a^k}^{z \\ \\prime }(0+) \\ge - 2 \\left[ 1 - \\beta (a^k) \\right] A(a^k) \\left\\Vert Q(a^k) - a^k \\right\\Vert \\left\\Vert \\widetilde{T}(a^k) - a^k \\right\\Vert .$ $\\Box $ Now we will prove an equivalence that characterizes the solution of (REF ) in terms of the iteration function $Q$ .",
"Moreover, if $x^*$ is a regular point that is not a vertex, then $x^*$ is a KKT point.",
"From now on, let us consider that $\\Omega = \\left\\lbrace y \\in \\mathbb {R}^n : g(y) \\le 0, h(y) = 0 \\right\\rbrace ,$ where $g : \\mathbb {R}^n \\rightarrow \\mathbb {R}^s$ is a convex function and $h : \\mathbb {R}^n \\rightarrow \\mathbb {R}^p$ is an affine function.",
"Theorem 21 Let $\\Omega $ be defined as in (REF ) and $x \\in \\Omega $ .",
"Consider the following propositions: $x$ is a KKT point.",
"$x$ is the minimizer of the problem (REF ).",
"$Q(x) = x$ .",
"If $x \\ne a^1, \\ldots , a^m$ , $g$ and $h$ are continuously differentiable, and $x$ is a regular point, then (a), (b) and (c) are equivalent.",
"If $x = a^k$ for some $k = 1, \\ldots , m$ , then (b) implies (c).",
"Let $x \\ne a^1, \\ldots , a^m$ be.",
"Since $f$ is strictly convex and $\\Omega $ is convex, the KKT optimality conditions are necessary and sufficient.",
"Therefore, it holds that (a) is equivalent to (b).",
"Now we will prove that (b) implies (c).",
"Let us suppose that $x$ is the minimizer of the problem (REF ).",
"If $x$ were not a fixed point of the iteration function $Q$ , we would have that $x \\ne Q(x)$ , which means that $f(Q(x)) < f(x)$ by Theorem REF .",
"This contradicts the hypothesis.",
"To demonstrate that (c) implies (a), we will assume that $x$ is a fixed point of $Q$ , that is, $x = Q(x)$ .",
"Since $Q(x) = P_{\\Omega } \\circ T(x)$ , $x$ is the solution of: $\\begin{array}{rl}\\displaystyle \\mathop { \\mathrm {argmin} }_z & \\displaystyle F(z) = \\frac{1}{2} \\left\\Vert z - T(x) \\right\\Vert ^2 \\\\ \\mathrm {subject \\, to} & g(z) \\le 0, \\\\ & h(z) = 0.\\end{array}$ Since $F$ and $g$ are convex, $h$ is affine, and $x$ is a regular point, the KKT optimality conditions hold at $x$ .",
"That is, there exist multipliers $\\left\\lbrace \\mu _j \\right\\rbrace _{j=1}^s$ and $\\left\\lbrace \\lambda _j \\right\\rbrace _{j=1}^p$ such that (see [31], [38]): $x - T(x) + \\sum _{j=1}^s \\mu _j \\nabla g_j(x) + \\sum _{j=1}^p \\lambda _j \\nabla h_j(x) & = & 0,\\\\\\mu _j g_j( x ) & = & 0, \\quad j = 1, \\ldots , s,\\\\\\mu _j & \\ge & 0, \\quad j = 1, \\ldots , s,\\\\g(x) & \\le & 0,\\\\h(x) & = & 0.$ Multiplying these equations by $2 A(x)$ , using equation (REF ), Lemma REF and Remark REF , we obtain: $\\nabla f(x) + \\sum _{j=1}^s \\left( 2 A(x) \\mu _j \\right) \\nabla g_j(x) + \\sum _{j=1}^p \\left( 2 A(x) \\lambda _j \\right) \\nabla h_j(x) & = & 0,\\\\\\left( 2 A(x) \\mu _j \\right) g_j( x ) & = & 0, \\quad j = 1, \\ldots , s,\\\\\\left( 2 A(x) \\mu _j \\right) & \\ge & 0, \\quad j = 1, \\ldots , s,\\\\g(x) & \\le & 0,\\\\h(x) & = & 0.$ where $\\left\\lbrace 2 A(x) \\mu _j \\right\\rbrace _{j=1}^s$ and $\\left\\lbrace 2 A(x) \\lambda _j \\right\\rbrace _{j=1}^p$ are multipliers.",
"Therefore, $x$ is a KKT point of the problem (REF ) (see [31], [38]).",
"Now, let us suppose that $x = a^k$ for some $k = 1, \\ldots , m$ .",
"As before, if $x$ is a minimizer of the problem (REF ), then $Q(a^k) = a^k$ , otherwise $f(Q(a^k)) < f(a^k)$ , which would be a contradiction.",
"$\\Box $"
],
[
"Numerical experiments.",
"The purpose of this section is to discuss the efficiency and robustness of the proposed algorithm versus a solver for nonlinear programming problems.",
"A prototype code of Algorithm REF was programmed in MATLAB (version R2011a) and executed in a PC running Linux OS, Intel(R) Core(TM) i7 CPU Q720, 1.60GHz.",
"We have considered a closed and convex set $\\Omega \\subset \\mathbb {R}^2$ defined by the set $\\Omega = \\left\\lbrace y \\in \\mathbb {R}^n : g(y) \\le 0 \\right\\rbrace $ , where $g$ is given by: $g( x ) = \\left[ \\begin{array}{c} \\displaystyle - 4 - \\frac{ 1 }{ 8 } x + \\frac{ 7 }{ 72 } x^2 + \\frac{ 1 }{ 216 } x^2 ( x - 3 ) + y \\\\[3mm] \\displaystyle \\frac{ 4 }{ 5 } x + y - \\frac{ 59 }{ 10 } \\\\[3mm] \\displaystyle x - \\frac{ 11 }{ 2 } \\\\[3mm] \\displaystyle \\frac{ 3 }{ 2 } x - y - \\frac{ 35 }{ 4 } \\\\[3mm] \\displaystyle x - y - \\frac{ 13 }{ 2 } \\\\[3mm] \\displaystyle - 4 + \\frac{ 1 }{ 8 } ( x - 1 ) + \\frac{ 1 }{ 16 } ( x - 1 )^2 + \\frac{ 1 }{ 32 } ( x - 1 )^2 ( x - 3 ) - y \\\\[3mm] \\displaystyle - \\frac{ 1 }{ 3 } x - y - \\frac{ 11 }{ 3 } \\\\[3mm] \\displaystyle - \\frac{ 2 }{ 3 } x - y - \\frac{ 13 }{ 3 } \\\\[3mm] \\displaystyle - 4 x + y - 19 \\end{array} \\right].$ The feasible set is defined by linear and nonlinear constraints, as it can be seen in Figure REF .",
"Figure: Feasible set Ω\\Omega We have built 1000 different experiments where for each one: The number of vertices was $m = 50$ .",
"The vertices were normally distributed random vectors, with mean equal to 0 and standard deviation equal to 10.",
"The weights were uniformly distributed random positive numbers between 0 and 10.",
"Tolerance was set to $\\varepsilon = 0.00001$ .",
"On one hand, each experiment was solved using Algorithm REF and, on the other hand, it was considered as a nonlinear programming problem and solved using function $fmincon$ (see [32] and references therein).",
"Since the Weber function (REF ) is not differentiable at the vertices, nonlinear programming solvers may fail.",
"Let $x_m(i)$ be the solution of (REF ) obtained by $fmincon$ in experiment $i$ , and $f_{m}(i) = f(x_m(i))$ .",
"Analogously, let $x_p(i)$ be the solution of (REF ) obtained by Algorithm REF in experiment $i$ , and $f_{p}(i) = f(x_p(i))$ .",
"Figure REF shows the difference between the arrays $f_{m}$ and $f_{p}$ .",
"Both methods finished succesfully in all cases, however, Algorithm REF found equal or better results for all experiments.",
"For example, the difference $f_{m} - f_{p}$ was greater than $0.01$ in 35 experiments (the maximum difference ocurred in experiment 506).",
"Figure: Difference between minimum values found by Algorithm and fminconfmincon.Feasibility of the solutions $x_p(i)$ can be checked computing $\\max (g(x_m(i)))$ .",
"Results can be seen in Figure REF Figure: Feasibility of the solution x p (i)x_p(i) obtained by Algorithm ."
],
[
"Conclusions",
"This paper proposes a Weiszfeld-like algorithm for solving the Weber problem constrained to a closed and convex set, and it is well defined even when an iterate is a vertex.",
"The algorithm consists of two stages: first, iterate using the fixed point modified Weiszfeld iteration (REF ), and second, either project onto the set $\\Omega $ when the iterate is different from the vertices, or, if the iterate is a vertex $a^k$ , take the point belonging to the line that joins $T(a^k)$ with $a^k$ as defined in (REF ).",
"It is proved that the constrained problem (REF ) has a unique solution.",
"Besides that, the definition of the iteration function $Q$ allows us to demonstrate that the proposed algorithm produces a sequence $\\left\\lbrace x^{(l)} \\right\\rbrace $ of feasible iterates.",
"Moreover, the sequence $\\left\\lbrace f \\left( x^{(l)} \\right) \\right\\rbrace $ is not increasing, and when $x^{(l)} \\ne Q\\left( x^{(l)} \\right)$ , the sequence decreases at the next iterate.",
"It can be seen that if a point $x^*$ is the solution of the problem (REF ) then $x^*$ is a fixed point of the iteration function $Q$ .",
"Even more, if $x^*$ is different from the vertices, the fact of being $x^*$ a fixed point of $Q$ is equivalent to the fact that $x^*$ satisfies the KKT optimality conditions, and equivalent to the fact that $x^*$ is the solution of the problem (REF ).",
"These properties allows us to connect the proposed algorithm with the minimization problem.",
"Numerical experiments showed that the proposed algorithm found equal or better solutions than a well-known standard solver, in a practical example with 1000 random choices of vertices and weights.",
"That is due to the fact that the proposed algorithm does not use of the existence of derivatives at the vertices, because the Weber function is not differentiable at the vertices."
]
] | 1204.1087 |
[
[
"p-Wave holographic superconductors with Weyl corrections"
],
[
"Abstract We study the (3+1) dimensional p-wave holographic superconductors with Weyl corrections both numerically and analytically.",
"We describe numerically the behavior of critical temperature $T_{c}$ with respect to charge density $\\rho$ in a limited range of Weyl coupling parameter $\\gamma$ and we find in general the condensation becomes harder with the increase of parameter $\\gamma$.",
"In strong coupling limit of Yang-Mills theory, we show that the minimum value of $T_{c}$ obtained from analytical approach is in good agreement with the numerical results, and finally show how we got remarkably a similar result in the critical exponent 1/2 of the chemical potential $\\mu$ and the order parameter$<J^1_x>$ with the numerical curves of superconductors."
],
[
"Introduction",
"Anti-de Sitter/ Conformal field theory (AdS/CFT) links a d- dimensional strongly coupled conformal field theory on the boundary to a $(d+1)$ - dimensional weakly coupled dual gravitational description in the bulk[1], [2], [3].",
"It is necessary to couple a complex scalar field with an Einstein- Maxwell theory to explain the simplest model for holographic superconductors.",
"In a holographic superconductor, below a critical temperature, the gauge symmetry breaks and a black hole is constructed by the unstable developing scalar hair near the horizon.",
"According to the AdS/CFT correspondence, the complex scalar field is dual to a charged operator at the boundary, therefore a superconductor phase transition will be occurred by both the $U(1)$ symmetry-breaking in the gravity and a global $U(1)$ symmetry-breaking in the dual boundary theory[4].",
"Holographic superconductors have been properly considered in two different models , one with an Abelian-Higgs model, which is the gravity dual of an $s$ -wave superconductor with a scalar order parameter.",
"This model has been studied by many authors see[2], [3], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14].",
"The other type uses a $SU(2)$ gauge theory [15], [16], [17], [18], [19], [20], [21], [22], [23].",
"Hartnoll et al.",
"[3] investigate further this Abelian-Higgs model of superconductivity, and according to AdS/CFT correspondence construct a $s$ -wave holographic superconductor solution with a scalar order parameter displaying the phase transition process at the critical temperature $T_{c}$ below which the charge condensate form.",
"The Einstein-Yang-Mills (EYM) model of holographic superconductors constructed later by Gubser[24], where spontaneous symmetry-breaking solutions through a condensate of non Abelian gauge fields is presented, and also p-wave and $(p+ip)$ -wave backgrounds have been studied [15].",
"In this case,the CFT have a global $SU(2)$ symmetry and hence three conserved currents .",
"The first effort on analytic methods in this topic was the Herzog's work[25], where critical exponent and the expectation values of the dual operators was attained.",
"Analytical studies of superconductors have been established in two major methods: the small parameter perturbation theory as in[25], and the variational method [26], [27].",
"In the presented paper, we study the Weyl corrected p-wave holographic superconductor composed of a non-Abelian SU(2) gauge field( the matter sector) and a black hole background( the gravity sector) by using the variational method giving only critical temperature $T_{c}^{Min}$ .",
"As it has been mentioned in[28], for an Abelian gauge field and large range of the Weyl coupling value $\\frac{-1}{16} < \\gamma < \\frac{1}{24}$ , the universal relation for the critical temperature $T_{c} \\approx \\@root 3 \\of {\\rho }$ has been found.",
"In this paper we have explored the same validity of the critical temperature relation in a non-Abelian gauge field with Weyl correction numerically and analytically.",
"The organization of this paper is as follows.",
"In section II we reconstruct the Weyl corrected superconductor's solution of the EYM theory, which is dual to a p-wave superconductor.",
"In section III we present numerical results for condensation and critical temperature of the holographic superconductor.",
"Then we investigate the behavior of critical temperature $T_{c}$ and dual of chemical potential $\\mu $ with respect to Weyl coupling parameter $\\gamma $ , dual of charge density $\\rho $ , and order parameter $\\langle J_{x}^{1} \\rangle $ analytically in section IV.",
"The conclusion and some discussion are given in section V."
],
[
" Weyl corrected P-wave superconductors",
"In this section we study the holographic phase transition for the probe $SU(2)$ Yang-Mills (YM) field $A^{a}_{\\mu }$ in a five dimensional space-time.",
"The bulk action of the Weyl gravity with an $SU(2)$ Yang-Mills in a five dimensional spacetime is: $S=\\int dt d^{4}x\\sqrt{-g}\\lbrace \\frac{1}{16\\pi G_{5}}(R+12)-\\frac{1}{4g^2}F^{a}_{\\mu \\nu }F^{a\\mu \\nu }+\\gamma C^{\\mu \\nu \\rho \\sigma }F^{a}_{\\mu \\nu }F^{a}_{\\rho \\sigma }\\rbrace $ Here $G_{5}$ is the five dimensional gravitational constant, $g$ is the Yang-Mills coupling constant, and the negative cosmological constant is satisfied by the factor $\\frac{12}{l^2},\\ \\ l=1$ in the first parenthesis.",
"The field strength component is given as below, where $A_{\\mu }^{a}$ 's are the non-Abelian gauge fields.",
"$F^{a}_{\\mu \\nu }=\\partial _{\\mu }A^{a}_{\\nu }-\\partial _{\\nu }A^{a}_{\\mu }+\\varepsilon ^{abc}A^{b}_{\\mu }A^{c}_{\\nu },\\lbrace a=1,2,3,\\mu =0,1,2,3\\rbrace $ The Weyl's coupling $\\gamma $ is limited such that its value is in the interval $-\\frac{1}{16}<\\gamma <\\frac{1}{24}$ , (for more precise details see [29]).",
"In the probe limit by neglecting the back reactions , the gravity sector is effectively decoupled from the matter field's sector.",
"In this probe limit, the background metric is given by an AdS-Schwarzschild black hole: $ds^2=r^2(-fdt^2+dx^idx_i)+\\frac{dr^2}{r^2f}$ where: $f=1-(\\frac{r_{+}}{r})^4$ The black hole horizon is $r=r_{+}$ .",
"The Smarr-Bekenstein-Hawking temperature of the black hole is determined by the Schwarzschild radius as $T=\\frac{r_{+}}{\\pi }$ .",
"This is the same as the temperature of the conformal field theory on the boundary of the AdS spacetime.",
"Applying the Euler-Lagrange equation, we can derive the generalized Yang-Mills equation as[28]: $\\nabla _{\\mu }\\left( F^{a\\mu \\nu } - 4\\gamma C^{\\mu \\nu \\rho \\sigma }F^{a}_{\\rho \\sigma } \\right) =-\\epsilon ^{a}_{bc}A^{b}_{\\mu }F^{c\\mu \\nu }+ 4\\gamma C^{\\mu \\nu \\rho \\sigma }\\epsilon ^{a}_{bc}A^{b}_{\\mu }F^{c}_{\\rho \\sigma }$ where $C_{\\mu \\nu \\rho \\sigma }$ is the Weyl tensor and has the following nonzero components in $AdS^{5}$ : $ C_{0i0j}=f(r)r_{+}^{4} \\delta _{ij},~~C_{0r0r}=-\\frac{3 r_{+}^{4}}{ r^{4}},~~ C_{irjr}=-\\frac{r_{+}^{4}}{r^{4} f(r)} \\delta _{ij},~~ C_{ijkl}=r_{+}^{4} \\delta _{ik}\\delta _{jl}.$ For realization of a holographic p-wave superconductor we take the following anstaz for Yang-Mills gauge field [30]: $A=\\varphi (r)\\sigma ^{3}dt+\\psi (r)\\sigma ^1 dx$ ($\\sigma ^{i}$ Pauli's matrixes).",
"The condensation of $\\psi (r)$ breaks the $SU(2)$ symmetry and the final state is the superconductor phase transition.",
"The gauge function $\\psi (r)$ is dual to the $J^1 _x$ operator on the boundary, choosing $x$ axis as a special direction, the condensation phase of $\\psi (r)$ breaks the symmetry and leads to a phase transition, which can be interpreted as a $p$ -wave superconductor phase transition on the boundary.",
"The resulting Yang-Mills equations for metric (3) are given by: $ \\left( 1 - \\frac{24 \\gamma r_{+}^{4}}{r^{4}}\\right)\\varphi ^{\\prime \\prime } + \\left( \\frac{3}{r} + \\frac{24 \\gamma r_{+}^{4}}{r^{5}} \\right)\\varphi ^{\\prime }- \\left( 1 + \\frac{8 \\gamma r_{+}^{4}}{r^{4}} \\right)\\frac{\\psi ^{2}\\varphi }{r^{4}f}=0\\\\ \\left( 1 - \\frac{8 \\gamma r_{+}^{4}}{r^{4}}\\right)\\psi ^{\\prime \\prime } + \\left[ \\frac{3}{r} + \\frac{f^{\\prime }}{f} - \\frac{8 \\gamma r_{+}^{4}}{r^{4}} \\left( -\\frac{1}{r} + \\frac{f^{\\prime }}{f} \\right) \\right]\\psi ^{\\prime } + \\left( 1 + \\frac{8 \\gamma r_{+}^{4}}{r^{4}} \\right)\\frac{\\varphi ^{2}\\psi }{r^{4}f^{2}}=0$ where the prime denotes derivative with respect to $r$ .",
"It is more conveint to work in terms of the dimensionless parameter $z=\\frac{r_{+}}{r}$ , in which at the horizon $z=1$ , and the boundary at the infinity locates at $z=0$ .",
"Then the the equations of motion (REF ) and () can be reexpressed as: $ \\left( 1 - 24 \\gamma z^{4} \\right)\\varphi ^{\\prime \\prime } -\\frac{1}{z} \\left( 1 + 72 \\gamma z^{4} \\right)\\varphi ^{\\prime }- \\left( 1 + 8 \\gamma z^{4} \\right)\\frac{\\psi ^{2}\\varphi }{f}=0\\\\ \\left( 1 - 8 \\gamma z^{4} \\right)\\psi ^{\\prime \\prime } + \\left[-\\frac{1}{z} + \\frac{f^{\\prime }}{f} - 8 \\gamma z^{4} \\left( \\frac{3}{z} +\\frac{f^{\\prime }}{f} \\right) \\right] \\psi ^{\\prime } + \\left( 1 + 8 \\gamma z^{4}\\right) \\frac{\\varphi ^{2}\\psi }{f^{2}}=0$ where the prime now denotes derivative with respect to $z$ .",
"The boundary conditions at infinity, i.e.",
"$z\\rightarrow 0 $ , are: $\\varphi \\simeq \\mu -\\rho z^2\\\\\\psi \\simeq \\psi ^{(0)}+\\psi ^{(2)}z^2$ $\\mu $ and $\\rho $ are dual to the chemical potential and charge density of the CFT boundary, $\\psi ^{(0)}$ and $\\psi ^{(2)}$ are dual to the source term and expectation value of the boundary operator $J^1_x$ respectively.",
"Further to have a normalizable solution, we always set the source $\\psi ^{(0)}$ to zero.",
"Figure: The condensation as a function oftemperature for the operators <J x 1 ><J_{x}^{1}>.",
"γ=-0.06,-0.04,-0.02,0,0.02,0.04\\gamma =-0.06, -0.04,-0.02, 0, 0.02, 0.04 from top to bottom and the dotted line is justthe case γ=0\\gamma =0.Table: The critical temperature T c T_{c} for differentvalues of Weyl coupling parameter γ\\gamma (Numerical results)."
],
[
"numerical treatment",
"In this section we will present numerical results for the condensation and critical temperature due to the shooting method.",
"From EOM's (REF ) and () and the asymptotic behavior of $\\psi $ and $\\varphi $ at infinity (12) and (13), we can obtain the regularity condition as: $\\varphi (1)=\\varphi ^{\\prime }(0)=\\psi ^{\\prime }(0)=\\psi ^{\\prime }(1)=0$ combining boundary condition (12) and (13) with regularity condition (REF ), we can solve EOM's (REF ) and () numerically by using a shooting method, and plot the FIG.1 to demonstrate the condensation as a function of temperature for the operator$< J^1 _x >$ .",
"The curve in FIG.1 is qualitatively similar to that obtained in the holographic superconductors[28], [32], where the condensation of $< J^1 _x >$ goes to a constant at zero temperature.",
"As we can see from the FIG.1, it is easy to find that the critical temperatures of Weyl corrected superconductors is increasing as the parameter $\\gamma $ varies in the range of $-0.06$ to $0.04$ , therefore we conclude that when $\\gamma < 0 $ the critical temperature is smaller and the formation of scalar hair is harder and vice versa when $\\gamma > 0$ .",
"We have also presented the critical temperature $T_{c}$ with different values of the parameter $\\gamma $ in the TABLE I.",
"According to the results in the TABLE I, we can conclude that by minimizing the coupling parameter $\\gamma $ , the critical temperature decrease smoothly ."
],
[
"Analytical treatment",
"In this section we compute the critical temperature and critical exponent via an analytical method, which has been proposed recently [26].",
"In this method by defining appropriate equation matching with field's boundary conditions, the field's EOM will be transformed to Sturm-Liouville self adjoint form.",
"Therefore according to the general variational method to solve the Sturm-Liouville problem ([33] or appendix of [26] ), the eigenvalue $\\lambda ^{2}$ minimizing the Sturm-Liouwille equation can be found.",
"Using this minimum value of $\\lambda $ , one can obtain the minimum critical temperature $T_{c}^{Min}$ .",
"In this section we calculate this $T_{c}^{Min}$ and discuss the critical exponent."
],
[
"Critical temperature $T_C^{Min}$",
"Considering the non linear system (10,11).",
"If there is a second order continuous phase transition at the critical temperature, the solution of the EOMs at the $T_C$ should be: $\\psi (z)=0,\\varphi (z)=\\lambda h_c(1-z^2)$ Here $\\lambda =\\frac{\\rho }{h_c^3}$ , $h_c$ is the radius of the horizon corresponding to $T=T_c$ .",
"At a temperature slightly below $T_c$ , the EOM for $\\psi $ becomes: $z^2\\frac{d}{dz}((\\frac{(1-z^4)}{2g^2z}+4\\gamma h_c^3z^3)\\frac{d\\psi }{dz})+\\lambda ^2[h_c^3(\\frac{z}{2g^2}+4\\gamma z^5)\\frac{1-z^4}{1+z^4}]\\psi =0$ It is appropriate to define: $\\psi (z)=\\frac{\\langle J^1 _x \\rangle }{h}z^2 F(z)$ Matching the boundary condition at the boundary $z=0$ , we normalize the function as $F(0)=1, F^{\\prime }(0)=0$ .",
"The equation for F(z) is: $\\frac{d}{dz}(k(z)\\frac{dF(z)}{dz})-p(z)F(z)+\\lambda ^2q(z)F(z)=0$ where: $k(z)=\\frac{z^3(1-z^4)}{2g^2 }+4\\gamma h_c^3 z^7\\\\p(z)=-2z^5(-\\frac{2}{g^2}+16\\gamma h_c^3)\\\\q(z)=h_c^3z^2(\\frac{h_cz}{2g^2}+4\\gamma z^5)\\frac{1-z^4}{1+z^4}$ The eigenvalue $\\lambda $ minimizes the expression (18) is obtained from the following functional: $\\lambda ^2=\\frac{\\int _{0}^{1}(k(z)F^{\\prime }(z)^2+p(z)F(z)^2)dz}{\\int _{0}^{1}q(z)F(z)^2dz}$ To estimate it, we use the trial function $F(z)=1-\\alpha z^2$ .",
"We then obtain: $\\lambda _{\\alpha }^2=\\frac{2 g^2 \\left(-1.6 h_c^3 \\alpha ^2 \\gamma +8.h_c^3 \\alpha \\gamma -5.33333 h_c^3 \\gamma +\\frac{0.533333 \\alpha ^2}{g ^2}-\\frac{\\alpha }{g ^2}+\\frac{0.666667}{g ^2}\\right)}{h_c^3 \\left(\\alpha ^2\\left(-0.21929 \\gamma g ^2-0.0151862\\right)+\\alpha \\left(0.628086\\gamma g ^2+0.052961\\right)-0.485957 \\gamma g^2-0.0568528\\right)}$ Which attains a minimum at $\\alpha =0.304936$ , and from the $\\lambda =\\frac{\\rho }{h_c^3}$ and $T_c=\\frac{h_c}{\\pi }$ the minimum value of the critical temperature can be read as: $T_c^{Min(\\pm )}=0.256926 \\@root 3 \\of {\\frac{-0.128539\\pm 0.3125 g ^2\\sqrt{\\frac{1.90164\\gamma g ^2 \\rho ^2 \\left(1.00743 \\gamma g^2+0.134769\\right)+0.169187}{g ^4}}}{\\gamma g ^2}}$ We know that in strong-coupling regime of the YM theory, the quantities can be expanded in series of $\\frac{1}{g}$ [34].",
"Since for some values of $g$ , we may be have $T_{c}^{Min(-)}<0$ , therefore only the $T_{c}^{Min(+)}$ is acceptable and can be read in strong limit of order $\\frac{1}{g^2}$ as: $T_{c}^{Min(+)}\\approx 0.1943040830 \\rho ^{\\frac{1}{3}}+\\frac{0.1497404(-0.128539\\gamma + 0.028931219\\gamma \\rho )}{\\gamma ^{2}\\rho ^\\frac{2}{3}g^{2}}$ In prob limit by neglecting the back reaction, the large values of the YM coupling is accessible.",
"Comparing equation (25) with the TABLE I we observe that the analytic value of the leading order $T_{c}^{Min}\\approx 0.1943040830 \\rho ^{\\frac{1}{3}}$ obeys the well known role $T_{c}\\propto \\rho ^{\\frac{1}{3}}$ and it is the lower bound for tabulated values of $T_{c}$ , given in TABLE I.",
"According to the numerical results of TABLE I, the minimum value of $T_{c}$ reads as: $T_{c(numeric)}^{Min(+)}\\approx 0.1701 \\rho ^{\\frac{1}{3}}$ which shows that the analytic values in relation (25) and the numerical estimate in equation (26) are in good agreement with each other.",
"It seems that there is a deep relation between the strong limit of YM part of the action and the analytical results of the p-wave superconductors ."
],
[
"Relation of $<J_x^1> -(\\mu -\\mu _c)$",
"Now we want to know the behavior of the order parameter at $T_c$ , by solving the equation for the scalar potential close to $T_c$ , therefore by substituting (13) in (10) we have: $\\frac{d}{dz}((-\\frac{1}{2z g^2}+12\\gamma z^3)\\frac{d\\varphi }{dz})+(\\frac{z}{h_c})^2(\\frac{ z}{2(1-z^4)g^2}+\\frac{4\\gamma z^5}{1-z^4 })(\\frac{<J^1 _x>}{h_c})^2F(z)^2\\varphi =0$ Since near the critical point, the order parameter $<J^1 _x>$ is small, we can expand $\\varphi $ as a series form of the small parameter as below: $\\varphi =\\mu _c+<J^1 _x>\\chi (z)+ ...$ The boundary condition imposes that $\\chi (z)$ to be $\\chi (1)= 0$ .The EOM for $\\chi (z)$ can be obtained as below: $\\frac{d}{dz}((-\\frac{1}{2z g^2}+12\\gamma z^3)\\frac{d\\chi (z)}{dz})+(\\frac{z}{h_c})^2(\\frac{ z}{2(1-z^4)g^2}+\\frac{4\\gamma z^5}{1-z^4 })\\frac{J^1 _x}{h_c^2}F(z)^2\\mu _c=0$ By integration from both sides of the (29), the EOM for $\\chi (z)$ can be reduced to: $(-\\frac{1}{2z\\zeta ^2}+12\\gamma z^3)\\frac{d\\chi (z)}{dz}=-\\mu _c\\frac{<J^1 _x>}{h_c^4}\\int z^2(\\frac{z}{2(1-z^4)g^2 }+\\frac{4\\gamma z^5}{1-z^4 })F(z)^2 dz$ By regularity condition, we have $\\chi ^{\\prime }(0)=0$ , so we take $F(z)=(1-z^2)(1-\\alpha z^2)$ .",
"Now $\\varphi (z)$ can be expanded as: $\\varphi (z)\\sim \\mu -\\rho z^2\\approx \\mu _c+<J^1_x>(\\chi (0)+\\chi ^{\\prime }(0)z+...)$ Now from (31), by comparing the coefficients of $z^0$ term in both sides of the above formula, we obtain: $\\mu -\\mu _c\\approx \\frac{(<J^1 _x>)^2 \\mu _c}{34560 h^4 \\gamma ^2 g^4}(3003.22 \\gamma ^{3/2} g ^3 \\left(8 \\gamma g ^2+1\\right)\\text{Li}_{2}\\left(\\frac{12.",
"\\sqrt{\\gamma } g }{12.\\sqrt{\\gamma } g -2.44949}\\right)+(9047.79+6359.31 ) \\gamma ^2 g ^4)$ Where $\\alpha =0.304936$ is a parameter minimizing the equation (23), and $\\text{Li}_{2}(z)$ gives the polylogarithm function.",
"This critical exponent $\\frac{1}{2}$ for the condensation value and $(\\mu -\\mu _c)$ qualitatively match the numerical curves for superconductors with Weyl corrections[28]."
],
[
"conclusions",
"In this letter we have investigated the Weyl corrected p-wave Holographic superconductors at the probe limit using numerical and analytical solutions.",
"We obtained the behavior of the critical temperature $T_{c}$ as a function of the dual charge density $\\rho $ in different values of Weyl coupling parameter $\\frac{-1}{16}< \\gamma <\\frac{1}{24}$ .",
"We have found that the critical temperature increases by growing the Weyl coupling parameter, therefore the condensation becomes harder when $\\gamma < 0 $ , and vice versa when $\\gamma > 0$ .",
"As a final point, obtaining the critical exponent $\\frac{1}{2}$ for the chemical potential $\\mu $ and order parameter$\\langle J^1 _x \\rangle $ show a good agreement with the numerical curves of Weyl corrected s-wave holographic superconductors.",
"Furthermore, we have shown that in the strong limit of YM theory, the analytical and numerical values of the minimum value of the critical temperature are in good agreement."
],
[
"acknowledgement",
"The authors would like to thank Jian-Pin Wu, from Beijing Normal University (China), for helpful suggestion and recommending useful references, and also we truthfully claim that without his valuable numerical codes and results the substantial improvements of this presentation and outcomes would not have been achieved.",
"Besides, we thank the referee for good observations and kind guidelines.",
"NM would like to acknowledge the financial support of University of Tehran for this research under grant No.",
"02/1/28450."
]
] | 1204.1246 |
[
[
"Imaging high-dimensional spatial entanglement with a camera"
],
[
"Abstract The light produced by parametric down-conversion shows strong spatial entanglement that leads to violations of EPR criteria for separability.",
"Historically, such studies have been performed by scanning a single-element, single-photon detector across a detection plane.",
"Here we show that modern electron-multiplying charge-coupled device cameras can measure correlations in both position and momentum across a multi-pixel field of view.",
"This capability allows us to observe entanglement of around 2,500 spatial states and demonstrate Einstein-Podolsky-Rosen type correlations by more than two orders of magnitude.",
"More generally, our work shows that cameras can lead to important new capabilities in quantum optics and quantum information science."
],
[
"Results",
"Imaging system.",
"The experimental setup used to measure spatial correlations in both position and momentum is shown in Fig.",
"REF .",
"A $150\\,\\rm {mW}$ , high-repetition-rate $355\\,\\rm {nm}$ laser is attenuated to $2\\,\\rm {mW}$ and using a simple telescope the Gaussian beam radius is expanded to $\\sigma _{p} = (0.66\\pm 0.05)\\rm {mm}$ .",
"A $5\\,\\rm {mm}$ long $\\beta $ -Barium Borate (BBO) crystal cut for type-I phase matching was angled to provide near-collinear output for degenerate down-converted photons at $710\\,{\\rm nm}$ .",
"The subsequent choice of lens configurations was chosen to maintain a constant separation ($700\\,\\rm {mm}$ ) between the BBO crystal and the EMCCD (Andor iXon3), thus providing an efficient system for manual switching between the image and far-field planes.",
"Figure: Experimental scheme used to measure position and momentum correlations.",
"(𝐚){\\rm ({\\bf a})} The imaging system used to measure position correlations consists of a telescope with lenses 100 mm 100\\,\\rm {mm} and 250 mm 250\\,\\rm {mm}.",
"(𝐛){\\rm ({\\bf b})} Momentum correlations are obtained with the implementation of three consecutive Fourier systems with lenses 50 mm 50\\,\\rm {mm}, 100 mm 100\\,\\rm {mm} and 200 mm 200\\,\\rm {mm}.As used by other authors [25], [13], we use a Gaussian model to describe the spatial structure of the two photon field at the crystal: $\\Psi (\\rho _1,\\rho _2) = N\\exp \\left[-\\frac{|\\rho _1+\\rho _2|^2}{4\\sigma _+^2}\\right]\\exp \\left[-\\frac{|\\rho _1-\\rho _2|^2}{4\\sigma _-^2}\\right],$ where $N = 1/(\\pi \\sigma _-\\sigma _+)$ is a normalization constant, $\\rho _i = (x_i,y_i)$ is the transverse position of photon $i$ ($i = 1,2$ ) and $\\sigma _{\\pm }$ are the standard deviations of the two Gaussians, giving the strength of the position and momentum correlations, $\\sigma _-$ and $\\sigma _+^{-1}$ , respectively.",
"We define $\\sigma _+$ to be equal to the standard deviation of the Gaussian field distribution of the pump laser, $\\sigma _{p}$ , with wavelength $\\lambda _{p}$ and $\\sigma _- = \\sqrt{\\frac{\\alpha L\\lambda _{p}}{2\\pi }},$ where $L$ is the length of the SPDC crystal and $\\alpha = 0.455$ is an adjustment constant [26].",
"Eq.",
"(REF ) is referred to as the transverse wave function of the post-selected two-photon field of SPDC for a Gaussian pump beam [27].",
"The joint detection probability is $\\mathcal {P}(\\rho _1,\\rho _2)\\propto |\\Psi (\\rho _1,\\rho _2)|^2$ .",
"According to the Gaussian model and the parameters of our system we predict the number of modes for joint detections in both position and momentum to be $\\left(\\sigma _+/\\sigma _-\\right)^2 \\approx 3500$ .",
"Figure: Position and momentum correlation functions in the x ' x^{\\prime } dimension for an increasing number of images.",
"(𝐚){\\rm ({\\bf a})} Measured position correlations from imaging plane scheme.",
"(𝐛){\\rm ({\\bf b})} Measured momentum correlations from far-field scheme.",
"Blue circles represents the spatial-correlation and red squares represent the reference-correlation.",
"The asymmetry of the far-field correlation function is the result of a non-uniform near collinear degenerate light cone.When measuring position correlations, we use an imaging system with a magnification of $M = 2.5$ , comprising two lenses of focal lengths $100\\,\\rm {mm}$ and $250\\,\\rm {mm}$ (Fig.",
"REF a).",
"The far-field image is obtained by using a composite Fourier system with an effective focal length of $f_{e} = 100\\,{\\rm mm}$ (Fig.",
"REF b).",
"Given our imaging and far-field system we predict a transverse correlation length $\\sigma _{{\\rm pos}} = M\\sigma _- \\approx 28\\,\\mu \\rm {m}$ and $\\sigma _{{\\rm mom}} = f_e/(k\\sigma _+) \\approx 17\\,\\mu \\rm {m}$ , where $k = k_{p}/2$ is the wavenumber of the down-converted photons.",
"Note that the position correlation length is larger than the pixel size of the camera ($16\\,\\mu \\rm {m}$ ) meaning that the position correlated photon pairs have a high probability of being detected in adjacent pixels which removes the need to count multiple photons within the same pixel.",
"A $10\\,\\rm {nm}$ bandpass filter centred at $710\\,\\rm {nm}$ with a transmission efficiency $\\eta _{{\\rm filter}} = 90\\%$ is located immediately in front of the EMCCD.",
"The EMCCD camera used in our experiment is a back-illuminated $512\\times 512$ array of $16\\times 16\\,\\mu \\rm {m^2}$ pixels optimized for visible wavelengths.",
"The quantum efficiency of detection is given as the product of the efficiency of the camera, filter and lenses.",
"Thus providing that either the signal or idler photon is detected, we predicted a heralding efficiency for detecting its entangled partner to be $\\eta \\approx 80\\%$ .",
"Image analysis.",
"A series of $100,000$ images was recorded for image and far-field configurations (see Methods).",
"The probability distributions $\\mathcal {P}(\\rho _-)$ and $\\mathcal {P}(p_+)$ were calculated respectively by counting the number of coincidences as a function of the difference (imaging plane) or sum (far-field) of the pixel coordinates in both $x^{\\prime }$ and $y^{\\prime }$ directions at the plane of the CCD array.",
"Here we define $\\rho _- \\equiv \\rho _1-\\rho _2$ and $p_+ \\equiv p_1 + p_2$ , where $p_i = (p_{x_i},p_{y_i})$ is the transverse momentum of photon $i$ .",
"The transformation from coordinates in the plane of the CCD array to the crystal is performed using the scaling factors $\\gamma _{\\rm {pos}} \\equiv 1/M$ for imaging plane and $\\gamma _{\\rm {mom}} \\equiv (k \\hbar )/f_{e}$ for far-field.",
"Our result is obtained by averaging the spatial correlation function over all recorded frames.",
"Within each linear dimension this processing reveals correlations between photon pairs, which are clearly visible on top of a broader correlation arising from uncorrelated events, as shown in Fig.",
"REF .",
"We measured the background correlation by performing the same analysis but between consecutive frames to give a reference correlation.",
"Fig.",
"REF a shows the measured coincidence counts in the image plane as a function of $x^{\\prime }_{-}$ coordinates, while coincidence counts measured in the far-field as a function of $x^{\\prime }_{+}$ are shown in Fig.",
"REF b.",
"A correlation peak in both position and momentum can already be seen in a few images and becomes significantly distinguishable above noise as more images are summed.",
"Figure: Background subtracted auto-correlation functions in position and momentum for high and low heralding efficiency configurations.",
"Auto-correlation functions for the image plane (𝐚){\\rm ({\\bf a})} and far-field (𝐛){\\rm ({\\bf b})} of the crystal, summed over 100 000 images and with the measured background correlation removed.",
"In (𝐜){\\rm ({\\bf c})} and (𝐝){\\rm ({\\bf d})} we show the same measurements performed with a lower heralding efficiency.",
"Variances along the x ' x^{\\prime }-direction of the high heralding efficiency configuration are indicated.",
"Due to increased coincidence counts attributed to smearing, the correlation function for y - ' =±1y^{\\prime }_{-} = \\pm 1 were set to 0 in (a){\\rm (a)} and (c){\\rm (c)} for clarity.By subtracting the reference from the spatial correlation function it is possible to remove unwanted pixel defects and obtain a strong correlation in position and anti-correlation in momentum as indicated in Fig.",
"REF a and Fig.",
"REF b respectively.",
"Since we cannot distinguish the detection of more than one photon on the same pixel, we are unable to measure the correlation function for ${\\rho }^{\\prime }_1 = {\\rho }^{\\prime }_2$ , for which a value equal to 0 has been assigned.",
"To confirm that these correlations arise between photon pairs the neutral density filter used to attenuate the pump beam was placed after the BBO crystal to reduce the heralding efficiency to approximately $2.0\\%$ , whilst ensuring the same photon flux.",
"As shown in Fig.",
"REF c and Fig.",
"REF d, we do not observe a strong correlation in these measurements; however along the $y^{\\prime }_{-}$ axis in the image plane measurements we observe a correlation indicative of charge smearing between pixels in the readout direction.",
"EPR-type correlations.",
"From the thresholded images we can calculate the joint probability distributions for both the $x^{\\prime }$ and $y^{\\prime }$ coordinates in the image plane and far-field, as shown in Fig.",
"REF .",
"Fig.",
"REF a and Fig.",
"REF b show the coincidence counts in the planes $(x_1,x_2)$ and $(y_1,y_2)$ , while Fig.",
"REF c and Fig.",
"REF d show the coincidence counts in the planes $(p_{x_1},p_{x_2})$ and $(p_{y_1},p_{y_2})$ .",
"In the $y^{\\prime }$ direction we observe a strong correlation resulting from charge smearing, thus preventing any meaningful analysis from image plane measurements (Fig.",
"REF b); however the expected anti-correlation in the far-field (Fig.",
"REF d) is still evident.",
"Note that background subtraction has been performed.",
"Figure: Joint probability distributions for x ' x^{\\prime } and y ' y^{\\prime }-coordinates.",
"The probability distributions for joint detections in both x ' x^{\\prime } and y ' y^{\\prime }-coordinates in the image plane (𝐚{\\rm ({\\bf a}} and 𝐛){\\rm {\\bf b})} and far-field (𝐜{\\rm ({\\bf c}} and 𝐝){\\rm {\\bf d})} are shown respectively.",
"Charge smearing gives rise to an artificial correlation in the y ' y^{\\prime } direction (𝐛{\\rm ({\\bf b}} and 𝐝){\\rm {\\bf d})}.It has been shown that EPR-like correlations can be identified by violating the inequality [28], [29] $\\Delta ^2_{\\rm {min}}(x_1|x_2)\\Delta ^2_{\\rm {min}}(p_{x_1}|p_{x_2})>\\frac{\\hbar ^2}{4},$ where $\\Delta ^2_{\\rm {min}}(r_1|r_2)$ is the minimum inferred variance, describing the minimum uncertainty in measuring the variable $r_1$ conditional on the measurement of variable $r_2$ .",
"The minimum inferred variance for $x$ -coordinates in the far-field measurement is defined as $\\Delta ^2_{\\rm {min}}(p_{x_1}|p_{x_2})=\\int \\mathcal {P}(p_{x_2})\\Delta ^2(p_{x_1}|p_{x_2})dp_{x_2}.$ We define the minimum inferred variance for the $x$ -coordinates in the image plane with a similar approach.",
"Fitting Gaussians to the joint probabilities in the $x$ -direction (Fig.",
"REF a and Fig.",
"REF c) we obtain the variances shown in Table REF .",
"Substituting these quantities into Eq.",
"(REF ) we find $\\Delta ^2_{\\rm {min}}(x_1|x_2)\\Delta ^2_{\\rm {min}}(p_{x_1}|p_{x_2})=&(6.6\\pm 1.0)\\times 10^{-4}\\hbar ^2,\\\\\\Delta ^2_{\\rm {min}}(x_2|x_1)\\Delta ^2_{\\rm {min}}(p_{x_2}|p_{x_1})=&(6.2\\pm 0.9)\\times 10^{-4}\\hbar ^2,$ indicating that the transverse DOF of the two photon field exhibits EPR non-locality.",
"This result is in good agreement with the theoretical prediction (from Eqs.",
"(REF ) and (REF )) of $\\approx 3\\times 10^{-4}\\hbar ^2$ and, although performed with background subtraction, is an order of magnitude smaller than those in Refs.",
"[4], [13].",
"High-dimensional entanglement.",
"From the joint distributions in Fig.",
"REF , we can estimate the ratio $(\\sigma _+/\\sigma _-)$ for both $x$ and $y$ transverse DOF, in the image plane (IP) and far-field (FF).",
"We define the maximum number of joint detections for our position and momentum measurements to be $D^{\\rm max}_{\\rm pos}=[(\\sigma _{x^{\\prime }_+}/\\sigma _{x^{\\prime }_-})(\\sigma _{y^{\\prime }_+}/\\sigma _{y^{\\prime }_-})]_{\\rm {IP}}$ and $D^{\\rm max}_{\\rm mom}=[(\\sigma _{x^{\\prime }_-}/\\sigma _{x^{\\prime }_+})(\\sigma _{y^{\\prime }_-}/\\sigma _{y^{\\prime }_+})]_{\\rm {FF}}$ .",
"We must also account for the proportion of the beam measured by the detector array which is calculated by integrating the fitted Gaussian functions over the array size.",
"In all cases we find this gives access to more than $85\\%$ of the available states.",
"Due to charge smearing in the image plane measurements we must estimate the dimensionality in $y$ based on that of the $x$ dimension, which is supported by the circular symmetry of Fig.",
"REF a.",
"Thus, the measured dimensionality in position and momentum is found to be $D_{\\rm pos} = D^{\\rm max}_{\\rm pos} \\times 0.95^2 \\approx 3200$ and $D_{\\rm mom} = D^{\\rm max}_{\\rm mom} \\times 0.95 \\times 0.85 \\approx 2500$ , respectively.",
"To the best of our knowledge this represents the largest dimensionality for any experiment using entangled spatial states of photons."
],
[
"Discussion",
"We have performed a multimode detection of around 2500 spatially entangled states of photon pairs produced by SPDC.",
"We have shown that it is possible to utilize an EMCCD camera to measure spatial correlations between photon pairs in both the image plane and far-field of the down-conversion crystal.",
"After background subtraction, we found that the spatial correlations violate EPR criterion.",
"We acknowledge that the use of background-subtracted measurements do not meet the strict requirements for tests of EPR non-locality.",
"If the background was included in the data, it would lead to an overestimate of the standard deviations of the correlations.",
"Furthermore, in order to satisfy non-locality, the measurement of each photon must be performed outside the light-cone of its entangled partner.",
"This issue could be solved in a modified setup in which the two-photons are split and sent to two spatially separated cameras.",
"However, our results are in close agreement with the theoretical predictions, providing evidence that EMCCD technology can already be used in the characterization of spatial correlations of photons.",
"The ability to access both transverse DOF of photons enables a large state space suggesting the use of spatial states in quantum communication and quantum information processing.",
"In this case, information should be encoded and processed in an alphabet defined as orthogonal spatial states and subsequently read by projecting the full transverse field of the single photon on to a complete set of orthogonal spatial states.",
"Thus, low-noise and efficient 2D detector arrays are essential in the use of spatial states in these applications, for which our work is a significant step forward.",
"Future developments in imaging technologies will likely result in 2D detector arrays with even lower noise and widen their application in quantum optics.",
"Table: Variances obtained from Gaussian fittings of background subtracted correlation functions shown Fig.",
"."
],
[
"Methods",
"Thresholding.",
"There are several noise sources that affect an EMCCD camera.",
"As well as stray photons, the camera is subject to dark noise; thermally excited electrons, clock induced noise resulting from spurious electrons created during charge transfers between pixels, and readout noise generated by the readout register.",
"We characterized the level of total noise in our camera when operated at the maximum gain by obtaining $100,000$ images while the laser is blocked but with the shutter on the camera open.",
"Performing an analysis similar to that of Ref.",
"[19] we find a Gaussian distribution of readout noise, centred at a value $390\\,e^-$ with a standard deviation $\\sigma _{\\rm noise} = 6\\,e^-$ , in addition to a long tail that decreases exponentially for higher readout values.",
"Our thresholding is performed by subtracting the mean background value for each pixel and assigning a 1 to pixels with values greater than one standard deviation of noise, and a 0 to all pixels having smaller values.",
"This binary thresholding was found to reduce the level of noise whilst maximizing the correlation signal strength.",
"Camera Settings.",
"The camera was operated at $-85^0\\rm {C}$ , with a horizontal pixel shift readout rate of $1\\,\\rm {MHz}$ , a vertical pixel shift every $0.3\\,\\mu \\rm {s}$ and a vertical clock amplitude voltage of $+4$ above the default factory setting.",
"The optimum photon flux incident upon the camera was chosen to maximize the strength of the correlation signal relative to noise sources.",
"In our experiment the image-plane measurements were made with an exposure time of $0.4\\,\\rm {ms}$ , while the far-field measurements are obtained using an exposure time of $1\\,\\rm {ms}$ , in both cases giving an average photon flux of approximately 0.02 photons/pixel, which is equivalent to the level of noise.",
"A region of interest measuring $201 \\times 201$ pixels was chosen to observe most of the down-converted field, which resulted in a frame rate of approximately $5\\,{\\rm Hz}$ .",
"Acknowledgements M.J.P.",
"would like to thank the Royal Society and the Wolfson Foundation.",
"M.J.P.",
"and R.W.B.",
"thank the US DARPA/DSO InPho program.",
"M.P.E., D.S.T., F.I., R.E.W., G.S.B.",
"and M.J.P.",
"acknowledge the financial support from the UK EPSRC.",
"J.L., M.A., and R.W.B thank Canada's CERC program.",
"The authors would like to thank N. Radwell for useful discussions.",
"Author contributions M.J.P., R.W.B., G.S.B.",
"and J.L.",
"conceived the experiment.",
"M.P.E., D.S.T., F.I.",
"and R.E.W.",
"designed and performed the experiment.",
"M.P.E., D.S.T., F.I., J.L., M.A.",
"and M.J.P.",
"designed the measurement and analysis programs.",
"M.P.E., D.S.T., F.I., J.L.",
"and M.J.P.",
"analysed the results.",
"All authors contributed to the writing of the manuscript."
]
] | 1204.1293 |
[
[
"Positive stationary solutions for p-Laplacian problems with nonpositive\n perturbation"
],
[
"Abstract The paper is devoted to the existence of positive solutions of nonlinear elliptic equations with $p$-Laplacian.",
"We provide a general topological degree that detects solutions of the problem $$ \\{{array}{l} A(u)=F(u) u\\in M {array}.",
"$$ where $A:X\\supset D(A)\\to X^*$ is a maximal monotone operator in a Banach space $X$ and $F:M\\to X^*$ is a continuous mapping defined on a closed convex cone $M\\subset X$.",
"Next, we apply this general framework to a class of partial differential equations with $p$-Laplacian under Dirichlet boundary conditions."
],
[
"Introduction",
"We shall be concerned with solutions to the following nonlinear boundary value problem $\\left\\lbrace \\begin{array}{l}-\\mathrm {div} (|\\nabla u(x)|^{p-2} \\nabla u(x)) = f(x,u (x)), \\ x\\in \\Omega ,\\\\\\ \\ u(x)\\ge 0,\\ x\\in \\Omega ,\\\\\\ \\ u(x) = 0,\\ x\\in \\partial \\Omega \\end{array} \\right.$ where $\\Omega \\subset \\mathbb {R}^N$ ($N\\ge 1$ ) is a bounded domain with the smooth boundary $\\partial \\Omega $ , $p\\ge 2$ and $f:\\Omega \\times [0,+\\infty ) \\rightarrow \\mathbb {R}$ is a Carathéodory function (Recall that we say that $f:\\Omega \\times [0,+\\infty )\\rightarrow \\mathbb {R}$ is a Carathéodory function if $f(\\cdot , s)$ is measurable for all $s\\in [0,+\\infty )$ and $f(x,\\cdot )$ is continuous for almost all $x\\in \\Omega $ .",
").",
"The differential term $\\mathrm {div} (|\\nabla u (x)|^{p-2}\\nabla u(x))$ is referred to as the $p$ -Laplacian of $u$ at a point $x\\in \\Omega $ .",
"We search for weak solutions in the Sobolev space $W_{0}^{1,p}(\\Omega )$ , i.e.",
"$u\\in W_{0}^{1,p} (\\Omega )$ such that $\\int _{\\Omega } |\\nabla u(x)|^{p-2} \\nabla u(x) \\cdot \\nabla v(x) \\,\\mathrm {d}x = \\int _{\\Omega } f(x, u(x)) v(x) \\,\\mathrm {d}x \\ \\ \\mbox{ for all } \\ v\\in W_{0}^{1,p} (\\Omega ).$ Such boundary problems with $p$ -Laplace were widely studied by many authors who used various methods.",
"Let us mention just a few.",
"Equations with the one dimensional $p$ -Laplacian, i.e.",
"when $N=1$ , were studied by Manásevich, Njoku i Zanolin [17], Drábek, García-Huidobro and Manásevich [7] and as well as by Kryszewski and the author [5].",
"In the general case, i.e.",
"when $N>1$ , positive solutions of $p$ -Laplace problems have been studied by a number of authors, e.g.",
"Huang [13], Drábek and Pohozaev [6], Cañada, Drábek and Gámez [3], Filippiakis, Gasiński and Papageorgiou [9] or Montreanu D., Montreanu V. V. and Papageorgiou [18], Väth [19].",
"Generally speaking, in the above mentioned papers, either $N=1$ or $N$ is arbitrary but the right has side of the equation - the function $f$ is assumed to be non-negative or satisfy some monotonicity assumptions.",
"This makes possible to apply Krasnosel'skii's fixed point theorem (in general, fixed point index in cones) or variational methods.",
"These assumptions on $f$ seem rather restrictive and sometimes unnatural, especially, when we take into account physical interpretation of the considered boundary value problem.",
"In this paper, we do not require $f$ to be non-negative or monotone.",
"A general tool for detection of nonnegative solutions is provided.",
"It is based on the geometric idea of tangency and using fixed point index in cones.",
"We construct a topological degree for perturbations of maximal monotone operators with respect to closed convex cones.",
"Next we prove appropriate index formulae, which together with the homotopy property, allow us to compute the topological degree in specific examples.",
"It is noteworthy, that this setting does not require variational structure and can be also used for systems of $p$ -Laplace problems.",
"In this paper, we apply the method to show the following existence criterion Theorem 1.1 Suppose that a Carathéodory function $f:\\Omega \\times [0,+\\infty )\\rightarrow \\mathbb {R}$ and $\\rho _0, \\rho _\\infty \\in L^\\infty (\\Omega )$ satisfy the following conditions $ & & \\mbox{there is } C>0 \\mbox{ such that }|f(x,s)| \\le C(1+s^{p-1}) \\mbox{ for all } s\\ge 0 \\mbox{ and a.a. } x\\in \\Omega ;\\\\ & & \\lim _{s\\rightarrow 0^+} \\frac{f(x,s)}{s^{p-1}} = \\rho _0 (x)\\mbox{ and } \\lim _{s\\rightarrow \\infty } \\frac{f(x,s)}{s^{p-1}} = \\rho _\\infty (x) \\mbox{ uniformly with respect to } x\\in \\Omega .$ If the principal eigenvalue $\\lambda _{1,p}$ of the $p$ -Laplace operator lies between $\\rho _0$ and $\\rho _\\infty $ , i.e.",
"either $\\rho _0(x) < \\lambda _{1,p} <\\rho _\\infty (x)$ , for a.a. $x\\in \\Omega $ , or $ \\rho _\\infty (x) < \\lambda _{1,p} < \\rho _0 (x)$ , for a.a. $x\\in \\Omega $ , then the problem (REF ) admits a nontrivial weak solution $u\\in W_{0}^{1,p} (\\Omega )$ such that $u(x)\\ge 0$ for a.e.",
"$x\\in \\Omega $ .",
"Here the principal eigenvalue $\\lambda _{1,p}$ is the smallest real number $\\lambda $ such that the problem $\\left\\lbrace \\begin{array}{l}-\\mathrm {div} (|\\nabla u(x)|^{p-2} \\nabla u(x)) = \\lambda |u (x)|^{p-2} u(x),\\ x\\in \\Omega \\\\\\ \\ u(x)\\ge 0,\\ x\\in \\Omega \\\\\\ \\ u(x) = 0,\\ x\\in \\partial \\Omega \\end{array} \\right.$ admits a nonzero weak solution (see Remark REF for more details).",
"Theorem REF corresponds directly to the result of [13], obtained by different methods (the sub-supersolution technique and the existence result for variational inequalities) and under different assumptions corresponding to the inequality $\\rho _\\infty < \\lambda _{1,p} <\\rho _0$ .",
"Our general method allows us to consider also the case $\\rho _\\infty > \\lambda _{1,p}> \\rho _0$ .",
"The paper is organized as follows.",
"In Section 2 we develop a topological degree detecting coincidence points of maximal monotone operators and continuous operators in closed convex cones.",
"This general tool will be useful if we rewrite the problem (REF ) in the form $\\left\\lbrace \\begin{array}{l} A_p u = N_f (u)\\\\u\\in M_p\\end{array}\\right.$ where $A_p:L^p(\\Omega ) \\supset D(A_p)\\rightarrow L^p (\\Omega )^*$ is the maximal monotone operator determined by the $p$ -Laplacian, $N_f:L^p(\\Omega )\\rightarrow L^p (\\Omega )^*$ is the Nemytzkii type operator associated with $f$ and $M_p$ is the closed convex cone of all non-negative elements in the space $L^p (\\Omega )$ .",
"Section 3 provides a general setting in which assumptions of Section 2 are verified.",
"Next, in Section 4 we show that the problem (REF ) falls into the setting and, using our topological degree together with spectral properties of $p$ -Laplacian, we derive topological index formulae.",
"They turn out to be essential in the proof of Theorem REF , which is provided at the end of Section 4.",
"Notation If $X$ is a metric space and $B \\subset X$ , then $\\partial B$ and $cl B$ stand for the boundary of $B$ and the closure of $B$ , respectively.",
"If $x_0\\in X$ and $r>0$ , then $B (x_0,r):=\\lbrace x\\in M\\mid d(x,x_0)<r \\rbrace $ .",
"If $E$ is a normed space, then by $\\Vert \\cdot \\Vert $ we denote its norm.",
"If $E$ is a normed space and $E^*$ its dual space (of all continuous linear functionals), then $\\langle \\cdot , \\cdot \\rangle = \\langle \\cdot , \\cdot \\rangle _{E}:E^*\\times E \\rightarrow \\mathbb {R}$ denotes the duality operator $\\langle p,u\\rangle :=p(u)$ , $p\\in E^*$ , $u\\in E$ .",
"If $V$ is another normed space then ${\\cal L}(V,E)$ stands for the space of all bounded linear operators with domain $V$ and values in $E$ with the operator norm denoted by $\\Vert \\cdot \\Vert _{{\\cal L}(V,E)}$ or simply $\\Vert \\cdot \\Vert $ if no confusion may appear.",
"For $x\\in \\mathbb {R}^N$ , $N\\ge 1$ , $|x|$ denotes the Euclidean norm of $x$ and $x\\cdot y$ is the Euclidean scalar product of $x,y\\in \\mathbb {R}^N$ ."
],
[
"Constrained topological degree for perturbations of maximal monotone operators ",
"In this section we provide a construction of a topological degree detecting solutions of the abstract constrained problem $\\left\\lbrace \\begin{array}{l}0 \\in -A u +F(u)\\\\u\\in M\\end{array}\\right.$ where $A:X\\supset D(A)\\multimap X^*$ is a densely defined maximal monotone operator, the constraint set $M$ is a subset of $X$ and $F: \\overline{U} \\rightarrow X^*$ is a continuous mapping defined on the closure of an open bounded $U\\subset M$ .",
"Throughout the whole section we make the following assumptions $({\\cal A}_1)$ there is a homeomorphism $N:X\\rightarrow X^*$ such that $N$ is bounded on bounded sets and the mappings $J_\\alpha : X^*\\rightarrow X$ , $\\alpha >0$ , $J_\\alpha (\\tau ):= u, \\ \\mbox{ where } u\\in D(A) \\mbox{ is the unique element such that } \\tau \\in (N+\\alpha A)(u),$ are well defined and continuous; $({\\cal A}_2)$ the mapping ${\\cal J}:X^* \\times (0,+\\infty )\\ni (\\tau , \\alpha )\\mapsto J_{\\alpha } (\\tau )\\in X$ is bounded on bounded sets and such that ${\\cal J}_{\\ |\\, X^* \\times [\\alpha _1, \\alpha _2]}$ is completely continuous if $0<\\alpha _1 \\le \\alpha _2$ ; $({\\cal A}_3)$ $M\\subset X$ is a neighborhood retract of $X$ , $J_\\alpha (N(M)) \\subset M$ for $\\alpha >0$ , and $M^* :=N(M)$ is an ${\\cal L}$ -retract (see [2] and [5]), i.e.",
"there exist a retraction $r:B(M^*,\\eta )\\rightarrow M^*$ with some $\\eta >0$ and a constant $L>0$ such that $\\Vert r(\\tau )- \\tau \\Vert \\le L d_{M^*} (\\tau ) \\ \\mbox{ for all } \\ \\tau \\in B(M^*, \\eta );$ $({\\cal A}_4)$ $F$ is continuous, bounded on bounded sets and satisfies the tangency condition $F(N^{-1}(\\tau ))\\in T_{M^{*}} (\\tau ), \\ \\mbox{ for } \\tau \\in N(\\overline{U}),$ where $T_{M^*} (\\tau )$ is the Bouligand tangent cone to $M^*$ at the point $\\tau $ , i.e.",
"$T_{M^*} (\\tau ): = \\left\\lbrace \\theta \\in X^* \\mid \\liminf _{\\alpha \\rightarrow 0^+} \\frac{d_{M^*} (\\tau + \\alpha \\theta ) }{\\alpha } =0 \\right\\rbrace .$ Remark 2.1 Since maximal monotone operators have closed graphs, it can be shown that in order to verify the continuity of the mapping ${\\cal J}_{\\ |\\, X^* \\times [\\alpha _1, \\alpha _2]}$ from condition $({\\cal A}_2)$ it is sufficient to know that it maps bounded sets into relatively compact ones.",
"Our goal is to transform the problem (REF ) into a fixed point one in $M$ and for which fixed point index theory can be used.",
"To this end define $\\Phi _{\\alpha }=\\Phi _{\\alpha }^{A,F}:\\overline{U} \\rightarrow M$ by $\\Phi _\\alpha (u):= J_{\\alpha } \\left(r \\left(N(u) + \\alpha F(u)\\right)\\right), \\ \\ u\\in \\overline{U},$ whenever $0<\\alpha < \\eta /\\sup \\lbrace \\Vert F(u)\\Vert \\mid u\\in \\overline{U} \\rbrace $ .",
"Obviously, it is well defined, since for such $\\alpha $ one has $(N+\\alpha F) (\\overline{U}) \\subset B(M^*,\\eta )$ .",
"Moreover, observe that due to the assumptions, the mapping $r \\circ (N+\\alpha F)$ is bounded on bounded sets and, by $({\\cal A}_2)$ , $\\Phi _\\alpha $ is compact.",
"Exploiting the tangency condition (REF ) and the inequality (REF ) together with compactness, we obtain the following localization of fixed points results.",
"Proposition 2.2 If $K\\subset \\overline{U}$ is a closed set such that $\\lbrace u\\in \\overline{U}\\cap D(A) \\mid 0\\in -Au + F(u)\\rbrace \\cap K = \\emptyset ,$ then, for sufficiently small $\\alpha >0$ , $\\lbrace u\\in \\overline{U} \\mid \\Phi _\\alpha (u)=u\\rbrace \\cap K = \\emptyset .$ Remark 2.3 Actually the tangency condition (REF ) and the continuity of $F\\circ N^{-1}$ imply $F\\left( N^{-1}(\\tau ) \\right) \\in C_{M^*} (\\tau ) := \\left\\lbrace \\theta \\in X^* \\mid \\lim _{\\alpha \\rightarrow 0^+, \\, \\varrho \\rightarrow \\tau , \\, \\varrho \\in M} \\frac{d_M (\\varrho + \\alpha \\theta )}{\\alpha } =0 \\right\\rbrace \\, \\mbox{ for all } \\tau \\in N(U).$ Indeed $F \\left( N^{-1}(\\tau ) \\right)=\\lim _{\\varrho \\rightarrow \\tau } F \\left( N^{-1}(\\varrho ) \\right) \\in \\mathop {\\mathrm {Liminf}\\,}_{\\varrho \\rightarrow \\tau , \\, \\varrho \\in M^*} T_{M^*} (\\varrho ) \\subset C_{M^*} (\\tau ).$ The proof of the latter inclusion can be found in [1].",
"Lemma 2.4 (i) The graph $\\mathrm {Gr}(A):= \\lbrace (u, \\tau )\\in X\\times X^* \\mid u\\in D(A)\\rbrace $ is closed; (ii) If a sequence of pairs $(u_n ,\\tau _n) \\in \\mathrm {Gr}(A)$ , $n\\ge 1$ , is bounded, then the sequence $(u_n)$ has a convergent subsequence.",
"Proof: (i) Take any sequence of points $(u_n, \\tau _n)\\in \\mathrm {Gr}(A)$ , $n\\ge 1$ , such that $(u_n, \\tau _n)\\rightarrow (u_0, \\tau _0)$ in $X\\times X^*$ , as $n\\rightarrow +\\infty $ , for some $(u_0, \\tau _0) \\in X\\times X^*$ .",
"Clearly, $\\tau _n \\in Au_n$ , and this gives $N(u_n) + \\tau _n \\in (N+A)(u_n)$ , which, by $({\\cal A}_1)$ , gives $u_n = J_{1} (N(u_n)+\\tau _n)$ , $n\\ge 1$ .",
"Hence, using the continuity of $N$ and $J_{1}$ yields $u_n = J_{1}(N(u_n)+\\tau _n)) \\rightarrow J_{1} (N(u_0) + \\tau _0)$ as $n\\rightarrow +\\infty $ , which implies $u_0 = J_{1} (N(u_0) + \\tau _0)$ , i.e.",
"$\\tau _0\\in Au_0$ .",
"This shows that $\\mathrm {Gr}(A)$ is closed.",
"(ii) Note that, for each $n\\ge 1$ , $u_n = J_{1} (N(u_n) + \\tau _n) \\in J_{1} (N(B(0,R))+B(0,R))$ , where $R>0$ is a constant such that $\\Vert u_n\\Vert _X \\le R$ and $\\Vert \\tau _n\\Vert _{X^*} \\le R$ for $n\\ge 1$ .",
"The boundedness of $N$ and $({\\cal A}_2)$ imply that the set $(u_n)$ is a sequence of elements of the relatively compact set $J_{1} (N(B(0,R))+B(0,R))$ .",
"$\\square $ Proof of Proposition REF : Suppose to the contrary that there exists a sequence $(\\alpha _n)$ such that $\\alpha _n\\rightarrow 0^+$ such that for each $n\\ge 1$ there is $u_n\\in K$ with $\\Phi _{\\alpha _n} (u_n) = u_n$ , that is $N(u_n)+ \\alpha _n \\tau _n = r(N(u_n) + \\alpha _n F(u_n)) \\mbox{ for some } \\tau _n \\in Au_n.$ In view of (REF ), one has $\\alpha _n \\Vert \\tau _n - F(u_n)\\Vert & = & \\Vert r(N(u_n) + \\alpha _n F(u_n)) - ( N (u_n) + \\alpha _n F (u_n) ) \\Vert \\\\& \\le & L d_{M^*} (N(u_n) + \\alpha _n F(u_n)) \\ \\ \\mbox{ for all } n\\ge 1.",
"\\nonumber $ This implies $\\Vert \\tau _n \\Vert \\le \\Vert F(u_n)\\Vert + L \\alpha _{n}^{-1} d_{M^*} (N(u_n) + \\alpha _n F(u_n)) \\le (1+L) \\Vert F(u_n)\\Vert , \\ n\\ge 1,$ which means that $(\\tau _n)$ is bounded.",
"Therefore, by use of Lemma REF (ii), we may assume without loss of generality that $u_n \\rightarrow u_0$ for some $u_0\\in M$ .",
"Now using (REF ) and putting $p_n:=N(u_n)$ , $n\\ge 0$ , we see that $\\Vert \\tau _n - F(u_n)\\Vert \\le L \\cdot \\frac{d_{M^*} (p_n + \\alpha _n F(N^{-1}(p_0))}{\\alpha _n} + L\\Vert F(u_n)-F(u_0)\\Vert , \\ \\ \\mbox{ for } n\\ge 1.$ By the tangency condition $({\\cal A}_4)$ and Remark REF together with the continuity of $F$ , we get that $\\tau _n \\rightarrow F(u_n)$ as $n\\rightarrow +\\infty $ .",
"Hence, we have obtained that $(u_n, \\tau _n)\\rightarrow (u_0, F(u_0))$ and, by Lemma REF (i), $(u_0,F(u_0)) \\in \\mathrm {Gr}(A)$ , i.e.",
"$F(u_0)\\in A u_0$ , a contradiction completing the proof.",
"$\\square $ Now we put $\\mathrm {Deg}_M ( A, F, U ) := \\lim _{\\alpha \\rightarrow 0^+} \\mathrm {ind}_M ( \\Phi _\\alpha , U ),$ where $\\mathrm {ind}_M$ stands for the fixed point index for compact mappings of absolute neighborhood retracts due to Granas – see [12] or [8] for details.",
"We call this number as the topological degree of coincidence (or just topological degree) of $A$ and $F$ with respect to $M$.",
"Theorem 2.5 The coincidence degree defined by (REF ) is well defined and has the following properties: (i) (existence) if $\\mathrm {Deg}_M (A,F, U)\\ne 0$ , then there exists $u\\in U\\cap D(A)$ such that $0\\in -Au+F(u)$ ; (ii) (additivity) if $U_1, U_2$ are open disjoint subsets of a bounded open $U\\subset M$ and $0\\notin (-A+F)(\\overline{U} \\setminus (U_1\\cup U_2))$ , then $\\mathrm {Deg}_M (A,F, U) = \\mathrm {Deg}_M (A,F, U_1) +\\mathrm {Deg}_M (A,F, U_2);$ (iii) (homotopy invariance) if $H:\\overline{U} \\times [0,1]\\rightarrow X^*$ is a continuous and bounded mapping such that $H( N^{-1}(\\tau ), t) \\in T_{M^*} (\\tau ) \\mbox{ for all } \\tau \\in N(\\overline{U}), \\, t>0,$ and $0\\notin -Au+H(u,t)$ for all $u\\in \\partial U\\cap D(A)$ and $t\\in [0,1]$ , then $\\mathrm {Deg}_M (A,H(0,\\cdot ), U) = \\mathrm {Deg}_M (A, H(1,\\cdot ), U);$ (iv) (normalization) if $M$ is bounded and the mapping $\\widetilde{\\cal J}: X^* \\times [0,+\\infty ) \\ni (\\tau , \\alpha ) \\mapsto J^{\\alpha } \\tau \\in X$ with $J^0 = N^{-1}$ is continuous, then $\\mathrm {Deg}_M (A,F, M) = \\chi (M)$ .",
"Proof: Note that for sufficiently small $\\alpha >0$ it follows from Propostion REF that $\\Phi _\\alpha $ has no fixed point in $\\partial U$ , i.e.",
"the fixed point index $\\mathrm {ind}_M (\\Phi _\\alpha , U)$ is well defined.",
"If $\\alpha _1, \\alpha _2>0$ are small enough, then, by $({\\cal A}_2)$ , $\\Phi _{\\alpha _1}$ is homotopic with $\\Phi _{\\alpha _2}$ , which gives $\\mathrm {ind}_M (\\Phi _{\\alpha _1}, U)= \\mathrm {ind}_M (\\Phi _{\\alpha _2}, U)$ , which means that the limit in (REF ) exists.",
"(i) Suppose to the contrary that there is no $u\\in U \\cap D(A)$ such that $0\\in - Au + F(u)$ .",
"Then, in view of Proposition REF , for sufficiently small $\\alpha >0$ the mappings $\\Phi _\\alpha $ have no fixed points in $\\overline{U}$ , i.e.",
"$\\mathrm {Deg}_M (A,F,U) = \\mathrm {ind}_M (\\Phi _\\alpha , U)=0$ , a contradiction.",
"(ii) Due to Proposition REF , for sufficiently small $\\alpha >0$ , $\\Phi _\\alpha $ has no fixed points in $\\overline{U} \\setminus (U_1\\cup U_2)$ .",
"Therefore, by the definition of the degree, $\\mathrm {Deg}_M (A, F, U)=\\mathrm {ind}_M (\\Phi _\\alpha , U) \\ \\ \\mbox{ and } \\ \\ \\mathrm {Deg}_M (A, F, U_k)=\\mathrm {ind}_M (\\Phi _\\alpha , U_k) \\mbox{ for } k=1,2.$ By the additivity property of the fixed point index $\\mathrm {ind}_M (\\Phi _\\alpha , U) = \\mathrm {ind}_M (\\Phi _\\alpha , U_1) + \\mathrm {ind}_M (\\Phi _\\alpha , U_2),$ which together with the earlier equalities gives the desired additivity of the degree.",
"(iii) For sufficiently small $\\alpha >0$ one can define $\\Phi _\\alpha :\\overline{U}\\times [0,1] \\rightarrow M$ by $\\Phi _\\alpha (u,t):= J_{\\alpha }\\left( r (N(u) + \\alpha H(u,t)) \\right), \\ u\\in \\overline{U}, \\ t\\in [0,1].$ Proceeding along the lines of the proof of Proposition REF we can prove that for sufficiently small $\\alpha >0$ $\\Phi _\\alpha (u,t)\\ne u \\mbox{ for all } u\\in \\partial U, \\ t\\in [0,1].$ Hence, by the homotopy invariance of the fixed point index and the formula defining the degree, $\\mathrm {Deg}_M (A,H(\\cdot , 0),U) = \\mathrm {ind}_M (\\Phi _\\alpha (\\cdot ,0), U) = \\mathrm {ind}_M (\\Phi _\\alpha (\\cdot ,1), U)= \\mathrm {Deg}_M (A,H(\\cdot , 1),U).$ (iv) Take small $\\alpha >0$ such that $\\Phi _\\alpha $ is well defined.",
"Then $\\mathrm {Deg}_M (A, F, M) = \\mathrm {ind}_M (\\Phi _\\alpha , M).$ Note that the normalization property for the fixed point index states that the homomorphism $H_* (\\Phi _\\alpha ):H_* (M)\\rightarrow H_*(M)$ induced on (singular) homology spaces is a Leray endomorphism and $\\mathrm {ind}_M (\\Phi _\\alpha , M) = \\Lambda (\\Phi _\\alpha )$ where $\\Lambda (\\Phi _\\alpha )$ is the generalized Leschetz number of the compact map $\\Phi _\\alpha $ – see [8] or [12].",
"Further, consider $\\Psi :M\\times [0,1]\\rightarrow M$ given by $\\Psi (u,t) := \\widetilde{\\cal J} (r(N(u)+t\\alpha F(u)),t\\alpha ), \\ \\ u\\in M, \\ \\ t\\in [0,1].$ By the assumption, $\\Psi $ is a continuous homotopy joining $\\Psi (\\cdot , 1) = \\Phi _\\alpha $ with the identity map $\\mathrm {id}_M: M\\rightarrow M$ .",
"Hence, for the maps induced on homology spaced one has $H_* (\\Phi _\\alpha ) = H_* (\\mathrm {id}_M)=\\mathrm {id}_{H_* (M)}$ and, since $H_* (\\Phi _\\alpha )$ is an endomorphism Leray, we infer that $\\Lambda (\\Phi _\\alpha )=\\sum _{n=0}^{\\infty } (-1)^n \\dim H_n (M) = \\chi (M)$ , which together with (REF ) ends the proof.",
"$\\square $ We end this section with a general result, which allows us to compute the degree is specific situations (comp.",
"[5]).",
"Theorem 2.6 Let $M$ and $M^*$ be closed convex cones and that the mappings $A$ and $N$ are homogeneous with the same degree (i.e.",
"there exists $\\gamma >0$ such that $A(a u) = a^{\\gamma } A(u)$ , $u\\in D(A)$ , $a>0$ , and $N(au)=a^\\gamma N(u)$ for all $u\\in X$ , $a>0$ .",
").",
"Suppose that there exists $\\lambda _1 \\ge 0$ satisfying the following conditions $({\\cal M}_1)$ $(A-\\lambda N)^{-1}(\\lbrace 0\\rbrace ) \\cap M = \\lbrace 0\\rbrace \\ \\mbox{ for } \\ \\lambda \\ne \\lambda _1;$ $({\\cal M}_2)$ there exists $\\tau _0 \\in M^*$ such that $(A-\\lambda N)^{-1} (\\lbrace \\tau _0 \\rbrace ) \\cap M =\\emptyset $ for $\\lambda >\\lambda _1$ .",
"Then $\\mathrm {Deg}_M (A, \\lambda N, B_M (0,\\delta )) = \\left\\lbrace \\begin{array}{ll} 1, & \\lambda < \\lambda _1,\\\\0, & \\lambda >\\lambda _1,\\end{array} \\right.$ for any $\\delta >0$ .",
"Proof: Note that in view of $({\\cal M}_1)$ the topological degree $\\mathrm {Deg}_M (A, \\lambda N, B_M (0, \\delta ))$ is well defined.",
"Now fix $\\lambda <\\lambda _1$ .",
"By the very construction, for sufficiently small $\\alpha >0$ , $\\mathrm {Deg}_M (A, \\lambda N, B_M (0,\\delta )) = \\mathrm {ind}_M (\\Phi _\\alpha , B_M (0,\\delta ))$ where $\\Phi _\\alpha :\\overline{B_M (0,\\delta )}\\rightarrow M$ is given by $\\Phi _\\alpha (u):= J_\\alpha (r(N(u) + \\alpha \\lambda N(u))), \\ u \\in \\overline{B_M (0,\\delta )}.$ Define $\\Theta : \\overline{B_M (0,\\delta )}\\times [0,1]\\rightarrow M$ by $\\Theta (u,t):= t \\Phi _\\alpha (u), \\ u \\in \\overline{B_M (0,\\delta )}, \\, t\\in [0,1].$ Suppose there are $u\\ne 0$ and $t\\in [0,1]$ such that $\\Theta (u,t)=u$ .",
"Then $0 \\in -A(u) + \\mu N(u)$ with $\\mu :=(t^\\gamma -1)/\\alpha + t^\\gamma \\lambda $ , i.e.",
"$u\\in (A-\\mu N )^{-1} (\\lbrace 0\\rbrace )\\cap M$ , and, since $\\mu =(t^\\gamma -1)/\\alpha + t^\\gamma \\lambda < \\lambda _1$ we get a contradiction with $({\\cal M}_1)$ .",
"Hence, we can use the homotopy invariance of fixed point index to see that $\\mathrm {ind}_M (\\Phi _\\alpha ,B_M (0,\\delta )) =\\mathrm {ind}_M (0, B_M (0,\\delta )) = 1$ .",
"This along with (REF ) implies the required equality.",
"Let us pass to the case when $\\lambda >\\lambda _1$ .",
"Define $H:M \\times [0,1] \\rightarrow X$ by $H(u,t):= \\lambda N (u) + t\\tau _0$ , $u\\in M$ , $t\\in [0,1]$ .",
"If $-A(u)+H(u,t)= 0$ , then either $t=0$ and, due to $({\\cal M}_1)$ , $u=0$ or, by the homogeneity $-A(t^{-1/\\gamma }u) + \\lambda N (t^{-1/\\gamma }u) + \\tau _0 = 0$ , where $\\gamma >0$ is the common homogeneity degree for $A$ and $N$ .",
"The latter equality contradicts $({\\cal M}_2)$ .",
"Hence, the degrees $\\mathrm {Deg}_M (A, H(\\cdot ,t), B_M (0,\\delta ) )$ , $t\\in [0,1]$ , are well defined and homotopy invariance can be used to obtain $\\mathrm {Deg}_M (A, \\lambda N, B_M (0,\\delta )) = \\mathrm {Deg}_M (A, \\lambda A +\\tau _0, B_M (0,\\delta ).$ Finally the existence property of the degree together with $({\\cal M}_2)$ implies $\\mathrm {Deg}_M (A, \\lambda A +\\tau _0, B_M (0,\\delta ))=0$ , which completes the proof.",
"$\\square $"
],
[
"Abstract setting for $p$ -Laplacian",
"Now we shall consider an abstract example falling into the setting of Section 2.",
"It will be used in the sequel for the $p$ -Laplace operator and the cone of positive functions in $L^{p}(\\Omega )$ .",
"Let $X$ and $Y$ be reflexive normed spaces with a dense and compact linear embedding $i:Y\\rightarrow X$ .",
"(That is the mapping $i$ is linear and completely continuous with its range $i(Y)$ dense in $X$ .)",
"Suppose that a closed convex cone $M\\subset X$ and functionals ${a}:Y\\rightarrow \\mathbb {R}$ and ${n}: X\\rightarrow \\mathbb {R}$ satisfy the following conditions: (a1) ${a}$ and ${n}$ are coercive $C^1$ functionals;(By coercivity we mean that counterimages of intervals $(-\\infty ,m)$ , with respect to a given functional, are bounded for all $m\\in \\mathbb {R}$ .)",
"(a2) there exists a continuous function $\\kappa :[0,+\\infty )\\rightarrow [0, +\\infty )$ such that $\\kappa ^{-1}(\\lbrace 0\\rbrace ) = \\lbrace 0\\rbrace $ , $\\lim \\limits _{s\\rightarrow +\\infty } \\kappa (s) = +\\infty $ and $\\langle D{a} (u_1) - D{a} (u_2), u_1-u_2 \\rangle _{Y} \\ge \\kappa (\\Vert u_1-u_2\\Vert _Y)\\Vert u_1-u_2\\Vert _Y \\ \\ \\mbox{ for all }u_1, u_2\\in Y,\\\\\\langle D{n} (u_1) - D{n} (u_2), u_1-u_2 \\rangle _{X} \\ge \\kappa (\\Vert u_1-u_2\\Vert _X)\\Vert u_1-u_2\\Vert _X \\ \\ \\mbox{ for all }u_1, u_2\\in X;$ (a3) for any $u\\in M$ there exist $u^+, u^-\\in M$ such that $u=u^+ - u^-$ and ${n}(u^+) \\le {n}(u)$ ; if $u\\in i(Y)$ , then $u^+, u^- \\in i(Y)$ and ${a} (i^{-1} u^+) \\le {a} (i^{-1} u)$ ; (a4) $n$ is bounded on bounded sets and monotone with respect to $M$ , i.e.",
"${n}(u+v) \\ge {n} (u)$ for any $u,v\\in M$ .",
"Let ${\\cal A}:Y \\rightarrow Y^*$ and $N:X\\rightarrow X^*$ be defined by by $ {\\cal A} : = D{a}$ and $N:=D {n}$ .",
"Note that that, due to (a2), both ${a}$ and ${n}$ are strictly convex and ${\\cal A}$ and $N$ are monotone operators.",
"Define $A:D(A)\\rightarrow X^*$ by $D(A):= i \\left( {\\cal A}^{-1} (i^*(X^*))\\right) \\mbox{ and } A u:= (i^*)^{-1}({\\cal A} i^{-1}u), \\mbox{ for } u\\in D(A).$ The above operation of restriction is a generalization of the analogical one that is usually considered in the case of a Gelfand triple $Y\\subset X \\subset Y^*$ where $X$ is a Hilbert space.",
"Below we show that assumptions $({\\cal A}_1)$ and $({\\cal A}_2)$ of Section 2 are satisfied.",
"Proposition 3.1 Under the above assumptions (i) $N$ is a homeomorphism which is bounded on bounded sets; (ii) $N(M) = M^*:=\\lbrace \\tau \\in X^* \\mid \\langle \\tau , u \\rangle \\ge 0 \\mbox{ for all } u\\in M \\rbrace $ ; (iii) $A$ is a densely defined maximal monotone operator; (iv) for any $\\alpha >0$ and $\\tau \\in X^*$ there is a unique $u\\in D(A)$ such that $\\tau = (N+\\alpha A) (u)$ ; (v) if $J_\\alpha : X^*\\rightarrow X$ , $\\alpha >0$ , is given by $J_{\\alpha } \\tau := u \\mbox{ where } u\\in D(A) \\mbox{ is such that } N(u)+\\alpha A (u) = \\tau ,$ and ${\\cal J}:X^* \\times [0,+\\infty ) \\rightarrow X$ by ${\\cal J} (u, \\alpha ):= J_{\\alpha } u,$ then ${\\cal J}$ is bounded on bounded sets and ${\\cal J}_{\\ |\\, X^* \\times [\\alpha _1, \\alpha _2]}$ with $0<\\alpha _1 \\le \\alpha _2$ is completely continuous; (vi) $J_{\\alpha }(M^*) \\subset M$ for all $\\alpha >0$ .",
"Proof: To see (i), first note that $N$ is continuous, since ${n}$ is $C^1$ .",
"Moreover, as a strictly convex coercive functional on the reflexive Banach space $X$ , for any $\\tau \\in X^{*}$ , ${n}-\\tau $ admits a unique minimum point $u\\in X$ , i.e.",
"$D{n} (u)-\\tau =0$ , which gives $N(u)=\\tau $ .",
"Conversely, if $u\\in X$ is such that $N(u)=\\tau $ , then, by the strict convexity, $u$ is the unique minimum point.",
"Hence, $N$ is bijective.",
"To see that $N^{-1}$ is continuous, take any $(\\tau _n)$ in $X$ with $\\tau _n \\rightarrow \\tau $ in $X^*$ as $n\\rightarrow +\\infty $ .",
"Observe that, by (a2), we get $\\langle \\tau _n - \\tau , N^{-1}(\\tau _n)-N^{-1}(\\tau )\\rangle _X \\ge \\kappa (\\Vert N^{-1}(\\tau _n) - N^{-1} (\\tau )\\Vert _X)\\Vert N^{-1}(\\tau _n) - N^{-1} (\\tau )\\Vert _X,$ which yields the inequality $\\Vert \\tau _n-\\tau \\Vert _{X^*}\\ge \\kappa \\left(\\Vert N^{-1}(\\tau _n) - N^{-1} (\\tau )\\Vert _X \\right).$ This in turn means that $N^{-1}(\\tau _n) \\rightarrow N^{-1} (\\tau )$ in $X$ as $n\\rightarrow +\\infty $ , that is $N^{-1}$ is continuous.",
"To show that $N$ is bounded on bounded sets, we suppose to the contrary that there exists a bounded sequence $(u_n)$ in $X$ such that $\\Vert N(u_n)\\Vert _{X^*}\\rightarrow +\\infty $ as $n\\rightarrow +\\infty $ .",
"Since $X$ is reflexive, for each $n\\ge 1$ one finds an element $v_n\\in X$ such that $\\Vert v_n - u_n\\Vert _{X} = 1$ and $\\Vert N(u_n)\\Vert _{X^*} = \\langle N(u_n), v_n-u_n\\rangle \\le {n} ( v_n) - {n} (u_n) \\le \\sup _{D_{X}(0, R+1)} {n} - \\inf _{D_X(0,R)} {n}$ where $R>0$ is such that $\\Vert u_n\\Vert \\le R$ for all $n\\ge 1$ .",
"Thus, a contradiction proving the claim.",
"To get (ii) take any $u\\in M$ and $v\\in M$ .",
"In view of (a4) ${n}(u+h v)-{n} (u) \\ge 0 \\ \\mbox{ for any } h>0,$ which, after a division by $h$ and passage to the limit with $h\\rightarrow 0^+$ , yields $\\langle N(u), v\\rangle \\ge 0$ .",
"Hence $N(M)\\subset M^*$ .",
"To prove the converse inclusion $M^*\\subset N(M)$ , we take any $\\tau \\in M^*$ .",
"As we mentioned ${n}-\\tau $ attains the minimum at some $u\\in X$ .",
"On the other hand, by (a2), ${n}(u^+) -\\tau (u^+) \\le {n}(u) - \\tau (u^+) + \\tau (u^-) = {n}(u) - \\tau (u).$ and, since the minimum point is unique, we infer that $u=u^+ \\in M$ .",
"To show (iii), take any $u_1, u_2\\in D(A)$ .",
"Clearly $(A u_k) \\circ i = {\\cal A} (\\tilde{u}_k)$ with $\\tilde{u}_k = i^{-1} (u_k)$ , for $k=1,2$ .",
"Therefore, by (a2), $\\langle A u_1-Au_2, u_1 - u_2 \\rangle _{X} = [Au_1 - Au_2] i(\\tilde{u}_1-\\tilde{u}_2)= \\langle {\\cal A} (\\tilde{u}_1) - {\\cal A} (\\tilde{u}_2), \\tilde{u}_1-\\tilde{u}_2\\rangle _Y \\ge 0.$ Hence $A$ is monotone and it is left to prove that $A$ is maximal monotone, i.e.",
"that additionally one has $A (D(A))=X^*$ .",
"To see it we choose any $\\tau \\in X^*$ and put $\\Phi :={a} - i^*(\\tau )$ .",
"$\\Phi $ is a convex coercive functional on the reflexive space $Y$ .",
"Hence it admits a miniumem, i.e.",
"there is a point $\\bar{u}\\in Y$ such that $D\\Phi (\\bar{u})=0$ , i.e.",
"${\\cal A}(\\bar{u}) = i^*(\\tau )$ .",
"This means that $u:=i(\\bar{u})\\in D(A)$ and that $A(u)=\\tau $ .",
"To show (iv) take any $\\tau \\in X^*$ and $\\alpha >0$ .",
"We proceed like in (iii), that is we consider a functional $\\Phi := {n}\\circ i + \\alpha {a} - i^*(\\tau )$ on $Y$ .",
"It is clear that $\\Phi $ – as a strictly convex and coercive functional on a reflexive Banach space – admits a minimum, i.e.",
"there exists $\\bar{u}\\in Y$ such that $D\\Phi (\\bar{u})=0$ .",
"This means that $i^*(N(i(\\bar{u}))) + \\alpha {\\cal A} (\\bar{u}) = i^*(\\tau )$ .",
"Subsequently, we deduce that ${\\cal A}(\\bar{u}) \\in i^*(X^*)$ , i.e.",
"$u:= i(\\bar{u})\\in D(A)$ and $N(u) + \\alpha A (u) = \\tau $ .",
"Moreover, observe that for each $u\\in D(A)$ such that $N(u) + \\alpha A (u) = \\tau $ , $i^{-1}(u)$ is a critical point of $\\Phi $ .",
"Since $\\Phi $ is strictly convex it has to be the unique minimum point.",
"(v) Suppose that a sequence $(\\tau _n)$ is bounded in $X^*$ and $(\\beta _n)$ is a sequence in $[\\alpha _1, \\alpha _2]$ .",
"Put $u_n:= J_{\\beta _n} (\\tau _n)$ , $n\\ge 1$ .",
"Then $i^*N(u_n) + \\beta _n {\\cal A} (\\bar{u}_n) = i^*(\\tau _n)$ , where $\\bar{u}_n:=i^{-1}(u_n)$ , $n\\ge 1$ .",
"Since $N$ is bounded and $\\beta _n > \\alpha _1>0$ for all $n\\ge 1$ , we infer that $({\\cal A}(\\bar{u}_n))$ is bounded.",
"Observe that, in view of (a2), $\\langle {\\cal A} (\\bar{u}_n) - {\\cal A}(0), \\bar{u}_n \\rangle _{Y} \\ge \\kappa (\\Vert \\bar{u}_n\\Vert _{Y})\\Vert \\bar{u}_n\\Vert _Y,$ i.e.",
"$\\Vert {\\cal A}(u_n) - {\\cal A}(0)\\Vert _{Y} \\ge \\kappa (\\Vert \\bar{u}_n\\Vert _{Y})$ .",
"Hence, by the boundedness of $({\\cal A}(\\bar{u}_n))$ and the assumed property of $\\kappa $ , $(\\bar{u}_n)$ is bounded.",
"Therefore $(u_n) = (i(\\bar{u}_n))$ is relatively compact, which together with Remark REF proves the assertion.",
"In order to prove (vi), take any $\\tau \\in M^*$ .",
"We need to show that $u:=J_{\\alpha } (\\tau ) \\in M$ .",
"In the proof of (iv) we have showed that $i^{-1} u$ is the unique minimum of the functional $\\Phi ={n}\\circ i + \\alpha {a} - i^*(\\tau )$ on $Y$ .",
"On the other hand, by use of (a3) and the definition of $M^*$ , one has $\\Phi (i^{-1} u^+ )\\!=\\!",
"{n} (u^+) \\!+\\!",
"\\alpha {a}(i^{-1} u^+)\\!",
"-\\!",
"\\tau (u^+)\\!\\le \\!",
"{n}(u) \\!+\\!",
"\\alpha {a} (i^{-1} u))\\!-\\!\\tau ( u^+) \\!+\\!",
"\\tau (u^-) = \\Phi (i^{-1}u).$ This means that $i^{-1} u=i^{-1} u^+$ and $u\\in M$ .",
"$\\square $"
],
[
"Elliptic problems with $p$ -Laplacian",
"Now we shall apply the above abstract setting from the previous section to the $p$ -Laplacian problem.",
"To this end fix $p>2$ , and put $X_p:= L^p (\\Omega ), \\ Y_p:=W_{0}^{1,p} (\\Omega ) \\mbox{ and } M_p:=\\lbrace u\\in X \\mid u(x)\\ge 0 \\mbox{ for a.e. }",
"x\\in \\Omega \\rbrace .$ Both, $X_p$ and $Y_p$ are reflexive and, by the Rellich-Kondrachov theorem, the natural embedding $i:Y_p\\rightarrow X_p$ is compact and dense.",
"It is easy to see that $M_p$ is a closed convex subset of $X_p$ .",
"Next define functionals ${a}:Y_p\\rightarrow \\mathbb {R}$ and ${n}:X_p\\rightarrow \\mathbb {R}$ by ${a} (u):= \\frac{1}{p} \\int _{\\Omega } |\\nabla u(x)|^p \\,\\mathrm {d}x, \\ u\\in Y_p,\\\\{n}(u):= \\frac{1}{p} \\int _{\\Omega } |u(x)|^p \\,\\mathrm {d}x, \\ u\\in X_p.$ We prove that these objects satisfy the abstract assumptions of the general setting.",
"Proposition 4.1 The functionals ${a}$ and ${n}$ with the cone $M_p$ satisfy all the assumptions (a1) – (a4) from Section and $\\langle D{a} (u), v\\rangle _{Y} = \\frac{1}{p} \\int _{\\Omega } |\\nabla u(x)|^{p-2} \\nabla u(x)\\cdot \\nabla v(x) \\,\\mathrm {d}x, \\ u, v \\in Y_p, \\\\\\langle D{a} (u) - D{a} (v), u-v \\rangle _Y \\ge 2^{2-p}\\Vert u-v\\Vert _{Y}^{p}, \\, u,v\\in Y_p, \\\\ \\langle D{n} (u), v\\rangle _{X} = \\frac{1}{p} \\int _{\\Omega } |u(x)|^{p-2} u(x) v(x) \\,\\mathrm {d}x, \\ u,v \\in X_p,\\\\\\langle D {n} (u) - D {n} (v), u-v \\rangle _X \\ge 2^{2-p}\\Vert u-v\\Vert _{X}^{p}, \\, u,v\\in X_p.$ Moreover, if $A_p:D(A_p) \\rightarrow X_p$ is defined, in analogy to (REF ), by $D(A_p):= i\\left( (D {a})^{-1} (i^*(X^*))\\right) \\mbox{ and } A_p u:= (i^*)^{-1}( D({a}) i^{-1} u), \\mbox{ for } u\\in D(A),$ then $A_p u= -\\mathrm {div} (|\\nabla u|^{p-2} \\nabla u), \\mbox{ for } u\\in D(A_p),$ where the divergence is meant in the distributional sense and $D(A_p)=\\lbrace u\\in W_{0}^{1,p} (\\Omega ) \\mid \\mathrm {div}(|\\nabla u|^{p-2}\\nabla u) \\mbox{ exists and belongs to } L^{p} (\\Omega )\\rbrace .$ Proof: In order to see (a1), note that the functionals ${a}$ and ${n}$ are clearly Gateaux differentiable with the formulas (REF ) and () satisfied.",
"Since these Gateaux derivatives are continuous the functionals are Frêchet differentiable.",
"The coercivity is immediate as ${a}(u) = (1/p)\\Vert u\\Vert _{Y_p}^{p}$ , $u\\in Y$ , and ${n}(u)=(1/p)\\Vert u\\Vert _{X_p}^{p}$ , $u\\in X$ .",
"One can check the condition (a2), i.e.",
"() and (), by use of the following inequality $(|x|^{p-2}x-|y|^{p-2}y)\\cdot (x-y)\\ge 2^{2-p} |x-y|^p \\mbox{ for any } x,y\\in \\mathbb {R}^M, \\, M\\ge 1.$ Obviously, for $\\kappa :[0,+\\infty )\\rightarrow [0, +\\infty )$ , given by $\\kappa (s):=2^{2-p} s^p$ , $s\\ge 0$ , one has $\\kappa ^{-1}(\\lbrace 0\\rbrace ) = \\lbrace 0\\rbrace $ , $\\lim \\limits _{s\\rightarrow +\\infty } \\kappa (s) = +\\infty $ .",
"As for (a3), take any $u\\in X$ .",
"Then taking $u_+ := \\max \\lbrace u , 0\\rbrace $ and $u_-:=\\max \\lbrace -u,0\\rbrace $ we have $u= u^+ - u^-$ and ${n}(u_+ ) = \\frac{1}{p} \\int _{\\Omega } |u_+ (x)|^p \\,\\mathrm {d}x \\le \\frac{1}{p} \\int _{\\Omega } |u (x)|^p \\,\\mathrm {d}x = {n}(u).$ If $u\\in Y_p = W_{0}^{1,p}(\\Omega )$ , then, due to Lemma 7.6 of [11], $\\nabla u_+(x) = 0$ if $u(x)\\le 0$ and $\\nabla u_+ (x)=\\nabla u(x)$ if $u(x)\\ge 0$ .",
"Therefore $u_+ \\in Y_p$ and ${a}(u_+) = \\frac{1}{p}\\int _{\\Omega }|\\nabla u_+(x)|^p \\,\\mathrm {d}x \\le \\frac{1}{p}\\int _{\\Omega }|\\nabla u (x)|^p \\,\\mathrm {d}x= {a}(u).$ Finally, (a4) is immediate as, for $u,v\\in M_p$ , $|u|^{p}= u^p \\le (u+v)^{p}=|u+v|^p$ .",
"$\\square $ In view of Section , the operators $A_p$ , $N_p:=D {n}$ together with $M_p$ and $M_p^*$ satisfy the assumptions made in Section and the topological degree can be applied for perturbations of $A_p$ .",
"Before we proceed further let us pay attention to the perturbation term.",
"Proposition 4.2 Let $f:\\Omega \\times [0,+\\infty )\\rightarrow \\mathbb {R}$ satisfy (REF ) and $f(x,0) \\ge 0$ for a.a. $x\\in \\Omega $ .",
"Then the mapping $F : X_p \\rightarrow X_{p}^{*}$ given by $\\langle F(u), v \\rangle _{X_p}:= \\int _{\\Omega } f(u(x))v(x) \\,\\mathrm {d}x, \\ \\ u\\in M_p, \\, v\\in X_p,$ is well defined, continuous, bounded on bounded sets and $F(N^{-1}(\\tau )) \\in T_{M_{p}^*} (\\tau ) \\ \\ \\mbox{ for any } \\ \\tau \\in M_p^{*}.$ Lemma 4.3 Let $1<q<\\infty $ , $\\Omega \\subset \\mathbb {R}^N$ , $N\\ge 1$ , be open and $M_q:= \\lbrace u \\in L^q(\\Omega ) \\mid u(x) \\ge 0 \\mbox{ for a.e. }",
"x\\in \\Omega \\rbrace .$ Then $\\ \\ T_{M_q} (u) = \\lbrace v \\in L^q (\\Omega ) \\mid v(x)\\ge 0 \\mbox{ for a.e. }",
"x\\in \\Omega \\mbox{ such that } u(x)=0 \\rbrace $ .",
"Proof: Put $T_u:=\\lbrace v \\in L^q (\\Omega ) \\mid v(x)\\ge 0 \\mbox{ for a.e. }",
"x\\in \\Omega \\mbox{ such that } u(x)=0 \\rbrace $ .",
"To see that $T_u\\subset T_{M_q} (u)$ take any $v\\in T_u$ and define $v_n\\in L^1(\\Omega )$ , $n\\ge 1$ , by $v_n (x):=\\left\\lbrace \\begin{array}{cl}v(x) & \\mbox{ if } v(x) + n u(x) \\ge 0,\\\\0 & \\mbox{ if } v(x) + n u (x) <0.",
"\\end{array}\\right.$ Clearly, $v_n \\in M_q - n u \\subset T_{M_q} (u)$ , for each $n\\ge 1$ .",
"Moreover it is clear that, for a.e.",
"$x\\in \\Omega $ and any $n\\ge 1$ , $v_n(x) = v(x)\\ge 0$ if $u(x)=0$ and $v_n (x) \\rightarrow v(x)$ if $u(x)>0$ .",
"This implies that $v_n \\rightarrow v$ in $L^q(\\Omega )$ , i.e.",
"$v\\in T_{M_q} (u)$ .",
"In order to show the converse inclusion, observe that $T_u$ is closed and, for any $h>0$ , $h(M-u) \\subset T_u$ .",
"This clearly implies that $T_{M_q} (u)\\subset T_u$ .",
"$\\square $ Proof of Proposition REF: Using the Riesz representation isomorphism $\\varrho $ between $L^p (\\Omega )^*$ and $L^q(\\Omega )$ , $1/p +1/q=1$ , the mapping $F\\circ N^{-1}$ can be treated as the mapping $L^q(\\Omega )\\ni u \\mapsto f(\\cdot , \\theta _q (u)) \\in L^q(\\Omega )$ where $\\theta _q:\\mathbb {R}\\rightarrow \\mathbb {R}$ is given by $\\theta _q (s)=|s|^{q-2}s$ , $s\\in \\mathbb {R}$ .",
"It is well defined as $|(f(x,\\theta _q(s))| \\le C(1+|\\theta _q (s)|^p) = C(1+|s|) \\mbox{ for } s\\ge 0 \\mbox{ and a.e. }",
"x\\in \\Omega .$ Observe that $f(x,\\theta _q (0))=f(x,0)\\ge 0$ for.",
"a.e.",
"$x\\in \\Omega $ , which, by use of Lemma REF , implies that $f(\\cdot , \\theta _q (u(\\cdot )))\\in T_{M_q}(u)$ for all $u\\in M_q$ .",
"Since $\\varrho (M_p^*)=M_q$ , we infer that (REF ) holds.",
"$\\square $ Hence we have showed that the problem (REF ) indeed can be formulated as an abstract problem $\\left\\lbrace \\begin{array}{l}A_p (u) = F (u),\\\\u\\in M_p \\cap D(A_p).",
"\\end{array}\\right.$ In order to take advantage of the topological degree effectively we need some methods of computing it.",
"Theorem 4.4 If $\\ 2<p<\\infty $ and $\\rho \\in L^{\\infty } (\\Omega )$ is such that either $\\rho (x)> \\lambda _{1,p}$ for a.e.",
"$x\\in \\Omega $ , or $\\rho (x)< \\lambda _{1,p}$ for a.e.",
"$x\\in \\Omega $ , then $\\mathrm {Deg}_{M_p} (A_p, \\rho N_{p}, B_{M_p} (0,R)) = \\left\\lbrace \\begin{array}{cl}1, & \\mbox{ if } \\ \\ \\rho (x) < \\lambda _{1,p} \\ \\ \\mbox{ for a.e. }",
"\\ \\ x\\in \\Omega ,\\\\0, & \\mbox{ if } \\ \\ \\rho (x) > \\lambda _{1,p} \\ \\ \\mbox{ for a.e. }",
"\\ \\ x\\in \\Omega .\\end{array}\\right.$ Remark 4.5 Before passing to the proof of Theorem REF , we need to make a comment on the eigenvalue problem relating to the $p$ -Laplace operator.",
"Solving the nonlinear eigenvalue problem $\\left\\lbrace \\begin{array}{l}A_p (u) = \\lambda N_p(u)\\\\u\\in M_p \\cap D(A_p)\\end{array} \\right.$ reduces to find nonnegative weak solutions $u\\in W^{1,p}(\\Omega )$ of $\\left\\lbrace \\begin{array}{cl}-\\mathrm {div} (|\\nabla u(x)|^{p-2} \\nabla u(x)) = \\lambda |u(x)|^{p-2} u(x), & x\\in \\Omega ,\\\\u(x) = 0, & x\\in \\partial \\Omega .",
"\\\\\\end{array}\\right.$ It appears that some properties of the eigenvalue problem for the Laplace operator are also valid for the $p$ -Laplace one.",
"For details we refer to [14], [15] and [16].",
"In particular, it is known that (REF ) does not admit any nonzero solutions if $\\lambda \\le 0$ , i.e.",
"the $p$ -Laplace has no nonpositive eigenvalues.",
"Moreover, there exists the smallest eigenvalue $\\lambda _{1,p}$ given by the Rayleigh formula $\\lambda _{1,p} = \\inf \\limits _{u\\in W_{0}^{1,p} (\\Omega ), u\\ne 0} \\frac{\\int _{\\Omega } |\\nabla u(x)|^p \\,\\mathrm {d}x}{\\int _{\\Omega } |u(x)|^p \\,\\mathrm {d}x}.$ The eigenfunctions corresponding to $\\lambda _{1,p}$ are either strictly positive or negative in $\\Omega $ and belong to $L^\\infty (\\Omega )$ .",
"Moreover, $\\lambda _{1,p}$ is an isolated eigenvalue and if there are two eigenfunctions $u,v$ for $\\lambda _{1,p}$ , then there exists $\\alpha \\in \\mathbb {R}$ such that $u=\\alpha v$ .",
"It is also known that if any eigenfunction does not change its sign in $\\Omega $ , then the corresponding eigenvalue must be equal to $\\lambda _{1,p}$ .",
"$\\square $ In the proof we shall use a few lemmata given below.",
"Lemma 4.6 There are $ C, s > 0$ such that $\\Vert u\\Vert _{L^p} \\le C |\\tilde{\\Omega }|^{s} \\Vert \\nabla u\\Vert _{L^p}$ for all $u\\in W_{0}^{1,p}(\\Omega )$ and measurable $\\tilde{\\Omega }\\subset \\Omega $ with the property $u(x)=0$ if $x\\notin \\tilde{\\Omega }$ .",
"Proof: By the Sobolev embedding theorem there exists $q>p$ such that $\\Vert u \\Vert _{L^q} \\le C\\Vert \\nabla u\\Vert _{p} \\mbox{ for all } u\\in W_{0}^{1,p}(\\Omega ).$ On the other hand, by the Hölder inequality, $\\Vert u\\Vert _{L^p} \\le \\Vert u \\Vert _{L^q} |\\tilde{\\Omega }|^{1/p-1/q}.$ Combining the two above inequalities we get the desired one with $s:=1/p-1/q$ .",
"$\\square $ Lemma 4.7 Let $v$ be a nonnegative weak solution of (REF ) with $\\lambda =\\lambda _{1,p}$ and $\\rho \\in L^{\\infty }(\\Omega )$ .",
"If $u\\in W_{0}^{1,p}(\\Omega )$ is a weak solution to $-\\mathrm {div} (|\\nabla u|^{p-2} \\nabla u) = \\rho |u|^{p-2} u + |v|^{p-2}v \\, \\mbox{ on } \\Omega ,$ then $u\\in L^{\\infty }(\\Omega )$ .",
"Proof: Here we adapt the arguments from [15].",
"Note that without loss of generality we can consider the equation $-\\mathrm {div} (|\\nabla u|^{p-2} \\nabla u) = \\rho |u|^{p-2} u + \\lambda _{1,p} |v|^{p-2} v \\, \\mbox{ on } \\Omega .$ Take any $k>0$ and put $\\eta :=\\max \\lbrace u-v-k,0\\rbrace $ .",
"Since $\\eta \\in W_{0}^{1,p}(\\Omega )$ , we get $\\int _{\\Omega _k} (|\\nabla u|^{p-2}\\nabla u - |\\nabla v|^{p-2} \\nabla v) \\cdot \\nabla (u-v)\\,\\mathrm {d}x \\le \\Vert \\rho \\Vert _{L^{\\infty }} \\int _{\\Omega _k} u^{p-1}(u-v-k) \\,\\mathrm {d}x$ with $\\Omega _k := \\lbrace x\\in \\Omega \\mid u(x)-v(x)-k > 0 \\rbrace $ .",
"This, by use of (REF ) and the convexity of the function $s\\mapsto |s |^{p-1}$ , gives $\\int _{\\Omega _k} |\\nabla (u-v)|^p \\,\\mathrm {d}x & \\le & C_1 \\int _{\\Omega _k} u^{p-1} (u-v-k) \\,\\mathrm {d}x \\\\& \\le & C_1 2^{p-2} \\left( \\int _{\\Omega _k} (u-v-k)^p \\,\\mathrm {d}x + \\int _{\\Omega _k} (v+k)^{p-1}(u-v-k) \\,\\mathrm {d}x\\right)$ for some constant $C_1>0$ (here all the constant are to be independent of $k$ ).",
"By applying Lemma REF , one gets $\\int _{\\Omega _k} (u-v-k)^p \\,\\mathrm {d}x \\le C|\\Omega _k|^{s} \\int _{\\Omega _k} |\\nabla (u-v)|^{p} \\,\\mathrm {d}x,$ which together with the previous inequality yields $(1- C_2|\\Omega _k|^{s}) \\int _{\\Omega _k} (u-v-k)^p \\,\\mathrm {d}x \\le C_2 |\\Omega _k|^{s} \\int _{\\Omega _k} (v+k)^{p-1}(u-v-k) \\,\\mathrm {d}x$ for some $C_2>0$ .",
"Since $|\\Omega _k|\\rightarrow 0$ as $k\\rightarrow +\\infty $ , there is $k_0$ such that for all $k\\ge k_0$ $1-C_2 |\\Omega _k|^{s} >1/2$ .",
"Further, for $k\\ge k_0$ , $\\int _{\\Omega _k} (u-v-k)^p \\,\\mathrm {d}x \\le 2 C_2 |\\Omega _k|^{s} (\\Vert v\\Vert _{L^{\\infty }} + k)^{p-1} \\int _{\\Omega _k} (u-v-k) \\,\\mathrm {d}x.$ Next we observe that the Hölder inequality yields $\\int _{\\Omega _k} (u-v-k) \\,\\mathrm {d}x \\le C_4 k |\\Omega _k|^{1+s(p-1)^{-1}} \\mbox{ for all } k\\ge k_0$ and some constant $C_4>0$ .",
"Now define $j:(0,+\\infty )\\rightarrow [0,+\\infty )$ by $j(k):= \\int _{\\Omega _k} (u-v-k) \\,\\mathrm {d}x, \\, \\, k>0.$ Note that by the Tonelli-Fubini theorem applied to the set $\\lbrace (x,t)\\in \\Omega \\times [0,+\\infty )\\mid u(x)-v(x)>t>k \\rbrace $ one has $j(k) = \\int _{k}^{+\\infty } |\\Omega _t| \\,\\mathrm {d}t, \\,\\, k> 0.$ Obviously, $j$ is nonincreasing and absolutely continuous with $j^{\\prime }(k)=-|\\Omega _k|$ for a.e.",
"$k\\ge 0$ .",
"We claim that $j(k)=0$ for some $k>0$ .",
"If it were not so, then (REF ) could be rewritten as $j(k)^{\\theta } \\le -C_{4}^{\\theta } k^{\\theta } j^{\\prime }(k) \\mbox{ for all } k\\ge k_0$ with $\\theta :=(1+s(p-1)^{-1})^{-1}$ , and consequently $k^{-\\theta } \\le - C_{4}^{\\theta } j(k)^{-\\theta } j^{\\prime }(k) \\mbox{ for all } k\\ge k_0.$ This after integration would give $k^{1-\\theta } + C_{4}^{\\theta } j(k)^{1-\\theta } \\le k_0^{1-\\theta } + C_{4}^{\\theta } j(k_0)^{1-\\theta } \\mbox{ for all } k\\ge k_0,$ which yields a contradiction proving the claim that $j(k)=0$ for some $k>0$ .",
"Then, for some $k>0$ , $|\\Omega _k|=0$ and $u\\le v+k$ a.e.",
"on $\\Omega $ .",
"This shows that $u\\in L^{\\infty }(\\Omega )$ , as $v\\in L^{\\infty }(\\Omega )$ (see Remark REF ).",
"$\\square $ Lemma 4.8 (see [10]) If $h\\in L^\\infty (\\Omega )$ is nonnegative and nonzero, then the equation $-\\mathrm {div} (|\\nabla u|^{p-2} \\nabla u) = \\lambda _{1,p} |u|^{p-2}u + h, \\, \\mbox{ on } \\Omega ,$ has no nonzero weak solution in $W_{0}^{1,p}(\\Omega )$ .",
"Lemma 4.9 If $\\rho \\in L^\\infty (\\Omega )$ and either $\\rho (x)> \\lambda _{1,p}$ for a.e.",
"$x\\in \\Omega $ or $\\rho (x)<\\lambda _{1,p}$ for a.e.",
"$x\\in \\Omega $ , then the problem $-\\mathrm {div}(|\\nabla u|^{p-2}\\nabla u) = \\rho |u|^{p-2}u \\mbox{ on } \\Omega $ does not admit a nonzero solution $u\\in W_{0}^{1,p}(\\Omega )$ such that $u\\ge 0$ .",
"Proof: If $\\rho <\\lambda _{1,p}$ a.e.",
"on $\\Omega $ and $u\\in W_{0}^{1,p}(\\Omega )$ is a nonzero weak solution of (REF ), then $\\int _{\\Omega } |\\nabla u|^{p} \\,\\mathrm {d}x = \\int _{\\Omega } \\rho |u|^{p-2}u \\,\\mathrm {d}x < \\lambda _{1,p} \\int _{\\Omega } |u|^{p-2}u \\,\\mathrm {d}x,$ which gives $\\lambda _{1,p}> \\int _{\\Omega } |\\nabla u|^p \\,\\mathrm {d}x / \\int _{\\Omega } |u|^p \\,\\mathrm {d}x$ , a contradiction with the Rayleigh formula.",
"In the case $\\rho >\\lambda _{1,p}$ a.e.",
"on $\\Omega $ , we observe that if $u$ is a weak solution of (REF ), then $u$ is a weak solution of $-\\mathrm {div}(|\\nabla u|^{p-2}\\nabla u) =\\lambda _{1,p} |u|^{p-2}u + h \\ \\mbox{ on } \\ \\Omega $ with $h:=(\\rho - \\lambda _{1,p})|u|^{p-2}u$ .",
"Clearly, $h\\ge 0$ and $h\\in L^{\\infty }(\\Omega )$ , since $u\\in L^{\\infty }(\\Omega )$ due to Lemma REF .",
"Hence, Lemma REF leads to a contradiction ending the proof.",
"$\\square $ Proof of Theorem REF : Assume that $\\rho >\\lambda _{1,p}$ a.e.",
"on $\\Omega $ and fix $\\tilde{\\lambda }>\\lambda _{1,p}$ .",
"Define $H:X_p \\times [0,1]\\rightarrow X_p$ by $H (u,t) := (t\\tilde{\\lambda }+ (1-t) \\rho ) N_p(u)$ , $u\\in X_p$ , $t\\in [0,1]$ .",
"In view of Lemma REF , $-A_p (u)+H(u,t)\\ne 0$ for all $u\\in D(A_p) \\setminus \\lbrace 0 \\rbrace $ and $t\\in [0,1]$ .",
"Therefore, we can use the homotopy invariance – Theorem REF (iii) to get $\\mathrm {Deg}_{M_p} (A_p, \\rho N_{p}, B_{M_p} (0,R) ) = \\mathrm {Deg}_{M_p} ( A_p, \\tilde{\\lambda }N_p, B_{M_p}(0,R) ).$ In a similar manner one can prove the same formula in the case $\\rho <\\lambda _{1,p}$ a.e.",
"on $\\Omega $ with $\\tilde{\\lambda }<\\lambda _{1,p}$ .",
"Now we shall prove that conditions $({\\cal M}_1)$ and $({\\cal M}_2)$ of Theorem REF are satisfied.",
"Observe that, in view of Lemma REF , for any $\\lambda \\ne \\lambda _{1,p}$ , the eigenvalue problem (REF ) has no nontrivial and nonnegative weak solutions, i.e.",
"$({\\cal M}_1)$ holds.",
"To show $({\\cal M}_2)$ let $\\tau _0\\in L^p(\\Omega )$ be the functional determined by $|u_0|^{p-2}u_0$ with $u_0$ being a fixed positive solution of the eigenvalue problem (REF ) with $\\lambda =\\lambda _{1,p}$ .",
"Suppose that there exists $u\\in (A_p-\\lambda N_p)^{-1}( \\lbrace \\tau _0 \\rbrace ) \\cap M_p$ for some $\\lambda > \\lambda _{1,p}$ .",
"This means that $u\\in W_{0}^{1,p}(\\Omega )$ is a nonnegative weak solution of $-\\mathrm {div} (|\\nabla u|^{p-2} \\nabla u) = \\lambda _{1,p} |u|^{p-2} u + h \\ \\mbox{ on } \\ \\Omega $ with $h:=(\\lambda -\\lambda _{1,p}) |u|^{p-2} u + |u_0|^{p-2}u_0$ .",
"It follows from Lemma REF that $h\\in L^{\\infty } (\\Omega )$ .",
"Since $h\\ge 0$ , Lemma REF implies that such a solution does not exist, a contradiction proving $({\\cal M}_2)$ .",
"Hence, by Theorem REF and (REF ), the desired formula follows.",
"$\\square $ The obtained formula results in the following general one.",
"Theorem 4.10 Let $f$ and $F$ be as in Proposition REF and suppose that (REF ) hold.",
"(i) If $\\rho _0$ is as in () and either $\\rho _0 (x) < \\lambda _{1,p}$ , for a.e.",
"$x\\in \\Omega $ , or $\\lambda _{1,p}<\\rho _0 (x)$ , for a.e.",
"$x\\in \\Omega $ , then there exists $\\delta >0$ such that $A_p (u) \\ne F (u) \\mbox{ for all } u \\in D(A_p) \\cap \\left( B_{M_p} (0,\\delta ) \\setminus \\lbrace 0 \\rbrace \\right)$ and $\\mathrm {Deg}_M (A, F, B_M (0,\\delta )) = \\left\\lbrace \\begin{array}{cl}1, & \\ \\ \\mbox{ if } \\ \\ \\rho _0(x) < \\lambda _{1,p} \\ \\ \\mbox{ for a.e. }",
"x\\in \\Omega ,\\\\0, & \\ \\ \\mbox{ if } \\ \\ \\rho _0(x) > \\lambda _{1,p} \\ \\ \\mbox{ for a.e. }",
"x\\in \\Omega .\\end{array}\\right.$ (ii) If $\\rho _\\infty $ is as in () either $\\rho _\\infty (x) < \\lambda _{1,p}$ , for a.e.",
"$x\\in \\Omega $ , or $\\lambda _{1,p}<\\rho _\\infty (x)$ , for a.e.",
"$x\\in \\Omega $ , then there exists $R>0$ such that $A_p (u) \\ne F(u) \\mbox{ for all } u \\in D(A_p) \\cap \\left( M_p\\setminus B_{M_p} (0,R)\\right)$ and $\\mathrm {Deg}_{M_p} (A_p, F, B_{M_p} (0,R)) = \\left\\lbrace \\begin{array}{cl}1, & \\mbox{ if } \\ \\ \\rho _\\infty (x)< \\lambda _{1,p} \\ \\ \\mbox{ for a.e. }",
"x\\in \\Omega ,\\\\0, & \\mbox{ if } \\ \\ \\rho _\\infty (x)> \\lambda _{1,p} \\ \\ \\mbox{ for a.e. }",
"x\\in \\Omega .\\end{array}\\right.$ Proof: (i) Define $H:M_p\\times [0,1]\\rightarrow X_p$ by $H(u,t):= t F(u) + (1-t) \\rho _0 N_{p} (u),$ $(u,t)\\in M_p \\times [0,1]$ .",
"By Proposition REF , $H$ is continuous and $F\\circ N_{p}^{-1}$ is tangent to $M^*$ .",
"Moreover we claim that $\\mbox{ there is $\\delta >0$such that $-A_p (u)+H(u,t) \\ne 0$ for all $u\\in M_p \\cap D(A_p), \\ t\\in [0,1]$.",
"}$ Suppose to the contrary that there exists $(u_n)$ in $(M_p\\cap D(A_p))\\setminus \\lbrace 0 \\rbrace $ and $(t_n)$ in $[0,1]$ such that $u_n \\rightarrow 0$ in $X_p$ and $-A_p (u_n) + H(u_n, t_n)=0$ , $n\\ge 1$ .",
"Then clearly, if we put $w_n:= \\Vert u_n\\Vert _{X_p}^{-1}u_n$ and $s_n:=\\Vert u_n\\Vert _{X_p}$ , then $A_p(w_n) = s_{n}^{1-p} H(s_n w_n, t_n)$ , which gives $w_n =J_{1} \\left( N_p (w_n) + s_{n}^{1-p} H(s_n w_n, t_n) \\right), \\ n\\ge 1.$ The growth condition (REF ) and the existence of the first limit in () imply that there exists $C_1>0$ such that $\\Vert N_p (w_n) + s_{n}^{1-p} H(s_n w_n, t_n)\\Vert _{X_{p}^{*}}\\le C_1$ for all $n\\ge 1$ .",
"Therefore we infer that $(w_n)$ has a subsequence convergent in $X_p$ , since, according to Proposition REF and Proposition REF (v), $J_1$ is completely continuous.",
"In the sequel, we may assume that $(w_n)$ converges almost everywhere to some $w_0\\in M_p \\setminus \\lbrace 0 \\rbrace $ and that one has $g\\in X_p$ such that $|w_n|\\le g$ a.e.",
"on $\\Omega $ .",
"Further, note that if $w_n (x)\\ne 0$ , then $\\frac{f(x,s_n w_n(x))}{s_{n}^{p-1}} = \\frac{f(x, s_n w_n (x))}{(s_n w_n (x))^{p-1}} (w_n(x))^{p-1} \\rightarrow \\rho _0 (x) (w_0(x))^{p-1} \\mbox{ as } n\\rightarrow +\\infty ,$ which, by the dominated convergence theorem, implies that $s_{n}^{1-p} H(s_n w_n, t_n)\\rightarrow \\rho _0 N_p(w_0)$ in $X_p^{*}$ .",
"Hence, a passage to the limit in (REF ) yields $w_0 = J_1 (N_p (w_0)+ \\rho _0 N_p (w_0) )$ , i.e.",
"$-A_p w_0 + \\rho _0 N_p (w_0)=0$ .",
"This is a contradiction due to Lemma REF and (REF ) is proved.",
"Clearly, (REF ) allows us to use the homotopy invariance – Theorem REF (iii) to see that $\\mathrm {Deg}_{M_p} (A_p, F, B_{M_p} (0,R)) = \\mathrm {Deg}_{M_p} (A_p, \\rho _0 N_p, B_{M_p} (0,R))$ , which together with Theorem REF provides the required formula.",
"(ii) The proof is analogical to that for part (i) and it is left to the reader.",
"$\\square $ Proof of Theorem REF: Let $\\delta >0$ and $R>\\delta $ be like in Theorem REF .",
"Then by use of the additivity property – Theorem REF (ii), we get $\\mathrm {Deg}_{M_p} (A_p, F, B_{M_p} (0,R) \\setminus \\overline{B_{M_p} (0,\\delta )})= \\mathrm {Deg}_{M_p} (A_p, F, B_{M_p} (0,R))\\!",
"-\\!",
"\\mathrm {Deg}_{M_p} (A_p, F, B_{M_p} (0,\\delta )) \\\\= \\left\\lbrace \\begin{array}{cl}1, & \\mbox{ if } \\ \\rho _0 (x) > \\lambda _{1,p} > \\rho _\\infty (x) \\ \\mbox{ for a.e. }",
"x\\in \\Omega ,\\\\-1, & \\mbox{ if } \\ \\rho _0 (x) < \\lambda _{1,p} < \\rho _\\infty (x) \\ \\mbox{ for a.e. }",
"x\\in \\Omega .\\end{array}\\right.$ Hence the existence property of the topological degree gives the existence of $u\\in B_{M_p} (0,R) \\setminus \\overline{B_{M_p} (0,\\delta )}$ such that $A_p (u) = F(u)$ , which is a required nonzero nonnegative weak solution of (REF ).",
"$\\square $"
]
] | 1204.1295 |
[
[
"Gauge and moduli hierarchy in a multiply warped braneworld scenario"
],
[
"Abstract A generalized Randall sundrum model in six dimensional bulk is studied in presence of non-flat 3-branes at the orbifold fixed points.",
"The warp factors for this model is determined in terms of multiple moduli and brane cosmological constant.",
"We show that the requirements of a vanishingly small cosmological constant on the visible brane along with non-hierarchical moduli, each with scale close to Planck length, lead to a scenario where the 3-branes can not have any intermediate scale and have energy scales either close to Tev or close to Planck scale.",
"Such a scenario can address both the gauge hierarchy as well as fermion mass hierarchy problem in standard model.",
"Thus simultaneous resolutions to these problems are closely linked with the near flatness condition of our universe without any intermediate hierarchical scale for the moduli."
],
[
"Introduction",
"Large hierarchy of mass scales between the Planck and the electroweak scales results into well-known fine tuning problem in connection with the mass of the Higgs, the only scalar particle in the standard model.",
"It has been shown that due to large radiative corrections the Higgs mass diverges quadratically and can not be confined within Tev scale unless some unnatural tuning is done order by order in the perturbation theory.",
"This problem has been addressed in different variants of extra dimensional models.",
"Among all extra dimensional models the scenario proposed by Randall and Sundrum has drawn lot of attention.",
"It assumes a warp geometry of the space-time in 5 dimensions [1].",
"The fifth dimension is compactified on a space $S^1/Z_2$ and the length scale of this extra dimension is of the order of Planck length $r_c$ .",
"Two 3-branes are located at the two orbifold fixed points.",
"The exponential warping of the length scale along the fifth dimension naturally suppresses Planck scale quantities of one 3-brane, which we call hidden brane/Planck brane, into electroweak scale on the second 3-brane, which is identified as TeV-brane/visible brane and can be interpreted as our universe without introducing any new hierarchical scale into the theory.",
"A generalization of RS model have been considered previously by introducing more than one warped extra dimension in the theory [2].",
"Such models have interesting implications in particle phenomenology in higher dimensional models which can be summarized as : Resolution of the well known fermion mass hierarchy problem among standard model fermions [3].",
"The localization of massless fermions with a definite chirality on the visible 3-brane [4].",
"A consistent description of a bulk Higgs and gauge fields with spontaneous symmetry breaking in the bulk, along with proper $W$ and $Z$ boson masses on the visible brane [5].",
"Provides a stack of branes picture similar to string inspired models [2].",
"Motivated by the proposal of a non-vanishing cosmological constant as dark energy model for our present universe, in this work we generalize the six dimensional multiple warped model to include non-flat 3-branes.",
"Such generalization was done earlier for five dimensional scenario in [6].",
"We find the warp factors solutions for both de-Sitter and anti de-Sitter 3-branes with appropriate brane tensions.",
"We organize our paper as follows.",
"In the following section we explain some features of the six dimensional doubly warped model with flat 3-branes.",
"In sec.II we describe the six dimensional doubly warped model with non vanishing cosmological constant on the 3-branes (i.e with non-flat branes).",
"In sec.III we explain the correlation between the brane induced cosmological constant, and the ratio of the two extra dimensional moduli in the de-sitter brane.",
"In sec.IV we present our result and show that the four important aspects, namely 1) the value of cosmological constant in the present universe 2) hierarchical warping along the two extra dimensions, 3) hierarchy between the two extra dimensional moduli and 4) the gauge hierarchy problem, have interesting correlations among themselves.",
"Here we briefly outline the mechanism for generalizing the 5-dimensional RS model to 6-dimensional doubly warped geometry model with flat 3-branes [2].",
"The 6-dimensional doubly warped model has six space-time dimensions.",
"The extra two spatial dimensions are orbifolded successively by $Z_2$ symmetry .",
"The manifold for such geometry is $[M^{(1,3)}\\times S^1/Z_2]\\times S^1/Z_2$ with four non-compact dimensions denoted by $x^\\mu $ , $\\mu =0,\\cdots ,3$ .",
"As we are interested in doubly warped model, the metric in this model can be chosen as $ds^2 = b^2(z)[a^2(y)\\eta _{\\mu \\nu }dx^\\mu dx^\\nu +R_y^2dy^2]+r_z^2dz^2$ The angular coordinates $y,z$ represent the extra spatial dimensions with moduli $R_y$ and $r_z$ respectively.",
"The Minkowski metric in the usual 4-dimensions has the form $\\eta _{\\mu \\nu } = {\\rm diag}(-1,1,1,1)$ .",
"The functions $a(y),b(z)$ provide the warp factors in the $y$ and $z$ directions respectively.",
"The total bulk-brane action of this model can be written as: $S &=& S_6+S_5+S_4\\nonumber \\\\S_6 &=& \\int d^4xdydz\\sqrt{-g_6}(R_6-\\Lambda ), \\quad \\nonumber \\\\S_5 &=& \\int d^4xdydz\\sqrt{-g_{5}}[V_1\\delta (y)+V_2\\delta (y-\\pi )]\\nonumber \\\\&&~~+ \\int d^4xdydz\\sqrt{-g_{5}}[V_3\\delta (z)+V_4\\delta (z-\\pi )]\\nonumber \\\\S_4&=& \\int d^4x \\sqrt{-g_{vis}}[{\\cal {L}}-\\hat{V}][\\delta (y)\\delta (z)+\\delta (y)\\delta (z-\\pi )\\nonumber \\\\&&~~+\\delta (y-\\pi )\\delta (z)+\\delta (y-\\pi )\\delta (z-\\pi )]$ Here, $V_{1,2}$ and $V_{3,4}$ are brane tensions of the branes located at $y=0,\\pi $ and $z=0,\\pi $ , respectively.",
"$\\Lambda $ is the cosmological constant in 6-dimensions.",
"After solving Einstein's equations, the solutions to the warp factors of the metric as given in eq.",
"(REF ) [2] are, $a(y) &=& \\exp (-c|y|), \\quad b(z) = \\frac{\\cosh (kz)}{\\cosh (k\\pi )}\\nonumber \\\\{\\rm where}~~ c&\\equiv & \\frac{R_yk}{r_z\\cosh (k\\pi )},\\quad k\\equiv r_z\\sqrt{\\frac{-\\Lambda }{10M_P^4}}$ Here, $M_P$ is the 4-dimensional Planck scale.",
"The 5-d RS model can be retrieved in the limit $r_{z}\\rightarrow 0$ .",
"The warp factors $a(y)$ and $b(z)$ provide maximum suppression at $y=\\pi $ and $z=0$ .",
"For this reason we can interpret the 3-brane formed out of the intersection of 4-branes at $y=\\pi $ and $z=0$ as our standard model brane.",
"The suppression on the standard model brane can be written as $f=\\frac{\\exp (-c\\pi )}{\\cosh (k\\pi )}$ The desired suppression of the order of $10^{-16}$ on the standard model brane can be obtained for different choices of the parameters $c$ and $k$ .",
"However from the relation for $c$ in eq.",
"(REF ) it can be shown that if we want to avoid large hierarchy in the moduli $R_y$ and $r_z$ , the warping in one direction must be large while that in the other direction must be small.",
"In our analysis we shall explore this feature for non-flat 3-branes over the entire parameter space of $c$ , $k$ and moduli ratio $R_{y}/r_{z}$ for different values of brane cosmological constant.",
"Previously a generalization of the 5-dimensional RS model has been considered with the non flat 3-branes [6].",
"In this work we consider doubly compactified six dimensional space-time with $Z_{2}$ orbifolding along each of the compact direction.",
"Thus the manifold under consideration is [$M^{(1,3)}\\times S^{1}/Z_2$ ]$\\times S^{1}/Z_{2}$ with four non-compact dimensions denoted by $x^\\mu $ , $\\mu =0,\\cdots ,3$ .",
"We choose a doubly warped general metric as: $ds^2 = b^2(z)[a^2(y)g_{\\mu \\nu }dx^\\mu dx^\\nu +R_y^2dy^2]+r_z^2dz^2$ Since orbifolding, in general, requires a localized concentration of energy we introduce 4-branes [(4+1)-dimensional space-time] at the orbifold fixed points namely at $y=0,\\pi $ and $z=0,\\pi $ .",
"The total bulk-brane action in this case is given by, $S&=&S_{6}+S_{5}+S_{4}\\nonumber \\\\S_6&=& \\int d^{4}xdydz\\sqrt{-g_{6}}(R_{6}-\\Lambda )\\nonumber \\\\S_{5}&=&\\int d^{4}xdydz\\sqrt{-g_{5}}[V_{1}\\delta (y)+V_{2}\\delta (y-\\pi )]\\nonumber \\\\&&~~+ \\int d^{4}xdydz\\sqrt{-g_{5}}[V_3\\delta (z)+V_{4}\\delta (z-\\pi )]\\nonumber \\\\S_4&=& \\int d^4x \\sqrt{-g_{vis}}[{\\cal {L}}-\\hat{V}][\\delta (y)\\delta (z)+\\delta (y)\\delta (z-\\pi )\\nonumber \\\\&&~~+\\delta (y-\\pi )\\delta (z)+\\delta (y-\\pi )\\delta (z-\\pi )]$ The brane potential terms may be coordinate dependent as, $V_{1,2}=V_{1,2}(z)$ and $V_{3,4}=V_{3,4}(y)$ .",
"The term $S_{4}$ is the action for 3-branes located at $(y,z)=(0,0),(0,\\pi ),(\\pi ,0),(\\pi ,\\pi )$ .",
"The full 6-dimensional Einstein's equation can be written as, $-M^{4}\\sqrt{-g_{6}}(R_{MN}-\\frac{R}{2}g_{MN})&=&\\Lambda _{6}\\sqrt{-g_{6}}g_{MN}+\\sqrt{-g_{5}}V_{1}(z)g_{\\alpha \\beta }\\delta ^{\\alpha }_{M}\\delta ^{\\beta }_{N}\\delta (y)\\nonumber \\\\&+&\\sqrt{-g_{5}}V_{2}(z)g_{\\alpha \\beta }\\delta ^{\\alpha }_{M}\\delta ^{\\beta }_{N}\\delta (y-\\pi )\\nonumber \\\\&+&\\sqrt{-\\tilde{g}_{5}}V_{3}(y) \\tilde{g}_{\\tilde{\\alpha }\\tilde{\\beta }}\\delta ^{\\tilde{\\alpha }}_{M}\\delta ^{\\tilde{\\beta }}_{N}\\delta (z)\\nonumber \\\\&+&\\sqrt{-\\tilde{g}_{5}}V_{4}(y) \\tilde{g}_{\\tilde{\\alpha }\\tilde{\\beta }}\\delta ^{\\tilde{\\alpha }}_{M}\\delta ^{\\tilde{\\beta }}_{N}\\delta (z-\\pi ) $ Here M,N are bulk indices, $\\alpha $ , $\\beta $ run over the usual four space-time coordinates ($x^{\\mu }$ ) and the compact coordinate $z$ while $\\tilde{\\alpha }$ , $\\tilde{\\beta }$ run over ($x^{\\mu }$ ) and the compact coordinate $y$ .",
"$g$ , $\\tilde{g}$ are the respective metrices in these (4+1)-dimensional spaces.",
"For the metric (REF ), the different components of Einstein's equations reduce to a set of three independent equations, $^{4}G_{\\mu \\nu }+g_{\\mu \\nu }[\\frac{3a^{\\prime }(y)^2}{R_{y}^2}+a(y){\\frac{3a^{\\prime \\prime }(y)}{R_{y}^2}+\\frac{2a(y)}{r_{z}^2}(3b^{\\prime }(z)^2+2b(z)b^{\\prime \\prime }(z))}]\\nonumber \\\\=-\\frac{\\Lambda _{6}}{M^{4}}a(y)^{2}b(z)^{2}g_{\\mu \\nu }$ $6a^{\\prime }(y)^2r_{z}^{2}+a(y)^2[6R_{y}^{2}b^{\\prime }(z)^2+4R_{y}^{2}b(z)b^{\\prime \\prime }(z)]-(1/2)~ ^{4}R~b(z)^{2}a(y)^2R_{y}^{2}r_{z}^{2}\\nonumber \\\\=\\frac{-\\Lambda _{6}}{M^{4}}b(z)^{2}a(y)^2R_{y}^{2}r_{z}^{2}\\\\6a^{\\prime }(y)^2r_{z}^{2}+4r_{z}^{2}a(y)a^{\\prime \\prime }(y)-(1/2)~ ^4R~r_{z}^{2}R_{y}^{2}a(y)^2b(z)^210R_{y}^{2}a(y)^2b^{\\prime }(z)^2\\nonumber \\\\=-\\frac{\\Lambda _{6}}{M^4}b(z)^2a(y)^2R_{y}^{2}r_{z}^{2}$ $^{4}G_{\\mu \\nu }$ and $^4R$ are the four dimensional Einstein tensor and Ricci scalar respectively, defined with respect to $g_{\\mu \\nu }$ .",
"Dividing both sides of the equation (REF ) by $g_{\\mu \\nu }$ for any $\\mu $ , $\\nu $ and rearranging terms it is seen that one side contains $a(y)$ and $b(z)$ and their derivatives, while the other side depends on the brane coordinates $x^{\\mu }$ only.",
"Thus we can equate each side to an arbitrary constant $\\Omega $ such that, $^4G_{\\mu \\nu }=-\\Omega g_{\\mu \\nu }\\\\a(y)^2[\\frac{3}{R_{y}^2}\\frac{a^{\\prime \\prime }(y)}{a(y)}+\\frac{3}{R_{y}^2}\\frac{a^{\\prime }(y)^2}{a(y)^2}+\\frac{2}{r_{z}^2}\\nonumber \\\\(3b^{\\prime }(z)^2+2b(z)b^{\\prime \\prime }(z))+\\frac{\\Lambda _{6}}{M^4}b(z)^2]=\\Omega $ From equation(REF ) we identify $\\Omega $ as the effective 4-D cosmological constant on the 3-branes.",
"We obtain the solutions for the warp factors $a(y)$ and $b(z)$ for de sitter brane from equation () as, $a(y)={\\omega ^{\\prime }{\\rm sinh}}\\left[ {\\rm ln}\\frac{c^{\\prime }_2}{{\\rm \\omega ^{\\prime }}}-cy\\right], \\quad b(z)=\\frac{{\\rm cosh}(kz)}{{\\rm cosh}(k\\pi )}$ Where $c_{2}^{^{\\prime }}$ is an integration constant.",
"Here $\\omega ^{\\prime }=\\omega {\\rm cosh}(k\\pi )$ , with $\\omega ^2=\\frac{\\Omega }{3k^{\\prime 2}}$ , where $k^{^{\\prime }}= \\sqrt{\\frac{-\\Lambda }{10M^4}}$ , $k=k^{^{\\prime }}r_z$ and $c=\\frac{R_{y}k}{r_{z}{\\rm cosh}(k\\pi )}$ .",
"Normalizing the warp factor to unity at the orbifold fixed point $y=0$ , we get $c^{\\prime }_2=\\left[ 1+(1+\\omega ^{\\prime 2})^{\\frac{1}{2}}\\right] $ .",
"Note that the result for RS model generalized to six dimensions with flat branes is recovered in the limit $\\omega \\rightarrow 0$ .",
"We now focus our attention to the boundary terms to determine the brane tensions.",
"Using the explicit form of $a(y)$ , $b(z)$ from equation(REF ) and implementing the boundary conditions across the two boundaries at $y=0$ , $y=\\pi $ and $z=0$ , $z=\\pi $ respectively, we obtain: $V_{2}(z)=8M^2\\sqrt{\\frac{-\\Lambda }{10}}\\frac{(\\frac{\\omega ^{^{\\prime }2}}{c^{^{\\prime }2}_{2}}e^{2c\\pi }+1)}{(\\frac{\\omega ^{^{\\prime }2}}{c^{^{\\prime }2}_{2}}e^{2c\\pi }-1)}{\\rm sech}(kz)\\\\V_{1}(z)=8M^2\\sqrt{\\frac{-\\Lambda }{10}}{\\rm sech}(kz)\\frac{(1+\\omega ^{^{\\prime }2}/c^{^{\\prime }2}_{2})}{(1-\\omega ^{^{\\prime }2}/c^{^{\\prime }2}_{})}$ The above two equations imply that the two 4-branes sitting at $y=0$ and $y=\\pi $ have $z$ dependent tensions.",
"Similarly we find $V_{3}(y)=0$ and $V_{4}(y)=-\\frac{8M^4k}{r_{z}}{\\rm tanh}(k\\pi )$ .",
"Thus we have determined the tensions for all the 4-branes in this model.",
"The intersection of two 4-branes give rise to 3-brane.",
"With this identification, the theory contains four 3-branes located at $(y,z)=(0,0),(0,\\pi )$ , $(\\pi ,0)$ , $(\\pi ,\\pi )$ .",
"The metric on the 3-brane located at $(y=0,z=\\pi )$ has no warping and can be identified with the Planck brane.",
"Similarly we identify the standard model brane with the one at $y=\\pi $ , $z=0$ where the warping is maximum.",
"Finally we obtain the expressions for 3-brane tensions in terms of $\\omega $ , the induced brane cosmological constant as, $V_{vis}=8M^2\\left( \\frac{\\frac{\\omega ^{2}{\\rm cosh}^{2}(k\\pi )e^{2c\\pi }}{(4+2\\omega ^{2}{\\rm cosh}^{2}(k\\pi ))}+1}{\\frac{\\omega ^{2}{\\rm cosh}^{2}(k\\pi )e^{2c\\pi }}{(4+2\\omega ^{2}{\\rm cosh}^{2}(k\\pi ))}-1}\\right)\\sqrt{-\\frac{\\Lambda }{10}}\\\\V_{hid}=8M^2\\left[\\left(\\frac{1+\\frac{\\omega ^{2}{\\rm cosh}^{2}(k\\pi )e^{2c\\pi }}{(4+2\\omega ^{2}{\\rm cosh}^{2}(k\\pi ))}}{1-\\frac{\\omega ^{2}{\\rm cosh}^{2}(k\\pi )e^{2c\\pi }}{(4+2\\omega ^{2}{\\rm cosh}^{2}(k\\pi ))}}\\right){\\rm sech}(k\\pi )-{\\rm tanh}(k\\pi )\\right]\\sqrt{-\\frac{\\Lambda }{10}}$ The two other 3-branes located at $(0,0)$ and $(\\pi ,\\pi )$ have tensions close to hidden brane and visible brane tensions respectively.",
"Once again limit $\\omega \\rightarrow 0$ reproduces the expression for the 6-D flat 3-brane tensions[2] and with $r_{z}\\rightarrow 0$ (i.e $k=k^{^{\\prime }}r_{z}\\rightarrow 0$ ) we recover 5-D RS brane tensions[1].",
"Now to solve the gauge hierarchy problem we equate the warp factors at $y=\\pi $ , $z=0$ to the ratio of the mass scale in the two 3-branes given by $10^{-n}$ such that, $a(y)b(z)|_{y=\\pi ,z=0}=10^{-n}\\nonumber \\\\{\\omega ^{\\prime }{\\rm sinh}}\\left[ {\\rm ln}\\frac{c^{\\prime }_2}{{\\rm \\omega ^{\\prime }}}-c \\pi \\right]\\frac{1}{{\\rm cosh}(k\\pi )}=10^{-n}$ At this point, we keep $n$ arbitrary but we will subsequently take it to be $\\simeq 16$ to achieve a Planck to Tev scale warping.",
"Defining $c\\pi =x$ , the above equation has a positive root for $e^{-x}$ as, $e^{-x}={\\rm cosh}(k\\pi )\\frac{10^{-n}}{c_{2}^{^{\\prime }}}\\left[1+\\left\\lbrace 1+\\omega ^210^{2n}\\right\\rbrace ^{1/2} \\right]$ We re-parametrize the brane cosmological constant $\\omega $ as: $\\omega ^{2}\\equiv 10^{-N}$ .",
"Equation (REF ) now simplifies to: $-N=\\frac{1}{{\\rm ln}10}{\\rm ln}\\left[ \\frac{4e^{-2x}-4{\\rm cosh}(k\\pi )10^{-n}e^{-x}}{{\\rm cosh}^{2}(k\\pi )[1+10^{-n}{\\rm cosh}(k\\pi )e^{-x}-2e^{-2x}]}\\right] $ This equation relates the three parameters present in this model, $\\omega $ (which has been re-parametrized as $10^{-N}$ ), $k$ , and $x(=c\\pi )$ .",
"The solution of $x$ , derived from expression (REF ), is obtained as, $x=16{\\rm ln}10-{\\rm ln}{\\rm cosh}(k\\pi )-{\\rm ln}2+{\\rm ln}[2+\\frac{1}{2}10^{-N}{\\rm cosh}^{2}(k\\pi )]-{\\rm ln}[1+\\frac{1}{4}10^{-(N-2n)}] $ Now from expression (REF ) we numerically calculate $x$ i.e $c\\pi $ for different values of $k\\pi $ taking cosmological constant $\\omega ^{2}$ as parameter for $n=16$ .",
"This value of $n$ ensures the resolution of the gauge hierarchy problem for all the determined values of the parameters in our subsequent analysis.",
"Also we calculate the corresponding values of the ratio of two moduli $R_{y}/r_{z}$ from the expression $c=\\frac{R_{y}k}{r_{z}{\\rm cosh}(k\\pi )}$ .",
"The numerical values are given in the following table (REF ): Table: Numerical values of cπc\\pi andR y /r z R_{y}/r_{z} for every kπk\\pi and fixed cosmological constant ω\\omega (in Planck unit)in dS space-timeFor given $k\\pi $ the corresponding values of $c\\pi $ and $R_{y}/r_{z}$ saturate below $\\omega ^{2}<10^{-40}$ .",
"It may be seen from table(REF ), when the cosmological constant is very large (i.e approximately of the order of 1), for a small $k\\pi \\approx 1.12$ , $c\\pi $ is also small $0.56$ and ratio of the two moduli is $0.84$ .",
"Hence it is evident that for a very large cosmological constant we can have equal warping along both the extra dimensions and the two extra dimensional moduli are approximately close to $l_{Planck}$ i.e.",
"without any hierarchical values.",
"Now if we fix a large $k\\pi =30.12$ , the value of $c\\pi $ turns out to be very very small ($\\approx 10^{-13}$ ) and also the ratio of the two moduli becomes hierarchical.",
"From table (REF ), it may be noted that when we decrease the value of cosmological constant, for small $k\\pi $ , values of $c\\pi $ become large and the two extra dimensional moduli are again reasonably close to each other with values close to $l_{Planck}$ .",
"But if we fix a large $k\\pi $ then we can see that for decreasing $\\omega ^{2}$ , $c\\pi $ values become small and the ratio of the two extra dimensional moduli become large.",
"Thus we conclude that a small but equal warping from both the extra dimensions and the minimum hierarchy between the two extra dimensional moduli can be achieved only when the induced cosmological constant on the brane is very large.",
"But if we keep on decreasing the brane induced cosmological constant towards its present observed value which is estimated to be of the order of $10^{-120}$ , we cannot have equal warping along both the extra dimensions.",
"Moreover if we demand that the two extra dimensional moduli are approximately of the same order, then in order to solve the gauge hierarchy problem the most favourable condition consistent with a small value of the brane cosmological constant is small $k\\pi $ and large $c\\pi $ i.e small warping along $z$ direction and large warping along $y$ direction.",
"This resembles to 5-dimensional RS model perturbed slightly by the additional warping along $z$ direction such that two 3-branes have scales close to Tev while the two other have scales close to Planck scale.",
"For Anti De-Sitter brane i.e $\\Omega <0$ , we find that there is an upper bound on the brane induced cosmological constant similar to the one found in [6] which is $10^{-32}$ in Planck units.",
"Repeating the entire analysis in the ADS sector for values of the cosmological constant lower than $10^{-32}$ we determine the correlations among the parameters illustrated in table (REF ) and (REF ).",
"Table: Numerical values of cπc\\pi andR y /r z R_{y}/r_{z} for different kπk\\pi with fixed cosmological constant ω\\omega in ADS space-time.",
"The numerical values of ω\\omega are all inPlanckian unitsTable: Numerical values of cπc\\pi andR y /r z R_{y}/r_{z} for different kπk\\pi with fixed cosmological constant ω\\omega in ADS space-time.",
"The numerical values of ω\\omega are all inPlanckian unitsHere also we come to a similar conclusion that for small cosmological constant, in order to keep minimum hierarchy between the two extra dimensional moduli, the most favourable condition is small $k\\pi \\approx 1.12$ and large $c\\pi \\approx 36.31$ .",
"In this work the generalization of RS model has been done for a 6-dimensional ADS bulk with non-flat 3- branes.",
"Requiring the warping from the hidden brane to visible brane $\\sim 10^{-16}$ we find that for de-Sitter 3-brane, the warping along both the directions can be nearly equal with very small hierarchy between the two moduli only when the brane cosmological constant is very large compared to the present value.",
"To achieve similar warping along both the directions for nearly vanishing cosmological constant we have to introduce large hierarchy between the moduli.",
"On the contrary for small value of the brane cosmological constant with non-hierarchical small moduli $\\sim l_{Planck}$ , the warping along one direction ( in our case along $y$ ) is very large while the other direction (i.e.",
"$z$ ) is nearly flat.",
"The corresponding values of the moduli $c$ and $k$ can be stabilized following the Goldberger-Wise stabilization mechanism [11] by introducing a 6-dimensional bulk scalar field.",
"This leads to a scenario where the 3-branes can not have any intermediate scale and have energy scales either close to Tev or close to Planck scale.",
"This remarkable correlations clearly point out that the most favoured condition for small cosmological constant, non-hierarchical moduli and the resolution of the gauge hierarchy problem correspond to a very large warping along $y$ direction with very small warping along z-direction.",
"This scenario is nothing but a weak perturbation of the original 5-dimensional RS model due to the presence of the sixth dimension which in turn leads to two 3-branes with energy scale close to Tev scale while two other 3-branes having energy scale close to Plank scale.",
"Such a feature of brane clustering with closely-spaced energy scales enhances when more and more extra warped dimensions are added leading to a stack of Tev scale 3-branes and a stack of Planck scale 3-branes.",
"It has been shown in [3] that such scenario offers a possible geometric resolution of the fermion mass hierarchy problem among the standard model fermions.",
"Our work thus explains that in a multiple warped geometry model the requirements of nearly flat 3-brane and non-hierarchical moduli lead naturally to a stack of closely clustered Tev 3-branes which in turn offers a geometric understanding of the fermion mass hierarchy as well as gauge hierarchy problem simultaneously."
]
] | 1204.1154 |
[
[
"Intrinsic instability of electronic interfaces with strong Rashba\n coupling"
],
[
"Abstract We consider a model for the two-dimensional electron gas formed at the interface of oxide heterostructures, which includes a Rashba spin-orbit coupling proportional to the electric field perpendicular to the interface.",
"Based on the standard mechanism of polarity catastrophe, we assume that the electric field is proportional to the electron density.",
"Under these simple and general assumptions, we show that a phase separation instability occurs for realistic values of the spin-orbit coupling and of the band parameters.",
"This could provide an intrinsic mechanism for the recently observed inhomogeneous phases at the LaAlO_3/SrTiO_3 or LaTiO_3/SrTiO_3 interfaces."
],
[
"The Rashba Spin-Orbit Coupling",
"A full ab initio derivation of the Rashba coupling is a quite challenging task, which goes beyond the scope of this Letter.",
"Here, based on standard textbook arguments [1], [2], we simply remind that $\\alpha \\propto \\left< \\psi (z)\\left| \\frac{d}{dz} \\left[\\frac{1}{\\varepsilon ^{\\prime }(z)}-\\frac{1}{\\varepsilon ^{\\prime }(z)+\\Delta } \\right] \\right| \\psi (z)\\right>$ where $\\varepsilon ^{\\prime }(z)=\\varepsilon +V(z)+E_{gap}$ , $\\varepsilon $ is the subband energy relative to the bulk conduction band, $V(z)$ is the band-bending potential, $E_{gap}$ is the band gap, and $\\Delta $ is a measure of the spin-orbit coupling within a Kane ${\\mathbf {k\\cdot p}}$ approach [3].",
"Thus, since for a uniform electric field $E$ along $z$ the potential is $V(z)=-zE$ , one obtains $\\alpha (E) &\\propto & \\left[\\frac{1}{\\left(\\varepsilon +V(z)+E_{gap}\\right)^2}-\\frac{1}{\\left(\\varepsilon +V(z)+E_{gap}+\\Delta \\right)^2}\\right] \\frac{dV(z)}{dz}\\approx \\frac{ E\\Delta }{\\left(\\varepsilon +E\\overline{z}+E_{gap}\\right)^3}\\nonumber \\\\&=& \\frac{ \\tilde{\\alpha }E}{\\left(1 +\\beta E \\right)^3},$ where $\\overline{z}\\sim 2-6$ nm is the width of the region where the surface reconstruction occurs [4].",
"This expression has the standard linear behavior at small field, but saturates and then decreases at large fields.",
"This latter behavior is important to stabilize the system against unphysical unlimited growth of the electric field.",
"Indeed, in the absence of the denominator, the system would be unstable because it would be energetically (too) convenient to attract large densities of electrons to increase enormously the electric field and the RSO coupling with the consequent formation of deeper and deeper minima in the bands.",
"This would eventually produce a negative compressibility.",
"We emphasize, that the negative compressibility found in the Letter is not of this type."
],
[
"The anisotropic Rashba band structure",
"The real STO substrate has three bands, arising from the three $d_{xy}$ , $d_{xz}$ , and $d_{yz}$ orbitals.",
"Of course the atomic spin-orbit mixes these $t_{2g}$ levels (see, e.g., Ref.",
"[5] for an insight into possible consequences of this mixing), but here, we will neglect this mixing for the sake of simplicity and because it is unessential for the physical effects at issue in this Letter.",
"Then the 2DEG may be represented by the prototypical model Hamiltonian $H=\\sum _{k,\\sigma \\varsigma }c_{k\\sigma }^\\dagger H_{\\sigma \\varsigma }(k)c_{k\\varsigma }$ , where $c_{k\\sigma }^{(\\dagger )}$ annihilates (creates) an electron with quasimomentum $k$ and spin projection $\\sigma $ , $H_{\\sigma \\varsigma }(k)=\\left(\\frac{\\hbar ^2k_x^2}{2m_x}+ \\frac{\\hbar ^2k_y^2}{2m_y}+ \\Delta _{xy,xz,yz}\\right)\\delta _{\\sigma \\varsigma }+\\alpha _x k_x\\sigma ^y_{\\sigma \\varsigma }-\\alpha _yk_y\\sigma ^x_{\\sigma \\varsigma },$ and $\\sigma ^{x,y}$ are the Pauli matrices.",
"The lowest band due to the $d_{xy}$ orbitals has a light isotropic mass, $m\\sim 0.7 m_0$ ($m_0$ is the bare electronic mass) and $\\Delta _{xy}=0$ , $E^{iso}_\\pm (k_x,k_y)=\\frac{\\hbar ^2k_x^2}{2m}+\\frac{\\hbar ^2k_y^2}{2m}\\pm \\sqrt{\\alpha ^2 k_x^2+\\alpha ^2k_y^2}$ while two anisotropic bands due to the $d_{xz}$ and $d_{yz}$ orbitals are present with a shift $\\Delta _{xz,yz}\\equiv \\Delta $ at $k_x=k_y=0$ .",
"These latter have a heavy mass (as large as $20 m_0$ ) in one direction and a light mass in the other.",
"These anisotropic bands have dispersions of the form $E^{aniso}_\\pm (k_x,k_y)=\\frac{\\hbar ^2k_x^2}{2m_x}+\\frac{\\hbar ^2k_y^2}{2m_y}\\pm \\sqrt{\\alpha _x^2 k_x^2+\\alpha _y^2k_y^2}+\\Delta ,$ where $\\Delta =50$ meV is the shift with respect to the $d_{xy}$ band and $m_x=20 m_y=20 m$ (for the $d_{yz}$ ) or $m_y=20 m_x=20 m$ (for the $d_{xz}$ ).",
"The lowest band has four stationary points, which can be minima or saddle points depending on the values of $\\nu \\ge 1$ and $\\frac{1}{\\nu }\\le \\eta \\le 1$ .",
"In the $\\eta =1/\\nu $ case the bottom of the band occurs at $-\\varepsilon _0+\\Delta $ , where $\\varepsilon _0\\equiv m\\alpha ^2/(2\\hbar ^2)$ , while for $\\eta =1$ it is at $-\\nu \\varepsilon _0+\\Delta $ .",
"The DOS of the anisotropic $d_{xz,yz}$ bands depends only on the ratio $\\zeta =(\\varepsilon -\\Delta )/\\varepsilon _0$ and its expression can be given analytically in terms of complete elliptic integrals of the first ($K$ ) and third ($\\Pi $ ) kind.",
"Specifically, when $\\eta ^2\\nu <1$ $N(\\zeta )=\\theta [-\\zeta (\\zeta + 1)]{\\cal {A}}_1(\\zeta )+\\theta [-(\\zeta +\\eta ^2 \\nu )(\\zeta +1)]{\\cal {A}}_2(\\zeta )+\\theta [\\zeta ] N_0 \\sqrt{\\nu }$ where ${\\cal {A}}_1(\\zeta )&=&\\frac{2N_0\\sqrt{\\nu }}{\\pi \\sqrt{\\eta ^2\\nu +\\zeta }}\\left[-\\zeta K\\left(-\\frac{\\zeta (\\eta ^2 \\nu -1)}{\\eta ^2 \\nu +\\zeta }\\right) +(1 + \\zeta ) \\Pi \\left(\\frac{\\eta ^2 \\nu - 1}{\\eta ^2 \\nu + \\zeta },\\frac{-\\zeta (\\eta ^2 \\nu -1)}{\\eta ^2 \\nu +\\zeta }\\right)\\right] \\\\{\\cal {A}}_2(\\zeta )&=& \\frac{2N_0\\sqrt{\\nu }}{\\pi \\sqrt{- \\zeta (\\eta ^2 \\nu -1)}}\\left[-\\zeta K\\left(-\\frac{\\eta ^2 \\nu +\\zeta }{\\zeta (\\eta ^2 \\nu - 1)}\\right) \\right.\\nonumber \\\\&&-\\left.",
"(1 + \\zeta ) \\Pi \\left(\\frac{\\eta ^2 \\nu + \\zeta }{\\eta ^2 \\nu - 1}, \\frac{\\eta ^2 \\nu + \\zeta }{-\\zeta (\\eta ^2 \\nu -1)}\\right)\\right]$ On the other hand, when $\\eta ^2\\nu >1$ $N(\\zeta )=\\theta [-\\zeta (\\zeta + \\eta ^2\\nu )]{\\cal {A}}_3(\\zeta )+\\theta [-(\\zeta + \\eta ^2 \\nu )(\\zeta +1)]{\\cal {A}}_4(\\zeta )+\\theta [\\zeta ] N_0 \\sqrt{\\nu }$ where ${\\cal {A}}_3(\\zeta )&=&\\frac{2N_0\\sqrt{\\nu }}{\\pi \\eta \\sqrt{1 + \\zeta }}\\Pi \\left(1-\\frac{ 1}{\\eta ^2 \\nu },\\frac{-\\zeta (1-\\eta ^2 \\nu )}{\\eta ^2 \\nu (1+ \\zeta )}\\right) \\\\{\\cal {A}}_4(\\zeta )&=& \\frac{2N_0\\sqrt{\\nu }}{\\pi \\sqrt{-\\zeta (1-\\eta ^2 \\nu )}}\\Pi \\left(\\frac{1+ \\zeta }{\\zeta }, \\frac{\\eta ^2 \\nu (1 + \\zeta )}{\\zeta (\\eta ^2 \\nu -1)}\\right)$ Fig.",
"REF reports the DOS for various values of $\\nu $ , in the extreme cases of $\\eta =1$ (a) and $\\eta =1/\\nu $ (b).",
"Figure: (Color online) Density of states for the isotropic case (black, solid line)and for the anisotropic cases (the anisotropic bands do not include theshift Δ\\Delta to allow an easier comparison between the isotropic and anisotropic bandDOS): a) Case with η=1\\eta =1 and ν=3\\nu =3 (red online, dashed),ν=10\\nu =10 (green online, dot-dashed), and ν=20\\nu =20 (blue online, dot-dot-dashed).",
"b)Case η=1/ν\\eta =1/\\nu and ν=3\\nu =3 (red online, dashed), ν=10\\nu =10 (green online,dot-dashed), and ν=20\\nu =20 (blue online, dot-dot-dashed).While the generic expressions for $\\mu (n)$ are cumbersome, for $\\mu >\\Delta $ the electron density inside each one of the anisotropic bands reads $n=(\\mu -\\Delta ) N_0\\sqrt{\\nu }+2N_0\\varepsilon _0f(\\nu ,\\eta ),$ where $2N_0 f(\\nu ,\\eta )\\equiv \\int ^0_{\\frac{\\varepsilon _{min}-\\Delta }{\\varepsilon _0}}N(\\zeta )d\\zeta $ $\\varepsilon _{min}(\\nu ,\\eta )$ being the bottom of the lowest Rashba split anisotropic band.",
"We insert the factor $2N_0$ to normalize this function to $ f (1, 1) = 1$ so that in the isotropic case with zero splitting $\\Delta $ the Rashba term is $2N_0\\varepsilon _0$ , as previously found.",
"$f(\\nu ,\\eta )$ is a (rapidly) increasing function of the mass anisotropy $\\nu $ (see Fig.",
"REF ).",
"For the chemical potential one obtains $\\mu =\\Delta +\\frac{1}{\\sqrt{\\nu }}\\left[\\frac{n}{N_0}-2\\varepsilon _0 f(\\nu ,\\eta )\\right].$ The negative term contains the function $f(\\nu ,\\eta )\\gg 1$ , which makes the condition $\\kappa <0$ and the related electronic phase separation much easier to occur.",
"As shown in Fig.",
"REF , $f(\\nu ,\\eta )$ depends on the anisotropy of the RSO term $\\eta $ .",
"To reconstruct the relativistic form of the SO, $({\\bf v}\\times {\\bf \\sigma } )\\cdot \\hat{\\bf E}$ , one should assume $\\alpha _{x,y}\\propto m_{x,y}^{-1}$ , i.e., $\\eta =1/\\nu $ .",
"In this case, $f(\\nu ,1/\\nu )\\sim \\sqrt{\\nu }$ grows rather slowly with $\\nu $ .",
"On the other hand, if the RSO term is isotropic ($\\eta =1$ ) despite the mass anisotropy, $f(\\nu ,1)\\sim \\nu ^{\\frac{3}{2}}$ is a rapidly increasing function of $\\nu $ .",
"The precise RSO coupling in real materials should be determined by first-principles calculations, which are beyond the scope of this Letter, but the behavior of $f(\\nu ,\\eta )$ should be intermediate between these two extreme cases.",
"Therefore, our analysis shows that the RSO-mediated instability can take a moderate or strong advantage from the mass anisotropy.",
"Figure: (Color online) Enhancement factor f(μ,ν)f(\\mu ,\\nu ) [cf.",
"Eq.",
"()]for different values of the RSO anisotropy parameter η\\eta ."
]
] | 1204.0962 |
[
[
"New Sum Rules from Low Energy Compton Scattering on Arbitrary Spin\n Target"
],
[
"Abstract We derive two sum rules by studying the low energy Compton scattering on a target of arbitrary (nonzero) spin j.",
"In the first sum rule, we consider the possibility that the intermediate state in the scattering can have spin |j \\pm 1| and the same mass as the target.",
"The second sum rule applies if the theory at hand possesses intermediate narrow resonances with masses different from the mass of the scatterer.",
"These sum rules are generalizations of the Gerasimov-Drell-Hearn-Weinberg sum rule.",
"Along with the requirement of tree level unitarity, they relate different low energy couplings in the theory.",
"Using these sum rules, we show that in certain cases the gyromagnetic ratio can differ from the \"natural\" value g=2, even at tree level, without spoiling perturbative unitarity.",
"These sum rules can be used as constraints applicable to all supergravity and higher-spin theories that contain particles charged under some U(1) gauge field.",
"In particular, applied to four dimensional N=8 supergravity in a spontaneously broken phase, these sum rules suggest that for the theory to have a good ultraviolet behavior, additional massive states need to be present, such as those coming from the embedding of the N=8 supergravity in type II superstring theory.",
"We also discuss the possible implications of the sum rules for QCD in the large-N_c limit."
],
[
"Introduction, Summary and an Application to $N=8$ Supergravity",
"Dispersion relations based on analyticity, unitarity and Lorentz invariance of scattering amplitudes can be interpreted as consistency conditions that constrain low-energy data, i.e.",
"parameters of an effective action.",
"Conversely, if we know the low energy effective action of a given theory, dispersion relations either constrain any of its possible UV completions, or show that no completion exists.",
"Beautiful examples of the latter phenomenon are given in Ref. [1].",
"In this paper we revisit topics addressed years ago by one of us [2] and find their implications on the existence and properties of UV completions of low energy effective theories.",
"In Ref.",
"[2] it was argued that when effective field theories of elementary particles admit a perturbative expansion (in some parametrically small dimensionless coupling constant), then the gyromagnetic ratio of the (weakly interacting) particles described by such theory had to be close to a preferred “natural” value: $g=2$ .",
"This result was obtained in a Lagrangian approach.",
"Many years earlier, Weinberg [3] also proposed an argument, based on a generalization of the Gerasimov-Drell-Hearn (GDH) [4] sum rule, which similarly selected $g = 2$ as the preferred value in weakly interacting theories.",
"Given a particle of mass $m$ and electric charge $e$ , the GDH-Weinberg sum rule connects the gyromagnetic ratio $g$ to a dispersion integral.",
"As usual, $g$ is defined as the ratio of the particle's magnetic moment $\\mu $ to its spin $J$ , so that (in the particle's rest frame): $\\vec{\\mu } = \\frac{e g}{2m}\\vec{J} \\ .$ The main ingredient of the Weinberg sum rule is the low energy forwardAssume that the photon propagates along some $z$ -direction with helicity $\\lambda = \\pm 1$ , and the target has a spin-$z$ projection $J_z$ .",
"For more details, see Appendix .",
"Compton scattering amplitude of a photon with energy $\\omega $ and helicity $\\lambda $ off a massive target of spin $J$ .",
"This amplitude, $f_{\\rm scat} (\\omega , \\lambda )$ , is a real analytic function of the photon's energy $\\omega $ away from the real $\\omega $ -axis, where cuts and poles may exist at $\\omega > 0 $ .",
"The imaginary part of $f_{\\rm scat}$ is given by the optical theorem: ${\\rm Im} f_{\\rm scat} (\\omega ,\\lambda ) = \\frac{\\omega }{4\\pi }\\sigma _{\\rm tot}(\\omega , \\lambda ) \\ ,$ where $\\sigma _{\\rm tot}$ is the total cross-section for a photon with helicity $\\lambda $ and energy $\\omega $ .",
"Define now the following function: $&f_-(\\omega ^2) \\equiv \\frac{f_{\\rm scat} (\\omega ,+1)-f_{\\rm scat} (\\omega ,-1)}{2\\omega } \\ .$ When no intermediate (one particle) state exists in the Compton scattering, with either mass or spin different from those of the target, then, it can be checked that (see, e.g., Appendix ): $&f_-(\\omega ^2 \\rightarrow 0) = \\frac{e^2J_z}{16\\pi m^2}(g-2)^2 \\ .$ Using the optical theorem (REF ) and definition (REF ), we have: ${\\rm Im}f_-(\\omega ^2) = \\frac{1}{8\\pi }\\Delta \\sigma (\\omega ) \\ , \\qquad \\Delta \\sigma (\\omega ) \\equiv \\sigma _{\\rm tot}(\\omega ,+1) - \\sigma _{\\rm tot}(\\omega ,-1) \\ .$ Assuming that $f_-(\\omega ^2)$ vanishes when $|\\omega ^2| \\rightarrow \\infty $ , one can write an unsubtracted dispersion relation: $f_-(\\omega ^2) = \\frac{1}{4\\pi ^2}\\int ^{\\infty }_0\\frac{\\Delta \\sigma (\\omega ^{\\prime })}{\\omega ^{\\prime 2} - \\omega ^2 - i \\epsilon } ~\\omega ^{\\prime } d\\omega ^{\\prime } \\ .$ The validity of this assumption is far from obvious and its justification requires some additional assumptions about the UV behavior of the theory.",
"We will argue below that unsubtracted dispersion relations hold in two important case: superstring theory and any unitary completion of $N=8$ supergravity.",
"When $\\omega ^2=0$ , Eqs.",
"(REF ) and (REF ), impliy the GDH-Weinberg sum rule [3], [4]: $\\frac{\\pi e^2 J_z}{4m^2} \\left(g-2\\right)^2 = \\int ^{\\infty }_0\\frac{\\Delta \\sigma (\\omega ^{\\prime })}{\\omega ^{\\prime }} d\\omega ^{\\prime } \\ .$ In weakly interacting systems, the RHS of Eq.",
"(REF ) is parametrically smaller than the LHS, since in the former the final state contains at least two particles and is thus $O(e^4)$ , while the latter is $O(e^2)$ ; hence $g = 2 + {\\cal O}(e^2)$ .",
"In truncated $N=2$ supergravity theory (where one drops the cosmological and quartic Fermionic terms from the Lagrangian given in Ref.",
"[5]) it can be checked that the gravitino has indeed $g=2$ , see also Ref. [6].",
"On the other hand, in spontaneously broken $N=8$ supergravity theory [7], see also [13], contrary to expectations, one finds that $g=1$ , for both spin-1/2 and spin-3/2 fields.",
"To show this, we use the dimensionally reduced actionThe $N=8$ supergravity theory with global E(6) and local USp(8) invariance was constructed in five dimensions by Cremmer, Scherk and Schwarz in Ref. [7].",
"Spontaneous symmetry breaking of $N=8$ supergravity theory is achieved by dimensional reduction to 4D via the Scherk-Schwarz mechanism [8].",
"given in Ref.",
"[9], and consider only the relevant terms of the Lagrangians for spin-1/2 and spin-3/2 fields, defined as $\\chi ^{abc}$ and $\\psi _{\\mu }^a$ , respectively.",
"In our notations (see Appendices), and in units, $\\kappa ^2 = 4\\pi G_N =1$ , where $G_N$ is the 4D Newton's constant, these terms, written in flat space-time, unitary gauge ($\\psi _5^a=0$ ) and near the ground state, take the following form: $&{\\cal L}_{4D}(\\chi ) = \\frac{1}{12}\\bar{\\chi }^{abc}\\biggl [i\\left(\\partial \\hspace{-7.0pt}\\diagup + 2i{\\cal M}_{abc}B\\hspace{-7.0pt}\\diagup \\right)- {\\cal M}_{abc} +\\frac{1}{4}\\sigma ^{\\mu \\nu }B_{\\mu \\nu }\\biggr ]\\chi _{abc} \\ , \\\\ &{\\cal L}_{4D}(\\psi ) = \\frac{1}{2}\\bar{\\psi }^{a}_{\\mu }\\biggl [i\\gamma ^{\\mu \\rho \\nu }\\left(\\partial _{\\nu } + 2i{\\cal M}_{a}B_{\\nu }\\right) - {\\cal M}_{a}\\gamma ^{\\mu \\rho } + \\frac{i}{2}\\left(B^{\\mu \\rho } - i\\gamma _5\\tilde{B}^{\\mu \\rho }\\right)\\biggr ]\\psi _{\\rho a} \\ ,$ where $B_{\\mu }$ is a graviphoton field, ${\\cal M}_{abc}$ and ${\\cal M}_a$ are the mass matrices of $\\chi ^{abc}$ and $\\psi _{\\mu }^a$ correspondingly.",
"These spinors are $USp(8)$ tensors, and both are charged under the graviphoton, with a charge $e=2\\kappa m$ , where $m$ is the mass of the spinor.",
"Notice, also that there is no $\\chi $ -$\\psi $ -$B$ mixing.",
"The Lagrangians above should be compared with the Dirac Lagrangian supplemented with a Pauli term, and the Lagrangian of a charged Rarita-Schwinger field with non-minimal terms [6]: $&{\\cal L}_{DP} = \\bar{\\chi }\\left[i(\\partial \\hspace{-7.0pt}\\diagup + ieB\\hspace{-7.0pt}\\diagup ) - m - \\frac{e(g_{1/2}-2)}{8m}\\sigma ^{\\mu \\nu }B_{\\mu \\nu }\\right]\\chi \\ , \\\\ &{\\cal L}_{RS} = \\bar{\\psi }_{\\mu }\\left[ i\\gamma ^{\\mu \\rho \\nu }(\\partial _{\\nu } + ie B_{\\nu }) - m\\gamma ^{\\mu \\rho } + \\frac{ie}{m}\\left\\lbrace \\frac{3}{4}\\left(g_{3/2}- \\frac{2}{3}\\right)B^{\\mu \\rho } - i\\alpha \\gamma _5\\tilde{B}^{\\mu \\rho }\\right\\rbrace \\right]\\psi _{\\rho } \\ ,$ where $\\alpha $ is a parameter unrelated to $g_{3/2}$ .As was observed in [6], the $\\gamma _5$ matrix in the $\\bar{\\psi }_{\\mu }\\gamma _5 \\tilde{B}^{\\mu \\nu }\\psi _{\\nu }$ term of () mixes the “large” and “small” components of $\\psi _{\\mu }$ , and thus gives contributions of higher order in $\\omega $ .",
"That is why this term does not contribute to $g_{3/2}$ .",
"Finally, comparing (REF –) with (REF –), we conclude that in case of the spontaneously broken $N=8$ supergravity: $g_{1/2} = g_{3/2}= 1$ and $\\alpha =1/4$ .Similarly, comparing () with the truncated $N=2$ supergravity [5], one can deduce that: $g_{3/2} = 2$ and $\\alpha = 1$ .",
"In $N=2$ supergravity with a gauged central charge [11], $g_{3/2}=2$ , however, $g_{1/2} \\ne 2$ or 1.",
"It appears that $g=1$ for all heavy particles in the Kaluza-Klein theory [10].",
"This observation was also confirmed within the string theory and on the example of D0-branes in [12].",
"It is often the case that in supergravity theories, $e \\sim m/M_P$ , where $M_P$ is the Planck mass, suggesting that the charge can be made arbitrarily small.",
"It is thus natural to ask: why Eq.",
"(REF ) fails so miserably in case of the spontaneously broken $N=8$ supergravity theory?",
"One could argue that this happens because $f_-({\\omega ^2})$ does not vanish sufficiently fast, when $|\\omega ^2| \\rightarrow \\infty $ , since supergravity is power counting non-renormalizable.",
"However, spontaneously broken $N=8$ supergravity can be embedded in type II superstring theory [14]Spontaneously broken $N=4,2,1$ theories can be embedded also in heterotic string theory [15]., where the condition $\\lim _{\\omega ^2\\rightarrow \\infty }f_-(\\omega ^2)= 0$ holds, order-by-order in string perturbation theory.",
"A more explicit and general argument supporting unsubtracted dispersion relations is that, as argued in Refs.",
"[16] (see also [17]), the gravitational 2-body elastic scattering amplitude $f(s,t)$ of any theory obeying Hermitian unitarity and crossing symmetry is polynomially bound in the complex-$s$ upper half plane for fixed Mandelstam variable $t$ .",
"Hermitian unitarity implies $f(s,t)^*=f(s^*,t^*)$ , thus polynomial boundedness in $s$ for the forward elastic scattering amplitude ($t=0$ ).",
"Furthermore, Eq.",
"(REF ) suggests that the scattering amplitude $f_-(s,0)$ is not only polynomially bound, but indeed vanishes at large positive $s$ as $\\sim 1/s$ .",
"Crossing symmetry, Hermitian unitarity, and the same argument used to prove polynomial boundedness, i.e.",
"the Phragmen Lindelöf theorem, then show that $f_-(s,0)$ vanishes at large $s$ in the whole complex plane.Eq.",
"(REF ) gives a tree-level $f_-(s,0)$ that does not vanish at $s=\\infty $ ; this fact alone shows that $N=8$ supergravity cannot be a perturbatively complete theory of gravity.",
"As we will argue, the solution to this puzzle is different and it is one of the main results of this paper.",
"The point is that the GDH-Weinberg sum rule is modified when the Compton scattering amplitude includes intermediate one-particle states of masses $M_n \\ne m$ .In this case, $\\Delta \\sigma $ becomes a sum of terms proportional to $\\delta (\\omega -\\omega _n)$ plus the contribution from the continuum.",
"Taking into account this possibility, we find the following generalization of the GDH-Weinberg sum rule: $&{\\qquad }{\\qquad } \\frac{e^2 J_z}{4m^2}(g-2)^2 = [K^{\\dagger },K]_{ii} +\\frac{1}{\\pi }\\int ^{\\infty }_0 \\frac{\\Delta \\sigma }{\\omega } d\\omega \\ , \\\\[5pt] \\nonumber [K^{\\dagger },K]_{ii} \\equiv \\sum _n &\\left[(K_{ni})^{\\dagger }K_{ni} -K_{in}(K_{in})^{\\dagger }\\right] , {\\qquad } K_{ni} \\equiv \\left[2\\omega _n\\sqrt{m(\\omega _n + m)}\\right]^{-1}~\\langle n |\\vec{\\epsilon }_{\\lambda =1}\\vec{{\\cal J}}|i \\rangle \\ ,$ where $m$ is the mass of the target, $M_n$ is the mass of the intermediate state, $\\omega _n = \\frac{1}{2m}(M^2_n - m^2)$ , $\\vec{{\\cal J}}$ is the electromagnetic current, and $\\vec{\\epsilon }_{\\lambda }$ is the polarization vector of the photon with helicity $\\lambda $ .",
"The matrix $K_{ni}$ describes transition between states of different mass, and possibly spin.",
"This sum rule should be obeyed not only in string theory, but also in any Lorentz invariant, causal, unitary UV completion of $N=8$ supergravity.",
"Because $e \\sim m/M_P$ , the LHS of (REF ) is ${\\cal O}(1/M_P^2)$ , independent of the mass $m$ , and thus is the RHS.",
"Generically this implies that the new intermediate states have masses $M_n$ independent of $m$ ; these masses thus define a cutoff, below which the theory is described by $N=8$ supergravity.",
"Type II superstring compactified on a 6-torus is a concrete example of this general situation.",
"In this case the relation between the string mass scale $M_S\\sim 1/\\sqrt{\\alpha ^{\\prime }}$ and the Planck scale is $M_P=M_S^4\\sqrt{V_6}/g_S$ , where $V_6$ is the volume of the 6-torus, and $g_S$ is the string coupling constant.",
"In the perturbative regime, where $V_6M_S^6 \\gg 1$ and $g_S\\ll 1$ , we have: $M_P\\gg M_S $ .",
"When all radii of the torus are ${\\cal O}(1/M_S)$ , $M_S$ becomes the only UV cutoff scale and $M_n={\\cal O}(M_S)$ .",
"The fact that new states, besides those already present in $N=8$ supergravity,In a spontaneously broken phase, where gravitino is massive and charged [7].",
"are necessary for $N=8$ to admit a UV completion has an immediate consequence.",
"It implies that $N=8$ supergravity is not perturbatively complete by and in itself.",
"Such a statement, that is at variance with other remarkable finiteness properties of the theory (see e.g [18]) warrants further discussion.",
"It follows from two assumptions, besides Lorentz invariance and unitarity, both satisfied by string theory but holding more generally in any perturbative regularization of gravity.",
"The first is that the cutoff scale $\\Lambda $ is parametrically smaller than $M_P$ : $\\Lambda = \\lambda _c M_P$ , where $\\lambda _c \\ll 1$ is the dimensionless coupling constant of the UV complete theory.",
"The second assumption is that scattering amplitudes are regular in the limit $m\\rightarrow 0$ ; more precisely, that the mass $m$ appears only with positive powers in scattering amplitudes.",
"One can then use the fact that forward scattering depends only on the relativistic invariant $s=m^2 + 2m\\omega $ to rewrite the sum rule (REF ) as $\\lambda ^2_c \\frac{\\pi J_z}{4\\Lambda ^2}(g-2)^2 =\\lambda ^2_c \\int _{m^2}^\\infty \\frac{ds}{s-m^2} \\sum _n f_n[m^2,(s-m^2)/\\Lambda ^2]\\delta (s-M_n^2) + O(\\lambda ^4_c).$ Terms ${\\cal O}(\\lambda ^4_c)$ come from the continuum part of the total cross section, in which the final state contains at least two particles.",
"By assumption, the $m\\rightarrow 0$ limit of Eq.",
"(REF ) is smooth, so the functions $f_n[0,x]$ are well defined, $m$ -independent and dimensionless.",
"In the limit $m\\rightarrow 0$ , dividing both sides of Eq.",
"(REF ) by $\\lambda ^2_c$ , we get: $\\frac{\\pi J_z}{4\\Lambda ^2}(g-2)^2 =\\int _{0}^\\infty \\frac{ds}{s} \\sum _n f_n[0,s/\\Lambda ^2]\\delta (s-M_n^2) + O(\\lambda ^2_c).$ If the LHS of (REF ) is nonzero, then the RHS must contain new states with masses $M_n = {\\cal O}(\\Lambda )$ , because, by assumption, the only massless states in the theory are those of $N=8$ supergravity.Because of $N=8$ supersymmetry, new massless states would necessarily contain an additional massless spin two particle interacting with supergravity particles, in contradiction with the no go theorem in [19].",
"One may worry that at energies above $M_P$ the cross section $\\Delta \\sigma $ would never be perturbative because contributions from black hole intermediate states would dominate.",
"In general, these black holes carry mass $M$ , charge $Q$ and angular momentum $J$ .",
"In such case, the cross section can be crudely approximated by the cross sectional area of a black hole [16]: $\\sigma _J \\sim \\pi r^2_+ \\approx 4\\pi G^2_N M^2-2\\pi G_NQ^2 - 2\\pi \\frac{J^2}{M^2}$ where $r_+ = G_N M + \\sqrt{G^2_NM^2 - J^2/M^2 -G_NQ^2}$ is the outer horizon of the Kerr-Newman black hole, and we assumed $J^2 + G_N M^2 Q^2 \\ll G^2_N M^4 $ or $J \\ll G_NM^2$ , since $Q^2 = e^2 \\sim G_N m^2$ and $m \\ll M$ .",
"Clearly, the above estimate of the cross section is only applicable in case $r_+ \\sim G_N M \\ge 1/\\Lambda $ , where $\\Lambda $ is a cutoff of the theory (as was mentioned above $\\Lambda = \\lambda _c M_P$ and $\\lambda _c \\ll 1$ ).",
"Therefore, our estimate is valid, only when $M > \\omega _c$ , where $ \\omega _c \\equiv M_{P}\\left(M_P/\\Lambda \\right) = \\Lambda /\\lambda ^2_c $ .",
"In this case, the difference between the spin aligned ($J=j+1$ ) and anti-aligned ($J=j-1$ ) cross sections is: $\\Delta \\sigma _{j}(\\omega > \\omega _c) \\equiv \\sigma _{j-1} -\\sigma _{j+1} \\sim \\frac{8\\pi j}{\\omega ^2} \\ ,$ and therefore, the dispersion integral can be divided into two parts as follows: $\\int _{0}^{\\infty }\\frac{\\Delta \\sigma _{j}}{\\omega } d\\omega = \\int _{0}^{\\omega _c}\\frac{\\Delta \\sigma _{j}}{\\omega } d\\omega + \\int _{\\omega _c}^{\\infty }\\frac{\\Delta \\sigma _{j}}{\\omega } d\\omega \\ , {\\qquad } \\ \\int _{\\omega _c}^{\\infty }\\frac{\\Delta \\sigma _{j}}{\\omega } d\\omega \\sim \\frac{4\\pi j}{\\omega ^2_c} \\sim {\\cal O}\\left(\\frac{\\lambda ^2_c}{M^2_P}\\right) \\ ,$ where the second part is due to the exchange of black hole states.",
"Since, in supergravity theories $e \\sim m/M_P$ , from Eq.",
"(REF ) it follows that the black hole contribution to the gyromagnetic ratio is: $(g-2)^2_{\\rm B.H.}",
"\\sim {\\cal O}\\left(\\lambda ^2_c\\right) \\ll 1 \\ .$ If $N=8$ supergravity is a self-complete theory by itself, our argument shows that, unsurprisingly, it must unitarize at the Planck scale, in which case $\\omega _c=M_P$ and the black hole contributions to the dispersive integral becomes ${\\cal O}(1)$ .",
"This paper also generalizes the Weinberg-GHD sum rule in another way: it allows for off-diagonal couplings in case particles of spin $j$ and $|j\\pm 1|$ are degenerate in mass.",
"The general sum rule for Compton scattering on a target of arbitrary nonzero spin-$j$ , with additional intermediate mass-degenerate states of spin $|j\\pm 1|$ and possible heavy narrow resonances, can be written as: $&\\frac{e^2J_z}{4m^2} \\left[(g_j -2)^2 - \\left(j+\\frac{3}{2}\\right)h_{j+1/2}^2 + \\left(j-\\frac{1}{2}\\right) h_{j-1/2}^2\\right] = [K^{\\dagger },K]_{ii} + \\frac{1}{\\pi }\\int ^{\\infty }_{0} \\frac{\\Delta \\sigma }{\\omega } d\\omega \\ .$ where $g_j$ is the gyromagnetic ratio for a particle of spin $j$ , and the parameters $h_{j\\pm 1/2}$ are couplings of the theory defined in Eq.",
"(REF ).Notice, that the LHS of Eq.",
"(REF ) can be also written as a commutator of some generator like $K$ .",
"When applied to theories where all states are either proportional to a light mass scale $m$ or to a higher scale $M$ and where charges are proportional to the mass, such as Kaluza-Klein theories or spontaneously broken extended supergravities, this sum rule generalizes Eq.",
"(REF ) to $\\frac{\\pi J_z}{4M^2}\\left[(g_j -2)^2 - \\left(j+\\frac{3}{2}\\right)h_{j+1/2}^2 + \\left(j-\\frac{1}{2}\\right) h_{j-1/2}^2\\right] =\\int _{0}^\\infty \\frac{ds}{s} \\sum _n f_n[0,s/M^2]\\delta (s-M_n^2) +O(\\lambda ^2).$ Since magnetic dipole couplings are already taken into account by the LHS of Eq.",
"(REF ), the dispersive integral at low $s$ contains only quadrupole or higher multipole interactions; therefore, $f_n[0,s/M^2]$ is at most $O(s^2/M^4)$ at low $s$ .",
"This implies that, if the LHS in (REF ) is nonzero, the RHS must contain contributions from the massive states, since the one-particle contribution to the dispersive integral due to states that become massless in the limit $m\\rightarrow 0$ vanishes.",
"This paper is organized as follows: In section 2, we describe a formalism used by Weinberg to study the low energy Compton scattering.",
"We also study scattering on a target of spin-1/2, with possible spin-3/2 intermediate state, and deduce some generalization of the GDH-Weinberg sum rule.",
"In section 3, we consider the generalization of the sum rule to Compton scattering on a target of arbitrary spin.",
"In section 4, we further generalize the sum rule, assuming that there are other states with masses different from the mass of the scatterer.",
"In section 5, we discuss possible implications of our sum rule for nucleon to delta electromagnetic transitions, in the large $N_c$ limit."
],
[
"Compton Scattering: Weinberg's Approach",
"Define the non-diagonal vertex for the emission of a single soft photon as follows: $\\langle \\textbf {p}^{\\prime }, s^{\\prime }, \\sigma ^{\\prime } | {\\cal J}^{\\mu }(0)| \\textbf {p}, s, \\sigma \\rangle = \\Gamma _{\\sigma ^{\\prime }\\sigma }^{\\mu }(\\textbf {p}^{\\prime },\\textbf {p}) \\ ,$ where ${\\cal J}^{\\mu }$ is the conserved electromagnetic (EM) current, $| \\textbf {p}, s, \\sigma \\rangle $ is a single-particle state with momentum $\\textbf {p}$ , spin-$s$ and spin $z$ -component $\\sigma $ , as well as energy $E(\\textbf {p}) = \\sqrt{\\textbf {p}^2+m^2}$ .",
"We adopt the following normalization: $\\langle \\textbf {p}^{\\prime }, s, \\sigma ^{\\prime } | \\textbf {p}, s, \\sigma \\rangle = (2\\pi )^32E_{\\textbf {p}}\\delta _{\\sigma ^{\\prime }\\sigma }\\delta ^{(3)}(\\textbf {p}^{\\prime }-\\textbf {p}) \\ .$ The S-matrix, describing the Compton scattering can be written as: $\\langle \\sigma ^{\\prime }\\textbf {p}^{\\prime };\\lambda ^{\\prime },\\textbf {k}^{\\prime } |S| \\sigma ,\\textbf {p};\\lambda ,\\textbf {k} \\rangle &=i\\epsilon ^{*^{\\prime }}_{\\nu }(\\textbf {k}^{\\prime },\\lambda ^{\\prime })\\epsilon _{\\mu }(\\textbf {k},\\lambda ) ~ (2\\pi )^4\\delta ^{(4)}(p+k-p^{\\prime }-k^{\\prime })M_{\\sigma ^{\\prime }\\sigma }^{\\nu \\mu }(\\textbf {k}; \\textbf {p}^{\\prime }, \\textbf {p}) \\ ,\\\\[5pt]M_{\\sigma ^{\\prime }\\sigma }^{\\nu \\mu }(\\textbf {k}; \\textbf {p}^{\\prime }, \\textbf {p}) &\\equiv i\\int d^4x~e^{ikx} \\left[\\langle \\textbf {p}^{\\prime }, s, \\sigma ^{\\prime } | T\\lbrace {\\cal J}^{\\nu }(0){\\cal J}^{\\mu }(x)\\rbrace |\\textbf {p}, s, \\sigma \\rangle + {\\rm C.T.}",
"\\right] \\ ,$ where by ${\\rm C.T.",
"}$ we mean other contact terms (seagulls), such as terms emerging from the interaction of the initial and final photon at a single point.",
"We will need the pole structure of $M^{\\nu \\mu }$ ($\\sigma $ , $\\sigma ^{\\prime }$ indices are dropped for convenience).",
"Inserting a complete set of states between the current operators, the time-ordered product can be written as: $&i\\int d^4x ~ e^{ikx}\\langle \\textbf {p}^{\\prime }, s, \\sigma ^{\\prime } |T\\lbrace {\\cal J}^{\\nu }(0){\\cal J}^{\\mu }(x) \\rbrace | \\textbf {p}, s, \\sigma \\rangle \\\\\\nonumber &= \\int \\frac{d^3p_n}{2E_n(\\textbf {p}_n)}\\sum _n \\biggl \\lbrace \\langle \\textbf {p}^{\\prime }, s, \\sigma ^{\\prime } | {\\cal J}^{\\nu }(0)| n \\rangle \\langle n| {\\cal J}^{\\mu }(x)| \\textbf {p}, s, \\sigma \\rangle \\frac{\\delta ^{(3)}(\\textbf {p}_n - \\textbf {p}-\\textbf {k})}{E_n(\\textbf {p}_n) - E(\\textbf {p})- \\omega - i\\epsilon } \\\\ \\nonumber &+\\langle \\textbf {p}^{\\prime }, s,\\sigma ^{\\prime } | {\\cal J}^{\\mu }(0)| n \\rangle \\langle n| {\\cal J}^{\\nu }(x)| \\textbf {p}, s, \\sigma \\rangle \\frac{\\delta ^{(3)}(\\textbf {p}_n - \\textbf {p}^{\\prime }+\\textbf {k})}{E_n(\\textbf {p}_n)-E^{\\prime }(\\textbf {p}^{\\prime }) + \\omega - i\\epsilon } \\biggr \\rbrace \\\\ \\nonumber &=\\sum _n \\biggl \\lbrace \\frac{\\Gamma ^{\\nu }(\\textbf {p}^{\\prime },\\textbf {p}+\\textbf {k})\\Gamma ^{\\mu }(\\textbf {p}+\\textbf {k},\\textbf {p})}{2E_n(\\textbf {p}+\\textbf {k})[E_n(\\textbf {p}+\\textbf {k})-E(\\textbf {p}) - \\omega - i\\epsilon ]}+ \\frac{\\Gamma ^{\\mu }(\\textbf {p}^{\\prime },\\textbf {p}^{\\prime }-\\textbf {k})\\Gamma ^{\\nu }(\\textbf {p}^{\\prime }-\\textbf {k},\\textbf {p})}{2E_n(\\textbf {p}^{\\prime }-\\textbf {k})[E_n(\\textbf {p}^{\\prime }-\\textbf {k})-E^{\\prime }(\\textbf {p}^{\\prime }) + \\omega - i\\epsilon ]} \\biggr \\rbrace \\ .$ If $|n\\rangle $ is a single-particle intermediate state with the same mass $m$ as the target, then both terms in the sum above have a pole at $k^{\\mu }=0$ .",
"Using Eq.",
"(REF ), in the forward scattering limit, and near the $\\omega =0$ pole, we get: $M^{\\nu \\mu }(\\textbf {k}; \\textbf {p}^{\\prime }, \\textbf {p}) \\approx &-\\frac{1}{2m\\omega }\\biggl [\\Gamma ^{\\nu }(\\textbf {p}^{\\prime },\\textbf {p}+\\textbf {k})\\Gamma ^{\\mu }(\\textbf {p}+\\textbf {k},\\textbf {p}) - \\Gamma ^{\\mu }(\\textbf {p}^{\\prime },\\textbf {p}^{\\prime }-\\textbf {k})\\Gamma ^{\\nu }(\\textbf {p}^{\\prime }-\\textbf {k},\\textbf {p})\\biggr ] + \\ {\\rm O.T.}",
"\\ ,$ where by O.T.",
"we mean other terms that do not contain poles at $\\omega =0$ .",
"As usual, the scattering amplitude will be defined as: $f_{\\rm scat}(\\textbf {k}^{\\prime },\\lambda ^{\\prime }; \\textbf {k}, \\lambda ) = \\frac{1}{8\\pi m}\\epsilon ^{*^{\\prime }}_j\\epsilon _i M^{ji}(\\textbf {k}^{\\prime },\\textbf {k}, \\omega ) \\ .$ The case when $s=s^{\\prime }$ was considered in detail by Weinberg [3].",
"Here, we are interested in the case when the intermediate state could be a particle of different spin (but of the same mass) as the target.",
"To be more specific, we will consider a situation when $|s^{\\prime }-s|= 1$ or 0.",
"In particular, when $s=1/2$ and $s^{\\prime }=3/2$ , the vertex function is [see also Eq.",
"(REF )]:This vertex clearly implies the conservation of EM current: $k_{\\nu }\\Gamma ^{\\nu }_{\\sigma \\sigma ^{\\prime }} = 0 $ .",
"$\\Gamma ^{\\nu }_{\\sigma \\sigma ^{\\prime }}(p^{\\prime },p^{\\prime }+k^{\\prime }) &= \\frac{ie \\kappa _{M}}{2m^2}\\epsilon ^{\\mu \\nu \\alpha \\beta }p^{\\prime }_{\\mu }k^{\\prime }_{\\alpha } \\bar{u}^{\\sigma }(p^{\\prime })\\psi ^{\\sigma ^{\\prime }}_{\\beta }(p^{\\prime }+k^{\\prime }) \\ .$ In case of spin-1/2 target, we take the intermediate states to be either spin-1/2 or spin-3/2 particle state.",
"Here, we will only need to compute the part of the amplitude with spin-3/2 intermediate state, since the result in case of the spin-1/2 intermediate state is already known.",
"Using Eqs.",
"(REF ) and (REF ) [also Eq.",
"(REF ) to sum over Rarita-Schwinger (RS) states] we compute:From Eq.",
"(REF ) and current conservation: $k_{\\mu }M^{\\nu \\mu }(\\textbf {k}; \\textbf {p}^{\\prime }, \\textbf {p}) = 0$ , we can deduce that O.T.",
"in Eq.",
"(REF ) do not contribute to this part of the amplitude.",
"$f_{RS}(\\textbf {k}^{\\prime },\\lambda ^{\\prime }; \\textbf {k}, \\lambda ) &= \\frac{i\\omega e^2\\kappa ^2_{M}}{12\\pi m^2} \\left[(\\vec{n}^{\\prime } \\times \\vec{\\epsilon }^{^{\\prime }*})\\times (\\vec{n} \\times \\vec{\\epsilon })\\right]\\vec{J} \\ .$ In the forward-scattering limit, $f_{RS} = f_{RS}(\\omega , \\lambda )$ , and we can define the following amplitude: $&g_{-}(\\omega ^2) \\equiv \\frac{f_{RS}(\\omega , +1) - f_{RS}(\\omega , -1)}{2\\omega } \\ ,$ in which case, it can be checked that: $&g_{-}(\\omega ^2 \\rightarrow 0) = -\\frac{e^2\\kappa _M^2}{12\\pi m^2}~J_z \\ .$ Therefore, assuming $g_-(|\\omega ^2| \\rightarrow \\infty ) \\rightarrow 0$ , the total scattering amplitude will satisfy the following generalized unsubtracted dispersion relation: $4\\pi ^2[f_-(0)+g_-(0)] &= \\frac{\\pi e^2}{m^2} \\left(\\kappa ^2_p - \\frac{1}{3}\\kappa _M^2\\right)J_z= \\int ^{\\infty }_0\\frac{\\sigma _{\\rm tot}(\\omega ^{\\prime },+1) - \\sigma _{\\rm tot}(\\omega ^{\\prime },-1)}{\\omega ^{\\prime }}d\\omega ^{\\prime } \\ ,$ where $\\kappa _p = (g-2)/2$ is the anomalous magnetic moment of the target (see Appendix ).",
"Eq.",
"(REF ) is a generalization of the GDH-Weinberg sum rule, when there is a spin-3/2 intermediate state in the Compton scattering process, that has the same mass as the spin-1/2 target.",
"One of the consequences of this sum rule is that in weakly coupled theories, the gyromagnetic ratio for spin-1/2 particle can be different from its “natural” value $g=2$ .",
"As can be deduced from Eq.",
"(REF ), $g = 2\\left(1 \\pm \\frac{1}{\\sqrt{3}}\\kappa _M\\right) + {\\cal O}(e^2) \\ .$ The same result can be obtained using Feynman diagrams in Appendix ."
],
[
"Generalization to Spin-$j$ Target: First Sum Rule",
"The non-diagonal transition vertex for the emission of a single soft photon can be written as in [21]: $&\\Gamma ^{\\mu }(\\textbf {k},\\textbf {0}) \\equiv \\langle \\textbf {k}, j^{\\prime }, \\sigma ^{\\prime } | {\\cal J}^{\\mu }| \\textbf {0}, j, \\sigma \\rangle = -2im~\\epsilon ^{0\\mu \\alpha \\beta }k_{\\alpha } \\langle j^{\\prime }, \\sigma ^{\\prime }|\\mu _{\\beta }|j,\\sigma \\rangle + {\\cal O}(\\omega ^2) \\ ,$ where $\\mu _i$ is the magnetic moment and $j \\ne j^{\\prime }$ .",
"Applying the Wigner-Eckart theorem, the matrix elements of the magnetic moment can be parametrized as follows: $\\langle j,\\sigma |\\mu _3|j,\\sigma \\rangle = e g_j \\sigma /2m$ and $&\\langle j \\pm 1/2, \\sigma - 1/2|\\mu ^{-}|j\\mp 1/2,\\sigma +1/2\\rangle = \\pm \\frac{e}{2m}h_j\\sqrt{(j\\mp \\sigma +1)(j \\mp \\sigma )} \\ ,$ where $\\mu ^{\\pm } \\equiv \\mu _1 \\pm i\\mu _2$ .",
"Using Eqs.",
"(REF ) and (REF ), which are valid for any spin, the non-diagonal contribution to the forward scattering amplitude on a target of arbitrary spin-$j$ ($\\ne 0$ ) is: $&\\tilde{f}_{\\rm scat}= -\\frac{1}{16\\pi m^2\\omega }\\epsilon ^{*^{\\prime }}_{\\nu }\\epsilon _{\\mu }\\biggl [\\Gamma ^{\\nu \\dagger }(\\textbf {k},\\textbf {0})\\Gamma ^{\\mu }(\\textbf {k},\\textbf {0}) - \\Gamma ^{\\mu \\dagger }(-\\textbf {k},0)\\Gamma ^{\\nu }(-\\textbf {k},0)\\biggr ] \\\\ \\nonumber &= -\\frac{\\omega }{4\\pi }\\sum _{j^{\\prime }\\sigma ^{\\prime }}2\\langle j, \\sigma |\\mu _{[i}|j^{\\prime },\\sigma ^{\\prime }\\rangle \\langle j^{\\prime },\\sigma ^{\\prime } |\\mu _{i^{\\prime }]}|j, \\sigma \\rangle (\\vec{n} \\times \\vec{\\epsilon }^{^{\\prime }*})_{i} (\\vec{n} \\times \\vec{\\epsilon })_{i^{\\prime }} \\ .$ Taking into account that: $&\\langle j , \\sigma |\\mu ^{-}|j\\mp 1,\\sigma +1\\rangle = \\pm \\frac{e}{2m}h_{j \\mp 1/2}\\sqrt{(j \\mp \\sigma \\mp 1 + 1)(j \\mp \\sigma \\mp 1)} \\ , \\\\ \\nonumber &\\langle j , \\sigma |\\mu ^{+}|j\\pm 1,\\sigma -1\\rangle = \\pm \\frac{e}{2m}h_{j \\pm 1/2}\\sqrt{(j \\mp \\sigma \\pm 1 + 1)(j \\mp \\sigma \\pm 1)} \\ ,$ and applying the definition in Eq.",
"(REF ) for $\\tilde{f}_{\\rm scat}(\\omega ,\\lambda )$ , we arrive at the following result in the forward scattering limit: $4\\pi ^2\\tilde{f}_-(0) &= -\\frac{\\pi e^2}{8 m^2}J_z\\left[ (2j+3)h_{j+1/2}^2 - (2j-1)h_{j-1/2}^2 \\right] \\ .$ Therefore, a more general form of the sum rule can be written as: $&\\frac{\\pi e^2J_z}{4m^2} \\left[(g_j -2)^2 - \\left(j+\\frac{3}{2}\\right)h_{j+1/2}^2 + \\left(j-\\frac{1}{2}\\right) h_{j-1/2}^2\\right]= \\int ^{\\infty }_0\\frac{\\Delta \\sigma (\\omega ^{\\prime })}{\\omega ^{\\prime }}d\\omega ^{\\prime } \\ .$ As before, tree-level unitarity demands: $&(g_j -2)^2 + \\frac{(2j-1)}{2}h_{j-1/2}^2 = \\frac{(2j+3)}{2}h_{j+1/2}^2 \\ .$ This is in agreement with the observation made, e.g., in Ref.",
"[21], proposing that $g=2$ , when $h_j=0$ , for all $j$ .",
"As a simple illustration, when $j=1/2$ : $(g_{1/2} - 2)^2 = 2h_{1}^2 \\ ,$ suggesting, $h_{1} = \\sqrt{\\frac{2}{3}}\\kappa _M = \\sqrt{2} \\kappa _p$ , where we recalled Eq.",
"(REF ).",
"Similarly, when $j=3/2$ , $&(g_{3/2} -2)^2 - 3h_{2}^2 + h_{1}^2 = 0 \\ .$"
],
[
"Generalization for Arbitrary Intermediate State: Second Sum Rule",
"Consider a situation when the intermediate state in the Compton scattering process is a very narrow resonance of mass $M_n$ ($\\ne m$ ).",
"Then, using Eqs.",
"(REF ) and (REF ), near each pole the forward scattering amplitude on a target at rest becomes (see also Ref.",
"[3]): $f^{(n)}(\\omega \\rightarrow \\omega _n, \\lambda ) \\rightarrow \\frac{1}{16\\pi m}\\frac{\\langle \\textbf {0}, s, \\sigma | \\vec{\\epsilon }^{\\ *}_{\\lambda }\\vec{{\\cal J}}|\\textbf {k}, s_n, \\sigma _n \\rangle \\langle \\textbf {k}, s_n, \\sigma _n | \\vec{\\epsilon }_{\\lambda }\\vec{J}|\\textbf {0}, s, \\sigma \\rangle }{\\left(\\sqrt{M^2_n + \\omega ^2} - m - \\omega \\right)\\sqrt{M^2_n + \\omega ^2_n}} \\ ,$ where $\\omega _n = \\frac{1}{2m}(M^2_n - m^2)$ , so that: $\\sqrt{M^2_n + \\omega ^2_n} = m+\\omega _n$ .",
"Now, defining: $K_{ni} \\equiv \\sqrt{\\frac{m}{2E_n}}~\\frac{\\langle \\textbf {k}, s_n, \\sigma _n | {\\cal J}_x +i {\\cal J}_y| \\textbf {0}, s, \\sigma \\rangle }{M^2_n - m^2} \\ , \\ \\ \\ \\ \\ K_{in} \\equiv \\sqrt{\\frac{m}{2E_n}}~\\frac{\\langle \\textbf {0}, s, \\sigma | {\\cal J}_x + i{\\cal J}_y|\\textbf {k}, s_n, \\sigma _n \\rangle }{M^2_n - m^2} \\ ,$ the amplitude $f^{(n)}_{-}(\\omega ^2)$ , determined using Eq.",
"(REF ), near each pole, becomes: $&f^{(n)}_{-}(\\omega ^2) = -\\frac{1}{4\\pi } ~\\frac{ \\omega ^2_n}{\\omega ^2 - \\omega ^2_n}\\left[(K_{ni})^{\\dagger }K_{ni} - K_{in}(K_{in})^{\\dagger }\\right] \\ .$ For the total forward scattering amplitude, $f^{\\rm tot}_-(\\omega ^2)$ , with an appropriate contour of integration, we will have: $\\frac{1}{2\\pi i}\\int _C \\frac{f^{\\rm tot}_{-}(\\omega ^{\\prime 2})}{\\omega ^{\\prime 2} - \\omega ^2}d\\omega ^{\\prime 2} &= \\frac{1}{\\pi }\\int ^{\\infty }_0 \\frac{\\mathop {\\rm Im}f^{\\rm tot}_-(\\omega ^{\\prime 2})}{\\omega ^{\\prime 2} - \\omega ^2 - i\\epsilon } d\\omega ^{\\prime 2} \\ .$ In the case $\\omega ^2=0$ and when the integration contour encircles all the particle poles we obtain: $&4\\pi [f_{-}(0)+g_-(0)] - [K^{\\dagger },K]_{ii} = \\frac{1}{\\pi }\\int ^{\\infty }_0 \\frac{\\Delta \\sigma (\\omega ^{\\prime })}{\\omega ^{\\prime }} d\\omega ^{\\prime } \\ , \\\\[5pt] \\nonumber &[K^{\\dagger },K]_{ii} \\equiv \\sum _n \\left[(K_{ni})^{\\dagger }K_{ni} - K_{in}(K_{in})^{\\dagger }\\right]^{\\omega ^2_n}_0 \\ .$ It is also interesting to consider the case when $\\omega ^2 = \\omega ^2_k$ in Eq.",
"(REF ).",
"Then one obtains a relation between the derivatives of the matrix elements $K_{ni}$ and the dispersion integral.",
"Notice that $f_-(0)$ and $g_-(0)$ can be expressed in terms of commutator of some generators like $K_{ni}$ , in which case the left hand side of (REF ) would take a more compact form.",
"Summarizing, when all other intermediate states in the Compton scattering have masses (and spins) different from the mass (or spin) of the scatterer, we have the second sum rule: $\\frac{e^2 J_z}{4m^2}(g-2)^2 &= \\frac{1}{\\pi }\\int ^{\\infty }_0 \\frac{\\Delta \\sigma }{\\omega } d\\omega + [K^{\\dagger },K]_{ii} \\ge 0 \\ .$ Since the left hand side of Eq.",
"(REF ) is always non negative, so should be the right hand side.",
"Ignoring the integral, we can rewrite Eq.",
"(REF ) at leading order in $e^2$ , as follows: $g-2 &= \\pm \\frac{2m}{e}\\sqrt{\\frac{[K^{\\dagger },K]_{ii}}{J_z}} \\ .$ This means that the gyromagnetic ratio may receive corrections even at tree level, if $[K^{\\dagger },K]_{ii} \\ge 0$ .",
"In supergravity theories, $e \\sim m/M_P$ , and we expect $g-2$ to be independent on $m$ , therefore, $[K^{\\dagger },K]_{ii} \\sim e^2/m^2$ , suggesting that $g-2 \\sim {\\cal O}(1)$ , as for spontaneously broken $N=8$ supergravity [7].",
"Finally, allowing for an intermediate state, in a Compton scattering off of a target of spin $j$ , to be any narrow resonance of arbitrary mass and spin, we arrive at the most general sum rule: $&\\frac{e^2J_z}{4m^2} \\left[(g_j -2)^2 - \\left(j+\\frac{3}{2}\\right)h_{j+1/2}^2 + \\left(j-\\frac{1}{2}\\right) h_{j-1/2}^2\\right] = [K^{\\dagger },K]_{ii} + \\frac{1}{\\pi }\\int ^{\\infty }_{0} \\frac{\\Delta \\sigma }{\\omega } d\\omega \\ .$ This sum rule can be extended by taking into account other global charges of the theory at hand.",
"We will not consider such an extension here, since it depends on the specific form of the theory.",
"Instead, below we will study a particular case, namely the example of the $N \\rightarrow \\Delta $ magnetic transition."
],
[
"Some Applications to QCD",
"Application of Eq.",
"(REF ) or (REF ) for $j=1/2$ , directly to Compton scattering on a nucleon ($N(938)$ ) with a possible delta ($\\Delta (1232)$ ) intermediate state would be wrong for two main reasons.",
"First of all, the mass difference $\\delta $ between $\\Delta $ and $N$ is around $300 \\ {\\rm MeV}$ and the forward scattering amplitude, in the $\\omega \\rightarrow 0 $ limit, would behave as $\\omega ^2/(\\delta - \\omega ) \\sim {\\cal O}(\\omega ^2)$ .",
"It is only when $\\delta \\rightarrow 0$ that the scattering amplitude becomes of order ${\\cal O}(\\omega )$ and contributes comparably to the forward amplitude.",
"The second reason is that these baryons are strongly coupled systems and loop corrections may be significant.",
"However, we can still make a formal use of the first sum rule if we work in the limit when $N_c \\rightarrow \\infty $ , in which case $\\delta \\sim {\\cal O}(1/N_c)$ (see, e.g.",
"Refs.",
"[24], [25]).",
"We also need to take $e \\rightarrow 0$ and $N_c \\rightarrow \\infty $ limits in such a way that if $\\Delta \\sigma \\sim N^k_c$ , then $e^2N^{k+2}_c \\ll 1$ , and the dispersive integral can be safely ignored.",
"Comparing Eqs.",
"(REF ) and (REF ) one can deduce that: $\\mu _{p\\rightarrow \\Delta ^+} = g_M(0) = \\sqrt{\\frac{2}{3}}~\\kappa _M = \\pm \\sqrt{2} \\kappa _p \\ ,$ where we work in units of nuclear Bohr magneton, and we took into account the appropriate isospin factor $T^3$ corresponding to the $p \\rightarrow \\Delta ^+$ transition.",
"Taking the experimental value of the proton's anomalous magnetic moment, $\\kappa _p \\approx 1.79$ , we will obtain $\\mu _{p\\rightarrow \\Delta ^+} \\approx \\pm 2.54$ .",
"Unfortunately, nothing can be said about the sign, since our sum rule relates the squares of the couplings.",
"Nevertheless, in absolute value, our result for $\\mu _{p\\rightarrow \\Delta ^+}$ is not very far from similar ones, obtained within the framework of other models, like the Skyrme model [25], and holographic QCD [26], which respectively give the values $2.3$ and $2.58$ (the experimental value is: $\\mu _{p\\rightarrow \\Delta ^+} =3.46 \\pm 0.03$ [27]).",
"In all models of baryons in the large-$N_c$ limit, baryons are finite size objects, whose sizes ($R$ ) do not scale with $N_c$ , while their masses ($M$ ) do.",
"As in the case of the GDH-Weinberg sum rule [28], in the zero radius limit, when $M R \\rightarrow 0$ , we expect the magnetic transition moment to approach its canonical value (REF ).",
"However, in the Skyrme model $R$ is fixed and $M R \\sim {\\cal O}(N_c) $ .",
"This is not surprising, since the Lagrangian of the Skyrme model [25] is known to behave badly at high energies.",
"Now, we want to apply the second sum rule to the same system.",
"In this case, we take the physical masses of $N$ and $\\Delta $ and only assume that these are narrow resonances.",
"Although in this case the scattering amplitude with $\\Delta $ intermediate state vanishes when $\\omega \\rightarrow 0$ , it contributes to the dispersion integral, when $\\omega \\rightarrow \\omega _n$ .",
"Employing Ref.",
"[23], we obtain: $&\\langle \\textbf {k}, 3/2, \\sigma ^{\\prime } | \\vec{\\epsilon }_{\\lambda }\\vec{{\\cal J}}| \\textbf {0}, 1/2, \\sigma \\rangle = \\sqrt{\\frac{3m_N}{2M_{\\Delta }}}~\\frac{ie \\mu _{p\\rightarrow \\Delta ^+} }{(m_N+M_{\\Delta })}\\omega ~\\bar{\\psi }^{\\sigma ^{\\prime }}_r(\\textbf {k})u^{\\sigma }(\\textbf {0})~(\\vec{n} \\times \\vec{\\epsilon }_{\\lambda })_r \\ .$ After straightforward computations, the second sum rule gives: $\\kappa _p = \\pm 2 \\frac{m_N^2}{(m_N+M_{\\Delta })}~\\frac{\\mu _{p\\rightarrow \\Delta ^+}}{\\sqrt{m^2_N + M^2_{\\Delta }}} \\ .$ Numerically, $\\mu _{p\\rightarrow \\Delta ^+} \\approx \\pm 1.91\\kappa _p$ ; therefore, taking $\\kappa _p=1.79$ , we obtain $\\mu _{p\\rightarrow \\Delta ^+} \\approx \\pm 3.42$ .",
"In absolute value this result is coincidentally close to the experimental value: $\\mu _{p\\rightarrow \\Delta ^+} =3.46 \\pm 0.03$ [27].",
"Consider another example (also relevant for large-$N_c$ QCD), when the target has spin-1/2 and the intermediate particle is an excited state with the same spin.",
"Then, using Ref.",
"[29], the general form of the transition matrix element can be written as: $&\\sqrt{\\frac{m}{2E_n}}~\\frac{\\langle \\textbf {k}, 1/2^*,\\sigma ^{\\prime } |{\\cal J}_x + i \\lambda {\\cal J}_y| \\textbf {0}, 1/2,\\sigma \\rangle }{M^2_n - m^2}= 2e G_{*} \\omega ~\\frac{m}{(M_n-m)}\\sqrt{\\frac{mM_n}{M_n^2+m^2}}~(\\sigma _1 + i\\lambda \\sigma _2)_{\\sigma \\sigma ^{\\prime }}\\ ,$ where $G_*$ is some dimensionful coupling.",
"Direct computations show that $[K^{\\dagger },K]_{ii} < 0$ , which means that for this theory with excited states to be unitary at tree level, we need $G_*=0$ .",
"However, this conclusion might change if we include intermediate states with spin-3/2.",
"This work is supported by NSF grant PHY-0758032 and by ERC Advanced Investigator Grant n.226455 Supersymmetry, Quantum Gravity and Gauge Fields (Superfields).",
"We would like to thank Sergio Ferrara, Gregory Gabadadze and Sergei Dubovsky for interesting discussions."
],
[
"Low Energy Compton Amplitude for a Spin-1/2 Target",
"Consider the Compton scattering process: $N\\lbrace p,\\sigma \\rbrace + \\gamma \\lbrace k,\\epsilon _{\\mu }(k)\\rbrace \\rightarrow N\\lbrace p^{\\prime },\\sigma ^{\\prime }\\rbrace + \\gamma \\lbrace k^{\\prime },\\epsilon ^{\\prime }_{\\nu }(k^{\\prime })\\rbrace \\ ,$ where $p$ and $p^{\\prime }$ are initial and final 4-momenta of scatterer (which is a spin-1/2 particle), $\\sigma $ and $\\sigma ^{\\prime }$ are projections of initial and final spin along the $z$ -direction.",
"Analogously, $k$ , $\\epsilon _{\\mu }(k)$ and $k^{\\prime }$ , $\\epsilon ^{\\prime }_{\\nu }(k^{\\prime })$ are the 4-momenta and polarizations of initial and final photons.",
"We take the external particles to be on-shell, that is: $k_0 = |\\textbf {k}|$ , $k_0^{\\prime } = |\\textbf {k}^{\\prime }|$ , $p_0 = \\sqrt{\\textbf {p}^2 + m^2}$ and $p^{\\prime }_0 = \\sqrt{\\textbf {p}^{\\prime 2} + m^2}$ .",
"The scattering amplitude corresponding to process in (REF ) is: $\\nonumber {\\cal M} &= -e^2\\bar{u}(p^{\\prime },\\sigma ^{\\prime })\\biggl \\lbrace \\Gamma ^{\\nu }(p^{\\prime },p+k)\\epsilon ^{*^{\\prime }}_{\\nu }(k^{\\prime })\\frac{p\\hspace{-7.0pt}\\diagup + k\\hspace{-7.0pt}\\diagup + m}{(p+k)^2 - m^2}\\Gamma ^{\\mu }(p+k,p)\\epsilon _{\\mu }(k) \\\\&+ \\Gamma ^{\\mu }(p^{\\prime },p-k^{\\prime })\\epsilon _{\\mu }(k)\\frac{p\\hspace{-7.0pt}\\diagup - k\\hspace{-7.0pt}\\diagup ^{\\prime } + m}{(p-k^{\\prime })^2 - m^2}\\Gamma ^{\\nu }(p-k^{\\prime },p)\\epsilon ^{*^{\\prime }}_{\\nu }(k^{\\prime })\\biggr \\rbrace u(p,\\sigma ) \\ , \\\\[8pt]&{\\qquad }{\\qquad } \\Gamma ^{\\mu }(p_2,p_1) \\equiv \\gamma ^{\\mu }F_D(q^2) + \\frac{i\\sigma ^{\\mu \\nu }}{2m}q_{\\nu }F_P(q^2) \\ ,$ where $e$ is the electric charge of scatterer, $m$ is its mass, and $q = p_2 -p_1$ .",
"Using the notations of Peskin & Schroeder, $a\\hspace{-7.0pt}\\diagup \\equiv a_{\\mu }\\gamma ^{\\mu }$ , $\\sigma ^{\\mu \\nu } =i[\\gamma ^{\\mu },\\gamma ^{\\nu }]/2$ .",
"Here, $F_D(q^2)$ and $F_P(q^2)$ are Dirac and Pauli form factors, which for $q^2=0$ are: $F_D(0) = 1$ and $F_P(0)=\\kappa _p$ , where $e\\kappa _p/(2m)$ is the anomalous magnetic moment of the scatterer.",
"The latter arises from the Pauli Lagrangian: ${\\cal L}_{\\rm P} = -\\frac{e\\kappa _p}{4m}~\\bar{u}~\\sigma ^{\\mu \\nu }u ~F_{\\mu \\nu } \\ .$ In what follows, we will adopt a `gauge' in which the initial and final photon are transverely polarized in the laboratory frame.",
"That is, we choose: $\\epsilon (k) \\cdot k = \\epsilon ^{*^{\\prime }}(k^{\\prime }) \\cdot k^{\\prime } = \\epsilon (k) \\cdot p = \\epsilon ^{*^{\\prime }}(k^{\\prime }) \\cdot p = 0 \\ ,$ where $p=(m,\\textbf {0})$ , implying that $\\epsilon ^0 = \\epsilon ^{\\prime 0} = 0$ and $\\vec{\\epsilon }~\\vec{k} = \\vec{\\epsilon }^{~^{\\prime }}\\vec{k}^{\\prime } = 0$ .",
"We also adopt the following normalization: $\\epsilon (k) \\cdot \\epsilon ^*(k) = \\epsilon ^{\\prime }(k^{\\prime })\\cdot \\epsilon ^{*^{\\prime }}(k^{\\prime }) = -1$ .",
"Using the Dirac equation: $(p\\hspace{-7.0pt}\\diagup -m)u(p)=0$ , and after some simplifications, we can rewrite the amplitude as follows: ${\\cal M} &=\\frac{e^2\\mu }{2m}\\bar{u}(p^{\\prime },\\sigma ^{\\prime })\\biggl [ \\left(\\epsilon \\hspace{-7.0pt}\\diagup ^{\\prime } + \\mu ^{\\prime } \\epsilon \\hspace{-7.0pt}\\diagup ^{\\prime }k\\hspace{-7.0pt}\\diagup ^{\\prime }\\right)\\left(\\frac{\\epsilon \\hspace{-7.0pt}\\diagup k\\hspace{-7.0pt}\\diagup }{\\omega } + \\frac{1-\\mu }{\\mu }\\epsilon \\hspace{-7.0pt}\\diagup \\right)+ \\left(\\epsilon \\hspace{-7.0pt}\\diagup - \\mu ^{\\prime } \\epsilon \\hspace{-7.0pt}\\diagup k\\hspace{-7.0pt}\\diagup \\right)\\left(\\frac{\\epsilon \\hspace{-7.0pt}\\diagup ^{\\prime }k\\hspace{-7.0pt}\\diagup ^{\\prime }}{\\omega ^{\\prime }} + \\frac{1-\\mu }{\\mu }\\epsilon \\hspace{-7.0pt}\\diagup ^{\\prime }\\right)\\biggr ] u(p,\\sigma ) \\ ,$ where $\\omega \\equiv pk/m $ , $\\omega ^{\\prime } \\equiv pk^{\\prime }/m$ , $\\epsilon \\hspace{-7.0pt}\\diagup ^{\\prime } \\equiv \\gamma ^{\\mu }\\epsilon ^{*^{\\prime }}_{\\mu }$ , $\\mu ^{\\prime } \\equiv \\kappa _p/(2m)$ and $\\mu = 1 + 2m\\mu ^{\\prime }$ is the magnetic moment.",
"Since we work in the frame where $p_0=m$ and $\\textbf {p}=\\textbf {0}$ , we have: $\\omega = k_0$ and $\\omega ^{\\prime } = k_0^{\\prime }$ .",
"The initial state is $u^T(p,\\sigma ) = \\sqrt{m}\\left\\lbrace \\xi ,\\xi \\right\\rbrace $ , where $\\xi $ is a spinor such that $\\xi ^{\\dagger }\\xi = 1$ .",
"Similarly, for $|\\textbf {p}^{\\prime }| \\ll m $ , $\\bar{u}(p^{\\prime },\\sigma ^{\\prime }) = \\sqrt{m}\\left\\lbrace \\xi ^{\\prime \\dagger }\\left(1 + \\frac{1}{2m}\\vec{\\sigma } \\textbf {p}^{\\prime }\\right), \\xi ^{\\prime \\dagger }\\left( 1 - \\frac{1}{2m}\\vec{\\sigma } \\textbf {p}^{\\prime }\\right)\\right\\rbrace \\ .$ Taking $\\omega = \\omega ^{\\prime }$ and defining $\\vec{n} \\equiv \\vec{k}/\\omega $ and $\\vec{n}^{\\prime } \\equiv \\vec{k}^{\\prime }/\\omega ^{\\prime }$ , we can perform direct matrix and vector multiplications to arrive to the final result.",
"After tedious but straightforward calculations, the answer can be written in a familiar form [20]: $f_{\\rm scat} &= \\frac{e^2}{4\\pi m}\\biggl \\lbrace -\\vec{\\epsilon }^{^{\\prime }*}\\vec{\\epsilon } +\\frac{i\\kappa _p}{m}\\omega ~\\vec{J}~(\\vec{\\epsilon }^{^{\\prime }*} \\times \\vec{\\epsilon }) - \\frac{i\\mu ^2}{m}\\omega ~\\vec{J}~\\left[(\\vec{n}^{\\prime } \\times \\vec{\\epsilon }^{^{\\prime }*}) \\times (\\vec{n} \\times \\vec{\\epsilon })\\right] \\\\ \\nonumber &- \\frac{i\\mu }{2m}~\\omega ~\\vec{J}~\\left[\\left\\lbrace \\vec{n}(\\vec{n}\\times \\vec{\\epsilon }) + (\\vec{n}\\times \\vec{\\epsilon })\\vec{n}\\right\\rbrace \\vec{\\epsilon }^{^{\\prime }*}-\\lbrace \\vec{n}^{\\prime }(\\vec{n}^{\\prime }\\times \\vec{\\epsilon }^{^{\\prime }*}) + (\\vec{n}^{\\prime }\\times \\vec{\\epsilon }^{^{\\prime }*})\\vec{n}^{\\prime }\\rbrace \\vec{\\epsilon }\\right]\\biggr \\rbrace \\ ,$ where $f_{\\rm scat} \\equiv {\\cal M}/(8\\pi m) $ and $\\vec{J}\\equiv \\delta _{\\sigma ^{\\prime }\\sigma } \\xi ^{\\dagger }_{\\sigma } \\vec{\\sigma } \\xi _{\\sigma }/2$ , with $\\vec{J}$ being the spin of the scatterer (when $\\sigma =\\sigma ^{\\prime }$ )."
],
[
"General Properties of Rarita-Schwinger Field",
"The Lagrangian for a free massive spin-3/2, Rarita-Schwinger (RS) field, $\\psi _{\\mu }$ , can be written as: ${\\cal L}_{\\rm RS} = \\bar{\\psi }_{\\mu }\\left(i \\gamma ^{\\mu \\nu \\rho }\\partial _{\\rho } - m \\gamma ^{\\mu \\nu } \\right)\\psi _{\\nu } \\ ,$ where $\\gamma ^{\\mu \\nu } \\equiv \\gamma ^{[\\mu }\\gamma ^{\\nu ]}$ and $\\gamma ^{\\mu \\nu \\rho } \\equiv \\gamma ^{[\\mu }\\gamma ^{\\nu }\\gamma ^{\\rho ]}$ .",
"The equations of motion that follow from this Lagrangian can be equivalently written as Dirac equations along with the transversality and tracelessness constraints: $\\left(i\\partial \\hspace{-7.0pt}\\diagup - m\\right)\\psi _{\\mu } = 0 \\ , \\ \\ \\ \\partial ^{\\mu }\\psi _{\\mu } = 0 \\ , \\ \\ \\ \\gamma ^{\\mu }\\psi _{\\mu } = 0 \\ .$ These constraints guarantee that among 16 independent components of RS field only $2s+1 = 4$ physical degrees of freedom will propagate.",
"The wave function for the RS field can be written as a product of massive spin-1 and spin-1/2 polarizations: $e_{\\mu }(\\vec{p},\\textit {n})$ with $\\textit {n}=-1,0,1$ , and $u(\\vec{p},\\sigma )$ with ($\\sigma =\\pm 1/2$ ), respectively.",
"More specifically (see, e.g.",
"Refs.",
"[30]): $\\psi _{\\mu }(\\vec{p},r) &= \\sum _{\\sigma ,\\textit {n}} \\langle \\left(\\frac{1}{2},\\sigma \\right)\\left(1,\\textit {n}\\right)|\\left(\\frac{3}{2},r\\right)\\rangle ~u(\\vec{p},\\sigma )~e_{\\mu }(\\vec{p},\\textit {n}) \\ ,$ where $u(\\vec{p},\\sigma )$ satisfies equations: $(p\\hspace{-7.0pt}\\diagup -m)u(\\vec{p},\\sigma )=0$ and $\\hat{\\sigma }_z u(\\vec{p},\\sigma ) = \\sigma u(\\vec{p},\\sigma ) $ .",
"The polarization vectors satisfy the following normalization and transversality conditions: $&e^*_{\\mu }(\\vec{p},\\textit {n})e^{\\mu }(\\vec{p},\\textit {n}^{\\prime }) = -\\delta _{\\textit {n}\\textit {n}^{\\prime }} \\ , {\\qquad }p^{\\mu }e_{\\mu }(\\vec{p},\\textit {n}) = p^{\\mu }e^*_{\\mu }(\\vec{p},\\textit {n}) = 0 \\ .$ Substituting the values of the Clebsch-Gordan coefficients in Eq.",
"(REF ), that are proportional to $\\delta _{r,\\textit {n}+\\sigma }$ , we will get: $\\psi _{\\alpha }^{\\pm 3/2} &= e_{\\alpha }^{\\pm 1}u^{\\pm 1/2} \\ , {\\qquad }\\psi _{\\alpha }^{\\pm 1/2} = \\frac{1}{\\sqrt{3}}\\left(e_{\\alpha }^{\\pm 1}u^{\\mp 1/2} + \\sqrt{2}e_{\\alpha }^0 u^{\\pm 1/2}\\right) \\ ,$ where $\\psi _{\\alpha }(\\vec{p},r) \\equiv \\psi _{\\alpha }^{r}$ , $u(\\vec{p},\\sigma ) \\equiv u^{\\sigma }$ and $e_{\\mu }(\\vec{p},\\textit {n}) \\equiv e_{\\mu }^\\textit {n}$ .",
"These solutions satisfy the Dirac equation, as well as transversality and tracelessness constraints.",
"Moreover, the RS states are normalized as: $&\\bar{\\psi }_{\\alpha }(\\vec{p},r)\\psi ^{\\alpha }(\\vec{p},r^{\\prime }) = -2m \\delta _{rr^{\\prime }} \\ , {\\qquad }\\bar{\\psi }_{\\alpha }(\\vec{p},r)\\gamma _{\\mu }\\psi ^{\\alpha }(\\vec{p},r^{\\prime }) = -2p_{\\mu } \\delta _{rr^{\\prime }} \\ .$ In what follows it would be useful to note that: $P_{\\mu \\nu }(p) &\\equiv \\sum _{r}\\psi _{\\nu }(\\vec{p},r)\\bar{\\psi }_{\\mu }(\\vec{p},r) = -(p\\hspace{-7.0pt}\\diagup + m)\\Pi _{\\mu \\nu }(p) \\ , \\\\ \\nonumber \\Pi _{\\mu \\nu }(p) &\\equiv \\left(g_{\\mu \\nu } -\\frac{p_{\\mu }p_{\\nu }}{m^2}\\right) - \\frac{1}{3}\\left(\\gamma _{\\mu } -\\frac{p_{\\mu }}{m}\\right)\\left(\\gamma _{\\nu } +\\frac{p_{\\nu }}{m}\\right) \\ .$ In the chiral representation of the $\\gamma $ -matrices (as in Peskin & Schroeder), it can be checked that, when $p = (m,0,0,0)$ , the solutions for $u^{\\sigma }$ and $e_{\\mu }^\\textit {n}$ are: $&u^{-1/2} = \\sqrt{m}(0,1,0,1)^T \\ , \\ \\ \\ \\ u^{+1/2} = \\sqrt{m}(1,0,1,0)^T \\ , \\\\ \\nonumber &e^+_{\\mu } = \\frac{1}{\\sqrt{2}}(0,1,i,0) \\ , \\ \\ \\ e^-_{\\mu } = -\\frac{1}{\\sqrt{2}}(0,1,-i,0) \\ , \\ \\ \\ e^0_{\\mu } = (0,0,0,-1) \\ .$ Writing the quantized RS field as, $&\\Psi _{\\mu }(x) = \\int \\frac{d^3p}{(2\\pi )^3}\\frac{1}{2E_{\\textbf {p}}}\\sum _{\\lambda }\\left\\lbrace e^{-ipx}\\psi _{\\mu }(\\vec{p},\\lambda )a_{\\vec{p},\\lambda } + e^{ipx}\\psi ^C_{\\mu }(\\vec{p},\\lambda )a^{\\dagger }_{\\vec{p},\\lambda }\\right\\rbrace \\ , \\\\&\\lbrace a_{\\vec{p},\\lambda }, a^{\\dagger }_{\\vec{p}^{\\prime },\\lambda ^{\\prime }} \\rbrace = (2\\pi )^32E_{\\textbf {p}}\\delta _{\\lambda \\lambda ^{\\prime }}\\delta (\\textbf {p}-\\textbf {p}^{\\prime }) \\ ,{\\qquad }\\lbrace a_{\\vec{p},\\lambda }, a_{\\vec{p}^{\\prime },\\lambda ^{\\prime }} \\rbrace = \\lbrace a^{\\dagger }_{\\vec{p},\\lambda }, a^{\\dagger }_{\\vec{p}^{\\prime },\\lambda ^{\\prime }} \\rbrace = 0 \\ ,$ and using Eq.",
"(REF ), it can be deduced that: $&\\langle 0| \\Psi _{\\nu }(x) \\bar{\\Psi }_{\\mu }(y) |0 \\rangle = \\int \\frac{d^3p}{(2\\pi )^3}\\frac{1}{2E_{\\textbf {p}}}\\sum _{\\lambda }\\psi _{\\nu }(\\vec{p},\\lambda )\\bar{\\psi }_{\\mu }(\\vec{p},\\lambda ) e^{-ip(x-y)} \\\\ \\nonumber &=-(i\\partial \\hspace{-7.0pt}\\diagup _x+m)\\Pi _{\\mu \\nu }(i\\partial _x) \\int \\frac{d^3p}{(2\\pi )^3}\\frac{1}{2E_{\\textbf {p}}}e^{-ip(x-y)} \\ , \\\\&\\langle 0| \\bar{\\Psi }_{\\mu }(y)\\Psi _{\\nu }(x) |0 \\rangle = \\int \\frac{d^3p}{(2\\pi )^3}\\frac{1}{2E_{\\textbf {p}}}\\sum _{\\lambda }\\psi ^C_{\\nu }(\\vec{p},\\lambda )\\bar{\\psi }^C_{\\mu }(\\vec{p},\\lambda ) e^{-ip(y-x)} \\\\ \\nonumber &=(i\\partial \\hspace{-7.0pt}\\diagup _x+m)\\Pi _{\\mu \\nu }(i\\partial _x) \\int \\frac{d^3p}{(2\\pi )^3}\\frac{1}{2E_{\\textbf {p}}}e^{-ip(y-x)} \\ .$ Since the Feynman propagator is defined as $S^F_{\\mu \\nu }(x-y) \\equiv \\langle 0 |T\\Psi _{\\nu }(x) \\bar{\\Psi }_{\\mu }(y) | 0 \\rangle $ , we have: $&S^F_{\\mu \\nu }(x-y) = -(i\\partial \\hspace{-7.0pt}\\diagup _x+m)\\Pi _{\\mu \\nu }(i\\partial _x) D_F(x-y) \\ , \\\\&D_F(x-y) \\equiv \\int \\frac{d^4p}{(2\\pi )^4}~\\frac{i}{p^2-m^2 +i\\epsilon } e^{-ip(x-y)} \\ ,$ where $D_F(x-y)$ is the Feynman propagator of a free scalar field.",
"More explicitly, $S^F_{\\alpha \\beta }(p) &= -i\\frac{\\left(p\\hspace{-7.0pt}\\diagup + m\\right)}{p^2-m^2 + i\\epsilon }\\left[g_{\\alpha \\beta } - \\frac{1}{3}\\gamma _{\\alpha }\\gamma _{\\beta } - \\frac{2p_{\\alpha }p_{\\beta }}{3m^2}-\\frac{\\left(\\gamma _{\\alpha }p_{\\beta }- \\gamma _{\\beta }p_{\\alpha }\\right)}{3m}\\right]\\ .$"
],
[
"First Sum Rule for Spin-1/2 Target: Alternative Approach",
"The most general form of the vertex function, describing interactions between photon, spin-1/2 and spin-3/2 particles of the same mass, following Jones and Scadron [22] can be written in terms of magnetic dipole ($g_M$ ), electric quadrupole ($g_E$ ), and Coulomb quadrupole ($g_C$ ) form factors.",
"This interaction vertex effectively emerges from the following Lagrangian (see, e.g.",
"[23]): ${\\cal L}_{\\rm int} = \\frac{3ie}{4m^2}~\\bar{u}T^3&\\biggl [g_M \\left(\\partial _{\\mu }\\psi _{\\nu }\\right) \\tilde{F}^{\\mu \\nu } + i g_E \\gamma _5 \\left(\\partial _{\\mu }\\psi _{\\nu }\\right)F^{\\mu \\nu } -\\frac{2g_C}{m}\\gamma _5\\gamma ^{\\alpha }\\partial _{[\\alpha }\\psi _{\\nu ]}\\partial _{\\mu }F^{\\mu \\nu } \\biggr ] + {\\rm h.c.} \\ ,$ where $\\tilde{F}^{\\mu \\nu } = \\epsilon ^{\\mu \\nu \\alpha \\beta }F_{\\alpha \\beta }/2$ and $T^3$ is an operator due to additional internal degrees of freedom (such as global or isospin charge) that the fields could carry.",
"In the soft momentum transfer (or near forward) limit that we are interested in, only the first term will matter.",
"We will rewrite the Lagrangian describing the magnetic dipole transition as: ${\\cal L}^{M1}_{\\rm int} &= \\frac{ie\\kappa _{M}}{2m^2}~\\bar{u}\\left(\\partial _{\\mu }\\psi _{\\nu }\\right) \\tilde{F}^{\\mu \\nu } + {\\rm h.c.} \\ .$ The interaction vertex describing the dominant magnetic dipole $3/2 \\rightarrow 1/2$ transition is: $\\Gamma ^{\\nu \\beta }(p^{\\prime },p^{\\prime }+k^{\\prime }) &= \\frac{e \\kappa _{M}}{2m^2}\\epsilon ^{\\mu \\nu \\alpha \\beta }p^{\\prime }_{\\mu }k^{\\prime }_{\\alpha } \\ ,$ where $p^{\\prime }$ is the momentum of spin-1/2 state and $k^{\\prime }$ is the photon momentum.",
"Similarly, the vertex of $1/2 \\rightarrow 3/2$ transition is: $\\Gamma ^{\\nu \\beta }(p+k,p) &= -\\frac{e\\kappa _{M}}{2m^2}\\epsilon ^{\\mu \\nu \\alpha \\beta }p_{\\mu }k_{\\alpha } \\ ,$ where $p$ is the momentum of spin-1/2 state and $k$ is the photon momentum.",
"For forward scattering, when $p_1 = p_2 = (m,0,0,0)$ and $\\omega = \\omega ^{\\prime }$ , we have: $\\Gamma ^{\\nu \\beta }(p^{\\prime },p^{\\prime }+k^{\\prime }) = -\\Gamma ^{\\nu \\beta }(p+k,p) &= \\frac{e\\kappa _{M}}{2m}\\omega ~\\epsilon ^{0\\nu \\beta k}n_{k} \\ .$ Consider the scattering in (REF ), where the intermediate state is a spin-3/2 Rarita-Schwinger particle with the same mass as the scatterer.",
"We want to find the amplitude corresponding to this process.",
"It can be written as: $\\nonumber i{\\cal M_{RS}} &= \\bar{u}(p^{\\prime },\\sigma ^{\\prime })\\biggl \\lbrace \\Gamma ^{\\nu \\alpha }(p^{\\prime },p+k)\\epsilon ^{*^{\\prime }}_{\\nu }(k^{\\prime })S_{\\alpha \\beta }(p+k)\\Gamma ^{\\mu \\beta }(p+k,p)\\epsilon _{\\mu }(k) \\\\&+ \\Gamma ^{\\mu \\beta }(p^{\\prime },p-k^{\\prime })\\epsilon _{\\mu }(k)S_{\\beta \\alpha }(p-k^{\\prime })\\Gamma ^{\\nu \\alpha }(p-k^{\\prime },p)\\epsilon ^{*^{\\prime }}_{\\nu }(k^{\\prime })\\biggr \\rbrace u(p,\\sigma ) \\ ,$ where vertices are defined in Eqs.",
"(REF )-(REF ), and propagator of RS field, $S_{\\alpha \\beta }$ , is given by Eq.",
"(REF ).",
"Since we are interested in the near forward scattering amplitude up to order ${\\cal O}(\\omega )$ , and using Eqs.",
"(REF ) and (REF ), direct computations for $f_{RS} = {\\cal M}_{RS}/(8\\pi m)$ give: $\\nonumber f_{RS}(\\omega , \\lambda ) &= \\frac{i}{12\\pi }\\frac{e^2\\kappa _M^2}{m^2}~\\omega \\left[(\\vec{\\epsilon }^{^{\\prime }*} \\times \\vec{n}^{\\prime }) \\times \\left(\\vec{\\epsilon } \\times \\vec{n}\\right)\\right]\\vec{J} = -\\frac{\\lambda \\omega }{12\\pi }\\frac{e^2\\kappa _M^2}{m^2}~J_z \\ .$ Finally, using the definition for $g_-(\\omega ^2)$ , given in Eq.",
"(REF ), in the forward limit we have: $4\\pi ^2g_-(0) &= -\\frac{\\pi e^2\\kappa ^2_{M}}{3 m^2}J_z \\ ,$ which is in agreement with the previous result (REF ), as should be expected."
]
] | 1204.1064 |
[
[
"Spectrum of Higher Derivative 6D Chiral Supergravity"
],
[
"Abstract Gauged off-shell Maxwell-Einstein supergravity in six dimensions with N=(1,0) supersymmetry has a higher derivative extension afforded by a supersymmetrized Riemann squared term.",
"This theory admits a supersymmetric Minkowski x S^2 compactification with a U(1) monopole of unit charge on S^2.",
"We determine the full spectrum of the theory on this background.",
"We also determine the spectrum on a non-supersymmetric version of this compactification in which the monopole charge is different from unity, and we find the peculiar feature that there are massless gravitini in a representation of the S^2 isometry group determined by the monopole charge."
],
[
"Introduction",
"Higher-derivative supergravities are of considerable interest, especially when they arise as low-energy effective actions of string theories with higher-derivative corrections proportional to powers of the slope parameter $\\alpha ^{\\prime }$ .",
"However, their construction is notoriously difficult, in part due to the fact that supergravities exist only on-shell in ten dimensions.",
"In view of this difficulty, the compactifications of these theories are rarely studied.",
"In order to gain insights into the compactification of higher-derivative theories, it is instructive to investigate the issue in the simpler situation of lower-dimensional supergravities with higher-derivative terms, postponing for the present the question of how they may arise from ten dimensions.",
"An important technical advantage is that in some lower-dimensional cases, off-shell formulations of the supergravity theories exist.",
"This leads us to consider in particular ${\\cal N}=(1,0)$ supergravity in six dimensions, which is the highest dimension, and the highest degree of supersymmetry, for which a supergravity with an off-shell formulation is known.",
"The off-shell formulation of this supergravity was constructed in [1], [2], and a higher-derivative extension with an off-shell supersymmetrized Riemann-squared term was obtained in [3], [4].",
"The gauging of the $U(1)$ R-symmetry in the presence of this higher-derivative extension has also recently been obtained [5].",
"The model has two parameters, namely an overall coefficient $M^{-2}$ in front of the higher-derivative superinvariant in the action, and the gauge-coupling constant $g$ .",
"In the present paper, we shall study the six-dimensional gauged ${\\cal N}=(1,0)$ theory with the Riemann-squared term constructed in [5].",
"In the absence of the curvature-squared terms the model is an (off-shell) version of the Salam-Sezgin theory constructed long ago [6].",
"It was shown in [6] that the model had the unusual feature of admitting a supersymmetric Minkowski$_4\\times S^2$ vacuum, in which there is a $U(1)$ monopole flux with charge $q=\\pm 1$ on the $S^2$ internal space.",
"A remarkable feature of the theory with the Riemann-squared extension is that the Minkowski$_4 \\times S^2$ background continues to be a supersymmetric solution [5].",
"It also admits non-supersymmetric Minkowski$_4\\times S^2$ backgrounds in which the quantised monopole charge $q$ is larger than 1.",
"Our focus in this paper is to study the spectrum of the Kaluza-Klein states in the fluctuations around the Minkowski$_4\\times S^2$ background.",
"As far as we are aware, such a Kaluza-Klein spectral analysis of a higher-derivative supergravity around a background with non-abelian symmetries has not previously been carried out.",
"Even in the much simpler $S^2$ reduction of the Salam-Sezgin model discussed in [6], the situation is of considerable interest because of the very unusual feature of obtaining non-abelian symmetries from a sphere reduction, whilst obtaining a Poincaré rather than AdS supergravity in the lower dimension.",
"As expected, the states assemble into ${\\cal N}=1$ four-dimensional supermultiplets.",
"In the model constructed in [5] with the higher-order Riemann-squared extension, we find a number of novel features associated with the occurrence of higher-order wave operators, and the fact that certain fields that were purely auxiliary prior to the inclusion of the higher-order terms now become dynamical.",
"In particular, we find that certain four-dimensional vector supermultiplets have wave operators that give rise to masses $m$ that are determined by a non-trivial polynomial of fourth order in $m^2$ .",
"This leads to mass-squared values that are not simply linear in the eigenvalues of the Laplace operators on the internal space, but, rather, involve non-trivial roots of the associated quartic equation.",
"One consequence of this is that the values of $m^2$ can be negative or even complex, thus implying that there will be instabilities.",
"The occurrence of such states might at first sight seem surprising in a supersymmetric vacuum.",
"A standard argument for positive semi-definiteness of the energy, first given in [7], uses the fact that if a state $|\\psi \\rangle $ is annihilated by the supercharge $Q$ , then the superalgebra $\\lbrace Q,Q\\rbrace \\sim P$ implies that $P_0\\ge 0$ .",
"However, a crucial ingredient in this argument is that the norm on the states $|\\psi \\rangle $ is positive definite [8] In our case, the higher-derivative terms in the six-dimensional theory lead to ghost modes in the spectrum, and thus the assumptions required for the positivity result in [7] are violated.",
"The detailed structure of the quartic polynomial in $m^2$ for the vector multiplets implies that two of the four roots are always real and positive, while the remaining two can be complex.",
"The conditions under which this occurs are governed by the ratio $M^2/g^2$ and by the Kaluza-Klein level number $\\ell $ of the harmonics on $S^2$ .",
"As $M^2$ becomes larger, the non-positivity and complexity of the two roots sets in at larger and larger values of the level number $\\ell $ .",
"$M^2$ must at least satisfy $M^2\\ge 8(5+2 \\sqrt{6})g^2$ in order for the roots to be real and positive even at the lowest level $\\ell =0$ .",
"We also study the spectrum of the modes in the non-supersymmetric Minkowski$_4\\times S^2$ vacua that arise for $S^2$ monopole charges $q$ greater than 1.",
"An interesting feature in these cases is that the spectrum includes an $SU(2)$ multiplet of massless spin-${\\textstyle {\\frac{\\scriptstyle 3}{\\scriptstyle 2} } }$ fields at level $\\ell ={\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } }(|q|-3)$ .",
"The organisation of the paper is as follows.",
"In section 2 we review the six-dimensional gauged ${\\cal N}=(1,0)$ off-shell $R+ |\\hbox{Riem}|^2$ supergravity that was recently constructed in [5].",
"In section 3 we study the complete linearised spectrum of Kaluza-Klein modes in the supersymmetric Minkowski$_4\\times S^2$ vacuum, which has a monopole charge $q=1$ on the $S^2$ internal space, and exists for any value of the coupling $M^{-2}$ of the Riemann-squared invariant.",
"In section 4 we repeat the analysis for the non-supersymmetric Minkowski$_4\\times S^2$ vacua, which have arbitrary integer monopole charge $|q|\\ge 2$ , and which exist only for a special value of the ratio $g^2/M^2$ .",
"For this analysis we need many results on the properties of spin-weighted spherical harmonics on $S^2$ , since these are needed for the expansions in the monopole background of the fermion fields and certain vector fields that carry charges.",
"We present a detailed discussion of these harmonics in appendix B.",
"In appendix A we give our spinor conventions, and in appendix C we summarise some results for spin projection operators in four dimensions."
],
[
"The Theory",
"The off-shell $6D$ $(1,0)$ supergravity multiplet consists of the fields [1] $\\left(e_\\mu {}^a,\\ V_\\mu ^{\\prime ij},\\ V_\\mu , \\ B_{\\mu \\nu },\\ L,\\ C_{\\mu \\nu \\rho \\sigma },\\ \\psi _\\mu ^i,\\ \\chi ^i \\right)$ where $V_\\mu ^{\\prime ij}$ is symmetric and traceless in its $Sp(1)$ doublet indices, $B$ and $C$ are antisymmetric tensor fields, $L$ is a real scalar, and the spinors are symplectic Majorana-Weyl.",
"The above fields have $(15,12,5,10,1,5, 40,8)$ degrees of freedom.",
"In addition, we shall consider the off-shell Maxwell multiplet consisting of the fields $\\left( A_\\mu , \\ Y^{ij},\\ \\lambda ^i\\right)\\ ,$ where $Y^{ij}$ is symmetric in its indices and the fermion is symplectic Majorana Weyl.",
"These fields have $(5,3,8)$ degrees of freedom.",
"The total Lagrangian we shall study is given by ${\\cal L}= {\\cal L}_R -\\frac{1}{8M^2} {\\cal L}_{R^2}\\ ,$ where the $U(1)_R$ gauged off-shell supergravity Lagrangian, up to quartic fermion terms, is [1], [5]We have let $g\\rightarrow 4g$ and $A_\\mu \\rightarrow A_\\mu /{\\sqrt{2}}$ in the results of [5].",
"$e^{-1}{\\cal L}_R &=&\\frac{1}{2} L R +\\frac{1}{2} L^{-1}\\partial _\\mu L\\partial ^\\mu L+2{\\sqrt{2}}g L \\delta ^{ij}Y_{ij} -\\frac{1}{24} L H_{\\mu \\nu \\rho }H^{\\mu \\nu \\rho }\\nonumber \\\\&& +L V^{\\prime }_\\mu {}^{ij}V^{^{\\prime }\\mu }{}_{ij} -\\frac{1}{4} L^{-1}E^\\mu E_\\mu +\\frac{1}{\\sqrt{2}} E^\\mu \\left( V_\\mu +2g A_\\mu \\right)\\nonumber \\\\&& +Y^{ij} Y_{ij}-\\frac{1}{8}F_{\\mu \\nu }F^{\\mu \\nu }-\\frac{1}{16}\\varepsilon ^{\\mu \\nu \\rho \\sigma \\lambda \\tau }B_{\\mu \\nu }F_{\\rho \\sigma }F_{\\lambda \\tau }\\nonumber \\\\[0.2cm]&& -\\frac{1}{2}L \\bar{\\psi }_{\\rho }\\gamma ^{\\mu \\nu \\rho }D_{\\mu }\\psi _{\\nu }-{\\sqrt{2}} {\\bar{\\chi }}_i\\gamma ^{\\mu \\nu }D_\\mu \\psi _{\\nu j} \\delta ^{ij}+ L^{-1}{\\bar{\\chi }} {D}\\chi \\nonumber \\\\&& -\\frac{1}{2} {\\bar{\\psi }}^\\mu \\gamma ^\\nu \\psi _\\nu \\partial _\\mu L-\\frac{1}{\\sqrt{2}} \\delta _{ij}{\\bar{\\psi }}_\\nu ^i\\gamma ^\\mu \\gamma ^\\nu \\chi ^j L^{-1}\\partial _\\mu L-2{\\sqrt{2}}g L {\\bar{\\lambda }}_i\\gamma ^\\mu \\psi _{\\mu j} \\delta ^{ij}\\nonumber \\\\[0.2cm]&&+ 2 g {\\bar{\\lambda }}\\chi +\\frac{1}{2} V_\\mu ^{^{\\prime }ij}\\left( {2\\sqrt{2}}{\\bar{\\chi }}^k\\psi ^\\mu _i\\delta _{jk}-3L^{-1}{\\bar{\\chi }}_i\\gamma ^\\mu \\chi _j\\right)\\nonumber \\\\&& -\\frac{1}{48}L H_{\\mu \\nu \\rho }\\left(\\bar{\\psi }^\\lambda \\gamma _{[\\lambda } \\gamma ^{\\mu \\nu \\rho }\\gamma _{\\tau ]}\\psi ^\\tau +2{\\sqrt{2}} L^{-1}{\\bar{\\psi }}_{\\lambda i}\\gamma ^{\\lambda \\mu \\nu \\rho }\\chi _j \\delta ^{ij}-2L^{-2}{\\bar{\\chi }}\\gamma ^{\\mu \\nu \\rho }\\chi \\right)\\nonumber \\\\&& -\\frac{1}{4\\sqrt{2}} E_\\rho \\left(\\psi _\\mu ^i \\gamma ^{\\rho \\mu \\nu }\\psi _\\nu ^j \\delta _{ij}-2{\\sqrt{2}} L^{-1} {\\bar{\\psi }}_\\sigma \\gamma ^{\\rho }\\gamma ^{\\sigma }\\chi +2 L^{-2}{\\bar{\\chi }}_i\\gamma ^\\rho \\chi _j \\delta ^{ij} \\right)\\nonumber \\\\&&-2{\\bar{\\lambda }}{D}\\lambda +\\frac{1}{12}H_{\\mu \\nu \\rho }{\\bar{\\lambda }}\\gamma ^{\\mu \\nu \\rho }\\lambda +\\frac{1}{2\\sqrt{2} } F_{\\mu \\nu } {\\bar{\\lambda }}\\gamma ^\\rho \\gamma ^{\\mu \\nu }\\psi _\\rho \\ ,$ where $H_{\\mu \\nu \\rho }=3\\partial _{[\\mu } B_{\\nu \\rho ]}$ and $E^\\mu &=& \\frac{1}{24}\\varepsilon ^{\\mu \\nu _1\\cdots \\nu _5} \\partial _{[\\nu _1}C_{\\nu _2\\cdots \\nu _5]}\\ .\\\\[0.2cm]D_\\mu \\psi _\\nu ^i &=& (\\partial _\\mu + \\frac{1}{4}\\omega _{\\mu }{}^{ab}\\gamma _{ab} )\\psi _\\nu ^i-\\frac{1}{2} V_\\mu \\delta ^{ij}\\psi _{\\nu j}\\ ,\\\\[0.2cm]D_\\mu \\chi ^i &=& (\\partial _\\mu + \\frac{1}{4}\\omega _{\\mu }{}^{ab}\\gamma _{ab})\\chi ^i-\\frac{1}{2} V_\\mu \\delta ^{ij}\\chi _j + V_\\mu {}^{^{\\prime }i}{}_j \\chi ^j\\ .$ Note the presence of arbitrary coupling constant in ${\\cal L}_R$ .",
"In fact, the sum of all the terms in this Lagrangian that depend on $g$ separately have the off-shell supersymmetry.",
"Thus, the total Lagrangian is a sum of three separately off-shell supersymmetric pieces.",
"The Lagrangian for the supersymmetrized Riemann squared term, up to quartic fermion terms, is given by [3], [4] $e^{-1} {\\mathcal {L}}_{\\rm R^2} &=& R_{\\mu \\nu }{}^{ab}(\\omega _-)R^{\\mu \\nu }{}_{ab}(\\omega _-)-2G^{ab}G_{ab} -4G_{\\mu \\nu }^{^{\\prime }ij}G^{^{\\prime }\\mu \\nu }_{ij}\\ ,\\nonumber \\\\&& +\\frac{1}{4} \\varepsilon ^{\\mu \\nu \\rho \\sigma \\lambda \\tau }B_{\\mu \\nu }R_{\\rho \\sigma ab}(\\omega _-)R_{\\lambda \\tau }{}_{ab}(\\omega _-)\\nonumber \\\\[0.2cm]&& +2{\\bar{\\psi }}^{ab}(\\omega _+)\\gamma ^\\mu D_\\mu (\\omega ,\\omega _-)\\psi _{ab}(\\omega _+)- R_{\\nu \\rho }{}^{ab}(\\omega _-){\\bar{\\psi }}_{ab}(\\omega _+)\\gamma ^\\mu \\gamma ^{\\nu \\rho }\\psi _\\mu \\nonumber \\\\[0.2cm]&& -8G_{\\mu \\nu }^{ij}\\left({\\bar{\\psi }}^\\mu _i \\gamma _\\lambda \\psi ^{\\lambda \\nu }_j(\\omega _+)+\\frac{1}{6} {\\bar{\\psi }}^\\mu _i \\gamma \\cdot H\\psi ^\\nu _j\\right)-\\frac{1}{12}{\\bar{\\psi }}^{ab}(\\omega _+)\\gamma \\cdot H \\psi _{ab}(\\omega _+)\\nonumber \\\\&&-\\frac{1}{2} \\Bigl [ D_\\mu (\\omega _-,\\Gamma _+) R^{\\mu \\nu ab} (\\omega _-)-2 H_{\\mu \\nu }{}^\\rho R^{\\mu \\nu ab} (\\omega _-)\\Bigr ] {\\bar{\\psi }}^a\\gamma _\\rho \\psi _b\\ ,$ where $G_{\\mu \\nu }^{^{\\prime }ij}$ and $G_{\\mu \\nu }$ are the field strengths associated with $V_\\mu ^{^{\\prime }ij}$ and $V_\\mu $ , which can be combined as $V_\\mu ^{ij} = V_\\mu ^{^{\\prime }ij} + {\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } } \\delta ^{ij} V_\\mu $ .",
"Furthermore $\\psi _{\\mu \\nu }(\\omega _+) =2D_{[\\mu }(\\omega _+) \\psi _{\\nu ]}$ and $D_\\mu (\\omega ,\\omega _-)\\psi ^{ab i} &=&(\\partial _\\mu +\\frac{1}{4}\\omega _\\mu {}^{cd}\\gamma _{cd} )\\psi ^{ab i}+2 \\omega _{\\mu -}{}^{c[a} \\psi ^{b]i}{}_c +V_\\mu ^i{}_j \\psi ^{ab i}\\ ,\\nonumber \\\\[0.2cm]\\omega _{\\mu \\pm }{}^{ab} &=& \\omega _\\mu {}^{ab} \\pm {\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } } H_\\mu {}^{ab}\\ ,\\qquad \\Gamma _{\\mu \\nu \\pm }^\\rho =\\Gamma _{\\mu \\nu }^\\rho \\pm {\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } } H_\\mu {}^{\\nu \\rho }\\ .$ The off-shell resulting supersymmetry transformations of the Poincaré multiplet, up to cubic fermion terms, are [1], [3], [5] $\\delta e_{\\mu }{}^a&=&\\frac{1}{2}{\\bar{\\epsilon }}\\gamma ^a\\psi _{\\mu }\\ ,\\nonumber \\\\\\delta \\psi _{\\mu }{}^i&=& (\\partial _{\\mu } +\\frac{1}{4}\\omega _{\\mu ab}\\gamma ^{ab})\\epsilon ^i+V_\\mu {}^i{}_j\\epsilon ^j +\\frac{1}{8} H_{\\mu \\nu \\rho }\\gamma ^{\\nu \\rho }\\epsilon ^i\\ ,\\nonumber \\\\\\delta B_{\\mu \\nu }&=&-{\\bar{\\epsilon }}\\gamma _{[\\mu }\\psi _{\\nu ]}\\ ,\\nonumber \\\\\\delta \\chi ^i &=& \\frac{1}{2\\sqrt{2}} \\gamma ^\\mu \\delta ^{ij}\\partial _\\mu L \\epsilon _j-\\frac{1}{4} \\gamma ^\\mu E_\\mu \\epsilon ^i+\\frac{1}{\\sqrt{2}}\\gamma ^\\mu V^{\\prime }{}_\\mu ^{(i}{}_k \\delta ^{j)k} L\\epsilon _j - \\frac{1}{12\\sqrt{2}}L\\delta ^{ij}\\gamma \\cdot H \\epsilon _j \\ ,\\nonumber \\\\[0.2cm]\\delta L &=& \\frac{1}{\\sqrt{2}} {\\bar{\\epsilon }}^i \\chi ^j\\delta _{ij} \\ ,\\nonumber \\\\[0.2cm]\\delta C_{\\mu \\nu \\rho \\sigma } &=& L{\\bar{\\epsilon }}^i\\gamma _{[\\mu \\nu \\rho }\\psi _{\\sigma ]}^j\\delta _{ij} -\\frac{1}{2\\sqrt{2}} {\\bar{\\epsilon }}\\gamma _{\\mu \\nu \\rho \\sigma }\\chi \\ ,\\nonumber \\\\[0.2cm]\\delta V_{\\mu }{}^{ij}&=& \\frac{1}{2}\\bar{\\epsilon }^{(i}\\gamma ^{\\rho }\\psi _{\\mu \\rho }^{j)}+\\frac{1}{12}\\bar{\\epsilon }^{(i}\\gamma \\cdot H\\psi _{\\mu }^{j)} +\\frac{1}{8}\\sigma ^{-1}\\bar{\\epsilon }^{(i}\\gamma ^{\\rho }\\Bigl (H_{[\\mu }{}^{ab}\\gamma _{ab}\\psi _{\\rho ]}^{j)}\\Bigr )$ and the off-shell supersymmetry transformations of the vector multiplet are $\\delta A_{\\mu }&=&-\\bar{\\epsilon }\\gamma _{\\mu }\\lambda \\ ,\\nonumber \\\\\\delta \\lambda ^{i}&=&\\frac{1}{8\\sqrt{2}}\\gamma ^{\\mu \\nu }F_{\\mu \\nu }\\epsilon ^i-\\frac{1}{2}Y^{ij}\\epsilon _j\\ ,\\nonumber \\\\\\delta Y^{ij} &=& -\\bar{\\epsilon }^{(i}\\gamma ^{\\mu }D_{\\mu }\\lambda ^{j)}+\\frac{1}{8} {\\bar{\\epsilon }}^{(i} \\gamma ^\\mu \\gamma \\cdot H \\psi _\\mu ^{j)}-\\frac{1}{24} {\\bar{\\lambda }}^i \\gamma \\cdot H \\lambda ^j-\\frac{1}{2} Y^{k(i} {\\bar{\\epsilon }}^{j)} \\gamma ^\\mu \\psi _{\\mu k}\\ .$ Of the auxiliary fields of the Poincaré supergravity, $V_\\mu ^{^{\\prime }ij}$ and $V_\\mu $ can no longer be eliminated algebraically due to the presence of the Riemann squared invariant but $Y^{ij}$ and $C_{\\mu \\nu \\rho \\sigma }$ can still be eliminated by means of their field equations as $Y^{ij} = -{\\sqrt{2}} g L\\delta ^{ij}\\ ,\\qquad E_\\mu = {\\sqrt{2}} L\\left(V_\\mu +2g A_\\mu \\right)\\ .$ The total Lagrangian we shall study here is given by ${\\cal L}= L_{\\rm R} -\\frac{1}{8M^2} {\\cal L}_{\\rm R^2}\\ ,$ where $M$ is an arbitrary mass parameter."
],
[
"Supersymmetric $\\rm {Minkowski}_4\\times S^2$ background",
"We shall study the compactification on the one half supersymmetric vacuum solution with the geometry of $\\rm {Minkowski}_4\\times S^2$ .",
"From here on, the 6D coordinates will be denoted by $x^M$ and they will be split as $(x^\\mu ,y^m)$ to denote the coordinates of 4D spacetime and the internal two-dimensional space.",
"The supersymmetric $\\rm {Minkowski}_4\\times S^2$ vacuum solution given by [5] $\\bar{R}_{\\mu \\nu \\lambda \\rho } &=0\\ , &\\qquad \\bar{R}_{mn} &= \\alpha ^2\\bar{g}_{mn}\\ , \\qquad \\bar{L} &=1\\ , \\nonumber \\\\\\bar{F}_{\\mu \\nu } &=0\\ , & \\qquad \\bar{F}_{mn} &= 4g\\epsilon _{mn}\\ , \\qquad & \\nonumber \\\\\\bar{G}_{\\mu \\nu } &=0\\ , & \\qquad \\bar{G}_{mn} &= -\\alpha ^2\\epsilon _{mn}\\ , \\qquad &$ where $\\alpha ^2\\equiv 8g^2$ , $\\bar{g}_{mn}$ is the metric on $S^2$ with radius 1/$\\alpha $ , and $\\epsilon _{mn}$ is the Levi-Civita tensor on the same $S^2$ .",
"We define the complex vectors ${\\hat{Z}}_M = {\\hat{V}}^{^{\\prime }11}_M+i {\\hat{V}}^{^{\\prime }12}_M\\ ,$ and parametrize the linearized fluctuations around above background as follows $\\hat{g}_{MN} &=\\bar{g}_{MN}+\\hat{h}_{MN}\\ ,&\\qquad \\hat{L} &=1+\\hat{\\phi }\\ ,&\\qquad \\hat{A}_{M} &=\\bar{A}_M+\\hat{a}_M\\ ,\\nonumber \\\\\\hat{V}_{M} &=\\bar{V}_M+\\hat{v}_{M}\\ , &\\qquad ~~{\\hat{Z}}_{M} &={\\hat{z}}_M\\ , &\\qquad \\hat{B}_{MN} &=\\hat{b}_{MN}\\ ,$ where we use “$\\rm {hat}$ ” to stand for six dimensional quantities and “$\\rm {bar}$ ” to denote quantities evaluated in the vacuum background.",
"In the background specified above, the linearized six dimensional bosonic and fermionic gauge symmetries are expressed asFor later convenience, starting from the USp(2) symplectic-Majorana-Weyl spinors we have defined Weyl spinors by complexifying as $\\psi =\\psi _1+i\\psi _2 $ and rescaled $\\hat{\\chi }$ and $\\hat{\\lambda }$ used in [5] by $\\hat{\\chi }\\rightarrow \\sqrt{2}\\hat{\\chi }$ , $\\hat{\\lambda }\\rightarrow \\sqrt{2}\\hat{\\lambda }$ .",
"$\\delta \\hat{h}_{MN} &= \\bar{\\nabla }_{M}\\hat{\\xi }_N+ \\bar{\\nabla }_{N}\\hat{\\xi }_M\\ , & \\qquad \\delta \\hat{ a}_{M} &=\\hat{\\xi }^{N}\\bar{F}_{NM}+\\partial _{M}\\hat{\\Lambda }, & \\nonumber \\\\\\delta \\hat{v}_{M}&=\\hat{\\xi }^{N}\\bar{G}_{NM}-2g\\partial _{M}\\hat{\\Lambda }\\ , &\\qquad \\delta \\hat{b}_{MN} &=\\partial _{M}\\hat{\\Lambda }_{N}-\\partial _{N}\\hat{\\Lambda }_{M}\\ , & \\nonumber \\\\\\delta \\hat{\\psi }_M &=\\bar{D}_M\\hat{\\epsilon }, &\\qquad \\delta \\hat{\\lambda } &={\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 16} } }{\\bar{\\Gamma }}^{MN}\\bar{F}_{MN}\\hat{\\epsilon }+{\\textstyle {\\frac{\\scriptstyle i}{\\scriptstyle 2} } }g\\hat{\\epsilon }\\ , & \\qquad \\delta \\hat{\\chi }& =0\\ .$ This background preserves half supersymmetry because it admits a Killing spinor $\\hat{\\eta }$ which has the following properties $\\delta \\hat{\\psi }_M=\\bar{D}_M\\hat{\\eta }=0,\\qquad \\delta \\hat{\\chi }=0,\\qquad \\delta \\hat{\\lambda }=({\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 16} } }{\\bar{\\Gamma }}^{MN}\\bar{F}_{MN}+{\\textstyle {\\frac{\\scriptstyle i}{\\scriptstyle 2} } }g)\\hat{\\eta }=0\\ ,$ and by choosing the six dimensional gamma matrices as in Appendix, it can be shown that $\\hat{\\eta }=\\epsilon \\otimes \\eta ,\\qquad \\eta =\\left(\\begin{array}{c}0 \\\\1 \\\\\\end{array}\\right),$ where $\\epsilon $ is a constant four dimensional Weyl spinor with appropriate chirality inherited from six dimensions."
],
[
"Bosonic Sector",
"In this section, we shall drop the “bar” on the covariant derivatives for simplicity in notation.",
"The linearized bosonic field equations are given as follows $(\\hat{R}^{(L)}_{MN}+\\hat{\\phi }\\bar{R}_{MN}) &=&{\\hat{\\nabla }}_M{\\hat{\\nabla }}_N\\hat{\\phi }+\\alpha ^2\\bar{g}_{MN}\\hat{\\phi }+{\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } }(\\hat{F}^{(L)}_{MP}\\bar{F}_{N}^{~P}+ \\hat{F}^{(L)}_{NP}\\bar{F}_{M}^{~P} -\\bar{F}_{MP}\\bar{F}_{NQ}\\hat{h}^{PQ})\\\\[0.2cm]&&+{\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } }\\hat{h}_{MN}(\\alpha ^2-{\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 4} } }\\bar{F}^{PQ}\\bar{F}_{PQ})-{\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 4} } }\\bar{g}_{MN}(\\hat{F}^{(L)}_{PQ}\\bar{F}^{PQ}-\\bar{F}_{P}^{~Q}\\bar{F}^{PT}\\hat{h}_{QT}),\\nonumber \\\\&&-{\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 8M^2} } }S^{(L)}_{MN}\\ ,\\nonumber \\\\[0.2cm]\\hat{R}^{(L)} &=& 2\\alpha ^2\\hat{\\phi }+2\\hat{\\Box }\\hat{\\phi }\\ ,\\\\[0.2cm]\\hat{\\nabla }^{P}\\hat{H}^{(L)}_{PMN} &=& {\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } }\\varepsilon _{MN}^{~~~~PQST}({\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } }\\hat{F}^{(L)}_{PQ}\\bar{F}_{ST}-\\frac{1}{2M^2}\\tilde{R}^{(L)J}_{~~~~KPQ}\\bar{R}^{K}_{~JST})+\\frac{1}{M^2}\\hat{\\nabla }^P\\hat{\\Box } \\hat{H}^{(L)}_{PMN}, \\nonumber \\\\&&+3{M^2}\\hat{\\nabla }^P(\\hat{H}^{(L)ST}_{+~~~~[P}\\bar{R}_{MN]ST})\\ ,$ $0&=& \\hat{\\nabla }^{P}\\hat{F}^{(L)}_{PM}-\\hat{\\nabla }^P\\hat{h}_{PQ}\\bar{F}^{Q}_{~M}-\\hat{\\nabla }_P\\hat{h}_{QM}\\bar{F}^{PQ}+{\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } }\\hat{\\nabla }_P\\hat{h}\\bar{F}^{P}_{~M}+4g(\\hat{v}_M+2g\\hat{a}_M)-{\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } }*\\hat{H}^{(L)}_{MPQ}\\bar{F}^{PQ}\\ ,\\nonumber \\\\&&\\\\0&=& \\hat{\\nabla }^P\\hat{G}^{(L)}_{PM}-\\hat{\\nabla }^P\\hat{h}_{PQ}\\bar{G}^{Q}_{~M}-\\hat{\\nabla }_P\\hat{h}_{QM}\\bar{G}^{PQ}+{\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } }\\hat{\\nabla }_P\\hat{h}\\bar{G}^{P}_{~M}-M^2({\\hat{v}}_M+2g {\\hat{a}}_M) \\ ,\\\\[0.2cm]0&=& \\left( {\\hat{\\nabla }}^P - i{\\bar{V}}^P\\right) \\hat{G}_{PM}^{^{\\prime }(L)}- i{\\bar{G}}_{MN} {\\hat{z}}^N-M^2 {\\hat{z}}_M \\ ,$ where ${\\hat{G}}_{MN}^{^{\\prime }(L)} &=& 2{\\hat{D}}_{[M}{\\hat{z}}_{N]}\\ ,\\qquad {\\hat{D}}_M {\\hat{z}}_N \\equiv ( {\\hat{\\nabla }}_M - i{\\bar{V}}_M ){\\hat{z}}_N\\ ,\\nonumber \\\\[0.2cm]R^{(L)P}_{~~~~MNQ} &=&\\hat{R}^{(L)P}_{~~~~MNQ}-\\hat{\\nabla }_{[N}H^{(L)P}_{Q]~~M}\\ ,\\nonumber \\\\[0.2cm]S^{(L)}_{MN} &=& 8\\left(\\hat{G}^{(L)}_{MP}\\bar{G}_{N}^{~P}+\\hat{G}^{(L)}_{NP}\\bar{G}_{M}^{~P}-\\bar{G}_{M}^{~P}\\bar{G}_{N}^{~Q}\\hat{h}_{PQ}\\right)-4\\bar{g}_{MN}(\\hat{G}^{(L)}_{PQ}\\bar{G}^{PQ}-\\bar{G}_{P}^{~Q}\\bar{G}^{PT}\\hat{h}_{QT})\\nonumber \\\\&& +4(\\tilde{R}^{(L)S}_{~~~~QMP}\\bar{R}^{Q~~~P}_{~SN}+\\bar{R}^{S~~~P}_{~QM}\\tilde{R}^{(L)Q}_{~~~~SNP}-\\hat{h}^{PQ}\\bar{R}^{S}_{~TMP}\\bar{R}^{T}_{~SNQ})-2\\hat{h}_{MN}\\bar{G}^{PQ}\\bar{G}_{PQ} \\nonumber \\\\&&+\\hat{h}_{MN}\\bar{R}^{PQST}\\bar{R}_{PQST}-2\\bar{g}_{MN}(\\tilde{R}^{(L)S}_{~~~~TPQ}\\bar{R}^{T~PQ}_{~S} +\\bar{R}_{JKSP}\\bar{R}^{JKS}_{~~~~Q}\\hat{h}^{PQ})\\nonumber \\\\&&+8(\\hat{\\nabla }^{P}\\tilde{\\nabla }^{Q}\\tilde{R}_{P(MN)Q})^{(L)}+8\\hat{\\nabla }^{S}(\\bar{R}_{S(M}^{~~~~PQ}\\hat{H}^{+{(L)}}_{N)PQ}),$ and the penultimate term takes the form $(\\hat{\\nabla }^{P}\\tilde{\\nabla }^{Q}\\tilde{R}_{P(MN)Q})^{(L)} &=&\\bar{R}^{P~~~Q}_{~(MN)}(\\bar{R}_{P}^{~S}\\hat{h}_{SQ}-\\bar{R}^{S~T}_{~P~Q}\\hat{h}_{ST}-{\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } }\\hat{\\Box } \\hat{h}_{PQ})+\\hat{\\nabla }_{P}\\hat{\\nabla }_{Q}\\hat{h}_{S(M}\\bar{R}^{P~~QS}_{~N)}\\nonumber \\\\&&-{\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } }(\\hat{\\nabla }^{P}\\hat{\\nabla }_{(M}\\hat{h}^{QS}\\bar{R}_{N)SPQ}+\\hat{\\nabla }^{P}\\hat{\\nabla }^{S}\\hat{h}^{Q}_{~(M}\\bar{R}_{N)SPQ}\\nonumber \\\\&&-{\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } }\\bar{R}^{PQS}_{~~~~~T}\\hat{h}^{T}_{~(M}\\bar{R}_{N)SPQ}+{\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } }\\bar{R}^{PQ}_{~~~T(M}\\bar{R}_{N)SPQ}\\hat{h}^{ST} )\\nonumber \\\\&& +{\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } }\\bar{R}_{PMNQ}\\hat{\\nabla }^{P}\\hat{\\nabla }^{Q}\\hat{h}+\\hat{\\nabla }_{P}\\hat{\\nabla }^{Q}\\tilde{R}^{(L)P}_{~~~~(MN)Q}\\nonumber \\\\&& -{\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } }(\\hat{\\nabla }_P \\hat{H}^{(L)}_{QS(M}\\bar{R}^{PQ~S}_{~~~N)}+\\hat{\\nabla }_P\\hat{H}^{(L)}_{QS(M}\\bar{R}^{P~QS}_{~N)}).$ The covariant derivative ${\\tilde{\\nabla }}_M$ is defined with respect to the connection ${\\widetilde{\\Gamma }}_{\\mu \\nu }^\\rho $ containing bosonic torsion as ${\\widetilde{\\Gamma }}_{\\mu \\nu }^\\rho = \\left\\lbrace \\begin{array}{c}\\rho \\\\\\mu \\nu \\\\\\end{array}\\right\\rbrace +\\frac{1}{2} H_{\\mu \\nu }{}^\\rho \\ .$ Note that we are using $\\hat{G}^{^{\\prime }(L)}_{MN}$ to denote the covariant field strength of the complex vector field $\\hat{z}_M$ , and $G^{(L)}_{MN}$ to denote the field strength of the real vector $v_M$ .",
"There are no transverse traceless spin-2 harmonics on $S^2$ , and the transverse spin-1 harmonics are related to spin-0 harmonics by $Y^{(\\ell )}_m=\\epsilon _m{}^n\\nabla _nY^{(\\ell )}\\ .$ We can expand the six-dimensional bosonic fields in terms of $S^2$ harmonics as follows $\\hat{h}_{\\mu \\nu }& =&\\sum _{\\ell \\ge 0 }h_{\\mu \\nu }^{(\\ell )}Y^{(\\ell )} \\ ,\\nonumber \\\\\\hat{h}_{mn}&=&\\sum _{\\ell \\ge 2 }\\left(L^{(\\ell )}\\nabla _{\\lbrace m}Y^{(\\ell )}_{n\\rbrace }+\\tilde{L}^{(\\ell )}\\nabla _{\\lbrace m}\\nabla _{n\\rbrace }Y^{(\\ell )}\\right)+ \\bar{g}_{mn}\\sum _{\\ell \\ge 0}N^{(\\ell )}Y^{(\\ell )}\\ ,\\nonumber \\\\\\hat{h}_{\\mu m}&=&\\sum _{\\ell \\ge 1}(k_{\\mu }^{(\\ell )}Y_{m}^{(\\ell )}+\\tilde{k}_{\\mu }^{(\\ell )}\\nabla _mY^{(\\ell )})\\ ,\\nonumber \\\\\\hat{\\phi } &=&\\sum _{\\ell \\ge 0}\\phi ^{(\\ell )}Y^{(\\ell )}\\ ,\\nonumber \\\\\\hat{a}_{\\mu }&=&\\sum _{\\ell \\ge 0 }a_{\\mu }^{(\\ell )}Y^{(\\ell )},\\qquad \\hat{a}_{m}=\\sum _{\\ell \\ge 1}\\left(a^{(\\ell )}Y_m^{(\\ell )}+\\tilde{a}^{(\\ell )}\\nabla _mY^{(\\ell )}\\right)\\ ,\\nonumber \\\\\\hat{v}_{\\mu }&=&\\sum _{\\ell \\ge 0}v^{(\\ell )}_{\\mu }Y^{(\\ell )},\\qquad \\hat{v}_{m}=\\sum _{\\ell \\ge 1}\\left(v^{(\\ell )}Y^{(\\ell )}_{m}+{\\tilde{v}}^{(\\ell )}\\nabla _m Y^{(\\ell )}\\right)\\ ,\\nonumber \\\\{\\hat{z}}_\\mu &=& \\sum _{\\ell \\ge 1} z^{(\\ell )}_\\mu {}_{-1}{Y}^{(\\ell )} \\ ,\\nonumber \\\\{\\hat{z}}_m =&& \\sum _{\\ell =0,1} z^{(\\ell )} {}_{-1}V_{m}^{(\\ell )}+\\sum _{\\ell \\ge 2} \\left(z^{(\\ell )} D_m {}_{-1}{ Y}^{(\\ell )} +i{\\tilde{z}}^{(\\ell )} \\epsilon _m{}^n D_n {}_{-1}{Y}^{(\\ell )}\\right)\\ ,\\nonumber \\\\\\hat{b}_{\\mu \\nu }&=&\\sum _{\\ell \\ge 0}b_{\\mu \\nu }^{(\\ell )}Y^{(\\ell )},\\qquad \\hat{b}_{mn}=\\epsilon _{mn}\\sum _{\\ell \\ge 0}b^{(\\ell )}Y^{(\\ell )}\\ ,\\nonumber \\\\\\hat{b}_{\\mu m}&=&\\sum _{\\ell \\ge 1}\\left(b_{\\mu }^{(\\ell )}Y^{(\\ell )}_{m}+\\tilde{b}_{\\mu }^{(\\ell )}\\nabla _mY^{(\\ell )}\\right)\\ ,$ where the notation $\\lbrace mn\\rbrace $ means “symmetric and traceless,” and in the $\\hat{z}_m$ expansion ${}_{-1}V_{m}^{(0)}$ and ${}_{-1}V_{m}^{(1)}$ are level $\\ell =0$ and $\\ell =1$ complex anti-self dual vector harmonics with charge $-1$ on the 2-sphere, whose explicit forms are given in Appendix B.2.",
"$D_m$ is the $U(1)$ covariant derivative on the 2-sphere, and ${}_{-1}{Y}^{(\\ell )}$ are the charged harmonics which are described in some detail in Appendix B.1.",
"Furthermore, the scalar harmonics $Y^{(\\ell )}$ employed above satisfy $\\Box _2Y^{(\\ell )}=- \\alpha ^2c_\\ell Y^{(\\ell )}\\ ,$ where $\\Box _2$ is the d'Alembertian on $S^2$ with radius $1/\\alpha $ and $\\boxed{{c_\\ell \\equiv \\ell (\\ell +1)}} \\ ,\\qquad \\boxed{\\alpha ^2\\equiv 8g^2}\\ .$ We have also used the spin-1 harmonics $Y_m^{(\\ell )}$ which satisfy the relations $\\Box _2Y^{(\\ell )}_n=-(c_\\ell -1)\\alpha ^2 Y^{(\\ell )}_n\\ ,\\qquad \\epsilon ^{mn}\\nabla _m Y^{(\\ell )}_n=\\alpha ^2c_\\ell \\, Y^{(\\ell )}\\ .$ Utilizing the six dimensional gauge symmetries (REF ), we impose the following gauge condition on the linearized fields [9] $\\hat{\\nabla }^{m}\\hat{h}_{\\lbrace mn\\rbrace }&=&0,\\qquad \\hat{\\nabla }^{m}\\hat{h}_{m\\mu }=0,\\nonumber \\\\\\hat{ \\nabla }^m\\hat{a}_m&=&0,\\qquad \\hat{ \\nabla }^m\\hat{b}_{mM}=0.$ Upon the use of these gauge conditions, the harmonic expansions (REF ) simplify to $\\hat{h}_{\\mu \\nu }&=&\\sum _{\\ell \\ge 0}h_{\\mu \\nu }^{(\\ell )}Y^{(\\ell )}\\ ,\\qquad \\ \\ \\hat{h}_{\\mu m}=\\sum _{\\ell \\ge 1}k_{\\mu }^{(\\ell )}Y_{m}^{(\\ell )}\\ ,\\nonumber \\\\\\hat{h}_{mn}&=&\\bar{g}_{mn}\\sum _{\\ell \\ge 0}N^{(\\ell )}Y^{(\\ell )},\\qquad \\hat{\\phi }=\\sum _{\\ell \\ge 0}\\phi ^{(\\ell )}Y^{(\\ell )}\\ ,\\nonumber \\\\\\hat{a}_{\\mu }&=&\\sum _{\\ell \\ge 0}a_{\\mu }^{(\\ell )}Y^{(\\ell )}\\ ,\\qquad \\quad \\hat{a}_{m}=\\sum _{\\ell \\ge 1}a^{(\\ell )}Y^{(\\ell )}_m\\ ,\\nonumber \\\\\\hat{v}_{\\mu } & =&\\sum _{\\ell \\ge 0}v^{(\\ell )}_{\\mu }Y^{(\\ell )}\\ ,\\qquad \\quad \\hat{v}_{m}=\\sum _{\\ell \\ge 1}\\left(v^{(\\ell )}Y^{(\\ell )}_{m}+{\\tilde{v}}^{(\\ell )}\\nabla _m Y^{(\\ell )}\\right)\\ ,\\nonumber \\\\{\\hat{z}}_\\mu &=& \\sum _{\\ell \\ge 1} z^{(\\ell )}_\\mu \\ _{-1} Y^{(\\ell )} \\ ,\\nonumber \\\\{\\hat{z}}_m =&& \\sum _{\\ell =0,1} z^{(\\ell )} {}_{-1}V_{m}^{(\\ell )}+\\sum _{\\ell \\ge 2} \\left(z^{(\\ell )} D_m {}_{-1}{Y}^{(\\ell )} +i{\\tilde{z}}^{(\\ell )} \\epsilon _m{}^n D_n{}_{-1}{Y}^{(\\ell )}\\right)\\ ,\\nonumber \\\\\\hat{b}_{\\mu \\nu }&=&\\sum _{\\ell \\ge 0}b_{\\mu \\nu }^{(\\ell )}Y^{(\\ell )}\\ ,\\qquad \\ \\ \\hat{ b}_{\\mu m}=\\sum _{\\ell \\ge 1}b_{\\mu }^{(\\ell )}Y^{(\\ell )}_{m}\\ ,\\qquad \\hat{b}_{mn}=\\epsilon _{mn}b^{(0)}Y^{(0)}\\ .$ The de Donder-Lorentz gauge (REF ) does not fix all the gauge symmetries, and consequently there are some residual ones generated by harmonic zero modes, $S^2$ Killing vector $Y^{(1)}_m$ and conformal Killing vectors $\\nabla _mY^{(1)}$ .",
"Specifically, these residual gauge symmetries are: The four dimensional coordinate transformation generated by $\\hat{\\xi }_{\\mu }=\\xi ^{(0)}_{\\mu }Y^{(0)}$ $\\delta h_{\\mu \\nu }^{(0)}=\\partial _{\\mu }\\xi ^{(0)}_{\\nu }+\\partial _{\\nu }\\xi ^{(0)}_{\\mu }\\ .$ The Stueckelberg shift symmetries generated by $\\hat{\\xi }_{m}=\\xi ^{(1)}\\nabla _{m}Y^{(1)}$ $\\delta h^{(1)}_{\\mu \\nu }&=&-\\partial _{\\mu }\\partial _{\\nu }\\xi ^{(1)},\\qquad \\delta N^{(1)}=-2\\xi ^{(1)} \\nonumber \\\\\\delta a^{(1)}&=&4g\\xi ^{(1)},\\qquad \\delta v^{(1)}=-\\alpha ^2\\xi ^{(1)}\\ .$ Linearized $SU(2)$ symmetry generated by $\\hat{\\xi }_{m}=\\xi ^{\\prime (1)}Y^{(1)}_m$ and $\\hat{\\Lambda }=-4g\\xi ^{\\prime (1)}Y^{(1)}$ $\\delta k_{\\mu }^{(1)}=\\partial _{\\mu }\\xi ^{\\prime (1)}\\ .$ Four dimensional $U(1)_{\\rm {R}}$ symmetry generated by $\\hat{\\Lambda }=\\Lambda ^{(0)} Y^{(0)}$ $\\delta a^{(0)}_{\\mu }=\\partial _{\\mu }\\Lambda ^{(0)}\\ .$ Abelian 2-form symmetry generated by $\\hat{\\Lambda }_{\\mu }=\\Lambda ^{(0)}_{\\mu } Y^{(0)}$ $\\delta b_{\\mu \\nu }^{(0)}=\\partial _{\\mu }\\Lambda _{\\nu }^{(0)}-\\partial _{\\nu }\\Lambda _{\\mu }^{(0)}\\ .$ We shall take into account these symmetries in the analysis of the spectrum below, where we treat the spin-2, spin-1 and spin-0 sectors separately.",
"In doing so we shall encounter the following wave operators ${\\cal O}_1 &\\equiv &\\hat{\\Box }_0 + \\alpha ^2 - M^2\\ ,\\nonumber \\\\[0.2cm]{\\cal O}_2 &\\equiv & \\hat{\\Box }_0^2 -M^2 \\hat{\\Box }_0 -\\alpha ^4\\,c_\\ell \\ ,\\nonumber \\\\[0.2cm]{\\cal O}_4 &\\equiv & \\hat{\\Box }_0^4 + (\\alpha ^2-M^2)\\, \\hat{\\Box }_0^3-2\\alpha ^2(\\alpha ^2c_\\ell - M^2)\\, \\hat{\\Box }_0^2 -4c_\\ell \\,\\alpha ^4 (\\alpha ^2 - M^2)\\, \\hat{\\Box }_0 -2\\alpha ^8\\, c_\\ell ^2\\ ,$ where $\\boxed{\\hat{\\Box }_0 \\equiv \\Box - \\alpha ^2c_\\ell }\\ .$ In particular, the operator ${\\cal O}_4$ has the property that for $\\ell =1$ it factorizes as ${\\cal O}_4|_{\\ell =1} &=& \\Box \\,{\\cal O}_3\\ ,\\nonumber \\\\[0.2cm]{\\cal O}_3 & \\equiv & \\Box ^3 -(M^2+7\\alpha ^2)\\Box ^2 +2\\alpha ^2 (4M^2+7\\alpha ^2)\\Box -12\\alpha ^4(M^2+\\alpha ^2)\\ .$ In the $\\ell =0$ sector, we will encounter the wave operator $\\widetilde{\\cal O}_2 = \\Box (\\Box +\\alpha ^2) -M^2(\\Box -2\\alpha ^2)\\ .$"
],
[
"The spin-2 sector contains only the transverse and traceless gravitons, which upon the use of the spin projector operators provided in the Appendix C, and for $\\ell \\ge 1$ , satisfy the following equation $\\ell \\ge 1:\\qquad {\\cal O}_2 \\left({{\\cal P}}^{2}h\\right)^{(\\ell )}_{\\mu \\nu }=0\\ ,$ where $P^2$ is the spin-2 projector defined in Appendix C. This equation describes two massive gravitons with mass squared $\\ell \\ge 1: \\qquad m^2_{\\pm } (\\ell )= \\frac{1}{2} \\left(M^2 +2\\alpha ^2c_\\ell \\pm \\sqrt{M^4+4\\alpha ^4 c_\\ell }\\right)\\ .$ The $\\ell =0$ needs to be treated separately, and in this case the gravitons satisfy $(\\Box -M^2)R^{L(0)}_{\\mu \\nu }=-M^2\\left( \\partial _{\\mu }\\partial _{\\nu }S^{(0)}+\\alpha ^2\\eta _{\\mu \\nu } S^{(0)}\\right)+\\partial _{\\mu }\\partial _{\\nu }(\\Box +\\alpha ^2) S^{(0)},$ where $S^{(0)}=\\phi ^{(0)}+N^{(0)}$ .",
"The solutions of this equation can be expressed as $h^{(0)}_{\\mu \\nu }=h^{\\prime (0)}_{\\mu \\nu }+h^{\\prime \\prime (0)}_{\\mu \\nu }$ , where $h^{\\prime \\prime (0)}_{\\mu \\nu }$ is completely determined by $S^{(0)}$ while $h^{\\prime (0)}_{\\mu \\nu }$ is the solution to the following equations modulo the gauge symmetry (REF ): $\\ell =0:\\qquad \\Box (\\Box -M^2)h^{\\prime (0)}_{\\mu \\nu }=0,\\qquad \\,R^{\\prime (0)}=0\\ ,$ which describe a massless graviton and massive graviton with $m^2=M^2$ .",
"Let $\\ell \\ge 2$ .",
"Then, the spin-1 sector consists of eight vectors $(P^{1}h)_{\\mu \\nu },\\partial ^{\\nu }b_{\\mu \\nu }, z^{T}_{\\mu }, k^{T}_{\\mu },a^{T}_{\\mu },v^{T}_{\\mu },b_{\\mu \\nu }^{T},b^{T}_{\\mu })$ , where “T” indicates the transverse part and $(P^{1}h)_{\\mu \\nu }=P^1_{\\mu \\nu }{}^{\\rho \\sigma }h_{\\rho \\sigma }$ (see Appendix B).",
"Of these eight vectors, $(k^{T}_{\\mu },a^{T}_{\\mu },v^{T}_{\\mu },b_{\\mu \\nu }^{T},b^{T}_{\\mu })$ have mixing with each other through the following equations $0&=&{\\cal O}_2 b^{T(\\ell )}_{\\mu \\nu }+4gM^2\\star F^{(\\ell )}_{\\mu \\nu }(a)-\\alpha ^4c_\\ell \\left(\\star F^{(\\ell )}_{\\mu \\nu }(k)-\\star F^{(\\ell )}_{\\mu \\nu }(b)\\right)\\ ,\\\\0&=& {\\cal O}_1 \\hat{\\Box }_0 b^{T(\\ell )}_{\\mu }+{\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } }\\alpha ^2\\epsilon _{\\mu }^{~\\nu \\lambda \\rho }\\partial _{\\nu }b_{\\lambda \\rho }^{T(\\ell )}\\ ,\\\\0&=&(\\hat{\\Box }_0+\\alpha ^2)a^{T(\\ell )}_{\\mu }-4g\\alpha ^2c_\\ell k^{T(\\ell )}_{\\mu }+4gv_{\\mu }^{T(\\ell )}-2g\\epsilon _{\\mu }^{~\\nu \\lambda \\rho }\\partial _{\\nu }b_{\\lambda \\rho }^{T(\\ell )}\\ ,\\\\0&=&(\\hat{\\Box }_0-M^2)v^{T(\\ell )}_{\\mu }+\\alpha ^2c_\\ell k^{T(\\ell )}_{\\mu }+{\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 4} } }gM^2a^{(\\ell )}_{\\mu }\\ ,\\\\0&=&\\biggl ( {\\cal O}_2+\\alpha ^2(\\hat{\\Box }_0-\\alpha ^2c_\\ell )\\biggr )k^{T(\\ell )}_{\\mu }+4gM^2a^{T(\\ell )}_{\\mu }+{\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } } \\alpha ^2\\left(4 v^{T(\\ell )}_{\\mu }-\\epsilon _{\\mu }^{~\\nu \\lambda \\rho }\\partial _{\\nu }b_{\\lambda \\rho }^{T(\\ell )}\\right)\\ .$ Diagonalising the associated $5\\times 5$ operator-valued matrix, we find that the modes are annihilated by ${\\cal O}_1^2\\,{\\cal O}_2\\,{\\cal O}_4$ .",
"In particular, the linear combinations with coefficients $(-1,-4g\\alpha ^2c_\\ell /M^2,0,0,1)$ and $(2,4g\\alpha ^2c_\\ell /M^2,1,0,0)$ are annihilated by ${\\cal O}_1$ .",
"The remaining vectors, namely, $ \\left( (P^{1} h)_{\\mu \\nu },\\partial ^{\\nu }b_{\\mu \\nu }, z_\\mu \\right)$ are separately annihilated by ${\\cal O}_1$ as well.",
"In summary, for $\\ell \\ge 2$ the total wave operator can be denoted by $\\ell \\ge 2:\\qquad {\\cal O}^{(1)} = {\\cal O}_1^6\\,{\\cal O}_2\\,{\\cal O}_4\\ ,$ implying six massive vectors with mass squared $\\ell \\ge 2:\\qquad m^2(\\ell ) = M^2+ \\alpha ^2 (c_\\ell -1)\\ ,$ two massive vectors with mass squared defined in Eq.",
"(REF ) and four massive vectors whose squared masses are given by the roots of the polynomial $&& x^4 + ax^3+bx^2+cx+d=0\\ ,\\nonumber \\\\[0.2cm]&& a= -M^2-(4\\ell ^2+4\\ell -1)\\alpha ^2\\ ,\\nonumber \\\\[0.2cm]&& b= \\alpha ^2\\left[ 2M^2+\\ell (\\ell +1)\\left(\\,(6\\ell ^2+6\\ell -5)\\alpha ^2+3M^2\\right)\\,\\right] \\ ,\\nonumber \\\\[0.2cm]&& c=-\\ell (\\ell +1)\\alpha ^2 \\left\\lbrace \\alpha ^2\\left[4+\\ell (\\ell +1)(4\\ell ^2+4\\ell -7)\\right]\\alpha ^2+3\\ell (\\ell +1)M^2\\right\\rbrace \\ ,\\nonumber \\\\[0.2cm]&& d=\\ell ^2(\\ell +1)^2(\\ell -1)(\\ell +2)\\alpha ^6 \\left[(\\ell ^2+\\ell -1)\\alpha ^2+M^2\\right]\\ .$ Next, consider the case $\\ell =1$ .",
"Recalling the factorization result given in (REF ), the total wave operator becomes $\\ell =1:\\qquad {\\cal O}^{(1)} = {\\cal O}_1^6 {\\cal O}_2|_{\\ell =1} \\Box {\\cal O}_3\\ .$ In particular the massless vector is a linear combination of $(k^{T(1)}_{\\mu },a^{T(1)}_{\\mu },v^{T(1)}_{\\mu },b_{\\mu \\nu }^{T(1)},b^{T(1)}_{\\mu })$ with mixing coefficients $(1,-4g,\\alpha ^2,1,0)$ .",
"The squared masses associated with ${\\cal O}_1^6 {\\cal O}_2|_{\\ell =1}$ can be read of from (REF ) and (REF ) by setting $\\ell =1$ , and those associated with ${\\cal O}_3$ are the roots of the following polynomial $x^3 -(M^2+7\\alpha ^2)x^2 +2\\alpha ^2 (4M^2+7\\alpha ^2)x -12\\alpha ^4(M^2+\\alpha ^2)=0\\ .$ There remains the case of $\\ell =0$ .",
"The only vector fluctuations at this level are $(b_{\\mu \\nu }^{T(0)},a^{T(0)}_{\\mu },v^{T(0)}_{\\mu })$ .",
"Upon diagonalising the associated $3\\times 3$ operator-valued matrix, we find that the modes are annihilated by the following partially-factorising operator polynomial $\\ell =0:\\qquad {\\cal O}^{(1)} = \\Box (\\Box -M^2) \\widetilde{\\cal O}_2 \\ ,$ where the would-be massless vector annihilated by $\\Box $ is eaten by the two form and the operator ${\\tilde{O}}_2$ is defined in (REF ).",
"Thus, for $\\ell =0$ there are no massless vector modes, a massive vector with mass $M$ and two massive vectors with squared masses given by $\\widetilde{m}^2_{\\pm } = \\frac{1}{2}\\left(M^2-\\alpha ^2 \\pm \\sqrt{M^4-10M^2\\alpha ^2 +\\alpha ^4}\\, \\right)\\ .$ We start with the case $\\ell \\ge 2$ .",
"Defining $\\tilde{\\varphi }=\\omega _{\\mu \\nu }h^{\\mu \\nu }$ and $\\varphi ={\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 3} } }\\theta _{\\mu \\nu }h^{\\mu \\nu }$ (see Appendix C), this sector consists of thirteen scalars $\\left(\\phi , N,\\varphi , \\tilde{\\varphi },a, v, \\partial ^{\\mu }k_{\\mu },\\partial ^\\mu b_{\\mu },\\partial ^\\mu a_{\\mu },\\partial ^\\mu v_{\\mu }, \\partial ^\\mu z_{\\mu }, z, \\tilde{z}\\right)$ .",
"The first six scalars $(\\phi ,N, \\varphi , \\tilde{\\varphi },a, v)$ mix as follows $\\ell \\ge 2:\\ \\ 0&=&2(\\hat{\\Box }_0+\\alpha ^2)\\phi ^{(\\ell )}+(2\\hat{\\Box }_0+2\\alpha ^2+ \\alpha ^2c_\\ell )N^{(\\ell )}+3\\hat{\\Box }_0\\varphi ^{(\\ell )}-\\alpha ^2c_\\ell \\tilde{\\varphi }^{(\\ell )}\\ ,\\\\0&=&{\\cal O}_2\\varphi ^{(\\ell )}+2\\alpha ^2M^2 \\left[\\phi ^{(\\ell )}-2gc_\\ell a^{(\\ell )}\\right]-2\\alpha ^2(M^2-\\alpha ^2c_\\ell )N^{(\\ell )}+2\\alpha ^3c_\\ell v^{(\\ell )}\\ ,\\\\0&=&\\biggl (M^2-(\\alpha ^2+\\Box )\\biggr )\\tilde{\\varphi }^{(\\ell )} +3(M^2-\\alpha ^2)\\varphi ^{(\\ell )}+2M^2\\phi ^{(\\ell )}+(2\\hat{\\Box }_0 +2\\alpha ^2+\\alpha ^2c_\\ell )N^{(\\ell )}\\ ,\\nonumber \\\\&& \\\\0&=& (\\hat{\\Box _0}+\\alpha ^2)a^{(\\ell )}+4gN^{(\\ell )}-2g(\\tilde{\\varphi }^{(\\ell )}+3\\varphi ^{(\\ell )})+4gv^{(\\ell )}\\ ,\\\\0&=&(\\hat{\\Box }_0-M^2)v^{(\\ell )}-\\alpha ^2 N^{(\\ell )}+\\frac{1}{2}\\alpha ^2(\\tilde{\\varphi }^{(\\ell )} +3\\varphi ^{(\\ell )})-2 gM^2a^{(\\ell )}\\ ,\\\\0&=&\\biggl (M^2-(\\alpha ^2-\\Box )\\biggr )N^{(\\ell )}+3M^2\\varphi ^{(\\ell )}+2M^2\\phi ^{(\\ell )}-4gM^2a^{(\\ell )}-2\\alpha ^2 v^{(\\ell )}-\\alpha ^2(c_\\ell -1)\\tilde{\\varphi }^{(\\ell )}\\ .\\nonumber \\\\&&$ Diagonalising the associated $6\\times 6$ operator-valued matrix, we find that the modes are annihilated by ${\\cal O}_1^3\\,{\\cal O}_4$ .",
"Of the remaining scalars, $(\\partial ^{\\mu }k_{\\mu },\\partial ^\\mu b_{\\mu },\\partial ^\\mu a_{\\mu },\\partial ^\\mu v_{\\mu },\\tilde{v})$ mix but only three of them are dynamical.",
"We choose these to be $(\\partial ^{\\mu }k_{\\mu },\\partial ^\\mu b_{\\mu },\\tilde{v})$ which are separately annihilated by ${\\cal O}_1$ , while $(\\partial ^\\mu a_{\\mu },\\partial ^\\mu v_{\\mu })$ are determined by $\\partial ^\\mu a_\\mu ^{(\\ell )} &=& \\frac{\\alpha ^2}{2g}\\left( {\\tilde{v}}^{(\\ell )} - \\partial ^\\mu k_\\mu ^{(\\ell )}\\right)\\ ,\\nonumber \\\\[0.2cm]\\partial ^\\mu v_\\mu ^{(\\ell )} &=& \\alpha ^2 (c_\\ell -1){\\tilde{v}}^{(\\ell )} +\\alpha ^2 \\partial ^\\mu k_\\mu ^{(\\ell )}\\ .$ Finally, the remaining scalars $(z, \\tilde{z})$ are annihilated by ${\\cal O}_1$ , and the longitudinal modes $\\partial ^{\\mu }z_{\\mu }$ are given in terms of $z$ and $\\tilde{z}$ , by virtue of the equation $\\hat{D}^{M}\\hat{z}_M=0$ .",
"Thus, for $\\ell \\ge 2$ the total wave operator is given by $\\ell \\ge 2:\\qquad {\\cal O}^{(0)} = {\\cal O}_1^{10}\\,{\\cal O}_4\\ .$ Of these, the three linear combinations of $(\\phi , N,\\varphi , \\tilde{\\varphi },a, v)$ with coefficients $(2+\\alpha ^2{c_\\ell }/{M^2},-2,0,0,0,2)$ , $(-2-\\alpha ^2c_\\ell /{M^2},2,0,0,-{8g}/{M^2},0)$ and $(-2+\\alpha ^2{c_\\ell }/{M^2},2,0,4,0,0)$ are annihilated by ${\\cal O}_1$ .",
"In the case of $\\ell =1$ , utilizing the residual symmetry (REF ) and (REF ), one can eliminate $N^{(1)}$ and $\\partial ^\\mu k_{\\mu }^{(1)}$ .",
"Taking into account the fact that the harmonic expansion of ${\\hat{z}}_m$ contributes only one complex scalar for $\\ell =1$ , namely $z^{(1)}$ , we find that for $\\ell =1$ the total wave operator for the scalar fields is given by $\\ell =1:\\qquad {\\cal O}^{(0)} = {\\cal O}_1^6|_{\\ell =1}\\,{\\cal O}_3\\ .$ There remains the case of $\\ell =0$ .",
"Of the remaining scalars, $(\\phi ^{(0)},N^{(0)},b^{(0)})$ satisfy the equations $ \\widetilde{\\cal O}_2\\,S^{(0)}=0$ where $S^{(0)}=\\phi ^{(0)}+N^{(0)}$ , $\\Box \\,{{\\cal O}_1}|_{\\ell =0}\\, b^{(0)}=0$ and $\\Box \\,{{\\cal O}_1}|_{\\ell =0}\\, N^{(0)}= 0$ .",
"Finally, there is a complex scalar $z^{(0)}$ annihilated by ${\\cal O}_1|_{\\ell =0}$ .",
"Thus, for $\\ell =0$ the total wave operator or the scalar fields is given by $\\ell =0: \\qquad {\\cal O}^{(0)} = \\Box ^2\\,{\\cal O}_1^4|_{\\ell =0}\\, \\widetilde{\\cal O}_2\\ ,$ implying two massless and six massive scalars in this sector."
],
[
"Fermionic sector",
"In terms of the complex spinor, the linearized equations of fermions around the background (REF ) are given by $0&=& {\\bar{\\Gamma }}^{MPQ}\\bar{D}_{P}\\psi _{Q}+2 i{\\bar{\\Gamma }}^{NM}\\bar{D}_N\\chi -4g i{\\bar{\\Gamma }}^M\\lambda -{\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } }\\bar{F}_{PQ}{\\bar{\\Gamma }}^{PQ}{\\bar{\\Gamma }}^{M}\\lambda -\\frac{1}{8M^2}\\Theta ^{M}\\ ,\\\\0&=&{\\bar{\\Gamma }}^{MN}\\bar{D}_M\\psi _{N}-2i{\\bar{\\Gamma }}^{M}\\bar{D}_{M}\\chi -8gi\\lambda ,\\\\0&=& 2gi{\\bar{\\Gamma }}^M\\psi _{M}+8g\\chi -4{\\bar{\\Gamma }}^M {\\bar{D}}_M\\lambda +{\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 4} } }\\bar{F}_{PQ}{\\bar{\\Gamma }}^{M}{\\bar{\\Gamma }}^{PQ}\\psi _{M}\\ ,$ where $\\bar{D}_M\\psi &=&(\\partial _M+{\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 4} } }\\bar{\\omega }_M^{~AB}{\\bar{\\Gamma }}_{AB})\\psi -{\\textstyle {\\frac{\\scriptstyle i}{\\scriptstyle 2} } }\\bar{V}_M\\psi \\ ,\\nonumber \\\\[0.2cm]\\Theta ^{M}&=& 8{\\bar{\\Gamma }}^P\\bar{D}_Q\\bar{ D}_P\\psi ^{QM} -2\\bar{R}^{PM}_{~~~ST}{\\bar{\\Gamma }}^Q{\\bar{\\Gamma }}^{ST}\\bar{D}_{P}\\psi _{Q}+2\\bar{R}^{PQ}_{~~~ST}{\\bar{\\Gamma }}^{ST}{\\bar{\\Gamma }}^{M}\\bar{D}_{P}\\psi _{Q}\\nonumber \\\\&& +8i\\bar{G}^{P[M}{\\bar{\\Gamma }}^{Q]}\\bar{D}_{Q}\\psi _{P}-8i\\bar{G}^{M[P}{\\bar{\\Gamma }}^{Q]}\\bar{D}_{Q}\\psi _{P}\\ .$ In the remainder of this section, we shall drop the “bar” on the covariant derivatives as well as the $\\Gamma $ -matrices for simplicity in notation.",
"Since the (REF ) contains gauge field, we adopt the spin-weighted harmonics $ _{s-{\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } }}\\eta ^{(\\ell )} $ , which are described in detail in appendix B, as the expansion basis.",
"In this section we will need the harmonics for $s=0$ , namely $ _{-{\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } }} \\eta ^{(\\ell )} $ which we will denote as $\\eta ^{(\\ell )}$ for brevity in notation.",
"These harmonics satisfy the relations, $\\eta _-^{(0)}=\\eta \\ ,\\qquad \\eta ^{(\\ell )}_+ = \\frac{1}{i\\alpha \\sqrt{c_\\ell }}\\nabla _n Y^{(\\ell )}\\sigma ^n\\eta \\ ,\\qquad \\eta ^{(\\ell )}_- = Y^{(\\ell )}\\eta ,\\qquad \\ell \\ge 1\\ ,$ and have the following properties $\\sigma _3 {\\eta _{\\pm }^{(\\ell )}}&=\\pm {\\eta _{\\pm }^{(\\ell )}}\\ ,&\\qquad \\quad \\sigma ^n {D}_n{\\eta _{\\pm }^{(\\ell )}}& =i\\alpha \\sqrt{c_\\ell }\\,{\\eta _{\\mp }^{(\\ell )}}\\ ,\\nonumber \\\\[{D}_m, {D}_n]{\\eta _-^{(\\ell )}}&=0\\ , &\\qquad \\quad [{D}_m,{D}_n]{\\eta _+^{(\\ell )}} &=i\\alpha ^2\\epsilon _{mn}{\\eta _+^{(\\ell )}} ,\\nonumber \\\\{D}^n{D}_n{\\eta _-^{(\\ell )}}&=-\\alpha ^2c_\\ell {\\eta _-^{(\\ell )}}, &\\qquad \\quad {D}^n{D}_n{\\eta _+^{(\\ell )}}&=-\\alpha ^2(c_\\ell -1){\\eta _+^{(\\ell )}}\\ .$ The Killing spinor $\\eta $ also has the property $\\sigma _3\\eta =-\\eta $ .",
"Furthermore, given that $\\Gamma _7=\\gamma _5\\times \\sigma _3$ (see Appendix A), the chirality property of a spinor in $6D$ correlates the $4D$ and $\\sigma _3$ chiralities.",
"Since there is no gamma traceless and transverse spin-3/2 harmonics on the $S^2$ , generically, the harmonic expansion are carried out as $\\hat{\\psi }_{\\mu } &=&\\psi _{\\mu -}^{(0)}\\otimes {\\eta ^{(0)}}+\\sum _{\\ell \\ge 1}\\left(\\psi _{\\mu +}^{(\\ell )}\\otimes {\\eta _+^{(\\ell )}}+\\psi _{\\mu -}^{(\\ell )}\\otimes {\\eta _-^{(\\ell )}}\\right)\\ ,\\nonumber \\\\\\hat{\\psi }_m &=& \\Gamma _m\\psi ^{(0)}_+\\otimes {\\eta ^{(0)}}+\\Gamma _m\\sum _{\\ell \\ge 1}\\left(\\psi _-^{(\\ell )}\\otimes {\\eta _+^{(\\ell )}}+\\psi _+^{(\\ell )}\\otimes {\\eta _-^{(\\ell )}}\\right)\\nonumber \\\\&&+\\sum _{\\ell \\ge 1}\\left(\\tilde{\\psi }_+^{(\\ell )}\\otimes D_{\\lbrace m\\rbrace }{\\eta _+^{(\\ell )}}+\\tilde{\\psi }_-^{(\\ell )}\\otimes D_{\\lbrace m\\rbrace }{\\eta _-^{(\\ell )}}\\right)\\ ,\\nonumber \\\\\\hat{\\chi }&=&\\chi ^{(0)}_+ \\otimes {\\eta ^{(0)}}+ \\sum _{\\ell \\ge 1} \\left(\\chi _-^{(\\ell )}\\otimes {\\eta _+^{(\\ell )}}+\\chi _+^{(\\ell )}\\otimes {\\eta _-^{(\\ell )}}\\right)\\ ,\\nonumber \\\\\\hat{\\lambda } &=& \\lambda ^{(0)}_-\\otimes \\eta ^{(0)}+\\sum _{\\ell \\ge 1}\\left( \\lambda _+^{(\\ell )} \\eta _+^{(\\ell )}+\\lambda _-^{(\\ell )} \\eta _-^{(\\ell )}\\right)\\ ,$ where $D_{\\lbrace m\\rbrace }$ is the gamma traceless covariant derivative and the $\\pm $ subscripts denote chirality property under $\\gamma _5$ .",
"Using the 6D linearized fermionic gauge symmetry (REF ), one can impose the following gauge condition $\\hat{\\psi }_{\\lbrace m\\rbrace }=0\\ ,$ where $\\lbrace m\\rbrace $ means $\\Gamma $ -traceless.",
"As a consequence, the expansion takes the following simpler forms $\\hat{\\psi }_{\\mu } &=&\\psi _{\\mu -}^{(0)}\\otimes {\\eta ^{(0)}}+\\sum _{\\ell \\ge 1}\\left(\\psi _{\\mu +}^{(\\ell )}\\otimes {\\eta _+^{(\\ell )}}+\\psi _{\\mu -}^{(\\ell )}\\otimes {\\eta _-^{(\\ell )}}\\right)\\ ,\\\\\\hat{\\psi }_m &=& \\Gamma _m\\psi ^{(0)}_+\\otimes {\\eta ^{(0)}}+\\Gamma _m\\sum _{\\ell \\ge 1}\\left(\\psi _-^{(\\ell )}\\otimes {\\eta _+^{(\\ell )}}+\\psi _+^{(\\ell )}\\otimes {\\eta _-^{(\\ell )}}\\right)\\ ,\\\\\\hat{\\chi } &=& \\chi ^{(0)}_+\\otimes {\\eta ^{(0)}}+ \\sum _{\\ell \\ge 1}\\left(\\chi _-^{(\\ell )}\\otimes {\\eta _+^{(\\ell )}} +\\chi _+^{(\\ell )}\\otimes {\\eta _-^{(\\ell )}}\\right)\\ ,\\\\\\hat{\\lambda } &=& \\lambda ^{(0)}_-\\otimes {\\eta ^{(0)}}+\\sum _{\\ell \\ge 1} \\left(\\lambda _+^{(\\ell )}\\otimes {\\eta _+^{(\\ell )}}+\\lambda _-^{(\\ell )}\\otimes {\\eta _-^{(\\ell )}}\\right)\\ .$ The gauge choice (REF ) does not fix all the gauge symmetries, we find the following residual symmetry transformations Generated by $\\hat{\\epsilon }^{(0)}=\\epsilon ^{(0)}{\\eta ^{(0)}}$ : $\\delta \\psi ^{(0)}_\\mu =\\partial _{\\mu }\\epsilon ^{(0)},$ Generated by $\\hat{\\epsilon }=\\epsilon _+^{(1)}{\\eta _+^{(1)}}$ : $\\delta \\psi _{\\mu +}^{(1)}&=&\\partial _{\\mu }\\epsilon _+^{(1)}+\\frac{i}{\\sqrt{2}}\\,\\alpha \\,\\epsilon _+^{(1)}\\ ,\\\\\\delta \\lambda _+^{(1)}&=&ig\\epsilon _+^{(1)}\\ .$ We shall take into account these symmetries in the analysis of the spectrum below, where we treat the spin-3/2, spin-1/2 sectors separately."
],
[
"Let us begin with the restriction $\\ell \\ge 1$ .",
"This sector contains only the gravitino fields which satisfy the following equations $\\ell \\ge 1:\\quad &&{\\partial }\\Big ((\\hat{\\Box }_0+\\alpha ^2)-M^2\\Big ) P^{3/2}\\psi _{\\mu +}^{(\\ell )}+i\\alpha \\sqrt{c_\\ell }\\Big ((\\hat{\\Box }_0+\\alpha ^2)-M^2\\Big )P^{3/2}\\psi _{\\mu -}^{(\\ell )}=0\\ ,\\nonumber \\\\&&i\\alpha \\sqrt{c_\\ell }\\Big ((\\hat{\\Box }_0+\\alpha ^2)-M^2\\Big )P^{3/2}\\psi _{\\mu +}^{(\\ell )}-{\\partial }\\Big (\\hat{\\Box }_0-M^2\\Big )P^{3/2}\\psi _{\\mu -}^{(\\ell )}=0\\ ,$ where $P^{3/2}$ is the spin-3/2 projector operator defined in Appendix C. Diagonalising the associated $2\\times 2$ operator-valued matrix, we find that the modes are annihilated by the partially-factorising operator polynomial, of sixth order in ${\\partial }$ , given by $\\ell \\ge 1:\\qquad {\\cal O}^{(3/2)} = {\\cal O}_1 {\\cal O}_2\\ .$ Next, consider the case of $\\ell =0$ .",
"By choosing the gauge $\\gamma ^{\\mu }\\psi ^{(0)}_{\\mu }=0$ , the gravitino equation can be written as ${\\partial }\\left(\\Box -M^2\\right)\\psi ^{(0)\\mu }=-\\left({\\partial }\\partial ^{\\mu } - M^2\\gamma ^{\\mu }\\right)\\Psi ^{(0)}-2M^2\\left( \\gamma ^{\\mu \\nu }\\partial _{\\nu }\\psi ^{(0)}-i\\gamma ^{\\mu \\nu }\\partial _{\\nu }\\chi ^{(0)}\\right)\\ .$ The solutions of above equation can be expressed as $\\psi ^{(0)}_{\\mu }=\\psi ^{\\prime (0)}_{\\mu }+\\psi ^{\\prime \\prime (0)}_{\\mu }$ where $\\psi ^{\\prime \\prime (0)}_{\\mu }$ is completely determined by $\\psi ^{(0)}$ and $\\chi ^{(0)}$ while $\\psi ^{\\prime (0)}_{\\mu }$ is the solution to the following equations modular gauge symmetry (REF ) $\\ell =0:\\qquad {\\partial }(\\Box -M^2)\\psi ^{\\prime (0)}_{\\mu }=0\\ ,\\qquad \\gamma ^{\\mu }\\psi ^{\\prime (0)}_{\\mu }=0\\ ,\\qquad \\partial ^{\\mu }\\psi ^{\\prime (0)}_{\\mu }=0\\ .$ It describes a massless and two massive gravitini.",
"The $\\ell \\ge 2$ sector consists of ten spin 1/2 fields $(\\Lambda _+,\\Psi _+,\\Lambda _-,\\Psi _-,$ $\\psi _+,\\psi _-,\\chi _+,\\chi _-,\\lambda _+,\\lambda _-)$ , where $\\Psi ^{(\\ell )}\\equiv \\partial ^{\\mu }\\psi _{\\mu }^{(\\ell )}$ and $\\gamma ^{\\mu }\\psi _{\\mu }^{(\\ell )}\\equiv \\Lambda ^{(\\ell )}$ .",
"The linearized equations describing their mixing are $0&=&{\\partial }\\Lambda ^{(\\ell )}_{-}-\\Psi _+^{(\\ell )} +i\\alpha \\sqrt{c_\\ell }\\psi _+^{(\\ell )}-i\\alpha \\sqrt{c_\\ell }\\Lambda _+^{(\\ell )} +2{\\partial }\\psi _-^{(\\ell )}-2i{\\partial }\\chi _-^{(\\ell )} +2\\alpha \\sqrt{c_\\ell }\\chi _+^{(\\ell )}-8gi\\lambda _+^{(\\ell )}\\ ,\\\\[0.2cm]0&=&{\\partial }\\Lambda _+^{(\\ell )} -\\Psi _-^{(\\ell )}+i\\alpha \\sqrt{c_\\ell }\\psi _-^{(\\ell )}+i\\alpha \\sqrt{c_\\ell }\\Lambda _-^{(\\ell )} -2{\\partial }\\psi _+^{(\\ell )}+2i{\\partial }\\chi _+^{(\\ell )} +2\\alpha \\sqrt{c_\\ell }\\chi _-^{(\\ell )}-8gi\\lambda _-^{(\\ell )}\\ ,\\\\[0.2cm]0&=&i g\\Lambda _-^{(\\ell )}+2g\\chi _-^{(\\ell )} -{\\partial }\\lambda _+^{(\\ell )}-i\\alpha \\sqrt{c_\\ell }\\lambda _-^{(\\ell )}\\ ,\\\\[0.2cm]0&=&2i g\\psi _+^{(\\ell )}+2g\\chi _+^{(\\ell )}+{\\partial }\\lambda _-^{(\\ell )}-i\\alpha \\sqrt{c_\\ell }\\lambda _+^{(\\ell )}\\ ,\\\\[0.2cm]0&=&i\\alpha \\sqrt{c_\\ell }{\\partial }(\\Box +M^2)\\psi _+^{(\\ell )}-iM^2\\alpha \\sqrt{c_\\ell }{\\partial }\\Lambda _+^{(\\ell )}+i\\alpha \\sqrt{c_\\ell }\\biggl (M^2-\\alpha ^2+\\alpha ^2c_\\ell \\biggr )\\Psi _-^{(\\ell )}+2M^2\\alpha \\sqrt{c_\\ell }{\\partial }\\chi _+^{(\\ell )}\\nonumber \\\\&&-8igM^2{\\partial }\\lambda _+^{(\\ell )}+\\alpha ^2(c_\\ell -1){\\partial }\\Psi _+^{(\\ell )}+\\alpha ^2(2-c_\\ell )\\Box \\psi _-^{(\\ell )}\\ ,\\\\[0.2cm]0&=&-i{\\partial }\\biggl (\\Box +M^2-\\alpha ^2)\\biggr )\\psi _-^{(\\ell )}-iM^2{\\partial }\\Lambda _-^{(\\ell )}+i\\biggl (\\alpha ^2+\\alpha ^2c_\\ell - M^2\\biggr )\\Psi _+^{(\\ell )}-2M^2{\\partial }\\chi _-^{(\\ell )}\\nonumber \\\\&&-\\alpha \\sqrt{c_\\ell }\\left( {\\partial }\\Psi _-^{(\\ell )} +\\Box \\psi _+^{(\\ell )}\\right)\\ ,\\\\[0.2cm]0&=&{\\partial }\\biggl (\\Box +\\alpha ^2-\\alpha ^2c_\\ell +2M^2\\biggr )\\Lambda _-^{(\\ell )}-\\biggl (\\Box +2\\alpha ^2-2\\alpha ^2c_\\ell +2M^2\\biggr )\\Psi _+^{(\\ell )}+i\\alpha \\sqrt{c_\\ell }(\\Box +4M^2)\\psi _+^{(\\ell )}\\nonumber \\\\&&+{\\partial }\\biggl (6M^2+2\\alpha ^2-\\alpha ^2c_\\ell )\\biggr )\\psi _-^{(\\ell )}-i\\alpha \\sqrt{c_\\ell }\\biggl (\\Box +\\alpha ^2-\\alpha ^2c_\\ell +3M^2\\biggr )\\Lambda _+^{(\\ell )}-6iM^2{\\partial }\\chi _-^{(\\ell )}\\nonumber \\\\&&+8\\alpha M^2\\sqrt{c_\\ell }\\chi _+^{(\\ell )}-32igM^2\\lambda _+^{(\\ell )}+i\\alpha \\sqrt{c_\\ell }{\\partial }\\Psi _-^{(\\ell )}\\ ,\\\\[0.2cm]0&=&-{\\partial }\\biggl (\\Box -\\alpha ^2c_\\ell +2M^2)\\biggr )\\Lambda _+^{(\\ell )}+\\biggl (\\Box -2\\alpha ^2c_\\ell +2M^2)\\biggr )\\Psi _-^{(\\ell )} -i\\alpha \\sqrt{c_\\ell }\\biggl (\\Box -4\\alpha ^2+4M^2)\\biggr )\\psi _-^{(\\ell )}\\nonumber \\\\&&+{\\partial }(6M^2-\\alpha ^2c_\\ell )\\psi _+^{(\\ell )}-i\\alpha \\sqrt{c_\\ell }\\biggl (\\Box +\\alpha ^2-\\alpha ^2c_\\ell +3M^2\\biggr )\\Lambda _-^{(\\ell )}-6iM^2{\\partial }\\chi _+^{(\\ell )}-8\\alpha M^2\\sqrt{c_\\ell }\\chi _-^{(\\ell )}\\nonumber \\\\&&+i\\alpha \\sqrt{c_\\ell }{\\partial }\\Psi _+^{(\\ell )}\\ ,\\\\[0.2cm]0&=& M^2\\left(-\\Lambda _-^{(\\ell )}+2i\\chi _-^{(\\ell )}\\right)-\\left(2\\Box -\\alpha ^2c_\\ell \\right)\\psi _-^{(\\ell )}+{\\partial }\\Psi _+^{(\\ell )} +i\\alpha \\sqrt{c_\\ell }\\Psi _-^{(\\ell )}+i\\alpha \\sqrt{c_\\ell }{\\partial }\\psi _+^{(\\ell )}\\ ,\\\\[0.2cm]0&=&(\\alpha ^2-M^2)\\Lambda _+^{(\\ell )}-2iM^2\\chi _+^{(\\ell )} +\\left(2\\Box +2\\alpha ^2- \\alpha ^2c_\\ell \\right)\\psi _+^{(\\ell )}-i\\alpha \\sqrt{c_\\ell }\\Psi _+^{(\\ell )}+{\\partial }\\Psi _-^{(\\ell )}+i\\alpha \\sqrt{c_\\ell }{\\partial }\\psi _-^{(\\ell )}\\ .\\nonumber \\\\&&$ Diagonalising the associated $10\\times 10$ operator-valued matrix, we find that the modes are annihilated by the partially-factorising operator polynomial, of eighteenth order in ${\\partial }$ , given by $\\ell \\ge 2:\\qquad {\\cal O}^{1/2} = {\\cal O}_1^5\\, {\\cal O}_4\\ .$ Next we consider the case of $\\ell =1$ .",
"In this case, one can use the fermionic shift symmetry (REF ) to eliminate $\\psi _+^{(1)}$ .",
"Consequently we get 9 by 9 mixing and we find that the modes are annihilated by the partially-factorising operator polynomial, of fifteenth order in ${\\partial }$ , given by $\\ell =1:\\qquad {\\cal O}^{1/2} = {\\cal O}_1^4|_{\\ell =1}\\, {\\partial }\\, {\\cal O}_3\\ ,$ where ${\\cal O}_3$ is defined in (REF ).",
"The factor ${\\partial }$ demonstrates that there is massless spin-1/2 mode.",
"This massless mode corresponds to linear combinations of $(\\Lambda ^{(1)}_+,\\Psi ^{(1)}_+,\\Lambda ^{(1)}_-,\\Psi ^{(1)}_-,\\psi ^{(1)}_-,\\chi ^{(1)}_+,\\chi ^{(1)}_-,\\lambda ^{(1)}_+,\\lambda ^{(1)}_-)$ with mixing coefficients $(8,0,0,0,-2,-2i,0,0,1)$ .",
"There remains the case of $\\ell =0$ .",
"In this case, we have $\\ell =0:\\qquad 0&=&-\\Psi ^{(0)}-2{\\partial }\\psi ^{(0)}+2i{\\partial }\\chi ^{(0)}-8gi\\lambda ^{(0)}\\ ,\\\\0&=& 2gi\\psi ^{(0)}+2g\\chi ^{(0)}+{\\partial }\\lambda ^{(0)}\\ ,\\\\0&=& (1+\\frac{1}{2M^2}\\Box )\\Psi ^{(0)}+3{\\partial }\\psi ^{(0)}-3i{\\partial }\\chi ^{(0)}\\ ,\\\\0&=& -\\Psi ^{(0)}-{\\partial }\\left(1+\\frac{1}{M^2} (\\Box +\\alpha ^2)\\right)\\psi ^{(0)}+2i{\\partial }\\chi ^{(0)}-8gi\\lambda ^{(0)}\\ .$ Diagonalising the associated $4\\times 4$ operator-valued matrix, we find that the modes are annihilated by the partially-factorising operator polynomial, of seventh order in ${\\partial }$ , given by $\\ell =0:\\qquad {\\cal O}^{(1/2)} = {\\cal O}_1|_{\\ell =0}\\,{\\partial }\\,\\widetilde{\\cal O}_2\\ .$ Thus, at the $\\ell =0$ level, there is only one massless spin-1/2 modes given by $\\Psi ^{(0)}=0,\\lambda ^{(0)}=0,i\\psi ^{(0)}+\\chi ^{(0)}=0$ ."
],
[
"The supermultiplet structure and stability",
"In arranging the full spectrum described above into a collection of supermultiplet structure, it is useful to recall that following massive supermultiplets: $&{\\rm massive\\ supergravity\\ multiplet:} & \\qquad (h_{\\mu \\nu }, A_\\mu , \\Psi _\\mu )\\ ,\\nonumber \\\\&{\\rm massive\\ gravitino \\ multiplet:} & \\qquad (\\psi _\\mu , Z_\\mu , \\chi )\\ ,\\nonumber \\\\&{\\rm massive\\ vector multiplet\\ multiplet:} & \\qquad (A_\\mu , \\phi , \\lambda )\\ ,\\nonumber \\\\&{\\rm massive\\ scalar\\ multiplet:} & \\qquad (Z, \\psi )\\ ,$ where $A_\\mu $ is a real and $Z_\\mu $ is a complex vector, $\\phi $ is a real and $Z$ is a complex scalar, and $\\Psi _\\mu $ is Dirac and $\\psi _\\mu ,\\chi , \\psi $ are Majorana.",
"The Dirac gravitino can be written as $\\Psi =\\psi _{\\mu +}^1 + \\psi _{\\mu -}^2$ , where the two terms represent Weyl spinors that are independent of each other, and consequently $\\Psi _\\mu $ on-shell describes 8 real degrees of freedom.",
"The Majorana gravitino, on the other hand can be written as $\\psi _\\mu =\\psi _{\\mu +} + \\psi _{\\mu -}$ where $\\psi _{\\mu -}= (\\psi _{\\mu +})^*$ .",
"Thus, on shell $\\psi _\\mu $ describes 4 real degrees of freedom.",
"With this information at hand, we can now tabulate the supermultiplet structure of the full spectrum.",
"It is convenient to do so by specifying the wave operators for different spin fields and consider the cases of $\\ell =0$ , $\\ell =1$ and $\\ell \\ge 2$ separately.",
"The results are given in Table 1, Table 2 and Table 3.",
"Table: The spectrum of wave operators for ℓ=0\\ell =0.",
"The operator 𝒪 1 {\\cal O}_1 is to be evaluated for ℓ=0\\ell =0.",
"There is one massless spin-2 and one massless spin-0 multiplet, a massive spin-2 multiplet with mass MM, two spin-0 multiplets with squared mass m 2 =M 2 -α 2 m^2=M^2-\\alpha ^2 and two massive spin-1 multiplets with mass 2 ^2 given in ().Table: The spectrum of wave operators for ℓ=1\\ell =1.",
"The operators 𝒪 1 {\\cal O}_1 and 𝒪 2 {\\cal O}_2 are to be evaluated for ℓ=1\\ell =1.",
"There are two spin-2 multiplets with squared masses given in () for ℓ=1\\ell =1, two spin-3/2, two spin-1 multiplets and two spin-0 multiplets with squared mass m 2 =M 2 +α 2 m^2=M^2+\\alpha ^2, three spin-1 multiplets with squared masses given by the roots of the polynomial given in () and a massless vector multiplet.Table: The spectrum of wave operators for ℓ≥2\\ell \\ge 2.",
"For each integer ℓ\\ell , there are two spin-2 multiplets with squared mass m ± 2 (ℓ)m^2_\\pm (\\ell ) given in (), two spin-3/2, two spin-1 multiplets and four spin-0 multiplets with squared mass m 2 (ℓ)m^2(\\ell ) given in (), and four spin-1 multiplets with squared masses given by the roots of the polynomial given in ().Having established the full spectrum of states in the four-dimensional theory, we may now examine the question of stability, which is governed by the mass values for the massive fields.",
"We begin with the $\\ell =0$ level, given in Table 1.",
"In addition to the massless graviton and scalar multiplets, there are massive graviton, vector and scalar multiplets at this level.",
"The massive graviton multiplet has $m^2=M^2$ , the scalar multiplet has $m^2=M^2-\\alpha ^2$ , and the vector multiplet has masses given by (REF ).",
"These imply, respectively, that stability requires $M^2>0$ , $M^2>\\alpha ^2$ and $M^2\\ge (5+2\\sqrt{6})\\alpha ^2 \\approx 9.89898 \\alpha ^2$ .",
"At the level $\\ell =1$ , in addition to the massless vector multiplet, there are massive graviton, gravitino, vector and scalar multiplets.",
"The gravitino multiplet, two of the massive vector multiplets and the scalar multiplet have masses given by the operator ${\\cal O}_1$ , implying $m^2=M^2+\\alpha ^2$ , which therefore impose no new conditions.",
"The massive graviton multiplet with mass operator ${\\cal O}_2$ has $m^2$ given by equation (REF ) with $\\ell $ set equal to 1.",
"This implies $m^2= (M^2+4\\alpha ^2\\pm \\sqrt{M^4+8\\alpha ^4})/2$ , and hence gives no further restriction.",
"There remains the massive vector multiplet with mass operator ${\\cal O}_3$ given in (REF ).",
"This gives a cubic polynomial in $m^2$ , and we find that this has three real (and positive) roots for $m^2$ provided that $\\mu \\equiv M^2/\\alpha ^2$ satisfies the condition $4\\mu ^4 - 64\\mu ^3 + 153 \\mu ^2 - 26\\mu - 139\\ge 0\\,.$ This implies we must have $\\mu \\ge \\mu _{\\rm min}$ , where $\\mu _{\\rm min}\\approx 13.1425$ .",
"In other words, at level $\\ell =1$ stability requires that $M^2$ should exceed approximately $13.1425 \\alpha ^2$ .",
"For levels $\\ell \\ge 2$ , the multiplets are given in Table 3.",
"The gravitino, vector and scalar multiplets with mass operator ${\\cal O}_1$ have $m^2$ given in equation (REF ), and these are always positive for all values of $\\ell \\ge 2$ .",
"Likewise, for the graviton multiplet with mass operator ${\\cal O}_2$ , $m^2$ , given in (REF ), is positive for all $\\ell \\ge 2$ .",
"There remains the vector multiplet with mass operator ${\\cal O}_4$ .",
"This leads to a quartic polynomial in $m^2$ , which can be read off from (REF ) and (REF ).",
"One can show that this polynomial necessarily has at least two real roots, which are positive, and that if the four roots for $m^2$ are real then they are also positive.",
"The condition for having four real roots is that a rather complicated discriminant of sixth order in $\\mu =M^2/\\alpha ^2$ should be positive.",
"This discriminant also depends on the level $\\ell $ .",
"For a few representative values of $\\ell $ , we find the requirement $\\mu \\ge \\mu _{\\rm min}(\\ell )$ : Table: Minimum values of μ=M 2 /α 2 \\mu =M^2/\\alpha ^2 necessary to achieve realpositive mass-squared values for the 𝒪 4 {\\cal O}_4 vector multiplet at levelℓ\\ell .In the limit of large $\\ell $ , we find that to leading order, $\\mu _{\\rm min}(\\ell )$ grows linearly with $\\ell $ , with $\\mu _{\\rm min}(\\ell ) \\sim 4.05874 \\, \\ell +\\cdots $ This implies that for any given ratio $\\mu =M^2/\\alpha ^2$ , there is a a critical level $\\ell _{\\rm max}$ beyond which the Kaluza-Klein tower must be truncated in order not to have modes with complex masses, which would be associated with instabilities."
],
[
"Non-supersymmetric $\\rm {Minkowski}_4\\times S^2$ background",
"In addition to supersymmetric vacuum solution discussed in previous section, the theory [5] also possesses non-supersymmetric $\\rm {Minkowski}_4\\times S^2$ vacua when $M^2=\\alpha ^2$ , with the curvature and flux given by $\\bar{R}_{\\mu \\nu \\lambda \\rho } &=0 \\ ,&\\qquad \\bar{R}_{mn} &= \\alpha ^2\\bar{g}_{mn}\\ , \\qquad \\bar{L} &=1\\ , \\nonumber \\\\\\bar{F}_{\\mu \\nu } &=0\\ , & \\qquad \\bar{F}_{mn} &= 4q g\\epsilon _{mn}\\ , \\qquad & \\nonumber \\\\\\bar{G}_{\\mu \\nu } &=0\\ , & \\qquad \\bar{G}_{mn} &= -q\\alpha ^2\\epsilon _{mn}\\ , \\qquad &$ where $q$ plays the role of monopole charge and is quantized to be $q=0,\\pm 1,\\pm 2\\ldots $ .",
"The supersymmetric vacua correspond to $q=\\pm 1$ ."
],
[
"Bosonic sector",
"We will perform a similar spectrum analysis around the non-supersymmetric background.",
"The harmonic expansion (REF ) for the uncharged fields after the gauge fixing is still valid, and the residual gauge symmetries are almost the same except that some terms related to background flux should be multiplied by the monopole charge.",
"We present the results for the spectrum below.",
"While we shall use the same notation for operators such as $\\hat{\\Box }_0$ and others, it is understood that they are to be evaluated for $M^2=\\alpha ^2$ ."
],
[
"The equations of motion satisfied by graviton for $\\ell \\ge 1$ is $\\ell \\ge 1:\\qquad ( \\hat{\\Box }_0^2-\\alpha ^2\\hat{\\Box }_0-\\alpha ^4\\, c_\\ell )({{\\cal P}}^{2} h)^{(\\ell )}_{\\mu \\nu }=0\\ ,$ describing massive gravitons with square masses $m^2_\\pm (\\ell ) = \\frac{1}{2} \\alpha ^2\\left(1+c_\\ell \\pm \\sqrt{1+4c_\\ell }\\right)\\ .$ For $\\ell =0$ , the linearized field equation is $\\ell =0:\\qquad (\\Box -\\alpha ^2)R^{L(0)}_{\\mu \\nu }=-\\alpha ^2\\partial _{\\mu }\\partial _{\\nu }S^{(0)}-\\alpha ^4\\eta _{\\mu \\nu }S^{(0)}+\\partial _{\\mu }\\partial _{\\nu }(\\Box +\\alpha ^2)S^{(0)}\\ ,$ where $S^{(0)}=\\phi ^{(0)}+N^{(0)}$ .",
"It describes a massless graviton and massive graviton with squared mass $m^2=\\alpha ^2$ .",
"For $\\ell \\ge 2$ the mixing among the five vector fields $(k^{T}_{\\mu },a^{T}_{\\mu },v^{T}_{\\mu }$ $,b_{\\mu \\nu }^{T},b^{T}_{\\mu })$ now have the following form $\\ell \\ge 2:\\quad 0&=&(2c_\\ell \\alpha ^4-\\hat{\\Box }_0^2)k^{T(\\ell )}_{\\mu }-4g\\alpha ^2q a^{T(\\ell )}_{\\mu }-\\frac{\\alpha ^2}{2}(4 q v^{T(\\ell )}_{\\mu }-\\epsilon _{\\mu }^{~\\nu \\lambda \\rho }\\partial _{\\nu }b_{\\lambda \\rho }^{T(\\ell )})\\ ,\\\\[0.2cm]0&=&(\\alpha ^2\\,\\Box -\\hat{\\Box }^2_0)b^{T(\\ell )}_{\\mu \\nu }-4q g\\alpha ^2\\star F^{(\\ell )}_{\\mu \\nu }(a)+\\alpha ^4c_\\ell (\\star F^{(\\ell )}_{\\mu \\nu }(k)-\\star F^{(\\ell )}_{\\mu \\nu }(b))\\ ,\\\\[0.2cm]0&=& \\hat{\\Box }_0^2b^{T(\\ell )}_{\\mu }+{\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } }\\alpha ^2\\epsilon _{\\mu }^{~\\nu \\lambda \\rho }\\partial _{\\nu }b_{\\lambda \\rho }^{T(\\ell )}\\ ,\\\\[0.2cm]0&=& (\\hat{\\Box }_0+\\alpha ^2)a^{T(\\ell )}_{\\mu } -4q g\\alpha ^2c_\\ell k^{T(\\ell )}_{\\mu }+4gv_{\\mu }^{T(\\ell )}-2q g\\epsilon _{\\mu }^{~\\nu \\lambda \\rho }\\partial _{\\nu }b_{\\lambda \\rho }^{T(\\ell )}\\ ,\\\\[0.2cm]0&=& (\\hat{\\Box }_0-\\alpha ^2)v^{T(\\ell )}_{\\mu }+\\alpha ^4qc_\\ell k^{T(\\ell )}_{\\mu }-2g\\alpha ^2 a^{(\\ell )}_{\\mu }\\ ,$ Diagonalising the associated $5\\times 5$ operator-valued matrix, we find that the modes are annihilated by the partially-factorising operator polynomial, of eighth order in $\\hat{\\Box }_0$ , given by $\\hat{\\Box }_0^2\\,{\\cal O}_6$ .",
"The explicit form of ${\\cal O}_6$ can be obtained straightforwardly from (REF ), and one finds that it is symmetric under $q\\rightarrow -q$ meaning that vector spectrum is symmetric under the sign change of monopole charge.",
"Of the remaining vectors $(({{\\cal P}}^{1}h)_{\\mu \\nu }, \\partial ^\\mu b_{\\mu \\nu })$ are annihilated by $\\hat{\\Box }_0$ .",
"Thus, apart from the charged vectors which will be treated separately below, the total wave operator for $\\ell \\ge 2$ is given by $\\ell \\ge 2:\\qquad {\\cal O}^{(1)} = \\hat{\\Box }_0^4|_{M^2=\\alpha ^2}\\,{\\cal O}_6\\ ,$ implying four massive vectors with squared masses $m^2=\\alpha ^2 c_\\ell $ , and six massive vectors whose squared masses $m^2$ correspond to the roots ${\\cal O}_6$ in which $\\Box $ is to be replaced by $m^2$ .",
"In the case of $\\ell =1$ , again excluding the charged vector, we find that a massless vector appears since for $c_\\ell =2$ , the operator ${\\cal O}_6 $ factorizes as ${\\cal O}_6= \\Box \\, {\\cal O}_5$ , and the total wave operator becomes $\\ell =1:\\qquad {\\cal O}^{(1)} = \\hat{\\Box }_0^4|_{\\ell =1}\\,\\Box \\, {\\cal O}_5\\ .$ The massless vector is composed from a linear combination of $(k^{T(1)}_{\\mu },a^{T(1)}_{\\mu },v^{T(1)}_{\\mu },b_{\\mu \\nu }^{T(1)},b^{T(1)}_{\\mu })$ with mixing coefficients $\\left({\\textstyle {\\frac{\\scriptstyle 2}{\\scriptstyle 1+q^2} } },-{\\textstyle {\\frac{\\scriptstyle 8q g}{\\scriptstyle 1+q^2} } },{\\textstyle {\\frac{\\scriptstyle 2q\\alpha ^2}{\\scriptstyle 1+q^2} } },1,0\\right)$ .",
"In the uncharged vector sector, there remains the case of $\\ell =0$ , for which the relevant vector fields are $(b_{\\mu \\nu }^{T(0)},a^{T(0)}_{\\mu },v^{T(0)}_{\\mu })$ .",
"Upon diagonalising the associated $3\\times 3$ operator-valued matrix, we find that the modes are annihilated by the following partially-factorising operator polynomial $\\ell =0: \\qquad {\\cal O}^{(1)}= \\Box (\\Box -\\alpha ^2)(\\Box ^2+2\\alpha ^4q^2)\\ .$ As before, we find that the would-be massless modes annihilated by $\\Box $ is eaten by the two form.",
"Thus there are no massless vector modes at $\\ell =0$ .",
"Finally, we turn to the treatment of the complex vector $\\hat{z}_{\\mu }$ .",
"This field $\\hat{z}_{\\mu }$ is expanded in terms of charge “-$q$ ” scalar harmonics starting from $\\ell =|q|$ as follows: ${\\hat{z}}_\\mu = \\sum _{\\ell \\ge q} z^{(\\ell )}_\\mu \\, {}_{-q}{Y}^{(\\ell )}\\ ,$ The resulting linearized field equation is $\\ell \\ge |q|:\\qquad \\left(\\Box -\\alpha ^2c_\\ell +\\alpha ^2q^2-M^2\\right)\\, z^{T(\\ell )}_{\\mu }=0\\ .$ For $\\ell \\ge 2$ , the equations describing the mixing between $(\\phi , N, \\varphi , \\tilde{\\varphi },a, v)$ take the following form $\\ell \\ge 2:\\quad 0&=&2(\\hat{\\Box }_0+\\alpha ^2)\\phi ^{(\\ell )}+(2\\hat{\\Box }_0+2\\alpha ^2+\\alpha ^2c_\\ell )N^{(\\ell )}+3\\hat{\\Box }_0\\varphi ^{(\\ell )}-\\alpha ^2c_\\ell \\tilde{\\varphi }^{(\\ell )}\\ ,\\\\[0.2cm]0&=&\\alpha ^2(3\\varphi ^{(\\ell )}+2\\phi ^{(\\ell )}-4gq a^{(\\ell )}-2q v^{(\\ell )})+\\alpha ^2(1-c_\\ell )\\tilde{\\varphi }^{(\\ell )}+\\Box \\,N^{(\\ell )}\\ ,\\\\[0.2cm]0&=&2\\alpha ^2\\phi ^{(\\ell )}+(2\\hat{\\Box }_0+2\\alpha ^2+\\alpha ^2c_\\ell )N^{(\\ell )}-\\Box \\tilde{\\varphi }^{(\\ell )}\\ ,\\\\[0.2cm]0&=& (\\hat{\\Box _0}+\\alpha ^2)a^{(\\ell )}+4gq N^{(\\ell )}-2gq (3\\varphi ^{(\\ell )}+\\tilde{\\varphi }^{(\\ell )})+4gv^{(\\ell )}\\ ,\\\\[0.2cm]0&=&(\\hat{ \\Box }_0-\\alpha ^2)v^{(\\ell )}-\\alpha ^2 q N^{(\\ell )}+\\frac{\\alpha ^2}{2}q(3\\varphi ^{(\\ell )}+\\tilde{\\varphi }^{\\ell })-2g\\alpha ^2 a^{(\\ell )}\\ ,\\\\[0.2cm]0&=&(\\alpha ^2\\Box -\\hat{\\Box }^2_0)\\varphi ^{(\\ell )}+2\\alpha ^4\\phi ^{(\\ell )}+\\alpha ^4(2-c_\\ell )N^{(\\ell )}-4q gc_\\ell \\alpha ^4 a^{(\\ell )}-2c_\\ell \\alpha ^4q v^{(\\ell )} \\ .$ Diagonalising the associated $6\\times 6$ operator-valued matrix, we find that the modes are annihilated by the partially-factorising operator polynomial, of seventh order in $\\hat{\\Box }_0$ , given by ${\\cal O}_4\\hat{\\Box }_0^3|_{M^2=\\alpha ^2}$ Of the remaining scalars, $(\\partial ^{\\mu }k_{\\mu },\\partial ^\\mu b_{\\mu },\\partial ^\\mu a_{\\mu },\\partial ^\\mu v_{\\mu },\\tilde{v})$ , three of them namely $(\\partial ^{\\mu }k_{\\mu },\\partial ^\\mu b_{\\mu },\\tilde{v}) $ are annihilated by ${\\hat{\\Box }}_0$ , and the remaining two are determined in terms of them.",
"Thus, in total, apart from the complex scalars which will be treated separately below, the wave operator for the scalar fields is given by ${\\cal O}^{(0)} = {\\hat{\\Box }}_0^6\\,{\\cal O}_4\\ .$ Next, consider the case $\\ell =1$ .",
"Utilizing the residual symmetry (REF ) and (REF ), one can eliminate $N^{(1)}$ and $\\partial ^\\mu k_{\\mu }^{(1)}$ .",
"The mass operator coming from the mixing among $(\\phi ,\\varphi , \\tilde{\\varphi },a, v)$ takes the form $\\hat{\\Box }_0^2|_{\\ell =1}{\\cal O}_3$ .",
"Taking into account $\\partial ^\\mu b_\\mu ^{(1)}$ and $\\widetilde{v}^{(1)}$ , the total wave operator, again, excluding the complex scalar sector, is given by $\\ell =1:\\qquad {\\cal O}^{(0)} = \\hat{\\Box }_0^4\\,{\\cal O}_3\\ .$ There remaining the case of $\\ell =0$ .",
"In this case, the relevant scalar fields are $(\\phi ^{(0)},N^{(0)},b^{(0)})$ , and they satisfy the following equations respectively $\\ell =0:\\quad &&(\\Box ^2+2\\alpha ^4)S^{(0)}=0\\ ,\\qquad S^{(0)}=\\phi ^{(0)}+N^{(0)}\\ ,\\nonumber \\\\&& \\Box ^2b^{(0)}=0\\ ,\\nonumber \\\\&&\\Box ^2N^{(0)}= 0\\ .$ Thus besides two massless modes, we also have modes with linear time coordinate dependence.",
"Finally, we discuss the complex scalars originating from $\\hat{z}_m$ .",
"For positive monopole charge, the harmonic expansion of $\\hat{z}_m$ is given by $\\hat{z}_m=z^{(q-1)} {}_{-q}V_m^{(q-1)}+z^{(q)} {}_{-q}V_m^{(q)}+\\sum _{\\ell >q}(z^{(\\ell )}D_m{}_{-q}Y^{(\\ell )}+\\tilde{z}^{(\\ell )}\\epsilon _m^{~n}D_n{}_{-q}Y^{(\\ell )})\\ .$ Thus we have $&&\\ell >q &:\\qquad z^{(\\ell )}\\ ,\\ \\tilde{z}^{(\\ell )} &\\qquad \\mbox{with} &\\qquad m^2=\\alpha ^2c_\\ell -\\alpha ^2q+M^2\\ ,\\nonumber \\\\&&\\ell =q &:\\qquad z^{(q)}\\ , &\\qquad \\mbox{with} &\\qquad m^2=\\alpha ^2q+M^2\\ ,\\nonumber \\\\&&\\ell =q-1 &:\\qquad z^{(q-1)}\\ , &\\qquad \\mbox{with} &\\qquad m^2=M^2-\\alpha ^2q\\ .$"
],
[
"Fermionic sector",
"The analysis of the fermionic spectrum in a non-supersymmetric background (REF ) is more subtle than that in supersymmetric background.",
"Since the non-supersymmetric background do not posses Killing spinor, we will use spin-weighted harmonics $ _{s-{\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } }}\\eta ^{(\\ell )}$ , described in detail in appendix B, as basis of expansion.",
"For brevity, we shall use the notation $_{s-{\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } }}\\eta ^{(\\ell )} \\equiv \\tilde{\\eta }^{(\\ell )}\\ ,$ where $s={\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } }(1-q)$ .",
"These harmonics satisfy the relations $&&\\tilde{\\eta }^{(\\ell )}_+=\\eta _+ \\left(_{s-1}Y^{(\\ell )} \\right)\\ , \\qquad \\tilde{\\eta }^{(\\ell )}_- = \\eta _- \\left( _{s}Y^{(\\ell )} \\right)\\ ,\\\\&& (\\frac{d}{d\\theta }+m\\csc \\theta +s\\cot \\theta ){}\\left(_{s}Y^{(\\ell )}\\right)=\\sqrt{(\\ell +s)(\\ell +1-s)}{}\\left(_{s-1}Y^{(\\ell )}\\right)\\ ,\\\\&&(\\frac{d}{d\\theta }-m\\csc \\theta -(s-1)\\cot \\theta ){}\\left(_{s-1}Y^{(\\ell )}\\right)=-\\sqrt{(\\ell +s)(\\ell +1-s)}{} \\left(_{s}Y^{(\\ell )}\\right)\\ .$ The lowest level would have definite chirality when $\\ell =-s$ for $q>0$ and $\\ell =s-1$ for $q<0$ .",
"The spin weighted harmonics satisfy the following properties $\\sigma _3\\tilde{\\eta }_{\\pm }^{(\\ell )} &=\\pm \\tilde{\\eta }_{\\pm }^{(\\ell )}\\ , &\\qquad \\sigma ^n D_n\\tilde{\\eta }_{\\pm }^{(\\ell )} &=i\\alpha \\sqrt{\\tilde{c_\\ell }}\\,\\tilde{\\eta }_{\\mp }^{(\\ell )}\\ ,\\nonumber \\\\[D_m, D_n]\\tilde{\\eta }_-^{(\\ell )} &=-is\\alpha ^2\\epsilon _{mn}\\tilde{\\eta }_-^{(\\ell )}\\ ,&\\qquad [D_m,D_n]{\\tilde{\\eta }}_+^{(\\ell )} &=i(1-s)\\alpha ^2\\epsilon _{mn}{\\eta _+^{(\\ell )}}\\ ,\\nonumber \\\\D^nD_n\\tilde{\\eta }_-^{(\\ell )} &=-\\alpha ^2(\\tilde{c_\\ell }-s)\\tilde{\\eta }_-^{(\\ell )}\\ ,&\\qquad D^nD_n\\tilde{\\eta }_+^{(\\ell )} &= \\alpha ^2 (1-s-\\tilde{c_\\ell })\\tilde{\\eta }_+^{(\\ell )}\\ ,$ where $\\tilde{c_\\ell }=\\biggl (c_\\ell -s(s-1)\\biggr )\\ .$ The harmonic expansion for 6D spin-$1/2$ fields follows the same procedure as in supersymmetric case by using the spin weighted harmonics, while it is more subtle when expanding the 6D gravitini.",
"It can be checked that the linearized equations have the following discreet symmetry $q\\rightarrow -q\\ , &\\qquad \\psi _{\\mu +}\\rightarrow -\\psi _{\\mu -},& \\qquad \\psi _{\\mu -}\\rightarrow \\psi _{\\mu +}\\ ,\\nonumber \\\\\\psi _-\\rightarrow \\psi _+\\ , &\\qquad \\psi _+\\rightarrow -\\psi _-\\ , &\\qquad \\chi _-\\rightarrow \\chi _+\\ ,\\nonumber \\\\\\chi _+\\rightarrow -\\chi _-\\ , &\\qquad \\lambda _+\\rightarrow -\\lambda _-\\ ,& \\qquad \\lambda _-\\rightarrow \\lambda _+\\ ,$ which implies that the spectrum keeps the same under the sign change of monopole charge.",
"In the following, we will focus on the case with positive monopole charge and use $|s|$ to denote ${\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } }(q-1)$ ."
],
[
"The gravitini satisfy $\\ell \\ge |s|+1:\\qquad &&{\\partial }\\Big (\\Box +\\alpha ^2|s|-\\alpha ^2\\tilde{c_\\ell }\\Big )\\psi _{\\mu +}^{(\\ell )}+i\\alpha \\sqrt{\\tilde{c_\\ell }}\\Big (\\Box -\\alpha ^2\\tilde{c_\\ell }\\Big )\\psi _{\\mu -}^{(\\ell )}=0\\ ,\\nonumber \\\\&&i\\alpha \\sqrt{\\tilde{c_\\ell }}\\Big (\\Box -\\alpha ^2\\tilde{c_\\ell }\\Big )\\psi _{\\mu +}^{(\\ell )}-{\\partial }\\Big (\\Box +(|s|+1-{\\tilde{c}}_\\ell )\\alpha ^2\\Big )\\psi _{\\mu -}^{(\\ell )}=0\\ .$ Diagonalising the associated $2\\times 2$ operator-valued matrix, we find that the modes are annihilated by the partially-factorising operator polynomial, of third order in $\\Box $ , given by ${\\cal O}^{(3/2)} = \\tilde{{\\cal O}}_3\\ ,$ where the explicit form of $\\tilde{{\\cal O}}_3$ can be deduced from (REF ).",
"Next, we consider the case of $\\ell =|s|$ .",
"In this case, we find that the quadratic action for the lowest level fermionic fields is proportional to ${\\cal L}^{(2)}&\\propto & -i\\bar{\\chi }\\gamma ^{\\mu \\nu }\\partial _{\\mu }\\psi _{\\nu }-i\\bar{\\psi }_{\\mu }\\gamma ^{\\mu \\nu }\\partial _{\\nu }\\chi +2i\\bar{\\chi }{\\partial }\\psi -2i\\bar{\\psi }{\\partial }\\chi +2\\bar{\\chi }{\\partial }\\chi \\nonumber \\\\&&-8g\\bar{\\chi }\\lambda -8g\\bar{\\lambda }\\chi -4\\bar{\\lambda }{\\partial }\\lambda -4i g|s|\\bar{\\lambda }\\gamma ^{\\mu }\\psi _{\\mu }-4i g|s|\\bar{\\psi }_{\\mu }\\gamma ^{\\mu }\\lambda \\nonumber \\\\&& -8g(1+|s|)i\\bar{\\lambda }\\psi +8g(1+|s|)i\\bar{\\psi }\\lambda -{\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } }\\bar{\\psi }_{\\mu }\\gamma ^{\\mu \\nu \\lambda }\\partial _{\\nu }\\psi _{\\lambda }+\\bar{\\psi }{\\partial }\\psi \\nonumber \\\\&&+\\bar{\\psi }_{\\mu }\\gamma ^{\\mu \\nu }\\partial _{\\nu }\\psi -\\psi \\gamma ^{\\mu \\nu }\\partial _{\\mu }\\psi _{\\nu }-\\frac{1}{\\alpha ^2}\\biggl ({\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 4} } }\\bar{\\psi }_{\\mu \\nu }{\\partial }\\psi ^{\\mu \\nu }-\\bar{\\psi }{\\partial }\\Box \\psi \\nonumber \\\\&& +{\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } }|s|\\alpha ^2\\bar{\\psi }_{\\mu }{\\partial }\\psi ^{\\mu }-|s|\\alpha ^2\\bar{\\psi }\\partial _{\\mu }\\psi ^{\\mu }+|s|\\alpha ^2\\bar{\\psi }_{\\mu }\\partial ^{\\mu }\\psi -(1+2|s|)\\alpha ^2\\bar{\\psi }{\\partial }\\psi \\biggr )\\ .$ Unlike the supersymmetric case, we see the appearance of the terms $\\bar{\\lambda }\\gamma ^{\\mu }\\psi _{\\mu }$ , $\\bar{\\psi }_{\\mu }{\\partial }\\psi ^{\\mu }$ and $\\bar{\\psi }_{\\mu }\\partial ^{\\mu }\\psi $ which break the fermionic gauge symmetry.",
"The homogeneous solutions for gravitini satisfy ${\\partial }(\\Box -\\alpha ^2-|s|\\alpha ^2)\\psi _{\\mu }^{(|s|)}=0,\\qquad \\gamma ^{\\mu }\\psi ^{(|s|)}_{\\mu }=0,\\qquad \\partial ^{\\mu }\\psi _{\\mu }^{(|s|)}=0.$ Thus, due to the lack of fermionic gauge symmetry, the longitudinal mode $\\psi _{\\mu }\\propto \\,p_{\\mu }e^{ipx}$ with $p^2=0$ becomes a dynamical degree of freedom.",
"The $\\ell \\ge |s|+2$ sector consists of ten spin-1/2 fields $(\\Lambda _+,\\Psi _+,\\Lambda _-,\\Psi _-,$ $\\psi _+,\\psi _-,\\chi _+,\\chi _-,\\lambda _+,\\lambda _-)$ .",
"The linearized equations describing their mixing are $0&=&{\\partial }\\Lambda ^{(\\ell )}_{-}-\\Psi _+^{(\\ell )}+i\\alpha \\sqrt{\\tilde{c_\\ell }}\\psi _+^{(\\ell )}-i\\alpha \\sqrt{\\tilde{c_\\ell }}\\Lambda _+^{(\\ell )}+2{\\partial }\\psi _-^{(\\ell )}-2i{\\partial }\\chi _-^{(\\ell )}+2\\alpha \\sqrt{\\tilde{c_\\ell }}\\chi _+^{(\\ell )}-8gi\\lambda _+^{(\\ell )}\\ ,\\\\0&=&{\\partial }\\Lambda _+^{(\\ell )}-\\Psi _-^{(\\ell )}+i\\alpha \\sqrt{\\tilde{c_\\ell }}\\psi _-^{(\\ell )}+i\\alpha \\sqrt{\\tilde{c_\\ell }}\\Lambda _-^{(\\ell )}-2{\\partial }\\psi _+^{(\\ell )}+2i{\\partial }\\chi _+^{(\\ell )} +2\\alpha \\sqrt{\\tilde{c_\\ell }}\\chi _-^{(\\ell )}-8gi\\lambda _-^{(\\ell )}\\ ,\\\\0&=& g(1+|s|)i\\Lambda _-^{(\\ell )}+2g\\chi _-^{(\\ell )}-{\\partial }\\lambda _+^{(\\ell )}-i\\alpha \\sqrt{\\tilde{c_\\ell }}\\lambda _-^{(\\ell )}\\ ,\\\\0&=&2i g|s|\\psi _+^{(\\ell )}+2g\\chi _+^{(\\ell )}+{\\partial }\\lambda _-^{(\\ell )}-i\\alpha \\sqrt{\\tilde{c_\\ell }}\\lambda _+^{(\\ell )}\\ ,\\\\0&=&i\\sqrt{\\tilde{c_\\ell }}{\\partial }(\\Box +\\alpha ^2+|s|\\alpha ^2)\\psi _+^{(\\ell )}-i\\alpha ^2\\sqrt{\\tilde{c_\\ell }}{\\partial }\\Lambda _+^{(\\ell )}+i\\alpha ^2\\tilde{c_\\ell }^{3/2}\\Psi _-^{(\\ell )}+2\\alpha ^2\\sqrt{\\tilde{c_\\ell }}{\\partial }\\chi _+^{(\\ell )}\\nonumber \\\\&& -8g(1+|s|)\\alpha \\,i{\\partial }\\lambda _+^{(\\ell )}+\\alpha (\\tilde{c_\\ell }-1-|s|){\\partial }\\Psi _+^{(\\ell )}+\\alpha (2+2|s|-\\tilde{c_\\ell })\\Box \\psi _-^{(\\ell )}\\ ,\\\\0&=& i\\sqrt{\\tilde{c_\\ell }}{\\partial }(\\Box -|s|\\alpha ^2)\\psi _-^{(\\ell )}+i\\alpha ^2\\sqrt{\\tilde{c_\\ell }}{\\partial }\\Lambda _-^{(\\ell )}-i\\alpha ^2\\tilde{c_\\ell }^{3/2}\\Psi _+^{(\\ell )}+2\\alpha ^2\\sqrt{\\tilde{c_\\ell }}{\\partial }\\chi _-^{(\\ell )}+8ig|s|\\alpha {\\partial }\\lambda _{-}^{(\\ell )}\\nonumber \\\\&&+\\alpha (\\tilde{c_\\ell }+|s|){\\partial }\\Psi _-^{(\\ell )}+\\alpha (\\tilde{c_\\ell }+2|s|)\\Box \\psi _+^{(\\ell )}\\ ,\\\\0&=&{\\partial }(\\Box +3\\alpha ^2+|s|\\alpha ^2-\\alpha ^2\\tilde{c_\\ell })\\Lambda _-^{(\\ell )}-(\\Box +4\\alpha ^2+2|s|\\alpha ^2-2\\alpha ^2\\tilde{c_\\ell })\\Psi _+^{(\\ell )}+i\\alpha \\sqrt{\\tilde{c_\\ell }}(\\Box +4\\alpha ^2+4|s|\\alpha ^2)\\psi _+^{(\\ell )}\\nonumber \\\\&&+\\alpha ^2{\\partial }(8+2|s|-\\tilde{c_\\ell })\\psi _-^{(\\ell )}-i\\alpha \\sqrt{\\tilde{c_\\ell }}(\\Box +4\\alpha ^2-\\alpha ^2\\tilde{c_\\ell })\\Lambda _+^{(\\ell )}-6i\\alpha ^2{\\partial }\\chi _-^{(\\ell )}+8\\alpha ^3\\sqrt{\\tilde{c_\\ell }}\\chi _+^{(\\ell )}\\nonumber \\\\&&-32ig\\alpha ^2\\,(1+|s|)\\lambda _+^{(\\ell )} +i\\alpha \\sqrt{\\tilde{c_\\ell }}{\\partial }\\Psi _-^{(\\ell )}\\ ,\\\\0&=&-{\\partial }(\\Box +2\\alpha ^2-|s|\\alpha ^2-\\alpha ^2\\tilde{c_\\ell })\\Lambda _+^{(\\ell )}+(\\Box +2\\alpha ^2-2|s|-2\\alpha ^2\\tilde{c_\\ell })\\Psi _-^{(\\ell )}- i\\alpha \\sqrt{\\tilde{c_\\ell }}(\\Box -4|s|\\alpha ^2)\\psi _-^{(\\ell )}\\nonumber \\\\&&+\\alpha ^2{\\partial }(6-2|s|-\\tilde{c_\\ell })\\psi _+^{(\\ell )}-i\\alpha \\sqrt{\\tilde{c_\\ell }}(\\Box +4\\alpha ^2-\\alpha ^2\\tilde{c_\\ell })\\Lambda _-^{(\\ell )}-6i\\alpha ^2 {\\partial }\\chi _+^{(\\ell )}-8\\alpha ^3\\sqrt{\\tilde{c_\\ell }}\\chi _-^{(\\ell )}\\nonumber \\\\&&-32ig|s|\\alpha ^2 \\lambda _-^{(\\ell )}+i\\alpha \\sqrt{\\tilde{c_\\ell }} {\\partial }\\Psi _+^{(\\ell )}\\ ,\\\\0&=&(1+|s|)\\alpha ^2\\Lambda _-^{(\\ell )}-2i\\alpha ^2 \\chi _-^{(\\ell )}+(2\\Box -2 |s|\\alpha ^2-\\alpha ^2\\tilde{c_\\ell })\\psi _-^{(\\ell )}-{\\partial }\\Psi _+^{(\\ell )}-i\\alpha \\sqrt{\\tilde{c_\\ell }}\\Psi _-^{(\\ell )}-i\\alpha \\sqrt{\\tilde{c_\\ell }}{\\partial }\\psi _+^{(\\ell )}\\ ,\\nonumber \\\\&& \\\\0&=&|s|\\alpha ^2\\Lambda _+^{(\\ell )}-2i\\alpha ^2 \\chi _+^{(\\ell )}+(2\\Box +2\\alpha ^2+2|s|\\alpha ^2-\\alpha ^2\\tilde{c_\\ell })\\psi _+^{(\\ell )} -i\\alpha \\sqrt{\\tilde{c_\\ell }}\\Psi _+^{(\\ell )}+{\\partial }\\Psi _-^{(\\ell )}+i\\alpha \\sqrt{\\tilde{c_\\ell }}{\\partial }\\psi _-^{(\\ell )}\\ .\\nonumber \\\\$ Diagonalising the associated $10\\times 10$ operator-valued matrix, we find that the modes are annihilated by the partially-factorising operator polynomial, of ninth order in $\\Box $ , given by $\\ell \\ge |s|+2:\\qquad {\\cal O}^{(0)} = \\hat{\\Box }_0^3\\,\\tilde{{\\cal O}}_6\\ ,$ where the explicit form of the operator ${\\cal O}_6$ can be determined from the linearized spin 1/2 field equations listed above.",
"$\\ell =|s|+1$ At this level, since $D_m{\\eta _+^{(|s|+1)}}=\\frac{i}{2}\\alpha \\sqrt{\\tilde{c_\\ell }}\\sigma _m{\\eta _-^{(|s|+1)}}$ , there emerges a fermionic gauge symmetry generated by $\\hat{\\epsilon }=\\epsilon _+^{(|s|+1)}{\\eta _+^{(|s|+1)}}$ $\\delta \\psi _{\\mu +}^{(|s|+1)}&=&\\partial _{\\mu }\\epsilon _+^{(|s|+1)}+i\\alpha \\sqrt{\\frac{(1+|s|)}{2}}\\,\\epsilon _+^{(|s|+1)}\\ ,\\nonumber \\\\\\delta \\lambda _+^{(|s|+1,m)}&=&ig(1+|s|)\\epsilon _+^{(|s|+1)}\\ .$ Using this gauge symmetry, one can eliminate $\\psi _+^{(|s|+1)}$ , such that the 10 by 10 mixing becomes 9 by 9 mixing.",
"Diagonalising this system, we find that the modes are determined by the partially-factorising operator polynomial, of fifteenth order in ${\\partial }$ , given by $\\ell =|s|+1:\\qquad {\\partial }\\Big (\\Box - (|s|+2)\\alpha ^2\\Big )^2 \\tilde{{\\cal O}}_5\\ ,$ where the explicit form of $\\tilde{\\cal O}_5$ can be deduced from the mixing equations.",
"It is clear that is a massless spin-1/2 mode.",
"Explicitly, it is a linear combination of $(\\Lambda ^{(|s|+1)}_+,\\Psi ^{(|s|+1)}_+,\\Lambda ^{(|s|+1)}_-,\\Psi ^{(|s|+1)}_-,\\psi ^{(|s|+1)}_-,$ $\\chi ^{(|s|+1)}_-, \\chi ^{(|s|+1)}_+,\\lambda ^{(|s|+1)}_+,\\lambda ^{(|s|+1)}_-)$ with mixing coefficients $(8(1+2|s|),0,0,0,-2,-2i(1+2|s|),0,0,\\sqrt{1+|s|}(1+4|s|).$ $\\ell =|s|$ The coupled system of linearized field equations for the spin-1/2 fields $(\\Lambda _+^{(|s|)},\\Psi _-^{(|s|)}, \\psi _+^{(|s|)},\\chi _+^{(|s|)},\\lambda _-^{(|s|)})$ are $0&=&{\\partial }\\Lambda _+^{(|s|)}-\\Psi _-^{(|s|)}-2{\\partial }\\psi _+^{(|s|)}+2i{\\partial }\\chi _+^{(|s|)}-8gi\\lambda _-^{(|s|)}\\ ,\\\\0&=&i g |s| \\Lambda _+^{(|s|)}+2g(1+|s|)i\\psi _+^{(|s|)}+2g\\chi _+^{(|s|)}+{\\partial }\\lambda _-^{(|s|)}=0,\\\\0&=&8gi{\\partial }\\lambda _-^{(|s|)}+{\\partial }\\Psi _-^{(|s|)}+2\\Box \\psi _+^{(|s|)}=0,\\\\0&=&(\\Box +2\\alpha ^2-|s|\\alpha ^2){\\partial }\\Lambda _+^{(|s| )}-(\\Box +2\\alpha ^2-2|s|\\alpha ^2)\\Psi _-^{(|s| )}+(2|s|-6)\\alpha ^2{\\partial }\\psi _+^{(|s| )}+6\\alpha ^2\\,i{\\partial }\\chi _+^{(|s| )}\\nonumber \\\\&&+32g|s|\\alpha ^2\\,i\\lambda _-^{(|s|) }\\ ,\\\\0&=&\\alpha ^2{\\partial }\\Lambda _+^{(|s| )}-(1+|s|)\\alpha ^2\\Psi _-^{(|s| )}-{\\partial }(\\Box +2\\alpha ^2+2|s|\\alpha ^2)\\psi _+^{|s|}-8g(1+|s|)\\alpha ^2\\,i\\lambda _-^{(|s|)}+2\\alpha ^2\\,i{\\partial }\\chi _+^{(|s|)}\\nonumber .\\\\$ Diagonalising this system, we find two massless modes which satisfy the relations $\\Psi _-^{(|s|)}=0\\ ,\\qquad \\lambda _-^{(|s|)}=0\\ ,\\qquad |s|\\Lambda _+^{(|s|)}-2i\\chi _+^{(|s|)}+2(1+|s|)\\psi _+^{(|s|)}=0\\ .$ $\\ell =|s|-1$ When monopole charge $q\\ge 3$ , there exist charge “$-q/2$ ” vector-spinor harmonics $\\eta _m$ on $S^2$ possessing following properties $D^m\\left( _{s-{\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } }}\\eta ^{(|s|-1)}\\right)_m=0\\ , \\qquad \\sigma ^m\\left( _{s-{\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } }}\\eta ^{(|s|-1)}\\right)_m =0\\ ,$ where $s=(1-q)/2$ .",
"It follows that $\\sigma ^m D_m \\left( _{s-{\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } }}\\eta ^{(|s|-1)}\\right)_n=0\\ .$ Associated to this harmonics, there is a new spin-1/2 field $\\tilde{\\psi }^{(|s|-1)}$ satisfying ${\\partial }\\Box \\tilde{\\psi }^{(|s|-1)}=0\\ .$"
],
[
"Remarks on the non-supersymmetric spectrum",
"We saw in the supersymmetric vacuum that even at the $\\ell =0$ level in the Kaluza-Klein harmonic expansions, avoiding tachyons in the four-dimensional spectrum required imposing the condition $M^2\\ge (5+2\\sqrt{6})\\alpha ^2$ on the parameter $M$ in the six-dimensional Lagrangian.",
"In the non-supersymmetric vacua we necessarily have $M^2=\\alpha ^2$ , and in fact having larger values for the background monopole charge $q$ then the $q=\\pm 1$ supersymmetric case only makes the tachyon problem worse, as can be seen from the vector mass operator (REF ).",
"For this reason, we shall not explore further the precise details of the occurrence of tachyonic states in the non-supersymmetric backgrounds.",
"One feature of interest that we shall, however, comment on is the occurrence of massless fermions in the non-supersymmetric backgrounds.",
"As can be seen from the spin-${\\textstyle {\\frac{\\scriptstyle 3}{\\scriptstyle 2} } }$ operator in (REF ), there will be massless spin-${\\textstyle {\\frac{\\scriptstyle 3}{\\scriptstyle 2} } }$ fields at level $\\ell =|s|={\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } }(q-1)$ ; thus these will occur in an $SU(2)$ multiplet of dimension $2\\ell +1 = q$ .",
"At $\\ell =|s|+1$ , $\\ell =|s|$ and $\\ell =|s|-1$ there will also be massless spin-${\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } }$ modes, as was discussed in the spin-${\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } }$ section above."
],
[
"Conclusions",
"In this paper we have studied the complete linearised spectrum in the $S^2$ Kaluza-Klein reduction of off-shell six-dimensional ${\\cal N}=(1,0)$ gauged supergravity extended by a Riemann-squared superinvariant.",
"The higher-derivative terms in the six-dimensional theory can be expected to imply the occurrence of ghosts.",
"As discussed in the introduction, the usual argument for the positivity of the energies of states in a supersymmetric background breaks down, and indeed we found that states in the Kaluza-Klein spectrum could now have complex energies, thus implying instabilities.",
"One way to understand the occurrence of complex masses is that there are mixings between four-dimensional ghostlike and non-ghostlike modes.",
"This may be illustrated by the following simple example.",
"Consider a set of fields $\\phi _i$ with the Lagrangian ${\\cal L}= -{\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } } K_{ij}\\, \\partial \\phi _i\\, \\partial \\phi _j -{\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } } V_{ij}\\phi _i\\phi _j\\,,$ where $K_{ij}$ and $V_{ij}$ are constant symmetric matrices.",
"If the eigenvalues of $K_{ij}$ are all positive, then $K_{ij}$ and $V_{ij}$ can be simultaneously diagonalised, by means of orthogonal transformations combined with rescalings of the fields.",
"However, if $K_{ij}$ has negative as well as positive eigenvalues, then the rescalings will introduce factors of $\\sqrt{-1}$ and the diagonalised mass matrix will be complex.",
"One way to avoid the ghost problems of the higher-derivative theory is to treat it not as an exact model in its own right, but rather as an effective theory valid at energy scales $\\sqrt{\\Lambda }$ much smaller than $M$ .",
"In this case the propagators are governed by the leading-order theory without the higher-derivative terms, and these terms are treated as interactions.",
"The Kaluza-Klein spectrum in the reduced four-dimensional theory would then be simply that of the original Salam-Sezgin model corresponding to the $M^2\\rightarrow \\infty $ limit.",
"This spectrum has been given in [6] and our results agree in the $M^2\\rightarrow \\infty $ limit.",
"However, the Kaluza-Klein level number $\\ell $ would have to be restricted to lie below some maximum value, in order to satisfy the $\\Lambda <<M^2$ limit.",
"Interestingly, this condition is sufficient to ensure that in the full, extended, theory, the $m^2$ values of the retained modes would all be real and positive.",
"This can be seen from (REF ), which indicates that the $m^2$ values will all be real and positive if $\\ell $ is less than about $M^2/(4\\alpha ^2)$ .",
"A couple of remarks about the consistency of the Kaluza-Klein reduction are in order.",
"Although we have restricted ourselves to a linearised analysis of the four-dimensional spectrum, it should be emphasised that provided one is keeping all the infinite towers of modes, then even at the full non-linear order the reduction would still be consistent.",
"The truncation of the spectrum at some maximum value of the level number $\\ell $ that we discussed in the previous paragraph would not, of course, be consistent beyond the linear order, since the higher modes that were being set to zero would be excited by sources involving the modes that are being retained.",
"Another more subtle question of consistency arises in this model also.",
"It was shown in [11] that the Salam-Sezgin theory admits a non-trivial consistent Pauli reduction on $S^2$ , in which a finite subset of fields including the $\\ell =1$ triplet of Yang Mills gauge bosons are retained.",
"It would be interesting to see whether such a Pauli reduction is still possible in the theory with the higher-derivative extension that we have been considering in this paper.",
"Another interesting question is whether the six dimensional model, with the auxiliary fields eliminated in an order by order expansion in inverse powers of $M^2$ , can be embedded into the ten-dimensional heterotic string.",
"In the gauged theory where $g\\ne 0$ , this continues to be a challenging problem even before the higher-derivative terms are considered (although some progress was made in a restricted sector of the theory in [12]).",
"For $g=0$ , on the other hand, it was conjectured in [13] that there is a relation with the 4-torus reduction of the heterotic theory with Riemann-squared corrections that were constructed in [14].",
"This relation holds upon making a suitable truncation and performing an S-duality transformation.",
"This conjecture was tested to lowest order in the bosonic sector in [13].",
"It would also be interesting to study exact solutions of the higher-derivative six-dimensional supergravity.",
"While many solutions of the Salam-Sezgin theory are known, exact solutions of the higher-derivative theory, beyond the vacuum solutions we have discussed in this paper, are scarce.",
"As far as we are aware, the only further example, which exists only in the ungauged theory, is the self-dual string that was found in [15].",
"A further question is whether there exist other quadratic-curvature superinvariants over and above the Riemann-squared invariant of the theory we have been considering.",
"This may have consequences for the embedding of the theory in ten dimensions."
],
[
"Acknowledgements",
"We are very grateful to F. Coomans and A. van Proeyen for many helpful discussions on the six-dimensional model.",
"The research of C.N.P.",
"is supported in part by DOE grant DE-FG03-95ER40917i, and that of E.S.",
"is supported by in part by NSF grant PHY-0906222."
],
[
"Conventions",
"We choose the 6$D$ gamma matrices to be $&& {\\Gamma }^0=\\gamma ^0\\otimes \\sigma _3\\ ,\\qquad {\\Gamma }^1=\\gamma ^1\\otimes \\sigma _3\\ ,\\qquad {\\Gamma }^2=\\gamma ^2\\otimes \\sigma _3\\ ,\\nonumber \\\\&&{\\Gamma }^3=\\gamma _3\\otimes \\sigma _3\\ ,\\qquad {\\Gamma }^4=1_{4\\times 4}\\otimes \\sigma _1\\ ,\\qquad {\\Gamma }^5=1_{4\\times 4}\\otimes \\sigma _2\\ .$ One can check that $&& {\\Gamma }^0{\\Gamma }^{\\mu \\dagger }{\\Gamma }^0={\\Gamma }^{\\mu }\\ ,\\qquad B={\\Gamma }^3\\hat{\\Gamma }^5\\ ,\\nonumber \\\\&& B^*B=-1\\ ,\\qquad B{\\Gamma }^{\\mu }B^{-1}={\\Gamma }^{\\mu *}\\ .$ The $SU$ (2) symplectic-Majorana-Weyl spinor is defined by $\\psi ^{*i}=(\\psi _{i})^*=\\epsilon ^{ij}B\\psi _{j}.$ A useful formula related to the $SU$ (2) symplectic-Majorana-Weyl spinor is $\\bar{\\lambda }^i{\\Gamma }^{(n)}\\psi ^j=t_n\\bar{\\psi }^j{\\Gamma }^{(n)}\\lambda ^i\\ ,\\qquad t_{n}=\\left\\lbrace \\begin{array}{ll}+, & n=1,2,5,6; \\\\-, & n=0,3,4.\\end{array}\\right.$"
],
[
"Spin-weighted Harmonics on $S^2$",
"In this appendix, we give an elementary construction of the spin-weighted spherical harmonics.",
"This is based on a specialisation of results for the analogous harmonics in the complex projective space $CP^n$ , which were discussed in [10].",
"Since the azimuthal label $m$ on the spin-weighted spherical harmonics $_sY_{\\ell m}$ plays an important role in the derivations in this appendix, we shall suspend our convention used in the body of the paper of suppressing the $m$ label.",
"In order to avoid confusion with coordinate indices, we shall use $i$ , $j$ ,... for coordinate indices on $S^2$ in this appendix."
],
[
"Scalar spin-weighted harmonics",
"The scalar spin-weighted spherical harmonics $_sY_{\\ell m}$ are the eigenfunctions of the charged scalar Laplacian $\\Box _{(s)}$ on the unit $S^2$ , carrying electric charge $s$ , in the presence of a Dirac monopole with potential $A=-\\cos \\theta \\, d\\phi $ : $\\Box _{(s)} \\equiv {\\frac{1}{\\sin \\theta }}\\, {\\frac{\\partial }{\\partial \\theta }}\\Big ( \\sin \\theta \\, {\\frac{\\partial }{\\partial \\theta }}\\Big ) +{\\frac{1}{\\sin ^2\\theta }}\\,\\Big ({\\frac{\\partial }{\\partial \\phi }} + {i}\\, s\\, \\cos \\theta \\Big )^2\\,.$ In the language of differential forms, the charged Laplacian operator on the unit $S^2$ with metric $d\\Omega _2^2=d\\theta ^2 + \\sin ^2\\theta \\, d\\phi ^2$ may be written in terms of the charged Hodge-de Rham operator $\\Delta \\equiv {*D*}D + D{*D*}\\,,$ as $-\\Box _{(s)} =\\Delta \\,,$ where $D$ is the charge-$s$ gauge-covariant exterior derivative $D= d + {i}s \\cos \\theta d\\phi \\,.$ The spin-weighted harmonics may be constructed by starting with the four-dimensional scalar Laplacian on ${C}^2$ , and then embedding the unit $S^3$ , viewed as a $U(1)$ bundle over $S^2$ , in ${C}^2$ .",
"Introducing complex coordinates $Z^a$ on ${C}^2$ , the four-dimensional Laplacian is $\\Box _4 = 4 {\\frac{\\partial ^2}{\\partial Z^a \\partial \\bar{Z}_a}}\\,.$ Clearly, if we define functions $f= T_{a_1\\cdots a_p}{}^{b_1\\cdots b_q}\\, Z^{a_1}\\cdots Z^{a_p}\\,\\bar{Z}_{b_1}\\cdots \\bar{Z}_{b_q}\\,,$ where $T_{a_1\\cdots a_p}{}^{b_1\\cdots b_q}$ is symmetric in its upper and its lower indices, and traceless with respect to any contraction of upper and lower indices, then they will satisfy $\\Box _4 f=0\\,.$ Writing $Z^1= r\\,e^{{i}(\\psi +\\phi )/2}\\, \\cos {\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } }\\theta \\,,\\qquad Z^2= r\\,e^{{i}(\\psi -\\phi )/2}\\, \\sin {\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } }\\theta \\,,$ the Euclidean metric on ${C}^2$ is expressible as $ds_4^2 = dr^2 + r^2\\, d\\Omega _3^2\\,,$ where $d\\Omega _3^2 = {\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 4} } } (d\\psi +\\cos \\theta \\, d\\phi )^2 + {\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 4} } }(d\\theta ^2 +\\sin ^2\\theta \\, d\\phi ^2)$ is the metric on the unit 3-sphere.",
"The four-dimensional Laplacian is given by $\\Box _4 = {\\frac{1}{r^3}}\\, {\\frac{\\partial }{\\partial r}}\\Big (r^3\\, {\\frac{\\partial }{\\partial r}}\\Big ) +{\\frac{1}{r^2}}\\, \\Box _3,$ where $\\Box _3$ is the Laplacian on the unit $S^3$ .",
"Noting from (REF ) and (REF ) that $f$ takes the form $f = r^{p+q}\\, e^{{i}(p-q)\\psi /2}\\, Y(\\theta ,\\phi )\\,,$ then (REF ) and (REF ) imply that $\\Box _3 \\Big (e^{{i}(p-q)\\psi /2}\\, Y(\\theta ,\\phi )\\Big ) =-(p+q)(p+q+2)\\, e^{{i}(p-q)\\psi /2}\\, Y(\\theta ,\\phi )\\,.$ From (REF ), $\\Box _3$ is given by ${\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 4} } }\\,\\Box _3 = {\\frac{1}{\\sin \\theta }}\\, {\\frac{\\partial }{\\partial \\theta }}\\Big ( \\sin \\theta \\, {\\frac{\\partial }{\\partial \\theta }}\\Big ) +{\\frac{1}{\\sin ^2\\theta }}\\,\\Big ({\\frac{\\partial }{\\partial \\phi }} - \\cos \\theta \\,{\\frac{\\partial }{\\partial \\psi }}\\Big )^2+{\\frac{\\partial ^2}{\\partial \\psi ^2}}\\,,$ and so if we define $p=\\ell -s\\,,\\qquad q=\\ell + s\\,,$ then (REF ) implies that $Y(\\theta ,\\phi )$ satisfies $-\\Box _{(s)}\\, Y = [\\ell (\\ell +1) -s^2]\\, Y\\,,$ where $\\Box _{(s)}$ is the charged scalar Laplacian on $S^2$ that we defined in equation (REF ).",
"Up to an overall conventional normalisation, we see that $Y(\\theta ,\\phi )$ constructed from (REF ) and (REF ) is nothing but a spin-weighted spherical harmonic $_sY_{\\ell m}$ .",
"Since $p$ and $q$ in (REF ) are non-negative integers, and they are related to $\\ell $ and $s$ by (REF ), it follows that $\\ell \\ge |s|\\,.$ It is easily seen that the number of independent traceless symmetric tensors $T_{a_1\\cdots a_p}{}^{b_1\\cdots b_q}$ in (REF ) is equal to $1+p+q$ , and hence we have constructed the $2\\ell +1$ spin-weighted spherical harmonics $_sY_{\\ell m}$ at level $\\ell $ satisfying $-\\Box _{(s)}\\, _sY_{\\ell m} = [\\ell (\\ell +1) -s^2]\\, _sY_{\\ell m}\\,,\\qquad \\ell \\ge |s|\\,,\\qquad -\\ell \\le m\\le \\ell \\,.$ Note that $s$ , $\\ell $ and $m$ are either all integers, or else all half-integers.",
"With the conventional normalisation, the spin-weighted spherical harmonics satisfy the relations ${\\cal D}_{\\!-} \\,\\, _sY_{\\ell m}\\equiv \\Big ({\\frac{\\partial }{\\partial \\theta }} + m\\csc \\theta + s \\cot \\theta \\Big ) \\, _sY_{\\ell m}&=&\\sqrt{(\\ell +s)(\\ell +1-s)}\\,\\, _{s-1}Y_{\\ell m}\\,,\\nonumber \\\\{\\cal D}_{\\!+}\\, \\,_{s-1}Y_{\\ell m}\\equiv \\Big ({\\frac{\\partial }{\\partial \\theta }} - m\\csc \\theta - (s-1) \\cot \\theta \\Big ) \\,_{s-1}Y_{\\ell m} &=&-\\sqrt{(\\ell +s)(\\ell +1-s)}\\,\\, _sY_{\\ell m}\\,.$"
],
[
"Vector spin-weighted harmonics",
"The spin-weighted vector harmonics are the eigenfunctions of the charged Hodge-de Rham operator (REF ) acting on 1-forms: $\\Delta V = \\tilde{\\lambda }\\, V \\ , \\qquad V=dy^i V_i\\ .$ Generically, these eigenfunctions can be constructed from the scalar spin-weighted harmonics $_sY_{\\ell m}$ (denoted simply as $Y$ below) by writing $V = DY + \\mu \\, {*D}Y\\,,$ where $D=d + {i}\\, s \\cos \\theta \\, d\\phi $ is the gauge-covariant exterior derivative.",
"We shall write the eigenvalues for the scalar spin-weighted harmonics, given by (REF ), simply as $\\lambda $ , so that $\\Delta Y =\\lambda Y\\,,\\qquad \\lambda = \\ell (\\ell +1) -s^2,.$ Noting that $D^2= -{i}s \\Omega _2\\,,$ where $\\Omega _2=\\sin \\theta \\, d\\theta \\wedge d\\phi $ is the volume form on the unit $S^2$ , that $D{*D*} (DY)= D\\Delta Y= \\lambda DY$ , ${*D*}D(DY) = -{i}s \\, {*D}({*\\Omega _2} Y) =-{i}s\\, {*D}Y$ and that ${\\Delta *} = {*\\Delta }$ , we see that $\\Delta V =(\\lambda + {i}\\mu s)DY + (\\mu \\lambda -{i}s){*DY}\\,.$ Thus $V$ is an eigenfunction, satisfying (REF ), if $\\mu =\\pm {i}$ , and so generically we get two distinct vector eigenfunctions $V^\\pm $ with corresponding eigenvalues $\\tilde{\\lambda }_\\pm $ from each scalar eigenfunction $Y$ , where $V^\\pm = DY \\mp {i}{*DY}\\,,\\qquad \\tilde{\\lambda }_\\pm = \\lambda \\pm s\\,.$ In terms of $\\ell $ and $s$ , these eigenvalues are given by $\\tilde{\\lambda }_+ = (\\ell +s)(\\ell +1-s)\\,,\\qquad \\tilde{\\lambda }_- = (\\ell -s)(\\ell +1+s)\\,.$ Note that the vector harmonics $V^\\pm $ obey the complex duality conditions ${* V^\\pm }= \\pm {i}V^\\pm \\,.$ A special case arises if the scalar eigenvalue $\\lambda $ is equal to $s$ or $-s$ .",
"(Since $\\lambda $ is necessarily non-negative, the former can only arise if $s$ is positive, and it implies $\\ell =s$ , while the latter arises if $s$ is negative, and implies $\\ell =-s$ .)",
"Calculating the norm of $V^-$ , we find $\\int {*{\\bar{V}^-}}\\wedge V^- &=&\\int ({*D}\\bar{Y} -{i}{*D}\\bar{Y} )\\wedge (DY +{i}\\, {*D}Y)= 2\\int ({*D}Y\\wedge D\\bar{Y} - {i}\\, DY\\wedge D\\bar{Y})\\nonumber \\\\&=& 2\\int ( (D{*D} Y)\\bar{Y} - {i}\\, (D^2 Y)\\bar{Y})= 2(\\lambda -s) \\int |Y|^2\\,,$ and so $V^-=0$ if $\\lambda =s$ .",
"A similar calculation shows $V^+=0$ if $\\lambda =-s$ .",
"Thus if $\\lambda =s$ then the mode $V^-$ , which from (REF ) would have had eigenvalue $\\tilde{\\lambda }_-=0$ , is absent.",
"Similarly, if $\\lambda =-s$ then $V^+$ , which would likewise have had eigenvalue $\\tilde{\\lambda }_+=0$ , is absent.",
"In fact vector spin-weighted zero modes of $\\Delta $ do arise, but they cannot be constructed from scalar harmonics in the manner described above.",
"If $\\Delta V=0$ then integrating ${*{\\bar{V}}} \\Delta V$ over the sphere implies $\\int (|D{*V}|^2 + |DV|^2)=0$ and hence $V$ is (gauge) closed and co-closed, $DV=0\\,,\\qquad D{*V}=0\\,.$ We can project into the self-dual and anti-self-dual subspaces, and thus seek 1-forms $V$ satisfying ${*V}= \\pm {i}V\\,,\\qquad DV=0\\,.$ Making the ansatz $V= e^{{i}m\\phi }\\, (fd\\theta + g d\\phi )\\,,$ where $f$ and $g$ are functions of $\\theta $ , we find ${*V}=e^{{i}m\\phi }\\, (-f\\sin \\theta \\, d\\phi +g\\, \\csc \\theta \\, d\\theta )$ and hence the duality condition implies $g=\\pm {i}\\, f\\sin \\theta \\,.$ The condition $DV=0$ implies $g^{\\prime }= {i}f(m + s \\cos \\theta )$ , and hence we obtain $f = c (\\sin \\theta )^{-1\\pm s}\\, (\\tan {\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } }\\theta )^{\\pm m}\\,.$ Thus if $s\\ge 1$ we obtain regular self-dual harmonics (${*V^+}= +{i}\\, V^+$ ) given by $V^+= (\\sin {\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } }\\theta )^{s-1+m}\\, (\\cos {\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } }\\theta )^{s-1-m}\\,e^{{i}m\\phi }\\, (d\\theta + {i}\\, \\sin \\theta \\, d\\phi )\\,, \\qquad -(s-1)\\le m \\le (s-1)\\,,$ while if $s\\le -1$ we obtain regular anti-self-dual harmonics (${*V^-}=-{i}\\, V^-$ ) given by $V^-= (\\sin {\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } }\\theta )^{-s-1-m}\\, (\\cos {\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } }\\theta )^{-s-1-m}\\,e^{{i}m\\phi }\\, (d\\theta - {i}\\, \\sin \\theta \\, d\\phi )\\,, \\qquad s+1\\le m \\le -s-1\\,.$ In each case, these charge-$s$ vector harmonics form an $\\ell = |s|-1$ representation of $SU(2)$ , as evidenced by the $(2|s|-1)$ -fold multiplet of $m$ values.",
"In summary, each spin-weighted scalar harmonic $Y$ with eigenvalue $\\lambda =\\ell (\\ell +1)-s^2$ and with with $\\ell \\ge |s|+1$ gives rise to two spin-weighted vector harmonics, namely a self-dual harmonic $V^+$ with eigenvalue $\\tilde{\\lambda }_+=(\\ell +s)(\\ell +1-s)$ for the charge$-s$ Hodge-de Rham operator $\\Delta $ , and an anti-self dual harmonic $V^-$ with eigenvalue $\\tilde{\\lambda }_- =(\\ell -s)(\\ell +1+s)$ .",
"However, the lowest-level spin-weighted scalar harmonic, with $\\ell =|s|$ , gives rise to only one spin-weighted vector harmonic, namely $V^+$ if $s$ is positive, or $V^-$ if $s$ is negative.",
"The “missing” vector harmonic when $\\ell =|s|$ would have been a zero-mode of $\\Delta $ .",
"In its place, a zero-mode harmonic satisfying $\\Delta V=0$ does occur, but it cannot be constructed from the scalar spin-weighted harmonics.",
"It corresponds to $\\ell =|s|-1$ , and therefore has multiplicity $2|s|-1$ ."
],
[
"Spin-${\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } }$ spin-weighted harmonics",
"We may define the spin-weighted spinor harmonics to be charged solutions of the Dirac equation in the monopole background.",
"They may, in general, be constructed from the scalar spin-weighted harmonics, as we now describe.",
"We first note that there exist two charged gauge-covariantly constant spinors on $S^2$ with the monopole background, satisfying $D\\eta =0$ where we now add a spin connection term to the gauge-covariant exterior derivative, $D= \\nabla + {i}\\, s\\, \\cos \\theta \\, d\\phi \\,,\\qquad \\nabla \\equiv d +{\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 4} } } \\omega _{ab}\\sigma ^{ab}\\,$ and, when acting on $\\eta ^\\pm $ , $s=\\pm {\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } }$ .",
"This can be seen from the integrability condition $0=[D_i,D_j]\\eta = {\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 4} } } R_{ijk\\ell }\\sigma ^{k\\ell }\\eta - {i}\\, s\\, \\epsilon _{ij}\\,\\eta = {i}\\, \\epsilon _{ij}\\, ({\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } }\\sigma _3 -s)\\eta \\,$ from which we see that there exist two solutions: $s={\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } }:&& D\\eta _+=0\\,,\\qquad \\sigma _3\\eta _+=\\eta _+\\,,\\nonumber \\\\s=-{\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } }:&& D\\eta _-=0\\,,\\qquad \\sigma _3\\eta _-= -\\eta _-\\,.$ Using the standard basis for the Pauli matrices $\\sigma _i$ , the solutions, we find that the gauge-covariantly constant spinors $\\eta _\\pm $ are given by $\\eta _+=\\begin{pmatrix} 1 \\\\ 0\\end{pmatrix} \\,,\\qquad \\eta _-=\\begin{pmatrix} 0 \\\\ 1\\end{pmatrix}\\,.$ From these spinors, which are normalised so that $\\bar{\\eta }_+\\eta _+=\\bar{\\eta }_-\\eta _-=1$ , we may construct the gauge-covariantly constant vector $U = \\bar{\\eta }_- \\sigma ^i\\eta _+\\, \\partial _i = {\\frac{\\partial }{\\partial \\theta }} +{i}\\csc \\theta \\, {\\frac{\\partial }{\\partial \\phi }}\\,,$ which has charge $s=1$ .",
"Its complex conjugate $\\bar{U}=\\partial /\\partial \\theta -{i}\\,\\csc \\theta \\, \\partial /\\partial \\phi $ has charge $s=-1$ .",
"In fact $U$ is the holormorphic $(1,0)$ -form on the Kähler manifold $S^2$ , satisfying $J_i{}^j\\, U_j= {i}\\, U_i$ , where $J_{ij}=\\epsilon _{ij}$ is the Kähler form.",
"Note that $\\sigma ^i\\eta _+ = U^i\\, \\eta _-\\,,\\qquad \\sigma ^i\\eta _-= \\bar{U}^i\\, \\eta _+\\,.$ The operators $U^i D_i$ and $\\bar{U}^i D_i$ give precisely ${\\cal D}_+$ and ${\\cal D}_-$ , defined in (REF ), when acting on $_sY_{\\ell m}$ and $_{s-1}Y_{\\ell m}$ respectively.",
"It is now clear why ${\\cal D}_+$ and ${\\cal D}_-$ raise and lower the charge of the spin-weighted scalar harmonics by one unit, since $U^i$ and $\\bar{U}^i$ are gauge-covariantly constant vectors carrying $+1$ and $-1$ charge respectively.",
"The solutions of the charged Dirac equation can be expressed in terms of chiral spinors $\\psi _+$ and anti-chiral spinors $\\psi _-$ , satisfying $\\sigma ^i D_i \\psi _+ = {i}\\, \\lambda _+\\, \\psi _-\\,,\\qquad \\sigma ^i D_i \\psi _- = {i}\\, \\lambda _-\\, \\psi _+\\,.$ Since $(\\sigma ^i D_i)^2\\psi = D^iD_i \\psi +(\\tilde{s}\\sigma _3-{\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } })\\psi $ where $\\psi $ is any spinor with charge $\\tilde{s}$ , we have $-D^iD_i\\psi _+ = (\\lambda _+\\lambda _- + \\tilde{s}-{\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } })\\psi _+\\,,\\qquad -D^iD_i\\psi _- = (\\lambda _+\\lambda _- - \\tilde{s}-{\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } })\\psi _-\\,.$ The product $\\lambda _+\\lambda _-$ is therefore uniquely determined, as a function of $\\tilde{s}$ and $\\ell $ , for each spinor eigenfunction of the second-order operator $D^iD_i$ .",
"The values of $\\lambda _+$ and $\\lambda _-$ separately are not determined, but depend upon the choice of relative normalisation for $\\psi _+$ and $\\psi _-$ in (REF ).",
"It is convenient to consider spinor eigenfunctions $\\psi _+$ and $\\psi _-$ with charge $\\tilde{s}=s-{\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } }$ .",
"We may construct these from the scalar spin-weighted harmonics $_sY_{\\ell m}$ discussed earlier by writing $\\psi _- = \\eta _- \\left(_sY_{\\ell m}\\right)\\ , \\qquad \\psi _+ = \\eta _+\\left(_{s-1}Y_{\\ell m}\\right)\\ .$ Note these $\\psi _\\pm $ are denoted by ${\\tilde{\\eta }}^{(\\ell )}_\\pm $ in (REF ).",
"For brevity in notation, however, we shall continue to use the notation $\\psi _\\pm $ instead in this section.",
"Acting on these with the Dirac operator, we find, using (REF ), (REF ) and (REF ), that $\\sigma ^iD_i\\psi _- &=& \\sigma ^i\\eta _-\\, D_i \\, _sY_{\\ell m} =\\eta _+\\, \\bar{U}^i D_i\\, _sY_{\\ell m} \\nonumber \\\\&=& \\eta _+\\,\\Big ({\\frac{\\partial }{\\partial \\theta }} + m\\csc \\theta + s \\cot \\theta \\Big ) \\, _sY_{\\ell m}=\\eta _+\\, \\sqrt{(\\ell +s)(\\ell +1-s)}\\, _{s-1}Y_{\\ell m}\\,,\\nonumber \\\\\\sigma ^iD_i\\psi _+ &=& \\sigma ^i\\eta _+\\, D_i \\, _{s-1}Y_{\\ell m} =\\eta _-\\, U^i D_i\\, _{s-1}Y_{\\ell m} \\\\&=& \\eta _-\\,\\Big ({\\frac{\\partial }{\\partial \\theta }} - m\\csc \\theta - (s-1) \\cot \\theta \\Big )\\, _{s-1}Y_{\\ell m}=-\\eta _-\\, \\sqrt{(\\ell +s)(\\ell +1-s)}\\, _sY_{\\ell m}\\,,\\nonumber $ and hence $\\sigma ^iD_i \\psi _- = \\sqrt{(\\ell +s)(\\ell +1-s)}\\,\\psi _+\\,,\\qquad \\ \\sigma ^iD_i \\psi _+ = - \\sqrt{(\\ell +s)(\\ell +1-s)}\\,\\psi _-\\,.$ It is worth remarking that there is an alternative procedure that in general constructs the charged spin-${\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } }$ harmonics from scalar harmonics, in which only one of the gauge-covariantly constant spinors is required.",
"For example, using only $\\eta _-$ we can construct the negative-chirality spin-${\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } }$ harmonics $\\psi _-$ as in the first equation in (REF ), while for the positive-chirality harmonics we take $\\psi _+^{\\prime } = \\sigma ^i\\eta _- \\, D_i \\,_sY_{\\ell m}\\,.$ A straightforward calculation shows that $\\sigma ^i D_i\\, \\psi _+^{\\prime } = -(\\ell +s)(\\ell +1-s)\\, \\eta _-\\, _sY_{\\ell m}\\,.$ The harmonics $\\psi _+^{\\prime }$ are in general proportional to the harmonics $\\psi _+$ given in (REF ).",
"However, the charge ${\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } }$ harmonic $\\psi _+=\\eta _+$ itself (which is a zero mode of the Dirac operator) cannot be constructed using (REF ), since it would require taking $s=1$ and $\\ell =0$ , for which $_sY_{\\ell m}$ does not exist.",
"It might also seem that charge $-s-{\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } }$ zero modes $\\psi _+^{\\prime }$ would be obtained if $s$ were negative and $\\ell =-s$ .",
"However, calculating the norm of $\\psi _+^{\\prime }$ , we find $\\int _{S^2} |\\psi _+^{\\prime }|^2 \\sqrt{g} d^2x = (\\ell +s)(\\ell +1-s)\\,\\int _{S^2} |_sY_{\\ell m}|^2\\, \\sqrt{g} d^2x\\,,$ and thus $\\psi _+^{\\prime }$ would actually be identically zero if $\\ell =-s$ .",
"These putative zero modes are in fact not obtained by the construction for $\\psi _+$ in (REF ) either, since this would require the use of scalar harmonics with $\\ell $ smaller than the magnitude of their spin weight."
],
[
"Spin-${\\textstyle {\\frac{\\scriptstyle 3}{\\scriptstyle 2} } }$ spin-weighted harmonics",
"The general spin-${\\textstyle {\\frac{\\scriptstyle 3}{\\scriptstyle 2} } }$ harmonics $\\eta _i$ can be decomposed into chiral and antichiral projections $\\eta _i^\\pm $ satisfying $\\sigma ^iD_i\\, \\eta ^+_j = \\lambda _+\\, \\eta _j^-\\,,\\qquad \\sigma ^iD_i\\, \\eta ^-_j = \\lambda _-\\, \\eta _j^+\\,.$ Each chiral projection admits a decomposition of the form $\\eta _i^\\pm = \\sigma _i \\psi ^\\mp + \\eta _{\\lbrace i\\rbrace }^\\pm + \\tilde{\\eta }_i^\\pm \\,,$ where $\\eta _{\\lbrace i\\rbrace }^\\pm $ is longitudinal and gamma traceless, $\\sigma ^i\\, \\eta _{\\lbrace i\\rbrace }^\\pm =0$ , and $\\tilde{\\eta }_i^\\pm $ is transverse and gamma traceless, satisfying $D^i\\tilde{\\eta }_i^\\pm =0$ and $\\sigma ^i\\, \\tilde{\\eta }_i^\\pm =0$ .",
"We can write $\\eta _{\\lbrace i\\rbrace }^\\pm $ in terms of spin-${\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } }$ modes $\\eta ^\\pm $ as $\\eta _{\\lbrace i\\rbrace }^\\pm = 2D_i\\psi ^\\pm - \\sigma _i \\sigma ^j D_j\\psi ^\\pm = (D_i \\mp {i}\\, \\epsilon _i{}^j\\, D_j)\\psi ^\\pm \\,.$ In fact $\\eta _{\\lbrace i\\rbrace }^\\pm $ can alternatively be written in terms of the vector harmonics $V^\\pm $ constructed from scalar harmonics as in (REF ), by taking $\\ \\eta ^\\pm _{\\lbrace i\\rbrace } = V^\\pm _i\\, \\eta ^{\\pm }\\,.$ The gamma-tracelessness of $\\eta ^\\pm _{\\lbrace i\\rbrace }$ follows immediately from the fact that $V^\\pm _i$ and $\\sigma ^i\\eta ^\\pm $ are either both self-dual or both anti-self dual.",
"The charge carried by $\\eta ^\\pm _{\\lbrace i\\rbrace }$ will, of course, be equal to $s\\pm {\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } }$ , where $s$ is the charge of $V_i^\\pm $ .",
"The transverse traceless spin-${\\textstyle {\\frac{\\scriptstyle 3}{\\scriptstyle 2} } }$ harmonics $\\tilde{\\eta }_i$ can be constructed in the same way, and are given by (REF ) except that now, $V^\\pm _i$ are the self-dual vector harmonics (REF ) or the anti-self dual harmonics (REF ) that cannot be constructed from scalar harmonics.",
"Since such $V^\\pm _i$ vectors arise only when $s\\ge 1$ or $s\\le -1$ respectively, the transverse traceless spin-${\\textstyle {\\frac{\\scriptstyle 3}{\\scriptstyle 2} } }$ harmonics $\\tilde{\\eta }^\\pm _i$ arise only for charges $\\tilde{s}\\ge {\\textstyle {\\frac{\\scriptstyle 3}{\\scriptstyle 2} } }$ or $\\tilde{s}\\le -{\\textstyle {\\frac{\\scriptstyle 3}{\\scriptstyle 2} } }$ respectively.",
"All necessary properties of the spin-${\\textstyle {\\frac{\\scriptstyle 3}{\\scriptstyle 2} } }$ harmonics follow from the properties of the lower-spin harmonics that we discussed previously."
],
[
"Spin Projection Operators",
"The well known spin projector operators associated with a second rank symmetric tensor field are given by [16] $&&{{\\cal P}}_{\\mu \\nu ,\\rho \\sigma }^{2}={\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } }(\\theta _{\\mu \\rho }\\theta _{\\nu \\sigma }+\\theta _{\\mu \\sigma }\\theta _{\\nu \\rho }-{\\textstyle {\\frac{\\scriptstyle 2}{\\scriptstyle 3} } }\\theta _{\\mu \\nu }\\theta _{\\rho \\sigma })\\ ,\\nonumber \\\\[0.2cm]&&{{\\cal P}}_{\\mu \\nu ,\\rho \\sigma }^{1}={\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 2} } }(\\theta _{\\mu \\rho }\\omega _{\\nu \\sigma }+\\theta _{\\mu \\sigma }\\omega _{\\nu \\rho }+\\theta _{\\nu \\rho }\\omega _{\\mu \\sigma }+\\theta _{\\nu \\sigma }\\omega _{\\mu \\rho })\\ ,\\nonumber \\\\[0.2cm]&&{{\\cal P}}_{\\mu \\nu ,\\rho \\sigma }^{(0,s)}={\\textstyle {\\frac{\\scriptstyle 1}{\\scriptstyle 3} } }\\theta _{\\mu \\nu }\\theta _{\\rho \\sigma }\\ ,\\nonumber \\\\[0.2cm]&&{{\\cal P}}_{\\mu \\nu ,\\rho \\sigma }^{(0,\\omega )}=\\omega _{\\mu \\nu }\\omega _{\\rho \\sigma }\\ ,$ where $\\theta _{\\mu \\nu }=\\eta _{\\mu \\nu }-\\Box ^{-1}\\partial _{\\mu }\\partial _{\\nu }\\ ,\\qquad \\omega _{\\mu \\nu }=\\Box ^{-1}\\partial _{\\mu }\\partial _{\\nu } \\ .$ Similarly, the spin projector operators associated with vector-spinor field take the form $P_{\\mu \\nu }^{3/2} &=& \\theta _{\\mu \\nu }-\\frac{1}{3} \\theta _\\mu \\theta _\\nu \\ ,\\nonumber \\\\[0.2cm](P_{11}^{1/2})_{\\mu \\nu } &=& \\frac{1}{3} \\theta _\\mu \\theta _\\nu \\ ,\\qquad (P_{12}^{1/2})_{\\mu \\nu } = \\frac{1}{\\sqrt{3}} \\theta _\\mu \\omega _\\nu \\ ,\\nonumber \\\\[0.2cm](P_{21}^{1/2})_{\\mu \\nu } &=& \\frac{1}{\\sqrt{3}} \\omega _\\mu \\theta _\\nu \\ ,\\qquad (P_{22}^{1/2})_{\\mu \\nu } =\\omega _\\mu \\omega _\\nu \\ ,$ where $\\theta _\\mu = \\theta _{\\mu \\nu } \\gamma ^\\nu \\ ,\\qquad \\omega _\\mu = \\omega _{\\mu \\nu }\\gamma ^\\nu \\ .$"
]
] | 1204.1060 |
[
[
"Second Eigenvalue of the Yamabe Operator and Applications"
],
[
"Abstract Let $(M, g)$ be a compact Riemannian manifold of dimension $n \\geq 3$.",
"In this paper, we give various properties of the eigenvalues of the Yamabe operator $L_g$.",
"In particular, we show how the second eigenvalue of $L_g$ is related to the existence of nodal solutions of the equation $L_g u = \\epsilon | u|^{N-2} u$, where $\\epsilon = +1, 0,$ or -1."
],
[
"Introduction",
"This paper is part of a Phd thesis whose purpose is to study the relationships between the eigenvalues of the Yamabe operator, in particular their sign, and analytic, geometrical or topological properties of compact manifolds of dimension $n \\ge 3$ : Let $(M,g)$ be a $n$ -dimensional compact Riemannian manifold ($n \\ge 3$ ).",
"The Yamabe operator or conformal Laplacian operator $L_g$ is defined by $L_g(u):=c_n\\Delta _gu+S_g u,$ where $\\Delta _g$ is the Laplace-Beltrami operator, $c_n=\\frac{4(n-1)}{n-2}$ and $S_g$ the scalar curvature of $g$ .",
"The Yamabe operator $L_g$ has discrete spectrum $\\rm spec(L_g)=\\left\\lbrace \\lambda _1(g), \\lambda _2(g),\\cdots \\right\\rbrace ,$ where the eigenvalues are such that ${\\lambda }_1(g) < {\\lambda }_2(g) \\le {\\lambda }_3(g) \\le \\cdots \\le {\\lambda }_k(g) \\cdots \\rightarrow +\\infty .$ The $i-$ th eigenvalue ${\\lambda }_i(g)$ is characterized by $ \\lambda _i (g) = \\inf _{V\\in Gr_{i}({H_1^2(M)})} \\sup _{v \\in V\\setminus \\lbrace 0\\rbrace }\\frac{\\int _M v L_{g} v \\ dv_g }{\\int _M v^2 \\ dv_g},$ where $Gr_{i}({H_1^2(M)})$ stands for the set of all $i$ -dimensional subspaces of $H_1^2(M).$ Our project is to understand what we can deduce from the sign of $\\lambda _i$ .",
"Now, we summarize what is known about this question and explain our motivations.",
"At first, it is straightforward to see that the sign of $\\lambda _1(g)$ is the same as the sign of the Yamabe constant $\\mu (M,g)$ of $(M,g)$ (and as a consequence is conformally invariant).",
"See Section for more informations.",
"Hence the positivity of $\\lambda _1(g)$ has many consequences usually stated in terms of positivity of the Yamabe constant.",
"For instance, we obtain Proposition 1.1 A compact manifold $M$ of dimension $n \\ge 3$ carries a metric with positive scalar curvature if and only if it carries a metric $g$ such that $\\lambda _1(g) >0$ .",
"We recall that classifying such compact manifolds is a challenging open problem, only solved for $n=3$ using Perelman's techniques.",
"We also mention [5] where M. Dahl and C. Bär deduce many topological properties of compact manifolds from a careful study of the eigenvalues $\\lambda _i$ of the Yamabe operator $L_g$ .",
"The sign of $\\lambda _1$ can also be read in terms of existence or non-existence of positive solutions of the Yamabe equation: $ L_g u = \\epsilon |u|^{N-2} u,$ where $N := \\frac{2n}{n-2}$ and $\\epsilon \\in \\lbrace -1;0;1\\rbrace $ .",
"Inspired by this observation, B. Ammann and E. Humbert [1] enlighted the role of $\\lambda _2$ in the existence of nodal solutions (i.e.",
"having a changing sign) of the Yamabe equation (REF ).",
"See again Section for more explanations.",
"In this paper, we establish various properties of the eigenvalues of the Yamabe operator.",
"First of all, we extend their definition to what we call generalized metrics when possible (see Paragraph ) and prove that their sign is a conformal invariant (see Paragraph REF ).",
"This paper initiates the study of the relationships between these conformal invariants and the topology of the manifold by showing that their negativity is not topologically obstructed (see Paragraph REF ).",
"These investigations will be treated much more deeply in [7].",
"The main point of this article is to complete the results of B. Ammann and E. Humbert [1] and to study how the sign of the second eigenvalue of the Yamabe operator can be related to the existence of nodal solutions of the Yamabe equation (REF ), in particular when the Yamabe constant of $(M,g)$ is negative.",
"Our main result is to prove that under this condition, such a solution always exists with $\\epsilon = sign (\\lambda _2(g))$ .",
"This is the object of Theorem REF .",
"The author would like to thank Emmanuel Humbert for his support and encouragements."
],
[
"Eigenvalues in conformal metrics",
"In the whole paper, we will deal with the behavior of the eigenvalues of the Yamabe operator in a fixed conformal class.",
"It will be usefull to express their definition relatively to a fixed metric.",
"This is the goal of this section."
],
[
"Smooth metrics",
"Let $(M,g)$ be a compact Riemannian manifold of dimension $n\\ge 3$ , we keep the notations of the introduction and for any metric $\\widetilde{g}$ , we will denote by ${\\lambda }_1(\\widetilde{g}) < {\\lambda }_2(\\widetilde{g}) \\le {\\lambda }_3(\\widetilde{g}) \\le \\cdots \\le {\\lambda }_k(\\widetilde{g}) \\cdots \\rightarrow +\\infty ,$ the eigenvalues of the Yamabe operator.",
"We will deal with the case where $\\widetilde{g}$ is conformal to $g$ , i.e.",
"when $\\widetilde{g} =u^{N-2} g,$ where $u$ is a positive function of class $C^\\infty $ .",
"By referring to [1], one sees that the $i-$ th eigenvalue ${\\lambda }_i(\\widetilde{g})$ is given by $\\lambda _i (\\widetilde{g}) = \\inf _{V\\in Gr_{i}({H_1^2(M)})} \\sup _{v \\in V\\setminus \\lbrace 0\\rbrace }\\frac{\\int _M c_n |v|^2 + S_g v^2 \\, dv_g }{\\int _M v^2 \\ u^{N-2} \\ dv_g},$ where $\\widetilde{g} = u^{N-2}g;$ $u\\in C^\\infty (M),$ $u>0$ and $Gr_{i}({H_1^2(M)})$ stands for the set of all $i$ -dimensional subspaces of $H_1^2(M).$"
],
[
"Generalized metrics",
"Reducing to smooth metrics will too restrictive for our investigations.",
"We will need to work with generalized metrics, i.e.",
"metrics of the form $\\widetilde{g} = u^{N-2}g$ with $u\\in L^N(M),$ $u\\ge 0$ and $u\\lnot \\equiv 0$ .",
"The Yamabe operator $L_{\\widetilde{g}}$ has no meaning any more but the definition of $\\lambda _i(\\widetilde{g})$ can anyway be extended to this case by using (REF ) as it was done in [1] when the Yamabe constant was non-negative i.e.",
"when $\\lambda _1(g) \\ge 0$ .",
"When $\\lambda _1(g) < 0$ , the situation is a little bit different: $\\lambda _i(\\widetilde{g})$ defined by (REF ) can be $-\\infty $ as proved by the following proposition.",
"Proposition 2.1 Assume that $\\lambda _1(g)<0$ , then there exists $u\\in L^N(M)$ , $u \\lnot \\equiv 0$ , $u\\ge 0$ such that $\\lambda _1(\\widetilde{g}) = -\\infty $ , where $\\widetilde{g} = u^{N-2}g$ .",
"This proposition will be proved in Paragraph REF .",
"To make sure that $\\lambda _1(\\widetilde{g})$ is finite, one has to assume in addition that $u$ is positive.",
"Proposition 2.2 Let $u$ be a positive function in $L^N(M)$ .",
"Suppose that ${\\lambda }_1(g)<0$ .",
"Then, we have ${\\lambda }_1(\\widetilde{g})> -\\infty .$ The proposition is proved in Paragraph REF .",
"Notation 2.3 The $i^{th}$ eigenvalue of $L_g,$ ${\\lambda }_i(\\widetilde{g}) = {\\lambda }_i(u^{N-2}g)$ will be denoted by ${\\lambda }_i(u)$ when there is no ambiguity about $g$ ."
],
[
"Proof of Proposition ",
"We have ${\\lambda }_1(g)<0$ , this implies that there exists a function $v\\in C^{\\infty }(M)$ such that $\\int _M (L_g v)v\\,dv_g < 0.$ Let $P$ be a point of $M$ .",
"For ${\\varepsilon }>0$ , we define $\\eta _{{\\varepsilon }}$ as follows $\\left\\lbrace \\begin{array}{c}0 \\le \\eta _{{\\varepsilon }} \\le 1,\\\\\\\\\\eta _{{\\varepsilon }} = 0 \\text{ on }B_{{\\varepsilon }}(P),\\\\\\\\\\eta _{{\\varepsilon }} = 1\\text{ on } M\\backslash B_{2{\\varepsilon }}(P),\\\\\\\\\\vert \\nabla \\eta _{{\\varepsilon }} \\vert \\le \\frac{2}{{\\varepsilon }}.\\end{array}\\right.$ where $B_{\\delta }(P)$ stands for the ball of center $P$ and radius $\\delta $ in the metric $g$ .",
"Then one easily cheks $\\lim _{{\\varepsilon }\\rightarrow 0}\\int _M (L_g(\\eta _{{\\varepsilon }}v))(\\eta _{{\\varepsilon }}v)\\,dv_g = \\int _M (L_g v)v\\,dv_g.$ We define $w:= \\eta _{{\\varepsilon }}v$ .",
"Therefore, for a fixed small ${\\varepsilon }> 0$ , we have $\\int _M (L_g w)w\\, dv_g < 0.$ Let $u\\ge 0$ , $u\\lnot \\equiv 0$ of class $C^{\\infty }$ with support in $B_{{\\varepsilon }}(P).$ For $\\alpha >0$ , since $(w+\\alpha )u \\lnot \\equiv 0$ , we can write ${\\lambda }_1(\\widetilde{g}) &=& \\inf _{v^{\\prime }} \\frac{\\int _M (L_g v^{\\prime })v^{\\prime }\\,dv_g}{\\int _M u^{N-2} v^{\\prime 2} \\,dv_g}\\\\&\\le & \\lim _{\\alpha \\rightarrow 0^+} \\frac{\\int _M (L_g (w+\\alpha ))(w+\\alpha )\\,dv_g}{\\int _M u^{N-2} (w+\\alpha )^2 \\,dv_g}.$ Moreover, we have $\\lim _{\\alpha \\rightarrow 0^+}\\int _M (L_g (w+\\alpha ))(w+\\alpha )\\,dv_g=\\int _M (L_g (w))w\\,dv_g <0,$ and $ \\lim _{\\alpha \\rightarrow 0^+} \\int _M u^{N-2} (w+\\alpha )^2 \\,dv_g = 0$ which gives that $\\lim _{\\alpha \\rightarrow 0}\\frac{\\int _M (L_g (w+\\alpha ))(w+\\alpha )\\,dv_g}{\\int _M u^{N-2} (w+\\alpha )^2 \\,dv_g} = -\\infty .$ This ends the proof of Proposition REF ."
],
[
"Proof of Proposition ",
"Let $(v_m)_m$ be a minimizing sequence for ${\\lambda }_1(u),$ i.e.",
"$v_m\\in H_1^2(M)$ such that $\\lim _{m\\longrightarrow \\infty } \\frac{\\int _M c_n\\vert \\nabla v_m\\vert ^2 + S_gv_m^2 \\ dv_g}{\\int _M \\vert u\\vert ^{N-2} v_m^2 \\ dv_g} = \\lim _{m\\longrightarrow \\infty } {\\lambda }_m = {\\lambda }_1(u) < 0.$ Since $(\\vert v_m\\vert )_m$ is also a minimizing sequence for ${\\lambda }_1(u),$ we can assume that $v_m\\ge 0.$ We normalize $v_m$ by $\\int _M \\vert u\\vert ^{N-2} v_m^2 \\ dv_g = 1.$ Here we show that $(v_m)_m$ is bounded in $H_1^2(M).$ Indeed, suppose that $(v_m)_m$ is not bounded in $H_1^2(M)$ and let $v_m^\\prime = \\frac{v_m}{\\parallel v_m\\parallel _{H_1^2(M)}}.$ $(v_m^\\prime )_m$ is bounded in $H_1^2(M),$ and his norm is equal to 1, then there exists $v^\\prime \\in H_1^2(M),$ (after restriction to a subsequence) such that $v_m^\\prime \\rightharpoonup v^\\prime \\text{ in } H_1^2(M),$ $v_m^\\prime \\longrightarrow v^\\prime \\text{ in }L^2(M).$ We have $c_n \\int _M \\left|\\nabla v_m^{\\prime }\\right|^2\\, dv_g + \\int _M S_g v_m^{\\prime 2} \\, dv_g = {\\lambda }_m \\int _M \\left|u\\right|^{N-2}v_m^{\\prime 2}\\, dv_g.$ Moreover, $\\int _M \\left|u\\right|^{N-2}v^{\\prime 2}\\, dv_g \\le \\int _M \\left|u\\right|^{N-2}v_m^{\\prime 2}\\, dv_g \\rightarrow _{m\\longrightarrow \\infty } 0$ since $\\left\\Vert v_m\\right\\Vert _{H_1^2(M)}\\longrightarrow \\infty .$ It follows that $\\int _M \\left|u\\right|^{N-2}v^{\\prime 2}\\, dv_g = 0$ and since $u$ is positive, $v^{\\prime } = 0.$ Now, we write $1 = \\int _M \\left|\\nabla v_m^{\\prime }\\right|^2\\, dv_g + \\underbrace{\\int _M \\left|v_m^{\\prime }\\right|^2 \\, dv_g}_{\\longrightarrow 0}.$ We deduce that $\\lim _{m\\rightarrow \\infty } \\int _M \\left|\\nabla v_m^{\\prime }\\right|^2\\, dv_g=1,$ giving the desired contradiction: $c_n \\underbrace{\\int _M \\left|\\nabla v_m^{\\prime }\\right|^2\\, dv_g}_{\\longrightarrow 1} + \\underbrace{\\int _M S_g v_m^{\\prime 2} \\, dv_g}_{\\longrightarrow 0} = {\\lambda }_m \\int _M \\left|u\\right|^{N-2}v_m^{\\prime 2}\\, dv_g \\le 0.$ This proves that $(v_m)_m$ is bounded in $H_1^2(M),$ and implies that ${\\lambda }_m \\ge C.$ We finally get ${\\lambda }_1(\\widetilde{g}) > -\\infty .$"
],
[
"PDE associated to ${\\lambda }_i$",
"Proposition 2.4 For any non-negative function $u\\in L^N(M),$ such that ${\\lambda }_1(u)>-\\infty $ , there exists functions $v_1>0, v_2, \\ldots , v_k \\in H_1^2(M)$ having a changing sign, such that in the sense of distributions, we have $L_g v_1 = {\\lambda }_1(u) \\vert u\\vert ^{N-2} v_1,$ and $L_g v_k = {\\lambda }_k(u) \\vert u\\vert ^{N-2} v_k.$ Moreover, we can normalize the $v_k$ by $ \\int _M \\vert u\\vert ^{N-2} v_k^2 \\ dv_g = 1 \\text{ and }\\int _M\\vert u\\vert ^{N-2} v_i v_j \\ dv_g =0 \\hspace{5.69046pt}\\forall i \\ne j.$ Proof: Let $(v_m)_m$ be a minimizing sequence for ${\\lambda }_1(u),$ i.e.",
"$v_m\\in H_1^2(M)$ such that $\\lim _{m\\longrightarrow \\infty } \\frac{\\int _M c_n\\vert \\nabla v_m\\vert ^2 + S_gv_m^2 \\ dv_g}{\\int _M \\vert u\\vert ^{N-2} v_m^2 \\ dv_g} = {\\lambda }_1(u).$ According to the Paragraph REF , we get that $(v_m)_m$ is bounded in $H_1^2(M)$ and there exists $v\\ge 0$ in $H_1^2(M)$ such that $v_m$ converges to $v$ weakly in $H_1^2(M)$ and strongly in $L^2(M)$ (after restriction to a subsequence).",
"We now want to prove $\\int _M \\vert u\\vert ^{N-2} v^2 \\ dv_g = \\lim _{m\\longrightarrow \\infty } \\int _M\\vert u\\vert ^{N-2} v_m^2 \\ dv_g = 1.$ If $u$ is smooth, this relation is clear.",
"So let us assume that $u\\in Ł^N(M),$ let $A$ be a large real number and set $u_A = \\inf \\left\\lbrace u,A\\right\\rbrace .$ By Hölder inequality, we write $\\left| \\int _M u^{N-2} \\left(v_m^2 - v^2\\right) \\,dv_g \\right| &=& \\left| \\int _M\\left(u^{N-2} -u_A^{N-2} +u_A^{N-2}\\right)\\left(v_m^2 - v^2\\right) \\,dv_g \\right|\\\\& \\le & \\left( \\int _M u_A^{N-2} |v_m^2 - v^2| \\,dv_g + \\int _M (u^{N-2}- u_A^{N-2}) (|v_m|+|v|)^2 \\,dv_g\\right) \\\\& \\le & A^{N-2} \\int _M |v_m^2 - v^2| \\,dv_g\\\\&&+ {\\left( \\int _M (u^{N-2}-u_A^{N-2})^\\frac{N}{N-2} \\,dv_g\\right)}^{\\frac{N-2}{N}} {\\left( \\int _M (|v_m|+|v|)^N \\,dv_g\\right)}^{\\frac{2}{N}}.$ $(v_m)_m$ is bounded in $H_1^2(M),$ it is bounded in $L^N(M).$ Hence there exists a constant $C$ such that $\\int _M \\left(|v_m|+|v|\\right)^N \\,dv_g \\le C.$ The convergence in $L^2(M)$ gives $\\lim _{m\\longrightarrow \\infty }\\int _M |v_m^2 - v^2| \\ dv_g = 0.$ By dominated convergence theorem, we have $\\lim _{A\\longrightarrow \\infty }\\int _M \\left(u^{N-2}-u_A^{N-2}\\right)^\\frac{N}{N-2}\\ dv_g= 0.$ Hence, we get (REF ).",
"Since $\\lim _m \\int _M \\langle \\nabla v_m, \\nabla {\\varphi }\\rangle \\ dv_g = \\int _M\\langle \\nabla v, \\nabla {\\varphi }\\rangle \\ dv_g,$ $\\lim _m \\int _M S_g v_m {\\varphi }\\ dv_g = \\int _M S_g v {\\varphi }\\ dv_g$ and $\\lim _m \\int _M \\vert u\\vert ^{N-2} v_m {\\varphi }\\ dv_g = \\int _M \\vert u\\vert ^{N-2} v{\\varphi }\\ dv_g,$ (by strong convergence in $L^2(M)$ ), we obtain that in the sense of distributions $v$ verifies $L_gv = {\\lambda }_1(u) \\vert u\\vert ^{N-2} v.$ Now we define ${\\lambda }_k^\\prime (u) = \\inf _{{\\scriptstyle v_k; \\vert u\\vert ^{\\frac{N-2}{2}} v_k\\lnot \\equiv 0\\atop \\scriptstyle \\int _M \\vert u\\vert ^{N-2} v_i v_k \\ dv_g = 0 \\forall i<k}} \\frac{\\int _M c_n\\vert \\nabla v_k\\vert ^2 + S_g v_k^2 \\ dv_g}{\\int _M \\vert u\\vert ^{N-2} \\vert v_k\\vert ^2 \\ dv_g}.$ we remark that ${\\lambda }_k^\\prime (u) = {\\lambda }_k(u)$ and $v_k$ is constructed by induction using the same method.",
"This ends the proof of Proposition REF .",
"$\\square $"
],
[
"The sign of ${\\lambda }_i$ is conformally invariant",
"Proposition 3.1 The sign of ${\\lambda }_i$ is independent of the metric selected in the conformal class.",
"More precisely, for any conformal metric $\\widetilde{g} = u^{N-2}g,$ where $u$ is a non-negative function in $L^N(M)$ , ${\\lambda }_i(u)$ and ${\\lambda }_i(1)$ have same sign.",
"Proof: We assume for example that ${\\lambda }_i(u) = 0$ and ${\\lambda }_i(1) > 0,$ we know that ${\\lambda }_i (u) = \\inf _{u_1,\\ldots , u_i}\\sup _{{\\lambda }_1,\\ldots ,{\\lambda }_i}\\frac{\\int _ML_g({\\lambda }_1 u_1+ \\cdots +{\\lambda }_i u_i)({\\lambda }_1 u_1+ \\cdots +{\\lambda }_i u_i) \\ dv_g}{\\int _M ({\\lambda }_1u_1+ \\cdots +{\\lambda }_i u_i)^2 u^{N-2} \\ dv_g},$ and ${\\lambda }_i (1) = \\inf _{u_1,\\ldots , u_i}\\sup _{{\\lambda }_1,\\ldots ,{\\lambda }_i}\\frac{\\int _ML_g({\\lambda }_1 u_1+ \\cdots +{\\lambda }_i u_i)({\\lambda }_1 u_1+ \\cdots +{\\lambda }_i u_i) \\ dv_g}{\\int _M ({\\lambda }_1u_1+ \\cdots +{\\lambda }_i u_i)^2 \\ dv_g}.$ Suppose that ${\\lambda }_i(u)$ is attained by $v_1,\\ldots ,v_i$ .",
"Since denominators of this expressions are positive, then $\\sup _{{\\lambda }_1,\\cdots ,{\\lambda }_i}\\int _M L_g({\\lambda }_1 v_1+ \\cdots +{\\lambda }_i v_i)({\\lambda }_1 v_1+\\cdots +{\\lambda }_i v_i) \\ dv_g = 0.$ So ${\\lambda }_i(1) \\le \\sup _{{\\lambda }_1,\\ldots ,{\\lambda }_i}\\frac{\\int _M L_g({\\lambda }_1 v_1+ \\cdots +{\\lambda }_iv_i)({\\lambda }_1 v_1+ \\cdots +{\\lambda }_i v_i) \\ dv_g}{\\int _M ({\\lambda }_1 v_1+ \\cdots +{\\lambda }_i v_i)^2u^{N-2} \\ dv_g} = 0,$ which gives a contradiction.",
"The remaining cases are treated similarly."
],
[
"The negativity of ${\\lambda }_k$ is not topologically obstructed",
"In this paragraph, we will see that on each manifold, there exists a metric which has a negative $k^{th}$ -eigenvalue.",
"Proposition 3.2 On any compact Riemannian manifold $M,$ and for all $k\\ge 1$ there exists a metric $g$ such that ${\\lambda }_k(g)<0.$ Proof: Let $M$ be a compact Riemannian manifold of dimension $n,$ and we take $k$ spheres of dimension $n=dim (M).$ We equip each sphere $\\mathbb {S}^n$ by the same metric $g,$ such that $\\mu (g)<0.$ We can do this by referring to [3] (Theorem [1] page 38).",
"Let $P\\in M,$ since $\\mu (g)<0,$ for all ${\\varepsilon }, \\delta >0$ we can find a function $u$ supported in $\\mathbb {S}^n\\backslash {B_{{\\varepsilon }}(P)}$ such that $\\frac{\\int _M c_n\\vert \\nabla u\\vert ^2 + S_g u^2 \\ dv_g}{\\int _M u^2 \\ dv_g}<-\\delta .$ Indeed, let $\\eta _{\\varepsilon }$ be a smooth cut-off function such that $0\\le \\eta _{\\varepsilon }\\le 1,$ $\\eta _{\\varepsilon }(B_{{\\varepsilon }}(P))=0,$ $\\eta _{\\varepsilon }(\\mathbb {S}^n\\backslash {B_{2{\\varepsilon }}(P)})=1,$ $\\vert \\nabla \\eta _{\\varepsilon }\\vert \\le \\frac{2}{{\\varepsilon }}$ and a function $v$ satisfying $I_g(v) = \\frac{\\int _Mc_n\\vert \\nabla v\\vert ^2 + S_g v^2 \\ dv_g}{\\int _M v^2 \\ dv_g}< -2\\delta .$ Note that the existence of $v$ is given by the fact that $\\mu (g)<0.$ The desired function $u$ will be given by $\\eta _{\\varepsilon }v,$ where ${\\varepsilon }>0$ is sufficiently small.",
"Indeed, it suffices to notice that, as easily checked, $\\lim _{{\\varepsilon }\\longrightarrow 0}I_g(\\eta _{\\varepsilon }v) = I_g(v).$ Let $P_1,\\cdots ,P_k$ be points of $M.$ We consider the following connected sum $M^\\prime = M \\# (\\mathbb {S}^n)_1 \\# \\ldots \\# (\\mathbb {S}^n)_k,$ where the $(\\mathbb {S}^n)_i$ are attached at $P$ on the spheres $\\mathbb {S}^n$ and at $P_i$ on $M$ so that the handles are attached in $B_{{\\varepsilon }}(P)$ and $B_{{\\varepsilon }}(P_i)$ .",
"Note that $M^\\prime $ is diffeomorphic to $M.$ Moreover, the above construction allows to see $(\\mathbb {S}^n)_i\\backslash B_{{\\varepsilon }}(P_i)$ as a part of $M^\\prime .$ We take on $M^\\prime $ any metric $h$ satisfying $h\\vert _{(\\mathbb {S}^n)_i\\backslash B_{{\\varepsilon }}(P_i)}= g.$ On $M^\\prime ,$ we define the following function $u_i=\\left|\\;\\begin{matrix}u\\hfill & \\hbox{on } (\\mathbb {S}^n)_i\\backslash B_{{\\varepsilon }}(P_i)\\\\\\\\0\\hfill & \\hbox{otherwise} .\\end{matrix}\\right.$ Since the $u_i$ have disjoint supports, we get ${\\lambda }_k(1)&\\le & \\sup _{{\\lambda }_1,\\ldots ,{\\lambda }_k}\\frac{\\int _{M^\\prime } L_g({\\lambda }_1 u_1+\\cdots +{\\lambda }_ku_k)({\\lambda }_1 u_1+ \\cdots +{\\lambda }_k u_k) \\ dv_h}{\\int _{M^\\prime } ({\\lambda }_1 u_1+ \\cdots +{\\lambda }_ku_k)^2 \\ dv_h}\\\\\\\\&\\le & \\sup _{{\\lambda }_1,\\ldots ,{\\lambda }_k}\\frac{({\\lambda }_1^2+ \\ldots + {\\lambda }_k^2)\\int _M (L_gu)u \\ dv_g}{({\\lambda }_1^2+ \\ldots + {\\lambda }_k^2)\\int _M u^2 \\ dv_g}\\\\\\\\&\\le & \\frac{\\int _M (L_gu)u \\ dv_g}{\\int _M u^2 \\ dv_g}< -\\delta .$"
],
[
"Nodal solutions of the Yamabe equations",
"A famous problem in Riemannian geometry is the Yamabe problem, solved between 1960 and 1984 by Yamabe, Trüdinger, Aubin and Schoen, [15], [13], [2], [12].",
"The reader can also refer to [11], [8], [3].",
"The Yamabe problem consists in finding a metric $\\widetilde{g}$ conformal to $g$ such that the scalar curvature $S_{\\widetilde{g}}$ of $\\widetilde{g}$ is constant.",
"Solving this problem is equivalent to finding a positive smooth function and a number $C_0 \\in \\mathbb {R}$ such that $L_g(u) = C_0 \\vert u\\vert ^{N-2} u,$ where $N = \\frac{2n}{n-2}.$ In order to obtain solutions of the Yamabe equation we define the Yamabe invariant by $\\mu (M,g):= \\inf _ {u\\ne 0, u\\in C^\\infty (M)}Y(u),$ where $Y(u) = \\frac{\\int _M c_n\\vert \\nabla u\\vert ^2+S_g u^2 \\ dv_g}{\\left(\\int _M \\vert u\\vert ^N \\ dv_g\\right)^\\frac{2}{N}}.$ The works of Yamabe, Trüdinger, Aubin and Schoen provides a positive smooth minimizer $u$ of $Y$ , satisfying, if normalized by $\\Vert u\\Vert _{L^N(M)} = 1,$ $L_gu = \\mu (M,g) \\vert u\\vert ^{N-2}u.$ The metric $\\widetilde{g} = u^{N-2} g$ is the desired metric: its scalar curvature is constant equal to $\\mu (M,g)$ .",
"If we set $u^{\\prime } = \\mu (M,g)^{\\frac{n-2}{4}} u$ , we obtain a positive solution of $L_gu^{\\prime } = {\\varepsilon }\\vert u^{\\prime }\\vert ^{N-2}u^{\\prime }$ where ${\\varepsilon }= \\hbox{ sign } (\\mu (M,g))= \\hbox{ sign}(\\lambda _1(g))$ .",
"Now, if $\\mu (M,g)\\ge 0,$ it is easy to chek that $\\mu (M,g) = \\inf _{\\widetilde{g}\\in \\left[ g\\right]} {\\lambda }_1(\\widetilde{g}) vol(M,\\widetilde{g})^\\frac{2}{n},$ where $\\left[ g\\right]$ is the conformal class of $g$ and ${\\lambda }_1$ is the first eigenvalue of the Yamabe operator $L_g$ .",
"Inspired by this approach, in their paper [1], B. Ammann et E. Humbert introduced the second Yamabe invariant defined by $\\mu _2(M,g)&=& \\inf _{\\widetilde{g}} {\\lambda }_2(\\widetilde{g}) vol(M,\\widetilde{g})^\\frac{2}{n}\\\\&=& \\inf _u {\\lambda }_2(u^{N-2}g) \\left(\\int _M u^N \\ dv_g\\right)^\\frac{2}{n}.$ They studied this invariant in the case where $\\mu (M,g)\\ge 0$ , and they proved that $\\mu _2$ is attained by a generalized metric, (i.e.",
"a metric of the form $u^{N-2}g$ where $u\\in L^N(M)$ , $u\\ge 0$ which may vanish), in the following two cases $\\bullet $ $\\mu (M,g)>0$ , $(M,g)$ is not locally conformally flat and $n\\ge 11$ .",
"$\\bullet $ $\\mu (M,g)=0$ , $(M,g)$ is not locally conformally flat and $n\\ge 9$ .",
"In this context, they proved that $u$ is the absolute value of a changing sign function $w$ of class $C^{3,\\alpha }(M)$ , which verifies the following equation $L_g w = \\mu _2(M,g)\\vert w\\vert ^{N-2} w.$ Many works are devoted to the study of this kind of solutions, for example [1], [6], [9], [10], [14].",
"See also [4] for an analogue study for the Paneitz-Branson operator.",
"Setting again $w^{\\prime } = \\mu _2(M,g)^{\\frac{n-2}{4}} w$ , we obtain a solution of $L_g w^{\\prime }= {\\varepsilon }\\vert w^{\\prime }\\vert ^{N-2} w^{\\prime }$ with $\\epsilon =1= \\hbox{ sign }(\\mu _2(M,g))= \\hbox{ sign }(\\lambda _2(g)).$ The goal of this section is to study if this result extends to metrics where the sign of ${\\lambda }_2(g)$ is arbitrary.",
"The answer is yes when ${\\lambda }_2 < 0$ without any other condition, we obtain this result by a method different than the one of [1].",
"Notice that this situation occurs for a large number of metrics (see Proposition REF ).",
"When ${\\lambda }_2\\ge 0$ , we show that the methods in [1] can be extended to the case where $\\mu (M,g)<0$ .",
"Namely, the main result of this paper is: Theorem 4.1 Let $(M,g)$ be a compact Riemannian manifold of dimension $n\\ge 3$ whose Yamabe invariant $\\mu (M,g)$ is strictly negative, we denote by ${\\lambda }_2$ the second eigenvalue of $L_g.$ Then, if ${\\lambda }_2\\le 0$ or if ${\\lambda }_2>0$ , $(M,g)$ not locally conformally flat and $n\\ge 6$ : There exists a function $w$ changing sign, solution of the equation $L_g w = {\\varepsilon }\\vert w \\vert ^{N-2}w,$ where ${\\varepsilon }=+1$ if ${\\lambda }_2>0,$ ${\\varepsilon }= -1$ if ${\\lambda }_2< 0$ and ${\\varepsilon }=0$ if ${\\lambda }_2 = 0.$ Moreover, $w \\in C^{3,\\alpha }(M)$ , for all $\\alpha < N-2.$"
],
[
"The case ${\\lambda }_2 = 0$",
"This case is obvious: indeed, Proposition REF provides the existence of a nodal solution $v$ of $L_g v = 0= {\\varepsilon }\\left|v\\right|^{N-2}v$ where ${\\varepsilon }= 0 = \\hbox{ sign }(\\lambda _2(g))$ ."
],
[
"The case ${\\lambda }_2 > 0$",
"As in [1], we introduce the second Yamabe invariant given by $\\mu _2(M,g)&=& \\inf _{\\widetilde{g}}{\\lambda }_2(\\widetilde{g}) vol(M,\\widetilde{g})^\\frac{2}{n}\\\\&=& \\inf _{u>0} {\\lambda }_2(u)\\left(\\int _M u^N \\ dv_g\\right)^\\frac{2}{n}.$ By Proposition REF below, the problem reduces to finding a minimizer of $\\mu _2(M,g)$ .",
"The case where $\\mu (M,g) \\ge 0$ have been treated in [1].",
"We will then focus on the case where $\\mu (M,g) < 0$ (i.e.",
"${\\lambda }_1(g) <0$ ).",
"We will see that the method of Ammann and Humbert remains valid in this case and the following three propositions answer our questions.",
"Proposition 4.2 Let $(M,g)$ be a compact Riemannian manifold of dimension $n\\ge 3$ , such that ${\\lambda }_2>0$ .",
"If $\\mu _2(M,g)<\\mu (\\mathbb {S}^n),$ with $\\mu (\\mathbb {S}^n)=n(n-1)\\omega _n^{\\frac{2}{n}}$ , where $\\omega _n$ stands for the volume of the standard sphere $\\mathbb {S}^n$ , then the second Yamabe invariant is attained by a non-negative function $u\\in L^N(M)$ that we normalize by $\\int _M u^N \\ dv_g = 1$ .",
"There exists a function $w$ having a changing sign which verifies in the sense of distributions the following equation $L_gw = \\mu _2(M,g) \\vert u\\vert ^{N-2} w.$ The functions $u$ and $w$ will be normalized by $\\int _M u^N \\ dv_g = 1, \\hspace{5.69046pt}\\int _M u^{N-2} \\ w^2 \\ dv_g = 1.$ Proposition 4.3 The two functions $u$ and $w$ given by Proposition REF satisfy $u = \\vert w\\vert .$ Finally, we give a condition under which assumption (REF ) is satisfied: Proposition 4.4 Let $(M,g)$ be a compact Riemannian manifold of dimension $n\\ge 6,$ suppose that $M$ is not locally conformally flat and his Yamabe invariant $\\mu (M,g)<0,$ then $\\mu _2(M,g)< \\mu (\\mathbb {S}^n).$"
],
[
"Proof of Proposition ",
"The case where $\\mu (M,g) \\ge 0$ is done in [1], hence we consider here the case where $\\mu (M,g) < 0$ .",
"By the solution of the Yamabe problem, we can assume without loss of generality, that $S_g = -1$ .",
"Let $(u_m)_m$ be a minimizing sequence for $\\mu _2(M,g),$ i.e., $u_m$ is positive, smooth and $\\lim _{m\\longrightarrow \\infty }\\lambda _2(u_m)\\left(\\int _Mu_m^Ndv_g\\right)^\\frac{2}{n} =\\mu _2(M,g).$ The sequence $(u_m)_m$ will be choosen such that $\\int _M u_m^N \\ dv_g=1$ , hence $\\mu _2(M,g)=\\lim _{m\\longrightarrow \\infty }\\lambda _2(u_m).$ For each $u_m,$ Proposition REF provides the existence of a function $w_m \\in H_1^2(M)$ such that $L_g w_m = {\\lambda }_2(u_m)\\vert u_m\\vert ^{N-2}w_m.$ Moreover, the sequence $(w_m)_m$ can be normalized by $\\int _M\\vert u_m\\vert ^{N-2}w_m^2 \\ dv_g=1.$ Since $\\int _Mu_m^Ndv_g=1,$ $(u_m)_m$ is bounded in $L^N(M)$ which is a reflexive space, there exists $u\\in L^N(M)$ such that $u_m$ converges weakly to $u$ in $L^N(M)$ , we have $u_m\\rightharpoonup u \\text{ in }L^N(M).$ $\\bullet $ The sequence $(w_m)_m$ is bounded in $H_1^2(M).$ We proceed by contradiction and assume that $\\Vert w_m\\Vert _{H_1^2(M)}\\longrightarrow \\infty .$ Let $w^\\prime _m=\\frac{w_m}{\\Vert w_m\\Vert _{H_1^2(M)}}.$ $\\Vert w^\\prime _m\\Vert _{H_1^2(M)}=1,$ hence $(w^\\prime _m)_m$ is bounded in $H_1^2(M).$ Since $H_1^2(M)$ is a reflexive space, this implies using Kondrakov and Banach-Alaoglu theorems, that there exists a subsequence $(w_m^\\prime )_m$ and $w^\\prime \\in H_1^2(M)$ such that $w^\\prime _m\\rightharpoonup w^\\prime \\text{ in } H_1^2(M),$ and $w^\\prime _m\\longrightarrow w^\\prime \\text{ in } L^2(M).$ Equation (REF ) is linear, so $w^\\prime _m$ satisfies $L_gw_m^{\\prime } = \\lambda _2(u_m)\\left|u_m \\right|^{N-2}w^\\prime _m.$ Hence for all $\\varphi \\in C^\\infty (M),$ we have: $c_n \\int _M \\left\\langle \\nabla w^\\prime _m,\\nabla \\varphi \\right\\rangle dv_g+\\int _MS_gw^\\prime _m\\varphi dv_g=\\int _M\\lambda _2(u_m)\\left|u_m\\right|^{N-2}w^\\prime _m \\varphi dv_g.$ Since $w^\\prime _m \\rightharpoonup w^\\prime $ in $H_1^2(M)$ and $w \\longmapsto \\left\\langle \\nabla w ,\\nabla \\varphi \\right\\rangle $ is a linear form on $H_1^2(M),$ then $c_n\\int _M \\left\\langle \\nabla w_m^\\prime ,\\nabla \\varphi \\right\\rangle dv_g\\longrightarrow c_n \\int _M\\left\\langle \\nabla w^{\\prime },\\nabla \\varphi \\right\\rangle dv_g.$ The sequence $w_m^\\prime $ converges strongly to $w^\\prime $ in $L^2(M)$ .",
"This gives that $\\int _M S_gw_m^\\prime \\varphi dv_g\\longrightarrow \\int _M S_g w^\\prime \\varphi dv_g.$ Using Hölder inequality, we obtain that $\\int _M\\left|u_m\\right|^{N-2}w^\\prime _m\\varphi \\ dv_g\\longrightarrow 0.$ Indeed, $\\left|\\int _M\\left|u_m\\right|^{N-2}w^\\prime _m\\varphi dv_g\\right|&\\le &\\left\\Vert \\varphi \\right\\Vert _{\\infty }\\int _M\\left|u_m\\right|^\\frac{N-2}{2}\\left|w_m^\\prime \\right|\\left|u_m\\right|^\\frac{N-2}{2}\\ dv_g\\\\&\\le &\\left\\Vert \\varphi \\right\\Vert _{\\infty }\\left(\\int _M\\left|u_m\\right|^{N-2}{w^\\prime _m}^2 \\ dv_g\\right)^\\frac{1}{2}\\left(\\int _M\\left|u_m\\right|^{N-2} \\ dv_g\\right)^\\frac{1}{2}\\\\&\\le &\\left\\Vert \\varphi \\right\\Vert _{\\infty }\\frac{\\left(\\int _M\\left|u_m\\right|^{N-2}{w_m}^2 \\ dv_g\\right)^\\frac{1}{2}}{\\left\\Vert w_m\\right\\Vert _{H_1^2(M)}}\\left(\\int _M\\left|u_m\\right|^{N}\\ dv_g\\right)^\\frac{N-2}{2N}\\left(\\rm vol(M,g)^{1-\\frac{N-2}{N}}\\right)^\\frac{1}{2}\\\\&\\le &\\left\\Vert \\varphi \\right\\Vert _{\\infty } \\frac{1}{\\left\\Vert w_m\\right\\Vert _{H_1^2(M)}}\\left(\\rm vol(M,g)\\right)^\\frac{1}{N} \\longrightarrow _{m\\longrightarrow +\\infty }0.$ Then $c_n \\int _M\\left\\langle \\nabla w^{\\prime },\\nabla \\varphi \\right\\rangle \\ dv_g+\\int _MS_g w^\\prime \\varphi \\ dv_g=0,$ which means that in the sense of distributions, we have $L_g w^\\prime =0.$ Since ${\\lambda }_1(1)<0$ and ${\\lambda }_2(1)$ is positive, $0\\notin Sp(L_g)$ .",
"It follows that $w^\\prime =0.$ Now, we also have $\\int _M c_n\\left|\\nabla w^\\prime _m\\right|^2 \\ dv_g+\\int _M S_g{w^\\prime _m}^2 \\ dv_g=\\lambda _2(u_m)\\int _M\\left|u_m\\right|^{N-2}{w_m^\\prime }^2 \\ dv_g,$ with $\\lambda _2(u_m)\\int _M\\left|u_m\\right|^{N-2}{w_m^\\prime }^2 \\ dv_g=\\frac{\\lambda _2(u_m)}{\\left\\Vert w_m\\right\\Vert ^2_{H_1^2(M)}}\\longrightarrow 0$ and $\\int _M S_g{w^\\prime _m}^2 \\ dv_g\\longrightarrow \\int _M S_g{w^\\prime }^2 \\ dv_g = 0.$ Hence $\\int _M\\left|\\nabla w^\\prime _m\\right|^2 \\ dv_g\\longrightarrow 0.$ Finally, we get that $\\left\\Vert w^\\prime _m\\right\\Vert ^2_{H_1^2(M)}=1=\\int _M\\left|\\nabla w_m^\\prime \\right|^2 \\ dv_g+\\int _M{w^\\prime _m}^2 \\ dv_g\\longrightarrow 0,$ which gives the desired contradiction.",
"We obtain that $(w_m)_m$ is a bounded sequence in $H_1^2(M).$ Then there exists $w\\in H_1^2(M)$ such that: $w_m\\rightharpoonup w \\text{ in } H_1^2(M),$ $w_m\\longrightarrow w \\text{ in }L^2(M).$ It follows that in the sense of distributions, we have $L_gw=\\mu _2(M,g)\\left|u\\right|^{N-2}w.$ It remains to show that $w$ changes sign and is different from zero.",
"$\\bullet $ Suppose that $w$ does not change sign.",
"Without loss of generality, we can assume that $w\\ge 0.$ In the sense of distributions, we have $c_n\\Delta _gw+S_gw=\\mu _2(M,g)\\left|u\\right|^{N-2}w.$ It was already mentioned at the beginning of this section that we can assume that $S_g<0,$ because $\\mu (M,g)<0.$ Integrating (REF ) over $M$ , we get: $\\underbrace{\\int _Mc_n\\Delta _gw \\ dv_g}_{=0}+\\underbrace{\\int _MS_gw \\ dv_g}_{<0}=\\underbrace{\\mu _2(M,g)\\int _M\\left|u\\right|^{N-2}w \\ dv_g}_{\\ge 0}.$ This gives a contradiction unless $w \\equiv 0$ which is prohibited by what follows.",
"$\\bullet $ Assume that $w=0.$ By referring to [8] and [2] we have the following theorem: If $(M,g)$ is a Riemannian manifold of dimension $n\\ge 3,$ for all $\\epsilon >0,$ there exists $B_{\\epsilon }$ such that for any $u\\in H_1^2(M),$ we have $\\left(\\int _M\\left|u\\right|^N \\ dv_g \\right)^\\frac{2}{N}\\le (\\mu (\\mathbb {S}^n)^{-1}+\\epsilon )\\left(\\int _M c_n\\left|\\nabla u\\right|^2 \\ dv_g+B_\\epsilon \\int _Mu^2 \\ dv_g\\right).$ We obtain $c_n\\int _M\\left|\\nabla w_m\\right|^2 \\ dv_g+S_g\\int _Mw_m^2 \\ dv_g&=&\\mu _2(M,g)\\int _M\\left|u_m\\right|^{N-2}w_m^2 \\ dv_g\\\\&\\le &\\mu _2(M,g)\\underbrace{\\left(\\int _M\\left|u_m\\right|^{N} \\ dv_g\\right)^\\frac{N-2}{N}}_{=1}\\left(\\int _M\\left|w_m\\right|^{N} \\ dv_g\\right)^\\frac{2}{N}\\\\&\\le &\\underbrace{\\mu _2(M,g)(\\mu (\\mathbb {S}^n)^{-1}+\\epsilon )}_{<1(\\text{if}\\hspace{2.84544pt}{\\varepsilon }\\hspace{2.84544pt}\\text{is small enough})}\\left(\\int _Mc_n\\left|\\nabla w_m\\right|^2 \\ dv_g+B_\\epsilon \\int _Mw_m^2 \\ dv_g\\right).$ Hence $c_n\\underbrace{\\left[1-\\mu _2(M,g)(\\mu (\\mathbb {S}^n)^{-1}+\\epsilon )\\right]}_{>0}\\int _M\\left|\\nabla w_m\\right|^2 \\ dv_g\\le c\\underbrace{\\int _Mw_m^2 \\ dv_g}_{\\longrightarrow 0},$ then $\\int _M\\left|\\nabla w_m\\right|^2 \\ dv_g\\longrightarrow 0,$ so $\\left\\Vert w_m\\right\\Vert _{H_1^2(M)}\\longrightarrow 0.$ This shows that $w_m\\longrightarrow 0$ in $H_1^2(M).$ We finally get that $1=\\int _M\\left|u_m\\right|^{N-2}w_m^2 \\ dv_g\\le \\left(\\int _M\\left|u_m\\right|^N \\ dv_g\\right)^\\frac{N-2}{2}\\underbrace{\\int _Mw_m^N \\ dv_g}_{\\longrightarrow 0}.$ This gives a contradiction, then $w\\ne 0.$"
],
[
"Proof of Proposition ",
"Since ${\\lambda }_2(g)>0$ , then $\\mu _2(M,g) = \\inf _{u>0} {\\lambda }_2(u)\\left(\\int _M u^N \\ dv_g\\right)^\\frac{2}{n} = \\inf _{u\\ge 0} {\\lambda }_2(u)\\left(\\int _M u^N \\ dv_g\\right)^\\frac{2}{n}.$ We mimic the proof of Lemma 3.3 in [1] by taking $w_1 = w_+ = \\sup \\left\\lbrace 0,w\\right\\rbrace $ and $w_2 = w_- = \\sup \\left\\lbrace 0, -w\\right\\rbrace $ .",
"This gives that $u = aw_+ + b w_-,$ where $a,b>0$ .",
"By Lemma 3.1, $w\\in C^{2,\\alpha }$ , $u\\in C^{0,\\alpha }$ and Step 4 of the proof of Theorem 3.4 in [1] then shows that $u = \\vert w\\vert .$ Since $w$ is in $H_1^2(M),$ Lemma 3.1 of [1] says that $w\\in L^{N + {\\varepsilon }}(M),$ because $w$ satisfies the equation $L_gw =\\mu _2\\vert w\\vert ^{N-2} w,$ and standard bootstrap arguments gives that $w\\in C^{3,\\alpha }(M)$ for all $\\alpha <N-2$ ."
],
[
"Proof of Proposition ",
"In this paragraph, we will see that if $M$ is not locally conformally flat of dimension $n\\ge 6,$ then we obtain that $\\mu _2(M,g)<\\mu (\\mathbb {S}^n).$ We still consider the case where $\\mu (M,g)<0$ .",
"Then there exists a positive function $v$ solution of the Yamabe equation $L_gv=\\mu (M,g)v^{N-1}.$ Let $x_0$ be a point of $M$ at which the Weyl tensor is not zero (such a point exists because the manifold is not locally conformally flat and $n\\ge 4$ ) and $(x_1,\\ldots , x_n)$ be a system of normal coordinates at $x_0.$ For $x\\in M,$ denote by $r=d(x,x_0)$ the distance to the point $x_0.$ If $\\delta $ is a small fixed number, let $\\eta $ be a cut-off function of class $C^\\infty $ defined by $\\left\\lbrace \\begin{array}{c}0 \\le \\eta \\le 1,\\\\\\\\\\eta =1 \\text{ on }B_{\\delta }(x_0),\\\\\\\\\\eta =0 \\text{ on } M\\backslash B_{2\\delta }(x_0),\\\\\\\\\\vert \\nabla \\eta \\vert \\le \\frac{2}{\\delta }.\\end{array}\\right.$ For all $\\epsilon >0$ we define the following function $v_{\\epsilon } =c_{\\epsilon } \\eta (\\epsilon +r^2)^\\frac{2-n}{2},$ where $c_\\epsilon $ is choosen such that $\\int _M v_{\\epsilon }^N \\ dv_g=1.$ By referring to [2] $\\lim _{\\epsilon \\longrightarrow 0}Y(v_{\\epsilon })=\\mu _1(\\mathbb {S}^n),$ where $Y(u)$ is the Yamabe functional defined by $Y(u)=\\frac{\\int _M c_n\\vert \\nabla u\\vert ^2+S_g u^2 \\ dv_g}{\\left(\\int _M u^N \\ dv_g\\right)^\\frac{2}{N}}.$ If $(M,g)$ is not locally conformally flat, by a calculation made in [2], there exists a constant $C(M)>0$ such that $Y(v_{{\\varepsilon }})=\\left|\\;\\begin{matrix}\\mu _1(\\doba {S}^n) - C(M) {\\varepsilon }^2 + o({\\varepsilon }^2)\\hfill & \\hbox{ if } n > 6 \\\\\\\\\\mu _1(\\doba {S}^n) - C(M) {\\varepsilon }^2 |\\ln ({\\varepsilon })| + o({\\varepsilon }^2 |\\ln ({\\varepsilon })|)\\hfill & \\hbox{ if } n = 6.\\end{matrix}\\right.$ Again from [2] there exists constants $a,$ $b,$ $C_1,$ $C_2>0$ , such that $a {\\varepsilon }^{\\frac{n-2}{4} } \\le c_{{\\varepsilon }} \\le b {\\varepsilon }^{\\frac{n-2}{4} },$ and $ C_1 {\\alpha }_{p,{\\varepsilon }} \\le \\int _M v_{{\\varepsilon }}^p \\,dv_g \\le C_2 {\\alpha }_{p,{\\varepsilon }}$ where $ {\\alpha }_{p,{\\varepsilon }} = \\left| \\begin{array}{lll}{\\varepsilon }^{\\frac{2n - (n-2) p}{4}} & \\hbox{if} & p> \\frac{n}{n-2};\\\\\\\\|\\ln ({\\varepsilon })| {\\varepsilon }^{\\frac{n}{4}} & \\hbox{if} & p= \\frac{n}{n-2};\\\\\\\\{\\varepsilon }^{ \\frac{(n-2) p}{4}} & \\hbox{if}& p< \\frac{n}{n-2}\\end{array} \\right.",
"$ We have $\\mu _2(M,g)&=&\\inf _u\\lambda _2(u)\\left(\\int _Mu^N \\ dv_g\\right)^{\\frac{2}{n}}\\\\\\\\&=& \\inf _{\\scriptstyle u \\atop \\scriptstyle w,w^\\prime } \\sup _{\\lambda ,\\mu } \\frac{\\int _M L_g(\\lambda w+\\mu w^\\prime )(\\lambda w+\\mu w^\\prime ) \\ dv_g}{\\int _M u^{N-2}(\\lambda w+\\mu w^\\prime )^2 \\ dv_g}\\left(\\int _Mu^N \\ dv_g\\right)^{\\frac{2}{n}}\\\\\\\\&=& \\inf _{\\scriptstyle u \\atop \\scriptstyle w,w^\\prime } \\sup _{\\lambda ,\\mu }F(u,\\lambda w+\\mu w^\\prime ).$ Let $\\lambda _\\epsilon ,$ $\\mu _\\epsilon $ such that $\\lambda _\\epsilon ^2+\\mu _\\epsilon ^2=1$ and $F(v_\\epsilon , \\lambda _\\epsilon v+\\mu _\\epsilon v_\\epsilon )=\\sup _{(\\lambda ,\\mu )\\in \\mathbb {R}^2\\backslash \\left\\lbrace (0,0)\\right\\rbrace } F(v_\\epsilon , \\lambda v +\\mu v_\\epsilon ),$ where $v$ is the function defined in the equation (REF ).",
"Calculating $F(v_\\epsilon , \\lambda _\\epsilon v+\\mu _\\epsilon v_\\epsilon ),$ we get $F(v_\\epsilon , \\lambda _\\epsilon v+\\mu _\\epsilon v_\\epsilon )&=& \\frac{\\int _ML_g(\\lambda _\\epsilon v+\\mu _\\epsilon v_\\epsilon )(\\lambda _\\epsilon v+\\mu _\\epsilon v_\\epsilon ) \\ dv_g}{\\int _M v_\\epsilon ^{N-2}(\\lambda _\\epsilon v+\\mu _\\epsilon v_\\epsilon )^2 \\ dv_g}\\left(\\int _Mv_\\epsilon ^N \\ dv_g\\right)^{\\frac{2}{n}}\\\\\\\\&=& \\frac{\\lambda _\\epsilon ^2 \\mu (M,g)+\\mu _\\epsilon ^2Y(v_\\epsilon )+2\\lambda _\\epsilon \\mu _\\epsilon \\mu (M,g)\\int _M v^{N-1}v_\\epsilon \\ dv_g}{\\lambda _\\epsilon ^2\\int _Mv_\\epsilon ^{N-2}v^2 \\ dv_g+\\mu _\\epsilon ^2+2\\lambda _\\epsilon \\mu _\\epsilon \\int _M v_\\epsilon ^{N-1}v \\ dv_g}\\\\\\\\&=&\\frac{A_\\epsilon }{B_\\epsilon }.$ If ${\\lambda }_{\\varepsilon }\\longrightarrow {\\lambda }\\ne 0, \\hspace{5.69046pt}\\mu _{\\varepsilon }\\longrightarrow \\mu \\ne 0,$ then $F(v_{\\varepsilon },{\\lambda }_{\\varepsilon }v + \\mu _{\\varepsilon }v_{\\varepsilon })\\longrightarrow \\frac{{\\lambda }^2 \\mu (M,g) + \\mu ^2\\mu (\\mathbb {S}^n)}{\\mu ^2}<\\mu (\\mathbb {S}^n).$ Similarly, if $\\mu = 0,$ ${\\lambda }^2 = 1,$ then the numerator $A_{\\varepsilon }\\sim \\mu (M,g)< 0,$ while the denominator $B_{\\varepsilon }$ remains positive, which gives again that $F(v_{\\varepsilon },{\\lambda }_{\\varepsilon }v + \\mu _{\\varepsilon }v_{\\varepsilon })\\le 0<\\mu (\\mathbb {S}^n),$ which gives the desired inequality.",
"Then, in the sequel, we assume that ${\\lambda }_{\\varepsilon }\\longrightarrow 0$ and $\\mu _{\\varepsilon }\\longrightarrow \\pm 1.$ The case $n>6$ Using (REF ) we have $ \\int _M v^{N-1} v_{{\\varepsilon }} \\,dv_g \\sim _{{\\varepsilon }\\rightarrow 0} C {\\varepsilon }^{\\frac{n-2}{4}},$ $\\int _M v_{{\\varepsilon }}^{N-2} v^2 \\,dv_g \\sim _{{\\varepsilon }\\rightarrow 0} C {\\varepsilon },$ and $\\int _M v_{{\\varepsilon }}^{N-1} v \\,dv_g \\sim _{{\\varepsilon }\\rightarrow 0} C {\\varepsilon }^{\\frac{n-2}{4}},$ where $C$ denotes a constant that might change its value from line to line.",
"We distinguish two cases $\\bullet $ there exists a constant $C>0$ such that $\\vert {\\lambda }_{\\varepsilon }\\vert \\le C {\\varepsilon }^\\frac{n-2}{4},$ or $\\bullet $ there exists $\\alpha _{\\varepsilon }$ such that $\\vert \\lambda _{\\varepsilon }\\vert = \\alpha _{\\varepsilon }{\\varepsilon }^{\\frac{n-2}{4}},$ and $\\alpha _{\\varepsilon }\\longrightarrow +\\infty .$ (possibly extracting a subsequence).",
"Suppose first that (REF ) is verified.",
"Then we have $\\vert {\\lambda }_{\\varepsilon }\\vert \\le C {\\varepsilon }^\\frac{n-2}{4}.$ Hence ${\\lambda }_{\\varepsilon }^2=O({\\varepsilon }^\\frac{n-2}{2}),$ so $\\mu ^2_{\\varepsilon }=1-{\\lambda }^2_{\\varepsilon }=1+O({\\varepsilon }^\\frac{n-2}{2}).$ Therefore $\\mu _{\\varepsilon }=1+O({\\varepsilon }^\\frac{n-2}{2}).$ This gives $A_\\epsilon &=& O({\\varepsilon }^{\\frac{n-2}{2}}) + ( 1 +O({\\varepsilon }^{\\frac{n-2}{2}})) \\Big ( \\mu (\\mathbb {S}^n) -C(M){\\varepsilon }^2 +o({\\varepsilon }^2)\\Big ) + O({\\varepsilon }^{\\frac{n-2}{2}})\\\\&=& \\mu (\\mathbb {S}^n)-C(M){\\varepsilon }^2 + O({\\varepsilon }^{\\frac{n-2}{2}}) +o({\\varepsilon }^2).$ Since $\\frac{n-2}{2} > 2$ , $A_{\\varepsilon }=\\mu (\\mathbb {S}^n) -C(M){\\varepsilon }^2 +o({\\varepsilon }^2),$ and $B_{\\varepsilon }= O({\\varepsilon }^{\\frac{n-2}{2}+1}) + 1 + O({\\varepsilon }^{\\frac{n-2}{2}})+O({\\varepsilon }^{\\frac{n-2}{2}}) = 1+ o({\\varepsilon }^2).$ Then, $\\frac{A_{\\varepsilon }}{B_{\\varepsilon }} = \\mu (\\mathbb {S}^n)- C(M) {\\varepsilon }^2 + o({\\varepsilon }^2) < \\mu (\\mathbb {S}^n).$ Assume now that (REF ) is fulfilled.",
"In this case $\\frac{A_{\\varepsilon }}{B_{\\varepsilon }} &=& \\frac{\\lambda ^2_{\\varepsilon }\\mu (M, g) + (1 - \\lambda ^2_{\\varepsilon })Y(v_{\\varepsilon }) + \\lambda _{\\varepsilon }O({\\varepsilon }^{\\frac{n-2}{4}})}{ \\lambda ^2_{\\varepsilon }O({\\varepsilon }) + (1- \\lambda ^2_{\\varepsilon }) + 2\\lambda _{\\varepsilon }\\mu _{\\varepsilon }O({\\varepsilon }^{\\frac{n-2}{4}})}\\\\&=& \\frac{\\lambda ^2_{\\varepsilon }\\mu (M, g) + (1 - \\lambda ^2_{\\varepsilon }) Y(v_{\\varepsilon }) +o({\\lambda }^2_{\\varepsilon })}{o({\\lambda }^2_{\\varepsilon }) + (1-{\\lambda }^2_{\\varepsilon }) + o({\\lambda }^2_{\\varepsilon })}\\\\&=& \\frac{{\\lambda }^2_{\\varepsilon }\\mu (M, g)}{1-{\\lambda }_{\\varepsilon }^2 + o({\\lambda }^2_{\\varepsilon })} + \\frac{Y(v_{\\varepsilon })}{ 1 +\\frac{o({\\lambda }_{\\varepsilon }^2)}{\\mu ^2_{\\varepsilon }}} + o({\\lambda }_{\\varepsilon }^2)\\\\\\\\&\\le & \\mu (M,g) {\\lambda }_{\\varepsilon }^2 + \\mu (\\mathbb {S}^n)(1 + o({\\lambda }_{\\varepsilon }^2)) + o({\\lambda }_{\\varepsilon }^2)\\\\&\\le & \\mu (\\mathbb {S}^n) + \\mu (M,g) {\\lambda }_{\\varepsilon }^2 + o({\\lambda }_{\\varepsilon }^2)\\\\&<& \\mu (\\mathbb {S}^n),$ because $\\mu (M,g)<0$ and $Y(v_{\\varepsilon }) \\le \\mu (\\mathbb {S}^n).$ The case $n =6$ Since $\\int _M v_{\\varepsilon }^{N-2} v^2 dv_g &\\sim _{{\\varepsilon }\\rightarrow 0}& C {\\varepsilon },\\\\\\int _M v^{N-1}v_{\\varepsilon }dv_g &\\sim _{{\\varepsilon }\\rightarrow 0}& C {\\varepsilon },\\\\\\int _M v_{\\varepsilon }^{N-1} v dv_g &\\sim _{{\\varepsilon }\\rightarrow 0}& C {\\varepsilon },$ then $A_{\\varepsilon }= {\\lambda }_{\\varepsilon }^2 \\mu (M,g) + \\mu ^2_{\\varepsilon }Y(v_{\\varepsilon }) + 2{\\lambda }_{\\varepsilon }\\mu _{\\varepsilon }O({\\varepsilon }),$ $B_{\\varepsilon }= {\\lambda }^2_{\\varepsilon }O({\\varepsilon }) + \\mu ^2_{\\varepsilon }+ 2{\\lambda }_{\\varepsilon }\\mu _{\\varepsilon }O({\\varepsilon }).$ Again, we have two cases to study If $\\vert {\\lambda }_{\\varepsilon }\\vert \\le C{\\varepsilon },$ then ${\\lambda }_{\\varepsilon }^2\\le C{\\varepsilon }^2.$ This implies $A_{\\varepsilon }= \\mu (\\mathbb {S}^n) - C{\\varepsilon }^2\\vert \\ln ({\\varepsilon })\\vert + o({\\varepsilon }^2\\vert \\ln ({\\varepsilon })\\vert )$ and $B_{\\varepsilon }= 1+O({\\varepsilon }^2) = 1 + o({\\varepsilon }^2 \\vert \\ln ({\\varepsilon })\\vert ).$ Hence $\\frac{A_{\\varepsilon }}{B_{\\varepsilon }}< \\mu (\\mathbb {S}^n).$ If $\\vert {\\lambda }_{\\varepsilon }\\vert = \\alpha _{\\varepsilon }{\\varepsilon },$ with $\\alpha _{\\varepsilon }\\longrightarrow +\\infty .$ Since $Y(v_{\\varepsilon }) \\le \\mu (\\mathbb {S}^n),$ therefore $A_{\\varepsilon }= \\alpha _{\\varepsilon }^2 {\\varepsilon }^2 \\mu (M,g) + \\mu ^2_{\\varepsilon }\\mu (\\mathbb {S}^n) + o(\\alpha _{\\varepsilon }^2{\\varepsilon }^2),$ and $B_{\\varepsilon }= \\mu ^2_{\\varepsilon }+ o(\\alpha _{\\varepsilon }^2 {\\varepsilon }^2).$ Therefore $\\frac{A_{\\varepsilon }}{B_{\\varepsilon }} &=& \\mu (M,g) \\frac{\\alpha _{\\varepsilon }^2 {\\varepsilon }^2}{1+o(1)}+ \\frac{\\mu (\\mathbb {S}^n)}{1 + o(\\alpha _{\\varepsilon }^2 {\\varepsilon }^2)} + \\frac{o(\\alpha _{\\varepsilon }^2 {\\varepsilon }^2)}{1+ o(1)}\\\\&=& \\mu (M,g)\\alpha _{\\varepsilon }^2 {\\varepsilon }^2 + \\mu (\\mathbb {S}^n) + o(\\alpha _{\\varepsilon }^2 {\\varepsilon }^2)\\\\&<& \\mu (\\mathbb {S}^n).$ This ends the proof of Propositon REF .",
"$\\square $ So we get a solution $w$ having a changing sign of the equation $L_g w = \\mu _2 \\vert w\\vert ^{N-2}w.$ Finally, to obtain the resultat announced in Theorem REF , it suffices to set $w^\\prime = \\mu _2^\\frac{n-2}{4} w,$ then $w^\\prime $ verifies $L_gw^\\prime = {\\varepsilon }\\vert w^\\prime \\vert ^{N-2} w^\\prime ,$ with ${\\varepsilon }= 1 = \\hbox{ sign}({\\lambda }_2(g))$ ."
],
[
"The case ${\\lambda }_2<0$",
"In this section, we will show that in all cases, there exists a nodal solution of the equation $L_g w = C_0 \\vert w\\vert ^{N-2}w,$ where $C_0$ is a negative constant.",
"First, since $\\mu <0,$ we assume in the whole section that the metric $g$ is such that $S_g=-1.$ In this context, the approach will be different.",
"Indeed, the second Yamabe invariant is not well defined as shown in the following proposition: Proposition 5.1 Let $M$ be a compact Riemannian manifold of dimension $n\\ge 3$ .",
"Suppose that ${\\lambda }_2< 0$ , then $\\inf _u {\\lambda }_2(u)\\left(\\int _M u^N \\ dv_g\\right)^\\frac{2}{n} = -\\infty .$ The proof will be detailed in Subsection REF .",
"We will use a new functional $I_g(u)= \\frac{\\left(\\int _M\\vert L_gu\\vert ^\\frac{2n}{n+2} \\ dv_g\\right)^\\frac{n+2}{n}}{\\vert \\int _M uL_gu \\ dv_g\\vert }.$ We study $\\alpha := \\inf I_g(u)$ where the infimum is taken over the functions $u\\in H_2^\\frac{2n}{n+2}(M)$ such that $\\int _M uL_gu \\ dv_g <0,$ and with the following constraint $\\int _M \\vert u \\vert ^{N-2} u \\ v \\ dv_g=0,$ for any function $v\\in \\ker L_g.$ We will show that $\\alpha $ is a conformal invariant.",
"We obtain also that the infimum of this functional is attained by a function $u.$ We set $v = \\vert L_gu\\vert ^\\frac{-4}{n+2} L_gu,$ and we will observe that $v$ has the following properties: $\\bullet $ $v$ is a solution of the equation $L_g v = \\alpha ^\\prime \\vert v\\vert ^{N-2}v,$ where $\\alpha ^\\prime <0$ (i.e.",
"has same sign than ${\\lambda }_2$ ).",
"$\\bullet $ $v$ has a changing sign.",
"$\\bullet $ $v$ is of class $C^{3,\\alpha }(M)$ ($\\alpha < N-2$ )."
],
[
"Conformal invariance of $\\alpha $",
"Let $\\widetilde{g} = {\\varphi }^\\frac{4}{n-2} g$ be a conformal metric, ${\\varphi }$ a smooth positive function.",
"Then $dv_{\\widetilde{g}} = {\\varphi }^\\frac{2n}{n-2} dv_g,$ and $L_{\\widetilde{g}}u = {\\varphi }^{-\\frac{n+2}{n-2}}L_g(u{\\varphi }),$ for all functions $u.$ Remark that $I_{\\widetilde{g}}(u) = I_g(u{\\varphi }).$ $I_{\\widetilde{g}} (u)&=& \\frac{\\left(\\int _M \\vert L_{\\widetilde{g}}u\\vert ^\\frac{2n}{n+2} \\ dv_{\\widetilde{g}}\\right)^\\frac{n+2}{n}}{\\vert \\int _M uL_{\\widetilde{g}}u \\ dv_{\\widetilde{g}}\\vert }\\\\\\\\&=& \\frac{\\left(\\int _M \\vert {\\varphi }\\vert ^\\frac{-2n}{n-2} \\vert L_g(u{\\varphi })\\vert ^\\frac{2n}{n+2} {\\varphi }^\\frac{2n}{n-2} \\ dv_g\\right)^\\frac{n+2}{n}}{\\vert \\int _M u{\\varphi }^\\frac{-(n+2)}{n-2} L_g (u{\\varphi }){\\varphi }^\\frac{2n}{n-2} \\ dv_g\\vert }\\\\\\\\&=& \\frac{\\left(\\int _M \\vert L_g(u{\\varphi })\\vert ^\\frac{2n}{n-2} \\ dv_g\\right)^\\frac{n+2}{n}}{\\vert \\int _M u{\\varphi }L_g(u{\\varphi }) \\ dv_g\\vert }\\\\\\\\&=& I_g(u{\\varphi }),$ where we have used $\\int _M uL_{\\widetilde{g}}u \\ dv_{\\widetilde{g}} &=& \\int _M u{\\varphi }^{-\\frac{n+2}{n-2}}L_g(u{\\varphi }){\\varphi }^\\frac{2n}{n-2} \\ dv_g\\\\&=& \\int _M (u{\\varphi })L_g(u{\\varphi }) \\ dv_g.$ Assume that for any $v\\in \\ker L_{\\widetilde{g}},$ we have $\\int _M \\vert u\\vert ^{N-2} uv \\ dv_{\\widetilde{g}} = 0.$ Then, for any $v^\\prime \\in \\ker L_g,$ we obtain $\\int _M \\vert u{\\varphi }\\vert ^{N-2} (u{\\varphi })v^\\prime \\ dv_g = \\int _M \\vert u\\vert ^{N-2} u (v^\\prime {\\varphi }^{-1}) \\ dv_{\\widetilde{g}} = 0,$ since $L_{\\widetilde{g}}(v^\\prime {\\varphi }^{-1}) = {\\varphi }^{-\\frac{n+2}{n-2}}L_g(v^\\prime ) = 0,$ i.e.",
"$v^\\prime {\\varphi }^{-1}\\in Ker L_{\\widetilde{g}}.$ $\\square $"
],
[
"Proof of Proposition ",
"Assume that ${\\lambda }_2(g)< 0,$ and choose $u>0.$ By Lemma REF , there exists two functions $v_1$ and $v_2$ solutions of the following equations $L_g v_1 = {\\lambda }_1(u) \\vert u\\vert ^{N-2} v_1,$ and $L_g v_2 = {\\lambda }_2(u) \\vert u\\vert ^{N-2} v_2,$ such that $\\int _M \\vert u\\vert ^{N-2} v_1v_2 \\ dv_g=0.$ Let $v_{\\varepsilon }$ the function defined in Section REF , and let $V=\\left\\lbrace v_1,v_2\\right\\rbrace .$ For all $v\\in V,$ we get $\\lim _{{\\varepsilon }\\longrightarrow 0}\\int _Mv_{\\varepsilon }^{N-2} v^2 \\ dv_g = 0.$ Since ${\\lambda }_1(u)<0$ and ${\\lambda }_2(u)<0,$ then for ${\\varepsilon }$ sufficiently small, we have $\\lim _{{\\varepsilon }\\rightarrow 0}\\left(\\sup _{v\\in V}\\frac{\\int _M (L_gv)(v) \\ dv_g}{\\int _M v_{\\varepsilon }^{N-2}v^2 \\ dv_g}\\right) =-\\infty ,$ hence $\\lim _{{\\varepsilon }\\rightarrow 0}\\left(\\inf _u {\\lambda }_2(u)\\left(\\int _M u^N \\ dv_g\\right)^\\frac{2}{n}\\right) =-\\infty .$ $\\square $"
],
[
"The infimum of the functional $I_g$ is attained",
"Let $(u_m)_m$ be a minimizing sequence, i.e., $\\lim _{m\\longrightarrow \\infty }I_g(u_m) = \\alpha ,$ with $\\int _M \\vert u_m \\vert ^{N-2} \\ u_m \\ v \\ dv_g=0, \\hspace{5.69046pt}\\forall \\ v\\in \\ker L_g.$ We can assume that $\\int _M u_mL_gu_m \\ dv_g = -1.$ Then $\\alpha = \\lim _{m\\longrightarrow \\infty } \\left(\\int _M\\vert L_gu_m\\vert ^\\frac{2n}{n+2} \\ dv_g\\right)^\\frac{n+2}{n}.$ Now we show that $(u_m)_m$ is a bounded sequence in $H_2^\\frac{2n}{n+2}(M).$ We proceed by contradiction and we assume that, up to a subsequence, $\\lim \\Vert u_m\\Vert _{H_2^\\frac{2n}{n+2}(M)}= +\\infty $ .",
"Let $v_m = \\frac{u_m}{\\Vert u_m\\Vert _{H_2^\\frac{2n}{n+2}(M)}}.$ Since $\\Vert v_m\\Vert _{H_2^\\frac{2n}{n+2}(M)} = 1,$ $(v_m)_m$ is a bounded sequence in $H_2^\\frac{2n}{n+2}(M),$ and therefore there exists $v \\in H_2^\\frac{2n}{n+2}(M)$ such that after restriction to a subsequence $v_m\\rightharpoonup v\\text{ in }H_2^\\frac{2n}{n+2}(M),$ $v_m\\longrightarrow v\\text{ in }L^2(M).$ By standard arguments, we get $\\left(\\int _M\\vert L_gv\\vert ^\\frac{2n}{n+2} \\ dv_g\\right)^\\frac{n+2}{n}\\le \\underbrace{\\liminf _m \\left(\\int _M\\vert L_gv_m\\vert ^\\frac{2n}{n+2} \\ dv_g\\right)^\\frac{n+2}{n}}_{=0}.$ This gives $L_gv = 0,$ hence $v\\in \\ker L_g.$ We have for all function $v^\\prime \\in \\ker L_g,$ $\\int _M \\vert v_m\\vert ^{N-2}v_m v^\\prime \\ dv_g = \\frac{\\int _M \\vert u_m\\vert ^{N-2}u_m v^\\prime \\ dv_g}{\\Vert u_m\\Vert ^{N-1}_{H_2^\\frac{2n}{n+2}}} = 0.$ In particular for $v^\\prime = v$ , $\\int _M \\vert v_m\\vert ^{N-2}v_m v \\ dv_g = 0\\longrightarrow _{m\\rightarrow \\infty }\\int _Mv^N \\ dv_g,$ so $v=0.$ According to the regularity Theorem 3.75 in [3], we have $1 = \\Vert v_m\\Vert _{H_2^\\frac{2n}{n+2}}\\le C\\left[\\underbrace{\\Vert L_gv_m\\Vert _{L^\\frac{2n}{n+2}}}_{\\longrightarrow 0} +\\Vert v_m\\Vert _{L^\\frac{2n}{n+2}}\\right].$ Passing to the limit, we obtain $\\int _M v^\\frac{2n}{n+2} \\ dv_g\\ge \\frac{1}{C},$ which gives a contradiction.",
"We deduce that $(u_m)_m$ is a bounded sequence in $H_2^\\frac{2n}{n+2}(M).$ Then, after restriction to a subsequence, there exists $u$ in $H_2^\\frac{2n}{n+2}(M)$ such that $u_m\\rightharpoonup u \\text{ in } H_2^\\frac{2n}{n+2}(M),$ $u_m\\rightharpoonup u \\text{ in }H_1^2(M),$ $u_m\\longrightarrow u \\text{ in } L^2(M).$ Further, we have $\\left(\\int _M \\vert L_gu\\vert ^\\frac{2n}{n+2} \\ dv_g\\right)^\\frac{n+2}{n}\\le \\liminf _m \\left(\\int _M \\vert L_gu_m\\vert ^\\frac{2n}{n+2} \\ dv_g\\right)^\\frac{n+2}{n}.$ Moreover, $\\int _M uL_gu \\ dv_g =& \\int _M \\vert \\nabla u\\vert ^2 \\ dv_g - \\int _M u^2 \\ dv_g \\nonumber \\\\\\le & \\liminf _m \\int _M \\vert \\nabla u_m\\vert ^2 \\ dv_g - \\int _M u_m^2 \\ dv_g\\nonumber \\\\=& \\liminf _m \\int _M u_mL_gu_m \\ dv_g = -1.$ Therefore $ \\int _M uL_gu \\ dv_g < 0,$ and $u\\ne 0.$ Finally, with (REF ) $I_g(u) &=& \\frac{\\left(\\int _M \\vert L_gu\\vert ^\\frac{2n}{n+2} \\ dv_g\\right)^\\frac{n+2}{n}}{\\vert \\int _M uL_gu \\ dv_g\\vert }\\\\&\\le & \\liminf _m \\frac{\\left(\\int _M \\vert L_gu_m\\vert ^\\frac{2n}{n+2} \\ dv_g\\right)^\\frac{n+2}{n}}{\\vert \\int _M u_mL_gu_m \\ dv_g\\vert }= \\liminf _m I_g(u_m) =\\alpha .$ Hence the result is proved i.e.",
"$I_g(u) = \\alpha $ .",
"Euler equation Notice that $\\int _M \\vert u\\vert ^{N-2} \\ u \\ v^\\prime \\ dv_g = 0, \\text{ for anyfunction }v^\\prime \\in \\ker L_g.$ In particular, ${\\alpha }\\ne 0$ .",
"Remark also that $\\int _M uL_gu \\ dv_g = -1.$ Indeed, the relation $\\int _M uL_gu \\ dv_g < -1$ would imply that $I_g(u)< \\lim I_g(u_m) = {\\alpha }$ .",
"We now write Euler equation of $u$ .",
"Let $\\left\\lbrace u_1, \\ldots ,u_k\\right\\rbrace $ be a base of $\\ker L_g.$ By the Lagrange multipliers theorem, there exists real numbers $ {\\lambda }_1,\\ldots ,{\\lambda }_k$ for which, for all function ${\\varphi }\\in C^\\infty (M),$ we get $\\frac{d}{dt}|_{t=0}I_g(u+t{\\varphi }) = \\Sigma _i {\\lambda }_i\\frac{d}{dt}|_{t=0} g_i(u+t{\\varphi }),$ where $g_i(u) = \\int _M \\vert u \\vert ^{N-2} \\ u \\ u_i \\ dv_g.$ Setting $a = \\left(\\int _M \\vert L_gu\\vert ^\\frac{2n}{n+2} \\ dv_g\\right)^\\frac{n+2}{n}$ , one checks $2 a^\\frac{2}{n+2}\\int _M\\vert L_gu\\vert ^\\frac{-4}{n+2}L_gu L_g{\\varphi }\\ dv_g + 2 a\\int _M {\\varphi }L_gu \\ dv_g = (N-1) \\Sigma _i{\\lambda }_i\\int _M \\vert u \\vert ^{N-2} \\ {\\varphi }\\ u_i \\ dv_g.$ If ${\\varphi }\\in \\ker L_g,$ this last equation implies that $\\Sigma _i {\\lambda }_i\\int _M \\vert u \\vert ^{N-2} \\ {\\varphi }\\ u_i \\ dv_g = 0.$ Then, for ${\\varphi }= \\Sigma _i{\\lambda }_i \\ u_i \\in \\ker L_g,$ we have $\\int _M \\vert u\\vert ^{N-2} \\ {\\varphi }^2 \\ dv_g = 0.$ Therefore $\\vert u\\vert ^{N-2} \\ {\\varphi }^2 = 0&\\Rightarrow & \\vert u\\vert ^{N-2} \\ {\\varphi }= 0\\\\&\\Rightarrow &\\Sigma _i \\ {\\lambda }_i \\ \\vert u\\vert ^{N-2} \\ u_i = 0.$ This gives, for any function ${\\varphi }$ (in $\\ker L_g$ or not), that $\\Sigma _i {\\lambda }_i\\int _M \\vert u\\vert ^{N-2} {\\varphi }\\ u_i \\ dv_g = 0.$ Then $u$ verifies in the sense of distributions the following equation $ L_g\\left(\\vert L_gu\\vert ^\\frac{-4}{n+2} L_gu\\right) = \\alpha ^\\prime L_gu,$ where $\\alpha ^\\prime = - \\alpha ^\\frac{n}{n+2}= -\\int _M \\vert L_gu\\vert ^\\frac{2n}{n+2} \\ dv_g.$ We set $v = \\vert L_gu\\vert ^\\frac{-4}{n+2} L_gu \\in L^N(M),$ then $\\vert v\\vert = \\vert L_gu\\vert ^{1-\\frac{4}{n+2}} = \\vert L_gu\\vert ^\\frac{n-2}{n+2}.$ Hence, $ L_gu = \\vert v\\vert ^{N-2} \\ v.$ Replacing each term by its value in Equation (REF ), we obtain $L_gv = \\alpha ^\\prime \\vert v\\vert ^{N-2} \\ v.$ Regularity of $v$ We have $u\\in H_2^\\frac{2n}{n+2}(M),$ then $L_gu \\in L^\\frac{2n}{n+2}(M).$ Therefore $v\\in L^N(M),$ since $\\vert v\\vert ^N = \\vert L_g u\\vert ^\\frac{2n}{n+2}.$ Moreover, in the sense of distributions $L_gv = \\alpha ^\\prime \\vert v\\vert ^{N-2} \\ v,$ this implies that $\\vert L_g v \\vert = \\vert \\alpha ^\\prime \\vert \\vert v\\vert ^{N-1},$ hence $L_gv \\in L^\\frac{N}{N-1} (M)= L^\\frac{2n}{n+2}(M),$ therefore $v \\in H_2^\\frac{2n}{n+2}(M)\\subset H_1^2(M).$ Using Lemma 3.1 of [1], we get $v\\in L^{N+{\\varepsilon }}(M),$ By a standard bootstrap argument, we show that $v\\in C^{3,\\alpha }(M) (\\alpha <N-2).$ Calculating now $I_g(v),$ using (REF ), we have $I_g(v) &=& \\frac{\\left(\\int _M\\vert L_g v\\vert ^\\frac{2n}{n+2} \\ dv_g\\right)^\\frac{n+2}{n}}{\\vert \\int _M vL_gv \\ dv_g\\vert }\\\\\\\\&=& \\frac{\\alpha ^{\\prime 2}\\left(\\int _M \\vert v\\vert ^{(N-1)\\times \\frac{2n}{n+2}} \\ dv_g\\right)^\\frac{n+2}{n}}{\\vert \\alpha ^\\prime \\vert \\int _M \\vert v\\vert ^N \\ dv_g}\\\\\\\\&=& \\alpha ^\\frac{n}{n+2} \\frac{\\left(\\int _M\\vert v \\vert ^\\frac{2n}{n-2} \\ dv_g\\right)^\\frac{n+2}{n}}{\\int _M \\vert v\\vert ^\\frac{2n}{n-2} \\ dv_g}\\\\\\\\&=& \\alpha ^\\frac{n}{n+2} \\left(\\int _M \\vert v\\vert ^\\frac{2n}{n-2} \\ dv_g\\right)^\\frac{2}{n}\\\\\\\\&=& \\alpha ^\\frac{n}{n+2} \\left(\\int _M \\vert L_gu\\vert ^\\frac{2n}{n+2} \\ dv_g\\right)^\\frac{2}{n}\\\\\\\\&=& \\alpha ^\\frac{n}{n+2} \\alpha ^\\frac{2}{n+2} = \\alpha .$ The function $v$ satisfies that, for any function $v^\\prime \\in \\ker L_g,$ $\\int _M \\vert v\\vert ^{N-2}v v^\\prime \\ dv_g = 0.$ Indeed, $\\int _M \\vert v\\vert ^{N-2} \\ v \\ v^\\prime \\ dv_g &=& \\int _M L_gu \\ v^\\prime \\ dv_g\\\\&=& \\int _M u \\ L_g v^\\prime \\ dv_g \\\\&=&0.$ $\\bullet $ $v$ has changing sign We proceed by contradiction and assume that $v\\ge 0$ .",
"Since $v \\ne 0$ , we deduce from the maximum principle that $v>0$ .",
"In addition, Equation (REF ) says that there exists an $i$ such that $\\alpha ^\\prime = {\\lambda }_i(v)$ .",
"The only positive eigenfunctions are the ones associated to $\\lambda _1$ and hence ${\\alpha }^{\\prime }=\\lambda _1(v)$ .",
"By Proposition REF , there exists a function $w$ solution of the following equation $L_gw = {\\lambda }_2(v) \\vert v\\vert ^{N-2} w.$ $I_g(w) &=& \\frac{\\left(\\int _M \\vert L_gw\\vert ^\\frac{2n}{n+2} \\ dv_g\\right)^\\frac{n+2}{n}}{\\vert \\int _M wL_gw \\ dv_g\\vert }\\\\\\\\&=& \\frac{\\vert {\\lambda }_2(v) \\vert ^2 \\left(\\int _M \\vert v\\vert ^{(N-2)\\times \\frac{2n}{n+2}} \\vert w\\vert ^\\frac{2n}{n+2} \\ dv_g\\right)^\\frac{n+2}{n}}{\\vert {\\lambda }_2(v)\\vert \\int _M \\vert v\\vert ^{N-2} \\ w^2 \\ dv_g}.$ By applying the Hölder inequality with $p = \\frac{n+2}{n}$ and $q = \\frac{n+2}{2},$ we get $\\int _M \\vert v\\vert ^{(N-2)\\times \\frac{2n}{n+2}} \\ \\vert w\\vert ^\\frac{2n}{n+2}\\ dv_g &=& \\int _M \\vert v\\vert ^{(N-2)\\times \\frac{n}{n+2}} \\ \\vert w\\vert ^\\frac{2n}{n+2} \\ \\vert v\\vert ^{(N-2)\\times \\frac{n}{n+2}} \\ dv_g\\\\\\\\&\\le & \\left(\\int _M \\vert v\\vert ^{N-2} \\ w^2 \\ dv_g\\right)^\\frac{n}{n+2}\\left(\\int _M \\vert v\\vert ^{\\frac{4}{n-2}\\times \\frac{n}{2}} \\ dv_g\\right)^\\frac{2}{n+2}.$ Therefore $I_g(w) &\\le & \\vert {\\lambda }_2(v) \\vert \\left(\\int _M \\vert v \\vert ^\\frac{2n}{n-2} \\ dv_g\\right)^\\frac{2}{n}\\\\&=& \\vert {\\lambda }_1(v) \\vert \\left(\\int _M \\vert L_gu \\vert ^\\frac{2n}{n+2} \\ dv_g\\right)^\\frac{2}{n}\\\\&=& \\alpha ^\\frac{n}{n+2}\\alpha ^\\frac{2}{n+2} = \\alpha ,$ since by assumption ${\\lambda }_1(v) = \\alpha ^\\prime = \\alpha ^\\frac{n}{n+2},$ which gives a contradiction.",
"Then $v$ is a nodal solution of the equation $L_g v = \\alpha ^{\\prime } \\vert v\\vert ^{N-2} v,$ where $\\alpha ^{\\prime } <0$ .",
"Setting $v^\\prime : = \\vert \\alpha \\vert ^\\frac{n-2}{4},$ we obtain that $v^\\prime $ is a solution of the equation $L_g v^\\prime = {\\varepsilon }\\vert v^\\prime \\vert ^{N-2} v^\\prime $ with ${\\varepsilon }= -1 \\hbox{ sign }(\\lambda _2(g))$ .",
"This ends the proof of Theorem REF ."
]
] | 1204.1268 |
[
[
"Experimental implementation of a fully controllable depolarizing quantum\n operation"
],
[
"Abstract The depolarizing quantum operation plays an important role in studying the quantum noise effect and implementing general quantum operations.",
"In this work, we report a scheme which implements a fully controllable input-state independent depolarizing quantum operation for a photonic polarization qubit."
],
[
"Experimental implementation of a fully controllable depolarizing quantum operation Youn-Chang [email protected] Jong-Chan Lee Yoon-Ho [email protected] Department of Physics, Pohang University of Science and Technology (POSTECH), Pohang, 790-784, Korea The depolarizing quantum operation plays an important role in studying the quantum noise effect and implementing general quantum operations.",
"In this work, we report a scheme which implements a fully controllable input-state independent depolarizing quantum operation for a photonic polarization qubit.",
"03.67.Dd, 03.67.Hk, 42.79.Sz Depolarizing quantum operation, Quantum channel, Quantum noise One of the main challenges in experimental quantum information is to deal with decoherence.",
"To preserve a qubit state, it is often necessary to isolate the qubit from the environment but to be able to do information processing tasks, interactions with an external (classical or quantum) system is necessary.",
"More often than not, unwanted interactions with the environment leave the qubit in a different quantum state or even cause the qubit to lose coherence.",
"Such unwanted quantum state transformation can be described by the quantum process due to a noisy quantum channel, in other words, a noisy quantum process, which can be understood with a few basic single-qubit noise operations including the bit-flip, phase-flip, depolarization, and amplitude damping [1].",
"Understanding and being able to implement each of the basic quantum noise processes are important in theoretical and experimental quantum information science.",
"First, it would allow us to simulate and quantify quantum noise processes.",
"Second, the bit-flip and the phase-flip operations are essential for quantum error correction protocols.",
"Third, the depolarization and the amplitude damping operations can describe decoherence.",
"The depolarization operation is of particular interest because, in addition to being relevant to a number of practical quantum communication and computation scenarios [2], [3], [4], [5], [6], it is also an essential operation for optimally approximating non-physical quantum operations [7], [8] and for generating exotic quantum states including Werner states [9] and bound entangled states [10].",
"Clearly, from the experimental point of view, it is important to develop a method to achieve a fully controllable input-independent depolarization quantum operation.",
"As such, there have been many reports on experimental implementation of a depolarization quantum channel.",
"In Ref.",
"[11], optical scattering media were used to achieve the depolarization quantum channel but the scheme naturally induces the spread in the photon momenta and it is difficult to control the degree of depolarization in this scheme.",
"A controllable depolarization channel was demonstrated in Ref.",
"[12], [13] but the output was dependent on the input polarization and the scheme also introduced the momentum spread for the photon.",
"Input-independent depolarizing quantum operation was demonstrated in Ref.",
"[3], [14], but here the depolarizing operation was achieved by time averaging of many `fast' operations, i.e., incoherent sum of many different pure quantum states.",
"In this paper, we report an experimental implementation of a fully-controllable depolarization quantum operation for a photonic polarization qubit.",
"The scheme is completely input-state independent so that it is possible to introduce any desired degree of depolarization regardless of the state of the input qubit.",
"Furthermore, our scheme does not rely on time- or spatial-averaging so that neither the measurement duration nor the measurement area affect the output quantum state.",
"In other words, our scheme achieves a truly observer-independent depolarizing quantum operation.",
"Figure: Experimental schematic for a fully-controllable input-independent depolarizing quantum operation for a photonic polarization qubit ρ\\rho .The depolarizing quantum operation is described as $\\mathcal {E}(\\rho ) = \\frac{p I}{2}+(1-p)\\rho ,$ where $p$ is the degree of decoherence ($0\\le p\\le 1$ ), $\\rho $ is the input state of the photonic polarization qubit, and $I$ is the two-dimensional identity matrix.",
"We implement the depolarizing operation $\\mathcal {E}(\\rho )$ by using a modified displaced Sagnac interferometer setup as shown in Fig.",
"REF .",
"The displaced Sagnac interferometer consists of a polarizing beam splitter (PBS) and two half-wave plates that can be oriented at an angle $\\theta $ (@$\\theta $ in Fig.",
"REF ).",
"Another half-wave plate fixed at 45$^\\circ $ at the output mode $B$ makes the polarization state of the two output modes $A$ and $B$ identical.",
"It is not difficult to see that the displaced Sagnac interferometer in Fig.",
"REF acts as a continuously variable non-polarizing beam splitter in which the output ratio $A:B=1-p:p$ can be linearly varied by setting the angle $\\theta $ of the two half-wave plates.",
"Note that the splitting parameter $p = \\sin ^{2}(2 \\theta )$ .",
"This is confirmed in the experiment as shown in Fig.",
"REF .",
"In the experiment, we prepared the single-photon polarization qubit $\\rho $ by using the heralded single-photon state generated from spontaneous parametric down-conversion (SPDC) in a 3 mm type-II BBO crystal.",
"The wavelengths of the pump and that of the SPDC photon are 405 nm and 810 nm.",
"The input polarization qubit was prepared in a pure state $\\rho = |\\psi \\rangle \\langle \\psi |$ by using a half-wave and a quarter-wave plate.",
"We then measured the count rates at the two single-photon detectors placed at the output ports $A$ and $B$ and the data are shown in Fig.",
"REF .",
"As expected, the data show the linear splitting ratio between two outputs $A$ and $B$ with the splitting parameter $p$ .",
"In Fig.",
"REF , we plot the averaged normalized outputs $A$ and $B$ for six input polarization states, $|H\\rangle $ (horizontal), $|V\\rangle $ (vertical), $|D\\rangle \\equiv (|H\\rangle + |V\\rangle )/\\sqrt{2}$ , $|A\\rangle \\equiv (|H\\rangle - |V\\rangle )/\\sqrt{2}$ , $|R\\rangle \\equiv (|H\\rangle - i|V\\rangle )/\\sqrt{2}$ , and $|L\\rangle \\equiv (|H\\rangle +i|V\\rangle )/\\sqrt{2}$ .",
"The normalized outputs $A$ and $B$ for each polarization state look almost identical to the averaged result shown in Fig.",
"REF .",
"We also point out that the linearity and the splitting ratio are very stable over time due to the Sagnac geometry.",
"Figure: Data showing the linear splitting ratio A:B=1-p:pA:B = 1-p:p where p=sin 2 (2θ)p = \\sin ^2(2\\theta ) for the displaced Sagnac.",
"Each data point represents the averaged normalized output of six input polarization states, |H〉|H\\rangle , |V〉|V\\rangle , |D〉|D\\rangle , |A〉|A\\rangle , |R〉|R\\rangle , and |L〉|L\\rangle .",
"The normalized output of each polarization state appears almost identical to the averaged result shown here.Previously reported continuously variable non-polarizing beam splitters by using a prism pair [17], a sapphire disk [18], and a phase grating [19] were all input-polarization dependent.",
"However, our scheme based on the displaced Sagnac interferometer is completely input-polarization independent as demonstrated in Fig.",
"REF .",
"To be sure that the displaced Sagnac splits the input probability amplitude into two spatial modes $A$ and $B$ without affecting the quantum state $\\rho $ , we have also performed the quantum state tomography of the input qubit as well as the output qubit in modes $A$ and $B$ [15].",
"We observed that the fidelity between the input and the output quantum states is better than $0.982 \\pm 0.003$ for all polarization qubit states we have tested.",
"Thus, the displaced Sagnac acts as a nearly ideal identity operation for the polarization qubit except that it diverts the amplitude into two different spatial modes $A$ and $B$ .",
"Figure: Experimental data.",
"As pp is increased, the qubit states become more mixed, hence moving toward the center of the Bloch sphere.",
"The outer sphere represents p=0p=0 (pure states) and the inner sphere representsp=0.5p=0.5.",
"The arrows point the data points (i.e., qubit states) on the inner sphere.Equation (1) clearly states that, to implement the depolarizing quantum operation $\\mathcal {E}(\\rho )$ , it is necessary to achieve a mixture of the input quantum state $\\rho $ and the unpolarized state $I/2$ with the weighting factors $1-p$ and $p$ , respectively.",
"Note now that the qubit states found at the outputs $A$ and $B$ are identical to the input $\\rho $ but with the probability amplitudes $1-p$ and $p$ , respectively.",
"Thus, we first couple the output mode $B$ into a 2-m long multi-mode fiber to transform a polarized input to a completely unpolarized state.",
"We then combine the output of the multi-mode fiber (after collimation) and that of $A$ at a beam splitter (BS).",
"The 2-m long multi-mode fiber ensures that the beam combination at BS is a completely incoherent process since the path length difference is orders of magnitude larger than the single-photon coherence time, which is on the order of hundreds of femtoseconds.",
"Therefore, the quantum state found at the two outputs of the BS is described precisely as $pI/2 + (1-p)\\rho $ , indicating that the the input state $\\rho $ has gone through the depolarizing quantum operation $\\mathcal {E}(\\rho )$ in eq. (1).",
"To demonstrate that the outputs of the BS indeed correspond to the quantum state after the depolarizing quantum operation, we performed quantum state tomography on the output states for six different input qubit states.",
"The experimental data are shown in Fig.",
"REF .",
"It is clear that, by increasing $p$ , the qubit states become more mixed, moving toward the center of the Bloch sphere.",
"It is important to note that the depolarizing quantum operation is an isotropic operation so that the output state purity $Tr[\\rho ^2]$ does not depend on the input state: it only depends on the choice $p$ .",
"As such, the depolarizing quantum operation should only shrink the size of the Bloch sphere, rather than making it asymmetric.",
"This feature is well demonstrated in Fig.",
"REF : all the data points corresponding to the depolarizing quantum operation of the same $p$ , regardless of the input state, should reside on the same sphere.",
"In Fig.",
"REF , the arrows represent the qubit states after the depolarizing quantum operation $\\mathcal {E}(\\rho )$ for $p=0.5$ and they all lie on the inner sphere representing all qubit states undergone $\\mathcal {E}(\\rho )$ with $p=0.5$ .",
"We also note that changing $p$ is quite easy in our setup as it requires only the rotation of the waveplates due to the relation $p=\\sin ^2(2\\theta )$ .",
"Figure: Real part of χ\\chi matrices of the depolarizing quantumchannel.",
"II refers to the two dimensional identity matrix.",
"XX, YY, and ZZ correspond to Pauli operators.The process fidelities (in comparison to the ideal operation) are (a) 0.966,(b) 0.994, (c) 0.998, and (d) 0.997.",
"The imaginary part of the χ\\chi matrices are almost zero, hence not shown.It is known that a quantum channel that implements a particular quantum operation can be fully characterized by performing quantum process tomography [16].",
"We have carried out quantum process tomography for the depolarizing quantum operation with various $p$ and the resulting $\\chi $ matrices are shown in Fig.",
"REF .",
"For a single-qubit operation as this one, it is often best to use the Pauli-basis for decomposing the quantum process as done in the figure.",
"Clearly, when $p=0$ , the quantum process corresponds to an identity operation as it should be, see Fig.",
"REF (a).",
"As $p$ is increased, contributions from the Pauli operations rise, see Fig.",
"REF (b) and REF (c), and when $p=1$ , it is clear that the output state will be a fully mixed state regardless of the input state, see Fig.",
"REF (d).",
"The high process fidelities for various $p$ values indicate the robustness of our setup to faithfully implement the fully controllable depolarizing quantum operation.",
"In summary, we have reported an experimental realization of a fully controllable depolarizing quantum operation for a single-photon polarization qubit.",
"Our scheme not only allows continuous adjustment of the degree of depolarization but also is independent of the input quantum state, as demonstrated with quantum state tomography and quantum process tomography.",
"A versatile depolarizing quantum channel like the one reported in this paper should find applications in many areas of photonic quantum information research, including generating exotic quantum states, studying the quantum noise processes, approximating non-physical quantum operations, etc.",
"This work was supported in part by the National Research Foundation of Korea (2009-0070668 and 2011-0021452)."
]
] | 1204.0850 |
[
[
"Measurements of the branching fractions of the decays $B^{0}_{s} \\to\n D^{\\mp}_{s} K^{\\pm} $ and $B^{0}_{s} \\to D^{-}_{s} \\pi^{+}$"
],
[
"Abstract The decay mode $\\B^{0}_{s} \\to D^{\\mp}_{s} K^{\\pm} $ allows for one of the theoretically cleanest measurements of the CKM angle $\\gamma$ through the study of time-dependent $\\ensuremath{CP}\\xspace$ violation.",
"This paper reports a measurement of its branching fraction relative to the Cabibbo-favoured mode $\\B^{0}_{s} \\to D^{-}_{s} \\pi^{+}$ based on a data sample of 0.37 fb$^{-1}$ proton-proton collisions at $\\sqrt{s} = 7$ TeV collected in 2011 with the LHCb detector.",
"In addition, the ratio of $\\ensuremath{\\mathrm{B}}\\xspace$ meson production fractions $\\ensuremath{f_s/f_d}$, determined from semileptonic decays, together with the known branching fraction of the control channel $B^{0} \\to D^{-} \\pi^{+}$, is used to perform an absolute measurement of the branching fractions: $B (\\B^0_s \\to D^-_s \\pi^+) \\;= (2.95 \\pm 0.05 \\pm 0.17^{\\,+\\,0.18}_{\\,-\\,0.22}) \\times 10^{-3}$, $B (\\B^0_s \\to D^\\mp_s K^\\pm) = (1.90 \\pm 0.12 \\pm 0.13^{\\,+\\,0.12}_{\\,-\\,0.14}) \\times 10^{-4}\\,$, where the first uncertainty is statistical, the second the experimental systematic uncertainty, and the third the uncertainty due to $f_s/f_d$."
],
[
"Introduction",
"Unlike the flavour-specific decay $B ^0_s \\!\\rightarrow D ^-_s \\pi ^+ $ , the Cabibbo-suppressed decay $B ^0_s \\!\\rightarrow D_s^\\mp K^\\pm $ proceeds through two different tree-level amplitudes of similar strength: a $\\bar{b} \\rightarrow \\bar{c} u \\bar{s}$ transition leading to $B^0_s \\rightarrow D^-_s K^+$ and a $\\bar{b} \\rightarrow \\bar{u} c \\bar{s}$ transition leading to $B^0_s \\rightarrow D^+_sK^-$ .",
"These two decay amplitudes can have a large $C\\!P$ -violating interference via $B^0_s-\\bar{B}^0_s$ mixing, allowing the determination of the CKM angle $\\gamma $ with negligible theoretical uncertainties through the measurement of tagged and untagged time-dependent decay rates to both the $D^-_s K^+$ and $D^+_s K^-$ final states [1].",
"Although the $B ^0_s \\!\\rightarrow D_s^\\mp K^\\pm $ decay mode has been observed by the CDF [2] and BELLE [3] collaborations, only the LHCb experiment has both the necessary decay time resolution and access to large enough signal yields to perform the time-dependent $C\\!P$ measurement.",
"In this analysis, the $B ^0_s \\!\\rightarrow D_s^\\mp K^\\pm $ branching fraction is determined relative to $B ^0_s \\!\\rightarrow D ^-_s \\pi ^+ $ , and the absolute $B ^0_s \\!\\rightarrow D ^-_s \\pi ^+ $ branching fraction is determined using the known branching fraction of $B ^0 \\!\\rightarrow D ^- \\pi ^+ $ and the production fraction ratio $f_s/f_d$ [4].",
"The two measurements are then combined to obtain the absolute branching fraction of the decay $B ^0_s \\!\\rightarrow D_s^\\mp K^\\pm $ .",
"Charge conjugate modes are implied throughout.",
"Our notation $B ^0 \\!\\rightarrow D ^- \\pi ^+ $ , which matches that of Ref.",
"[5], encompasses both the Cabibbo-favoured $B^0 \\rightarrow D^-\\pi ^+$ mode and the doubly-Cabibbo-suppressed $B^0 \\rightarrow D^+\\pi ^-$ mode.",
"The LHCb detector [6] is a single-arm forward spectrometer covering the pseudo-rapidity range $2<\\eta <5$ , designed for studing particles containing $b $ or $c $ quarks.",
"The detector includes a high-precision tracking system consisting of a silicon-strip vertex detector surrounding the $pp$ interaction region, a large-area silicon-strip detector located upstream of a dipole magnet with a bending power of about $4{\\rm \\,Tm}$ , and three stations of silicon-strip detectors and straw drift tubes placed downstream.",
"The combined tracking system has a momentum resolution $\\Delta p/p$ that varies from 0.4% at 5${\\mathrm {\\,Ge\\hspace{-1.00006pt}V\\!/}c}$ to 0.6% at 100${\\mathrm {\\,Ge\\hspace{-1.00006pt}V\\!/}c}$ , an impact parameter resolution of 20$\\,\\rm m$ for tracks with high transverse momentum, and a decay time resolution of 50 fs.",
"Charged hadrons are identified using two ring-imaging Cherenkov detectors.",
"Photon, electron and hadron candidates are identified by a calorimeter system consisting of scintillating-pad and pre-shower detectors, an electromagnetic calorimeter, and a hadronic calorimeter.",
"Muons are identified by a muon system composed of alternating layers of iron and multiwire proportional chambers.",
"The LHCb trigger consists of a hardware stage, based on information from the calorimeter and muon systems, followed by a software stage which applies a full event reconstruction.",
"Two categories of events are recognised based on the hardware trigger decision.",
"The first category are events triggered by tracks from signal decays which have an associated cluster in the calorimeters, and the second category are events triggered independently of the signal decay particles.",
"Events which do not fall into either of these two categories are not used in the subsequent analysis.",
"The second, software, trigger stage requires a two-, three- or four-track secondary vertex with a large value of the scalar sum of the transverse momenta ($p_{\\rm T}$ ) of the tracks, and a significant displacement from the primary interaction.",
"At least one of the tracks used to form this vertex is required to have $p_{\\rm T}> 1.7$ GeV$/c$ , an impact parameter $\\chi ^2$ $>16$ , and a track fit $\\chi ^2 $ per degree of freedom $\\chi ^2/\\textrm {ndf}$ $< 2$ .",
"A multivariate algorithm is used for the identification of the secondary vertices [7].",
"Each input variable is binned to minimise the effect of systematic differences between the trigger behaviour on data and simulated events.",
"The samples of simulated events used in this analysis are based on the Pythia 6.4 generator [8], with a choice of parameters specifically configured for LHCb [9].",
"The EvtGen package [10] describes the decay of the $B $ mesons, and the Geant4 package [11] simulates the detector response.",
"QED radiative corrections are generated with the Photos package [12].",
"The analysis is based on a sample of $pp$ collisions corresponding to an integrated luminosity of 0.37 fb$^{-1}$ , collected at the LHC in 2011 at a centre-of-mass energy $\\sqrt{s} = 7$ TeV.",
"The decay modes $B ^0_s \\!\\rightarrow D ^-_s \\pi ^+ $ and $B ^0_s \\!\\rightarrow D_s^\\mp K^\\pm $ are topologically identical and are selected using identical geometric and kinematic criteria, thereby minimising efficiency corrections in the ratio of branching fractions.",
"The decay mode $B ^0 \\!\\rightarrow D ^- \\pi ^+ $ has a similar topology to the other two, differing only in the Dalitz plot structure of the $D $ decay and the lifetime of the $D $ meson.",
"These differences are verified, using simulated events, to alter the selection efficiency at the level of a few percent, and are taken into account.",
"$B ^0_s$ ($B ^0$ ) candidates are reconstructed from a $D ^-_s$ ($D ^-$ ) candidate and an additional pion or kaon (the “bachelor” particle), with the $D ^-_s$ ($D ^-$ ) meson decaying in the $K^+ K^- \\pi ^-$ ($K^+ \\pi ^-\\pi ^-$ ) mode.",
"All selection criteria will now be specified for the $B ^0_s$ decays, and are implied to be identical for the $B ^0$ decay unless explicitly stated otherwise.",
"All final-state particles are required to satisfy a track fit $\\chi ^2 / \\textrm {ndf} < 4$ and to have a high transverse momentum and a large impact parameter $\\chi ^2$ with respect to all primary vertices in the event.",
"In order to remove backgrounds which contain the same final-state particles as the signal decay, and therefore have the same mass lineshape, but do not proceed through the decay of a charmed meson, the flight distance $\\chi ^2$ of the $D ^-_s$ from the $B ^0_s$ is required to be larger than 2.",
"Only $D ^-_s$ and bachelor candidates forming a vertex with a $\\chi ^2 / \\textrm {ndf} < 9$ are considered as $B ^0_s$ candidates.",
"The same vertex quality criterion is applied to the $D ^-_s$ candidates.",
"The $B ^0_s$ candidate is further required to point to the primary vertex imposing $\\theta _{\\textrm {flight}} < 0.8$ degrees, where $\\theta _{\\textrm {flight}}$ is the angle between the candidate momentum vector and the line between the primary vertex and the $B ^0_s$ vertex.",
"The $B ^0_s$ candidates are also required to have a $\\chi ^2$ of their impact parameter with respect to the primary vertex less than 16.",
"Further suppression of combinatorial backgrounds is achieved using a gradient boosted decision tree technique [13] identical to the decision tree used in the previously published determination of $f_s/f_d$ with the hadronic decays [14].",
"The optimal working point is evaluated directly from a sub-sample of $B ^0_s \\!\\rightarrow D ^-_s \\pi ^+ $ events, corresponding to 10$\\%$ of the full dataset used, distributed evenly over the data taking period and selected using particle identification and trigger requirements.",
"The chosen figure of merit is the significance of the $B ^0_s \\!\\rightarrow D_s^\\mp K^\\pm $ signal, scaled according to the Cabibbo suppression relative to the $B ^0_s \\!\\rightarrow D ^-_s \\pi ^+ $ signal, with respect to the combinatorial background.",
"The significance exhibits a wide plateau around its maximum, and the optimal working point is chosen at the point in the plateau which maximizes the signal yield.",
"Multiple candidates occur in about $2\\%$ of the events and in such cases a single candidate is selected at random."
],
[
"Particle identification",
"Particle identification (PID) criteria serve two purposes in the selection of the three signal decays $B ^0 \\!\\rightarrow D ^- \\pi ^+ $ , $B ^0_s \\!\\rightarrow D ^-_s \\pi ^+ $ and $B ^0_s \\!\\rightarrow D_s^\\mp K^\\pm $ .",
"When applied to the decay products of the $D ^-_s$ or $D ^-$ , they suppress misidentified backgrounds which have the same bachelor particle as the signal mode under consideration, henceforth the “cross-feed” backgrounds.",
"When applied to the bachelor particle (pion or kaon) they separate the Cabibbo-favoured from the Cabibbo-suppressed decay modes.",
"All PID criteria are based on the differences in log-likelihood (DLL) between the kaon, proton, or pion hypotheses.",
"Their efficiencies are obtained from calibration samples of $D ^{*+} \\rightarrow (D ^0 \\rightarrow K^- \\pi ^+) \\pi ^+ $ and $焃 \\rightarrow p\\pi ^- $ signals, which are themselves selected without any PID requirements.",
"These samples are split according to the magnet polarity, binned in momentum and $p_{\\rm T}$ , and then reweighted to have the same momentum and $p_{\\rm T}$ distributions as the signal decays under study.",
"The selection of a pure $B ^0 \\!\\rightarrow D ^- \\pi ^+ $ sample can be accomplished with minimal PID requirements since all cross-feed backgrounds are less abundant than the signal.",
"The $\\overline{焃} ^0_b \\!\\rightarrow \\overline{焃} _c^- \\pi ^+ $ background is suppressed by requiring that both pions produced in the $D ^-$ decay satisfy $\\textrm {DLL}_{\\pi -p}>-10$ , and the $B ^0 \\!\\rightarrow D ^- K ^+ $ background is suppressed by requiring that the bachelor pion satisfies $\\textrm {DLL}_{K-\\pi }<0$ .",
"The selection of a pure $B ^0_s \\!\\rightarrow D ^-_s \\pi ^+ $ or $B ^0_s \\!\\rightarrow D_s^\\mp K^\\pm $ sample requires the suppression of the $B ^0 \\!\\rightarrow D ^- \\pi ^+ $ and $\\overline{焃} ^0_b \\!\\rightarrow \\overline{焃} _c^- \\pi ^+ $ backgrounds, whereas the combinatorial background contributes to a lesser extent.",
"The $D^{-}$ contamination in the $D^{-}_{s}$ data sample is reduced by requiring that the kaon which has the same charge as the pion in $D ^-_s \\rightarrow K^+ K^- \\pi ^- $ satisfies $\\textrm {DLL}_{K-\\pi }>5$ .",
"In addition, the other kaon is required to satisfy $\\textrm {DLL}_{K-\\pi }>0$ .",
"This helps to suppress combinatorial as well as doubly misidentified backgrounds.",
"For the same reason the pion is required to have $\\textrm {DLL}_{K-\\pi }<5$ .",
"The contamination of $\\overline{焃} ^0_b \\!\\rightarrow \\overline{焃} _c^- \\pi ^+ $ , $\\overline{焃} _c^- \\rightarrow \\overline{p } K^+ \\pi ^-$ is reduced by applying a requirement of $\\textrm {DLL}_{K-p}>0$ to the candidates that, when reconstructed under the $\\overline{焃} _c^- \\rightarrow \\overline{p } K^+ \\pi ^-$ mass hypothesis, lie within $\\pm 21$ MeV$/c^2$ of the $\\overline{焃} _c^- $ mass.",
"Because of its larger branching fraction, $B ^0_s \\!\\rightarrow D ^-_s \\pi ^+ $ is a significant background to $B ^0_s \\!\\rightarrow D_s^\\mp K^\\pm $ .",
"It is suppressed by demanding that the bachelor satisfies the criterion $\\textrm {DLL}_{K-\\pi }>5$ .",
"Conversely, a sample of $B ^0_s \\!\\rightarrow D ^-_s \\pi ^+ $ , free of $B ^0_s \\!\\rightarrow D_s^\\mp K^\\pm $ contamination, is obtained by requiring that the bachelor satisfies $\\textrm {DLL}_{K-\\pi }<0$ .",
"The efficiency and misidentification probabilities for the PID criterion used to select the bachelor, $D ^-$ , and $D ^-_s$ candidates are summarised in Table REF .",
"Table: PID efficiency and misidentification probabilities, separated according to the up (U) and down (D) magnetpolarities.",
"The first two lines refer to the bachelor track selection,the third line is the D - D ^- efficiency and the fourth the D s - D ^-_s efficiency.",
"Probabilitiesare obtained from the efficiencies in the D *+ D ^{*+} calibration sample,binned in momentum and p T p_{\\rm T}.",
"Only bachelor tracks with momentum below100 GeV/c/c are considered.",
"The uncertainties shown are the statistical uncertainties due tothe finite number of signal events in the PID calibration samples."
],
[
"Mass fits",
"The fits to the invariant mass distributions of the $B ^0_s \\!\\rightarrow D ^-_s \\pi ^+ $ and $B ^0_s \\!\\rightarrow D_s^\\mp K^\\pm $ candidates require knowledge of the signal and background shapes.",
"The signal lineshape is taken from a fit to simulated signal events which had the full trigger, reconstruction, and selection chain applied to them.",
"Various lineshape parameterisations have been examined.",
"The best fit to the simulated event distributions is obtained with the sum of two Crystal Ball functions [15] with a common peak position and width, and opposite side power-law tails.",
"Mass shifts in the signal peaks relative to world average values [5], arising from an imperfect detector alignment [16], are observed in the data and are accounted for.",
"A constraint on the $D ^-_s$ meson mass is used to improve the $B ^0_s$ mass resolution.",
"Three kinds of backgrounds need to be considered: fully reconstructed (misidentified) backgrounds, partially reconstructed backgrounds with or without misidentification (e.g.",
"$B ^0_s \\!\\rightarrow D ^{*-}_s K ^+ $ or $B ^0_s \\!\\rightarrow D ^-_s \\rho ^{+}$ ), and combinatorial backgrounds.",
"The three most important fully reconstructed backgrounds are $B ^0 \\!\\rightarrow D ^-_s K ^+ $ and $B ^0_s \\!\\rightarrow D ^-_s \\pi ^+ $ for $B ^0_s \\!\\rightarrow D_s^\\mp K^\\pm $ , and $B ^0 \\!\\rightarrow D ^- \\pi ^+ $ for $B ^0_s \\!\\rightarrow D ^-_s \\pi ^+ $ .",
"The mass distribution of the $B ^0 \\!\\rightarrow D ^- \\pi ^+ $ events does not suffer from fully reconstructed backgrounds.",
"In the case of the $B ^0 \\!\\rightarrow D ^-_s K ^+ $ decay, which is fully reconstructed under its own mass hypothesis, the signal shape is fixed to be the same as for $B ^0_s \\!\\rightarrow D_s^\\mp K^\\pm $ and the peak position is varied.",
"The shapes of the misidentified backgrounds $B ^0 \\!\\rightarrow D ^- \\pi ^+ $ and $B ^0_s \\!\\rightarrow D ^-_s \\pi ^+ $ are taken from data using a reweighting procedure.",
"First, a clean signal sample of $B ^0 \\!\\rightarrow D ^- \\pi ^+ $ and $B ^0_s \\!\\rightarrow D ^-_s \\pi ^+ $ decays is obtained by applying the PID selection for the bachelor track given in Sect. .",
"The invariant mass of these decays under the wrong mass hypothesis ($B ^0_s \\!\\rightarrow D ^-_s \\pi ^+ $ or $B ^0_s \\!\\rightarrow D_s^\\mp K^\\pm $ ) depends on the momentum of the misidentified particle.",
"This momentum distribution must therefore be reweighted by taking into account the momentum dependence of the misidentifaction rate.",
"This dependence is obtained using a dedicated calibration sample of prompt $D ^{*+}$ decays.",
"The mass distributions under the wrong mass hypothesis are then reweighted using this momentum distribution to obtain the $B ^0 \\!\\rightarrow D ^- \\pi ^+ $ and $B ^0_s \\!\\rightarrow D ^-_s \\pi ^+ $ mass shapes under the $B ^0_s \\!\\rightarrow D ^-_s \\pi ^+ $ and $B ^0_s \\!\\rightarrow D_s^\\mp K^\\pm $ mass hypotheses, respectively.",
"For partially reconstructed backgrounds, the probability density functions (PDFs) of the invariant mass distributions are taken from samples of simulated events generated in specific exclusive modes and are corrected for mass shifts, momentum spectra, and PID efficiencies in data.",
"The use of simulated events is justified by the observed good agreement between data and simulation.",
"The combinatorial background in the $B ^0_s \\!\\rightarrow D ^-_s \\pi ^+ $ and $B ^0 \\!\\rightarrow D ^- \\pi ^+ $ fits is modelled by an exponential function where the exponent is allowed to vary in the fit.",
"The resulting shape and normalisation of the combinatorial backgrounds are in agreement within one standard deviation with the distribution of a wrong-sign control sample (where the $D^{-}_{s}$ and the bachelor track have the same charges).",
"The shape of the combinatorial background in the $B ^0_s \\!\\rightarrow D_s^\\mp K^\\pm $ fit cannot be left free because of the partially reconstructed backgrounds which dominate in the mass region below the signal peak.",
"In this case, therefore, the combinatorial slope is fixed to be flat, as measured from the wrong sign events.",
"In the $B ^0_s \\!\\rightarrow D_s^\\mp K^\\pm $ fit, an additional complication arises due to backgrounds from $焃 ^0_b \\!\\rightarrow D ^-_s p$ and $焃 ^0_b \\!\\rightarrow D ^{*-}_s p$ , which fall in the signal region when misreconstructed.",
"To avoid a loss of $B ^0_s \\!\\rightarrow D_s^\\mp K^\\pm $ signal, no requirement is made on the $\\textrm {DLL}_{K-p}$ of the bachelor particle.",
"Instead, the $焃 ^0_b \\!\\rightarrow D ^-_s p$ mass shape is obtained from simulated $焃 ^0_b \\!\\rightarrow D ^-_s p$ decays, which are reweighted in momentum using the efficiency of the $\\textrm {DLL}_{K-\\pi }> 5$ requirement on protons.",
"The $焃 ^0_b \\!\\rightarrow D ^{*-}_s p$ mass shape is obtained by shifting the $焃 ^0_b \\!\\rightarrow D ^-_s p$ mass shape downwards by 200 MeV$/c^2$ .",
"The branching fractions of $焃 ^0_b \\!\\rightarrow D ^-_s p$ and $焃 ^0_b \\!\\rightarrow D ^{*-}_s p$ are assumed to be equal, motivated by the fact that the decays $B^0\\rightarrow D^-D_s^+$ and $B^0\\rightarrow D^-D_s^{*+}$ (dominated by similar tree topologies) have almost equal branching fractions.",
"Therefore the overall mass shape is formed by summing the $焃 ^0_b \\!\\rightarrow D ^-_s p$ and $焃 ^0_b \\!\\rightarrow D ^{*-}_s p$ shapes with equal weight.",
"The signal yields are obtained from unbinned extended maximum likelihood fits to the data.",
"In order to achieve the highest sensitivity, the sample is separated according to the two magnet polarities, allowing for possible differences in PID performance and in running conditions.",
"A simultaneous fit to the two magnet polarities is performed for each decay, with the peak position and width of each signal, as well as the combinatorial background shape, shared between the two.",
"The fit under the $B ^0_s \\!\\rightarrow D ^-_s \\pi ^+ $ hypothesis requires a description of the $B ^0 \\!\\rightarrow D ^- \\pi ^+ $ background.",
"A fit to the $B ^0 \\!\\rightarrow D ^- \\pi ^+ $ spectrum is first performed to determine the yield of signal $B ^0 \\!\\rightarrow D ^- \\pi ^+ $ events, shown in Fig.",
"REF .",
"The expected $B ^0 \\!\\rightarrow D ^- \\pi ^+ $ contribution under the $B ^0_s \\!\\rightarrow D ^-_s \\pi ^+ $ hypothesis is subsequently constrained with a $10\\%$ uncertainty to account for uncertainties on the PID efficiencies.",
"The fits to the $B ^0_s \\!\\rightarrow D ^-_s \\pi ^+ $ candidates are shown in Fig.",
"REF and the fit results for both decay modes are summarised in Table REF .",
"The peak position of the signal shape is varied, as are the yields of the different partially reconstructed backgrounds (except $B ^0 \\!\\rightarrow D ^- \\pi ^+ $ ) and the shape of the combinatorial background.",
"The width of the signal is fixed to the values found in the $B ^0 \\!\\rightarrow D ^- \\pi ^+ $ fit ($17.2$ MeV$/c^2$ ), scaled by the ratio of widths observed in simulated events between $B ^0 \\!\\rightarrow D ^- \\pi ^+ $ and $B ^0_s \\!\\rightarrow D ^-_s \\pi ^+ $ decays ($0.987$ ).",
"The accuracy of these fixed parameters is evaluated using ensembles of simulated experiments described in Sect. .",
"The yield of $B ^0 \\!\\rightarrow D ^-_s \\pi ^+ $ is fixed to be 2.9% of the $B ^0_s \\!\\rightarrow D ^-_s \\pi ^+ $ signal yield, based on the world average branching fraction of $B ^0 \\!\\rightarrow D ^-_s \\pi ^+ $ of $(2.16 \\pm 0.26) \\times 10^{-5}$ , the value of $f_s/f_d$ given in [4], and the value of the branching fraction computed in this paper.",
"The shape used to fit this component is the sum of two Crystal Ball functions obtained from the $B ^0_s \\!\\rightarrow D ^-_s \\pi ^+ $ sample with the peak position fixed to the value obtained with the fit of the $B ^0 \\!\\rightarrow D ^- \\pi ^+ $ data sample and the width fixed to the width of the $B ^0_s \\!\\rightarrow D ^-_s \\pi ^+ $ peak.",
"Figure: Mass distribution of the B 0 →D - π + B ^0 \\!\\rightarrow D ^- \\pi ^+ candidates (top) and B s 0 →D s - π + B ^0_s \\!\\rightarrow D ^-_s \\pi ^+ candidates (bottom).",
"The stacked background shapes follow the same top-to-bottom order in the legend and the plot.",
"For illustration purposes the plot includes events from both magnet polarities, but they are fitted separately as described in the text.The $\\overline{焃} ^0_b \\!\\rightarrow \\overline{焃} _c^- \\pi ^+ $ background is negligible in this fit owing to the effectiveness of the veto procedure described earlier.",
"Nevertheless, a $\\overline{焃} ^0_b \\!\\rightarrow \\overline{焃} _c^- \\pi ^+ $ component, whose yield is allowed to vary, is included in the fit (with the mass shape obtained using the reweighting procedure on simulated events described previously) and results in a negligible contribution, as expected.",
"Table: Results of the mass fits to the B 0 →D - π + B ^0 \\!\\rightarrow D ^- \\pi ^+ , B s 0 →D s - π + B ^0_s \\!\\rightarrow D ^-_s \\pi ^+ , and B s 0 →D s ∓ K ± B ^0_s \\!\\rightarrow D_s^\\mp K^\\pm candidates separated according to the up (U) and down (D) magnetpolarities.",
"In the B s 0 →D s ∓ K ± B ^0_s \\!\\rightarrow D_s^\\mp K^\\pm case,the number quoted for B s 0 →D s - π + B ^0_s \\!\\rightarrow D ^-_s \\pi ^+ also includes a smallnumber of B 0 →D - π + B ^0 \\!\\rightarrow D ^- \\pi ^+ events which have the same mass shape (20 events from the expected misidentification).See Table 3 for the constrained values used in the B s 0 →D s ∓ K ± B ^0_s \\!\\rightarrow D_s^\\mp K^\\pm decay fit for the partially reconstructed backgrounds and the B 0 →D - K + B ^0 \\!\\rightarrow D ^- K ^+ decay channel.The fits for the $B ^0_s \\!\\rightarrow D_s^\\mp K^\\pm $ candidates are shown in Fig.",
"REF and the fit results are collected in Table REF .",
"There are numerous reflections which contribute to the mass distribution.",
"The most important reflection is $B ^0_s \\!\\rightarrow D ^-_s \\pi ^+ $ , whose shape is taken from the earlier $B ^0_s \\!\\rightarrow D ^-_s \\pi ^+ $ signal fit, reweighted according to the efficiencies of the applied PID requirements.",
"Furthermore, the yield of the $B ^0 \\!\\rightarrow D ^- K ^+ $ reflection is constrained to the values in Table REF .",
"In addition, there is potential cross-feed from partially reconstructed modes with a misidentified pion such as $B ^0_s \\!\\rightarrow D ^-_s \\rho ^{+}$ , as well as several small contributions from partially reconstructed backgrounds with similar mass shapes.",
"The yields of these modes, whose branching fractions are known or can be estimated (e.g.",
"$B ^0_s \\!\\rightarrow D ^-_s \\rho ^{+}$ , $B ^0_s \\!\\rightarrow D ^-_s K^{*+}$ ), are constrained to the values in Table REF , based on criteria such as relative branching fractions and reconstruction efficiencies and PID probabilities.",
"An important cross-check is performed by comparing the fitted value of the yield of misidentified $B ^0_s \\!\\rightarrow D ^-_s \\pi ^+ $ events ($318\\pm 30$ ) to the yield expected from PID efficiencies ($370\\pm 11$ ) and an agreement is found.",
"Figure: Mass distribution of the B s 0 →D s ∓ K ± B ^0_s \\!\\rightarrow D_s^\\mp K^\\pm candidates.",
"The stacked background shapes follow the same top-to-bottom order in the legend and the plot.",
"For illustration purposes the plot includesevents from both magnet polarities, but they are fitted separately as described in the text.Table: Gaussian constraints on the yields of partially reconstructed and misidentified backgrounds applied in the B s 0 →D s ∓ K ± B ^0_s \\!\\rightarrow D_s^\\mp K^\\pm fit,separated according to the up (U) and down (D) magnet polarities."
],
[
"Systematic uncertainties",
"The major systematic uncertainities on the measurement of the relative branching fraction of $B ^0_s \\!\\rightarrow D_s^\\mp K^\\pm $ and $B ^0_s \\!\\rightarrow D ^-_s \\pi ^+ $ are related to the fit, PID calibration, and trigger and offline selection efficiency corrections.",
"Systematic uncertainties related to the fit are evaluated by generating large sets of simulated experiments using the nominal fit, and then fitting them with a model where certain parameters are varied.",
"To give two examples, the signal width is deliberately fixed to a value different from the width used in the generation, or the combinatorial background slope in the $B ^0_s \\!\\rightarrow D_s^\\mp K^\\pm $ fit is fixed to the combinatorial background slope found in the $B ^0_s \\!\\rightarrow D ^-_s \\pi ^+ $ fit.",
"The deviations of the peak position of the pull distributions from zero are then included in the systematic uncertainty.",
"Table: Relative systematic uncertainities on the branching fraction ratios.In the case of the $B ^0_s \\!\\rightarrow D_s^\\mp K^\\pm $ fit the presence of constraints for the partially reconstructed backgrounds must be considered.",
"The generic extended likelihood function can be written as $\\mathcal {L} = e^{-N} N^{N_{\\rm obs}}\\times \\prod _j G(N^j; N^j_c, \\sigma _{N^j_0})\\times \\prod _{i=1}^{N_{\\rm obs}} P(m_i; \\vec{\\lambda })\\,,$ where the first factor is the extended Poissonian likelihood in which $N$ is the total number of fitted events, given by the sum of the fitted component yields $N = \\sum _k N_k$ .",
"The fitted data sample contains $N_{\\rm obs}$ events.",
"The second factor is the product of the $j$ external constraints on the yields, $j<k$ , where $G$ stands for a Gaussian PDF, and $N_c \\pm \\sigma _{N_0}$ is the constraint value.",
"The third factor is a product over all events in the sample, $P$ is the total PDF of the fit, $P(m_i; \\vec{\\lambda }) = \\sum _k N_k P_k(m_i; \\vec{\\lambda }_k)$ , and $\\vec{\\lambda }$ is the vector of parameters that define the mass shape and are not fixed in the fit.",
"Each simulated dataset is generated by first varing the component yield $N_k$ using a Poissonian PDF, then sampling the resulting number of events from $P_k$ , and repeating the procedure for all components.",
"In addition, constraint values $N_c^j$ used when fitting the simulated dataset are generated by drawing from $G(N; N^j_0, \\sigma _{N^j_0})$ , where $N^j_0$ is the true central value of the constraint, while in the nominal fit to the data $N_c^j = N^j_0$ .",
"The sources of systematic uncertainty considered for the fit are signal widths, the slope of the combinatorial backgrounds, and constraints placed on specific backgrounds.",
"The largest deviations are due to the signal widths and the fixed slope of the combinatorial background in the $B ^0_s \\!\\rightarrow D_s^\\mp K^\\pm $ fit.",
"The systematic uncertainty related to PID enters in two ways: firstly as an uncertainty on the overall efficiencies and misidentification probabilities, and secondly from the shape for the misidentified backgrounds which relies on correct reweighting of PID efficiency versus momentum.",
"The absolute errors on the individual $K$ and $\\pi $ efficiencies, after reweighting of the $D ^{*+}$ calibration sample, have been determined for the momentum spectra that are relevant for this analysis, and are found to be 0.5% for $\\textrm {DLL}_{K-\\pi }< 0$ and 0.5% for $\\textrm {DLL}_{K-\\pi }> 5$ .",
"The observed signal yields are corrected by the difference observed in the (non-PID) selection efficiencies of different modes as measured from simulated events: $\\epsilon (B ^0_s \\!\\rightarrow D ^-_s \\pi ^+ )/\\epsilon (B ^0 \\!\\rightarrow D ^- \\pi ^+ ) &=& 1.015\\;,\\\\\\epsilon (B ^0_s \\!\\rightarrow D ^-_s \\pi ^+ )/\\epsilon (B ^0_s \\!\\rightarrow D_s^\\mp K^\\pm ) &=& 1.061\\;.$ A systematic uncertainty is assigned on the ratio to account for percent level differences between the data and the simulation.",
"These are dominated by the simulation of the hardware trigger.",
"All sources of systematic uncertainty are summarized in Table REF ."
],
[
"Determination of the branching fractions",
"The $B ^0_s \\!\\rightarrow D_s^\\mp K^\\pm $ branching fraction relative to $B ^0_s \\!\\rightarrow D ^-_s \\pi ^+ $ is obtained by correcting the raw signal yields for PID and selection efficiency differences $\\frac{{\\cal B}\\left(B ^0_s \\!\\rightarrow D_s^\\mp K^\\pm \\right)}{{\\cal B}\\left(B ^0_s \\!\\rightarrow D ^-_s \\pi ^+ \\right)} =\\frac{N_{B ^0_s \\!\\rightarrow D_s^\\mp K^\\pm }}{N_{B ^0_s \\!\\rightarrow D ^-_s \\pi ^+ }}\\frac{\\epsilon ^{\\textrm {PID}}_{B ^0_s \\!\\rightarrow D ^-_s \\pi ^+ }}{\\epsilon ^{\\textrm {PID}}_{B ^0_s \\!\\rightarrow D_s^\\mp K^\\pm }}\\frac{\\epsilon ^{\\textrm {Sel}}_{B ^0_s \\!\\rightarrow D ^-_s \\pi ^+ }}{\\epsilon ^{\\textrm {Sel}}_{B ^0_s \\!\\rightarrow D_s^\\mp K^\\pm }}\\;,$ where $\\epsilon _{X}$ is the efficiency to reconstruct decay mode $X$ and $N_{X}$ is the number of observed events in this decay mode.",
"The PID efficiencies are given in Table REF , and the ratio of the two selection efficiencies is $0.943\\pm 0.013$ .",
"The ratio of the branching fractions of $B ^0_s \\!\\rightarrow D_s^\\mp K^\\pm $ relative to $B ^0_s \\!\\rightarrow D ^-_s \\pi ^+ $ is determined separately for the down ($0.0601\\pm 0.0056$ ) and up ($0.0694\\pm 0.0066$ ) magnet polarities and the two results are in good agreement.",
"The quoted errors are purely statistical.",
"The combined result is $\\frac{{\\cal B}\\left(B ^0_s \\!\\rightarrow D_s^\\mp K^\\pm \\right)}{{\\cal B}\\left(B ^0_s \\!\\rightarrow D ^-_s \\pi ^+ \\right)} = 0.0646\\pm 0.0043 \\pm 0.0025 \\;,$ where the first uncertainty is statistical and the second is the total systematic uncertainty from Table REF .",
"The relative yields of $B ^0_s \\!\\rightarrow D ^-_s \\pi ^+ $ and $B ^0 \\!\\rightarrow D ^- \\pi ^+ $ are used to extract the branching fraction of $B ^0_s \\!\\rightarrow D ^-_s \\pi ^+ $ from the following relation ${\\cal B}(B ^0_s \\!\\rightarrow D ^-_s \\pi ^+ )= {\\cal B}\\left(B ^0 \\!\\rightarrow D ^- \\pi ^+ \\right)\\frac{\\epsilon _{B ^0 \\!\\rightarrow D ^- \\pi ^+ }}{\\epsilon _{B ^0_s \\!\\rightarrow D ^-_s \\pi ^+ }}\\frac{N_{B ^0_s \\!\\rightarrow D ^-_s \\pi ^+ }{\\cal B}\\left(D^- \\rightarrow K^+ \\pi ^-\\pi ^-\\right)}{\\frac{f_s}{f_d}N_{B ^0 \\!\\rightarrow D ^- \\pi ^+ }{\\cal B}\\left(D^-_s \\rightarrow K^- K^+ \\pi ^-\\right)}\\;,$ using the recent $f_s/f_d$ measurement from semileptonic decays [4] $\\frac{f_s}{f_d}= 0.268 \\pm 0.008^{+0.022}_{-0.020}\\;,$ where the first uncertainty is statistical and the second systematic.",
"Only the semileptonic result is used since the hadronic determination of $f_s/f_d$ relies on theoretical assumptions about the ratio of the branching fractions of the $B ^0_s \\!\\rightarrow D ^-_s \\pi ^+ $ and $B ^0 \\!\\rightarrow D ^- \\pi ^+ $ decays.",
"In addition, the following world average values [5] for the $B$ and $D$ branching fractions are used ${\\cal B}(B ^0 \\!\\rightarrow D ^- \\pi ^+ ) &=& \\left(2.68\\pm 0.13\\right)\\times 10^{-3}\\;,\\\\{\\cal B}(D ^- \\rightarrow K^+ \\pi ^- \\pi ^-) &=& \\left(9.13\\pm 0.19\\right)\\times 10^{-2}\\;,\\\\{\\cal B}(D ^-_s \\rightarrow K^+ K^- \\pi ^-) &=& \\left(5.49\\pm 0.27\\right)\\times 10^{-2}\\;,$ leading to ${\\cal B}(B ^0_s \\!\\rightarrow D ^-_s \\pi ^+ ) &=& (2.95 \\pm 0.05 \\pm 0.17^{+0.18}_{-0.22})\\times 10^{-3}\\;,\\\\{\\cal B}(B ^0_s \\!\\rightarrow D_s^\\mp K^\\pm ) &=& (1.90 \\pm 0.12 \\pm 0.13^{+0.12}_{-0.14})\\times 10^{-4}\\;,$ where the first uncertainty is statistical, the second is the experimental systematics (as listed in Table REF ) plus the uncertainty arising from the $B ^0 \\!\\rightarrow D ^- \\pi ^+ $ branching fraction, and the third is the uncertainty (statistical and systematic) from the semileptonic $f_s/f_d$ measurement.",
"Both measurements are significantly more precise than the existing world averages [5]."
],
[
"Acknowledgments",
"We express our gratitude to our colleagues in the CERN accelerator departments for the excellent performance of the LHC.",
"We thank the technical and administrative staff at CERN and at the LHCb institutes, and acknowledge support from the National Agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); CERN; NSFC (China); CNRS/IN2P3 (France); BMBF, DFG, HGF and MPG (Germany); SFI (Ireland); INFN (Italy); FOM and NWO (The Netherlands); SCSR (Poland); ANCS (Romania); MinES of Russia and Rosatom (Russia); MICINN, XuntaGal and GENCAT (Spain); SNSF and SER (Switzerland); NAS Ukraine (Ukraine); STFC (United Kingdom); NSF (USA).",
"We also acknowledge the support received from the ERC under FP7 and the Region Auvergne.",
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[
[
"Diffusion along transition chains of invariant tori and Aubry-Mather\n sets"
],
[
"Abstract We describe a topological mechanism for the existence of diffusing orbits in a dynamical system satisfying the following assumptions: (i) the phase space contains a normally hyperbolic invariant manifold diffeomorphic to a two-dimensional annulus, (ii) the restriction of the dynamics to the annulus is an area preserving monotone twist map, (iii) the annulus contains sequences of invariant one-dimensional tori that form transition chains, i.e., the unstable manifold of each torus has a topologically transverse intersection with the stable manifold of the next torus in the sequence, (iv) the transition chains of tori are interspersed with gaps created by resonances, (v) within each gap there is prescribed a finite collection of Aubry-Mather sets.",
"Under these assumptions, there exist trajectories that follow the transition chains, cross over the gaps, and follow the Aubry-Mather sets within each gap, in any specified order.",
"This mechanism is related to the Arnold diffusion problem in Hamiltonian systems.",
"In particular, we prove the existence of diffusing trajectories in the large gap problem of Hamiltonian systems.",
"The argument is topological and constructive."
],
[
"Introduction",
"In this paper we study a topological mechanism of diffusion and chaotic orbits related to the Arnold diffusion problem.",
"We consider a normally hyperbolic invariant manifold diffeomorphic to a two-dimensional annulus.",
"We assume that the dynamics restricted to the annulus is given by an area preserving monotone twist map.",
"We assume that inside the annulus there exist invariant primary one-dimensional tori (homotopically nontrivial invariant closed curves) with the property that the unstable manifold of each torus has topologically transverse intersections with the stable manifolds of all sufficiently close tori.",
"Sequences of such tori and their heteroclinic connections form transition chains of tori.",
"The successive transition chains of tori are interspersed with gaps.",
"We assume that each gap is a region in the annulus bounded by two invariant primary tori of one of the following types: (i) a Birkhoff Zone of Instability, i.e., it contains no invariant primary torus in its interior (ii) it contains only finitely many invariant primary tori in its interior, such that each torus is either isolated, or it consists of a hyperbolic periodic orbit together with its stable and unstable manifolds that are assumed to coincide.",
"Within each gap we prescribe a finite collection of Aubry-Mather sets.",
"We prove the existence of orbits that shadow the primary tori in each transition chain, cross over the gaps that separate the successive transition chains, and also shadow the specified Aubry-Mather sets within each gap.",
"The motivation for this result is the Arnold diffusion problem of Hamiltonian systems.",
"This problem asserts that all sufficiently small perturbations of generic, integrable Hamiltonian systems exhibit orbits along which the action variable changes substantially; also, there exist chaotic orbits that can be coded through symbolic dynamics.",
"A classification of nearly integrable systems proposed in [13] distinguishes between a priori stable systems, in which the unperturbed system can be expressed in terms of action-angle variables only, and a priori unstable systems, in which the unperturbed system contains both action-angle and hyperbolic variables.",
"In the a priori stable case analytical results have been announced in [47].",
"In the a priori unstable case there have been several analytical results in the last several years (see [13], [54], [51], [52], [11], [18], [3], [12]).",
"Some of the approaches involve variational methods, geometric methods, or topological methods.",
"See [20] for an overview on the Arnold diffusion problem, applications, and additional references.",
"It is relevant in applications to detect, combine, and compare different mechanisms of diffusion displayed by concrete systems.",
"In many models, as well as in numerical experiments, diffusion can only be observed for perturbations of sizes much larger than those considered by the analytical approaches [14], [40], [39], [36], [28].",
"For these types of problems, the geometric and topological approaches are particularly advantageous as they yield constructive methods to detect diffusion, quantitative estimates on diffusing orbits, and explicit conditions that can be verified in concrete examples.",
"In this paper we describe a general method to establish the existence of diffusing orbits for a large class of dynamical systems.",
"The dynamical systems under consideration are not assumed to be small perturbations of integrable Hamiltonians.",
"Moreover, some systems that are not Hamiltonian can be considered.",
"Our method requires the existence of certain geometric objects that organize the dynamics, and employs topological tools to establish the existence of diffusing orbits.",
"The existence of the geometric objects can be verified in concrete systems through analytical methods, or through numerical methods, or through a combination of thereof.",
"We illustrate our method in the case of a Hamiltonian system consisting of a pendulum and a rotator with a small periodic coupling.",
"We give an analytic argument for the existence of diffusing orbits.",
"In [15] the topological method is applied to show, with the aid of a computer, the existence of diffusing orbits in the spatial restricted three-body problem, where the two primaries are the Sun and the Earth; this model is not nearly integrable.",
"Similar ideas appear in [10], [21], [22].",
"The diffusing orbits detected by this approach follow transition chains of invariant tori up to the gaps between these transition chains, and then cross the gaps following the inner dynamics restricted to the annulus.",
"The orbits that cross the gaps follow Birkhoff connecting orbits that go from one boundary of the gap to the other, or Mather connecting orbits, that shadow a prescribed sequence of Aubry-Mather set inside the gap, or homoclinic orbits.",
"Related approaches, that use Birkhoff's ideas to understand generic drift properties of random iterations of exact-symplectic twist maps, appear in [48], [38].",
"These yield the concept of a polysystem, which is a locally constant skew-product over a Bernoulli shift [43].",
"Polysystems are a key ingredient in the approach of the instability problem in a priori unstable systems in [6].",
"Arnold diffusion-type orbits for random iterations of flows of families of Tonelli Hamiltonians are studied in [42], based on an extension of the pseudograph method from [3].",
"The diffusing orbits that we obtain in this paper are similar to those found through variational methods, as in [11], [12]; they are however not necessarily action minimizing.",
"Also, we do not need the unperturbed Hamiltonian to have positive-definite normal torsion.",
"This paper completes the investigations undertaken in [25], [26] where some non-generic conditions, or some conditions difficult to verify in concrete systems, were assumed.",
"The present paper assumes very general conditions and provides a new topological mechanism of diffusion based on Aubry-Mather sets."
],
[
"Main result",
"In this section we state the assumptions and the main result of this paper.",
"After the statement, the assumptions and the main result are explained and exemplified.",
"(A1) $M$ is a $n$ -dimensional $C^r$ -differentiable Riemannian manifold, and $f:M\\rightarrow M$ is a $C^r$ -smooth map, for some $r\\ge 2$ .",
"(A2) There exists a 2-dimensional submanifold $\\Lambda $ in $M$ , diffeormorphic to an annulus $\\Lambda \\simeq \\mathbb {T}^1\\times [0,1]$ .",
"We assume that $f$ is 2-normally hyperbolic to $\\Lambda $ in $M$ (see Subsection REF for the definition).",
"Since $f$ is 2-normally hyperbolic to $\\Lambda $ , $W^s(\\Lambda )$ , $W^u(\\Lambda )$ , and $\\Lambda $ are all $C^2$ -differentiable.",
"Denote the dimensions of the stable and unstable manifolds of a point $x \\in \\Lambda $ by $\\text{dim}(W^s(x)) = n_s$ and $\\text{dim}(W^u(x)) = n_u$ .",
"Then, $n = 2 + n_s + n_u$ .",
"(A3) On $\\Lambda $ there is a system of angle-action coordinates $(\\phi ,I)$ , with $\\phi \\in {\\mathbb {T}}^1$ and $I \\in [0,1]$ .",
"The restriction $f|_{\\Lambda }$ of $f$ to $\\Lambda $ is a boundary component preserving, area preserving, monotone twist map, with respect to the angle-action coordinates $(\\phi ,I)$ .",
"(A4) The stable and unstable manifolds of $\\Lambda $ , $W^s(\\Lambda )$ and $W^u(\\Lambda )$ , have a differentiably transverse intersection along a 2-dimensional homoclinic channel $\\Gamma $ .",
"Since the manifolds $W^s(\\Lambda )$ and $W^u(\\Lambda )$ are $C^2$ and transverse, $\\Gamma $ is $C^2$ .",
"We assume that the scattering map $S$ associated to $\\Gamma $ is well defined, and hence is $C^1$ .",
"See Subsection REF .",
"(A5) There exists a bi-infinite sequence of Lipschitz primary invariant tori $\\lbrace T_i\\rbrace _{i\\in \\mathbb {Z}}$ in $\\Lambda $ , and a bi-infinite, increasing sequence of integers $\\lbrace i_k\\rbrace _{k\\in \\mathbb {Z}}$ with the following properties: (i) Each torus $T_i$ intersects the domain $U^-$ and the range $U^+$ of the scattering map $S$ associated to $\\Gamma $ .",
"(ii) For each $i\\in \\lbrace {i_{k}+1}, \\ldots , {i_{k+1}-1}\\rbrace $ , the image of $T_i\\cap U^-$ under the scattering map $S$ is topologically transverse to $T_{i+1}$ .",
"(iii) For each torus $T_i$ with $i\\in \\lbrace {i_{k}+2}, \\ldots , {i_{k+1}-1}\\rbrace $ , the restriction of $f$ to $T_i$ is topologically transitive.",
"(iv) Each torus $T_i$ with $i\\in \\lbrace {i_{k}+2}, \\ldots , {i_{k+1}-1}\\rbrace $ , can be $C^0$ -approximated from both sides by other primary invariant tori from $\\Lambda $ .",
"We will refer to a finite sequence $\\lbrace T_i\\rbrace _{i=i_k+1,\\ldots , i_{k+1}}$ as above as a transition chain of tori.",
"(A6) The region in $\\Lambda $ between $T_{i_k}$ and $T_{i_{k}+1}$ contains no invariant primary torus in its interior.",
"(A7) Inside each region between $T_{i_k}$ and $T_{i_{k}+1}$ there is prescribed a finite collection of Aubry-Mather sets $\\lbrace \\Sigma _{\\omega ^k_1}, \\Sigma _{\\omega ^k_2},\\ldots , \\Sigma _{\\omega ^k_{s_k}}\\rbrace $ , where $s_k\\ge 1$ , and $\\omega ^k_s$ denotes the rotation number of $\\Sigma _{\\omega ^k_s}$ .",
"Each Aubry-Mather set $\\Sigma _{\\omega ^k_s}$ is assumed to lie on some essential circle $C_{\\omega ^k_s}$ , with the circles $C_{\\omega ^k_s}$ mutually disjoint for all $s\\in \\lbrace 1,\\ldots ,s_k\\rbrace $ , and $C_{\\omega ^k_s}$ below $C_{\\omega ^k_{s^{\\prime }}}$ for all $\\omega ^k_s<\\omega ^k_{s^{\\prime }}$ .",
"The vertical ordering of these circles is relative to the $I$ -coordinate on the annulus.",
"Instead of (A6) we can consider the following condition: (A6$^{\\prime }$ ) The region $\\Lambda _k$ in $\\Lambda $ between $T_{i_k}$ and $T_{i_{k}+1}$ contains finitely many invariant primary tori $\\lbrace \\Upsilon _{h^k_1}, \\ldots , \\Upsilon _{h^k_{l_k}}\\rbrace $ , where $l_k\\ge 1$ , satisfying the following properties: (i) Each $\\Upsilon _{h^k_j}$ falls in one of the following two cases: (a) $\\Upsilon _{h^k_j}$ is an isolated invariant primary torus, i.e., an invariant torus that has a neighborhood in the annulus that does not contain any other invariant primary torus inside.",
"(b) There exists a hyperbolic periodic orbit in $\\Lambda $ such that its stable and unstable manifolds coincide.",
"Each invariant manifold has two branches, an upper branch and a lower branch (where the vertical ordering is relative to the $I$ -coordinate of the annulus).",
"Then $\\Upsilon _{h^k_j}$ is an invariant primary torus consisting of the hyperbolic periodic orbit together with the upper branches of the invariant manifolds, or consisting of the hyperbolic periodic orbit together with the lower branches of the invariant manifolds.",
"(ii) The invariant primary tori $\\lbrace \\Upsilon _{h^k_1}, \\ldots , \\Upsilon _{h^k_{l_k}}\\rbrace $ are vertically ordered, in the sense that $\\Upsilon _{h^k_j}$ is below $\\Upsilon _{h^k_{j+1}}$ , for all $j=1,\\ldots , l_k-1$ .",
"The vertical ordering of these tori is relative to the $I$ -coordinate on the annulus.",
"(iii) For each $\\Upsilon _{h^k_j}$ , $j=1,\\ldots , l_k$ , the inverse image $S^{-1}(\\Upsilon _{h^k_j}\\cap U^+)$ forms with $\\Upsilon _{h^k_j}$ a topological disk $D_{h^k_j} \\subseteq U^-$ below $\\Upsilon _{h^k_j}$ , such that $S(D_{h^k_j})\\subseteq U^+$ is a topological disk above $\\Upsilon _{h^k_j}$ , which is bounded by $\\Upsilon _{h^k_j}$ and $S(\\Upsilon _{h^k_j}\\cap U^-)$ .",
"See Fig.",
"REF .",
"Now we state the main result of the paper.",
"Theorem 2.1 Let $f:M\\rightarrow M$ be a $C^r$ -differentiable map, and let $(T_i)_{i\\in \\mathbb {Z}}$ be a sequence of invariant primary tori in $\\Lambda $ , satisfying the properties (A1) – (A6), or (A1)-(A5) and (A6 $^{\\prime }$ ), from above.",
"Then for each sequence $(\\epsilon _i)_{i\\in \\mathbb {Z}}$ of positive real numbers, there exist a point $z\\in M$ and a bi-infinite increasing sequence of integers $(N_i)_{i\\in \\mathbb {Z}}$ such that $ d(f^{N_i}(z), T_{i})<\\epsilon _i, \\textrm { for all }i\\in \\mathbb {Z}.$ In addition, if condition (A7) is assumed, and some positive integers $\\lbrace n^k_s\\rbrace _{s=1,\\ldots , s_k}$ , $k\\in \\mathbb {Z}$ are given, then there exist $z\\in M$ and $(N_i)_{i\\in \\mathbb {Z}}$ as in (REF ), and positive integers $\\lbrace m^k_s\\rbrace _{s=1,\\ldots , s_k}$ , $k\\in \\mathbb {Z}$ , such that, for each $k$ and each $s\\in \\lbrace 1,\\ldots , s_k\\rbrace $ , we have $\\pi _\\phi (f^j(w^k_s))<\\pi _\\phi (f^j(z))<\\pi _\\phi (f^j(\\bar{w}^k_s)),$ for some $w^k_s,\\bar{w}^k_s\\in \\Sigma _{\\omega ^k_s}$ and for all $j$ with $N_{i_k}+\\sum _{t=0}^{s-1} n^k_t+\\sum _{t=0}^{s-1}m^k_t\\le j\\le N_{i_k}+\\sum _{t=0}^{s} n^k_t+\\sum _{t=0}^{s-1}m^k_t.$ Figure: An illustration of the condition (A6 ' ^{\\prime }).",
"In the figure, inside the region between T i k T_{i_k} and T i k +1 T_{i_{k}+1}, three invariant primary tori are shown, Υ h j k \\Upsilon _{h^k_j}, Υ h j+1 k \\Upsilon _{h^k_{j+1}}, Υ h j+2 k \\Upsilon _{h^k_{j+2}}, where Υ h j k \\Upsilon _{h^k_j} is as in (A6 ' ^{\\prime }-i-a), and Υ h j+1 k \\Upsilon _{h^k_{j+1}}, Υ h j+2 k \\Upsilon _{h^k_{j+2}} are as in (A6 ' ^{\\prime }-i-b).",
"The vertical ordering of the tori stated condition (A6 ' ^{\\prime }-ii) is depicted.",
"For each of the tori Υ h j k \\Upsilon _{h^k_j}, Υ h j+1 k \\Upsilon _{h^k_{j+1}}, Υ h j+2 k \\Upsilon _{h^k_{j+2}}, the condition (A6 ' ^{\\prime }-iii) is also illustrated.Theorem REF asserts that if the conditions (A1)-(A6), or (A1)-(A5) and (A6$^{\\prime }$ ) are satisfied, then there exists an orbit that shadows all tori in the transition chains in the prescribed order, and also crosses over the large gaps that separate the successive transition chains.",
"In particular, there exists an orbit that travels arbitrarily far with respect to the action variable, and there also exists an orbit that executes chaotic excursions.",
"Additionally, if some Aubry-Mather sets are prescribed inside each gap that separates the successive transition chains, as in condition (A7), then there exists an orbit that, besides shadowing the transition chains, it also shadows the Aubry-Mather sets in the prescribed order.",
"Note that the tori in the transition chains are shadowed in the sense that the diffusing orbit gets arbitrarily close to these tori.",
"However, the Aubry-Mather sets are shadowed in the sense of the cyclical ordering: for each prescribed Aubry-Mather set, the diffusing orbit stays between the orbits of two points in the Aubry-Mather set, relative to the $\\phi $ -coordinate, for any prescribed time interval.",
"Now we explain each assumption.",
"Assumption (A1) describes a $C^r$ -differentiable, discrete dynamical system.",
"In applications, the map $f$ represents the first return map to a Poincaré section associated to a flow.",
"In many examples of interest the flow is a Hamiltonian flow.",
"Assumption (A2) prescribes the existence of a normally hyperbolic invariant manifold $\\Lambda $ for $f$ , which is diffeomorphic to an annulus.",
"The differentiability class $r$ and the contraction and expansion rates along the stable and unstable bundles on $\\Lambda $ are chosen so that the manifolds $\\Lambda $ , $W^u(\\Lambda )$ , $W^s(\\Lambda )$ , $\\Gamma $ are at least $C^2$ -differentiable, and the scattering map $S$ associated to $\\Gamma $ is at least $C^1$ -differentiable.",
"The relations between the rates and the differentiability of these objects is given explicitly in Subsection REF .",
"In many examples of nearly integrable Hamiltonian systems, e.g.",
"[2], one can identify a normally hyperbolic invariant manifold $\\Lambda _0$ in the unperturbed system, and use the standard theory of normal hyperbolicity to establish the persistence of a normally hyperbolic invariant manifold $\\Lambda _\\varepsilon $ diffeomorphic to $\\Lambda _0$ for the perturbed system, for all sufficiently small perturbation parameters $\\varepsilon \\ne 0$ .",
"There also exist examples, e.g.",
"from celestial mechanics, were the existence of a normally hyperbolic manifold can be established through a computer assisted proof (see [9]).",
"We note that the assumption (A2) does not require that the stable and unstable manifolds of $\\Lambda $ have equal dimensions, thus our setting includes dynamical systems that are not Hamiltonian.",
"Assumption (A3) is satisfied automatically in examples like the weakly-coupled pendulum-rotator system considered in [18], or the periodically perturbed geodesic flow on a torus considered in [17], [33].",
"Some properties of area preserving, monotone twist maps of the annulus are reviewed in Section .",
"Assumption (A4) asserts that the stable and unstable manifolds of $\\Lambda $ have a transverse intersection $\\Gamma $ along a homoclinic manifold $\\Gamma $ , such that the scattering map associated to $\\Gamma $ is well defined, and hence is $C^1$ .",
"See Subsection REF .",
"In perturbed systems one often uses a Melnikov method to establish the existence, and the persistence for all sufficiently small values of the perturbation, of a transverse intersection of the invariant manifolds.",
"Assumption (A5) prescribes the existence of a bi-infinite collection $\\lbrace T_{i}\\rbrace _{i\\in \\mathbb {Z}}$ of invariant primary Lipschitz tori that can be grouped into transition chains of the type $\\lbrace T_i\\rbrace _{i=i_k+1,\\ldots ,i_{k+1}}$ .",
"Assumption (A5)-(i) requires that each torus $T_i$ intersects the domain and the range of the scattering map.",
"Assumption (A5)-(ii) requires that each torus in a transition chain$\\lbrace T_i\\rbrace _{i=i_k+1,\\ldots ,i_{k+1}}$ is mapped by the scattering map topologically transversally across the next torus in the chain.",
"Since the tori are only Lipschitz, topological transversality (topological crossing) is used in place of differentiable transversality.",
"The definition of topological crossing can be found in [8].",
"Roughly speaking, two manifolds are topologically crossing if they can be made differentiably transverse with non-zero oriented intersection number by the means of a sufficiently small homotopy.",
"In our case the two manifolds are two 1-dimensional arcs of $S(T_i)$ and of $T_{i+1}$ in $\\Lambda $ .",
"From (A5)-(ii), it follows as in [19] that $W^u(T_i)$ has a topologically transverse intersection point with $W^s(T_{i+1})$ .",
"Condition (A5)-(iii) requires that each torus in $\\lbrace T_{i}\\rbrace _{i=i_{k}+1, \\ldots , i_{k+1}}$ , except for the end tori, are topologically transitive.",
"Assumption (A5)-(iv) says that all tori in the transition chain except for the end ones can be $C^0$ -approximated from both ways by some other invariant primary tori in $\\Lambda $ , not necessarily from the transition chain.",
"This means that for each $i\\in \\lbrace i_{k}+2, \\ldots , i_{k+1}-1\\rbrace $ there exist two sequences of invariant primary tori $(T_{j^-_l(i)})_{l\\ge 1},(T_{j^+_l(i)})_{l\\ge 1}$ in $\\Lambda $ that approach $T_i$ in the $C^0$ -topology, such that the annulus bounded by $T_{j^-_l(i)}$ and $T_{j^+_l(i)}$ contains $T_i$ in its interior for all $l$ .",
"Assumption (A6) says that every pair of successive transition chains$\\lbrace T_{i}\\rbrace _{i=i_{k-1}+1, \\ldots , i_{k}}$ and $\\lbrace T_{i}\\rbrace _{i=i_{k}+1, \\ldots , i_{k+1}}$ is separated by the region between $T_{i_{k}}$ and $T_{i_{k}+1}$ which contains no invariant primary torus in its interior.",
"A region in an annulus that is bounded by two invariant primary tori and contains no invariant primary torus in its interior is referred as a Birkhoff Zone of Instability (BZI).",
"The boundary tori have in general only Lipschitz regularity.",
"Assumptions (A5) and (A6) describe a geometric structure that is typical for the large gap problem for a priori unstable Hamiltonian systems.",
"In such systems, Melnikov theory implies that $W^u(\\Lambda )$ intersects transversally $W^s(\\Lambda )$ at an angle of order $\\varepsilon $ , where $\\varepsilon $ is the size of the perturbation.",
"The KAM theorem yields a Cantor family of smooth, invariant primary tori that survives the perturbation.",
"The family of tori is interrupted by `large gaps' of order $\\varepsilon ^{1/2}$ located at the resonant regions.",
"Using the transverse intersection between $W^u(\\Lambda )$ and $W^s(\\Lambda )$ , one can find heteroclinic connections between KAM tori that are sufficiently close, within order $\\varepsilon $ , from one another, and thus form transition chains of tori.",
"Since the large gaps are of order $\\varepsilon ^{1/2}$ and the splitting size of $W^u(\\Lambda )$ , $W^s(\\Lambda )$ is only order $\\varepsilon $ , the transition chain mechanism cannot be extended across the large gaps.",
"In our model, the large gaps are modeled by BZI's, as in assumption (A6).",
"This can be achieved by extending the transition chains to maximal transition chains, that go from the boundary of one large gap to the boundary of the next large gap.",
"The intermediate tori in the chain can be chosen as KAM tori: therefore the assumption that these tori are topologically transitive and are $C^0$ -approximable from both sides by other tori is satisfied in such cases.",
"This may not be the case for the tori at the ends of the transition chains.",
"Assumption (A7) says that inside each BZI between $T_{i_k}$ and $T_{i_{k}+1}$ there is a prescribed collection of Aubry-Mather sets $\\lbrace \\Sigma _{\\omega ^k_1}, \\Sigma _{\\omega ^k_2},\\ldots , \\Sigma _{\\omega ^k_{s_k}}\\rbrace $ that is vertically ordered.",
"The vertical ordering means that the Aubry-Mather sets lie on essential (non-invariant) circles $C_{\\omega ^k_s}$ that are graphs over the $\\phi $ -coordinate of the annulus, and with $C_{\\omega ^k_s}$ below $C_{\\omega ^k_{s^{\\prime }}}$ provided $\\omega ^k_s<\\omega ^k_{s^{\\prime }}$ ; we write $C_{\\omega ^k_s}\\prec C_{\\omega ^k_{s^{\\prime }}}$ .",
"The vertical ordering of the Aubry-Mather sets is shown for example in [27].",
"Assumption (A6$^{\\prime }$ ) is a relaxation of (A6).",
"Instead of requiring that the region in $\\Lambda $ between $T_{i_{k}}$ and $T_{i_{k}+1}$ is a BZI, it allows the existence of finitely many invariant primary tori $\\lbrace \\Upsilon _{h^k_j}\\rbrace _{j=1, \\ldots , l_k}$ that separate the region into disjoint components.",
"These invariant primary tori are either isolated or else they consist of hyperbolic periodic points together with branches of their stable and unstable manifolds which are assumed to coincide.",
"We require that the image of each $\\Upsilon _{h^k_j}$ under $S$ satisfies a certain transversality condition with $\\Upsilon _{h^k_{j+1}}$ that allows one to use the scattering map in order to move points from one side of the set to the other side of the set.",
"We note that isolated invariant tori and hyperbolic periodic points whose stable and unstable manifolds coincide do not occur in generic systems."
],
[
"Application",
"We apply Theorem REF to show the existence of diffusing orbits in an example of a nearly integrable Hamiltonian system.",
"Let $H_\\varepsilon (p, q, I,\\phi ,t)&=&h_0(I)\\pm \\left(\\frac{1}{2}p^2+V(q)\\right)+\\varepsilon h(p, q,I,\\phi , t; \\varepsilon ),$ where $(p, q, I,\\phi , t)\\in \\mathbb {R}\\times \\mathbb {T}^1\\times \\mathbb {R}\\times \\mathbb {T}^1\\times \\mathbb {T}^1$ and $h$ is a trigonometric polynomial in $(\\phi ,t)$ .",
"Here $h_0(I)$ represents a rotator, $P_{\\pm }(p,q)=\\pm (\\frac{1}{2}p^2+V(q))$ represents a pendulum, and $\\varepsilon h$ a small, periodic coupling.",
"We assume that $V$ , $h_0$ and $h$ are uniformly $C^r$ for some $r$ sufficiently large.",
"We assume that $V$ is periodic in $q$ of period 1 and has a unique non-degenerate global maximum; this implies that the pendulum has a homoclinic orbit $(p^0 (\\sigma ),q^0 (\\sigma ))$ to $(0,0)$ , with $\\sigma \\in \\mathbb {R}$ .",
"We also assume that $h_0$ satisfies a uniform twist condition $\\partial ^2h_0/\\partial I^2>\\theta $ , for some $\\theta >0$ , and for all $I$ in some interval $(I^-, I^+)$ , with $I^-<I^+$ independent of $\\varepsilon $ .",
"The Melnikov potential for the homoclinic orbit $(p^0 (\\sigma ),q^0 (\\sigma ))$ is defined by $\\mathcal {L}(I,\\phi ,t)=&\\displaystyle -\\int _{-\\infty }^{\\infty }&\\left[h(p^0(\\sigma ),q^0(\\sigma ),I,\\phi +\\omega (I)\\sigma ,t+\\sigma ;0)\\right.\\\\& &\\left.-h(0,0,I,\\phi +\\omega (I)\\sigma ,t+\\sigma ;0)\\right]d\\sigma ,$ where $\\omega (I)=(\\partial h_0/\\partial I)(I)$ .",
"We assume the following non-degeneracy conditions on the Melnikov potential: (i) For each $I\\in (I^-,I^+)$ , and each $(\\phi , t)$ in some open set in $\\mathbb {T}^1\\times \\mathbb {T}^1$ , the map $\\tau \\in \\mathbb {R} \\rightarrow \\mathcal {L}(I,\\phi -\\omega (I)\\tau ,t-\\tau )\\in \\mathbb {R}$ has a non-degenerate critical point $\\tau ^*$ , which can be parameterized as $\\tau ^*=\\tau ^*(I,\\phi ,t).$ (ii) For each $(I,\\phi , t)$ as above, the function $(I,\\phi ,t)\\rightarrow \\frac{\\partial {\\mathcal {L}}}{\\partial \\phi }(I,\\phi -\\omega (I)\\tau ^*,t-\\tau ^*)$ is non-constant, negative in the case of $P_{-}$ , and positive in the case of $P_{+}$ .",
"This example and the above conditions are considered in [18].",
"There are some additional non-degeneracy conditions on $h$ and $\\partial h/\\partial \\varepsilon $ that are required in [18]; we do not need to assume those conditions here.",
"Now we verify the conditions (A1)-(A6) from Section for this model.",
"We will rely heavily on the estimates from [18].",
"Condition (A1).",
"The time-dependent Hamiltonian in (REF ) is transformed into an autonomous Hamiltonian by introducing a new variable $A$ , symplectically conjugate with $t$ obtaining $ \\tilde{H}_\\varepsilon (p, q, I,\\phi ,A,t)=h_0(I)\\pm (\\frac{1}{2}p^2+V(q))+A+\\varepsilon h(p, q,I,\\phi , t; \\varepsilon ),$ where $(p, q, I,\\phi , A,t)\\in (\\mathbb {R}\\times \\mathbb {T}^1)^3$ .",
"We fix an energy manifold $\\lbrace \\tilde{H}_\\varepsilon =\\tilde{h}\\rbrace $ for some $\\tilde{h}$ , and restrict to the Poincaré section $\\lbrace t=1\\rbrace $ for the Hamiltonian flow.",
"The resulting manifold is a 4-dimensional manifold $M_\\epsilon $ parametrized by some coordinates $(p_\\varepsilon ,q_\\varepsilon ,I_\\varepsilon ,\\phi _\\varepsilon )$ .",
"The first return map to $M_\\varepsilon $ of the Hamiltonian flow is a $C^r$ -differentiable map $f_\\epsilon $ .",
"Condition (A2).",
"In the unperturbed case $\\varepsilon =0$ , the manifold $\\Lambda _0:=\\lbrace (p,q,I,\\phi )\\,|\\, p=q=0 \\rbrace $ is a normally hyperbolic invariant manifold for $f_0$ .",
"The dynamics on $\\Lambda _0$ is given by an integrable twist map, and $\\Lambda _0$ is foliated by invariant 1-dimensional tori.",
"For the perturbed system, $\\Lambda _0$ can be continued to a manifold $\\Lambda _\\varepsilon $ diffeomorphic to $\\Lambda _0$ , that is locally invariant for the perturbed flow for all $\\varepsilon $ sufficiently small.",
"The theory of normal hyperbolicity [30] can be used as in [18] to show that $\\Lambda _\\varepsilon $ can be extended so that it is a normally hyperbolic invariant manifold for the flow.",
"Each point in $\\Lambda _\\varepsilon $ has 1-dimensional stable and unstable manifolds.",
"The regularity of $h_0$ and the uniform twist condition allows one to apply the KAM theorem and conclude the existence of a KAM family of primary invariant tori in $\\Lambda _\\varepsilon $ that survives the perturbation, for all $\\varepsilon $ sufficiently small.",
"Condition (A3).",
"The map $f_\\varepsilon $ is symplectic.",
"This plus the twist condition on $h_0$ implies that $f_\\varepsilon $ restricted to $\\Lambda _\\varepsilon $ is an area preserving, monotone twist map.",
"Condition (A4).",
"The non-degeneracy conditions on the Melnikov function imply that $W^u(\\Lambda _\\varepsilon )$ and $W^s(\\Lambda _\\varepsilon )$ have a transverse intersection along a homoclinic manifold $\\Gamma _\\varepsilon $ , provided $\\varepsilon $ is sufficiently small.",
"Moreover, it is shown in [18], [19] that by restricting to some convenient homoclinic manifold $\\Gamma _\\varepsilon $ one can ensure that the maps $\\Omega _\\varepsilon ^\\pm :\\Gamma _\\varepsilon \\rightarrow \\Lambda _\\varepsilon $ are diffeomorphisms onto their images; thus the scattering map $S_\\varepsilon : U^- _\\varepsilon \\rightarrow U^+ _\\varepsilon $ is a diffeomorphism between some two open sets $U^-_\\varepsilon , U^+_\\varepsilon \\subseteq \\Lambda _\\varepsilon $ , of size $O(1)$ .",
"For $\\varepsilon $ fixed to some sufficiently small value, we let $\\Gamma :=\\Gamma _\\varepsilon $ and $S:=S_\\varepsilon $ .",
"Conditions (A5),(A6) and (A6$^{\\prime }$ ) The paper [18] applies an averaging procedure to reduce the dynamics on $\\Lambda _\\varepsilon $ to a normal form up to $O(\\varepsilon ^2)$ away from resonances.",
"The averaging procedure fails within the resonant regions, corresponding to the values $I_\\varepsilon (k,l)$ of the action variable where $k\\omega (I)+l=0$ .",
"A resonance is said to be of order $j$ if the $j$ -th order averaging cannot be applied about the corresponding action level set.",
"Since $h$ is a trigonometric polynomial, one has to deal with only finitely many resonant regions.",
"Outside the resonant regions one applies the KAM theorem and obtain KAM tori that are at a distance of order $O(\\varepsilon ^{3/2})$ from one another.",
"The resonant regions yield gaps between KAM tori of size $O(\\varepsilon ^{j/2})$ , where $j$ is the order of the resonance.",
"Only the resonances of order 1 and 2 are of interest, as they produce gaps of size $O(\\varepsilon )$ and $O(\\varepsilon ^{1/2})$ respectively.",
"Inside each resonant region, the system can be approximated by a system similar to a pendulum.",
"In such a region, under appropriate non-degeneracy conditions, it is shown that there exist primary KAM tori close to the separatrices of the pendulum, secondary KAM tori (homotopically trivial), and stable and unstable manifolds of hyperbolic periodic orbits that pass close to the separatrices of the pendulum.",
"Moreover, these objects can be chosen to be $O(\\varepsilon ^{3/2})$ from one another.",
"In the generic case when the stable and unstable manifolds of hyperbolic periodic orbit intersect transversally, and there are no isolated invariant tori, a resonant region determines a BZI as in (A6).",
"In the non-generic case when the stable and unstable manifolds of a hyperbolic periodic orbit coincide, or there exist isolated invariant tori, the resonant region is as described as in (A6$^{\\prime }$ ).",
"The estimates from [18] imply that there exist primary KAM tori that are within $O(\\varepsilon ^{3/2})$ from the boundaries of the gap, or to the stable and unstable manifolds of the hyperbolic periodic orbits inside the resonant regions.",
"These estimates do not allow one to precisely locate the boundaries of the BZI's or to say anything about their dynamics.",
"The Melnikov conditions imply that the scattering map $S_\\varepsilon $ associated to this homoclinic channel $\\Gamma _\\varepsilon $ can be computed in terms of the Melnikov potential $\\mathcal {L}$ .",
"If $S_\\varepsilon (x^-)=x^+$ , then the change in the $I_\\varepsilon $ -coordinate under $S_\\varepsilon $ is given by $ I_\\varepsilon (x^+)-I_\\varepsilon (x^-)=-\\varepsilon \\frac{\\partial \\mathcal {L}}{\\partial \\phi }(I_\\varepsilon ,\\phi _\\varepsilon -\\omega (I_\\varepsilon )\\tau ^*, t-\\tau ^*)+O_{C^1}(\\varepsilon ^{1+\\varrho }),$ for some $\\varrho >0$ .",
"Condition (ii) implies that there are points in the domain of the scattering map $S$ whose $I_\\varepsilon $ -coordinate is increased by $O(\\varepsilon )$ under $S$ .",
"We can use these estimates to construct transition chains of invariant primary tori alternating with gaps, as in (A5) and (A6), or as in (A5) and (A6$^{\\prime }$ ).",
"For $\\varepsilon $ sufficiently small and fixed, we let $M:=M_\\varepsilon $ , $f:=f_\\varepsilon $ , and $\\Lambda $ be the annulus in $\\Lambda _\\varepsilon $ bounded by a pair of tori $T_{I_a}$ , $T_{I_b}$ with $I^-<I_a<I_b<I^+$ .",
"First, we choose a sequence of resonant regions and non-resonant regions that intersect the domain $U^-$ and the range $U^+$ of the scattering map.",
"Since the KAM primary tori are within $O(\\varepsilon ^{3/2})$ from one another, and the scattering map makes jumps of order $O(\\varepsilon )$ in the increasing direction of $I_\\varepsilon $ , then we can find smooth KAM primary tori $\\lbrace T_{i_{k}+2}, T_{i_{k}+2}, \\ldots , T_{i_{k+1}-1}\\rbrace $ such that $W^u(T_i)$ has a transverse intersection with $W^s(T_{i+1})$ for all $i\\in \\lbrace {i_{k}+2}, {i_{k}+3}, \\ldots , {i_{k+1}-1}\\rbrace $ , and that $T_{i_{k}+2}$ and $T_{i_{k+1}-1}$ are within $O(\\varepsilon ^{3/2})$ from the separatrices of the penduli corresponding to two consecutive resonant gaps of orderer 1 or 2.",
"The dynamics on each such a torus is quasi-periodic, so is topologically transitive.",
"This ensures condition (A5)-(iii).",
"Moreover, we can choose these KAM tori so that they are `interior' to the Cantor family of tori, i.e.",
"they can be approximated from both sides by other KAM primary tori.",
"This ensures condition (A5)-(iv).",
"To the transition chain $\\lbrace T_{i_{k}+2}, T_{i_{k}+3}, \\ldots , T_{i_{k+1}-1}\\rbrace $ we add, at each end, a torus $T_{i_{k}+1}$ and a torus $T_{i_{k+1}}$ .",
"These end tori bound resonant gaps that are either BZI's or consist of hyperbolic periodic orbits together with their invariant manifolds.",
"Since $T_{i_{k}+1},T_{i_{k+1}}$ are within $O(\\varepsilon ^{3/2})$ from $T_{i_{k}+2},T_{i_{k+1}-1}$ , respectively, and the scattering map $S_\\varepsilon $ makes jumps by order $O(\\varepsilon )$ , it follows that $S(T_{i_{k}+1})$ topologically crosses $T_{i_{k}+2}$ , and $S(T_{i_{k+1}-1})$ topologically crosses $T_{i_{k+1}}$ .",
"This ensures condition (A5)-(ii).",
"Condition (A1)-(i) is ensured automatically by our initial choice of the resonant regions and the non-resonant regions so that they intersect the domain $U^-$ and the range $U^+$ of $S_\\varepsilon $ .",
"The end tori $T_{i_{k}+1}$ and $T_{i_{k+1}}$ are at the boundaries of two consecutive resonant gaps.",
"This construction is continued for all resonant and non-resonant regions.",
"Thus, for $\\varepsilon $ fixed and sufficiently small, we obtain sequences of tori $\\lbrace T_{i_{k}+1}, T_{i_{k}+2}, \\ldots , T_{i_{k+1}}\\rbrace $ as in (A5), interspersed with gaps between $T_{i_{k}}$ and $T_{i_{k}+1}$ , and also between $T_{i_{k+1}}$ and $T_{i_{k+1}+1}$ , as in (A6).",
"We are under the assumption of Theorem REF .",
"Then there exists a diffusing orbit that shadows the transition chains of invariant primary tori and crosses the prescribed gaps.",
"In particular, if we choose an initial torus $T_{I_a}$ and a final torus $T_{I_b}$ so that they are $O(1)$ apart, we obtain a diffusing orbit whose action variable changes by $O(1)$ .",
"We note our theorem applies even for the choice of the pendulum $P_-$ , when the unperturbed Hamiltonian does not have positive-definite normal torsion.",
"The assumption of positive definiteness seems to be very important for variational methods.",
"We emphasize that, although we are using many of the estimates from [18], we obtain a different mechanism of diffusion.",
"Our mechanism still involves transition chains of invariant primary tori, but uses the inner dynamics restricted to the normally hyperbolic invariant manifold to cross over the large gaps.",
"The paper [18] identifies secondary tori and hyperbolic invariant manifolds of lower dimensional tori inside the large gaps, and forms transition chains of such objects that can be joined with the transition chains of primary tori.",
"Hence, it still uses the outer dynamics to cross over those gaps.",
"Since in our approach we do not use transition chains of secondary tori or of hyperbolic invariant manifolds of lower dimensional tori, we do not need to assume the additional non-degeneracy conditions on the scattering map acting on these objects as in [18].",
"Moreover, one can combine the topological mechanism in this paper with the one in [23] and obtain diffusing orbits that visit any given collection of primary tori, secondary tori, invariant manifolds of lower dimensional tori, and Aubry-Mather sets, in any prescribed order."
],
[
"Background on twist maps and Aubry-Mather sets",
"Let $\\tilde{A} = \\mathbb {T}^1 \\times [0,1]= \\lbrace (x, y)\\in \\mathbb {T}^1 \\times [0,1]\\rbrace $ be an annulus, and let $A=\\mathbb {R}\\times [0,1]$ be its universal cover with the natural projection $\\pi :A\\rightarrow \\tilde{A}$ given by $\\pi (x,y)=(\\tilde{x},\\tilde{y})$ , where $\\tilde{x}=x (\\textrm {mod } 1)$ and $\\tilde{y}=y$ .",
"Let $\\pi _x$ be the projection onto the first component, and $\\pi _y$ be the projection onto the second component.",
"Let $\\tilde{f}:\\tilde{A}\\rightarrow \\tilde{A}$ be a $C^1$ mapping on $\\tilde{A}$ , and let $f: A\\rightarrow A$ be the unique lift of $\\tilde{f}$ to $A$ satisfying $\\pi _x(f(0,0))\\in [0,1)$ and $\\pi \\circ f=\\tilde{f}\\circ \\pi $ .",
"In order to simplify the notation, below we will not make distinction between $\\tilde{A}$ and $A$ , and between $\\tilde{f}$ and $f$ .",
"We assume that $f$ is orientation preserving, boundary preserving, area preserving, and its satisfies a monotone twist condition, i.e., $|\\partial (\\pi _{x}\\circ f)/\\partial y|>0$ at all points in the annulus.",
"We note that the above properties imply that $f$ is exact symplectic, i.e.",
"$ f$ has zero flux, meaning that for any rotational curve $\\gamma $ the area of the regions above $\\gamma $ and below $f(\\gamma )$ equals the area below $\\gamma $ and above $f(\\gamma )$ .",
"In the sequel we will assume that $f$ is a positive twist, meaning that$\\partial (\\pi _{x}\\circ f)/\\partial y>0$ at all points.",
"The map $ f$ restricted to the boundary components $\\mathbb {T}^1 \\times \\lbrace 0\\rbrace $ , $\\mathbb {T}^1 \\times \\lbrace 1\\rbrace $ of the annulus has well defined rotation numbers $\\omega _-,\\omega _+$ , respectively, with $\\omega ^-<\\omega ^+$ .",
"We will assume that $\\omega ^-,\\omega ^+>0$ .",
"By an invariant primary torus (essential invariant circle) we mean a 1-dimensional torus $T$ invariant under $f$ that cannot be homotopically deformed into a point inside the annulus.",
"Since $f$ is a monotone twist map, each invariant primary torus $T$ is the graph of some Lipschitz function (see [4], [5]).",
"A region in $A$ between two invariant primary tori $T_1$ and $T_2$ is called a Birkhoff Zone of Instability (BZI) provided that there is no invariant primary torus in the interior of the region.",
"It is known that, for an area preserving monotone twist map $f$ of $A$ , given a BZI, there exist Birkhoff connecting orbits that go from any neighborhood of one boundary torus to any neighborhood of the other boundary torus (see [4], [5], [37]).",
"We have the following results: Theorem 4.1 (Birkhoff Connecting Theorem) Suppose that $T_1$ and $T_2$ bound a BZI.",
"For every pair of neighborhoods $U$ of $T_1$ and $V$ of $T_2$ there exist a point $z\\in U$ and an integer $N>0$ such that $f^{N}(z)\\in V$ .",
"Corollary 4.2 Suppose that $T_1$ and $T_2$ bound a BZI, and that the restrictions of $f$ to $T_1$ and $T_2$ are topologically transitive.",
"For every $\\zeta _1\\in T_1,\\zeta _2\\in T_2$ and every pair of neighborhoods $U$ of $\\zeta _1$ and $V$ of $\\zeta _2$ , there exist a point $z\\in U$ and an integer $N>0$ such that $f^{N}(z)\\in V$ .",
"A subset $M\\subseteq A$ is said to be monotone (cyclically ordered) if $\\pi _x(z_1)<\\pi _x(z_2)$ implies $\\pi _x(f(z_1))<\\pi _x(f(z_2))$ for all $z_1,z_2\\in M$ .",
"For $z\\in A$ the extended orbit of $z$ is the set $EO(z)=\\lbrace f^n(z)+(j,0)\\,:\\,n,j\\in \\mathbb {Z}\\rbrace $ .",
"The orbit of $z$ is said to be monotone (cyclically ordered) if the set $EO(z)$ is monotone.",
"If the orbit of $z\\in A$ is monotone, then the rotation number $\\rho (z)=\\lim _{n\\rightarrow \\infty }(\\pi _x(f^n(z))/n)$ exists.",
"We denote $\\textrm {Rot}(\\omega )=\\lbrace z\\in A\\,:\\,\\rho (z)=\\omega \\rbrace $ .",
"All points in the same monotone set have the same rotation number.",
"Definition 4.3 An Aubry-Mather set for $\\omega \\in \\mathbb {T}^1$ is a minimal, monotone, $f$ -invariant subset of $\\textrm {Rot}(\\omega )$ .",
"Here by a minimal set we mean a closed invariant set that does not contain any proper closed invariant subsets.",
"(Equivalently, the orbit of every point in the set is dense in the set.)",
"This should not be confused with action-minimizing or $h$ -minimal sets, where $h$ is a generating function for $f$ .",
"Theorem 4.4 (Aubry-Mather Theorem) For every $\\omega \\in [\\omega ^-,\\omega ^+]$ , there exists a non-empty Aubry-Mather set $\\Sigma _\\omega $ in $\\textrm {Rot}(\\omega )$ .",
"Aubry-Mather sets defined as above can be obtained as limits of monotone Birkhoff periodic orbits [34].",
"There may be many Aubry-Mather sets with the same rotation number [45].",
"On the other hand, if one requires Aubry-Mather sets to be action minizing, there exists a unique recurrent Aubry-Mather set for any given irrational rotation number.",
"In the sequel we will use the following result on the vertical ordering of Aubry-Mather sets from [27].",
"Theorem 4.5 There exists a family of essential circles $C_\\omega $ in $A$ for $\\omega \\in [\\omega ^-,\\omega ^+]$ such that: (i) Each $C_\\omega $ is a graph over $y=0$ ; (ii) The circles $C_\\omega $ are mutually disjoint, and if $\\omega ^{\\prime }>\\omega $ then $C_{\\omega ^{\\prime }}$ is above $C_{\\omega }$ ; (iii) Each $C_\\omega $ contains an Aubry-Mather set $\\Sigma _\\omega $ .",
"The above circles have Lipschitz regularity, and are projections of so called `ghost circles' that are objects in $\\mathbb {R}^\\mathbb {Z}$ .",
"See [27] for details.",
"A similar result to Theorem REF appears in [35] who find Aubry-Mather sets lying on pseudo-graphs that are (not strictly) vertically ordered.",
"There are some analogues of the Birkhoff Connecting Theorem for Aubry-Mather sets.",
"The following lemma is used in [33] to provide a topological proof for Mather Connecting Theorem stated below.",
"Lemma 4.6 Suppose that $T_1$ and $T_2$ bound a BZI.",
"Let $\\Sigma _\\omega $ be an Aubry-Mather set of rotation number $\\omega $ inside the BZI.",
"Let $p$ be a recurrent point in $\\Sigma _\\omega $ and $W(p)$ be a neighborhood of $p$ inside the BZI.",
"The following hold true: (i) For some positive number $n^+$ (resp.",
"$n^-$ ) depending on $W(p)$ the set$\\bigcup _{j=0}^{n^+}f^j(W(p))$ (resp.",
"$\\bigcup _{j=0}^{n^-}f^{-j}(W(p))$ ) separates the cylinder.",
"(ii) The set $W^{+\\infty }:=\\bigcup _{j=0}^{\\infty }f^j(W (p))$ (resp.",
"the set $W^{-\\infty }:=\\bigcup _{j=0}^{\\infty }f^{-j}(W(p) )$ ), is connected and open.",
"(iii) The closure of $W^{+\\infty }$ (resp.",
"$W ^{-\\infty }$ ) contains both boundary tori $T_1$ and $T_2$ .",
"(iv) The set $W ^\\infty :=\\bigcup _{j=-\\infty }^{\\infty }f^j(W(p))$ is invariant, and both $W^{+\\infty } $ and $W^{-\\infty } $ are open and dense in $W ^\\infty $ .",
"The following result says that there exist orbits that visit any prescribed bi-infinite sequence of Aubry-Mather sets inside a BZI (see [46], [53], [29], [35], [33]).",
"Theorem 4.7 (Mather Connecting Theorem) Suppose that $T_1$ and $T_2$ bound a BZI, and $\\lbrace \\Sigma _{\\omega _i}\\rbrace _{i\\in \\mathbb {Z}}$ is a bi-infinite sequence of Aubry-Mather sets inside the BZI.",
"Let $\\varepsilon _i>0$ for $i\\in \\mathbb {Z}$ .",
"Then there exist a point $z$ inside the BZI and an increasing bi-infinite sequence of integers $\\lbrace j_i\\rbrace _{i\\in \\mathbb {Z}}$ such that $f^{j_i}(z)$ is within $\\varepsilon _i$ from $\\Sigma _{\\omega _i}$ for all $i\\in \\mathbb {Z}$ .",
"The Aubry-Mather sets in Theorem REF are action minimizing.",
"The following topological version of Mather Connecting Theorem, due to Hall [29], provides shadowing orbits of Aubry-Mather sets that are not necessarily action minimizing.",
"This approach can be implemented in rigorous computer experiments [32].",
"Theorem 4.8 Suppose that $T_1$ and $T_2$ bound a BZI, and $\\lbrace z_s\\rbrace _{s\\in \\mathbb {Z}}$ is a bi-infinite sequence of monotone $(p_s/q_s)$ -periodic points, with the rotation numbers $p_s/q_s$ mutually distinct, inside the BZI.",
"Given a bi-infinite sequence $\\lbrace n_s\\rbrace _{s\\in \\mathbb {Z}}$ of positive integers, then there exist a point $z$ and a bi-infinite sequence $\\lbrace m_s\\rbrace _{s\\in \\mathbb {Z}}$ of positive integers such that, for each $s\\ge 0$ , there exist some points $w_s,\\bar{w}_s$ in the extended orbit of $z_s$ such that $\\begin{split}\\pi _x(f^j(w_s))<\\pi _x(f^j(z))<\\pi _x(f^j(\\bar{w}_s))\\textrm { for }\\\\\\sum _{t=0}^{s-1} n_t+\\sum _{t=0}^{s-1}m_t\\le j\\le \\sum _{t=0}^{s} n_t+\\sum _{t=0}^{s-1}m_t.\\end{split}$ A similar statement holds for each $s<0$ .",
"In the above, $n_s$ represents the number of iterates for which the orbit of $z$ shadows – in the sense of the cyclical ordering – the extended orbit of $z_s$ , and $m_s$ represents the number of iterates it takes the orbit of $z$ to pass from the extended orbit of $z_s$ to the extended orbit of $z_{s+1}$ .",
"They main tool used in Hall's arguments is that of a positive (negative) diagonal.",
"Denote by $\\mathcal {Z}$ the BZI bounded by the tori $T_1$ and $T_2$ .",
"Let $I_z&=&\\lbrace w\\in \\mathcal {Z}\\,|\\, \\pi _x(w)=\\pi _x(z)\\rbrace ,\\\\I^+_z&=&\\lbrace w\\in I_z\\,|\\, \\pi _y(w)\\ge \\pi _y(z)\\rbrace ,\\\\I^-_z&=&\\lbrace w\\in I_z\\,|\\, \\pi _y(w)\\le \\pi _y(z)\\rbrace ,\\\\B_{z_0,z_1}&=&\\lbrace w\\in \\mathcal {Z}\\,|\\, \\pi _x(z_0)< \\pi _x(w)< \\pi _x(z_1)\\rbrace ,$ where $z,z_0,z_1$ are points in the annulus.",
"A positive diagonal $D$ in $B_{z_0,z_1}$ is a set $D\\subseteq \\textrm {cl}(B_{z_0,z_1})$ such that (i) $D$ is simply connected and the closure of its interior; (ii) $\\partial D \\cap \\textrm {cl}(B_{z_0,z_1})\\subseteq I^-_{z_0}\\cup I^+_{z_1} \\cup T_1\\cup T_2$ ; (iii) $\\partial D\\cap I^-_{z_0}\\ne \\emptyset $ and $\\partial D\\cap I^+_{z_1}\\ne \\emptyset $ .",
"The set $\\partial D\\cap B_{z_0,z_1}$ has exactly two components connecting $I^-_{z_0}\\cup T_1$ to $I^+_{z_1}\\cup T_2$ , which are called the upper and lower edges of $D$ , respectively.",
"We informally say that these components `stretch across' $B_{z_0,z_1}$ .",
"See Fig.",
"REF .",
"A negative diagonal and its upper and lower edges are defined similarly.",
"Figure: Two positive diagonal sets.",
"The positive diagonal set on the left has its upper and lower edges marked.",
"The positive diagonal set on the right is obtained by intersecting f(B w 0 ,w 1 )f(B_{w_0,w_1}) with B f(w 0 ),f(w 1 ) B_{f(w_0),f(w_1)}.",
"The upper edge of this diagonal set is contained in f(I w 0 + )f(I^+_{w_0}) and the lower edge is contained in f(I w 1 - )f(I^-_{w_1}).An important feature of positive diagonals is the following hereditary property.",
"Given $z_0,z_0$ such that $\\pi _x(z_0)<\\pi _x(z_1)$ and $\\pi _x(f(z_0))<\\pi _x(f(z_1))$ , if $D$ is a positive diagonal in $B_{z_0,z_1}$ , then $f(D)\\cap B_{f(z_0),f(z_1)}$ has a component $D^{\\prime }$ that is a positive diagonal in $B_{f(z_0),f(z_1)}$ .",
"One way to generate a positive diagonal set is by taking a component of the intersection between $f^k(B_{w_0,w_1})$ and $B_{f^k(w_0),f^k(w_1)}$ .",
"In this case, there exists a positive diagonal in $B_{f^k(w_0),f^k(w_1)}$ whose upper edge is contained on $f^k(I_{w_0}^+)$ and lower edge is contained in $f^k(I_{w_1}^-)$ .",
"More general, one has the following important property.",
"If $D$ has the upper edge contained in $f^k(I^+_{w_0})$ and the lower edge contained in $f^k(I^-_{w_1})$ , and $\\partial D\\cap B_{z_0,z_1}\\subseteq f^k(I^+_{w_0}\\cup I^-_{w_1})$ , for some $w_0,w_1$ with $\\pi _x(w_0)<\\pi _x(w_1)$ and some $k>0$ , then then $f(D)\\cap B_{f(z_0),f(z_1)}$ has a component $D^{\\prime }$ which can be chosen so that its upper edge is contained in $f^{k+1}(I^+_{w_0})$ and its lower edge is contained in $f^{k+1}(I^-_{w_1})$ .",
"See Fig.",
"REF .",
"A similar property holds for negative diagonals.",
"The proof of Theorem REF in [29] is an inductive argument which, for a given pair of adjacent points $w_0,\\bar{w}_0$ in the extended orbit of $z_0$ , and for each $\\sigma \\ge 0$ , produces a nested sequence $D_0\\supseteq D_1\\supseteq \\ldots \\supseteq D_{\\sigma }$ of negative diagonals of $B_{w_0,\\bar{w}_0}$ such that, for each $s\\in \\lbrace 0,\\ldots ,\\sigma \\rbrace $ , the following hold: (a) the orbit of each point $z \\in D_s$ satisfies the ordering relation (REF ), and (b) there is a sufficiently large $j_s>0$ such that $f^{j_s+j}(D_s)$ contains a component that is a positive diagonal in $B_{f^j(w_s),f^j(\\bar{w}_s)}$ , for some adjacent points $w_s,\\bar{w}_s\\in EO(z_s)$ , and for all $j=1,\\ldots , n_s$ .",
"In the above, $j_s=\\sum _{t=0}^{s-1} n_t+\\sum _{t=0}^{s-1}m_t$ .",
"Moreover, in this inductive argument one can choose the diagonal sets $D_s$ so that $f^{j_s+j}(D_s)$ has the upper edge contained in $f^{j_s+j}(I^+_{w_0})$ , lower edge contained in $f^{j_s+j}(I^-_{\\bar{w}_0})$ , and $\\partial f^{j_s+j}(D_s)\\cap B_{f^{j_s+j}(w_0),f^{j_s+j}(\\bar{w}_0)} \\subseteq f^{j_s+j} (I^+_{w_0}\\cup I^-_{\\bar{w}_0})$ .",
"For the basis step, starting with $w_0,\\bar{w}_0$ and applying the hereditary property from above $n_0$ times, one obtains a negative diagonal set $D_0$ of $B_{w_0,\\bar{w}_0}$ with the properties that each point $z \\in D_0$ satisfies the ordering relation (REF ) for $s=0$ , and $f^{n_0}(D_0)$ has a component that is a positive diagonal of $B_{f^{n_0}(w_0),f^{n_0}(\\bar{w}_0)}$ .",
"For the inductive step, one assumes a negative diagonal $D_\\sigma $ of $B_{w_0,\\bar{w}_0}$ as above, and wants to produce a negative diagonal $D_{\\sigma +1}\\subseteq D_\\sigma $ of $B_{w_0,\\bar{w}_0}$ which fulfils the corresponding properties.",
"The key idea is to use the existence of points near $y=0$ that get near $y=1$ , and of points near $y=1$ that get near $y=0$ , as provided by Theorem REF , in order to show that for some $j_\\sigma $ sufficiently large $f^{j_\\sigma }(D_\\sigma )$ contains a component that stretches all the way across a fundamental interval of the annulus.",
"Hence $f^{j_\\sigma }(D_\\sigma )$ contains a subset that is a positive diagonal of $B_{w_{\\sigma +1}, \\bar{w}_{\\sigma +1}}$ for two adjacent points $w_{\\sigma +1}, \\bar{w}_{\\sigma +1}\\in EO(z_{\\sigma +1})$ .",
"From this it follows that $f^{j_\\sigma +j}(D_\\sigma )$ contains a component that is a positive diagonal in $B_{f^j(w_{\\sigma +1}),f^j(\\bar{w}_{\\sigma +1})}$ for all $j=1,\\ldots , n_{\\sigma +1}$ .",
"This completes the inductive step.",
"Applying a similar argument for the negative iterates of $f$ produces a nested sequence of positive diagonals of $B_{w_0,\\bar{w}_0}$ .",
"A positive diagonal of $B_{w_0,\\bar{w}_0}$ always has a non-empty intersection with a negative diagonal of $B_{w_0,\\bar{w}_0}$ .",
"This implies the existence of points $z$ whose forward orbits satisfy the ordering conditions in (REF ) and whose backwards orbits satisfy similar ordering conditions.",
"Using limit arguments as in [34], one can obtain shadowing of Aubry-Mather sets of irrational rotation numbers as well.",
"These topological ideas will be used in the proof of Theorem REF below.",
"Remark 4.9 An immediate consequence of Lemma REF is that, given a neighborhood $W$ of a point $p\\in \\Sigma _\\omega $ , where $\\Sigma _\\omega $ is an Aubry-Mather set inside a BZI bounded by $T_1$ and T$_2$ , and given a neighborhood $U$ of $T_1$ or $T_2$ , there exists an arbitrarily large $j>0$ such that $f^j(W(p))\\cap U\\ne \\emptyset $ .",
"Also, there exists an arbitrarily large $j^{\\prime }>0$ such that $f^{-j^{\\prime }}(W(p))\\cap U\\ne \\emptyset $ .",
"Remark 4.10 The results in this section hold if we replace conditions (i) and (ii) from the definition of an area preserving, monotone twist map with the following weaker conditions: (i') $f$ satisfies the following `condition B': for every pair of neighborhoods $U_1$ of $T_1$ and $U_2$ of $T_2$ , there exist $z_1,z_2\\in A$ and $n_1,n_2>0$ such that $z_1\\in U_1$ and $f^{n_1}(z_1)\\in U_2$ , and $z_2\\in U_2$ and $f^{n_2}(z_2)\\in U_1$ .",
"Note that this condition only makes sense if we restrict the dynamics to a BZI.",
"(ii') $f$ satisfies the following positive tilt condition: if we denote by $\\theta _{z}$ the angle deviation from the vertical, measured from the vertical vector $(0,1)$ to $Df_{z}(0,1)$ , with the clockwise direction taken as the positive direction, and defined in such a way that $\\theta _{(x,0)}\\in [-\\pi /2,\\pi /2]$ and $\\theta $ is continuous, then $\\theta _{z}> 0$ at all points.",
"See Fig.",
"REF .",
"Compositions of positive twist maps, are for example, positive tilt maps.",
"We shall note that the Aubry-Mather theory applies to positive tilt maps as well (see [31]).",
"Figure: Positively tilted map.Remark 4.11 Aubry-Mather theory and the above shadowing result also hold for generalized twist maps of the higher dimensional annulus $S^1\\times \\mathbb {R}^n$ .",
"See [1].",
"Remark 4.12 The topological approach in this section does not yield trajectories that get close to each set in the prescribed collection of Aubry-Mather sets, as in Theorem REF .",
"It seems possible, however, that these topological methods can be combined with the variational methods in [46] to obtain trajectories that go very close to each Aubry-Mather set in the given collection."
],
[
"Scattering map",
"The scattering map acts on the normally hyperbolic invariant manifold $\\Lambda $ and relates the past asymptotic trajectory of each orbit in the homoclinic manifold to its future asymptotic behavior.",
"We review its properties following [19].",
"For the general case, we consider a $C^r$ -differentiable manifold $M$ , a $C^r$ -differentiable map $f:M\\rightarrow M$ , and an $l$ -dimensional normally hyperbolic invariant manifold $\\Lambda $ for $f$ .",
"By the definition of normal hyperbolicity, there exists a splitting of the tangent bundle of $TM$ into sub-bundles $TM=E^u\\oplus E^s\\oplus T\\Lambda ,$ that are invariant under $df$ , and there exist a constant $C>0$ and rates $0<\\lambda <\\mu ^{-1}<1$ , such that for all $x\\in \\Lambda $ we have $\\begin{split}v\\in E^s_x \\Leftrightarrow \\Vert Df^k_x(v)\\Vert \\le C\\lambda ^k\\Vert v\\Vert \\textrm { for all } k\\ge 0,\\\\v\\in E^u_x \\Leftrightarrow \\Vert Df^k_x(v)\\Vert \\le C\\lambda ^{-k}\\Vert v\\Vert \\textrm { for all } k\\le 0,\\\\v\\in T_x\\Lambda \\Leftrightarrow \\Vert Df^k_x(v)\\Vert \\le C\\mu ^{|k|}\\Vert v\\Vert \\textrm { for all } k\\in \\mathbb {Z}.\\end{split}$ The smoothness of the invariant objects defined by the normally hyperbolic structure depends on the rates $\\lambda $ and $\\mu $ .",
"The map $f$ is said to be $\\ell $ -normally hyperbolic along $\\Lambda $ provided that $1 \\le \\ell \\le r$ is an integer satisfying $\\lambda \\,\\mu ^\\ell <1$ , i.e., $\\ell < (\\log \\lambda ^{-1})(\\log \\mu )^{-1}$ .",
"Then the stable and unstable manifolds $W^s(\\Lambda )$ and $W^u(\\Lambda )$ and the normally hyperbolic manifold $\\Lambda $ are all $C^\\ell $ -differentiable.",
"The splitting $E^s_z=T_z(W^s(x))$ depends $C^{\\ell -1}$ smoothly on $z$ in $W^s(\\Lambda )$ so $\\lbrace \\,W^s(x)\\, |\\, x \\in \\Lambda \\,\\rbrace $ is a $C^{\\ell -1}$ foliation of $W^s(\\Lambda )$ .",
"(This is stated explicitly in [50] and follows from the $C^r$ Section Theorem of [30].)",
"Similarly, $\\lbrace \\,W^u(x)\\, |\\, x \\in \\Lambda \\,\\rbrace $ is a $C^{\\ell -1}$ foliation of $W^u(\\Lambda )$ .",
"Since the stable (resp.",
"unstable) manifolds of $\\Lambda $ are foliated by stable (resp.",
"unstable) manifolds of points, we have that for each $x\\in W^s(\\Lambda )$ (resp.",
"$x\\in W^u(\\Lambda )$ ), there exists a unique $x^+ \\in \\Lambda $ (resp.",
"$x^- \\in \\Lambda $ ) such that $x \\in W^s(x^+)$ (resp.",
"$x \\in W^u(x^-)$ ).",
"We define the maps $\\Omega ^+ : W^s(\\Lambda ) \\rightarrow \\Lambda $ by $\\Omega ^+(x) = x^+$ and $\\Omega ^- : W^u(\\Lambda ) \\rightarrow \\Lambda $ by $\\Omega ^-(x) = x^-$ .",
"The maps $\\Omega ^+$ and $\\Omega ^-$ are $C^{\\ell -1}$ -smooth since the foliations are smooth.",
"We now describe the scattering map.",
"Assume that $W^u(\\Lambda )$ and $W^s(\\Lambda )$ have a differentiably transverse intersection along a homoclinic $l$ -dimensional $C^{\\ell -1}$ -differentiable manifold $\\Gamma $ .",
"This means that $\\Gamma \\subseteq W^u(\\Lambda ) \\cap W^s(\\Lambda )$ and, for each $x\\in \\Gamma $ , we have $ \\begin{split}T_xM=T_xW^u(\\Lambda )+T_xW^s(\\Lambda ),\\\\T_x\\Gamma =T_xW^u(\\Lambda )\\cap T_xW^s(\\Lambda ).\\end{split} $ We assume the additional condition that for each $x\\in \\Gamma $ we have $\\begin{split}T_xW^s(\\Lambda )=T_xW^s(x^+)\\oplus T_x(\\Gamma ),\\\\T_xW^u(\\Lambda )=T_xW^u(x^-)\\oplus T_x(\\Gamma ),\\end{split} $ where $x^-,x^+$ are the uniquely defined points in $\\Lambda $ corresponding to $x$ .",
"The restrictions $\\Omega ^+_\\Gamma ,\\Omega ^-_\\Gamma $ of $\\Omega ^+,\\Omega ^-$ to $\\Gamma $ are local $C^{\\ell -1}$ -diffeomorphisms.",
"By replacing $\\Gamma $ to a submanifold of it (which, with an abuse of notation, we still denote $\\Gamma $ ) we can ensure that $\\Omega ^+_\\Gamma :\\Gamma \\rightarrow U^+,\\Omega ^-_\\Gamma :\\Gamma \\rightarrow U^-$ are $C^{\\ell -1}$ -diffeomorphisms from $\\Gamma $ to the open sets $U^+,U^-$ in $\\Lambda $ , respectively.",
"Definition 5.1 A homoclinic manifold $\\Gamma $ satisfying (REF ) and (REF ), and for which the corresponding restrictions of the wave maps are $C^{\\ell -1}$ -diffeomorphisms, is referred as a homoclinic channel.",
"Definition 5.2 Given a homoclinic channel $\\Gamma $ , the scattering map associated to $\\Gamma $ is the $C^{\\ell -1}$ -diffeomorphism $S_\\Gamma =\\Omega _\\Gamma ^+\\circ (\\Omega _\\Gamma ^-)^{-1}$ from the open subset $U^-:=\\Omega _\\Gamma ^-(\\Gamma )$ in $\\Lambda $ to the open subset $U^+:=\\Omega _\\Gamma ^+(\\Gamma )$ in $\\Lambda $ .",
"In the sequel we will regard $S_\\Gamma $ as a partially defined map, so the image of a set $A$ by $S_\\Gamma $ means the set $S_\\Gamma (A\\cap U^-)$ .",
"In this paper, we need the following property of the scattering map.",
"Proposition 5.3 Assume that $T_1$ and $T_2$ are two invariant submanifolds of complementary dimensions in $\\Lambda $ .",
"Then $W^u(T_1)$ has a topologically transverse intersection with $W^s(T_2)$ inside $\\Gamma $ if and only if $S_\\Gamma (T_1\\cap U^-)$ has a topologically transverse intersection with $T_2\\cap U^+$ in $\\Lambda $ .",
"Let $\\bar{T}_1=(\\Omega ^-_\\Gamma )^{-1}(T_1\\cap U^-)\\subseteq \\Gamma $ and $\\bar{T}_2=(\\Omega ^+_\\Gamma )^{-1}(T_2\\cap U^+)\\subseteq \\Gamma $ .",
"A topologically transverse intersection of $W^u(T_1)$ with $W^s(T_2)$ in $\\Gamma $ occurs if and only if $\\bar{T}_1$ intersects $\\bar{T}_2$ topologically transversally in $\\Gamma $ , which is equivalent to $S_\\Gamma (T_1\\cap U^-)$ intersects topologically transversally $T_2\\cap U^+$ in $\\Lambda $ .",
"For the definition of topological transversality (topological crossing) see [8].",
"For the main result of this paper the normally hyperbolic invariant manifold $\\Lambda $ is assumed to be 2-dimensional, i.e., $l=2$ , and the invariant submanifolds $T_1,T_2$ in Proposition REF are 1-dimensional invariant tori."
],
[
"Topological method of correctly aligned windows",
"We describe briefly the topological method of correctly aligned windows.",
"We follow [55].",
"See also [24], [23], [41].",
"Definition 5.4 An $(n_1,n_2)$ -window in an $n$ -dimensional manifold $M$ , where $n_1+n_2=n$ , is a compact subset $W$ of $M$ together with a homeomorphism $\\chi $ from some open neighborhood of $[0,1]^{n_1}\\times [0,1]^{n_2}$ in $\\mathbb {R}^{n_1}\\times \\mathbb {R}^{n_2}$ to an open subset of $M$ , such that $W=\\chi ([0,1]^{n_1}\\times [0,1]^{n_2}),$ and with a choice of an `exit set' $W^{\\rm exit} =\\chi \\left(\\partial [0,1]^{n_1}\\times [0,1]^{n_2} \\right)$ and of an `entry set' $W^{\\rm entry}=\\chi \\left([0,1]^{n_1}\\times \\partial [0,1]^{n_2}\\right).$ Denote by $\\pi _{1}: \\mathbb {R}^{n_1}\\times \\mathbb {R}^{n_2}\\rightarrow \\mathbb {R}^{n_1}$ the projection onto the first component, and by $\\pi _{2}: \\mathbb {R}^{n_1}\\times \\mathbb {R}^{n_2}\\rightarrow \\mathbb {R}^{n_2}$ the projection onto the second component.",
"Definition 5.5 Let $W_1$ and $W_2$ be $(n_1,n_2)$ -windows, and let $\\chi _1$ and $\\chi _2$ be the corresponding local parametrizations.",
"Let $f$ be a continuous map on $M$ with $f(\\textrm {im}(\\chi _1))\\subseteq \\textrm {im}(\\chi _2)$ .",
"We say that $W_1$ is correctly aligned with $W_2$ under $f$ if the following conditions are satisfied: (i) $f(W^{\\rm exit}_1) \\cap (W_2) = \\emptyset $ and $f(W_1) \\cap W_2^{\\rm entry} = \\emptyset $ ; (ii) There exists $y_0 \\in [0,1]^{n_2}$ such that the curve $x\\in [0,1]^{n_1} \\mapsto \\hat{f} (x,y_0)$ , where $\\hat{f}:=\\chi _2^{-1}\\circ f\\circ \\chi _1$ , has the following properties: $\\hat{f}_{y_0}\\left( [0,1]^{n_1}\\right)\\subseteq \\mathbb {R}^{n_1}\\times (0,1)^{n_2},\\\\\\hat{f}_{y_0}\\left( \\partial [0,1]^{n_1}\\right)\\subseteq (\\mathbb {R}^{n_1}\\setminus [0,1]^{n_1})\\times (0,1)^{n_2}=\\emptyset ,\\\\\\deg (\\pi _{1}\\circ \\hat{f}_{y_0},0)=w\\ne 0.$ We call the integer $w\\ne 0$ in the above definition the degree of the alignment.",
"The following result is a topological version of the Shadowing Lemma.",
"Theorem 5.6 Let $\\lbrace W_i\\rbrace _{i\\in \\mathbb {Z}}$ , be a collection of $(n_1,n_2)$ -windows in $M$ , and let $f_i$ be a collection of continuous maps on $M$ .",
"If for each $i\\in \\mathbb {Z}$ , $W_i$ is correctly aligned with $W_{i+1}$ under $f_i$ , then there exists a point $p\\in W_0$ such that $(f_{i}\\circ \\cdots \\circ f_{0})(p)\\in W_{i+1}, \\textrm { for all } i\\in \\mathbb {Z}.",
"$ Moreover, assuming that there exists $k>0$ such that $W_{i}=W_{(i\\,{\\rm mod}\\, k)}$ and $f_{i}=f_{(i\\,{\\rm mod}\\, k)}$ for all $i\\in \\mathbb {Z}$ , then there exists a point $p$ as above that is periodic in the sense $(f_{k-1}\\circ \\cdots \\circ f_{0})(p)=p.$ The correct alignment of windows is robust, in the sense that if two windows are correctly aligned under a map, then they remain correctly aligned under a sufficiently small $C^0$ -perturbation of the map.",
"Robustness makes the method of correctly aligned windows appropriate for perturbative arguments, as well as for rigorous numerical experiments.",
"Also, the correct alignment satisfies a natural product property.",
"Given two windows and a map, if each window can be written as a product of window components, and if the components of the first window are correctly aligned with the corresponding components of the second window under the appropriate components of the map, then the first window is correctly aligned with the second window under the given map.",
"For example, if we consider a pair of windows in a neighborhood of a normally hyperbolic invariant manifold, if the center components of the windows are correctly aligned and the hyperbolic components of the windows are also correctly aligned, then the windows are correctly aligned.",
"Although the product property is quite intuitive, its rigorous statement is rather technical, so we will omit it here.",
"The details can be found in [23].",
"In Sections and , we will consider various windows lying on one of the manifolds $\\Lambda $ , $W^u(\\Lambda )$ , $W^s(\\Lambda )$ , or $M$ .",
"Without explicit mention, every such a window will be represented by the image of a rectangle through a local parametrization of the appropriate manifold.",
"We will also consider correct alignment relations of windows lying on the same manifold.",
"All the correct alignment relations in the arguments presented later in this paper have degree $w=1$ ."
],
[
"Existence of Birkhoff connecting orbits",
"In this section we state and prove an extension of the Corollary REF of the Birkhoff connecting orbit theorem, and an extension of Mather's theorem on shadowing of Aubry-Mather sets, specifically of Theorem REF .",
"The methodology is based on the topological approach of Hall and on the Jordan Curve Theorem.",
"The statements below will be used in the proof of the main theorem.",
"Throughout the section we assume that $f$ is an orientation preserving, boundary preserving, area preserving, monotone twist map of the annulus $A$ , as in Section , and we adopt the notation conventions from that section.",
"In Corollary REF , it was assumed that the restrictions of the map to the boundary tori of the BZI are topologically transitive, and it was inferred the existence of connecting orbits from an arbitrarily small neighborhood of some prescribed point on one boundary torus to an arbitrarily small neighborhood of some prescribed point on the other boundary torus.",
"In the statements below we prove the same result without the topological transitivity assumption.",
"Theorem 6.1 Suppose that $T_1$ and $T_2$ bound a BZI $\\mathcal {Z}$ .",
"Assume that $\\zeta _1\\in T_1$ and $\\zeta _2\\in T_2$ .",
"Fix a pair of neighborhoods $U$ of $\\zeta _1$ and $V$ of $\\zeta _2$ .",
"Then there exists a point $z\\in U$ and an integer $N>0$ such that $f^{N}(z)\\in V$ for some $N>0$ that can be chosen arbitrarily large.",
"Moreover, if we assume that $U,V$ are chosen so that $U\\cap \\mathcal {Z},V\\cap \\mathcal {Z}$ are topological disks, then there exists a point $z^{\\prime }\\in \\partial U$ such that $f^N(z^{\\prime })\\in \\partial V$ .",
"Consider the neighborhood $U$ of $\\zeta _1\\in T_1$ .",
"Choose two points $\\zeta ^{\\prime }_1,\\zeta ^{\\prime \\prime }_1\\in T_1\\cap U$ such that $\\zeta _1$ is between $\\zeta ^{\\prime }_1$ and $\\zeta ^{\\prime \\prime }_1$ and the portion of $T_1$ between $\\zeta ^{\\prime }_1$ and $\\zeta ^{\\prime \\prime }_1$ is contained in $\\textrm {int}(U)$ .",
"Choose a simple curve $\\gamma _0$ inside $U$ , with endpoints at $\\zeta ^{\\prime }_1$ and $\\zeta ^{\\prime \\prime }_1$ .",
"The curve $\\gamma _0$ together with the portion of $T_1$ between $\\zeta ^{\\prime }_1$ and $\\zeta ^{\\prime \\prime }_1$ determines a closed topological disk $U_0\\subseteq U$ , which is a one-sided compact neighborhood of $\\zeta _1$ in $\\mathcal {Z}$ .",
"Similarly, we can choose a one-sided compact neighborhood $V_0\\subseteq V$ of $\\zeta _2\\in T_2$ , whose boundary consists of a simple curve $\\eta _0$ connecting two points $\\zeta ^{\\prime }_2,\\zeta ^{\\prime \\prime }_2\\in T_2$ , with $\\zeta _2$ between $\\zeta ^{\\prime }_2$ and $\\zeta ^{\\prime \\prime }_2$ , and the portion of $T_2$ between $\\zeta ^{\\prime }_2$ and $\\zeta ^{\\prime \\prime }_2$ .",
"In this way, for proving the theorem we can consider the one-sided neighborhoods $U_0,V_0$ instead of $U,V$ , respectively.",
"Assume first that the interior of $U_0$ meets some Aubry-Mather set $\\Sigma _{\\rho _1}\\subseteq \\mathcal {Z}$ , and that the interior of $V_0$ meets some Aubry-Mather set $\\Sigma _{\\rho _2}\\subseteq \\mathcal {Z}$ .",
"Since $U_0$ and $V_0$ are neighborhoods of points in the Aubry-Mather sets $\\Sigma _{\\rho _1}$ and $\\Sigma _{\\rho _2}$ respectively, Theorem REF yields the existence of a forward orbit that goes from $U_0$ to $V_0$ .",
"Hence there exists $N>0$ such that $f^N(U_0)\\cap V_0\\ne \\emptyset $ .",
"Since $f^N(U_0)$ and $V_0$ are open topological disks that have common points as well as non-common points (e.g., the points in $T_1$ and $T_2$ , respectively), the Jordan Curve Theorem implies that $f^N(\\partial U_0)\\cap \\partial V_0\\ne \\emptyset $ .",
"Assume now that the interiors of $U_0,V_0$ do not meet any Aubry-Mather set.",
"We choose three Aubry-Mather sets $\\Sigma _{\\rho _1}$ , $\\Sigma _{\\rho _1^{\\prime }}$ , $\\Sigma _{\\rho _1^{\\prime \\prime }}$ in $\\mathcal {Z}$ , lying on three essential circles $C_{\\rho _1},C_{\\rho ^{\\prime }_1},C_{\\rho ^{\\prime \\prime }_1}$ , respectively, with ${\\rho _1}<{\\rho ^{\\prime }_1}<{\\rho ^{\\prime \\prime }_1}$ irrational rotation numbers, and $C_{\\rho _1}\\prec C_{\\rho ^{\\prime }_1}\\prec C_{\\rho ^{\\prime \\prime }_1}$ .",
"The existence of such vertically ordered Aubry-Mather sets follows from Theorem REF .",
"The proof of the theorem uses the following intermediate step.",
"Claim.",
"There exist $j^{\\prime }_*>j_*>0$ such that $\\mathcal {Z}\\setminus [f^{j_*}(U_0)\\cup f^{j^{\\prime }_*}(U_0)]$ contains a component $\\mathcal {U}$ which is an open topological disk that is a neighborhood of some point in $\\Sigma _{\\rho _1}$ .",
"Moreover, $j_*,j^{\\prime }_*$ can be chosen arbitrarily large.",
"A similar statement holds for $T_2$ .",
"Proof of the claim.",
"Let $p_1$ be a point in $\\Sigma _{\\rho ^{\\prime \\prime }_1}$ .",
"Let $W(p_1)$ be a small neighborhood of $p_1$ inside the BZI, which does not intersect $\\Sigma _{\\rho _1}$ and $\\Sigma _{\\rho _1^{\\prime }}$ .",
"By assumption, $U_0$ does not meet any of the sets $\\Sigma _{\\rho _1}$ , $\\Sigma _{\\rho ^{\\prime }_1}$ , $\\Sigma _{\\rho ^{\\prime \\prime }_1}$ .",
"By Lemma REF (iii) the closure of $\\bigcup _{j=0}^{\\infty }f^{-j}(W(p_1))$ contains $T_1$ , and in particular $\\zeta _1$ .",
"Since $U_0$ is a one-sided neighborhood of $\\zeta _1$ , there exists $j_1>0$ such that $f^{j_1}(U_0)\\cap W(p_1)\\ne \\emptyset $ .",
"In the covering space of the annulus, $f^{j_1}(U_0)$ intersects some copy $W^{h_1}:=W(p_1)+(h_1,0)$ of $W(p_1)$ , where $h_1$ is some positive integer.",
"Since $U_0$ does not intersect $\\Sigma _{\\rho _1}$ and $\\Sigma _{\\rho ^{\\prime }_1}$ , it follows that $f^{j_1}(U_0)$ does not intersect $\\Sigma _{\\rho _1}$ and $\\Sigma _{\\rho ^{\\prime }_1}$ .",
"On the other hand, $f^{j_1}(U_0)$ intersects the essential circles $C_{\\rho _1},C_{\\rho ^{\\prime }_1}$ , containing the Aubry-Mather sets $\\Sigma _{\\rho _1},\\Sigma _{\\rho ^{\\prime }_1}$ , respectively.",
"Let $\\gamma _{1}:[0,1]\\rightarrow U_0$ be a vertical curve, i.e., $\\gamma _1(0)\\in T_1$ and $\\pi _x(\\gamma _1(t))=\\pi _x(\\gamma _1(0))$ for all $t$ , such that $f^{j_1}(\\gamma _1(1))$ is an intersection point of $f^{j_1}(U_0)$ with $W^{h_1}$ .",
"The curve $f^{j_1}(\\gamma _1)$ is a positively tilted curve which crosses both essential circles $C_{\\rho _1}$ and $C_{\\rho ^{\\prime }_1}$ .",
"(See Remark REF .)",
"Since $f^{j_1}(U_0)$ is disjoint from $\\Sigma _{\\rho _1}$ , the intersections between $f^{j_1}(\\gamma _1)$ and $C_{\\rho _1}$ occur within the `gaps' of $\\Sigma _{\\rho _1}$ , i.e., within the open interval components of $C_{\\rho _1}\\setminus \\Sigma _{\\rho _1}$ .",
"We can assign an oriented intersection number between $f^{j_1}(\\gamma _1 )$ and each gap of $\\Sigma _{\\rho _1}$ .",
"Consider a homotopy $h_s:A\\rightarrow A$ , $s\\in [0,1]$ , such that $f^{j_1}(h_s(\\gamma _1))$ keeps the endpoints of $f^{j_1}(\\gamma _1)$ fixed for all $s$ , $f^{j_1}(h_s(\\gamma _1))$ does not intersect $\\Sigma _{\\rho _1}$ for any $s\\in [0,1]$ , and $f^{j_1}(h_1(\\gamma _1))$ is transverse to $C_{\\rho _1}$ (see [8]).",
"We set the oriented intersection number of $f^{j_1}(h_1(\\gamma _1))$ with $C_{\\rho _1}$ to be $+1$ at a point where the curve moves from below $C_{\\rho _1}$ to above $C_{\\rho _1}$ as $t$ increases, and to be $-1$ at a point where the curve moves from above $C_{\\rho _1}$ to below $C_{\\rho _1}$ as $t$ increases.",
"Then we assign an oriented intersection number between $f^{j_1}(h_1(\\gamma _1))$ and a gap of $\\Sigma _{\\rho _1}$ , by adding the oriented intersection numbers for all of the intersection points within that gap.",
"Since the oriented intersection number is preserved by homotopy, the oriented intersection number between $f^{j_1}( \\gamma _1)$ and a gap is, by definition, the oriented intersection number between $f^{j_1}(h_1(\\gamma _1))$ and that gap.",
"Then the oriented intersection number between $f^{j_1}( \\gamma _1 )$ and $C_{\\rho _1}$ is the sum of the oriented intersection numbers over all gaps.",
"Since the curve $f^{j_1}(\\gamma _1)$ starts from below $C_{\\rho _1}$ and ends above $C_{\\rho _1}$ , there exists a gap for which the oriented intersection number with $f^{j_1}(\\gamma _1)$ is positive.",
"We follow the curve $t\\mapsto f^{j_1}(\\gamma _1(t))$ starting with $t=0$ and we mark the first gap of $\\Sigma _{\\rho _1}$ that is crossed by $f^{j_1}(\\gamma _1)$ with a positive oriented intersection number; we denote by $a^1_{\\rho _1},b^1_{\\rho _1}$ the endpoints of this gap.",
"This means that if $f^{j_1}(\\gamma _1)$ crosses other gaps of $\\Sigma _{\\rho _1}$ that are to the left of this gap, it does so with 0 oriented intersection number.",
"(If the first gap crossed would be crossed with negative oriented intersection number, it would violate the positive tilt condition of $f^{j_1}(\\gamma _1)$ .)",
"Thus, when the curve $f^{j_1}(\\gamma _1)$ crosses the gap between $a^1_{\\rho _1},b^1_{\\rho _1}$ , it comes from below the circle $C_{\\rho _1}$ .",
"Following the curve segment of $f^{j_1}(\\gamma _1(t))$ after crossing the gap of endpoints $a^1_{\\rho _1},b^1_{\\rho _1}$ , we mark the first gap of $\\Sigma _{\\rho ^{\\prime }_1}$ that is crossed by $f^{j_1}(\\gamma _1)$ with positive oriented intersection number, and we denote by $a^1_{\\rho ^{\\prime }_1},b^1_{\\rho ^{\\prime }_1}$ its endpoints.",
"Similarly, when the curve $f^{j_1}(\\gamma _1)$ crosses the gap between $a^1_{\\rho ^{\\prime }_1},b^1_{\\rho ^{\\prime }_1}$ , it comes from below the circle $C_{\\rho ^{\\prime }_1}$ .",
"We claim that the left endpoint of the lower gap is to the right of the right endpoint of the upper gap, i.e., $\\pi _x(a^1_{\\rho _1})<\\pi _x(b^1_{\\rho ^{\\prime }_1})$ .",
"Otherwise, if $\\pi _x(a^1_{\\rho _1})\\ge \\pi _x(b^1_{\\rho ^{\\prime }_1})$ , then there exists an arc $f^{j_1}(\\gamma _1(s))$ , $s\\in [s_1,s_2]$ , such that $\\pi _x(f^{j_1}(\\gamma _1(s_1))=\\pi _x(f^{j_1}(\\gamma _1(s_2))=\\pi _x(a^1_{\\rho _1})$ , $\\pi _y(f^{j_1}(\\gamma _1(s_1))<\\pi _y(f^{j_1}(\\gamma _1(s_2))$ , and $\\pi _x(f^{j_1}(\\gamma _1(s))\\ge \\pi _x(a^1_{\\rho _1})$ for all $s\\in (s_1,s_2)$ .",
"This implies that either the angle deviation from the vertical $\\theta (s)$ along the curve $f^{j_1}(\\gamma _1)$ becomes non-positive for some $s\\in (s_1,s_2)$ , or that there exists another arc $f^{j_1}(\\gamma _1(\\tau ))$ , $\\tau \\in [\\tau _1,\\tau _2]$ , with $\\tau _1<\\tau _2<s_1<s_2$ , such that $\\pi _x(f^{j_1}(\\gamma _1(\\tau _1))=\\pi _x(f^{j_1}(\\gamma _1(\\tau _2))=\\pi _x(a^1_{\\rho _1})$ , $\\pi _y(f^{j_1}(\\gamma _1(\\tau _1))>\\pi _y(f^{j_1}(\\gamma _1(\\tau _2))$ , and $\\pi _x(f^{j_1}(\\gamma _1(s))\\le \\pi _x(f^{j_1}(\\gamma _1(\\tau ))$ for all $s\\in (s_1,s_2)$ and $\\tau \\in (\\tau _1,\\tau _2)$ .",
"In the first case we obtain a contradiction with the positive tilt condition on the curve $f^{j_1}(\\gamma _1)$ .",
"See Fig.",
"REF .",
"In the second case we obtain a contradiction with the fact that $f^{j_1}(\\gamma _1)$ comes from below $C_{\\rho _1}$ before crossing the gap of endpoints $a^1_{\\rho _1},b^1_{\\rho _1}$ .",
"The conclusion of this step is that the curve $f^{j_1}(\\gamma _1(t))$ passes through the gap between $a^1_{\\rho _1}$ and $b^1_{\\rho _1}$ of $\\Sigma _{\\rho _1}$ , from below $C_{\\rho _1}$ to above $C_{\\rho _1}$ , then it passes through the gap between $a^1_{\\rho ^{\\prime }_1}$ and $b^1_{\\rho ^{\\prime }_1}$ of $\\Sigma _{\\rho ^{\\prime }_1}$ , from below $C_{\\rho ^{\\prime }_1}$ to above $C_{\\rho ^{\\prime }_1}$ , and $\\pi _x(a^1_{\\rho _1})<\\pi _x(b^1_{\\rho ^{\\prime }_1})$ .",
"Figure: Violation of the positive tilt condition.Now we consider a one-sided rectangular neighborhood $U_1\\subseteq U_0$ of some point in $T_1$ , bounded below by $T_1$ , to the left by $\\gamma _1$ , and to the right by some other vertical curve segment $\\gamma _1^{\\prime }$ .",
"If $\\gamma ^{\\prime }_1$ is sufficiently close to $\\gamma _1$ , then, by continuity, the image of each vertical curve in $U_1$ under $f^{j_1}$ crosses the gap between $a^1_{\\rho _1}$ and $b^1_{\\rho _1}$ of $\\Sigma _{\\rho _1}$ with positive oriented intersection number, and crosses the gap between $a^1_{\\rho ^{\\prime }_1}$ and $b^1_{\\rho ^{\\prime }_1}$ of $\\Sigma _{\\rho ^{\\prime }_1}$ with positive oriented intersection number.",
"We choose and fix a set $U_1\\subseteq U_0$ with these properties.",
"By Lemma REF (iii) (see also Remark REF ) the closure of $\\bigcup _{j=0}^{\\infty }f^{-j}(W(p_1))$ contains $T_1$ , so there exists $j_2>j_1$ such that, in the annulus, $f^{j_2}(U_1)\\cap W(p_1)\\ne \\emptyset $ , and, in the covering space of the annulus, $f^{j_2}(U_1)$ intersects some copy $W^{h_2}:=W(p_1)+(h_2,0)$ of $W(p_1)$ for some positive integer $h_2>h_1$ .",
"(Due to the positive twist condition on $f$ and the assumption that the rotation numbers on the boundary components of the annulus are positive, the vertical line $\\lbrace x=\\pi _x(p_1)+h_1\\rbrace $ is mapped by $f^{j_2-j_1}$ to a positive tilted map to the right of $\\lbrace x=\\pi _x(p_1)+h_1\\rbrace $ , hence, if $j_2$ is large enough, we can choose $h_2>h_1$ .)",
"Then there exists a vertical curve $\\gamma _2:[0,1]\\rightarrow U_1$ , such that $f^{j_2}(\\gamma _2(1))$ is an intersection point of $f^{j_2}(U_1)$ with $W^{h_2}$ .",
"The curve $f^{j_2}(\\gamma _2(t))$ crosses $C_{\\rho _1}$ and $C_{\\rho ^{\\prime }_1}$ .",
"Let $a^2_{\\rho _1},b^2_{\\rho _1}$ be the endpoints of the leftmost gap of $\\Sigma _{\\rho _1}$ that is crossed by $f^{j_2}(\\gamma _2(t))$ with positive oriented intersection number equal, and let $a^2_{\\rho ^{\\prime }_1},b^2_{\\rho ^{\\prime }_1}$ be the endpoints of the leftmost gap of $C_{\\rho ^{\\prime }_1}$ that is crossed by $f^{j_2}(\\gamma _2(t))$ with positive oriented intersection number.",
"The image curve $f^{j_2}(\\gamma _2)$ is a positively tilted curve located on the `right side' of the positively tilted curve $f^{j_2}(\\gamma _1)$ , in the sense that any graph over $x$ that intersects both $f^{j_2}(\\gamma _1)$ and $f^{j_2}(\\gamma _2)$ has the leftmost intersection point with the $f^{j_1}(\\gamma _1)$ .",
"Therefore, the gap endpoints $a^2_{\\rho _1},b^2_{\\rho _1}$ are either the image under $f^{j_2-j_1}$ of the gap endpoints $a^1_{\\rho _1},b^1_{\\rho _1}$ found at the previous step, or are the image under $f^{j_2-j_1}$ of some other gap endpoints of $\\Sigma _{\\rho _1}$ located to the right of the gap between $a^1_{\\rho _1}$ and $b^1_{\\rho _1}$ .",
"Similarly, the gap endpoints $a^2_{\\rho ^{\\prime }_1},b^2_{\\rho ^{\\prime }_1}$ are either the image under $f^{j_2-j_1}$ of the gap endpoints $a^1_{\\rho ^{\\prime }_1},b^1_{\\rho ^{\\prime }_1}$ from the previous step, or are the image under $f^{j_2-j_1}$ of some other gap of $\\Sigma _{\\rho ^{\\prime }_1}$ located to the right of the gap between $a^1_{\\rho ^{\\prime }_1}$ and $b^1_{\\rho ^{\\prime }_1}$ .",
"Then, there exists a one-sided rectangular neighborhood $U_2\\subseteq U_1$ of some point in $T_1$ , bounded below by $T_1$ , to the left by $\\gamma _2$ , and to the right by some other vertical curve segment $\\gamma ^{\\prime }_2$ , such that the image of each vertical curve in $U_2$ under $f^{j_2}$ crosses the gap between $a^2_{\\rho _1}$ and $b^2_{\\rho _1}$ of $\\Sigma _{\\rho _2}$ with positive oriented intersection number, and crosses the gap between $a^2_{\\rho ^{\\prime }_1}$ and $b^2_{\\rho ^{\\prime }_1}$ of $\\Sigma _{\\rho ^{\\prime }_1}$ with positive oriented intersection number.",
"Recursively, we obtain a nested sequence of one-sided neighborhoods of points in $T_1$ , denoted $U_1\\supseteq U_2\\supseteq \\ldots U_m\\supseteq \\ldots $ , all contained in $U_0$ , and two sequences of positive integers $j_1<j_2<\\ldots <j_m<\\ldots $ and $h_1<h_2<\\ldots <h_m<\\ldots $ with the following properties: (i) each set $U_m$ is a topological rectangle consisting of vertical curves starting from $T_1$ , bounded on the left-side by a vertical curve $\\gamma _m$ and on the right by a vertical curve $\\gamma ^{\\prime }_m$ ; (ii) $f^{j_m}(U_m)\\cap W^{h_m}\\ne \\emptyset $ , where $W^{h_m}:=W(p_1)+(h_m,0)$ ; (iii) the image of each vertical curve in $U_m$ under $f^{j_m}$ crosses $C_{\\rho _1}$ with positive oriented intersection number through a gap between $a^m_{\\rho _1}$ and $b^m_{\\rho _1}$ of $\\Sigma _{\\rho _1}$ , and it crosses $C_{\\rho ^{\\prime }_1}$ with positive oriented intersection number through a gap between $a^m_{\\rho ^{\\prime }_1}$ and $b^m_{\\rho ^{\\prime }_1}$ of $\\Sigma _{\\rho ^{\\prime }_1}$ ; (iv) $\\pi _x(a^m_{\\rho _1})<\\pi _x(b^m_{\\rho ^{\\prime }_1})$ ; (v) the gap endpoints $ a^m_{\\rho _1},b^m_{\\rho _1} $ are the images under $f^{j_m-j_{m-1}}$ of the gap endpoints $ a^{m-1}_{\\rho _1},b^{m-1}_{\\rho _1}$ , or of the endpoints of some other gap in $\\Sigma _{\\rho _1}$ located to the right side of this gap; the gap endpoints $ a^m_{\\rho ^{\\prime }_1},b^m_{\\rho ^{\\prime }_1}$ are the images under $f^{j_m-j_{m-1}}$ of the gap endpoints $a^{m-1}_{\\rho _1}$ , $b^{m-1}_{\\rho ^{\\prime }_1}$ , or of the endpoints of some other gap in $\\Sigma _{\\rho ^{\\prime }_1}$ located to the right side of this gap.",
"The endpoints of a gap of $\\Sigma _{\\rho _1}$ or $\\Sigma _{\\rho ^{\\prime }_1}$ are mapped by $f$ into the endpoints of some other gap of $\\Sigma _{\\rho _1}$ or $\\Sigma _{\\rho ^{\\prime }_1}$ , respectively.",
"Also, the order of the gaps is preserved under iteration.",
"The endpoints of the gap in $\\Sigma _{\\rho _1}$ are iterated with rotation number $\\rho _1$ , and the endpoints of the gap in $\\Sigma _{\\rho ^{\\prime }_1}$ are iterated with rotation number $\\rho ^{\\prime }_1>\\rho _1$ .",
"Then, for some sufficiently large iterate $j_m$ the order of the gaps gets reversed in the annulus.",
"That is, we get the following ordering in terms of the angle coordinate in the covering space of the annulus: (i) $\\pi _x(a^m_{\\rho _1})<\\pi _x(b^m_{\\rho _1})< \\pi _x(a^1_{\\rho _1})+h_m<\\pi _x(b^1_{\\rho _1})+h_m$ , (ii) $\\pi _x(a^1_{\\rho ^{\\prime }_1})+h_m<\\pi _x(b^1_{\\rho ^{\\prime }_1})+h_m<\\pi _x(a^m_{\\rho ^{\\prime }_1})<\\pi _x(b^m_{\\rho ^{\\prime }_1})$ .",
"Since $f^{j_1}(U_1)$ and $f^{j_m}(U_m)$ are connected, the above ordering of the crossings with $C_{\\rho _1}$ and $C_{\\rho ^{\\prime }_1}$ implies that $f^{j_1}(U_1)$ and $f^{j_m}(U_m)$ have an intersection point above $C_{\\rho _1}$ .",
"As $U_1, U_m\\subseteq U_0$ , it follows that $f^{j_1}(U_0)$ and $f^{j_m}(U_0)$ have an intersection point above $C_{\\rho _1}$ .",
"Since $f^{j_1}(U_0)$ and $f^{j_m}(U_0)$ are connected and disjoint from the Aubry-Mather set $\\Sigma _{\\rho _1}\\subseteq C_{\\rho _1}$ , there exists a component $\\mathcal {U}$ of the complement $\\mathcal {Z}\\setminus [f^{j_1}(U_0)\\cup f^{j_m}(U_0)]$ which is an open topological disk containing some point $\\xi _1\\in \\Sigma _{\\rho _1}$ .",
"The boundary of $\\mathcal {U}$ consists of a finite union of sub-arcs of the boundaries of $f^{j_1}(U_0)$ and $f^{j_m}(U_0)$ , and possibly of curve segments of $T_1$ .",
"See Figure REF .",
"Letting $j_*=j_1$ and $j^{\\prime }_*=j_m$ ends the proof of the claim.",
"Figure: An open topological disk forming a neighborhood of a point in the Aubry-Mather set Σ ρ 1 \\Sigma _{\\rho _1}.We now apply the statement of the claim to $T_2$ , starting with the one-sided neighborhood $V_0$ of $\\zeta _2\\in T_2$ .",
"We choose three Aubry-Mather sets $\\Sigma _{\\rho _2}$ , $\\Sigma _{\\rho _2^{\\prime }}$ , $\\Sigma _{\\rho _2^{\\prime \\prime }}$ in $\\mathcal {Z}$ , lying on three essential circles $C_{\\rho _2},C_{\\rho ^{\\prime }_2},C_{\\rho ^{\\prime \\prime }_2}$ , respectively, with ${\\rho _2}>{\\rho ^{\\prime }_2}>{\\rho ^{\\prime \\prime }_2}$ irrational rotation numbers, and $C_{\\rho _2} \\succ C_{\\rho ^{\\prime }_2} \\succ C_{\\rho ^{\\prime \\prime }_2}$ .",
"The statement in the claim implies that there exists a neighborhood $\\mathcal {V}$ of some point $\\xi _2\\in \\Sigma _{\\rho _2}$ , homoeomorphic to an open disk, whose boundary consists of a finite union of sub-arcs in the boundaries of $f^{-l_*}(V_0)$ and $f^{-l^{\\prime }_*}(V_0)$ , for some $l_*<l^{\\prime }_*$ , and possibly a finite union of curve segments of $T_2$ .",
"By Theorem REF there is an orbit that goes from $\\mathcal {U}$ to $\\mathcal {V}$ , i.e., there exists $k_*$ such that $f^{k_*}(\\mathcal {U})\\cap \\mathcal {V}\\ne \\emptyset $ .",
"The boundary of $\\mathcal {U}$ (resp.",
"$\\mathcal {V}$ ) is a simple closed curved separating the annulus into two connected components.",
"Also, the boundary of each of $f^{j_*}(U_0),f^{j^{\\prime }_*}(U_0),f^{-l_*}(V_0),f^{-l^{\\prime }_*}(V_0)$ is a simple closed curve.",
"Since $f^{k_*}(\\mathcal {U})\\cap \\mathcal {V}\\ne \\emptyset $ , the Jordan Curve Theorem implies that either $\\partial f^{k_*}(\\mathcal {U}) \\cap \\textrm {int}[f^{-l_*}(V_0)\\cup f^{-l^{\\prime }_*}(V_0)] \\ne \\emptyset $ or $\\partial \\mathcal {V} \\cap \\textrm {int}[f^{j_*}(U_0)\\cup f^{j^{\\prime }_*}(U_0)]\\ne \\emptyset $ .",
"It follows that $\\textrm {int}(f^{k_*+j_*}(U_0))$ or $\\textrm {int}(f^{k_*+j^{\\prime }_*}(U_0))$ has a non-empty intersection with $\\textrm {int}(f^{-l_*}(V_0))$ or $\\textrm {int}(f^{-l^{\\prime }_*}(V_0))$ .",
"Thus, some forward iterate of $\\textrm {int}(U_0)$ intersects $\\textrm {int}(V_0)$ .",
"Hence there exists $N>0$ such that $f^N(\\textrm {int}(U _0))\\cap \\textrm {int}(V_0) \\ne \\emptyset $ .",
"Since the sets $f^{N}(U_0)$ and $V_0$ are topological disks have interior points in common, but also points that are not in common (namely the points lying on $T_1$ and $T_2$ , respectively), the Jordan Curve Theorem implies that $f^{N}(\\partial U _0)\\cap \\partial V_0\\ne \\emptyset $ .",
"The remaining case of the proof, when the interior of $U_0$ does intersect some Aubry-Mather set and the interior of $V_0$ does not, or when the interior of $U_0$ does intersect some Aubry-Mather set and the interior of $V_0$ does not, follows easily from the above arguments.",
"The next statement says that given two points on the boundary tori of a BZI, and a finite sequence of Aubry-Mather sets inside the zone, there exists an orbit that starts in a prescribed neighborhood of the point on the lower boundary torus, then moves on and shadows, in the sense of the ordering of the orbit, each Aubry-Mather set in the sequence, and ends in a prescribed neighborhood of the point on the upper boundary torus.",
"This result extends Theorem REF , and relies on the topological argument of Hall.",
"As in the previous theorem, we do not need any extra conditions on the dynamics on the boundary tori.",
"The resulting shadowing orbits are not necessarily minimal.",
"Theorem 6.2 Suppose that $T_1$ and $T_2$ bound a BZI $\\mathcal {Z}$ .",
"Let $\\zeta _1\\in T_1,\\zeta _2\\in T_2$ , $U$ be a neighborhood of $\\zeta _1$ , and $V$ a neighborhood of $\\zeta _2$ .",
"Let $\\lbrace \\Sigma _{\\omega _s}\\rbrace _{s\\in \\lbrace 1,\\ldots ,\\sigma \\rbrace }$ be a finite collection of Aubry-Mather sets inside $\\mathcal {Z}$ such that each $\\Sigma _{\\omega _s}$ lies on some essential circle $C_{\\omega _s}$ that is a graph over the $x$ -coordinate, with $C_{\\omega _s}\\prec C_{\\omega _{s^{\\prime }}}$ provided $\\omega _s<\\omega _{s^{\\prime }}$ .",
"Let $\\lbrace n_s\\rbrace _{s=1,\\ldots , \\sigma }$ be sequence of positive integers.",
"Then there exist a point $z\\in U$ , and a sequence of positive integers $\\lbrace m_s\\rbrace _{s=0,\\ldots ,\\sigma }$ , such that, for each $s\\in \\lbrace 1,\\ldots ,\\sigma \\rbrace $ , $\\begin{split}\\pi _x(f^j(w_s))<\\pi _x(f^j(z))<\\pi _x(f^j(\\bar{w}_s))\\textrm { for } \\\\ \\sum _{t=1}^{s-1} n_t+\\sum _{t=0}^{s-1}m_t\\le j\\le \\sum _{t=1}^{s} n_t+\\sum _{t=0}^{s-1}m_t,\\end{split}$ where $w_s$ and $\\bar{w}_s$ are some points in the Aubry-Mather set $\\Sigma _{\\omega _s}$ , and $f^{N}(z)\\in V$ for $N=\\sum _{t=1}^{\\sigma } n_t+\\sum _{t=0}^{\\sigma }m_t$ .",
"The number $N$ can be chosen arbitrarily large.",
"Moreover, if $U,V$ are chosen so that $U\\cap \\mathcal {Z},V\\cap \\mathcal {Z}$ are topological disks, then there exists a point $z^{\\prime }\\in \\partial U$ satisfying the ordering condition (REF ) such that $f^N(z^{\\prime })\\in \\partial V$ .",
"We use the construction of diagonal sets described in the sketch of the proof of Theorem REF ; for details see [29].",
"Part 1.",
"As in the proof of Theorem REF we can choose one sided compact neighborhoods $U_0$ of $\\zeta _1\\in T_1$ and $V_0$ of $\\zeta _2\\in T_2$ , such that $U_0\\cap \\mathcal {Z}$ and $V_0\\cap \\mathcal {Z}$ are topological disks.",
"We first prove the existence of a point $z\\in U_0$ satisfying (REF ) and such that $f^N(z)\\in V_0$ .",
"Let $C_{\\omega _1}$ be the essential circle containing $\\Sigma _{\\omega _1}$ .",
"We choose an Aubry-Mather set $\\Sigma _{\\rho _1}$ lying on some essential circle $C_{\\rho _1}$ , such that $C_{\\omega _1}\\prec C_{\\rho _1}$ .",
"We choose a point $p_1\\in \\Sigma _{\\rho _1}$ and a small neighborhood $W(p_1)$ of $p_1$ which does not intersect $\\Sigma _{\\omega _1}$ .",
"We assume that the interior of $U_0$ does not meet $\\Sigma _{\\omega _1}$ and $\\Sigma _{\\rho _1}$ , otherwise the proof follows as in [29].",
"By Lemma REF (iii) the closure of $\\bigcup _{j=0}^{\\infty }f^{-j}(W(p_1))$ contains $T_1$ , and in particular $\\zeta _1$ .",
"Proceeding as in the proof of Theorem REF , we obtain a nested sequence $U_1\\supseteq U_2\\supseteq \\ldots \\supseteq U_i$ of one-sided neighborhoods of points in $T_1$ , all contained in $U_0$ , and two sequences of positive integers $j_1<j_2<\\ldots <j_i<\\ldots $ and $h_1<h_2<\\ldots <h_i<\\ldots $ with the following properties: (i) each set $U_i$ is a topological rectangle consisting of vertical curves starting from $T_1$ , bounded on the left-side by a vertical curve $\\gamma _i$ and on the right by a vertical curve $\\gamma ^{\\prime }_i$ ; (ii) $f^{j_i}(U_i)\\cap W^{h_i}\\ne \\emptyset $ , where $W^{h_i}:=W(p_1)+(h_i,0)$ ; (iii) the image of each vertical curve in $U_i$ under $f^{j_i}$ crosses $C_{\\omega _1}$ with positive oriented intersection number through a gap in $\\Sigma _{\\omega _1}$ of endpoints $a^i_{\\omega _1}$ and $b^i_{\\omega _1}$ ; the gap is chosen to be the first gap that is crossed over with positive oriented intersection number; (iv) the endpoints of the gap between $a^i_{\\omega _1}$ and $b^i_{\\omega _1}$ are the images under $f^{j_i-j_{i-1}}$ of either the endpoints of the gap between $a^{i-1}_{\\omega _1}$ and $b^{i-1}_{\\omega _1}$ , or of a gap in $\\Sigma _{\\omega _1}$ located to the right side of that gap.",
"Since the rotation number of $\\Sigma _{\\omega _1}$ is smaller than the rotation number of $\\Sigma _{\\rho _1}$ , any pair of points chosen on these two sets shift apart from one another under positive iterations.",
"Therefore there exists some $i$ large enough so that the gap of endpoints $a^i_{\\omega _1}$ and $b^i_{\\omega _1}$ is on the left side of $W^{h_i}$ , in the sense that $\\pi _x(b^i_{\\omega _1})<\\pi _x(z)$ for all $z\\in W^{h_i}$ .",
"We claim that, by choosing $i$ large enough and $\\gamma ^{\\prime }_i$ sufficiently close to $\\gamma _i$ , we can ensure that the set $f^{j_i}(U_i)$ has a part which is a positive diagonal set in $B_{a^{i}_{\\omega _ 1},b^{i}_{\\omega _1}}$ .",
"Now we justify the claim.",
"The image of the left-side $\\gamma _i$ of $U_i$ is mapped by $f^{j_i}$ onto a positively tilted curve that crosses the gap between $a^i_{\\omega _1}$ and $b^i_{\\omega _1}$ with positive oriented intersection number.",
"Cutting the curve $f^{j_1}(\\gamma _1)$ with the vertical strip $B_{a^i_{\\omega _1},b^i_{\\omega _1}}$ yields at least one component that connects $I_{a^i_{\\omega _1}}=\\lbrace x=a^i_{\\omega _1}\\rbrace $ to $I_{b^i_{\\omega _1}}=\\lbrace x=b^i_{\\omega _1}\\rbrace $ with positive oriented intersection number.",
"Take the first such a component and follow it in the direction of the increase of the parameter $t$ .",
"The intersection of this component with $I_{a^i_{\\omega _1}}$ needs to occur at a point $r_1$ below $a^i_{\\omega _1}$ , i.e.",
"$r_1\\in I^{-}_{a^i_{\\omega _1}}$ , otherwise this component does not come from below $C_{\\omega _1}$ .",
"Following this component forward starting from $r_1$ , the curve cannot intersect $I_{a^i_{\\omega _1}}$ above $a^i_{\\omega _1}$ as this would violate the positive tilt condition, or the choice of the gap between $a^i_{\\omega _1}$ and $b^i_{\\omega _1}$ being the first gap that is crossed with positive oriented intersection number.",
"Following the component starting from $r_1$ , it must first meet $I_{b^i_{\\omega _1}}$ at a point $r^{\\prime }_1$ above $b^i_{\\omega _1}$ , i.e.",
"$r^{\\prime }_1\\in I^{+}_{b^i_{\\omega _1}}$ ; otherwise this component does not have positive oriented intersection number with the gap between $a^i_{\\omega _1}$ and $b^i_{\\omega _1}$ .",
"Therefore, the component of $f^{j_i}(\\gamma _i)$ between $r_1$ and $r^{\\prime }_1$ goes from $I^-_{a^i_{\\omega _1}}$ to $I^+_{b^i_{\\omega _1}}$ without intersecting again $I^+_{a^i_{\\omega _1}}$ or $I^-_{b^i_{\\omega _1}}$ .",
"Now taking a curve $\\gamma ^{\\prime }_i$ sufficiently close to $\\gamma _i$ results in a set $U_i$ with the property that $f^{j_i}(U_i)$ has a part which is a positive diagonal set in $B_{a^{i}_{\\omega _ 1},b^{i}_{\\omega _1}}$ .",
"We change notation at this point: we denote $m_0:=j_i$ , $w_1:=a^{i}_{\\omega _1}$ , and $\\bar{w}_1:=b^{i}_{\\omega _1}$ , $U^{\\prime }=U_i$ for $i$ fixed as above.",
"Thus, the points $w_1$ and $\\bar{w}_1$ are the endpoints of a gap in $\\Sigma _{\\omega _1}$ , and $f^{m_0}(U^{\\prime })$ has a part which is a positive diagonal in $B_{w_1,\\bar{w}_1}$ .",
"The positive integer $m_0$ is the first term of the sequence $\\lbrace m_s\\rbrace _{s=0,\\ldots ,\\sigma }$ in the statement of the theorem.",
"Note that $U^{\\prime }$ consists of a union of vertical segments emerging from $T_1$ .",
"See Figure REF .",
"Figure: A positive diagonal set.Using the construction described in the sketch of the proof of Theorem REF , we obtain a nested sequence $D_0\\supseteq D_1\\supseteq \\ldots \\supseteq D_\\sigma $ of negative diagonals of $B_{w_1,\\bar{w}_1}$ and a sequence of positive integers $\\lbrace m_s\\rbrace _{s=0,\\ldots ,\\sigma -1}$ such that for each $s\\in \\lbrace 1,\\ldots ,\\sigma \\rbrace $ and each $z \\in D_s$ we have $\\pi _x(f^j(w_s))<\\pi _x(f^j(z))<\\pi _x(f^j(\\bar{w}_s)) \\textrm { for } j_s\\le j\\le j_s+n_s,$ where $j_s:=\\sum _{t=1}^{s}n_t+\\sum _{t=0}^{s-1}m_t$ , $w_s$ and $\\bar{w}_s$ are the endpoints of some gap in the Aubry-Mather set $\\Sigma _{\\omega _s}$ , and $f^{(j_s+n_s)}(D_s)$ is a positive diagonal in $B_{f^{n_s}(w_s),f^{n_s}(\\bar{w}_s)}$ .",
"In particular, $f^{(j_\\sigma +n_\\sigma )}(D_\\sigma )$ is a positive diagonal in $B_{f^{n_\\sigma }(w_\\sigma ),f^{n_\\sigma }(\\bar{w}_\\sigma )}$ , where $w_\\sigma $ and $\\bar{w}_\\sigma $ are the endpoints of some gap in the Aubry-Mather set $\\Sigma _{\\omega _\\sigma }$ .",
"Since $f^{m_0}(U^{\\prime })$ has a part which is a positive diagonal in $B_{w_1,\\bar{w}_1}$ and $D_{\\sigma }$ is a negative diagonal of $B_{w_1,\\bar{w}_1}$ , then $f^{m_0}(U^{\\prime })$ and $D_{\\sigma }$ have a non-empty intersection.",
"Also, $f^{(j_\\sigma +n_\\sigma )}(U^{\\prime })$ has a part which is a positive diagonal in $B_{f^{n_\\sigma }(w_\\sigma ),f^{n_\\sigma }(\\bar{w}_\\sigma )}$ that stretches across $f^{(j_\\sigma +n_\\sigma )}(D_\\sigma )$ .",
"Now we start with the one-sided neighborhood $V_0$ of $\\zeta _2\\in T_2$ .",
"Let $C_{\\omega _\\sigma }$ be an essential circle containing $\\Sigma _{\\omega _\\sigma }$ .",
"We choose an Aubry-Mather set $\\Sigma _{\\rho _2}$ lying on an essential circle $C_{\\rho _2}$ , such that $C_{\\rho _2}$ is below $C_{\\omega _\\sigma }$ .",
"We assume that the interior of $V_0$ does not meet $\\Sigma _{\\omega _\\sigma }$ and $\\Sigma _{\\rho _2}$ , otherwise the proof follows as in [29].",
"Using Lemma REF (iii) and following the procedure described above for negative iterations, we produce a one-sided neighborhood $V^{\\prime }$ of a point in $T_2$ , with $V^{\\prime }\\subseteq V_0$ , and a positive integer $m^{\\prime }_\\sigma $ such that $f^{-m^{\\prime }_\\sigma }(V^{\\prime })$ contains a part which is a negative diagonal in $B_{w^{\\prime }_\\sigma ,\\bar{w}^{\\prime }_\\sigma }$ , where $w^{\\prime }_\\sigma $ and $\\bar{w}^{\\prime }_\\sigma $ are the endpoints of a gap in $\\Sigma _{\\omega _\\sigma }$ .",
"By using the existence of orbits passing from near $T_2$ to near $T_1$ , and of orbits passing from near $T_1$ to near $T_2$ , as in the sketch of the proof of Theorem REF , we can further iterate the positive diagonal $f^{(j_\\sigma +n_\\sigma )}(U^{\\prime })$ from above, so that we obtain an iterate $f^{(j_\\sigma +n_\\sigma +m^{\\prime \\prime }_\\sigma )}(U^{\\prime })$ which contains a component that stretches all the way across a fundamental interval of the annulus.",
"In particular, $f^{(j_\\sigma +n_\\sigma +m^{\\prime \\prime }_\\sigma )}(U^{\\prime })$ contains a component that is a positive diagonal of $B_{w^{\\prime }_\\sigma ,\\bar{w}^{\\prime }_\\sigma }$ .",
"Since $f^{-m^{\\prime }_\\sigma }(V^{\\prime })$ is a negative diagonal in $B_{w^{\\prime }_\\sigma ,\\bar{w}^{\\prime }_\\sigma }$ , then $f^{(j_\\sigma +n_\\sigma +m^{\\prime \\prime }_\\sigma )}(U^{\\prime })$ has a nonempty intersection with $f^{-m^{\\prime }_\\sigma }(V^{\\prime })$ .",
"Equivalently, $f^{(j_\\sigma +n_\\sigma +m_\\sigma )}(U^{\\prime })$ has a non-empty intersection with $V^{\\prime }$ , where $m_\\sigma :=m^{\\prime }_\\sigma +m^{\\prime \\prime }_\\sigma $ .",
"Thus, each point $z \\in U^{\\prime }\\cap f^{-(j_\\sigma +n_\\sigma +m_\\sigma )}(V^{\\prime })$ goes from the neighborhood $U_0$ of $\\zeta _1$ to the neighborhood $V_0$ of $\\zeta _2$ and it shadows, in the sense of the ordering, each of the Aubry-Mather set $\\Sigma _{\\omega _s}$ , $s=1,\\ldots ,\\sigma $ , along the way.",
"Part 2.",
"Now we explain how to modify the above proof to show that there exists a point $z^{\\prime }\\in \\partial U_0$ that satisfies (REF ) and $f^{N}(z^{\\prime })\\in \\partial V_0$ .",
"This does not follow immediately from the above argument since the image of $\\partial U_0$ under iteration may fail being a positively tilted curve; hence we cannot infer that $f^{m_0}(\\partial U_0)$ intersects the negative diagonal set $D_\\sigma $ from above.",
"By Theorem REF , there exist $l>0$ and a point $q\\in U_0$ depending on $l$ such that $f^l(q)$ is in some prescribed neighborhood of a point $r\\in T_2$ , where $l$ can be chosen arbitrarily large.",
"Since the points of $T_1$ and $T_2$ have different rotation numbers hence move apart under iteration, there exists $l_0$ sufficiently large such that $f^{l_0}(T_1\\cap U_0)$ and $f^{l_0}(q)$ are separated by a fundamental interval of the annulus, i.e., $\\pi _x(f^{l_0}(r))-\\pi _x(f^{l_0}(q))>1$ for all $r\\in T_1\\cap U_0$ .",
"Let $x_0,\\bar{x}_0\\in \\mathbb {R}$ be such that $\\pi _x(f^{l_0}(r))<x_0<\\bar{x}_0 <\\pi _x(f^{l_0}(q))$ and $1<\\bar{x}_0-x_0$ .",
"Let $w_0$ be a point on the vertical line $\\lbrace x=x_0\\rbrace $ whose $y$ -coordinate is larger than that of any point in $f^{l_0}(U_0)\\cap \\lbrace x={x_0}\\rbrace $ .",
"Similarly, let $\\bar{w}_0$ be a point on the vertical line $\\lbrace x=\\bar{x}_0\\rbrace $ whose $y$ -coordinate is smaller than that of any point in $f^{l_0}(U_0)\\cap \\lbrace x=\\bar{x}_0\\rbrace $ .",
"Then $f^{l_0}(U_0)\\cap \\textrm {cl}(B_{w_0, \\bar{w}_0})$ has a component that is a positive diagonal in $B_{w_0, \\bar{w}_0}$ .",
"Let $\\Sigma _{\\omega _1}$ be the first set in the prescribed sequence of Aubry-Mather sets.",
"Now we want to show that, by slightly adjusting the vertical strip $B_{w_0, \\bar{w}_0}$ to a new vertical strip $B_{z_0, \\bar{z}_0}$ , there exists a diagonal set in $f^{l_0}(U_0)\\cap B_{z_0, \\bar{z}_0}$ such that a sufficiently large iterate of this diagonal set has a component that is a positive diagonal in $B_{w_1,\\bar{w}_1}$ , for some points $w_1,\\bar{w}_1\\in \\Sigma _{\\omega _1}$ .",
"There exists $x^{\\prime }_0$ sufficiently close to $x_0$ such that for all $x^{\\prime \\prime }_0$ between $x_0$ and $x^{\\prime }_0$ , the point $(x^{\\prime \\prime }_0, \\pi _y(w_0))$ has the $y$ -coordinate larger than that of any point in $f^{l_0}(\\textrm {cl}(U_0))\\cap \\lbrace x=x^{\\prime \\prime }_0\\rbrace $ .",
"Then the set $W_0=\\lbrace (x^{\\prime \\prime }_0,y^{\\prime \\prime }_0)\\,|\\, x_0<x^{\\prime \\prime }_0<x^{\\prime }_0, \\pi _y(w_0)<y^{\\prime \\prime }_0\\rbrace $ is a neighborhood of an arc in $T_2$ , with the property that each point $(x^{\\prime \\prime }_0,y^{\\prime \\prime }_0)\\in W_0$ has the $y$ -coordinate larger than that of any point in $f^{l_0}(\\textrm {cl}(U_0))\\cap \\lbrace x=x^{\\prime \\prime }_0\\rbrace $ .",
"Similarly, there is $\\bar{x}^{\\prime }_0$ sufficiently close to $\\bar{x}_0$ such that for all $\\bar{x}^{\\prime \\prime }_0$ between $\\bar{x}_0$ and $\\bar{x}^{\\prime }_0$ , the point $(\\bar{x}^{\\prime \\prime }_0, \\pi _y(\\bar{w}_0))$ has the $y$ -coordinate smaller than that of any point in $f^{l_0}(\\textrm {cl}(U_0))\\cap \\lbrace x=\\bar{x}^{\\prime \\prime }_0\\rbrace $ .",
"Then the set $\\bar{W}_0=\\lbrace (\\bar{x}^{\\prime \\prime }_0,\\bar{y}^{\\prime \\prime }_0)\\,|\\, \\bar{x}_0<\\bar{x}^{\\prime \\prime }_0<\\bar{x}^{\\prime }_0, \\pi _y(\\bar{w}_0)>\\bar{y}^{\\prime \\prime }_0\\rbrace $ is a neighborhood of an arc in $T_1$ , with the property that each point $(\\bar{x}^{\\prime \\prime }_0,\\bar{y}^{\\prime \\prime }_0)\\in \\bar{W}_0$ has the $y$ -coordinate smaller than that of any point in $f^{l_0}(\\textrm {cl}(U_0))\\cap \\lbrace x=\\bar{x}^{\\prime \\prime }_0\\rbrace $ .",
"Let $\\Sigma _{\\rho _1}$ be an Aubry-Mather set lying on an essential circle $C_{\\rho _1}$ that is below the essential circle $C_{\\omega _1}$ containing $\\Sigma _{\\omega _1}$ , let $p_1\\in C_{\\rho _1}$ , and let $W(p_1)$ be a small neighborhood of $p_1$ that does not intersect $\\Sigma _{\\omega _1}$ .",
"Then there exists $z_0\\in W_0$ and ${j_0}$ sufficiently large such that $f^{j_0}(z_0)\\in W(p_1)$ .",
"Since the curve $f^{j_0}(I^+_{z_0})$ is a positively tilted curve emerging from $T_2$ , the arguments used in Part 1 show that there is a gap of the Aubry-Mather set $\\Sigma _{\\omega _1}$ , between a pair of points $w_1,w^{\\prime }_1\\in \\Sigma _{\\omega _1}$ , such that $f^{j_0}(I^+_{z_0})$ crosses this gap with negative intersection number (where the parametrization of $f^{j_0}(I^+_{z_0})$ is chosen so that to $t=0$ it corresponds a point on $T_2$ ) and has its first intersection with $I_{w_1}$ below the point $w_1$ , provided ${j_0}$ is chosen large enough.",
"This implies that the image of the diagonal component of $f^{l_0}(\\textrm {cl}(U_0))\\cap B_{z_0,\\bar{w}_0}$ under $f^{j_0}$ has a component $\\Delta _0$ in $B_{w_1,w^{\\prime }_1}$ that satisfies the positive diagonal set conditions relative to its left side.",
"See Fig.",
"REF .",
"Figure: Construction of a positive diagonal set.Now, for all $j^{\\prime }>0$ , the image of $\\Delta _0$ under $f^{j^{\\prime }}$ also has a component that satisfies the positive diagonal set conditions relative to the left side of $B_{f^{j^{\\prime }}(w_1),f^{j^{\\prime }}(w^{\\prime }_1)}$ .",
"(This follows from the hereditary property of diagonal sets, see Section .)",
"In a similar fashion, there exist a point $\\bar{z}_0\\in \\bar{W}_0$ and $j^{\\prime }_0$ sufficiently large such that the positively tilted curve $f^{j^{\\prime }_0+j_0}(I^-_{\\bar{z}_0})$ , emerging from $T_1$ , crosses a gap of the Aubry-Mather set $\\Sigma _{\\omega _1}$ between a pair of points $\\bar{w}_1,\\bar{w}^{\\prime }_1\\in \\Sigma _{\\omega _1}$ , the oriented intersection number between $f^{j^{\\prime }_0+j_0}(I^-_{\\bar{z}_0})$ and this gap is positive, and the first intersection with $I_{\\bar{w}^{\\prime }_1}$ occurs above the point $\\bar{w}^{\\prime }_1$ .",
"Thus, the image of $\\Delta _0$ under $f^{j^{\\prime }_0}$ has a component that satisfies the positive diagonal set conditions relative to its right side.",
"In summary, the image of the diagonal set component of $f^{l_0}(\\textrm {cl}(U_0))\\cap B_{z_0, \\bar{z}_0}$ under $f^{j_0+j^{\\prime }_0}$ contains a component $\\Delta ^{\\prime }_0$ that is a positive diagonal in $B_{w_1,\\bar{w}^{\\prime }_1}$ , where $w_1,\\bar{w}^{\\prime }_1$ are two points in the Aubry-Mather set $\\Sigma _{\\omega _1}$ .",
"Moreover, the upper edge and the lower edge of $\\Delta ^{\\prime }_0$ are contained in $f^{j^{\\prime }_0+j_0+l_0}(\\partial U_0)$ .",
"We apply an analogous argument at the other boundary torus $T_2$ .",
"Given $V_0$ a neighborhood of a point $\\zeta _2\\in T_2$ , there exist $L>0$ and a pair of points $w_\\sigma ,\\bar{w}_\\sigma \\in \\Sigma _{\\omega _\\sigma }$ such that $f^{-L}(\\textrm {cl}(V_0))\\cap B_{w_\\sigma ,\\bar{w}_\\sigma }$ has a component $D^{\\prime \\prime }_\\sigma $ that is a negative diagonal in $B_{w_\\sigma ,\\bar{w}_\\sigma }$ .",
"The upper edge and the lower edge of this positive diagonal set are contained in $f^{-L}(\\partial V_0)$ .",
"Now we apply the argument from Part 1.",
"There exists a negative diagonal set $D_\\sigma $ in $B_{w_1,\\bar{w}_1}$ such that all points $z\\in D_\\sigma $ satisfy (REF ).",
"By the argument for Theorem REF , the upper and lower edge of $D_{\\sigma }$ lie on $f^{-j_\\sigma }(I^-_{w_\\sigma }),f^{-j_\\sigma }(I^+_{\\bar{w}_\\sigma })$ where $j_\\sigma =\\sum _{s=0}^{\\sigma }n_s+\\sum _{s=0}^{\\sigma -1}m_s$ .",
"Since negative and positive diagonal sets in the same vertical strip always intersect, the negative diagonal set $D_\\sigma $ intersects $\\Delta ^{\\prime }_0$ , and in particular it intersects its upper and lower edges that are contained in $f^{j^{\\prime }_0+j_0+l_0}(\\partial U_0)$ .",
"Iterating $\\Delta ^{\\prime }_0$ forward for $j_\\sigma $ times yields a positive diagonal set $D^{\\prime }_\\sigma $ in $B_{w_\\sigma ,\\bar{w}_\\sigma }$ .",
"The upper and lower edges of $D^{\\prime }_\\sigma $ are contained in $f^{j_\\sigma +j^{\\prime }_0+j_0+l_0}(\\partial U_0)$ .",
"The positive diagonal set $D^{\\prime }_\\sigma $ intersects the negative diagonal component $D^{\\prime \\prime }_\\sigma $ of $f^{-L}(\\textrm {cl}(V_0))\\cap B_{w_\\sigma ,\\bar{w}_\\sigma }$ .",
"In particular the upper and lower edges of $D^{\\prime }_{\\sigma }$ , that are contained in $f^{j_\\sigma +j^{\\prime }_0+j_0+l_0}(\\partial U_0)$ , intersect the upper and lower edges of $D^{\\prime \\prime }_\\sigma $ that are contained in $f^{-L}(\\partial V_0)$ .",
"Thus, there exists a point $z^{\\prime }\\in \\partial U_0$ that is taken by $f^{j_\\sigma +j^{\\prime }_0+j_0+l_0}$ to $\\partial V_0$ and satisfies the ordering relations (REF )."
],
[
"A shadowing lemma in normally hyperbolic invariant manifolds",
"In this section we present a shadowing lemma-type of result saying that, given a sequence of windows within a normally hyperbolic invariant manifold, consisting of pairs of windows correctly aligned under the scattering map, alternating with pairs of windows correctly aligned under some iterate of the inner map, then there exists a true orbit in the full space dynamics that follows these windows.",
"This result reduces the construction of windows within the full dimensional phase space to the construction of lower dimensional windows within the normally hyperbolic invariant manifold.",
"For this section, we assume a diffeomorphism $f:M\\rightarrow M$ on a manifold $M$ , and an $l$ -dimensional normally hyperbolic invariant manifold $\\Lambda \\subseteq M$ as in Subsection REF .",
"In the subsequent sections, we will apply Lemma REF only in the case when $\\Lambda $ is a 2-dimensional normally hyperbolic invariant manifold in $M$ , i.e., $l=2$ .",
"A more general version of this lemma which does not involve the scattering map, and some additional details and applications, appear in [16].",
"Lemma 7.1 Let $\\lbrace R_i,R^{\\prime }_i\\rbrace _{i\\in \\mathbb {Z}}$ be a bi-infinite sequence of $l$ -dimensional windows contained in $\\Lambda $ .",
"Assume that the following properties hold for all $i\\in \\mathbb {Z}$ : (i) $R_{i}\\subseteq U^-$ and $R^{\\prime }_{i}\\subseteq U^+$ .",
"(ii) $R_{i}$ is correctly aligned with $R^{\\prime }_{i+1}$ under the scattering map $S$ .",
"(iii) for each pair $R^{\\prime }_{i+1},R_{i+1}$ and for each $L>0$ there exists $L^{\\prime }>L$ such that $R^{\\prime }_{i+1}$ is correctly aligned with $R_{i+1}$ under the iterate $f_{\\mid \\Lambda }^{L^{\\prime }}$ of the restriction $f_{\\mid \\Lambda }$ of $f$ to $\\Lambda $ .",
"Fix any bi-infinite sequence of positive real numbers $\\lbrace \\varepsilon _i\\rbrace _{i\\in \\mathbb {Z}}$ .",
"Then there exist an orbit $(f^{n}(z))_{n\\in \\mathbb {Z}}$ of some point $z\\in M$ , an increasing sequence of integers $(n_i)_{i\\in \\mathbb {Z}}$ , and some sequences of positive integers $\\lbrace N_i\\rbrace _{i\\in \\mathbb {Z}}, \\lbrace K_i\\rbrace _{i\\in \\mathbb {Z}},\\lbrace M_i\\rbrace _{i\\in \\mathbb {Z}}$ , such that, for all $i\\in \\mathbb {Z}$ : $ \\begin{split}d(f^{n_i}(z),\\Gamma )<\\varepsilon _i,\\\\d(f^{n_i+N_{i+1}}(z), f_{\\mid \\Lambda }^{N_{i+1}}(R^{\\prime }_{i+1}))<\\varepsilon _{i+1},\\\\d(f^{n_{i}-M_{i}}(z), f_{\\mid \\Lambda }^{-M_{i}}(R_{i}))<\\varepsilon _{i},\\\\n_{i+1}=n_i+N_{i+1}+K_{i+1}+M_{i+1}.\\end{split}$ The idea of this proof is to `thicken' some appropriate iterates of the windows $R_{i},R^{\\prime }_{i}$ in $\\Lambda $ to full dimensional windows $W_i,W^{\\prime }_i$ in $M$ , so that $\\lbrace W_i,W^{\\prime }_i\\rbrace _{i\\in \\mathbb {Z}}$ form a sequence of windows that are correctly aligned under some appropriate maps.",
"We start with a brief sketch of the construction before we proceed to the formal proof.",
"We constructs some copies $\\bar{R}_{i},\\bar{R}^{\\prime }_{i+1}$ in $\\Gamma $ of $R_{i},R^{\\prime }_{i+1}$ , respectively, through the inverses of the wave maps (see Section REF ).",
"We then expand the rectangles $\\bar{R}_{i},\\bar{R}^{\\prime }_{i+1}$ into the hyperbolic directions to produce a pair of windows $\\bar{W}_{i},\\bar{W}^{\\prime }_{i+1}$ , respectively, which are correctly aligned under the identity map.",
"Then we take a backwards iterate of $\\bar{W}_{i}$ such that $f^{-M_{i}} (\\bar{W}_{i})$ is sufficiently close to $\\Lambda $ , and we construct a new window $W_i$ about $f^{-M_i}(R_i)$ such that $W_i$ is correctly aligned with $\\bar{W}_i$ under $f^{M_i}$ .",
"Similarly, we construct a window $W^{\\prime }_{i+1}$ about $f^{N_i}(R^{\\prime }_{i+1})$ such that $\\bar{W}^{\\prime }_{i+1}$ is correctly aligned with $W^{\\prime }_{i+1}$ under $f^{N_{i+1}}$ .",
"We are given that we can align $R^{\\prime }_{i+1}$ with $R_{i+1}$ under some high enough iterate.",
"Hence we can align $W^{\\prime }_{i+1}$ with $W_{i+1}$ under some iterate $f^{K_{i+1}}$ .",
"This construction can be continued inductively.",
"Notationwise, the $N_i$ 's are associated to forward iterations along the stable manifold, the $M_i$ 's to backwards iterations along the unstable manifold, and the $K_i$ 's to iterations following the inner dynamics of $f$ restricted to $\\Lambda $ .",
"See Fig.",
"REF .",
"Figure: Schematic illustration of the construction of windows for the shadowing lemma.",
"The windows R i ,R i+1 ' R_i,R^{\\prime }_{i+1} are depicted as large dots.Step 1.",
"Let $\\lbrace R,R^{\\prime }\\rbrace $ be a pair of $l$ -dimensional windows of the type $\\lbrace R_{i},R^{\\prime }_{i+1}\\rbrace $ , and let $\\lbrace \\varepsilon ,\\varepsilon ^{\\prime }\\rbrace $ stand for the corresponding $\\lbrace \\varepsilon _i,\\varepsilon _{i+1}\\rbrace $ .",
"Let $\\bar{R} =(\\Omega ^-_\\Gamma )^{-1}(R )$ and $\\bar{R}^{\\prime } =(\\Omega ^+_\\Gamma )^{-1}(R^{\\prime } )$ be the copies of $R $ and $R^{\\prime } $ , respectively, in the homoclinic channel $\\Gamma $ .",
"By making some arbitrarily small changes in the sizes of their exit and entry directions, we can alter the windows $\\bar{R} $ and $\\bar{R}^{\\prime } $ such that $R $ is correctly aligned with $\\bar{R} $ under $(\\Omega ^-_\\Gamma )^{-1}$ , $\\bar{R}$ is correctly aligned with $\\bar{R}^{\\prime } $ under the identity mapping, and $\\bar{R}^{\\prime } $ is correctly aligned with $R $ under $\\Omega ^+_\\Gamma $ .",
"We `thicken' the $l$ -dimensional windows $\\bar{R} $ and $\\bar{R}^{\\prime } $ in $\\Gamma $ , which are correctly aligned under the identity mapping, to $(l+n_u+n_s)$ -dimensional windows $\\bar{W} $ and $\\bar{W}^{\\prime } $ , respectively, that are correctly aligned in $M$ under the identity mapping as well.",
"We now explain the `thickening' procedure.",
"First, we describe how to thicken $\\bar{R} $ to a full dimensional window $\\bar{W} $ .",
"We choose some $0<\\bar{\\delta }<\\varepsilon $ and $0<\\bar{\\eta }<\\varepsilon $ .",
"At each point $x \\in \\bar{R} $ we choose an $n_u$ -dimensional closed ball $\\bar{B}_{\\bar{\\delta }}(x )$ of radius $\\bar{\\delta }$ centered at $x $ and contained in $W^u({x^- })$ , where $x^- =\\Omega ^-_\\Gamma (x )$ .",
"Let $\\bar{\\Delta }:=\\bigcup _{x \\in \\bar{R} }\\bar{B}^{u}_{\\bar{\\delta }}(x )$ .",
"Note that $\\bar{\\Delta }$ is contained in $W^u(\\Lambda )$ and is homeomorphic to a $(l+n_u)$ -dimensional rectangle.",
"We define the exit set and the entry set of this rectangle as follows: $\\begin{split}(\\bar{\\Delta })^{\\rm exit}:=\\bigcup _{x \\in (\\bar{R} )^{\\rm exit}}\\bar{B}^{u}_{\\bar{\\delta }}(x ) \\cup \\bigcup _{x \\in \\bar{R} }\\partial \\bar{B}^{u}_{\\bar{\\delta }}(x ),\\\\(\\bar{\\Delta })^{\\rm entry}:=\\bigcup _{x \\in (\\bar{R} )^{\\rm entry}}\\bar{B}^{u}_{\\bar{\\delta }}(x ).\\end{split}$ We consider the normal bundle $N$ to $W^u(\\Lambda )$ .",
"At each point $y \\in \\bar{\\Delta }$ , we choose an $n_s$ -dimensional closed ball $\\bar{B}^s_{\\bar{\\eta }}(y)$ centered at $y$ and contained in the image of $N_y$ under the exponential map $\\exp _{y }:T_{y }M\\rightarrow M$ .",
"We let $\\bar{W} :=\\bigcup _{y \\in \\bar{\\Delta }}\\bar{B}^{s}_{\\bar{\\eta }}(y )$ .",
"By the Tubular Neighborhood Theorem (see, e.g., [7]), we have that for $\\bar{\\eta }$ sufficiently small, $\\bar{W} $ is a homeomorphic copy of a $(l+n_u+n_s)$ -dimensional rectangle.",
"We now define the exit set and the entry set of $\\bar{W} $ as follows: $\\begin{split} (\\bar{W} )^{\\rm exit}:=\\bigcup _{y \\in (\\bar{\\Delta })^{\\rm exit}}\\bar{B}^{s}_{\\bar{\\eta }}(y ),\\\\(\\bar{W} )^{\\rm entry}:=\\bigcup _{y \\in (\\bar{\\Delta })^{\\rm entry}}\\bar{B}^{s}_{\\bar{\\eta }}(y ) \\cup \\bigcup _{y \\in (\\bar{\\Delta })}\\partial \\bar{B}^{s}_{\\bar{\\eta }}(y ).\\end{split}$ Second, we describe in a similar fashion how to thicken $\\bar{R}^{\\prime } $ to a full dimensional window $\\bar{W}^{\\prime } $ .",
"We choose $0<\\bar{\\delta }^{\\prime } <\\varepsilon ^{\\prime } $ and $0<\\bar{\\eta }^{\\prime } <\\varepsilon ^{\\prime }$ .",
"We consider the $(l+n_s)$ -dimensional rectangle $\\bar{\\Delta }^{\\prime } :=\\bigcup _{x^{\\prime } \\in \\bar{R}^{\\prime } }\\bar{B}^{s}_{\\bar{\\eta }^{\\prime } }(x^{\\prime })\\subseteq W^s(\\Lambda )$ , where $\\bar{B}^s_{\\bar{\\eta }^{\\prime } }(x^{\\prime } )$ is the $n$ -dimensional closed ball of radius $\\bar{\\eta }^{\\prime } $ centered at $x^{\\prime } $ and contained in $W^s(x^+ )$ , with $x^+ =\\Omega ^+_\\Gamma (x^{\\prime } )$ .",
"Its exit set and entry sets are defined as follows: $\\begin{split}(\\bar{\\Delta }^{\\prime } )^{\\rm exit}:=\\bigcup _{x \\in (\\bar{R}^{\\prime } )^{\\rm exit}}\\bar{B}^{s}_{\\bar{\\eta }^{\\prime } }(x^{\\prime } ),\\\\(\\bar{\\Delta }^{\\prime } )^{\\rm entry}:=\\bigcup _{x^{\\prime } \\in (\\bar{R}^{\\prime } )^{\\rm entry}}\\bar{B}^{s}_{\\bar{\\eta }^{\\prime } }(x^{\\prime } )\\cup \\bigcup _{x\\in (\\bar{R}^{\\prime } )}\\partial \\bar{B}^{s}_{\\bar{\\eta }^{\\prime } }(x^{\\prime } ).\\end{split}$ We define $\\bar{W}^{\\prime } :=\\bigcup _{y^{\\prime } \\in \\bar{\\Delta }^{\\prime } }\\bar{B}^{u}_{\\bar{\\delta }^{\\prime } }(y^{\\prime } )$ , where $\\bar{B}^u_{\\bar{\\delta }^{\\prime } }(y^{\\prime } )$ is the $n$ -dimensional closed ball centered at $y^{\\prime } $ and contained in the image of $N^{\\prime }_{y^{\\prime } }$ under the exponential map $\\exp _{y^{\\prime } }:T_{y^{\\prime } }M\\rightarrow M$ , with $N^{\\prime }$ being the normal bundle to $W^s(\\Lambda )$ .",
"For $\\bar{\\delta }^{\\prime } >0$ sufficiently small $\\bar{W}^{\\prime }$ is a homeomorphic copy of a $(l+n_u+n_s)$ -dimensional rectangle.",
"The exit set and the entry set of $\\bar{W}^{\\prime } $ are defined by: $\\begin{split} (\\bar{W}^{\\prime } )^{\\rm exit}:=\\bigcup _{y^{\\prime } \\in (\\bar{\\Delta }^{\\prime } )^{\\rm exit}}\\bar{B}^{u}_{\\bar{\\delta }^{\\prime } }(y^{\\prime } )\\cup \\bigcup _{y^{\\prime } \\in (\\bar{\\Delta }^{\\prime } )}\\partial \\bar{B}^{u}_{\\bar{\\delta }^{\\prime } }(y^{\\prime } ),\\\\(W^{\\prime } )^{\\rm entry}:=\\bigcup _{y^{\\prime } \\in (\\bar{\\Delta }^{\\prime } )^{\\rm entry}}\\bar{B}^{u}_{\\bar{\\delta }^{\\prime }_{i+1}}(y^{\\prime } ).\\end{split}$ This completes the description of the thickening of the $l$ -dimensional window $\\bar{R} $ into a $(l+n_u+n_s)$ -dimensional window $\\bar{W} $ , and of the thickening of the $l$ -dimensional window $\\bar{R}^{\\prime } $ into a $(l+n_u+n_s)$ -dimensional window $\\bar{W}^{\\prime } $ .",
"Note that by construction $\\bar{W} $ is contained in an $\\varepsilon $ -neighborhood of $\\Lambda $ and $\\bar{W}^{\\prime } $ is contained in an $\\varepsilon ^{\\prime }$ -neighborhood of $\\Gamma $ .",
"In order to make $\\bar{W} $ correctly aligned with $\\bar{W}^{\\prime } $ under the identity map, we choose $\\bar{\\delta }^{\\prime } $ sufficiently small relative to $\\bar{\\delta }$ , and $\\bar{\\eta }$ sufficiently small relative to $\\bar{\\eta }^{\\prime } $ .",
"Step 2.",
"We take a negative iterate $f^{-M}(\\bar{R})$ of $\\bar{R}$ , where $M>0$ .",
"We have that $f^{-M}(\\Gamma )$ is $\\varepsilon $ -close to $\\Lambda $ in the $C^1$ -topology, for all $M$ sufficiently large.",
"The vectors tangent to the fibers $W^u(x^-)$ in $\\bar{R}$ are contracted, and the vectors transverse to $W^u(\\Lambda )$ along $\\bar{R}$ are expanded by the derivative of $f^{-M}$ .",
"We choose $M$ sufficiently large so that $f^{-M}(\\bar{R})$ is $\\varepsilon $ -close to $f^{-M}(R)$ .",
"We now construct a window $W$ about $f^{-M}(R)$ that is correctly aligned with $f^{-M}(\\bar{W})$ under the identity.",
"Note that each closed ball $\\bar{B}^{u}_{\\delta }(x)$ , which is a part of $\\bar{\\Delta }$ , gets exponentially contracted as it is mapped onto $W^u(f^{-{M }}(x^-))$ by $f^{-{M }}$ .",
"By the Lambda Lemma (see the version in [44]), each closed ball $\\bar{B}^s_{\\eta }(y)$ , $y\\in \\bar{\\Delta }$ , $C^1$ -approaches some subset of $W^s(f^{-{M }}(y^-))$ under $f^{-{M }}$ , as $M \\rightarrow \\infty $ .",
"For $M $ sufficiently large, we can assume that $f^{-{M }}(\\bar{B}^s_{\\bar{\\eta }}(y))$ is $\\varepsilon $ -close to some ball in $W^s(f^{-{M }}(y^-))$ in the $C^1$ -topology, for all $y\\in \\bar{\\Delta }$ .",
"We fix $M$ with the above properties.",
"As $R $ is correctly aligned with $\\bar{R}$ under $(\\Omega ^-_\\Gamma )^{-1}$ , we have that $f^{-M }(R )$ is correctly aligned with $f^{-M }(\\bar{R} )$ under $({\\Omega ^-}_{f^{-M }(\\Gamma )})^{-1}$ .",
"In other words, $f^{-M }(R )$ is correctly aligned under the identity mapping with the projection of $f^{-M }(\\bar{R} )$ onto $\\Lambda $ along the unstable fibers.",
"To define the window $W$ , we use a local linearization of the normally hyperbolic invariant manifold.",
"By Theorem 1 in [49], there exists a homeomorphism $h$ of an open neighborhood of $(E^u\\oplus E^s)_{ \\mid \\Lambda }$ to an open neighborhood of $\\Lambda $ in $M$ such that $h\\circ Df = f\\circ h$ .",
"We select a point $x \\in f^{-M}( R )$ .",
"Since $R$ is contractible the bundles are trivial on $f^{-M}(R)$ and we can identify $(E^u\\oplus E^s)_{\\mid f^{-M }(R)}$ with $f^{-M }(R) \\times E^u_{x } \\times E^s_{x }$ .",
"Let us consider $0<\\delta <\\varepsilon $ and $0<\\eta <\\varepsilon $ .",
"We define a window $W $ as $W =h(f^{-M }( R ) \\times \\bar{B}^u_{\\delta }(0) \\times \\bar{B}^s_{\\eta }(0)),$ where $\\bar{B}^u_{\\delta }(0)$ is the closed ball centered at 0 of radius $\\delta $ in $E^u_{x }$ and $\\bar{B}^s_{\\eta }(0)$ is the closed ball centered at 0 of radius $\\eta $ in $E^s_{x }$ .",
"We define the exit set of $W $ as $(W )^{\\text{exit}}= h(f^{-M }(\\bar{R} ) \\times \\partial \\bar{B}^u_{\\delta }(0) \\times \\bar{B}^s_{\\eta }(0))\\cup h(f^{-M}(\\bar{R}^{\\text{exit}}) \\times B^u_{\\delta }(0) \\times \\bar{B}^s_{\\eta }(0)).$ Similarly, the entry set of $W$ is defined as $(W)^{\\text{entry}}= h(f^{-M}(\\bar{R}) \\times \\bar{B}^u_{\\delta }(0) \\times \\partial \\bar{B}^s_{\\eta }(0))\\cup h(f^{-M}(\\bar{R}^{\\text{entry}}) \\times B^u_{\\delta }(0) \\times \\bar{B}^s_{\\eta }(0)).$ In order to ensure the correct alignment of $W$ with $f^{-M}(\\bar{W})$ under the identity map, we choose $\\delta ,\\eta $ such that $h(f^{-M}( R) \\times \\bar{B}^u_{\\delta }(0) \\times \\lbrace 0\\rbrace )$ is correctly aligned with $f^{-M }(\\bar{\\Delta })$ under the identity map (the exit sets of both windows being in the unstable directions), and that each closed ball $f^{-M }(\\bar{B}^s_{\\eta })$ intersects $W$ in a closed ball that is contained in the interior of $f^{-M }(\\bar{B}^s_{\\eta })$ .",
"The existence of suitable $\\delta ,\\eta $ follows from the exponential contraction of $\\bar{\\Delta }$ under negative iteration, and from the Lambda Lemma applied to $\\bar{B}^s_{\\eta }(y)$ under negative iteration.",
"In a similar fashion, we construct a window $W^{\\prime }$ contained in an $\\varepsilon ^{\\prime }$ -neighborhood of $\\Lambda $ such that $\\bar{W}^{\\prime }$ is correctly aligned with $W^{\\prime }$ under $f^{N}$ .",
"The window $W^{\\prime }$ , and its entry and exit sets, are defined by: $\\begin{split}W^{\\prime } =&h(f^{N}( R^{\\prime }) \\times \\bar{B}^u_{\\delta ^{\\prime }}(0) \\times \\bar{B}^s_{\\eta ^{\\prime }}(0)),\\\\(W^{\\prime } )^{\\text{exit}}=& h(f^{N}( R^{\\prime }) \\times \\partial \\bar{B}^u_{\\delta ^{\\prime }}(0) \\times \\bar{B}^s_{\\eta ^{\\prime }}(0))\\\\&\\cup h(f^{N}( (R^{\\prime })^\\text{exit}) \\times \\bar{B}^u_{\\delta ^{\\prime }}(0) \\times \\bar{B}^s_{\\eta ^{\\prime }}(0)),\\\\(W^{\\prime } )^{\\text{entry}}=&h(f^{N}( R^{\\prime }) \\times \\bar{B}^u_{\\delta ^{\\prime }}(0) \\times \\partial \\bar{B}^s_{\\eta ^{\\prime }}(0))\\\\&\\cup h(f^{N}(( R^{\\prime })^\\text{entry}) \\times \\bar{B}^u_{\\delta ^{\\prime }}(0) \\times \\bar{B}^s_{\\eta ^{\\prime }}(0)),\\end{split}$ for some appropriate choices of radii $0<\\delta ^{\\prime } ,\\eta ^{\\prime } <\\varepsilon ^{\\prime }$ .",
"Step 3.",
"Suppose that we have constructed, as in Step 2, a window $W^{\\prime } $ about the $l$ -dimensional rectangle $f^{N }(R^{\\prime } )\\subseteq \\Lambda $ , and a window $W $ about the $l$ -dimensional rectangle $f^{-M }(R )\\subseteq \\Lambda $ .",
"Under positive iterations, the rectangle $\\bar{B}^u_{\\delta ^{\\prime } }(0)\\times \\bar{B}^s_{\\eta ^{\\prime }}(0)\\subseteq E^u\\oplus E^s$ gets exponentially expanded in the unstable direction and exponentially contracted in the stable direction by $Df$ .",
"Thus $\\bar{B}^u_{\\delta ^{\\prime } }(0)\\times \\bar{B}^s_{\\eta ^{\\prime } }(0)$ gets correctly aligned with $\\bar{B}^u_{\\delta }(0)\\times \\bar{B}^s_{\\eta }(0)$ under some power $Df^{L}$ of $Df$ , provided $L $ is sufficiently large.",
"This implies that $f^{L }(h(\\lbrace x \\rbrace \\times \\bar{B}^u_{\\delta ^{\\prime } }(0)\\times \\bar{B}^s_{\\eta ^{\\prime } }(0)))$ is correctly aligned with $h({f^{L }(x )}\\times \\bar{B}^u_{\\delta }(0)\\times \\bar{B}^s_{\\eta }(0))$ under the identity map (both rectangles are contained in $h({f^{L }(x )}\\times E^u\\times E^s)$ .",
"By assumption (iii), there exists $L^{\\prime } >\\max \\lbrace L ,N +M \\rbrace $ such that $R^{\\prime } $ is correctly aligned with $R $ under $f^{L^{\\prime } }$ .",
"This means that $f^{N }(R^{\\prime } )$ is correctly aligned with $f^{-M }(R )$ under $f^{K }$ with $K :=L^{\\prime } -N -M >0$ .",
"The product property of correctly aligned windows implies that $W^{\\prime } $ is correctly aligned with $W $ under $f^{K }$ , provided that $K $ is chosen as above.",
"Step 4.",
"We will now describe the process of constructing, based on Steps 1, 2, and 3, two bi-infinite sequences of windows $\\lbrace W_{i},W^{\\prime }_{i}\\rbrace _{i\\in \\mathbb {Z}}$ and $\\lbrace \\bar{W}_{i},\\bar{W}^{\\prime }_{i}\\rbrace _{i\\in \\mathbb {Z}}$ such that, for all $i\\in \\mathbb {Z}$ , $W_i$ is correctly aligned with $\\bar{W}_i$ under $f^{N_i}$ , $\\bar{W}_i$ is correctly aligned with $\\bar{W}^{\\prime }_{i+1}$ under the identity mapping, $\\bar{W}^{\\prime }_{i+1}$ is correctly aligned with $W^{\\prime }_{i+1}$ under $f^{M_i}$ , and $W^{\\prime }_{i+1}$ is correctly aligned with $W_{i+1}$ under $f^{K_{i+1}}$ .",
"The point of this step is that we can repeatedly choose the rectangles and the parameters at Steps 1, 2, and 3, in a consistent way, in order to produce infinite sequences of correctly aligned windows.",
"Staring with $i=0$ and continuing for all $i\\ge 0$ , we do the following.",
"For a given pair of $l$ -dimensional windows $\\lbrace R_i,R^{\\prime }_{i+1}\\rbrace $ , we consider the corresponding copies $\\lbrace \\bar{R}_i,\\bar{R}^{\\prime }_{i+1}\\rbrace $ in $\\Gamma $ such that $\\bar{R}_i$ is correctly aligned with $\\bar{R}^{\\prime }_{i+1}$ under the identity.",
"As in Step 1, we construct a pair of $(l+n_u+n_s)$ -dimensional windows $\\bar{W}_i$ about $\\bar{R}_i$ , and $\\bar{W}^{\\prime }_{i+1}$ about $\\bar{R}^{\\prime }_{i+1}$ .",
"The size of $\\bar{W}_i$ in the hyperbolic directions is given by some disks radii $\\bar{\\delta }_i,\\bar{\\eta }_i<\\varepsilon _i$ , and that of $\\bar{W}^{\\prime }_{i+1}$ by some disk radii $\\bar{\\delta }^{\\prime }_{i+1}, \\bar{\\eta }^{\\prime }_{i+1}<\\varepsilon _{i+1}$ .",
"We choose the quantities $\\bar{\\delta }_i,\\bar{\\eta }_i, \\bar{\\delta }^{\\prime }_{i+1}, \\bar{\\eta }^{\\prime }_{i+1}$ such that $\\bar{W}_i$ is correctly aligned with $\\bar{W}^{\\prime }_{i+1}$ under the identity.",
"Then, we consider the pair of $l$ -dimensional windows $\\lbrace R_{i+1},R^{\\prime }_{i+2}\\rbrace $ in $\\Lambda $ , and their corresponding copies $\\lbrace \\bar{R}_{i+1},\\bar{R}^{\\prime }_{i+2}\\rbrace $ in $\\Gamma $ .",
"As in Step 1, we construct a pair of $(l+n_u+n_s)$ -dimensional windows $\\bar{W}_{i+1}$ about $\\bar{R}_{i+1}$ , and $\\bar{W}^{\\prime }_{i+2}$ about $\\bar{R}^{\\prime }_{i+2}$ .",
"By choosing the quantities $\\bar{\\delta }_{i+1},\\bar{\\eta }_{i+1}, \\bar{\\delta }^{\\prime }_{i+2}, \\bar{\\eta }^{\\prime }_{i+2}$ as in Step 1 we can ensure that $\\bar{W}_{i+1}$ is correctly aligned with $\\bar{W}^{\\prime }_{i+2}$ .",
"We choose $N_{i+1},M_{i+1}$ large enough so that $f^{N_{i+1}}(\\bar{W}^{\\prime }_{i+1})$ is contained in an $\\varepsilon _{i+1}$ -neighborhood of $f^{N_{i+1}}(R^{\\prime }_{i+1})$ , and $f^{-M_{i+1}}(\\bar{W}_{i+1})$ is contained in an $\\varepsilon _{i+1}$ -neighborhood of $f^{-M_{i+1}}(R_{i+1})$ .",
"As in Step 2, we construct a window $W^{\\prime }_{i+1}$ about $f^{N_1}(R^{\\prime }_{i+1})$ such that $\\bar{W}^{\\prime }_{i+1}$ correctly aligned with $W^{\\prime }_{i+1}$ under $f^{N_{i+1}}$ , and a window $W_{i+1}$ about $f^{-M_1}(R_{i+1})$ such that $W_{i+1}$ correctly aligned with $\\bar{W}_{i+1}$ under $f^{M_{i+1}}$ .",
"This amounts to choosing the quantities $\\delta ^{\\prime }_{i+1}, \\eta ^{\\prime }_{i+1},\\delta _{i+1},\\eta _{i+1}$ as in Step 2 in order to ensure the correct alignment of the windows.",
"Then, we choose $K_{i+1}$ sufficiently large, and at least as large as $N_{i+1}+M_{i+1}$ , such that $W^{\\prime }_{i+1}$ is correctly aligned with $W_{i+1}$ under $f^{K_{i+1}}$ .",
"At this point, we have that $\\bar{W}^{\\prime }_{i+1}$ is correctly aligned with $W^{\\prime }_{i+1}$ under $f^{N_{i+1}}$ , $ W^{\\prime }_{i+1}$ is correctly aligned with $W_{i+1}$ under $f^{K_{i+1}}$ , and $W_{i+1}$ is correctly aligned with $\\bar{W}_{i+1}$ under $f^{M_{i+1}}$ .",
"Inductively, we obtain two sequences of windows $\\lbrace W_{i},W^{\\prime }_{i}\\rbrace _{i\\ge 0}$ and $\\lbrace \\bar{W}_{i},\\bar{W}^{\\prime }_{i}\\rbrace _{i\\ge 0}$ that satisfy the desired correct alignment conditions for all $i\\ge 0$ .",
"A similar inductive construction of windows can be done backwards starting with $W_0, W^{\\prime }_1$ .",
"In the end, we obtain two bi-infinite sequences of windows $\\lbrace W_{i},W^{\\prime }_{i}\\rbrace _{i\\in \\mathbb {Z}}$ and $\\lbrace \\bar{W}_{i},\\bar{W}^{\\prime }_{i}\\rbrace _{i\\ge 0}$ that satisfy the desired correct alignment conditions for all $i\\in \\mathbb {Z}$ .",
"By Theorem REF , there exists an orbit $\\lbrace f^n(z)\\rbrace _{n\\in \\mathbb {Z}}$ with $f^{n_i}(z)\\in \\bar{W}_i\\cap \\bar{W}^{\\prime }_{i+1}$ , $f^{n_i+N_{i+1}}(z)\\in W^{\\prime }_{i+1}$ , $f^{n_i+N_{i+1}+K_{i+1}}(z)\\in W_{i+1}$ , $f^{n_i+N_{i+1}+K_{i+1}+M_{i+1}}(z)\\in \\bar{W}_{i+1}\\cap \\bar{W}^{\\prime }_{i+2}$ , for all $i\\in \\mathbb {Z}$ .",
"Thus $n_{i+1}=n_i+N_{i+1}+K_{i+1}+M_{i+1}$ .",
"Such an orbit $\\lbrace f^n(z)\\rbrace _{n\\in \\mathbb {Z}}$ satisfies the properties required by Lemma REF ."
],
[
"Construction of correctly aligned windows",
"In this section we prove Theorem REF .",
"The methodology consists of constructing 2-dimensional windows in $\\Lambda $ about the prescribed invariant primary tori, BZI's, and Aubry-Mather sets inside the BZI's.",
"The successive pairs of windows are correctly aligned under the scattering map alternatively with powers of the inner map.",
"Lemma REF implies that there exist trajectories that follow these windows."
],
[
"Construction of correctly aligned windows across a BZI",
"In this section we will construct correctly aligned windows across a BZI between two successive transition chains of tori.",
"On each side of the BZI we will choose a one-sided neighborhood of a point on the boundary torus, and we will use Theorem REF or Theorem REF to cross over the BZI.",
"These one-sided neighborhoods are of a special type: their boundaries are images of some transition tori under the inner or outer dynamics.",
"We will construct some windows about the boundaries of these one-sided neighborhoods.",
"Then we will construct some other windows about the corresponding transition tori; these windows will be chosen so that they have a pair of sides lying on some nearby tori.",
"We will use this feature later to connect sequence of windows across the BZI's with sequences of windows along the transition chains.",
"Consider an annular region $\\Lambda _k$ in $\\Lambda $ that is a BZI, and is between two transition chains of invariant tori, as in (A5).",
"To simplify notation, we denote the tori at the boundary of $\\Lambda _k$ by $T_{a}$ and $T_{b}$ .",
"We choose a pair of transition tori $T_{i}, T_{j}$ in $\\Lambda $ as in (A5), ordered as follows: $T_{j}\\prec T_{i}\\prec T_{a}$ .",
"These tori are outside of the BZI $\\Lambda _k$ and on the same side of it as $T_{a}$ .",
"By (A5-iv) there exist $T_{i^{\\prime }}\\prec T_i$ and $T_{j^{\\prime }}\\prec T_j$ such that $T_{i^{\\prime }}$ is $\\varepsilon _i$ -close to $T_i$ and $T_{j^{\\prime }}$ is $\\varepsilon _j$ -close to $T_j$ , in the $C^0$ topology.",
"By (A5-ii) $S(T_j)$ intersects $T_i$ in a topologically transverse manner, so both $S(T_j)$ and $S(T_{j^{\\prime }})$ intersect both $T_i$ and $T_{i^{\\prime }}$ in a topologically transverse manner, provided $T_{i^{\\prime }}, T_{j^{\\prime }}$ are sufficiently close to $T_i,T_j$ , respectively.",
"Since $S$ is a diffeomorphism, $T_{i}, T_{i^{\\prime }}$ and the images of $T_{j}, T_{j^{\\prime }}$ under $S$ form a topological rectangle $D_{iji^{\\prime }j^{\\prime }}$ in $\\Lambda $ .",
"This rectangle may not be contained in the domain $U^-$ of the scattering map $S$ .",
"Provided that we choose the tori $T_{i}, T_{i^{\\prime }}$ sufficiently $C^0$ -close to one another, and also $T_{j}, T_{j^{\\prime }}$ sufficiently $C^0$ -close to one another, the rectangle $D_{iji^{\\prime }j^{\\prime }}$ will be sufficiently small so that some iterate $f_{\\mid \\Lambda }^{K_a}$ of $f_{\\mid \\Lambda }$ takes the rectangle $D_{iji^{\\prime }j^{\\prime }}$ into a rectangle $f_{\\mid \\Lambda }^{K_a}(D_{iji^{\\prime }j^{\\prime }})$ inside $U^-$ .",
"This is possible since each torus intersects $U^-$ by (A5-i), and the motion on the tori is topologically transitive by (A5-iii).",
"The rectangle $f_{\\mid \\Lambda }^{K_a}(D_{iji^{\\prime }j^{\\prime }})$ has a pair of sides lying on the tori $T_{i}, T_{i^{\\prime }}$ , and the other pair of sides on $(f_{\\mid \\Lambda }^{K_a}\\circ S)(T_{j}),(f_{\\mid \\Lambda }^{K_a}\\circ S)(T_{j^{\\prime }})$ .",
"The curves $(f_{\\mid \\Lambda }^{K_a}\\circ S)(T_{j}),(f_{\\mid \\Lambda }^{K_a}\\circ S)(T_{j^{\\prime }})$ are topologically transverse to both $T_{i},T_{i^{\\prime }}$ .",
"By assumption (A5-ii), $S(T_i)$ topologically crosses $T_{a}$ , so $S^{-1}(T_{a})$ topologically crosses $T_{i}$ .",
"We can ensure that the interior of $f_{\\mid \\Lambda }^{K_a}(D_{iji^{\\prime }j^{\\prime }})$ intersects $S^{-1}(T_{a})$ by choosing $K_a$ sufficiently large, and the tori in each pair $T_{i}, T_{i^{\\prime }}$ and $T_{j}, T_{j^{\\prime }}$ sufficiently $C^0$ -close to one another.",
"This implies that the image of $f_{\\mid \\Lambda }^{K_a}(D_{iji^{\\prime }j^{\\prime }})$ under $S$ is a topological rectangle in $\\Lambda $ which intersects $T_{a}$ , and the intersection of $S(f_{\\mid \\Lambda }^{K_a}(D_{iji^{\\prime }j^{\\prime }}))$ with $\\Lambda _k$ forms a one-sided neighborhood in $\\Lambda _k$ of some part of $T_{a}$ .",
"The boundary of $S(f_{\\mid \\Lambda }^{K_a}(D_{iji^{\\prime }j^{\\prime }}))\\cap \\Lambda _k$ consists of arcs of the curves $T_{a}, S(T_i), S(T_{i^{\\prime }}),S\\circ f_{\\mid \\Lambda }^{K_a}\\circ S(T_j),S\\circ f_{\\mid \\Lambda }^{K_a}\\circ S(T_{j^{\\prime }})$ .",
"We make a similar construction on the other side of the BZI $\\Lambda _k$ .",
"We choose a pair of transition tori $T_{k}, T_{l}$ with $ T_{b}\\prec T_{k}\\prec T_{l}$ , outside of the BZI $\\Lambda _k$ and on the same side as $T_{b}$ .",
"We also choose $T_{k}\\prec T_{k^{\\prime }}$ and $T_{l}\\prec T_{l^{\\prime }}$ such that $T_{k^{\\prime }}$ is $\\varepsilon _k$ -close to $T_k$ and $T_{l^{\\prime }}$ is $\\varepsilon _l$ -close to $T_l$ , and $S^{-1}(T_l), S^{-1}(T_{l^{\\prime }})$ are topologically transverse to both $T_k,T_{k^{\\prime }}$ , and so they form a a topological rectangle $D_{klk^{\\prime }l^{\\prime }}$ .",
"There exists $K_b$ sufficiently large such that $f_{\\mid \\Lambda }^{-K_b}(D_{klk^{\\prime }l^{\\prime }})\\subseteq U^-$ , its interior intersects $S(T_{b})$ , and $S^{-1}(f_{\\mid \\Lambda }^{-K_b}(D_{klk^{\\prime }l^{\\prime }}))$ forms a one-sided neighborhood in $\\Lambda _k$ of some part of $T_{b}$ .",
"The boundary of $S^{-1}(f_{\\mid \\Lambda }^{K_b}(D_{klk^{\\prime }l^{\\prime }}))\\cap \\Lambda _k$ consists of arcs of the curves $T_{b}, S(T_k), S(T_{k^{\\prime }}),S^{-1}\\circ f_{\\mid \\Lambda }^{K_b}\\circ S^{-1}(T_l),S^{-1}\\circ f_{\\mid \\Lambda }^{K_b}\\circ S^{-1}(T_{l^{\\prime }})$ .",
"Figure: Orbits across a BZIAt this stage we have obtained in $\\Lambda _k$ a one-sided neighborhood $(S\\circ f_{\\mid \\Lambda }^{K_a})(D_{iji^{\\prime }j^{\\prime }})$ of an arc in $T_{a}$ , and a one-sided neighborhood $(S^{-1}\\circ f_{\\mid \\Lambda }^{-K_b})(D_{klk^{\\prime }l^{\\prime }})$ of an arc in $T_{b}$ .",
"If we are under the assumptions (A1)-(A6), Theorem REF yields a point $x_a\\in \\partial (S\\circ f_{\\mid \\Lambda }^{K_a})(D_{iji^{\\prime }j^{\\prime }})$ whose image $x_b=f_{\\mid \\Lambda }^{K_{ab}}(x_a)$ under some power $f_{\\mid \\Lambda }^{K_{ab}}$ lies on $\\partial (S^{-1}\\circ f_{\\mid \\Lambda }^{-K_b})(D_{klk^{\\prime }l^{\\prime }})$ .",
"If we also assume (A7), Theorem REF yields a point $x_a\\in \\partial (S\\circ f_{\\mid \\Lambda }^{K_a})(D_{iji^{\\prime }j^{\\prime }})$ with $x_b=f_{\\mid \\Lambda }^{K_{ab}}(x_a)\\in \\partial (S^{-1}\\circ f_{\\mid \\Lambda }^{-K_b})(D_{klk^{\\prime }l^{\\prime }})$ as above, satisfying the additional conditions $\\pi _\\phi (f^j(w^k_s))<\\pi _\\phi (f^j(x_a))<\\pi _\\phi (f^j(\\bar{w}^k_s)), $ for each $s\\in \\lbrace 1,\\ldots , s_k\\rbrace $ , where $w^k_s,\\bar{w}^k_s\\in \\Sigma _{\\omega ^k_s}$ , and for all $j$ within a certain interval of integers.",
"The trajectories of all points sufficiently close to $x_a$ will satisfy these conditions as well.",
"In either case, there exist an arc $\\bar{e}^{\\prime }_a\\subseteq \\partial (S\\circ f_{\\mid \\Lambda }^{K_a})(D_{iji^{\\prime }j^{\\prime }})$ containing $x_a$ , and another arc $\\bar{e}_b\\subseteq \\partial (S^{-1}\\circ f_{\\mid \\Lambda }^{-K_b})(D_{klk^{\\prime }l^{\\prime }})$ containing $x_b$ , such that $f_{\\mid \\Lambda }^{K_{ab}}(\\bar{e}^{\\prime }_a)$ is topologically transverse to $\\bar{e}_b$ at $x_b$ .",
"The arc $\\bar{e}^{\\prime }_a$ lies on one of the sets $S({T_{i}}), S({T_{i^{\\prime }}}), (S\\circ f_{\\mid \\Lambda }^{K_{a}}\\circ S)(T_{j}), (S\\circ f_{\\mid \\Lambda }^{K_{a}}\\circ S)(T_{j^{\\prime }})$ .",
"Similarly, the arc $\\bar{e}_b$ lies on one of the sets $S^{-1}(T_{k}), S^{-1}({T_{k^{\\prime }}}), (S^{-1}\\circ f_{\\mid \\Lambda }^{-K_{b}}\\circ S^{-1})(T_{l}), (S^{-1}\\circ f_{\\mid \\Lambda }^{-K_{b}}\\circ S^{-1})(T_{l^{\\prime }})$ .",
"We define a 2-dimensional window $R^{\\prime }_a$ about $\\bar{e}^{\\prime }_a$ , and a 2-dimensional window $R_b$ about $\\bar{e}_b$ such that $R^{\\prime }_a$ is correctly aligned with $R_b$ under $f_{\\mid \\Lambda }^{K_{ab}}$ .",
"Informally, the exit direction of $R^{\\prime }_a$ is along $\\bar{e}^{\\prime }_a$ , and the exit direction of $R_b$ is across $\\bar{e}_b$ .",
"The formal construction now follows.",
"Since the arc $\\bar{e}^{\\prime }_a$ is an embedded 1-dimensional $C^0$ -submanifold of $\\Lambda $ , there exists a $C^0$ -local parametrization $\\chi ^{\\prime }_a:\\mathbb {R}^2\\rightarrow \\Lambda $ such that $\\chi ^{\\prime }_a([0,1]\\times \\lbrace 0\\rbrace )=\\bar{e}^{\\prime }_a$ , provided $\\bar{e}^{\\prime }_a$ is sufficiently small.",
"Then $R^{\\prime }_a=\\chi _a([0,1]\\times [-\\eta ^{\\prime }_a,\\eta ^{\\prime }_a]),$ is a topological rectangle.",
"We define the exit set of $R^{\\prime }_a$ as ${R^{\\prime }}_a^\\textrm {exit}=\\chi _a(\\partial [0,1]\\times [-\\eta ^{\\prime }_a,\\eta ^{\\prime }_a]).$ The definition of the entry set of of $R^{\\prime }_a$ follows by default.",
"Similarly, there exists a $C^0$ -local parametrization $\\chi _b:\\mathbb {R}^2\\rightarrow \\Lambda $ such that $\\chi _b(\\lbrace 0\\rbrace \\times [0,1])=\\bar{e}_b$ , and $R_b=\\chi _b([-\\delta _b,\\delta _b]\\times [0,1]),$ is a topological rectangle.",
"We define the exit set of $R_b$ as ${R}_b^\\textrm {exit}=\\chi _b(\\partial [-\\delta _b,\\delta _b]\\times [0,1]).$ By choosing $\\eta ^{\\prime }_a, \\delta _b$ sufficiently small, we ensure that $R^{\\prime }_a$ is correctly aligned with $R_b$ under $f_{\\mid \\Lambda }^{K_{ab}}$ (see Definition REF ).",
"See Figure REF .",
"We now construct other windows outside the BZI $\\Lambda _k$ .",
"We consider two cases: first case, when the arc $\\bar{e}^{\\prime }_{a}$ is a part of $S(T_{i})$ or $S(T_{i^{\\prime }})$ ; second case, when $\\bar{e}^{\\prime }_a$ is a part of $(S\\circ f_{\\mid \\Lambda }^{K_a}\\circ S)(T_{j})$ or $(S\\circ f_{\\mid \\Lambda }^{K_a}\\circ S)(T_{j^{\\prime }})$ .",
"Case 1.",
"In the first case, when $\\bar{e}^{\\prime }_a$ is a part of $S(T_{i})$ or $S(T_{i^{\\prime }})$ we proceed with the construction as follows.",
"Let us say that $\\bar{e}^{\\prime }_a$ is a part of $S(T_{i})$ .",
"We take the inverse image of $R^{\\prime }_a$ by $S$ .",
"This is a topological rectangle $S^{-1}(R^{\\prime }_a)$ about the torus $T_{i}$ .",
"We construct a window $R_{i}$ about an arc $\\bar{e}_{i}$ in $T_{i}$ , such that $R_{i}$ is correctly aligned with $S^{-1}(R^{\\prime }_{a})$ under the identity map, and with the exit direction of $R_{i}$ in the direction of $T_i$ .",
"Let $\\bar{e}_{i}$ be an arc in $T_{i}$ , and $\\chi _{i}:\\mathbb {R}^2\\rightarrow \\Lambda $ be a local parametrization with $\\chi _{i}([0,1]\\times \\lbrace 0\\rbrace )= \\bar{e}_{i}$ .",
"We define $R_{i}=\\chi _{i}([0,1]\\times [-\\eta _i,\\eta _i])\\\\{R}^\\textrm {exit}_{i}=\\chi _{i}(\\partial [0,1]\\times [-\\eta _i,\\eta _i]),$ where $\\eta _{i}>0$ is sufficiently small.",
"By choosing the arc $\\bar{e}_{i}$ sufficiently large so that $\\bar{e}_{i}\\supseteq S^{-1}(\\bar{e}^{\\prime }_a)$ , and by choosing $\\eta _{i}>0$ sufficiently small, we can ensure that $R_{i}$ is correctly aligned with $S^{-1}(R^{\\prime }_{a})$ under the identity map, or, equivalently, $R_{i}$ is correctly aligned with $R^{\\prime }_{a}$ under $S$ .",
"See Figure REF .",
"Figure: Construction of windows near the boundaries of a BZI – case 1We construct another window $R^{\\prime }_{i}$ about $T_{i}$ such that $R^{\\prime }_{i}$ is correctly aligned with $R_{i}$ under some power $f_{\\mid \\Lambda }^{K_{i}}$ of $f_{\\mid \\Lambda }$ , with $K_{i}>0$ .",
"The window $R^{\\prime }_{i}$ will have its exit direction across the torus $T_{i}$ .",
"We choose a pair of invariant primary tori $T^{\\rm lower}_{i}\\prec T_{i}\\prec T^{\\rm upper}_{i}$ , with $T^{\\rm lower}_{i}, T^{\\rm upper}_{i}$ located $\\varepsilon _{i}$ -close to $T_{i}$ in the $C^0$ -topology.",
"Such tori exist due to the assumption that $T_{i}$ is not an end torus in the transition chain, as in (A5-iv).",
"Moreover, we choose the tori $T^{\\rm lower}_{i}, T^{\\rm upper}_{i}$ sufficiently close to $T_{i}$ so that the components of $R_{i}^{\\rm entry}$ are outside the annulus between $T^{\\rm lower}_{i}$ and $ T^{\\rm upper}_{i}$ , with one component on one side and the other component on the other side of this annulus.",
"We choose an arc $\\bar{e}^{\\prime }_i\\subseteq T_i$ that lies on an edge of the topological rectangle $D_{iji^{\\prime }j^{\\prime }}$ .",
"Let $\\chi ^{\\prime }_{i}:\\mathbb {R}^2\\rightarrow \\Lambda $ be a $C^0$ -local parametrization with $\\chi ^{\\prime }_{i}(\\lbrace 0\\rbrace \\times [0,1])= \\bar{e}^{\\prime }_{i}$ , such that for some $\\delta ^{\\prime }_i$ sufficiently small, $\\chi ^{\\prime }_i (\\lbrace -\\delta ^{\\prime }_i\\rbrace \\times [0,1])\\subseteq T_i^{\\rm lower}$ and $\\chi ^{\\prime }_i (\\lbrace \\delta ^{\\prime }_i\\rbrace \\times [0,1])\\subseteq T_i^{\\rm upper}$ .",
"We define $R^{\\prime }_{i}$ by: $R^{\\prime }_{i}=\\chi ^{\\prime }_i ([-\\delta ^{\\prime }_i,\\delta ^{\\prime }_i]\\times [0,1]),\\\\{R^{\\prime }}_{i}^\\textrm {exit}=\\chi ^{\\prime }_i (\\partial [-\\delta ^{\\prime }_i,\\delta ^{\\prime }_i]\\times [0,1]).$ The exit set components of $R^{\\prime }_i$ lie on the tori $T_i^{\\rm upper}, T_i^{\\rm lower}$ neighboring $T_i$ .",
"Since the motion on the tori is topologically transitive, by (A5-iii), there exists $K_{i}>0$ such that $R^{\\prime }_{i}$ is correctly aligned with $R_{i}$ under $f_{\\mid \\Lambda }^{K_{i}}$ .",
"Indeed, to achieve correct alignment of these windows in the covering space of the annulus we only have to choose $K_{i}$ sufficiently large so that the two components of ${R^{\\prime }}_{i}^\\textrm {exit}$ , which are two arcs in $T^{\\rm lower}_{i}$ and $T^{\\rm upper}_{i}$ , are mapped by $f_{\\mid \\Lambda }^{K_{i}}$ on the opposite sides of the part of $R_{i}$ between $T^{\\rm lower}_{i}$ and $T^{\\rm upper}_{i}$ .",
"We note that the number of iterates $K_{i}$ of $f_{\\mid \\Lambda }$ needed to make $R^{\\prime }_{i}$ correctly aligned with $R_{i}$ may be different than the number of iterates $K_a$ which takes the topological rectangle $D_{iji^{\\prime }j^{\\prime }}$ onto $f_{\\mid \\Lambda }^{K_a}(D_{ihi^{\\prime }j^{\\prime }})$ .",
"See Figure REF .",
"The conclusion of this step is that we obtain the window $R^{\\prime }_{i}$ around $T_{i}$ , with its exit direction across $T_{i}$ , such that $R^{\\prime }_{i}$ is correctly aligned with $R_{i}$ under $f_{\\mid \\Lambda }^{K_{i}}$ .",
"Both windows $R^{\\prime }_i,R_i$ are contained in an $\\varepsilon _i$ -neighborhood of $T_i$ .",
"In the case when the edge $\\bar{e}^{\\prime }_a$ of $D_{iji^{\\prime }j^{\\prime }}$ is a part of $S(T_{i^{\\prime }})$ instead of $S(T_{i})$ , the construction goes similarly to the one above.",
"Case 2.",
"We now consider the second case, when the arc $\\bar{e}^{\\prime }_a$ of $\\partial (S\\circ f_{\\mid \\Lambda }^{K_a})(D_{iji^{\\prime }j^{\\prime }})$ is a part of $(S\\circ f_{\\mid \\Lambda }^{K_a}\\circ S)(T_{j})$ or $(S\\circ f_{\\mid \\Lambda }^{K_a}\\circ S)(T_{j^{\\prime }})$ .",
"Let us say that $\\bar{e}^{\\prime }_a$ is a part of $(S\\circ f_{\\mid \\Lambda }^{K_a}\\circ S)(T_{j})$ .",
"We construct a window $R^{\\prime }_a$ in $\\Lambda $ about $\\bar{e}^{\\prime }_a$ as before; the exit set of $R^{\\prime }_a$ is in a direction along $\\bar{e}^{\\prime }_a$ , and the size of $R^{\\prime }_a$ in the direction across $R_a$ is given by some parameter $\\eta ^{\\prime }_a$ .",
"See Figure REF .",
"We consider the inverse image $S^{-1}(R^{\\prime }_a)$ of $R^{\\prime }_a$ by $S$ , which is a topological rectangle about $(f_{\\mid \\Lambda }^{K_a}\\circ S)(T_{j})$ .",
"Let $\\bar{e}^{\\prime \\prime }_j$ be an arc in $(f_{\\mid \\Lambda }^{K_a}\\circ S)(T_{j})$ such that $S(\\bar{e}^{\\prime \\prime }_j)\\supset \\bar{e}^{\\prime }_a$ , and let $\\chi ^{\\prime \\prime }_j :\\mathbb {R}^2\\rightarrow \\Lambda $ be a local parametrization with $\\chi ^{\\prime \\prime }_{j}([0,1]\\times \\lbrace 0\\rbrace )= \\bar{e}^{\\prime \\prime }_{j}$ .",
"We define $R^{\\prime \\prime }_{j}$ by: $R^{\\prime \\prime }_{j}=\\chi ^{\\prime \\prime }_j ([0,1]\\times [-\\eta ^{\\prime \\prime }_j,\\eta ^{\\prime \\prime }_j]) ,\\\\{R^{\\prime \\prime }}_{j}^\\textrm {exit}=\\chi ^{\\prime \\prime }_j (\\partial [0,1]\\times [-\\eta ^{\\prime \\prime }_j,\\eta ^{\\prime \\prime }_j]).$ By choosing $\\eta ^{\\prime \\prime }_{j}>0$ sufficiently small, we can ensure that $R^{\\prime \\prime }_{j}$ is correctly aligned with aligned with $R^{\\prime }_{a}$ under $S$ .",
"The window $R^{\\prime \\prime }_{j}$ is in a neighborhood of an edge of the topological rectangle $f_{\\mid \\Lambda }^{K_a}(D_{iji^{\\prime }j^{\\prime }})$ .",
"The exit set of $R^{\\prime \\prime }_{j}$ is in the direction of $(f_{\\mid \\Lambda }^{K_a}\\circ S)(T_{j})$ .",
"See Figure REF .",
"Figure: Construction of windows near the boundaries of a BZI – case 2We construct a window $R^{\\prime \\prime \\prime }_{j}$ about $S(T_j)$ such that $R^{\\prime \\prime \\prime }_{j}$ is correctly aligned with $R^{\\prime \\prime }_{j}$ under $f_{\\mid \\Lambda }^{K_{a}}$ .",
"We define $R^{\\prime \\prime \\prime }_{j} =\\chi ^{\\prime \\prime \\prime }_j ([0,1]\\times [-\\eta ^{\\prime \\prime \\prime }_j,\\eta ^{\\prime \\prime \\prime }_j]) ,\\\\{R^{\\prime \\prime \\prime }}_{j}^\\textrm {exit}=\\chi ^{\\prime \\prime \\prime }_j (\\partial [0,1]\\times [-\\eta ^{\\prime \\prime \\prime }_j,\\eta ^{\\prime \\prime \\prime }_j]),$ where $\\bar{e}^{\\prime \\prime \\prime }_{j}$ is an arc of $S(T_{j})$ with $f_{\\mid \\Lambda }^{K_a}(\\bar{e}^{\\prime \\prime \\prime }_{j})\\supseteq \\bar{e}^{\\prime \\prime }_j$ , $\\chi ^{\\prime \\prime \\prime }_j :\\mathbb {R}^2\\rightarrow \\Lambda $ is a local parametrization with $\\chi ^{\\prime \\prime \\prime }_{j}([0,1]\\times \\lbrace 0\\rbrace )= \\bar{e}^{\\prime \\prime \\prime }_{j}$ , and $\\eta ^{\\prime \\prime \\prime }_j>0$ is sufficiently small.",
"The arc $\\bar{e}^{\\prime \\prime \\prime }_{j}$ is contained in one of the edges of the topological rectangle $D_{iji^{\\prime }j^{\\prime }}$ .",
"For suitable $\\eta ^{\\prime \\prime \\prime }_{j}>0$ , we can ensure that $R^{\\prime \\prime \\prime }_{j}$ is correctly aligned with aligned with $R^{\\prime \\prime }_{j}$ under $f_{\\mid \\Lambda }^{K_a}$ .",
"Moreover, we choose $R^{\\prime \\prime \\prime }_{j},R^{\\prime \\prime }_{j}$ so that these windows are both contained in an $\\varepsilon _j$ neighborhood of $T_j$ .",
"The exit set of $R^{\\prime \\prime \\prime }_{j}$ is in a direction along $S(T_{j})$ .",
"We take the inverse image $S^{-1}(R^{\\prime \\prime \\prime }_{j})$ of $R^{\\prime \\prime \\prime }_{j}$ under $S$ , which is a topological rectangle about an arc in $T_{j}$ .",
"In a fashion similar to Case 1, we construct two windows $R^{\\prime }_j,R_j$ about $T_j$ such that $R^{\\prime }_j$ is correctly aligned with $R_j$ under some power $f_{\\mid \\Lambda }^{K_j}$ of $f_{\\mid \\Lambda }$ , and $R_j$ is correctly aligned with $R^{\\prime \\prime \\prime }_j$ under $S$ .",
"The exit of $R^{\\prime }_j$ is chosen in a direction across $T_j$ , and the exit set components ${R^{\\prime }}^{\\rm exit}_j$ of $R^{\\prime }_j$ lie on two invariant primary tori $T_j^{\\rm upper}$ , $T_j^{\\rm lower}$ neighboring $T_j$ , that are $\\varepsilon _j$ -close to $T_j$ and contain $T_j$ between them.",
"The exit of $R_j$ is in a direction along $T_j$ .",
"The windows $R^{\\prime }_j,R_j$ are chosen to lie in an $\\varepsilon _j$ neighborhood of $T_j$ relative to the $C^0$ -topology.",
"The case when the arc $\\bar{e}^{\\prime }_a$ is a part of $(S\\circ f_{\\mid \\Lambda }^{K_a}\\circ S)(T_{j^{\\prime }})$ is treated similarly.",
"This concludes the construction of correctly aligned windows in $\\Lambda $ , starting with the window $R^{\\prime }_{a}$ about $T_{a}$ , and moving backwards along transition tori that are on the same side as $T_{a}$ of the BZI $\\Lambda _k$ .",
"This construction yields a sequence of windows of the type $R^{\\prime }_{i}, R_{i}, R^{\\prime }_{a},$ in the first case, or $R^{\\prime }_{j}, R_{j}, R^{\\prime \\prime \\prime }_{j}, R^{\\prime \\prime }_{j}, R^{\\prime }_{a},$ in the second case.",
"In the first case, $R^{\\prime }_{i}$ is correctly aligned with $R_{i}$ under some power $f_{\\mid \\Lambda }^{K_{i}}$ of the inner map, and $R_{i}$ is correctly aligned with $R^{\\prime }_{a}$ under the outer map $S$ .",
"The exit direction of $R^{\\prime }_{i}$ is across the torus $T_{i}$ , and its exit set components lie on some invariant primary tori that are $\\varepsilon _i$ -close to $T_{i}$ .",
"The windows $R^{\\prime }_i,R_i$ are contained in an $\\varepsilon _i$ neighborhood of $T_i$ .",
"In the second case, $R^{\\prime }_{j}$ is correctly aligned with $R_{j}$ under some power $f_{\\mid \\Lambda }^{K_{j}}$ of the inner map, $R_{j}$ is correctly aligned with $R^{\\prime \\prime \\prime }_{j}$ under the outer map $S$ , $R^{\\prime \\prime \\prime }_{j}$ is correctly aligned with $R^{\\prime \\prime }_{j}$ under some power $f_{\\mid \\Lambda }^{K_{a}}$ of the inner map, and $R^{\\prime \\prime }_{j}$ is correctly aligned with $R^{\\prime }_{a}$ under the outer map $S$ .",
"The exit direction of $R_{j}$ is across the torus $T_{j}$ , and its exit set components lie on some invariant primary tori that are $\\varepsilon _j$ -close to $T_{j}$ .",
"The windows $R^{\\prime }_j,R_j$ are contained in an $\\varepsilon _j$ neighborhood of $T_j$ , and the windows $R^{\\prime \\prime \\prime }_j,R^{\\prime \\prime }_j$ are contained in an $\\varepsilon _i$ neighborhood of $T_i$ .",
"We proceed with a similar construction on the other side of the BZI between $\\Lambda _k$ , that is, on the same side of the BZI as $T_b$ .",
"We have already defined the window $R_b$ about $T_b$ that $R^{\\prime }_a$ is correctly aligned with $R_b$ under $f_{\\mid \\Lambda }^{K_{ab}}$ .",
"Starting with the window $R_{b}$ and moving forward along the transition chain $T_b,T_k,T_l$ , we construct a sequence of windows of the type $R_{b}, R^{\\prime }_{k} $ or of the type $R_{b}, R^{\\prime \\prime \\prime }_{k}, R^{\\prime \\prime }_{k}, R^{\\prime }_{l},$ satisfying the correct alignment conditions below.",
"In the first case, $R_{b}$ is correctly aligned with $R^{\\prime }_{k}$ under the outer map $S$ .",
"The exit direction of $R^{\\prime }_k$ is across $T_k$ , and the exit set components lie on two invariant primary tori $\\varepsilon _k$ -close to $T_k$ .",
"Moreover, $R^{\\prime }_{k}$ is contained in an $\\varepsilon _k$ -neighborhood of $T_k$ .",
"In the second case, $R^{\\prime }_{b}$ is correctly aligned with $R^{\\prime \\prime \\prime }_{k}$ under the outer map $S$ , $R^{\\prime \\prime \\prime }_{k}$ is correctly aligned with $R^{\\prime \\prime }_{k}$ under some power $f_{\\mid \\Lambda }^{K_b}$ of the inner map, and $R^{\\prime \\prime }_{k}$ is correctly aligned with $R^{\\prime }_{l}$ under the outer map $S$ .",
"The exit direction of $R^{\\prime }_{l}$ is across $T_{l}$ , and its exit set components lie on some invariant primary tori that are $\\varepsilon _l$ -close to $T_{l}$ .",
"The windows $R^{\\prime \\prime \\prime }_{k}, R^{\\prime \\prime }_{k}$ are contained in an $\\varepsilon _k$ -neighborhood of $T_k$ , and $R^{\\prime }_l$ is contained in an $\\varepsilon _l$ -neighborhood of $T_l$ .",
"Similar statements apply when instead of windows around $T_{i},T_{j},T_{k},T_{l}$ we construct windows about $T_{i^{\\prime }},T_{j^{\\prime }},T_{k^{\\prime }},T_{l^{\\prime }}$ , respectively.",
"The conclusion of this section is that, by combining a sequence of correctly aligned window of the type (REF ) or (REF ) with a sequence of correctly aligned window of the type (REF ) or (REF ) we obtain a finite sequence of correctly aligned windows that crosses the BZI $\\Lambda _k$ .",
"The shadowing lemma-type of result Lemma REF yields an orbit that visits some prescribed neighborhood in the phase space of each window in $\\Lambda $ .",
"In particular, the shadowing orbit has points that go close to the transition tori."
],
[
"Construction of correctly aligned windows across annular regions separated by invariant tori",
"We consider an annular region in $\\Lambda $ between two transition chains of invariant tori.",
"Inside this annular region, we assume the existence of a finite collection of invariant tori that separate the region, as in (A6$^{\\prime }$ -i), and are vertically ordered as in (A6$^{\\prime }$ -ii).",
"Thus, the annular region between the transition chains is not a BZI.",
"We also assume that the scattering map satisfies the non-degeneracy condition (A6$^{\\prime }$ -iii) on these invariant tori.",
"The situation presented in this section is non-generic.",
"Assume that the annular region in $\\Lambda $ is bounded by a pair of invariant Lipschitz tori $T_{a}$ and $T_{b}$ .",
"Let $T_a$ be the end torus of the transition chain of one side, and $T_b$ be the end torus of the transition chain on the other side.",
"We assume that inside the region in the annulus bounded by $T_a$ and $T_b$ there exist a finite collection of invariant tori $\\lbrace \\Upsilon _{h}\\rbrace _{h=1,\\ldots ,k-1}$ .",
"Each $\\Upsilon _{h}$ is either an isolated invariant primary torus, or consists of a hyperbolic periodic orbit together with the upper branches, or with the lower branches, of its stable and unstable manifolds; the stable and unstable manifolds are assumed to coincide.",
"In the second case, there is another invariant torus, say $\\Upsilon _{h+1}$ , formed by the remaining branches of the same hyperbolic periodic orbit as for $\\Upsilon _{h}$ .",
"$\\Upsilon _{h+1}$ is assumed to be above $\\Upsilon _{h}$ , relative to the $I$ -coordinate, and is assumed to share with $\\Upsilon _{h}$ only the points of the periodic orbit.",
"The region in the annulus bounded by $\\Upsilon _{h}$ and $\\Upsilon _{h+1}$ is referred to as a resonant region.",
"Thus, the region in the annulus between $T_a$ and $T_b$ is divided into a finite number of BZI's and resonant regions.",
"The constructions in Subsection REF provide a one-sided neighborhood of the type $(S\\circ f_{\\mid \\Lambda }^{K_a})(D_{iji^{\\prime }j^{\\prime }})$ of some point in $T_a$ , and a one-sided neighborhood $(S^{-1}\\circ f_{\\mid \\Lambda }^{-K_b})(D_{klk^{\\prime }l^{\\prime }})$ of some point in $T_b$ .",
"Let us denote $D_0=(S\\circ f_{\\mid \\Lambda }^{K_a})(D_{iji^{\\prime }j^{\\prime }})$ and $D_{k+1}=(S^{-1}\\circ f_{\\mid \\Lambda }^{-K_b})(D_{klk^{\\prime }l^{\\prime }})$ .",
"Assume that $\\Upsilon _1$ is an isolated invariant primary torus.",
"We have that $S^{-1}(\\Upsilon _{1})$ forms with $\\Upsilon _{1}$ a topological disk $D_1$ between $T_a$ and $\\Upsilon _1$ , which is mapped by $S$ onto a topological disk $S(D_{1})$ between $\\Upsilon _1$ and $\\Upsilon _{2}$ .",
"Theorem REF provides us a trajectory that starts from $\\partial D_0$ and ends at $\\partial D_1$ .",
"In particular, there exist $K_1>0$ and a component of $f^{K_1}(D_{0})\\cap D_1$ that is a topological disk $D^{\\prime }_1$ whose boundary contains an arc of $S^{-1}(T_1)$ .",
"The image of $D^{\\prime }_1$ under $S$ is a one-sided neighborhood $D^{\\prime \\prime }_1\\subseteq S(D_1)$ of some point in $\\Upsilon _1$ , such that $D^{\\prime \\prime }_1$ is contained in the region between $\\Upsilon _1$ and $\\Upsilon _2$ .",
"The boundary of $S(D^{\\prime }_1)$ consists of an arc in $\\Upsilon _1$ and an arc in $(S\\circ f^{K_1}(\\partial D_{0}))$ .",
"See Fig.",
"REF .",
"Figure: Crossing over an isolated invariant primary torus.Now assume that $\\Upsilon _1$ consists of a hyperbolic periodic orbit, together with the lower branches of its stable and unstable manifolds, and that $\\Upsilon _2$ consists of the same hyperbolic periodic orbit, together with the upper branches of the stable and unstable manifolds.",
"The stable and unstable manifolds are assumed to coincide.",
"Excepting for the common points, $\\Upsilon _{1}$ is below $\\Upsilon _{2}$ .",
"Thus, $\\Upsilon _1$ and $\\Upsilon _2$ enclose a resonant region within the annulus.",
"We have that $S^{-1}(\\Upsilon _{1})$ forms with $\\Upsilon _{1}$ a topological disk $D_1$ between $T_a$ and $\\Upsilon _1$ , which is mapped by $S$ onto a topological disk $S(D_{1})$ between $\\Upsilon _1$ and $\\Upsilon _{2}$ .",
"We also have that $S^{-1}(\\Upsilon _{2})$ forms with $\\Upsilon _{2}$ a topological disk $D_2$ between $\\Upsilon _1$ and $\\Upsilon _2$ , which is mapped by $S$ onto a topological disk $S(D_{2})$ between $\\Upsilon _2$ and $\\Upsilon _3$ .",
"By Theorem REF there exist $K_1>0$ and a component of $f^{K_1}(D_{0})\\cap D_1$ that is a topological disk $D^{\\prime }_1 $ whose boundary contains an arc of $S^{-1}(\\Upsilon _1)$ .",
"The image of $D^{\\prime }_1$ under $S$ is a one-sided neighborhood $D^{\\prime \\prime }_1\\subseteq S(D_1)$ of some point in $\\Upsilon _1$ , contained in the region between $\\Upsilon _1 $ and $\\Upsilon _2$ .",
"The boundary of ${D^{\\prime \\prime }}_1$ consists of an arc in $\\Upsilon _1$ and an arc in $S(f^{K_1}(\\partial D_{0}))$ .",
"By the argument in the Homoclinic Orbit Theorem (see e.g.",
"[8]), there exists $N_1$ such that $f^{N_1}({D^{\\prime \\prime }}_1)$ intersects $D_2$ .",
"Let ${D^{\\prime }}_2$ be a component of this intersection that is a topological disk, and whose boundary contains an arc of $S^{-1}(\\Upsilon _2)$ .",
"Then the image of ${D^{\\prime }}_2$ under $S$ is a one sided neighborhood $D^{\\prime \\prime }_2$ of some point in $\\Upsilon _2$ that is contained in the region between $\\Upsilon _2$ and $\\Upsilon _3$ .",
"The boundary of $D^{\\prime \\prime }_2$ consists of an arc in $\\Upsilon _2$ and an arc in $(S\\circ f^{N_1}\\circ S\\circ f^{K_1})(\\partial D_{0})$ .",
"See Figure REF .",
"Figure: Construction of windows across a resonance.The main point of this construction is that, using the inner and outer dynamics, one can cross the successive BZI's and resonance regions determined by $\\lbrace \\Upsilon _h\\rbrace _{h=1,\\ldots ,k}$ one at a time, and obtain at each step a one-sided neighborhood of some point in some $\\Upsilon _h$ , which is between $\\Upsilon _h$ and $\\Upsilon _{h+1}$ , such that the boundary of that neighborhood contains the image of $\\partial D_0$ under a suitable composition of $S$ and powers of $f$ .",
"This argument can be repeated for each invariant torus $\\Upsilon _h$ with $2\\le h\\le k$ , yielding a point $x_a\\in \\partial D_{0}$ that is mapped by some appropriate composition of $S$ and powers of $f$ onto a point $x_b\\in \\partial D_{k+1}$ .",
"Then the constructions from Subsection REF can be applied to obtain a window $R^{\\prime }_a$ about $\\partial D_{0}$ , and a window $R^{\\prime }_b$ about $\\partial D_{k+1}$ , such that $R^{\\prime }_a$ is correctly aligned with $R_b$ under the appropriate composition of $S$ and powers of $f$ .",
"Then there exists a shadowing orbit that goes from a neighborhood of $R_a$ in $M$ to a neighborhood of $R^{\\prime }_b$ in $M$ .",
"If the region between $T_a$ and $T_b$ contains some prescribed collection of Aubry-Mather sets, then Theorem REF yields a shadowing orbit that crosses the region between $T_a$ and $T_b$ and follows the prescribed Aubry-Mather sets as in (REF )."
],
[
"Construction of correctly aligned windows along transition chains of tori",
"We consider a finite transition chain of invariant primary tori $\\lbrace T_1,T_2,\\ldots , T_n\\rbrace $ in the annulus $\\Lambda $ satisfying (A5).",
"All tori $T_k$ , $k=1, \\ldots , n$ , intersect the subset $U^-_+$ of the domain of the scattering map $S$ where $S$ moves points upwards in the annulus $\\Lambda $ .",
"The motion on each torus $T_k$ is topologically transitive.",
"Each torus $T_k$ with $2\\le k\\le n-1$ is an `interior' torus, i.e.",
"it can be $C^0$ -approximated from both sides in $\\Lambda $ by other invariant primary tori.",
"For each $k\\in \\lbrace 1,\\ldots ,n-1\\rbrace $ , the image $S(T_k)$ of the torus $T_k$ under the scattering map $S$ has a topologically transverse intersection with the torus $T_{k+1}$ .",
"We would like to show that there exists an orbit that visits some $\\varepsilon _k$ -neighborhood of each torus $T_k$ , $k=1,\\ldots ,n$ , in the prescribed order.",
"To this end we will construct a sequence of 2-dimensional windows in $\\Lambda $ that are correctly aligned one with another under the scattering map alternatively with some iterates of the inner map, as in Lemma REF .",
"Then the lemma will imply the existence of a shadowing orbit to the transition chain.",
"For each $k\\in \\lbrace 1,\\ldots ,n-1\\rbrace $ , we choose and fix a pair of points $x^-_{k,k+1}\\in T_k$ and $x^+_{k,k+1}\\in T_{k+1}$ such that $S(x^-_{k,k+1})=x^+_{k,k+1}$ and $S(T_{k})$ intersects $T_{k+1}$ transversally at $x^+_{k,k+1}$ .",
"We construct inductively a sequence of correctly aligned windows in $\\Lambda $ along the tori $\\lbrace T_1,T_2,\\ldots , T_n\\rbrace $ , such that each window is correctly aligned with the next window in the sequence either by the outer map or by some sufficiently large power of the inner map.",
"Moreover, each window will be contained in some $\\varepsilon $ -neighborhood of some transition torus.",
"We start the inductive construction at $T_1$ .",
"Consider the point $x^-_{1,2}\\in T_1$ with $S(x^-_{1,2})=x^+_{1,2}\\in T_{2}$ as above.",
"We construct a window $R^{\\prime }_1$ about $T_1$ as follows.",
"Let $\\bar{e}^{\\prime }_1$ be an arc contained in $T_1$ , and $\\chi ^{\\prime }_1:\\mathbb {R}^2\\rightarrow \\Lambda $ a $C^0$ -local parametrization such that $\\chi ^{\\prime }_1([0,1]\\times \\lbrace 0\\rbrace )=\\bar{e}^{\\prime }_1$ .",
"Then we define $R^{\\prime }_{1}=\\chi ^{\\prime }_1([0,1]\\times [-\\delta ^{\\prime }_1,\\delta ^{\\prime }_1]),\\\\ {R^{\\prime }}_{1}^\\textrm {exit}=\\chi ^{\\prime }_1(\\partial [0,1]\\times [-\\delta ^{\\prime }_1,\\delta ^{\\prime }_1]),$ where $0<\\delta ^{\\prime }_1<\\varepsilon _1$ .",
"We choose $\\bar{e}^{\\prime }_1$ and $\\delta ^{\\prime }_1$ sufficiently small so that and $S(\\bar{e}^{\\prime }_1)$ intersects $T_2$ only at $x^+_{1,2}$ , and also so that $R^{\\prime }_1$ defined as above is contained in ${U}^-_+$ .",
"The exit direction of $R^{\\prime }_1$ is in the direction of the torus $T_1$ .",
"The image $S(R^{\\prime }_1)$ of $R^{\\prime }_1$ under the scattering map is a topological rectangle.",
"Since $S(\\bar{e}^{\\prime }_1)$ is transverse to $T_2$ at $x^+_{1,2}$ , then by choosing $\\delta ^{\\prime }_1$ sufficiently small we ensure that the components of $S({R^{\\prime }}_{1}^\\textrm {exit})$ lie on opposite sides of $T_2$ .",
"Thus, the exit direction of $S(R^{\\prime }_1)$ is across the torus $T_2$ .",
"Next we consider the point $x^-_{2,3}\\in T_2$ with $S(x^-_{2,3})=x^+_{2,3}\\in T_{3}$ as above.",
"We construct a window $R^{\\prime }_2$ about $T_2$ in a manner similar to $R^{\\prime }_1$ : $R^{\\prime }_{2}=\\chi ^{\\prime }_2([0,1]\\times [-\\delta ^{\\prime }_2,\\delta ^{\\prime }_2]),\\\\ {R^{\\prime }}_{2}^\\textrm {exit}=\\chi ^{\\prime }_2(\\partial [0,1]\\times [-\\delta ^{\\prime }_2,\\delta ^{\\prime }_2]),$ where $\\bar{e}^{\\prime }_2$ is an arc contained in $T_2$ , $\\chi ^{\\prime }_2:\\mathbb {R}^2\\rightarrow \\Lambda $ is a $C^0$ -local parametrization such that $\\chi ^{\\prime }_2([0,1]\\times \\lbrace 0\\rbrace )=\\bar{e}^{\\prime }_2$ , and $0<\\delta ^{\\prime }_2<\\varepsilon _2$ is chosen sufficiently small.",
"Now we construct a window $R_2$ about the point $x^+_{1,2}$ such that $R^{\\prime }_1$ is correctly aligned with $R_2$ under $S$ and $R_2$ is correctly aligned with $R^{\\prime }_2$ under some power of $f_{\\mid \\Lambda }$ .",
"We choose an arc $\\bar{e}_2$ in $T_2$ containing $x^+_{1,2}$ such that $\\bar{e}_2 \\supseteq S(R^{\\prime }_1)\\cap T_2$ .",
"We choose a pair of invariant tori $T_2^{\\rm lower}$ and $T_2^{\\rm upper}$ such that $T_2^{\\rm lower} \\prec T_2 \\prec T_2^{\\rm upper}$ and the exit set components of $S(R^{\\prime }_1)$ , as well as the entry set components of $R^{\\prime }_2$ , are outside of the annulus bounded by $T_2^{\\rm lower}$ and $T_2^{\\rm upper}$ , on the both sides of the annulus.",
"Furthermore, we require that $T_2^{\\rm lower}$ and $T_2^{\\rm upper}$ should be $\\varepsilon _2$ -close to $T_2$ in the $C^0$ -topology.",
"The existence of such neighboring tori $T_2^{\\rm lower}$ and $T_2^{\\rm upper}$ to $T_2$ is guaranteed by (A5-iv).",
"Let $\\chi _2:\\mathbb {R}^2\\rightarrow \\Lambda $ be a $C^0$ -local parametrization such that $\\chi _2( \\lbrace 0\\rbrace \\times [0,1])=\\bar{e}_2$ , $\\chi _2( \\lbrace -\\delta _2\\rbrace \\times [0,1])\\subseteq T_2^{\\rm lower}$ and $\\chi _2( \\lbrace \\delta _2\\rbrace \\times [0,1])\\subseteq T_2^{\\rm upper}$ , for some $0<\\delta _2<\\varepsilon _2$ sufficiently small.",
"Define $R_{2}=\\chi _2([-\\delta _2,\\delta _2]\\times [0,1]),\\\\ {R}_{2}^\\textrm {exit}=\\chi _2(\\partial [-\\delta _2,\\delta _2]\\times [0,1]).$ The exit set components of $R_2$ lie on the invariant tori $T_2^{\\rm lower},T_2^{\\rm upper}$ neighboring $T_2$ .",
"Since the motion on the torus $T_2$ is topological transitive, there exists a power $f_{\\mid \\Lambda }^{K_2}$ of $f_{\\mid \\Lambda }$ such that $R_2$ is correctly aligned with $R^{\\prime }_2$ under $f_{\\mid \\Lambda }^{K_2}$ .",
"See Figure REF .",
"We have obtained the windows $R^{\\prime }_1$ about $T_1$ and $R^{\\prime }_2,R_2$ about $T_2$ such that $R^{\\prime }_1$ is correctly aligned with $R_2$ under $S$ and $R_2$ is correctly aligned with $R^{\\prime }_2$ under some power of $f_{\\mid \\Lambda }$ .",
"This ends the initial step of the inductive construction.",
"The inductive step goes on similarly, yielding the windows $R^{\\prime }_k$ about $T_k$ and $R^{\\prime }_{k+1},R_{k+1}$ about $T_{k+1}$ such that $R^{\\prime }_k$ is correctly aligned with $R_{k+1}$ under $S$ and $R_{k+1}$ is correctly aligned with $R^{\\prime }_{k+1}$ under some power of $f_{\\mid \\Lambda }$ .",
"See Figure REF .",
"Figure: Construction of windows along a transition chain.The construction proceeds inductively until a sequence of windows in $\\Lambda $ of the following type is obtained $ R^{\\prime }_1,R_2,R^{\\prime }_2, \\ldots , R_k,R^{\\prime }_k, R_{k+1}, R^{\\prime }_{k+1}, \\ldots , R_n, $ where for each $k\\in \\lbrace 1,\\ldots ,n-1\\rbrace $ we have that $R^{\\prime }_k$ is correctly aligned with $R_{k+1}$ under $S$ , and $R_{k+1}$ is correctly aligned with $R^{\\prime }_{k+1}$ under $f_{\\mid \\Lambda }^{K_{k+1}}$ for some $K_{k+1}>0$ .",
"Each window $R_k,R^{\\prime }_k$ is contained in an $\\varepsilon _k$ -neighborhood of the torus $T_k$ .",
"Applying the shadowing lemma-type of result Lemma REF provides an orbit that visits an $\\varepsilon _k$ -neighborhood in the phase space of each window in the sequence, and in particular of each torus in the transition chain."
],
[
"Gluing correctly aligned windows across BZI's with correctly aligned windows along transition chains of tori",
"We consider three transition chains of tori $\\lbrace T_i\\rbrace _{i=i_{k-1}+1,\\ldots , i_{{k}}}$ , $\\lbrace T_i\\rbrace _{i=i_k+1,\\ldots , i_{k+1}}$ , $\\lbrace T_i\\rbrace _{i=i_{k+1}+1,\\ldots , i_{{k+2}}}$ , with the property that each one of the regions between $T_{i_{k}}$ and $T_{i_{{k}}+1}$ , and between $T_{i_{k+1}}$ and $T_{i_{k+1}+1}$ , is either a BZI as in (A6), or it contains a finite number of invariant tori that separate the region as in (A6$^{\\prime }$ ).",
"Inside each region there is a prescribed collection of Aubry-Mather sets as in (A7) The constructions in Subsection REF and in Subsection REF yield correctly aligned windows in $\\Lambda $ that cross the region between $T_{i_{k}}$ and $T_{i_{{k}}+1}$ , and correctly aligned windows in $\\Lambda $ that cross the region between $T_{i_{k+1}}$ and $T_{i_{k+1}+1}$ .",
"The construction in Subsection REF yield sequences of correctly aligned windows along the adjacent transition chains.",
"The choices of the windows constructed along the transition chains depend on the choices of the windows that cross the region between $T_{i_{k}}$ and $T_{i_{{k}}+1}$ , and of the windows that cross the region between $T_{i_{k+1}}$ and $T_{i_{k+1}+1}$ .",
"Propagating the construction of windows starting from $T_{i_k+1}$ and moving forward along the transition chain $\\lbrace T_i\\rbrace _{i=i_k+1,\\ldots , i_{k+1}}$ , and the construction of windows starting from $T_{i_{k+1}}$ and moving backward along the same transition chain $\\lbrace T_i\\rbrace _{i=i_k+1,\\ldots , i_{k+1}}$ , may result in a pair of windows about some intermediate torus that are not correctly aligned.",
"We would like to glue this sequences of windows in a manner that is correctly aligned, without having to revise the windows constructed to that point.",
"Assume that $T_j$ is one of the tori $\\lbrace T_i\\rbrace _{i=i_k+1,\\ldots , i_{k+1}}$ , with $j\\in \\lbrace i_k+2,\\ldots , i_{k+1}-1\\rbrace $ .",
"Assume that one has already constructed a window $R_j$ about $T_j$ by propagating the construction from $T_{i_k+1}$ and moving forward along the transition chain, and another window $R^{\\prime }_j$ about $T_j$ by propagating the construction from $T_{i_{k+1}}$ and moving backwards along the transition chain.",
"The window $R_j$ is of the form $R _{j}=\\chi _j ([-\\delta _j,\\delta _j]\\times [0,1]),\\\\{R }_{j}^\\textrm {exit}=\\chi _j (\\partial [-\\delta _j,\\delta _j]\\times [0,1]),$ where $\\chi _{j}:\\mathbb {R}^2\\rightarrow \\Lambda $ is a $C^0$ -local parametrization with $\\chi _{j}(\\lbrace 0\\rbrace \\times [0,1])\\subseteq T_j$ , and $\\chi _j (\\lbrace -\\delta _j\\rbrace \\times [0,1])\\subseteq T_j^{\\rm lower}$ and $\\chi _j (\\lbrace \\delta _j\\rbrace \\times [0,1])\\subseteq T_j^{\\rm upper}$ , where $T_j^{\\rm lower}$ and $T_j^{\\rm upper}$ are two primary invariant tori on the opposite sides of $T_j$ .",
"The window $R^{\\prime }_j$ is of the form $R^{\\prime }_{j}=\\chi ^{\\prime }_j ([0,1]\\times [-\\delta ^{\\prime }_j,\\delta ^{\\prime }_j]),\\\\{R^{\\prime }}_{j}^\\textrm {exit}=\\chi ^{\\prime }_j (\\partial [0,1]\\times [-\\delta ^{\\prime }_j,\\delta ^{\\prime }_j]),$ where $\\chi ^{\\prime }_{j}:\\mathbb {R}^2\\rightarrow \\Lambda $ is a $C^0$ -local parametrization with $\\chi ^{\\prime }_{j}([0,1]\\times \\lbrace 0\\rbrace )\\subseteq T_j$ , and $\\chi ^{\\prime }_j ([0,1]\\times \\lbrace -\\delta ^{\\prime }_j\\rbrace )\\subseteq T_j^{\\rm lower}$ and $\\chi ^{\\prime }_j ([0,1]\\times \\lbrace \\delta ^{\\prime }_j\\rbrace )\\subseteq {T^{\\prime }}_j^{\\rm upper}$ , where ${T^{\\prime }}_j^{\\rm lower}$ and ${T^{\\prime }}_j^{\\rm upper}$ are two primary invariant tori on the opposite sides of $T_j$ .",
"Let us assume that the annular region between ${T^{\\prime }}_j^{\\rm lower}$ and ${T^{\\prime }}_j^{\\rm upper}$ is inside the region between $T_j^{\\rm lower}$ and $T_j^{\\rm upper}$ .",
"We construct a new window $R^{\\prime \\prime }_j$ about $T_j$ , such that $R_j$ is correctly aligned with $R^{\\prime \\prime }_j$ under the identity map, and $R^{\\prime \\prime }_j$ is correctly aligned with $R^{\\prime }_j$ under some power of $f$ .",
"We let $R^{\\prime \\prime }_j$ is of the form $R^{\\prime \\prime }_{j}=\\chi ^{\\prime \\prime }_j ([-\\delta ^{\\prime \\prime }_j,\\delta ^{\\prime \\prime }_j]\\times [0,1]),\\\\{R^{\\prime \\prime }}_{j}^\\textrm {exit}=\\chi ^{\\prime \\prime }_j (\\partial [-\\delta ^{\\prime \\prime }_j,\\delta ^{\\prime \\prime }_j]\\times [0,1]),$ where $\\chi ^{\\prime \\prime }_{j}:\\mathbb {R}^2\\rightarrow \\Lambda $ is a $C^0$ -local parametrization with $\\chi ^{\\prime \\prime }_{j}(\\lbrace 0\\rbrace \\times [0,1])\\supseteq \\chi ^{\\prime }_{j}(\\lbrace 0\\rbrace \\times [0,1])$ , and $\\chi ^{\\prime \\prime }_j (\\lbrace -\\delta ^{\\prime \\prime }_j\\rbrace \\times [0,1])\\subseteq {T^{\\prime }}_j^{\\rm lower}$ and $\\chi ^{\\prime \\prime }_j (\\lbrace \\delta ^{\\prime \\prime }_j\\rbrace \\times [0,1])\\subseteq {T^{\\prime }}_j^{\\rm upper}$ , for some $\\delta ^{\\prime \\prime }_j>0$ .",
"Since the motion on the torus $T_j$ is topological transitive, there exists a power $f_{\\mid \\Lambda }^{K_j}$ of $f_{\\mid \\Lambda }$ such that $R^{\\prime \\prime }_j$ is correctly aligned with $R^{\\prime }_j$ under $f_{\\mid \\Lambda }^{K_j}$ .",
"See Figure REF .",
"We have obtained that $R_j$ is correctly aligned with $R^{\\prime \\prime }_j$ under the identity map, and $R^{\\prime \\prime }_j$ is correctly aligned with $R^{\\prime }_j$ under some power $f_{\\mid \\Lambda }^{K_j}$ of $f_{\\mid \\Lambda }$ .",
"Figure: Gluing sequences of correctly aligned windows along a transition chain.The case when the annular region between $T_j^{\\rm lower}$ and $T_j^{\\rm upper}$ is inside the region between ${T^{\\prime }}_j^{\\rm lower}$ and ${T^{\\prime }}_j^{\\rm upper}$ results in the window $R_j$ correctly aligned with $R^{\\prime }_j$ under some power of $f$ , without the need of creating an intermediate window $R^{\\prime \\prime }_j$ .",
"The case when neither annular region is contained in the other annular region can be dealt similarly by constructing an intermediate window $R^{\\prime \\prime }_j$ .",
"Through this process, a sequence of correctly windows constructed forward along a transition chain can be glued, in a correctly aligned manner, with a sequence of correctly windows constructed backwards along the same transition chain."
],
[
"Proof of Theorem ",
"To summarize, in Subsection REF and in Subsection REF , we described the construction of correctly aligned windows that cross an annular regions between two consecutive transition chains of invariant tori.",
"These annular regions are BZI's or else they are separated by some finite collection of invariant tori.",
"If each annular region has some prescribed collection od Aubry-Mather sets, the construction yields windows that follow these Aubry-Mather sets.",
"In Subsection REF yield sequences of correctly aligned windows each transition chain.",
"In Subsection REF the construction of a sequence of correctly windows constructed forward along a transition chain can be glued, in a correctly aligned manner, with a the construction of a sequence of correctly windows constructed backwards along the same transition chain.",
"Thus, starting from some initial annular region, one can construct sequences of correctly aligned windows, forward and backwards, along infinitely many topological transition chains interspersed with annular regions.",
"The Shadowing Lemma-type of result Theorem REF implies the existence of an orbit that gets $\\varepsilon _i$ -close to each transition torus $T_i$ , and also follows each Aubry-Mather set $\\Sigma _i$ for a prescribed time $n_i$ ."
]
] | 1204.1503 |
[
[
"Stochastic Turing Patterns on a Network"
],
[
"Abstract The process of stochastic Turing instability on a network is discussed for a specific case study, the stochastic Brusselator model.",
"The system is shown to spontaneously differentiate into activator-rich and activator-poor nodes, outside the region of parameters classically deputed to the deterministic Turing instability.",
"This phenomenon, as revealed by direct stochastic simulations, is explained analytically, and eventually traced back to the finite size corrections stemming from the inherent graininess of the scrutinized medium."
],
[
"At the next-to-leading order approximation in the van Kampen expansion, one obtains a Fokker Planck equation for the probability distribution of fluctuations, which is equivalent to the Langevin equation (REF ).",
"For each fixed pair of nodes, $i$ , $j$ , the $2 \\times 2$ matrix $\\mathbf {\\mathcal {M}_{sr,ij}}$ may be decomposed as the sum of two contributions, one relative to the activator-inhibitor reactions (non spatial components (NS)), and the other associated to the diffusion process (spatial components (SP)): $\\mathbf {\\mathcal {M}_{sr,ij}}=\\mathbf {\\mathcal {M}_{sr}}^{(NS)}+\\mathbf {\\mathcal {M}_{sr}}^{(SP)}\\Delta _{ij}$ .",
"The above matrices are evaluated at the mean-field fixed points $\\phi ^{*}$ , $\\psi ^{*}$ .",
"After a lengthy calculation [16] one obtains the following entries for matrix $\\mathbf {\\mathcal {M}}^{(NS)}$ : $\\mathcal {M}^{(NS)}_{11}&=&-a-b-d+2c\\phi ^*\\psi ^*, \\nonumber \\\\\\mathcal {M}^{(NS)}_{12}&=&-a+c\\phi ^{*^2}, \\nonumber \\\\\\mathcal {M}^{(NS)}_{21}&=&b-2c\\phi ^*\\psi ^*, \\nonumber \\\\\\mathcal {M}^{(NS)}_{22}&=&-c\\phi ^{*^2}.",
"\\nonumber $ The elements of matrix $\\mathbf {\\mathcal {M}}^{(SP)}$ read instead: $\\mathcal {M}^{(SP)}_{11}&=&2\\mu (1-\\psi ^*), \\\\\\mathcal {M}^{(SP)}_{12}&=&2\\mu \\phi ^*, \\\\\\mathcal {M}^{(SP)}_{21}&=&2\\delta \\psi ^*, \\\\\\mathcal {M}^{(SP)}_{22}&=&2\\delta (1-\\phi ^*).$ As concerns the matrix $\\mathbf {\\mathcal {B}}$ , one finds: $\\mathcal {B}_{11,ii} &=& D_1 + \\tilde{k}_i H_1, \\\\\\mathcal {B}_{12,ii}&=&\\mathcal {B}_{21, ii}=C \\\\\\mathcal {B}_{22,ii}&=& D_2 + \\tilde{k}_i H_2 \\\\\\mathcal {B}_{11, ij}&=&-\\left(\\frac{1}{k_i}+\\frac{1}{k_j}\\right) H_1 \\\\\\mathcal {B}_{12, ij}&=&\\mathcal {B}_{21, ij}=0.",
"\\\\\\mathcal {B}_{22, ij}&=&-\\left(\\frac{1}{k_i}+\\frac{1}{k_j}\\right) H_2$ where: $D_1 &=& a(1-\\phi ^*-\\psi ^*)+\\phi ^*\\left(b+c\\phi ^*\\psi ^*+d\\right) \\\\H_1 &=& 4\\mu \\phi ^*(1-\\phi ^*-\\psi ^*) \\nonumber \\\\C &=& -\\phi ^*\\left(b+c\\phi ^*\\psi ^*\\right) \\nonumber \\\\D_2 &=& \\phi ^*\\left(b+c\\phi ^*\\psi ^*\\right) \\nonumber \\\\H_2 &=& 4\\delta \\psi ^*(1-\\phi ^*-\\psi ^*) \\nonumber \\\\$ Matrix $\\mathcal {B}$ can be hence cast in the equivalent form: $\\mathcal {B}_{ss,ij} &=& (D_s+\\tilde{k}_i H_s) \\delta _{ij} - \\left(\\frac{1}{k_i}+\\frac{1}{k_j}\\right) W_{ij} H_s \\nonumber \\\\\\mathcal {B}_{rs,ij} &=& \\mathcal {B}_{sr,ij} = C \\delta _{ij}$ for $r,s=1,2$ .",
"Performing the transformation (REF ) on both sides of the Langevin equation (REF ), we get $\\mathtt {j} \\omega \\tilde{\\xi }_{s}^\\alpha = \\sum _{r=1}^2\\left(\\mathcal {M}_{sr}^{(NS)}+\\mathcal {M}_{sr}^{(SP)}\\Lambda _\\alpha \\right)\\tilde{\\xi }_r^\\alpha +\\tilde{\\eta }_s^\\alpha $ where the term $\\tilde{\\eta }_s^\\alpha $ denotes the transform of the noise.",
"Introducing the matrix $\\Phi _{sr}=\\mathtt {j} \\omega \\delta _{sr}-\\left(\\mathcal {M}_{sr}^{(NS)}+\\mathcal {M}_{sr}^{(SP)}\\Lambda _\\alpha \\right)$ we get $\\tilde{\\xi }_s^\\alpha =\\sum _{r=1}^2 \\Phi _{sr}^{-1}\\tilde{\\eta }_r^\\alpha $ , and thus the power spectrum of the $s$ -th species is given by $P_s(\\omega ,\\Lambda _\\alpha )=\\langle |\\tilde{\\xi }^\\alpha _s|^2\\rangle =\\sum _{r,k=1}^2 \\Phi _{sr}^{-1}\\langle \\tilde{\\eta }^\\alpha _r \\tilde{\\eta }_k^{\\alpha }\\rangle (\\Phi _{ks}^{\\dag })^{-1}$ It can be shown that $\\langle \\tilde{\\eta }^\\alpha _r \\tilde{\\eta }_k^{\\alpha } \\rangle = \\sum _{i,j}\\mathcal {B}_{rk,ij}v_i^{(\\alpha )}v_{j}^{(\\alpha )} $ .",
"An explicit form for the $2\\times 2$ matrix $\\langle \\tilde{\\eta }^\\alpha _r \\tilde{\\eta }_k^{\\alpha } \\rangle $ can be derived by making use of Eqs (REF ).",
"Let us focus on the non trivial contribution $r=k$ : $&&\\sum _{i,j}\\mathcal {B}_{rr,ij}v_i^{(\\alpha )}v_{j}^{(\\alpha )} =\\\\ &&\\sum _{i,j}\\left[ (D_r+\\tilde{k}_i H_r) \\delta _{ij} - \\left(\\frac{1}{k_i}+\\frac{1}{k_j}\\right) W_{ij} H_r \\right]v_i^{(\\alpha )}v_{j}^{(\\alpha )} =\\\\&& D_r \\sum _{i} v_i^{(\\alpha )} v_{i}^{(\\alpha )} - H_r \\sum _{i,j} \\Delta _{i,j} v_i^{(\\alpha )} v_{i}^{(\\alpha )} =D_r - H_r \\Lambda _{\\alpha }$ where in the last step we made use of $\\sum _{i} v_i^{(\\alpha )} v_{i}^{(\\alpha )}=1$ and $\\sum _j \\Delta _{ij} v_j^{(\\alpha )}=\\Lambda _\\alpha v_i^{\\alpha }$ .",
"The component $r \\ne k$ is trivially equal to $C$ .",
"The power spectrum is fully specified as function of $\\Lambda _\\alpha $ and $\\omega $ .",
"Figure REF is obtained by setting $\\omega =0$ in the above formulae.",
"We emphasize that the same result can be recovered from the continuum medium power spectrum [16] provided $-k^2$ is replaced by the discrete eigenvalue $\\Lambda _{\\alpha }$ .",
"We end this Appendix by providing a list of explanatory captions to the movies annexed as supplementary material.",
"mf_inside_X.mov and mf_inside_Y.mov show respectively the time evolution of the mean field concentrations $\\phi _i$ and $\\psi _i$ .",
"The data are obtained by integrating the governing mean field equations (REF ).",
"Parameters are chosen so to yield a Turing instability (magenta circle in Fig.",
"REF ).",
"At time $t=0$ a small perturbation is applied to perturb the homogeneous fixed point.",
"Then, the system evolves toward a stable, non-homogeneous stationary configuration.",
"st_inside_X.mov and st_inside_Y.mov show the result of the stochastic simulations (fluctuating blue circles), for respectively $n_i/N$ and $m_i/N$ .",
"The red symbols refer to a late time snapshot of the deterministic dynamics.",
"Parameters are set as specified above: the system is hence inside the region of Turing instability.",
"Notice that the noise that takes the system away from the homogeneous fixed point is now endogenous to the system and not externally imposed.",
"st_outside_X.mov and st_outside_Y.mov report on the stochastic simulations (blue symbols), for respectively $n_i/N$ and $m_i/N$ .",
"The parameters are now assigned so to fall outside the region of Turing order (blue diamond in Fig.",
"REF ).",
"The mean field solutions are not destabilized and converge to the stable fixed point.",
"At variance, the stochastic simulations evolve toward a non homogeneous state.",
"Financial support from Ente Cassa di Risparmio di Firenze and the Program Prin2009 funded by Italian MIUR is acknowledged.",
"D.F.",
"thanks Tommaso Biancalani for pointing out reference [18] and for stimulating discussions.",
"F.D.P.",
"thanks Alessio Cardillo for providing the code to generate the network."
]
] | 1204.1475 |
[
[
"Metabasins - a State Space Aggregation for highly disordered Energy\n Landscapes"
],
[
"Abstract Glass-forming systems, which are characterized by a highly disordered energy landscape, have been studied in physics by a simulation-based state space aggregation.",
"The purpose of this article is to develop a path-independent approach within the framework of aperiodic, reversible Markov chains with exponentially small transition probabilities which depend on some energy function.",
"This will lead to the definition of certain metastates, also called metabasins in physics.",
"More precisely, our aggregation procedure will provide a sequence of state space partitions such that on an appropriate aggregation level certain properties (see Properties 1--4 of the Introduction) are fulfilled.",
"Roughly speaking, this will be the case for the finest aggregation such that transitions back to an already visited (meta-)state are very unlikely within a moderate time frame."
],
[
"Introduction",
"tocsectionIntroduction Supercooled liquids of glass forming systems are typical examples of high-dimensional systems with highly disordered energy landscapes and our main concern behind this work.",
"Simulations have shown that many important characteristics of such a system are better described by a process on the set of so-called metabasins (MB) than by the more common process on the set of visited minima of the energy landscape (see [12]).",
"Those MB are formed in the following way by aggregation of suitable states of the describing process $(X_{n})_{n\\ge 0}$ along a simulated trajectory: Fixing a reasonable observation time $T$ , define $\\chi _0\\equiv 0$ and then recursively for $n\\ge 1$ $\\chi _n:=\\inf \\big \\lbrace k>\\chi _{n-1}\\,|\\,\\lbrace X_{k}{,}...,X_{T}\\rbrace \\cap \\lbrace X_{0}{,}...,X_{k-1}\\rbrace =\\emptyset \\big \\rbrace .$ Then the MB up to $\\upsilon :=\\sup \\lbrace n\\ge 0\\,|\\,\\chi _n\\le T\\rbrace $ are chosen as $\\mathcal {V}_{n}:=\\lbrace X_{\\chi _n}{,}...,X_{\\chi _{n+1}-1}\\rbrace ,\\quad 0\\le n\\le \\upsilon .$ Simulation studies have shown that local sampling within a MB does not affect typical parameters of the process like the diffusion coefficient or the time to reach equilibrium.",
"Dynamical aspects are therefore fully characterized by the MB-valued process.",
"Furthermore, this model reduction by aggregation, as proposed in [12] and [18], offers several advantages (referred to as Properties 1–5 hereafter): The probability of a transition from one MB to any other one does not depend on the state at which this MB is entered.",
"There are basically no reciprocating jumps between two MB.",
"This is in strong contrast to the unaggregated process where such jumps occur very often: The system falls back to a minimum many times before eventually cresting a high energy barrier and then falling into a new valley, where it will again take many unsuccessful trials to escape.",
"These reciprocating jumps are not only irrelevant for the actual motion on the state space but also complicating the estimation of parameters like the diffusion coefficient or the relaxation time.",
"The expected time spent in a MB is proportional to its depth.",
"Thus there is a strong and explicit relation between dynamics and thermodynamics, not in terms of the absolute but the relative energy.",
"The energy barriers between any two MB are approximately of the same height, that is, there is an energy threshold $E_0$ such that, for a small $\\varepsilon $ , it requires a crossing of at least $E_0-\\varepsilon $ and at most $E_0+\\varepsilon $ to make a transition from one MB to another.",
"Such systems with $\\varepsilon =0$ are called trap models (see [3]).",
"The sojourn times and the jump distances between successively visited MB (measured in Euclidean distance) form sequences of weakly or even uncorrelated random variables, and are also mutually independent, at least approximately.",
"Therefore, the aggregated process can be well approximated by a continuous time random walk, which in turn simplifies its analysis and thus the analysis of the whole process.",
"Despite these advantages, the suggested definition of MB has the obvious blemish that it depends on the realization of the considered process and may thus vary from simulation to simulation.",
"To provide a mathematically stringent definition of a path-independent aggregation of the state space, which maintains the above properties and is based on the well-established notion of metastable states, is therefore our principal concern here with the main results being Theorem REF and Theorem REF .",
"In this endeavor, we will draw on some of the ideas developed by Bovier in [4] and by Scoppola in [21], most notably her definition of metastable states.",
"Metastability, a phenomenon of ongoing interest for complex physical systems described by finite Markov processes on very large state spaces, can be defined and dealt with in several ways.",
"It has been derived from a renormalization procedure in [20], by a pathwise approach in [6], and via energy landscapes in [4], the latter being also our approach hereafter.",
"To characterize a supercooled liquid, i.e.",
"a glass forming system at low temperature, via its energy landscape was first done by Goldstein in 1969 [11] and has by now become a common method.",
"The general task when studying metastability, as well originally raised in physics ([13], [19] or [18]), is to provide mathematical tools for an analysis of the property of thermodynamical systems to evolve in state space along a trajectory of unstable or metastable states with very long sojourn times.",
"Inspired by simulations of glass forming systems at very low temperatures with the Metropolis algorithm, we will study (as in [21]) finite Markov chains with exponentially small transition probabilities which are determined by an energy function and a parameter $\\beta >0$ .",
"This parameter can be understood as the inverse temperature and we are thus interested in the behavior of the process as $\\beta \\rightarrow \\infty $ .",
"We envisage an energy function of highly complex order and without the hierarchical ordering that is typical in spin glass models.",
"A good picture is provided by randomly chosen energies with correlations between neighbors or by an energy landscape that looks like a real mountain landscape.",
"We will show that, towards an aggregation outlined above, the metastable states as defined in [21] are quite appropriate because they have an ordering from a kind of “weak” to a kind of “strong” metastability.",
"Around those states we will define and then study connected valleys [Definition REF ] characterized by minimal energy barriers.",
"In the limit of low temperatures, any such barrier will determine the speed, respectively probability of a transition between the two valleys it separates.",
"More precisely, in the limit $\\beta \\rightarrow \\infty $ , the process, when starting in a state $x$ , will almost surely reach a state with lower barrier earlier than a state with higher barrier [Theorem REF ].",
"In the limit of low temperatures, the bottom (minimum) of an entered valley will therefore almost surely be reached before that valley is left again [Proposition REF ].",
"As a consequence, the probability for a transition from one valley to another is asymptotically independent of the state where the valley is entered.",
"This is Property 1 above.",
"Furthermore, since valleys as well as metastable states have a hierarchical ordering, we can build valleys of higher order by a successive merger of valleys of lower order [Proposition REF ].",
"Given an appropriate energy landscape, this procedure can annihilate (on the macroscopic scale) the accumulation of reciprocating jumps by merging valleys exhibiting such jumps into a single valley [Subsection REF ].",
"Hence, valleys of sufficiently high order will have Property 2.",
"Beside the macroscopic process [Section ], which describes the transitions between valleys, one can also analyze the microscopic process [Section ], that is, the system behavior when moving within a fixed valley.",
"Here we will give a formula for the exit time and connect it with its parameters [Theorem REF ].",
"This will confirm Property 3.",
"Having thus established Properties 1–4 [Theorem REF ], we will finally proceed to a comparison of our path-independent definition of MB with the path-dependent one given above.",
"It will be shown [Theorem REF ] that both coincide with high probability under some reasonable conditions on the connectivity of valleys which, in essence, ensure the existence of reasonable path-dependent MB.",
"We will also briefly touch on the phenomenon of quasi-stationarity [Proposition REF ] which is a large area [17] but to our best knowledge less studied in connection with the aggregation of states of large physical systems driven by energy landscapes.",
"Let us mention two further publications which, despite having a different thrust, provide definitions of valleys, called basins of attraction or metastates there, to deal with related questions.",
"Olivieri & Scoppola [16] fully describe the tube of exit from a domain in terms of which basins of attraction of increasing order are visited during a stay in that domain and for how long these basins are visited.",
"In a very recent publication, Beltrán & Landim [2], by working with transition rates instead of energies, aim at finding a universal depth (and time scale) for all metastates.",
"However, we rather aim at the finest aggregation such that transitions back to an already visited metastate are very unlikely within a time frame used in simulations.",
"This finest aggregation will lead to valleys of very variable depth just as simulations do not exhibit a universal depth or timescale."
],
[
"Valleys",
"Let $X$ be a Markov chain on a finite set $\\mathcal {S}$ with transition matrix $\\mathbf {P}=(p(r,s))_{r,s\\in \\mathcal {S}}$ and stationary distribution $\\pi $ , and let $E:\\mathcal {S}\\rightarrow \\mathbb {R}$ be an energy function such that the following conditions hold: Irreducibility: $\\mathbf {P}$ is irreducible with $p(s,s)>0$ and $p(r,s)>0$ iff $p(s,r)>0$ for all $r,s\\in \\mathcal {S}$ .",
"Transition Probabilities: There exist parameters $\\beta >0$ and $\\gamma _{\\beta }>0$ with $\\gamma _{\\beta }\\rightarrow 0, \\beta \\gamma _{\\beta }\\rightarrow \\infty $ as $\\beta \\rightarrow \\infty $ such that $e^{-\\beta ((E(s)-E(r))^++\\gamma _{\\beta }/|\\mathcal {S}|)}\\le p(r,s)\\le e^{-\\beta ((E(s)-E(r))^+-\\gamma _{\\beta }/|\\mathcal {S}|)}$ for all distinct $r,s\\in \\mathcal {S}$ with $p(r,s)>0$ .",
"Furthermore, $p^{*}(r,s):=\\lim _{\\beta \\rightarrow \\infty }p(r,s)$ exists for all $r,s\\in \\mathcal {S}$ , is positive if $E(r)\\ge E(s)$ and $=0$ otherwise.",
"Reversibility: The pair $(\\pi ,\\mathbf {P})$ satisfies the detailed balance equations, i.e.",
"$\\pi (r)p(r,s)=\\pi (s)p(s,r)$ for all $r,s\\in \\mathcal {S}$ .",
"Non-Degeneracy: $E(r)\\ne E(s)$ for all $r,s \\in \\mathcal {S}, r\\ne s$ .",
"We are thus dealing with a reversible Markov chain with exponentially small transition probabilities driven by an energy landscape.",
"As an example, which is also the main motivation behind this work, one can think of a Metropolis chain with transition probabilities of the form $p(r,s)=\\frac{1}{C(r)}e^{-\\beta (E(s)-E(r))^+}.$ Here $\\beta $ is the inverse temperature and $C(r), r\\in \\mathcal {S}$ , is a parameter, independent from $\\beta $ , giving the number of neighbors of $r$ .",
"For $\\gamma _{\\beta }/|\\mathcal {S}|:=\\max _{r\\in \\mathcal {S}}\\ln (C(r))(\\beta +1)^{-1/2}$ the above conditions are fulfilled.",
"Let us start with the following basic result for the ratios of the stationary distribution.",
"Lemma 1.1 For any two states $r,s\\in \\mathcal {S}$ with $E(r)>E(s)$ , we have $e^{-\\beta (E(r)-E(s)+2\\gamma _{\\beta })}\\le \\frac{\\pi (r)}{\\pi (s)}\\le e^{-\\beta (E(r)-E(s)-2\\gamma _{\\beta })}.$ [Proof:] To start with, assume $r\\sim s$ .",
"Reversibility and the assumptions on the transition probabilities imply $\\frac{\\pi (r)}{\\pi (s)}\\ &=\\ \\frac{p(s,r)}{p(r,s)}\\ \\le \\ \\frac{e^{-\\beta ((E(r)-E(s))^+-\\gamma _{\\beta }/|\\mathcal {S}|)}}{e^{-\\beta ((E(s)-E(r))^++\\gamma _{\\beta }/|\\mathcal {S}|)}}\\ =\\ e^{-\\beta (E(r)-E(s)-2\\gamma _{\\beta }/|\\mathcal {S}|)}.$ and $\\frac{\\pi (r)}{\\pi (s)}\\ &=\\ \\frac{p(s,r)}{p(r,s)}\\ \\ge \\ \\frac{e^{-\\beta ((E(r)-E(s))^++\\gamma _{\\beta }/|\\mathcal {S}|)}}{e^{-\\beta ((E(s)-E(r))^+-\\gamma _{\\beta }/|\\mathcal {S}|)}}\\ =\\ e^{-\\beta (E(r)-E(s)+2\\gamma _{\\beta }/|\\mathcal {S}|)}.$ Now let $r$ and $s$ be arbitrary.",
"By the irreducibility, there is a path $r=r_0,r_1{,}...,r_n=s$ from $r$ to $s$ of neighboring states with $\\pi (r_i)/\\pi (r_{i+1})\\in [e^{-\\beta (E(r_i)-E(r_{i+1})+2\\gamma _{\\beta }/|\\mathcal {S}|)},e^{-\\beta (E(r_i)-E(r_{i+1})-2\\gamma _{\\beta }/|\\mathcal {S}|)}],\\, 0\\le i\\le n-1$ .",
"This finishes the proof.",
"Under the stated assumptions, Scoppola [21] has shown the existence of a successive filtration (aggregation) $\\mathcal {S}=M^{(0)}\\supset M^{(1)}\\supset ... \\supset M^{(\\mathfrak {n})}=\\lbrace s_0\\rbrace $ , $\\mathfrak {n}\\in \\mathbb {N}$ , of the state space such that the elements of each set $M^{(i)}, 1\\le i\\le \\mathfrak {n}$ , can be called metastable in the following sense: They arise from the local minima of the energy function or certain modifications of it.",
"There is a lower bound on the expected time needed for a transition from $m_1$ to $m_2$ for any $m_1,m_2\\in M^{(i)}$ which increases very fast with $i$ .",
"There exists a constant $C$ such that $\\mathbb {P}_m(X_t\\notin M^{(i+1)})\\le e^{-C\\beta }$ for all $m\\in M^{(i)}, 0\\le i\\le \\mathfrak {n}-1,$ and sufficiently large $t$ .",
"This filtration starts with $M^{(1)}:=\\lbrace s\\in \\mathcal {S}| E(s)<E(r) \\textrm { for all $ s$}\\rbrace ,$$and deletes one local minimum at each step.",
"In fact, the local minimum with minimal activation energy for a transition to another minimum is deleted, see \\cite {Scop} and \\cite {Scop2} for further details.$ Figure: Example of an energy landscape with minima shown as black dots (•\\bullet )Example 1.2 For the simple energy function depicted in Figure REF , a successive application of the algorithm from [21] as illustrated in Figure REF leads to the following decomposition into subsets of metastable states: $M^{(1)}&=\\lbrace 2,4,6,8,10,12,14\\rbrace , \\ &M^{(2)}&=\\lbrace 2,4,6,10,12,14\\rbrace , \\\\M^{(3)}&=\\lbrace 2,4,6,10,14\\rbrace , &M^{(4)}&=\\lbrace 2,4,10,14\\rbrace ,\\\\M^{(5)}&=\\lbrace 4,10,14\\rbrace , \\ &M^{(6)}&=\\lbrace 4,14\\rbrace ,\\\\M^{(7)}&=\\lbrace 4\\rbrace .$ Figure: Successive application of the algorithm in to the energy landscape in Figure .",
"For each step ii, the metastable states as well as the corresponding valleys are shown.Based on the filtration of $\\mathcal {S}$ just described, we now proceed to a definition of a sequence of metastable sets associated with the metastable states which will induce the MB.",
"In order to do so, we must study first minimal paths between two states and maximal energies along such paths.",
"Definition 1.3 (a) For any two distinct states $r,s\\in \\mathcal {S}$ , let $\\Gamma (r,s):=\\lbrace (x_0{,}...,x_k)|\\,k\\in \\mathbb {N},\\,x_0=r,\\,x_k=s,\\,x_i\\ne x_j,\\, p(x_i,x_{i+1})>0 \\textrm { for } 0\\le i\\le k-1,\\,i\\ne j\\rbrace $ be the set of all finite self-avoiding paths from $r$ to $s$ having positive probability.",
"For any such path $\\gamma =(\\gamma _0{,}...,\\gamma _k)\\in \\Gamma (r,s)$ , let $|\\gamma |:=k$ be its length.",
"We further write $t\\in \\gamma $ if $t\\in \\lbrace \\gamma _{1}{,}...,\\gamma _{k}\\rbrace $ .",
"(b) A self-avoiding path $\\gamma =(\\gamma _{1}{,}...,\\gamma _{k})$ from $r$ to $s$ is called minimal if its maximal energy $\\max _{1\\le i\\le k} E(\\gamma _i)$ is minimal among all $\\gamma ^{\\prime }\\in \\Gamma (r,s)$ .",
"The set of these paths is denoted $\\Gamma ^*(r,s)$ .",
"(c) The essential saddle $z^*(r,s)$ between $r$ and $s$ is then defined as $z^*(r,s):=\\mathop {\\mathrm {argmax}}_{t\\in \\gamma }E(t)\\in \\mathcal {S}$ for any $\\gamma \\in \\Gamma ^{*}(r,s)$ .",
"As for (c), it is to be noted that, due to the assumed non-degeneracy of the energy function, the essential saddle is unique, which means that it does not depend on (as it must) which minimal path we choose in the definition of $z^*(r,s)$ .",
"There may indeed be several minimal paths, every single one thus crossing the saddle at some time.",
"With the help of these notions the valleys can now be defined in a quite concrete way.",
"Let us label the local minima as $m^{(1)}{,}...,m^{(\\mathfrak {n})}$ , so that $M^{(i)}=\\lbrace m^{(i)}{,}...,m^{(\\mathfrak {n})}\\rbrace $ for each $i=1{,}...,\\mathfrak {n}$ .",
"Definition 1.4 For each $m \\in M^{(i)}$ , $1\\le i\\le \\mathfrak {n}$ , let $V^{(i)}_<(m):=\\left\\lbrace s\\in \\mathcal {S}\\Big |E(z^*(s,m))< E(z^*(s,m^{\\prime }))\\text{ for all }m^{\\prime }\\in M^{(i)}\\backslash \\lbrace m\\rbrace \\right\\rbrace .$ We say that state $s$ is attracted by $m$ at level $i$, expressed as $s\\leadsto m$ at level $i$ , if $ E(z^*(s,m))=\\min _{n\\in M^{(i)}}E(z^*(s,n))$ and every minimal path from $s$ to a state $m^{\\prime }\\in M^{(i)}\\backslash \\lbrace m\\rbrace $ with $E(z^*(s,m^{\\prime }))=E(z^*(s,m))$ hits $V^{(i)}_{<}(m)$ at some time.",
"Finally, let $l(i):=\\inf \\big \\lbrace i<j\\le \\mathfrak {n}|m^{(i)}\\leadsto m\\text{ at level }j\\text{ for some }m\\in M^{(j)}\\big \\rbrace $ denote the minimal level at which the minimal state $m^{(i)}$ becomes attracted by a minimal state of superior level.",
"Definition 1.5 (a) Initialization: For each $m \\in M^{(1)}$ , define $V^{(1)}(m):=\\left\\lbrace s\\in \\mathcal {S}\\,\\Big |\\,s\\leadsto m\\text{ at level }1\\right\\rbrace .$ as the valley of order 1 containing $m$ and let $N^{(1)}:=\\left(\\bigcup _{j=1}^{\\mathfrak {n}}V^{(1)}(m^{(j)})\\right)^{c}$ be the set of non-assigned states at level 1.",
"(b) Recursion: For $2\\le i\\le \\mathfrak {n}$ and $m\\in M^{(i)}$ , define $V^{(i)}(m):=V^{(i-1)}(m)\\,\\cup \\,\\left\\lbrace s\\in N^{(i-1)}\\,\\Big |\\,s\\leadsto m\\text{ at level }i\\right\\rbrace \\,\\cup \\,\\bigcup _{j:l(j)=i,m^{(j)}\\leadsto m\\text{ at level }i}\\hspace{-6.0pt}V^{(j)}(m^{(j)})$ as the valley of order $i$ containing $m$ and let $N^{(i)}:=\\left(\\bigcup _{j=1}^{\\mathfrak {n}}V^{(i\\wedge j)}(m^{(j)})\\right)^{c}$ be the set of non-assigned states at level i.",
"Here is a more intuitive description of what the previous two definitions render in a formal way: First, we define, for each level $i$ and $m\\in M^{(i)}$ , the set $V^{(i)}_{<}(m)$ of those states $s$ that are strongly attracted by $m$ in the sense that $E(z^{*}(s,m))$ is strictly smaller than $E(z^{*}(s,m^{\\prime }))$ for any other $m^{\\prime }\\in M^{(i)}$ .",
"Then, starting at level one, each valley $V^{(1)}(m)$ , $m\\in M^{(1)}$ , is formed from $V^{(1)}_{<}(m)$ by adjoining all further states $s$ attracted by $m$ at this level.",
"This leaves us with a set of non-assigned states, denoted $N^{(1)}$ .",
"In the next step (level 2), any $V^{(2)}(m)$ for $m\\in M^{(2)}$ is obtained by adjoining to $V^{(1)}(m)$ all those $s\\in N^{(1)}$ which are attracted by $m$ at level 2.",
"Observe that this ensures $V^{(2)}_{<}(m)\\subset V^{(2)}(m)$ .",
"Moreover, if $m^{(1)}$ is attracted by $m$ at level 2, then $V^{(1)}(m^{(1)})$ is merged into $V^{(2)}(m)$ as well.",
"If no such $m$ exists (thus $l(1)>2$ ), it remains untouched until reaching level $l(1)$ where its bottom state $m^{(1)}$ becomes attracted by some $m^{\\prime }\\in M^{(l(1))}$ causing its valley to be merged into $V^{(l(1))}(m^{\\prime })$ .",
"This procedure continues in the now obvious recursive manner until at level $\\mathfrak {n}$ all states have been merged into one valley.",
"Obviously, valleys of the same order are pairwise disjoint.",
"Also, valleys once formed at some level can only be merged as a whole and will thus never be ripped apart during the recursive construction.",
"For the energy function depicted in Figure REF , the successively derived valleys of order $i=1{,}...,7$ are shown in Figure REF .",
"Before proceeding to results on the general shape of valleys, we collect some basic, mostly technical properties of essential saddles which will be useful thereafter.",
"Proposition 1.6 For any $r,s,u\\in \\mathcal {S}$ , $0\\le i\\le \\mathfrak {n}$ , $m_1,m_2\\in M^{(i)}, m_1\\ne m_2,$ and $x_1,x_2\\in \\mathcal {S}$ with $x_1\\in V^{(i)}_<(m_1)$ and $x_2\\in V^{(i)}(m_2)$ , we have (a) $z^*(r,s)=z^*(s,r)$ .",
"(b) $E(z^*(r,s))\\le E(z^*(r,u))\\vee E(z^*(u,s))$ .",
"(c) $E(z^*(x_2,m_2))\\le E(z^*(x_2,m^{\\prime }))$ for all $m^{\\prime }\\in M^{(i)}$ .",
"(d) $E(z^*(x_1,m_2))=E(z^*(m_1,m_2))$ .",
"(e) $E(z^*(x_1,x_2))\\ge E(z^*(m_1,m_2))$ .",
"(f) $z^*(x_1,x_2)\\ne x_1$ .",
"[Proof:] Parts (a) and (b) are obvious.",
"For (c) we use an induction over $i$ and note that there is nothing to show when $i=1$ .",
"For general $i$ , we must only verify that $E(z^{*}(x_{2},m_{2}))\\le E(z^{*}(x_{2},m^{\\prime }))$ for all $m^{\\prime }\\in M^{(i)}$ if $x_{2}\\in V^{(j)}(m^{(j)})$ for some $j<i$ such that $l(j)=i$ and $m^{(j)}\\leadsto m$ at level $i$ (due to the recursive definition of $V^{(i)}(m)$ ).",
"But the latter ensures that $E(z^{*}(x_{2},m^{(j)}))\\le E(z^{*}(x_{2},n))$ for all $n\\in M^{(j)}\\supset M^{(i)}$ (inductive hypothesis) as well as $E(z^{*}(m^{(j)},m_{2}))\\le E(z^{*}(m^{(j)},m^{\\prime }))$ for all $m^{\\prime }\\in M^{(i)}$ .",
"Consequently, for any such $m^{\\prime }$ , $E(z^{*}(x_2,m_2))&\\le E(z^{*}(x_{2},m^{(j)}))\\vee E(z^{*}(m^{(j)},m_{2}))\\\\&\\le E(z^{*}(x_{2},m^{(j)}))\\vee E(z^{*}(m^{(j)},m^{\\prime }))\\\\&\\le E(z^{*}(x_{2},m^{(j)}))\\vee E(z^{*}(x_{2},m^{\\prime })))\\\\&=E(z^{*}(x_{2},m^{\\prime }))$ as asserted.",
"For assertion (d), note that $E(z^*(x_1,m_2))>E(z^*(x_1,m_1))$ , which in combination with (a) and (b) implies $E(z^*(m_1,m_2))\\le E(z^*(m_1,x_1))\\vee E(z^*(x_1,m_2))=E(z^*(x_1,m_2))$ and then further $E(z^*(x_1,m_2))\\le \\underbrace{E(z^*(x_1,m_1))}_{<E(z^*(x_1,m_2))}\\vee \\underbrace{E(z^*(m_1,m_2))}_{\\le E(z^*(x_1,m_2))}\\le E(z^*(x_1,m_2)).$ So the above must be an identity, i.e.",
"$E(z^*(x_1,m_2))=E(z^*(m_1,m_2))$ .",
"Turning to part (e), we first infer with the help of (c) and (d) that $E(z^*(x_1,m_1))&< E(z^*(x_1,m_2)) \\nonumber \\\\&=E(z^*(m_1,m_2)) \\nonumber \\\\&\\le E(z^*(m_1,x_2))\\vee E(z^*(x_2,m_2))\\\\&= E(z^*(x_2,m_1)) \\nonumber \\\\&\\le E(z^*(x_2,x_1))\\vee E(z^*(x_1,m_1)), \\nonumber $ thus $E(z^*(x_1,m_1))<E(z^*(x_1,x_2)).$ Together with the just shown inequality $E(z^*(m_1,m_2))\\le E(z^*(x_2,m_1))$ (see (REF )) and another use of (c), this yields $E(z^*(x_1,x_2))=E(z^*(x_2,x_1))\\vee E(z^*(x_1,m_1))\\ge E(z^*(x_2,m_1))\\ge E(z^*(m_1,m_2)).$ Finally, we infer with the help of (REF ) that $E(z^*(x_1,x_2))&>E(z^*(x_1,m_1))\\ge E(x_1)$ and thus $z^*(x_1,x_2)\\ne x_1$ as claimed in (f).",
"Remark 1.7 It is useful to point out the following consequence of the previous proposition.",
"If, for an arbitrary state $s$ and any two distinct metastable states $m,n\\in M^{(i)}$ , there exists a minimal path $\\gamma $ from $s$ to $n$ that hits a state $r$ with $E(z^*(r,m))<E(z^*(r,n)$ , then there is also a minimal path from $s$ to $n$ that passes through $m$ .",
"Namely, if we replace the segment from $r$ to $n$ of the former path by the concatenation of two minimal paths from $r$ to $m$ and from $m$ to $n$ , then the maximal energy of this new path is $E(z^{*}(s,n))\\vee E(z^{*}(r,m))\\vee E(z^{*}(m,n))&\\le E(z^{*}(s,n))\\vee E(z^{*}(r,m))\\vee E(z^*(r,n))\\\\&=E(z^{*}(s,n))\\vee E(z^{*}(r,n))\\\\&=E(z^*(s,n)),$ by Proposition REF (b), whence the new path has to be minimal from $s$ to $n$ as well.",
"This yields two facts: (a) A minimal path from $s$ to $n$ , where $s\\leadsto n$ at level $i$ , hits $V^{(i)}_<(n)$ before it hits any $r$ with $E(z^*(r,m))<E(z^*(r,n))$ for some $m\\in M^{(i)}$ .",
"Otherwise, since the subpath from $r$ to $m$ can be chosen to stay in $\\lbrace t|E(z^*(t,m))<E(z^*(t,n))\\rbrace $ and thus $E(z^*(s,m))=E(z^*(s,n))$ , there would be a path from $s$ to $m$ not hitting $V^{(i)}_<(n)$ .",
"(b) If $s\\leadsto n$ at level $i$ and $m\\in M^{(i)}\\backslash \\lbrace n\\rbrace $ with $E(z^*(s,n))=E(z^*(s,m))$ , then a minimal path from $s$ to $m$ does not only hit $V^{(i)}_<(n)$ at some time, but in fact earlier than any other valley $V^{(i)}_<(m^{\\prime }), m^{\\prime }\\in M^{(i)}\\backslash \\lbrace n\\rbrace $ .",
"Lemma 1.8 Let $1\\le i<j\\le \\mathfrak {n}$ , $m=m^{(i)}$ and $s\\in V^{(i)}(m)$ .",
"Then $s\\in V^{(j)}_{<}(m^{\\prime })$ for some $m^{\\prime }\\in M^{(j)}$ implies $l(i)\\le j$ , $m\\in V^{(j)}_{<}(m^{\\prime })$ and thus $V^{(i)}(m)\\subset V^{(j)}(m^{\\prime })$ .",
"In other words, whenever $V^{(i)}(m^{(i)})$ contains an element $s$ which at some higher level $j$ belongs to some $V^{(j)}_{<}(m^{\\prime })$ , $m^{\\prime }\\in M^{(j)}$ , the same must hold true for $m^{(i)}$ itself implying $V^{(i)}(m^{(i)})\\subset V^{(j)}(m^{\\prime })$ .",
"Conversely, this guarantees that $V^{(i)}(m^{(i)})$ will have no common elements with any $V^{(j)}_{<}(m^{\\prime })$ at levels $j<l(i)$ where it has not yet been merged into a valley of higher order.",
"[Proof:] Let us first note that, under the given assumptions, $E(z^{*}(s,m))\\le E(z^{*}(s,m^{\\prime }))<E(z^{*}(s,n))$ for all $n\\in M^{(j)}\\backslash \\lbrace m^{\\prime }\\rbrace $ , whence $E(z^{*}(m,n))\\le E(z^{*}(s,m))\\vee E(z^{*}(s,n))=E(z^{*}(s,n))\\le E(z^{*}(s,m))\\vee E(z^{*}(m,n))$ entails $E(z^{*}(m,n))=E(z^{*}(s,n))$ for all such $n$ .",
"Using this fact, we find that $E(z^{*}(m,m^{\\prime }))&\\le E(z^{*}(s,m))\\vee E(z^{*}(s,m^{\\prime }))<E(z^{*}(s,n))=E(z^{*}(m,n))$ for all $n\\in M^{(j)}\\backslash \\lbrace m^{\\prime }\\rbrace $ , which implies $m\\leadsto m^{\\prime }$ at level $j$ and thus $l(i)\\le j$ as well as the other assertions.",
"Proposition 1.9 For every $m\\in M^{(i)}$ and $1\\le i \\le \\mathfrak {n}$ , $V_{<}^{(i)}(m)$ is connected.",
"[Proof:] Pick any $s\\in V^{(i)}_<(m)$ , any minimal path from $s$ to $m$ and finally any intermediate state $r$ along this path for which $r\\in V^{(i)}_<(m)$ must be verified.",
"For every $m^{\\prime }\\in M^{(i)}\\backslash \\lbrace m\\rbrace $ , we find $E(z^*(r,m))&\\le E(z^*(r,s))\\vee E(z^*(s,m))\\\\&=E(z^*(s,m))\\\\&<E(z^*(s,m^{\\prime }))\\\\&\\le \\underbrace{E(z^*(s,r))}_{<E(z^*(s,m^{\\prime }))}\\vee E(z^*(r,m^{\\prime }))\\\\&=E(z^*(r,m^{\\prime })),$ which shows $r\\in V^{(i)}_<(m)$ as required.",
"Note that we have even shown that a minimal path from a state in $V^{(i)}_<(m)$ to $m$ will never leave this set.",
"We may expect and will indeed show as Proposition REF below that $V^{(i)}(m)$ is connected as well.",
"The following lemma is needed for its proof.",
"Lemma 1.10 Given $1\\le i\\le \\mathfrak {n},\\,m\\in M^{(i)}$ and $s\\leadsto m$ at level $i$ , let $\\gamma =(\\gamma _{1}{,}...,\\gamma _{k})\\in \\Gamma ^{*}(s,m)$ be a path such that $E(z^*(\\gamma _i,m))\\le E(z^*(\\gamma _i,n))$ for all $n\\in M^{(i)}\\backslash \\lbrace m\\rbrace $ , and which stays in $V_{<}^{(i)}(m)$ once hitting this set (such a $\\gamma $ exists by Remark REF (a)).",
"Then $\\gamma _{j}\\leadsto m$ at level $i$ for each $j=1{,}...,k$ .",
"[Proof:] There is nothing to prove for $\\gamma _{1}=s$ and any $\\gamma _{j}\\in V_{<}^{(i)}(m)$ .",
"So let $r$ be any other state visited by $\\gamma $ , pick an arbitrary $n\\in M^{(i)}\\backslash \\lbrace m\\rbrace $ with $E(z^*(r,n))=E(z^*(r,m))$ and then any minimal path $\\tau $ from $r$ to $n$ .",
"Let $\\sigma $ be the subpath of $\\gamma $ from $s$ to $r$ .",
"We must show that $\\tau $ hits $V^{(i)}_<(m)$ .",
"First, we point out that the maximal energy $E(z^*(s,r))\\vee E(z^*(r,n))$ of $\\sigma \\tau $ , the concatenation of $\\sigma $ and $\\tau $ , satisfies $E(z^*(s,n))\\le E(z^*(s,r))\\vee E(z^*(r,n))\\le E(z^*(s,m))\\vee E(z^*(r,m))=E(z^*(s,m))\\le E(z^*(s,n)),$ implying $\\sigma \\tau \\in \\Gamma ^{*}(s,n)$ and, furthermore, $E(z^*(s,r))\\vee E(z^*(r,n))=E(z^*(s,m))\\vee E(z^*(r,n))=E(z^*(s,m))\\vee E(z^*(r,m))=E(z^*(s,m)).$ Thus $\\sigma \\tau $ must hit $V^{(i)}_<(m)$ .",
"But since $\\sigma $ does not hit $V_{<}^{(i)}(m)$ by assumption, we conclude that $\\tau $ must hit $V_{<}^{(i)}(m)$ .",
"Since $\\tau \\in \\Gamma ^{*}(r,n)$ was arbitrary, we infer $r\\leadsto m$ at level $i$ .",
"The next two propositions provide information on the shape of the valleys and their nested structure.",
"Proposition 1.11 For every $m\\in M^{(i)}$ and $1\\le i \\le \\mathfrak {n}$ , $V^{(i)}(m)$ is connected.",
"[Proof:] We use an inductive argument.",
"If $i=1$ , the assertion follows directly from the definition of the level-one valleys because any $s\\in V^{(1)}(m)$ , $m\\in M^{(1)}$ , may be connected to $m$ by a minimal path that eventually enters $V_{<}^{(1)}(m)$ without hitting any other $V_{<}^{(1)}(n)$ and is therefore completely contained in $V^{(1)}(m)$ by the previous lemma.",
"Turning to the inductive step, suppose the assertion holds true up to level $i-1$ .",
"Fix any $m\\in M^{(i)}$ and notice that, by the inductive hypothesis, $V^{(i-1)}(m)$ as well as all $V^{(j)}(m^{(j)})$ with $l(j)=i$ and $m^{(j)}\\leadsto m$ at level $i$ are connected.",
"Now, since these $m^{(j)}$ as well as all $s\\in N^{(i-1)}$ attracted by $m$ at level $i$ may be connected to $m$ by minimal paths as assumed in Lemma REF , we conclude that $V^{(i)}(m)$ is also connected.",
"The second proposition shows the nested structure of our construction of valleys.",
"Proposition 1.12 The following inclusions hold true: $V^{(1)}(m)\\subseteq ...\\subseteq V^{(i)}(m)$ for each $m\\in M^{(i)},\\,1\\le i\\le \\mathfrak {n}$ .",
"$V^{(i)}(m)\\subseteq V^{(j)}(n)$ for each $1\\le i<j\\le \\mathfrak {n}$ , $n\\in M^{(j)}$ and $m\\in M^{(i)}\\cap V^{(j)}(n)$ .",
"[Proof:] Since there is nothing to show for (a) we move directly to (b).",
"But if $m\\in M^{(i)}\\cap V^{(j)}(n)$ , then the definition of valleys ensures the existence of $1\\le k\\le j-i$ and of $n_{1}{,}...,n_{k-1}\\in M^{(j)}\\backslash M^{(i)}$ such that $n_{p-1}\\leadsto n_{p}$ at level $l_{p}$ for each $p=1{,}...,k$ and levels $i<l_{1}<...<l_{k}=j$ , where $n_{0}:=m$ and $n_{k}:=n$ .",
"As a consequence, $V^{(i)}(m)\\subseteq V^{(l_{1})}(n_{1})\\subseteq ...\\subseteq V^{(l_{k-1})}(n_{k-1})\\subseteq V^{(j)}(n)$ which proves the asserted inclusion.",
"To finish the analysis of the shape of the valleys we show that they have the following important property: a special class of minimal paths from the inside of any $V^{(i)}(m)$ to the outside of it must hit its interior $V^{(i)}_<(m)$ .",
"But in order to show this we must first verify that all states attracted by $m$ at level $i$ belong to $V^{(i)}(m)$ .",
"Lemma 1.13 For each $1\\le i\\le \\mathfrak {n}$ and $m\\in M^{(i)}$ , we have that $\\left\\lbrace s\\in \\mathcal {S}\\Big | s\\leadsto m \\textrm { at level i}\\right\\rbrace \\subset V^{(i)}(m)\\subset \\left\\lbrace s\\in \\mathcal {S}\\Big |E(z^{*}(s,m))\\le E(z^{*}(s,m^{\\prime }))\\text{ for all }m^{\\prime }\\in M^{(i)}\\right\\rbrace .$ [Proof:] For the second inclusion it suffices to refer to Proposition REF (c).",
"The first inclusion being obviously true for $s\\in N^{(i-1)}$ , we turn directly to the case when $s\\leadsto n_1 \\textrm { at level } l_1, \\quad n_1\\leadsto n_2 \\textrm { at level }l_2, \\quad ... \\quad n_{k-1}\\leadsto n_k \\textrm { at level }l_k$ with $k\\ge 1$ and $1\\le l_1\\le ... \\le l_k\\le i-1$ .",
"Here, $n_1$ denotes the first minimum to which $s$ is attracted (thus $s\\in V^{(l_{1})}(n_{1})$ ), while $n_k$ is the last minimum of this kind in the sequence.",
"We may assume without loss of generality that $n_j\\ne m$ for all $j$ , for otherwise the assertion is clear.",
"We show now that $n_1\\leadsto m$ at level $i$ which in turn implies $n_j\\leadsto m$ at level $i$ for all $1\\le j\\le k$ .",
"As a consequence, $n_k\\in V^{(i)}(m)$ and thus $s\\in V^{(i)}(m)$ .",
"If $E(z^*(n_1,m))<E(z^*(n_1,m^{\\prime }))$ for all $m^{\\prime }\\in M^{(i)}\\backslash \\lbrace m\\rbrace $ , the assertion is proved.",
"Hence suppose $E(z^*(n_1,m))\\ge E(z^*(n_1,m^{\\prime }))$ for some $m^{\\prime }\\in M^{(i)}\\backslash \\lbrace m\\rbrace $ .",
"Then $E(z^*(s,m^{\\prime }))&\\le E(z^*(s,n_1))\\vee E(z^*(n_1,m^{\\prime }))\\\\&\\le E(z^*(s,n_1))\\vee E(z^*(n_1,m))\\\\&\\le E(z^*(s,n_1))\\vee E(z^*(s,m))\\\\&=E(z^*(s,m))\\\\&\\le E(z^*(s,m^{\\prime })),$ implies $E(z^*(s,m))=E(z^*(s,m^{\\prime }))$ and also that the concatenation of any minimal path $\\gamma $ from $s$ to $n_1$ and any minimal path $\\tau $ from $n_1$ to $m^{\\prime }$ (with maximal energy $E(z^*(s,n_1))\\vee E(z^*(n_1,m^{\\prime }))$ ) constitutes a minimal path from $s$ to $m^{\\prime }$ and must therefore hit $V^{(i)}_<(m)$ .",
"Note that we can choose $\\gamma $ to stay in $V^{(l_{1})}(n_1)$ since $s\\in V^{(l_{1})}(n_{1})$ and $V^{(l_{1})}(n_{1})$ is connected.",
"Now, if $\\tau $ hits $V^{(i)}_<(m)$ , then $E(z^*(n_1,m))=E(z^*(n_1,m^{\\prime }))$ and we are done.",
"Otherwise, $\\gamma $ hits $V^{(i)}_<(m)$ implying $V^{(j)}(n_1)\\cap V^{(i)}_<(m)\\ne \\emptyset $ .",
"Now use Lemma REF to conclude $n_1\\in V^{(i)}_<(m)$ and therefore $n_1\\leadsto m$ at level $i$ .",
"This completes the argument for the first inclusion.",
"We provide too further lemmata that will be needed later on.",
"Lemma 1.14 Let $m\\in M^{(i)}, x\\leadsto m$ at level $i$ and $y\\notin V^{(i)}(m)$ .",
"Then either every minimal path from $x$ to $y$ hits the set $V^{(i)}_<(m)$ , or $E(z^*(x,y))>E(z^*(x,m))$ .",
"[Proof:] Suppose there is a minimal path $\\gamma $ from $x$ to $y$ avoiding $V^{(i)}_<(m)$ .",
"Since $y\\notin V^{(i)}(m)$ , it is not attracted by $m$ at level $i$ implying the existence of some $m^{\\prime }\\in M^{(i)}$ with $E(z^*(y,m^{\\prime }))\\le E(z^*(y,m))$ and of some $\\tau \\in \\Gamma ^*(y,m^{\\prime })$ avoiding $V^{(i)}_<(m)$ .",
"Hence, the concatenation $\\gamma \\tau $ avoids $V^{(i)}_<(m)$ and must therefore have maximal energy larger than $E(z^*(x,m))$ .",
"Consequently, $E(z^*(x,m))&<E(z^*(x,y))\\vee E(z^*(y,m^{\\prime }))\\\\&\\le E(z^*(x,y))\\vee E(z^*(y,m))\\\\&\\le E(z^*(x,y))\\vee E(z^*(x,m)),$ and thus $E(z^*(x,y))>E(z^*(x,m))$ .",
"In order to state the second lemma, let us define the outer part $\\partial ^{+}V$ of a valley $V$ to be the set of those states outside $V$ which are adjacent to a state in $V$ .",
"With the help of the previous result, we can easily show that $\\partial ^{+}V$ contains only non-assigned states at any level where $V$ has not yet been merged into a larger valley.",
"Lemma 1.15 For any $1\\le i, j\\le \\mathfrak {n}$ and $m=m^{(i)}$ with $l(i)> j$ , the outer part $\\partial ^{+}V$ of the valley $V:=V^{(j\\wedge i)}(m)$ is a subset of $N^{(j)}$ and $E(z^*(s,m))=E(s)$ for every $s\\in \\partial ^+V$ .",
"[Proof:] First, let $s\\in \\partial ^{+}V$ and suppose that $s\\notin N^{(j)}$ .",
"Then $s\\leadsto m^{\\prime }$ at level $k$ , in particular $s\\in V^{(k)}(m^{\\prime })$ for some $m^{\\prime }\\in M^{(k)}$ and $k\\le j$ .",
"Pick any $r\\in V$ with $r\\sim s$ and note that $r\\in \\partial ^{+}V^{(k)}(m^{\\prime })$ .",
"W.l.o.g.",
"we may assume that $r\\leadsto m$ at level $j\\wedge i$ .",
"Then Lemma REF (with $x=r$ and $y=s$ ) ensures that either $E(z^{*}(r,s))>E(z^{*}(r,m)\\ge E(r)$ , thus $z^{*}(r,s)=s$ and $E(r)<E(s)$ , or $r\\in V_<(m)$ and, for some $n\\in M^{(j\\wedge i)}$ , $\\begin{split}E(z^*(r,m))&<E(z^*(r,n))\\\\&\\le E(z^*(r,s))\\vee E(z^*(s,n))\\\\&\\le E(z^*(r,s))\\vee E(z^*(s,m))\\\\&\\le E(z^*(r,s))\\vee E(z^*(r,m))\\\\&=E(s)\\vee E(z^*(r,m)),\\end{split}$ and thus again $E(r)<E(s)$ .",
"On the other hand, by the very the same lemma (now with $x=s$ and $y=r$ ), we infer $E(r)>E(s)$ which is clearly impossible.",
"Consequently, $s$ must be non-assigned at level $j$ as claimed.",
"For the second assertion take again $s\\in \\partial ^+V$ and a minimal path $\\gamma =(s,r{,}...,m)$ from $s$ to $m$ with $r\\in V$ .",
"Again, by use of Lemma REF , we find either $E(z^*(r,m))<E(z^*(r,s))=E(s)$ or $r\\in V_<(m)$ , leading analogously to equation (REF ) to $E(z^*(r,m))< E(s)\\vee E(z^*(r,m))$ and thus $E(z^*(r,m))<E(s)$ .",
"In conclusion, both cases result in $E(z^*(s,m))=E(s)\\vee E(z^*(r,m))=E(s),$ finishing the proof.",
"The reader may wonder why valleys are defined here via essential saddles and not via the at first glance more natural overall energy barriers, viz.",
"$I(s,m):=\\inf _{\\gamma \\in \\Gamma (s,m)}I(\\gamma _1{,}...,\\gamma _{|\\gamma |})$ with $I(s_1{,}...,s_n):=\\sum _{i=1}^{n}(E(s_i)-E(s_{i-1}))^+$ for a state $s$ in a valley and the pertinent minimum $m$ .",
"This latter quantity, also called activation energy, is indeed an important parameter in [21].",
"The reason for our definition becomes clear when regarding the last proposition which shows the nested structure of valleys of increasing order and which may fail to hold when choosing an alternative definition based on the activation energy.",
"Valleys that are formed in one step may then be ripped apart in the next one.",
"This happens, for instance, if there is just one large saddle along the path to a metastable state and several small ones, lower than the essential saddle, along the paths to another minimum such that their total sum is larger than the big saddle.",
"In further support of our approach, it will be seen later that the essential saddles are the critical parameters for the behavior of the aggregated chain (see Theorem REF ).",
"In a nutshell, by going from $(V^{(i)}(m))_{m\\in M^{(i)}}$ to $(V^{(i+1)}(m))_{m\\in M^{(i+1)}}$ , some valleys are merged into one (with only the smaller minima retained as metastable states) and additionally those states from $N^{(i)}$ are added which at level $i$ were attracted by metastable states now all belonging to the same valley.",
"This induces the following tree-structure on the state space: Fix $\\varnothing =s_0$ .",
"The first generation of the tree consists of all $m\\in M^{(\\mathfrak {n}-1)}\\cup N^{(\\mathfrak {n}-1)}$ and are thus connected to the root.",
"The second generation of the tree consists of all $m\\in M^{(\\mathfrak {n}-2)}\\cup N^{(\\mathfrak {n}-2)}$ , and $m$ is connected to the node $k$ of the first generation for which $E(z^*(m,k))$ is minimal or to itself (in the obvious sense).",
"This continues until in the $\\mathfrak {n}^{\\textrm {th}}$ generation each state is listed and connected either with its unique point of attraction in the previous generation or with itself.",
"Example 1.16 For the energy function of Example REF and depicted in Figure REF , the described tree is shown in Figure REF .",
"Figure: The tree belonging to Figure The sets of non-assigned states at the different levels are $N^{(1)}&=\\lbrace 3,5,7,9,11,13\\rbrace , \\ &N^{(2)}&=\\lbrace 3,5,7,11,13\\rbrace , &\\\\N^{(3)}&=\\lbrace 3,5,7,11\\rbrace , &N^{(4)}&=\\lbrace 3,7,11\\rbrace , &\\\\N^{(5)}&=\\lbrace 7,11\\rbrace , \\ &N^{(6)}&=\\lbrace 11\\rbrace , \\\\N^{(7)}&=\\emptyset .",
"&&$ At each level $i$ of such a tree the subtree rooted at a node $m\\in M^{(i)}$ consists of the states in the valley $V^{(i)}(m)$ .",
"A similar graph-theoretical modeling in order to visualize high dimensional energy landscapes has been used, for example, by Okushima et al.",
"in [15].",
"These authors work with saddles of paths as well.",
"In contrast to our approach, every possible path, that is every possible saddle is represented as a node in the tree.",
"But as we will see, in the limit of low temperatures $(\\beta \\rightarrow \\infty )$ the essential saddle is all we need.",
"Now there are two fundamental directions for further investigations: (1) Microscopic View: What happens while the process visits a fixed valley $V$ (of arbitrary level)?",
"In Section , we will show that during each visit of a valley $V$ its minimum will be reached with probability tending to 1 as $\\beta \\rightarrow \\infty $ , and we also calculate the expected residence time in $V$ , establish Property 3 stated in the Introduction and comment briefly on quasi-stationarity.",
"(2) Macroscopic View: How does the process jump between the valleys?",
"In Section , by drawing on the results of Section , we will show that an appropriate aggregated chain is Markovian in the limit as $\\beta \\rightarrow \\infty $ and calculate its transition probabilities.",
"With this we will finally be able to provide the definition of MB and establish Properties 1, 2 and 4 listed in the Introduction."
],
[
"Microscopic View: Fixing a Valley",
"Based on the provided definition of valleys of different orders, we are now going to study the process when moving in a fixed valley."
],
[
"Trajectories for $\\beta \\rightarrow \\infty $",
"The first goal in our study of the microscopic process and also the basic result for the subsequent analysis of the macroscopic process deals with the probabilities of reaching certain states earlier than others.",
"From this we will conclude that in the limit $\\beta \\rightarrow \\infty $ (which is the low temperature limit in the Metropolis Algorithm) the process, when starting somewhere in a valley, will visit its minimum before leaving it.",
"For $A\\subset \\mathcal {S}$ and $x\\in \\mathcal {S}$ , we define $\\tau _A:=\\inf \\lbrace n\\ge 1|X_n\\in A\\rbrace ,\\quad \\tau _x:=\\tau _{\\lbrace x\\rbrace }\\quad \\text{and}\\quad \\mathcal {N}(x):=\\lbrace y\\in \\mathcal {S}|p(x,y)>0\\rbrace .$ Two states $x,y$ with $p(x,y)>0$ are called neighbors ($x\\sim y$ ) and $\\mathcal {N}(x)$ the neighborhood of $x$ .",
"Hence, $\\mathcal {S}$ may (and will) be viewed as a graph hereafter with edge set $\\lbrace (x,y)|x\\sim y\\rbrace $ .",
"Given any subgraph $\\Delta $ , we will write $\\widetilde{\\mathbf {P}}$ for the transition matrix of the chain restricted to $\\Delta $ ($\\tilde{p}(r,s)=p(r,s)$ for all distinct $r,s\\in \\Delta $ , $\\tilde{p}(r,r)=1-\\sum _{s\\in \\Delta , s\\ne r}\\tilde{p}(r,s)$ ) and $\\widetilde{\\mathbb {P}}_{x}$ for probabilities when regarding this restricted chain starting at $x\\in \\Delta $ .",
"Theorem 2.1 Let $x, y, z\\in \\mathcal {S}$ be any pairwise distinct states satisfying $E(z^*(x,z))>E(z^*(x,y))$ and $z^*(x,z)\\ne x$ .",
"Then there exist nonnegative constants $K(\\beta )$ satisfying $\\sup _{\\beta >0}K(\\beta )<\\infty $ (and given more explicitly in Proposition REF below) such that $\\mathbb {P}_{x}(\\tau _{z}<\\tau _{y})\\le K(\\beta )\\,e^{-\\beta (E(z^*(x,z))-E(z^*(x,y))-7\\gamma _{\\beta })}=:\\varepsilon (x,y,z,\\beta ) \\overset{\\beta \\rightarrow \\infty }{\\longrightarrow } 0.$ Thus in the limit of low temperatures $(\\beta \\rightarrow \\infty )$ , only the smallest of all possible energy barriers affects the speed of a transition.",
"In particular, we have the following result which is preliminary to the subsequent one.",
"Theorem 2.2 Given distinct $x,y\\in \\mathcal {S}$ and $m\\in M^{(i)}$ such that $x\\leadsto m$ at level $i$ and $y\\notin V^{(i)}(m)$ , let $B:=\\lbrace z\\in \\mathcal {S}|E(z^*(x,z))>E(z^*(x,m))\\rbrace $ .",
"Then it holds true that $\\mathbb {P}_{x}&(\\tau _{y}<\\tau _{m})\\\\&\\le \\ \\varepsilon (x,m,y,\\beta )\\,{1}_{B}(y)+\\left(\\sum _{z:E(z)>E(z^*(x,y))}\\varepsilon (x,y,z,\\beta )+\\sum _{z\\in V^{(i)}_<(m)}\\varepsilon (z,m,y,\\beta )\\right)\\,{1}_{B^{c}}(y)\\\\&=:\\ \\tilde{\\varepsilon }(x,m,y,\\beta )\\ \\overset{\\beta \\rightarrow \\infty }{\\longrightarrow }\\ 0.$ Theorem 2.3 Given $m\\in M^{(i)}$ , $x\\in V^{(i)}(m)$ and $y\\notin V^{(i)}(m)$ , let $k\\le i$ be such that $&m_{0}:=x\\leadsto m_{1}\\text{ at level }l_{1},\\quad m_{1}\\leadsto m_{2}\\text{ at level }l_{2},\\quad ...\\quad m_{k-1}\\leadsto m_{k}=m\\text{ at level }l_{k}$ for suitable $1\\le l_{1}<...<l_{k}\\le i$ , $m_{j}\\in M^{(l_{j})}$ for $j=1{,}...,k$ determined by the construction in Definition REF .",
"Then $\\mathbb {P}_{x}(\\tau _{y}<\\tau _{m})\\ &\\le \\ \\sum _{j=1}^{k}\\mathbb {P}_{m_{j-1}}(\\tau _{y}<\\tau _{m_{j}})\\le \\ \\sum _{j=1}^{k}\\tilde{\\varepsilon }(m_{j-1},m_{j},y,\\beta )\\ \\overset{\\beta \\rightarrow \\infty }{\\longrightarrow }\\ 0.$ For the valleys as defined here this confirms Property 1 stated in the Introduction: If $\\beta $ is sufficiently large, then with high probability the minimum of a valley is visited before this valley is left.",
"All three theorems are proved at end of the next subsection after a number of auxiliary results."
],
[
"Auxiliary results and proofs",
"The proof of Theorem REF will be accomplished by a combination of two propositions due to Bovier et al.",
"[5] for a more special situation.",
"We proceed with a reformulation of the first one in a weaker form and under weaker assumptions.",
"Proposition 2.4 (compare Theorem 1.8 in [5]) Let $x, y, z\\in \\mathcal {S}$ be pairwise distinct such that $z^*(x,z)\\ne x$ .",
"Then $\\mathbb {P}_x(\\tau _z<\\tau _x)&\\le K(\\beta )\\,|\\mathcal {S}|^{-1}\\,e^{-\\beta (E(z^*(x,z))-E(x)-2\\gamma _{\\beta })},\\\\\\mathbb {P}_x(\\tau _y<\\tau _x)&\\ge |\\mathcal {S}|^{-1}\\,e^{-\\beta (E(z^*(x,y))-E(x)+5\\gamma _{\\beta })},$ where $K(\\beta ):=|\\mathcal {S}|\\max _{r\\in \\mathcal {S}}|\\mathcal {N}(r)|\\left(|\\mathcal {S}|e^{-\\beta (\\min _{a\\ne b:E(a)>E(b)}(E(a)-E(b))-2\\gamma _{\\beta })}+1\\right)$ .",
"The proof requires several lemmata, the first of which may already be found in [14] and is stated here in the notation of [5].",
"Lemma 2.5 (see Theorem 2.1 in [5]) Defining $\\mathcal {H}^x_z:=\\lbrace h:\\mathcal {S}\\rightarrow [0,1]\\,|\\, h(x)=0,h(z)=1\\rbrace $ and the Dirichlet form $\\mathcal {E}(h):=\\sum _{r\\sim s\\in \\mathcal {S}}\\pi (r)p(r,s) (h(r)-h(s))^2,$ we have $\\mathbb {P}_x(\\tau _z<\\tau _x)=\\frac{1}{2\\pi (x)}\\inf _{h\\in \\mathcal {H}^x_z}\\mathcal {E}(h).$ Lemma 2.6 (see Lemma 2.2 in [5]) For any subgraph $\\Delta \\subset \\mathcal {S}$ containing $x,y$ and corresponding transition matrix $\\widetilde{\\mathbf {P}}$ , we have $\\mathbb {P}_x(\\tau _y<\\tau _x)\\ge \\widetilde{\\mathbb {P}}_x(\\tau _y<\\tau _x).$ Lemma 2.7 (see Lemma 2.5 in [5]) If $\\Delta =(\\omega _0{,}...,\\omega _k)$ is any one-dimensional subgraph of $\\mathcal {S}$ , then $\\widetilde{\\mathbb {P}}_{\\omega _0}(\\tau _{\\omega _k}<\\tau _{\\omega _0}) =\\left(\\sum _{i=1}^k\\frac{\\pi (\\omega _0)}{\\pi (\\omega _i)}\\frac{1}{p(\\omega _i,\\omega _{i-1})}\\right)^{-1}.$ [Proof:](of Proposition REF ) In view of Lemma REF , we must find an appropriate function $h$ for the upper bound.",
"Let us define $\\mathcal {R}:=\\lbrace s\\in \\mathcal {S}\\,|\\,\\exists ~ \\gamma \\in \\Gamma (x,z):\\mathop {\\mathrm {argmax}}_{i=1{,}... , |\\gamma |}E(\\gamma _i)=s\\rbrace ,$ the set of all peak states (with respect to the energy function) along self-avoiding paths from $x$ to $z$ .",
"Obviously $z^*(x,z)\\in \\mathcal {R}$ and, by non-degeneracy, $E(z^*(x,z))<E(s)$ for all $s\\in \\mathcal {R}\\backslash \\lbrace z^*(x,z)\\rbrace $ .",
"The set $\\mathcal {R}$ divides $\\mathcal {S}$ into a set $\\mathcal {R}_x$ containing $x$ (and consisting of those $y$ that can be reached from $x$ without hitting $\\mathcal {R}$ ) and a set $\\mathcal {R}_z$ containing $z$ .",
"Now choose $h(r):={1}_{\\mathcal {R}_z\\cup \\mathcal {R}}(r), ~r\\in \\mathcal {S}.$ Note that only neighboring states contribute to the Dirichlet form $\\mathcal {E}(h)$ and that, for any two such $r\\sim s$ , the term $(h(r)-h(s))^2$ is positive iff one of these states is in $\\mathcal {R}$ and the other one in $\\mathcal {R}_x$ , in which case the squared difference equals 1.",
"By invoking Lemma REF , we obtain $\\mathbb {P}_x(\\tau _z<\\tau _x)\\ &=\\ \\frac{1}{2\\pi (x)}\\inf _{h\\in \\mathcal {H}^x_z}\\mathcal {E}(h)\\\\&\\le \\ \\frac{\\pi (z^*(x,z))}{\\pi (x)}\\left(\\sum _{\\mathcal {R}\\ni r\\ne z^*(x,z)}\\,\\sum _{s\\sim r}\\frac{\\pi (r)}{\\pi (z^*(x,z))}\\,p(r,s)+|\\mathcal {N}(z^*(x,z))|\\right)\\\\&\\le \\ e^{-\\beta (E(z^*(x,z))-E(x)-2\\gamma _{\\beta })}\\\\&\\hspace*{28.45274pt}\\times \\left(\\sum _{\\mathcal {R}\\ni r\\ne z^*(x,z)}\\,\\sum _{s\\sim r}e^{-\\beta (\\min _{a\\ne b:E(a)>E(b)} (E(a)- E(b))-2\\gamma _{\\beta })}+|\\mathcal {N}(z^*(x,z))|\\right)\\\\&\\le \\ e^{-\\beta (E(z^*(x,z))-E(x)-2\\gamma _{\\beta })}\\\\&\\hspace*{28.45274pt}\\times \\,\\max _{r\\in \\mathcal {S}}|\\mathcal {N}(r)|\\,\\left(|\\mathcal {S}|e^{-\\beta (\\min _{a\\ne b:E(a)>E(b)}(E(a)- E(b))-2\\gamma _{\\beta })}+1\\right).$ Lemmata REF and REF will enter in the proof of the lower bound.",
"Consider the chain restricted to the one-dimensional subgraph given by a minimal path $\\rho =(s_{1}{,}...,s_{|\\rho |})$ from $x$ to $y$ .",
"Then $\\mathbb {P}_x(\\tau _y<\\tau _x)\\ &\\ge \\ \\left(\\sum _{i=1}^{|\\rho |}\\frac{\\pi (x)}{\\pi (s_i)}\\frac{1}{p(s_i,s_{i-1})}\\right)^{-1}\\\\&\\ge \\ \\frac{\\pi (z^*(x,y))}{\\pi (x)}\\left(\\sum _{i=1}^{|\\rho |}\\frac{\\pi (z^*(x,y))}{\\pi (s_i)}e^{\\beta ((E(s_{i-1})-E(s_i))^{+} +\\gamma _{\\beta })}\\right)^{-1}\\\\&\\ge \\ e^{-\\beta (E(z^*(x,y))-E(x) +2\\gamma _{\\beta })}\\left(\\sum _{i=1}^{|\\rho |}e^{-\\beta (E(z^*(x,y))-E(s_i)-(E(s_{i-1})-E(s_i))^{+}-3\\gamma _{\\beta })}\\right)^{-1}\\\\&\\ge \\ e^{-\\beta (E(z^*(x,y))-E(x)+5\\gamma _{\\beta })}\\,\\frac{1}{|\\mathcal {S}|}.$ This completes the proof of Proposition REF .",
"We proceed to the second proposition needed to prove Theorem REF .",
"Proposition 2.8 (see Corollary 1.6 in [5]) Given $I\\subset \\mathcal {S}$ and distinct $x,z\\in \\mathcal {S}\\backslash I$ , $\\mathbb {P}_x(\\tau _z<\\tau _I)=\\frac{\\mathbb {P}_x(\\tau _z<\\tau _{I\\cup \\lbrace x\\rbrace })}{\\mathbb {P}_x(\\tau _{I\\cup \\lbrace z\\rbrace }<\\tau _x)}$ holds true.",
"With the help of Propositions REF and REF , the proof of Theorem REF can now be given quite easily.",
"[Proof:](of Theorem REF ) By first using the previous result and then Proposition REF (with $K(\\beta )$ as defined there), we find $\\mathbb {P}_{x}(\\tau _{z}<\\tau _{y})\\ &=\\ \\frac{\\mathbb {P}_{x}(\\tau _{z}<\\tau _{\\lbrace x,y\\rbrace })}{\\mathbb {P}_{x}(\\tau _{\\lbrace z,y\\rbrace }<\\tau _{x})}\\\\&\\le \\ \\frac{\\mathbb {P}_{x}(\\tau _{z}<\\tau _{x})}{\\mathbb {P}_{x}(\\tau _{y}<\\tau _{x})}\\\\&\\le \\ K(\\beta )\\,e^{-\\beta (E(z^*(x,z))-E(z^*(x,y))-7\\gamma _{\\beta })}.$ The argument is completed by noting that $E(z^*(x,z))>E(z^*(x,y))$ and $K(\\beta )\\ge 0$ converges to $|\\mathcal {S}|\\max _{r\\in \\mathcal {S}}|\\mathcal {N}(r)|$ as $\\beta \\rightarrow \\infty $ .",
"[Proof:](of Theorem REF ) If $B$ occurs, the asserted bound follows directly from Theorem REF .",
"Proceeding to the case when $B^{c}$ occurs, i.e.",
"$E(z^*(x,y))\\le E(z^*(x,m))$ , we first point out that $\\mathbb {P}_{x}(\\tau _{y}<\\tau _{m})\\ &=\\ \\mathbb {P}_{x}(\\tau _{y}<\\tau _{m}, E(X_n)>E(z^*(x,y)) \\text{ for some }n\\le \\tau _{y})\\\\&\\qquad +\\ \\mathbb {P}_{x}(\\tau _{y}<\\tau _{m},E(X_n)\\le E(z^*(x,y))\\text{ for all }n\\le \\tau _{y})\\\\&=:\\ P_{1}+P_{2}.$ For all $z\\in \\mathcal {S}$ with $E(z)>E(z^*(x,y))$ , we have $z^*(x,z)\\ne x$ and $E(z^*(x,z))>E(z^*(x,y))$ , for $E(z^*(x,z))\\ge E(z)>E(z^*(x,y))\\ge E(x).$ Therefore, by an appeal to Theorem REF , $P_{1}\\ &\\le \\ \\mathbb {P}_{x}(\\tau _{z}<\\tau _{y}\\text{ for some $z$ with $E(z)>E(z^*(x,y))$})\\\\&\\le \\ \\sum _{z:E(z)>E(z^*(x,y))}\\mathbb {P}_{x}(\\tau _z<\\tau _{y})\\\\&\\le \\ \\sum _{z:E(z)>E(z^*(x,y))}\\varepsilon (x,y,z,\\beta ).$ To get an estimate for $P_{2}$ , note that every minimal path from $x$ to $y$ must pass through $V^{(i)}_<(m)$ (Lemma REF ).",
"With this observation and by another appeal to Theorem REF , we infer $\\mathbb {P}_{x}(\\tau _{y}<\\tau _{m},E(X_n)\\le E(z^*(x,y))\\text{ for all }n\\le \\tau _{y})\\ &\\le \\sum _{z\\in V^{(i)}_<(m)}\\mathbb {P}_z(\\tau _{y}<\\tau _ {m})\\\\&\\le \\ \\sum _{z\\in V^{(i)}_<(m)}\\varepsilon (z,m,y,\\beta ),$ having further utilized that (by Proposition REF (f) and (e)) $z^*(z,y)\\ne z$ and $E(z^*(z,m))&<E(z^*(z,m^{\\prime }))\\\\&\\le E(z^*(z,y))\\vee E(z^*(y,m^{\\prime }))\\\\&\\le E(z^*(z,y))\\vee E(z^*(y,m))\\\\&\\le E(z^*(z,y))\\vee E(z^*(z,m))\\\\&=E(z^*(z,y))$ for some $m^{\\prime }\\in M^{(i)}\\backslash \\lbrace m\\rbrace $ with $E(z^*(y,m))\\ge E(z^*(y,m^{\\prime }))$ , which must exist since $y\\notin V^{(i)}(m)$ .",
"[Proof:](of Theorem REF ) We first note that $y\\notin V^{(l_j)}(m_j)$ for all $1\\le j\\le k$ .",
"With $m_{0}{,}...,m_{k}$ as stated in the theorem (recall $m_{0}=x$ and $m_{k}=m$ ), we obtain $\\mathbb {P}_{x}(\\tau _{y}<\\tau _{m})&=\\mathbb {P}_{x}(\\tau _{m_{1}}<\\tau _{y}<\\tau _{m})+\\mathbb {P}_{x}(\\tau _{y}<\\tau _{m_1}\\wedge \\tau _{m})\\\\&\\le \\mathbb {P}_{m_{1}}(\\tau _{y}<\\tau _{m})+\\mathbb {P}_{x}(\\tau _{y}<\\tau _{m_{1}})\\\\&\\le \\mathbb {P}_{m_{2}}(\\tau _{y}<\\tau _{m})+\\mathbb {P}_{m_{1}}(\\tau _{y}<\\tau _{m_{2}})+\\mathbb {P}_{x}(\\tau _{y}<\\tau _{m_{1}})\\\\&\\hspace{28.45274pt}\\vdots \\\\&\\le \\sum _{j=1}^{k}\\mathbb {P}_{m_{j-1}}(\\tau _{y}<\\tau _{m_{j}}).$ Finally use Theorem REF to infer $\\mathbb {P}_{m_{j-1}}(\\tau _{y}<\\tau _{m_{j}})\\le \\tilde{\\varepsilon }(m_{j-1},m_{j},y,\\beta )$ for each $j=1{,}...,k$ .",
"Naturally, several other questions concerning the behavior of the process when moving in a fixed valley are of interest, and quasi-stationarity may appear as one to come up with first.",
"For a given valley $V$ (of any level), a quasi-stationary distribution $\\nu =(\\nu (j))_{j\\in V}$ is characterized by the quasi-invariance, viz.",
"$\\mathbb {P}_{\\nu }(X_{n}=j|\\tau _{\\mathcal {S}\\backslash V}>n)=\\nu (j)\\quad \\text{for all }j\\in V,$ but also satisfies $\\lim _{n\\rightarrow \\infty }\\,\\mathbb {P}_{\\mu }(X_{n}=j|\\tau _{\\mathcal {S}\\backslash V}>n)=\\nu (j)\\quad \\text{for all }j\\in V$ if $\\mu $ is an arbitrary distribution with $\\mu (V)=1$ .",
"The latter property renders uniqueness of $\\nu $ .",
"Since $\\mathcal {S}$ is finite, existence of $\\nu $ follows by an old result due to Darroch & Seneta [7].",
"It is obtained as the normalized eigenvector of the Perron-Frobenius eigenvalue $\\lambda =\\lambda (V)$ of a modification of $\\mathbf {P}$ .",
"This eigenvalue $\\lambda $ is also the probability for the chain to stay in $V$ at least one step when started with $\\nu $ , thus $\\mathbb {P}_{\\nu }(\\tau _{V^{c}}>1)=\\lambda $ .",
"As an immediate consequence, one finds that the exit time $\\tau _{V^{c}}$ has a geometric distribution with parameter $1-\\lambda $ under $\\mathbb {P}_{\\nu }$ .",
"In the present context, this naturally raises the question how the parameter $\\lambda $ relates to the transition probabilities or the energies of the valley $V$ .",
"A simple probabilistic argument shows the following basic and intuitively obvious result concerning the eigenvalues associated with the nesting $V^{(1)}(m)\\subset ...\\subset V^{(i)}(m)$ (Proposition REF ) for any $1\\le i\\le \\mathfrak {n}$ and $m\\in M^{(i)}$ .",
"Proposition 2.9 Fixing any $1\\le i\\le \\mathfrak {n}$ and $m\\in M^{(i)}$ , let $\\lambda ^{(j)}:=\\lambda (V^{(j)}(m))$ for $j=1{,}...,i$ .",
"Then $\\lambda ^{(1)}\\le ...\\le \\lambda ^{(i)}$ .",
"[Proof:] Write $\\nu _{j}$ as shorthand for the quasi-stationary distribution on $V^{(j)}(m)$ and $T_{j}$ for $\\tau _{\\mathcal {S}\\backslash V^{(j)}(m)}$ .",
"Plainly, $T_{j}\\le T_{j+1}$ $(\\lambda ^{(j)})^{n}\\ =\\ \\mathbb {P}_{\\nu _{j}}(T_{j}>n)\\ &\\le \\ \\mathbb {P}_{\\nu _{j}}(T_{j+1}>n)\\nonumber \\\\&=\\ \\int _{\\lbrace T_{j+1}>k\\rbrace }\\mathbb {P}_{X_{k}}(T_{j+1}>n-k)\\ d\\mathbb {P}_{\\nu _{j}}\\nonumber \\\\&=\\ \\mathbb {P}_{\\nu _{j}}(T_{j+1}>k)\\,\\mathbb {P}_{\\mu _{k}}(T_{j+1}>n-k),$ where $\\mu _{k}(x):=\\mathbb {P}_{\\nu _{j}}(X_{k}=x|T_{j+1}>k)$ for $x\\in V^{(j+1)}$ .",
"Since $\\mathcal {S}$ is finite and by virtue of (REF ), we have that $\\mu _{k}\\le 2\\nu _{j+1}$ when choosing $k$ sufficiently large.",
"For any such $k$ , we find that (REF ) has upper bound $2\\,\\mathbb {P}_{\\nu _{j}}(T_{j+1}>k)\\,\\mathbb {P}_{\\nu _{j+1}}(T_{j+1}>n-k)\\ =\\ 2\\,\\mathbb {P}_{\\nu _{j}}(T_{j+1}>k)\\,(\\lambda ^{(j+1)})^{n-k}.$ Hence, we finally conclude $\\lambda ^{(j)}\\ \\le \\ \\Big (2\\,\\mathbb {P}_{\\nu _{j}}(T_{j+1}>k)\\,(\\lambda ^{(j+1)})^{-k}\\Big )^{1/n}\\,\\lambda ^{(j+1)}$ and thereby the assertion upon letting $n\\rightarrow \\infty $ .",
"An alternative matrix-analytic proof draws on an old result by Frobenius [9], here cited from [10].",
"Lemma 2.10 If $A=(a_{ij})$ and $C=(c_{ij})$ denote two real $k\\times k$ -matrices such that $A$ is nonnegative and irreducible with maximal eigenvalue $\\lambda _A^*$ and $|c_{ij}|\\le a_{ij}$ for all $1\\le i,j\\le k$ , then $|\\lambda |\\le \\lambda _A^*$ for all eigenvalues $\\lambda $ of $C$ .",
"Second proof of Proposition REF : For any fixed valley $V$ , collaps all states $s\\notin V$ into an absorbing state (grave) $\\Delta $ which leaves transition probabilities between states in $V$ unchanged.",
"A proper rearrangement of states allows us to assume that the new transition matrix has the form $\\mathbf {P}=\\begin{pmatrix} 1 & {\\bf 0} \\\\ \\mathbf {p} & \\mathbf {Q}\\end{pmatrix}$ for a $|V|\\times 1$ -column vector $\\mathbf {p}\\ne 0$ and a nonnegative, substochastic and irreducible $|V|\\times |V|$ -matrix $\\mathbf {Q}$ .",
"Now, for any $2\\le j\\le i$ , let $A$ be this matrix $\\mathbf {Q}$ when $V=V^{(j)}(m)$ , and $D$ be this matrix when $V=V^{(j-1)}(m)$ .",
"Then, obviously, $A:=\\begin{pmatrix} A_1 & A_2 \\\\ A_3 & D \\end{pmatrix}$ and $A$ is irreducible and nonnegative with maximal eigenvalue $\\lambda ^{(j)}$ .",
"Defining further $C:=\\begin{pmatrix} 0 & 0 \\\\ 0 & D \\end{pmatrix}.$ the largest eigenvalue of $C$ equals the largest eigenvalue of $D$ , thus $\\lambda ^{(j-1)}$ .",
"Finally, the desired conclusion follows from the previous lemma, since $|c_{ij}|=c_{ij}\\le a_{ij}$ for all $i\\le i,j\\le k$ .$\\square $ Another question is how long a given valley is visited and thus about its exit time.",
"There is an extensive literature on exit problems for different kinds of stochastic processes.",
"We mention [22] and [8] as two related to our work.",
"The latter one studies perturbed systems on a continuous space.",
"We can discretize their argument to get, with use of the main theorem in [21], a nice result on the time needed to leave a valley $V^{(i)}(m)$ for any fixed $1\\le i\\le \\mathfrak {n}$ and $m\\in M^{(i)}$ .",
"This result is more explicit than the one in [16].",
"Definition 2.11 For $1\\le i\\le \\mathfrak {n},\\, N:=N^{(i)}$ , we define the following stopping (entrance/exit) times: $\\xi _0^{(i)}&:=\\tau _{N^c}\\\\\\zeta _{n}^{(i)}&:=\\inf \\left\\lbrace k\\ge \\xi _n^{(i)}|X_k\\in N\\right\\rbrace \\\\\\xi _{n+1}^{(i)}&:=\\inf \\left\\lbrace k\\ge \\zeta _{n}^{(i)}|X_k\\in N^c\\right\\rbrace , ~n\\ge 0.$ The entrance times $\\xi ^{(i)}_n$ mark the epochs when a new valley is visited, while the exit times $\\zeta ^{(i)}_n$ are the epochs at which a valley is left.",
"The reader should notice that we do not restrict ourselves to valleys of order $i$ but include those valleys which up to order $i$ have not yet been absorbed by some larger valley.",
"Exit and entrance times never coincide since there is no way to go from one valley to another without hitting a non-assigned state - crests are always non-assigned (see Lemma REF ).",
"In this section, we will focus on $\\zeta _0^{(i)}$ for any fixed $i$ , thus writing $\\zeta _0:=\\zeta _0^{(i)}$ hereafter, but later for the macroscopic process the other times will be needed as well.",
"For each valley $V^{(i)}(m),\\,m\\in M^{(i)}$ , let us define $s_m\\ =\\ s_m^{(i)}\\ :=\\ \\mathop {\\mathrm {argmin}}_{s\\in \\partial ^+V^{(i)}(m)}E(s)\\ =\\ \\mathop {\\mathrm {argmin}}_{s\\in \\partial ^+V^{(i)}(m)}E(z^*(m,s)),$ where the second equality follows from Lemma REF .",
"Theorem 2.12 Let $m\\in M^{(i)}$ .",
"Then $\\lim _{\\beta \\rightarrow \\infty }\\frac{1}{\\beta }\\ln \\mathbb {E}_{r}\\zeta _0\\ =\\ E(s_m)-E(m)$ for any $r\\in V^{(i)}(m)$ .",
"For the upper bound, we need a result from [21], which in our notation is: Proposition 2.13 (see Main Theorem (iv) in [21]) For any $1\\le i\\le \\mathfrak {n},\\, \\beta $ sufficiently large and $t> 2^{i-1}\\exp (\\beta (E(s_{m^{(i-1)}})-E(m^{(i-1)})+2i|\\mathcal {S}|\\gamma _{\\beta }))$ , $\\sup _{x\\in M^{(i-1)}}\\mathbb {P}_x(\\tau _{M^{(i)}}>t)\\le \\exp (-\\Delta \\beta )$ holds true with a positive constant $\\Delta $ , where $M^{(0)}=\\mathcal {S}$ should be recalled.",
"This will now be used to show the following result.",
"Lemma 2.14 Fix $1\\le i\\le \\mathfrak {n},\\, m\\in M^{(i)}$ and $r\\in V^{(i)}(m)$ .",
"Then, for any $\\beta $ sufficiently large and $t>2^{i}\\exp (\\beta (E(s_{m})-E(m)+2(i+1)|\\mathcal {S}|\\gamma _{\\beta }))$ , it holds true that $\\mathbb {P}_r(\\zeta _0<(i+1)t)\\ \\ge \\ \\frac{1}{4}.$ [Proof:] Let us first note that we can always arrange for $m$ being equal to $m^{(i)}$ by sufficiently decreasing the energy function at any $m^{\\prime }\\in M^{(i)}\\backslash \\lbrace m\\rbrace $ so as to make $E(s_{m})-E(m)$ minimal among all states in $M^{(i)}$ .",
"This affects neither the valley $V^{(i)}(m)$ and its outer boundary nor the distribution of $\\zeta _0$ when starting in $m$ , for this distribution does not depend on the energy landscape outside of $V^{(i)}(m)\\cup \\partial ^+V^{(i)}(m)$ .",
"When applying the previous proposition, the constant $\\Delta $ may have changed but is still positive which suffices for our purposes.",
"So let $m=m^{(i)}$ hereafter.",
"Fix $t>2^i\\exp (\\beta (E(s_{m})-E(m)+2(i+1)|\\mathcal {S}|\\gamma _{\\beta }))$ and $T:=it$ .",
"Since $E(s_{m})-E(m)\\ge E(s_{m^{(j)}})-E(m^{(j)})$ for every $1\\le j\\le i$ , we infer $\\mathbb {P}_r(\\tau _{M^{(i)}}\\le T)\\ &\\ge \\ \\mathbb {P}_r(\\tau _{M^{(i)}}\\le T, \\tau _{M^{(1)}}\\le t)\\\\&\\ge \\ \\mathbb {P}_r(\\tau _{M^{(1)}}\\le t)\\,\\inf _{x\\in M^{(1)}}\\mathbb {P}_{x}(\\tau _{M^{(i)}}\\le (i-1)t)\\\\&\\ge \\ \\mathbb {P}_r(\\tau _{M^{(1)}}\\le t)\\,\\inf _{x\\in M^{(1)}}\\mathbb {P}_{x}(\\tau _{M^{(2)}}\\le t)\\,\\inf _{x\\in M^{(2)}}\\mathbb {P}_{x}(\\tau _{M^{(i)}}\\le (i-2)t)\\\\&\\hspace*{28.45274pt}\\vdots \\\\&\\ge \\ \\prod _{j=1}^{i}\\inf _{x\\in M^{(j-1)}}\\mathbb {P}_{x}(\\tau _{M^{(j)}}\\le t)\\\\&\\ge \\ \\big (1-\\exp (-\\Delta \\beta )\\big )^i\\\\&\\ge \\ \\frac{3}{4}$ for $\\beta $ sufficiently large.",
"Furthermore, for $\\beta $ so large that $\\mathbb {P}_r(\\tau _{M^{(i)}}<\\tau _m)\\le 1/4$ , we find that $\\mathbb {P}_r(\\tau _{M^{(i)}}\\le T)\\ &=\\ \\mathbb {P}_r(\\tau _{M^{(i)}}=\\tau _m\\le T)+\\mathbb {P}_r(\\tau _{M^{(i)}}\\le T, \\tau _{M^{(i)}}<\\tau _m)\\\\&\\le \\ \\mathbb {P}_r(\\tau _m\\le T)+\\mathbb {P}_r(\\tau _{M^{(i)}}<\\tau _m)\\\\&\\le \\ \\mathbb {P}_r(\\tau _m\\le T)+\\frac{1}{4}.$ By combining both estimates, we obtain $\\mathbb {P}_r(\\tau _m\\le T)\\ \\ge \\ \\mathbb {P}_r(\\tau _{M^{(i)}}\\le T)-\\frac{1}{4}\\ \\ge \\ \\frac{1}{2}.$ Hence, state $m$ is hit in time $T$ with at least probability $1/2$ when starting in $r$ .",
"Since $m=m^{(i)}$ , we further have $\\mathbb {P}_m(\\zeta _0\\le t)\\ \\ge \\ \\mathbb {P}_m(\\tau _{M^{(i+1)}}\\le t)\\ \\ge \\ 1-\\exp (-\\Delta \\beta )\\ \\ge \\ \\frac{1}{2}$ for $\\beta $ sufficiently large.",
"Hence, state $s_m$ is hit in time $t$ with at least probability $1/2$ when starting in $m$ .",
"By combining the estimates, we finally obtain $\\mathbb {P}_{r}(\\zeta _0\\le (i+1)t)\\ &\\ge \\ \\mathbb {P}_{r}(\\zeta _0\\le T+t|\\tau _{m}\\le T)\\,\\mathbb {P}_{r}(\\tau _{m}\\le T)\\\\&\\ge \\ \\mathbb {P}_{r}(\\tau _{m}\\le T)\\,\\mathbb {P}_{m}(\\zeta _0\\le t)\\\\&\\ge \\ \\frac{1}{4},$ which proves our claim.",
"[Proof:](of Theorem REF ) Using the lemma just shown, we infer $\\mathbb {E}_{r}(\\zeta _0)\\ &\\le \\ (i+1)t\\sum _{n\\ge 0}(n+1)\\,\\mathbb {P}_{r}\\left(n(i+1)t\\le \\zeta _0<(n+1)(i+1)t\\right)\\\\&=\\ (i+1)t\\sum _{n\\ge 0}(n+1)\\Big (\\mathbb {P}_{r}\\left(\\zeta _0\\ge n(i+1)t\\right)-\\mathbb {P}_{r}\\left(\\zeta _0\\ge (n+1)(i+1)t\\right)\\Big )\\\\&=\\ (i+1)t\\sum _{n\\ge 0}\\mathbb {P}_{r}\\left(\\zeta _0\\ge n(i+1)t\\right)\\\\&\\le \\ (i+1)t\\sum _{n\\ge 0}\\left(\\max _{x\\in V}\\mathbb {P}_x\\left(\\zeta _0\\ge (i+1)t\\right)\\right)^n\\\\&\\le \\ (i+1)t\\sum _{n\\ge 0}\\left(\\frac{3}{4}\\right)^n\\\\&=\\ 4(i+1)t,$ where $t:=2^i\\exp (\\beta (E(s_{m})-E(m)+2(i+1)|\\mathcal {S}|\\gamma _{\\beta }))+1$ .",
"Since $\\gamma _{\\beta }\\rightarrow 0$ , we get in the limit $\\lim _{\\beta \\rightarrow \\infty }\\frac{1}{\\beta }\\ln \\mathbb {E}_{r}\\zeta _0\\ \\le \\ E(s_m)-E(m)$ for all $r\\in V^{(i)}(m)$ .",
"Turning to the lower bound, define a sequence of stopping times, viz.",
"$\\rho _0:=0$ and $\\rho _n:=\\inf \\lbrace k>\\rho _{n-1}|X_k=m\\text{ or }X_{k}\\in \\partial ^+ V\\rbrace $ for $n\\ge 1$ .",
"Then $Z_n:=X_{\\rho _n}$ , $n\\ge 0$ , forms a Markov chain the transition probabilities of which when starting in $m$ can be estimated with the help of Proposition REF , namely $\\mathbb {P}(Z_1\\in \\partial ^+ V^{(i)}(m)|Z_0=m)\\ &=\\ \\mathbb {P}_{m}(\\rho _1=\\zeta _0)\\\\&=\\ \\mathbb {P}_{m}(\\zeta _0<\\tau _m)\\\\&\\le \\ \\sum _{s\\in \\partial ^+V}\\mathbb {P}_m(\\tau _s<\\tau _m)\\\\&\\le \\ K(\\beta )\\,e^{-\\beta (\\min _{s\\in \\partial ^+V}E(z^*(m,s))-E(m)-2\\gamma _{\\beta })}\\\\&=\\ K(\\beta )\\,e^{-\\beta (E(s_m)-E(m)-2\\gamma _{\\beta })}$ where $K(\\beta )\\rightarrow K\\in (0,\\infty )$ as $\\beta \\rightarrow \\infty $ .",
"Further defining $\\nu :=\\inf \\lbrace k\\ge 0| Z_k\\in \\partial ^+ V\\rbrace $ , this implies in combination with a geometric trials argument that $\\mathbb {P}_{m}(\\nu >n)\\ \\ge \\ \\left(1-K(\\beta )\\,e^{-\\beta (E(s_m)-E(m)-2\\gamma _{\\beta })}\\right)^{n-1}.$ As a consequence, $\\mathbb {E}_{m}\\zeta _0\\ &=\\ \\sum _{n\\ge 1} \\mathbb {E}_{m}(\\underbrace{\\rho _n-\\rho _{n-1}}_{\\ge 1}){1}_{\\lbrace \\nu \\ge n\\rbrace }\\ \\ge \\ \\sum _{n\\ge 1} \\mathbb {P}_{m}(\\nu \\ge n)\\ \\ge \\ K(\\beta )^{-1}\\,e^{\\beta (E(s_m)-E(m)-2\\gamma _{\\beta })}.$ For arbitrary $r\\in V$ , we now infer $\\mathbb {E}_{r}\\zeta _0\\ &=\\ \\mathbb {E}_{r}\\zeta _0{1}_{\\lbrace \\zeta _0\\le \\rho _1\\rbrace }+\\mathbb {E}_{r}\\zeta _0{1}_{\\lbrace \\zeta _0>\\rho _1\\rbrace }\\\\&\\ge \\ \\mathbb {E}_{r}\\big (\\mathbb {E}_r(\\zeta _0{1}_{\\lbrace \\zeta _0>\\rho _1\\rbrace }|X_{\\rho _{1}}=m)\\big )\\\\&\\ge \\ \\mathbb {E}_{r}{1}_{\\lbrace \\zeta _0>\\rho _1\\rbrace }\\mathbb {E}_{m}\\zeta _0\\\\&\\ge \\ \\mathbb {P}_{r}(\\zeta _0>\\rho _1)K(\\beta )^{-1}\\,e^{\\beta (E(s_m)-E(m)-2\\gamma _{\\beta })}\\\\&\\ge \\ \\frac{1}{2}\\,K(\\beta )^{-1}\\,e^{\\beta (E(s_m)-E(m)-2\\gamma _{\\beta })}$ for all sufficiently large $\\beta $ , because $\\lim _{\\beta \\rightarrow \\infty }\\mathbb {P}_{r}(\\zeta _0>\\rho _1)=1$ (Theorem REF ).",
"Finally, by taking logarithms and letting $\\beta $ tend to $\\infty $ , we arrive at the inequality $\\lim _{\\beta \\rightarrow \\infty }\\frac{1}{\\beta }\\ln \\mathbb {E}_{r}\\zeta _0\\ \\ge \\ E(s_m)-E(m)$ which completes the proof.",
"In [12], $E(s_m)-E(m),\\,m\\in M^{(i)},$ is referred to as the depth of the valley $V(m)$ .",
"Therefore, Property 3 from the Introduction holds true and we can relate thermodynamics of the system (energies) to dynamics of the chain (holding times) in a very precise way.",
"Especially, there is no universal scale for the times spent in different valleys because in general they differ exponentially."
],
[
"Macroscopic View: Transitions between valleys",
"With the help of the nested state space decompositions into valleys of different orders and around bottom states of different stability, we will now be able to provide an appropriate definition of the metabasins (MB) that has been announced and to some extent discussed in the Introduction.",
"We will further define and study macroscopic versions of the original process $X=(X_{n})_{n\\ge 0}$ .",
"These are obtained by choosing different levels of aggregation in the sense that they keep track only of the valleys of a chosen level that are visited by $X$ .",
"The motivation behind this approach is, on the one hand, to exhibit strong relations between properties of the energy landscape and the behavior of $X$ (as in Theorem REF ) and, on the other hand, to describe essential features of this process by looking at suitable macroscopic scales.",
"In the subsequent definition of aggregated versions of $X$ , we will distinguish between two variants: A time-scale preserving aggregation that, for a fixed level and each $n$ , keeps track of the valley the original chain visits at time $n$ and thus only blinds its exact location within a valley.",
"An accelerated version that, while also keeping track of the visited valleys, further blinds the sojourn times within a valley by counting a visit just once.",
"Actually, the definition of these aggregations at a chosen level $i$ is a little more complicated because their state space, denoted $\\mathcal {S}^{(i)}$ below and the elements of which we call level $i$ metastates, also comprises the non-assigned states at level $i$ as well as the minima of those valleys that were formed at an earlier level and whose merger is pending at level $i$ because their minima are not attracted at this level.",
"Definition 3.1 Fix $1\\le i\\le \\mathfrak {n}$ , let $\\mathcal {S}^{(i)}:=\\lbrace m^{(j)}\\in M^{(1)}|\\,l(j)>i\\rbrace \\cup N^{(i)}$ and $V^{(i)}(s):={\\left\\lbrace \\begin{array}{ll}V^{(i)}(m^{(j)}),&\\text{if }s=m^{(j)} \\textrm { for some j\\ge i}\\\\V^{(j)}(m^{(j)}),&\\text{if }s=m^{(j)} \\textrm { for some j<i}\\\\\\hfill \\lbrace s\\rbrace ,&\\textrm {if s\\in N^{(i)}}\\end{array}\\right.",
"}$ for $s\\in \\mathcal {S}^{(i)}$ .",
"Then define $\\overline{Y}_n^{(i)}\\ &:=\\ \\sum _{s\\in \\mathcal {S}^{(i)}}s\\,{1}_{\\lbrace X_{n}\\in V^{(i)}(s)\\rbrace },\\quad n\\ge 0,\\\\\\text{and}\\quad Y^{(i)}_n\\ &:=\\ \\overline{Y}^{(i)}_{\\sigma _n},\\quad n\\ge 0,$ where $\\sigma _{0}=\\sigma _{0}^{(i)}:\\equiv 0$ and $\\sigma _{n} = \\sigma _{n}^{(i)} := \\inf \\left\\lbrace k>\\sigma _{n-1}\\Big |\\overline{Y}_k^{(i)}\\ne \\overline{Y}_{k-1}^{(i)}\\right\\rbrace $ for $n\\ge 1$ .",
"We call $\\overline{Y}^{(i)}=(\\overline{Y}_n^{(i)})_{n\\ge 0}$ and $Y^{(i)}=(Y_{n}^{(i)})_{n\\ge 0}$ the aggregated chain (AC) and the accelerated aggregated chain (AAC) (at level $i$ ) associated with $X=(X_n)_{n\\ge 0}$ .",
"So, starting in an arbitrary valley, the original chain stays there for a time $\\zeta _0=\\zeta _{0}^{(i)}$ (as defined in Definition REF ) before it jumps via some non-assigned states $k_1{,}..., k_l$ (staying a geometric time in each of these states) to another valley at time $\\xi _1=\\xi _{1}^{(i)}$ .",
"There it stays for $\\zeta _1-\\xi _1$ time units before it moves on in a similar manner.",
"By going from $X$ to its aggregation $\\overline{Y}^{(i)}$ at level $i$ , we regard the whole valley $V^{(i)}(s)$ for $s\\in \\mathcal {S}^{(i)}$ as one single metastate and therefore give up information about the exact location of $X$ within a valley.",
"$\\overline{Y}^{(i)}$ is a jump process on $\\mathcal {S}^{(i)}$ with successive sojourn times $\\sigma _{n+1}-\\sigma _n,\\,n\\ge 0$ , which do not only depend on the valley but also on the states of entrance and exit.",
"The AAC then is the embedded chain, viz.",
"$\\overline{Y}^{(i)}_n=\\sum _{j\\ge 0}Y^{(i)}_j{1}_{\\lbrace \\sigma _j\\le n<\\sigma _{j+1}\\rbrace },$ giving the states only at jumps epochs: starting from the minimum of a first valley it moves to states $k_1{,}...,k_l\\in N^{(i)}$ and then proceeds to the minimum of a second valley, and so on.",
"Of course, at small temperatures the time spent in a non-assigned state or in a valley around a low order metastable state is very small compared to the time spent in a valley around a metastable state of higher order.",
"Thus, such states can be seen as instantaneous and of little importance for the evolution of the process.",
"We account for them nonetheless for two reasons.",
"First, in the path-dependent definition mentioned in the Introduction and used in Physics, they build small MB of great transitional activity of the process and are thus relevant in view of our goal to provide a definition of MB that conforms as much as possible to a path-dependent one.",
"Second, a complete partitioning of the state space that is an assignment of every $s\\in \\mathcal {S}$ to a metastate via a global algorithm fails when merely focusing on $\\left\\lbrace V^{(i)}(m), m\\in M^{(i)}\\right\\rbrace $ because there is neither an obvious nor natural way how to assign non-assigned states to them.",
"The incoherent scattering function and its associated relaxation time, for $X$ defined by $S(q,n):=\\mathbb {E}_{\\pi }\\cos \\big (q|X_n-X_0|\\big )$ (with $|\\cdot |$ being Euclidean distance in phase space) and $\\tau _q(\\varepsilon ):=\\inf \\lbrace n|S(q,n)\\le \\varepsilon \\rbrace ,\\quad \\varepsilon >0,$ respectively, may serve as an example which shows the strong relation between the behavior of the original process and its macroscopic versions.",
"For more detailed information on the meaning and relevance of $S(q,n)$ as a measure of incoherent scattering between the initial state of a glass-forming system and its state $n$ time steps onward, we refer to the survey by Heuer [12].",
"Proposition 3.2 For each $1\\le i\\le \\mathfrak {n}$ there is a constant $\\Delta (i)$ such that $\\sup _{n\\ge 0}\\,\\mathbb {P}_{\\pi }\\left(X_n\\ne \\overline{Y}^{(i)}_n\\right)\\le e^{-\\Delta (i)\\beta }.$ As a consequence, for any given $\\varepsilon >0$ , the incoherent scattering functions of $X$ and $\\overline{Y}^{(i)}$ differ by at most $4\\varepsilon $ for $\\beta $ sufficiently large.",
"[Proof:] Use Lemma REF to infer $\\mathbb {P}_{\\pi }(X_n\\ne \\overline{Y}^{(i)}_n)&=\\sum _{s\\in \\mathcal {S}^{(i)}\\backslash N^{(i)}}\\mathbb {P}_{\\pi }(\\overline{Y}^{(i)}_n=s)\\sum _{x\\in V^{(i)}(s)\\backslash \\lbrace s\\rbrace }\\frac{\\pi (x)}{\\pi (V^{(i)}(s))}\\\\&\\le \\sum _{s\\in \\mathcal {S}^{(i)}\\backslash N^{(i)}}\\mathbb {P}_{\\pi }(\\overline{Y}^{(i)}_n=s)\\sum _{x\\in V^{(i)}(s)\\backslash \\lbrace s\\rbrace }\\frac{\\pi (x)}{\\pi (s)}\\\\&\\le \\sum _{s\\in \\mathcal {S}^{(i)}\\backslash N^{(i)}}\\mathbb {P}_{\\pi }(\\overline{Y}^{(i)}_n=s)\\sum _{x\\in V^{(i)}(s)\\backslash \\lbrace s\\rbrace }e^{-\\beta (E(x)-E(s)-2\\gamma _{\\beta })}\\\\&\\le \\max _{s\\in \\mathcal {S}^{(i)}\\backslash N^{(i)}}|V^{(i)}(s)|\\max _{x\\in V^{(i)}(s)\\backslash \\lbrace s\\rbrace }e^{-\\beta (E(x)-E(s)-2\\gamma _{\\beta })}.$ This proves equation (REF ) because $E(x)>E(s)$ for each $x\\in V^{(i)}(s)\\backslash \\lbrace s\\rbrace , s\\in \\mathcal {S}^{(i)}\\backslash N^{(i)}$ .",
"Now let $\\beta $ be so large that $e^{-\\Delta (i)\\beta }\\le \\varepsilon $ for a given $\\varepsilon >0$ and observe that $\\mathbb {E}_{\\pi }&\\cos \\big (q|X_n-X_0|\\big )\\\\&=\\ \\int _{\\lbrace X_n=\\overline{Y}_{n}^{(i)},X_0=\\overline{Y}^{(i)}_0\\rbrace }\\cos \\big (q|X_n-X_0|\\big )\\ d\\mathbb {P}_{\\pi }\\ +\\ \\int _{\\lbrace X_n\\ne \\overline{Y}_n^{(i)}\\rbrace \\cup \\lbrace X_0\\ne \\overline{Y}^{(i)}_0\\rbrace }\\cos \\big (q|X_n-X_0|\\big )\\ d\\mathbb {P}_{\\pi }\\\\&\\le \\ \\int _{\\lbrace X_n=\\overline{Y}_n^{(i)},X_0=\\overline{Y}^{(i)}_0\\rbrace }\\cos \\big (q|X_n-X_0|\\big )\\ d\\mathbb {P}_{\\pi }\\ +\\ \\mathbb {P}_{\\pi }(X_n\\ne \\overline{Y}_n^{(i)})\\ +\\ \\mathbb {P}_{\\pi }(X_0\\ne \\overline{Y}^{(i)}_0)\\\\&=\\ \\mathbb {E}_{\\pi }\\cos \\left(q|\\overline{Y}_n^{(i)}-\\overline{Y}_0^{(i)}|\\right)\\ -\\ \\int _{\\lbrace X_n\\ne \\overline{Y}_n^{(i)}\\rbrace \\cup \\lbrace X_0\\ne \\overline{Y}^{(i)}_0\\rbrace }\\cos \\left(q|\\overline{Y}_n^{(i)}-\\overline{Y}_0^{(i)}|\\right)\\ d\\mathbb {P}_{\\pi } +\\ 2\\varepsilon \\\\&\\le \\ \\mathbb {E}_{\\pi }\\cos \\left(q|\\overline{Y}_n^{(i)}-\\overline{Y}_0^{(i)}|\\right)\\ +\\ 4\\varepsilon $ and, by a similar argument, $\\mathbb {E}_{\\pi }\\cos \\left(q|\\overline{Y}_n^{(i)}-\\overline{Y}_0^{(i)}|\\right)\\ \\le \\ \\mathbb {E}_{\\pi }&\\cos \\big (q|X_n-X_0|\\big )\\ +\\ 4\\varepsilon .$ This completes the proof."
],
[
"(Semi-)Markov Property",
"In general, both aggregated chains are no longer Markovian.",
"Transition probabilities of the AAC not only depend on the current state, i.e.",
"the current valley, but also on the entrance state into that valley, whereas transition probabilities of the AC depend on the current sojourn times which in turn depend on the previous, the present and the next state.",
"On the other hand, since valleys are defined in such a way that asymptotically almost surely (a.a.s.",
"), i.e., with probability tending to one as $\\beta \\rightarrow \\infty $ , the minimum will be reached from anywhere inside the valley before the valley is left, and since, furthermore, the exit state on the outer boundary a.a.s.",
"equals the one with the smallest energy, the AAC will be shown below to converge to a certain Markov chain on $\\mathcal {S}^{(i)}$ .",
"Also, the sojourn times depend on the past only via the last and the current state.",
"This means that the AC converges to a semi-Markov chain (for semi-Markov chains see for example [1]): Definition 3.3 Given any nonempty countable set $\\mathcal {S}$ , let $(M_{n},T_{n})_{n\\ge 0}$ be a bivariate temporally homogeneous Markov chain on $\\mathcal {S}\\times \\mathbb {N}$ , with transition kernel $Q(s,\\cdot )$ only depending on the first component, viz., for all $n\\ge 0, s\\in S$ and $t\\ge 0$ , $\\mathbb {P}(M_{n+1}=s, T_{n+1}\\le t|M_{n},T_{n})=Q(M_{n},\\lbrace s\\rbrace \\times [0,t])$ holds.",
"Put $S_n:=\\sum _{i=0}^{n}T_i$ for $n\\ge 0$ and $\\nu (t):=\\max \\lbrace n\\ge 0|S_n\\le t\\rbrace \\ (\\max \\emptyset :=0)$ for $t\\ge 0$ .",
"Then $Z_n:=M_{\\nu (n)}, n\\ge 0,$ is called semi-Markov chain with embedded Markov chain $(M_{n})_{n\\ge 0}$ and sojourn or holding times $T_{0},T_{1}{,}...$ .",
"Note that equation (REF ) holds iff $M=(M_{n})_{n\\ge 0}$ forms a temporally homogeneous Markov chain and the $(T_{n})_{n\\ge 0}$ are conditionally independent given $M$ such that the distribution of $T_{n}$ only depends on $M_{n-1},M_{n}$ for $n\\ge 1$ (in a temporally homogeneous manner), and on $M_{0}$ for $n=0$ .",
"Note further that we have specialized to the case where holding times take values in $\\mathbb {N}$ only (instead of $(0,\\infty )$ ).",
"Recall from (REF ) the definition of $s_m$ for $m\\in \\mathcal {S}^{(i)}\\backslash N^{(i)}$ and notice that the second equality there entails $E(z^{*}(m,s_{m}))<E(z^*(m,s)$ for any $s\\in \\partial ^+V^{(i)}(m) \\backslash \\lbrace s_{m}\\rbrace $ .",
"Further recall from our basic assumptions that $p^{*}(r,s)=\\lim _{\\beta \\rightarrow \\infty }p(r,s)$ exists for all $r,s\\in \\mathcal {S}$ and is positive if $E(r)\\ge E(s)$ .",
"The following result, revealing the announced convergence for AAC, confirms in particular that a valley $V(m)$ , $m\\in \\mathcal {S}^{(i)}\\backslash N^{(i)}$ , is a.a.s.",
"to be left via $s_{m}$ .",
"Proposition 3.4 For each $1\\le i\\le \\mathfrak {n}$ and as $\\beta \\rightarrow \\infty $ , the level $i$ AAC $Y^{(i)}$ converges to a Markov chain ${\\widehat{Y}}^{(i)}=({\\widehat{Y}}_{n}^{(i)})_{n\\ge 0}$ on $\\mathcal {S}^{(i)}$ with transition probabilities ${\\widehat{p}}(r,s)={\\widehat{p}}_{i}(r,s)$ stated below, that is $\\lim _{\\beta \\rightarrow \\infty }\\mathbb {P}(Y^{(i)}_{n+1}=s|Y_{n}^{(i)}=r,Y^{(i)}_{n-1}=m_{n-1}{,}...,Y^{(i)}_0=m_0)\\ =\\ {\\widehat{p}}(r,s)$ for all $m_{0}{,}...,m_{n-1},r,s\\in \\mathcal {S}^{(i)}$ and $n\\ge 0$ .",
"We have ${\\widehat{p}}(r,\\cdot ):=\\delta _{s_{r}}$ if $r\\in \\mathcal {S}^{(i)}\\backslash N^{(i)}$ , and ${\\widehat{p}}(r,\\cdot )\\ :=\\ \\frac{1}{1-p^*(r,r)}\\left(\\sum _{s\\in \\mathcal {N}(r)\\cap N^{(i)}}p^*(r,s)\\,\\delta _s+\\sum _{s\\in \\mathcal {S}^{(i)}\\backslash N^{(i)}}\\!\\!\\left(\\sum _{r^{\\prime }\\in \\mathcal {N}(r)\\cap V^{(i)}(s)}p^*(r,r^{\\prime })\\right)\\delta _{s}\\right),$ if $r\\in N^{(i)}$ .",
"${\\widehat{Y}}^{(i)}=({\\widehat{Y}}_{n}^{(i)})_{n\\ge 0}$ is called the asymptotic jump chain at level $i$ hereafter.",
"Note that, typically, it is not irreducible.",
"It may have transient states, not necessarily non-assigned, and its irreducibility classes are of the form $\\lbrace m_{1}{,}...,m_{k},s\\rbrace $ for a collection $m_{1}{,}...,m_{k}\\in \\mathcal {S}^{(i)}\\backslash N^{(i)}$ and some $s\\in N^{(i)}$ satisfying $s=s_{m_{1}}=...=s_{m_{k}}$ .",
"[Proof:] Fix $1\\le i\\le \\mathfrak {n}$ and write $Y_{n}$ for $Y_{n}^{(i)}$ .",
"The first step is to verify that, as $\\beta \\rightarrow \\infty $ , $\\mathbb {P}(Y_{n+1}=s|Y_{n}=r,Y_{n-1}=m_{n-1}{,}...,Y_0=m_0)=\\mathbb {P}_{r}(Y_{1}=s)+o(1)$ for all $m_{0}{,}...,m_{n-1},r,s\\in \\mathcal {S}^{(i)}$ and $n\\ge 0$ .",
"If $r\\in N^{(i)}$ , then $Y_n=X_{\\sigma _n}$ and the Markov property of $X$ provide us with the even stronger result $\\mathbb {P}(Y_{n+1}=s|Y_{n}=r,Y_{n-1}=m_{n-1}{,}...,Y_0=m_0)=\\mathbb {P}_{r}(Y_{1}=s).$ A little more care is needed if $r\\in \\mathcal {S}^{(i)}\\backslash N^{(i)}$ .",
"For any $s\\in \\mathcal {S}^{(i)},\\,x\\in V^{(i)}(r)$ and $n\\ge 0$ , we have $\\mathbb {P}(Y_{n+1}=s&|Y_n=r,X_{\\sigma _n}=x)\\\\&=\\ \\mathbb {P}_x(Y_1=s,\\tau _{r}<\\sigma _1)+\\mathbb {P}_x(Y_1=s,\\tau _{r}>\\sigma _1)\\\\&=\\ \\mathbb {P}_{r}(Y_1=s)\\,\\mathbb {P}_x(\\tau _{r}<\\sigma _1)+\\mathbb {P}_x(Y_1=s,\\tau _{r}>\\sigma _1).$ The last two summands can further be bounded by $\\mathbb {P}_{r}(Y_1=s)\\,\\mathbb {P}_x(\\tau _{r}<\\sigma _1)\\ \\le \\ \\mathbb {P}_{r}(Y_1=s)\\quad \\text{and}\\quad \\mathbb {P}_x(Y_1=s,\\tau _{r}>\\sigma _1)\\ \\le \\ \\mathbb {P}_x(\\sigma _1<\\tau _{r}).$ For the last probability, Theorem REF ensures $\\mathbb {P}_x(\\sigma _1<\\tau _{r})\\ &\\le \\ \\sum _{z\\in \\partial ^+ V^{(i)}(r)}\\mathbb {P}_x(\\tau _z<\\tau _{r})\\ \\le \\ \\sum _{z\\in \\partial ^+ V^{(i)}(r)} \\tilde{\\varepsilon }(x,r,z,\\beta )\\ \\stackrel{\\beta \\rightarrow \\infty }{\\longrightarrow }\\ 0.$ Consequently, as $\\beta \\rightarrow \\infty $ , $\\mathbb {P}_{r}(Y_1=s)\\ &=\\ \\left(1-\\sum _{z\\in \\partial ^+ V^{(i)}(r)} \\tilde{\\varepsilon }(x,r,z,\\beta )\\right)\\mathbb {P}_{r}(Y_1=s)+o(1)\\\\&\\le \\ \\big (1-\\mathbb {P}_x(\\sigma _1<\\tau _{r})\\big )\\,\\mathbb {P}_{r}(Y_1=s)+o(1)\\\\&\\le \\ \\mathbb {P}(Y_{n+1}=s|X_{\\sigma _n}=x,Y_n=r)+o(1)\\\\&\\le \\ \\mathbb {P}_{r}(Y_1=s)+\\sum _{z\\in \\partial ^+ V^{(i)}(r)} \\tilde{\\varepsilon }(x,r,z,\\beta )+o(1)\\\\&=\\ \\mathbb {P}_{r}(Y_1=s)+o(1),$ and therefore $\\mathbb {P}&(Y_{n+1}=s|Y_{n}=r,Y_{n-1}=m_{n-1}{,}...,Y_0=m_0)\\\\&=\\ \\sum _{x\\in V^{(i)}(r)}\\mathbb {P}(Y_{n+1}=s|X_{\\sigma _n}=x, Y_n=r)\\,\\mathbb {P}(X_{\\sigma _n}=x|Y_{n}=r,Y_{n-1}=m_{n-1}{,}...,Y_0=m_0)\\\\&=\\ \\mathbb {P}_{r}(Y_1=s)+o(1).$ It remains to verify that $\\mathbb {P}_{r}(Y_1=s)={\\widehat{p}}(r,s)+o(1)$ for any $r,s\\in \\mathcal {S}^{(i)}$ .",
"If $r\\in \\mathcal {S}^{(i)}\\backslash N^{(i)}$ , then $\\sigma _{1}=\\tau _{N^{(i)}}$ and $Y_{1}=X_{\\tau _{N^{(i)}}}$ .",
"Since $E(z^*(r,s_r))<E(z^*(r,s))$ for each $s_r\\ne s\\in N^{(i)}\\cap \\partial ^+V^{(i)}(r)$ , we now infer with the help of Theorem REF $\\mathbb {P}_{r}(Y_{1}\\ne s_{r})\\ &=\\ \\mathbb {P}_{r}\\big (\\tau _{s}<\\tau _{s_{r}}\\text{ for some }s\\in N^{(i)}\\backslash \\lbrace s_{r}\\rbrace \\big )\\\\&\\ \\le \\ \\sum _{s_{r}\\ne s\\in N^{(i)}}\\mathbb {P}_{r}\\big (\\tau _{s}<\\tau _{s_{r}})\\\\&\\le \\ \\sum _{s_{r}\\ne s\\in N^{(i)}}\\tilde{\\varepsilon }(r,s_{r},s,\\beta )\\ \\\\&=\\ o(1),$ as $\\beta \\rightarrow \\infty $ and thus $\\mathbb {P}_{r}(Y_{1}\\in \\cdot )\\rightarrow \\delta _{s_{r}}={\\widehat{p}}(r,\\cdot )$ as claimed.",
"If $r\\in N^{(i)}$ , then either $Y_{1}=s\\in \\mathcal {N}(r)\\cap N^{(i)}$ , or $Y_{1}=s\\in \\mathcal {S}^{(i)}\\backslash N^{(i)}$ and $X_{\\sigma _{1}}=r^{\\prime }$ for some $r^{\\prime }\\in \\mathcal {N}(r)\\cap V^{(i)}(s)$ .",
"It thus follows that $\\mathbb {P}_{r}(Y_{1}=s)\\ =\\ \\mathbb {P}_{r}(X_{\\sigma _{1}}=s)\\ =\\ \\frac{p(r,s)}{1-p(r,r)}\\ =\\ \\frac{p^{*}(r,s)}{1-p^{*}(r,r)}+o(1)$ if $s\\in \\mathcal {N}(r)\\cap N^{(i)}$ , while $\\mathbb {P}_{r}(Y_{1}=s)\\ =\\ \\sum _{r^{\\prime }\\in \\mathcal {N}(r)\\cap V^{(i)}(s)}\\mathbb {P}_{r}(X_{\\sigma _{1}}=r^{\\prime })\\ =\\ \\sum _{r^{\\prime }\\in \\mathcal {N}(r)\\cap V^{(i)}(s)}\\frac{p^{*}(r,r^{\\prime })}{1-p^{*}(r,r)}+o(1)$ in the second case.",
"Having shown that $Y^{(i)}$ behaves asymptotically as a Markov chain, viz.",
"the jump chain ${\\widehat{Y}}^{(i)}$ , it is fairly easy to verify with the help of the next simple lemma that the augmented bivariate AC $\\big (\\overline{Y}_{n}^{(i)},\\overline{Y}_{n+1}^{(i)}\\big )_{n\\ge 0}$ is asymptotically semi-Markovian.",
"Lemma 3.5 For each $\\beta >0$ , the sojourn times $\\sigma _{n+1}-\\sigma _{n}$ , $n\\ge 0$ , of the AC $\\overline{Y}^{(i)}$ are conditionally independent given $Y^{(i)}$ .",
"The conditional law of $\\sigma _{n+1}-\\sigma _{n}$ depends only on $(Y_{n-1}^{(i)},Y_{n}^{(i)},Y_{n+1}^{(i)})$ and satisfies $&\\mathbb {P}\\big (\\sigma _{n+1}-\\sigma _{n}\\in \\cdot ~|Y^{(i)}_{n-1}=x,Y^{(i)}_{n}=y,Y^{(i)}_{n+1}=z\\big )\\nonumber \\\\&\\hspace{14.22636pt}=\\ Q((x,y,z),\\cdot \\, )\\ :=\\ {\\left\\lbrace \\begin{array}{ll}\\hfill \\textit {Geom}(1-p(y,y)),&\\text{if }y\\in N^{(i)}\\\\\\sum _{s\\in V^{(i)}(y),s\\sim x}\\mathbb {P}_s(\\sigma _1\\in \\cdot ~|Y^{(i)}_1=z)\\,\\mathbb {P}_{x}(X_{\\sigma _1}=s),&\\text{if }y\\notin N^{(i)}\\end{array}\\right.",
"}$ for all $x,y,z\\in \\mathcal {S}^{(i)}$ with $\\mathbb {P}(Y^{(i)}_{n-1}=x,Y^{(i)}_{n}=y,Y^{(i)}_{n+1}=z)>0$ and $n\\ge 1$ .",
"[Proof:] The assertions follow easily when observing that, on the one hand, at least one state $y\\in N^{(i)}$ must be visited between two states $x,z\\in \\mathcal {S}^{(i)}\\backslash N^{(i)}$ (Lemma REF ) and that, on the other hand, the original chain $X$ and its aggregation $\\overline{Y}^{(i)}$ coincide at any epoch where a non-assigned state is hit, which renders the Markov property of $\\overline{Y}^{(i)}$ at these epochs.",
"Further details are omitted.",
"In order to formulate the next result, let $0={\\widehat{\\sigma }}_{0}<{\\widehat{\\sigma }}_{1}<...$ be an increasing sequence of random variables such that its increments ${\\widehat{\\sigma }}_{n+1}-{\\widehat{\\sigma }}_{n}, n\\ge 0$ , are conditionally independent given the asymptotic jump chain ${\\widehat{Y}}^{(i)}$ .",
"Moreover, let the conditional law of ${\\widehat{\\sigma }}_{n+1}-{\\widehat{\\sigma }}_{n}$ depend only on $\\big ({\\widehat{Y}}^{(i)}_{n-1},{\\widehat{Y}}^{(i)}_{n},{\\widehat{Y}}^{(i)}_{n+1}\\big )$ and be equal to $Q\\big (\\big ({\\widehat{Y}}^{(i)}_{n-1},{\\widehat{Y}}^{(i)}_{n},{\\widehat{Y}}^{(i)}_{n+1}\\big ),\\cdot \\,\\big )$ , with $Q$ as defined in (REF ).",
"Then $(({\\widehat{Y}}^{(i)}_{n},{\\widehat{Y}}^{(i)}_{n+1}),{\\widehat{\\sigma }}_{n+1})_{n\\ge 0}$ forms a Markov renewal process and $\\big ({\\widehat{Y}}^{(i)}_{{\\widehat{\\nu }}(n)},{\\widehat{Y}}^{(i)}_{{\\widehat{\\nu }}(n+1)}\\big )_{n\\ge 0}$ a semi-Markov chain, where ${\\widehat{\\nu }}(n):=\\sup \\lbrace k\\ge 0|{\\widehat{\\sigma }}_{k}\\le n\\rbrace $ .",
"Proposition 3.6 For each $1\\le i\\le \\mathfrak {n}$ , $((Y^{(i)}_{n},Y^{(i)}_{n+1}),\\sigma _{n+1})_{n\\ge 0}$ converges to the Markov renewal process $(({\\widehat{Y}}^{(i)}_{n},{\\widehat{Y}}^{(i)}_{n+1}),{\\widehat{\\sigma }}_{n+1})_{n\\ge 0}$ in the sense that $\\lim _{\\beta \\rightarrow \\infty }\\frac{\\mathbb {P}_{y_0}\\big (\\big (Y^{(i)}_{k},Y^{(i)}_{k+1}\\big )=(y_{k},y_{k+1}),\\,\\sigma _{k+1}=i_{k+1},\\,0\\le k\\le n\\big )}{\\mathbb {P}_{y_0}\\big (\\big ({\\widehat{Y}}^{(i)}_{k},{\\widehat{Y}}^{(i)}_{k+1}\\big )=(y_{k},y_{k+1}),\\,{\\widehat{\\sigma }}_{k+1}=i_{k+1},\\,0\\le k\\le n\\big )}\\ =\\ 1$ for all $y_{0}{,}...,y_{n+1}\\in \\mathcal {S}^{(i)}$ , $0<i_{1}<...<i_{n+1}$ and $n\\ge 0$ such that the denominator is positive.",
"Furthermore, $(\\overline{Y}^{(i)}_{n},\\overline{Y}^{(i)}_{n+1})_{n\\ge 0}$ is asymptotically semi-Markovian in the sense that $\\lim _{\\beta \\rightarrow \\infty }\\frac{\\mathbb {P}_{y_0}\\big (\\big (\\overline{Y}_{k}^{(i)},\\overline{Y}_{k+1}^{(i)}\\big )=(y_{k},y_{k+1}),0\\le k\\le n\\big )}{\\mathbb {P}_{y_0}\\big (\\big ({\\widehat{Y}}^{(i)}_{{\\widehat{\\nu }}(k)},{\\widehat{Y}}^{(i)}_{{\\widehat{\\nu }}(k+1)}\\big )=(y_{k},y_{k+1}),0\\le k\\le n\\big )}\\ =\\ 1$ for all $y_{0}{,}...,y_{n+1}\\in \\mathcal {S}^{(i)}$ and $n\\ge 0$ such that the denominator is positive.",
"[Proof:] The first assertion being obvious by Proposition REF , note that it implies, with $\\nu (n):=\\sup \\lbrace k\\ge 0|\\sigma _{k}\\le n\\rbrace $ , $\\lim _{\\beta \\rightarrow \\infty }\\frac{\\mathbb {P}_{y_0}\\big (\\big (Y_{\\nu (k)}^{(i)},Y_{\\nu (k+1)}^{(i)}\\big )=(y_{k},y_{k+1}),0\\le k\\le n\\big )}{\\mathbb {P}_{y_0}\\big (\\big ({\\widehat{Y}}^{(i)}_{{\\widehat{\\nu }}(k)},{\\widehat{Y}}^{(i)}_{{\\widehat{\\nu }}(k+1)}\\big )=(y_{k},y_{k+1}),0\\le k\\le n\\big )}\\ =\\ 1$ for all $y_{0}{,}...,y_{n+1}\\in \\mathcal {S}^{(i)}$ and $n\\ge 0$ such that the denominator is positive.",
"Therefore the second assertion follows when finally noting that $Y^{(i)}_{\\nu (n)}=\\sum _{j\\ge 0}Y^{(i)}_j{1}_{\\lbrace \\sigma _j\\le n<\\sigma _{j+1}\\rbrace }=\\overline{Y}^{(i)}_n$ for each $n\\ge 0$ .",
"So we have shown that, although aggregation generally entails the loss of the Markov property, here it leads back to processes of this kind (Markov or semi-Markov chains) in an asymptotic sense at low temperature regimes."
],
[
"Reciprocating Jumps",
"As discussed to some extent in the Introduction, we want to find an aggregation level at which reciprocating jumps appear to be very unlikely so as to obtain a better picture of essential features of the observed process.",
"To render precision to this informal statement requires to further specify the term “reciprocating jump” and to provide a measure of likelihood for its occurrence.",
"It is useful to point out first that the original chain $X$ exhibits two types of reciprocating jumps: Intra-valley jumps which occur between states inside a valley (starting in a minimum the process falls back to it many times before leaving the valley).",
"Inter-valley jumps which occur between two valleys (typically, when the energy barrier between these valleys is much lower then the barrier to any other valley).",
"Figure: Illustration of intra-valley jumps (left panel) versus inter-valley jumps (right panel).Clearly, intra-valley jumps disappear by aggregating valleys into metastates, while inter-valley jumps may also be viewed as intra-valley jumps for higher order valleys and do occur when transitions between any two of them are much more likely than those to other valleys in which case they should be aggregated into one valley.",
"This motivates the following definition.",
"Definition 3.7 We say the process $(Y^{(i)}_n)_{n\\in \\mathbb {N}}$ exhibits reciprocating jumps of order $\\varepsilon >0$ if there exists a nonempty subset $A\\mathcal {S}^{(i)}\\backslash N^{(i)}$ with the following property: For each $m_{1}\\in A$ , there exists $m_{2}\\in A$ such that $\\lim _{\\beta \\rightarrow \\infty }\\frac{1}{\\beta }\\left(\\ln \\left(\\mathbb {P}_{m_1}\\left(X_{\\xi _1}\\in V^{(i)}(m_2)\\right)\\right)-\\ln \\left(\\mathbb {P}_{m_1}\\left(X_{\\xi _1}\\in V^{(i)}(m)\\right)\\right)\\right)\\ge \\varepsilon $ for all $m\\in \\mathcal {S}^{(i)}\\backslash (N^{(i)}\\cup A)$ .",
"In other words, it is exponentially more likely to stay in $A$ than to leave it (ignoring intermediate visits to non-assigned states).",
"In view of our principal goal to give a path-independent definition of MBs, we must point out that, by irreducibility, reciprocating jumps always occur with positive probability at any nontrivial level of aggregation and can therefore never be ruled out completely.",
"This is in contrast to the path-dependent version by Heuer [12] in which the non-occurrence of reciprocating jumps appears to be the crucial requirement.",
"As a consequence, Definition REF provides an alternative, probabilistic and verifiable criterion for reciprocating jumps to be sufficiently unlikely in a chosen aggregation.",
"The following proposition contains further information on which valleys are visited successively by providing the probabilities of making a transition from $V^{(i)}(m)$ to $V^{(i)}(m^{\\prime })$ for any $m,m^{\\prime }\\in \\mathcal {S}^{(i)}\\backslash N^{(i)}$ .",
"It is a direct consequence of the asymptotic results in the previous subsection, notably Proposition REF .",
"Proposition 3.8 Let $m\\in \\mathcal {S}^{(i)}\\backslash N^{(i)}, s_m\\in \\partial ^+V^{(i)}(m)$ be as defined in (REF ) (i.e., the state on the outer boundary of $V^{(i)}(m)$ with minimal energy).",
"Then $\\lim _{\\beta \\rightarrow \\infty }\\mathbb {P}_m(X_{\\xi _1}\\in V^{(i)}(m))\\ =\\ {\\widehat{p}}(s_m,m)\\ =\\ \\frac{\\sum _{r\\in \\mathcal {N}(s_m)\\cap V^{(i)}(m)}p^{*}(s_m,r)}{1-p^{*}(s_m,s_m)},$ while $\\lim _{\\beta \\rightarrow \\infty }\\mathbb {P}_m(X_{\\xi _1}\\in V^{(i)}(m^{\\prime }))\\ =\\ {\\widehat{p}}(s_{m},m^{\\prime })+\\sum _{n\\ge 1}\\sum _{r_{1}{,}...,r_{n}\\in N^{(i)}}{\\widehat{p}}(s,r_{1})\\cdot ...\\cdot {\\widehat{p}}(r_{n-1},r_{n})\\,{\\widehat{p}}(r_{n},m^{\\prime })$ for any other $m^{\\prime }\\in \\mathcal {S}^{(i)}\\backslash N^{(i)}$ .",
"The reader should notice that, as ${\\widehat{p}}(s,r)=0$ whenever $E(s)<E(r)$ , the last sum actually ranges only over those non-assigned $r_{1}{,}...,r_{n}$ with $E(s_{m})>E(r_{1})>...>E(r_{n})>E(m^{\\prime })$ .",
"[Proof:] Let us first point out that $\\mathbb {P}_{r}(X_{\\xi _{0}}\\in V^{(i)}(m))=o(1)$ as $\\beta \\rightarrow \\infty $ for any $r\\in N^{(i)}$ such that $E(r)<E(s_{m})$ .",
"Namely, since the last property implies $r\\notin \\partial ^{+}V^{(i)}(m)$ , any path from $r$ into $V^{(i)}(m)$ must traverse a state $s\\in \\partial ^{+}V^{(i)}(m)$ with $E(s)\\ge E(s_{m})>E(r)$ , whence the probability for such a path goes to zero as $\\beta \\rightarrow \\infty $ .",
"Noting further that $\\mathbb {P}_m(Y_1^{(i)}\\ne s_m)=o(1)$ as $\\beta \\rightarrow \\infty $ by Proposition REF , we now infer (with $\\xi _{n}=\\xi _{n}^{(i)}$ ) $\\mathbb {P}_m(X_{\\xi _1}\\in V^{(i)}(m))\\ &=\\ \\mathbb {P}_{s_m}(X_{\\xi _0}\\in V^{(i)}(m))+o(1)\\\\&=\\ \\mathbb {P}_{s_m}(Y^{(i)}_1=m)+\\sum _{r\\in \\mathcal {N}(s_{m})\\cap N^{(i)}}\\frac{p(s_{m},r)}{1-p(s_m,s_m)}\\,\\mathbb {P}_{r}(X_{\\xi _0}\\in V^{(i)}(m))+o(1)\\\\&=\\ {\\widehat{p}}(s_{m},m)+o(1).$ The expression for ${\\widehat{p}}(s_{m},m)$ in terms of the $p^{*}(s_{m},r)$ may be read off directly from the formula given in Proposition REF .",
"For $m^{\\prime }\\ne m$ , $m^{\\prime }\\in \\mathcal {S}^{(i)}\\backslash N^{(i)}$ , we obtain in a similar manner $\\mathbb {P}_m&(X_{\\xi _1}\\in V^{(i)}(m^{\\prime }))\\\\&=\\ \\mathbb {P}_{s_{m}}(Y_{1}^{(i)}=m^{\\prime })\\ +\\ \\sum _{n\\ge 1}\\sum _{r_{1}{,}...,r_{n}\\in N^{(i)}} \\mathbb {P}_{s_{m}}(Y_{1}^{(i)}=r_{1}{,}...,Y_{n}^{(i)}=r_{n},Y_{n+1}^{(i)}=m^{\\prime })+o(1)\\\\&=\\ {\\widehat{p}}(s_{m},m^{\\prime })+\\sum _{n\\ge 1}\\sum _{r_{1}{,}...,r_{n}\\in N^{(i)}}{\\widehat{p}}(s_m,r_{1})\\cdot ...\\cdot {\\widehat{p}}(r_{n-1},r_{n})\\,{\\widehat{p}}(r_{n},m^{\\prime })+o(1),$ the last line by another appeal to the afore-mentioned proposition.",
"In essence, the previous result tells us that a valley $V^{(i)}(m^{\\prime })$ is neighbored to $V^{(i)}(m)$ , that is, reachable with positive probability by the asymptotic jump chain ${\\widehat{Y}}^{(i)}$ (and thus by $Y^{(i)}$ at any temperature level $\\beta $ ) without intermediately hitting any other valley, iff there exists at least one (in terms of energies) decreasing path in $N^{(i)}$ from $s_{m}$ to $m^{\\prime }$ .",
"For any other such pair of valleys, connected by a path through states in $N^{(i)}$ , the transition probability decreases to zero exponentially in $\\beta $ .",
"If this path can be chosen to be unimodal, here called uphill-downhill-path, this can be stated in a very precise way as the next result shows.",
"Lemma 3.9 Let $m_0,m_1\\in \\mathcal {S}^{(i)}\\backslash N^{(i)}$ be two distinct local minima for some $0\\le i\\le \\mathfrak {n}$ .",
"Suppose there exists a minimal path $\\gamma =(\\gamma _0{,}...,\\gamma _k)$ from $s_{m_0}$ to $m_1$ not hitting any other valley but $V^{(i)}(m_{1})$ and such that $I(\\gamma _0{,}...,\\gamma _{k})=E(z^*(s,m_1))-E(s_{m_0})$ .",
"Then $\\lim _{\\beta \\rightarrow \\infty }\\frac{1}{\\beta }\\ln \\mathbb {P}_{m_0}(X_{\\xi _1}\\in V^{(i)}(m_1))\\ =\\ -\\left(E(z^*(m_0,m_1))-E(s_{m_0})\\right).$ Without assuming the existence of $\\gamma $ as stated, the result remains valid when replacing $=$ with $\\le $ .",
"Note that $I(\\gamma _0{,}... ,\\gamma _{k})=E(z^*(s_{m_0},m_1))-E(s_{m_0})$ does indeed imply the already mentioned property that $E(\\gamma _i)&>E(\\gamma _{i-1})\\quad \\textrm {for 1\\le i\\le j}\\\\\\text{and}\\quad E(\\gamma _i)&<E(\\gamma _{i-1})\\quad \\textrm {for j+1\\le i\\le k}$ if $\\gamma _j=z^*(s_{m_0},m_1)$ .",
"We call such a path an uphill-downhill-path because it first straddles the energy barrier $E(z^{*}(s_{m_0},m_{1}))$ and then falls down to the local minimum $m_{1}$ .",
"The existence of such a path can be found in most 2- or higher dimensional energy landscapes.",
"[Proof:] With $\\gamma $ as stated, a lower bound for $\\mathbb {P}_{m_0}(X_{\\xi _1}\\in V^{(i)}(m_1))$ is easily obtained as follows: $\\mathbb {P}_{m_0}(X_{\\xi _1}\\in V^{(i)}(m_1))\\ &\\ge \\ \\mathbb {P}_{m_0}(X_{\\zeta _{0}+i}=\\gamma _i,\\, 0\\le i\\le k)\\\\&\\ge \\ \\mathbb {P}_{m_0}(X_{\\zeta _{0}}=s_{m_0})\\,e^{-\\beta I(\\gamma _0{,}..., \\gamma _{k})-\\gamma _{\\beta }\\beta |\\mathcal {S}|}\\\\&=\\ (1+o(1))\\,e^{-\\beta \\left(E\\left(z^*\\left(s_{m_0},m_1\\right)\\right)-E(s_{m_0})\\right)-\\gamma _{\\beta }\\beta |\\mathcal {S}|}\\\\&=\\ (1+o(1))\\,e^{-\\beta \\left(E\\left(z^*\\left(m_0,m_1\\right)\\right)-E(s_{m_0})\\right)-\\gamma _{\\beta }\\beta |\\mathcal {S}|}.$ For an upper bound, which does not require the existence of a $\\gamma $ as claimed, we decompose the event into disjoint sets depending on the number of visits $N$ , say, to $m_0$ between 1 and $\\zeta _{0}=\\zeta _{0}^{(i)}$ .",
"This leads to $\\mathbb {P}_{m_0}(X_{\\xi _1}\\in V^{(i)}(m_1),N=0)=\\mathbb {P}_{m_0}(\\xi _1=\\tau _{V^{(i)}(m_1)}<\\tau _{m_0})$ and, for $k\\ge 1$ , $\\mathbb {P}_{m_0}&(X_{\\xi _1}\\in V^{(i)}(m_1),N=k)\\\\&=\\mathbb {P}_{m_0}(X_{\\xi _1}\\in V^{(i)}(m_1),\\,|\\lbrace \\tau _{m_0}< n\\le \\zeta _0|X_n=m_0\\rbrace |=k-1,\\,\\tau _{m_0}<\\zeta _0)\\\\&=\\mathbb {P}_{m_0}(X_{\\xi _1}\\in V^{(i)}(m_1),N=k-1)\\,\\mathbb {P}_{m_0}(\\tau _{m_0}<\\zeta _0)\\\\&~~\\vdots \\\\&=\\mathbb {P}_{m_0}(X_{\\xi _1}\\in V^{(i)}(m_1),N=0)\\,\\mathbb {P}_{m_0}(\\tau _{m_0}<\\zeta _0)^k\\\\&=\\mathbb {P}_{m_0}(\\xi _1=\\tau _{V^{(i)}(m_1)}<\\tau _{m_0})\\,\\mathbb {P}_{m_0}(\\tau _{m_0}<\\zeta _0)^k.$ Consequently, $\\mathbb {P}_{m_0}(X_{\\xi _1}\\in V^{(i)}(m_1))&=\\sum _{k\\ge 0}\\mathbb {P}_{m_0}(X_{\\xi _1}\\in V^{(i)}(m_1),N=k)\\\\&=\\sum _{k\\ge 0}\\mathbb {P}_{m_0}(\\xi _1=\\tau _{V^{(i)}(m_1)}<\\tau _{m_0})\\,\\mathbb {P}_{m_0}(\\tau _{m_0}<\\zeta _0)^k\\\\&=\\frac{\\mathbb {P}_{m_0}(\\xi _1=\\tau _{V^{(i)}(m_1)}<\\tau _{m_0})}{\\mathbb {P}_{m_0}(\\zeta _0<\\tau _{m_0})}.$ By invoking Proposition REF , we infer $\\frac{\\mathbb {P}_{m_0}(\\xi _1=\\tau _{V^{(i)}(m_1)}<\\tau _{m_0})}{\\mathbb {P}_{m_0}(\\zeta _0<\\tau _{m_0})}\\ &\\le \\ \\frac{\\sum _{r\\in V^{(i)}(m_1)}\\mathbb {P}_{m_0}(\\tau _{r}<\\tau _{m_0})}{\\mathbb {P}_{m_0}(\\tau _x<\\tau _{m_0})}\\nonumber \\\\&\\le \\ K(\\beta )\\sum _{r\\in V^{(i)}(m_1)}e^{-\\beta (E(z^*(m_0,r))-E(z^*(m_0,x))-7\\gamma _{\\beta })}$ for all $x\\in V^{(i)}(m_0)^c$ , where $K(\\beta )\\ =\\ |\\mathcal {S}|\\left(|\\mathcal {S}|\\exp \\left(-\\beta \\min _{a\\ne b:E(a)>E(b)}(E(a)-E(b))+2\\gamma _{\\beta }\\beta \\right)+1\\right)\\max _{r\\in \\mathcal {S}}|\\mathcal {N}(r)|.$ For any $r\\in V^{(i)}(m_{1})$ , we have $E(z^{*}(m_{0},r))\\ge E(z^{*}(m_{1},r))$ and therefore $E(z^{*}(m_{0},m_{1}))\\ \\le \\ E(z^{*}(m_{0},r))\\vee E(z^{*}(r,m_{1}))\\ =\\ E(z^{*}(m_{0},r)).$ Using this in (REF ), we obtain $\\frac{\\mathbb {P}_{m_0}(\\xi ^{(i)}_1=\\tau _{V^{(i)}(m_1)}<\\tau _{m_0})}{\\mathbb {P}_{m_0}(\\zeta _0<\\tau _{m_0})}\\ \\le \\ K(\\beta )\\, |\\mathcal {S}|\\, e^{-\\beta (E(z^*(m_0,m_1))-E(z^*(m_0,x))-7\\gamma _{\\beta })}$ and then, upon choosing $x=s_{m_0}$ and noting that $E(z^{*}(m_{0},s_{m_0}))=E(s_{m_0})$ , $\\mathbb {P}_{m_0}(X_{\\xi _1}\\in V^{(i)}(m_1))\\ \\le \\ K(\\beta )\\,|\\mathcal {S}|\\,e^{-\\beta \\left(E\\left(z^*\\left(m_0,m_1\\right)\\right)-E(s_{m_0})-7\\gamma _{\\beta }\\right)}.$ By combining all previous results, we finally conclude $\\lim _{\\beta \\rightarrow \\infty }\\frac{1}{\\beta }\\ln \\mathbb {P}_{m_0}(X_{\\xi _1}\\in V^{(i)}(m_1))\\ =\\ -\\left(E(z^*(m_0,m_1))-E(s_{m_0})\\right)$ as asserted.",
"To summarize, which valleys are visited consecutively depends on (a) their spatial arrangement and (b) the energy barriers between them: A transition from one valley $V^{(i)}(m_0)$ to another valley $V^{(i)}(m_1)$ is only possible, if there exists a path from $s_{m_0}$ to $V^{(i)}(m_1)$ , not hitting any other valley.",
"This transition is made at small temperatures (i.e.",
"large $\\beta $ ) if the additional energy barrier $E(z^*(s_{m_0},m_1))-E(s_{m_0})$ is sufficiently small or in other words the energy barrier $E(z^*(s_{m_0},m_1))$ is approximately of the same height as all other energy barriers, including the barrier $E(z^*(s_{m_0},m_0))=E(s_{m_0})$ .",
"A result similar to the previous lemma holds true for transitions from $m\\in \\mathcal {S}^{(i)}\\backslash N^{(i)}$ to any $s\\in \\partial ^+V(m)$ .",
"Lemma 3.10 Let $m\\in \\mathcal {S}^{(i)}\\backslash N^{(i)}$ and $s\\in \\partial ^+V^{(i)}(m)$ .",
"Then $\\lim _{\\beta \\rightarrow \\infty }\\frac{1}{\\beta }\\ln \\mathbb {P}_m(Y_1=s)=-(E(s)-E(s_m)).$ [Proof:] For the proof, decompose again the event $\\lbrace Y_1=s\\rbrace $ with respect to the number of visits to $m$ before $V(m)=V^{(i)}(m)$ is left (or use Proposition REF ), giving $\\mathbb {P}_m(Y_1=s)\\ =\\ \\frac{\\mathbb {P}_m(\\sigma _1=\\tau _s<\\tau _m)}{\\mathbb {P}_m(\\sigma _1<\\tau _m)}.$ The proof of the lower bound is much more technical.",
"Let $\\gamma =(\\gamma _1{,}...,\\gamma _n)\\in \\Gamma ^*(m,s)$ be a minimal path which leaves $V(m)$ only in the last step and such that for any $\\gamma _i,\\gamma _j\\in \\gamma $ both, the subpath from $\\gamma _i$ to $\\gamma _j$ , and the inversed path from $\\gamma _j$ to $\\gamma _i$ , are minimal.",
"Define $r_0:=m \\quad \\textrm {and}\\quad r_1:=\\gamma _{i_0}\\quad \\textrm {with}\\quad i_0:=\\inf \\lbrace 0\\le i\\le n-1|E(\\gamma _{i+1})\\ge E(s_m)\\rbrace .$ In particular, $E(r_1)<E(s_m)$ and $E(z^*(r_1,r_0))<E(s_m)$ .",
"Define furthermore the first record by $s_1:=\\gamma _{i_1}$ with $i_1\\ :=\\ \\inf \\Big \\lbrace i_0< i\\le n\\,\\Big |\\,&E(\\gamma _i)\\ge E(s_m),\\,\\inf \\lbrace j\\ge i|E(\\gamma _j)<E(\\gamma _{i})\\rbrace <\\inf \\lbrace j\\ge i|E(\\gamma _j)>E(\\gamma _{i})\\rbrace \\Big \\rbrace ,$ and then successively for $k\\ge 1$ with $s_k=\\gamma _{i_k}\\ne s$ the records $s_{k+1}:=\\gamma _{i_{k+1}}$ with $i_{k+1}\\ :=\\ \\inf \\Big \\lbrace n\\ge i\\ge \\inf &\\lbrace j\\ge i_k|E(\\gamma _j)<E(s_k)\\rbrace \\,\\Big |\\,E(\\gamma _i)\\ge E(s_k),\\\\& \\inf \\lbrace j\\ge i|E(\\gamma _j)<E(\\gamma _{i})\\rbrace <\\inf \\lbrace j\\ge i|E(\\gamma _j)>E(\\gamma _{i})\\rbrace \\Big \\rbrace .$ Note that the energy of these records is increasing.",
"Let $s_{k-1}$ be the last record defined in this way and $s_k:=s$ .",
"Since $E(z^*(m,s))=E(s)$ , $s_k$ is as well a record.",
"Given the records $s_1{,}...,s_k$ , for $1\\le i\\le k-1$ let $r_{2i}$ be the first minimum along $\\gamma $ after $s_i$ and $r_{2i+1}$ the last minimum along $\\gamma $ before $s_{i+1}$ .",
"Here a minimum along $\\gamma $ is some $\\gamma _i\\in \\gamma $ such that it is a minimum of $E$ restricted to $\\gamma $ .",
"Finally, let $r_{2k}:=s_k=s$ .",
"In the following we will proof that (a) $\\mathbb {P}_{r_{2j}}(\\tau _{r_{2j+1}}<\\zeta _0)\\rightarrow 1$ as $\\beta \\rightarrow \\infty $ for any $0\\le j\\le k-1$ , (b) $\\mathbb {P}_{r_1}(\\tau _{r_2}<\\zeta _0)\\ge e^{-\\beta (E(z^*(r_1,r_2))-E(s_m)+o(1))}$ , (c) $\\mathbb {P}_{r_{2j+1}}(\\tau _{r_{2j+2}}<\\zeta _0)\\ge e^{-\\beta (E(z^*(r_{2j+1},r_{2j+2}))-E(z^*(r_{2j-1},r_{2j}))+o(1))}$ for any $1\\le j\\le k-2$ , (d) $\\mathbb {P}_{r_{2k-1}}(\\tau _{r_{2k}}=\\zeta _0)\\ge e^{-\\beta (E(z^*(r_{2k-1},r_{2k}))-E(z^*(r_{2k-3},r_{2k-2}))+o(1))}$ .",
"This gives for $\\beta $ large enough $\\mathbb {P}_m(Y_1=s)\\ &\\ge \\ \\left(\\prod _{j=0}^{2k-2}\\mathbb {P}_{r_j}(\\tau _{r_{j+1}}<\\zeta _0)\\right)\\mathbb {P}_{r_{2k-1}}(\\tau _{r_{2k}}=\\zeta _0)\\\\&\\ge \\ \\frac{1}{2}\\left(\\prod _{j=0}^{k-2}\\mathbb {P}_{r_{2j+1}}(\\tau _{r_{2j+2}}<\\zeta _0)\\right)\\mathbb {P}_{r_{2k-1}}(\\tau _{r_{2k}}=\\zeta _0)\\\\&\\ge \\ \\frac{1}{2} e^{-\\beta (E(z^*(r_1,r_2))-E(s_m)+o(1))}\\cdot \\prod _{j=1}^{k-1}e^{-\\beta (E(z^*(r_{2j+1},r_{2j+2}))-E(z^*(r_{2j-1},r_{2j}))+o(1))}\\\\&=\\ \\frac{1}{2}e^{-\\beta (E(z^*(r_{2k-1},r_{2k}))-E(s_m)+o(1))}\\\\&=\\ e^{-\\beta (E(s)-E(s_m)+o(1))},$ and thus the assertion.",
"(a) For $j=0$ this is obvious since $E(z^*(r_0,r_1))<E(s_m)$ .",
"For $1\\le j\\le k-1$ and any $r^{\\prime }\\in \\partial ^+V(m)$ it holds true that $E(z^*(r_{2j},r_{2j+1}))\\ <\\ E(z^*(r_{2j},r_{2j-1}))\\ \\le \\ E(z^*(r_{2j},m))\\ \\le \\ E(z^*(r_{2j},r^{\\prime })),$ where we make use of the fact that between $r_{2j}$ and $r_{2j+1}$ the energy stays below the last record, and that all subpaths of $\\gamma $ are minimal as well.",
"(b) By the definition of $r_1$ and $r_2$ , there is a unimodal path between them so that the cumulative activation energy along this path equals $E(z^*(r_1,r_2))-E(r_1)$ .",
"Therefore $\\mathbb {P}_{r_1}(\\tau _{r_2}<(\\zeta _0\\wedge \\tau _{r_1}))\\ \\ge \\ e^{-\\beta (E(z^*(r_1,r_2))-E(r_1)+o(1))}.$ Furthermore, for any $r^{\\prime }\\in \\partial ^+V(m)$ we have $E(z^*(r_1,r^{\\prime }))\\ &\\ge \\ E(s_m)>E(r_1)\\quad \\textrm { and }\\quad E(z^*(r_1,r_2))\\ \\ge \\ E(s_m)>E(r_1).$ Therefore, $\\mathbb {P}_{r_1}((\\tau _{r_2}\\wedge \\zeta _0)<\\tau _{r_1})\\ &\\le \\ \\sum _{r^{\\prime }\\in \\partial ^+V(m)\\cup \\lbrace r_2\\rbrace }\\mathbb {P}_{r_1}(\\tau _{r^{\\prime }}<\\tau _{r_1})\\\\&\\le \\ \\sum _{r^{\\prime }\\in \\partial ^+V(m)\\cup \\lbrace r_2\\rbrace }e^{-\\beta (E(z^*(r_1,r^{\\prime }))-E(r_1)+o(1))}\\\\&\\le \\ e^{-\\beta (E(s_m)-E(r_1)+o(1))}.$ Combining the two estimates, we get $\\mathbb {P}_{r_1}(\\tau _{r_2}<\\zeta _0)\\ &=\\ \\frac{\\mathbb {P}_{r_1}(\\tau _{r_2}<(\\zeta _0\\wedge \\tau _{r_1}))}{\\mathbb {P}_{r_1}((\\tau _{r_2}\\wedge \\zeta _0)<\\tau _{r_1})}\\ \\ge \\ e^{-\\beta (E(z^*(r_1,r_2))-E(s_m)+o(1))}.$ (c) Let $1\\le j\\le k-2$ .",
"We use the same strategy as in the proof of (b).",
"So, again, by the definition of $r_{2j+1}$ and $r_{2j+2}$ , there is a unimodal path between them with cumulative activation energy $E(z^*(r_{2j+1},r_{2j+2}))-E(r_{2j+1})$ along this path, and $\\mathbb {P}_{r_{2j+1}}(\\tau _{r_{2j+2}}<(\\zeta _0\\wedge \\tau _{r_{2j+1}}))\\ \\ge \\ e^{-\\beta (E(z^*(r_{2j+1},r_{2j+2}))-E(r_{2j+1})+o(1))}.$ Furthermore, $E(z^*(r_{2j-1},r_{2j}))\\ \\le \\ E(z^*(r_{2j+1},r_{2j+2}))\\quad \\textrm { and }\\quad E(r_{2j+1})\\ <\\ E(z^*(r_{2j+1},r_{2j+2})).$ Finally, for any $r^{\\prime }\\in \\partial ^+V(m)$ , by use of Equation (REF ), $E(r_{2j+1})\\ &\\le E(z^*(r_{2j+1},r_{2j}))\\\\&<\\ E(z^*(r_{2j},r^{\\prime }))\\\\&\\le \\ E(z^*(r_{2j},r_{2j+1}))\\vee E(z^*(r_{2j+1},r^{\\prime }))\\\\&=\\ E(z^*(r_{2j+1},r^{\\prime })).$ Thus, $\\mathbb {P}_{r_{2j+1}}((\\tau _{r_{2j+2}}\\wedge \\zeta _0)<\\tau _{r_{2j+1}})\\ &\\le \\ \\sum _{r^{\\prime }\\in \\partial ^+V(m)\\cup \\lbrace r_{2j+2}\\rbrace }\\mathbb {P}_{r_{2j+1}}(\\tau _{r^{\\prime }}<\\tau _{r_{2j+1}})\\\\&\\le \\ \\sum _{r^{\\prime }\\in \\partial ^+V(m)\\cup \\lbrace r_{2j+2}\\rbrace }e^{-\\beta (E(z^*(r_{2j+1},r^{\\prime }))-E(r_{2j+1})+o(1))}\\\\&\\le \\ e^{-\\beta (E(r_{2j-1},r_{2j})-E(r_{2j+1})+o(1))}.$ Combining the two estimates, we get $\\mathbb {P}_{r_{2j+1}}(\\tau _{r_{2j+2}}<\\zeta _0)\\ &=\\ \\frac{\\mathbb {P}_{r_{2j+1}}(\\tau _{r_{2j+2}}<(\\zeta _0\\wedge \\tau _{r_{2j+1}}))}{\\mathbb {P}_{r_{2j+1}}((\\tau _{r_{2j+2}}\\wedge \\zeta _0)<\\tau _{r_{2j+1}})}\\ \\ge \\ e^{-\\beta (E(z^*(r_{2j+1},r_{2j+2}))-E(r_{2j-1},r_{2j})+o(1))}.$ (d) All bounds for energies in (c) can be proved in the very same way (here $r^{\\prime }\\in \\partial ^+V(m)\\backslash \\lbrace s\\rbrace $ ), so that $\\mathbb {P}_{r_{2k-1}}(\\tau _{r_{2k}}=\\zeta _0)\\ &=\\ \\frac{\\mathbb {P}_{r_{2k-1}}(\\tau _{r_{2k}}=(\\zeta _0\\wedge \\tau _{r_{2k-1}}))}{\\mathbb {P}_{r_{2k-1}}((\\tau _{r_{2k}}\\wedge \\zeta _0)<\\tau _{r_{2k-1}})}\\ \\ge \\ e^{-\\beta (E(z^*(r_{2k-1},r_{2k}))-E(r_{2k-3},r_{2k-2})+o(1))}.$ Now we see for the reciprocating jumps in the accelerated chain: Proposition 3.11 Fix $1\\le i\\le \\mathfrak {n}$ and $\\varepsilon >0$ .",
"Then the AAC at level $i$ exhibits no reciprocating jumps of order $\\varepsilon $ if the following three conditions hold true: (1) $E(z^*(m_0,m_1))-E(s_{m_0})\\le \\varepsilon $ for all distinct $m_0,m_1\\in \\mathcal {S}^{(i)}\\backslash N^{(i)}$ .",
"(2) For each $m\\in \\mathcal {S}^{(i)}\\backslash N^{(i)}$ , there exist at least two distinct $m_1,m_2\\in \\mathcal {S}^{(i)}\\backslash N^{(i)}, m\\ne m_1,m_2,$ such that $\\mathbb {P}_m(X_{\\xi _1}\\in V^{(i)}(m_j))>0$ for $j=1,2$ .",
"(3) For each pair $m_0,m_1\\in \\mathcal {S}^{(i)}\\backslash N^{(i)}$ with $\\mathbb {P}_{m_0}(X_{\\xi _1}\\in V^{(i)}(m_1))>0$ , there exists a minimal uphill-downhill-path from $s_{m_0}$ to $m_1$ not hitting any valley but $V^{(i)}(m_1)$ .",
"The origin of our endeavor to define aggregations with no reciprocating jumps of an order larger than a small $\\varepsilon $ is to obtain an associated process with (almost) decorrelated increments (in Euclidean state space), for this and a proper centering causes the variance up to the $n$ -th jump to grow with $n$ instead of $n^2$ .",
"This is known as diffusive behavior in physics.",
"Without aggregation increments are highly correlated due to the following argument: at any given time, the process is with high probability in a minimum and when leaving it, say by making a positive jump, the next increment is most likely negative because there is a drift back to the minimum.",
"Likewise, the increments of the asymptotic jump chain are neither uncorrelated nor having mean zero since trajectories of ${\\widehat{Y}}$ on an irreducibility class are almost surely of the form $m_1\\rightarrow s\\rightarrow m_2\\rightarrow s\\rightarrow m_3\\rightarrow ...$ , where $s\\in N^{(i)}$ and $m_1,m_2{,}...\\in \\mathcal {S}^{(i)}\\backslash N^{(i)}$ .",
"Thus, given the previous increments, it is in general easy to predict the next increment and they do not have mean zero.",
"On the other hand, if we can choose $\\beta $ and $i$ such that, for any $m_0,m_1$ , we have $\\mathbb {P}_{m_0}(Y_1=m_1)\\in \\lbrace p_{m_0}\\pm \\varepsilon \\rbrace \\cup [0,\\varepsilon ]$ for $\\varepsilon \\ll p_{m_0}$ , we obtain an AAC which behaves roughly like a RW on a graph.",
"Such a RW is diffusive if we assume periodic boundary conditions (or sufficiently large state space compared to the observation time $n$ ) and an energy landscape $E$ which is homogeneous enough to ensure zero-mean increments.",
"In particular $\\lbrace m|\\mathbb {P}_{m_0}(Y_1=m)=p_{m_0}\\pm \\varepsilon \\rbrace $ has to comprise at least two states."
],
[
"Metabasins",
"A path-independent definition of metabasins can now be given on the basis of the previous considerations.",
"Definition 3.12 A finite Markov chain $X$ driven by an energy function $E$ satisfying the assumptions stated at the beginning of Section has metabasins of order $\\varepsilon >0$ if there exists an aggregation level $i<\\mathfrak {n}-1$ such that the following conditions are fulfilled for each $m\\in \\mathcal {S}^{(i)}\\backslash N^{(i)}$ : (MB4) (MB1) $\\sup _{m^{\\prime }\\in \\mathcal {S}^{(i)}\\backslash (N^{(i)}\\cup \\lbrace m\\rbrace )}E(z^*(m,m^{\\prime }))-E(s_m)\\le \\varepsilon $ .",
"(MB2) There are at least two distinct $m_1,m_2\\in \\mathcal {S}^{(i)}\\backslash (N^{(i)}\\cup \\lbrace m\\rbrace )$ with a minimal uphill-downhill-path from $s_m$ to $m_k$ not hitting any other valley but $V^{(i)}(m_{k})$ for $k=1,2$ .",
"In this case, the valleys $(V^{(i)}(m))_{m\\in \\mathcal {S}^{(i)}}$ are called metabasins (MB) of order $\\varepsilon $.",
"The reader should notice that each singleton set $\\lbrace s\\rbrace $ consisting of a non-assigned state $s\\in \\mathcal {N}^{(i)}$ forms a MB.",
"The conditions (MB1) and (MB2) ensure the good nature of (a) the energy barriers and (b) the spatial arrangement of minima.",
"As already pointed out, this determines which valleys are visited consecutively.",
"Properties of MB which can be concluded from the results of the previous sections are summarized in the next theorem.",
"The reader is reminded of Properties 1–4 stated in the Introduction.",
"Theorem 3.13 For MB as defined in Definition REF we have (4) (1) The transition probabilities for jumps between MB do not depend on the point of entrance as $\\beta \\rightarrow \\infty $ (Property 1).",
"(2) There are no reciprocating jumps of order $\\varepsilon $ (Property 2).",
"(3) The expected residence time in a MB depends on $E$ only via the depth of the MB as $\\beta \\rightarrow \\infty $ (Property 3).",
"(4) Regarding only MB pertaining to local minima, the system is a trap model (Property 4).",
"[Proof:] (1) follows from Proposition REF , (2) from Proposition REF , (3) from Theorem REF , and (4) directly from the definition.",
"It should not be surprising that the path-dependent definition of MB by Heuer [12] and stated in the Introduction differs from our path-independent one.",
"Figure: Example of an energy landscape with a tree-like structure.For example, the energy landscape in Figure REF has no reasonable path-dependent MB because every transition between two branches of the shown tree must pass through the state $x$ .",
"For a typical trajectory, there will be at most three MB: the states visited before the first occurrence of $x$ , the states visited between the first and the last occurrence of $x$ , and the states visited after the last occurrence of $x$ .",
"The reason for this poor performance is the tree-like structure of the energy landscape or, more generally, the fact that the connectivity between the branches is too small to allow a selfavoiding walk through more than two branches.",
"This results in a small recurrence time for $x$ (compared to the number of states visited in between).",
"However, every branch constitutes a MB when using the path-independent definition for sufficiently small $\\varepsilon $ , in which case the AAC forms a Markov chain and, given the Metropolis algorithm, even a RW on the graph.",
"Having thus exemplified that the two definitions of MB do not necessarily coincide, where the path-independent approach applies to a wider class of energy landscapes, we turn to the question about conditions for them to coincide with high probability.",
"As already pointed out, we have to assume a sufficient connectivity between the metastates to ensure the existence of reasonable path-dependent MB.",
"In terms of this connectivity (for a precise definition see Definition REF ) and the parameter $\\beta $ and $\\varepsilon $ , our last result, Theorem REF below, provides lower bounds for the probability that both definitions yield the same partition of the state space.",
"The first step towards this end is to identify and count, for each $m\\in \\mathcal {S}^{(i)}$ and a given $\\beta $ , the states $s\\in \\mathcal {S}^{(i)}$ for which a transition of $Y$ from $m$ to $s$ is likely.",
"This leads to the announced connectivity parameter.",
"Definition 3.14 Let $\\varepsilon >0$ and suppose that $X$ has MB of order $\\varepsilon >0$ at level $i$ .",
"Define the connectivity parameters $\\eta _1\\ &=\\ \\eta _{1,\\varepsilon }\\ :=\\ \\min _{n\\in N^{(i)},r\\in \\mathcal {S}^{(i)}:V^{(i)}(r)\\cap \\mathcal {N}(n)\\ne \\emptyset }\\left|\\left\\lbrace s\\in \\mathcal {N}(n)\\backslash V^{(i)}(r)\\big |\\,E(s)\\le E(n)+\\varepsilon \\right\\rbrace \\right|,\\nonumber \\\\\\eta _2\\ &=\\ \\eta _{2,\\varepsilon }\\ :=\\ \\min _{m\\in \\mathcal {S}^{(i)}\\backslash N^{(i)}}\\big |\\lbrace s\\in \\partial ^+V^{(i)}(m)|E(s)\\le E(s_m)+\\varepsilon \\rbrace \\big |,\\\\\\eta _3\\ &=\\ \\eta _{3,\\varepsilon }\\ :=\\ \\min _{n\\in N^{(i)}}\\big |\\lbrace s\\in \\mathcal {S}^{(i)}| E(x)\\le E(n)+\\varepsilon \\textrm { for some x\\in V^{(i)}(s)\\cap \\mathcal {N}(n)}\\rbrace \\big |.\\nonumber $ $\\eta _1$ is the minimal number of neighboring sites of a non-assigned state $n$ which do not belong to a particular neighboring valley and whose energy is at most $\\varepsilon $ plus the energy of $n$ .",
"$\\eta _2$ is the minimal number of neighboring sites/valleys of a valley $V^{(i)}(m)$ whose energy is at most $\\varepsilon $ plus the energy of $s_m$ and which can be reached via an uphill-path from $m$ .",
"Finally, $\\eta _3$ is the minimal number of neighboring valleys of a non-assigned state $n$ which comprise a state with energy of at most $\\varepsilon $ plus the energy of $n$ .",
"$\\eta _1$ and $\\eta _3$ are always at least 2 and in fact quite large in the very complex energy landscapes of structural glasses.",
"For very small $\\varepsilon $ , $\\eta _2$ may be 1, but if $X$ has MB of order $\\varepsilon $ in a high dimensional energy landscape, then $\\eta _2$ can be assumed to be quite large as well.",
"That transitions to states counted above have reasonable large probabilities is content of the following lemma.",
"Thus, the defined parameters do in fact measure the connectivity of the MB.",
"Lemma 3.15 Let $\\varepsilon >0$ and suppose that $X$ has MB of order $\\varepsilon >0$ at level $i$ with connectivity parameters defined in (REF ).",
"Writing $Y_{k}$ for $Y_{k}^{(i)}$ and $V(m)$ for $V^{(i)}(m)$ , $m\\in \\mathcal {S}^{(i)}$ , the following assertions hold true for all sufficiently large $\\beta $ : (a) If $m\\in \\mathcal {S}^{(i)}\\backslash N^{(i)}$ and $s\\in \\partial ^{+}V(m)\\cap \\lbrace x|E(x)-E(s_{m})\\le \\varepsilon \\rbrace $ , or $m\\in N^{(i)}$ and $s\\in \\mathcal {S}^{(i)}$ satisfies $V(s)\\cap \\lbrace x\\in \\mathcal {N}(m)|E(x)-E(m)\\le \\varepsilon \\rbrace \\ne \\emptyset $ , then $\\mathbb {P}_{m}(Y_{1}=s)\\ \\ge \\ e^{-2\\beta \\varepsilon }.$ (b) For any distinct $m\\in N^{(i)}$ and $s\\in \\mathcal {S}^{(i)}$ , $\\mathbb {P}_{m}(Y_1\\ne s)\\ \\ge \\ \\eta _1\\,e^{-2\\beta \\varepsilon },$ (c) For any distinct $m\\in \\mathcal {S}^{(i)}\\backslash N^{(i)}$ and $s\\in \\mathcal {S}^{(i)}$ , $\\mathbb {P}_{m}(Y_1\\ne s)\\ \\ge \\ (\\eta _2-1)\\,e^{-2\\beta \\varepsilon },$ We see that, for $\\varepsilon $ small enough compared to $\\beta $ , transitions with an energy barrier of at most $\\varepsilon $ are still quite likely and thus a jump to a particular valley quite unlikely in the case of high connectivity.",
"[Proof:] (a) Choose $\\beta _{0}>0$ so large that, for $\\beta \\ge \\beta _{0}$ , $\\gamma _{\\beta }\\le \\varepsilon $ and $\\mathbb {P}_m(Y_1=s)\\ge e^{-2\\beta (E(s)-E(s_m))}$ for any $m\\in \\mathcal {S}^{(i)}\\backslash N^{(i)}$ and $s\\in \\partial ^{+}V(m)$ , the latter being possible by Lemma REF .",
"Then for any such $m$ and $s$ , we infer $\\mathbb {P}_{m}(Y_1=s)\\ge e^{-2\\varepsilon \\beta }$ provided that additionally $E(s)\\le E(s_m)+\\varepsilon $ holds true.",
"If $m\\in N^{(i)}$ , then $\\mathbb {P}_{m}(Y_1=s)\\ge e^{-2\\varepsilon \\beta }$ for any $s\\in \\mathcal {S}^{(i)}$ such that $E(x)\\le E(s)+\\varepsilon $ for some $x\\in V(s)\\cap \\mathcal {N}(m)$ , for $\\mathbb {P}_{m}(Y_1=s)\\ \\ge \\ \\mathbb {P}_{m}(X_{\\sigma _{1}}=x)\\ =\\ p(m,x)\\ \\ge \\ e^{-\\beta ((E(x)-E(m))^{+}+\\gamma _{\\beta })}.$ (b) Pick again $\\beta _{0}$ so large that $\\gamma _{\\beta }\\le \\varepsilon $ for all $\\beta \\ge \\beta _{0}$ .",
"Then, $\\mathbb {P}_{m}(Y_1\\ne s)\\ &\\ge \\ \\sum _{x\\in \\mathcal {N}(m),x\\notin V(s)}p(m,x)\\\\&\\ge \\ \\sum _{x\\in \\mathcal {N}(m),x\\notin V(s), E(x)\\le E(m)+\\varepsilon }\\exp \\big (-\\beta ((E(x)-E(m))^{+}+\\gamma _{\\beta })\\big )\\\\&\\ge \\ \\eta _1\\exp (-2\\beta \\varepsilon ),$ by definition of $\\eta _1$ .",
"(c) Fix $\\beta _{0}$ so large that $\\mathbb {P}_m(Y_1=x)\\ge e^{-2\\beta (E(x)-E(s_m))}$ for any $x\\in \\partial ^{+}V(m)$ and $\\beta \\ge \\beta _{0}$ .",
"In the very same way as in part (b), we then get for all $\\beta \\ge \\beta _0$ $\\mathbb {P}_{m}(Y_1\\ne s)\\ &\\ge \\ \\sum _{x\\in \\partial ^+V(m),x\\ne s}\\mathbb {P}_m(Y_1=x)\\\\&\\ge \\ \\sum _{x\\in \\partial ^+V(m),x\\ne s, E(x)\\le E(s_m)+\\varepsilon }\\exp \\big (-2\\beta (E(x)-E(s_m))\\big )\\\\&\\ge \\ (\\eta _2-1)\\exp (-2\\beta \\varepsilon ),$ by definition of $\\eta _2$ .",
"The above result motivates that in the case of high connectivity the probability to revisit a particular valley within a fixed time $T$ is quite small, or in other words, the probability for the AAC to jump along a selfavoiding path is quite high.",
"This is the main step towards the announced theorem and stated below.",
"The observation time $T$ of course has to be small compared to the cover time of the process.",
"Lemma 3.16 Let $\\varepsilon >0$ and suppose that $X$ has MB of order $\\varepsilon >0$ at level $i$ with connectivity parameters defined in (REF ).",
"Writing $Y_{k}$ for $Y_{k}^{(i)}$ and $V(m)$ for $V^{(i)}(m)$ , $m\\in \\mathcal {S}^{(i)}$ , define $\\tau _{V(m)}^{(i)}:=\\inf \\lbrace k\\ge 1|Y_{k}=m\\rbrace .$ Then the following assertions hold true for all sufficiently large $\\beta $ : (a) For any $0<\\delta <1-\\mathbb {P}_{m}(Y_{2}=m)$ and $1\\le T\\le T(m,\\beta )+1$ , $\\mathbb {P}_{m}\\left(\\tau _{V(m)}^{(i)}>T\\right)\\ \\ge \\ \\delta ,$ where $T(m,\\beta )\\ :=\\ \\frac{\\ln \\delta }{\\ln \\!\\big (\\min _{m^{\\prime }\\ne m}\\mathbb {P}_{m^{\\prime }}(Y_1\\ne m)(1-{1}_{\\lbrace m^{\\prime }\\notin N^{(i)}\\rbrace }\\,\\delta (m^{\\prime },\\beta ))\\big )}$ and $\\delta (m^{\\prime },\\beta )\\ :=\\ \\max _{x\\in V(m^{\\prime })}\\sum _{z\\in \\partial ^+ V(m^{\\prime })}\\tilde{\\varepsilon }(x,m^{\\prime },z,\\beta ).$ In particular, if $\\delta \\le \\left((\\eta _1\\wedge (\\eta _2-2))e^{-2\\beta \\varepsilon }\\right)^{T}$ for some $T>0$ , then $T(m,\\beta )\\ge T$ .",
"(b) For each $k\\ge 1$ and $m_{0}\\in \\mathcal {S}^{(i)}$ , $\\sum _{m_{1}{,}...,m_{k}}\\prod _{j=0}^{k-1}\\mathbb {P}_{m_j}(Y_1=m_{j+1})\\ \\ge \\ [\\eta _2\\wedge \\eta _3]_{k}\\,e^{-2k\\varepsilon \\beta }$ where summation ranges over all pairwise distinct $m_{1}{,}...,m_{k}\\in \\mathcal {S}^{(i)}\\backslash \\lbrace m_0\\rbrace $ and for $N\\in \\mathbb {N}$ we write $[N]_{k}:=N(N-1)\\cdot ...\\cdot (N-k+1)$ .",
"It should be noticed that $\\mathbb {P}_{m}(\\tau _{V(m)}^{(i)}>1)=1$ (the AAC never stays put) and $\\mathbb {P}_{m}(\\tau _{V(m)}^{(i)}>T)\\ \\le \\ \\mathbb {P}_{m}(\\tau _{V(m)}^{(i)}>2)\\ =\\ 1-\\mathbb {P}_{m}(Y^{(i)}_2=m)$ for every $T\\ge 2$ with equality holding only if $T=2$ .",
"We thus see that $\\mathbb {P}_{m}(\\tau _{V(m)}^{(i)}>T)\\ge \\delta $ entails $\\delta <1-\\mathbb {P}_{m}(Y_2=m)$ , the latter being typically large.",
"Furthermore, the bound on the number of self-avoiding path of length $k$ is very crude and can be improved when knowing more about the spatial arrangement of the metastable states.",
"[Proof:] (a) Recall from the first part of the proof of Proposition REF that $\\mathbb {P}_{m}(Y_{n+1}\\ne z|Y_n=y,X_{\\sigma _n}=x)\\ &\\ge \\ \\mathbb {P}_{y}(Y_1\\ne z)\\left(1-{1}_{\\lbrace y\\notin N^{(i)}\\rbrace }\\sum _{r\\in \\partial ^{+}V(y)}\\mathbb {P}_{x}(\\tau _{r}<\\tau _{y})\\right)\\\\&\\ge \\ \\mathbb {P}_{y}(Y_1\\ne z)\\left(1-{1}_{\\lbrace y\\notin N^{(i)}\\rbrace } \\sum _{r\\in \\partial ^+V(y)}\\tilde{\\varepsilon }(x,y,r,\\beta )\\right)\\\\&\\ge \\ \\mathbb {P}_{y}(Y_1\\ne z)\\left(1-{1}_{\\lbrace y\\notin N^{(i)}\\rbrace }\\,\\delta (y,\\beta )\\right)\\\\$ holds true for all $y,z\\in \\mathcal {S}^{(i)},\\,x\\in V(y)$ and $\\beta >0$ .",
"This will now be used repeatedly to show that $\\mathbb {P}_{m}\\Big (\\tau _{V(m)}^{(i)}>T\\Big )\\ \\ge \\ \\left(\\min _{m^{\\prime }\\ne m}\\mathbb {P}_{m^{\\prime }}(Y_1\\ne m)(1-{1}_{\\lbrace m^{\\prime }\\notin N^{(i)}\\rbrace }\\,\\delta (m^{\\prime },\\beta ))\\right)^{T-1}$ for each $T>2$ .",
"Putting $\\mathfrak {m}(x):=m^{\\prime }$ if $x\\in V(m^{\\prime })$ for $m^{\\prime }\\in \\mathcal {S}^{(i)}$ , we obtain $\\mathbb {P}_{m}&\\Big (\\tau _{V(m)}^{(i)}>T\\Big )\\\\&=\\ \\mathbb {P}_{m}(Y_{1}\\ne m{,}...,Y_{T}\\ne m)\\\\&=\\ \\sum _{x_1{,}...,x_{T-1}\\notin V^{(i)}(m)}\\mathbb {P}_{m}(X_{\\sigma _1}=x_1)\\prod _{k=1}^{T-2}\\mathbb {P}_{m}\\big (X_{\\sigma _{k+1}}=x_{k+1}|X_{\\sigma _{k}}=x_{k}, Y_{k}=\\mathfrak {m}(x_k)\\big )\\\\&\\hspace*{76.82234pt}\\times \\mathbb {P}_{m}(Y_{T}\\ne m|X_{\\sigma _{T-1}}=x_{T-1}, Y_{T-1}=\\mathfrak {m}(x_{T-1}))\\\\&\\ge \\ \\sum _{x_1{,}...,x_{T-1}\\notin V^{(i)}(m)}\\mathbb {P}_{m}(X_{\\sigma _1}=x_1)\\prod _{k=1}^{T-2}\\mathbb {P}_{m}\\big (X_{\\sigma _{k+1}}=x_{k+1}|X_{\\sigma _{k}}=x_{k}, Y_{k}=\\mathfrak {m}(x_k)\\big )\\\\&\\hspace{48.36958pt}\\times \\left(\\min _{m^{\\prime }\\ne m}\\left(\\mathbb {P}_{m^{\\prime }}(Y_1\\ne m)(1-{1}_{\\lbrace m^{\\prime }\\notin N^{(i)}\\rbrace }\\,\\delta (m^{\\prime },\\beta ))\\right)\\right)\\\\&\\hspace{5.69046pt}\\vdots \\\\&\\ge \\ \\min _{m^{\\prime }\\ne m}\\Big (\\mathbb {P}_{m^{\\prime }}(Y_1\\ne m)(1-{1}_{\\lbrace m^{\\prime }\\notin N^{(i)}\\rbrace }\\,\\delta (m^{\\prime },\\beta ))\\Big )^{T-1}.\\\\$ But this establishes the asserted inequality when finally observing that the last expression is $\\ge \\delta $ iff $T\\le T(m,\\beta )+1$ .",
"Having just said that $T(m,\\beta )\\ge T$ holds iff $\\min _{m^{\\prime }\\ne m}\\Big (\\mathbb {P}_{m^{\\prime }}(Y_1\\ne m)(1-{1}_{\\lbrace m^{\\prime }\\notin N^{(i)}\\rbrace }\\,\\delta (m^{\\prime },\\beta ))\\Big )^{T}\\ \\ge \\ \\delta ,$ it suffices to note that, as $\\beta \\rightarrow \\infty $ , $\\delta (m^{\\prime },\\beta )\\rightarrow 0$ holds true if $m^{\\prime }\\in \\mathcal {S}^{(i)}\\backslash N^{(i)}$ , giving $1-\\delta (m^{\\prime },\\beta )\\ \\ge \\ \\frac{\\eta _2-2}{\\eta _2-1}$ for sufficiently large $\\beta $ .",
"Together with Lemma REF (b), this further yields $\\min _{m^{\\prime }\\ne m}\\Big (\\mathbb {P}_{m^{\\prime }}(Y_1\\ne m)(1-{1}_{\\lbrace m^{\\prime }\\notin N^{(i)}\\rbrace }\\,\\delta (m^{\\prime },\\beta ))\\Big )^{T}&\\ge \\ \\left((\\eta _1\\wedge (\\eta _2-2))e^{-2\\beta \\varepsilon }\\right)^{T}$ and then the assertion.",
"(c) Here it suffices to notice that, by (a), $[\\eta _2\\wedge \\eta _3]_{k}$ forms a lower bound for the number of self-avoiding paths $(m_{0}{,}...,m_{k})$ such that $\\mathbb {P}_{m_{j}}(Y_{1}=m_{j+1})\\ge e^{-2\\beta \\varepsilon }$ for each $j=0{,}...,k-1$ .",
"We proceed with the announced result about the relation between path-dependent and path-independent MB.",
"To this end, we fix $T=\\sigma _{K}$ for some $K\\in \\mathbb {N}$ .",
"Let $\\mathcal {V}_{k}$ for $k=1{,}...,\\upsilon $ denote the random number of MB obtained from $X_{0}{,}...,X_{T}$ as defined in the Introduction.",
"For $x\\in \\mathcal {S}$ , we further let $\\mathcal {V}(x)$ denote the MB $\\mathcal {V}_{k}$ containing $x$ and put $\\mathcal {V}(x):=\\emptyset $ if no such MB exists which is the case iff $x\\notin \\lbrace X_{0}{,}...,X_{T}\\rbrace $ .",
"Theorem 3.17 Let $\\varepsilon >0$ and suppose that $X$ has MB of order $\\varepsilon >0$ at level $i$ with connectivity parameters defined in (REF ).",
"Fix $K\\in \\mathbb {N}$ , $T=\\sigma _{K}$ and $0<\\delta \\le \\left((\\eta _1\\wedge (\\eta _2-1)-1)e^{-2\\beta \\varepsilon }\\right)^{K}$ .",
"Then, for each $0\\le k<K$ and $m_0\\in \\mathcal {S}^{(i)}$ , there exists $\\beta _0>0$ such that for all $\\beta \\ge \\beta _0$ (a) $\\mathbb {P}_{m_0}\\!\\left(V^{(i)}_<(Y_{k})\\subseteq \\mathcal {V}(Y_{k})\\right)\\ge 1-(\\max _{m\\in \\mathcal {S}^{(i)}\\backslash N^{(i)}}|V_<(m)|+2)\\max _{m\\in \\mathcal {S}^{(i)}\\backslash N^{(i)}}\\delta (m,\\beta )$ , where $V_{<}^{(i)}(s):=\\lbrace s\\rbrace $ if $s\\in N^{(i)}$ .",
"(b) $\\mathbb {P}_{m_0}(\\mathcal {V}(Y_{j})\\subseteq V^{(i)}(Y_{j}),\\,0\\le j<k)\\ge 1-k(\\max _{m\\in \\mathcal {S}^{(i)}\\backslash N^{(i)}}\\delta (m,\\beta )+(1-\\delta ))$ .",
"(c) If $\\eta _2\\wedge \\eta _3>K-1$ , then $\\mathbb {P}_{m_0}(\\mathcal {V}(Y_j)\\subseteq V^{(i)}(Y_j),\\,0\\le j\\le K-1)\\ \\ge \\ [\\eta _2\\wedge \\eta _3]_{K}\\left(1-\\max _{m\\in \\mathcal {S}^{(i)}\\backslash N^{(i)}}\\delta (m,\\beta )\\right)^{K-1}e^{-2K\\varepsilon \\beta }.$ For the occurring bounds to be significant, two requirements must be met.",
"First, $K$ must be small compared to the cover time of the AAC and $\\varepsilon $ must be small compared to $\\beta _0$ to ensure $\\exp (-2\\beta \\varepsilon )\\gg 0$ .",
"Second, the connectivity must be high to ensure $1-\\delta \\ll 1$ and $[\\eta _2\\wedge \\eta _3]_{K}e^{-2K\\varepsilon \\beta }\\gg 0$ .",
"Typically, the inclusions in parts (b) and (c) are strict because of high energy states within a valley that will probably be missed during one simulation run and therefore not belong to any path-dependent MB.",
"On the other hand, since our approach strives to cover the state space as completely as possible by valleys the latter comprise such high energy states whenever they are assignable in the sense described in Section .",
"[Proof:] With $i$ being fixed, let us write as earlier $V(m)$ for $V^{(i)}(m)$ , and also $V_{<}(m)$ for $V_{<}^{(i)}(m)$ .",
"(a) For a given $0\\le k<K$ , define $A_k&:=\\lbrace \\sigma _k\\le \\tau _{Y_k}<\\sigma _{k+1}\\rbrace ,\\\\B_k&:=\\lbrace \\textrm { for every x\\in V_{<}(m) exists \\tau _{Y_k}\\le l_x<\\sigma _k such that X_l=x}\\rbrace ,\\\\C_k&:=\\lbrace X_l=Y_k \\textrm { for some \\max _{x\\in V_<(Y_k)}\\tau _x\\le l<\\sigma _k}\\rbrace .$ With $\\delta (m,\\beta )$ as defined in Lemma REF and using $\\mathbb {P}_{x}(\\sigma _1<\\tau _m)\\ &\\le \\ \\sum _{y\\in \\partial ^+V(m)}\\tilde{\\varepsilon }(x,m,y,\\beta )\\ \\le \\ \\max _{m\\in \\mathcal {S}^{(i)}\\backslash N^{(i)}}\\delta (m,\\beta )\\ =:\\ \\delta _{\\max }$ for $x\\in V(m),\\,m\\in \\mathcal {S}^{(i)}\\backslash N^{(i)}$ , we obtain $\\mathbb {P}_{m_0}\\!&\\left(V^{(i)}_<(Y_{k})\\subseteq \\mathcal {V}(Y_{k})\\right)\\nonumber \\\\&\\ge \\ \\mathbb {P}_{m_0}(A_k\\cap B_k\\cap C_k)\\nonumber \\\\&=\\ \\sum _{m\\in \\mathcal {S}^{(i)},r\\in V(m)}\\mathbb {P}_{m_0}(\\lbrace X_{\\sigma _k}=r\\rbrace \\cap A_k\\cap B_k\\cap C_k)\\nonumber \\\\&=\\ \\sum _{m\\in \\mathcal {S}^{(i)},r\\in V(m)}\\mathbb {P}_{m_0}(X_{\\sigma _k}=r)\\mathbb {P}_{m_0}(A_k|X_{\\sigma _k}=r)\\mathbb {P}_{m_0}(B_k\\cap C_k|\\lbrace X_{\\sigma _k}=r\\rbrace \\cap A_k)\\nonumber \\\\&=\\ \\sum _{m\\in \\mathcal {S}^{(i)},r\\in V(m)}\\mathbb {P}_{m_0}(X_{\\sigma _k}=r)\\mathbb {P}_r(\\tau _m<\\sigma _1)\\nonumber \\\\&\\hspace{28.45274pt}\\times \\mathbb {P}_m(\\tau _x<\\sigma _1 \\textrm { for every x\\in V_<(m), X_l=m for some \\max _{x\\in V_<(m)}\\tau _x\\le l<\\sigma _1})\\nonumber \\\\&\\ge \\ \\sum _{m\\in \\mathcal {S}^{(i)},r\\in V(m)}\\mathbb {P}_{m_0}(X_{\\sigma _k}=r)(1-\\delta _{\\max })\\nonumber \\\\&\\hspace{28.45274pt}\\times \\mathbb {P}_m(\\tau _x<\\sigma _1 \\textrm { for every x\\in V_<(m), X_l=m for some \\max _{x\\in V_<(m)}\\tau _x\\le l<\\sigma _1})\\nonumber \\\\&=\\ (1-\\delta _{\\max })\\sum _{m\\in \\mathcal {S}^{(i)}}\\mathbb {P}_{m_0}(Y_k=m)\\nonumber \\\\&\\hspace{28.45274pt}\\times \\mathbb {P}_m(\\tau _x<\\sigma _1 \\textrm { for every x\\in V_<(m), X_l=m for some \\max _{x\\in V_<(m)}\\tau _x\\le l<\\sigma _1}).$ Thus, in order to show that with high probability a path-dependent MB comprises the inner part of a valley, we show that with high probability, when starting in its minimum, the whole inner part will be visited and the process will return to the minimum once more before the valley is left.",
"This is trivial if $m\\in N^{(i)}$ and thus $V_{<}^{(i)}(m)=\\lbrace m\\rbrace $ , for then $\\mathbb {P}_m(\\tau _x<\\sigma _1 \\textrm { for every $ V<(m)$, $ Xl=m$ for some $ xV<(m)xl<1$})=1.$$$ More needs to be done if $m\\in \\mathcal {S}^{(i)}\\backslash N^{(i)}$ , where $\\mathbb {P}_{m}&\\left(\\tau _x<\\sigma _1 \\textrm { for every \\right.x\\in V_<(m), X_l=m for some \\max _{x\\in V_<(m)}\\tau _x\\le l<\\sigma _1}\\\\&\\ge \\ 1-\\mathbb {P}_{m}\\left(\\tau _x> \\sigma _1 \\textrm { for some \\right.x\\in V_<(m)}-\\mathbb {P}_m\\big (\\textrm {X_l\\ne m for each \\max _{x\\in V_<(m)}\\tau _x\\le l<\\sigma _1}\\big ).$ The second probability in the preceding line can further be bounded with the help of (REF ), viz.",
"$\\mathbb {P}_m&(\\textrm {X_l\\ne m for each \\max _{x\\in V_<(m)}\\tau _x\\le l<\\sigma _1})\\\\&=\\ \\sum _{y\\in V_<(m)}\\mathbb {P}_m(\\max _{x\\in V_<(m)}\\tau _x=\\tau _y,\\textrm { X_l\\ne m for every \\max _{x\\in V_<(m)}\\tau _x\\le l<\\sigma _1})\\\\&\\le \\ \\sum _{y\\in V_<(m)}\\mathbb {P}_m(\\max _{x\\in V_<(m)}\\tau _x=\\tau _y)\\mathbb {P}_y(\\tau _m>\\sigma _1)\\\\&\\le \\ \\delta _{\\max },$ while for the first probability, we obtain with the help of Theorem REF $\\mathbb {P}_{m}\\left(\\tau _x> \\sigma _1 \\textrm { for some \\right.x\\in V_<(m)}\\ &\\le \\sum _{x\\in V_<(m)}\\mathbb {P}_{m}(\\sigma _1<\\tau _x)\\nonumber \\\\&\\le \\sum _{x\\in V_<(m)}\\sum _{y\\in \\partial ^+V(m)}\\mathbb {P}_m(\\tau _y<\\tau _x)\\nonumber \\\\&\\le \\sum _{x\\in V_<(m)}\\sum _{y\\in \\partial ^+V(m)}\\varepsilon (m,x,y,\\beta ),$ because $E(z^*(m,y))>E(z^*(m,x))$ for $x\\in V_<(m)$ and $y\\in \\partial ^+V(m)$ .",
"The latter can be seen as follows: It has been shown in the proof of Theorem REF that $E(z^*(x,m))<E(z^*(x,y))$ .",
"Hence, $E(z^*(x,m))<E(z^*(x,y))\\le E(z^*(x,m))\\vee E(z^*(y,m))=E(z^{*}(y,m))$ as asserted.",
"Next, we infer $E(z^*(x,y))\\le E(z^*(x,m))\\vee E(z^*(m,y))\\le E(z^*(x,m))\\vee E(z^*(x,y))=E(z^*(x,y)),$ thus $E(z^*(x,y))=E(z^*(m,y))$ .",
"Recalling the definition of $\\varepsilon (m,x,y,\\beta )$ , the last equality implies $\\varepsilon (m,x,y,\\beta )=\\varepsilon (x,m,y,\\beta )$ which will now be used to further bound the expression in (REF ), namely $\\sum _{x\\in V_<(m)}\\sum _{y\\in \\partial ^+V(m)}\\varepsilon (m,x,y,\\beta )\\ &=\\ \\sum _{x\\in V_<(m)}\\sum _{y\\in \\partial ^+V(m)}\\varepsilon (x,m,y,\\beta )\\\\&=\\ \\sum _{x\\in V_<(m)}\\sum _{y\\in \\partial ^+V(m)}\\tilde{\\varepsilon }(x,m,y,\\beta )\\\\&\\le \\ |V_<(m)|\\,\\delta _{\\max }\\\\&\\le \\ \\max _{m\\in \\mathcal {S}^{(i)}\\backslash N^{(i)}}|V_<(m)|\\,\\delta _{\\max }.$ Together with (REF ) this yields as asserted $\\mathbb {P}_{m_0}\\!&\\left(V^{(i)}_<(Y_{k})\\subseteq \\mathcal {V}(Y_{k})\\right)\\\\&\\ge \\ (1-\\delta _{\\max })\\sum _{m\\in \\mathcal {S}^{(i)}}\\mathbb {P}_{m_0}(Y_k=m)\\\\&\\hspace{56.9055pt}\\times \\mathbb {P}_m(\\tau _x<\\sigma _1 \\textrm { for every x\\in V_<(m), X_l=m for some \\max _{x\\in V_<(m)}\\tau _x\\le l<\\sigma _1})\\\\&\\ge \\ \\left(1-\\delta _{\\max }\\right)\\left(1-\\left(\\max _{m\\in \\mathcal {S}^{(i)}\\backslash N^{(i)}}|V_<(m)|+1\\right)\\delta _{\\max }\\right)\\\\&\\ge \\ 1-\\left(\\max _{m\\in \\mathcal {S}^{(i)}\\backslash N^{(i)}}|V_<(m)|+2\\right)\\delta _{\\max }.$ (b) According to Lemma REF , choose $\\beta _{0}>0$ such that $\\max _{m\\in \\mathcal {S}^{(i)}}\\mathbb {P}_{m}\\left(\\tau _{V(m)}^{(i)}\\le K\\right)\\ \\le \\ 1-\\delta $ for each $\\beta \\ge \\beta _{0}$ .",
"By using (REF ) and (REF ), we now infer $\\mathbb {P}_{m_0}\\!&\\left(Y_l= Y_k\\textrm { for some }k+1\\le l\\le K\\right)\\\\&=\\ \\sum _{s\\in \\mathcal {S}}\\mathbb {P}_{m_0}\\!\\left(Y_l= Y_k\\textrm { for some }k+1\\le l\\le K,X_{\\sigma _{k}}=s\\right)\\\\&\\le \\ \\sum _{s\\in \\mathcal {S}}\\mathbb {P}_{m_0}(X_{\\sigma _{k}}=s)\\Big (\\mathbb {P}_{s}(\\tau _{V(Y_0)}^{(i)}\\le K-k,\\,X_{j}=\\mathfrak {m}(s) \\text{ for some }0\\le j< \\sigma _1)\\\\&\\hspace*{48.36958pt}+{1}_{\\lbrace s\\notin N^{(i)}\\rbrace }\\,\\mathbb {P}_{s}(\\tau _{V(Y_0)}^{(i)}\\le K-k, X_{j}\\ne \\mathfrak {m}(s) \\text{ for all }0\\le j< \\sigma _1)\\Big )\\\\&\\le \\ \\sum _{s\\in \\mathcal {S}}\\mathbb {P}_{m_0}(X_{\\sigma _{k}}=s)\\,\\mathbb {P}_{\\mathfrak {m}(s)}(\\tau _{V(Y_0)}^{(i)}\\le K-k)+\\sum _{s\\notin N^{(i)}}\\mathbb {P}_{m_0}(X_{\\sigma _{k}}=s)\\,\\mathbb {P}_{s}(\\sigma _1<\\tau _{\\mathfrak {m}(s)})\\\\&\\le \\ 1-\\delta +\\delta _{\\max },$ and finally $\\mathbb {P}_{m_0}(\\mathcal {V}(Y_j)\\subset V^{(i)}(Y_j),\\,0\\le j< k)\\ &\\ge \\ \\mathbb {P}_{m_0}\\left(\\bigcap _{j=0}^{k-1}\\left\\lbrace Y_l\\ne Y_j,j+1\\le l\\le K\\right\\rbrace \\right)\\\\&\\ge \\ 1-\\sum _{j=0}^{k-1}\\mathbb {P}_{m_0}(Y_l= Y_j\\textrm { for some }j+1\\le l\\le K)\\\\&\\ge \\ 1-k(\\delta _{\\max }+(1-\\delta )).$ (c) In the following calculation, let $r_{0}=m_{0}$ , $\\sum _{m_{j}}$ range over all $K$ -vectors $(m_{1},...,m_{K})$ with pairwise distinct components in $\\mathcal {S}^{(i)}\\backslash \\lbrace m_0\\rbrace $ and, for each $k<K$ , let $\\sum _{r_{1}{,}...,r_{k}}$ range over all $k$ -vectors $(r_{1}{,}...,r_{k})$ such that $r_{j}\\in V(m_{j})$ for each $j=1{,}...,k$ .",
"As in part (b), use (REF ) repeatedly to infer $&\\mathbb {P}_{m_0}(\\mathcal {V}(Y_j)\\subset V^{(i)}(Y_j),\\,0\\le j\\le K-1)\\\\&\\ge \\ \\sum _{m_{j}}\\sum _{r_{1}{,}...,r_{K-1}}\\mathbb {P}_{m_0}\\left(\\bigcap _{j=0}^{K-1}\\lbrace Y_j=m_j, X_{\\sigma _j}=r_j,\\,\\tau _{m_j}<\\sigma _{j+1}\\rbrace \\cap \\lbrace Y_K=m_K\\rbrace \\right)\\\\&=\\ \\sum _{m_{j}}\\sum _{r_{1}{,}...,r_{K-1}}\\prod _{j=0}^{K-2}\\mathbb {P}_{r_{j}}\\left(Y_0=m_j,X_{\\sigma _{1}}=r_{j+1},\\,\\tau _{m_{j}}<\\sigma _1\\right)\\,\\mathbb {P}_{r_{K-1}}\\left(\\tau _{m_{K-1}}<\\sigma _1\\right)\\,\\mathbb {P}_{m_{K-1}}(Y_1=m_K)\\\\&\\ge \\ (1-\\delta _{\\max })\\sum _{m_{j}}\\sum _{r_{1}{,}...,r_{K-1}}\\prod _{j=0}^{K-2}\\mathbb {P}_{r_{j}}\\left(Y_0=m_j, X_{\\sigma _{1}}=r_{j+1},\\,\\tau _{m_{j}}<\\sigma _1\\right)\\mathbb {P}_{m_{K-1}}(Y_1=m_K)$ $&=\\ (1-\\delta _{\\max })\\sum _{m_{j}}\\sum _{r_{1}{,}...,r_{K-2}}\\prod _{j=0}^{K-3}\\mathbb {P}_{r_{j}}\\left(Y_0=m_j, X_{\\sigma _{1}}=r_{j+1},\\,\\tau _{m_{j}}<\\sigma _1\\right)\\\\&\\hspace{79.6678pt}\\times \\mathbb {P}_{r_{K-2}}\\left(Y_0=m_{K-2}, Y_1=m_{K-1},\\,\\tau _{m_{K-2}}<\\sigma _1\\right)\\,\\mathbb {P}_{m_{K-1}}(Y_1=m_K)\\\\&=\\ (1-\\delta _{\\max })\\sum _{m_{j}}\\sum _{r_{1}{,}...,r_{K-2}}\\prod _{j=0}^{K-3}\\mathbb {P}_{r_{j}}\\left(Y_0=m_j, X_{\\sigma _{1}}=r_{j+1},\\,\\tau _{m_{j}}<\\sigma _1\\right)\\\\&\\hspace{79.6678pt}\\times \\mathbb {P}_{r_{K-2}}\\left(Y_1=m_{K-1},\\,\\tau _{m_{K-2}}<\\sigma _1\\right)\\,\\mathbb {P}_{m_{K-1}}(Y_1=m_K)\\\\&\\ge \\ (1-\\delta _{\\max })\\sum _{m_{j}}\\sum _{r_{1}{,}...,r_{K-2}}\\prod _{j=0}^{K-3}\\mathbb {P}_{r_{j}}\\left(Y_0=m_j, X_{\\sigma _{1}}=r_{j+1},\\,\\tau _{m_{j}}<\\sigma _1\\right)\\\\&\\hspace{79.6678pt}\\times \\mathbb {P}_{m_{K-2}}\\left(Y_1=m_{K-1}\\right)\\,\\mathbb {P}_{r_{K-2}}(\\tau _{m_{K-2}}<\\sigma _1)\\,\\mathbb {P}_{m_{K-1}}(Y_1=m_K)\\\\&\\ge \\ (1-\\delta _{\\max })^{2}\\sum _{m_{j}}\\sum _{r_{1}{,}...,r_{K-2}}\\prod _{j=0}^{K-3}\\mathbb {P}_{r_{j}}\\left(Y_0=m_j, X_{\\sigma _{1}}=r_{j+1},\\,\\tau _{m_{j}}<\\sigma _1\\right)\\\\&\\hspace{142.26378pt}\\times \\mathbb {P}_{m_{K-2}}\\left(Y_1=m_{K-1}\\right)\\,\\mathbb {P}_{m_{K-1}}(Y_1=m_K)\\\\&\\hspace{5.69046pt}\\vdots \\\\&\\ge \\ (1-\\delta _{\\max })^{K-1}\\sum _{m_{j}}\\prod _{j=0}^{K-1}\\mathbb {P}_{m_j}(Y_1=m_{j+1})\\\\&\\ge (1-\\delta _{\\max })^{K-1}\\,[\\eta _2\\wedge \\eta _3]_{K}\\,e^{-2K\\varepsilon \\beta },$ the last line following from Lemma REF .",
"Figure: (a) 2-dimensional modification of the energy landscape from Example .",
"(b) sup m ' ∈𝒮 (i) ∖N (i) |E(z * (m,m ' ))-E(s m )|\\sup _{m^{\\prime }\\in \\mathcal {S}^{(i)}\\backslash N^{(i)}}|E(z^*(m,m^{\\prime }))-E(s_m)| for the various metastable states in 𝒮 (i) ∖N (i) \\mathcal {S}^{(i)}\\backslash N^{(i)} in dependence of the level 1≤i≤𝔫1\\le i\\le \\mathfrak {n}.Example 3.18 We return to Example REF given in Section , but modify the energy landscape by allowing direct transitions between some saddles (see Figure REF (a)) because (MB2) can clearly not be fulfilled in a one-dimensional model.",
"While having no effect on the metastable states $m\\in M^{(i)}$ , valleys change in the way that, for levels $i\\in \\lbrace 5,6\\rbrace $ , the states $\\lbrace 1,2,3\\rbrace $ do no longer belong to the valley around state 4 and $\\lbrace 1,2\\rbrace $ forms its own valley.",
"The energy-differences $\\sup _{m^{\\prime }\\in \\mathcal {S}^{(i)}\\backslash N^{(i)}}E(z^*(m,m^{\\prime }))-E(s_m)$ of the various metastable states $m$ at each level $1\\le i\\le \\mathfrak {n}=7$ are shown in Figure REF (b).",
"The supremum of these energy differences decreases in $i$ , and we obtain MB of order 1 for $i\\ge 4$ , and of order $0.5$ for $i\\ge 6$ .",
"To illustrate the behavior, we have run a Metropolis Algorithm on this energy landscape.",
"For initial state $s=4$ and $\\beta =0.75$ , the energies of the trajectories of the original chain as well of the aggregated chain at levels $i=3,4,5$ are shown in Figure REF .",
"The following observations are worth to be pointed out: The number of reciprocating jumps decreases with increasing level of aggregation.",
"The deeper the valley, the longer the residence time.",
"The motion in state space is well described by the aggregated process.",
"Due to the very small size of the state space and a long simulation time, valleys are revisited.",
"Figure: Energies of the true trajectory and of the trajectories of the aggregated chain at levels i=3,4,5i=3,4,5."
],
[
"Acknowledgment",
"tocsectionAcknowledgment We are very indebted to Andreas Heuer for sharing his insight about glass forming structures with us and also for his advice and many stimulating discussions that helped to improve the presentation of this article."
]
] | 1204.0884 |
[
[
"Perturbations in loop quantum cosmology"
],
[
"Abstract The era of precision cosmology has allowed us to accurately determine many important cosmological parameters, in particular via the CMB.",
"Confronting Loop Quantum Cosmology with these observations provides us with a powerful test of the theory.",
"For this to be possible we need a detailed understanding of the generation and evolution of inhomogeneous perturbations during the early, Quantum Gravity, phase of the universe.",
"Here we describe how Loop Quantum Cosmology provides a completion of the inflationary paradigm, that is consistent with the observed power spectra of the CMB."
],
[
"Open issues with inflation",
"The standard picture we have of early universe cosmology is that of inflation.",
"This beautiful idea contains very few basic assumptions, yet it produces, with spectacular agreement, the power spectra of density fluctuations observed in the Cosmic Microwave Background (CMB).",
"Nevertheless, inflation remains a paradigm in search of a model and there are several key questions that remain unanswered.",
"Broadly speaking, these can be split into two categories: difficulties facing the particles physics interpretation of inflation, and those we expect to be related to quantum gravity.",
"Examples of each type are given in Table (REF ).",
"Inflation is a period of quasi-de Sitter expansion and typically the assumption is that slow-roll inflation began with the quantum state describing perturbations in its `natural' vacuum state: the Bunch-Davies vacuum.",
"A priori this is an odd assumption, since it says that the quantum state is tuned to the subsequent (quasi-de Sitter) evolution of the geometry.",
"Essentially this implies that the pre-inflationary dynamics of the universe and the `true' initial state conspired in such a way as to ensure that we arrived at the onset of slow-roll inflation with no particles present (relative to the Bunch-Davies vacuum).",
"The intuition behind this assumption was that even if there were particles present at the onset of inflation, the exponential expansion would rapidly dilute them and hence their consequences can safely be ignored.",
"This intuition misses the important fact that quantum fields in a dynamical space-time experience both spontaneous and simulated creation of particles [2].",
"The latter effect actually compensates for the exponential growth of the volume in such a way that the particle number density remains approximately constant.",
"In classical general relativity inflation is inevitably preceded by the Big-Bang and the only natural place to give initial conditions is at this singularity.",
"An important open question then is: can one find a quantum gravity completion to the inflationary paradigm?",
"To be viably, such a completion should be non-singular and agree with current observations.",
"It would also open up the exciting possibility that we can directly observe the pre-inflationary universe.",
"If it were possible to see observational consequences of the quantum state at the onset of slow-roll, we would be able to probe the dynamics of the quantum gravity era of the universe.",
"Table: Some examples of the unresolved issues facing inflation."
],
[
"Loop Quantum Cosmology",
"Loop Quantum Gravity is a particularly well developed approach to quantising gravity [3].",
"It is a Hamiltonian quantisation that maintains the fundamental relationship between geometry and gravity and is fully non-perturbative.",
"However a rigorous understanding of the dynamics of the theory are still lacking (see Madhavan Varadarajan's talk in this session).",
"One useful way to make progress is to consider simplified (truncated) systems of the full theory, which, on the one hand, can be completely understood, and on the other are physically interesting.",
"This approach has been applied with great success to the study of black-holes (see Fernando Barbero's talk in this session) and graviton propagators (see Carlo Rovelli's talk in this session) within Loop Quantum Gravity.",
"One can also consider the truncation of full general relativity to homogeneous systems and then use Loop Quantum Gravity techniques to quantise these.",
"This leads to Loop Quantum Cosmology (LQC)[4], which has turned out to be very successful at answering many of the difficulties facing classical cosmology.",
"In particular, it has been shown in detail how the classical singularity is replaced by a `Big-Bounce' and how the late time, low energy, limit reproduces the standard expectations of general relativity.",
"It has also been shown that the probability of having a sufficiently long phase of (single scalar field driven) slow-roll inflation, within LQC, is very close to one [5].",
"However in order to make predictions about LQC effects on the CMB, one first has to extend the underlying approach to include (peturbative) inhomogeneities.",
"There have been several promising attempts to include such inhomogeneities (see Jakub Mielczarek's talk in this session) based on consistent alterations of the homogeneous formulation of LQC.",
"Here we describe a systematic approach to extending the underlying formulation."
],
[
"LQC and perturbations",
"The first step is to find a suitable truncation of the phase-space of classical general relativity that allows for cosmologies with perturbative inhomogeneities.",
"Since we work in the Hamiltonian theory, we restrict our attention to cosmologies whose spatial slices are (flat) tori.",
"One can then show that the full phase-space decomposes into homogeneous and purely inhomogeneous parts, $\\Gamma _{\\rm Full} = \\Gamma _{\\rm H} \\times \\Gamma _{\\rm IH}~.$ The canonical variables of the homogeneous phase-space ($\\Gamma _{\\rm H}$ ) are $\\left( a,\\pi _a,\\phi ,\\pi _\\phi \\right)$ , i.e.",
"the scale factor $a$ and the scalar field $\\phi $ , and their conjugate momenta.",
"The canonical variables of the inhomogeneous phase-space ($\\Gamma _{\\rm IH}$ ) are the corresponding perturbations, $\\left( h_{ab}\\left(x\\right),\\pi ^{ab}\\left(x\\right),\\varphi \\left(x\\right),\\pi _\\varphi \\left(x\\right)\\right)$ , all of which are purely inhomogeneous.",
"All the canonical structures of the phase-space decompose in this way, in particular the symplectic structure and the Possion brackets factor.",
"One can now consider the inhomogeneous fields as perturbations, expand the constraints and find gauge invariant degrees of freedom [6].",
"In particular the gauge invariant scalar modes of the perturbations satisfy the constraint, ${\\cal C}= \\frac{\\pi _\\phi ^2}{2l^3} + \\frac{m^2V^2}{2l^3} - \\frac{3}{8\\pi G l^3} b^2 V^2+\\int {\\rm d}^3 k\\left( \\frac{1}{2} P_k^2 + f\\left( V,b,\\phi ,\\pi _\\phi ; k\\right) Q^2_k\\right) \\approx 0~,$ where $V\\sim a^3$ and $b\\sim \\pi _a/a^2$ are canonically conjugate background variables, $l$ is the coordinate size of the 3-torus and $\\left( P_k, Q_k \\right)$ are the gauge invariant, scalar degree of freedom (related to the Mukhanov variable).",
"The important point is that gauge invariant scalar (and tensor) perturbations behave exactly as test scalar fields with a time dependent mass $f\\left( V,b,\\phi ,\\pi _\\phi ; k\\right)$ .",
"In [7] it was shown how to quantise such a system, and we schematically sketch this procedure here.",
"Writing the background and perturbation parts of Eq.",
"(REF ) as, ${\\cal C} = \\frac{\\pi _\\phi ^2}{2 l^3}+H^2_0 + \\int {\\rm d}^3 k H_{\\tau ,k}~,$ where $H_0$ is the Hamiltonian for the background geometry (and the potential term) and $H_{\\tau ,k}$ is the (time dependent) Hamiltonian for the perturbations.",
"These are then promoted to operators and we deparameterise with respect to the scalar field $\\phi $ , to find, $-i \\hbar \\partial _\\phi \\Psi = \\left( \\widehat{H_0^2} + \\int {\\rm d}^3 k\\widehat{H}_{\\tau ,k} \\right)^{1/2} \\Psi \\approx \\left(\\widehat{H}_0 + \\widehat{H}_0^{-1/2}\\left( \\int {\\rm d}^3 k\\widehat{H}_{\\tau ,k}\\right) \\widehat{H}_0^{-1/2} \\right) \\Psi ~,$ where $\\widehat{H_0^2} = \\hbar ^2 \\widehat{\\Theta }_0$ is the operator that governs the LQC evolution of the background [4], and the right hand side has been approximated via a series expansion.",
"This is the only approximation that is made in the procedure and it can be viewed as a test-field approximation.",
"Essentially, one restrictions ones attention to those states in which the $\\widehat{H}_0$ is dominant and $\\widehat{H}_{\\tau ,k}$ does not alter the background dynamicsThere are many important issues to do with ensuring that the integral over $k$ results in a well defined operator that are not being described here.",
"See [1] for details..",
"The second term on the right hand side of Eq.",
"(REF ) is precisely the Hamiltonian for the perturbations associated to the relational time defined by $\\phi $ .",
"Finally, one decomposes the wave-function as a tensor product of back-ground and perturbation pieces, $\\Psi \\left( V, Q_k,\\phi \\right) = \\Psi _0\\left( V,\\phi \\right) \\otimes \\psi \\left( Q_k, \\phi \\right)~,$ and considers states for which $\\Psi _0$ is sharply peaked with respect to some particular classical background geometry at late times i.e.",
"a semi-classical background geometry.",
"Taking expectation values of Eq.",
"(REF ) with respect to $\\Psi _0$ , one arrives the standard Hamiltonian for test quantum fields in a curved space-time, with the background scale factor $a\\left(\\phi \\right)$ replaced by $\\langle \\Psi _0\\big |\\hat{a}\\big |\\Psi _0\\rangle \\big |_{\\phi }$ ."
],
[
"Results and conclusions",
"In the previous section we briefly sketched how one can extend the formulation of LQC to include (perturbative) inhomogeneities.",
"With this in hand, one can now look for consequences of the pre-inflationary era and in particular, we can look for deviations from the standard initial conditions for inflation.",
"The standard approach in inflation is to assume the existence of a scalar field with a suitable potential, give oneself some initial conditions for the quantum state of the perturbations and hence calculate the late time power spectra.",
"Here we will take exactly the same approach: we specify initial conditions at the Big-Bounce and calculate the resulting power-spectra (here we concentrate only on tensor perturbations).",
"Since, at the bounce, the wavelength of all the observable modes is much smaller than the curvature scale, we take the initial state (for these modes) to be the Minkowski vacuum (see [1] for a discussion on this point).",
"Figure (REF ) shows the resulting late time (dimensionless) power spectra, $\\Delta ^2_R$ , defined as, $\\delta \\left({\\bf k} + {\\bf k}^{\\prime }\\right) \\frac{\\Delta ^2_R\\left(k\\right)}{4\\pi |{\\bf k}|^3} =\\langle 0| \\hat{R}_{\\bf k} \\hat{R}_{\\bf k^{\\prime }}|0\\rangle ~,$ where the variable $R_{\\bf k}$ is related to the the Mukhanov variable $Q_{\\bf k}$ by $R_{\\bf k} = \\frac{\\dot{\\phi }}{H} Q_{\\bf k}$ , and represents the gravitational potential on constant $\\phi $ hyper-surfaces.",
"The important result here is that provided the background scalar field, $\\phi $ , at the bounce is large enough, i.e.",
"$\\phi \\left(t_{\\rm bounce}\\right) \\ge 1.2$ , the power spectra is entirely consistent with that of standard inflation.",
"Hence we have a completion of the inflationary paradigm, including the quantum gravity era, which agree with the observations.",
"Note that there remains a (small but important) window ($0.95 < \\phi \\left(t_{\\rm bounce}\\right)< 1.2$ ) in which we may hope to see corrections to the largest scales (smallest $k$ ) of the observed power spectra, and hence directly observe pre-inflationary (i.e.",
"quantum gravity) physics.",
"Figure: The power-spectra for tensor perturbations at the end of slow-roll inflation, given an initialvacuum at the bounce.",
"We have set at bounce =1a\\left(t_{\\rm bounce}\\right) =1 and hence the physicalscale at (for example) decoupling depend on the amount of inflation that has occurred.",
"Note in particular the agreementwith the predictions of standard inflation provided φt bounce >1.2\\phi \\left(t_{\\rm bounce}\\right) > 1.2."
],
[
"Acknowledgments",
"This work was supported in part by the NSF grants PHY0854743 and PHY1068743."
]
] | 1204.1288 |
Subsets and Splits